Formalization: Efficient Estimation of Binary Treatment Effects under Local Differential Privacy
The complete Lean development behind this paper — every definition, lemma, and theorem of its module, including helpers the paper text never cites. Identifiers link within this page, into the Causalean library, or out to the official Mathlib docs. Note: this index was generated at a different commit than the paper's audited pin (f9001b6) — statements may have moved since the audit.
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Basic 72 declarations This file fixes the four input symbols of a randomized binary trial and gives the fourteen ray staircase program for its private information surface.
Binary trial and finite staircase program
This file fixes the four input symbols of a randomized binary trial and gives the fourteen ray staircase program for its private information surface.
def PrivateAlphabet
the private alphabet is the mathematical object specified below.
Definition (Lean source)
def
PrivateAlphabet :
Type
given by
Fin 4
def StaircasePattern
Coordinates of the fourteen nonconstant rays used in the finite program. The stated result follows.
Definition (Lean source)
def
StaircasePattern :
Type
given by
Fin 14
def StaircaseSubset
The declared pattern carrier is the full power set, including the empty and full sets. The stated result follows.
Definition (Lean source)
def
StaircaseSubset :
Type
given by
def staircasePatternSpace
the staircase pattern space is the mathematical object specified below.
Definition (Lean source)
def
given by
theorem staircasePatternSpace_eq_univ
the staircase pattern space eq univ assertion holds.
Formal statement
conclusion
Proof (Lean source)
lemma staircasePatternSpace_eq_univ : staircasePatternSpace = univ := by
ext s
simp [staircasePatternSpace]
theorem staircasePatternSpace_card
the staircase pattern space card assertion holds.
Formal statement
conclusion
staircasePatternSpace.card = 16
Proof (Lean source)
lemma staircasePatternSpace_card : staircasePatternSpace.card = 16 := by
simp [staircasePatternSpace]
theorem staircasePatternSpace_contains_constants
the staircase pattern space contains constants assertion holds.
Formal statement
conclusion
conclusion 1
(∅ : StaircaseSubset) ∈ staircasePatternSpace
conclusion 2
(Finset.univ : StaircaseSubset) ∈ staircasePatternSpace
Proof (Lean source)
lemma staircasePatternSpace_contains_constants :
(∅ : StaircaseSubset) ∈ staircasePatternSpace ∧
(Finset.univ : StaircaseSubset) ∈ staircasePatternSpace := by
simp [staircasePatternSpace]
def TrialParameter
the trial parameter is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The Trial Parameter is the displayed object.
Definition (Lean source)
def
TrialParameter :
Type
given by
Fin 2 → ℝ
the staircase weight is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The Staircase Weight is the displayed object.
Definition (Lean source)
def
StaircaseWeight :
Type
given by
StaircasePattern → ℝ
the contrast is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
def contrastVector
the contrast vector is the mathematical object specified below.
Definition (Lean source)
def
given by
fun k => if k = 0 then -1 else 1
def controlProb
For the supplied quantities and conditions, the control prob is the mathematical object specified below.
Definition (Lean source)
parameters
p :
ℝ
def
controlProb p :
ℝ
given by
1 - p
def piTheta
For the supplied quantities and conditions, the pi theta is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
θ :
p :
ℝ
def
given by
clause 1
| ⟨0, _⟩ => controlProb p * (1 - θ 0)
clause 2
| ⟨1, _⟩ => controlProb p * θ 0
clause 3
| ⟨2, _⟩ => p * (1 - θ 1)
clause 4
| _ => p * θ 1
def inputLaw
For the supplied quantities and conditions, the input law is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
θ :
p :
ℝ
def
def inputScore
the input score is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
θ :
def
given by
clause 1
| ⟨0, _⟩ => fun k => if k = 0 then -(1 - θ 0)⁻¹ else 0
clause 2
| ⟨1, _⟩ => fun k => if k = 0 then (θ 0)⁻¹ else 0
clause 3
| ⟨2, _⟩ => fun k => if k = 1 then -(1 - θ 1)⁻¹ else 0
clause 4
| _ => fun k => if k = 1 then (θ 1)⁻¹ else 0
For the supplied quantities and conditions, the private record is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
Ω :
Sort u_1
W Y :
ℕ → Ω → ℝ
i :
ℕ
ω :
Ω
def
given by
if W i ω = 0 then if Y i ω = 0 then 0 else 1 else if Y i ω = 0 then 2 else 3
def armMeanParameter
For the supplied quantities and conditions, the arm mean parameter is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
given by
fun k => if k = 0 then ∫ ω, Y0 0 ω ∂μ else ∫ ω, Y1 0 ω ∂μ
the iid subjects is the mathematical object specified below. The Iid Subjects is determined by the displayed parameters.
Definition (Lean source)
parameters
Ω :
Type u_1
Y0 :
ℕ → Ω → ℝ
Y1 :
ℕ → Ω → ℝ
W :
ℕ → Ω → ℝ
def
IidSubjects μ Y0 Y1 W :
Prop
given by
clause 1
iIndepFun (fun i ω => (Y0 i ω, Y1 i ω, W i ω)) μ
the random assignment is the mathematical object specified below. The Random Assignment is determined by the displayed parameters.
the known assignment is the mathematical object specified below. The Known Assignment is determined by the displayed parameters.
the outcome consistency is the mathematical object specified below. The Outcome Consistency is determined by the displayed parameters.
Definition (Lean source)
parameters
Ω :
Type u_1
Y0 :
ℕ → Ω → ℝ
Y1 :
ℕ → Ω → ℝ
W :
ℕ → Ω → ℝ
Y :
ℕ → Ω → ℝ
i ω :
def
OutcomeConsistency Y0 Y1 W Y :
Prop
given by
Y i ω = W i ω * Y1 i ω + (1 - W i ω) * Y0 i ω
the binary outcomes is the mathematical object specified below. The Binary Outcomes is determined by the displayed parameters.
the interior assignment is the mathematical object specified below. The Interior Assignment is determined by the displayed parameters.
Definition (Lean source)
parameters
p :
ℝ
def
InteriorAssignment p :
Prop
given by
clause 1
0 < p
clause 2
p < 1
the interior means is the mathematical object specified below. The Interior Means is determined by the displayed parameters.
Definition (Lean source)
parameters
def
InteriorMeans θ :
Prop
given by
clause 1
0 < θ 0
clause 2
θ 0 < 1
clause 3
0 < θ 1
clause 4
θ 1 < 1
For the supplied quantities and conditions, the assignment range is the mathematical object specified below.
Definition (Lean source)
parameters
p :
ℝ
def
AssignmentRange p :
Prop
given by
clause 1
0 ≤ p
clause 2
p ≤ 1
the mean range is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
θ :
def
MeanRange θ :
Prop
given by
clause 1
0 ≤ θ 0
clause 2
θ 0 ≤ 1
clause 3
0 ≤ θ 1
clause 4
θ 1 ≤ 1
def ContrastRange
the contrast range is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
τ :
ℝ
def
ContrastRange τ :
Prop
given by
clause 1
-1 ≤ τ
clause 2
τ ≤ 1
For the supplied quantities and conditions, the control range is the mathematical object specified below.
Definition (Lean source)
parameters
p :
ℝ
def
ControlRange p :
Prop
theorem assignmentRange
Under the supplied quantities and conditions, the assignment range assertion holds.
Formal statement
Proof (Lean source)
lemma InteriorAssignment.assignmentRange (hp : InteriorAssignment p) :
AssignmentRange p := ⟨hp.1.le, hp.2.le⟩
theorem meanRange
Under the supplied quantities and conditions, the mean range assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the mean Range.
Formal statement
Proof (Lean source)
lemma InteriorMeans.meanRange (hθ : InteriorMeans θ) : MeanRange θ :=
⟨hθ.1.le, hθ.2.1.le, hθ.2.2.1.le, hθ.2.2.2.le⟩
theorem contrastRange
Under the supplied quantities and conditions, the contrast range assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the contrast Range.
Formal statement
Proof (Lean source)
lemma MeanRange.contrastRange (hθ : MeanRange θ) : ContrastRange (contrast θ) := by
unfold MeanRange at hθ
unfold ContrastRange contrast
constructor <;> linarith [hθ.1, hθ.2.1, hθ.2.2.1, hθ.2.2.2]
theorem controlRange
Under the supplied quantities and conditions, the control range assertion holds.
Formal statement
Proof (Lean source)
lemma AssignmentRange.controlRange (hp : AssignmentRange p) : ControlRange p := by
unfold AssignmentRange at hp
unfold ControlRange controlProb
constructor <;> linarith [hp.1, hp.2]
def PositiveSampleSize
the positive sample size is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The Positive Sample Size is the displayed object.
Definition (Lean source)
def
PositiveSampleSize :
Type
given by
{n : ℕ // 0 < n}
The public binary-trial model is the family of IID laws of the observed treatment-outcome record. Its nth member is the n-fold product of the four-point law at (θ,p). The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. For the displayed quantities and conditions, these specify the stated inputs. The Binary Trial Model is determined by the displayed parameters.
Definition (Lean source)
structure LatentBinaryTrialRepresentation
A latent representation of the public binary-trial model by potential outcomes, assignment, and the observed outcome. The Latent Binary Trial Representation is determined by the displayed parameters.
Definition (Lean source)
parameters
Ω :
Type u_1
Y0 :
ℕ → Ω → ℝ
Y1 :
ℕ → Ω → ℝ
W :
ℕ → Ω → ℝ
Y :
ℕ → Ω → ℝ
p :
ℝ
assumes
measurable :
∀ i, Measurable (fun ω => (Y0 i ω, Y1 i ω, W i ω))
iid :
IidSubjects μ Y0 Y1 W
randomized :
RandomAssignment μ Y0 Y1 W
assigned :
KnownAssignment μ W p
consistent :
OutcomeConsistency Y0 Y1 W Y
binary :
BinaryOutcomes μ Y0 Y1
assignmentInterior :
meansInterior :
parameterLaw :
θ = armMeanParameter μ Y0 Y1
Under the supplied quantities and conditions, the binary event probs assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
conclusion 1
μ.real {ω | X ω = 1} = ∫ ω, X ω ∂μ
conclusion 2
μ.real {ω | X ω = 0} = 1 - ∫ ω, X ω ∂μ
Proof (Lean source)
lemma binary_event_probs [IsProbabilityMeasure μ] (X : Ω → ℝ)
(hXmeas : AEMeasurable X μ)
(hX : ∀ᵐ ω ∂μ, X ω = 0 ∨ X ω = 1) :
μ.real {ω | X ω = 1} = ∫ ω, X ω ∂μ ∧
μ.real {ω | X ω = 0} = 1 - ∫ ω, X ω ∂μ := by
constructor
· exact (integral_of_ae_eq_zero_or_one hXmeas hX).symm
· rw [← integral_one_sub_of_ae_eq_zero_or_one hXmeas hX]
have hXint : Integrable X μ :=
Integrable.of_mem_Icc (0 : ℝ) 1 hXmeas (by
filter_upwards [hX] with ω hω
rcases hω with hω | hω <;> simp [hω])
rw [integral_sub (integrable_const 1) hXint]
simp
theorem recordLaw_eq_piTheta
Under the supplied quantities and conditions, the record law eq pi theta assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Ω :
Type u_1
Y0 :
ℕ → Ω → ℝ
Y1 :
ℕ → Ω → ℝ
W :
ℕ → Ω → ℝ
Y :
ℕ → Ω → ℝ
p :
ℝ
hmodel :
LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ
i :
ℕ
conclusion
μ.map (privateRecord W Y i) = inputLaw θ p
Proof (Lean source)
lemma recordLaw_eq_piTheta (hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ)
(i : ℕ) :
μ.map (privateRecord W Y i) = inputLaw θ p := by
letI : IsProbabilityMeasure μ := hmodel.iid.1.isProbabilityMeasure
have hY0dist : IdentDistrib (Y0 0) (Y0 i) μ μ := by
convert (hmodel.iid.2 i).comp (by fun_prop :
Measurable (fun x : ℝ × ℝ × ℝ => x.1)) using 1 <;> rfl
have hY1dist : IdentDistrib (Y1 0) (Y1 i) μ μ := by
convert (hmodel.iid.2 i).comp (by fun_prop :
Measurable (fun x : ℝ × ℝ × ℝ => x.2.1)) using 1 <;> rfl
have hWdist : IdentDistrib (W 0) (W i) μ μ := by
convert (hmodel.iid.2 i).comp (by fun_prop :
Measurable (fun x : ℝ × ℝ × ℝ => x.2.2)) using 1 <;> rfl
have hmean0 : ∫ ω, Y0 i ω ∂μ = θ 0 := by
rw [← hY0dist.integral_eq, hmodel.parameterLaw]
rfl
have hmean1 : ∫ ω, Y1 i ω ∂μ = θ 1 := by
rw [← hY1dist.integral_eq, hmodel.parameterLaw]
rfl
have hbinary0 : ∀ᵐ ω ∂μ, Y0 i ω = 0 ∨ Y0 i ω = 1 := by
filter_upwards [(hmodel.binary i)] with ω hω
simpa using hω.1
have hbinary1 : ∀ᵐ ω ∂μ, Y1 i ω = 0 ∨ Y1 i ω = 1 := by
filter_upwards [(hmodel.binary i)] with ω hω
simpa using hω.2
have hbinaryW : ∀ᵐ ω ∂μ, W i ω = 0 ∨ W i ω = 1 := by
filter_upwards [(hmodel.assigned i).1] with ω hω
simpa using hω
have hy0 := binary_event_probs μ (Y0 i) hY0dist.aemeasurable_snd hbinary0
have hy1 := binary_event_probs μ (Y1 i) hY1dist.aemeasurable_snd hbinary1
have hw := binary_event_probs μ (W i) hWdist.aemeasurable_snd hbinaryW
have hm0 : Measurable (Y0 i) := (hmodel.measurable i).fst
have hm1 : Measurable (Y1 i) := (hmodel.measurable i).snd.fst
have hmw : Measurable (W i) := (hmodel.measurable i).snd.snd
have hmy : Measurable (Y i) := by
have hf : Measurable (fun ω => W i ω * Y1 i ω +
(1 - W i ω) * Y0 i ω) := by fun_prop
convert hf using 1
funext ω
exact hmodel.consistent i ω
have hmr : Measurable (privateRecord W Y i) := by
unfold privateRecord
apply Measurable.ite (hmw (measurableSet_singleton 0))
· exact Measurable.ite (hmy (measurableSet_singleton 0))
measurable_const measurable_const
· exact Measurable.ite (hmy (measurableSet_singleton 0))
measurable_const measurable_const
have hmassY0one : μ {ω | Y0 i ω = 1} = ofReal (θ 0) := by
rw [← ENNReal.ofReal_toReal (measure_ne_top μ _)]
change ofReal (μ.real {ω | Y0 i ω = 1}) = _
rw [hy0.1, hmean0]
have hmassY0zero : μ {ω | Y0 i ω = 0} = ofReal (1 - θ 0) := by
rw [← ENNReal.ofReal_toReal (measure_ne_top μ _)]
change ofReal (μ.real {ω | Y0 i ω = 0}) = _
rw [hy0.2, hmean0]
have hmassY1one : μ {ω | Y1 i ω = 1} = ofReal (θ 1) := by
rw [← ENNReal.ofReal_toReal (measure_ne_top μ _)]
change ofReal (μ.real {ω | Y1 i ω = 1}) = _
rw [hy1.1, hmean1]
have hmassY1zero : μ {ω | Y1 i ω = 0} = ofReal (1 - θ 1) := by
rw [← ENNReal.ofReal_toReal (measure_ne_top μ _)]
change ofReal (μ.real {ω | Y1 i ω = 0}) = _
rw [hy1.2, hmean1]
have hmassWone : μ {ω | W i ω = 1} = ofReal p := (hmodel.assigned i).2
have hmeanW : ∫ ω, W i ω ∂μ = p := by
rw [← hw.1]
change (μ {ω | W i ω = 1}).toReal = p
rw [hmassWone]
exact ENNReal.toReal_ofReal hmodel.assignmentInterior.1.le
have hmassWzero : μ {ω | W i ω = 0} = ofReal (controlProb p) := by
rw [← ENNReal.ofReal_toReal (measure_ne_top μ _)]
change ofReal (μ.real {ω | W i ω = 0}) = _
rw [hw.2, hmeanW]
rfl
have hprod0 (a b : ℝ) :
μ {ω | W i ω = a ∧ Y0 i ω = b} =
μ {ω | W i ω = a} * μ {ω | Y0 i ω = b} := by
have h := (hmodel.randomized i).measure_inter_preimage_eq_mul
({a} : Set ℝ) ({x : ℝ × ℝ | x.1 = b})
(measurableSet_singleton a)
(measurable_fst (measurableSet_singleton b))
change μ {ω | W i ω = a ∧ Y0 i ω = b} =
μ {ω | W i ω = a} * μ {ω | Y0 i ω = b} at h
exact h
have hprod1 (a b : ℝ) :
μ {ω | W i ω = a ∧ Y1 i ω = b} =
μ {ω | W i ω = a} * μ {ω | Y1 i ω = b} := by
have h := (hmodel.randomized i).measure_inter_preimage_eq_mul
({a} : Set ℝ) ({x : ℝ × ℝ | x.2 = b})
(measurableSet_singleton a)
(measurable_snd (measurableSet_singleton b))
change μ {ω | W i ω = a ∧ Y1 i ω = b} =
μ {ω | W i ω = a} * μ {ω | Y1 i ω = b} at h
exact h
have hrec (ω : Ω) (hwω : W i ω = 0 ∨ W i ω = 1) :
privateRecord W Y i ω =
if W i ω = 0 then
(if Y0 i ω = 0 then (0 : Fin 4) else 1)
else if Y1 i ω = 0 then 2 else 3 := by
unfold privateRecord
rw [hmodel.consistent i ω]
rcases hwω with h0 | h1
· simp [h0]
· simp [h1]
have hatom0 : μ ((privateRecord W Y i) ⁻¹' {0}) =
μ {ω | W i ω = 0 ∧ Y0 i ω = 0} := by
apply measure_congr
filter_upwards [hbinaryW] with ω hwω
change (privateRecord W Y i ω = 0) = (W i ω = 0 ∧ Y0 i ω = 0)
apply propext
rw [hrec ω hwω]
by_cases h0 : W i ω = 0
· simp [h0]
· simp [h0]
split_ifs <;> decide
have hatom1 : μ ((privateRecord W Y i) ⁻¹' {1}) =
μ {ω | W i ω = 0 ∧ Y0 i ω = 1} := by
apply measure_congr
filter_upwards [hbinaryW, hbinary0, hbinary1] with ω hwω hy0ω hy1ω
change (privateRecord W Y i ω = 1) = (W i ω = 0 ∧ Y0 i ω = 1)
apply propext
rw [hrec ω hwω]
rcases hwω with h0 | h1
· rcases hy0ω with h | h <;> simp [h0, h]
· rcases hy1ω with h | h <;> simp [h1, h]
have hatom2 : μ ((privateRecord W Y i) ⁻¹' {2}) =
μ {ω | W i ω = 1 ∧ Y1 i ω = 0} := by
apply measure_congr
filter_upwards [hbinaryW, hbinary0, hbinary1] with ω hwω hy0ω hy1ω
change (privateRecord W Y i ω = 2) = (W i ω = 1 ∧ Y1 i ω = 0)
apply propext
rw [hrec ω hwω]
rcases hwω with h0 | h1
· rcases hy0ω with h | h <;> simp [h0, h]
· rcases hy1ω with h | h <;> simp [h1, h]
have hatom3 : μ ((privateRecord W Y i) ⁻¹' {3}) =
μ {ω | W i ω = 1 ∧ Y1 i ω = 1} := by
apply measure_congr
filter_upwards [hbinaryW, hbinary0, hbinary1] with ω hwω hy0ω hy1ω
change (privateRecord W Y i ω = 3) = (W i ω = 1 ∧ Y1 i ω = 1)
apply propext
rw [hrec ω hwω]
rcases hwω with h0 | h1
· rcases hy0ω with h | h <;> simp [h0, h]
· rcases hy1ω with h | h <;> simp [h1, h]
have hinput (j : Fin 4) :
inputLaw θ p {j} = ofReal (piTheta θ p j) := by
unfold inputLaw
rw [withDensity_apply _ (measurableSet_singleton j), lintegral_singleton]
simp
apply Measure.ext_of_singleton
intro j
fin_cases j
· change (μ.map (privateRecord W Y i)) {0} = inputLaw θ p {0}
rw [Measure.map_apply hmr (measurableSet_singleton 0), hatom0,
hprod0, hmassWzero, hmassY0zero, hinput 0]
change ofReal (controlProb p) * ofReal (1 - θ 0) =
ofReal (controlProb p * (1 - θ 0))
exact (ENNReal.ofReal_mul (le_of_lt (by
dsimp [controlProb]
exact sub_pos.mpr hmodel.assignmentInterior.2))).symm
· change (μ.map (privateRecord W Y i)) {1} = inputLaw θ p {1}
rw [Measure.map_apply hmr (measurableSet_singleton 1), hatom1,
hprod0, hmassWzero, hmassY0one, hinput 1]
change ofReal (controlProb p) * ofReal (θ 0) =
ofReal (controlProb p * θ 0)
exact (ENNReal.ofReal_mul (le_of_lt (by
dsimp [controlProb]
exact sub_pos.mpr hmodel.assignmentInterior.2))).symm
· change (μ.map (privateRecord W Y i)) {2} = inputLaw θ p {2}
rw [Measure.map_apply hmr (measurableSet_singleton 2), hatom2,
hprod1, hmassWone, hmassY1zero, hinput 2]
change ofReal p * ofReal (1 - θ 1) =
ofReal (p * (1 - θ 1))
exact (ENNReal.ofReal_mul hmodel.assignmentInterior.1.le).symm
· change (μ.map (privateRecord W Y i)) {3} = inputLaw θ p {3}
rw [Measure.map_apply hmr (measurableSet_singleton 3), hatom3,
hprod1, hmassWone, hmassY1one, hinput 3]
change ofReal p * ofReal (θ 1) =
ofReal (p * θ 1)
exact (ENNReal.ofReal_mul hmodel.assignmentInterior.1.le).symm
theorem latentRepresentation_prefixLaw
A latent representation realizes every finite prefix of the public observed-law model. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Ω :
Type u_1
Y0 :
ℕ → Ω → ℝ
Y1 :
ℕ → Ω → ℝ
W :
ℕ → Ω → ℝ
Y :
ℕ → Ω → ℝ
p :
ℝ
hmodel :
LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ
conclusion
μ.map (fun ω => fun i : Fin n.val => privateRecord W Y i ω)
= BinaryTrialModel θ p hmodel.meansInterior hmodel.assignmentInterior n
Proof (Lean source)
lemma latentRepresentation_prefixLaw
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ)
(n : PositiveSampleSize) :
μ.map (fun ω => fun i : Fin n.val => privateRecord W Y i ω) =
BinaryTrialModel θ p hmodel.meansInterior hmodel.assignmentInterior n := by
have hmeas (i : Fin n.val) : Measurable (privateRecord W Y i) := by
have hm := hmodel.measurable i.val
have hmw : Measurable (W i) := hm.snd.snd
have hmy : Measurable (Y i) := by
have hf : Measurable (fun ω => W i ω * Y1 i ω +
(1 - W i ω) * Y0 i ω) := by
exact hmw.mul hm.snd.fst |>.add ((measurable_const.sub hmw).mul hm.fst)
convert hf using 1
funext ω
exact hmodel.consistent i ω
unfold privateRecord
exact Measurable.ite (hmw (measurableSet_singleton 0))
(Measurable.ite (hmy (measurableSet_singleton 0)) measurable_const measurable_const)
(Measurable.ite (hmy (measurableSet_singleton 0)) measurable_const measurable_const)
let g : (Fin n.val) → (ℝ × ℝ × ℝ) → PrivateAlphabet := fun _ x =>
if x.2.2 = 0 then (if x.1 = 0 then 0 else 1)
else if x.2.1 = 0 then 2 else 3
have hg (i : Fin n.val) : Measurable (g i) := by
unfold g
apply Measurable.ite (measurable_snd.snd (measurableSet_singleton 0))
· exact Measurable.ite (measurable_fst (measurableSet_singleton 0))
measurable_const measurable_const
· exact Measurable.ite (measurable_snd.fst (measurableSet_singleton 0))
measurable_const measurable_const
have hindTriple : iIndepFun
(fun i : Fin n.val => fun ω => (Y0 i ω, Y1 i ω, W i ω)) μ :=
hmodel.iid.1.precomp Fin.val_injective
have hindG := hindTriple.comp g hg
have hcomp (i : Fin n.val) :
g i ∘ (fun ω => (Y0 i ω, Y1 i ω, W i ω)) =ᵐ[μ]
privateRecord W Y i := by
filter_upwards [(hmodel.assigned i).1] with ω hw
have hw' : W i ω = 0 ∨ W i ω = 1 := by simpa using hw
simp only [Function.comp_apply, g, privateRecord]
rw [hmodel.consistent i ω]
rcases hw' with hw | hw <;> simp [hw]
have hind : iIndepFun (fun i : Fin n.val => privateRecord W Y i) μ := by
apply hindG.congr
exact hcomp
rw [hind.map_fun_eq_pi_map (fun i => (hmeas i).aemeasurable)]
unfold BinaryTrialModel
congr 1
funext i
exact recordLaw_eq_piTheta μ Y0 Y1 W Y p θ hmodel i
theorem binaryTrialModel_iff_latentRepresentation
A family belongs to the public observed-law model exactly when some latent binary-trial representation realizes all of its finite-prefix laws. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the binary Trial Model iff latent Representation.
Formal statement
hypotheses
p :
ℝ
hp :
hθ :
P :
conclusion
P = BinaryTrialModel θ p hθ hp ↔ ∃ (Ω : Type) (mΩ : MeasurableSpace Ω), letI
Proof (Lean source)
theorem binaryTrialModel_iff_latentRepresentation
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(P : (n : PositiveSampleSize) → Measure (Fin n.val → PrivateAlphabet)) :
P = BinaryTrialModel θ p hθ hp ↔
∃ (Ω : Type) (mΩ : MeasurableSpace Ω),
letI := mΩ
∃ (μ : Measure Ω) (Y0 Y1 W Y : ℕ → Ω → ℝ),
LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ ∧
∀ n, μ.map (fun ω => fun i : Fin n.val => privateRecord W Y i ω) = P n := by
constructor
· rintro rfl
let ν := canonicalSubjectLaw θ p hθ hp
let μ := Measure.infinitePi (fun _ : ℕ => ν)
have hrep : LatentBinaryTrialRepresentation μ canonicalY0 canonicalY1
canonicalW canonicalY p θ := canonicalLatentRepresentation θ p hθ hp
exact ⟨ℕ → ((Bool × Bool) × Bool), inferInstance, μ,
canonicalY0, canonicalY1, canonicalW, canonicalY, hrep,
latentRepresentation_prefixLaw μ canonicalY0 canonicalY1 canonicalW canonicalY
p θ hrep⟩
· rintro ⟨Ω, mΩ, hrep⟩
letI := mΩ
rcases hrep with ⟨μ, Y0, Y1, W, Y, hmodel, hP⟩
funext n
rw [← hP n]
exact latentRepresentation_prefixLaw μ Y0 Y1 W Y p θ hmodel n
def privacyRatio
the privacy ratio is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
def privacyIncrement
the privacy increment is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
def admissiblePrivacyRatio
The paper's likelihood-ratio parameter is evaluated only at a positive privacy budget. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The admissible Privacy Ratio is determined by the displayed parameters.
Definition (Lean source)
def admissiblePrivacyIncrement
The paper's staircase increment is evaluated only at a positive privacy budget. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The admissible Privacy Increment is determined by the displayed parameters.
Definition (Lean source)
theorem admissiblePrivacyRatio_gt_one
Under the supplied quantities and conditions, the admissible privacy ratio gt one assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the admissible Privacy Ratio gt one.
Formal statement
Proof (Lean source)
lemma admissiblePrivacyRatio_gt_one (ε : ℝ) (hε : 0 < ε) :
1 < admissiblePrivacyRatio ε hε := by
exact Real.one_lt_exp_iff.mpr hε
theorem admissiblePrivacyIncrement_pos
Under the supplied quantities and conditions, the admissible privacy increment pos assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the admissible Privacy Increment pos.
Formal statement
Proof (Lean source)
lemma admissiblePrivacyIncrement_pos (ε : ℝ) (hε : 0 < ε) :
0 < admissiblePrivacyIncrement ε hε := by
exact sub_pos.mpr (admissiblePrivacyRatio_gt_one ε hε)
theorem admissiblePrivacyRatio_covers
Under the supplied quantities and conditions, the admissible privacy ratio covers assertion holds.
Formal statement
Proof (Lean source)
lemma admissiblePrivacyRatio_covers (r : ℝ) (hr : 1 < r) :
∃ (ε : ℝ) (hε : 0 < ε), admissiblePrivacyRatio ε hε = r := by
exact ⟨Real.log r, Real.log_pos hr, Real.exp_log (by linarith)⟩
theorem admissiblePrivacyIncrement_covers
Under the supplied quantities and conditions, the admissible privacy increment covers assertion holds.
Formal statement
Proof (Lean source)
lemma admissiblePrivacyIncrement_covers (d : ℝ) (hd : 0 < d) :
∃ (ε : ℝ) (hε : 0 < ε), admissiblePrivacyIncrement ε hε = d := by
refine ⟨Real.log (d + 1), Real.log_pos (by linarith), ?_⟩
simp [admissiblePrivacyIncrement, privacyIncrement, privacyRatio,
Real.exp_log (show 0 < d + 1 by linarith)]
def patternContains
For the supplied quantities and conditions, the pattern contains is the mathematical object specified below.
def staircasePatternSubset
The fixed enumeration embeds the fourteen nonconstant masks into the full power set. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
def patternRay
For the supplied quantities and conditions, the pattern ray is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
patternRay ε s j :
ℝ
given by
1 + privacyIncrement ε * if patternContains s j then 1 else 0
def patternMass
For the supplied quantities and conditions, the pattern mass is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
patternMass θ p ε s :
ℝ
given by
def admissiblePatternMass
The paper's mass factor is computed on the declared probability-parameter domains. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. For the displayed quantities and conditions, these specify the stated inputs. The admissible Pattern Mass is determined by the displayed parameters.
Definition (Lean source)
parameters
θ :
p ε :
ℝ
_hθ :
_hp :
_hε :
0 < ε
s :
def
admissiblePatternMass θ p ε _hθ _hp _hε s :
ℝ
given by
patternMass θ p ε s
theorem admissiblePatternMass_pos
Under the supplied quantities and conditions, the admissible pattern mass pos assertion holds. For the displayed quantities and conditions, these specify the stated inputs. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the admissible Pattern Mass pos.
Formal statement
hypotheses
conclusion
0 < admissiblePatternMass θ p ε hθ hp hε s
Proof (Lean source)
lemma admissiblePatternMass_pos (θ : TrialParameter) (p ε : ℝ)
(hθ : MeanRange θ) (hp : AssignmentRange p) (hε : 0 < ε)
(s : StaircasePattern) : 0 < admissiblePatternMass θ p ε hθ hp hε s := by
have hπ (j : Fin 4) : 0 ≤ piTheta θ p j := by
rcases hp with ⟨hp0, hp1⟩
rcases hθ with ⟨h00, h01, h10, h11⟩
fin_cases j <;> simp only [piTheta, controlProb] <;>
apply mul_nonneg <;> linarith
have hr (j : Fin 4) : 1 ≤ patternRay ε s j := by
have hd := (admissiblePrivacyIncrement_pos ε hε).le
change 0 ≤ privacyIncrement ε at hd
unfold patternRay
split_ifs <;> simp_all
have hsum : (∑ j : Fin 4, piTheta θ p j) = 1 := by
rw [Fin.sum_univ_four]
change controlProb p * (1 - θ 0) + controlProb p * θ 0 +
p * (1 - θ 1) + p * θ 1 = 1
dsimp [controlProb]
ring
have hbound : 1 ≤ patternMass θ p ε s := by
rw [← hsum]
unfold patternMass
exact Finset.sum_le_sum fun j _ => by
simpa using mul_le_mul_of_nonneg_left (hr j) (hπ j)
exact lt_of_lt_of_le (by norm_num) hbound
def patternGradient
For the supplied quantities and conditions, the pattern gradient is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
p ε :
ℝ
s :
Fin 14
def
given by
fun k
=> if k = 0 then privacyIncrement ε * controlProb p * ((if patternContains s 1 then 1 else 0) - (if patternContains s 0 then 1 else 0)) else privacyIncrement ε * p * ((if patternContains s 3 then 1 else 0) - (if patternContains s 2 then 1 else 0))
def staircaseMatrix
For the supplied quantities and conditions, the staircase matrix is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
the staircase feasible is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
ε :
ℝ
α :
def
staircaseFeasible ε α :
Prop
def direction
For the supplied quantities and conditions, the direction is the mathematical object specified below.
Definition (Lean source)
def projectedGradient
For the supplied quantities and conditions, the projected gradient is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
projectedGradient p ε s t :
ℝ
given by
def patternInformation
For the supplied quantities and conditions, the pattern information is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
patternInformation θ p ε s t :
ℝ
given by
(projectedGradient p ε s t) ^ 2 / patternMass θ p ε s
For the supplied quantities and conditions, the information matrix is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
given by
fun a b
def outputScore
For the supplied quantities and conditions, the output score is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
def projectedScore
For the supplied quantities and conditions, the projected score is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
projectedScore θ p ε s t :
ℝ
given by
projectedGradient p ε s t / patternMass θ p ε s
For the supplied quantities and conditions, the information objective is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
informationObjective θ p ε α t :
ℝ
given by
∑ s : Fin 14, α s * patternInformation θ p ε s t
def Jstar
The optimized Fisher-information value is determined by the displayed parameters.
Definition (Lean source)
parameters
θ :
p ε :
ℝ
def
Jstar θ p ε :
ℝ
given by
sSup {z : ℝ | ∃ α : StaircaseWeight, staircaseFeasible ε α ∧ z = sInf {y : ℝ | ∃ t : ℝ, y = informationObjective θ p ε α t}}
The reciprocal optimized contrast variance is determined by the displayed parameters.
Definition (Lean source)
def PrivateHistory
For the supplied quantities and conditions, the private history is the mathematical object specified below. The Private History is determined by the displayed parameters.
def SequentialKernel
the sequential kernel is the mathematical object specified below.
Definition (Lean source)
parameters
def
SequentialKernel :
Type u_2
given by
For the supplied quantities and conditions, the fixed privacy is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
epsSeq :
ℕ → ℝ
ε :
ℝ
def
FixedPrivacy epsSeq ε :
Prop
given by
clause 1
0 < ε
clause 2
k :
epsSeq k = ε
For the supplied quantities and conditions, the sequential ldp is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
SequentialLDP ε Q :
Prop
given by
structure SequentialChannels
For the supplied quantities and conditions, this sequential channels packages the stated mathematical data. For the displayed quantities and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
assumes
fixed :
FixedPrivacy epsSeq ε
ldp :
SequentialLDP (Z := Z) (epsSeq n) Q
For the supplied quantities and conditions, the pilot diverges is the mathematical object specified below.
For the supplied quantities and conditions, the pilot sublinear is the mathematical object specified below.
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.AttainmentRigidity 3 declarations Variance attainment forces equality in every certified information direction.
Directional equality and postprocessing nonmerger
Variance attainment forces equality in every certified information direction. After staircase refinement, distinct projected scores then force the active postprocessing rows to be pairwise mutually singular, giving an output cardinality lower bound for arbitrary measurable output spaces.
theorem attainment_forces_direction_lower
If a stationary channel attains the oracle variance, its Fisher information in every affine contrast direction is at least the oracle information. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the attainment forces direction lower.
Formal statement
Proof (Lean source)
-- @node: attainment_forces_direction_lower
lemma attainment_forces_direction_lower
{Z : Type*} [MeasurableSpace Z]
(θ : TrialParameter) (p ε t : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(Q : Kernel (Fin 4) Z) (hQ : StationaryLDP ε Q)
(hattain : contrastVariance (channelFisherInfo θ p Q) =
ofReal (Vstar θ p ε)) :
Jstar θ p ε ≤
informationQuadratic (channelFisherInfo θ p Q) (direction t) := by
let I := channelFisherInfo θ p Q
have hI : I.PosSemidef :=
channelFisherInfo_posSemidef p θ hp hθ ε Q hQ
have hJ : 0 < Jstar θ p ε :=
Jstar_pos_interior θ p ε hp hθ hε
have hrange : contrastInRange I := by
by_contra hn
have htop : contrastVariance I = ⊤ := by simp [contrastVariance, hn]
rw [htop, Vstar] at hattain
exact ENNReal.top_ne_ofReal hattain
obtain ⟨w, hw⟩ := hrange
let r := ∑ k : Fin 2, contrastVector k * w k
have hr : 0 < r := contrastSolution_value_pos I hI w hw
have her : ofReal r = ofReal (Jstar θ p ε)⁻¹ := by
rw [← contrastVariance_eq_solution_value I hI w hw]
simpa [I, Vstar] using hattain
have hr_eq : r = (Jstar θ p ε)⁻¹ :=
(ENNReal.ofReal_eq_ofReal_iff hr.le (inv_nonneg.mpr hJ.le)).mp her
have hc := contrastInformation_cauchy I hI w hw t
dsimp [r] at hr_eq
rw [hr_eq] at hc
calc
Jstar θ p ε = 1 * Jstar θ p ε := by ring
_ ≤ (informationQuadratic I (direction t) * (Jstar θ p ε)⁻¹) *
Jstar θ p ε := mul_le_mul_of_nonneg_right hc hJ.le
_ = informationQuadratic I (direction t) := by
field_simp [ne_of_gt hJ]
theorem attaining_staircase_refinement_direction_eq
Under a uniform directional upper bound, an attaining channel has a staircase refinement with equality in that direction, so the refinement's equality-rigidity conclusion is available. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the attaining staircase refinement direction eq.
Formal statement
hypotheses
Z :
Type*
θ :
p ε t :
ℝ
hp :
hθ :
hε :
0 < ε
hQ :
StationaryLDP ε Q
hattain :
hupper :
∀ β : StaircaseWeight
if
then
informationObjective θ p ε β t ≤ Jstar θ p ε
conclusion
conclusion 1
projectedScore θ p ε s t = projectedScore θ p ε u t
Proof (Lean source)
-- @node: attaining_staircase_refinement_direction_eq
lemma attaining_staircase_refinement_direction_eq
{Z : Type*} [MeasurableSpace Z]
(θ : TrialParameter) (p ε t : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(Q : Kernel (Fin 4) Z) (hQ : StationaryLDP ε Q)
(hattain : contrastVariance (channelFisherInfo θ p Q) =
ofReal (Vstar θ p ε))
(hupper : ∀ β : StaircaseWeight, staircaseFeasible ε β →
informationObjective θ p ε β t ≤ Jstar θ p ε) :
∃ α : StaircaseWeight, ∃ K : Kernel (Fin 14) Z,
staircaseFeasible ε α ∧ IsMarkovKernel K ∧
Q = K.comp (staircaseChannel ε α) ∧
informationObjective θ p ε α t = Jstar θ p ε ∧
informationQuadratic (informationMatrix θ p ε α) (direction t) =
informationQuadratic (channelFisherInfo θ p Q) (direction t) ∧
(∀ s u : Fin 14, α s ≠ 0 → α u ≠ 0 → ¬ (K s) ⟂ₘ (K u) →
projectedScore θ p ε s t = projectedScore θ p ε u t) := by
let epsSeq : ℕ → ℝ := fun _ ↦ ε
have hfixed : FixedPrivacy epsSeq ε := ⟨hε, fun _ ↦ rfl⟩
obtain ⟨α, K, hα, hK, hfactor, hdiff, hrigid⟩ :=
staircase_refinement_interior p θ ε hp hθ epsSeq hfixed Q hQ
have horder :
informationQuadratic (channelFisherInfo θ p Q) (direction t) ≤
informationQuadratic (informationMatrix θ p ε α) (direction t) := by
have hn := hdiff.dotProduct_mulVec_nonneg (direction t)
simp [informationQuadratic, dotProduct, mulVec, star_trivial] at hn ⊢
linarith
have hαupper :
informationQuadratic (informationMatrix θ p ε α) (direction t) ≤
Jstar θ p ε := by
have heq : informationObjective θ p ε α t =
informationQuadratic (informationMatrix θ p ε α) (direction t) := by
rw [informationObjective_eq_informationQuadratic θ p ε α t]
rfl
rw [← heq]
exact hupper α hα
have hQlower : Jstar θ p ε ≤
informationQuadratic (channelFisherInfo θ p Q) (direction t) :=
attainment_forces_direction_lower θ p ε t hp hθ hε Q hQ hattain
have hQeq :
informationQuadratic (channelFisherInfo θ p Q) (direction t) =
Jstar θ p ε := le_antisymm (horder.trans hαupper) hQlower
have hαeq :
informationQuadratic (informationMatrix θ p ε α) (direction t) =
Jstar θ p ε := le_antisymm hαupper (hQlower.trans horder)
refine ⟨α, K, hα, hK, hfactor, ?_, hαeq.trans hQeq.symm, ?_⟩
· calc
informationObjective θ p ε α t =
informationQuadratic (informationMatrix θ p ε α) (direction t) := by
rw [informationObjective_eq_informationQuadratic θ p ε α t]
rfl
_ = Jstar θ p ε := hαeq
· exact hrigid t (hαeq.trans hQeq.symm)
theorem outputCardinality_ge_of_attainment_direction
A certified attaining direction with positive, pairwise distinct active scores requires at least as many output atoms as active staircase rays. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the output Cardinality ge of attainment direction.
Formal statement
hypotheses
Z :
Type*
θ :
p ε t :
ℝ
hp :
hθ :
hε :
0 < ε
hupper :
∀ β : StaircaseWeight
if
then
informationObjective θ p ε β t ≤ Jstar θ p ε
hpositive :
∀ β : StaircaseWeight
hdistinct :
∀ s ∈ A, ∀ u ∈ A, s ≠ u → projectedScore θ p ε s t ≠ projectedScore θ p ε u t
hQ :
StationaryLDP ε Q
hattain :
conclusion
(A.card : ℕ∞) ≤ outputCardinality θ p Q
Proof (Lean source)
-- @node: outputCardinality_ge_of_attainment_direction
lemma outputCardinality_ge_of_attainment_direction
{Z : Type*} [MeasurableSpace Z]
(θ : TrialParameter) (p ε t : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(A : Finset (Fin 14))
(hupper : ∀ β : StaircaseWeight, staircaseFeasible ε β →
informationObjective θ p ε β t ≤ Jstar θ p ε)
(hpositive : ∀ β : StaircaseWeight, staircaseFeasible ε β →
informationObjective θ p ε β t = Jstar θ p ε →
∀ s ∈ A, 0 < β s)
(hdistinct : ∀ s ∈ A, ∀ u ∈ A, s ≠ u →
projectedScore θ p ε s t ≠ projectedScore θ p ε u t)
(Q : Kernel (Fin 4) Z) (hQ : StationaryLDP ε Q)
(hattain : contrastVariance (channelFisherInfo θ p Q) =
ofReal (Vstar θ p ε)) :
(A.card : ℕ∞) ≤ outputCardinality θ p Q := by
obtain ⟨α, K, hα, hK, hfactor, hαeq, heq, hrigid⟩ :=
attaining_staircase_refinement_direction_eq θ p ε t hp hθ hε Q hQ
hattain hupper
have hpos : ∀ s ∈ A, 0 < α s := hpositive α hα hαeq
have hsing : ∀ s ∈ A, ∀ u ∈ A, s ≠ u → K s ⟂ₘ K u := by
intro s hs u hu hne
by_contra hn
exact hdistinct s hs u hu hne
(hrigid s u (ne_of_gt (hpos s hs)) (ne_of_gt (hpos u hu)) hn)
exact active_card_le_outputCardinality_of_pairwise_singular θ p ε hp hθ
A α hpos K hK Q hfactor hsing
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.BalancedBinary 20 declarations This file groups the four trial cells by the signed treatment--outcome bit.
Binary submodel for the balanced trial
This file groups the four trial cells by the signed treatment--outcome bit. The grouped experiment is the one-dimensional Bernoulli submodel used in the balanced randomized-response calculation.
def halfNN
the half nn is the mathematical object specified below.
Definition (Lean source)
def
given by
⟨(1 / 2 : ℝ), by norm_num⟩
def half
the half is the mathematical object specified below.
Definition (Lean source)
def
half :
ℝ≥0∞
given by
def groupedChannel
Average the two full-record rows having the same signed bit. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
theorem groupedChannel_apply
Under the supplied quantities and conditions, the grouped channel apply assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma groupedChannel_apply (Q : Kernel (Fin 4) Z) (b : Bool) :
groupedChannel Q b =
if b then half • (Q 0 + Q 3) else half • (Q 1 + Q 2) := by
rfl
theorem groupedChannel_markov
the grouped channel markov assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
Type u_1
Q :
conclusion
Proof (Lean source)
lemma groupedChannel_markov (Q : Kernel (Fin 4) Z) [IsMarkovKernel Q] :
IsMarkovKernel (groupedChannel Q) := by
have hhalf : half * 2 = (1 : ENNReal) := by
change (↑halfNN : ENNReal) * ↑(2 : NNReal) = ↑(1 : NNReal)
rw [← ENNReal.coe_mul]
congr 1
apply NNReal.eq
change (1 / 2 : ℝ) * 2 = 1
norm_num
constructor
intro b
constructor
cases b <;> simp [groupedChannel_apply]
all_goals
calc
half + half = half * 2 := by ring
_ = 1 := hhalf
theorem groupedChannel_private
the grouped channel private assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
SetwisePrivate (groupedChannel Q) ε
Proof (Lean source)
lemma groupedChannel_private (Q : Kernel (Fin 4) Z) {ε : ℝ}
(hQ : StationaryLDP ε Q) :
SetwisePrivate (groupedChannel Q) ε := by
intro b c A hA
cases b <;> cases c
· simp only [groupedChannel_apply, Measure.smul_apply, hA, Measure.add_apply]
calc
half * (Q 1 A + Q 2 A) ≤
half * (ofReal (exp ε) * Q 1 A +
ofReal (exp ε) * Q 2 A) := by
gcongr
exact hQ.2 A hA 1 1
exact hQ.2 A hA 2 2
_ = ofReal (exp ε) * (half * (Q 1 A + Q 2 A)) := by ring
· simp only [groupedChannel_apply, Measure.smul_apply, hA, Measure.add_apply]
calc
half * (Q 1 A + Q 2 A) ≤
half * (ofReal (exp ε) * Q 0 A +
ofReal (exp ε) * Q 3 A) := by
gcongr
exact hQ.2 A hA 1 0
exact hQ.2 A hA 2 3
_ = ofReal (exp ε) * (half * (Q 0 A + Q 3 A)) := by
ring
· simp only [groupedChannel_apply, Measure.smul_apply, hA, Measure.add_apply]
calc
half * (Q 0 A + Q 3 A) ≤
half * (ofReal (exp ε) * Q 1 A +
ofReal (exp ε) * Q 2 A) := by
gcongr
exact hQ.2 A hA 0 1
exact hQ.2 A hA 3 2
_ = ofReal (exp ε) * (half * (Q 1 A + Q 2 A)) := by
ring
· simp only [groupedChannel_apply, Measure.smul_apply, hA, Measure.add_apply]
calc
half * (Q 0 A + Q 3 A) ≤
half * (ofReal (exp ε) * Q 0 A +
ofReal (exp ε) * Q 3 A) := by
gcongr
exact hQ.2 A hA 0 0
exact hQ.2 A hA 3 3
_ = ofReal (exp ε) * (half * (Q 0 A + Q 3 A)) := by ring
theorem reference_groupedChannel
the reference grouped channel assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma reference_groupedChannel (Q : Kernel (Fin 4) Z) :
reference (groupedChannel Q) =
half • dominatingMeasure Q := by
ext A hA
simp [reference, groupedChannel_apply,
dominatingMeasure, Measure.sum_apply, hA, Fin.sum_univ_succ]
ring
theorem groupedDensity_true
the grouped density true assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma groupedDensity_true (Q : Kernel (Fin 4) Z) [IsMarkovKernel Q] :
∀ᵐ z ∂dominatingMeasure Q,
rowDensity (groupedChannel Q) true z =
channelDensity Q 0 z + channelDensity Q 3 z := by
let D := dominatingMeasure Q
let A := Q 0 + Q 3
letI : IsFiniteMeasure D := by dsimp [D, dominatingMeasure]; infer_instance
letI : IsFiniteMeasure A := by dsimp [A]; infer_instance
have hs := Measure.rnDeriv_smul_same A D (r := halfNN) (by
apply ne_of_gt
change 0 < (1 / 2 : ℝ)
norm_num)
have ha := Measure.rnDeriv_add (Q 0) (Q 3) D
filter_upwards [hs, ha, (Q 0).rnDeriv_ne_top D, (Q 3).rnDeriv_ne_top D] with z hs ha h0 h3
unfold rowDensity channelDensity
rw [reference_groupedChannel, groupedChannel_apply]
change (((halfNN • A).rnDeriv (halfNN • D) z).toReal) = _
rw [hs, ha]
change (((Q 0).rnDeriv D z + (Q 3).rnDeriv D z).toReal) = _
rw [ENNReal.toReal_add h0 h3]
theorem groupedDensity_false
the grouped density false assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma groupedDensity_false (Q : Kernel (Fin 4) Z) [IsMarkovKernel Q] :
∀ᵐ z ∂dominatingMeasure Q,
rowDensity (groupedChannel Q) false z =
channelDensity Q 1 z + channelDensity Q 2 z := by
let D := dominatingMeasure Q
let A := Q 1 + Q 2
letI : IsFiniteMeasure D := by dsimp [D, dominatingMeasure]; infer_instance
letI : IsFiniteMeasure A := by dsimp [A]; infer_instance
have hs := Measure.rnDeriv_smul_same A D (r := halfNN) (by
apply ne_of_gt
change 0 < (1 / 2 : ℝ)
norm_num)
have ha := Measure.rnDeriv_add (Q 1) (Q 2) D
filter_upwards [hs, ha, (Q 1).rnDeriv_ne_top D, (Q 2).rnDeriv_ne_top D] with z hs ha h1 h2
unfold rowDensity channelDensity
rw [reference_groupedChannel, groupedChannel_apply]
change (((halfNN • A).rnDeriv (halfNN • D) z).toReal) = _
rw [hs, ha]
change (((Q 1).rnDeriv D z + (Q 2).rnDeriv D z).toReal) = _
rw [ENNReal.toReal_add h1 h2]
theorem abs_contrast_lt_one
Under the supplied quantities and conditions, the abs contrast lt one assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the abs contrast lt one.
Formal statement
hypotheses
conclusion
|contrast θ| < 1
Proof (Lean source)
lemma abs_contrast_lt_one (θ : TrialParameter) (hθ : InteriorMeans θ)
(hbalance : θ 0 + θ 1 = 1) :
|contrast θ| < 1 := by
rcases hθ with ⟨h00, h01, h10, h11⟩
rw [abs_lt]
constructor <;> simp [contrast] <;> linarith
theorem piTheta_balanced_by_val
Under the supplied quantities and conditions, the pi theta balanced by val assertion holds.
Formal statement
hypotheses
θ :
j :
Fin 4
conclusion
piTheta θ (1 / 2) j
= if j.val = 0 then controlProb (1 / 2) * (1 - θ 0) else if j.val = 1 then controlProb (1 / 2) * θ 0 else if j.val = 2 then (1 / 2) * (1 - θ 1) else (1 / 2) * θ 1
Proof (Lean source)
lemma piTheta_balanced_by_val (θ : TrialParameter) (j : Fin 4) :
piTheta θ (1 / 2) j =
if j.val = 0 then controlProb (1 / 2) * (1 - θ 0)
else if j.val = 1 then controlProb (1 / 2) * θ 0
else if j.val = 2 then (1 / 2) * (1 - θ 1)
else (1 / 2) * θ 1 := by
rcases j with ⟨j, hj⟩
interval_cases j <;> rfl
theorem channelOutputDensity_balanced
the channel output density balanced assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
channelOutputDensity θ (1 / 2) Q z
= ((1 - θ 0) * (channelDensity Q 0 z + channelDensity Q 3 z) + θ 0 * (channelDensity Q 1 z + channelDensity Q 2 z)) / 2
Proof (Lean source)
lemma channelOutputDensity_balanced (θ : TrialParameter)
(hbalance : θ 0 + θ 1 = 1) (Q : Kernel (Fin 4) Z) (z : Z) :
channelOutputDensity θ (1 / 2) Q z =
((1 - θ 0) * (channelDensity Q 0 z + channelDensity Q 3 z) +
θ 0 * (channelDensity Q 1 z + channelDensity Q 2 z)) / 2 := by
have hθ1 : θ 1 = 1 - θ 0 := by linarith
unfold channelOutputDensity
rw [show (∑ j : Fin 4, piTheta θ (1 / 2) j * channelDensity Q j z) =
piTheta θ (1 / 2) 0 * channelDensity Q 0 z +
piTheta θ (1 / 2) 1 * channelDensity Q 1 z +
piTheta θ (1 / 2) 2 * channelDensity Q 2 z +
piTheta θ (1 / 2) 3 * channelDensity Q 3 z by
simp [Fin.sum_univ_succ]
ring]
have hp0 : piTheta θ (1 / 2) 0 = (1 - θ 0) / 2 := by
rw [piTheta_balanced_by_val]
norm_num [controlProb]
ring
have hp1 : piTheta θ (1 / 2) 1 = θ 0 / 2 := by
rw [piTheta_balanced_by_val]
norm_num [controlProb]
ring
have hp2 : piTheta θ (1 / 2) 2 = θ 0 / 2 := by
rw [piTheta_balanced_by_val, hθ1]
norm_num
ring
have hp3 : piTheta θ (1 / 2) 3 = (1 - θ 0) / 2 := by
rw [piTheta_balanced_by_val, hθ1]
norm_num
ring
rw [hp0, hp1, hp2, hp3]
ring
theorem directionalDerivative_balanced
Under the supplied quantities and conditions, the directional derivative balanced assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
(∑ k, direction (-1 / 2) k * channelDerivativeDensity (1 / 2) Q k z)
Proof (Lean source)
lemma directionalDerivative_balanced (Q : Kernel (Fin 4) Z) (z : Z) :
(∑ k, direction (-1 / 2) k * channelDerivativeDensity (1 / 2) Q k z) =
(channelDensity Q 0 z + channelDensity Q 3 z -
(channelDensity Q 1 z + channelDensity Q 2 z)) / 4 := by
simp +decide [channelDerivativeDensity, inputDerivative, direction,
controlProb, Fin.sum_univ_succ]
ring
theorem grouped_information_eq_channelQuadratic
Under the supplied quantities and conditions, the grouped information eq channel quadratic assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the grouped information eq channel Quadratic.
Formal statement
hypotheses
conclusion
information (groupedChannel Q) (contrast θ)
Proof (Lean source)
lemma grouped_information_eq_channelQuadratic
(θ : TrialParameter) (hθ : InteriorMeans θ)
(hbalance : θ 0 + θ 1 = 1) (ε : ℝ)
(Q : Kernel (Fin 4) Z) (hQ : StationaryLDP ε Q) :
information (groupedChannel Q) (contrast θ) =
informationQuadratic (channelFisherInfo θ (1 / 2) Q) (direction (-1 / 2)) := by
letI : IsMarkovKernel Q := hQ.1
letI : IsMarkovKernel (groupedChannel Q) := groupedChannel_markov Q
have ht := abs_contrast_lt_one θ hθ hbalance
rw [information_eq_contrast_integral
(groupedChannel Q) (contrast θ) ht]
rw [informationQuadratic_eq_channelIntegral (1 / 2) θ
(by constructor <;> norm_num) hθ ε Q hQ (direction (-1 / 2))]
rw [reference_groupedChannel]
rw [show half • dominatingMeasure Q = halfNN • dominatingMeasure Q by rfl]
rw [integral_smul_nnreal_measure]
change (halfNN : ℝ) * (∫ x,
balancedDensity (groupedChannel Q) x *
contrast (groupedChannel Q) x ^ 2 /
(1 + contrast θ *
contrast (groupedChannel Q) x)
∂dominatingMeasure Q) = _
rw [← integral_const_mul]
apply integral_congr_ae
filter_upwards [groupedDensity_true Q, groupedDensity_false Q] with z hzT hzF
rw [show (halfNN : ℝ) = 1 / 2 by rfl]
unfold balancedDensity
contrast
rw [hzT, hzF]
rw [directionalDerivative_balanced, channelOutputDensity_balanced θ hbalance]
have hcontrast : contrast θ = 1 - 2 * θ 0 := by
simp [contrast]
linarith
rw [hcontrast]
by_cases hs : channelDensity Q 0 z + channelDensity Q 3 z +
(channelDensity Q 1 z + channelDensity Q 2 z) = 0
· have h0 : channelDensity Q 0 z = 0 := by
have hn (j : Fin 4) : 0 ≤ channelDensity Q j z := ENNReal.toReal_nonneg
nlinarith [hn 0, hn 1, hn 2, hn 3]
have h1 : channelDensity Q 1 z = 0 := by
have hn (j : Fin 4) : 0 ≤ channelDensity Q j z := ENNReal.toReal_nonneg
nlinarith [hn 0, hn 1, hn 2, hn 3]
have h2 : channelDensity Q 2 z = 0 := by
have hn (j : Fin 4) : 0 ≤ channelDensity Q j z := ENNReal.toReal_nonneg
nlinarith [hn 0, hn 1, hn 2, hn 3]
have h3 : channelDensity Q 3 z = 0 := by
have hn (j : Fin 4) : 0 ≤ channelDensity Q j z := ENNReal.toReal_nonneg
nlinarith [hn 0, hn 1, hn 2, hn 3]
simp [h0, h1, h2, h3]
· have hn (j : Fin 4) : 0 ≤ channelDensity Q j z := ENNReal.toReal_nonneg
have hsumpos : 0 < channelDensity Q 0 z + channelDensity Q 3 z +
(channelDensity Q 1 z + channelDensity Q 2 z) := by
exact lt_of_le_of_ne
(add_nonneg (add_nonneg (hn 0) (hn 3)) (add_nonneg (hn 1) (hn 2)))
(Ne.symm hs)
have hθ0 := hθ.1
have hθ0' := hθ.2.1
have hdenpos : 0 <
(1 - θ 0) * (channelDensity Q 0 z + channelDensity Q 3 z) +
θ 0 * (channelDensity Q 1 z + channelDensity Q 2 z) := by
by_cases ha : channelDensity Q 0 z + channelDensity Q 3 z = 0
· have hb : 0 < channelDensity Q 1 z + channelDensity Q 2 z := by
linarith
rw [ha]
simpa using mul_pos hθ0 hb
· have ha' : 0 < channelDensity Q 0 z + channelDensity Q 3 z :=
lt_of_le_of_ne (add_nonneg (hn 0) (hn 3)) (Ne.symm ha)
exact add_pos_of_pos_of_nonneg (mul_pos (by linarith) ha')
(mul_nonneg hθ0.le (add_nonneg (hn 1) (hn 2)))
have hden2 : channelDensity Q 0 z * 2 - channelDensity Q 0 z * θ 0 * 2 +
channelDensity Q 3 z * 2 - channelDensity Q 3 z * θ 0 * 2 +
channelDensity Q 1 z * θ 0 * 2 + channelDensity Q 2 z * θ 0 * 2 ≠ 0 := by
intro hz
nlinarith
field_simp [hs, hden2]
rw [show
channelDensity Q 0 z + channelDensity Q 3 z +
(channelDensity Q 1 z + channelDensity Q 2 z) +
(channelDensity Q 0 z + channelDensity Q 3 z -
(channelDensity Q 1 z + channelDensity Q 2 z)) * (1 - 2 * θ 0) =
2 * ((1 - θ 0) * (channelDensity Q 0 z + channelDensity Q 3 z) +
θ 0 * (channelDensity Q 1 z + channelDensity Q 2 z)) by ring]
field_simp [ne_of_gt hdenpos]
ring
theorem groupedStaircase_atom
the grouped staircase atom assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the grouped Staircase atom.
Formal statement
hypotheses
conclusion
= if b then α s * (patternRay ε s 0 + patternRay ε s 3) / 2 else α s * (patternRay ε s 1 + patternRay ε s 2) / 2
Proof (Lean source)
lemma groupedStaircase_atom (ε : ℝ) (α : StaircaseWeight)
(hα : staircaseFeasible ε α) (b : Bool) (s : Fin 14) :
((groupedChannel (staircaseChannel ε α) b) {s}).toReal =
if b then
α s * (patternRay ε s 0 + patternRay ε s 3) / 2
else
α s * (patternRay ε s 1 + patternRay ε s 2) / 2 := by
have hnonneg (j : Fin 4) : 0 ≤ α s * patternRay ε s j := by
apply mul_nonneg (hα.1 s)
unfold patternRay privacyIncrement privacyRatio
split_ifs <;> simp <;> positivity
have hrow (j : Fin 4) :
((staircaseChannel ε α) j {s}).toReal = α s * patternRay ε s j := by
change (Measure.count.withDensity
(fun u : Fin 14 => ofReal (α u * patternRay ε u j)) {s}).toReal = _
rw [withDensity_apply _ (measurableSet_singleton s), lintegral_singleton]
simp
exact hnonneg j
letI : IsMarkovKernel (staircaseChannel ε α) :=
staircaseChannel_markov ε α hα
have hfinite (j : Fin 4) : (staircaseChannel ε α) j {s} ≠ ∞ :=
measure_ne_top _ _
have hhalfReal : half.toReal = (1 / 2 : ℝ) := by
rw [half, ENNReal.coe_toReal]
rfl
cases b
· change (half * ((staircaseChannel ε α) 1 {s} +
(staircaseChannel ε α) 2 {s})).toReal = _
rw [ENNReal.toReal_mul]
rw [ENNReal.toReal_add (hfinite 1) (hfinite 2)]
rw [hrow, hrow]
rw [hhalfReal]
simp
ring
· change (half * ((staircaseChannel ε α) 0 {s} +
(staircaseChannel ε α) 3 {s})).toReal = _
rw [ENNReal.toReal_mul]
rw [ENNReal.toReal_add (hfinite 0) (hfinite 3)]
rw [hrow, hrow]
rw [hhalfReal]
simp
ring
theorem patternMass_balanced
the pattern mass balanced assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
patternMass θ (1 / 2) ε s
= ((1 - θ 0) * (patternRay ε s 0 + patternRay ε s 3) + θ 0 * (patternRay ε s 1 + patternRay ε s 2)) / 2
Proof (Lean source)
lemma patternMass_balanced (θ : TrialParameter)
(hbalance : θ 0 + θ 1 = 1) (ε : ℝ) (s : Fin 14) :
patternMass θ (1 / 2) ε s =
((1 - θ 0) * (patternRay ε s 0 + patternRay ε s 3) +
θ 0 * (patternRay ε s 1 + patternRay ε s 2)) / 2 := by
unfold patternMass
rw [show (∑ j : Fin 4, piTheta θ (1 / 2) j * patternRay ε s j) =
piTheta θ (1 / 2) 0 * patternRay ε s 0 +
piTheta θ (1 / 2) 1 * patternRay ε s 1 +
piTheta θ (1 / 2) 2 * patternRay ε s 2 +
piTheta θ (1 / 2) 3 * patternRay ε s 3 by
simp [Fin.sum_univ_succ]; ring]
have hθ1 : θ 1 = 1 - θ 0 := by linarith
rw [piTheta_balanced_by_val, piTheta_balanced_by_val,
piTheta_balanced_by_val, piTheta_balanced_by_val, hθ1]
norm_num [controlProb]
ring
theorem projectedGradient_balanced
Under the supplied quantities and conditions, the projected gradient balanced assertion holds.
Formal statement
Proof (Lean source)
lemma projectedGradient_balanced (ε : ℝ) (s : Fin 14) :
projectedGradient (1 / 2) ε s (-1 / 2) =
(patternRay ε s 0 + patternRay ε s 3 -
(patternRay ε s 1 + patternRay ε s 2)) / 4 := by
cases h0 : patternContains s 0 <;>
cases h1 : patternContains s 1 <;>
cases h2 : patternContains s 2 <;>
cases h3 : patternContains s 3 <;>
simp [projectedGradient, patternGradient, patternRay, privacyIncrement,
direction, controlProb, Fin.sum_univ_two, h0, h1, h2, h3] <;> ring
theorem groupedStaircase_information_eq_objective
Under the supplied quantities and conditions, the grouped staircase information eq objective assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the grouped Staircase information eq objective.
Formal statement
hypotheses
θ :
hθ :
hbalance :
θ 0 + θ 1 = 1
ε :
ℝ
hε :
0 < ε
α :
hα :
conclusion
= informationObjective θ (1 / 2) ε α (-1 / 2)
Proof (Lean source)
lemma groupedStaircase_information_eq_objective
(θ : TrialParameter) (hθ : InteriorMeans θ)
(hbalance : θ 0 + θ 1 = 1) (ε : ℝ) (hε : 0 < ε)
(α : StaircaseWeight) (hα : staircaseFeasible ε α) :
information
(groupedChannel (staircaseChannel ε α)) (contrast θ) =
informationObjective θ (1 / 2) ε α (-1 / 2) := by
letI : IsMarkovKernel (staircaseChannel ε α) :=
staircaseChannel_markov ε α hα
letI : IsMarkovKernel (groupedChannel (staircaseChannel ε α)) :=
groupedChannel_markov _
rw [information_eq_discrete _ _
(abs_contrast_lt_one θ hθ hbalance)]
unfold discreteInformation informationObjective
rw [tsum_fintype]
apply Finset.sum_congr rfl
intro s hs
rw [groupedStaircase_atom ε α hα true s,
groupedStaircase_atom ε α hα false s]
simp
rw [show (2 : ℝ)⁻¹ = (1 / 2 : ℝ) by norm_num]
unfold patternInformation
rw [patternMass_balanced θ hbalance ε s,
projectedGradient_balanced ε s]
have hcontrast : contrast θ = 1 - 2 * θ 0 := by
simp [contrast]
linarith
rw [hcontrast]
by_cases ha : α s = 0
· simp [ha]
· have hm := patternMass_pos_interior θ (1 / 2) ε
(by constructor <;> norm_num) hθ hε.le s
have hmassne : ((1 - θ 0) * (patternRay ε s 0 + patternRay ε s 3) +
θ 0 * (patternRay ε s 1 + patternRay ε s 2)) ≠ 0 := by
intro h
rw [patternMass_balanced θ hbalance ε s, h] at hm
norm_num at hm
have hd : (1 + (1 - 2 * θ 0)) *
(α s * (patternRay ε s 0 + patternRay ε s 3) / 2) +
(1 - (1 - 2 * θ 0)) *
(α s * (patternRay ε s 1 + patternRay ε s 2) / 2) ≠ 0 := by
rw [show
(1 + (1 - 2 * θ 0)) *
(α s * (patternRay ε s 0 + patternRay ε s 3) / 2) +
(1 - (1 - 2 * θ 0)) *
(α s * (patternRay ε s 1 + patternRay ε s 2) / 2) =
α s * ((1 - θ 0) * (patternRay ε s 0 + patternRay ε s 3) +
θ 0 * (patternRay ε s 1 + patternRay ε s 2)) by ring]
exact mul_ne_zero ha hmassne
field_simp [ha, hd, ne_of_gt hm]
rw [show
(patternRay ε s 0 + patternRay ε s 3) * (1 + (1 - 2 * θ 0)) +
(patternRay ε s 1 + patternRay ε s 2) * (1 - (1 - 2 * θ 0)) =
2 * ((1 - θ 0) * (patternRay ε s 0 + patternRay ε s 3) +
θ 0 * (patternRay ε s 1 + patternRay ε s 2)) by ring]
field_simp [hmassne]
ring
theorem informationObjective_balanced_le_sharp
Under the supplied quantities and conditions, the information objective balanced le sharp assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the information Objective balanced le sharp.
Formal statement
hypotheses
θ :
hθ :
hbalance :
θ 0 + θ 1 = 1
ε :
ℝ
hε :
0 < ε
α :
hα :
conclusion
informationObjective θ (1 / 2) ε α (-1 / 2)
≤ contraction ε ^ 2 / (1 - contraction ε ^ 2 * contrast θ ^ 2)
Proof (Lean source)
lemma informationObjective_balanced_le_sharp
(θ : TrialParameter) (hθ : InteriorMeans θ)
(hbalance : θ 0 + θ 1 = 1) (ε : ℝ) (hε : 0 < ε)
(α : StaircaseWeight) (hα : staircaseFeasible ε α) :
informationObjective θ (1 / 2) ε α (-1 / 2) ≤
contraction ε ^ 2 /
(1 - contraction ε ^ 2 *
contrast θ ^ 2) := by
letI : IsMarkovKernel (staircaseChannel ε α) :=
staircaseChannel_markov ε α hα
letI : IsMarkovKernel (groupedChannel (staircaseChannel ε α)) :=
groupedChannel_markov _
rw [← groupedStaircase_information_eq_objective θ hθ hbalance ε hε α hα]
exact information_le_sharp
(groupedChannel (staircaseChannel ε α)) ε (contrast θ) hε
(abs_contrast_lt_one θ hθ hbalance)
(groupedChannel_private _ ⟨inferInstance, fun A hA j j' =>
staircaseChannel_private ε hε α hα A j j'⟩)
theorem Jstar_balanced_le_sharp
Under the supplied quantities and conditions, the jstar balanced le sharp assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the Jstar balanced le sharp.
Formal statement
hypotheses
conclusion
Jstar θ (1 / 2) ε
≤ contraction ε ^ 2 / (1 - contraction ε ^ 2 * contrast θ ^ 2)
Proof (Lean source)
lemma Jstar_balanced_le_sharp
(θ : TrialParameter) (hθ : InteriorMeans θ)
(hbalance : θ 0 + θ 1 = 1) (ε : ℝ) (hε : 0 < ε) :
Jstar θ (1 / 2) ε ≤
contraction ε ^ 2 /
(1 - contraction ε ^ 2 *
contrast θ ^ 2) := by
refine (Jstar_le_upperEnvelope θ (1 / 2) ε
(by constructor <;> norm_num) hθ hε.le (-1 / 2)).trans ?_
obtain ⟨α, hα, hupper⟩ :=
upperEnvelope_attained θ (1 / 2) ε (-1 / 2) hε.le
rw [hupper]
exact informationObjective_balanced_le_sharp θ hθ hbalance ε hε α hα
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.CenteredR5Certificate 42 declarations Explicit centered five-ray certificate
Explicit centered five-ray certificate
def centeredR5Scale
For the supplied quantities and conditions, the centered r5 scale is the mathematical object specified below.
Definition (Lean source)
def centeredR5Denom
For the supplied quantities and conditions, the centered r5 denom is the mathematical object specified below.
Definition (Lean source)
def centeredR5Direction
For the supplied quantities and conditions, the centered r5 direction is the mathematical object specified below.
Definition (Lean source)
def centeredR5Mix
For the supplied quantities and conditions, the centered r5 mix is the mathematical object specified below.
Definition (Lean source)
def centeredR5Value
For the supplied quantities and conditions, the centered r5 value is the mathematical object specified below.
Definition (Lean source)
parameters
p :
ℝ
def
centeredR5Value p :
ℝ
given by
centeredR5Scale p ^ 2 * (1 - p) ^ 2 / centeredR5Denom p ^ 2
def centeredR5Dual
For the supplied quantities and conditions, the centered r5 dual is the mathematical object specified below.
Definition (Lean source)
parameters
p :
ℝ
def
given by
fun j => if j.val < 2 then -centeredR5Value p / 4 else 3 * centeredR5Value p / 4
def centeredR5LeftScore
For the supplied quantities and conditions, the centered r5 left score is the mathematical object specified below.
Definition (Lean source)
def centeredR5RightScore
For the supplied quantities and conditions, the centered r5 right score is the mathematical object specified below.
Definition (Lean source)
parameters
p :
ℝ
def
centeredR5RightScore p :
ℝ
given by
2 * p * (centeredR5Direction p + 1) / (1 + p)
def centeredR5InformationTable
For the supplied quantities and conditions, the centered r5 information table is the mathematical object specified below.
Definition (Lean source)
parameters
p t :
ℝ
s :
Fin 14
def
centeredR5InformationTable p t s :
ℝ
given by
match s.val with | 0 | 1
=> 4 * (1 - p) ^ 2 * t ^ 2 / (2 - p) | 2 => 0 | 3 | 7 => 4 * p ^ 2 * (t + 1) ^ 2 / (1 + p) | 4 | 9 => 2 * (p + t) ^ 2 | 5 | 8 => 2 * (p - (1 - 2 * p) * t) ^ 2 | 6 | 10 => 4 * p ^ 2 * (t + 1) ^ 2 / (3 - p) | 11 => 0 | _ => 4 * (1 - p) ^ 2 * t ^ 2 / (2 + p)
def centeredR5DualCoeff
For the supplied quantities and conditions, the centered r5 dual coeff is the mathematical object specified below.
Definition (Lean source)
parameters
s :
Fin 14
def
centeredR5DualCoeff s :
ℝ
given by
match s.val with | 0 | 1
=> 1 / 2 | 2 => 0 | 3 | 7 => 5 / 2 | 4 | 9 => 2 | 5 | 8 => 2 | 6 | 10 => 3 / 2 | 11 => 4 | _ => 7 / 2
def centeredR5Weight
For the supplied quantities and conditions, the centered r5 weight is the mathematical object specified below.
Definition (Lean source)
parameters
p :
ℝ
def
given by
fun s
=> if s = 5 ∨ s = 8 then centeredR5Mix p / 4 else if s = 2 ∨ s = 3 ∨ s = 7 then (1 - centeredR5Mix p) / 5 else 0
theorem centered_piTheta
Under the supplied quantities and conditions, the centered pi theta assertion holds.
Formal statement
Proof (Lean source)
lemma centered_piTheta (p : ℝ) (j : Fin 4) :
piTheta (fun _ => (1 / 2 : ℝ)) p j =
if j.val < 2 then (1 - p) / 2 else p / 2 := by
rw [centered_piTheta_by_val]
rcases j with ⟨j, hj⟩
interval_cases j <;> norm_num [controlProb] <;> ring
theorem centered_patternMass
Under the supplied quantities and conditions, the centered pattern mass assertion holds.
Formal statement
hypotheses
p ε :
ℝ
s :
Fin 14
conclusion
patternMass (fun _ => (1 / 2 : ℝ)) p ε s
= ∑ j : Fin 4, (if j.val < 2 then (1 - p) / 2 else p / 2) * patternRay ε s j
Proof (Lean source)
lemma centered_patternMass (p ε : ℝ) (s : Fin 14) :
patternMass (fun _ => (1 / 2 : ℝ)) p ε s =
∑ j : Fin 4, (if j.val < 2 then (1 - p) / 2 else p / 2) *
patternRay ε s j := by
unfold patternMass
simp_rw [centered_piTheta]
theorem centered_patternMass_inv
Under the supplied quantities and conditions, the centered pattern mass inv assertion holds.
Formal statement
hypotheses
p ε :
ℝ
s :
Fin 14
conclusion
patternMass (fun _ => (2 : ℝ)⁻¹) p ε s
= ∑ j : Fin 4, (if j.val < 2 then (1 - p) / 2 else p / 2) * patternRay ε s j
Proof (Lean source)
lemma centered_patternMass_inv (p ε : ℝ) (s : Fin 14) :
patternMass (fun _ => (2 : ℝ)⁻¹) p ε s =
∑ j : Fin 4, (if j.val < 2 then (1 - p) / 2 else p / 2) *
patternRay ε s j := by
simpa only [one_div] using centered_patternMass p ε s
theorem centered_patternInformation
Under the supplied quantities and conditions, the centered pattern information assertion holds.
Formal statement
hypotheses
p ε t :
ℝ
s :
Fin 14
conclusion
patternInformation (fun _ => (2 : ℝ)⁻¹) p ε s t
= projectedGradient p ε s t ^ 2 / (∑ j : Fin 4, (if j.val < 2 then (1 - p) / 2 else p / 2) * patternRay ε s j)
Proof (Lean source)
lemma centered_patternInformation (p ε t : ℝ) (s : Fin 14) :
patternInformation (fun _ => (2 : ℝ)⁻¹) p ε s t =
projectedGradient p ε s t ^ 2 /
(∑ j : Fin 4, (if j.val < 2 then (1 - p) / 2 else p / 2) *
patternRay ε s j) := by
rw [patternInformation, centered_patternMass_inv]
theorem centered_projectedScore
Under the supplied quantities and conditions, the centered projected score assertion holds.
Formal statement
hypotheses
p ε t :
ℝ
s :
Fin 14
conclusion
projectedScore (fun _ => (2 : ℝ)⁻¹) p ε s t
= projectedGradient p ε s t / (∑ j : Fin 4, (if j.val < 2 then (1 - p) / 2 else p / 2) * patternRay ε s j)
Proof (Lean source)
lemma centered_projectedScore (p ε t : ℝ) (s : Fin 14) :
projectedScore (fun _ => (2 : ℝ)⁻¹) p ε s t =
projectedGradient p ε s t /
(∑ j : Fin 4, (if j.val < 2 then (1 - p) / 2 else p / 2) *
patternRay ε s j) := by
rw [projectedScore, centered_patternMass_inv]
theorem centeredR5_information_table
Under the supplied quantities and conditions, the centered r5 information table assertion holds.
Formal statement
hypotheses
p t :
ℝ
s :
Fin 14
conclusion
patternInformation (fun _ => (1 / 2 : ℝ)) p (log 3) s t
= centeredR5InformationTable p t s
Proof (Lean source)
lemma centeredR5_information_table (p t : ℝ) (s : Fin 14) :
patternInformation (fun _ => (1 / 2 : ℝ)) p (log 3) s t =
centeredR5InformationTable p t s := by
have he : exp (log (3 : ℝ)) = 3 := Real.exp_log (by norm_num)
simp only [one_div]
rw [centered_patternInformation]
rcases s with ⟨s, hs⟩
interval_cases s <;>
simp +decide [centeredR5InformationTable, projectedGradient,
patternGradient, patternRay, patternContains, controlProb, direction,
privacyIncrement, privacyRatio, he, Fin.sum_univ_succ] <;> try ring
theorem centeredR5_dual_table
Under the supplied quantities and conditions, the centered r5 dual table assertion holds.
Formal statement
Proof (Lean source)
lemma centeredR5_dual_table (p : ℝ) (s : Fin 14) :
dualRay (log 3) (centeredR5Dual p) s =
centeredR5DualCoeff s * centeredR5Value p := by
have he : exp (log (3 : ℝ)) = 3 := Real.exp_log (by norm_num)
rcases s with ⟨s, hs⟩
interval_cases s <;>
simp +decide [centeredR5DualCoeff, dualRay, centeredR5Dual, patternRay,
patternContains, privacyIncrement, privacyRatio, he,
Fin.sum_univ_succ] <;> ring
theorem centeredR5Scale_sq
Under the supplied quantities and conditions, the centered r5 scale sq assertion holds.
Formal statement
Proof (Lean source)
lemma centeredR5Scale_sq (p : ℝ) (hp0 : 0 < p) :
centeredR5Scale p ^ 2 = 8 * p ^ 2 / (5 * (1 + p)) := by
unfold centeredR5Scale
rw [Real.sq_sqrt]
positivity
theorem centeredR5Scale_bounds
Under the supplied quantities and conditions, the centered r5 scale bounds assertion holds.
Formal statement
hypotheses
p :
ℝ
hp0 :
0 < p
hp1 :
p < 1 / 2
Proof (Lean source)
lemma centeredR5Scale_bounds (p : ℝ) (hp0 : 0 < p) (hp1 : p < 1 / 2) :
p < centeredR5Scale p ∧ centeredR5Scale p < 2 * p := by
have hpden : 0 < 5 * (1 + p) := by positivity
have hs0 : 0 ≤ centeredR5Scale p := Real.sqrt_nonneg _
have hs2 := centeredR5Scale_sq p hp0
field_simp [ne_of_gt hpden] at hs2
constructor
· nlinarith [sq_nonneg (centeredR5Scale p - p)]
· nlinarith [sq_nonneg (2 * p - centeredR5Scale p)]
theorem centeredR5Denom_pos
Under the supplied quantities and conditions, the centered r5 denom pos assertion holds.
Formal statement
Proof (Lean source)
lemma centeredR5Denom_pos (p : ℝ) (hp0 : 0 < p) (hp1 : p < 1 / 2) :
0 < centeredR5Denom p := by
have hs0 : 0 ≤ centeredR5Scale p := Real.sqrt_nonneg _
unfold centeredR5Denom
linarith
theorem centeredR5Mix_mem_Ioo
Under the supplied quantities and conditions, the centered r5 mix mem ioo assertion holds.
Formal statement
hypotheses
p :
ℝ
hp0 :
0 < p
hp1 :
p < 1 / 2
Proof (Lean source)
lemma centeredR5Mix_mem_Ioo (p : ℝ) (hp0 : 0 < p) (hp1 : p < 1 / 2) :
0 < centeredR5Mix p ∧ centeredR5Mix p < 1 := by
have hd := centeredR5Denom_pos p hp0 hp1
have hs := (centeredR5Scale_bounds p hp0 hp1).1
unfold centeredR5Mix
constructor
· exact div_pos (hp0.trans hs) hd
· rw [div_lt_one hd]
unfold centeredR5Denom
linarith
theorem centeredR5Direction_neg
Under the supplied quantities and conditions, the centered r5 direction neg assertion holds.
Formal statement
Proof (Lean source)
lemma centeredR5Direction_neg (p : ℝ) (hp0 : 0 < p) (hp1 : p < 1 / 2) :
centeredR5Direction p < 0 := by
exact div_neg_of_neg_of_pos
(sub_neg.mpr (centeredR5Scale_bounds p hp0 hp1).1)
(centeredR5Denom_pos p hp0 hp1)
theorem centeredR5LeftScore_pos
Under the supplied quantities and conditions, the centered r5 left score pos assertion holds.
Formal statement
Proof (Lean source)
lemma centeredR5LeftScore_pos (p : ℝ) (hp0 : 0 < p) (hp1 : p < 1 / 2) :
0 < centeredR5LeftScore p := by
have ht := centeredR5Direction_neg p hp0 hp1
unfold centeredR5LeftScore
nlinarith
theorem centeredR5RightScore_pos
Under the supplied quantities and conditions, the centered r5 right score pos assertion holds.
Formal statement
Proof (Lean source)
lemma centeredR5RightScore_pos (p : ℝ) (hp0 : 0 < p) (hp1 : p < 1 / 2) :
0 < centeredR5RightScore p := by
have hd := centeredR5Denom_pos p hp0 hp1
have htp1 : 0 < centeredR5Direction p + 1 := by
unfold centeredR5Direction centeredR5Denom
have hd' : 1 - 2 * p + centeredR5Scale p ≠ 0 := by
exact ne_of_gt hd
have hd'' : 1 - p * 2 + centeredR5Scale p ≠ 0 := by
intro h
apply hd'
nlinarith
rw [show (p - centeredR5Scale p) /
(1 - 2 * p + centeredR5Scale p) + 1 =
(1 - p) / (1 - 2 * p + centeredR5Scale p) by
field_simp [hd', hd'']
ring]
exact div_pos (by linarith) hd
unfold centeredR5RightScore
positivity
theorem centeredR5Direction_add_one
Under the supplied quantities and conditions, the centered r5 direction add one assertion holds.
Formal statement
hypotheses
p :
ℝ
hp0 :
0 < p
hp1 :
p < 1 / 2
conclusion
centeredR5Direction p + 1 = (1 - p) / centeredR5Denom p
Proof (Lean source)
lemma centeredR5Direction_add_one (p : ℝ) (hp0 : 0 < p) (hp1 : p < 1 / 2) :
centeredR5Direction p + 1 = (1 - p) / centeredR5Denom p := by
have hd := ne_of_gt (centeredR5Denom_pos p hp0 hp1)
unfold centeredR5Direction
field_simp [hd]
unfold centeredR5Denom
ring
theorem centeredR5_common_score
Under the supplied quantities and conditions, the centered r5 common score assertion holds.
Formal statement
hypotheses
p :
ℝ
hp0 :
0 < p
hp1 :
p < 1 / 2
conclusion
Proof (Lean source)
lemma centeredR5_common_score (p : ℝ) (hp0 : 0 < p) (hp1 : p < 1 / 2) :
p - (1 - 2 * p) * centeredR5Direction p =
centeredR5Scale p * (1 - p) / centeredR5Denom p := by
have hd := ne_of_gt (centeredR5Denom_pos p hp0 hp1)
unfold centeredR5Direction
field_simp [hd]
unfold centeredR5Denom
ring
theorem centeredR5Value_eq_common_score_sq
Under the supplied quantities and conditions, the centered r5 value eq common score sq assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p :
ℝ
hp0 :
0 < p
hp1 :
p < 1 / 2
conclusion
centeredR5Value p = (p - (1 - 2 * p) * centeredR5Direction p) ^ 2
Proof (Lean source)
lemma centeredR5Value_eq_common_score_sq (p : ℝ) (hp0 : 0 < p)
(hp1 : p < 1 / 2) :
centeredR5Value p =
(p - (1 - 2 * p) * centeredR5Direction p) ^ 2 := by
rw [centeredR5_common_score p hp0 hp1]
unfold centeredR5Value
ring
theorem centeredR5Value_pos
Under the supplied quantities and conditions, the centered r5 value pos assertion holds.
Formal statement
Proof (Lean source)
lemma centeredR5Value_pos (p : ℝ) (hp0 : 0 < p) (hp1 : p < 1 / 2) :
0 < centeredR5Value p := by
unfold centeredR5Value
have hs : 0 < centeredR5Scale p :=
hp0.trans (centeredR5Scale_bounds p hp0 hp1).1
have hd := centeredR5Denom_pos p hp0 hp1
exact div_pos (mul_pos (sq_pos_of_pos hs) (sq_pos_of_pos (by linarith)))
(sq_pos_of_pos hd)
theorem centeredR5_slack_mask12
Under the supplied quantities and conditions, the centered r5 slack mask12 assertion holds.
Formal statement
hypotheses
p :
ℝ
hp0 :
0 < p
hp1 :
p < 1 / 2
conclusion
4 * (1 - p) ^ 2 * centeredR5Direction p ^ 2 / (2 - p) < centeredR5Value p / 2
Proof (Lean source)
lemma centeredR5_slack_mask12 (p : ℝ) (hp0 : 0 < p) (hp1 : p < 1 / 2) :
4 * (1 - p) ^ 2 * centeredR5Direction p ^ 2 / (2 - p) <
centeredR5Value p / 2 := by
have hd := centeredR5Denom_pos p hp0 hp1
have hd' := ne_of_gt hd
have hp2 : 0 < 2 - p := by linarith
have hr := centeredR5_mask12_reduced p hp0 hp1
have hq : 0 < (1 - p) ^ 2 := sq_pos_of_pos (by linarith)
unfold centeredR5Direction centeredR5Value
rw [div_lt_div_iff₀ hp2 (by positivity : (0 : ℝ) < 2)]
field_simp [hd']
nlinarith [mul_lt_mul_of_pos_left hr hq]
theorem centeredR5_slack_mask5_10
Under the supplied quantities and conditions, the centered r5 slack mask5 10 assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p :
ℝ
hp0 :
0 < p
hp1 :
p < 1 / 2
conclusion
2 * (p + centeredR5Direction p) ^ 2 < 2 * centeredR5Value p
Proof (Lean source)
lemma centeredR5_slack_mask5_10 (p : ℝ) (hp0 : 0 < p)
(hp1 : p < 1 / 2) :
2 * (p + centeredR5Direction p) ^ 2 < 2 * centeredR5Value p := by
have hd := centeredR5Denom_pos p hp0 hp1
have hd' := ne_of_gt hd
have hsb := centeredR5Scale_bounds p hp0 hp1
have hq : 0 < 1 - p := by linarith
have hpt : p + centeredR5Direction p =
(1 - p) * (2 * p - centeredR5Scale p) / centeredR5Denom p := by
unfold centeredR5Direction
field_simp [hd']
unfold centeredR5Denom
ring
have hl := centeredR5_common_score p hp0 hp1
have hpt0 : 0 < p + centeredR5Direction p := by
rw [hpt]
exact div_pos (mul_pos hq (sub_pos.mpr hsb.2)) hd
have hlt : p + centeredR5Direction p <
p - (1 - 2 * p) * centeredR5Direction p := by
rw [hpt, hl, div_lt_div_iff_of_pos_right hd]
nlinarith [mul_pos hq (sub_pos.mpr hsb.1)]
have hl0 : 0 < p - (1 - 2 * p) * centeredR5Direction p := by
rw [hl]
exact div_pos (mul_pos (hp0.trans hsb.1) hq) hd
rw [centeredR5Value_eq_common_score_sq p hp0 hp1]
nlinarith [mul_pos (add_pos hpt0 hl0) (sub_pos.mpr hlt)]
theorem centeredR5_slack_mask7_11
Under the supplied quantities and conditions, the centered r5 slack mask7 11 assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p :
ℝ
hp0 :
0 < p
hp1 :
p < 1 / 2
conclusion
4 * p ^ 2 * (centeredR5Direction p + 1) ^ 2 / (3 - p) < (3 / 2 : ℝ) * centeredR5Value p
Proof (Lean source)
lemma centeredR5_slack_mask7_11 (p : ℝ) (hp0 : 0 < p)
(hp1 : p < 1 / 2) :
4 * p ^ 2 * (centeredR5Direction p + 1) ^ 2 / (3 - p) <
(3 / 2 : ℝ) * centeredR5Value p := by
have hp3 : 0 < 3 - p := by linarith
have hd := centeredR5Denom_pos p hp0 hp1
have hd' := ne_of_gt hd
have hs2 := centeredR5Scale_sq p hp0
rw [centeredR5Direction_add_one p hp0 hp1]
unfold centeredR5Value
rw [div_lt_iff₀ hp3]
field_simp [hd']
field_simp at hs2
have hspos : 0 < centeredR5Scale p :=
hp0.trans (centeredR5Scale_bounds p hp0 hp1).1
have hc : 8 * p ^ 2 < 3 * centeredR5Scale p ^ 2 * (3 - p) := by
nlinarith [sq_pos_of_pos hspos]
have hq : 0 < (1 - p) ^ 2 := sq_pos_of_pos (by linarith)
nlinarith [mul_lt_mul_of_pos_right hc hq]
theorem centeredR5_slack_mask12_zero
Under the supplied quantities and conditions, the centered r5 slack mask12 zero assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma centeredR5_slack_mask12_zero (p : ℝ) (hp0 : 0 < p)
(hp1 : p < 1 / 2) : (0 : ℝ) < 4 * centeredR5Value p := by
exact mul_pos (by norm_num) (centeredR5Value_pos p hp0 hp1)
theorem centeredR5_slack_mask13_14
Under the supplied quantities and conditions, the centered r5 slack mask13 14 assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p :
ℝ
hp0 :
0 < p
hp1 :
p < 1 / 2
conclusion
4 * (1 - p) ^ 2 * centeredR5Direction p ^ 2 / (2 + p) < (7 / 2 : ℝ) * centeredR5Value p
Proof (Lean source)
lemma centeredR5_slack_mask13_14 (p : ℝ) (hp0 : 0 < p)
(hp1 : p < 1 / 2) :
4 * (1 - p) ^ 2 * centeredR5Direction p ^ 2 / (2 + p) <
(7 / 2 : ℝ) * centeredR5Value p := by
have hd := centeredR5Denom_pos p hp0 hp1
have hd' := ne_of_gt hd
have hp2 : 0 < 2 + p := by linarith
have hr := centeredR5_mask12_reduced p hp0 hp1
have hq : 0 < (1 - p) ^ 2 := sq_pos_of_pos (by linarith)
have hlarge : 8 * (p - centeredR5Scale p) ^ 2 <
7 * (2 + p) * centeredR5Scale p ^ 2 := by
have hspos : 0 < centeredR5Scale p :=
hp0.trans (centeredR5Scale_bounds p hp0 hp1).1
nlinarith [sq_pos_of_pos hspos]
unfold centeredR5Direction centeredR5Value
rw [div_lt_iff₀ hp2]
field_simp [hd']
nlinarith [mul_lt_mul_of_pos_left hlarge hq]
theorem centeredR5Weight_feasible
Under the supplied quantities and conditions, the centered r5 weight feasible assertion holds.
Formal statement
hypotheses
p :
ℝ
hp0 :
0 < p
hp1 :
p < 1 / 2
conclusion
Proof (Lean source)
lemma centeredR5Weight_feasible (p : ℝ) (hp0 : 0 < p) (hp1 : p < 1 / 2) :
staircaseFeasible (log 3) (centeredR5Weight p) := by
have he : exp (log (3 : ℝ)) = 3 := Real.exp_log (by norm_num)
obtain ⟨hmix0, hmix1⟩ := centeredR5Mix_mem_Ioo p hp0 hp1
constructor
· intro s
simp only [centeredR5Weight]
split_ifs <;> positivity
· intro j
fin_cases j <;>
simp +decide [staircaseMatrix, centeredR5Weight, patternRay,
patternContains, privacyRatio, privacyIncrement, he,
Fin.sum_univ_succ] <;> ring
theorem centeredR5Weight_active
Under the supplied quantities and conditions, the centered r5 weight active assertion holds.
Formal statement
hypotheses
p :
ℝ
hp0 :
0 < p
hp1 :
p < 1 / 2
conclusion
conclusion 1
activeSupport (centeredR5Weight p) r5Active
Proof (Lean source)
lemma centeredR5Weight_active (p : ℝ) (hp0 : 0 < p) (hp1 : p < 1 / 2) :
activeSupport (centeredR5Weight p) r5Active ∧
∀ s ∈ r5Active, 0 < centeredR5Weight p s := by
obtain ⟨hm0, hm1⟩ := centeredR5Mix_mem_Ioo p hp0 hp1
constructor
· intro s
rcases s with ⟨s, hs⟩
interval_cases s <;> simp_all [centeredR5Weight, r5Active] <;> linarith
· intro s hs
rcases s with ⟨s, hslt⟩
interval_cases s <;> simp_all [centeredR5Weight, r5Active] <;> positivity
theorem centeredR5Weight_stationary
Under the supplied quantities and conditions, the centered r5 weight stationary assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p :
ℝ
hp0 :
0 < p
hp1 :
p < 1 / 2
conclusion
(∑ s : Fin 14, centeredR5Weight p s * patternInformationSlope (fun _ => (1 / 2 : ℝ)) p (log 3) s (centeredR5Direction p))
= 0
Proof (Lean source)
lemma centeredR5Weight_stationary (p : ℝ) (hp0 : 0 < p)
(hp1 : p < 1 / 2) :
(∑ s : Fin 14, centeredR5Weight p s *
patternInformationSlope (fun _ => (1 / 2 : ℝ)) p (log 3) s
(centeredR5Direction p)) = 0 := by
have he : exp (log (3 : ℝ)) = 3 :=
Real.exp_log (by norm_num)
have hd := centeredR5Denom_pos p hp0 hp1
have hd' : 1 + centeredR5Scale p - p * 2 ≠ 0 := by
intro hz
unfold centeredR5Denom at hd
nlinarith
have hs2 := centeredR5Scale_sq p hp0
unfold patternInformationSlope patternMass
simp_rw [centered_piTheta]
simp +decide [centeredR5Weight, centeredR5Direction, centeredR5Mix,
centeredR5Denom, projectedGradient, patternGradient, patternRay,
patternContains, controlProb, direction, privacyIncrement,
privacyRatio, he, Fin.sum_univ_succ]
ring_nf
field_simp [hd']
field_simp at hs2
nlinarith
theorem centeredR5_active_dual_equalities
Under the supplied quantities and conditions, the centered r5 active dual equalities assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p :
ℝ
hp0 :
0 < p
hp1 :
p < 1 / 2
s ∈ r5Active :
conclusion
Proof (Lean source)
lemma centeredR5_active_dual_equalities (p : ℝ) (hp0 : 0 < p)
(hp1 : p < 1 / 2) :
∀ s ∈ r5Active,
dualRay (log 3) (centeredR5Dual p) s =
patternInformation (fun _ => (1 / 2 : ℝ)) p (log 3) s
(centeredR5Direction p) := by
intro s hs
rw [centeredR5_dual_table, centeredR5_information_table]
have hs2 := centeredR5Scale_sq p hp0
have ht := centeredR5Direction_add_one p hp0 hp1
have hv := centeredR5Value_eq_common_score_sq p hp0 hp1
rcases s with ⟨s, hsl⟩
interval_cases s <;>
simp_all [r5Active, centeredR5DualCoeff, centeredR5InformationTable]
all_goals
have hp1' : p < 1 / 2 := by simpa only [one_div] using hp1
rw [centeredR5_common_score p hp0 hp1']
have hd := ne_of_gt (centeredR5Denom_pos p hp0 hp1')
field_simp [hd]
field_simp at hs2
nlinarith
theorem centeredR5_inactive_dual_slacks
Under the supplied quantities and conditions, the centered r5 inactive dual slacks assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p :
ℝ
hp0 :
0 < p
hp1 :
p < 1 / 2
s ∉ r5Active :
conclusion
patternInformation (fun _ => (1 / 2 : ℝ)) p (log 3) s (centeredR5Direction p) < dualRay (log 3) (centeredR5Dual p) s
Proof (Lean source)
lemma centeredR5_inactive_dual_slacks (p : ℝ) (hp0 : 0 < p)
(hp1 : p < 1 / 2) :
∀ s ∉ r5Active,
patternInformation (fun _ => (1 / 2 : ℝ)) p (log 3) s
(centeredR5Direction p) <
dualRay (log 3) (centeredR5Dual p) s := by
intro s hs
rw [centeredR5_dual_table, centeredR5_information_table]
have hA := centeredR5_slack_mask12 p hp0 hp1
have hB := centeredR5_slack_mask5_10 p hp0 hp1
have hC := centeredR5_slack_mask7_11 p hp0 hp1
have hD := centeredR5_slack_mask12_zero p hp0 hp1
have hE := centeredR5_slack_mask13_14 p hp0 hp1
rcases s with ⟨s, hsl⟩
interval_cases s <;>
simp_all [r5Active, centeredR5DualCoeff, centeredR5InformationTable]
all_goals
convert hA using 1
ring
theorem centeredR5_active_scores_pairwise_ne
Under the supplied quantities and conditions, the centered r5 active scores pairwise ne assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p :
ℝ
hp0 :
0 < p
hp1 :
p < 1 / 2
s ∈ r5Active :
u ∈ r5Active :
s ≠ u
conclusion
projectedScore (fun _ => (1 / 2 : ℝ)) p (log 3) s (centeredR5Direction p)
≠ projectedScore (fun _ => (1 / 2 : ℝ)) p (log 3) u (centeredR5Direction p)
Proof (Lean source)
lemma centeredR5_active_scores_pairwise_ne (p : ℝ) (hp0 : 0 < p)
(hp1 : p < 1 / 2) :
∀ s ∈ r5Active, ∀ u ∈ r5Active, s ≠ u →
projectedScore (fun _ => (1 / 2 : ℝ)) p (log 3) s
(centeredR5Direction p) ≠
projectedScore (fun _ => (1 / 2 : ℝ)) p (log 3) u
(centeredR5Direction p) := by
intro s hs u hu hne
wlog hlt : s.val < u.val generalizing s u
· apply Ne.symm
apply this u hu s hs (Ne.symm hne)
omega
have he : exp (log (3 : ℝ)) = 3 := Real.exp_log (by norm_num)
have hd := centeredR5Denom_pos p hp0 hp1
have hd' : 1 + centeredR5Scale p - p * 2 ≠ 0 := by
intro hz
unfold centeredR5Denom at hd
nlinarith
have hd2 : 1 - p * 2 + centeredR5Scale p ≠ 0 := by
intro hz
apply hd'
nlinarith
have hsb := centeredR5Scale_bounds p hp0 hp1
have hs2 := centeredR5Scale_sq p hp0
rcases s with ⟨s, hslt⟩
rcases u with ⟨u, hult⟩
interval_cases s <;> interval_cases u <;> simp_all [r5Active]
all_goals
unfold projectedScore
simp_rw [centered_patternMass_inv]
simp +decide [centeredR5Direction, centeredR5Denom, projectedGradient,
patternGradient, patternRay, patternContains, controlProb, direction,
privacyIncrement, privacyRatio, he, Fin.sum_univ_succ]
ring_nf
field_simp [hd', hd2]
ring_nf
field_simp [hd', hd2]
field_simp at hs2
simp only [inv_eq_one_div] at *
field_simp [hd', hd2] at *
ring_nf at *
nlinarith
theorem centeredR5_supported_stationary_eq
Under the supplied quantities and conditions, the centered r5 supported stationary eq assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the centered R5 supported stationary eq.
Formal statement
hypotheses
p :
ℝ
hp0 :
0 < p
hp1 :
p < 1 / 2
β :
hβ :
staircaseFeasible (log 3) β
hout :
∀ s ∉ r5Active, β s = 0
hstat :
∑ s : Fin 14, β s * patternInformationSlope (fun _ => (1 / 2 : ℝ)) p (log 3) s (centeredR5Direction p)
= 0
conclusion
β = centeredR5Weight p
Proof (Lean source)
lemma centeredR5_supported_stationary_eq (p : ℝ) (hp0 : 0 < p) (hp1 : p < 1 / 2)
(β : StaircaseWeight) (hβ : staircaseFeasible (log 3) β)
(hout : ∀ s ∉ r5Active, β s = 0)
(hstat : ∑ s : Fin 14, β s *
patternInformationSlope (fun _ => (1 / 2 : ℝ)) p (log 3) s
(centeredR5Direction p) = 0) :
β = centeredR5Weight p := by
obtain ⟨h23, h37, h58, hnorm⟩ :=
r5_supported_feasible_coordinates (log 3) (Real.log_pos (by norm_num)) β hβ hout
have he : exp (log (3 : ℝ)) = 3 := Real.exp_log (by norm_num)
have hd := centeredR5Denom_pos p hp0 hp1
have hd' : 1 + centeredR5Scale p - p * 2 ≠ 0 := by
intro hz
unfold centeredR5Denom at hd
nlinarith
have hd2 : 1 - p * 2 + centeredR5Scale p ≠ 0 := by
intro hz
apply hd'
nlinarith
have hs2 := centeredR5Scale_sq p hp0
have hstat' := hstat
rw [he] at hnorm
unfold patternInformationSlope patternMass at hstat'
simp_rw [centered_piTheta] at hstat'
simp +decide [hout, r5Active, centeredR5Direction, centeredR5Denom,
projectedGradient, patternGradient, patternRay, patternContains,
controlProb, direction, privacyIncrement, privacyRatio, he,
Fin.sum_univ_succ] at hstat'
rw [← h37, ← h23, ← h58] at hstat'
ring_nf at hstat'
field_simp [hd', hd2] at hstat'
ring_nf at hstat'
field_simp at hs2
let C := 10 * centeredR5Scale p * p ^ 2 +
5 * centeredR5Scale p * p - 5 * centeredR5Scale p - 8 * p ^ 2
have hfac : (-8 / 5 : ℝ) * (p - 1) * (C * β 5 + 2 * p ^ 2) = 0 := by
rw [← hstat']
dsimp [C]
rw [show β 2 = (1 - 4 * β 5) / 5 by nlinarith [hnorm]]
ring
have hX : C * β 5 + 2 * p ^ 2 = 0 := by
rcases mul_eq_zero.mp hfac with hleft | hX
· rcases mul_eq_zero.mp hleft with hnum | hp
· norm_num at hnum
· linarith
· exact hX
have hCS : C * centeredR5Scale p =
-8 * p ^ 2 * centeredR5Denom p := by
dsimp [C, centeredR5Denom]
nlinarith [hs2]
have hspos : 0 < centeredR5Scale p :=
hp0.trans (centeredR5Scale_bounds p hp0 hp1).1
have hC : C ≠ 0 := by
intro hC
rw [hC, zero_mul] at hCS
nlinarith [sq_pos_of_pos hp0]
have htarget : 4 * centeredR5Denom p * β 5 = centeredR5Scale p := by
apply (mul_left_cancel₀ hC)
calc
C * (4 * centeredR5Denom p * β 5) =
4 * centeredR5Denom p * (C * β 5) := by ring
_ = C * centeredR5Scale p := by rw [hCS]; nlinarith [hX]
have hfive : β 5 = centeredR5Weight p 5 := by
simp [centeredR5Weight, centeredR5Mix, centeredR5Denom]
field_simp [hd', hd2]
apply (eq_div_iff (by
unfold centeredR5Denom at hd
exact ne_of_gt hd)).2
unfold centeredR5Denom at htarget
nlinarith [htarget]
exact r5_supported_feasible_eq_of_five_eq (log 3)
(Real.log_pos (by norm_num)) β (centeredR5Weight p) hβ
(centeredR5Weight_feasible p hp0 hp1) hout
(fun s hs => by
by_contra hn
exact hs (((centeredR5Weight_active p hp0 hp1).1 s).mp hn)) hfive
theorem centeredPoint_mem_R5
Under the supplied quantities and conditions, the centered point mem r5 assertion holds.
Formal statement
Proof (Lean source)
lemma centeredPoint_mem_R5 (p : ℝ) (hp0 : 0 < p) (hp1 : p < 1 / 2) :
(p, (1 / 2 : ℝ), (1 / 2 : ℝ), log 3) ∈ R5 := by
simp only [R5, certifiedRegion, Set.mem_ofPred_eq]
refine ⟨hp0, by linarith, by norm_num, by norm_num, by norm_num,
by norm_num, Real.log_pos (by norm_num), ?_⟩
have hθ : (fun k : Fin 2 => if k = 0 then (1 / 2 : ℝ) else 1 / 2) =
(fun _ => (1 / 2 : ℝ)) := by
funext k
split <;> rfl
simp only [hθ]
refine ⟨centeredR5Weight p, centeredR5Direction p, centeredR5Dual p,
centeredR5Weight_feasible p hp0 hp1, ?_, ?_, ?_, ?_, ?_, ?_⟩
· exact (centeredR5Weight_active p hp0 hp1).1
· exact (centeredR5Weight_active p hp0 hp1).2
· simpa only [one_div] using centeredR5Weight_stationary p hp0 hp1
· intro s hs
simpa only [one_div] using centeredR5_active_dual_equalities p hp0 hp1 s hs
· intro s hs
simpa only [one_div] using centeredR5_inactive_dual_slacks p hp0 hp1 s hs
· intro _ s hs u hu hne
simpa only [one_div] using
centeredR5_active_scores_pairwise_ne p hp0 hp1 s hs u hu hne
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.CenteredR5Rigidity 2 declarations Centered five-ray profile rigidity
Centered five-ray profile rigidity
theorem centeredR5_profile_optimal_unique
Under the supplied quantities and conditions, the centered r5 profile optimal unique assertion holds.
Formal statement
hypotheses
p :
ℝ
hp0 :
0 < p
hp1 :
p < 1 / 2
conclusion
conclusion 1
conclusion 2
= sInf {u : ℝ | ∃ t : ℝ, u = informationObjective (fun _ => (1 / 2 : ℝ)) p (log 3) (centeredR5Weight p) t}
Proof (Lean source)
lemma centeredR5_profile_optimal_unique (p : ℝ) (hp0 : 0 < p) (hp1 : p < 1 / 2) :
staircaseFeasible (log 3) (centeredR5Weight p) ∧
Jstar (fun _ => (1 / 2 : ℝ)) p (log 3) =
sInf {u : ℝ | ∃ t : ℝ, u = informationObjective
(fun _ => (1 / 2 : ℝ)) p (log 3) (centeredR5Weight p) t} ∧
∀ β : StaircaseWeight, staircaseFeasible (log 3) β →
Jstar (fun _ => (1 / 2 : ℝ)) p (log 3) =
sInf {u : ℝ | ∃ t : ℝ, u = informationObjective
(fun _ => (1 / 2 : ℝ)) p (log 3) β t} →
β = centeredR5Weight p := by
let θ : TrialParameter := fun _ => (1 / 2 : ℝ)
let α := centeredR5Weight p
let t := centeredR5Direction p
let η := centeredR5Dual p
have hp : InteriorAssignment p := ⟨hp0, by linarith⟩
have hθ : InteriorMeans θ := by
dsimp [θ, InteriorMeans]
norm_num
have hε : 0 ≤ log 3 := (Real.log_pos (by norm_num)).le
have hα : staircaseFeasible (log 3) α := centeredR5Weight_feasible p hp0 hp1
have hsupp : activeSupport α r5Active := (centeredR5Weight_active p hp0 hp1).1
have hstat : ∑ s : Fin 14, α s * patternInformationSlope θ p
(log 3) s t = 0 := by
simpa [θ, α, t] using centeredR5Weight_stationary p hp0 hp1
have hactive : ∀ s ∈ r5Active,
dualRay (log 3) η s = patternInformation θ p (log 3) s t := by
intro s hs
simpa [θ, η, t] using centeredR5_active_dual_equalities p hp0 hp1 s hs
have hinactive : ∀ s ∉ r5Active,
patternInformation θ p (log 3) s t < dualRay (log 3) η s := by
intro s hs
simpa [θ, η, t] using centeredR5_inactive_dual_slacks p hp0 hp1 s hs
obtain ⟨hopt, hrigid⟩ := strictCertificate_profile_optimal_and_rigid
θ p (log 3) hp hθ hε r5Active α t η hα hsupp hstat hactive hinactive
refine ⟨hα, hopt, ?_⟩
intro β hβ hβopt
have hout := hrigid β hβ hβopt
have hbelow : BddBelow {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p (log 3) β v} := by
refine ⟨0, ?_⟩
rintro u ⟨v, rfl⟩
exact informationObjective_nonneg_interior θ p (log 3) hp hθ hε β hβ v
have hvalue : informationObjective θ p (log 3) α t = ∑ j : Fin 4, η j := by
calc
_ = ∑ s : Fin 14, α s * dualRay (log 3) η s := by
apply Finset.sum_congr rfl
intro s _
by_cases hs : s ∈ r5Active
· rw [hactive s hs]
· have hz : α s = 0 := by
by_contra hn
exact hs ((hsupp s).mp hn)
simp [hz]
_ = _ := feasible_dualRay_sum_general (log 3) α hα η
have hprofile : sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p (log 3) α v} =
informationObjective θ p (log 3) α t := by
apply IsLeast.csInf_eq
refine ⟨⟨t, rfl⟩, ?_⟩
rintro _ ⟨v, rfl⟩
have hslope := centeredR5Weight_stationary p hp0 hp1
exact informationObjective_min_of_stationary θ p (log 3) hp hθ hε α hα t
(by
rw [← hslope]
apply Finset.sum_congr rfl
intro s _
simp [θ, α, t, patternInformationSlope]
ring) v
have hβat : informationObjective θ p (log 3) β t =
Jstar θ p (log 3) := by
apply le_antisymm
· calc
_ ≤ ∑ j : Fin 4, η j :=
certificate_dual_upper_bound θ p (log 3) t r5Active η
hactive hinactive β hβ
_ = _ := by rw [← hvalue, ← hprofile, ← hopt]
· rw [hβopt]
exact csInf_le hbelow ⟨t, rfl⟩
have hmin (u : ℝ) : informationObjective θ p (log 3) β t ≤
informationObjective θ p (log 3) β u := by
rw [hβat, hβopt]
exact csInf_le hbelow ⟨u, rfl⟩
have hraw := informationObjective_stationary_of_min θ p (log 3) β t hmin
have hslope : ∑ s : Fin 14, β s * patternInformationSlope θ p
(log 3) s t = 0 := by
rw [← hraw]
apply Finset.sum_congr rfl
intro s _
simp [patternInformationSlope]
ring
exact centeredR5_supported_stationary_eq p hp0 hp1 β hβ
(by simpa [r5Active] using hout) (by simpa [θ, t] using hslope)
theorem centeredR5_outputCardinality_ge_five
Under the supplied quantities and conditions, the centered r5 output cardinality ge five assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p :
ℝ
hp0 :
0 < p
hp1 :
p < 1 / 2
Z :
Type*
hQ :
StationaryLDP (log 3) Q
hattain :
contrastVariance (channelFisherInfo (fun _ => (1 / 2 : ℝ)) p Q)
conclusion
5 ≤ outputCardinality (fun _ => (1 / 2 : ℝ)) p Q
Proof (Lean source)
lemma centeredR5_outputCardinality_ge_five (p : ℝ) (hp0 : 0 < p) (hp1 : p < 1 / 2)
(Z : Type*) [MeasurableSpace Z] (Q : Kernel (Fin 4) Z)
(hQ : StationaryLDP (log 3) Q)
(hattain : contrastVariance
(channelFisherInfo (fun _ => (1 / 2 : ℝ)) p Q) =
ofReal (Vstar (fun _ => (1 / 2 : ℝ)) p (log 3))) :
5 ≤ outputCardinality (fun _ => (1 / 2 : ℝ)) p Q := by
let θ : TrialParameter := fun _ => (1 / 2 : ℝ)
let α0 := centeredR5Weight p
let t0 := centeredR5Direction p
let η := centeredR5Dual p
have hp : InteriorAssignment p := ⟨hp0, by linarith⟩
have hθ : InteriorMeans θ := by dsimp [θ, InteriorMeans]; norm_num
have hε : 0 < log 3 := Real.log_pos (by norm_num)
have hα0 : staircaseFeasible (log 3) α0 :=
centeredR5Weight_feasible p hp0 hp1
have hsupp0 : activeSupport α0 r5Active :=
(centeredR5Weight_active p hp0 hp1).1
have hactive : ∀ s ∈ r5Active,
dualRay (log 3) η s = patternInformation θ p (log 3) s t0 := by
intro s hs
simpa [θ, η, t0] using centeredR5_active_dual_equalities p hp0 hp1 s hs
have hinactive : ∀ s ∉ r5Active,
patternInformation θ p (log 3) s t0 < dualRay (log 3) η s := by
intro s hs
simpa [θ, η, t0] using centeredR5_inactive_dual_slacks p hp0 hp1 s hs
have hvalue : informationObjective θ p (log 3) α0 t0 =
∑ j : Fin 4, η j := by
calc
_ = ∑ s : Fin 14, α0 s * dualRay (log 3) η s := by
apply Finset.sum_congr rfl
intro s _
by_cases hs : s ∈ r5Active
· rw [hactive s hs]
· have hz : α0 s = 0 := by
by_contra hn
exact hs ((hsupp0 s).mp hn)
simp [hz]
_ = _ := feasible_dualRay_sum_general (log 3) α0 hα0 η
obtain ⟨_, hopt, _⟩ := centeredR5_profile_optimal_unique p hp0 hp1
have hmin0 (u : ℝ) : informationObjective θ p (log 3) α0 t0 ≤
informationObjective θ p (log 3) α0 u := by
have hslope := centeredR5Weight_stationary p hp0 hp1
exact informationObjective_min_of_stationary θ p (log 3) hp hθ hε.le
α0 hα0 t0 (by
rw [← hslope]
apply Finset.sum_congr rfl
intro s _
simp [θ, α0, t0, patternInformationSlope]
ring) u
have hleast : sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p (log 3) α0 v} =
informationObjective θ p (log 3) α0 t0 := by
apply IsLeast.csInf_eq
exact ⟨⟨t0, rfl⟩, by rintro _ ⟨u, rfl⟩; exact hmin0 u⟩
have hJsum : Jstar θ p (log 3) = ∑ j : Fin 4, η j := by
rw [hopt, hleast, hvalue]
have hupper (β : StaircaseWeight) (hβ : staircaseFeasible (log 3) β) :
informationObjective θ p (log 3) β t0 ≤ Jstar θ p (log 3) := by
rw [hJsum]
exact certificate_dual_upper_bound θ p (log 3) t0 r5Active η
hactive hinactive β hβ
let epsSeq : ℕ → ℝ := fun _ => log 3
have hfixed : FixedPrivacy epsSeq (log 3) := ⟨hε, fun _ => rfl⟩
obtain ⟨β, K, hβ, hK, hfactor, hdiff, hrigid⟩ :=
staircase_refinement_interior p θ (log 3) hp hθ epsSeq hfixed Q hQ
have horder (u : ℝ) :
informationQuadratic (channelFisherInfo θ p Q) (direction u) ≤
informationQuadratic (informationMatrix θ p (log 3) β) (direction u) := by
have hn := hdiff.dotProduct_mulVec_nonneg (direction u)
simp [informationQuadratic, dotProduct, mulVec, star_trivial] at hn ⊢
linarith
have hlower (u : ℝ) : Jstar θ p (log 3) ≤
informationObjective θ p (log 3) β u := by
calc
_ ≤ informationQuadratic (channelFisherInfo θ p Q) (direction u) :=
attainment_forces_direction_lower θ p (log 3) u hp hθ hε Q hQ
(by simpa [θ] using hattain)
_ ≤ informationQuadratic (informationMatrix θ p (log 3) β)
(direction u) := horder u
_ = informationObjective θ p (log 3) β u := by
rw [informationObjective_eq_informationQuadratic]
rfl
have hβat : informationObjective θ p (log 3) β t0 =
Jstar θ p (log 3) := le_antisymm (hupper β hβ) (hlower t0)
have hout : ∀ s ∉ r5Active, β s = 0 :=
certificate_equality_forces_inactive_zero θ p (log 3) t0 r5Active η
hactive hinactive β hβ (hβat.trans hJsum)
have hmin (u : ℝ) : informationObjective θ p (log 3) β t0 ≤
informationObjective θ p (log 3) β u := by rw [hβat]; exact hlower u
have hraw := informationObjective_stationary_of_min θ p (log 3) β t0 hmin
have hslope : ∑ s : Fin 14, β s * patternInformationSlope θ p
(log 3) s t0 = 0 := by
rw [← hraw]
apply Finset.sum_congr rfl
intro s _
simp [patternInformationSlope]
ring
have hβeq : β = α0 := centeredR5_supported_stationary_eq p hp0 hp1 β hβ
(by simpa [r5Active] using hout) (by simpa [θ, t0] using hslope)
have hpos : ∀ s ∈ r5Active, 0 < β s := by
rw [hβeq]
exact (centeredR5Weight_active p hp0 hp1).2
have heq : informationQuadratic (informationMatrix θ p (log 3) β)
(direction t0) = informationQuadratic (channelFisherInfo θ p Q)
(direction t0) := by
have hQlower := attainment_forces_direction_lower θ p (log 3) t0
hp hθ hε Q hQ (by simpa [θ] using hattain)
have hβupper : informationQuadratic (informationMatrix θ p (log 3) β)
(direction t0) ≤ Jstar θ p (log 3) := by
calc
_ = informationObjective θ p (log 3) β t0 := by
rw [informationObjective_eq_informationQuadratic]
rfl
_ ≤ _ := hupper β hβ
exact le_antisymm (hβupper.trans hQlower) (horder t0)
have hsing : ∀ s ∈ r5Active, ∀ u ∈ r5Active, s ≠ u → K s ⟂ₘ K u := by
intro s hs u hu hne
by_contra hn
have hscore := hrigid t0 heq s u (ne_of_gt (hpos s hs))
(ne_of_gt (hpos u hu)) hn
exact centeredR5_active_scores_pairwise_ne p hp0 hp1 s hs u hu hne
(by simpa [θ, t0] using hscore)
have hcard := active_card_le_outputCardinality_of_pairwise_singular
θ p (log 3) hp hθ r5Active β hpos K hK Q hfactor hsing
norm_num [r5Active] at hcard ⊢
simpa [θ] using hcard
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.CertificateRigidity 5 declarations These lemmas isolate the finite duality step used by the certified support regions.
Consequences of a strict staircase dual certificate
These lemmas isolate the finite duality step used by the certified support regions. Equality in the certificate bound forces every inactive staircase weight to vanish.
theorem feasible_dualRay_sum_general
the feasible dual ray sum general assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the feasible dual Ray sum general.
Formal statement
hypotheses
Proof (Lean source)
lemma feasible_dualRay_sum_general (ε : ℝ) (α : StaircaseWeight)
(hα : staircaseFeasible ε α) (η : Fin 4 → ℝ) :
(∑ s : Fin 14, α s * dualRay ε η s) = ∑ j : Fin 4, η j := by
simp only [dualRay, Finset.mul_sum]
rw [Finset.sum_comm]
simp_rw [← mul_assoc, ← Finset.sum_mul]
change (∑ j : Fin 4, staircaseMatrix ε α j * η j) = ∑ j : Fin 4, η j
simp only [hα.2, one_mul]
theorem certificate_dual_upper_bound
Under the supplied quantities and conditions, the certificate dual upper bound assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the certificate dual upper bound.
Formal statement
hypotheses
θ :
p ε t :
ℝ
η :
Fin 4 → ℝ
hactive :
∀ s ∈ K, dualRay ε η s = patternInformation θ p ε s t
hinactive :
∀ s ∉ K, patternInformation θ p ε s t < dualRay ε η s
β :
hβ :
conclusion
informationObjective θ p ε β t ≤ ∑ j : Fin 4, η j
Proof (Lean source)
lemma certificate_dual_upper_bound (θ : TrialParameter) (p ε t : ℝ)
(K : Finset (Fin 14)) (η : Fin 4 → ℝ)
(hactive : ∀ s ∈ K,
dualRay ε η s = patternInformation θ p ε s t)
(hinactive : ∀ s ∉ K,
patternInformation θ p ε s t < dualRay ε η s)
(β : StaircaseWeight) (hβ : staircaseFeasible ε β) :
informationObjective θ p ε β t ≤ ∑ j : Fin 4, η j := by
calc
informationObjective θ p ε β t =
∑ s : Fin 14, β s * patternInformation θ p ε s t := rfl
_ ≤ ∑ s : Fin 14, β s * dualRay ε η s := by
apply Finset.sum_le_sum
intro s _
apply mul_le_mul_of_nonneg_left _ (hβ.1 s)
by_cases hs : s ∈ K
· exact le_of_eq (hactive s hs).symm
· exact le_of_lt (hinactive s hs)
_ = ∑ j : Fin 4, η j := feasible_dualRay_sum_general ε β hβ η
theorem certificate_equality_forces_inactive_zero
Under the supplied quantities and conditions, the certificate equality forces inactive zero assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the certificate equality forces inactive zero.
Formal statement
hypotheses
θ :
p ε t :
ℝ
η :
Fin 4 → ℝ
hactive :
∀ s ∈ K, dualRay ε η s = patternInformation θ p ε s t
hinactive :
∀ s ∉ K, patternInformation θ p ε s t < dualRay ε η s
β :
hβ :
heq :
informationObjective θ p ε β t = ∑ j : Fin 4, η j
s ∉ K :
conclusion
β s = 0
Proof (Lean source)
lemma certificate_equality_forces_inactive_zero
(θ : TrialParameter) (p ε t : ℝ)
(K : Finset (Fin 14)) (η : Fin 4 → ℝ)
(hactive : ∀ s ∈ K,
dualRay ε η s = patternInformation θ p ε s t)
(hinactive : ∀ s ∉ K,
patternInformation θ p ε s t < dualRay ε η s)
(β : StaircaseWeight) (hβ : staircaseFeasible ε β)
(heq : informationObjective θ p ε β t = ∑ j : Fin 4, η j) :
∀ s ∉ K, β s = 0 := by
have hsum :
(∑ s : Fin 14, β s * patternInformation θ p ε s t) =
∑ s : Fin 14, β s * dualRay ε η s := by
calc
_ = informationObjective θ p ε β t := rfl
_ = ∑ j : Fin 4, η j := heq
_ = _ := (feasible_dualRay_sum_general ε β hβ η).symm
have hterm (s : Fin 14) :
β s * patternInformation θ p ε s t ≤ β s * dualRay ε η s := by
apply mul_le_mul_of_nonneg_left _ (hβ.1 s)
by_cases hs : s ∈ K
· exact le_of_eq (hactive s hs).symm
· exact le_of_lt (hinactive s hs)
have hterm_eq (s : Fin 14) :
β s * patternInformation θ p ε s t = β s * dualRay ε η s :=
(Finset.sum_eq_sum_iff_of_le (fun s _ ↦ hterm s)).mp hsum s
(Finset.mem_univ s)
intro s hs
have hstrict := hinactive s hs
have hnonneg := hβ.1 s
have he := hterm_eq s
nlinarith
theorem strictCertificate_profile_optimal_and_rigid
Under the supplied quantities and conditions, the strict certificate profile optimal and rigid assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the strict Certificate profile optimal and rigid.
Formal statement
hypotheses
θ :
p ε :
ℝ
hp :
hθ :
hε :
0 ≤ ε
α :
t :
ℝ
η :
Fin 4 → ℝ
hα :
hsupport :
activeSupport α K
hstationary :
∑ s : Fin 14, α s * patternInformationSlope θ p ε s t = 0
hactive :
∀ s ∈ K, dualRay ε η s = patternInformation θ p ε s t
hinactive :
∀ s ∉ K, patternInformation θ p ε s t < dualRay ε η s
conclusion
conclusion 1
conclusion 2
β :
s ∉ K :
β s = 0
Proof (Lean source)
lemma strictCertificate_profile_optimal_and_rigid
(θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 ≤ ε)
(K : Finset (Fin 14)) (α : StaircaseWeight) (t : ℝ)
(η : Fin 4 → ℝ) (hα : staircaseFeasible ε α)
(hsupport : activeSupport α K)
(hstationary : ∑ s : Fin 14,
α s * patternInformationSlope θ p ε s t = 0)
(hactive : ∀ s ∈ K,
dualRay ε η s = patternInformation θ p ε s t)
(hinactive : ∀ s ∉ K,
patternInformation θ p ε s t < dualRay ε η s) :
Jstar θ p ε =
sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε α v} ∧
∀ β : StaircaseWeight, staircaseFeasible ε β →
Jstar θ p ε =
sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε β v} →
∀ s ∉ K, β s = 0 := by
have hstat : ∑ s : Fin 14,
2 * α s * projectedGradient p ε s t *
(patternGradient p ε s 0 + patternGradient p ε s 1) /
patternMass θ p ε s = 0 := by
rw [← hstationary]
apply Finset.sum_congr rfl
intro s _
simp only [patternInformationSlope]
ring
have hmin (u : ℝ) :
informationObjective θ p ε α t ≤ informationObjective θ p ε α u :=
informationObjective_min_of_stationary θ p ε hp hθ hε α hα t hstat u
have hprofile :
sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε α v} =
informationObjective θ p ε α t := by
apply IsLeast.csInf_eq
exact ⟨⟨t, rfl⟩, by rintro u ⟨v, rfl⟩; exact hmin v⟩
have hvalue : informationObjective θ p ε α t = ∑ j : Fin 4, η j := by
calc
informationObjective θ p ε α t =
∑ s : Fin 14, α s * patternInformation θ p ε s t := rfl
_ = ∑ s : Fin 14, α s * dualRay ε η s := by
apply Finset.sum_congr rfl
intro s _
by_cases hs : s ∈ K
· rw [hactive s hs]
· have hz : α s = 0 := by
by_contra hn
exact hs ((hsupport s).mp hn)
simp [hz]
_ = ∑ j : Fin 4, η j :=
feasible_dualRay_sum_general ε α hα η
have hJ : Jstar θ p ε =
sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε α v} := by
unfold Jstar
apply IsGreatest.csSup_eq
refine ⟨⟨α, hα, rfl⟩, ?_⟩
rintro u ⟨β, hβ, rfl⟩
have hbelow : BddBelow {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε β v} := by
refine ⟨0, ?_⟩
rintro u ⟨v, rfl⟩
exact informationObjective_nonneg_interior θ p ε hp hθ hε β hβ v
calc
sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε β v} ≤
informationObjective θ p ε β t := csInf_le hbelow ⟨t, rfl⟩
_ ≤ ∑ j : Fin 4, η j :=
certificate_dual_upper_bound θ p ε t K η hactive hinactive β hβ
_ = informationObjective θ p ε α t := hvalue.symm
_ = sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε α v} := hprofile.symm
refine ⟨hJ, ?_⟩
intro β hβ hβopt
have hbelow : BddBelow {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε β v} := by
refine ⟨0, ?_⟩
rintro u ⟨v, rfl⟩
exact informationObjective_nonneg_interior θ p ε hp hθ hε β hβ v
have hβeq : informationObjective θ p ε β t = ∑ j : Fin 4, η j := by
apply le_antisymm
· exact certificate_dual_upper_bound θ p ε t K η hactive hinactive β hβ
· rw [← hvalue, ← hprofile, ← hJ, hβopt]
exact csInf_le hbelow ⟨t, rfl⟩
exact certificate_equality_forces_inactive_zero θ p ε t K η hactive
hinactive β hβ hβeq
theorem r5Certificate_profile_optimal_and_rigid
Under the supplied quantities and conditions, the r5 certificate profile optimal and rigid assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p μ0 μ1 ε :
ℝ
hx :
(p, μ0, μ1, ε) ∈ R5
conclusion
conclusion 1
s ∈ ({2, 3, 5, 7, 8} : Finset (Fin 14)) :
0 < α s
conclusion 2
conclusion 3
β :
s ∉ ({2, 3, 5, 7, 8} : Finset (Fin 14)) :
β s = 0
Proof (Lean source)
lemma r5Certificate_profile_optimal_and_rigid
(p μ0 μ1 ε : ℝ) (hx : (p, μ0, μ1, ε) ∈ R5) :
let θ : TrialParameter := fun k ↦ if k = 0 then μ0 else μ1
∃ α : StaircaseWeight,
staircaseFeasible ε α ∧
activeSupport α ({2, 3, 5, 7, 8} : Finset (Fin 14)) ∧
(∀ s ∈ ({2, 3, 5, 7, 8} : Finset (Fin 14)), 0 < α s) ∧
Jstar θ p ε = sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε α v} ∧
∀ β : StaircaseWeight, staircaseFeasible ε β →
Jstar θ p ε = sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε β v} →
∀ s ∉ ({2, 3, 5, 7, 8} : Finset (Fin 14)), β s = 0 := by
dsimp only
change 0 < p ∧ p < 1 ∧
0 < μ0 ∧ μ0 < 1 ∧ 0 < μ1 ∧ μ1 < 1 ∧ 0 < ε ∧
∃ α : StaircaseWeight, ∃ t : ℝ, ∃ η : Fin 4 → ℝ,
staircaseFeasible ε α ∧
activeSupport α ({2, 3, 5, 7, 8} : Finset (Fin 14)) ∧
(∀ s ∈ ({2, 3, 5, 7, 8} : Finset (Fin 14)), 0 < α s) ∧
(∑ s : Fin 14, α s *
patternInformationSlope (fun k ↦ if k = 0 then μ0 else μ1)
p ε s t) = 0 ∧
(∀ s ∈ ({2, 3, 5, 7, 8} : Finset (Fin 14)),
dualRay ε η s =
patternInformation (fun k ↦ if k = 0 then μ0 else μ1) p ε s t) ∧
(∀ s ∉ ({2, 3, 5, 7, 8} : Finset (Fin 14)),
patternInformation (fun k ↦ if k = 0 then μ0 else μ1) p ε s t <
dualRay ε η s) ∧
(true → ∀ s ∈ ({2, 3, 5, 7, 8} : Finset (Fin 14)),
∀ u ∈ ({2, 3, 5, 7, 8} : Finset (Fin 14)), s ≠ u →
projectedScore (fun k ↦ if k = 0 then μ0 else μ1) p ε s t ≠
projectedScore (fun k ↦ if k = 0 then μ0 else μ1) p ε u t) at hx
rcases hx with ⟨hp0, hp1, hμ0, hμ0', hμ1, hμ1', hε,
α, t, η, hα, hsupp, hpos, hstat, hactive, hinactive, _⟩
have hp : InteriorAssignment p := ⟨hp0, hp1⟩
have hθ : InteriorMeans (fun k ↦ if k = 0 then μ0 else μ1) := by
change 0 < μ0 ∧ μ0 < 1 ∧ 0 < μ1 ∧ μ1 < 1
exact ⟨hμ0, hμ0', hμ1, hμ1'⟩
obtain ⟨hopt, hrigid⟩ := strictCertificate_profile_optimal_and_rigid
(fun k ↦ if k = 0 then μ0 else μ1) p ε hp hθ hε.le
({2, 3, 5, 7, 8} : Finset (Fin 14)) α t η hα hsupp hstat
hactive hinactive
exact ⟨α, hα, hsupp, hpos, hopt, hrigid⟩
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.Channels 16 declarations Radon–Nikodym densities are taken against the finite sum of the four channel rows, so no countable-generation condition is imposed on the output space.
Stationary private channels and their information
Radon–Nikodym densities are taken against the finite sum of the four channel rows, so no countable-generation condition is imposed on the output space.
the stationary ldp is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The Stationary LDP is determined by the displayed parameters.
Definition (Lean source)
parameters
def
StationaryLDP ε Q :
Prop
given by
clause 1
def dominatingMeasure
the dominating measure is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
def channelDensity
For the supplied quantities and conditions, the channel density is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
channelDensity Q j :
Z → ℝ
given by
fun z => (Q j |>.rnDeriv (dominatingMeasure Q) z).toReal
def inputDerivative
For the supplied quantities and conditions, the input derivative is the mathematical object specified below.
Definition (Lean source)
parameters
def
inputDerivative p j k :
ℝ
given by
if k
= 0 then if j = 0 then -controlProb p else if j = 1 then controlProb p else 0 else if j = 2 then -p else if j = 3 then p else 0
def channelOutputDensity
For the supplied quantities and conditions, the channel output density is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The channel Output Density is determined by the displayed parameters.
Definition (Lean source)
parameters
def
channelOutputDensity θ p Q z :
ℝ
given by
def channelDerivativeDensity
For the supplied quantities and conditions, the channel derivative density is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
channelDerivativeDensity p Q k z :
ℝ
given by
For the supplied quantities and conditions, the channel fisher info is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The channel Fisher Info is determined by the displayed parameters.
Definition (Lean source)
parameters
given by
fun a b
=> ∫ z, (channelDerivativeDensity p Q a z * channelDerivativeDensity p Q b z) / channelOutputDensity θ p Q z ∂dominatingMeasure Q
def outputLaw
For the supplied quantities and conditions, the output law is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The output Law is determined by the displayed parameters.
For the supplied quantities and conditions, the output cardinality is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The output Cardinality is determined by the displayed parameters.
Definition (Lean source)
parameters
def
outputCardinality θ p Q :
ℕ∞
given by
if h : Finite Z then letI := h encard {z : Z | outputLaw θ p Q {z} ≠ 0} else ⊤
the staircase channel is the mathematical object specified below. The staircase Channel is determined by the displayed parameters.
Definition (Lean source)
parameters
ε :
ℝ
α :
given by
Kernel.ofFunOfCountable fun j
theorem channelDensity_staircase_ae
the channel density staircase ae assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the channel Density staircase ae.
Formal statement
hypotheses
conclusion
channelDensity (staircaseChannel ε α) j
=ᵐ[ dominatingMeasure (staircaseChannel ε α)] fun s => if α s = 0 then 0 else patternRay ε s j / ∑ i : Fin 4, patternRay ε s i
Proof (Lean source)
lemma channelDensity_staircase_ae (ε : ℝ) (α : StaircaseWeight)
(hε : 0 ≤ ε) (hα : ∀ s, 0 ≤ α s) (j : Fin 4) :
channelDensity (staircaseChannel ε α) j =ᵐ[
dominatingMeasure (staircaseChannel ε α)]
fun s => if α s = 0 then 0 else
patternRay ε s j / ∑ i : Fin 4, patternRay ε s i := by
filter_upwards [channelDensity_staircase_raw_ae ε α j] with s hs
rw [hs]
have hray (i : Fin 4) : 0 ≤ patternRay ε s i := by
unfold patternRay privacyIncrement privacyRatio
cases patternContains s i <;> simp <;> positivity
have htotal : staircaseTotalMass ε α s =
ofReal (α s * ∑ i : Fin 4, patternRay ε s i) := by
unfold staircaseTotalMass staircaseRowMass
rw [← ENNReal.ofReal_sum_of_nonneg]
· congr 1
rw [Finset.mul_sum]
· intro i hi
exact mul_nonneg (hα s) (hray i)
rw [htotal]
unfold staircaseRowMass
by_cases ha : α s = 0
· simp [ha]
· have ha_pos : 0 < α s := lt_of_le_of_ne (hα s) (Ne.symm ha)
have hR_pos : 0 < ∑ i : Fin 4, patternRay ε s i := by
have hone : 0 < patternRay ε s (0 : Fin 4) := lt_of_lt_of_le zero_lt_one (by
unfold patternRay
split_ifs <;> simp [privacyIncrement, privacyRatio,
(Real.one_le_exp_iff).2 hε])
exact lt_of_lt_of_le hone (Finset.single_le_sum (fun i hi => hray i)
(Finset.mem_univ 0))
rw [ENNReal.toReal_div, ENNReal.toReal_ofReal (mul_nonneg ha_pos.le (hray j)),
ENNReal.toReal_ofReal (mul_nonneg ha_pos.le hR_pos.le)]
simp [ha]
field_simp
theorem dominatingMeasure_staircase_singleton_real
the dominating measure staircase singleton real assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the dominating Measure staircase singleton real.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma dominatingMeasure_staircase_singleton_real (ε : ℝ)
(α : StaircaseWeight) (hε : 0 ≤ ε) (hα : ∀ s, 0 ≤ α s)
(s : Fin 14) :
(dominatingMeasure (staircaseChannel ε α)).real {s} =
α s * ∑ j : Fin 4, patternRay ε s j := by
rw [dominatingMeasure_staircaseChannel]
change ((Measure.count.withDensity (staircaseTotalMass ε α)) {s}).toReal = _
rw [withDensity_apply _ (measurableSet_singleton s)]
simp only [lintegral_singleton]
have hray (j : Fin 4) : 0 ≤ patternRay ε s j := by
unfold patternRay privacyIncrement privacyRatio
cases patternContains s j <;> simp <;> positivity
have htotal : staircaseTotalMass ε α s =
ofReal (α s * ∑ j : Fin 4, patternRay ε s j) := by
unfold staircaseTotalMass staircaseRowMass
rw [← ENNReal.ofReal_sum_of_nonneg]
· congr 1
rw [Finset.mul_sum]
· intro j hj
exact mul_nonneg (hα s) (hray j)
have hRnonneg : 0 ≤ ∑ j : Fin 4, patternRay ε s j :=
Finset.sum_nonneg fun j hj => hray j
rw [htotal]
simp [ENNReal.toReal_ofReal (mul_nonneg (hα s) hRnonneg)]
Under the supplied quantities and conditions, the pattern ray ldp assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pattern Ray ldp.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma patternRay_ldp (ε : ℝ) (hε : 0 < ε)
(s : Fin 14) (j j' : Fin 4) :
patternRay ε s j ≤ exp ε * patternRay ε s j' := by
have he : 1 ≤ exp ε := (Real.one_le_exp_iff).2 (le_of_lt hε)
cases h : patternContains s j <;> cases h' : patternContains s j' <;>
simp [patternRay, privacyIncrement, privacyRatio, h, h'] <;>
nlinarith [sq_nonneg (exp ε - 1)]
theorem staircaseChannel_markov
the staircase channel markov assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the staircase Channel markov.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma staircaseChannel_markov (ε : ℝ) (α : StaircaseWeight)
(hα : staircaseFeasible ε α) : IsMarkovKernel (staircaseChannel ε α) := by
constructor
intro j
constructor
have hsum : (∑ s : Fin 14, α s * patternRay ε s j) = 1 := hα.2 j
rw [← ENNReal.ofReal_one, ← hsum]
change (Measure.count.withDensity
(fun s : Fin 14 => ofReal (α s * patternRay ε s j))) univ = _
simp only [withDensity_apply _ MeasurableSet.univ, setLIntegral_univ,
lintegral_count]
rw [tsum_fintype, ENNReal.ofReal_sum_of_nonneg]
intro s hs
apply mul_nonneg (hα.1 s)
unfold patternRay privacyIncrement privacyRatio
cases patternContains s j <;> simp <;> positivity
theorem staircaseChannel_private
Under the supplied quantities and conditions, the staircase channel private assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the staircase Channel private.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma staircaseChannel_private (ε : ℝ) (hε : 0 < ε)
(α : StaircaseWeight) (hα : staircaseFeasible ε α)
(A : Set (Fin 14)) (j j' : Fin 4) :
staircaseChannel ε α j A ≤
ofReal (exp ε) * staircaseChannel ε α j' A := by
let r := ofReal (exp ε)
have hr : r ≠ ∞ := ENNReal.ofReal_ne_top
have hm : Measure.count.withDensity
(fun s : Fin 14 => ofReal (α s * patternRay ε s j)) ≤
r • Measure.count.withDensity
(fun s : Fin 14 => ofReal (α s * patternRay ε s j')) := by
rw [← withDensity_smul' r _ hr]
apply withDensity_mono
filter_upwards [] with s
dsimp [r]
rw [← ENNReal.ofReal_mul (le_of_lt (Real.exp_pos ε))]
apply ENNReal.ofReal_le_ofReal
have h := mul_le_mul_of_nonneg_left (patternRay_ldp ε hε s j j') (hα.1 s)
nlinarith
exact hm A
theorem staircaseChannel_stationaryLDP
Under the supplied quantities and conditions, the staircase channel stationary ldp assertion holds. Under the stated assumptions, the staircase Channel stationary LDP.
Formal statement
hypotheses
conclusion
StationaryLDP ε (staircaseChannel ε α)
Proof (Lean source)
lemma staircaseChannel_stationaryLDP (ε : ℝ) (hε : 0 < ε)
(α : StaircaseWeight) (hα : staircaseFeasible ε α) :
StationaryLDP ε (staircaseChannel ε α) := by
exact ⟨staircaseChannel_markov ε α hα,
fun A _ j j' => staircaseChannel_private ε hε α hα A j j'⟩
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.GeneralR3Cardinality 3 declarations General three-ray cardinality in the distinct-score neighborhood
General three-ray cardinality in the distinct-score neighborhood
theorem r3Certificate_direction_eq
Under the supplied quantities and conditions, the r3 certificate direction eq assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the r3 Certificate direction eq.
Formal statement
hypotheses
p μ0 μ1 ε :
ℝ
hp0 :
0 < p
hp1 :
p < 1
hμ00 :
0 < μ0
hμ01 :
μ0 < 1
hμ10 :
0 < μ1
hμ11 :
μ1 < 1
hε :
0 < ε
α :
t :
ℝ
hα :
hsupp :
hstat :
∑ s : Fin 14, α s * patternInformationSlope (fun k => if k = 0 then μ0 else μ1) p ε s t
= 0
conclusion
t = r3Direction p μ0 μ1 ε
Proof (Lean source)
lemma r3Certificate_direction_eq (p μ0 μ1 ε : ℝ)
(hp0 : 0 < p) (hp1 : p < 1) (hμ00 : 0 < μ0) (hμ01 : μ0 < 1)
(hμ10 : 0 < μ1) (hμ11 : μ1 < 1) (hε : 0 < ε)
(α : StaircaseWeight) (t : ℝ) (hα : staircaseFeasible ε α)
(hsupp : activeSupport α ({0, 5, 7} : Finset (Fin 14)))
(hstat : ∑ s : Fin 14, α s * patternInformationSlope
(fun k => if k = 0 then μ0 else μ1) p ε s t = 0) :
t = r3Direction p μ0 μ1 ε := by
have hsupp' : staircaseSupport α = ({0, 5, 7} : Finset (Fin 14)) := by
ext s
simpa [staircaseSupport] using hsupp s
have hαeq := threeMaskWeights_unique ε hε α hα hsupp'
have hstat' : r3Stationarity p μ0 μ1 ε t = 0 := by
rw [r3Stationarity, ← hstat]
apply Finset.sum_congr rfl
intro s _
rw [hαeq]
rfl
have hc : r3StationarityCoeff p μ0 μ1 ε ≠ 0 :=
ne_of_gt (r3StationarityCoeff_pos p μ0 μ1 ε
hp0 hp1 hμ00 hμ01 hμ10 hμ11 hε)
have hdir := r3Stationarity_r3Direction_eq_zero p μ0 μ1 ε hc
rw [r3Stationarity_affine] at hstat' hdir
have hm : (t - r3Direction p μ0 μ1 ε) *
r3StationarityCoeff p μ0 μ1 ε = 0 := by linarith
exact sub_eq_zero.mp ((mul_eq_zero.mp hm).resolve_right hc)
theorem r3Certificate_staircase_attains_three
Under the supplied quantities and conditions, the r3 certificate staircase attains three assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
θ :
fun k => if k = 0 then μ0 else μ1 (: TrialParameter)
∃ α : StaircaseWeight,
conclusion 1
staircaseFeasible (log 3) α
conclusion 2
conclusion 3
Proof (Lean source)
lemma r3Certificate_staircase_attains_three
(p μ0 μ1 : ℝ) (hx : (p, μ0, μ1, log 3) ∈ R3) :
let θ : TrialParameter := fun k => if k = 0 then μ0 else μ1
∃ α : StaircaseWeight, staircaseFeasible (log 3) α ∧
contrastVariance (informationMatrix θ p (log 3) α) =
ofReal (Vstar θ p (log 3)) ∧
outputCardinality θ p (staircaseChannel (log 3) α) = 3 := by
dsimp only
obtain ⟨α, hα, hsupp, hopt, _⟩ :=
r3Certificate_unique_optimum_log3 p μ0 μ1 hx
change 0 < p ∧ p < 1 ∧ 0 < μ0 ∧ μ0 < 1 ∧ 0 < μ1 ∧ μ1 < 1 ∧
0 < log 3 ∧ _ at hx
rcases hx with ⟨hp0, hp1, hμ00, hμ01, hμ10, hμ11, hε,
α0, t, η, hα0, hsupp0, _, hstat, hactive, hinactive, _⟩
let θ : TrialParameter := fun k => if k = 0 then μ0 else μ1
have hp : InteriorAssignment p := ⟨hp0, hp1⟩
have hθ : InteriorMeans θ := ⟨hμ00, hμ01, hμ10, hμ11⟩
have hαeq : α = α0 := by
have ha := r3_feasible_eq_canonical α hα (by
intro s hs
by_contra hn
exact hs ((hsupp s).mp hn))
have hb := r3_feasible_eq_canonical α0 hα0 (by
intro s hs
by_contra hn
exact hs ((hsupp0 s).mp hn))
exact ha.trans hb.symm
subst α0
have hmin (u : ℝ) : informationObjective θ p (log 3) α t ≤
informationObjective θ p (log 3) α u :=
informationObjective_min_of_stationary θ p (log 3) hp hθ hε.le α hα t
(by rw [← hstat]; apply Finset.sum_congr rfl; intro s _;
simp [patternInformationSlope]; ring) u
have hprofile : sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p (log 3) α v} =
informationObjective θ p (log 3) α t := by
apply IsLeast.csInf_eq
exact ⟨⟨t, rfl⟩, by rintro _ ⟨u, rfl⟩; exact hmin u⟩
have hvalue : informationObjective θ p (log 3) α t =
Jstar θ p (log 3) := by rw [← hprofile, ← hopt]
have hquad (u : ℝ) : informationObjective θ p (log 3) α u =
informationQuadratic (informationMatrix θ p (log 3) α) (direction u) := by
rw [informationObjective_eq_informationQuadratic]
rfl
have hqmin (u : ℝ) := show
informationQuadratic (informationMatrix θ p (log 3) α) (direction t) ≤
informationQuadratic (informationMatrix θ p (log 3) α) (direction u) by
rw [← hquad t, ← hquad u]
exact hmin u
have hqpos : 0 < informationQuadratic
(informationMatrix θ p (log 3) α) (direction t) := by
rw [← hquad t, hvalue]
exact Jstar_pos_interior θ p (log 3) hp hθ hε
have hvariance : contrastVariance (informationMatrix θ p (log 3) α) =
ofReal (Vstar θ p (log 3)) := by
rw [contrastVariance_eq_reciprocal_of_minimizer
(informationMatrix θ p (log 3) α)
(informationMatrix_posSemidef_interior θ p (log 3) hp hθ hε.le α hα)
t hqmin hqpos]
congr 1
rw [← hquad t, hvalue]
rfl
have hsuppcard : staircaseSupport α = ({0, 5, 7} : Finset (Fin 14)) := by
ext s
simpa [staircaseSupport] using hsupp s
have hcard : outputCardinality θ p (staircaseChannel (log 3) α) = 3 := by
rw [outputCardinality_staircase_eq_support_card θ p (log 3) hp hθ α hα.1,
hsuppcard]
have hc : ({0, 5, 7} : Finset (Fin 14)).card = 3 := by decide
rw [hc]
norm_num
exact ⟨α, hα, hvariance, hcard⟩
theorem r3Certificate_outputCardinality_ge_three_of_distinct
Under the supplied quantities and conditions, the r3 certificate output cardinality ge three of distinct assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p μ0 μ1 :
ℝ
hdistinct :
∀ i k : Fin 3
if
i ≠ k
then
r3ProjectedScoreAt (p, μ0, μ1, log 3) i
≠ r3ProjectedScoreAt (p, μ0, μ1, log 3) k
Z :
Type*
hQ :
StationaryLDP (log 3) Q
hattain :
contrastVariance (channelFisherInfo (fun k => if k = 0 then μ0 else μ1) p Q)
conclusion
3 ≤ outputCardinality (fun k => if k = 0 then μ0 else μ1) p Q
Proof (Lean source)
lemma r3Certificate_outputCardinality_ge_three_of_distinct
(p μ0 μ1 : ℝ) (hx : (p, μ0, μ1, log 3) ∈ R3)
(hdistinct : ∀ i k : Fin 3, i ≠ k →
r3ProjectedScoreAt (p, μ0, μ1, log 3) i ≠
r3ProjectedScoreAt (p, μ0, μ1, log 3) k)
{Z : Type*} [MeasurableSpace Z] (Q : Kernel (Fin 4) Z)
(hQ : StationaryLDP (log 3) Q)
(hattain : contrastVariance
(channelFisherInfo (fun k => if k = 0 then μ0 else μ1) p Q) =
ofReal (Vstar (fun k => if k = 0 then μ0 else μ1)
p (log 3))) :
3 ≤ outputCardinality (fun k => if k = 0 then μ0 else μ1) p Q := by
have hxcopy := hx
change 0 < p ∧ p < 1 ∧ 0 < μ0 ∧ μ0 < 1 ∧ 0 < μ1 ∧ μ1 < 1 ∧
0 < log 3 ∧ _ at hxcopy
rcases hxcopy with ⟨hp0, hp1, hμ00, hμ01, hμ10, hμ11, hε,
α, t, η, hα, hsupp, _, hstat, hactive, hinactive, _⟩
let θ : TrialParameter := fun k => if k = 0 then μ0 else μ1
let A : Finset (Fin 14) := {0, 5, 7}
have hp : InteriorAssignment p := ⟨hp0, hp1⟩
have hθ : InteriorMeans θ := ⟨hμ00, hμ01, hμ10, hμ11⟩
have ht : t = r3Direction p μ0 μ1 (log 3) :=
r3Certificate_direction_eq p μ0 μ1 (log 3) hp0 hp1 hμ00 hμ01
hμ10 hμ11 hε α t hα hsupp hstat
obtain ⟨hopt, _⟩ := strictCertificate_profile_optimal_and_rigid
θ p (log 3) hp hθ hε.le A α t η hα hsupp hstat hactive hinactive
have hmin (u : ℝ) : informationObjective θ p (log 3) α t ≤
informationObjective θ p (log 3) α u :=
informationObjective_min_of_stationary θ p (log 3) hp hθ hε.le α hα t
(by rw [← hstat]; apply Finset.sum_congr rfl; intro s _;
simp [patternInformationSlope]; ring) u
have hprofile : sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p (log 3) α v} =
informationObjective θ p (log 3) α t := by
apply IsLeast.csInf_eq
exact ⟨⟨t, rfl⟩, by rintro _ ⟨u, rfl⟩; exact hmin u⟩
have hvalue : informationObjective θ p (log 3) α t = ∑ j : Fin 4, η j := by
calc
_ = ∑ s : Fin 14, α s * dualRay (log 3) η s := by
apply Finset.sum_congr rfl
intro s _
by_cases hs : s ∈ A
· rw [hactive s hs]
· have hz : α s = 0 := by by_contra hn; exact hs ((hsupp s).mp hn)
simp [hz]
_ = _ := feasible_dualRay_sum_general (log 3) α hα η
have hJsum : Jstar θ p (log 3) = ∑ j : Fin 4, η j := by
rw [hopt, hprofile, hvalue]
have hupper : ∀ β : StaircaseWeight, staircaseFeasible (log 3) β →
informationObjective θ p (log 3) β t ≤ Jstar θ p (log 3) := by
intro β hβ
rw [hJsum]
exact certificate_dual_upper_bound θ p (log 3) t A η
hactive hinactive β hβ
have hpositive : ∀ β : StaircaseWeight, staircaseFeasible (log 3) β →
informationObjective θ p (log 3) β t = Jstar θ p (log 3) →
∀ s ∈ A, 0 < β s := by
intro β hβ heq
have hout := certificate_equality_forces_inactive_zero θ p (log 3) t A η
hactive hinactive β hβ (heq.trans hJsum)
have hcanonical := r3_feasible_eq_canonical β hβ hout
intro s hs
rw [hcanonical]
change s ∈ ({0, 5, 7} : Finset (Fin 14)) at hs
simp [hs]
have hmaskinj : Injective r3Mask := by
intro i k hik
fin_cases i <;> fin_cases k <;> simp_all [r3Mask]
have htheta : regionTheta (p, μ0, μ1, log 3) = θ := by
funext k
fin_cases k <;> rfl
have hscores : ∀ s ∈ A, ∀ u ∈ A, s ≠ u →
projectedScore θ p (log 3) s t ≠
projectedScore θ p (log 3) u t := by
intro s hs u hu hne
obtain ⟨i, rfl⟩ : ∃ i : Fin 3, r3Mask i = s := by
have hs' : s = 0 ∨ s = 5 ∨ s = 7 := by simpa [A] using hs
rcases hs' with rfl | rfl | rfl
· exact ⟨0, by simp [r3Mask]⟩
· exact ⟨1, by simp [r3Mask]⟩
· exact ⟨2, by simp [r3Mask]⟩
obtain ⟨k, rfl⟩ : ∃ k : Fin 3, r3Mask k = u := by
have hu' : u = 0 ∨ u = 5 ∨ u = 7 := by simpa [A] using hu
rcases hu' with rfl | rfl | rfl
· exact ⟨0, by simp [r3Mask]⟩
· exact ⟨1, by simp [r3Mask]⟩
· exact ⟨2, by simp [r3Mask]⟩
have hik : i ≠ k := fun e => hne (congrArg r3Mask e)
have hd := hdistinct i k hik
rw [r3ProjectedScoreAt_eq, r3ProjectedScoreAt_eq, htheta] at hd
simpa only [θ, ht] using hd
have hcard := outputCardinality_ge_of_attainment_direction
θ p (log 3) t hp hθ hε A hupper hpositive hscores Q hQ
(by simpa only [θ] using hattain)
have hAcard : A.card = 3 := by decide
rw [hAcard] at hcard
norm_num at hcard ⊢
simpa only [θ] using hcard
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.GeneralR5Rigidity 3 declarations General five-ray certificate rigidity
General five-ray certificate rigidity
theorem r5Certificate_unique_optimum
Every R5 certificate has a unique optimal staircase weight, including away from the centered explicit family. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p μ0 μ1 ε :
ℝ
hx :
(p, μ0, μ1, ε) ∈ R5
conclusion
θ :
fun k => if k = 0 then μ0 else μ1 (: TrialParameter)
∃ α : StaircaseWeight,
activeSupport α r5Active
conclusion 1
s ∈ r5Active :
0 < α s
conclusion 2
conclusion 3
β :
β = α
Proof (Lean source)
lemma r5Certificate_unique_optimum
(p μ0 μ1 ε : ℝ) (hx : (p, μ0, μ1, ε) ∈ R5) :
let θ : TrialParameter := fun k => if k = 0 then μ0 else μ1
∃ α : StaircaseWeight,
staircaseFeasible ε α ∧
activeSupport α r5Active ∧
(∀ s ∈ r5Active, 0 < α s) ∧
Jstar θ p ε = sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε α v} ∧
∀ β : StaircaseWeight, staircaseFeasible ε β →
Jstar θ p ε = sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε β v} →
β = α := by
dsimp only
change 0 < p ∧ p < 1 ∧
0 < μ0 ∧ μ0 < 1 ∧ 0 < μ1 ∧ μ1 < 1 ∧ 0 < ε ∧
∃ α : StaircaseWeight, ∃ t : ℝ, ∃ η : Fin 4 → ℝ,
staircaseFeasible ε α ∧ activeSupport α r5Active ∧
(∀ s ∈ r5Active, 0 < α s) ∧
(∑ s : Fin 14, α s * patternInformationSlope
(fun k => if k = 0 then μ0 else μ1) p ε s t) = 0 ∧
(∀ s ∈ r5Active, dualRay ε η s =
patternInformation (fun k => if k = 0 then μ0 else μ1) p ε s t) ∧
(∀ s ∉ r5Active, patternInformation
(fun k => if k = 0 then μ0 else μ1) p ε s t < dualRay ε η s) ∧
(true → ∀ s ∈ r5Active, ∀ u ∈ r5Active, s ≠ u →
projectedScore (fun k => if k = 0 then μ0 else μ1) p ε s t ≠
projectedScore (fun k => if k = 0 then μ0 else μ1) p ε u t) at hx
rcases hx with ⟨hp0, hp1, hμ00, hμ01, hμ10, hμ11, hε,
α, t, η, hα, hsupp, hpos, hstat, hactive, hinactive, _⟩
let θ : TrialParameter := fun k => if k = 0 then μ0 else μ1
let x : RegionParameter := (p, μ0, μ1, ε)
have hxθ : regionTheta x = θ := by
funext k
fin_cases k <;> rfl
have hp : InteriorAssignment p := ⟨hp0, hp1⟩
have hθ : InteriorMeans θ := by
change 0 < μ0 ∧ μ0 < 1 ∧ 0 < μ1 ∧ μ1 < 1
exact ⟨hμ00, hμ01, hμ10, hμ11⟩
have hm (s : Fin 14) : patternMass θ p ε s ≠ 0 :=
ne_of_gt (patternMass_pos_interior θ p ε hp hθ hε.le s)
have hαout : ∀ s ∉ r5Active, α s = 0 := by
intro s hs
by_contra hn
exact hs ((hsupp s).mp hn)
obtain ⟨hopt, hrigid⟩ := strictCertificate_profile_optimal_and_rigid
θ p ε hp hθ hε.le r5Active α t η hα hsupp hstat hactive hinactive
have hcoeff : 0 < r5BinaryCoeff x ∧ 0 < r5TernaryCoeff x :=
r5Coeffs_pos_of_interior x hp0 hp1 hμ00 hμ01 hμ10 hμ11 hε
have hobj : r5BinaryObjective x t = r5TernaryObjective x t := by
apply r5_objectives_eq_of_activeDual x η t
intro s hs
simpa only [hxθ] using hactive s hs
have heq : r5BinaryCoeff x * (p - (1 - 2 * p) * t) ^ 2 =
r5TernaryCoeff x * (t + 1) ^ 2 := by
rw [← r5BinaryObjective_eq_quadratic x t (by simpa [hxθ] using hm 5)
(by simpa [hxθ] using hm 8),
← r5TernaryObjective_eq_quadratic x t (by simpa [hxθ] using hm 3)
(by simpa [hxθ] using hm 7)]
exact hobj
have hR : r5TernaryDerivative x t ≠ 0 :=
r5TernaryDerivative_ne_zero_of_objectives_eq x t hp1 hcoeff.1 hcoeff.2
(by simpa [hxθ] using hm 3) (by simpa [hxθ] using hm 7) heq
have hαmix : r5MixtureCoordinate ε α =
-r5TernaryDerivative x t /
(r5BinaryDerivative x t - r5TernaryDerivative x t) := by
have hs := r5_stationarity_eq_mixture x α hε hα hαout t (by
simpa only [hxθ] using hstat)
exact (mixture_stationarity_solve hR hs).2
refine ⟨α, hα, hsupp, hpos, hopt, ?_⟩
intro β hβ hβopt
have hβout : ∀ s ∉ r5Active, β s = 0 := hrigid β hβ hβopt
have hbelow : BddBelow {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε β v} := by
refine ⟨0, ?_⟩
rintro u ⟨v, rfl⟩
exact informationObjective_nonneg_interior θ p ε hp hθ hε.le β hβ v
have hαvalue : informationObjective θ p ε α t = ∑ j : Fin 4, η j := by
calc
_ = ∑ s : Fin 14, α s * dualRay ε η s := by
apply Finset.sum_congr rfl
intro s _
by_cases hs : s ∈ r5Active
· rw [hactive s hs]
· simp [hαout s hs]
_ = _ := feasible_dualRay_sum_general ε α hα η
have hαprofile : sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε α v} =
informationObjective θ p ε α t := by
apply IsLeast.csInf_eq
refine ⟨⟨t, rfl⟩, ?_⟩
rintro _ ⟨v, rfl⟩
exact informationObjective_min_of_stationary θ p ε hp hθ hε.le α hα t
(by
rw [← hstat]
apply Finset.sum_congr rfl
intro s _
simp [patternInformationSlope]
ring) v
have hβat : informationObjective θ p ε β t = Jstar θ p ε := by
apply le_antisymm
· calc
_ ≤ ∑ j : Fin 4, η j :=
certificate_dual_upper_bound θ p ε t r5Active η hactive hinactive β hβ
_ = _ := by rw [← hαvalue, ← hαprofile, ← hopt]
· rw [hβopt]
exact csInf_le hbelow ⟨t, rfl⟩
have hβmin (u : ℝ) : informationObjective θ p ε β t ≤
informationObjective θ p ε β u := by
rw [hβat, hβopt]
exact csInf_le hbelow ⟨u, rfl⟩
have hraw := informationObjective_stationary_of_min θ p ε β t hβmin
have hβstat : ∑ s : Fin 14, β s * patternInformationSlope θ p ε s t = 0 := by
rw [← hraw]
apply Finset.sum_congr rfl
intro s _
simp [patternInformationSlope]
ring
have hβmix : r5MixtureCoordinate ε β =
-r5TernaryDerivative x t /
(r5BinaryDerivative x t - r5TernaryDerivative x t) := by
have hs := r5_stationarity_eq_mixture x β hε hβ hβout t (by
simpa only [hxθ] using hβstat)
exact (mixture_stationarity_solve hR hs).2
have hfive : β 5 = α 5 := by
unfold r5MixtureCoordinate at hαmix hβmix
have hd : 0 < exp ε + 1 := by positivity
nlinarith
exact r5_supported_feasible_eq_of_five_eq ε hε β α hβ hα hβout hαout hfive
theorem r5Certificate_outputCardinality_ge_five
Every channel attaining an oracle certified by R5 has at least five positive output atoms. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p μ0 μ1 ε :
ℝ
hx :
(p, μ0, μ1, ε) ∈ R5
Z :
Type*
hQ :
StationaryLDP ε Q
hattain :
contrastVariance (channelFisherInfo (fun k => if k = 0 then μ0 else μ1) p Q)
conclusion
5 ≤ outputCardinality (fun k => if k = 0 then μ0 else μ1) p Q
Proof (Lean source)
lemma r5Certificate_outputCardinality_ge_five
(p μ0 μ1 ε : ℝ) (hx : (p, μ0, μ1, ε) ∈ R5)
{Z : Type*} [MeasurableSpace Z] (Q : Kernel (Fin 4) Z)
(hQ : StationaryLDP ε Q)
(hattain : contrastVariance
(channelFisherInfo (fun k => if k = 0 then μ0 else μ1) p Q) =
ofReal (Vstar (fun k => if k = 0 then μ0 else μ1) p ε)) :
5 ≤ outputCardinality (fun k => if k = 0 then μ0 else μ1) p Q := by
have hxcopy := hx
change 0 < p ∧ p < 1 ∧
0 < μ0 ∧ μ0 < 1 ∧ 0 < μ1 ∧ μ1 < 1 ∧ 0 < ε ∧ _ at hxcopy
rcases hxcopy with ⟨hp0, hp1, hμ00, hμ01, hμ10, hμ11, hε,
α0, t, η, hα0, hsupp0, hpos0, hstat0, hactive, hinactive, hdistinct⟩
let θ : TrialParameter := fun k => if k = 0 then μ0 else μ1
have hp : InteriorAssignment p := ⟨hp0, hp1⟩
have hθ : InteriorMeans θ := by
change 0 < μ0 ∧ μ0 < 1 ∧ 0 < μ1 ∧ μ1 < 1
exact ⟨hμ00, hμ01, hμ10, hμ11⟩
obtain ⟨αu, hαu, _, hposu, hoptu, hunique⟩ :=
r5Certificate_unique_optimum p μ0 μ1 ε hx
obtain ⟨hopt0, _⟩ := strictCertificate_profile_optimal_and_rigid
θ p ε hp hθ hε.le r5Active α0 t η hα0 hsupp0 hstat0 hactive hinactive
have hα0value : informationObjective θ p ε α0 t = ∑ j : Fin 4, η j := by
calc
_ = ∑ s : Fin 14, α0 s * dualRay ε η s := by
apply Finset.sum_congr rfl
intro s _
by_cases hs : s ∈ r5Active
· rw [hactive s hs]
· have hz : α0 s = 0 := by
by_contra hn
exact hs ((hsupp0 s).mp hn)
simp [hz]
_ = _ := feasible_dualRay_sum_general ε α0 hα0 η
have hα0min (u : ℝ) : informationObjective θ p ε α0 t ≤
informationObjective θ p ε α0 u :=
informationObjective_min_of_stationary θ p ε hp hθ hε.le α0 hα0 t
(by
rw [← hstat0]
apply Finset.sum_congr rfl
intro s _
simp [patternInformationSlope]
ring) u
have hα0profile : sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε α0 v} =
informationObjective θ p ε α0 t := by
apply IsLeast.csInf_eq
exact ⟨⟨t, rfl⟩, by rintro _ ⟨u, rfl⟩; exact hα0min u⟩
have hJsum : Jstar θ p ε = ∑ j : Fin 4, η j := by
rw [hopt0, hα0profile, hα0value]
have hupper (β : StaircaseWeight) (hβ : staircaseFeasible ε β) :
informationObjective θ p ε β t ≤ Jstar θ p ε := by
rw [hJsum]
exact certificate_dual_upper_bound θ p ε t r5Active η
hactive hinactive β hβ
let epsSeq : ℕ → ℝ := fun _ => ε
have hfixed : FixedPrivacy epsSeq ε := ⟨hε, fun _ => rfl⟩
obtain ⟨β, K, hβ, hK, hfactor, hdiff, hrigid⟩ :=
staircase_refinement_interior p θ ε hp hθ epsSeq hfixed Q hQ
have horder (u : ℝ) :
informationQuadratic (channelFisherInfo θ p Q) (direction u) ≤
informationQuadratic (informationMatrix θ p ε β) (direction u) := by
have hn := hdiff.dotProduct_mulVec_nonneg (direction u)
simp [informationQuadratic, dotProduct, mulVec, star_trivial] at hn ⊢
linarith
have hlower (u : ℝ) : Jstar θ p ε ≤
informationObjective θ p ε β u := by
calc
_ ≤ informationQuadratic (channelFisherInfo θ p Q) (direction u) :=
attainment_forces_direction_lower θ p ε u hp hθ hε Q hQ (by
simpa only [θ] using hattain)
_ ≤ informationQuadratic (informationMatrix θ p ε β) (direction u) := horder u
_ = informationObjective θ p ε β u := by
rw [informationObjective_eq_informationQuadratic]
rfl
have hβat : informationObjective θ p ε β t = Jstar θ p ε :=
le_antisymm (hupper β hβ) (hlower t)
have hβprofile : sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε β v} =
informationObjective θ p ε β t := by
apply IsLeast.csInf_eq
exact ⟨⟨t, rfl⟩, by rintro _ ⟨u, rfl⟩; rw [hβat]; exact hlower u⟩
have hβopt : Jstar θ p ε = sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε β v} := by
rw [hβprofile, hβat]
have hβeq : β = αu := hunique β hβ hβopt
have hpos : ∀ s ∈ r5Active, 0 < β s := by
rw [hβeq]
exact hposu
have heq : informationQuadratic (informationMatrix θ p ε β) (direction t) =
informationQuadratic (channelFisherInfo θ p Q) (direction t) := by
have hQlower := attainment_forces_direction_lower θ p ε t hp hθ hε Q hQ
(by simpa only [θ] using hattain)
have hβupper : informationQuadratic (informationMatrix θ p ε β)
(direction t) ≤ Jstar θ p ε := by
calc
_ = informationObjective θ p ε β t := by
rw [informationObjective_eq_informationQuadratic]
rfl
_ ≤ _ := hupper β hβ
exact le_antisymm (hβupper.trans hQlower) (horder t)
have hsing : ∀ s ∈ r5Active, ∀ u ∈ r5Active, s ≠ u → K s ⟂ₘ K u := by
intro s hs u hu hne
by_contra hn
have hscore := hrigid t heq s u (ne_of_gt (hpos s hs))
(ne_of_gt (hpos u hu)) hn
exact hdistinct rfl s hs u hu hne (by simpa only [θ] using hscore)
have hcard := active_card_le_outputCardinality_of_pairwise_singular
θ p ε hp hθ r5Active β hpos K hK Q hfactor hsing
have hAcard : r5Active.card = 5 := by decide
rw [hAcard] at hcard
norm_num at hcard ⊢
simpa only [θ] using hcard
theorem r5Certificate_staircase_attains_five
The primal witness carried by an R5 certificate attains the oracle and has exactly five positive staircase outputs. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p μ0 μ1 ε :
ℝ
hx :
(p, μ0, μ1, ε) ∈ R5
conclusion
θ :
fun k => if k = 0 then μ0 else μ1 (: TrialParameter)
∃ α : StaircaseWeight,
conclusion 1
conclusion 2
conclusion 3
outputCardinality θ p (staircaseChannel ε α) = 5
Proof (Lean source)
lemma r5Certificate_staircase_attains_five
(p μ0 μ1 ε : ℝ) (hx : (p, μ0, μ1, ε) ∈ R5) :
let θ : TrialParameter := fun k => if k = 0 then μ0 else μ1
∃ α : StaircaseWeight, staircaseFeasible ε α ∧
contrastVariance (informationMatrix θ p ε α) =
ofReal (Vstar θ p ε) ∧
outputCardinality θ p (staircaseChannel ε α) = 5 := by
dsimp only
change 0 < p ∧ p < 1 ∧
0 < μ0 ∧ μ0 < 1 ∧ 0 < μ1 ∧ μ1 < 1 ∧ 0 < ε ∧ _ at hx
rcases hx with ⟨hp0, hp1, hμ00, hμ01, hμ10, hμ11, hε,
α, t, η, hα, hsupp, hpos, hstat, hactive, hinactive, _⟩
let θ : TrialParameter := fun k => if k = 0 then μ0 else μ1
have hp : InteriorAssignment p := ⟨hp0, hp1⟩
have hθ : InteriorMeans θ := by
change 0 < μ0 ∧ μ0 < 1 ∧ 0 < μ1 ∧ μ1 < 1
exact ⟨hμ00, hμ01, hμ10, hμ11⟩
obtain ⟨hopt, _⟩ := strictCertificate_profile_optimal_and_rigid
θ p ε hp hθ hε.le r5Active α t η hα hsupp hstat hactive hinactive
have hmin (u : ℝ) : informationObjective θ p ε α t ≤
informationObjective θ p ε α u :=
informationObjective_min_of_stationary θ p ε hp hθ hε.le α hα t
(by
rw [← hstat]
apply Finset.sum_congr rfl
intro s _
simp [patternInformationSlope]
ring) u
have hprofile : sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε α v} =
informationObjective θ p ε α t := by
apply IsLeast.csInf_eq
exact ⟨⟨t, rfl⟩, by rintro _ ⟨u, rfl⟩; exact hmin u⟩
have hvalue : informationObjective θ p ε α t = Jstar θ p ε := by
rw [← hprofile, ← hopt]
have hquad (u : ℝ) : informationObjective θ p ε α u =
informationQuadratic (informationMatrix θ p ε α) (direction u) := by
rw [informationObjective_eq_informationQuadratic]
rfl
have hqmin (u : ℝ) :
informationQuadratic (informationMatrix θ p ε α) (direction t) ≤
informationQuadratic (informationMatrix θ p ε α) (direction u) := by
rw [← hquad t, ← hquad u]
exact hmin u
have hqpos : 0 < informationQuadratic
(informationMatrix θ p ε α) (direction t) := by
rw [← hquad t, hvalue]
exact Jstar_pos_interior θ p ε hp hθ hε
have hvariance : contrastVariance (informationMatrix θ p ε α) =
ofReal (Vstar θ p ε) := by
rw [contrastVariance_eq_reciprocal_of_minimizer
(informationMatrix θ p ε α)
(informationMatrix_posSemidef_interior θ p ε hp hθ hε.le α hα)
t hqmin hqpos]
congr 1
rw [← hquad t, hvalue]
rfl
have hsuppcard : (staircaseSupport α).card = 5 := by
have hs : staircaseSupport α = r5Active := by
ext s
simp only [staircaseSupport, mem_filter, Finset.mem_univ, true_and]
exact hsupp s
rw [hs]
decide
have hcard : outputCardinality θ p (staircaseChannel ε α) = 5 := by
rw [outputCardinality_staircase_eq_support_card θ p ε hp hθ α hα.1,
hsuppcard]
norm_num
exact ⟨α, hα, hvariance, hcard⟩
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.Laplace 9 declarations The custom Horvitz–Thompson score is released with independent centered Laplace noise at the published sensitivity scale.
Published custom Laplace comparator
The custom Horvitz–Thompson score is released with independent centered Laplace noise at the published sensitivity scale.
def sensitivity
For the supplied quantities and conditions, the sensitivity is the mathematical object specified below.
Definition (Lean source)
def comparatorNoiseLaw
For the supplied quantities and conditions, the comparator noise law is the mathematical object specified below. The comparator Noise Law is determined by the displayed parameters.
Definition (Lean source)
def ComparatorNoise
The observed-record/noise pairs are jointly independent across subjects; each noise has the published Laplace marginal and is independent of its observed private record. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The Comparator Noise is determined by the displayed parameters.
Definition (Lean source)
parameters
def
ComparatorNoise μ W Y L p ε :
Prop
given by
clause 1
iIndepFun (fun i ω => (privateRecord W Y i ω, L i ω)) μ
def oaReleaseValue
For the supplied quantities and conditions, the oa release value is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The oa Release Value is determined by the displayed parameters.
Definition (Lean source)
parameters
Ω :
Type*
W Y L :
ℕ → Ω → ℝ
p :
ℝ
i :
ℕ
ω :
Ω
def
oaReleaseValue W Y L p i ω :
ℝ
given by
W i ω * Y i ω / p - (1 - W i ω) * Y i ω / controlProb p + L i ω
the oa release is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The oa Release is determined by the displayed parameters.
Definition (Lean source)
parameters
def
oaRelease μ W Y L A τhat p ε Δ :
Prop
given by
clause 1
clause 2
0 < ε
clause 3
Δ = sensitivity p
clause 4
Δ = 1 / (p * controlProb p)
clause 5
ComparatorNoise μ W Y L p ε
For the supplied quantities and conditions, the tau oa is the mathematical object specified below. The tau OA is determined by the displayed parameters.
Definition (Lean source)
parameters
Ω :
Type*
W Y L :
ℕ → Ω → ℝ
p :
ℝ
n :
ℕ
ω :
Ω
def
tauOA W Y L p n ω :
ℝ
given by
(n : ℝ)⁻¹ * ∑ i ∈ range n, oaReleaseValue W Y L p i ω
For the supplied quantities and conditions, the vhat oa is the mathematical object specified below. The Vhat OA is determined by the displayed parameters.
Definition (Lean source)
parameters
Ω :
Type*
W Y L :
ℕ → Ω → ℝ
p :
ℝ
n :
ℕ
ω :
Ω
def
VhatOA W Y L p n ω :
ℝ
given by
theorem VhatOA_eq_second_moment_sub_sq
the vhat oa eq second moment sub sq assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Ω :
Type*
W Y L :
ℕ → Ω → ℝ
p :
ℝ
n :
ℕ
hn :
0 < n
ω :
Ω
conclusion
VhatOA W Y L p n ω
= (n : ℝ)⁻¹ * ∑ i ∈ range n, (oaReleaseValue W Y L p i ω) ^ 2
- (tauOA W Y L p n ω) ^ 2
Proof (Lean source)
lemma VhatOA_eq_second_moment_sub_sq {Ω : Type*}
(W Y L : ℕ → Ω → ℝ) (p : ℝ) (n : ℕ) (hn : 0 < n) (ω : Ω) :
VhatOA W Y L p n ω =
(n : ℝ)⁻¹ * ∑ i ∈ range n,
(oaReleaseValue W Y L p i ω) ^ 2 -
(tauOA W Y L p n ω) ^ 2 := by
let a : ℕ → ℝ := fun i => oaReleaseValue W Y L p i ω
let m : ℝ := tauOA W Y L p n ω
have hn' : (n : ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hn)
have hm : (n : ℝ)⁻¹ * ∑ i ∈ range n, a i = m := rfl
simp only [VhatOA, a, m, sub_sq]
rw [Finset.sum_add_distrib, Finset.sum_sub_distrib]
have hcross :
(∑ i ∈ range n, 2 * a i * m) =
2 * m * ∑ i ∈ range n, a i := by
rw [← Finset.sum_mul, ← Finset.mul_sum]
ring
simp only [Finset.sum_const, Finset.card_range]
change (n : ℝ)⁻¹ *
((∑ i ∈ range n, a i ^ 2) -
(∑ i ∈ range n, 2 * a i * m) + n • m ^ 2) =
(n : ℝ)⁻¹ * ∑ i ∈ range n, a i ^ 2 - m ^ 2
rw [hcross]
simp only [nsmul_eq_mul]
field_simp at hm ⊢
rw [hm]
ring
For the supplied quantities and conditions, the voa is the mathematical object specified below. The VOA is determined by the displayed parameters.
Definition (Lean source)
parameters
θ :
p ε :
ℝ
def
VOA θ p ε :
ℝ
given by
θ 1 / p + θ 0 / controlProb p - (contrast θ) ^ 2 + 2 * (sensitivity p) ^ 2 / ε ^ 2
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.LaplaceMoments 5 declarations This file records the first two moments and square integrability of the positive-scale centered Laplace distribution used by the comparator release.
Moments of the centered Laplace law
This file records the first two moments and square integrability of the positive-scale centered Laplace distribution used by the comparator release.
theorem laplaceMeasure_integral_id
A centered Laplace draw has mean zero at every positive scale. For the displayed inputs and conditions, the stated result follows.
Formal statement
Proof (Lean source)
lemma laplaceMeasure_integral_id (b : ℝ) (hb : 0 < b) :
∫ x, x ∂(laplaceMeasure b) = 0 := by
rw [laplaceMeasure, integral_withDensity_eq_integral_toReal_smul
(measurable_laplacePDF b).ennreal_ofReal
(ae_of_all _ fun _ => ENNReal.ofReal_lt_top)]
simp_rw [ENNReal.toReal_ofReal (laplacePDF_nonneg' hb _), smul_eq_mul]
have hneg := integral_neg_eq_self (μ := (volume : Measure ℝ))
(f := fun x : ℝ => laplacePDF b x * x)
have hodd : (fun x : ℝ => laplacePDF b (-x) * (-x)) =
fun x => -(laplacePDF b x * x) := by
funext x
simp [laplacePDF]
rw [hodd, integral_neg] at hneg
linarith
theorem laplaceMeasure_integrable_sq
The square of a centered Laplace draw is integrable at every positive scale. For the displayed inputs and conditions, the stated result follows.
Formal statement
Proof (Lean source)
lemma laplaceMeasure_integrable_sq (b : ℝ) (hb : 0 < b) :
Integrable (fun x : ℝ => x ^ 2) (laplaceMeasure b) := by
rw [laplaceMeasure,
integrable_withDensity_iff_integrable_smul' (measurable_laplacePDF b).ennreal_ofReal
(ae_of_all _ fun _ => ENNReal.ofReal_lt_top)]
simpa only [ENNReal.toReal_ofReal (laplacePDF_nonneg' hb _), smul_eq_mul, mul_comm] using
integrable_sq_mul_laplacePDF b hb
theorem laplaceMeasure_integral_sq
The second moment of a centered Laplace draw with positive scale b is 2 * b ^ 2. For the displayed inputs and conditions, the stated result follows.
Formal statement
Proof (Lean source)
lemma laplaceMeasure_integral_sq (b : ℝ) (hb : 0 < b) :
∫ x, x ^ 2 ∂(laplaceMeasure b) = 2 * b ^ 2 := by
rw [laplaceMeasure, integral_withDensity_eq_integral_toReal_smul
(measurable_laplacePDF b).ennreal_ofReal
(ae_of_all _ fun _ => ENNReal.ofReal_lt_top)]
simp_rw [ENNReal.toReal_ofReal (laplacePDF_nonneg' hb _), smul_eq_mul]
let f : ℝ → ℝ := fun x => (2 * b)⁻¹ * (x ^ 2 * exp (-x / b))
rw [show (fun x : ℝ => laplacePDF b x * x ^ 2) = fun x => f |x| by
funext x
simp only [f, laplacePDF, sq_abs]
ring, integral_comp_abs]
have hplain : (∫ x in Ioi (0 : ℝ), x ^ 2 * exp (-x / b)) = 2 * b ^ 3 := by
calc
_ = ∫ x in Ioi (0 : ℝ), x ^ ((3 : ℝ) - 1) *
exp (-((1 / b) * x)) := by
apply setIntegral_congr_fun measurableSet_Ioi
intro x hx
simp only [show (3 : ℝ) - 1 = 2 by norm_num, Real.rpow_two]
rw [show -x / b = -(1 / b * x) by ring]
_ = (1 / (1 / b)) ^ (3 : ℝ) * Gamma (3 : ℝ) :=
Real.integral_rpow_mul_exp_neg_mul_Ioi (by norm_num) (one_div_pos.mpr hb)
_ = 2 * b ^ 3 := by
rw [show Gamma (3 : ℝ) = 2 by norm_num]
field_simp [hb.ne']
exact Real.rpow_natCast b 3
rw [show (∫ x in Ioi (0 : ℝ), f x) = (2 * b)⁻¹ * (2 * b ^ 3) by
rw [integral_const_mul, hplain]]
field_simp [hb.ne']
theorem aemeasurable
Each comparator noise coordinate is almost everywhere measurable. This is recovered from its stipulated nonzero Laplace pushforward law, so no separate measurability assumption is needed. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the aemeasurable.
Formal statement
hypotheses
conclusion
AEMeasurable (L i) μ
Proof (Lean source)
lemma ComparatorNoise.aemeasurable {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} {W Y L : ℕ → Ω → ℝ} {p ε : ℝ}
(hnoise : ComparatorNoise μ W Y L p ε)
(hp : InteriorAssignment p) (hε : 0 < ε) (i : ℕ) :
AEMeasurable (L i) μ := by
have hsens : 0 < sensitivity p := by
unfold sensitivity controlProb
exact add_pos (inv_pos.mpr hp.1) (inv_pos.mpr (sub_pos.mpr hp.2))
have hscale : 0 < sensitivity p / ε := div_pos hsens hε
letI : IsProbabilityMeasure (comparatorNoiseLaw p ε) := by
unfold comparatorNoiseLaw
exact laplaceMeasure_isProbabilityMeasure _ hscale
have hne : μ.map (L i) ≠ 0 := by
rw [hnoise.2.1 i]
exact IsProbabilityMeasure.ne_zero _
exact AEMeasurable.of_map_ne_zero hne
theorem oaReleaseValue_aemeasurable
Every noisy release coordinate is almost everywhere measurable under the binary-trial and comparator-noise assumptions. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the oa Release Value aemeasurable.
Formal statement
hypotheses
Ω :
θ :
p ε :
ℝ
hmodel :
LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ
hnoise :
ComparatorNoise μ W Y L p ε
hε :
0 < ε
i :
ℕ
conclusion
AEMeasurable (oaReleaseValue W Y L p i) μ
Proof (Lean source)
lemma oaReleaseValue_aemeasurable {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} {Y0 Y1 W Y L : ℕ → Ω → ℝ} {θ : TrialParameter} {p ε : ℝ}
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ)
(hnoise : ComparatorNoise μ W Y L p ε) (hε : 0 < ε) (i : ℕ) :
AEMeasurable (oaReleaseValue W Y L p i) μ := by
have hY0 : Measurable (Y0 i) := (hmodel.measurable i).fst
have hY1 : Measurable (Y1 i) := (hmodel.measurable i).snd.fst
have hW : Measurable (W i) := (hmodel.measurable i).snd.snd
have hY : Measurable (Y i) := by
have hformula : Y i = fun ω =>
W i ω * Y1 i ω + (1 - W i ω) * Y0 i ω := by
funext ω
exact hmodel.consistent i ω
rw [hformula]
exact (hW.mul hY1).add ((measurable_const.sub hW).mul hY0)
have hL : AEMeasurable (L i) μ :=
hnoise.aemeasurable hmodel.assignmentInterior hε i
unfold oaReleaseValue
exact (((hW.mul hY).div_const p).sub
((measurable_const.sub hW).mul hY |>.div_const (controlProb p))).aemeasurable.add hL
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.LaplaceRatio 1 declarations A nonnegative estimated denominator makes the ordinary square-root ratio agree with the total fallback map.
Coupled variance-ratio continuity
A nonnegative estimated denominator makes the ordinary square-root ratio agree with the total fallback map. Consequently marginal convergence in probability passes through every coupling of the two rows.
theorem VhatStar_nonneg
Under the supplied quantities and conditions, the vhat star nonneg assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p ε :
ℝ
m :
ℕ → ℕ
select :
TrialParameter → StaircaseWeight × ℝ × ℝ
n :
ℕ
z :
Transcript (pilotOutputFamily n)
conclusion
0 ≤ VhatStar p ε m select z
Proof (Lean source)
lemma VhatStar_nonneg (p ε : ℝ) (m : ℕ → ℕ)
(select : TrialParameter → StaircaseWeight × ℝ × ℝ)
{n : ℕ} (z : Transcript (pilotOutputFamily n)) :
0 ≤ VhatStar p ε m select z := by
unfold VhatStar
exact mul_nonneg (inv_nonneg.mpr (Nat.cast_nonneg _))
(Finset.sum_nonneg fun i _ => by
split_ifs
· exact sq_nonneg _
· exact le_rfl)
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.OAConsistency 28 declarations IID and moment adapters for the custom Laplace release
IID and moment adapters for the custom Laplace release
def oaTrialValue
For the supplied quantities and conditions, the oa trial value is the mathematical object specified below.
Definition (Lean source)
parameters
p :
ℝ
j :
Fin 4
def
oaTrialValue p j :
ℝ
given by
if j = 0 then 0 else if j = 1 then -(controlProb p)⁻¹ else if j = 2 then 0 else p⁻¹
theorem oaTrialValue_mean_inputLaw
Under the supplied quantities and conditions, the oa trial value mean input law assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the oa Trial Value mean input Law.
Formal statement
hypotheses
conclusion
∫ j, oaTrialValue p j ∂(inputLaw θ p) = contrast θ
Proof (Lean source)
lemma oaTrialValue_mean_inputLaw (θ : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
∫ j, oaTrialValue p j ∂(inputLaw θ p) = contrast θ := by
have hpi (j : Fin 4) : 0 ≤ piTheta θ p j := by
rcases hp with ⟨hp0, hp1⟩
rcases hθ with ⟨h00, h01, h10, h11⟩
fin_cases j <;> simp only [piTheta, controlProb] <;>
positivity
have hpi0 : piTheta θ p (0 : Fin 4) = controlProb p * (1 - θ 0) := rfl
have hpi1 : piTheta θ p (1 : Fin 4) = controlProb p * θ 0 := rfl
have hpi2 : piTheta θ p (2 : Fin 4) = p * (1 - θ 1) := rfl
have hpi3 : piTheta θ p (3 : Fin 4) = p * θ 1 := rfl
rw [inputLaw, integral_withDensity_eq_integral_toReal_smul]
· simp_rw [ENNReal.toReal_ofReal (hpi _), smul_eq_mul]
rw [integral_count]
simp +decide [oaTrialValue, hpi0, hpi1, hpi2, hpi3, controlProb, contrast,
Fin.sum_univ_succ]
field_simp [ne_of_gt hp.1, ne_of_gt (sub_pos.mpr hp.2)]
ring
· exact (measurable_of_finite _).ennreal_ofReal
· exact ae_of_all _ fun _ => ENNReal.ofReal_lt_top
theorem oaTrialValue_sq_mean_inputLaw
Under the supplied quantities and conditions, the oa trial value sq mean input law assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the oa Trial Value sq mean input Law.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma oaTrialValue_sq_mean_inputLaw (θ : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
∫ j, (oaTrialValue p j) ^ 2 ∂(inputLaw θ p) =
θ 1 / p + θ 0 / controlProb p := by
have hpi (j : Fin 4) : 0 ≤ piTheta θ p j := by
rcases hp with ⟨hp0, hp1⟩
rcases hθ with ⟨h00, h01, h10, h11⟩
fin_cases j <;> simp only [piTheta, controlProb] <;>
positivity
have hpi0 : piTheta θ p (0 : Fin 4) = controlProb p * (1 - θ 0) := rfl
have hpi1 : piTheta θ p (1 : Fin 4) = controlProb p * θ 0 := rfl
have hpi2 : piTheta θ p (2 : Fin 4) = p * (1 - θ 1) := rfl
have hpi3 : piTheta θ p (3 : Fin 4) = p * θ 1 := rfl
rw [inputLaw, integral_withDensity_eq_integral_toReal_smul]
· simp_rw [ENNReal.toReal_ofReal (hpi _), smul_eq_mul]
rw [integral_count]
simp +decide [oaTrialValue, hpi0, hpi1, hpi2, hpi3, controlProb,
Fin.sum_univ_succ]
field_simp [ne_of_gt hp.1, ne_of_gt (sub_pos.mpr hp.2)]
ring
· exact (measurable_of_finite _).ennreal_ofReal
· exact ae_of_all _ fun _ => ENNReal.ofReal_lt_top
def oaTrialPart
For the supplied quantities and conditions, the oa trial part is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The oa Trial Part is determined by the displayed parameters.
Definition (Lean source)
parameters
Ω :
Type*
W Y :
ℕ → Ω → ℝ
p :
ℝ
i :
ℕ
ω :
Ω
def
oaTrialPart W Y p i ω :
ℝ
given by
W i ω * Y i ω / p - (1 - W i ω) * Y i ω / controlProb p
theorem privateRecord_measurable_of_model
the private record measurable of model assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (privateRecord W Y i)
Proof (Lean source)
lemma privateRecord_measurable_of_model {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} {Y0 Y1 W Y : ℕ → Ω → ℝ} {p : ℝ} {θ : TrialParameter}
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ) (i : ℕ) :
Measurable (privateRecord W Y i) := by
have hW : Measurable (W i) := (hmodel.measurable i).snd.snd
have hY0 : Measurable (Y0 i) := (hmodel.measurable i).fst
have hY1 : Measurable (Y1 i) := (hmodel.measurable i).snd.fst
have hY : Measurable (Y i) := by
rw [show Y i = fun ω => W i ω * Y1 i ω +
(1 - W i ω) * Y0 i ω by funext ω; exact hmodel.consistent i ω]
exact (hW.mul hY1).add ((measurable_const.sub hW).mul hY0)
unfold privateRecord
exact Measurable.ite (hW (measurableSet_singleton 0))
(Measurable.ite (hY (measurableSet_singleton 0)) measurable_const measurable_const)
(Measurable.ite (hY (measurableSet_singleton 0)) measurable_const measurable_const)
theorem oaTrialPart_ae_eq_record
the oa trial part ae eq record assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma oaTrialPart_ae_eq_record {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} {Y0 Y1 W Y : ℕ → Ω → ℝ} {p : ℝ} {θ : TrialParameter}
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ) (i : ℕ) :
oaTrialPart W Y p i =ᵐ[μ]
fun ω => oaTrialValue p (privateRecord W Y i ω) := by
filter_upwards [(hmodel.assigned i).1, hmodel.binary i] with ω hW hY
simp only [Set.mem_insert_iff, Set.mem_singleton_iff] at hW hY
rcases hW with hW | hW
· rcases hY.1 with hY0 | hY0 <;>
simp [oaTrialPart, oaTrialValue, privateRecord, hmodel.consistent i ω,
hW, hY0]
· rcases hY.2 with hY1 | hY1 <;>
simp [oaTrialPart, oaTrialValue, privateRecord, hmodel.consistent i ω,
hW, hY1]
theorem oaTrialPart_integral
the oa trial part integral assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
∫ ω, oaTrialPart W Y p i ω ∂μ = contrast θ
Proof (Lean source)
lemma oaTrialPart_integral {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ]
{Y0 Y1 W Y : ℕ → Ω → ℝ} {p : ℝ} {θ : TrialParameter}
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ) (i : ℕ) :
∫ ω, oaTrialPart W Y p i ω ∂μ = contrast θ := by
rw [integral_congr_ae (oaTrialPart_ae_eq_record hmodel i)]
rw [← integral_map (privateRecord_measurable_of_model hmodel i).aemeasurable
(measurable_of_finite _).aestronglyMeasurable,
recordLaw_eq_piTheta μ Y0 Y1 W Y p θ hmodel i]
exact oaTrialValue_mean_inputLaw θ p hmodel.assignmentInterior hmodel.meansInterior
theorem oaTrialPart_sq_integral
the oa trial part sq integral assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
∫ ω, (oaTrialPart W Y p i ω) ^ 2 ∂μ = θ 1 / p + θ 0 / controlProb p
Proof (Lean source)
lemma oaTrialPart_sq_integral {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ]
{Y0 Y1 W Y : ℕ → Ω → ℝ} {p : ℝ} {θ : TrialParameter}
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ) (i : ℕ) :
∫ ω, (oaTrialPart W Y p i ω) ^ 2 ∂μ =
θ 1 / p + θ 0 / controlProb p := by
have hsq : (fun ω => (oaTrialPart W Y p i ω) ^ 2) =ᵐ[μ]
fun ω => (oaTrialValue p (privateRecord W Y i ω)) ^ 2 := by
filter_upwards [oaTrialPart_ae_eq_record hmodel i] with ω hω
rw [hω]
rw [integral_congr_ae hsq]
rw [← integral_map (privateRecord_measurable_of_model hmodel i).aemeasurable
((measurable_of_finite _).pow_const 2).aestronglyMeasurable,
recordLaw_eq_piTheta μ Y0 Y1 W Y p θ hmodel i]
exact oaTrialValue_sq_mean_inputLaw θ p hmodel.assignmentInterior hmodel.meansInterior
theorem oaTrialPart_integrable
the oa trial part integrable assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Integrable (oaTrialPart W Y p i) μ
Proof (Lean source)
lemma oaTrialPart_integrable {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ]
{Y0 Y1 W Y : ℕ → Ω → ℝ} {p : ℝ} {θ : TrialParameter}
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ) (i : ℕ) :
Integrable (oaTrialPart W Y p i) μ := by
have hm : Measurable (fun ω => oaTrialValue p (privateRecord W Y i ω)) :=
(show Measurable (oaTrialValue p) from measurable_of_finite _).comp
(privateRecord_measurable_of_model hmodel i)
have hi : Integrable (fun ω => oaTrialValue p (privateRecord W Y i ω)) μ :=
Integrable.of_bound hm.aestronglyMeasurable
(|(controlProb p)⁻¹| + |p⁻¹|)
(Filter.Eventually.of_forall fun ω => by
simpa [Real.norm_eq_abs] using oaTrialValue_abs_le p (privateRecord W Y i ω))
exact hi.congr (oaTrialPart_ae_eq_record hmodel i).symm
theorem oaTrialPart_sq_integrable
the oa trial part sq integrable assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Integrable (fun ω => (oaTrialPart W Y p i ω) ^ 2) μ
Proof (Lean source)
lemma oaTrialPart_sq_integrable {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ]
{Y0 Y1 W Y : ℕ → Ω → ℝ} {p : ℝ} {θ : TrialParameter}
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ) (i : ℕ) :
Integrable (fun ω => (oaTrialPart W Y p i ω) ^ 2) μ := by
have hm : Measurable (fun ω => (oaTrialValue p (privateRecord W Y i ω)) ^ 2) :=
((show Measurable (oaTrialValue p) from measurable_of_finite _).comp
(privateRecord_measurable_of_model hmodel i)).pow_const 2
have hi : Integrable (fun ω => (oaTrialValue p (privateRecord W Y i ω)) ^ 2) μ :=
Integrable.of_bound hm.aestronglyMeasurable
((|(controlProb p)⁻¹| + |p⁻¹|) ^ 2)
(Filter.Eventually.of_forall fun ω => by
have h := oaTrialValue_abs_le p (privateRecord W Y i ω)
rw [Real.norm_eq_abs, abs_sq]
have hs := mul_self_le_mul_self (abs_nonneg _) h
nlinarith [sq_abs (oaTrialValue p (privateRecord W Y i ω))])
have hsq : (fun ω => (oaTrialValue p (privateRecord W Y i ω)) ^ 2) =ᵐ[μ]
fun ω => (oaTrialPart W Y p i ω) ^ 2 := by
filter_upwards [oaTrialPart_ae_eq_record hmodel i] with ω hω
rw [hω]
exact hi.congr hsq
theorem comparatorNoise_integral
the comparator noise integral assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the comparator Noise integral.
Formal statement
hypotheses
conclusion
∫ ω, L i ω ∂μ = 0
Proof (Lean source)
lemma comparatorNoise_integral {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ]
{W Y L : ℕ → Ω → ℝ} {p ε : ℝ}
(hp : InteriorAssignment p)
(hnoise : ComparatorNoise μ W Y L p ε) (hε : 0 < ε) (i : ℕ) :
∫ ω, L i ω ∂μ = 0 := by
have hL := hnoise.aemeasurable hp hε i
have hscale : 0 < sensitivity p / ε := by
apply div_pos _ hε
unfold sensitivity controlProb
exact add_pos (inv_pos.mpr hp.1) (inv_pos.mpr (sub_pos.mpr hp.2))
change (∫ ω, id (L i ω) ∂μ) = 0
rw [← integral_map hL measurable_id.aestronglyMeasurable, hnoise.2.1 i]
exact laplaceMeasure_integral_id (sensitivity p / ε) hscale
theorem comparatorNoise_sq_integral
the comparator noise sq integral assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the comparator Noise sq integral.
Formal statement
hypotheses
conclusion
∫ ω, (L i ω) ^ 2 ∂μ = 2 * (sensitivity p) ^ 2 / ε ^ 2
Proof (Lean source)
lemma comparatorNoise_sq_integral {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ]
{W Y L : ℕ → Ω → ℝ} {p ε : ℝ}
(hp : InteriorAssignment p)
(hnoise : ComparatorNoise μ W Y L p ε) (hε : 0 < ε) (i : ℕ) :
∫ ω, (L i ω) ^ 2 ∂μ = 2 * (sensitivity p) ^ 2 / ε ^ 2 := by
have hL := hnoise.aemeasurable hp hε i
have hscale : 0 < sensitivity p / ε := by
apply div_pos _ hε
unfold sensitivity controlProb
exact add_pos (inv_pos.mpr hp.1) (inv_pos.mpr (sub_pos.mpr hp.2))
change (∫ ω, (id (L i ω)) ^ 2 ∂μ) = _
rw [← integral_map hL (measurable_id.pow_const 2).aestronglyMeasurable,
hnoise.2.1 i]
simp only [comparatorNoiseLaw, id_eq]
rw [laplaceMeasure_integral_sq (sensitivity p / ε) hscale]
ring
theorem comparatorNoise_sq_integrable
the comparator noise sq integrable assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the comparator Noise sq integrable.
Formal statement
hypotheses
conclusion
Integrable (fun ω => (L i ω) ^ 2) μ
Proof (Lean source)
lemma comparatorNoise_sq_integrable {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ]
{W Y L : ℕ → Ω → ℝ} {p ε : ℝ}
(hp : InteriorAssignment p)
(hnoise : ComparatorNoise μ W Y L p ε) (hε : 0 < ε) (i : ℕ) :
Integrable (fun ω => (L i ω) ^ 2) μ := by
have hL := hnoise.aemeasurable hp hε i
have hscale : 0 < sensitivity p / ε := by
apply div_pos _ hε
unfold sensitivity controlProb
exact add_pos (inv_pos.mpr hp.1) (inv_pos.mpr (sub_pos.mpr hp.2))
have hmap : Integrable (fun x : ℝ => x ^ 2) (μ.map (L i)) := by
rw [hnoise.2.1 i]
exact laplaceMeasure_integrable_sq (sensitivity p / ε) hscale
exact (integrable_map_measure (measurable_id.pow_const 2).aestronglyMeasurable hL).mp hmap
theorem comparatorNoise_integrable
the comparator noise integrable assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the comparator Noise integrable.
Formal statement
hypotheses
conclusion
Integrable (L i) μ
Proof (Lean source)
lemma comparatorNoise_integrable {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ]
{W Y L : ℕ → Ω → ℝ} {p ε : ℝ}
(hp : InteriorAssignment p)
(hnoise : ComparatorNoise μ W Y L p ε) (hε : 0 < ε) (i : ℕ) :
Integrable (L i) μ := by
have hL := hnoise.aemeasurable hp hε i
exact ((memLp_two_iff_integrable_sq hL.aestronglyMeasurable).2
(comparatorNoise_sq_integrable hp hnoise hε i)).integrable (by norm_num)
theorem oaTrialPart_indep_noise
the oa trial part indep noise assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Ω :
p ε :
ℝ
θ :
hmodel :
LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ
hnoise :
ComparatorNoise μ W Y L p ε
i :
ℕ
conclusion
IndepFun (L i) (oaTrialPart W Y p i) μ
Proof (Lean source)
lemma oaTrialPart_indep_noise {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} {Y0 Y1 W Y L : ℕ → Ω → ℝ} {p ε : ℝ} {θ : TrialParameter}
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ)
(hnoise : ComparatorNoise μ W Y L p ε) (i : ℕ) :
IndepFun (L i) (oaTrialPart W Y p i) μ := by
have h := (hnoise.2.2 i).comp measurable_id
(show Measurable (oaTrialValue p) from measurable_of_finite _)
apply h.congr (Filter.Eventually.of_forall fun _ => rfl)
simpa [Function.comp_def] using (oaTrialPart_ae_eq_record hmodel i).symm
theorem oaReleaseValue_eq_trial_add
the oa release value eq trial add assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Ω :
Type*
W Y L :
ℕ → Ω → ℝ
p :
ℝ
i :
ℕ
ω :
Ω
conclusion
oaReleaseValue W Y L p i ω = oaTrialPart W Y p i ω + L i ω
Proof (Lean source)
lemma oaReleaseValue_eq_trial_add {Ω : Type*}
(W Y L : ℕ → Ω → ℝ) (p : ℝ) (i : ℕ) (ω : Ω) :
oaReleaseValue W Y L p i ω = oaTrialPart W Y p i ω + L i ω := rfl
theorem oaReleaseValue_integral
the oa release value integral assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the oa Release Value integral.
Formal statement
hypotheses
Ω :
Y0 Y1 W Y L :
ℕ → Ω → ℝ
p ε :
ℝ
θ :
hmodel :
LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ
hnoise :
ComparatorNoise μ W Y L p ε
hε :
0 < ε
i :
ℕ
conclusion
∫ ω, oaReleaseValue W Y L p i ω ∂μ = contrast θ
Proof (Lean source)
lemma oaReleaseValue_integral {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ]
{Y0 Y1 W Y L : ℕ → Ω → ℝ} {p ε : ℝ} {θ : TrialParameter}
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ)
(hnoise : ComparatorNoise μ W Y L p ε) (hε : 0 < ε) (i : ℕ) :
∫ ω, oaReleaseValue W Y L p i ω ∂μ = contrast θ := by
rw [show (fun ω => oaReleaseValue W Y L p i ω) =
fun ω => oaTrialPart W Y p i ω + L i ω by funext ω; rfl,
integral_add (oaTrialPart_integrable hmodel i)
(comparatorNoise_integrable hmodel.assignmentInterior hnoise hε i),
oaTrialPart_integral hmodel i,
comparatorNoise_integral hmodel.assignmentInterior hnoise hε i, add_zero]
theorem oaReleaseValue_sq_integrable
the oa release value sq integrable assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the oa Release Value sq integrable.
Formal statement
hypotheses
Ω :
Y0 Y1 W Y L :
ℕ → Ω → ℝ
p ε :
ℝ
θ :
hmodel :
LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ
hnoise :
ComparatorNoise μ W Y L p ε
hε :
0 < ε
i :
ℕ
conclusion
Integrable (fun ω => (oaReleaseValue W Y L p i ω) ^ 2) μ
Proof (Lean source)
lemma oaReleaseValue_sq_integrable {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ]
{Y0 Y1 W Y L : ℕ → Ω → ℝ} {p ε : ℝ} {θ : TrialParameter}
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ)
(hnoise : ComparatorNoise μ W Y L p ε) (hε : 0 < ε) (i : ℕ) :
Integrable (fun ω => (oaReleaseValue W Y L p i ω) ^ 2) μ := by
have hb : MemLp (oaTrialPart W Y p i) 2 μ :=
(memLp_two_iff_integrable_sq
(oaTrialPart_integrable hmodel i).aestronglyMeasurable).2
(oaTrialPart_sq_integrable hmodel i)
have hn : MemLp (L i) 2 μ :=
(memLp_two_iff_integrable_sq
(hnoise.aemeasurable hmodel.assignmentInterior hε i).aestronglyMeasurable).2
(comparatorNoise_sq_integrable hmodel.assignmentInterior hnoise hε i)
have ha := hb.add hn
have heq : (oaTrialPart W Y p i + L i) =
fun ω => oaReleaseValue W Y L p i ω := by funext ω; rfl
rw [heq] at ha
exact ha.integrable_sq
theorem oaReleaseValue_sq_integral
the oa release value sq integral assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the oa Release Value sq integral.
Formal statement
hypotheses
Ω :
Y0 Y1 W Y L :
ℕ → Ω → ℝ
p ε :
ℝ
θ :
hmodel :
LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ
hnoise :
ComparatorNoise μ W Y L p ε
hε :
0 < ε
i :
ℕ
conclusion
∫ ω, (oaReleaseValue W Y L p i ω) ^ 2 ∂μ
= θ 1 / p + θ 0 / controlProb p + 2 * (sensitivity p) ^ 2 / ε ^ 2
Proof (Lean source)
lemma oaReleaseValue_sq_integral {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ]
{Y0 Y1 W Y L : ℕ → Ω → ℝ} {p ε : ℝ} {θ : TrialParameter}
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ)
(hnoise : ComparatorNoise μ W Y L p ε) (hε : 0 < ε) (i : ℕ) :
∫ ω, (oaReleaseValue W Y L p i ω) ^ 2 ∂μ =
θ 1 / p + θ 0 / controlProb p +
2 * (sensitivity p) ^ 2 / ε ^ 2 := by
have hb2 := oaTrialPart_sq_integrable hmodel i
have hn2 := comparatorNoise_sq_integrable hmodel.assignmentInterior hnoise hε i
have hb : MemLp (oaTrialPart W Y p i) 2 μ :=
(memLp_two_iff_integrable_sq
(oaTrialPart_integrable hmodel i).aestronglyMeasurable).2 hb2
have hn : MemLp (L i) 2 μ :=
(memLp_two_iff_integrable_sq
(hnoise.aemeasurable hmodel.assignmentInterior hε i).aestronglyMeasurable).2 hn2
have hcrossInt : Integrable (fun ω => L i ω * oaTrialPart W Y p i ω) μ := by
change Integrable (L i * oaTrialPart W Y p i) μ
exact hn.integrable_mul hb
have hcross : (∫ ω, L i ω * oaTrialPart W Y p i ω ∂μ) = 0 := by
rw [IndepFun.integral_fun_mul_eq_mul_integral
(oaTrialPart_indep_noise hmodel hnoise i)
(hnoise.aemeasurable hmodel.assignmentInterior hε i).aestronglyMeasurable
(oaTrialPart_integrable hmodel i).aestronglyMeasurable,
comparatorNoise_integral hmodel.assignmentInterior hnoise hε i, zero_mul]
calc
_ = ∫ ω, ((oaTrialPart W Y p i ω) ^ 2 +
2 * (L i ω * oaTrialPart W Y p i ω)) + (L i ω) ^ 2 ∂μ := by
apply integral_congr_ae
filter_upwards with ω
rw [oaReleaseValue_eq_trial_add]
ring
_ = (∫ ω, ((oaTrialPart W Y p i ω) ^ 2 +
2 * (L i ω * oaTrialPart W Y p i ω)) ∂μ) +
∫ ω, (L i ω) ^ 2 ∂μ :=
integral_add (hb2.add (hcrossInt.const_mul 2)) hn2
_ = ((∫ ω, (oaTrialPart W Y p i ω) ^ 2 ∂μ) +
∫ ω, 2 * (L i ω * oaTrialPart W Y p i ω) ∂μ) +
∫ ω, (L i ω) ^ 2 ∂μ := by
rw [integral_add hb2 (hcrossInt.const_mul 2)]
_ = (∫ ω, (oaTrialPart W Y p i ω) ^ 2 ∂μ) +
2 * (∫ ω, L i ω * oaTrialPart W Y p i ω ∂μ) +
∫ ω, (L i ω) ^ 2 ∂μ := by rw [integral_const_mul]
_ = _ := by
rw [hcross, oaTrialPart_sq_integral hmodel i,
comparatorNoise_sq_integral hmodel.assignmentInterior hnoise hε i]
ring
theorem oaReleaseValue_centered_sq_integral
the oa release value centered sq integral assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the oa Release Value centered sq integral.
Formal statement
hypotheses
Ω :
Y0 Y1 W Y L :
ℕ → Ω → ℝ
p ε :
ℝ
θ :
hmodel :
LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ
hnoise :
ComparatorNoise μ W Y L p ε
hε :
0 < ε
i :
ℕ
conclusion
∫ ω, (oaReleaseValue W Y L p i ω - contrast θ) ^ 2 ∂μ = VOA θ p ε
Proof (Lean source)
lemma oaReleaseValue_centered_sq_integral {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ]
{Y0 Y1 W Y L : ℕ → Ω → ℝ} {p ε : ℝ} {θ : TrialParameter}
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ)
(hnoise : ComparatorNoise μ W Y L p ε) (hε : 0 < ε) (i : ℕ) :
∫ ω, (oaReleaseValue W Y L p i ω - contrast θ) ^ 2 ∂μ = VOA θ p ε := by
have hA2 := oaReleaseValue_sq_integrable hmodel hnoise hε i
have hA : Integrable (fun ω => oaReleaseValue W Y L p i ω) μ := by
change Integrable (oaTrialPart W Y p i + L i) μ
exact (oaTrialPart_integrable hmodel i).add
(comparatorNoise_integrable hmodel.assignmentInterior hnoise hε i)
calc
_ = (∫ ω, (oaReleaseValue W Y L p i ω) ^ 2 ∂μ) -
2 * contrast θ * (∫ ω, oaReleaseValue W Y L p i ω ∂μ) +
(contrast θ) ^ 2 := by
rw [show (fun ω => (oaReleaseValue W Y L p i ω - contrast θ) ^ 2) =
fun ω => (oaReleaseValue W Y L p i ω) ^ 2 -
2 * contrast θ * oaReleaseValue W Y L p i ω + (contrast θ) ^ 2 by
funext ω; ring]
calc
_ = (∫ ω, (oaReleaseValue W Y L p i ω) ^ 2 -
2 * contrast θ * oaReleaseValue W Y L p i ω ∂μ) +
∫ _ : Ω, (contrast θ) ^ 2 ∂μ :=
integral_add (hA2.sub (hA.const_mul (2 * contrast θ))) (integrable_const _)
_ = ((∫ ω, (oaReleaseValue W Y L p i ω) ^ 2 ∂μ) -
∫ ω, 2 * contrast θ * oaReleaseValue W Y L p i ω ∂μ) +
∫ _ : Ω, (contrast θ) ^ 2 ∂μ := by
rw [integral_sub hA2 (hA.const_mul (2 * contrast θ))]
_ = _ := by rw [integral_const_mul, integral_const]; simp
_ = _ := by
rw [oaReleaseValue_sq_integral hmodel hnoise hε i,
oaReleaseValue_integral hmodel hnoise hε i]
unfold VOA
ring
def oaCombinedValue
For the supplied quantities and conditions, the oa combined value is the mathematical object specified below.
Definition (Lean source)
parameters
p :
ℝ
x :
PrivateAlphabet × ℝ
def
oaCombinedValue p x :
ℝ
given by
oaTrialValue p x.1 + x.2
theorem oaCombinedValue_measurable
Under the supplied quantities and conditions, the oa combined value measurable assertion holds.
Formal statement
Proof (Lean source)
lemma oaCombinedValue_measurable (p : ℝ) : Measurable (oaCombinedValue p) := by
unfold oaCombinedValue
exact ((show Measurable (oaTrialValue p) from measurable_of_finite _).comp
measurable_fst).add measurable_snd
theorem oaReleaseValue_ae_eq_combined
the oa release value ae eq combined assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma oaReleaseValue_ae_eq_combined {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} {Y0 Y1 W Y L : ℕ → Ω → ℝ} {p : ℝ} {θ : TrialParameter}
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ) (i : ℕ) :
oaReleaseValue W Y L p i =ᵐ[μ] fun ω =>
oaCombinedValue p (privateRecord W Y i ω, L i ω) := by
filter_upwards [oaTrialPart_ae_eq_record hmodel i] with ω hω
rw [oaReleaseValue_eq_trial_add, hω]
rfl
theorem oaReleaseValue_iIndep
the oa release value i indep assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Ω :
p ε :
ℝ
θ :
hmodel :
LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ
hnoise :
ComparatorNoise μ W Y L p ε
conclusion
iIndepFun (fun i => oaReleaseValue W Y L p i) μ
Proof (Lean source)
lemma oaReleaseValue_iIndep {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} {Y0 Y1 W Y L : ℕ → Ω → ℝ} {p ε : ℝ} {θ : TrialParameter}
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ)
(hnoise : ComparatorNoise μ W Y L p ε) :
iIndepFun (fun i => oaReleaseValue W Y L p i) μ := by
have hcomp := hnoise.1.comp
(fun _ : ℕ => oaCombinedValue p)
(fun _ => oaCombinedValue_measurable p)
exact hcomp.congr fun i => (oaReleaseValue_ae_eq_combined hmodel i).symm
theorem oaCombined_identDistrib
the oa combined ident distrib assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the oa Combined ident Distrib.
Formal statement
hypotheses
Ω :
p ε :
ℝ
θ :
hmodel :
LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ
hnoise :
ComparatorNoise μ W Y L p ε
hε :
0 < ε
i :
ℕ
conclusion
IdentDistrib (fun ω => (privateRecord W Y 0 ω, L 0 ω)) (fun ω => (privateRecord W Y i ω, L i ω)) μ μ
Proof (Lean source)
lemma oaCombined_identDistrib {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} {Y0 Y1 W Y L : ℕ → Ω → ℝ} {p ε : ℝ} {θ : TrialParameter}
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ)
(hnoise : ComparatorNoise μ W Y L p ε) (hε : 0 < ε) (i : ℕ) :
IdentDistrib
(fun ω => (privateRecord W Y 0 ω, L 0 ω))
(fun ω => (privateRecord W Y i ω, L i ω)) μ μ := by
letI : IsProbabilityMeasure μ := hmodel.iid.1.isProbabilityMeasure
let B : ℕ → Ω → PrivateAlphabet := privateRecord W Y
have hB (k : ℕ) : Measurable (B k) := privateRecord_measurable_of_model hmodel k
have hL (k : ℕ) : AEMeasurable (L k) μ :=
hnoise.aemeasurable hmodel.assignmentInterior hε k
have hprod (k : ℕ) : μ.map (fun ω => (B k ω, L k ω)) =
(μ.map (B k)).prod (μ.map (L k)) := by
exact (indepFun_iff_map_prod_eq_prod_map_map (hB k).aemeasurable (hL k)).mp
(hnoise.2.2 k).symm
refine ⟨(hB 0).aemeasurable.prodMk (hL 0),
(hB i).aemeasurable.prodMk (hL i), ?_⟩
change μ.map (fun ω => (B 0 ω, L 0 ω)) = μ.map (fun ω => (B i ω, L i ω))
rw [hprod 0, hprod i,
show μ.map (B 0) = inputLaw θ p from
recordLaw_eq_piTheta μ Y0 Y1 W Y p θ hmodel 0,
show μ.map (B i) = inputLaw θ p from
recordLaw_eq_piTheta μ Y0 Y1 W Y p θ hmodel i,
hnoise.2.1 0, hnoise.2.1 i]
theorem oaReleaseValue_identDistrib
the oa release value ident distrib assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the oa Release Value ident Distrib.
Formal statement
hypotheses
Ω :
p ε :
ℝ
θ :
hmodel :
LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ
hnoise :
ComparatorNoise μ W Y L p ε
hε :
0 < ε
i :
ℕ
conclusion
Proof (Lean source)
lemma oaReleaseValue_identDistrib {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} {Y0 Y1 W Y L : ℕ → Ω → ℝ} {p ε : ℝ} {θ : TrialParameter}
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ)
(hnoise : ComparatorNoise μ W Y L p ε) (hε : 0 < ε) (i : ℕ) :
IdentDistrib (oaReleaseValue W Y L p 0) (oaReleaseValue W Y L p i) μ μ := by
have hc := (oaCombined_identDistrib hmodel hnoise hε i).comp
(oaCombinedValue_measurable p)
refine ⟨oaReleaseValue_aemeasurable hmodel hnoise hε 0,
oaReleaseValue_aemeasurable hmodel hnoise hε i, ?_⟩
rw [Measure.map_congr (oaReleaseValue_ae_eq_combined hmodel 0),
Measure.map_congr (oaReleaseValue_ae_eq_combined hmodel i)]
exact hc.map_eq
theorem oaRelease_clt
the oa release clt assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the oa Release clt.
Formal statement
hypotheses
Ω :
Y0 Y1 W Y L :
ℕ → Ω → ℝ
p ε :
ℝ
θ :
hmodel :
LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ
hnoise :
ComparatorNoise μ W Y L p ε
hε :
0 < ε
conclusion
Modes.TendstoInLaw (fun _ : ℕ => μ) (fun n ω => sqrt n * (tauOA W Y L p n ω - contrast θ)) atTop (gaussianMeasure 0 (VOA θ p ε))
Proof (Lean source)
lemma oaRelease_clt {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ]
{Y0 Y1 W Y L : ℕ → Ω → ℝ} {p ε : ℝ} {θ : TrialParameter}
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ)
(hnoise : ComparatorNoise μ W Y L p ε) (hε : 0 < ε) :
Modes.TendstoInLaw (fun _ : ℕ => μ)
(fun n ω => sqrt n * (tauOA W Y L p n ω - contrast θ))
atTop (gaussianMeasure 0 (VOA θ p ε)) := by
let A : ℕ → Ω → ℝ := fun i => oaReleaseValue W Y L p i
have hA (i : ℕ) : AEMeasurable (A i) μ :=
oaReleaseValue_aemeasurable hmodel hnoise hε i
have hAI : iIndepFun A μ := oaReleaseValue_iIndep hmodel hnoise
have hAD (i : ℕ) : IdentDistrib (A 0) (A i) μ μ :=
oaReleaseValue_identDistrib hmodel hnoise hε i
let Am : ℕ → Ω → ℝ := fun i => (hA i).mk (A i)
have hAmMeas (i : ℕ) : Measurable (Am i) := (hA i).measurable_mk
have hAmAE (i : ℕ) : Am i =ᵐ[μ] A i := (hA i).ae_eq_mk.symm
have hAmI : iIndepFun Am μ := hAI.congr fun i => (hAmAE i).symm
have hAmD (i : ℕ) : IdentDistrib (Am 0) (Am i) μ μ := by
refine ⟨(hAmMeas 0).aemeasurable, (hAmMeas i).aemeasurable, ?_⟩
rw [Measure.map_congr (hAmAE 0), Measure.map_congr (hAmAE i)]
exact (hAD i).map_eq
let S : IIDSample Ω ℝ μ (μ.map (A 0)) := {
Z := Am
meas := hAmMeas
indep := hAmI
identDist := hAmD
law := Measure.map_congr (hAmAE 0) }
let ψ : ℝ → ℝ := id - fun _ => contrast θ
letI : IsProbabilityMeasure (μ.map (A 0)) := Measure.isProbabilityMeasure_map (hA 0)
have hmean : ∫ x, ψ x ∂(μ.map (A 0)) = 0 := by
change (∫ x, x - contrast θ ∂(μ.map (A 0))) = 0
rw [integral_map (hA 0)
(show AEStronglyMeasurable (fun x : ℝ => x - contrast θ) (μ.map (A 0)) by
fun_prop)]
have hAint : Integrable (A 0) μ := by
change Integrable (oaTrialPart W Y p 0 + L 0) μ
exact (oaTrialPart_integrable hmodel 0).add
(comparatorNoise_integrable hmodel.assignmentInterior hnoise hε 0)
rw [integral_sub hAint (integrable_const _),
oaReleaseValue_integral hmodel hnoise hε 0,
integral_const]
simp
have hsq : Integrable (fun x => (ψ x) ^ 2) (μ.map (A 0)) := by
change Integrable (fun x : ℝ => (x - contrast θ) ^ 2) (μ.map (A 0))
rw [integrable_map_measure
(show AEStronglyMeasurable (fun x : ℝ => (x - contrast θ) ^ 2)
(μ.map (A 0)) by fun_prop) (hA 0)]
have hrel2 := oaReleaseValue_sq_integrable hmodel hnoise hε 0
have hrel : Integrable (fun ω => oaReleaseValue W Y L p 0 ω) μ := by
change Integrable (oaTrialPart W Y p 0 + L 0) μ
exact (oaTrialPart_integrable hmodel 0).add
(comparatorNoise_integrable hmodel.assignmentInterior hnoise hε 0)
have hexpand : Integrable (fun ω =>
(oaReleaseValue W Y L p 0 ω) ^ 2 -
2 * contrast θ * oaReleaseValue W Y L p 0 ω + (contrast θ) ^ 2) μ :=
(hrel2.sub (hrel.const_mul (2 * contrast θ))).add (integrable_const _)
exact hexpand.congr (Filter.Eventually.of_forall fun ω => by
change _ = (oaReleaseValue W Y L p 0 ω - contrast θ) ^ 2
ring)
have hvar : ∫ x, (ψ x) ^ 2 ∂(μ.map (A 0)) = VOA θ p ε := by
change (∫ x : ℝ, (x - contrast θ) ^ 2 ∂(μ.map (A 0))) = VOA θ p ε
rw [integral_map (hA 0)
(show AEStronglyMeasurable (fun x : ℝ => (x - contrast θ) ^ 2)
(μ.map (A 0)) by fun_prop)]
simpa only [Function.comp_apply, A] using
oaReleaseValue_centered_sq_integral hmodel hnoise hε 0
have hclt0 := S.clt_normalized_sum (measurable_id.sub measurable_const) hmean hsq
have hclt : Modes.TendstoInLaw (fun _ : ℕ => μ)
(IsAsymLinear.normalizedSum S ψ (fun m => range m)) atTop
(gaussianMeasure 0 (VOA θ p ε)) := by
change Modes.TendstoInLaw (fun _ : ℕ => μ)
(IsAsymLinear.normalizedSum S ψ (fun m => range m)) atTop
(gaussianMeasure 0 (∫ x, (ψ x) ^ 2 ∂(μ.map (A 0)))) at hclt0
rw [hvar] at hclt0
exact hclt0
have htargetMeas (n : ℕ) : AEMeasurable
(fun ω => sqrt n * (tauOA W Y L p n ω - contrast θ)) μ := by
have hsum (s : Finset ℕ) : AEMeasurable (fun ω => ∑ i ∈ s, A i ω) μ := by
induction s using Finset.induction_on with
| empty => simp
| @insert a s ha ih =>
have heq : (fun ω => ∑ i ∈ insert a s, A i ω) =
A a + fun ω => ∑ i ∈ s, A i ω := by
funext ω
rw [Finset.sum_insert ha]
rfl
rw [heq]
change AEMeasurable (A a + fun ω => ∑ i ∈ s, A i ω) μ
exact (hA a).add ih
unfold tauOA
exact aemeasurable_const.mul
((aemeasurable_const.mul (hsum (range n))).sub aemeasurable_const)
have hnormMeas (n : ℕ) : AEMeasurable
(IsAsymLinear.normalizedSum S ψ (fun m => range m) n) μ := by
exact ((Finset.measurable_sum _ fun i _ =>
(measurable_id.sub measurable_const).comp (S.meas i)).const_mul _).aemeasurable
apply Modes.TendstoInLaw.congr_ae hclt htargetMeas
filter_upwards with n
have hall : ∀ᵐ ω ∂μ, ∀ i ∈ range n, S.Z i ω = A i ω := by
induction range n using Finset.induction_on with
| empty => simp
| @insert a s ha ih =>
filter_upwards [hAmAE a, ih] with ω haω hsω
intro i hi
simp only [Finset.mem_insert] at hi
rcases hi with rfl | hi
· exact haω
· exact hsω i hi
filter_upwards [hall] with ω hω
unfold IsAsymLinear.normalizedSum tauOA ψ
simp only [Finset.card_range, Pi.sub_apply, id_eq]
change (sqrt n)⁻¹ *
(∑ i ∈ range n, (S.Z i ω - contrast θ)) =
sqrt n * ((n : ℝ)⁻¹ *
∑ i ∈ range n, oaReleaseValue W Y L p i ω - contrast θ)
have hsumEq : (∑ i ∈ range n, (S.Z i ω - contrast θ)) =
∑ i ∈ range n, (A i ω - contrast θ) := by
apply Finset.sum_congr rfl
intro i hi
rw [hω i hi]
rw [hsumEq]
simp only [A]
rcases n with _ | n
· simp
· have hn : (0 : ℝ) < (n + 1 : ℕ) := by positivity
have hspos : 0 < sqrt (n + 1 : ℕ) := Real.sqrt_pos.2 hn
have hrsq : (sqrt (n + 1 : ℕ)) ^ 2 = ((n + 1 : ℕ) : ℝ) :=
Real.sq_sqrt hn.le
rw [Finset.sum_sub_distrib, Finset.sum_const, Finset.card_range]
simp only [nsmul_eq_mul]
field_simp [ne_of_gt hspos, ne_of_gt hn]
rw [hrsq]
theorem oaVariance_consistency
the oa variance consistency assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the oa Variance consistency.
Formal statement
hypotheses
Ω :
Y0 Y1 W Y L :
ℕ → Ω → ℝ
p ε :
ℝ
θ :
hmodel :
LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ
hnoise :
ComparatorNoise μ W Y L p ε
hε :
0 < ε
δ :
ℝ
0 < δ
Proof (Lean source)
lemma oaVariance_consistency {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ]
{Y0 Y1 W Y L : ℕ → Ω → ℝ} {p ε : ℝ} {θ : TrialParameter}
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ)
(hnoise : ComparatorNoise μ W Y L p ε) (hε : 0 < ε) :
∀ δ : ℝ, 0 < δ →
Tendsto (fun n => μ {ω | δ < |VhatOA W Y L p n ω - VOA θ p ε|})
atTop (nhds 0) := by
let A : ℕ → Ω → ℝ := fun i => oaReleaseValue W Y L p i
have hA (i : ℕ) : AEMeasurable (A i) μ :=
oaReleaseValue_aemeasurable hmodel hnoise hε i
have hAI : iIndepFun A μ := oaReleaseValue_iIndep hmodel hnoise
have hAD (i : ℕ) : IdentDistrib (A 0) (A i) μ μ :=
oaReleaseValue_identDistrib hmodel hnoise hε i
let Am : ℕ → Ω → ℝ := fun i => (hA i).mk (A i)
have hAmMeas (i : ℕ) : Measurable (Am i) := (hA i).measurable_mk
have hAmAE (i : ℕ) : Am i =ᵐ[μ] A i := (hA i).ae_eq_mk.symm
have hAmI : iIndepFun Am μ := hAI.congr fun i => (hAmAE i).symm
have hAmD (i : ℕ) : IdentDistrib (Am 0) (Am i) μ μ := by
refine ⟨(hAmMeas 0).aemeasurable, (hAmMeas i).aemeasurable, ?_⟩
rw [Measure.map_congr (hAmAE 0), Measure.map_congr (hAmAE i)]
exact (hAD i).map_eq
let S : IIDSample Ω ℝ μ (μ.map (A 0)) := {
Z := Am
meas := hAmMeas
indep := hAmI
identDist := hAmD
law := Measure.map_congr (hAmAE 0) }
let ψ : ℝ → ℝ := fun x => x - contrast θ
letI : IsProbabilityMeasure (μ.map (A 0)) := Measure.isProbabilityMeasure_map (hA 0)
have hψmean : ∫ x, ψ x ∂(μ.map (A 0)) = 0 := by
change (∫ x, x - contrast θ ∂(μ.map (A 0))) = 0
rw [integral_map (hA 0)
(show AEStronglyMeasurable (fun x : ℝ => x - contrast θ) (μ.map (A 0)) by
fun_prop)]
have hAint : Integrable (A 0) μ := by
change Integrable (oaTrialPart W Y p 0 + L 0) μ
exact (oaTrialPart_integrable hmodel 0).add
(comparatorNoise_integrable hmodel.assignmentInterior hnoise hε 0)
rw [integral_sub hAint (integrable_const _),
oaReleaseValue_integral hmodel hnoise hε 0, integral_const]
simp
have hψint : Integrable (fun ω => ψ (S.Z 0 ω)) μ := by
have hAint : Integrable (A 0) μ := by
change Integrable (oaTrialPart W Y p 0 + L 0) μ
exact (oaTrialPart_integrable hmodel 0).add
(comparatorNoise_integrable hmodel.assignmentInterior hnoise hε 0)
have hi : Integrable (fun ω => A 0 ω - contrast θ) μ :=
hAint.sub (integrable_const _)
exact hi.congr (by
filter_upwards [hAmAE 0] with ω hω
simp only [ψ, S]
rw [hω])
have hψsq : Integrable (fun ω => (ψ (S.Z 0 ω)) ^ 2) μ := by
have hi : Integrable (fun ω => (A 0 ω - contrast θ) ^ 2) μ := by
have hrel2 := oaReleaseValue_sq_integrable hmodel hnoise hε 0
have hrel : Integrable (A 0) μ := by
change Integrable (oaTrialPart W Y p 0 + L 0) μ
exact (oaTrialPart_integrable hmodel 0).add
(comparatorNoise_integrable hmodel.assignmentInterior hnoise hε 0)
exact ((hrel2.sub (hrel.const_mul (2 * contrast θ))).add
(integrable_const ((contrast θ) ^ 2))).congr
(Filter.Eventually.of_forall fun ω => by
change (A 0 ω) ^ 2 - 2 * contrast θ * A 0 ω + (contrast θ) ^ 2 =
(A 0 ω - contrast θ) ^ 2
ring)
exact hi.congr (by
filter_upwards [hAmAE 0] with ω hω
simp only [ψ, S]
rw [hω])
have hpop : ∫ x, (ψ x) ^ 2 ∂(μ.map (A 0)) = VOA θ p ε := by
change (∫ x : ℝ, (x - contrast θ) ^ 2 ∂(μ.map (A 0))) = VOA θ p ε
rw [integral_map (hA 0)
(show AEStronglyMeasurable (fun x : ℝ => (x - contrast θ) ^ 2)
(μ.map (A 0)) by fun_prop)]
simpa only [Function.comp_apply, A] using
oaReleaseValue_centered_sq_integral hmodel hnoise hε 0
have hcons := S.empiricalVar_tendsto_inProb
(measurable_id.sub measurable_const) hψint hψsq hψmean
change Modes.TendstoInProbability (fun _ : ℕ => μ) (S.empiricalVar ψ) atTop
(fun _ _ => ∫ x, (ψ x) ^ 2 ∂(μ.map (A 0))) at hcons
rw [hpop] at hcons
have hemp (n : ℕ) : S.empiricalVar ψ n =ᵐ[μ] VhatOA W Y L p n := by
have hall : ∀ᵐ ω ∂μ, ∀ i ∈ range n, S.Z i ω = A i ω := by
induction range n using Finset.induction_on with
| empty => simp
| @insert a s ha ih =>
filter_upwards [hAmAE a, ih] with ω haω hsω
intro i hi
simp only [Finset.mem_insert] at hi
rcases hi with rfl | hi
· exact haω
· exact hsω i hi
filter_upwards [hall] with ω hω
rcases n with _ | n
· simp [IIDSample.empiricalVar, IIDSample.sampleMean, VhatOA, tauOA]
· rw [VhatOA_eq_second_moment_sub_sq W Y L p (n + 1) (by omega)]
unfold IIDSample.empiricalVar IIDSample.sampleMean ψ tauOA
have hsum1 : (∑ i ∈ range (n + 1), (S.Z i ω - contrast θ)) =
∑ i ∈ range (n + 1), (A i ω - contrast θ) := by
apply Finset.sum_congr rfl
intro i hi
rw [hω i hi]
have hsum2 : (∑ i ∈ range (n + 1), (S.Z i ω - contrast θ) ^ 2) =
∑ i ∈ range (n + 1), (A i ω - contrast θ) ^ 2 := by
apply Finset.sum_congr rfl
intro i hi
rw [hω i hi]
have hsqexpand : (∑ i ∈ range (n + 1),
(A i ω - contrast θ) ^ 2) =
(∑ i ∈ range (n + 1), (A i ω) ^ 2) -
2 * contrast θ * (∑ i ∈ range (n + 1), A i ω) +
(n + 1 : ℝ) * (contrast θ) ^ 2 := by
have hpoint (x : ℝ) : (x - contrast θ) ^ 2 =
x ^ 2 - (2 * contrast θ) * x + (contrast θ) ^ 2 := by ring
simp_rw [hpoint]
rw [Finset.sum_add_distrib, Finset.sum_sub_distrib,
← Finset.mul_sum, Finset.sum_const, Finset.card_range]
simp only [nsmul_eq_mul]
norm_num [Nat.cast_add, Nat.cast_one]
rw [hsum1, hsum2, hsqexpand, Finset.sum_sub_distrib, Finset.sum_const,
Finset.card_range]
simp only [nsmul_eq_mul, A]
have hn : ((n + 1 : ℕ) : ℝ) ≠ 0 := by positivity
field_simp [hn]
norm_num [Nat.cast_add, Nat.cast_one]
ring
intro δ hδ
have htail := (Modes.tendstoInProbability_iff_norm
(fun _ : ℕ => μ) (S.empiricalVar ψ) atTop (fun _ _ => VOA θ p ε)).mp hcons δ hδ
have htail' : Tendsto (fun n => μ {ω | δ ≤
|VhatOA W Y L p n ω - VOA θ p ε|}) atTop (nhds 0) := by
apply htail.congr'
filter_upwards with n
apply measure_congr
filter_upwards [hemp n] with ω hω
simp only [Real.norm_eq_abs]
change (δ ≤ |S.empiricalVar ψ n ω - VOA θ p ε|) =
(δ ≤ |VhatOA W Y L p n ω - VOA θ p ε|)
rw [hω]
exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds htail'
(fun n => (bot_le : (0 : ENNReal) ≤ μ {ω | δ <
|VhatOA W Y L p n ω - VOA θ p ε|}))
(fun n => measure_mono (by
intro ω hω
change δ < |VhatOA W Y L p n ω - VOA θ p ε| at hω
change δ ≤ |VhatOA W Y L p n ω - VOA θ p ε|
exact le_of_lt hω))
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.OracleEnvelope 9 declarations Staircase information envelope
Staircase information envelope
theorem informationObjective_affine_weights
Under the supplied quantities and conditions, the information objective affine weights assertion holds.
Formal statement
hypotheses
conclusion
informationObjective θ p ε (a • α + b • β) t
= a * informationObjective θ p ε α t + b * informationObjective θ p ε β t
Proof (Lean source)
lemma informationObjective_affine_weights (θ : TrialParameter) (p ε t a b : ℝ)
(α β : StaircaseWeight) :
informationObjective θ p ε (a • α + b • β) t =
a * informationObjective θ p ε α t +
b * informationObjective θ p ε β t := by
unfold informationObjective
rw [Finset.mul_sum, Finset.mul_sum, ← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro s _
change (a * α s + b * β s) * patternInformation θ p ε s t =
a * (α s * patternInformation θ p ε s t) +
b * (β s * patternInformation θ p ε s t)
ring
theorem informationObjective_exists_maximizer
the information objective exists maximizer assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the information Objective exists maximizer.
Formal statement
hypotheses
conclusion
∃ α ∈ feasibleSet ε,
Proof (Lean source)
lemma informationObjective_exists_maximizer (θ : TrialParameter)
(p ε t : ℝ) (hε : 0 ≤ ε) :
∃ α ∈ feasibleSet ε, ∀ β ∈ feasibleSet ε,
informationObjective θ p ε β t ≤ informationObjective θ p ε α t := by
obtain ⟨α, hα, hmax⟩ :=
(staircaseFeasible_compact ε hε).exists_isMaxOn
(staircaseFeasible_nonempty ε)
(informationObjective_continuous_weight θ p ε t).continuousOn
exact ⟨α, hα, fun β hβ => hmax hβ⟩
theorem upperEnvelope_attained
Under the supplied quantities and conditions, the upper envelope attained assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the upper Envelope attained.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma upperEnvelope_attained (θ : TrialParameter) (p ε t : ℝ)
(hε : 0 ≤ ε) :
∃ α ∈ feasibleSet ε,
upperEnvelope θ p ε t = informationObjective θ p ε α t := by
obtain ⟨α, hα, hmax⟩ :=
informationObjective_exists_maximizer θ p ε t hε
refine ⟨α, hα, ?_⟩
unfold upperEnvelope
apply IsGreatest.csSup_eq
constructor
· exact ⟨α, hα, rfl⟩
· rintro u ⟨β, hβ, rfl⟩
exact hmax β hβ
theorem upperEnvelope_quadratic_lower_bound
Under the supplied quantities and conditions, the upper envelope quadratic lower bound assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the upper Envelope quadratic lower bound.
Formal statement
hypotheses
Proof (Lean source)
lemma upperEnvelope_quadratic_lower_bound (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε) :
∃ c : ℝ, 0 < c ∧ ∀ t : ℝ, c * t ^ 2 ≤ upperEnvelope θ p ε t := by
obtain ⟨α, hα, hα0⟩ := staircaseFeasible_positive_first_pattern ε
let c := α 0 * ((privacyIncrement ε * controlProb p) ^ 2 /
patternMass θ p ε 0)
have hc : 0 < c := by
have hd : 0 < privacyIncrement ε := privacyIncrement_pos hε
have hq : 0 < controlProb p := by dsimp [controlProb]; linarith [hp.2]
have hm := patternMass_pos_interior θ p ε hp hθ hε.le 0
dsimp [c]
positivity
refine ⟨c, hc, fun t => ?_⟩
obtain ⟨β, hβ, hmax⟩ :=
informationObjective_exists_maximizer θ p ε t hε.le
have hterm : α 0 * patternInformation θ p ε 0 t ≤
informationObjective θ p ε α t := by
unfold informationObjective
exact Finset.single_le_sum
(fun s _ => show 0 ≤ α s * patternInformation θ p ε s t from
mul_nonneg (hα.1 s)
(by unfold patternInformation
exact div_nonneg (sq_nonneg _)
(le_of_lt (patternMass_pos_interior θ p ε hp hθ hε.le s))))
(Finset.mem_univ 0)
calc
c * t ^ 2 = α 0 * patternInformation θ p ε 0 t := by
rw [first_pattern_information_quadratic]
dsimp [c]
ring
_ ≤ informationObjective θ p ε α t := hterm
_ ≤ upperEnvelope θ p ε t := by
unfold upperEnvelope
exact le_csSup ⟨informationObjective θ p ε β t, by
rintro u ⟨γ, hγ, rfl⟩
exact hmax γ hγ⟩ ⟨α, hα, rfl⟩
theorem upperEnvelope_continuous
Under the supplied quantities and conditions, the upper envelope continuous assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the upper Envelope continuous.
Formal statement
hypotheses
conclusion
Continuous (fun t : ℝ => upperEnvelope θ p ε t)
Proof (Lean source)
lemma upperEnvelope_continuous (θ : TrialParameter) (p ε : ℝ)
(hε : 0 ≤ ε) :
Continuous (fun t : ℝ => upperEnvelope θ p ε t) := by
have hf : Continuous (fun z : ℝ × StaircaseWeight =>
informationObjective θ p ε z.2 z.1) := by
unfold informationObjective
apply continuous_finsetSum
intro s _
unfold patternInformation projectedGradient direction
simp +decide only [Fin.sum_univ_succ, Fin.sum_univ_zero, ite_true,
ite_false, add_zero]
fun_prop
have hc := (staircaseFeasible_compact ε hε).continuous_sSup
(f := fun t α => informationObjective θ p ε α t) hf
simpa [upperEnvelope, image, eq_comm] using hc
For the supplied quantities and conditions, the vertex envelope is the mathematical object specified below. The vertex Envelope is determined by the displayed parameters.
Definition (Lean source)
parameters
θ :
p ε t :
ℝ
def
vertexEnvelope θ p ε t :
ℝ
given by
theorem upperEnvelope_eq_vertexEnvelope
Under the supplied quantities and conditions, the upper envelope eq vertex envelope assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the upper Envelope eq vertex Envelope.
Formal statement
hypotheses
conclusion
upperEnvelope θ p ε t = vertexEnvelope θ p ε t
Proof (Lean source)
lemma upperEnvelope_eq_vertexEnvelope (θ : TrialParameter) (p ε t : ℝ)
(hε : 0 ≤ ε) :
upperEnvelope θ p ε t = vertexEnvelope θ p ε t := by
let S := feasibleSet ε
let f : StaircaseWeight → ℝ := fun α => informationObjective θ p ε α t
have hcompact : IsCompact S := staircaseFeasible_compact ε hε
have hconvex : Convex ℝ S := staircaseFeasible_convex ε
have hne : S.Nonempty := staircaseFeasible_nonempty ε
obtain ⟨v, hv⟩ := hcompact.extremePoints_nonempty hne
obtain ⟨a, ha, hmax⟩ :=
informationObjective_exists_maximizer θ p ε t hε
have hsup : upperEnvelope θ p ε t = f a := by
unfold upperEnvelope
apply IsGreatest.csSup_eq
exact ⟨⟨a, ha, rfl⟩, by
rintro z ⟨b, hb, rfl⟩
exact hmax b hb⟩
have hvBound : ∀ x ∈ S.extremePoints ℝ, f x ≤ f a := by
intro x hx
exact hmax x (extremePoints_subset hx)
have hvertex : vertexEnvelope θ p ε t ≤ f a := by
unfold vertexEnvelope
apply csSup_le
· exact ⟨f v, v, hv, rfl⟩
· rintro z ⟨x, hx, rfl⟩
exact hvBound x hx
have hsub : S.extremePoints ℝ ⊆ {x | f x ≤ vertexEnvelope θ p ε t} := by
intro x hx
unfold vertexEnvelope
exact le_csSup ⟨f a, by
rintro z ⟨y, hy, rfl⟩
exact hvBound y hy⟩ ⟨x, hx, rfl⟩
have hclosed : IsClosed {x : StaircaseWeight | f x ≤ vertexEnvelope θ p ε t} :=
isClosed_le (informationObjective_continuous_weight θ p ε t) continuous_const
have hconvexBound : Convex ℝ {x : StaircaseWeight | f x ≤ vertexEnvelope θ p ε t} := by
intro x hx y hy a b ha hb hab
change f (a • x + b • y) ≤ vertexEnvelope θ p ε t
rw [show f (a • x + b • y) = a * f x + b * f y from
informationObjective_affine_weights θ p ε t a b x y]
calc
a * f x + b * f y ≤ a * vertexEnvelope θ p ε t +
b * vertexEnvelope θ p ε t :=
add_le_add (mul_le_mul_of_nonneg_left hx ha)
(mul_le_mul_of_nonneg_left hy hb)
_ = vertexEnvelope θ p ε t := by rw [← add_mul, hab, one_mul]
have hclosure : S ⊆ {x | f x ≤ vertexEnvelope θ p ε t} := by
rw [← closure_convexHull_extremePoints hcompact hconvex]
exact closure_minimal (convexHull_min hsub hconvexBound) hclosed
exact le_antisymm (hsup ▸ hclosure ha) (hsup ▸ hvertex)
theorem Jstar_le_upperEnvelope
Under the supplied quantities and conditions, the jstar le upper envelope assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the Jstar le upper Envelope.
Formal statement
hypotheses
conclusion
Jstar θ p ε ≤ upperEnvelope θ p ε t
Proof (Lean source)
lemma Jstar_le_upperEnvelope (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(hε : 0 ≤ ε) (t : ℝ) :
Jstar θ p ε ≤ upperEnvelope θ p ε t := by
obtain ⟨β, hβ, hmax⟩ :=
informationObjective_exists_maximizer θ p ε t hε
have hH : upperEnvelope θ p ε t = informationObjective θ p ε β t := by
unfold upperEnvelope
apply IsGreatest.csSup_eq
exact ⟨⟨β, hβ, rfl⟩, by
rintro u ⟨γ, hγ, rfl⟩
exact hmax γ hγ⟩
unfold Jstar
apply csSup_le
· obtain ⟨α, hα⟩ := staircaseFeasible_nonempty ε
exact ⟨_, α, hα, rfl⟩
· rintro z ⟨α, hα, rfl⟩
have hbdd : BddBelow {y : ℝ | ∃ u : ℝ,
y = informationObjective θ p ε α u} := by
refine ⟨0, ?_⟩
rintro y ⟨u, rfl⟩
exact informationObjective_nonneg_interior θ p ε hp hθ hε α hα u
exact (csInf_le hbdd ⟨t, rfl⟩).trans (hH ▸ hmax α hα)
theorem upperEnvelope_exists_minimizer
Under the supplied quantities and conditions, the upper envelope exists minimizer assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the upper Envelope exists minimizer.
Formal statement
hypotheses
conclusion
∃ t : ℝ, ∀ u : ℝ, upperEnvelope θ p ε t ≤ upperEnvelope θ p ε u
Proof (Lean source)
lemma upperEnvelope_exists_minimizer (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε) :
∃ t : ℝ, ∀ u : ℝ, upperEnvelope θ p ε t ≤ upperEnvelope θ p ε u := by
obtain ⟨c, hc, hbound⟩ :=
upperEnvelope_quadratic_lower_bound θ p ε hp hθ hε
have hnonneg : 0 ≤ upperEnvelope θ p ε 0 := by
have h := hbound 0
simpa using h
have hbounded : Bornology.IsBounded
{t : ℝ | upperEnvelope θ p ε t ≤ upperEnvelope θ p ε 0} := by
rw [isBounded_iff_forall_norm_le]
refine ⟨Real.sqrt (upperEnvelope θ p ε 0 / c), ?_⟩
intro t ht
have hs : t ^ 2 ≤ upperEnvelope θ p ε 0 / c :=
(le_div_iff₀ hc).2 (by rw [mul_comm]; exact (hbound t).trans ht)
have hdiv : 0 ≤ upperEnvelope θ p ε 0 / c := div_nonneg hnonneg hc.le
rw [Real.norm_eq_abs]
nlinarith [Real.sq_sqrt hdiv, sq_abs t, abs_nonneg t,
Real.sqrt_nonneg (upperEnvelope θ p ε 0 / c)]
exact (upperEnvelope_continuous θ p ε hε.le).exists_forall_le_of_isBounded
0 hbounded
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.OracleGeometry 31 declarations The contrast variance is the information-matrix quadratic solution value, with infinity when the contrast is outside the information range.
Exact finite staircase oracle
The contrast variance is the information-matrix quadratic solution value, with infinity when the contrast is outside the information range. The theorem states both the arbitrary-channel comparison and the finite saddle identity.
the contrast in range is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
the contrast variance is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
contrastVariance I :
ℝ≥0∞
given by
if contrastInRange I then ofReal (sInf {u : ℝ | ∃ v : TrialParameter, I.mulVec v = contrastVector ∧ u = ∑ k : Fin 2, contrastVector k * v k}) else ⊤
def feasibleSet
the feasible set is the mathematical object specified below. The feasible Set is determined by the displayed parameters.
Definition (Lean source)
theorem piTheta_pos_interior
Under the supplied quantities and conditions, the pi theta pos interior assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pi Theta pos interior.
Formal statement
hypotheses
conclusion
0 < piTheta θ p j
Proof (Lean source)
lemma piTheta_pos_interior (θ : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (j : Fin 4) :
0 < piTheta θ p j := by
rcases hp with ⟨hp0, hp1⟩
rcases hθ with ⟨h00, h01, h10, h11⟩
fin_cases j <;> dsimp [piTheta, controlProb] <;> apply mul_pos <;> linarith
Under the supplied quantities and conditions, the pattern ray one le interior assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pattern Ray one le interior.
Formal statement
Proof (Lean source)
lemma patternRay_one_le_interior (ε : ℝ) (hε : 0 ≤ ε)
(s : Fin 14) (j : Fin 4) : 1 ≤ patternRay ε s j := by
unfold patternRay privacyIncrement privacyRatio
split_ifs <;> simp_all [Real.one_le_exp_iff]
theorem patternMass_pos_interior
Under the supplied quantities and conditions, the pattern mass pos interior assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pattern Mass pos interior.
Formal statement
hypotheses
conclusion
0 < patternMass θ p ε s
Proof (Lean source)
lemma patternMass_pos_interior (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(hε : 0 ≤ ε) (s : Fin 14) :
0 < patternMass θ p ε s := by
unfold patternMass
apply Finset.sum_pos'
· intro j _
exact le_of_lt (mul_pos (piTheta_pos_interior θ p hp hθ j)
(lt_of_lt_of_le (by norm_num) (patternRay_one_le_interior ε hε s j)))
· exact ⟨0, Finset.mem_univ _, mul_pos
(piTheta_pos_interior θ p hp hθ 0)
(lt_of_lt_of_le (by norm_num) (patternRay_one_le_interior ε hε s 0))⟩
theorem informationObjective_nonneg_interior
Under the supplied quantities and conditions, the information objective nonneg interior assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the information Objective nonneg interior.
Formal statement
hypotheses
θ :
p ε :
ℝ
hp :
hθ :
hε :
0 ≤ ε
α :
hα :
t :
ℝ
conclusion
0 ≤ informationObjective θ p ε α t
Proof (Lean source)
lemma informationObjective_nonneg_interior (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 ≤ ε)
(α : StaircaseWeight) (hα : staircaseFeasible ε α) (t : ℝ) :
0 ≤ informationObjective θ p ε α t := by
unfold informationObjective patternInformation
apply Finset.sum_nonneg
intro s _
exact mul_nonneg (hα.1 s)
(div_nonneg (sq_nonneg _) (le_of_lt (patternMass_pos_interior θ p ε hp hθ hε s)))
theorem informationQuadratic_nonneg_staircase
Under the supplied quantities and conditions, the information quadratic nonneg staircase assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the information Quadratic nonneg staircase.
Formal statement
hypotheses
θ :
p ε :
ℝ
hp :
hθ :
hε :
0 ≤ ε
α :
hα :
v :
conclusion
0 ≤ informationQuadratic (informationMatrix θ p ε α) v
Proof (Lean source)
lemma informationQuadratic_nonneg_staircase (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 ≤ ε)
(α : StaircaseWeight) (hα : staircaseFeasible ε α)
(v : TrialParameter) :
0 ≤ informationQuadratic (informationMatrix θ p ε α) v := by
rw [informationQuadratic_eq_pattern_sum]
apply Finset.sum_nonneg
intro s _
exact div_nonneg (mul_nonneg (hα.1 s) (sq_nonneg _))
(le_of_lt (patternMass_pos_interior θ p ε hp hθ hε s))
theorem informationMatrix_posSemidef_interior
Under the supplied quantities and conditions, the information matrix pos semidef interior assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the information Matrix pos Semidef interior.
Formal statement
hypotheses
θ :
p ε :
ℝ
hp :
hθ :
hε :
0 ≤ ε
α :
hα :
conclusion
(informationMatrix θ p ε α).PosSemidef
Proof (Lean source)
lemma informationMatrix_posSemidef_interior (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 ≤ ε)
(α : StaircaseWeight) (hα : staircaseFeasible ε α) :
(informationMatrix θ p ε α).PosSemidef := by
rw [Matrix.posSemidef_iff_dotProduct_mulVec]
constructor
· ext i j
simp only [Matrix.conjTranspose_apply, star_trivial]
unfold informationMatrix
apply Finset.sum_congr rfl
intro s _
ring
· intro v
have hq := informationQuadratic_nonneg_staircase θ p ε hp hθ hε α hα v
have heq : star v ⬝ᵥ (informationMatrix θ p ε α).mulVec v =
informationQuadratic (informationMatrix θ p ε α) v := by
simp [informationQuadratic, dotProduct, mulVec,
Fin.sum_univ_succ, star_trivial]
ring
rw [heq]
exact hq
the contrast solution value pos assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
0 < ∑ k : Fin 2, contrastVector k * w k
Proof (Lean source)
lemma contrastSolution_value_pos (I : Matrix (Fin 2) (Fin 2) ℝ)
(hI : I.PosSemidef) (w : TrialParameter)
(hw : I.mulVec w = contrastVector) :
0 < ∑ k : Fin 2, contrastVector k * w k := by
let r := ∑ k : Fin 2, contrastVector k * w k
have hq : informationQuadratic I w = r := by
dsimp [r]
calc
informationQuadratic I w = ∑ k, w k * (I.mulVec w) k := by
simp [informationQuadratic, mulVec, Finset.mul_sum]
ring
_ = ∑ k, w k * contrastVector k := by rw [hw]
_ = ∑ k, contrastVector k * w k := by
apply Finset.sum_congr rfl
intro k _
ring
have hr : 0 ≤ r := by
rw [← hq]
have hn := hI.dotProduct_mulVec_nonneg w
rw [show star w ⬝ᵥ I.mulVec w = informationQuadratic I w by
simp [informationQuadratic, dotProduct, mulVec, star_trivial]
ring] at hn
exact hn
refine lt_of_le_of_ne hr ?_
intro hr0
have hzero : I.mulVec w = 0 := by
apply (hI.dotProduct_mulVec_zero_iff w).mp
rw [show star w ⬝ᵥ I.mulVec w = informationQuadratic I w by
simp [informationQuadratic, dotProduct, mulVec, star_trivial]
ring]
rw [hq, hr0]
rw [hw] at hzero
have := congrFun hzero 0
norm_num [contrastVector] at this
the contrast solution value unique assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma contrastSolution_value_unique (I : Matrix (Fin 2) (Fin 2) ℝ)
(hI : I.PosSemidef) (w v : TrialParameter)
(hw : I.mulVec w = contrastVector) (hv : I.mulVec v = contrastVector) :
(∑ k : Fin 2, contrastVector k * v k) =
∑ k : Fin 2, contrastVector k * w k := by
have hsym := hI.isHermitian
have hs := congrFun₂ hsym 0 1
calc
(∑ k, contrastVector k * v k) =
∑ k, (I.mulVec w) k * v k := by rw [hw]
_ = ∑ k, w k * (I.mulVec v) k := by
simp [mulVec, Fin.sum_univ_succ, star_trivial] at hs ⊢
rw [hs]
ring
_ = ∑ k, w k * contrastVector k := by rw [hv]
_ = ∑ k, contrastVector k * w k := by
apply Finset.sum_congr rfl
intro k _
ring
theorem contrastVariance_eq_solution_value
the contrast variance eq solution value assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma contrastVariance_eq_solution_value (I : Matrix (Fin 2) (Fin 2) ℝ)
(hI : I.PosSemidef) (w : TrialParameter)
(hw : I.mulVec w = contrastVector) :
contrastVariance I =
ofReal (∑ k : Fin 2, contrastVector k * w k) := by
have hrange : contrastInRange I := ⟨w, hw⟩
rw [contrastVariance, if_pos hrange]
congr 1
apply IsLeast.csInf_eq
refine ⟨⟨w, hw, rfl⟩, ?_⟩
rintro u ⟨v, hv, rfl⟩
exact le_of_eq (contrastSolution_value_unique I hI w v hw hv).symm
the contrast information cauchy assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma contrastInformation_cauchy (I : Matrix (Fin 2) (Fin 2) ℝ)
(hI : I.PosSemidef) (w : TrialParameter)
(hw : I.mulVec w = contrastVector) (t : ℝ) :
1 ≤ informationQuadratic I (direction t) *
(∑ k : Fin 2, contrastVector k * w k) := by
let r := ∑ k : Fin 2, contrastVector k * w k
let x := direction t
let y : TrialParameter := fun k => r * x k - w k
have hr : 0 < r := contrastSolution_value_pos I hI w hw
have hy : 0 ≤ informationQuadratic I y := by
have hn := hI.dotProduct_mulVec_nonneg y
rw [show star y ⬝ᵥ I.mulVec y = informationQuadratic I y by
simp [informationQuadratic, dotProduct, mulVec, star_trivial]
ring] at hn
exact hn
have hsym := hI.isHermitian
have hs := congrFun₂ hsym 0 1
have hqw : informationQuadratic I w = r := by
dsimp [r]
calc
informationQuadratic I w = ∑ k, w k * (I.mulVec w) k := by
simp [informationQuadratic, mulVec]
ring
_ = ∑ k, w k * contrastVector k := by rw [hw]
_ = ∑ k, contrastVector k * w k := by
apply Finset.sum_congr rfl
intro k _
ring
have hcross : (∑ k, x k * (I.mulVec w) k) = 1 := by
rw [hw]
simp [x, direction, contrastVector, Fin.sum_univ_succ]
have hid : informationQuadratic I y =
r * (r * informationQuadratic I x - 1) := by
have hexpand : informationQuadratic I y =
r ^ 2 * informationQuadratic I x -
2 * r * (∑ k, x k * (I.mulVec w) k) +
informationQuadratic I w := by
dsimp [y]
simp [informationQuadratic, mulVec, Fin.sum_univ_succ,
star_trivial] at hs ⊢
rw [hs]
ring
rw [hexpand, hcross, hqw]
ring
rw [hid] at hy
dsimp [r, x] at hy hr ⊢
nlinarith
theorem reciprocal_le_contrastVariance
the reciprocal le contrast variance assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
ofReal j⁻¹ ≤ contrastVariance I
Proof (Lean source)
lemma reciprocal_le_contrastVariance (I : Matrix (Fin 2) (Fin 2) ℝ)
(hI : I.PosSemidef) (j : ℝ) (hj : 0 < j)
(t : ℝ) (ht : informationQuadratic I (direction t) ≤ j) :
ofReal j⁻¹ ≤ contrastVariance I := by
by_cases hrange : contrastInRange I
· obtain ⟨w, hw⟩ := hrange
rw [contrastVariance_eq_solution_value I hI w hw]
apply ENNReal.ofReal_le_ofReal
apply (inv_le_iff_one_le_mul₀' hj).2
exact (contrastInformation_cauchy I hI w hw t).trans
(mul_le_mul_of_nonneg_right ht
(contrastSolution_value_pos I hI w hw).le)
· simp [contrastVariance, hrange]
theorem contrastVariance_eq_reciprocal_of_minimizer
Under the supplied quantities and conditions, the contrast variance eq reciprocal of minimizer assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma contrastVariance_eq_reciprocal_of_minimizer
(I : Matrix (Fin 2) (Fin 2) ℝ) (hI : I.PosSemidef) (t : ℝ)
(hmin : ∀ u : ℝ,
informationQuadratic I (direction t) ≤
informationQuadratic I (direction u))
(hpos : 0 < informationQuadratic I (direction t)) :
contrastVariance I =
ofReal (informationQuadratic I (direction t))⁻¹ := by
let j := informationQuadratic I (direction t)
let A := I 0 0 + I 0 1 + I 1 0 + I 1 1
let B := 2 * (I 0 0 * t + I 0 1 * (t + 1) +
I 1 0 * t + I 1 1 * (t + 1))
have hsym := hI.isHermitian
have hs := congrFun₂ hsym 0 1
have hpoly (x : ℝ) : 0 ≤ x * B + x ^ 2 * A := by
have h := hmin (t + x)
have hid : informationQuadratic I (direction (t + x)) =
informationQuadratic I (direction t) + x * B + x ^ 2 * A := by
dsimp [A, B]
simp [informationQuadratic, direction, Fin.sum_univ_succ,
star_trivial] at hs ⊢
rw [hs]
ring
rw [hid] at h
linarith
have hB : B = 0 := by
by_contra hB0
have hden : 0 < |A| + 1 := by positivity
have hAle : A ≤ |A| := le_abs_self A
have h := hpoly (-B / (2 * (|A| + 1)))
have hsq : 0 < B ^ 2 := sq_pos_of_ne_zero hB0
field_simp at h
nlinarith [mul_nonneg hsq.le (sub_nonneg.mpr hAle)]
let w : TrialParameter := fun k => direction t k / j
have hsum :
(I 0 0 * t + I 0 1 * (t + 1)) +
(I 1 0 * t + I 1 1 * (t + 1)) = 0 := by
dsimp [B] at hB
linarith
have hj_eq : j = I 1 0 * t + I 1 1 * (t + 1) := by
dsimp [j]
simp [informationQuadratic, direction, Fin.sum_univ_succ]
linear_combination t * hsum
have hw : I.mulVec w = contrastVector := by
have hj : 0 < j := hpos
funext k
fin_cases k <;>
simp [mulVec, dotProduct, w, direction, contrastVector,
Fin.sum_univ_succ] at hs ⊢ <;>
field_simp [ne_of_gt hj] <;> nlinarith
rw [contrastVariance_eq_solution_value I hI w hw]
congr 1
dsimp [w, j]
simp [direction, contrastVector, Fin.sum_univ_succ]
field_simp
ring
theorem first_pattern_information_quadratic Lemma first_pattern_information_quadratic in the paper ↗
Under the supplied quantities and conditions, the first pattern information quadratic assertion holds.
Formal statement
Proof (Lean source)
lemma first_pattern_information_quadratic (θ : TrialParameter) (p ε t : ℝ) :
patternInformation θ p ε 0 t =
((privacyIncrement ε * controlProb p) ^ 2 /
patternMass θ p ε 0) * t ^ 2 := by
simp +decide [patternInformation, projectedGradient, patternGradient,
direction, Fin.sum_univ_succ]
ring
theorem informationObjective_eq_informationQuadratic
Under the supplied quantities and conditions, the information objective eq information quadratic assertion holds.
Formal statement
hypotheses
conclusion
informationObjective θ p ε α t
Proof (Lean source)
lemma informationObjective_eq_informationQuadratic
(θ : TrialParameter) (p ε : ℝ) (α : StaircaseWeight) (t : ℝ) :
informationObjective θ p ε α t =
∑ a : Fin 2, ∑ b : Fin 2,
direction t a * informationMatrix θ p ε α a b * direction t b := by
have h :
(∑ a : Fin 2, ∑ b : Fin 2,
direction t a * informationMatrix θ p ε α a b * direction t b) =
∑ s : Fin 14, α s *
(∑ k : Fin 2, patternGradient p ε s k * direction t k) ^ 2 /
patternMass θ p ε s := by
unfold informationMatrix
simp_rw [Finset.mul_sum, Finset.sum_mul, pow_two]
calc
(∑ a, ∑ b, ∑ s,
direction t a * (α s * patternGradient p ε s a * patternGradient p ε s b /
patternMass θ p ε s) * direction t b) =
∑ a, ∑ s, ∑ b,
direction t a * (α s * patternGradient p ε s a * patternGradient p ε s b /
patternMass θ p ε s) * direction t b := by
apply Finset.sum_congr rfl
intro a _
rw [Finset.sum_comm]
_ = ∑ s, ∑ a, ∑ b,
direction t a * (α s * patternGradient p ε s a * patternGradient p ε s b /
patternMass θ p ε s) * direction t b := by rw [Finset.sum_comm]
_ = _ := by
apply Finset.sum_congr rfl
intro s _
simp_rw [Finset.mul_sum, Finset.sum_mul, Finset.sum_div]
apply Finset.sum_congr rfl
intro a _
simp_rw [Finset.mul_sum, Finset.sum_div]
apply Finset.sum_congr rfl
intro b _
ring
calc
informationObjective θ p ε α t =
∑ s : Fin 14, α s *
(∑ k : Fin 2, patternGradient p ε s k * direction t k) ^ 2 /
patternMass θ p ε s := by
unfold informationObjective patternInformation projectedGradient
apply Finset.sum_congr rfl
intro s _
ring
_ = _ := h.symm
theorem staircaseFeasible_weight_le_one
Under the supplied quantities and conditions, the staircase feasible weight le one assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the staircase Feasible weight le one.
Formal statement
hypotheses
conclusion
α s ≤ 1
Proof (Lean source)
lemma staircaseFeasible_weight_le_one (ε : ℝ) (hε : 0 ≤ ε)
(α : StaircaseWeight) (hα : staircaseFeasible ε α) (s : Fin 14) :
α s ≤ 1 := by
have hr (u : Fin 14) : 1 ≤ patternRay ε u 0 := by
unfold patternRay privacyIncrement privacyRatio
split_ifs <;> simp_all [Real.one_le_exp_iff]
have hterm (u : Fin 14) : 0 ≤ α u * patternRay ε u 0 :=
mul_nonneg (hα.1 u) (by linarith [hr u])
have hle : α s ≤ α s * patternRay ε s 0 := by
nlinarith [mul_nonneg (hα.1 s) (sub_nonneg.mpr (hr s))]
have hsingle : α s * patternRay ε s 0 ≤
∑ u : Fin 14, α u * patternRay ε u 0 :=
Finset.single_le_sum (fun u _ => hterm u) (Finset.mem_univ s)
rw [← staircaseMatrix, hα.2 0] at hsingle
exact hle.trans hsingle
theorem staircaseFeasible_sum_le_one
Under the supplied quantities and conditions, the staircase feasible sum le one assertion holds. Under the stated assumptions, the staircase Feasible sum le one.
Formal statement
hypotheses
conclusion
(∑ s : Fin 14, α s) ≤ 1
Proof (Lean source)
lemma staircaseFeasible_sum_le_one (ε : ℝ) (hε : 0 ≤ ε)
(α : StaircaseWeight) (hα : staircaseFeasible ε α) :
(∑ s : Fin 14, α s) ≤ 1 := by
have hr (s : Fin 14) : 1 ≤ patternRay ε s 0 := by
unfold patternRay privacyIncrement privacyRatio
split_ifs <;> simp_all [Real.one_le_exp_iff]
calc
(∑ s : Fin 14, α s) = ∑ s : Fin 14, α s * 1 := by simp
_ ≤ ∑ s : Fin 14, α s * patternRay ε s 0 := by
apply Finset.sum_le_sum
intro s hs
exact mul_le_mul_of_nonneg_left (hr s) (hα.1 s)
_ = 1 := hα.2 0
the staircase feasible nonempty assertion holds.
Formal statement
Proof (Lean source)
lemma staircaseFeasible_nonempty (ε : ℝ) :
(feasibleSet ε).Nonempty := by
let α : StaircaseWeight :=
fun s => if s = 0 ∨ s = 13 then 1 / (exp ε + 1) else 0
refine ⟨α, ?_⟩
change staircaseFeasible ε α
constructor
· intro s
dsimp [α]
split_ifs <;> positivity
· intro j
fin_cases j <;>
simp +decide [staircaseMatrix, α, patternRay, patternContains,
privacyIncrement, privacyRatio, Fin.sum_univ_succ] <;>
field_simp <;> ring
theorem staircaseFeasible_singleton_design
the staircase feasible singleton design assertion holds.
Formal statement
hypotheses
ε :
ℝ
conclusion
staircaseFeasible ε (fun s : Fin 14 => if s ∈ ({0, 1, 3, 7} : Finset (Fin 14)) then 1 / (exp ε + 3) else 0)
Proof (Lean source)
lemma staircaseFeasible_singleton_design (ε : ℝ) :
staircaseFeasible ε (fun s : Fin 14 =>
if s ∈ ({0, 1, 3, 7} : Finset (Fin 14)) then
1 / (exp ε + 3) else 0) := by
constructor
· intro s
dsimp
split_ifs <;> positivity
· intro j
fin_cases j <;>
simp +decide [staircaseMatrix, patternRay, patternContains,
privacyIncrement, privacyRatio, Fin.sum_univ_succ] <;>
field_simp <;> ring
theorem staircaseFeasible_positive_first_pattern
the staircase feasible positive first pattern assertion holds.
Formal statement
hypotheses
ε :
ℝ
Proof (Lean source)
lemma staircaseFeasible_positive_first_pattern (ε : ℝ) :
∃ α : StaircaseWeight, staircaseFeasible ε α ∧ 0 < α 0 := by
let α : StaircaseWeight :=
fun s => if s = 0 ∨ s = 13 then 1 / (exp ε + 1) else 0
refine ⟨α, ?_, by dsimp [α]; positivity⟩
constructor
· intro s
dsimp [α]
split_ifs <;> positivity
· intro j
fin_cases j <;>
simp +decide [staircaseMatrix, α, patternRay, patternContains,
privacyIncrement, privacyRatio, Fin.sum_univ_succ] <;>
field_simp <;> ring
the staircase feasible closed assertion holds.
Formal statement
Proof (Lean source)
lemma staircaseFeasible_closed (ε : ℝ) : IsClosed (feasibleSet ε) := by
have hcoord (s : Fin 14) : Continuous (fun α : StaircaseWeight => α s) :=
continuous_apply s
have hrow (j : Fin 4) :
Continuous (fun α : StaircaseWeight => staircaseMatrix ε α j) := by
unfold staircaseMatrix
fun_prop
have hnonneg : IsClosed {α : StaircaseWeight | ∀ s, 0 ≤ α s} := by
simpa only [Set.ofPred_forall] using
(isClosed_iInter (fun s : Fin 14 =>
isClosed_le continuous_const (hcoord s)))
have hnormal : IsClosed {α : StaircaseWeight | ∀ j, staircaseMatrix ε α j = 1} := by
simpa only [Set.ofPred_forall] using
(isClosed_iInter (fun j : Fin 4 =>
isClosed_eq (hrow j) continuous_const))
convert hnonneg.inter hnormal using 1
ext α
simp only [feasibleSet, Set.mem_ofPred_eq, Set.mem_inter_iff,
staircaseFeasible]
Under the supplied quantities and conditions, the staircase feasible compact assertion holds. Under the stated assumptions, the staircase Feasible compact.
Formal statement
Proof (Lean source)
lemma staircaseFeasible_compact (ε : ℝ) (hε : 0 ≤ ε) :
IsCompact (feasibleSet ε) := by
have hbox : IsCompact {α : StaircaseWeight | ∀ s, α s ∈ Icc (0 : ℝ) 1} :=
isCompact_pi_infinite (fun _ => isCompact_Icc)
apply hbox.of_isClosed_subset (staircaseFeasible_closed ε)
intro α hα s
have hsingle : α s ≤ ∑ u : Fin 14, α u := by
exact Finset.single_le_sum (fun u _ => hα.1 u) (Finset.mem_univ s)
exact ⟨hα.1 s, hsingle.trans (staircaseFeasible_sum_le_one ε hε α hα)⟩
the staircase feasible convex assertion holds.
Formal statement
Proof (Lean source)
lemma staircaseFeasible_convex (ε : ℝ) : Convex ℝ (feasibleSet ε) := by
intro α hα β hβ a b ha hb hab
constructor
· intro s
change 0 ≤ a * α s + b * β s
exact add_nonneg (mul_nonneg ha (hα.1 s))
(mul_nonneg hb (hβ.1 s))
· intro j
change (∑ s : Fin 14, (a * α s + b * β s) * patternRay ε s j) = 1
calc
_ = a * staircaseMatrix ε α j + b * staircaseMatrix ε β j := by
simp only [staircaseMatrix, Finset.mul_sum]
rw [← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro s hs
ring
_ = 1 := by rw [hα.2 j, hβ.2 j]; linarith
For the supplied quantities and conditions, the upper envelope is the mathematical object specified below. The upper Envelope is determined by the displayed parameters.
Definition (Lean source)
parameters
θ :
p ε t :
ℝ
def
upperEnvelope θ p ε t :
ℝ
given by
theorem informationObjective_continuous_weight
the information objective continuous weight assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
θ :
p ε t :
ℝ
conclusion
Proof (Lean source)
lemma informationObjective_continuous_weight (θ : TrialParameter)
(p ε t : ℝ) :
Continuous (fun α : StaircaseWeight => informationObjective θ p ε α t) := by
unfold informationObjective
fun_prop
theorem informationObjective_quadratic_difference
Under the supplied quantities and conditions, the information objective quadratic difference assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
informationObjective θ p ε α u
= informationObjective θ p ε α t
+ (u - t) * (∑ s : Fin 14, 2 * α s * projectedGradient p ε s t * (patternGradient p ε s 0 + patternGradient p ε s 1) / patternMass θ p ε s)
+ (u - t) ^ 2 * (∑ s : Fin 14, α s * (patternGradient p ε s 0 + patternGradient p ε s 1) ^ 2 / patternMass θ p ε s)
Proof (Lean source)
lemma informationObjective_quadratic_difference (θ : TrialParameter) (p ε : ℝ)
(α : StaircaseWeight) (t u : ℝ) :
informationObjective θ p ε α u =
informationObjective θ p ε α t +
(u - t) * (∑ s : Fin 14,
2 * α s * projectedGradient p ε s t *
(patternGradient p ε s 0 + patternGradient p ε s 1) /
patternMass θ p ε s) +
(u - t) ^ 2 * (∑ s : Fin 14,
α s * (patternGradient p ε s 0 + patternGradient p ε s 1) ^ 2 /
patternMass θ p ε s) := by
simp only [informationObjective, Finset.mul_sum, ← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro s _
simp [patternInformation, projectedGradient, direction, Fin.sum_univ_succ]
ring
theorem informationObjective_min_of_stationary
Under the supplied quantities and conditions, the information objective min of stationary assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the information Objective min of stationary.
Formal statement
hypotheses
θ :
p ε :
ℝ
hp :
hθ :
hε :
0 ≤ ε
α :
hα :
t :
ℝ
hstationary :
(∑ s : Fin 14, 2 * α s * projectedGradient p ε s t * (patternGradient p ε s 0 + patternGradient p ε s 1) / patternMass θ p ε s)
= 0
u :
ℝ
conclusion
informationObjective θ p ε α t ≤ informationObjective θ p ε α u
Proof (Lean source)
lemma informationObjective_min_of_stationary (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 ≤ ε)
(α : StaircaseWeight) (hα : staircaseFeasible ε α) (t : ℝ)
(hstationary : (∑ s : Fin 14,
2 * α s * projectedGradient p ε s t *
(patternGradient p ε s 0 + patternGradient p ε s 1) /
patternMass θ p ε s) = 0) :
∀ u : ℝ, informationObjective θ p ε α t ≤
informationObjective θ p ε α u := by
intro u
conv_rhs => rw [informationObjective_quadratic_difference θ p ε α t u]
rw [hstationary, mul_zero, add_zero]
have hnonneg : 0 ≤ ∑ s : Fin 14,
α s * (patternGradient p ε s 0 + patternGradient p ε s 1) ^ 2 /
patternMass θ p ε s := by
apply Finset.sum_nonneg
intro s _
exact div_nonneg (mul_nonneg (hα.1 s) (sq_nonneg _))
(le_of_lt (patternMass_pos_interior θ p ε hp hθ hε s))
exact le_add_of_nonneg_right (mul_nonneg (sq_nonneg _) hnonneg)
theorem informationObjective_stationary_of_min
Under the supplied quantities and conditions, the information objective stationary of min assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
θ :
p ε :
ℝ
α :
t :
ℝ
hmin :
∀ u : ℝ, informationObjective θ p ε α t ≤ informationObjective θ p ε α u
conclusion
(∑ s : Fin 14, 2 * α s * projectedGradient p ε s t * (patternGradient p ε s 0 + patternGradient p ε s 1) / patternMass θ p ε s)
= 0
Proof (Lean source)
lemma informationObjective_stationary_of_min (θ : TrialParameter) (p ε : ℝ)
(α : StaircaseWeight) (t : ℝ)
(hmin : ∀ u : ℝ, informationObjective θ p ε α t ≤
informationObjective θ p ε α u) :
(∑ s : Fin 14,
2 * α s * projectedGradient p ε s t *
(patternGradient p ε s 0 + patternGradient p ε s 1) /
patternMass θ p ε s) = 0 := by
let A := ∑ s : Fin 14,
2 * α s * projectedGradient p ε s t *
(patternGradient p ε s 0 + patternGradient p ε s 1) /
patternMass θ p ε s
let B := ∑ s : Fin 14,
α s * (patternGradient p ε s 0 + patternGradient p ε s 1) ^ 2 /
patternMass θ p ε s
have hpoly (x : ℝ) : 0 ≤ x * A + x ^ 2 * B := by
have h := hmin (t + x)
have hdiff := informationObjective_quadratic_difference θ p ε α t (t + x)
rw [hdiff] at h
simp only [add_sub_cancel_left] at h
dsimp [A, B]
linarith
have hp (x : ℝ) : 0 ≤ x * A + x ^ 2 * B := hpoly x
by_contra hn
have hA : A ≠ 0 := hn
have hden : 0 < |B| + 1 := by positivity
have hB : B ≤ |B| := le_abs_self B
have h := hp (-A / (2 * (|B| + 1)))
have hsq : 0 < A ^ 2 := sq_pos_of_ne_zero hA
have hden2 : 0 < (|B| + 1) ^ 2 := sq_pos_of_pos hden
field_simp at h
nlinarith [mul_nonneg hsq.le (sub_nonneg.mpr hB)]
theorem informationObjective_exists_minimizer
Under the supplied quantities and conditions, the information objective exists minimizer assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the information Objective exists minimizer.
Formal statement
hypotheses
θ :
p ε :
ℝ
hp :
hθ :
hε :
0 ≤ ε
α :
hα :
conclusion
∃ t : ℝ, ∀ u : ℝ, informationObjective θ p ε α t ≤ informationObjective θ p ε α u
Proof (Lean source)
lemma informationObjective_exists_minimizer (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 ≤ ε)
(α : StaircaseWeight) (hα : staircaseFeasible ε α) :
∃ t : ℝ, ∀ u : ℝ, informationObjective θ p ε α t ≤
informationObjective θ p ε α u := by
let A := ∑ s : Fin 14,
2 * α s * projectedGradient p ε s 0 *
(patternGradient p ε s 0 + patternGradient p ε s 1) /
patternMass θ p ε s
let B := ∑ s : Fin 14,
α s * (patternGradient p ε s 0 + patternGradient p ε s 1) ^ 2 /
patternMass θ p ε s
have hB : 0 ≤ B := by
dsimp [B]
apply Finset.sum_nonneg
intro s _
exact div_nonneg (mul_nonneg (hα.1 s) (sq_nonneg _))
(le_of_lt (patternMass_pos_interior θ p ε hp hθ hε s))
have hquad (u : ℝ) :
informationObjective θ p ε α u =
informationObjective θ p ε α 0 + u * A + u ^ 2 * B := by
convert informationObjective_quadratic_difference θ p ε α 0 u using 1
simp [A, B]
by_cases hB0 : B = 0
· have hA0 : A = 0 := by
by_contra hA
have hval := informationObjective_nonneg_interior θ p ε hp hθ hε α hα
(-(informationObjective θ p ε α 0 + 1) / A)
rw [hquad, hB0] at hval
have hident : informationObjective θ p ε α 0 +
(-(informationObjective θ p ε α 0 + 1) / A) * A +
(-(informationObjective θ p ε α 0 + 1) / A) ^ 2 * 0 = -1 := by
field_simp [hA]
ring
rw [hident] at hval
linarith
refine ⟨0, fun u => ?_⟩
rw [hquad u, hA0, hB0]
simp
· have hBpos : 0 < B := lt_of_le_of_ne hB (Ne.symm hB0)
let t := -A / (2 * B)
refine ⟨t, fun u => ?_⟩
rw [hquad u, hquad t]
have hsq : 0 ≤ B * (u - t) ^ 2 := mul_nonneg hB (sq_nonneg _)
have hid :
u * A + u ^ 2 * B - (t * A + t ^ 2 * B) =
B * (u - t) ^ 2 := by
dsimp [t]
field_simp
ring
linarith
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.OracleValue 5 declarations Staircase profile values
Staircase profile values
theorem informationObjective_profile_upperSemicontinuous
Under the supplied quantities and conditions, the information objective profile upper semicontinuous assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the information Objective profile upper Semicontinuous.
Formal statement
hypotheses
conclusion
UpperSemicontinuousOn (fun α : StaircaseWeight => ⨅ t : ℝ, informationObjective θ p ε α t) (feasibleSet ε)
Proof (Lean source)
lemma informationObjective_profile_upperSemicontinuous
(θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 ≤ ε) :
UpperSemicontinuousOn
(fun α : StaircaseWeight =>
⨅ t : ℝ, informationObjective θ p ε α t) (feasibleSet ε) := by
apply upperSemicontinuousOn_ciInf
· intro α hα
refine ⟨0, ?_⟩
rintro u ⟨t, rfl⟩
exact informationObjective_nonneg_interior θ p ε hp hθ hε α hα t
· intro t
exact (informationObjective_continuous_weight θ p ε t).continuousOn.upperSemicontinuousOn
theorem informationObjective_profile_exists_maximizer
Under the supplied quantities and conditions, the information objective profile exists maximizer assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the information Objective profile exists maximizer.
Formal statement
hypotheses
conclusion
∃ α : StaircaseWeight,
conclusion 1
β :
(⨅ t : ℝ, informationObjective θ p ε β t) ≤ ⨅ t : ℝ, informationObjective θ p ε α t
Proof (Lean source)
lemma informationObjective_profile_exists_maximizer
(θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 ≤ ε) :
∃ α : StaircaseWeight, staircaseFeasible ε α ∧
∀ β : StaircaseWeight, staircaseFeasible ε β →
(⨅ t : ℝ, informationObjective θ p ε β t) ≤
⨅ t : ℝ, informationObjective θ p ε α t := by
obtain ⟨α, hα, hmax⟩ :=
(informationObjective_profile_upperSemicontinuous θ p ε hp hθ hε).exists_isMaxOn
(staircaseFeasible_nonempty ε) (staircaseFeasible_compact ε hε)
exact ⟨α, hα, fun β hβ => hmax hβ⟩
theorem Jstar_exists_optimal_weight
Under the supplied quantities and conditions, the jstar exists optimal weight assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the Jstar exists optimal weight.
Formal statement
hypotheses
Proof (Lean source)
lemma Jstar_exists_optimal_weight (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 ≤ ε) :
∃ α : StaircaseWeight, staircaseFeasible ε α ∧
Jstar θ p ε =
sInf {u : ℝ | ∃ t : ℝ,
u = informationObjective θ p ε α t} := by
obtain ⟨α, hα, hmax⟩ :=
informationObjective_profile_exists_maximizer θ p ε hp hθ hε
refine ⟨α, hα, ?_⟩
have hprofile (β : StaircaseWeight) :
(⨅ t : ℝ, informationObjective θ p ε β t) =
sInf {u : ℝ | ∃ t : ℝ,
u = informationObjective θ p ε β t} := by
rw [show {u : ℝ | ∃ t : ℝ, u = informationObjective θ p ε β t} =
range (informationObjective θ p ε β) from by ext u; simp [eq_comm]]
rfl
apply IsGreatest.csSup_eq
refine ⟨⟨α, hα, rfl⟩, ?_⟩
rintro u ⟨β, hβ, rfl⟩
rw [← hprofile β, ← hprofile α]
exact hmax β hβ
theorem Jstar_pos_interior
Under the supplied quantities and conditions, the jstar pos interior assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the Jstar pos interior.
Formal statement
hypotheses
conclusion
0 < Jstar θ p ε
Proof (Lean source)
lemma Jstar_pos_interior (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε) :
0 < Jstar θ p ε := by
let α : StaircaseWeight := fun s =>
if s ∈ ({0, 1, 3, 7} : Finset (Fin 14)) then
1 / (exp ε + 3) else 0
have hα : staircaseFeasible ε α := staircaseFeasible_singleton_design ε
obtain ⟨t, ht⟩ := informationObjective_exists_minimizer θ p ε
hp hθ hε.le α hα
have hmass (s : Fin 14) : 0 < patternMass θ p ε s :=
patternMass_pos_interior θ p ε hp hθ hε.le s
have hterm (s : Fin 14) :
α s * patternInformation θ p ε s t ≤
informationObjective θ p ε α t := by
unfold informationObjective
apply Finset.single_le_sum (s := univ) (a := s)
(fun u hu => ?_) (Finset.mem_univ _)
exact mul_nonneg (hα.1 u)
(div_nonneg (sq_nonneg _ ) (hmass u).le)
have h0 : 0 < α 0 * ((privacyIncrement ε * controlProb p) ^ 2 /
patternMass θ p ε 0) := by
have hd := privacyIncrement_pos hε
have hq : 0 < controlProb p := by dsimp [controlProb]; linarith [hp.2]
have hm := hmass 0
dsimp [α]
positivity
have h7form : patternInformation θ p ε 7 t =
((privacyIncrement ε * p) ^ 2 / patternMass θ p ε 7) * (t + 1) ^ 2 := by
simp +decide [patternInformation, projectedGradient, patternGradient,
direction, Fin.sum_univ_succ]
ring
have h7 : 0 < α 7 * ((privacyIncrement ε * p) ^ 2 /
patternMass θ p ε 7) := by
have hd := privacyIncrement_pos hε
have hm := hmass 7
have hp0 := hp.1
dsimp [α]
positivity
have hvalue : 0 < informationObjective θ p ε α t := by
have hnonneg := informationObjective_nonneg_interior θ p ε hp hθ hε.le α hα t
by_contra hn
have hz : informationObjective θ p ε α t = 0 := le_antisymm (le_of_not_gt hn) hnonneg
have hterm0 := hterm 0
have hterm7 := hterm 7
rw [first_pattern_information_quadratic, hz] at hterm0
rw [h7form, hz] at hterm7
have ht0 : t = 0 := by
by_contra htne
nlinarith [mul_pos h0 (sq_pos_of_ne_zero htne)]
have ht7 : t + 1 = 0 := by
by_contra htne
nlinarith [mul_pos h7 (sq_pos_of_ne_zero htne)]
linarith
have hleast :
sInf {u : ℝ | ∃ v : ℝ, u = informationObjective θ p ε α v} =
informationObjective θ p ε α t := by
apply IsLeast.csInf_eq
exact ⟨⟨t, rfl⟩, by rintro u ⟨v, rfl⟩; exact ht v⟩
have hbound : informationObjective θ p ε α t ≤ Jstar θ p ε := by
obtain ⟨γ, hγ, hmax⟩ :=
informationObjective_profile_exists_maximizer θ p ε hp hθ hε.le
have hprofile (β : StaircaseWeight) :
(⨅ v : ℝ, informationObjective θ p ε β v) =
sInf {u : ℝ | ∃ v : ℝ, u = informationObjective θ p ε β v} := by
rw [show {u : ℝ | ∃ v : ℝ, u = informationObjective θ p ε β v} =
range (informationObjective θ p ε β) from by ext u; simp [eq_comm]]
rfl
rw [Jstar, ← hleast]
apply le_csSup
· refine ⟨sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε γ v}, ?_⟩
rintro u ⟨β, hβ, rfl⟩
rw [← hprofile β, ← hprofile γ]
exact hmax β hβ
· exact ⟨α, hα, rfl⟩
exact lt_of_lt_of_le hvalue hbound
theorem Jstar_eq_envelope_of_stationary_maximizer
Under the supplied quantities and conditions, the jstar eq envelope of stationary maximizer assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the Jstar eq envelope of stationary maximizer.
Formal statement
hypotheses
θ :
p ε :
ℝ
hp :
hθ :
hε :
0 ≤ ε
α :
hα :
t :
ℝ
hmax :
∀ β : StaircaseWeight
if
then
informationObjective θ p ε β t ≤ informationObjective θ p ε α t
hstationary :
(∑ s : Fin 14, 2 * α s * projectedGradient p ε s t * (patternGradient p ε s 0 + patternGradient p ε s 1) / patternMass θ p ε s)
= 0
conclusion
Proof (Lean source)
lemma Jstar_eq_envelope_of_stationary_maximizer
(θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 ≤ ε)
(α : StaircaseWeight) (hα : staircaseFeasible ε α) (t : ℝ)
(hmax : ∀ β : StaircaseWeight, staircaseFeasible ε β →
informationObjective θ p ε β t ≤ informationObjective θ p ε α t)
(hstationary : (∑ s : Fin 14,
2 * α s * projectedGradient p ε s t *
(patternGradient p ε s 0 + patternGradient p ε s 1) /
patternMass θ p ε s) = 0) :
Jstar θ p ε =
sInf {u : ℝ | ∃ v : ℝ, u = upperEnvelope θ p ε v} := by
have hleast : sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε α v} =
informationObjective θ p ε α t := by
apply IsLeast.csInf_eq
exact ⟨⟨t, rfl⟩, by
rintro u ⟨v, rfl⟩
exact informationObjective_min_of_stationary θ p ε hp hθ hε α hα t
hstationary v⟩
have hbound : informationObjective θ p ε α t ≤ Jstar θ p ε := by
obtain ⟨γ, hγ, hγmax⟩ :=
informationObjective_profile_exists_maximizer θ p ε hp hθ hε
have hprofile (β : StaircaseWeight) :
(⨅ v : ℝ, informationObjective θ p ε β v) =
sInf {u : ℝ | ∃ v : ℝ, u = informationObjective θ p ε β v} := by
rw [show {u : ℝ | ∃ v : ℝ, u = informationObjective θ p ε β v} =
range (informationObjective θ p ε β) from by ext u; simp [eq_comm]]
rfl
rw [Jstar, ← hleast]
apply le_csSup
· refine ⟨sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε γ v}, ?_⟩
rintro u ⟨β, hβ, rfl⟩
rw [← hprofile β, ← hprofile γ]
exact hγmax β hβ
· exact ⟨α, hα, rfl⟩
have henvelope : upperEnvelope θ p ε t =
informationObjective θ p ε α t := by
unfold upperEnvelope
apply IsGreatest.csSup_eq
exact ⟨⟨α, hα, rfl⟩, by
rintro u ⟨β, hβ, rfl⟩
exact hmax β hβ⟩
apply le_antisymm
· apply le_csInf
· exact ⟨upperEnvelope θ p ε 0, 0, rfl⟩
· rintro u ⟨v, rfl⟩
exact Jstar_le_upperEnvelope θ p ε hp hθ hε v
· have hlow : BddBelow {u : ℝ | ∃ v : ℝ,
u = upperEnvelope θ p ε v} := by
refine ⟨0, ?_⟩
rintro u ⟨v, rfl⟩
obtain ⟨β, hβ, hβmax⟩ :=
informationObjective_exists_maximizer θ p ε v hε
have heq : upperEnvelope θ p ε v =
informationObjective θ p ε β v := by
unfold upperEnvelope
apply IsGreatest.csSup_eq
exact ⟨⟨β, hβ, rfl⟩, by
rintro w ⟨δ, hδ, rfl⟩
exact hβmax δ hδ⟩
rw [heq]
exact informationObjective_nonneg_interior θ p ε hp hθ hε β hβ v
exact (csInf_le hlow ⟨t, rfl⟩).trans (henvelope.trans_le hbound)
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.Pilot 32 declarations Four-category randomized response estimates the input probabilities.
Private pilot and affine main-sample estimator
Four-category randomized response estimates the input probabilities. A measurable saddle selector chooses the main staircase channel and influence values from that pilot, with the pilot discarded from the final average.
def PilotOutput
the pilot output is the mathematical object specified below.
the pilot output family is the mathematical object specified below.
Definition (Lean source)
def
given by
fun _ _ => PilotOutput
instance instMeasurableSpacePilotOutputFamily
The measurable-space instance for pilot outputs is determined by the displayed parameters.
Definition (Lean source)
instance : ∀ n i, MeasurableSpace (pilotOutputFamily n i) :=
fun _ _ => inferInstanceAs (MeasurableSpace PilotOutput)
instance instMeasurableSingletonClassPilotOutput Definition instMeasurableSingletonClassPilotOutput in the paper ↗
the inst measurable singleton class pilot output is the mathematical object specified below.
Definition (Lean source)
instance instMeasurableSingletonClassPilotOutput : MeasurableSingletonClass PilotOutput := by
refine ⟨fun x => ?_⟩
cases x with
| inl a => simpa only [Set.image_singleton] using (measurableSet_singleton a).inl_image
| inr b => simpa only [Set.image_singleton] using (measurableSet_singleton b).inr_image
instance instMeasurableSingletonClassPilotOutputFamily
The measurable-singleton instance for pilot outputs is determined by the displayed parameters.
Definition (Lean source)
instance : ∀ n i, MeasurableSingletonClass (pilotOutputFamily n i) :=
fun _ _ => inferInstanceAs (MeasurableSingletonClass PilotOutput)
instance instCountablePrivateHistoryPilotOutputFamily
The countability instance for private pilot histories is determined by the displayed parameters.
Definition (Lean source)
instance (n : ℕ) (i : Fin n) :
Countable (PrivateHistory (Z := pilotOutputFamily n) i) := inferInstance
instance instMeasurableSingletonClassPilotHistory
For the supplied quantities and conditions, the inst measurable singleton class pilot history is the mathematical object specified below.
Definition (Lean source)
instance instMeasurableSingletonClassPilotHistory (n : ℕ) (i : Fin n) :
MeasurableSingletonClass (PrivateHistory (Z := pilotOutputFamily n) i) := by
infer_instance
def rrPilotProbability
For the supplied quantities and conditions, the rr pilot probability is the mathematical object specified below. The rr Pilot Probability is determined by the displayed parameters.
Under the supplied quantities and conditions, the rr pilot probability ldp assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the rr Pilot Probability ldp.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma rrPilotProbability_ldp (ε : ℝ) (hε : 0 < ε)
(j j' k : Fin 4) :
rrPilotProbability ε j k ≤ exp ε * rrPilotProbability ε j' k := by
have he : 1 ≤ exp ε := (Real.one_le_exp_iff).2 (le_of_lt hε)
have hd : 0 < exp ε + 3 := by positivity
unfold rrPilotProbability
rw [← mul_div_assoc]
apply (div_le_div_iff_of_pos_right hd).2
split_ifs <;> nlinarith [sq_nonneg (exp ε - 1)]
def rrPilotKernel
the rr pilot kernel is the mathematical object specified below. The rr Pilot Kernel is determined by the displayed parameters.
the rr pilot kernel markov assertion holds.
Formal statement
Proof (Lean source)
lemma rrPilotKernel_markov (ε : ℝ) : IsMarkovKernel (rrPilotKernel ε) := by
constructor
intro j
constructor
have he : 0 < exp ε := Real.exp_pos ε
have hd : exp ε + 3 ≠ 0 := by positivity
have hsum : (∑ k : Fin 4, rrPilotProbability ε j k) = 1 := by
fin_cases j <;> simp +decide [rrPilotProbability, Fin.sum_univ_succ] <;>
field_simp <;> ring
rw [← ENNReal.ofReal_one, ← hsum]
change (Measure.count.withDensity
(fun k : Fin 4 => ofReal (rrPilotProbability ε j k))) univ = _
simp only [withDensity_apply _ MeasurableSet.univ, setLIntegral_univ,
lintegral_count]
rw [tsum_fintype, ENNReal.ofReal_sum_of_nonneg]
intro k hk
unfold rrPilotProbability
split_ifs <;> positivity
Under the supplied quantities and conditions, the rr pilot kernel private assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the rr Pilot Kernel private.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma rrPilotKernel_private (ε : ℝ) (hε : 0 < ε)
(A : Set (Fin 4)) (j j' : Fin 4) :
rrPilotKernel ε j A ≤ ofReal (exp ε) * rrPilotKernel ε j' A := by
let r := ofReal (exp ε)
have hr : r ≠ ∞ := ENNReal.ofReal_ne_top
have hm : Measure.count.withDensity
(fun k : Fin 4 => ofReal (rrPilotProbability ε j k)) ≤
r • Measure.count.withDensity
(fun k : Fin 4 => ofReal (rrPilotProbability ε j' k)) := by
rw [← withDensity_smul' r _ hr]
apply withDensity_mono
filter_upwards [] with k
dsimp [r]
rw [← ENNReal.ofReal_mul (le_of_lt (Real.exp_pos ε))]
exact ENNReal.ofReal_le_ofReal (rrPilotProbability_ldp ε hε j j' k)
exact hm A
def clipInterior
For the supplied quantities and conditions, the clip interior is the mathematical object specified below. The clip Interior is determined by the displayed parameters.
def pilotFrequency
For the supplied quantities and conditions, the pilot frequency is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
pilotFrequency m z k :
ℝ
def pilotTheta
For the supplied quantities and conditions, the pilot theta is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The pilot Theta is determined by the displayed parameters.
Definition (Lean source)
parameters
def
given by
δ :
(m n + 2 : ℝ)⁻¹
invFreq :
fun a
def pilotHistoryFrequency
For the supplied quantities and conditions, the pilot history frequency is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
pilotHistoryFrequency m i h k :
ℝ
def pilotHistoryTheta
For the supplied quantities and conditions, the pilot history theta is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The pilot History Theta is determined by the displayed parameters.
Definition (Lean source)
parameters
def
given by
δ :
(m n + 2 : ℝ)⁻¹
invFreq :
fun a
Under the supplied quantities and conditions, the clip interior interior assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the clip Interior interior.
Formal statement
hypotheses
δ x :
ℝ
hδ :
0 < δ
hhalf :
δ ≤ 1 / 2
Proof (Lean source)
lemma clipInterior_interior (δ x : ℝ) (hδ : 0 < δ)
(hhalf : δ ≤ 1 / 2) : 0 < clipInterior δ x ∧ clipInterior δ x < 1 := by
unfold clipInterior
constructor
· exact lt_of_lt_of_le hδ (le_max_left _ _)
· have hbound : max δ (min (1 - δ) x) ≤ 1 - δ :=
max_le (by linarith) (min_le_left _ _)
linarith
theorem pilotHistoryTheta_interior
Under the supplied quantities and conditions, the pilot history theta interior assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
InteriorMeans (pilotHistoryTheta p ε m i h)
Proof (Lean source)
lemma pilotHistoryTheta_interior (p ε : ℝ) (m : ℕ → ℕ) {n : ℕ}
(i : Fin n) (h : PrivateHistory (Z := pilotOutputFamily n) i) :
InteriorMeans (pilotHistoryTheta p ε m i h) := by
have hδ : 0 < ((m n + 2 : ℝ)⁻¹) := by positivity
have hhalf : ((m n + 2 : ℝ)⁻¹) ≤ 1 / 2 := by
rw [one_div]
have hden : (2 : ℝ) ≤ m n + 2 := by
exact_mod_cast Nat.le_add_left 2 (m n)
simpa [one_div] using one_div_le_one_div_of_le
(by norm_num : (0 : ℝ) < 2) hden
have h0 := clipInterior_interior ((m n + 2 : ℝ)⁻¹)
(((exp ε + 3) * pilotHistoryFrequency m i h 1 - 1) /
(exp ε - 1) / controlProb p) hδ hhalf
have h1 := clipInterior_interior ((m n + 2 : ℝ)⁻¹)
(((exp ε + 3) * pilotHistoryFrequency m i h 3 - 1) /
(exp ε - 1) / p) hδ hhalf
simpa [InteriorMeans, pilotHistoryTheta] using
intro h0.1 (intro h0.2 (intro h1.1 h1.2))
For the supplied quantities and conditions, the saddle selection is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The Saddle Selection is determined by the displayed parameters.
Definition (Lean source)
parameters
p ε :
ℝ
select :
TrialParameter → StaircaseWeight × ℝ × ℝ
def
SaddleSelection p ε select :
Prop
given by
clause 1
Measurable select
clause 2
θ :
staircaseFeasible ε (select θ).1
u :
informationObjective θ p ε (select θ).1 (select θ).2.1
≤ informationObjective θ p ε (select θ).1 u
Jstar θ p ε = (select θ).2.2
(select θ).2.2 = informationObjective θ p ε (select θ).1 (select θ).2.1
For the supplied quantities and conditions, the strong saddle selection is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The Strong Saddle Selection is determined by the displayed parameters.
Definition (Lean source)
parameters
p ε :
ℝ
select :
TrialParameter → StaircaseWeight × ℝ × ℝ
def
StrongSaddleSelection p ε select :
Prop
given by
clause 1
SaddleSelection p ε select
clause 2
informationObjective θ p ε β (select θ).2.1
≤ informationObjective θ p ε (select θ).1 (select θ).2.1
theorem exists_measurable_strong_saddle_selector
Under the supplied quantities and conditions, the exists measurable strong saddle selector assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the exists measurable strong saddle selector.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma exists_measurable_strong_saddle_selector (p ε : ℝ)
(hp : InteriorAssignment p) (hε : 0 < ε) :
∃ select : TrialParameter → StaircaseWeight × ℝ × ℝ,
StrongSaddleSelection p ε select := by
obtain ⟨select, hm, hs⟩ :=
exists_measurable_saddle_selector_bridge_strong p ε hp hε
refine ⟨select, ⟨⟨hm, ?_⟩, ?_⟩⟩
· intro θ hθ
have h := hs θ hθ
exact ⟨h.1, h.2.1, h.2.2.1, h.2.2.2.1⟩
· intro θ hθ
exact (hs θ hθ).2.2.2.2
theorem exists_measurable_saddle_selector
Under the supplied quantities and conditions, the exists measurable saddle selector assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the exists measurable saddle selector.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma exists_measurable_saddle_selector (p ε : ℝ)
(hp : InteriorAssignment p) (hε : 0 < ε) :
∃ select : TrialParameter → StaircaseWeight × ℝ × ℝ,
SaddleSelection p ε select := by
exact exists_measurable_saddle_selector_bridge p ε hp hε
def phiTilde
For the supplied quantities and conditions, the phi tilde is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The phi Tilde is determined by the displayed parameters.
Definition (Lean source)
parameters
def
phiTilde θ p ε select s :
ℝ
given by
projectedGradient p ε s (select θ).2.1 / ((select θ).2.2 * patternMass θ p ε s)
def phiOutput
For the supplied quantities and conditions, the phi output is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The phi Output is determined by the displayed parameters.
Definition (Lean source)
parameters
def
phiOutput θ p ε select z :
ℝ
def tauStar
For the supplied quantities and conditions, the tau star is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The tau Star is determined by the displayed parameters.
Definition (Lean source)
parameters
p ε :
ℝ
m :
ℕ → ℕ
select :
TrialParameter → StaircaseWeight × ℝ × ℝ
n :
ℕ
z :
Transcript (pilotOutputFamily n)
def
tauStar p ε m select z :
ℝ
given by
θtilde :
pilotTheta p ε m z
N :
n - m n
For the supplied quantities and conditions, the vhat star is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The Vhat Star is determined by the displayed parameters.
Definition (Lean source)
parameters
p ε :
ℝ
m :
ℕ → ℕ
select :
TrialParameter → StaircaseWeight × ℝ × ℝ
n :
ℕ
z :
Transcript (pilotOutputFamily n)
def
VhatStar p ε m select z :
ℝ
given by
θtilde :
pilotTheta p ε m z
N :
n - m n
def pilotProtocol
For the supplied quantities and conditions, the pilot protocol is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The pilot Protocol is determined by the displayed parameters.
Definition (Lean source)
parameters
def
given by
fun i
=> Kernel.ofFunOfCountable fun xh => if i.val < m n then (rrPilotKernel ε xh.1).map inl else ((staircaseChannel ε (select (pilotHistoryTheta p ε m i xh.2)).1) xh.1).map inr
theorem pilotProtocol_private
Under the supplied quantities and conditions, the pilot protocol private assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Protocol private.
Formal statement
hypotheses
p ε :
ℝ
m :
ℕ → ℕ
select :
TrialParameter → StaircaseWeight × ℝ × ℝ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
conclusion
Proof (Lean source)
lemma pilotProtocol_private (p ε : ℝ) (m : ℕ → ℕ)
(select : TrialParameter → StaircaseWeight × ℝ × ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε) (n : ℕ) :
SequentialLDP (Z := pilotOutputFamily n) ε
(pilotProtocol p ε m select n) := by
constructor
· intro i
constructor
intro xh
change IsProbabilityMeasure
(if i.val < m n then (rrPilotKernel ε xh.1).map inl
else (staircaseChannel ε
(select (pilotHistoryTheta p ε m i xh.2)).1 xh.1).map inr)
split_ifs with hi
· haveI : IsMarkovKernel (rrPilotKernel ε) := rrPilotKernel_markov ε
exact Measure.isProbabilityMeasure_map (by fun_prop)
· have hsel := hselect.1.2
(pilotHistoryTheta p ε m i xh.2)
(pilotHistoryTheta_interior p ε m i xh.2)
haveI : IsMarkovKernel (staircaseChannel ε
(select (pilotHistoryTheta p ε m i xh.2)).1) :=
staircaseChannel_markov ε _ hsel.1
exact Measure.isProbabilityMeasure_map (by fun_prop)
· intro i A hA x x' h
change (if i.val < m n then (rrPilotKernel ε x).map inl
else (staircaseChannel ε
(select (pilotHistoryTheta p ε m i h)).1 x).map inr) A ≤
ofReal (exp ε) *
(if i.val < m n then (rrPilotKernel ε x').map inl
else (staircaseChannel ε
(select (pilotHistoryTheta p ε m i h)).1 x').map inr) A
split_ifs with hi
· rw [Measure.map_apply (by fun_prop) hA,
Measure.map_apply (by fun_prop) hA]
exact rrPilotKernel_private ε hε _ x x'
· rw [Measure.map_apply (by fun_prop) hA,
Measure.map_apply (by fun_prop) hA]
have hsel := hselect.1.2
(pilotHistoryTheta p ε m i h)
(pilotHistoryTheta_interior p ε m i h)
exact staircaseChannel_private ε hε _ hsel.1 _ x x'
theorem tauStar_measurable
Under the supplied quantities and conditions, the tau star measurable assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (tauStar p ε m select (n := n))
Proof (Lean source)
lemma tauStar_measurable (p ε : ℝ) (m : ℕ → ℕ)
(select : TrialParameter → StaircaseWeight × ℝ × ℝ) (n : ℕ) :
Measurable (tauStar p ε m select (n := n)) := by
exact measurable_of_finite _
theorem exists_pilot_transcript
Under the supplied quantities and conditions, the exists pilot transcript assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the exists pilot transcript.
Formal statement
hypotheses
p ε :
ℝ
m :
ℕ → ℕ
select :
TrialParameter → StaircaseWeight × ℝ × ℝ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
conclusion
TranscriptFactorizes (pilotProtocol p ε m select n) R
Proof (Lean source)
lemma exists_pilot_transcript (p ε : ℝ) (m : ℕ → ℕ)
(select : TrialParameter → StaircaseWeight × ℝ × ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε) (n : ℕ) :
∃ R : (Fin n → Fin 4) → Measure (Transcript (pilotOutputFamily n)),
TranscriptFactorizes
(pilotProtocol p ε m select n) R := by
exact exists_transcript_factorization
(pilotProtocol p ε m select n)
(pilotProtocol_private p ε m select hselect hp hε n).1
def pilotEstimator
For the supplied quantities and conditions, the pilot estimator is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The pilot Estimator is determined by the displayed parameters.
Definition (Lean source)
parameters
p ε :
ℝ
m :
ℕ → ℕ
select :
TrialParameter → StaircaseWeight × ℝ × ℝ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
def
given by
clause 1
channel n := pilotProtocol p ε m select n
clause 2
transcript n := finiteSequenceTranscript (pilotProtocol p ε m select n)
clause 3
factorizes n := finiteSequenceTranscript_factorizes (pilotProtocol p ε m select n) (pilotProtocol_private p ε m select hselect hp hε n).1
clause 4
privacy n := pilotProtocol_private p ε m select hselect hp hε n
clause 6
estimate_measurable n := tauStar_measurable p ε m select n
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotAdaptiveAveraging 7 declarations Finite pilot averaging and main-row scale transfer
Finite pilot averaging and main-row scale transfer
theorem finitePilot_average_error_le
the finite pilot average error le assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the finite Pilot average error le.
Formal statement
hypotheses
P :
Type*
Fintype P
w f :
P → ℝ
L δ :
ℝ
hw :
∀ a, 0 ≤ w a
hsum :
∑ a, w a = 1
hf :
∀ a, |f a| ≤ 1
hL :
|L| ≤ 1
hδ :
0 ≤ δ
conclusion
|∑ a, w a * f a - L| ≤ δ + 2 * ∑ a, w a * if δ ≤ |f a - L| then 1 else 0
Proof (Lean source)
lemma finitePilot_average_error_le {P : Type*} [Fintype P]
(w f : P → ℝ) (L δ : ℝ)
(hw : ∀ a, 0 ≤ w a) (hsum : ∑ a, w a = 1)
(hf : ∀ a, |f a| ≤ 1) (hL : |L| ≤ 1) (hδ : 0 ≤ δ) :
|∑ a, w a * f a - L| ≤
δ + 2 * ∑ a, w a * if δ ≤ |f a - L| then 1 else 0 := by
have hpoint (a : P) :
w a * |f a - L| ≤ w a *
(δ + 2 * if δ ≤ |f a - L| then 1 else 0) := by
apply mul_le_mul_of_nonneg_left _ (hw a)
split_ifs with hbad
· have hdiff : |f a - L| ≤ 2 := by
calc
|f a - L| ≤ |f a| + |L| := abs_sub _ _
_ ≤ 2 := by linarith [hf a, hL]
linarith
· simpa [hbad] using (le_of_lt (lt_of_not_ge hbad))
calc
|∑ a, w a * f a - L| = |∑ a, w a * (f a - L)| := by
congr 1
calc
∑ a, w a * f a - L = ∑ a, w a * f a - L * ∑ a, w a := by rw [hsum]; ring
_ = ∑ a, w a * (f a - L) := by
rw [Finset.mul_sum, ← Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro a _
ring
_ ≤ ∑ a, |w a * (f a - L)| := Finset.abs_sum_le_sum_abs _ _
_ = ∑ a, w a * |f a - L| := by
apply Finset.sum_congr rfl
intro a _
rw [abs_mul, abs_of_nonneg (hw a)]
_ ≤ ∑ a, w a *
(δ + 2 * if δ ≤ |f a - L| then 1 else 0) :=
Finset.sum_le_sum fun a _ => hpoint a
_ = δ + 2 * ∑ a, w a *
if δ ≤ |f a - L| then 1 else 0 := by
simp_rw [mul_add]
rw [Finset.sum_add_distrib, ← Finset.sum_mul]
simp_rw [show ∀ a : P, w a * (2 * if δ ≤ |f a - L| then 1 else 0) =
2 * (w a * if δ ≤ |f a - L| then 1 else 0) by intro a; ring]
rw [← Finset.mul_sum, hsum]
ring
theorem finitePilot_average_tendsto_of_badMass
the finite pilot average tendsto of bad mass assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
P :
ℕ → Type*
∀ n, Fintype (P n)
w :
(n : ℕ) → P n → ℝ
f :
(n : ℕ) → P n → ℝ
L :
ℝ
hw :
∀ n a, 0 ≤ w n a
hsum :
∀ n, ∑ a, w n a = 1
hf :
∀ n a, |f n a| ≤ 1
hL :
|L| ≤ 1
Proof (Lean source)
lemma finitePilot_average_tendsto_of_badMass
{P : ℕ → Type*} [∀ n, Fintype (P n)]
(w : (n : ℕ) → P n → ℝ) (f : (n : ℕ) → P n → ℝ) (L : ℝ)
(hw : ∀ n a, 0 ≤ w n a) (hsum : ∀ n, ∑ a, w n a = 1)
(hf : ∀ n a, |f n a| ≤ 1) (hL : |L| ≤ 1)
(hbad : ∀ δ : ℝ, 0 < δ → Tendsto (fun n =>
∑ a, w n a * if δ ≤ |f n a - L| then 1 else 0) atTop (𝓝 0)) :
Tendsto (fun n => ∑ a, w n a * f n a) atTop (𝓝 L) := by
rw [Metric.tendsto_atTop]
intro η hη
have hbadη := hbad (η / 2) (half_pos hη)
have hevent : ∀ᶠ n in atTop,
(∑ a, w n a * if η / 2 ≤ |f n a - L| then 1 else 0) < η / 4 :=
(tendsto_order.1 hbadη).2 _ (by linarith)
rcases (eventually_atTop.1 hevent) with ⟨N, hN⟩
exact ⟨N, fun n hn => by
rw [Real.dist_eq]
have hbound := finitePilot_average_error_le (w n) (f n) L (η / 2)
(hw n) (hsum n) (hf n) hL (by positivity)
exact lt_of_le_of_lt hbound (by linarith [hN n hn])⟩
def adaptiveMainSize
For the supplied quantities and conditions, the adaptive main size is the mathematical object specified below.
Definition (Lean source)
theorem adaptiveMainSize_add_pilot
Under the supplied quantities and conditions, the adaptive main size add pilot assertion holds.
Formal statement
Proof (Lean source)
lemma adaptiveMainSize_add_pilot (m : ℕ → ℕ) (n : ℕ) :
adaptiveMainSize m n + adaptivePilotSize m n = n := by
exact Nat.sub_add_cancel (adaptivePilotSize_le m n)
theorem adaptiveMainSize_ratio_tendsto_one
Under the supplied quantities and conditions, the adaptive main size ratio tendsto one assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
m :
ℕ → ℕ
hsub :
Proof (Lean source)
lemma adaptiveMainSize_ratio_tendsto_one (m : ℕ → ℕ)
(hsub : PilotSublinear m) :
Tendsto (fun n => (adaptiveMainSize m n : ℝ) / n) atTop (𝓝 1) := by
have hpilot := adaptivePilotSize_sublinear m hsub
have ht : Tendsto (fun n => (1 : ℝ) - (adaptivePilotSize m n : ℝ) / n)
atTop (𝓝 1) := by simpa using tendsto_const_nhds.sub hpilot
apply ht.congr'
filter_upwards [eventually_ne_atTop 0] with n hn
rw [adaptiveMainSize, Nat.cast_sub (adaptivePilotSize_le m n)]
field_simp
theorem adaptiveMainSize_tendsto_atTop
Under the supplied quantities and conditions, the adaptive main size tendsto at top assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma adaptiveMainSize_tendsto_atTop (m : ℕ → ℕ)
(hsub : PilotSublinear m) :
Tendsto (adaptiveMainSize m) atTop atTop := by
have hratio := adaptiveMainSize_ratio_tendsto_one m hsub
have hreal : Tendsto (fun n => (adaptiveMainSize m n : ℝ)) atTop atTop := by
have hprod := hratio.pos_mul_atTop zero_lt_one tendsto_natCast_atTop_atTop
apply hprod.congr'
filter_upwards [eventually_ne_atTop 0] with n hn
change (adaptiveMainSize m n : ℝ) / n * n = adaptiveMainSize m n
rw [div_mul_cancel₀]
exact_mod_cast hn
exact tendsto_natCast_atTop_iff.mp hreal
theorem adaptive_row_main_sqrt_ratio_tendsto_one
Under the supplied quantities and conditions, the adaptive row main sqrt ratio tendsto one assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
m :
ℕ → ℕ
hsub :
Proof (Lean source)
lemma adaptive_row_main_sqrt_ratio_tendsto_one (m : ℕ → ℕ)
(hsub : PilotSublinear m) :
Tendsto (fun n : ℕ => sqrt ((n : ℝ) / (adaptiveMainSize m n : ℝ)))
atTop (𝓝 1) := by
have hratio := adaptiveMainSize_ratio_tendsto_one m hsub
have hinv : Tendsto (fun n : ℕ => ((adaptiveMainSize m n : ℝ) / n)⁻¹)
atTop (𝓝 (1 : ℝ)) := by
simpa using hratio.inv₀ one_ne_zero
have hinv' : Tendsto
(fun n : ℕ => (n : ℝ) / (adaptiveMainSize m n : ℝ))
atTop (𝓝 (1 : ℝ)) := by
simpa only [inv_div] using hinv
convert (Real.continuous_sqrt.tendsto 1).comp hinv' using 1 <;>
simp [Function.comp_def]
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotAdaptiveContinuity 6 declarations Continuity of the private-pilot oracle value and adaptive variance
Continuity of the private-pilot oracle value and adaptive variance
def adaptivePilotPolytope
For the supplied quantities and conditions, the adaptive pilot polytope is the mathematical object specified below. The adaptive Pilot Polytope is determined by the displayed parameters.
Definition (Lean source)
parameters
ε :
ℝ
hε :
0 ≤ ε
given by
clause 1
A j s := patternRay ε s j
clause 2
b _ := 1
clause 3
compact := by have hc := (staircaseFeasible_compact ε hε).image (EuclideanSpace.equiv (Fin 14) ℝ).symm.continuous convert hc using 1 ext α simp only [Set.mem_setOf_eq, Set.mem_image, feasibleSet] constructor · rintro ⟨hn, heq⟩ refine ⟨adaptiveToWeight α, ⟨hn, ?_⟩, adaptiveOfWeight_adaptiveToWeight α⟩ intro j simpa [adaptiveToWeight, staircaseMatrix, mul_comm] using heq j · rintro ⟨β, ⟨hn, heq⟩, rfl⟩ refine ⟨?_, ?_⟩ · simpa [adaptiveOfWeight, adaptiveToWeight] using hn · intro j simpa [adaptiveOfWeight, adaptiveToWeight, staircaseMatrix, mul_comm] using heq j
clause 4
nonempty := by obtain ⟨α, hα⟩ := staircaseFeasible_nonempty ε refine ⟨adaptiveOfWeight α, ?_, ?_⟩ · simpa [adaptiveOfWeight, adaptiveToWeight] using hα.1 · intro j simpa [adaptiveOfWeight, adaptiveToWeight, staircaseMatrix, mul_comm] using hα.2 j
def adaptivePilotQuadratics
For the supplied quantities and conditions, the adaptive pilot quadratics is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The adaptive Pilot Quadratics is determined by the displayed parameters.
Definition (Lean source)
parameters
def
given by
clause 1
clause 2
b s θ := 2 * (patternGradient p ε s 0 + patternGradient p ε s 1) * patternGradient p ε s 1 / patternMass θ.1 p ε s
clause 3
c s θ := (patternGradient p ε s 1) ^ 2 / patternMass θ.1 p ε s
clause 4
measurable_a := by intro s exact measurable_const.div (measurable_adaptivePatternMass_subtype p ε s)
clause 5
measurable_b := by intro s exact measurable_const.div (measurable_adaptivePatternMass_subtype p ε s)
clause 6
measurable_c := by intro s exact measurable_const.div (measurable_adaptivePatternMass_subtype p ε s)
clause 7
nonneg_a := by intro θ s exact div_nonneg (sq_nonneg _) (patternMass_pos_interior θ.1 p ε hp θ.2 hε s).le
theorem Jstar_eq_adaptiveQuadraticValue
Under the supplied quantities and conditions, the jstar eq adaptive quadratic value assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the Jstar eq adaptive Quadratic Value.
Formal statement
hypotheses
p ε :
ℝ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
θ :
{θ : TrialParameter // InteriorMeans θ}
conclusion
Jstar θ.1 p ε
= value (adaptivePilotPolytope ε hε.le) (adaptivePilotQuadratics p ε hp hε.le) θ
Proof (Lean source)
lemma Jstar_eq_adaptiveQuadraticValue (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε)
(θ : {θ : TrialParameter // InteriorMeans θ}) :
Jstar θ.1 p ε = value (adaptivePilotPolytope ε hε.le)
(adaptivePilotQuadratics p ε hp hε.le) θ := by
let select := select
have hsaddle := adaptivePilot_selected_isSaddle select p ε hselect hp hε θ
rw [value_eq_of_isSaddle _ _ _ _ _ hsaddle,
adaptivePilotQuadratics_objective]
simp only [adaptiveToWeight_adaptiveOfWeight]
have hs := (hselect.1).2 θ.1 θ.2
exact hs.2.2.1.trans hs.2.2.2
theorem continuousAt_Jstar
Under the supplied quantities and conditions, the continuous at jstar assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the continuous At Jstar.
Formal statement
hypotheses
conclusion
ContinuousAt (fun η => Jstar η p ε) θ
Proof (Lean source)
lemma continuousAt_Jstar (θ : TrialParameter) (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε) :
ContinuousAt (fun η => Jstar η p ε) θ := by
let Θ := {η : TrialParameter // InteriorMeans η}
let θ0 : Θ := ⟨θ, hθ⟩
let B := pilotTUpper p ε
let K : Set ℝ := Icc (-B) B
have hB : 0 ≤ B := by
unfold B pilotTUpper
exact Real.sqrt_nonneg _
have hatt : {η : Θ | ∃ α : EuclideanSpace ℝ (Fin 14),
∃ t ∈ K, IsSaddle (adaptivePilotPolytope ε hε.le)
(adaptivePilotQuadratics p ε hp hε.le) η α t} ∈ nhds θ0 := by
filter_upwards [] with η
let select := select
refine ⟨adaptiveOfWeight (select η.1).1, (select η.1).2.1, ?_, ?_⟩
· have ht := strongSelector_t_bound select
(hselect) hp hε η.1 η.2
exact (abs_le.mp ht)
· exact adaptivePilot_selected_isSaddle select p ε hselect hp hε η
have hvalue : ContinuousAt
(value (adaptivePilotPolytope ε hε.le)
(adaptivePilotQuadratics p ε hp hε.le)) θ0 := by
apply continuousAt_quadratic_value
(adaptivePilotPolytope ε hε.le)
(adaptivePilotQuadratics p ε hp hε.le) K
(isCompact_Icc) ⟨0, by simpa [K] using hB⟩ θ0 univ
isOpen_univ (Set.mem_univ θ0)
· exact (continuous_adaptivePilotQuadratics_objective p ε hp hε).continuousOn
· exact hatt
have hsub : ContinuousAt (fun η : Θ => Jstar η.1 p ε) θ0 := by
apply hvalue.congr_of_eventuallyEq
filter_upwards [] with η
exact Jstar_eq_adaptiveQuadraticValue select p ε hselect hp hε η
have hwithin : ContinuousWithinAt (fun η => Jstar η p ε)
{η | InteriorMeans η} θ := by
apply (continuousWithinAt_iff_continuousAt_domRestrict
(f := fun η => Jstar η p ε)
(s := {η | InteriorMeans η}) hθ).2
change ContinuousAt
(fun η : {η : TrialParameter // InteriorMeans η} => Jstar η.1 p ε)
⟨θ, hθ⟩
simpa only [Θ, θ0] using hsub
exact hwithin.continuousAt (isOpen_interiorMeans.mem_nhds hθ)
theorem continuousAt_Vstar
Under the supplied quantities and conditions, the continuous at vstar assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the continuous At Vstar.
Formal statement
hypotheses
conclusion
ContinuousAt (fun η => Vstar η p ε) θ
Proof (Lean source)
lemma continuousAt_Vstar (θ : TrialParameter) (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε) :
ContinuousAt (fun η => Vstar η p ε) θ := by
change ContinuousAt ((fun η => Jstar η p ε)⁻¹) θ
exact (continuousAt_Jstar select θ p ε hselect hp hθ hε).inv₀
(ne_of_gt (Jstar_pos_interior θ p ε hp hθ hε))
theorem continuousAt_adaptiveScoreVariance_diag
Under the supplied quantities and conditions, the continuous at adaptive score variance diag assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the continuous At adaptive Score Variance diag.
Formal statement
hypotheses
conclusion
ContinuousAt (fun pair : TrialParameter × TrialParameter => adaptiveScoreVariance select pair.1 pair.2 p ε) (θ, θ)
Proof (Lean source)
lemma continuousAt_adaptiveScoreVariance_diag
(θ : TrialParameter) (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε) :
ContinuousAt (fun pair : TrialParameter × TrialParameter =>
adaptiveScoreVariance select pair.1 pair.2 p ε) (θ, θ) := by
let B := pilotPhiUpper p ε
let g : TrialParameter × TrialParameter → ℝ := fun q =>
B ^ 2 * ∑ s : Fin 14,
|patternMass q.1 p ε s - patternMass q.2 p ε s| +
(contrast q.1 - contrast q.2) ^ 2
have hgcont : Continuous g := by
unfold g
apply Continuous.add
· apply Continuous.mul continuous_const
apply continuous_finset_sum
intro s _
exact (((continuous_adaptivePatternMass p ε s).comp continuous_fst).sub
((continuous_adaptivePatternMass p ε s).comp continuous_snd)).abs
· unfold contrast
fun_prop
have hgzero : g (θ, θ) = 0 := by simp [g]
have hinterior : {q : TrialParameter × TrialParameter |
InteriorMeans q.1 ∧ InteriorMeans q.2} ∈ nhds (θ, θ) := by
have hopen := isOpen_interiorMeans.prod isOpen_interiorMeans
exact hopen.mem_nhds ⟨hθ, hθ⟩
let f : TrialParameter × TrialParameter → ℝ := fun q =>
adaptiveScoreVariance select q.1 q.2 p ε - Vstar q.2 p ε
have hbound : {q : TrialParameter × TrialParameter | ‖f q‖ ≤ g q} ∈
nhds (θ, θ) := by
filter_upwards [hinterior] with q hq
rw [Real.norm_eq_abs]
exact adaptiveScoreVariance_diag_error_le select q.1 q.2 p ε hselect hp hq.1 hq.2 hε
have hgtend : Tendsto g (nhds (θ, θ)) (nhds 0) := by
rw [← hgzero]
exact hgcont.continuousAt
have hftend : Tendsto f (nhds (θ, θ)) (nhds 0) :=
squeeze_zero_norm' hbound hgtend
have hv : ContinuousAt (fun q : TrialParameter × TrialParameter =>
Vstar q.2 p ε) (θ, θ) :=
by
have hsnd : ContinuousAt
(fun q : TrialParameter × TrialParameter => q.2) (θ, θ) :=
continuousAt_snd
change ContinuousAt ((fun η => Vstar η p ε) ∘
(fun q : TrialParameter × TrialParameter => q.2)) (θ, θ)
exact (continuousAt_Vstar select θ p ε hselect hp hθ hε).comp hsnd
have hsum := hftend.add hv
change Tendsto (fun pair : TrialParameter × TrialParameter =>
adaptiveScoreVariance select pair.1 pair.2 p ε) (nhds (θ, θ))
(nhds (adaptiveScoreVariance select θ θ p ε))
rw [adaptiveScoreVariance_diag select θ p ε hselect hp hθ hε]
simpa only [f, zero_add, sub_add_cancel] using hsum
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotAdaptiveInputs 19 declarations Finite adaptive main-release inputs
Finite adaptive main-release inputs
def adaptiveMainMass
For the supplied quantities and conditions, the adaptive main mass is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The adaptive Main Mass is determined by the displayed parameters.
Definition (Lean source)
def adaptiveSelectedScore
For the supplied quantities and conditions, the adaptive selected score is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The adaptive Selected Score is determined by the displayed parameters.
Definition (Lean source)
theorem adaptiveMainMass_nonneg
Under the supplied quantities and conditions, the adaptive main mass nonneg assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Main Mass nonneg.
Formal statement
hypotheses
conclusion
0 ≤ adaptiveMainMass select θ η p ε s
Proof (Lean source)
lemma adaptiveMainMass_nonneg (θ η : TrialParameter) (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hη : InteriorMeans η)
(hε : 0 < ε) (s : Fin 14) :
0 ≤ adaptiveMainMass select θ η p ε s := by
exact mul_nonneg (((hselect.1).2 η hη).1.1 s)
(patternMass_pos_interior θ p ε hp hθ hε.le s).le
theorem staircaseWeightedMass_sum
Under the supplied quantities and conditions, the staircase weighted mass sum assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the staircase Weighted Mass sum.
Formal statement
hypotheses
conclusion
∑ s : Fin 14, α s * patternMass θ p ε s = 1
Proof (Lean source)
lemma staircaseWeightedMass_sum (θ : TrialParameter) (p ε : ℝ)
(α : StaircaseWeight) (hα : staircaseFeasible ε α) :
∑ s : Fin 14, α s * patternMass θ p ε s = 1 := by
unfold patternMass
simp_rw [Finset.mul_sum]
calc
(∑ s : Fin 14, ∑ j : Fin 4,
α s * (piTheta θ p j * patternRay ε s j)) =
∑ j : Fin 4, piTheta θ p j *
∑ s : Fin 14, α s * patternRay ε s j := by
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro j _
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro s _
ring
_ = ∑ j : Fin 4, piTheta θ p j * 1 := by
apply Finset.sum_congr rfl
intro j _
have hj := hα.2 j
unfold staircaseMatrix at hj
rw [hj]
_ = 1 := by simp [piTheta_sum_eq_one]
theorem adaptiveMainMass_sum
Under the supplied quantities and conditions, the adaptive main mass sum assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Main Mass sum.
Formal statement
hypotheses
conclusion
∑ s : Fin 14, adaptiveMainMass select θ η p ε s = 1
Proof (Lean source)
lemma adaptiveMainMass_sum (θ η : TrialParameter) (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hη : InteriorMeans η) (hε : 0 < ε) :
∑ s : Fin 14, adaptiveMainMass select θ η p ε s = 1 := by
exact staircaseWeightedMass_sum θ p ε _
(((hselect.1).2 η hη).1)
theorem adaptiveSelectedScore_mean
Under the supplied quantities and conditions, the adaptive selected score mean assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Selected Score mean.
Formal statement
hypotheses
conclusion
= contrast θ - contrast η
Proof (Lean source)
lemma adaptiveSelectedScore_mean (θ η : TrialParameter) (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hη : InteriorMeans η) (hε : 0 < ε) :
∑ s : Fin 14, adaptiveMainMass select θ η p ε s *
adaptiveSelectedScore select η p ε s = contrast θ - contrast η := by
exact phiTilde_mean_identity θ η p ε _ hp hη hε
(hselect.1)
theorem adaptiveSelectedScore_secondMoment_diag
Under the supplied quantities and conditions, the adaptive selected score second moment diag assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Selected Score second Moment diag.
Formal statement
hypotheses
conclusion
= Vstar η p ε
Proof (Lean source)
lemma adaptiveSelectedScore_secondMoment_diag (η : TrialParameter) (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hη : InteriorMeans η) (hε : 0 < ε) :
∑ s : Fin 14, adaptiveMainMass select η η p ε s *
(adaptiveSelectedScore select η p ε s) ^ 2 = Vstar η p ε := by
exact phiTilde_secondMoment_identity η p ε _ hp hη hε
(hselect.1)
def adaptiveScoreVariance
For the supplied quantities and conditions, the adaptive score variance is the mathematical object specified below. The adaptive Score Variance is determined by the displayed parameters.
Definition (Lean source)
parameters
def
adaptiveScoreVariance select θ η p ε :
ℝ
given by
- (contrast θ - contrast η) ^ 2
def adaptiveScoreMoment
For the supplied quantities and conditions, the adaptive score moment is the mathematical object specified below. The adaptive Score Moment is determined by the displayed parameters.
Definition (Lean source)
parameters
def
adaptiveScoreMoment select r θ η p ε :
ℝ
given by
theorem adaptiveScoreMoment_nonneg
Under the supplied quantities and conditions, the adaptive score moment nonneg assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Score Moment nonneg.
Formal statement
hypotheses
conclusion
0 ≤ adaptiveScoreMoment select r θ η p ε
Proof (Lean source)
lemma adaptiveScoreMoment_nonneg (r : ℝ) (θ η : TrialParameter) (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hη : InteriorMeans η)
(hε : 0 < ε) : 0 ≤ adaptiveScoreMoment select r θ η p ε := by
unfold adaptiveScoreMoment
exact Finset.sum_nonneg fun s _ => mul_nonneg
(adaptiveMainMass_nonneg select θ η p ε hselect hp hθ hη hε s)
(Real.rpow_nonneg (abs_nonneg _) _)
theorem adaptiveScoreVariance_eq_centered_sum
Under the supplied quantities and conditions, the adaptive score variance eq centered sum assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Score Variance eq centered sum.
Formal statement
hypotheses
conclusion
adaptiveScoreVariance select θ η p ε
= ∑ s : Fin 14, adaptiveMainMass select θ η p ε s * (adaptiveSelectedScore select η p ε s - (contrast θ - contrast η)) ^ 2
Proof (Lean source)
lemma adaptiveScoreVariance_eq_centered_sum (θ η : TrialParameter) (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hη : InteriorMeans η) (hε : 0 < ε) :
adaptiveScoreVariance select θ η p ε =
∑ s : Fin 14, adaptiveMainMass select θ η p ε s *
(adaptiveSelectedScore select η p ε s -
(contrast θ - contrast η)) ^ 2 := by
rw [adaptiveScoreVariance]
have hm := adaptiveSelectedScore_mean select θ η p ε hselect hp hη hε
have hmass := adaptiveMainMass_sum select θ η p ε hselect hp hη hε
have hpoint (s : Fin 14) :
adaptiveMainMass select θ η p ε s *
(adaptiveSelectedScore select η p ε s -
(contrast θ - contrast η)) ^ 2 =
adaptiveMainMass select θ η p ε s *
(adaptiveSelectedScore select η p ε s) ^ 2 -
2 * (contrast θ - contrast η) *
(adaptiveMainMass select θ η p ε s *
adaptiveSelectedScore select η p ε s) +
(contrast θ - contrast η) ^ 2 *
adaptiveMainMass select θ η p ε s := by
ring
simp_rw [hpoint]
rw [Finset.sum_add_distrib, Finset.sum_sub_distrib,
← Finset.mul_sum, ← Finset.mul_sum, hm, hmass]
ring
theorem adaptiveScoreVariance_nonneg
Under the supplied quantities and conditions, the adaptive score variance nonneg assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Score Variance nonneg.
Formal statement
hypotheses
conclusion
0 ≤ adaptiveScoreVariance select θ η p ε
Proof (Lean source)
lemma adaptiveScoreVariance_nonneg (θ η : TrialParameter) (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hη : InteriorMeans η)
(hε : 0 < ε) : 0 ≤ adaptiveScoreVariance select θ η p ε := by
rw [adaptiveScoreVariance_eq_centered_sum select θ η p ε hselect hp hη hε]
exact Finset.sum_nonneg fun s _ => mul_nonneg
(adaptiveMainMass_nonneg select θ η p ε hselect hp hθ hη hε s) (sq_nonneg _)
theorem adaptiveScoreVariance_diag
Under the supplied quantities and conditions, the adaptive score variance diag assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Score Variance diag.
Formal statement
hypotheses
conclusion
adaptiveScoreVariance select η η p ε = Vstar η p ε
Proof (Lean source)
lemma adaptiveScoreVariance_diag (η : TrialParameter) (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hη : InteriorMeans η) (hε : 0 < ε) :
adaptiveScoreVariance select η η p ε = Vstar η p ε := by
rw [adaptiveScoreVariance, adaptiveSelectedScore_secondMoment_diag select η p ε hselect hp hη hε]
simp
theorem adaptiveSelectedScore_abs_le
Under the supplied quantities and conditions, the adaptive selected score abs le assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Selected Score abs le.
Formal statement
hypotheses
conclusion
|adaptiveSelectedScore select η p ε s| ≤ pilotPhiUpper p ε
Proof (Lean source)
lemma adaptiveSelectedScore_abs_le (η : TrialParameter) (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hη : InteriorMeans η) (hε : 0 < ε)
(s : Fin 14) :
|adaptiveSelectedScore select η p ε s| ≤ pilotPhiUpper p ε :=
strongSelector_phiTilde_bound select p ε hselect hp hε η hη s
theorem adaptiveSelectedScore_secondMoment_le
Under the supplied quantities and conditions, the adaptive selected score second moment le assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Selected Score second Moment le.
Formal statement
hypotheses
conclusion
≤ (pilotPhiUpper p ε) ^ 2
Proof (Lean source)
lemma adaptiveSelectedScore_secondMoment_le (θ η : TrialParameter) (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hη : InteriorMeans η)
(hε : 0 < ε) :
∑ s : Fin 14, adaptiveMainMass select θ η p ε s *
(adaptiveSelectedScore select η p ε s) ^ 2 ≤
(pilotPhiUpper p ε) ^ 2 := by
calc
_ ≤ ∑ s : Fin 14, adaptiveMainMass select θ η p ε s *
(pilotPhiUpper p ε) ^ 2 := by
apply Finset.sum_le_sum
intro s _
have hab := adaptiveSelectedScore_abs_le select η p ε hselect hp hη hε s
have hB : 0 ≤ pilotPhiUpper p ε := (abs_nonneg _).trans hab
apply mul_le_mul_of_nonneg_left
· simpa [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hB).2 hab
· exact adaptiveMainMass_nonneg select θ η p ε hselect hp hθ hη hε s
_ = _ := by rw [← Finset.sum_mul, adaptiveMainMass_sum select θ η p ε hselect hp hη hε, one_mul]
theorem adaptiveSelectedScore_absMoment_le
Under the supplied quantities and conditions, the adaptive selected score abs moment le assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Selected Score abs Moment le.
Formal statement
hypotheses
conclusion
≤ (pilotPhiUpper p ε) ^ r
Proof (Lean source)
lemma adaptiveSelectedScore_absMoment_le (θ η : TrialParameter) (p ε r : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hη : InteriorMeans η)
(hε : 0 < ε) (hr : 0 < r) :
∑ s : Fin 14, adaptiveMainMass select θ η p ε s *
|adaptiveSelectedScore select η p ε s| ^ r ≤
(pilotPhiUpper p ε) ^ r := by
calc
_ ≤ ∑ s : Fin 14, adaptiveMainMass select θ η p ε s *
(pilotPhiUpper p ε) ^ r := by
apply Finset.sum_le_sum
intro s _
apply mul_le_mul_of_nonneg_left
· exact Real.rpow_le_rpow (abs_nonneg _)
(adaptiveSelectedScore_abs_le select η p ε hselect hp hη hε s) hr.le
· exact adaptiveMainMass_nonneg select θ η p ε hselect hp hθ hη hε s
_ = _ := by
rw [← Finset.sum_mul, adaptiveMainMass_sum select θ η p ε hselect hp hη hε, one_mul]
theorem adaptiveScoreMoment_le
Under the supplied quantities and conditions, the adaptive score moment le assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Score Moment le.
Formal statement
hypotheses
conclusion
adaptiveScoreMoment select r θ η p ε ≤ (pilotPhiUpper p ε) ^ r
Proof (Lean source)
lemma adaptiveScoreMoment_le (θ η : TrialParameter) (p ε r : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hη : InteriorMeans η)
(hε : 0 < ε) (hr : 0 < r) :
adaptiveScoreMoment select r θ η p ε ≤ (pilotPhiUpper p ε) ^ r := by
exact adaptiveSelectedScore_absMoment_le select θ η p ε r hselect hp hθ hη hε hr
theorem adaptiveScoreVariance_le
Under the supplied quantities and conditions, the adaptive score variance le assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Score Variance le.
Formal statement
hypotheses
conclusion
adaptiveScoreVariance select θ η p ε ≤ (pilotPhiUpper p ε) ^ 2
Proof (Lean source)
lemma adaptiveScoreVariance_le (θ η : TrialParameter) (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hη : InteriorMeans η)
(hε : 0 < ε) :
adaptiveScoreVariance select θ η p ε ≤ (pilotPhiUpper p ε) ^ 2 := by
unfold adaptiveScoreVariance
nlinarith [adaptiveSelectedScore_secondMoment_le select θ η p ε hselect hp hθ hη hε,
sq_nonneg (contrast θ - contrast η)]
theorem adaptiveRareEventIntegrand_bounds
Under the supplied quantities and conditions, the adaptive rare event integrand bounds assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Rare Event Integrand bounds.
Formal statement
hypotheses
conclusion
conclusion 1
1 ≤ 1 + adaptiveScoreVariance select θ η p ε + adaptiveScoreMoment select r θ η p ε
conclusion 2
1 + adaptiveScoreVariance select θ η p ε + adaptiveScoreMoment select r θ η p ε
≤ 1 + (pilotPhiUpper p ε) ^ 2 + (pilotPhiUpper p ε) ^ r
Proof (Lean source)
lemma adaptiveRareEventIntegrand_bounds (θ η : TrialParameter) (p ε r : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hη : InteriorMeans η)
(hε : 0 < ε) (hr : 0 < r) :
1 ≤ 1 + adaptiveScoreVariance select θ η p ε +
adaptiveScoreMoment select r θ η p ε ∧
1 + adaptiveScoreVariance select θ η p ε +
adaptiveScoreMoment select r θ η p ε ≤
1 + (pilotPhiUpper p ε) ^ 2 + (pilotPhiUpper p ε) ^ r := by
constructor
· nlinarith [adaptiveScoreVariance_nonneg select θ η p ε hselect hp hθ hη hε,
adaptiveScoreMoment_nonneg select r θ η p ε hselect hp hθ hη hε]
· linarith [adaptiveScoreVariance_le select θ η p ε hselect hp hθ hη hε,
adaptiveScoreMoment_le select θ η p ε r hselect hp hθ hη hε hr]
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotAdaptiveLaw 15 declarations Exact finite-law bridges for the adaptive pilot protocol
Exact finite-law bridges for the adaptive pilot protocol
theorem fin_prod_split
Under the supplied quantities and conditions, the fin prod split assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma fin_prod_split {M : Type*} [CommMonoid M] {m n : ℕ} (hmn : m ≤ n)
(f : Fin m → M) (g : Fin (n - m) → M) :
(∏ i : Fin n, if hi : i.val < m then f ⟨i.val, hi⟩
else g ⟨i.val - m, by omega⟩) =
(∏ j : Fin m, f j) * ∏ j : Fin (n - m), g j := by
let e : Fin (m + (n - m)) ≃ Fin n := Fin.castOrderIso (Nat.add_sub_of_le hmn)
rw [← e.prod_comp]
rw [Fin.prod_univ_add]
congr 1
· apply Finset.prod_congr rfl
intro j _
simp [e]
· apply Finset.prod_congr rfl
intro j _
simp [e]
def pilotAdaptiveEncode
For the supplied quantities and conditions, the pilot adaptive encode is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
theorem pilotAdaptiveEncode_pilot
Under the supplied quantities and conditions, the pilot adaptive encode pilot assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma pilotAdaptiveEncode_pilot {m n : ℕ} (hmn : m ≤ n)
(u : Fin m → Fin 4) (v : Fin (n - m) → Fin 14) (j : Fin m) :
pilotAdaptiveEncode hmn u v (castLE hmn j) = inl (u j) := by
simp [pilotAdaptiveEncode]
theorem pilotAdaptiveEncode_main
Under the supplied quantities and conditions, the pilot adaptive encode main assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma pilotAdaptiveEncode_main {m n : ℕ} (hmn : m ≤ n)
(u : Fin m → Fin 4) (v : Fin (n - m) → Fin 14) (j : Fin (n - m)) :
pilotAdaptiveEncode hmn u v
⟨m + j.val, by omega⟩ = inr (v j) := by
simp [pilotAdaptiveEncode]
def pilotAdaptivePrefixTheta
For the supplied quantities and conditions, the pilot adaptive prefix theta is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The pilot Adaptive Prefix Theta is determined by the displayed parameters.
Definition (Lean source)
theorem pilotTheta_pilotAdaptiveEncode
Under the supplied quantities and conditions, the pilot theta pilot adaptive encode assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
pilotTheta p ε mseq (pilotAdaptiveEncode hmn u v)
= pilotAdaptivePrefixTheta p ε mseq hmn u
Proof (Lean source)
lemma pilotTheta_pilotAdaptiveEncode (p ε : ℝ) (mseq : ℕ → ℕ) {n : ℕ}
(hmn : mseq n ≤ n) (u : Fin (mseq n) → Fin 4)
(v : Fin (n - mseq n) → Fin 14) :
pilotTheta p ε mseq (pilotAdaptiveEncode hmn u v) =
pilotAdaptivePrefixTheta p ε mseq hmn u := by
have hfreq (k : Fin 4) :
pilotFrequency mseq (pilotAdaptiveEncode hmn u v) k =
pilotFrequency mseq (pilotAdaptiveEncode hmn u (fun _ => 0)) k := by
unfold pilotFrequency
congr 1
apply Finset.sum_congr rfl
intro i _
by_cases hi : i.val < mseq n <;> simp [pilotAdaptiveEncode, hi]
funext a
simp only [pilotTheta, pilotAdaptivePrefixTheta]
rw [hfreq 1, hfreq 3]
theorem finiteSequenceTranscript_singleton_apply
the finite sequence transcript singleton apply assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
n :
ℕ
Fin n → Type
∀ i, MeasurableSpace (Z i)
∀ i, MeasurableSingletonClass (Z i)
Q :
SequentialKernel (Z := Z)
hQ :
∀ i, IsMarkovKernel (Q i)
t :
Transcript Z
conclusion
finiteSequenceTranscript Q x {t} = ∏ i : Fin n, Q i (x i, fun j => t j.1) {t i}
Proof (Lean source)
lemma finiteSequenceTranscript_singleton_apply {n : ℕ} {Z : Fin n → Type}
[∀ i, MeasurableSpace (Z i)] [∀ i, MeasurableSingletonClass (Z i)]
(Q : SequentialKernel (Z := Z)) (hQ : ∀ i, IsMarkovKernel (Q i))
(x : Fin n → Fin 4) (t : Transcript Z) :
finiteSequenceTranscript Q x {t} =
∏ i : Fin n, Q i (x i, fun j => t j.1) {t i} := by
have hQa : ∀ i, IsMarkovKernel (finiteSequenceKernel Q i) := by
intro i
letI : IsMarkovKernel (Q i) := hQ i
dsimp [finiteSequenceKernel]
infer_instance
unfold finiteSequenceTranscript
rw [Measure.map_apply measurable_fsTranscriptToTranscript
(measurableSet_singleton t)]
have hpre : fsTranscriptToTranscript ⁻¹' ({t} : Set (Transcript Z)) =
{transcriptToFSTranscript t} := by
ext y
simp only [Set.mem_preimage, Set.mem_singleton_iff]
constructor
· intro hy
funext i
exact congrFun hy i
· intro hy
subst y
rfl
rw [hpre, transcriptLaw_singleton (finiteSequenceKernel Q) hQa]
apply Finset.prod_congr rfl
intro i _
rfl
theorem pilotHistoryFrequency_restrict_eq
Under the supplied quantities and conditions, the pilot history frequency restrict eq assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
pilotHistoryFrequency m i (fun j => z j.1) k = pilotFrequency m z k
Proof (Lean source)
lemma pilotHistoryFrequency_restrict_eq (m : ℕ → ℕ) {n : ℕ}
(i : Fin n) (hi : m n ≤ i.val)
(z : Transcript (pilotOutputFamily n)) (k : Fin 4) :
pilotHistoryFrequency m i (fun j => z j.1) k = pilotFrequency m z k := by
have hmn : m n ≤ n := hi.trans (le_of_lt i.isLt)
unfold pilotHistoryFrequency pilotFrequency
congr 1
have hleft :
(∑ j : {j : Fin n // j < i},
if j.1.val < m n ∧ z j.1 = inl k then (1 : ℝ) else 0) =
∑ a : Fin (m n),
if z (castLE hmn a) = inl k then (1 : ℝ) else 0 := by
have hfilter :
(∑ j : {j : Fin n // j < i},
if j.1.val < m n ∧ z j.1 = inl k then (1 : ℝ) else 0) =
∑ j ∈ (Finset.univ.filter fun j : {j : Fin n // j < i} =>
j.1.val < m n), if z j.1 = inl k then (1 : ℝ) else 0 := by
rw [Finset.sum_filter]
apply Finset.sum_congr rfl
intro j _
by_cases hj : j.1.val < m n <;> simp [hj]
rw [hfilter]
symm
apply Finset.sum_bij (fun a _ =>
⟨Fin.castLE hmn a, by
change a.val < i.val
exact lt_of_lt_of_le a.isLt hi⟩)
· intro a _
simp
· intro a _ b _ hab
exact Fin.ext (congrArg (fun x => x.1.val) hab)
· intro j hj
have hjlt : j.1.val < m n := (Finset.mem_filter.mp hj).2
let a : Fin (m n) := ⟨j.1.val, hjlt⟩
refine ⟨a, Finset.mem_univ _, ?_⟩
apply Subtype.ext
exact Fin.ext rfl
· intro a _
simp
rw [hleft]
symm
rw [← fin_sum_ite_lt hmn (fun a : Fin (m n) =>
if z (castLE hmn a) = inl k then (1 : ℝ) else 0)]
apply Finset.sum_congr rfl
intro j _
by_cases hj : j.val < m n <;> simp [hj]
theorem pilotHistoryTheta_restrict_eq
Under the supplied quantities and conditions, the pilot history theta restrict eq assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
pilotHistoryTheta p ε m i (fun j => z j.1) = pilotTheta p ε m z
Proof (Lean source)
lemma pilotHistoryTheta_restrict_eq (p ε : ℝ) (m : ℕ → ℕ) {n : ℕ}
(i : Fin n) (hi : m n ≤ i.val)
(z : Transcript (pilotOutputFamily n)) :
pilotHistoryTheta p ε m i (fun j => z j.1) = pilotTheta p ε m z := by
funext a
simp only [pilotHistoryTheta, pilotTheta]
rw [pilotHistoryFrequency_restrict_eq m i hi z 1,
pilotHistoryFrequency_restrict_eq m i hi z 3]
theorem pilotProtocol_main_singleton
Under the supplied quantities and conditions, the pilot protocol main singleton assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Protocol main singleton.
Formal statement
hypotheses
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
i :
Fin n
hi :
m n ≤ i.val
x :
Fin 4
z :
Transcript (pilotOutputFamily n)
s :
Fin 14
conclusion
pilotProtocol p ε m (select) n i (x, fun j => z j.1) {inr s}
Proof (Lean source)
lemma pilotProtocol_main_singleton (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε) {n : ℕ}
(i : Fin n) (hi : m n ≤ i.val) (x : Fin 4)
(z : Transcript (pilotOutputFamily n)) (s : Fin 14) :
pilotProtocol p ε m (select) n i
(x, fun j => z j.1) {Sum.inr s} =
ofReal
((select (pilotTheta p ε m z)).1 s *
patternRay ε s x) := by
simp only [pilotProtocol]
change (if i.val < m n then
((rrPilotKernel ε x).map inl)
else
((staircaseChannel ε
(select
(pilotHistoryTheta p ε m i (fun j => z j.1))).1 x).map Sum.inr))
{Sum.inr s} = _
rw [if_neg (not_lt_of_ge hi), pilotHistoryTheta_restrict_eq p ε m i hi z]
rw [Measure.map_apply (by fun_prop) (measurableSet_singleton _)]
have hpre : (Sum.inr : Fin 14 → PilotOutput) ⁻¹'
({Sum.inr s} : Set PilotOutput) = ({s} : Set (Fin 14)) := by
ext y
simp
rw [hpre]
change (Measure.count.withDensity (fun u : Fin 14 => ofReal
((select (pilotTheta p ε m z)).1 u *
patternRay ε u x))) {s} = _
rw [withDensity_apply _ (measurableSet_singleton s)]
simp
theorem pilotProtocol_main_output_singleton
Under the supplied quantities and conditions, the pilot protocol main output singleton assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Protocol main output singleton.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
n :
ℕ
i :
Fin n
hi :
m n ≤ i.val
z :
Transcript (pilotOutputFamily n)
s :
Fin 14
conclusion
Proof (Lean source)
lemma pilotProtocol_main_output_singleton
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε) {n : ℕ}
(i : Fin n) (hi : m n ≤ i.val)
(z : Transcript (pilotOutputFamily n)) (s : Fin 14) :
∑ x : Fin 4, ofReal (piTheta θ p x) *
pilotProtocol p ε m (select) n i
(x, fun j => z j.1) {inr s} =
ofReal
(adaptiveMainMass select θ (pilotTheta p ε m z) p ε s) := by
let α := (select (pilotTheta p ε m z)).1
have hη : InteriorMeans (pilotTheta p ε m z) := by
rw [← pilotHistoryTheta_restrict_eq p ε m i hi z]
exact pilotHistoryTheta_interior p ε m i (fun j => z j.1)
have hα : staircaseFeasible ε α :=
((hselect.1).2 _ hη).1
simp_rw [pilotProtocol_main_singleton select p ε m hselect hp hε i hi _ z s]
have hpi (x : Fin 4) : 0 ≤ piTheta θ p x :=
(piTheta_pos_interior θ p hp hθ x).le
simp_rw [← ENNReal.ofReal_mul (hpi _)]
rw [← ENNReal.ofReal_sum_of_nonneg]
· congr 1
simp only [adaptiveMainMass, patternMass]
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro x _
ring
· intro x _
exact mul_nonneg (hpi x)
(mul_nonneg (hα.1 s) (by
unfold patternRay privacyIncrement privacyRatio
cases patternContains s x <;> simp <;> positivity))
theorem pilotProtocol_pilot_output_singleton
Under the supplied quantities and conditions, the pilot protocol pilot output singleton assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Protocol pilot output singleton.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
n :
ℕ
i :
Fin n
hi :
i.val < m n
h :
PrivateHistory (Z := pilotOutputFamily n) i
k :
Fin 4
conclusion
Proof (Lean source)
lemma pilotProtocol_pilot_output_singleton
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε) {n : ℕ}
(i : Fin n) (hi : i.val < m n)
(h : PrivateHistory (Z := pilotOutputFamily n) i) (k : Fin 4) :
∑ x : Fin 4, ofReal (piTheta θ p x) *
pilotProtocol p ε m (select) n i (x, h)
{Sum.inl k} =
ofReal
(∑ x : Fin 4, piTheta θ p x * rrPilotProbability ε x k) := by
have hstage (x : Fin 4) :
pilotProtocol p ε m (select) n i (x, h)
{Sum.inl k} = ofReal (rrPilotProbability ε x k) := by
simp only [pilotProtocol]
change (if i.val < m n then ((rrPilotKernel ε x).map Sum.inl)
else _) {Sum.inl k} = _
rw [if_pos hi, Measure.map_apply (by fun_prop) (measurableSet_singleton _)]
have hpre : (Sum.inl : Fin 4 → PilotOutput) ⁻¹'
({Sum.inl k} : Set PilotOutput) = {k} := by
ext y
simp
rw [hpre, rrPilotKernel_singleton]
simp_rw [hstage]
have hpi (x : Fin 4) : 0 ≤ piTheta θ p x :=
(piTheta_pos_interior θ p hp hθ x).le
simp_rw [← ENNReal.ofReal_mul (hpi _)]
rw [ENNReal.ofReal_sum_of_nonneg]
intro x _
exact mul_nonneg (hpi x) (by
unfold rrPilotProbability
split_ifs <;> positivity)
theorem pilotAdaptive_joint_singleton
Under the supplied quantities and conditions, the pilot adaptive joint singleton assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Adaptive joint singleton.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
n :
ℕ
hmn :
m n ≤ n
conclusion
transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n {pilotAdaptiveEncode hmn u v}
= (∏ j : Fin (m n), ofReal (∑ a : Fin 4, piTheta θ p a * rrPilotProbability ε a (u j))) * ∏ j : Fin (n - m n), ofReal (adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u) p ε (v j))
Proof (Lean source)
lemma pilotAdaptive_joint_singleton
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε) {n : ℕ}
(hmn : m n ≤ n) (u : Fin (m n) → Fin 4)
(v : Fin (n - m n) → Fin 14) :
transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n
{pilotAdaptiveEncode hmn u v} =
(∏ j : Fin (m n), ofReal
(∑ a : Fin 4, piTheta θ p a * rrPilotProbability ε a (u j))) *
∏ j : Fin (n - m n), ofReal
(adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)) := by
let t := pilotAdaptiveEncode hmn u v
let Q := pilotProtocol p ε m (select) n
let F : (i : Fin n) → Fin 4 → ℝ≥0∞ := fun i a =>
ofReal (piTheta θ p a) *
Q i (a, fun j => t j.1) {t i}
have hQ : ∀ i, IsMarkovKernel (Q i) := by
intro i
exact (pilotProtocol_private p ε m select hselect hp hε n).1 i
have hpi (a : Fin 4) : 0 ≤ piTheta θ p a :=
(piTheta_pos_interior θ p hp hθ a).le
have hpath (x : Fin n → Fin 4) :
ofReal (inputPathProbability θ p x) =
∏ i : Fin n, ofReal (piTheta θ p (x i)) := by
change ofReal (∏ i : Fin n, piTheta θ p (x i)) = _
rw [ENNReal.ofReal_prod_of_nonneg]
intro i _
exact hpi (x i)
have hstage (i : Fin n) :
(∑ a : Fin 4, F i a) =
if hi : i.val < m n then
ofReal
(∑ a : Fin 4, piTheta θ p a *
rrPilotProbability ε a (u ⟨i.val, hi⟩))
else
ofReal
(adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v ⟨i.val - m n, by omega⟩)) := by
by_cases hi : i.val < m n
· rw [dif_pos hi]
have ht : t i = inl (u ⟨i.val, hi⟩) := by
simp [t, pilotAdaptiveEncode, hi]
simp only [F, Q]
rw [ht]
exact pilotProtocol_pilot_output_singleton select θ p ε m hselect hp hθ hε i hi
(fun j => t j.1) (u ⟨i.val, hi⟩)
· rw [dif_neg hi]
have himain : m n ≤ i.val := Nat.le_of_not_gt hi
have ht : t i = inr (v ⟨i.val - m n, by omega⟩) := by
simp [t, pilotAdaptiveEncode, hi]
simp only [F, Q]
rw [ht]
rw [pilotProtocol_main_output_singleton select θ p ε m hselect hp hθ hε i himain t]
rw [pilotTheta_pilotAdaptiveEncode p ε m hmn u v]
unfold transcriptLaw
simp only [Measure.coe_finsetSum, Finset.sum_apply, Measure.smul_apply,
smul_eq_mul]
change (∑ x : Fin n → Fin 4,
ofReal (inputPathProbability θ p x) *
finiteSequenceTranscript Q x {t}) = _
simp_rw [finiteSequenceTranscript_singleton_apply Q hQ]
calc
(∑ x : Fin n → Fin 4,
ofReal (inputPathProbability θ p x) *
∏ i : Fin n, Q i (x i, fun j => t j.1) {t i}) =
∑ x : Fin n → Fin 4, ∏ i : Fin n, F i (x i) := by
apply Finset.sum_congr rfl
intro x _
rw [hpath x, ← Finset.prod_mul_distrib]
_ = ∏ i : Fin n, ∑ a : Fin 4, F i a :=
(Fintype.prod_sum F).symm
_ = ∏ i : Fin n, if hi : i.val < m n then
ofReal
(∑ a : Fin 4, piTheta θ p a *
rrPilotProbability ε a (u ⟨i.val, hi⟩))
else
ofReal
(adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v ⟨i.val - m n, by omega⟩)) := by
apply Finset.prod_congr rfl
intro i _
exact hstage i
_ = _ := by
simpa using fin_prod_split hmn
(fun j : Fin (m n) => ofReal
(∑ a : Fin 4, piTheta θ p a * rrPilotProbability ε a (u j)))
(fun j : Fin (n - m n) => ofReal
(adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)))
theorem pilotProtocol_main_wrongTag
Under the supplied quantities and conditions, the pilot protocol main wrong tag assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Protocol main wrong Tag.
Formal statement
hypotheses
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
i :
Fin n
hi :
m n ≤ i.val
x :
Fin 4
h :
PrivateHistory (Z := pilotOutputFamily n) i
k :
Fin 4
conclusion
pilotProtocol p ε m (select) n i (x, h) {inl k} = 0
Proof (Lean source)
lemma pilotProtocol_main_wrongTag (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε) {n : ℕ}
(i : Fin n) (hi : m n ≤ i.val) (x : Fin 4)
(h : PrivateHistory (Z := pilotOutputFamily n) i) (k : Fin 4) :
pilotProtocol p ε m (select) n i (x, h)
{inl k} = 0 := by
simp only [pilotProtocol]
change (if i.val < m n then
((rrPilotKernel ε x).map inl)
else
((staircaseChannel ε
(select (pilotHistoryTheta p ε m i h)).1 x).map Sum.inr))
{inl k} = 0
rw [if_neg (not_lt_of_ge hi), Measure.map_apply (by fun_prop)
(measurableSet_singleton _)]
have hpre : (Sum.inr : Fin 14 → PilotOutput) ⁻¹'
({inl k} : Set PilotOutput) = ∅ := by
ext y
simp
rw [hpre]
simp
theorem pilotProtocol_pilot_wrongTag
Under the supplied quantities and conditions, the pilot protocol pilot wrong tag assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Protocol pilot wrong Tag.
Formal statement
hypotheses
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
i :
Fin n
hi :
i.val < m n
x :
Fin 4
h :
PrivateHistory (Z := pilotOutputFamily n) i
s :
Fin 14
conclusion
pilotProtocol p ε m (select) n i (x, h) {inr s} = 0
Proof (Lean source)
lemma pilotProtocol_pilot_wrongTag (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε) {n : ℕ}
(i : Fin n) (hi : i.val < m n) (x : Fin 4)
(h : PrivateHistory (Z := pilotOutputFamily n) i) (s : Fin 14) :
pilotProtocol p ε m (select) n i (x, h)
{inr s} = 0 := by
simp only [pilotProtocol]
change (if i.val < m n then ((rrPilotKernel ε x).map Sum.inl)
else _) {inr s} = 0
rw [if_pos hi, Measure.map_apply (by fun_prop) (measurableSet_singleton _)]
have hpre : (Sum.inl : Fin 4 → PilotOutput) ⁻¹'
({inr s} : Set PilotOutput) = ∅ := by
ext y
simp
rw [hpre]
simp
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotAdaptiveLocalization 2 declarations Localization bounds for adaptive pilot inputs
Localization bounds for adaptive pilot inputs
theorem pilotTheta_interior
Under the supplied quantities and conditions, the pilot theta interior assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
InteriorMeans (pilotTheta p ε m z)
Proof (Lean source)
lemma pilotTheta_interior (p ε : ℝ) (m : ℕ → ℕ) {n : ℕ}
(z : Transcript (pilotOutputFamily n)) :
InteriorMeans (pilotTheta p ε m z) := by
have hδ : 0 < ((m n + 2 : ℝ)⁻¹) := by positivity
have hhalf : ((m n + 2 : ℝ)⁻¹) ≤ 1 / 2 := by
rw [one_div]
have hden : (2 : ℝ) ≤ m n + 2 := by
exact_mod_cast Nat.le_add_left 2 (m n)
simpa [one_div] using one_div_le_one_div_of_le
(by norm_num : (0 : ℝ) < 2) hden
have h0 := clipInterior_interior ((m n + 2 : ℝ)⁻¹)
(((exp ε + 3) * pilotFrequency m z 1 - 1) /
(exp ε - 1) / controlProb p) hδ hhalf
have h1 := clipInterior_interior ((m n + 2 : ℝ)⁻¹)
(((exp ε + 3) * pilotFrequency m z 3 - 1) /
(exp ε - 1) / p) hδ hhalf
simpa [InteriorMeans, pilotTheta] using
intro h0.1 (intro h0.2 (intro h1.1 h1.2))
theorem adaptivePilotRareEvent_integral_le
Under the supplied quantities and conditions, the adaptive pilot rare event integral le assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Rare Event integral le.
Formal statement
hypotheses
θ :
p ε r q :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hr :
0 < r
hq :
0 < q
n :
ℕ
hmn :
m n ≤ n
hmpos :
0 < m n
hδ0l :
(m n + 2 : ℝ)⁻¹ + r ≤ θ 0
hδ0r :
(m n + 2 : ℝ)⁻¹ + r ≤ 1 - θ 0
hδ1l :
(m n + 2 : ℝ)⁻¹ + r ≤ θ 1
hδ1r :
(m n + 2 : ℝ)⁻¹ + r ≤ 1 - θ 1
conclusion
a0 :
μ :
transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n
∫ z in {z | r ≤ |pilotTheta p ε m z 0 - θ 0| ∨ r ≤ |pilotTheta p ε m z 1 - θ 1|}, (1 + adaptiveScoreVariance select θ (pilotTheta p ε m z) p ε + adaptiveScoreMoment select q θ (pilotTheta p ε m z) p ε) ∂μ
≤ (1 + (pilotPhiUpper p ε) ^ 2 + (pilotPhiUpper p ε) ^ q) * (ofReal (1 / ((m n : ℝ) * a0 ^ 2)) + ofReal (1 / ((m n : ℝ) * a1 ^ 2))).toReal
Proof (Lean source)
lemma adaptivePilotRareEvent_integral_le
(θ : TrialParameter) (p ε r q : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hr : 0 < r) (hq : 0 < q) {n : ℕ} (hmn : m n ≤ n) (hmpos : 0 < m n)
(hδ0l : (m n + 2 : ℝ)⁻¹ + r ≤ θ 0)
(hδ0r : (m n + 2 : ℝ)⁻¹ + r ≤ 1 - θ 0)
(hδ1l : (m n + 2 : ℝ)⁻¹ + r ≤ θ 1)
(hδ1r : (m n + 2 : ℝ)⁻¹ + r ≤ 1 - θ 1) :
let a0 := r * controlProb p * (exp ε - 1) / (exp ε + 3)
let a1 := r * p * (exp ε - 1) / (exp ε + 3)
let μ := transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n
∫ z in {z | r ≤ |pilotTheta p ε m z 0 - θ 0| ∨
r ≤ |pilotTheta p ε m z 1 - θ 1|},
(1 + adaptiveScoreVariance select θ (pilotTheta p ε m z) p ε +
adaptiveScoreMoment select q θ (pilotTheta p ε m z) p ε) ∂μ ≤
(1 + (pilotPhiUpper p ε) ^ 2 + (pilotPhiUpper p ε) ^ q) *
(ofReal (1 / ((m n : ℝ) * a0 ^ 2)) +
ofReal (1 / ((m n : ℝ) * a1 ^ 2))).toReal := by
dsimp only
let μ := transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n
let A : Set (Transcript (pilotOutputFamily n)) :=
{z | r ≤ |pilotTheta p ε m z 0 - θ 0| ∨
r ≤ |pilotTheta p ε m z 1 - θ 1|}
let C := 1 + (pilotPhiUpper p ε) ^ 2 + (pilotPhiUpper p ε) ^ q
let f : Transcript (pilotOutputFamily n) → ℝ := fun z =>
1 + adaptiveScoreVariance select θ (pilotTheta p ε m z) p ε +
adaptiveScoreMoment select q θ (pilotTheta p ε m z) p ε
letI : IsProbabilityMeasure μ :=
transcriptLaw_isProbability (pilotEstimator p ε m select hselect hp hε) θ p hp hθ n
have hfint : IntegrableOn f A μ := Integrable.of_finite
have hcint : IntegrableOn (fun _ : Transcript (pilotOutputFamily n) => C) A μ :=
integrableOn_const
have hpoint (z : Transcript (pilotOutputFamily n)) : f z ≤ C := by
exact (adaptiveRareEventIntegrand_bounds select θ (pilotTheta p ε m z) p ε q
hselect hp hθ
(pilotTheta_interior p ε m z) hε hq).2
have hdev : μ A ≤
ofReal (1 / ((m n : ℝ) *
(r * controlProb p * (exp ε - 1) / (exp ε + 3)) ^ 2)) +
ofReal (1 / ((m n : ℝ) *
(r * p * (exp ε - 1) / (exp ε + 3)) ^ 2)) := by
simpa [μ, A] using pilotTheta_deviation_le select θ p ε r m hselect hp hθ hε hr hmn hmpos
hδ0l hδ0r hδ1l hδ1r
have hdevReal : μ.real A ≤
(ofReal (1 / ((m n : ℝ) *
(r * controlProb p * (exp ε - 1) / (exp ε + 3)) ^ 2)) +
ofReal (1 / ((m n : ℝ) *
(r * p * (exp ε - 1) / (exp ε + 3)) ^ 2))).toReal := by
rw [measureReal_def]
exact ENNReal.toReal_mono (by simp) hdev
have hC : 0 ≤ C := by
dsimp [C]
have hB := pilotPhiUpper_nonneg p ε hp hε
positivity
change ∫ z in A, f z ∂μ ≤ C * _
calc
(∫ z in A, f z ∂μ) ≤ ∫ _z in A, C ∂μ := by
apply integral_mono_ae hfint hcint
exact Filter.Eventually.of_forall hpoint
_ = C * μ.real A := by
rw [setIntegral_const]
simp only [smul_eq_mul]
ring
_ ≤ C * _ := mul_le_mul_of_nonneg_left hdevReal hC
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotAdaptiveModelFields 19 declarations This file packages the finite pilot sample, its selected parameter, and fixed compact interior neighborhoods used by the adaptive asymptotic model.
Routine model fields for adaptive pilot asymptotics
This file packages the finite pilot sample, its selected parameter, and fixed compact interior neighborhoods used by the adaptive asymptotic model. It does not depend on the generic adaptive limit theorem.
def AdaptivePilotSample
For the supplied quantities and conditions, the adaptive pilot sample is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The Adaptive Pilot Sample is determined by the displayed parameters.
Definition (Lean source)
parameters
m :
ℕ → ℕ
n :
ℕ
def
AdaptivePilotSample m n :
Type
given by
def adaptivePilotSelector
For the supplied quantities and conditions, the adaptive pilot selector is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The adaptive Pilot Selector is determined by the displayed parameters.
Definition (Lean source)
parameters
def
given by
theorem adaptivePilotSelector_interior
Under the supplied quantities and conditions, the adaptive pilot selector interior assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
InteriorMeans (adaptivePilotSelector p ε m n u)
Proof (Lean source)
lemma adaptivePilotSelector_interior (p ε : ℝ) (m : ℕ → ℕ) (n : ℕ)
(u : AdaptivePilotSample m n) :
InteriorMeans (adaptivePilotSelector p ε m n u) := by
exact pilotTheta_interior p ε (adaptivePilotSize m)
(pilotAdaptiveEncode (adaptivePilotSize_le m n) u (fun _ => 0))
theorem measurable_adaptivePilotSelector
Under the supplied quantities and conditions, the measurable adaptive pilot selector assertion holds.
Formal statement
Proof (Lean source)
lemma measurable_adaptivePilotSelector (p ε : ℝ) (m : ℕ → ℕ) (n : ℕ) :
Measurable (adaptivePilotSelector p ε m n) := by
exact measurable_of_finite _
theorem measurable_adaptiveSelectedScore_pair
Under the supplied quantities and conditions, the measurable adaptive selected score pair assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the measurable adaptive Selected Score pair.
Formal statement
Proof (Lean source)
lemma measurable_adaptiveSelectedScore_pair (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε) :
Measurable (fun pair : TrialParameter × Fin 14 =>
adaptiveSelectedScore select pair.1 p ε pair.2) := by
have hs : Measurable (select) :=
(hselect.1).1
apply measurable_from_prod_countable_left
intro s
have ht : Measurable (fun η => (select η).2.1) :=
measurable_fst.comp (measurable_snd.comp hs)
have hJ : Measurable (fun η => (select η).2.2) :=
measurable_snd.comp (measurable_snd.comp hs)
have hg : Measurable (fun η =>
projectedGradient p ε s (select η).2.1) := by
exact measurable_projectedGradient_comp p ε s _ ht
have hm : Measurable (fun η => patternMass η p ε s) :=
measurable_patternMass_arg p ε s
unfold adaptiveSelectedScore phiTilde
exact hg.div (hJ.mul hm)
theorem measurable_adaptiveMainMass_pair
Under the supplied quantities and conditions, the measurable adaptive main mass pair assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the measurable adaptive Main Mass pair.
Formal statement
hypotheses
conclusion
Measurable (fun pair : (TrialParameter × TrialParameter) × Fin 14 => adaptiveMainMass select pair.1.1 pair.1.2 p ε pair.2)
Proof (Lean source)
lemma measurable_adaptiveMainMass_pair (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε) :
Measurable (fun pair : (TrialParameter × TrialParameter) × Fin 14 =>
adaptiveMainMass select pair.1.1 pair.1.2 p ε pair.2) := by
have hs : Measurable (select) :=
(hselect.1).1
apply measurable_from_prod_countable_left
intro s
have hα : Measurable (fun η => (select η).1 s) :=
measurable_pi_apply s |>.comp (measurable_fst.comp hs)
have hm : Measurable (fun θ => patternMass θ p ε s) :=
measurable_patternMass_arg p ε s
unfold adaptiveMainMass
exact (hα.comp measurable_snd).mul (hm.comp measurable_fst)
def adaptiveInteriorMargin
the adaptive interior margin is the mathematical object specified below. The adaptive Interior Margin is determined by the displayed parameters.
Definition (Lean source)
parameters
θ :
def
adaptiveInteriorMargin θ :
ℝ
def adaptiveNeighborhood
the adaptive neighborhood is the mathematical object specified below.
Definition (Lean source)
def
given by
{η | InteriorMeans η}
def adaptiveCompactValues
the adaptive compact values is the mathematical object specified below. The adaptive Compact Values is determined by the displayed parameters.
Definition (Lean source)
theorem adaptiveInteriorMargin_pos
Under the supplied quantities and conditions, the adaptive interior margin pos assertion holds. Under the stated assumptions, the adaptive Interior Margin pos.
Formal statement
Proof (Lean source)
lemma adaptiveInteriorMargin_pos (θ : TrialParameter) (hθ : InteriorMeans θ) :
0 < adaptiveInteriorMargin θ := by
simp only [adaptiveInteriorMargin, lt_min_iff]
exact ⟨⟨hθ.1, sub_pos.mpr hθ.2.1⟩,
⟨hθ.2.2.1, sub_pos.mpr hθ.2.2.2⟩⟩
theorem adaptiveNeighborhood_isOpen
the adaptive neighborhood is open assertion holds.
Formal statement
conclusion
Proof (Lean source)
lemma adaptiveNeighborhood_isOpen : IsOpen adaptiveNeighborhood := by
change IsOpen { η : TrialParameter |
0 < η 0 ∧ η 0 < 1 ∧ 0 < η 1 ∧ η 1 < 1 }
have h0l : IsOpen { η : TrialParameter | (0 : ℝ) < η 0 } :=
isOpen_lt continuous_const (continuous_apply 0)
have h0r : IsOpen { η : TrialParameter | η 0 < (1 : ℝ) } :=
isOpen_lt (continuous_apply 0) continuous_const
have h1l : IsOpen { η : TrialParameter | (0 : ℝ) < η 1 } :=
isOpen_lt continuous_const (continuous_apply 1)
have h1r : IsOpen { η : TrialParameter | η 1 < (1 : ℝ) } :=
isOpen_lt (continuous_apply 1) continuous_const
convert ((h0l.inter h0r).inter h1l).inter h1r using 1 <;>
ext η <;> simp [and_assoc]
theorem adaptiveCompactValues_isCompact
the adaptive compact values is compact assertion holds.
Formal statement
Proof (Lean source)
lemma adaptiveCompactValues_isCompact (θ : TrialParameter) :
IsCompact (adaptiveCompactValues θ) := by
exact isCompact_closedBall _ _
theorem adaptiveCompactValues_mem
Under the supplied quantities and conditions, the adaptive compact values mem assertion holds. Under the stated assumptions, the adaptive Compact Values mem.
Formal statement
Proof (Lean source)
lemma adaptiveCompactValues_mem (θ : TrialParameter) (hθ : InteriorMeans θ) :
θ ∈ adaptiveCompactValues θ := by
exact Metric.mem_closedBall_self
(div_nonneg (adaptiveInteriorMargin_pos θ hθ).le (by norm_num))
theorem adaptiveCompactValues_mem_interior
the adaptive compact values mem interior assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Compact Values mem interior.
Formal statement
Proof (Lean source)
lemma adaptiveCompactValues_mem_interior (θ : TrialParameter)
(hθ : InteriorMeans θ) :
θ ∈ interior (adaptiveCompactValues θ) := by
rw [mem_interior_iff_mem_nhds]
apply mem_of_superset
(Metric.ball_mem_nhds θ (half_pos (adaptiveInteriorMargin_pos θ hθ)))
exact Metric.ball_subset_closedBall
theorem adaptiveCompactValues_interiorMeans
the adaptive compact values interior means assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Compact Values interior Means.
Formal statement
hypotheses
θ :
hθ :
conclusion
Proof (Lean source)
lemma adaptiveCompactValues_interiorMeans (θ : TrialParameter)
(hθ : InteriorMeans θ) :
adaptiveCompactValues θ ⊆ adaptiveNeighborhood := by
intro η hη
have hmpos := adaptiveInteriorMargin_pos θ hθ
have hrnonneg : 0 ≤ adaptiveInteriorMargin θ / 2 := by positivity
have hd := (dist_pi_le_iff hrnonneg).mp (Metric.mem_closedBall.mp hη)
have hd0 := hd 0
have hd1 := hd 1
rw [Real.dist_eq] at hd0 hd1
have hm0l : adaptiveInteriorMargin θ ≤ θ 0 := by
exact (min_le_left _ _).trans (min_le_left _ _)
have hm0r : adaptiveInteriorMargin θ ≤ 1 - θ 0 := by
exact (min_le_left _ _).trans (min_le_right _ _)
have hm1l : adaptiveInteriorMargin θ ≤ θ 1 := by
exact (min_le_right _ _).trans (min_le_left _ _)
have hm1r : adaptiveInteriorMargin θ ≤ 1 - θ 1 := by
exact (min_le_right _ _).trans (min_le_right _ _)
change 0 < η 0 ∧ η 0 < 1 ∧ 0 < η 1 ∧ η 1 < 1
have habs0 := abs_le.mp hd0
have habs1 := abs_le.mp hd1
constructor
· linarith
constructor
· linarith
constructor <;> linarith
def adaptiveGoodPilot
For the supplied quantities and conditions, the adaptive good pilot is the mathematical object specified below. The adaptive Good Pilot is determined by the displayed parameters.
Definition (Lean source)
theorem adaptiveGoodPilot_measurable
Under the supplied quantities and conditions, the adaptive good pilot measurable assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
MeasurableSet (adaptiveGoodPilot θ p ε m n)
Proof (Lean source)
lemma adaptiveGoodPilot_measurable (θ : TrialParameter) (p ε : ℝ)
(m : ℕ → ℕ) (n : ℕ) :
MeasurableSet (adaptiveGoodPilot θ p ε m n) := by
change MeasurableSet
((adaptivePilotSelector p ε m n) ⁻¹' closedBall θ
(adaptiveInteriorMargin θ / 2))
exact Metric.isClosed_closedBall.measurableSet.preimage
(measurable_adaptivePilotSelector p ε m n)
theorem adaptiveGoodPilot_selector_mem
Under the supplied quantities and conditions, the adaptive good pilot selector mem assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
adaptivePilotSelector p ε m n u ∈ adaptiveCompactValues θ
Proof (Lean source)
lemma adaptiveGoodPilot_selector_mem (θ : TrialParameter) (p ε : ℝ)
(m : ℕ → ℕ) (n : ℕ) {u : AdaptivePilotSample m n}
(hu : u ∈ adaptiveGoodPilot θ p ε m n) :
adaptivePilotSelector p ε m n u ∈ adaptiveCompactValues θ := by
exact hu
theorem adaptiveScoreMoment_uniform_on_compact
Under the supplied quantities and conditions, the adaptive score moment uniform on compact assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Score Moment uniform on compact.
Formal statement
hypotheses
Proof (Lean source)
lemma adaptiveScoreMoment_uniform_on_compact (θ : TrialParameter) (p ε r : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hr : 0 < r) :
∃ C : ℝ, 0 ≤ C ∧ ∀ θ' η,
θ' ∈ adaptiveNeighborhood → η ∈ adaptiveCompactValues θ →
adaptiveScoreMoment select r θ' η p ε ≤ C := by
refine ⟨(pilotPhiUpper p ε) ^ r,
Real.rpow_nonneg (pilotPhiUpper_nonneg p ε hp hε) r, ?_⟩
intro θ' η hθ' hη
exact adaptiveScoreMoment_le select θ' η p ε r hselect hp hθ'
(adaptiveCompactValues_interiorMeans θ hθ hη) hε hr
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotAdaptiveRegularization 20 declarations Finite-row regularization for the adaptive pilot
Finite-row regularization for the adaptive pilot
def adaptivePilotSize
For the supplied quantities and conditions, the adaptive pilot size is the mathematical object specified below.
Definition (Lean source)
theorem adaptivePilotSize_le
Under the supplied quantities and conditions, the adaptive pilot size le assertion holds.
Formal statement
Proof (Lean source)
lemma adaptivePilotSize_le (m : ℕ → ℕ) (n : ℕ) :
adaptivePilotSize m n ≤ n := by
exact min_le_right _ _
theorem adaptivePilotSize_eq
Under the supplied quantities and conditions, the adaptive pilot size eq assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
adaptivePilotSize m n = m n
Proof (Lean source)
lemma adaptivePilotSize_eq (m : ℕ → ℕ) (hsub : PilotSublinear m)
{n : ℕ} (hn : 2 ≤ n) : adaptivePilotSize m n = m n := by
exact min_eq_left (hsub.2 n hn).2.le
theorem adaptivePilotSize_tendsto_atTop
Under the supplied quantities and conditions, the adaptive pilot size tendsto at top assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma adaptivePilotSize_tendsto_atTop (m : ℕ → ℕ)
(hdiv : PilotDiverges m) (hsub : PilotSublinear m) :
Tendsto (adaptivePilotSize m) atTop atTop := by
apply hdiv.congr'
filter_upwards [eventually_ge_atTop 2] with n hn
exact (adaptivePilotSize_eq m hsub hn).symm
theorem adaptivePilotSize_sublinear
Under the supplied quantities and conditions, the adaptive pilot size sublinear assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
m :
ℕ → ℕ
hsub :
Proof (Lean source)
lemma adaptivePilotSize_sublinear (m : ℕ → ℕ)
(hsub : PilotSublinear m) :
Tendsto (fun n => (adaptivePilotSize m n : ℝ) / n) atTop (nhds 0) := by
apply hsub.1.congr'
filter_upwards [eventually_ge_atTop 2] with n hn
rw [adaptivePilotSize_eq m hsub hn]
def adaptiveTrueParam
For the supplied quantities and conditions, the adaptive true param is the mathematical object specified below. The adaptive True Param is determined by the displayed parameters.
Definition (Lean source)
parameters
θ h :
n :
ℕ
def
given by
theorem adaptiveTrueParam_interior
Under the supplied quantities and conditions, the adaptive true param interior assertion holds. Under the stated assumptions, the adaptive True Param interior.
Formal statement
hypotheses
conclusion
InteriorMeans (adaptiveTrueParam θ h n)
Proof (Lean source)
lemma adaptiveTrueParam_interior (θ h : TrialParameter) (n : ℕ)
(hθ : InteriorMeans θ) : InteriorMeans (adaptiveTrueParam θ h n) := by
unfold adaptiveTrueParam
split_ifs with hlocal
· exact hlocal
· exact hθ
theorem adaptiveTrueParam_eq_localAlternative
Under the supplied quantities and conditions, the adaptive true param eq local alternative assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
adaptiveTrueParam θ h n = localAlternative θ h n
Proof (Lean source)
lemma adaptiveTrueParam_eq_localAlternative (θ h : TrialParameter) (n : ℕ)
(hlocal : InteriorMeans (localAlternative θ h n)) :
adaptiveTrueParam θ h n = localAlternative θ h n := by
simp [adaptiveTrueParam, hlocal]
theorem localAlternative_tendsto
Under the supplied quantities and conditions, the local alternative tendsto assertion holds.
Formal statement
Proof (Lean source)
lemma localAlternative_tendsto (θ h : TrialParameter) :
Tendsto (localAlternative θ h) atTop (nhds θ) := by
have hsqrt : Tendsto (fun n : ℕ => sqrt (n : ℝ)) atTop atTop :=
Real.tendsto_sqrt_atTop.comp tendsto_natCast_atTop_atTop
have hinv : Tendsto (fun n : ℕ => (sqrt (n : ℝ))⁻¹) atTop (nhds 0) :=
tendsto_inv_atTop_zero.comp hsqrt
rw [tendsto_pi_nhds]
intro k
have hpert : Tendsto (fun n : ℕ => h k * (sqrt (n : ℝ))⁻¹)
atTop (nhds 0) := by
convert (tendsto_const_nhds.mul hinv :
Tendsto (fun n : ℕ => h k * (sqrt (n : ℝ))⁻¹)
atTop (nhds (h k * 0))) using 1 <;> simp
simpa [localAlternative, div_eq_mul_inv] using tendsto_const_nhds.add hpert
theorem eventually_localAlternative_interior
Under the supplied quantities and conditions, the eventually local alternative interior assertion holds. Under the stated assumptions, the eventually local Alternative interior.
Formal statement
hypotheses
θ h :
hθ :
conclusion
Proof (Lean source)
lemma eventually_localAlternative_interior (θ h : TrialParameter)
(hθ : InteriorMeans θ) :
∀ᶠ n in atTop, InteriorMeans (localAlternative θ h n) := by
have ht := localAlternative_tendsto θ h
have h0 := tendsto_pi_nhds.1 ht 0
have h1 := tendsto_pi_nhds.1 ht 1
filter_upwards [
(tendsto_order.1 h0).1 0 hθ.1,
(tendsto_order.1 h0).2 1 hθ.2.1,
(tendsto_order.1 h1).1 0 hθ.2.2.1,
(tendsto_order.1 h1).2 1 hθ.2.2.2] with n hn0l hn0r hn1l hn1r
exact ⟨hn0l, hn0r, hn1l, hn1r⟩
theorem eventually_adaptiveTrueParam_eq
Under the supplied quantities and conditions, the eventually adaptive true param eq assertion holds. Under the stated assumptions, the eventually adaptive True Param eq.
Formal statement
hypotheses
θ h :
hθ :
conclusion
Proof (Lean source)
lemma eventually_adaptiveTrueParam_eq (θ h : TrialParameter)
(hθ : InteriorMeans θ) :
∀ᶠ n in atTop, adaptiveTrueParam θ h n = localAlternative θ h n := by
filter_upwards [eventually_localAlternative_interior θ h hθ] with n hn
exact adaptiveTrueParam_eq_localAlternative θ h n hn
theorem direction_coord_abs_le
the direction coord abs le assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
|h k| ≤ H
Proof (Lean source)
lemma direction_coord_abs_le {h : TrialParameter} {H : ℝ}
(hH : sqrt ((h 0) ^ 2 + (h 1) ^ 2) ≤ H) (k : Fin 2) :
|h k| ≤ H := by
have hsqrt : 0 ≤ sqrt ((h 0) ^ 2 + (h 1) ^ 2) := Real.sqrt_nonneg _
have hsum : 0 ≤ (h 0) ^ 2 + (h 1) ^ 2 := add_nonneg (sq_nonneg _) (sq_nonneg _)
fin_cases k
· have hsq : |h 0| ^ 2 ≤ (sqrt ((h 0) ^ 2 + (h 1) ^ 2)) ^ 2 := by
rw [sq_abs, Real.sq_sqrt hsum]
exact le_add_of_nonneg_right (sq_nonneg _)
exact ((sq_le_sq₀ (abs_nonneg _) hsqrt).mp hsq).trans hH
· have hsq : |h 1| ^ 2 ≤ (sqrt ((h 0) ^ 2 + (h 1) ^ 2)) ^ 2 := by
rw [sq_abs, Real.sq_sqrt hsum]
exact le_add_of_nonneg_left (sq_nonneg _)
exact ((sq_le_sq₀ (abs_nonneg _) hsqrt).mp hsq).trans hH
theorem eventually_localAlternative_dist_lt_uniform
Under the supplied quantities and conditions, the eventually local alternative dist lt uniform assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the eventually local Alternative dist lt uniform.
Formal statement
hypotheses
conclusion
≤ H → dist (localAlternative θ h n) θ < δ
Proof (Lean source)
lemma eventually_localAlternative_dist_lt_uniform (θ : TrialParameter) (H δ : ℝ)
(hδ : 0 < δ) :
∀ᶠ n in atTop, ∀ h : TrialParameter,
sqrt ((h 0) ^ 2 + (h 1) ^ 2) ≤ H →
dist (localAlternative θ h n) θ < δ := by
have hsqrt : Tendsto (fun n : ℕ => sqrt (n : ℝ)) atTop atTop :=
Real.tendsto_sqrt_atTop.comp tendsto_natCast_atTop_atTop
have hev := hsqrt.eventually_gt_atTop (max 0 (H / δ))
filter_upwards [hev] with n hn h hH
have hsqrtpos : 0 < sqrt (n : ℝ) :=
lt_of_le_of_lt (le_max_left 0 (H / δ)) hn
rw [dist_pi_lt_iff hδ]
intro k
rw [Real.dist_eq]
change |θ k + h k / sqrt (n : ℝ) - θ k| < δ
rw [add_sub_cancel_left, abs_div, abs_of_pos hsqrtpos]
have hk := direction_coord_abs_le hH k
have hratio : H / δ < sqrt (n : ℝ) :=
lt_of_le_of_lt (le_max_right 0 (H / δ)) hn
calc
|h k| / sqrt (n : ℝ) ≤ H / sqrt (n : ℝ) := by
exact div_le_div_of_nonneg_right hk hsqrtpos.le
_ < δ := by
apply (div_lt_iff₀ hsqrtpos).2
have hmul := (div_lt_iff₀ hδ).1 hratio
nlinarith
theorem eventually_localAlternative_interior_uniform
Under the supplied quantities and conditions, the eventually local alternative interior uniform assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the eventually local Alternative interior uniform.
Formal statement
hypotheses
conclusion
≤ H → InteriorMeans (localAlternative θ h n)
Proof (Lean source)
lemma eventually_localAlternative_interior_uniform (θ : TrialParameter) (H : ℝ)
(hθ : InteriorMeans θ) :
∀ᶠ n in atTop, ∀ h : TrialParameter,
sqrt ((h 0) ^ 2 + (h 1) ^ 2) ≤ H →
InteriorMeans (localAlternative θ h n) := by
let δ := min (min (θ 0) (1 - θ 0)) (min (θ 1) (1 - θ 1))
have hδ : 0 < δ := by
simp only [δ, lt_min_iff]
exact ⟨⟨hθ.1, sub_pos.mpr hθ.2.1⟩, ⟨hθ.2.2.1, sub_pos.mpr hθ.2.2.2⟩⟩
filter_upwards [eventually_localAlternative_dist_lt_uniform θ H δ hδ] with n hn
intro h hH
have hd := (dist_pi_lt_iff hδ).mp (hn h hH)
have hd0 := hd 0
have hd1 := hd 1
rw [Real.dist_eq] at hd0 hd1
dsimp [localAlternative] at hd0 hd1 ⊢
have hδ0 : δ ≤ θ 0 ∧ δ ≤ 1 - θ 0 := by
dsimp [δ]
exact ⟨(min_le_left _ _).trans (min_le_left _ _),
(min_le_left _ _).trans (min_le_right _ _)⟩
have hδ1 : δ ≤ θ 1 ∧ δ ≤ 1 - θ 1 := by
dsimp [δ]
exact ⟨(min_le_right _ _).trans (min_le_left _ _),
(min_le_right _ _).trans (min_le_right _ _)⟩
constructor
· change 0 < θ 0 + h 0 / sqrt (n : ℝ)
linarith [(abs_lt.mp hd0).1]
constructor
· change θ 0 + h 0 / sqrt (n : ℝ) < 1
linarith [(abs_lt.mp hd0).2]
constructor
· change 0 < θ 1 + h 1 / sqrt (n : ℝ)
linarith [(abs_lt.mp hd1).1]
· change θ 1 + h 1 / sqrt (n : ℝ) < 1
linarith [(abs_lt.mp hd1).2]
theorem eventually_adaptiveTrueParam_eq_uniform
Under the supplied quantities and conditions, the eventually adaptive true param eq uniform assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the eventually adaptive True Param eq uniform.
Formal statement
hypotheses
conclusion
≤ H → adaptiveTrueParam θ h n = localAlternative θ h n
Proof (Lean source)
lemma eventually_adaptiveTrueParam_eq_uniform (θ : TrialParameter) (H : ℝ)
(hθ : InteriorMeans θ) :
∀ᶠ n in atTop, ∀ h : TrialParameter,
sqrt ((h 0) ^ 2 + (h 1) ^ 2) ≤ H →
adaptiveTrueParam θ h n = localAlternative θ h n := by
filter_upwards [eventually_localAlternative_interior_uniform θ H hθ] with n hn
intro h hH
exact adaptiveTrueParam_eq_localAlternative θ h n (hn h hH)
theorem eventually_adaptiveTrueParam_dist_lt_uniform
Under the supplied quantities and conditions, the eventually adaptive true param dist lt uniform assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the eventually adaptive True Param dist lt uniform.
Formal statement
hypotheses
conclusion
≤ H → dist (adaptiveTrueParam θ h n) θ < δ
Proof (Lean source)
lemma eventually_adaptiveTrueParam_dist_lt_uniform (θ : TrialParameter) (H δ : ℝ)
(hθ : InteriorMeans θ) (hδ : 0 < δ) :
∀ᶠ n in atTop, ∀ h : TrialParameter,
sqrt ((h 0) ^ 2 + (h 1) ^ 2) ≤ H →
dist (adaptiveTrueParam θ h n) θ < δ := by
filter_upwards [eventually_adaptiveTrueParam_eq_uniform θ H hθ,
eventually_localAlternative_dist_lt_uniform θ H δ hδ] with n heq hdist
intro h hH
rw [heq h hH]
exact hdist h hH
theorem eventually_adaptiveTrueParam_mem_open_uniform
Under the supplied quantities and conditions, the eventually adaptive true param mem open uniform assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the eventually adaptive True Param mem open uniform.
Formal statement
hypotheses
conclusion
≤ H → adaptiveTrueParam θ h n ∈ N
Proof (Lean source)
lemma eventually_adaptiveTrueParam_mem_open_uniform
(θ : TrialParameter) (H : ℝ) (N : Set TrialParameter)
(hθ : InteriorMeans θ) (hNopen : IsOpen N) (hθN : θ ∈ N) :
∀ᶠ n in atTop, ∀ h : TrialParameter,
sqrt ((h 0) ^ 2 + (h 1) ^ 2) ≤ H →
adaptiveTrueParam θ h n ∈ N := by
obtain ⟨δ, hδ, hball⟩ := Metric.isOpen_iff.mp hNopen θ hθN
filter_upwards [eventually_adaptiveTrueParam_dist_lt_uniform θ H δ hθ hδ] with n hn
intro h hH
exact hball (by simpa [Metric.mem_ball, dist_comm] using hn h hH)
def adaptivePilotPrefixMass
For the supplied quantities and conditions, the adaptive pilot prefix mass is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The adaptive Pilot Prefix Mass is determined by the displayed parameters.
Definition (Lean source)
parameters
def
adaptivePilotPrefixMass θ h p ε m n u :
ℝ
given by
∏ j : Fin (adaptivePilotSize m n), ∑ a : Fin 4, piTheta (adaptiveTrueParam θ h n) p a * rrPilotProbability ε a (u j)
theorem adaptivePilotPrefixMass_nonneg
Under the supplied quantities and conditions, the adaptive pilot prefix mass nonneg assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Prefix Mass nonneg.
Formal statement
hypotheses
θ h :
p ε :
ℝ
m :
ℕ → ℕ
n :
ℕ
hp :
hθ :
u :
conclusion
0 ≤ adaptivePilotPrefixMass θ h p ε m n u
Proof (Lean source)
lemma adaptivePilotPrefixMass_nonneg (θ h : TrialParameter) (p ε : ℝ)
(m : ℕ → ℕ) (n : ℕ) (hp : InteriorAssignment p)
(hθ : InteriorMeans θ) (u : Fin (adaptivePilotSize m n) → Fin 4) :
0 ≤ adaptivePilotPrefixMass θ h p ε m n u := by
unfold adaptivePilotPrefixMass
apply Finset.prod_nonneg
intro j _
apply Finset.sum_nonneg
intro a _
exact mul_nonneg
(piTheta_pos_interior (adaptiveTrueParam θ h n) p hp
(adaptiveTrueParam_interior θ h n hθ) a).le
(by
unfold rrPilotProbability
split_ifs <;> positivity)
theorem adaptivePilotPrefixMass_sum
Under the supplied quantities and conditions, the adaptive pilot prefix mass sum assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma adaptivePilotPrefixMass_sum (θ h : TrialParameter) (p ε : ℝ)
(m : ℕ → ℕ) (n : ℕ) :
∑ u : Fin (adaptivePilotSize m n) → Fin 4,
adaptivePilotPrefixMass θ h p ε m n u = 1 := by
unfold adaptivePilotPrefixMass
calc
(∑ u : Fin (adaptivePilotSize m n) → Fin 4,
∏ j : Fin (adaptivePilotSize m n),
∑ a : Fin 4, piTheta (adaptiveTrueParam θ h n) p a *
rrPilotProbability ε a (u j)) =
∏ _j : Fin (adaptivePilotSize m n), ∑ k : Fin 4,
∑ a : Fin 4, piTheta (adaptiveTrueParam θ h n) p a *
rrPilotProbability ε a k :=
(Fintype.prod_sum (fun _j : Fin (adaptivePilotSize m n) => fun k : Fin 4 =>
∑ a : Fin 4, piTheta (adaptiveTrueParam θ h n) p a *
rrPilotProbability ε a k)).symm
_ = 1 := by simp [rrPilot_marginal_sum_eq_one]
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotAdaptiveTransfer 6 declarations Finite pilot transfer lemmas for local limits and risks
Finite pilot transfer lemmas for local limits and risks
theorem finitePilot_conditionalCDF_tendsto
the finite pilot conditional cdf tendsto assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
P :
ℕ → Type*
∀ n, Fintype (P n)
w :
(n : ℕ) → P n → ℝ
F :
(n : ℕ) → P n → ℝ
L :
ℝ
hw :
∀ n a, 0 ≤ w n a
hsum :
∀ n, ∑ a, w n a = 1
hF0 :
∀ n a, 0 ≤ F n a
hF1 :
∀ n a, F n a ≤ 1
hL0 :
0 ≤ L
hL1 :
L ≤ 1
Proof (Lean source)
lemma finitePilot_conditionalCDF_tendsto
{P : ℕ → Type*} [∀ n, Fintype (P n)]
(w : (n : ℕ) → P n → ℝ) (F : (n : ℕ) → P n → ℝ) (L : ℝ)
(hw : ∀ n a, 0 ≤ w n a) (hsum : ∀ n, ∑ a, w n a = 1)
(hF0 : ∀ n a, 0 ≤ F n a) (hF1 : ∀ n a, F n a ≤ 1)
(hL0 : 0 ≤ L) (hL1 : L ≤ 1)
(hbad : ∀ δ : ℝ, 0 < δ → Tendsto (fun n =>
∑ a, w n a * if δ ≤ |F n a - L| then 1 else 0) atTop (𝓝 0)) :
Tendsto (fun n => ∑ a, w n a * F n a) atTop (𝓝 L) := by
apply finitePilot_average_tendsto_of_badMass w F L hw hsum
· intro n a
rw [abs_le]
exact ⟨by linarith [hF0 n a], hF1 n a⟩
· rw [abs_le]
exact ⟨by linarith, hL1⟩
· exact hbad
theorem finitePilot_good_bad_average_error_le
the finite pilot good bad average error le assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the finite Pilot good bad average error le.
Formal statement
hypotheses
P :
Type*
Fintype P
w g :
P → ℝ
good :
P → Prop
DecidablePred good
L δ :
ℝ
hw :
∀ a, 0 ≤ w a
hsum :
∑ a, w a = 1
hL :
0 ≤ L
hδ :
0 ≤ δ
hgood :
∀ a
if
good a
then
|g a - L| ≤ δ
conclusion
|∑ a, w a * g a - L| ≤ δ + ∑ a, w a * if good a then 0 else |g a - L|
Proof (Lean source)
lemma finitePilot_good_bad_average_error_le {P : Type*} [Fintype P]
(w g : P → ℝ) (good : P → Prop) [DecidablePred good]
(L δ : ℝ) (hw : ∀ a, 0 ≤ w a) (hsum : ∑ a, w a = 1)
(hL : 0 ≤ L) (hδ : 0 ≤ δ)
(hgood : ∀ a, good a → |g a - L| ≤ δ) :
|∑ a, w a * g a - L| ≤
δ + ∑ a, w a * if good a then 0 else |g a - L| := by
have hpoint (a : P) :
w a * |g a - L| ≤
w a * (δ + if good a then 0 else |g a - L|) := by
apply mul_le_mul_of_nonneg_left _ (hw a)
split_ifs with ha
· simpa using hgood a ha
· linarith [abs_nonneg (g a - L)]
calc
|∑ a, w a * g a - L| = |∑ a, w a * (g a - L)| := by
congr 1
calc
∑ a, w a * g a - L = ∑ a, w a * g a - L * ∑ a, w a := by
rw [hsum]
ring
_ = ∑ a, w a * (g a - L) := by
rw [Finset.mul_sum, ← Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro a _
ring
_ ≤ ∑ a, |w a * (g a - L)| := Finset.abs_sum_le_sum_abs _ _
_ = ∑ a, w a * |g a - L| := by
apply Finset.sum_congr rfl
intro a _
rw [abs_mul, abs_of_nonneg (hw a)]
_ ≤ ∑ a, w a * (δ + if good a then 0 else |g a - L|) :=
Finset.sum_le_sum fun a _ => hpoint a
_ = δ + ∑ a, w a * if good a then 0 else |g a - L| := by
simp_rw [mul_add]
rw [Finset.sum_add_distrib, ← Finset.sum_mul, hsum]
ring
theorem finitePilot_good_bad_average_tendsto
the finite pilot good bad average tendsto assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
P :
ℕ → Type*
∀ n, Fintype (P n)
w g :
(n : ℕ) → P n → ℝ
good :
(n : ℕ) → P n → Prop
∀ n, DecidablePred (good n)
L :
ℝ
hw :
∀ n a, 0 ≤ w n a
hsum :
∀ n, ∑ a, w n a = 1
hL :
0 ≤ L
hgood :
∀ δ : ℝ, 0 < δ → ∀ᶠ n in atTop, ∀ a, good n a → |g n a - L| ≤ δ
Proof (Lean source)
lemma finitePilot_good_bad_average_tendsto
{P : ℕ → Type*} [∀ n, Fintype (P n)]
(w g : (n : ℕ) → P n → ℝ) (good : (n : ℕ) → P n → Prop)
[∀ n, DecidablePred (good n)] (L : ℝ)
(hw : ∀ n a, 0 ≤ w n a) (hsum : ∀ n, ∑ a, w n a = 1)
(hL : 0 ≤ L)
(hgood : ∀ δ : ℝ, 0 < δ → ∀ᶠ n in atTop,
∀ a, good n a → |g n a - L| ≤ δ)
(hbad : Tendsto (fun n =>
∑ a, w n a * if good n a then 0 else |g n a - L|) atTop (𝓝 0)) :
Tendsto (fun n => ∑ a, w n a * g n a) atTop (𝓝 L) := by
rw [Metric.tendsto_atTop]
intro η hη
have hgoodη := hgood (η / 2) (half_pos hη)
have hbadη : ∀ᶠ n in atTop,
(∑ a, w n a * if good n a then 0 else |g n a - L|) < η / 2 :=
(tendsto_order.1 hbad).2 _ (half_pos hη)
have hevent := hgoodη.and hbadη
rcases (eventually_atTop.1 hevent) with ⟨N, hN⟩
exact ⟨N, fun n hn => by
rw [Real.dist_eq]
have hpair := hN n hn
have hbound := finitePilot_good_bad_average_error_le
(w n) (g n) (good n) L (η / 2) (hw n) (hsum n) hL
(by positivity) hpair.1
exact lt_of_le_of_lt hbound (by linarith [hpair.2])⟩
theorem adaptive_mainScale_tendsto_one
Under the supplied quantities and conditions, the adaptive main scale tendsto one assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
m :
ℕ → ℕ
hsub :
Proof (Lean source)
lemma adaptive_mainScale_tendsto_one (m : ℕ → ℕ)
(hsub : PilotSublinear m) :
Tendsto (fun n : ℕ => (n : ℝ) / (adaptiveMainSize m n : ℝ))
atTop (𝓝 1) := by
have hratio := adaptiveMainSize_ratio_tendsto_one m hsub
simpa only [inv_div, inv_one] using hratio.inv₀ one_ne_zero
theorem adaptive_mainScale_mul_tendsto
the adaptive main scale mul tendsto assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Proof (Lean source)
lemma adaptive_mainScale_mul_tendsto {m : ℕ → ℕ}
(hsub : PilotSublinear m) {a : ℕ → ℝ} {L : ℝ}
(ha : Tendsto a atTop (𝓝 L)) :
Tendsto (fun n : ℕ => ((n : ℝ) / (adaptiveMainSize m n : ℝ)) * a n)
atTop (𝓝 L) := by
simpa using (adaptive_mainScale_tendsto_one m hsub).mul ha
theorem adaptive_sqrtScale_mul_tendsto
the adaptive sqrt scale mul tendsto assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma adaptive_sqrtScale_mul_tendsto {m : ℕ → ℕ}
(hsub : PilotSublinear m) {a : ℕ → ℝ} {L : ℝ}
(ha : Tendsto a atTop (𝓝 L)) :
Tendsto (fun n : ℕ =>
sqrt ((n : ℝ) / (adaptiveMainSize m n : ℝ)) * a n)
atTop (𝓝 L) := by
simpa using (adaptive_row_main_sqrt_ratio_tendsto_one m hsub).mul ha
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotAttainmentSupport 23 declarations This file supplies the finite pilot-prefix integral decomposition and the outer asymptotic packaging used by the private-pilot theorem.
Assembly lemmas for private-pilot attainment
This file supplies the finite pilot-prefix integral decomposition and the outer asymptotic packaging used by the private-pilot theorem. It deliberately does not depend on the generic triangular-array limit theorem.
def pilotObservedPrefix
The observed coordinates lying in the pilot portion of a transcript. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
given by
fun i => z i.1
theorem finiteProduct_weighted_sum_expectation
Under a finite product law with normalized one-coordinate weights, the weighted expectation of a coordinate sum is the number of coordinates times the one-coordinate weighted expectation. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma finiteProduct_weighted_sum_expectation {A : Type*} [Fintype A]
(N : ℕ) (w q : A → ℝ) (hw : ∑ a, w a = 1) :
∑ v : Fin N → A, (∏ j, w (v j)) * (∑ j, q (v j)) =
(N : ℝ) * (∑ a, w a * q a) := by
classical
rw [show (∑ v : Fin N → A, (∏ j, w (v j)) * (∑ j, q (v j))) =
∑ v : Fin N → A, ∑ j, (∏ k, w (v k)) * q (v j) by
apply Finset.sum_congr rfl
intro v _
exact Finset.mul_sum _ _ _]
rw [Finset.sum_comm]
calc
(∑ j : Fin N, ∑ v : Fin N → A, (∏ k, w (v k)) * q (v j)) =
∑ j : Fin N, ∑ a, w a * q a := by
apply Finset.sum_congr rfl
intro j _
let F : Fin N → A → ℝ := fun k a =>
w a * if k = j then q a else 1
calc
(∑ v : Fin N → A, (∏ k, w (v k)) * q (v j)) =
∑ v : Fin N → A, ∏ k, F k (v k) := by
apply Finset.sum_congr rfl
intro v _
rw [show (∏ k, F k (v k)) =
(∏ k, w (v k)) * q (v j) by
unfold F
rw [Finset.prod_mul_distrib]
simp]
_ = ∏ k : Fin N, ∑ a, F k a := (Fintype.prod_sum F).symm
_ = ∑ a, w a * q a := by
rw [Finset.prod_eq_single j]
· simp [F]
· intro k _ hkj
simp [F, hkj, hw]
· simp
_ = (N : ℝ) * (∑ a, w a * q a) := by
simp [Finset.sum_const, nsmul_eq_mul]
theorem adaptiveMain_centered_product_sum
For a fixed pilot selection and a nonempty main row, the product-law expectation of the corrected main-sample estimator is centered at the true contrast. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Main centered product sum.
Formal statement
hypotheses
conclusion
∑ v : Fin N → Fin 14, (∏ j, adaptiveMainMass select θ η p ε (v j)) * (contrast η + (N : ℝ)⁻¹ * ∑ j, adaptiveSelectedScore select η p ε (v j) - contrast θ)
= 0
Proof (Lean source)
lemma adaptiveMain_centered_product_sum
(θ η : TrialParameter) (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hη : InteriorMeans η) (hε : 0 < ε)
(N : ℕ) (hN : 0 < N) :
∑ v : Fin N → Fin 14,
(∏ j, adaptiveMainMass select θ η p ε (v j)) *
(contrast η + (N : ℝ)⁻¹ *
∑ j, adaptiveSelectedScore select η p ε (v j) - contrast θ) = 0 := by
let w : Fin 14 → ℝ := adaptiveMainMass select θ η p ε
let q : Fin 14 → ℝ := adaptiveSelectedScore select η p ε
have hw : ∑ s, w s = 1 := adaptiveMainMass_sum select θ η p ε hselect hp hη hε
have hq : ∑ s, w s * q s = contrast θ - contrast η :=
adaptiveSelectedScore_mean select θ η p ε hselect hp hη hε
have hprod : ∑ v : Fin N → Fin 14, ∏ j, w (v j) = 1 := by
rw [← Fintype.prod_sum]
simp [hw]
have hsum := finiteProduct_weighted_sum_expectation N w q hw
change (∑ v : Fin N → Fin 14, (∏ j, w (v j)) *
(contrast η + (N : ℝ)⁻¹ * ∑ j, q (v j) - contrast θ)) = 0
calc
_ = (contrast η - contrast θ) * (∑ v : Fin N → Fin 14, ∏ j, w (v j)) +
(N : ℝ)⁻¹ *
(∑ v : Fin N → Fin 14, (∏ j, w (v j)) * ∑ j, q (v j)) := by
rw [Finset.mul_sum, Finset.mul_sum]
rw [← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro v _
ring
_ = (contrast η - contrast θ) +
(N : ℝ)⁻¹ * ((N : ℝ) * (contrast θ - contrast η)) := by
rw [hprod, hsum, hq]
ring
_ = 0 := by
have hNr : (N : ℝ) ≠ 0 := by exact_mod_cast (ne_of_gt hN)
field_simp
ring
theorem fin_sum_split
Split a finite sum at a bounded initial segment. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma fin_sum_split {M : Type*} [AddCommMonoid M] {m n : ℕ} (hmn : m ≤ n)
(f : Fin m → M) (g : Fin (n - m) → M) :
(∑ i : Fin n, if hi : i.val < m then f ⟨i.val, hi⟩
else g ⟨i.val - m, by omega⟩) =
(∑ j : Fin m, f j) + ∑ j : Fin (n - m), g j := by
let e : Fin (m + (n - m)) ≃ Fin n := Fin.castOrderIso (Nat.add_sub_of_le hmn)
rw [← e.sum_comp]
rw [Fin.sum_univ_add]
congr 1
· apply Finset.sum_congr rfl
intro j _
simp [e]
· apply Finset.sum_congr rfl
intro j _
simp [e]
theorem tauStar_pilotAdaptiveEncode
On an encoded pilot/main transcript, tauStar is the pilot contrast plus the average of the selected main-release scores. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the tau Star pilot Adaptive Encode.
Formal statement
hypotheses
p ε :
ℝ
m :
ℕ → ℕ
hp :
hε :
0 < ε
n :
ℕ
hmn :
m n ≤ n
conclusion
tauStar p ε m (select) (pilotAdaptiveEncode hmn u v)
= contrast (pilotAdaptivePrefixTheta p ε m hmn u)
+ ((n - m n : ℕ) : ℝ)⁻¹ * ∑ j, adaptiveSelectedScore select (pilotAdaptivePrefixTheta p ε m hmn u) p ε (v j)
Proof (Lean source)
lemma tauStar_pilotAdaptiveEncode
(p ε : ℝ) (m : ℕ → ℕ) (hp : InteriorAssignment p) (hε : 0 < ε)
{n : ℕ} (hmn : m n ≤ n) (u : Fin (m n) → Fin 4)
(v : Fin (n - m n) → Fin 14) :
tauStar p ε m (select) (pilotAdaptiveEncode hmn u v) =
contrast (pilotAdaptivePrefixTheta p ε m hmn u) +
((n - m n : ℕ) : ℝ)⁻¹ *
∑ j, adaptiveSelectedScore select
(pilotAdaptivePrefixTheta p ε m hmn u) p ε (v j) := by
unfold tauStar
rw [pilotTheta_pilotAdaptiveEncode p ε m hmn u v]
dsimp only
congr 1
rw [show (∑ i : Fin n,
if m n ≤ i.val then
phiOutput (pilotAdaptivePrefixTheta p ε m hmn u) p ε
(select) (pilotAdaptiveEncode hmn u v i)
else 0) =
∑ i : Fin n, if hi : i.val < m n then (0 : ℝ)
else adaptiveSelectedScore select (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v ⟨i.val - m n, by omega⟩) by
apply Finset.sum_congr rfl
intro i _
by_cases hi : i.val < m n
· simp [hi]
· have him : m n ≤ i.val := Nat.le_of_not_gt hi
simp [hi, him, pilotAdaptiveEncode, phiOutput, adaptiveSelectedScore]]
rw [fin_sum_split hmn (fun _ : Fin (m n) => (0 : ℝ))
(fun j : Fin (n - m n) => adaptiveSelectedScore select
(pilotAdaptivePrefixTheta p ε m hmn u) p ε (v j))]
simp
theorem pilotAdaptive_encoded_centered_sum
Conditional on a proper pilot prefix, the exact joint-law atom masses center the encoded main-row estimator at the true contrast. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Adaptive encoded centered sum.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hsub :
n :
ℕ
hn :
2 ≤ n
conclusion
∑ v : Fin (n - m n) → Fin 14, (transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n {pilotAdaptiveEncode (le_of_lt (hsub.2 n hn).2) u v}).toReal * ((pilotEstimator p ε m select hselect hp hε).estimate n (pilotAdaptiveEncode (le_of_lt (hsub.2 n hn).2) u v) - contrast θ)
= 0
Proof (Lean source)
lemma pilotAdaptive_encoded_centered_sum
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hsub : PilotSublinear m) (n : ℕ) (hn : 2 ≤ n)
(u : Fin (m n) → Fin 4) :
∑ v : Fin (n - m n) → Fin 14,
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n
{pilotAdaptiveEncode (le_of_lt (hsub.2 n hn).2) u v}).toReal *
((pilotEstimator p ε m select hselect hp hε).estimate n
(pilotAdaptiveEncode (le_of_lt (hsub.2 n hn).2) u v) - contrast θ) = 0 := by
let hmn : m n ≤ n := le_of_lt (hsub.2 n hn).2
let η := pilotAdaptivePrefixTheta p ε m hmn u
have hη : InteriorMeans η := by
exact pilotTheta_interior p ε m
(pilotAdaptiveEncode hmn u (fun _ => 0))
have hN : 0 < n - m n := Nat.sub_pos_of_lt (hsub.2 n hn).2
let a : ℝ := ∏ j : Fin (m n),
∑ x : Fin 4, piTheta θ p x * rrPilotProbability ε x (u j)
have hpilot_nonneg (j : Fin (m n)) :
0 ≤ ∑ x : Fin 4, piTheta θ p x * rrPilotProbability ε x (u j) := by
apply Finset.sum_nonneg
intro x _
exact mul_nonneg (piTheta_pos_interior θ p hp hθ x).le (by
unfold rrPilotProbability
split_ifs <;> positivity)
have hmain_nonneg (s : Fin 14) :
0 ≤ adaptiveMainMass select θ η p ε s :=
adaptiveMainMass_nonneg select θ η p ε hselect hp hθ hη hε s
have hmass (v : Fin (n - m n) → Fin 14) :
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n
{pilotAdaptiveEncode hmn u v}).toReal =
a * ∏ j, adaptiveMainMass select θ η p ε (v j) := by
have hpProd : 0 ≤ ∏ j : Fin (m n),
∑ x : Fin 4, piTheta θ p x * rrPilotProbability ε x (u j) :=
Finset.prod_nonneg fun j _ => hpilot_nonneg j
have hmProd : 0 ≤ ∏ j : Fin (n - m n),
adaptiveMainMass select θ η p ε (v j) :=
Finset.prod_nonneg fun j _ => hmain_nonneg (v j)
rw [pilotAdaptive_joint_singleton select θ p ε m hselect hp hθ hε hmn u v]
rw [ENNReal.toReal_mul]
rw [← ENNReal.ofReal_prod_of_nonneg (fun j _ => hpilot_nonneg j)]
rw [← ENNReal.ofReal_prod_of_nonneg (fun j _ => hmain_nonneg (v j))]
rw [ENNReal.toReal_ofReal hpProd, ENNReal.toReal_ofReal hmProd]
change (∑ v : Fin (n - m n) → Fin 14,
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n
{pilotAdaptiveEncode hmn u v}).toReal *
(tauStar p ε m (select) (pilotAdaptiveEncode hmn u v) -
contrast θ)) = 0
simp_rw [hmass, tauStar_pilotAdaptiveEncode select p ε m hp hε hmn u]
change (∑ x : Fin (n - m n) → Fin 14,
(a * ∏ j, adaptiveMainMass select θ η p ε (x j)) *
(contrast η + ((n - m n : ℕ) : ℝ)⁻¹ *
∑ j, adaptiveSelectedScore select η p ε (x j) - contrast θ)) = 0
rw [show (∑ x : Fin (n - m n) → Fin 14,
(a * ∏ j, adaptiveMainMass select θ η p ε (x j)) *
(contrast η + ((n - m n : ℕ) : ℝ)⁻¹ *
∑ j, adaptiveSelectedScore select η p ε (x j) - contrast θ)) =
a * ∑ x : Fin (n - m n) → Fin 14,
(∏ j, adaptiveMainMass select θ η p ε (x j)) *
(contrast η + ((n - m n : ℕ) : ℝ)⁻¹ *
∑ j, adaptiveSelectedScore select η p ε (x j) - contrast θ) by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro x _
ring]
rw [adaptiveMain_centered_product_sum select θ η p ε hselect hp hη hε (n - m n) hN]
simp
theorem pilotAdaptive_singleton_zero_of_not_range
A transcript outside the correctly tagged pilot/main encoding has zero mass under the exact private-pilot law. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Adaptive singleton zero of not range.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
hmn :
m n ≤ n
t :
Transcript (pilotOutputFamily n)
ht :
=> pilotAdaptiveEncode hmn uv.1 uv.2
conclusion
transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n {t} = 0
Proof (Lean source)
lemma pilotAdaptive_singleton_zero_of_not_range
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε)
{n : ℕ} (hmn : m n ≤ n) (t : Transcript (pilotOutputFamily n))
(ht : t ∉ Set.range fun uv :
(Fin (m n) → Fin 4) × (Fin (n - m n) → Fin 14) =>
pilotAdaptiveEncode hmn uv.1 uv.2) :
transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n {t} = 0 := by
have hwrong :
(∃ i : Fin n, i.val < m n ∧ ∃ s : Fin 14, t i = inr s) ∨
(∃ i : Fin n, m n ≤ i.val ∧ ∃ k : Fin 4, t i = inl k) := by
by_contra hnone
push Not at hnone
let u : Fin (m n) → Fin 4 := fun j =>
elim id (fun _ => 0) (t (castLE hmn j) : PilotOutput)
let v : Fin (n - m n) → Fin 14 := fun j =>
elim (fun _ => 0) id
(t ⟨m n + j.val, by omega⟩ : PilotOutput)
apply ht
refine ⟨(u, v), ?_⟩
funext i
by_cases hi : i.val < m n
· cases hti : t i with
| inl k => simp [pilotAdaptiveEncode, hi, u, hti]
| inr s => exact False.elim (hnone.1 i hi s hti)
· have him : m n ≤ i.val := Nat.le_of_not_gt hi
have hjlt : i.val - m n < n - m n := by omega
have hidx : (⟨m n + (⟨i.val - m n, hjlt⟩ : Fin (n - m n)).val,
by omega⟩ : Fin n) = i := Fin.ext (Nat.add_sub_of_le him)
cases hti : t i with
| inl k => exact False.elim (hnone.2 i him k hti)
| inr s =>
have htidx : t ⟨m n + (⟨i.val - m n, hjlt⟩ : Fin (n - m n)).val,
by omega⟩ = inr s := by rw [hidx]; exact hti
simp [pilotAdaptiveEncode, hi, v, htidx]
unfold transcriptLaw
simp only [Measure.coe_finsetSum, Finset.sum_apply, Measure.smul_apply,
smul_eq_mul]
apply Finset.sum_eq_zero
intro x _
suffices (pilotEstimator p ε m select hselect hp hε).transcript n x {t} = 0 by simp [this]
change finiteSequenceTranscript
(pilotProtocol p ε m (select) n) x {t} = 0
rw [finiteSequenceTranscript_singleton_apply]
· rcases hwrong with ⟨i, hi, s, hs⟩ | ⟨i, hi, k, hk⟩
· apply Finset.prod_eq_zero (Finset.mem_univ i)
rw [hs]
exact pilotProtocol_pilot_wrongTag select p ε m hselect hp hε i hi (x i)
(fun j => t j.1) s
· apply Finset.prod_eq_zero (Finset.mem_univ i)
rw [hk]
exact pilotProtocol_main_wrongTag select p ε m hselect hp hε i hi (x i)
(fun j => t j.1) k
· exact (pilotProtocol_private p ε m select hselect hp hε n).1
theorem pilotAdaptiveEncode_pair_injective
The joint pilot/main transcript encoding is injective. For the displayed inputs and conditions, the stated result follows.
Formal statement
Proof (Lean source)
lemma pilotAdaptiveEncode_pair_injective {m n : ℕ} (hmn : m ≤ n) :
Injective (fun uv : (Fin m → Fin 4) × (Fin (n - m) → Fin 14) =>
pilotAdaptiveEncode hmn uv.1 uv.2) := by
intro uv uv' h
apply Prod.ext
· funext j
have hj := congrFun h (castLE hmn j)
dsimp only at hj
rw [pilotAdaptiveEncode_pilot hmn uv.1 uv.2 j,
pilotAdaptiveEncode_pilot hmn uv'.1 uv'.2 j] at hj
exact Sum.inl_injective hj
· funext j
have hj := congrFun h (⟨m + j.val, by omega⟩ : Fin n)
dsimp only at hj
rw [pilotAdaptiveEncode_main hmn uv.1 uv.2 j,
pilotAdaptiveEncode_main hmn uv'.1 uv'.2 j] at hj
exact Sum.inr_injective hj
theorem sum_eq_sum_image_of_zero_off_range
A finite sum supported on an injectively parameterized range can be reindexed by the parameters. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
∑ x : Ω, F x = ∑ a : A, F (e a)
Proof (Lean source)
lemma sum_eq_sum_image_of_zero_off_range
{A Ω M : Type*} [Fintype A] [Fintype Ω] [AddCommMonoid M]
(e : A → Ω) (he : Injective e) (F : Ω → M)
(hzero : ∀ x, x ∉ range e → F x = 0) :
∑ x : Ω, F x = ∑ a : A, F (e a) := by
classical
rw [← Finset.sum_image (fun a _ b _ hab => he hab)]
symm
apply Finset.sum_subset (by simp)
intro x _ hx
apply hzero x
simpa using hx
theorem pilot_prefix_cell_centered
The centered private-pilot estimator has integral zero on every observed pilot-prefix fiber. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot prefix cell centered.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hsub :
n :
ℕ
hn :
2 ≤ n
r :
{i : Fin n // i.val < m n} → PilotOutput
conclusion
(∫ z in (pilotObservedPrefix (n := n) m) ⁻¹' {r}, ((pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ) ∂transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
= 0
Proof (Lean source)
lemma pilot_prefix_cell_centered
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hsub : PilotSublinear m) (n : ℕ) (hn : 2 ≤ n)
(r : {i : Fin n // i.val < m n} → PilotOutput) :
(∫ z in (pilotObservedPrefix (n := n) m) ⁻¹' {r},
((pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ)
∂transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) = 0 := by
classical
let μ := transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n
let H := pilotObservedPrefix (n := n) m
let g : Transcript (pilotOutputFamily n) → ℝ := fun z =>
(pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ
let hmn : m n ≤ n := le_of_lt (hsub.2 n hn).2
let e := fun uv : (Fin (m n) → Fin 4) × (Fin (n - m n) → Fin 14) =>
pilotAdaptiveEncode hmn uv.1 uv.2
letI : IsProbabilityMeasure μ :=
transcriptLaw_isProbability (pilotEstimator p ε m select hselect hp hε) θ p hp hθ n
have hprefix (u : Fin (m n) → Fin 4) (v : Fin (n - m n) → Fin 14) :
H (e (u, v)) = fun i => inl (u ⟨i.val, i.property⟩) := by
funext i
change (if hi : i.val < m n then inl (u ⟨i.val, hi⟩)
else inr (v ⟨i.val - m n, by omega⟩)) = _
simp [i.property]
rw [MeasureTheory.integral_fintype Integrable.of_finite]
change (∑ z, (μ.restrict (H ⁻¹' {r}) {z}).toReal * g z) = 0
rw [sum_eq_sum_image_of_zero_off_range e
(pilotAdaptiveEncode_pair_injective hmn)]
· rw [Fintype.sum_prod_type]
apply Finset.sum_eq_zero
intro u _
by_cases hur : H (e (u, fun _ => 0)) = r
· have huv (v : Fin (n - m n) → Fin 14) : H (e (u, v)) = r := by
calc
H (e (u, v)) = (fun i => inl (u ⟨i.val, i.property⟩)) := hprefix u v
_ = H (e (u, fun _ => 0)) := (hprefix u (fun _ => 0)).symm
_ = r := hur
simp_rw [show ∀ v : Fin (n - m n) → Fin 14,
(μ.restrict (H ⁻¹' {r}) {e (u, v)}).toReal =
(μ {e (u, v)}).toReal by
intro v
rw [MeasureTheory.Measure.restrict_apply (measurableSet_singleton _)]
simp [huv v]]
exact pilotAdaptive_encoded_centered_sum select θ p ε m hselect hp hθ hε hsub n hn u
· have huv (v : Fin (n - m n) → Fin 14) : H (e (u, v)) ≠ r := by
intro h
apply hur
calc
H (e (u, fun _ => 0)) =
(fun i => inl (u ⟨i.val, i.property⟩)) := hprefix u (fun _ => 0)
_ = H (e (u, v)) := (hprefix u v).symm
_ = r := h
apply Finset.sum_eq_zero
intro v _
rw [MeasureTheory.Measure.restrict_apply (measurableSet_singleton _)]
simp [huv v]
· intro t ht
have hzero := pilotAdaptive_singleton_zero_of_not_range
select θ p ε m hselect hp hε hmn t ht
rw [MeasureTheory.Measure.restrict_apply (measurableSet_singleton _)]
have hsubsingleton : {t} ∩ H ⁻¹' {r} ⊆ ({t} : Set _) := Set.inter_subset_left
have hz : μ ({t} ∩ H ⁻¹' {r}) = 0 := measure_mono_null hsubsingleton hzero
simp [hz]
def waldFallbackPivot
A Wald pivot with an explicit zero-variance fallback. The fallback makes the closed-interval event exactly equivalent to the unstudentized Wald event, including samples on which the variance estimate is zero. The wald Fallback Pivot is determined by the displayed parameters.
theorem mem_Icc_waldFallbackPivot_iff
Membership of the fallback pivot in the symmetric cutoff interval is exactly the corresponding Wald inequality. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
cutoff :
ℝ
hcutoff :
0 < cutoff
n :
ℕ
hn :
0 < n
e v :
ℝ
hv :
0 ≤ v
conclusion
waldFallbackPivot cutoff n e v ∈ Icc (-cutoff) cutoff
↔ |e| ≤ cutoff * sqrt (v / n)
Proof (Lean source)
lemma mem_Icc_waldFallbackPivot_iff
{cutoff : ℝ} (hcutoff : 0 < cutoff) {n : ℕ} (hn : 0 < n)
{e v : ℝ} (hv : 0 ≤ v) :
waldFallbackPivot cutoff n e v ∈ Icc (-cutoff) cutoff ↔
|e| ≤ cutoff * sqrt (v / n) := by
by_cases hv0 : v = 0
· subst v
by_cases he0 : e = 0
· subst e
simp [waldFallbackPivot, hcutoff.le]
· have habs : 0 < |e| := abs_pos.mpr he0
simp [waldFallbackPivot, he0, not_le_of_gt habs]
· have hvpos : 0 < v := lt_of_le_of_ne hv (Ne.symm hv0)
have hnreal : 0 < (n : ℝ) := by exact_mod_cast hn
have hsn : 0 < sqrt (n : ℝ) := Real.sqrt_pos.2 hnreal
have hsv : 0 < sqrt v := Real.sqrt_pos.2 hvpos
rw [show waldFallbackPivot cutoff n e v =
sqrt n * e / sqrt v by simp [waldFallbackPivot, hv0]]
rw [show (sqrt n * e / sqrt v ∈ Icc (-cutoff) cutoff) ↔
|sqrt n * e / sqrt v| ≤ cutoff by simp [abs_le]]
rw [Real.sqrt_div hv]
simp only [abs_div, abs_mul]
rw [abs_of_nonneg (Real.sqrt_nonneg _),
abs_of_nonneg (Real.sqrt_nonneg _)]
rw [div_le_iff₀ hsv]
rw [show cutoff * (sqrt v / sqrt (n : ℝ)) =
(cutoff * sqrt v) / sqrt (n : ℝ) by ring]
rw [le_div_iff₀ hsn]
ring_nf
theorem pilotVhatStar_nonneg
The main-sample empirical variance is nonnegative on every transcript. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p ε :
ℝ
m :
ℕ → ℕ
select :
TrialParameter → StaircaseWeight × ℝ × ℝ
n :
ℕ
z :
Transcript (pilotOutputFamily n)
conclusion
0 ≤ VhatStar p ε m select z
Proof (Lean source)
lemma pilotVhatStar_nonneg (p ε : ℝ) (m : ℕ → ℕ)
(select : TrialParameter → StaircaseWeight × ℝ × ℝ)
{n : ℕ} (z : Transcript (pilotOutputFamily n)) :
0 ≤ VhatStar p ε m select z := by
unfold VhatStar
exact mul_nonneg (inv_nonneg.mpr (Nat.cast_nonneg _))
(Finset.sum_nonneg fun i _ => by split_ifs <;> positivity)
def pilotWaldPivot
The fallback-studentized private-pilot estimation error. The pilot Wald Pivot is determined by the displayed parameters.
Definition (Lean source)
parameters
cutoff :
ℝ
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
z :
Transcript (pilotOutputFamily n)
def
pilotWaldPivot select cutoff θ p ε m hselect hp hε n z :
ℝ
given by
waldFallbackPivot cutoff n ((pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ) (VhatStar p ε m (select) z)
theorem pilot_wald_event_eq
At every positive sample size, the private-pilot Wald coverage event is the symmetric closed-interval event for the fallback pivot. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot wald event eq.
Formal statement
hypotheses
cutoff :
ℝ
hcutoff :
0 < cutoff
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
hn :
0 < n
conclusion
{z : Transcript (pilotOutputFamily n) | |(pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ| ≤ cutoff * sqrt (VhatStar p ε m (select) z / n)}
= {z | pilotWaldPivot select cutoff θ p ε m hselect hp hε n z ∈ Icc (-cutoff) cutoff}
Proof (Lean source)
lemma pilot_wald_event_eq
(cutoff : ℝ) (hcutoff : 0 < cutoff)
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε) (n : ℕ) (hn : 0 < n) :
{z : Transcript (pilotOutputFamily n) |
|(pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ| ≤
cutoff * sqrt
(VhatStar p ε m (select) z / n)} =
{z | pilotWaldPivot select cutoff θ p ε m hselect hp hε n z ∈
Icc (-cutoff) cutoff} := by
ext z
exact (mem_Icc_waldFallbackPivot_iff hcutoff hn
(pilotVhatStar_nonneg p ε m (select) z)).symm
theorem pilot_wald_coverage_of_pivot_Icc
Convergence of the fallback-pivot interval probability transfers directly to the private-pilot Wald coverage probability. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot wald coverage of pivot Icc.
Formal statement
hypotheses
cutoff :
ℝ
hcutoff :
0 < cutoff
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
L :
ℝ
hpivot :
Tendsto (fun n => ((transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) {z | pilotWaldPivot select cutoff θ p ε m hselect hp hε n z ∈ Icc (-cutoff) cutoff}).toReal) atTop (nhds L)
conclusion
Tendsto (fun n => ((transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) {z | |(pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ| ≤ cutoff * sqrt (VhatStar p ε m (select) z / n)}).toReal) atTop (nhds L)
Proof (Lean source)
lemma pilot_wald_coverage_of_pivot_Icc
(cutoff : ℝ) (hcutoff : 0 < cutoff)
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε) (L : ℝ)
(hpivot : Tendsto (fun n =>
((transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
{z | pilotWaldPivot select cutoff θ p ε m hselect hp hε n z ∈
Icc (-cutoff) cutoff}).toReal) atTop (nhds L)) :
Tendsto (fun n =>
((transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
{z | |(pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ| ≤
cutoff * sqrt
(VhatStar p ε m (select) z / n)}).toReal)
atTop (nhds L) := by
refine hpivot.congr' ?_
filter_upwards [eventually_gt_atTop (0 : ℕ)] with n hn
rw [pilot_wald_event_eq select cutoff hcutoff θ p ε m hselect hp hε n hn]
theorem pilot_adapted_centered_integral
If a centered integrand has integral zero on every finite pilot-prefix fiber, multiplying it by any statistic determined by the pilot prefix remains integrable and has integral zero. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
n :
ℕ
m :
ℕ → ℕ
μ :
Measure (Transcript (pilotOutputFamily n))
g f :
Transcript (pilotOutputFamily n) → ℝ
hcenter :
∀ u, ∫ z in (pilotObservedPrefix (n := n) m) ⁻¹' {u}, g z ∂μ = 0
hadapt :
∀ z z'
if
(∀ i : Fin n, i.val < m n → z i = z' i)
then
f z = f z'
Proof (Lean source)
lemma pilot_adapted_centered_integral {n : ℕ} (m : ℕ → ℕ)
(μ : Measure (Transcript (pilotOutputFamily n))) [IsFiniteMeasure μ]
(g f : Transcript (pilotOutputFamily n) → ℝ)
(hcenter : ∀ u, ∫ z in
(pilotObservedPrefix (n := n) m) ⁻¹' {u}, g z ∂μ = 0)
(hadapt : ∀ z z',
(∀ i : Fin n, i.val < m n → z i = z' i) → f z = f z') :
Integrable (fun z => g z * f z) μ ∧
(∫ z, g z * f z ∂μ) = 0 := by
classical
let extend : ({i : Fin n // i.val < m n} → PilotOutput) →
Transcript (pilotOutputFamily n) := fun u i =>
if hi : i.val < m n then u ⟨i, hi⟩ else inl 0
let c : ({i : Fin n // i.val < m n} → PilotOutput) → ℝ :=
fun u => f (extend u)
have hf_eq : ∀ z, f z = c (pilotObservedPrefix m z) := by
intro z
apply hadapt z
intro i hi
simp [extend, pilotObservedPrefix, hi]
constructor
· exact Integrable.of_finite
· have hfiber : ∀ u : ({i : Fin n // i.val < m n} → PilotOutput),
MeasurableSet
((pilotObservedPrefix (n := n) m) ⁻¹' {u}) := by
intro u
exact (measurable_of_finite _)
(measurableSet_singleton u)
have hdecomp := integral_cellConst_mul
(μ := μ) (H := pilotObservedPrefix (n := n) m)
hfiber c (f := g) (hf := Integrable.of_finite)
rw [show (∫ z, g z * f z ∂μ) =
∫ z, c (pilotObservedPrefix m z) • g z ∂μ by
apply integral_congr_ae
filter_upwards with z
rw [hf_eq z]
simp [mul_comm]]
rw [hdecomp]
simp [hcenter]
theorem pilot_conditional_unbiasedness_of_prefix_cells
Fiberwise centering of the pilot estimator implies the full conditional unbiasedness clause against every bounded measurable pilot statistic. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot conditional unbiasedness of prefix cells.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hcenter :
∀ n ≥ 2,
∀ u : ({i : Fin n // i.val < m n} → PilotOutput),
(∫ z in (pilotObservedPrefix (n := n) m) ⁻¹' {u}, ((pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ) ∂transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
= 0
n ≥ 2 :
f :
Transcript (pilotOutputFamily n) → ℝ
(∃ C : ℝ, ∀ z, |f z| ≤ C)
(∀ z z', (∀ i : Fin n, i.val < m n → z i = z' i) → f z = f z')
conclusion
conclusion 1
Integrable (fun z => ((pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ) * f z) (transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
conclusion 2
(∫ z, ((pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ) * f z ∂transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
= 0
Proof (Lean source)
lemma pilot_conditional_unbiasedness_of_prefix_cells
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hcenter : ∀ n ≥ 2,
∀ u : ({i : Fin n // i.val < m n} → PilotOutput),
(∫ z in (pilotObservedPrefix (n := n) m) ⁻¹' {u},
((pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ)
∂transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) = 0) :
∀ n ≥ 2, ∀ f : Transcript (pilotOutputFamily n) → ℝ,
Measurable f →
(∃ C : ℝ, ∀ z, |f z| ≤ C) →
(∀ z z', (∀ i : Fin n, i.val < m n → z i = z' i) → f z = f z') →
Integrable (fun z =>
((pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ) * f z)
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) ∧
(∫ z, ((pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ) * f z
∂transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) = 0 := by
intro n hn f _ _ hadapt
haveI : IsProbabilityMeasure
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) :=
transcriptLaw_isProbability _ θ p hp hθ n
exact pilot_adapted_centered_integral m
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
(fun z => (pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ)
f (hcenter n hn) hadapt
theorem pilot_conditional_unbiasedness
The private-pilot estimator satisfies the headline conditional unbiasedness clause under the stated sublinear pilot condition. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot conditional unbiasedness.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hsub :
n ≥ 2 :
f :
Transcript (pilotOutputFamily n) → ℝ
(∃ C : ℝ, ∀ z, |f z| ≤ C)
(∀ z z', (∀ i : Fin n, i.val < m n → z i = z' i) → f z = f z')
conclusion
conclusion 1
Integrable (fun z => ((pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ) * f z) (transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
conclusion 2
(∫ z, ((pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ) * f z ∂transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
= 0
Proof (Lean source)
lemma pilot_conditional_unbiasedness
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hsub : PilotSublinear m) :
∀ n ≥ 2, ∀ f : Transcript (pilotOutputFamily n) → ℝ,
Measurable f →
(∃ C : ℝ, ∀ z, |f z| ≤ C) →
(∀ z z', (∀ i : Fin n, i.val < m n → z i = z' i) → f z = f z') →
Integrable (fun z =>
((pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ) * f z)
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) ∧
(∫ z, ((pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ) * f z
∂transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) = 0 := by
apply pilot_conditional_unbiasedness_of_prefix_cells select θ p ε m hselect hp hθ hε
intro n hn r
exact pilot_prefix_cell_centered select θ p ε m hselect hp hθ hε hsub n hn r
theorem gaussian_second_lintegral_lt_top
A centered Gaussian law has a finite second moment in the extended-real form required by RegularProcedure. For the displayed inputs and conditions, the stated result follows.
Formal statement
Proof (Lean source)
lemma gaussian_second_lintegral_lt_top (v : ℝ) :
(∫⁻ u, ofReal (u ^ 2) ∂gaussianMeasure 0 v) < ⊤ := by
have hmem : MemLp id 2 (gaussianMeasure 0 v) := by
unfold gaussianMeasure
exact ProbabilityTheory.memLp_id_gaussianReal 2
exact hmem.integrable_sq.lintegral_lt_top
theorem regular_of_weakLocalLimit_and_uniformTails
A weak local Gaussian limit and the defining uniform squared-tail condition assemble into the regular-procedure interface. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ :
p v :
ℝ
hweak :
WeakLocalLimit P θ p (gaussianMeasure 0 v)
htails :
∀ H : ℝ
if
0 < H
then
Tendsto (fun M : ℕ => limsup (fun n => sSup {u : ℝ≥0∞ | ∃ h ∈ localIndexSet θ n H, u = ∫⁻ z, ofReal ((scaledError P θ h n z) ^ 2 * if M < |scaledError P θ h n z| then 1 else 0) ∂transcriptLaw P (localAlternative θ h n) p n}) atTop) atTop (nhds 0)
conclusion
RegularProcedure P θ p
Proof (Lean source)
lemma regular_of_weakLocalLimit_and_uniformTails
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)]
{ε : ℝ} (P : ProcedureSequence Z ε)
(θ : TrialParameter) (p v : ℝ)
(hweak : WeakLocalLimit P θ p (gaussianMeasure 0 v))
(htails : ∀ H : ℝ, 0 < H →
Tendsto (fun M : ℕ =>
limsup (fun n =>
sSup {u : ℝ≥0∞ | ∃ h ∈ localIndexSet θ n H,
u = ∫⁻ z, ofReal
((scaledError P θ h n z) ^ 2 *
if M < |scaledError P θ h n z| then 1 else 0)
∂transcriptLaw P (localAlternative θ h n) p n})
atTop) atTop (nhds 0)) :
RegularProcedure P θ p := by
refine ⟨gaussianMeasure 0 v, inferInstance, ?_, hweak, htails⟩
exact gaussian_second_lintegral_lt_top v
theorem pilot_limit_variance_regular_of_core
The pilot weak limit, plug-in variance consistency, and uniform squared tails package the three asymptotic interfaces consumed together by the attainment theorem. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot limit variance regular of core.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
hweak :
hvhat :
htails :
∀ H : ℝ
if
0 < H
then
Tendsto (fun M : ℕ => limsup (fun n => sSup {u : ℝ≥0∞ | ∃ h ∈ localIndexSet θ n H, u = ∫⁻ z, ofReal ((scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z) ^ 2 * if M < |scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z| then 1 else 0) ∂transcriptLaw (pilotEstimator p ε m select hselect hp hε) (localAlternative θ h n) p n}) atTop) atTop (nhds 0)
conclusion
conclusion 1
conclusion 2
δ :
ℝ
0 < δ
conclusion 3
RegularProcedure (pilotEstimator p ε m select hselect hp hε) θ p
Proof (Lean source)
-- keep: reusable finite-pilot construction, law, localization, or limit API for related adaptive procedures
lemma pilot_limit_variance_regular_of_core
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε)
(hweak : WeakLocalLimit (pilotEstimator p ε m select hselect hp hε) θ p
(gaussianMeasure 0 (Vstar θ p ε)))
(hvhat : ∀ δ : ℝ, 0 < δ →
Tendsto (fun n =>
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
{z | δ < |VhatStar p ε m (select) z -
Vstar θ p ε|}) atTop (nhds 0))
(htails : ∀ H : ℝ, 0 < H →
Tendsto (fun M : ℕ =>
limsup (fun n =>
sSup {u : ℝ≥0∞ | ∃ h ∈ localIndexSet θ n H,
u = ∫⁻ z, ofReal
((scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z) ^ 2 *
if M < |scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z|
then 1 else 0)
∂transcriptLaw (pilotEstimator p ε m select hselect hp hε)
(localAlternative θ h n) p n})
atTop) atTop (nhds 0)) :
WeakLocalLimit (pilotEstimator p ε m select hselect hp hε) θ p
(gaussianMeasure 0 (Vstar θ p ε)) ∧
(∀ δ : ℝ, 0 < δ →
Tendsto (fun n =>
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
{z | δ < |VhatStar p ε m (select) z -
Vstar θ p ε|}) atTop (nhds 0)) ∧
RegularProcedure (pilotEstimator p ε m select hselect hp hε) θ p := by
exact ⟨hweak, hvhat,
regular_of_weakLocalLimit_and_uniformTails
(pilotEstimator p ε m select hselect hp hε) θ p (Vstar θ p ε) hweak htails⟩
theorem localAsymptoticRisk_eq_of_liminf_localWorstRisk
If every positive-radius local worst-risk sequence has the same liminf, the outer radius supremum in localAsymptoticRisk equals that value. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
localAsymptoticRisk P θ p = v
Proof (Lean source)
lemma localAsymptoticRisk_eq_of_liminf_localWorstRisk
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)]
{ε : ℝ} (P : ProcedureSequence Z ε)
(θ : TrialParameter) (p : ℝ) (v : ℝ≥0∞)
(hlim : ∀ H : ℝ, 0 < H →
liminf (fun n => localWorstRisk P θ p H n) atTop = v) :
localAsymptoticRisk P θ p = v := by
unfold localAsymptoticRisk
apply le_antisymm
· apply sSup_le
intro u hu
obtain ⟨H, hH, rfl⟩ := hu
exact le_of_eq (hlim H hH)
· apply le_sSup
exact ⟨1, by norm_num, (hlim 1 (by norm_num)).symm⟩
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotBridge 20 declarations This file isolates two interfaces needed by the pilot construction: reducing conditional cylinder factorization to a joint-law identity, and the pointwise existence part of the saddle selector.
Bridges for the private-pilot construction
This file isolates two interfaces needed by the pilot construction: reducing conditional cylinder factorization to a joint-law identity, and the pointwise existence part of the saddle selector.
theorem transcriptFactorizes_of_joint
the transcript factorizes of joint assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
n :
ℕ
Fin n → Type
∀ i, MeasurableSpace (Z i)
Q :
SequentialKernel (Z := Z)
hmarkov :
∀ i, IsMarkovKernel (Q i)
hprob :
∀ x, IsProbabilityMeasure (R x)
hjoint :
(R x).map (fun z ↦ (historyPrefix i z, z i))
= (R x).map (historyPrefix i) ⊗ₘ (Q i).comap (fun h ↦ (x i, h)) (by fun_prop)
conclusion
Proof (Lean source)
lemma transcriptFactorizes_of_joint {n : ℕ} {Z : Fin n → Type}
[∀ i, MeasurableSpace (Z i)]
(Q : SequentialKernel (Z := Z))
(R : (Fin n → Fin 4) → Measure (Transcript Z))
(hmarkov : ∀ i, IsMarkovKernel (Q i))
(hprob : ∀ x, IsProbabilityMeasure (R x))
(hjoint : ∀ (x : Fin n → Fin 4) (i : Fin n),
(R x).map (fun z ↦ (historyPrefix i z, z i)) =
(R x).map (historyPrefix i) ⊗ₘ
(Q i).comap (fun h ↦ (x i, h)) (by fun_prop)) :
TranscriptFactorizes Q R := by
intro x
refine ⟨hprob x, ?_⟩
intro i B A hB hA
letI : IsMarkovKernel (Q i) := hmarkov i
have hhist : Measurable (historyPrefix i : Transcript Z → PrivateHistory i) :=
measurable_pi_lambda _ fun j ↦ measurable_pi_apply j.1
have hpair : Measurable (fun z : Transcript Z ↦
(historyPrefix i z, z i)) := hhist.prodMk (measurable_pi_apply i)
change (R x) ((fun z ↦ (historyPrefix i z, z i)) ⁻¹' (B ×ˢ A)) = _
rw [← Measure.map_apply hpair (hB.prod hA), hjoint x i,
Measure.compProd_apply_prod hB hA]
rfl
theorem exists_saddle_pointwise
Under the supplied quantities and conditions, the exists saddle pointwise assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the exists saddle pointwise.
Formal statement
hypotheses
Proof (Lean source)
lemma exists_saddle_pointwise (p ε : ℝ)
(hp : InteriorAssignment p) (hε : 0 < ε) (θ : TrialParameter)
(hθ : InteriorMeans θ) :
∃ α : StaircaseWeight, ∃ t : ℝ,
staircaseFeasible ε α ∧
(∀ u, informationObjective θ p ε α t ≤
informationObjective θ p ε α u) ∧
Jstar θ p ε = informationObjective θ p ε α t := by
obtain ⟨α, hα, hopt⟩ :=
Jstar_exists_optimal_weight θ p ε hp hθ hε.le
obtain ⟨t, ht⟩ :=
informationObjective_exists_minimizer θ p ε hp hθ hε.le α hα
refine ⟨α, t, hα, ht, ?_⟩
apply hopt.trans
apply IsLeast.csInf_eq
exact ⟨⟨t, rfl⟩, by rintro _ ⟨u, rfl⟩; exact ht u⟩
def fsHistoryToPrivate
the fs history to private is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
given by
fun j => h ⟨j.1.val, j.2⟩
theorem measurable_fsHistoryToPrivate
the measurable fs history to private assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (fsHistoryToPrivate (Z := Z) i)
Proof (Lean source)
lemma measurable_fsHistoryToPrivate {n : ℕ} {Z : Fin n → Type} [∀ i, MeasurableSpace (Z i)]
(i : Fin n) : Measurable (fsHistoryToPrivate (Z := Z) i) := by
apply measurable_pi_iff.mpr
intro j
let k : Fin i.val := ⟨j.1.val, j.2⟩
exact measurable_pi_apply k
def privateToFSHistory
the private to fs history is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
theorem measurable_privateToFSHistory
the measurable private to fs history assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (privateToFSHistory (Z := Z) i)
Proof (Lean source)
lemma measurable_privateToFSHistory {n : ℕ} {Z : Fin n → Type}
[∀ i, MeasurableSpace (Z i)] (i : Fin n) :
Measurable (privateToFSHistory (Z := Z) i) := by
apply measurable_pi_iff.mpr
intro j
let k : {k : Fin n // k < i} :=
⟨Fin.castLE (le_of_lt i.isLt) j, j.isLt⟩
exact measurable_pi_apply k
theorem fsHistoryToPrivate_left_inv
the fs history to private left inv assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
fsHistoryToPrivate i (privateToFSHistory i h) = h
Proof (Lean source)
lemma fsHistoryToPrivate_left_inv {n : ℕ} {Z : Fin n → Type}
(i : Fin n) (h : PrivateHistory (Z := Z) i) :
fsHistoryToPrivate i (privateToFSHistory i h) = h := by
funext j
rfl
theorem fsHistoryToPrivate_right_inv
the fs history to private right inv assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
privateToFSHistory i (fsHistoryToPrivate i h) = h
Proof (Lean source)
lemma fsHistoryToPrivate_right_inv {n : ℕ} {Z : Fin n → Type}
(i : Fin n)
(h : History Z i.val
(le_of_lt i.isLt)) :
privateToFSHistory i (fsHistoryToPrivate i h) = h := by
funext j
rfl
def historyMeasurableEquiv
the history measurable equiv is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
given by
clause 1
toEquiv := { toFun := fsHistoryToPrivate i invFun := privateToFSHistory i left_inv := fsHistoryToPrivate_right_inv i right_inv := fsHistoryToPrivate_left_inv i }
clause 2
measurable_toFun := measurable_fsHistoryToPrivate i
clause 3
measurable_invFun := measurable_privateToFSHistory i
def fsTranscriptToTranscript
the fs transcript to transcript is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
n :
ℕ
Fin n → Type
z :
def
fsTranscriptToTranscript z :
Transcript Z
given by
fun i => z i
def transcriptToFSTranscript
the transcript to fs transcript is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
theorem measurable_fsTranscriptToTranscript
the measurable fs transcript to transcript assertion holds.
Formal statement
hypotheses
n :
ℕ
Fin n → Type
∀ i, MeasurableSpace (Z i)
conclusion
Measurable (fsTranscriptToTranscript (Z := Z))
Proof (Lean source)
lemma measurable_fsTranscriptToTranscript {n : ℕ} {Z : Fin n → Type}
[∀ i, MeasurableSpace (Z i)] :
Measurable (fsTranscriptToTranscript (Z := Z)) := by
apply measurable_pi_iff.mpr
intro i
exact measurable_pi_apply i
theorem measurable_transcriptToFSTranscript
the measurable transcript to fs transcript assertion holds.
Formal statement
hypotheses
n :
ℕ
Fin n → Type
∀ i, MeasurableSpace (Z i)
conclusion
Measurable (transcriptToFSTranscript (Z := Z))
Proof (Lean source)
lemma measurable_transcriptToFSTranscript {n : ℕ} {Z : Fin n → Type}
[∀ i, MeasurableSpace (Z i)] :
Measurable (transcriptToFSTranscript (Z := Z)) := by
apply measurable_pi_iff.mpr
intro i
exact measurable_pi_apply i
def finiteSequenceKernel
The finite-sequence kernel family is determined by the displayed parameters.
Definition (Lean source)
parameters
def
given by
fun i
=> (Q i).comap (fun xh => (xh.1, fsHistoryToPrivate i xh.2)) (by apply prodMk measurable_fst exact (measurable_fsHistoryToPrivate i).comp measurable_snd)
def finiteSequenceTranscript
The transcript law generated by a finite sequence of kernels is determined by the displayed parameters.
Definition (Lean source)
parameters
def
given by
def transcriptPrefix
For the supplied quantities and conditions, the transcript prefix is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
theorem measurable_transcriptPrefix
the measurable transcript prefix assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (transcriptPrefix (Z := Z) hk)
Proof (Lean source)
lemma measurable_transcriptPrefix {n k : ℕ} {Z : Fin n → Type}
[∀ i, MeasurableSpace (Z i)] (hk : k ≤ n) :
Measurable (transcriptPrefix (Z := Z) hk) := by
apply measurable_pi_iff.mpr
intro j
exact measurable_pi_apply (castLE hk j)
theorem finiteSequenceTranscript_map_prefix
The explicit finite transcript construction has exactly the library finite-sequence prefix law. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
n k :
ℕ
Fin n → Type
∀ i, MeasurableSpace (Z i)
Q :
SequentialKernel (Z := Z)
hQ :
∀ i, IsMarkovKernel (Q i)
hk :
k ≤ n
conclusion
Measure.map (transcriptPrefix (Z := Z) hk) (finiteSequenceTranscript Q x)
= prefixLaw (finiteSequenceKernel Q) x k hk
Proof (Lean source)
lemma finiteSequenceTranscript_map_prefix {n k : ℕ} {Z : Fin n → Type}
[∀ i, MeasurableSpace (Z i)] (Q : SequentialKernel (Z := Z))
(hQ : ∀ i, IsMarkovKernel (Q i)) (x : Fin n → Fin 4) (hk : k ≤ n) :
Measure.map (transcriptPrefix (Z := Z) hk) (finiteSequenceTranscript Q x) =
prefixLaw (finiteSequenceKernel Q) x k hk := by
have hQa : ∀ i, IsMarkovKernel (finiteSequenceKernel Q i) := by
intro i
letI : IsMarkovKernel (Q i) := hQ i
dsimp [finiteSequenceKernel]
infer_instance
unfold finiteSequenceTranscript
rw [Measure.map_map (measurable_transcriptPrefix hk)
measurable_fsTranscriptToTranscript]
have hcomp : transcriptPrefix (Z := Z) hk ∘ fsTranscriptToTranscript =
take hk := by rfl
rw [hcomp]
exact transcriptLaw_map_prefix
(finiteSequenceKernel Q) hQa x k hk
theorem finiteSequenceTranscript_factorizes
the finite sequence transcript factorizes assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
n :
ℕ
Fin n → Type
∀ i, MeasurableSpace (Z i)
Q :
SequentialKernel (Z := Z)
hmarkov :
∀ i, IsMarkovKernel (Q i)
conclusion
Proof (Lean source)
lemma finiteSequenceTranscript_factorizes {n : ℕ} {Z : Fin n → Type}
[∀ i, MeasurableSpace (Z i)]
(Q : SequentialKernel (Z := Z))
(hmarkov : ∀ i, IsMarkovKernel (Q i)) :
TranscriptFactorizes Q (finiteSequenceTranscript Q) := by
let Qa := finiteSequenceKernel Q
have hQa : ∀ i, IsMarkovKernel (Qa i) := by
intro i
letI : IsMarkovKernel (Q i) := hmarkov i
dsimp [Qa, finiteSequenceKernel]
infer_instance
let R : (Fin n → Fin 4) → Measure (Transcript Z) := finiteSequenceTranscript Q
change TranscriptFactorizes Q R
refine transcriptFactorizes_of_joint Q R hmarkov ?_ ?_
· intro x
letI := Causalean.Mathlib.Probability.Kernel.FiniteSequence.isProbabilityMeasure_transcriptLaw
Qa hQa x
exact Measure.isProbabilityMeasure_map measurable_fsTranscriptToTranscript.aemeasurable
· intro x i
have hi : i.val + 1 ≤ n := i.isLt
have hidx : nextIndex hi = i := Fin.ext (by rfl)
let e := historyMeasurableEquiv (Z := Z) i
let ez : Z (nextIndex hi) ≃ᵐ Z i :=
MeasurableEquiv.cast (congrArg Z hidx) (by cases hidx; rfl)
let law := transcriptLaw Qa x
let μ := prefixLaw Qa x i.val (le_of_succ_le hi)
let κ : Kernel (History Z i.val (le_of_succ_le hi)) (Z (nextIndex hi)) :=
(Qa (nextIndex hi)).comap
(fun h => (x (nextIndex hi), h)) (by fun_prop)
let f := fun z : Transcript Z =>
(take (le_of_succ_le hi) z, z (nextIndex hi))
have hf : Measurable f := by
dsimp only [f]
apply Measurable.prodMk
· apply measurable_pi_iff.mpr
intro j
exact measurable_pi_apply (castLE (le_of_succ_le hi) j)
· exact measurable_pi_apply (nextIndex hi)
have hbase : Measure.map f law = μ ⊗ₘ κ := by
exact Causalean.Mathlib.Probability.Kernel.FiniteSequence.transcriptLaw_map_take_next
Qa hQa x i.val hi
haveI : IsProbabilityMeasure μ :=
Causalean.Mathlib.Probability.Kernel.FiniteSequence.isProbabilityMeasure_prefixLaw
Qa hQa x i.val _
haveI : IsMarkovKernel κ := by dsimp [κ]; infer_instance
haveI : IsMarkovKernel (κ.map ez) :=
ProbabilityTheory.Kernel.IsMarkovKernel.map κ ez.measurable
have hpair :
(fun z : Transcript Z => (historyPrefix i z, z i)) ∘
fsTranscriptToTranscript = map e ez ∘ f := by
funext z
apply Prod.ext
· funext j
rfl
· cases hidx
rfl
have hownPair : Measurable (fun z : Transcript Z =>
(historyPrefix i z, z i)) := by
apply Measurable.prodMk
· apply measurable_pi_iff.mpr
intro j
exact measurable_pi_apply j.1
· exact measurable_pi_apply i
have hjointRaw : Measure.map (map e ez) (μ ⊗ₘ κ) =
Measure.map e μ ⊗ₘ (κ.map ez).comap e.symm e.symm.measurable := by
calc
_ = Measure.map (map e (id : Z i → Z i))
(Measure.map (map (id : _ → _) ez) (μ ⊗ₘ κ)) := by
rw [Measure.map_map (by fun_prop) (by fun_prop)]
congr 1
_ = Measure.map (map e (id : Z i → Z i)) (μ ⊗ₘ κ.map ez) := by
rw [← Measure.compProd_map ez.measurable]
_ = _ := map_compProd_prodMap_left_eq_compProd_comap
μ e (κ.map ez)
have hpref : Measurable (historyPrefix i : Transcript Z → PrivateHistory i) := by
apply measurable_pi_iff.mpr
intro j
exact measurable_pi_apply j.1
have hμ : Measure.map e μ = Measure.map (historyPrefix i) (R x) := by
have hprefix := transcriptLaw_map_prefix
Qa hQa x i.val (le_of_succ_le hi)
calc
Measure.map e μ = Measure.map e (Measure.map (take (le_of_succ_le hi)) law) := by
rw [hprefix]
_ = Measure.map (e ∘ take (le_of_succ_le hi)) law := by
rw [Measure.map_map e.measurable (by
apply measurable_pi_iff.mpr
intro j
exact measurable_pi_apply (castLE (le_of_succ_le hi) j))]
_ = Measure.map (historyPrefix i ∘ fsTranscriptToTranscript) law := by
congr 1
_ = Measure.map (historyPrefix i) (Measure.map fsTranscriptToTranscript law) := by
rw [Measure.map_map hpref measurable_fsTranscriptToTranscript]
_ = _ := by rfl
have hκ : (κ.map ez).comap e.symm e.symm.measurable =
(Q i).comap (fun h => (x i, h)) (by fun_prop) := by
ext h A hA
rw [ProbabilityTheory.Kernel.comap_apply,
ProbabilityTheory.Kernel.map_apply' _ _ _ hA,
ProbabilityTheory.Kernel.comap_apply,
ProbabilityTheory.Kernel.comap_apply]
dsimp only [Qa, finiteSequenceKernel]
rw [ProbabilityTheory.Kernel.comap_apply]
cases hidx
· change ((Q i) (x i,
fsHistoryToPrivate i (privateToFSHistory i h))) A =
((Q i) (x i, h)) A
rw [fsHistoryToPrivate_left_inv]
· exact ez.measurable
dsimp only [R]
unfold finiteSequenceTranscript
rw [Measure.map_map hownPair measurable_fsTranscriptToTranscript, hpair,
← Measure.map_map (by fun_prop) hf, hbase, hjointRaw, hμ, hκ]
rfl
theorem exists_transcript_factorization
the exists transcript factorization assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
n :
ℕ
Fin n → Type
∀ i, MeasurableSpace (Z i)
Q :
SequentialKernel (Z := Z)
hmarkov :
∀ i, IsMarkovKernel (Q i)
conclusion
Proof (Lean source)
lemma exists_transcript_factorization {n : ℕ} {Z : Fin n → Type}
[∀ i, MeasurableSpace (Z i)]
(Q : SequentialKernel (Z := Z))
(hmarkov : ∀ i, IsMarkovKernel (Q i)) :
∃ R : (Fin n → Fin 4) → Measure (Transcript Z),
TranscriptFactorizes Q R :=
⟨finiteSequenceTranscript Q, finiteSequenceTranscript_factorizes Q hmarkov⟩
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotCDFIdentity 6 declarations This file separates the finite-sample distributional step from the later triangular-row Gaussian limit.
Exact conditional CDF identity for the private pilot estimator
This file separates the finite-sample distributional step from the later
triangular-row Gaussian limit. The conditional CDF below is the literal iid
main-row product mass evaluated on the centered root-n estimator error.
theorem finite_sum_cast_eq
A finite sum is invariant under the canonical transport along an equality of its row lengths. For the displayed inputs and conditions, the stated result follows.
Formal statement
def adaptivePilotRowScaledCDF
The conditional CDF of the centered root-n error in one adaptive iid main row. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The adaptive Pilot Row Scaled CDF is determined by the displayed parameters.
Definition (Lean source)
parameters
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
x :
ℝ
def
adaptivePilotRowScaledCDF select θ h η p ε m hselect hp hε n x :
ℝ
given by
∑ v : Fin (adaptiveMainSize m n) → Fin 14, adaptivePilotRowProductMass select θ h η p ε m hselect hp hε n v * if sqrt (n : ℝ) * (adaptiveMainSize m n : ℝ)⁻¹ * (∑ j, adaptivePilotRowScore select θ h η p ε hselect hp hε n (v j))
≤ x then 1 else 0
def pilotPrefixScaledErrorCDF
The genuine released-prefix mixture of the conditional adaptive-row CDF. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The pilot Prefix Scaled Error CDF is determined by the displayed parameters.
Definition (Lean source)
parameters
θ h :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
hmn :
m n ≤ n
x :
ℝ
def
pilotPrefixScaledErrorCDF select θ h p ε m hselect hp hε hmn x :
ℝ
given by
∑ u : Fin (m n) → Fin 4, pilotPrefixWeight (localAlternative θ h n) p ε m n u * adaptivePilotRowScaledCDF select θ h (fun _ => pilotAdaptivePrefixTheta p ε m hmn u) p ε m hselect hp hε n x
theorem scaledError_pilotAdaptiveEncode_eq_adaptivePilotRow
On a correctly encoded transcript, the centered scaled estimator error is the scaled average of the literal centered adaptive main-row scores. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the scaled Error pilot Adaptive Encode eq adaptive Pilot Row.
Formal statement
hypotheses
θ h :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
hmn :
m n ≤ n
hlocal :
InteriorMeans (localAlternative θ h n)
hN :
0 < adaptiveMainSize m n
hmain :
adaptiveMainSize m n = n - m n
conclusion
= sqrt (n : ℝ) * (adaptiveMainSize m n : ℝ)⁻¹ * ∑ i, adaptivePilotRowScore select θ h (fun _ => pilotAdaptivePrefixTheta p ε m hmn u) p ε hselect hp hε n (v (cast hmain i))
Proof (Lean source)
lemma scaledError_pilotAdaptiveEncode_eq_adaptivePilotRow
(θ h : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε) {n : ℕ}
(hmn : m n ≤ n) (hlocal : InteriorMeans (localAlternative θ h n))
(hN : 0 < adaptiveMainSize m n)
(hmain : adaptiveMainSize m n = n - m n)
(u : Fin (m n) → Fin 4) (v : Fin (n - m n) → Fin 14) :
scaledError (pilotEstimator p ε m select hselect hp hε) θ h n
(pilotAdaptiveEncode hmn u v) =
sqrt (n : ℝ) * (adaptiveMainSize m n : ℝ)⁻¹ *
∑ i, adaptivePilotRowScore select θ h
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε hselect hp hε n (v (cast hmain i)) := by
let η := pilotAdaptivePrefixTheta p ε m hmn u
have htrue : adaptiveTrueParam θ h n = localAlternative θ h n :=
adaptiveTrueParam_eq_localAlternative θ h n hlocal
have herr := adaptivePilotRow_estimatorError_eq select θ h (fun _ => η)
p ε m hselect hp hε n hN (fun i => v (cast hmain i))
have hcast : (adaptiveMainSize m n : ℝ) = (n - m n : ℕ) := by
exact_mod_cast hmain
have hsum :
(∑ i : Fin (adaptiveMainSize m n),
adaptiveSelectedScore select η p ε (v (cast hmain i))) =
∑ j : Fin (n - m n), adaptiveSelectedScore select η p ε (v j) :=
finite_sum_cast_eq hmain
(fun j => adaptiveSelectedScore select η p ε (v j))
unfold scaledError
change sqrt (n : ℝ) *
(tauStar p ε m (select)
(pilotAdaptiveEncode hmn u v) - contrast (localAlternative θ h n)) = _
rw [tauStar_pilotAdaptiveEncode select p ε m hp hε hmn u v]
calc
sqrt (n : ℝ) *
(contrast η + ((n - m n : ℕ) : ℝ)⁻¹ *
(∑ j, adaptiveSelectedScore select η p ε (v j)) -
contrast (localAlternative θ h n)) =
sqrt (n : ℝ) *
(contrast η + (adaptiveMainSize m n : ℝ)⁻¹ *
(∑ i, adaptiveSelectedScore select η p ε
(v (cast hmain i))) - contrast (adaptiveTrueParam θ h n)) := by
rw [htrue, hcast, hsum]
_ = sqrt (n : ℝ) *
((adaptiveMainSize m n : ℝ)⁻¹ *
∑ i, adaptivePilotRowScore select θ h (fun _ => η)
p ε hselect hp hε n (v (cast hmain i))) := by rw [herr]
_ = _ := by simp only [η]; ring
theorem transcriptLaw_scaledError_le_eq_ofReal_pilotPrefixScaledErrorCDF
At every proper pilot split with an interior local alternative, the actual scaled-error transcript CDF is exactly the released-prefix mixture of the conditional centered iid-row CDFs. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the transcript Law scaled Error le eq of Real pilot Prefix Scaled Error CDF.
Formal statement
hypotheses
θ h :
p ε x :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
hsub :
n :
ℕ
hn :
2 ≤ n
hlocal :
InteriorMeans (localAlternative θ h n)
conclusion
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) (localAlternative θ h n) p n) {z | scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z ≤ x}
Proof (Lean source)
lemma transcriptLaw_scaledError_le_eq_ofReal_pilotPrefixScaledErrorCDF
(θ h : TrialParameter) (p ε x : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε)
(hsub : PilotSublinear m) {n : ℕ} (hn : 2 ≤ n)
(hlocal : InteriorMeans (localAlternative θ h n)) :
(transcriptLaw (pilotEstimator p ε m select hselect hp hε)
(localAlternative θ h n) p n)
{z | scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z ≤ x} =
ofReal
(pilotPrefixScaledErrorCDF select θ h p ε m hselect hp hε
(le_of_lt (hsub.2 n hn).2) x) := by
classical
let θn := localAlternative θ h n
let hmn : m n ≤ n := le_of_lt (hsub.2 n hn).2
let μ := transcriptLaw (pilotEstimator p ε m select hselect hp hε) θn p n
let A : Set (Transcript (pilotOutputFamily n)) :=
{z | scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z ≤ x}
let e := fun uv : (Fin (m n) → Fin 4) × (Fin (n - m n) → Fin 14) =>
pilotAdaptiveEncode hmn uv.1 uv.2
have hN : 0 < adaptiveMainSize m n := by
rw [adaptiveMainSize, adaptivePilotSize_eq m hsub hn]
exact Nat.sub_pos_of_lt (hsub.2 n hn).2
have hmain : adaptiveMainSize m n = n - m n := by
simp [adaptiveMainSize, adaptivePilotSize_eq m hsub hn]
have htrue : adaptiveTrueParam θ h n = θn :=
adaptiveTrueParam_eq_localAlternative θ h n hlocal
have hw (u : Fin (m n) → Fin 4) :
0 ≤ pilotPrefixWeight θn p ε m n u :=
pilotPrefixWeight_nonneg θn p ε m n hp hlocal u
have hm (u : Fin (m n) → Fin 4) (v : Fin (n - m n) → Fin 14) :
0 ≤ ∏ j, adaptiveMainMass select θn
(pilotAdaptivePrefixTheta p ε m hmn u) p ε (v j) := by
apply Finset.prod_nonneg
intro j _
exact adaptiveMainMass_nonneg select θn _ p ε hselect hp hlocal
(pilotTheta_interior p ε m (pilotAdaptiveEncode hmn u (fun _ => 0)))
hε (v j)
have hatom (u : Fin (m n) → Fin 4) (v : Fin (n - m n) → Fin 14) :
μ {e (u, v)} = ofReal
(pilotPrefixWeight θn p ε m n u *
∏ j, adaptiveMainMass select θn (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)) := by
rw [pilotAdaptive_joint_singleton_eq_prefix_mul_mainProduct
select θn p ε m hselect hp hlocal hε hmn u v]
change ofReal (pilotPrefixWeight θn p ε m n u) *
ofReal (∏ j, adaptiveMainMass select θn
(pilotAdaptivePrefixTheta p ε m hmn u) p ε (v j)) = _
exact (ENNReal.ofReal_mul (hw u)).symm
have hA : MeasurableSet A := MeasurableSet.of_discrete
rw [show μ A = ∫⁻ z, A.indicator (fun _ => (1 : ENNReal)) z ∂μ by
rw [lintegral_indicator hA, setLIntegral_one]]
rw [MeasureTheory.lintegral_fintype]
rw [sum_eq_sum_image_of_zero_off_range e (pilotAdaptiveEncode_pair_injective hmn)]
· rw [Fintype.sum_prod_type]
simp_rw [hatom]
have hind (u : Fin (m n) → Fin 4) (v : Fin (n - m n) → Fin 14) :
A.indicator (fun _ => (1 : ENNReal)) (e (u, v)) =
if sqrt (n : ℝ) * (adaptiveMainSize m n : ℝ)⁻¹ *
(∑ i, adaptivePilotRowScore select θ h
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε hselect hp hε n (v (cast hmain i))) ≤ x
then 1 else 0 := by
rw [Set.indicator_apply]
simp only [A, Set.mem_ofPred_eq, e,
scaledError_pilotAdaptiveEncode_eq_adaptivePilotRow
select θ h p ε m hselect hp hε hmn hlocal hN hmain u v]
simp_rw [hind]
have hindOfReal (u : Fin (m n) → Fin 4) (v : Fin (n - m n) → Fin 14) :
(if sqrt (n : ℝ) * (adaptiveMainSize m n : ℝ)⁻¹ *
(∑ i, adaptivePilotRowScore select θ h
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε hselect hp hε n (v (cast hmain i))) ≤ x
then (1 : ENNReal) else 0) =
ofReal
(if sqrt (n : ℝ) * (adaptiveMainSize m n : ℝ)⁻¹ *
(∑ i, adaptivePilotRowScore select θ h
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε hselect hp hε n (v (cast hmain i))) ≤ x
then (1 : ℝ) else 0) := by
split_ifs <;> simp
simp_rw [hindOfReal]
have hmul (u : Fin (m n) → Fin 4) (v : Fin (n - m n) → Fin 14) :
ofReal
(if sqrt (n : ℝ) * (adaptiveMainSize m n : ℝ)⁻¹ *
(∑ i, adaptivePilotRowScore select θ h
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε hselect hp hε n (v (cast hmain i))) ≤ x
then (1 : ℝ) else 0) *
ofReal (pilotPrefixWeight θn p ε m n u *
∏ j, adaptiveMainMass select θn (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)) =
ofReal
((if sqrt (n : ℝ) * (adaptiveMainSize m n : ℝ)⁻¹ *
(∑ i, adaptivePilotRowScore select θ h
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε hselect hp hε n (v (cast hmain i))) ≤ x
then (1 : ℝ) else 0) *
(pilotPrefixWeight θn p ε m n u *
∏ j, adaptiveMainMass select θn (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j))) := by
exact (ENNReal.ofReal_mul (by split_ifs <;> positivity)).symm
simp_rw [hmul]
have hofReal_sum {B : Type} [Fintype B] (f : B → ℝ)
(hf : ∀ b, 0 ≤ f b) :
(∑ b, ofReal (f b)) = ofReal (∑ b, f b) := by
simpa using (ENNReal.ofReal_sum_of_nonneg
(s := univ) (fun b _ => hf b)).symm
rw [show (∑ u : Fin (m n) → Fin 4, ∑ v : Fin (n - m n) → Fin 14,
ofReal
((if sqrt (n : ℝ) * (adaptiveMainSize m n : ℝ)⁻¹ *
(∑ i, adaptivePilotRowScore select θ h
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε hselect hp hε n (v (cast hmain i))) ≤ x
then (1 : ℝ) else 0) *
(pilotPrefixWeight θn p ε m n u *
∏ j, adaptiveMainMass select θn (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)))) =
ofReal (∑ u : Fin (m n) → Fin 4,
∑ v : Fin (n - m n) → Fin 14,
(if sqrt (n : ℝ) * (adaptiveMainSize m n : ℝ)⁻¹ *
(∑ i, adaptivePilotRowScore select θ h
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε hselect hp hε n (v (cast hmain i))) ≤ x
then (1 : ℝ) else 0) *
(pilotPrefixWeight θn p ε m n u *
∏ j, adaptiveMainMass select θn (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j))) by
rw [show (∑ u : Fin (m n) → Fin 4, ∑ v : Fin (n - m n) → Fin 14,
ofReal
((if sqrt (n : ℝ) * (adaptiveMainSize m n : ℝ)⁻¹ *
(∑ i, adaptivePilotRowScore select θ h
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε hselect hp hε n (v (cast hmain i))) ≤ x
then (1 : ℝ) else 0) *
(pilotPrefixWeight θn p ε m n u *
∏ j, adaptiveMainMass select θn (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)))) =
∑ u : Fin (m n) → Fin 4, ofReal
(∑ v : Fin (n - m n) → Fin 14,
(if sqrt (n : ℝ) * (adaptiveMainSize m n : ℝ)⁻¹ *
(∑ i, adaptivePilotRowScore select θ h
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε hselect hp hε n (v (cast hmain i))) ≤ x
then (1 : ℝ) else 0) *
(pilotPrefixWeight θn p ε m n u *
∏ j, adaptiveMainMass select θn (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j))) by
apply Finset.sum_congr rfl
intro u _
exact hofReal_sum _ (fun v => mul_nonneg
(by split_ifs <;> positivity) (mul_nonneg (hw u) (hm u v))) ]
exact hofReal_sum _ (fun u => Finset.sum_nonneg fun v _ => mul_nonneg
(by split_ifs <;> positivity) (mul_nonneg (hw u) (hm u v))) ]
congr 1
rw [show hmn = le_of_lt (hsub.2 n hn).2 from Subsingleton.elim _ _]
unfold pilotPrefixScaledErrorCDF adaptivePilotRowScaledCDF
apply Finset.sum_congr rfl
intro u _
rw [Finset.mul_sum]
let rowEquiv : (Fin (adaptiveMainSize m n) → Fin 14) ≃
(Fin (n - m n) → Fin 14) :=
Equiv.piCongrLeft (fun _ : Fin (n - m n) => Fin 14) (finCongr hmain)
symm
apply Fintype.sum_equiv rowEquiv
intro v
have happly (i : Fin (adaptiveMainSize m n)) :
rowEquiv v (cast hmain i) = v i := by rfl
have happly' (i : Fin (adaptiveMainSize m n)) :
rowEquiv v ((finCongr hmain) i) = v i := by
change rowEquiv v (cast hmain i) = v i
exact happly i
have hscores :
(∑ i : Fin (adaptiveMainSize m n),
adaptivePilotRowScore select θ h
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε hselect hp hε n (rowEquiv v (cast hmain i))) =
∑ i : Fin (adaptiveMainSize m n),
adaptivePilotRowScore select θ h
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε hselect hp hε n (v i) := by
apply Finset.sum_congr rfl
intro i _
rw [happly]
have hprod :
(∏ i : Fin (adaptiveMainSize m n),
adaptiveMainMass select θn (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v i)) =
∏ j : Fin (n - m n),
adaptiveMainMass select θn (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (rowEquiv v j) := by
simpa only [happly'] using (Equiv.prod_comp (finCongr hmain)
(fun j : Fin (n - m n) =>
adaptiveMainMass select θn (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (rowEquiv v j)))
simp only [adaptivePilotRowProductMass, adaptivePilotRowMass, htrue]
rw [hprod, hscores]
ring
· intro z hz
have hzμ := pilotAdaptive_singleton_zero_of_not_range
select θn p ε m hselect hp hε hmn z hz
apply mul_eq_zero.mpr
right
simpa only [μ] using hzμ
theorem transcriptLaw_real_scaledError_le_eq_pilotPrefixScaledErrorCDF
The real-valued CDF of the actual scaled estimator error is the genuine prefix-weighted conditional row CDF. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the transcript Law real scaled Error le eq pilot Prefix Scaled Error CDF.
Formal statement
hypotheses
θ h :
p ε x :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
hsub :
n :
ℕ
hn :
2 ≤ n
hlocal :
InteriorMeans (localAlternative θ h n)
conclusion
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) (localAlternative θ h n) p n).real {z | scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z ≤ x}
= pilotPrefixScaledErrorCDF select θ h p ε m hselect hp hε (le_of_lt (hsub.2 n hn).2) x
Proof (Lean source)
lemma transcriptLaw_real_scaledError_le_eq_pilotPrefixScaledErrorCDF
(θ h : TrialParameter) (p ε x : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε)
(hsub : PilotSublinear m) {n : ℕ} (hn : 2 ≤ n)
(hlocal : InteriorMeans (localAlternative θ h n)) :
(transcriptLaw (pilotEstimator p ε m select hselect hp hε)
(localAlternative θ h n) p n).real
{z | scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z ≤ x} =
pilotPrefixScaledErrorCDF select θ h p ε m hselect hp hε
(le_of_lt (hsub.2 n hn).2) x := by
let hmn : m n ≤ n := le_of_lt (hsub.2 n hn).2
have htrue : adaptiveTrueParam θ h n = localAlternative θ h n :=
adaptiveTrueParam_eq_localAlternative θ h n hlocal
have hw (u : Fin (m n) → Fin 4) :
0 ≤ pilotPrefixWeight (localAlternative θ h n) p ε m n u :=
pilotPrefixWeight_nonneg (localAlternative θ h n) p ε m n hp hlocal u
have hmix : 0 ≤ pilotPrefixScaledErrorCDF select θ h p ε m hselect hp hε hmn x := by
unfold pilotPrefixScaledErrorCDF adaptivePilotRowScaledCDF
apply Finset.sum_nonneg
intro u _
apply mul_nonneg (hw u)
apply Finset.sum_nonneg
intro v _
apply mul_nonneg
· unfold adaptivePilotRowProductMass adaptivePilotRowMass
rw [htrue]
apply Finset.prod_nonneg
intro j _
exact adaptiveMainMass_nonneg select (localAlternative θ h n) _ p ε hselect hp hlocal
(pilotTheta_interior p ε m (pilotAdaptiveEncode hmn u (fun _ => 0)))
hε (v j)
· split_ifs <;> positivity
rw [Measure.real_def,
transcriptLaw_scaledError_le_eq_ofReal_pilotPrefixScaledErrorCDF
select θ h p ε x m hselect hp hε hsub hn hlocal]
exact ENNReal.toReal_ofReal hmix
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotGaussianTransfer 7 declarations This module transfers an explicit conditional Gaussian CDF limit for every convergent sequence of genuine pilot selectors to the unconditional CDF of the actual private-pilot transcript.
Transfer of conditional pilot-row Gaussian limits
This module transfers an explicit conditional Gaussian CDF limit for every convergent sequence of genuine pilot selectors to the unconditional CDF of the actual private-pilot transcript. It contains no triangular-array CLT: that analytic input remains a premise of the main theorem.
def pilotGaussianRadius
The shrinking localization radius used to average conditional row limits. It goes to zero slowly enough that the pilot Chebyshev bound still vanishes. For the displayed inputs and conditions, the stated result follows. The pilot Gaussian Radius is determined by the displayed parameters.
Definition (Lean source)
parameters
def
pilotGaussianRadius θ h m n :
ℝ
given by
theorem pilotGaussianBaseRadius_tendsto_zero
The pilot-size component of pilotGaussianRadius tends to zero. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma pilotGaussianBaseRadius_tendsto_zero (m : ℕ → ℕ)
(hdiv : PilotDiverges m) :
Tendsto (fun n => sqrt ((sqrt (m n : ℝ))⁻¹))
atTop (nhds 0) := by
have hm : Tendsto (fun n => (m n : ℝ)) atTop atTop :=
tendsto_natCast_atTop_atTop.comp hdiv
have hsqrt : Tendsto (fun n => sqrt (m n : ℝ)) atTop atTop :=
Real.tendsto_sqrt_atTop.comp hm
have hinv : Tendsto (fun n => (sqrt (m n : ℝ))⁻¹) atTop (nhds 0) :=
tendsto_inv_atTop_zero.comp hsqrt
simpa using hinv.sqrt
theorem pilotGaussianRadius_tendsto_zero
The localization radius tends to zero. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma pilotGaussianRadius_tendsto_zero (θ h : TrialParameter)
(m : ℕ → ℕ) (hdiv : PilotDiverges m) :
Tendsto (pilotGaussianRadius θ h m) atTop (nhds 0) := by
have hbase := pilotGaussianBaseRadius_tendsto_zero m hdiv
have hdist : Tendsto (fun n => dist (localAlternative θ h n) θ)
atTop (nhds 0) := by
simpa using (localAlternative_tendsto θ h).dist
(tendsto_const_nhds : Tendsto (fun _ : ℕ => θ) atTop (nhds θ))
have hshift : Tendsto (fun n => sqrt
(dist (localAlternative θ h n) θ)) atTop (nhds 0) := by
simpa using hdist.sqrt
change Tendsto (fun n => 2 * max
(sqrt ((sqrt (m n : ℝ))⁻¹))
(sqrt (dist (localAlternative θ h n) θ))) atTop (nhds 0)
convert ((tendsto_const_nhds : Tendsto (fun _ : ℕ => (2 : ℝ)) atTop (nhds 2)).mul
(hbase.max hshift)) using 1 <;> norm_num
theorem pilotPrefixCDF_uniform_of_subsequential_tendsto
Convergence along every selected subsequence gives uniform control over raw released prefixes in the shrinking Gaussian-transfer neighborhood. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Prefix CDF uniform of subsequential tendsto.
Formal statement
hypotheses
conclusion
∀ᶠ n in atTop, ∀ (hmn : m n ≤ n) (u : Fin (m n) → Fin 4), dist (pilotAdaptivePrefixTheta p ε m hmn u) θ
≤ ρ n → |E n u - L| < δ
Proof (Lean source)
lemma pilotPrefixCDF_uniform_of_subsequential_tendsto
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ) (ρ : ℕ → ℝ)
(E : (n : ℕ) → (Fin (m n) → Fin 4) → ℝ) (L : ℝ)
(hρ : Tendsto ρ atTop (nhds 0))
(hseq : ∀ (φ : ℕ → ℕ), StrictMono φ →
∀ (hφlarge : ∀ k, m (φ k) < φ k)
(u : (k : ℕ) → Fin (m (φ k)) → Fin 4),
Tendsto (fun k => pilotAdaptivePrefixTheta p ε m
(le_of_lt (hφlarge k)) (u k))
atTop (nhds θ) →
Tendsto (fun k => E (φ k) (u k)) atTop (nhds L))
(hlt : ∀ᶠ n in atTop, m n < n) :
∀ δ : ℝ, 0 < δ → ∀ᶠ n in atTop,
∀ (hmn : m n ≤ n) (u : Fin (m n) → Fin 4),
dist (pilotAdaptivePrefixTheta p ε m hmn u) θ ≤ ρ n →
|E n u - L| < δ := by
intro δ hδ
by_contra huniform
have hfrequent : ∃ᶠ n in atTop,
¬(∀ (hmn : m n ≤ n) (u : Fin (m n) → Fin 4),
dist (pilotAdaptivePrefixTheta p ε m hmn u) θ ≤ ρ n →
|E n u - L| < δ) := not_eventually.mp huniform
have hbad : ∃ᶠ n in atTop, ∃ hmn : m n < n,
∃ (u : Fin (m n) → Fin 4),
dist (pilotAdaptivePrefixTheta p ε m
(le_of_lt hmn) u) θ ≤ ρ n ∧
δ ≤ |E n u - L| := by
apply (hfrequent.and_eventually hlt).mono
intro n hpair
rcases hpair with ⟨hn, hmn⟩
push Not at hn
obtain ⟨hmn', u, hu, hE⟩ := hn
refine ⟨hmn, u, ?_, hE⟩
simpa only [Subsingleton.elim hmn' (le_of_lt hmn)] using hu
obtain ⟨φ, hφ, hφbad⟩ := extraction_of_frequently_atTop hbad
choose u hu using fun k => (hφbad k).2
have hselector : Tendsto
(fun k => pilotAdaptivePrefixTheta p ε m
(le_of_lt (hφbad k).1) (u k)) atTop (nhds θ) := by
rw [Metric.tendsto_atTop]
intro η hη
have hρsub : Tendsto (fun k => ρ (φ k)) atTop (nhds 0) :=
hρ.comp hφ.tendsto_atTop
have hρlt : ∀ᶠ k in atTop, ρ (φ k) < η :=
(tendsto_order.1 hρsub).2 η hη
rcases eventually_atTop.1 hρlt with ⟨N, hN⟩
exact ⟨N, fun k hk => lt_of_le_of_lt (hu k).1 (hN k hk)⟩
have hE := hseq φ hφ (fun k => (hφbad k).1) u hselector
have hclose : ∀ᶠ k in atTop, |E (φ k) (u k) - L| < δ := by
exact eventually_atTop.2 (Metric.tendsto_atTop.1 hE δ hδ)
rcases eventually_atTop.1 hclose with ⟨N, hN⟩
exact (not_lt_of_ge (hu N).2 (hN N le_rfl)).elim
theorem pilotGaussianBadEnvelope_tendsto_zero
The varying-radius Chebyshev envelope used in Gaussian transfer vanishes. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Gaussian Bad Envelope tendsto zero.
Formal statement
hypotheses
Proof (Lean source)
lemma pilotGaussianBadEnvelope_tendsto_zero
(p ε : ℝ) (m : ℕ → ℕ) (hp : InteriorAssignment p)
(hε : 0 < ε) (hdiv : PilotDiverges m) :
Tendsto (fun n => pilotBadMassEnvelope p ε
(sqrt ((sqrt (m n : ℝ))⁻¹)) m n) atTop (nhds 0) := by
let c0 := controlProb p * (exp ε - 1) / (exp ε + 3)
let c1 := p * (exp ε - 1) / (exp ε + 3)
have hc0 : 0 < c0 := by
dsimp [c0, controlProb]
have hep : 1 < exp ε := by
simpa using (Real.exp_lt_exp.mpr hε : exp 0 < exp ε)
have hcp : 0 < 1 - p := by linarith [hp.2]
exact div_pos (mul_pos hcp (sub_pos.mpr hep)) (by positivity)
have hc1 : 0 < c1 := by
dsimp [c1]
have hep : 1 < exp ε := by
simpa using (Real.exp_lt_exp.mpr hε : exp 0 < exp ε)
exact div_pos (mul_pos hp.1 (sub_pos.mpr hep)) (by positivity)
have hm : Tendsto (fun n => (m n : ℝ)) atTop atTop :=
tendsto_natCast_atTop_atTop.comp hdiv
have hsqrt : Tendsto (fun n => sqrt (m n : ℝ)) atTop atTop :=
Real.tendsto_sqrt_atTop.comp hm
have hpos : ∀ᶠ n in atTop, 0 < (m n : ℝ) := hm.eventually_gt_atTop 0
have hrewrite : ∀ᶠ n in atTop,
pilotBadMassEnvelope p ε
(sqrt ((sqrt (m n : ℝ))⁻¹)) m n =
1 / (sqrt (m n : ℝ) * c0 ^ 2) +
1 / (sqrt (m n : ℝ) * c1 ^ 2) := by
filter_upwards [hpos] with n hn
have hs : 0 < sqrt (m n : ℝ) := Real.sqrt_pos.2 hn
have hsq : (sqrt ((sqrt (m n : ℝ))⁻¹)) ^ 2 =
(sqrt (m n : ℝ))⁻¹ := Real.sq_sqrt (inv_nonneg.mpr hs.le)
have hcancel : (m n : ℝ) * (sqrt (m n : ℝ))⁻¹ =
sqrt (m n : ℝ) := by
nth_rewrite 1 [← Real.sq_sqrt hn.le]
field_simp
unfold pilotBadMassEnvelope
have heq0 : sqrt ((sqrt (m n : ℝ))⁻¹) * controlProb p *
(exp ε - 1) / (exp ε + 3) =
sqrt ((sqrt (m n : ℝ))⁻¹) * c0 := by
dsimp only [c0]
ring
have heq1 : sqrt ((sqrt (m n : ℝ))⁻¹) * p *
(exp ε - 1) / (exp ε + 3) =
sqrt ((sqrt (m n : ℝ))⁻¹) * c1 := by
dsimp only [c1]
ring
rw [heq0, heq1]
simp only [mul_pow, hsq, ← mul_assoc, hcancel]
have h0 : Tendsto (fun n => 1 / (sqrt (m n : ℝ) * c0 ^ 2))
atTop (nhds 0) := by
convert tendsto_inv_atTop_zero.comp
(hsqrt.const_mul_atTop (sq_pos_of_pos hc0)) using 1 <;>
simp [Function.comp_def, one_div, mul_comm]
have h1 : Tendsto (fun n => 1 / (sqrt (m n : ℝ) * c1 ^ 2))
atTop (nhds 0) := by
convert tendsto_inv_atTop_zero.comp
(hsqrt.const_mul_atTop (sq_pos_of_pos hc1)) using 1 <;>
simp [Function.comp_def, one_div, mul_comm]
have ht : Tendsto (fun n =>
1 / (sqrt (m n : ℝ) * c0 ^ 2) +
1 / (sqrt (m n : ℝ) * c1 ^ 2)) atTop (nhds 0) := by
convert h0.add h1 using 1 <;> norm_num
exact ht.congr' (hrewrite.mono fun _ hn => hn.symm)
theorem pilot_scaledError_cdf_tendsto_of_conditionalRows
An explicit conditional adaptive-row Gaussian CDF limit along every subsequence and every convergent genuine selector sequence implies the exact unconditional scaled-error CDF limit of the private-pilot transcript. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot scaled Error cdf tendsto of conditional Rows.
Formal statement
hypotheses
θ h :
p ε x :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hdiv :
hsub :
hrow :
∀ (φ : ℕ → ℕ),
StrictMono φ → ∀ η : ℕ → TrialParameter, (∀ k, InteriorMeans (η k)) → Tendsto η atTop (nhds θ) → Tendsto (fun k => adaptivePilotRowScaledCDF select θ h (fun _ => η k) p ε m hselect hp hε (φ k) x) atTop (nhds ((gaussianMeasure 0 (Vstar θ p ε)).real (Iic x)))
conclusion
Tendsto (fun n => (transcriptLaw (pilotEstimator p ε m select hselect hp hε) (localAlternative θ h n) p n).real {z | scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z ≤ x}) atTop (nhds ((gaussianMeasure 0 (Vstar θ p ε)).real (Iic x)))
Proof (Lean source)
theorem pilot_scaledError_cdf_tendsto_of_conditionalRows
(θ h : TrialParameter) (p ε x : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hdiv : PilotDiverges m) (hsub : PilotSublinear m)
(hrow : ∀ (φ : ℕ → ℕ), StrictMono φ →
∀ η : ℕ → TrialParameter,
(∀ k, InteriorMeans (η k)) → Tendsto η atTop (nhds θ) →
Tendsto (fun k => adaptivePilotRowScaledCDF select θ h
(fun _ => η k) p ε m hselect hp hε (φ k) x) atTop
(nhds ((gaussianMeasure 0 (Vstar θ p ε)).real (Iic x)))) :
Tendsto (fun n =>
(transcriptLaw (pilotEstimator p ε m select hselect hp hε)
(localAlternative θ h n) p n).real
{z | scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z ≤ x})
atTop (nhds ((gaussianMeasure 0 (Vstar θ p ε)).real (Iic x))) := by
classical
let L := (gaussianMeasure 0 (Vstar θ p ε)).real (Iic x)
let base : ℕ → ℝ := fun n => sqrt ((sqrt (m n : ℝ))⁻¹)
let ρ := pilotGaussianRadius θ h m
let E : (n : ℕ) → (Fin (m n) → Fin 4) → ℝ := fun n u =>
if hn : 2 ≤ n then
adaptivePilotRowScaledCDF select θ h
(fun _ => pilotAdaptivePrefixTheta p ε m
(le_of_lt (hsub.2 n hn).2) u)
p ε m hselect hp hε n x else L
have huniform : ∀ δ : ℝ, 0 < δ → ∀ᶠ n in atTop,
∀ (hmn : m n ≤ n) (u : Fin (m n) → Fin 4),
dist (pilotAdaptivePrefixTheta p ε m hmn u) θ ≤ ρ n →
|E n u - L| < δ := by
apply pilotPrefixCDF_uniform_of_subsequential_tendsto θ p ε m ρ E L
(pilotGaussianRadius_tendsto_zero θ h m hdiv)
· intro φ hφ hφlarge u hu
let η : ℕ → TrialParameter := fun k => pilotAdaptivePrefixTheta p ε m
(le_of_lt (hφlarge k)) (u k)
have hηint : ∀ k, InteriorMeans (η k) := by
intro k
exact pilotTheta_interior p ε m
(pilotAdaptiveEncode _ (u k) (fun _ => 0))
have hev : ∀ᶠ k in atTop, 2 ≤ φ k :=
hφ.tendsto_atTop.eventually_ge_atTop 2
apply (hrow φ hφ η hηint hu).congr'
filter_upwards [hev] with k hk
simp only [E, dif_pos hk, η, Subsingleton.elim]
· filter_upwards [eventually_ge_atTop 2] with n hn
exact (hsub.2 n hn).2
have htarget0 : 0 ≤ L := measureReal_nonneg
have htarget1 : L ≤ 1 := by
letI : IsProbabilityMeasure (gaussianMeasure 0 (Vstar θ p ε)) := by
infer_instance
exact measureReal_le_one
let rowParam : ℕ → TrialParameter := fun n =>
if InteriorMeans (localAlternative θ h n) then localAlternative θ h n else θ
have hrowParamInterior (n : ℕ) : InteriorMeans (rowParam n) := by
dsimp only [rowParam]
split_ifs with hn
· exact hn
· exact hθ
let w : (n : ℕ) → (Fin (m n) → Fin 4) → ℝ := fun n =>
pilotPrefixWeight (rowParam n) p ε m n
have havg : Tendsto (fun n => ∑ u, w n u * E n u) atTop (nhds L) := by
apply finitePilot_conditionalCDF_tendsto w E L
· intro n u
exact pilotPrefixWeight_nonneg (rowParam n) p ε m n hp
(hrowParamInterior n) u
· intro n
exact pilotPrefixWeight_sum (rowParam n) p ε m n
· intro n u
by_cases hn : 2 ≤ n
· simp only [E, dif_pos hn]
let η : ℕ → TrialParameter := fun _ => pilotAdaptivePrefixTheta p ε m
(le_of_lt (hsub.2 n hn).2) u
have hηint : ∀ k, InteriorMeans (η k) := fun _ =>
pilotTheta_interior p ε m
(pilotAdaptiveEncode _ u (fun _ => 0))
unfold adaptivePilotRowScaledCDF
apply Finset.sum_nonneg
intro v _
apply mul_nonneg
· simpa only [η, Subsingleton.elim] using
adaptivePilotRowProductMass_nonneg select θ h η p ε m hselect hp hθ
hηint hε n v
· split_ifs <;> positivity
· simp only [E, dif_neg hn]
exact htarget0
· intro n u
by_cases hn : 2 ≤ n
· simp only [E, dif_pos hn]
let η : ℕ → TrialParameter := fun _ => pilotAdaptivePrefixTheta p ε m
(le_of_lt (hsub.2 n hn).2) u
have hηint : ∀ k, InteriorMeans (η k) := fun _ =>
pilotTheta_interior p ε m
(pilotAdaptiveEncode _ u (fun _ => 0))
unfold adaptivePilotRowScaledCDF
calc
_ ≤ ∑ v, adaptivePilotRowProductMass select θ h η p ε m hselect hp hε n v := by
apply Finset.sum_le_sum
intro v _
have hm0 := adaptivePilotRowProductMass_nonneg select θ h η p ε m hselect hp hθ
hηint hε n v
split_ifs
· simpa only [mul_one, η] using
(le_refl (adaptivePilotRowProductMass select θ h η p ε m hselect hp hε n v))
· simpa only [mul_zero] using hm0
_ = 1 := adaptivePilotRowProductMass_sum select θ h η p ε m hselect hp
hηint hε n
· simp only [E, dif_neg hn]
exact htarget1
· exact htarget0
· exact htarget1
· intro δ hδ
have hu := huniform δ hδ
have hbase0 : Tendsto base atTop (nhds 0) :=
pilotGaussianBaseRadius_tendsto_zero m hdiv
have hbasepos : ∀ᶠ n in atTop, 0 < base n := by
have hmpos : ∀ᶠ n in atTop, 0 < (m n : ℝ) :=
(tendsto_natCast_atTop_atTop.comp hdiv).eventually_gt_atTop 0
filter_upwards [hmpos] with n hn
exact Real.sqrt_pos.2 (inv_pos.mpr (Real.sqrt_pos.2 hn))
have hdist0 : Tendsto (fun n => dist (localAlternative θ h n) θ)
atTop (nhds 0) := by
simpa using (localAlternative_tendsto θ h).dist
(tendsto_const_nhds : Tendsto (fun _ : ℕ => θ) atTop (nhds θ))
have hdistle : ∀ᶠ n in atTop,
dist (localAlternative θ h n) θ ≤
sqrt (dist (localAlternative θ h n) θ) := by
have hdlt : ∀ᶠ n in atTop,
dist (localAlternative θ h n) θ < 1 :=
(tendsto_order.1 hdist0).2 1 zero_lt_one
filter_upwards [hdlt] with n hn
let d := dist (localAlternative θ h n) θ
have hd0 : 0 ≤ d := dist_nonneg
have hs0 : 0 ≤ sqrt d := Real.sqrt_nonneg d
have hs2 : (sqrt d) ^ 2 = d := Real.sq_sqrt hd0
have hd1 : d ≤ 1 := hn.le
nlinarith [sq_nonneg (d - sqrt d)]
let M := adaptiveInteriorMargin θ
have hM : 0 < M := adaptiveInteriorMargin_pos θ hθ
have hbaseSmall : ∀ᶠ n in atTop, base n < M / 4 :=
(tendsto_order.1 hbase0).2 _ (by positivity)
have hregSmall := pilotRegularizer_eventually_lt m hdiv (M / 4) (by positivity)
have hlocalClose : ∀ᶠ n in atTop,
dist (localAlternative θ h n) θ < M / 4 :=
(Metric.tendsto_atTop.1 (localAlternative_tendsto θ h) (M / 4)
(by positivity)) |> eventually_atTop.2
have hlocal := eventually_localAlternative_interior θ h hθ
have hn2 : ∀ᶠ n in atTop, 2 ≤ n := eventually_ge_atTop 2
have hprefixBad : Tendsto (fun n => if hn : 2 ≤ n then
pilotPrefixBadMass (localAlternative θ h n) p ε (base n) m
(le_of_lt (hsub.2 n hn).2) else 0) atTop (nhds 0) := by
have henv := pilotGaussianBadEnvelope_tendsto_zero p ε m hp hε hdiv
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds henv
· filter_upwards [hn2, hlocal] with n hn hln
rw [dif_pos hn]
unfold pilotPrefixBadMass
apply Finset.sum_nonneg
intro u _
exact mul_nonneg
(pilotPrefixWeight_nonneg _ p ε m n hp hln u)
(by split_ifs <;> positivity)
· filter_upwards [hn2, hlocal, hbasepos, hbaseSmall, hregSmall,
hlocalClose] with n hn hln hbpos hbsmall hreg hclose
rw [dif_pos hn]
let hmn : m n ≤ n := le_of_lt (hsub.2 n hn).2
have hmpos : 0 < m n := lt_of_lt_of_le Nat.zero_lt_one
(hsub.2 n hn).1
have hcoord : ∀ k : Fin 2,
|localAlternative θ h n k - θ k| < M / 4 := by
have hd := (dist_pi_lt_iff (by positivity : 0 < M / 4)).mp hclose
intro k
simpa only [Real.dist_eq] using hd k
have hM0l : M ≤ θ 0 := (min_le_left _ _).trans (min_le_left _ _)
have hM0r : M ≤ 1 - θ 0 :=
(min_le_left _ _).trans (min_le_right _ _)
have hM1l : M ≤ θ 1 := (min_le_right _ _).trans (min_le_left _ _)
have hM1r : M ≤ 1 - θ 1 :=
(min_le_right _ _).trans (min_le_right _ _)
have hδ0l : (m n + 2 : ℝ)⁻¹ + base n ≤
localAlternative θ h n 0 := by
have hc := (abs_lt.mp (hcoord 0)).1
dsimp only [M] at hM0l hreg hbsmall hc ⊢
linarith
have hδ0r : (m n + 2 : ℝ)⁻¹ + base n ≤
1 - localAlternative θ h n 0 := by
have hc := (abs_lt.mp (hcoord 0)).2
dsimp only [M] at hM0r hreg hbsmall hc ⊢
linarith
have hδ1l : (m n + 2 : ℝ)⁻¹ + base n ≤
localAlternative θ h n 1 := by
have hc := (abs_lt.mp (hcoord 1)).1
dsimp only [M] at hM1l hreg hbsmall hc ⊢
linarith
have hδ1r : (m n + 2 : ℝ)⁻¹ + base n ≤
1 - localAlternative θ h n 1 := by
have hc := (abs_lt.mp (hcoord 1)).2
dsimp only [M] at hM1r hreg hbsmall hc ⊢
linarith
simpa only [pilotBadMassEnvelope, base, Subsingleton.elim] using
pilotPrefixBadMass_le
select (localAlternative θ h n) p ε (base n) m hselect hp hln hε hbpos
hmn hmpos hδ0l hδ0r hδ1l hδ1r
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hprefixBad
· filter_upwards [] with n
apply Finset.sum_nonneg
intro u _
exact mul_nonneg (pilotPrefixWeight_nonneg (rowParam n) p ε m n hp
(hrowParamInterior n) u) (by split_ifs <;> positivity)
· filter_upwards [hn2, hlocal, hu, hbasepos, hdistle] with
n hn hln hunif hbpos hdle
rw [dif_pos hn]
apply Finset.sum_le_sum
intro u _
simp only [w, rowParam, if_pos hln]
apply mul_le_mul_of_nonneg_left _
(pilotPrefixWeight_nonneg (localAlternative θ h n) p ε m n hp hln u)
by_cases hgood :
|pilotAdaptivePrefixTheta p ε m
(le_of_lt (hsub.2 n hn).2) u 0 -
localAlternative θ h n 0| < base n ∧
|pilotAdaptivePrefixTheta p ε m
(le_of_lt (hsub.2 n hn).2) u 1 -
localAlternative θ h n 1| < base n
· have hnearRow : dist (pilotAdaptivePrefixTheta p ε m
(le_of_lt (hsub.2 n hn).2) u)
(localAlternative θ h n) < base n := by
rw [dist_pi_lt_iff hbpos]
intro k
fin_cases k
· rw [Real.dist_eq]
change |pilotAdaptivePrefixTheta p ε m
(le_of_lt (hsub.2 n hn).2) u 0 -
localAlternative θ h n 0| < base n
exact hgood.1
· rw [Real.dist_eq]
change |pilotAdaptivePrefixTheta p ε m
(le_of_lt (hsub.2 n hn).2) u 1 -
localAlternative θ h n 1| < base n
exact hgood.2
have hnear : dist (pilotAdaptivePrefixTheta p ε m
(le_of_lt (hsub.2 n hn).2) u) θ ≤ ρ n := by
calc
_ ≤ dist (pilotAdaptivePrefixTheta p ε m
(le_of_lt (hsub.2 n hn).2) u)
(localAlternative θ h n) +
dist (localAlternative θ h n) θ := dist_triangle _ _ _
_ ≤ max (base n) (sqrt
(dist (localAlternative θ h n) θ)) +
-- … truncated; follow the source link for the rest …
theorem pilot_scaledError_cdf_tendsto_of_all_conditionalRows
A conditional adaptive-row Gaussian CDF limit for every local direction and threshold transfers to the corresponding family of actual transcript CDF limits. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot scaled Error cdf tendsto of all conditional Rows.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hdiv :
hsub :
hrow :
∀ (h : TrialParameter) (x : ℝ) (φ : ℕ → ℕ),
StrictMono φ → ∀ η : ℕ → TrialParameter, (∀ k, InteriorMeans (η k)) → Tendsto η atTop (nhds θ) → Tendsto (fun k => adaptivePilotRowScaledCDF select θ h (fun _ => η k) p ε m hselect hp hε (φ k) x) atTop (nhds ((gaussianMeasure 0 (Vstar θ p ε)).real (Iic x)))
h :
x :
ℝ
conclusion
Tendsto (fun n => (transcriptLaw (pilotEstimator p ε m select hselect hp hε) (localAlternative θ h n) p n).real {z | scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z ≤ x}) atTop (nhds ((gaussianMeasure 0 (Vstar θ p ε)).real (Iic x)))
Proof (Lean source)
theorem pilot_scaledError_cdf_tendsto_of_all_conditionalRows
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hdiv : PilotDiverges m) (hsub : PilotSublinear m)
(hrow : ∀ (h : TrialParameter) (x : ℝ) (φ : ℕ → ℕ), StrictMono φ →
∀ η : ℕ → TrialParameter,
(∀ k, InteriorMeans (η k)) → Tendsto η atTop (nhds θ) →
Tendsto (fun k => adaptivePilotRowScaledCDF select θ h
(fun _ => η k) p ε m hselect hp hε (φ k) x) atTop
(nhds ((gaussianMeasure 0 (Vstar θ p ε)).real (Iic x)))) :
∀ h : TrialParameter, ∀ x : ℝ,
Tendsto (fun n =>
(transcriptLaw (pilotEstimator p ε m select hselect hp hε)
(localAlternative θ h n) p n).real
{z | scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z ≤ x})
atTop (nhds ((gaussianMeasure 0 (Vstar θ p ε)).real (Iic x))) := by
intro h x
exact pilot_scaledError_cdf_tendsto_of_conditionalRows select θ h p ε x m hselect hp hθ hε
hdiv hsub (hrow h x)
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotGrowth 29 declarations Uniform growth bounds for the pilot saddle score
Uniform growth bounds for the pilot saddle score
theorem patternRay_le_exp
Under the supplied quantities and conditions, the pattern ray le exp assertion holds. Under the stated assumptions, the pattern Ray le exp.
Formal statement
Proof (Lean source)
lemma patternRay_le_exp (ε : ℝ) (hε : 0 ≤ ε) (s : Fin 14) (j : Fin 4) :
patternRay ε s j ≤ exp ε := by
unfold patternRay privacyIncrement privacyRatio
split_ifs <;> simp_all [Real.one_le_exp_iff]
theorem patternMass_one_le
Under the supplied quantities and conditions, the pattern mass one le assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pattern Mass one le.
Formal statement
hypotheses
conclusion
1 ≤ patternMass θ p ε s
Proof (Lean source)
lemma patternMass_one_le (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 ≤ ε)
(s : Fin 14) :
1 ≤ patternMass θ p ε s := by
rw [← piTheta_sum_eq_one θ p]
unfold patternMass
apply Finset.sum_le_sum
intro j hj
rw [← mul_one (piTheta θ p j)]
simpa only [mul_one] using
mul_le_mul_of_nonneg_left (patternRay_one_le_interior ε hε s j)
(piTheta_pos_interior θ p hp hθ j).le
theorem patternMass_le_exp
Under the supplied quantities and conditions, the pattern mass le exp assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pattern Mass le exp.
Formal statement
hypotheses
conclusion
patternMass θ p ε s ≤ exp ε
Proof (Lean source)
lemma patternMass_le_exp (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 ≤ ε)
(s : Fin 14) :
patternMass θ p ε s ≤ exp ε := by
rw [← one_mul (exp ε), ← piTheta_sum_eq_one θ p,
Finset.sum_mul]
unfold patternMass
apply Finset.sum_le_sum
intro j hj
exact mul_le_mul_of_nonneg_left (patternRay_le_exp ε hε s j)
(piTheta_pos_interior θ p hp hθ j).le
def pilotReferenceWeight
the pilot reference weight is the mathematical object specified below. The pilot Reference Weight is determined by the displayed parameters.
Definition (Lean source)
parameters
ε :
ℝ
def
theorem pilotReferenceWeight_feasible
the pilot reference weight feasible assertion holds.
Formal statement
Proof (Lean source)
lemma pilotReferenceWeight_feasible (ε : ℝ) :
staircaseFeasible ε (pilotReferenceWeight ε) := by
exact staircaseFeasible_singleton_design ε
def pilotLowerA
For the supplied quantities and conditions, the pilot lower a is the mathematical object specified below. The pilot Lower A is determined by the displayed parameters.
Definition (Lean source)
parameters
p ε :
ℝ
def
pilotLowerA p ε :
ℝ
given by
def pilotLowerB
For the supplied quantities and conditions, the pilot lower b is the mathematical object specified below. The pilot Lower B is determined by the displayed parameters.
Definition (Lean source)
parameters
p ε :
ℝ
def
pilotLowerB p ε :
ℝ
given by
theorem pilotLowerA_pos
Under the supplied quantities and conditions, the pilot lower a pos assertion holds. Under the stated assumptions, the pilot Lower A pos.
Formal statement
Proof (Lean source)
lemma pilotLowerA_pos (p ε : ℝ) (hp : InteriorAssignment p) (hε : 0 < ε) :
0 < pilotLowerA p ε := by
have hq : 0 < controlProb p := by dsimp [controlProb]; linarith [hp.2]
have hd := privacyIncrement_pos hε
dsimp [pilotLowerA]
positivity
theorem pilotLowerB_pos
Under the supplied quantities and conditions, the pilot lower b pos assertion holds. Under the stated assumptions, the pilot Lower B pos.
Formal statement
Proof (Lean source)
lemma pilotLowerB_pos (p ε : ℝ) (hp : InteriorAssignment p) (hε : 0 < ε) :
0 < pilotLowerB p ε := by
have hd := privacyIncrement_pos hε
have hp0 := hp.1
dsimp [pilotLowerB]
positivity
theorem seventh_pattern_information_quadratic
Under the supplied quantities and conditions, the seventh pattern information quadratic assertion holds.
Formal statement
hypotheses
θ :
p ε t :
ℝ
conclusion
patternInformation θ p ε 7 t
= ((privacyIncrement ε * p) ^ 2 / patternMass θ p ε 7) * (t + 1) ^ 2
Proof (Lean source)
lemma seventh_pattern_information_quadratic
(θ : TrialParameter) (p ε t : ℝ) :
patternInformation θ p ε 7 t =
((privacyIncrement ε * p) ^ 2 / patternMass θ p ε 7) * (t + 1) ^ 2 := by
simp +decide [patternInformation, projectedGradient, patternGradient,
direction, Fin.sum_univ_succ]
ring
theorem pilotReferenceObjective_lower
Under the supplied quantities and conditions, the pilot reference objective lower assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Reference Objective lower.
Formal statement
hypotheses
conclusion
pilotLowerA p ε * t ^ 2 + pilotLowerB p ε * (t + 1) ^ 2
≤ informationObjective θ p ε (pilotReferenceWeight ε) t
Proof (Lean source)
lemma pilotReferenceObjective_lower (θ : TrialParameter) (p ε t : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε) :
pilotLowerA p ε * t ^ 2 + pilotLowerB p ε * (t + 1) ^ 2 ≤
informationObjective θ p ε (pilotReferenceWeight ε) t := by
have hm0pos := patternMass_pos_interior θ p ε hp hθ hε.le 0
have hm7pos := patternMass_pos_interior θ p ε hp hθ hε.le 7
have hm0le := patternMass_le_exp θ p ε hp hθ hε.le 0
have hm7le := patternMass_le_exp θ p ε hp hθ hε.le 7
have hterm0 : pilotLowerA p ε * t ^ 2 ≤
pilotReferenceWeight ε 0 * patternInformation θ p ε 0 t := by
rw [first_pattern_information_quadratic]
simp only [pilotReferenceWeight, Finset.mem_insert, Finset.mem_singleton,
true_or, if_true]
dsimp [pilotLowerA]
have hn : 0 ≤ (privacyIncrement ε * controlProb p) ^ 2 := sq_nonneg _
have hd := (div_le_div_iff₀ (Real.exp_pos ε) hm0pos).2
(mul_le_mul_of_nonneg_left hm0le hn)
have hc : 0 ≤ 1 / (exp ε + 3) := by positivity
calc
1 / (exp ε + 3) * (privacyIncrement ε * controlProb p) ^ 2 /
exp ε * t ^ 2 =
1 / (exp ε + 3) *
((privacyIncrement ε * controlProb p) ^ 2 / exp ε * t ^ 2) := by ring
_ ≤ 1 / (exp ε + 3) *
((privacyIncrement ε * controlProb p) ^ 2 /
patternMass θ p ε 0 * t ^ 2) :=
mul_le_mul_of_nonneg_left
(mul_le_mul_of_nonneg_right hd (sq_nonneg t)) hc
have hterm7 : pilotLowerB p ε * (t + 1) ^ 2 ≤
pilotReferenceWeight ε 7 * patternInformation θ p ε 7 t := by
rw [seventh_pattern_information_quadratic]
simp only [pilotReferenceWeight, Finset.mem_insert, Finset.mem_singleton,
OfNat.ofNat_ne_zero, OfNat.ofNat_ne_one, or_false, true_or, if_true]
dsimp [pilotLowerB]
have hn : 0 ≤ (privacyIncrement ε * p) ^ 2 := sq_nonneg _
have hd := (div_le_div_iff₀ (Real.exp_pos ε) hm7pos).2
(mul_le_mul_of_nonneg_left hm7le hn)
have hc : 0 ≤ 1 / (exp ε + 3) := by positivity
calc
1 / (exp ε + 3) * (privacyIncrement ε * p) ^ 2 /
exp ε * (t + 1) ^ 2 =
1 / (exp ε + 3) *
((privacyIncrement ε * p) ^ 2 / exp ε * (t + 1) ^ 2) := by ring
_ ≤ 1 / (exp ε + 3) *
((privacyIncrement ε * p) ^ 2 /
patternMass θ p ε 7 * (t + 1) ^ 2) :=
mul_le_mul_of_nonneg_left
(mul_le_mul_of_nonneg_right hd (sq_nonneg (t + 1))) hc
calc
pilotLowerA p ε * t ^ 2 + pilotLowerB p ε * (t + 1) ^ 2 ≤
pilotReferenceWeight ε 0 * patternInformation θ p ε 0 t +
pilotReferenceWeight ε 7 * patternInformation θ p ε 7 t :=
add_le_add hterm0 hterm7
_ = ∑ s ∈ ({0, 7} : Finset (Fin 14)),
pilotReferenceWeight ε s * patternInformation θ p ε s t := by
simp
_ ≤ informationObjective θ p ε (pilotReferenceWeight ε) t := by
unfold informationObjective
apply Finset.sum_le_sum_of_subset_of_nonneg (by simp)
intro s hs hnot
exact mul_nonneg ((pilotReferenceWeight_feasible ε).1 s)
(div_nonneg (sq_nonneg _)
(patternMass_pos_interior θ p ε hp hθ hε.le s).le)
def pilotJLower
For the supplied quantities and conditions, the pilot j lower is the mathematical object specified below. The pilot JLower is determined by the displayed parameters.
Definition (Lean source)
parameters
p ε :
ℝ
def
pilotJLower p ε :
ℝ
given by
theorem pilotJLower_pos
Under the supplied quantities and conditions, the pilot j lower pos assertion holds. Under the stated assumptions, the pilot JLower pos.
Formal statement
Proof (Lean source)
lemma pilotJLower_pos (p ε : ℝ) (hp : InteriorAssignment p) (hε : 0 < ε) :
0 < pilotJLower p ε := by
exact div_pos (mul_pos (pilotLowerA_pos p ε hp hε)
(pilotLowerB_pos p ε hp hε))
(add_pos (pilotLowerA_pos p ε hp hε) (pilotLowerB_pos p ε hp hε))
theorem pilotJLower_le_quadratic
Under the supplied quantities and conditions, the pilot j lower le quadratic assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot JLower le quadratic.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma pilotJLower_le_quadratic (p ε t : ℝ)
(hp : InteriorAssignment p) (hε : 0 < ε) :
pilotJLower p ε ≤
pilotLowerA p ε * t ^ 2 + pilotLowerB p ε * (t + 1) ^ 2 := by
let A := pilotLowerA p ε
let B := pilotLowerB p ε
have hA : 0 < A := pilotLowerA_pos p ε hp hε
have hB : 0 < B := pilotLowerB_pos p ε hp hε
have hsq : 0 ≤ (A * t + B * (t + 1)) ^ 2 := sq_nonneg _
dsimp [pilotJLower, A, B]
apply (div_le_iff₀ (add_pos hA hB)).2
nlinarith
theorem staircaseWeight_le_one
Under the supplied quantities and conditions, the staircase weight le one assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the staircase Weight le one.
Formal statement
hypotheses
conclusion
α s ≤ 1
Proof (Lean source)
lemma staircaseWeight_le_one (ε : ℝ) (hε : 0 ≤ ε)
(α : StaircaseWeight) (hα : staircaseFeasible ε α) (s : Fin 14) :
α s ≤ 1 := by
have hterm : α s * patternRay ε s 0 ≤
∑ u : Fin 14, α u * patternRay ε u 0 := by
apply Finset.single_le_sum (s := univ) (a := s)
· intro u hu
exact mul_nonneg (hα.1 u)
(le_trans (by norm_num) (patternRay_one_le_interior ε hε u 0))
· exact Finset.mem_univ s
calc
α s = α s * 1 := by ring
_ ≤ α s * patternRay ε s 0 :=
mul_le_mul_of_nonneg_left (patternRay_one_le_interior ε hε s 0) (hα.1 s)
_ ≤ _ := hterm
_ = 1 := hα.2 0
def pilotObjectiveZeroUpper
For the supplied quantities and conditions, the pilot objective zero upper is the mathematical object specified below. The pilot Objective Zero Upper is determined by the displayed parameters.
Definition (Lean source)
parameters
p ε :
ℝ
def
pilotObjectiveZeroUpper p ε :
ℝ
given by
∑ s : Fin 14, projectedGradient p ε s 0 ^ 2
theorem informationObjective_zero_le_upper
Under the supplied quantities and conditions, the information objective zero le upper assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the information Objective zero le upper.
Formal statement
hypotheses
θ :
p ε :
ℝ
hp :
hθ :
hε :
0 ≤ ε
α :
hα :
conclusion
informationObjective θ p ε α 0 ≤ pilotObjectiveZeroUpper p ε
Proof (Lean source)
lemma informationObjective_zero_le_upper (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 ≤ ε)
(α : StaircaseWeight) (hα : staircaseFeasible ε α) :
informationObjective θ p ε α 0 ≤ pilotObjectiveZeroUpper p ε := by
unfold informationObjective patternInformation pilotObjectiveZeroUpper
apply Finset.sum_le_sum
intro s hs
have hm := patternMass_one_le θ p ε hp hθ hε s
have ha := staircaseWeight_le_one ε hε α hα s
have hdiv : projectedGradient p ε s 0 ^ 2 / patternMass θ p ε s ≤
projectedGradient p ε s 0 ^ 2 := by
apply (div_le_iff₀ (lt_of_lt_of_le (by norm_num) hm)).2
nlinarith [sq_nonneg (projectedGradient p ε s 0)]
calc
α s * (projectedGradient p ε s 0 ^ 2 / patternMass θ p ε s) ≤
1 * (projectedGradient p ε s 0 ^ 2 / patternMass θ p ε s) := by
exact mul_le_mul_of_nonneg_right ha
(div_nonneg (sq_nonneg _) (le_trans (by norm_num) hm))
_ ≤ projectedGradient p ε s 0 ^ 2 := by simpa using hdiv
theorem strongSelector_J_bounds
Under the supplied quantities and conditions, the strong selector j bounds assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the strong Selector J bounds.
Formal statement
hypotheses
p :
ℝ
ε :
ℝ
select :
TrialParameter → StaircaseWeight × ℝ × ℝ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
θ :
hθ :
conclusion
conclusion 1
pilotJLower p ε ≤ (select θ).2.2
conclusion 2
(select θ).2.2 ≤ pilotObjectiveZeroUpper p ε
Proof (Lean source)
lemma strongSelector_J_bounds (select : TrialParameter → StaircaseWeight × ℝ × ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε)
(θ : TrialParameter) (hθ : InteriorMeans θ) :
pilotJLower p ε ≤ (select θ).2.2 ∧
(select θ).2.2 ≤ pilotObjectiveZeroUpper p ε := by
have hs := hselect.1.2 θ hθ
have hmax := hselect.2 θ hθ
have href := pilotReferenceObjective_lower θ p ε (select θ).2.1 hp hθ hε
have hquad := pilotJLower_le_quadratic p ε (select θ).2.1 hp hε
constructor
· calc
pilotJLower p ε ≤ pilotLowerA p ε * (select θ).2.1 ^ 2 +
pilotLowerB p ε * ((select θ).2.1 + 1) ^ 2 := hquad
_ ≤ informationObjective θ p ε (pilotReferenceWeight ε) (select θ).2.1 := href
_ ≤ informationObjective θ p ε (select θ).1 (select θ).2.1 :=
hmax (pilotReferenceWeight ε) (pilotReferenceWeight_feasible ε)
_ = (select θ).2.2 := hs.2.2.2.symm
· calc
(select θ).2.2 =
informationObjective θ p ε (select θ).1 (select θ).2.1 := hs.2.2.2
_ ≤ informationObjective θ p ε (select θ).1 0 := hs.2.1 0
_ ≤ pilotObjectiveZeroUpper p ε :=
informationObjective_zero_le_upper θ p ε hp hθ hε.le _ hs.1
def pilotTUpper
For the supplied quantities and conditions, the pilot t upper is the mathematical object specified below. The pilot TUpper is determined by the displayed parameters.
Definition (Lean source)
parameters
p ε :
ℝ
def
pilotTUpper p ε :
ℝ
given by
theorem strongSelector_t_bound
Under the supplied quantities and conditions, the strong selector t bound assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the strong Selector t bound.
Formal statement
hypotheses
p :
ℝ
ε :
ℝ
select :
TrialParameter → StaircaseWeight × ℝ × ℝ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
θ :
hθ :
conclusion
|(select θ).2.1| ≤ pilotTUpper p ε
Proof (Lean source)
lemma strongSelector_t_bound (select : TrialParameter → StaircaseWeight × ℝ × ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε)
(θ : TrialParameter) (hθ : InteriorMeans θ) :
|(select θ).2.1| ≤ pilotTUpper p ε := by
have hs := hselect.1.2 θ hθ
have hmax := hselect.2 θ hθ
have href := pilotReferenceObjective_lower θ p ε (select θ).2.1 hp hθ hε
have hJupper := (strongSelector_J_bounds select hselect hp hε θ hθ).2
have hAt : pilotLowerA p ε * (select θ).2.1 ^ 2 ≤
pilotObjectiveZeroUpper p ε := by
calc
pilotLowerA p ε * (select θ).2.1 ^ 2 ≤
pilotLowerA p ε * (select θ).2.1 ^ 2 +
pilotLowerB p ε * ((select θ).2.1 + 1) ^ 2 := by
exact le_add_of_nonneg_right (mul_nonneg
(pilotLowerB_pos p ε hp hε).le (sq_nonneg _))
_ ≤ informationObjective θ p ε (pilotReferenceWeight ε) (select θ).2.1 := href
_ ≤ informationObjective θ p ε (select θ).1 (select θ).2.1 :=
hmax (pilotReferenceWeight ε) (pilotReferenceWeight_feasible ε)
_ = (select θ).2.2 := hs.2.2.2.symm
_ ≤ pilotObjectiveZeroUpper p ε := hJupper
have hratio : (select θ).2.1 ^ 2 ≤
pilotObjectiveZeroUpper p ε / pilotLowerA p ε := by
exact (le_div_iff₀ (pilotLowerA_pos p ε hp hε)).2 (by simpa [mul_comm] using hAt)
rw [pilotTUpper, Real.le_sqrt (abs_nonneg _)
(div_nonneg (by unfold pilotObjectiveZeroUpper; positivity)
(pilotLowerA_pos p ε hp hε).le)]
simpa [sq_abs] using hratio
def pilotScoreGrowthUpper
For the supplied quantities and conditions, the pilot score growth upper is the mathematical object specified below. The pilot Score Growth Upper is determined by the displayed parameters.
Definition (Lean source)
parameters
p ε :
ℝ
def
pilotScoreGrowthUpper p ε :
ℝ
given by
(1 + pilotTUpper p ε) / pilotJLower p ε
theorem strongSelector_scoreGrowth_bound
Under the supplied quantities and conditions, the strong selector score growth bound assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the strong Selector score Growth bound.
Formal statement
hypotheses
conclusion
(1 + |(select θ).2.1|) / (select θ).2.2 ≤ pilotScoreGrowthUpper p ε
Proof (Lean source)
lemma strongSelector_scoreGrowth_bound (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε)
(θ : TrialParameter) (hθ : InteriorMeans θ) :
(1 + |(select θ).2.1|) / (select θ).2.2 ≤
pilotScoreGrowthUpper p ε := by
have hJ := strongSelector_J_bounds select hselect hp hε θ hθ
have ht := strongSelector_t_bound select hselect hp hε θ hθ
have hT : 0 ≤ pilotTUpper p ε := by
unfold pilotTUpper
exact Real.sqrt_nonneg _
apply div_le_div₀
· exact add_nonneg (by norm_num) hT
· linarith
· exact pilotJLower_pos p ε hp hε
· exact hJ.1
def pilotGradientUpper
For the supplied quantities and conditions, the pilot gradient upper is the mathematical object specified below. The pilot Gradient Upper is determined by the displayed parameters.
Definition (Lean source)
parameters
p ε :
ℝ
def
pilotGradientUpper p ε :
ℝ
given by
theorem pilotGradientUpper_nonneg
Under the supplied quantities and conditions, the pilot gradient upper nonneg assertion holds.
Formal statement
Proof (Lean source)
lemma pilotGradientUpper_nonneg (p ε : ℝ) : 0 ≤ pilotGradientUpper p ε := by
unfold pilotGradientUpper
positivity
theorem projectedGradient_abs_le
Under the supplied quantities and conditions, the projected gradient abs le assertion holds.
Formal statement
hypotheses
p ε t :
ℝ
s :
Fin 14
conclusion
|projectedGradient p ε s t| ≤ pilotGradientUpper p ε * (|t| + 1)
Proof (Lean source)
lemma projectedGradient_abs_le (p ε t : ℝ) (s : Fin 14) :
|projectedGradient p ε s t| ≤
pilotGradientUpper p ε * (|t| + 1) := by
have hlocal :
|patternGradient p ε s 0| + |patternGradient p ε s 1| ≤
pilotGradientUpper p ε := by
unfold pilotGradientUpper
apply Finset.single_le_sum (s := univ) (a := s)
· intro u hu
positivity
· exact Finset.mem_univ s
have ht1 : |t + 1| ≤ |t| + 1 := by
calc
|t + 1| ≤ |t| + |(1 : ℝ)| := abs_add_le _ _
_ = |t| + 1 := by norm_num
unfold projectedGradient
simp only [Fin.sum_univ_two, direction, ↓reduceIte, Fin.isValue]
calc
|patternGradient p ε s 0 * t + patternGradient p ε s 1 * (t + 1)| ≤
|patternGradient p ε s 0 * t| +
|patternGradient p ε s 1 * (t + 1)| := abs_add_le _ _
_ = |patternGradient p ε s 0| * |t| +
|patternGradient p ε s 1| * |t + 1| := by rw [abs_mul, abs_mul]
_ ≤ |patternGradient p ε s 0| * (|t| + 1) +
|patternGradient p ε s 1| * (|t| + 1) := by
exact add_le_add
(mul_le_mul_of_nonneg_left (le_add_of_nonneg_right (by norm_num)) (abs_nonneg _))
(mul_le_mul_of_nonneg_left ht1 (abs_nonneg _))
_ = (|patternGradient p ε s 0| + |patternGradient p ε s 1|) *
(|t| + 1) := by ring
_ ≤ pilotGradientUpper p ε * (|t| + 1) := by
exact mul_le_mul_of_nonneg_right hlocal (by positivity)
def pilotPhiUpper
For the supplied quantities and conditions, the pilot phi upper is the mathematical object specified below. The pilot Phi Upper is determined by the displayed parameters.
Definition (Lean source)
parameters
p ε :
ℝ
def
pilotPhiUpper p ε :
ℝ
given by
theorem strongSelector_phiTilde_bound
Under the supplied quantities and conditions, the strong selector phi tilde bound assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the strong Selector phi Tilde bound.
Formal statement
hypotheses
conclusion
|phiTilde θ p ε (select) s| ≤ pilotPhiUpper p ε
Proof (Lean source)
lemma strongSelector_phiTilde_bound (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε)
(θ : TrialParameter) (hθ : InteriorMeans θ) (s : Fin 14) :
|phiTilde θ p ε (select) s| ≤ pilotPhiUpper p ε := by
have hJ := strongSelector_J_bounds select hselect hp hε θ hθ
have ht := strongSelector_t_bound select hselect hp hε θ hθ
have hT : 0 ≤ pilotTUpper p ε := by
unfold pilotTUpper
exact Real.sqrt_nonneg _
have hm := patternMass_one_le θ p ε hp hθ hε.le s
have hJpos : 0 < (select θ).2.2 :=
lt_of_lt_of_le (pilotJLower_pos p ε hp hε) hJ.1
have hden : pilotJLower p ε ≤
(select θ).2.2 * patternMass θ p ε s := by
calc
pilotJLower p ε ≤ (select θ).2.2 := hJ.1
_ = (select θ).2.2 * 1 := by ring
_ ≤ (select θ).2.2 * patternMass θ p ε s :=
mul_le_mul_of_nonneg_left hm hJpos.le
rw [phiTilde, abs_div, abs_of_pos
(mul_pos hJpos (lt_of_lt_of_le (by norm_num) hm))]
apply div_le_div₀
· exact mul_nonneg (pilotGradientUpper_nonneg p ε)
(add_nonneg hT (by norm_num))
· calc
|projectedGradient p ε s (select θ).2.1| ≤
pilotGradientUpper p ε * (|(select θ).2.1| + 1) :=
projectedGradient_abs_le p ε _ s
_ ≤ pilotGradientUpper p ε * (pilotTUpper p ε + 1) := by
exact mul_le_mul_of_nonneg_left (by linarith)
(pilotGradientUpper_nonneg p ε)
· exact pilotJLower_pos p ε hp hε
· exact hden
theorem pilotPhiUpper_nonneg
Under the supplied quantities and conditions, the pilot phi upper nonneg assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Phi Upper nonneg.
Formal statement
Proof (Lean source)
lemma pilotPhiUpper_nonneg (p ε : ℝ)
(hp : InteriorAssignment p) (hε : 0 < ε) :
0 ≤ pilotPhiUpper p ε := by
unfold pilotPhiUpper
exact div_nonneg
(mul_nonneg (pilotGradientUpper_nonneg p ε)
(add_nonneg (by unfold pilotTUpper; exact Real.sqrt_nonneg _) (by norm_num)))
(pilotJLower_pos p ε hp hε).le
theorem strongSelector_phiOutput_bound
Under the supplied quantities and conditions, the strong selector phi output bound assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the strong Selector phi Output bound.
Formal statement
hypotheses
conclusion
|phiOutput θ p ε (select) z| ≤ pilotPhiUpper p ε
Proof (Lean source)
lemma strongSelector_phiOutput_bound (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε)
(θ : TrialParameter) (hθ : InteriorMeans θ) (z : PilotOutput) :
|phiOutput θ p ε (select) z| ≤ pilotPhiUpper p ε := by
cases z with
| inl k =>
simp [phiOutput, pilotPhiUpper_nonneg p ε hp hε]
| inr s =>
simpa [phiOutput] using
strongSelector_phiTilde_bound select p ε hselect hp hε θ hθ s
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotLocalRiskAttainment 10 declarations Exact local-risk attainment for the private-pilot estimator
Exact local-risk attainment for the private-pilot estimator
def pilotBadMassEnvelope
The parameter-free Chebyshev envelope appearing in the bad-prefix bound. For the displayed inputs and conditions, the stated result follows. The pilot Bad Mass Envelope is determined by the displayed parameters.
Definition (Lean source)
parameters
p ε r :
ℝ
m :
ℕ → ℕ
n :
ℕ
def
pilotBadMassEnvelope p ε r m n :
ℝ
given by
theorem pilotBadMassEnvelope_tendsto_zero
For every fixed positive localization radius, pilot divergence makes the uniform Chebyshev envelope vanish. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Bad Mass Envelope tendsto zero.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma pilotBadMassEnvelope_tendsto_zero
(p ε r : ℝ) (m : ℕ → ℕ) (hp : InteriorAssignment p)
(hε : 0 < ε) (hr : 0 < r) (hdiv : PilotDiverges m) :
Tendsto (pilotBadMassEnvelope p ε r m) atTop (nhds 0) := by
let a0 := r * controlProb p * (exp ε - 1) / (exp ε + 3)
let a1 := r * p * (exp ε - 1) / (exp ε + 3)
have hep : 1 < exp ε := by
simpa using (Real.exp_lt_exp.mpr hε : exp 0 < exp ε)
have ha0 : 0 < a0 := by
dsimp [a0, controlProb]
have hcp : 0 < 1 - p := by linarith [hp.2]
exact div_pos (mul_pos (mul_pos hr hcp) (sub_pos.mpr hep)) (by positivity)
have ha1 : 0 < a1 := by
dsimp [a1]
exact div_pos (mul_pos (mul_pos hr hp.1) (sub_pos.mpr hep)) (by positivity)
have hmreal : Tendsto (fun n => (m n : ℝ)) atTop atTop :=
tendsto_natCast_atTop_atTop.comp hdiv
have hinv : Tendsto (fun n => ((m n : ℝ))⁻¹) atTop (nhds 0) :=
tendsto_inv_atTop_zero.comp hmreal
have h0 : Tendsto (fun n => 1 / ((m n : ℝ) * a0 ^ 2)) atTop (nhds 0) := by
have ht := hinv.mul_const (a0 ^ 2)⁻¹
convert ht using 1
· funext n
rw [one_div, mul_inv_rev]
ring
· simp
have h1 : Tendsto (fun n => 1 / ((m n : ℝ) * a1 ^ 2)) atTop (nhds 0) := by
have ht := hinv.mul_const (a1 ^ 2)⁻¹
convert ht using 1
· funext n
rw [one_div, mul_inv_rev]
ring
· simp
change Tendsto (fun n => 1 / ((m n : ℝ) * a0 ^ 2) +
1 / ((m n : ℝ) * a1 ^ 2)) atTop (nhds 0)
simpa using h0.add h1
theorem pilotRegularizer_eventually_lt
The clipping regularizer is eventually smaller than every fixed positive margin when the pilot size diverges. For the displayed inputs and conditions, the stated result follows.
Formal statement
hypotheses
conclusion
∀ᶠ n in atTop, (m n + 2 : ℝ)⁻¹ < c
Proof (Lean source)
lemma pilotRegularizer_eventually_lt
(m : ℕ → ℕ) (hdiv : PilotDiverges m) (c : ℝ) (hc : 0 < c) :
∀ᶠ n in atTop, (m n + 2 : ℝ)⁻¹ < c := by
have hmreal : Tendsto (fun n => (m n : ℝ)) atTop atTop :=
tendsto_natCast_atTop_atTop.comp hdiv
have hadd : Tendsto (fun n => (m n : ℝ) + 2) atTop atTop :=
by
rw [tendsto_atTop]
intro b
filter_upwards [hmreal.eventually_ge_atTop b] with n hn
linarith
have hinv : Tendsto (fun n => ((m n : ℝ) + 2)⁻¹) atTop (nhds 0) :=
tendsto_inv_atTop_zero.comp hadd
have hev := (tendsto_order.1 hinv).2 c hc
simpa only [Nat.cast_add, Nat.cast_ofNat] using hev
theorem eventually_uniform_pilotPrefixVarianceAverage
For every bounded local direction, the genuine prefix-averaged selected variance converges uniformly to the oracle variance. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the eventually uniform pilot Prefix Variance Average.
Formal statement
hypotheses
θ :
p ε H :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hdiv :
hsub :
δ :
ℝ
0 < δ
conclusion
≤ H → InteriorMeans (localAlternative θ h n) → |pilotPrefixVarianceAverage select (localAlternative θ h n) p ε m (le_of_lt (hsub.2 n hn).2) hselect hp hε
- Vstar θ p ε| < δ
Proof (Lean source)
lemma eventually_uniform_pilotPrefixVarianceAverage
(θ : TrialParameter) (p ε H : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hdiv : PilotDiverges m) (hsub : PilotSublinear m) :
∀ δ : ℝ, 0 < δ → ∀ᶠ n in atTop,
∀ (hn : 2 ≤ n) (h : TrialParameter),
sqrt ((h 0) ^ 2 + (h 1) ^ 2) ≤ H →
InteriorMeans (localAlternative θ h n) →
|pilotPrefixVarianceAverage select (localAlternative θ h n) p ε m
(le_of_lt (hsub.2 n hn).2) hselect hp hε - Vstar θ p ε| < δ := by
intro δ hδ
obtain ⟨R, hR, hcont⟩ :=
adaptiveScoreVariance_uniform_near_diag select θ p ε hselect hp hθ hε (δ / 2) (half_pos hδ)
let M := adaptiveInteriorMargin θ
have hM : 0 < M := adaptiveInteriorMargin_pos θ hθ
let r := min (R / 3) (M / 8)
have hr : 0 < r := by dsimp [r]; positivity
have hrR : r ≤ R / 3 := min_le_left _ _
have hrM : r ≤ M / 8 := min_le_right _ _
let C := (pilotPhiUpper p ε) ^ 2 + |Vstar θ p ε|
have hC : 0 ≤ C := by dsimp [C]; positivity
have hloc := eventually_localAlternative_dist_lt_uniform θ H r hr
have hreg := pilotRegularizer_eventually_lt m hdiv (M / 4) (by positivity)
have henv : ∀ᶠ n in atTop,
pilotBadMassEnvelope p ε r m n < δ / (2 * (C + 1)) :=
(tendsto_order.1 (pilotBadMassEnvelope_tendsto_zero
p ε r m hp hε hr hdiv)).2 _ (by positivity)
filter_upwards [hloc, hreg, henv] with n hnloc hnreg hnenv
intro hn h hH hlocal
let hmn : m n ≤ n := le_of_lt (hsub.2 n hn).2
have hmpos : 0 < m n := lt_of_lt_of_le Nat.zero_lt_one (hsub.2 n hn).1
have hd : ∀ k : Fin 2, dist (localAlternative θ h n k) (θ k) < r :=
(dist_pi_lt_iff hr).mp (hnloc h hH)
have hM0l : M ≤ θ 0 := (min_le_left _ _).trans (min_le_left _ _)
have hM0r : M ≤ 1 - θ 0 := (min_le_left _ _).trans (min_le_right _ _)
have hM1l : M ≤ θ 1 := (min_le_right _ _).trans (min_le_left _ _)
have hM1r : M ≤ 1 - θ 1 := (min_le_right _ _).trans (min_le_right _ _)
have hd0 := hd 0
have hd1 := hd 1
rw [Real.dist_eq] at hd0 hd1
have hδ0l : (m n + 2 : ℝ)⁻¹ + r ≤ localAlternative θ h n 0 := by
have := (abs_lt.mp hd0).1
dsimp [M] at hM0l
dsimp [M] at hnreg hrM
linarith
have hδ0r : (m n + 2 : ℝ)⁻¹ + r ≤ 1 - localAlternative θ h n 0 := by
have := (abs_lt.mp hd0).2
dsimp [M] at hM0r
dsimp [M] at hnreg hrM
linarith
have hδ1l : (m n + 2 : ℝ)⁻¹ + r ≤ localAlternative θ h n 1 := by
have := (abs_lt.mp hd1).1
dsimp [M] at hM1l
dsimp [M] at hnreg hrM
linarith
have hδ1r : (m n + 2 : ℝ)⁻¹ + r ≤ 1 - localAlternative θ h n 1 := by
have := (abs_lt.mp hd1).2
dsimp [M] at hM1r
dsimp [M] at hnreg hrM
linarith
have hbad := pilotPrefixBadMass_le select (localAlternative θ h n) p ε r m
hselect hp hlocal hε hr hmn hmpos hδ0l hδ0r hδ1l hδ1r
have hbad' : pilotPrefixBadMass (localAlternative θ h n) p ε r m hmn ≤
pilotBadMassEnvelope p ε r m n := by
exact hbad
have hgood (η : TrialParameter) (hη : InteriorMeans η)
(hηclose : ∀ k : Fin 2, |η k - localAlternative θ h n k| < r) :
|adaptiveScoreVariance select (localAlternative θ h n) η p ε -
Vstar θ p ε| ≤ δ / 2 := by
have hηrow : dist η (localAlternative θ h n) < r := by
rw [dist_pi_lt_iff hr]
intro k
simpa only [Real.dist_eq] using hηclose k
have hθrow : dist (localAlternative θ h n) θ < R :=
lt_of_lt_of_le (hnloc h hH) (by linarith [hrR, hR])
have hηθ : dist η θ < R := by
calc
dist η θ ≤ dist η (localAlternative θ h n) +
dist (localAlternative θ h n) θ := dist_triangle _ _ _
_ < r + r := add_lt_add hηrow (hnloc h hH)
_ < R := by linarith [hrR, hR]
exact (hcont _ _ hθrow hηθ).le
have havg := pilotPrefixVarianceAverage_error_le select θ (localAlternative θ h n)
p ε r (δ / 2) m hselect hp hθ hlocal hε (half_pos hδ).le hmn hgood
have hCbad : C * pilotPrefixBadMass (localAlternative θ h n) p ε r m hmn <
δ / 2 := by
calc
C * pilotPrefixBadMass (localAlternative θ h n) p ε r m hmn ≤
C * pilotBadMassEnvelope p ε r m n :=
mul_le_mul_of_nonneg_left hbad' hC
_ ≤ (C + 1) * pilotBadMassEnvelope p ε r m n := by
apply mul_le_mul_of_nonneg_right (by linarith)
unfold pilotBadMassEnvelope
positivity
_ < (C + 1) * (δ / (2 * (C + 1))) :=
mul_lt_mul_of_pos_left hnenv (by linarith)
_ = δ / 2 := by
have hCp : 0 < C + 1 := by linarith
field_simp [ne_of_gt hCp]
dsimp only [C] at hCbad
exact lt_of_le_of_lt havg (by linarith)
theorem pilotPrefixVarianceAverage_bounds
Under the supplied quantities and conditions, the pilot prefix variance average bounds assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Prefix Variance Average bounds.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
n :
ℕ
hmn :
m n ≤ n
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
conclusion
conclusion 1
0 ≤ pilotPrefixVarianceAverage select θ p ε m hmn hselect hp hε
conclusion 2
pilotPrefixVarianceAverage select θ p ε m hmn hselect hp hε ≤ (pilotPhiUpper p ε) ^ 2
Proof (Lean source)
lemma pilotPrefixVarianceAverage_bounds
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ) {n : ℕ}
(hmn : m n ≤ n) (hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε) :
0 ≤ pilotPrefixVarianceAverage select θ p ε m hmn hselect hp hε ∧
pilotPrefixVarianceAverage select θ p ε m hmn hselect hp hε ≤
(pilotPhiUpper p ε) ^ 2 := by
have hw (u : Fin (m n) → Fin 4) :
0 ≤ pilotPrefixWeight θ p ε m n u :=
pilotPrefixWeight_nonneg θ p ε m n hp hθ u
have hη (u : Fin (m n) → Fin 4) :
InteriorMeans (pilotAdaptivePrefixTheta p ε m hmn u) := by
exact pilotTheta_interior p ε m (pilotAdaptiveEncode hmn u (fun _ => 0))
constructor
· unfold pilotPrefixVarianceAverage
exact Finset.sum_nonneg fun u _ => mul_nonneg (hw u)
(adaptiveScoreVariance_nonneg select θ _ p ε hselect hp hθ (hη u) hε)
· calc
pilotPrefixVarianceAverage select θ p ε m hmn hselect hp hε ≤
∑ u : Fin (m n) → Fin 4,
pilotPrefixWeight θ p ε m n u * (pilotPhiUpper p ε) ^ 2 := by
unfold pilotPrefixVarianceAverage
apply Finset.sum_le_sum
intro u _
exact mul_le_mul_of_nonneg_left
(adaptiveScoreVariance_le select θ _ p ε hselect hp hθ (hη u) hε) (hw u)
_ = (pilotPhiUpper p ε) ^ 2 := by
rw [← Finset.sum_mul, pilotPrefixWeight_sum θ p ε m n, one_mul]
theorem eventually_uniform_scaledPrefixVarianceAverage
The finite main-row correction preserves the uniform prefix-risk limit. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the eventually uniform scaled Prefix Variance Average.
Formal statement
hypotheses
θ :
p ε H :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hdiv :
hsub :
δ :
ℝ
0 < δ
conclusion
≤ H → InteriorMeans (localAlternative θ h n) → |(n : ℝ) / (adaptiveMainSize m n : ℝ) * pilotPrefixVarianceAverage select (localAlternative θ h n) p ε m (le_of_lt (hsub.2 n hn).2) hselect hp hε
- Vstar θ p ε| < δ
Proof (Lean source)
lemma eventually_uniform_scaledPrefixVarianceAverage
(θ : TrialParameter) (p ε H : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hdiv : PilotDiverges m) (hsub : PilotSublinear m) :
∀ δ : ℝ, 0 < δ → ∀ᶠ n in atTop,
∀ (hn : 2 ≤ n) (h : TrialParameter),
sqrt ((h 0) ^ 2 + (h 1) ^ 2) ≤ H →
InteriorMeans (localAlternative θ h n) →
|(n : ℝ) / (adaptiveMainSize m n : ℝ) *
pilotPrefixVarianceAverage select (localAlternative θ h n) p ε m
(le_of_lt (hsub.2 n hn).2) hselect hp hε - Vstar θ p ε| < δ := by
intro δ hδ
let K := (pilotPhiUpper p ε) ^ 2 + 1
have hK : 0 < K := by dsimp [K]; positivity
have havg := eventually_uniform_pilotPrefixVarianceAverage
select θ p ε H m hselect hp hθ hε hdiv hsub (δ / 2) (half_pos hδ)
have hscale : ∀ᶠ n : ℕ in atTop,
|(n : ℝ) / (adaptiveMainSize m n : ℝ) - 1| < δ / (2 * K) := by
have ht := adaptive_mainScale_tendsto_one m hsub
obtain ⟨N, hN⟩ := Metric.tendsto_atTop.1 ht
(δ / (2 * K)) (by positivity)
exact eventually_atTop.2 ⟨N, fun n hn => by
simpa only [Real.dist_eq] using hN n hn⟩
filter_upwards [havg, hscale] with n hnavg hnscale
intro hn h hH hlocal
let hmn : m n ≤ n := le_of_lt (hsub.2 n hn).2
let A := pilotPrefixVarianceAverage select (localAlternative θ h n) p ε m hmn hselect hp hε
have hA := pilotPrefixVarianceAverage_bounds
select (localAlternative θ h n) p ε m hmn hselect hp hlocal hε
have hAabs : |A| ≤ (pilotPhiUpper p ε) ^ 2 := by
rw [abs_of_nonneg hA.1]
exact hA.2
have havg' : |A - Vstar θ p ε| < δ / 2 := by
simpa only [A, hmn] using hnavg hn h hH hlocal
calc
|(n : ℝ) / (adaptiveMainSize m n : ℝ) * A - Vstar θ p ε| ≤
|((n : ℝ) / (adaptiveMainSize m n : ℝ) - 1) * A| +
|A - Vstar θ p ε| := by
have hid : (n : ℝ) / (adaptiveMainSize m n : ℝ) * A - Vstar θ p ε =
((n : ℝ) / (adaptiveMainSize m n : ℝ) - 1) * A +
(A - Vstar θ p ε) := by ring
rw [hid]
exact abs_add_le _ _
_ ≤ |(n : ℝ) / (adaptiveMainSize m n : ℝ) - 1| *
(pilotPhiUpper p ε) ^ 2 + |A - Vstar θ p ε| := by
rw [abs_mul]
gcongr
_ ≤ |(n : ℝ) / (adaptiveMainSize m n : ℝ) - 1| * K +
|A - Vstar θ p ε| := by
gcongr
dsimp [K]
linarith
_ < δ / (2 * K) * K + δ / 2 :=
add_lt_add_of_lt_of_lt
(mul_lt_mul_of_pos_right hnscale hK) havg'
_ = δ := by
field_simp [ne_of_gt hK]
norm_num
theorem eventually_localWorstRisk_sandwich
Uniform real-risk control sandwiches the local worst risk between two finite ENNReal.ofReal bounds. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the eventually local Worst Risk sandwich.
Formal statement
hypotheses
θ :
p ε H :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hdiv :
hsub :
hH :
0 < H
δ :
ℝ
0 < δ
conclusion
≤ localWorstRisk (pilotEstimator p ε m select hselect hp hε) θ p H n ∧
localWorstRisk (pilotEstimator p ε m select hselect hp hε) θ p H n
Proof (Lean source)
lemma eventually_localWorstRisk_sandwich
(θ : TrialParameter) (p ε H : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hdiv : PilotDiverges m) (hsub : PilotSublinear m) (hH : 0 < H) :
∀ δ : ℝ, 0 < δ → ∀ᶠ n in atTop,
ofReal (Vstar θ p ε - δ) ≤
localWorstRisk (pilotEstimator p ε m select hselect hp hε) θ p H n ∧
localWorstRisk (pilotEstimator p ε m select hselect hp hε) θ p H n ≤
ofReal (Vstar θ p ε + δ) := by
intro δ hδ
have hunif := eventually_uniform_scaledPrefixVarianceAverage
select θ p ε H m hselect hp hθ hε hdiv hsub δ hδ
filter_upwards [hunif, eventually_ge_atTop 2] with n hnif hn
have hlower : ofReal (Vstar θ p ε - δ) ≤
localWorstRisk (pilotEstimator p ε m select hselect hp hε) θ p H n := by
let h0 : TrialParameter := fun _ => 0
have hh0 : h0 ∈ localIndexSet θ n H := by
constructor
· dsimp [h0]
simpa using hH.le
· have hz : localAlternative θ h0 n = θ := by
funext k
simp [h0, localAlternative]
rw [hz]
exact hθ
have hid := pilotEstimator_localRisk_eq_scaled_prefixVarianceAverage
select θ h0 p ε H m hselect hp hθ hε hsub n hn hh0
have hclose := hnif hn h0 hh0.1 hh0.2
have hreal : Vstar θ p ε - δ <
(n : ℝ) / (adaptiveMainSize m n : ℝ) *
pilotPrefixVarianceAverage select (localAlternative θ h0 n) p ε m
(le_of_lt (hsub.2 n hn).2) hselect hp hε := by
linarith [(abs_lt.mp hclose).1]
have hmember : localRisk (pilotEstimator p ε m select hselect hp hε) θ h0 p n ∈
{u : ℝ≥0∞ | ∃ h ∈ localIndexSet θ n H,
u = localRisk (pilotEstimator p ε m select hselect hp hε) θ h p n} :=
⟨h0, hh0, rfl⟩
calc
ofReal (Vstar θ p ε - δ) ≤
localRisk (pilotEstimator p ε m select hselect hp hε) θ h0 p n := by
rw [hid]
exact ENNReal.ofReal_le_ofReal hreal.le
_ ≤ localWorstRisk (pilotEstimator p ε m select hselect hp hε) θ p H n := by
unfold localWorstRisk
exact le_sSup hmember
have hupper : localWorstRisk (pilotEstimator p ε m select hselect hp hε) θ p H n ≤
ofReal (Vstar θ p ε + δ) := by
unfold localWorstRisk
apply sSup_le
intro u hu
obtain ⟨h, hh, rfl⟩ := hu
rw [pilotEstimator_localRisk_eq_scaled_prefixVarianceAverage
select θ h p ε H m hselect hp hθ hε hsub n hn hh]
apply ENNReal.ofReal_le_ofReal
linarith [(abs_lt.mp (hnif hn h hh.1 hh.2)).2]
exact ⟨hlower, hupper⟩
theorem pilot_localWorstRisk_tendsto
Under the supplied quantities and conditions, the pilot local worst risk tendsto assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot local Worst Risk tendsto.
Formal statement
hypotheses
θ :
p ε H :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hdiv :
hsub :
hH :
0 < H
conclusion
Tendsto (fun n => localWorstRisk (pilotEstimator p ε m select hselect hp hε) θ p H n) atTop (nhds (ofReal (Vstar θ p ε)))
Proof (Lean source)
lemma pilot_localWorstRisk_tendsto
(θ : TrialParameter) (p ε H : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hdiv : PilotDiverges m) (hsub : PilotSublinear m) (hH : 0 < H) :
Tendsto (fun n =>
localWorstRisk (pilotEstimator p ε m select hselect hp hε) θ p H n) atTop
(nhds (ofReal (Vstar θ p ε))) := by
rw [tendsto_order]
constructor
· intro a ha
have haTop : a ≠ ⊤ := by
intro hat
rw [hat] at ha
exact (not_lt_of_ge le_top) ha
have hV0 : 0 ≤ Vstar θ p ε :=
(inv_pos.mpr (Jstar_pos_interior θ p ε hp hθ hε)).le
have haV : a.toReal < Vstar θ p ε := by
have := (ENNReal.toReal_lt_toReal haTop ENNReal.ofReal_ne_top).2 ha
simpa [ENNReal.toReal_ofReal hV0] using this
let δ := (Vstar θ p ε - a.toReal) / 2
have hδ : 0 < δ := by dsimp [δ]; linarith
have hsand := eventually_localWorstRisk_sandwich
select θ p ε H m hselect hp hθ hε hdiv hsub hH δ hδ
filter_upwards [hsand] with n hn
have hreal : a.toReal < Vstar θ p ε - δ := by
dsimp [δ]
linarith
have hlt : a < ofReal (Vstar θ p ε - δ) :=
(ENNReal.lt_ofReal_iff_toReal_lt haTop).2 hreal
exact hlt.trans_le hn.1
· intro b hb
by_cases hbTop : b = ⊤
· have hsand := eventually_localWorstRisk_sandwich
select θ p ε H m hselect hp hθ hε hdiv hsub hH 1 (by norm_num)
filter_upwards [hsand] with n hn
rw [hbTop]
exact lt_top_iff_ne_top.2 (ne_top_of_le_ne_top ENNReal.ofReal_ne_top hn.2)
· have hV0 : 0 ≤ Vstar θ p ε :=
(inv_pos.mpr (Jstar_pos_interior θ p ε hp hθ hε)).le
have hVb : Vstar θ p ε < b.toReal := by
have := (ENNReal.toReal_lt_toReal ENNReal.ofReal_ne_top hbTop).2 hb
simpa [ENNReal.toReal_ofReal hV0] using this
let δ := (b.toReal - Vstar θ p ε) / 2
have hδ : 0 < δ := by dsimp [δ]; linarith
have hsand := eventually_localWorstRisk_sandwich
select θ p ε H m hselect hp hθ hε hdiv hsub hH δ hδ
filter_upwards [hsand] with n hn
have hsum0 : 0 ≤ Vstar θ p ε + δ := by positivity
have hreal : Vstar θ p ε + δ < b.toReal := by
dsimp [δ]
linarith
have hlt : ofReal (Vstar θ p ε + δ) < b := by
apply (ENNReal.toReal_lt_toReal ENNReal.ofReal_ne_top hbTop).1
simpa [ENNReal.toReal_ofReal hsum0] using hreal
exact hn.2.trans_lt hlt
theorem pilot_liminf_localWorstRisk_eq
Under the supplied quantities and conditions, the pilot liminf local worst risk eq assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot liminf local Worst Risk eq.
Formal statement
hypotheses
θ :
p ε H :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hdiv :
hsub :
hH :
0 < H
conclusion
Proof (Lean source)
lemma pilot_liminf_localWorstRisk_eq
(θ : TrialParameter) (p ε H : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hdiv : PilotDiverges m) (hsub : PilotSublinear m) (hH : 0 < H) :
liminf (fun n =>
localWorstRisk (pilotEstimator p ε m select hselect hp hε) θ p H n) atTop =
ofReal (Vstar θ p ε) :=
(pilot_localWorstRisk_tendsto select θ p ε H m hselect hp hθ hε hdiv hsub hH).liminf_eq
theorem pilot_localAsymptoticRisk_eq
The private-pilot estimator attains the exact local asymptotic risk. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot local Asymptotic Risk eq.
Formal statement
hypotheses
conclusion
localAsymptoticRisk (pilotEstimator p ε m select hselect hp hε) θ p
Proof (Lean source)
lemma pilot_localAsymptoticRisk_eq
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hdiv : PilotDiverges m) (hsub : PilotSublinear m) :
localAsymptoticRisk (pilotEstimator p ε m select hselect hp hε) θ p =
ofReal (Vstar θ p ε) := by
apply localAsymptoticRisk_eq_of_liminf_localWorstRisk
intro H hH
exact pilot_liminf_localWorstRisk_eq
select θ p ε H m hselect hp hθ hε hdiv hsub hH
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotLocalRiskConvergence 10 declarations Uniform finite-prefix control for private-pilot local risk
Uniform finite-prefix control for private-pilot local risk
def pilotPrefixWeight
The literal probability mass of a released pilot prefix. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The pilot Prefix Weight is determined by the displayed parameters.
Definition (Lean source)
parameters
def
pilotPrefixWeight θ p ε m n u :
ℝ
given by
def pilotPrefixBadMass
Prefix mass on which the clipped pilot selector leaves a coordinatewise r-neighborhood of the row parameter. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The pilot Prefix Bad Mass is determined by the displayed parameters.
Definition (Lean source)
parameters
def
pilotPrefixBadMass θ p ε r m hmn :
ℝ
given by
≤ |pilotAdaptivePrefixTheta p ε m hmn u 0
- θ 0| ∨ r ≤ |pilotAdaptivePrefixTheta p ε m hmn u 1
- θ 1| then 1 else 0
def pilotPrefixVarianceAverage
The selected-score variance averaged over the genuine pilot-prefix law. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The pilot Prefix Variance Average is determined by the displayed parameters.
Definition (Lean source)
parameters
θ :
p ε :
ℝ
m :
ℕ → ℕ
n :
ℕ
hmn :
m n ≤ n
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
def
pilotPrefixVarianceAverage select θ p ε m hmn hselect hp hε :
ℝ
given by
∑ u : Fin (m n) → Fin 4, pilotPrefixWeight θ p ε m n u * adaptiveScoreVariance select θ (pilotAdaptivePrefixTheta p ε m hmn u) p ε
theorem pilotPrefixWeight_nonneg
Under the supplied quantities and conditions, the pilot prefix weight nonneg assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Prefix Weight nonneg.
Formal statement
Proof (Lean source)
lemma pilotPrefixWeight_nonneg (θ : TrialParameter) (p ε : ℝ)
(m : ℕ → ℕ) (n : ℕ) (hp : InteriorAssignment p)
(hθ : InteriorMeans θ) (u : Fin (m n) → Fin 4) :
0 ≤ pilotPrefixWeight θ p ε m n u := by
apply Finset.prod_nonneg
intro j _
apply Finset.sum_nonneg
intro a _
exact mul_nonneg (piTheta_pos_interior θ p hp hθ a).le (by
unfold rrPilotProbability
split_ifs <;> positivity)
theorem pilotPrefixWeight_sum
Under the supplied quantities and conditions, the pilot prefix weight sum assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma pilotPrefixWeight_sum (θ : TrialParameter) (p ε : ℝ)
(m : ℕ → ℕ) (n : ℕ) :
∑ u : Fin (m n) → Fin 4, pilotPrefixWeight θ p ε m n u = 1 := by
unfold pilotPrefixWeight
calc
_ = ∏ _j : Fin (m n), ∑ k : Fin 4,
∑ a : Fin 4, piTheta θ p a * rrPilotProbability ε a k := by
simpa using (Fintype.prod_sum (fun _j : Fin (m n) => fun k : Fin 4 =>
∑ a : Fin 4, piTheta θ p a * rrPilotProbability ε a k)).symm
_ = 1 := by simp [rrPilot_marginal_sum_eq_one]
theorem transcriptLaw_pilotTheta_bad_eq_ofReal_prefixBadMass
The real bad-prefix mass is exactly the corresponding event probability under the full transcript law. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the transcript Law pilot Theta bad eq of Real prefix Bad Mass.
Formal statement
hypotheses
θ :
p ε r :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
n :
ℕ
hmn :
m n ≤ n
conclusion
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) {z | r ≤ |pilotTheta p ε m z 0 - θ 0| ∨ r ≤ |pilotTheta p ε m z 1 - θ 1|}
= ofReal (pilotPrefixBadMass θ p ε r m hmn)
Proof (Lean source)
lemma transcriptLaw_pilotTheta_bad_eq_ofReal_prefixBadMass
(θ : TrialParameter) (p ε r : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
{n : ℕ} (hmn : m n ≤ n) :
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
{z | r ≤ |pilotTheta p ε m z 0 - θ 0| ∨
r ≤ |pilotTheta p ε m z 1 - θ 1|} =
ofReal (pilotPrefixBadMass θ p ε r m hmn) := by
classical
let μ := transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n
let A : Set (Transcript (pilotOutputFamily n)) :=
{z | r ≤ |pilotTheta p ε m z 0 - θ 0| ∨
r ≤ |pilotTheta p ε m z 1 - θ 1|}
let e := fun uv : (Fin (m n) → Fin 4) × (Fin (n - m n) → Fin 14) =>
pilotAdaptiveEncode hmn uv.1 uv.2
have hmainSum (u : Fin (m n) → Fin 4) :
∑ v : Fin (n - m n) → Fin 14,
∏ j, adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j) = 1 := by
calc
_ = ∏ _j : Fin (n - m n), ∑ s : Fin 14,
adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε s := by
simpa using (Fintype.prod_sum (fun _j : Fin (n - m n) => fun s : Fin 14 =>
adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε s)).symm
_ = ∏ _j : Fin (n - m n), (1 : ℝ) := by
apply Finset.prod_congr rfl
intro j _
have hη : InteriorMeans (pilotAdaptivePrefixTheta p ε m hmn u) := by
exact pilotTheta_interior p ε m
(pilotAdaptiveEncode hmn u (fun _ => 0))
exact adaptiveMainMass_sum select θ _ p ε hselect hp hη hε
_ = 1 := by simp
have hw (u : Fin (m n) → Fin 4) :
0 ≤ pilotPrefixWeight θ p ε m n u :=
pilotPrefixWeight_nonneg θ p ε m n hp hθ u
have hm (u : Fin (m n) → Fin 4) (v : Fin (n - m n) → Fin 14) :
0 ≤ ∏ j, adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j) := by
apply Finset.prod_nonneg
intro j _
exact adaptiveMainMass_nonneg select θ _ p ε hselect hp hθ
(pilotTheta_interior p ε m (pilotAdaptiveEncode hmn u (fun _ => 0))) hε (v j)
have hatom (u : Fin (m n) → Fin 4) (v : Fin (n - m n) → Fin 14) :
μ {e (u, v)} = ofReal
(pilotPrefixWeight θ p ε m n u *
∏ j, adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)) := by
rw [pilotAdaptive_joint_singleton_eq_prefix_mul_mainProduct
select θ p ε m hselect hp hθ hε hmn u v]
change ofReal (pilotPrefixWeight θ p ε m n u) *
ofReal (∏ j, adaptiveMainMass select θ
(pilotAdaptivePrefixTheta p ε m hmn u) p ε (v j)) = _
exact (ENNReal.ofReal_mul (hw u)).symm
have hA : MeasurableSet A := MeasurableSet.of_discrete
rw [show μ A = ∫⁻ z, A.indicator (fun _ => (1 : ENNReal)) z ∂μ by
rw [lintegral_indicator hA, setLIntegral_one]]
rw [MeasureTheory.lintegral_fintype]
rw [sum_eq_sum_image_of_zero_off_range e
(pilotAdaptiveEncode_pair_injective hmn)]
· rw [Fintype.sum_prod_type]
simp_rw [hatom]
change (∑ u, ∑ v,
A.indicator (fun _ => (1 : ENNReal)) (e (u, v)) *
ofReal (pilotPrefixWeight θ p ε m n u *
∏ j, adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j))) = _
have hind (u : Fin (m n) → Fin 4) (v : Fin (n - m n) → Fin 14) :
A.indicator (fun _ => (1 : ENNReal)) (e (u, v)) =
if r ≤ |pilotAdaptivePrefixTheta p ε m hmn u 0 - θ 0| ∨
r ≤ |pilotAdaptivePrefixTheta p ε m hmn u 1 - θ 1| then 1 else 0 := by
rw [Set.indicator_apply]
simp only [A, Set.mem_ofPred_eq, e,
pilotTheta_pilotAdaptiveEncode p ε m hmn u v]
simp_rw [hind]
unfold pilotPrefixBadMass
have hofReal_sum {B : Type} [Fintype B] (f : B → ℝ)
(hf : ∀ b, 0 ≤ f b) :
(∑ b, ofReal (f b)) = ofReal (∑ b, f b) := by
simpa using (ENNReal.ofReal_sum_of_nonneg
(s := univ) (fun b _ => hf b)).symm
have hinner (u : Fin (m n) → Fin 4) :
(∑ v : Fin (n - m n) → Fin 14,
(if r ≤ |pilotAdaptivePrefixTheta p ε m hmn u 0 - θ 0| ∨
r ≤ |pilotAdaptivePrefixTheta p ε m hmn u 1 - θ 1| then 1 else 0) *
ofReal (pilotPrefixWeight θ p ε m n u *
∏ j, adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j))) =
ofReal (pilotPrefixWeight θ p ε m n u *
if r ≤ |pilotAdaptivePrefixTheta p ε m hmn u 0 - θ 0| ∨
r ≤ |pilotAdaptivePrefixTheta p ε m hmn u 1 - θ 1| then 1 else 0) := by
split_ifs with hu
· simp only [one_mul, mul_one]
rw [hofReal_sum]
· congr 1
rw [← Finset.mul_sum, hmainSum u, mul_one]
· intro v
exact mul_nonneg (hw u) (hm u v)
· simp
simp_rw [hinner]
apply hofReal_sum
intro u
exact mul_nonneg (hw u) (by split_ifs <;> positivity)
· intro z hz
have hzμ := pilotAdaptive_singleton_zero_of_not_range
select θ p ε m hselect hp hε hmn z hz
apply mul_eq_zero.mpr
right
simpa only [μ] using hzμ
theorem pilotPrefixBadMass_le
Uniform Chebyshev control of the bad prefix mass. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Prefix Bad Mass le.
Formal statement
hypotheses
θ :
p ε r :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hr :
0 < r
n :
ℕ
hmn :
m n ≤ n
hmpos :
0 < m n
hδ0l :
(m n + 2 : ℝ)⁻¹ + r ≤ θ 0
hδ0r :
(m n + 2 : ℝ)⁻¹ + r ≤ 1 - θ 0
hδ1l :
(m n + 2 : ℝ)⁻¹ + r ≤ θ 1
hδ1r :
(m n + 2 : ℝ)⁻¹ + r ≤ 1 - θ 1
conclusion
pilotPrefixBadMass θ p ε r m hmn
Proof (Lean source)
lemma pilotPrefixBadMass_le
(θ : TrialParameter) (p ε r : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hr : 0 < r) {n : ℕ} (hmn : m n ≤ n) (hmpos : 0 < m n)
(hδ0l : (m n + 2 : ℝ)⁻¹ + r ≤ θ 0)
(hδ0r : (m n + 2 : ℝ)⁻¹ + r ≤ 1 - θ 0)
(hδ1l : (m n + 2 : ℝ)⁻¹ + r ≤ θ 1)
(hδ1r : (m n + 2 : ℝ)⁻¹ + r ≤ 1 - θ 1) :
pilotPrefixBadMass θ p ε r m hmn ≤
1 / ((m n : ℝ) *
(r * controlProb p * (exp ε - 1) / (exp ε + 3)) ^ 2) +
1 / ((m n : ℝ) *
(r * p * (exp ε - 1) / (exp ε + 3)) ^ 2) := by
have hmeasure := pilotTheta_deviation_le select θ p ε r m hselect hp hθ hε hr hmn hmpos
hδ0l hδ0r hδ1l hδ1r
rw [transcriptLaw_pilotTheta_bad_eq_ofReal_prefixBadMass
select θ p ε r m hselect hp hθ hε hmn] at hmeasure
let x := 1 / ((m n : ℝ) *
(r * controlProb p * (exp ε - 1) / (exp ε + 3)) ^ 2)
let y := 1 / ((m n : ℝ) *
(r * p * (exp ε - 1) / (exp ε + 3)) ^ 2)
have hx : 0 ≤ x := by dsimp [x]; positivity
have hy : 0 ≤ y := by dsimp [y]; positivity
change ofReal (pilotPrefixBadMass θ p ε r m hmn) ≤
ofReal x + ofReal y at hmeasure
rw [← ENNReal.ofReal_add hx hy] at hmeasure
exact (ENNReal.ofReal_le_ofReal_iff (add_nonneg hx hy)).mp hmeasure
theorem pilotPrefixVarianceAverage_error_le
A deterministic good/bad decomposition for the prefix-averaged variance. It uses the exact prefix weights and leaves only the genuine bad-prefix mass. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Prefix Variance Average error le.
Formal statement
hypotheses
θ θ' :
p ε r δ :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hθ' :
hε :
0 < ε
hδ :
0 ≤ δ
n :
ℕ
hmn :
m n ≤ n
hgood :
∀ η : TrialParameter
conclusion
|pilotPrefixVarianceAverage select θ' p ε m hmn hselect hp hε - Vstar θ p ε|
Proof (Lean source)
lemma pilotPrefixVarianceAverage_error_le
(θ θ' : TrialParameter) (p ε r δ : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(hθ' : InteriorMeans θ') (hε : 0 < ε)
(hδ : 0 ≤ δ) {n : ℕ} (hmn : m n ≤ n)
(hgood : ∀ η : TrialParameter, InteriorMeans η →
(∀ k : Fin 2, |η k - θ' k| < r) →
|adaptiveScoreVariance select θ' η p ε - Vstar θ p ε| ≤ δ) :
|pilotPrefixVarianceAverage select θ' p ε m hmn hselect hp hε - Vstar θ p ε| ≤
δ + ((pilotPhiUpper p ε) ^ 2 + |Vstar θ p ε|) *
pilotPrefixBadMass θ' p ε r m hmn := by
let good : (Fin (m n) → Fin 4) → Prop := fun u =>
|pilotAdaptivePrefixTheta p ε m hmn u 0 - θ' 0| < r ∧
|pilotAdaptivePrefixTheta p ε m hmn u 1 - θ' 1| < r
let w : (Fin (m n) → Fin 4) → ℝ := pilotPrefixWeight θ' p ε m n
let g : (Fin (m n) → Fin 4) → ℝ := fun u =>
adaptiveScoreVariance select θ' (pilotAdaptivePrefixTheta p ε m hmn u)
p ε
have hw (u : Fin (m n) → Fin 4) : 0 ≤ w u :=
pilotPrefixWeight_nonneg θ' p ε m n hp hθ' u
have hsum : ∑ u, w u = 1 := pilotPrefixWeight_sum θ' p ε m n
have hgood' (u : Fin (m n) → Fin 4) (hu : good u) :
|g u - Vstar θ p ε| ≤ δ := by
apply hgood _
· exact pilotTheta_interior p ε m
(pilotAdaptiveEncode hmn u (fun _ => 0))
· intro k
fin_cases k
· exact hu.1
· exact hu.2
have hdecomp := finitePilot_good_bad_average_error_le
w g good (Vstar θ p ε) δ hw hsum
(le_of_lt (inv_pos.mpr (Jstar_pos_interior θ p ε hp hθ hε))) hδ hgood'
calc
|pilotPrefixVarianceAverage select θ' p ε m hmn hselect hp hε - Vstar θ p ε| ≤
δ + ∑ u, w u * if good u then 0 else |g u - Vstar θ p ε| := by
simpa [pilotPrefixVarianceAverage, w, g] using hdecomp
_ ≤ δ + ∑ u, w u * if good u then 0 else
((pilotPhiUpper p ε) ^ 2 + |Vstar θ p ε|) := by
apply add_le_add le_rfl
apply Finset.sum_le_sum
intro u _
apply mul_le_mul_of_nonneg_left _ (hw u)
split_ifs with hu
· exact le_rfl
· calc
|g u - Vstar θ p ε| ≤ |g u| + |Vstar θ p ε| := abs_sub _ _
_ ≤ (pilotPhiUpper p ε) ^ 2 + |Vstar θ p ε| := by
gcongr
have hη : InteriorMeans (pilotAdaptivePrefixTheta p ε m hmn u) := by
exact pilotTheta_interior p ε m
(pilotAdaptiveEncode hmn u (fun _ => 0))
have hv0 : 0 ≤ g u :=
adaptiveScoreVariance_nonneg select θ' _ p ε hselect hp hθ' hη hε
rw [abs_of_nonneg hv0]
exact adaptiveScoreVariance_le select θ' _ p ε hselect hp hθ' hη hε
_ = δ + ((pilotPhiUpper p ε) ^ 2 + |Vstar θ p ε|) *
pilotPrefixBadMass θ' p ε r m hmn := by
unfold pilotPrefixBadMass good w
congr 1
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro u _
by_cases hbad : r ≤ |pilotAdaptivePrefixTheta p ε m hmn u 0 - θ' 0| ∨
r ≤ |pilotAdaptivePrefixTheta p ε m hmn u 1 - θ' 1|
· have hngood : ¬(|pilotAdaptivePrefixTheta p ε m hmn u 0 - θ' 0| < r ∧
|pilotAdaptivePrefixTheta p ε m hmn u 1 - θ' 1| < r) := by
intro hg
exact hbad.elim (fun h => (not_le_of_gt hg.1) h)
(fun h => (not_le_of_gt hg.2) h)
simp [hbad, hngood]
ring
· have hgood : |pilotAdaptivePrefixTheta p ε m hmn u 0 - θ' 0| < r ∧
|pilotAdaptivePrefixTheta p ε m hmn u 1 - θ' 1| < r := by
push Not at hbad
exact hbad
simp [hbad, hgood]
theorem adaptiveScoreVariance_uniform_near_diag
Continuity at the diagonal gives one common neighborhood controlling both the local row parameter and the selected pilot parameter. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Score Variance uniform near diag.
Formal statement
hypotheses
Proof (Lean source)
lemma adaptiveScoreVariance_uniform_near_diag
(θ : TrialParameter) (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select) (hp : InteriorAssignment p)
(hθ : InteriorMeans θ) (hε : 0 < ε) :
∀ δ : ℝ, 0 < δ → ∃ r : ℝ, 0 < r ∧
∀ θ' η : TrialParameter,
dist θ' θ < r → dist η θ < r →
|adaptiveScoreVariance select θ' η p ε - Vstar θ p ε| < δ := by
intro δ hδ
have hc := continuousAt_adaptiveScoreVariance_diag select θ p ε hselect hp hθ hε
rw [Metric.continuousAt_iff] at hc
obtain ⟨r, hr, hcontrol⟩ := hc δ hδ
refine ⟨r, hr, ?_⟩
intro θ' η hθ' hη
have hpair : dist (θ', η) (θ, θ) < r := by
simpa only [Prod.dist_eq, max_lt_iff] using intro hθ' hη
simpa only [Real.dist_eq, adaptiveScoreVariance_diag select θ p ε hselect hp hθ hε] using
hcontrol hpair
theorem pilotEstimator_localRisk_eq_scaled_prefixVarianceAverage
Compact notation for the exact local-risk identity on the local index set. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Estimator local Risk eq scaled prefix Variance Average.
Formal statement
hypotheses
θ h :
p ε H :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hsub :
n :
ℕ
hn :
2 ≤ n
hh :
h ∈ localIndexSet θ n H
conclusion
localRisk (pilotEstimator p ε m select hselect hp hε) θ h p n
= ofReal ((n : ℝ) / (adaptiveMainSize m n : ℝ) * pilotPrefixVarianceAverage select (localAlternative θ h n) p ε m (le_of_lt (hsub.2 n hn).2) hselect hp hε)
Proof (Lean source)
lemma pilotEstimator_localRisk_eq_scaled_prefixVarianceAverage
(θ h : TrialParameter) (p ε H : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hsub : PilotSublinear m) (n : ℕ) (hn : 2 ≤ n)
(hh : h ∈ localIndexSet θ n H) :
localRisk (pilotEstimator p ε m select hselect hp hε) θ h p n =
ofReal ((n : ℝ) / (adaptiveMainSize m n : ℝ) *
pilotPrefixVarianceAverage select (localAlternative θ h n) p ε m
(le_of_lt (hsub.2 n hn).2) hselect hp hε) := by
simpa only [pilotPrefixVarianceAverage, pilotPrefixWeight] using
pilotEstimator_localRisk_eq_prefixVariance_of_mem_localIndexSet
select θ h p ε H m hselect hp hθ hε hsub n hn hh
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotLocalRiskIdentity 2 declarations Exact transcript-law identity for private-pilot local risk
Exact transcript-law identity for private-pilot local risk
theorem pilotEstimator_localRisk_eq_prefixVariance
At every proper finite pilot split for which the local alternative is interior, the transcript-law local risk is exactly the prefix average of the conditional selected-score variances, including the finite main-size factor. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Estimator local Risk eq prefix Variance.
Formal statement
hypotheses
θ h :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hsub :
n :
ℕ
hn :
2 ≤ n
hlocal :
InteriorMeans (localAlternative θ h n)
conclusion
localRisk (pilotEstimator p ε m select hselect hp hε) θ h p n
= ofReal ((n : ℝ) / (adaptiveMainSize m n : ℝ) * ∑ u : Fin (m n) → Fin 4, (∏ j : Fin (m n), ∑ a : Fin 4, piTheta (localAlternative θ h n) p a * rrPilotProbability ε a (u j)) * adaptiveScoreVariance select (localAlternative θ h n) (pilotAdaptivePrefixTheta p ε m (le_of_lt (hsub.2 n hn).2) u) p ε)
Proof (Lean source)
lemma pilotEstimator_localRisk_eq_prefixVariance
(θ h : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hsub : PilotSublinear m) (n : ℕ) (hn : 2 ≤ n)
(hlocal : InteriorMeans (localAlternative θ h n)) :
localRisk (pilotEstimator p ε m select hselect hp hε) θ h p n =
ofReal
((n : ℝ) / (adaptiveMainSize m n : ℝ) *
∑ u : Fin (m n) → Fin 4,
(∏ j : Fin (m n), ∑ a : Fin 4,
piTheta (localAlternative θ h n) p a *
rrPilotProbability ε a (u j)) *
adaptiveScoreVariance select (localAlternative θ h n)
(pilotAdaptivePrefixTheta p ε m
(le_of_lt (hsub.2 n hn).2) u) p ε) := by
classical
let θn := localAlternative θ h n
let P := pilotEstimator p ε m select hselect hp hε
let μ := transcriptLaw P θn p n
let hmn : m n ≤ n := le_of_lt (hsub.2 n hn).2
let e := fun uv : (Fin (m n) → Fin 4) × (Fin (n - m n) → Fin 14) =>
pilotAdaptiveEncode hmn uv.1 uv.2
have htrue : adaptiveTrueParam θ h n = θn :=
adaptiveTrueParam_eq_localAlternative θ h n hlocal
have hmain : adaptiveMainSize m n = n - m n := by
simp [adaptiveMainSize, adaptivePilotSize_eq m hsub hn]
have hrisk := adaptivePilotPrefix_rootN_meanSquare
select θ h p ε m hselect hp hθ hε hsub n hn
rw [htrue] at hrisk
rw [hmain] at hrisk
unfold localRisk
rw [MeasureTheory.lintegral_fintype]
change (∑ z : Transcript (pilotOutputFamily n),
ofReal ((scaledError P θ h n z) ^ 2) * μ {z}) = _
rw [sum_eq_sum_image_of_zero_off_range e
(pilotAdaptiveEncode_pair_injective hmn)]
· rw [Fintype.sum_prod_type]
have hpilot_nonneg (u : Fin (m n) → Fin 4) :
0 ≤ ∏ j : Fin (m n), ∑ a : Fin 4,
piTheta θn p a * rrPilotProbability ε a (u j) := by
apply Finset.prod_nonneg
intro j _
apply Finset.sum_nonneg
intro a _
exact mul_nonneg (piTheta_pos_interior θn p hp hlocal a).le (by
unfold rrPilotProbability
split_ifs <;> positivity)
have hmain_nonneg (u : Fin (m n) → Fin 4)
(v : Fin (n - m n) → Fin 14) :
0 ≤ ∏ j : Fin (n - m n),
adaptiveMainMass select θn (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j) := by
apply Finset.prod_nonneg
intro j _
exact adaptiveMainMass_nonneg select θn _ p ε hselect hp hlocal
(pilotTheta_interior p ε m (pilotAdaptiveEncode hmn u (fun _ => 0)))
hε (v j)
have hatom (u : Fin (m n) → Fin 4)
(v : Fin (n - m n) → Fin 14) :
μ {e (u, v)} =
ofReal ((∏ j : Fin (m n), ∑ a : Fin 4,
piTheta θn p a * rrPilotProbability ε a (u j)) *
∏ j : Fin (n - m n),
adaptiveMainMass select θn (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)) := by
rw [pilotAdaptive_joint_singleton_eq_prefix_mul_mainProduct
select θn p ε m hselect hp hlocal hε hmn u v]
rw [← ENNReal.ofReal_mul (hpilot_nonneg u)]
simp_rw [hatom]
have herr (u : Fin (m n) → Fin 4)
(v : Fin (n - m n) → Fin 14) :
scaledError P θ h n (e (u, v)) =
sqrt (n : ℝ) *
(contrast (pilotAdaptivePrefixTheta p ε m hmn u) +
((n - m n : ℕ) : ℝ)⁻¹ *
(∑ j, adaptiveSelectedScore select
(pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)) - contrast θn) := by
unfold scaledError P
change sqrt (n : ℝ) *
(tauStar p ε m (select) (e (u, v)) -
contrast θn) = _
rw [tauStar_pilotAdaptiveEncode select p ε m hp hε hmn u v]
simp_rw [herr]
simp_rw [← ENNReal.ofReal_mul (sq_nonneg _)]
have hofReal_sum {A : Type} [Fintype A] (f : A → ℝ)
(hf : ∀ a, 0 ≤ f a) :
(∑ a, ofReal (f a)) = ofReal (∑ a, f a) := by
simpa using (ENNReal.ofReal_sum_of_nonneg
(s := univ) (fun a _ => hf a)).symm
calc
(∑ u : Fin (m n) → Fin 4,
∑ v : Fin (n - m n) → Fin 14,
ofReal
((sqrt (n : ℝ) *
(contrast (pilotAdaptivePrefixTheta p ε m hmn u) +
((n - m n : ℕ) : ℝ)⁻¹ *
(∑ j, adaptiveSelectedScore select
(pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)) - contrast θn)) ^ 2 *
((∏ j : Fin (m n), ∑ a : Fin 4,
piTheta θn p a * rrPilotProbability ε a (u j)) *
∏ j : Fin (n - m n),
adaptiveMainMass select θn
(pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)))) =
∑ u : Fin (m n) → Fin 4, ofReal
(∑ v : Fin (n - m n) → Fin 14,
(sqrt (n : ℝ) *
(contrast (pilotAdaptivePrefixTheta p ε m hmn u) +
((n - m n : ℕ) : ℝ)⁻¹ *
(∑ j, adaptiveSelectedScore select
(pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)) - contrast θn)) ^ 2 *
((∏ j : Fin (m n), ∑ a : Fin 4,
piTheta θn p a * rrPilotProbability ε a (u j)) *
∏ j : Fin (n - m n),
adaptiveMainMass select θn
(pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j))) := by
apply Finset.sum_congr rfl
intro u _
refine hofReal_sum (A := Fin (n - m n) → Fin 14) _ ?_
intro v
exact mul_nonneg (sq_nonneg _)
(mul_nonneg (hpilot_nonneg u) (hmain_nonneg u v))
_ = ofReal
(∑ u : Fin (m n) → Fin 4,
∑ v : Fin (n - m n) → Fin 14,
(sqrt (n : ℝ) *
(contrast (pilotAdaptivePrefixTheta p ε m hmn u) +
((n - m n : ℕ) : ℝ)⁻¹ *
(∑ j, adaptiveSelectedScore select
(pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)) - contrast θn)) ^ 2 *
((∏ j : Fin (m n), ∑ a : Fin 4,
piTheta θn p a * rrPilotProbability ε a (u j)) *
∏ j : Fin (n - m n),
adaptiveMainMass select θn
(pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j))) := by
refine hofReal_sum (A := Fin (m n) → Fin 4) _ ?_
intro u
exact Finset.sum_nonneg fun v _ => mul_nonneg (sq_nonneg _)
(mul_nonneg (hpilot_nonneg u) (hmain_nonneg u v))
_ = _ := by
congr 1
calc
(∑ u : Fin (m n) → Fin 4,
∑ v : Fin (n - m n) → Fin 14,
(sqrt (n : ℝ) *
(contrast (pilotAdaptivePrefixTheta p ε m hmn u) +
((n - m n : ℕ) : ℝ)⁻¹ *
(∑ j, adaptiveSelectedScore select
(pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)) - contrast θn)) ^ 2 *
((∏ j : Fin (m n), ∑ a : Fin 4,
piTheta θn p a * rrPilotProbability ε a (u j)) *
∏ j : Fin (n - m n),
adaptiveMainMass select θn
(pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j))) =
∑ u : Fin (m n) → Fin 4,
(∏ j : Fin (m n), ∑ a : Fin 4,
piTheta θn p a * rrPilotProbability ε a (u j)) *
∑ v : Fin (n - m n) → Fin 14,
(∏ j, adaptiveMainMass select θn
(pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)) *
(sqrt (n : ℝ) *
(contrast (pilotAdaptivePrefixTheta p ε m hmn u) +
((n - m n : ℕ) : ℝ)⁻¹ *
(∑ j, adaptiveSelectedScore select
(pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)) - contrast θn)) ^ 2 := by
apply Finset.sum_congr rfl
intro u _
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro v _
ring
_ = ∑ u : Fin (m n) → Fin 4,
(∏ j : Fin (m n), ∑ a : Fin 4,
piTheta θn p a * rrPilotProbability ε a (u j)) *
((n : ℝ) / (n - m n : ℕ) *
adaptiveScoreVariance select θn
(pilotAdaptivePrefixTheta p ε m hmn u) p ε) := by
simpa only [hmn] using hrisk
_ = (n : ℝ) / (adaptiveMainSize m n : ℝ) *
∑ u : Fin (m n) → Fin 4,
(∏ j : Fin (m n), ∑ a : Fin 4,
piTheta (localAlternative θ h n) p a *
rrPilotProbability ε a (u j)) *
adaptiveScoreVariance select (localAlternative θ h n)
(pilotAdaptivePrefixTheta p ε m
(le_of_lt (hsub.2 n hn).2) u) p ε := by
rw [hmain]
simp only [θn]
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro u _
ring
· intro z hz
have hzμ := pilotAdaptive_singleton_zero_of_not_range
select θn p ε m hselect hp hε hmn z hz
apply mul_eq_zero.mpr
right
simpa only [μ, P] using hzμ
theorem pilotEstimator_localRisk_eq_prefixVariance_of_mem_localIndexSet
The exact risk identity applies to every direction in the finite-sample local index set; its membership condition supplies precisely the interiority needed by the transcript-law factorization. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Estimator local Risk eq prefix Variance of mem local Index Set.
Formal statement
hypotheses
θ h :
p ε H :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hsub :
n :
ℕ
hn :
2 ≤ n
hh :
h ∈ localIndexSet θ n H
conclusion
localRisk (pilotEstimator p ε m select hselect hp hε) θ h p n
= ofReal ((n : ℝ) / (adaptiveMainSize m n : ℝ) * ∑ u : Fin (m n) → Fin 4, (∏ j : Fin (m n), ∑ a : Fin 4, piTheta (localAlternative θ h n) p a * rrPilotProbability ε a (u j)) * adaptiveScoreVariance select (localAlternative θ h n) (pilotAdaptivePrefixTheta p ε m (le_of_lt (hsub.2 n hn).2) u) p ε)
Proof (Lean source)
lemma pilotEstimator_localRisk_eq_prefixVariance_of_mem_localIndexSet
(θ h : TrialParameter) (p ε H : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hsub : PilotSublinear m) (n : ℕ) (hn : 2 ≤ n)
(hh : h ∈ localIndexSet θ n H) :
localRisk (pilotEstimator p ε m select hselect hp hε) θ h p n =
ofReal
((n : ℝ) / (adaptiveMainSize m n : ℝ) *
∑ u : Fin (m n) → Fin 4,
(∏ j : Fin (m n), ∑ a : Fin 4,
piTheta (localAlternative θ h n) p a *
rrPilotProbability ε a (u j)) *
adaptiveScoreVariance select (localAlternative θ h n)
(pilotAdaptivePrefixTheta p ε m
(le_of_lt (hsub.2 n hn).2) u) p ε) := by
exact pilotEstimator_localRisk_eq_prefixVariance
select θ h p ε m hselect hp hθ hε hsub n hn hh.2
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotLocalRiskSupport 5 declarations Exact conditional local risk of the adaptive main row
Exact conditional local risk of the adaptive main row
theorem expect_scoreSum_sq
The squared sum of a centered finite iid row has expectation equal to the row length times the one-coordinate second moment. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma RowModel.expect_scoreSum_sq {Y : Type*} [Fintype Y]
(M : RowModel Y) (n : ℕ) :
M.expect n (fun ys => (M.scoreSum n ys) ^ 2) =
(M.N n : ℝ) * M.oneSecond n := by
classical
have hsumsq (ys : Fin (M.N n) → Y) :
(∑ i, M.f n (ys i)) ^ 2 =
∑ i, ∑ j, M.f n (ys i) * M.f n (ys j) := by
rw [pow_two, Finset.sum_mul]
apply Finset.sum_congr rfl
intro i _
rw [Finset.mul_sum]
simp only [scoreSum]
simp_rw [hsumsq]
rw [expect]
simp_rw [Finset.mul_sum]
calc
(∑ ys : Fin (M.N n) → Y, ∑ i, ∑ j,
M.mass n ys * (M.f n (ys i) * M.f n (ys j))) =
∑ i, ∑ ys : Fin (M.N n) → Y, ∑ j,
M.mass n ys * (M.f n (ys i) * M.f n (ys j)) := by
rw [Finset.sum_comm]
_ = ∑ i, ∑ j, ∑ ys : Fin (M.N n) → Y,
M.mass n ys * (M.f n (ys i) * M.f n (ys j)) := by
apply Finset.sum_congr rfl
intro i _
rw [Finset.sum_comm]
_ = ∑ i, ∑ j, M.expect n (fun ys => M.f n (ys i) * M.f n (ys j)) := by
rfl
_ = ∑ i, M.expect n (fun ys => (M.f n (ys i)) ^ 2) := by
apply Finset.sum_congr rfl
intro i _
rw [Finset.sum_eq_single i]
· simp [pow_two]
· intro j _ hji
rw [M.expect_coordinate_mul n i j (Ne.symm hji)]
simp [M.centered]
· simp
_ = ∑ _i : Fin (M.N n), M.oneSecond n := by
apply Finset.sum_congr rfl
intro i _
simpa [oneSecond] using M.expect_coordinate n i (fun y => (M.f n y) ^ 2)
_ = (M.N n : ℝ) * M.oneSecond n := by simp
theorem adaptivePilotRow_conditional_meanSquare
Conditional on a convergent sequence of interior pilot selectors, the exact squared main-row estimation error is the adaptive score variance divided by the main-row size. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Row conditional mean Square.
Formal statement
hypotheses
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hη :
∀ n, InteriorMeans (η n)
hε :
0 < ε
hsub :
n :
ℕ
hN :
0 < adaptiveMainSize m n
conclusion
(∑ v : Fin (adaptiveMainSize m n) → Fin 14, adaptivePilotRowProductMass select θ h η p ε m hselect hp hε n v * (contrast (η n) + (adaptiveMainSize m n : ℝ)⁻¹ * (∑ j, adaptiveSelectedScore select (η n) p ε (v j)) - contrast (adaptiveTrueParam θ h n)) ^ 2)
= (adaptiveMainSize m n : ℝ)⁻¹ * adaptiveScoreVariance select (adaptiveTrueParam θ h n) (η n) p ε
Proof (Lean source)
lemma adaptivePilotRow_conditional_meanSquare
(θ h : TrialParameter) (η : ℕ → TrialParameter)
(p ε : ℝ) (m : ℕ → ℕ) (hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p)
(hθ : InteriorMeans θ) (hη : ∀ n, InteriorMeans (η n)) (hε : 0 < ε)
(hηlim : Tendsto η atTop (nhds θ)) (hsub : PilotSublinear m)
(n : ℕ) (hN : 0 < adaptiveMainSize m n) :
(∑ v : Fin (adaptiveMainSize m n) → Fin 14,
adaptivePilotRowProductMass select θ h η p ε m hselect hp hε n v *
(contrast (η n) + (adaptiveMainSize m n : ℝ)⁻¹ *
(∑ j, adaptiveSelectedScore select (η n) p ε (v j)) -
contrast (adaptiveTrueParam θ h n)) ^ 2) =
(adaptiveMainSize m n : ℝ)⁻¹ *
adaptiveScoreVariance select (adaptiveTrueParam θ h n) (η n) p ε := by
let M : RowModel (Fin 14) :=
{ N := adaptiveMainSize m
w := adaptivePilotRowMass select θ h η p ε hselect hp hε
f := adaptivePilotRowScore select θ h η p ε hselect hp hε
B := 2 * pilotPhiUpper p ε
v₀ := adaptivePilotLimitVariance select θ p ε hselect hp hθ hε
mass_nonneg := adaptivePilotRowMass_nonneg select θ h η p ε hselect hp hθ hη hε
mass_one := adaptivePilotRowMass_sum select θ h η p ε hselect hp hη hε
centered := adaptivePilotRowScore_centered select θ h η p ε hselect hp hη hε
bound := adaptivePilotRowScore_abs_le select θ h η p ε hselect hp hθ hη hε
variance_tendsto := adaptivePilotRow_variance_tendsto
select θ h η p ε hselect hp hθ hη hε hηlim
variance_pos := adaptivePilotLimitVariance_pos select θ p ε hselect hp hθ hε
length_tendsto := adaptiveMainSize_tendsto_atTop m hsub }
have hsquare := M.expect_scoreSum_sq n
have hNr : (adaptiveMainSize m n : ℝ) ≠ 0 := by exact_mod_cast ne_of_gt hN
simp only [M, RowModel.expect, RowModel.mass, RowModel.scoreSum,
RowModel.oneSecond] at hsquare
simp_rw [adaptivePilotRow_estimatorError_eq select θ h η p ε m hselect hp hε n hN]
calc
_ = ((adaptiveMainSize m n : ℝ)⁻¹) ^ 2 *
(∑ v : Fin (adaptiveMainSize m n) → Fin 14,
adaptivePilotRowProductMass select θ h η p ε m hselect hp hε n v *
(∑ j, adaptivePilotRowScore select θ h η p ε hselect hp hε n (v j)) ^ 2) := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro v _
ring
_ = ((adaptiveMainSize m n : ℝ)⁻¹) ^ 2 *
((adaptiveMainSize m n : ℝ) *
adaptiveScoreVariance select (adaptiveTrueParam θ h n) (η n) p ε) := by
congr 1
simpa [adaptivePilotRowProductMass,
adaptivePilotRow_secondMoment_eq select θ h η p ε hselect hp hη hε n] using hsquare
_ = _ := by field_simp
theorem adaptivePilotRow_conditional_rootN_meanSquare
The exact conditional mean-square error after root-n scaling. The finite-sample discrepancy from the selected-score variance is precisely the main-row fraction n / (n - m n). For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Row conditional root N mean Square.
Formal statement
hypotheses
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hη :
∀ n, InteriorMeans (η n)
hε :
0 < ε
hsub :
n :
ℕ
hN :
0 < adaptiveMainSize m n
conclusion
(∑ v : Fin (adaptiveMainSize m n) → Fin 14, adaptivePilotRowProductMass select θ h η p ε m hselect hp hε n v * (sqrt (n : ℝ) * (contrast (η n) + (adaptiveMainSize m n : ℝ)⁻¹ * (∑ j, adaptiveSelectedScore select (η n) p ε (v j)) - contrast (adaptiveTrueParam θ h n))) ^ 2)
= (n : ℝ) / (adaptiveMainSize m n : ℝ) * adaptiveScoreVariance select (adaptiveTrueParam θ h n) (η n) p ε
Proof (Lean source)
lemma adaptivePilotRow_conditional_rootN_meanSquare
(θ h : TrialParameter) (η : ℕ → TrialParameter)
(p ε : ℝ) (m : ℕ → ℕ) (hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p)
(hθ : InteriorMeans θ) (hη : ∀ n, InteriorMeans (η n)) (hε : 0 < ε)
(hηlim : Tendsto η atTop (nhds θ)) (hsub : PilotSublinear m)
(n : ℕ) (hN : 0 < adaptiveMainSize m n) :
(∑ v : Fin (adaptiveMainSize m n) → Fin 14,
adaptivePilotRowProductMass select θ h η p ε m hselect hp hε n v *
(sqrt (n : ℝ) *
(contrast (η n) + (adaptiveMainSize m n : ℝ)⁻¹ *
(∑ j, adaptiveSelectedScore select (η n) p ε (v j)) -
contrast (adaptiveTrueParam θ h n))) ^ 2) =
(n : ℝ) / (adaptiveMainSize m n : ℝ) *
adaptiveScoreVariance select (adaptiveTrueParam θ h n) (η n) p ε := by
have hsqrt : (sqrt (n : ℝ)) ^ 2 = (n : ℝ) :=
Real.sq_sqrt (Nat.cast_nonneg n)
have hrisk := adaptivePilotRow_conditional_meanSquare
select θ h η p ε m hselect hp hθ hη hε hηlim hsub n hN
have hNr : (adaptiveMainSize m n : ℝ) ≠ 0 := by exact_mod_cast ne_of_gt hN
calc
_ = (n : ℝ) *
(∑ v : Fin (adaptiveMainSize m n) → Fin 14,
adaptivePilotRowProductMass select θ h η p ε m hselect hp hε n v *
(contrast (η n) + (adaptiveMainSize m n : ℝ)⁻¹ *
(∑ j, adaptiveSelectedScore select (η n) p ε (v j)) -
contrast (adaptiveTrueParam θ h n)) ^ 2) := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro v _
rw [mul_pow, hsqrt]
ring
_ = (n : ℝ) * ((adaptiveMainSize m n : ℝ)⁻¹ *
adaptiveScoreVariance select (adaptiveTrueParam θ h n) (η n) p ε) := by
rw [hrisk]
_ = _ := by
field_simp
theorem adaptivePilotRow_rootN_riskValue_tendsto
Along every convergent sequence of interior pilot selectors, the exact conditional root-n risk value converges to the oracle variance. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Row root N risk Value tendsto.
Formal statement
hypotheses
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hη :
∀ n, InteriorMeans (η n)
hε :
0 < ε
hsub :
conclusion
Tendsto (fun n : ℕ => (n : ℝ) / (adaptiveMainSize m n : ℝ) * adaptiveScoreVariance select (adaptiveTrueParam θ h n) (η n) p ε) atTop (nhds (Vstar θ p ε))
Proof (Lean source)
-- keep: reusable finite-pilot construction, law, localization, or limit API for related adaptive procedures
lemma adaptivePilotRow_rootN_riskValue_tendsto
(θ h : TrialParameter) (η : ℕ → TrialParameter)
(p ε : ℝ) (m : ℕ → ℕ) (hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p)
(hθ : InteriorMeans θ) (hη : ∀ n, InteriorMeans (η n)) (hε : 0 < ε)
(hηlim : Tendsto η atTop (nhds θ)) (hsub : PilotSublinear m) :
Tendsto (fun n : ℕ =>
(n : ℝ) / (adaptiveMainSize m n : ℝ) *
adaptiveScoreVariance select (adaptiveTrueParam θ h n) (η n) p ε)
atTop (nhds (Vstar θ p ε)) := by
have hvar : Tendsto (fun n =>
adaptiveScoreVariance select (adaptiveTrueParam θ h n) (η n) p ε)
atTop (nhds (Vstar θ p ε)) := by
have ht := adaptivePilotRow_variance_tendsto
select θ h η p ε hselect hp hθ hη hε hηlim
convert ht using 1
· funext n
rw [adaptivePilotRow_secondMoment_eq select θ h η p ε hselect hp hη hε n]
· rfl
exact adaptive_mainScale_mul_tendsto hsub hvar
theorem adaptivePilotPrefix_rootN_meanSquare
Averaging the exact conditional risk over every finite pilot prefix leaves only the prefix average of the selected-score variances. This is the exact finite-law risk decomposition, before any pilot-localization limit is used. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Prefix root N mean Square.
Formal statement
hypotheses
θ h :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hsub :
n :
ℕ
hn :
2 ≤ n
conclusion
(∑ u : Fin (m n) → Fin 4, (∏ j : Fin (m n), ∑ a : Fin 4, piTheta (adaptiveTrueParam θ h n) p a * rrPilotProbability ε a (u j)) * (∑ v : Fin (adaptiveMainSize m n) → Fin 14, (∏ j, adaptiveMainMass select (adaptiveTrueParam θ h n) (pilotAdaptivePrefixTheta p ε m (le_of_lt (hsub.2 n hn).2) u) p ε (v j)) * (sqrt (n : ℝ) * (contrast (pilotAdaptivePrefixTheta p ε m (le_of_lt (hsub.2 n hn).2) u) + (adaptiveMainSize m n : ℝ)⁻¹ * (∑ j, adaptiveSelectedScore select (pilotAdaptivePrefixTheta p ε m (le_of_lt (hsub.2 n hn).2) u) p ε (v j)) - contrast (adaptiveTrueParam θ h n))) ^ 2))
= ∑ u : Fin (m n) → Fin 4, (∏ j : Fin (m n), ∑ a : Fin 4, piTheta (adaptiveTrueParam θ h n) p a * rrPilotProbability ε a (u j)) * ((n : ℝ) / (adaptiveMainSize m n : ℝ) * adaptiveScoreVariance select (adaptiveTrueParam θ h n) (pilotAdaptivePrefixTheta p ε m (le_of_lt (hsub.2 n hn).2) u) p ε)
Proof (Lean source)
lemma adaptivePilotPrefix_rootN_meanSquare
(θ h : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hsub : PilotSublinear m) (n : ℕ) (hn : 2 ≤ n) :
(∑ u : Fin (m n) → Fin 4,
(∏ j : Fin (m n), ∑ a : Fin 4,
piTheta (adaptiveTrueParam θ h n) p a * rrPilotProbability ε a (u j)) *
(∑ v : Fin (adaptiveMainSize m n) → Fin 14,
(∏ j, adaptiveMainMass select (adaptiveTrueParam θ h n)
(pilotAdaptivePrefixTheta p ε m
(le_of_lt (hsub.2 n hn).2) u) p ε (v j)) *
(sqrt (n : ℝ) *
(contrast (pilotAdaptivePrefixTheta p ε m
(le_of_lt (hsub.2 n hn).2) u) +
(adaptiveMainSize m n : ℝ)⁻¹ *
(∑ j, adaptiveSelectedScore select
(pilotAdaptivePrefixTheta p ε m
(le_of_lt (hsub.2 n hn).2) u)
p ε (v j)) -
contrast (adaptiveTrueParam θ h n))) ^ 2)) =
∑ u : Fin (m n) → Fin 4,
(∏ j : Fin (m n), ∑ a : Fin 4,
piTheta (adaptiveTrueParam θ h n) p a * rrPilotProbability ε a (u j)) *
((n : ℝ) / (adaptiveMainSize m n : ℝ) *
adaptiveScoreVariance select (adaptiveTrueParam θ h n)
(pilotAdaptivePrefixTheta p ε m
(le_of_lt (hsub.2 n hn).2) u) p ε) := by
classical
let hmn : m n ≤ n := le_of_lt (hsub.2 n hn).2
have hN : 0 < adaptiveMainSize m n := by
rw [adaptiveMainSize, adaptivePilotSize_eq m hsub hn]
exact Nat.sub_pos_of_lt (hsub.2 n hn).2
apply Finset.sum_congr rfl
intro u _
congr 1
let η₀ := pilotAdaptivePrefixTheta p ε m hmn u
let η : ℕ → TrialParameter := fun k => if k = n then η₀ else θ
have hη₀ : InteriorMeans η₀ := by
exact pilotTheta_interior p ε m (pilotAdaptiveEncode hmn u (fun _ => 0))
have hη : ∀ k, InteriorMeans (η k) := by
intro k
simp only [η]
split_ifs <;> assumption
have hηlim : Tendsto η atTop (nhds θ) := by
apply tendsto_const_nhds.congr'
filter_upwards [eventually_gt_atTop n] with k hk
simp [η, ne_of_gt hk]
have hrisk := adaptivePilotRow_conditional_rootN_meanSquare
select θ h η p ε m hselect hp hθ hη hε hηlim hsub n hN
simpa only [adaptivePilotRowProductMass, adaptivePilotRowMass, η, if_pos, η₀,
hmn] using hrisk
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotLocalization 7 declarations Quantitative localization of the randomized-response pilot
Quantitative localization of the randomized-response pilot
def pilotInverseFrequency
The affine inverse applied to one released-category frequency. For the displayed inputs and conditions, the stated result follows. The pilot Inverse Frequency is determined by the displayed parameters.
theorem pilotInverseFrequency_sub_privateMass
Under the supplied quantities and conditions, the pilot inverse frequency sub private mass assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Inverse Frequency sub private Mass.
Formal statement
hypotheses
conclusion
pilotInverseFrequency ε x - piTheta θ p k
Proof (Lean source)
lemma pilotInverseFrequency_sub_privateMass (θ : TrialParameter) (p ε x : ℝ)
(hε : 0 < ε) (k : Fin 4) :
pilotInverseFrequency ε x - piTheta θ p k =
(exp ε + 3) / (exp ε - 1) *
(x - ∑ j : Fin 4, piTheta θ p j * rrPilotProbability ε j k) := by
have he : exp ε - 1 ≠ 0 := by
have : 1 < exp ε := by
simpa using (Real.exp_lt_exp.mpr hε : exp 0 < exp ε)
linarith
rw [pilotInverseFrequency, rrPilot_marginal_probability]
field_simp [he, ne_of_gt (by positivity : 0 < exp ε + 3)]
ring
theorem pilotInverseFrequency_error_abs
Under the supplied quantities and conditions, the pilot inverse frequency error abs assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Inverse Frequency error abs.
Formal statement
hypotheses
conclusion
|pilotInverseFrequency ε x - piTheta θ p k|
Proof (Lean source)
lemma pilotInverseFrequency_error_abs (θ : TrialParameter) (p ε x : ℝ)
(hε : 0 < ε) (k : Fin 4) :
|pilotInverseFrequency ε x - piTheta θ p k| =
(exp ε + 3) / (exp ε - 1) *
|x - ∑ j : Fin 4, piTheta θ p j * rrPilotProbability ε j k| := by
rw [pilotInverseFrequency_sub_privateMass θ p ε x hε k, abs_mul]
have hn : 0 ≤ (exp ε + 3) / (exp ε - 1) := by
have : 1 < exp ε := by
simpa using (Real.exp_lt_exp.mpr hε : exp 0 < exp ε)
positivity
rw [abs_of_nonneg hn]
theorem clipInterior_eq_of_abs_sub_lt
Clipping does not change a raw inverse which lies in its clipping interval. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the clip Interior eq of abs sub lt.
Formal statement
hypotheses
δ x θ r :
ℝ
hδleft :
δ + r ≤ θ
hδright :
δ + r ≤ 1 - θ
hx :
|x - θ| < r
conclusion
clipInterior δ x = x
Proof (Lean source)
lemma clipInterior_eq_of_abs_sub_lt {δ x θ r : ℝ}
(hδleft : δ + r ≤ θ) (hδright : δ + r ≤ 1 - θ)
(hx : |x - θ| < r) :
clipInterior δ x = x := by
apply clipInterior_eq_self
· linarith [(abs_lt.mp hx).1]
· linarith [(abs_lt.mp hx).2]
theorem pilotTheta_localized_of_frequency
A sufficiently accurate pair of calibrated pilot frequencies puts the clipped pilot estimate in the fixed coordinatewise r-neighborhood of the truth. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Theta localized of frequency.
Formal statement
hypotheses
θ :
p ε r :
ℝ
m :
ℕ → ℕ
n :
ℕ
z :
Transcript (pilotOutputFamily n)
hp :
hθ :
hε :
0 < ε
hr :
0 < r
hδ0l :
(m n + 2 : ℝ)⁻¹ + r ≤ θ 0
hδ0r :
(m n + 2 : ℝ)⁻¹ + r ≤ 1 - θ 0
hδ1l :
(m n + 2 : ℝ)⁻¹ + r ≤ θ 1
hδ1r :
(m n + 2 : ℝ)⁻¹ + r ≤ 1 - θ 1
hfreq0 :
|pilotFrequency m z 1
- ∑ j : Fin 4, piTheta θ p j * rrPilotProbability ε j 1| < r * controlProb p * (exp ε - 1) / (exp ε + 3)
hfreq1 :
|pilotFrequency m z 3
Proof (Lean source)
lemma pilotTheta_localized_of_frequency
(θ : TrialParameter) (p ε r : ℝ) (m : ℕ → ℕ) {n : ℕ}
(z : Transcript (pilotOutputFamily n))
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hr : 0 < r)
(hδ0l : (m n + 2 : ℝ)⁻¹ + r ≤ θ 0)
(hδ0r : (m n + 2 : ℝ)⁻¹ + r ≤ 1 - θ 0)
(hδ1l : (m n + 2 : ℝ)⁻¹ + r ≤ θ 1)
(hδ1r : (m n + 2 : ℝ)⁻¹ + r ≤ 1 - θ 1)
(hfreq0 : |pilotFrequency m z 1 -
∑ j : Fin 4, piTheta θ p j * rrPilotProbability ε j 1| <
r * controlProb p * (exp ε - 1) / (exp ε + 3))
(hfreq1 : |pilotFrequency m z 3 -
∑ j : Fin 4, piTheta θ p j * rrPilotProbability ε j 3| <
r * p * (exp ε - 1) / (exp ε + 3)) :
|pilotTheta p ε m z 0 - θ 0| < r ∧
|pilotTheta p ε m z 1 - θ 1| < r := by
have hep : 1 < exp ε := by
simpa using (Real.exp_lt_exp.mpr hε : exp 0 < exp ε)
have hcp : 0 < controlProb p := by
dsimp [controlProb]
linarith [hp.2]
have hraw0 :
|pilotInverseFrequency ε (pilotFrequency m z 1) / controlProb p - θ 0| < r := by
have hpi : piTheta θ p 1 = controlProb p * θ 0 := rfl
have htheta : θ 0 = piTheta θ p 1 / controlProb p := by
rw [hpi]
field_simp
rw [htheta, ← sub_div]
have herr := pilotInverseFrequency_error_abs θ p ε (pilotFrequency m z 1) hε 1
rw [abs_div, abs_of_pos hcp, herr]
apply (div_lt_iff₀ hcp).2
calc
(exp ε + 3) / (exp ε - 1) *
|pilotFrequency m z 1 -
∑ j : Fin 4, piTheta θ p j * rrPilotProbability ε j 1| <
(exp ε + 3) / (exp ε - 1) *
(r * controlProb p * (exp ε - 1) / (exp ε + 3)) := by
gcongr <;> positivity
_ = r * controlProb p := by
field_simp [ne_of_gt (sub_pos.mpr hep), ne_of_gt (by positivity : 0 < exp ε + 3)]
have hraw1 :
|pilotInverseFrequency ε (pilotFrequency m z 3) / p - θ 1| < r := by
have hp0 : 0 < p := hp.1
have hpi : piTheta θ p 3 = p * θ 1 := rfl
have htheta : θ 1 = piTheta θ p 3 / p := by
rw [hpi]
field_simp
rw [htheta, ← sub_div]
have herr := pilotInverseFrequency_error_abs θ p ε (pilotFrequency m z 3) hε 3
rw [abs_div, abs_of_pos hp0, herr]
apply (div_lt_iff₀ hp0).2
calc
(exp ε + 3) / (exp ε - 1) *
|pilotFrequency m z 3 -
∑ j : Fin 4, piTheta θ p j * rrPilotProbability ε j 3| <
(exp ε + 3) / (exp ε - 1) *
(r * p * (exp ε - 1) / (exp ε + 3)) := by
gcongr <;> positivity
_ = r * p := by
field_simp [ne_of_gt (sub_pos.mpr hep), ne_of_gt (by positivity : 0 < exp ε + 3)]
have hclip0 := clipInterior_eq_of_abs_sub_lt hδ0l hδ0r hraw0
have hclip1 := clipInterior_eq_of_abs_sub_lt hδ1l hδ1r hraw1
constructor
· change |clipInterior (m n + 2 : ℝ)⁻¹
(pilotInverseFrequency ε (pilotFrequency m z 1) / controlProb p) - θ 0| < r
rw [hclip0]
exact hraw0
· change |clipInterior (m n + 2 : ℝ)⁻¹
(pilotInverseFrequency ε (pilotFrequency m z 3) / p) - θ 1| < r
rw [hclip1]
exact hraw1
theorem pilotFrequency_deviation_le
Uniform Chebyshev concentration of one released pilot-cell frequency. The bound is parameter free apart from the exact centering probability. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Frequency deviation le.
Formal statement
hypotheses
θ :
p ε a :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
n :
ℕ
hmn :
m n ≤ n
hmpos :
0 < m n
ha :
0 < a
k :
Fin 4
conclusion
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) {z | a ≤ |pilotFrequency m z k - ∑ j : Fin 4, piTheta θ p j * rrPilotProbability ε j k|}
≤ ofReal (1 / ((m n : ℝ) * a ^ 2))
Proof (Lean source)
lemma pilotFrequency_deviation_le
(θ : TrialParameter) (p ε a : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
{n : ℕ} (hmn : m n ≤ n) (hmpos : 0 < m n) (ha : 0 < a) (k : Fin 4) :
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
{z | a ≤ |pilotFrequency m z k -
∑ j : Fin 4, piTheta θ p j * rrPilotProbability ε j k|} ≤
ofReal (1 / ((m n : ℝ) * a ^ 2)) := by
let P := rrPilotMarginal θ p ε
letI : IsProbabilityMeasure P := rrPilotMarginal_isProbability θ p ε hp hθ
let g := pilotCellIndicator k
have hg : MemLp g 2 P := by
apply MemLp.of_bound (measurable_pilotCellIndicator k).aestronglyMeasurable 1
filter_upwards [] with x
dsimp [g, pilotCellIndicator]
split_ifs <;> norm_num
have hcheb := iid_sum_chebyshev
(N := m n) P g hg (mul_pos (Nat.cast_pos.mpr hmpos) ha)
have hvar : Var[g; P] ≤ 1 := by
have hg2 : Integrable (fun x => (g x) ^ 2) P := by
apply Integrable.of_bound
((measurable_pilotCellIndicator k).pow_const 2).aestronglyMeasurable 1
filter_upwards [] with x
dsimp [g, pilotCellIndicator]
split_ifs <;> norm_num
calc
Var[g; P] ≤ ∫ x, (g x) ^ 2 ∂P := variance_le_expectation_sq hg.aestronglyMeasurable
_ ≤ 1 := by
have hpoint : ∀ x, (g x) ^ 2 ≤ (1 : ℝ) := by
intro x
dsimp [g, pilotCellIndicator]
split_ifs <;> norm_num
calc
(∫ x, (g x) ^ 2 ∂P) ≤ ∫ _x, (1 : ℝ) ∂P := by
apply integral_mono_ae hg2 (integrable_const 1)
exact Filter.Eventually.of_forall hpoint
_ = 1 := by simp
have hlaw := pilotFrequency_map_eq_pi select θ p ε m hselect hp hθ hε hmn k
have hfreq : Measurable (fun z : Transcript (pilotOutputFamily n) =>
pilotFrequency m z k) := measurable_pilotFrequency m k
have hsum : Measurable (fun u : Fin (m n) → Fin 4 =>
∑ i : Fin (m n), g (u i)) := by fun_prop
let q := ∑ j : Fin 4, piTheta θ p j * rrPilotProbability ε j k
have hmean : ∫ x, g x ∂P = q := integral_pilotCellIndicator θ p ε hp hθ k
let B : Set ℝ := {x | a ≤ |x - q|}
have hB : MeasurableSet B := by
exact measurableSet_le measurable_const
((measurable_id.sub measurable_const).abs)
change (transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
((fun z => pilotFrequency m z k) ⁻¹' B) ≤ _
rw [← Measure.map_apply hfreq hB, hlaw, Measure.map_apply (by fun_prop) hB]
have hevent : {u : Fin (m n) → Fin 4 | a ≤ |pilotPrefixFrequency u k - q|} =
{u | (m n : ℝ) * a ≤
|(∑ i : Fin (m n), g (u i)) - (m n : ℝ) * ∫ x, g x ∂P|} := by
ext u
simp only [Set.mem_setOf_eq, pilotPrefixFrequency]
have hmne : (m n : ℝ) ≠ 0 := ne_of_gt (Nat.cast_pos.mpr hmpos)
change a ≤ |(m n : ℝ)⁻¹ * ∑ j : Fin (m n), g (u j) - q| ↔
(m n : ℝ) * a ≤
|(∑ j : Fin (m n), g (u j)) - (m n : ℝ) * ∫ x, g x ∂P|
rw [hmean, inv_eq_one_div]
have hid : (m n : ℝ) *
((1 / (m n : ℝ) * ∑ j : Fin (m n), g (u j)) - q) =
(∑ j : Fin (m n), g (u j)) - (m n : ℝ) * q := by
field_simp [hmne]
have habs : |(∑ j : Fin (m n), g (u j)) - (m n : ℝ) * q| =
(m n : ℝ) *
|(1 / (m n : ℝ) * ∑ j : Fin (m n), g (u j)) - q| := by
rw [← hid, abs_mul, abs_of_pos (Nat.cast_pos.mpr hmpos)]
constructor <;> intro h
· rw [habs]
exact (mul_le_mul_iff_of_pos_left (Nat.cast_pos.mpr hmpos)).2 h
· rw [habs] at h
exact (mul_le_mul_iff_of_pos_left (Nat.cast_pos.mpr hmpos)).1 h
change (Measure.pi fun _ : Fin (m n) => rrPilotMarginal θ p ε)
{u | a ≤ |pilotPrefixFrequency u k - q|} ≤ _
rw [hevent]
calc
_ ≤ ofReal
((m n : ℝ) * Var[g; P] / ((m n : ℝ) * a) ^ 2) := hcheb
_ ≤ ofReal (1 / ((m n : ℝ) * a ^ 2)) := by
apply ENNReal.ofReal_le_ofReal
have hm0 : 0 < (m n : ℝ) := Nat.cast_pos.mpr hmpos
calc
(m n : ℝ) * Var[g; P] / ((m n : ℝ) * a) ^ 2 ≤
(m n : ℝ) * 1 / ((m n : ℝ) * a) ^ 2 := by
gcongr
_ = 1 / ((m n : ℝ) * a ^ 2) := by field_simp
theorem pilotTheta_deviation_le
The two-coordinate clipped pilot leaves an r-neighborhood of its true parameter with probability at most the sum of two parameter-uniform Chebyshev bounds. This form applies row by row to local parameter sequences. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Theta deviation le.
Formal statement
hypotheses
θ :
p ε r :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hr :
0 < r
n :
ℕ
hmn :
m n ≤ n
hmpos :
0 < m n
hδ0l :
(m n + 2 : ℝ)⁻¹ + r ≤ θ 0
hδ0r :
(m n + 2 : ℝ)⁻¹ + r ≤ 1 - θ 0
hδ1l :
(m n + 2 : ℝ)⁻¹ + r ≤ θ 1
hδ1r :
(m n + 2 : ℝ)⁻¹ + r ≤ 1 - θ 1
conclusion
a0 :
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) {z | r ≤ |pilotTheta p ε m z 0 - θ 0| ∨ r ≤ |pilotTheta p ε m z 1 - θ 1|}
Proof (Lean source)
lemma pilotTheta_deviation_le
(θ : TrialParameter) (p ε r : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hr : 0 < r) {n : ℕ} (hmn : m n ≤ n) (hmpos : 0 < m n)
(hδ0l : (m n + 2 : ℝ)⁻¹ + r ≤ θ 0)
(hδ0r : (m n + 2 : ℝ)⁻¹ + r ≤ 1 - θ 0)
(hδ1l : (m n + 2 : ℝ)⁻¹ + r ≤ θ 1)
(hδ1r : (m n + 2 : ℝ)⁻¹ + r ≤ 1 - θ 1) :
let a0 := r * controlProb p * (exp ε - 1) / (exp ε + 3)
let a1 := r * p * (exp ε - 1) / (exp ε + 3)
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
{z | r ≤ |pilotTheta p ε m z 0 - θ 0| ∨
r ≤ |pilotTheta p ε m z 1 - θ 1|} ≤
ofReal (1 / ((m n : ℝ) * a0 ^ 2)) +
ofReal (1 / ((m n : ℝ) * a1 ^ 2)) := by
dsimp only
let q0 := ∑ j : Fin 4, piTheta θ p j * rrPilotProbability ε j 1
let q1 := ∑ j : Fin 4, piTheta θ p j * rrPilotProbability ε j 3
let a0 := r * controlProb p * (exp ε - 1) / (exp ε + 3)
let a1 := r * p * (exp ε - 1) / (exp ε + 3)
have ha0 : 0 < a0 := by
dsimp [a0, controlProb]
have hep : 1 < exp ε := by
simpa using (Real.exp_lt_exp.mpr hε : exp 0 < exp ε)
exact div_pos (mul_pos (mul_pos hr (by linarith [hp.2])) (by linarith)) (by positivity)
have ha1 : 0 < a1 := by
dsimp [a1]
have hep : 1 < exp ε := by
simpa using (Real.exp_lt_exp.mpr hε : exp 0 < exp ε)
exact div_pos (mul_pos (mul_pos hr hp.1) (by linarith)) (by positivity)
let A0 : Set (Transcript (pilotOutputFamily n)) :=
{z | a0 ≤ |pilotFrequency m z 1 - q0|}
let A1 : Set (Transcript (pilotOutputFamily n)) :=
{z | a1 ≤ |pilotFrequency m z 3 - q1|}
have hsub : {z | r ≤ |pilotTheta p ε m z 0 - θ 0| ∨
r ≤ |pilotTheta p ε m z 1 - θ 1|} ⊆ A0 ∪ A1 := by
intro z hz
by_contra hzunion
have hz0 : |pilotFrequency m z 1 - q0| < a0 := by
exact lt_of_not_ge (fun h => hzunion (Set.mem_union_left A1 h))
have hz1 : |pilotFrequency m z 3 - q1| < a1 := by
exact lt_of_not_ge (fun h => hzunion (Set.mem_union_right A0 h))
have hloc := pilotTheta_localized_of_frequency θ p ε r m z hp hθ hε hr
hδ0l hδ0r hδ1l hδ1r (by simpa [a0, q0] using hz0)
(by simpa [a1, q1] using hz1)
exact hz.elim (fun h => (not_le_of_gt hloc.1) h)
(fun h => (not_le_of_gt hloc.2) h)
let μ := transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n
calc
μ {z | r ≤ |pilotTheta p ε m z 0 - θ 0| ∨
r ≤ |pilotTheta p ε m z 1 - θ 1|} ≤ μ (A0 ∪ A1) := measure_mono hsub
_ ≤ μ A0 + μ A1 := measure_union_le A0 A1
_ ≤ ofReal (1 / ((m n : ℝ) * a0 ^ 2)) +
ofReal (1 / ((m n : ℝ) * a1 ^ 2)) := by
apply add_le_add
· simpa [μ, A0, a0, q0] using
pilotFrequency_deviation_le select θ p ε a0 m hselect hp hθ hε hmn hmpos ha0 1
· simpa [μ, A1, a1, q1] using
pilotFrequency_deviation_le select θ p ε a1 m hselect hp hθ hε hmn hmpos ha1 3
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotMoments 14 declarations Algebraic moment identities for the pilot score
Algebraic moment identities for the pilot score
theorem rrPilot_marginal_probability
The randomized-response pilot output has the affine categorical probability used by the inverse-frequency estimator. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Proof (Lean source)
lemma rrPilot_marginal_probability (θ : TrialParameter) (p ε : ℝ)
(k : Fin 4) :
∑ j : Fin 4, piTheta θ p j * rrPilotProbability ε j k =
((exp ε - 1) * piTheta θ p k + 1) / (exp ε + 3) := by
have hpi0 : piTheta θ p 0 = controlProb p * (1 - θ 0) := rfl
have hpi1 : piTheta θ p 1 = controlProb p * θ 0 := rfl
have hpi2 : piTheta θ p 2 = p * (1 - θ 1) := rfl
have hpi3 : piTheta θ p 3 = p * θ 1 := rfl
fin_cases k <;>
simp +decide [rrPilotProbability, controlProb, Fin.sum_univ_succ,
hpi0, hpi1, hpi2, hpi3] <;>
field_simp [ne_of_gt (by positivity : 0 < exp ε + 3)] <;> ring
theorem rrPilot_inverse_marginal_probability
Inverting the exact randomized-response pilot probability recovers the corresponding private-input cell probability. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the rr Pilot inverse marginal probability.
Formal statement
hypotheses
Proof (Lean source)
-- keep: reusable finite-pilot construction, law, localization, or limit API for related adaptive procedures
lemma rrPilot_inverse_marginal_probability (θ : TrialParameter) (p ε : ℝ)
(hε : 0 < ε) (k : Fin 4) :
((exp ε + 3) *
(∑ j : Fin 4, piTheta θ p j * rrPilotProbability ε j k) - 1) /
(exp ε - 1) = piTheta θ p k := by
rw [rrPilot_marginal_probability]
have he : exp ε - 1 ≠ 0 := by
have : 1 < exp ε := by
simpa using (Real.exp_lt_exp.mpr hε : exp 0 < exp ε)
linarith
field_simp [he, ne_of_gt (by positivity : 0 < exp ε + 3)]
ring
theorem clipInterior_eq_self
Clipping is inactive once its margin is inside both distances to the boundary. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma clipInterior_eq_self {δ x : ℝ} (hleft : δ ≤ x)
(hright : δ ≤ 1 - x) :
clipInterior δ x = x := by
rw [clipInterior, min_eq_right (by linarith), max_eq_right hleft]
theorem matrix_mulVec_direction_eq_value_contrast
At a minimizing contrast direction, the information matrix sends that direction to its quadratic value times the contrast vector. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
hI :
I.IsHermitian
t :
ℝ
hmin :
conclusion
Proof (Lean source)
lemma matrix_mulVec_direction_eq_value_contrast
(I : Matrix (Fin 2) (Fin 2) ℝ) (hI : I.IsHermitian) (t : ℝ)
(hmin : ∀ u : ℝ,
informationQuadratic I (direction t) ≤
informationQuadratic I (direction u)) :
I.mulVec (direction t) = fun k =>
informationQuadratic I (direction t) * contrastVector k := by
let A := I 0 0 + I 0 1 + I 1 0 + I 1 1
let B := 2 * (I 0 0 * t + I 0 1 * (t + 1) +
I 1 0 * t + I 1 1 * (t + 1))
have hs := congrFun₂ hI 0 1
have hpoly (x : ℝ) : 0 ≤ x * B + x ^ 2 * A := by
have h := hmin (t + x)
have hid : informationQuadratic I (direction (t + x)) =
informationQuadratic I (direction t) + x * B + x ^ 2 * A := by
dsimp [A, B]
simp [informationQuadratic, direction, Fin.sum_univ_succ,
star_trivial] at hs ⊢
rw [hs]
ring
rw [hid] at h
linarith
have hB : B = 0 := by
by_contra hB0
have hden : 0 < |A| + 1 := by positivity
have hAle : A ≤ |A| := le_abs_self A
have h := hpoly (-B / (2 * (|A| + 1)))
have hsq : 0 < B ^ 2 := sq_pos_of_ne_zero hB0
field_simp at h
nlinarith [mul_nonneg hsq.le (sub_nonneg.mpr hAle)]
have hsum :
(I 0 0 * t + I 0 1 * (t + 1)) +
(I 1 0 * t + I 1 1 * (t + 1)) = 0 := by
dsimp [B] at hB
linarith
have hj : informationQuadratic I (direction t) =
I 1 0 * t + I 1 1 * (t + 1) := by
simp [informationQuadratic, direction, Fin.sum_univ_succ]
linear_combination t * hsum
funext k
fin_cases k <;>
simp [mulVec, direction, contrastVector, Fin.sum_univ_succ] at hs ⊢ <;>
rw [hj] <;> linarith
theorem saddle_information_normalEquation
The saddle direction satisfies the exact score normal equation. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
θ :
p ε :
ℝ
α :
t J :
ℝ
hmin :
∀ u, informationObjective θ p ε α t ≤ informationObjective θ p ε α u
hJ :
J = informationObjective θ p ε α t
conclusion
Proof (Lean source)
lemma saddle_information_normalEquation (θ : TrialParameter) (p ε : ℝ)
(α : StaircaseWeight) (t J : ℝ)
(hmin : ∀ u, informationObjective θ p ε α t ≤
informationObjective θ p ε α u)
(hJ : J = informationObjective θ p ε α t) :
(informationMatrix θ p ε α).mulVec (direction t) =
fun k => J * contrastVector k := by
have hquad (u : ℝ) :
informationQuadratic (informationMatrix θ p ε α) (direction u) =
informationObjective θ p ε α u := by
rw [informationObjective_eq_informationQuadratic]
simp [informationQuadratic, dotProduct, mulVec, star_trivial]
have hmin' : ∀ u : ℝ,
informationQuadratic (informationMatrix θ p ε α) (direction t) ≤
informationQuadratic (informationMatrix θ p ε α) (direction u) := by
intro u
rw [hquad, hquad]
exact hmin u
have hHermitian : (informationMatrix θ p ε α).IsHermitian := by
ext i j
simp only [Matrix.conjTranspose_apply, star_trivial]
unfold informationMatrix
apply Finset.sum_congr rfl
intro s _
ring
have hnormal := matrix_mulVec_direction_eq_value_contrast
(informationMatrix θ p ε α)
hHermitian t hmin'
calc
(informationMatrix θ p ε α).mulVec (direction t) =
fun k => informationObjective θ p ε α t * contrastVector k := by
rw [← hquad]
exact hnormal
_ = fun k => J * contrastVector k := by rw [hJ]
theorem patternMass_affine_difference
Under the supplied quantities and conditions, the pattern mass affine difference assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
patternMass θ p ε s
Proof (Lean source)
lemma patternMass_affine_difference (θ η : TrialParameter) (p ε : ℝ)
(s : Fin 14) :
patternMass θ p ε s = patternMass η p ε s +
∑ k : Fin 2, (θ k - η k) * patternGradient p ε s k := by
have hpi (ξ : TrialParameter) (j : Fin 4) : piTheta ξ p j =
if j.val = 0 then controlProb p * (1 - ξ 0)
else if j.val = 1 then controlProb p * ξ 0
else if j.val = 2 then p * (1 - ξ 1) else p * ξ 1 := by
fin_cases j <;> rfl
cases h0 : patternContains s 0 <;>
cases h1 : patternContains s 1 <;>
cases h2 : patternContains s 2 <;>
cases h3 : patternContains s 3 <;>
simp [patternMass, hpi, patternRay, patternGradient,
privacyIncrement, controlProb, Fin.sum_univ_two, Fin.sum_univ_four,
h0, h1, h2, h3] <;> ring
theorem patternGradient_eq_ray_sub
Under the supplied quantities and conditions, the pattern gradient eq ray sub assertion holds.
Formal statement
Proof (Lean source)
lemma patternGradient_eq_ray_sub (p ε : ℝ) (s : Fin 14) :
patternGradient p ε s 0 =
controlProb p * (patternRay ε s 1 - patternRay ε s 0) ∧
patternGradient p ε s 1 =
p * (patternRay ε s 3 - patternRay ε s 2) := by
cases h0 : patternContains s 0 <;>
cases h1 : patternContains s 1 <;>
cases h2 : patternContains s 2 <;>
cases h3 : patternContains s 3 <;>
simp [patternGradient, patternRay, privacyIncrement,
h0, h1, h2, h3] <;> ring <;> simp
theorem staircase_weightedGradient_sum_eq_zero
Under the supplied quantities and conditions, the staircase weighted gradient sum eq zero assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the staircase weighted Gradient sum eq zero.
Formal statement
hypotheses
conclusion
∑ s : Fin 14, α s * patternGradient p ε s k = 0
Proof (Lean source)
lemma staircase_weightedGradient_sum_eq_zero (p ε : ℝ)
(α : StaircaseWeight) (hα : staircaseFeasible ε α) (k : Fin 2) :
∑ s : Fin 14, α s * patternGradient p ε s k = 0 := by
fin_cases k
· calc
(∑ s : Fin 14, α s * patternGradient p ε s 0) =
∑ s : Fin 14, α s *
(controlProb p * (patternRay ε s 1 - patternRay ε s 0)) := by
apply Finset.sum_congr rfl
intro s _
rw [(patternGradient_eq_ray_sub p ε s).1]
_ = controlProb p *
((∑ s : Fin 14, α s * patternRay ε s 1) -
∑ s : Fin 14, α s * patternRay ε s 0) := by
rw [← Finset.sum_sub_distrib, Finset.mul_sum]
apply Finset.sum_congr rfl
intro s _
ring
_ = controlProb p *
(staircaseMatrix ε α 1 - staircaseMatrix ε α 0) := rfl
_ = 0 := by rw [hα.2 1, hα.2 0]; ring
· calc
(∑ s : Fin 14, α s * patternGradient p ε s 1) =
∑ s : Fin 14, α s *
(p * (patternRay ε s 3 - patternRay ε s 2)) := by
apply Finset.sum_congr rfl
intro s _
rw [(patternGradient_eq_ray_sub p ε s).2]
_ = p * ((∑ s : Fin 14, α s * patternRay ε s 3) -
∑ s : Fin 14, α s * patternRay ε s 2) := by
rw [← Finset.sum_sub_distrib, Finset.mul_sum]
apply Finset.sum_congr rfl
intro s _
ring
_ = p * (staircaseMatrix ε α 3 - staircaseMatrix ε α 2) := rfl
_ = 0 := by rw [hα.2 3, hα.2 2]; ring
theorem staircase_weightedProjectedGradient_sum_eq_zero
Under the supplied quantities and conditions, the staircase weighted projected gradient sum eq zero assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the staircase weighted Projected Gradient sum eq zero.
Formal statement
hypotheses
conclusion
∑ s : Fin 14, α s * projectedGradient p ε s t = 0
Proof (Lean source)
lemma staircase_weightedProjectedGradient_sum_eq_zero (p ε t : ℝ)
(α : StaircaseWeight) (hα : staircaseFeasible ε α) :
∑ s : Fin 14, α s * projectedGradient p ε s t = 0 := by
have h0 := staircase_weightedGradient_sum_eq_zero p ε α hα 0
have h1 := staircase_weightedGradient_sum_eq_zero p ε α hα 1
calc
(∑ s : Fin 14, α s * projectedGradient p ε s t) =
(∑ s : Fin 14, α s * patternGradient p ε s 0) * direction t 0 +
(∑ s : Fin 14, α s * patternGradient p ε s 1) * direction t 1 := by
unfold projectedGradient
simp only [Fin.sum_univ_two]
rw [Finset.sum_mul, Finset.sum_mul, ← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro s _
ring
_ = 0 := by rw [h0, h1]; ring
theorem informationMatrix_mulVec_direction_apply
Under the supplied quantities and conditions, the information matrix mul vec direction apply assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma informationMatrix_mulVec_direction_apply (θ : TrialParameter) (p ε : ℝ)
(α : StaircaseWeight) (t : ℝ) (k : Fin 2) :
(informationMatrix θ p ε α).mulVec (direction t) k =
∑ s : Fin 14, α s * patternGradient p ε s k *
projectedGradient p ε s t / patternMass θ p ε s := by
fin_cases k <;>
simp [informationMatrix, mulVec, projectedGradient,
Fin.sum_univ_two] <;>
rw [Finset.sum_mul, Finset.sum_mul, ← Finset.sum_add_distrib] <;>
apply Finset.sum_congr rfl <;>
intro s _ <;> ring
theorem affineCorrection_mean_identity
Exact one-step affine correction identity. If η selects a feasible saddle (α,t,J), then averaging its influence score under the true parameter θ corrects contrast η exactly to contrast θ. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the affine Correction mean identity.
Formal statement
hypotheses
θ η :
p ε :
ℝ
hp :
hη :
hε :
0 ≤ ε
α :
t J :
ℝ
hα :
hJ :
J ≠ 0
hnormal :
conclusion
= contrast θ - contrast η
Proof (Lean source)
lemma affineCorrection_mean_identity (θ η : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hη : InteriorMeans η) (hε : 0 ≤ ε)
(α : StaircaseWeight) (t J : ℝ) (hα : staircaseFeasible ε α)
(hJ : J ≠ 0)
(hnormal : (informationMatrix η p ε α).mulVec (direction t) =
fun k => J * contrastVector k) :
∑ s : Fin 14, α s * patternMass θ p ε s *
(projectedGradient p ε s t / (J * patternMass η p ε s)) =
contrast θ - contrast η := by
have hm (s : Fin 14) : patternMass η p ε s ≠ 0 :=
ne_of_gt (patternMass_pos_interior η p ε hp hη hε s)
have hexpand :
(∑ s : Fin 14, α s * patternMass θ p ε s *
(projectedGradient p ε s t / (J * patternMass η p ε s))) =
J⁻¹ * ((∑ s : Fin 14, α s * projectedGradient p ε s t) +
∑ k : Fin 2, (θ k - η k) *
(informationMatrix η p ε α).mulVec (direction t) k) := by
simp only [Fin.sum_univ_two]
rw [informationMatrix_mulVec_direction_apply η p ε α t 0,
informationMatrix_mulVec_direction_apply η p ε α t 1]
rw [Finset.mul_sum, Finset.mul_sum,
← Finset.sum_add_distrib, ← Finset.sum_add_distrib,
Finset.mul_sum]
apply Finset.sum_congr rfl
intro s _
rw [patternMass_affine_difference θ η p ε s]
simp only [Fin.sum_univ_two]
field_simp [hJ, hm s]
rw [hexpand, staircase_weightedProjectedGradient_sum_eq_zero p ε t α hα,
zero_add, hnormal]
simp only [Fin.sum_univ_two]
simp [contrastVector, contrast]
field_simp [hJ]
ring
theorem phiTilde_mean_identity
The selected affine correction has the exact mean shift under every interior parameter. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the phi Tilde mean identity.
Formal statement
hypotheses
θ η :
p ε :
ℝ
select :
TrialParameter → StaircaseWeight × ℝ × ℝ
hp :
hη :
hε :
0 < ε
hselect :
SaddleSelection p ε select
conclusion
= contrast θ - contrast η
Proof (Lean source)
lemma phiTilde_mean_identity (θ η : TrialParameter) (p ε : ℝ)
(select : TrialParameter → StaircaseWeight × ℝ × ℝ)
(hp : InteriorAssignment p) (hη : InteriorMeans η) (hε : 0 < ε)
(hselect : SaddleSelection p ε select) :
∑ s : Fin 14, (select η).1 s * patternMass θ p ε s *
phiTilde η p ε select s = contrast θ - contrast η := by
have hs := hselect.2 η hη
have hJpos : 0 < (select η).2.2 := by
rw [← hs.2.2.1]
exact Jstar_pos_interior η p ε hp hη hε
have hnormal := saddle_information_normalEquation η p ε
(select η).1 (select η).2.1 (select η).2.2 hs.2.1 hs.2.2.2
simpa [phiTilde] using affineCorrection_mean_identity θ η p ε
hp hη hε.le (select η).1 (select η).2.1 (select η).2.2
hs.1 hJpos.ne' hnormal
theorem affineCorrection_secondMoment_identity
At its construction parameter, a saddle affine correction has second moment equal to reciprocal saddle information. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the affine Correction second Moment identity.
Formal statement
hypotheses
conclusion
∑ s : Fin 14, α s * patternMass η p ε s * (projectedGradient p ε s t / (J * patternMass η p ε s)) ^ 2
= J⁻¹
Proof (Lean source)
lemma affineCorrection_secondMoment_identity (η : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hη : InteriorMeans η) (hε : 0 ≤ ε)
(α : StaircaseWeight) (t J : ℝ) (hJ : J ≠ 0)
(hvalue : J = informationObjective η p ε α t) :
∑ s : Fin 14, α s * patternMass η p ε s *
(projectedGradient p ε s t / (J * patternMass η p ε s)) ^ 2 =
J⁻¹ := by
have hm (s : Fin 14) : patternMass η p ε s ≠ 0 :=
ne_of_gt (patternMass_pos_interior η p ε hp hη hε s)
calc
(∑ s : Fin 14, α s * patternMass η p ε s *
(projectedGradient p ε s t / (J * patternMass η p ε s)) ^ 2) =
(J ^ 2)⁻¹ * informationObjective η p ε α t := by
unfold informationObjective patternInformation
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro s _
field_simp [hJ, hm s]
_ = J⁻¹ := by rw [← hvalue]; field_simp [hJ]
theorem phiTilde_secondMoment_identity
The selected influence value has exact conditional second moment Vstar at the selection parameter. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the phi Tilde second Moment identity.
Formal statement
hypotheses
η :
p ε :
ℝ
select :
TrialParameter → StaircaseWeight × ℝ × ℝ
hp :
hη :
hε :
0 < ε
hselect :
SaddleSelection p ε select
conclusion
= Vstar η p ε
Proof (Lean source)
lemma phiTilde_secondMoment_identity (η : TrialParameter) (p ε : ℝ)
(select : TrialParameter → StaircaseWeight × ℝ × ℝ)
(hp : InteriorAssignment p) (hη : InteriorMeans η) (hε : 0 < ε)
(hselect : SaddleSelection p ε select) :
∑ s : Fin 14, (select η).1 s * patternMass η p ε s *
(phiTilde η p ε select s) ^ 2 = Vstar η p ε := by
have hs := hselect.2 η hη
have hJpos : 0 < (select η).2.2 := by
rw [← hs.2.2.1]
exact Jstar_pos_interior η p ε hp hη hε
have hmoment := affineCorrection_secondMoment_identity η p ε hp hη hε.le
(select η).1 (select η).2.1 (select η).2.2 hJpos.ne' hs.2.2.2
simpa [phiTilde, Vstar, hs.2.2.1] using hmoment
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotPrefix 29 declarations Finite-prefix laws for the randomized-response pilot
Finite-prefix laws for the randomized-response pilot
theorem fin_prod_ite_lt
the fin prod ite lt assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the fin prod ite lt.
Formal statement
hypotheses
Proof (Lean source)
lemma fin_prod_ite_lt {M : Type*} [CommMonoid M] {n ℓ : ℕ}
(hℓn : ℓ ≤ n) (f : Fin ℓ → M) :
∏ i : Fin n, (if hi : i.val < ℓ then f ⟨i.val, hi⟩ else 1) =
∏ j : Fin ℓ, f j := by
let g : Fin n → M := fun i =>
if hi : i.val < ℓ then f ⟨i.val, hi⟩ else 1
change (∏ i : Fin n, g i) = _
have hfilter : (∏ i : Fin n, g i) =
∏ i ∈ (Finset.univ.filter fun i : Fin n => i.val < ℓ), g i := by
rw [Finset.prod_filter]
apply Finset.prod_congr rfl
intro i _
by_cases hi : i.val < ℓ <;> simp [g, hi]
rw [hfilter]
symm
apply Finset.prod_bij (fun j _ => castLE hℓn j)
· intro j _
simp
· intro a _ b _ hab
exact Fin.castLE_injective hℓn hab
· intro i hi
have hil : i.val < ℓ := (Finset.mem_filter.mp hi).2
exact ⟨⟨i.val, hil⟩, Finset.mem_univ _, Fin.ext rfl⟩
· intro j _
simp [g]
theorem fin_sum_ite_lt
the fin sum ite lt assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the fin sum ite lt.
Formal statement
hypotheses
Proof (Lean source)
lemma fin_sum_ite_lt {M : Type*} [AddCommMonoid M] {n ℓ : ℕ}
(hℓn : ℓ ≤ n) (f : Fin ℓ → M) :
∑ i : Fin n, (if hi : i.val < ℓ then f ⟨i.val, hi⟩ else 0) =
∑ j : Fin ℓ, f j := by
let g : Fin n → M := fun i =>
if hi : i.val < ℓ then f ⟨i.val, hi⟩ else 0
change (∑ i : Fin n, g i) = _
have hfilter : (∑ i : Fin n, g i) =
∑ i ∈ (Finset.univ.filter fun i : Fin n => i.val < ℓ), g i := by
rw [Finset.sum_filter]
apply Finset.sum_congr rfl
intro i _
by_cases hi : i.val < ℓ <;> simp [g, hi]
rw [hfilter]
symm
apply Finset.sum_bij (fun j _ => castLE hℓn j)
· intro j _
simp
· intro a _ b _ hab
exact Fin.castLE_injective hℓn hab
· intro i hi
have hil : i.val < ℓ := (Finset.mem_filter.mp hi).2
exact ⟨⟨i.val, hil⟩, Finset.mem_univ _, Fin.ext rfl⟩
· intro j _
simp [g]
theorem rrPilotKernel_singleton
Under the supplied quantities and conditions, the rr pilot kernel singleton assertion holds.
Formal statement
Proof (Lean source)
lemma rrPilotKernel_singleton (ε : ℝ) (j k : Fin 4) :
rrPilotKernel ε j {k} = ofReal (rrPilotProbability ε j k) := by
change (Measure.count.withDensity
(fun l : Fin 4 => ofReal (rrPilotProbability ε j l))) {k} = _
rw [withDensity_apply _ (measurableSet_singleton k)]
simp
theorem piTheta_sum_eq_one
Under the supplied quantities and conditions, the pi theta sum eq one assertion holds.
Formal statement
Proof (Lean source)
lemma piTheta_sum_eq_one (θ : TrialParameter) (p : ℝ) :
∑ j : Fin 4, piTheta θ p j = 1 := by
have hpi0 : piTheta θ p 0 = controlProb p * (1 - θ 0) := rfl
have hpi1 : piTheta θ p 1 = controlProb p * θ 0 := rfl
have hpi2 : piTheta θ p 2 = p * (1 - θ 1) := rfl
have hpi3 : piTheta θ p 3 = p * θ 1 := rfl
simp [Fin.sum_univ_four, hpi0, hpi1, hpi2, hpi3, controlProb]
ring
theorem inputPathProbability_sum_eq_one
Under the supplied quantities and conditions, the input path probability sum eq one assertion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma inputPathProbability_sum_eq_one (θ : TrialParameter) (p : ℝ) (n : ℕ) :
∑ x : Fin n → Fin 4, inputPathProbability θ p x = 1 := by
change (∑ x : Fin n → Fin 4, ∏ i : Fin n, piTheta θ p (x i)) = 1
calc
(∑ x : Fin n → Fin 4, ∏ i : Fin n, piTheta θ p (x i)) =
∏ _i : Fin n, ∑ a : Fin 4, piTheta θ p a :=
(Fintype.prod_sum (fun _i : Fin n => fun a : Fin 4 => piTheta θ p a)).symm
_ = 1 := by simp [piTheta_sum_eq_one]
theorem transcriptLaw_isProbability
the transcript law is probability assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the transcript Law is Probability.
Formal statement
hypotheses
conclusion
IsProbabilityMeasure (transcriptLaw P θ p n)
Proof (Lean source)
lemma transcriptLaw_isProbability {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (n : ℕ) :
IsProbabilityMeasure (transcriptLaw P θ p n) := by
rw [isProbabilityMeasure_iff]
unfold transcriptLaw
simp only [Measure.coe_finsetSum, Finset.sum_apply, Measure.smul_apply,
smul_eq_mul]
have hpath (x : Fin n → Fin 4) : 0 ≤ inputPathProbability θ p x := by
unfold inputPathProbability
exact Finset.prod_nonneg fun i _ =>
(piTheta_pos_interior θ p hp hθ (x i)).le
rw [show (∑ x : Fin n → Fin 4,
ofReal (inputPathProbability θ p x) *
P.transcript n x univ) =
∑ x : Fin n → Fin 4,
ofReal (inputPathProbability θ p x) by
apply Finset.sum_congr rfl
intro x _
rw [(P.factorizes n x).1.measure_univ, mul_one]]
rw [← ENNReal.ofReal_sum_of_nonneg]
· rw [inputPathProbability_sum_eq_one]
simp
· intro x _
exact hpath x
theorem inputPath_prefix_rr_sum
Marginalizing independent private inputs turns the deterministic-path pilot prefix product into the product of calibrated RR marginals. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the input Path prefix rr sum.
Formal statement
hypotheses
conclusion
∑ x : Fin n → Fin 4, inputPathProbability θ p x * (∏ j : Fin ℓ, rrPilotProbability ε (x (castLE hℓn j)) (u j))
Proof (Lean source)
lemma inputPath_prefix_rr_sum (θ : TrialParameter) (p ε : ℝ)
{n ℓ : ℕ} (hℓn : ℓ ≤ n) (u : Fin ℓ → Fin 4) :
∑ x : Fin n → Fin 4, inputPathProbability θ p x *
(∏ j : Fin ℓ,
rrPilotProbability ε (x (castLE hℓn j)) (u j)) =
∏ j : Fin ℓ,
(∑ a : Fin 4, piTheta θ p a * rrPilotProbability ε a (u j)) := by
let F : (i : Fin n) → Fin 4 → ℝ := fun i a =>
piTheta θ p a *
if hi : i.val < ℓ then rrPilotProbability ε a (u ⟨i.val, hi⟩) else 1
calc
(∑ x : Fin n → Fin 4, inputPathProbability θ p x *
(∏ j : Fin ℓ,
rrPilotProbability ε (x (castLE hℓn j)) (u j))) =
∑ x : Fin n → Fin 4, ∏ i : Fin n, F i (x i) := by
apply Finset.sum_congr rfl
intro x _
rw [show inputPathProbability θ p x =
∏ i : Fin n, piTheta θ p (x i) by rfl]
rw [← fin_prod_ite_lt hℓn (fun j =>
rrPilotProbability ε (x (castLE hℓn j)) (u j))]
rw [← Finset.prod_mul_distrib]
apply Finset.prod_congr rfl
intro i _
simp only [F]
split_ifs with hi <;> rfl
_ = ∏ i : Fin n, ∑ a : Fin 4, F i a := (Fintype.prod_sum F).symm
_ = ∏ i : Fin n, (if hi : i.val < ℓ then
∑ a : Fin 4, piTheta θ p a * rrPilotProbability ε a (u ⟨i.val, hi⟩)
else 1) := by
apply Finset.prod_congr rfl
intro i _
simp only [F]
split_ifs with hi
· rfl
· simp [piTheta_sum_eq_one]
_ = _ := fin_prod_ite_lt hℓn (fun j =>
∑ a : Fin 4, piTheta θ p a * rrPilotProbability ε a (u j))
theorem rrPilot_marginal_sum_eq_one
Under the supplied quantities and conditions, the rr pilot marginal sum eq one assertion holds.
Formal statement
hypotheses
θ :
p ε :
ℝ
conclusion
Proof (Lean source)
lemma rrPilot_marginal_sum_eq_one (θ : TrialParameter) (p ε : ℝ) :
∑ k : Fin 4, (∑ a : Fin 4,
piTheta θ p a * rrPilotProbability ε a k) = 1 := by
simp_rw [rrPilot_marginal_probability]
rw [← Finset.sum_div, Finset.sum_add_distrib, ← Finset.mul_sum,
piTheta_sum_eq_one]
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin,
Nat.cast_ofNat]
field_simp [ne_of_gt (by positivity : 0 < exp ε + 3)]
ring
def rrPilotMarginal
The one-coordinate randomized-response marginal after integrating out the private four-cell input. For the displayed inputs and conditions, the stated result follows. The rr Pilot Marginal is determined by the displayed parameters.
Definition (Lean source)
parameters
θ :
p ε :
ℝ
given by
Measure.count.withDensity fun k
theorem rrPilotMarginal_singleton
Under the supplied quantities and conditions, the rr pilot marginal singleton assertion holds.
Formal statement
hypotheses
conclusion
rrPilotMarginal θ p ε {k}
Proof (Lean source)
lemma rrPilotMarginal_singleton (θ : TrialParameter) (p ε : ℝ) (k : Fin 4) :
rrPilotMarginal θ p ε {k} = ofReal
(∑ a : Fin 4, piTheta θ p a * rrPilotProbability ε a k) := by
rw [rrPilotMarginal, withDensity_apply _ (measurableSet_singleton k)]
simp
theorem rrPilotMarginal_isProbability
Under the supplied quantities and conditions, the rr pilot marginal is probability assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the rr Pilot Marginal is Probability.
Formal statement
hypotheses
conclusion
IsProbabilityMeasure (rrPilotMarginal θ p ε)
Proof (Lean source)
lemma rrPilotMarginal_isProbability (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
IsProbabilityMeasure (rrPilotMarginal θ p ε) := by
rw [isProbabilityMeasure_iff, rrPilotMarginal, withDensity_apply _ MeasurableSet.univ,
setLIntegral_univ, lintegral_count, tsum_fintype,
← ENNReal.ofReal_sum_of_nonneg]
· rw [rrPilot_marginal_sum_eq_one]
simp
· intro k _
exact Finset.sum_nonneg fun a _ => mul_nonneg
(piTheta_pos_interior θ p hp hθ a).le (by
unfold rrPilotProbability
split_ifs <;> positivity)
theorem pilotPrefix_component_singleton
For a deterministic private-input path, every initial segment lying wholly inside the pilot has the product randomized-response atom probability. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Prefix component singleton.
Formal statement
hypotheses
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n ℓ :
ℕ
hℓn :
ℓ ≤ n
hℓm :
ℓ ≤ m n
conclusion
prefixLaw (finiteSequenceKernel (pilotProtocol p ε m (select) n)) x ℓ hℓn {fun j => (inl (u j) : PilotOutput)}
Proof (Lean source)
lemma pilotPrefix_component_singleton (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε)
{n ℓ : ℕ} (hℓn : ℓ ≤ n) (hℓm : ℓ ≤ m n)
(x : Fin n → Fin 4) (u : Fin ℓ → Fin 4) :
prefixLaw
(finiteSequenceKernel
(pilotProtocol p ε m (select) n)) x ℓ hℓn
{fun j => (Sum.inl (u j) : PilotOutput)} =
∏ j : Fin ℓ, ofReal
(rrPilotProbability ε (x (castLE hℓn j)) (u j)) := by
induction ℓ with
| zero =>
have he : ({fun j : Fin 0 => (Sum.inl (u j) : PilotOutput)} :
Set (History (pilotOutputFamily n) 0 hℓn)) = univ := by
ext v
have hv : v = fun j : Fin 0 => (Sum.inl (u j) : PilotOutput) := by
funext j
exact j.elim0
constructor <;> intro
· exact Set.mem_univ _
· exact Set.mem_singleton_iff.mpr hv
rw [he]
have hQ : ∀ i, IsMarkovKernel
(finiteSequenceKernel
(pilotProtocol p ε m (select) n) i) := by
intro i
letI : IsMarkovKernel
(pilotProtocol p ε m (select) n i) :=
(pilotProtocol_private p ε m select hselect hp hε n).1 i
dsimp [finiteSequenceKernel]
infer_instance
haveI := isProbabilityMeasure_prefixLaw
(finiteSequenceKernel
(pilotProtocol p ε m (select) n)) hQ x 0 hℓn
simp
| succ r ih =>
have hrn : r ≤ n := Nat.le_of_succ_le hℓn
have hrm : r ≤ m n := le_trans (le_succ r) hℓm
have hQ : ∀ i, IsMarkovKernel
(finiteSequenceKernel
(pilotProtocol p ε m (select) n) i) := by
intro i
letI : IsMarkovKernel
(pilotProtocol p ε m (select) n i) :=
(pilotProtocol_private p ε m select hselect hp hε n).1 i
dsimp [finiteSequenceKernel]
infer_instance
let up : Fin r → Fin 4 := fun j => u j.castSucc
let uf : History (pilotOutputFamily n) (r + 1) hℓn :=
fun j => (Sum.inl (u j) : PilotOutput)
let ufp : History (pilotOutputFamily n) r (Nat.le_of_succ_le hℓn) :=
Fin.init uf
have hu : uf =
snoc (Z := pilotOutputFamily n) hℓn
ufp
(Sum.inl (u (last r))) := by
exact (Fin.snoc_init_self uf).symm
change (prefixLaw _ x (r + 1) hℓn) {uf} = _
rw [hu, prefixLaw_succ_singleton _ hQ]
change (prefixLaw _ x r (Nat.le_of_succ_le hℓn))
({((fun j => (Sum.inl (up j) : PilotOutput)) :
History (pilotOutputFamily n) r (Nat.le_of_succ_le hℓn))} :
Set (History (pilotOutputFamily n) r (Nat.le_of_succ_le hℓn))) * _ = _
rw [ih (Nat.le_of_succ_le hℓn) hrm up]
rw [Fin.prod_univ_castSucc]
congr 1
have hidx : nextIndex hℓn = castLE hℓn (last r) := Fin.ext rfl
cases hidx
dsimp [finiteSequenceKernel, pilotProtocol]
simp only [nextIndex, castLE, last, Nat.succ_eq_add_one]
change (if r < m n then
((rrPilotKernel ε) (x ⟨r, by omega⟩)).map Sum.inl
else _) {Sum.inl (u (last r))} = _
rw [if_pos (by exact_mod_cast (lt_of_lt_of_le (lt_succ_self r) hℓm))]
rw [Measure.map_apply (by fun_prop) (measurableSet_singleton _)]
have hpre : (Sum.inl : Fin 4 → PilotOutput) ⁻¹'
({Sum.inl (u (last r))} : Set PilotOutput) =
({u (last r)} : Set (Fin 4)) := by ext y; simp
rw [hpre]
convert rrPilotKernel_singleton ε (x ⟨r, by omega⟩) (u (last r))
rfl
theorem pilotPrefix_marginal_singleton
After marginalizing the IID private-input path, every all-pilot prefix atom has the product of the calibrated RR marginal probabilities. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Prefix marginal singleton.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
n :
ℕ
hmn :
m n ≤ n
conclusion
Measure.map (transcriptPrefix (Z := pilotOutputFamily n) hmn) (transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) {fun j => (inl (u j) : PilotOutput)}
Proof (Lean source)
lemma pilotPrefix_marginal_singleton (θ : TrialParameter) (p ε : ℝ)
(m : ℕ → ℕ) (hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(hε : 0 < ε) {n : ℕ} (hmn : m n ≤ n) (u : Fin (m n) → Fin 4) :
Measure.map (transcriptPrefix (Z := pilotOutputFamily n) hmn)
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
{fun j => (inl (u j) : PilotOutput)} =
∏ j : Fin (m n), ofReal
(∑ a : Fin 4, piTheta θ p a * rrPilotProbability ε a (u j)) := by
have hs : MeasurableSet
({fun j => (inl (u j) : PilotOutput)} :
Set (History (pilotOutputFamily n) (m n) hmn)) := by
change MeasurableSet ({fun j : Fin (m n) =>
(inl (u j) : PilotOutput)} : Set (Fin (m n) → PilotOutput))
have h := MeasurableSet.univ_pi (fun j =>
measurableSet_singleton (inl (u j) : PilotOutput))
exact (Set.univ_pi_singleton _).symm ▸ h
rw [Measure.map_apply (measurable_transcriptPrefix hmn)
hs]
unfold transcriptLaw
simp only [Measure.coe_finsetSum, Finset.sum_apply, Measure.smul_apply,
smul_eq_mul]
have hpath (x : Fin n → Fin 4) : 0 ≤ inputPathProbability θ p x := by
unfold inputPathProbability
exact Finset.prod_nonneg fun i _ =>
(piTheta_pos_interior θ p hp hθ (x i)).le
have hrr (a k : Fin 4) : 0 ≤ rrPilotProbability ε a k := by
unfold rrPilotProbability
split_ifs <;> positivity
calc
(∑ x : Fin n → Fin 4,
ofReal (inputPathProbability θ p x) *
(pilotEstimator p ε m select hselect hp hε).transcript n x
(transcriptPrefix hmn ⁻¹' {fun j => inl (u j)})) =
∑ x : Fin n → Fin 4,
ofReal (inputPathProbability θ p x) *
∏ j : Fin (m n), ofReal
(rrPilotProbability ε (x (castLE hmn j)) (u j)) := by
apply Finset.sum_congr rfl
intro x _
congr 1
rw [show (pilotEstimator p ε m select hselect hp hε).transcript n x =
finiteSequenceTranscript
(pilotProtocol p ε m (select) n) x by rfl]
rw [← Measure.map_apply (measurable_transcriptPrefix hmn)
hs,
finiteSequenceTranscript_map_prefix _
(pilotProtocol_private p ε m select hselect hp hε n).1 x hmn]
exact pilotPrefix_component_singleton select p ε m hselect hp hε hmn le_rfl x u
_ = ofReal (∑ x : Fin n → Fin 4,
inputPathProbability θ p x *
∏ j : Fin (m n),
rrPilotProbability ε (x (castLE hmn j)) (u j)) := by
rw [ENNReal.ofReal_sum_of_nonneg]
· apply Finset.sum_congr rfl
intro x _
rw [ENNReal.ofReal_mul (hpath x), ENNReal.ofReal_prod_of_nonneg]
intro j _
exact hrr _ _
· intro x _
exact mul_nonneg (hpath x) (Finset.prod_nonneg fun j _ => hrr _ _)
_ = ofReal (∏ j : Fin (m n),
∑ a : Fin 4, piTheta θ p a * rrPilotProbability ε a (u j)) := by
rw [inputPath_prefix_rr_sum θ p ε hmn u]
_ = _ := by
rw [ENNReal.ofReal_prod_of_nonneg]
intro j _
exact Finset.sum_nonneg fun a _ =>
mul_nonneg (piTheta_pos_interior θ p hp hθ a).le (hrr _ _)
def pilotPrefixEmbed
For the supplied quantities and conditions, the pilot prefix embed is the mathematical object specified below.
Definition (Lean source)
theorem pilotPrefixEmbed_injective
the pilot prefix embed injective assertion holds.
Formal statement
Proof (Lean source)
lemma pilotPrefixEmbed_injective {ℓ : ℕ} :
Injective (@pilotPrefixEmbed ℓ) := by
intro u v huv
funext j
exact Sum.inl_injective (congrFun huv j)
theorem pilotPrefix_range_measure
The marginalized pilot prefix is concentrated on four-category releases; the masses of those atoms sum to one. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Prefix range measure.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
n :
ℕ
hmn :
m n ≤ n
conclusion
Measure.map (transcriptPrefix (Z := pilotOutputFamily n) hmn) (transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) (range (fun u : Fin (m n) → Fin 4 => ((fun j => inl (u j)) : History (pilotOutputFamily n) (m n) hmn)))
= 1
Proof (Lean source)
lemma pilotPrefix_range_measure (θ : TrialParameter) (p ε : ℝ)
(m : ℕ → ℕ) (hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(hε : 0 < ε) {n : ℕ} (hmn : m n ≤ n) :
Measure.map (transcriptPrefix (Z := pilotOutputFamily n) hmn)
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
(range (fun u : Fin (m n) → Fin 4 =>
((fun j => inl (u j)) :
History (pilotOutputFamily n) (m n) hmn))) = 1 := by
classical
let μ := Measure.map (transcriptPrefix (Z := pilotOutputFamily n) hmn)
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
let e : (Fin (m n) → Fin 4) →
History (pilotOutputFamily n) (m n) hmn :=
fun u j => inl (u j)
have he : Injective e := by
intro u v huv
funext j
exact Sum.inl_injective (congrFun huv j)
letI : MeasurableSingletonClass
(History (pilotOutputFamily n) (m n) hmn) := by
constructor
intro v
change MeasurableSet ({v} : Set (Fin (m n) → PilotOutput))
have hv := MeasurableSet.univ_pi (fun j => measurableSet_singleton (v j))
exact (Set.univ_pi_singleton v).symm ▸ hv
have hrange : range e =
(↑(Finset.univ.image e) :
Set (History (pilotOutputFamily n) (m n) hmn)) := by
ext z
simp
change μ (range e) = 1
rw [hrange, ← MeasureTheory.sum_measure_singleton]
rw [Finset.sum_image]
· rw [show (∑ u ∈ (Finset.univ : Finset (Fin (m n) → Fin 4)), μ {e u}) =
∑ u : Fin (m n) → Fin 4,
∏ j : Fin (m n), ofReal
(∑ a : Fin 4, piTheta θ p a * rrPilotProbability ε a (u j)) by
apply Finset.sum_congr rfl
intro u _
change Measure.map (transcriptPrefix (Z := pilotOutputFamily n) hmn)
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
{((fun j => inl (u j)) :
History (pilotOutputFamily n) (m n) hmn)} = _
exact pilotPrefix_marginal_singleton select θ p ε m hselect hp hθ hε hmn u]
calc
(∑ u : Fin (m n) → Fin 4,
∏ j : Fin (m n), ofReal
(∑ a : Fin 4,
piTheta θ p a * rrPilotProbability ε a (u j))) =
∏ j : Fin (m n), ∑ k : Fin 4, ofReal
(∑ a : Fin 4,
piTheta θ p a * rrPilotProbability ε a k) :=
(Fintype.prod_sum (fun _j : Fin (m n) => fun k : Fin 4 =>
ofReal (∑ a : Fin 4,
piTheta θ p a * rrPilotProbability ε a k))).symm
_ = 1 := by
apply Finset.prod_eq_one
intro j _
rw [← ENNReal.ofReal_sum_of_nonneg]
· rw [rrPilot_marginal_sum_eq_one]
simp
· intro k _
exact Finset.sum_nonneg fun a _ => mul_nonneg
(piTheta_pos_interior θ p hp hθ a).le (by
unfold rrPilotProbability
split_ifs <;> positivity)
· exact Set.injOn_of_injective he
theorem pilotPrefix_law_eq_map_pi
The finite pilot prefix has exactly the IID product law of the calibrated four-category randomized-response marginal, embedded into pilot outputs. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Prefix law eq map pi.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
n :
ℕ
hmn :
m n ≤ n
conclusion
Measure.map (transcriptPrefix (Z := pilotOutputFamily n) hmn) (transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
= Measure.map (fun u : Fin (m n) → Fin 4 => ((fun j => inl (u j)) : History (pilotOutputFamily n) (m n) hmn)) (Measure.pi fun _ : Fin (m n) => rrPilotMarginal θ p ε)
Proof (Lean source)
lemma pilotPrefix_law_eq_map_pi (θ : TrialParameter) (p ε : ℝ)
(m : ℕ → ℕ) (hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(hε : 0 < ε) {n : ℕ} (hmn : m n ≤ n) :
Measure.map (transcriptPrefix (Z := pilotOutputFamily n) hmn)
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) =
Measure.map
(fun u : Fin (m n) → Fin 4 =>
((fun j => inl (u j)) :
History (pilotOutputFamily n) (m n) hmn))
(Measure.pi fun _ : Fin (m n) => rrPilotMarginal θ p ε) := by
classical
let e : (Fin (m n) → Fin 4) →
History (pilotOutputFamily n) (m n) hmn :=
fun u j => inl (u j)
have he : Injective e := by
intro u v huv
funext j
exact Sum.inl_injective (congrFun huv j)
have hme : Measurable e := by fun_prop
letI : IsProbabilityMeasure (rrPilotMarginal θ p ε) :=
rrPilotMarginal_isProbability θ p ε hp hθ
letI : IsProbabilityMeasure
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) :=
transcriptLaw_isProbability _ θ p hp hθ n
let μ := Measure.map (transcriptPrefix (Z := pilotOutputFamily n) hmn)
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
letI : IsProbabilityMeasure μ :=
Measure.isProbabilityMeasure_map (measurable_transcriptPrefix hmn).aemeasurable
letI : Countable (History (pilotOutputFamily n) (m n) hmn) :=
inferInstanceAs (Countable (Fin (m n) → PilotOutput))
letI : MeasurableSingletonClass
(History (pilotOutputFamily n) (m n) hmn) := by
constructor
intro v
change MeasurableSet ({v} : Set (Fin (m n) → PilotOutput))
have hv := MeasurableSet.univ_pi (fun j => measurableSet_singleton (v j))
exact (Set.univ_pi_singleton v).symm ▸ hv
have hrangeMeas : MeasurableSet (range e) := by
exact Set.Finite.measurableSet (Set.toFinite (range e))
have hrange : μ (range e) = 1 := by
exact pilotPrefix_range_measure select θ p ε m hselect hp hθ hε hmn
have hcompl : μ (range e)ᶜ = 0 := by
rw [MeasureTheory.measure_compl hrangeMeas (by simp [hrange]),
measure_univ, hrange]
simp
apply Measure.ext_of_singleton
intro z
by_cases hz : z ∈ range e
· obtain ⟨u, rfl⟩ := hz
change μ {e u} = (Measure.map e (Measure.pi fun _ : Fin (m n) =>
rrPilotMarginal θ p ε)) {e u}
rw [Measure.map_apply hme (measurableSet_singleton _)]
have hpre : e ⁻¹' ({e u} : Set (History
(pilotOutputFamily n) (m n) hmn)) = {u} := by
ext v
simp [he.eq_iff]
rw [hpre, Measure.pi_singleton]
change Measure.map (transcriptPrefix (Z := pilotOutputFamily n) hmn)
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
{((fun j => inl (u j)) :
History (pilotOutputFamily n) (m n) hmn)} = _
rw [pilotPrefix_marginal_singleton select θ p ε m hselect hp hθ hε hmn]
apply Finset.prod_congr rfl
intro j _
exact (rrPilotMarginal_singleton θ p ε (u j)).symm
· change μ {z} = (Measure.map e (Measure.pi fun _ : Fin (m n) =>
rrPilotMarginal θ p ε)) {z}
have hzsub : ({z} : Set (History
(pilotOutputFamily n) (m n) hmn)) ⊆ (range e)ᶜ := by
intro y hy
have hyz : y = z := Set.mem_singleton_iff.mp hy
subst y
exact hz
rw [MeasureTheory.measure_mono_null hzsub hcompl]
rw [Measure.map_apply hme (measurableSet_singleton _)]
have hpre : e ⁻¹' ({z} : Set (History
(pilotOutputFamily n) (m n) hmn)) = ∅ := by
ext u
change (e u = z) ↔ False
constructor
· intro heu
exact hz ⟨u, heu⟩
· intro hf
exact hf.elim
rw [hpre]
simp
def pilotPrefixFrequency
For the supplied quantities and conditions, the pilot prefix frequency is the mathematical object specified below.
def pilotPrefixOutputFrequency
For the supplied quantities and conditions, the pilot prefix output frequency is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
pilotPrefixOutputFrequency v k :
ℝ
theorem pilotFrequency_eq_prefixFrequency
Under the supplied quantities and conditions, the pilot frequency eq prefix frequency assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma pilotFrequency_eq_prefixFrequency (m : ℕ → ℕ) {n : ℕ}
(hmn : m n ≤ n) (z : Transcript (pilotOutputFamily n)) (k : Fin 4) :
pilotFrequency m z k =
pilotPrefixOutputFrequency (transcriptPrefix hmn z) k := by
unfold pilotFrequency pilotPrefixOutputFrequency transcriptPrefix
congr 1
calc
(∑ i : Fin n,
if i.val < m n ∧ z i = inl k then (1 : ℝ) else 0) =
∑ j : Fin (m n),
if z (castLE hmn j) = inl k then (1 : ℝ) else 0 := by
rw [← fin_sum_ite_lt hmn (fun j : Fin (m n) =>
if z (castLE hmn j) = inl k then (1 : ℝ) else 0)]
apply Finset.sum_congr rfl
intro i _
by_cases hi : i.val < m n <;> simp [hi]
_ = _ := rfl
theorem measurable_pilotFrequency
Each empirical pilot-cell frequency is measurable.
Formal statement
hypotheses
conclusion
Proof (Lean source)
@[fun_prop] lemma measurable_pilotFrequency (m : ℕ → ℕ) {n : ℕ} (k : Fin 4) :
Measurable (fun z : Transcript (pilotOutputFamily n) => pilotFrequency m z k) := by
unfold pilotFrequency
apply Measurable.const_mul
apply Finset.measurable_sum
intro i _
have hi : MeasurableSet {z : Transcript (pilotOutputFamily n) |
i.val < m n ∧ z i = inl k} := by
by_cases hlt : i.val < m n
· have heval : Measurable (fun z : Transcript (pilotOutputFamily n) => z i) :=
measurable_pi_apply i
have heq : {z : Transcript (pilotOutputFamily n) |
i.val < m n ∧ z i = inl k} =
(fun z : Transcript (pilotOutputFamily n) => z i) ⁻¹'
({inl k} : Set PilotOutput) := by
ext z
simp [hlt]
rw [heq]
exact heval (measurableSet_singleton (inl k))
· simp [hlt]
exact measurable_const.ite hi measurable_const
theorem pilotPrefixFrequency_embed
Under the supplied quantities and conditions, the pilot prefix frequency embed assertion holds.
Formal statement
hypotheses
conclusion
= pilotPrefixFrequency u k
Proof (Lean source)
lemma pilotPrefixFrequency_embed {ℓ : ℕ} (u : Fin ℓ → Fin 4) (k : Fin 4) :
pilotPrefixOutputFrequency (fun j => (inl (u j) : PilotOutput)) k =
pilotPrefixFrequency u k := by
unfold pilotPrefixOutputFrequency pilotPrefixFrequency
simp
theorem pilotFrequency_map_eq_pi
Under the supplied quantities and conditions, the pilot frequency map eq pi assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Frequency map eq pi.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
n :
ℕ
hmn :
m n ≤ n
k :
Fin 4
conclusion
Measure.map (fun z => pilotFrequency m z k) (transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
= Measure.map (fun u : Fin (m n) → Fin 4 => pilotPrefixFrequency u k) (Measure.pi fun _ : Fin (m n) => rrPilotMarginal θ p ε)
Proof (Lean source)
lemma pilotFrequency_map_eq_pi (θ : TrialParameter) (p ε : ℝ)
(m : ℕ → ℕ) (hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(hε : 0 < ε) {n : ℕ} (hmn : m n ≤ n) (k : Fin 4) :
Measure.map (fun z => pilotFrequency m z k)
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) =
Measure.map (fun u : Fin (m n) → Fin 4 => pilotPrefixFrequency u k)
(Measure.pi fun _ : Fin (m n) => rrPilotMarginal θ p ε) := by
let prefixFreq : History (pilotOutputFamily n) (m n) hmn → ℝ :=
fun v => pilotPrefixOutputFrequency v k
have hprefixFreq : Measurable prefixFreq := by
unfold prefixFreq pilotPrefixOutputFrequency
apply Measurable.const_mul
apply Finset.measurable_sum
intro j _
have hj : Measurable (fun v : History
(pilotOutputFamily n) (m n) hmn => v j) := measurable_pi_apply j
have hs : MeasurableSet {v : History
(pilotOutputFamily n) (m n) hmn | v j = inl k} :=
hj (measurableSet_singleton (inl k))
exact measurable_const.ite hs measurable_const
have hcomp : (fun z : Transcript (pilotOutputFamily n) =>
pilotFrequency m z k) =
prefixFreq ∘ transcriptPrefix hmn := by
funext z
exact pilotFrequency_eq_prefixFrequency m hmn z k
rw [hcomp, ← Measure.map_map hprefixFreq (measurable_transcriptPrefix hmn),
pilotPrefix_law_eq_map_pi select θ p ε m hselect hp hθ hε hmn]
let e : (Fin (m n) → Fin 4) →
History (pilotOutputFamily n) (m n) hmn :=
fun u j => inl (u j)
have hme : Measurable e := by fun_prop
change Measure.map prefixFreq
(Measure.map e (Measure.pi fun _ : Fin (m n) => rrPilotMarginal θ p ε)) = _
rw [Measure.map_map hprefixFreq hme]
congr 1
funext u
exact pilotPrefixFrequency_embed u k
def pilotCellIndicator
For the supplied quantities and conditions, the pilot cell indicator is the mathematical object specified below.
Definition (Lean source)
theorem measurable_pilotCellIndicator
Each pilot-cell indicator is measurable.
Formal statement
Proof (Lean source)
@[fun_prop] lemma measurable_pilotCellIndicator (k : Fin 4) :
Measurable (pilotCellIndicator k) := by
unfold pilotCellIndicator
exact measurable_const.ite (measurableSet_singleton k) measurable_const
theorem integrable_pilotCellIndicator
Under the supplied quantities and conditions, the integrable pilot cell indicator assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the integrable pilot Cell Indicator.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma integrable_pilotCellIndicator (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (k : Fin 4) :
Integrable (pilotCellIndicator k) (rrPilotMarginal θ p ε) := by
letI : IsProbabilityMeasure (rrPilotMarginal θ p ε) :=
rrPilotMarginal_isProbability θ p ε hp hθ
exact Integrable.of_bound (measurable_pilotCellIndicator k).aestronglyMeasurable
1 (ae_of_all _ fun a => by
unfold pilotCellIndicator
split_ifs <;> norm_num)
theorem integral_pilotCellIndicator
Under the supplied quantities and conditions, the integral pilot cell indicator assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the integral pilot Cell Indicator.
Formal statement
hypotheses
conclusion
∫ a, pilotCellIndicator k a ∂rrPilotMarginal θ p ε
Proof (Lean source)
lemma integral_pilotCellIndicator (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (k : Fin 4) :
∫ a, pilotCellIndicator k a ∂rrPilotMarginal θ p ε =
∑ b : Fin 4, piTheta θ p b * rrPilotProbability ε b k := by
letI : IsProbabilityMeasure (rrPilotMarginal θ p ε) :=
rrPilotMarginal_isProbability θ p ε hp hθ
rw [show pilotCellIndicator k =
({k} : Set (Fin 4)).indicator (fun _ => (1 : ℝ)) by
funext a
by_cases ha : a = k <;> simp [pilotCellIndicator, ha]]
rw [integral_indicator (measurableSet_singleton k), integral_const]
rw [MeasureTheory.measureReal_def, Measure.restrict_apply_univ,
rrPilotMarginal_singleton]
simp only [smul_eq_mul, mul_one]
rw [ENNReal.toReal_ofReal]
exact Finset.sum_nonneg fun b _ => mul_nonneg
(piTheta_pos_interior θ p hp hθ b).le (by
unfold rrPilotProbability
split_ifs <;> positivity)
theorem pilotPrefixFrequency_map_pi_eq_sampleMean
Under the supplied quantities and conditions, the pilot prefix frequency map pi eq sample mean assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Prefix Frequency map pi eq sample Mean.
Formal statement
hypotheses
conclusion
P :
rrPilotMarginal θ p ε
= Measure.map (fun ω : ℕ → Fin 4 => (N : ℝ)⁻¹ * ∑ i ∈ range N, pilotCellIndicator k (ω i)) (Measure.infinitePi fun _ : ℕ => P)
Proof (Lean source)
lemma pilotPrefixFrequency_map_pi_eq_sampleMean (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (N : ℕ) (k : Fin 4) :
let P := rrPilotMarginal θ p ε
Measure.map (fun u : Fin N → Fin 4 => pilotPrefixFrequency u k)
(Measure.pi fun _ : Fin N => P) =
Measure.map (fun ω : ℕ → Fin 4 =>
(N : ℝ)⁻¹ * ∑ i ∈ range N, pilotCellIndicator k (ω i))
(Measure.infinitePi fun _ : ℕ => P) := by
dsimp
let P := rrPilotMarginal θ p ε
letI : IsProbabilityMeasure P := rrPilotMarginal_isProbability θ p ε hp hθ
let S := iidSample_infinitePi P
have hpush := iidSample_finN_pushforward S N
rw [← hpush]
have hmeanMeas : Measurable (fun u : Fin N → Fin 4 =>
pilotPrefixFrequency u k) := by
unfold pilotPrefixFrequency
apply Measurable.const_mul
apply Finset.measurable_sum
intro j _
have hj : Measurable (fun u : Fin N → Fin 4 => u j) := measurable_pi_apply j
exact measurable_const.ite (hj (measurableSet_singleton k)) measurable_const
rw [Measure.map_map hmeanMeas (iidSample_finN_measurable S N)]
congr 1
funext ω
change pilotPrefixFrequency (fun j : Fin N => ω j) k = _
unfold pilotPrefixFrequency pilotCellIndicator
congr 1
exact Fin.sum_univ_eq_sum_range (fun i => if ω i = k then (1 : ℝ) else 0) N
theorem pilotFrequency_tendstoInProbability
Each released pilot-cell frequency obeys the categorical weak law under the actual finite transcript experiment. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Frequency tendsto In Probability.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hdiverges :
hsublinear :
k :
Fin 4
conclusion
TendstoInProbability (fun n => transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) (fun n z => pilotFrequency m z k) atTop (fun _ _ => ∑ b : Fin 4, piTheta θ p b * rrPilotProbability ε b k)
Proof (Lean source)
-- keep: reusable finite-pilot construction, law, localization, or limit API for related adaptive procedures
lemma pilotFrequency_tendstoInProbability (θ : TrialParameter) (p ε : ℝ)
(m : ℕ → ℕ) (hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(hε : 0 < ε) (hdiverges : PilotDiverges m)
(hsublinear : PilotSublinear m) (k : Fin 4) :
TendstoInProbability
(fun n => transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
(fun n z => pilotFrequency m z k) atTop
(fun _ _ => ∑ b : Fin 4,
piTheta θ p b * rrPilotProbability ε b k) := by
let P := rrPilotMarginal θ p ε
letI : IsProbabilityMeasure P := rrPilotMarginal_isProbability θ p ε hp hθ
let S := iidSample_infinitePi P
have hw := S.sampleMean_tendsto_inProb
(measurable_pilotCellIndicator k)
(integrable_pilotCellIndicator θ p ε hp hθ k)
rw [integral_pilotCellIndicator θ p ε hp hθ k] at hw
intro δ hδ
have hwδ := hw δ hδ
have hwcomp := hwδ.comp hdiverges
have hwcomp' : Tendsto (fun n => (Measure.infinitePi fun _ : ℕ => P)
{ω | δ ≤ edist (S.sampleMean (pilotCellIndicator k) (m n) ω)
(∑ b : Fin 4, piTheta θ p b * rrPilotProbability ε b k)})
atTop (nhds 0) := by
convert hwcomp using 1 <;> rfl
have heq : (fun n => (Measure.infinitePi fun _ : ℕ => P)
{ω | δ ≤ edist (S.sampleMean (pilotCellIndicator k) (m n) ω)
(∑ b : Fin 4, piTheta θ p b * rrPilotProbability ε b k)}) =ᶠ[atTop]
(fun n => (transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
{z | δ ≤ edist (pilotFrequency m z k)
(∑ b : Fin 4, piTheta θ p b * rrPilotProbability ε b k)}) := by
change ∀ᶠ n in atTop,
(Measure.infinitePi fun _ : ℕ => P)
{ω | δ ≤ edist (S.sampleMean (pilotCellIndicator k) (m n) ω)
(∑ b : Fin 4, piTheta θ p b * rrPilotProbability ε b k)} =
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
{z | δ ≤ edist (pilotFrequency m z k)
(∑ b : Fin 4, piTheta θ p b * rrPilotProbability ε b k)}
apply (Filter.eventually_atTop (α := ℕ)).2
refine ⟨2, ?_⟩
intro n hn
have hmn : m n ≤ n := (hsublinear.2 n hn).2.le
let B : Set ℝ := {r | δ ≤ edist r
(∑ b : Fin 4, piTheta θ p b * rrPilotProbability ε b k)}
have hB : MeasurableSet B := by
exact measurableSet_le measurable_const (measurable_id.edist measurable_const)
have hfreq := measurable_pilotFrequency m (n := n) k
have hsample := S.measurable_sampleMean (measurable_pilotCellIndicator k) (m n)
have hlaw : Measure.map (fun z => pilotFrequency m z k)
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) =
Measure.map (S.sampleMean (pilotCellIndicator k) (m n))
(Measure.infinitePi fun _ : ℕ => P) := by
calc
Measure.map (fun z => pilotFrequency m z k)
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) =
Measure.map (fun u : Fin (m n) → Fin 4 => pilotPrefixFrequency u k)
(Measure.pi fun _ : Fin (m n) => P) :=
pilotFrequency_map_eq_pi select θ p ε m hselect hp hθ hε hmn k
_ = Measure.map (S.sampleMean (pilotCellIndicator k) (m n))
(Measure.infinitePi fun _ : ℕ => P) := by
have hbase := pilotPrefixFrequency_map_pi_eq_sampleMean
θ p ε hp hθ (m n) k
have hfun : S.sampleMean (pilotCellIndicator k) (m n) =
fun ω : ℕ → Fin 4 => (m n : ℝ)⁻¹ *
∑ i ∈ range (m n), pilotCellIndicator k (ω i) := by
funext ω
rfl
rw [hfun]
simpa [P] using hbase
change (Measure.infinitePi fun _ : ℕ => P)
((S.sampleMean (pilotCellIndicator k) (m n)) ⁻¹' B) =
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
((fun z => pilotFrequency m z k) ⁻¹' B)
rw [← Measure.map_apply hsample hB, ← hlaw, Measure.map_apply hfreq hB]
exact Filter.Tendsto.congr' heq hwcomp'
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotRates 2 declarations The integer square-root pilot size diverges, remains asymptotically negligible, and lies strictly between zero and the full sample size once n ≥ 2.
Square-root pilot sample size
The integer square-root pilot size diverges, remains asymptotically negligible,
and lies strictly between zero and the full sample size once n ≥ 2.
theorem natSqrt_pilotDiverges
The integer square-root pilot sample size tends to infinity. The stated result follows.
Formal statement
conclusion
Proof (Lean source)
-- @node: natSqrt_pilotDiverges
lemma natSqrt_pilotDiverges : PilotDiverges sqrt := by
rw [PilotDiverges, tendsto_atTop_atTop]
intro b
refine ⟨b * b, ?_⟩
intro a ha
exact Nat.le_sqrt.mpr ha
theorem natSqrt_pilotSublinear
The integer square-root pilot is sublinear and is a nonempty proper split for every sample size at least two. The stated result follows.
Formal statement
conclusion
Proof (Lean source)
-- @node: natSqrt_pilotSublinear
lemma natSqrt_pilotSublinear : PilotSublinear Nat.sqrt := by
constructor
· have hsqrt : Tendsto (fun n : ℕ => sqrt (n : ℝ)) atTop atTop :=
Real.tendsto_sqrt_atTop.comp tendsto_natCast_atTop_atTop
apply squeeze_zero'
(Eventually.of_forall fun n =>
div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _))
?_ (tendsto_inv_atTop_zero.comp hsqrt)
filter_upwards [eventually_gt_atTop 0] with n hn
have hnreal : 0 < (n : ℝ) := by exact_mod_cast hn
have hspos : 0 < sqrt (n : ℝ) := Real.sqrt_pos.mpr hnreal
calc
(Nat.sqrt n : ℝ) / n ≤ sqrt (n : ℝ) / n :=
div_le_div_of_nonneg_right Real.nat_sqrt_le_real_sqrt hnreal.le
_ = (sqrt (n : ℝ))⁻¹ := by
nth_rewrite 2 [← Real.mul_self_sqrt hnreal.le]
field_simp
· intro n hn
constructor
· exact Nat.le_sqrt.mpr (by omega)
· exact Nat.sqrt_lt_self (by omega)
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotRowGaussianLimit 1 declarations This module instantiates the finite bounded iid triangular-row central limit theorem with the genuine adaptive main-release mass and centered score.
Gaussian limit for the conditional adaptive pilot row
This module instantiates the finite bounded iid triangular-row central limit theorem with the genuine adaptive main-release mass and centered score. It also transfers the generic row-length normalization to the paper's root-total-sample normalization.
theorem adaptivePilotRowScaledCDF_subsequence_tendsto_gaussian
Along every strict sample-size subsequence and every convergent sequence of interior pilot selectors, the actual centered conditional main-row CDF converges to the oracle Gaussian CDF under root-total-sample scaling. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
theta h :
p eps :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p eps select
hp :
htheta :
InteriorMeans theta
heps :
0 < eps
hsub :
x :
ℝ
phi :
ℕ → ℕ
hphi :
StrictMono phi
eta :
ℕ → TrialParameter
heta :
∀ k, InteriorMeans (eta k)
conclusion
Tendsto (fun k => adaptivePilotRowScaledCDF select theta h (fun _ => eta k) p eps m hselect hp heps (phi k) x) atTop (nhds ((gaussianMeasure 0 (Vstar theta p eps)).real (Iic x)))
Proof (Lean source)
theorem adaptivePilotRowScaledCDF_subsequence_tendsto_gaussian
(theta h : TrialParameter) (p eps : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p eps select)
(hp : InteriorAssignment p) (htheta : InteriorMeans theta)
(heps : 0 < eps) (hsub : PilotSublinear m)
(x : ℝ) (phi : ℕ → ℕ) (hphi : StrictMono phi)
(eta : ℕ → TrialParameter) (heta : ∀ k, InteriorMeans (eta k))
(hetalim : Tendsto eta atTop (nhds theta)) :
Tendsto (fun k => adaptivePilotRowScaledCDF select theta h
(fun _ => eta k) p eps m hselect hp heps (phi k) x) atTop
(nhds ((gaussianMeasure 0 (Vstar theta p eps)).real (Iic x))) := by
let M := pilotSubsequenceRowModel select theta h eta p eps m phi hselect hp htheta heta heps
hsub hphi hetalim
let a : ℕ → ℝ := fun k =>
sqrt (((phi k : ℕ) : ℝ) / (adaptiveMainSize m (phi k) : ℝ))
let t : ℕ → ℝ := fun k => x / a k
have ha : Tendsto a atTop (nhds 1) := by
exact (adaptive_row_main_sqrt_ratio_tendsto_one m hsub).comp hphi.tendsto_atTop
have ht : Tendsto t atTop (nhds x) := by
change Tendsto ((fun _ : ℕ => x) / a) atTop (nhds x)
simpa using tendsto_const_nhds.div ha one_ne_zero
have hMpoint (y : ℝ) : Tendsto (fun k => M.normalizedCDF k y) atTop
(nhds (cdf (gaussianReal 0 M.v₀) y)) :=
M.normalizedCDF_tendsto_gaussian y
have hMmono (k : ℕ) : Monotone (M.normalizedCDF k) := by
intro y z hyz
unfold RowModel.normalizedCDF RowModel.expect
apply Finset.sum_le_sum
intro v _
apply mul_le_mul_of_nonneg_left _ (M.mass_nonnegative k v)
change (if M.normalizedSum k v ≤ y then 1 else 0) ≤
if M.normalizedSum k v ≤ z then 1 else 0
split_ifs with hy hz
· exact le_rfl
· exact (hz (hy.trans hyz)).elim
· exact zero_le_one
· exact le_rfl
have hmoving : Tendsto (fun k => M.normalizedCDF k (t k)) atTop
(nhds (cdf (gaussianReal 0 M.v₀) x)) :=
tendsto_moving_of_monotone_pointwise (fun k => M.normalizedCDF k)
(cdf (gaussianReal 0 M.v₀)) t x hMmono
(continuous_cdf_gaussianReal_zero M.variance_pos).continuousAt
hMpoint ht
have heq : ∀ᶠ k in atTop,
adaptivePilotRowScaledCDF select theta h (fun _ => eta k)
p eps m hselect hp heps (phi k) x = M.normalizedCDF k (t k) := by
have hN : ∀ᶠ k in atTop, 0 < adaptiveMainSize m (phi k) := by
exact (M.length_tendsto.eventually (eventually_ge_atTop 1)).mono
(fun k hk => hk)
filter_upwards [hN] with k hNk
classical
unfold adaptivePilotRowScaledCDF RowModel.normalizedCDF RowModel.expect
apply Finset.sum_congr rfl
intro v _
dsimp only [adaptivePilotRowProductMass, adaptivePilotRowMass,
M, pilotSubsequenceRowModel, RowModel.mass,
RowModel.normalizedSum, RowModel.scoreSum, t, a]
have hNr : 0 < (adaptiveMainSize m (phi k) : ℝ) := by exact_mod_cast hNk
have hphir : 0 < (phi k : ℝ) := by
have hle : adaptiveMainSize m (phi k) ≤ phi k := by
unfold adaptiveMainSize
omega
exact_mod_cast lt_of_lt_of_le hNk hle
have hsN : 0 < sqrt (adaptiveMainSize m (phi k) : ℝ) :=
Real.sqrt_pos.2 hNr
have hsa : 0 < sqrt ((phi k : ℝ) /
(adaptiveMainSize m (phi k) : ℝ)) := Real.sqrt_pos.2 (div_pos hphir hNr)
let S : ℝ := ∑ j, adaptivePilotRowScore select theta h (fun _ => eta k)
p eps hselect hp heps (phi k) (v j)
change
(∏ j, adaptiveMainMass select (adaptiveTrueParam theta h (phi k)) (eta k)
p eps (v j)) *
(if sqrt (phi k : ℝ) *
(adaptiveMainSize m (phi k) : ℝ)⁻¹ * S ≤ x then 1 else 0) =
(∏ j, adaptiveMainMass select (adaptiveTrueParam theta h (phi k)) (eta k)
p eps (v j)) *
(if (sqrt (adaptiveMainSize m (phi k) : ℝ))⁻¹ * S ≤
x / sqrt ((phi k : ℝ) / (adaptiveMainSize m (phi k) : ℝ))
then 1 else 0)
have hscale : sqrt (phi k : ℝ) *
(adaptiveMainSize m (phi k) : ℝ)⁻¹ * S ≤ x ↔
(sqrt (adaptiveMainSize m (phi k) : ℝ))⁻¹ * S ≤
x / sqrt ((phi k : ℝ) /
(adaptiveMainSize m (phi k) : ℝ)) := by
rw [le_div_iff₀ hsa]
rw [show sqrt (phi k : ℝ) *
(adaptiveMainSize m (phi k) : ℝ)⁻¹ * S =
(sqrt (adaptiveMainSize m (phi k) : ℝ))⁻¹ * S *
sqrt ((phi k : ℝ) / (adaptiveMainSize m (phi k) : ℝ)) by
rw [Real.sqrt_div (Nat.cast_nonneg (phi k))]
field_simp
rw [Real.sq_sqrt hNr.le]
ring]
by_cases hc : sqrt (phi k : ℝ) *
(adaptiveMainSize m (phi k) : ℝ)⁻¹ * S ≤ x
· rw [if_pos hc, if_pos (hscale.mp hc)]
· rw [if_neg hc, if_neg (fun hz => hc (hscale.mpr hz))]
have hfinal := hmoving.congr' (heq.mono fun _ hk => hk.symm)
have htarget : cdf (gaussianReal 0 M.v₀) x =
(gaussianMeasure 0 (Vstar theta p eps)).real (Iic x) := by
change cdf (gaussianReal 0 (adaptivePilotLimitVariance select theta p eps hselect hp htheta heps)) x = _
rw [ProbabilityTheory.cdf_eq_real]
unfold gaussianMeasure
have hv : 0 ≤ Vstar theta p eps :=
(inv_pos.mpr (Jstar_pos_interior theta p eps hp htheta heps)).le
have hvar : adaptivePilotLimitVariance select theta p eps hselect hp htheta heps =
(Vstar theta p eps).toNNReal := by
apply NNReal.eq
change Vstar theta p eps = max (Vstar theta p eps) 0
exact (max_eq_left hv).symm
rw [hvar]
rw [htarget] at hfinal
exact hfinal
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotRowInputs 17 declarations This module packages the exact finite alphabet mass and centered score used by the adaptive main sample.
Finite triangular-row inputs for private-pilot attainment
This module packages the exact finite alphabet mass and centered score used by the adaptive main sample. It stays independent of the generic triangular-array substrate, so a later adapter can instantiate that substrate without rebuilding the paper-specific mass, centering, boundedness, variance, and finite-product algebra.
def adaptivePilotRowMass
The one-coordinate mass in the adaptive main row along a local alternative and a chosen sequence of pilot selectors. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The adaptive Pilot Row Mass is determined by the displayed parameters.
Definition (Lean source)
parameters
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
s :
Fin 14
def
adaptivePilotRowMass select θ h η p ε hselect hp hε n s :
ℝ
given by
adaptiveMainMass select (adaptiveTrueParam θ h n) (η n) p ε s
def adaptivePilotRowScore
The selected main-release score centered by its exact one-coordinate mean. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The adaptive Pilot Row Score is determined by the displayed parameters.
Definition (Lean source)
parameters
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
s :
Fin 14
def
adaptivePilotRowScore select θ h η p ε hselect hp hε n s :
ℝ
given by
adaptiveSelectedScore select (η n) p ε s
- (contrast (adaptiveTrueParam θ h n) - contrast (η n))
def adaptivePilotLimitVariance
The limiting row variance as a nonnegative real. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The adaptive Pilot Limit Variance is determined by the displayed parameters.
Definition (Lean source)
parameters
def
given by
⟨Vstar θ p ε, (inv_pos.mpr (Jstar_pos_interior θ p ε hp hθ hε)).le⟩
theorem adaptivePilotRowMass_nonneg
The adaptive main row has nonnegative one-coordinate masses. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Row Mass nonneg.
Formal statement
hypotheses
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hη :
∀ n, InteriorMeans (η n)
hε :
0 < ε
n :
ℕ
s :
Fin 14
conclusion
0 ≤ adaptivePilotRowMass select θ h η p ε hselect hp hε n s
Proof (Lean source)
lemma adaptivePilotRowMass_nonneg (θ h : TrialParameter)
(η : ℕ → TrialParameter) (p ε : ℝ) (hselect : StrongSaddleSelection p ε select) (hp : InteriorAssignment p)
(hθ : InteriorMeans θ) (hη : ∀ n, InteriorMeans (η n)) (hε : 0 < ε)
(n : ℕ) (s : Fin 14) :
0 ≤ adaptivePilotRowMass select θ h η p ε hselect hp hε n s := by
exact adaptiveMainMass_nonneg select _ _ p ε hselect hp
(adaptiveTrueParam_interior θ h n hθ) (hη n) hε s
theorem adaptivePilotRowMass_sum
The adaptive main row masses sum to one. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Row Mass sum.
Formal statement
hypotheses
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
hselect :
StrongSaddleSelection p ε select
hp :
hη :
∀ n, InteriorMeans (η n)
hε :
0 < ε
n :
ℕ
conclusion
∑ s : Fin 14, adaptivePilotRowMass select θ h η p ε hselect hp hε n s = 1
Proof (Lean source)
lemma adaptivePilotRowMass_sum (θ h : TrialParameter)
(η : ℕ → TrialParameter) (p ε : ℝ) (hselect : StrongSaddleSelection p ε select) (hp : InteriorAssignment p)
(hη : ∀ n, InteriorMeans (η n)) (hε : 0 < ε) (n : ℕ) :
∑ s : Fin 14, adaptivePilotRowMass select θ h η p ε hselect hp hε n s = 1 := by
exact adaptiveMainMass_sum select _ _ p ε hselect hp (hη n) hε
theorem adaptivePilotRowScore_centered
The adaptive main-row score is exactly centered under its row mass. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Row Score centered.
Formal statement
hypotheses
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
hselect :
StrongSaddleSelection p ε select
hp :
hη :
∀ n, InteriorMeans (η n)
hε :
0 < ε
n :
ℕ
conclusion
∑ s : Fin 14, adaptivePilotRowMass select θ h η p ε hselect hp hε n s * adaptivePilotRowScore select θ h η p ε hselect hp hε n s
= 0
Proof (Lean source)
lemma adaptivePilotRowScore_centered (θ h : TrialParameter)
(η : ℕ → TrialParameter) (p ε : ℝ) (hselect : StrongSaddleSelection p ε select) (hp : InteriorAssignment p)
(hη : ∀ n, InteriorMeans (η n)) (hε : 0 < ε) (n : ℕ) :
∑ s : Fin 14, adaptivePilotRowMass select θ h η p ε hselect hp hε n s *
adaptivePilotRowScore select θ h η p ε hselect hp hε n s = 0 := by
rw [show (∑ s : Fin 14, adaptivePilotRowMass select θ h η p ε hselect hp hε n s *
adaptivePilotRowScore select θ h η p ε hselect hp hε n s) =
(∑ s : Fin 14, adaptiveMainMass select (adaptiveTrueParam θ h n) (η n)
p ε s * adaptiveSelectedScore select (η n) p ε s) -
(contrast (adaptiveTrueParam θ h n) - contrast (η n)) *
∑ s : Fin 14, adaptiveMainMass select (adaptiveTrueParam θ h n) (η n)
p ε s by
simp only [adaptivePilotRowMass, adaptivePilotRowScore]
simp_rw [mul_sub]
rw [Finset.sum_sub_distrib, Finset.mul_sum]
apply congrArg₂ (· - ·) rfl
apply Finset.sum_congr rfl
intro s _
ring]
rw [adaptiveSelectedScore_mean select _ _ p ε hselect hp (hη n) hε,
adaptiveMainMass_sum select _ _ p ε hselect hp (hη n) hε]
ring
theorem adaptivePilotRowMean_abs_le
The exact mean correction is no larger than the common selected-score bound. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Row Mean abs le.
Formal statement
hypotheses
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hη :
∀ n, InteriorMeans (η n)
hε :
0 < ε
n :
ℕ
conclusion
|contrast (adaptiveTrueParam θ h n) - contrast (η n)| ≤ pilotPhiUpper p ε
Proof (Lean source)
lemma adaptivePilotRowMean_abs_le (θ h : TrialParameter)
(η : ℕ → TrialParameter) (p ε : ℝ) (hselect : StrongSaddleSelection p ε select) (hp : InteriorAssignment p)
(hθ : InteriorMeans θ) (hη : ∀ n, InteriorMeans (η n)) (hε : 0 < ε)
(n : ℕ) :
|contrast (adaptiveTrueParam θ h n) - contrast (η n)| ≤
pilotPhiUpper p ε := by
rw [← adaptiveSelectedScore_mean select (adaptiveTrueParam θ h n) (η n)
p ε hselect hp (hη n) hε]
calc
|∑ s : Fin 14, adaptiveMainMass select (adaptiveTrueParam θ h n) (η n)
p ε s * adaptiveSelectedScore select (η n) p ε s| ≤
∑ s : Fin 14, |adaptiveMainMass select (adaptiveTrueParam θ h n) (η n)
p ε s * adaptiveSelectedScore select (η n) p ε s| :=
Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ s : Fin 14, adaptiveMainMass select (adaptiveTrueParam θ h n) (η n)
p ε s * pilotPhiUpper p ε := by
apply Finset.sum_le_sum
intro s _
rw [abs_mul, abs_of_nonneg (adaptiveMainMass_nonneg select _ _ p ε hselect hp
(adaptiveTrueParam_interior θ h n hθ) (hη n) hε s)]
exact mul_le_mul_of_nonneg_left
(adaptiveSelectedScore_abs_le select (η n) p ε hselect hp (hη n) hε s)
(adaptiveMainMass_nonneg select _ _ p ε hselect hp
(adaptiveTrueParam_interior θ h n hθ) (hη n) hε s)
_ = pilotPhiUpper p ε := by
rw [← Finset.sum_mul, adaptiveMainMass_sum select _ _ p ε hselect hp (hη n) hε,
one_mul]
theorem adaptivePilotRowScore_abs_le
Every centered adaptive row score is bounded by twice the common selected-score bound. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Row Score abs le.
Formal statement
hypotheses
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hη :
∀ n, InteriorMeans (η n)
hε :
0 < ε
n :
ℕ
s :
Fin 14
conclusion
|adaptivePilotRowScore select θ h η p ε hselect hp hε n s| ≤ 2 * pilotPhiUpper p ε
Proof (Lean source)
lemma adaptivePilotRowScore_abs_le (θ h : TrialParameter)
(η : ℕ → TrialParameter) (p ε : ℝ) (hselect : StrongSaddleSelection p ε select) (hp : InteriorAssignment p)
(hθ : InteriorMeans θ) (hη : ∀ n, InteriorMeans (η n)) (hε : 0 < ε)
(n : ℕ) (s : Fin 14) :
|adaptivePilotRowScore select θ h η p ε hselect hp hε n s| ≤
2 * pilotPhiUpper p ε := by
unfold adaptivePilotRowScore
calc
|_ - _| ≤ |adaptiveSelectedScore select (η n) p ε s| +
|contrast (adaptiveTrueParam θ h n) - contrast (η n)| :=
abs_sub _ _
_ ≤ pilotPhiUpper p ε + pilotPhiUpper p ε :=
add_le_add
(adaptiveSelectedScore_abs_le select (η n) p ε hselect hp (hη n) hε s)
(adaptivePilotRowMean_abs_le select θ h η p ε hselect hp hθ hη hε n)
_ = 2 * pilotPhiUpper p ε := by ring
theorem adaptivePilotRow_secondMoment_eq
The one-coordinate second moment of the centered row score is exactly the adaptive score variance. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Row second Moment eq.
Formal statement
hypotheses
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
hselect :
StrongSaddleSelection p ε select
hp :
hη :
∀ n, InteriorMeans (η n)
hε :
0 < ε
n :
ℕ
conclusion
(∑ s : Fin 14, adaptivePilotRowMass select θ h η p ε hselect hp hε n s * (adaptivePilotRowScore select θ h η p ε hselect hp hε n s) ^ 2)
= adaptiveScoreVariance select (adaptiveTrueParam θ h n) (η n) p ε
Proof (Lean source)
lemma adaptivePilotRow_secondMoment_eq (θ h : TrialParameter)
(η : ℕ → TrialParameter) (p ε : ℝ) (hselect : StrongSaddleSelection p ε select) (hp : InteriorAssignment p)
(hη : ∀ n, InteriorMeans (η n)) (hε : 0 < ε) (n : ℕ) :
(∑ s : Fin 14, adaptivePilotRowMass select θ h η p ε hselect hp hε n s *
(adaptivePilotRowScore select θ h η p ε hselect hp hε n s) ^ 2) =
adaptiveScoreVariance select (adaptiveTrueParam θ h n) (η n) p ε := by
exact (adaptiveScoreVariance_eq_centered_sum select
(adaptiveTrueParam θ h n) (η n) p ε hselect hp (hη n) hε).symm
theorem adaptiveTrueParam_tendsto
The regularized local parameter converges to the base parameter. For the displayed inputs and conditions, the stated result follows. Under the stated assumptions, the adaptive True Param tendsto.
Formal statement
hypotheses
θ h :
hθ :
conclusion
Proof (Lean source)
lemma adaptiveTrueParam_tendsto (θ h : TrialParameter) (hθ : InteriorMeans θ) :
Tendsto (adaptiveTrueParam θ h) atTop (nhds θ) := by
apply (localAlternative_tendsto θ h).congr'
filter_upwards [eventually_adaptiveTrueParam_eq θ h hθ] with n hn
exact hn.symm
theorem adaptivePilotRow_variance_tendsto
If the selected pilot parameters converge to the base point, the centered adaptive row variances converge to the oracle variance. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Row variance tendsto.
Formal statement
hypotheses
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hη :
∀ n, InteriorMeans (η n)
hε :
0 < ε
conclusion
Tendsto (fun n => ∑ s : Fin 14, adaptivePilotRowMass select θ h η p ε hselect hp hε n s * (adaptivePilotRowScore select θ h η p ε hselect hp hε n s) ^ 2) atTop (nhds ((adaptivePilotLimitVariance select θ p ε hselect hp hθ hε : NNReal) : ℝ))
Proof (Lean source)
lemma adaptivePilotRow_variance_tendsto (θ h : TrialParameter)
(η : ℕ → TrialParameter) (p ε : ℝ) (hselect : StrongSaddleSelection p ε select) (hp : InteriorAssignment p)
(hθ : InteriorMeans θ) (hη : ∀ n, InteriorMeans (η n)) (hε : 0 < ε)
(hηlim : Tendsto η atTop (nhds θ)) :
Tendsto (fun n => ∑ s : Fin 14,
adaptivePilotRowMass select θ h η p ε hselect hp hε n s *
(adaptivePilotRowScore select θ h η p ε hselect hp hε n s) ^ 2)
atTop (nhds ((adaptivePilotLimitVariance select θ p ε hselect hp hθ hε : NNReal) : ℝ)) := by
have hpair : Tendsto (fun n => (adaptiveTrueParam θ h n, η n)) atTop
(nhds (θ, θ)) := (adaptiveTrueParam_tendsto θ h hθ).prodMk_nhds hηlim
have hv := (continuousAt_adaptiveScoreVariance_diag select θ p ε hselect hp hθ hε).tendsto.comp hpair
change Tendsto (fun n => ∑ s : Fin 14,
adaptivePilotRowMass select θ h η p ε hselect hp hε n s *
(adaptivePilotRowScore select θ h η p ε hselect hp hε n s) ^ 2)
atTop (nhds (Vstar θ p ε))
convert hv using 1
· funext n
exact adaptivePilotRow_secondMoment_eq select θ h η p ε hselect hp hη hε n
· simp [adaptiveScoreVariance_diag select θ p ε hselect hp hθ hε]
theorem adaptivePilotLimitVariance_pos
The limiting adaptive row variance is strictly positive. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Limit Variance pos.
Formal statement
hypotheses
conclusion
0 < adaptivePilotLimitVariance select θ p ε hselect hp hθ hε
Proof (Lean source)
lemma adaptivePilotLimitVariance_pos (θ : TrialParameter) (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε) :
0 < adaptivePilotLimitVariance select θ p ε hselect hp hθ hε := by
change 0 < Vstar θ p ε
unfold Vstar
exact inv_pos.mpr (Jstar_pos_interior θ p ε hp hθ hε)
def adaptivePilotRowProductMass
The literal iid product mass of an adaptive main row. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The adaptive Pilot Row Product Mass is determined by the displayed parameters.
Definition (Lean source)
parameters
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
v :
def
adaptivePilotRowProductMass select θ h η p ε m hselect hp hε n v :
ℝ
given by
∏ j, adaptivePilotRowMass select θ h η p ε hselect hp hε n (v j)
theorem adaptivePilotRowProductMass_nonneg
Every conditional main-row product atom has nonnegative mass. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Row Product Mass nonneg.
Formal statement
hypotheses
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hη :
∀ n, InteriorMeans (η n)
hε :
0 < ε
n :
ℕ
v :
conclusion
0 ≤ adaptivePilotRowProductMass select θ h η p ε m hselect hp hε n v
Proof (Lean source)
lemma adaptivePilotRowProductMass_nonneg (θ h : TrialParameter)
(η : ℕ → TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(hη : ∀ n, InteriorMeans (η n)) (hε : 0 < ε) (n : ℕ)
(v : Fin (adaptiveMainSize m n) → Fin 14) :
0 ≤ adaptivePilotRowProductMass select θ h η p ε m hselect hp hε n v := by
exact Finset.prod_nonneg fun j _ =>
adaptivePilotRowMass_nonneg select θ h η p ε hselect hp hθ hη hε n (v j)
theorem adaptivePilotRowProductMass_sum
The conditional iid main-row product masses sum to one. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Row Product Mass sum.
Formal statement
hypotheses
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hη :
∀ n, InteriorMeans (η n)
hε :
0 < ε
n :
ℕ
conclusion
∑ v : Fin (adaptiveMainSize m n) → Fin 14, adaptivePilotRowProductMass select θ h η p ε m hselect hp hε n v
= 1
Proof (Lean source)
lemma adaptivePilotRowProductMass_sum (θ h : TrialParameter)
(η : ℕ → TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hη : ∀ n, InteriorMeans (η n))
(hε : 0 < ε) (n : ℕ) :
∑ v : Fin (adaptiveMainSize m n) → Fin 14,
adaptivePilotRowProductMass select θ h η p ε m hselect hp hε n v = 1 := by
classical
calc
_ = ∏ _j : Fin (adaptiveMainSize m n),
∑ s : Fin 14, adaptivePilotRowMass select θ h η p ε hselect hp hε n s := by
simpa [adaptivePilotRowProductMass] using
(Fintype.prod_sum (fun _j : Fin (adaptiveMainSize m n) =>
adaptivePilotRowMass select θ h η p ε hselect hp hε n)).symm
_ = 1 := by simp [adaptivePilotRowMass_sum select θ h η p ε hselect hp hη hε n]
theorem adaptivePilotRowScore_sum_eq
Summing the centered row score subtracts exactly the row length times the paper estimator's mean correction. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Row Score sum eq.
Formal statement
hypotheses
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
conclusion
∑ j, adaptivePilotRowScore select θ h η p ε hselect hp hε n (v j)
= (∑ j, adaptiveSelectedScore select (η n) p ε (v j))
- (adaptiveMainSize m n : ℝ) * (contrast (adaptiveTrueParam θ h n) - contrast (η n))
Proof (Lean source)
lemma adaptivePilotRowScore_sum_eq (θ h : TrialParameter)
(η : ℕ → TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε) (n : ℕ)
(v : Fin (adaptiveMainSize m n) → Fin 14) :
∑ j, adaptivePilotRowScore select θ h η p ε hselect hp hε n (v j) =
(∑ j, adaptiveSelectedScore select (η n) p ε (v j)) -
(adaptiveMainSize m n : ℝ) *
(contrast (adaptiveTrueParam θ h n) - contrast (η n)) := by
simp only [adaptivePilotRowScore, Finset.sum_sub_distrib,
Finset.sum_const, nsmul_eq_mul]
have hcard : (Finset.univ : Finset (Fin (adaptiveMainSize m n))).card =
adaptiveMainSize m n := by simp
rw [hcard]
ring
theorem adaptivePilotRow_estimatorError_eq
For a nonempty main row, its corrected sample average minus the true contrast is the average centered row score. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Row estimator Error eq.
Formal statement
hypotheses
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
hn :
0 < adaptiveMainSize m n
conclusion
contrast (η n)
+ (adaptiveMainSize m n : ℝ)⁻¹ * (∑ j, adaptiveSelectedScore select (η n) p ε (v j))
- contrast (adaptiveTrueParam θ h n)
= (adaptiveMainSize m n : ℝ)⁻¹ * ∑ j, adaptivePilotRowScore select θ h η p ε hselect hp hε n (v j)
Proof (Lean source)
lemma adaptivePilotRow_estimatorError_eq (θ h : TrialParameter)
(η : ℕ → TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε) (n : ℕ)
(hn : 0 < adaptiveMainSize m n)
(v : Fin (adaptiveMainSize m n) → Fin 14) :
contrast (η n) + (adaptiveMainSize m n : ℝ)⁻¹ *
(∑ j, adaptiveSelectedScore select (η n) p ε (v j)) -
contrast (adaptiveTrueParam θ h n) =
(adaptiveMainSize m n : ℝ)⁻¹ *
∑ j, adaptivePilotRowScore select θ h η p ε hselect hp hε n (v j) := by
rw [adaptivePilotRowScore_sum_eq select θ h η p ε m hselect hp hε n v]
have hN : (adaptiveMainSize m n : ℝ) ≠ 0 := by
exact_mod_cast (ne_of_gt hn)
field_simp
ring
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotRowUniformization 1 declarations This module turns convergence along every selected bad subsequence of finite pilot prefixes into eventual uniform control over all prefixes whose selected parameter lies in a shrinking neighborhood of the base point.
Sequential uniformization over shrinking pilot neighborhoods
This module turns convergence along every selected bad subsequence of finite pilot prefixes into eventual uniform control over all prefixes whose selected parameter lies in a shrinking neighborhood of the base point.
theorem adaptivePilotSelector_uniform_of_subsequential_tendsto
If every subsequence of finite pilot prefixes whose selected parameters converge to the base point has the same scalar limit, then the scalar errors are eventually uniform over all pilot prefixes selected inside any shrinking radius. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Selector uniform of subsequential tendsto.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
ρ :
ℕ → ℝ
E :
(n : ℕ) → AdaptivePilotSample m n → ℝ
L :
ℝ
hseq :
∀ (φ : ℕ → ℕ),
StrictMono φ → ∀ u : (k : ℕ) → AdaptivePilotSample m (φ k), Tendsto (fun k => adaptivePilotSelector p ε m (φ k) (u k)) atTop (nhds θ) → Tendsto (fun k => E (φ k) (u k)) atTop (nhds L)
δ :
ℝ
0 < δ
conclusion
≤ ρ n → |E n u - L| < δ
Proof (Lean source)
-- keep: reusable finite-pilot construction, law, localization, or limit API for related adaptive procedures
lemma adaptivePilotSelector_uniform_of_subsequential_tendsto
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(ρ : ℕ → ℝ)
(E : (n : ℕ) → AdaptivePilotSample m n → ℝ) (L : ℝ)
(hρ : Tendsto ρ atTop (nhds 0))
(hseq : ∀ (φ : ℕ → ℕ), StrictMono φ →
∀ u : (k : ℕ) → AdaptivePilotSample m (φ k),
Tendsto (fun k => adaptivePilotSelector p ε m (φ k) (u k))
atTop (nhds θ) →
Tendsto (fun k => E (φ k) (u k)) atTop (nhds L)) :
∀ δ : ℝ, 0 < δ → ∀ᶠ n in atTop,
∀ u : AdaptivePilotSample m n,
dist (adaptivePilotSelector p ε m n u) θ ≤ ρ n →
|E n u - L| < δ := by
intro δ hδ
by_contra huniform
have hfrequent : ∃ᶠ n in atTop,
¬(∀ u : AdaptivePilotSample m n,
dist (adaptivePilotSelector p ε m n u) θ ≤ ρ n →
|E n u - L| < δ) := not_eventually.mp huniform
have hbad : ∃ᶠ n in atTop, ∃ u : AdaptivePilotSample m n,
dist (adaptivePilotSelector p ε m n u) θ ≤ ρ n ∧
δ ≤ |E n u - L| := by
apply hfrequent.mono
intro n hn
push Not at hn
exact hn
obtain ⟨φ, hφ, hφbad⟩ := extraction_of_frequently_atTop hbad
choose u hu using hφbad
have hselector : Tendsto
(fun k => adaptivePilotSelector p ε m (φ k) (u k))
atTop (nhds θ) := by
rw [Metric.tendsto_atTop]
intro η hη
have hρsub : Tendsto (fun k => ρ (φ k)) atTop (nhds 0) :=
hρ.comp hφ.tendsto_atTop
have hρlt : ∀ᶠ k in atTop, ρ (φ k) < η :=
(tendsto_order.1 hρsub).2 η hη
rcases eventually_atTop.1 hρlt with ⟨N, hN⟩
exact ⟨N, fun k hk => lt_of_le_of_lt (hu k).1 (hN k hk)⟩
have hE := hseq φ hφ u hselector
have hclose : ∀ᶠ k in atTop, |E (φ k) (u k) - L| < δ := by
rcases (Metric.tendsto_atTop.1 hE) δ hδ with ⟨N, hN⟩
exact eventually_atTop.2 ⟨N, fun k hk => by
simpa [Real.dist_eq] using hN k hk⟩
rcases eventually_atTop.1 hclose with ⟨N, hN⟩
exact (not_lt_of_ge (hu N).2 (hN N le_rfl)).elim
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotRowVariance 12 declarations This module identifies the paper's main-sample variance estimator with the literal empirical variance of the centered finite row.
Empirical variance identities for adaptive pilot rows
This module identifies the paper's main-sample variance estimator with the literal empirical variance of the centered finite row. It also records the encoded-transcript support factorization and the sample-size scaling used for root-n errors.
def adaptivePilotRowEmpiricalMean
The empirical mean of the centered adaptive row score. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The adaptive Pilot Row Empirical Mean is determined by the displayed parameters.
Definition (Lean source)
parameters
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
def
adaptivePilotRowEmpiricalMean select θ h η p ε m hselect hp hε n v :
ℝ
given by
(adaptiveMainSize m n : ℝ)⁻¹ * ∑ j, adaptivePilotRowScore select θ h η p ε hselect hp hε n (v j)
def adaptivePilotRowEmpiricalSecond
The empirical second moment of the centered adaptive row score. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The adaptive Pilot Row Empirical Second is determined by the displayed parameters.
Definition (Lean source)
parameters
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
def
adaptivePilotRowEmpiricalSecond select θ h η p ε m hselect hp hε n v :
ℝ
given by
(adaptiveMainSize m n : ℝ)⁻¹ * ∑ j, (adaptivePilotRowScore select θ h η p ε hselect hp hε n (v j)) ^ 2
def adaptivePilotRowEmpiricalVariance
The empirical variance of the centered adaptive row score. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The adaptive Pilot Row Empirical Variance is determined by the displayed parameters.
Definition (Lean source)
parameters
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
v :
def
adaptivePilotRowEmpiricalVariance select θ h η p ε m hselect hp hε n v :
ℝ
given by
adaptivePilotRowEmpiricalSecond select θ h η p ε m hselect hp hε n v
- (adaptivePilotRowEmpiricalMean select θ h η p ε m hselect hp hε n v) ^ 2
theorem finite_empiricalVariance_eq_average_sq_deviation
A finite nonempty sample's second-moment-minus-mean-square formula equals its average squared deviation from the empirical mean. For the displayed inputs and conditions, the stated result follows.
Formal statement
hypotheses
conclusion
(N : ℝ)⁻¹ * ∑ j, (x j) ^ 2 - ((N : ℝ)⁻¹ * ∑ j, x j) ^ 2
= (N : ℝ)⁻¹ * ∑ j, (x j - (N : ℝ)⁻¹ * ∑ k, x k) ^ 2
Proof (Lean source)
lemma finite_empiricalVariance_eq_average_sq_deviation
{N : ℕ} (hN : 0 < N) (x : Fin N → ℝ) :
(N : ℝ)⁻¹ * ∑ j, (x j) ^ 2 - ((N : ℝ)⁻¹ * ∑ j, x j) ^ 2 =
(N : ℝ)⁻¹ * ∑ j, (x j - (N : ℝ)⁻¹ * ∑ k, x k) ^ 2 := by
have hNr : (N : ℝ) ≠ 0 := by exact_mod_cast (ne_of_gt hN)
have hcard : (Finset.univ : Finset (Fin N)).card = N := by simp
simp_rw [sub_sq]
rw [Finset.sum_add_distrib, Finset.sum_sub_distrib, Finset.mul_sum,
Finset.sum_const, nsmul_eq_mul, hcard]
have hsq : (∑ i : Fin N, (N : ℝ)⁻¹ * x i ^ 2) =
(N : ℝ)⁻¹ * ∑ i, x i ^ 2 := (Finset.mul_sum _ _ _).symm
have hcross :
(∑ i : Fin N, 2 * x i * ((N : ℝ)⁻¹ * ∑ j, x j)) =
(2 * ((N : ℝ)⁻¹ * ∑ j, x j)) * ∑ j, x j := by
calc
_ = ∑ i : Fin N, (2 * ((N : ℝ)⁻¹ * ∑ j, x j)) * x i := by
apply Finset.sum_congr rfl
intro i _
ring
_ = _ := (Finset.mul_sum _ _ _).symm
rw [hsq, hcross]
field_simp
ring
theorem adaptivePilotRowEmpiricalVariance_eq_average_sq_deviation
The adaptive row empirical variance is its average squared deviation from the empirical row mean. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Row Empirical Variance eq average sq deviation.
Formal statement
hypotheses
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
hN :
0 < adaptiveMainSize m n
conclusion
adaptivePilotRowEmpiricalVariance select θ h η p ε m hselect hp hε n v
= (adaptiveMainSize m n : ℝ)⁻¹ * ∑ j, (adaptivePilotRowScore select θ h η p ε hselect hp hε n (v j) - adaptivePilotRowEmpiricalMean select θ h η p ε m hselect hp hε n v) ^ 2
Proof (Lean source)
lemma adaptivePilotRowEmpiricalVariance_eq_average_sq_deviation
(θ h : TrialParameter) (η : ℕ → TrialParameter) (p ε : ℝ)
(m : ℕ → ℕ) (hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε) (n : ℕ)
(hN : 0 < adaptiveMainSize m n)
(v : Fin (adaptiveMainSize m n) → Fin 14) :
adaptivePilotRowEmpiricalVariance select θ h η p ε m hselect hp hε n v =
(adaptiveMainSize m n : ℝ)⁻¹ * ∑ j,
(adaptivePilotRowScore select θ h η p ε hselect hp hε n (v j) -
adaptivePilotRowEmpiricalMean select θ h η p ε m hselect hp hε n v) ^ 2 := by
exact finite_empiricalVariance_eq_average_sq_deviation hN
(fun j => adaptivePilotRowScore select θ h η p ε hselect hp hε n (v j))
theorem pilotAdaptiveEncode_main_score_sum
On an encoded transcript, summing any transformed main-release score over the full transcript is the corresponding sum over the main row. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Adaptive Encode main score sum.
Formal statement
hypotheses
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
hmn :
m n ≤ n
g :
ℝ → ℝ
conclusion
(∑ i : Fin n, if m n ≤ i.val then g (phiOutput (pilotAdaptivePrefixTheta p ε m hmn u) p ε (select) (pilotAdaptiveEncode hmn u v i)) else 0)
= ∑ j, g (adaptiveSelectedScore select (pilotAdaptivePrefixTheta p ε m hmn u) p ε (v j))
Proof (Lean source)
lemma pilotAdaptiveEncode_main_score_sum
(p ε : ℝ) (m : ℕ → ℕ) (hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε)
{n : ℕ} (hmn : m n ≤ n) (u : Fin (m n) → Fin 4)
(v : Fin (n - m n) → Fin 14) (g : ℝ → ℝ) :
(∑ i : Fin n, if m n ≤ i.val then
g (phiOutput (pilotAdaptivePrefixTheta p ε m hmn u) p ε
(select) (pilotAdaptiveEncode hmn u v i))
else 0) =
∑ j, g (adaptiveSelectedScore select
(pilotAdaptivePrefixTheta p ε m hmn u) p ε (v j)) := by
rw [show (∑ i : Fin n, if m n ≤ i.val then
g (phiOutput (pilotAdaptivePrefixTheta p ε m hmn u) p ε
(select) (pilotAdaptiveEncode hmn u v i))
else 0) =
∑ i : Fin n, if hi : i.val < m n then (0 : ℝ)
else g (adaptiveSelectedScore
select (pilotAdaptivePrefixTheta p ε m hmn u) p ε
(v ⟨i.val - m n, by omega⟩)) by
apply Finset.sum_congr rfl
intro i _
by_cases hi : i.val < m n
· simp [hi]
· have him : m n ≤ i.val := Nat.le_of_not_gt hi
simp [hi, him, pilotAdaptiveEncode, phiOutput, adaptiveSelectedScore]]
rw [fin_sum_split hmn (fun _ : Fin (m n) => (0 : ℝ))
(fun j : Fin (n - m n) => g (adaptiveSelectedScore
select (pilotAdaptivePrefixTheta p ε m hmn u) p ε (v j)))]
simp
theorem VhatStar_pilotAdaptiveEncode
On a properly encoded transcript, VhatStar is exactly the empirical variance of the selected main-release scores. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the Vhat Star pilot Adaptive Encode.
Formal statement
hypotheses
conclusion
VhatStar p ε m (select) (pilotAdaptiveEncode hmn u v)
= let η := pilotAdaptivePrefixTheta p ε m hmn u let q : Fin 14 → ℝ := adaptiveSelectedScore select η p ε let qbar := ((n - m n : ℕ) : ℝ)⁻¹ * ∑ j, q (v j) ((n - m n : ℕ) : ℝ)⁻¹ * ∑ j, (q (v j) - qbar) ^ 2
Proof (Lean source)
lemma VhatStar_pilotAdaptiveEncode
(p ε : ℝ) (m : ℕ → ℕ) (hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε)
{n : ℕ} (hmn : m n ≤ n) (u : Fin (m n) → Fin 4)
(v : Fin (n - m n) → Fin 14) :
VhatStar p ε m (select)
(pilotAdaptiveEncode hmn u v) =
let η := pilotAdaptivePrefixTheta p ε m hmn u
let q : Fin 14 → ℝ := adaptiveSelectedScore select η p ε
let qbar := ((n - m n : ℕ) : ℝ)⁻¹ * ∑ j, q (v j)
((n - m n : ℕ) : ℝ)⁻¹ * ∑ j, (q (v j) - qbar) ^ 2 := by
rw [VhatStar, pilotTheta_pilotAdaptiveEncode p ε m hmn u v]
dsimp only
have hsum := pilotAdaptiveEncode_main_score_sum select p ε m hselect hp hε hmn u v id
simp only [id_eq] at hsum
rw [hsum]
rw [pilotAdaptiveEncode_main_score_sum select p ε m hselect hp hε hmn u v
(fun x => (x - ((n - m n : ℕ) : ℝ)⁻¹ *
∑ j, adaptiveSelectedScore select (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)) ^ 2)]
theorem adaptivePilotRowEmpiricalVariance_eq_selected
Centering every selected score by the same population correction does not change its empirical variance. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Row Empirical Variance eq selected.
Formal statement
hypotheses
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
hN :
0 < adaptiveMainSize m n
conclusion
adaptivePilotRowEmpiricalVariance select θ h η p ε m hselect hp hε n v
= let q : Fin 14 → ℝ := adaptiveSelectedScore select (η n) p ε let qbar := (adaptiveMainSize m n : ℝ)⁻¹ * ∑ j, q (v j) (adaptiveMainSize m n : ℝ)⁻¹ * ∑ j, (q (v j) - qbar) ^ 2
Proof (Lean source)
lemma adaptivePilotRowEmpiricalVariance_eq_selected
(θ h : TrialParameter) (η : ℕ → TrialParameter) (p ε : ℝ)
(m : ℕ → ℕ) (hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε) (n : ℕ)
(hN : 0 < adaptiveMainSize m n)
(v : Fin (adaptiveMainSize m n) → Fin 14) :
adaptivePilotRowEmpiricalVariance select θ h η p ε m hselect hp hε n v =
let q : Fin 14 → ℝ := adaptiveSelectedScore select (η n) p ε
let qbar := (adaptiveMainSize m n : ℝ)⁻¹ * ∑ j, q (v j)
(adaptiveMainSize m n : ℝ)⁻¹ * ∑ j, (q (v j) - qbar) ^ 2 := by
rw [adaptivePilotRowEmpiricalVariance_eq_average_sq_deviation
select θ h η p ε m hselect hp hε n hN v]
have hNr : (adaptiveMainSize m n : ℝ) ≠ 0 := by
exact_mod_cast (ne_of_gt hN)
have hmean : adaptivePilotRowEmpiricalMean select θ h η p ε m hselect hp hε n v =
(adaptiveMainSize m n : ℝ)⁻¹ *
∑ j, adaptiveSelectedScore select (η n) p ε (v j) -
(contrast (adaptiveTrueParam θ h n) - contrast (η n)) := by
unfold adaptivePilotRowEmpiricalMean
rw [adaptivePilotRowScore_sum_eq select θ h η p ε m hselect hp hε n v]
field_simp
rw [hmean]
apply congrArg ((adaptiveMainSize m n : ℝ)⁻¹ * ·)
apply Finset.sum_congr rfl
intro j _
unfold adaptivePilotRowScore
ring
def adaptiveEncodedRowEmpiricalVariance
The empirical variance of the population-centered scores in an encoded main row. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The adaptive Encoded Row Empirical Variance is determined by the displayed parameters.
Definition (Lean source)
parameters
θ h :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
hmn :
m n ≤ n
def
adaptiveEncodedRowEmpiricalVariance select θ h p ε m hselect hp hε hmn u v :
ℝ
given by
η :
pilotAdaptivePrefixTheta p ε m hmn u
f :
fun s
=> adaptiveSelectedScore select η p ε s
- (contrast (adaptiveTrueParam θ h n) - contrast η) (: Fin 14 → ℝ)
((n - m n : ℕ) : ℝ)⁻¹ * ∑ j, (f (v j)) ^ 2 - (((n - m n : ℕ) : ℝ)⁻¹ * ∑ j, f (v j)) ^ 2
theorem VhatStar_pilotAdaptiveEncode_eq_rowVariance
Under a proper nonempty pilot split, VhatStar on an encoded transcript is exactly the empirical variance of the population-centered main-row score. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the Vhat Star pilot Adaptive Encode eq row Variance.
Formal statement
hypotheses
θ h :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
hmn :
m n ≤ n
hN :
0 < n - m n
conclusion
VhatStar p ε m (select) (pilotAdaptiveEncode hmn u v)
= adaptiveEncodedRowEmpiricalVariance select θ h p ε m hselect hp hε hmn u v
Proof (Lean source)
-- keep: reusable finite-pilot construction, law, localization, or limit API for related adaptive procedures
lemma VhatStar_pilotAdaptiveEncode_eq_rowVariance
(θ h : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε) {n : ℕ}
(hmn : m n ≤ n) (hN : 0 < n - m n) (u : Fin (m n) → Fin 4)
(v : Fin (n - m n) → Fin 14) :
VhatStar p ε m (select)
(pilotAdaptiveEncode hmn u v) =
adaptiveEncodedRowEmpiricalVariance select θ h p ε m hselect hp hε hmn u v := by
rw [VhatStar_pilotAdaptiveEncode select p ε m hselect hp hε hmn u v]
unfold adaptiveEncodedRowEmpiricalVariance
dsimp only
rw [finite_empiricalVariance_eq_average_sq_deviation hN]
let q : Fin 14 → ℝ :=
adaptiveSelectedScore select (pilotAdaptivePrefixTheta p ε m hmn u) p ε
let c := contrast (adaptiveTrueParam θ h n) -
contrast (pilotAdaptivePrefixTheta p ε m hmn u)
have hNr : ((n - m n : ℕ) : ℝ) ≠ 0 := by
exact_mod_cast (ne_of_gt hN)
have hmean : ((n - m n : ℕ) : ℝ)⁻¹ * ∑ j, (q (v j) - c) =
((n - m n : ℕ) : ℝ)⁻¹ * ∑ j, q (v j) - c := by
rw [Finset.sum_sub_distrib, Finset.sum_const, nsmul_eq_mul]
have hcard : (Finset.univ : Finset (Fin (n - m n))).card = n - m n := by simp
rw [hcard]
field_simp
rw [hmean]
apply congrArg (((n - m n : ℕ) : ℝ)⁻¹ * ·)
apply Finset.sum_congr rfl
intro j _
ring
theorem adaptivePilotRow_rootN_error_sq
Root-n squared main-estimation error is the row-normalized squared sum multiplied by the exact sample-size ratio n / adaptiveMainSize m n. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the adaptive Pilot Row root N error sq.
Formal statement
hypotheses
θ h :
η :
ℕ → TrialParameter
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
hN :
0 < adaptiveMainSize m n
conclusion
(n : ℝ) * (contrast (η n) + (adaptiveMainSize m n : ℝ)⁻¹ * (∑ j, adaptiveSelectedScore select (η n) p ε (v j)) - contrast (adaptiveTrueParam θ h n)) ^ 2
= ((n : ℝ) / (adaptiveMainSize m n : ℝ)) * ((sqrt (adaptiveMainSize m n : ℝ))⁻¹ * ∑ j, adaptivePilotRowScore select θ h η p ε hselect hp hε n (v j)) ^ 2
Proof (Lean source)
-- keep: reusable finite-pilot construction, law, localization, or limit API for related adaptive procedures
lemma adaptivePilotRow_rootN_error_sq
(θ h : TrialParameter) (η : ℕ → TrialParameter) (p ε : ℝ)
(m : ℕ → ℕ) (hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε) (n : ℕ)
(hN : 0 < adaptiveMainSize m n)
(v : Fin (adaptiveMainSize m n) → Fin 14) :
(n : ℝ) *
(contrast (η n) + (adaptiveMainSize m n : ℝ)⁻¹ *
(∑ j, adaptiveSelectedScore select (η n) p ε (v j)) -
contrast (adaptiveTrueParam θ h n)) ^ 2 =
((n : ℝ) / (adaptiveMainSize m n : ℝ)) *
((sqrt (adaptiveMainSize m n : ℝ))⁻¹ *
∑ j, adaptivePilotRowScore select θ h η p ε hselect hp hε n (v j)) ^ 2 := by
rw [adaptivePilotRow_estimatorError_eq select θ h η p ε m hselect hp hε n hN v]
have hNr : 0 < (adaptiveMainSize m n : ℝ) := by exact_mod_cast hN
have hsqrt : (sqrt (adaptiveMainSize m n : ℝ)) ^ 2 =
adaptiveMainSize m n := Real.sq_sqrt hNr.le
have hsqrt_ne : sqrt (adaptiveMainSize m n : ℝ) ≠ 0 :=
ne_of_gt (Real.sqrt_pos.2 hNr)
field_simp
rw [hsqrt]
ring
theorem pilotAdaptive_joint_singleton_eq_prefix_mul_mainProduct
The exact transcript atom is its pilot-prefix mass times the conditional iid main-row product mass. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Adaptive joint singleton eq prefix mul main Product.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
n :
ℕ
hmn :
m n ≤ n
conclusion
transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n {pilotAdaptiveEncode hmn u v}
= ofReal (∏ j : Fin (m n), ∑ a : Fin 4, piTheta θ p a * rrPilotProbability ε a (u j)) * ofReal (∏ j : Fin (n - m n), adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u) p ε (v j))
Proof (Lean source)
lemma pilotAdaptive_joint_singleton_eq_prefix_mul_mainProduct
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
{n : ℕ} (hmn : m n ≤ n) (u : Fin (m n) → Fin 4)
(v : Fin (n - m n) → Fin 14) :
transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n
{pilotAdaptiveEncode hmn u v} =
ofReal (∏ j : Fin (m n),
∑ a : Fin 4, piTheta θ p a * rrPilotProbability ε a (u j)) *
ofReal (∏ j : Fin (n - m n),
adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)) := by
rw [pilotAdaptive_joint_singleton select θ p ε m hselect hp hθ hε hmn u v]
rw [ENNReal.ofReal_prod_of_nonneg]
· rw [ENNReal.ofReal_prod_of_nonneg]
intro j _
exact adaptiveMainMass_nonneg select θ _ p ε hselect hp hθ
(pilotTheta_interior p ε m (pilotAdaptiveEncode hmn u (fun _ => 0))) hε (v j)
· intro j _
apply Finset.sum_nonneg
intro a _
exact mul_nonneg (piTheta_pos_interior θ p hp hθ a).le (by
unfold rrPilotProbability
split_ifs <;> positivity)
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotSelectorBridge 2 declarations Measurable finite-quadratic selector adapter
Measurable finite-quadratic selector adapter
theorem exists_measurable_saddle_selector_bridge_strong
Under the supplied quantities and conditions, the exists measurable saddle selector bridge strong assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the exists measurable saddle selector bridge strong.
Formal statement
hypotheses
conclusion
∃ select : TrialParameter → StaircaseWeight × ℝ × ℝ,
Measurable select
conclusion 1
θ :
staircaseFeasible ε (select θ).1
u :
informationObjective θ p ε (select θ).1 (select θ).2.1
≤ informationObjective θ p ε (select θ).1 u
Jstar θ p ε = (select θ).2.2
(select θ).2.2 = informationObjective θ p ε (select θ).1 (select θ).2.1
β :
informationObjective θ p ε β (select θ).2.1
≤ informationObjective θ p ε (select θ).1 (select θ).2.1
Proof (Lean source)
lemma exists_measurable_saddle_selector_bridge_strong (p ε : ℝ)
(hp : InteriorAssignment p) (hε : 0 < ε) :
∃ select : TrialParameter → StaircaseWeight × ℝ × ℝ,
Measurable select ∧
∀ θ, InteriorMeans θ →
staircaseFeasible ε (select θ).1 ∧
(∀ u, informationObjective θ p ε (select θ).1 (select θ).2.1 ≤
informationObjective θ p ε (select θ).1 u) ∧
Jstar θ p ε = (select θ).2.2 ∧
(select θ).2.2 =
informationObjective θ p ε (select θ).1 (select θ).2.1 ∧
(∀ β, staircaseFeasible ε β →
informationObjective θ p ε β (select θ).2.1 ≤
informationObjective θ p ε (select θ).1 (select θ).2.1) := by
classical
let Θ := {θ : TrialParameter // InteriorMeans θ}
letI : StandardBorelSpace Θ := measurableSet_interiorMeans.standardBorel
have hatt : ∀ θ : Θ, θ ∈ Set.univ → ∃ α t,
IsSaddle (pilotPolytope ε hε.le) (pilotQuadratics p ε hp hε.le) θ α t :=
fun θ _ => exists_quadratic_saddle p ε hp hε θ
obtain ⟨α, t, hαm, htm, hs⟩ := exists_measurable_saddle_on
(pilotPolytope ε hε.le) (pilotQuadratics p ε hp hε.le)
Set.univ MeasurableSet.univ hatt
let lift : Θ → (Set.univ : Set Θ) := fun x => ⟨x, Set.mem_univ x⟩
have hlift : Measurable lift := by
dsimp [lift]
fun_prop
let αD : Θ → EuclideanSpace ℝ (Fin 14) := α ∘ lift
let tD : Θ → ℝ := t ∘ lift
have hαD : Measurable αD := hαm.comp hlift
have htD : Measurable tD := htm.comp hlift
let policyD : Θ → StaircaseWeight × ℝ × ℝ := fun x =>
(toWeight (αD x), tD x,
(pilotQuadratics p ε hp hε.le).objective x (αD x) (tD x))
have hpolicyD : Measurable policyD := by
apply prodMk
· apply measurable_pi_lambda
intro s
exact ((EuclideanSpace.proj (𝕜 := ℝ) s).continuous.measurable.comp hαD)
· exact htD.prodMk ((pilotQuadratics p ε hp hε.le).measurable_objective.comp
(measurable_id.prodMk (hαD.prodMk htD)))
let select : TrialParameter → StaircaseWeight × ℝ × ℝ := fun θ =>
if hθ : InteriorMeans θ then
policyD ⟨θ, hθ⟩
else (0, 0, 0)
have hselect : Measurable select := by
exact Measurable.dite hpolicyD measurable_const measurableSet_interiorMeans
refine ⟨select, hselect, ?_⟩
intro θ hθ
let x : Θ := ⟨θ, hθ⟩
let xu : (Set.univ : Set Θ) := ⟨x, Set.mem_univ x⟩
have hsx := hs xu
have hfeasSel : staircaseFeasible ε (toWeight (α xu)) :=
(pilotPolytope_weights ε hε.le _).1 hsx.1
have hupper : ∀ z ∈ {z : ℝ | ∃ β : StaircaseWeight,
staircaseFeasible ε β ∧
z = sInf {y : ℝ | ∃ u : ℝ, y = informationObjective θ p ε β u}},
z ≤ informationObjective θ p ε (toWeight (α xu)) (t xu) := by
rintro _ ⟨β, hβ, rfl⟩
have hb : BddBelow {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε β v} := by
refine ⟨0, ?_⟩
rintro _ ⟨u, rfl⟩
exact informationObjective_nonneg_interior θ p ε hp hθ hε.le β hβ u
calc
sInf {u : ℝ | ∃ v : ℝ, u = informationObjective θ p ε β v} ≤
informationObjective θ p ε β (t xu) := csInf_le hb ⟨t xu, rfl⟩
_ ≤ informationObjective θ p ε (toWeight (α xu)) (t xu) := by
have hh := hsx.2.2 (ofWeight β)
((pilotPolytope_weights ε hε.le _).2 (by simpa))
rw [pilotQuadratics_objective, pilotQuadratics_objective] at hh
simpa [x, xu] using hh
have hne : {z : ℝ | ∃ β : StaircaseWeight, staircaseFeasible ε β ∧
z = sInf {y : ℝ | ∃ u : ℝ, y = informationObjective θ p ε β u}}.Nonempty :=
⟨sInf {y : ℝ | ∃ u : ℝ,
y = informationObjective θ p ε (toWeight (α xu)) u},
toWeight (α xu), hfeasSel, rfl⟩
have hj : Jstar θ p ε = (pilotQuadratics p ε hp hε.le).objective x
(α xu) (t xu) := by
rw [pilotQuadratics_objective]
apply le_antisymm
· rw [Jstar]
apply csSup_le
· exact hne
· exact hupper
· rw [Jstar]
apply le_csSup
· exact ⟨_, hupper⟩
· refine ⟨toWeight (α xu), (pilotPolytope_weights ε hε.le _).1 hsx.1, ?_⟩
symm
apply IsLeast.csInf_eq
refine ⟨⟨t xu, rfl⟩, ?_⟩
rintro _ ⟨u, rfl⟩
have hh := hsx.2.1 u
rw [pilotQuadratics_objective, pilotQuadratics_objective] at hh
simpa [x, xu] using hh
simp only [select, hθ, dite_true, policyD, αD, tD, lift, Function.comp_apply]
refine ⟨(pilotPolytope_weights ε hε.le _).1 hsx.1, ?_, hj, ?_, ?_⟩
· intro u
have hh := hsx.2.1 u
rw [pilotQuadratics_objective, pilotQuadratics_objective] at hh
simpa [x, xu] using hh
· exact pilotQuadratics_objective p ε hp hε.le x (α xu) (t xu)
· intro β hβ
have hh := hsx.2.2 (ofWeight β)
((pilotPolytope_weights ε hε.le _).2 (by simpa))
rw [pilotQuadratics_objective, pilotQuadratics_objective] at hh
simpa [x, xu] using hh
theorem exists_measurable_saddle_selector_bridge
Under the supplied quantities and conditions, the exists measurable saddle selector bridge assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the exists measurable saddle selector bridge.
Formal statement
hypotheses
conclusion
∃ select : TrialParameter → StaircaseWeight × ℝ × ℝ,
Measurable select
conclusion 1
θ :
staircaseFeasible ε (select θ).1
u :
informationObjective θ p ε (select θ).1 (select θ).2.1
≤ informationObjective θ p ε (select θ).1 u
Jstar θ p ε = (select θ).2.2
(select θ).2.2 = informationObjective θ p ε (select θ).1 (select θ).2.1
Proof (Lean source)
lemma exists_measurable_saddle_selector_bridge (p ε : ℝ)
(hp : InteriorAssignment p) (hε : 0 < ε) :
∃ select : TrialParameter → StaircaseWeight × ℝ × ℝ,
Measurable select ∧
∀ θ, InteriorMeans θ →
staircaseFeasible ε (select θ).1 ∧
(∀ u, informationObjective θ p ε (select θ).1 (select θ).2.1 ≤
informationObjective θ p ε (select θ).1 u) ∧
Jstar θ p ε = (select θ).2.2 ∧
(select θ).2.2 =
informationObjective θ p ε (select θ).1 (select θ).2.1 := by
obtain ⟨select, hm, hs⟩ :=
exists_measurable_saddle_selector_bridge_strong p ε hp hε
refine ⟨select, hm, ?_⟩
intro θ hθ
have h := hs θ hθ
exact ⟨h.1, h.2.1, h.2.2.1, h.2.2.2.1⟩
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotStudentizationSupport 3 declarations This file derives the Slutsky approximation required by the varying-law Wald adapter from the genuine CDF and plug-in variance consistency premises.
Row-varying studentization for the private pilot
This file derives the Slutsky approximation required by the varying-law Wald adapter from the genuine CDF and plug-in variance consistency premises.
theorem varyingLaw_tight_of_standardNormal_cdf
A row-varying statistic whose CDF converges to the standard normal is tight. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the varying Law tight of standard Normal cdf.
Formal statement
hypotheses
Ω :
ℕ → Type*
∀ n, MeasurableSpace (Ω n)
μ :
∀ n, Measure (Ω n)
∀ n, IsProbabilityMeasure (μ n)
T :
∀ n
if
Ω n
then
ℝ
hTmeas :
∀ n, Measurable (T n)
hcdf :
η :
ℝ
0 < η
Proof (Lean source)
lemma varyingLaw_tight_of_standardNormal_cdf
{Ω : ℕ → Type*} [∀ n, MeasurableSpace (Ω n)]
(μ : ∀ n, Measure (Ω n)) [∀ n, IsProbabilityMeasure (μ n)]
(T : ∀ n, Ω n → ℝ) (hTmeas : ∀ n, Measurable (T n))
(hcdf : ∀ x : ℝ,
Tendsto (fun n => (μ n).real {z | T n z ≤ x}) atTop
(nhds ((gaussianMeasure 0 1).real (Iic x)))) :
∀ η : ℝ, 0 < η → ∃ M : ℝ, 0 < M ∧
∀ᶠ n in atTop, (μ n).real {z | M ≤ |T n z|} < η := by
intro η hη
let η₀ : ℝ := min (η / 8) (1 / 8)
have hη₀ : 0 < η₀ := lt_min (by linarith) (by norm_num)
obtain ⟨R, hR, hleft, hright⟩ :=
gaussian_tail_small_gaussian (m := 0) (v := 1) hη₀
have hleftR : (gaussianMeasure 0 1).real (Iic (-R)) ≤ η₀ := by
have ht := ENNReal.toReal_mono ENNReal.ofReal_ne_top hleft
simpa [MeasureTheory.measureReal_def, ENNReal.toReal_ofReal hη₀.le] using ht
have hrightR : (gaussianMeasure 0 1).real (Ici R) ≤ η₀ := by
have ht := ENNReal.toReal_mono ENNReal.ofReal_ne_top hright
simpa [MeasureTheory.measureReal_def, ENNReal.toReal_ofReal hη₀.le] using ht
have hcomp : (gaussianMeasure 0 1).real (Icc (-R) R)ᶜ ≤ 2 * η₀ := by
calc
_ ≤ (gaussianMeasure 0 1).real (Iic (-R) ∪ Ici R) := by
apply MeasureTheory.measureReal_mono (h₂ := measure_ne_top _ _)
intro x hx
simp only [mem_compl_iff, mem_Icc, not_and_or, not_le] at hx
exact hx.imp le_of_lt le_of_lt
_ ≤ (gaussianMeasure 0 1).real (Iic (-R)) +
(gaussianMeasure 0 1).real (Ici R) :=
MeasureTheory.measureReal_union_le _ _
_ ≤ 2 * η₀ := by linarith
have hzeroApprox : ∀ δ : ℝ, 0 < δ →
Tendsto (fun n => (μ n).real {z | δ ≤ |T n z - T n z|})
atTop (nhds 0) := by
intro δ hδ
convert tendsto_const_nhds (x := (0 : ℝ)) using 1
funext n
simp [not_le_of_gt hδ]
have hinter := varyingLaw_Icc_tendsto_of_cdf_and_approx μ T T
(gaussianMeasure 0 1) hTmeas hzeroApprox hcdf
(by simpa [gaussianMeasure] using
(continuous_cdf_gaussianReal_zero (v := (1 : NNReal)) (by norm_num)))
(gaussianMeasure_zero_one_frontier_Icc hR)
have hcompl : Tendsto (fun n => (μ n).real {z | T n z ∉ Icc (-R) R})
atTop (nhds ((gaussianMeasure 0 1).real (Icc (-R) R)ᶜ)) := by
have hsub :=
(tendsto_const_nhds : Tendsto (fun _ : ℕ => (1 : ℝ)) atTop (nhds 1)).sub hinter
have hq := MeasureTheory.measureReal_add_measureReal_compl
(μ := gaussianMeasure 0 1)
(show MeasurableSet (Icc (-R) R) from measurableSet_Icc)
have hq' : (gaussianMeasure 0 1).real (Icc (-R) R) +
(gaussianMeasure 0 1).real (Icc (-R) R)ᶜ = 1 := by
simpa [MeasureTheory.measureReal_def] using hq
rw [show (gaussianMeasure 0 1).real (Icc (-R) R)ᶜ =
1 - (gaussianMeasure 0 1).real (Icc (-R) R) by linarith [hq']]
apply hsub.congr'
filter_upwards with n
have hn := MeasureTheory.measureReal_add_measureReal_compl
(μ := μ n) (hTmeas n
(show MeasurableSet (Icc (-R) R) from measurableSet_Icc))
have hn' : (μ n).real (T n ⁻¹' Icc (-R) R) +
(μ n).real (T n ⁻¹' Icc (-R) R)ᶜ = 1 := by
simpa [MeasureTheory.measureReal_def] using hn
rw [show {z | T n z ∈ Icc (-R) R} = T n ⁻¹' Icc (-R) R by rfl,
show {z | T n z ∉ Icc (-R) R} = (T n ⁻¹' Icc (-R) R)ᶜ by rfl]
linarith [hn']
refine ⟨R + 1, by linarith, ?_⟩
have hev := hcompl.eventually_lt_const (by
have h2η : 2 * η₀ < η := by
dsimp [η₀]
have hm := min_le_left (η / 8) (1 / 8)
linarith
exact lt_of_le_of_lt hcomp h2η)
filter_upwards [hev] with n hn
apply lt_of_le_of_lt (MeasureTheory.measureReal_mono (h₂ := measure_ne_top _ _) ?_) hn
intro z hz
simp only [mem_Icc, not_and_or, not_le]
change R + 1 ≤ |T n z| at hz
rcases (le_abs.mp hz) with hz | hz
· exact inr (by linarith)
· exact inl (by linarith)
theorem pilotWaldPivot_approx_standardizedScaledError
Plug-in variance consistency and the Gaussian CDF limit imply that the fallback-studentized pilot pivot is asymptotically indistinguishable from the oracle-standardized scaled error. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Wald Pivot approx standardized Scaled Error.
Formal statement
hypotheses
cutoff :
ℝ
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hstandardCDF :
∀ x : ℝ,
Tendsto (fun n => (transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n).real {z | scaledError (pilotEstimator p ε m select hselect hp hε) θ 0 n z / sqrt (Vstar θ p ε) ≤ x}) atTop (nhds ((gaussianMeasure 0 1).real (Iic x)))
hvhat :
δ :
ℝ
0 < δ
conclusion
Tendsto (fun n => (transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n).real {z | δ ≤ |pilotWaldPivot select cutoff θ p ε m hselect hp hε n z - scaledError (pilotEstimator p ε m select hselect hp hε) θ 0 n z / sqrt (Vstar θ p ε)|}) atTop (nhds 0)
Proof (Lean source)
lemma pilotWaldPivot_approx_standardizedScaledError
(cutoff : ℝ) (θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hstandardCDF : ∀ x : ℝ,
Tendsto (fun n =>
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n).real
{z | scaledError (pilotEstimator p ε m select hselect hp hε) θ 0 n z /
Real.sqrt (Vstar θ p ε) ≤ x}) atTop
(nhds ((gaussianMeasure 0 1).real (Iic x))))
(hvhat : ∀ r : ℝ, 0 < r →
Tendsto (fun n =>
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n).real
{z | r ≤ |VhatStar p ε m (select) z -
Vstar θ p ε|}) atTop (nhds 0)) :
∀ δ : ℝ, 0 < δ →
Tendsto (fun n =>
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n).real
{z | δ ≤ |pilotWaldPivot select cutoff θ p ε m hselect hp hε n z -
scaledError (pilotEstimator p ε m select hselect hp hε) θ 0 n z /
Real.sqrt (Vstar θ p ε)|}) atTop (nhds 0) := by
let μ : ∀ n, Measure (Transcript (pilotOutputFamily n)) := fun n =>
transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n
let T : ∀ n, Transcript (pilotOutputFamily n) → ℝ := fun n z =>
scaledError (pilotEstimator p ε m select hselect hp hε) θ 0 n z /
Real.sqrt (Vstar θ p ε)
let _ (n : ℕ) : IsProbabilityMeasure (μ n) :=
transcriptLaw_isProbability _ θ p hp hθ n
have hTmeas (n : ℕ) : Measurable (T n) := measurable_of_finite _
have hV : 0 < Vstar θ p ε :=
inv_pos.mpr (Jstar_pos_interior θ p ε hp hθ hε)
let g : ℝ → ℝ := fun v => Real.sqrt (Vstar θ p ε) / Real.sqrt v
have hg : ContinuousAt g (Vstar θ p ε) := by
dsimp [g]
exact continuousAt_const.div₀ Real.continuous_sqrt.continuousAt
(ne_of_gt (Real.sqrt_pos.2 hV))
have hgV : g (Vstar θ p ε) = 1 := by
dsimp [g]
field_simp [ne_of_gt (Real.sqrt_pos.2 hV)]
intro δ hδ
rw [Metric.tendsto_atTop]
intro ζ hζ
obtain ⟨M, hM, hMtight⟩ :=
varyingLaw_tight_of_standardNormal_cdf μ T hTmeas hstandardCDF
(ζ / 2) (by linarith)
obtain ⟨r₀, hr₀, hgr₀⟩ :=
(Metric.continuousAt_iff.mp hg) (δ / M) (div_pos hδ hM)
let r : ℝ := min (r₀ / 2) (Vstar θ p ε / 2)
have hr : 0 < r := lt_min (half_pos hr₀) (half_pos hV)
have hrV : r < Vstar θ p ε :=
(min_le_right _ _).trans_lt (half_lt_self hV)
have hvarEv := (hvhat r hr).eventually_lt_const (half_pos hζ)
refine Filter.eventually_atTop.1 ?_
filter_upwards [hMtight, hvarEv] with n htight hvar
rw [Real.dist_eq, sub_zero, abs_of_nonneg MeasureTheory.measureReal_nonneg]
have hsubset :
{z | δ ≤ |pilotWaldPivot select cutoff θ p ε m hselect hp hε n z - T n z|} ⊆
{z | r ≤ |VhatStar p ε m (select) z -
Vstar θ p ε|} ∪ {z | M ≤ |T n z|} := by
intro z hz
by_contra hbad
simp only [mem_union, mem_ofPred_eq, not_or, not_le] at hbad
have hVn : 0 < VhatStar p ε m (select) z := by
linarith [abs_lt.mp hbad.1 |>.1]
have hdist : dist
(VhatStar p ε m (select) z)
(Vstar θ p ε) < r₀ := by
rw [Real.dist_eq]
exact hbad.1.trans_le (min_le_left _ _ |>.trans (half_le_self hr₀.le))
have hgclose := hgr₀ hdist
rw [Real.dist_eq, hgV] at hgclose
have hpivot : pilotWaldPivot select cutoff θ p ε m hselect hp hε n z =
T n z * g (VhatStar p ε m (select) z) := by
have hlocal : localAlternative θ 0 n = θ := by
funext k
simp [localAlternative]
simp [pilotWaldPivot, waldFallbackPivot, ne_of_gt hVn, T, g,
scaledError, hlocal]
field_simp [ne_of_gt (Real.sqrt_pos.2 hV),
ne_of_gt (Real.sqrt_pos.2 hVn)]
change δ ≤ |pilotWaldPivot select cutoff θ p ε m hselect hp hε n z - T n z| at hz
rw [hpivot, show T n z * g
(VhatStar p ε m (select) z) - T n z =
T n z * (g (VhatStar p ε m
(select) z) - 1) by ring,
abs_mul] at hz
have hprod : |T n z| *
|g (VhatStar p ε m (select) z) - 1| < δ := by
calc
_ ≤ M * |g (VhatStar p ε m (select) z) - 1| :=
mul_le_mul_of_nonneg_right hbad.2.le (abs_nonneg _)
_ < M * (δ / M) := mul_lt_mul_of_pos_left hgclose hM
_ = δ := by field_simp
exact (not_lt_of_ge hz) hprod
have hmono := MeasureTheory.measureReal_mono (μ := μ n)
(h₂ := measure_ne_top _ _) hsubset
have hunion := MeasureTheory.measureReal_union_le (μ := μ n)
{z | r ≤ |VhatStar p ε m (select) z -
Vstar θ p ε|} {z | M ≤ |T n z|}
exact lt_of_le_of_lt (hmono.trans hunion) (by linarith)
theorem pilot_wald_coverage_of_standardized_cdf_and_VhatStar_consistent
The actual standardized-error CDF limit and consistent VhatStar imply the private-pilot Wald coverage limit, with the zero-variance fallback handled inside the proved Slutsky approximation. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot wald coverage of standardized cdf and Vhat Star consistent.
Formal statement
hypotheses
cutoff :
ℝ
hcutoff :
0 < cutoff
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hstandardCDF :
∀ x : ℝ,
Tendsto (fun n => (transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n).real {z | scaledError (pilotEstimator p ε m select hselect hp hε) θ 0 n z / sqrt (Vstar θ p ε) ≤ x}) atTop (nhds ((gaussianMeasure 0 1).real (Iic x)))
conclusion
Tendsto (fun n => ((transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) {z | |(pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ| ≤ cutoff * sqrt (VhatStar p ε m (select) z / n)}).toReal) atTop (nhds ((gaussianMeasure 0 1).real (Icc (-cutoff) cutoff)))
Proof (Lean source)
lemma pilot_wald_coverage_of_standardized_cdf_and_VhatStar_consistent
(cutoff : ℝ) (hcutoff : 0 < cutoff)
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hstandardCDF : ∀ x : ℝ,
Tendsto (fun n =>
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n).real
{z | scaledError (pilotEstimator p ε m select hselect hp hε) θ 0 n z /
sqrt (Vstar θ p ε) ≤ x}) atTop
(nhds ((gaussianMeasure 0 1).real (Iic x))))
(hvhat : ∀ r : ℝ, 0 < r →
Tendsto (fun n =>
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n).real
{z | r ≤ |VhatStar p ε m (select) z -
Vstar θ p ε|}) atTop (nhds 0)) :
Tendsto (fun n =>
((transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
{z | |(pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ| ≤
cutoff * sqrt
(VhatStar p ε m (select) z / n)}).toReal)
atTop (nhds ((gaussianMeasure 0 1).real (Icc (-cutoff) cutoff))) := by
apply pilot_wald_coverage_of_standardized_cdf_and_approx
select cutoff hcutoff θ p ε m hselect hp hθ hε
· exact pilotWaldPivot_approx_standardizedScaledError
select cutoff θ p ε m hselect hp hθ hε hstandardCDF hvhat
· exact hstandardCDF
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotUniformTails 1 declarations Uniform squared tails for the private-pilot estimator — finite-product fourth moments.
Uniform squared tails for the private-pilot estimator — finite-product fourth moments.
theorem pilot_scaledError_uniform_squared_tails
The private-pilot scaled error has uniformly vanishing squared tails over bounded local directions. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot scaled Error uniform squared tails.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hsub :
H :
ℝ
0 < H
conclusion
Tendsto (fun M : ℕ => limsup (fun n => sSup {u : ℝ≥0∞ | ∃ h ∈ localIndexSet θ n H, u = ∫⁻ z, ofReal ((scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z) ^ 2 * if M < |scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z| then 1 else 0) ∂transcriptLaw (pilotEstimator p ε m select hselect hp hε) (localAlternative θ h n) p n}) atTop) atTop (nhds 0)
Proof (Lean source)
lemma pilot_scaledError_uniform_squared_tails
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select) (hp : InteriorAssignment p)
(hθ : InteriorMeans θ) (hε : 0 < ε) (hsub : PilotSublinear m) :
∀ H : ℝ, 0 < H →
Tendsto (fun M : ℕ =>
limsup (fun n =>
sSup {u : ℝ≥0∞ | ∃ h ∈ localIndexSet θ n H,
u = ∫⁻ z, ofReal
((scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z) ^ 2 *
if M < |scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z|
then 1 else 0)
∂transcriptLaw (pilotEstimator p ε m select hselect hp hε)
(localAlternative θ h n) p n})
Filter.atTop) Filter.atTop (nhds 0) := by
intro H _hH
let C : ℝ := 16 * (2 * pilotPhiUpper p ε) ^ 4
let F : ℕ → ℝ≥0∞ := fun M =>
limsup (fun n =>
sSup {u : ℝ≥0∞ | ∃ h ∈ localIndexSet θ n H,
u = ∫⁻ z, ofReal
((scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z) ^ 2 *
if M < |scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z|
then 1 else 0)
∂transcriptLaw (pilotEstimator p ε m select hselect hp hε)
(localAlternative θ h n) p n}) Filter.atTop
let D : ℕ → ℝ≥0∞ := fun M =>
ofReal C / ofReal ((M : ℝ) ^ 2)
have hratio : ∀ᶠ n : ℕ in Filter.atTop,
(n : ℝ) / (adaptiveMainSize m n : ℝ) ≤ 2 := by
have ht := adaptive_mainScale_tendsto_one m hsub
exact ((tendsto_order.1 ht).2 2 (by norm_num)).mono fun _ hn => hn.le
have hn2 : ∀ᶠ n : ℕ in Filter.atTop, 2 ≤ n :=
Filter.eventually_atTop.2 ⟨2, fun _ hn => hn⟩
have hFD : ∀ᶠ M : ℕ in Filter.atTop, F M ≤ D M := by
filter_upwards [Filter.eventually_gt_atTop 0] with M hM
dsimp only [F, D]
apply Filter.limsup_le_of_le (hf := by
change ∃ b : ℝ≥0∞, ∀ a, (∀ᶠ n : ℕ in Filter.atTop,
sSup {u : ℝ≥0∞ | ∃ h ∈ localIndexSet θ n H,
u = ∫⁻ z, ofReal
((scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z) ^ 2 *
if M < |scaledError (pilotEstimator p ε m select hselect hp hε) θ h n z|
then 1 else 0)
∂transcriptLaw (pilotEstimator p ε m select hselect hp hε)
(localAlternative θ h n) p n} ≤ a) → b ≤ a
exact ⟨0, fun _ _ => bot_le⟩)
filter_upwards [hratio, hn2] with n hnratio hn
apply sSup_le
intro u hu
obtain ⟨h, hh, rfl⟩ := hu
have hfour := pilot_scaledError_fourth_lintegral_le
select θ h p ε m hselect hp hθ hε hsub n hn hh.2 hnratio
exact squaredTail_lintegral_le_of_fourth
(transcriptLaw (pilotEstimator p ε m select hselect hp hε)
(localAlternative θ h n) p n)
(scaledError (pilotEstimator p ε m select hselect hp hε) θ h n)
(M : ℝ) C (by exact_mod_cast hM) hfour
have hD : Tendsto D Filter.atTop (nhds 0) := by
have hcast : Tendsto (fun M : ℕ => (M : ℝ)) Filter.atTop Filter.atTop := tendsto_natCast_atTop_atTop
have hden : Tendsto (fun M : ℕ => (M : ℝ) * (M : ℝ))
Filter.atTop Filter.atTop := by
rw [Filter.tendsto_atTop]
intro b
filter_upwards [(Filter.tendsto_atTop.1 hcast) b,
Filter.eventually_gt_atTop 0] with M hb hM
have hMreal : 1 ≤ (M : ℝ) := by exact_mod_cast hM
nlinarith
have hreal : Tendsto (fun M : ℕ => C / ((M : ℝ) ^ 2))
Filter.atTop (nhds 0) := by
simpa only [pow_two] using tendsto_const_nhds.div_atTop hden
have heq : (fun M : ℕ => ofReal (C / ((M : ℝ) ^ 2))) =ᶠ[
Filter.atTop] D := by
filter_upwards [Filter.eventually_gt_atTop 0] with M hM
dsimp only [D]
rw [ENNReal.ofReal_div_of_pos]
positivity
simpa using (ENNReal.tendsto_ofReal hreal).congr' heq
exact tendsto_of_tendsto_of_tendsto_of_le_of_le'
tendsto_const_nhds hD (Filter.Eventually.of_forall fun _ => bot_le) hFD
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotVarianceConsistency 5 declarations The arguments here use only finite-product second-moment bounds.
Consistency of the pilot plug-in variance
The arguments here use only finite-product second-moment bounds. In particular, they are independent of the triangular-array central limit theorem.
theorem eventProbability_empiricalVariance_sub_oneSecond_le
A finite-row Chebyshev bound for empirical variance around its own one-coordinate second moment. For the displayed inputs and conditions, the stated result follows. Under the stated assumptions, the event Probability empirical Variance sub one Second le.
Formal statement
hypotheses
conclusion
M.eventProbability n (fun ys => δ ≤ |M.empiricalVariance n ys - M.oneSecond n|)
≤ M.B ^ 4 / (M.N n : ℝ) / (δ / 2) ^ 2 + M.B ^ 2 / (M.N n : ℝ) / (δ / 2)
Proof (Lean source)
theorem eventProbability_empiricalVariance_sub_oneSecond_le
(n : ℕ) (hn : 0 < M.N n) (δ : ℝ) (hδ : 0 < δ) :
M.eventProbability n (fun ys =>
δ ≤ |M.empiricalVariance n ys - M.oneSecond n|) ≤
M.B ^ 4 / (M.N n : ℝ) / (δ / 2) ^ 2 +
M.B ^ 2 / (M.N n : ℝ) / (δ / 2) := by
classical
have hhalf : 0 < δ / 2 := half_pos hδ
have hsqrt : 0 < sqrt (δ / 2) := Real.sqrt_pos.2 hhalf
rw [M.eventProbability_eq_designPr]
have hmono : (M.productDesign n).Pr
(fun ys => δ ≤ |M.empiricalVariance n ys - M.oneSecond n|) ≤
(M.productDesign n).Pr (fun ys =>
δ / 2 ≤ |M.empiricalSecond n ys - M.oneSecond n| ∨
sqrt (δ / 2) ≤ |M.empiricalMean n ys|) := by
apply (M.productDesign n).Pr_mono
intro ys hys
by_contra hc
push Not at hc
obtain ⟨hsecond, hmean⟩ := hc
have hmean_sq : (M.empiricalMean n ys) ^ 2 < δ / 2 := by
have hpow := (sq_lt_sq₀ (abs_nonneg (M.empiricalMean n ys)) hsqrt.le).2 hmean
rw [sq_abs, Real.sq_sqrt hhalf.le] at hpow
exact hpow
have htri : |M.empiricalVariance n ys - M.oneSecond n| ≤
|M.empiricalSecond n ys - M.oneSecond n| +
(M.empiricalMean n ys) ^ 2 := by
rw [empiricalVariance]
have heq : M.empiricalSecond n ys - (M.empiricalMean n ys) ^ 2 -
M.oneSecond n =
(M.empiricalSecond n ys - M.oneSecond n) -
(M.empiricalMean n ys) ^ 2 := by ring
rw [heq]
calc
_ ≤ |M.empiricalSecond n ys - M.oneSecond n| +
|(M.empiricalMean n ys) ^ 2| := abs_sub _ _
_ = _ := by
rw [abs_of_nonneg (sq_nonneg (M.empiricalMean n ys))]
linarith
have hEsecond : (M.productDesign n).E (M.empiricalSecond n) =
M.oneSecond n := M.expect_empiricalSecond n hn
have hsecondCheb := (M.productDesign n).chebyshev
(M.empiricalSecond n) hhalf
rw [hEsecond] at hsecondCheb
have hsecondVar : (M.productDesign n).Var (M.empiricalSecond n) =
M.expect n (fun ys =>
(M.empiricalSecond n ys - M.oneSecond n) ^ 2) := by
rw [show (M.productDesign n).Var (M.empiricalSecond n) =
M.expect n (fun ys => (M.empiricalSecond n ys -
(M.productDesign n).E (M.empiricalSecond n)) ^ 2) from rfl,
hEsecond]
rw [hsecondVar] at hsecondCheb
have hsecondBound : (M.productDesign n).Pr
(fun ys => δ / 2 ≤ |M.empiricalSecond n ys - M.oneSecond n|) ≤
M.B ^ 4 / (M.N n : ℝ) / (δ / 2) ^ 2 :=
hsecondCheb.trans
(div_le_div_of_nonneg_right (M.empiricalSecond_mse_le n hn)
(sq_nonneg _))
have hEmean : (M.productDesign n).E (M.empiricalMean n) = 0 := by
change M.expect n (M.empiricalMean n) = 0
simp only [expect, empiricalMean, scoreSum, Finset.mul_sum]
rw [Finset.sum_comm]
have hcoord (i : Fin (M.N n)) :
∑ ys : Fin (M.N n) → Y, M.mass n ys * M.f n (ys i) = 0 := by
change M.expect n (fun ys => M.f n (ys i)) = 0
rw [M.expect_coordinate]
exact M.centered n
calc
(∑ i : Fin (M.N n), ∑ ys : Fin (M.N n) → Y,
M.mass n ys * ((M.N n : ℝ)⁻¹ * M.f n (ys i))) =
∑ i : Fin (M.N n), (M.N n : ℝ)⁻¹ *
(∑ ys : Fin (M.N n) → Y, M.mass n ys * M.f n (ys i)) := by
apply Finset.sum_congr rfl
intro i _
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro ys _
ring
_ = 0 := by simp [hcoord]
have hmeanCheb := (M.productDesign n).chebyshev (M.empiricalMean n) hsqrt
rw [hEmean] at hmeanCheb
simp only [sub_zero, Real.sq_sqrt hhalf.le] at hmeanCheb
have hmeanVar : (M.productDesign n).Var (M.empiricalMean n) =
M.expect n (fun ys => (M.empiricalMean n ys) ^ 2) := by
rw [show (M.productDesign n).Var (M.empiricalMean n) =
M.expect n (fun ys => (M.empiricalMean n ys -
(M.productDesign n).E (M.empiricalMean n)) ^ 2) from rfl, hEmean]
simp
rw [hmeanVar] at hmeanCheb
have hmeanBound : (M.productDesign n).Pr
(fun ys => sqrt (δ / 2) ≤ |M.empiricalMean n ys|) ≤
M.B ^ 2 / (M.N n : ℝ) / (δ / 2) :=
hmeanCheb.trans
(div_le_div_of_nonneg_right (M.empiricalMean_mse_le n hn)
hhalf.le)
calc
_ ≤ (M.productDesign n).Pr
(fun ys => δ / 2 ≤ |M.empiricalSecond n ys - M.oneSecond n|) +
(M.productDesign n).Pr
(fun ys => sqrt (δ / 2) ≤ |M.empiricalMean n ys|) :=
hmono.trans (Pr_or_le
(M.productDesign n) _ _)
_ ≤ M.B ^ 4 / (M.N n : ℝ) / (δ / 2) ^ 2 +
M.B ^ 2 / (M.N n : ℝ) / (δ / 2) :=
add_le_add hsecondBound hmeanBound
def pilotConditionalVarianceBadMass
Conditional main-row probability that the encoded empirical variance misses the oracle variance. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The pilot Conditional Variance Bad Mass is determined by the displayed parameters.
Definition (Lean source)
parameters
θ :
p ε δ :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
hmn :
m n ≤ n
def
pilotConditionalVarianceBadMass select θ p ε δ m hselect hp hε hmn u :
ℝ
given by
η :
fun _ : ℕ => pilotAdaptivePrefixTheta p ε m hmn u
∑ v : Fin (adaptiveMainSize m n) → Fin 14, adaptivePilotRowProductMass select θ (fun _ => 0) η p ε m hselect hp hε n v * if δ < |adaptivePilotRowEmpiricalVariance select θ (fun _ => 0) η p ε m hselect hp hε n v
- Vstar θ p ε| then 1 else 0
def pilotVarianceBadMass
Prefix mixture of the conditional bad-variance probabilities. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The pilot Variance Bad Mass is determined by the displayed parameters.
Definition (Lean source)
parameters
θ :
p ε δ :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
hmn :
m n ≤ n
def
pilotVarianceBadMass select θ p ε δ m hselect hp hε hmn :
ℝ
given by
∑ u : Fin (m n) → Fin 4, pilotPrefixWeight θ p ε m n u * pilotConditionalVarianceBadMass select θ p ε δ m hselect hp hε hmn u
theorem pilotConditionalVarianceBadMass_le
The finite-product Chebyshev bound, specialized to one genuine adaptive main row and centered at its population variance. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Conditional Variance Bad Mass le.
Formal statement
hypotheses
θ :
p ε δ :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hsub :
n :
ℕ
hn :
2 ≤ n
hmn :
m n ≤ n
hδ :
0 < δ
hclose :
|adaptiveScoreVariance select θ (pilotAdaptivePrefixTheta p ε m hmn u) p ε
- Vstar θ p ε| < δ / 2
conclusion
pilotConditionalVarianceBadMass select θ p ε δ m hselect hp hε hmn u
≤ (2 * pilotPhiUpper p ε) ^ 4 / (adaptiveMainSize m n : ℝ) / (δ / 4) ^ 2
+ (2 * pilotPhiUpper p ε) ^ 2 / (adaptiveMainSize m n : ℝ) / (δ / 4)
Proof (Lean source)
lemma pilotConditionalVarianceBadMass_le
(θ : TrialParameter) (p ε δ : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hsub : PilotSublinear m) {n : ℕ} (hn : 2 ≤ n)
(hmn : m n ≤ n) (u : Fin (m n) → Fin 4) (hδ : 0 < δ)
(hclose : |adaptiveScoreVariance select θ
(pilotAdaptivePrefixTheta p ε m hmn u) p ε -
Vstar θ p ε| < δ / 2) :
pilotConditionalVarianceBadMass select θ p ε δ m hselect hp hε hmn u ≤
(2 * pilotPhiUpper p ε) ^ 4 /
(adaptiveMainSize m n : ℝ) / (δ / 4) ^ 2 +
(2 * pilotPhiUpper p ε) ^ 2 /
(adaptiveMainSize m n : ℝ) / (δ / 4) := by
classical
let η₀ := pilotAdaptivePrefixTheta p ε m hmn u
let η : ℕ → TrialParameter := fun k => if k = n then η₀ else θ
have hη₀ : InteriorMeans η₀ := by
exact pilotTheta_interior p ε m (pilotAdaptiveEncode hmn u (fun _ => 0))
have hη : ∀ k, InteriorMeans (η k) := by
intro k
simp only [η]
split_ifs <;> assumption
have hηlim : Tendsto η atTop (nhds θ) := by
apply tendsto_const_nhds.congr'
filter_upwards [eventually_gt_atTop n] with k hk
simp [η, ne_of_gt hk]
let M : RowModel (Fin 14) :=
{ N := adaptiveMainSize m
w := adaptivePilotRowMass select θ (fun _ => 0) η p ε hselect hp hε
f := adaptivePilotRowScore select θ (fun _ => 0) η p ε hselect hp hε
B := 2 * pilotPhiUpper p ε
v₀ := adaptivePilotLimitVariance select θ p ε hselect hp hθ hε
mass_nonneg := adaptivePilotRowMass_nonneg
select θ (fun _ => 0) η p ε hselect hp hθ hη hε
mass_one := adaptivePilotRowMass_sum select θ (fun _ => 0) η p ε hselect hp hη hε
centered := adaptivePilotRowScore_centered
select θ (fun _ => 0) η p ε hselect hp hη hε
bound := adaptivePilotRowScore_abs_le
select θ (fun _ => 0) η p ε hselect hp hθ hη hε
variance_tendsto := adaptivePilotRow_variance_tendsto
select θ (fun _ => 0) η p ε hselect hp hθ hη hε hηlim
variance_pos := adaptivePilotLimitVariance_pos select θ p ε hselect hp hθ hε
length_tendsto := adaptiveMainSize_tendsto_atTop m hsub }
have hN : 0 < M.N n := by
change 0 < adaptiveMainSize m n
rw [adaptiveMainSize, adaptivePilotSize_eq m hsub hn]
exact Nat.sub_pos_of_lt (hsub.2 n hn).2
have hmain : adaptiveMainSize m n = n - m n := by
simp [adaptiveMainSize, adaptivePilotSize_eq m hsub hn]
have hzero : adaptiveTrueParam θ (fun _ => 0) n = θ := by
have hlocal : localAlternative θ (fun _ => 0) n = θ := by
funext k
simp [localAlternative]
rw [adaptiveTrueParam_eq_localAlternative]
· exact hlocal
· simpa only [hlocal] using hθ
have honeSecond : M.oneSecond n =
adaptiveScoreVariance select θ η₀ p ε := by
change (∑ s : Fin 14,
adaptivePilotRowMass select θ (fun _ => 0) η p ε hselect hp hε n s *
(adaptivePilotRowScore select θ (fun _ => 0) η p ε hselect hp hε n s) ^ 2) = _
rw [adaptivePilotRow_secondMoment_eq
select θ (fun _ => 0) η p ε hselect hp hη hε n, hzero]
simp only [η, if_pos, η₀]
have hrow (v : Fin (adaptiveMainSize m n) → Fin 14) :
M.empiricalVariance n v =
adaptivePilotRowEmpiricalVariance select θ (fun _ => 0)
(fun _ => η₀) p ε m hselect hp hε n v := by
simp only [M, RowModel.empiricalVariance, RowModel.empiricalSecond,
RowModel.empiricalMean, RowModel.scoreSum,
adaptivePilotRowEmpiricalVariance, adaptivePilotRowEmpiricalSecond,
adaptivePilotRowEmpiricalMean, adaptivePilotRowScore, η]
simp
have hmass (v : Fin (adaptiveMainSize m n) → Fin 14) :
adaptivePilotRowProductMass select θ (fun _ => 0) (fun _ => η₀)
p ε m hselect hp hε n v = M.mass n v := by
simp only [M, RowModel.mass, adaptivePilotRowProductMass,
adaptivePilotRowMass, η, if_pos]
have hmono : pilotConditionalVarianceBadMass select θ p ε δ m hselect hp hε hmn u ≤
(M.productDesign n).Pr (fun v =>
δ / 2 ≤ |M.empiricalVariance n v - M.oneSecond n|) := by
unfold pilotConditionalVarianceBadMass
dsimp only
rw [Pr,
M.productDesign_expect]
unfold RowModel.expect ind
change (∑ v : Fin (adaptiveMainSize m n) → Fin 14,
adaptivePilotRowProductMass select θ (fun _ => 0) (fun _ => η₀)
p ε m hselect hp hε n v *
(if δ < |adaptivePilotRowEmpiricalVariance select θ (fun _ => 0)
(fun _ => η₀) p ε m hselect hp hε n v - Vstar θ p ε| then 1 else 0)) ≤ _
apply Finset.sum_le_sum
intro v _
rw [hmass v]
have hind :
(if δ < |adaptivePilotRowEmpiricalVariance select θ (fun _ => 0)
(fun _ => η₀) p ε m hselect hp hε n v - Vstar θ p ε|
then (1 : ℝ) else 0) ≤
if δ / 2 ≤ |M.empiricalVariance n v - M.oneSecond n|
then (1 : ℝ) else 0 := by
by_cases hv : δ < |adaptivePilotRowEmpiricalVariance select θ (fun _ => 0)
(fun _ => η₀) p ε m hselect hp hε n v - Vstar θ p ε|
· have hgood : δ / 2 ≤ |M.empiricalVariance n v - M.oneSecond n| := by
rw [hrow v, honeSecond]
have htri := abs_sub_le
(adaptivePilotRowEmpiricalVariance select θ (fun _ => 0)
(fun _ => η₀) p ε m hselect hp hε n v)
(adaptiveScoreVariance select θ η₀ p ε) (Vstar θ p ε)
by_contra hc
have hc' : |adaptivePilotRowEmpiricalVariance select θ (fun _ => 0)
(fun _ => η₀) p ε m hselect hp hε n v -
adaptiveScoreVariance select θ η₀ p ε| < δ / 2 :=
lt_of_not_ge hc
linarith
simp [hv, hgood]
· by_cases hg : δ / 2 ≤ |M.empiricalVariance n v - M.oneSecond n|
· simp [hv, hg]
· simp [hv, hg]
exact mul_le_mul_of_nonneg_left hind (M.mass_nonnegative n v)
rw [← M.eventProbability_eq_designPr] at hmono
have hbound := M.eventProbability_empiricalVariance_sub_oneSecond_le
n hN (δ / 2) (half_pos hδ)
have hrhs : M.B ^ 4 / (M.N n : ℝ) / (δ / 2 / 2) ^ 2 +
M.B ^ 2 / (M.N n : ℝ) / (δ / 2 / 2) =
(2 * pilotPhiUpper p ε) ^ 4 /
(adaptiveMainSize m n : ℝ) / (δ / 4) ^ 2 +
(2 * pilotPhiUpper p ε) ^ 2 /
(adaptiveMainSize m n : ℝ) / (δ / 4) := by
dsimp only [M]
ring
exact (hmono.trans hbound).trans_eq hrhs
theorem pilotVarianceBadMass_tendsto_zero
Averaging the uniform finite-row bound over the genuine pilot prefixes gives vanishing bad variance mass. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Variance Bad Mass tendsto zero.
Formal statement
hypotheses
θ :
p ε δ :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hdiv :
hsub :
hδ :
0 < δ
conclusion
Tendsto (fun n => if hn : 2 ≤ n then pilotVarianceBadMass select θ p ε δ m hselect hp hε (le_of_lt (hsub.2 n hn).2) else 0) atTop (nhds 0)
Proof (Lean source)
lemma pilotVarianceBadMass_tendsto_zero
(θ : TrialParameter) (p ε δ : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hdiv : PilotDiverges m) (hsub : PilotSublinear m) (hδ : 0 < δ) :
Tendsto (fun n => if hn : 2 ≤ n then
pilotVarianceBadMass select θ p ε δ m hselect hp hε
(le_of_lt (hsub.2 n hn).2) else 0) atTop (nhds 0) := by
obtain ⟨R, hR, hcont⟩ := adaptiveScoreVariance_uniform_near_diag
select θ p ε hselect hp hθ hε (δ / 2) (half_pos hδ)
let M₀ := adaptiveInteriorMargin θ
have hM₀ : 0 < M₀ := adaptiveInteriorMargin_pos θ hθ
let r := min (R / 2) (M₀ / 4)
have hr : 0 < r := by dsimp [r]; positivity
have hrR : r ≤ R / 2 := min_le_left _ _
have hrM : r ≤ M₀ / 4 := min_le_right _ _
let rowEnvelope : ℕ → ℝ := fun n =>
(2 * pilotPhiUpper p ε) ^ 4 /
(adaptiveMainSize m n : ℝ) / (δ / 4) ^ 2 +
(2 * pilotPhiUpper p ε) ^ 2 /
(adaptiveMainSize m n : ℝ) / (δ / 4)
have hrowEnv : Tendsto rowEnvelope atTop (nhds 0) := by
have hlen := adaptiveMainSize_tendsto_atTop m hsub
have hfirst := (tendsto_const_div_atTop_nhds_zero_nat
((2 * pilotPhiUpper p ε) ^ 4)).comp hlen
have hsecond := (tendsto_const_div_atTop_nhds_zero_nat
((2 * pilotPhiUpper p ε) ^ 2)).comp hlen
dsimp only [rowEnvelope]
simpa only [Function.comp_def, zero_div, add_zero] using
(hfirst.div_const ((δ / 4) ^ 2)).add
(hsecond.div_const (δ / 4))
have hreg := pilotRegularizer_eventually_lt m hdiv (M₀ / 2) (by positivity)
have hbadEnv := pilotBadMassEnvelope_tendsto_zero
p ε r m hp hε hr hdiv
have hbound : ∀ᶠ n in atTop,
(if hn : 2 ≤ n then pilotVarianceBadMass select θ p ε δ m hselect hp hε
(le_of_lt (hsub.2 n hn).2) else 0) ≤
pilotBadMassEnvelope p ε r m n + rowEnvelope n := by
filter_upwards [eventually_ge_atTop 2, hreg] with n hn hnreg
let hmn : m n ≤ n := le_of_lt (hsub.2 n hn).2
rw [dif_pos hn]
have hproof : le_of_lt (hsub.2 n hn).2 = hmn := Subsingleton.elim _ _
rw [show le_of_lt (hsub.2 n hn).2 = hmn from hproof]
have hmpos : 0 < m n := lt_of_lt_of_le Nat.zero_lt_one (hsub.2 n hn).1
have hM0l : M₀ ≤ θ 0 := (min_le_left _ _).trans (min_le_left _ _)
have hM0r : M₀ ≤ 1 - θ 0 := (min_le_left _ _).trans (min_le_right _ _)
have hM1l : M₀ ≤ θ 1 := (min_le_right _ _).trans (min_le_left _ _)
have hM1r : M₀ ≤ 1 - θ 1 := (min_le_right _ _).trans (min_le_right _ _)
have hδ0l : (m n + 2 : ℝ)⁻¹ + r ≤ θ 0 := by
dsimp only [M₀] at hnreg hrM hM0l
linarith
have hδ0r : (m n + 2 : ℝ)⁻¹ + r ≤ 1 - θ 0 := by
dsimp only [M₀] at hnreg hrM hM0r
linarith
have hδ1l : (m n + 2 : ℝ)⁻¹ + r ≤ θ 1 := by
dsimp only [M₀] at hnreg hrM hM1l
linarith
have hδ1r : (m n + 2 : ℝ)⁻¹ + r ≤ 1 - θ 1 := by
dsimp only [M₀] at hnreg hrM hM1r
linarith
have hbad := pilotPrefixBadMass_le select θ p ε r m hselect hp hθ hε hr hmn hmpos
hδ0l hδ0r hδ1l hδ1r
have hw (u : Fin (m n) → Fin 4) :
0 ≤ pilotPrefixWeight θ p ε m n u :=
pilotPrefixWeight_nonneg θ p ε m n hp hθ u
have hcond (u : Fin (m n) → Fin 4) :
pilotConditionalVarianceBadMass select θ p ε δ m hselect hp hε hmn u ≤
(if r ≤ |pilotAdaptivePrefixTheta p ε m hmn u 0 - θ 0| ∨
r ≤ |pilotAdaptivePrefixTheta p ε m hmn u 1 - θ 1|
then 1 else rowEnvelope n) := by
by_cases hu : r ≤ |pilotAdaptivePrefixTheta p ε m hmn u 0 - θ 0| ∨
r ≤ |pilotAdaptivePrefixTheta p ε m hmn u 1 - θ 1|
· rw [if_pos hu]
let η := fun _ : ℕ => pilotAdaptivePrefixTheta p ε m hmn u
have hη : ∀ k, InteriorMeans (η k) := fun _ =>
pilotTheta_interior p ε m (pilotAdaptiveEncode hmn u (fun _ => 0))
unfold pilotConditionalVarianceBadMass
dsimp only
calc
_ ≤ ∑ v : Fin (adaptiveMainSize m n) → Fin 14,
adaptivePilotRowProductMass select θ (fun _ => 0) η
p ε m hselect hp hε n v := by
apply Finset.sum_le_sum
intro v _
apply mul_le_of_le_one_right
· exact adaptivePilotRowProductMass_nonneg
select θ (fun _ => 0) η p ε m hselect hp hθ hη hε n v
· split_ifs <;> norm_num
_ = 1 := adaptivePilotRowProductMass_sum
select θ (fun _ => 0) η p ε m hselect hp hη hε n
· rw [if_neg hu]
push Not at hu
have hηdist : dist (pilotAdaptivePrefixTheta p ε m hmn u) θ < R := by
rw [dist_pi_lt_iff hR]
intro k
have hrltR : r < R := lt_of_le_of_lt hrR (by linarith)
fin_cases k
· rw [Real.dist_eq]
change |pilotAdaptivePrefixTheta p ε m hmn u 0 - θ 0| < R
linarith [hu.1]
· rw [Real.dist_eq]
change |pilotAdaptivePrefixTheta p ε m hmn u 1 - θ 1| < R
linarith [hu.2]
have hclose : |adaptiveScoreVariance select θ
(pilotAdaptivePrefixTheta p ε m hmn u) p ε -
Vstar θ p ε| < δ / 2 := by
exact hcont θ (pilotAdaptivePrefixTheta p ε m hmn u)
(by simpa using hR) hηdist
exact pilotConditionalVarianceBadMass_le
select θ p ε δ m hselect hp hθ hε hsub hn hmn u hδ hclose
unfold pilotVarianceBadMass
calc
_ ≤ ∑ u : Fin (m n) → Fin 4,
pilotPrefixWeight θ p ε m n u *
(if r ≤ |pilotAdaptivePrefixTheta p ε m hmn u 0 - θ 0| ∨
r ≤ |pilotAdaptivePrefixTheta p ε m hmn u 1 - θ 1|
then 1 else rowEnvelope n) :=
Finset.sum_le_sum fun u _ => mul_le_mul_of_nonneg_left (hcond u) (hw u)
_ ≤ ∑ u : Fin (m n) → Fin 4,
pilotPrefixWeight θ p ε m n u *
((if r ≤ |pilotAdaptivePrefixTheta p ε m hmn u 0 - θ 0| ∨
r ≤ |pilotAdaptivePrefixTheta p ε m hmn u 1 - θ 1|
then 1 else 0) + rowEnvelope n) := by
apply Finset.sum_le_sum
intro u _
apply mul_le_mul_of_nonneg_left _ (hw u)
have hrow0 : 0 ≤ rowEnvelope n := by
dsimp only [rowEnvelope]
positivity
split_ifs <;> linarith
_ = pilotPrefixBadMass θ p ε r m hmn + rowEnvelope n := by
simp_rw [mul_add]
rw [Finset.sum_add_distrib, ← Finset.sum_mul,
pilotPrefixWeight_sum θ p ε m n, one_mul]
rfl
_ ≤ pilotBadMassEnvelope p ε r m n + rowEnvelope n :=
add_le_add hbad le_rfl
have hnonneg : ∀ n, 0 ≤ (if hn : 2 ≤ n then
pilotVarianceBadMass select θ p ε δ m hselect hp hε
(le_of_lt (hsub.2 n hn).2) else 0) := by
intro n
by_cases hn : 2 ≤ n
· rw [dif_pos hn]
let hmn : m n ≤ n := le_of_lt (hsub.2 n hn).2
have hproof : le_of_lt (hsub.2 n hn).2 = hmn := Subsingleton.elim _ _
rw [show le_of_lt (hsub.2 n hn).2 = hmn from hproof]
unfold pilotVarianceBadMass pilotConditionalVarianceBadMass
exact Finset.sum_nonneg fun u _ => mul_nonneg
(pilotPrefixWeight_nonneg θ p ε m n hp hθ u)
(Finset.sum_nonneg fun v _ => mul_nonneg
(adaptivePilotRowProductMass_nonneg select θ (fun _ => 0)
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε m hselect hp hθ (fun _ => pilotTheta_interior p ε m
(pilotAdaptiveEncode hmn u (fun _ => 0))) hε n v)
(by split_ifs <;> positivity))
· simp [hn]
exact squeeze_zero' (Eventually.of_forall hnonneg) hbound
(by simpa only [add_zero] using hbadEnv.add hrowEnv)
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotVhatConsistency 5 declarations Transcript-law wrapper for pilot variance consistency
Transcript-law wrapper for pilot variance consistency
theorem finite_empiricalVariance_cast_eq
Empirical variance is invariant under transport along an equality of finite row lengths. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
-- keep: reusable finite-pilot construction, law, localization, or limit API for related adaptive procedures
lemma finite_empiricalVariance_cast_eq {N K : ℕ} (h : N = K)
(q : Fin K → ℝ) :
(N : ℝ)⁻¹ * ∑ i : Fin N, (q (cast h i)) ^ 2 -
((N : ℝ)⁻¹ * ∑ i : Fin N, q (cast h i)) ^ 2 =
(K : ℝ)⁻¹ * ∑ j : Fin K, (q j) ^ 2 -
((K : ℝ)⁻¹ * ∑ j : Fin K, q j) ^ 2 := by
subst K
rfl
theorem finite_average_sq_deviation_cast_eq
Average squared deviations are invariant under transport along an equality of finite row lengths. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
theorem VhatStar_pilotAdaptiveEncode_eq_adaptiveEmpiricalVariance
On an encoded row, VhatStar equals the adaptive empirical variance after the canonical main-size reindexing. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the Vhat Star pilot Adaptive Encode eq adaptive Empirical Variance.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hε :
0 < ε
n :
ℕ
hmn :
m n ≤ n
hN :
0 < adaptiveMainSize m n
hmain :
adaptiveMainSize m n = n - m n
conclusion
VhatStar p ε m (select) (pilotAdaptiveEncode hmn u v)
= adaptivePilotRowEmpiricalVariance select θ (fun _ => 0) (fun _ => pilotAdaptivePrefixTheta p ε m hmn u) p ε m hselect hp hε n (fun i => v (cast hmain i))
Proof (Lean source)
lemma VhatStar_pilotAdaptiveEncode_eq_adaptiveEmpiricalVariance
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hε : 0 < ε) {n : ℕ}
(hmn : m n ≤ n) (hN : 0 < adaptiveMainSize m n)
(hmain : adaptiveMainSize m n = n - m n)
(u : Fin (m n) → Fin 4) (v : Fin (n - m n) → Fin 14) :
VhatStar p ε m (select)
(pilotAdaptiveEncode hmn u v) =
adaptivePilotRowEmpiricalVariance select θ (fun _ => 0)
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε m hselect hp hε n (fun i => v (cast hmain i)) := by
rw [VhatStar_pilotAdaptiveEncode select p ε m hselect hp hε hmn u v]
rw [adaptivePilotRowEmpiricalVariance_eq_selected
select θ (fun _ => 0) (fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε m hselect hp hε n hN]
exact (finite_average_sq_deviation_cast_eq hmain
(fun j => adaptiveSelectedScore select (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j))).symm
theorem transcriptLaw_VhatStar_bad_eq_ofReal_pilotVarianceBadMass
The actual bad-variance transcript event is exactly the real prefix mixture proved consistent in PilotVarianceConsistency. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the transcript Law Vhat Star bad eq of Real pilot Variance Bad Mass.
Formal statement
hypotheses
θ :
p ε δ :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hsub :
n :
ℕ
hn :
2 ≤ n
conclusion
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) {z | δ < |VhatStar p ε m (select) z - Vstar θ p ε|}
Proof (Lean source)
lemma transcriptLaw_VhatStar_bad_eq_ofReal_pilotVarianceBadMass
(θ : TrialParameter) (p ε δ : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hsub : PilotSublinear m) {n : ℕ} (hn : 2 ≤ n) :
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
{z | δ < |VhatStar p ε m (select) z -
Vstar θ p ε|} =
ofReal (pilotVarianceBadMass select θ p ε δ m hselect hp hε
(le_of_lt (hsub.2 n hn).2)) := by
classical
let hmn : m n ≤ n := le_of_lt (hsub.2 n hn).2
let μ := transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n
let A : Set (Transcript (pilotOutputFamily n)) :=
{z | δ < |VhatStar p ε m (select) z - Vstar θ p ε|}
let e := fun uv : (Fin (m n) → Fin 4) × (Fin (n - m n) → Fin 14) =>
pilotAdaptiveEncode hmn uv.1 uv.2
have hN : 0 < adaptiveMainSize m n := by
rw [adaptiveMainSize, adaptivePilotSize_eq m hsub hn]
exact Nat.sub_pos_of_lt (hsub.2 n hn).2
have hmain : adaptiveMainSize m n = n - m n := by
simp [adaptiveMainSize, adaptivePilotSize_eq m hsub hn]
have hzero : adaptiveTrueParam θ (fun _ => 0) n = θ := by
have hlocal : localAlternative θ (fun _ => 0) n = θ := by
funext k
simp [localAlternative]
rw [adaptiveTrueParam_eq_localAlternative]
· exact hlocal
· simpa only [hlocal] using hθ
have hw (u : Fin (m n) → Fin 4) : 0 ≤ pilotPrefixWeight θ p ε m n u :=
pilotPrefixWeight_nonneg θ p ε m n hp hθ u
have hm (u : Fin (m n) → Fin 4) (v : Fin (n - m n) → Fin 14) :
0 ≤ ∏ j, adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j) := by
apply Finset.prod_nonneg
intro j _
exact adaptiveMainMass_nonneg select θ _ p ε hselect hp hθ
(pilotTheta_interior p ε m (pilotAdaptiveEncode hmn u (fun _ => 0)))
hε (v j)
have hatom (u : Fin (m n) → Fin 4) (v : Fin (n - m n) → Fin 14) :
μ {e (u, v)} = ofReal
(pilotPrefixWeight θ p ε m n u *
∏ j, adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)) := by
rw [pilotAdaptive_joint_singleton_eq_prefix_mul_mainProduct
select θ p ε m hselect hp hθ hε hmn u v]
change ofReal (pilotPrefixWeight θ p ε m n u) *
ofReal (∏ j, adaptiveMainMass select θ
(pilotAdaptivePrefixTheta p ε m hmn u) p ε (v j)) = _
exact (ENNReal.ofReal_mul (hw u)).symm
have hA : MeasurableSet A := MeasurableSet.of_discrete
rw [show μ A = ∫⁻ z, A.indicator (fun _ => (1 : ENNReal)) z ∂μ by
rw [lintegral_indicator hA, setLIntegral_one]]
rw [MeasureTheory.lintegral_fintype]
rw [sum_eq_sum_image_of_zero_off_range e (pilotAdaptiveEncode_pair_injective hmn)]
· rw [Fintype.sum_prod_type]
simp_rw [hatom]
have hind (u : Fin (m n) → Fin 4) (v : Fin (n - m n) → Fin 14) :
A.indicator (fun _ => (1 : ENNReal)) (e (u, v)) =
if δ < |adaptivePilotRowEmpiricalVariance select θ (fun _ => 0)
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε m hselect hp hε n (fun i => v (cast hmain i)) - Vstar θ p ε|
then 1 else 0 := by
rw [Set.indicator_apply]
simp only [A, Set.mem_ofPred_eq, e,
VhatStar_pilotAdaptiveEncode_eq_adaptiveEmpiricalVariance
select θ p ε m hselect hp hε hmn hN hmain u v]
simp_rw [hind]
have hindOfReal (u : Fin (m n) → Fin 4) (v : Fin (n - m n) → Fin 14) :
(if δ < |adaptivePilotRowEmpiricalVariance select θ (fun _ => 0)
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε m hselect hp hε n (fun i => v (cast hmain i)) - Vstar θ p ε|
then (1 : ENNReal) else 0) =
ofReal (if δ < |adaptivePilotRowEmpiricalVariance select θ (fun _ => 0)
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε m hselect hp hε n (fun i => v (cast hmain i)) - Vstar θ p ε|
then (1 : ℝ) else 0) := by
split_ifs <;> simp
simp_rw [hindOfReal]
have hmul (u : Fin (m n) → Fin 4) (v : Fin (n - m n) → Fin 14) :
ofReal (if δ < |adaptivePilotRowEmpiricalVariance select θ (fun _ => 0)
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε m hselect hp hε n (fun i => v (cast hmain i)) - Vstar θ p ε|
then (1 : ℝ) else 0) *
ofReal (pilotPrefixWeight θ p ε m n u *
∏ j, adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)) =
ofReal ((if δ < |adaptivePilotRowEmpiricalVariance select θ (fun _ => 0)
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε m hselect hp hε n (fun i => v (cast hmain i)) - Vstar θ p ε|
then (1 : ℝ) else 0) *
(pilotPrefixWeight θ p ε m n u *
∏ j, adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j))) := by
exact (ENNReal.ofReal_mul (by split_ifs <;> positivity)).symm
simp_rw [hmul]
have hofReal_sum {B : Type} [Fintype B] (f : B → ℝ)
(hf : ∀ b, 0 ≤ f b) :
(∑ b, ofReal (f b)) = ofReal (∑ b, f b) := by
simpa using (ENNReal.ofReal_sum_of_nonneg
(s := univ) (fun b _ => hf b)).symm
calc
(∑ u : Fin (m n) → Fin 4, ∑ v : Fin (n - m n) → Fin 14,
ofReal ((if δ < |adaptivePilotRowEmpiricalVariance select θ (fun _ => 0)
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε m hselect hp hε n (fun i => v (cast hmain i)) - Vstar θ p ε|
then (1 : ℝ) else 0) *
(pilotPrefixWeight θ p ε m n u *
∏ j, adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j)))) =
ofReal (∑ u : Fin (m n) → Fin 4,
∑ v : Fin (n - m n) → Fin 14,
(if δ < |adaptivePilotRowEmpiricalVariance select θ (fun _ => 0)
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε m hselect hp hε n (fun i => v (cast hmain i)) - Vstar θ p ε|
then (1 : ℝ) else 0) *
(pilotPrefixWeight θ p ε m n u *
∏ j, adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v j))) := by
rw [show (∑ u : Fin (m n) → Fin 4,
∑ v : Fin (n - m n) → Fin 14,
ofReal ((if δ < |adaptivePilotRowEmpiricalVariance select θ
(fun _ => 0) (fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε m hselect hp hε n (fun i => v (cast hmain i)) - Vstar θ p ε|
then (1 : ℝ) else 0) *
(pilotPrefixWeight θ p ε m n u *
∏ j, adaptiveMainMass select θ
(pilotAdaptivePrefixTheta p ε m hmn u) p ε (v j)))) =
∑ u : Fin (m n) → Fin 4, ofReal
(∑ v : Fin (n - m n) → Fin 14,
(if δ < |adaptivePilotRowEmpiricalVariance select θ (fun _ => 0)
(fun _ => pilotAdaptivePrefixTheta p ε m hmn u)
p ε m hselect hp hε n (fun i => v (cast hmain i)) - Vstar θ p ε|
then (1 : ℝ) else 0) *
(pilotPrefixWeight θ p ε m n u *
∏ j, adaptiveMainMass select θ
(pilotAdaptivePrefixTheta p ε m hmn u) p ε (v j))) by
apply Finset.sum_congr rfl
intro u _
exact hofReal_sum _ (fun v => mul_nonneg
(by split_ifs <;> positivity) (mul_nonneg (hw u) (hm u v)))]
exact hofReal_sum _ (fun u => Finset.sum_nonneg fun v _ => mul_nonneg
(by split_ifs <;> positivity) (mul_nonneg (hw u) (hm u v)))
_ = ofReal (pilotVarianceBadMass select θ p ε δ m hselect hp hε
(le_of_lt (hsub.2 n hn).2)) := by
congr 1
rw [show hmn = le_of_lt (hsub.2 n hn).2 from Subsingleton.elim _ _]
unfold pilotVarianceBadMass pilotConditionalVarianceBadMass
apply Finset.sum_congr rfl
intro u _
rw [Finset.mul_sum]
let rowEquiv : (Fin (adaptiveMainSize m n) → Fin 14) ≃
(Fin (n - m n) → Fin 14) :=
Equiv.piCongrLeft (fun _ : Fin (n - m n) => Fin 14) (finCongr hmain)
symm
apply Fintype.sum_equiv rowEquiv
intro v
have happly (i : Fin (adaptiveMainSize m n)) :
rowEquiv v (cast hmain i) = v i := by
rfl
have happly' (i : Fin (adaptiveMainSize m n)) :
rowEquiv v ((finCongr hmain) i) = v i := by
change rowEquiv v (cast hmain i) = v i
exact happly i
have hfun : (fun i => rowEquiv v (cast hmain i)) = v :=
funext happly
have hprod :
(∏ i : Fin (adaptiveMainSize m n),
adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (v i)) =
∏ j : Fin (n - m n),
adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (rowEquiv v j) := by
simpa only [happly'] using (Equiv.prod_comp (finCongr hmain)
(fun j : Fin (n - m n) =>
adaptiveMainMass select θ (pilotAdaptivePrefixTheta p ε m hmn u)
p ε (rowEquiv v j)))
simp only [adaptivePilotRowProductMass, adaptivePilotRowMass, hzero,
hfun, hprod]
ring
· intro z hz
have hzμ := pilotAdaptive_singleton_zero_of_not_range
select θ p ε m hselect hp hε hmn z hz
apply mul_eq_zero.mpr
right
simpa only [μ] using hzμ
theorem pilot_VhatStar_consistent
Under the pilot growth and interiority conditions, the probability under the actual transcript law that VhatStar differs from Vstar by more than any positive tolerance tends to zero. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot Vhat Star consistent.
Formal statement
hypotheses
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hdiv :
hsub :
δ :
ℝ
0 < δ
Proof (Lean source)
lemma pilot_VhatStar_consistent
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hdiv : PilotDiverges m) (hsub : PilotSublinear m) :
∀ δ : ℝ, 0 < δ → Tendsto (fun n =>
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
{z | δ < |VhatStar p ε m (select) z -
Vstar θ p ε|}) atTop (nhds 0) := by
intro δ hδ
have hmass := pilotVarianceBadMass_tendsto_zero
select θ p ε δ m hselect hp hθ hε hdiv hsub hδ
have hof := ENNReal.continuous_ofReal.continuousAt.tendsto.comp hmass
simpa only [ENNReal.ofReal_zero] using hof.congr' (by
filter_upwards [eventually_ge_atTop 2] with n hn
rw [Function.comp_apply, dif_pos hn,
transcriptLaw_VhatStar_bad_eq_ofReal_pilotVarianceBadMass
select θ p ε δ m hselect hp hθ hε hsub hn])
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotWaldLimitSupport 3 declarations The standard studentization API uses one fixed sample space.
Varying-law support for pilot Wald limits
The standard studentization API uses one fixed sample space. Pilot transcripts change with the sample size, so this file supplies the corresponding CDF-level converging-together and interval-probability adapters.
theorem varyingLaw_cdf_converging_together
CDF-level converging together for row-varying probability spaces. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the varying Law cdf converging together.
Formal statement
hypotheses
Ω :
ℕ → Type*
∀ n, MeasurableSpace (Ω n)
μ :
∀ n, Measure (Ω n)
∀ n, IsProbabilityMeasure (μ n)
S T :
∀ n
if
Ω n
then
ℝ
Φ :
ℝ → ℝ
hΦ :
x :
ℝ
Proof (Lean source)
lemma varyingLaw_cdf_converging_together
{Ω : ℕ → Type*} [∀ n, MeasurableSpace (Ω n)]
(μ : ∀ n, Measure (Ω n)) [∀ n, IsProbabilityMeasure (μ n)]
(S T : ∀ n, Ω n → ℝ) (Φ : ℝ → ℝ)
(hApprox : ∀ η : ℝ, 0 < η →
Tendsto (fun n => (μ n).real {z | η ≤ |S n z - T n z|})
atTop (nhds 0))
(hT : ∀ x : ℝ,
Tendsto (fun n => (μ n).real {z | T n z ≤ x}) atTop (nhds (Φ x)))
(hΦ : Continuous Φ) :
∀ x : ℝ,
Tendsto (fun n => (μ n).real {z | S n z ≤ x}) atTop (nhds (Φ x)) := by
intro x
rw [Metric.tendsto_atTop]
intro ε hε
have hε4 : 0 < ε / 4 := by linarith
obtain ⟨η, hηpos, hη⟩ :=
(Metric.continuousAt_iff.mp hΦ.continuousAt) (ε / 4) hε4
let γ : ℝ := η / 2
have hγpos : 0 < γ := by dsimp [γ]; linarith
have hγ_lt_η : γ < η := by dsimp [γ]; linarith
have hΦplus : dist (Φ (x + γ)) (Φ x) < ε / 4 := by
apply hη
rw [Real.dist_eq]
simpa [abs_of_pos hγpos] using hγ_lt_η
have hΦminus : dist (Φ (x - γ)) (Φ x) < ε / 4 := by
apply hη
rw [Real.dist_eq]
simpa [abs_of_pos hγpos] using hγ_lt_η
rw [Real.dist_eq] at hΦplus hΦminus
have hR := hApprox γ hγpos
have hTplus := hT (x + γ)
have hTminus := hT (x - γ)
have hRev : ∀ᶠ n in atTop,
(μ n).real {z | γ ≤ |S n z - T n z|} < ε / 4 := by
have h := (Metric.tendsto_nhds.mp hR) (ε / 4) hε4
filter_upwards [h] with n hn
rw [Real.dist_eq, sub_zero,
abs_of_nonneg MeasureTheory.measureReal_nonneg] at hn
exact hn
have hTplusEv : ∀ᶠ n in atTop,
(μ n).real {z | T n z ≤ x + γ} < Φ (x + γ) + ε / 4 := by
have h := (Metric.tendsto_nhds.mp hTplus) (ε / 4) hε4
filter_upwards [h] with n hn
rw [Real.dist_eq] at hn
linarith [(abs_lt.mp hn).2]
have hTminusEv : ∀ᶠ n in atTop,
Φ (x - γ) - ε / 4 < (μ n).real {z | T n z ≤ x - γ} := by
have h := (Metric.tendsto_nhds.mp hTminus) (ε / 4) hε4
filter_upwards [h] with n hn
rw [Real.dist_eq] at hn
linarith [(abs_lt.mp hn).1]
refine Filter.eventually_atTop.1 ?_
filter_upwards [hRev, hTplusEv, hTminusEv] with n hRn hTpn hTmn
rw [Real.dist_eq, abs_sub_lt_iff]
constructor
· have hmono :
(μ n).real {z | S n z ≤ x} ≤
(μ n).real ({z | T n z ≤ x + γ} ∪
{z | γ ≤ |S n z - T n z|}) := by
apply MeasureTheory.measureReal_mono (h₂ := measure_ne_top _ _)
intro z hz
change S n z ≤ x at hz
by_cases hfar : γ ≤ |S n z - T n z|
· exact inr hfar
· apply inl
rw [not_le, abs_sub_lt_iff] at hfar
change T n z ≤ x + γ
linarith
have hunion := MeasureTheory.measureReal_union_le (μ := μ n)
{z | T n z ≤ x + γ} {z | γ ≤ |S n z - T n z|}
linarith [hmono.trans hunion, (abs_lt.mp hΦplus).2]
· have hmono :
(μ n).real {z | T n z ≤ x - γ} ≤
(μ n).real ({z | S n z ≤ x} ∪
{z | γ ≤ |S n z - T n z|}) := by
apply MeasureTheory.measureReal_mono (h₂ := measure_ne_top _ _)
intro z hz
change T n z ≤ x - γ at hz
by_cases hfar : γ ≤ |S n z - T n z|
· exact inr hfar
· apply inl
rw [not_le, abs_sub_lt_iff] at hfar
change S n z ≤ x
linarith
have hunion := MeasureTheory.measureReal_union_le (μ := μ n)
{z | S n z ≤ x} {z | γ ≤ |S n z - T n z|}
linarith [hmono.trans hunion, (abs_lt.mp hΦminus).1]
theorem varyingLaw_Icc_tendsto_of_cdf_and_approx
A negligible row-varying perturbation of a statistic with continuous-limit CDF has the expected probability limit on every continuity interval. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the varying Law Icc tendsto of cdf and approx.
Formal statement
hypotheses
Ω :
ℕ → Type*
∀ n, MeasurableSpace (Ω n)
μ :
∀ n, Measure (Ω n)
∀ n, IsProbabilityMeasure (μ n)
S T :
∀ n
if
Ω n
then
ℝ
Q :
hS :
∀ n, Measurable (S n)
hQcdf :
Continuous (cdf Q)
a b :
ℝ
hboundary :
Q (frontier (Icc a b)) = 0
Proof (Lean source)
lemma varyingLaw_Icc_tendsto_of_cdf_and_approx
{Ω : ℕ → Type*} [∀ n, MeasurableSpace (Ω n)]
(μ : ∀ n, Measure (Ω n)) [∀ n, IsProbabilityMeasure (μ n)]
(S T : ∀ n, Ω n → ℝ) (Q : Measure ℝ) [IsProbabilityMeasure Q]
(hS : ∀ n, Measurable (S n))
(hApprox : ∀ η : ℝ, 0 < η →
Tendsto (fun n => (μ n).real {z | η ≤ |S n z - T n z|})
atTop (nhds 0))
(hT : ∀ x : ℝ,
Tendsto (fun n => (μ n).real {z | T n z ≤ x})
atTop (nhds (Q.real (Iic x))))
(hQcdf : Continuous (cdf Q)) {a b : ℝ}
(hboundary : Q (frontier (Icc a b)) = 0) :
Tendsto (fun n => (μ n).real {z | S n z ∈ Icc a b})
atTop (nhds (Q.real (Icc a b))) := by
let νs : ℕ → ProbabilityMeasure ℝ := fun n =>
⟨Measure.map (S n) (μ n), Measure.isProbabilityMeasure_map (hS n).aemeasurable⟩
let ν : ProbabilityMeasure ℝ := ⟨Q, inferInstance⟩
have hScdf (x : ℝ) :
Tendsto (fun n => (cdf (νs n : Measure ℝ) x : ℝ)) atTop
(nhds (cdf (ν : Measure ℝ) x : ℝ)) := by
have hraw := varyingLaw_cdf_converging_together μ S T
(fun x => Q.real (Iic x)) hApprox hT (by
simpa only [← ProbabilityTheory.cdf_eq_real] using hQcdf)
have htarget : (cdf (ν : Measure ℝ) x : ℝ) = Q.real (Iic x) := by
rw [ProbabilityTheory.cdf_eq_real]
rfl
rw [htarget]
apply (hraw x).congr'
filter_upwards with n
rw [ProbabilityTheory.cdf_eq_real]
change (μ n).real {z | S n z ≤ x} =
(Measure.map (S n) (μ n)).real (Iic x)
rw [MeasureTheory.measureReal_def, MeasureTheory.measureReal_def,
Measure.map_apply (hS n) measurableSet_Iic]
rfl
have hν : Tendsto νs atTop (nhds ν) :=
probabilityMeasure_tendsto_of_cdf_tendsto νs ν hScdf
have hprob := ProbabilityMeasure.tendsto_measure_of_null_frontier_of_tendsto
hν (E := Icc a b) (by
change toNNReal (Q (frontier (Icc a b))) = 0
simp [hboundary])
have hreal := NNReal.continuous_coe.tendsto _ |>.comp hprob
apply hreal.congr'
filter_upwards with n
change (((νs n) (Icc a b) : NNReal) : ℝ) =
(μ n).real {z | S n z ∈ Icc a b}
rw [show (((νs n) (Icc a b) : NNReal) : ℝ) =
(Measure.map (S n) (μ n)).real (Icc a b) by
change ((toNNReal
((Measure.map (S n) (μ n)) (Icc a b)) : NNReal) : ℝ) = _
rw [ENNReal.coe_toNNReal_eq_toReal]
rfl]
rw [MeasureTheory.measureReal_def, MeasureTheory.measureReal_def,
Measure.map_apply (hS n) measurableSet_Icc]
rfl
theorem pilot_wald_coverage_of_standardized_cdf_and_approx
Exact pilot Wald coverage once the standardized oracle CDF and the row-varying Slutsky approximation to the fallback pivot have been proved. The fallback is retained verbatim, so samples with zero estimated variance are handled by pilot_wald_event_eq. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot wald coverage of standardized cdf and approx.
Formal statement
hypotheses
cutoff :
ℝ
hcutoff :
0 < cutoff
θ :
p ε :
ℝ
m :
ℕ → ℕ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hApprox :
∀ η : ℝ
if
0 < η
then
Tendsto (fun n => (transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n).real {z | η ≤ |pilotWaldPivot select cutoff θ p ε m hselect hp hε n z - scaledError (pilotEstimator p ε m select hselect hp hε) θ 0 n z / sqrt (Vstar θ p ε)|}) atTop (nhds 0)
hstandardCDF :
∀ x : ℝ,
Tendsto (fun n => (transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n).real {z | scaledError (pilotEstimator p ε m select hselect hp hε) θ 0 n z / sqrt (Vstar θ p ε) ≤ x}) atTop (nhds ((gaussianMeasure 0 1).real (Iic x)))
conclusion
Tendsto (fun n => ((transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n) {z | |(pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ| ≤ cutoff * sqrt (VhatStar p ε m (select) z / n)}).toReal) atTop (nhds ((gaussianMeasure 0 1).real (Icc (-cutoff) cutoff)))
Proof (Lean source)
lemma pilot_wald_coverage_of_standardized_cdf_and_approx
(cutoff : ℝ) (hcutoff : 0 < cutoff)
(θ : TrialParameter) (p ε : ℝ) (m : ℕ → ℕ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hApprox : ∀ η : ℝ, 0 < η →
Tendsto (fun n =>
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n).real
{z | η ≤ |pilotWaldPivot select cutoff θ p ε m hselect hp hε n z -
scaledError (pilotEstimator p ε m select hselect hp hε) θ 0 n z /
sqrt (Vstar θ p ε)|}) atTop (nhds 0))
(hstandardCDF : ∀ x : ℝ,
Tendsto (fun n =>
(transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n).real
{z | scaledError (pilotEstimator p ε m select hselect hp hε) θ 0 n z /
sqrt (Vstar θ p ε) ≤ x}) atTop
(nhds ((gaussianMeasure 0 1).real (Iic x)))) :
Tendsto (fun n =>
((transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n)
{z | |(pilotEstimator p ε m select hselect hp hε).estimate n z - contrast θ| ≤
cutoff * sqrt
(VhatStar p ε m (select) z / n)}).toReal)
atTop (nhds ((gaussianMeasure 0 1).real (Icc (-cutoff) cutoff))) := by
let μ : ∀ n, Measure (Transcript (pilotOutputFamily n)) := fun n =>
transcriptLaw (pilotEstimator p ε m select hselect hp hε) θ p n
let S : ∀ n, Transcript (pilotOutputFamily n) → ℝ := fun n =>
pilotWaldPivot select cutoff θ p ε m hselect hp hε n
let T : ∀ n, Transcript (pilotOutputFamily n) → ℝ := fun n z =>
scaledError (pilotEstimator p ε m select hselect hp hε) θ 0 n z /
sqrt (Vstar θ p ε)
let _ (n : ℕ) : IsProbabilityMeasure (μ n) :=
transcriptLaw_isProbability _ θ p hp hθ n
have hS (n : ℕ) : Measurable (S n) := measurable_of_finite _
have hpivot : Tendsto (fun n =>
(μ n).real {z | S n z ∈ Icc (-cutoff) cutoff}) atTop
(nhds ((gaussianMeasure 0 1).real (Icc (-cutoff) cutoff))) := by
apply varyingLaw_Icc_tendsto_of_cdf_and_approx μ S T
(gaussianMeasure 0 1) hS hApprox hstandardCDF
· simpa [gaussianMeasure] using
(continuous_cdf_gaussianReal_zero (v := (1 : NNReal)) (by norm_num))
· exact gaussianMeasure_zero_one_frontier_Icc hcutoff
apply pilot_wald_coverage_of_pivot_Icc select cutoff hcutoff θ p ε m hselect hp hε
((gaussianMeasure 0 1).real (Icc (-cutoff) cutoff))
simpa only [μ, S, MeasureTheory.measureReal_def] using hpivot
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PilotWeakLimitSupport 2 declarations This file supplies the measure-theoretic bridge from convergence of real CDFs to weak convergence.
CDF support for the pilot weak limit
This file supplies the measure-theoretic bridge from convergence of real CDFs to weak convergence. The paper-specific theorem below deliberately leaves the CDF convergence of the actual centered estimator as an explicit hypothesis.
theorem probabilityMeasure_tendsto_of_cdf_tendsto
Pointwise convergence of the CDFs of real probability measures implies weak convergence. This all-points form is convenient for triangular-array adapters. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the probability Measure tendsto of cdf tendsto.
Formal statement
Proof (Lean source)
lemma probabilityMeasure_tendsto_of_cdf_tendsto
(μs : ℕ → ProbabilityMeasure ℝ) (μ : ProbabilityMeasure ℝ)
(hcdf : ∀ x : ℝ,
Tendsto (fun n => (cdf (μs n : Measure ℝ) x : ℝ)) atTop
(nhds (cdf (μ : Measure ℝ) x : ℝ))) :
Tendsto μs atTop (nhds μ) := by
let S : Set (Set ℝ) := {s | ∃ a b : ℝ, a < b ∧ Ioc a b = s}
have hpi : IsPiSystem S := by
simpa only [S, id_eq] using (isPiSystem_Ioc (id : ℝ → ℝ) id)
apply hpi.tendsto_probabilityMeasure_of_tendsto_of_mem
· rintro s ⟨a, b, hab, rfl⟩
exact measurableSet_Ioc
· intro u hu x hxu
obtain ⟨r, hr, hball⟩ := Metric.isOpen_iff.mp hu x hxu
refine ⟨Ioc (x - r / 2) (x + r / 2), ?_, ?_, ?_⟩
· exact ⟨x - r / 2, x + r / 2, by linarith, rfl⟩
· exact Ioc_mem_nhds (by linarith) (by linarith)
· intro y hy
apply hball
rw [Metric.mem_ball, Real.dist_eq]
rw [abs_lt]
constructor <;> linarith [hy.1, hy.2]
· rintro s ⟨a, b, hab, rfl⟩
rw [← NNReal.tendsto_coe]
have hmass (ν : ProbabilityMeasure ℝ) :
((ν (Ioc a b) : NNReal) : ℝ) =
(cdf (ν : Measure ℝ) b : ℝ) - (cdf (ν : Measure ℝ) a : ℝ) := by
change (ν : Measure ℝ).real (Ioc a b) = _
rw [show Ioc a b = Iic b \ Iic a by ext y; simp]
rw [MeasureTheory.measureReal_sdiff (μ := (ν : Measure ℝ))
(Iic_subset_Iic.mpr hab.le) measurableSet_Iic]
rw [← ProbabilityTheory.cdf_eq_real, ← ProbabilityTheory.cdf_eq_real]
simpa only [hmass] using (hcdf b).sub (hcdf a)
theorem weakLocalLimit_of_scaledError_cdf_tendsto
CDF convergence of the actual centered, root-scaled estimator laws implies the bounded-continuous-test formulation of the local weak limit. The transcript spaces may vary with n; the hypothesis is stated directly using their induced laws. No pilot CLT or CDF convergence is hidden in this adapter. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the weak Local Limit of scaled Error cdf tendsto.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ :
p :
ℝ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
hcdf :
∀ h : TrialParameter,
∀ x : ℝ,
Tendsto (fun n => (transcriptLaw P (localAlternative θ h n) p n).real {z | scaledError P θ h n z ≤ x}) atTop (nhds ((gaussianMeasure 0 (Vstar θ p ε)).real (Iic x)))
conclusion
Proof (Lean source)
lemma weakLocalLimit_of_scaledError_cdf_tendsto
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hcdf : ∀ h : TrialParameter, ∀ x : ℝ,
Tendsto (fun n =>
(transcriptLaw P (localAlternative θ h n) p n).real
{z | scaledError P θ h n z ≤ x}) atTop
(nhds ((gaussianMeasure 0 (Vstar θ p ε)).real (Iic x)))) :
WeakLocalLimit P θ p (gaussianMeasure 0 (Vstar θ p ε)) := by
classical
have hvpos : 0 < Vstar θ p ε := by
exact_mod_cast adaptivePilotLimitVariance_pos select θ p ε hselect hp hθ hε
let L : Measure ℝ := gaussianMeasure 0 (Vstar θ p ε)
let _ : IsProbabilityMeasure L := by
dsimp [L, gaussianMeasure]
rw [Real.toNNReal_of_nonneg hvpos.le]
infer_instance
intro h f hf ⟨C, hC⟩
have hscaled (n : ℕ) : Measurable (scaledError P θ h n) := by
exact measurable_const.mul ((P.estimate_measurable n).sub measurable_const)
let ν : ℕ → ProbabilityMeasure ℝ := fun n =>
if hn : InteriorMeans (localAlternative θ h n) then
letI : IsProbabilityMeasure
(transcriptLaw P (localAlternative θ h n) p n) :=
transcriptLaw_isProbability P _ p hp hn n
⟨Measure.map (scaledError P θ h n)
(transcriptLaw P (localAlternative θ h n) p n),
Measure.isProbabilityMeasure_map (hscaled n).aemeasurable⟩
else ⟨L, inferInstance⟩
let νlim : ProbabilityMeasure ℝ := ⟨L, inferInstance⟩
have hνcdf (x : ℝ) :
Tendsto (fun n => (cdf (ν n : Measure ℝ) x : ℝ)) atTop
(nhds (cdf (νlim : Measure ℝ) x : ℝ)) := by
have htarget : (cdf (νlim : Measure ℝ) x : ℝ) =
(gaussianMeasure 0 (Vstar θ p ε)).real (Iic x) := by
rw [ProbabilityTheory.cdf_eq_real]
rfl
rw [htarget]
apply (hcdf h x).congr'
filter_upwards [eventually_localAlternative_interior θ h hθ] with n hn
have hmap : (ν n : Measure ℝ) =
Measure.map (scaledError P θ h n)
(transcriptLaw P (localAlternative θ h n) p n) := by
simp only [ν, hn, ↓reduceDIte]
rfl
rw [ProbabilityTheory.cdf_eq_real, hmap]
rw [MeasureTheory.measureReal_def, MeasureTheory.measureReal_def]
congr 1
change (transcriptLaw P (localAlternative θ h n) p n)
((scaledError P θ h n) ⁻¹' Iic x) = _
exact (Measure.map_apply (hscaled n) measurableSet_Iic).symm
have hν : Tendsto ν atTop (nhds νlim) :=
probabilityMeasure_tendsto_of_cdf_tendsto ν νlim hνcdf
let fb : BoundedContinuousFunction ℝ ℝ :=
BoundedContinuousFunction.ofNormedAddCommGroup f hf C (by
intro x
simpa only [Real.norm_eq_abs] using hC x)
have hint :=
(ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hν) fb
apply hint.congr'
filter_upwards [eventually_localAlternative_interior θ h hθ] with n hn
have hmap : (ν n : Measure ℝ) =
Measure.map (scaledError P θ h n)
(transcriptLaw P (localAlternative θ h n) p n) := by
simp only [ν, hn, ↓reduceDIte]
rfl
rw [hmap]
change (∫ y, f y ∂Measure.map (scaledError P θ h n)
(transcriptLaw P (localAlternative θ h n) p n)) = _
rw [MeasureTheory.integral_map (hscaled n).aemeasurable
hf.aestronglyMeasurable]
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.PostprocessCardinality 3 declarations Positive active staircase rays whose postprocessing laws are pairwise mutually singular require distinct positive-mass output atoms.
Output cardinality under a singular postprocessing refinement
Positive active staircase rays whose postprocessing laws are pairwise mutually singular require distinct positive-mass output atoms. The argument works for arbitrary measurable structures on a finite output type and returns the definitional infinite bound for nonfinite output types.
theorem postprocess_row_ac_input
the postprocess row ac input assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
K s ≪ Q 0
Proof (Lean source)
lemma postprocess_row_ac_input
{Z : Type*} [MeasurableSpace Z]
(ε : ℝ) (α : StaircaseWeight) (K : Kernel (Fin 14) Z)
(hK : IsMarkovKernel K) (Q : Kernel (Fin 4) Z)
(hfactor : Q = K.comp (staircaseChannel ε α))
(s : Fin 14) (hs : 0 < α s) : K s ≪ Q 0 := by
letI : IsMarkovKernel K := hK
apply Measure.AbsolutelyContinuous.mk
intro A hA hQA
rw [hfactor, Kernel.comp_apply' _ _ _ hA, lintegral_fintype] at hQA
have hsum : ∑ u : Fin 14,
K u A * (staircaseChannel ε α) 0 {u} = 0 := hQA
have hterm : K s A * (staircaseChannel ε α) 0 {s} = 0 :=
(Finset.sum_eq_zero_iff_of_nonneg (fun _ _ ↦ zero_le)).mp hsum s
(Finset.mem_univ s)
have hT : (staircaseChannel ε α) 0 {s} ≠ 0 := by
change (Measure.count.withDensity (fun u : Fin 14 ↦
ofReal (α u * patternRay ε u 0))) {s} ≠ 0
simp [withDensity_apply _ (measurableSet_singleton s)]
have hr : 0 < patternRay ε s 0 := by
simp only [patternRay, privacyIncrement, privacyRatio]
split <;> simp_all <;> positivity
exact mul_pos hs hr
exact (mul_eq_zero.mp hterm).resolve_right hT
the input zero ac output law assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Q 0 ≪ outputLaw θ p Q
Proof (Lean source)
lemma input_zero_ac_outputLaw
{Z : Type*} [MeasurableSpace Z]
(θ : TrialParameter) (p : ℝ) (Q : Kernel (Fin 4) Z)
(hpi : 0 < piTheta θ p 0) : Q 0 ≪ outputLaw θ p Q := by
apply Measure.AbsolutelyContinuous.mk
intro A _ hout
have hsum : ∑ j : Fin 4,
ofReal (piTheta θ p j) * Q j A = 0 := by
simpa only [outputLaw, Measure.coe_finsetSum, Measure.coe_smul,
Finset.sum_apply, Pi.smul_apply, smul_eq_mul] using hout
have hterm : ofReal (piTheta θ p 0) * Q 0 A = 0 :=
(Finset.sum_eq_zero_iff_of_nonneg (fun _ _ ↦ zero_le)).mp hsum 0
(Finset.mem_univ 0)
have hcoef : ofReal (piTheta θ p 0) ≠ 0 := by simp [hpi]
exact (mul_eq_zero.mp hterm).resolve_left hcoef
theorem active_card_le_outputCardinality_of_pairwise_singular
the active card le output cardinality of pairwise singular assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the active card le output Cardinality of pairwise singular.
Formal statement
hypotheses
Z :
Type*
θ :
p ε :
ℝ
hp :
hθ :
α :
hpos :
∀ s ∈ A, 0 < α s
hK :
hfactor :
Q = K.comp (staircaseChannel ε α)
hsing :
∀ s ∈ A, ∀ u ∈ A, s ≠ u → K s ⟂ₘ K u
conclusion
(A.card : ℕ∞) ≤ outputCardinality θ p Q
Proof (Lean source)
lemma active_card_le_outputCardinality_of_pairwise_singular
{Z : Type*} [MeasurableSpace Z]
(θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(A : Finset (Fin 14)) (α : StaircaseWeight)
(hpos : ∀ s ∈ A, 0 < α s)
(K : Kernel (Fin 14) Z) (hK : IsMarkovKernel K)
(Q : Kernel (Fin 4) Z)
(hfactor : Q = K.comp (staircaseChannel ε α))
(hsing : ∀ s ∈ A, ∀ u ∈ A, s ≠ u → K s ⟂ₘ K u) :
(A.card : ℕ∞) ≤ outputCardinality θ p Q := by
have hpi : 0 < piTheta θ p 0 := by
rcases hp with ⟨hp0, hp1⟩
rcases hθ with ⟨h00, h01, h10, h11⟩
simp only [piTheta, controlProb]
exact mul_pos (by linarith) (by linarith)
have hac (s : Fin 14) (hs : s ∈ A) : K s ≪ outputLaw θ p Q :=
(postprocess_row_ac_input ε α K hK Q hfactor s (hpos s hs)).trans
(input_zero_ac_outputLaw θ p Q hpi)
unfold outputCardinality
split_ifs with hfinite
· letI := hfinite
letI : IsMarkovKernel K := hK
exact pairwise_singular_card_le_positive_atoms
A (fun s ↦ K s)
(outputLaw θ p Q) hac hsing
· exact le_top
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.Procedures 18 declarations A transcript experiment carries its history-dependent channels and the induced finite transcript laws.
Sequential experiments and local procedures
A transcript experiment carries its history-dependent channels and the induced finite transcript laws. The laws are pinned by the conditional cylinder factorization, which is valid on arbitrary measurable output spaces.
For the supplied quantities and conditions, the output family is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The Output Family is the displayed object.
Definition (Lean source)
def
OutputFamily :
Type 1
given by
(n : ℕ) → Fin n → Type
For the supplied quantities and conditions, the transcript is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The Transcript is determined by the displayed parameters.
Definition (Lean source)
def historyPrefix
the history prefix is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
given by
fun j => z j.1
the transcript factorizes is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
TranscriptFactorizes Q R :
Prop
given by
clause 1
IsProbabilityMeasure (R x)
clause 2
R x {z | historyPrefix i z ∈ B ∧ z i ∈ A}
= ∫⁻ h in B, Q i (x i, h) A ∂(R x).map (historyPrefix i)
For the supplied quantities and conditions, this procedure sequence packages the stated mathematical data. The Procedure Sequence is determined by the displayed parameters.
Definition (Lean source)
parameters
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
fields
@realizes (\mathcal Z_i,\mathcal G_i)(arbitrary output measurable spaces)
channel :
∀ n, SequentialKernel (Z := Z n)
factorizes :
∀ n, TranscriptFactorizes (channel n) (transcript n)
privacy :
∀ n, SequentialLDP (Z := Z n) ε (channel n)
estimate :
∀ n
if
Transcript (Z n)
then
ℝ
estimate_measurable :
∀ n, Measurable (estimate n)
def inputPathProbability
For the supplied quantities and conditions, the input path probability is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The input Path Probability is determined by the displayed parameters.
Definition (Lean source)
parameters
def
inputPathProbability θ p x :
ℝ
the transcript law is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The transcript Law is determined by the displayed parameters.
Definition (Lean source)
For the supplied quantities and conditions, the local alternative is the mathematical object specified below. The local Alternative is determined by the displayed parameters.
Definition (Lean source)
def localIndexSet
For the supplied quantities and conditions, the local index set is the mathematical object specified below. The local Index Set is determined by the displayed parameters.
Definition (Lean source)
parameters
def
given by
the scaled error is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The scaled Error is determined by the displayed parameters.
Definition (Lean source)
parameters
def
scaledError P θ0 h n z :
ℝ
given by
sqrt n * (P.estimate n z - contrast (localAlternative θ0 h n))
def localRisk
the local risk is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The local Risk is determined by the displayed parameters.
Definition (Lean source)
parameters
def
localRisk P θ0 h p n :
ℝ≥0∞
given by
def localWorstRisk
the local worst risk is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The local Worst Risk is determined by the displayed parameters.
Definition (Lean source)
parameters
def
localWorstRisk P θ0 p H n :
ℝ≥0∞
given by
the local asymptotic risk is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The local Asymptotic Risk is determined by the displayed parameters.
Definition (Lean source)
parameters
def
localAsymptoticRisk P θ0 p :
ℝ≥0∞
given by
the weak local limit is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The Weak Local Limit is determined by the displayed parameters.
Definition (Lean source)
parameters
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ0 :
p :
ℝ
L :
Measure ℝ
h :
f :
ℝ → ℝ
(∃ C : ℝ, ∀ u, |f u| ≤ C)
def
WeakLocalLimit P θ0 p L :
Prop
given by
Tendsto (fun n => ∫ z, f (scaledError P θ0 h n z) ∂transcriptLaw P (localAlternative θ0 h n) p n) atTop (nhds (∫ u, f u ∂L))
the regular procedure is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The Regular Procedure is determined by the displayed parameters.
Definition (Lean source)
parameters
def
RegularProcedure P θ0 p :
Prop
given by
∃ L : Measure ℝ,
clause 1
H :
ℝ
0 < H
Tendsto (fun M : ℕ => limsup (fun n => sSup {u : ℝ≥0∞ | ∃ h ∈ localIndexSet θ0 n H, u = ∫⁻ z, ofReal ((scaledError P θ0 h n z) ^ 2 * if M < |scaledError P θ0 h n z| then 1 else 0) ∂transcriptLaw P (localAlternative θ0 h n) p n}) atTop) atTop (nhds 0)
theorem exists_selectedDirection
Under the supplied quantities and conditions, the exists selected direction assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the exists selected Direction.
Formal statement
hypotheses
conclusion
∃ t : ℝ, upperEnvelope θ p ε t = Jstar θ p ε
Proof (Lean source)
lemma exists_selectedDirection (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(hε : 0 < ε) :
∃ t : ℝ, upperEnvelope θ p ε t = Jstar θ p ε := by
have hfixed : FixedPrivacy (fun _ => ε) ε := ⟨hε, fun _ => rfl⟩
exact (finite_oracle.{0} θ p ε hp hθ (fun _ => ε) hfixed).2.2.2.2
def selectedDirection
For the supplied quantities and conditions, the selected direction is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The selected Direction is determined by the displayed parameters.
Definition (Lean source)
parameters
def
selectedDirection θ p ε hp hθ hε :
ℝ
given by
choose (exists_selectedDirection θ p ε hp hθ hε)
def vtHandle
For the supplied quantities and conditions, the vt handle is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The vt Handle is determined by the displayed parameters.
Definition (Lean source)
parameters
def
given by
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.QuadraticConvexity 2 declarations The profile is a nonnegative weighted sum of squared affine scores.
Convexity of the staircase information profile
The profile is a nonnegative weighted sum of squared affine scores.
theorem informationObjective_convex_time
Under the supplied quantities and conditions, the information objective convex time assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the information Objective convex time.
Formal statement
hypotheses
θ :
p ε :
ℝ
α :
hα :
∀ s, 0 ≤ α s
hmass :
∀ s, 0 < patternMass θ p ε s
t u a b :
ℝ
ha :
0 ≤ a
hb :
0 ≤ b
hab :
a + b = 1
conclusion
informationObjective θ p ε α (a * t + b * u)
≤ a * informationObjective θ p ε α t + b * informationObjective θ p ε α u
Proof (Lean source)
lemma informationObjective_convex_time (θ : TrialParameter) (p ε : ℝ)
(α : StaircaseWeight) (hα : ∀ s, 0 ≤ α s)
(hmass : ∀ s, 0 < patternMass θ p ε s)
(t u a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
informationObjective θ p ε α (a * t + b * u) ≤
a * informationObjective θ p ε α t +
b * informationObjective θ p ε α u := by
unfold informationObjective
rw [Finset.mul_sum, Finset.mul_sum, ← Finset.sum_add_distrib]
apply Finset.sum_le_sum
intro s _
have hslope : projectedGradient p ε s (a * t + b * u) =
a * projectedGradient p ε s t +
b * projectedGradient p ε s u := by
simp [projectedGradient, direction, Fin.sum_univ_succ]
linear_combination -(patternGradient p ε s 1) * hab
have hsq :
(a * projectedGradient p ε s t +
b * projectedGradient p ε s u) ^ 2 ≤
a * (projectedGradient p ε s t) ^ 2 +
b * (projectedGradient p ε s u) ^ 2 := by
have h := mul_nonneg (mul_nonneg ha hb)
(sq_nonneg (projectedGradient p ε s t - projectedGradient p ε s u))
nlinarith [h]
unfold patternInformation
rw [hslope]
have hc : 0 ≤ α s / patternMass θ p ε s :=
div_nonneg (hα s) (hmass s).le
calc
α s * ((a * projectedGradient p ε s t +
b * projectedGradient p ε s u) ^ 2 /
patternMass θ p ε s) =
(α s / patternMass θ p ε s) *
(a * projectedGradient p ε s t +
b * projectedGradient p ε s u) ^ 2 := by ring
_ ≤ (α s / patternMass θ p ε s) *
(a * (projectedGradient p ε s t) ^ 2 +
b * (projectedGradient p ε s u) ^ 2) :=
mul_le_mul_of_nonneg_left hsq hc
_ = a * (α s * ((projectedGradient p ε s t) ^ 2 /
patternMass θ p ε s)) +
b * (α s * ((projectedGradient p ε s u) ^ 2 /
patternMass θ p ε s)) := by ring
theorem informationObjective_convexOn_time
Under the supplied quantities and conditions, the information objective convex on time assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the information Objective convex On time.
Formal statement
Proof (Lean source)
lemma informationObjective_convexOn_time (θ : TrialParameter) (p ε : ℝ)
(α : StaircaseWeight) (hα : ∀ s, 0 ≤ α s)
(hmass : ∀ s, 0 < patternMass θ p ε s) :
ConvexOn ℝ univ (informationObjective θ p ε α) := by
refine ⟨convex_univ, ?_⟩
intro t _ u _ a b ha hb hab
simpa only [smul_eq_mul] using
informationObjective_convex_time θ p ε α hα hmass t u a b ha hb hab
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.R3Continuity 19 declarations This file supplies the canonical stationarity direction, continuity of the rational information data on the interior, and an explicit right inverse for the three active dual-ray equations.
Continuous certificate data for the three-output region
This file supplies the canonical stationarity direction, continuity of the rational information data on the interior, and an explicit right inverse for the three active dual-ray equations. It also gives the local form of the promoted anchored linear solver needed when coefficients are only continuous at the certificate point.
def r3Direction
The scalar score direction used in the three-output region is determined by the displayed parameters.
Definition (Lean source)
parameters
p μ0 μ1 ε :
ℝ
def
r3Direction p μ0 μ1 ε :
ℝ
given by
-r3Stationarity p μ0 μ1 ε 0 / r3StationarityCoeff p μ0 μ1 ε
theorem r3Stationarity_affine
Under the supplied quantities and conditions, the r3 stationarity affine assertion holds.
Formal statement
hypotheses
p μ0 μ1 ε t :
ℝ
conclusion
r3Stationarity p μ0 μ1 ε t
= r3Stationarity p μ0 μ1 ε 0 + t * r3StationarityCoeff p μ0 μ1 ε
Proof (Lean source)
lemma r3Stationarity_affine (p μ0 μ1 ε t : ℝ) :
r3Stationarity p μ0 μ1 ε t =
r3Stationarity p μ0 μ1 ε 0 +
t * r3StationarityCoeff p μ0 μ1 ε := by
unfold r3StationarityCoeff r3Stationarity
rw [← Finset.sum_sub_distrib]
rw [Finset.mul_sum, ← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro s _
simp [patternInformationSlope, projectedGradient, direction,
Fin.sum_univ_succ]
ring
theorem r3Stationarity_r3Direction_eq_zero
Under the supplied quantities and conditions, the r3 stationarity r3 direction eq zero assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p μ0 μ1 ε :
ℝ
hcoeff :
r3StationarityCoeff p μ0 μ1 ε ≠ 0
conclusion
r3Stationarity p μ0 μ1 ε (r3Direction p μ0 μ1 ε) = 0
Proof (Lean source)
lemma r3Stationarity_r3Direction_eq_zero (p μ0 μ1 ε : ℝ)
(hcoeff : r3StationarityCoeff p μ0 μ1 ε ≠ 0) :
r3Stationarity p μ0 μ1 ε (r3Direction p μ0 μ1 ε) = 0 := by
rw [r3Stationarity_affine]
unfold r3Direction
field_simp
ring
def RegionParameter
the region parameter is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The Region Parameter is the displayed object.
Definition (Lean source)
def
RegionParameter :
Type
given by
ℝ × ℝ × ℝ × ℝ
def regionTheta
For the supplied quantities and conditions, the region theta is the mathematical object specified below.
Definition (Lean source)
parameters
x :
def
given by
fun k => if k = 0 then x.2.1 else x.2.2.1
theorem continuous_patternRay_region
Under the supplied quantities and conditions, the continuous pattern ray region assertion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma continuous_patternRay_region (s : Fin 14) (j : Fin 4) :
Continuous (fun x : RegionParameter => patternRay x.2.2.2 s j) := by
unfold patternRay privacyIncrement privacyRatio
by_cases h : patternContains s j
· simp [h]
fun_prop
· simp [h]
exact continuous_const
theorem continuous_patternMass_region
Under the supplied quantities and conditions, the continuous pattern mass region assertion holds.
Formal statement
Proof (Lean source)
lemma continuous_patternMass_region (s : Fin 14) :
Continuous (fun x : RegionParameter =>
patternMass (regionTheta x) x.1 x.2.2.2 s) := by
unfold patternMass
apply continuous_finsetSum
intro j _
apply Continuous.mul
· fin_cases j <;> simp +decide [regionTheta, piTheta, controlProb] <;> fun_prop
· exact continuous_patternRay_region s j
theorem continuousAt_projectedGradient_region
Under the supplied quantities and conditions, the continuous at projected gradient region assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma continuousAt_projectedGradient_region (s : Fin 14)
(t : RegionParameter → ℝ) (x : RegionParameter)
(ht : ContinuousAt t x) :
ContinuousAt (fun y : RegionParameter =>
projectedGradient y.1 y.2.2.2 s (t y)) x := by
simp +decide [projectedGradient, patternGradient, direction,
privacyIncrement, privacyRatio, controlProb, Fin.sum_univ_succ]
fun_prop
theorem continuousAt_patternInformation_region
Under the supplied quantities and conditions, the continuous at pattern information region assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
s :
Fin 14
t :
RegionParameter → ℝ
x :
ht :
ContinuousAt t x
hm :
patternMass (regionTheta x) x.1 x.2.2.2 s ≠ 0
conclusion
Proof (Lean source)
lemma continuousAt_patternInformation_region (s : Fin 14)
(t : RegionParameter → ℝ) (x : RegionParameter)
(ht : ContinuousAt t x)
(hm : patternMass (regionTheta x) x.1 x.2.2.2 s ≠ 0) :
ContinuousAt (fun y : RegionParameter =>
patternInformation (regionTheta y) y.1 y.2.2.2 s (t y)) x := by
unfold patternInformation
apply ContinuousAt.div
· exact (continuousAt_projectedGradient_region s t x ht).pow 2
· exact (continuous_patternMass_region s).continuousAt
· exact hm
theorem continuousAt_projectedScore_region
Under the supplied quantities and conditions, the continuous at projected score region assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
s :
Fin 14
t :
RegionParameter → ℝ
x :
ht :
ContinuousAt t x
hm :
patternMass (regionTheta x) x.1 x.2.2.2 s ≠ 0
conclusion
Proof (Lean source)
lemma continuousAt_projectedScore_region (s : Fin 14)
(t : RegionParameter → ℝ) (x : RegionParameter)
(ht : ContinuousAt t x)
(hm : patternMass (regionTheta x) x.1 x.2.2.2 s ≠ 0) :
ContinuousAt (fun y : RegionParameter =>
projectedScore (regionTheta y) y.1 y.2.2.2 s (t y)) x := by
unfold projectedScore
exact (continuousAt_projectedGradient_region s t x ht).div
(continuous_patternMass_region s).continuousAt hm
theorem continuousAt_patternInformationSlope_region
Under the supplied quantities and conditions, the continuous at pattern information slope region assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
s :
Fin 14
t :
RegionParameter → ℝ
x :
ht :
ContinuousAt t x
hm :
patternMass (regionTheta x) x.1 x.2.2.2 s ≠ 0
conclusion
ContinuousAt (fun y : RegionParameter => patternInformationSlope (regionTheta y) y.1 y.2.2.2 s (t y)) x
Proof (Lean source)
lemma continuousAt_patternInformationSlope_region (s : Fin 14)
(t : RegionParameter → ℝ) (x : RegionParameter)
(ht : ContinuousAt t x)
(hm : patternMass (regionTheta x) x.1 x.2.2.2 s ≠ 0) :
ContinuousAt (fun y : RegionParameter =>
patternInformationSlope (regionTheta y) y.1 y.2.2.2 s (t y)) x := by
unfold patternInformationSlope
apply ContinuousAt.div
· apply ContinuousAt.mul
· apply ContinuousAt.mul continuousAt_const
(continuousAt_projectedGradient_region s t x ht)
· simp +decide [patternGradient, privacyIncrement, privacyRatio,
controlProb]
fun_prop
· exact (continuous_patternMass_region s).continuousAt
· exact hm
theorem continuousAt_r3Stationarity_region
Under the supplied quantities and conditions, the continuous at r3 stationarity region assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma continuousAt_r3Stationarity_region
(t : RegionParameter → ℝ) (x : RegionParameter)
(ht : ContinuousAt t x)
(hm : ∀ s : Fin 14,
patternMass (regionTheta x) x.1 x.2.2.2 s ≠ 0) :
ContinuousAt (fun y : RegionParameter =>
r3Stationarity y.1 y.2.1 y.2.2.1 y.2.2.2 (t y)) x := by
unfold r3Stationarity
let f := fun s : Fin 14 => fun y : RegionParameter =>
r3Weight y.2.2.2 s *
patternInformationSlope (regionTheta y) y.1 y.2.2.2 s (t y)
have hf (s : Fin 14) : ContinuousAt (f s) x := by
apply ContinuousAt.mul
· unfold r3Weight
by_cases hs : s ∈ ({0, 5, 7} : Finset (Fin 14))
· simp [hs]
exact (((Real.continuous_exp.comp continuous_snd.snd.snd).add
continuous_const).continuousAt.inv₀
(show exp x.2.2.2 + 2 ≠ 0 by positivity))
· simp [hs]
exact continuousAt_const
· exact continuousAt_patternInformationSlope_region s t x ht (hm s)
have hsum : ∀ S : Finset (Fin 14),
ContinuousAt (fun y => ∑ s ∈ S, f s y) x := by
intro S
induction S using Finset.induction_on with
| empty => simpa using (continuousAt_const : ContinuousAt (fun _ : RegionParameter => (0 : ℝ)) x)
| @insert a S ha ih =>
simp only [Finset.sum_insert ha]
change ContinuousAt (fun y => f a y + ∑ s ∈ S, f s y) x
exact (hf a).add ih
change ContinuousAt (fun y => ∑ s : Fin 14,
r3Weight y.2.2.2 s *
patternInformationSlope (regionTheta y) y.1 y.2.2.2 s (t y)) x
simpa only [f] using hsum univ
theorem continuousAt_r3Direction_region
Under the supplied quantities and conditions, the continuous at r3 direction region assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma continuousAt_r3Direction_region (x : RegionParameter)
(hm : ∀ s : Fin 14,
patternMass (regionTheta x) x.1 x.2.2.2 s ≠ 0)
(hc : r3StationarityCoeff x.1 x.2.1 x.2.2.1 x.2.2.2 ≠ 0) :
ContinuousAt (fun y : RegionParameter =>
r3Direction y.1 y.2.1 y.2.2.1 y.2.2.2) x := by
unfold r3Direction r3StationarityCoeff
apply ContinuousAt.div
· exact (continuousAt_r3Stationarity_region (fun _ => 0) x continuousAt_const hm).neg
· exact (continuousAt_r3Stationarity_region (fun _ => 1) x continuousAt_const hm).sub
(continuousAt_r3Stationarity_region (fun _ => 0) x continuousAt_const hm)
· exact hc
def r3Mask
For the supplied quantities and conditions, the r3 mask is the mathematical object specified below.
def r3DualMatrix
The dual constraint matrix for the three-output region is determined by the displayed parameters.
Definition (Lean source)
def r3DualRightInverse
A right inverse for the three-output dual matrix is determined by the displayed parameters.
theorem r3DualMatrix_mul_rightInverse
Under the supplied quantities and conditions, the r3 dual matrix mul right inverse assertion holds. Under the stated assumptions, the r3 Dual Matrix mul right Inverse.
Formal statement
Proof (Lean source)
lemma r3DualMatrix_mul_rightInverse (x : RegionParameter) (hε : 0 < x.2.2.2) :
r3DualMatrix x * r3DualRightInverse x.2.2.2 = 1 := by
have hr : 1 < exp x.2.2.2 := (Real.one_lt_exp_iff).2 hε
have hd : (exp x.2.2.2 - 1) * (exp x.2.2.2 + 2) ≠ 0 := by
positivity
have hd1 : exp x.2.2.2 - 1 ≠ 0 := by positivity
ext i k
change (∑ j : Fin 4, r3DualMatrix x i j *
r3DualRightInverse x.2.2.2 j k) = (1 : Matrix (Fin 3) (Fin 3) ℝ) i k
fin_cases i <;> fin_cases k <;>
simp +decide [r3DualMatrix, r3DualRightInverse, r3Mask,
patternRay, patternContains, privacyIncrement, privacyRatio,
Fin.sum_univ_succ] <;> field_simp [hd, hd1] <;> ring
theorem continuous_r3DualMatrix
the continuous r3 dual matrix assertion holds.
Formal statement
conclusion
Proof (Lean source)
lemma continuous_r3DualMatrix : Continuous r3DualMatrix := by
apply continuous_matrix
intro i j
exact continuous_patternRay_region (r3Mask i) j
theorem exists_eventually_anchoredSolution_of_rightInverse
the exists eventually anchored solution of right inverse assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the exists eventually anchored Solution of right Inverse.
Formal statement
hypotheses
X :
Type*
k n :
ℕ
b :
X → Fin k → ℝ
x₀ :
X
y₀ :
Fin n → ℝ
hA :
ContinuousAt A x₀
hb :
ContinuousAt b x₀
hR :
A x₀ * R₀ = 1
hy :
mulVec (A x₀) y₀ = b x₀
conclusion
y :
anchoredSolution A b R₀ y₀
Proof (Lean source)
lemma exists_eventually_anchoredSolution_of_rightInverse
{X : Type*} [TopologicalSpace X] {k n : ℕ}
(A : X → Matrix (Fin k) (Fin n) ℝ) (b : X → Fin k → ℝ)
(R₀ : Matrix (Fin n) (Fin k) ℝ) (x₀ : X) (y₀ : Fin n → ℝ)
(hA : ContinuousAt A x₀) (hb : ContinuousAt b x₀)
(hR : A x₀ * R₀ = 1) (hy : mulVec (A x₀) y₀ = b x₀) :
let y := anchoredSolution A b R₀ y₀
ContinuousAt y x₀ ∧ y x₀ = y₀ ∧
∀ᶠ x in nhds x₀, mulVec (A x) (y x) = b x := by
dsimp only
have hM : ContinuousAt (fun x => A x * R₀) x₀ :=
(continuous_fst.matrix_mul continuous_snd).continuousAt.comp
(hA.prodMk continuousAt_const)
have hdet : ContinuousAt (fun x => (A x * R₀).det) x₀ :=
continuous_id.matrix_det.continuousAt.comp hM
have hdet0 : (A x₀ * R₀).det ≠ 0 := by rw [hR]; simp
have hev : ∀ᶠ x in nhds x₀, (A x * R₀).det ≠ 0 :=
hdet.eventually_ne hdet0
have hInv : ContinuousAt (fun x => (A x * R₀)⁻¹) x₀ := by
have hringInv : ContinuousAt Ring.inverse (A x₀ * R₀).det := by
rw [show (Ring.inverse : ℝ → ℝ) = inv from funext Ring.inverse_eq_inv]
exact continuousAt_inv₀ hdet0
exact ContinuousAt.comp' (f := fun x => A x * R₀)
(continuousAt_matrix_inv (A x₀ * R₀) hringInv)
hM
have hResidual : ContinuousAt (fun x => b x - mulVec (A x) y₀) x₀ :=
hb.sub ((continuous_fst.matrix_mulVec continuous_snd).continuousAt.comp
(hA.prodMk continuousAt_const))
have hInner : ContinuousAt
(fun x => mulVec (A x * R₀)⁻¹ (b x - mulVec (A x) y₀)) x₀ :=
(continuous_fst.matrix_mulVec continuous_snd).continuousAt.comp
(hInv.prodMk hResidual)
have hcont : ContinuousAt (anchoredSolution A b R₀ y₀) x₀ := by
change ContinuousAt
(fun x => y₀ + mulVec R₀
(mulVec (A x * R₀)⁻¹ (b x - mulVec (A x) y₀))) x₀
exact continuousAt_const.add
((continuous_fst.matrix_mulVec continuous_snd).continuousAt.comp
(continuousAt_const.prodMk hInner))
refine ⟨hcont, anchoredSolution_at_base A b R₀ y₀ x₀ hR hy, ?_⟩
filter_upwards [hev] with x hx
exact anchoredSolution_eq_of_det_ne_zero A b R₀ y₀ x hx
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.R3Rigidity 3 declarations The strict certificate confines every optimal staircase design to the three active masks.
Rigidity of the three-ray certificate
The strict certificate confines every optimal staircase design to the three
active masks. At privacy level log 3, normalization then fixes all three
weights to 1/5.
theorem r3Certificate_profile_optimal_and_rigid
Under the supplied quantities and conditions, the r3 certificate profile optimal and rigid assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p μ0 μ1 ε :
ℝ
hx :
(p, μ0, μ1, ε) ∈ R3
conclusion
conclusion 1
s ∈ ({0, 5, 7} : Finset (Fin 14)) :
0 < α s
conclusion 2
conclusion 3
β :
s ∉ ({0, 5, 7} : Finset (Fin 14)) :
β s = 0
Proof (Lean source)
lemma r3Certificate_profile_optimal_and_rigid
(p μ0 μ1 ε : ℝ) (hx : (p, μ0, μ1, ε) ∈ R3) :
let θ : TrialParameter := fun k ↦ if k = 0 then μ0 else μ1
∃ α : StaircaseWeight,
staircaseFeasible ε α ∧
activeSupport α ({0, 5, 7} : Finset (Fin 14)) ∧
(∀ s ∈ ({0, 5, 7} : Finset (Fin 14)), 0 < α s) ∧
Jstar θ p ε = sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε α v} ∧
∀ β : StaircaseWeight, staircaseFeasible ε β →
Jstar θ p ε = sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε β v} →
∀ s ∉ ({0, 5, 7} : Finset (Fin 14)), β s = 0 := by
dsimp only
change 0 < p ∧ p < 1 ∧
0 < μ0 ∧ μ0 < 1 ∧ 0 < μ1 ∧ μ1 < 1 ∧ 0 < ε ∧
∃ α : StaircaseWeight, ∃ t : ℝ, ∃ η : Fin 4 → ℝ,
staircaseFeasible ε α ∧
activeSupport α ({0, 5, 7} : Finset (Fin 14)) ∧
(∀ s ∈ ({0, 5, 7} : Finset (Fin 14)), 0 < α s) ∧
(∑ s : Fin 14, α s *
patternInformationSlope (fun k ↦ if k = 0 then μ0 else μ1)
p ε s t) = 0 ∧
(∀ s ∈ ({0, 5, 7} : Finset (Fin 14)),
dualRay ε η s =
patternInformation (fun k ↦ if k = 0 then μ0 else μ1) p ε s t) ∧
(∀ s ∉ ({0, 5, 7} : Finset (Fin 14)),
patternInformation (fun k ↦ if k = 0 then μ0 else μ1) p ε s t <
dualRay ε η s) ∧
(false → ∀ s ∈ ({0, 5, 7} : Finset (Fin 14)),
∀ u ∈ ({0, 5, 7} : Finset (Fin 14)), s ≠ u →
projectedScore (fun k ↦ if k = 0 then μ0 else μ1) p ε s t ≠
projectedScore (fun k ↦ if k = 0 then μ0 else μ1) p ε u t) at hx
rcases hx with ⟨hp0, hp1, hμ0, hμ0', hμ1, hμ1', hε,
α, t, η, hα, hsupp, hpos, hstat, hactive, hinactive, _⟩
have hp : InteriorAssignment p := ⟨hp0, hp1⟩
have hθ : InteriorMeans (fun k ↦ if k = 0 then μ0 else μ1) := by
change 0 < μ0 ∧ μ0 < 1 ∧ 0 < μ1 ∧ μ1 < 1
exact ⟨hμ0, hμ0', hμ1, hμ1'⟩
obtain ⟨hopt, hrigid⟩ := strictCertificate_profile_optimal_and_rigid
(fun k ↦ if k = 0 then μ0 else μ1) p ε hp hθ hε.le
({0, 5, 7} : Finset (Fin 14)) α t η hα hsupp hstat hactive hinactive
exact ⟨α, hα, hsupp, hpos, hopt, hrigid⟩
theorem r3_feasible_eq_canonical
the r3 feasible eq canonical assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the r3 feasible eq canonical.
Formal statement
hypotheses
β :
hβ :
staircaseFeasible (log 3) β
Proof (Lean source)
lemma r3_feasible_eq_canonical (β : StaircaseWeight)
(hβ : staircaseFeasible (log 3) β)
(hout : ∀ s ∉ ({0, 5, 7} : Finset (Fin 14)), β s = 0) :
β = fun s ↦ if s ∈ ({0, 5, 7} : Finset (Fin 14)) then 1 / 5 else 0 := by
have he : exp (log (3 : ℝ)) = 3 := Real.exp_log (by norm_num)
have h0 := hβ.2 0
have h1 := hβ.2 1
have h2 := hβ.2 2
have h3 := hβ.2 3
simp +decide [staircaseMatrix, patternRay, patternContains,
privacyIncrement, privacyRatio, he, Fin.sum_univ_succ, hout] at h0 h1 h2 h3
funext s
by_cases hs : s ∈ ({0, 5, 7} : Finset (Fin 14))
· have hs' : s = 0 ∨ s = 5 ∨ s = 7 := by simpa using hs
rcases hs' with rfl | rfl | rfl <;> simp +decide <;> linarith
· rw [hout s hs]
simp [hs]
theorem r3Certificate_unique_optimum_log3
Under the supplied quantities and conditions, the r3 certificate unique optimum log3 assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
θ :
fun k ↦ if k = 0 then μ0 else μ1 (: TrialParameter)
∃ α : StaircaseWeight,
Proof (Lean source)
lemma r3Certificate_unique_optimum_log3
(p μ0 μ1 : ℝ) (hx : (p, μ0, μ1, log 3) ∈ R3) :
let θ : TrialParameter := fun k ↦ if k = 0 then μ0 else μ1
∃ α : StaircaseWeight,
staircaseFeasible (log 3) α ∧
activeSupport α ({0, 5, 7} : Finset (Fin 14)) ∧
Jstar θ p (log 3) = sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p (log 3) α v} ∧
∀ β : StaircaseWeight, staircaseFeasible (log 3) β →
Jstar θ p (log 3) = sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p (log 3) β v} →
β = α := by
dsimp only
obtain ⟨α, hα, hsupp, _, hopt, hrigid⟩ :=
r3Certificate_profile_optimal_and_rigid p μ0 μ1 (log 3) hx
have houtα : ∀ s ∉ ({0, 5, 7} : Finset (Fin 14)), α s = 0 := by
intro s hs
by_contra hn
exact hs ((hsupp s).mp hn)
have hαcanonical := r3_feasible_eq_canonical α hα houtα
refine ⟨α, hα, hsupp, hopt, ?_⟩
intro β hβ hβopt
have hβcanonical := r3_feasible_eq_canonical β hβ (hrigid β hβ hβopt)
exact hβcanonical.trans hαcanonical.symm
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.R5Continuity 48 declarations The four active rows used to determine the dual certificate have an explicit right inverse throughout the positive-privacy parameter region.
Continuous active-dual solver for the five-output region
The four active rows used to determine the dual certificate have an explicit right inverse throughout the positive-privacy parameter region. The fifth active equality is supplied separately by the two constituent-objective identity in the R5 continuation argument.
def r5DualMask
For the supplied quantities and conditions, the r5 dual mask is the mathematical object specified below.
def r5DualMatrix
The dual constraint matrix for the five-output region is determined by the displayed parameters.
Definition (Lean source)
def r5DualRightInverse
A right inverse for the five-output dual matrix is determined by the displayed parameters.
theorem r5DualMatrix_mul_rightInverse
Under the supplied quantities and conditions, the r5 dual matrix mul right inverse assertion holds. Under the stated assumptions, the r5 Dual Matrix mul right Inverse.
Formal statement
Proof (Lean source)
lemma r5DualMatrix_mul_rightInverse (x : RegionParameter) (hε : 0 < x.2.2.2) :
r5DualMatrix x * r5DualRightInverse x.2.2.2 = 1 := by
have hr : 1 < exp x.2.2.2 := (Real.one_lt_exp_iff).2 hε
have hd1 : exp x.2.2.2 - 1 ≠ 0 := by positivity
have hda : (exp x.2.2.2 - 1) * (exp x.2.2.2 + 2) ≠ 0 := by
positivity
ext i k
change (∑ j : Fin 4, r5DualMatrix x i j *
r5DualRightInverse x.2.2.2 j k) = (1 : Matrix (Fin 4) (Fin 4) ℝ) i k
fin_cases i <;> fin_cases k <;>
simp +decide [r5DualMatrix, r5DualRightInverse, r5DualMask,
patternRay, patternContains, privacyIncrement, privacyRatio,
Fin.sum_univ_succ] <;> field_simp [hd1, hda] <;> ring
theorem continuous_r5DualMatrix
the continuous r5 dual matrix assertion holds.
Formal statement
conclusion
Proof (Lean source)
lemma continuous_r5DualMatrix : Continuous r5DualMatrix := by
apply continuous_matrix
intro i j
exact continuous_patternRay_region (r5DualMask i) j
def r5BinaryObjective
The normalized objective contributed by the two binary active rays. The r5 Binary Objective is determined by the displayed parameters.
Definition (Lean source)
parameters
x :
t :
ℝ
def
r5BinaryObjective x t :
ℝ
given by
(patternInformation (regionTheta x) x.1 x.2.2.2 5 t + patternInformation (regionTheta x) x.1 x.2.2.2 8 t) / (exp x.2.2.2 + 1)
def r5TernaryObjective
The normalized objective contributed by the three ternary active rays. The r5 Ternary Objective is determined by the displayed parameters.
Definition (Lean source)
parameters
x :
t :
ℝ
def
r5TernaryObjective x t :
ℝ
given by
(patternInformation (regionTheta x) x.1 x.2.2.2 2 t + patternInformation (regionTheta x) x.1 x.2.2.2 3 t + patternInformation (regionTheta x) x.1 x.2.2.2 7 t) / (exp x.2.2.2 + 2)
def r5BinaryCoeff
The binary component coefficient in the five-output continuation argument is determined by the displayed parameters.
Definition (Lean source)
parameters
x :
def
r5BinaryCoeff x :
ℝ
given by
privacyIncrement x.2.2.2 ^ 2 / (exp x.2.2.2 + 1) * (1 / patternMass (regionTheta x) x.1 x.2.2.2 5 + 1 / patternMass (regionTheta x) x.1 x.2.2.2 8)
def r5TernaryCoeff
The ternary component coefficient in the five-output continuation argument is determined by the displayed parameters.
Definition (Lean source)
parameters
x :
def
r5TernaryCoeff x :
ℝ
given by
privacyIncrement x.2.2.2 ^ 2 * x.1 ^ 2 / (exp x.2.2.2 + 2) * (1 / patternMass (regionTheta x) x.1 x.2.2.2 3 + 1 / patternMass (regionTheta x) x.1 x.2.2.2 7)
theorem r5BinaryObjective_eq_quadratic
Under the supplied quantities and conditions, the r5 binary objective eq quadratic assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
x :
t :
ℝ
hm5 :
patternMass (regionTheta x) x.1 x.2.2.2 5 ≠ 0
hm8 :
patternMass (regionTheta x) x.1 x.2.2.2 8 ≠ 0
conclusion
r5BinaryObjective x t = r5BinaryCoeff x * (x.1 - (1 - 2 * x.1) * t) ^ 2
Proof (Lean source)
lemma r5BinaryObjective_eq_quadratic (x : RegionParameter) (t : ℝ)
(hm5 : patternMass (regionTheta x) x.1 x.2.2.2 5 ≠ 0)
(hm8 : patternMass (regionTheta x) x.1 x.2.2.2 8 ≠ 0) :
r5BinaryObjective x t =
r5BinaryCoeff x * (x.1 - (1 - 2 * x.1) * t) ^ 2 := by
have hd : exp x.2.2.2 + 1 ≠ 0 := by positivity
unfold r5BinaryObjective r5BinaryCoeff patternInformation
simp +decide [projectedGradient, patternGradient, patternContains,
controlProb, direction, Fin.sum_univ_succ]
field_simp [hm5, hm8, hd]
ring
theorem r5TernaryObjective_eq_quadratic
Under the supplied quantities and conditions, the r5 ternary objective eq quadratic assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
x :
t :
ℝ
hm3 :
patternMass (regionTheta x) x.1 x.2.2.2 3 ≠ 0
hm7 :
patternMass (regionTheta x) x.1 x.2.2.2 7 ≠ 0
conclusion
r5TernaryObjective x t = r5TernaryCoeff x * (t + 1) ^ 2
Proof (Lean source)
lemma r5TernaryObjective_eq_quadratic (x : RegionParameter) (t : ℝ)
(hm3 : patternMass (regionTheta x) x.1 x.2.2.2 3 ≠ 0)
(hm7 : patternMass (regionTheta x) x.1 x.2.2.2 7 ≠ 0) :
r5TernaryObjective x t = r5TernaryCoeff x * (t + 1) ^ 2 := by
have hd : exp x.2.2.2 + 2 ≠ 0 := by positivity
unfold r5TernaryObjective r5TernaryCoeff patternInformation
simp +decide [projectedGradient, patternGradient, patternContains,
controlProb, direction, Fin.sum_univ_succ]
field_simp [hm3, hm7, hd]
theorem r5Coeffs_pos_of_interior
Under the supplied quantities and conditions, the r5 coeffs pos of interior assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the r5 Coeffs pos of interior.
Formal statement
hypotheses
x :
hp0 :
0 < x.1
hp1 :
x.1 < 1
hμ00 :
0 < x.2.1
hμ01 :
x.2.1 < 1
hμ10 :
0 < x.2.2.1
hμ11 :
x.2.2.1 < 1
hε :
0 < x.2.2.2
Proof (Lean source)
lemma r5Coeffs_pos_of_interior (x : RegionParameter)
(hp0 : 0 < x.1) (hp1 : x.1 < 1)
(hμ00 : 0 < x.2.1) (hμ01 : x.2.1 < 1)
(hμ10 : 0 < x.2.2.1) (hμ11 : x.2.2.1 < 1)
(hε : 0 < x.2.2.2) :
0 < r5BinaryCoeff x ∧ 0 < r5TernaryCoeff x := by
have hd : 0 < privacyIncrement x.2.2.2 := by
unfold privacyIncrement privacyRatio
exact sub_pos.mpr ((Real.one_lt_exp_iff).2 hε)
have hm (s : Fin 14) :
0 < patternMass (regionTheta x) x.1 x.2.2.2 s := by
exact patternMass_pos_of_interior x.1 x.2.1 x.2.2.1 x.2.2.2
hp0 hp1 hμ00 hμ01 hμ10 hμ11 s
constructor
· unfold r5BinaryCoeff
exact mul_pos (div_pos (sq_pos_of_pos hd) (by positivity))
(add_pos (one_div_pos.mpr (hm 5)) (one_div_pos.mpr (hm 8)))
· unfold r5TernaryCoeff
exact mul_pos
(div_pos (mul_pos (sq_pos_of_pos hd) (sq_pos_of_pos hp0)) (by positivity))
(add_pos (one_div_pos.mpr (hm 3)) (one_div_pos.mpr (hm 7)))
theorem continuousAt_r5BinaryCoeff
Under the supplied quantities and conditions, the continuous at r5 binary coeff assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
x :
hm5 :
patternMass (regionTheta x) x.1 x.2.2.2 5 ≠ 0
hm8 :
patternMass (regionTheta x) x.1 x.2.2.2 8 ≠ 0
conclusion
Proof (Lean source)
lemma continuousAt_r5BinaryCoeff (x : RegionParameter)
(hm5 : patternMass (regionTheta x) x.1 x.2.2.2 5 ≠ 0)
(hm8 : patternMass (regionTheta x) x.1 x.2.2.2 8 ≠ 0) :
ContinuousAt r5BinaryCoeff x := by
unfold r5BinaryCoeff privacyIncrement privacyRatio
apply ContinuousAt.mul
· apply ContinuousAt.div
· exact (((Real.continuous_exp.comp continuous_snd.snd.snd).sub
continuous_const).pow 2).continuousAt
· exact ((Real.continuous_exp.comp continuous_snd.snd.snd).add
continuous_const).continuousAt
· positivity
· apply ContinuousAt.add
· exact continuousAt_const.div
(continuous_patternMass_region 5).continuousAt hm5
· exact continuousAt_const.div
(continuous_patternMass_region 8).continuousAt hm8
theorem continuousAt_r5TernaryCoeff
Under the supplied quantities and conditions, the continuous at r5 ternary coeff assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
x :
hm3 :
patternMass (regionTheta x) x.1 x.2.2.2 3 ≠ 0
hm7 :
patternMass (regionTheta x) x.1 x.2.2.2 7 ≠ 0
conclusion
Proof (Lean source)
lemma continuousAt_r5TernaryCoeff (x : RegionParameter)
(hm3 : patternMass (regionTheta x) x.1 x.2.2.2 3 ≠ 0)
(hm7 : patternMass (regionTheta x) x.1 x.2.2.2 7 ≠ 0) :
ContinuousAt r5TernaryCoeff x := by
unfold r5TernaryCoeff privacyIncrement privacyRatio
apply ContinuousAt.mul
· apply ContinuousAt.div
· exact ((((Real.continuous_exp.comp continuous_snd.snd.snd).sub
continuous_const).pow 2).mul (continuous_fst.pow 2)).continuousAt
· exact ((Real.continuous_exp.comp continuous_snd.snd.snd).add
continuous_const).continuousAt
· positivity
· apply ContinuousAt.add
· exact continuousAt_const.div
(continuous_patternMass_region 3).continuousAt hm3
· exact continuousAt_const.div
(continuous_patternMass_region 7).continuousAt hm7
def r5Scale
The scaling factor in the five-output continuation argument is determined by the displayed parameters.
Definition (Lean source)
def r5DirectionBranch
The selected directional branch for the five-output region is determined by the displayed parameters.
Definition (Lean source)
parameters
σ :
ℝ
x :
def
r5DirectionBranch σ x :
ℝ
theorem r5Scale_sq
Under the supplied quantities and conditions, the r5 scale sq assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma r5Scale_sq (x : RegionParameter)
(hA : 0 < r5BinaryCoeff x) (hC : 0 ≤ r5TernaryCoeff x) :
r5Scale x ^ 2 = r5TernaryCoeff x / r5BinaryCoeff x := by
unfold r5Scale
rw [Real.sq_sqrt]
exact div_nonneg hC hA.le
theorem r5DirectionBranch_objectives_eq
Under the supplied quantities and conditions, the r5 direction branch objectives eq assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the r5 Direction Branch objectives eq.
Formal statement
hypotheses
x :
σ :
ℝ
hσ :
σ ^ 2 = 1
hA :
0 < r5BinaryCoeff x
hC :
0 ≤ r5TernaryCoeff x
hden :
1 - 2 * x.1 + σ * r5Scale x ≠ 0
conclusion
r5BinaryCoeff x * (x.1 - (1 - 2 * x.1) * r5DirectionBranch σ x) ^ 2
= r5TernaryCoeff x * (r5DirectionBranch σ x + 1) ^ 2
Proof (Lean source)
lemma r5DirectionBranch_objectives_eq (x : RegionParameter) (σ : ℝ)
(hσ : σ ^ 2 = 1)
(hA : 0 < r5BinaryCoeff x) (hC : 0 ≤ r5TernaryCoeff x)
(hden : 1 - 2 * x.1 + σ * r5Scale x ≠ 0) :
r5BinaryCoeff x *
(x.1 - (1 - 2 * x.1) * r5DirectionBranch σ x) ^ 2 =
r5TernaryCoeff x * (r5DirectionBranch σ x + 1) ^ 2 := by
have hs := r5Scale_sq x hA hC
have hden' : 1 - x.1 * 2 + σ * r5Scale x ≠ 0 := by
convert hden using 1 <;> ring
have hleft :
x.1 - (1 - 2 * x.1) * r5DirectionBranch σ x =
σ * r5Scale x * (1 - x.1) /
(1 - 2 * x.1 + σ * r5Scale x) := by
unfold r5DirectionBranch
field_simp [hden, hden']
ring
have hright : r5DirectionBranch σ x + 1 =
(1 - x.1) / (1 - 2 * x.1 + σ * r5Scale x) := by
unfold r5DirectionBranch
field_simp [hden, hden']
ring
have hCA : r5BinaryCoeff x * r5Scale x ^ 2 = r5TernaryCoeff x := by
rw [hs, mul_div_cancel₀ _ (ne_of_gt hA)]
have hcore : r5BinaryCoeff x * σ ^ 2 * r5Scale x ^ 2 =
r5TernaryCoeff x := by
rw [hσ, mul_one]
exact hCA
rw [hleft, hright]
field_simp [hden]
rw [hcore]
ring
theorem exists_r5DirectionBranch_eq
Under the supplied quantities and conditions, the exists r5 direction branch eq assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
x :
t :
ℝ
hp1 :
x.1 < 1
hA :
0 < r5BinaryCoeff x
hC :
0 ≤ r5TernaryCoeff x
heq :
r5BinaryCoeff x * (x.1 - (1 - 2 * x.1) * t) ^ 2 = r5TernaryCoeff x * (t + 1) ^ 2
conclusion
∃ σ : ℝ,
conclusion 1
σ = 1 ∨ σ = -1
conclusion 2
1 - 2 * x.1 + σ * r5Scale x ≠ 0
conclusion 3
r5DirectionBranch σ x = t
Proof (Lean source)
lemma exists_r5DirectionBranch_eq (x : RegionParameter) (t : ℝ)
(hp1 : x.1 < 1)
(hA : 0 < r5BinaryCoeff x) (hC : 0 ≤ r5TernaryCoeff x)
(heq : r5BinaryCoeff x * (x.1 - (1 - 2 * x.1) * t) ^ 2 =
r5TernaryCoeff x * (t + 1) ^ 2) :
∃ σ : ℝ, (σ = 1 ∨ σ = -1) ∧
1 - 2 * x.1 + σ * r5Scale x ≠ 0 ∧
r5DirectionBranch σ x = t := by
have hs := r5Scale_sq x hA hC
have hCA : r5BinaryCoeff x * r5Scale x ^ 2 = r5TernaryCoeff x := by
rw [hs, mul_div_cancel₀ _ (ne_of_gt hA)]
have habsq : (x.1 - (1 - 2 * x.1) * t) ^ 2 =
(r5Scale x * (t + 1)) ^ 2 := by
rw [← hCA] at heq
nlinarith [sq_nonneg (x.1 - (1 - 2 * x.1) * t),
sq_nonneg (r5Scale x * (t + 1))]
rcases (sq_eq_sq_iff_eq_or_eq_neg.mp habsq) with hrel | hrel
· refine ⟨1, inl rfl, ?_, ?_⟩
· intro hden
have hnum : x.1 - r5Scale x = 0 := by
linear_combination hrel + t * hden
nlinarith
· unfold r5DirectionBranch
rw [div_eq_iff (by
intro hden
have hnum : x.1 - r5Scale x = 0 := by
linear_combination hrel + t * hden
nlinarith : 1 - 2 * x.1 + 1 * r5Scale x ≠ 0)]
nlinarith
· refine ⟨-1, inr rfl, ?_, ?_⟩
· intro hden
have hnum : x.1 + r5Scale x = 0 := by
linear_combination hrel + t * hden
nlinarith
· unfold r5DirectionBranch
rw [div_eq_iff (by
intro hden
have hnum : x.1 + r5Scale x = 0 := by
linear_combination hrel + t * hden
nlinarith : 1 - 2 * x.1 + (-1) * r5Scale x ≠ 0)]
nlinarith
theorem continuousAt_r5Scale
Under the supplied quantities and conditions, the continuous at r5 scale assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
x :
hA :
r5BinaryCoeff x ≠ 0
hm3 :
patternMass (regionTheta x) x.1 x.2.2.2 3 ≠ 0
hm5 :
patternMass (regionTheta x) x.1 x.2.2.2 5 ≠ 0
hm7 :
patternMass (regionTheta x) x.1 x.2.2.2 7 ≠ 0
hm8 :
patternMass (regionTheta x) x.1 x.2.2.2 8 ≠ 0
conclusion
Proof (Lean source)
lemma continuousAt_r5Scale (x : RegionParameter)
(hA : r5BinaryCoeff x ≠ 0)
(hm3 : patternMass (regionTheta x) x.1 x.2.2.2 3 ≠ 0)
(hm5 : patternMass (regionTheta x) x.1 x.2.2.2 5 ≠ 0)
(hm7 : patternMass (regionTheta x) x.1 x.2.2.2 7 ≠ 0)
(hm8 : patternMass (regionTheta x) x.1 x.2.2.2 8 ≠ 0) :
ContinuousAt r5Scale x := by
unfold r5Scale
exact Real.continuous_sqrt.continuousAt.comp
((continuousAt_r5TernaryCoeff x hm3 hm7).div
(continuousAt_r5BinaryCoeff x hm5 hm8) hA)
theorem continuousAt_r5DirectionBranch
Under the supplied quantities and conditions, the continuous at r5 direction branch assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
x :
σ :
ℝ
hA :
r5BinaryCoeff x ≠ 0
hm3 :
patternMass (regionTheta x) x.1 x.2.2.2 3 ≠ 0
hm5 :
patternMass (regionTheta x) x.1 x.2.2.2 5 ≠ 0
hm7 :
patternMass (regionTheta x) x.1 x.2.2.2 7 ≠ 0
hm8 :
patternMass (regionTheta x) x.1 x.2.2.2 8 ≠ 0
hden :
1 - 2 * x.1 + σ * r5Scale x ≠ 0
conclusion
Proof (Lean source)
lemma continuousAt_r5DirectionBranch (x : RegionParameter) (σ : ℝ)
(hA : r5BinaryCoeff x ≠ 0)
(hm3 : patternMass (regionTheta x) x.1 x.2.2.2 3 ≠ 0)
(hm5 : patternMass (regionTheta x) x.1 x.2.2.2 5 ≠ 0)
(hm7 : patternMass (regionTheta x) x.1 x.2.2.2 7 ≠ 0)
(hm8 : patternMass (regionTheta x) x.1 x.2.2.2 8 ≠ 0)
(hden : 1 - 2 * x.1 + σ * r5Scale x ≠ 0) :
ContinuousAt (r5DirectionBranch σ) x := by
have hs := continuousAt_r5Scale x hA hm3 hm5 hm7 hm8
unfold r5DirectionBranch
apply ContinuousAt.div
· exact continuous_fst.continuousAt.sub (continuousAt_const.mul hs)
· exact (continuousAt_const.sub (continuousAt_const.mul continuous_fst.continuousAt)).add
(continuousAt_const.mul hs)
· exact hden
def r5MixtureCoordinate
The mixture coordinate induced by a staircase weight is determined by the displayed parameters.
Definition (Lean source)
theorem r5_supported_feasible_parameterization
Under the supplied quantities and conditions, the r5 supported feasible parameterization assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the r5 supported feasible parameterization.
Formal statement
hypotheses
conclusion
conclusion 1
α 5 = r5MixtureCoordinate ε α / (exp ε + 1)
conclusion 2
α 8 = r5MixtureCoordinate ε α / (exp ε + 1)
conclusion 3
α 2 = (1 - r5MixtureCoordinate ε α) / (exp ε + 2)
conclusion 4
α 3 = (1 - r5MixtureCoordinate ε α) / (exp ε + 2)
conclusion 5
α 7 = (1 - r5MixtureCoordinate ε α) / (exp ε + 2)
Proof (Lean source)
lemma r5_supported_feasible_parameterization (ε : ℝ) (hε : 0 < ε)
(α : StaircaseWeight) (hα : staircaseFeasible ε α)
(hout : ∀ s ∉ r5Active, α s = 0) :
α 5 = r5MixtureCoordinate ε α / (exp ε + 1) ∧
α 8 = r5MixtureCoordinate ε α / (exp ε + 1) ∧
α 2 = (1 - r5MixtureCoordinate ε α) / (exp ε + 2) ∧
α 3 = (1 - r5MixtureCoordinate ε α) / (exp ε + 2) ∧
α 7 = (1 - r5MixtureCoordinate ε α) / (exp ε + 2) := by
obtain ⟨h23, h37, h58, hnorm⟩ :=
r5_supported_feasible_coordinates ε hε α hα hout
have hd1 : exp ε + 1 ≠ 0 := by positivity
have hd2 : exp ε + 2 ≠ 0 := by positivity
have h5 : α 5 = r5MixtureCoordinate ε α / (exp ε + 1) := by
unfold r5MixtureCoordinate
field_simp [hd1]
have h2 : α 2 = (1 - r5MixtureCoordinate ε α) / (exp ε + 2) := by
unfold r5MixtureCoordinate
apply (eq_div_iff hd2).2
nlinarith
exact ⟨h5, h58 ▸ h5, h2, h23 ▸ h2, h37 ▸ h23 ▸ h2⟩
theorem r5MixtureCoordinate_mem_Ioo
Under the supplied quantities and conditions, the r5 mixture coordinate mem ioo assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the r5 Mixture Coordinate mem Ioo.
Formal statement
hypotheses
conclusion
r5MixtureCoordinate ε α ∈ Ioo 0 1
Proof (Lean source)
lemma r5MixtureCoordinate_mem_Ioo (ε : ℝ) (hε : 0 < ε)
(α : StaircaseWeight) (hα : staircaseFeasible ε α)
(hout : ∀ s ∉ r5Active, α s = 0)
(hpos : ∀ s ∈ r5Active, 0 < α s) :
r5MixtureCoordinate ε α ∈ Ioo 0 1 := by
obtain ⟨_, _, h2, _, _⟩ :=
r5_supported_feasible_parameterization ε hε α hα hout
have hu0 : 0 < r5MixtureCoordinate ε α := by
unfold r5MixtureCoordinate
exact mul_pos (by positivity) (hpos 5 (by simp [r5Active]))
have h2pos := hpos 2 (by simp [r5Active])
have hd2 : 0 < exp ε + 2 := by positivity
rw [h2] at h2pos
rcases div_pos_iff.mp h2pos with h | h
· exact ⟨hu0, by linarith [h.1]⟩
· linarith [h.2]
def r5BinaryDerivative
The derivative of the binary continuation component is determined by the displayed parameters.
Definition (Lean source)
parameters
x :
t :
ℝ
def
r5BinaryDerivative x t :
ℝ
given by
(patternInformationSlope (regionTheta x) x.1 x.2.2.2 5 t + patternInformationSlope (regionTheta x) x.1 x.2.2.2 8 t) / (exp x.2.2.2 + 1)
def r5TernaryDerivative
The derivative of the ternary continuation component is determined by the displayed parameters.
Definition (Lean source)
parameters
x :
t :
ℝ
def
r5TernaryDerivative x t :
ℝ
given by
(patternInformationSlope (regionTheta x) x.1 x.2.2.2 2 t + patternInformationSlope (regionTheta x) x.1 x.2.2.2 3 t + patternInformationSlope (regionTheta x) x.1 x.2.2.2 7 t) / (exp x.2.2.2 + 2)
theorem r5BinaryDerivative_eq
Under the supplied quantities and conditions, the r5 binary derivative eq assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
x :
t :
ℝ
hm5 :
patternMass (regionTheta x) x.1 x.2.2.2 5 ≠ 0
hm8 :
patternMass (regionTheta x) x.1 x.2.2.2 8 ≠ 0
conclusion
r5BinaryDerivative x t = -2 * r5BinaryCoeff x * (1 - 2 * x.1) * (x.1 - (1 - 2 * x.1) * t)
Proof (Lean source)
lemma r5BinaryDerivative_eq (x : RegionParameter) (t : ℝ)
(hm5 : patternMass (regionTheta x) x.1 x.2.2.2 5 ≠ 0)
(hm8 : patternMass (regionTheta x) x.1 x.2.2.2 8 ≠ 0) :
r5BinaryDerivative x t =
-2 * r5BinaryCoeff x * (1 - 2 * x.1) *
(x.1 - (1 - 2 * x.1) * t) := by
have hd : exp x.2.2.2 + 1 ≠ 0 := by positivity
unfold r5BinaryDerivative r5BinaryCoeff patternInformationSlope
simp +decide [projectedGradient, patternGradient, patternContains,
controlProb, direction, Fin.sum_univ_succ]
field_simp [hm5, hm8, hd]
ring
theorem r5TernaryDerivative_eq
Under the supplied quantities and conditions, the r5 ternary derivative eq assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
x :
t :
ℝ
hm3 :
patternMass (regionTheta x) x.1 x.2.2.2 3 ≠ 0
hm7 :
patternMass (regionTheta x) x.1 x.2.2.2 7 ≠ 0
conclusion
r5TernaryDerivative x t = 2 * r5TernaryCoeff x * (t + 1)
Proof (Lean source)
lemma r5TernaryDerivative_eq (x : RegionParameter) (t : ℝ)
(hm3 : patternMass (regionTheta x) x.1 x.2.2.2 3 ≠ 0)
(hm7 : patternMass (regionTheta x) x.1 x.2.2.2 7 ≠ 0) :
r5TernaryDerivative x t = 2 * r5TernaryCoeff x * (t + 1) := by
have hd : exp x.2.2.2 + 2 ≠ 0 := by positivity
unfold r5TernaryDerivative r5TernaryCoeff patternInformationSlope
simp +decide [projectedGradient, patternGradient, patternContains,
controlProb, direction, Fin.sum_univ_succ]
field_simp [hm3, hm7, hd]
theorem r5_stationarity_eq_mixture
Under the supplied quantities and conditions, the r5 stationarity eq mixture assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the r5 stationarity eq mixture.
Formal statement
hypotheses
x :
α :
hε :
0 < x.2.2.2
hα :
staircaseFeasible x.2.2.2 α
hout :
∀ s ∉ r5Active, α s = 0
t :
ℝ
hstationary :
conclusion
r5MixtureCoordinate x.2.2.2 α * r5BinaryDerivative x t
+ (1 - r5MixtureCoordinate x.2.2.2 α) * r5TernaryDerivative x t
= 0
Proof (Lean source)
lemma r5_stationarity_eq_mixture (x : RegionParameter) (α : StaircaseWeight)
(hε : 0 < x.2.2.2) (hα : staircaseFeasible x.2.2.2 α)
(hout : ∀ s ∉ r5Active, α s = 0) (t : ℝ)
(hstationary :
(∑ s : Fin 14, α s *
patternInformationSlope (regionTheta x) x.1 x.2.2.2 s t) = 0) :
r5MixtureCoordinate x.2.2.2 α * r5BinaryDerivative x t +
(1 - r5MixtureCoordinate x.2.2.2 α) * r5TernaryDerivative x t = 0 := by
obtain ⟨h5, h8, h2, h3, h7⟩ :=
r5_supported_feasible_parameterization x.2.2.2 hε α hα hout
have hd1 : exp x.2.2.2 + 1 ≠ 0 := by positivity
have hd2 : exp x.2.2.2 + 2 ≠ 0 := by positivity
unfold r5BinaryDerivative r5TernaryDerivative
simp only [Fin.sum_univ_succ] at hstationary
simp +decide [r5Active, hout] at hstationary
rw [h2, h3, h5, h7, h8] at hstationary
field_simp [hd1, hd2] at hstationary ⊢
nlinarith
theorem r5TernaryDerivative_ne_zero_of_objectives_eq
Under the supplied quantities and conditions, the r5 ternary derivative ne zero of objectives eq assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
x :
t :
ℝ
hp1 :
x.1 < 1
hA :
0 < r5BinaryCoeff x
hC :
0 < r5TernaryCoeff x
hm3 :
patternMass (regionTheta x) x.1 x.2.2.2 3 ≠ 0
hm7 :
patternMass (regionTheta x) x.1 x.2.2.2 7 ≠ 0
heq :
r5BinaryCoeff x * (x.1 - (1 - 2 * x.1) * t) ^ 2 = r5TernaryCoeff x * (t + 1) ^ 2
conclusion
r5TernaryDerivative x t ≠ 0
Proof (Lean source)
lemma r5TernaryDerivative_ne_zero_of_objectives_eq (x : RegionParameter)
(t : ℝ) (hp1 : x.1 < 1)
(hA : 0 < r5BinaryCoeff x) (hC : 0 < r5TernaryCoeff x)
(hm3 : patternMass (regionTheta x) x.1 x.2.2.2 3 ≠ 0)
(hm7 : patternMass (regionTheta x) x.1 x.2.2.2 7 ≠ 0)
(heq : r5BinaryCoeff x * (x.1 - (1 - 2 * x.1) * t) ^ 2 =
r5TernaryCoeff x * (t + 1) ^ 2) :
r5TernaryDerivative x t ≠ 0 := by
rw [r5TernaryDerivative_eq x t hm3 hm7]
intro hz
have ht : t + 1 = 0 := by nlinarith
rw [ht, zero_pow (by norm_num : (2 : ℕ) ≠ 0), mul_zero] at heq
have hpzero : x.1 - (1 - 2 * x.1) * t = 0 := by
have hsquare : (x.1 - (1 - 2 * x.1) * t) ^ 2 = 0 :=
(mul_eq_zero.mp heq).resolve_left (ne_of_gt hA)
exact (sq_eq_zero_iff).mp hsquare
nlinarith
theorem mixture_stationarity_solve
the mixture stationarity solve assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
u P R :
ℝ
hR :
R ≠ 0
hstationary :
u * P + (1 - u) * R = 0
conclusion
conclusion 1
P - R ≠ 0
conclusion 2
u = -R / (P - R)
Proof (Lean source)
lemma mixture_stationarity_solve {u P R : ℝ}
(hR : R ≠ 0) (hstationary : u * P + (1 - u) * R = 0) :
P - R ≠ 0 ∧ u = -R / (P - R) := by
have hden : P - R ≠ 0 := by
intro h
have : P = R := by linarith
rw [this] at hstationary
apply hR
linear_combination hstationary
refine ⟨hden, ?_⟩
apply (eq_div_iff hden).2
nlinarith
def r5ContinuedMixture
The continued mixture coordinate for the five-output region is determined by the displayed parameters.
Definition (Lean source)
parameters
σ :
ℝ
x :
def
r5ContinuedMixture σ x :
ℝ
given by
t :
theorem r5ContinuedMixture_eq
Under the supplied quantities and conditions, the r5 continued mixture eq assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the r5 Continued Mixture eq.
Formal statement
hypotheses
x :
α :
t σ :
ℝ
ht :
r5DirectionBranch σ x = t
hε :
0 < x.2.2.2
hα :
staircaseFeasible x.2.2.2 α
hout :
∀ s ∉ r5Active, α s = 0
hstationary :
hR :
r5TernaryDerivative x t ≠ 0
conclusion
r5ContinuedMixture σ x = r5MixtureCoordinate x.2.2.2 α
Proof (Lean source)
lemma r5ContinuedMixture_eq (x : RegionParameter) (α : StaircaseWeight)
(t σ : ℝ) (ht : r5DirectionBranch σ x = t)
(hε : 0 < x.2.2.2) (hα : staircaseFeasible x.2.2.2 α)
(hout : ∀ s ∉ r5Active, α s = 0)
(hstationary :
(∑ s : Fin 14, α s *
patternInformationSlope (regionTheta x) x.1 x.2.2.2 s t) = 0)
(hR : r5TernaryDerivative x t ≠ 0) :
r5ContinuedMixture σ x = r5MixtureCoordinate x.2.2.2 α := by
have hmix := r5_stationarity_eq_mixture x α hε hα hout t hstationary
obtain ⟨_, hu⟩ := mixture_stationarity_solve hR hmix
unfold r5ContinuedMixture
rw [ht]
exact hu.symm
theorem continuousAt_r5BinaryDerivative_comp
Under the supplied quantities and conditions, the continuous at r5 binary derivative comp assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
x :
t :
RegionParameter → ℝ
ht :
ContinuousAt t x
hm5 :
patternMass (regionTheta x) x.1 x.2.2.2 5 ≠ 0
hm8 :
patternMass (regionTheta x) x.1 x.2.2.2 8 ≠ 0
conclusion
ContinuousAt (fun y ↦ r5BinaryDerivative y (t y)) x
Proof (Lean source)
lemma continuousAt_r5BinaryDerivative_comp (x : RegionParameter)
(t : RegionParameter → ℝ) (ht : ContinuousAt t x)
(hm5 : patternMass (regionTheta x) x.1 x.2.2.2 5 ≠ 0)
(hm8 : patternMass (regionTheta x) x.1 x.2.2.2 8 ≠ 0) :
ContinuousAt (fun y ↦ r5BinaryDerivative y (t y)) x := by
unfold r5BinaryDerivative
apply ContinuousAt.div
· exact (continuousAt_patternInformationSlope_region 5 t x ht hm5).add
(continuousAt_patternInformationSlope_region 8 t x ht hm8)
· exact ((Real.continuous_exp.comp continuous_snd.snd.snd).add
continuous_const).continuousAt
· positivity
theorem continuousAt_r5TernaryDerivative_comp
Under the supplied quantities and conditions, the continuous at r5 ternary derivative comp assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
x :
t :
RegionParameter → ℝ
ht :
ContinuousAt t x
hm2 :
patternMass (regionTheta x) x.1 x.2.2.2 2 ≠ 0
hm3 :
patternMass (regionTheta x) x.1 x.2.2.2 3 ≠ 0
hm7 :
patternMass (regionTheta x) x.1 x.2.2.2 7 ≠ 0
conclusion
ContinuousAt (fun y ↦ r5TernaryDerivative y (t y)) x
Proof (Lean source)
lemma continuousAt_r5TernaryDerivative_comp (x : RegionParameter)
(t : RegionParameter → ℝ) (ht : ContinuousAt t x)
(hm2 : patternMass (regionTheta x) x.1 x.2.2.2 2 ≠ 0)
(hm3 : patternMass (regionTheta x) x.1 x.2.2.2 3 ≠ 0)
(hm7 : patternMass (regionTheta x) x.1 x.2.2.2 7 ≠ 0) :
ContinuousAt (fun y ↦ r5TernaryDerivative y (t y)) x := by
unfold r5TernaryDerivative
apply ContinuousAt.div
· exact ((continuousAt_patternInformationSlope_region 2 t x ht hm2).add
(continuousAt_patternInformationSlope_region 3 t x ht hm3)).add
(continuousAt_patternInformationSlope_region 7 t x ht hm7)
· exact ((Real.continuous_exp.comp continuous_snd.snd.snd).add
continuous_const).continuousAt
· positivity
theorem continuousAt_r5ContinuedMixture
Under the supplied quantities and conditions, the continuous at r5 continued mixture assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
x :
σ :
ℝ
ht :
hm2 :
patternMass (regionTheta x) x.1 x.2.2.2 2 ≠ 0
hm3 :
patternMass (regionTheta x) x.1 x.2.2.2 3 ≠ 0
hm5 :
patternMass (regionTheta x) x.1 x.2.2.2 5 ≠ 0
hm7 :
patternMass (regionTheta x) x.1 x.2.2.2 7 ≠ 0
hm8 :
patternMass (regionTheta x) x.1 x.2.2.2 8 ≠ 0
hden :
conclusion
Proof (Lean source)
lemma continuousAt_r5ContinuedMixture (x : RegionParameter) (σ : ℝ)
(ht : ContinuousAt (r5DirectionBranch σ) x)
(hm2 : patternMass (regionTheta x) x.1 x.2.2.2 2 ≠ 0)
(hm3 : patternMass (regionTheta x) x.1 x.2.2.2 3 ≠ 0)
(hm5 : patternMass (regionTheta x) x.1 x.2.2.2 5 ≠ 0)
(hm7 : patternMass (regionTheta x) x.1 x.2.2.2 7 ≠ 0)
(hm8 : patternMass (regionTheta x) x.1 x.2.2.2 8 ≠ 0)
(hden : r5BinaryDerivative x (r5DirectionBranch σ x) -
r5TernaryDerivative x (r5DirectionBranch σ x) ≠ 0) :
ContinuousAt (r5ContinuedMixture σ) x := by
have hP := continuousAt_r5BinaryDerivative_comp x
(r5DirectionBranch σ) ht hm5 hm8
have hR := continuousAt_r5TernaryDerivative_comp x
(r5DirectionBranch σ) ht hm2 hm3 hm7
unfold r5ContinuedMixture
exact hR.neg.div (hP.sub hR) hden
def r5WeightFromCoordinate
The staircase weight reconstructed from a mixture coordinate is determined by the displayed parameters.
Definition (Lean source)
parameters
ε u :
ℝ
def
theorem r5WeightFromCoordinate_feasible
Under the supplied quantities and conditions, the r5 weight from coordinate feasible assertion holds.
Formal statement
Proof (Lean source)
lemma r5WeightFromCoordinate_feasible (ε u : ℝ) :
staircaseFeasible ε (r5WeightFromCoordinate ε u) ↔ u ∈ Icc 0 1 := by
have hd1 : 0 < exp ε + 1 := by positivity
have hd2 : 0 < exp ε + 2 := by positivity
constructor
· intro h
have h5 := h.1 5
have h2 := h.1 2
simp +decide [r5WeightFromCoordinate] at h5 h2
constructor
· rcases div_nonneg_iff.mp h5 with hpos | hneg
· exact hpos.1
· exact False.elim ((not_le_of_gt hd1) hneg.2)
· have hn : 0 ≤ 1 - u := by
rcases div_nonneg_iff.mp h2 with hpos | hneg
· exact hpos.1
· exact False.elim ((not_le_of_gt hd2) hneg.2)
linarith
· rintro ⟨hu0, hu1⟩
constructor
· intro s
unfold r5WeightFromCoordinate
split_ifs <;> positivity
· intro j
rcases j with ⟨j, hj⟩
interval_cases j <;>
simp +decide [staircaseMatrix, r5WeightFromCoordinate, patternRay,
patternContains, privacyIncrement, privacyRatio, Fin.sum_univ_succ] <;>
field_simp <;> ring
theorem r5WeightFromCoordinate_eq
Under the supplied quantities and conditions, the r5 weight from coordinate eq assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the r5 Weight From Coordinate eq.
Formal statement
hypotheses
conclusion
r5WeightFromCoordinate ε (r5MixtureCoordinate ε α) = α
Proof (Lean source)
lemma r5WeightFromCoordinate_eq (ε : ℝ) (hε : 0 < ε)
(α : StaircaseWeight) (hα : staircaseFeasible ε α)
(hout : ∀ s ∉ r5Active, α s = 0) :
r5WeightFromCoordinate ε (r5MixtureCoordinate ε α) = α := by
obtain ⟨h5, h8, h2, h3, h7⟩ :=
r5_supported_feasible_parameterization ε hε α hα hout
funext s
rcases s with ⟨s, hs⟩
interval_cases s <;>
simp_all +decide [r5WeightFromCoordinate, r5Active]
theorem r5WeightFromCoordinate_active
Under the supplied quantities and conditions, the r5 weight from coordinate active assertion holds.
Formal statement
Proof (Lean source)
lemma r5WeightFromCoordinate_active (ε u : ℝ) (hu : u ∈ Ioo 0 1) :
activeSupport (r5WeightFromCoordinate ε u) r5Active := by
rcases hu with ⟨hu0, hu1⟩
intro s
rcases s with ⟨s, hs⟩
interval_cases s <;>
simp +decide [r5WeightFromCoordinate, r5Active] <;>
try (constructor <;> positivity) <;> try positivity
theorem r5WeightFromCoordinate_pos
Under the supplied quantities and conditions, the r5 weight from coordinate pos assertion holds.
Formal statement
Proof (Lean source)
lemma r5WeightFromCoordinate_pos (ε u : ℝ) (hu : u ∈ Ioo 0 1) :
∀ s ∈ r5Active, 0 < r5WeightFromCoordinate ε u s := by
rcases hu with ⟨hu0, hu1⟩
intro s hs
rcases s with ⟨s, hslt⟩
interval_cases s <;>
simp_all +decide [r5WeightFromCoordinate, r5Active] <;> try positivity
theorem r5WeightFromCoordinate_stationarity
Under the supplied quantities and conditions, the r5 weight from coordinate stationarity assertion holds.
Formal statement
hypotheses
x :
u t :
ℝ
conclusion
(∑ s : Fin 14, r5WeightFromCoordinate x.2.2.2 u s * patternInformationSlope (regionTheta x) x.1 x.2.2.2 s t)
= u * r5BinaryDerivative x t + (1 - u) * r5TernaryDerivative x t
Proof (Lean source)
lemma r5WeightFromCoordinate_stationarity (x : RegionParameter) (u t : ℝ) :
(∑ s : Fin 14, r5WeightFromCoordinate x.2.2.2 u s *
patternInformationSlope (regionTheta x) x.1 x.2.2.2 s t) =
u * r5BinaryDerivative x t + (1 - u) * r5TernaryDerivative x t := by
have hd1 : exp x.2.2.2 + 1 ≠ 0 := by positivity
have hd2 : exp x.2.2.2 + 2 ≠ 0 := by positivity
unfold r5BinaryDerivative r5TernaryDerivative
simp +decide [r5WeightFromCoordinate, Fin.sum_univ_succ]
field_simp [hd1, hd2]
ring
theorem r5ContinuedWeight_stationary
Under the supplied quantities and conditions, the r5 continued weight stationary assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
x :
σ :
ℝ
hden :
conclusion
(∑ s : Fin 14, r5WeightFromCoordinate x.2.2.2 (r5ContinuedMixture σ x) s * patternInformationSlope (regionTheta x) x.1 x.2.2.2 s (r5DirectionBranch σ x))
= 0
Proof (Lean source)
lemma r5ContinuedWeight_stationary (x : RegionParameter) (σ : ℝ)
(hden : r5BinaryDerivative x (r5DirectionBranch σ x) -
r5TernaryDerivative x (r5DirectionBranch σ x) ≠ 0) :
(∑ s : Fin 14,
r5WeightFromCoordinate x.2.2.2 (r5ContinuedMixture σ x) s *
patternInformationSlope (regionTheta x) x.1 x.2.2.2 s
(r5DirectionBranch σ x)) = 0 := by
rw [r5WeightFromCoordinate_stationarity]
unfold r5ContinuedMixture
field_simp [hden]
ring
def r5DualTarget
The target vector in the five-output dual system is determined by the displayed parameters.
Definition (Lean source)
theorem continuousAt_r5DualTarget
Under the supplied quantities and conditions, the continuous at r5 dual target assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
ContinuousAt (r5DualTarget t) x
Proof (Lean source)
lemma continuousAt_r5DualTarget (t : RegionParameter → ℝ)
(x : RegionParameter) (ht : ContinuousAt t x)
(hm : ∀ s : Fin 14,
patternMass (regionTheta x) x.1 x.2.2.2 s ≠ 0) :
ContinuousAt (r5DualTarget t) x := by
apply continuousAt_pi.mpr
intro i
exact continuousAt_patternInformation_region (r5DualMask i) t x ht
(hm (r5DualMask i))
theorem exists_local_r5Dual
Under the supplied quantities and conditions, the exists local r5 dual assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the exists local r5 Dual.
Formal statement
hypotheses
x :
t :
RegionParameter → ℝ
η0 :
Fin 4 → ℝ
hε :
0 < x.2.2.2
ht :
ContinuousAt t x
hm :
hactive :
∀ s ∈ r5Active,
conclusion
∃ η : RegionParameter → Fin 4 → ℝ,
conclusion 1
ContinuousAt η x
conclusion 2
η x = η0
conclusion 3
Proof (Lean source)
lemma exists_local_r5Dual (x : RegionParameter) (t : RegionParameter → ℝ)
(η0 : Fin 4 → ℝ) (hε : 0 < x.2.2.2)
(ht : ContinuousAt t x)
(hm : ∀ s : Fin 14,
patternMass (regionTheta x) x.1 x.2.2.2 s ≠ 0)
(hactive : ∀ s ∈ r5Active, dualRay x.2.2.2 η0 s =
patternInformation (regionTheta x) x.1 x.2.2.2 s (t x)) :
∃ η : RegionParameter → Fin 4 → ℝ,
ContinuousAt η x ∧ η x = η0 ∧
∀ᶠ y in nhds x,
mulVec (r5DualMatrix y) (η y) = r5DualTarget t y := by
have hbase : mulVec (r5DualMatrix x) η0 = r5DualTarget t x := by
funext i
have hi := hactive (r5DualMask i) (by
fin_cases i <;> simp [r5DualMask, r5Active])
simpa [r5DualMatrix, r5DualTarget, mulVec, dotProduct,
dualRay, mul_comm] using hi
let η := anchoredSolution
r5DualMatrix (r5DualTarget t) (r5DualRightInverse x.2.2.2) η0
obtain ⟨hcont, hbaseη, hev⟩ :=
exists_eventually_anchoredSolution_of_rightInverse
r5DualMatrix (r5DualTarget t) (r5DualRightInverse x.2.2.2)
x η0 continuous_r5DualMatrix.continuousAt
(continuousAt_r5DualTarget t x ht hm)
(r5DualMatrix_mul_rightInverse x hε) hbase
exact ⟨η, hcont, hbaseη, hev⟩
theorem r5_fifthDual_eq
Under the supplied quantities and conditions, the r5 fifth dual eq assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma r5_fifthDual_eq (x : RegionParameter) (η : Fin 4 → ℝ) (t : ℝ)
(hobj : r5BinaryObjective x t = r5TernaryObjective x t)
(h2 : dualRay x.2.2.2 η 2 =
patternInformation (regionTheta x) x.1 x.2.2.2 2 t)
(h3 : dualRay x.2.2.2 η 3 =
patternInformation (regionTheta x) x.1 x.2.2.2 3 t)
(h5 : dualRay x.2.2.2 η 5 =
patternInformation (regionTheta x) x.1 x.2.2.2 5 t)
(h7 : dualRay x.2.2.2 η 7 =
patternInformation (regionTheta x) x.1 x.2.2.2 7 t) :
dualRay x.2.2.2 η 8 =
patternInformation (regionTheta x) x.1 x.2.2.2 8 t := by
have hd1 : exp x.2.2.2 + 1 ≠ 0 := by positivity
have hd2 : exp x.2.2.2 + 2 ≠ 0 := by positivity
have hdualavg :
(dualRay x.2.2.2 η 5 + dualRay x.2.2.2 η 8) /
(exp x.2.2.2 + 1) =
(dualRay x.2.2.2 η 2 + dualRay x.2.2.2 η 3 +
dualRay x.2.2.2 η 7) / (exp x.2.2.2 + 2) := by
simp +decide [dualRay, patternRay, patternContains, privacyIncrement,
privacyRatio, Fin.sum_univ_succ]
field_simp [hd1, hd2]
ring
unfold r5BinaryObjective r5TernaryObjective at hobj
rw [← h5, ← h2, ← h3, ← h7] at hobj
field_simp [hd1, hd2] at hobj hdualavg
have hprod : (exp x.2.2.2 + 2) *
(patternInformation (regionTheta x) x.1 x.2.2.2 8 t -
dualRay x.2.2.2 η 8) = 0 := by
linear_combination hobj - hdualavg
have hz := (mul_eq_zero.mp hprod).resolve_left hd2
linarith
theorem r5_objectives_eq_of_activeDual
The five active dual equalities force the two normalized constituent objectives to agree. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
r5BinaryObjective x t = r5TernaryObjective x t
Proof (Lean source)
lemma r5_objectives_eq_of_activeDual (x : RegionParameter)
(η : Fin 4 → ℝ) (t : ℝ)
(hdual : ∀ s ∈ r5Active,
dualRay x.2.2.2 η s =
patternInformation (regionTheta x) x.1 x.2.2.2 s t) :
r5BinaryObjective x t = r5TernaryObjective x t := by
have hr1 : exp x.2.2.2 + 1 ≠ 0 := by positivity
have hr2 : exp x.2.2.2 + 2 ≠ 0 := by positivity
have h2 := hdual 2 (by simp [r5Active])
have h3 := hdual 3 (by simp [r5Active])
have h5 := hdual 5 (by simp [r5Active])
have h7 := hdual 7 (by simp [r5Active])
have h8 := hdual 8 (by simp [r5Active])
unfold r5BinaryObjective r5TernaryObjective
rw [← h5, ← h8, ← h2, ← h3, ← h7]
simp +decide [dualRay, patternRay, patternContains, privacyIncrement,
privacyRatio, Fin.sum_univ_succ]
field_simp [hr1, hr2]
ring
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.R5Rigidity 3 declarations Linear rigidity of the five-ray feasible family
Linear rigidity of the five-ray feasible family
def r5Active
the r5 active is the mathematical object specified below.
theorem r5_supported_feasible_coordinates
The four normalization equations leave exactly the expected one-dimensional family on the five active rays. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the r5 supported feasible coordinates.
Formal statement
hypotheses
Proof (Lean source)
lemma r5_supported_feasible_coordinates (ε : ℝ) (hε : 0 < ε)
(β : StaircaseWeight) (hβ : staircaseFeasible ε β)
(hout : ∀ s ∉ r5Active, β s = 0) :
β 2 = β 3 ∧ β 3 = β 7 ∧ β 5 = β 8 ∧
(exp ε + 2) * β 2 + (exp ε + 1) * β 5 = 1 := by
have h0 := hβ.2 0
have h1 := hβ.2 1
have h2 := hβ.2 2
have h3 := hβ.2 3
simp +decide [staircaseMatrix, patternRay, patternContains, privacyRatio,
privacyIncrement, r5Active, hout, Fin.sum_univ_succ] at h0 h1 h2 h3
have he : 1 < exp ε := (Real.one_lt_exp_iff).2 hε
have h23 : β 2 = β 3 := by nlinarith
have h37 : β 3 = β 7 := by nlinarith
have h58 : β 5 = β 8 := by nlinarith
refine ⟨h23, h37, h58, ?_⟩
nlinarith
theorem r5_supported_feasible_eq_of_five_eq
A single binary-ray coordinate determines a feasible weight supported on the five-ray family. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the r5 supported feasible eq of five eq.
Formal statement
Proof (Lean source)
lemma r5_supported_feasible_eq_of_five_eq (ε : ℝ) (hε : 0 < ε)
(β γ : StaircaseWeight) (hβ : staircaseFeasible ε β)
(hγ : staircaseFeasible ε γ)
(hβout : ∀ s ∉ r5Active, β s = 0)
(hγout : ∀ s ∉ r5Active, γ s = 0)
(hfive : β 5 = γ 5) : β = γ := by
obtain ⟨hβ23, hβ37, hβ58, hβnorm⟩ :=
r5_supported_feasible_coordinates ε hε β hβ hβout
obtain ⟨hγ23, hγ37, hγ58, hγnorm⟩ :=
r5_supported_feasible_coordinates ε hε γ hγ hγout
have he : 0 < exp ε + 2 := by positivity
have htwo : β 2 = γ 2 := by nlinarith
funext s
by_cases hs : s ∈ r5Active
· rcases s with ⟨s, hslt⟩
interval_cases s <;> simp_all [r5Active]
· rw [hβout s hs, hγout s hs]
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.Refinement 17 declarations The refinement splits each output density into fourteen extreme rays.
Measurable staircase refinement
The refinement splits each output density into fourteen extreme rays. Equality in one information direction constrains rays mixed by a common output law.
theorem channelFisherInfo_staircase_eq_informationMatrix
Under the supplied quantities and conditions, the channel fisher info staircase eq information matrix assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the channel Fisher Info staircase eq information Matrix.
Formal statement
hypotheses
θ :
p ε :
ℝ
hp :
hθ :
hε :
0 < ε
α :
hα :
conclusion
Proof (Lean source)
lemma channelFisherInfo_staircase_eq_informationMatrix
(θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(α : StaircaseWeight) (hα : staircaseFeasible ε α) :
channelFisherInfo θ p (staircaseChannel ε α) =
informationMatrix θ p ε α := by
let Q := staircaseChannel ε α
let R : Fin 14 → ℝ := fun s => ∑ j : Fin 4, patternRay ε s j
have hray (s : Fin 14) (j : Fin 4) : 0 < patternRay ε s j := by
unfold patternRay privacyIncrement privacyRatio
cases patternContains s j <;> simp <;> positivity
have hR (s : Fin 14) : 0 < R s := by
exact lt_of_lt_of_le (hray s 0)
(Finset.single_le_sum (fun j hj => (hray s j).le) (Finset.mem_univ 0))
have hmass (s : Fin 14) : 0 < patternMass θ p ε s := by
unfold patternMass
apply Finset.sum_pos
· intro j hj
exact mul_pos (piTheta_pos p θ ⟨hp, hθ⟩ j) (hray s j)
· simp
have hdensity (j : Fin 4) :
channelDensity Q j =ᵐ[dominatingMeasure Q]
fun s => if α s = 0 then 0 else patternRay ε s j / R s := by
exact channelDensity_staircase_ae ε α hε.le hα.1 j
let _ : IsMarkovKernel Q := staircaseChannel_markov ε α hα
let _ : IsFiniteMeasure (dominatingMeasure Q) :=
⟨by simp [dominatingMeasure]⟩
ext a b
unfold channelFisherInfo informationMatrix
have hintegrand :
(fun s => channelDerivativeDensity p Q a s *
channelDerivativeDensity p Q b s / channelOutputDensity θ p Q s) =ᵐ[
dominatingMeasure Q]
fun s => if α s = 0 then 0 else
patternGradient p ε s a * patternGradient p ε s b /
(patternMass θ p ε s * R s) := by
filter_upwards [hdensity 0, hdensity 1, hdensity 2, hdensity 3] with
s h0 h1 h2 h3
have hd (j : Fin 4) : channelDensity Q j s =
if α s = 0 then 0 else patternRay ε s j / R s := by
fin_cases j <;> assumption
simp_rw [channelDerivativeDensity, channelOutputDensity, hd]
by_cases ha : α s = 0
· simp [ha]
· simp only [ha, ↓reduceIte]
have hder (k : Fin 2) :
(∑ j : Fin 4, inputDerivative p j k *
(patternRay ε s j / R s)) = patternGradient p ε s k / R s := by
simp only [div_eq_mul_inv, ← mul_assoc, ← Finset.sum_mul]
rw [sum_inputDerivative_mul_patternRay]
have hout :
(∑ j : Fin 4, piTheta θ p j * (patternRay ε s j / R s)) =
patternMass θ p ε s / R s := by
simp only [div_eq_mul_inv, ← mul_assoc, ← Finset.sum_mul]
rfl
rw [hder a, hder b, hout]
field_simp [ne_of_gt (hR s), ne_of_gt (hmass s)]
rw [integral_congr_ae hintegrand]
rw [integral_fintype Integrable.of_finite]
apply Finset.sum_congr rfl
intro s hs
rw [show (dominatingMeasure Q).real {s} = α s * R s by
exact dominatingMeasure_staircase_singleton_real ε α hε.le hα.1 s]
by_cases ha : α s = 0
· simp [ha]
· simp only [ha, ↓reduceIte, smul_eq_mul]
field_simp [ne_of_gt (hR s), ne_of_gt (hmass s)]
def informationQuadratic
the information quadratic is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
theorem informationQuadratic_eq_pattern_sum
Under the supplied quantities and conditions, the information quadratic eq pattern sum assertion holds.
Formal statement
hypotheses
conclusion
informationQuadratic (informationMatrix θ p eps α) v
= ∑ s, α s * (∑ k, patternGradient p eps s k * v k) ^ 2 / patternMass θ p eps s
Proof (Lean source)
lemma informationQuadratic_eq_pattern_sum
(θ : TrialParameter) (p eps : ℝ) (α : StaircaseWeight) (v : TrialParameter) :
informationQuadratic (informationMatrix θ p eps α) v =
∑ s, α s * (∑ k, patternGradient p eps s k * v k) ^ 2 /
patternMass θ p eps s := by
unfold informationQuadratic informationMatrix
simp_rw [Finset.mul_sum, Finset.sum_mul, pow_two]
calc
(∑ a, ∑ b, ∑ s,
v a * (α s * patternGradient p eps s a * patternGradient p eps s b /
patternMass θ p eps s) * v b) =
∑ a, ∑ s, ∑ b,
v a * (α s * patternGradient p eps s a * patternGradient p eps s b /
patternMass θ p eps s) * v b := by
apply Finset.sum_congr rfl
intro a _
rw [Finset.sum_comm]
_ = ∑ s, ∑ a, ∑ b,
v a * (α s * patternGradient p eps s a * patternGradient p eps s b /
patternMass θ p eps s) * v b := by rw [Finset.sum_comm]
_ = _ := by
apply Finset.sum_congr rfl
intro s _
simp_rw [Finset.mul_sum, Finset.sum_mul, Finset.sum_div]
apply Finset.sum_congr rfl
intro a _
simp_rw [Finset.mul_sum, Finset.sum_div]
apply Finset.sum_congr rfl
intro b _
ring
theorem channelRow_le_dominating
Every row of a finite-input channel is bounded by its dominating measure.
Formal statement
hypotheses
conclusion
Q j ≤ dominatingMeasure Q
Proof (Lean source)
lemma channelRow_le_dominating {Z : Type*} [MeasurableSpace Z]
(Q : Kernel (Fin 4) Z) (j : Fin 4) :
Q j ≤ dominatingMeasure Q := by
simp only [dominatingMeasure]
exact Finset.single_le_sum (fun _ _ ↦ bot_le) (Finset.mem_univ j)
theorem channelRow_absolutelyContinuous_dominating
the channel row absolutely continuous dominating assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Q j ≪ dominatingMeasure Q
Proof (Lean source)
lemma channelRow_absolutelyContinuous_dominating {Z : Type*} [MeasurableSpace Z]
(Q : Kernel (Fin 4) Z) (j : Fin 4) :
Q j ≪ dominatingMeasure Q :=
Measure.absolutelyContinuous_of_le (channelRow_le_dominating Q j)
theorem channelDensity_measurable
the channel density measurable assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (channelDensity Q j)
Proof (Lean source)
lemma channelDensity_measurable {Z : Type*} [MeasurableSpace Z]
(Q : Kernel (Fin 4) Z) (j : Fin 4) :
Measurable (channelDensity Q j) := by
exact (Measure.measurable_rnDeriv (Q j) (dominatingMeasure Q)).ennreal_toReal
theorem channelDensity_nonneg
the channel density nonneg assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
0 ≤ channelDensity Q j z
Proof (Lean source)
lemma channelDensity_nonneg {Z : Type*} [MeasurableSpace Z]
(Q : Kernel (Fin 4) Z) (j : Fin 4) (z : Z) :
0 ≤ channelDensity Q j z :=
ENNReal.toReal_nonneg
theorem channelDensity_le_one_ae
the channel density le one ae assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
∀ᵐ z ∂dominatingMeasure Q, channelDensity Q j z ≤ 1
Proof (Lean source)
lemma channelDensity_le_one_ae {Z : Type*} [MeasurableSpace Z] {eps : ℝ}
(Q : Kernel (Fin 4) Z) (hQ : StationaryLDP eps Q) (j : Fin 4) :
∀ᵐ z ∂dominatingMeasure Q, channelDensity Q j z ≤ 1 := by
let _ : IsMarkovKernel Q := hQ.1
let _ : IsFiniteMeasure (dominatingMeasure Q) := ⟨by simp [dominatingMeasure]⟩
filter_upwards [Measure.rnDeriv_le_one_of_le (channelRow_le_dominating Q j)] with z hz
exact ENNReal.toReal_mono (by simp) hz
theorem informationQuadratic_eq_channelIntegral
Directional information as a single row-sum Radon--Nikodym integral. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the information Quadratic eq channel Integral.
Formal statement
hypotheses
conclusion
informationQuadratic (channelFisherInfo θ p Q) v
= ∫ z, (∑ k, v k * channelDerivativeDensity p Q k z) ^ 2 / channelOutputDensity θ p Q z ∂dominatingMeasure Q
Proof (Lean source)
lemma informationQuadratic_eq_channelIntegral {Z : Type*} [MeasurableSpace Z]
(p : ℝ) (θ : TrialParameter) (hp : InteriorAssignment p)
(hθ : InteriorMeans θ) (ε : ℝ)
(Q : Kernel (Fin 4) Z) (hQ : StationaryLDP ε Q) (v : TrialParameter) :
informationQuadratic (channelFisherInfo θ p Q) v =
∫ z, (∑ k, v k * channelDerivativeDensity p Q k z) ^ 2 /
channelOutputDensity θ p Q z ∂dominatingMeasure Q := by
exact (outputFisher_eq_informationQuadratic p θ ⟨hp, hθ⟩ ε Q hQ v).symm.trans
(outputFisher_eq_integral p θ Q ⟨hp, hθ⟩ v)
theorem ldp_row_le
the ldp row le assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Proof (Lean source)
lemma ldp_row_le {Z : Type*} [MeasurableSpace Z] {eps : ℝ}
(Q : Kernel (Fin 4) Z) (hQ : StationaryLDP eps Q) (x x' : Fin 4) :
Q x ≤ ofReal (exp eps) • Q x' := by
rw [Measure.le_iff]
intro A hA
simpa [Measure.smul_apply, smul_eq_mul] using hQ.2 A hA x x'
theorem integral_channelDensity
the integral channel density assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma integral_channelDensity {Z : Type*} [MeasurableSpace Z] {eps : ℝ}
(Q : Kernel (Fin 4) Z) (hQ : StationaryLDP eps Q) (j : Fin 4) :
∫ z, channelDensity Q j z ∂dominatingMeasure Q = (Q j).real univ := by
let _ : IsMarkovKernel Q := hQ.1
let _ : IsFiniteMeasure (dominatingMeasure Q) := ⟨by simp [dominatingMeasure]⟩
exact Measure.integral_toReal_rnDeriv
(channelRow_absolutelyContinuous_dominating Q j)
theorem integral_channelDensity_eq_one
the integral channel density eq one assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
∫ z, channelDensity Q j z ∂dominatingMeasure Q = 1
Proof (Lean source)
lemma integral_channelDensity_eq_one {Z : Type*} [MeasurableSpace Z] {eps : ℝ}
(Q : Kernel (Fin 4) Z) (hQ : StationaryLDP eps Q) (j : Fin 4) :
∫ z, channelDensity Q j z ∂dominatingMeasure Q = 1 := by
rw [integral_channelDensity Q hQ j]
let _ : IsMarkovKernel Q := hQ.1
simp
theorem privacyIncrement_pos
Under the supplied quantities and conditions, the privacy increment pos assertion holds.
Formal statement
Proof (Lean source)
lemma privacyIncrement_pos {eps : ℝ} (heps : 0 < eps) :
0 < privacyIncrement eps := by
rw [privacyIncrement, privacyRatio, sub_pos]
exact Real.one_lt_exp_iff.mpr heps
theorem complementary_patternRay
Under the supplied quantities and conditions, the complementary pattern ray assertion holds.
Formal statement
Proof (Lean source)
lemma complementary_patternRay (eps : ℝ) (j : Fin 4) :
patternRay eps 0 j + patternRay eps 13 j = exp eps + 1 := by
have h : patternContains 0 j = !(patternContains 13 j) := by
fin_cases j <;> decide
rw [patternRay, patternRay, h]
cases h13 : patternContains 13 j <;>
simp [privacyIncrement, privacyRatio] <;> ring
theorem staircase_refinement_interior
the staircase refinement interior assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the staircase refinement interior.
Formal statement
hypotheses
Z :
Type*
p :
ℝ
θ :
ε :
ℝ
hp :
hθ :
epsSeq :
ℕ → ℝ
hfixed :
FixedPrivacy epsSeq ε
hQ :
StationaryLDP ε Q
conclusion
Q = K.comp (staircaseChannel ε α)
conclusion 1
projectedScore θ p ε s t = projectedScore θ p ε u t
Proof (Lean source)
lemma staircase_refinement_interior {Z : Type*} [MeasurableSpace Z]
(p : ℝ) (θ : TrialParameter) (ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(epsSeq : ℕ → ℝ) (hfixed : FixedPrivacy epsSeq ε)
(Q : Kernel (Fin 4) Z) (hQ : StationaryLDP ε Q) :
∃ α : StaircaseWeight, ∃ K : Kernel (Fin 14) Z,
staircaseFeasible ε α ∧ IsMarkovKernel K ∧
Q = K.comp (staircaseChannel ε α) ∧
(informationMatrix θ p ε α - channelFisherInfo θ p Q).PosSemidef ∧
(∀ t : ℝ,
informationQuadratic (informationMatrix θ p ε α) (direction t) =
informationQuadratic (channelFisherInfo θ p Q) (direction t) →
∀ s u : Fin 14, α s ≠ 0 → α u ≠ 0 → ¬ (K s) ⟂ₘ (K u) →
projectedScore θ p ε s t = projectedScore θ p ε u t) := by
let hmodel : InteriorData p θ := ⟨hp, hθ⟩
letI : IsMarkovKernel Q := hQ.1
have hr : 1 < exp ε := Real.one_lt_exp_iff.mpr hfixed.1
obtain ⟨R⟩ := Staircase.exists_refinement Q (exp ε) hr (by
intro i j A hA
exact hQ.2 A hA i j)
refine ⟨reindexedWeight R, reindexedPostprocess R,
reindexedWeight_feasible ε R, reindexedPostprocess_markov R,
reindexed_factorizes ε R,
informationDifference_posSemidef p θ ε Q hQ R hmodel, ?_⟩
intro t heq s u hs hu hnot
have hgap :
Staircase.finiteFisher R.T (piTheta θ p)
(Staircase.projectedScore (trialScore θ p) (direction t)) -
Staircase.outputFisher Q (piTheta θ p)
(Staircase.projectedScore (trialScore θ p) (direction t)) = 0 := by
rw [finiteFisher_reindex p θ R hmodel (direction t),
outputFisher_eq_informationQuadratic p θ hmodel ε Q hQ
(direction t)]
linarith
exact reindexed_rigidity p θ Q R hmodel t hgap s u hs hu hnot
theorem channelFisherInfo_posSemidef
the channel fisher info pos semidef assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the channel Fisher Info pos Semidef.
Formal statement
hypotheses
conclusion
(channelFisherInfo θ p Q).PosSemidef
Proof (Lean source)
lemma channelFisherInfo_posSemidef {Z : Type*} [MeasurableSpace Z]
(p : ℝ) (θ : TrialParameter) (hp : InteriorAssignment p)
(hθ : InteriorMeans θ) (ε : ℝ) (Q : Kernel (Fin 4) Z)
(hQ : StationaryLDP ε Q) :
(channelFisherInfo θ p Q).PosSemidef := by
exact channelFisherInfo_posSemidef_interior p θ ⟨hp, hθ⟩ ε Q hQ
theorem staircase_refinement
the staircase refinement assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Ω Z :
μ :
Measure Ω
Y0 Y1 W Y :
ℕ → Ω → ℝ
p :
ℝ
θ :
ε :
ℝ
hmodel :
LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ
epsSeq :
ℕ → ℝ
hfixed :
FixedPrivacy epsSeq ε
hQ :
StationaryLDP ε Q
conclusion
Q = K.comp (staircaseChannel ε α)
conclusion 1
projectedScore θ p ε s t = projectedScore θ p ε u t
Proof (Lean source)
lemma staircase_refinement {Ω Z : Type*} [MeasurableSpace Ω]
[MeasurableSpace Z] (μ : Measure Ω) (Y0 Y1 W Y : ℕ → Ω → ℝ)
(p : ℝ) (θ : TrialParameter) (ε : ℝ)
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ)
(epsSeq : ℕ → ℝ) (hfixed : FixedPrivacy epsSeq ε)
(Q : Kernel (Fin 4) Z) (hQ : StationaryLDP ε Q) :
∃ α : StaircaseWeight, ∃ K : Kernel (Fin 14) Z,
staircaseFeasible ε α ∧ IsMarkovKernel K ∧
Q = K.comp (staircaseChannel ε α) ∧
(informationMatrix θ p ε α - channelFisherInfo θ p Q).PosSemidef ∧
(∀ t : ℝ,
informationQuadratic (informationMatrix θ p ε α) (direction t) =
informationQuadratic (channelFisherInfo θ p Q) (direction t) →
∀ s u : Fin 14, α s ≠ 0 → α u ≠ 0 → ¬ (K s) ⟂ₘ (K u) →
projectedScore θ p ε s t = projectedScore θ p ε u t) := by
exact staircase_refinement_interior p θ ε hmodel.assignmentInterior
hmodel.meansInterior epsSeq hfixed Q hQ
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.RegionOpenness 6 declarations This file provides the positivity facts and affine stationarity coefficient used to construct local certificates for the three-mask support regime.
Continuity substrate for the three-output region
This file provides the positivity facts and affine stationarity coefficient used to construct local certificates for the three-mask support regime.
theorem patternMass_pos_of_interior
Every staircase-pattern mass is positive at interior trial parameters. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pattern Mass pos of interior.
Formal statement
hypotheses
p μ0 μ1 ε :
ℝ
hp0 :
0 < p
hp1 :
p < 1
hμ00 :
0 < μ0
hμ01 :
μ0 < 1
hμ10 :
0 < μ1
hμ11 :
μ1 < 1
s :
Fin 14
conclusion
0 < patternMass (fun k => if k = 0 then μ0 else μ1) p ε s
Proof (Lean source)
lemma patternMass_pos_of_interior (p μ0 μ1 ε : ℝ)
(hp0 : 0 < p) (hp1 : p < 1) (hμ00 : 0 < μ0) (hμ01 : μ0 < 1)
(hμ10 : 0 < μ1) (hμ11 : μ1 < 1) (s : Fin 14) :
0 < patternMass (fun k => if k = 0 then μ0 else μ1) p ε s := by
have hr (j : Fin 4) : 0 < patternRay ε s j := by
simp only [patternRay, privacyIncrement, privacyRatio]
split <;> simp_all <;> positivity
have hpi (j : Fin 4) :
0 ≤ piTheta (fun k => if k = 0 then μ0 else μ1) p j := by
fin_cases j <;> simp only [piTheta, controlProb] <;> simp_all <;> try positivity
all_goals linarith
unfold patternMass
apply Finset.sum_pos'
· intro j _
exact mul_nonneg (hpi j) (hr j).le
· refine ⟨0, Finset.mem_univ _, ?_⟩
change 0 < ((1 - p) * (1 - μ0)) * patternRay ε s 0
exact mul_pos (mul_pos (sub_pos.mpr hp1) (sub_pos.mpr hμ01)) (hr 0)
def r3Weight
The canonical weights on masks 0, 5, and 7. The stated result follows. The r3 Weight is determined by the displayed parameters.
Definition (Lean source)
parameters
ε :
ℝ
def
def r3Stationarity
The stationarity equation evaluated at the canonical three-mask weights. For the displayed inputs and conditions, the stated result follows. The r3 Stationarity is determined by the displayed parameters.
Definition (Lean source)
parameters
p μ0 μ1 ε t :
ℝ
def
r3Stationarity p μ0 μ1 ε t :
ℝ
given by
def r3StationarityCoeff
The linear coefficient of the canonical three-mask stationarity equation. For the displayed inputs and conditions, the stated result follows. The r3 Stationarity Coeff is determined by the displayed parameters.
Definition (Lean source)
parameters
p μ0 μ1 ε :
ℝ
def
r3StationarityCoeff p μ0 μ1 ε :
ℝ
given by
r3Stationarity p μ0 μ1 ε 1 - r3Stationarity p μ0 μ1 ε 0
theorem patternInformationSlope_one_sub_zero
The unit finite difference of a pattern's information slope is its positive quadratic coefficient. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
patternInformationSlope θ p ε s 1 - patternInformationSlope θ p ε s 0
Proof (Lean source)
lemma patternInformationSlope_one_sub_zero (θ : TrialParameter) (p ε : ℝ)
(s : Fin 14) :
patternInformationSlope θ p ε s 1 - patternInformationSlope θ p ε s 0 =
2 * (patternGradient p ε s 0 + patternGradient p ε s 1) ^ 2 /
patternMass θ p ε s := by
simp [patternInformationSlope, projectedGradient, direction, Fin.sum_univ_succ]
ring
theorem r3StationarityCoeff_pos
The canonical three-mask stationarity equation has positive linear coefficient at interior trial parameters and positive privacy budget. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the r3 Stationarity Coeff pos.
Formal statement
hypotheses
p μ0 μ1 ε :
ℝ
hp0 :
0 < p
hp1 :
p < 1
hμ00 :
0 < μ0
hμ01 :
μ0 < 1
hμ10 :
0 < μ1
hμ11 :
μ1 < 1
hε :
0 < ε
conclusion
0 < r3StationarityCoeff p μ0 μ1 ε
Proof (Lean source)
lemma r3StationarityCoeff_pos (p μ0 μ1 ε : ℝ)
(hp0 : 0 < p) (hp1 : p < 1) (hμ00 : 0 < μ0) (hμ01 : μ0 < 1)
(hμ10 : 0 < μ1) (hμ11 : μ1 < 1) (hε : 0 < ε) :
0 < r3StationarityCoeff p μ0 μ1 ε := by
have hm (s : Fin 14) :
0 < patternMass (fun k => if k = 0 then μ0 else μ1) p ε s :=
patternMass_pos_of_interior p μ0 μ1 ε hp0 hp1 hμ00 hμ01 hμ10 hμ11 s
rw [show r3StationarityCoeff p μ0 μ1 ε =
∑ s : Fin 14, r3Weight ε s *
(patternInformationSlope (fun k => if k = 0 then μ0 else μ1) p ε s 1 -
patternInformationSlope (fun k => if k = 0 then μ0 else μ1) p ε s 0) by
unfold r3StationarityCoeff r3Stationarity
rw [← Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro s _
ring]
simp_rw [patternInformationSlope_one_sub_zero]
apply Finset.sum_pos'
· intro s _
apply mul_nonneg
· simp only [r3Weight]
split <;> positivity
· exact div_nonneg (mul_nonneg (by norm_num) (sq_nonneg _)) (hm s).le
· refine ⟨0, Finset.mem_univ _, ?_⟩
apply mul_pos
· simp [r3Weight]
positivity
· apply div_pos
· apply mul_pos
· norm_num
· simp +decide [patternGradient, controlProb, privacyIncrement, privacyRatio]
have : 1 < exp ε := (Real.one_lt_exp_iff).2 hε
positivity
· exact hm 0
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.Regions 6 declarations The two support regimes are instances of one finite primal and strict dual certificate predicate.
Certified staircase support regions
The two support regimes are instances of one finite primal and strict dual certificate predicate. The pattern numbers follow the four-bit enumeration.
def patternInformationSlope
For the supplied quantities and conditions, the pattern information slope is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The pattern Information Slope is determined by the displayed parameters.
Definition (Lean source)
parameters
def
patternInformationSlope θ p ε s t :
ℝ
given by
2 * projectedGradient p ε s t * (patternGradient p ε s 0 + patternGradient p ε s 1) / patternMass θ p ε s
def dualRay
For the supplied quantities and conditions, the dual ray is the mathematical object specified below. The dual Ray is determined by the displayed parameters.
Definition (Lean source)
the active support is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The active Support is determined by the displayed parameters.
Definition (Lean source)
parameters
def
activeSupport α K :
Prop
given by
α s ≠ 0 ↔ s ∈ K
def certifiedRegion
For the supplied quantities and conditions, the certified region is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
given by
{x | let p := x.1 let μ0 := x.2.1 let μ1 := x.2.2.1 let ε := x.2.2.2 let θ : TrialParameter := fun k => if k = 0 then μ0 else μ1 0 < p ∧ p < 1 ∧ 0 < μ0 ∧ μ0 < 1 ∧ 0 < μ1 ∧ μ1 < 1 ∧ 0 < ε ∧ ∃ α : StaircaseWeight, ∃ t : ℝ, ∃ η : Fin 4 → ℝ, staircaseFeasible ε α ∧ activeSupport α K ∧ (∀ s ∈ K, 0 < α s) ∧ (∑ s : Fin 14, α s * patternInformationSlope θ p ε s t) = 0 ∧ (∀ s ∈ K, dualRay ε η s = patternInformation θ p ε s t) ∧ (∀ s ∉ K, patternInformation θ p ε s t < dualRay ε η s) ∧ (distinct → ∀ s ∈ K, ∀ u ∈ K, s ≠ u → projectedScore θ p ε s t ≠ projectedScore θ p ε u t)}
the r3 is the mathematical object specified below.
Definition (Lean source)
def
given by
the r5 is the mathematical object specified below.
Definition (Lean source)
def
given by
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialAdjacentScore 16 declarations Actual adjacent-prefix score characterization
Actual adjacent-prefix score characterization
def canonicalInitLast
Canonical splitting coordinates, with the last coordinate indexed by the literal successor index rather than by an opaque proof term. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
def canonicalSnoc
The inverse of canonicalInitLast. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
theorem measurable_canonicalInitLast
the measurable canonical init last assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (canonicalInitLast (Z := Z) hk)
Proof (Lean source)
lemma measurable_canonicalInitLast {n k : ℕ} {Z : Fin n → Type*}
[∀ i, MeasurableSpace (Z i)] (hk : k + 1 ≤ n) :
Measurable (canonicalInitLast (Z := Z) hk) := by
cases nextIndex_eq_canonicalNextIndex hk
exact measurable_init_last hk
theorem measurable_canonicalSnoc
the measurable canonical snoc assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (canonicalSnoc (Z := Z) hk)
Proof (Lean source)
lemma measurable_canonicalSnoc {n k : ℕ} {Z : Fin n → Type*}
[∀ i, MeasurableSpace (Z i)] (hk : k + 1 ≤ n) :
Measurable (canonicalSnoc (Z := Z) hk) := by
cases nextIndex_eq_canonicalNextIndex hk
exact measurable_snoc hk
theorem canonicalSnoc_initLast
Under the stated assumptions, canonically rejoining a history with its last observation recovers the original history.
Formal statement
hypotheses
conclusion
canonicalSnoc hk (canonicalInitLast hk u) = u
Proof (Lean source)
@[simp] lemma canonicalSnoc_initLast {n k : ℕ} {Z : Fin n → Type*}
(hk : k + 1 ≤ n) (u : History Z (k + 1) hk) :
canonicalSnoc hk (canonicalInitLast hk u) = u := by
exact Fin.snoc_init_self u
theorem canonicalInitLast_snoc
Under the stated assumptions, splitting a canonically extended history recovers its original history and last observation.
Formal statement
hypotheses
conclusion
canonicalInitLast hk (canonicalSnoc hk u) = u
Proof (Lean source)
@[simp] lemma canonicalInitLast_snoc {n k : ℕ} {Z : Fin n → Type*}
(hk : k + 1 ≤ n)
(u : History Z k (le_of_succ_le hk) × Z (canonicalNextIndex hk)) :
canonicalInitLast hk (canonicalSnoc hk u) = u := by
cases u with
| mk h z =>
apply Prod.ext
· funext i
exact congrFun (Fin.init_snoc
(α := fun j : Fin (k + 1) => Z (castLE hk j))
(p := h) (x := z)) i
· exact Fin.snoc_last
(α := fun j : Fin (k + 1) => Z (castLE hk j))
(p := h) (x := z)
theorem integrable_transcriptPrefixMixtureRealDerivative
the integrable transcript prefix mixture real derivative assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Integrable (transcriptPrefixMixtureRealDerivative P θ v p n k hk) (transcriptPrefixReferenceMeasure P n k hk)
Proof (Lean source)
lemma integrable_transcriptPrefixMixtureRealDerivative
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(n k : ℕ) (hk : k ≤ n) :
Integrable (transcriptPrefixMixtureRealDerivative P θ v p n k hk)
(transcriptPrefixReferenceMeasure P n k hk) := by
let ν := transcriptPrefixReferenceMeasure P n k hk
let μ (x : Fin n → Fin 4) :=
(P.transcript n x).map (transcriptPrefix hk)
letI (x : Fin n → Fin 4) : IsProbabilityMeasure (P.transcript n x) :=
(P.factorizes n x).1
letI (x : Fin n → Fin 4) : IsProbabilityMeasure (μ x) :=
Measure.isProbabilityMeasure_map (measurable_transcriptPrefix hk).aemeasurable
letI : IsFiniteMeasure ν := by
dsimp [ν, transcriptPrefixReferenceMeasure]
infer_instance
have hac (x : Fin n → Fin 4) : μ x ≪ ν := by
dsimp [μ, ν, transcriptPrefixReferenceMeasure]
exact (transcriptComponent_ac_reference P n x).map
(measurable_transcriptPrefix hk)
have hcomponent (x : Fin n → Fin 4) : Integrable
(transcriptPrefixComponentRealDensity P n k hk x) ν := by
have h := (integrable_toReal_rnDeriv_mul_iff (hac x)
(f := fun _ => (1 : ℝ))).2 (integrable_const (1 : ℝ))
change Integrable (fun y => ((μ x).rnDeriv ν y).toReal) ν
simpa only [mul_one] using h
unfold transcriptPrefixMixtureRealDerivative
exact integrable_finsetSum _ fun x _ => (hcomponent x).const_mul _
theorem conditionalTranscriptPrefixDirectionalScore_integrable_general
the conditional transcript prefix directional score integrable general assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the conditional Transcript Prefix Directional Score integrable general.
Formal statement
hypotheses
conclusion
Integrable (conditionalTranscriptPrefixDirectionalScore P θ v p n k hk) (transcriptPrefixLaw P θ p n k hk)
Proof (Lean source)
lemma conditionalTranscriptPrefixDirectionalScore_integrable_general
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k ≤ n) :
Integrable (conditionalTranscriptPrefixDirectionalScore P θ v p n k hk)
(transcriptPrefixLaw P θ p n k hk) := by
let ν := transcriptPrefixReferenceMeasure P n k hk
let μ := transcriptPrefixLaw P θ p n k hk
letI : IsFiniteMeasure ν := by
dsimp [ν, transcriptPrefixReferenceMeasure]
infer_instance
letI : IsFiniteMeasure μ := by
dsimp [μ, transcriptPrefixLaw]
infer_instance
rw [← transcriptPrefixLaw_eq_withDensity_reference P θ p n k hk]
rw [integrable_withDensity_iff_integrable_smul'
(Measure.measurable_rnDeriv μ ν) (Measure.rnDeriv_lt_top μ ν)]
refine (integrable_transcriptPrefixMixtureRealDerivative P θ v p n k hk).congr ?_
filter_upwards [transcriptPrefixDensity_toReal_ae_eq_mixture_of_interior
P θ p n k hk hp hθ] with h hdensity
rw [smul_eq_mul, hdensity]
rw [← conditionalTranscriptPrefixDirectionalScore_mul_density
P θ v p hp hθ n k hk h]
theorem setIntegral_conditionalTranscriptPrefixDirectionalScore
the set integral conditional transcript prefix directional score assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the set Integral conditional Transcript Prefix Directional Score.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
hp :
hθ :
n k :
ℕ
hk :
k ≤ n
A :
Set (History (Z n) k hk)
hA :
conclusion
∫ h in A, conditionalTranscriptPrefixDirectionalScore P θ v p n k hk h ∂transcriptPrefixLaw P θ p n k hk
= ∫ h in A, transcriptPrefixMixtureRealDerivative P θ v p n k hk h ∂transcriptPrefixReferenceMeasure P n k hk
Proof (Lean source)
lemma setIntegral_conditionalTranscriptPrefixDirectionalScore
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k ≤ n) (A : Set (History (Z n) k hk))
(hA : MeasurableSet A) :
∫ h in A, conditionalTranscriptPrefixDirectionalScore P θ v p n k hk h
∂transcriptPrefixLaw P θ p n k hk =
∫ h in A, transcriptPrefixMixtureRealDerivative P θ v p n k hk h
∂transcriptPrefixReferenceMeasure P n k hk := by
let ν := transcriptPrefixReferenceMeasure P n k hk
let μ := transcriptPrefixLaw P θ p n k hk
letI : IsFiniteMeasure ν := by
dsimp [ν, transcriptPrefixReferenceMeasure]
infer_instance
letI : IsFiniteMeasure μ := by
dsimp [μ, transcriptPrefixLaw]
infer_instance
rw [← transcriptPrefixLaw_eq_withDensity_reference P θ p n k hk]
simp_rw [← integral_indicator hA]
rw [integral_withDensity_eq_integral_toReal_smul
(Measure.measurable_rnDeriv μ ν) (Measure.rnDeriv_lt_top μ ν)]
apply integral_congr_ae
filter_upwards [transcriptPrefixDensity_toReal_ae_eq_mixture_of_interior
P θ p n k hk hp hθ] with h hdensity
by_cases hh : h ∈ A
· simp only [hh, indicator_of_mem, smul_eq_mul]
rw [show (μ.rnDeriv ν h).toReal =
transcriptPrefixMixtureRealDensity P θ p n k hk h by
simpa [μ, ν] using hdensity]
simpa only [mul_comm] using
(conditionalTranscriptPrefixDirectionalScore_mul_density
P θ v p hp hθ n k hk h)
· simp [hh]
theorem transcriptSuccessorLaw_map_canonicalInitLast
the transcript successor law map canonical init last assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the transcript Successor Law map canonical Init Last.
Formal statement
hypotheses
conclusion
= outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))
Proof (Lean source)
lemma transcriptSuccessorLaw_map_canonicalInitLast
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n) :
(transcriptPrefixLaw P θ p n (k + 1) hk).map
(canonicalInitLast (Z := Z n) hk) =
outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) := by
cases nextIndex_eq_canonicalNextIndex hk
exact transcriptSuccessorLaw_eq_jointChannel P θ p hp hθ n k hk
def actualSuccessorScoreInCoordinates
The actual successor-prefix score, written in canonical split coordinates. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The actual Successor Score In Coordinates is determined by the displayed parameters.
Definition (Lean source)
parameters
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
n k :
ℕ
hk :
k + 1 ≤ n
z :
History (Z n) k (le_of_succ_le hk) × Z n (canonicalNextIndex hk)
def
actualSuccessorScoreInCoordinates P θ v p n k hk z :
ℝ
given by
conditionalTranscriptPrefixDirectionalScore P θ v p n (k + 1) hk (canonicalSnoc hk z)
theorem measurable_actualSuccessorScoreInCoordinates
the measurable actual successor score in coordinates assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (actualSuccessorScoreInCoordinates P θ v p n k hk)
Proof (Lean source)
lemma measurable_actualSuccessorScoreInCoordinates
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(n k : ℕ) (hk : k + 1 ≤ n) :
Measurable (actualSuccessorScoreInCoordinates P θ v p n k hk) := by
unfold actualSuccessorScoreInCoordinates
conditionalTranscriptPrefixDirectionalScore
exact ((measurable_transcriptPrefixMixtureRealDerivative
P θ v p n (k + 1) hk).div
(measurable_transcriptPrefixMixtureRealDensity
P θ p n (k + 1) hk)).comp (measurable_canonicalSnoc hk)
theorem actualSuccessorScoreInCoordinates_integrable
the actual successor score in coordinates integrable assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the actual Successor Score In Coordinates integrable.
Formal statement
hypotheses
conclusion
Integrable (actualSuccessorScoreInCoordinates P θ v p n k hk) (outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk))))
Proof (Lean source)
lemma actualSuccessorScoreInCoordinates_integrable
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n) :
Integrable (actualSuccessorScoreInCoordinates P θ v p n k hk)
(outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))) := by
rw [← transcriptSuccessorLaw_map_canonicalInitLast
P θ p hp hθ n k hk]
rw [integrable_map_measure
(measurable_actualSuccessorScoreInCoordinates P θ v p n k hk).aestronglyMeasurable
(measurable_canonicalInitLast hk).aemeasurable]
simpa [Function.comp_def, actualSuccessorScoreInCoordinates] using
(conditionalTranscriptPrefixDirectionalScore_integrable_general
P θ v p hp hθ n (k + 1) hk)
theorem hasDerivAt_transcriptPrefixEventMass
the has deriv at transcript prefix event mass assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the has Deriv At transcript Prefix Event Mass.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
hp :
hθ :
n k :
ℕ
hk :
k ≤ n
A :
Set (History (Z n) k hk)
hA :
conclusion
HasDerivAt (fun u => (transcriptPrefixLaw P (parameterPath θ v u) p n k hk A).toReal) (∫ h in A, transcriptPrefixMixtureRealDerivative P θ v p n k hk h ∂transcriptPrefixReferenceMeasure P n k hk) 0
Proof (Lean source)
lemma hasDerivAt_transcriptPrefixEventMass
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k ≤ n) (A : Set (History (Z n) k hk))
(hA : MeasurableSet A) :
HasDerivAt (fun u =>
(transcriptPrefixLaw P (parameterPath θ v u) p n k hk A).toReal)
(∫ h in A, transcriptPrefixMixtureRealDerivative P θ v p n k hk h
∂transcriptPrefixReferenceMeasure P n k hk) 0 := by
let g : History (Z n) k hk → ℝ := A.indicator (fun _ => 1)
have hg : Measurable g := measurable_const.indicator hA
have hgInt (x : Fin n → Fin 4) : Integrable g
((P.transcript n x).map (transcriptPrefix hk)) := by
letI : IsProbabilityMeasure (P.transcript n x) := (P.factorizes n x).1
letI : IsProbabilityMeasure
((P.transcript n x).map (transcriptPrefix hk)) :=
Measure.isProbabilityMeasure_map (measurable_transcriptPrefix hk).aemeasurable
exact (integrable_const (1 : ℝ)).indicator hA
have h := hasDerivAt_integral_transcriptPrefixLaw
P θ v p hp hθ n k hk g hg hgInt
have hcoef : (∫ h, transcriptPrefixMixtureRealDerivative P θ v p n k hk h * g h
∂transcriptPrefixReferenceMeasure P n k hk) =
∫ h in A, transcriptPrefixMixtureRealDerivative P θ v p n k hk h
∂transcriptPrefixReferenceMeasure P n k hk := by
rw [← integral_indicator hA]
apply integral_congr_ae
filter_upwards with y
by_cases hy : y ∈ A <;> simp [g, hy]
rw [hcoef] at h
simpa [g, integral_indicator hA, measureReal_def] using h
theorem actualSuccessorScoreInCoordinates_ae_eq_candidate
The actual adjacent prefix score is the inherited prefix score plus the current channel score, almost everywhere under the actual successor law. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the actual Successor Score In Coordinates ae eq candidate.
Formal statement
hypotheses
conclusion
actualSuccessorScoreInCoordinates P θ v p n k hk
=ᵐ[ outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))] successorDirectionalScoreCandidate P θ v p n k hk
Proof (Lean source)
lemma actualSuccessorScoreInCoordinates_ae_eq_candidate
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n) :
actualSuccessorScoreInCoordinates P θ v p n k hk =ᵐ[
outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))]
successorDirectionalScoreCandidate P θ v p n k hk := by
cases nextIndex_eq_canonicalNextIndex hk
let μ := transcriptPrefixLaw P θ p n (k + 1) hk
let split := canonicalInitLast (Z := Z n) hk
let ν := outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))
have hmap : μ.map split = ν :=
transcriptSuccessorLaw_map_canonicalInitLast P θ p hp hθ n k hk
have hf := actualSuccessorScoreInCoordinates_integrable
P θ v p hp hθ n k hk
have hg := successorDirectionalScoreCandidate_integrable
P θ v p hp hθ n k hk
refine Integrable.ae_eq_of_forall_setIntegral_eq _ _ hf hg ?_
intro A hA _
have hprefix := hasDerivAt_transcriptPrefixEventMass P θ v p hp hθ
n (k + 1) hk (split ⁻¹' A) (hA.preimage (measurable_canonicalInitLast hk))
have hsuccessor := hasDerivAt_successorPrefixEventMass
P θ v p hp hθ n k hk A hA
have hevent : (fun u =>
(transcriptPrefixLaw P (parameterPath θ v u) p n (k + 1) hk
(split ⁻¹' A)).toReal) =
fun u => successorPrefixEventMass P θ v p n k hk u A := by
funext u
unfold successorPrefixEventMass
change ((transcriptPrefixLaw P (parameterPath θ v u) p n (k + 1) hk)
(split ⁻¹' A)).toReal =
((Measure.map split
(transcriptPrefixLaw P (parameterPath θ v u) p n (k + 1) hk)) A).toReal
rw [Measure.map_apply (measurable_canonicalInitLast hk) hA]
rw [hevent] at hprefix
have hcoeff := hprefix.unique hsuccessor
change (∫ x in A, actualSuccessorScoreInCoordinates P θ v p n k hk x ∂ν) =
∫ x in A, successorDirectionalScoreCandidate P θ v p n k hk x ∂ν
rw [← hmap]
simp_rw [← integral_indicator hA]
have hactualMeas : AEStronglyMeasurable
(A.indicator (actualSuccessorScoreInCoordinates P θ v p n k hk))
(Measure.map split μ) := by
have hm : Measurable
(A.indicator (actualSuccessorScoreInCoordinates P θ v p n k hk)) :=
(measurable_actualSuccessorScoreInCoordinates P θ v p n k hk).indicator hA
exact hm.aestronglyMeasurable
rw [integral_map (measurable_canonicalInitLast hk).aemeasurable
hactualMeas]
have hactual : (fun u => A.indicator
(actualSuccessorScoreInCoordinates P θ v p n k hk) (split u)) =
(split ⁻¹' A).indicator
(conditionalTranscriptPrefixDirectionalScore P θ v p n (k + 1) hk) := by
funext u
by_cases hu : split u ∈ A <;>
simp [hu, split, actualSuccessorScoreInCoordinates]
rw [hactual, integral_indicator
(hA.preimage (measurable_canonicalInitLast hk))]
rw [setIntegral_conditionalTranscriptPrefixDirectionalScore
P θ v p hp hθ n (k + 1) hk (split ⁻¹' A)
(hA.preimage (measurable_canonicalInitLast hk))]
rw [hmap, integral_indicator hA]
rw [← successorEventDerivative_eq_setIntegral_successorDirectionalScoreCandidate
P θ v p hp hθ n k hk A hA]
exact hcoeff
theorem actualSuccessorScoreIncrement_ae_eq_channelDirectionalScore
Consequently, the adjacent score increment is exactly the current joint-channel directional score almost everywhere. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the actual Successor Score Increment ae eq channel Directional Score.
Formal statement
hypotheses
conclusion
(fun z => actualSuccessorScoreInCoordinates P θ v p n k hk z - liftedPrefixDirectionalScore P θ v p n k hk z)
=ᵐ[ outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))] channelDirectionalScore θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) v
Proof (Lean source)
lemma actualSuccessorScoreIncrement_ae_eq_channelDirectionalScore
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n) :
(fun z => actualSuccessorScoreInCoordinates P θ v p n k hk z -
liftedPrefixDirectionalScore P θ v p n k hk z) =ᵐ[
outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))]
channelDirectionalScore θ p
(sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) v := by
filter_upwards [actualSuccessorScoreInCoordinates_ae_eq_candidate
P θ v p hp hθ n k hk] with z hz
rw [hz]
simp [successorDirectionalScoreCandidate]
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialBayesMeasurability 4 declarations Measurability of scalar-direction local risks
Measurability of scalar-direction local risks
theorem measurable_scaledError_sq_scaledDirection
Squared scaled error is jointly measurable in a scalar local direction and an arbitrary released transcript. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (fun az : ℝ × Transcript (Z n) => ofReal ((scaledError P theta (scaledDirection az.1 v) n az.2) ^ 2))
Proof (Lean source)
lemma measurable_scaledError_sq_scaledDirection
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (theta v : TrialParameter) (n : ℕ) :
Measurable (fun az : ℝ × Transcript (Z n) =>
ofReal ((scaledError P theta (scaledDirection az.1 v) n az.2) ^ 2)) := by
apply ENNReal.measurable_ofReal.comp
apply Measurable.pow_const
unfold scaledError contrast localAlternative
simp only [scaledDirection]
exact measurable_const.mul ((P.estimate_measurable n).comp measurable_snd |>.sub (by fun_prop))
theorem measurable_localRisk_scaledDirection
For every procedure, the local risk along a scalar direction is measurable without any countability assumption on the output spaces. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma measurable_localRisk_scaledDirection
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (theta v : TrialParameter) (p : ℝ) (n : ℕ) :
Measurable (fun a => localRisk P theta (scaledDirection a v) p n) := by
let loss : ℝ → Transcript (Z n) → ℝ≥0∞ := fun a z =>
ofReal ((scaledError P theta (scaledDirection a v) n z) ^ 2)
have hloss : Measurable (uncurry loss) := by
exact measurable_scaledError_sq_scaledDirection P theta v n
have hcomponent (x : Fin n → Fin 4) : Measurable (fun a =>
∫⁻ z, loss a z ∂P.transcript n x) := by
letI : IsProbabilityMeasure (P.transcript n x) := (P.factorizes n x).1
exact hloss.lintegral_prod_right
have hweight (x : Fin n → Fin 4) : Measurable (fun a =>
ofReal (inputPathProbability
(localAlternative theta (scaledDirection a v) n) p x)) := by
apply ENNReal.measurable_ofReal.comp
have heq (a : ℝ) : localAlternative theta (scaledDirection a v) n =
parameterPath theta v (a / sqrt n) := by
funext k
simp only [localAlternative, scaledDirection, parameterPath]
ring
simp_rw [heq]
have hc : Continuous (fun u => inputPathProbability
(parameterPath theta v u) p x) := by
rw [continuous_iff_continuousAt]
intro u
exact (hasDerivAt_inputPathProbability_parameterPath theta v p u x).continuousAt
exact hc.measurable.comp (by fun_prop)
have hsum : Measurable (fun a => ∑ x : Fin n → Fin 4,
ofReal (inputPathProbability
(localAlternative theta (scaledDirection a v) n) p x) *
∫⁻ z, loss a z ∂P.transcript n x) := by
apply Finset.measurable_fun_sum
intro x hx
exact (hweight x).mul (hcomponent x)
convert hsum using 1
funext a
unfold localRisk transcriptLaw
rw [MeasureTheory.lintegral_finset_sum_measure]
apply Finset.sum_congr rfl
intro x hx
rw [MeasureTheory.lintegral_smul_measure]
rfl
theorem aemeasurable_localRisk_scaledDirection
The scalar local-risk measurability hypothesis used by compact-prior Bayes ordering is therefore automatic. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma aemeasurable_localRisk_scaledDirection
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (theta v : TrialParameter) (p ell upper : ℝ)
(n : ℕ) :
AEMeasurable (fun a => localRisk P theta (scaledDirection a v) p n)
(parameterMeasure ell upper) :=
(measurable_localRisk_scaledDirection P theta v p n).aemeasurable
theorem eventually_compactPrior_bayesRisk_le_localWorstRisk_auto
The compact-prior Bayes ordering theorem has no remaining measurability premise for a paper procedure. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
theta v :
p H ell upper :
ℝ
htheta :
InteriorMeans theta
w :
ℝ → ℝ
hw :
hwint :
Integrable w (parameterMeasure ell upper)
hw_nonneg :
∀ a, 0 ≤ w a
hnorm :
∫ a, w a ∂Causalean.Stat.Minimax.ObservationDependentVanTrees.parameterMeasure ell upper
= 1
hcover :
∀ a ∈ Icc ell upper,
conclusion
∀ᶠ n in atTop, AEMeasurable (fun a => ofReal (w a) * localRisk P theta (scaledDirection a v) p n) (parameterMeasure ell upper) ∧ (∫⁻ a, ofReal (w a) * localRisk P theta (scaledDirection a v) p n ∂Causalean.Stat.Minimax.ObservationDependentVanTrees.parameterMeasure ell upper)
≤ localWorstRisk P theta p H n
Proof (Lean source)
-- keep: reusable sequential-law, Fisher-information, or van-Trees bridge for related adaptive experiments
lemma eventually_compactPrior_bayesRisk_le_localWorstRisk_auto
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (theta v : TrialParameter) (p H ell upper : ℝ)
(htheta : InteriorMeans theta)
(w : ℝ → ℝ) (hw : Measurable w)
(hwint : Integrable w
(parameterMeasure ell upper))
(hw_nonneg : ∀ a, 0 ≤ w a)
(hnorm : ∫ a, w a
∂Causalean.Stat.Minimax.ObservationDependentVanTrees.parameterMeasure ell upper = 1)
(hcover : ∀ a ∈ Icc ell upper,
sqrt (((scaledDirection a v) 0) ^ 2 +
((scaledDirection a v) 1) ^ 2) ≤ H) :
∀ᶠ n in atTop,
AEMeasurable (fun a => ofReal (w a) *
localRisk P theta (scaledDirection a v) p n)
(parameterMeasure ell upper) ∧
(∫⁻ a, ofReal (w a) *
localRisk P theta (scaledDirection a v) p n
∂Causalean.Stat.Minimax.ObservationDependentVanTrees.parameterMeasure ell upper) ≤
localWorstRisk P theta p H n := by
exact eventually_compactPrior_bayesRisk_le_localWorstRisk
P theta v p H ell upper htheta w hw hwint hw_nonneg hnorm hcover
(fun n ↦ aemeasurable_localRisk_scaledDirection P theta v p ell upper n)
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialBayesOrdering 6 declarations Compact-prior Bayes risk versus local worst risk
Compact-prior Bayes risk versus local worst risk
def scaledDirection
Scalar multiplication of a trial-parameter direction. For the displayed inputs and conditions, the stated result follows.
Definition (Lean source)
parameters
a :
ℝ
v :
def
given by
fun k => a * v k
theorem norm_scaledDirection
Euclidean norm scales by the absolute scalar along a trial direction. For the displayed inputs and conditions, the stated result follows.
Formal statement
Proof (Lean source)
lemma norm_scaledDirection (a : ℝ) (v : TrialParameter) :
sqrt (((scaledDirection a v) 0) ^ 2 +
((scaledDirection a v) 1) ^ 2) =
|a| * sqrt ((v 0) ^ 2 + (v 1) ^ 2) := by
unfold scaledDirection
rw [show (a * v 0) ^ 2 + (a * v 1) ^ 2 =
a ^ 2 * ((v 0) ^ 2 + (v 1) ^ 2) by ring]
rw [Real.sqrt_mul (sq_nonneg a), Real.sqrt_sq_eq_abs]
theorem scaledDirection_mem_cover
The scalar directions indexed by [-R,R] fit in the radius obtained by multiplying R by the Euclidean norm of the base direction. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
≤ R * sqrt ((v 0) ^ 2 + (v 1) ^ 2)
Proof (Lean source)
lemma scaledDirection_mem_cover {R a : ℝ} (v : TrialParameter)
(ha : a ∈ Icc (-R) R) :
sqrt (((scaledDirection a v) 0) ^ 2 +
((scaledDirection a v) 1) ^ 2) ≤
R * sqrt ((v 0) ^ 2 + (v 1) ^ 2) := by
rw [norm_scaledDirection]
apply mul_le_mul_of_nonneg_right _ (Real.sqrt_nonneg _)
rw [abs_le]
exact ⟨ha.1, ha.2⟩
theorem lintegral_ofReal_eq_one_of_integral_eq_one
An integrable nonnegative real density normalized in the Bochner integral is also normalized after embedding into ENNReal. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
A :
Type*
mu :
Measure A
w :
A → ℝ
hwint :
Integrable w mu
hw_nonneg :
∀ᵐ a ∂mu, 0 ≤ w a
hnorm :
∫ a, w a ∂mu = 1
conclusion
∫⁻ a, ofReal (w a) ∂mu = 1
Proof (Lean source)
lemma lintegral_ofReal_eq_one_of_integral_eq_one {A : Type*} [MeasurableSpace A]
(mu : Measure A) (w : A → ℝ) (hwint : Integrable w mu)
(hw_nonneg : ∀ᵐ a ∂mu, 0 ≤ w a) (hnorm : ∫ a, w a ∂mu = 1) :
∫⁻ a, ofReal (w a) ∂mu = 1 := by
rw [← ofReal_integral_eq_lintegral_ofReal hwint hw_nonneg, hnorm,
ENNReal.ofReal_one]
theorem weighted_lintegral_le_uniform
A normalized nonnegative weight preserves a uniform ENNReal upper bound under integration. The measurability conclusion records the exact hypothesis needed by later product and iterated-integral rewrites. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
A :
Type*
mu :
Measure A
w :
A → ℝ
f :
A → ℝ≥0∞
B :
ℝ≥0∞
hw :
hf :
AEMeasurable f mu
hwint :
Integrable w mu
hw_nonneg :
∀ᵐ a ∂mu, 0 ≤ w a
hnorm :
∫ a, w a ∂mu = 1
hle :
∀ᵐ a ∂mu, f a ≤ B
conclusion
conclusion 1
AEMeasurable (fun a => ofReal (w a) * f a) mu
conclusion 2
(∫⁻ a, ofReal (w a) * f a ∂mu) ≤ B
Proof (Lean source)
lemma weighted_lintegral_le_uniform {A : Type*} [MeasurableSpace A]
(mu : Measure A) (w : A → ℝ) (f : A → ℝ≥0∞) (B : ℝ≥0∞)
(hw : Measurable w) (hf : AEMeasurable f mu)
(hwint : Integrable w mu) (hw_nonneg : ∀ᵐ a ∂mu, 0 ≤ w a)
(hnorm : ∫ a, w a ∂mu = 1)
(hle : ∀ᵐ a ∂mu, f a ≤ B) :
AEMeasurable (fun a => ofReal (w a) * f a) mu ∧
(∫⁻ a, ofReal (w a) * f a ∂mu) ≤ B := by
have hwm : AEMeasurable (fun a => ofReal (w a)) mu :=
(ENNReal.measurable_ofReal.comp hw).aemeasurable
constructor
· exact hwm.mul hf
· calc
(∫⁻ a, ofReal (w a) * f a ∂mu) ≤
∫⁻ a, ofReal (w a) * B ∂mu := by
apply lintegral_mono_ae
filter_upwards [hle] with a ha
gcongr
_ = (∫⁻ a, ofReal (w a) ∂mu) * B :=
lintegral_mul_const B (ENNReal.measurable_ofReal.comp hw)
_ = B := by rw [lintegral_ofReal_eq_one_of_integral_eq_one
mu w hwint hw_nonneg hnorm, one_mul]
theorem eventually_compactPrior_bayesRisk_le_localWorstRisk
If a compact family of scalar directions fits inside radius H, then at all sufficiently large sample sizes its normalized Bayes average risk is at most the local worst risk at that radius. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
theta v :
p H ell upper :
ℝ
htheta :
InteriorMeans theta
w :
ℝ → ℝ
hw :
hwint :
Integrable w (parameterMeasure ell upper)
hw_nonneg :
∀ a, 0 ≤ w a
hnorm :
∫ a, w a ∂parameterMeasure ell upper = 1
hcover :
∀ a ∈ Icc ell upper,
hriskMeas :
∀ n,
conclusion
∀ᶠ n in atTop, AEMeasurable (fun a => ofReal (w a) * localRisk P theta (scaledDirection a v) p n) (parameterMeasure ell upper) ∧ (∫⁻ a, ofReal (w a) * localRisk P theta (scaledDirection a v) p n ∂parameterMeasure ell upper)
≤ localWorstRisk P theta p H n
Proof (Lean source)
lemma eventually_compactPrior_bayesRisk_le_localWorstRisk
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (theta v : TrialParameter) (p H ell upper : ℝ)
(htheta : InteriorMeans theta)
(w : ℝ → ℝ) (hw : Measurable w)
(hwint : Integrable w (parameterMeasure ell upper))
(hw_nonneg : ∀ a, 0 ≤ w a)
(hnorm : ∫ a, w a ∂parameterMeasure ell upper = 1)
(hcover : ∀ a ∈ Icc ell upper,
sqrt (((scaledDirection a v) 0) ^ 2 +
((scaledDirection a v) 1) ^ 2) ≤ H)
(hriskMeas : ∀ n, AEMeasurable
(fun a => localRisk P theta (scaledDirection a v) p n)
(parameterMeasure ell upper)) :
∀ᶠ n in atTop,
AEMeasurable (fun a => ofReal (w a) *
localRisk P theta (scaledDirection a v) p n)
(parameterMeasure ell upper) ∧
(∫⁻ a, ofReal (w a) *
localRisk P theta (scaledDirection a v) p n
∂parameterMeasure ell upper) ≤ localWorstRisk P theta p H n := by
filter_upwards [eventually_localAlternative_interior_uniform theta H htheta]
with n hinterior
have hmem : ∀ᵐ a ∂parameterMeasure ell upper, a ∈ Icc ell upper := by
unfold parameterMeasure
exact ae_restrict_mem measurableSet_Icc
apply weighted_lintegral_le_uniform
(parameterMeasure ell upper) w
(fun a => localRisk P theta (scaledDirection a v) p n)
(localWorstRisk P theta p H n) hw (hriskMeas n) hwint
(Filter.Eventually.of_forall hw_nonneg) hnorm
filter_upwards [hmem] with a ha
apply localRisk_le_localWorstRisk
exact ⟨hcover a ha, hinterior (scaledDirection a v) (hcover a ha)⟩
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialDomination 8 declarations For a fixed procedure and sample size, summing the deterministic-input transcript laws gives one finite reference measure.
Common domination of sequential transcript experiments
For a fixed procedure and sample size, summing the deterministic-input transcript laws gives one finite reference measure. Every parameterized transcript mixture is absolutely continuous with respect to this measure and therefore has a Radon--Nikodym density on arbitrary measurable output spaces.
def transcriptReferenceMeasure
The sum of all deterministic-input transcript laws is a common reference measure for the fixed procedure and sample size. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
theorem transcriptComponent_ac_reference
Every deterministic-input transcript law is dominated by the common reference measure. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
P.transcript n x ≪ transcriptReferenceMeasure P n
Proof (Lean source)
-- @node: transcriptComponent_ac_reference
lemma transcriptComponent_ac_reference {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (n : ℕ) (x : Fin n → Fin 4) :
P.transcript n x ≪ transcriptReferenceMeasure P n := by
apply Measure.AbsolutelyContinuous.mk
intro A hA href
simp only [transcriptReferenceMeasure, Measure.coe_finsetSum,
Finset.sum_apply] at href
exact (Finset.sum_eq_zero_iff_of_nonneg (fun _ _ ↦ zero_le)).mp href x
(Finset.mem_univ x)
theorem transcriptLaw_ac_reference
Every parameterized transcript law is dominated by the same reference measure, without any countability assumption on the output sigma-fields. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
transcriptLaw P θ p n ≪ transcriptReferenceMeasure P n
Proof (Lean source)
-- @node: transcriptLaw_ac_reference
lemma transcriptLaw_ac_reference {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ) (n : ℕ) :
transcriptLaw P θ p n ≪ transcriptReferenceMeasure P n := by
unfold transcriptLaw
apply Measure.AbsolutelyContinuous.mk
intro A hA href
simp only [Measure.coe_finsetSum, Finset.sum_apply]
apply Finset.sum_eq_zero
intro x _
rw [Measure.smul_apply, smul_eq_mul]
exact mul_eq_zero_of_right _ (transcriptComponent_ac_reference P n x href)
theorem transcriptReferenceMeasure_apply_univ
The reference measure has total mass equal to the number of deterministic input paths. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: transcriptReferenceMeasure_apply_univ
lemma transcriptReferenceMeasure_apply_univ {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (n : ℕ) :
transcriptReferenceMeasure P n univ = card (Fin n → Fin 4) := by
simp only [transcriptReferenceMeasure, Measure.coe_finsetSum,
Finset.sum_apply]
simp only [(P.factorizes n _).1.measure_univ, Finset.sum_const, nsmul_eq_mul,
mul_one]
norm_cast
instance transcriptReferenceMeasure_isFinite
the transcript reference measure is finite is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
instance transcriptReferenceMeasure_isFinite {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (n : ℕ) :
IsFiniteMeasure (transcriptReferenceMeasure P n) := by
constructor
rw [transcriptReferenceMeasure_apply_univ]
exact ENNReal.natCast_lt_top _
instance transcriptLaw_isFinite
the transcript law is finite is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The transcript Law is Finite is determined by the displayed parameters.
Definition (Lean source)
instance transcriptLaw_isFinite {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ) (n : ℕ) :
IsFiniteMeasure (transcriptLaw P θ p n) := by
constructor
simp only [transcriptLaw, Measure.coe_finsetSum, Finset.sum_apply,
Measure.smul_apply, smul_eq_mul, (P.factorizes n _).1.measure_univ, mul_one]
exact ENNReal.sum_lt_top.mpr fun _ _ ↦ ENNReal.ofReal_lt_top
def transcriptDensity
The Radon--Nikodym density of a transcript law against its common finite reference measure. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The transcript Density is determined by the displayed parameters.
Definition (Lean source)
parameters
def
transcriptDensity P θ p n :
Transcript (Z n) → ℝ≥0∞
given by
(transcriptLaw P θ p n).rnDeriv (transcriptReferenceMeasure P n)
theorem transcriptLaw_eq_withDensity_reference
Integrating the transcript density against the common reference measure recovers the parameterized transcript law. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
(transcriptReferenceMeasure P n).withDensity (transcriptDensity P θ p n)
= transcriptLaw P θ p n
Proof (Lean source)
-- @node: transcriptLaw_eq_withDensity_reference
lemma transcriptLaw_eq_withDensity_reference {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ) (n : ℕ) :
(transcriptReferenceMeasure P n).withDensity
(transcriptDensity P θ p n) = transcriptLaw P θ p n := by
exact Measure.withDensity_rnDeriv_eq (transcriptLaw P θ p n)
(transcriptReferenceMeasure P n) (transcriptLaw_ac_reference P θ p n)
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialDoob 11 declarations This file equips a full released transcript with its coordinate-prefix filtration and instantiates the generic finite-horizon Doob information decomposition for the paper's global finite-mixture score.
Doob decomposition of the sequential transcript score
This file equips a full released transcript with its coordinate-prefix filtration and instantiates the generic finite-horizon Doob information decomposition for the paper's global finite-mixture score. Identifying the conditional expectations with the explicit prefix scores is kept separate until the score-projection substrate is available.
def transcriptPrefixFiltration
The prefix filtration on a length-n transcript is generated at time k by the coordinates whose indices are strictly below k. The stated result follows.
Definition (Lean source)
theorem transcriptPrefixFiltration_zero
the transcript prefix filtration zero assertion holds.
Formal statement
hypotheses
n :
ℕ
Fin n → Type
∀ i, MeasurableSpace (Z i)
conclusion
transcriptPrefixFiltration (Z := Z) 0 = ⊥
Proof (Lean source)
lemma transcriptPrefixFiltration_zero {n : ℕ} {Z : Fin n → Type}
[∀ i, MeasurableSpace (Z i)] :
transcriptPrefixFiltration (Z := Z) 0 = ⊥ := by
apply le_antisymm
· change (⨆ i : Fin n, ⨆ _h : i.val < 0,
comap (fun z : Transcript Z ↦ z i) inferInstance) ≤ ⊥
apply iSup₂_le
intro i hi
exact (Nat.not_lt_zero _ hi).elim
· exact bot_le
theorem transcriptPrefixFiltration_terminal
the transcript prefix filtration terminal assertion holds.
Formal statement
hypotheses
n :
ℕ
Fin n → Type
∀ i, MeasurableSpace (Z i)
conclusion
transcriptPrefixFiltration (Z := Z) n = (inferInstance : MeasurableSpace (Transcript Z))
Proof (Lean source)
lemma transcriptPrefixFiltration_terminal {n : ℕ} {Z : Fin n → Type}
[∀ i, MeasurableSpace (Z i)] :
transcriptPrefixFiltration (Z := Z) n =
(inferInstance : MeasurableSpace (Transcript Z)) := by
apply le_antisymm (Filtration.le _ _)
change (⨆ i : Fin n,
comap (fun z : Transcript Z ↦ z i) inferInstance) ≤
(⨆ i : Fin n, ⨆ _h : i.val < n,
comap (fun z : Transcript Z ↦ z i) inferInstance)
apply iSup_le
intro i
exact le_iSup_of_le i (le_iSup_of_le i.isLt le_rfl)
theorem transcriptPrefixFiltration_eq_comap
The coordinate-prefix filtration agrees with the sigma-algebra pulled back by the explicit prefix map. This is the bridge needed to apply measurable-statistic score projection to the Doob increments. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
transcriptPrefixFiltration (Z := Z) k
Proof (Lean source)
-- keep: reusable sequential-law, Fisher-information, or van-Trees bridge for related adaptive experiments
lemma transcriptPrefixFiltration_eq_comap {n k : ℕ} {Z : Fin n → Type}
[∀ i, MeasurableSpace (Z i)] (hk : k ≤ n) :
transcriptPrefixFiltration (Z := Z) k =
comap (transcriptPrefix (Z := Z) hk)
(inferInstance : MeasurableSpace (History Z k hk)) := by
apply le_antisymm
· apply iSup₂_le
intro i hi
let j : Fin k := ⟨i.val, hi⟩
have hij : castLE hk j = i := Fin.ext (by rfl)
rw [← hij]
have hj : Measurable (fun h : History Z k hk ↦ h j) := measurable_pi_apply j
change comap
(fun z : Transcript Z ↦ (transcriptPrefix (Z := Z) hk z) j) inferInstance ≤
comap (transcriptPrefix (Z := Z) hk) inferInstance
rw [show (fun z : Transcript Z ↦ (transcriptPrefix (Z := Z) hk z) j) =
(fun h : History Z k hk ↦ h j) ∘ transcriptPrefix (Z := Z) hk by rfl]
rw [← MeasurableSpace.comap_comp]
exact comap_mono (g := transcriptPrefix (Z := Z) hk) hj.comap_le
· rw [show (inferInstance : MeasurableSpace (History Z k hk)) =
⨆ j : Fin k, comap (fun h : History Z k hk ↦ h j) inferInstance by
rfl]
rw [MeasurableSpace.comap_iSup]
apply iSup_le
intro j
rw [MeasurableSpace.comap_comp]
exact le_iSup_of_le (castLE hk j) (le_iSup_of_le j.isLt le_rfl)
theorem condExp_globalScore_terminal
At the terminal time, conditioning the global transcript score on the prefix filtration returns the score itself. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the cond Exp global Score terminal.
Formal statement
hypotheses
conclusion
(transcriptLaw P θ p n)[conditionalTranscriptDirectionalScore P θ v p n | transcriptPrefixFiltration (Z := Z n) n]
=ᵐ[transcriptLaw P θ p n] conditionalTranscriptDirectionalScore P θ v p n
Proof (Lean source)
lemma condExp_globalScore_terminal {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
(transcriptLaw P θ p n)[conditionalTranscriptDirectionalScore P θ v p n |
transcriptPrefixFiltration (Z := Z n) n] =ᵐ[transcriptLaw P θ p n]
conditionalTranscriptDirectionalScore P θ v p n := by
letI : IsProbabilityMeasure (transcriptLaw P θ p n) :=
transcriptLaw_isProbability P θ p hp hθ n
rw [transcriptPrefixFiltration_terminal]
exact Filter.Eventually.of_forall (fun z ↦ congrFun (condExp_of_stronglyMeasurable le_rfl
(measurable_conditionalTranscriptDirectionalScore P θ v p n).stronglyMeasurable
(conditionalTranscriptDirectionalScore_integrable P θ v p n hp hθ)) z)
theorem condExp_globalScore_initial
At time zero, conditioning the centered global transcript score on the prefix filtration is zero. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the cond Exp global Score initial.
Formal statement
hypotheses
conclusion
(transcriptLaw P θ p n)[conditionalTranscriptDirectionalScore P θ v p n | transcriptPrefixFiltration (Z := Z n) 0]
=ᵐ[transcriptLaw P θ p n] (0 : Transcript (Z n) → ℝ)
Proof (Lean source)
lemma condExp_globalScore_initial {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
(transcriptLaw P θ p n)[conditionalTranscriptDirectionalScore P θ v p n |
transcriptPrefixFiltration (Z := Z n) 0] =ᵐ[transcriptLaw P θ p n]
(0 : Transcript (Z n) → ℝ) := by
letI : IsProbabilityMeasure (transcriptLaw P θ p n) :=
transcriptLaw_isProbability P θ p hp hθ n
rw [transcriptPrefixFiltration_zero, condExp_bot]
rw [integral_conditionalTranscriptDirectionalScore_eq_zero P θ v p n hp hθ]
rfl
theorem globalScore_fisher_eq_doob_sum
The paper's global transcript Fisher information is the sum of the squared Doob prefix increments. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the global Score fisher eq doob sum.
Formal statement
hypotheses
conclusion
(∫ z, conditionalTranscriptDirectionalScore P θ v p n z ^ 2 ∂transcriptLaw P θ p n)
= ∑ k ∈ range n, ∫ z, increment (transcriptLaw P θ p n) (transcriptPrefixFiltration (Z := Z n)) (conditionalTranscriptDirectionalScore P θ v p n) k z ^ 2 ∂transcriptLaw P θ p n
Proof (Lean source)
-- keep: reusable sequential-law, Fisher-information, or van-Trees bridge for related adaptive experiments
lemma globalScore_fisher_eq_doob_sum {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
(∫ z, conditionalTranscriptDirectionalScore P θ v p n z ^ 2
∂transcriptLaw P θ p n) =
∑ k ∈ range n, ∫ z,
increment
(transcriptLaw P θ p n) (transcriptPrefixFiltration (Z := Z n))
(conditionalTranscriptDirectionalScore P θ v p n) k z ^ 2
∂transcriptLaw P θ p n := by
letI : IsProbabilityMeasure (transcriptLaw P θ p n) :=
transcriptLaw_isProbability P θ p hp hθ n
exact fisher_eq_sum
(transcriptPrefixFiltration (Z := Z n))
(conditionalTranscriptDirectionalScore P θ v p n)
(conditionalTranscriptDirectionalScore_memLp_two P θ v p n hp hθ) n
(condExp_globalScore_terminal P θ v p n hp hθ)
(condExp_globalScore_initial P θ v p n hp hθ)
theorem globalScore_doobIncrement_memLp_two
Every prefix increment of the global transcript score is square-integrable. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the global Score doob Increment mem Lp two.
Formal statement
hypotheses
conclusion
MemLp (increment (transcriptLaw P θ p n) (transcriptPrefixFiltration (Z := Z n)) (conditionalTranscriptDirectionalScore P θ v p n) k) 2 (transcriptLaw P θ p n)
Proof (Lean source)
-- keep: reusable sequential-law, Fisher-information, or van-Trees bridge for related adaptive experiments
lemma globalScore_doobIncrement_memLp_two {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n k : ℕ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
MemLp (increment
(transcriptLaw P θ p n) (transcriptPrefixFiltration (Z := Z n))
(conditionalTranscriptDirectionalScore P θ v p n) k) 2
(transcriptLaw P θ p n) := by
letI : IsProbabilityMeasure (transcriptLaw P θ p n) :=
transcriptLaw_isProbability P θ p hp hθ n
exact increment_memLp
(transcriptPrefixFiltration (Z := Z n))
(conditionalTranscriptDirectionalScore P θ v p n)
(conditionalTranscriptDirectionalScore_memLp_two P θ v p n hp hθ) k
theorem condExp_globalScore_doobIncrement_zero
Every prefix increment of the global transcript score has conditional mean zero given the preceding prefix sigma-algebra. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the cond Exp global Score doob Increment zero.
Formal statement
hypotheses
conclusion
(transcriptLaw P θ p n)[ increment (transcriptLaw P θ p n) (transcriptPrefixFiltration (Z := Z n)) (conditionalTranscriptDirectionalScore P θ v p n) k | transcriptPrefixFiltration (Z := Z n) k]
=ᵐ[transcriptLaw P θ p n] (0 : Transcript (Z n) → ℝ)
Proof (Lean source)
-- keep: reusable sequential-law, Fisher-information, or van-Trees bridge for related adaptive experiments
lemma condExp_globalScore_doobIncrement_zero {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n k : ℕ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
(transcriptLaw P θ p n)[
increment
(transcriptLaw P θ p n) (transcriptPrefixFiltration (Z := Z n))
(conditionalTranscriptDirectionalScore P θ v p n) k |
transcriptPrefixFiltration (Z := Z n) k] =ᵐ[transcriptLaw P θ p n]
(0 : Transcript (Z n) → ℝ) := by
letI : IsProbabilityMeasure (transcriptLaw P θ p n) :=
transcriptLaw_isProbability P θ p hp hθ n
exact increment_condExp_zero
(transcriptPrefixFiltration (Z := Z n))
(conditionalTranscriptDirectionalScore P θ v p n)
(conditionalTranscriptDirectionalScore_memLp_two P θ v p n hp hθ) k
theorem globalScore_eq_sum_doobIncrement
The finite sum of the prefix increments recovers the centered global transcript score almost everywhere. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the global Score eq sum doob Increment.
Formal statement
hypotheses
conclusion
(fun z ↦ ∑ k ∈ range n, increment (transcriptLaw P θ p n) (transcriptPrefixFiltration (Z := Z n)) (conditionalTranscriptDirectionalScore P θ v p n) k z)
=ᵐ[ transcriptLaw P θ p n] conditionalTranscriptDirectionalScore P θ v p n
Proof (Lean source)
-- keep: reusable sequential-law, Fisher-information, or van-Trees bridge for related adaptive experiments
lemma globalScore_eq_sum_doobIncrement {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
(fun z ↦ ∑ k ∈ range n,
increment
(transcriptLaw P θ p n) (transcriptPrefixFiltration (Z := Z n))
(conditionalTranscriptDirectionalScore P θ v p n) k z) =ᵐ[
transcriptLaw P θ p n]
conditionalTranscriptDirectionalScore P θ v p n := by
letI : IsProbabilityMeasure (transcriptLaw P θ p n) :=
transcriptLaw_isProbability P θ p hp hθ n
filter_upwards [increment_sum
(transcriptPrefixFiltration (Z := Z n))
(conditionalTranscriptDirectionalScore P θ v p n)
(conditionalTranscriptDirectionalScore_memLp_two P θ v p n hp hθ) n,
condExp_globalScore_terminal P θ v p n hp hθ,
condExp_globalScore_initial P θ v p n hp hθ] with z hsum hterminal hinitial
rw [hsum, hterminal, hinitial]
simp
theorem globalScore_doobIncrement_orthogonal
Distinct prefix increments of the global transcript score are orthogonal. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the global Score doob Increment orthogonal.
Formal statement
hypotheses
conclusion
∫ z, increment (transcriptLaw P θ p n) (transcriptPrefixFiltration (Z := Z n)) (conditionalTranscriptDirectionalScore P θ v p n) i z * increment (transcriptLaw P θ p n) (transcriptPrefixFiltration (Z := Z n)) (conditionalTranscriptDirectionalScore P θ v p n) j z ∂transcriptLaw P θ p n
= 0
Proof (Lean source)
-- keep: reusable sequential-law, Fisher-information, or van-Trees bridge for related adaptive experiments
lemma globalScore_doobIncrement_orthogonal {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
{i j : ℕ} (hij : i ≠ j) :
∫ z,
increment
(transcriptLaw P θ p n) (transcriptPrefixFiltration (Z := Z n))
(conditionalTranscriptDirectionalScore P θ v p n) i z *
increment
(transcriptLaw P θ p n) (transcriptPrefixFiltration (Z := Z n))
(conditionalTranscriptDirectionalScore P θ v p n) j z
∂transcriptLaw P θ p n = 0 := by
letI : IsProbabilityMeasure (transcriptLaw P θ p n) :=
transcriptLaw_isProbability P θ p hp hθ n
exact increment_orthogonal
(transcriptPrefixFiltration (Z := Z n))
(conditionalTranscriptDirectionalScore P θ v p n)
(conditionalTranscriptDirectionalScore_memLp_two P θ v p n hp hθ) hij
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialFisherInformation 13 declarations Square integrability and one-step Fisher information
Square integrability and one-step Fisher information
theorem transcriptPrefixDerivative_eq_completeDataScoreMixture
the transcript prefix derivative eq complete data score mixture assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
n k :
ℕ
hk :
k ≤ n
h :
History (Z n) k hk
hx :
conclusion
transcriptPrefixMixtureRealDerivative P θ v p n k hk h
= ∑ x : Fin n → Fin 4, inputPathProbability θ p x * inputPathDirectionalScore θ v p x * transcriptPrefixComponentRealDensity P n k hk x h
Proof (Lean source)
lemma transcriptPrefixDerivative_eq_completeDataScoreMixture
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(n k : ℕ) (hk : k ≤ n) (h : History (Z n) k hk)
(hx : ∀ x : Fin n → Fin 4, inputPathProbability θ p x ≠ 0) :
transcriptPrefixMixtureRealDerivative P θ v p n k hk h =
∑ x : Fin n → Fin 4,
inputPathProbability θ p x * inputPathDirectionalScore θ v p x *
transcriptPrefixComponentRealDensity P n k hk x h := by
unfold transcriptPrefixMixtureRealDerivative
apply Finset.sum_congr rfl
intro x _
rw [inputPathDerivative_eq_probability_mul_score θ v p x (hx x)]
theorem abs_conditionalTranscriptPrefixDirectionalScore_le
the abs conditional transcript prefix directional score le assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the abs conditional Transcript Prefix Directional Score le.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
hp :
hθ :
n k :
ℕ
hk :
k ≤ n
h :
History (Z n) k hk
conclusion
|conditionalTranscriptPrefixDirectionalScore P θ v p n k hk h|
≤ inputPathScoreBound (n := n) θ v p
Proof (Lean source)
lemma abs_conditionalTranscriptPrefixDirectionalScore_le
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k ≤ n) (h : History (Z n) k hk) :
|conditionalTranscriptPrefixDirectionalScore P θ v p n k hk h| ≤
inputPathScoreBound (n := n) θ v p := by
let B := inputPathScoreBound (n := n) θ v p
let d := transcriptPrefixMixtureRealDensity P θ p n k hk h
have hB : 0 ≤ B := inputPathScoreBound_nonneg θ v p
have hd : 0 ≤ d := transcriptPrefixMixtureRealDensity_nonneg
P θ p n k hk hp hθ h
have hx (x : Fin n → Fin 4) : inputPathProbability θ p x ≠ 0 :=
ne_of_gt (inputPathProbability_pos_of_interior θ p hp hθ x)
have hn : |transcriptPrefixMixtureRealDerivative P θ v p n k hk h| ≤ B * d := by
rw [transcriptPrefixDerivative_eq_completeDataScoreMixture
P θ v p n k hk h hx]
calc
|∑ x : Fin n → Fin 4, inputPathProbability θ p x *
inputPathDirectionalScore θ v p x *
transcriptPrefixComponentRealDensity P n k hk x h| ≤
∑ x : Fin n → Fin 4, |inputPathProbability θ p x *
inputPathDirectionalScore θ v p x *
transcriptPrefixComponentRealDensity P n k hk x h| :=
Finset.abs_sum_le_sum_abs _ _
_ = ∑ x : Fin n → Fin 4,
(inputPathProbability θ p x *
transcriptPrefixComponentRealDensity P n k hk x h) *
|inputPathDirectionalScore θ v p x| := by
apply Finset.sum_congr rfl
intro x _
have hc : 0 ≤ transcriptPrefixComponentRealDensity P n k hk x h :=
ENNReal.toReal_nonneg
rw [abs_mul, abs_mul, abs_of_pos
(inputPathProbability_pos_of_interior θ p hp hθ x),
abs_of_nonneg hc]
ring
_ ≤ ∑ x : Fin n → Fin 4,
(inputPathProbability θ p x *
transcriptPrefixComponentRealDensity P n k hk x h) * B := by
apply Finset.sum_le_sum
intro x _
exact mul_le_mul_of_nonneg_left
(abs_inputPathDirectionalScore_le_bound θ v p x)
(mul_nonneg (inputPathProbability_pos_of_interior θ p hp hθ x).le
ENNReal.toReal_nonneg)
_ = B * d := by
simp only [d, transcriptPrefixMixtureRealDensity]
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro x _
ring
by_cases hdz : d = 0
· have hn0 : transcriptPrefixMixtureRealDerivative P θ v p n k hk h = 0 := by
rw [hdz, mul_zero] at hn
exact abs_eq_zero.mp (le_antisymm hn (abs_nonneg _))
rw [conditionalTranscriptPrefixDirectionalScore, hn0,
show transcriptPrefixMixtureRealDensity P θ p n k hk h = 0 from hdz]
simpa using hB
· have hdpos : 0 < d := lt_of_le_of_ne hd (Ne.symm hdz)
rw [conditionalTranscriptPrefixDirectionalScore, abs_div,
abs_of_nonneg hd]
exact (div_le_iff₀ hdpos).2 (by simpa [d] using hn)
theorem conditionalTranscriptPrefixDirectionalScore_memLp_two
the conditional transcript prefix directional score mem lp two assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the conditional Transcript Prefix Directional Score mem Lp two.
Formal statement
hypotheses
conclusion
MemLp (conditionalTranscriptPrefixDirectionalScore P θ v p n k hk) 2 (transcriptPrefixLaw P θ p n k hk)
Proof (Lean source)
lemma conditionalTranscriptPrefixDirectionalScore_memLp_two
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k ≤ n) :
MemLp (conditionalTranscriptPrefixDirectionalScore P θ v p n k hk) 2
(transcriptPrefixLaw P θ p n k hk) := by
letI : IsProbabilityMeasure (transcriptPrefixLaw P θ p n k hk) :=
transcriptPrefixLaw_isProbability P θ p hp hθ n k hk
apply MemLp.of_bound
((measurable_transcriptPrefixMixtureRealDerivative P θ v p n k hk).div
(measurable_transcriptPrefixMixtureRealDensity P θ p n k hk)).aestronglyMeasurable
(inputPathScoreBound (n := n) θ v p)
filter_upwards with h
rw [Real.norm_eq_abs]
exact abs_conditionalTranscriptPrefixDirectionalScore_le
P θ v p hp hθ n k hk h
theorem conditionalTranscriptPrefixDirectionalScore_sq_integrable
the conditional transcript prefix directional score sq integrable assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the conditional Transcript Prefix Directional Score sq integrable.
Formal statement
hypotheses
conclusion
Integrable (fun h => conditionalTranscriptPrefixDirectionalScore P θ v p n k hk h ^ 2) (transcriptPrefixLaw P θ p n k hk)
Proof (Lean source)
lemma conditionalTranscriptPrefixDirectionalScore_sq_integrable
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k ≤ n) :
Integrable (fun h =>
conditionalTranscriptPrefixDirectionalScore P θ v p n k hk h ^ 2)
(transcriptPrefixLaw P θ p n k hk) :=
(conditionalTranscriptPrefixDirectionalScore_memLp_two
P θ v p hp hθ n k hk).integrable_sq
theorem actualSuccessorScoreInCoordinates_sq_integrable
the actual successor score in coordinates sq integrable assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the actual Successor Score In Coordinates sq integrable.
Formal statement
hypotheses
conclusion
Integrable (fun z => actualSuccessorScoreInCoordinates P θ v p n k hk z ^ 2) (outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk))))
Proof (Lean source)
lemma actualSuccessorScoreInCoordinates_sq_integrable
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n) :
Integrable (fun z => actualSuccessorScoreInCoordinates P θ v p n k hk z ^ 2)
(outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))) := by
rw [← transcriptSuccessorLaw_map_canonicalInitLast P θ p hp hθ n k hk]
have hm : Measurable (fun z =>
actualSuccessorScoreInCoordinates P θ v p n k hk z ^ 2) :=
(measurable_actualSuccessorScoreInCoordinates P θ v p n k hk).pow_const 2
rw [integrable_map_measure
hm.aestronglyMeasurable
(measurable_canonicalInitLast hk).aemeasurable]
simpa [Function.comp_def, actualSuccessorScoreInCoordinates] using
(conditionalTranscriptPrefixDirectionalScore_sq_integrable
P θ v p hp hθ n (k + 1) hk)
theorem liftedPrefixDirectionalScore_sq_integrable_successorLaw
the lifted prefix directional score sq integrable successor law assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the lifted Prefix Directional Score sq integrable successor Law.
Formal statement
hypotheses
conclusion
Integrable (fun z => liftedPrefixDirectionalScore P θ v p n k hk z ^ 2) (outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk))))
Proof (Lean source)
lemma liftedPrefixDirectionalScore_sq_integrable_successorLaw
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n) :
Integrable (fun z => liftedPrefixDirectionalScore P θ v p n k hk z ^ 2)
(outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))) := by
let i := canonicalNextIndex hk
let μ := transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
let K : Kernel (History (Z n) k (le_of_succ_le hk)) (Z n i) :=
(observedStageKernel P θ p n i).comap
(fun h => ((), h)) (measurable_const.prodMk measurable_id)
letI : IsProbabilityMeasure μ :=
transcriptPrefixLaw_isProbability P θ p hp hθ n k (le_of_succ_le hk)
letI : IsMarkovKernel (observedStageKernel P θ p n i) :=
observedStageKernel_isMarkov P θ p hp hθ n i
letI : IsMarkovKernel K := by
dsimp [K]
infer_instance
have hout : outputLaw θ p (sequentialJointChannel P n i μ) = μ ⊗ₘ K := by
simpa [K] using outputLaw_sequentialJointChannel P θ p hp hθ n i μ
rw [hout]
have hmeas : AEStronglyMeasurable
(fun z => liftedPrefixDirectionalScore P θ v p n k hk z ^ 2) (μ ⊗ₘ K) := by
have hm : Measurable (fun z =>
liftedPrefixDirectionalScore P θ v p n k hk z ^ 2) :=
(measurable_liftedPrefixDirectionalScore P θ v p n k hk).pow_const 2
exact hm.aestronglyMeasurable
rw [Measure.integrable_compProd_iff hmeas]
constructor
· filter_upwards with h
simpa [liftedPrefixDirectionalScore] using
(integrable_const
(conditionalTranscriptPrefixDirectionalScore P θ v p n k
(le_of_succ_le hk) h ^ 2) : Integrable _ (K h))
· simpa [liftedPrefixDirectionalScore, μ] using
(conditionalTranscriptPrefixDirectionalScore_sq_integrable
P θ v p hp hθ n k (le_of_succ_le hk)).norm
theorem channelDirectionalScore_sq_integrable_successorLaw
the channel directional score sq integrable successor law assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the channel Directional Score sq integrable successor Law.
Formal statement
hypotheses
conclusion
Integrable (fun z => channelDirectionalScore θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) v z ^ 2) (outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk))))
Proof (Lean source)
lemma channelDirectionalScore_sq_integrable_successorLaw
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n) :
Integrable (fun z => channelDirectionalScore θ p
(sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) v z ^ 2)
(outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))) := by
let ν := outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))
have ha := actualSuccessorScoreInCoordinates_sq_integrable
P θ v p hp hθ n k hk
have hl := liftedPrefixDirectionalScore_sq_integrable_successorLaw
P θ v p hp hθ n k hk
have hmaMeas : AEStronglyMeasurable
(actualSuccessorScoreInCoordinates P θ v p n k hk) ν := by
exact (measurable_actualSuccessorScoreInCoordinates P θ v p n k hk)
|>.aestronglyMeasurable
have hmlMeas : AEStronglyMeasurable
(liftedPrefixDirectionalScore P θ v p n k hk) ν := by
exact (measurable_liftedPrefixDirectionalScore P θ v p n k hk)
|>.aestronglyMeasurable
have hma : MemLp (actualSuccessorScoreInCoordinates P θ v p n k hk) 2 ν :=
(memLp_two_iff_integrable_sq hmaMeas).2 ha
have hml : MemLp (liftedPrefixDirectionalScore P θ v p n k hk) 2 ν :=
(memLp_two_iff_integrable_sq hmlMeas).2 hl
have hsub := hma.sub hml
have hae := actualSuccessorScoreIncrement_ae_eq_channelDirectionalScore
P θ v p hp hθ n k hk
exact (((memLp_congr_ae hae).mp hsub).integrable_sq)
theorem integral_channelDirectionalScore_sq_successorLaw
the integral channel directional score sq successor law assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the integral channel Directional Score sq successor Law.
Formal statement
hypotheses
conclusion
∫ z, channelDirectionalScore θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) v z ^ 2 ∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))
= informationQuadratic (channelFisherInfo θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))) v
Proof (Lean source)
lemma integral_channelDirectionalScore_sq_successorLaw
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n) :
∫ z, channelDirectionalScore θ p
(sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) v z ^ 2
∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) =
informationQuadratic (channelFisherInfo θ p
(sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))) v := by
let μ := transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
let Q := sequentialJointChannel P n (canonicalNextIndex hk) μ
let ν := dominatingMeasure Q
let ρ := outputLaw θ p Q
letI : IsProbabilityMeasure μ :=
transcriptPrefixLaw_isProbability P θ p hp hθ n k (le_of_succ_le hk)
letI : IsMarkovKernel Q :=
sequentialJointChannel_isMarkov P n (canonicalNextIndex hk) μ
letI : IsFiniteMeasure ν := ⟨by simp [ν, dominatingMeasure]⟩
letI : IsFiniteMeasure ρ := by
constructor
dsimp [ρ, outputLaw]
simp only [Measure.coe_finsetSum, Finset.sum_apply, Measure.smul_apply,
smul_eq_mul, measure_univ, mul_one]
exact ENNReal.sum_lt_top.mpr fun _ _ => ENNReal.ofReal_lt_top
change (∫ z, channelDirectionalScore θ p Q v z ^ 2 ∂ρ) = _
rw [← Measure.withDensity_rnDeriv_eq ρ ν
(outputLaw_ac_dominatingMeasure θ p Q)]
rw [integral_withDensity_eq_integral_toReal_smul
(Measure.measurable_rnDeriv ρ ν) (Measure.rnDeriv_lt_top ρ ν)]
rw [show (∫ z, (ρ.rnDeriv ν z).toReal •
channelDirectionalScore θ p Q v z ^ 2 ∂ν) =
∫ z, channelDirectionalScore θ p Q v z ^ 2 *
channelOutputDensity θ p Q z ∂ν by
apply integral_congr_ae
filter_upwards [outputLawDensity_toReal_ae_eq_channelOutputDensity
θ p Q hp hθ] with z hz
rw [smul_eq_mul, hz]
ring]
exact @integral_sequentialJoint_channelDirectionalScore_sq
Z _ ε P θ v p hp hθ n (canonicalNextIndex hk) μ _
theorem successor_channelFisher_le_upperEnvelope
the successor channel fisher le upper envelope assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the successor channel Fisher le upper Envelope.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ :
p t :
ℝ
hp :
hθ :
hε :
0 < ε
n k :
ℕ
hk :
k + 1 ≤ n
conclusion
∫ z, channelDirectionalScore θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) (direction t) z ^ 2 ∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))
≤ upperEnvelope θ p ε t
Proof (Lean source)
lemma successor_channelFisher_le_upperEnvelope
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p t : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(n k : ℕ) (hk : k + 1 ≤ n) :
∫ z, channelDirectionalScore θ p
(sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))
(direction t) z ^ 2
∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) ≤
upperEnvelope θ p ε t := by
rw [integral_channelDirectionalScore_sq_successorLaw
P θ (direction t) p hp hθ n k hk]
letI : IsProbabilityMeasure
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) :=
transcriptPrefixLaw_isProbability P θ p hp hθ n k (le_of_succ_le hk)
exact sequentialJointChannel_fisher_le_upperEnvelope P n
(canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))
θ p t hp hθ hε
theorem sum_inputDirectionalDerivative_eq_zero
Under the supplied quantities and conditions, the sum input directional derivative eq zero assertion holds.
Formal statement
Proof (Lean source)
lemma sum_inputDirectionalDerivative_eq_zero (p : ℝ) (v : TrialParameter) :
∑ a : Fin 4, inputDirectionalDerivative p v a = 0 := by
unfold inputDirectionalDerivative inputDerivative
simp [Fin.sum_univ_succ]
theorem setIntegral_channelDirectionalScore_fst_preimage_eq_zero
The current channel score has zero integral on every event determined by the predecessor history. This is the exact first-marginal centering input for the remaining cross-term argument. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the set Integral channel Directional Score fst preimage eq zero.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
hp :
hθ :
n k :
ℕ
hk :
k + 1 ≤ n
B :
Set (History (Z n) k (le_of_succ_le hk))
hB :
conclusion
∫ z in fst ⁻¹' B, channelDirectionalScore θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) v z ∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))
= 0
Proof (Lean source)
lemma setIntegral_channelDirectionalScore_fst_preimage_eq_zero
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n)
(B : Set (History (Z n) k (le_of_succ_le hk)))
(hB : MeasurableSet B) :
∫ z in Prod.fst ⁻¹' B, channelDirectionalScore θ p
(sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) v z
∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) = 0 := by
let μ := transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
let i := canonicalNextIndex hk
let Q := sequentialJointChannel P n i μ
letI : IsProbabilityMeasure μ :=
transcriptPrefixLaw_isProbability P θ p hp hθ n k (le_of_succ_le hk)
letI : IsMarkovKernel Q := sequentialJointChannel_isMarkov P n i μ
have hA : MeasurableSet (Prod.fst ⁻¹' B :
Set (History (Z n) k (le_of_succ_le hk) × Z n i)) :=
hB.preimage measurable_fst
rw [setIntegral_channelDirectionalScore_outputLaw θ v p Q hp hθ _ hA]
rw [@setIntegral_sequentialJoint_channelDirectionalScore
Z _ ε P θ v p hp hθ n i μ _ (Prod.fst ⁻¹' B)]
unfold sequentialJointEventDerivative
change (∑ a : Fin 4,
inputDirectionalDerivative p v a * (Q a (Prod.fst ⁻¹' B)).toReal) = 0
have hrow (a : Fin 4) : (Q a (Prod.fst ⁻¹' B)).toReal = (μ B).toReal := by
letI : IsMarkovKernel (P.channel n i) := (P.privacy n).1 i
rw [show Prod.fst ⁻¹' B = B ×ˢ univ by ext z; simp]
dsimp [Q, sequentialJointChannel, releasedHistoryRowKernel]
rw [Kernel.compProd_apply_prod hB MeasurableSet.univ]
simp_rw [Kernel.comap_apply', measure_univ]
rw [Kernel.const_apply, setLIntegral_one]
simp_rw [hrow]
rw [← Finset.sum_mul]
rw [sum_inputDirectionalDerivative_eq_zero]
simp
theorem condExp_channelDirectionalScore_fst_eq_zero
The score increment is conditionally centered given the predecessor history. This formulation is the one needed to kill the Fisher cross term. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the cond Exp channel Directional Score fst eq zero.
Formal statement
hypotheses
conclusion
condExp (comap fst inferInstance) (outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))) (channelDirectionalScore θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) v)
=ᵐ[ outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))] 0
Proof (Lean source)
lemma condExp_channelDirectionalScore_fst_eq_zero
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n) :
condExp (comap fst inferInstance)
(outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))))
(channelDirectionalScore θ p
(sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) v) =ᵐ[
outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))] 0 := by
let i := canonicalNextIndex hk
let H := History (Z n) i.val i.isLt.le
let μ : Measure H := transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
let Q := sequentialJointChannel P n i μ
let ρ := outputLaw θ p Q
letI : IsProbabilityMeasure μ :=
transcriptPrefixLaw_isProbability P θ p hp hθ n k (le_of_succ_le hk)
letI : IsMarkovKernel Q :=
sequentialJointChannel_isMarkov P n (canonicalNextIndex hk) μ
letI : IsFiniteMeasure ρ := by
constructor
dsimp [ρ, outputLaw]
simp only [Measure.coe_finsetSum, Finset.sum_apply, Measure.smul_apply,
smul_eq_mul, measure_univ, mul_one]
exact ENNReal.sum_lt_top.mpr fun _ _ => ENNReal.ofReal_lt_top
have hc2 := channelDirectionalScore_sq_integrable_successorLaw
P θ v p hp hθ n k hk
have hcMeas : Measurable (channelDirectionalScore θ p Q v) := by
have hdens (j : Fin 4) : Measurable (channelDensity Q j) :=
(Measure.measurable_rnDeriv (Q j) (dominatingMeasure Q)).ennreal_toReal
have hout : Measurable (channelOutputDensity θ p Q) := by
unfold channelOutputDensity
exact Finset.measurable_sum _ fun j _ => measurable_const.mul (hdens j)
have hder : Measurable (channelDirectionalDerivativeDensity p Q v) := by
unfold channelDirectionalDerivativeDensity channelDerivativeDensity
exact Finset.measurable_sum _ fun j _ => measurable_const.mul
(Finset.measurable_sum _ fun a _ => measurable_const.mul (hdens a))
unfold channelDirectionalScore
exact (hder.div hout).ite (measurableSet_lt measurable_const hout) measurable_const
have hcMem : MemLp (channelDirectionalScore θ p Q v) 2 ρ :=
(memLp_two_iff_integrable_sq hcMeas.aestronglyMeasurable).2 hc2
have hc : Integrable (channelDirectionalScore θ p Q v) ρ :=
hcMem.integrable (by norm_num)
have hm : comap (@fst H (Z n i)) inferInstance ≤
(inferInstance : MeasurableSpace (H × Z n i)) :=
(measurable_fst (α := H) (β := Z n i)).comap_le
have hz : (fun _ : H × Z n i => (0 : ℝ)) =ᵐ[ρ]
condExp (comap (@fst H (Z n i)) inferInstance) ρ
(channelDirectionalScore θ p Q v) := by
refine ae_eq_condExp_of_forall_setIntegral_eq hm hc (fun s _ _ => ?_) ?_ ?_
· exact integrableOn_zero
· intro s hs _
obtain ⟨B, hB, rfl⟩ := hs
simpa [Q, ρ] using (setIntegral_channelDirectionalScore_fst_preimage_eq_zero
P θ v p hp hθ n k hk B hB).symm
· exact aestronglyMeasurable_zero
exact hz.symm
theorem integral_liftedPrefix_mul_channelDirectionalScore_eq_zero
The lifted predecessor score is orthogonal to the current channel-score increment under the actual successor law. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the integral lifted Prefix mul channel Directional Score eq zero.
Formal statement
hypotheses
conclusion
∫ z, liftedPrefixDirectionalScore P θ v p n k hk z * channelDirectionalScore θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) v z ∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))
= 0
Proof (Lean source)
lemma integral_liftedPrefix_mul_channelDirectionalScore_eq_zero
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n) :
∫ z, liftedPrefixDirectionalScore P θ v p n k hk z *
channelDirectionalScore θ p
(sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) v z
∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) = 0 := by
let i := canonicalNextIndex hk
let H := History (Z n) i.val i.isLt.le
let μ : Measure H := transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
let Q := sequentialJointChannel P n i μ
let ρ := outputLaw θ p Q
let past : H × Z n i → ℝ := liftedPrefixDirectionalScore P θ v p n k hk
let current : H × Z n i → ℝ := channelDirectionalScore θ p Q v
letI : IsProbabilityMeasure μ :=
transcriptPrefixLaw_isProbability P θ p hp hθ n k (le_of_succ_le hk)
letI : IsMarkovKernel Q := sequentialJointChannel_isMarkov P n i μ
letI : IsFiniteMeasure ρ := by
constructor
dsimp [ρ, outputLaw]
simp only [Measure.coe_finsetSum, Finset.sum_apply, Measure.smul_apply,
smul_eq_mul, measure_univ, mul_one]
exact ENNReal.sum_lt_top.mpr fun _ _ => ENNReal.ofReal_lt_top
have hm : comap (@fst H (Z n i)) inferInstance ≤
(inferInstance : MeasurableSpace (H × Z n i)) :=
(measurable_fst (α := H) (β := Z n i)).comap_le
have hpastMeas : @Measurable (H × Z n i) ℝ
(comap (@fst H (Z n i)) inferInstance) (borel ℝ) past := by
dsimp [past, liftedPrefixDirectionalScore]
exact (measurable_transcriptPrefixMixtureRealDerivative
P θ v p n k (le_of_succ_le hk)).div
(measurable_transcriptPrefixMixtureRealDensity
P θ p n k (le_of_succ_le hk)) |>.comp
(Measurable.of_comap_le le_rfl)
have hpastSq := liftedPrefixDirectionalScore_sq_integrable_successorLaw
P θ v p hp hθ n k hk
have hcurrentSq := channelDirectionalScore_sq_integrable_successorLaw
P θ v p hp hθ n k hk
have hpastMem : MemLp past 2 ρ :=
(memLp_two_iff_integrable_sq
(measurable_liftedPrefixDirectionalScore P θ v p n k hk).aestronglyMeasurable).2
hpastSq
have hcurrentMeas : Measurable current := by
dsimp [current]
have hdens (j : Fin 4) : Measurable (channelDensity Q j) :=
(Measure.measurable_rnDeriv (Q j) (dominatingMeasure Q)).ennreal_toReal
have hout : Measurable (channelOutputDensity θ p Q) := by
unfold channelOutputDensity
exact Finset.measurable_sum _ fun j _ => measurable_const.mul (hdens j)
have hder : Measurable (channelDirectionalDerivativeDensity p Q v) := by
unfold channelDirectionalDerivativeDensity channelDerivativeDensity
exact Finset.measurable_sum _ fun j _ => measurable_const.mul
(Finset.measurable_sum _ fun a _ => measurable_const.mul (hdens a))
unfold channelDirectionalScore
exact (hder.div hout).ite (measurableSet_lt measurable_const hout) measurable_const
have hcurrentMem : MemLp current 2 ρ :=
(memLp_two_iff_integrable_sq hcurrentMeas.aestronglyMeasurable).2 hcurrentSq
have hprod : Integrable (past * current) ρ :=
hpastMem.integrable_mul hcurrentMem
have hcurrentInt : Integrable current ρ := hcurrentMem.integrable (by norm_num)
have hpull := condExp_mul_of_aestronglyMeasurable_left
hpastMeas.aestronglyMeasurable hprod hcurrentInt
have hcenter := condExp_channelDirectionalScore_fst_eq_zero
P θ v p hp hθ n k hk
change ∫ z, past z * current z ∂ρ = 0
calc
∫ z, past z * current z ∂ρ =
∫ z, condExp (comap (@fst H (Z n i)) inferInstance)
ρ (past * current) z ∂ρ := (integral_condExp hm).symm
_ = ∫ z, past z * condExp
(comap (@fst H (Z n i)) inferInstance)
ρ current z ∂ρ := integral_congr_ae hpull
_ = 0 := by
apply integral_eq_zero_of_ae
filter_upwards [hcenter] with z hz
have hz' : condExp
(comap (@fst H (Z n i)) inferInstance)
ρ current z = 0 := by
simpa [current, ρ, Q, μ, i] using hz
simp [hz']
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialFisherRecursion 13 declarations Finite sequential Fisher recursion
Finite sequential Fisher recursion
theorem integral_liftedPrefixScore_sq_eq_prefixScore_sq
the integral lifted prefix score sq eq prefix score sq assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the integral lifted Prefix Score sq eq prefix Score sq.
Formal statement
hypotheses
conclusion
∫ z, liftedPrefixDirectionalScore P θ v p n k hk z ^ 2 ∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))
= ∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n k (le_of_succ_le hk) h ^ 2 ∂transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
Proof (Lean source)
lemma integral_liftedPrefixScore_sq_eq_prefixScore_sq
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n) :
∫ z, liftedPrefixDirectionalScore P θ v p n k hk z ^ 2
∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) =
∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n k
(le_of_succ_le hk) h ^ 2
∂transcriptPrefixLaw P θ p n k (le_of_succ_le hk) := by
let i := canonicalNextIndex hk
let μ := transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
let K : Kernel (History (Z n) k (le_of_succ_le hk)) (Z n i) :=
(observedStageKernel P θ p n i).comap
(fun h => ((), h)) (measurable_const.prodMk measurable_id)
letI : IsProbabilityMeasure μ :=
transcriptPrefixLaw_isProbability P θ p hp hθ n k (le_of_succ_le hk)
letI : IsMarkovKernel (observedStageKernel P θ p n i) :=
observedStageKernel_isMarkov P θ p hp hθ n i
letI : IsMarkovKernel K := by
dsimp [K]
infer_instance
have hout : outputLaw θ p (sequentialJointChannel P n i μ) = μ ⊗ₘ K := by
simpa [K] using outputLaw_sequentialJointChannel P θ p hp hθ n i μ
have hint := liftedPrefixDirectionalScore_sq_integrable_successorLaw
P θ v p hp hθ n k hk
have hint' : Integrable
(fun z => liftedPrefixDirectionalScore P θ v p n k hk z ^ 2) (μ ⊗ₘ K) := by
rw [← hout]
simpa [i, μ] using hint
rw [hout, Measure.integral_compProd hint']
simp [liftedPrefixDirectionalScore, μ]
theorem integral_actualSuccessorScore_sq_eq_prefixSuccessorScore_sq
the integral actual successor score sq eq prefix successor score sq assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the integral actual Successor Score sq eq prefix Successor Score sq.
Formal statement
hypotheses
conclusion
∫ z, actualSuccessorScoreInCoordinates P θ v p n k hk z ^ 2 ∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))
= ∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n (k + 1) hk h ^ 2 ∂transcriptPrefixLaw P θ p n (k + 1) hk
Proof (Lean source)
lemma integral_actualSuccessorScore_sq_eq_prefixSuccessorScore_sq
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n) :
∫ z, actualSuccessorScoreInCoordinates P θ v p n k hk z ^ 2
∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) =
∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n (k + 1) hk h ^ 2
∂transcriptPrefixLaw P θ p n (k + 1) hk := by
rw [← transcriptSuccessorLaw_map_canonicalInitLast P θ p hp hθ n k hk]
have hm : Measurable (fun z =>
actualSuccessorScoreInCoordinates P θ v p n k hk z ^ 2) :=
(measurable_actualSuccessorScoreInCoordinates P θ v p n k hk).pow_const 2
rw [integral_map (measurable_canonicalInitLast hk).aemeasurable hm.aestronglyMeasurable]
simp [Function.comp_def, actualSuccessorScoreInCoordinates]
theorem measurable_channelDirectionalScore_recursion
the measurable channel directional score recursion assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (channelDirectionalScore θ p Q v)
Proof (Lean source)
lemma measurable_channelDirectionalScore_recursion
{Y : Type*} [MeasurableSpace Y]
(θ v : TrialParameter) (p : ℝ) (Q : Kernel (Fin 4) Y) :
Measurable (channelDirectionalScore θ p Q v) := by
have hdens (j : Fin 4) : Measurable (channelDensity Q j) :=
(Measure.measurable_rnDeriv (Q j) (dominatingMeasure Q)).ennreal_toReal
have hout : Measurable (channelOutputDensity θ p Q) := by
unfold channelOutputDensity
exact Finset.measurable_sum _ fun j _ => measurable_const.mul (hdens j)
have hder : Measurable (channelDirectionalDerivativeDensity p Q v) := by
unfold channelDirectionalDerivativeDensity channelDerivativeDensity
exact Finset.measurable_sum _ fun j _ => measurable_const.mul
(Finset.measurable_sum _ fun a _ => measurable_const.mul (hdens a))
unfold channelDirectionalScore
exact (hder.div hout).ite (measurableSet_lt measurable_const hout) measurable_const
theorem prefixScoreEnergy_succ
the prefix score energy succ assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the prefix Score Energy succ.
Formal statement
hypotheses
conclusion
(∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n (k + 1) hk h ^ 2 ∂transcriptPrefixLaw P θ p n (k + 1) hk)
= (∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n k (le_of_succ_le hk) h ^ 2 ∂transcriptPrefixLaw P θ p n k (le_of_succ_le hk))
+ ∫ z, channelDirectionalScore θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) v z ^ 2 ∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))
Proof (Lean source)
lemma prefixScoreEnergy_succ
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n) :
(∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n (k + 1) hk h ^ 2
∂transcriptPrefixLaw P θ p n (k + 1) hk) =
(∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n k
(le_of_succ_le hk) h ^ 2
∂transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) +
∫ z, channelDirectionalScore θ p
(sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) v z ^ 2
∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) := by
rw [← integral_actualSuccessorScore_sq_eq_prefixSuccessorScore_sq
P θ v p hp hθ n k hk]
rw [← integral_liftedPrefixScore_sq_eq_prefixScore_sq
P θ v p hp hθ n k hk]
let i := canonicalNextIndex hk
let H := History (Z n) i.val i.isLt.le
let ρ : Measure (H × Z n i) :=
outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))
let A : H × Z n i → ℝ := actualSuccessorScoreInCoordinates P θ v p n k hk
let L : H × Z n i → ℝ := liftedPrefixDirectionalScore P θ v p n k hk
let C : H × Z n i → ℝ := channelDirectionalScore θ p
(sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) v
have hL2 := liftedPrefixDirectionalScore_sq_integrable_successorLaw
P θ v p hp hθ n k hk
have hC2 := channelDirectionalScore_sq_integrable_successorLaw
P θ v p hp hθ n k hk
have hLMem : MemLp L 2 ρ :=
(memLp_two_iff_integrable_sq
(measurable_liftedPrefixDirectionalScore P θ v p n k hk).aestronglyMeasurable).2 hL2
have hCMem : MemLp C 2 ρ :=
(memLp_two_iff_integrable_sq
(measurable_channelDirectionalScore_recursion θ v p _).aestronglyMeasurable).2 hC2
have hLC : Integrable (L * C) ρ := hLMem.integrable_mul hCMem
have hae := actualSuccessorScoreInCoordinates_ae_eq_candidate
P θ v p hp hθ n k hk
have hcross := integral_liftedPrefix_mul_channelDirectionalScore_eq_zero
P θ v p hp hθ n k hk
change (∫ z, A z ^ 2 ∂ρ) = (∫ z, L z ^ 2 ∂ρ) + ∫ z, C z ^ 2 ∂ρ
calc
∫ z, A z ^ 2 ∂ρ = ∫ z, (L z + C z) ^ 2 ∂ρ := by
apply integral_congr_ae
filter_upwards [hae] with z hz
simpa [A, L, C, successorDirectionalScoreCandidate] using
congrArg (fun x => x ^ 2) hz
_ = ∫ z, (L z ^ 2 + 2 * (L z * C z)) + C z ^ 2 ∂ρ := by
congr 1
funext z
ring
_ = (∫ z, L z ^ 2 + 2 * (L z * C z) ∂ρ) +
∫ z, C z ^ 2 ∂ρ :=
integral_add (hLMem.integrable_sq.add (hLC.const_mul 2))
hCMem.integrable_sq
_ = ((∫ z, L z ^ 2 ∂ρ) + ∫ z, 2 * (L z * C z) ∂ρ) +
∫ z, C z ^ 2 ∂ρ := by
congr 1
exact integral_add hLMem.integrable_sq (hLC.const_mul 2)
_ = ((∫ z, L z ^ 2 ∂ρ) + 2 * ∫ z, L z * C z ∂ρ) +
∫ z, C z ^ 2 ∂ρ := by rw [integral_const_mul]
_ = (∫ z, L z ^ 2 ∂ρ) + ∫ z, C z ^ 2 ∂ρ := by
rw [show (∫ z, L z * C z ∂ρ) = 0 by simpa [L, C, ρ] using hcross]
ring
theorem integral_transcriptPrefixMixtureRealDerivative_eq_zero
the integral transcript prefix mixture real derivative eq zero assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
∫ h, transcriptPrefixMixtureRealDerivative P θ v p n k hk h ∂transcriptPrefixReferenceMeasure P n k hk
= 0
Proof (Lean source)
lemma integral_transcriptPrefixMixtureRealDerivative_eq_zero
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(n k : ℕ) (hk : k ≤ n) :
∫ h, transcriptPrefixMixtureRealDerivative P θ v p n k hk h
∂transcriptPrefixReferenceMeasure P n k hk = 0 := by
let ν := transcriptPrefixReferenceMeasure P n k hk
let μ (x : Fin n → Fin 4) := (P.transcript n x).map (transcriptPrefix hk)
letI (x : Fin n → Fin 4) : IsProbabilityMeasure (P.transcript n x) :=
(P.factorizes n x).1
letI (x : Fin n → Fin 4) : IsProbabilityMeasure (μ x) :=
Measure.isProbabilityMeasure_map (measurable_transcriptPrefix hk).aemeasurable
letI : IsFiniteMeasure ν := by
dsimp [ν, transcriptPrefixReferenceMeasure]
infer_instance
have hac (x : Fin n → Fin 4) : μ x ≪ ν := by
dsimp [μ, ν]
exact (transcriptComponent_ac_reference P n x).map
(measurable_transcriptPrefix hk)
have hint (x : Fin n → Fin 4) : Integrable
(transcriptPrefixComponentRealDensity P n k hk x) ν := by
have h := (integrable_toReal_rnDeriv_mul_iff (hac x)
(f := fun _ => (1 : ℝ))).2 (integrable_const (1 : ℝ))
change Integrable (fun z => ((μ x).rnDeriv ν z).toReal) ν
simpa only [mul_one] using h
unfold transcriptPrefixMixtureRealDerivative
rw [integral_finset_sum _ (fun x _ => (hint x).const_mul _)]
calc
∑ x : Fin n → Fin 4,
∫ h, inputPathDirectionalDerivative θ v p x *
transcriptPrefixComponentRealDensity P n k hk x h ∂ν =
∑ x : Fin n → Fin 4, inputPathDirectionalDerivative θ v p x := by
apply Finset.sum_congr rfl
intro x _
rw [integral_const_mul]
rw [show (∫ h, transcriptPrefixComponentRealDensity P n k hk x h ∂ν) = 1 by
unfold transcriptPrefixComponentRealDensity
rw [Measure.integral_toReal_rnDeriv (hac x)]
simp [measureReal_def, μ]]
ring
_ = 0 := sum_inputPathDirectionalDerivative_eq_zero θ v p
theorem prefixScoreEnergy_zero
the prefix score energy zero assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the prefix Score Energy zero.
Formal statement
hypotheses
conclusion
∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n 0 h0 h ^ 2 ∂transcriptPrefixLaw P θ p n 0 h0
= 0
Proof (Lean source)
lemma prefixScoreEnergy_zero
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (n : ℕ)
(h0 : 0 ≤ n) :
∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n 0 h0 h ^ 2
∂transcriptPrefixLaw P θ p n 0 h0 = 0 := by
let μ := transcriptPrefixLaw P θ p n 0 h0
let emptyHistory : History (Z n) 0 h0 := fun j => j.elim0
let score := conditionalTranscriptPrefixDirectionalScore P θ v p n 0 h0
letI : IsProbabilityMeasure μ :=
transcriptPrefixLaw_isProbability P θ p hp hθ n 0 h0
have hscoreInt : ∫ h, score h ∂μ = 0 := by
have hset := setIntegral_conditionalTranscriptPrefixDirectionalScore
P θ v p hp hθ n 0 h0 univ MeasurableSet.univ
simpa [score, μ, integral_transcriptPrefixMixtureRealDerivative_eq_zero
P θ v p n 0 h0] using hset
have hconst : score = fun _ => score emptyHistory := by
funext h
congr 1
funext j
exact j.elim0
rw [hconst] at hscoreInt
have hvalue : score emptyHistory = 0 := by
simpa [μ] using hscoreInt
change ∫ h, score h ^ 2 ∂μ = 0
rw [hconst]
simp [hvalue]
theorem prefixScoreEnergy_le_mul_upperEnvelope
the prefix score energy le mul upper envelope assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the prefix Score Energy le mul upper Envelope.
Formal statement
hypotheses
conclusion
(∫ h, conditionalTranscriptPrefixDirectionalScore P θ (direction t) p n k hk h ^ 2 ∂transcriptPrefixLaw P θ p n k hk)
≤ (k : ℝ) * upperEnvelope θ p ε t
Proof (Lean source)
lemma prefixScoreEnergy_le_mul_upperEnvelope
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p t : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(n k : ℕ) (hk : k ≤ n) :
(∫ h, conditionalTranscriptPrefixDirectionalScore P θ (direction t) p n k hk h ^ 2
∂transcriptPrefixLaw P θ p n k hk) ≤
(k : ℝ) * upperEnvelope θ p ε t := by
induction k with
| zero =>
rw [prefixScoreEnergy_zero P θ (direction t) p hp hθ n hk]
simp
| succ k ih =>
have hprev : k ≤ n := le_of_succ_le hk
rw [prefixScoreEnergy_succ P θ (direction t) p hp hθ n k hk]
calc
(∫ h, conditionalTranscriptPrefixDirectionalScore P θ (direction t) p n k hprev h ^ 2
∂transcriptPrefixLaw P θ p n k hprev) +
∫ z, channelDirectionalScore θ p
(sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k hprev)) (direction t) z ^ 2
∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k hprev)) ≤
(k : ℝ) * upperEnvelope θ p ε t + upperEnvelope θ p ε t :=
add_le_add (ih hprev)
(successor_channelFisher_le_upperEnvelope
P θ p t hp hθ hε n k hk)
_ = ((k + 1 : ℕ) : ℝ) * upperEnvelope θ p ε t := by
push_cast
ring
theorem terminalPrefixScoreEnergy_eq_transcriptScoreEnergy
the terminal prefix score energy eq transcript score energy assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
(∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n n (le_refl n) h ^ 2 ∂transcriptPrefixLaw P θ p n n (le_refl n))
= ∫ z, conditionalTranscriptDirectionalScore P θ v p n z ^ 2 ∂transcriptLaw P θ p n
Proof (Lean source)
lemma terminalPrefixScoreEnergy_eq_transcriptScoreEnergy
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ) :
(∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n n (le_refl n) h ^ 2
∂transcriptPrefixLaw P θ p n n (le_refl n)) =
∫ z, conditionalTranscriptDirectionalScore P θ v p n z ^ 2
∂transcriptLaw P θ p n := by
have hmap (μ : Measure (Transcript (Z n))) :
μ.map (transcriptPrefix (Z := Z n) (le_refl n)) = μ := by
have hf : transcriptPrefix (Z := Z n) (le_refl n) =
(fun z : Transcript (Z n) => z) := by
funext z
funext j
rfl
rw [hf]
exact Measure.map_id'
unfold conditionalTranscriptPrefixDirectionalScore
transcriptPrefixMixtureRealDerivative transcriptPrefixMixtureRealDensity
transcriptPrefixComponentRealDensity transcriptPrefixLaw
transcriptPrefixReferenceMeasure
simp_rw [hmap]
rfl
theorem transcriptDirectionalFisher_le_mul_upperEnvelope
the transcript directional fisher le mul upper envelope assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the transcript Directional Fisher le mul upper Envelope.
Formal statement
hypotheses
conclusion
(∫ z, conditionalTranscriptDirectionalScore P θ (direction t) p n z ^ 2 ∂transcriptLaw P θ p n)
≤ (n : ℝ) * upperEnvelope θ p ε t
Proof (Lean source)
lemma transcriptDirectionalFisher_le_mul_upperEnvelope
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p t : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(n : ℕ) :
(∫ z, conditionalTranscriptDirectionalScore P θ (direction t) p n z ^ 2
∂transcriptLaw P θ p n) ≤
(n : ℝ) * upperEnvelope θ p ε t := by
rw [← terminalPrefixScoreEnergy_eq_transcriptScoreEnergy
P θ (direction t) p n]
exact prefixScoreEnergy_le_mul_upperEnvelope
P θ p t hp hθ hε n n (le_refl n)
theorem inputDirectionalDerivative_scaledDirection
Under the supplied quantities and conditions, the input directional derivative scaled direction assertion holds.
Formal statement
hypotheses
conclusion
inputDirectionalDerivative p (scaledDirection c v) j
= c * inputDirectionalDerivative p v j
Proof (Lean source)
lemma inputDirectionalDerivative_scaledDirection
(p c : ℝ) (v : TrialParameter) (j : Fin 4) :
inputDirectionalDerivative p (scaledDirection c v) j =
c * inputDirectionalDerivative p v j := by
unfold inputDirectionalDerivative scaledDirection
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro k _
ring
theorem inputPathDirectionalDerivative_scaledDirection
the input path directional derivative scaled direction assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
inputPathDirectionalDerivative θ (scaledDirection c v) p x
= c * inputPathDirectionalDerivative θ v p x
Proof (Lean source)
lemma inputPathDirectionalDerivative_scaledDirection {n : ℕ}
(θ v : TrialParameter) (p c : ℝ) (x : Fin n → Fin 4) :
inputPathDirectionalDerivative θ (scaledDirection c v) p x =
c * inputPathDirectionalDerivative θ v p x := by
unfold inputPathDirectionalDerivative
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro i _
rw [inputDirectionalDerivative_scaledDirection]
ring
theorem conditionalTranscriptDirectionalScore_scaledDirection
the conditional transcript directional score scaled direction assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
conditionalTranscriptDirectionalScore P θ (scaledDirection c v) p n
= fun z => c * conditionalTranscriptDirectionalScore P θ v p n z
Proof (Lean source)
lemma conditionalTranscriptDirectionalScore_scaledDirection
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p c : ℝ) (n : ℕ) :
conditionalTranscriptDirectionalScore P θ (scaledDirection c v) p n =
fun z => c * conditionalTranscriptDirectionalScore P θ v p n z := by
funext z
unfold conditionalTranscriptDirectionalScore transcriptMixtureRealDerivative
simp_rw [inputPathDirectionalDerivative_scaledDirection]
simp_rw [mul_assoc]
rw [← Finset.mul_sum]
ring
theorem transcriptScaledDirectionalFisher_le
the transcript scaled directional fisher le assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the transcript Scaled Directional Fisher le.
Formal statement
hypotheses
conclusion
(∫ z, conditionalTranscriptDirectionalScore P θ (scaledDirection c (direction t)) p n z ^ 2 ∂transcriptLaw P θ p n)
≤ c ^ 2 * ((n : ℝ) * upperEnvelope θ p ε t)
Proof (Lean source)
lemma transcriptScaledDirectionalFisher_le
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p t c : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(n : ℕ) :
(∫ z, conditionalTranscriptDirectionalScore P θ
(scaledDirection c (direction t)) p n z ^ 2
∂transcriptLaw P θ p n) ≤
c ^ 2 * ((n : ℝ) * upperEnvelope θ p ε t) := by
rw [conditionalTranscriptDirectionalScore_scaledDirection]
rw [show (∫ z, (c * conditionalTranscriptDirectionalScore P θ (direction t) p n z) ^ 2
∂transcriptLaw P θ p n) =
c ^ 2 * ∫ z, conditionalTranscriptDirectionalScore P θ (direction t) p n z ^ 2
∂transcriptLaw P θ p n by
rw [← integral_const_mul]
apply integral_congr_ae
filter_upwards with z
ring]
exact mul_le_mul_of_nonneg_left
(transcriptDirectionalFisher_le_mul_upperEnvelope
P θ p t hp hθ hε n) (sq_nonneg c)
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialGlobalScore 17 declarations Global finite-mixture transcript scores
Global finite-mixture transcript scores
def inputPathScoreBound
A finite bound for the complete-data path scores. For the displayed inputs and conditions, the stated result follows. The input Path Score Bound is determined by the displayed parameters.
Definition (Lean source)
parameters
def
inputPathScoreBound θ v p :
ℝ
given by
theorem inputPathScoreBound_nonneg
Under the supplied quantities and conditions, the input path score bound nonneg assertion holds.
Formal statement
Proof (Lean source)
lemma inputPathScoreBound_nonneg {n : ℕ} (θ v : TrialParameter) (p : ℝ) :
0 ≤ inputPathScoreBound (n := n) θ v p := by
exact Finset.sum_nonneg fun _ _ => abs_nonneg _
theorem abs_inputPathDirectionalScore_le_bound
the abs input path directional score le bound assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
|inputPathDirectionalScore θ v p x| ≤ inputPathScoreBound (n := n) θ v p
Proof (Lean source)
lemma abs_inputPathDirectionalScore_le_bound {n : ℕ}
(θ v : TrialParameter) (p : ℝ) (x : Fin n → Fin 4) :
|inputPathDirectionalScore θ v p x| ≤ inputPathScoreBound (n := n) θ v p := by
unfold inputPathScoreBound
exact Finset.single_le_sum
(fun y _ => abs_nonneg (inputPathDirectionalScore θ v p y))
(Finset.mem_univ x)
theorem measurable_transcriptComponentRealDensity
the measurable transcript component real density assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma measurable_transcriptComponentRealDensity {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (n : ℕ) (x : Fin n → Fin 4) :
Measurable (transcriptComponentRealDensity P n x) := by
exact (Measure.measurable_rnDeriv _ _).ennreal_toReal
theorem measurable_transcriptMixtureRealDensity
the measurable transcript mixture real density assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (transcriptMixtureRealDensity P θ p n)
Proof (Lean source)
lemma measurable_transcriptMixtureRealDensity {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ) (n : ℕ) :
Measurable (transcriptMixtureRealDensity P θ p n) := by
unfold transcriptMixtureRealDensity
exact Finset.measurable_fun_sum _ fun x _ =>
measurable_const.mul (measurable_transcriptComponentRealDensity P n x)
theorem measurable_transcriptMixtureRealDerivative
the measurable transcript mixture real derivative assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (transcriptMixtureRealDerivative P θ v p n)
Proof (Lean source)
lemma measurable_transcriptMixtureRealDerivative {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ) :
Measurable (transcriptMixtureRealDerivative P θ v p n) := by
unfold transcriptMixtureRealDerivative
exact Finset.measurable_fun_sum _ fun x _ =>
measurable_const.mul (measurable_transcriptComponentRealDensity P n x)
theorem measurable_conditionalTranscriptDirectionalScore
the measurable conditional transcript directional score assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (conditionalTranscriptDirectionalScore P θ v p n)
Proof (Lean source)
lemma measurable_conditionalTranscriptDirectionalScore {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ) :
Measurable (conditionalTranscriptDirectionalScore P θ v p n) := by
exact (measurable_transcriptMixtureRealDerivative P θ v p n).div
(measurable_transcriptMixtureRealDensity P θ p n)
theorem transcriptMixtureRealDensity_nonneg
the transcript mixture real density nonneg assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the transcript Mixture Real Density nonneg.
Formal statement
hypotheses
conclusion
0 ≤ transcriptMixtureRealDensity P θ p n z
Proof (Lean source)
lemma transcriptMixtureRealDensity_nonneg {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ) (n : ℕ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(z : Transcript (Z n)) :
0 ≤ transcriptMixtureRealDensity P θ p n z := by
unfold transcriptMixtureRealDensity
exact Finset.sum_nonneg fun x _ => mul_nonneg
(inputPathProbability_pos_of_interior θ p hp hθ x).le ENNReal.toReal_nonneg
theorem abs_conditionalTranscriptDirectionalScore_le
The observed transcript score is a finite weighted average of the complete-data path scores and is therefore uniformly bounded. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the abs conditional Transcript Directional Score le.
Formal statement
hypotheses
conclusion
|conditionalTranscriptDirectionalScore P θ v p n z| ≤ inputPathScoreBound (n := n) θ v p
Proof (Lean source)
lemma abs_conditionalTranscriptDirectionalScore_le {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(z : Transcript (Z n)) :
|conditionalTranscriptDirectionalScore P θ v p n z| ≤
inputPathScoreBound (n := n) θ v p := by
let B := inputPathScoreBound (n := n) θ v p
let d := transcriptMixtureRealDensity P θ p n z
have hB : 0 ≤ B := inputPathScoreBound_nonneg θ v p
have hd : 0 ≤ d := transcriptMixtureRealDensity_nonneg P θ p n hp hθ z
have hx (x : Fin n → Fin 4) : inputPathProbability θ p x ≠ 0 :=
ne_of_gt (inputPathProbability_pos_of_interior θ p hp hθ x)
have hn : |transcriptMixtureRealDerivative P θ v p n z| ≤ B * d := by
rw [transcriptDerivative_eq_completeDataScoreMixture P θ v p n z hx]
calc
|∑ x : Fin n → Fin 4,
inputPathProbability θ p x * inputPathDirectionalScore θ v p x *
transcriptComponentRealDensity P n x z| ≤
∑ x : Fin n → Fin 4,
|inputPathProbability θ p x * inputPathDirectionalScore θ v p x *
transcriptComponentRealDensity P n x z| :=
Finset.abs_sum_le_sum_abs _ _
_ = ∑ x : Fin n → Fin 4,
(inputPathProbability θ p x *
transcriptComponentRealDensity P n x z) *
|inputPathDirectionalScore θ v p x| := by
apply Finset.sum_congr rfl
intro x _
rw [abs_mul, abs_mul, abs_of_pos
(inputPathProbability_pos_of_interior θ p hp hθ x),
abs_of_nonneg (show 0 ≤ transcriptComponentRealDensity P n x z by
exact ENNReal.toReal_nonneg)]
ring
_ ≤ ∑ x : Fin n → Fin 4,
(inputPathProbability θ p x *
transcriptComponentRealDensity P n x z) * B := by
apply Finset.sum_le_sum
intro x _
exact mul_le_mul_of_nonneg_left
(abs_inputPathDirectionalScore_le_bound θ v p x)
(mul_nonneg (inputPathProbability_pos_of_interior θ p hp hθ x).le
ENNReal.toReal_nonneg)
_ = B * d := by
simp only [d, transcriptMixtureRealDensity]
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro x _
ring
by_cases hdz : d = 0
· have hn0 : transcriptMixtureRealDerivative P θ v p n z = 0 := by
rw [hdz, mul_zero] at hn
exact abs_eq_zero.mp (le_antisymm hn (abs_nonneg _))
rw [conditionalTranscriptDirectionalScore, hn0, show
transcriptMixtureRealDensity P θ p n z = 0 by exact hdz]
simp only [zero_div, abs_zero]
exact hB
· have hdpos : 0 < d := lt_of_le_of_ne hd (Ne.symm hdz)
rw [conditionalTranscriptDirectionalScore, abs_div,
abs_of_nonneg (show 0 ≤ transcriptMixtureRealDensity P θ p n z by exact hd)]
exact (div_le_iff₀ hdpos).2 (by simpa [d] using hn)
theorem conditionalTranscriptDirectionalScore_memLp_two
the conditional transcript directional score mem lp two assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the conditional Transcript Directional Score mem Lp two.
Formal statement
hypotheses
conclusion
MemLp (conditionalTranscriptDirectionalScore P θ v p n) 2 (transcriptLaw P θ p n)
Proof (Lean source)
lemma conditionalTranscriptDirectionalScore_memLp_two {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
MemLp (conditionalTranscriptDirectionalScore P θ v p n) 2
(transcriptLaw P θ p n) := by
apply MemLp.of_bound
(measurable_conditionalTranscriptDirectionalScore P θ v p n).aestronglyMeasurable
(inputPathScoreBound (n := n) θ v p)
filter_upwards [] with z
simpa [Real.norm_eq_abs] using
abs_conditionalTranscriptDirectionalScore_le P θ v p n hp hθ z
theorem transcriptReference_ac_transcriptLaw_of_interior
the transcript reference ac transcript law of interior assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the transcript Reference ac transcript Law of interior.
Formal statement
hypotheses
conclusion
transcriptReferenceMeasure P n ≪ transcriptLaw P θ p n
Proof (Lean source)
lemma transcriptReference_ac_transcriptLaw_of_interior {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ) (n : ℕ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
transcriptReferenceMeasure P n ≪ transcriptLaw P θ p n := by
classical
apply Measure.AbsolutelyContinuous.mk
intro A hA hlaw
have hsum : ∑ x : Fin n → Fin 4,
ofReal (inputPathProbability θ p x) * P.transcript n x A = 0 := by
simpa [transcriptLaw, Measure.finsetSum_apply, Measure.smul_apply,
smul_eq_mul] using hlaw
have hcomponent (x : Fin n → Fin 4) : P.transcript n x A = 0 := by
have hx := (Finset.sum_eq_zero_iff.mp hsum x (Finset.mem_univ x))
exact (mul_eq_zero.mp hx).resolve_left
(ne_of_gt (ENNReal.ofReal_pos.mpr
(inputPathProbability_pos_of_interior θ p hp hθ x)))
simp [transcriptReferenceMeasure, Measure.finsetSum_apply, hcomponent]
theorem transcriptMixtureRealDensity_pos_ae
the transcript mixture real density pos ae assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the transcript Mixture Real Density pos ae.
Formal statement
hypotheses
conclusion
∀ᵐ z ∂transcriptReferenceMeasure P n, 0 < transcriptMixtureRealDensity P θ p n z
Proof (Lean source)
lemma transcriptMixtureRealDensity_pos_ae {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ) (n : ℕ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
∀ᵐ z ∂transcriptReferenceMeasure P n,
0 < transcriptMixtureRealDensity P θ p n z := by
have hac := transcriptLaw_ac_reference P θ p n
have hrev := transcriptReference_ac_transcriptLaw_of_interior P θ p n hp hθ
have hposENN : ∀ᵐ z ∂transcriptReferenceMeasure P n,
0 < transcriptDensity P θ p n z :=
hrev.ae_le (Measure.rnDeriv_pos hac)
have hneTop : ∀ᵐ z ∂transcriptReferenceMeasure P n,
transcriptDensity P θ p n z ≠ ∞ :=
Measure.rnDeriv_ne_top _ _
have hpos : ∀ᵐ z ∂transcriptReferenceMeasure P n,
0 < (transcriptDensity P θ p n z).toReal := by
filter_upwards [hposENN, hneTop] with z hz htop
exact ENNReal.toReal_pos hz.ne' htop
filter_upwards [hpos,
transcriptDensity_toReal_ae_eq_mixture_of_interior P θ p n hp hθ]
with z hz heq
simpa [heq] using hz
theorem sum_inputPathDirectionalDerivative_eq_zero
the sum input path directional derivative eq zero assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma sum_inputPathDirectionalDerivative_eq_zero {n : ℕ}
(θ v : TrialParameter) (p : ℝ) :
∑ x : Fin n → Fin 4, inputPathDirectionalDerivative θ v p x = 0 := by
have hzero : parameterPath θ v 0 = θ := by
funext k
simp [parameterPath]
have hder : HasDerivAt
(fun a => ∑ x : Fin n → Fin 4,
inputPathProbability (parameterPath θ v a) p x)
(∑ x : Fin n → Fin 4, inputPathDirectionalDerivative θ v p x) 0 := by
simpa only [hzero] using HasDerivAt.fun_sum
(u := (Finset.univ : Finset (Fin n → Fin 4)))
(fun x _ => hasDerivAt_inputPathProbability_parameterPath θ v p 0 x)
have heq : (fun a => ∑ x : Fin n → Fin 4,
inputPathProbability (parameterPath θ v a) p x) = fun _ : ℝ => 1 := by
funext a
exact inputPathProbability_sum_eq_one _ p n
rw [heq] at hder
exact hder.unique (hasDerivAt_const (x := (0 : ℝ)) (c := (1 : ℝ)))
theorem integral_transcriptMixtureRealDerivative_eq_zero
the integral transcript mixture real derivative eq zero assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
∫ z, transcriptMixtureRealDerivative P θ v p n z ∂transcriptReferenceMeasure P n = 0
Proof (Lean source)
lemma integral_transcriptMixtureRealDerivative_eq_zero {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ) :
∫ z, transcriptMixtureRealDerivative P θ v p n z
∂transcriptReferenceMeasure P n = 0 := by
letI (x : Fin n → Fin 4) : IsProbabilityMeasure (P.transcript n x) :=
(P.factorizes n x).1
have hint (x : Fin n → Fin 4) : Integrable
(transcriptComponentRealDensity P n x) (transcriptReferenceMeasure P n) := by
have h := (integrable_toReal_rnDeriv_mul_iff
(transcriptComponent_ac_reference P n x)
(f := fun _ => (1 : ℝ))).2 (integrable_const (1 : ℝ))
change Integrable (fun z =>
((P.transcript n x).rnDeriv (transcriptReferenceMeasure P n) z).toReal)
(transcriptReferenceMeasure P n)
simpa only [mul_one] using h
simp_rw [transcriptMixtureRealDerivative]
rw [integral_finset_sum _ (fun x _ => (hint x).const_mul _)]
calc
∑ x : Fin n → Fin 4,
∫ z, inputPathDirectionalDerivative θ v p x *
transcriptComponentRealDensity P n x z
∂transcriptReferenceMeasure P n =
∑ x : Fin n → Fin 4, inputPathDirectionalDerivative θ v p x := by
apply Finset.sum_congr rfl
intro x _
rw [integral_const_mul]
rw [show (∫ z, transcriptComponentRealDensity P n x z
∂transcriptReferenceMeasure P n) = 1 by
unfold transcriptComponentRealDensity
rw [Measure.integral_toReal_rnDeriv
(transcriptComponent_ac_reference P n x)]
simp [measureReal_def]]
ring
_ = 0 := sum_inputPathDirectionalDerivative_eq_zero θ v p
theorem conditionalTranscriptDirectionalScore_integrable
the conditional transcript directional score integrable assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the conditional Transcript Directional Score integrable.
Formal statement
hypotheses
conclusion
Integrable (conditionalTranscriptDirectionalScore P θ v p n) (transcriptLaw P θ p n)
Proof (Lean source)
lemma conditionalTranscriptDirectionalScore_integrable {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
Integrable (conditionalTranscriptDirectionalScore P θ v p n)
(transcriptLaw P θ p n) :=
(conditionalTranscriptDirectionalScore_memLp_two P θ v p n hp hθ).integrable
(by norm_num)
theorem integral_conditionalTranscriptDirectionalScore_eq_zero
The global observed transcript score is centered under the transcript law. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the integral conditional Transcript Directional Score eq zero.
Formal statement
hypotheses
conclusion
∫ z, conditionalTranscriptDirectionalScore P θ v p n z ∂transcriptLaw P θ p n = 0
Proof (Lean source)
lemma integral_conditionalTranscriptDirectionalScore_eq_zero {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
∫ z, conditionalTranscriptDirectionalScore P θ v p n z
∂transcriptLaw P θ p n = 0 := by
rw [← transcriptLaw_eq_withDensity_reference P θ p n]
change (∫ z, conditionalTranscriptDirectionalScore P θ v p n z
∂(transcriptReferenceMeasure P n).withDensity
((transcriptLaw P θ p n).rnDeriv (transcriptReferenceMeasure P n))) = 0
rw [integral_withDensity_eq_integral_toReal_smul
(Measure.measurable_rnDeriv _ _)
(Measure.rnDeriv_lt_top _ _)]
change (∫ z, (transcriptDensity P θ p n z).toReal •
conditionalTranscriptDirectionalScore P θ v p n z
∂transcriptReferenceMeasure P n) = 0
rw [show (∫ z,
(transcriptDensity P θ p n z).toReal •
conditionalTranscriptDirectionalScore P θ v p n z
∂transcriptReferenceMeasure P n) =
∫ z, transcriptMixtureRealDerivative P θ v p n z
∂transcriptReferenceMeasure P n by
apply integral_congr_ae
filter_upwards
[transcriptDensity_toReal_ae_eq_mixture_of_interior P θ p n hp hθ,
transcriptMixtureRealDensity_pos_ae P θ p n hp hθ]
with z heq hpos
simpa [smul_eq_mul, heq] using
(transcriptDerivative_eq_density_mul_conditionalScore
P θ v p n z hpos.ne').symm]
exact integral_transcriptMixtureRealDerivative_eq_zero P θ v p n
theorem conditionalTranscriptDirectionalScore_sq_integrable
the conditional transcript directional score sq integrable assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the conditional Transcript Directional Score sq integrable.
Formal statement
hypotheses
conclusion
Integrable (fun z => (conditionalTranscriptDirectionalScore P θ v p n z) ^ 2) (transcriptLaw P θ p n)
Proof (Lean source)
lemma conditionalTranscriptDirectionalScore_sq_integrable {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
Integrable (fun z => (conditionalTranscriptDirectionalScore P θ v p n z) ^ 2)
(transcriptLaw P θ p n) :=
(conditionalTranscriptDirectionalScore_memLp_two P θ v p n hp hθ).integrable_sq
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialHistoryTransport 3 declarations Canonical successor history index
Canonical successor history index
def canonicalNextIndex
The successor index is the canonical Fin n value with numeral part k. Rewriting by this equality before forming a dependent History also aligns its measurable-space instance; the remaining bound proofs are propositionally irrelevant. For the displayed inputs and conditions, the stated result follows.
Definition (Lean source)
parameters
n k :
ℕ
hk :
k + 1 ≤ n
def
given by
⟨k, Nat.lt_of_succ_le hk⟩
theorem nextIndex_eq_canonicalNextIndex
Under the supplied quantities and conditions, the next index eq canonical next index assertion holds.
Formal statement
Proof (Lean source)
lemma nextIndex_eq_canonicalNextIndex {n k : ℕ} (hk : k + 1 ≤ n) :
nextIndex hk = canonicalNextIndex hk := by
apply Fin.ext
rfl
theorem canonicalNextIndex_val
Under the stated assumptions, the canonical next index has natural-number value equal to the current history length.
Formal statement
Proof (Lean source)
@[simp] lemma canonicalNextIndex_val {n k : ℕ} (hk : k + 1 ≤ n) :
(canonicalNextIndex hk).val = k := rfl
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialInformationEnvelope 6 declarations This file identifies the least-favorable contrast direction and proves local continuity of its stationary staircase information envelope.
Local sequential information envelope
This file identifies the least-favorable contrast direction and proves local continuity of its stationary staircase information envelope. The resulting sequential criterion is uniform over arbitrary scalar sequences staying in a fixed compact interval.
theorem contrast_selectedDirection
The selected least-favorable direction has unit contrast. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the contrast selected Direction.
Formal statement
hypotheses
conclusion
contrast (direction (selectedDirection θ p ε hp hθ hε)) = 1
Proof (Lean source)
lemma contrast_selectedDirection (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε) :
contrast (direction (selectedDirection θ p ε hp hθ hε)) = 1 := by
simp [contrast, direction]
theorem upperEnvelope_selectedDirection
At the base parameter, the selected direction attains the stationary staircase envelope value Jstar. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the upper Envelope selected Direction.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma upperEnvelope_selectedDirection (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε) :
upperEnvelope θ p ε (selectedDirection θ p ε hp hθ hε) =
Jstar θ p ε :=
Classical.choose_spec (exists_selectedDirection θ p ε hp hθ hε)
theorem sequentialJointChannel_fisher_selectedDirection_le_upperEnvelope
Every history-conditioned sequential channel has directional Fisher information at most the stationary envelope in the selected direction. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the sequential Joint Channel fisher selected Direction le upper Envelope.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
n :
ℕ
i :
Fin n
μ :
η θ :
p :
ℝ
hp :
hη :
hθ :
hε :
0 < ε
conclusion
informationQuadratic (channelFisherInfo η p (sequentialJointChannel P n i μ)) (direction (selectedDirection θ p ε hp hθ hε))
≤ upperEnvelope η p ε (selectedDirection θ p ε hp hθ hε)
Proof (Lean source)
-- keep: reusable sequential-law, Fisher-information, or van-Trees bridge for related adaptive experiments
lemma sequentialJointChannel_fisher_selectedDirection_le_upperEnvelope
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (n : ℕ) (i : Fin n)
(μ : Measure (History (Z n) i.val (le_of_lt i.isLt)))
[IsProbabilityMeasure μ] (η θ : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hη : InteriorMeans η)
(hθ : InteriorMeans θ) (hε : 0 < ε) :
informationQuadratic
(channelFisherInfo η p (sequentialJointChannel P n i μ))
(direction (selectedDirection θ p ε hp hθ hε)) ≤
upperEnvelope η p ε (selectedDirection θ p ε hp hθ hε) := by
exact sequentialJointChannel_fisher_le_upperEnvelope
P n i μ η p (selectedDirection θ p ε hp hθ hε) hp hη hε
theorem continuous_upperEnvelope_interior
At fixed direction, the staircase upper envelope is continuous on the interior trial-parameter space. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the continuous upper Envelope interior.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma continuous_upperEnvelope_interior (p ε t : ℝ)
(hp : InteriorAssignment p) (hε : 0 ≤ ε) :
Continuous (fun θ : {η : TrialParameter // InteriorMeans η} =>
upperEnvelope θ.1 p ε t) := by
let Θ := {η : TrialParameter // InteriorMeans η}
have hmass (s : Fin 14) : Continuous (fun θ : Θ =>
patternMass θ.1 p ε s) := by
unfold patternMass
apply continuous_finsetSum
intro j _
apply Continuous.mul
· fin_cases j <;> simp only [piTheta]
· exact continuous_const.mul
(continuous_const.sub ((continuous_apply 0).comp continuous_subtype_val))
· exact continuous_const.mul
((continuous_apply 0).comp continuous_subtype_val)
· exact continuous_const.mul
(continuous_const.sub ((continuous_apply 1).comp continuous_subtype_val))
· exact continuous_const.mul
((continuous_apply 1).comp continuous_subtype_val)
· exact continuous_const
have hobj : Continuous (fun z : Θ × StaircaseWeight =>
informationObjective z.1.1 p ε z.2 t) := by
unfold informationObjective patternInformation
apply continuous_finsetSum
intro s _
apply Continuous.mul
· exact (continuous_apply s).comp continuous_snd
· apply Continuous.div continuous_const
((hmass s).comp continuous_fst)
intro z
exact ne_of_gt (patternMass_pos_interior z.1.1 p ε hp
z.1.2 hε s)
have hc := (staircaseFeasible_compact ε hε).continuous_sSup
(f := fun (θ : Θ) (α : StaircaseWeight) =>
informationObjective θ.1 p ε α t) hobj
simpa [upperEnvelope, image, eq_comm] using hc
theorem tendsto_upperEnvelope_selectedDirection_of_tendsto
Any interior parameter sequence converging to the base point has its selected-direction envelope converge to Jstar at that point. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the tendsto upper Envelope selected Direction of tendsto.
Formal statement
hypotheses
conclusion
Tendsto (fun n => upperEnvelope (η n) p ε (selectedDirection θ p ε hp hθ hε)) atTop (nhds (Jstar θ p ε))
Proof (Lean source)
lemma tendsto_upperEnvelope_selectedDirection_of_tendsto
(θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(η : ℕ → TrialParameter) (hη : ∀ n, InteriorMeans (η n))
(hηθ : Tendsto η atTop (nhds θ)) :
Tendsto (fun n => upperEnvelope (η n) p ε
(selectedDirection θ p ε hp hθ hε)) atTop
(nhds (Jstar θ p ε)) := by
let Θ := {x : TrialParameter // InteriorMeans x}
let θsub : Θ := ⟨θ, hθ⟩
let ηsub : ℕ → Θ := fun n => ⟨η n, hη n⟩
have hlift : Tendsto ηsub atTop (nhds θsub) := by
rw [tendsto_subtype_rng]
exact hηθ
have hc := (continuous_upperEnvelope_interior p ε
(selectedDirection θ p ε hp hθ hε) hp hε.le).continuousAt
(x := θsub)
have ht := Filter.Tendsto.comp hc hlift
have heq : (fun x : Θ => upperEnvelope x.1 p ε
(selectedDirection θ p ε hp hθ hε)) ∘ ηsub =
fun n => upperEnvelope (η n) p ε
(selectedDirection θ p ε hp hθ hε) := by
funext n
rfl
rw [heq] at ht
simpa only [θsub,
upperEnvelope_selectedDirection θ p ε hp hθ hε] using ht
theorem tendsto_upperEnvelope_compactScaled_selectedDirection
Uniform sequential form: along any scalar sequence contained in a fixed compact interval, the inverse-root-sample local envelope in the selected direction converges to the base Jstar. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the tendsto upper Envelope compact Scaled selected Direction.
Formal statement
hypotheses
θ :
p ε R :
ℝ
hp :
hθ :
hε :
0 < ε
_hR :
0 ≤ R
a :
ℕ → ℝ
ha :
∀ n, a n ∈ Icc (-R) R
hinterior :
∀ n,
conclusion
Tendsto (fun n => upperEnvelope (parameterPath θ (direction (selectedDirection θ p ε hp hθ hε)) (a n / sqrt n)) p ε (selectedDirection θ p ε hp hθ hε)) atTop (nhds (Jstar θ p ε))
Proof (Lean source)
-- keep: reusable sequential-law, Fisher-information, or van-Trees bridge for related adaptive experiments
lemma tendsto_upperEnvelope_compactScaled_selectedDirection
(θ : TrialParameter) (p ε R : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(_hR : 0 ≤ R) (a : ℕ → ℝ) (ha : ∀ n, a n ∈ Icc (-R) R)
(hinterior : ∀ n, InteriorMeans
(parameterPath θ (direction (selectedDirection θ p ε hp hθ hε))
(a n / sqrt n))) :
Tendsto (fun n => upperEnvelope
(parameterPath θ (direction (selectedDirection θ p ε hp hθ hε))
(a n / sqrt n)) p ε
(selectedDirection θ p ε hp hθ hε)) atTop
(nhds (Jstar θ p ε)) := by
have hinv : Tendsto (fun n : ℕ => (sqrt (n : ℝ))⁻¹)
atTop (nhds 0) :=
tendsto_inv_atTop_zero.comp
(Real.tendsto_sqrt_atTop.comp tendsto_natCast_atTop_atTop)
have habounded : ∀ᶠ n : ℕ in atTop, |a n| ≤ R :=
Filter.Eventually.of_forall fun n => abs_le.2 (ha n)
have hscalar : Tendsto (fun n => a n / sqrt n) atTop (nhds 0) := by
simpa only [div_eq_mul_inv] using
bdd_le_mul_tendsto_zero' R habounded hinv
have hpath : Tendsto (fun n =>
parameterPath θ (direction (selectedDirection θ p ε hp hθ hε))
(a n / sqrt n)) atTop (nhds θ) := by
rw [tendsto_pi_nhds]
intro k
simpa [parameterPath] using
tendsto_const_nhds.add
(hscalar.mul tendsto_const_nhds)
exact tendsto_upperEnvelope_selectedDirection_of_tendsto
θ p ε hp hθ hε _ hinterior hpath
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialJointChannel 6 declarations Joint prefix and next-release channels
Joint prefix and next-release channels
def releasedHistoryRowKernel
The original next-release channel, with a finite input row and a released finite-sequence history. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
given by
(P.channel n i).comap (fun jh => (jh.1, fsHistoryToPrivate i jh.2)) (measurable_fst.prodMk ((measurable_fsHistoryToPrivate i).comp measurable_snd))
def sequentialJointChannel
Adjoin a common random released prefix to each of the four current-input rows. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The sequential Joint Channel is determined by the displayed parameters.
Definition (Lean source)
parameters
given by
(Kernel.const (Fin 4) μ).compProd (releasedHistoryRowKernel P n i)
theorem sequentialJointChannel_isMarkov
the sequential joint channel is markov assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
IsMarkovKernel (sequentialJointChannel P n i μ)
Proof (Lean source)
lemma sequentialJointChannel_isMarkov {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (n : ℕ) (i : Fin n)
(μ : Measure (History (Z n) i.val (le_of_lt i.isLt)))
[IsProbabilityMeasure μ] : IsMarkovKernel (sequentialJointChannel P n i μ) := by
letI : IsMarkovKernel (P.channel n i) := (P.privacy n).1 i
letI : IsMarkovKernel (releasedHistoryRowKernel P n i) := by
unfold releasedHistoryRowKernel
infer_instance
unfold sequentialJointChannel releasedHistoryRowKernel
infer_instance
theorem sequentialJointChannel_stationaryLDP
the sequential joint channel stationary ldp assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
StationaryLDP (ε) (sequentialJointChannel P n i μ)
Proof (Lean source)
lemma sequentialJointChannel_stationaryLDP {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (n : ℕ) (i : Fin n)
(μ : Measure (History (Z n) i.val (le_of_lt i.isLt)))
[IsProbabilityMeasure μ] :
StationaryLDP (ε) (sequentialJointChannel P n i μ) := by
letI : IsMarkovKernel (P.channel n i) := (P.privacy n).1 i
letI : IsMarkovKernel (releasedHistoryRowKernel P n i) := by
unfold releasedHistoryRowKernel
infer_instance
refine ⟨sequentialJointChannel_isMarkov P n i μ, ?_⟩
intro A hA j j'
rw [sequentialJointChannel, Kernel.compProd_apply hA,
Kernel.compProd_apply hA]
have hsection (a : Fin 4) : Measurable (fun h =>
releasedHistoryRowKernel P n i (a, h) (Prod.mk h ⁻¹' A)) :=
((releasedHistoryRowKernel P n i).comap (fun h => (a, h))
(measurable_const.prodMk measurable_id)).measurable_kernel_prodMk_left hA
calc
(∫⁻ h, releasedHistoryRowKernel P n i (j, h) (Prod.mk h ⁻¹' A) ∂μ) ≤
∫⁻ h, ofReal (exp (ε)) *
releasedHistoryRowKernel P n i (j', h) (Prod.mk h ⁻¹' A) ∂μ := by
apply lintegral_mono
intro h
exact (P.privacy n).2 i (Prod.mk h ⁻¹' A)
(hA.preimage (measurable_const.prodMk measurable_id)) j j'
(fsHistoryToPrivate i h)
_ = ofReal (exp (ε)) *
∫⁻ h, releasedHistoryRowKernel P n i (j', h) (Prod.mk h ⁻¹' A) ∂μ := by
rw [lintegral_const_mul _ (hsection j')]
theorem outputLaw_sequentialJointChannel
Mixing the four joint rows gives the prefix measure composed with the observed next-release kernel. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the output Law sequential Joint Channel.
Formal statement
hypotheses
conclusion
outputLaw θ p (sequentialJointChannel P n i μ)
= μ ⊗ₘ ((observedStageKernel P θ p n i).comap (fun h => ((), h)) (measurable_const.prodMk measurable_id))
Proof (Lean source)
lemma outputLaw_sequentialJointChannel {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n : ℕ) (i : Fin n)
(μ : Measure (History (Z n) i.val (le_of_lt i.isLt)))
[IsProbabilityMeasure μ] :
outputLaw θ p (sequentialJointChannel P n i μ) =
μ ⊗ₘ ((observedStageKernel P θ p n i).comap
(fun h => ((), h)) (measurable_const.prodMk measurable_id)) := by
classical
letI : IsMarkovKernel (P.channel n i) := (P.privacy n).1 i
letI : IsMarkovKernel (releasedHistoryRowKernel P n i) := by
unfold releasedHistoryRowKernel
infer_instance
letI : IsMarkovKernel (observedStageKernel P θ p n i) :=
observedStageKernel_isMarkov P θ p hp hθ n i
ext A hA
have hsection (j : Fin 4) : Measurable (fun h =>
P.channel n i (j, fsHistoryToPrivate i h) (Prod.mk h ⁻¹' A)) :=
((releasedHistoryRowKernel P n i).comap (fun h => (j, h))
(measurable_const.prodMk measurable_id)).measurable_kernel_prodMk_left hA
simp_rw [outputLaw, Measure.finsetSum_apply, Measure.smul_apply, smul_eq_mul,
sequentialJointChannel, Kernel.compProd_apply hA]
rw [Measure.compProd_apply hA]
change (∑ j : Fin 4, ofReal (piTheta θ p j) *
∫⁻ h, P.channel n i (j, fsHistoryToPrivate i h) (Prod.mk h ⁻¹' A) ∂μ) =
∫⁻ h, ∑ j : Fin 4, ofReal (piTheta θ p j) *
P.channel n i (j, fsHistoryToPrivate i h) (Prod.mk h ⁻¹' A) ∂μ
rw [lintegral_finset_sum]
· apply Finset.sum_congr rfl
intro j _
rw [lintegral_const_mul _ (hsection j)]
· intro j _
exact measurable_const.mul (hsection j)
theorem sequentialJointChannel_fisher_le_upperEnvelope
The joint channel's directional Fisher information is bounded by the finite staircase envelope. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the sequential Joint Channel fisher le upper Envelope.
Formal statement
hypotheses
conclusion
≤ upperEnvelope θ p (ε) t
Proof (Lean source)
lemma sequentialJointChannel_fisher_le_upperEnvelope {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (n : ℕ) (i : Fin n)
(μ : Measure (History (Z n) i.val (le_of_lt i.isLt)))
[IsProbabilityMeasure μ]
(θ : TrialParameter) (p t : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(heps : 0 < ε) :
informationQuadratic
(channelFisherInfo θ p (sequentialJointChannel P n i μ)) (direction t) ≤
upperEnvelope θ p (ε) t := by
let Q := sequentialJointChannel P n i μ
have hQ : StationaryLDP (ε) Q :=
sequentialJointChannel_stationaryLDP P n i μ
obtain ⟨α, K, hα, hK, hfac, hpsd, hrigid⟩ :=
staircase_refinement_interior p θ (ε) hp hθ
(fun _ => ε) ⟨heps, fun _ => rfl⟩ Q hQ
have hquad : informationQuadratic (channelFisherInfo θ p Q) (direction t) ≤
informationQuadratic (informationMatrix θ p (ε) α)
(direction t) := by
have hz := hpsd.dotProduct_mulVec_nonneg (direction t)
have hnonneg : 0 ≤ informationQuadratic
(informationMatrix θ p ε α - channelFisherInfo θ p Q) (direction t) := by
rw [show informationQuadratic
(informationMatrix θ p ε α - channelFisherInfo θ p Q) (direction t) =
star (direction t) ⬝ᵥ
(informationMatrix θ p ε α - channelFisherInfo θ p Q).mulVec
(direction t) by
simp [informationQuadratic, mulVec, dotProduct,
Fin.sum_univ_two]
ring]
exact hz
have heq : informationQuadratic
(informationMatrix θ p ε α - channelFisherInfo θ p Q) (direction t) =
informationQuadratic (informationMatrix θ p ε α) (direction t) -
informationQuadratic (channelFisherInfo θ p Q) (direction t) := by
unfold informationQuadratic
simp [Fin.sum_univ_two]
ring
linarith
calc
informationQuadratic (channelFisherInfo θ p Q) (direction t) ≤
informationQuadratic (informationMatrix θ p (ε) α)
(direction t) := hquad
_ = informationObjective θ p (ε) α t :=
(informationObjective_eq_informationQuadratic θ p ε α t).symm
_ ≤ upperEnvelope θ p (ε) t := by
obtain ⟨β, hβ, hmax⟩ :=
informationObjective_exists_maximizer θ p ε t heps.le
have hub : upperEnvelope θ p ε t = informationObjective θ p ε β t := by
unfold upperEnvelope
apply IsGreatest.csSup_eq
exact ⟨⟨β, hβ, rfl⟩, by
rintro u ⟨γ, hγ, rfl⟩
exact hmax γ hγ⟩
rw [hub]
exact hmax α hα
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialLocalRiskOrder 9 declarations Order and limit calculus for the sequential local minimax lower bound
Order and limit calculus for the sequential local minimax lower bound
theorem localRisk_le_localWorstRisk
Every admissible local alternative's risk is bounded by the corresponding local worst risk. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
localRisk P θ h p n ≤ localWorstRisk P θ p H n
Proof (Lean source)
lemma localRisk_le_localWorstRisk {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p H : ℝ) (n : ℕ)
(h : TrialParameter) (hh : h ∈ localIndexSet θ n H) :
localRisk P θ h p n ≤ localWorstRisk P θ p H n := by
unfold localWorstRisk
apply le_sSup
exact ⟨h, hh, rfl⟩
theorem eventually_mem_localIndexSet
Every fixed direction whose Euclidean norm is at most the radius is eventually admissible at an interior base parameter. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
∀ᶠ n in atTop, h ∈ localIndexSet theta n H
Proof (Lean source)
lemma eventually_mem_localIndexSet (theta h : TrialParameter) (H : ℝ)
(htheta : InteriorMeans theta)
(hh : sqrt ((h 0) ^ 2 + (h 1) ^ 2) ≤ H) :
∀ᶠ n in atTop, h ∈ localIndexSet theta n H := by
filter_upwards [eventually_localAlternative_interior theta h htheta] with n hn
exact ⟨hh, hn⟩
theorem eventually_localRisk_le_localWorstRisk
Consequently, the risk at any fixed bounded direction is eventually below the local worst-risk sequence. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
theta :
p H :
ℝ
h :
htheta :
InteriorMeans theta
hh :
sqrt ((h 0) ^ 2 + (h 1) ^ 2) ≤ H
conclusion
Proof (Lean source)
-- keep: reusable sequential-law, Fisher-information, or van-Trees bridge for related adaptive experiments
lemma eventually_localRisk_le_localWorstRisk {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (theta : TrialParameter) (p H : ℝ)
(h : TrialParameter) (htheta : InteriorMeans theta)
(hh : sqrt ((h 0) ^ 2 + (h 1) ^ 2) ≤ H) :
∀ᶠ n in atTop,
localRisk P theta h p n ≤ localWorstRisk P theta p H n := by
filter_upwards [eventually_mem_localIndexSet theta h H htheta hh] with n hn
exact localRisk_le_localWorstRisk P theta p H n h hn
theorem le_localAsymptoticRisk_of_le_liminf
A lower bound on the liminf at one positive radius is already a lower bound on the outer-radius local asymptotic risk. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
v ≤ localAsymptoticRisk P θ p
Proof (Lean source)
lemma le_localAsymptoticRisk_of_le_liminf {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p H : ℝ)
(hH : 0 < H) (v : ℝ≥0∞)
(hlim : v ≤ liminf (fun n => localWorstRisk P θ p H n) atTop) :
v ≤ localAsymptoticRisk P θ p := by
unfold localAsymptoticRisk
exact hlim.trans (le_sSup ⟨H, hH, rfl⟩)
theorem le_localAsymptoticRisk_of_eventually_ge_of_tendsto
An eventually valid finite-sample lower envelope transfers through the liminf and the outer radius supremum. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ :
p H :
ℝ
hH :
0 < H
b :
ℕ → ℝ≥0∞
v :
ℝ≥0∞
hlower :
∀ᶠ n in atTop, b n ≤ localWorstRisk P θ p H n
conclusion
v ≤ localAsymptoticRisk P θ p
Proof (Lean source)
lemma le_localAsymptoticRisk_of_eventually_ge_of_tendsto {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p H : ℝ)
(hH : 0 < H) (b : ℕ → ℝ≥0∞) (v : ℝ≥0∞)
(hb : Tendsto b atTop (nhds v))
(hlower : ∀ᶠ n in atTop, b n ≤ localWorstRisk P θ p H n) :
v ≤ localAsymptoticRisk P θ p := by
apply le_localAsymptoticRisk_of_le_liminf P θ p H hH v
rw [← hb.liminf_eq]
exact liminf_le_liminf hlower (by isBoundedDefault) (by isBoundedDefault)
theorem ofReal_le_localAsymptoticRisk_of_bayes_envelope
A real finite-sample Bayes envelope converging to v gives the exact ENNReal.ofReal v local asymptotic lower bound. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
ofReal v ≤ localAsymptoticRisk P θ p
Proof (Lean source)
lemma ofReal_le_localAsymptoticRisk_of_bayes_envelope {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p H : ℝ)
(hH : 0 < H) (b : ℕ → ℝ) (v : ℝ)
(hb : Tendsto b atTop (nhds v))
(hlower : ∀ᶠ n in atTop,
ofReal (b n) ≤ localWorstRisk P θ p H n) :
ofReal v ≤ localAsymptoticRisk P θ p := by
apply le_localAsymptoticRisk_of_eventually_ge_of_tendsto
P θ p H hH (fun n => ofReal (b n)) (ofReal v)
· exact ENNReal.continuous_ofReal.continuousAt.tendsto.comp hb
· exact hlower
theorem tendsto_nat_div_const_add_nat_mul
The result identifies the relation called tendsto nat div const add nat mul: given the declared inputs and assumptions, the stated mathematical conclusion holds. Under the stated assumptions, the tendsto nat div const add nat mul.
Formal statement
Proof (Lean source)
lemma tendsto_nat_div_const_add_nat_mul (c : ℝ) (info : ℕ → ℝ) (I : ℝ) (hI : 0 < I)
(hinfo : Tendsto info atTop (nhds I)) :
Tendsto (fun n : ℕ => (n : ℝ) / (c + (n : ℝ) * info n))
atTop (nhds (1 / I)) := by
have hn : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop :=
tendsto_natCast_atTop_atTop
have hinv : Tendsto (fun n : ℕ => ((n : ℝ))⁻¹) atTop (nhds 0) :=
tendsto_inv_atTop_zero.comp hn
have hc : Tendsto (fun n : ℕ => c / (n : ℝ)) atTop (nhds 0) := by
simpa only [div_eq_mul_inv, mul_zero] using tendsto_const_nhds.mul hinv
have hquot : Tendsto (fun n : ℕ => 1 / (c / (n : ℝ) + info n))
atTop (nhds (1 / I)) := by
change Tendsto ((fun _ : ℕ ↦ (1 : ℝ)) /
(fun n : ℕ ↦ c / (n : ℝ) + info n)) atTop (nhds (1 / I))
simpa only [zero_add] using
tendsto_const_nhds.div (hc.add hinfo) (by simpa using ne_of_gt hI)
apply hquot.congr'
filter_upwards [eventually_atTop.2 ⟨1, fun n hn1 => hn1⟩] with n hn1
have hn0 : (n : ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hn1)
field_simp
theorem tendsto_ofReal_nat_div_const_add_nat_mul
The preceding real van Trees envelope also converges after embedding into nonnegative extended reals. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma tendsto_ofReal_nat_div_const_add_nat_mul
(c : ℝ) (info : ℕ → ℝ) (I : ℝ) (hI : 0 < I)
(hinfo : Tendsto info atTop (nhds I)) :
Tendsto (fun n : ℕ => ofReal
((n : ℝ) / (c + (n : ℝ) * info n)))
atTop (nhds (ofReal (1 / I))) := by
exact ENNReal.continuous_ofReal.continuousAt.tendsto.comp
(tendsto_nat_div_const_add_nat_mul c info I hI hinfo)
theorem inv_le_localAsymptoticRisk_of_finite_vanTrees
A finite-n van Trees lower bound with a convergent per-sample information envelope yields the exact local asymptotic reciprocal-information bound. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ :
p H c I :
ℝ
hH :
0 < H
hI :
0 < I
info :
ℕ → ℝ
hbayes :
≤ localWorstRisk P θ p H n
conclusion
ofReal (1 / I) ≤ localAsymptoticRisk P θ p
Proof (Lean source)
-- keep: reusable sequential-law, Fisher-information, or van-Trees bridge for related adaptive experiments
lemma inv_le_localAsymptoticRisk_of_finite_vanTrees {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p H c I : ℝ)
(hH : 0 < H) (hI : 0 < I) (info : ℕ → ℝ)
(hinfo : Tendsto info atTop (nhds I))
(hbayes : ∀ᶠ n : ℕ in atTop,
ofReal ((n : ℝ) / (c + (n : ℝ) * info n)) ≤
localWorstRisk P θ p H n) :
ofReal (1 / I) ≤ localAsymptoticRisk P θ p := by
exact le_localAsymptoticRisk_of_eventually_ge_of_tendsto
P θ p H hH _ _
(tendsto_ofReal_nat_div_const_add_nat_mul c info I hI hinfo) hbayes
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialLowerBound 6 declarations Universal sequential local asymptotic lower bound
Universal sequential local asymptotic lower bound
theorem likelihoodScore_vtDensity_eq_transcriptScore
the likelihood score vt density eq transcript score assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the likelihood Score vt Density eq transcript Score.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ d :
p ell upper a :
ℝ
n :
ℕ
hp :
hη :
InteriorMeans (parameterPath θ d a)
ha :
a ∈ Icc ell upper
z :
Transcript (Z n)
conclusion
= conditionalTranscriptDirectionalScore P (parameterPath θ d a) d p n z
Proof (Lean source)
lemma likelihoodScore_vtDensity_eq_transcriptScore
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ d : TrialParameter) (p ell upper a : ℝ)
(n : ℕ) (hp : InteriorAssignment p)
(hη : InteriorMeans (parameterPath θ d a)) (ha : a ∈ Icc ell upper)
(z : Transcript (Z n)) :
likelihoodScore (vtDensity P θ d p n ell upper)
(vtDensityDeriv P θ d p n ell upper) a z =
conditionalTranscriptDirectionalScore P (parameterPath θ d a) d p n z := by
rw [conditionalTranscriptDirectionalScore_eq_guarded
P (parameterPath θ d a) d p n hp hη z]
unfold likelihoodScore vtDensity vtDensityDeriv
simp only [ha, ↓reduceIte]
theorem fisherInformation_vtDensity_eq_transcriptScoreEnergy
the fisher information vt density eq transcript score energy assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the fisher Information vt Density eq transcript Score Energy.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ d :
p ell upper a :
ℝ
n :
ℕ
hp :
hη :
InteriorMeans (parameterPath θ d a)
ha :
a ∈ Icc ell upper
conclusion
fisherInformation (transcriptReferenceMeasure P n) (vtDensity P θ d p n ell upper) (vtDensityDeriv P θ d p n ell upper) a
= ∫ z, conditionalTranscriptDirectionalScore P (parameterPath θ d a) d p n z ^ 2 ∂transcriptLaw P (parameterPath θ d a) p n
Proof (Lean source)
lemma fisherInformation_vtDensity_eq_transcriptScoreEnergy
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ d : TrialParameter) (p ell upper a : ℝ)
(n : ℕ) (hp : InteriorAssignment p)
(hη : InteriorMeans (parameterPath θ d a)) (ha : a ∈ Icc ell upper) :
fisherInformation (transcriptReferenceMeasure P n)
(vtDensity P θ d p n ell upper)
(vtDensityDeriv P θ d p n ell upper) a =
∫ z, conditionalTranscriptDirectionalScore P
(parameterPath θ d a) d p n z ^ 2
∂transcriptLaw P (parameterPath θ d a) p n := by
let q := transcriptMixtureRealDensity P (parameterPath θ d a) p n
let s := conditionalTranscriptDirectionalScore P (parameterPath θ d a) d p n
have hqnonneg (z : Transcript (Z n)) : 0 ≤ q z :=
transcriptMixtureRealDensity_nonneg P (parameterPath θ d a) p n hp hη z
unfold fisherInformation
rw [show (fun z => vtDensity P θ d p n ell upper a z *
likelihoodScore (vtDensity P θ d p n ell upper)
(vtDensityDeriv P θ d p n ell upper) a z ^ 2) =
fun z => q z * s z ^ 2 by
funext z
rw [likelihoodScore_vtDensity_eq_transcriptScore
P θ d p ell upper a n hp hη ha z]
simp [q, s, vtDensity, ha]]
rw [← transcriptLaw_eq_withDensity_mixture
P (parameterPath θ d a) p n hp hη]
rw [integral_withDensity_eq_integral_toReal_smul
(measurable_transcriptMixtureRealDensity
P (parameterPath θ d a) p n).ennreal_ofReal
(Filter.Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)]
apply integral_congr_ae
filter_upwards with z
rw [ENNReal.toReal_ofReal (hqnonneg z), smul_eq_mul]
theorem fisherInformation_scaledSelected_le_upperEnvelope
the fisher information scaled selected le upper envelope assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the fisher Information scaled Selected le upper Envelope.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ :
p R a :
ℝ
hp :
hθ :
hε :
0 < ε
n :
ℕ
hn :
0 < n
ha :
a ∈ Icc (-R) R
hη :
conclusion
fisherInformation (transcriptReferenceMeasure P n) (vtDensity P θ (scaledDirection ((sqrt n)⁻¹) (direction (selectedDirection θ p ε hp hθ hε))) p n (-R) R) (vtDensityDeriv P θ (scaledDirection ((sqrt n)⁻¹) (direction (selectedDirection θ p ε hp hθ hε))) p n (-R) R) a
≤ upperEnvelope (parameterPath θ (direction (selectedDirection θ p ε hp hθ hε)) (a / sqrt n)) p ε (selectedDirection θ p ε hp hθ hε)
Proof (Lean source)
lemma fisherInformation_scaledSelected_le_upperEnvelope
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p R a : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(n : ℕ) (hn : 0 < n)
(ha : a ∈ Icc (-R) R)
(hη : InteriorMeans (parameterPath θ
(direction (selectedDirection θ p ε hp hθ hε)) (a / sqrt n))) :
fisherInformation (transcriptReferenceMeasure P n)
(vtDensity P θ
(scaledDirection ((sqrt n)⁻¹)
(direction (selectedDirection θ p ε hp hθ hε)))
p n (-R) R)
(vtDensityDeriv P θ
(scaledDirection ((sqrt n)⁻¹)
(direction (selectedDirection θ p ε hp hθ hε)))
p n (-R) R) a ≤
upperEnvelope
(parameterPath θ
(direction (selectedDirection θ p ε hp hθ hε))
(a / sqrt n)) p ε
(selectedDirection θ p ε hp hθ hε) := by
let v := direction (selectedDirection θ p ε hp hθ hε)
let c := (sqrt n)⁻¹
let η := parameterPath θ v (a / sqrt n)
have hpath : parameterPath θ (scaledDirection c v) a = η := by
funext k
simp [η, v, c, parameterPath, scaledDirection]
ring
rw [fisherInformation_vtDensity_eq_transcriptScoreEnergy
P θ (scaledDirection c v) p (-R) R a n hp (hpath ▸ hη) ha]
rw [hpath]
have hb := transcriptScaledDirectionalFisher_le
P η p (selectedDirection θ p ε hp hθ hε) c hp hη hε n
calc
(∫ z, conditionalTranscriptDirectionalScore P η
(scaledDirection c v) p n z ^ 2 ∂transcriptLaw P η p n) ≤
c ^ 2 * ((n : ℝ) * upperEnvelope η p ε
(selectedDirection θ p ε hp hθ hε)) := by simpa [v] using hb
_ = upperEnvelope η p ε
(selectedDirection θ p ε hp hθ hε) := by
have hnreal : (0 : ℝ) < n := by exact_mod_cast hn
have hsqrt : sqrt (n : ℝ) ≠ 0 := ne_of_gt (Real.sqrt_pos.2 hnreal)
dsimp [c]
rw [inv_pow]
rw [show (sqrt (n : ℝ)) ^ 2 = n by
exact Real.sq_sqrt (Nat.cast_nonneg n)]
field_simp
_ = _ := rfl
theorem integrable_vtLikelihoodFisher_section
the integrable vt likelihood fisher section assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ d :
p ell upper a :
ℝ
n :
ℕ
hp :
hinterior :
∀ a ∈ Icc ell upper, InteriorMeans (parameterPath θ d a)
conclusion
Integrable (fun z => vtDensity P θ d p n ell upper a z * likelihoodScore (vtDensity P θ d p n ell upper) (vtDensityDeriv P θ d p n ell upper) a z ^ 2) (transcriptReferenceMeasure P n)
Proof (Lean source)
lemma integrable_vtLikelihoodFisher_section
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ d : TrialParameter) (p ell upper a : ℝ)
(n : ℕ) (hp : InteriorAssignment p)
(hinterior : ∀ a ∈ Icc ell upper,
InteriorMeans (parameterPath θ d a)) :
Integrable (fun z => vtDensity P θ d p n ell upper a z *
likelihoodScore (vtDensity P θ d p n ell upper)
(vtDensityDeriv P θ d p n ell upper) a z ^ 2)
(transcriptReferenceMeasure P n) := by
by_cases ha : a ∈ Icc ell upper
· let η := parameterPath θ d a
let q := transcriptMixtureRealDensity P η p n
let score := conditionalTranscriptDirectionalScore P η d p n
have hη := hinterior a ha
have hs : Integrable (fun z => score z ^ 2) (transcriptLaw P η p n) :=
(conditionalTranscriptDirectionalScore_memLp_two
P η d p n hp hη).integrable_sq
rw [← transcriptLaw_eq_withDensity_mixture P η p n hp hη] at hs
have hs' := (integrable_withDensity_iff_integrable_smul'
(measurable_transcriptMixtureRealDensity P η p n).ennreal_ofReal
(Filter.Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)).mp hs
convert hs' using 1
funext z
rw [likelihoodScore_vtDensity_eq_transcriptScore
P θ d p ell upper a n hp hη ha z]
have hq := transcriptMixtureRealDensity_nonneg P η p n hp hη z
simp [vtDensity, ha, η, score, ENNReal.toReal_ofReal hq, smul_eq_mul]
· simp [vtDensity, likelihoodScore, ha]
theorem scaledSelected_fisher_integrable_and_control
the scaled selected fisher integrable and control assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the scaled Selected fisher integrable and control.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ :
p R :
ℝ
hp :
hθ :
hε :
0 < ε
hR :
0 < R
n :
ℕ
hn :
0 < n
hinterior :
∀ a ∈ Icc (-R) R,
henvInt :
Integrable (fun a => sequentialVTPrior R a * upperEnvelope (parameterPath θ (direction (selectedDirection θ p ε hp hθ hε)) (a / sqrt n)) p ε (selectedDirection θ p ε hp hθ hε)) (parameterMeasure (-R) R)
conclusion
conclusion 1
Integrable (fun z : ℝ × Transcript (Z n) => sequentialVTPrior R z.1 * vtDensity P θ (scaledDirection ((sqrt n)⁻¹) (direction (selectedDirection θ p ε hp hθ hε))) p n (-R) R z.1 z.2 * (likelihoodScore (vtDensity P θ (scaledDirection ((sqrt n)⁻¹) (direction (selectedDirection θ p ε hp hθ hε))) p n (-R) R) (vtDensityDeriv P θ (scaledDirection ((sqrt n)⁻¹) (direction (selectedDirection θ p ε hp hθ hε))) p n (-R) R) z.1 z.2) ^ 2) ((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n))
conclusion 2
(∫ a, sequentialVTPrior R a * fisherInformation (transcriptReferenceMeasure P n) (vtDensity P θ (scaledDirection ((sqrt n)⁻¹) (direction (selectedDirection θ p ε hp hθ hε))) p n (-R) R) (vtDensityDeriv P θ (scaledDirection ((sqrt n)⁻¹) (direction (selectedDirection θ p ε hp hθ hε))) p n (-R) R) a ∂parameterMeasure (-R) R)
≤ sequentialPriorInformationEnvelope θ p ε R hp hθ hε n
Proof (Lean source)
lemma scaledSelected_fisher_integrable_and_control
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p R : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hR : 0 < R) (n : ℕ) (hn : 0 < n)
(hinterior : ∀ a ∈ Icc (-R) R, InteriorMeans
(parameterPath θ
(direction (selectedDirection θ p ε hp hθ hε))
(a / sqrt n)))
(henvInt : Integrable (fun a => sequentialVTPrior R a *
upperEnvelope
(parameterPath θ
(direction (selectedDirection θ p ε hp hθ hε))
(a / sqrt n)) p ε
(selectedDirection θ p ε hp hθ hε))
(parameterMeasure (-R) R)) :
Integrable (fun z : ℝ × Transcript (Z n) =>
sequentialVTPrior R z.1 *
vtDensity P θ
(scaledDirection ((sqrt n)⁻¹)
(direction (selectedDirection θ p ε hp hθ hε)))
p n (-R) R z.1 z.2 *
(likelihoodScore
(vtDensity P θ
(scaledDirection ((sqrt n)⁻¹)
(direction (selectedDirection θ p ε hp hθ hε)))
p n (-R) R)
(vtDensityDeriv P θ
(scaledDirection ((sqrt n)⁻¹)
(direction (selectedDirection θ p ε hp hθ hε)))
p n (-R) R) z.1 z.2) ^ 2)
((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n)) ∧
(∫ a, sequentialVTPrior R a *
fisherInformation (transcriptReferenceMeasure P n)
(vtDensity P θ
(scaledDirection ((sqrt n)⁻¹)
(direction (selectedDirection θ p ε hp hθ hε)))
p n (-R) R)
(vtDensityDeriv P θ
(scaledDirection ((sqrt n)⁻¹)
(direction (selectedDirection θ p ε hp hθ hε)))
p n (-R) R) a
∂parameterMeasure (-R) R) ≤
sequentialPriorInformationEnvelope θ p ε R hp hθ hε n := by
let v := direction (selectedDirection θ p ε hp hθ hε)
let d := scaledDirection ((sqrt n)⁻¹) v
let q := vtDensity P θ d p n (-R) R
let dq := vtDensityDeriv P θ d p n (-R) R
let w := sequentialVTPrior R
let ν := transcriptReferenceMeasure P n
let π := parameterMeasure (-R) R
let F : ℝ × Transcript (Z n) → ℝ := fun z =>
w z.1 * q z.1 z.2 * (likelihoodScore q dq z.1 z.2) ^ 2
let env : ℝ → ℝ := fun a => upperEnvelope
(parameterPath θ v (a / sqrt n)) p ε
(selectedDirection θ p ε hp hθ hε)
let info : ℝ → ℝ := fun a => fisherInformation ν q dq a
have hpath (a : ℝ) : parameterPath θ d a =
parameterPath θ v (a / sqrt n) := by
funext k
simp [d, v, parameterPath, scaledDirection, div_eq_mul_inv]
ring
have hmeas := sequentialVanTreesProductMeasurable
P θ d p R n hR (P.estimate n) (P.estimate_measurable n)
have hFmeas : AEStronglyMeasurable F (π.prod ν) := by
exact hmeas.fisherSq.aestronglyMeasurable
have hFnonneg (z : ℝ × Transcript (Z n)) : 0 ≤ F z := by
dsimp [F]
exact mul_nonneg
(mul_nonneg (sequentialVTPrior_facts hR |>.nonneg z.1)
(vtDensity_nonneg P θ d p n (-R) R hp
(by
intro a ha
exact hpath a ▸ hinterior a ha) z.1 z.2))
(sq_nonneg _)
have hsection (a : ℝ) : Integrable (fun z => F (a, z)) ν := by
have hs := integrable_vtLikelihoodFisher_section
P θ d p (-R) R a n hp (by
intro b hb
exact hpath b ▸ hinterior b hb)
exact (hs.const_mul (w a)).congr (Filter.Eventually.of_forall fun z => by
dsimp [F]
ring)
let G : ℝ → ℝ := fun a => ∫ z, ‖F (a, z)‖ ∂ν
have hGmeas : AEStronglyMeasurable G π := by
exact hFmeas.norm.integral_prod_right'
have hG_eq (a : ℝ) : G a = w a * info a := by
calc
G a = ∫ z, F (a, z) ∂ν := by
apply integral_congr_ae
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (hFnonneg (a, z))]
_ = w a * ∫ z, q a z * (likelihoodScore q dq a z) ^ 2 ∂ν := by
rw [← integral_const_mul]
apply integral_congr_ae
filter_upwards with z
dsimp [F]
ring
_ = w a * info a := rfl
have hGInt : Integrable G π := by
apply henvInt.mono' hGmeas
have hmem : ∀ᵐ a ∂π, a ∈ Icc (-R) R := by
dsimp [π, parameterMeasure]
exact ae_restrict_mem measurableSet_Icc
filter_upwards [hmem] with a ha
rw [hG_eq, Real.norm_eq_abs, abs_of_nonneg
(mul_nonneg (sequentialVTPrior_facts hR |>.nonneg a)
(by
dsimp [info]
rw [fisherInformation_vtDensity_eq_transcriptScoreEnergy
P θ d p (-R) R a n hp
(by
exact hpath a ▸ hinterior a ha) ha]
exact integral_nonneg fun _ => sq_nonneg _))]
exact mul_le_mul_of_nonneg_left
(by
dsimp [info, env]
simpa [d, v] using fisherInformation_scaledSelected_le_upperEnvelope
P θ p R a hp hθ hε n hn ha (hinterior a ha))
(sequentialVTPrior_facts hR |>.nonneg a)
have hFInt : Integrable F (π.prod ν) :=
(integrable_prod_iff hFmeas).2
⟨Filter.Eventually.of_forall hsection, hGInt⟩
constructor
· simpa [F, π, ν, w, q, dq, d, v] using hFInt
· have hweightedInt : Integrable (fun a => w a * info a) π := by
exact hGInt.congr (Filter.Eventually.of_forall fun a => hG_eq a)
calc
(∫ a, sequentialVTPrior R a *
fisherInformation (transcriptReferenceMeasure P n)
(vtDensity P θ d p n (-R) R)
(vtDensityDeriv P θ d p n (-R) R) a ∂π) ≤
∫ a, w a * env a ∂π := by
apply integral_mono_ae
(by simpa [w, info, q, dq, ν, d, v] using hweightedInt)
(by simpa [w, env, π, v] using henvInt)
have hmem : ∀ᵐ a ∂π, a ∈ Icc (-R) R := by
dsimp [π, parameterMeasure]
exact ae_restrict_mem measurableSet_Icc
filter_upwards [hmem] with a ha
exact mul_le_mul_of_nonneg_left
(by
dsimp [info, env, ν, q, dq, d, v]
exact fisherInformation_scaledSelected_le_upperEnvelope
P θ p R a hp hθ hε n hn ha (hinterior a ha))
(sequentialVTPrior_facts hR |>.nonneg a)
_ = sequentialPriorInformationEnvelope θ p ε R hp hθ hε n := by
rfl
theorem sequential_localAsymptoticRisk_lower_bound
Every sequentially private procedure has local asymptotic risk at least the stationary minimax reciprocal information. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the sequential local Asymptotic Risk lower bound.
Formal statement
Proof (Lean source)
lemma sequential_localAsymptoticRisk_lower_bound
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε) :
ofReal (Vstar θ p ε) ≤ localAsymptoticRisk P θ p := by
let t := selectedDirection θ p ε hp hθ hε
let v := direction t
let normv := sqrt ((v 0) ^ 2 + (v 1) ^ 2)
let H : ℝ → ℝ := fun R => R * normv
let info : ℝ → ℕ → ℝ := fun R n =>
sequentialPriorInformationEnvelope θ p ε R hp hθ hε n
have hnormv : 0 < normv := by
have hsum : 0 < (v 0) ^ 2 + (v 1) ^ 2 := by
dsimp [v]
simp [direction]
nlinarith [sq_nonneg t, sq_nonneg (t + 1)]
exact Real.sqrt_pos.2 hsum
have hH : ∀ R, 0 < R → 0 < H R := by
intro R hR
exact mul_pos hR hnormv
have hinfo : ∀ R, 0 < R → Tendsto (info R) atTop (nhds (Jstar θ p ε)) := by
intro R hR
exact tendsto_sequentialPriorInformationEnvelope θ p ε R hp hθ hε hR
have hlower : ∀ R, 0 < R → ∀ᶠ n : ℕ in atTop,
ofReal (1 / (info R n + 40 / R ^ 2)) ≤
localWorstRisk P θ p (H R) n := by
intro R hR
have hinterior := eventually_parameterPath_compactScaled_interior
θ v R hθ
have henvInt := eventually_integrable_priorWeighted_upperEnvelope
θ p ε R hp hθ hε hR
filter_upwards [eventually_gt_atTop (0 : ℕ), hinterior, henvInt]
with n hn hnInterior hnEnvInt
have hcontrol := scaledSelected_fisher_integrable_and_control
P θ p R hp hθ hε hR n hn hnInterior hnEnvInt
have hpath (a : ℝ) : parameterPath θ
(scaledDirection ((sqrt n)⁻¹) v) a =
parameterPath θ v (a / sqrt n) := by
funext k
simp [parameterPath, scaledDirection, div_eq_mul_inv]
ring
have hactualInterior : ∀ a ∈ Icc (-R) R, InteriorMeans
(parameterPath θ (scaledDirection ((sqrt n)⁻¹) v) a) := by
intro a ha
exact hpath a ▸ hnInterior a ha
have havgNonneg : 0 ≤ ∫ a, sequentialVTPrior R a *
fisherInformation (transcriptReferenceMeasure P n)
(vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R)
(vtDensityDeriv P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R) a
∂parameterMeasure (-R) R := by
apply integral_nonneg_of_ae
have hmem : ∀ᵐ a ∂parameterMeasure (-R) R, a ∈ Icc (-R) R := by
unfold parameterMeasure
exact ae_restrict_mem measurableSet_Icc
filter_upwards [hmem] with a ha
apply mul_nonneg (sequentialVTPrior_facts hR |>.nonneg a)
rw [fisherInformation_vtDensity_eq_transcriptScoreEnergy
P θ (scaledDirection ((sqrt n)⁻¹) v) p (-R) R a n hp
(hactualInterior a ha) ha]
exact integral_nonneg fun _ => sq_nonneg _
have hinfoPos : 0 < priorInformation (-R) R
(sequentialVTPrior R) (sequentialVTPriorDeriv R) +
∫ a, sequentialVTPrior R a *
fisherInformation (transcriptReferenceMeasure P n)
(vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R)
(vtDensityDeriv P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R) a
∂parameterMeasure (-R) R := by
rw [(sequentialVTPrior_facts hR).information]
have hprior : 0 < 40 / R ^ 2 := div_pos (by norm_num) (sq_pos_of_pos hR)
linarith
have hbound := ofReal_vanTreesEnvelope_le_localWorstRisk_of_fisher_control_allErrors
P θ v p R (H R) (info R n) n hn hR hp hactualInterior
(le_rfl : R * sqrt ((v 0) ^ 2 + (v 1) ^ 2) ≤ H R)
hcontrol.1 hcontrol.2 hinfoPos
have hcontrast : contrast v = 1 := by
simpa [v, t] using contrast_selectedDirection θ p ε hp hθ hε
simpa [hcontrast] using hbound
have hrecip := inv_information_le_localAsymptoticRisk_of_priorRadius_family
P θ p (Jstar θ p ε) H info
(Jstar_pos_interior θ p ε hp hθ hε) hH hinfo hlower
simpa [Vstar, one_div] using hrecip
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialMinimaxAssembly 5 declarations This file packages the elementary order-theoretic and pilot-rate steps used to assemble the sequential local asymptotic minimax theorem.
Structural assembly for sequential minimaxity
This file packages the elementary order-theoretic and pilot-rate steps used to assemble the sequential local asymptotic minimax theorem. In particular, the infimum argument only needs a universal risk lower bound and one regular procedure attaining that bound.
def regularProcedureRiskSet
The risks achieved by regular sequential procedures over arbitrary measurable output families. For the displayed inputs and conditions, the stated result follows. The regular Procedure Risk Set is determined by the displayed parameters.
Definition (Lean source)
parameters
θ :
p ε :
ℝ
def
given by
{u | ∃ Z : OutputFamily, ∃ mZ : ∀ n i, MeasurableSpace (Z n i), letI := mZ ∃ P : ProcedureSequence Z ε, RegularProcedure P θ p ∧ u = localAsymptoticRisk P θ p}
theorem natSqrt_pilotRates
The integer square-root pilot simultaneously diverges and is sublinear. The stated result follows.
Formal statement
Proof (Lean source)
lemma natSqrt_pilotRates :
PilotDiverges sqrt ∧ PilotSublinear sqrt :=
⟨natSqrt_pilotDiverges, natSqrt_pilotSublinear⟩
theorem lowerBound_regularProcedureRiskSet
A universal lower bound for procedure risk is a lower bound for the risks of regular procedures. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
ofReal v ≤ u
Proof (Lean source)
lemma lowerBound_regularProcedureRiskSet {θ : TrialParameter} {p ε v : ℝ}
(hlower : ∀ (Z : OutputFamily) [∀ n i, MeasurableSpace (Z n i)]
(P : ProcedureSequence Z ε),
ofReal v ≤ localAsymptoticRisk P θ p) :
∀ u ∈ regularProcedureRiskSet θ p ε, ofReal v ≤ u := by
rintro u ⟨Z, mZ, P, -, rfl⟩
let _ : ∀ n i, MeasurableSpace (Z n i) := mZ
exact hlower Z P
theorem localAsymptoticRisk_mem_regularProcedureRiskSet
The risk of any displayed regular procedure belongs to the regular risk set. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
θ :
p ε :
ℝ
Z :
mZ :
∀ n i, MeasurableSpace (Z n i)
P :
@ProcedureSequence Z mZ ε
hregular :
RegularProcedure P θ p
conclusion
localAsymptoticRisk P θ p ∈ regularProcedureRiskSet θ p ε
Proof (Lean source)
lemma localAsymptoticRisk_mem_regularProcedureRiskSet
{θ : TrialParameter} {p ε : ℝ}
(Z : OutputFamily) (mZ : ∀ n i, MeasurableSpace (Z n i))
(P : @ProcedureSequence Z mZ ε) (hregular : RegularProcedure P θ p) :
localAsymptoticRisk P θ p ∈ regularProcedureRiskSet θ p ε := by
refine ⟨Z, mZ, ?_⟩
let _ : ∀ n i, MeasurableSpace (Z n i) := mZ
exact ⟨P, hregular, rfl⟩
theorem sInf_regularProcedureRiskSet_eq_of_lowerBound_of_attained
If every procedure has risk at least the target and one regular procedure attains it, then the infimum over regular risks equals the target. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
θ :
p ε v :
ℝ
hlower :
ofReal v ≤ localAsymptoticRisk P θ p
Z :
mZ :
∀ n i, MeasurableSpace (Z n i)
P :
@ProcedureSequence Z mZ ε
hregular :
RegularProcedure P θ p
hattains :
localAsymptoticRisk P θ p = ofReal v
conclusion
Proof (Lean source)
-- keep: reusable sequential-law, Fisher-information, or van-Trees bridge for related adaptive experiments
lemma sInf_regularProcedureRiskSet_eq_of_lowerBound_of_attained
{θ : TrialParameter} {p ε v : ℝ}
(hlower : ∀ (Z : OutputFamily) [∀ n i, MeasurableSpace (Z n i)]
(P : ProcedureSequence Z ε),
ofReal v ≤ localAsymptoticRisk P θ p)
(Z : OutputFamily) (mZ : ∀ n i, MeasurableSpace (Z n i))
(P : @ProcedureSequence Z mZ ε) (hregular : RegularProcedure P θ p)
(hattains : localAsymptoticRisk P θ p = ofReal v) :
sInf (regularProcedureRiskSet θ p ε) = ofReal v := by
apply le_antisymm
· apply sInf_le
rw [← hattains]
exact localAsymptoticRisk_mem_regularProcedureRiskSet Z mZ P hregular
· apply le_sInf
exact lowerBound_regularProcedureRiskSet hlower
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialObservedKernel 7 declarations Observed kernels for finite IID latent inputs
Observed kernels for finite IID latent inputs
def observedStageMeasure
Average one private release row over the fresh IID four-cell input. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The observed Stage Measure is determined by the displayed parameters.
Definition (Lean source)
theorem measurable_observedStageMeasure
the measurable observed stage measure assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (observedStageMeasure P θ p n i)
Proof (Lean source)
lemma measurable_observedStageMeasure {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ)
(n : ℕ) (i : Fin n) :
Measurable (observedStageMeasure P θ p n i) := by
refine Measure.measurable_of_measurable_coe _ fun A hA => ?_
simp only [observedStageMeasure, Measure.finsetSum_apply,
Measure.smul_apply, smul_eq_mul]
exact Finset.measurable_fun_sum _ fun j _ =>
measurable_const.mul ((P.channel n i).measurable_coe hA |>.comp
(measurable_const.prodMk measurable_id))
def observedStageKernel
The observed next-output kernel. The unit coordinate matches the generic finite-horizon score API; the history is the released history only. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The observed Stage Kernel is determined by the displayed parameters.
Definition (Lean source)
parameters
given by
(⟨observedStageMeasure P θ p n i, measurable_observedStageMeasure P θ p n i⟩ : Kernel (PrivateHistory (Z := Z n) i) (Z n i)).comap (fun uh => fsHistoryToPrivate i uh.2) ((measurable_fsHistoryToPrivate i).comp measurable_snd)
theorem observedStageKernel_apply
the observed stage kernel apply assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
observedStageKernel P θ p n i uh
Proof (Lean source)
lemma observedStageKernel_apply {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ)
(n : ℕ) (i : Fin n)
(uh : Unit × History (Z n) i.val (le_of_lt i.isLt)) :
observedStageKernel P θ p n i uh =
∑ j : Fin 4, ofReal (piTheta θ p j) •
P.channel n i (j, fsHistoryToPrivate i uh.2) := by
rfl
theorem observedStageKernel_isMarkov
the observed stage kernel is markov assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the observed Stage Kernel is Markov.
Formal statement
hypotheses
conclusion
IsMarkovKernel (observedStageKernel P θ p n i)
Proof (Lean source)
lemma observedStageKernel_isMarkov {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n : ℕ) (i : Fin n) :
IsMarkovKernel (observedStageKernel P θ p n i) := by
refine ⟨fun uh => ⟨?_⟩⟩
rw [observedStageKernel_apply]
simp only [Measure.finsetSum_apply, Measure.smul_apply, smul_eq_mul]
have hmarkov := (P.privacy n).1 i
letI : IsMarkovKernel (P.channel n i) := hmarkov
simp only [measure_univ, mul_one]
rw [← ENNReal.ofReal_sum_of_nonneg]
· rw [piTheta_sum_eq_one]
norm_num
· intro j _
exact (piTheta_pos_interior θ p hp hθ j).le
def observedKernelFamily
Parameterize the observed release kernels along a local affine path. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The observed Kernel Family is determined by the displayed parameters.
Definition (Lean source)
parameters
def
given by
fun i => observedStageKernel P (parameterPath θ v u) p n i
theorem observedKernelFamily_isMarkov
the observed kernel family is markov assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
hp :
n :
ℕ
u :
ℝ
hu :
InteriorMeans (parameterPath θ v u)
i :
Fin n
conclusion
IsMarkovKernel (observedKernelFamily P θ v p n u i)
Proof (Lean source)
lemma observedKernelFamily_isMarkov {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (n : ℕ) (u : ℝ)
(hu : InteriorMeans (parameterPath θ v u)) (i : Fin n) :
IsMarkovKernel (observedKernelFamily P θ v p n u i) :=
observedStageKernel_isMarkov P _ p hp hu n i
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialObservedLaw 1 declarations This file marginalizes the finite IID four-cell input path and identifies the resulting transcript law with the law generated by the observed kernels.
Exact law of the observed sequential experiment
This file marginalizes the finite IID four-cell input path and identifies the resulting transcript law with the law generated by the observed kernels.
theorem transcriptLaw_eq_observedKernelLaw
Marginalizing the IID four-cell input path in the paper transcript law gives exactly the promoted finite-sequence law of the observed kernels. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
hp :
n :
ℕ
u :
ℝ
hu :
InteriorMeans (parameterPath θ v u)
conclusion
transcriptLaw P (parameterPath θ v u) p n
= law (observedKernelFamily P θ v p n) u
Proof (Lean source)
theorem transcriptLaw_eq_observedKernelLaw {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (n : ℕ) (u : ℝ)
(hu : InteriorMeans (parameterPath θ v u)) :
transcriptLaw P (parameterPath θ v u) p n =
law
(observedKernelFamily P θ v p n) u := by
let θu := parameterPath θ v u
have hQ (i : Fin n) : IsMarkovKernel (P.channel n i) := (P.privacy n).1 i
have hdet (x : Fin n → Fin 4) :
P.transcript n x = finiteSequenceTranscript (P.channel n) x :=
(P.factorizes n).eq_finiteSequenceTranscript hQ x
unfold transcriptLaw
simp_rw [hdet]
unfold finiteSequenceTranscript
simp_rw [← Measure.map_smul]
rw [← Measure.map_finset_sum' measurable_fsTranscriptToTranscript.aemeasurable]
change Measure.map fsTranscriptToTranscript
(prefixMixtureLaw (finiteSequenceKernel (P.channel n)) θu p n le_rfl) = _
rw [prefixMixtureLaw_eq_observed P θu p hp hu n n le_rfl]
unfold law observedKernelFamily
dsimp only [θu]
ext A hA
rw [Measure.map_apply measurable_fsTranscriptToTranscript hA]
congr 1
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialPastScoreIntegral 7 declarations This file rewrites the explicit prefix-mixture derivative from SequentialSuccessorDerivative as an integral of the prefix directional score against the actual prefix law.
The predecessor score in a successor event derivative
This file rewrites the explicit prefix-mixture derivative from
SequentialSuccessorDerivative as an integral of the prefix directional score
against the actual prefix law. It uses only finite-mixture domination and does
not invoke score projection.
theorem integrable_transcriptPrefixMixtureRealDerivative_mul_successorRowKernelSection
The explicit prefix derivative remains integrable after multiplication by a measurable successor-row event section. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Integrable (fun h => transcriptPrefixMixtureRealDerivative P θ v p n k (le_of_succ_le hk) h * successorRowKernelSection P n k hk a A h) (transcriptPrefixReferenceMeasure P n k (le_of_succ_le hk))
Proof (Lean source)
lemma integrable_transcriptPrefixMixtureRealDerivative_mul_successorRowKernelSection
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(n k : ℕ) (hk : k + 1 ≤ n) (a : Fin 4)
(A : Set (History (Z n) k (le_of_succ_le hk) ×
Z n (nextIndex hk))) (hA : MeasurableSet A) :
Integrable (fun h =>
transcriptPrefixMixtureRealDerivative P θ v p n k
(le_of_succ_le hk) h *
successorRowKernelSection P n k hk a A h)
(transcriptPrefixReferenceMeasure P n k (le_of_succ_le hk)) := by
let ν := transcriptPrefixReferenceMeasure P n k (le_of_succ_le hk)
let μ (x : Fin n → Fin 4) :=
(P.transcript n x).map (transcriptPrefix (le_of_succ_le hk))
letI (x : Fin n → Fin 4) : IsProbabilityMeasure (P.transcript n x) :=
(P.factorizes n x).1
letI (x : Fin n → Fin 4) : IsProbabilityMeasure (μ x) := by
dsimp [μ]
exact Measure.isProbabilityMeasure_map
(measurable_transcriptPrefix (le_of_succ_le hk)).aemeasurable
letI : IsFiniteMeasure ν := by
dsimp [ν, transcriptPrefixReferenceMeasure]
infer_instance
have hac (x : Fin n → Fin 4) : μ x ≪ ν := by
dsimp [μ, ν, transcriptPrefixReferenceMeasure]
exact (transcriptComponent_ac_reference P n x).map
(measurable_transcriptPrefix (le_of_succ_le hk))
have hcomponent (x : Fin n → Fin 4) : Integrable (fun h =>
transcriptPrefixComponentRealDensity P n k (le_of_succ_le hk) x h *
successorRowKernelSection P n k hk a A h) ν := by
have h := (integrable_toReal_rnDeriv_mul_iff (hac x)).2
(integrable_successorRowKernelSection_component P n k hk a A hA x)
simpa [transcriptPrefixComponentRealDensity, μ, ν] using h
unfold transcriptPrefixMixtureRealDerivative
simp_rw [Finset.sum_mul]
exact integrable_finsetSum _ fun x _ => by
simpa only [mul_assoc] using (hcomponent x).const_mul
(inputPathDirectionalDerivative θ v p x)
theorem transcriptPrefixLaw_eq_withDensity_reference
A prefix law is its common reference measure weighted by its Radon--Nikodym derivative. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
(transcriptPrefixReferenceMeasure P n k hk).withDensity ((transcriptPrefixLaw P θ p n k hk).rnDeriv (transcriptPrefixReferenceMeasure P n k hk))
= transcriptPrefixLaw P θ p n k hk
Proof (Lean source)
lemma transcriptPrefixLaw_eq_withDensity_reference
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ)
(n k : ℕ) (hk : k ≤ n) :
(transcriptPrefixReferenceMeasure P n k hk).withDensity
((transcriptPrefixLaw P θ p n k hk).rnDeriv
(transcriptPrefixReferenceMeasure P n k hk)) =
transcriptPrefixLaw P θ p n k hk := by
letI : IsFiniteMeasure (transcriptPrefixReferenceMeasure P n k hk) := by
unfold transcriptPrefixReferenceMeasure
infer_instance
letI : IsFiniteMeasure (transcriptPrefixLaw P θ p n k hk) := by
unfold transcriptPrefixLaw
infer_instance
exact Measure.withDensity_rnDeriv_eq (transcriptPrefixLaw P θ p n k hk)
(transcriptPrefixReferenceMeasure P n k hk)
(transcriptPrefixLaw_ac_reference P θ p n k hk)
theorem conditionalTranscriptPrefixDirectionalScore_mul_density
Multiplying the guarded prefix score by the prefix density recovers the explicit real derivative density everywhere. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the conditional Transcript Prefix Directional Score mul density.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
hp :
hθ :
n k :
ℕ
hk :
k ≤ n
h :
History (Z n) k hk
conclusion
transcriptPrefixMixtureRealDensity P θ p n k hk h * conditionalTranscriptPrefixDirectionalScore P θ v p n k hk h
= transcriptPrefixMixtureRealDerivative P θ v p n k hk h
Proof (Lean source)
lemma conditionalTranscriptPrefixDirectionalScore_mul_density
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k ≤ n) (h : History (Z n) k hk) :
transcriptPrefixMixtureRealDensity P θ p n k hk h *
conditionalTranscriptPrefixDirectionalScore P θ v p n k hk h =
transcriptPrefixMixtureRealDerivative P θ v p n k hk h := by
rw [conditionalTranscriptPrefixDirectionalScore_eq_guarded
P θ v p n k hk hp hθ h]
split_ifs with hpos
· field_simp
· have hnonneg := transcriptPrefixMixtureRealDensity_nonneg
P θ p n k hk hp hθ h
have hz : transcriptPrefixMixtureRealDensity P θ p n k hk h = 0 :=
le_antisymm (le_of_not_gt hpos) hnonneg
rw [hz, zero_mul,
transcriptPrefixMixtureRealDerivative_eq_zero_of_density_eq_zero
P θ v p n k hk hp hθ h hz]
theorem predecessorRowEventDerivative_eq_integral_pastScore
The predecessor-row derivative is the integral of the actual prefix score times the bounded row-event section under the actual prefix law. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the predecessor Row Event Derivative eq integral past Score.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
hp :
hθ :
n k :
ℕ
hk :
k + 1 ≤ n
a :
Fin 4
A :
hA :
conclusion
predecessorRowEventDerivative P θ v p n k hk a A
= ∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n k (le_of_succ_le hk) h * successorRowKernelSection P n k hk a A h ∂transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
Proof (Lean source)
lemma predecessorRowEventDerivative_eq_integral_pastScore
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n) (a : Fin 4)
(A : Set (History (Z n) k (le_of_succ_le hk) ×
Z n (nextIndex hk))) (hA : MeasurableSet A) :
predecessorRowEventDerivative P θ v p n k hk a A =
∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n k
(le_of_succ_le hk) h *
successorRowKernelSection P n k hk a A h
∂transcriptPrefixLaw P θ p n k (le_of_succ_le hk) := by
let ν := transcriptPrefixReferenceMeasure P n k (le_of_succ_le hk)
let μ := transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
letI : IsFiniteMeasure ν := by
dsimp [ν, transcriptPrefixReferenceMeasure]
infer_instance
letI : IsFiniteMeasure μ := by
dsimp [μ, transcriptPrefixLaw]
infer_instance
rw [← transcriptPrefixLaw_eq_withDensity_reference P θ p n k
(le_of_succ_le hk)]
rw [integral_withDensity_eq_integral_toReal_smul
(Measure.measurable_rnDeriv μ ν) (Measure.rnDeriv_lt_top μ ν)]
unfold predecessorRowEventDerivative
apply integral_congr_ae
filter_upwards
[transcriptPrefixDensity_toReal_ae_eq_mixture_of_interior
P θ p n k (le_of_succ_le hk) hp hθ]
with h hdensity
rw [smul_eq_mul, hdensity]
rw [← conditionalTranscriptPrefixDirectionalScore_mul_density
P θ v p hp hθ n k (le_of_succ_le hk) h]
ring
theorem integrable_conditionalTranscriptPrefixDirectionalScore_mul_successorRowKernelSection
The past-score integrand used above is integrable under the actual prefix law. This is the integrability input needed by a subsequent projection or conditional-expectation uniqueness argument. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the integrable conditional Transcript Prefix Directional Score mul successor Row Kernel Section.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
hp :
hθ :
n k :
ℕ
hk :
k + 1 ≤ n
a :
Fin 4
A :
hA :
conclusion
Integrable (fun h => conditionalTranscriptPrefixDirectionalScore P θ v p n k (le_of_succ_le hk) h * successorRowKernelSection P n k hk a A h) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk))
Proof (Lean source)
lemma integrable_conditionalTranscriptPrefixDirectionalScore_mul_successorRowKernelSection
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n) (a : Fin 4)
(A : Set (History (Z n) k (le_of_succ_le hk) ×
Z n (nextIndex hk))) (hA : MeasurableSet A) :
Integrable (fun h =>
conditionalTranscriptPrefixDirectionalScore P θ v p n k
(le_of_succ_le hk) h *
successorRowKernelSection P n k hk a A h)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) := by
let ν := transcriptPrefixReferenceMeasure P n k (le_of_succ_le hk)
let μ := transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
letI : IsFiniteMeasure ν := by
dsimp [ν, transcriptPrefixReferenceMeasure]
infer_instance
letI : IsFiniteMeasure μ := by
dsimp [μ, transcriptPrefixLaw]
infer_instance
rw [← transcriptPrefixLaw_eq_withDensity_reference P θ p n k
(le_of_succ_le hk)]
rw [integrable_withDensity_iff_integrable_smul'
(Measure.measurable_rnDeriv μ ν) (Measure.rnDeriv_lt_top μ ν)]
have hderiv :=
integrable_transcriptPrefixMixtureRealDerivative_mul_successorRowKernelSection
P θ v p n k hk a A hA
refine hderiv.congr ?_
filter_upwards
[transcriptPrefixDensity_toReal_ae_eq_mixture_of_interior
P θ p n k (le_of_succ_le hk) hp hθ]
with h hdensity
rw [smul_eq_mul, hdensity,
← conditionalTranscriptPrefixDirectionalScore_mul_density
P θ v p hp hθ n k (le_of_succ_le hk) h]
ring
theorem predecessorSuccessorEventDerivative_eq_sum_integral_pastScore
Summing the row identities with the base four-cell probabilities gives the complete predecessor contribution in the successor product rule. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the predecessor Successor Event Derivative eq sum integral past Score.
Formal statement
hypotheses
conclusion
predecessorSuccessorEventDerivative θ p (fun a => predecessorRowEventDerivative P θ v p n k hk a A)
= ∑ a : Fin 4, piTheta θ p a * ∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n k (le_of_succ_le hk) h * successorRowKernelSection P n k hk a A h ∂transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
Proof (Lean source)
lemma predecessorSuccessorEventDerivative_eq_sum_integral_pastScore
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n)
(A : Set (History (Z n) k (le_of_succ_le hk) ×
Z n (nextIndex hk))) (hA : MeasurableSet A) :
predecessorSuccessorEventDerivative θ p
(fun a => predecessorRowEventDerivative P θ v p n k hk a A) =
∑ a : Fin 4, piTheta θ p a *
∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n k
(le_of_succ_le hk) h *
successorRowKernelSection P n k hk a A h
∂transcriptPrefixLaw P θ p n k (le_of_succ_le hk) := by
unfold predecessorSuccessorEventDerivative
apply Finset.sum_congr rfl
intro a _
change piTheta θ p a * predecessorRowEventDerivative P θ v p n k hk a A = _
rw [predecessorRowEventDerivative_eq_integral_pastScore
P θ v p hp hθ n k hk a A hA]
theorem transcriptSuccessorLaw_pastScore_characterization
Paper-side successor characterization before score projection: the mapped successor law is the joint-channel output law, and every predecessor row derivative is represented by the integrable prefix-score kernel integral. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the transcript Successor Law past Score characterization.
Formal statement
hypotheses
conclusion
μ :
transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
= outputLaw θ p (sequentialJointChannel P n (nextIndex hk) μ)
a :
Fin 4
Integrable (fun h => conditionalTranscriptPrefixDirectionalScore P θ v p n k (le_of_succ_le hk) h * successorRowKernelSection P n k hk a A h) μ
predecessorRowEventDerivative P θ v p n k hk a A
= ∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n k (le_of_succ_le hk) h * successorRowKernelSection P n k hk a A h ∂μ
Proof (Lean source)
-- keep: reusable sequential-law, Fisher-information, or van-Trees bridge for related adaptive experiments
lemma transcriptSuccessorLaw_pastScore_characterization
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n)
(A : Set (History (Z n) k (le_of_succ_le hk) ×
Z n (nextIndex hk))) (hA : MeasurableSet A) :
let μ := transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
(transcriptPrefixLaw P θ p n (k + 1) hk).map
(fun u => (init hk u, last hk u)) =
outputLaw θ p (sequentialJointChannel P n (nextIndex hk) μ) ∧
∀ a : Fin 4,
Integrable (fun h =>
conditionalTranscriptPrefixDirectionalScore P θ v p n k
(le_of_succ_le hk) h *
successorRowKernelSection P n k hk a A h) μ ∧
predecessorRowEventDerivative P θ v p n k hk a A =
∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n k
(le_of_succ_le hk) h *
successorRowKernelSection P n k hk a A h ∂μ := by
dsimp only
refine ⟨transcriptSuccessorLaw_eq_jointChannel P θ p hp hθ n k hk, ?_⟩
intro a
exact ⟨
integrable_conditionalTranscriptPrefixDirectionalScore_mul_successorRowKernelSection
P θ v p hp hθ n k hk a A hA,
predecessorRowEventDerivative_eq_integral_pastScore
P θ v p hp hθ n k hk a A hA⟩
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialPrefixDensity 13 declarations This file constructs explicit common references, densities, and directional derivative densities after projecting a full transcript to a finite prefix.
Dominated finite-mixture scores on transcript prefixes
This file constructs explicit common references, densities, and directional derivative densities after projecting a full transcript to a finite prefix. The identities are paper-side inputs for score projection and do not use a conditional expectation theorem.
def transcriptPrefixReferenceMeasure
the transcript prefix reference measure is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
given by
def transcriptPrefixLaw
the transcript prefix law is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The transcript Prefix Law is determined by the displayed parameters.
Definition (Lean source)
parameters
def
given by
(transcriptLaw P θ p n).map (transcriptPrefix hk)
def transcriptPrefixComponentRealDensity
the transcript prefix component real density is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
transcriptPrefixComponentRealDensity P n k hk x h :
ℝ
given by
(((P.transcript n x).map (transcriptPrefix hk)).rnDeriv (transcriptPrefixReferenceMeasure P n k hk) h).toReal
def transcriptPrefixMixtureRealDensity
the transcript prefix mixture real density is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The transcript Prefix Mixture Real Density is determined by the displayed parameters.
Definition (Lean source)
parameters
def
transcriptPrefixMixtureRealDensity P θ p n k hk h :
ℝ
given by
def transcriptPrefixMixtureRealDerivative
the transcript prefix mixture real derivative is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The transcript Prefix Mixture Real Derivative is determined by the displayed parameters.
Definition (Lean source)
parameters
def
transcriptPrefixMixtureRealDerivative P θ v p n k hk h :
ℝ
given by
∑ x : Fin n → Fin 4, inputPathDirectionalDerivative θ v p x * transcriptPrefixComponentRealDensity P n k hk x h
def conditionalTranscriptPrefixDirectionalScore
the conditional transcript prefix directional score is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The conditional Transcript Prefix Directional Score is determined by the displayed parameters.
Definition (Lean source)
parameters
def
conditionalTranscriptPrefixDirectionalScore P θ v p n k hk h :
ℝ
given by
transcriptPrefixMixtureRealDerivative P θ v p n k hk h / transcriptPrefixMixtureRealDensity P θ p n k hk h
theorem transcriptPrefixReferenceMeasure_eq_sum
the transcript prefix reference measure eq sum assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
transcriptPrefixReferenceMeasure P n k hk
Proof (Lean source)
lemma transcriptPrefixReferenceMeasure_eq_sum {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (n k : ℕ) (hk : k ≤ n) :
transcriptPrefixReferenceMeasure P n k hk =
∑ x : Fin n → Fin 4, (P.transcript n x).map (transcriptPrefix hk) := by
unfold transcriptPrefixReferenceMeasure transcriptReferenceMeasure
exact map_finset_sum_measure (transcriptPrefix hk)
(measurable_transcriptPrefix hk) univ _
theorem transcriptPrefixLaw_eq_sum
the transcript prefix law eq sum assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
transcriptPrefixLaw P θ p n k hk
= ∑ x : Fin n → Fin 4, ofReal (inputPathProbability θ p x) • (P.transcript n x).map (transcriptPrefix hk)
Proof (Lean source)
lemma transcriptPrefixLaw_eq_sum {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ)
(n k : ℕ) (hk : k ≤ n) :
transcriptPrefixLaw P θ p n k hk =
∑ x : Fin n → Fin 4, ofReal (inputPathProbability θ p x) •
(P.transcript n x).map (transcriptPrefix hk) := by
unfold transcriptPrefixLaw transcriptLaw
rw [map_finset_sum_measure (transcriptPrefix hk)
(measurable_transcriptPrefix hk) univ]
simp_rw [Measure.map_smul]
theorem transcriptPrefixLaw_ac_reference
the transcript prefix law ac reference assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
transcriptPrefixLaw P θ p n k hk ≪ transcriptPrefixReferenceMeasure P n k hk
Proof (Lean source)
lemma transcriptPrefixLaw_ac_reference {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ)
(n k : ℕ) (hk : k ≤ n) :
transcriptPrefixLaw P θ p n k hk ≪
transcriptPrefixReferenceMeasure P n k hk := by
rw [transcriptPrefixLaw_eq_sum, transcriptPrefixReferenceMeasure_eq_sum]
apply Measure.AbsolutelyContinuous.mk
intro A hA href
simp only [Measure.coe_finsetSum, Finset.sum_apply] at href ⊢
apply Finset.sum_eq_zero
intro x _
rw [Measure.smul_apply, smul_eq_mul]
apply mul_eq_zero_of_right
exact (Finset.sum_eq_zero_iff_of_nonneg (fun _ _ ↦ zero_le)).mp href x
(Finset.mem_univ x)
theorem transcriptPrefixDensity_toReal_ae_eq_mixture_of_interior
the transcript prefix density to real ae eq mixture of interior assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the transcript Prefix Density to Real ae eq mixture of interior.
Formal statement
hypotheses
conclusion
(fun h => ((transcriptPrefixLaw P θ p n k hk).rnDeriv (transcriptPrefixReferenceMeasure P n k hk) h).toReal)
=ᵐ[ transcriptPrefixReferenceMeasure P n k hk] transcriptPrefixMixtureRealDensity P θ p n k hk
Proof (Lean source)
lemma transcriptPrefixDensity_toReal_ae_eq_mixture_of_interior {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ)
(n k : ℕ) (hk : k ≤ n)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
(fun h => ((transcriptPrefixLaw P θ p n k hk).rnDeriv
(transcriptPrefixReferenceMeasure P n k hk) h).toReal) =ᵐ[
transcriptPrefixReferenceMeasure P n k hk]
transcriptPrefixMixtureRealDensity P θ p n k hk := by
classical
let ν := transcriptPrefixReferenceMeasure P n k hk
let μ (x : Fin n → Fin 4) :=
(P.transcript n x).map (transcriptPrefix hk)
let w (x : Fin n → Fin 4) : ℝ≥0∞ :=
ofReal (inputPathProbability θ p x)
letI (x : Fin n → Fin 4) : IsProbabilityMeasure (P.transcript n x) :=
(P.factorizes n x).1
letI (x : Fin n → Fin 4) : IsProbabilityMeasure (μ x) :=
Measure.isProbabilityMeasure_map (measurable_transcriptPrefix hk).aemeasurable
letI : IsFiniteMeasure ν := by
dsimp [ν, transcriptPrefixReferenceMeasure]
infer_instance
have hw (x : Fin n → Fin 4) : w x ≠ ∞ := by simp [w]
haveI (x : Fin n → Fin 4) : IsFiniteMeasure (w x • μ x) :=
(μ x).smul_finite (hw x)
have hrn (t : Finset (Fin n → Fin 4)) :
(t.sum fun x ↦ w x • μ x).rnDeriv ν =ᵐ[ν]
fun h ↦ t.sum fun x ↦ w x * (μ x).rnDeriv ν h := by
induction t using Finset.induction_on with
| empty =>
filter_upwards [Measure.rnDeriv_zero ν] with h hh
simpa using hh
| @insert x t hx ih =>
haveI : IsFiniteMeasure (t.sum fun y ↦ w y • μ y) := inferInstance
have hadd := Measure.rnDeriv_add (w x • μ x)
(t.sum fun y ↦ w y • μ y) ν
have hsmul := Measure.rnDeriv_smul_left_of_ne_top (μ x) ν (hw x)
filter_upwards [hadd, hsmul, ih] with h hh ha hb
simpa [Finset.sum_insert hx, Pi.add_apply, Pi.smul_apply,
smul_eq_mul, ha, hb] using hh
have hfinite : ∀ᵐ h ∂ν, ∀ x : Fin n → Fin 4,
(μ x).rnDeriv ν h ≠ ∞ := by
rw [Filter.eventually_all]
intro x
exact (Measure.rnDeriv_lt_top (μ x) ν).mono (fun _ hh ↦ ne_of_lt hh)
filter_upwards [hrn univ, hfinite] with h hh hfin
change ((transcriptPrefixLaw P θ p n k hk).rnDeriv ν h).toReal =
transcriptPrefixMixtureRealDensity P θ p n k hk h
rw [transcriptPrefixLaw_eq_sum P θ p n k hk]
rw [hh, ENNReal.toReal_sum
(fun x _ ↦ ENNReal.mul_ne_top (hw x) (hfin x))]
simp [transcriptPrefixMixtureRealDensity,
transcriptPrefixComponentRealDensity, μ, w, ν, ENNReal.toReal_mul,
(inputPathProbability_pos_of_interior θ p hp hθ _).le]
theorem measurable_transcriptPrefixMixtureRealDensity
the measurable transcript prefix mixture real density assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (transcriptPrefixMixtureRealDensity P θ p n k hk)
Proof (Lean source)
lemma measurable_transcriptPrefixMixtureRealDensity {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ)
(n k : ℕ) (hk : k ≤ n) :
Measurable (transcriptPrefixMixtureRealDensity P θ p n k hk) := by
unfold transcriptPrefixMixtureRealDensity transcriptPrefixComponentRealDensity
exact Finset.measurable_fun_sum _ fun x _ ↦ measurable_const.mul
(Measure.measurable_rnDeriv _ _).ennreal_toReal
theorem measurable_transcriptPrefixMixtureRealDerivative
the measurable transcript prefix mixture real derivative assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (transcriptPrefixMixtureRealDerivative P θ v p n k hk)
Proof (Lean source)
lemma measurable_transcriptPrefixMixtureRealDerivative {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(n k : ℕ) (hk : k ≤ n) :
Measurable (transcriptPrefixMixtureRealDerivative P θ v p n k hk) := by
unfold transcriptPrefixMixtureRealDerivative transcriptPrefixComponentRealDensity
exact Finset.measurable_fun_sum _ fun x _ ↦ measurable_const.mul
(Measure.measurable_rnDeriv _ _).ennreal_toReal
theorem transcriptPrefix_derivative_pushforward
the transcript prefix derivative pushforward assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
n k :
ℕ
hk :
k ≤ n
A :
Set (History (Z n) k hk)
hA :
conclusion
∫ z in transcriptPrefix hk ⁻¹' A, transcriptMixtureRealDerivative P θ v p n z ∂transcriptReferenceMeasure P n
= ∫ h in A, transcriptPrefixMixtureRealDerivative P θ v p n k hk h ∂transcriptPrefixReferenceMeasure P n k hk
Proof (Lean source)
lemma transcriptPrefix_derivative_pushforward {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(n k : ℕ) (hk : k ≤ n) (A : Set (History (Z n) k hk))
(hA : MeasurableSet A) :
∫ z in transcriptPrefix hk ⁻¹' A,
transcriptMixtureRealDerivative P θ v p n z
∂transcriptReferenceMeasure P n =
∫ h in A, transcriptPrefixMixtureRealDerivative P θ v p n k hk h
∂transcriptPrefixReferenceMeasure P n k hk := by
letI (x : Fin n → Fin 4) : IsProbabilityMeasure (P.transcript n x) :=
(P.factorizes n x).1
letI (x : Fin n → Fin 4) : IsProbabilityMeasure
((P.transcript n x).map (transcriptPrefix hk)) :=
Measure.isProbabilityMeasure_map (measurable_transcriptPrefix hk).aemeasurable
letI : IsFiniteMeasure (transcriptPrefixReferenceMeasure P n k hk) := by
unfold transcriptPrefixReferenceMeasure
infer_instance
have hpac (x : Fin n → Fin 4) :
(P.transcript n x).map (transcriptPrefix hk) ≪
transcriptPrefixReferenceMeasure P n k hk := by
unfold transcriptPrefixReferenceMeasure
exact (transcriptComponent_ac_reference P n x).map
(measurable_transcriptPrefix hk)
have hintFull (x : Fin n → Fin 4) : Integrable
(transcriptComponentRealDensity P n x)
(transcriptReferenceMeasure P n) := by
have h := (integrable_toReal_rnDeriv_mul_iff
(transcriptComponent_ac_reference P n x)
(f := fun _ ↦ (1 : ℝ))).2 (integrable_const (1 : ℝ))
change Integrable (fun z ↦
((P.transcript n x).rnDeriv (transcriptReferenceMeasure P n) z).toReal)
(transcriptReferenceMeasure P n)
simpa only [mul_one] using h
have hintPrefix (x : Fin n → Fin 4) : Integrable
(transcriptPrefixComponentRealDensity P n k hk x)
(transcriptPrefixReferenceMeasure P n k hk) := by
have h := (integrable_toReal_rnDeriv_mul_iff (hpac x)
(f := fun _ ↦ (1 : ℝ))).2 (integrable_const (1 : ℝ))
change Integrable (fun z ↦
(((P.transcript n x).map (transcriptPrefix hk)).rnDeriv
(transcriptPrefixReferenceMeasure P n k hk) z).toReal)
(transcriptPrefixReferenceMeasure P n k hk)
simpa only [mul_one] using h
simp_rw [transcriptMixtureRealDerivative,
transcriptPrefixMixtureRealDerivative]
rw [integral_finset_sum _ (fun x _ ↦
((hintFull x).const_mul _).integrableOn),
integral_finset_sum _ (fun x _ ↦
((hintPrefix x).const_mul _).integrableOn)]
apply Finset.sum_congr rfl
intro x _
rw [MeasureTheory.integral_const_mul,
MeasureTheory.integral_const_mul]
unfold transcriptComponentRealDensity transcriptPrefixComponentRealDensity
rw [
Measure.setIntegral_toReal_rnDeriv
(transcriptComponent_ac_reference P n x),
Measure.setIntegral_toReal_rnDeriv (hpac x)]
simp only [measureReal_def]
rw [Measure.map_apply (measurable_transcriptPrefix hk) hA]
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialPriorInformationEnvelope 8 declarations This file upgrades the local stationary-envelope limit to uniform convergence over the compact scalar prior and integrates that envelope against the exact normalized quartic prior used by the sequential van Trees argumen
Compact-prior sequential information envelope
This file upgrades the local stationary-envelope limit to uniform convergence over the compact scalar prior and integrates that envelope against the exact normalized quartic prior used by the sequential van Trees argument.
theorem parameterPath_compactScaled_eq_localAlternative
The inverse-root-sample parameter path agrees with the local alternative indexed by the corresponding scalar multiple of the direction. For the displayed inputs and conditions, the stated result follows.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma parameterPath_compactScaled_eq_localAlternative
(θ v : TrialParameter) (a : ℝ) (n : ℕ) :
parameterPath θ v (a / sqrt n) =
localAlternative θ (scaledDirection a v) n := by
funext k
simp [parameterPath, localAlternative, scaledDirection]
ring
theorem eventually_parameterPath_compactScaled_interior
A compact scalar family of inverse-root-sample paths is eventually contained in the interior parameter space. For the displayed inputs and conditions, the stated result follows. Under the stated assumptions, the eventually parameter Path compact Scaled interior.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma eventually_parameterPath_compactScaled_interior
(θ v : TrialParameter) (R : ℝ) (hθ : InteriorMeans θ) :
∀ᶠ n : ℕ in atTop, ∀ a ∈ Icc (-R) R,
InteriorMeans (parameterPath θ v (a / sqrt n)) := by
let H := R * sqrt ((v 0) ^ 2 + (v 1) ^ 2)
have hev := eventually_localAlternative_interior_uniform θ H hθ
refine hev.mono ?_
intro n hn a ha
rw [parameterPath_compactScaled_eq_localAlternative]
exact hn (scaledDirection a v) (scaledDirection_mem_cover v ha)
theorem tendstoUniformlyOn_upperEnvelope_compactScaled_selectedDirection
The selected stationary information envelope converges uniformly over a fixed compact scalar family of inverse-root-sample local paths. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the tendsto Uniformly On upper Envelope compact Scaled selected Direction.
Formal statement
hypotheses
conclusion
TendstoUniformlyOn (fun (n : ℕ) a => upperEnvelope (parameterPath θ (direction (selectedDirection θ p ε hp hθ hε)) (a / sqrt n)) p ε (selectedDirection θ p ε hp hθ hε)) (fun _ => Jstar θ p ε) atTop (Icc (-R) R)
Proof (Lean source)
lemma tendstoUniformlyOn_upperEnvelope_compactScaled_selectedDirection
(θ : TrialParameter) (p ε R : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε) :
TendstoUniformlyOn
(fun (n : ℕ) a => upperEnvelope
(parameterPath θ
(direction (selectedDirection θ p ε hp hθ hε))
(a / sqrt n)) p ε
(selectedDirection θ p ε hp hθ hε))
(fun _ => Jstar θ p ε) atTop (Icc (-R) R) := by
rw [Metric.tendstoUniformlyOn_iff]
intro δ hδ
let v := direction (selectedDirection θ p ε hp hθ hε)
let Θ := {η : TrialParameter // InteriorMeans η}
let θsub : Θ := ⟨θ, hθ⟩
have hc := (continuous_upperEnvelope_interior p ε
(selectedDirection θ p ε hp hθ hε) hp hε.le).continuousAt
(x := θsub)
obtain ⟨r, hr, hcontrol⟩ := (Metric.continuousAt_iff.mp hc) δ hδ
let H := R * sqrt ((v 0) ^ 2 + (v 1) ^ 2)
have hevDist := eventually_localAlternative_dist_lt_uniform θ H r hr
have hevInterior := eventually_localAlternative_interior_uniform θ H hθ
refine (hevDist.and hevInterior).mono ?_
intro n hn a ha
rcases hn with ⟨hdist, hinterior⟩
have hcover : sqrt (((scaledDirection a v) 0) ^ 2 +
((scaledDirection a v) 1) ^ 2) ≤ H :=
scaledDirection_mem_cover v ha
have heq : parameterPath θ v (a / sqrt n) =
localAlternative θ (scaledDirection a v) n :=
parameterPath_compactScaled_eq_localAlternative θ v a n
have hη : InteriorMeans (parameterPath θ v (a / sqrt n)) := by
rw [heq]
exact hinterior (scaledDirection a v) hcover
have hclose : dist (parameterPath θ v (a / sqrt n)) θ < r := by
rw [heq]
exact hdist (scaledDirection a v) hcover
have hsubclose : dist (⟨parameterPath θ v
(a / sqrt n), hη⟩ : Θ) θsub < r := by
change dist (parameterPath θ v (a / sqrt n)) θ < r
exact hclose
have hout := hcontrol (x := (⟨parameterPath θ v
(a / sqrt n), hη⟩ : Θ)) hsubclose
simpa [v, θsub, upperEnvelope_selectedDirection θ p ε hp hθ hε,
dist_comm] using hout
def sequentialPriorInformationEnvelope
The prior-averaged per-sample stationary information envelope. The sequential Prior Information Envelope is determined by the displayed parameters.
Definition (Lean source)
parameters
def
sequentialPriorInformationEnvelope θ p ε R hp hθ hε n :
ℝ
given by
∫ a, sequentialVTPrior R a * upperEnvelope (parameterPath θ (direction (selectedDirection θ p ε hp hθ hε)) (a / sqrt n)) p ε (selectedDirection θ p ε hp hθ hε) ∂parameterMeasure (-R) R
theorem aestronglyMeasurable_priorWeighted_upperEnvelope
Once the compact path lies in the interior, its selected envelope is continuous on the scalar interval and hence measurable for the prior measure. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the aestrongly Measurable prior Weighted upper Envelope.
Formal statement
hypotheses
θ :
p ε R :
ℝ
hp :
hθ :
hε :
0 < ε
hR :
0 < R
n :
ℕ
hinterior :
∀ a ∈ Icc (-R) R,
conclusion
AEStronglyMeasurable (fun a => sequentialVTPrior R a * upperEnvelope (parameterPath θ (direction (selectedDirection θ p ε hp hθ hε)) (a / sqrt n)) p ε (selectedDirection θ p ε hp hθ hε)) (parameterMeasure (-R) R)
Proof (Lean source)
lemma aestronglyMeasurable_priorWeighted_upperEnvelope
(θ : TrialParameter) (p ε R : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hR : 0 < R)
(n : ℕ)
(hinterior : ∀ a ∈ Icc (-R) R, InteriorMeans
(parameterPath θ
(direction (selectedDirection θ p ε hp hθ hε))
(a / sqrt n))) :
AEStronglyMeasurable (fun a => sequentialVTPrior R a *
upperEnvelope
(parameterPath θ
(direction (selectedDirection θ p ε hp hθ hε))
(a / sqrt n)) p ε
(selectedDirection θ p ε hp hθ hε))
(parameterMeasure (-R) R) := by
let v := direction (selectedDirection θ p ε hp hθ hε)
let Θ := {η : TrialParameter // InteriorMeans η}
let path : ℝ → TrialParameter := fun a =>
parameterPath θ v (a / sqrt n)
let pathSub : ↥(Icc (-R) R) → Θ := fun a =>
⟨path a.1, hinterior a.1 a.2⟩
have hpath : Continuous path := by
apply continuous_pi
intro k
dsimp [path, parameterPath]
exact continuous_const.add
((continuous_id.div_const _).mul continuous_const)
have hpathSub : Continuous pathSub := by
apply Continuous.subtype_mk
exact hpath.comp continuous_subtype_val
have henv : Continuous fun a : ↥(Icc (-R) R) =>
upperEnvelope (pathSub a).1 p ε
(selectedDirection θ p ε hp hθ hε) :=
(continuous_upperEnvelope_interior p ε
(selectedDirection θ p ε hp hθ hε) hp hε.le).comp hpathSub
have henvOn : ContinuousOn (fun a => upperEnvelope (path a) p ε
(selectedDirection θ p ε hp hθ hε)) (Icc (-R) R) := by
rw [continuousOn_iff_continuous_domRestrict]
change Continuous (fun a : ↥(Icc (-R) R) =>
upperEnvelope (path a.1) p ε
(selectedDirection θ p ε hp hθ hε))
simpa only [pathSub] using henv
have hprior : Continuous (sequentialVTPrior R) := by
unfold sequentialVTPrior
exact (smoothPrior_contDiff (by linarith)).continuous
unfold parameterMeasure
exact (hprior.continuousOn.mul henvOn).aestronglyMeasurable measurableSet_Icc
theorem eventually_integrable_priorWeighted_upperEnvelope
For all sufficiently large sample sizes, the prior-weighted stationary information envelope is integrable under the compact parameter measure. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the eventually integrable prior Weighted upper Envelope.
Formal statement
hypotheses
conclusion
∀ᶠ n : ℕ in atTop, Integrable (fun a => sequentialVTPrior R a * upperEnvelope (parameterPath θ (direction (selectedDirection θ p ε hp hθ hε)) (a / sqrt n)) p ε (selectedDirection θ p ε hp hθ hε)) (parameterMeasure (-R) R)
Proof (Lean source)
lemma eventually_integrable_priorWeighted_upperEnvelope
(θ : TrialParameter) (p ε R : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hR : 0 < R) :
∀ᶠ n : ℕ in atTop, Integrable (fun a => sequentialVTPrior R a *
upperEnvelope
(parameterPath θ
(direction (selectedDirection θ p ε hp hθ hε))
(a / sqrt n)) p ε
(selectedDirection θ p ε hp hθ hε))
(parameterMeasure (-R) R) := by
let v := direction (selectedDirection θ p ε hp hθ hε)
let F : ℕ → ℝ → ℝ := fun n a => sequentialVTPrior R a *
upperEnvelope (parameterPath θ v (a / sqrt n)) p ε
(selectedDirection θ p ε hp hθ hε)
let B : ℝ → ℝ := fun a =>
(|Jstar θ p ε| + 1) * sequentialVTPrior R a
have hpriorInt : Integrable (sequentialVTPrior R)
(parameterMeasure (-R) R) := by
unfold sequentialVTPrior
exact smoothPrior_integrable_parameterMeasure (by linarith)
have hBint : Integrable B (parameterMeasure (-R) R) := by
exact hpriorInt.const_mul (|Jstar θ p ε| + 1)
have hu := tendstoUniformlyOn_upperEnvelope_compactScaled_selectedDirection
θ p ε R hp hθ hε
rw [Metric.tendstoUniformlyOn_iff] at hu
have hclose := hu 1 zero_lt_one
have hinterior := eventually_parameterPath_compactScaled_interior
θ v R hθ
refine (hclose.and hinterior).mono ?_
intro n hn
rcases hn with ⟨hnclose, hninterior⟩
have hmeas : AEStronglyMeasurable (F n)
(parameterMeasure (-R) R) := by
exact aestronglyMeasurable_priorWeighted_upperEnvelope
θ p ε R hp hθ hε hR n hninterior
apply hBint.mono' hmeas
have hmem : ∀ᵐ a ∂parameterMeasure (-R) R, a ∈ Icc (-R) R := by
unfold parameterMeasure
exact ae_restrict_mem measurableSet_Icc
filter_upwards [hmem] with a ha
have henv : |upperEnvelope (parameterPath θ v
(a / sqrt n)) p ε
(selectedDirection θ p ε hp hθ hε)| < |Jstar θ p ε| + 1 := by
have hd := hnclose a ha
rw [Real.dist_eq] at hd
calc
|upperEnvelope (parameterPath θ v (a / sqrt n)) p ε
(selectedDirection θ p ε hp hθ hε)| =
|Jstar θ p ε +
(upperEnvelope (parameterPath θ v (a / sqrt n)) p ε
(selectedDirection θ p ε hp hθ hε) - Jstar θ p ε)| := by
congr 1
ring
_ ≤
|Jstar θ p ε| +
|upperEnvelope (parameterPath θ v (a / sqrt n)) p ε
(selectedDirection θ p ε hp hθ hε) - Jstar θ p ε| :=
abs_add_le _ _
_ < |Jstar θ p ε| + 1 := by
rw [abs_sub_comm]
linarith
have hpriorNonneg := (sequentialVTPrior_facts hR).nonneg a
dsimp [F, B]
rw [abs_mul, abs_of_nonneg hpriorNonneg]
simpa [mul_comm] using
(mul_le_mul_of_nonneg_left henv.le hpriorNonneg)
theorem integral_priorWeighted_le_nat_mul_sequentialPriorInformationEnvelope
A pointwise compact-prior Fisher bound by n times the stationary envelope integrates to n times the per-sample prior information envelope. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the integral prior Weighted le nat mul sequential Prior Information Envelope.
Formal statement
hypotheses
θ :
p ε R :
ℝ
hp :
hθ :
hε :
0 < ε
hR :
0 < R
n :
ℕ
f :
ℝ → ℝ
hfInt :
henvInt :
Integrable (fun a => sequentialVTPrior R a * upperEnvelope (parameterPath θ (direction (selectedDirection θ p ε hp hθ hε)) (a / sqrt n)) p ε (selectedDirection θ p ε hp hθ hε)) (parameterMeasure (-R) R)
hle :
∀ a ∈ Icc (-R) R,
f a
≤ (n : ℝ) * upperEnvelope (parameterPath θ (direction (selectedDirection θ p ε hp hθ hε)) (a / sqrt n)) p ε (selectedDirection θ p ε hp hθ hε)
conclusion
(∫ a, sequentialVTPrior R a * f a ∂parameterMeasure (-R) R)
≤ (n : ℝ) * sequentialPriorInformationEnvelope θ p ε R hp hθ hε n
Proof (Lean source)
-- keep: reusable sequential-law, Fisher-information, or van-Trees bridge for related adaptive experiments
lemma integral_priorWeighted_le_nat_mul_sequentialPriorInformationEnvelope
(θ : TrialParameter) (p ε R : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hR : 0 < R) (n : ℕ) (f : ℝ → ℝ)
(hfInt : Integrable (fun a => sequentialVTPrior R a * f a)
(parameterMeasure (-R) R))
(henvInt : Integrable (fun a => sequentialVTPrior R a *
upperEnvelope
(parameterPath θ
(direction (selectedDirection θ p ε hp hθ hε))
(a / sqrt n)) p ε
(selectedDirection θ p ε hp hθ hε))
(parameterMeasure (-R) R))
(hle : ∀ a ∈ Icc (-R) R, f a ≤ (n : ℝ) *
upperEnvelope
(parameterPath θ
(direction (selectedDirection θ p ε hp hθ hε))
(a / sqrt n)) p ε
(selectedDirection θ p ε hp hθ hε)) :
(∫ a, sequentialVTPrior R a * f a
∂parameterMeasure (-R) R) ≤
(n : ℝ) * sequentialPriorInformationEnvelope
θ p ε R hp hθ hε n := by
let env : ℝ → ℝ := fun a => upperEnvelope
(parameterPath θ
(direction (selectedDirection θ p ε hp hθ hε))
(a / sqrt n)) p ε
(selectedDirection θ p ε hp hθ hε)
have hrightInt : Integrable (fun a =>
sequentialVTPrior R a * ((n : ℝ) * env a))
(parameterMeasure (-R) R) := by
convert henvInt.const_mul (n : ℝ) using 1
funext a
ring
calc
(∫ a, sequentialVTPrior R a * f a
∂parameterMeasure (-R) R) ≤
∫ a, sequentialVTPrior R a * ((n : ℝ) * env a)
∂parameterMeasure (-R) R := by
apply integral_mono_ae hfInt hrightInt
have hmem : ∀ᵐ a ∂parameterMeasure (-R) R,
a ∈ Icc (-R) R := by
unfold parameterMeasure
exact ae_restrict_mem measurableSet_Icc
filter_upwards [hmem] with a ha
exact mul_le_mul_of_nonneg_left (hle a ha)
((sequentialVTPrior_facts hR).nonneg a)
_ = (n : ℝ) * sequentialPriorInformationEnvelope
θ p ε R hp hθ hε n := by
rw [show (fun a => sequentialVTPrior R a * ((n : ℝ) * env a)) =
fun a => (n : ℝ) * (sequentialVTPrior R a * env a) by
funext a
ring]
rw [integral_const_mul]
rfl
theorem tendsto_sequentialPriorInformationEnvelope
The normalized compact-prior average of the selected per-sample envelope converges to the least-favorable stationary information Jstar. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the tendsto sequential Prior Information Envelope.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma tendsto_sequentialPriorInformationEnvelope
(θ : TrialParameter) (p ε R : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε)
(hR : 0 < R) :
Tendsto (sequentialPriorInformationEnvelope θ p ε R hp hθ hε)
atTop (nhds (Jstar θ p ε)) := by
let v := direction (selectedDirection θ p ε hp hθ hε)
let F : ℕ → ℝ → ℝ := fun n a => sequentialVTPrior R a *
upperEnvelope (parameterPath θ v (a / sqrt n)) p ε
(selectedDirection θ p ε hp hθ hε)
let limit : ℝ → ℝ := fun a => sequentialVTPrior R a * Jstar θ p ε
let B : ℝ → ℝ := fun a =>
(|Jstar θ p ε| + 1) * sequentialVTPrior R a
have hpriorInt : Integrable (sequentialVTPrior R)
(parameterMeasure (-R) R) := by
unfold sequentialVTPrior
exact smoothPrior_integrable_parameterMeasure (by linarith)
have hBint : Integrable B (parameterMeasure (-R) R) :=
hpriorInt.const_mul (|Jstar θ p ε| + 1)
have hu := tendstoUniformlyOn_upperEnvelope_compactScaled_selectedDirection
θ p ε R hp hθ hε
have huMetric := Metric.tendstoUniformlyOn_iff.mp hu
have hclose := huMetric 1 zero_lt_one
have hinterior := eventually_parameterPath_compactScaled_interior
θ v R hθ
have hmeas : ∀ᶠ n : ℕ in atTop,
AEStronglyMeasurable (F n) (parameterMeasure (-R) R) := by
refine hinterior.mono ?_
intro n hn
exact aestronglyMeasurable_priorWeighted_upperEnvelope
θ p ε R hp hθ hε hR n hn
have hmem : ∀ᵐ a ∂parameterMeasure (-R) R, a ∈ Icc (-R) R := by
unfold parameterMeasure
exact ae_restrict_mem measurableSet_Icc
have hbound : ∀ᶠ n : ℕ in atTop, ∀ᵐ a ∂parameterMeasure (-R) R,
‖F n a‖ ≤ B a := by
refine hclose.mono ?_
intro n hn
filter_upwards [hmem] with a ha
have henv : |upperEnvelope (parameterPath θ v
(a / sqrt n)) p ε
(selectedDirection θ p ε hp hθ hε)| ≤ |Jstar θ p ε| + 1 := by
have hd := hn a ha
rw [Real.dist_eq] at hd
calc
|upperEnvelope (parameterPath θ v (a / sqrt n)) p ε
(selectedDirection θ p ε hp hθ hε)| =
|Jstar θ p ε +
(upperEnvelope (parameterPath θ v (a / sqrt n)) p ε
(selectedDirection θ p ε hp hθ hε) - Jstar θ p ε)| := by
congr 1
ring
_ ≤
|Jstar θ p ε| +
|upperEnvelope (parameterPath θ v (a / sqrt n)) p ε
(selectedDirection θ p ε hp hθ hε) - Jstar θ p ε| :=
abs_add_le _ _
_ ≤ |Jstar θ p ε| + 1 := by
rw [abs_sub_comm]
linarith
have hpriorNonneg := (sequentialVTPrior_facts hR).nonneg a
dsimp [F, B]
rw [abs_mul, abs_of_nonneg hpriorNonneg]
simpa [mul_comm] using
(mul_le_mul_of_nonneg_left henv hpriorNonneg)
have hlim : ∀ᵐ a ∂parameterMeasure (-R) R,
Tendsto (fun n => F n a) atTop (nhds (limit a)) := by
filter_upwards [hmem] with a ha
exact (hu.tendsto_at ha).const_mul (sequentialVTPrior R a)
have hIntegral := tendsto_integral_filter_of_dominated_convergence
B hmeas hbound hBint hlim
have hlimitIntegral : ∫ a, limit a ∂parameterMeasure (-R) R =
Jstar θ p ε := by
dsimp [limit]
rw [integral_mul_const, (sequentialVTPrior_facts hR).normalized, one_mul]
rw [hlimitIntegral] at hIntegral
change Tendsto (fun n =>
sequentialPriorInformationEnvelope θ p ε R hp hθ hε n)
atTop (nhds (Jstar θ p ε))
simpa only [sequentialPriorInformationEnvelope, F, v] using hIntegral
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialPriorRadiusLimit 3 declarations This file separates the two limits in a local van Trees argument.
Sequential removal of the van Trees prior-radius penalty
This file separates the two limits in a local van Trees argument. At a fixed
prior radius R, the prior Fisher-information term 40 / R ^ 2 remains after
the sample-size limit. Only after transferring that fixed-radius bound to the
local asymptotic risk may one send R to infinity.
theorem tendsto_inv_info_add_priorRadius
At fixed positive prior radius, convergence of the experiment information gives convergence of the full reciprocal denominator, including the prior information term. For the displayed inputs and conditions, the stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma tendsto_inv_info_add_priorRadius
(info : ℕ → ℝ) (I R : ℝ) (hI : 0 < I)
(hinfo : Tendsto info atTop (nhds I)) :
Tendsto (fun n => 1 / (info n + 40 / R ^ 2)) atTop
(nhds (1 / (I + 40 / R ^ 2))) := by
have hdenom : Tendsto (fun n => info n + 40 / R ^ 2) atTop
(nhds (I + 40 / R ^ 2)) :=
hinfo.add tendsto_const_nhds
have hdenom_ne : I + 40 / R ^ 2 ≠ 0 := by
have hprior : 0 ≤ 40 / R ^ 2 := div_nonneg (by norm_num) (sq_nonneg R)
exact ne_of_gt (hI.trans_le (le_add_of_nonneg_right hprior))
exact tendsto_const_nhds.div hdenom hdenom_ne
theorem priorRadius_bound_le_localAsymptoticRisk
A finite-sample lower bound with a fixed prior radius first yields the corresponding penalized reciprocal-information local asymptotic bound. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ :
p H :
ℝ
hH :
0 < H
info :
ℕ → ℝ
I R :
ℝ
hI :
0 < I
hlower :
conclusion
ofReal (1 / (I + 40 / R ^ 2)) ≤ localAsymptoticRisk P θ p
Proof (Lean source)
lemma priorRadius_bound_le_localAsymptoticRisk {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p H : ℝ)
(hH : 0 < H) (info : ℕ → ℝ) (I R : ℝ) (hI : 0 < I)
(hinfo : Tendsto info atTop (nhds I))
(hlower : ∀ᶠ n : ℕ in atTop,
ofReal (1 / (info n + 40 / R ^ 2)) ≤
localWorstRisk P θ p H n) :
ofReal (1 / (I + 40 / R ^ 2)) ≤
localAsymptoticRisk P θ p := by
exact ofReal_le_localAsymptoticRisk_of_bayes_envelope
P θ p H hH _ _ (tendsto_inv_info_add_priorRadius info I R hI hinfo) hlower
theorem inv_information_le_localAsymptoticRisk_of_priorRadius_family
Correct two-step assembly for a family of finite-sample van Trees bounds. Under the stated assumptions, the inv information le local Asymptotic Risk of prior Radius family.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ :
p I :
ℝ
H :
ℝ → ℝ
info :
ℝ → ℕ → ℝ
hI :
0 < I
hH :
∀ R
if
0 < R
then
0 < H R
hlower :
∀ R
if
0 < R
then
≤ localWorstRisk P θ p (H R) n
conclusion
ofReal (1 / I) ≤ localAsymptoticRisk P θ p
Proof (Lean source)
lemma inv_information_le_localAsymptoticRisk_of_priorRadius_family
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p I : ℝ)
(H : ℝ → ℝ) (info : ℝ → ℕ → ℝ) (hI : 0 < I)
(hH : ∀ R, 0 < R → 0 < H R)
(hinfo : ∀ R, 0 < R → Tendsto (info R) atTop (nhds I))
(hlower : ∀ R, 0 < R → ∀ᶠ n : ℕ in atTop,
ofReal (1 / (info R n + 40 / R ^ 2)) ≤
localWorstRisk P θ p (H R) n) :
ofReal (1 / I) ≤ localAsymptoticRisk P θ p := by
let radius : ℕ → ℝ := fun k => (k : ℝ) + 1
have hradius_pos (k : ℕ) : 0 < radius k := by
dsimp [radius]
positivity
have hfixed (k : ℕ) :
ofReal (1 / (I + 40 / (radius k) ^ 2)) ≤
localAsymptoticRisk P θ p :=
priorRadius_bound_le_localAsymptoticRisk
P θ p (H (radius k)) (hH _ (hradius_pos k))
(info (radius k)) I (radius k) hI
(hinfo _ (hradius_pos k)) (hlower _ (hradius_pos k))
have hinv_radius : Tendsto (fun k : ℕ => 1 / radius k) atTop (nhds 0) := by
simpa only [radius] using
(tendsto_one_div_add_atTop_nhds_zero_nat :
Tendsto (fun k : ℕ => (1 : ℝ) / ((k : ℝ) + 1)) atTop (nhds 0))
have hinv_radius' : Tendsto (fun k : ℕ => (radius k)⁻¹) atTop (nhds 0) := by
simpa only [one_div] using hinv_radius
have hprior : Tendsto (fun k : ℕ => 40 / (radius k) ^ 2) atTop (nhds 0) := by
simpa [div_eq_mul_inv, inv_pow] using
(tendsto_const_nhds.mul (hinv_radius'.pow 2) :
Tendsto (fun k : ℕ => (40 : ℝ) * ((radius k)⁻¹) ^ 2)
atTop (nhds ((40 : ℝ) * 0 ^ 2)))
have hrecip : Tendsto (fun k : ℕ => 1 / (I + 40 / (radius k) ^ 2))
atTop (nhds (1 / I)) := by
have hone : Tendsto (fun _ : ℕ => (1 : ℝ)) atTop (nhds 1) :=
tendsto_const_nhds
have hdenom : Tendsto (fun k : ℕ => I + 40 / (radius k) ^ 2)
atTop (nhds I) := by
simpa only [add_zero] using tendsto_const_nhds.add hprior
change Tendsto ((fun _ : ℕ => (1 : ℝ)) /
(fun k : ℕ => I + 40 / (radius k) ^ 2)) atTop (nhds (1 / I))
exact hone.div hdenom (ne_of_gt hI)
have hofReal : Tendsto
(fun k : ℕ => ofReal (1 / (I + 40 / (radius k) ^ 2)))
atTop (nhds (ofReal (1 / I))) :=
ENNReal.continuous_ofReal.continuousAt.tendsto.comp hrecip
exact le_of_tendsto' hofReal hfixed
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialScore 19 declarations This file differentiates the finite latent-input mixture against the common transcript reference measure.
Directional scores for finite sequential transcript mixtures
This file differentiates the finite latent-input mixture against the common transcript reference measure. It identifies the complete-data input-path score and the induced conditional transcript score pointwise.
def parameterPath
For the supplied quantities and conditions, the parameter path is the mathematical object specified below. The parameter Path is determined by the displayed parameters.
Definition (Lean source)
parameters
θ v :
u :
ℝ
def
given by
fun k ↦ θ k + u * v k
def inputDirectionalDerivative
For the supplied quantities and conditions, the input directional derivative is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
inputDirectionalDerivative p v j :
ℝ
given by
∑ k : Fin 2, inputDerivative p j k * v k
theorem piTheta_by_val_score
Under the supplied quantities and conditions, the pi theta by val score assertion holds.
Formal statement
hypotheses
conclusion
piTheta θ p j
= if j.val = 0 then controlProb p * (1 - θ 0) else if j.val = 1 then controlProb p * θ 0 else if j.val = 2 then p * (1 - θ 1) else p * θ 1
Proof (Lean source)
lemma piTheta_by_val_score (θ : TrialParameter) (p : ℝ) (j : Fin 4) :
piTheta θ p j =
if j.val = 0 then controlProb p * (1 - θ 0)
else if j.val = 1 then controlProb p * θ 0
else if j.val = 2 then p * (1 - θ 1)
else p * θ 1 := by
rcases j with ⟨j, hj⟩
interval_cases j <;> rfl
theorem piTheta_pos_of_interior
Under the supplied quantities and conditions, the pi theta pos of interior assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pi Theta pos of interior.
Formal statement
hypotheses
conclusion
0 < piTheta θ p j
Proof (Lean source)
lemma piTheta_pos_of_interior (θ : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (j : Fin 4) :
0 < piTheta θ p j := by
rcases hp with ⟨hp0, hp1⟩
rcases hθ with ⟨h00, h01, h10, h11⟩
rw [piTheta_by_val_score]
rcases j with ⟨j, hj⟩
interval_cases j <;> simp +decide only [Fin.isValue, ↓reduceIte, controlProb] <;>
nlinarith
theorem inputPathProbability_pos_of_interior
the input path probability pos of interior assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the input Path Probability pos of interior.
Formal statement
hypotheses
conclusion
0 < inputPathProbability θ p x
Proof (Lean source)
lemma inputPathProbability_pos_of_interior {n : ℕ}
(θ : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(x : Fin n → Fin 4) : 0 < inputPathProbability θ p x := by
unfold inputPathProbability
exact Finset.prod_pos fun i _ ↦ piTheta_pos_of_interior θ p hp hθ (x i)
theorem hasDerivAt_piTheta_parameterPath
Under the supplied quantities and conditions, the has deriv at pi theta parameter path assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma hasDerivAt_piTheta_parameterPath (θ v : TrialParameter) (p u : ℝ)
(j : Fin 4) :
HasDerivAt (fun a ↦ piTheta (parameterPath θ v a) p j)
(inputDirectionalDerivative p v j) u := by
have heq : (fun a ↦ piTheta (parameterPath θ v a) p j) =
fun a ↦ piTheta θ p j + a * inputDirectionalDerivative p v j := by
funext a
rw [piTheta_by_val_score, piTheta_by_val_score]
rcases j with ⟨j, hj⟩
interval_cases j <;>
simp [parameterPath, inputDirectionalDerivative, inputDerivative,
controlProb, Fin.sum_univ_succ] <;> ring
have hder : HasDerivAt
(fun a ↦ piTheta θ p j + a * inputDirectionalDerivative p v j)
(inputDirectionalDerivative p v j) u := by
have hi : HasDerivAt (fun a : ℝ ↦ a)
1 u := hasDerivAt_id u
have hm : HasDerivAt
(fun a : ℝ ↦ a * inputDirectionalDerivative p v j)
(inputDirectionalDerivative p v j) u := by
simpa using hi.mul_const (inputDirectionalDerivative p v j)
exact hm.const_add (piTheta θ p j)
exact heq.symm ▸ hder
def inputPathDirectionalDerivative
For the supplied quantities and conditions, the input path directional derivative is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The input Path Directional Derivative is determined by the displayed parameters.
Definition (Lean source)
parameters
def
inputPathDirectionalDerivative θ v p x :
ℝ
given by
theorem hasDerivAt_inputPathProbability_parameterPath
the has deriv at input path probability parameter path assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
HasDerivAt (fun a ↦ inputPathProbability (parameterPath θ v a) p x) (inputPathDirectionalDerivative (parameterPath θ v u) v p x) u
Proof (Lean source)
lemma hasDerivAt_inputPathProbability_parameterPath {n : ℕ}
(θ v : TrialParameter) (p u : ℝ) (x : Fin n → Fin 4) :
HasDerivAt (fun a ↦ inputPathProbability (parameterPath θ v a) p x)
(inputPathDirectionalDerivative (parameterPath θ v u) v p x) u := by
simpa [inputPathProbability, inputPathDirectionalDerivative, mul_comm] using
HasDerivAt.fun_finsetProd (u := (Finset.univ : Finset (Fin n)))
(fun i _ ↦ hasDerivAt_piTheta_parameterPath θ v p u (x i))
def transcriptComponentRealDensity
The real-valued density of one transcript component is determined by the displayed parameters.
Definition (Lean source)
parameters
def
transcriptComponentRealDensity P n x z :
ℝ
given by
((P.transcript n x).rnDeriv (transcriptReferenceMeasure P n) z).toReal
def transcriptMixtureRealDensity
The real-valued density of the transcript mixture is determined by the displayed parameters.
Definition (Lean source)
parameters
def
transcriptMixtureRealDensity P θ p n z :
ℝ
given by
def transcriptMixtureRealDerivative
The directional derivative of the real-valued transcript-mixture density is determined by the displayed parameters.
Definition (Lean source)
parameters
def
transcriptMixtureRealDerivative P θ v p n z :
ℝ
given by
∑ x : Fin n → Fin 4, inputPathDirectionalDerivative θ v p x * transcriptComponentRealDensity P n x z
theorem transcriptDensity_toReal_ae_eq_mixture
On a parameter range where the four-cell path probabilities are nonnegative, the explicit finite-mixture density is the Radon--Nikodym density of the transcript law. This is the paper-side interface between the finite latent-input expansion and score calculus on an arbitrary transcript measurable space. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
(fun z ↦ (transcriptDensity P θ p n z).toReal)
=ᵐ[transcriptReferenceMeasure P n] transcriptMixtureRealDensity P θ p n
Proof (Lean source)
lemma transcriptDensity_toReal_ae_eq_mixture {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ) (n : ℕ)
(hprob : ∀ x : Fin n → Fin 4, 0 ≤ inputPathProbability θ p x) :
(fun z ↦ (transcriptDensity P θ p n z).toReal) =ᵐ[transcriptReferenceMeasure P n]
transcriptMixtureRealDensity P θ p n := by
classical
let ν := transcriptReferenceMeasure P n
let w (x : Fin n → Fin 4) : ℝ≥0∞ :=
ofReal (inputPathProbability θ p x)
letI componentProbability (x : Fin n → Fin 4) :
IsProbabilityMeasure (P.transcript n x) := (P.factorizes n x).1
have hw (x : Fin n → Fin 4) : w x ≠ ∞ := by
simp [w]
haveI (x : Fin n → Fin 4) : IsFiniteMeasure (w x • P.transcript n x) :=
(P.transcript n x).smul_finite (hw x)
have hrn (t : Finset (Fin n → Fin 4)) :
(t.sum fun x ↦ w x • P.transcript n x).rnDeriv ν =ᵐ[ν]
fun z ↦ t.sum fun x ↦ w x * (P.transcript n x).rnDeriv ν z := by
induction t using Finset.induction_on with
| empty =>
filter_upwards [Measure.rnDeriv_zero ν] with z hz
simpa using hz
| @insert x t hx ih =>
haveI : IsFiniteMeasure (t.sum fun y ↦ w y • P.transcript n y) := inferInstance
have hadd := Measure.rnDeriv_add (w x • P.transcript n x)
(t.sum fun y ↦ w y • P.transcript n y) ν
have hsmul := Measure.rnDeriv_smul_left_of_ne_top (P.transcript n x) ν (hw x)
filter_upwards [hadd, hsmul, ih] with z hz ha hb
simpa [Finset.sum_insert hx, Pi.add_apply, Pi.smul_apply, smul_eq_mul, ha, hb]
using hz
have hfinite : ∀ᵐ z ∂ν, ∀ x : Fin n → Fin 4,
(P.transcript n x).rnDeriv ν z ≠ ∞ := by
rw [Filter.eventually_all]
intro x
exact (Measure.rnDeriv_lt_top (P.transcript n x) ν).mono
(fun _ hz ↦ ne_of_lt hz)
filter_upwards [hrn univ, hfinite] with z hz hfin
change ((transcriptLaw P θ p n).rnDeriv ν z).toReal =
transcriptMixtureRealDensity P θ p n z
rw [show transcriptLaw P θ p n =
Finset.univ.sum (fun x ↦ w x • P.transcript n x) by
simp [transcriptLaw, w]]
rw [hz, ENNReal.toReal_sum (fun x _ ↦ ENNReal.mul_ne_top (hw x) (hfin x))]
simp [transcriptMixtureRealDensity, transcriptComponentRealDensity, w, ν,
ENNReal.toReal_mul, hprob]
theorem transcriptDensity_toReal_ae_eq_mixture_of_interior
the transcript density to real ae eq mixture of interior assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the transcript Density to Real ae eq mixture of interior.
Formal statement
hypotheses
conclusion
(fun z ↦ (transcriptDensity P θ p n z).toReal)
=ᵐ[transcriptReferenceMeasure P n] transcriptMixtureRealDensity P θ p n
Proof (Lean source)
lemma transcriptDensity_toReal_ae_eq_mixture_of_interior {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ) (n : ℕ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
(fun z ↦ (transcriptDensity P θ p n z).toReal) =ᵐ[transcriptReferenceMeasure P n]
transcriptMixtureRealDensity P θ p n :=
transcriptDensity_toReal_ae_eq_mixture P θ p n fun x ↦
(inputPathProbability_pos_of_interior θ p hp hθ x).le
theorem hasDerivAt_transcriptMixtureRealDensity_parameterPath
the has deriv at transcript mixture real density parameter path assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
HasDerivAt (fun a ↦ transcriptMixtureRealDensity P (parameterPath θ v a) p n z) (transcriptMixtureRealDerivative P (parameterPath θ v u) v p n z) u
Proof (Lean source)
lemma hasDerivAt_transcriptMixtureRealDensity_parameterPath {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p u : ℝ) (n : ℕ)
(z : Transcript (Z n)) :
HasDerivAt
(fun a ↦ transcriptMixtureRealDensity P (parameterPath θ v a) p n z)
(transcriptMixtureRealDerivative P (parameterPath θ v u) v p n z) u := by
simpa [transcriptMixtureRealDensity, transcriptMixtureRealDerivative] using
HasDerivAt.fun_sum (u := (Finset.univ : Finset (Fin n → Fin 4)))
(fun x _ ↦
(hasDerivAt_inputPathProbability_parameterPath θ v p u x).mul_const
(transcriptComponentRealDensity P n x z))
def inputPathDirectionalScore
The complete-data directional score of one latent input path. The input Path Directional Score is determined by the displayed parameters.
Definition (Lean source)
parameters
def
inputPathDirectionalScore θ v p x :
ℝ
given by
inputPathDirectionalDerivative θ v p x / inputPathProbability θ p x
theorem inputPathDerivative_eq_probability_mul_score
the input path derivative eq probability mul score assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
inputPathDirectionalDerivative θ v p x
= inputPathProbability θ p x * inputPathDirectionalScore θ v p x
Proof (Lean source)
lemma inputPathDerivative_eq_probability_mul_score {n : ℕ}
(θ v : TrialParameter) (p : ℝ) (x : Fin n → Fin 4)
(hx : inputPathProbability θ p x ≠ 0) :
inputPathDirectionalDerivative θ v p x =
inputPathProbability θ p x * inputPathDirectionalScore θ v p x := by
rw [inputPathDirectionalScore, mul_div_cancel₀ _ hx]
def conditionalTranscriptDirectionalScore
The conditional transcript score: the derivative of the finite complete-data mixture density divided by that mixture density. The conditional Transcript Directional Score is determined by the displayed parameters.
Definition (Lean source)
parameters
def
conditionalTranscriptDirectionalScore P θ v p n z :
ℝ
given by
transcriptMixtureRealDerivative P θ v p n z / transcriptMixtureRealDensity P θ p n z
theorem transcriptDerivative_eq_density_mul_conditionalScore
the transcript derivative eq density mul conditional score assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
n :
ℕ
z :
Transcript (Z n)
hz :
transcriptMixtureRealDensity P θ p n z ≠ 0
conclusion
transcriptMixtureRealDerivative P θ v p n z
= transcriptMixtureRealDensity P θ p n z * conditionalTranscriptDirectionalScore P θ v p n z
Proof (Lean source)
lemma transcriptDerivative_eq_density_mul_conditionalScore {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(z : Transcript (Z n))
(hz : transcriptMixtureRealDensity P θ p n z ≠ 0) :
transcriptMixtureRealDerivative P θ v p n z =
transcriptMixtureRealDensity P θ p n z *
conditionalTranscriptDirectionalScore P θ v p n z := by
rw [conditionalTranscriptDirectionalScore, mul_div_cancel₀ _ hz]
theorem transcriptDerivative_eq_completeDataScoreMixture
the transcript derivative eq complete data score mixture assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
transcriptMixtureRealDerivative P θ v p n z
= ∑ x : Fin n → Fin 4, inputPathProbability θ p x * inputPathDirectionalScore θ v p x * transcriptComponentRealDensity P n x z
Proof (Lean source)
lemma transcriptDerivative_eq_completeDataScoreMixture {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(z : Transcript (Z n))
(hx : ∀ x : Fin n → Fin 4, inputPathProbability θ p x ≠ 0) :
transcriptMixtureRealDerivative P θ v p n z =
∑ x : Fin n → Fin 4,
inputPathProbability θ p x * inputPathDirectionalScore θ v p x *
transcriptComponentRealDensity P n x z := by
rw [transcriptMixtureRealDerivative]
apply Finset.sum_congr rfl
intro x _
rw [inputPathDerivative_eq_probability_mul_score θ v p x (hx x)]
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialScoreProjectionInputs 5 declarations Paper inputs for measurable-statistic score projection
Paper inputs for measurable-statistic score projection
theorem transcriptMixtureRealDerivative_eq_zero_of_density_eq_zero
A full transcript mixture derivative vanishes wherever its positive-weight finite-mixture density vanishes. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
theta v :
p :
ℝ
n :
ℕ
hp :
htheta :
InteriorMeans theta
z :
Transcript (Z n)
hz :
transcriptMixtureRealDensity P theta p n z = 0
conclusion
transcriptMixtureRealDerivative P theta v p n z = 0
Proof (Lean source)
lemma transcriptMixtureRealDerivative_eq_zero_of_density_eq_zero
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (theta v : TrialParameter) (p : ℝ) (n : ℕ)
(hp : InteriorAssignment p) (htheta : InteriorMeans theta)
(z : Transcript (Z n))
(hz : transcriptMixtureRealDensity P theta p n z = 0) :
transcriptMixtureRealDerivative P theta v p n z = 0 := by
have hterms : ∀ x : Fin n → Fin 4,
inputPathProbability theta p x * transcriptComponentRealDensity P n x z = 0 := by
have hnonneg : ∀ x ∈ (Finset.univ : Finset (Fin n → Fin 4)),
0 ≤ inputPathProbability theta p x * transcriptComponentRealDensity P n x z := by
intro x hx
exact mul_nonneg (inputPathProbability_pos_of_interior theta p hp htheta x).le
ENNReal.toReal_nonneg
exact fun x => (Finset.sum_eq_zero_iff_of_nonneg hnonneg).mp hz x (Finset.mem_univ x)
unfold transcriptMixtureRealDerivative
apply Finset.sum_eq_zero
intro x hx
have hcomponent : transcriptComponentRealDensity P n x z = 0 := by
have hprob := inputPathProbability_pos_of_interior theta p hp htheta x
exact (mul_eq_zero.mp (hterms x)).resolve_left (ne_of_gt hprob)
rw [hcomponent, mul_zero]
theorem conditionalTranscriptDirectionalScore_eq_guarded
The raw quotient defining the full transcript score agrees with the guarded zero convention used by score projection. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
theta v :
p :
ℝ
n :
ℕ
hp :
htheta :
InteriorMeans theta
z :
Transcript (Z n)
conclusion
conditionalTranscriptDirectionalScore P theta v p n z
= if 0 < transcriptMixtureRealDensity P theta p n z then transcriptMixtureRealDerivative P theta v p n z / transcriptMixtureRealDensity P theta p n z else 0
Proof (Lean source)
lemma conditionalTranscriptDirectionalScore_eq_guarded
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (theta v : TrialParameter) (p : ℝ) (n : ℕ)
(hp : InteriorAssignment p) (htheta : InteriorMeans theta)
(z : Transcript (Z n)) :
conditionalTranscriptDirectionalScore P theta v p n z =
if 0 < transcriptMixtureRealDensity P theta p n z then
transcriptMixtureRealDerivative P theta v p n z /
transcriptMixtureRealDensity P theta p n z
else 0 := by
unfold conditionalTranscriptDirectionalScore
split_ifs with hpos
· rfl
· have hnonneg := transcriptMixtureRealDensity_nonneg P theta p n hp htheta z
have hz : transcriptMixtureRealDensity P theta p n z = 0 :=
le_antisymm (le_of_not_gt hpos) hnonneg
rw [hz, transcriptMixtureRealDerivative_eq_zero_of_density_eq_zero
P theta v p n hp htheta z hz, zero_div]
theorem transcriptPrefixMixtureRealDensity_nonneg
Prefix finite-mixture densities are nonnegative under interior parameters. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
theta :
p :
ℝ
n k :
ℕ
hk :
k ≤ n
hp :
htheta :
InteriorMeans theta
h :
History (Z n) k hk
conclusion
0 ≤ transcriptPrefixMixtureRealDensity P theta p n k hk h
Proof (Lean source)
lemma transcriptPrefixMixtureRealDensity_nonneg
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (theta : TrialParameter) (p : ℝ)
(n k : ℕ) (hk : k ≤ n)
(hp : InteriorAssignment p) (htheta : InteriorMeans theta)
(h : History (Z n) k hk) :
0 ≤ transcriptPrefixMixtureRealDensity P theta p n k hk h := by
unfold transcriptPrefixMixtureRealDensity
exact Finset.sum_nonneg fun x _ => mul_nonneg
(inputPathProbability_pos_of_interior theta p hp htheta x).le ENNReal.toReal_nonneg
theorem transcriptPrefixMixtureRealDerivative_eq_zero_of_density_eq_zero
A prefix mixture derivative vanishes wherever its positive-weight finite-mixture density vanishes. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
theta v :
p :
ℝ
n k :
ℕ
hk :
k ≤ n
hp :
htheta :
InteriorMeans theta
h :
History (Z n) k hk
hz :
transcriptPrefixMixtureRealDensity P theta p n k hk h = 0
conclusion
transcriptPrefixMixtureRealDerivative P theta v p n k hk h = 0
Proof (Lean source)
lemma transcriptPrefixMixtureRealDerivative_eq_zero_of_density_eq_zero
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (theta v : TrialParameter) (p : ℝ)
(n k : ℕ) (hk : k ≤ n)
(hp : InteriorAssignment p) (htheta : InteriorMeans theta)
(h : History (Z n) k hk)
(hz : transcriptPrefixMixtureRealDensity P theta p n k hk h = 0) :
transcriptPrefixMixtureRealDerivative P theta v p n k hk h = 0 := by
have hterms : ∀ x : Fin n → Fin 4,
inputPathProbability theta p x *
transcriptPrefixComponentRealDensity P n k hk x h = 0 := by
have hnonneg : ∀ x ∈ (Finset.univ : Finset (Fin n → Fin 4)),
0 ≤ inputPathProbability theta p x *
transcriptPrefixComponentRealDensity P n k hk x h := by
intro x hx
exact mul_nonneg (inputPathProbability_pos_of_interior theta p hp htheta x).le
ENNReal.toReal_nonneg
exact fun x => (Finset.sum_eq_zero_iff_of_nonneg hnonneg).mp hz x (Finset.mem_univ x)
unfold transcriptPrefixMixtureRealDerivative
apply Finset.sum_eq_zero
intro x hx
have hcomponent : transcriptPrefixComponentRealDensity P n k hk x h = 0 := by
have hprob := inputPathProbability_pos_of_interior theta p hp htheta x
exact (mul_eq_zero.mp (hterms x)).resolve_left (ne_of_gt hprob)
rw [hcomponent, mul_zero]
theorem conditionalTranscriptPrefixDirectionalScore_eq_guarded
The raw quotient defining a prefix score agrees with the guarded zero convention used by score projection. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
theta v :
p :
ℝ
n k :
ℕ
hk :
k ≤ n
hp :
htheta :
InteriorMeans theta
h :
History (Z n) k hk
conclusion
conditionalTranscriptPrefixDirectionalScore P theta v p n k hk h
= if 0 < transcriptPrefixMixtureRealDensity P theta p n k hk h then transcriptPrefixMixtureRealDerivative P theta v p n k hk h / transcriptPrefixMixtureRealDensity P theta p n k hk h else 0
Proof (Lean source)
lemma conditionalTranscriptPrefixDirectionalScore_eq_guarded
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (theta v : TrialParameter) (p : ℝ)
(n k : ℕ) (hk : k ≤ n)
(hp : InteriorAssignment p) (htheta : InteriorMeans theta)
(h : History (Z n) k hk) :
conditionalTranscriptPrefixDirectionalScore P theta v p n k hk h =
if 0 < transcriptPrefixMixtureRealDensity P theta p n k hk h then
transcriptPrefixMixtureRealDerivative P theta v p n k hk h /
transcriptPrefixMixtureRealDensity P theta p n k hk h
else 0 := by
unfold conditionalTranscriptPrefixDirectionalScore
split_ifs with hpos
· rfl
· have hnonneg := transcriptPrefixMixtureRealDensity_nonneg
P theta p n k hk hp htheta h
have hz : transcriptPrefixMixtureRealDensity P theta p n k hk h = 0 :=
le_antisymm (le_of_not_gt hpos) hnonneg
rw [hz, transcriptPrefixMixtureRealDerivative_eq_zero_of_density_eq_zero
P theta v p n k hk hp htheta h hz, zero_div]
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialSuccessor 6 declarations Successor-prefix and finite joint-channel identities
Successor-prefix and finite joint-channel identities
theorem transcriptPrefixLaw_eq_observedPrefixLaw
the transcript prefix law eq observed prefix law assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the transcript Prefix Law eq observed Prefix Law.
Formal statement
hypotheses
conclusion
transcriptPrefixLaw P θ p n k hk
= prefixLaw (fun i => observedStageKernel P θ p n i) (fun _ => ()) k hk
Proof (Lean source)
lemma transcriptPrefixLaw_eq_observedPrefixLaw {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k ≤ n) :
transcriptPrefixLaw P θ p n k hk =
prefixLaw (fun i => observedStageKernel P θ p n i)
(fun _ => ()) k hk := by
have hQ (i : Fin n) : IsMarkovKernel (observedStageKernel P θ p n i) :=
observedStageKernel_isMarkov P θ p hp hθ n i
have hpath : parameterPath θ (fun _ => 0) 0 = θ := by
funext j
simp [parameterPath]
have hfull := transcriptLaw_eq_observedKernelLaw P θ (fun _ => 0) p hp n 0
(hpath.symm ▸ hθ)
rw [hpath] at hfull
have hfamily : observedKernelFamily P θ (fun _ => 0) p n 0 =
fun i => observedStageKernel P θ p n i := by
funext i
simp only [observedKernelFamily]
rw [hpath]
unfold transcriptPrefixLaw
rw [hfull]
unfold law
rw [hfamily]
have hprefix : (transcriptPrefix hk : Transcript (Z n) → _) = take hk := by
funext z j
rfl
rw [hprefix]
exact transcriptLaw_map_prefix
(fun i => observedStageKernel P θ p n i) hQ (fun _ => ()) k hk
theorem transcriptSuccessorLaw_eq_jointChannel
the transcript successor law eq joint channel assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the transcript Successor Law eq joint Channel.
Formal statement
hypotheses
conclusion
= outputLaw θ p (sequentialJointChannel P n (nextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))
Proof (Lean source)
lemma transcriptSuccessorLaw_eq_jointChannel {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n) :
(transcriptPrefixLaw P θ p n (k + 1) hk).map
(fun u => (init hk u, last hk u)) =
outputLaw θ p (sequentialJointChannel P n (nextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) := by
let Q := fun i => observedStageKernel P θ p n i
let μ := transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
have hQ (i : Fin n) : IsMarkovKernel (Q i) :=
observedStageKernel_isMarkov P θ p hp hθ n i
have hμ : μ = prefixLaw Q (fun _ => ()) k (le_of_succ_le hk) :=
transcriptPrefixLaw_eq_observedPrefixLaw P θ p hp hθ n k
(le_of_succ_le hk)
letI : IsProbabilityMeasure μ := by
rw [hμ]
exact isProbabilityMeasure_prefixLaw Q hQ (fun _ => ()) k _
letI : IsProbabilityMeasure
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) := by
change IsProbabilityMeasure μ
infer_instance
have hμprob : IsProbabilityMeasure μ := inferInstance
rw [transcriptPrefixLaw_eq_observedPrefixLaw P θ p hp hθ n (k + 1) hk]
rw [prefixLaw_succ_map_init_last Q (fun _ => ()) k hk]
rw [← hμ]
exact (@outputLaw_sequentialJointChannel Z _ ε P θ p hp hθ n
(nextIndex hk) μ hμprob).symm
def sequentialJointEventDerivative
the sequential joint event derivative is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The sequential Joint Event Derivative is determined by the displayed parameters.
Definition (Lean source)
parameters
def
sequentialJointEventDerivative P v p n i μ A :
ℝ
given by
theorem hasDerivAt_sequentialJoint_realMixture
the has deriv at sequential joint real mixture assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
HasDerivAt (fun u => ∑ a : Fin 4, piTheta (parameterPath θ v u) p a * (sequentialJointChannel P n i μ a A).toReal) (sequentialJointEventDerivative P v p n i μ A) 0
Proof (Lean source)
lemma hasDerivAt_sequentialJoint_realMixture {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(n : ℕ) (i : Fin n)
(μ : Measure (History (Z n) i.val (le_of_lt i.isLt)))
(A : Set (History (Z n) i.val (le_of_lt i.isLt) × Z n i)) :
HasDerivAt (fun u => ∑ a : Fin 4,
piTheta (parameterPath θ v u) p a *
(sequentialJointChannel P n i μ a A).toReal)
(sequentialJointEventDerivative P v p n i μ A) 0 := by
simpa [sequentialJointEventDerivative] using
HasDerivAt.fun_sum (u := (Finset.univ : Finset (Fin 4)))
(fun a _ ↦ (hasDerivAt_piTheta_parameterPath θ v p 0 a).mul_const
(sequentialJointChannel P n i μ a A).toReal)
theorem sequentialJoint_realMixture_eq_outputLaw_toReal
the sequential joint real mixture eq output law to real assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the sequential Joint real Mixture eq output Law to Real.
Formal statement
hypotheses
conclusion
= (outputLaw θ p (sequentialJointChannel P n i μ) A).toReal
Proof (Lean source)
lemma sequentialJoint_realMixture_eq_outputLaw_toReal {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n : ℕ) (i : Fin n)
(μ : Measure (History (Z n) i.val (le_of_lt i.isLt)))
[IsProbabilityMeasure μ]
(A : Set (History (Z n) i.val (le_of_lt i.isLt) × Z n i))
(hA : MeasurableSet A) :
∑ a : Fin 4, piTheta θ p a *
(sequentialJointChannel P n i μ a A).toReal =
(outputLaw θ p (sequentialJointChannel P n i μ) A).toReal := by
let Q := sequentialJointChannel P n i μ
have hQ : IsMarkovKernel Q := sequentialJointChannel_isMarkov P n i μ
have hfinite (a : Fin 4) : Q a A ≠ ∞ := by
letI : IsProbabilityMeasure (Q a) := hQ.isProbabilityMeasure a
exact measure_ne_top (Q a) A
unfold outputLaw
simp only [Measure.finsetSum_apply, Measure.smul_apply, smul_eq_mul]
rw [ENNReal.toReal_sum (fun a _ ↦
ENNReal.mul_ne_top ENNReal.ofReal_ne_top (hfinite a))]
simp_rw [ENNReal.toReal_mul,
ENNReal.toReal_ofReal (piTheta_pos_of_interior θ p hp hθ _).le]
rfl
theorem hasDerivAt_sequentialJoint_outputLaw
the has deriv at sequential joint output law assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the has Deriv At sequential Joint output Law.
Formal statement
hypotheses
conclusion
HasDerivAt (fun u => (outputLaw (parameterPath θ v u) p (sequentialJointChannel P n i μ) A).toReal) (sequentialJointEventDerivative P v p n i μ A) 0
Proof (Lean source)
lemma hasDerivAt_sequentialJoint_outputLaw {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n : ℕ) (i : Fin n)
(μ : Measure (History (Z n) i.val (le_of_lt i.isLt)))
[IsProbabilityMeasure μ]
(A : Set (History (Z n) i.val (le_of_lt i.isLt) × Z n i))
(hA : MeasurableSet A) :
HasDerivAt (fun u =>
(outputLaw (parameterPath θ v u) p
(sequentialJointChannel P n i μ) A).toReal)
(sequentialJointEventDerivative P v p n i μ A) 0 := by
have hc0 : ContinuousAt (fun u => parameterPath θ v u 0) 0 := by
unfold parameterPath
fun_prop
have hc1 : ContinuousAt (fun u => parameterPath θ v u 1) 0 := by
unfold parameterPath
fun_prop
have h00 : (0 : ℝ) < parameterPath θ v 0 0 := by
simpa [parameterPath] using hθ.1
have h01 : parameterPath θ v 0 0 < (1 : ℝ) := by
simpa [parameterPath] using hθ.2.1
have h10 : (0 : ℝ) < parameterPath θ v 0 1 := by
simpa [parameterPath] using hθ.2.2.1
have h11 : parameterPath θ v 0 1 < (1 : ℝ) := by
simpa [parameterPath] using hθ.2.2.2
have hinterior : {u : ℝ | InteriorMeans (parameterPath θ v u)} ∈ nhds 0 := by
filter_upwards [continuousAt_const.eventually_lt hc0 h00,
hc0.eventually_lt continuousAt_const h01,
continuousAt_const.eventually_lt hc1 h10,
hc1.eventually_lt continuousAt_const h11] with u hu00 hu01 hu10 hu11
exact ⟨hu00, hu01, hu10, hu11⟩
have heq : EventuallyEq (nhds 0) (fun u =>
(outputLaw (parameterPath θ v u) p
(sequentialJointChannel P n i μ) A).toReal) (fun u =>
∑ a : Fin 4, piTheta (parameterPath θ v u) p a *
(sequentialJointChannel P n i μ a A).toReal) := by
filter_upwards [hinterior] with u hu
exact (sequentialJoint_realMixture_eq_outputLaw_toReal P
(parameterPath θ v u) p hp hu n i μ A hA).symm
exact (hasDerivAt_sequentialJoint_realMixture P θ v p n i μ A).congr_of_eventuallyEq heq
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialSuccessorDerivative 12 declarations This file records the exact product rule for the successor-prefix law.
Derivative decomposition for a sequential successor event
This file records the exact product rule for the successor-prefix law. It keeps the derivative caused by the varying predecessor prefix law separate from the derivative of the current four-row mixture.
def varyingPredecessorRowEventMass
The mass of a successor event in one fixed current-input row while the predecessor prefix law varies along the parameter path. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The varying Predecessor Row Event Mass is determined by the displayed parameters.
Definition (Lean source)
parameters
def
varyingPredecessorRowEventMass P θ v p n k hk u a A :
ℝ
given by
(sequentialJointChannel P n (nextIndex hk) (transcriptPrefixLaw P (parameterPath θ v u) p n k (le_of_succ_le hk)) a A).toReal
def predecessorSuccessorEventDerivative
The derivative contribution from the varying predecessor law, after mixing the four current-input rows at the base parameter. For the displayed inputs and conditions, the stated result follows. The predecessor Successor Event Derivative is determined by the displayed parameters.
Definition (Lean source)
parameters
def
predecessorSuccessorEventDerivative θ p pastRowDerivative :
ℝ
def successorPrefixEventMass
The real mass of a successor-prefix event after splitting the successor history into its predecessor and final output coordinates. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The successor Prefix Event Mass is determined by the displayed parameters.
Definition (Lean source)
parameters
def
successorPrefixEventMass P θ v p n k hk u A :
ℝ
given by
(((transcriptPrefixLaw P (parameterPath θ v u) p n (k + 1) hk).map (fun w => (init hk w, last hk w))) A).toReal
theorem hasDerivAt_integral_transcriptPrefixLaw
Integrating a fixed measurable function against the varying prefix law differentiates by integrating it against the explicit finite-mixture prefix derivative density. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the has Deriv At integral transcript Prefix Law.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
hp :
hθ :
n k :
ℕ
hk :
k ≤ n
g :
History (Z n) k hk → ℝ
hg :
hgInt :
conclusion
HasDerivAt (fun u => ∫ h, g h ∂transcriptPrefixLaw P (parameterPath θ v u) p n k hk) (∫ h, transcriptPrefixMixtureRealDerivative P θ v p n k hk h * g h ∂transcriptPrefixReferenceMeasure P n k hk) 0
Proof (Lean source)
lemma hasDerivAt_integral_transcriptPrefixLaw
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k ≤ n) (g : History (Z n) k hk → ℝ)
(hg : Measurable g)
(hgInt : ∀ x : Fin n → Fin 4, Integrable g
((P.transcript n x).map (transcriptPrefix hk))) :
HasDerivAt (fun u => ∫ h, g h
∂transcriptPrefixLaw P (parameterPath θ v u) p n k hk)
(∫ h, transcriptPrefixMixtureRealDerivative P θ v p n k hk h * g h
∂transcriptPrefixReferenceMeasure P n k hk) 0 := by
let ν := transcriptPrefixReferenceMeasure P n k hk
let μ (x : Fin n → Fin 4) :=
(P.transcript n x).map (transcriptPrefix hk)
letI (x : Fin n → Fin 4) : IsProbabilityMeasure (P.transcript n x) :=
(P.factorizes n x).1
letI (x : Fin n → Fin 4) : IsProbabilityMeasure (μ x) :=
Measure.isProbabilityMeasure_map (measurable_transcriptPrefix hk).aemeasurable
letI : IsFiniteMeasure ν := by
dsimp [ν, transcriptPrefixReferenceMeasure]
infer_instance
have hzero : parameterPath θ v 0 = θ := by
funext j
simp [parameterPath]
have hc0 : ContinuousAt (fun u => parameterPath θ v u 0) 0 := by
unfold parameterPath
fun_prop
have hc1 : ContinuousAt (fun u => parameterPath θ v u 1) 0 := by
unfold parameterPath
fun_prop
have hinterior : {u : ℝ | InteriorMeans (parameterPath θ v u)} ∈ nhds 0 := by
filter_upwards
[continuousAt_const.eventually_lt hc0 (by simpa [hzero] using hθ.1),
hc0.eventually_lt continuousAt_const (by simpa [hzero] using hθ.2.1),
continuousAt_const.eventually_lt hc1 (by simpa [hzero] using hθ.2.2.1),
hc1.eventually_lt continuousAt_const (by simpa [hzero] using hθ.2.2.2)]
with u hu00 hu01 hu10 hu11
exact ⟨hu00, hu01, hu10, hu11⟩
have heq : (fun u => ∫ h, g h
∂transcriptPrefixLaw P (parameterPath θ v u) p n k hk) =ᶠ[nhds 0]
fun u => ∑ x : Fin n → Fin 4,
inputPathProbability (parameterPath θ v u) p x * ∫ h, g h ∂μ x := by
filter_upwards [hinterior] with u hu
rw [transcriptPrefixLaw_eq_sum, integral_finset_sum_measure]
· apply Finset.sum_congr rfl
intro x _
rw [integral_smul_measure, ENNReal.toReal_ofReal
(inputPathProbability_pos_of_interior
(parameterPath θ v u) p hp hu x).le]
rfl
· intro x _
exact (hgInt x).smul_measure ENNReal.ofReal_ne_top
have hfiniteDerivative : HasDerivAt (fun u => ∑ x : Fin n → Fin 4,
inputPathProbability (parameterPath θ v u) p x * ∫ h, g h ∂μ x)
(∑ x : Fin n → Fin 4,
inputPathDirectionalDerivative θ v p x * ∫ h, g h ∂μ x) 0 := by
simpa only [hzero] using HasDerivAt.fun_sum
(u := (Finset.univ : Finset (Fin n → Fin 4)))
(fun x _ => (hasDerivAt_inputPathProbability_parameterPath
θ v p 0 x).mul_const (∫ h, g h ∂μ x))
have hpac (x : Fin n → Fin 4) : μ x ≪ ν := by
dsimp [μ, ν, transcriptPrefixReferenceMeasure]
exact (transcriptComponent_ac_reference P n x).map
(measurable_transcriptPrefix hk)
have htermInt (x : Fin n → Fin 4) : Integrable (fun h =>
inputPathDirectionalDerivative θ v p x *
transcriptPrefixComponentRealDensity P n k hk x h * g h) ν := by
have hi : Integrable (fun h =>
transcriptPrefixComponentRealDensity P n k hk x h * g h) ν := by
change Integrable (fun h => ((μ x).rnDeriv ν h).toReal * g h) ν
exact (integrable_toReal_rnDeriv_mul_iff (hpac x)).2 (hgInt x)
convert hi.const_mul (inputPathDirectionalDerivative θ v p x) using 1 <;>
ring
have hderivative : (∫ h,
transcriptPrefixMixtureRealDerivative P θ v p n k hk h * g h ∂ν) =
∑ x : Fin n → Fin 4,
inputPathDirectionalDerivative θ v p x * ∫ h, g h ∂μ x := by
simp_rw [transcriptPrefixMixtureRealDerivative, Finset.sum_mul]
rw [integral_finset_sum _ (fun x _ => htermInt x)]
apply Finset.sum_congr rfl
intro x _
rw [show (fun h => inputPathDirectionalDerivative θ v p x *
transcriptPrefixComponentRealDensity P n k hk x h * g h) =
fun h => inputPathDirectionalDerivative θ v p x *
(transcriptPrefixComponentRealDensity P n k hk x h * g h) by
funext h
ring]
rw [integral_const_mul]
change inputPathDirectionalDerivative θ v p x *
(∫ h, ((μ x).rnDeriv ν h).toReal * g h ∂ν) = _
rw [integral_toReal_rnDeriv_mul (hpac x)]
rw [hderivative]
exact hfiniteDerivative.congr_of_eventuallyEq heq
def successorRowKernelSection
The real event section of one fixed current-input row as a function of the released predecessor history. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
n k :
ℕ
hk :
k + 1 ≤ n
a :
Fin 4
A :
h :
History (Z n) k (le_of_succ_le hk)
def
successorRowKernelSection P n k hk a A h :
ℝ
given by
theorem measurable_successorRowKernelSection
A fixed-row successor event section is measurable on the predecessor history space, with no regularity assumption on the output space. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (successorRowKernelSection P n k hk a A)
Proof (Lean source)
lemma measurable_successorRowKernelSection {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (n k : ℕ) (hk : k + 1 ≤ n)
(a : Fin 4)
(A : Set (History (Z n) k (le_of_succ_le hk) ×
Z n (nextIndex hk))) (hA : MeasurableSet A) :
Measurable (successorRowKernelSection P n k hk a A) := by
letI : IsMarkovKernel (P.channel n (nextIndex hk)) :=
(P.privacy n).1 (nextIndex hk)
let K : Kernel (History (Z n) k (le_of_succ_le hk))
(Z n (nextIndex hk)) :=
(P.channel n (nextIndex hk)).comap
(fun h => (a, fsHistoryToPrivate (nextIndex hk) h))
(measurable_const.prodMk
(measurable_fsHistoryToPrivate (nextIndex hk)))
have hm : Measurable (fun h => K h (Prod.mk h ⁻¹' A)) :=
K.measurable_kernel_prodMk_left hA
exact hm.ennreal_toReal
theorem integrable_successorRowKernelSection_component
The row section is integrable under every deterministic-input predecessor prefix component. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Integrable (successorRowKernelSection P n k hk a A) ((P.transcript n x).map (transcriptPrefix (le_of_succ_le hk)))
Proof (Lean source)
lemma integrable_successorRowKernelSection_component {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (n k : ℕ) (hk : k + 1 ≤ n)
(a : Fin 4)
(A : Set (History (Z n) k (le_of_succ_le hk) ×
Z n (nextIndex hk))) (hA : MeasurableSet A)
(x : Fin n → Fin 4) :
Integrable (successorRowKernelSection P n k hk a A)
((P.transcript n x).map
(transcriptPrefix (le_of_succ_le hk))) := by
letI : IsProbabilityMeasure (P.transcript n x) := (P.factorizes n x).1
letI : IsProbabilityMeasure ((P.transcript n x).map
(transcriptPrefix (le_of_succ_le hk))) :=
Measure.isProbabilityMeasure_map
(measurable_transcriptPrefix (le_of_succ_le hk)).aemeasurable
apply Integrable.of_bound
(measurable_successorRowKernelSection P n k hk a A hA).aestronglyMeasurable 1
filter_upwards [] with h
change |((P.channel n (nextIndex hk))
(a, fsHistoryToPrivate (nextIndex hk) h) (Prod.mk h ⁻¹' A)).toReal| ≤ 1
rw [abs_of_nonneg ENNReal.toReal_nonneg]
have hQ : IsMarkovKernel (P.channel n (nextIndex hk)) :=
(P.privacy n).1 (nextIndex hk)
letI : IsProbabilityMeasure
((P.channel n (nextIndex hk))
(a, fsHistoryToPrivate (nextIndex hk) h)) :=
hQ.isProbabilityMeasure _
have hle : (P.channel n (nextIndex hk))
(a, fsHistoryToPrivate (nextIndex hk) h) (Prod.mk h ⁻¹' A) ≤ 1 := by
have hm : (P.channel n (nextIndex hk))
(a, fsHistoryToPrivate (nextIndex hk) h) (Prod.mk h ⁻¹' A) ≤
(P.channel n (nextIndex hk))
(a, fsHistoryToPrivate (nextIndex hk) h) univ :=
MeasureTheory.measure_mono (Set.subset_univ _)
simpa using hm
simpa using ENNReal.toReal_mono ENNReal.one_ne_top hle
theorem sequentialJointChannel_apply_toReal_eq_integral
the sequential joint channel apply to real eq integral assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
(sequentialJointChannel P n i μ a A).toReal
= ∫ h, ((P.channel n i) (a, fsHistoryToPrivate i h) (Prod.mk h ⁻¹' A)).toReal ∂μ
Proof (Lean source)
lemma sequentialJointChannel_apply_toReal_eq_integral {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (n : ℕ) (i : Fin n)
(μ : Measure (History (Z n) i.val (le_of_lt i.isLt)))
[IsFiniteMeasure μ]
(a : Fin 4)
(A : Set (History (Z n) i.val (le_of_lt i.isLt) × Z n i))
(hA : MeasurableSet A) :
(sequentialJointChannel P n i μ a A).toReal =
∫ h, ((P.channel n i)
(a, fsHistoryToPrivate i h) (Prod.mk h ⁻¹' A)).toReal ∂μ := by
letI : IsMarkovKernel (P.channel n i) := (P.privacy n).1 i
unfold sequentialJointChannel releasedHistoryRowKernel
rw [Kernel.compProd_apply hA, integral_toReal]
· rfl
· have hm := ((P.channel n i).comap
(fun h => (a, fsHistoryToPrivate i h))
(measurable_const.prodMk
(measurable_fsHistoryToPrivate i))).measurable_kernel_prodMk_left hA
exact hm.aemeasurable
· filter_upwards [] with h
exact measure_lt_top _ _
def predecessorRowEventDerivative
The predecessor contribution for one current-input row is the prefix derivative density integrated against that row's bounded event section. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The predecessor Row Event Derivative is determined by the displayed parameters.
Definition (Lean source)
parameters
def
predecessorRowEventDerivative P θ v p n k hk a A :
ℝ
given by
∫ h, transcriptPrefixMixtureRealDerivative P θ v p n k (le_of_succ_le hk) h * successorRowKernelSection P n k hk a A h ∂transcriptPrefixReferenceMeasure P n k (le_of_succ_le hk)
theorem hasDerivAt_varyingPredecessorRowEventMass
The derivative of a fixed current-input row is exactly the predecessor prefix derivative contribution; the current row itself is held fixed here. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the has Deriv At varying Predecessor Row Event Mass.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
hp :
hθ :
n k :
ℕ
hk :
k + 1 ≤ n
a :
Fin 4
A :
hA :
conclusion
HasDerivAt (fun u => varyingPredecessorRowEventMass P θ v p n k hk u a A) (predecessorRowEventDerivative P θ v p n k hk a A) 0
Proof (Lean source)
lemma hasDerivAt_varyingPredecessorRowEventMass
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n) (a : Fin 4)
(A : Set (History (Z n) k (le_of_succ_le hk) ×
Z n (nextIndex hk))) (hA : MeasurableSet A) :
HasDerivAt (fun u => varyingPredecessorRowEventMass
P θ v p n k hk u a A)
(predecessorRowEventDerivative P θ v p n k hk a A) 0 := by
let g := successorRowKernelSection P n k hk a A
have hbase := hasDerivAt_integral_transcriptPrefixLaw P θ v p hp hθ
n k (le_of_succ_le hk) g
(measurable_successorRowKernelSection P n k hk a A hA)
(fun x => integrable_successorRowKernelSection_component
P n k hk a A hA x)
have hzero : parameterPath θ v 0 = θ := by
funext j
simp [parameterPath]
have hc0 : ContinuousAt (fun u => parameterPath θ v u 0) 0 := by
unfold parameterPath
fun_prop
have hc1 : ContinuousAt (fun u => parameterPath θ v u 1) 0 := by
unfold parameterPath
fun_prop
have hinterior : {u : ℝ | InteriorMeans (parameterPath θ v u)} ∈ nhds 0 := by
filter_upwards
[continuousAt_const.eventually_lt hc0 (by simpa [hzero] using hθ.1),
hc0.eventually_lt continuousAt_const (by simpa [hzero] using hθ.2.1),
continuousAt_const.eventually_lt hc1 (by simpa [hzero] using hθ.2.2.1),
hc1.eventually_lt continuousAt_const (by simpa [hzero] using hθ.2.2.2)]
with u hu00 hu01 hu10 hu11
exact ⟨hu00, hu01, hu10, hu11⟩
have heq : (fun u => varyingPredecessorRowEventMass
P θ v p n k hk u a A) =ᶠ[nhds 0] fun u => ∫ h, g h
∂transcriptPrefixLaw P (parameterPath θ v u) p n k
(le_of_succ_le hk) := by
filter_upwards [hinterior] with u hu
have hprob : IsProbabilityMeasure
(transcriptPrefixLaw P (parameterPath θ v u) p n k
(le_of_succ_le hk)) :=
transcriptPrefixLaw_isProbability P (parameterPath θ v u) p hp hu
n k (le_of_succ_le hk)
let μu : Measure (History (Z n) (nextIndex hk).val
(le_of_lt (nextIndex hk).isLt)) :=
transcriptPrefixLaw P (parameterPath θ v u) p n k
(le_of_succ_le hk)
letI : IsProbabilityMeasure μu := by
constructor
change transcriptPrefixLaw P (parameterPath θ v u) p n k
(le_of_succ_le hk) univ = 1
exact @IsProbabilityMeasure.measure_univ _ _ _ hprob
letI : IsFiniteMeasure μu := ⟨by
rw [IsProbabilityMeasure.measure_univ]
exact ENNReal.one_lt_top⟩
unfold varyingPredecessorRowEventMass
change (sequentialJointChannel P n (nextIndex hk) μu a A).toReal = _
rw [sequentialJointChannel_apply_toReal_eq_integral P n (nextIndex hk)
μu a A hA]
rfl
exact hbase.congr_of_eventuallyEq heq
theorem hasDerivAt_successorPrefixEventMass_of_predecessor_rows
The successor-event derivative is the sum of the varying-predecessor contribution and the current joint-channel directional derivative. The row derivative premise concerns only variation of the predecessor prefix law; the current four-cell weights are differentiated explicitly here. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the has Deriv At successor Prefix Event Mass of predecessor rows.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
hp :
hθ :
n k :
ℕ
hk :
k + 1 ≤ n
A :
hA :
pastRowDerivative :
Fin 4 → ℝ
hrow :
∀ a,
HasDerivAt (fun u => varyingPredecessorRowEventMass P θ v p n k hk u a A) (pastRowDerivative a) 0
conclusion
HasDerivAt (fun u => successorPrefixEventMass P θ v p n k hk u A) (predecessorSuccessorEventDerivative θ p pastRowDerivative + sequentialJointEventDerivative P v p n (nextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) A) 0
Proof (Lean source)
lemma hasDerivAt_successorPrefixEventMass_of_predecessor_rows
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n)
(A : Set (History (Z n) k (le_of_succ_le hk) ×
Z n (nextIndex hk))) (hA : MeasurableSet A)
(pastRowDerivative : Fin 4 → ℝ)
(hrow : ∀ a, HasDerivAt
(fun u => varyingPredecessorRowEventMass
P θ v p n k hk u a A) (pastRowDerivative a) 0) :
HasDerivAt (fun u => successorPrefixEventMass
P θ v p n k hk u A)
(predecessorSuccessorEventDerivative θ p pastRowDerivative +
sequentialJointEventDerivative P v p n (nextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) A) 0 := by
have hzero : parameterPath θ v 0 = θ := by
funext j
simp [parameterPath]
have hc0 : ContinuousAt (fun u => parameterPath θ v u 0) 0 := by
unfold parameterPath
fun_prop
have hc1 : ContinuousAt (fun u => parameterPath θ v u 1) 0 := by
unfold parameterPath
fun_prop
have hinterior : {u : ℝ | InteriorMeans (parameterPath θ v u)} ∈ nhds 0 := by
filter_upwards
[continuousAt_const.eventually_lt hc0 (by simpa [hzero] using hθ.1),
hc0.eventually_lt continuousAt_const (by simpa [hzero] using hθ.2.1),
continuousAt_const.eventually_lt hc1 (by simpa [hzero] using hθ.2.2.1),
hc1.eventually_lt continuousAt_const (by simpa [hzero] using hθ.2.2.2)]
with u hu00 hu01 hu10 hu11
exact ⟨hu00, hu01, hu10, hu11⟩
have heq : (fun u => successorPrefixEventMass P θ v p n k hk u A) =ᶠ[nhds 0]
fun u => ∑ a : Fin 4,
piTheta (parameterPath θ v u) p a *
varyingPredecessorRowEventMass P θ v p n k hk u a A := by
filter_upwards [hinterior] with u hu
have hμprob : IsProbabilityMeasure
(transcriptPrefixLaw P (parameterPath θ v u) p n k
(le_of_succ_le hk)) :=
transcriptPrefixLaw_isProbability P (parameterPath θ v u) p hp hu
n k (le_of_succ_le hk)
have hlaw := transcriptSuccessorLaw_eq_jointChannel P
(parameterPath θ v u) p hp hu n k hk
have hmix := @sequentialJoint_realMixture_eq_outputLaw_toReal Z _ ε P
(parameterPath θ v u) p hp hu n (nextIndex hk)
(transcriptPrefixLaw P (parameterPath θ v u) p n k
(le_of_succ_le hk)) hμprob A hA
unfold successorPrefixEventMass varyingPredecessorRowEventMass
rw [hlaw]
exact hmix.symm
have hsum : HasDerivAt (fun u => ∑ a : Fin 4,
piTheta (parameterPath θ v u) p a *
varyingPredecessorRowEventMass P θ v p n k hk u a A)
(∑ a : Fin 4,
(inputDirectionalDerivative p v a *
varyingPredecessorRowEventMass P θ v p n k hk 0 a A +
piTheta θ p a * pastRowDerivative a)) 0 := by
simpa only [Pi.mul_apply, hzero] using HasDerivAt.fun_sum
(u := (Finset.univ : Finset (Fin 4)))
(fun a _ => (hasDerivAt_piTheta_parameterPath θ v p 0 a).mul (hrow a))
have hderivative : (∑ a : Fin 4,
(inputDirectionalDerivative p v a *
varyingPredecessorRowEventMass P θ v p n k hk 0 a A +
piTheta θ p a * pastRowDerivative a)) =
predecessorSuccessorEventDerivative θ p pastRowDerivative +
sequentialJointEventDerivative P v p n (nextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) A := by
simp only [predecessorSuccessorEventDerivative,
sequentialJointEventDerivative, Finset.sum_add_distrib]
rw [add_comm]
congr 1
apply Finset.sum_congr rfl
intro a _
simp [varyingPredecessorRowEventMass, hzero]
rw [hderivative] at hsum
exact hsum.congr_of_eventuallyEq heq
theorem hasDerivAt_successorPrefixEventMass
The exact successor-event derivative obtained by differentiating both the varying predecessor prefix law and the current observation-dependent row. The first summand is the finite-mixture pushforward of the prefix score; the second is the directional derivative of the current joint channel with its predecessor law fixed at the base parameter. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the has Deriv At successor Prefix Event Mass.
Formal statement
hypotheses
conclusion
HasDerivAt (fun u => successorPrefixEventMass P θ v p n k hk u A) (predecessorSuccessorEventDerivative θ p (fun a => predecessorRowEventDerivative P θ v p n k hk a A) + sequentialJointEventDerivative P v p n (nextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) A) 0
Proof (Lean source)
lemma hasDerivAt_successorPrefixEventMass
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n)
(A : Set (History (Z n) k (le_of_succ_le hk) ×
Z n (nextIndex hk))) (hA : MeasurableSet A) :
HasDerivAt (fun u => successorPrefixEventMass
P θ v p n k hk u A)
(predecessorSuccessorEventDerivative θ p
(fun a => predecessorRowEventDerivative P θ v p n k hk a A) +
sequentialJointEventDerivative P v p n (nextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) A) 0 := by
exact hasDerivAt_successorPrefixEventMass_of_predecessor_rows
P θ v p hp hθ n k hk A hA
(fun a => predecessorRowEventDerivative P θ v p n k hk a A)
(fun a => hasDerivAt_varyingPredecessorRowEventMass
P θ v p hp hθ n k hk a A hA)
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialSuccessorScore 13 declarations This file identifies the directional score of the four-row joint channel which generates a released prefix together with its next output.
Directional scores for successor joint channels
This file identifies the directional score of the four-row joint channel which generates a released prefix together with its next output. The score is expressed against the finite row-sum dominating measure, so the result does not require countable generation of the output spaces.
def channelDirectionalDerivativeDensity
the channel directional derivative density is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
def
channelDirectionalDerivativeDensity p Q v z :
ℝ
given by
∑ k : Fin 2, v k * channelDerivativeDensity p Q k z
def channelDirectionalScore
the channel directional score is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The channel Directional Score is determined by the displayed parameters.
Definition (Lean source)
parameters
def
channelDirectionalScore θ p Q v z :
ℝ
given by
if 0 < channelOutputDensity θ p Q z then channelDirectionalDerivativeDensity p Q v z / channelOutputDensity θ p Q z else 0
theorem channelDirectionalDerivativeDensity_eq_inputSum
the channel directional derivative density eq input sum assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma channelDirectionalDerivativeDensity_eq_inputSum
{Z : Type*} [MeasurableSpace Z]
(p : ℝ) (Q : Kernel (Fin 4) Z) (v : TrialParameter) (z : Z) :
channelDirectionalDerivativeDensity p Q v z =
∑ j : Fin 4, inputDirectionalDerivative p v j * channelDensity Q j z := by
unfold channelDirectionalDerivativeDensity channelDerivativeDensity
simp_rw [Finset.mul_sum]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro j hj
unfold inputDirectionalDerivative
simp_rw [Finset.sum_mul]
apply Finset.sum_congr rfl
intro k hk
ring
theorem channelOutputDensity_nonneg_of_interior
the channel output density nonneg of interior assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the channel Output Density nonneg of interior.
Formal statement
Proof (Lean source)
lemma channelOutputDensity_nonneg_of_interior
{Z : Type*} [MeasurableSpace Z]
(θ : TrialParameter) (p : ℝ) (Q : Kernel (Fin 4) Z)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (z : Z) :
0 ≤ channelOutputDensity θ p Q z := by
unfold channelOutputDensity
exact Finset.sum_nonneg fun j _ => mul_nonneg
(piTheta_pos_of_interior θ p hp hθ j).le ENNReal.toReal_nonneg
theorem channelDirectionalDerivativeDensity_eq_zero_of_outputDensity_eq_zero
the channel directional derivative density eq zero of output density eq zero assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the channel Directional Derivative Density eq zero of output Density eq zero.
Formal statement
hypotheses
conclusion
channelDirectionalDerivativeDensity p Q v z = 0
Proof (Lean source)
lemma channelDirectionalDerivativeDensity_eq_zero_of_outputDensity_eq_zero
{Z : Type*} [MeasurableSpace Z]
(θ v : TrialParameter) (p : ℝ) (Q : Kernel (Fin 4) Z)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (z : Z)
(hz : channelOutputDensity θ p Q z = 0) :
channelDirectionalDerivativeDensity p Q v z = 0 := by
have hterms : ∀ j : Fin 4,
piTheta θ p j * channelDensity Q j z = 0 := by
have hnonneg : ∀ j ∈ (Finset.univ : Finset (Fin 4)),
0 ≤ piTheta θ p j * channelDensity Q j z := by
intro j hj
exact mul_nonneg (piTheta_pos_of_interior θ p hp hθ j).le
ENNReal.toReal_nonneg
exact fun j => (Finset.sum_eq_zero_iff_of_nonneg hnonneg).mp hz j
(Finset.mem_univ j)
rw [channelDirectionalDerivativeDensity_eq_inputSum]
apply Finset.sum_eq_zero
intro j hj
have hdensity : channelDensity Q j z = 0 := by
exact (mul_eq_zero.mp (hterms j)).resolve_left
(ne_of_gt (piTheta_pos_of_interior θ p hp hθ j))
rw [hdensity, mul_zero]
theorem channelDirectionalScore_mul_outputDensity
the channel directional score mul output density assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the channel Directional Score mul output Density.
Formal statement
hypotheses
Z :
Type*
θ v :
p :
ℝ
hp :
hθ :
z :
Z
conclusion
channelDirectionalScore θ p Q v z * channelOutputDensity θ p Q z
= channelDirectionalDerivativeDensity p Q v z
Proof (Lean source)
lemma channelDirectionalScore_mul_outputDensity
{Z : Type*} [MeasurableSpace Z]
(θ v : TrialParameter) (p : ℝ) (Q : Kernel (Fin 4) Z)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (z : Z) :
channelDirectionalScore θ p Q v z * channelOutputDensity θ p Q z =
channelDirectionalDerivativeDensity p Q v z := by
unfold channelDirectionalScore
split_ifs with hpos
· field_simp
· have hnonneg := channelOutputDensity_nonneg_of_interior θ p Q hp hθ z
have hz : channelOutputDensity θ p Q z = 0 :=
le_antisymm (le_of_not_gt hpos) hnonneg
rw [zero_mul, channelDirectionalDerivativeDensity_eq_zero_of_outputDensity_eq_zero
θ v p Q hp hθ z hz]
theorem integrable_channelDensity_dominating
the integrable channel density dominating assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma integrable_channelDensity_dominating
{Z : Type*} [MeasurableSpace Z]
(Q : Kernel (Fin 4) Z) [IsMarkovKernel Q] (j : Fin 4) :
Integrable (channelDensity Q j) (dominatingMeasure Q) := by
letI : IsFiniteMeasure (Q j) := inferInstance
letI : IsFiniteMeasure (dominatingMeasure Q) :=
⟨by simp [dominatingMeasure]⟩
have h := (integrable_toReal_rnDeriv_mul_iff
(channelRow_absolutelyContinuous_dominating Q j)
(f := fun _ => (1 : ℝ))).2 (integrable_const (1 : ℝ))
change Integrable
(fun z => ((Q j).rnDeriv (dominatingMeasure Q) z).toReal)
(dominatingMeasure Q)
simpa only [mul_one] using h
theorem setIntegral_channelDirectionalDerivativeDensity
the set integral channel directional derivative density assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
∫ z in A, channelDirectionalDerivativeDensity p Q v z ∂dominatingMeasure Q
Proof (Lean source)
lemma setIntegral_channelDirectionalDerivativeDensity
{Z : Type*} [MeasurableSpace Z]
(p : ℝ) (Q : Kernel (Fin 4) Z) [IsMarkovKernel Q]
(v : TrialParameter) (A : Set Z) :
∫ z in A, channelDirectionalDerivativeDensity p Q v z
∂dominatingMeasure Q =
∑ j : Fin 4, inputDirectionalDerivative p v j * (Q j A).toReal := by
letI : IsFiniteMeasure (dominatingMeasure Q) :=
⟨by simp [dominatingMeasure]⟩
calc
(∫ z in A, channelDirectionalDerivativeDensity p Q v z
∂dominatingMeasure Q) =
∫ z in A, ∑ j : Fin 4,
inputDirectionalDerivative p v j * channelDensity Q j z
∂dominatingMeasure Q := by
apply integral_congr_ae
filter_upwards [] with z
exact channelDirectionalDerivativeDensity_eq_inputSum p Q v z
_ = ∑ j : Fin 4, ∫ z in A,
inputDirectionalDerivative p v j * channelDensity Q j z
∂dominatingMeasure Q := by
rw [integral_finset_sum]
intro j hj
exact (integrable_channelDensity_dominating Q j).integrableOn.const_mul _
_ = ∑ j : Fin 4,
inputDirectionalDerivative p v j * (Q j A).toReal := by
apply Finset.sum_congr rfl
intro j hj
rw [integral_const_mul]
rw [show (∫ z in A, channelDensity Q j z ∂dominatingMeasure Q) =
(Q j).real A by
exact Measure.setIntegral_toReal_rnDeriv
(channelRow_absolutelyContinuous_dominating Q j) A]
rfl
theorem setIntegral_sequentialJoint_channelDirectionalScore
the set integral sequential joint channel directional score assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the set Integral sequential Joint channel Directional Score.
Formal statement
hypotheses
conclusion
∫ z in A, channelDirectionalScore θ p (sequentialJointChannel P n i μ) v z * channelOutputDensity θ p (sequentialJointChannel P n i μ) z ∂dominatingMeasure (sequentialJointChannel P n i μ)
= sequentialJointEventDerivative P v p n i μ A
Proof (Lean source)
lemma setIntegral_sequentialJoint_channelDirectionalScore
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n : ℕ) (i : Fin n)
(μ : Measure (History (Z n) i.val (le_of_lt i.isLt)))
[IsProbabilityMeasure μ]
(A : Set (History (Z n) i.val (le_of_lt i.isLt) × Z n i)) :
∫ z in A,
channelDirectionalScore θ p (sequentialJointChannel P n i μ) v z *
channelOutputDensity θ p (sequentialJointChannel P n i μ) z
∂dominatingMeasure (sequentialJointChannel P n i μ) =
sequentialJointEventDerivative P v p n i μ A := by
let Q := sequentialJointChannel P n i μ
letI : IsMarkovKernel Q := sequentialJointChannel_isMarkov P n i μ
have hpoint (z) : channelDirectionalScore θ p Q v z *
channelOutputDensity θ p Q z =
channelDirectionalDerivativeDensity p Q v z :=
channelDirectionalScore_mul_outputDensity θ v p Q hp hθ z
change (∫ z in A, channelDirectionalScore θ p Q v z *
channelOutputDensity θ p Q z ∂dominatingMeasure Q) = _
simp_rw [hpoint]
exact setIntegral_channelDirectionalDerivativeDensity p Q v A
theorem channelDirectionalScore_sq_mul_outputDensity
the channel directional score sq mul output density assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the channel Directional Score sq mul output Density.
Formal statement
hypotheses
Z :
Type*
θ v :
p :
ℝ
hp :
hθ :
z :
Z
conclusion
channelDirectionalScore θ p Q v z ^ 2 * channelOutputDensity θ p Q z
= channelDirectionalDerivativeDensity p Q v z ^ 2 / channelOutputDensity θ p Q z
Proof (Lean source)
lemma channelDirectionalScore_sq_mul_outputDensity
{Z : Type*} [MeasurableSpace Z]
(θ v : TrialParameter) (p : ℝ) (Q : Kernel (Fin 4) Z)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (z : Z) :
channelDirectionalScore θ p Q v z ^ 2 * channelOutputDensity θ p Q z =
channelDirectionalDerivativeDensity p Q v z ^ 2 /
channelOutputDensity θ p Q z := by
unfold channelDirectionalScore
split_ifs with hpos
· field_simp
· have hnonneg := channelOutputDensity_nonneg_of_interior θ p Q hp hθ z
have hz : channelOutputDensity θ p Q z = 0 :=
le_antisymm (le_of_not_gt hpos) hnonneg
have hder := channelDirectionalDerivativeDensity_eq_zero_of_outputDensity_eq_zero
θ v p Q hp hθ z hz
simp [hz, hder]
theorem integral_sequentialJoint_channelDirectionalScore_sq
the integral sequential joint channel directional score sq assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the integral sequential Joint channel Directional Score sq.
Formal statement
hypotheses
conclusion
∫ z, channelDirectionalScore θ p (sequentialJointChannel P n i μ) v z ^ 2 * channelOutputDensity θ p (sequentialJointChannel P n i μ) z ∂dominatingMeasure (sequentialJointChannel P n i μ)
Proof (Lean source)
lemma integral_sequentialJoint_channelDirectionalScore_sq
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n : ℕ) (i : Fin n)
(μ : Measure (History (Z n) i.val (le_of_lt i.isLt)))
[IsProbabilityMeasure μ] :
∫ z,
channelDirectionalScore θ p (sequentialJointChannel P n i μ) v z ^ 2 *
channelOutputDensity θ p (sequentialJointChannel P n i μ) z
∂dominatingMeasure (sequentialJointChannel P n i μ) =
informationQuadratic
(channelFisherInfo θ p (sequentialJointChannel P n i μ)) v := by
let Q := sequentialJointChannel P n i μ
have hQ : StationaryLDP ε Q := sequentialJointChannel_stationaryLDP P n i μ
change (∫ z, channelDirectionalScore θ p Q v z ^ 2 *
channelOutputDensity θ p Q z ∂dominatingMeasure Q) = _
calc
(∫ z, channelDirectionalScore θ p Q v z ^ 2 *
channelOutputDensity θ p Q z ∂dominatingMeasure Q) =
∫ z, channelDirectionalDerivativeDensity p Q v z ^ 2 /
channelOutputDensity θ p Q z ∂dominatingMeasure Q := by
apply integral_congr_ae
filter_upwards [] with z
exact channelDirectionalScore_sq_mul_outputDensity θ v p Q hp hθ z
_ = informationQuadratic (channelFisherInfo θ p Q) v := by
rw [informationQuadratic_eq_channelIntegral p θ hp hθ ε Q hQ v]
rfl
theorem transcriptPrefixLaw_isProbability
the transcript prefix law is probability assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the transcript Prefix Law is Probability.
Formal statement
hypotheses
conclusion
IsProbabilityMeasure (transcriptPrefixLaw P θ p n k hk)
Proof (Lean source)
lemma transcriptPrefixLaw_isProbability
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k ≤ n) :
IsProbabilityMeasure (transcriptPrefixLaw P θ p n k hk) := by
rw [transcriptPrefixLaw_eq_observedPrefixLaw P θ p hp hθ n k hk]
exact isProbabilityMeasure_prefixLaw
(fun i => observedStageKernel P θ p n i)
(fun i => observedStageKernel_isMarkov P θ p hp hθ n i)
(fun _ => ()) k hk
theorem transcriptSuccessorLaw_jointScore_characterization
Under the successor coordinates (init,last), the exact joint-channel score identities apply to the pushed-forward prefix law. The derivative here holds the predecessor prefix law fixed; differentiation of the whole successor law additionally contains the predecessor score. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the transcript Successor Law joint Score characterization.
Formal statement
hypotheses
conclusion
μ :
transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
Q :
sequentialJointChannel P n (nextIndex hk) μ
= outputLaw θ p Q
(∫ z in A, channelDirectionalScore θ p Q v z * channelOutputDensity θ p Q z ∂dominatingMeasure Q)
= sequentialJointEventDerivative P v p n (nextIndex hk) μ A
(∫ z, channelDirectionalScore θ p Q v z ^ 2 * channelOutputDensity θ p Q z ∂dominatingMeasure Q)
= informationQuadratic (channelFisherInfo θ p Q) v
Proof (Lean source)
lemma transcriptSuccessorLaw_jointScore_characterization
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n)
(A : Set (History (Z n) k (le_of_succ_le hk) × Z n (nextIndex hk))) :
let μ := transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
let Q := sequentialJointChannel P n (nextIndex hk) μ
(transcriptPrefixLaw P θ p n (k + 1) hk).map
(fun u => (init hk u, last hk u)) = outputLaw θ p Q ∧
(∫ z in A, channelDirectionalScore θ p Q v z *
channelOutputDensity θ p Q z ∂dominatingMeasure Q) =
sequentialJointEventDerivative P v p n (nextIndex hk) μ A ∧
(∫ z, channelDirectionalScore θ p Q v z ^ 2 *
channelOutputDensity θ p Q z ∂dominatingMeasure Q) =
informationQuadratic (channelFisherInfo θ p Q) v := by
dsimp only
have hμ : IsProbabilityMeasure
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) :=
transcriptPrefixLaw_isProbability P θ p hp hθ n k (le_of_succ_le hk)
refine ⟨transcriptSuccessorLaw_eq_jointChannel P θ p hp hθ n k hk, ?_, ?_⟩
· exact @setIntegral_sequentialJoint_channelDirectionalScore
Z _ ε P θ v p hp hθ n (nextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) hμ A
· exact @integral_sequentialJoint_channelDirectionalScore_sq
Z _ ε P θ v p hp hθ n (nextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) hμ
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialSuccessorScoreCharacterization 15 declarations Event characterization of the successor score candidate
Event characterization of the successor score candidate
theorem outputLaw_ac_dominatingMeasure
the output law ac dominating measure assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
outputLaw θ p Q ≪ dominatingMeasure Q
Proof (Lean source)
lemma outputLaw_ac_dominatingMeasure {Y : Type*} [MeasurableSpace Y]
(θ : TrialParameter) (p : ℝ) (Q : Kernel (Fin 4) Y) :
outputLaw θ p Q ≪ dominatingMeasure Q := by
rw [outputLaw, dominatingMeasure]
apply Measure.AbsolutelyContinuous.mk
intro A hA href
simp only [Measure.coe_finsetSum, Finset.sum_apply] at href ⊢
apply Finset.sum_eq_zero
intro a _
rw [Measure.smul_apply, smul_eq_mul]
apply mul_eq_zero_of_right
exact (Finset.sum_eq_zero_iff_of_nonneg (fun _ _ => zero_le)).mp href a
(Finset.mem_univ a)
theorem outputLawDensity_toReal_ae_eq_channelOutputDensity
the output law density to real ae eq channel output density assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the output Law Density to Real ae eq channel Output Density.
Formal statement
hypotheses
Y :
Type*
θ :
p :
ℝ
Q :
hp :
hθ :
conclusion
(fun y => ((outputLaw θ p Q).rnDeriv (dominatingMeasure Q) y).toReal)
=ᵐ[dominatingMeasure Q] channelOutputDensity θ p Q
Proof (Lean source)
lemma outputLawDensity_toReal_ae_eq_channelOutputDensity
{Y : Type*} [MeasurableSpace Y]
(θ : TrialParameter) (p : ℝ) (Q : Kernel (Fin 4) Y)
[IsMarkovKernel Q] (hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
(fun y => ((outputLaw θ p Q).rnDeriv (dominatingMeasure Q) y).toReal)
=ᵐ[dominatingMeasure Q] channelOutputDensity θ p Q := by
classical
let ν := dominatingMeasure Q
let μ (a : Fin 4) := Q a
let w (a : Fin 4) : ℝ≥0∞ := ofReal (piTheta θ p a)
letI : IsFiniteMeasure ν := ⟨by simp [ν, dominatingMeasure]⟩
have hw (a : Fin 4) : w a ≠ ∞ := by simp [w]
haveI (a : Fin 4) : IsFiniteMeasure (w a • μ a) :=
(μ a).smul_finite (hw a)
have hrn (s : Finset (Fin 4)) :
(s.sum fun a => w a • μ a).rnDeriv ν =ᵐ[ν]
fun y => s.sum fun a => w a * (μ a).rnDeriv ν y := by
induction s using Finset.induction_on with
| empty =>
filter_upwards [Measure.rnDeriv_zero ν] with y hy
simpa using hy
| @insert a s ha ih =>
haveI : IsFiniteMeasure (s.sum fun b => w b • μ b) := inferInstance
have hadd := Measure.rnDeriv_add (w a • μ a)
(s.sum fun b => w b • μ b) ν
have hsmul := Measure.rnDeriv_smul_left_of_ne_top (μ a) ν (hw a)
filter_upwards [hadd, hsmul, ih] with y hy hs hi
simpa [Finset.sum_insert ha, Pi.add_apply, Pi.smul_apply,
smul_eq_mul, hs, hi] using hy
have hfinite : ∀ᵐ y ∂ν, ∀ a : Fin 4, (μ a).rnDeriv ν y ≠ ∞ := by
rw [Filter.eventually_all]
intro a
exact (Measure.rnDeriv_lt_top (μ a) ν).mono (fun _ h => ne_of_lt h)
filter_upwards [hrn univ, hfinite] with y hy hfin
change ((outputLaw θ p Q).rnDeriv ν y).toReal =
channelOutputDensity θ p Q y
rw [show outputLaw θ p Q = ∑ a : Fin 4, w a • μ a by rfl]
rw [hy, ENNReal.toReal_sum
(fun a _ => ENNReal.mul_ne_top (hw a) (hfin a))]
simp [channelOutputDensity, channelDensity, μ, w, ν, ENNReal.toReal_mul,
(piTheta_pos_of_interior θ p hp hθ _).le]
theorem channelDirectionalDerivativeDensity_integrable_dominating
the channel directional derivative density integrable dominating assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma channelDirectionalDerivativeDensity_integrable_dominating
{Y : Type*} [MeasurableSpace Y]
(p : ℝ) (Q : Kernel (Fin 4) Y) [IsMarkovKernel Q]
(v : TrialParameter) :
Integrable (channelDirectionalDerivativeDensity p Q v)
(dominatingMeasure Q) := by
have hsum : Integrable (fun y => ∑ a : Fin 4,
inputDirectionalDerivative p v a * channelDensity Q a y)
(dominatingMeasure Q) :=
integrable_finsetSum _ fun a _ =>
(integrable_channelDensity_dominating Q a).const_mul _
exact hsum.congr (ae_of_all _ fun y =>
(channelDirectionalDerivativeDensity_eq_inputSum p Q v y).symm)
theorem channelDirectionalScore_integrable_outputLaw
the channel directional score integrable output law assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the channel Directional Score integrable output Law.
Formal statement
hypotheses
Y :
Type*
θ v :
p :
ℝ
Q :
hp :
hθ :
conclusion
Integrable (channelDirectionalScore θ p Q v) (outputLaw θ p Q)
Proof (Lean source)
lemma channelDirectionalScore_integrable_outputLaw
{Y : Type*} [MeasurableSpace Y]
(θ v : TrialParameter) (p : ℝ) (Q : Kernel (Fin 4) Y)
[IsMarkovKernel Q] (hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
Integrable (channelDirectionalScore θ p Q v) (outputLaw θ p Q) := by
let ν := dominatingMeasure Q
let μ := outputLaw θ p Q
letI : IsFiniteMeasure ν := ⟨by simp [ν, dominatingMeasure]⟩
letI : IsFiniteMeasure μ := by
constructor
dsimp [μ, outputLaw]
simp only [Measure.coe_finsetSum, Finset.sum_apply, Measure.smul_apply,
smul_eq_mul, measure_univ, mul_one]
exact ENNReal.sum_lt_top.mpr fun _ _ => ENNReal.ofReal_lt_top
change Integrable (channelDirectionalScore θ p Q v) μ
rw [← Measure.withDensity_rnDeriv_eq μ ν
(outputLaw_ac_dominatingMeasure θ p Q)]
rw [integrable_withDensity_iff_integrable_smul'
(Measure.measurable_rnDeriv μ ν) (Measure.rnDeriv_lt_top μ ν)]
have hderiv := channelDirectionalDerivativeDensity_integrable_dominating p Q v
refine hderiv.congr ?_
filter_upwards [outputLawDensity_toReal_ae_eq_channelOutputDensity
θ p Q hp hθ] with y hdensity
rw [smul_eq_mul, hdensity, mul_comm,
channelDirectionalScore_mul_outputDensity θ v p Q hp hθ y]
theorem setIntegral_channelDirectionalScore_outputLaw
the set integral channel directional score output law assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the set Integral channel Directional Score output Law.
Formal statement
hypotheses
conclusion
∫ y in A, channelDirectionalScore θ p Q v y ∂outputLaw θ p Q
= ∫ y in A, channelDirectionalScore θ p Q v y * channelOutputDensity θ p Q y ∂dominatingMeasure Q
Proof (Lean source)
lemma setIntegral_channelDirectionalScore_outputLaw
{Y : Type*} [MeasurableSpace Y]
(θ v : TrialParameter) (p : ℝ) (Q : Kernel (Fin 4) Y)
[IsMarkovKernel Q] (hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(A : Set Y) (hA : MeasurableSet A) :
∫ y in A, channelDirectionalScore θ p Q v y ∂outputLaw θ p Q =
∫ y in A, channelDirectionalScore θ p Q v y *
channelOutputDensity θ p Q y ∂dominatingMeasure Q := by
let ν := dominatingMeasure Q
let μ := outputLaw θ p Q
letI : IsFiniteMeasure ν := ⟨by simp [ν, dominatingMeasure]⟩
letI : IsFiniteMeasure μ := by
constructor
dsimp [μ, outputLaw]
simp only [Measure.coe_finsetSum, Finset.sum_apply, Measure.smul_apply,
smul_eq_mul, measure_univ, mul_one]
exact ENNReal.sum_lt_top.mpr fun _ _ => ENNReal.ofReal_lt_top
change (∫ y in A, channelDirectionalScore θ p Q v y ∂μ) = _
rw [← Measure.withDensity_rnDeriv_eq μ ν
(outputLaw_ac_dominatingMeasure θ p Q)]
simp_rw [← integral_indicator hA]
rw [integral_withDensity_eq_integral_toReal_smul
(Measure.measurable_rnDeriv μ ν) (Measure.rnDeriv_lt_top μ ν)]
apply integral_congr_ae
filter_upwards [outputLawDensity_toReal_ae_eq_channelOutputDensity
θ p Q hp hθ] with y hdensity
have hdensity' : (μ.rnDeriv ν y).toReal =
channelOutputDensity θ p Q y := by
simpa [μ, ν] using hdensity
by_cases hy : y ∈ A
· simp [hy, smul_eq_mul, hdensity']
ring
· simp [hy]
def liftedPrefixDirectionalScore
The prefix score lifted to successor coordinates. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The lifted Prefix Directional Score is determined by the displayed parameters.
Definition (Lean source)
parameters
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
n k :
ℕ
hk :
k + 1 ≤ n
z :
History (Z n) k (le_of_succ_le hk) × Z n (canonicalNextIndex hk)
def
liftedPrefixDirectionalScore P θ v p n k hk z :
ℝ
given by
conditionalTranscriptPrefixDirectionalScore P θ v p n k (le_of_succ_le hk) z.1
theorem measurable_liftedPrefixDirectionalScore
the measurable lifted prefix directional score assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (liftedPrefixDirectionalScore P θ v p n k hk)
Proof (Lean source)
lemma measurable_liftedPrefixDirectionalScore
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(n k : ℕ) (hk : k + 1 ≤ n) :
Measurable (liftedPrefixDirectionalScore P θ v p n k hk) := by
unfold liftedPrefixDirectionalScore conditionalTranscriptPrefixDirectionalScore
exact ((measurable_transcriptPrefixMixtureRealDerivative P θ v p n k
(le_of_succ_le hk)).div
(measurable_transcriptPrefixMixtureRealDensity P θ p n k
(le_of_succ_le hk))).comp measurable_fst
theorem conditionalTranscriptPrefixDirectionalScore_integrable
the conditional transcript prefix directional score integrable assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the conditional Transcript Prefix Directional Score integrable.
Formal statement
hypotheses
conclusion
Integrable (conditionalTranscriptPrefixDirectionalScore P θ v p n k (le_of_succ_le hk)) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk))
Proof (Lean source)
lemma conditionalTranscriptPrefixDirectionalScore_integrable
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n) :
Integrable (conditionalTranscriptPrefixDirectionalScore P θ v p n k
(le_of_succ_le hk))
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) := by
letI : IsMarkovKernel (P.channel n (nextIndex hk)) :=
(P.privacy n).1 (nextIndex hk)
have h :=
integrable_conditionalTranscriptPrefixDirectionalScore_mul_successorRowKernelSection
P θ v p hp hθ n k hk (0 : Fin 4) univ MeasurableSet.univ
simpa [successorRowKernelSection] using h
theorem successorEventDerivative_eq_pastRows_add_currentScoreIntegral
Exact event characterization before the final projection step. The past part is in its verified row-disintegrated form, while the current channel score is integrated against the actual joint-channel output law. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the successor Event Derivative eq past Rows add current Score Integral.
Formal statement
hypotheses
conclusion
predecessorSuccessorEventDerivative θ p (fun a => predecessorRowEventDerivative P θ v p n k hk a A)
+ sequentialJointEventDerivative P v p n (nextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) A
= (∑ a : Fin 4, piTheta θ p a * ∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n k (le_of_succ_le hk) h * successorRowKernelSection P n k hk a A h ∂transcriptPrefixLaw P θ p n k (le_of_succ_le hk))
+ ∫ z in A, channelDirectionalScore θ p (sequentialJointChannel P n (nextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) v z ∂outputLaw θ p (sequentialJointChannel P n (nextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))
Proof (Lean source)
-- keep: reusable sequential-law, Fisher-information, or van-Trees bridge for related adaptive experiments
lemma successorEventDerivative_eq_pastRows_add_currentScoreIntegral
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n)
(A : Set (History (Z n) k (le_of_succ_le hk) ×
Z n (nextIndex hk))) (hA : MeasurableSet A) :
predecessorSuccessorEventDerivative θ p
(fun a => predecessorRowEventDerivative P θ v p n k hk a A) +
sequentialJointEventDerivative P v p n (nextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) A =
(∑ a : Fin 4, piTheta θ p a *
∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n k
(le_of_succ_le hk) h *
successorRowKernelSection P n k hk a A h
∂transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) +
∫ z in A, channelDirectionalScore θ p
(sequentialJointChannel P n (nextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) v z
∂outputLaw θ p (sequentialJointChannel P n (nextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) := by
rw [predecessorSuccessorEventDerivative_eq_sum_integral_pastScore
P θ v p hp hθ n k hk A hA]
let μ := transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
let Q := sequentialJointChannel P n (nextIndex hk) μ
have hμ : IsProbabilityMeasure μ :=
transcriptPrefixLaw_isProbability P θ p hp hθ n k (le_of_succ_le hk)
have hQ : IsMarkovKernel Q := @sequentialJointChannel_isMarkov
Z _ ε P n (nextIndex hk) μ hμ
have hcurrent := (transcriptSuccessorLaw_jointScore_characterization
P θ v p hp hθ n k hk A).2.1
have hchange := @setIntegral_channelDirectionalScore_outputLaw
_ _ θ v p Q hQ hp hθ A hA
change _ + sequentialJointEventDerivative P v p n (nextIndex hk) μ A =
_ + ∫ z in A, channelDirectionalScore θ p Q v z ∂outputLaw θ p Q
rw [hchange, hcurrent]
theorem liftedPrefixDirectionalScore_integrable_successorLaw
the lifted prefix directional score integrable successor law assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the lifted Prefix Directional Score integrable successor Law.
Formal statement
hypotheses
conclusion
Integrable (liftedPrefixDirectionalScore P θ v p n k hk) (outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk))))
Proof (Lean source)
lemma liftedPrefixDirectionalScore_integrable_successorLaw
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n) :
Integrable (liftedPrefixDirectionalScore P θ v p n k hk)
(outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))) := by
let i := canonicalNextIndex hk
let μ := transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
let K : Kernel (History (Z n) k (le_of_succ_le hk)) (Z n i) :=
(observedStageKernel P θ p n i).comap
(fun h => ((), h)) (measurable_const.prodMk measurable_id)
letI : IsProbabilityMeasure μ :=
transcriptPrefixLaw_isProbability P θ p hp hθ n k (le_of_succ_le hk)
letI : IsMarkovKernel (observedStageKernel P θ p n i) :=
observedStageKernel_isMarkov P θ p hp hθ n i
letI : IsMarkovKernel (P.channel n i) := (P.privacy n).1 i
letI : IsMarkovKernel K := by
dsimp [K]
infer_instance
have hout : outputLaw θ p (sequentialJointChannel P n i μ) = μ ⊗ₘ K := by
simpa [K] using outputLaw_sequentialJointChannel P θ p hp hθ n i μ
rw [hout]
have hscore := conditionalTranscriptPrefixDirectionalScore_integrable
P θ v p hp hθ n k hk
have hmeas : AEStronglyMeasurable
(liftedPrefixDirectionalScore P θ v p n k hk) (μ ⊗ₘ K) :=
(measurable_liftedPrefixDirectionalScore P θ v p n k hk).aestronglyMeasurable
rw [Measure.integrable_compProd_iff hmeas]
constructor
· filter_upwards with h
simpa [liftedPrefixDirectionalScore] using
(integrable_const
(conditionalTranscriptPrefixDirectionalScore P θ v p n k
(le_of_succ_le hk) h) : Integrable _ (K h))
· simpa [liftedPrefixDirectionalScore, μ] using hscore.norm
theorem setIntegral_liftedPrefixDirectionalScore_successorLaw
the set integral lifted prefix directional score successor law assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the set Integral lifted Prefix Directional Score successor Law.
Formal statement
hypotheses
conclusion
∫ z in A, liftedPrefixDirectionalScore P θ v p n k hk z ∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))
= ∑ a : Fin 4, piTheta θ p a * ∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n k (le_of_succ_le hk) h * ((P.channel n (canonicalNextIndex hk)) (a, fsHistoryToPrivate (canonicalNextIndex hk) h) (Prod.mk h ⁻¹' A)).toReal ∂transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
Proof (Lean source)
lemma setIntegral_liftedPrefixDirectionalScore_successorLaw
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n)
(A : Set (History (Z n) k (le_of_succ_le hk) ×
Z n (canonicalNextIndex hk))) (hA : MeasurableSet A) :
∫ z in A, liftedPrefixDirectionalScore P θ v p n k hk z
∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) =
∑ a : Fin 4, piTheta θ p a *
∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n k
(le_of_succ_le hk) h *
((P.channel n (canonicalNextIndex hk))
(a, fsHistoryToPrivate (canonicalNextIndex hk) h)
(Prod.mk h ⁻¹' A)).toReal
∂transcriptPrefixLaw P θ p n k (le_of_succ_le hk) := by
let i := canonicalNextIndex hk
let μ := transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
let K : Kernel (History (Z n) k (le_of_succ_le hk)) (Z n i) :=
(observedStageKernel P θ p n i).comap
(fun h => ((), h)) (measurable_const.prodMk measurable_id)
letI : IsProbabilityMeasure μ :=
transcriptPrefixLaw_isProbability P θ p hp hθ n k (le_of_succ_le hk)
letI : IsMarkovKernel (observedStageKernel P θ p n i) :=
observedStageKernel_isMarkov P θ p hp hθ n i
letI : IsMarkovKernel (P.channel n i) := (P.privacy n).1 i
letI : IsMarkovKernel K := by
dsimp [K]
infer_instance
have hout : outputLaw θ p (sequentialJointChannel P n i μ) = μ ⊗ₘ K := by
simpa [K] using outputLaw_sequentialJointChannel P θ p hp hθ n i μ
rw [hout, ← integral_indicator hA]
have hint : Integrable (A.indicator
(liftedPrefixDirectionalScore P θ v p n k hk)) (μ ⊗ₘ K) := by
rw [← hout]
simpa only [nextIndex_eq_canonicalNextIndex] using
(liftedPrefixDirectionalScore_integrable_successorLaw
P θ v p hp hθ n k hk).indicator hA
rw [Measure.integral_compProd hint]
have hsection (h : History (Z n) k (le_of_succ_le hk)) :
MeasurableSet (Prod.mk h ⁻¹' A) :=
hA.preimage (measurable_const.prodMk measurable_id)
have hinner (h : History (Z n) k (le_of_succ_le hk)) :
(∫ y, A.indicator (liftedPrefixDirectionalScore P θ v p n k hk)
(h, y) ∂K h) =
conditionalTranscriptPrefixDirectionalScore P θ v p n k
(le_of_succ_le hk) h *
∑ a : Fin 4, piTheta θ p a *
((P.channel n i) (a, fsHistoryToPrivate i h)
(Prod.mk h ⁻¹' A)).toReal := by
rw [show (fun y => A.indicator
(liftedPrefixDirectionalScore P θ v p n k hk) (h, y)) =
(Prod.mk h ⁻¹' A).indicator (fun _ =>
conditionalTranscriptPrefixDirectionalScore P θ v p n k
(le_of_succ_le hk) h) by
funext y
by_cases hy : (h, y) ∈ A <;> simp [hy, liftedPrefixDirectionalScore]]
rw [integral_indicator (hsection h), setIntegral_const]
dsimp [K]
simp only [observedStageKernel_apply, measureReal_def]
rw [Measure.finsetSum_apply, ENNReal.toReal_sum]
· simp only [Measure.smul_apply, smul_eq_mul, ENNReal.toReal_mul,
ENNReal.toReal_ofReal (piTheta_pos_of_interior θ p hp hθ _).le]
ring
· intro a _
exact ENNReal.mul_ne_top ENNReal.ofReal_ne_top (measure_ne_top _ _)
rw [show (fun h => ∫ y, A.indicator
(liftedPrefixDirectionalScore P θ v p n k hk) (h, y) ∂K h) =
fun h => conditionalTranscriptPrefixDirectionalScore P θ v p n k
(le_of_succ_le hk) h *
∑ a : Fin 4, piTheta θ p a *
((P.channel n i) (a, fsHistoryToPrivate i h)
(Prod.mk h ⁻¹' A)).toReal by
funext h
exact hinner h]
simp only [Finset.mul_sum]
rw [integral_finset_sum]
· apply Finset.sum_congr rfl
intro a _
rw [← integral_const_mul]
apply integral_congr_ae
filter_upwards with h
ring
· intro a _
let Krow : Kernel (History (Z n) k (le_of_succ_le hk)) (Z n i) :=
(P.channel n i).comap
(fun h => (a, fsHistoryToPrivate i h))
(measurable_const.prodMk (measurable_fsHistoryToPrivate i))
have hmeas : Measurable (fun h : History (Z n) k (le_of_succ_le hk) =>
((P.channel n i) (a, fsHistoryToPrivate i h)
(Prod.mk h ⁻¹' A)).toReal) :=
(Krow.measurable_kernel_prodMk_left hA).ennreal_toReal
have hbounded : ∀ h, ‖((P.channel n i)
(a, fsHistoryToPrivate i h) (Prod.mk h ⁻¹' A)).toReal‖ ≤ 1 := by
intro h
rw [Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg]
apply ENNReal.toReal_mono ENNReal.one_ne_top
calc
_ ≤ (P.channel n i) (a, fsHistoryToPrivate i h) univ :=
measure_mono (Set.subset_univ _)
_ = 1 := measure_univ
simpa only [mul_assoc, mul_left_comm, mul_comm] using
((conditionalTranscriptPrefixDirectionalScore_integrable
P θ v p hp hθ n k hk).bdd_mul hmeas.aestronglyMeasurable
(ae_of_all _ hbounded)).const_mul (piTheta θ p a)
def successorDirectionalScoreCandidate
The full successor score candidate: the inherited prefix score plus the directional score of the current privatization channel. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. The successor Directional Score Candidate is determined by the displayed parameters.
Definition (Lean source)
parameters
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
n k :
ℕ
hk :
k + 1 ≤ n
z :
History (Z n) k (le_of_succ_le hk) × Z n (canonicalNextIndex hk)
def
successorDirectionalScoreCandidate P θ v p n k hk z :
ℝ
given by
liftedPrefixDirectionalScore P θ v p n k hk z
+ channelDirectionalScore θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) v z
theorem successorDirectionalScoreCandidate_integrable
the successor directional score candidate integrable assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the successor Directional Score Candidate integrable.
Formal statement
hypotheses
conclusion
Integrable (successorDirectionalScoreCandidate P θ v p n k hk) (outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk))))
Proof (Lean source)
lemma successorDirectionalScoreCandidate_integrable
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n) :
Integrable (successorDirectionalScoreCandidate P θ v p n k hk)
(outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))) := by
let μ := transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
let Q := sequentialJointChannel P n (canonicalNextIndex hk) μ
letI : IsProbabilityMeasure μ :=
transcriptPrefixLaw_isProbability P θ p hp hθ n k (le_of_succ_le hk)
letI : IsMarkovKernel Q := sequentialJointChannel_isMarkov
P n (canonicalNextIndex hk) μ
exact (liftedPrefixDirectionalScore_integrable_successorLaw
P θ v p hp hθ n k hk).add
(channelDirectionalScore_integrable_outputLaw θ v p Q hp hθ)
theorem setIntegral_successorDirectionalScoreCandidate_successorLaw
Canonical successor-coordinate packaging of the entire event derivative coefficient as one score integral. The first term is the derivative carried by the predecessor law and the second is the current-channel derivative. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the set Integral successor Directional Score Candidate successor Law.
Formal statement
hypotheses
conclusion
∫ z in A, successorDirectionalScoreCandidate P θ v p n k hk z ∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))
= (∑ a : Fin 4, piTheta θ p a * ∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n k (le_of_succ_le hk) h * ((P.channel n (canonicalNextIndex hk)) (a, fsHistoryToPrivate (canonicalNextIndex hk) h) (Prod.mk h ⁻¹' A)).toReal ∂transcriptPrefixLaw P θ p n k (le_of_succ_le hk))
+ sequentialJointEventDerivative P v p n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) A
Proof (Lean source)
lemma setIntegral_successorDirectionalScoreCandidate_successorLaw
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n)
(A : Set (History (Z n) k (le_of_succ_le hk) ×
Z n (canonicalNextIndex hk))) (hA : MeasurableSet A) :
∫ z in A, successorDirectionalScoreCandidate P θ v p n k hk z
∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) =
(∑ a : Fin 4, piTheta θ p a *
∫ h, conditionalTranscriptPrefixDirectionalScore P θ v p n k
(le_of_succ_le hk) h *
((P.channel n (canonicalNextIndex hk))
(a, fsHistoryToPrivate (canonicalNextIndex hk) h)
(Prod.mk h ⁻¹' A)).toReal
∂transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) +
sequentialJointEventDerivative P v p n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) A := by
let μ := transcriptPrefixLaw P θ p n k (le_of_succ_le hk)
let Q := sequentialJointChannel P n (canonicalNextIndex hk) μ
letI : IsProbabilityMeasure μ :=
transcriptPrefixLaw_isProbability P θ p hp hθ n k (le_of_succ_le hk)
letI : IsMarkovKernel Q := sequentialJointChannel_isMarkov
P n (canonicalNextIndex hk) μ
have hpast := liftedPrefixDirectionalScore_integrable_successorLaw
P θ v p hp hθ n k hk
have hcurrent := channelDirectionalScore_integrable_outputLaw θ v p Q hp hθ
change (∫ z in A, liftedPrefixDirectionalScore P θ v p n k hk z +
channelDirectionalScore θ p Q v z ∂outputLaw θ p Q) = _
rw [integral_add hpast.integrableOn hcurrent.integrableOn]
rw [setIntegral_liftedPrefixDirectionalScore_successorLaw
P θ v p hp hθ n k hk A hA]
rw [setIntegral_channelDirectionalScore_outputLaw
θ v p Q hp hθ A hA]
exact congrArg (fun x => _ + x)
(@setIntegral_sequentialJoint_channelDirectionalScore
Z _ ε P θ v p hp hθ n (canonicalNextIndex hk) μ _ A)
theorem successorEventDerivative_eq_setIntegral_successorDirectionalScoreCandidate
The actual predecessor-plus-current successor derivative equals the single integral of the full candidate score. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the successor Event Derivative eq set Integral successor Directional Score Candidate.
Formal statement
hypotheses
conclusion
predecessorSuccessorEventDerivative θ p (fun a => predecessorRowEventDerivative P θ v p n k hk a A)
+ sequentialJointEventDerivative P v p n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) A
= ∫ z in A, successorDirectionalScoreCandidate P θ v p n k hk z ∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk) (transcriptPrefixLaw P θ p n k (le_of_succ_le hk)))
Proof (Lean source)
lemma successorEventDerivative_eq_setIntegral_successorDirectionalScoreCandidate
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(n k : ℕ) (hk : k + 1 ≤ n)
(A : Set (History (Z n) k (le_of_succ_le hk) ×
Z n (canonicalNextIndex hk))) (hA : MeasurableSet A) :
predecessorSuccessorEventDerivative θ p
(fun a => predecessorRowEventDerivative P θ v p n k hk a A) +
sequentialJointEventDerivative P v p n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk)) A =
∫ z in A, successorDirectionalScoreCandidate P θ v p n k hk z
∂outputLaw θ p (sequentialJointChannel P n (canonicalNextIndex hk)
(transcriptPrefixLaw P θ p n k (le_of_succ_le hk))) := by
rw [setIntegral_successorDirectionalScoreCandidate_successorLaw
P θ v p hp hθ n k hk A hA]
rw [predecessorSuccessorEventDerivative_eq_sum_integral_pastScore
P θ v p hp hθ n k hk A hA]
unfold successorRowKernelSection
cases nextIndex_eq_canonicalNextIndex hk
rfl
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialVanTrees 14 declarations This file packages the finite-mixture transcript density along a scalar parameter path into the globally nonnegative likelihood fields required by the observation-dependent van Trees inequality.
Arbitrary-output van Trees likelihood fields
This file packages the finite-mixture transcript density along a scalar parameter path into the globally nonnegative likelihood fields required by the observation-dependent van Trees inequality. No countability or stagewise Radon--Nikodym selector is used.
theorem transcriptLaw_eq_withDensity_mixture
the transcript law eq with density mixture assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the transcript Law eq with Density mixture.
Formal statement
hypotheses
conclusion
(transcriptReferenceMeasure P n).withDensity (fun z ↦ ofReal (transcriptMixtureRealDensity P θ p n z))
= transcriptLaw P θ p n
Proof (Lean source)
lemma transcriptLaw_eq_withDensity_mixture {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ) (n : ℕ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
(transcriptReferenceMeasure P n).withDensity
(fun z ↦ ofReal (transcriptMixtureRealDensity P θ p n z)) =
transcriptLaw P θ p n := by
rw [← transcriptLaw_eq_withDensity_reference P θ p n]
apply withDensity_congr_ae
filter_upwards [transcriptDensity_toReal_ae_eq_mixture_of_interior
P θ p n hp hθ,
Measure.rnDeriv_lt_top (transcriptLaw P θ p n)
(transcriptReferenceMeasure P n)] with z hz htop
rw [← hz]
exact ENNReal.ofReal_toReal (ne_of_lt htop)
theorem integral_transcriptMixtureRealDensity_eq_one
the integral transcript mixture real density eq one assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the integral transcript Mixture Real Density eq one.
Formal statement
hypotheses
conclusion
∫ z, transcriptMixtureRealDensity P θ p n z ∂transcriptReferenceMeasure P n = 1
Proof (Lean source)
lemma integral_transcriptMixtureRealDensity_eq_one {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ : TrialParameter) (p : ℝ) (n : ℕ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) :
∫ z, transcriptMixtureRealDensity P θ p n z
∂transcriptReferenceMeasure P n = 1 := by
have hnonneg := transcriptMixtureRealDensity_nonneg P θ p n hp hθ
letI (x : Fin n → Fin 4) : IsProbabilityMeasure (P.transcript n x) :=
(P.factorizes n x).1
have hcomp (x : Fin n → Fin 4) : Integrable
(transcriptComponentRealDensity P n x) (transcriptReferenceMeasure P n) := by
have h := (integrable_toReal_rnDeriv_mul_iff
(transcriptComponent_ac_reference P n x)
(f := fun _ ↦ (1 : ℝ))).2 (integrable_const (1 : ℝ))
change Integrable (fun z ↦
((P.transcript n x).rnDeriv (transcriptReferenceMeasure P n) z).toReal)
(transcriptReferenceMeasure P n)
simpa only [mul_one] using h
have hint : Integrable (transcriptMixtureRealDensity P θ p n)
(transcriptReferenceMeasure P n) := by
unfold transcriptMixtureRealDensity
exact integrable_finsetSum _ fun x _ ↦ (hcomp x).const_mul _
have hlaw := transcriptLaw_eq_withDensity_mixture P θ p n hp hθ
have hmass := congrArg (fun μ : Measure (Transcript (Z n)) ↦ μ univ) hlaw
rw [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ,
← ofReal_integral_eq_lintegral_ofReal hint (Filter.Eventually.of_forall hnonneg)] at hmass
have hprob : transcriptLaw P θ p n univ = 1 := by
letI : IsProbabilityMeasure (transcriptLaw P θ p n) :=
transcriptLaw_isProbability P θ p hp hθ n
exact measure_univ
rw [hprob] at hmass
exact ENNReal.ofReal_eq_one.mp (by simpa using hmass)
def vtDensity
the vt density is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The vt Density is determined by the displayed parameters.
Definition (Lean source)
parameters
def
vtDensity P θ v p n ell upper a z :
ℝ
given by
def vtDensityDeriv
the vt density deriv is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The vt Density Deriv is determined by the displayed parameters.
Definition (Lean source)
parameters
def
vtDensityDeriv P θ v p n ell upper a z :
ℝ
given by
theorem vtDensity_nonneg
the vt density nonneg assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
0 ≤ vtDensity P θ v p n ell upper a z
Proof (Lean source)
lemma vtDensity_nonneg {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(ell upper : ℝ) (hp : InteriorAssignment p)
(hinterior : ∀ a ∈ Icc ell upper, InteriorMeans (parameterPath θ v a)) :
∀ a z, 0 ≤ vtDensity P θ v p n ell upper a z := by
intro a z
rw [vtDensity]
split_ifs with ha
· exact transcriptMixtureRealDensity_nonneg P _ p n hp (hinterior a ha) z
· exact le_rfl
theorem vtDensity_hasDerivAt
the vt density has deriv at assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
n :
ℕ
ell upper a :
ℝ
ha :
a ∈ Ioo ell upper
z :
Transcript (Z n)
conclusion
Proof (Lean source)
lemma vtDensity_hasDerivAt {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
{ell upper a : ℝ} (ha : a ∈ Ioo ell upper) (z : Transcript (Z n)) :
HasDerivAt (fun b ↦ vtDensity P θ v p n ell upper b z)
(vtDensityDeriv P θ v p n ell upper a z) a := by
have hnhds : Icc ell upper ∈ nhds a :=
Filter.mem_of_superset (isOpen_Ioo.mem_nhds ha) Set.Ioo_subset_Icc_self
have heq : (fun b ↦ vtDensity P θ v p n ell upper b z) =ᶠ[nhds a]
fun b ↦ transcriptMixtureRealDensity P (parameterPath θ v b) p n z := by
filter_upwards [hnhds] with b hb
simp [vtDensity, hb]
have hder := hasDerivAt_transcriptMixtureRealDensity_parameterPath
P θ v p a n z
have haIcc : a ∈ Icc ell upper := ⟨ha.1.le, ha.2.le⟩
rw [vtDensityDeriv, if_pos haIcc]
exact hder.congr_of_eventuallyEq heq
theorem ae_hasDerivAt_vtDensity
The interval-extended likelihood has its declared derivative jointly almost everywhere under parameter measure times the arbitrary-output reference measure. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
∀ᵐ az ∂((parameterMeasure ell upper).prod (transcriptReferenceMeasure P n)), HasDerivAt (fun b ↦ vtDensity P θ v p n ell upper b az.2) (vtDensityDeriv P θ v p n ell upper az.1 az.2) az.1
Proof (Lean source)
lemma ae_hasDerivAt_vtDensity {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(ell upper : ℝ) :
∀ᵐ az ∂((Causalean.Stat.Minimax.ObservationDependentVanTrees.parameterMeasure
ell upper).prod (transcriptReferenceMeasure P n)),
HasDerivAt (fun b ↦ vtDensity P θ v p n ell upper b az.2)
(vtDensityDeriv P θ v p n ell upper az.1 az.2) az.1 := by
let π := Causalean.Stat.Minimax.ObservationDependentVanTrees.parameterMeasure ell upper
have hmem : ∀ᵐ a ∂π, a ∈ Ioo ell upper := by
unfold π Causalean.Stat.Minimax.ObservationDependentVanTrees.parameterMeasure
filter_upwards [ae_restrict_mem measurableSet_Icc,
ae_restrict_of_ae (volume.ae_ne ell),
ae_restrict_of_ae (volume.ae_ne upper)] with a ha haell haupper
exact ⟨lt_of_le_of_ne ha.1 (Ne.symm haell), lt_of_le_of_ne ha.2 haupper⟩
have hlift : ∀ᵐ az ∂(π.prod (transcriptReferenceMeasure P n)),
az.1 ∈ Ioo ell upper :=
Measure.quasiMeasurePreserving_fst.tendsto_ae.eventually hmem
filter_upwards [hlift] with az haz
exact vtDensity_hasDerivAt P θ v p n haz az.2
theorem vtDensity_integral_eq_one
the vt density integral eq one assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
n :
ℕ
ell upper :
ℝ
hp :
hinterior :
a :
ℝ
ha :
a ∈ Icc ell upper
conclusion
∫ z, vtDensity P θ v p n ell upper a z ∂transcriptReferenceMeasure P n = 1
Proof (Lean source)
lemma vtDensity_integral_eq_one {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(ell upper : ℝ) (hp : InteriorAssignment p)
(hinterior : ∀ a ∈ Icc ell upper, InteriorMeans (parameterPath θ v a))
{a : ℝ} (ha : a ∈ Icc ell upper) :
∫ z, vtDensity P θ v p n ell upper a z
∂transcriptReferenceMeasure P n = 1 := by
simp only [vtDensity, if_pos ha]
exact integral_transcriptMixtureRealDensity_eq_one P _ p n hp (hinterior a ha)
theorem vtDensity_integral_hasDerivAt
the vt density integral has deriv at assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
n :
ℕ
ell upper :
ℝ
hp :
hinterior :
a :
ℝ
ha :
a ∈ Ioo ell upper
conclusion
HasDerivAt (fun b ↦ ∫ z, vtDensity P θ v p n ell upper b z ∂transcriptReferenceMeasure P n) 0 a
Proof (Lean source)
lemma vtDensity_integral_hasDerivAt {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(ell upper : ℝ) (hp : InteriorAssignment p)
(hinterior : ∀ a ∈ Icc ell upper, InteriorMeans (parameterPath θ v a))
{a : ℝ} (ha : a ∈ Ioo ell upper) :
HasDerivAt (fun b ↦ ∫ z, vtDensity P θ v p n ell upper b z
∂transcriptReferenceMeasure P n) 0 a := by
have hnhds : Icc ell upper ∈ nhds a :=
Filter.mem_of_superset (isOpen_Ioo.mem_nhds ha) Set.Ioo_subset_Icc_self
have heq : (fun b ↦ ∫ z, vtDensity P θ v p n ell upper b z
∂transcriptReferenceMeasure P n) =ᶠ[nhds a] fun _ ↦ (1 : ℝ) := by
filter_upwards [hnhds] with b hb
exact vtDensity_integral_eq_one P θ v p n ell upper hp hinterior hb
exact (hasDerivAt_const (x := a) (c := (1 : ℝ))).congr_of_eventuallyEq heq
theorem integrable_transcriptMixtureRealDerivative
the integrable transcript mixture real derivative assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma integrable_transcriptMixtureRealDerivative {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ) :
Integrable (transcriptMixtureRealDerivative P θ v p n)
(transcriptReferenceMeasure P n) := by
letI (x : Fin n → Fin 4) : IsProbabilityMeasure (P.transcript n x) :=
(P.factorizes n x).1
have hcomp (x : Fin n → Fin 4) : Integrable
(transcriptComponentRealDensity P n x) (transcriptReferenceMeasure P n) := by
have h := (integrable_toReal_rnDeriv_mul_iff
(transcriptComponent_ac_reference P n x)
(f := fun _ ↦ (1 : ℝ))).2 (integrable_const (1 : ℝ))
change Integrable (fun z ↦
((P.transcript n x).rnDeriv (transcriptReferenceMeasure P n) z).toReal)
(transcriptReferenceMeasure P n)
simpa only [mul_one] using h
unfold transcriptMixtureRealDerivative
exact integrable_finsetSum _ fun x _ ↦ (hcomp x).const_mul _
theorem vtDensity_integrable
the vt density integrable assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
n :
ℕ
ell upper :
ℝ
a :
ℝ
ha :
a ∈ Icc ell upper
conclusion
Proof (Lean source)
lemma vtDensity_integrable {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(ell upper : ℝ) {a : ℝ} (ha : a ∈ Icc ell upper) :
Integrable (vtDensity P θ v p n ell upper a)
(transcriptReferenceMeasure P n) := by
unfold vtDensity
letI (x : Fin n → Fin 4) : IsProbabilityMeasure (P.transcript n x) :=
(P.factorizes n x).1
have hraw : Integrable
(transcriptMixtureRealDensity P (parameterPath θ v a) p n)
(transcriptReferenceMeasure P n) := by
unfold transcriptMixtureRealDensity
exact integrable_finsetSum _ fun x _ ↦ (by
have h := (integrable_toReal_rnDeriv_mul_iff
(transcriptComponent_ac_reference P n x)
(f := fun _ ↦ (1 : ℝ))).2 (integrable_const (1 : ℝ))
have hc : Integrable (transcriptComponentRealDensity P n x)
(transcriptReferenceMeasure P n) := by
change Integrable (fun z ↦
((P.transcript n x).rnDeriv (transcriptReferenceMeasure P n) z).toReal)
(transcriptReferenceMeasure P n)
simpa only [mul_one] using h
exact hc.const_mul _)
simpa only [if_pos ha] using hraw
theorem vtDensityDeriv_integrable
the vt density deriv integrable assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
n :
ℕ
ell upper :
ℝ
a :
ℝ
ha :
a ∈ Icc ell upper
conclusion
Proof (Lean source)
lemma vtDensityDeriv_integrable {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(ell upper : ℝ) {a : ℝ} (ha : a ∈ Icc ell upper) :
Integrable (vtDensityDeriv P θ v p n ell upper a)
(transcriptReferenceMeasure P n) := by
unfold vtDensityDeriv
simpa only [if_pos ha] using integrable_transcriptMixtureRealDerivative P
(parameterPath θ v a) v p n
theorem transcriptDensityPath_contDiff
the transcript density path cont diff assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma transcriptDensityPath_contDiff {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(z : Transcript (Z n)) :
ContDiff ℝ 1 (fun a ↦
transcriptMixtureRealDensity P (parameterPath θ v a) p n z) := by
unfold transcriptMixtureRealDensity
apply ContDiff.sum
intro x hx
apply ContDiff.mul
· unfold inputPathProbability
apply contDiff_prod
intro i hi
generalize hxi : x i = j
fin_cases j <;> simp [piTheta, parameterPath, hxi] <;> fun_prop
· fun_prop
theorem vtDensity_absolutelyContinuous
the vt density absolutely continuous assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p :
ℝ
n :
ℕ
ell upper :
ℝ
hellu :
ell < upper
z :
Transcript (Z n)
conclusion
AbsolutelyContinuousOnInterval (fun a ↦ vtDensity P θ v p n ell upper a z) ell upper
Proof (Lean source)
lemma vtDensity_absolutelyContinuous {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(ell upper : ℝ) (hellu : ell < upper) (z : Transcript (Z n)) :
AbsolutelyContinuousOnInterval
(fun a ↦ vtDensity P θ v p n ell upper a z) ell upper := by
have hraw : AbsolutelyContinuousOnInterval
(fun a ↦ transcriptMixtureRealDensity P (parameterPath θ v a) p n z)
ell upper :=
(transcriptDensityPath_contDiff P θ v p n z).contDiffOn
|>.absolutelyContinuousOnInterval
rw [absolutelyContinuousOnInterval_iff] at hraw ⊢
intro δ hδ
obtain ⟨η, hη, hraw⟩ := hraw δ hδ
refine ⟨η, hη, ?_⟩
rintro ⟨m, I⟩ hI hlen
have hbound : ∀ i ∈ range m,
(I i).1 ∈ Icc ell upper ∧ (I i).2 ∈ Icc ell upper := by
intro i hi
have hi' := hI.1 i hi
simpa [Set.uIcc_of_le hellu.le] using hi'
convert hraw (m, I) hI hlen using 1
apply Finset.sum_congr rfl
intro i hi
rw [vtDensity, if_pos (hbound i hi).1,
vtDensity, if_pos (hbound i hi).2]
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialVanTreesAssembly 2 declarations This file assembles the paper's already verified likelihood, prior, target, and product-integral adapters into a finite-sample Bayes inequality.
Finite-sample sequential van Trees assembly
This file assembles the paper's already verified likelihood, prior, target, and product-integral adapters into a finite-sample Bayes inequality. Transcript Fisher integrability and its numerical upper bound remain explicit premises; they are the score-projection and Doob-information obligations.
theorem integrable_scoreSqField_of_prior_fisher_cross
Prior-score, likelihood-score, and cross-term integrability imply integrability of the guarded joint-score square. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
X :
Type*
w dw :
ℝ → ℝ
q dq :
ℝ → X → ℝ
mu :
Measure (ℝ × X)
hw :
∀ a, 0 ≤ w a
hq :
∀ a x, 0 ≤ q a x
hprior :
Integrable (fun z : ℝ × X => w z.1 * q z.1 z.2 * (priorScore w dw z.1) ^ 2) mu
hfisher :
Integrable (fun z : ℝ × X => w z.1 * q z.1 z.2 * (likelihoodScore q dq z.1 z.2) ^ 2) mu
hcross :
Integrable (fun z : ℝ × X => w z.1 * q z.1 z.2 * (priorScore w dw z.1 * likelihoodScore q dq z.1 z.2)) mu
conclusion
Integrable (scoreSqField w dw q dq) mu
Proof (Lean source)
lemma integrable_scoreSqField_of_prior_fisher_cross
{X : Type*} [MeasurableSpace X]
(w dw : ℝ → ℝ) (q dq : ℝ → X → ℝ) (mu : Measure (ℝ × X))
(hw : ∀ a, 0 ≤ w a) (hq : ∀ a x, 0 ≤ q a x)
(hprior : Integrable (fun z : ℝ × X =>
w z.1 * q z.1 z.2 * (priorScore w dw z.1) ^ 2) mu)
(hfisher : Integrable (fun z : ℝ × X =>
w z.1 * q z.1 z.2 * (likelihoodScore q dq z.1 z.2) ^ 2) mu)
(hcross : Integrable (fun z : ℝ × X =>
w z.1 * q z.1 z.2 *
(priorScore w dw z.1 * likelihoodScore q dq z.1 z.2)) mu) :
Integrable (scoreSqField w dw q dq) mu := by
have hscore_expand (a : ℝ) (x : X) :
scoreSqField w dw q dq (a, x) =
w a * q a x * (priorScore w dw a) ^ 2 +
(w a * q a x * (likelihoodScore q dq a x) ^ 2 +
2 * (w a * q a x *
(priorScore w dw a * likelihoodScore q dq a x))) := by
by_cases hwpos : 0 < w a
· by_cases hqpos : 0 < q a x
· simp only [scoreSqField, jointScore, jointDensity, priorScore,
likelihoodScore, hwpos, hqpos, mul_pos, if_true]
field_simp
ring
· have hqzero : q a x = 0 :=
le_antisymm (le_of_not_gt hqpos) (hq a x)
simp [scoreSqField, jointScore, jointDensity, hqzero]
· have hwzero : w a = 0 :=
le_antisymm (le_of_not_gt hwpos) (hw a)
simp [scoreSqField, jointScore, jointDensity, hwzero]
have hsum := hprior.add (hfisher.add (hcross.const_mul 2))
exact hsum.congr (Filter.Eventually.of_forall fun z =>
(hscore_expand z.1 z.2).symm)
theorem sequential_vanTrees_bayes_lower_bound_of_fisher_control
For a compact scalar path, finite weighted transcript Fisher information and an explicit bound on its prior average give the finite-sample Bayes risk lower bound. The Fisher premises are precisely the part supplied later by score projection and the sequential Doob bound. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p R I :
ℝ
n :
ℕ
hR :
0 < R
hp :
hinterior :
∀ a ∈ Icc (-R) R, InteriorMeans (parameterPath θ v a)
T :
Transcript (Z n) → ℝ
hT :
herrorSqInt :
Integrable (errorSqField (sequentialVTPrior R) (vtDensity P θ v p n (-R) R) (vtTarget θ v) T) ((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n))
hfisherSqInt :
Integrable (fun z : ℝ × Transcript (Z n) => sequentialVTPrior R z.1 * vtDensity P θ v p n (-R) R z.1 z.2 * (likelihoodScore (vtDensity P θ v p n (-R) R) (vtDensityDeriv P θ v p n (-R) R) z.1 z.2) ^ 2) ((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n))
hfisher_le :
(∫ a, sequentialVTPrior R a * fisherInformation (transcriptReferenceMeasure P n) (vtDensity P θ v p n (-R) R) (vtDensityDeriv P θ v p n (-R) R) a ∂parameterMeasure (-R) R)
≤ I
hinfoPos :
+ ∫ a, sequentialVTPrior R a * fisherInformation (transcriptReferenceMeasure P n) (vtDensity P θ v p n (-R) R) (vtDensityDeriv P θ v p n (-R) R) a ∂parameterMeasure (-R) R
conclusion
contrast v ^ 2 / (I + 40 / R ^ 2)
≤ ∫ z, errorSqField (sequentialVTPrior R) (vtDensity P θ v p n (-R) R) (vtTarget θ v) T z ∂((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n))
Proof (Lean source)
lemma sequential_vanTrees_bayes_lower_bound_of_fisher_control
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p R I : ℝ) (n : ℕ)
(hR : 0 < R) (hp : InteriorAssignment p)
(hinterior : ∀ a ∈ Icc (-R) R,
InteriorMeans (parameterPath θ v a))
(T : Transcript (Z n) → ℝ) (hT : Measurable T)
(herrorSqInt : Integrable
(errorSqField (sequentialVTPrior R)
(vtDensity P θ v p n (-R) R) (vtTarget θ v) T)
((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n)))
(hfisherSqInt : Integrable (fun z : ℝ × Transcript (Z n) =>
sequentialVTPrior R z.1 * vtDensity P θ v p n (-R) R z.1 z.2 *
(likelihoodScore (vtDensity P θ v p n (-R) R)
(vtDensityDeriv P θ v p n (-R) R) z.1 z.2) ^ 2)
((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n)))
(hfisher_le :
(∫ a, sequentialVTPrior R a *
fisherInformation (transcriptReferenceMeasure P n)
(vtDensity P θ v p n (-R) R)
(vtDensityDeriv P θ v p n (-R) R) a
∂parameterMeasure (-R) R) ≤ I)
(hinfoPos : 0 < priorInformation (-R) R
(sequentialVTPrior R) (sequentialVTPriorDeriv R) +
∫ a, sequentialVTPrior R a *
fisherInformation (transcriptReferenceMeasure P n)
(vtDensity P θ v p n (-R) R)
(vtDensityDeriv P θ v p n (-R) R) a
∂parameterMeasure (-R) R) :
contrast v ^ 2 / (I + 40 / R ^ 2) ≤
∫ z, errorSqField (sequentialVTPrior R)
(vtDensity P θ v p n (-R) R) (vtTarget θ v) T z
∂((parameterMeasure (-R) R).prod
(transcriptReferenceMeasure P n)) := by
let mu := transcriptReferenceMeasure P n
let q := vtDensity P θ v p n (-R) R
let dq := vtDensityDeriv P θ v p n (-R) R
let w := sequentialVTPrior R
let dw := sequentialVTPriorDeriv R
let g := vtTarget (X := Transcript (Z n)) θ v
let dg := vtTargetDeriv (X := Transcript (Z n)) v
have hprior := sequentialVTPrior_facts hR
have hmeas := sequentialVanTreesProductMeasurable P θ v p R n hR T hT
have hmem : ∀ᵐ a ∂parameterMeasure (-R) R, a ∈ Icc (-R) R := by
unfold parameterMeasure
exact ae_restrict_mem measurableSet_Icc
have hnormAE : ∀ᵐ a ∂parameterMeasure (-R) R,
∫ z, q a z ∂mu = 1 := by
filter_upwards [hmem] with a ha
exact vtDensity_integral_eq_one P θ v p n (-R) R hp hinterior ha
have hqintAE : ∀ᵐ a ∂parameterMeasure (-R) R,
Integrable (q a) mu := by
filter_upwards [hmem] with a ha
exact vtDensity_integrable P θ v p n (-R) R ha
have hjointInt : Integrable (jointDensity w q)
((parameterMeasure (-R) R).prod mu) := by
apply integrable_jointDensity_of_normalization w q
· exact hprior.nonneg
· exact vtDensity_nonneg P θ v p n (-R) R hp hinterior
· exact hqintAE
· exact hnormAE
· unfold w sequentialVTPrior
exact smoothPrior_integrable_parameterMeasure (by linarith)
· exact ((hprior.contDiff.continuous.measurable.comp measurable_fst).mul
(measurable_vtDensity_joint P θ v p n (-R) R)).aestronglyMeasurable
have hsensitivityInt : Integrable (sensitivityField w q dg)
((parameterMeasure (-R) R).prod mu) :=
integrable_sequentialSensitivity_of_jointDensity w q v _ hjointInt
have hpriorJointInt : Integrable (fun z : ℝ × Transcript (Z n) =>
w z.1 * q z.1 z.2 * (priorScore w dw z.1) ^ 2)
((parameterMeasure (-R) R).prod mu) := by
apply integrable_priorJointSq_of_normalization w dw q
· exact hprior.nonneg
· exact vtDensity_nonneg P θ v p n (-R) R hp hinterior
· exact hqintAE
· exact hnormAE
· exact hprior.scoreSqIntegrable
· exact hmeas.priorJointSq.aestronglyMeasurable
have hwmeas : Measurable w := hprior.contDiff.continuous.measurable
have hdwmeas : Measurable dw := by
unfold dw sequentialVTPriorDeriv smoothPriorDeriv
apply Measurable.ite
· exact measurableSet_lt
(continuous_abs.comp (continuous_id.sub continuous_const)).measurable
measurable_const
· fun_prop
· exact measurable_const
have hqmeas : Measurable (fun z : ℝ × Transcript (Z n) => q z.1 z.2) :=
measurable_vtDensity_joint P θ v p n (-R) R
have hdqmeas : Measurable (fun z : ℝ × Transcript (Z n) => dq z.1 z.2) :=
measurable_vtDensityDeriv_joint P θ v p n (-R) R
have hcrossInt : Integrable (fun z : ℝ × Transcript (Z n) =>
w z.1 * q z.1 z.2 *
(priorScore w dw z.1 * likelihoodScore q dq z.1 z.2))
((parameterMeasure (-R) R).prod mu) := by
exact integrable_priorLikelihoodCross_of_squares w dw q dq _
hwmeas hdwmeas hqmeas hdqmeas hprior.nonneg
(vtDensity_nonneg P θ v p n (-R) R hp hinterior)
hpriorJointInt hfisherSqInt
have hscoreSqInt : Integrable (scoreSqField w dw q dq)
((parameterMeasure (-R) R).prod mu) :=
integrable_scoreSqField_of_prior_fisher_cross w dw q dq _
hprior.nonneg (vtDensity_nonneg P θ v p n (-R) R hp hinterior)
hpriorJointInt hfisherSqInt hcrossInt
have hscoreMeas : Measurable (jointScore w dw q dq) := by
unfold jointScore jointDensity
exact Measurable.ite
(measurableSet_lt measurable_const
((hwmeas.comp measurable_fst).mul hqmeas))
(((hdwmeas.comp measurable_fst).mul hqmeas |>.add
((hwmeas.comp measurable_fst).mul hdqmeas)).div
((hwmeas.comp measurable_fst).mul hqmeas)) measurable_const
have herrorScoreInt : Integrable (errorScoreField w dw q dq g T)
((parameterMeasure (-R) R).prod mu) :=
integrable_errorScoreField_of_squares w dw q dq g T _ hwmeas hqmeas
(by unfold g vtTarget; fun_prop) hT hscoreMeas hprior.nonneg
(vtDensity_nonneg P θ v p n (-R) R hp hinterior)
herrorSqInt hscoreSqInt
have hwzero : ∀ a, w a = 0 → dw a = 0 := by
intro a ha
exact derivative_eq_zero_of_nonnegative_of_eq_zero hprior.nonneg
(hprior.hasDeriv a) ha
have hqzero : ∀ᵐ z ∂((parameterMeasure (-R) R).prod mu),
q z.1 z.2 = 0 → dq z.1 z.2 = 0 := by
filter_upwards [ae_hasDerivAt_vtDensity P θ v p n (-R) R] with z hz
exact derivative_eq_zero_of_nonnegative_of_eq_zero
(fun a => vtDensity_nonneg P θ v p n (-R) R hp hinterior a z.2)
hz
have hbalanceInt : Integrable (derivativeBalanceField w dw q dq g dg T)
((parameterMeasure (-R) R).prod mu) := by
apply (herrorScoreInt.sub hsensitivityInt).congr
filter_upwards [hqzero] with z hz
have heq : derivativeBalanceField w dw q dq g dg T z =
errorScoreField w dw q dq g T z - sensitivityField w q dg z := by
rw [errorScoreField_eq_numerator (hprior.nonneg z.1)
(vtDensity_nonneg P θ v p n (-R) R hp hinterior z.1 z.2)
(hwzero z.1) hz]
unfold derivativeBalanceField sensitivityField jointDensity
ring
exact heq.symm
have hboundary : ∀ᵐ z ∂mu,
w R * q R z * (T z - g R z) = 0 ∧
w (-R) * q (-R) z * (T z - g (-R) z) = 0 := by
have hend := sequentialVTPrior_endpoints hR
filter_upwards with z
simp [w, hend.1, hend.2]
have hvan := observation_dependent_van_trees mu (by linarith : -R < R)
w dw q dq g dg T hprior.contDiff hprior.hasDeriv
hprior.nonneg
(vtDensity_nonneg P θ v p n (-R) R hp hinterior)
(fun a ha => vtDensity_integral_eq_one P θ v p n (-R) R hp hinterior ha)
(fun a ha => by
have hzero : ∫ z, dq a z ∂mu = 0 := by
have haIcc : a ∈ Icc (-R) R := ⟨ha.1.le, ha.2.le⟩
simpa [dq, mu, vtDensityDeriv, haIcc] using
integral_transcriptMixtureRealDerivative_eq_zero P
(parameterPath θ v a) v p n
rw [hzero]
exact vtDensity_integral_hasDerivAt P θ v p n (-R) R hp hinterior ha)
(Filter.Eventually.of_forall
(vtDensity_absolutelyContinuous P θ v p n (-R) R (by linarith)))
(Filter.Eventually.of_forall (fun z => vtTarget_absolutelyContinuous θ v z (-R) R))
(ae_hasDerivAt_vtDensity P θ v p n (-R) R)
(Filter.Eventually.of_forall (fun z => hasDerivAt_vtTarget θ v z.1 z.2))
hboundary hbalanceInt
hmeas.errorScore.aestronglyMeasurable herrorScoreInt
hsensitivityInt
hmeas.errorSq.aestronglyMeasurable herrorSqInt
hmeas.scoreSq.aestronglyMeasurable hscoreSqInt
hpriorJointInt
hfisherSqInt
hcrossInt hinfoPos
have hsensitivity :
(∫ z, sensitivityField w q dg z
∂((parameterMeasure (-R) R).prod mu)) = contrast v :=
integral_sequentialSensitivity_eq w q v hprior.normalized hnormAE hsensitivityInt
have hdenom : priorInformation (-R) R w dw +
∫ a, w a * fisherInformation mu q dq a ∂parameterMeasure (-R) R ≤
I + 40 / R ^ 2 := by
rw [hprior.information]
linarith
have hcompare : contrast v ^ 2 / (I + 40 / R ^ 2) ≤
contrast v ^ 2 / (priorInformation (-R) R w dw +
∫ a, w a * fisherInformation mu q dq a ∂parameterMeasure (-R) R) :=
div_le_div₀ (sq_nonneg _) le_rfl hinfoPos hdenom
apply hcompare.trans
simpa [mu, q, dq, w, dw, g, dg, hsensitivity] using hvan
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialVanTreesIntegrals 10 declarations Product integral identities for the sequential van Trees adapter
Product integral identities for the sequential van Trees adapter
theorem integrable_jointDensity_of_normalization
Integrable normalized likelihood sections lift an integrable nonnegative prior density to an integrable joint density. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
X :
ell upper :
ℝ
w :
ℝ → ℝ
p :
ℝ → X → ℝ
hw_nonneg :
∀ a, 0 ≤ w a
hp_nonneg :
∀ a x, 0 ≤ p a x
hpint :
∀ᵐ a ∂parameterMeasure ell upper, Integrable (fun x ↦ p a x) mu
hnorm :
∀ᵐ a ∂parameterMeasure ell upper, ∫ x, p a x ∂mu = 1
hwint :
Integrable w (parameterMeasure ell upper)
hmeas :
conclusion
Proof (Lean source)
lemma integrable_jointDensity_of_normalization {X : Type*} [MeasurableSpace X]
{mu : Measure X} [SigmaFinite mu] {ell upper : ℝ}
(w : ℝ → ℝ) (p : ℝ → X → ℝ)
(hw_nonneg : ∀ a, 0 ≤ w a) (hp_nonneg : ∀ a x, 0 ≤ p a x)
(hpint : ∀ᵐ a ∂parameterMeasure ell upper, Integrable (fun x ↦ p a x) mu)
(hnorm : ∀ᵐ a ∂parameterMeasure ell upper, ∫ x, p a x ∂mu = 1)
(hwint : Integrable w (parameterMeasure ell upper))
(hmeas : AEStronglyMeasurable (jointDensity w p)
((parameterMeasure ell upper).prod mu)) :
Integrable (jointDensity w p) ((parameterMeasure ell upper).prod mu) := by
rw [integrable_prod_iff hmeas]
constructor
· filter_upwards [hpint] with a ha
exact (ha.const_mul (w a)).congr (by
filter_upwards with x
rfl)
· apply hwint.congr
filter_upwards [hnorm] with a ha
symm
unfold jointDensity
calc
∫ x, ‖w a * p a x‖ ∂mu = ∫ x, w a * p a x ∂mu := by
apply integral_congr_ae
filter_upwards with x
rw [Real.norm_of_nonneg (mul_nonneg (hw_nonneg a) (hp_nonneg a x))]
_ = w a * ∫ x, p a x ∂mu :=
integral_const_mul (μ := mu) (w a) (fun x ↦ p a x)
_ = w a := by rw [ha, mul_one]
theorem integrable_sequentialSensitivity_of_jointDensity
Joint-density integrability gives integrability of the affine target sensitivity field. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
X :
Type*
w :
ℝ → ℝ
p :
ℝ → X → ℝ
v :
mu :
Measure (ℝ × X)
hjoint :
Integrable (jointDensity w p) mu
conclusion
Proof (Lean source)
lemma integrable_sequentialSensitivity_of_jointDensity {X : Type*}
[MeasurableSpace X] (w : ℝ → ℝ) (p : ℝ → X → ℝ)
(v : TrialParameter) (mu : Measure (ℝ × X))
(hjoint : Integrable (jointDensity w p) mu) :
Integrable (sensitivityField w p (vtTargetDeriv (X := X) v)) mu := by
have h := hjoint.const_mul (contrast v)
apply h.congr
filter_upwards with z
unfold sensitivityField vtTargetDeriv
rfl
theorem integrable_priorJointSq_of_normalization
Prior-score integrability and normalized integrable likelihood sections imply integrability of the lifted prior-score square on product space. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
X :
ell upper :
ℝ
w dw :
ℝ → ℝ
p :
ℝ → X → ℝ
hw_nonneg :
∀ a, 0 ≤ w a
hp_nonneg :
∀ a x, 0 ≤ p a x
hpint :
∀ᵐ a ∂parameterMeasure ell upper, Integrable (fun x ↦ p a x) mu
hnorm :
∀ᵐ a ∂parameterMeasure ell upper, ∫ x, p a x ∂mu = 1
hprior :
hmeas :
AEStronglyMeasurable (fun z : ℝ × X ↦ w z.1 * p z.1 z.2 * (priorScore w dw z.1) ^ 2) ((parameterMeasure ell upper).prod mu)
conclusion
Integrable (fun z : ℝ × X ↦ w z.1 * p z.1 z.2 * (priorScore w dw z.1) ^ 2) ((parameterMeasure ell upper).prod mu)
Proof (Lean source)
lemma integrable_priorJointSq_of_normalization {X : Type*} [MeasurableSpace X]
{mu : Measure X} [SigmaFinite mu] {ell upper : ℝ}
(w dw : ℝ → ℝ) (p : ℝ → X → ℝ)
(hw_nonneg : ∀ a, 0 ≤ w a) (hp_nonneg : ∀ a x, 0 ≤ p a x)
(hpint : ∀ᵐ a ∂parameterMeasure ell upper, Integrable (fun x ↦ p a x) mu)
(hnorm : ∀ᵐ a ∂parameterMeasure ell upper, ∫ x, p a x ∂mu = 1)
(hprior : Integrable (fun a ↦ w a * (priorScore w dw a) ^ 2)
(parameterMeasure ell upper))
(hmeas : AEStronglyMeasurable (fun z : ℝ × X ↦
w z.1 * p z.1 z.2 * (priorScore w dw z.1) ^ 2)
((parameterMeasure ell upper).prod mu)) :
Integrable (fun z : ℝ × X ↦
w z.1 * p z.1 z.2 * (priorScore w dw z.1) ^ 2)
((parameterMeasure ell upper).prod mu) := by
rw [integrable_prod_iff hmeas]
constructor
· filter_upwards [hpint] with a ha
have hc := ha.const_mul (w a * (priorScore w dw a) ^ 2)
apply hc.congr
filter_upwards with x
ring
· apply hprior.congr
filter_upwards [hnorm] with a ha
symm
calc
∫ x, ‖w a * p a x * (priorScore w dw a) ^ 2‖ ∂mu =
∫ x, w a * p a x * (priorScore w dw a) ^ 2 ∂mu := by
apply integral_congr_ae
filter_upwards with x
rw [Real.norm_of_nonneg (mul_nonneg
(mul_nonneg (hw_nonneg a) (hp_nonneg a x)) (sq_nonneg _))]
_ =
(w a * (priorScore w dw a) ^ 2) * ∫ x, p a x ∂mu := by
rw [← integral_const_mul (μ := mu)
(w a * (priorScore w dw a) ^ 2) (fun x ↦ p a x)]
apply integral_congr_ae
filter_upwards with x
ring
_ = w a * (priorScore w dw a) ^ 2 := by rw [ha, mul_one]
theorem integral_priorJointSq_eq_priorInformation
A normalized likelihood lifts the prior score square from parameter space to parameter--observation product space without changing its integral. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
X :
ell upper :
ℝ
w dw :
ℝ → ℝ
p :
ℝ → X → ℝ
hnorm :
∀ᵐ a ∂parameterMeasure ell upper, ∫ x, p a x ∂mu = 1
hint :
Integrable (fun z : ℝ × X => w z.1 * p z.1 z.2 * (priorScore w dw z.1) ^ 2) ((parameterMeasure ell upper).prod mu)
conclusion
= priorInformation ell upper w dw
Proof (Lean source)
-- keep: reusable sequential-law, Fisher-information, or van-Trees bridge for related adaptive experiments
lemma integral_priorJointSq_eq_priorInformation {X : Type*} [MeasurableSpace X]
{mu : Measure X} [SigmaFinite mu] {ell upper : ℝ}
(w dw : ℝ → ℝ) (p : ℝ → X → ℝ)
(hnorm : ∀ᵐ a ∂parameterMeasure ell upper, ∫ x, p a x ∂mu = 1)
(hint : Integrable (fun z : ℝ × X =>
w z.1 * p z.1 z.2 * (priorScore w dw z.1) ^ 2)
((parameterMeasure ell upper).prod mu)) :
∫ z, w z.1 * p z.1 z.2 * (priorScore w dw z.1) ^ 2
∂((parameterMeasure ell upper).prod mu) =
priorInformation ell upper w dw := by
letI : IsFiniteMeasure (parameterMeasure ell upper) := by
unfold parameterMeasure
infer_instance
rw [integral_prod _ hint]
unfold priorInformation
apply integral_congr_ae
filter_upwards [hnorm] with a ha
calc
∫ x, w a * p a x * (priorScore w dw a) ^ 2 ∂mu =
∫ x, (w a * (priorScore w dw a) ^ 2) * p a x ∂mu := by
apply integral_congr_ae
filter_upwards with x
ring
_ = (w a * (priorScore w dw a) ^ 2) * ∫ x, p a x ∂mu :=
integral_const_mul (μ := mu) (w a * (priorScore w dw a) ^ 2)
(fun x ↦ p a x)
_ = w a * (priorScore w dw a) ^ 2 := by rw [ha, mul_one]
theorem integral_fisherJointSq_eq
The product integral of the weighted likelihood-score square is the prior average of conditional Fisher information. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
X :
ell upper :
ℝ
w :
ℝ → ℝ
p dp :
ℝ → X → ℝ
hint :
Integrable (fun z : ℝ × X => w z.1 * p z.1 z.2 * (likelihoodScore p dp z.1 z.2) ^ 2) ((parameterMeasure ell upper).prod mu)
conclusion
= ∫ a, w a * fisherInformation mu p dp a ∂parameterMeasure ell upper
Proof (Lean source)
-- keep: reusable sequential-law, Fisher-information, or van-Trees bridge for related adaptive experiments
lemma integral_fisherJointSq_eq {X : Type*} [MeasurableSpace X]
{mu : Measure X} [SigmaFinite mu] {ell upper : ℝ}
(w : ℝ → ℝ) (p dp : ℝ → X → ℝ)
(hint : Integrable (fun z : ℝ × X =>
w z.1 * p z.1 z.2 * (likelihoodScore p dp z.1 z.2) ^ 2)
((parameterMeasure ell upper).prod mu)) :
∫ z, w z.1 * p z.1 z.2 * (likelihoodScore p dp z.1 z.2) ^ 2
∂((parameterMeasure ell upper).prod mu) =
∫ a, w a * fisherInformation mu p dp a ∂parameterMeasure ell upper := by
letI : IsFiniteMeasure (parameterMeasure ell upper) := by
unfold parameterMeasure
infer_instance
rw [integral_prod _ hint]
apply integral_congr_ae
filter_upwards with a
unfold fisherInformation
calc
∫ x, w a * p a x * (likelihoodScore p dp a x) ^ 2 ∂mu =
∫ x, w a * (p a x * (likelihoodScore p dp a x) ^ 2) ∂mu := by
apply integral_congr_ae
filter_upwards with x
ring
_ = w a * ∫ x, p a x * (likelihoodScore p dp a x) ^ 2 ∂mu :=
integral_const_mul (μ := mu) (w a)
(fun x ↦ p a x * (likelihoodScore p dp a x) ^ 2)
theorem integral_priorLikelihoodCross_eq_zero
Conditional centering of the likelihood score makes the prior--likelihood score cross term integrate to zero. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
X :
ell upper :
ℝ
w dw :
ℝ → ℝ
p dp :
ℝ → X → ℝ
hcenter :
∀ᵐ a ∂parameterMeasure ell upper, ∫ x, likelihoodScore p dp a x * p a x ∂mu = 0
hint :
Integrable (fun z : ℝ × X => w z.1 * p z.1 z.2 * (priorScore w dw z.1 * likelihoodScore p dp z.1 z.2)) ((parameterMeasure ell upper).prod mu)
conclusion
∫ z, w z.1 * p z.1 z.2 * (priorScore w dw z.1 * likelihoodScore p dp z.1 z.2) ∂((parameterMeasure ell upper).prod mu)
= 0
Proof (Lean source)
-- keep: reusable sequential-law, Fisher-information, or van-Trees bridge for related adaptive experiments
lemma integral_priorLikelihoodCross_eq_zero {X : Type*} [MeasurableSpace X]
{mu : Measure X} [SigmaFinite mu] {ell upper : ℝ}
(w dw : ℝ → ℝ) (p dp : ℝ → X → ℝ)
(hcenter : ∀ᵐ a ∂parameterMeasure ell upper,
∫ x, likelihoodScore p dp a x * p a x ∂mu = 0)
(hint : Integrable (fun z : ℝ × X =>
w z.1 * p z.1 z.2 *
(priorScore w dw z.1 * likelihoodScore p dp z.1 z.2))
((parameterMeasure ell upper).prod mu)) :
∫ z, w z.1 * p z.1 z.2 *
(priorScore w dw z.1 * likelihoodScore p dp z.1 z.2)
∂((parameterMeasure ell upper).prod mu) = 0 := by
letI : IsFiniteMeasure (parameterMeasure ell upper) := by
unfold parameterMeasure
infer_instance
rw [integral_prod _ hint]
calc
∫ a, (∫ x, w a * p a x *
(priorScore w dw a * likelihoodScore p dp a x) ∂mu)
∂parameterMeasure ell upper =
∫ a, 0 ∂parameterMeasure ell upper := by
apply integral_congr_ae
filter_upwards [hcenter] with a ha
calc
∫ x, w a * p a x *
(priorScore w dw a * likelihoodScore p dp a x) ∂mu =
∫ x, (w a * priorScore w dw a) *
(likelihoodScore p dp a x * p a x) ∂mu := by
apply integral_congr_ae
filter_upwards with x
ring
_ = (w a * priorScore w dw a) *
∫ x, likelihoodScore p dp a x * p a x ∂mu :=
integral_const_mul (μ := mu) (w a * priorScore w dw a)
(fun x ↦ likelihoodScore p dp a x * p a x)
_ = 0 := by rw [ha, mul_zero]
_ = 0 := integral_zero ℝ ℝ
theorem integral_sequentialSensitivity_eq
For the affine contrast target, normalized prior and likelihood make the integrated sensitivity equal the directional contrast. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
X :
ell upper :
ℝ
w :
ℝ → ℝ
p :
ℝ → X → ℝ
v :
hprior :
∫ a, w a ∂parameterMeasure ell upper = 1
hnorm :
∀ᵐ a ∂parameterMeasure ell upper, ∫ x, p a x ∂mu = 1
hint :
conclusion
= contrast v
Proof (Lean source)
lemma integral_sequentialSensitivity_eq {X : Type*} [MeasurableSpace X]
{mu : Measure X} [SigmaFinite mu] {ell upper : ℝ}
(w : ℝ → ℝ) (p : ℝ → X → ℝ) (v : TrialParameter)
(hprior : ∫ a, w a ∂parameterMeasure ell upper = 1)
(hnorm : ∀ᵐ a ∂parameterMeasure ell upper, ∫ x, p a x ∂mu = 1)
(hint : Integrable (sensitivityField w p (vtTargetDeriv (X := X) v))
((parameterMeasure ell upper).prod mu)) :
∫ z, sensitivityField w p (vtTargetDeriv (X := X) v) z
∂((parameterMeasure ell upper).prod mu) = contrast v := by
letI : IsFiniteMeasure (parameterMeasure ell upper) := by
unfold parameterMeasure
infer_instance
rw [integral_prod _ hint]
have hinner : (fun a => ∫ x, sensitivityField w p (vtTargetDeriv (X := X) v) (a,x) ∂mu)
=ᵐ[parameterMeasure ell upper] fun a => contrast v * w a := by
filter_upwards [hnorm] with a ha
unfold sensitivityField vtTargetDeriv jointDensity
calc
∫ x, contrast v * (w a * p a x) ∂mu =
∫ x, (contrast v * w a) * p a x ∂mu := by
apply integral_congr_ae
filter_upwards with x
ring
_ = (contrast v * w a) * ∫ x, p a x ∂mu :=
integral_const_mul (μ := mu) (contrast v * w a) (fun x ↦ p a x)
_ = contrast v * w a := by rw [ha, mul_one]
rw [integral_congr_ae hinner]
calc
∫ a, contrast v * w a ∂parameterMeasure ell upper =
contrast v * ∫ a, w a ∂parameterMeasure ell upper :=
integral_const_mul (μ := parameterMeasure ell upper) (contrast v) w
_ = contrast v := by rw [hprior, mul_one]
theorem derivativeBalanceField_eq_errorScore_sub_sensitivity
Under the usual zero-density derivative conditions, the derivative-balance field is exactly error-score minus sensitivity. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
X :
Type*
w dw :
ℝ → ℝ
p dp g dg :
ℝ → X → ℝ
T :
X → ℝ
hw :
∀ a, 0 ≤ w a
hp :
∀ a x, 0 ≤ p a x
hwzero :
∀ a
if
w a = 0
then
dw a = 0
hpzero :
∀ a x
if
p a x = 0
then
dp a x = 0
conclusion
derivativeBalanceField w dw p dp g dg T
= fun z => errorScoreField w dw p dp g T z - sensitivityField w p dg z
Proof (Lean source)
lemma derivativeBalanceField_eq_errorScore_sub_sensitivity {X : Type*}
(w dw : ℝ → ℝ) (p dp g dg : ℝ → X → ℝ) (T : X → ℝ)
(hw : ∀ a, 0 ≤ w a) (hp : ∀ a x, 0 ≤ p a x)
(hwzero : ∀ a, w a = 0 → dw a = 0)
(hpzero : ∀ a x, p a x = 0 → dp a x = 0) :
derivativeBalanceField w dw p dp g dg T =
fun z => errorScoreField w dw p dp g T z - sensitivityField w p dg z := by
funext z
rw [errorScoreField_eq_numerator (hw z.1) (hp z.1 z.2)
(hwzero z.1) (hpzero z.1 z.2)]
unfold derivativeBalanceField sensitivityField jointDensity
ring
theorem integrable_priorLikelihoodCross_of_squares
Prior and likelihood weighted square integrability implies integrability of their weighted score cross term. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
X :
Type*
w dw :
ℝ → ℝ
p dp :
ℝ → X → ℝ
mu :
Measure (ℝ × X)
hw :
hdw :
Measurable dw
hp :
Measurable (fun z : ℝ × X ↦ p z.1 z.2)
hdp :
Measurable (fun z : ℝ × X ↦ dp z.1 z.2)
hw_nonneg :
∀ a, 0 ≤ w a
hp_nonneg :
∀ a x, 0 ≤ p a x
hprior :
Integrable (fun z : ℝ × X ↦ w z.1 * p z.1 z.2 * (priorScore w dw z.1) ^ 2) mu
hfisher :
Integrable (fun z : ℝ × X ↦ w z.1 * p z.1 z.2 * (likelihoodScore p dp z.1 z.2) ^ 2) mu
conclusion
Integrable (fun z : ℝ × X ↦ w z.1 * p z.1 z.2 * (priorScore w dw z.1 * likelihoodScore p dp z.1 z.2)) mu
Proof (Lean source)
lemma integrable_priorLikelihoodCross_of_squares {X : Type*} [MeasurableSpace X]
(w dw : ℝ → ℝ) (p dp : ℝ → X → ℝ) (mu : Measure (ℝ × X))
(hw : Measurable w) (hdw : Measurable dw)
(hp : Measurable (fun z : ℝ × X ↦ p z.1 z.2))
(hdp : Measurable (fun z : ℝ × X ↦ dp z.1 z.2))
(hw_nonneg : ∀ a, 0 ≤ w a) (hp_nonneg : ∀ a x, 0 ≤ p a x)
(hprior : Integrable (fun z : ℝ × X ↦
w z.1 * p z.1 z.2 * (priorScore w dw z.1) ^ 2) mu)
(hfisher : Integrable (fun z : ℝ × X ↦
w z.1 * p z.1 z.2 * (likelihoodScore p dp z.1 z.2) ^ 2) mu) :
Integrable (fun z : ℝ × X ↦
w z.1 * p z.1 z.2 *
(priorScore w dw z.1 * likelihoodScore p dp z.1 z.2)) mu := by
have hwj : Measurable (fun z : ℝ × X ↦ w z.1) := hw.comp measurable_fst
have hdwj : Measurable (fun z : ℝ × X ↦ dw z.1) := hdw.comp measurable_fst
have hpriorScore : Measurable (fun z : ℝ × X ↦ priorScore w dw z.1) := by
unfold priorScore
exact Measurable.ite (measurableSet_lt measurable_const hwj)
(hdwj.div hwj) measurable_const
have hlikeScore : Measurable
(fun z : ℝ × X ↦ likelihoodScore p dp z.1 z.2) := by
unfold likelihoodScore
exact Measurable.ite (measurableSet_lt measurable_const hp)
(hdp.div hp) measurable_const
have h := integrable_mul_mul_of_integrable_sq_mul
(fun z : ℝ × X ↦ priorScore w dw z.1)
(fun z : ℝ × X ↦ likelihoodScore p dp z.1 z.2)
(jointDensity w p) mu hpriorScore hlikeScore (hwj.mul hp)
(fun z ↦ mul_nonneg (hw_nonneg z.1) (hp_nonneg z.1 z.2))
(by
apply hprior.congr
filter_upwards with z
unfold jointDensity
ring)
(by
apply hfisher.congr
filter_upwards with z
unfold jointDensity
ring)
convert h using 1
funext z
unfold jointDensity
ring
theorem integrable_derivativeBalance_of_rewrite
Once the error-score and sensitivity fields are integrable, their exact balance identity supplies derivative-balance integrability. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
X :
Type*
w dw :
ℝ → ℝ
p dp g dg :
ℝ → X → ℝ
T :
X → ℝ
mu :
Measure (ℝ × X)
hw :
∀ a, 0 ≤ w a
hp :
∀ a x, 0 ≤ p a x
hwzero :
∀ a
if
w a = 0
then
dw a = 0
hpzero :
∀ a x
if
p a x = 0
then
dp a x = 0
herr :
Integrable (errorScoreField w dw p dp g T) mu
hsens :
Integrable (sensitivityField w p dg) mu
conclusion
Integrable (derivativeBalanceField w dw p dp g dg T) mu
Proof (Lean source)
-- keep: reusable sequential-law, Fisher-information, or van-Trees bridge for related adaptive experiments
lemma integrable_derivativeBalance_of_rewrite {X : Type*}
[MeasurableSpace X] (w dw : ℝ → ℝ)
(p dp g dg : ℝ → X → ℝ) (T : X → ℝ) (mu : Measure (ℝ × X))
(hw : ∀ a, 0 ≤ w a) (hp : ∀ a x, 0 ≤ p a x)
(hwzero : ∀ a, w a = 0 → dw a = 0)
(hpzero : ∀ a x, p a x = 0 → dp a x = 0)
(herr : Integrable (errorScoreField w dw p dp g T) mu)
(hsens : Integrable (sensitivityField w p dg) mu) :
Integrable (derivativeBalanceField w dw p dp g dg T) mu := by
rw [derivativeBalanceField_eq_errorScore_sub_sensitivity
w dw p dp g dg T hw hp hwzero hpzero]
exact herr.sub hsens
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialVanTreesRegularity 19 declarations Remaining measurable fields for the sequential van Trees adapter
Remaining measurable fields for the sequential van Trees adapter
def sequentialVTPrior
The centered quartic prior used on the scalar local interval [-R,R]. The sequential VTPrior is determined by the displayed parameters.
Definition (Lean source)
def sequentialVTPriorDeriv
The declared derivative of the centered local prior. The sequential VTPrior Deriv is determined by the displayed parameters.
Definition (Lean source)
structure SequentialVTPriorFacts
All paper-independent smooth-prior fields needed by van Trees on [-R,R]. For the displayed inputs and conditions, this structure records the stated data. The Sequential VTPrior Facts is determined by the displayed parameters.
Definition (Lean source)
parameters
R :
ℝ
assumes
contDiff :
ContDiff ℝ 1 (sequentialVTPrior R)
support :
hasDeriv :
nonneg :
∀ a, 0 ≤ sequentialVTPrior R a
normalized :
∫ a, sequentialVTPrior R a ∂parameterMeasure (-R) R = 1
scoreSqMeasurable :
AEStronglyMeasurable (fun a ↦ sequentialVTPrior R a * (priorScore (sequentialVTPrior R) (sequentialVTPriorDeriv R) a) ^ 2) (parameterMeasure (-R) R)
scoreSqIntegrable :
Integrable (fun a ↦ sequentialVTPrior R a * (priorScore (sequentialVTPrior R) (sequentialVTPriorDeriv R) a) ^ 2) (parameterMeasure (-R) R)
information :
theorem sequentialVTPrior_facts
Under the supplied quantities and conditions, the sequential vt prior facts assertion holds.
Formal statement
Proof (Lean source)
lemma sequentialVTPrior_facts {R : ℝ} (hR : 0 < R) :
SequentialVTPriorFacts R := by
have ha : 0 < R / 2 := by linarith
have hleft : -R ≤ (0 : ℝ) - R / 2 := by linarith
have hright : (0 : ℝ) + R / 2 ≤ R := by linarith
refine ⟨smoothPrior_contDiff ha, ?_, hasDerivAt_smoothPrior ha,
smoothPrior_nonneg ha, integral_smoothPrior_parameterMeasure ha hleft hright,
smoothPrior_scoreSq_aestronglyMeasurable ha,
smoothPrior_scoreSq_integrable ha, ?_⟩
· exact support_smoothPrior_subset_Icc ha (by linarith) (by linarith)
· unfold sequentialVTPrior sequentialVTPriorDeriv
rw [priorInformation_smoothPrior ha hleft hright]
field_simp [hR.ne']
ring
def vtTarget
For the supplied quantities and conditions, the vt target is the mathematical object specified below. The vt Target is determined by the displayed parameters.
Definition (Lean source)
parameters
def
vtTarget θ v a _x :
ℝ
given by
contrast θ + a * contrast v
def vtTargetDeriv
For the supplied quantities and conditions, the vt target deriv is the mathematical object specified below.
Definition (Lean source)
parameters
def
vtTargetDeriv v _a _x :
ℝ
given by
contrast v
theorem vtTarget_eq_parameterPath
Under the supplied quantities and conditions, the vt target eq parameter path assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
vtTarget θ v a x = contrast (parameterPath θ v a)
Proof (Lean source)
lemma vtTarget_eq_parameterPath {X : Type*} (θ v : TrialParameter)
(a : ℝ) (x : X) :
vtTarget θ v a x = contrast (parameterPath θ v a) := by
simp [vtTarget, contrast, parameterPath]
ring
theorem hasDerivAt_vtTarget
Under the supplied quantities and conditions, the has deriv at vt target assertion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma hasDerivAt_vtTarget {X : Type*} (θ v : TrialParameter) (a : ℝ) (x : X) :
HasDerivAt (fun b ↦ vtTarget θ v b x) (vtTargetDeriv v a x) a := by
unfold vtTarget vtTargetDeriv
simpa only [id_eq, one_mul] using
((hasDerivAt_id a).mul_const (contrast v)).const_add (contrast θ)
theorem vtTarget_absolutelyContinuous
Under the supplied quantities and conditions, the vt target absolutely continuous assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
AbsolutelyContinuousOnInterval (fun a ↦ vtTarget θ v a x) ell upper
Proof (Lean source)
lemma vtTarget_absolutelyContinuous {X : Type*} (θ v : TrialParameter)
(x : X) (ell upper : ℝ) :
AbsolutelyContinuousOnInterval (fun a ↦ vtTarget θ v a x) ell upper := by
apply ContDiffOn.absolutelyContinuousOnInterval
unfold vtTarget
exact (by fun_prop : ContDiff ℝ 1 (fun a ↦ contrast θ + a * contrast v)).contDiffOn
theorem measurable_vtDensity_joint
the measurable vt density joint assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (fun az : ℝ × Transcript (Z n) ↦ vtDensity P θ v p n ell upper az.1 az.2)
Proof (Lean source)
lemma measurable_vtDensity_joint {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(ell upper : ℝ) :
Measurable (fun az : ℝ × Transcript (Z n) ↦
vtDensity P θ v p n ell upper az.1 az.2) := by
unfold vtDensity transcriptMixtureRealDensity
apply Measurable.ite
· exact measurableSet_Icc.preimage measurable_fst
· apply Finset.measurable_fun_sum
intro x hx
have hc : Continuous (fun a ↦
inputPathProbability (parameterPath θ v a) p x) := by
rw [continuous_iff_continuousAt]
intro a
exact (hasDerivAt_inputPathProbability_parameterPath θ v p a x).continuousAt
exact (hc.measurable.comp measurable_fst).mul
((Measure.measurable_rnDeriv _ _).ennreal_toReal.comp measurable_snd)
· exact measurable_const
theorem continuous_transcriptMixtureRealDerivative_path
the continuous transcript mixture real derivative path assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma continuous_transcriptMixtureRealDerivative_path {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(z : Transcript (Z n)) :
Continuous (fun a ↦ transcriptMixtureRealDerivative P
(parameterPath θ v a) v p n z) := by
have hcd := (transcriptDensityPath_contDiff P θ v p n z).continuous_deriv (by norm_num)
have heq : deriv (fun a ↦
transcriptMixtureRealDensity P (parameterPath θ v a) p n z) =
fun a ↦ transcriptMixtureRealDerivative P (parameterPath θ v a) v p n z := by
funext a
exact (hasDerivAt_transcriptMixtureRealDensity_parameterPath
P θ v p a n z).deriv
rwa [heq] at hcd
theorem measurable_vtDensityDeriv_joint
the measurable vt density deriv joint assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Measurable (fun az : ℝ × Transcript (Z n) ↦ vtDensityDeriv P θ v p n ell upper az.1 az.2)
Proof (Lean source)
lemma measurable_vtDensityDeriv_joint {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p : ℝ) (n : ℕ)
(ell upper : ℝ) :
Measurable (fun az : ℝ × Transcript (Z n) ↦
vtDensityDeriv P θ v p n ell upper az.1 az.2) := by
unfold vtDensityDeriv transcriptMixtureRealDerivative
apply Measurable.ite
· exact measurableSet_Icc.preimage measurable_fst
· apply Finset.measurable_fun_sum
intro x hx
have hc : Continuous (fun a ↦
inputPathDirectionalDerivative (parameterPath θ v a) v p x) := by
unfold inputPathDirectionalDerivative
apply continuous_finsetSum
intro i hi
apply Continuous.mul continuous_const
apply continuous_finsetProd
intro j hj
rw [continuous_iff_continuousAt]
intro a
exact (hasDerivAt_piTheta_parameterPath θ v p a (x j)).continuousAt
exact (hc.measurable.comp measurable_fst).mul
((Measure.measurable_rnDeriv _ _).ennreal_toReal.comp measurable_snd)
· exact measurable_const
theorem integrable_mul_mul_of_integrable_sq_mul
A weighted cross term is integrable whenever both weighted square terms are integrable. This is the Cauchy–Schwarz step used for the van Trees error-score field. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
A :
Type*
f s q :
A → ℝ
mu :
Measure A
hf :
hs :
hq :
hq_nonneg :
∀ x, 0 ≤ q x
hf_sq :
Integrable (fun x ↦ f x ^ 2 * q x) mu
hs_sq :
Integrable (fun x ↦ s x ^ 2 * q x) mu
conclusion
Integrable (fun x ↦ f x * s x * q x) mu
Proof (Lean source)
lemma integrable_mul_mul_of_integrable_sq_mul {A : Type*} [MeasurableSpace A]
(f s q : A → ℝ) (mu : Measure A)
(hf : Measurable f) (hs : Measurable s) (hq : Measurable q)
(hq_nonneg : ∀ x, 0 ≤ q x)
(hf_sq : Integrable (fun x ↦ f x ^ 2 * q x) mu)
(hs_sq : Integrable (fun x ↦ s x ^ 2 * q x) mu) :
Integrable (fun x ↦ f x * s x * q x) mu := by
let qroot : A → ℝ := fun x ↦ sqrt (q x)
have hqroot : Measurable qroot :=
Real.continuous_sqrt.measurable.comp hq
have hfroot_sq : Integrable (fun x ↦ (f x * qroot x) ^ 2) mu := by
convert hf_sq using 1
funext x
dsimp [qroot]
rw [mul_pow, Real.sq_sqrt (hq_nonneg x)]
have hsroot_sq : Integrable (fun x ↦ (s x * qroot x) ^ 2) mu := by
convert hs_sq using 1
funext x
dsimp [qroot]
rw [mul_pow, Real.sq_sqrt (hq_nonneg x)]
have hfroot : MemLp (fun x ↦ f x * qroot x) 2 mu :=
(memLp_two_iff_integrable_sq
(hf.mul hqroot).aestronglyMeasurable).2 hfroot_sq
have hsroot : MemLp (fun x ↦ s x * qroot x) 2 mu :=
(memLp_two_iff_integrable_sq
(hs.mul hqroot).aestronglyMeasurable).2 hsroot_sq
have hcross := hfroot.integrable_mul hsroot
convert hcross using 1
funext x
dsimp [qroot]
have hroot : sqrt (q x) * sqrt (q x) = q x :=
Real.mul_self_sqrt (hq_nonneg x)
calc
f x * s x * q x = f x * s x *
(sqrt (q x) * sqrt (q x)) := by rw [hroot]
_ = f x * sqrt (q x) * (s x * sqrt (q x)) := by ring
theorem integrable_or_lintegral_ofReal_eq_top
A nonnegative measurable risk is either integrable or has infinite Lebesgue integral. This supplies the harmless infinite-risk branch before applying the finite-risk van Trees argument. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Integrable F mu ∨ ∫⁻ x, ofReal (F x) ∂mu = ∞
Proof (Lean source)
lemma integrable_or_lintegral_ofReal_eq_top {A : Type*} [MeasurableSpace A]
(F : A → ℝ) (mu : Measure A) (hF : Measurable F)
(hF_nonneg : ∀ x, 0 ≤ F x) :
Integrable F mu ∨ ∫⁻ x, ofReal (F x) ∂mu = ∞ := by
by_cases h : Integrable F mu
· exact Or.inl h
· right
have hnotfinite : ¬ HasFiniteIntegral F mu := by
intro hfinite
exact h ⟨hF.aestronglyMeasurable, hfinite⟩
have hnotlt : ¬ (∫⁻ x, ofReal ‖F x‖ ∂mu) < ∞ := by
exact fun hlt ↦ hnotfinite ((hasFiniteIntegral_iff_norm (μ := mu) F).2 hlt)
have htop : (∫⁻ x, ofReal ‖F x‖ ∂mu) = ∞ :=
top_unique (not_lt.mp hnotlt)
convert htop using 1
apply lintegral_congr
intro x
rw [Real.norm_eq_abs, abs_of_nonneg (hF_nonneg x)]
structure VanTreesProductMeasurable
The product-measurability fields required by the observation-dependent van Trees theorem. For the displayed quantities and conditions, the stated result follows.
Definition (Lean source)
parameters
assumes
balance :
Measurable (derivativeBalanceField w dw p dp g dg T)
errorScore :
Measurable (errorScoreField w dw p dp g T)
sensitivity :
Measurable (sensitivityField w p dg)
errorSq :
Measurable (errorSqField w p g T)
scoreSq :
Measurable (scoreSqField w dw p dp)
priorJointSq :
Measurable (fun z : ℝ × X ↦ w z.1 * p z.1 z.2 * (priorScore w dw z.1) ^ 2)
fisherSq :
Measurable (fun z : ℝ × X ↦ w z.1 * p z.1 z.2 * (likelihoodScore p dp z.1 z.2) ^ 2)
cross :
Measurable (fun z : ℝ × X ↦ w z.1 * p z.1 z.2 * (priorScore w dw z.1 * likelihoodScore p dp z.1 z.2))
theorem vanTreesProductMeasurable_of_joint
Joint measurability of the primitive density, derivative, target, and estimator functions supplies every product-measurability side condition in van Trees. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
X :
Type*
w dw :
ℝ → ℝ
p dp g dg :
ℝ → X → ℝ
T :
X → ℝ
hw :
hdw :
Measurable dw
hp :
Measurable (fun z : ℝ × X ↦ p z.1 z.2)
hdp :
Measurable (fun z : ℝ × X ↦ dp z.1 z.2)
hg :
Measurable (fun z : ℝ × X ↦ g z.1 z.2)
hdg :
Measurable (fun z : ℝ × X ↦ dg z.1 z.2)
hT :
conclusion
VanTreesProductMeasurable w dw p dp g dg T
Proof (Lean source)
lemma vanTreesProductMeasurable_of_joint {X : Type*} [MeasurableSpace X]
(w dw : ℝ → ℝ) (p dp g dg : ℝ → X → ℝ) (T : X → ℝ)
(hw : Measurable w) (hdw : Measurable dw)
(hp : Measurable (fun z : ℝ × X ↦ p z.1 z.2))
(hdp : Measurable (fun z : ℝ × X ↦ dp z.1 z.2))
(hg : Measurable (fun z : ℝ × X ↦ g z.1 z.2))
(hdg : Measurable (fun z : ℝ × X ↦ dg z.1 z.2))
(hT : Measurable T) :
VanTreesProductMeasurable w dw p dp g dg T := by
have hwj : Measurable (fun z : ℝ × X ↦ w z.1) := hw.comp measurable_fst
have hdwj : Measurable (fun z : ℝ × X ↦ dw z.1) := hdw.comp measurable_fst
have hTj : Measurable (fun z : ℝ × X ↦ T z.2) := hT.comp measurable_snd
have hjoint : Measurable (jointDensity w p) := by
exact hwj.mul hp
have hprior : Measurable (fun z : ℝ × X ↦ priorScore w dw z.1) := by
unfold priorScore
exact Measurable.ite (measurableSet_lt measurable_const hwj)
(hdwj.div hwj) measurable_const
have hlike : Measurable (fun z : ℝ × X ↦ likelihoodScore p dp z.1 z.2) := by
unfold likelihoodScore
exact Measurable.ite (measurableSet_lt measurable_const hp)
(hdp.div hp) measurable_const
have hscore : Measurable (jointScore w dw p dp) := by
unfold jointScore
exact Measurable.ite (measurableSet_lt measurable_const hjoint)
((hdwj.mul hp |>.add (hwj.mul hdp)).div hjoint) measurable_const
refine ⟨?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩
· unfold derivativeBalanceField
fun_prop
· unfold errorScoreField
exact (hTj.sub hg).mul hscore |>.mul hjoint
· unfold sensitivityField
exact hdg.mul hjoint
· unfold errorSqField
exact ((hTj.sub hg).pow_const 2).mul hjoint
· unfold scoreSqField
exact (hscore.pow_const 2).mul hjoint
· exact (hwj.mul hp).mul (hprior.pow_const 2)
· exact (hwj.mul hp).mul (hlike.pow_const 2)
· exact (hwj.mul hp).mul (hprior.mul hlike)
theorem sequentialVanTreesProductMeasurable
The sequential mixture likelihood, affine contrast target, and smooth prior satisfy all van Trees product-measurability conditions for every measurable transcript estimator. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p R :
ℝ
n :
ℕ
hR :
0 < R
T :
Transcript (Z n) → ℝ
hT :
conclusion
VanTreesProductMeasurable (sequentialVTPrior R) (sequentialVTPriorDeriv R) (vtDensity P θ v p n (-R) R) (vtDensityDeriv P θ v p n (-R) R) (vtTarget θ v) (vtTargetDeriv v) T
Proof (Lean source)
lemma sequentialVanTreesProductMeasurable {Z : OutputFamily}
[∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p R : ℝ) (n : ℕ)
(hR : 0 < R)
(T : Transcript (Z n) → ℝ) (hT : Measurable T) :
VanTreesProductMeasurable
(sequentialVTPrior R) (sequentialVTPriorDeriv R)
(vtDensity P θ v p n (-R) R) (vtDensityDeriv P θ v p n (-R) R)
(vtTarget θ v) (vtTargetDeriv v) T := by
apply vanTreesProductMeasurable_of_joint
· exact (smoothPrior_contDiff (a := R / 2) (c := 0) (by linarith)).continuous.measurable
· unfold sequentialVTPriorDeriv smoothPriorDeriv
apply Measurable.ite
· exact measurableSet_lt
(continuous_abs.comp (continuous_id.sub continuous_const)).measurable
measurable_const
· fun_prop
· exact measurable_const
· exact measurable_vtDensity_joint P θ v p n (-R) R
· exact measurable_vtDensityDeriv_joint P θ v p n (-R) R
· unfold vtTarget
fun_prop
· unfold vtTargetDeriv
exact measurable_const
· exact hT
theorem integrable_errorScoreField_of_squares
Finite weighted squared error and finite joint-score information imply integrability of the signed error-score field. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
X :
Type*
w dw :
ℝ → ℝ
p dp g :
ℝ → X → ℝ
T :
X → ℝ
mu :
Measure (ℝ × X)
hw :
hp :
Measurable (fun z : ℝ × X ↦ p z.1 z.2)
hg :
Measurable (fun z : ℝ × X ↦ g z.1 z.2)
hT :
hscore :
Measurable (jointScore w dw p dp)
hw_nonneg :
∀ a, 0 ≤ w a
hp_nonneg :
∀ a x, 0 ≤ p a x
herrorSq :
Integrable (errorSqField w p g T) mu
hscoreSq :
Integrable (scoreSqField w dw p dp) mu
conclusion
Integrable (errorScoreField w dw p dp g T) mu
Proof (Lean source)
lemma integrable_errorScoreField_of_squares {X : Type*} [MeasurableSpace X]
(w dw : ℝ → ℝ) (p dp g : ℝ → X → ℝ) (T : X → ℝ)
(mu : Measure (ℝ × X))
(hw : Measurable w)
(hp : Measurable (fun z : ℝ × X ↦ p z.1 z.2))
(hg : Measurable (fun z : ℝ × X ↦ g z.1 z.2))
(hT : Measurable T)
(hscore : Measurable (jointScore w dw p dp))
(hw_nonneg : ∀ a, 0 ≤ w a) (hp_nonneg : ∀ a x, 0 ≤ p a x)
(herrorSq : Integrable (errorSqField w p g T) mu)
(hscoreSq : Integrable (scoreSqField w dw p dp) mu) :
Integrable (errorScoreField w dw p dp g T) mu := by
have h := integrable_mul_mul_of_integrable_sq_mul
(fun z : ℝ × X ↦ T z.2 - g z.1 z.2)
(jointScore w dw p dp) (jointDensity w p) mu
((hT.comp measurable_snd).sub hg) hscore
((hw.comp measurable_fst).mul hp)
(fun z ↦ mul_nonneg (hw_nonneg z.1) (hp_nonneg z.1 z.2))
herrorSq hscoreSq
change Integrable (fun z : ℝ × X ↦
(T z.2 - g z.1 z.2) * jointScore w dw p dp z * jointDensity w p z) mu
exact h
theorem sequentialVTPrior_endpoints
The centered smooth prior vanishes at both endpoints of the wider local parameter interval. For the displayed inputs and conditions, the stated result follows.
Formal statement
hypotheses
R :
ℝ
hR :
0 < R
Proof (Lean source)
lemma sequentialVTPrior_endpoints {R : ℝ} (hR : 0 < R) :
sequentialVTPrior R R = 0 ∧ sequentialVTPrior R (-R) = 0 := by
unfold sequentialVTPrior
have h := smoothPrior_ambient_endpoints
(ell := -R) (u := R) (c := 0) (a := R / 2)
(by linarith) (by linarith)
exact ⟨h.2, h.1⟩
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SequentialVanTreesRiskBridge 10 declarations This file converts the native-real product-space Bayes error in the finite van Trees inequality into the weighted ENNReal local risk used by the sequential minimax statement.
Sequential van Trees risk and scaling bridge
This file converts the native-real product-space Bayes error in the finite
van Trees inequality into the weighted ENNReal local risk used by the
sequential minimax statement.
theorem parameterPath_sampleScaledDirection_eq_localAlternative
Scaling a parameter direction by inverse root sample size makes its scalar path exactly the corresponding local alternative. For the displayed inputs and conditions, the stated result follows.
Formal statement
hypotheses
conclusion
= localAlternative θ (scaledDirection a v) n
Proof (Lean source)
lemma parameterPath_sampleScaledDirection_eq_localAlternative
(θ v : TrialParameter) (a : ℝ) (n : ℕ) :
parameterPath θ (scaledDirection ((sqrt n)⁻¹) v) a =
localAlternative θ (scaledDirection a v) n := by
funext k
simp [parameterPath, scaledDirection, localAlternative]
ring
theorem scaledError_sq_eq_nat_mul_vtTarget_error_sq
Along the sample-scaled van Trees path, the squared scaled error is sample size times the unscaled target error. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
(scaledError P θ (scaledDirection a v) n z) ^ 2
Proof (Lean source)
lemma scaledError_sq_eq_nat_mul_vtTarget_error_sq
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (a : ℝ)
(n : ℕ) (z : Transcript (Z n)) :
(scaledError P θ (scaledDirection a v) n z) ^ 2 =
(n : ℝ) * (P.estimate n z -
vtTarget θ (scaledDirection ((sqrt n)⁻¹) v) a z) ^ 2 := by
rw [vtTarget_eq_parameterPath,
parameterPath_sampleScaledDirection_eq_localAlternative]
unfold scaledError
rw [mul_pow, Real.sq_sqrt (Nat.cast_nonneg n)]
theorem contrast_scaledDirection
Contrast is linear along scalar multiples of a trial direction. For the displayed inputs and conditions, the stated result follows.
Formal statement
Proof (Lean source)
lemma contrast_scaledDirection (a : ℝ) (v : TrialParameter) :
contrast (scaledDirection a v) = a * contrast v := by
simp [contrast, scaledDirection]
ring
theorem nat_mul_contrast_sampleScaledDirection_sq
For positive sample size, sample scaling cancels after multiplying the squared contrast sensitivity by the sample size. For the displayed inputs and conditions, the stated result follows.
Formal statement
hypotheses
conclusion
(n : ℝ) * contrast (scaledDirection ((sqrt n)⁻¹) v) ^ 2 = contrast v ^ 2
Proof (Lean source)
lemma nat_mul_contrast_sampleScaledDirection_sq
(v : TrialParameter) {n : ℕ} (hn : 0 < n) :
(n : ℝ) * contrast (scaledDirection ((sqrt n)⁻¹) v) ^ 2 =
contrast v ^ 2 := by
rw [contrast_scaledDirection]
have hsqrt : sqrt (n : ℝ) ≠ 0 := by positivity
have hsqrtSq : (sqrt (n : ℝ)) ^ 2 = n :=
Real.sq_sqrt (Nat.cast_nonneg n)
field_simp
nlinarith
theorem localRisk_scaledDirection_eq_lintegral_vtDensity
On the compact scalar interval, the local risk is the reference-measure integral weighted by the interval-extended transcript density. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p R a :
ℝ
n :
ℕ
hp :
hinterior :
ha :
a ∈ Icc (-R) R
conclusion
localRisk P θ (scaledDirection a v) p n
= ∫⁻ z, ofReal (vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v) p n (-R) R a z) * ofReal ((scaledError P θ (scaledDirection a v) n z) ^ 2) ∂transcriptReferenceMeasure P n
Proof (Lean source)
lemma localRisk_scaledDirection_eq_lintegral_vtDensity
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p R a : ℝ) (n : ℕ)
(hp : InteriorAssignment p)
(hinterior : ∀ b ∈ Icc (-R) R,
InteriorMeans (parameterPath θ
(scaledDirection ((sqrt n)⁻¹) v) b))
(ha : a ∈ Icc (-R) R) :
localRisk P θ (scaledDirection a v) p n =
∫⁻ z, ofReal
(vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R a z) *
ofReal ((scaledError P θ (scaledDirection a v) n z) ^ 2)
∂transcriptReferenceMeasure P n := by
have hpath := parameterPath_sampleScaledDirection_eq_localAlternative θ v a n
have hlaw := transcriptLaw_eq_withDensity_mixture P
(parameterPath θ (scaledDirection ((sqrt n)⁻¹) v) a) p n hp
(hinterior a ha)
have hq : (fun z => ofReal
(vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R a z)) =
fun z => ofReal
(transcriptMixtureRealDensity P
(parameterPath θ (scaledDirection ((sqrt n)⁻¹) v) a) p n z) := by
funext z
simp [vtDensity, ha]
have hlawq : (transcriptReferenceMeasure P n).withDensity
(fun z => ofReal
(vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R a z)) =
transcriptLaw P
(parameterPath θ (scaledDirection ((sqrt n)⁻¹) v) a) p n := by
rw [hq]
exact hlaw
unfold localRisk
rw [← hpath, ← hlawq]
rw [lintegral_withDensity_eq_lintegral_mul]
· rfl
· apply ENNReal.measurable_ofReal.comp
have hpair : Measurable (fun z : Transcript (Z n) => (a, z)) :=
Measurable.prod (f := fun z : Transcript (Z n) => (a, z))
measurable_const measurable_id
exact (measurable_vtDensity_joint P θ
(scaledDirection ((sqrt n)⁻¹) v) p n (-R) R).comp hpair
· apply ENNReal.measurable_ofReal.comp
exact (measurable_const.mul
((P.estimate_measurable n).sub measurable_const)).pow_const 2
theorem lintegral_scaledSequentialBayesError_eq_weighted_localRisk
The scaled nonnegative product-space error integral equals the weighted compact-prior local risk, without any finiteness assumption. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
(∫⁻ z : ℝ × Transcript (Z n), ofReal ((n : ℝ) * errorSqField (sequentialVTPrior R) (vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v) p n (-R) R) (vtTarget θ (scaledDirection ((sqrt n)⁻¹) v)) (P.estimate n) z) ∂((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n)))
= ∫⁻ a, ofReal (sequentialVTPrior R a) * localRisk P θ (scaledDirection a v) p n ∂parameterMeasure (-R) R
Proof (Lean source)
lemma lintegral_scaledSequentialBayesError_eq_weighted_localRisk
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p R : ℝ) (n : ℕ)
(hR : 0 < R) (hp : InteriorAssignment p)
(hinterior : ∀ a ∈ Icc (-R) R,
InteriorMeans (parameterPath θ
(scaledDirection ((sqrt n)⁻¹) v) a)) :
(∫⁻ z : ℝ × Transcript (Z n), ofReal ((n : ℝ) *
errorSqField (sequentialVTPrior R)
(vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R)
(vtTarget θ (scaledDirection ((sqrt n)⁻¹) v))
(P.estimate n) z)
∂((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n))) =
∫⁻ a, ofReal (sequentialVTPrior R a) *
localRisk P θ (scaledDirection a v) p n
∂parameterMeasure (-R) R := by
rw [lintegral_prod]
· apply lintegral_congr_ae
have hmem : ∀ᵐ a ∂parameterMeasure (-R) R, a ∈ Icc (-R) R := by
unfold parameterMeasure
exact ae_restrict_mem measurableSet_Icc
filter_upwards [hmem] with a ha
rw [localRisk_scaledDirection_eq_lintegral_vtDensity
P θ v p R a n hp hinterior ha]
rw [← lintegral_const_mul' _ _ ENNReal.ofReal_ne_top]
apply lintegral_congr
intro z
rw [scaledError_sq_eq_nat_mul_vtTarget_error_sq P θ v a n z]
unfold errorSqField jointDensity
rw [ENNReal.ofReal_mul (Nat.cast_nonneg n),
ENNReal.ofReal_mul (sq_nonneg _),
ENNReal.ofReal_mul ((sequentialVTPrior_facts hR).nonneg a),
ENNReal.ofReal_mul (Nat.cast_nonneg n)]
ac_rfl
· exact (ENNReal.measurable_ofReal.comp
(measurable_const.mul
(sequentialVanTreesProductMeasurable P θ
(scaledDirection ((sqrt n)⁻¹) v) p R n hR
(P.estimate n) (P.estimate_measurable n)).errorSq)).aemeasurable
theorem weightedLocalRisk_le_localWorstRisk
Compact-prior averaging is bounded by local worst risk whenever the scalar directions fit in the chosen radius and remain interior. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p R H :
ℝ
n :
ℕ
hR :
0 < R
hinterior :
hcover :
R * sqrt ((v 0) ^ 2 + (v 1) ^ 2) ≤ H
conclusion
(∫⁻ a, ofReal (sequentialVTPrior R a) * localRisk P θ (scaledDirection a v) p n ∂parameterMeasure (-R) R)
≤ localWorstRisk P θ p H n
Proof (Lean source)
lemma weightedLocalRisk_le_localWorstRisk
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter)
(p R H : ℝ) (n : ℕ) (hR : 0 < R)
(hinterior : ∀ a ∈ Icc (-R) R,
InteriorMeans (parameterPath θ
(scaledDirection ((sqrt n)⁻¹) v) a))
(hcover : R * sqrt ((v 0) ^ 2 + (v 1) ^ 2) ≤ H) :
(∫⁻ a, ofReal (sequentialVTPrior R a) *
localRisk P θ (scaledDirection a v) p n
∂parameterMeasure (-R) R) ≤ localWorstRisk P θ p H n := by
have hprior := sequentialVTPrior_facts hR
exact (weighted_lintegral_le_uniform
(parameterMeasure (-R) R) (sequentialVTPrior R)
(fun a => localRisk P θ (scaledDirection a v) p n)
(localWorstRisk P θ p H n)
hprior.contDiff.continuous.measurable
(measurable_localRisk_scaledDirection P θ v p n).aemeasurable
(by
unfold sequentialVTPrior
exact smoothPrior_integrable_parameterMeasure (by linarith))
(Filter.Eventually.of_forall hprior.nonneg) hprior.normalized
(by
have hmem : ∀ᵐ a ∂parameterMeasure (-R) R, a ∈ Icc (-R) R := by
unfold parameterMeasure
exact ae_restrict_mem measurableSet_Icc
filter_upwards [hmem] with a ha
apply localRisk_le_localWorstRisk
refine ⟨(scaledDirection_mem_cover v ha).trans hcover, ?_⟩
rw [← parameterPath_sampleScaledDirection_eq_localAlternative]
exact hinterior a ha)).2
theorem ofReal_nat_mul_sequentialBayesError_eq_weighted_localRisk
In the finite-error branch, multiplying the native-real Bayes error by the sample size gives exactly the compact-prior weighted local risk. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p R :
ℝ
n :
ℕ
hR :
0 < R
hp :
hinterior :
herrorSqInt :
Integrable (errorSqField (sequentialVTPrior R) (vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v) p n (-R) R) (vtTarget θ (scaledDirection ((sqrt n)⁻¹) v)) (P.estimate n)) ((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n))
conclusion
ofReal ((n : ℝ) * ∫ z, errorSqField (sequentialVTPrior R) (vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v) p n (-R) R) (vtTarget θ (scaledDirection ((sqrt n)⁻¹) v)) (P.estimate n) z ∂((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n)))
= ∫⁻ a, ofReal (sequentialVTPrior R a) * localRisk P θ (scaledDirection a v) p n ∂parameterMeasure (-R) R
Proof (Lean source)
lemma ofReal_nat_mul_sequentialBayesError_eq_weighted_localRisk
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter) (p R : ℝ) (n : ℕ)
(hR : 0 < R) (hp : InteriorAssignment p)
(hinterior : ∀ a ∈ Icc (-R) R,
InteriorMeans (parameterPath θ
(scaledDirection ((sqrt n)⁻¹) v) a))
(herrorSqInt : Integrable
(errorSqField (sequentialVTPrior R)
(vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R)
(vtTarget θ (scaledDirection ((sqrt n)⁻¹) v))
(P.estimate n))
((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n))) :
ofReal ((n : ℝ) *
∫ z, errorSqField (sequentialVTPrior R)
(vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R)
(vtTarget θ (scaledDirection ((sqrt n)⁻¹) v))
(P.estimate n) z
∂((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n))) =
∫⁻ a, ofReal (sequentialVTPrior R a) *
localRisk P θ (scaledDirection a v) p n
∂parameterMeasure (-R) R := by
let d := scaledDirection ((sqrt n)⁻¹) v
let q := vtDensity P θ d p n (-R) R
let g := vtTarget (X := Transcript (Z n)) θ d
let w := sequentialVTPrior R
let mu := transcriptReferenceMeasure P n
have hscaledInt : Integrable (fun z : ℝ × Transcript (Z n) =>
(n : ℝ) * errorSqField w q g (P.estimate n) z)
((parameterMeasure (-R) R).prod mu) := herrorSqInt.const_mul n
have hscaledNonneg : ∀ z : ℝ × Transcript (Z n),
0 ≤ (n : ℝ) * errorSqField w q g (P.estimate n) z := by
intro z
unfold errorSqField jointDensity
exact mul_nonneg (Nat.cast_nonneg n)
(mul_nonneg (sq_nonneg _)
(mul_nonneg ((sequentialVTPrior_facts hR).nonneg z.1)
(vtDensity_nonneg P θ d p n (-R) R hp hinterior z.1 z.2)))
calc
ofReal ((n : ℝ) *
∫ z, errorSqField (sequentialVTPrior R)
(vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R)
(vtTarget θ (scaledDirection ((sqrt n)⁻¹) v))
(P.estimate n) z
∂((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n))) =
ofReal (∫ z, (n : ℝ) *
errorSqField w q g (P.estimate n) z
∂((parameterMeasure (-R) R).prod mu)) := by
congr 1
rw [integral_const_mul]
_ = ∫⁻ z, ofReal ((n : ℝ) *
errorSqField w q g (P.estimate n) z)
∂((parameterMeasure (-R) R).prod mu) :=
ofReal_integral_eq_lintegral_ofReal hscaledInt
(Filter.Eventually.of_forall hscaledNonneg)
_ = ∫⁻ a, ofReal (sequentialVTPrior R a) *
localRisk P θ (scaledDirection a v) p n
∂parameterMeasure (-R) R := by
rw [lintegral_prod]
· apply lintegral_congr_ae
have hmem : ∀ᵐ a ∂parameterMeasure (-R) R, a ∈ Icc (-R) R := by
unfold parameterMeasure
exact ae_restrict_mem measurableSet_Icc
filter_upwards [hmem] with a ha
rw [localRisk_scaledDirection_eq_lintegral_vtDensity
P θ v p R a n hp hinterior ha]
rw [← lintegral_const_mul' _ _ ENNReal.ofReal_ne_top]
apply lintegral_congr
intro z
rw [scaledError_sq_eq_nat_mul_vtTarget_error_sq P θ v a n z]
unfold errorSqField jointDensity
dsimp [w, q, g, d, mu]
rw [ENNReal.ofReal_mul (Nat.cast_nonneg n),
ENNReal.ofReal_mul (sq_nonneg _),
ENNReal.ofReal_mul ((sequentialVTPrior_facts hR).nonneg a),
ENNReal.ofReal_mul (Nat.cast_nonneg n)]
ac_rfl
· fun_prop
theorem ofReal_vanTreesEnvelope_le_localWorstRisk_of_fisher_control
The finite van Trees bound, after sample scaling and compact-prior ordering, yields a finite-sample lower bound on local worst risk. Transcript Fisher control remains explicit. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the of Real van Trees Envelope le local Worst Risk of fisher control.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p R H I :
ℝ
n :
ℕ
hn :
0 < n
hR :
0 < R
hp :
hinterior :
hcover :
R * sqrt ((v 0) ^ 2 + (v 1) ^ 2) ≤ H
herrorSqInt :
Integrable (errorSqField (sequentialVTPrior R) (vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v) p n (-R) R) (vtTarget θ (scaledDirection ((sqrt n)⁻¹) v)) (P.estimate n)) ((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n))
hfisherSqInt :
Integrable (fun z : ℝ × Transcript (Z n) => sequentialVTPrior R z.1 * vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v) p n (-R) R z.1 z.2 * (likelihoodScore (vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v) p n (-R) R) (vtDensityDeriv P θ (scaledDirection ((sqrt n)⁻¹) v) p n (-R) R) z.1 z.2) ^ 2) ((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n))
hfisher_le :
(∫ a, sequentialVTPrior R a * fisherInformation (transcriptReferenceMeasure P n) (vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v) p n (-R) R) (vtDensityDeriv P θ (scaledDirection ((sqrt n)⁻¹) v) p n (-R) R) a ∂parameterMeasure (-R) R)
≤ I
hinfoPos :
+ ∫ a, sequentialVTPrior R a * fisherInformation (transcriptReferenceMeasure P n) (vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v) p n (-R) R) (vtDensityDeriv P θ (scaledDirection ((sqrt n)⁻¹) v) p n (-R) R) a ∂parameterMeasure (-R) R
conclusion
ofReal (contrast v ^ 2 / (I + 40 / R ^ 2)) ≤ localWorstRisk P θ p H n
Proof (Lean source)
lemma ofReal_vanTreesEnvelope_le_localWorstRisk_of_fisher_control
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter)
(p R H I : ℝ) (n : ℕ) (hn : 0 < n) (hR : 0 < R)
(hp : InteriorAssignment p)
(hinterior : ∀ a ∈ Icc (-R) R,
InteriorMeans (parameterPath θ
(scaledDirection ((sqrt n)⁻¹) v) a))
(hcover : R * sqrt ((v 0) ^ 2 + (v 1) ^ 2) ≤ H)
(herrorSqInt : Integrable
(errorSqField (sequentialVTPrior R)
(vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R)
(vtTarget θ (scaledDirection ((sqrt n)⁻¹) v))
(P.estimate n))
((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n)))
(hfisherSqInt : Integrable (fun z : ℝ × Transcript (Z n) =>
sequentialVTPrior R z.1 *
vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R z.1 z.2 *
(likelihoodScore
(vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R)
(vtDensityDeriv P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R) z.1 z.2) ^ 2)
((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n)))
(hfisher_le :
(∫ a, sequentialVTPrior R a *
fisherInformation (transcriptReferenceMeasure P n)
(vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R)
(vtDensityDeriv P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R) a
∂parameterMeasure (-R) R) ≤ I)
(hinfoPos : 0 < priorInformation (-R) R
(sequentialVTPrior R) (sequentialVTPriorDeriv R) +
∫ a, sequentialVTPrior R a *
fisherInformation (transcriptReferenceMeasure P n)
(vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R)
(vtDensityDeriv P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R) a
∂parameterMeasure (-R) R) :
ofReal (contrast v ^ 2 / (I + 40 / R ^ 2)) ≤
localWorstRisk P θ p H n := by
let d := scaledDirection ((sqrt n)⁻¹) v
let bayesError := ∫ z, errorSqField (sequentialVTPrior R)
(vtDensity P θ d p n (-R) R) (vtTarget θ d) (P.estimate n) z
∂((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n))
have hvt := sequential_vanTrees_bayes_lower_bound_of_fisher_control
P θ d p R I n hR hp hinterior (P.estimate n) (P.estimate_measurable n)
herrorSqInt hfisherSqInt hfisher_le hinfoPos
have hreal : contrast v ^ 2 / (I + 40 / R ^ 2) ≤
(n : ℝ) * bayesError := by
calc
contrast v ^ 2 / (I + 40 / R ^ 2) =
((n : ℝ) * contrast d ^ 2) / (I + 40 / R ^ 2) := by
rw [nat_mul_contrast_sampleScaledDirection_sq v hn]
_ = (n : ℝ) * (contrast d ^ 2 / (I + 40 / R ^ 2)) := by ring
_ ≤ (n : ℝ) * bayesError :=
mul_le_mul_of_nonneg_left hvt (Nat.cast_nonneg n)
have hweighted : ofReal ((n : ℝ) * bayesError) =
∫⁻ a, ofReal (sequentialVTPrior R a) *
localRisk P θ (scaledDirection a v) p n
∂parameterMeasure (-R) R := by
exact ofReal_nat_mul_sequentialBayesError_eq_weighted_localRisk
P θ v p R n hR hp hinterior herrorSqInt
have hprior := sequentialVTPrior_facts hR
have hbayesLe :
(∫⁻ a, ofReal (sequentialVTPrior R a) *
localRisk P θ (scaledDirection a v) p n
∂parameterMeasure (-R) R) ≤ localWorstRisk P θ p H n := by
exact (weighted_lintegral_le_uniform
(parameterMeasure (-R) R) (sequentialVTPrior R)
(fun a => localRisk P θ (scaledDirection a v) p n)
(localWorstRisk P θ p H n)
hprior.contDiff.continuous.measurable
(measurable_localRisk_scaledDirection P θ v p n).aemeasurable
(by
unfold sequentialVTPrior
exact smoothPrior_integrable_parameterMeasure (by linarith))
(Filter.Eventually.of_forall hprior.nonneg) hprior.normalized
(by
have hmem : ∀ᵐ a ∂parameterMeasure (-R) R, a ∈ Icc (-R) R := by
unfold parameterMeasure
exact ae_restrict_mem measurableSet_Icc
filter_upwards [hmem] with a ha
apply localRisk_le_localWorstRisk
refine ⟨(scaledDirection_mem_cover v ha).trans hcover, ?_⟩
rw [← parameterPath_sampleScaledDirection_eq_localAlternative]
exact hinterior a ha)).2
exact (ENNReal.ofReal_le_ofReal hreal).trans (hweighted.le.trans hbayesLe)
theorem ofReal_vanTreesEnvelope_le_localWorstRisk_of_fisher_control_allErrors
The finite-sample local-worst-risk lower bound holds for every estimator: the finite-error branch uses van Trees, while a nonintegrable weighted squared error forces the compact-prior Bayes risk and local worst risk to be infinite. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
∀ n i, MeasurableSpace (Z n i)
ε :
ℝ
P :
θ v :
p R H I :
ℝ
n :
ℕ
hn :
0 < n
hR :
0 < R
hp :
hinterior :
hcover :
R * sqrt ((v 0) ^ 2 + (v 1) ^ 2) ≤ H
hfisherSqInt :
Integrable (fun z : ℝ × Transcript (Z n) => sequentialVTPrior R z.1 * vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v) p n (-R) R z.1 z.2 * (likelihoodScore (vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v) p n (-R) R) (vtDensityDeriv P θ (scaledDirection ((sqrt n)⁻¹) v) p n (-R) R) z.1 z.2) ^ 2) ((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n))
hfisher_le :
(∫ a, sequentialVTPrior R a * fisherInformation (transcriptReferenceMeasure P n) (vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v) p n (-R) R) (vtDensityDeriv P θ (scaledDirection ((sqrt n)⁻¹) v) p n (-R) R) a ∂parameterMeasure (-R) R)
≤ I
hinfoPos :
+ ∫ a, sequentialVTPrior R a * fisherInformation (transcriptReferenceMeasure P n) (vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v) p n (-R) R) (vtDensityDeriv P θ (scaledDirection ((sqrt n)⁻¹) v) p n (-R) R) a ∂parameterMeasure (-R) R
conclusion
ofReal (contrast v ^ 2 / (I + 40 / R ^ 2)) ≤ localWorstRisk P θ p H n
Proof (Lean source)
lemma ofReal_vanTreesEnvelope_le_localWorstRisk_of_fisher_control_allErrors
{Z : OutputFamily} [∀ n i, MeasurableSpace (Z n i)] {ε : ℝ}
(P : ProcedureSequence Z ε) (θ v : TrialParameter)
(p R H I : ℝ) (n : ℕ) (hn : 0 < n) (hR : 0 < R)
(hp : InteriorAssignment p)
(hinterior : ∀ a ∈ Icc (-R) R,
InteriorMeans (parameterPath θ
(scaledDirection ((sqrt n)⁻¹) v) a))
(hcover : R * sqrt ((v 0) ^ 2 + (v 1) ^ 2) ≤ H)
(hfisherSqInt : Integrable (fun z : ℝ × Transcript (Z n) =>
sequentialVTPrior R z.1 *
vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R z.1 z.2 *
(likelihoodScore
(vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R)
(vtDensityDeriv P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R) z.1 z.2) ^ 2)
((parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n)))
(hfisher_le :
(∫ a, sequentialVTPrior R a *
fisherInformation (transcriptReferenceMeasure P n)
(vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R)
(vtDensityDeriv P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R) a
∂parameterMeasure (-R) R) ≤ I)
(hinfoPos : 0 < priorInformation (-R) R
(sequentialVTPrior R) (sequentialVTPriorDeriv R) +
∫ a, sequentialVTPrior R a *
fisherInformation (transcriptReferenceMeasure P n)
(vtDensity P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R)
(vtDensityDeriv P θ (scaledDirection ((sqrt n)⁻¹) v)
p n (-R) R) a
∂parameterMeasure (-R) R) :
ofReal (contrast v ^ 2 / (I + 40 / R ^ 2)) ≤
localWorstRisk P θ p H n := by
let d := scaledDirection ((sqrt n)⁻¹) v
let q := vtDensity P θ d p n (-R) R
let g := vtTarget (X := Transcript (Z n)) θ d
let w := sequentialVTPrior R
let mu := (parameterMeasure (-R) R).prod (transcriptReferenceMeasure P n)
let F := errorSqField w q g (P.estimate n)
have hFmeas : Measurable F :=
(sequentialVanTreesProductMeasurable P θ d p R n hR
(P.estimate n) (P.estimate_measurable n)).errorSq
have hFnonneg : ∀ z, 0 ≤ F z := by
intro z
unfold F errorSqField jointDensity w q
exact mul_nonneg (sq_nonneg _)
(mul_nonneg ((sequentialVTPrior_facts hR).nonneg z.1)
(vtDensity_nonneg P θ d p n (-R) R hp hinterior z.1 z.2))
rcases integrable_or_lintegral_ofReal_eq_top F mu hFmeas hFnonneg with
hfinite | hinfinite
· exact ofReal_vanTreesEnvelope_le_localWorstRisk_of_fisher_control
P θ v p R H I n hn hR hp hinterior hcover hfinite
hfisherSqInt hfisher_le hinfoPos
· have hscaledTop :
(∫⁻ z, ofReal ((n : ℝ) * F z) ∂mu) = ∞ := by
have hnTop : ofReal (n : ℝ) ≠ ∞ := ENNReal.ofReal_ne_top
simp_rw [ENNReal.ofReal_mul (Nat.cast_nonneg n)]
rw [lintegral_const_mul' _ _ hnTop, hinfinite]
simp [hn.ne']
have hweightedTop :
(∫⁻ a, ofReal (sequentialVTPrior R a) *
localRisk P θ (scaledDirection a v) p n
∂parameterMeasure (-R) R) = ∞ := by
rw [← lintegral_scaledSequentialBayesError_eq_weighted_localRisk
P θ v p R n hR hp hinterior]
simpa [F, mu, w, q, g, d] using hscaledTop
have hbayesLe := weightedLocalRisk_le_localWorstRisk
P θ v p R H n hR hinterior hcover
have hworstTop : localWorstRisk P θ p H n = ∞ := by
apply top_unique
rw [← hweightedTop]
exact hbayesLe
rw [hworstTop]
exact le_top
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.SparseSaddle 1 declarations The finite linear-program support theorem is applied at an attained minimizer of the upper envelope.
Sparse stationary staircase saddles
The finite linear-program support theorem is applied at an attained minimizer of the upper envelope. It preserves the four normalization moments, the stationarity moment, and the information value, leaving at most five active staircase rays.
theorem exists_sparse_staircase_saddle
Under the supplied quantities and conditions, the exists sparse staircase saddle assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the exists sparse staircase saddle.
Formal statement
hypotheses
Proof (Lean source)
lemma exists_sparse_staircase_saddle (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε) :
∃ α : StaircaseWeight,
staircaseFeasible ε α ∧
Jstar θ p ε =
sInf {u : ℝ | ∃ t : ℝ, u = informationObjective θ p ε α t} ∧
(weightSupport α).card ≤ 5 := by
obtain ⟨t, ht⟩ := upperEnvelope_exists_minimizer θ p ε hp hθ hε
have hleast :
sInf {u : ℝ | ∃ v : ℝ, u = upperEnvelope θ p ε v} =
upperEnvelope θ p ε t := by
apply IsLeast.csInf_eq
exact ⟨⟨t, rfl⟩, by rintro _ ⟨u, rfl⟩; exact ht u⟩
have hJ : upperEnvelope θ p ε t = Jstar θ p ε := by
rw [← hleast, ← finiteOracle_minimax θ p ε hp hθ hε]
obtain ⟨x, hx, hxopt⟩ :=
Jstar_exists_optimal_weight θ p ε hp hθ hε.le
have hxbdd : BddBelow {v : ℝ | ∃ w : ℝ,
v = informationObjective θ p ε x w} := by
refine ⟨0, ?_⟩
rintro _ ⟨u, rfl⟩
exact informationObjective_nonneg_interior θ p ε hp hθ hε.le x hx u
have hxlower (u : ℝ) : Jstar θ p ε ≤ informationObjective θ p ε x u := by
rw [hxopt]
exact csInf_le hxbdd ⟨u, rfl⟩
obtain ⟨zmax, hzmax, hzmaximal⟩ :=
informationObjective_exists_maximizer θ p ε t hε.le
have hzenvelope : upperEnvelope θ p ε t =
informationObjective θ p ε zmax t := by
unfold upperEnvelope
apply IsGreatest.csSup_eq
exact ⟨⟨zmax, hzmax, rfl⟩, by
rintro _ ⟨z, hz, rfl⟩
exact hzmaximal z hz⟩
have hxvalue : informationObjective θ p ε x t = Jstar θ p ε := by
apply le_antisymm
· exact (hzmaximal x hx).trans (by rw [← hzenvelope, hJ])
· exact hxlower t
have hxmax : ∀ z : StaircaseWeight, (∀ s, 0 ≤ z s) →
(∀ j, moment (patternRay ε · j) z = 1) →
moment (patternInformation θ p ε · t) z ≤
moment (patternInformation θ p ε · t) x := by
intro z hz hzmom
have hzfeas : staircaseFeasible ε z := by
refine ⟨hz, ?_⟩
intro j
simpa [moment, staircaseMatrix, mul_comm] using hzmom j
have hzx := hzmaximal z hzfeas
rw [← hzenvelope, hJ, ← hxvalue] at hzx
simpa [moment, informationObjective, mul_comm] using hzx
have hmoment_slope (a : StaircaseWeight) :
moment (patternInformationSlope θ p ε · t) a =
∑ s : Fin 14, 2 * a s * projectedGradient p ε s t *
(patternGradient p ε s 0 + patternGradient p ε s 1) /
patternMass θ p ε s := by
unfold moment patternInformationSlope
apply Finset.sum_congr rfl
intro s _
ring
have hxstationary : moment (patternInformationSlope θ p ε · t) x = 0 := by
have hraw := informationObjective_stationary_of_min θ p ε x t (by
intro u
rw [hxvalue]
exact hxlower u)
rw [hmoment_slope]
exact hraw
obtain ⟨y, hy, hymom, hystat, hyvalue, hycard, _⟩ :=
exists_sparse_maximizer
(fun j s => patternRay ε s j) (fun _ => 1)
(fun s => patternInformationSlope θ p ε s t)
(fun s => patternInformation θ p ε s t) x hx.1
(by
intro j
simpa [moment, staircaseMatrix, mul_comm] using hx.2 j)
hxmax
have hyfeas : staircaseFeasible ε y := by
refine ⟨hy, ?_⟩
intro j
simpa [moment, staircaseMatrix, mul_comm] using hymom j
have hystationary :
∑ s : Fin 14, 2 * y s * projectedGradient p ε s t *
(patternGradient p ε s 0 + patternGradient p ε s 1) /
patternMass θ p ε s = 0 := by
calc
_ = moment (patternInformationSlope θ p ε · t) y :=
(hmoment_slope y).symm
_ = moment (patternInformationSlope θ p ε · t) x := hystat
_ = 0 := hxstationary
have hymin := informationObjective_min_of_stationary θ p ε hp hθ hε.le
y hyfeas t hystationary
have hyat : informationObjective θ p ε y t = Jstar θ p ε := by
calc
informationObjective θ p ε y t =
moment (patternInformation θ p ε · t) y := by
simp [informationObjective, moment, mul_comm]
_ = moment (patternInformation θ p ε · t) x := hyvalue
_ = informationObjective θ p ε x t := by
simp [informationObjective, moment, mul_comm]
_ = Jstar θ p ε := hxvalue
refine ⟨y, hyfeas, ?_, ?_⟩
· rw [← hyat]
symm
apply IsLeast.csInf_eq
exact ⟨⟨t, rfl⟩, by rintro _ ⟨u, rfl⟩; exact hymin u⟩
· norm_num at hycard ⊢
exact hycard
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.TranscriptUniqueness 3 declarations This file shows that the conditional cylinder factorization determines every finite transcript law uniquely.
Uniqueness of finite transcript factorization
This file shows that the conditional cylinder factorization determines every finite transcript law uniquely.
theorem map_historyPrefix_prod_apply
The cylinder factorization determines the joint law of each private history and its next output. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
n :
ℕ
Fin n → Type
∀ i, MeasurableSpace (Z i)
Q :
SequentialKernel (Z := Z)
hQ :
∀ i, IsMarkovKernel (Q i)
hR :
i :
Fin n
conclusion
(R x).map (fun z ↦ (historyPrefix i z, z i))
= (R x).map (historyPrefix i) ⊗ₘ (Q i).comap (fun h ↦ (x i, h)) (by fun_prop)
Proof (Lean source)
lemma TranscriptFactorizes.map_historyPrefix_prod_apply {n : ℕ} {Z : Fin n → Type}
[∀ i, MeasurableSpace (Z i)]
{Q : SequentialKernel (Z := Z)}
{R : (Fin n → Fin 4) → Measure (Transcript Z)}
(hQ : ∀ i, IsMarkovKernel (Q i))
(hR : TranscriptFactorizes Q R) (x : Fin n → Fin 4) (i : Fin n) :
(R x).map (fun z ↦ (historyPrefix i z, z i)) =
(R x).map (historyPrefix i) ⊗ₘ
(Q i).comap (fun h ↦ (x i, h)) (by fun_prop) := by
letI : IsProbabilityMeasure (R x) := (hR x).1
letI : IsMarkovKernel (Q i) := hQ i
have hhist : Measurable (historyPrefix i : Transcript Z → PrivateHistory i) := by
apply measurable_pi_iff.mpr
intro j
exact measurable_pi_apply j.1
have hpair : Measurable (fun z : Transcript Z ↦ (historyPrefix i z, z i)) :=
hhist.prodMk (measurable_pi_apply i)
haveI : IsFiniteMeasure ((R x).map (fun z ↦ (historyPrefix i z, z i))) :=
Measure.isFiniteMeasure_map (R x) _
refine Measure.ext_prod (fun {B A} hB hA ↦ ?_)
rw [Measure.map_apply hpair (hB.prod hA), Measure.compProd_apply_prod hB hA]
change R x {z | historyPrefix i z ∈ B ∧ z i ∈ A} = _
exact (hR x).2 i B A hB hA
theorem map_transcriptPrefix
Every prefix marginal of a factorizing transcript law is the recursively constructed finite-sequence prefix law. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
n k :
ℕ
Fin n → Type
∀ i, MeasurableSpace (Z i)
Q :
SequentialKernel (Z := Z)
hQ :
∀ i, IsMarkovKernel (Q i)
hR :
hk :
k ≤ n
conclusion
Proof (Lean source)
lemma TranscriptFactorizes.map_transcriptPrefix {n k : ℕ} {Z : Fin n → Type}
[∀ i, MeasurableSpace (Z i)]
{Q : SequentialKernel (Z := Z)}
{R : (Fin n → Fin 4) → Measure (Transcript Z)}
(hQ : ∀ i, IsMarkovKernel (Q i)) (hR : TranscriptFactorizes Q R)
(x : Fin n → Fin 4) (hk : k ≤ n) :
(R x).map (transcriptPrefix (Z := Z) hk) =
prefixLaw (finiteSequenceKernel Q) x k hk := by
induction k with
| zero =>
letI : IsProbabilityMeasure (R x) := (hR x).1
let emptyHistory : History Z 0 hk := fun j ↦ j.elim0
have hfun : transcriptPrefix (Z := Z) hk =
(fun _ ↦ emptyHistory) := by
funext z j
exact j.elim0
rw [hfun, Measure.map_const, measure_univ, one_smul]
rfl
| succ k ih =>
have hprev : k ≤ n := Nat.le_of_succ_le hk
let i : Fin n := nextIndex hk
let hpriv : PrivateHistory (Z := Z) i ≃ᵐ History Z k hprev :=
(historyMeasurableEquiv (Z := Z) i).symm
have hjoint := hR.map_historyPrefix_prod_apply hQ x i
have hjoint_fs := congrArg (Measure.map (map hpriv (id : Z i → Z i))) hjoint
have hpair : Measurable (fun z : Transcript Z ↦
(historyPrefix i z, z i)) := by
apply Measurable.prodMk
· apply measurable_pi_iff.mpr
intro j
exact measurable_pi_apply j.1
· exact measurable_pi_apply i
have hleft : map hpriv (id : Z i → Z i) ∘
(fun z : Transcript Z ↦ (historyPrefix i z, z i)) =
(fun z ↦ (transcriptPrefix (Z := Z) hprev z, z i)) := by
funext z
apply Prod.ext
· funext j
dsimp only [hpriv, i]
rfl
· rfl
rw [Measure.map_map (by fun_prop) hpair, hleft] at hjoint_fs
have hright : Measure.map (map hpriv (id : Z i → Z i))
((R x).map (historyPrefix i) ⊗ₘ
(Q i).comap (fun h ↦ (x i, h)) (by fun_prop)) =
(R x).map (transcriptPrefix (Z := Z) hprev) ⊗ₘ
(finiteSequenceKernel Q i).comap
(fun h ↦ (x i, h)) (by fun_prop) := by
let μ := (R x).map (historyPrefix i)
let κ := (Q i).comap (fun h ↦ (x i, h)) (by fun_prop)
letI : IsProbabilityMeasure (R x) := (hR x).1
letI : IsMarkovKernel (Q i) := hQ i
letI : IsMarkovKernel κ := by dsimp [κ]; infer_instance
rw [map_compProd_prodMap_left_eq_compProd_comap
μ hpriv κ]
congr 1
· dsimp only [μ]
rw [Measure.map_map hpriv.measurable (by
apply measurable_pi_iff.mpr
intro j
exact measurable_pi_apply j.1)]
congr 1
rw [hright] at hjoint_fs
calc
(R x).map (transcriptPrefix (Z := Z) hk) =
((R x).map (fun z ↦
(transcriptPrefix (Z := Z) hprev z, z i))).map
(fun p ↦ snoc hk p.1 p.2) := by
rw [Measure.map_map (measurable_snoc hk)
((measurable_transcriptPrefix hprev).prodMk (measurable_pi_apply i))]
congr 1
funext z j
refine Fin.lastCases ?_ (fun l ↦ ?_) j
· exact (Fin.snoc_last
(α := fun j : Fin (k + 1) => Z (castLE hk j))
(p := transcriptPrefix hprev z) (x := z i)).symm
· exact (Fin.snoc_castSucc
(α := fun j : Fin (k + 1) => Z (castLE hk j))
(p := transcriptPrefix hprev z) (x := z i) l).symm
_ = ((R x).map (transcriptPrefix (Z := Z) hprev) ⊗ₘ
(finiteSequenceKernel Q i).comap
(fun h ↦ (x i, h)) (by fun_prop)).map
(fun p ↦ snoc hk p.1 p.2) := by rw [hjoint_fs]
_ = (prefixLaw (finiteSequenceKernel Q) x k hprev ⊗ₘ
(finiteSequenceKernel Q i).comap
(fun h ↦ (x i, h)) (by fun_prop)).map
(fun p ↦ snoc hk p.1 p.2) := by rw [ih hprev]
_ = prefixLaw (finiteSequenceKernel Q) x (k + 1) hk := by
exact (prefixLaw_succ (finiteSequenceKernel Q) x k hk).symm
theorem eq_finiteSequenceTranscript
A factorizing deterministic transcript law equals the canonical finite sequence construction. The stated result follows. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
n :
ℕ
Fin n → Type
∀ i, MeasurableSpace (Z i)
Q :
SequentialKernel (Z := Z)
hQ :
∀ i, IsMarkovKernel (Q i)
hR :
conclusion
R x = finiteSequenceTranscript Q x
Proof (Lean source)
theorem TranscriptFactorizes.eq_finiteSequenceTranscript {n : ℕ} {Z : Fin n → Type}
[∀ i, MeasurableSpace (Z i)]
{Q : SequentialKernel (Z := Z)}
{R : (Fin n → Fin 4) → Measure (Transcript Z)}
(hQ : ∀ i, IsMarkovKernel (Q i)) (hR : TranscriptFactorizes Q R)
(x : Fin n → Fin 4) :
R x = finiteSequenceTranscript Q x := by
have hprefix := hR.map_transcriptPrefix hQ x (k := n) le_rfl
have hcanon := finiteSequenceTranscript_map_prefix Q hQ x le_rfl
have hmaps := hprefix.trans hcanon.symm
have hid : transcriptPrefix (Z := Z) le_rfl =
(id : Transcript Z → Transcript Z) := by
funext z i
rfl
rw [hid] at hmaps
change Measure.map (id : Transcript Z → Transcript Z) (R x) =
Measure.map (id : Transcript Z → Transcript Z) (finiteSequenceTranscript Q x) at hmaps
simpa only [Measure.map_id] using hmaps
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.Helpers.TreatmentSwap 47 declarations Treatment-arm swap symmetry for centered staircase designs
Treatment-arm swap symmetry for centered staircase designs
def swapTreatmentPattern
Transpose the two-by-two treatment/outcome membership table of a staircase ray. For the displayed inputs and conditions, the stated result follows.
the swapped support is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
given by
K.image swapTreatmentPattern
def swapTreatmentInput
Exchange the two treatment arms in the four-point input alphabet. The stated result follows.
def swapTreatmentWeight
the swap treatment weight is the mathematical object specified below. The swap Treatment Weight is determined by the displayed parameters.
Definition (Lean source)
theorem swapTreatmentPattern_involutive
the swap treatment pattern involutive assertion holds.
Formal statement
conclusion
Proof (Lean source)
lemma swapTreatmentPattern_involutive : Involutive swapTreatmentPattern := by
intro s
fin_cases s <;> rfl
theorem swapTreatmentInput_involutive
the swap treatment input involutive assertion holds.
Formal statement
conclusion
Proof (Lean source)
lemma swapTreatmentInput_involutive : Involutive swapTreatmentInput := by
intro j
fin_cases j <;> rfl
theorem swapTreatmentInput_val
Under the supplied quantities and conditions, the swap treatment input val assertion holds.
Formal statement
Proof (Lean source)
lemma swapTreatmentInput_val (j : Fin 4) :
(swapTreatmentInput j).val = (j.val + 2) % 4 := by
fin_cases j <;> decide
def swapTreatmentPatternEquiv
the swap treatment pattern equiv is the mathematical object specified below.
Definition (Lean source)
given by
clause 1
toFun := swapTreatmentPattern
clause 2
invFun := swapTreatmentPattern
clause 3
left_inv := swapTreatmentPattern_involutive
clause 4
right_inv := swapTreatmentPattern_involutive
def swapTreatmentInputEquiv
the swap treatment input equiv is the mathematical object specified below.
Definition (Lean source)
given by
clause 1
toFun := swapTreatmentInput
clause 2
invFun := swapTreatmentInput
clause 3
left_inv := swapTreatmentInput_involutive
clause 4
right_inv := swapTreatmentInput_involutive
theorem patternRay_swapTreatment
Under the supplied quantities and conditions, the pattern ray swap treatment assertion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma patternRay_swapTreatment (ε : ℝ) (s : Fin 14) (j : Fin 4) :
patternRay ε (swapTreatmentPattern s) (swapTreatmentInput j) = patternRay ε s j := by
fin_cases s <;> fin_cases j <;>
simp [swapTreatmentPattern, swapTreatmentInput, patternRay, patternContains,
Nat.testBit]
theorem swapTreatmentWeight_involutive
the swap treatment weight involutive assertion holds.
Formal statement
conclusion
Proof (Lean source)
lemma swapTreatmentWeight_involutive : Involutive swapTreatmentWeight := by
intro α
funext s
simp [swapTreatmentWeight, swapTreatmentPattern_involutive s]
theorem staircaseMatrix_swapTreatment
Under the supplied quantities and conditions, the staircase matrix swap treatment assertion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma staircaseMatrix_swapTreatment (ε : ℝ) (α : StaircaseWeight) (j : Fin 4) :
staircaseMatrix ε (swapTreatmentWeight α) j =
staircaseMatrix ε α (swapTreatmentInput j) := by
unfold staircaseMatrix swapTreatmentWeight
rw [← swapTreatmentPatternEquiv.sum_comp
(fun s => α s * patternRay ε s (swapTreatmentInput j))]
apply Finset.sum_congr rfl
intro s _
change α (swapTreatmentPattern s) * patternRay ε s j =
α (swapTreatmentPattern s) *
patternRay ε (swapTreatmentPattern s) (swapTreatmentInput j)
rw [patternRay_swapTreatment]
theorem staircaseFeasible_swapTreatment
the staircase feasible swap treatment assertion holds.
Formal statement
hypotheses
ε :
ℝ
α :
conclusion
Proof (Lean source)
lemma staircaseFeasible_swapTreatment (ε : ℝ) (α : StaircaseWeight) :
staircaseFeasible ε (swapTreatmentWeight α) ↔ staircaseFeasible ε α := by
constructor <;> intro h
· constructor
· intro s
have := h.1 (swapTreatmentPattern s)
simpa [swapTreatmentWeight, swapTreatmentPattern_involutive s] using this
· intro j
have hm := h.2 (swapTreatmentInput j)
rw [staircaseMatrix_swapTreatment, swapTreatmentInput_involutive j] at hm
exact hm
· constructor
· intro s
exact h.1 (swapTreatmentPattern s)
· intro j
rw [staircaseMatrix_swapTreatment]
exact h.2 (swapTreatmentInput j)
theorem staircaseSupport_swapTreatment
the staircase support swap treatment assertion holds.
Formal statement
hypotheses
α :
conclusion
Proof (Lean source)
lemma staircaseSupport_swapTreatment (α : StaircaseWeight) :
staircaseSupport (swapTreatmentWeight α) = swappedSupport (staircaseSupport α) := by
ext s
simp only [staircaseSupport, mem_filter, Finset.mem_univ, true_and,
swappedSupport, mem_image]
constructor
· intro hs
refine ⟨swapTreatmentPattern s, ?_, swapTreatmentPattern_involutive s⟩
simpa [swapTreatmentWeight, swapTreatmentPattern_involutive s] using hs
· rintro ⟨u, hu, rfl⟩
simpa [swapTreatmentWeight, swapTreatmentPattern_involutive u] using hu
theorem centered_piTheta_swapTreatment
Under the supplied quantities and conditions, the centered pi theta swap treatment assertion holds.
Formal statement
hypotheses
p :
ℝ
j :
Fin 4
conclusion
piTheta (fun _ => (1 / 2 : ℝ)) p (swapTreatmentInput j)
= piTheta (fun _ => (1 / 2 : ℝ)) (1 - p) j
Proof (Lean source)
lemma centered_piTheta_swapTreatment (p : ℝ) (j : Fin 4) :
piTheta (fun _ => (1 / 2 : ℝ)) p (swapTreatmentInput j) =
piTheta (fun _ => (1 / 2 : ℝ)) (1 - p) j := by
rw [centered_piTheta, centered_piTheta]
rw [swapTreatmentInput_val]
fin_cases j <;> norm_num <;> ring
theorem centered_patternMass_swapTreatment
Under the supplied quantities and conditions, the centered pattern mass swap treatment assertion holds.
Formal statement
hypotheses
p ε :
ℝ
s :
Fin 14
conclusion
patternMass (fun _ => (1 / 2 : ℝ)) p ε (swapTreatmentPattern s)
= patternMass (fun _ => (1 / 2 : ℝ)) (1 - p) ε s
Proof (Lean source)
lemma centered_patternMass_swapTreatment (p ε : ℝ) (s : Fin 14) :
patternMass (fun _ => (1 / 2 : ℝ)) p ε (swapTreatmentPattern s) =
patternMass (fun _ => (1 / 2 : ℝ)) (1 - p) ε s := by
unfold patternMass
rw [← swapTreatmentInputEquiv.sum_comp
(fun j => piTheta (fun _ => (1 / 2 : ℝ)) p j *
patternRay ε (swapTreatmentPattern s) j)]
apply Finset.sum_congr rfl
intro j _
change piTheta (fun _ => (1 / 2 : ℝ)) p (swapTreatmentInput j) *
patternRay ε (swapTreatmentPattern s) (swapTreatmentInput j) = _
rw [centered_piTheta_swapTreatment, patternRay_swapTreatment]
theorem patternGradient_swapTreatment
Under the supplied quantities and conditions, the pattern gradient swap treatment assertion holds.
Formal statement
Proof (Lean source)
lemma patternGradient_swapTreatment (p ε : ℝ) (s : Fin 14) :
patternGradient p ε (swapTreatmentPattern s) 0 =
patternGradient (1 - p) ε s 1 ∧
patternGradient p ε (swapTreatmentPattern s) 1 =
patternGradient (1 - p) ε s 0 := by
fin_cases s <;>
simp [patternGradient, swapTreatmentPattern, patternContains, Nat.testBit,
controlProb] <;> ring
theorem projectedGradient_swapTreatment
Under the supplied quantities and conditions, the projected gradient swap treatment assertion holds.
Formal statement
Proof (Lean source)
lemma projectedGradient_swapTreatment (p ε t : ℝ) (s : Fin 14) :
projectedGradient p ε (swapTreatmentPattern s) t =
-projectedGradient (1 - p) ε s (-1 - t) := by
obtain ⟨h0, h1⟩ := patternGradient_swapTreatment p ε s
simp [projectedGradient, direction, Fin.sum_univ_two, h0, h1]
ring
theorem centered_patternInformation_swapTreatment
Under the supplied quantities and conditions, the centered pattern information swap treatment assertion holds.
Formal statement
hypotheses
p ε t :
ℝ
s :
Fin 14
conclusion
patternInformation (fun _ => (1 / 2 : ℝ)) p ε (swapTreatmentPattern s) t
= patternInformation (fun _ => (1 / 2 : ℝ)) (1 - p) ε s (-1 - t)
Proof (Lean source)
lemma centered_patternInformation_swapTreatment (p ε t : ℝ) (s : Fin 14) :
patternInformation (fun _ => (1 / 2 : ℝ)) p ε
(swapTreatmentPattern s) t =
patternInformation (fun _ => (1 / 2 : ℝ)) (1 - p) ε s (-1 - t) := by
rw [patternInformation, patternInformation, centered_patternMass_swapTreatment,
projectedGradient_swapTreatment]
ring
theorem centered_informationObjective_swapTreatment
Under the supplied quantities and conditions, the centered information objective swap treatment assertion holds.
Formal statement
hypotheses
p ε t :
ℝ
α :
conclusion
informationObjective (fun _ => (1 / 2 : ℝ)) p ε (swapTreatmentWeight α) t
= informationObjective (fun _ => (1 / 2 : ℝ)) (1 - p) ε α (-1 - t)
Proof (Lean source)
lemma centered_informationObjective_swapTreatment
(p ε t : ℝ) (α : StaircaseWeight) :
informationObjective (fun _ => (1 / 2 : ℝ)) p ε
(swapTreatmentWeight α) t =
informationObjective (fun _ => (1 / 2 : ℝ)) (1 - p) ε α (-1 - t) := by
unfold informationObjective swapTreatmentWeight
rw [← swapTreatmentPatternEquiv.sum_comp
(fun s => α s * patternInformation
(fun _ => (1 / 2 : ℝ)) (1 - p) ε s (-1 - t))]
apply Finset.sum_congr rfl
intro s _
change α (swapTreatmentPattern s) *
patternInformation (fun _ => (1 / 2 : ℝ)) p ε s t =
α (swapTreatmentPattern s) *
patternInformation (fun _ => (1 / 2 : ℝ)) (1 - p) ε
(swapTreatmentPattern s) (-1 - t)
rw [← centered_patternInformation_swapTreatment p ε t
(swapTreatmentPattern s), swapTreatmentPattern_involutive]
theorem centered_profileSet_swapTreatment
Under the supplied quantities and conditions, the centered profile set swap treatment assertion holds.
Formal statement
hypotheses
p ε :
ℝ
α :
conclusion
{u : ℝ | ∃ t : ℝ, u = informationObjective (fun _ => (1 / 2 : ℝ)) p ε (swapTreatmentWeight α) t}
= {u : ℝ | ∃ t : ℝ, u = informationObjective (fun _ => (1 / 2 : ℝ)) (1 - p) ε α t}
Proof (Lean source)
lemma centered_profileSet_swapTreatment (p ε : ℝ) (α : StaircaseWeight) :
{u : ℝ | ∃ t : ℝ, u = informationObjective
(fun _ => (1 / 2 : ℝ)) p ε (swapTreatmentWeight α) t} =
{u : ℝ | ∃ t : ℝ, u = informationObjective
(fun _ => (1 / 2 : ℝ)) (1 - p) ε α t} := by
ext u
constructor
· rintro ⟨t, rfl⟩
exact ⟨-1 - t, centered_informationObjective_swapTreatment p ε t α⟩
· rintro ⟨t, rfl⟩
refine ⟨-1 - t, ?_⟩
rw [centered_informationObjective_swapTreatment]
congr 2
ring
theorem centered_Jstar_swapTreatment
Under the supplied quantities and conditions, the centered jstar swap treatment assertion holds.
Formal statement
Proof (Lean source)
lemma centered_Jstar_swapTreatment (p ε : ℝ) :
Jstar (fun _ => (1 / 2 : ℝ)) p ε =
Jstar (fun _ => (1 / 2 : ℝ)) (1 - p) ε := by
unfold Jstar
congr 1
ext z
constructor
· rintro ⟨α, hα, rfl⟩
refine ⟨swapTreatmentWeight α,
(staircaseFeasible_swapTreatment ε α).2 hα, ?_⟩
have hs := congrArg sInf
(centered_profileSet_swapTreatment (1 - p) ε α)
convert hs.symm using 1 <;> ring
· rintro ⟨α, hα, rfl⟩
refine ⟨swapTreatmentWeight α,
(staircaseFeasible_swapTreatment ε α).2 hα, ?_⟩
exact (congrArg sInf
(centered_profileSet_swapTreatment p ε α)).symm
theorem centered_Vstar_swapTreatment
Under the supplied quantities and conditions, the centered vstar swap treatment assertion holds.
Formal statement
Proof (Lean source)
lemma centered_Vstar_swapTreatment (p ε : ℝ) :
Vstar (fun _ => (1 / 2 : ℝ)) p ε =
Vstar (fun _ => (1 / 2 : ℝ)) (1 - p) ε := by
rw [Vstar, Vstar, centered_Jstar_swapTreatment]
def swapTreatmentChannel
the swap treatment channel is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
given by
Q.comap swapTreatmentInput (measurable_of_finite _)
theorem swapTreatmentChannel_apply
Under the supplied quantities and conditions, the swap treatment channel apply assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
swapTreatmentChannel Q j = Q (swapTreatmentInput j)
Proof (Lean source)
lemma swapTreatmentChannel_apply (Q : Kernel (Fin 4) Z) (j : Fin 4) :
swapTreatmentChannel Q j = Q (swapTreatmentInput j) := rfl
theorem stationaryLDP_swapTreatment
the stationary ldp swap treatment assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma stationaryLDP_swapTreatment (ε : ℝ) (Q : Kernel (Fin 4) Z)
(hQ : StationaryLDP ε Q) : StationaryLDP ε (swapTreatmentChannel Q) := by
refine ⟨⟨fun j => hQ.1.isProbabilityMeasure (swapTreatmentInput j)⟩, ?_⟩
intro A hA j k
exact hQ.2 A hA (swapTreatmentInput j) (swapTreatmentInput k)
theorem dominatingMeasure_swapTreatment
the dominating measure swap treatment assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma dominatingMeasure_swapTreatment (Q : Kernel (Fin 4) Z) :
dominatingMeasure (swapTreatmentChannel Q) = dominatingMeasure Q := by
unfold dominatingMeasure
rw [← swapTreatmentInputEquiv.sum_comp (fun j => Q j)]
rfl
theorem channelDensity_swapTreatment
Under the supplied quantities and conditions, the channel density swap treatment assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma channelDensity_swapTreatment (Q : Kernel (Fin 4) Z) (j : Fin 4) :
channelDensity (swapTreatmentChannel Q) j =
channelDensity Q (swapTreatmentInput j) := by
funext z
simp only [channelDensity, swapTreatmentChannel_apply,
dominatingMeasure_swapTreatment]
theorem centered_channelOutputDensity_swapTreatment
Under the supplied quantities and conditions, the centered channel output density swap treatment assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
channelOutputDensity (fun _ => (1 / 2 : ℝ)) (1 - p) (swapTreatmentChannel Q) z
= channelOutputDensity (fun _ => (1 / 2 : ℝ)) p Q z
Proof (Lean source)
lemma centered_channelOutputDensity_swapTreatment (p : ℝ)
(Q : Kernel (Fin 4) Z) (z : Z) :
channelOutputDensity (fun _ => (1 / 2 : ℝ)) (1 - p)
(swapTreatmentChannel Q) z =
channelOutputDensity (fun _ => (1 / 2 : ℝ)) p Q z := by
unfold channelOutputDensity
rw [← swapTreatmentInputEquiv.sum_comp
(fun j => piTheta (fun _ => (1 / 2 : ℝ)) p j * channelDensity Q j z)]
apply Finset.sum_congr rfl
intro j _
rw [channelDensity_swapTreatment]
change piTheta (fun _ => (1 / 2 : ℝ)) (1 - p) j *
channelDensity Q (swapTreatmentInput j) z = _
rw [← centered_piTheta_swapTreatment]
rfl
theorem centered_channelOutputDensity_swapTreatment_invTwo
Under the supplied quantities and conditions, the centered channel output density swap treatment inv two assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
channelOutputDensity (fun _ => (2 : ℝ)⁻¹) (1 - p) (swapTreatmentChannel Q) z
= channelOutputDensity (fun _ => (2 : ℝ)⁻¹) p Q z
Proof (Lean source)
lemma centered_channelOutputDensity_swapTreatment_invTwo (p : ℝ)
(Q : Kernel (Fin 4) Z) (z : Z) :
channelOutputDensity (fun _ => (2 : ℝ)⁻¹) (1 - p)
(swapTreatmentChannel Q) z =
channelOutputDensity (fun _ => (2 : ℝ)⁻¹) p Q z := by
convert centered_channelOutputDensity_swapTreatment p Q z <;> norm_num
theorem inputDerivative_swapTreatment
Under the supplied quantities and conditions, the input derivative swap treatment assertion holds.
Formal statement
Proof (Lean source)
lemma inputDerivative_swapTreatment (p : ℝ) (j : Fin 4) :
inputDerivative (1 - p) j 0 = inputDerivative p (swapTreatmentInput j) 1 ∧
inputDerivative (1 - p) j 1 = inputDerivative p (swapTreatmentInput j) 0 := by
fin_cases j <;> simp [inputDerivative, swapTreatmentInput, controlProb] <;> ring
theorem channelDerivativeDensity_swapTreatment
Under the supplied quantities and conditions, the channel derivative density swap treatment assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
conclusion 1
channelDerivativeDensity (1 - p) (swapTreatmentChannel Q) 0 z
= channelDerivativeDensity p Q 1 z
conclusion 2
channelDerivativeDensity (1 - p) (swapTreatmentChannel Q) 1 z
= channelDerivativeDensity p Q 0 z
Proof (Lean source)
lemma channelDerivativeDensity_swapTreatment (p : ℝ) (Q : Kernel (Fin 4) Z)
(z : Z) :
channelDerivativeDensity (1 - p) (swapTreatmentChannel Q) 0 z =
channelDerivativeDensity p Q 1 z ∧
channelDerivativeDensity (1 - p) (swapTreatmentChannel Q) 1 z =
channelDerivativeDensity p Q 0 z := by
constructor
· unfold channelDerivativeDensity
rw [← swapTreatmentInputEquiv.sum_comp
(fun j => inputDerivative p j 1 * channelDensity Q j z)]
apply Finset.sum_congr rfl
intro j _
rw [channelDensity_swapTreatment]
change inputDerivative (1 - p) j 0 *
channelDensity Q (swapTreatmentInput j) z =
inputDerivative p (swapTreatmentInput j) 1 *
channelDensity Q (swapTreatmentInput j) z
rw [(inputDerivative_swapTreatment p j).1]
· unfold channelDerivativeDensity
rw [← swapTreatmentInputEquiv.sum_comp
(fun j => inputDerivative p j 0 * channelDensity Q j z)]
apply Finset.sum_congr rfl
intro j _
rw [channelDensity_swapTreatment]
change inputDerivative (1 - p) j 1 *
channelDensity Q (swapTreatmentInput j) z =
inputDerivative p (swapTreatmentInput j) 0 *
channelDensity Q (swapTreatmentInput j) z
rw [(inputDerivative_swapTreatment p j).2]
def swapTreatmentParameter
For the supplied quantities and conditions, the swap treatment parameter is the mathematical object specified below.
def swapTreatmentMatrix
the swap treatment matrix is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
given by
fun a b => I (swapTreatmentParameter a) (swapTreatmentParameter b)
theorem channelDerivativeDensity_swapTreatment_zero
Under the supplied quantities and conditions, the channel derivative density swap treatment zero assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
channelDerivativeDensity (1 - p) (swapTreatmentChannel Q) 0
= channelDerivativeDensity p Q 1
Proof (Lean source)
lemma channelDerivativeDensity_swapTreatment_zero (p : ℝ) (Q : Kernel (Fin 4) Z) :
channelDerivativeDensity (1 - p) (swapTreatmentChannel Q) 0 =
channelDerivativeDensity p Q 1 := by
funext z
exact (channelDerivativeDensity_swapTreatment p Q z).1
theorem channelDerivativeDensity_swapTreatment_one
Under the supplied quantities and conditions, the channel derivative density swap treatment one assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
channelDerivativeDensity (1 - p) (swapTreatmentChannel Q) 1
= channelDerivativeDensity p Q 0
Proof (Lean source)
lemma channelDerivativeDensity_swapTreatment_one (p : ℝ) (Q : Kernel (Fin 4) Z) :
channelDerivativeDensity (1 - p) (swapTreatmentChannel Q) 1 =
channelDerivativeDensity p Q 0 := by
funext z
exact (channelDerivativeDensity_swapTreatment p Q z).2
theorem channelFisherInfo_swapTreatment
Under the supplied quantities and conditions, the channel fisher info swap treatment assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
channelFisherInfo (fun _ => (1 / 2 : ℝ)) (1 - p) (swapTreatmentChannel Q)
= swapTreatmentMatrix (channelFisherInfo (fun _ => (1 / 2 : ℝ)) p Q)
Proof (Lean source)
lemma channelFisherInfo_swapTreatment (p : ℝ) (Q : Kernel (Fin 4) Z) :
channelFisherInfo (fun _ => (1 / 2 : ℝ)) (1 - p)
(swapTreatmentChannel Q) =
swapTreatmentMatrix
(channelFisherInfo (fun _ => (1 / 2 : ℝ)) p Q) := by
ext a b
fin_cases a <;> fin_cases b <;>
simp [channelFisherInfo, swapTreatmentMatrix, swapTreatmentParameter,
dominatingMeasure_swapTreatment,
channelDerivativeDensity_swapTreatment_zero,
channelDerivativeDensity_swapTreatment_one] <;>
apply MeasureTheory.integral_congr_ae <;>
filter_upwards with z <;>
rw [centered_channelOutputDensity_swapTreatment_invTwo]
theorem centered_outputLaw_swapTreatment
Under the supplied quantities and conditions, the centered output law swap treatment assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
outputLaw (fun _ => (1 / 2 : ℝ)) (1 - p) (swapTreatmentChannel Q)
= outputLaw (fun _ => (1 / 2 : ℝ)) p Q
Proof (Lean source)
lemma centered_outputLaw_swapTreatment (p : ℝ) (Q : Kernel (Fin 4) Z) :
outputLaw (fun _ => (1 / 2 : ℝ)) (1 - p) (swapTreatmentChannel Q) =
outputLaw (fun _ => (1 / 2 : ℝ)) p Q := by
unfold outputLaw
rw [← swapTreatmentInputEquiv.sum_comp
(fun j => ofReal (piTheta (fun _ => (1 / 2 : ℝ)) p j) • Q j)]
apply Finset.sum_congr rfl
intro j _
change ofReal (piTheta (fun _ => (1 / 2 : ℝ)) (1 - p) j) •
Q (swapTreatmentInput j) = _
rw [← centered_piTheta_swapTreatment]
rfl
theorem centered_outputCardinality_swapTreatment
Under the supplied quantities and conditions, the centered output cardinality swap treatment assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
outputCardinality (fun _ => (1 / 2 : ℝ)) (1 - p) (swapTreatmentChannel Q)
= outputCardinality (fun _ => (1 / 2 : ℝ)) p Q
Proof (Lean source)
lemma centered_outputCardinality_swapTreatment (p : ℝ) (Q : Kernel (Fin 4) Z) :
outputCardinality (fun _ => (1 / 2 : ℝ)) (1 - p)
(swapTreatmentChannel Q) =
outputCardinality (fun _ => (1 / 2 : ℝ)) p Q := by
classical
unfold outputCardinality
split <;> simp_all only [centered_outputLaw_swapTreatment]
def reflectTreatmentParameter
For the supplied quantities and conditions, the reflect treatment parameter is the mathematical object specified below.
Definition (Lean source)
theorem reflectTreatmentParameter_involutive
the reflect treatment parameter involutive assertion holds.
Formal statement
conclusion
Proof (Lean source)
lemma reflectTreatmentParameter_involutive :
Involutive reflectTreatmentParameter := by
intro v
funext k
fin_cases k <;> simp [reflectTreatmentParameter, swapTreatmentParameter]
theorem swapTreatmentMatrix_mulVec
the swap treatment matrix mul vec assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
= -(I.mulVec v (swapTreatmentParameter k))
Proof (Lean source)
lemma swapTreatmentMatrix_mulVec (I : Matrix (Fin 2) (Fin 2) ℝ)
(v : TrialParameter) (k : Fin 2) :
(swapTreatmentMatrix I).mulVec (reflectTreatmentParameter v) k =
-(I.mulVec v (swapTreatmentParameter k)) := by
fin_cases k <;>
simp [swapTreatmentMatrix, reflectTreatmentParameter, swapTreatmentParameter,
mulVec, Fin.sum_univ_two] <;> ring
theorem neg_contrastVector_swap
Under the supplied quantities and conditions, the neg contrast vector swap assertion holds.
Formal statement
Proof (Lean source)
lemma neg_contrastVector_swap (k : Fin 2) :
-contrastVector (swapTreatmentParameter k) = contrastVector k := by
fin_cases k <;> simp [contrastVector, swapTreatmentParameter]
theorem contrastInRange_swapTreatment
the contrast in range swap treatment assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma contrastInRange_swapTreatment (I : Matrix (Fin 2) (Fin 2) ℝ) :
contrastInRange (swapTreatmentMatrix I) ↔ contrastInRange I := by
constructor
· rintro ⟨v, hv⟩
refine ⟨reflectTreatmentParameter v, ?_⟩
funext k
have h0 := congrFun hv 0
have h1 := congrFun hv 1
fin_cases k <;>
simp [swapTreatmentMatrix, reflectTreatmentParameter, swapTreatmentParameter,
mulVec, Fin.sum_univ_two, contrastVector] at h0 h1 ⊢ <;> linarith
· rintro ⟨v, hv⟩
refine ⟨reflectTreatmentParameter v, ?_⟩
funext k
rw [swapTreatmentMatrix_mulVec, hv, neg_contrastVector_swap]
theorem contrastPairing_reflect
Under the supplied quantities and conditions, the contrast pairing reflect assertion holds.
Formal statement
Proof (Lean source)
lemma contrastPairing_reflect (v : TrialParameter) :
∑ k : Fin 2, contrastVector k * reflectTreatmentParameter v k =
∑ k : Fin 2, contrastVector k * v k := by
simp [contrastVector, reflectTreatmentParameter, swapTreatmentParameter,
Fin.sum_univ_two]
theorem contrastSolutionValues_swapTreatment
the contrast solution values swap treatment assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
{u : ℝ | ∃ v : TrialParameter, (swapTreatmentMatrix I).mulVec v = contrastVector ∧ u = ∑ k : Fin 2, contrastVector k * v k}
= {u : ℝ | ∃ v : TrialParameter, I.mulVec v = contrastVector ∧ u = ∑ k : Fin 2, contrastVector k * v k}
Proof (Lean source)
lemma contrastSolutionValues_swapTreatment (I : Matrix (Fin 2) (Fin 2) ℝ) :
{u : ℝ | ∃ v : TrialParameter,
(swapTreatmentMatrix I).mulVec v = contrastVector ∧
u = ∑ k : Fin 2, contrastVector k * v k} =
{u : ℝ | ∃ v : TrialParameter, I.mulVec v = contrastVector ∧
u = ∑ k : Fin 2, contrastVector k * v k} := by
ext u
constructor
· rintro ⟨v, hv, rfl⟩
refine ⟨reflectTreatmentParameter v, ?_, contrastPairing_reflect v |>.symm⟩
funext k
have h0 := congrFun hv 0
have h1 := congrFun hv 1
fin_cases k <;>
simp [swapTreatmentMatrix, reflectTreatmentParameter, swapTreatmentParameter,
mulVec, Fin.sum_univ_two, contrastVector] at h0 h1 ⊢ <;> linarith
· rintro ⟨v, hv, rfl⟩
refine ⟨reflectTreatmentParameter v, ?_, contrastPairing_reflect v |>.symm⟩
funext k
rw [swapTreatmentMatrix_mulVec, hv, neg_contrastVector_swap]
theorem contrastVariance_swapTreatment
the contrast variance swap treatment assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma contrastVariance_swapTreatment (I : Matrix (Fin 2) (Fin 2) ℝ) :
contrastVariance (swapTreatmentMatrix I) = contrastVariance I := by
classical
unfold contrastVariance
by_cases h : contrastInRange I
· have hs : contrastInRange (swapTreatmentMatrix I) :=
(contrastInRange_swapTreatment I).2 h
rw [if_pos h, if_pos hs, contrastSolutionValues_swapTreatment]
· have hs : ¬ contrastInRange (swapTreatmentMatrix I) := by
simpa [contrastInRange_swapTreatment I] using h
rw [if_neg h, if_neg hs]
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.TBalancedReduction 20 declarations When assignment is balanced and the arm means sum to one, randomized response to the signed treatment–outcome bit attains the information oracle.
Balanced randomized-response reduction
When assignment is balanced and the arm means sum to one, randomized response to the signed treatment–outcome bit attains the information oracle.
def signedBit
the signed bit is the mathematical object specified below.
the balanced rr is the mathematical object specified below. The balanced RR is determined by the displayed parameters.
theorem balancedRR_atom
Under the supplied quantities and conditions, the balanced rr atom assertion holds.
Formal statement
Proof (Lean source)
lemma balancedRR_atom (ε : ℝ) (j : Fin 4) (z : Fin 2) :
((balancedRR ε j) {z}).toReal =
if z = signedBit j then exp ε / (exp ε + 1)
else 1 / (exp ε + 1) := by
have he : 0 < exp ε := Real.exp_pos ε
have hd : 0 < exp ε + 1 := by positivity
change (Measure.count.withDensity (fun u : Fin 2 => ofReal
(if u = signedBit j then exp ε / (exp ε + 1)
else 1 / (exp ε + 1))) {z}).toReal = _
rw [withDensity_apply _ (measurableSet_singleton z), lintegral_singleton]
have hc : Measure.count {z} = 1 := by simp
rw [hc, mul_one]
split_ifs <;> rw [ENNReal.toReal_ofReal] <;> positivity
theorem grouped_balancedRR_atom
Under the supplied quantities and conditions, the grouped balanced rr atom assertion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma grouped_balancedRR_atom (ε : ℝ) (b : Bool) (z : Fin 2) :
((groupedChannel (balancedRR ε) b) {z}).toReal =
if b = (z = 1) then exp ε / (exp ε + 1)
else 1 / (exp ε + 1) := by
have hfinite (j : Fin 4) : balancedRR ε j {z} ≠ ∞ := by
intro hj
have h := balancedRR_atom ε j z
rw [hj] at h
simp only [ENNReal.toReal_top] at h
split_ifs at h
· have hp : 0 < exp ε / (exp ε + 1) := by positivity
linarith
· have hq : 0 < 1 / (exp ε + 1) := by positivity
linarith
have hhalfReal : half.toReal = (1 / 2 : ℝ) := by rfl
cases b
· change (half * (balancedRR ε 1 {z} + balancedRR ε 2 {z})).toReal = _
rw [ENNReal.toReal_mul,
ENNReal.toReal_add (hfinite 1) (hfinite 2)]
rw [balancedRR_atom, balancedRR_atom]
fin_cases z <;> rw [hhalfReal] <;> simp +decide [signedBit] <;>
field_simp <;> ring
· change (half * (balancedRR ε 0 {z} + balancedRR ε 3 {z})).toReal = _
rw [ENNReal.toReal_mul,
ENNReal.toReal_add (hfinite 0) (hfinite 3)]
rw [balancedRR_atom, balancedRR_atom]
fin_cases z <;> rw [hhalfReal] <;> simp +decide [signedBit] <;>
field_simp <;> ring
theorem randomizedResponse_atom
Under the supplied quantities and conditions, the randomized response atom assertion holds.
Formal statement
Proof (Lean source)
lemma randomizedResponse_atom (ε : ℝ) (b z : Bool) :
((randomizedResponse ε b) {z}).toReal =
if b = z then exp ε / (exp ε + 1)
else 1 / (exp ε + 1) := by
let e := exp ε
let p := e / (e + 1)
let q := 1 / (e + 1)
have he : 0 < e := by dsimp [e]; positivity
have he1 : 0 < e + 1 := by linarith
have hp : 0 < p := div_pos he he1
have hq : 0 < q := div_pos (by norm_num) he1
have hcq : 1 - q = p := by
dsimp [p, q]
field_simp
ring
have hcp : 1 - p = q := by
dsimp [p, q]
field_simp
ring
have hqi : (exp ε + 1)⁻¹ = q := by
dsimp [q, e]
rw [one_div]
change ((bernoulliBool
(responseProbability ε b)) {z}).toReal = _
cases b <;> cases z <;>
simp [responseProbability,
bernoulliBool,
Measure.add_apply, Measure.smul_apply]
all_goals try rw [hqi]
all_goals try positivity
all_goals first
| (change (ofReal (1 - q)).toReal = p
rw [hcq, ENNReal.toReal_ofReal hp.le])
| (change (ofReal q).toReal = q
rw [ENNReal.toReal_ofReal hq.le])
| (change (ofReal (1 - p)).toReal = q
rw [hcp, ENNReal.toReal_ofReal hq.le])
| (change (ofReal p).toReal = p
rw [ENNReal.toReal_ofReal hp.le])
Under the supplied quantities and conditions, the pi theta balanced by val assertion holds.
Formal statement
hypotheses
θ :
j :
Fin 4
conclusion
piTheta θ (1 / 2) j
= if j.val = 0 then controlProb (1 / 2) * (1 - θ 0) else if j.val = 1 then controlProb (1 / 2) * θ 0 else if j.val = 2 then (1 / 2) * (1 - θ 1) else (1 / 2) * θ 1
Proof (Lean source)
lemma piTheta_balanced_by_val (θ : TrialParameter) (j : Fin 4) :
piTheta θ (1 / 2) j =
if j.val = 0 then controlProb (1 / 2) * (1 - θ 0)
else if j.val = 1 then controlProb (1 / 2) * θ 0
else if j.val = 2 then (1 / 2) * (1 - θ 1)
else (1 / 2) * θ 1 := by
rcases j with ⟨j, hj⟩
interval_cases j <;> rfl
theorem signedBit_probability_one
the signed bit probability one assertion holds.
Formal statement
hypotheses
θ :
Proof (Lean source)
lemma signedBit_probability_one (θ : TrialParameter) :
(∑ j : Fin 4, if signedBit j = 1 then piTheta θ (1 / 2) j else 0) =
(1 + contrast θ) / 2 := by
simp +decide [signedBit, contrast, Fin.sum_univ_succ]
rw [← one_div]
rw [piTheta_balanced_by_val, piTheta_balanced_by_val]
norm_num [controlProb]
ring
theorem signedBit_probability_zero
the signed bit probability zero assertion holds.
Formal statement
hypotheses
θ :
Proof (Lean source)
lemma signedBit_probability_zero (θ : TrialParameter) :
(∑ j : Fin 4, if signedBit j = 0 then piTheta θ (1 / 2) j else 0) =
(1 - contrast θ) / 2 := by
simp +decide [signedBit, contrast, Fin.sum_univ_succ]
rw [← one_div]
rw [piTheta_balanced_by_val, piTheta_balanced_by_val]
norm_num [controlProb]
ring
theorem balancedRR_output_probability_one
the balanced rr output probability one assertion holds.
Formal statement
Proof (Lean source)
lemma balancedRR_output_probability_one (θ : TrialParameter) (ε : ℝ) :
(∑ j : Fin 4, piTheta θ (1 / 2) j *
(if (1 : Fin 2) = signedBit j then
exp ε / (exp ε + 1)
else 1 / (exp ε + 1))) =
(1 + (exp ε - 1) / (exp ε + 1) * contrast θ) / 2 := by
have hd : exp ε + 1 ≠ 0 := by positivity
simp +decide [signedBit, contrast, Fin.sum_univ_succ]
rw [← one_div]
rw [piTheta_balanced_by_val, piTheta_balanced_by_val,
piTheta_balanced_by_val, piTheta_balanced_by_val]
norm_num [controlProb]
field_simp
ring
theorem balancedRR_output_probability_zero
the balanced rr output probability zero assertion holds.
Formal statement
Proof (Lean source)
lemma balancedRR_output_probability_zero (θ : TrialParameter) (ε : ℝ) :
(∑ j : Fin 4, piTheta θ (1 / 2) j *
(if (0 : Fin 2) = signedBit j then
exp ε / (exp ε + 1)
else 1 / (exp ε + 1))) =
(1 - (exp ε - 1) / (exp ε + 1) * contrast θ) / 2 := by
have hd : exp ε + 1 ≠ 0 := by positivity
simp +decide [signedBit, contrast, Fin.sum_univ_succ]
rw [← one_div]
rw [piTheta_balanced_by_val, piTheta_balanced_by_val,
piTheta_balanced_by_val, piTheta_balanced_by_val]
norm_num [controlProb]
field_simp
ring
theorem balancedRR_stationary
Under the supplied quantities and conditions, the balanced rr stationary assertion holds. Under the stated assumptions, the balanced RR stationary.
Formal statement
Proof (Lean source)
lemma balancedRR_stationary (ε : ℝ) (hε : 0 < ε) :
StationaryLDP ε (balancedRR ε) := by
constructor
· constructor
intro j
constructor
have hsum : (∑ z : Fin 2,
(if z = signedBit j then exp ε / (exp ε + 1)
else 1 / (exp ε + 1))) = 1 := by
fin_cases j <;> simp +decide [signedBit, Fin.sum_univ_succ] <;>
field_simp <;> ring
rw [← ENNReal.ofReal_one, ← hsum]
change (Measure.count.withDensity
(fun z : Fin 2 => ofReal
(if z = signedBit j then exp ε / (exp ε + 1)
else 1 / (exp ε + 1)))) univ = _
simp only [withDensity_apply _ MeasurableSet.univ, setLIntegral_univ,
lintegral_count]
rw [tsum_fintype, ENNReal.ofReal_sum_of_nonneg]
intro z hz
split_ifs <;> positivity
· intro A hA j j'
let r := ofReal (exp ε)
have hr : r ≠ ∞ := ENNReal.ofReal_ne_top
have hm : Measure.count.withDensity
(fun z : Fin 2 => ofReal
(if z = signedBit j then exp ε / (exp ε + 1)
else 1 / (exp ε + 1))) ≤
r • Measure.count.withDensity
(fun z : Fin 2 => ofReal
(if z = signedBit j' then exp ε / (exp ε + 1)
else 1 / (exp ε + 1))) := by
rw [← withDensity_smul' r _ hr]
apply withDensity_mono
filter_upwards [] with z
dsimp [r]
rw [← ENNReal.ofReal_mul (le_of_lt (Real.exp_pos ε))]
apply ENNReal.ofReal_le_ofReal
have he : 1 ≤ exp ε := (Real.one_le_exp_iff).2 (le_of_lt hε)
have hd : 0 < exp ε + 1 := by positivity
split_ifs <;> rw [← mul_div_assoc] <;>
apply (div_le_div_iff_of_pos_right hd).2 <;>
nlinarith [sq_nonneg (exp ε - 1)]
exact hm A
theorem grouped_balancedRR_information_eq_randomizedResponse
Under the supplied quantities and conditions, the grouped balanced rr information eq randomized response assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the grouped balanced RR information eq randomized Response.
Formal statement
hypotheses
conclusion
= information (randomizedResponse ε) (contrast θ)
Proof (Lean source)
lemma grouped_balancedRR_information_eq_randomizedResponse
(θ : TrialParameter) (hθ : InteriorMeans θ)
(hbalance : θ 0 + θ 1 = 1) (ε : ℝ) (hε : 0 < ε) :
information
(groupedChannel (balancedRR ε)) (contrast θ) =
information
(randomizedResponse ε) (contrast θ) := by
letI : IsMarkovKernel (balancedRR ε) := (balancedRR_stationary ε hε).1
letI : IsMarkovKernel (groupedChannel (balancedRR ε)) :=
groupedChannel_markov _
letI : IsMarkovKernel
(randomizedResponse ε) :=
randomizedResponse_markov ε
have hτ := abs_contrast_lt_one θ hθ hbalance
rw [information_eq_discrete _ _ hτ,
information_eq_discrete _ _ hτ]
unfold discreteInformation
rw [tsum_fintype, tsum_fintype]
simp +decide [grouped_balancedRR_atom, randomizedResponse_atom,
Fin.sum_univ_succ]
ring
theorem grouped_balancedRR_information_eq
the grouped balanced rr information eq assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the grouped balanced RR information eq.
Formal statement
hypotheses
conclusion
= contraction ε ^ 2 / (1 - contraction ε ^ 2 * contrast θ ^ 2)
Proof (Lean source)
lemma grouped_balancedRR_information_eq (θ : TrialParameter)
(hθ : InteriorMeans θ) (hbalance : θ 0 + θ 1 = 1)
(ε : ℝ) (hε : 0 < ε) :
information
(groupedChannel (balancedRR ε)) (contrast θ) =
contraction ε ^ 2 /
(1 - contraction ε ^ 2 *
contrast θ ^ 2) := by
rw [grouped_balancedRR_information_eq_randomizedResponse
θ hθ hbalance ε hε]
exact randomizedResponse_information_eq
ε (contrast θ) hε (abs_contrast_lt_one θ hθ hbalance)
theorem balancedRR_directionalDerivative
Under the supplied quantities and conditions, the balanced rr directional derivative assertion holds.
Formal statement
Proof (Lean source)
lemma balancedRR_directionalDerivative (ε t : ℝ) (z : Fin 2) :
(∑ k, direction t k *
channelDerivativeDensity (1 / 2) (balancedRR ε) k z) =
(∑ k, direction (-1 / 2) k *
channelDerivativeDensity (1 / 2) (balancedRR ε) k z) := by
have h03 : balancedRR ε 0 = balancedRR ε 3 := by rfl
have h12 : balancedRR ε 1 = balancedRR ε 2 := by rfl
have hd03 : channelDensity (balancedRR ε) 0 z =
channelDensity (balancedRR ε) 3 z := by
unfold channelDensity
rw [h03]
have hd12 : channelDensity (balancedRR ε) 1 z =
channelDensity (balancedRR ε) 2 z := by
unfold channelDensity
rw [h12]
simp +decide [channelDerivativeDensity, inputDerivative, direction,
controlProb, Fin.sum_univ_succ]
rw [hd03, hd12]
ring
theorem balancedRR_quadratic_eq
Under the supplied quantities and conditions, the balanced rr quadratic eq assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the balanced RR quadratic eq.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma balancedRR_quadratic_eq (θ : TrialParameter) (hθ : InteriorMeans θ)
(ε : ℝ) (hε : 0 < ε) (t : ℝ) :
informationQuadratic (channelFisherInfo θ (1 / 2) (balancedRR ε))
(direction t) =
informationQuadratic (channelFisherInfo θ (1 / 2) (balancedRR ε))
(direction (-1 / 2)) := by
have hRR := balancedRR_stationary ε hε
rw [informationQuadratic_eq_channelIntegral (1 / 2) θ
(by constructor <;> norm_num) hθ ε (balancedRR ε) hRR (direction t),
informationQuadratic_eq_channelIntegral (1 / 2) θ
(by constructor <;> norm_num) hθ ε (balancedRR ε) hRR
(direction (-1 / 2))]
apply integral_congr_ae
filter_upwards [] with z
rw [balancedRR_directionalDerivative ε t z]
theorem sharp_information_pos
Under the supplied quantities and conditions, the sharp information pos assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the sharp information pos.
Formal statement
hypotheses
conclusion
0 < contraction ε ^ 2 / (1 - contraction ε ^ 2 * contrast θ ^ 2)
Proof (Lean source)
lemma sharp_information_pos (θ : TrialParameter) (hθ : InteriorMeans θ)
(hbalance : θ 0 + θ 1 = 1) (ε : ℝ) (hε : 0 < ε) :
0 < Causalean.Stat.Privacy.Binary.contraction ε ^ 2 /
(1 - Causalean.Stat.Privacy.Binary.contraction ε ^ 2 *
contrast θ ^ 2) := by
let κ := Causalean.Stat.Privacy.Binary.contraction ε
have hκ : 0 < κ := by
dsimp [κ, Causalean.Stat.Privacy.Binary.contraction]
have he : 1 < exp ε := (Real.one_lt_exp_iff).2 hε
positivity
have hκ1 : κ < 1 :=
(contraction_mem ε hε).2
have hκSq : κ ^ 2 < 1 := (sq_lt_one_iff₀ hκ.le).2 hκ1
have hτSq : contrast θ ^ 2 < 1 :=
(sq_lt_one_iff_abs_lt_one (contrast θ)).2
(abs_contrast_lt_one θ hθ hbalance)
have hprod : κ ^ 2 * contrast θ ^ 2 < 1 :=
mul_lt_one_of_nonneg_of_lt_one_left (sq_nonneg κ) hκSq hτSq.le
exact div_pos (sq_pos_of_pos hκ) (sub_pos.2 hprod)
theorem balancedRR_variance_eq_sharp
the balanced rr variance eq sharp assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the balanced RR variance eq sharp.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma balancedRR_variance_eq_sharp (θ : TrialParameter)
(hθ : InteriorMeans θ) (hbalance : θ 0 + θ 1 = 1)
(ε : ℝ) (hε : 0 < ε) :
contrastVariance (channelFisherInfo θ (1 / 2) (balancedRR ε)) =
ofReal
(contraction ε ^ 2 /
(1 - contraction ε ^ 2 *
contrast θ ^ 2))⁻¹ := by
have hRR := balancedRR_stationary ε hε
have hinfo := grouped_balancedRR_information_eq θ hθ hbalance ε hε
have hquad :
informationQuadratic
(channelFisherInfo θ (1 / 2) (balancedRR ε)) (direction (-1 / 2)) =
contraction ε ^ 2 /
(1 - contraction ε ^ 2 *
contrast θ ^ 2) := by
rw [← hinfo]
exact grouped_information_eq_channelQuadratic θ hθ hbalance ε
(balancedRR ε) hRR |>.symm
rw [contrastVariance_eq_reciprocal_of_minimizer
(channelFisherInfo θ (1 / 2) (balancedRR ε))
(channelFisherInfo_posSemidef (1 / 2) θ (by constructor <;> norm_num)
hθ ε (balancedRR ε) hRR)
(-1 / 2)
(fun u => by rw [balancedRR_quadratic_eq θ hθ ε hε u])
(hquad.symm ▸ sharp_information_pos θ hθ hbalance ε hε),
hquad]
theorem Vstar_balanced_eq_sharp_reciprocal
the vstar balanced eq sharp reciprocal assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the Vstar balanced eq sharp reciprocal.
Formal statement
hypotheses
conclusion
Vstar θ (1 / 2) ε
= (contraction ε ^ 2 / (1 - contraction ε ^ 2 * contrast θ ^ 2))⁻¹
Proof (Lean source)
lemma Vstar_balanced_eq_sharp_reciprocal (θ : TrialParameter)
(hθ : InteriorMeans θ) (hbalance : θ 0 + θ 1 = 1)
(ε : ℝ) (hε : 0 < ε) :
Vstar θ (1 / 2) ε =
(contraction ε ^ 2 /
(1 - contraction ε ^ 2 *
contrast θ ^ 2))⁻¹ := by
let B := contraction ε ^ 2 /
(1 - contraction ε ^ 2 * contrast θ ^ 2)
have hB : 0 < B := sharp_information_pos θ hθ hbalance ε hε
have hJ : 0 < Jstar θ (1 / 2) ε :=
Jstar_pos_interior θ (1 / 2) ε (by constructor <;> norm_num) hθ hε
have hJle : Jstar θ (1 / 2) ε ≤ B := by
simpa [B] using Jstar_balanced_le_sharp θ hθ hbalance ε hε
have hlower : B⁻¹ ≤ Vstar θ (1 / 2) ε := by
rw [Vstar]
exact (inv_le_inv₀ hB hJ).2 hJle
have huniv := (finite_oracle θ (1 / 2) ε
(by constructor <;> norm_num) hθ (fun _ => ε) ⟨hε, fun _ => rfl⟩).1
(Fin 2) (balancedRR ε) (balancedRR_stationary ε hε)
rw [balancedRR_variance_eq_sharp θ hθ hbalance ε hε] at huniv
have hupper : Vstar θ (1 / 2) ε ≤ B⁻¹ := by
rw [ENNReal.ofReal_le_ofReal_iff (inv_nonneg.mpr hB.le)] at huniv
simpa [B] using huniv
exact le_antisymm hupper hlower
theorem contraction_inv_eq_coth
Under the supplied quantities and conditions, the contraction inv eq coth assertion holds. Under the stated assumptions, the contraction inv eq coth.
Formal statement
Proof (Lean source)
lemma contraction_inv_eq_coth (ε : ℝ) (hε : 0 < ε) :
(contraction ε)⁻¹ =
cosh (ε / 2) / sinh (ε / 2) := by
have hs : sinh (ε / 2) ≠ 0 :=
ne_of_gt ((Real.sinh_pos_iff).2 (by linarith))
have he : exp (ε / 2) ≠ 0 := Real.exp_ne_zero _
have hexp : exp ε = exp (ε / 2) * exp (ε / 2) := by
rw [← Real.exp_add]
congr 1
ring
rw [contraction, Real.cosh_eq, Real.sinh_eq,
Real.exp_neg, hexp]
field_simp
theorem balanced_reduction
the balanced reduction assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the balanced reduction.
Formal statement
hypotheses
Proof (Lean source)
theorem balanced_reduction {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) (Y0 Y1 W Y : ℕ → Ω → ℝ)
(θ : TrialParameter) (ε : ℝ)
(hθ : InteriorMeans θ)
(hbalance : θ 0 + θ 1 = 1) (hε : 0 < ε) :
Vstar θ (1 / 2) ε =
(cosh (ε / 2) / sinh (ε / 2)) ^ 2 -
(contrast θ) ^ 2 ∧
StationaryLDP ε (balancedRR ε) ∧
contrastVariance (channelFisherInfo θ (1 / 2) (balancedRR ε)) =
ofReal (Vstar θ (1 / 2) ε) := by
refine ⟨?_, ?_, ?_⟩
· rw [Vstar_balanced_eq_sharp_reciprocal θ hθ hbalance ε hε]
let κ := Causalean.Stat.Privacy.Binary.contraction ε
have hκ : 0 < κ := by
dsimp [κ, Causalean.Stat.Privacy.Binary.contraction]
have he : 1 < exp ε := (Real.one_lt_exp_iff).2 hε
positivity
have hden : 1 - κ ^ 2 * contrast θ ^ 2 ≠ 0 := by
have hκ1 : κ < 1 :=
(contraction_mem ε hε).2
have hκSq : κ ^ 2 < 1 := (sq_lt_one_iff₀ hκ.le).2 hκ1
have hτSq : contrast θ ^ 2 < 1 :=
(sq_lt_one_iff_abs_lt_one (contrast θ)).2
(abs_contrast_lt_one θ hθ hbalance)
exact ne_of_gt (sub_pos.2
(mul_lt_one_of_nonneg_of_lt_one_left
(sq_nonneg κ) hκSq hτSq.le))
rw [show (κ ^ 2 / (1 - κ ^ 2 * contrast θ ^ 2))⁻¹ =
κ⁻¹ ^ 2 - contrast θ ^ 2 by
field_simp [hκ.ne', hden]
]
rw [contraction_inv_eq_coth ε hε]
· exact balancedRR_stationary ε hε
· rw [balancedRR_variance_eq_sharp θ hθ hbalance ε hε,
Vstar_balanced_eq_sharp_reciprocal θ hθ hbalance ε hε]
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.TFiniteOracle 2 declarations Exact finite staircase oracle
Exact finite staircase oracle
theorem finiteOracle_minimax
Under the supplied quantities and conditions, the finite oracle minimax assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the finite Oracle minimax.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma finiteOracle_minimax (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε) :
Jstar θ p ε =
sInf {u : ℝ | ∃ t : ℝ, u = upperEnvelope θ p ε t} := by
let S := feasibleSet ε
let f : StaircaseWeight → ℝ → ℝ :=
fun α t => -informationObjective θ p ε α t
have hS : S.Nonempty := staircaseFeasible_nonempty ε
have hcS : IsCompact S := staircaseFeasible_compact ε hε.le
have hconvS : Convex ℝ S := staircaseFeasible_convex ε
have hcontα (t : ℝ) : Continuous (fun α => f α t) := by
dsimp [f]
exact (informationObjective_continuous_weight θ p ε t).neg
have hquasiα (t : ℝ) : QuasiconvexOn ℝ S (fun α => f α t) := by
apply ConvexOn.quasiconvexOn
refine ⟨hconvS, ?_⟩
intro α hα β hβ a b ha hb hab
change -(informationObjective θ p ε (a • α + b • β) t) ≤
a * -informationObjective θ p ε α t +
b * -informationObjective θ p ε β t
rw [informationObjective_affine_weights]
simp
have hcontT (α : StaircaseWeight) :
Continuous (fun t => f α t) := by
dsimp [f]
unfold informationObjective patternInformation projectedGradient direction
simp +decide only [Fin.sum_univ_succ, Fin.sum_univ_zero, ite_true,
ite_false, add_zero]
fun_prop
have hquasiT (α : StaircaseWeight) (hα : α ∈ S) :
QuasiconcaveOn ℝ Set.univ (f α) := by
have hc := informationObjective_convexOn_time θ p ε α hα.1
(patternMass_pos_interior θ p ε hp hθ hε.le)
apply ConcaveOn.quasiconcaveOn
exact hc.neg
let M : StaircaseWeight → ℝ := fun α =>
sInf {u : ℝ | ∃ t : ℝ, u = informationObjective θ p ε α t}
let H : ℝ → ℝ := upperEnvelope θ p ε
have hM (α : StaircaseWeight) (hα : α ∈ S) :
∃ t : ℝ, M α = informationObjective θ p ε α t ∧
∀ u, informationObjective θ p ε α t ≤
informationObjective θ p ε α u := by
obtain ⟨t, ht⟩ := informationObjective_exists_minimizer θ p ε
hp hθ hε.le α hα
refine ⟨t, ?_, ht⟩
apply IsLeast.csInf_eq
exact ⟨⟨t, rfl⟩, by rintro _ ⟨u, rfl⟩; exact ht u⟩
have hH (t : ℝ) :
∃ α ∈ S, H t = informationObjective θ p ε α t ∧
∀ β ∈ S, informationObjective θ p ε β t ≤
informationObjective θ p ε α t := by
obtain ⟨α, hα, hmax⟩ :=
informationObjective_exists_maximizer θ p ε t hε.le
refine ⟨α, hα, ?_, hmax⟩
unfold H upperEnvelope
apply IsGreatest.csSup_eq
exact ⟨⟨α, hα, rfl⟩, by
rintro _ ⟨β, hβ, rfl⟩
exact hmax β hβ⟩
have hsup_y (α : StaircaseWeight) (hα : α ∈ S) :
IsLUB {u : ℝ | ∃ t ∈ (Set.univ : Set ℝ), f α t = u} (-M α) := by
obtain ⟨t, hMt, hmin⟩ := hM α hα
apply IsGreatest.isLUB
constructor
· exact ⟨t, Set.mem_univ _, by dsimp [f]; rw [hMt]⟩
· rintro u ⟨v, _, rfl⟩
dsimp [f]
rw [hMt]
exact neg_le_neg (hmin v)
have hinf_sup :
IsGLB {u : ℝ | ∃ α ∈ S, -M α = u} (-Jstar θ p ε) := by
obtain ⟨γ, hγ, hmax⟩ :=
informationObjective_profile_exists_maximizer θ p ε hp hθ hε.le
have hprofile (β : StaircaseWeight) :
(⨅ t : ℝ, informationObjective θ p ε β t) = M β := by
change (⨅ t : ℝ, informationObjective θ p ε β t) =
sInf {u : ℝ | ∃ t : ℝ, u = informationObjective θ p ε β t}
rw [show {u : ℝ | ∃ t : ℝ,
u = informationObjective θ p ε β t} =
range (informationObjective θ p ε β) from by ext u; simp [eq_comm]]
rfl
have hJ : Jstar θ p ε = M γ := by
apply IsGreatest.csSup_eq
exact ⟨⟨γ, hγ, rfl⟩, by
rintro _ ⟨β, hβ, rfl⟩
change M β ≤ M γ
rw [← hprofile β, ← hprofile γ]
exact hmax β hβ⟩
apply IsLeast.isGLB
constructor
· exact ⟨γ, hγ, by rw [hJ]⟩
· rintro _ ⟨β, hβ, rfl⟩
rw [hJ, ← hprofile γ, ← hprofile β]
exact neg_le_neg (hmax β hβ)
have hinf_x (t : ℝ) :
IsGLB {u : ℝ | ∃ α ∈ S, f α t = u} (-H t) := by
obtain ⟨α, hα, hHt, hmax⟩ := hH t
apply IsLeast.isGLB
constructor
· exact ⟨α, hα, by dsimp [f]; rw [hHt]⟩
· rintro _ ⟨β, hβ, rfl⟩
dsimp [f]
rw [hHt]
exact neg_le_neg (hmax β hβ)
have hsup_inf :
IsLUB {u : ℝ | ∃ t ∈ (Set.univ : Set ℝ), -H t = u}
(-sInf {u : ℝ | ∃ t : ℝ, u = H t}) := by
obtain ⟨t, ht⟩ := upperEnvelope_exists_minimizer θ p ε hp hθ hε
have hleast : sInf {u : ℝ | ∃ v : ℝ, u = H v} = H t := by
apply IsLeast.csInf_eq
exact ⟨⟨t, rfl⟩, by rintro _ ⟨v, rfl⟩; exact ht v⟩
apply IsGreatest.isLUB
constructor
· exact ⟨t, Set.mem_univ _, by rw [hleast]⟩
· rintro _ ⟨v, _, rfl⟩
rw [hleast]
exact neg_le_neg (ht v)
have hsion := Sion.minimax
(X := S) (Y := Set.univ) (f := f)
hS hcS
(fun t _ => (hcontα t).continuousOn.lowerSemicontinuousOn)
(fun t _ => hquasiα t)
convex_univ
(fun α hα => (hcontT α).continuousOn.upperSemicontinuousOn)
(fun α hα => hquasiT α hα)
hconvS
(fun α => -M α) hsup_y (-Jstar θ p ε) hinf_sup
(fun t => -H t) (fun t _ => hinf_x t)
(-sInf {u : ℝ | ∃ t : ℝ, u = H t}) hsup_inf
exact neg_injective hsion
theorem finite_oracle
Under the supplied quantities and conditions, the finite oracle assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
conclusion 1
conclusion 3
conclusion 4
conclusion 5
∃ t : ℝ, upperEnvelope θ p ε t = Jstar θ p ε
Proof (Lean source)
theorem finite_oracle (θ : TrialParameter) (p ε : ℝ)
(hassignment : InteriorAssignment p)
(hmeans : InteriorMeans θ)
(epsSeq : ℕ → ℝ) (hfixed : FixedPrivacy epsSeq ε) :
(∀ (Z : Type*) [MeasurableSpace Z] (Q : Kernel (Fin 4) Z),
StationaryLDP ε Q →
ofReal (Vstar θ p ε) ≤
contrastVariance (channelFisherInfo θ p Q)) ∧
(∃ α : StaircaseWeight, staircaseFeasible ε α ∧
StationaryLDP ε (staircaseChannel ε α) ∧
contrastVariance (channelFisherInfo θ p (staircaseChannel ε α)) =
ofReal (Vstar θ p ε) ∧
contrastVariance (informationMatrix θ p ε α) =
ofReal (Vstar θ p ε)) ∧
Jstar θ p ε =
sInf {u : ℝ | ∃ t : ℝ, u = upperEnvelope θ p ε t} ∧
Jstar θ p ε =
sInf {u : ℝ | ∃ t : ℝ, u = vertexEnvelope θ p ε t} ∧
(∃ t : ℝ, upperEnvelope θ p ε t = Jstar θ p ε) := by
have hminimax : Jstar θ p ε =
sInf {u : ℝ | ∃ t : ℝ, u = upperEnvelope θ p ε t} :=
finiteOracle_minimax θ p ε hassignment hmeans hfixed.1
have henvelopes :
{u : ℝ | ∃ t : ℝ, u = upperEnvelope θ p ε t} =
{u : ℝ | ∃ t : ℝ, u = vertexEnvelope θ p ε t} := by
ext u
constructor
· rintro ⟨t, rfl⟩
exact ⟨t, upperEnvelope_eq_vertexEnvelope θ p ε t hfixed.1.le⟩
· rintro ⟨t, rfl⟩
exact ⟨t, (upperEnvelope_eq_vertexEnvelope θ p ε t hfixed.1.le).symm⟩
refine ⟨?_, ?_, hminimax, hminimax.trans (congrArg sInf henvelopes), ?_⟩
· intro Z _ Q hQ
obtain ⟨α, K, hα, hK, hfactor, hdiff, hrigid⟩ :=
staircase_refinement_interior p θ ε hassignment hmeans epsSeq hfixed Q hQ
obtain ⟨t, ht⟩ := informationObjective_exists_minimizer θ p ε
hassignment hmeans hfixed.1.le α hα
have hleast :
sInf {u : ℝ | ∃ v : ℝ, u = informationObjective θ p ε α v} =
informationObjective θ p ε α t := by
apply IsLeast.csInf_eq
exact ⟨⟨t, rfl⟩, by rintro u ⟨v, rfl⟩; exact ht v⟩
have hprofile_le : informationObjective θ p ε α t ≤ Jstar θ p ε := by
obtain ⟨γ, hγ, hmax⟩ :=
informationObjective_profile_exists_maximizer θ p ε
hassignment hmeans hfixed.1.le
have hprofile (β : StaircaseWeight) :
(⨅ v : ℝ, informationObjective θ p ε β v) =
sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε β v} := by
rw [show {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε β v} =
range (informationObjective θ p ε β) from by
ext u
simp [eq_comm]]
rfl
rw [Jstar, ← hleast]
apply le_csSup
· refine ⟨sInf {u : ℝ | ∃ v : ℝ,
u = informationObjective θ p ε γ v}, ?_⟩
rintro u ⟨β, hβ, rfl⟩
rw [← hprofile β, ← hprofile γ]
exact hmax β hβ
· exact ⟨α, hα, rfl⟩
have horder : informationQuadratic (channelFisherInfo θ p Q) (direction t) ≤
informationQuadratic (informationMatrix θ p ε α) (direction t) := by
have hn := hdiff.dotProduct_mulVec_nonneg (direction t)
simp [informationQuadratic, dotProduct, mulVec, Finset.mul_sum,
Finset.sum_sub_distrib, star_trivial] at hn ⊢
linarith
have htQ : informationQuadratic (channelFisherInfo θ p Q) (direction t) ≤
Jstar θ p ε := by
refine horder.trans ?_
have heq : informationObjective θ p ε α t =
informationQuadratic (informationMatrix θ p ε α) (direction t) := by
rw [informationObjective_eq_informationQuadratic θ p ε α t]
rfl
rw [← heq]
exact hprofile_le
have hJ : 0 < Jstar θ p ε :=
Jstar_pos_interior θ p ε hassignment hmeans hfixed.1
simpa [Vstar] using reciprocal_le_contrastVariance
(channelFisherInfo θ p Q)
(channelFisherInfo_posSemidef p θ hassignment hmeans ε Q hQ)
(Jstar θ p ε) hJ t htQ
· obtain ⟨α, hα, hopt⟩ :=
Jstar_exists_optimal_weight θ p ε hassignment hmeans hfixed.1.le
obtain ⟨t, ht⟩ := informationObjective_exists_minimizer θ p ε
hassignment hmeans hfixed.1.le α hα
have hleast :
sInf {u : ℝ | ∃ v : ℝ, u = informationObjective θ p ε α v} =
informationObjective θ p ε α t := by
apply IsLeast.csInf_eq
exact ⟨⟨t, rfl⟩, by rintro u ⟨v, rfl⟩; exact ht v⟩
have hvalue : informationObjective θ p ε α t = Jstar θ p ε := by
rw [← hleast, ← hopt]
have hqmin : ∀ u : ℝ,
informationQuadratic (informationMatrix θ p ε α) (direction t) ≤
informationQuadratic (informationMatrix θ p ε α) (direction u) := by
intro u
have heqt : informationObjective θ p ε α t =
informationQuadratic (informationMatrix θ p ε α) (direction t) := by
rw [informationObjective_eq_informationQuadratic θ p ε α t]
rfl
have hequ : informationObjective θ p ε α u =
informationQuadratic (informationMatrix θ p ε α) (direction u) := by
rw [informationObjective_eq_informationQuadratic θ p ε α u]
rfl
rw [← heqt, ← hequ]
exact ht u
have hqpos : 0 <
informationQuadratic (informationMatrix θ p ε α) (direction t) := by
have heq : informationObjective θ p ε α t =
informationQuadratic (informationMatrix θ p ε α) (direction t) := by
rw [informationObjective_eq_informationQuadratic θ p ε α t]
rfl
rw [← heq, hvalue]
exact Jstar_pos_interior θ p ε hassignment hmeans hfixed.1
have hattain : contrastVariance (informationMatrix θ p ε α) =
ofReal (Vstar θ p ε) := by
rw [contrastVariance_eq_reciprocal_of_minimizer
(informationMatrix θ p ε α)
(informationMatrix_posSemidef_interior θ p ε hassignment hmeans
hfixed.1.le α hα) t hqmin hqpos]
congr 1
have heq : informationObjective θ p ε α t =
informationQuadratic (informationMatrix θ p ε α) (direction t) := by
rw [informationObjective_eq_informationQuadratic θ p ε α t]
rfl
rw [← heq, hvalue]
rfl
refine ⟨α, hα, staircaseChannel_stationaryLDP ε hfixed.1 α hα, ?_, hattain⟩
rw [channelFisherInfo_staircase_eq_informationMatrix θ p ε
hassignment hmeans hfixed.1 α hα]
exact hattain
· obtain ⟨t, ht⟩ :=
upperEnvelope_exists_minimizer θ p ε hassignment hmeans hfixed.1
have hleast : sInf {u : ℝ | ∃ v : ℝ, u = upperEnvelope θ p ε v} =
upperEnvelope θ p ε t := by
apply IsLeast.csInf_eq
exact ⟨⟨t, rfl⟩, by
rintro u ⟨v, rfl⟩
exact ht v⟩
exact ⟨t, (hminimax.trans hleast).symm⟩
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.TFiveOutputRegion 6 declarations The centered family has a unique five-ray optimum; distinct projected scores prevent any attaining stationary channel from merging those rays.
Five-output certificate region
The centered family has a unique five-ray optimum; distinct projected scores prevent any attaining stationary channel from merging those rays.
theorem R5_nonempty_centered
the r5 nonempty centered assertion holds.
Formal statement
conclusion
R5.Nonempty
Proof (Lean source)
lemma R5_nonempty_centered : R5.Nonempty := by
refine ⟨((1 / 4 : ℝ), (1 / 2 : ℝ), (1 / 2 : ℝ), log 3), ?_⟩
exact centeredPoint_mem_R5 (1 / 4) (by norm_num) (by norm_num)
theorem centered_five_output_region
the centered five output region assertion holds.
Formal statement
hypotheses
p :
ℝ
p :
ℝ
0 < p
p < (1 / 2 : ℝ)
conclusion
conclusion 3
5 ≤ outputCardinality (fun _ => (1 / 2 : ℝ)) p Q
Proof (Lean source)
lemma centered_five_output_region :
∀ p : ℝ, 0 < p → p < (1 / 2 : ℝ) →
(p, (1 / 2 : ℝ), (1 / 2 : ℝ), log 3) ∈ R5 ∧
(∃ α : StaircaseWeight,
staircaseFeasible (log 3) α ∧
staircaseSupport α = ({2, 3, 5, 7, 8} : Finset (Fin 14)) ∧
Jstar (fun _ => (1 / 2 : ℝ)) p (log 3) =
sInf {u : ℝ | ∃ t : ℝ,
u = informationObjective (fun _ => (1 / 2 : ℝ)) p
(log 3) α t} ∧
(∀ β : StaircaseWeight, staircaseFeasible (log 3) β →
Jstar (fun _ => (1 / 2 : ℝ)) p (log 3) =
sInf {u : ℝ | ∃ t : ℝ,
u = informationObjective (fun _ => (1 / 2 : ℝ)) p
(log 3) β t} → β = α)) ∧
(∀ (Z : Type*) [MeasurableSpace Z] (Q : Kernel (Fin 4) Z),
StationaryLDP (log 3) Q →
contrastVariance (channelFisherInfo (fun _ => (1 / 2 : ℝ)) p Q) =
ofReal (Vstar (fun _ => (1 / 2 : ℝ)) p (log 3)) →
5 ≤ outputCardinality (fun _ => (1 / 2 : ℝ)) p Q) := by
intro p hp0 hp1
refine ⟨centeredPoint_mem_R5 p hp0 hp1, ?_, ?_⟩
· obtain ⟨hfeas, hopt, hunique⟩ := centeredR5_profile_optimal_unique p hp0 hp1
refine ⟨centeredR5Weight p, hfeas, ?_, hopt, hunique⟩
ext s
simp only [staircaseSupport, mem_filter, Finset.mem_univ, true_and]
simpa only [r5Active] using (centeredR5Weight_active p hp0 hp1).1 s
· intro Z _ Q hQ hattain
exact centeredR5_outputCardinality_ge_five p hp0 hp1 Z Q hQ hattain
theorem reflectedPoint_mem_R5
Under the supplied quantities and conditions, the reflected point mem r5 assertion holds.
Formal statement
Proof (Lean source)
lemma reflectedPoint_mem_R5 (p : ℝ) (hp0 : 1 / 2 < p) (hp1 : p < 1) :
(1 - p, (1 / 2 : ℝ), (1 / 2 : ℝ), log 3) ∈ R5 := by
exact centeredPoint_mem_R5 (1 - p) (by linarith) (by linarith)
theorem reflected_five_output_region
Under the supplied quantities and conditions, the reflected five output region assertion holds.
Formal statement
hypotheses
p :
ℝ
hp0 :
1 / 2 < p
hp1 :
p < 1
conclusion
conclusion 2
5 ≤ outputCardinality (fun _ => (1 / 2 : ℝ)) p Q
Proof (Lean source)
lemma reflected_five_output_region (p : ℝ) (hp0 : 1 / 2 < p) (hp1 : p < 1) :
(∃ α : StaircaseWeight,
staircaseFeasible (log 3) α ∧
staircaseSupport α =
swappedSupport ({2, 3, 5, 7, 8} : Finset (Fin 14)) ∧
Jstar (fun _ => (1 / 2 : ℝ)) p (log 3) =
sInf {u : ℝ | ∃ t : ℝ,
u = informationObjective (fun _ => (1 / 2 : ℝ)) p
(log 3) α t} ∧
∀ β : StaircaseWeight, staircaseFeasible (log 3) β →
Jstar (fun _ => (1 / 2 : ℝ)) p (log 3) =
sInf {u : ℝ | ∃ t : ℝ,
u = informationObjective (fun _ => (1 / 2 : ℝ)) p
(log 3) β t} → β = α) ∧
∀ (Z : Type*) [MeasurableSpace Z] (Q : Kernel (Fin 4) Z),
StationaryLDP (log 3) Q →
contrastVariance (channelFisherInfo (fun _ => (1 / 2 : ℝ)) p Q) =
ofReal (Vstar (fun _ => (1 / 2 : ℝ)) p (log 3)) →
5 ≤ outputCardinality (fun _ => (1 / 2 : ℝ)) p Q := by
let q := 1 - p
have hq0 : 0 < q := by dsimp [q]; linarith
have hq1 : q < 1 / 2 := by dsimp [q]; linarith
obtain ⟨hfeas, hopt, hunique⟩ := centeredR5_profile_optimal_unique q hq0 hq1
let α0 := centeredR5Weight q
let α := swapTreatmentWeight α0
have hα : staircaseFeasible (log 3) α :=
(staircaseFeasible_swapTreatment _ _).2 hfeas
have hsupp0 : staircaseSupport α0 = ({2, 3, 5, 7, 8} : Finset (Fin 14)) := by
ext s
simp only [staircaseSupport, mem_filter, Finset.mem_univ, true_and]
simpa only [α0, r5Active] using (centeredR5Weight_active q hq0 hq1).1 s
have hsupp : staircaseSupport α =
swappedSupport ({2, 3, 5, 7, 8} : Finset (Fin 14)) := by
change staircaseSupport (swapTreatmentWeight α0) = _
rw [staircaseSupport_swapTreatment, hsupp0]
have hopt' : Jstar (fun _ => (1 / 2 : ℝ)) p (log 3) =
sInf {u : ℝ | ∃ t : ℝ, u = informationObjective
(fun _ => (1 / 2 : ℝ)) p (log 3) α t} := by
change Jstar (fun _ => (1 / 2 : ℝ)) p (log 3) =
sInf {u : ℝ | ∃ t : ℝ, u = informationObjective
(fun _ => (1 / 2 : ℝ)) p (log 3) (swapTreatmentWeight α0) t}
rw [centered_Jstar_swapTreatment p, centered_profileSet_swapTreatment]
convert hopt using 1 <;> dsimp [q] <;> ring
have hunique' : ∀ β : StaircaseWeight, staircaseFeasible (log 3) β →
Jstar (fun _ => (1 / 2 : ℝ)) p (log 3) =
sInf {u : ℝ | ∃ t : ℝ, u = informationObjective
(fun _ => (1 / 2 : ℝ)) p (log 3) β t} → β = α := by
intro β hβ hβopt
have hswapopt : Jstar (fun _ => (1 / 2 : ℝ)) q (log 3) =
sInf {u : ℝ | ∃ t : ℝ, u = informationObjective
(fun _ => (1 / 2 : ℝ)) q (log 3)
(swapTreatmentWeight β) t} := by
rw [centered_profileSet_swapTreatment]
calc
Jstar (fun _ => (1 / 2 : ℝ)) q (log 3) =
Jstar (fun _ => (1 / 2 : ℝ)) p (log 3) := by
rw [centered_Jstar_swapTreatment]
congr 2 <;> dsimp [q] <;> ring
_ = _ := by
convert hβopt using 1 <;> dsimp [q] <;> ring
have heq := hunique (swapTreatmentWeight β)
((staircaseFeasible_swapTreatment _ _).2 hβ) hswapopt
apply swapTreatmentWeight_involutive.injective
change swapTreatmentWeight β = swapTreatmentWeight α
have hσα : swapTreatmentWeight α = α0 := by
change swapTreatmentWeight (swapTreatmentWeight α0) = α0
exact swapTreatmentWeight_involutive α0
rw [hσα]
exact heq
refine ⟨⟨α, hα, hsupp, hopt', hunique'⟩, ?_⟩
intro Z _ Q hQ hattain
have hsQ := stationaryLDP_swapTreatment (log 3) Q hQ
have hsattain : contrastVariance
(channelFisherInfo (fun _ => (1 / 2 : ℝ)) q (swapTreatmentChannel Q)) =
ofReal (Vstar (fun _ => (1 / 2 : ℝ)) q (log 3)) := by
rw [show q = 1 - p by rfl, channelFisherInfo_swapTreatment,
contrastVariance_swapTreatment]
rw [← centered_Vstar_swapTreatment p]
exact hattain
have hcard := centeredR5_outputCardinality_ge_five q hq0 hq1 Z
(swapTreatmentChannel Q) hsQ hsattain
rw [show q = 1 - p by rfl, centered_outputCardinality_swapTreatment] at hcard
exact hcard
theorem isOpen_R5_continuousCertificate
the is open r5 continuous certificate assertion holds.
Proof (Lean source)
lemma isOpen_R5_continuousCertificate : IsOpen R5 := by
rw [isOpen_iff_mem_nhds]
intro x hx
rcases x with ⟨p, μ0, μ1, ε⟩
change 0 < p ∧ p < 1 ∧ 0 < μ0 ∧ μ0 < 1 ∧ 0 < μ1 ∧ μ1 < 1 ∧ 0 < ε ∧ _ at hx
rcases hx with ⟨hp0,hp1,hm00,hm01,hm10,hm11,hε,α,t,η0,
hα,hsupp,hpos,hstat,hactive,hinactive,hdistinct⟩
let x0 : RegionParameter := (p,μ0,μ1,ε)
have hxTheta : regionTheta x0 = (fun k => if k = 0 then μ0 else μ1) := by
funext k
fin_cases k <;> rfl
have hm (s : Fin 14) :
patternMass (regionTheta x0) x0.1 x0.2.2.2 s ≠ 0 :=
ne_of_gt (patternMass_pos_of_interior p μ0 μ1 ε
hp0 hp1 hm00 hm01 hm10 hm11 s)
have hout : ∀ s ∉ r5Active, α s = 0 := by
intro s hs
by_contra hn
exact hs ((hsupp s).mp hn)
have hcoeff := r5Coeffs_pos_of_interior x0 hp0 hp1 hm00 hm01 hm10 hm11 hε
have hobj := r5_objectives_eq_of_activeDual x0 η0 t hactive
have heq : r5BinaryCoeff x0 * (p - (1 - 2 * p) * t) ^ 2 =
r5TernaryCoeff x0 * (t + 1) ^ 2 := by
rw [← r5BinaryObjective_eq_quadratic x0 t (hm 5) (hm 8),
← r5TernaryObjective_eq_quadratic x0 t (hm 3) (hm 7)]
exact hobj
obtain ⟨σ, hσ, hbranchden, ht0⟩ :=
exists_r5DirectionBranch_eq x0 t hp1 hcoeff.1 hcoeff.2.le heq
have ht := continuousAt_r5DirectionBranch x0 σ (ne_of_gt hcoeff.1)
(hm 3) (hm 5) (hm 7) (hm 8) hbranchden
have hR : r5TernaryDerivative x0 t ≠ 0 :=
r5TernaryDerivative_ne_zero_of_objectives_eq x0 t hp1
hcoeff.1 hcoeff.2 (hm 3) (hm 7) heq
have hmix := r5_stationarity_eq_mixture x0 α hε hα hout t hstat
obtain ⟨hderivden0, huformula⟩ := mixture_stationarity_solve hR hmix
have hderivden : r5BinaryDerivative x0 (r5DirectionBranch σ x0) -
r5TernaryDerivative x0 (r5DirectionBranch σ x0) ≠ 0 := by
simpa [ht0] using hderivden0
have hu := continuousAt_r5ContinuedMixture x0 σ ht
(hm 2) (hm 3) (hm 5) (hm 7) (hm 8) hderivden
have hu0 : r5ContinuedMixture σ x0 = r5MixtureCoordinate ε α :=
r5ContinuedMixture_eq x0 α t σ ht0 hε hα hout hstat hR
obtain ⟨η, hη, hη0, hdual4⟩ :=
exists_local_r5Dual x0 (r5DirectionBranch σ) η0 hε ht hm (by
intro s hs
rw [hxTheta, ht0]
exact hactive s (by simpa [r5Active] using hs))
have hubase := r5MixtureCoordinate_mem_Ioo ε hε α hα hout hpos
have huIoo : ∀ᶠ y in nhds x0, r5ContinuedMixture σ y ∈ Ioo 0 1 := by
filter_upwards [continuousAt_const.eventually_lt hu (hu0.symm ▸ hubase.1),
hu.eventually_lt continuousAt_const (hu0.symm ▸ hubase.2)] with y hy0 hy1
exact ⟨hy0,hy1⟩
have hbranchdenEv : ∀ᶠ y in nhds x0,
1 - 2 * y.1 + σ * r5Scale y ≠ 0 := by
have hc : ContinuousAt (fun y : RegionParameter =>
1 - 2 * y.1 + σ * r5Scale y) x0 :=
(continuousAt_const.sub (continuousAt_const.mul continuousAt_fst)).add
(continuousAt_const.mul
(continuousAt_r5Scale x0 (ne_of_gt hcoeff.1)
(hm 3) (hm 5) (hm 7) (hm 8)))
exact hc.eventually_ne hbranchden
have hbasic : ∀ᶠ y in nhds x0,
0 < y.1 ∧ y.1 < 1 ∧ 0 < y.2.1 ∧ y.2.1 < 1 ∧
0 < y.2.2.1 ∧ y.2.2.1 < 1 ∧ 0 < y.2.2.2 := by
filter_upwards [continuousAt_const.eventually_lt continuousAt_fst hp0,
continuousAt_fst.eventually_lt continuousAt_const hp1,
continuousAt_const.eventually_lt continuousAt_snd.fst hm00,
continuousAt_snd.fst.eventually_lt continuousAt_const hm01,
continuousAt_const.eventually_lt continuousAt_snd.snd.fst hm10,
continuousAt_snd.snd.fst.eventually_lt continuousAt_const hm11,
continuousAt_const.eventually_lt continuousAt_snd.snd.snd hε] with y
yp0 yp1 ym00 ym01 ym10 ym11 yε
exact ⟨yp0,yp1,ym00,ym01,ym10,ym11,yε⟩
have hinfo (s : Fin 14) : ContinuousAt (fun y : RegionParameter =>
patternInformation (regionTheta y) y.1 y.2.2.2 s
(r5DirectionBranch σ y)) x0 :=
continuousAt_patternInformation_region s _ x0 ht (hm s)
have hscore (s : Fin 14) : ContinuousAt (fun y : RegionParameter =>
projectedScore (regionTheta y) y.1 y.2.2.2 s
(r5DirectionBranch σ y)) x0 :=
continuousAt_projectedScore_region s _ x0 ht (hm s)
have hdualcont (s : Fin 14) : ContinuousAt (fun y : RegionParameter =>
dualRay y.2.2.2 (η y) s) x0 := by
unfold dualRay
let f := fun j : Fin 4 => fun y : RegionParameter =>
patternRay y.2.2.2 s j * η y j
have hf (j : Fin 4) : ContinuousAt (f j) x0 :=
(continuous_patternRay_region s j).continuousAt.mul
((continuousAt_pi.mp hη) j)
have hsum : ∀ S : Finset (Fin 4),
ContinuousAt (fun y => ∑ j ∈ S, f j y) x0 := by
intro S
induction S using Finset.induction_on with
| empty => simpa using
(continuousAt_const : ContinuousAt (fun _ : RegionParameter => (0 : ℝ)) x0)
| @insert a S ha ih =>
simp only [Finset.sum_insert ha]
exact (hf a).add ih
simpa [f] using hsum Finset.univ
have hslack : ∀ᶠ y in nhds x0, ∀ s : Fin 14, s ∉ r5Active →
patternInformation (regionTheta y) y.1 y.2.2.2 s
(r5DirectionBranch σ y) < dualRay y.2.2.2 (η y) s := by
have hall : ∀ᶠ y in nhds x0, ∀ s ∈ (Finset.univ : Finset (Fin 14)),
s ∉ r5Active → patternInformation (regionTheta y) y.1 y.2.2.2 s
(r5DirectionBranch σ y) < dualRay y.2.2.2 (η y) s :=
(Filter.eventually_all_finset (Finset.univ : Finset (Fin 14))).2
fun s _ => by
by_cases hs : s ∈ r5Active
· exact Filter.Eventually.of_forall (fun _ h => False.elim (h hs))
· exact ((hinfo s).eventually_lt (hdualcont s)
(by rw [hxTheta, ht0, hη0]; exact hinactive s hs)).mono
(fun _ h _ => h)
exact hall.mono fun _ h s hs => h s (Finset.mem_univ s) hs
have hdistinctEv : ∀ᶠ y in nhds x0, ∀ s ∈ r5Active, ∀ u ∈ r5Active,
s ≠ u → projectedScore (regionTheta y) y.1 y.2.2.2 s
(r5DirectionBranch σ y) ≠ projectedScore (regionTheta y) y.1 y.2.2.2 u
(r5DirectionBranch σ y) := by
have hall : ∀ᶠ y in nhds x0, ∀ s ∈ r5Active, ∀ u ∈ r5Active,
s ≠ u → projectedScore (regionTheta y) y.1 y.2.2.2 s
(r5DirectionBranch σ y) ≠ projectedScore (regionTheta y) y.1 y.2.2.2 u
(r5DirectionBranch σ y) :=
(Filter.eventually_all_finset r5Active).2 fun s hs =>
(Filter.eventually_all_finset r5Active).2 fun u hu => by
by_cases hsu : s = u
· exact Filter.Eventually.of_forall (fun _ h => False.elim (h hsu))
· have hne := hdistinct rfl s (by simpa [r5Active] using hs)
u (by simpa [r5Active] using hu) hsu
have hne' : projectedScore (regionTheta x0) x0.1 x0.2.2.2 s
(r5DirectionBranch σ x0) ≠
projectedScore (regionTheta x0) x0.1 x0.2.2.2 u
(r5DirectionBranch σ x0) := by
change projectedScore (regionTheta x0) p ε s
(r5DirectionBranch σ x0) ≠
projectedScore (regionTheta x0) p ε u
(r5DirectionBranch σ x0)
rw [hxTheta, ht0]
exact hne
exact ((hscore s).sub (hscore u)).eventually_ne
(sub_ne_zero.mpr hne') |>.mono
(fun _ h _ => sub_ne_zero.mp h)
exact hall
have hderivdenEv : ∀ᶠ y in nhds x0,
r5BinaryDerivative y (r5DirectionBranch σ y) -
r5TernaryDerivative y (r5DirectionBranch σ y) ≠ 0 := by
have hP := continuousAt_r5BinaryDerivative_comp x0
(r5DirectionBranch σ) ht (hm 5) (hm 8)
have hRcont := continuousAt_r5TernaryDerivative_comp x0
(r5DirectionBranch σ) ht (hm 2) (hm 3) (hm 7)
exact (hP.sub hRcont).eventually_ne hderivden
filter_upwards [hbasic, huIoo, hbranchdenEv, hderivdenEv,
hdual4, hslack, hdistinctEv]
with y hybasic hyu hyden hyderivden hydual hyslack hydistinct
rcases y with ⟨q,ν0,ν1,δ⟩
rcases hybasic with ⟨hq0,hq1,hν00,hν01,hν10,hν11,hδ⟩
change 0 < q ∧ q < 1 ∧ 0 < ν0 ∧ ν0 < 1 ∧ 0 < ν1 ∧ ν1 < 1 ∧ 0 < δ ∧ _
let y0 : RegionParameter := (q,ν0,ν1,δ)
let ty := r5DirectionBranch σ y0
let uy := r5ContinuedMixture σ y0
let αy := r5WeightFromCoordinate δ uy
have hyTheta : regionTheta (q,ν0,ν1,δ) =
(fun k => if k = 0 then ν0 else ν1) := by
funext k
fin_cases k <;> rfl
rw [hyTheta] at hyslack hydistinct
refine ⟨hq0,hq1,hν00,hν01,hν10,hν11,hδ,αy,ty,η y0, ?_⟩
have hmy (s : Fin 14) : patternMass (regionTheta y0) q δ s ≠ 0 :=
ne_of_gt (patternMass_pos_of_interior q ν0 ν1 δ
hq0 hq1 hν00 hν01 hν10 hν11 s)
have hcy := r5Coeffs_pos_of_interior y0 hq0 hq1 hν00 hν01 hν10 hν11 hδ
have hobjy : r5BinaryObjective y0 ty = r5TernaryObjective y0 ty := by
rw [r5BinaryObjective_eq_quadratic y0 ty (hmy 5) (hmy 8),
r5TernaryObjective_eq_quadratic y0 ty (hmy 3) (hmy 7)]
exact r5DirectionBranch_objectives_eq y0 σ
(by rcases hσ with rfl | rfl <;> norm_num) hcy.1 hcy.2.le hyden
unfold r5DualTarget at hydual
rw [hyTheta] at hydual
refine ⟨(r5WeightFromCoordinate_feasible δ uy).2
⟨hyu.1.le, hyu.2.le⟩, ?_, ?_, ?_, ?_, ?_, ?_⟩
· exact r5WeightFromCoordinate_active δ uy hyu
· exact r5WeightFromCoordinate_pos δ uy hyu
· exact r5ContinuedWeight_stationary y0 σ (by simpa [y0, ty] using hyderivden)
· intro s hs
have hcases : s = 2 ∨ s = 3 ∨ s = 5 ∨ s = 7 ∨ s = 8 := by
simpa [r5Active] using hs
rcases hcases with rfl | rfl | rfl | rfl | rfl
· simpa [y0, ty, r5DualMatrix, r5DualTarget, r5DualMask, mulVec,
dotProduct, dualRay, mul_comm] using congrFun hydual 0
· simpa [y0, ty, r5DualMatrix, r5DualTarget, r5DualMask, mulVec,
dotProduct, dualRay, mul_comm] using congrFun hydual 1
· simpa [y0, ty, r5DualMatrix, r5DualTarget, r5DualMask, mulVec,
dotProduct, dualRay, mul_comm] using congrFun hydual 2
· simpa [y0, ty, r5DualMatrix, r5DualTarget, r5DualMask, mulVec,
dotProduct, dualRay, mul_comm] using congrFun hydual 3
· apply r5_fifthDual_eq y0 (η y0) ty hobjy
· simpa [y0, ty, hyTheta, r5DualMatrix, r5DualTarget, r5DualMask, mulVec,
dotProduct, dualRay, mul_comm] using congrFun hydual 0
· simpa [y0, ty, hyTheta, r5DualMatrix, r5DualTarget, r5DualMask, mulVec,
dotProduct, dualRay, mul_comm] using congrFun hydual 1
· simpa [y0, ty, hyTheta, r5DualMatrix, r5DualTarget, r5DualMask, mulVec,
dotProduct, dualRay, mul_comm] using congrFun hydual 2
· simpa [y0, ty, hyTheta, r5DualMatrix, r5DualTarget, r5DualMask, mulVec,
dotProduct, dualRay, mul_comm] using congrFun hydual 3
· simpa [y0, ty, r5Active] using hyslack
· intro _
simpa [y0, ty, r5Active] using hydistinct
theorem five_output_region
Under the supplied quantities and conditions, the five output region assertion holds. The five output region.
Formal statement
conclusion
conclusion 2
R5.Nonempty
conclusion 3
p :
ℝ
0 < p
p < (1 / 2 : ℝ)
∃ α : StaircaseWeight,
5 ≤ outputCardinality (fun _ => (1 / 2 : ℝ)) p Q
conclusion 4
p :
ℝ
(1 / 2 : ℝ) < p
p < 1
∃ α : StaircaseWeight,
5 ≤ outputCardinality (fun _ => (1 / 2 : ℝ)) p Q
conclusion 5
p μ0 μ1 ε :
ℝ
(p, μ0, μ1, ε) ∈ R5
Z :
Type*
StationaryLDP ε Q
contrastVariance (channelFisherInfo (fun k => if k = 0 then μ0 else μ1) p Q)
5 ≤ outputCardinality (fun k => if k = 0 then μ0 else μ1) p Q
Proof (Lean source)
theorem five_output_region :
IsOpen R5 ∧ R5.Nonempty ∧
(∀ p : ℝ, 0 < p → p < (1 / 2 : ℝ) →
(p, (1 / 2 : ℝ), (1 / 2 : ℝ), log 3) ∈ R5 ∧
(∃ α : StaircaseWeight,
staircaseFeasible (log 3) α ∧
staircaseSupport α = ({2, 3, 5, 7, 8} : Finset (Fin 14)) ∧
Jstar (fun _ => (1 / 2 : ℝ)) p (log 3) =
sInf {u : ℝ | ∃ t : ℝ,
u = informationObjective (fun _ => (1 / 2 : ℝ)) p
(log 3) α t} ∧
(∀ β : StaircaseWeight, staircaseFeasible (log 3) β →
Jstar (fun _ => (1 / 2 : ℝ)) p (log 3) =
sInf {u : ℝ | ∃ t : ℝ,
u = informationObjective (fun _ => (1 / 2 : ℝ)) p
(log 3) β t} → β = α)) ∧
(∀ (Z : Type*) [MeasurableSpace Z] (Q : Kernel (Fin 4) Z),
StationaryLDP (log 3) Q →
contrastVariance (channelFisherInfo (fun _ => (1 / 2 : ℝ)) p Q) =
ofReal (Vstar (fun _ => (1 / 2 : ℝ)) p (log 3)) →
5 ≤ outputCardinality (fun _ => (1 / 2 : ℝ)) p Q)) ∧
(∀ p : ℝ, (1 / 2 : ℝ) < p → p < 1 →
(1 - p, (1 / 2 : ℝ), (1 / 2 : ℝ), log 3) ∈ R5 ∧
(∃ α : StaircaseWeight,
staircaseFeasible (log 3) α ∧
staircaseSupport α =
swappedSupport ({2, 3, 5, 7, 8} : Finset (Fin 14)) ∧
Jstar (fun _ => (1 / 2 : ℝ)) p (log 3) =
sInf {u : ℝ | ∃ t : ℝ,
u = informationObjective (fun _ => (1 / 2 : ℝ)) p
(log 3) α t} ∧
∀ β : StaircaseWeight, staircaseFeasible (log 3) β →
Jstar (fun _ => (1 / 2 : ℝ)) p (log 3) =
sInf {u : ℝ | ∃ t : ℝ,
u = informationObjective (fun _ => (1 / 2 : ℝ)) p
(log 3) β t} → β = α) ∧
∀ (Z : Type*) [MeasurableSpace Z] (Q : Kernel (Fin 4) Z),
StationaryLDP (log 3) Q →
contrastVariance (channelFisherInfo (fun _ => (1 / 2 : ℝ)) p Q) =
ofReal (Vstar (fun _ => (1 / 2 : ℝ)) p (log 3)) →
5 ≤ outputCardinality (fun _ => (1 / 2 : ℝ)) p Q) ∧
(∀ p μ0 μ1 ε : ℝ, (p, μ0, μ1, ε) ∈ R5 →
∀ (Z : Type*) [MeasurableSpace Z] (Q : Kernel (Fin 4) Z),
StationaryLDP ε Q →
contrastVariance
(channelFisherInfo (fun k => if k = 0 then μ0 else μ1) p Q) =
ofReal (Vstar (fun k => if k = 0 then μ0 else μ1) p ε) →
5 ≤ outputCardinality (fun k => if k = 0 then μ0 else μ1) p Q) := by
refine ⟨isOpen_R5_continuousCertificate, R5_nonempty_centered,
centered_five_output_region, ?_, ?_⟩
· intro p hp0 hp1
exact ⟨reflectedPoint_mem_R5 p hp0 hp1,
reflected_five_output_region p hp0 hp1⟩
· intro p μ0 μ1 ε hx Z _ Q hQ hattain
exact r5Certificate_outputCardinality_ge_five p μ0 μ1 ε hx Q hQ hattain
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.TFiveOutputUpperBound 6 declarations An optimal finite staircase design has at most five positive pattern weights, and therefore at most five active released symbols.
Five-output upper bound
An optimal finite staircase design has at most five positive pattern weights, and therefore at most five active released symbols.
the staircase support is the mathematical object specified below. The staircase Support is determined by the displayed parameters.
Definition (Lean source)
parameters
α :
given by
Finset.univ.filter (fun s => α s ≠ 0)
theorem staircase_output_zero_of_weight_zero
Under the supplied quantities and conditions, the staircase output zero of weight zero assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
outputLaw θ p (staircaseChannel ε α) {s} = 0
Proof (Lean source)
lemma staircase_output_zero_of_weight_zero (θ : TrialParameter) (p ε : ℝ)
(α : StaircaseWeight) (s : Fin 14) (hs : α s = 0) :
outputLaw θ p (staircaseChannel ε α) {s} = 0 := by
simp only [outputLaw]
simp only [Measure.coe_finsetSum, Measure.coe_smul, Finset.sum_apply,
Pi.smul_apply, smul_eq_mul, Finset.sum_eq_zero_iff, Finset.mem_univ,
mul_eq_zero, ENNReal.ofReal_eq_zero, forall_const]
intro j
have hrow : (staircaseChannel ε α) j {s} = 0 := by
change (Measure.count.withDensity (fun u : Fin 14 =>
ofReal (α u * patternRay ε u j))) {s} = 0
simp [withDensity_apply _ (measurableSet_singleton s), hs]
exact inr hrow
theorem outputCardinality_staircase_le_support_card
the output cardinality staircase le support card assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma outputCardinality_staircase_le_support_card (θ : TrialParameter)
(p ε : ℝ) (α : StaircaseWeight) :
outputCardinality θ p (staircaseChannel ε α) ≤
(staircaseSupport α).card := by
let A : Set (Fin 14) :=
{s | outputLaw θ p (staircaseChannel ε α) {s} ≠ 0}
have hsubset : A ⊆ (staircaseSupport α : Set (Fin 14)) := by
intro s hs
simp only [staircaseSupport, Finset.mem_coe, mem_filter,
Finset.mem_univ, true_and]
by_contra hz
exact hs (staircase_output_zero_of_weight_zero θ p ε α s hz)
have hbound := Set.encard_le_encard hsubset
simp only [Set.encard_coe_eq_coe_finsetCard] at hbound
change A.encard ≤ _ at hbound
simpa only [outputCardinality, dif_pos (inferInstance : Finite (Fin 14))] using hbound
theorem staircase_output_pos_of_weight_pos
Under the supplied quantities and conditions, the staircase output pos of weight pos assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the staircase output pos of weight pos.
Formal statement
hypotheses
conclusion
outputLaw θ p (staircaseChannel ε α) {s} ≠ 0
Proof (Lean source)
lemma staircase_output_pos_of_weight_pos (θ : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(α : StaircaseWeight) (s : Fin 14) (hs : 0 < α s) :
outputLaw θ p (staircaseChannel ε α) {s} ≠ 0 := by
have hπ : 0 < piTheta θ p 0 := by
rcases hp with ⟨hp0, hp1⟩
rcases hθ with ⟨h0, h0', h1, h1'⟩
simp only [piTheta, controlProb]
exact mul_pos (by linarith) (by linarith)
have hr : 0 < patternRay ε s 0 := by
unfold patternRay privacyIncrement privacyRatio
split_ifs <;> simp_all <;> positivity
have hrow : (staircaseChannel ε α) 0 {s} ≠ 0 := by
change (Measure.count.withDensity (fun u : Fin 14 =>
ofReal (α u * patternRay ε u 0))) {s} ≠ 0
simp [withDensity_apply _ (measurableSet_singleton s)]
exact mul_pos hs hr
have hterm : ofReal (piTheta θ p 0) •
(staircaseChannel ε α) 0 {s} ≠ 0 := by
simp [hrow]
exact hπ
intro hz
have hle : ofReal (piTheta θ p 0) •
(staircaseChannel ε α) 0 {s} ≤
outputLaw θ p (staircaseChannel ε α) {s} := by
simp only [outputLaw, Measure.coe_finsetSum, Finset.sum_apply]
exact Finset.single_le_sum
(f := fun j : Fin 4 =>
(ofReal (piTheta θ p j) • (staircaseChannel ε α) j) {s})
(s := univ) (a := (0 : Fin 4))
(fun j hj => zero_le) (Finset.mem_univ _)
exact hterm (le_antisymm (hz ▸ hle) zero_le)
theorem outputCardinality_staircase_eq_support_card
the output cardinality staircase eq support card assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the output Cardinality staircase eq support card.
Formal statement
hypotheses
θ :
p ε :
ℝ
hp :
hθ :
α :
hα :
∀ s, 0 ≤ α s
conclusion
Proof (Lean source)
lemma outputCardinality_staircase_eq_support_card (θ : TrialParameter)
(p ε : ℝ) (hp : InteriorAssignment p) (hθ : InteriorMeans θ)
(α : StaircaseWeight) (hα : ∀ s, 0 ≤ α s) :
outputCardinality θ p (staircaseChannel ε α) =
(staircaseSupport α).card := by
let A : Set (Fin 14) :=
{s | outputLaw θ p (staircaseChannel ε α) {s} ≠ 0}
have hsets : A = (staircaseSupport α : Set (Fin 14)) := by
ext s
constructor
· intro hs
simp only [staircaseSupport, Finset.mem_coe, mem_filter,
Finset.mem_univ, true_and]
by_contra hz
exact hs (staircase_output_zero_of_weight_zero θ p ε α s hz)
· intro hs
have hne : α s ≠ 0 := by simpa [staircaseSupport] using hs
exact staircase_output_pos_of_weight_pos θ p ε hp hθ α s
(lt_of_le_of_ne (hα s) (Ne.symm hne))
simp only [outputCardinality, dif_pos (inferInstance : Finite (Fin 14))]
change A.encard = ↑(staircaseSupport α).card
rw [hsets]
exact Set.encard_coe_eq_coe_finsetCard _
theorem five_output_upper_bound
Under the supplied quantities and conditions, the five output upper bound assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the five output upper bound.
Formal statement
hypotheses
conclusion
∃ α : StaircaseWeight,
conclusion 1
conclusion 2
conclusion 3
(staircaseSupport α).card ≤ 5
conclusion 4
outputCardinality θ p (staircaseChannel ε α) ≤ 5
Proof (Lean source)
theorem five_output_upper_bound (θ : TrialParameter) (p ε : ℝ)
(hp : 0 < p ∧ p < 1)
(hθ : 0 < θ 0 ∧ θ 0 < 1 ∧ 0 < θ 1 ∧ θ 1 < 1)
(hε : 0 < ε) :
∃ α : StaircaseWeight,
staircaseFeasible ε α ∧
Jstar θ p ε =
sInf {u : ℝ | ∃ t : ℝ, u = informationObjective θ p ε α t} ∧
(staircaseSupport α).card ≤ 5 ∧
outputCardinality θ p (staircaseChannel ε α) ≤ 5 := by
obtain ⟨α, hfeasible, hoptimal, hsupport'⟩ :=
exists_sparse_staircase_saddle θ p ε hp hθ hε
have hsupport : (staircaseSupport α).card ≤ 5 := by
simpa [staircaseSupport,
weightSupport] using hsupport'
exact ⟨α, hfeasible, hoptimal, hsupport,
(outputCardinality_staircase_le_support_card θ p ε α).trans
(by exact_mod_cast hsupport)⟩
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.TLaplaceComparison 5 declarations The custom known-assignment sample mean has the displayed asymptotic variance.
Comparison with the published custom Laplace release
The custom known-assignment sample mean has the displayed asymptotic variance. Its Wald half-length constant exceeds the private-pilot oracle's at the balanced binary-trial point.
For the supplied quantities and conditions, the asymptotic wald half length is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs.
Definition (Lean source)
parameters
γ variance :
ℝ
n :
ℕ
def
asymptoticWaldHalfLength γ variance n :
ℝ
given by
waldCriticalValue γ * sqrt (variance / n)
theorem Vstar_nonneg
Under the supplied quantities and conditions, the vstar nonneg assertion holds. For the displayed quantities and conditions, these specify the stated inputs. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the Vstar nonneg.
Formal statement
hypotheses
select :
TrialParameter → StaircaseWeight × ℝ × ℝ
θ :
p ε :
ℝ
hselect :
StrongSaddleSelection p ε select
hp :
hθ :
hε :
0 < ε
conclusion
0 ≤ Vstar θ p ε
Proof (Lean source)
lemma Vstar_nonneg (select : TrialParameter → StaircaseWeight × ℝ × ℝ)
(θ : TrialParameter) (p ε : ℝ)
(hselect : StrongSaddleSelection p ε select)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ) (hε : 0 < ε) :
0 ≤ Vstar θ p ε := by
have hπ (j : Fin 4) : 0 ≤ piTheta θ p j := by
rcases hp with ⟨hp0, hp1⟩
rcases hθ with ⟨h0, h0', h1, h1'⟩
fin_cases j <;> simp only [piTheta, controlProb] <;>
apply mul_nonneg <;> linarith
have hr (s : Fin 14) (j : Fin 4) : 0 ≤ patternRay ε s j := by
have he : 1 ≤ exp ε := (Real.one_le_exp_iff).2 (le_of_lt hε)
unfold patternRay privacyIncrement privacyRatio
split_ifs <;> simp_all <;> linarith [Real.exp_pos ε]
have hm (s : Fin 14) : 0 ≤ patternMass θ p ε s := by
unfold patternMass
exact Finset.sum_nonneg fun j _ => mul_nonneg (hπ j) (hr s j)
have hF (α : StaircaseWeight) (hα : staircaseFeasible ε α) (t : ℝ) :
0 ≤ informationObjective θ p ε α t := by
unfold informationObjective
exact Finset.sum_nonneg fun s _ =>
mul_nonneg (hα.1 s) (div_nonneg (sq_nonneg _) (hm s))
have hJ : 0 ≤ Jstar θ p ε := by
unfold Jstar
apply Real.sSup_nonneg
intro z hz
rcases hz with ⟨α, hα, rfl⟩
apply Real.sInf_nonneg
intro y hy
rcases hy with ⟨t, rfl⟩
exact hF α hα t
exact inv_nonneg.mpr hJ
theorem VOA_balanced_value
the voa balanced value assertion holds.
Formal statement
conclusion
VOA (fun _ => (1 / 2 : ℝ)) (1 / 2) 1 = 34
Proof (Lean source)
lemma VOA_balanced_value :
VOA (fun _ => (1 / 2 : ℝ)) (1 / 2) 1 = 34 := by
norm_num [VOA, sensitivity, contrast, controlProb]
the balanced coth sq lt 34 assertion holds.
Proof (Lean source)
lemma balanced_coth_sq_lt_34 :
(cosh (1 / 2) / sinh (1 / 2)) ^ 2 < (34 : ℝ) := by
have hs : (1 / 2 : ℝ) < sinh (1 / 2) :=
Real.self_lt_sinh_iff.mpr (by norm_num)
have hspos : 0 < sinh (1 / 2) := by linarith
have hsq := Real.cosh_sq_sub_sinh_sq (1 / 2 : ℝ)
rw [div_pow]
apply (div_lt_iff₀ (sq_pos_of_pos hspos)).2
nlinarith [sq_nonneg (sinh (1 / 2) - 1 / 2)]
theorem laplace_comparison
the laplace comparison assertion holds. For the displayed quantities and conditions, these specify the stated inputs. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the laplace comparison.
Formal statement
hypotheses
Ω :
Type*
μ :
Y0 Y1 W Y L :
ℕ → Ω → ℝ
θ :
p ε γ :
ℝ
hmodel :
LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ
P :
hpublic :
P = BinaryTrialModel θ p hmodel.meansInterior hmodel.assignmentInterior
hrep :
∀ n, μ.map (fun ω => fun i : Fin n.val => privateRecord W Y i ω) = P n
hnoise :
ComparatorNoise μ W Y L p ε
hε :
0 < ε
hγ :
0 < γ ∧ γ < 1
hcritical :
0 < waldCriticalValue γ
select :
TrialParameter → StaircaseWeight × ℝ × ℝ
hselect :
StrongSaddleSelection p ε select
conclusion
conclusion 1
P = BinaryTrialModel θ p hmodel.meansInterior hmodel.assignmentInterior
n :
μ.map (fun ω => fun i : Fin n.val => privateRecord W Y i ω) = P n
conclusion 2
Modes.TendstoInLaw (fun _ : ℕ => μ) (fun n ω => sqrt n * (tauOA W Y L p n ω - contrast θ)) atTop (gaussianMeasure 0 (VOA θ p ε))
conclusion 3
Tendsto (fun n => asymptoticWaldHalfLength γ (VOA θ p ε) n / asymptoticWaldHalfLength γ (Vstar θ p ε) n) atTop (nhds (sqrt (VOA θ p ε / Vstar θ p ε)))
conclusion 4
δ :
ℝ
0 < δ
conclusion 5
∃ m : ℕ → ℕ,
m = sqrt
δ :
ℝ
0 < δ
ν :
(n : ℕ) → Measure (Ω × Transcript (pilotOutputFamily n))
(∀ n, (ν n).map fst = μ ∧ (ν n).map snd = transcriptLaw (pilotEstimator p ε m select hselect hmodel.assignmentInterior hε) θ p n)
conclusion 6
VOA (fun _ => (1 / 2 : ℝ)) (1 / 2) 1 = 34
Proof (Lean source)
theorem laplace_comparison {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(Y0 Y1 W Y L : ℕ → Ω → ℝ)
(θ : TrialParameter) (p ε γ : ℝ)
(hmodel : LatentBinaryTrialRepresentation μ Y0 Y1 W Y p θ)
(P : (n : PositiveSampleSize) → Measure (Fin n.val → PrivateAlphabet))
(hpublic : P = BinaryTrialModel θ p hmodel.meansInterior hmodel.assignmentInterior)
(hrep : ∀ n, μ.map (fun ω => fun i : Fin n.val => privateRecord W Y i ω) = P n)
(hnoise : ComparatorNoise μ W Y L p ε)
(hε : 0 < ε) (hγ : 0 < γ ∧ γ < 1)
(hcritical : 0 < waldCriticalValue γ)
(select : TrialParameter → StaircaseWeight × ℝ × ℝ)
(hselect : StrongSaddleSelection p ε select) :
(P = BinaryTrialModel θ p hmodel.meansInterior hmodel.assignmentInterior ∧
∀ n, μ.map (fun ω => fun i : Fin n.val => privateRecord W Y i ω) = P n) ∧
Modes.TendstoInLaw (fun _ : ℕ => μ)
(fun n ω => sqrt n *
(tauOA W Y L p n ω - contrast θ))
atTop (gaussianMeasure 0 (VOA θ p ε)) ∧
Tendsto (fun n =>
asymptoticWaldHalfLength γ (VOA θ p ε) n /
asymptoticWaldHalfLength γ (Vstar θ p ε) n)
atTop (nhds (sqrt (VOA θ p ε / Vstar θ p ε))) ∧
(∀ δ : ℝ, 0 < δ →
Tendsto (fun n => μ {ω | δ <
|VhatOA W Y L p n ω - VOA θ p ε|})
atTop (nhds 0)) ∧
(∃ m : ℕ → ℕ, m = Nat.sqrt ∧ PilotDiverges m ∧ PilotSublinear m ∧
∀ δ : ℝ, 0 < δ →
∀ ν : (n : ℕ) →
Measure (Ω × Transcript (pilotOutputFamily n)),
(∀ n, (ν n).map fst = μ ∧
(ν n).map snd =
transcriptLaw
(pilotEstimator p ε m select hselect hmodel.assignmentInterior hε)
θ p n) →
Tendsto (fun n => (ν n)
{u | δ < |sqrt
(VhatOA W Y L p n u.1 /
VhatStar p ε m
(select)
u.2) -
sqrt (VOA θ p ε / Vstar θ p ε)|})
atTop (nhds 0)) ∧
VOA (fun _ => (1 / 2 : ℝ)) (1 / 2) 1 = 34 ∧
Vstar (fun _ => (1 / 2 : ℝ)) (1 / 2) 1 =
(cosh (1 / 2) / sinh (1 / 2)) ^ 2 ∧
Vstar (fun _ => (1 / 2 : ℝ)) (1 / 2) 1 <
VOA (fun _ => (1 / 2 : ℝ)) (1 / 2) 1 := by
have hbalanced :
Vstar (fun _ => (1 / 2 : ℝ)) (1 / 2) 1 =
(cosh (1 / 2) / sinh (1 / 2)) ^ 2 := by
have h := (balanced_reduction μ Y0 Y1 W Y
(fun _ => (1 / 2 : ℝ)) 1
(by norm_num [InteriorMeans]) (by norm_num) (by norm_num)).1
simpa [contrast] using h
refine ⟨?_, ?_, ?_, ?_, ?_, VOA_balanced_value, hbalanced, ?_⟩
· exact ⟨hpublic, hrep⟩
· exact oaRelease_clt hmodel hnoise hε
· have hV : 0 ≤ Vstar θ p ε :=
Vstar_nonneg select θ p ε hselect hmodel.assignmentInterior hmodel.meansInterior hε
have hpoint (n : ℕ) (hn : 0 < n) :
asymptoticWaldHalfLength γ (VOA θ p ε) n /
asymptoticWaldHalfLength γ (Vstar θ p ε) n =
sqrt (VOA θ p ε / Vstar θ p ε) := by
have hnreal : (0 : ℝ) < n := Nat.cast_pos.mpr hn
have hsqrt : sqrt (n : ℝ) ≠ 0 := ne_of_gt (Real.sqrt_pos.2 hnreal)
rw [asymptoticWaldHalfLength, asymptoticWaldHalfLength,
Real.sqrt_div' _ (le_of_lt hnreal),
Real.sqrt_div' _ (le_of_lt hnreal), Real.sqrt_div' _ hV]
field_simp
apply Filter.Tendsto.congr' ?_ tendsto_const_nhds
filter_upwards [eventually_gt_atTop 0] with n hn
exact (hpoint n hn).symm
· exact oaVariance_consistency hmodel hnoise hε
· refine ⟨Nat.sqrt, rfl, natSqrt_pilotDiverges, natSqrt_pilotSublinear, ?_⟩
intro δ hδ ν hν
have hfixed : FixedPrivacy (fun _ => ε) ε := ⟨hε, fun _ => rfl⟩
have hpilot := pilot_attainment θ p ε γ
hmodel.assignmentInterior hmodel.meansInterior (fun _ => ε) hfixed select hselect
hγ Nat.sqrt natSqrt_pilotDiverges natSqrt_pilotSublinear
have hpilotVar := hpilot.2.2.2.1
have hcoupling (n : ℕ) : IsCoupling (ν n) μ
(transcriptLaw
(pilotEstimator p ε Nat.sqrt select hselect hmodel.assignmentInterior hε) θ p n) := by
letI : IsProbabilityMeasure (ν n) := by
letI : IsProbabilityMeasure (Measure.map fst (ν n)) :=
(hν n).1 ▸ inferInstance
exact Measure.isProbabilityMeasure_of_map fst
exact ⟨inferInstance, (hν n).1, (hν n).2⟩
have hVpos : 0 < Vstar θ p ε := by
unfold Vstar
exact inv_pos.mpr (Jstar_pos_interior θ p ε
hmodel.assignmentInterior hmodel.meansInterior hε)
exact Modes.coupled_sqrtRatio_tendsto_strict_abs
(fun _ : ℕ => μ)
(fun n => transcriptLaw
(pilotEstimator p ε Nat.sqrt select hselect hmodel.assignmentInterior hε) θ p n)
ν hcoupling
(fun n ω => VhatOA W Y L p n ω)
(fun _ z => VhatStar p ε Nat.sqrt
(select) z)
(VOA θ p ε) (Vstar θ p ε) hVpos atTop
(fun _ z => VhatStar_nonneg p ε Nat.sqrt
(select) z)
(oaVariance_consistency hmodel hnoise hε) hpilotVar δ hδ
· rw [hbalanced, VOA_balanced_value]
exact balanced_coth_sq_lt_34
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.TPilotAttainment 7 declarations The complete pilot procedure is private, conditionally unbiased, regular and asymptotically Gaussian.
Attainment by the private-pilot estimator
The complete pilot procedure is private, conditionally unbiased, regular and asymptotically Gaussian. Its main-sample variance gives Wald coverage at every interior parameter, including channel-support boundaries.
def standardNormalCDF
For the supplied quantities and conditions, the standard normal cdf is the mathematical object specified below.
Definition (Lean source)
the wald critical value is the mathematical object specified below. The wald Critical Value is determined by the displayed parameters.
Definition (Lean source)
parameters
γ :
ℝ
def
waldCriticalValue γ :
ℝ
given by
sInf {z : ℝ | 1 - γ / 2 ≤ standardNormalCDF z}
theorem standardNormalCDF_eq_stdNormalCDF
Under the supplied quantities and conditions, the standard normal cdf eq std normal cdf assertion holds.
Formal statement
Proof (Lean source)
lemma standardNormalCDF_eq_stdNormalCDF (x : ℝ) :
standardNormalCDF x = stdNormalCDF x := by
simp [standardNormalCDF, stdNormalCDF_def,
ProbabilityTheory.cdf_eq_real, MeasureTheory.measureReal_def, gaussianMeasure]
theorem waldCriticalValue_eq_probit
the wald critical value eq probit assertion holds.
Formal statement
Proof (Lean source)
lemma waldCriticalValue_eq_probit (γ : ℝ) :
waldCriticalValue γ = probit (1 - γ / 2) := by
unfold waldCriticalValue probit
congr 1
ext z
simp only [Set.mem_setOf_eq]
rw [standardNormalCDF_eq_stdNormalCDF]
theorem waldCriticalValue_pos
Under the supplied quantities and conditions, the wald critical value pos assertion holds. Under the stated assumptions, the wald Critical Value pos.
Formal statement
Proof (Lean source)
lemma waldCriticalValue_pos {γ : ℝ} (hγ : 0 < γ ∧ γ < 1) :
0 < waldCriticalValue γ := by
have hq0 : 0 < 1 - γ / 2 := by linarith
have hq1 : 1 - γ / 2 < 1 := by linarith
have hzero : stdNormalCDF 0 = 1 / 2 := by
have hsym := stdNormalCDF_neg 0
norm_num at hsym ⊢
linarith
rw [waldCriticalValue_eq_probit]
apply Causalean.Mathlib.stdNormalCDF_strictMono.lt_iff_lt.mp
rw [stdNormalCDF_probit hq0 hq1, hzero]
linarith
theorem gaussianMeasure_waldCriticalValue_Icc
the gaussian measure wald critical value icc assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the gaussian Measure wald Critical Value Icc.
Formal statement
hypotheses
γ :
ℝ
hγ :
0 < γ ∧ γ < 1
conclusion
= 1 - γ
Proof (Lean source)
lemma gaussianMeasure_waldCriticalValue_Icc {γ : ℝ}
(hγ : 0 < γ ∧ γ < 1) :
(gaussianMeasure 0 1
(Icc (-waldCriticalValue γ) (waldCriticalValue γ))).toReal = 1 - γ := by
have hp0 : 0 < γ / 2 := by linarith
have hp1 : γ / 2 < 1 := by linarith
have hq0 : 0 < 1 - γ / 2 := by linarith
have hq1 : 1 - γ / 2 < 1 := by linarith
have hsym : probit (γ / 2) =
-probit (1 - γ / 2) := by
apply Causalean.Mathlib.stdNormalCDF_strictMono.injective
rw [stdNormalCDF_probit hp0 hp1,
stdNormalCDF_neg,
stdNormalCDF_probit hq0 hq1]
ring
have h := gaussianReal_quantile_Icc_toReal
(v := (1 : NNReal)) (by norm_num) hγ.1 hγ.2
simpa [gaussianMeasure, waldCriticalValue_eq_probit, hsym,
MeasureTheory.measureReal_def] using h
theorem pilot_attainment
Under the supplied quantities and conditions, the pilot attainment assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the pilot attainment.
Formal statement
hypotheses
θ :
p ε γ :
ℝ
hassignment :
hmeans :
epsSeq :
ℕ → ℝ
hfixed :
FixedPrivacy epsSeq ε
select :
TrialParameter → StaircaseWeight × ℝ × ℝ
hselect :
StrongSaddleSelection p ε select
hγ :
0 < γ ∧ γ < 1
@realizes \gamma(Wald error in (0,1))
m :
ℕ → ℕ
hdiverges :
hsublinear :
conclusion
P :
pilotEstimator p ε m select hselect hassignment hfixed.1
n :
SequentialLDP (Z := pilotOutputFamily n) ε (P.channel n)
n ≥ 2 :
f :
Transcript (pilotOutputFamily n) → ℝ
(∃ C : ℝ, ∀ z, |f z| ≤ C)
(∀ z z', (∀ i : Fin n, i.val < m n → z i = z' i) → f z = f z')
Integrable (fun z => (P.estimate n z - contrast θ) * f z) (transcriptLaw P θ p n)
(∫ z, (P.estimate n z - contrast θ) * f z ∂transcriptLaw P θ p n) = 0
δ :
ℝ
0 < δ
RegularProcedure P θ p
Proof (Lean source)
theorem pilot_attainment
(θ : TrialParameter) (p ε γ : ℝ)
(hassignment : InteriorAssignment p)
(hmeans : InteriorMeans θ)
(epsSeq : ℕ → ℝ) (hfixed : FixedPrivacy epsSeq ε)
(select : TrialParameter → StaircaseWeight × ℝ × ℝ)
(hselect : StrongSaddleSelection p ε select)
(hγ : 0 < γ ∧ γ < 1) -- @realizes \gamma(Wald error in (0,1))
(m : ℕ → ℕ) (hdiverges : PilotDiverges m)
(hsublinear : PilotSublinear m) :
let P := pilotEstimator p ε m select hselect hassignment hfixed.1
(∀ n, SequentialLDP (Z := pilotOutputFamily n) ε (P.channel n)) ∧
(∀ n ≥ 2, ∀ f : Transcript (pilotOutputFamily n) → ℝ,
Measurable f →
(∃ C : ℝ, ∀ z, |f z| ≤ C) →
(∀ z z', (∀ i : Fin n, i.val < m n → z i = z' i) → f z = f z') →
Integrable (fun z => (P.estimate n z - contrast θ) * f z)
(transcriptLaw P θ p n) ∧
(∫ z, (P.estimate n z - contrast θ) * f z
∂transcriptLaw P θ p n) = 0) ∧
WeakLocalLimit P θ p (gaussianMeasure 0 (Vstar θ p ε)) ∧
(∀ δ : ℝ, 0 < δ →
Tendsto (fun n =>
(transcriptLaw P θ p n)
{z | δ < |VhatStar p ε m
(select) z -
Vstar θ p ε|}) atTop (nhds 0)) ∧
RegularProcedure P θ p ∧
Tendsto (fun n =>
((transcriptLaw P θ p n)
{z | |P.estimate n z - contrast θ| ≤
waldCriticalValue γ *
sqrt (VhatStar p ε m
(select) z / n)}).toReal)
atTop (nhds (1 - γ)) := by
let P := pilotEstimator p ε m select hselect hassignment hfixed.1
have hgaussianCDF : ∀ h : TrialParameter, ∀ x : ℝ,
Tendsto (fun n =>
(transcriptLaw P (localAlternative θ h n) p n).real
{z | scaledError P θ h n z ≤ x}) atTop
(nhds ((gaussianMeasure 0 (Vstar θ p ε)).real (Iic x))) := by
exact pilot_scaledError_cdf_tendsto_of_all_conditionalRows
select θ p ε m hselect hassignment hmeans hfixed.1 hdiverges hsublinear
(fun h x φ hφ η hη hηlim =>
adaptivePilotRowScaledCDF_subsequence_tendsto_gaussian
select θ h p ε m hselect hassignment hmeans hfixed.1 hsublinear
x φ hφ η hη hηlim)
have hweak : WeakLocalLimit P θ p
(gaussianMeasure 0 (Vstar θ p ε)) :=
weakLocalLimit_of_scaledError_cdf_tendsto
select P θ p hselect hassignment hmeans hfixed.1 hgaussianCDF
have hvhat : ∀ δ : ℝ, 0 < δ →
Tendsto (fun n =>
(transcriptLaw P θ p n)
{z | δ < |VhatStar p ε m
(select) z -
Vstar θ p ε|}) atTop (nhds 0) := by
exact pilot_VhatStar_consistent select θ p ε m hselect hassignment hmeans
hfixed.1 hdiverges hsublinear
have htails := pilot_scaledError_uniform_squared_tails
select θ p ε m hselect hassignment hmeans hfixed.1 hsublinear
have hregular : RegularProcedure P θ p :=
regular_of_weakLocalLimit_and_uniformTails
P θ p (Vstar θ p ε) hweak htails
have hV : 0 < Vstar θ p ε :=
inv_pos.mpr (Jstar_pos_interior θ p ε hassignment hmeans hfixed.1)
have hsqrtV : 0 < sqrt (Vstar θ p ε) := Real.sqrt_pos.2 hV
have hstandardCDF : ∀ x : ℝ,
Tendsto (fun n =>
(transcriptLaw P θ p n).real
{z | scaledError P θ 0 n z / sqrt (Vstar θ p ε) ≤ x}) atTop
(nhds ((gaussianMeasure 0 1).real (Iic x))) := by
intro x
have hraw := hgaussianCDF 0 (x * sqrt (Vstar θ p ε))
have hevent (n : ℕ) :
{z | scaledError P θ 0 n z / sqrt (Vstar θ p ε) ≤ x} =
{z | scaledError P θ 0 n z ≤ x * sqrt (Vstar θ p ε)} := by
ext z
exact div_le_iff₀ hsqrtV
have htarget :
(gaussianMeasure 0 (Vstar θ p ε)).real
(Iic (x * sqrt (Vstar θ p ε))) =
(gaussianMeasure 0 1).real (Iic x) := by
rw [← ProbabilityTheory.cdf_eq_real, ← ProbabilityTheory.cdf_eq_real]
simp only [gaussianMeasure]
rw [cdf_gaussianReal_zero
(Real.toNNReal_pos.mpr hV),
cdf_gaussianReal_zero (v := toNNReal 1)
(by norm_num [Real.toNNReal_pos])]
congr 1
rw [Real.coe_toNNReal (Vstar θ p ε) hV.le,
Real.coe_toNNReal 1 (by norm_num)]
simp only [Real.sqrt_one, div_one]
field_simp
rw [← htarget]
apply hraw.congr'
filter_upwards with n
rw [hevent n]
have hlocal0 : localAlternative θ 0 n = θ := by
funext k
simp [localAlternative]
rw [hlocal0]
letI (n : ℕ) : IsProbabilityMeasure (transcriptLaw P θ p n) :=
transcriptLaw_isProbability P θ p hassignment hmeans n
have hvhat_ge : ∀ r : ℝ, 0 < r →
Tendsto (fun n =>
(transcriptLaw P θ p n).real
{z | r ≤ |VhatStar p ε m
(select) z - Vstar θ p ε|})
atTop (nhds 0) := by
intro r hr
have hstrict := hvhat (r / 2) (half_pos hr)
have hstrictReal : Tendsto (fun n =>
(transcriptLaw P θ p n).real
{z | r / 2 < |VhatStar p ε m
(select) z - Vstar θ p ε|})
atTop (nhds 0) := by
change Tendsto (fun n => toReal
((transcriptLaw P θ p n)
{z | r / 2 < |VhatStar p ε m
(select) z - Vstar θ p ε|}))
atTop (nhds (toReal 0))
exact (ENNReal.tendsto_toReal (by simp)).comp hstrict
apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hstrictReal
· intro n
exact MeasureTheory.measureReal_nonneg
· intro n
apply MeasureTheory.measureReal_mono (h₂ := measure_ne_top _ _)
intro z hz
exact lt_of_lt_of_le (half_lt_self hr) hz
have hwald :=
pilot_wald_coverage_of_standardized_cdf_and_VhatStar_consistent
select (waldCriticalValue γ) (waldCriticalValue_pos hγ)
θ p ε m hselect hassignment hmeans hfixed.1 hstandardCDF hvhat_ge
refine ⟨?_, ?_, hweak, hvhat, hregular, ?_⟩
· intro n
exact pilotProtocol_private p ε m select hselect hassignment hfixed.1 n
· intro n hn f hf hbound hpref
exact pilot_conditional_unbiasedness select θ p ε m hselect hassignment hmeans
hfixed.1 hsublinear n hn f hf hbound hpref
· have hcoverage :
(gaussianMeasure 0 1).real
(Icc (-waldCriticalValue γ) (waldCriticalValue γ)) = 1 - γ := by
simpa only [MeasureTheory.measureReal_def] using
gaussianMeasure_waldCriticalValue_Icc hγ
rw [hcoverage] at hwald
simpa only [P] using hwald
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.TSequentialMinimax 2 declarations The lower bound covers every history-dependent private procedure on arbitrary measurable output spaces, whether regular or not.
Sequential local asymptotic minimax equality
The lower bound covers every history-dependent private procedure on arbitrary measurable output spaces, whether regular or not. A single private-pilot sequence is regular and attains the bound, giving the infimum equality.
For the supplied quantities and conditions, the regular risk set is the mathematical object specified below. The regular Risk Set is determined by the displayed parameters.
Definition (Lean source)
parameters
θ :
p ε :
ℝ
def
given by
{u | ∃ Z : OutputFamily, ∃ mZ : ∀ n i, MeasurableSpace (Z n i), letI := mZ ∃ P : ProcedureSequence Z ε, RegularProcedure P θ p ∧ u = localAsymptoticRisk P θ p}
theorem sequential_minimax
Under the supplied quantities and conditions, the sequential minimax assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the sequential minimax.
Formal statement
hypotheses
θ0 :
p ε :
ℝ
hp :
hθ :
@realizes \theta_0(interior local parameter point)
epsSeq :
ℕ → ℝ
hfixed :
FixedPrivacy epsSeq ε
select :
TrialParameter → StaircaseWeight × ℝ × ℝ
hselect :
StrongSaddleSelection p ε select
conclusion
conclusion 1
conclusion 2
∃ m : ℕ → ℕ,
m = sqrt
RegularProcedure (pilotEstimator p ε m select hselect hp hfixed.1) θ0 p
localAsymptoticRisk (pilotEstimator p ε m select hselect hp hfixed.1) θ0 p
conclusion 3
Proof (Lean source)
theorem sequential_minimax
(θ0 : TrialParameter) (p ε : ℝ)
(hp : InteriorAssignment p) (hθ : InteriorMeans θ0)
-- @realizes \theta_0(interior local parameter point)
(epsSeq : ℕ → ℝ) (hfixed : FixedPrivacy epsSeq ε)
(select : TrialParameter → StaircaseWeight × ℝ × ℝ)
(hselect : StrongSaddleSelection p ε select) :
(∀ (Z : OutputFamily) [∀ n i, MeasurableSpace (Z n i)]
(P : ProcedureSequence Z ε),
ofReal (Vstar θ0 p ε) ≤
localAsymptoticRisk P θ0 p) ∧
(∃ m : ℕ → ℕ,
m = Nat.sqrt ∧ PilotDiverges m ∧ PilotSublinear m ∧
RegularProcedure (pilotEstimator p ε m select hselect hp hfixed.1) θ0 p ∧
localAsymptoticRisk (pilotEstimator p ε m select hselect hp hfixed.1) θ0 p =
ofReal (Vstar θ0 p ε)) ∧
sInf (regularRiskSet θ0 p ε) =
ofReal (Vstar θ0 p ε) := by
have hlower : ∀ (Z : OutputFamily) [∀ n i, MeasurableSpace (Z n i)]
(P : ProcedureSequence Z ε),
ofReal (Vstar θ0 p ε) ≤ localAsymptoticRisk P θ0 p := by
intro Z mZ P
exact sequential_localAsymptoticRisk_lower_bound P θ0 p hp hθ hfixed.1
have hrates := natSqrt_pilotRates
have hpilot := pilot_attainment θ0 p ε (1 / 2) hp hθ
epsSeq hfixed select hselect (by norm_num) Nat.sqrt hrates.1 hrates.2
have hregular : RegularProcedure
(pilotEstimator p ε Nat.sqrt select hselect hp hfixed.1) θ0 p :=
hpilot.2.2.2.2.1
have hrisk : localAsymptoticRisk
(pilotEstimator p ε Nat.sqrt select hselect hp hfixed.1) θ0 p =
ofReal (Vstar θ0 p ε) :=
pilot_localAsymptoticRisk_eq select θ0 p ε Nat.sqrt hselect hp hθ hfixed.1
hrates.1 hrates.2
refine ⟨hlower, ?_, ?_⟩
· exact ⟨Nat.sqrt, rfl, hrates.1, hrates.2, hregular, hrisk⟩
· apply le_antisymm
· apply sInf_le
refine ⟨pilotOutputFamily, (fun _ _ => inferInstance), ?_⟩
let _ : ∀ n i, MeasurableSpace (pilotOutputFamily n i) :=
fun _ _ => inferInstance
exact ⟨pilotEstimator p ε Nat.sqrt select hselect hp hfixed.1,
hregular, hrisk.symm⟩
· apply le_sInf
rintro u ⟨Z, mZ, P, -, rfl⟩
let _ : ∀ n i, MeasurableSpace (Z n i) := mZ
exact hlower Z P
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.TSupportTransition 13 declarations Disjoint neighborhoods of the balanced and low-mean points have unique optimal staircase supports of sizes five and three, respectively.
Transition between optimal support sizes
Disjoint neighborhoods of the balanced and low-mean points have unique optimal staircase supports of sizes five and three, respectively.
For the supplied quantities and conditions, the unique optimal support is the mathematical object specified below. For the displayed quantities and conditions, these specify the stated inputs. The unique Optimal Support is determined by the displayed parameters.
Definition (Lean source)
parameters
def
uniqueOptimalSupport θ p ε K :
Prop
For the supplied quantities and conditions, the optimal cardinality is the mathematical object specified below. The optimal Cardinality is determined by the displayed parameters.
Definition (Lean source)
parameters
def
optimalCardinality θ p ε k :
Prop
given by
Under the supplied quantities and conditions, the pi theta by val transition assertion holds.
Formal statement
hypotheses
conclusion
piTheta θ p j
= if j.val = 0 then controlProb p * (1 - θ 0) else if j.val = 1 then controlProb p * θ 0 else if j.val = 2 then p * (1 - θ 1) else p * θ 1
Proof (Lean source)
lemma piTheta_by_val_transition (θ : TrialParameter) (p : ℝ) (j : Fin 4) :
piTheta θ p j =
if j.val = 0 then controlProb p * (1 - θ 0)
else if j.val = 1 then controlProb p * θ 0
else if j.val = 2 then p * (1 - θ 1)
else p * θ 1 := by
rcases j with ⟨j, hj⟩
interval_cases j <;> rfl
theorem lowMean_informationMatrix_value
the low mean information matrix value assertion holds.
Formal statement
conclusion
informationMatrix (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3) (fun s => if s ∈ ({0, 5, 7} : Finset (Fin 14)) then 1 / 5 else 0)
= !![(1379 / 3390 : ℝ), -(1 / 10 : ℝ); -(1 / 10 : ℝ), (411 / 3710 : ℝ)]
Proof (Lean source)
lemma lowMean_informationMatrix_value :
informationMatrix (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3)
(fun s => if s ∈ ({0, 5, 7} : Finset (Fin 14)) then 1 / 5 else 0) =
!![(1379 / 3390 : ℝ), -(1 / 10 : ℝ);
-(1 / 10 : ℝ), (411 / 3710 : ℝ)] := by
have he : exp (log (3 : ℝ)) = 3 := Real.exp_log (by norm_num)
ext a b
fin_cases a <;> fin_cases b <;>
simp +decide [informationMatrix, patternGradient, patternMass,
patternRay, patternContains, piTheta_by_val_transition,
privacyIncrement, privacyRatio, controlProb, he,
Fin.sum_univ_succ] <;> norm_num
theorem lowMean_informationMatrix_direction
the low mean information matrix direction assertion holds.
Formal statement
conclusion
(informationMatrix (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3) (fun s => if s ∈ ({0, 5, 7} : Finset (Fin 14)) then 1 / 5 else 0)).mulVec (direction (-(339 / 9985 : ℝ)))
= fun k => (441 / 3994 : ℝ) * contrastVector k
Proof (Lean source)
lemma lowMean_informationMatrix_direction :
(informationMatrix (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3)
(fun s => if s ∈ ({0, 5, 7} : Finset (Fin 14)) then 1 / 5 else 0)).mulVec
(direction (-(339 / 9985 : ℝ))) =
fun k => (441 / 3994 : ℝ) * contrastVector k := by
rw [lowMean_informationMatrix_value]
funext k
fin_cases k <;>
norm_num [mulVec, dotProduct, direction, contrastVector,
Fin.sum_univ_succ]
the low mean projected scores distinct assertion holds.
Formal statement
Proof (Lean source)
lemma lowMean_projectedScores_distinct :
∀ s ∈ ({0, 5, 7} : Finset (Fin 14)),
∀ u ∈ ({0, 5, 7} : Finset (Fin 14)), s ≠ u →
projectedScore (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3)
s (-(339 / 9985 : ℝ)) ≠
projectedScore (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3)
u (-(339 / 9985 : ℝ)) := by
intro s hs u hu hne
have he : exp (log (3 : ℝ)) = 3 := Real.exp_log (by norm_num)
have hs : s = 0 ∨ s = 5 ∨ s = 7 := by simpa using hs
have hu : u = 0 ∨ u = 5 ∨ u = 7 := by simpa using hu
rcases hs with rfl | rfl | rfl
all_goals rcases hu with rfl | rfl | rfl
all_goals first | contradiction | simp +decide [projectedScore, projectedGradient,
patternMass, patternGradient, patternRay, patternContains, piTheta_by_val_transition,
privacyIncrement, privacyRatio, controlProb, direction, he,
Fin.sum_univ_succ] <;> norm_num
theorem lowMean_outputCardinality_of_attainment
the low mean output cardinality of attainment assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Z :
Type*
hQ :
StationaryLDP (log 3) Q
hattain :
contrastVariance (channelFisherInfo (fun _ ↦ (1 / 10 : ℝ)) (3 / 10) Q)
conclusion
3 ≤ outputCardinality (fun _ ↦ (1 / 10 : ℝ)) (3 / 10) Q
Proof (Lean source)
lemma lowMean_outputCardinality_of_attainment
{Z : Type*} [MeasurableSpace Z] (Q : Kernel (Fin 4) Z)
(hQ : StationaryLDP (log 3) Q)
(hattain :
contrastVariance
(channelFisherInfo (fun _ ↦ (1 / 10 : ℝ)) (3 / 10) Q) =
ofReal
(Vstar (fun _ ↦ (1 / 10 : ℝ)) (3 / 10) (log 3))) :
3 ≤ outputCardinality (fun _ ↦ (1 / 10 : ℝ)) (3 / 10) Q := by
let θ : TrialParameter := fun _ ↦ (1 / 10 : ℝ)
let A : Finset (Fin 14) := {0, 5, 7}
have hp : InteriorAssignment (3 / 10 : ℝ) := ⟨by norm_num, by norm_num⟩
have hθ : InteriorMeans θ := by
simp only [InteriorMeans, θ]
norm_num
have hε : 0 < log 3 := Real.log_pos (by norm_num)
have hupper : ∀ β : StaircaseWeight,
staircaseFeasible (log 3) β →
informationObjective θ (3 / 10) (log 3) β
(-(339 / 9985 : ℝ)) ≤
Jstar θ (3 / 10) (log 3) := by
intro β hβ
dsimp only [θ]
rw [lowMean_Jstar_value]
exact lowMean_dual_upper_bound β hβ
have hpositive : ∀ β : StaircaseWeight,
staircaseFeasible (log 3) β →
informationObjective θ (3 / 10) (log 3) β
(-(339 / 9985 : ℝ)) =
Jstar θ (3 / 10) (log 3) →
∀ s ∈ A, 0 < β s := by
intro β hβ heq
have hactive : ∀ s ∈ A,
dualRay (log 3) lowMeanDual s =
patternInformation θ (3 / 10) (log 3) s
(-(339 / 9985 : ℝ)) := by
intro s hs
simpa only [A, θ, if_pos hs] using lowMeanDual_slack s
have hinactive : ∀ s ∉ A,
patternInformation θ (3 / 10) (log 3) s
(-(339 / 9985 : ℝ)) <
dualRay (log 3) lowMeanDual s := by
intro s hs
simpa only [A, θ, if_neg hs] using lowMeanDual_slack s
have hvalue : informationObjective θ (3 / 10) (log 3) β
(-(339 / 9985 : ℝ)) = ∑ j : Fin 4, lowMeanDual j := by
rw [heq]
dsimp only [θ]
rw [lowMean_Jstar_value, lowMeanDual_value]
have hout : ∀ s ∉ A, β s = 0 :=
certificate_equality_forces_inactive_zero θ (3 / 10) (log 3)
(-(339 / 9985 : ℝ)) A lowMeanDual hactive hinactive β hβ hvalue
have hcanonical := r3_feasible_eq_canonical β hβ (by
simpa only [A] using hout)
intro s hs
rw [hcanonical]
simp only [A] at hs
simp [hs]
have hcard := outputCardinality_ge_of_attainment_direction
θ (3 / 10) (log 3) (-(339 / 9985 : ℝ)) hp hθ hε A
hupper hpositive (by
simpa only [θ, A] using lowMean_projectedScores_distinct) Q hQ (by
simpa only [θ] using hattain)
norm_num [A] at hcard ⊢
exact hcard
theorem supportRegion_slices
Under the supplied quantities and conditions, the support region slices assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
Proof (Lean source)
lemma supportRegion_slices
(hR5 : IsOpen R5)
(h5 : ((3 / 10 : ℝ), (1 / 2 : ℝ), (1 / 2 : ℝ), log 3) ∈ R5)
(hR3 : IsOpen R3)
(h3 : ((3 / 10 : ℝ), (1 / 10 : ℝ), (1 / 10 : ℝ), log 3) ∈ R3) :
∃ U5 U3 : Set (ℝ × ℝ),
IsOpen U5 ∧ IsOpen U3 ∧
((1 / 2 : ℝ), (1 / 2 : ℝ)) ∈ U5 ∧
((1 / 10 : ℝ), (1 / 10 : ℝ)) ∈ U3 ∧
Disjoint U5 U3 ∧
(∀ μ ∈ U5,
((3 / 10 : ℝ), μ.1, μ.2, log 3) ∈ R5) ∧
∀ μ ∈ U3,
((3 / 10 : ℝ), μ.1, μ.2, log 3) ∈ R3 := by
let slice : ℝ × ℝ → ℝ × ℝ × ℝ × ℝ :=
fun μ => ((3 / 10 : ℝ), μ.1, μ.2, log 3)
let U5 : Set (ℝ × ℝ) := slice ⁻¹' R5 ∩ {μ | (3 / 10 : ℝ) < μ.1}
let U3 : Set (ℝ × ℝ) := slice ⁻¹' R3 ∩ {μ | μ.1 < (3 / 10 : ℝ)}
have hslice : Continuous slice := by
fun_prop
refine ⟨U5, U3, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩
· exact (hR5.preimage hslice).inter (isOpen_lt continuous_const continuous_fst)
· exact (hR3.preimage hslice).inter (isOpen_lt continuous_fst continuous_const)
· exact ⟨h5, by norm_num⟩
· exact ⟨h3, by norm_num⟩
· rw [Set.disjoint_left]
intro μ hμ5 hμ3
have hm5 := hμ5.2
have hm3 := hμ3.2
change (3 / 10 : ℝ) < μ.1 at hm5
change μ.1 < (3 / 10 : ℝ) at hm3
exact (not_lt_of_ge (le_of_lt hm5)) hm3
· intro μ hμ
exact hμ.1
· intro μ hμ
exact hμ.1
theorem optimalCardinality_of_mem_R5
Under the supplied quantities and conditions, the optimal cardinality of mem r5 assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p μ0 μ1 ε :
ℝ
hx :
(p, μ0, μ1, ε) ∈ R5
conclusion
optimalCardinality (fun k ↦ if k = 0 then μ0 else μ1) p ε 5
Proof (Lean source)
lemma optimalCardinality_of_mem_R5 (p μ0 μ1 ε : ℝ)
(hx : (p, μ0, μ1, ε) ∈ R5) :
optimalCardinality (fun k ↦ if k = 0 then μ0 else μ1) p ε 5 := by
refine ⟨r5Certificate_staircase_attains_five p μ0 μ1 ε hx, ?_⟩
intro Z _ Q hQ hattain
exact r5Certificate_outputCardinality_ge_five p μ0 μ1 ε hx Q hQ hattain
theorem uniqueOptimalSupport_of_mem_R5
Under the supplied quantities and conditions, the unique optimal support of mem r5 assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p μ0 μ1 ε :
ℝ
hx :
(p, μ0, μ1, ε) ∈ R5
conclusion
uniqueOptimalSupport (fun k ↦ if k = 0 then μ0 else μ1) p ε r5Active
Proof (Lean source)
lemma uniqueOptimalSupport_of_mem_R5 (p μ0 μ1 ε : ℝ)
(hx : (p, μ0, μ1, ε) ∈ R5) :
uniqueOptimalSupport (fun k ↦ if k = 0 then μ0 else μ1)
p ε r5Active := by
obtain ⟨α, hα, hsupp, _, hopt, hunique⟩ :=
r5Certificate_unique_optimum p μ0 μ1 ε hx
refine ⟨α, hα, ?_, hopt, hunique⟩
ext s
simp only [staircaseSupport, mem_filter, Finset.mem_univ, true_and]
exact hsupp s
theorem uniqueOptimalSupport_of_mem_R3
Under the supplied quantities and conditions, the unique optimal support of mem r3 assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
Proof (Lean source)
lemma uniqueOptimalSupport_of_mem_R3 (p μ0 μ1 : ℝ)
(hx : (p, μ0, μ1, log 3) ∈ R3) :
uniqueOptimalSupport (fun k ↦ if k = 0 then μ0 else μ1)
p (log 3) ({0, 5, 7} : Finset (Fin 14)) := by
obtain ⟨α, hα, hsupp, hopt, hunique⟩ :=
r3Certificate_unique_optimum_log3 p μ0 μ1 hx
refine ⟨α, hα, ?_, hopt, hunique⟩
ext s
simp only [staircaseSupport, mem_filter, Finset.mem_univ, true_and]
exact hsupp s
theorem optimalCardinality_of_mem_R3_of_distinct
Under the supplied quantities and conditions, the optimal cardinality of mem r3 of distinct assertion holds. For the displayed quantities and conditions, these specify the stated inputs.
Formal statement
hypotheses
p μ0 μ1 :
ℝ
hdistinct :
∀ i k : Fin 3
if
i ≠ k
then
r3ProjectedScoreAt (p, μ0, μ1, log 3) i
≠ r3ProjectedScoreAt (p, μ0, μ1, log 3) k
conclusion
optimalCardinality (fun k ↦ if k = 0 then μ0 else μ1) p (log 3) 3
Proof (Lean source)
lemma optimalCardinality_of_mem_R3_of_distinct (p μ0 μ1 : ℝ)
(hx : (p, μ0, μ1, log 3) ∈ R3)
(hdistinct : ∀ i k : Fin 3, i ≠ k →
r3ProjectedScoreAt (p, μ0, μ1, log 3) i ≠
r3ProjectedScoreAt (p, μ0, μ1, log 3) k) :
optimalCardinality (fun k ↦ if k = 0 then μ0 else μ1)
p (log 3) 3 := by
refine ⟨r3Certificate_staircase_attains_three p μ0 μ1 hx, ?_⟩
intro Z _ Q hQ hattain
exact r3Certificate_outputCardinality_ge_three_of_distinct
p μ0 μ1 hx hdistinct Q hQ hattain
theorem support_transition
the support transition assertion holds.
Formal statement
conclusion
∃ U5 U3 : Set (ℝ × ℝ),
conclusion 1
μ ∈ U5 :
uniqueOptimalSupport (fun k => if k = 0 then μ.1 else μ.2) (3 / 10) (log 3) ({2, 3, 5, 7, 8} : Finset (Fin 14))
optimalCardinality (fun k => if k = 0 then μ.1 else μ.2) (3 / 10) (log 3) 5
conclusion 2
μ ∈ U3 :
uniqueOptimalSupport (fun k => if k = 0 then μ.1 else μ.2) (3 / 10) (log 3) ({0, 5, 7} : Finset (Fin 14))
optimalCardinality (fun k => if k = 0 then μ.1 else μ.2) (3 / 10) (log 3) 3
Proof (Lean source)
theorem support_transition :
∃ U5 U3 : Set (ℝ × ℝ),
IsOpen U5 ∧ IsOpen U3 ∧
((1 / 2 : ℝ), (1 / 2 : ℝ)) ∈ U5 ∧
((1 / 10 : ℝ), (1 / 10 : ℝ)) ∈ U3 ∧
Disjoint U5 U3 ∧
(∀ μ ∈ U5,
uniqueOptimalSupport (fun k => if k = 0 then μ.1 else μ.2)
(3 / 10) (log 3) ({2, 3, 5, 7, 8} : Finset (Fin 14)) ∧
optimalCardinality (fun k => if k = 0 then μ.1 else μ.2)
(3 / 10) (log 3) 5) ∧
(∀ μ ∈ U3,
uniqueOptimalSupport (fun k => if k = 0 then μ.1 else μ.2)
(3 / 10) (log 3) ({0, 5, 7} : Finset (Fin 14)) ∧
optimalCardinality (fun k => if k = 0 then μ.1 else μ.2)
(3 / 10) (log 3) 3) := by
obtain ⟨V5, V3, hV5open, hV3open, hV5center, hV3center,
hVdisjoint, hV5mem, hV3mem⟩ := supportRegion_slices
isOpen_R5_continuousCertificate
(centeredPoint_mem_R5 (3 / 10) (by norm_num) (by norm_num))
three_output_region.1 three_output_region.2.1
obtain ⟨D3, hD3open, hD3center, hD3distinct⟩ :=
exists_lowMean_r3_distinctScore_neighborhood
let slice : ℝ × ℝ → RegionParameter :=
fun μ => ((3 / 10 : ℝ), μ.1, μ.2, log 3)
let U3 : Set (ℝ × ℝ) := V3 ∩ slice ⁻¹' D3
have hslice : Continuous slice := by fun_prop
refine ⟨V5, U3, hV5open, hV3open.inter (hD3open.preimage hslice),
hV5center, ⟨hV3center, hD3center⟩, ?_, ?_, ?_⟩
· rw [Set.disjoint_left]
intro μ hμ5 hμ3
exact Set.disjoint_left.mp hVdisjoint hμ5 hμ3.1
· intro μ hμ
have hx := hV5mem μ hμ
refine ⟨?_, optimalCardinality_of_mem_R5 (3 / 10) μ.1 μ.2
(log 3) hx⟩
simpa [r5Active] using uniqueOptimalSupport_of_mem_R5
(3 / 10) μ.1 μ.2 (log 3) hx
· intro μ hμ
have hx := hV3mem μ hμ.1
have hd := hD3distinct (slice μ) hμ.2
refine ⟨uniqueOptimalSupport_of_mem_R3 (3 / 10) μ.1 μ.2 hx,
optimalCardinality_of_mem_R3_of_distinct (3 / 10) μ.1 μ.2 hx ?_⟩
simpa [slice] using hd
CausalSmith.Stat.STAT_LdpAteEfficiencySurface_Research.TThreeOutputRegion 27 declarations At the low-mean point, the exact primal and dual values certify a unique three-ray optimum and its open neighborhood.
Three-output rational certificate
At the low-mean point, the exact primal and dual values certify a unique three-ray optimum and its open neighborhood.
theorem threeMaskWeights_feasible
Under the supplied quantities and conditions, the three mask weights feasible assertion holds. Under the stated assumptions, the three Mask Weights feasible.
Formal statement
hypotheses
ε :
ℝ
hε :
0 < ε
conclusion
Proof (Lean source)
lemma threeMaskWeights_feasible (ε : ℝ) (hε : 0 < ε) :
staircaseFeasible ε
(fun s => if s ∈ ({0, 5, 7} : Finset (Fin 14)) then
(exp ε + 2)⁻¹ else 0) := by
constructor
· intro s
change 0 ≤ (if s ∈ ({0, 5, 7} : Finset (Fin 14)) then
(exp ε + 2)⁻¹ else 0)
split_ifs <;> positivity
· intro j
have hd : exp ε + 2 ≠ 0 := by positivity
fin_cases j <;>
simp +decide [staircaseMatrix, patternRay, patternContains,
privacyIncrement, privacyRatio, Fin.sum_univ_succ] <;>
field_simp <;> ring
theorem threeMaskWeights_unique
Under the supplied quantities and conditions, the three mask weights unique assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the three Mask Weights unique.
Formal statement
hypotheses
ε :
ℝ
hε :
0 < ε
α :
hα :
hsupp :
Proof (Lean source)
lemma threeMaskWeights_unique (ε : ℝ) (hε : 0 < ε) (α : StaircaseWeight)
(hα : staircaseFeasible ε α)
(hsupp : staircaseSupport α = ({0, 5, 7} : Finset (Fin 14))) :
α = fun s => if s ∈ ({0, 5, 7} : Finset (Fin 14)) then
(exp ε + 2)⁻¹ else 0 := by
have hzero (s : Fin 14)
(hs : s ∉ ({0, 5, 7} : Finset (Fin 14))) : α s = 0 := by
by_contra hn
have hm : s ∈ staircaseSupport α := by
simp [staircaseSupport, hn]
rw [hsupp] at hm
exact hs hm
have h0 := hα.2 0
have h1 := hα.2 1
have h3 := hα.2 3
simp +decide [staircaseMatrix, patternRay, patternContains,
privacyIncrement, privacyRatio, Fin.sum_univ_succ, hzero] at h0 h1 h3
have hr : exp ε ≠ 1 := ne_of_gt ((Real.one_lt_exp_iff).2 hε)
have heq05 : α 0 = α 5 := by
have hx : (exp ε - 1) * (α 0 - α 5) = 0 := by
linear_combination h0 - h1
exact sub_eq_zero.mp ((mul_eq_zero.mp hx).resolve_left (sub_ne_zero.mpr hr))
have heq57 : α 5 = α 7 := by
have hx : (exp ε - 1) * (α 5 - α 7) = 0 := by
linear_combination h1 - h3
exact sub_eq_zero.mp ((mul_eq_zero.mp hx).resolve_left (sub_ne_zero.mpr hr))
have he : exp ε + 2 ≠ 0 := by positivity
have hα0 : α 0 = (exp ε + 2)⁻¹ := by
field_simp
rw [← heq05, ← heq57, ← heq05] at h0
nlinarith
have hα5 : α 5 = (exp ε + 2)⁻¹ := by
exact heq05.symm.trans hα0
have hα7 : α 7 = (exp ε + 2)⁻¹ := by
exact heq57.symm.trans hα5
funext s
by_cases hs : s ∈ ({0, 5, 7} : Finset (Fin 14))
· have hcases : s = 0 ∨ s = 5 ∨ s = 7 := by simpa using hs
rcases hcases with rfl | rfl | rfl
· simpa using hα0
· simpa using hα5
· simpa using hα7
· simp [hs, hzero s hs]
theorem threeMaskWeights_support
the three mask weights support assertion holds.
Formal statement
Proof (Lean source)
lemma threeMaskWeights_support (ε : ℝ) :
staircaseSupport
(fun s => if s ∈ ({0, 5, 7} : Finset (Fin 14)) then
(exp ε + 2)⁻¹ else 0) =
({0, 5, 7} : Finset (Fin 14)) := by
ext s
simp only [staircaseSupport, mem_filter, Finset.mem_univ, true_and]
by_cases hs : s ∈ ({0, 5, 7} : Finset (Fin 14))
· simp [hs, ne_of_gt (inv_pos.mpr (by positivity : 0 < exp ε + 2))]
· simp [hs]
the three mask weights continuous assertion holds.
Formal statement
conclusion
Continuous (fun ε : ℝ => (fun s : Fin 14 => if s ∈ ({0, 5, 7} : Finset (Fin 14)) then (exp ε + 2)⁻¹ else (0 : ℝ)))
Proof (Lean source)
lemma threeMaskWeights_continuous :
Continuous (fun ε : ℝ =>
(fun s : Fin 14 => if s ∈ ({0, 5, 7} : Finset (Fin 14)) then
(exp ε + 2)⁻¹ else (0 : ℝ))) := by
apply continuous_pi
intro s
by_cases hs : s ∈ ({0, 5, 7} : Finset (Fin 14))
· simp only [hs, ite_true]
convert ((Real.continuous_exp.add continuous_const).inv₀
(fun ε => ne_of_gt (by positivity : 0 < exp ε + 2))) using 1
funext ε
rfl
· simpa [hs] using (continuous_const : Continuous fun _ : ℝ => (0 : ℝ))
theorem lowMeanCandidate_feasible
the low mean candidate feasible assertion holds.
Formal statement
conclusion
∃ α : StaircaseWeight,
staircaseFeasible (log 3) α
conclusion 1
s ∈ ({0, 5, 7} : Finset (Fin 14)) :
α s = (1 / 5 : ℝ)
Proof (Lean source)
lemma lowMeanCandidate_feasible :
∃ α : StaircaseWeight,
staircaseFeasible (log 3) α ∧
staircaseSupport α = ({0, 5, 7} : Finset (Fin 14)) ∧
∀ s ∈ ({0, 5, 7} : Finset (Fin 14)), α s = (1 / 5 : ℝ) := by
let α : StaircaseWeight := fun s => if s ∈ ({0, 5, 7} : Finset (Fin 14)) then 1 / 5 else 0
refine ⟨α, ⟨?_, ?_⟩, ?_, ?_⟩
· have h := (threeMaskWeights_feasible (log 3)
(Real.log_pos (by norm_num : (1 : ℝ) < 3))).1
simpa [α, Real.exp_log (by norm_num : (0 : ℝ) < 3),
show (3 + 2 : ℝ)⁻¹ = 1 / 5 by norm_num] using h
· have h := (threeMaskWeights_feasible (log 3)
(Real.log_pos (by norm_num : (1 : ℝ) < 3))).2
simpa [α, Real.exp_log (by norm_num : (0 : ℝ) < 3),
show (3 + 2 : ℝ)⁻¹ = 1 / 5 by norm_num] using h
· simpa [α, Real.exp_log (by norm_num : (0 : ℝ) < 3),
show (3 + 2 : ℝ)⁻¹ = 1 / 5 by norm_num] using
threeMaskWeights_support (log 3)
· intro s hs
simp only [α, if_pos hs]
theorem lowMeanCandidate_information_value
the low mean candidate information value assertion holds.
Formal statement
conclusion
informationObjective (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3) (fun s => if s ∈ ({0, 5, 7} : Finset (Fin 14)) then 1 / 5 else 0) (-(339 / 9985 : ℝ))
= 441 / 3994
Proof (Lean source)
lemma lowMeanCandidate_information_value :
informationObjective (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3)
(fun s => if s ∈ ({0, 5, 7} : Finset (Fin 14)) then 1 / 5 else 0)
(-(339 / 9985 : ℝ)) = 441 / 3994 := by
have he : exp (log (3 : ℝ)) = 3 := Real.exp_log (by norm_num)
have hpi (j : Fin 4) :
piTheta (fun _ => (1 / 10 : ℝ)) (3 / 10) j =
if j.val = 0 then 63 / 100
else if j.val = 1 then 7 / 100
else if j.val = 2 then 27 / 100 else 3 / 100 := by
rcases j with ⟨j, hj⟩
interval_cases j <;> norm_num [piTheta, controlProb] <;> rfl
have hpi0 := hpi 0
have hpi1 := hpi 1
have hpi2 := hpi 2
have hpi3 := hpi 3
norm_num at hpi0 hpi1 hpi2 hpi3
simp +decide [informationObjective, patternInformation, projectedGradient,
patternGradient, patternMass, patternRay, patternContains,
privacyIncrement, privacyRatio, controlProb, direction, he,
Fin.sum_univ_succ]
simp only [show (10⁻¹ : ℝ) = 1 / 10 by norm_num,
hpi0, hpi1, hpi2, hpi3]
norm_num
the low mean candidate stationary assertion holds.
Formal statement
conclusion
(∑ s : Fin 14, (if s ∈ ({0, 5, 7} : Finset (Fin 14)) then (1 / 5 : ℝ) else 0) * patternInformationSlope (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3) s (-(339 / 9985 : ℝ)))
= 0
Proof (Lean source)
lemma lowMeanCandidate_stationary :
(∑ s : Fin 14,
(if s ∈ ({0, 5, 7} : Finset (Fin 14)) then (1 / 5 : ℝ) else 0) *
patternInformationSlope (fun _ => (1 / 10 : ℝ)) (3 / 10)
(log 3) s (-(339 / 9985 : ℝ))) = 0 := by
have he : exp (log (3 : ℝ)) = 3 := Real.exp_log (by norm_num)
have hpi (j : Fin 4) :
piTheta (fun _ => (1 / 10 : ℝ)) (3 / 10) j =
if j.val = 0 then 63 / 100
else if j.val = 1 then 7 / 100
else if j.val = 2 then 27 / 100 else 3 / 100 := by
rcases j with ⟨j, hj⟩
interval_cases j <;> norm_num [piTheta, controlProb] <;> rfl
have hpi0 := hpi 0
have hpi1 := hpi 1
have hpi2 := hpi 2
have hpi3 := hpi 3
norm_num at hpi0 hpi1 hpi2 hpi3
simp +decide [patternInformationSlope, projectedGradient, patternGradient,
patternMass, patternRay, patternContains, privacyIncrement,
privacyRatio, controlProb, direction, he, Fin.sum_univ_succ]
simp only [show (10⁻¹ : ℝ) = 1 / 10 by norm_num,
hpi0, hpi1, hpi2, hpi3]
norm_num
theorem lowMeanCandidate_global_min
Under the supplied quantities and conditions, the low mean candidate global min assertion holds.
Formal statement
hypotheses
t :
ℝ
conclusion
(441 / 3994 : ℝ)
≤ informationObjective (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3) (fun s => if s ∈ ({0, 5, 7} : Finset (Fin 14)) then 1 / 5 else 0) t
Proof (Lean source)
lemma lowMeanCandidate_global_min (t : ℝ) :
(441 / 3994 : ℝ) ≤
informationObjective (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3)
(fun s => if s ∈ ({0, 5, 7} : Finset (Fin 14)) then 1 / 5 else 0) t := by
have he : exp (log (3 : ℝ)) = 3 := Real.exp_log (by norm_num)
have hpi (j : Fin 4) :
piTheta (fun _ => (1 / 10 : ℝ)) (3 / 10) j =
if j.val = 0 then 63 / 100
else if j.val = 1 then 7 / 100
else if j.val = 2 then 27 / 100 else 3 / 100 := by
rcases j with ⟨j, hj⟩
interval_cases j <;> norm_num [piTheta, controlProb] <;> rfl
have hpi0 := hpi 0
have hpi1 := hpi 1
have hpi2 := hpi 2
have hpi3 := hpi 3
norm_num at hpi0 hpi1 hpi2 hpi3
simp +decide [informationObjective, patternInformation, projectedGradient,
patternGradient, patternMass, patternRay, patternContains,
privacyIncrement, privacyRatio, controlProb, direction, he,
Fin.sum_univ_succ]
simp only [show (10⁻¹ : ℝ) = 1 / 10 by norm_num,
hpi0, hpi1, hpi2, hpi3]
norm_num at *
nlinarith [sq_nonneg (t + 339 / 9985)]
The fixed dual certificate for the low-mean three-output region is the displayed object.
Definition (Lean source)
def
given by
![-(21817593 / 398800900 : ℝ), (32820291 / 8773619800 : ℝ), (509870781 / 8773619800 : ℝ), (41183667 / 398800900 : ℝ)]
theorem lowMeanDual_value
Under the supplied quantities and conditions, the low mean dual value assertion holds. The low Mean Dual value.
Formal statement
conclusion
(∑ j : Fin 4, lowMeanDual j) = (441 / 3994 : ℝ)
Proof (Lean source)
lemma lowMeanDual_value :
(∑ j : Fin 4, lowMeanDual j) = (441 / 3994 : ℝ) := by
norm_num [lowMeanDual, Fin.sum_univ_succ]
theorem lowMeanDual_slack
Under the supplied quantities and conditions, the low mean dual slack assertion holds.
Formal statement
hypotheses
s :
Fin 14
conclusion
= patternInformation (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3) s (-(339 / 9985 : ℝ)) else patternInformation (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3) s (-(339 / 9985 : ℝ)) < dualRay (log 3) lowMeanDual s
Proof (Lean source)
lemma lowMeanDual_slack (s : Fin 14) :
(if s ∈ ({0, 5, 7} : Finset (Fin 14)) then
dualRay (log 3) lowMeanDual s =
patternInformation (fun _ => (1 / 10 : ℝ)) (3 / 10)
(log 3) s (-(339 / 9985 : ℝ))
else
patternInformation (fun _ => (1 / 10 : ℝ)) (3 / 10)
(log 3) s (-(339 / 9985 : ℝ)) <
dualRay (log 3) lowMeanDual s) := by
have he : exp (log (3 : ℝ)) = 3 := Real.exp_log (by norm_num)
have hpi (j : Fin 4) :
piTheta (fun _ => (1 / 10 : ℝ)) (3 / 10) j =
if j.val = 0 then 63 / 100
else if j.val = 1 then 7 / 100
else if j.val = 2 then 27 / 100 else 3 / 100 := by
rcases j with ⟨j, hj⟩
interval_cases j <;> norm_num [piTheta, controlProb] <;> rfl
have hpi0 := hpi 0
have hpi1 := hpi 1
have hpi2 := hpi 2
have hpi3 := hpi 3
norm_num at hpi0 hpi1 hpi2 hpi3
fin_cases s <;>
simp +decide [dualRay, lowMeanDual, patternInformation, projectedGradient,
patternGradient, patternMass, patternRay, patternContains,
controlProb, privacyIncrement, privacyRatio, direction,
he, Fin.sum_univ_succ] <;>
(try simp only [show (10⁻¹ : ℝ) = 1 / 10 by norm_num,
hpi0, hpi1, hpi2, hpi3]) <;> norm_num
theorem lowMeanPoint_mem_R3
the low mean point mem r3 assertion holds.
Proof (Lean source)
lemma lowMeanPoint_mem_R3 :
((3 / 10 : ℝ), (1 / 10 : ℝ), (1 / 10 : ℝ), log 3) ∈ R3 := by
simp only [R3, certifiedRegion, Set.mem_ofPred_eq]
refine ⟨by norm_num, by norm_num, by norm_num, by norm_num,
by norm_num, by norm_num, Real.log_pos (by norm_num), ?_⟩
obtain ⟨α, hfeas, hsupp, hweight⟩ := lowMeanCandidate_feasible
refine ⟨α, -(339 / 9985 : ℝ), lowMeanDual, ?_⟩
have hθ : (fun k : Fin 2 => if k = 0 then (1 / 10 : ℝ) else 1 / 10) =
(fun _ => (1 / 10 : ℝ)) := by
funext k
split_ifs <;> rfl
rw [hθ]
refine ⟨hfeas, ?_, ?_, ?_, ?_, ?_, ?_⟩
· intro s
change α s ≠ 0 ↔ s ∈ ({0, 5, 7} : Finset (Fin 14))
have h := congrArg (fun K : Finset (Fin 14) => s ∈ K) hsupp
simpa [staircaseSupport] using h
· intro s hs
rw [hweight s hs]
norm_num
· have hα : α = (fun s => if s ∈ ({0, 5, 7} : Finset (Fin 14)) then
(1 / 5 : ℝ) else 0) := by
funext s
by_cases hs : s ∈ ({0, 5, 7} : Finset (Fin 14))
· simpa [hs] using hweight s hs
· have hzero : α s = 0 := by
by_contra hn
have hm : s ∈ staircaseSupport α := by
simp [staircaseSupport, hn]
rw [hsupp] at hm
exact hs hm
simp [hs, hzero]
rw [hα]
exact lowMeanCandidate_stationary
· intro s hs
simpa [hs] using lowMeanDual_slack s
· intro s hs
simpa [hs] using lowMeanDual_slack s
· simp
theorem feasible_dualRay_sum
the feasible dual ray sum assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the feasible dual Ray sum.
Formal statement
hypotheses
Proof (Lean source)
lemma feasible_dualRay_sum (ε : ℝ) (α : StaircaseWeight)
(hα : staircaseFeasible ε α) (η : Fin 4 → ℝ) :
(∑ s : Fin 14, α s * dualRay ε η s) = ∑ j : Fin 4, η j := by
simp only [dualRay, Finset.mul_sum]
rw [Finset.sum_comm]
simp_rw [← mul_assoc, ← Finset.sum_mul]
change (∑ j : Fin 4, staircaseMatrix ε α j * η j) = ∑ j : Fin 4, η j
simp only [hα.2, one_mul]
theorem lowMean_dual_upper_bound
the low mean dual upper bound assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the low Mean dual upper bound.
Formal statement
hypotheses
β :
hβ :
staircaseFeasible (log 3) β
conclusion
informationObjective (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3) β (-(339 / 9985 : ℝ))
≤ 441 / 3994
Proof (Lean source)
lemma lowMean_dual_upper_bound (β : StaircaseWeight)
(hβ : staircaseFeasible (log 3) β) :
informationObjective (fun _ => (1 / 10 : ℝ)) (3 / 10)
(log 3) β (-(339 / 9985 : ℝ)) ≤ 441 / 3994 := by
have hpoint (s : Fin 14) :
patternInformation (fun _ => (1 / 10 : ℝ)) (3 / 10)
(log 3) s (-(339 / 9985 : ℝ)) ≤
dualRay (log 3) lowMeanDual s := by
by_cases hs : s ∈ ({0, 5, 7} : Finset (Fin 14))
· have h := lowMeanDual_slack s
simp only [if_pos hs] at h
exact le_of_eq h.symm
· exact le_of_lt (by simpa [hs] using lowMeanDual_slack s)
calc
informationObjective (fun _ => (1 / 10 : ℝ)) (3 / 10)
(log 3) β (-(339 / 9985 : ℝ))
= ∑ s : Fin 14, β s * patternInformation
(fun _ => (1 / 10 : ℝ)) (3 / 10)
(log 3) s (-(339 / 9985 : ℝ)) := rfl
_ ≤ ∑ s : Fin 14, β s * dualRay (log 3) lowMeanDual s := by
apply Finset.sum_le_sum
intro s _
exact mul_le_mul_of_nonneg_left (hpoint s) (hβ.1 s)
_ = 441 / 3994 := by rw [feasible_dualRay_sum _ _ hβ, lowMeanDual_value]
theorem lowMean_information_nonneg
the low mean information nonneg assertion holds. For the displayed quantities and conditions, these specify the stated inputs. Under the stated assumptions, the low Mean information nonneg.
Formal statement
hypotheses
conclusion
0 ≤ informationObjective (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3) β t
Proof (Lean source)
lemma lowMean_information_nonneg (β : StaircaseWeight)
(hβ : staircaseFeasible (log 3) β) (t : ℝ) :
0 ≤ informationObjective (fun _ => (1 / 10 : ℝ)) (3 / 10)
(log 3) β t := by
have he : exp (log (3 : ℝ)) = 3 := Real.exp_log (by norm_num)
have hpi (j : Fin 4) :
piTheta (fun _ => (1 / 10 : ℝ)) (3 / 10) j =
if j.val = 0 then 63 / 100
else if j.val = 1 then 7 / 100
else if j.val = 2 then 27 / 100 else 3 / 100 := by
rcases j with ⟨j, hj⟩
interval_cases j <;> norm_num [piTheta, controlProb] <;> rfl
have hm (s : Fin 14) :
0 ≤ patternMass (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3) s := by
unfold patternMass
apply Finset.sum_nonneg
intro j _
apply mul_nonneg
· rw [hpi]
split_ifs <;> norm_num
· simp only [patternRay, privacyIncrement, privacyRatio, he]
split_ifs <;> norm_num
unfold informationObjective
apply Finset.sum_nonneg
intro s _
exact mul_nonneg (hβ.1 s)
(div_nonneg (sq_nonneg _) (hm s))
theorem lowMeanCandidate_inf_value
the low mean candidate inf value assertion holds.
Formal statement
conclusion
sInf {u : ℝ | ∃ t : ℝ, u = informationObjective (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3) (fun s => if s ∈ ({0, 5, 7} : Finset (Fin 14)) then 1 / 5 else 0) t}
= 441 / 3994
Proof (Lean source)
lemma lowMeanCandidate_inf_value :
sInf {u : ℝ | ∃ t : ℝ,
u = informationObjective (fun _ => (1 / 10 : ℝ))
(3 / 10) (log 3)
(fun s => if s ∈ ({0, 5, 7} : Finset (Fin 14)) then 1 / 5 else 0) t} =
441 / 3994 := by
let α : StaircaseWeight :=
fun s => if s ∈ ({0, 5, 7} : Finset (Fin 14)) then 1 / 5 else 0
let S : Set ℝ := {u | ∃ t : ℝ,
u = informationObjective (fun _ => (1 / 10 : ℝ))
(3 / 10) (log 3) α t}
have hnonempty : S.Nonempty := by
refine ⟨informationObjective (fun _ => (1 / 10 : ℝ))
(3 / 10) (log 3) α 0, 0, rfl⟩
have hbelow : BddBelow S := by
refine ⟨441 / 3994, ?_⟩
rintro u ⟨t, rfl⟩
exact lowMeanCandidate_global_min t
change sInf S = 441 / 3994
apply le_antisymm
· have hmem : (441 / 3994 : ℝ) ∈ S := by
refine ⟨-(339 / 9985 : ℝ), ?_⟩
exact lowMeanCandidate_information_value.symm
exact csInf_le hbelow hmem
· apply le_csInf hnonempty
rintro u ⟨t, rfl⟩
exact lowMeanCandidate_global_min t
theorem lowMean_Jstar_value
the low mean jstar value assertion holds.
Proof (Lean source)
lemma lowMean_Jstar_value :
Jstar (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3) =
(441 / 3994 : ℝ) := by
let θ : TrialParameter := fun _ => (1 / 10 : ℝ)
let E : Set ℝ := {z | ∃ β : StaircaseWeight,
staircaseFeasible (log 3) β ∧
z = sInf {u : ℝ | ∃ t : ℝ,
u = informationObjective θ (3 / 10) (log 3) β t}}
have hupper : ∀ z ∈ E, z ≤ (441 / 3994 : ℝ) := by
rintro z ⟨β, hβ, rfl⟩
have hbelow : BddBelow {u : ℝ | ∃ t : ℝ,
u = informationObjective θ (3 / 10) (log 3) β t} := by
refine ⟨0, ?_⟩
rintro u ⟨t, rfl⟩
exact lowMean_information_nonneg β hβ t
exact (csInf_le hbelow ⟨-(339 / 9985 : ℝ), rfl⟩).trans
(lowMean_dual_upper_bound β hβ)
obtain ⟨α, hα, hsupp, hweight⟩ := lowMeanCandidate_feasible
have hαeq : α = (fun s => if s ∈ ({0, 5, 7} : Finset (Fin 14)) then
(1 / 5 : ℝ) else 0) := by
funext s
by_cases hs : s ∈ ({0, 5, 7} : Finset (Fin 14))
· simpa [hs] using hweight s hs
· have hzero : α s = 0 := by
by_contra hn
have hm : s ∈ staircaseSupport α := by
simp [staircaseSupport, hn]
rw [hsupp] at hm
exact hs hm
simp [hs, hzero]
have hmem : (441 / 3994 : ℝ) ∈ E := by
refine ⟨α, hα, ?_⟩
rw [hαeq]
exact lowMeanCandidate_inf_value.symm
change sSup E = 441 / 3994
exact le_antisymm (csSup_le ⟨_, hmem⟩ hupper)
(le_csSup ⟨_, hupper⟩ hmem)
def r3DualTarget
The target vector in the three-output dual system is determined by the displayed parameters.
Definition (Lean source)
parameters
x :
def
given by
fun i
=> patternInformation (regionTheta x) x.1 x.2.2.2 (r3Mask i) (r3Direction x.1 x.2.1 x.2.2.1 x.2.2.2)
theorem r3Mask_mem
Under the supplied quantities and conditions, the r3 mask mem assertion holds.
theorem exists_local_r3Dual_of_mem
Under the supplied quantities and conditions, the exists local r3 dual of mem assertion holds.
Formal statement
hypotheses
x :
hx :
x ∈ R3
conclusion
∃ η : RegionParameter → Fin 4 → ℝ,
ContinuousAt η x
conclusion 1
s ∉ ({0,5,7} : Finset (Fin 14)) :
patternInformation (regionTheta x) x.1 x.2.2.2 s (r3Direction x.1 x.2.1 x.2.2.1 x.2.2.2) < dualRay x.2.2.2 (η x) s
conclusion 2
Proof (Lean source)
lemma exists_local_r3Dual_of_mem (x : RegionParameter) (hx : x ∈ R3) :
∃ η : RegionParameter → Fin 4 → ℝ,
ContinuousAt η x ∧
(∀ s ∉ ({0,5,7} : Finset (Fin 14)),
patternInformation (regionTheta x) x.1 x.2.2.2 s
(r3Direction x.1 x.2.1 x.2.2.1 x.2.2.2) <
dualRay x.2.2.2 (η x) s) ∧
∀ᶠ y in nhds x, mulVec (r3DualMatrix y) (η y) = r3DualTarget y := by
rcases x with ⟨p, μ0, μ1, ε⟩
change 0 < p ∧ p < 1 ∧ 0 < μ0 ∧ μ0 < 1 ∧ 0 < μ1 ∧ μ1 < 1 ∧ 0 < ε ∧
∃ α, ∃ t, ∃ η,
staircaseFeasible ε α ∧ activeSupport α ({0,5,7} : Finset (Fin 14)) ∧
(∀ s ∈ ({0,5,7} : Finset (Fin 14)), 0 < α s) ∧
(∑ s, α s * patternInformationSlope (fun k => if k=0 then μ0 else μ1) p ε s t) = 0 ∧
(∀ s ∈ ({0,5,7} : Finset (Fin 14)), dualRay ε η s =
patternInformation (fun k => if k=0 then μ0 else μ1) p ε s t) ∧
(∀ s ∉ ({0,5,7} : Finset (Fin 14)), patternInformation
(fun k => if k=0 then μ0 else μ1) p ε s t < dualRay ε η s) ∧ _ at hx
rcases hx with ⟨hp0,hp1,hm00,hm01,hm10,hm11,hε,α,t,η,hα,hsupp,_,hstat,hactive,hinactive,_⟩
have hsupp' : staircaseSupport α = ({0,5,7} : Finset (Fin 14)) := by
ext s
simpa [staircaseSupport] using hsupp s
have hαeq := threeMaskWeights_unique ε hε α hα hsupp'
have hstat' : r3Stationarity p μ0 μ1 ε t = 0 := by
rw [r3Stationarity]
rw [← hstat]
apply Finset.sum_congr rfl
intro s _
rw [hαeq]
rfl
have hc : r3StationarityCoeff p μ0 μ1 ε ≠ 0 :=
ne_of_gt (r3StationarityCoeff_pos p μ0 μ1 ε hp0 hp1 hm00 hm01 hm10 hm11 hε)
have hdir0 := r3Stationarity_r3Direction_eq_zero p μ0 μ1 ε hc
rw [r3Stationarity_affine] at hstat' hdir0
have ht : t = r3Direction p μ0 μ1 ε := by
have hm : (t - r3Direction p μ0 μ1 ε) * r3StationarityCoeff p μ0 μ1 ε = 0 := by
linarith
exact sub_eq_zero.mp ((mul_eq_zero.mp hm).resolve_right hc)
let x : RegionParameter := (p,μ0,μ1,ε)
have hmass (s : Fin 14) : patternMass (regionTheta x) x.1 x.2.2.2 s ≠ 0 :=
ne_of_gt (patternMass_pos_of_interior p μ0 μ1 ε hp0 hp1 hm00 hm01 hm10 hm11 s)
have htdir := continuousAt_r3Direction_region x hmass hc
have hb : ContinuousAt r3DualTarget x := by
apply (continuousAt_pi).2
intro i
exact continuousAt_patternInformation_region (r3Mask i)
(fun y => r3Direction y.1 y.2.1 y.2.2.1 y.2.2.2) x htdir (hmass _)
have hy : mulVec (r3DualMatrix x) η = r3DualTarget x := by
funext i
have ha := hactive (r3Mask i) (r3Mask_mem i)
have htheta : regionTheta x = (fun k => if k = 0 then μ0 else μ1) := by
funext k
fin_cases k <;> rfl
change (r3DualMatrix x).mulVec η i =
patternInformation (regionTheta x) p ε (r3Mask i)
(r3Direction p μ0 μ1 ε)
rw [htheta]
simpa [x, r3DualTarget, r3DualMatrix, mulVec,
dotProduct, dualRay, ht, mul_comm] using ha
have hR := r3DualMatrix_mul_rightInverse x hε
obtain ⟨hcont, hbase, hev⟩ := exists_eventually_anchoredSolution_of_rightInverse
r3DualMatrix r3DualTarget (r3DualRightInverse ε) x η
continuous_r3DualMatrix.continuousAt hb hR hy
let ηlocal := anchoredSolution r3DualMatrix r3DualTarget
(r3DualRightInverse ε) η
refine ⟨ηlocal, hcont, ?_, hev⟩
intro s hs
have htheta : regionTheta x = (fun k => if k = 0 then μ0 else μ1) := by
funext k
fin_cases k <;> rfl
dsimp only [ηlocal]
rw [hbase, htheta]
simpa [x, ht] using hinactive s hs
theorem lowMean_r3Direction_eq
the low mean r3 direction eq assertion holds.
Formal statement
conclusion
r3Direction (3 / 10) (1 / 10) (1 / 10) (log 3) = -(339 / 9985 : ℝ)
Proof (Lean source)
lemma lowMean_r3Direction_eq :
r3Direction (3 / 10) (1 / 10) (1 / 10) (log 3) =
-(339 / 9985 : ℝ) := by
have hc : r3StationarityCoeff (3 / 10) (1 / 10) (1 / 10) (log 3) ≠ 0 :=
ne_of_gt (r3StationarityCoeff_pos _ _ _ _ (by norm_num) (by norm_num)
(by norm_num) (by norm_num) (by norm_num) (by norm_num)
(Real.log_pos (by norm_num)))
have hnew := r3Stationarity_r3Direction_eq_zero _ _ _ _ hc
have hold : r3Stationarity (3 / 10) (1 / 10) (1 / 10) (log 3)
(-(339 / 9985 : ℝ)) = 0 := by
convert lowMeanCandidate_stationary using 1 <;>
norm_num [r3Stationarity, r3Weight,
Real.exp_log (by norm_num : (0 : ℝ) < 3)]
rw [r3Stationarity_affine] at hnew hold
have hm : (r3Direction (3 / 10) (1 / 10) (1 / 10) (log 3) -
(-(339 / 9985 : ℝ))) *
r3StationarityCoeff (3 / 10) (1 / 10) (1 / 10) (log 3) = 0 := by
linarith
exact sub_eq_zero.mp ((mul_eq_zero.mp hm).resolve_right hc)
theorem lowMean_r3_projectedScores_distinct
Under the supplied quantities and conditions, the low mean r3 projected scores distinct assertion holds.
Formal statement
hypotheses
i k :
Fin 3
hik :
i ≠ k
conclusion
projectedScore (regionTheta ((3 / 10 : ℝ), (1 / 10 : ℝ), (1 / 10 : ℝ), log 3)) (3 / 10) (log 3) (r3Mask i) (r3Direction (3 / 10) (1 / 10) (1 / 10) (log 3))
≠ projectedScore (regionTheta ((3 / 10 : ℝ), (1 / 10 : ℝ), (1 / 10 : ℝ), log 3)) (3 / 10) (log 3) (r3Mask k) (r3Direction (3 / 10) (1 / 10) (1 / 10) (log 3))
Proof (Lean source)
lemma lowMean_r3_projectedScores_distinct (i k : Fin 3) (hik : i ≠ k) :
projectedScore (regionTheta ((3 / 10 : ℝ), (1 / 10 : ℝ),
(1 / 10 : ℝ), log 3)) (3 / 10) (log 3) (r3Mask i)
(r3Direction (3 / 10) (1 / 10) (1 / 10) (log 3)) ≠
projectedScore (regionTheta ((3 / 10 : ℝ), (1 / 10 : ℝ),
(1 / 10 : ℝ), log 3)) (3 / 10) (log 3) (r3Mask k)
(r3Direction (3 / 10) (1 / 10) (1 / 10) (log 3)) := by
have hpi (j : Fin 4) :
piTheta (fun _ => (1 / 10 : ℝ)) (3 / 10) j =
if j.val = 0 then 63 / 100
else if j.val = 1 then 7 / 100
else if j.val = 2 then 27 / 100 else 3 / 100 := by
rcases j with ⟨j, hj⟩
interval_cases j <;> norm_num [piTheta, controlProb] <;> rfl
have hpi0 := hpi 0
have hpi1 := hpi 1
have hpi2 := hpi 2
have hpi3 := hpi 3
norm_num at hpi0 hpi1 hpi2 hpi3
have htheta : regionTheta ((3 / 10 : ℝ), (1 / 10 : ℝ),
(1 / 10 : ℝ), log 3) = (fun _ => (1 / 10 : ℝ)) := by
funext j
fin_cases j <;> rfl
rw [htheta]
rw [lowMean_r3Direction_eq]
simp only [projectedScore, patternMass, Fin.sum_univ_succ]
simp_rw [hpi]
fin_cases i <;> fin_cases k
all_goals simp only [Fin.isValue] at hik ⊢
all_goals try contradiction
all_goals
simp (disch := decide) [regionTheta, r3Mask, projectedScore,
projectedGradient, patternGradient, patternMass, patternRay,
patternContains, Nat.testBit, privacyIncrement, privacyRatio, controlProb, direction,
Real.exp_log (by norm_num : (0 : ℝ) < 3), Fin.sum_univ_succ] <;>
norm_num [piTheta, controlProb]
def r3ProjectedScoreAt
The projected score coordinate in the three-output region is determined by the displayed parameters.
Definition (Lean source)
theorem r3ProjectedScoreAt_eq
Under the supplied quantities and conditions, the r3 projected score at eq assertion holds.
Formal statement
Proof (Lean source)
lemma r3ProjectedScoreAt_eq (x : RegionParameter) (i : Fin 3) :
r3ProjectedScoreAt x i = projectedScore (regionTheta x) x.1 x.2.2.2
(r3Mask i) (r3Direction x.1 x.2.1 x.2.2.1 x.2.2.2) := rfl
theorem exists_lowMean_r3_distinctScore_neighborhood
the exists low mean r3 distinct score neighborhood assertion holds.
Formal statement
conclusion
∃ U : Set RegionParameter,
Proof (Lean source)
lemma exists_lowMean_r3_distinctScore_neighborhood :
∃ U : Set RegionParameter, IsOpen U ∧
((3 / 10 : ℝ), (1 / 10 : ℝ), (1 / 10 : ℝ), log 3) ∈ U ∧
∀ x ∈ U, ∀ i k : Fin 3, i ≠ k →
r3ProjectedScoreAt x i ≠ r3ProjectedScoreAt x k := by
let x0 : RegionParameter :=
((3 / 10 : ℝ), (1 / 10 : ℝ), (1 / 10 : ℝ), log 3)
have hm (s : Fin 14) : patternMass (regionTheta x0) x0.1 x0.2.2.2 s ≠ 0 :=
ne_of_gt (patternMass_pos_of_interior _ _ _ _ (by norm_num) (by norm_num)
(by norm_num) (by norm_num) (by norm_num) (by norm_num) s)
have hc : r3StationarityCoeff x0.1 x0.2.1 x0.2.2.1 x0.2.2.2 ≠ 0 :=
ne_of_gt (r3StationarityCoeff_pos _ _ _ _ (by norm_num) (by norm_num)
(by norm_num) (by norm_num) (by norm_num) (by norm_num)
(Real.log_pos (by norm_num)))
have ht := continuousAt_r3Direction_region x0 hm hc
have hs (i : Fin 3) : ContinuousAt (fun x => r3ProjectedScoreAt x i) x0 := by
exact continuousAt_projectedScore_region (r3Mask i)
(fun x => r3Direction x.1 x.2.1 x.2.2.1 x.2.2.2) x0 ht (hm (r3Mask i))
have hp (i k : Fin 3) (hik : i ≠ k) :
∀ᶠ x in nhds x0, r3ProjectedScoreAt x i ≠ r3ProjectedScoreAt x k := by
have hne : r3ProjectedScoreAt x0 i ≠ r3ProjectedScoreAt x0 k := by
exact lowMean_r3_projectedScores_distinct i k hik
have hd := (hs i).sub (hs k)
have hdiff : r3ProjectedScoreAt x0 i - r3ProjectedScoreAt x0 k ≠ 0 :=
sub_ne_zero.mpr hne
filter_upwards [hd.eventually_ne hdiff] with x hx
exact sub_ne_zero.mp hx
have hev : ∀ᶠ x in nhds x0, ∀ i k : Fin 3, i ≠ k →
r3ProjectedScoreAt x i ≠ r3ProjectedScoreAt x k := by
filter_upwards [hp 0 1 (by decide), hp 0 2 (by decide),
hp 1 0 (by decide), hp 1 2 (by decide), hp 2 0 (by decide),
hp 2 1 (by decide)] with x h01 h02 h10 h12 h20 h21
intro i k hik
fin_cases i <;> fin_cases k <;> simp_all
rcases mem_nhds_iff.mp hev with ⟨U, hUsub, hUopen, hx0⟩
exact ⟨U, hUopen, hx0, fun x hx => hUsub hx⟩
theorem isOpen_R3_continuousCertificate
the is open r3 continuous certificate assertion holds.
Proof (Lean source)
lemma isOpen_R3_continuousCertificate : IsOpen R3 := by
rw [isOpen_iff_mem_nhds]
intro x hx
obtain ⟨η, hη, hbaseslack, heq⟩ := exists_local_r3Dual_of_mem x hx
rcases x with ⟨p, μ0, μ1, ε⟩
change 0 < p ∧ p < 1 ∧ 0 < μ0 ∧ μ0 < 1 ∧ 0 < μ1 ∧ μ1 < 1 ∧ 0 < ε ∧ _ at hx
rcases hx with ⟨hp0,hp1,hm00,hm01,hm10,hm11,hε,α,t,η0,hα,hsupp,hpos,hstat,hactive,hinactive,_⟩
let x0 : RegionParameter := (p,μ0,μ1,ε)
have hmass (s : Fin 14) : patternMass (regionTheta x0) x0.1 x0.2.2.2 s ≠ 0 :=
ne_of_gt (patternMass_pos_of_interior p μ0 μ1 ε hp0 hp1 hm00 hm01 hm10 hm11 s)
have hc : r3StationarityCoeff p μ0 μ1 ε ≠ 0 :=
ne_of_gt (r3StationarityCoeff_pos p μ0 μ1 ε hp0 hp1 hm00 hm01 hm10 hm11 hε)
have ht := continuousAt_r3Direction_region x0 hmass hc
have hinfo (s : Fin 14) : ContinuousAt (fun y : RegionParameter =>
patternInformation (regionTheta y) y.1 y.2.2.2 s
(r3Direction y.1 y.2.1 y.2.2.1 y.2.2.2)) x0 :=
continuousAt_patternInformation_region s _ x0 ht (hmass s)
have hdual (s : Fin 14) : ContinuousAt (fun y : RegionParameter =>
dualRay y.2.2.2 (η y) s) x0 := by
have hηj (j : Fin 4) : ContinuousAt (fun y => η y j) x0 :=
(continuousAt_pi.mp hη) j
unfold dualRay
let f := fun j : Fin 4 => fun y : RegionParameter =>
patternRay y.2.2.2 s j * η y j
have hf (j : Fin 4) : ContinuousAt (f j) x0 :=
(continuous_patternRay_region s j).continuousAt.mul (hηj j)
have hsum : ∀ S : Finset (Fin 4),
ContinuousAt (fun y => ∑ j ∈ S, f j y) x0 := by
intro S
induction S using Finset.induction_on with
| empty => simpa using (continuousAt_const : ContinuousAt (fun _ : RegionParameter => (0 : ℝ)) x0)
| @insert a S ha ih =>
simp only [Finset.sum_insert ha]
change ContinuousAt (fun y => f a y + ∑ j ∈ S, f j y) x0
exact (hf a).add ih
simpa [f] using hsum Finset.univ
have hslack : ∀ᶠ y in nhds x0, ∀ s : Fin 14,
s ∉ ({0,5,7} : Finset (Fin 14)) →
patternInformation (regionTheta y) y.1 y.2.2.2 s
(r3Direction y.1 y.2.1 y.2.2.1 y.2.2.2) < dualRay y.2.2.2 (η y) s := by
have hall : ∀ᶠ y in nhds x0, ∀ s ∈ (Finset.univ : Finset (Fin 14)),
s ∉ ({0,5,7} : Finset (Fin 14)) →
patternInformation (regionTheta y) y.1 y.2.2.2 s
(r3Direction y.1 y.2.1 y.2.2.1 y.2.2.2) < dualRay y.2.2.2 (η y) s :=
(eventually_all_finset (Finset.univ : Finset (Fin 14))).2 fun s _ => by
by_cases hs : s ∈ ({0,5,7} : Finset (Fin 14))
· exact Filter.Eventually.of_forall (fun _ h => False.elim (h hs))
· have hraw := (hinfo s).eventually_lt (hdual s)
(by simpa [x0] using hbaseslack s hs)
exact hraw.mono fun _ h _ => h
exact hall.mono fun _ hy s hs => hy s (Finset.mem_univ s) hs
have hbasic : ∀ᶠ y in nhds x0,
0 < y.1 ∧ y.1 < 1 ∧ 0 < y.2.1 ∧ y.2.1 < 1 ∧
0 < y.2.2.1 ∧ y.2.2.1 < 1 ∧ 0 < y.2.2.2 := by
filter_upwards [continuousAt_const.eventually_lt continuousAt_fst hp0,
continuousAt_fst.eventually_lt continuousAt_const hp1,
continuousAt_const.eventually_lt continuousAt_snd.fst hm00,
continuousAt_snd.fst.eventually_lt continuousAt_const hm01,
continuousAt_const.eventually_lt continuousAt_snd.snd.fst hm10,
continuousAt_snd.snd.fst.eventually_lt continuousAt_const hm11,
continuousAt_const.eventually_lt continuousAt_snd.snd.snd hε] with y
yp0 yp1 ym00 ym01 ym10 ym11 yε
exact ⟨yp0,yp1,ym00,ym01,ym10,ym11,yε⟩
filter_upwards [heq, hslack, hbasic] with y hyeq hyslack hybasic
rcases y with ⟨q,ν0,ν1,δ⟩
rcases hybasic with ⟨hq0,hq1,hν00,hν01,hν10,hν11,hδ⟩
change 0 < q ∧ q < 1 ∧ 0 < ν0 ∧ ν0 < 1 ∧ 0 < ν1 ∧ ν1 < 1 ∧ 0 < δ ∧ _
refine ⟨hq0,hq1,hν00,hν01,hν10,hν11,hδ, r3Weight δ,
r3Direction q ν0 ν1 δ, η (q,ν0,ν1,δ), ?_⟩
have hfeas := threeMaskWeights_feasible δ hδ
have hsuppw := threeMaskWeights_support δ
refine ⟨hfeas, ?_, ?_, ?_, ?_, ?_, by simp⟩
· intro s
by_cases hs : s ∈ ({0,5,7} : Finset (Fin 14))
· simp [r3Weight, hs]
positivity
· simp [r3Weight, hs]
· intro s hs
simp [r3Weight, hs]
positivity
· exact r3Stationarity_r3Direction_eq_zero q ν0 ν1 δ
(ne_of_gt (r3StationarityCoeff_pos q ν0 ν1 δ hq0 hq1 hν00 hν01 hν10 hν11 hδ))
· intro s hs
obtain ⟨i, rfl⟩ : ∃ i : Fin 3, r3Mask i = s := by
have hcases : s = 0 ∨ s = 5 ∨ s = 7 := by simpa using hs
rcases hcases with rfl | rfl | rfl
· exact ⟨0, by simp [r3Mask]⟩
· exact ⟨1, by simp [r3Mask]⟩
· exact ⟨2, by simp [r3Mask]⟩
have hi := congrFun hyeq i
have htheta : regionTheta (q,ν0,ν1,δ) =
(fun k => if k = 0 then ν0 else ν1) := by
funext k
fin_cases k <;> rfl
change (r3DualMatrix (q,ν0,ν1,δ)).mulVec (η (q,ν0,ν1,δ)) i =
patternInformation (regionTheta (q,ν0,ν1,δ)) q δ (r3Mask i)
(r3Direction q ν0 ν1 δ) at hi
rw [htheta] at hi
change dualRay δ (η (q,ν0,ν1,δ)) (r3Mask i) =
patternInformation (fun k => if k = 0 then ν0 else ν1) q δ
(r3Mask i) (r3Direction q ν0 ν1 δ)
simpa [r3DualMatrix, r3DualTarget, mulVec, dotProduct,
dualRay, mul_comm] using hi
· exact hyslack
theorem three_output_region
the three output region assertion holds.
Formal statement
conclusion
conclusion 3
∃ α : StaircaseWeight,
staircaseFeasible (log 3) α
s ∈ ({0, 5, 7} : Finset (Fin 14)) :
α s = (1 / 5 : ℝ)
β = α
Proof (Lean source)
theorem three_output_region :
IsOpen R3 ∧
((3 / 10 : ℝ), (1 / 10 : ℝ), (1 / 10 : ℝ), log 3) ∈ R3 ∧
(∃ α : StaircaseWeight,
staircaseFeasible (log 3) α ∧
staircaseSupport α = ({0, 5, 7} : Finset (Fin 14)) ∧
(∀ s ∈ ({0, 5, 7} : Finset (Fin 14)), α s = (1 / 5 : ℝ)) ∧
Jstar (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3) =
sInf {u : ℝ | ∃ t : ℝ,
u = informationObjective (fun _ => (1 / 10 : ℝ))
(3 / 10) (log 3) α t} ∧
(∀ β : StaircaseWeight, staircaseFeasible (log 3) β →
Jstar (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3) =
sInf {u : ℝ | ∃ t : ℝ,
u = informationObjective (fun _ => (1 / 10 : ℝ))
(3 / 10) (log 3) β t} → β = α)) ∧
Jstar (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3) =
(441 / 3994 : ℝ) ∧
Vstar (fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3) =
(3994 / 441 : ℝ) := by
refine ⟨?_, lowMeanPoint_mem_R3, ?_, lowMean_Jstar_value, ?_⟩
· exact isOpen_R3_continuousCertificate
· obtain ⟨α, hα, hsupp, hweight⟩ := lowMeanCandidate_feasible
refine ⟨α, hα, hsupp, hweight, ?_, ?_⟩
· have hαeq : α = (fun s => if s ∈ ({0, 5, 7} : Finset (Fin 14)) then
(1 / 5 : ℝ) else 0) := by
funext s
by_cases hs : s ∈ ({0, 5, 7} : Finset (Fin 14))
· simpa [hs] using hweight s hs
· have hzero : α s = 0 := by
by_contra hn
have hm : s ∈ staircaseSupport α := by
simp [staircaseSupport, hn]
rw [hsupp] at hm
exact hs hm
simp [hs, hzero]
rw [lowMean_Jstar_value, hαeq]
exact lowMeanCandidate_inf_value.symm
· intro β hβ hopt
let t : ℝ := -(339 / 9985)
have hbelow : BddBelow {u : ℝ | ∃ x : ℝ,
u = informationObjective (fun _ => (1 / 10 : ℝ))
(3 / 10) (log 3) β x} := by
refine ⟨0, ?_⟩
rintro u ⟨x, rfl⟩
exact lowMean_information_nonneg β hβ x
have hge : (441 / 3994 : ℝ) ≤
informationObjective (fun _ => (1 / 10 : ℝ))
(3 / 10) (log 3) β t := by
rw [← lowMean_Jstar_value, hopt]
exact csInf_le hbelow ⟨t, rfl⟩
have hvalue : informationObjective (fun _ => (1 / 10 : ℝ))
(3 / 10) (log 3) β t = 441 / 3994 :=
le_antisymm (lowMean_dual_upper_bound β hβ) hge
have hsum :
(∑ s : Fin 14, β s * patternInformation
(fun _ => (1 / 10 : ℝ)) (3 / 10) (log 3) s t) =
∑ s : Fin 14, β s * dualRay (log 3) lowMeanDual s := by
calc
_ = 441 / 3994 := hvalue
_ = ∑ s : Fin 14, β s * dualRay (log 3) lowMeanDual s := by
rw [feasible_dualRay_sum _ _ hβ, lowMeanDual_value]
have hterm (s : Fin 14) :
β s * patternInformation (fun _ => (1 / 10 : ℝ))
(3 / 10) (log 3) s t ≤
β s * dualRay (log 3) lowMeanDual s := by
have h := lowMeanDual_slack s
by_cases hs : s ∈ ({0, 5, 7} : Finset (Fin 14))
· simp only [if_pos hs] at h
rw [h]
· simp only [if_neg hs] at h
exact mul_le_mul_of_nonneg_left (le_of_lt h) (hβ.1 s)
have hterm_eq (s : Fin 14) :
β s * patternInformation (fun _ => (1 / 10 : ℝ))
(3 / 10) (log 3) s t =
β s * dualRay (log 3) lowMeanDual s :=
(Finset.sum_eq_sum_iff_of_le (fun s _ => hterm s)).mp hsum s
(Finset.mem_univ s)
have hzero (s : Fin 14)
(hs : s ∉ ({0, 5, 7} : Finset (Fin 14))) : β s = 0 := by
have h := lowMeanDual_slack s
simp only [if_neg hs] at h
have hn := hβ.1 s
by_contra hz
have hp : 0 < β s := lt_of_le_of_ne hn (Ne.symm hz)
have hlt := mul_lt_mul_of_pos_left h hp
exact (ne_of_lt hlt) (hterm_eq s)
have he : exp (log (3 : ℝ)) = 3 := Real.exp_log (by norm_num)
have h0 := hβ.2 0
have h1 := hβ.2 1
have h3 := hβ.2 3
simp +decide [staircaseMatrix, patternRay, patternContains,
privacyIncrement, privacyRatio, he, Fin.sum_univ_succ,
hzero] at h0 h1 h3
have hβ0 : β 0 = 1 / 5 := by linarith
have hβ5 : β 5 = 1 / 5 := by linarith
have hβ7 : β 7 = 1 / 5 := by linarith
funext s
by_cases hs : s ∈ ({0, 5, 7} : Finset (Fin 14))
· have hcases : s = 0 ∨ s = 5 ∨ s = 7 := by simpa using hs
rcases hcases with rfl | rfl | rfl
· exact hβ0.trans (hweight 0 (by simp)).symm
· exact hβ5.trans (hweight 5 (by simp)).symm
· exact hβ7.trans (hweight 7 (by simp)).symm
· rw [hzero s hs]
have hαzero : α s = 0 := by
by_contra hz
have hm : s ∈ staircaseSupport α := by simp [staircaseSupport, hz]
rw [hsupp] at hm
exact hs hm
exact hαzero.symm
· norm_num [Vstar, lowMean_Jstar_value]