Formalization: Assignment Probabilities Without Row-level Links: Identification, Label Design, and Inference in Randomized Trials
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 (9135ac1) — statements may have moved since the audit.
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Basic 53 declarations Law-level primitives for the unlinked propensity-score ATE problem.
Law-level primitives for the unlinked propensity-score ATE problem.
For the specified mathematical inputs, this definition introduces the corresponding object.
def OutcomeSpace
This definition introduces the corresponding mathematical object.
def ArmSpace
This definition introduces the corresponding mathematical object.
Definition (Lean source)
def
ArmSpace :
Type
given by
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
def
FullRow ε J :
Type
given by
def Observation
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
def score
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
def outcome0
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
def outcome1
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
def arm
For the specified mathematical inputs, this definition introduces the corresponding object.
def observed
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
ω.2.2.2.2.1
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
ω.2.2.2.2.2
def releasedRecord
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
For the specified mathematical inputs, this definition introduces the corresponding object.
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
def
Overlap ε :
Prop
given by
clause 1
0 < ε
clause 2
ε < 1 / 2
def ScoreMarginal
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
ScoreMarginal P H :
Prop
given by
P.map score = H
def RandomizedAssignment
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
B :
def
RandomizedAssignment P :
Prop
def Consistency
For the specified mathematical inputs, this definition introduces the corresponding object.
def DeterministicRelease
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
def ReleasedLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
structure RandomizedCausalLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
assumes
probability :
measurableRelease :
scoreMarginal :
ScoreMarginal P H
randomizedAssignment :
consistency :
Consistency P
deterministicRelease :
structure CompatibleCausalLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
assumes
probability :
measurableRelease :
scoreMarginal :
ScoreMarginal P H
randomizedAssignment :
consistency :
Consistency P
deterministicRelease :
releasedLaw :
ReleasedLaw P Prel
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
given by
{P | IsProbabilityMeasure P ∧ RandomizedCausalLaw H g P}
def CompatibleLaws
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
given by
{P | IsProbabilityMeasure P ∧ CompatibleCausalLaw H g Prel P}
def DensityLower
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
DensityLower H f mf :
Prop
given by
clause 1
clause 2
H
clause 3
∀ᵐ e ∂((volume : Measure ℝ).comap (Subtype.val : ScoreSpace ε → ℝ)), mf ≤ f e
def DensityUpper
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
DensityUpper f Mf :
Prop
given by
∀ᵐ e ∂((volume : Measure ℝ).comap (Subtype.val : ScoreSpace ε → ℝ)), f e ≤ Mf
structure LowerScoreDensity
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
assumes
probability :
densityLower :
DensityLower H f mf
structure BoundedScoreDensity
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
assumes
def armProb
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
armProb a e :
ℝ
given by
if a then (e : ℝ) else 1 - (e : ℝ)
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
{e | g e = r}
def armCellMass
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
armCellMass H g a r :
ℝ
given by
∫ e in cell g r, armProb a e ∂H
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
def
CompatibleCellMasses H g Prel :
Prop
given by
clause 1
clause 2
IsProbabilityMeasure Prel
clause 3
clause 4
a r :
Prel.real {z | z.1 = r ∧ z.2.1 = a} = armCellMass H g a r
theorem compatibleCellMasses_of_randomizedCausalLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
hP :
RandomizedCausalLaw H g P
conclusion
CompatibleCellMasses H g (releasedLaw P)
Proof (Lean source)
lemma compatibleCellMasses_of_randomizedCausalLaw
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(P : Measure (FullRow ε J)) (hP : RandomizedCausalLaw H g P) :
CompatibleCellMasses H g (releasedLaw P) := by
have hH : IsProbabilityMeasure H := by
rw [← hP.scoreMarginal]
exact @Measure.isProbabilityMeasure_map _ _ _ _ P hP.probability _
(by unfold score; fun_prop : Measurable (score (ε := ε) (J := J))).aemeasurable
have hRel : IsProbabilityMeasure (releasedLaw P) := by
unfold releasedLaw
exact @Measure.isProbabilityMeasure_map _ _ _ _ P hP.probability _
(by unfold releasedRecord label arm observed; fun_prop :
Measurable (releasedRecord (ε := ε) (J := J))).aemeasurable
refine ⟨hH, hRel, hP.measurableRelease, ?_⟩
intro a r
let B : Set (ScoreSpace ε × OutcomeSpace × OutcomeSpace) :=
{x | g x.1 = r}
have hB : MeasurableSet B := by
dsimp [B]
exact measurableSet_eq_fun (hP.measurableRelease.comp measurable_fst) measurable_const
have hscore : Measurable (score (ε := ε) (J := J)) := by
unfold score
fun_prop
have hrecord : Measurable (releasedRecord (ε := ε) (J := J)) := by
unfold releasedRecord label arm observed
fun_prop
have hrel : MeasurableSet {z : Observation J | z.1 = r ∧ z.2.1 = a} := by
exact (measurableSet_eq_fun measurable_fst measurable_const).inter
(measurableSet_eq_fun (measurable_fst.comp measurable_snd) measurable_const)
have hlabel : ∀ᵐ ω ∂P, label ω = g (score ω) := hP.deterministicRelease
have htreated :
(releasedLaw P).real {z | z.1 = r ∧ z.2.1 = true} =
∫ e in cell g r, (e : ℝ) ∂H := by
have hrand := hP.randomizedAssignment B hB
have hsetRel : MeasurableSet {z : Observation J | z.1 = r ∧ z.2.1 = true} :=
(measurableSet_eq_fun measurable_fst measurable_const).inter
(measurableSet_eq_fun (measurable_fst.comp measurable_snd) measurable_const)
have hmap := MeasureTheory.map_measureReal_apply (μ := P) hrecord hsetRel
change (P.map releasedRecord).real _ = _
rw [hmap]
have hset :
P.real (releasedRecord ⁻¹' {z : Observation J | z.1 = r ∧ z.2.1 = true}) =
P.real {ω | arm ω = true ∧ (score ω, outcome0 ω, outcome1 ω) ∈ B} := by
apply measureReal_congr
filter_upwards [hlabel] with ω hω
change (label ω = r ∧ arm ω = true) =
(arm ω = true ∧ g (score ω) = r)
simp [hω, and_comm]
rw [hset, hrand]
rw [← hP.scoreMarginal]
have hcell : MeasurableSet (cell g r) :=
measurableSet_eq_fun hP.measurableRelease measurable_const
have hmap := MeasureTheory.setIntegral_map (μ := P) (g := score)
(f := fun e : ScoreSpace ε => (e : ℝ)) hcell
(by fun_prop : AEStronglyMeasurable (fun e : ScoreSpace ε => (e : ℝ)) (P.map score))
hscore.aemeasurable
have hpre : {ω : FullRow ε J | (score ω, outcome0 ω, outcome1 ω) ∈ B} =
(score (ε := ε) (J := J)) ⁻¹' cell g r := by
ext ω
rfl
rw [hpre]
exact hmap.symm
cases a
· letI : IsProbabilityMeasure P := hP.probability
let C : Set (FullRow ε J) := score ⁻¹' cell g r
let T : Set (FullRow ε J) := {ω | ω ∈ C ∧ arm ω = true}
have hC : MeasurableSet C :=
(measurableSet_eq_fun hP.measurableRelease measurable_const).preimage hscore
have hT : MeasurableSet T :=
hC.inter (measurableSet_eq_fun (by unfold arm; fun_prop) measurable_const)
have hTmass : P.real T = ∫ e in cell g r, (e : ℝ) ∂H := by
rw [← htreated]
have hsetRel : MeasurableSet {z : Observation J | z.1 = r ∧ z.2.1 = true} :=
(measurableSet_eq_fun measurable_fst measurable_const).inter
(measurableSet_eq_fun (measurable_fst.comp measurable_snd) measurable_const)
have hmap := MeasureTheory.map_measureReal_apply (μ := P) hrecord hsetRel
rw [show releasedLaw P = P.map releasedRecord from rfl, hmap]
apply measureReal_congr
filter_upwards [hlabel] with ω hω
change (ω ∈ C ∧ arm ω = true) = (label ω = r ∧ arm ω = true)
simp [C, cell, hω]
have hCmass : P.real C = H.real (cell g r) := by
rw [← hP.scoreMarginal]
exact (MeasureTheory.map_measureReal_apply hscore
(measurableSet_eq_fun hP.measurableRelease measurable_const)).symm
have hFmass :
(releasedLaw P).real {z | z.1 = r ∧ z.2.1 = false} = P.real (C \ T) := by
have hsetRel : MeasurableSet {z : Observation J | z.1 = r ∧ z.2.1 = false} :=
(measurableSet_eq_fun measurable_fst measurable_const).inter
(measurableSet_eq_fun (measurable_fst.comp measurable_snd) measurable_const)
have hmap := MeasureTheory.map_measureReal_apply (μ := P) hrecord hsetRel
rw [show releasedLaw P = P.map releasedRecord from rfl, hmap]
apply measureReal_congr
filter_upwards [hlabel] with ω hω
change (label ω = r ∧ arm ω = false) = (ω ∈ C ∧ ω ∉ T)
cases hArm : arm ω <;> simp [C, T, cell, hω, hArm]
rw [hFmass, measureReal_sdiff (show T ⊆ C from fun _ h => h.1) hT, hCmass,
hTmass]
letI : IsProbabilityMeasure H := hH
have hint : IntegrableOn (fun e : ScoreSpace ε => (e : ℝ)) (cell g r) H := by
apply Integrable.of_bound (by fun_prop)
(|ε| + |1 - ε| + 1)
filter_upwards [] with e
change |(e : ℝ)| ≤ |ε| + |1 - ε| + 1
rcases e.property with ⟨he₁, he₂⟩
have h₁ := neg_le_abs ε
have h₂ := le_abs_self (1 - ε)
have h₃ := neg_le_abs (1 - ε)
have h₄ := le_abs_self ε
exact abs_le.mpr ⟨by linarith, by linarith⟩
rw [armCellMass]
simp only [armProb, Bool.false_eq_true, ↓reduceIte]
rw [integral_sub (integrable_const 1) hint]
simp
· simpa [armCellMass, armProb] using htreated
theorem compatibleCellMasses_of_compatibleCausalLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hP :
CompatibleCausalLaw H g Prel P
conclusion
CompatibleCellMasses H g Prel
Proof (Lean source)
lemma compatibleCellMasses_of_compatibleCausalLaw
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (P : Measure (FullRow ε J))
(hP : CompatibleCausalLaw H g Prel P) :
CompatibleCellMasses H g Prel := by
have hRandom : RandomizedCausalLaw H g P :=
⟨hP.probability, hP.measurableRelease, hP.scoreMarginal, hP.randomizedAssignment,
hP.consistency, hP.deterministicRelease⟩
rw [← hP.releasedLaw]
exact compatibleCellMasses_of_randomizedCausalLaw H g P hRandom
theorem compatibleCellMasses_of_nonempty
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hcomp :
(CompatibleLaws H g Prel).Nonempty
conclusion
CompatibleCellMasses H g Prel
Proof (Lean source)
lemma compatibleCellMasses_of_nonempty
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J))
(hcomp : (CompatibleLaws H g Prel).Nonempty) :
CompatibleCellMasses H g Prel := by
obtain ⟨P, hP⟩ := hcomp
exact compatibleCellMasses_of_compatibleCausalLaw H g Prel P hP.2
def armCellOutcomeLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
J :
ℕ
Prel :
Measure (Observation J)
_hPrel :
IsProbabilityMeasure Prel
a :
r :
_hpos :
0 < Prel.real {z | z.1 = r ∧ z.2.1 = a}
def
given by
(Prel {z | z.1 = r ∧ z.2.1 = a})⁻¹ • ((Prel.restrict {z | z.1 = r ∧ z.2.1 = a}).map (fun z => (z.2.2 : ℝ)))
def armCellScoreLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
_hH :
g :
ScoreSpace ε → LabelSpace J
_hg :
a :
r :
_hq :
0 < armCellMass H g a r
def
def armCellWeightLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
hH :
g :
ScoreSpace ε → LabelSpace J
hg :
a :
r :
hq :
0 < armCellMass H g a r
def
given by
def generalizedQuantile
For the specified mathematical inputs, this definition introduces the corresponding object.
def armCell
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
a :
r :
hH :
hg :
hPrel :
IsProbabilityMeasure Prel
def
armCell H g Prel a r hH hg hPrel :
ℝ × (0 < Prel.real {z | z.1 = r ∧ z.2.1 = a} → Measure ℝ) × (0 < armCellMass H g a r → Measure (ScoreSpace ε)) × (0 < armCellMass H g a r → Measure ℝ)
given by
(armCellMass H g a r, armCellOutcomeLaw Prel hPrel a r, armCellScoreLaw H hH g hg a r, armCellWeightLaw H hH g hg a r)
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
a :
def
muLower H g Prel hMass a :
ℝ
given by
∑ r : LabelSpace J, if hq : 0 < armCellMass H g a r then armCellMass H g a r * ∫ u in (0 : ℝ)..1, generalizedQuantile (armCellOutcomeLaw Prel hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq)) u * generalizedQuantile (armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) (1 - u) else 0
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
a :
def
muUpper H g Prel hMass a :
ℝ
given by
∑ r : LabelSpace J, if hq : 0 < armCellMass H g a r then armCellMass H g a r * ∫ u in (0 : ℝ)..1, generalizedQuantile (armCellOutcomeLaw Prel hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq)) u * generalizedQuantile (armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) u else 0
def armMeanEndpoints
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
a :
def
armMeanEndpoints H g Prel hMass a :
ℝ × ℝ
def sharpATESet
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
def
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
def
sharpATELength H g Prel hMass :
ℝ
def worstCaseAmbiguity
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
worstCaseAmbiguity H g :
ℝ
given by
sSup {d : ℝ | ∃ P : Measure (FullRow ε J), ∃ (_hprob : IsProbabilityMeasure P), ∃ hP : RandomizedCausalLaw H g P, d = sharpATELength H g (releasedLaw P) (compatibleCellMasses_of_randomizedCausalLaw H g P hP)}
def kLabelReleases
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
def optimalKAmbiguity
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
optimalKAmbiguity ε K H :
ℝ
given by
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
def compandingMass
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
compandingMass f x :
ℝ
given by
∫ t in ε..(x : ℝ), sqrt (densityExtension f t / (t * (1 - t)))
def compandingRelease
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
⟨min (Nat.floor ((K : ℝ) * compandingMass f e / (∫ t in ε..(1 - ε), sqrt (densityExtension f t / (t * (1 - t)))))) (K - 1), by omega⟩
def highResolutionHandle
This definition introduces the corresponding mathematical object.
Definition (Lean source)
def
given by
"Proof strategy: uncross arbitrary measurable release cells for the paired \ reciprocal-score absolute-deviation objective D_H(g); then use local scalar \ quantization on ordered cells to derive matching lower and upper limits for \ K Δ_K(H), with an ordered attaining sequence. Inputs: H, its density f, \ measurable releases g, D_H(g), and Δ_K(H)."
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.AllLabelAmbiguityBridge 1 declarations Structural endpoint bridge for all-label ambiguity.
Structural endpoint bridge for all-label ambiguity.
theorem compatible_endpointGap_eq_sharpATELength
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
conclusion
conclusion 2
IsProbabilityMeasure Pminus
IsProbabilityMeasure Pplus
CompatibleCausalLaw H g Prel Pminus
CompatibleCausalLaw H g Prel Pplus
ate Pplus - ate Pminus = sharpATELength H g Prel hMass
Proof (Lean source)
lemma compatible_endpointGap_eq_sharpATELength {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) :
(∃ P : Measure (FullRow ε J), IsProbabilityMeasure P ∧
CompatibleCausalLaw H g Prel P) ∧
(∃ Pminus Pplus : Measure (FullRow ε J),
IsProbabilityMeasure Pminus ∧ IsProbabilityMeasure Pplus ∧
CompatibleCausalLaw H g Prel Pminus ∧
CompatibleCausalLaw H g Prel Pplus ∧
ate Pminus = muLower H g Prel hMass true -
muUpper H g Prel hMass false ∧
ate Pplus = muUpper H g Prel hMass true -
muLower H g Prel hMass false ∧
ate Pplus - ate Pminus = sharpATELength H g Prel hMass) := by
let Pminus := endpointFullLaw H g Prel hMass hOverlap true false
let Pplus := endpointFullLaw H g Prel hMass hOverlap false true
have hminus : CompatibleCausalLaw H g Prel Pminus :=
endpointFullLaw_compatibleCausalLaw H g Prel hMass hOverlap true false
have hplus : CompatibleCausalLaw H g Prel Pplus :=
endpointFullLaw_compatibleCausalLaw H g Prel hMass hOverlap false true
have hminusAte : ate Pminus = muLower H g Prel hMass true -
muUpper H g Prel hMass false := by
letI : IsProbabilityMeasure Pminus := hminus.probability
rw [ate_eq_potentialOutcomeMeans_sub]
have h1 := endpointFullLaw_armMean H g Prel hMass hOverlap true false true
have h0 := endpointFullLaw_armMean H g Prel hMass hOverlap true false false
simpa [Pminus] using congrArg₂ (· - ·) h1 h0
have hplusAte : ate Pplus = muUpper H g Prel hMass true -
muLower H g Prel hMass false := by
letI : IsProbabilityMeasure Pplus := hplus.probability
rw [ate_eq_potentialOutcomeMeans_sub]
have h1 := endpointFullLaw_armMean H g Prel hMass hOverlap false true true
have h0 := endpointFullLaw_armMean H g Prel hMass hOverlap false true false
simpa [Pplus] using congrArg₂ (· - ·) h1 h0
refine ⟨⟨Pminus, hminus.probability, hminus⟩,
Pminus, Pplus, hminus.probability, hplus.probability,
hminus, hplus, hminusAte, hplusAte, ?_⟩
rw [hminusAte, hplusAte]
unfold sharpATELength
ring
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.AllLabelCellScaling 1 declarations Scaling identity for normalized inverse propensity cell laws.
Scaling identity for normalized inverse propensity cell laws.
theorem armCellWeightLaw_absoluteDeviation
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
r :
hq :
0 < armCellMass H g a r
c :
ℝ
conclusion
armCellMass H g a r * (∫ w, |w - c| ∂(armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq))
= ∫ e in cell g r, |1 - c * armProb a e| ∂H
Proof (Lean source)
lemma armCellWeightLaw_absoluteDeviation {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) (c : ℝ) :
armCellMass H g a r *
(∫ w, |w - c| ∂(armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq)) =
∫ e in cell g r, |1 - c * armProb a e| ∂H := by
unfold armCellWeightLaw armCellScoreLaw
rw [MeasureTheory.integral_map]
· rw [integral_smul_measure]
rw [ENNReal.toReal_inv, ENNReal.toReal_ofReal hq.le]
simp only [smul_eq_mul]
rw [← mul_assoc, mul_inv_cancel₀ (ne_of_gt hq), one_mul]
unfold cell
rw [setIntegral_withDensity_eq_setIntegral_toReal_smul
(by cases a <;> simp [armProb] <;> fun_prop)
(Filter.Eventually.of_forall fun _ => by finiteness)
_ (measurableSet_eq_fun hMass.2.2.1 measurable_const)]
apply integral_congr_ae
filter_upwards [] with e
have hp : 0 < armProb a e := by
rcases e.property with ⟨he0, he1⟩
cases a <;> simp [armProb] <;> linarith [hOverlap.1]
rw [ENNReal.toReal_ofReal hp.le]
simp only [smul_eq_mul]
rw [abs_sub_comm]
calc
armProb a e * |c - (armProb a e)⁻¹| =
|armProb a e| * |c - (armProb a e)⁻¹| := by rw [abs_of_pos hp]
_ = |armProb a e * (c - (armProb a e)⁻¹)| := by rw [abs_mul]
_ = |c * armProb a e - 1| := by
congr 1
field_simp
_ = |1 - c * armProb a e| := abs_sub_comm _ _
· exact (by cases a <;> simp [armProb] <;> fun_prop)
· fun_prop
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.AllLabelCouplingRange 8 declarations This file isolates the analytic cell inequality used by the all-label ambiguity calculation.
A coupling range bound for bounded outcomes
This file isolates the analytic cell inequality used by the all-label
ambiguity calculation. If the first marginal is supported on [0,1], the
difference between product expectations under any two couplings is controlled
by the absolute deviation of the second marginal about every fixed center.
def fairBernoulliLaw
This definition introduces the corresponding mathematical object.
instance fairBernoulliLaw_isProbabilityMeasure
This definition introduces the corresponding mathematical object.
Definition (Lean source)
instance fairBernoulliLaw_isProbabilityMeasure :
IsProbabilityMeasure fairBernoulliLaw := by
refine ⟨?_⟩
simp [fairBernoulliLaw, smul_eq_mul]
exact ENNReal.inv_two_add_inv_two
theorem fairBernoulliLaw_quantile
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
u :
ℝ
hu0 :
0 < u
hu1 :
u < 1
conclusion
quantile fairBernoulliLaw u = if u ≤ 1 / 2 then 0 else 1
Proof (Lean source)
lemma fairBernoulliLaw_quantile (u : ℝ) (hu0 : 0 < u) (hu1 : u < 1) :
quantile fairBernoulliLaw u = if u ≤ 1 / 2 then 0 else 1 := by
split_ifs with hu
· apply le_antisymm
· apply quantile_le_of_le_cdf hu0
rw [fairBernoulliLaw_cdf_between 0 (le_refl 0) (by norm_num)]
exact hu
· by_contra hn
have hneg : quantile fairBernoulliLaw u < 0 := lt_of_not_ge hn
have hle := (quantile_le_iff hu0 hu1).mp
(le_refl (quantile fairBernoulliLaw u))
rw [fairBernoulliLaw_cdf_of_lt_zero _ hneg] at hle
linarith
· apply le_antisymm
· apply quantile_le_of_le_cdf hu0
rw [fairBernoulliLaw_cdf_one]
exact hu1.le
· by_contra hn
have hlt : quantile fairBernoulliLaw u < 1 := lt_of_not_ge hn
let x := (quantile fairBernoulliLaw u + 1) / 2
have hqx : quantile fairBernoulliLaw u ≤ x := by
dsimp [x]
linarith
have hx0 : 0 ≤ x := by
have hq0 : 0 ≤ quantile fairBernoulliLaw u := by
by_contra hn0
have hqneg : quantile fairBernoulliLaw u < 0 := lt_of_not_ge hn0
have hle := (quantile_le_iff hu0 hu1).mp
(le_refl (quantile fairBernoulliLaw u))
rw [fairBernoulliLaw_cdf_of_lt_zero _ hqneg] at hle
linarith
dsimp [x]
linarith
have hx1 : x < 1 := by dsimp [x]; linarith
have hle := (quantile_le_iff hu0 hu1).mp hqx
rw [fairBernoulliLaw_cdf_between x hx0 hx1] at hle
exact hu hle
theorem coupling_product_gap_le_absoluteDeviation
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
Y W :
πminus πplus :
Measure (ℝ × ℝ)
hminus :
IsCoupling πminus Y W
hplus :
IsCoupling πplus Y W
hYsupport :
Y (Icc (0 : ℝ) 1)ᶜ = 0
hY2 :
MemLp (fun y : ℝ => y) 2 Y
hW2 :
MemLp (fun w : ℝ => w) 2 W
c :
ℝ
conclusion
(∫ p, p.1 * p.2 ∂πplus) - (∫ p, p.1 * p.2 ∂πminus) ≤ ∫ w, |w - c| ∂W
Proof (Lean source)
lemma coupling_product_gap_le_absoluteDeviation
{Y W : Measure ℝ} [IsProbabilityMeasure Y] [IsProbabilityMeasure W]
{πminus πplus : Measure (ℝ × ℝ)}
(hminus : IsCoupling πminus Y W) (hplus : IsCoupling πplus Y W)
(hYsupport : Y (Icc (0 : ℝ) 1)ᶜ = 0)
(hY2 : MemLp (fun y : ℝ => y) 2 Y)
(hW2 : MemLp (fun w : ℝ => w) 2 W) (c : ℝ) :
(∫ p, p.1 * p.2 ∂πplus) - (∫ p, p.1 * p.2 ∂πminus)
≤ ∫ w, |w - c| ∂W := by
letI : IsProbabilityMeasure πminus := hminus.isProbabilityMeasure
letI : IsProbabilityMeasure πplus := hplus.isProbabilityMeasure
let pos : ℝ → ℝ := fun x => (x + |x|) / 2
let neg : ℝ → ℝ := fun x => (-x + |x|) / 2
have hYae (π : Measure (ℝ × ℝ)) (hπ : IsCoupling π Y W) :
∀ᵐ p ∂π, p.1 ∈ Icc (0 : ℝ) 1 := by
have hh : ∀ᵐ y ∂Y, y ∈ Icc (0 : ℝ) 1 := by
rw [ae_iff]
exact hYsupport
rw [← hπ.map_fst] at hh
exact (ae_map_iff measurable_fst.aemeasurable measurableSet_Icc).mp hh
have hposInt : Integrable (fun w : ℝ => pos (w - c)) W := by
apply Integrable.div_const
apply Integrable.add
· exact (hW2.integrable (by norm_num)).sub (integrable_const c)
· exact ((hW2.integrable (by norm_num)).sub (integrable_const c)).abs
have hnegInt : Integrable (fun w : ℝ => neg (w - c)) W := by
apply Integrable.div_const
apply Integrable.add
· exact ((hW2.integrable (by norm_num)).sub (integrable_const c)).neg
· exact ((hW2.integrable (by norm_num)).sub (integrable_const c)).abs
have hposMap (π : Measure (ℝ × ℝ)) (hπ : IsCoupling π Y W) :
(∫ p, pos (p.2 - c) ∂π) = ∫ w, pos (w - c) ∂W := by
calc
(∫ p, pos (p.2 - c) ∂π) =
∫ w, pos (w - c) ∂(π.map snd) := by
rw [integral_map measurable_snd.aemeasurable]
fun_prop
_ = ∫ w, pos (w - c) ∂W := by rw [hπ.map_snd]
have hnegMap (π : Measure (ℝ × ℝ)) (hπ : IsCoupling π Y W) :
(∫ p, neg (p.2 - c) ∂π) = ∫ w, neg (w - c) ∂W := by
calc
(∫ p, neg (p.2 - c) ∂π) =
∫ w, neg (w - c) ∂(π.map snd) := by
rw [integral_map measurable_snd.aemeasurable]
fun_prop
_ = ∫ w, neg (w - c) ∂W := by rw [hπ.map_snd]
have hplusProd := coupling_integrable_mul hplus hY2 hW2
have hminusProd := coupling_integrable_mul hminus hY2 hW2
have hplusFst : Integrable (fun p : ℝ × ℝ => p.1) πplus :=
(coupling_fst_memLp hplus hY2).integrable (by norm_num)
have hminusFst : Integrable (fun p : ℝ × ℝ => p.1) πminus :=
(coupling_fst_memLp hminus hY2).integrable (by norm_num)
have hplusPos : Integrable (fun p : ℝ × ℝ => pos (p.2 - c)) πplus := by
apply Integrable.div_const
apply Integrable.add
· exact ((coupling_snd_memLp hplus hW2).integrable (by norm_num)).sub
(integrable_const c)
· exact (((coupling_snd_memLp hplus hW2).integrable (by norm_num)).sub
(integrable_const c)).abs
have hminusNeg : Integrable (fun p : ℝ × ℝ => neg (p.2 - c)) πminus := by
apply Integrable.div_const
apply Integrable.add
· exact (((coupling_snd_memLp hminus hW2).integrable (by norm_num)).sub
(integrable_const c)).neg
· exact (((coupling_snd_memLp hminus hW2).integrable (by norm_num)).sub
(integrable_const c)).abs
have hu :
(∫ p, p.1 * p.2 ∂πplus) ≤
c * (∫ y, y ∂Y) + ∫ w, pos (w - c) ∂W := by
rw [← coupling_integral_fst hplus hY2, ← hposMap πplus hplus,
← integral_const_mul, ← integral_add (hplusFst.const_mul c) hplusPos]
apply integral_mono_ae hplusProd ((hplusFst.const_mul c).add hplusPos)
filter_upwards [hYae πplus hplus] with p hp
dsimp [pos]
have habs : -(p.2 - c) ≤ |p.2 - c| := neg_le_abs (p.2 - c)
have habs' : p.2 - c ≤ |p.2 - c| := le_abs_self (p.2 - c)
by_cases hw : 0 ≤ p.2 - c
· rw [abs_of_nonneg hw]
nlinarith [hp.2]
· rw [abs_of_nonpos (le_of_not_ge hw)]
nlinarith [hp.1]
have hl :
c * (∫ y, y ∂Y) - ∫ w, neg (w - c) ∂W ≤
∫ p, p.1 * p.2 ∂πminus := by
rw [← coupling_integral_fst hminus hY2, ← hnegMap πminus hminus,
← integral_const_mul, ← integral_sub (hminusFst.const_mul c) hminusNeg]
apply integral_mono_ae ((hminusFst.const_mul c).sub hminusNeg) hminusProd
filter_upwards [hYae πminus hminus] with p hp
dsimp [neg]
by_cases hw : 0 ≤ p.2 - c
· rw [abs_of_nonneg hw]
nlinarith [hp.1]
· rw [abs_of_nonpos (le_of_not_ge hw)]
nlinarith [hp.2]
have hsplit :
(∫ w, pos (w - c) ∂W) + (∫ w, neg (w - c) ∂W) =
∫ w, |w - c| ∂W := by
rw [← integral_add hposInt hnegInt]
apply integral_congr_ae
filter_upwards with w
dsimp [pos, neg]
ring
linarith
theorem coupling_product_gap_eq_absoluteDeviation
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
Y W :
πminus πplus :
Measure (ℝ × ℝ)
hminus :
IsCoupling πminus Y W
hplus :
IsCoupling πplus Y W
hY2 :
MemLp (fun y : ℝ => y) 2 Y
hW2 :
MemLp (fun w : ℝ => w) 2 W
c :
ℝ
hupper :
∀ᵐ p ∂πplus, p.1 * p.2 = c * p.1 + ((p.2 - c) + |p.2 - c|) / 2
hlower :
∀ᵐ p ∂πminus, p.1 * p.2 = c * p.1 - (-(p.2 - c) + |p.2 - c|) / 2
conclusion
(∫ p, p.1 * p.2 ∂πplus) - (∫ p, p.1 * p.2 ∂πminus) = ∫ w, |w - c| ∂W
Proof (Lean source)
lemma coupling_product_gap_eq_absoluteDeviation
{Y W : Measure ℝ} [IsProbabilityMeasure Y] [IsProbabilityMeasure W]
{πminus πplus : Measure (ℝ × ℝ)}
(hminus : IsCoupling πminus Y W) (hplus : IsCoupling πplus Y W)
(hY2 : MemLp (fun y : ℝ => y) 2 Y)
(hW2 : MemLp (fun w : ℝ => w) 2 W) (c : ℝ)
(hupper : ∀ᵐ p ∂πplus,
p.1 * p.2 = c * p.1 + ((p.2 - c) + |p.2 - c|) / 2)
(hlower : ∀ᵐ p ∂πminus,
p.1 * p.2 = c * p.1 - (-(p.2 - c) + |p.2 - c|) / 2) :
(∫ p, p.1 * p.2 ∂πplus) - (∫ p, p.1 * p.2 ∂πminus)
= ∫ w, |w - c| ∂W := by
letI : IsProbabilityMeasure πminus := hminus.isProbabilityMeasure
letI : IsProbabilityMeasure πplus := hplus.isProbabilityMeasure
let pos : ℝ → ℝ := fun x => (x + |x|) / 2
let neg : ℝ → ℝ := fun x => (-x + |x|) / 2
have hposInt : Integrable (fun w : ℝ => pos (w - c)) W := by
exact (((hW2.integrable (by norm_num)).sub (integrable_const c)).add
((hW2.integrable (by norm_num)).sub (integrable_const c)).abs).div_const 2
have hnegInt : Integrable (fun w : ℝ => neg (w - c)) W := by
exact ((((hW2.integrable (by norm_num)).sub (integrable_const c)).neg).add
((hW2.integrable (by norm_num)).sub (integrable_const c)).abs).div_const 2
have hposMap :
(∫ p, pos (p.2 - c) ∂πplus) = ∫ w, pos (w - c) ∂W := by
calc
_ = ∫ w, pos (w - c) ∂(πplus.map snd) := by
rw [integral_map measurable_snd.aemeasurable]
fun_prop
_ = _ := by rw [hplus.map_snd]
have hnegMap :
(∫ p, neg (p.2 - c) ∂πminus) = ∫ w, neg (w - c) ∂W := by
calc
_ = ∫ w, neg (w - c) ∂(πminus.map snd) := by
rw [integral_map measurable_snd.aemeasurable]
fun_prop
_ = _ := by rw [hminus.map_snd]
have hplusFst : Integrable (fun p : ℝ × ℝ => p.1) πplus :=
(coupling_fst_memLp hplus hY2).integrable (by norm_num)
have hminusFst : Integrable (fun p : ℝ × ℝ => p.1) πminus :=
(coupling_fst_memLp hminus hY2).integrable (by norm_num)
have hplusPos : Integrable (fun p : ℝ × ℝ => pos (p.2 - c)) πplus := by
apply Integrable.div_const
apply Integrable.add
· exact ((coupling_snd_memLp hplus hW2).integrable (by norm_num)).sub
(integrable_const c)
· exact (((coupling_snd_memLp hplus hW2).integrable (by norm_num)).sub
(integrable_const c)).abs
have hminusNeg : Integrable (fun p : ℝ × ℝ => neg (p.2 - c)) πminus := by
apply Integrable.div_const
apply Integrable.add
· exact (((coupling_snd_memLp hminus hW2).integrable (by norm_num)).sub
(integrable_const c)).neg
· exact (((coupling_snd_memLp hminus hW2).integrable (by norm_num)).sub
(integrable_const c)).abs
have hu :
(∫ p, p.1 * p.2 ∂πplus) =
c * (∫ y, y ∂Y) + ∫ w, pos (w - c) ∂W := by
rw [← coupling_integral_fst hplus hY2, ← hposMap,
← integral_const_mul, ← integral_add (hplusFst.const_mul c) hplusPos]
exact integral_congr_ae (hupper.mono fun p hp => by simpa [pos] using hp)
have hl :
(∫ p, p.1 * p.2 ∂πminus) =
c * (∫ y, y ∂Y) - ∫ w, neg (w - c) ∂W := by
rw [← coupling_integral_fst hminus hY2, ← hnegMap,
← integral_const_mul, ← integral_sub (hminusFst.const_mul c) hminusNeg]
exact integral_congr_ae (hlower.mono fun p hp => by simpa [neg] using hp)
have hsplit :
(∫ w, pos (w - c) ∂W) + (∫ w, neg (w - c) ∂W) =
∫ w, |w - c| ∂W := by
rw [← integral_add hposInt hnegInt]
apply integral_congr_ae
filter_upwards with w
dsimp [pos, neg]
ring
rw [hu, hl]
linarith
theorem fairBernoulli_monotoneCoupling_gap_eq_medianAbsoluteDeviation
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
W :
hW2 :
MemLp (fun w : ℝ => w) 2 W
conclusion
(∫ p, p.1 * p.2 ∂ (comonotoneCoupling fairBernoulliLaw W))
- (∫ p, p.1 * p.2 ∂ (countermonotoneCoupling fairBernoulliLaw W))
= ∫ w, |w - quantile W (1 / 2)| ∂W
Proof (Lean source)
lemma fairBernoulli_monotoneCoupling_gap_eq_medianAbsoluteDeviation
(W : Measure ℝ) [IsProbabilityMeasure W]
(hW2 : MemLp (fun w : ℝ => w) 2 W) :
(∫ p, p.1 * p.2 ∂
(Causalean.Stat.comonotoneCoupling fairBernoulliLaw W)) -
(∫ p, p.1 * p.2 ∂
(Causalean.Stat.countermonotoneCoupling fairBernoulliLaw W)) =
∫ w, |w - quantile W (1 / 2)| ∂W := by
let m := quantile W (1 / 2)
have hYae : ∀ᵐ y ∂fairBernoulliLaw, y ∈ Icc (0 : ℝ) 1 := by
rw [ae_iff]
simp [fairBernoulliLaw]
have hY2 : MemLp (fun y : ℝ => y) 2 fairBernoulliLaw := by
apply memLp_of_bounded
· filter_upwards [hYae] with y hy
exact hy
· fun_prop
have hupper : ∀ᵐ p ∂(Causalean.Stat.comonotoneCoupling fairBernoulliLaw W),
p.1 * p.2 =
m * p.1 + ((p.2 - m) + |p.2 - m|) / 2 := by
unfold Causalean.Stat.comonotoneCoupling
let φ : ℝ → ℝ × ℝ :=
fun u => (quantile fairBernoulliLaw u, quantile W u)
have hφ : AEMeasurable φ unifOI :=
(aemeasurable_quantile_unifOI fairBernoulliLaw).prodMk
(aemeasurable_quantile_unifOI W)
apply (ae_map_iff hφ (by
exact measurableSet_eq_fun (by fun_prop) (by fun_prop))).2
filter_upwards [ae_restrict_mem measurableSet_Ioo] with u hu
change quantile fairBernoulliLaw u * quantile W u =
m * quantile fairBernoulliLaw u +
((quantile W u - m) + |quantile W u - m|) / 2
rw [fairBernoulliLaw_quantile u hu.1 hu.2]
by_cases huh : u ≤ 1 / 2
· have hqm : quantile W u ≤ m := by
dsimp [m]
exact quantile_mono hu.1 (by norm_num) huh
norm_num [div_eq_mul_inv] at huh ⊢
have hstep : u ≤ (2 : ℝ)⁻¹ := by
convert huh using 1 <;> norm_num
simp [hstep, abs_of_nonpos (sub_nonpos.mpr hqm)]
· have hmq : m ≤ quantile W u := by
dsimp [m]
exact quantile_mono (by norm_num) hu.2 (le_of_not_ge huh)
norm_num [div_eq_mul_inv] at huh ⊢
have hstep : ¬u ≤ (1 / 2 : ℝ) := not_le_of_gt huh
rw [if_neg hstep, if_neg hstep,
abs_of_nonneg (sub_nonneg.mpr hmq)]
ring
have hlower :
∀ᵐ p ∂(Causalean.Stat.countermonotoneCoupling fairBernoulliLaw W),
p.1 * p.2 =
m * p.1 - (-(p.2 - m) + |p.2 - m|) / 2 := by
unfold Causalean.Stat.countermonotoneCoupling
let r : ℝ → ℝ := fun u => 1 - u
have hr : Measurable r := by fun_prop
have hWmap : AEMeasurable (quantile W)
(Causalean.Stat.unifOI.map r) := by
simpa [r, map_one_sub_unifOI] using
(aemeasurable_quantile_unifOI W)
have hWr : AEMeasurable (fun u : ℝ => quantile W (1 - u))
unifOI := by
simpa [r, Function.comp_def] using hWmap.comp_measurable hr
let φ : ℝ → ℝ × ℝ :=
fun u => (quantile fairBernoulliLaw u, quantile W (1 - u))
have hφ : AEMeasurable φ unifOI :=
(aemeasurable_quantile_unifOI fairBernoulliLaw).prodMk hWr
apply (ae_map_iff hφ (by
exact measurableSet_eq_fun (by fun_prop) (by fun_prop))).2
filter_upwards [ae_restrict_mem measurableSet_Ioo] with u hu
have hv0 : 0 < 1 - u := by linarith [hu.2]
have hv1 : 1 - u < 1 := by linarith [hu.1]
change quantile fairBernoulliLaw u * quantile W (1 - u) =
m * quantile fairBernoulliLaw u -
(-(quantile W (1 - u) - m) + |quantile W (1 - u) - m|) / 2
rw [fairBernoulliLaw_quantile u hu.1 hu.2]
by_cases huh : u ≤ 1 / 2
· have hmq : m ≤ quantile W (1 - u) := by
dsimp [m]
apply quantile_mono (by norm_num) hv1
linarith
norm_num [div_eq_mul_inv] at huh ⊢
have hstep : u ≤ (2 : ℝ)⁻¹ := by
convert huh using 1 <;> norm_num
simp [hstep, abs_of_nonneg (sub_nonneg.mpr hmq)]
· have hqm : quantile W (1 - u) ≤ m := by
dsimp [m]
apply quantile_mono hv0 (by norm_num)
linarith
norm_num [div_eq_mul_inv] at huh ⊢
have hstep : ¬u ≤ (1 / 2 : ℝ) := not_le_of_gt huh
rw [if_neg hstep, if_neg hstep,
abs_of_nonpos (sub_nonpos.mpr hqm)]
ring
simpa [m] using coupling_product_gap_eq_absoluteDeviation
(isCoupling_countermonotoneCoupling fairBernoulliLaw W)
(isCoupling_comonotoneCoupling fairBernoulliLaw W)
hY2 hW2 m hupper hlower
theorem fairBernoulli_monotoneCoupling_gap_eq_sInf_absoluteDeviation
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(∫ p, p.1 * p.2 ∂ (comonotoneCoupling fairBernoulliLaw W))
- (∫ p, p.1 * p.2 ∂ (countermonotoneCoupling fairBernoulliLaw W))
= sInf {v : ℝ | ∃ c ∈ C, v = ∫ w, |w - c| ∂W}
Proof (Lean source)
lemma fairBernoulli_monotoneCoupling_gap_eq_sInf_absoluteDeviation
(W : Measure ℝ) [IsProbabilityMeasure W]
(hW2 : MemLp (fun w : ℝ => w) 2 W)
(C : Set ℝ) (hC : C.Nonempty) (hmedian : quantile W (1 / 2) ∈ C) :
(∫ p, p.1 * p.2 ∂
(comonotoneCoupling fairBernoulliLaw W)) -
(∫ p, p.1 * p.2 ∂
(countermonotoneCoupling fairBernoulliLaw W)) =
sInf {v : ℝ | ∃ c ∈ C, v = ∫ w, |w - c| ∂W} := by
have hYsupport : fairBernoulliLaw (Icc (0 : ℝ) 1)ᶜ = 0 := by
simp [fairBernoulliLaw]
have hYae : ∀ᵐ y ∂fairBernoulliLaw, y ∈ Icc (0 : ℝ) 1 := by
rw [ae_iff]
exact hYsupport
have hY2 : MemLp (fun y : ℝ => y) 2 fairBernoulliLaw := by
apply memLp_of_bounded
· filter_upwards [hYae] with y hy
exact hy
· fun_prop
have hle :
(∫ p, p.1 * p.2 ∂
(comonotoneCoupling fairBernoulliLaw W)) -
(∫ p, p.1 * p.2 ∂
(countermonotoneCoupling fairBernoulliLaw W)) ≤
sInf {v : ℝ | ∃ c ∈ C, v = ∫ w, |w - c| ∂W} := by
apply le_csInf
· rcases hC with ⟨c, hc⟩
exact ⟨∫ w, |w - c| ∂W, c, hc, rfl⟩
· intro v hv
rcases hv with ⟨c, hc, rfl⟩
exact coupling_product_gap_le_absoluteDeviation
(isCoupling_countermonotoneCoupling fairBernoulliLaw W)
(isCoupling_comonotoneCoupling fairBernoulliLaw W)
hYsupport hY2 hW2 c
have heq :=
fairBernoulli_monotoneCoupling_gap_eq_medianAbsoluteDeviation W hW2
have hbelow : BddBelow {v : ℝ | ∃ c ∈ C, v = ∫ w, |w - c| ∂W} := by
refine ⟨0, ?_⟩
intro v hv
rcases hv with ⟨c, hc, rfl⟩
exact integral_nonneg fun w => abs_nonneg _
have hinf :
sInf {v : ℝ | ∃ c ∈ C, v = ∫ w, |w - c| ∂W} ≤
∫ w, |w - quantile W (1 / 2)| ∂W :=
csInf_le hbelow ⟨quantile W (1 / 2), hmedian, rfl⟩
exact le_antisymm hle (heq.symm ▸ hinf)
theorem monotoneCoupling_product_gap_le_sInf_absoluteDeviation
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
Y W :
hYsupport :
Y (Icc (0 : ℝ) 1)ᶜ = 0
hY2 :
MemLp (fun y : ℝ => y) 2 Y
hW2 :
MemLp (fun w : ℝ => w) 2 W
C :
Set ℝ
hC :
C.Nonempty
conclusion
(∫ p, p.1 * p.2 ∂(comonotoneCoupling Y W))
- (∫ p, p.1 * p.2 ∂(countermonotoneCoupling Y W))
≤ sInf {v : ℝ | ∃ c ∈ C, v = ∫ w, |w - c| ∂W}
Proof (Lean source)
lemma monotoneCoupling_product_gap_le_sInf_absoluteDeviation
(Y W : Measure ℝ) [IsProbabilityMeasure Y] [IsProbabilityMeasure W]
(hYsupport : Y (Icc (0 : ℝ) 1)ᶜ = 0)
(hY2 : MemLp (fun y : ℝ => y) 2 Y)
(hW2 : MemLp (fun w : ℝ => w) 2 W)
(C : Set ℝ) (hC : C.Nonempty) :
(∫ p, p.1 * p.2 ∂(comonotoneCoupling Y W)) -
(∫ p, p.1 * p.2 ∂(countermonotoneCoupling Y W))
≤ sInf {v : ℝ | ∃ c ∈ C, v = ∫ w, |w - c| ∂W} := by
apply le_csInf
· rcases hC with ⟨c, hc⟩
exact ⟨∫ w, |w - c| ∂W, c, hc, rfl⟩
· intro v hv
rcases hv with ⟨c, hc, rfl⟩
exact coupling_product_gap_le_absoluteDeviation
(isCoupling_countermonotoneCoupling Y W)
(isCoupling_comonotoneCoupling Y W)
hYsupport hY2 hW2 c
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.AllLabelDefinitions 6 declarations Definitions shared by the all-label ambiguity proof leaves.
Definitions shared by the all-label ambiguity proof leaves.
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
cellAbsoluteDeviation H g a r :
ℝ
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
∑ r : LabelSpace J, ∑ a : ArmSpace, ((ofReal (armCellMass H g a r)) / 2) • (Measure.dirac (r, a, (⟨0, by norm_num⟩ : OutcomeSpace)) + Measure.dirac (r, a, (⟨1, by norm_num⟩ : OutcomeSpace)))
theorem armCellMass_nonneg
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
0 ≤ armCellMass H g a r
Proof (Lean source)
lemma armCellMass_nonneg {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(hOverlap : Overlap ε) (a : ArmSpace) (r : LabelSpace J) :
0 ≤ armCellMass H g a r := by
unfold armCellMass
apply integral_nonneg
intro e
rcases e.property with ⟨he₁, he₂⟩
rcases hOverlap with ⟨hε, _⟩
cases a <;> simp [armProb] <;> linarith
theorem armCellMass_sum_arms
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(∑ a : ArmSpace, armCellMass H g a r) = H.real (cell g r)
Proof (Lean source)
lemma armCellMass_sum_arms {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (r : LabelSpace J) :
(∑ a : ArmSpace, armCellMass H g a r) = H.real (cell g r) := by
have hint : IntegrableOn (fun e : ScoreSpace ε => (e : ℝ)) (cell g r) H := by
apply Integrable.of_bound (by fun_prop) (|ε| + |1 - ε| + 1)
filter_upwards [] with e
change |(e : ℝ)| ≤ |ε| + |1 - ε| + 1
rcases e.property with ⟨he₁, he₂⟩
have h₁ := neg_le_abs ε
have h₂ := le_abs_self (1 - ε)
have h₃ := neg_le_abs (1 - ε)
have h₄ := le_abs_self ε
exact abs_le.mpr ⟨by linarith, by linarith⟩
simp only [Fintype.sum_bool, armCellMass, armProb, Bool.false_eq_true,
↓reduceIte]
rw [integral_sub (integrable_const 1) hint]
simp
theorem armCellMass_total
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(∑ r : LabelSpace J, ∑ a : ArmSpace, armCellMass H g a r) = 1
Proof (Lean source)
lemma armCellMass_total {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hg : Measurable g) :
(∑ r : LabelSpace J, ∑ a : ArmSpace, armCellMass H g a r) = 1 := by
simp_rw [armCellMass_sum_arms H g]
have h := MeasureTheory.sum_measureReal_preimage_singleton (μ := H)
(s := univ) (f := g)
(by intro r hr; exact measurableSet_eq_fun hg measurable_const)
simpa [cell, preimage, Set.mem_singleton_iff] using h
theorem halfOutcomeReleasedLaw_cellMasses
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
CompatibleCellMasses H g (halfOutcomeReleasedLaw H g)
Proof (Lean source)
lemma halfOutcomeReleasedLaw_cellMasses {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε) (hg : Measurable g) :
CompatibleCellMasses H g (halfOutcomeReleasedLaw H g) := by
unfold CompatibleCellMasses
refine ⟨inferInstance, ?_, hg, ?_⟩
· let t (a : ArmSpace) : ENNReal :=
∑ r : LabelSpace J, ofReal (armCellMass H g a r) / 2
have htfinite (a : ArmSpace) : t a ≠ ⊤ := by
dsimp [t]
apply ENNReal.sum_ne_top.mpr
intro r hr
exact ENNReal.div_ne_top ENNReal.ofReal_ne_top (by norm_num)
have htReal (a : ArmSpace) : (t a).toReal =
(∑ r : LabelSpace J, armCellMass H g a r) / 2 := by
dsimp [t]
rw [ENNReal.toReal_sum]
· simp_rw [ENNReal.toReal_div, ENNReal.toReal_ofReal
(armCellMass_nonneg H g hOverlap a _)]
simp [← Finset.sum_div]
· intro r hr
exact ENNReal.div_ne_top ENNReal.ofReal_ne_top (by norm_num)
apply isProbabilityMeasure_iff_real.mpr
simp [halfOutcomeReleasedLaw, Measure.real_def, Finset.sum_add_distrib]
change (t true + t false + (t true + t false)).toReal = 1
have htf : t true + t false ≠ ⊤ :=
ENNReal.add_ne_top.mpr ⟨htfinite true, htfinite false⟩
rw [ENNReal.toReal_add htf htf,
ENNReal.toReal_add (htfinite true) (htfinite false),
htReal true, htReal false]
have htotal := armCellMass_total H g hg
simp only [Fintype.sum_bool, Finset.sum_add_distrib] at htotal
linarith
· intro a r
simp [halfOutcomeReleasedLaw, Measure.real_def, indicator]
cases a <;> simp [Finset.sum_add_distrib]
all_goals exact armCellMass_nonneg H g hOverlap _ r
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.AllLabelFairCell 14 declarations Fair-outcome cell identities for the all-label ambiguity witness.
Fair-outcome cell identities for the all-label ambiguity witness.
theorem halfOutcomeReleasedLaw_restrictedOutcomeMap
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
a :
r :
conclusion
= (ofReal (armCellMass H g a r) / 2) • (Measure.dirac 0 + Measure.dirac 1)
Proof (Lean source)
lemma halfOutcomeReleasedLaw_restrictedOutcomeMap {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε) (hg : Measurable g)
(a : ArmSpace) (r : LabelSpace J) :
((halfOutcomeReleasedLaw H g).restrict
{z | z.1 = r ∧ z.2.1 = a}).map (fun z => (z.2.2 : ℝ)) =
(ofReal (armCellMass H g a r) / 2) •
(Measure.dirac 0 + Measure.dirac 1) := by
classical
ext B hB
rw [Measure.map_apply (by fun_prop) hB]
rw [Measure.restrict_apply (hB.preimage (by fun_prop))]
simp [halfOutcomeReleasedLaw, hB]
rw [Finset.sum_eq_single r]
· cases a <;>
by_cases h0 : (0 : ℝ) ∈ B <;>
by_cases h1 : (1 : ℝ) ∈ B <;>
simp [indicator, h0, h1]
· intro b hb hbr
simp [hbr]
· simp
theorem halfOutcomeReleasedLaw_armCellOutcomeLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
a :
r :
hq :
0 < armCellMass H g a r
conclusion
armCellOutcomeLaw (halfOutcomeReleasedLaw H g) (halfOutcomeReleasedLaw_cellMasses H g hOverlap hg).2.1 a r (by rw [(halfOutcomeReleasedLaw_cellMasses H g hOverlap hg).2.2.2 a r] exact hq)
Proof (Lean source)
lemma halfOutcomeReleasedLaw_armCellOutcomeLaw {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε) (hg : Measurable g)
(a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) :
armCellOutcomeLaw (halfOutcomeReleasedLaw H g)
(halfOutcomeReleasedLaw_cellMasses H g hOverlap hg).2.1 a r
(by
rw [(halfOutcomeReleasedLaw_cellMasses H g hOverlap hg).2.2.2 a r]
exact hq) =
fairBernoulliLaw := by
let Prel := halfOutcomeReleasedLaw H g
let hMass := halfOutcomeReleasedLaw_cellMasses H g hOverlap hg
letI : IsProbabilityMeasure Prel := hMass.2.1
have hmassReal :
Prel.real {z | z.1 = r ∧ z.2.1 = a} = armCellMass H g a r :=
hMass.2.2.2 a r
have hmass :
Prel {z | z.1 = r ∧ z.2.1 = a} =
ofReal (armCellMass H g a r) := by
calc
_ = ofReal
(Prel.real {z | z.1 = r ∧ z.2.1 = a}) :=
(ENNReal.ofReal_toReal (measure_ne_top _ _)).symm
_ = _ := by rw [hmassReal]
have hq0 : ofReal (armCellMass H g a r) ≠ 0 :=
by simp [ENNReal.ofReal_eq_zero, not_le.mpr hq]
unfold armCellOutcomeLaw
rw [halfOutcomeReleasedLaw_restrictedOutcomeMap H g hOverlap hg a r]
rw [show (halfOutcomeReleasedLaw H g) {z | z.1 = r ∧ z.2.1 = a} =
ofReal (armCellMass H g a r) by exact hmass]
rw [smul_smul]
unfold fairBernoulliLaw
congr 1
rw [div_eq_mul_inv]
rw [← mul_assoc, ENNReal.inv_mul_cancel hq0 ENNReal.ofReal_ne_top]
simp
theorem inverseArmProb_mem_admissibleCenters
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(armProb a e)⁻¹ ∈ Icc ((1 - ε)⁻¹) ε⁻¹
Proof (Lean source)
lemma inverseArmProb_mem_admissibleCenters {ε : ℝ}
(hOverlap : Overlap ε) (a : ArmSpace) (e : ScoreSpace ε) :
(armProb a e)⁻¹ ∈ Icc ((1 - ε)⁻¹) ε⁻¹ := by
have hb := projectionArmProb_bounds hOverlap a e
have hp : 0 < armProb a e := lt_of_lt_of_le hOverlap.1 hb.1
have hOne : 0 < 1 - ε := by linarith [hOverlap.1, hOverlap.2]
have hpUpper : armProb a e ≤ 1 - ε := by
rcases e.property with ⟨he0, he1⟩
cases a <;> simp [armProb] <;> linarith
exact ⟨(inv_le_inv₀ hOne hp).2 hpUpper,
(inv_le_inv₀ hp hOverlap.1).2 hb.1⟩
theorem armCellWeightLaw_support_admissibleCenters
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
a :
r :
hq :
0 < armCellMass H g a r
conclusion
(armCellWeightLaw H inferInstance g hg a r hq) (Icc ((1 - ε)⁻¹) ε⁻¹)ᶜ = 0
Proof (Lean source)
lemma armCellWeightLaw_support_admissibleCenters {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε) (hg : Measurable g)
(a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) :
(armCellWeightLaw H inferInstance g hg a r hq)
(Icc ((1 - ε)⁻¹) ε⁻¹)ᶜ = 0 := by
let G := armCellScoreLaw H inferInstance g hg a r hq
letI : IsProbabilityMeasure G :=
armCellScoreLaw_isProbabilityMeasure H g hg hOverlap a r hq
unfold armCellWeightLaw
rw [Measure.map_apply (measurable_inverseArmProb a) measurableSet_Icc.compl]
have hpre :
(fun e : ScoreSpace ε => (armProb a e)⁻¹) ⁻¹'
(Icc ((1 - ε)⁻¹) ε⁻¹)ᶜ = ∅ := by
ext e
simp only [mem_preimage, mem_compl_iff, mem_Icc, mem_empty_iff_false,
iff_false]
exact fun he => he (inverseArmProb_mem_admissibleCenters hOverlap a e)
rw [hpre]
simp
theorem quantile_mem_Icc_of_support
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
quantile μ τ ∈ Icc a b
Proof (Lean source)
lemma quantile_mem_Icc_of_support
(μ : Measure ℝ) [IsProbabilityMeasure μ] {a b τ : ℝ}
(hμ : μ (Icc a b)ᶜ = 0) (hτ0 : 0 < τ) (hτ1 : τ < 1) :
quantile μ τ ∈ Icc a b := by
have hcompb : μ (Iic b)ᶜ = 0 := measure_mono_null (by
intro x hx hxab
exact hx hxab.2) hμ
have hmu_b : μ (Iic b) = 1 := by
have hh := measure_compl (μ := μ) (s := (Iic b)ᶜ)
(by measurability) (by simp)
simpa only [compl_compl, hcompb, tsub_zero, measure_univ] using hh
have hcdfb : cdf μ b = 1 := by
rw [cdf_eq_real, measureReal_def, hmu_b]
norm_num
have hlow (t : ℝ) (ht : t < a) : cdf μ t = 0 := by
have hh : μ (Iic t) = 0 := measure_mono_null (by
intro x hx hxab
exact (not_le_of_gt (lt_of_le_of_lt hx ht)) hxab.1) hμ
rw [cdf_eq_real, measureReal_def, hh]
norm_num
constructor
· by_contra hn
have hqa : quantile μ τ < a := lt_of_not_ge hn
have hle := le_cdf_quantile (μ := μ) hτ1
rw [hlow _ hqa] at hle
linarith
· exact (quantile_le_iff hτ0 hτ1).mpr
(by rw [hcdfb]; exact hτ1.le)
theorem armCellWeightLaw_median_mem_admissibleCenters
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
a :
r :
hq :
0 < armCellMass H g a r
conclusion
Proof (Lean source)
lemma armCellWeightLaw_median_mem_admissibleCenters {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε) (hg : Measurable g)
(a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) :
quantile
(armCellWeightLaw H inferInstance g hg a r hq) (1 / 2) ∈
Icc ((1 - ε)⁻¹) ε⁻¹ := by
let G := armCellScoreLaw H inferInstance g hg a r hq
letI : IsProbabilityMeasure G :=
armCellScoreLaw_isProbabilityMeasure H g hg hOverlap a r hq
letI : IsProbabilityMeasure
(armCellWeightLaw H inferInstance g hg a r hq) :=
Measure.isProbabilityMeasure_map (measurable_inverseArmProb a).aemeasurable
exact quantile_mem_Icc_of_support _
(armCellWeightLaw_support_admissibleCenters H g hOverlap hg a r hq)
(by norm_num) (by norm_num)
theorem armCellWeightLaw_memLp_two
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
a :
r :
hq :
0 < armCellMass H g a r
conclusion
Proof (Lean source)
lemma armCellWeightLaw_memLp_two {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε) (hg : Measurable g)
(a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) :
MemLp (fun w : ℝ => w) 2
(armCellWeightLaw H inferInstance g hg a r hq) := by
let G := armCellScoreLaw H inferInstance g hg a r hq
letI : IsProbabilityMeasure G :=
armCellScoreLaw_isProbabilityMeasure H g hg hOverlap a r hq
letI : IsProbabilityMeasure
(armCellWeightLaw H inferInstance g hg a r hq) :=
Measure.isProbabilityMeasure_map (measurable_inverseArmProb a).aemeasurable
have hs := armCellWeightLaw_support_admissibleCenters
H g hOverlap hg a r hq
have hlower : 0 ≤ (1 - ε)⁻¹ := by
apply inv_nonneg.mpr
linarith [hOverlap.1, hOverlap.2]
have hs' :
(armCellWeightLaw H inferInstance g hg a r hq)
(Icc 0 ε⁻¹)ᶜ = 0 := by
apply measure_mono_null _ hs
intro w hw hmem
exact hw ⟨hlower.trans hmem.1, hmem.2⟩
apply memLp_of_bounded
· rw [ae_iff]
exact hs'
· fun_prop
theorem fairOutcome_normalizedQuantileGap_eq_sInf_absoluteDeviation
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
a :
r :
hq :
0 < armCellMass H g a r
conclusion
(∫ u in (0 : ℝ)..1, generalizedQuantile fairBernoulliLaw u * generalizedQuantile (armCellWeightLaw H inferInstance g hg a r hq) u)
- (∫ u in (0 : ℝ)..1, generalizedQuantile fairBernoulliLaw u * generalizedQuantile (armCellWeightLaw H inferInstance g hg a r hq) (1 - u))
= sInf {v : ℝ | ∃ c ∈ Icc ((1 - ε)⁻¹) ε⁻¹, v = ∫ w, |w - c| ∂ (armCellWeightLaw H inferInstance g hg a r hq)}
Proof (Lean source)
lemma fairOutcome_normalizedQuantileGap_eq_sInf_absoluteDeviation
{ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε) (hg : Measurable g)
(a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) :
(∫ u in (0 : ℝ)..1,
generalizedQuantile fairBernoulliLaw u *
generalizedQuantile
(armCellWeightLaw H inferInstance g hg a r hq) u) -
(∫ u in (0 : ℝ)..1,
generalizedQuantile fairBernoulliLaw u *
generalizedQuantile
(armCellWeightLaw H inferInstance g hg a r hq) (1 - u)) =
sInf {v : ℝ | ∃ c ∈ Icc ((1 - ε)⁻¹) ε⁻¹,
v = ∫ w, |w - c| ∂
(armCellWeightLaw H inferInstance g hg a r hq)} := by
let W := armCellWeightLaw H inferInstance g hg a r hq
let G := armCellScoreLaw H inferInstance g hg a r hq
letI : IsProbabilityMeasure G :=
armCellScoreLaw_isProbabilityMeasure H g hg hOverlap a r hq
letI : IsProbabilityMeasure W :=
Measure.isProbabilityMeasure_map (measurable_inverseArmProb a).aemeasurable
have hYae : ∀ᵐ y ∂fairBernoulliLaw, y ∈ Icc (0 : ℝ) 1 := by
rw [ae_iff]
simp [fairBernoulliLaw]
have hY2 : MemLp (fun y : ℝ => y) 2 fairBernoulliLaw := by
apply memLp_of_bounded
· exact hYae
· fun_prop
have hW2 : MemLp (fun w : ℝ => w) 2 W :=
armCellWeightLaw_memLp_two H g hOverlap hg a r hq
have hC : (Icc ((1 - ε)⁻¹) ε⁻¹).Nonempty := by
refine ⟨Causalean.Stat.quantile W (1 / 2), ?_⟩
exact armCellWeightLaw_median_mem_admissibleCenters
H g hOverlap hg a r hq
have heq :=
fairBernoulli_monotoneCoupling_gap_eq_sInf_absoluteDeviation
W hW2 (Icc ((1 - ε)⁻¹) ε⁻¹) hC
(armCellWeightLaw_median_mem_admissibleCenters
H g hOverlap hg a r hq)
rw [product_expectation_comonotoneCoupling_interval
fairBernoulliLaw W hY2 hW2,
product_expectation_countermonotoneCoupling_interval
fairBernoulliLaw W] at heq
simpa [W] using heq
theorem fairOutcome_positiveCell_gap_eq_cellAbsoluteDeviation
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
a :
r :
hq :
0 < armCellMass H g a r
conclusion
armCellMass H g a r * ((∫ u in (0 : ℝ)..1, generalizedQuantile fairBernoulliLaw u * generalizedQuantile (armCellWeightLaw H inferInstance g hg a r hq) u) - (∫ u in (0 : ℝ)..1, generalizedQuantile fairBernoulliLaw u * generalizedQuantile (armCellWeightLaw H inferInstance g hg a r hq) (1 - u)))
= cellAbsoluteDeviation H g a r
Proof (Lean source)
lemma fairOutcome_positiveCell_gap_eq_cellAbsoluteDeviation
{ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε) (hg : Measurable g)
(a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) :
armCellMass H g a r *
((∫ u in (0 : ℝ)..1,
generalizedQuantile fairBernoulliLaw u *
generalizedQuantile
(armCellWeightLaw H inferInstance g hg a r hq) u) -
(∫ u in (0 : ℝ)..1,
generalizedQuantile fairBernoulliLaw u *
generalizedQuantile
(armCellWeightLaw H inferInstance g hg a r hq) (1 - u))) =
cellAbsoluteDeviation H g a r := by
let Prel := halfOutcomeReleasedLaw H g
let hMass := halfOutcomeReleasedLaw_cellMasses H g hOverlap hg
let W := armCellWeightLaw H inferInstance g hg a r hq
let A : Set ℝ :=
{v : ℝ | ∃ c ∈ Icc ((1 - ε)⁻¹) ε⁻¹,
v = ∫ w, |w - c| ∂W}
let B : Set ℝ :=
{v : ℝ | ∃ c ∈ Icc ((1 - ε)⁻¹) ε⁻¹,
v = ∫ e in cell g r, |1 - c * armProb a e| ∂H}
have hscale (c : ℝ) :
armCellMass H g a r * (∫ w, |w - c| ∂W) =
∫ e in cell g r, |1 - c * armProb a e| ∂H := by
exact armCellWeightLaw_absoluteDeviation
H g Prel hMass hOverlap a r hq c
have hset : B = armCellMass H g a r • A := by
ext v
constructor
· rintro ⟨c, hc, rfl⟩
rw [Set.mem_smul_set]
refine ⟨∫ w, |w - c| ∂W, ⟨c, hc, rfl⟩, ?_⟩
simpa [smul_eq_mul] using hscale c
· intro hv
rw [Set.mem_smul_set] at hv
rcases hv with ⟨x, ⟨c, hc, rfl⟩, rfl⟩
refine ⟨c, hc, ?_⟩
simpa [smul_eq_mul] using hscale c
rw [fairOutcome_normalizedQuantileGap_eq_sInf_absoluteDeviation
H g hOverlap hg a r hq]
unfold cellAbsoluteDeviation
change armCellMass H g a r * sInf A = sInf B
rw [hset, Real.sInf_smul_of_nonneg hq.le, smul_eq_mul]
theorem halfOutcomeReleasedLaw_positiveCell_gap_eq_cellAbsoluteDeviation
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
a :
r :
hq :
0 < armCellMass H g a r
conclusion
hMass :
halfOutcomeReleasedLaw_cellMasses H g hOverlap hg
armCellMass H g a r * ((∫ u in (0 : ℝ)..1, generalizedQuantile (armCellOutcomeLaw (halfOutcomeReleasedLaw H g) hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq)) u * generalizedQuantile (armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) u) - (∫ u in (0 : ℝ)..1, generalizedQuantile (armCellOutcomeLaw (halfOutcomeReleasedLaw H g) hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq)) u * generalizedQuantile (armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) (1 - u)))
= cellAbsoluteDeviation H g a r
Proof (Lean source)
lemma halfOutcomeReleasedLaw_positiveCell_gap_eq_cellAbsoluteDeviation
{ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε) (hg : Measurable g)
(a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) :
let hMass := halfOutcomeReleasedLaw_cellMasses H g hOverlap hg
armCellMass H g a r *
((∫ u in (0 : ℝ)..1,
generalizedQuantile
(armCellOutcomeLaw (halfOutcomeReleasedLaw H g) hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq)) u *
generalizedQuantile
(armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) u) -
(∫ u in (0 : ℝ)..1,
generalizedQuantile
(armCellOutcomeLaw (halfOutcomeReleasedLaw H g) hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq)) u *
generalizedQuantile
(armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) (1 - u))) =
cellAbsoluteDeviation H g a r := by
dsimp only
rw [halfOutcomeReleasedLaw_armCellOutcomeLaw H g hOverlap hg a r hq]
exact fairOutcome_positiveCell_gap_eq_cellAbsoluteDeviation
H g hOverlap hg a r hq
theorem cellAbsoluteDeviation_eq_zero_of_armCellMass_nonpos
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
a :
r :
hq :
¬0 < armCellMass H g a r
conclusion
cellAbsoluteDeviation H g a r = 0
Proof (Lean source)
lemma cellAbsoluteDeviation_eq_zero_of_armCellMass_nonpos
{ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε) (hg : Measurable g)
(a : ArmSpace) (r : LabelSpace J)
(hq : ¬0 < armCellMass H g a r) :
cellAbsoluteDeviation H g a r = 0 := by
have hInt : IntegrableOn (fun e : ScoreSpace ε => armProb a e)
(cell g r) H := by
apply Integrable.of_bound
(by cases a <;> simp [armProb] <;> fun_prop) 1
filter_upwards [] with e
rcases e.property with ⟨he0, he1⟩
cases a <;> simp [armProb, abs_le] <;> constructor <;>
linarith [hOverlap.1, hOverlap.2]
have hlower :
ε * H.real (cell g r) ≤ armCellMass H g a r := by
have hmono : (∫ _ in cell g r, ε ∂H) ≤
∫ e in cell g r, armProb a e ∂H := by
apply integral_mono (integrable_const ε) hInt
intro e
rcases e.property with ⟨he0, he1⟩
cases a <;> simp [armProb] <;> linarith
simpa [armCellMass, Measure.real_def, mul_comm] using hmono
have hq0 : armCellMass H g a r ≤ 0 := le_of_not_gt hq
have hreal0 : H.real (cell g r) = 0 := by
have hrealNonneg : 0 ≤ H.real (cell g r) := measureReal_nonneg
nlinarith [hOverlap.1]
have hmeasure0 : H (cell g r) = 0 := by
exact ((ENNReal.toReal_eq_zero_iff _).mp hreal0).resolve_right
(measure_ne_top H (cell g r))
have hrestrict : H.restrict (cell g r) = 0 := by
ext B hB
rw [Measure.restrict_apply hB]
exact measure_mono_null inter_subset_right hmeasure0
have hzero (c : ℝ) :
(∫ e in cell g r, |1 - c * armProb a e| ∂H) = 0 := by
change (∫ e, |1 - c * armProb a e| ∂(H.restrict (cell g r))) = 0
rw [hrestrict]
simp
have hC : (Icc ((1 - ε)⁻¹) ε⁻¹).Nonempty := by
refine ⟨ε⁻¹, ?_, le_rfl⟩
have hOne : 0 < 1 - ε := by linarith [hOverlap.1, hOverlap.2]
exact (inv_le_inv₀ hOne hOverlap.1).2 (by linarith [hOverlap.2])
unfold cellAbsoluteDeviation
have hset :
{v : ℝ | ∃ c ∈ Icc ((1 - ε)⁻¹) ε⁻¹,
v = ∫ e in cell g r, |1 - c * armProb a e| ∂H} = {0} := by
ext v
constructor
· rintro ⟨c, hc, rfl⟩
simp [hzero c]
· intro hv
have hv0 : v = 0 := by simpa using hv
rcases hC with ⟨c, hc⟩
exact ⟨c, hc, by rw [hzero c, hv0]⟩
rw [hset]
simp
theorem halfOutcomeReleasedLaw_cellGap_eq_cellAbsoluteDeviation
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
a :
r :
conclusion
hMass :
halfOutcomeReleasedLaw_cellMasses H g hOverlap hg
(if hq : 0 < armCellMass H g a r then armCellMass H g a r * ∫ u in (0 : ℝ)..1, generalizedQuantile (armCellOutcomeLaw (halfOutcomeReleasedLaw H g) hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq)) u * generalizedQuantile (armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) u else 0)
- (if hq : 0 < armCellMass H g a r then armCellMass H g a r * ∫ u in (0 : ℝ)..1, generalizedQuantile (armCellOutcomeLaw (halfOutcomeReleasedLaw H g) hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq)) u * generalizedQuantile (armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) (1 - u) else 0)
= cellAbsoluteDeviation H g a r
Proof (Lean source)
lemma halfOutcomeReleasedLaw_cellGap_eq_cellAbsoluteDeviation
{ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε) (hg : Measurable g)
(a : ArmSpace) (r : LabelSpace J) :
let hMass := halfOutcomeReleasedLaw_cellMasses H g hOverlap hg
(if hq : 0 < armCellMass H g a r then
armCellMass H g a r *
∫ u in (0 : ℝ)..1,
generalizedQuantile
(armCellOutcomeLaw (halfOutcomeReleasedLaw H g) hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq)) u *
generalizedQuantile
(armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) u
else 0) -
(if hq : 0 < armCellMass H g a r then
armCellMass H g a r *
∫ u in (0 : ℝ)..1,
generalizedQuantile
(armCellOutcomeLaw (halfOutcomeReleasedLaw H g) hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq)) u *
generalizedQuantile
(armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) (1 - u)
else 0) =
cellAbsoluteDeviation H g a r := by
dsimp only
by_cases hq : 0 < armCellMass H g a r
· simp only [hq, dif_pos]
have hpos :=
halfOutcomeReleasedLaw_positiveCell_gap_eq_cellAbsoluteDeviation
H g hOverlap hg a r hq
dsimp only at hpos
ring_nf at hpos ⊢
exact hpos
· simp only [hq, dif_neg, zero_sub]
simpa [hq] using
(cellAbsoluteDeviation_eq_zero_of_armCellMass_nonpos
H g hOverlap hg a r hq).symm
theorem halfOutcome_muUpper_sub_muLower_eq_sum_cellAbsoluteDeviation
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
a :
conclusion
hMass :
halfOutcomeReleasedLaw_cellMasses H g hOverlap hg
muUpper H g (halfOutcomeReleasedLaw H g) hMass a
- muLower H g (halfOutcomeReleasedLaw H g) hMass a
= ∑ r : LabelSpace J, cellAbsoluteDeviation H g a r
Proof (Lean source)
lemma halfOutcome_muUpper_sub_muLower_eq_sum_cellAbsoluteDeviation
{ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε) (hg : Measurable g)
(a : ArmSpace) :
let hMass := halfOutcomeReleasedLaw_cellMasses H g hOverlap hg
muUpper H g (halfOutcomeReleasedLaw H g) hMass a -
muLower H g (halfOutcomeReleasedLaw H g) hMass a =
∑ r : LabelSpace J, cellAbsoluteDeviation H g a r := by
dsimp only
unfold muUpper muLower
rw [← Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro r hr
exact halfOutcomeReleasedLaw_cellGap_eq_cellAbsoluteDeviation
H g hOverlap hg a r
theorem halfOutcome_sharpATELength_eq_sum_cellAbsoluteDeviation
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma halfOutcome_sharpATELength_eq_sum_cellAbsoluteDeviation
{ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε) (hg : Measurable g) :
sharpATELength H g (halfOutcomeReleasedLaw H g)
(halfOutcomeReleasedLaw_cellMasses H g hOverlap hg) =
∑ r : LabelSpace J,
(cellAbsoluteDeviation H g true r +
cellAbsoluteDeviation H g false r) := by
have htrue :=
halfOutcome_muUpper_sub_muLower_eq_sum_cellAbsoluteDeviation
H g hOverlap hg true
have hfalse :=
halfOutcome_muUpper_sub_muLower_eq_sum_cellAbsoluteDeviation
H g hOverlap hg false
dsimp only at htrue hfalse
unfold sharpATELength
rw [Finset.sum_add_distrib]
linarith
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.AllLabelSupremumUpper 2 declarations The cellwise reciprocal-score deviation bounds are aggregated over both arms and then lifted through the supremum over randomized causal laws.
Universal upper bound for worst case all-label ambiguity
The cellwise reciprocal-score deviation bounds are aggregated over both arms and then lifted through the supremum over randomized causal laws.
theorem randomizedCausalLaw_sharpATELength_le_sum_cellAbsoluteDeviation
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hP :
RandomizedCausalLaw H g P
conclusion
Proof (Lean source)
lemma randomizedCausalLaw_sharpATELength_le_sum_cellAbsoluteDeviation
{ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(hOverlap : Overlap ε) (P : Measure (FullRow ε J))
(hP : RandomizedCausalLaw H g P) :
sharpATELength H g (releasedLaw P)
(compatibleCellMasses_of_randomizedCausalLaw H g P hP) ≤
∑ r : LabelSpace J,
(cellAbsoluteDeviation H g true r +
cellAbsoluteDeviation H g false r) := by
let hMass := compatibleCellMasses_of_randomizedCausalLaw H g P hP
have htrue := muUpper_sub_muLower_le_sum_cellAbsoluteDeviation
H g (releasedLaw P) hMass hOverlap true
have hfalse := muUpper_sub_muLower_le_sum_cellAbsoluteDeviation
H g (releasedLaw P) hMass hOverlap false
unfold sharpATELength
rw [Finset.sum_add_distrib]
linarith
theorem worstCaseAmbiguity_le_sum_cellAbsoluteDeviation
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma worstCaseAmbiguity_le_sum_cellAbsoluteDeviation {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε) (hg : Measurable g) :
worstCaseAmbiguity H g ≤
∑ r : LabelSpace J,
(cellAbsoluteDeviation H g true r +
cellAbsoluteDeviation H g false r) := by
unfold worstCaseAmbiguity
apply csSup_le
· let Prel := halfOutcomeReleasedLaw H g
let hMass : CompatibleCellMasses H g Prel :=
halfOutcomeReleasedLaw_cellMasses H g hOverlap hg
letI : IsProbabilityMeasure Prel := hMass.2.1
let P := reconstructedFullLaw H g Prel hg
have hComp : CompatibleCausalLaw H g Prel P :=
reconstructedFullLaw_compatibleCausalLaw H g Prel hg hMass hOverlap
have hRandom : RandomizedCausalLaw H g P :=
{ probability := hComp.probability
measurableRelease := hComp.measurableRelease
scoreMarginal := hComp.scoreMarginal
randomizedAssignment := hComp.randomizedAssignment
consistency := hComp.consistency
deterministicRelease := hComp.deterministicRelease }
exact ⟨sharpATELength H g (releasedLaw P)
(compatibleCellMasses_of_randomizedCausalLaw H g P hRandom),
P, hRandom.probability, hRandom, rfl⟩
· intro d hd
rcases hd with ⟨P, hprob, hP, rfl⟩
exact randomizedCausalLaw_sharpATELength_le_sum_cellAbsoluteDeviation
H g hOverlap P hP
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.AllLabelUpperCell 2 declarations This file bounds each normalized quantile-product cell range by the corresponding unnormalized reciprocal-score absolute deviation, including cells of zero arm mass.
Universal upper bound for all-label cell endpoint gaps
This file bounds each normalized quantile-product cell range by the corresponding unnormalized reciprocal-score absolute deviation, including cells of zero arm mass.
theorem armCell_quantileEndpoint_gap_le_absoluteDeviation
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
r :
conclusion
(if hq : 0 < armCellMass H g a r then armCellMass H g a r * ∫ u in (0 : ℝ)..1, generalizedQuantile (armCellOutcomeLaw Prel hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq)) u * generalizedQuantile (armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) u else 0)
- (if hq : 0 < armCellMass H g a r then armCellMass H g a r * ∫ u in (0 : ℝ)..1, generalizedQuantile (armCellOutcomeLaw Prel hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq)) u * generalizedQuantile (armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) (1 - u) else 0)
≤ cellAbsoluteDeviation H g a r
Proof (Lean source)
lemma armCell_quantileEndpoint_gap_le_absoluteDeviation {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) (r : LabelSpace J) :
(if hq : 0 < armCellMass H g a r then
armCellMass H g a r *
∫ u in (0 : ℝ)..1,
generalizedQuantile
(armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq)) u *
generalizedQuantile
(armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) u
else 0) -
(if hq : 0 < armCellMass H g a r then
armCellMass H g a r *
∫ u in (0 : ℝ)..1,
generalizedQuantile
(armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq)) u *
generalizedQuantile
(armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) (1 - u)
else 0) ≤ cellAbsoluteDeviation H g a r := by
by_cases hq : 0 < armCellMass H g a r
· simp only [hq, dif_pos]
let Y := armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq)
let W := armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq
letI : IsProbabilityMeasure H := hMass.1
letI : IsProbabilityMeasure Prel := hMass.2.1
letI : IsProbabilityMeasure Y :=
armCellOutcomeLaw_isProbabilityMeasure Prel a r
(by rw [hMass.2.2.2 a r]; exact hq)
letI : IsProbabilityMeasure
(armCellScoreLaw H hMass.1 g hMass.2.2.1 a r hq) :=
armCellScoreLaw_isProbabilityMeasure H g hMass.2.2.1 hOverlap a r hq
letI : IsProbabilityMeasure W := by
dsimp [W, armCellWeightLaw]
exact Measure.isProbabilityMeasure_map (by
cases a <;> simp [armProb] <;> fun_prop)
have hYsup : Y (Icc 0 1)ᶜ = 0 := by
dsimp [Y, armCellOutcomeLaw]
rw [Measure.map_apply (by fun_prop) measurableSet_Icc.compl]
have hpre : (fun z : Observation J => (z.2.2 : ℝ)) ⁻¹' (Icc 0 1)ᶜ = ∅ := by
ext z
exact iff_false_intro (fun hz => hz z.2.2.property)
rw [hpre]
simp
have hWsup : W (Icc 0 ε⁻¹)ᶜ = 0 := by
dsimp [W, armCellWeightLaw]
rw [Measure.map_apply (by
cases a <;> simp [armProb] <;> fun_prop) measurableSet_Icc.compl]
have hpre : (fun e : ScoreSpace ε => (armProb a e)⁻¹) ⁻¹'
(Icc 0 ε⁻¹)ᶜ = ∅ := by
ext e
apply iff_false_intro
intro he
apply he
have hp : ε ≤ armProb a e := by
rcases e.property with ⟨he0, he1⟩
cases a <;> simp [armProb] <;> linarith
have hpa : 0 < armProb a e := lt_of_lt_of_le hOverlap.1 hp
exact ⟨inv_nonneg.mpr hpa.le, (inv_le_inv₀ hpa hOverlap.1).mpr hp⟩
rw [hpre]
simp
have hY2 : MemLp (fun y : ℝ => y) 2 Y :=
allLabel_memLp_id_of_support_Icc Y 1 hYsup
have hW2 : MemLp (fun w : ℝ => w) 2 W :=
allLabel_memLp_id_of_support_Icc W ε⁻¹ hWsup
let C : Set ℝ := Icc ((1 - ε)⁻¹) ε⁻¹
have hC : C.Nonempty := allLabel_centerSet_nonempty hOverlap
have hgap := monotoneCoupling_product_gap_le_sInf_absoluteDeviation
Y W hYsup hY2 hW2 C hC
rw [product_expectation_comonotoneCoupling_interval Y W hY2 hW2,
product_expectation_countermonotoneCoupling_interval Y W] at hgap
have hbelow : BddBelow
{v : ℝ | ∃ c ∈ C, v = ∫ w, |w - c| ∂W} := by
refine ⟨0, ?_⟩
rintro v ⟨c, hc, rfl⟩
exact integral_nonneg fun _ => abs_nonneg _
unfold cellAbsoluteDeviation
apply le_csInf
· rcases hC with ⟨c, hc⟩
exact ⟨∫ e in cell g r, |1 - c * armProb a e| ∂H, c, hc, rfl⟩
· intro v hv
rcases hv with ⟨c, hc, rfl⟩
calc
armCellMass H g a r *
(∫ u in (0 : ℝ)..1,
generalizedQuantile Y u * generalizedQuantile W u) -
armCellMass H g a r *
(∫ u in (0 : ℝ)..1,
generalizedQuantile Y u * generalizedQuantile W (1 - u)) =
armCellMass H g a r *
((∫ u in (0 : ℝ)..1,
generalizedQuantile Y u * generalizedQuantile W u) -
∫ u in (0 : ℝ)..1,
generalizedQuantile Y u * generalizedQuantile W (1 - u)) := by ring
_ ≤ armCellMass H g a r *
sInf {v : ℝ | ∃ d ∈ C, v = ∫ w, |w - d| ∂W} :=
mul_le_mul_of_nonneg_left hgap hq.le
_ ≤ armCellMass H g a r * (∫ w, |w - c| ∂W) :=
mul_le_mul_of_nonneg_left (csInf_le hbelow ⟨c, hc, rfl⟩) hq.le
_ = ∫ e in cell g r, |1 - c * armProb a e| ∂H :=
armCellWeightLaw_absoluteDeviation
H g Prel hMass hOverlap a r hq c
· simp [hq]
unfold cellAbsoluteDeviation
apply le_csInf
· rcases allLabel_centerSet_nonempty hOverlap with ⟨c, hc⟩
exact ⟨∫ e in cell g r, |1 - c * armProb a e| ∂H, c, hc, rfl⟩
· intro v hv
rcases hv with ⟨c, hc, rfl⟩
exact integral_nonneg fun _ => abs_nonneg _
theorem muUpper_sub_muLower_le_sum_cellAbsoluteDeviation
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
conclusion
≤ ∑ r : LabelSpace J, cellAbsoluteDeviation H g a r
Proof (Lean source)
lemma muUpper_sub_muLower_le_sum_cellAbsoluteDeviation {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) :
muUpper H g Prel hMass a - muLower H g Prel hMass a ≤
∑ r : LabelSpace J, cellAbsoluteDeviation H g a r := by
unfold muUpper muLower
rw [← Finset.sum_sub_distrib]
exact Finset.sum_le_sum fun r _ =>
armCell_quantileEndpoint_gap_le_absoluteDeviation
H g Prel hMass hOverlap a r
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.CompandingReleaseGeometry 5 declarations Geometry of the ordered companding release used in the unlinked propensity-score ATE analysis.
Geometry of the ordered companding release used in the unlinked propensity-score ATE analysis.
theorem continuousOn_densityExtension
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma continuousOn_densityExtension
{f : ScoreSpace ε → ℝ} (hf : Continuous f) :
ContinuousOn (densityExtension f) (Icc ε (1 - ε)) := by
rw [continuousOn_iff_continuous_domRestrict]
convert hf using 1
funext x
simp [densityExtension, x.property]
theorem continuousOn_compandingWeight
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma continuousOn_compandingWeight
(hOverlap : 0 < ε ∧ ε < 1 / 2) {f : ScoreSpace ε → ℝ}
(hf : Continuous f) :
ContinuousOn
(fun t : ℝ => sqrt (densityExtension f t / (t * (1 - t))))
(Icc ε (1 - ε)) := by
have hden : ContinuousOn (fun t : ℝ => t * (1 - t)) (Icc ε (1 - ε)) := by
fun_prop
have hden_ne : ∀ t ∈ Icc ε (1 - ε), t * (1 - t) ≠ 0 := by
intro t ht
have ht_pos : 0 < t := hOverlap.1.trans_le ht.1
have ht_lt_one : t < 1 := by linarith [ht.2, hOverlap.1]
exact (mul_pos ht_pos (sub_pos.mpr ht_lt_one)).ne'
exact ((continuousOn_densityExtension hf).div hden hden_ne).sqrt
theorem compandingWeight_intervalIntegrable
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
IntervalIntegrable (fun t : ℝ => sqrt (densityExtension f t / (t * (1 - t)))) volume (x : ℝ) (y : ℝ)
Proof (Lean source)
lemma compandingWeight_intervalIntegrable
(hOverlap : 0 < ε ∧ ε < 1 / 2) {f : ScoreSpace ε → ℝ}
(hf : Continuous f) (x y : ScoreSpace ε) :
IntervalIntegrable
(fun t : ℝ => sqrt (densityExtension f t / (t * (1 - t))))
volume (x : ℝ) (y : ℝ) := by
apply ContinuousOn.intervalIntegrable
exact (continuousOn_compandingWeight hOverlap hf).mono
(uIcc_subset_Icc x.property y.property)
theorem monotone_compandingMass
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma monotone_compandingMass
(hOverlap : 0 < ε ∧ ε < 1 / 2) {f : ScoreSpace ε → ℝ}
(hf : Continuous f) : Monotone (compandingMass f) := by
intro x y hxy
unfold compandingMass
have hε_mem : ε ∈ Icc ε (1 - ε) := by
constructor
· rfl
· linarith
apply intervalIntegral.integral_mono_interval le_rfl x.property.1 hxy
· exact Eventually.of_forall fun _ => Real.sqrt_nonneg _
· exact compandingWeight_intervalIntegrable hOverlap hf ⟨ε, hε_mem⟩ y
theorem measurable_compandingRelease
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
hOverlap :
0 < ε ∧ ε < 1 / 2
f :
ScoreSpace ε → ℝ
hf :
_hf_pos :
∀ e, 0 < f e
K :
ℕ
hK :
0 < K
conclusion
Measurable (compandingRelease f K hK)
Proof (Lean source)
lemma measurable_compandingRelease
(hOverlap : 0 < ε ∧ ε < 1 / 2) {f : ScoreSpace ε → ℝ}
(hf : Continuous f) (_hf_pos : ∀ e, 0 < f e) (K : ℕ) (hK : 0 < K) :
Measurable (compandingRelease f K hK) := by
have hmass : Measurable (compandingMass f) :=
(monotone_compandingMass hOverlap hf).measurable
have hnat : Measurable fun e =>
min (Nat.floor ((K : ℝ) * compandingMass f e /
(∫ t in ε..(1 - ε), sqrt (densityExtension f t / (t * (1 - t))))))
(K - 1) := by
exact ((measurable_const.mul hmass).div_const
(∫ t in ε..(1 - ε), sqrt (densityExtension f t / (t * (1 - t)))))
|>.nat_floor.min measurable_const
unfold compandingRelease
apply measurable_to_countable'
intro r
convert hnat (measurableSet_singleton (r : ℕ)) using 1
ext e
exact Fin.ext_iff
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.CompatibilityFullLawConstruction 23 declarations Full-law reconstruction from compatible released arm-cell masses.
Full-law reconstruction from compatible released arm-cell masses.
def reconstructedArmPotentialLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
IsProbabilityMeasure Prel
hg :
a :
def
given by
H ⊗ₘ Kernel.comap (releasedPairLaw Prel).condKernel (fun e => (g e, a)) (hg.prodMk measurable_const)
instance reconstructedArmPotentialLaw_isProbabilityMeasure
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
instance reconstructedArmPotentialLaw_isProbabilityMeasure {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hg : Measurable g) (a : ArmSpace)
[IsProbabilityMeasure H] [IsProbabilityMeasure Prel] :
IsProbabilityMeasure (reconstructedArmPotentialLaw H g Prel hg a) := by
unfold reconstructedArmPotentialLaw
infer_instance
theorem reconstructedArmPotentialLaw_fst
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hg :
a :
IsProbabilityMeasure Prel
conclusion
(reconstructedArmPotentialLaw H g Prel hg a).fst = H
Proof (Lean source)
lemma reconstructedArmPotentialLaw_fst {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hg : Measurable g) (a : ArmSpace)
[IsProbabilityMeasure H] [IsProbabilityMeasure Prel] :
(reconstructedArmPotentialLaw H g Prel hg a).fst = H := by
unfold reconstructedArmPotentialLaw
exact Measure.fst_compProd H _
def reconstructedLatentLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
IsProbabilityMeasure Prel
hg :
def
given by
glueAlongFst (reconstructedArmPotentialLaw H g Prel hg false) (reconstructedArmPotentialLaw H g Prel hg true)
instance reconstructedLatentLaw_isProbabilityMeasure
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
instance reconstructedLatentLaw_isProbabilityMeasure {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hg : Measurable g)
[IsProbabilityMeasure H] [IsProbabilityMeasure Prel] :
IsProbabilityMeasure (reconstructedLatentLaw H g Prel hg) := by
unfold reconstructedLatentLaw
exact glueAlongFst_isProbabilityMeasure _ _
theorem reconstructedLatentLaw_map_control
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hg :
IsProbabilityMeasure Prel
conclusion
(reconstructedLatentLaw H g Prel hg).map (fun p => (p.1, p.2.1))
= reconstructedArmPotentialLaw H g Prel hg false
Proof (Lean source)
lemma reconstructedLatentLaw_map_control {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hg : Measurable g)
[IsProbabilityMeasure H] [IsProbabilityMeasure Prel] :
(reconstructedLatentLaw H g Prel hg).map (fun p => (p.1, p.2.1)) =
reconstructedArmPotentialLaw H g Prel hg false := by
unfold reconstructedLatentLaw
exact glueAlongFst_map_left _ _
theorem reconstructedLatentLaw_map_treated
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hg :
IsProbabilityMeasure Prel
conclusion
(reconstructedLatentLaw H g Prel hg).map (fun p => (p.1, p.2.2))
= reconstructedArmPotentialLaw H g Prel hg true
Proof (Lean source)
lemma reconstructedLatentLaw_map_treated {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hg : Measurable g)
[IsProbabilityMeasure H] [IsProbabilityMeasure Prel] :
(reconstructedLatentLaw H g Prel hg).map (fun p => (p.1, p.2.2)) =
reconstructedArmPotentialLaw H g Prel hg true := by
unfold reconstructedLatentLaw
apply glueAlongFst_map_right
rw [reconstructedArmPotentialLaw_fst, reconstructedArmPotentialLaw_fst]
theorem reconstructedLatentLaw_fst
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hg :
IsProbabilityMeasure Prel
conclusion
(reconstructedLatentLaw H g Prel hg).fst = H
Proof (Lean source)
lemma reconstructedLatentLaw_fst {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hg : Measurable g)
[IsProbabilityMeasure H] [IsProbabilityMeasure Prel] :
(reconstructedLatentLaw H g Prel hg).fst = H := by
unfold Measure.fst
calc
(reconstructedLatentLaw H g Prel hg).map fst =
((reconstructedLatentLaw H g Prel hg).map
(fun p => (p.1, p.2.1))).map fst := by
rw [Measure.map_map measurable_fst (by fun_prop)]
rfl
_ = (reconstructedArmPotentialLaw H g Prel hg false).map fst := by
rw [reconstructedLatentLaw_map_control]
_ = H := reconstructedArmPotentialLaw_fst H g Prel hg false
def reconstructedAssignmentDensity
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
def reconstructedFullRow
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
(p.1, p.2.1, p.2.2, a, if a then p.2.2 else p.2.1, g p.1)
theorem reconstructedFullRow_measurable
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma reconstructedFullRow_measurable {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hg : Measurable g) (a : ArmSpace) :
Measurable (reconstructedFullRow g a) := by
unfold reconstructedFullRow
cases a
· simp only [Bool.false_eq_true, ↓reduceIte]
fun_prop
· simp only [↓reduceIte]
fun_prop
def reconstructedFullLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
IsProbabilityMeasure Prel
hg :
given by
((reconstructedLatentLaw H g Prel hg).withDensity (reconstructedAssignmentDensity false)).map (reconstructedFullRow g false)
+ ((reconstructedLatentLaw H g Prel hg).withDensity (reconstructedAssignmentDensity true)).map (reconstructedFullRow g true)
theorem reconstructedAssignedPotentialLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
IsProbabilityMeasure Prel
hg :
a :
conclusion
((reconstructedLatentLaw H g Prel hg).withDensity (reconstructedAssignmentDensity a)).map (fun p => (p.1, if a then p.2.2 else p.2.1))
Proof (Lean source)
lemma reconstructedAssignedPotentialLaw { ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) [IsProbabilityMeasure H]
[IsProbabilityMeasure Prel] (hg : Measurable g) (a : ArmSpace) :
((reconstructedLatentLaw H g Prel hg).withDensity
(reconstructedAssignmentDensity a)).map
(fun p => (p.1, if a then p.2.2 else p.2.1)) =
(reconstructedArmPotentialLaw H g Prel hg a).withDensity
(fun p => ofReal (armProb a p.1)) := by
cases a
· simp only [Bool.false_eq_true, ↓reduceIte]
change ((reconstructedLatentLaw H g Prel hg).withDensity
((fun q : ScoreSpace ε × OutcomeSpace =>
ofReal (1 - (q.1 : ℝ))) ∘
(fun p => (p.1, p.2.1)))).map (fun p => (p.1, p.2.1)) = _
rw [map_withDensity_of_comp _ _ (by fun_prop) _ (by fun_prop),
reconstructedLatentLaw_map_control H g Prel hg]
rfl
· simp only [↓reduceIte]
change ((reconstructedLatentLaw H g Prel hg).withDensity
((fun q : ScoreSpace ε × OutcomeSpace =>
ofReal (q.1 : ℝ)) ∘
(fun p => (p.1, p.2.2)))).map (fun p => (p.1, p.2.2)) = _
rw [map_withDensity_of_comp _ _ (by fun_prop) _ (by fun_prop),
reconstructedLatentLaw_map_treated H g Prel hg]
rfl
theorem reconstructedAssignedReleasedPairLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
IsProbabilityMeasure Prel
hg :
a :
conclusion
((reconstructedLatentLaw H g Prel hg).withDensity (reconstructedAssignmentDensity a)).map (fun p => ((g p.1, a), if a then p.2.2 else p.2.1))
Proof (Lean source)
lemma reconstructedAssignedReleasedPairLaw {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) [IsProbabilityMeasure H]
[IsProbabilityMeasure Prel] (hg : Measurable g) (a : ArmSpace) :
((reconstructedLatentLaw H g Prel hg).withDensity
(reconstructedAssignmentDensity a)).map
(fun p => ((g p.1, a), if a then p.2.2 else p.2.1)) =
scoreAssignedKeyLaw H g a ⊗ₘ (releasedPairLaw Prel).condKernel := by
let k : ScoreSpace ε → LabelSpace J × ArmSpace := fun e => (g e, a)
let d : ScoreSpace ε → ENNReal :=
fun e => ofReal (armProb a e)
let q : ScoreSpace ε × (OutcomeSpace × OutcomeSpace) →
ScoreSpace ε × OutcomeSpace :=
fun p => (p.1, if a then p.2.2 else p.2.1)
let v : ScoreSpace ε × OutcomeSpace →
(LabelSpace J × ArmSpace) × OutcomeSpace :=
fun p => (k p.1, p.2)
have hk : Measurable k := hg.prodMk measurable_const
have hq : Measurable q := by
cases a <;> simp [q] <;> fun_prop
have hv : Measurable v := by
dsimp [v]
exact (hk.comp measurable_fst).prodMk measurable_snd
have hd : Measurable d := by
cases a <;> simp [d, armProb] <;> fun_prop
calc
((reconstructedLatentLaw H g Prel hg).withDensity
(reconstructedAssignmentDensity a)).map
(fun p => ((g p.1, a), if a then p.2.2 else p.2.1)) =
(((reconstructedLatentLaw H g Prel hg).withDensity
(reconstructedAssignmentDensity a)).map q).map v := by
rw [Measure.map_map hv hq]
rfl
_ = ((reconstructedArmPotentialLaw H g Prel hg a).withDensity
(fun p => d p.1)).map v := by
rw [reconstructedAssignedPotentialLaw H g Prel hg a]
_ = (((H.withDensity d) ⊗ₘ
Kernel.comap (releasedPairLaw Prel).condKernel k hk).map v) := by
unfold reconstructedArmPotentialLaw
rw [withDensity_fst_compProd H _ d hd]
_ = (H.withDensity d).map k ⊗ₘ
(releasedPairLaw Prel).condKernel := by
exact map_compProd_comap (H.withDensity d)
(releasedPairLaw Prel).condKernel k hk
_ = scoreAssignedKeyLaw H g a ⊗ₘ
(releasedPairLaw Prel).condKernel := by
rfl
theorem reconstructedAssignmentMeasures_sum
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
IsProbabilityMeasure Prel
hg :
hOverlap :
Overlap ε
conclusion
= reconstructedLatentLaw H g Prel hg
Proof (Lean source)
lemma reconstructedAssignmentMeasures_sum {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) [IsProbabilityMeasure H]
[IsProbabilityMeasure Prel] (hg : Measurable g) (hOverlap : Overlap ε) :
(reconstructedLatentLaw H g Prel hg).withDensity
(reconstructedAssignmentDensity false) +
(reconstructedLatentLaw H g Prel hg).withDensity
(reconstructedAssignmentDensity true) =
reconstructedLatentLaw H g Prel hg := by
rw [← withDensity_add_left (reconstructedAssignmentDensity_measurable false)]
have hfun : reconstructedAssignmentDensity false +
reconstructedAssignmentDensity true =
(1 : ScoreSpace ε × (OutcomeSpace × OutcomeSpace) → ENNReal) := by
funext p
exact reconstructedAssignmentDensity_sum hOverlap p
rw [hfun, withDensity_one]
theorem reconstructedFullLaw_isProbabilityMeasure
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
IsProbabilityMeasure Prel
hg :
hOverlap :
Overlap ε
conclusion
IsProbabilityMeasure (reconstructedFullLaw H g Prel hg)
Proof (Lean source)
lemma reconstructedFullLaw_isProbabilityMeasure {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) [IsProbabilityMeasure H]
[IsProbabilityMeasure Prel] (hg : Measurable g) (hOverlap : Overlap ε) :
IsProbabilityMeasure (reconstructedFullLaw H g Prel hg) := by
rw [isProbabilityMeasure_iff]
unfold reconstructedFullLaw
rw [Measure.add_apply,
Measure.map_apply (reconstructedFullRow_measurable g hg false) MeasurableSet.univ,
Measure.map_apply (reconstructedFullRow_measurable g hg true) MeasurableSet.univ]
simp only [preimage_univ]
rw [← Measure.add_apply, reconstructedAssignmentMeasures_sum H g Prel hg hOverlap]
exact isProbabilityMeasure_iff.mp inferInstance
theorem reconstructedFullLaw_scoreMarginal
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
IsProbabilityMeasure Prel
hg :
hOverlap :
Overlap ε
conclusion
ScoreMarginal (reconstructedFullLaw H g Prel hg) H
Proof (Lean source)
lemma reconstructedFullLaw_scoreMarginal {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) [IsProbabilityMeasure H]
[IsProbabilityMeasure Prel] (hg : Measurable g) (hOverlap : Overlap ε) :
ScoreMarginal (reconstructedFullLaw H g Prel hg) H := by
unfold ScoreMarginal reconstructedFullLaw
rw [Measure.map_add _ _ (by unfold score; fun_prop),
Measure.map_map (by unfold score; fun_prop)
(reconstructedFullRow_measurable g hg false),
Measure.map_map (by unfold score; fun_prop)
(reconstructedFullRow_measurable g hg true)]
change Measure.map fst
((reconstructedLatentLaw H g Prel hg).withDensity
(reconstructedAssignmentDensity false)) +
Measure.map fst
((reconstructedLatentLaw H g Prel hg).withDensity
(reconstructedAssignmentDensity true)) = H
rw [← Measure.map_add _ _ measurable_fst,
reconstructedAssignmentMeasures_sum H g Prel hg hOverlap]
exact reconstructedLatentLaw_fst H g Prel hg
theorem reconstructedFullLaw_consistency
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
IsProbabilityMeasure Prel
hg :
conclusion
Consistency (reconstructedFullLaw H g Prel hg)
Proof (Lean source)
lemma reconstructedFullLaw_consistency {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) [IsProbabilityMeasure H]
[IsProbabilityMeasure Prel] (hg : Measurable g) :
Consistency (reconstructedFullLaw H g Prel hg) := by
unfold Consistency reconstructedFullLaw
rw [ae_add_measure_iff]
constructor
· have hrow := reconstructedFullRow_measurable g hg false
rw [ae_map_iff hrow.aemeasurable (by
exact measurableSet_eq_fun (by unfold observed; fun_prop)
(by
have hcond : MeasurableSet {ω : FullRow ε J | arm ω = true} :=
measurableSet_eq_fun (by unfold arm; fun_prop) measurable_const
exact Measurable.ite hcond (by unfold outcome1; fun_prop)
(by unfold outcome0; fun_prop)))]
filter_upwards [] with p
rfl
· have hrow := reconstructedFullRow_measurable g hg true
rw [ae_map_iff hrow.aemeasurable (by
exact measurableSet_eq_fun (by unfold observed; fun_prop)
(by
have hcond : MeasurableSet {ω : FullRow ε J | arm ω = true} :=
measurableSet_eq_fun (by unfold arm; fun_prop) measurable_const
exact Measurable.ite hcond (by unfold outcome1; fun_prop)
(by unfold outcome0; fun_prop)))]
filter_upwards [] with p
rfl
theorem reconstructedFullLaw_deterministicRelease
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
IsProbabilityMeasure Prel
hg :
conclusion
DeterministicRelease g (reconstructedFullLaw H g Prel hg)
Proof (Lean source)
lemma reconstructedFullLaw_deterministicRelease {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) [IsProbabilityMeasure H]
[IsProbabilityMeasure Prel] (hg : Measurable g) :
DeterministicRelease g (reconstructedFullLaw H g Prel hg) := by
unfold DeterministicRelease reconstructedFullLaw
rw [ae_add_measure_iff]
constructor
· rw [ae_map_iff (reconstructedFullRow_measurable g hg false).aemeasurable (by
exact measurableSet_eq_fun (by unfold label; fun_prop)
(hg.comp (by unfold score; fun_prop)))]
filter_upwards [] with p
rfl
· rw [ae_map_iff (reconstructedFullRow_measurable g hg true).aemeasurable (by
exact measurableSet_eq_fun (by unfold label; fun_prop)
(hg.comp (by unfold score; fun_prop)))]
filter_upwards [] with p
rfl
theorem reconstructedFullLaw_randomizedAssignment
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
IsProbabilityMeasure Prel
hg :
hOverlap :
Overlap ε
conclusion
RandomizedAssignment (reconstructedFullLaw H g Prel hg)
Proof (Lean source)
lemma reconstructedFullLaw_randomizedAssignment {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) [IsProbabilityMeasure H]
[IsProbabilityMeasure Prel] (hg : Measurable g) (hOverlap : Overlap ε) :
RandomizedAssignment (reconstructedFullLaw H g Prel hg) := by
intro B hB
let L := reconstructedLatentLaw H g Prel hg
let M₀ := L.withDensity (reconstructedAssignmentDensity false)
let M₁ := L.withDensity (reconstructedAssignmentDensity true)
let D : Set (ScoreSpace ε × (OutcomeSpace × OutcomeSpace)) :=
{p | (p.1, p.2.1, p.2.2) ∈ B}
have hD : MeasurableSet D := hB.preimage (by fun_prop)
let S : Set (FullRow ε J) :=
{ω | arm ω = true ∧ (score ω, outcome0 ω, outcome1 ω) ∈ B}
let T : Set (FullRow ε J) :=
{ω | (score ω, outcome0 ω, outcome1 ω) ∈ B}
have hS : MeasurableSet S :=
(measurableSet_eq_fun (by unfold arm; fun_prop) measurable_const).inter
(hB.preimage (by unfold score outcome0 outcome1; fun_prop))
have hT : MeasurableSet T :=
hB.preimage (by unfold score outcome0 outcome1; fun_prop)
have hpre0S : reconstructedFullRow g false ⁻¹' S = ∅ := by
ext p
simp [S, reconstructedFullRow, arm]
have hpre1S : reconstructedFullRow g true ⁻¹' S = D := by
ext p
simp [S, D, reconstructedFullRow, arm, score, outcome0, outcome1]
have hpre0T : reconstructedFullRow g false ⁻¹' T = D := by
ext p
simp [T, D, reconstructedFullRow, score, outcome0, outcome1]
have hpre1T : reconstructedFullRow g true ⁻¹' T = D := by
ext p
simp [T, D, reconstructedFullRow, score, outcome0, outcome1]
letI : IsFiniteMeasure M₀ := reconstructedComponent_finite H g Prel hg hOverlap false
letI : IsFiniteMeasure M₁ := reconstructedComponent_finite H g Prel hg hOverlap true
have hmap0 : IsFiniteMeasure (M₀.map (reconstructedFullRow g false)) := ⟨by
rw [Measure.map_apply (reconstructedFullRow_measurable g hg false)
MeasurableSet.univ]
simpa using measure_lt_top M₀ Set.univ
⟩
have hmap1 : IsFiniteMeasure (M₁.map (reconstructedFullRow g true)) := ⟨by
rw [Measure.map_apply (reconstructedFullRow_measurable g hg true)
MeasurableSet.univ]
simpa using measure_lt_top M₁ Set.univ
⟩
change (reconstructedFullLaw H g Prel hg).real S =
∫ ω in T, (score ω : ℝ) ∂reconstructedFullLaw H g Prel hg
have hleft : (reconstructedFullLaw H g Prel hg).real S =
∫ p in D, (p.1 : ℝ) ∂L := by
unfold reconstructedFullLaw
rw [Measure.real_def, Measure.add_apply,
Measure.map_apply (reconstructedFullRow_measurable g hg false) hS,
Measure.map_apply (reconstructedFullRow_measurable g hg true) hS,
hpre0S, hpre1S, measure_empty, zero_add]
change M₁.real D = _
exact assignmentTrue_real H g Prel hg hOverlap D hD
rw [hleft]
unfold reconstructedFullLaw
letI : IsFiniteMeasure (M₀.map (reconstructedFullRow g false)) := hmap0
letI : IsFiniteMeasure (M₁.map (reconstructedFullRow g true)) := hmap1
rw [Measure.restrict_add,
integral_add_measure (fullScore_integrable_finite _).integrableOn
(fullScore_integrable_finite _).integrableOn,
setIntegral_map hT (by unfold score; fun_prop)
(reconstructedFullRow_measurable g hg false).aemeasurable,
setIntegral_map hT (by unfold score; fun_prop)
(reconstructedFullRow_measurable g hg true).aemeasurable,
hpre0T, hpre1T]
change (∫ p in D, (p.1 : ℝ) ∂L) =
(∫ p in D, (p.1 : ℝ) ∂M₀) + (∫ p in D, (p.1 : ℝ) ∂M₁)
rw [← integral_add_measure
(latentScore_integrable_finite M₀).integrableOn
(latentScore_integrable_finite M₁).integrableOn]
rw [← Measure.restrict_add, reconstructedAssignmentMeasures_sum H g Prel hg hOverlap]
theorem reconstructedFullLaw_releasedPairLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
IsProbabilityMeasure Prel
hg :
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
conclusion
= releasedPairLaw Prel
Proof (Lean source)
lemma reconstructedFullLaw_releasedPairLaw {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) [IsProbabilityMeasure H]
[IsProbabilityMeasure Prel] (hg : Measurable g)
(hMass : CompatibleCellMasses H g Prel) (hOverlap : Overlap ε) :
(reconstructedFullLaw H g Prel hg).map
(fun ω => ((label ω, arm ω), observed ω)) =
releasedPairLaw Prel := by
have hp : Measurable
(fun ω : FullRow ε J => ((label ω, arm ω), observed ω)) := by
unfold label arm observed
fun_prop
unfold reconstructedFullLaw
rw [Measure.map_add _ _ hp,
Measure.map_map hp (reconstructedFullRow_measurable g hg false),
Measure.map_map hp (reconstructedFullRow_measurable g hg true)]
change ((reconstructedLatentLaw H g Prel hg).withDensity
(reconstructedAssignmentDensity false)).map
(fun p => ((g p.1, false), p.2.1)) +
((reconstructedLatentLaw H g Prel hg).withDensity
(reconstructedAssignmentDensity true)).map
(fun p => ((g p.1, true), p.2.2)) = _
have hfalse := reconstructedAssignedReleasedPairLaw H g Prel hg false
have htrue := reconstructedAssignedReleasedPairLaw H g Prel hg true
simp only [Bool.false_eq_true, ↓reduceIte] at hfalse
simp only [↓reduceIte] at htrue
rw [hfalse, htrue, ← Measure.compProd_add_left]
change randomizedKeyLaw H g ⊗ₘ (releasedPairLaw Prel).condKernel = _
rw [randomizedKeyLaw_eq_releasedKeyLaw H g Prel hMass hOverlap,
releasedPairLaw_disintegrate Prel]
theorem reconstructedFullLaw_releasedLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
IsProbabilityMeasure Prel
hg :
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
conclusion
ReleasedLaw (reconstructedFullLaw H g Prel hg) Prel
Proof (Lean source)
lemma reconstructedFullLaw_releasedLaw {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) [IsProbabilityMeasure H]
[IsProbabilityMeasure Prel] (hg : Measurable g)
(hMass : CompatibleCellMasses H g Prel) (hOverlap : Overlap ε) :
ReleasedLaw (reconstructedFullLaw H g Prel hg) Prel := by
unfold ReleasedLaw releasedLaw
have hp : Measurable
(fun ω : FullRow ε J => ((label ω, arm ω), observed ω)) := by
unfold label arm observed
fun_prop
have hu : Measurable
(fun p : (LabelSpace J × ArmSpace) × OutcomeSpace =>
(p.1.1, p.1.2, p.2)) := by fun_prop
calc
(reconstructedFullLaw H g Prel hg).map releasedRecord =
((reconstructedFullLaw H g Prel hg).map
(fun ω => ((label ω, arm ω), observed ω))).map
(fun p => (p.1.1, p.1.2, p.2)) := by
rw [Measure.map_map hu hp]
rfl
_ = (releasedPairLaw Prel).map
(fun p => (p.1.1, p.1.2, p.2)) := by
rw [reconstructedFullLaw_releasedPairLaw H g Prel hg hMass hOverlap]
_ = Prel := releasedPairLaw_map_unpair Prel
theorem reconstructedFullLaw_compatibleCausalLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
IsProbabilityMeasure Prel
hg :
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
conclusion
CompatibleCausalLaw H g Prel (reconstructedFullLaw H g Prel hg)
Proof (Lean source)
lemma reconstructedFullLaw_compatibleCausalLaw {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) [IsProbabilityMeasure H]
[IsProbabilityMeasure Prel] (hg : Measurable g)
(hMass : CompatibleCellMasses H g Prel) (hOverlap : Overlap ε) :
CompatibleCausalLaw H g Prel (reconstructedFullLaw H g Prel hg) := by
exact
{ probability := reconstructedFullLaw_isProbabilityMeasure H g Prel hg hOverlap
measurableRelease := hg
scoreMarginal := reconstructedFullLaw_scoreMarginal H g Prel hg hOverlap
randomizedAssignment :=
reconstructedFullLaw_randomizedAssignment H g Prel hg hOverlap
consistency := reconstructedFullLaw_consistency H g Prel hg
deterministicRelease := reconstructedFullLaw_deterministicRelease H g Prel hg
releasedLaw := reconstructedFullLaw_releasedLaw H g Prel hg hMass hOverlap }
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.CompatibilityReconstruction 12 declarations Measure-level reconstruction leaves for the compatibility converse.
Measure-level reconstruction leaves for the compatibility converse.
def scoreAssignedKeyLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
def randomizedKeyLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
def releasedKeyLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
def releasedPairLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
J :
ℕ
Prel :
Measure (Observation J)
def
given by
Prel.map (fun z => ((z.1, z.2.1), z.2.2))
instance releasedPairLaw_isProbabilityMeasure
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
instance releasedPairLaw_isProbabilityMeasure {J : ℕ}
(Prel : Measure (Observation J)) [IsProbabilityMeasure Prel] :
IsProbabilityMeasure (releasedPairLaw Prel) := by
unfold releasedPairLaw
exact Measure.isProbabilityMeasure_map releasedPairing_measurable.aemeasurable
theorem releasedPairLaw_fst
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
J :
ℕ
Prel :
Measure (Observation J)
conclusion
(releasedPairLaw Prel).fst = releasedKeyLaw Prel
Proof (Lean source)
lemma releasedPairLaw_fst {J : ℕ}
(Prel : Measure (Observation J)) :
(releasedPairLaw Prel).fst = releasedKeyLaw Prel := by
unfold Measure.fst releasedPairLaw releasedKeyLaw
rw [Measure.map_map measurable_fst releasedPairing_measurable]
rfl
theorem releasedPairLaw_map_unpair
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
J :
ℕ
Prel :
Measure (Observation J)
conclusion
(releasedPairLaw Prel).map (fun p => (p.1.1, p.1.2, p.2)) = Prel
Proof (Lean source)
lemma releasedPairLaw_map_unpair {J : ℕ}
(Prel : Measure (Observation J)) :
(releasedPairLaw Prel).map (fun p => (p.1.1, p.1.2, p.2)) = Prel := by
unfold releasedPairLaw
rw [Measure.map_map (by fun_prop) releasedPairing_measurable]
have hfun : ((fun p : (LabelSpace J × ArmSpace) × OutcomeSpace =>
(p.1.1, p.1.2, p.2)) ∘
(fun z : Observation J => ((z.1, z.2.1), z.2.2))) = id := by
funext z
rfl
rw [hfun, Measure.map_id]
theorem releasedPairLaw_disintegrate
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma releasedPairLaw_disintegrate {J : ℕ}
(Prel : Measure (Observation J)) [IsProbabilityMeasure Prel] :
(releasedKeyLaw Prel) ⊗ₘ (releasedPairLaw Prel).condKernel =
releasedPairLaw Prel := by
rw [← releasedPairLaw_fst Prel]
exact (releasedPairLaw Prel).disintegrate (releasedPairLaw Prel).condKernel
theorem propensityTilt_real_cell
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
Proof (Lean source)
lemma propensityTilt_real_cell {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace J) (hg : Measurable g)
(hOverlap : Overlap ε) (a : ArmSpace) (r : LabelSpace J) :
(H.withDensity (fun e => ofReal (armProb a e))).real (cell g r) =
armCellMass H g a r := by
let F : ScoreSpace ε → ℝ := (cell g r).indicator (armProb a)
have hcell : MeasurableSet (cell g r) :=
measurableSet_eq_fun hg measurable_const
have hF : Measurable F := (armProb_measurable a).indicator hcell
have hbound : ∀ e, F e ∈ Icc (0 : ℝ) 1 := by
intro e
by_cases he : e ∈ cell g r
· simpa [F, indicator, he] using armProb_mem_Icc hOverlap a e
· simp [F, indicator, he]
have hInt : Integrable F H := by
apply Integrable.of_bound hF.aestronglyMeasurable 1
filter_upwards [] with e
simpa [Real.norm_eq_abs, abs_of_nonneg (hbound e).1] using (hbound e).2
unfold Measure.real armCellMass
rw [withDensity_apply _ hcell]
have hlin :
(∫⁻ e in cell g r, ofReal (armProb a e) ∂H) =
∫⁻ e, ofReal (F e) ∂H := by
rw [← lintegral_indicator hcell]
apply lintegral_congr
intro e
by_cases he : e ∈ cell g r <;> simp [F, indicator, he]
rw [hlin, ← ofReal_integral_eq_lintegral_ofReal hInt
(Filter.Eventually.of_forall (fun e => (hbound e).1))]
rw [ENNReal.toReal_ofReal (integral_nonneg (fun e => (hbound e).1))]
rw [← integral_indicator hcell]
theorem randomizedKeyLaw_real_singleton
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(randomizedKeyLaw H g).real {(r, a)} = armCellMass H g a r
Proof (Lean source)
lemma randomizedKeyLaw_real_singleton {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace J) (hg : Measurable g)
(hOverlap : Overlap ε) (r : LabelSpace J) (a : ArmSpace) :
(randomizedKeyLaw H g).real {(r, a)} = armCellMass H g a r := by
letI : IsFiniteMeasure
(H.withDensity (fun e => ofReal (armProb false e))) :=
isFiniteMeasure_withDensity_ofReal (armProb_integrable H hOverlap false).hasFiniteIntegral
letI : IsFiniteMeasure
(H.withDensity (fun e => ofReal (armProb true e))) :=
isFiniteMeasure_withDensity_ofReal (armProb_integrable H hOverlap true).hasFiniteIntegral
have hkey (b : ArmSpace) : Measurable (fun e : ScoreSpace ε => (g e, b)) := by
fun_prop
have hsingle : MeasurableSet ({(r, a)} : Set (LabelSpace J × ArmSpace)) :=
measurableSet_singleton _
letI : IsFiniteMeasure (scoreAssignedKeyLaw H g false) := ⟨by
have hs : (H.withDensity (fun e => ofReal (armProb false e))) univ < ∞ :=
measure_lt_top _ _
unfold scoreAssignedKeyLaw
rw [Measure.map_apply (hkey false) MeasurableSet.univ]
exact hs
⟩
letI : IsFiniteMeasure (scoreAssignedKeyLaw H g true) := ⟨by
have hs : (H.withDensity (fun e => ofReal (armProb true e))) univ < ∞ :=
measure_lt_top _ _
unfold scoreAssignedKeyLaw
rw [Measure.map_apply (hkey true) MeasurableSet.univ]
exact hs
⟩
unfold randomizedKeyLaw
rw [Measure.real_def, Measure.add_apply]
rw [ENNReal.toReal_add (measure_ne_top _ _) (measure_ne_top _ _)]
unfold scoreAssignedKeyLaw
rw [Measure.map_apply (hkey false) hsingle, Measure.map_apply (hkey true) hsingle]
cases a
· have hpreFalse : (fun e : ScoreSpace ε => (g e, false)) ⁻¹' {(r, false)} =
cell g r := by ext e; simp [cell]
have hpreTrue : (fun e : ScoreSpace ε => (g e, true)) ⁻¹' {(r, false)} = ∅ := by
ext e
simp
rw [hpreFalse, hpreTrue, measure_empty, ENNReal.toReal_zero, add_zero]
exact propensityTilt_real_cell H g hg hOverlap false r
· have hpreFalse : (fun e : ScoreSpace ε => (g e, false)) ⁻¹' {(r, true)} = ∅ := by
ext e
simp
have hpreTrue : (fun e : ScoreSpace ε => (g e, true)) ⁻¹' {(r, true)} =
cell g r := by ext e; simp [cell]
rw [hpreFalse, hpreTrue, measure_empty, ENNReal.toReal_zero, zero_add]
exact propensityTilt_real_cell H g hg hOverlap true r
theorem releasedKeyLaw_real_singleton
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(releasedKeyLaw Prel).real {(r, a)} = Prel.real {z | z.1 = r ∧ z.2.1 = a}
Proof (Lean source)
lemma releasedKeyLaw_real_singleton {J : ℕ}
(Prel : Measure (Observation J)) [IsProbabilityMeasure Prel]
(r : LabelSpace J) (a : ArmSpace) :
(releasedKeyLaw Prel).real {(r, a)} =
Prel.real {z | z.1 = r ∧ z.2.1 = a} := by
have hkey : Measurable (fun z : Observation J => (z.1, z.2.1)) := by fun_prop
have hsingle : MeasurableSet ({(r, a)} : Set (LabelSpace J × ArmSpace)) :=
measurableSet_singleton _
unfold releasedKeyLaw
rw [MeasureTheory.map_measureReal_apply hkey hsingle]
congr 1
ext z
simp
theorem randomizedKeyLaw_eq_releasedKeyLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
conclusion
randomizedKeyLaw H g = releasedKeyLaw Prel
Proof (Lean source)
lemma randomizedKeyLaw_eq_releasedKeyLaw {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) :
randomizedKeyLaw H g = releasedKeyLaw Prel := by
letI : IsProbabilityMeasure H := hMass.1
letI : IsProbabilityMeasure Prel := hMass.2.1
letI : IsFiniteMeasure
(H.withDensity (fun e => ofReal (armProb false e))) :=
isFiniteMeasure_withDensity_ofReal
(armProb_integrable H hOverlap false).hasFiniteIntegral
letI : IsFiniteMeasure
(H.withDensity (fun e => ofReal (armProb true e))) :=
isFiniteMeasure_withDensity_ofReal
(armProb_integrable H hOverlap true).hasFiniteIntegral
letI : IsFiniteMeasure (scoreAssignedKeyLaw H g false) := ⟨by
have hs : (H.withDensity (fun e => ofReal (armProb false e))) univ < ∞ :=
measure_lt_top _ _
unfold scoreAssignedKeyLaw
rw [Measure.map_apply (by
exact hMass.2.2.1.prodMk measurable_const) MeasurableSet.univ]
exact hs
⟩
letI : IsFiniteMeasure (scoreAssignedKeyLaw H g true) := ⟨by
have hs : (H.withDensity (fun e => ofReal (armProb true e))) univ < ∞ :=
measure_lt_top _ _
unfold scoreAssignedKeyLaw
rw [Measure.map_apply (by
exact hMass.2.2.1.prodMk measurable_const) MeasurableSet.univ]
exact hs
⟩
letI : IsFiniteMeasure (randomizedKeyLaw H g) := by
unfold randomizedKeyLaw scoreAssignedKeyLaw
infer_instance
letI : IsProbabilityMeasure (releasedKeyLaw Prel) := by
unfold releasedKeyLaw
have hkey : Measurable (fun z : Observation J => (z.1, z.2.1)) := by fun_prop
exact Measure.isProbabilityMeasure_map hkey.aemeasurable
apply Measure.ext_of_measureReal_singleton
rintro ⟨r, a⟩
rw [randomizedKeyLaw_real_singleton H g hMass.2.2.1 hOverlap r a,
releasedKeyLaw_real_singleton Prel r a, hMass.2.2.2 a r]
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.External 16 declarations The independent external score-log experiment and its honest excess-length risk.
The independent external score-log experiment and its honest excess-length risk.
def ExternalSample
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J n m :
ℕ
def
ExternalSample ε J n m :
Type
given by
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
assumes
probabilityP :
probabilityH :
overlap :
Overlap ε
measurableRelease :
scoreMarginal :
ScoreMarginal P H
randomizedAssignment :
consistency :
Consistency P
deterministicRelease :
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
theorem externalLawCellMasses
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
CompatibleCellMasses PH.2 g (releasedLaw PH.1)
Proof (Lean source)
lemma externalLawCellMasses
(g : ScoreSpace ε → LabelSpace J)
(PH : Measure (FullRow ε J) × Measure (ScoreSpace ε))
(hPH : PH ∈ ExternalLaws g) :
CompatibleCellMasses PH.2 g (releasedLaw PH.1) := by
have hX : ExternalScoreLaw g PH.1 PH.2 := hPH.2.2
have hRandom : RandomizedCausalLaw PH.2 g PH.1 :=
⟨hX.probabilityP, hX.measurableRelease, hX.scoreMarginal,
hX.randomizedAssignment,
hX.consistency, hX.deterministicRelease⟩
exact compatibleCellMasses_of_randomizedCausalLaw PH.2 g PH.1 hRandom
def ExternalLogIID
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
n :
ℕ
m :
ℕ
g :
ScoreSpace ε → LabelSpace J
Qj :
PH ∈ ExternalLaws g :
def
ExternalLogIID g Qj :
Prop
def ExternalLogIndependent
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
n :
ℕ
m :
ℕ
g :
ScoreSpace ε → LabelSpace J
Qj :
PH ∈ ExternalLaws g :
def
ExternalLogIndependent g Qj :
Prop
def JointTrialIID
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
n :
ℕ
m :
ℕ
g :
ScoreSpace ε → LabelSpace J
Qj :
PH ∈ ExternalLaws g :
def
JointTrialIID g Qj :
Prop
given by
def externalExperiment
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
theorem jointTrialLog_eq_externalExperiment
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
n :
ℕ
m :
ℕ
g :
ScoreSpace ε → LabelSpace J
Qj :
hTrial :
JointTrialIID g Qj
hLog :
ExternalLogIID g Qj
hInd :
PH :
hPH :
PH ∈ ExternalLaws g
conclusion
Qj PH.1 PH.2 = externalExperiment n m PH.1 PH.2
Proof (Lean source)
lemma jointTrialLog_eq_externalExperiment
(g : ScoreSpace ε → LabelSpace J)
(Qj : Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : JointTrialIID g Qj) (hLog : ExternalLogIID g Qj)
(hInd : ExternalLogIndependent g Qj)
(PH : Measure (FullRow ε J) × Measure (ScoreSpace ε))
(hPH : PH ∈ ExternalLaws g) :
Qj PH.1 PH.2 = externalExperiment n m PH.1 PH.2 := by
rw [hInd PH hPH, hTrial PH hPH, hLog PH hPH]
rfl
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J n m :
ℕ
fields
lo :
ExternalSample ε J n m → ℝ
hi :
ExternalSample ε J n m → ℝ
measurable_lo :
Measurable lo
measurable_hi :
Measurable hi
lower_bound :
∀ x, -1 ≤ lo x
ordered :
∀ x, lo x ≤ hi x
upper_bound :
∀ x, hi x ≤ 1
def contains
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
n :
ℕ
m :
ℕ
C :
ExternalIntervalProcedure ε J n m
x :
ExternalSample ε J n m
θ :
ℝ
def
contains C x θ :
Prop
given by
θ ∈ Icc (C.lo x) (C.hi x)
def length
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
n :
ℕ
m :
ℕ
C :
ExternalIntervalProcedure ε J n m
x :
ExternalSample ε J n m
def
length C x :
ℝ
given by
C.hi x - C.lo x
def externalHonestProcedures
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
{C | ∀ PH ∈ ExternalLaws g, 1 - α ≤ (externalExperiment n m PH.1 PH.2).real {x | C.contains x (ate PH.1)}}
def externalExcessRisk
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
externalExcessRisk g Qj α :
ℝ
given by
sInf {v : ℝ | ∃ C ∈ externalHonestProcedures g Qj α, v = sSup {u : ℝ | ∃ PH, ∃ hPH : PH ∈ ExternalLaws g, u = ∫ x, max 0 (C.length x - sharpATELength PH.2 g (releasedLaw PH.1) (externalLawCellMasses g PH hPH)) ∂(externalExperiment n m PH.1 PH.2)}}
def externalNormalizer
For the specified mathematical inputs, this definition introduces the corresponding object.
def externalNormalizedRisk
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
externalNormalizedRisk g Qj α :
ℝ
given by
externalNormalizer n m * externalExcessRisk g Qj α
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerBasic 4 declarations Basic interval bounds used by the lower certificate and projection risk.
Basic interval bounds used by the lower certificate and projection risk.
theorem externalATE_mem_Icc
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
ate P ∈ Icc (-1 : ℝ) 1
Proof (Lean source)
lemma externalATE_mem_Icc {ε : ℝ} {J : ℕ}
(P : Measure (FullRow ε J)) [IsProbabilityMeasure P] :
ate P ∈ Icc (-1 : ℝ) 1 := by
have h₀ : Integrable (fun ω => ((outcome0 ω : OutcomeSpace) : ℝ)) P := by
simpa using potentialOutcome_integrable P false
have h₁ : Integrable (fun ω => ((outcome1 ω : OutcomeSpace) : ℝ)) P := by
simpa using potentialOutcome_integrable P true
have h₀lo : 0 ≤ ∫ ω, ((outcome0 ω : OutcomeSpace) : ℝ) ∂P :=
integral_nonneg (fun ω => (outcome0 ω).property.1)
have h₁lo : 0 ≤ ∫ ω, ((outcome1 ω : OutcomeSpace) : ℝ) ∂P :=
integral_nonneg (fun ω => (outcome1 ω).property.1)
have h₀hi : (∫ ω, ((outcome0 ω : OutcomeSpace) : ℝ) ∂P) ≤ 1 := by
calc
_ ≤ ∫ _ω, (1 : ℝ) ∂P := integral_mono h₀ (integrable_const 1)
(fun ω => (outcome0 ω).property.2)
_ = 1 := by simp
have h₁hi : (∫ ω, ((outcome1 ω : OutcomeSpace) : ℝ) ∂P) ≤ 1 := by
calc
_ ≤ ∫ _ω, (1 : ℝ) ∂P := integral_mono h₁ (integrable_const 1)
(fun ω => (outcome1 ω).property.2)
_ = 1 := by simp
rw [ate_eq_potentialOutcomeMeans_sub]
exact ⟨by linarith, by linarith⟩
def externalFullInterval
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J n m :
ℕ
def
given by
clause 1
lo := fun _ => -1
clause 2
hi := fun _ => 1
clause 3
measurable_lo := measurable_const
clause 4
measurable_hi := measurable_const
clause 5
lower_bound := by intro; norm_num
clause 6
ordered := by intro; norm_num
clause 7
upper_bound := by intro; norm_num
theorem externalFullInterval_honest
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
hα :
0 ≤ α
Qj :
hTrial :
JointTrialIID g Qj
hLog :
ExternalLogIID g Qj
hInd :
conclusion
externalFullInterval ε J n m ∈ externalHonestProcedures g Qj α
Proof (Lean source)
lemma externalFullInterval_honest {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ) (hα : 0 ≤ α)
(Qj : Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : JointTrialIID g Qj) (hLog : ExternalLogIID g Qj)
(hInd : ExternalLogIndependent g Qj) :
externalFullInterval ε J n m ∈ externalHonestProcedures g Qj α := by
intro PH hPH
letI : IsProbabilityMeasure (Qj PH.1 PH.2) :=
externalLaw_jointExperiment_probability g Qj hTrial hLog hInd PH hPH
letI : IsProbabilityMeasure (externalExperiment n m PH.1 PH.2) := by
rw [← jointTrialLog_eq_externalExperiment g Qj hTrial hLog hInd PH hPH]
infer_instance
letI : IsProbabilityMeasure PH.1 := hPH.1
have hmem := externalATE_mem_Icc PH.1
have hset : {x : ExternalSample ε J n m |
(externalFullInterval ε J n m).contains x (ate PH.1)} = univ := by
ext x
simp [ExternalIntervalProcedure.contains, externalFullInterval, hmem]
rw [hset]
simp only [Measure.real_def, measure_univ, ENNReal.toReal_one]
linarith
theorem externalExcessIntegral_le_two
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
Qj :
hTrial :
JointTrialIID g Qj
hLog :
ExternalLogIID g Qj
hInd :
C :
ExternalIntervalProcedure ε J n m
PH :
hPH :
PH ∈ ExternalLaws g
conclusion
(∫ x, max 0 (C.length x - sharpATELength PH.2 g (releasedLaw PH.1) (externalLawCellMasses g PH hPH)) ∂Qj PH.1 PH.2)
≤ 2
Proof (Lean source)
lemma externalExcessIntegral_le_two {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J)
(Qj : Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : JointTrialIID g Qj) (hLog : ExternalLogIID g Qj)
(hInd : ExternalLogIndependent g Qj)
(C : ExternalIntervalProcedure ε J n m)
(PH : Measure (FullRow ε J) × Measure (ScoreSpace ε))
(hPH : PH ∈ ExternalLaws g) :
(∫ x, max 0 (C.length x -
sharpATELength PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH)) ∂Qj PH.1 PH.2) ≤ 2 := by
letI : IsProbabilityMeasure (Qj PH.1 PH.2) :=
externalLaw_jointExperiment_probability g Qj hTrial hLog hInd PH hPH
have hsharp : 0 ≤ sharpATELength PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH) :=
sharpATELength_nonneg_of_overlap PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH) hPH.2.2.overlap
have hpoint : ∀ x, max 0 (C.length x -
sharpATELength PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH)) ≤ 2 := by
intro x
have hlo := C.lower_bound x
have hhi := C.upper_bound x
dsimp [ExternalIntervalProcedure.length]
exact max_le (by norm_num) (by linarith)
calc
_ ≤ ∫ _x, (2 : ℝ) ∂Qj PH.1 PH.2 := by
apply integral_mono_of_nonneg
· filter_upwards [] with x
exact le_max_left _ _
· exact integrable_const 2
· filter_upwards [] with x
exact hpoint x
_ = 2 := by simp
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerCertificate 18 declarations Trial and external-log testing certificates for excess confidence length.
Trial and external-log testing certificates for excess confidence length.
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
fiberTripleSeparation g :
ℝ
given by
sSup {v : ℝ | ∃ (r : LabelSpace J) (e₁ e₂ e₃ : ScoreSpace ε), e₁ < e₂ ∧ e₂ < e₃ ∧ g e₁ = r ∧ g e₂ = r ∧ g e₃ = r ∧ v = (((e₃ : ℝ) - e₂) * (1 - (e₁ : ℝ) / e₂)) / sqrt (((e₃ : ℝ) - e₂) ^ 2 + ((e₃ : ℝ) - e₁) ^ 2 + ((e₂ : ℝ) - e₁) ^ 2)}
For the specified mathematical inputs, this definition introduces the corresponding object.
theorem ratioFilter_neBot
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma ratioFilter_neBot (ρ : ℝ) (hρ : 0 < ρ) : (ratioFilter ρ).NeBot := by
let seq : ℕ → ℕ × ℕ := fun m => (⌈ρ * (m : ℝ)⌉₊, m)
have hmul : Tendsto (fun m : ℕ => ρ * (m : ℝ)) atTop atTop :=
(tendsto_const_mul_atTop_of_pos hρ).2 tendsto_natCast_atTop_atTop
have hceil : Tendsto (fun m : ℕ => ⌈ρ * (m : ℝ)⌉₊) atTop atTop :=
tendsto_nat_ceil_atTop.comp hmul
have hpair : Tendsto seq atTop (atTop ×ˢ atTop) :=
hceil.prodMk tendsto_id
have hratio : Tendsto (fun m : ℕ =>
(⌈ρ * (m : ℝ)⌉₊ : ℝ) / (m : ℝ)) atTop (nhds ρ) :=
(tendsto_nat_ceil_mul_div_atTop hρ.le).comp tendsto_natCast_atTop_atTop
have hseq : Tendsto seq atTop (ratioFilter ρ) := by
unfold ratioFilter
apply tendsto_inf.mpr
refine ⟨hpair, ?_⟩
apply tendsto_comap_iff.mpr
exact hratio
exact (Filter.map_neBot_iff seq).2 inferInstance |>.mono hseq
For the specified mathematical inputs, this definition introduces the corresponding object.
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
logCertificate g α :
ℝ
given by
theorem fiberTriple_exists
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
∃ r : LabelSpace J, ∃ e₁ e₂ e₃ : ScoreSpace ε,
conclusion 1
e₁ < e₂
conclusion 2
e₂ < e₃
conclusion 3
g e₁ = r
conclusion 4
g e₂ = r
conclusion 5
g e₃ = r
Proof (Lean source)
lemma fiberTriple_exists {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε) :
∃ r : LabelSpace J, ∃ e₁ e₂ e₃ : ScoreSpace ε,
e₁ < e₂ ∧ e₂ < e₃ ∧ g e₁ = r ∧ g e₂ = r ∧ g e₃ = r := by
have hε : ε < 1 - ε := by rcases hOverlap with ⟨_, h⟩; linarith
let : Infinite (ScoreSpace ε) := Set.Icc.infinite hε
obtain ⟨r, hr⟩ := Finite.exists_infinite_fiber g
have hset : (g ⁻¹' {r}).Infinite := Set.infinite_coe_iff.mp hr
by_contra hn
push Not at hn
have hfinite : (g ⁻¹' {r}).Finite := by
apply Set.finite_of_forall_not_lt_lt
intro e₁ he₁ e₂ he₂ e₃ he₃ h₁₂ h₂₃
exact hn r e₁ e₂ e₃ h₁₂ h₂₃ he₁ he₂ he₃
exact hset hfinite
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma fiberTriple_ratio_bounds (x y z : ℝ)
(hx : 0 < x) (hxy : x < y) (hyz : y < z) :
0 < ((z - y) * (1 - x / y)) /
sqrt ((z - y) ^ 2 + (z - x) ^ 2 + (y - x) ^ 2) ∧
((z - y) * (1 - x / y)) /
sqrt ((z - y) ^ 2 + (z - x) ^ 2 + (y - x) ^ 2) ≤ 1 := by
have hy : 0 < y := lt_trans hx hxy
have hd : 0 < z - y := sub_pos.mpr hyz
have hfrac₀ : 0 < x / y := div_pos hx hy
have hfrac₁ : x / y < 1 := (div_lt_one hy).mpr hxy
have hsquare : 0 ≤ (z - y) ^ 2 + (z - x) ^ 2 + (y - x) ^ 2 := by positivity
have hsquare_pos : 0 < (z - y) ^ 2 + (z - x) ^ 2 + (y - x) ^ 2 := by
nlinarith [sq_nonneg (z - x), sq_nonneg (y - x), sq_pos_of_pos hd]
have hden : 0 < sqrt ((z - y) ^ 2 + (z - x) ^ 2 + (y - x) ^ 2) :=
Real.sqrt_pos.2 hsquare_pos
have hden_lower : z - y ≤
sqrt ((z - y) ^ 2 + (z - x) ^ 2 + (y - x) ^ 2) := by
have hroot := Real.sq_sqrt hsquare
nlinarith [Real.sqrt_nonneg ((z - y) ^ 2 + (z - x) ^ 2 + (y - x) ^ 2),
sq_nonneg (z - x), sq_nonneg (y - x)]
constructor
· exact div_pos (mul_pos hd (by linarith)) hden
· apply (div_le_one hden).mpr
nlinarith [mul_nonneg (le_of_lt hd) (le_of_lt hfrac₀)]
theorem fiberTripleSeparation_bounds
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
Proof (Lean source)
lemma fiberTripleSeparation_bounds {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε) :
0 < fiberTripleSeparation g ∧ fiberTripleSeparation g ≤ 1 := by
let T : Set ℝ := {v | ∃ (r : LabelSpace J) (e₁ e₂ e₃ : ScoreSpace ε),
e₁ < e₂ ∧ e₂ < e₃ ∧ g e₁ = r ∧ g e₂ = r ∧ g e₃ = r ∧
v = (((e₃ : ℝ) - e₂) * (1 - (e₁ : ℝ) / e₂)) /
sqrt (((e₃ : ℝ) - e₂) ^ 2 +
((e₃ : ℝ) - e₁) ^ 2 + ((e₂ : ℝ) - e₁) ^ 2)}
have hbound : ∀ v ∈ T, 0 < v ∧ v ≤ 1 := by
rintro v ⟨r, e₁, e₂, e₃, h₁₂, h₂₃, _, _, _, rfl⟩
exact fiberTriple_ratio_bounds _ _ _
(lt_of_lt_of_le hOverlap.1 e₁.property.1) h₁₂ h₂₃
obtain ⟨r, e₁, e₂, e₃, h₁₂, h₂₃, hr₁, hr₂, hr₃⟩ :=
fiberTriple_exists g hOverlap
let v : ℝ := (((e₃ : ℝ) - e₂) * (1 - (e₁ : ℝ) / e₂)) /
sqrt (((e₃ : ℝ) - e₂) ^ 2 +
((e₃ : ℝ) - e₁) ^ 2 + ((e₂ : ℝ) - e₁) ^ 2)
have hv : v ∈ T := ⟨r, e₁, e₂, e₃, h₁₂, h₂₃, hr₁, hr₂, hr₃, rfl⟩
have hbdd : BddAbove T := ⟨1, fun v hv => (hbound v hv).2⟩
change 0 < sSup T ∧ sSup T ≤ 1
exact ⟨lt_of_lt_of_le (hbound v hv).1 (le_csSup hbdd hv),
csSup_le ⟨v, hv⟩ (fun v hv => (hbound v hv).2)⟩
theorem fiberTriple_perturbation_identities Lemma fiberTriple_perturbation_identities in the paper ↗
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
x y z :
ℝ
hxy :
x < y
hyz :
y < z
conclusion
conclusion 1
(z - y) - (z - x) + (y - x) = 0
conclusion 2
x * (z - y) - y * (z - x) + z * (y - x) = 0
conclusion 3
0 < z - y
conclusion 4
0 < y - x
Proof (Lean source)
lemma fiberTriple_perturbation_identities (x y z : ℝ) (hxy : x < y) (hyz : y < z) :
(z - y) - (z - x) + (y - x) = 0 ∧
x * (z - y) - y * (z - x) + z * (y - x) = 0 ∧
0 < z - y ∧ 0 < y - x := by
constructor
· ring
constructor
· ring
exact ⟨sub_pos.mpr hyz, sub_pos.mpr hxy⟩
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
x y z t u :
ℝ
hy :
y ≠ 0
conclusion
((1 / 3 + u * (z - y)) + (t - x * (1 / 3 + u * (z - y))) / y)
- ((1 / 3 : ℝ) + (t - x * (1 / 3)) / y)
= u * (z - y) * (1 - x / y)
Proof (Lean source)
lemma fiberTriple_upperEndpoint_shift (x y z t u : ℝ) (hy : y ≠ 0) :
((1 / 3 + u * (z - y)) +
(t - x * (1 / 3 + u * (z - y))) / y) -
((1 / 3 : ℝ) + (t - x * (1 / 3)) / y) =
u * (z - y) * (1 - x / y) := by
field_simp
ring
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
x y z u :
ℝ
conclusion
((u * (z - y)) ^ 2 + (u * (z - x)) ^ 2 + (u * (y - x)) ^ 2) / (1 / 3 : ℝ)
= 3 * u ^ 2 * ((z - y) ^ 2 + (z - x) ^ 2 + (y - x) ^ 2)
Proof (Lean source)
lemma fiberTriple_chisq_identity (x y z u : ℝ) :
((u * (z - y)) ^ 2 + (u * (z - x)) ^ 2 +
(u * (y - x)) ^ 2) / (1 / 3 : ℝ) =
3 * u ^ 2 * ((z - y) ^ 2 + (z - x) ^ 2 + (y - x) ^ 2) := by
ring
theorem externalCertificate_constants_pos
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
Proof (Lean source)
lemma externalCertificate_constants_pos {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(hOverlap : Overlap ε) (hα : 0 < α ∧ α < 1 / 2) :
0 < trialCertificate α ∧ 0 < logCertificate g α := by
have hgap : 0 < 1 - 2 * α := by linarith [hα.2]
have hlog : 0 < log (1 + (1 - 2 * α) ^ 2) := by
apply Real.log_pos
nlinarith [sq_pos_of_pos hgap]
have hroot : 0 < sqrt (log (1 + (1 - 2 * α) ^ 2)) :=
Real.sqrt_pos.2 hlog
have hsep := (fiberTripleSeparation_bounds g hOverlap).1
constructor
· unfold trialCertificate
positivity
· unfold logCertificate
positivity
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
≤ externalNormalizer n m * R
Proof (Lean source)
lemma normalizedCertificate_trial (n m : ℕ) (hn : 0 < n) (hm : 0 < m)
(c R : ℝ) (hR : c * (sqrt (n : ℝ))⁻¹ ≤ R) :
c * sqrt (m : ℝ) /
(sqrt (n : ℝ) + sqrt (m : ℝ)) ≤
externalNormalizer n m * R := by
have hn' : 0 < sqrt (n : ℝ) := Real.sqrt_pos.2 (Nat.cast_pos.mpr hn)
have hm' : 0 < sqrt (m : ℝ) := Real.sqrt_pos.2 (Nat.cast_pos.mpr hm)
have hnorm : 0 ≤ externalNormalizer n m := by
unfold externalNormalizer
positivity
have hid : externalNormalizer n m * (c * (sqrt (n : ℝ))⁻¹) =
c * sqrt (m : ℝ) /
(sqrt (n : ℝ) + sqrt (m : ℝ)) := by
unfold externalNormalizer
field_simp
ring
rw [← hid]
exact mul_le_mul_of_nonneg_left hR hnorm
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
≤ externalNormalizer n m * R
Proof (Lean source)
lemma normalizedCertificate_log (n m : ℕ) (hn : 0 < n) (hm : 0 < m)
(c R : ℝ) (hR : c * (sqrt (m : ℝ))⁻¹ ≤ R) :
c * sqrt (n : ℝ) /
(sqrt (n : ℝ) + sqrt (m : ℝ)) ≤
externalNormalizer n m * R := by
have hn' : 0 < sqrt (n : ℝ) := Real.sqrt_pos.2 (Nat.cast_pos.mpr hn)
have hm' : 0 < sqrt (m : ℝ) := Real.sqrt_pos.2 (Nat.cast_pos.mpr hm)
have hnorm : 0 ≤ externalNormalizer n m := by
unfold externalNormalizer
positivity
have hid : externalNormalizer n m * (c * (sqrt (m : ℝ))⁻¹) =
c * sqrt (n : ℝ) /
(sqrt (n : ℝ) + sqrt (m : ℝ)) := by
unfold externalNormalizer
field_simp
ring
rw [← hid]
exact mul_le_mul_of_nonneg_left hR hnorm
theorem ratioFilter_eventually_pos
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma ratioFilter_eventually_pos (ρ : ℝ) :
∀ᶠ nm : ℕ × ℕ in ratioFilter ρ, 0 < nm.1 ∧ 0 < nm.2 := by
apply Filter.Eventually.filter_mono (by unfold ratioFilter; exact inf_le_left)
exact (eventually_gt_atTop (0 : ℕ)).prod_mk (eventually_gt_atTop (0 : ℕ))
theorem ratioFilter_ratio_tendsto
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ρ :
ℝ
conclusion
Proof (Lean source)
lemma ratioFilter_ratio_tendsto (ρ : ℝ) :
Tendsto (fun nm : ℕ × ℕ => (nm.1 : ℝ) / (nm.2 : ℝ))
(ratioFilter ρ) (nhds ρ) := by
exact (tendsto_iff_comap).2 (by
unfold ratioFilter
exact inf_le_right)
theorem normalizedCertificate_ratio_trial
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma normalizedCertificate_ratio_trial (ρ : ℝ) (hρ : 0 < ρ) :
Tendsto (fun nm : ℕ × ℕ =>
sqrt (nm.2 : ℝ) /
(sqrt (nm.1 : ℝ) + sqrt (nm.2 : ℝ)))
(ratioFilter ρ) (nhds (1 / (1 + sqrt ρ))) := by
have hr := ratioFilter_ratio_tendsto ρ
have hcomp : Tendsto (fun nm : ℕ × ℕ =>
1 / (1 + sqrt ((nm.1 : ℝ) / (nm.2 : ℝ))))
(ratioFilter ρ) (nhds (1 / (1 + sqrt ρ))) := by
have hcont : ContinuousAt (fun x : ℝ => 1 / (1 + sqrt x)) ρ := by
have hden : 1 + sqrt ρ ≠ 0 := ne_of_gt (by positivity)
fun_prop (disch := assumption)
exact hcont.tendsto.comp hr
apply hcomp.congr'
filter_upwards [hr.eventually (eventually_gt_nhds hρ)] with nm hratio
have hm : 0 < nm.2 := by
by_contra h
have hzero : nm.2 = 0 := by omega
simp [hzero] at hratio
have hsm : 0 < sqrt (nm.2 : ℝ) :=
Real.sqrt_pos.2 (by exact_mod_cast hm)
rw [Real.sqrt_div (by positivity : 0 ≤ (nm.1 : ℝ)) (nm.2 : ℝ)]
field_simp
ring
theorem normalizedCertificate_ratio_log
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma normalizedCertificate_ratio_log (ρ : ℝ) (hρ : 0 < ρ) :
Tendsto (fun nm : ℕ × ℕ =>
sqrt (nm.1 : ℝ) /
(sqrt (nm.1 : ℝ) + sqrt (nm.2 : ℝ)))
(ratioFilter ρ)
(nhds (sqrt ρ / (1 + sqrt ρ))) := by
have hr := ratioFilter_ratio_tendsto ρ
have hcomp : Tendsto (fun nm : ℕ × ℕ =>
sqrt ((nm.1 : ℝ) / (nm.2 : ℝ)) /
(1 + sqrt ((nm.1 : ℝ) / (nm.2 : ℝ))))
(ratioFilter ρ)
(nhds (sqrt ρ / (1 + sqrt ρ))) := by
have hcont : ContinuousAt (fun x : ℝ =>
sqrt x / (1 + sqrt x)) ρ := by
have hden : 1 + sqrt ρ ≠ 0 := ne_of_gt (by positivity)
fun_prop (disch := assumption)
exact hcont.tendsto.comp hr
apply hcomp.congr'
filter_upwards [hr.eventually (eventually_gt_nhds hρ)] with nm hratio
have hm : 0 < nm.2 := by
by_contra h
have hzero : nm.2 = 0 := by omega
simp [hzero] at hratio
have hsm : 0 < sqrt (nm.2 : ℝ) :=
Real.sqrt_pos.2 (by exact_mod_cast hm)
rw [Real.sqrt_div (by positivity : 0 ≤ (nm.1 : ℝ)) (nm.2 : ℝ)]
field_simp
ring
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerCertificateAssembly 2 declarations Assembly of the two-source lower certificate.
Assembly of the two-source lower certificate.
theorem externalExcessRisk_normalized_upper
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
hOverlap :
Overlap ε
hg :
hα :
0 < α ∧ α < 1 / 2
Qj :
∀ n m
hTrial :
∀ (n m : ℕ), JointTrialIID g (Qj n m)
hLog :
∀ (n m : ℕ), ExternalLogIID g (Qj n m)
hInd :
∀ (n m : ℕ), ExternalLogIndependent g (Qj n m)
conclusion
∃ C : ℝ,
∀ n m : ℕ, 0 < n → 0 < m → externalNormalizer n m * externalExcessRisk g (Qj n m) α ≤ C
Proof (Lean source)
lemma externalExcessRisk_normalized_upper {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(hOverlap : Overlap ε) (hg : Measurable g)
(hα : 0 < α ∧ α < 1 / 2)
(Qj : ∀ n m, Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : ∀ (n m : ℕ), JointTrialIID g (Qj n m))
(hLog : ∀ (n m : ℕ), ExternalLogIID g (Qj n m))
(hInd : ∀ (n m : ℕ), ExternalLogIndependent g (Qj n m)) :
∃ C : ℝ, ∀ n m : ℕ, 0 < n → 0 < m →
externalNormalizer n m * externalExcessRisk g (Qj n m) α ≤ C := by
let L : ℝ := 4 * (ε⁻¹ + (ε ^ 2)⁻¹)
let A : ℝ := L * (4 * J / α + 2 * J)
let B : ℝ := L * (2 * J * (2 - 2 * ε) / α + J * (2 - 2 * ε))
let C : ℝ := A + B + 1
have hε : 0 < ε := hOverlap.1
have hαpos : 0 < α := hα.1
have hw : 0 < 2 - 2 * ε := by linarith [hOverlap.2]
have hL : 0 ≤ L := by dsimp [L]; positivity
have hA : 0 ≤ A := by dsimp [A]; exact mul_nonneg hL (by positivity)
have hB : 0 ≤ B := by dsimp [B]; exact mul_nonneg hL (by positivity)
have hC : 0 ≤ C := by dsimp [C]; linarith
refine ⟨C, ?_⟩
intro n m hn hm
have hscale :
4 * (ε⁻¹ + (ε ^ 2)⁻¹) *
(4 * J / (α * sqrt n) + 2 * J / sqrt n +
2 * J * (2 - 2 * ε) / (α * sqrt m) +
J * (2 - 2 * ε) / sqrt m) ≤
C * ((sqrt (n : ℝ))⁻¹ + (sqrt (m : ℝ))⁻¹) := by
have hAn : A ≤ C := by dsimp [C]; linarith
have hBm : B ≤ C := by dsimp [C]; linarith
calc
_ = A * (sqrt (n : ℝ))⁻¹ +
B * (sqrt (m : ℝ))⁻¹ := by
dsimp [A, B, L]
simp only [div_eq_mul_inv, mul_inv_rev]
ring
_ ≤ C * (sqrt (n : ℝ))⁻¹ +
C * (sqrt (m : ℝ))⁻¹ :=
add_le_add (mul_le_mul_of_nonneg_right hAn (by positivity))
(mul_le_mul_of_nonneg_right hBm (by positivity))
_ = _ := by ring
let CI : ExternalIntervalProcedure ε J n m :=
externalProjectionIntervalProcedure g α hOverlap hg
have hCI : CI ∈ externalHonestProcedures g (Qj n m) α :=
externalProjectionIntervalProcedure_honest g α hOverlap hg hn hm hα.1
(Qj n m) (hTrial n m) (hLog n m) (hInd n m)
have hExp : ∀ PH, ∀ hPH : PH ∈ ExternalLaws g,
(∫ x, max 0 (CI.length x -
sharpATELength PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH)) ∂(Qj n m PH.1 PH.2)) ≤
C * ((sqrt (n : ℝ))⁻¹ + (sqrt (m : ℝ))⁻¹) := by
intro PH hPH
exact (externalProjectionIntervalProcedure_population_meanExcessLength_le
g (Qj n m) (hTrial n m) (hLog n m) (hInd n m)
PH hPH hn hm α hα.1).trans hscale
have hRisk := externalExcessRisk_le_of_honest_uniform_bound g α hOverlap hg
(Qj n m) (hTrial n m) (hLog n m) (hInd n m) CI hCI _
(by positivity) hExp
exact (externalNormalizedRisk_le_of_rootSum_bound g (Qj n m) α C hn hm hRisk)
theorem externalExcessRisk_lower_certificates
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
hOverlap :
Overlap ε
hg :
hα :
0 < α ∧ α < 1 / 2
Qj :
∀ n m
hTrial :
∀ (n m : ℕ), JointTrialIID g (Qj n m)
hLog :
∀ (n m : ℕ), ExternalLogIID g (Qj n m)
hInd :
∀ (n m : ℕ), ExternalLogIndependent g (Qj n m)
conclusion
conclusion 1
0 < fiberTripleSeparation g
fiberTripleSeparation g ≤ 1
conclusion 2
n m :
ℕ
0 < n
0 < m
conclusion 3
logCertificate g α
≤ liminf (fun m : ℕ => sInf {v : ℝ | ∃ n : ℕ, 0 < n ∧ v = sqrt (m : ℝ) * externalExcessRisk g (Qj n m) α}) atTop
0 < logCertificate g α
conclusion 4
ρ :
ℝ
0 < ρ
Proof (Lean source)
lemma externalExcessRisk_lower_certificates {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(hOverlap : Overlap ε) (hg : Measurable g)
(hα : 0 < α ∧ α < 1 / 2)
(Qj : ∀ n m, Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : ∀ (n m : ℕ), JointTrialIID g (Qj n m))
(hLog : ∀ (n m : ℕ), ExternalLogIID g (Qj n m))
(hInd : ∀ (n m : ℕ), ExternalLogIndependent g (Qj n m)) :
(0 < fiberTripleSeparation g ∧ fiberTripleSeparation g ≤ 1) ∧
(∀ (n m : ℕ), 0 < n → 0 < m →
-- @realizes n(positive trial size) @realizes m(positive log size)
trialCertificate α * (sqrt (n : ℝ))⁻¹ ≤
externalExcessRisk g (Qj n m) α) ∧
(logCertificate g α ≤
liminf (fun m : ℕ => sInf {v : ℝ | ∃ n : ℕ, 0 < n ∧
v = sqrt (m : ℝ) * externalExcessRisk g (Qj n m) α}) atTop ∧
0 < logCertificate g α) ∧
(∀ ρ : ℝ, 0 < ρ →
-- @realizes rho(positive asymptotic ratio)
max (trialCertificate α / (1 + sqrt ρ))
(logCertificate g α * sqrt ρ / (1 + sqrt ρ)) ≤
liminf (fun nm : ℕ × ℕ =>
externalNormalizedRisk g (Qj nm.1 nm.2) α) (ratioFilter ρ) ∧
0 < max (trialCertificate α / (1 + sqrt ρ))
(logCertificate g α * sqrt ρ / (1 + sqrt ρ))) := by
have hconstants := externalCertificate_constants_pos g α hOverlap hα
have hLogCert : logCertificate g α ≤
liminf (fun m : ℕ => sInf {v : ℝ | ∃ n : ℕ, 0 < n ∧
v = sqrt (m : ℝ) * externalExcessRisk g (Qj n m) α}) atTop := by
obtain ⟨C, hUpper⟩ := externalExcessRisk_normalized_upper g α
hOverlap hg hα Qj hTrial hLog hInd
exact scoreLog_logCertificate_liminf g hOverlap hg α hα Qj
hTrial hLog hInd C hUpper
have hTrialCert : ∀ (n m : ℕ), 0 < n → 0 < m →
trialCertificate α * (sqrt (n : ℝ))⁻¹ ≤
externalExcessRisk g (Qj n m) α := by
intro n m hn hm
let H : Measure (ScoreSpace ε) := Measure.dirac (trialCenterScore hOverlap)
let P : Measure (FullRow ε J) := trialCenterFullLaw g hOverlap (1 / 2)
have hPH : (P, H) ∈ ExternalLaws g :=
⟨trialCenterFullLaw_isProbabilityMeasure g hOverlap _ (by norm_num) (by norm_num),
inferInstance,
trialCenterFullLaw_externalScoreLaw g hOverlap hg _ (by norm_num) (by norm_num)⟩
have hbase (C : ExternalIntervalProcedure ε J n m)
(hC : C ∈ externalHonestProcedures g (Qj n m) α) :
trialCertificate α * (sqrt (n : ℝ))⁻¹ ≤
sSup {u : ℝ | ∃ PH, ∃ hPH : PH ∈ ExternalLaws g,
u = ∫ x, max 0 (C.length x -
sharpATELength PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH))
∂(externalExperiment n m PH.1 PH.2)} := by
let U : Set ℝ := {u : ℝ | ∃ PH, ∃ hPH : PH ∈ ExternalLaws g,
u = ∫ x, max 0 (C.length x -
sharpATELength PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH))
∂(externalExperiment n m PH.1 PH.2)}
have hUbd : BddAbove U := by
refine ⟨2, ?_⟩
rintro u ⟨PH, hPH', rfl⟩
simpa only [jointTrialLog_eq_externalExperiment g (Qj n m)
(hTrial n m) (hLog n m) (hInd n m) PH hPH'] using
externalExcessIntegral_le_two g (Qj n m) (hTrial n m)
(hLog n m) (hInd n m) C PH hPH'
have hzero : sharpATELength H g (releasedLaw P)
(externalLawCellMasses g (P, H) hPH) = 0 :=
trialCenter_sharpATELength_eq_zero g hOverlap (releasedLaw P)
(externalLawCellMasses g (P, H) hPH)
have hcenter : trialCertificate α * (sqrt (n : ℝ))⁻¹ ≤
∫ x, max 0 (C.length x -
sharpATELength H g (releasedLaw P)
(externalLawCellMasses g (P, H) hPH))
∂(externalExperiment n m P H) := by
have hlen := trialCenter_honest_expectedLength g hOverlap hg α hα hn
(Qj n m) (hTrial n m) (hLog n m) (hInd n m) C hC
change trialCertificate α * (sqrt (n : ℝ))⁻¹ ≤
∫ x, C.length x ∂(Qj n m P H) at hlen
rw [← jointTrialLog_eq_externalExperiment g (Qj n m)
(hTrial n m) (hLog n m) (hInd n m) (P, H) hPH]
convert hlen using 1
apply integral_congr_ae
filter_upwards [] with x
rw [hzero, sub_zero]
exact max_eq_right (sub_nonneg.mpr (C.ordered x))
exact hcenter.trans (le_csSup hUbd ⟨(P, H), hPH, rfl⟩)
unfold externalExcessRisk
apply le_csInf
· refine ⟨sSup {u : ℝ | ∃ PH, ∃ hPH : PH ∈ ExternalLaws g,
u = ∫ x, max 0 ((externalFullInterval ε J n m).length x -
sharpATELength PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH))
∂(externalExperiment n m PH.1 PH.2)}, ?_⟩
exact ⟨externalFullInterval ε J n m,
externalFullInterval_honest g α hα.1.le (Qj n m)
(hTrial n m) (hLog n m) (hInd n m), rfl⟩
· rintro v ⟨C, hC, rfl⟩
exact hbase C hC
refine ⟨fiberTripleSeparation_bounds g hOverlap, hTrialCert, ?_, ?_⟩
· exact ⟨hLogCert, hconstants.2⟩
· intro ρ hρ
refine ⟨?_, ?_⟩
· obtain ⟨C, hUpper⟩ := externalExcessRisk_normalized_upper g α
hOverlap hg hα Qj hTrial hLog hInd
let R : ℕ → ℕ → ℝ := fun n m => externalExcessRisk g (Qj n m) α
have hTrialRatio : trialCertificate α / (1 + sqrt ρ) ≤
liminf (fun nm : ℕ × ℕ => externalNormalizedRisk g (Qj nm.1 nm.2) α)
(ratioFilter ρ) := by
simpa only [R, externalNormalizedRisk] using
trialCertificate_ratio_liminf_of_upperBound ρ hρ (trialCertificate α) C R
(fun n m hn hm => by simpa only [R] using hTrialCert n m hn hm)
(fun n m hn hm => hUpper n m hn hm)
have hLogRatio : logCertificate g α * sqrt ρ / (1 + sqrt ρ) ≤
liminf (fun nm : ℕ × ℕ => externalNormalizedRisk g (Qj nm.1 nm.2) α)
(ratioFilter ρ) := by
have hNonneg : ∀ n m : ℕ, 0 < n → 0 < m → 0 ≤ R n m := by
intro n m hn hm
have hfactor : 0 ≤ (sqrt (n : ℝ))⁻¹ := by positivity
have hzero : 0 ≤ trialCertificate α * (sqrt (n : ℝ))⁻¹ :=
mul_nonneg hconstants.1.le hfactor
exact hzero.trans (by simpa only [R] using hTrialCert n m hn hm)
simpa only [R, externalNormalizedRisk] using
logCertificate_ratio_liminf_of_infimum ρ hρ (logCertificate g α) C R
hNonneg (by simpa only [R] using hLogCert) hUpper
exact max_le hTrialRatio hLogRatio
have hden : 0 < 1 + sqrt ρ := by positivity
have hfirst : 0 < trialCertificate α / (1 + sqrt ρ) :=
div_pos hconstants.1 hden
exact lt_of_lt_of_le hfirst (le_max_left _ _)
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerCertificateTransfer 3 declarations Transfer of pointwise two-sample certificates to a ratio-filter lower bound.
Transfer of pointwise two-sample certificates to a ratio-filter lower bound.
theorem trialCertificate_ratio_liminf_of_upperBound
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ρ :
ℝ
hρ :
0 < ρ
c C :
ℝ
R :
ℕ → ℕ → ℝ
hTrial :
∀ (n m : ℕ)
hUpper :
∀ (n m : ℕ)
conclusion
c / (1 + sqrt ρ)
Proof (Lean source)
lemma trialCertificate_ratio_liminf_of_upperBound
(ρ : ℝ) (hρ : 0 < ρ) (c C : ℝ) (R : ℕ → ℕ → ℝ)
(hTrial : ∀ (n m : ℕ), 0 < n → 0 < m →
c * (sqrt (n : ℝ))⁻¹ ≤ R n m)
(hUpper : ∀ (n m : ℕ), 0 < n → 0 < m →
externalNormalizer n m * R n m ≤ C) :
c / (1 + sqrt ρ) ≤
liminf (fun nm : ℕ × ℕ => externalNormalizer nm.1 nm.2 * R nm.1 nm.2)
(ratioFilter ρ) := by
letI : (ratioFilter ρ).NeBot := ratioFilter_neBot ρ hρ
let u : ℕ × ℕ → ℝ := fun nm =>
externalNormalizer nm.1 nm.2 * R nm.1 nm.2
let v : ℕ × ℕ → ℝ := fun nm =>
c * (sqrt (nm.2 : ℝ) /
(sqrt (nm.1 : ℝ) + sqrt (nm.2 : ℝ)))
have hvlim : Tendsto v (ratioFilter ρ)
(nhds (c / (1 + sqrt ρ))) := by
have hconst : Tendsto (fun _ : ℕ × ℕ => c) (ratioFilter ρ) (nhds c) :=
tendsto_const_nhds
have h := hconst.mul (normalizedCertificate_ratio_trial ρ hρ)
simpa [v, div_eq_mul_inv] using h
have hle : ∀ᶠ nm : ℕ × ℕ in ratioFilter ρ, v nm ≤ u nm := by
filter_upwards [ratioFilter_eventually_pos ρ] with nm hnm
simpa [v, u, mul_div_assoc] using
normalizedCertificate_trial nm.1 nm.2 hnm.1 hnm.2 c (R nm.1 nm.2)
(hTrial nm.1 nm.2 hnm.1 hnm.2)
have hub : ∀ᶠ nm : ℕ × ℕ in ratioFilter ρ, u nm ≤ C := by
filter_upwards [ratioFilter_eventually_pos ρ] with nm hnm
exact hUpper nm.1 nm.2 hnm.1 hnm.2
have hvu : (ratioFilter ρ).IsBoundedUnder (· ≥ ·) v := hvlim.isBoundedUnder_ge
have huu : (ratioFilter ρ).IsCoboundedUnder (· ≥ ·) u :=
(isBoundedUnder_of_eventually_le hub).isCoboundedUnder_ge
calc
c / (1 + sqrt ρ) = liminf v (ratioFilter ρ) := hvlim.liminf_eq.symm
_ ≤ liminf u (ratioFilter ρ) := liminf_le_liminf hle hvu huu
_ = liminf (fun nm : ℕ × ℕ =>
externalNormalizer nm.1 nm.2 * R nm.1 nm.2) (ratioFilter ρ) := rfl
theorem logCertificate_ratio_liminf_of_upperBound
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ρ :
ℝ
hρ :
0 < ρ
c C :
ℝ
R :
ℕ → ℕ → ℝ
hUpper :
∀ (n m : ℕ)
conclusion
Proof (Lean source)
lemma logCertificate_ratio_liminf_of_upperBound
(ρ : ℝ) (hρ : 0 < ρ) (c C : ℝ) (R : ℕ → ℕ → ℝ)
(hLog : ∀ᶠ m : ℕ in atTop, ∀ n : ℕ, 0 < n →
c * (sqrt (m : ℝ))⁻¹ ≤ R n m)
(hUpper : ∀ (n m : ℕ), 0 < n → 0 < m →
externalNormalizer n m * R n m ≤ C) :
c * sqrt ρ / (1 + sqrt ρ) ≤
liminf (fun nm : ℕ × ℕ => externalNormalizer nm.1 nm.2 * R nm.1 nm.2)
(ratioFilter ρ) := by
letI : (ratioFilter ρ).NeBot := ratioFilter_neBot ρ hρ
let u : ℕ × ℕ → ℝ := fun nm =>
externalNormalizer nm.1 nm.2 * R nm.1 nm.2
let v : ℕ × ℕ → ℝ := fun nm =>
c * (sqrt (nm.1 : ℝ) /
(sqrt (nm.1 : ℝ) + sqrt (nm.2 : ℝ)))
have hvlim : Tendsto v (ratioFilter ρ)
(nhds (c * sqrt ρ / (1 + sqrt ρ))) := by
have hconst : Tendsto (fun _ : ℕ × ℕ => c) (ratioFilter ρ) (nhds c) :=
tendsto_const_nhds
simpa only [v, mul_div_assoc] using
hconst.mul (normalizedCertificate_ratio_log ρ hρ)
have hlog' : ∀ᶠ nm : ℕ × ℕ in ratioFilter ρ,
∀ n : ℕ, 0 < n → c * (sqrt (nm.2 : ℝ))⁻¹ ≤ R n nm.2 := by
apply Filter.Eventually.filter_mono (by unfold ratioFilter; exact inf_le_left)
have hfirst : ∀ᶠ _ : ℕ in atTop, True :=
Filter.Eventually.of_forall (fun _ => True.intro)
exact (hfirst.prod_mk hLog).mono (fun _ h => h.2)
have hle : ∀ᶠ nm : ℕ × ℕ in ratioFilter ρ, v nm ≤ u nm := by
filter_upwards [ratioFilter_eventually_pos ρ, hlog'] with nm hnm hlognm
simpa [v, u, mul_div_assoc] using
normalizedCertificate_log nm.1 nm.2 hnm.1 hnm.2 c (R nm.1 nm.2)
(hlognm nm.1 hnm.1)
have hub : ∀ᶠ nm : ℕ × ℕ in ratioFilter ρ, u nm ≤ C := by
filter_upwards [ratioFilter_eventually_pos ρ] with nm hnm
exact hUpper nm.1 nm.2 hnm.1 hnm.2
have hvu : (ratioFilter ρ).IsBoundedUnder (· ≥ ·) v := hvlim.isBoundedUnder_ge
have huu : (ratioFilter ρ).IsCoboundedUnder (· ≥ ·) u :=
(isBoundedUnder_of_eventually_le hub).isCoboundedUnder_ge
calc
c * sqrt ρ / (1 + sqrt ρ) = liminf v (ratioFilter ρ) := hvlim.liminf_eq.symm
_ ≤ liminf u (ratioFilter ρ) := liminf_le_liminf hle hvu huu
_ = liminf (fun nm : ℕ × ℕ =>
externalNormalizer nm.1 nm.2 * R nm.1 nm.2) (ratioFilter ρ) := rfl
theorem logCertificate_ratio_liminf_of_infimum
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ρ :
ℝ
hρ :
0 < ρ
c C :
ℝ
R :
ℕ → ℕ → ℝ
hNonneg :
∀ n m : ℕ
if
0 < n
and
0 < m
then
0 ≤ R n m
hUpper :
∀ n m : ℕ
conclusion
Proof (Lean source)
lemma logCertificate_ratio_liminf_of_infimum
(ρ : ℝ) (hρ : 0 < ρ) (c C : ℝ) (R : ℕ → ℕ → ℝ)
(hNonneg : ∀ n m : ℕ, 0 < n → 0 < m → 0 ≤ R n m)
(hLog : c ≤ liminf (fun m : ℕ =>
sInf {v : ℝ | ∃ n : ℕ, 0 < n ∧ v = sqrt (m : ℝ) * R n m}) atTop)
(hUpper : ∀ n m : ℕ, 0 < n → 0 < m →
externalNormalizer n m * R n m ≤ C) :
c * sqrt ρ / (1 + sqrt ρ) ≤
liminf (fun nm : ℕ × ℕ => externalNormalizer nm.1 nm.2 * R nm.1 nm.2)
(ratioFilter ρ) := by
let F : ℕ → ℝ := fun m =>
sInf {v : ℝ | ∃ n : ℕ, 0 < n ∧ v = sqrt (m : ℝ) * R n m}
have hFnonneg : ∀ᶠ m : ℕ in atTop, 0 ≤ F m := by
filter_upwards [eventually_gt_atTop (0 : ℕ)] with m hm
apply le_csInf
· exact ⟨Real.sqrt (m : ℝ) * R 1 m, 1, by omega, rfl⟩
· rintro v ⟨n, hn, rfl⟩
exact mul_nonneg (Real.sqrt_nonneg _) (hNonneg n m hn hm)
have hFbelow : atTop.IsBoundedUnder (· ≥ ·) F :=
isBoundedUnder_of_eventually_ge hFnonneg
have hweight : 0 < sqrt ρ / (1 + sqrt ρ) := by positivity
apply le_of_forall_lt
intro x hx
have hxdiv : x / (sqrt ρ / (1 + sqrt ρ)) < c :=
(div_lt_iff₀ hweight).mpr (by simpa only [mul_div_assoc] using hx)
let β : ℝ := (x / (sqrt ρ / (1 + sqrt ρ)) + c) / 2
have hβ : β < c := by dsimp [β]; linarith
have hxβ : x < β * sqrt ρ / (1 + sqrt ρ) := by
have h : x / (sqrt ρ / (1 + sqrt ρ)) < β := by
dsimp [β]
linarith
simpa only [mul_div_assoc] using (div_lt_iff₀ hweight).mp h
have hFevent : ∀ᶠ m : ℕ in atTop, β < F m :=
eventually_lt_of_lt_liminf (lt_of_lt_of_le hβ hLog) hFbelow
have hEvent : ∀ᶠ m : ℕ in atTop, ∀ n : ℕ, 0 < n →
β * (sqrt (m : ℝ))⁻¹ ≤ R n m := by
filter_upwards [hFevent, eventually_gt_atTop (0 : ℕ)] with m hβF hm
intro n hn
have hSbd : BddBelow {v : ℝ | ∃ n' : ℕ, 0 < n' ∧
v = sqrt (m : ℝ) * R n' m} := by
refine ⟨0, ?_⟩
rintro v ⟨n', hn', rfl⟩
exact mul_nonneg (Real.sqrt_nonneg _) (hNonneg n' m hn' hm)
have hFle : F m ≤ sqrt (m : ℝ) * R n m :=
csInf_le hSbd ⟨n, hn, rfl⟩
have hsm : 0 < sqrt (m : ℝ) :=
Real.sqrt_pos.2 (Nat.cast_pos.mpr hm)
have hprod : β ≤ R n m * sqrt (m : ℝ) := by
simpa only [mul_comm] using (le_trans hβF.le hFle)
simpa only [div_eq_mul_inv] using (div_le_iff₀ hsm).mpr hprod
have hTransfer := logCertificate_ratio_liminf_of_upperBound ρ hρ β C R
hEvent hUpper
exact lt_of_lt_of_le hxβ hTransfer
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerScoreLogArmCell 5 declarations Arm-cell normalization leaves for the score-log construction.
Arm-cell normalization leaves for the score-log construction.
def scoreLogReleasedCellLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
theorem scoreLogReleasedCellLaw_isProbabilityMeasure
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
J :
ℕ
r :
q t c :
ℝ
hq0 :
0 ≤ q
hc0 :
0 ≤ c
hs0 :
0 ≤ t / q
hs1 :
t / q ≤ 1
hsum :
q + c = 1
conclusion
IsProbabilityMeasure (scoreLogReleasedCellLaw r q t c)
Proof (Lean source)
lemma scoreLogReleasedCellLaw_isProbabilityMeasure {J : ℕ}
(r : LabelSpace J) (q t c : ℝ)
(hq0 : 0 ≤ q) (hc0 : 0 ≤ c) (hs0 : 0 ≤ t / q) (hs1 : t / q ≤ 1)
(hsum : q + c = 1) :
IsProbabilityMeasure (scoreLogReleasedCellLaw r q t c) := by
letI := trialOutcomeLaw_isProbabilityMeasure (t / q) hs0 hs1
rw [isProbabilityMeasure_iff]
unfold scoreLogReleasedCellLaw
rw [Measure.add_apply, Measure.smul_apply, Measure.smul_apply,
Measure.map_apply (by fun_prop) MeasurableSet.univ]
simp only [Set.preimage_univ, measure_univ, Measure.dirac_apply, Set.mem_univ,
Set.indicator_of_mem, Pi.one_apply, smul_eq_mul, mul_one]
rw [← ENNReal.ofReal_add hq0 hc0, hsum]
norm_num
theorem scoreLogReleasedCellLaw_treated_mass
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma scoreLogReleasedCellLaw_treated_mass {J : ℕ}
(r : LabelSpace J) (q t c : ℝ)
(hq0 : 0 ≤ q) (hs0 : 0 ≤ t / q) (hs1 : t / q ≤ 1) :
(scoreLogReleasedCellLaw r q t c).real
{z | z.1 = r ∧ z.2.1 = true} = q := by
letI := trialOutcomeLaw_isProbabilityMeasure (t / q) hs0 hs1
unfold scoreLogReleasedCellLaw
rw [Measure.real_def, Measure.add_apply, Measure.smul_apply, Measure.smul_apply,
Measure.map_apply (by fun_prop)]
· simp [hq0, trialZeroOutcome]
· exact (measurableSet_eq_fun measurable_fst measurable_const).inter
(measurableSet_eq_fun (measurable_fst.comp measurable_snd) measurable_const)
theorem scoreLogReleasedCellLaw_treated_mass_ennreal
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma scoreLogReleasedCellLaw_treated_mass_ennreal {J : ℕ}
(r : LabelSpace J) (q t c : ℝ)
(hs0 : 0 ≤ t / q) (hs1 : t / q ≤ 1) :
scoreLogReleasedCellLaw r q t c {z | z.1 = r ∧ z.2.1 = true} =
ofReal q := by
letI := trialOutcomeLaw_isProbabilityMeasure (t / q) hs0 hs1
unfold scoreLogReleasedCellLaw
rw [Measure.add_apply, Measure.smul_apply, Measure.smul_apply,
Measure.map_apply (by fun_prop)]
· simp [trialZeroOutcome]
· exact (measurableSet_eq_fun measurable_fst measurable_const).inter
(measurableSet_eq_fun (measurable_fst.comp measurable_snd) measurable_const)
theorem scoreLogReleasedCellLaw_armCellOutcomeLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
J :
ℕ
r :
q t c :
ℝ
hq :
0 < q
hc0 :
0 ≤ c
hs0 :
0 ≤ t / q
hs1 :
t / q ≤ 1
hsum :
q + c = 1
conclusion
Prel :
scoreLogReleasedCellLaw r q t c
hPrel :
scoreLogReleasedCellLaw_isProbabilityMeasure r q t c hq.le hc0 hs0 hs1 hsum (: IsProbabilityMeasure Prel)
armCellOutcomeLaw Prel hPrel true r (by simpa [Prel, scoreLogReleasedCellLaw_treated_mass r q t c hq.le hs0 hs1] using hq)
= threeScoreOutcomeLaw q t
Proof (Lean source)
lemma scoreLogReleasedCellLaw_armCellOutcomeLaw {J : ℕ}
(r : LabelSpace J) (q t c : ℝ)
(hq : 0 < q) (hc0 : 0 ≤ c) (hs0 : 0 ≤ t / q) (hs1 : t / q ≤ 1)
(hsum : q + c = 1) :
let Prel := scoreLogReleasedCellLaw r q t c
let hPrel : IsProbabilityMeasure Prel :=
scoreLogReleasedCellLaw_isProbabilityMeasure r q t c hq.le hc0 hs0 hs1 hsum
armCellOutcomeLaw Prel hPrel true r
(by simpa [Prel, scoreLogReleasedCellLaw_treated_mass r q t c hq.le hs0 hs1]
using hq) =
threeScoreOutcomeLaw q t := by
dsimp only
letI : IsProbabilityMeasure (scoreLogReleasedCellLaw r q t c) :=
scoreLogReleasedCellLaw_isProbabilityMeasure r q t c hq.le hc0 hs0 hs1 hsum
apply Measure.ext fun B hB => ?_
rw [armCellOutcomeLaw_apply]
rw [scoreLogReleasedCellLaw_treated_mass_ennreal r q t c hs0 hs1]
unfold scoreLogReleasedCellLaw threeScoreOutcomeLaw trialOutcomeLaw
rw [Measure.add_apply, Measure.smul_apply, Measure.smul_apply]
rw [Measure.map_apply (by fun_prop)]
· rw [Measure.add_apply, Measure.add_apply,
Measure.smul_apply, Measure.smul_apply, Measure.smul_apply,
Measure.smul_apply]
simp only [Measure.dirac_apply,
Set.indicator_of_mem, smul_eq_mul]
simp [trialOneOutcome, trialZeroOutcome]
have hqE : ofReal q ≠ 0 := by positivity
have hqT : ofReal q ≠ ⊤ := ENNReal.ofReal_ne_top
rw [← mul_assoc, ENNReal.inv_mul_cancel hqE hqT, one_mul]
by_cases h1 : (1 : ℝ) ∈ B <;> by_cases h0 : (0 : ℝ) ∈ B <;>
simp [indicator, h1, h0, trialOneOutcome, trialZeroOutcome]
· exact (measurableSet_eq_fun measurable_fst measurable_const).inter
((measurableSet_eq_fun (measurable_fst.comp measurable_snd) measurable_const).inter
((by fun_prop : Measurable fun z : Observation J => (z.2.2 : ℝ)) hB))
· exact hB
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerScoreLogAttainment 5 declarations Paper-specific endpoint-law instantiations for the score-log pair.
Paper-specific endpoint-law instantiations for the score-log pair.
theorem endpointFullLaw_externalScoreLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
upper0 upper1 :
conclusion
ExternalScoreLaw g (endpointFullLaw H g Prel hMass hOverlap upper0 upper1) H where probabilityP
Proof (Lean source)
lemma endpointFullLaw_externalScoreLaw
{ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (upper0 upper1 : Bool) :
ExternalScoreLaw g
(endpointFullLaw H g Prel hMass hOverlap upper0 upper1) H where
probabilityP := endpointFullLaw_isProbabilityMeasure
H g Prel hMass hOverlap upper0 upper1
probabilityH := hMass.1
overlap := hOverlap
measurableRelease := hMass.2.2.1
scoreMarginal := endpointFullLaw_scoreMarginal
H g Prel hMass hOverlap upper0 upper1
randomizedAssignment := endpointFullLaw_randomizedAssignment
H g Prel hMass hOverlap upper0 upper1
consistency := endpointFullLaw_consistency
H g Prel hMass hOverlap upper0 upper1
deterministicRelease := endpointFullLaw_deterministicRelease
H g Prel hMass hOverlap upper0 upper1
theorem endpointFullLaw_mem_externalLaws
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
upper0 upper1 :
conclusion
(endpointFullLaw H g Prel hMass hOverlap upper0 upper1, H) ∈ ExternalLaws g
Proof (Lean source)
lemma endpointFullLaw_mem_externalLaws
{ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (upper0 upper1 : Bool) :
(endpointFullLaw H g Prel hMass hOverlap upper0 upper1, H) ∈
ExternalLaws g :=
⟨endpointFullLaw_isProbabilityMeasure H g Prel hMass hOverlap upper0 upper1,
hMass.1,
endpointFullLaw_externalScoreLaw H g Prel hMass hOverlap upper0 upper1⟩
theorem endpointFullLaw_ate
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
upper0 upper1 :
Proof (Lean source)
lemma endpointFullLaw_ate
{ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (upper0 upper1 : Bool) :
ate (endpointFullLaw H g Prel hMass hOverlap upper0 upper1) =
(if upper1 then muUpper H g Prel hMass true
else muLower H g Prel hMass true) -
(if upper0 then muUpper H g Prel hMass false
else muLower H g Prel hMass false) := by
letI : IsProbabilityMeasure
(endpointFullLaw H g Prel hMass hOverlap upper0 upper1) :=
endpointFullLaw_isProbabilityMeasure H g Prel hMass hOverlap upper0 upper1
rw [ate_eq_potentialOutcomeMeans_sub]
rw [show (∫ ω, ((outcome1 ω : OutcomeSpace) : ℝ)
∂endpointFullLaw H g Prel hMass hOverlap upper0 upper1) =
(if upper1 then muUpper H g Prel hMass true
else muLower H g Prel hMass true) by
simpa using endpointFullLaw_armMean H g Prel hMass hOverlap
upper0 upper1 true]
rw [show (∫ ω, ((outcome0 ω : OutcomeSpace) : ℝ)
∂endpointFullLaw H g Prel hMass hOverlap upper0 upper1) =
(if upper0 then muUpper H g Prel hMass false
else muLower H g Prel hMass false) by
simpa using endpointFullLaw_armMean H g Prel hMass hOverlap
upper0 upper1 false]
theorem scoreLogTripleBaseline_lowerEndpoint_ate
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
r :
x y z :
hxy :
x < y
hyz :
y < z
hx :
g x = r
hy :
g y = r
hz :
g z = r
t :
ℝ
ht₁ :
(x : ℝ) * (1 / 3) < t
ht₂ :
t < min ((x : ℝ) * (1 / 3) + (y : ℝ) * (1 / 3)) ((z : ℝ) * (1 / 3))
hMass :
CompatibleCellMasses (scoreLogTripleBaseline x y z) g (releasedLaw (scoreLogTripleBaselineFullLaw g x y z (t / (((x : ℝ) + y + z) / 3))))
conclusion
ate (endpointFullLaw (scoreLogTripleBaseline x y z) g (releasedLaw (scoreLogTripleBaselineFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)))) hMass hOverlap true false)
= threeScoreLowerEndpoint (z : ℝ) t
Proof (Lean source)
lemma scoreLogTripleBaseline_lowerEndpoint_ate
{ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(r : LabelSpace J) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z)
(hx : g x = r) (hy : g y = r) (hz : g z = r)
(t : ℝ)
(ht₁ : (x : ℝ) * (1 / 3) < t)
(ht₂ : t < min ((x : ℝ) * (1 / 3) + (y : ℝ) * (1 / 3))
((z : ℝ) * (1 / 3)))
(hMass : CompatibleCellMasses (scoreLogTripleBaseline x y z) g
(releasedLaw (scoreLogTripleBaselineFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3))))) :
ate (endpointFullLaw (scoreLogTripleBaseline x y z) g
(releasedLaw (scoreLogTripleBaselineFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)))) hMass hOverlap true false) =
threeScoreLowerEndpoint (z : ℝ) t := by
let q : ℝ := ((x : ℝ) + y + z) / 3
have hx0 : 0 < (x : ℝ) := lt_of_lt_of_le hOverlap.1 x.property.1
have hy0 : 0 < (y : ℝ) := lt_trans hx0 hxy
have hz0 : 0 < (z : ℝ) := lt_trans hy0 hyz
have hq : 0 < q := by dsimp [q]; linarith
have ht0 : 0 < t := lt_trans (mul_pos hx0 (by norm_num)) ht₁
have htq : t < q := by
have hleft := lt_of_lt_of_le ht₂ (min_le_left _ _)
dsimp [q]
nlinarith [mul_pos hz0 (by norm_num : (0 : ℝ) < 1 / 3)]
have hs0 : 0 ≤ t / q := (div_pos ht0 hq).le
have hs1 : t / q ≤ 1 := (div_lt_one hq).2 htq |>.le
have hq1 : q < 1 := by
have hx1 : (x : ℝ) < 1 :=
lt_of_le_of_lt x.property.2 (by linarith [hOverlap.1])
have hy1 : (y : ℝ) < 1 :=
lt_of_le_of_lt y.property.2 (by linarith [hOverlap.1])
have hz1 : (z : ℝ) < 1 :=
lt_of_le_of_lt z.property.2 (by linarith [hOverlap.1])
dsimp [q]
linarith
have hrel := scoreLogTripleBaseline_releasedLaw_cell
g hOverlap r x y z hx hy hz t hs0 hs1
have hc := muLower_muUpper_false_eq_zero_of_scoreLogReleasedCellLaw
(scoreLogTripleBaseline x y z) g
(releasedLaw (scoreLogTripleBaselineFullLaw g x y z (t / q))) hMass
r q t (1 - q) hrel hq.le (sub_pos.mpr hq1) hs0 hs1 (by ring)
rw [endpointFullLaw_ate]
simp only [Bool.false_eq_true, ↓reduceIte]
rw [scoreLogTripleBaseline_muLower_true g hOverlap r x y z hxy hyz
hx hy hz t ht₁ ht₂ hMass,
hc.2, sub_zero]
theorem scoreLogTriplePerturbed_upperEndpoint_ate
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
r :
x y z :
hxy :
x < y
hyz :
y < z
hx :
g x = r
hy :
g y = r
hz :
g z = r
t u :
ℝ
hu0 :
0 ≤ u
hu :
u * ((z : ℝ) - x) ≤ 1 / 3
ht₁ :
(x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) < t
ht₂ :
t < min ((x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) + (y : ℝ) * (1 / 3 - u * ((z : ℝ) - x))) ((z : ℝ) * (1 / 3 + u * ((y : ℝ) - x)))
hMass :
CompatibleCellMasses (scoreLogTriplePerturbed x y z u) g (releasedLaw (scoreLogTriplePerturbedFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)) u))
conclusion
ate (endpointFullLaw (scoreLogTriplePerturbed x y z u) g (releasedLaw (scoreLogTriplePerturbedFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)) u)) hMass hOverlap false true)
= threeScoreUpperEndpoint (x : ℝ) y (1 / 3 + u * ((z : ℝ) - y)) t
Proof (Lean source)
lemma scoreLogTriplePerturbed_upperEndpoint_ate
{ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(r : LabelSpace J) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z)
(hx : g x = r) (hy : g y = r) (hz : g z = r)
(t u : ℝ) (hu0 : 0 ≤ u) (hu : u * ((z : ℝ) - x) ≤ 1 / 3)
(ht₁ : (x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) < t)
(ht₂ : t < min
((x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) +
(y : ℝ) * (1 / 3 - u * ((z : ℝ) - x)))
((z : ℝ) * (1 / 3 + u * ((y : ℝ) - x))))
(hMass : CompatibleCellMasses (scoreLogTriplePerturbed x y z u) g
(releasedLaw (scoreLogTriplePerturbedFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)) u))) :
ate (endpointFullLaw (scoreLogTriplePerturbed x y z u) g
(releasedLaw (scoreLogTriplePerturbedFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)) u)) hMass hOverlap false true) =
threeScoreUpperEndpoint (x : ℝ) y
(1 / 3 + u * ((z : ℝ) - y)) t := by
let q : ℝ := ((x : ℝ) + y + z) / 3
have hx0 : 0 < (x : ℝ) := lt_of_lt_of_le hOverlap.1 x.property.1
have hy0 : 0 < (y : ℝ) := lt_trans hx0 hxy
have hz0 : 0 < (z : ℝ) := lt_trans hy0 hyz
have hq : 0 < q := by dsimp [q]; linarith
have hw := scoreLogTriple_weights_nonneg x y z hxy hyz u hu0 hu
have hsum := (scoreLogTriple_weight_identities x y z hxy hyz u).1
have hmoment := (scoreLogTriple_weight_identities x y z hxy hyz u).2
let wx : ℝ := 1 / 3 + u * ((z : ℝ) - y)
let wy : ℝ := 1 / 3 - u * ((z : ℝ) - x)
let wz : ℝ := 1 / 3 + u * ((y : ℝ) - x)
have ht0 : 0 < t := by
have hxwx : 0 ≤ (x : ℝ) * wx :=
mul_nonneg hx0.le (by simpa [wx] using hw.1)
exact lt_of_le_of_lt hxwx (by simpa [wx] using ht₁)
have htq : t < q := by
have hleft := lt_of_lt_of_le ht₂ (min_le_left _ _)
have hwz : 0 ≤ wz := by simpa [wz] using hw.2.2
have hmom : (x : ℝ) * wx + (y : ℝ) * wy + (z : ℝ) * wz = q := by
dsimp [wx, wy, wz, q]
calc
_ = (x : ℝ) / 3 + (y : ℝ) / 3 + (z : ℝ) / 3 := hmoment
_ = ((x : ℝ) + y + z) / 3 := by ring
have hleft' : t < (x : ℝ) * wx + (y : ℝ) * wy := by
simpa [wx, wy] using hleft
nlinarith [mul_nonneg hz0.le hwz]
have hs0 : 0 ≤ t / q := (div_pos ht0 hq).le
have hs1 : t / q ≤ 1 := (div_lt_one hq).2 htq |>.le
have hq1 : q < 1 := by
have hx1 : (x : ℝ) < 1 :=
lt_of_le_of_lt x.property.2 (by linarith [hOverlap.1])
have hy1 : (y : ℝ) < 1 :=
lt_of_le_of_lt y.property.2 (by linarith [hOverlap.1])
have hz1 : (z : ℝ) < 1 :=
lt_of_le_of_lt z.property.2 (by linarith [hOverlap.1])
dsimp [q]
linarith
have hrel := scoreLogTriplePerturbed_releasedLaw_cell
g hOverlap r x y z hxy hyz hx hy hz t u hs0 hs1 hu0 hu
have hc := muLower_muUpper_false_eq_zero_of_scoreLogReleasedCellLaw
(scoreLogTriplePerturbed x y z u) g
(releasedLaw (scoreLogTriplePerturbedFullLaw g x y z (t / q) u)) hMass
r q t (1 - q) hrel hq.le (sub_pos.mpr hq1) hs0 hs1 (by ring)
rw [endpointFullLaw_ate]
simp only [↓reduceIte, Bool.false_eq_true]
rw [scoreLogTriplePerturbed_muUpper_true g hOverlap r x y z hxy hyz
hx hy hz t u hu0 hu ht₁ ht₂ hMass,
hc.1, sub_zero]
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerScoreLogCalibration 9 declarations Numerical calibration of the three-score external-log perturbation.
Numerical calibration of the three-score external-log perturbation.
def threeScoreGapSq
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
x y z :
def
threeScoreGapSq x y z :
ℝ
given by
((z : ℝ) - y) ^ 2 + ((z : ℝ) - x) ^ 2 + ((y : ℝ) - x) ^ 2
def threeScoreCalibratedU
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
threeScoreCalibratedU α x y z m :
ℝ
theorem threeScoreGapSq_pos
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma threeScoreGapSq_pos {ε : ℝ} (x y z : ScoreSpace ε)
(_hxy : x < y) (hyz : y < z) :
0 < threeScoreGapSq x y z := by
unfold threeScoreGapSq
have hzy : 0 < (z : ℝ) - y := sub_pos.mpr hyz
nlinarith [sq_pos_of_pos hzy, sq_nonneg ((z : ℝ) - x), sq_nonneg ((y : ℝ) - x)]
theorem threeScoreCalibratedU_pos
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
0 < threeScoreCalibratedU α x y z m
Proof (Lean source)
lemma threeScoreCalibratedU_pos {ε : ℝ} (α : ℝ)
(hα : 0 < α ∧ α < 1 / 2) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z) (m : ℕ) (hm : 0 < m) :
0 < threeScoreCalibratedU α x y z m := by
unfold threeScoreCalibratedU
have hβ : 0 < 1 - 2 * α := by linarith [hα.2]
have hlog : 0 < log (1 + (1 - 2 * α) ^ 2) :=
Real.log_pos (by nlinarith [sq_pos_of_pos hβ])
have hmR : (0 : ℝ) < m := by exact_mod_cast hm
positivity [threeScoreGapSq_pos x y z hxy hyz]
theorem threeScoreCalibratedU_chisq_scalar
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
3 * (threeScoreCalibratedU α x y z m) ^ 2 * threeScoreGapSq x y z
= log (1 + (1 - 2 * α) ^ 2) / (m : ℝ)
Proof (Lean source)
lemma threeScoreCalibratedU_chisq_scalar {ε : ℝ} (α : ℝ)
(hα : 0 < α ∧ α < 1 / 2) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z) (m : ℕ) (hm : 0 < m) :
3 * (threeScoreCalibratedU α x y z m) ^ 2 * threeScoreGapSq x y z =
log (1 + (1 - 2 * α) ^ 2) / (m : ℝ) := by
let c : ℝ := log (1 + (1 - 2 * α) ^ 2)
let D : ℝ := threeScoreGapSq x y z
have hβ : 0 < 1 - 2 * α := by linarith [hα.2]
have hc : 0 < c := by
dsimp [c]
exact Real.log_pos (by nlinarith [sq_pos_of_pos hβ])
have hD : 0 < D := threeScoreGapSq_pos x y z hxy hyz
have hmR : (0 : ℝ) < m := by exact_mod_cast hm
have hroot3 : (sqrt 3) ^ 2 = 3 := Real.sq_sqrt (by norm_num)
have hrootc : (sqrt c) ^ 2 = c := Real.sq_sqrt hc.le
have hrootD : (sqrt D) ^ 2 = D := Real.sq_sqrt hD.le
have hrootm : (sqrt (m : ℝ)) ^ 2 = (m : ℝ) := Real.sq_sqrt hmR.le
change 3 * (sqrt c /
(sqrt 3 * sqrt D * sqrt (m : ℝ))) ^ 2 * D = c / (m : ℝ)
rw [div_pow, mul_pow, mul_pow, hroot3, hrootc, hrootD, hrootm]
field_simp
theorem threeScoreCalibratedU_pow_bound
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(1 + 3 * (threeScoreCalibratedU α x y z m) ^ 2 * threeScoreGapSq x y z) ^ m
≤ 1 + (1 - 2 * α) ^ 2
Proof (Lean source)
lemma threeScoreCalibratedU_pow_bound {ε : ℝ} (α : ℝ)
(hα : 0 < α ∧ α < 1 / 2) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z) (m : ℕ) (hm : 0 < m) :
(1 + 3 * (threeScoreCalibratedU α x y z m) ^ 2 *
threeScoreGapSq x y z) ^ m ≤ 1 + (1 - 2 * α) ^ 2 := by
rw [threeScoreCalibratedU_chisq_scalar α hα x y z hxy hyz m hm]
let c : ℝ := log (1 + (1 - 2 * α) ^ 2)
have hc : 0 < c := by
dsimp [c]
have hβ : 0 < 1 - 2 * α := by linarith [hα.2]
exact Real.log_pos (by nlinarith [sq_pos_of_pos hβ])
have hmR : (0 : ℝ) < m := by exact_mod_cast hm
have h := Real.one_sub_div_pow_le_exp_neg (n := m) (t := -c)
(by linarith : -c ≤ (m : ℝ))
have hexp : exp c = 1 + (1 - 2 * α) ^ 2 := Real.exp_log (by positivity)
simpa only [c, neg_div, sub_neg_eq_add, neg_neg, hexp] using h
theorem threeScoreCalibratedU_tendsto_zero
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma threeScoreCalibratedU_tendsto_zero {ε : ℝ} (α : ℝ)
(x y z : ScoreSpace ε) :
Tendsto (fun m : ℕ => threeScoreCalibratedU α x y z m) atTop (nhds 0) := by
have hinv : Tendsto (fun m : ℕ => (sqrt (m : ℝ))⁻¹) atTop (nhds 0) :=
tendsto_inv_atTop_zero.comp
(Real.tendsto_sqrt_atTop.comp tendsto_natCast_atTop_atTop)
let C : ℝ := sqrt (log (1 + (1 - 2 * α) ^ 2)) /
(sqrt 3 * sqrt (threeScoreGapSq x y z))
have h : Tendsto (fun m : ℕ => C * (sqrt (m : ℝ))⁻¹)
atTop (nhds (C * 0)) := tendsto_const_nhds.mul hinv
convert h using 1
· funext m
dsimp [C, threeScoreCalibratedU]
rw [div_eq_mul_inv, div_eq_mul_inv, mul_inv]
ring
· simp
theorem threeScoreCalibratedU_eventually_weights_pos
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
∀ᶠ m : ℕ in atTop, 0 < 1 / 3
+ threeScoreCalibratedU α x y z m * ((z : ℝ) - y) ∧ 0 < 1 / 3
- threeScoreCalibratedU α x y z m * ((z : ℝ) - x) ∧ 0 < 1 / 3
+ threeScoreCalibratedU α x y z m * ((y : ℝ) - x)
Proof (Lean source)
lemma threeScoreCalibratedU_eventually_weights_pos {ε : ℝ} (α : ℝ)
(x y z : ScoreSpace ε) :
∀ᶠ m : ℕ in atTop,
0 < 1 / 3 + threeScoreCalibratedU α x y z m * ((z : ℝ) - y) ∧
0 < 1 / 3 - threeScoreCalibratedU α x y z m * ((z : ℝ) - x) ∧
0 < 1 / 3 + threeScoreCalibratedU α x y z m * ((y : ℝ) - x) := by
have hu := threeScoreCalibratedU_tendsto_zero α x y z
have hxlim : Tendsto (fun m : ℕ =>
1 / 3 + threeScoreCalibratedU α x y z m * ((z : ℝ) - y))
atTop (nhds (1 / 3)) := by
simpa using tendsto_const_nhds.add (hu.mul_const ((z : ℝ) - y))
have hylim : Tendsto (fun m : ℕ =>
1 / 3 - threeScoreCalibratedU α x y z m * ((z : ℝ) - x))
atTop (nhds (1 / 3)) := by
simpa using tendsto_const_nhds.sub (hu.mul_const ((z : ℝ) - x))
have hzlim : Tendsto (fun m : ℕ =>
1 / 3 + threeScoreCalibratedU α x y z m * ((y : ℝ) - x))
atTop (nhds (1 / 3)) := by
simpa using tendsto_const_nhds.add (hu.mul_const ((y : ℝ) - x))
filter_upwards
[hxlim.eventually (Ioi_mem_nhds (by norm_num : (0 : ℝ) < 1 / 3)),
hylim.eventually (Ioi_mem_nhds (by norm_num : (0 : ℝ) < 1 / 3)),
hzlim.eventually (Ioi_mem_nhds (by norm_num : (0 : ℝ) < 1 / 3))]
with m hxm hym hzm
exact ⟨hxm, hym, hzm⟩
theorem threeScoreCalibratedU_commonTrial_tv_bound
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
T :
Type*
ρ :
α :
ℝ
hα :
0 < α ∧ α < 1 / 2
x y z :
hxy :
x < y
hyz :
y < z
m :
ℕ
hm :
0 < m
hx :
0 < 1 / 3 + threeScoreCalibratedU α x y z m * ((z : ℝ) - y)
hy :
0 < 1 / 3 - threeScoreCalibratedU α x y z m * ((z : ℝ) - x)
hz :
0 < 1 / 3 + threeScoreCalibratedU α x y z m * ((y : ℝ) - x)
conclusion
tvDist (ρ.prod (Measure.pi (fun _ : Fin m => threeScorePerturbedLaw x y z (threeScoreCalibratedU α x y z m)))) (ρ.prod (Measure.pi (fun _ : Fin m => threeScoreBaseLaw x y z)))
≤ (1 - 2 * α) / 2
Proof (Lean source)
lemma threeScoreCalibratedU_commonTrial_tv_bound {ε : ℝ} {T : Type*}
[MeasurableSpace T] (ρ : Measure T) [IsProbabilityMeasure ρ]
(α : ℝ) (hα : 0 < α ∧ α < 1 / 2) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z) (m : ℕ) (hm : 0 < m)
(hx : 0 < 1 / 3 + threeScoreCalibratedU α x y z m * ((z : ℝ) - y))
(hy : 0 < 1 / 3 - threeScoreCalibratedU α x y z m * ((z : ℝ) - x))
(hz : 0 < 1 / 3 + threeScoreCalibratedU α x y z m * ((y : ℝ) - x)) :
tvDist
(ρ.prod (Measure.pi (fun _ : Fin m =>
threeScorePerturbedLaw x y z (threeScoreCalibratedU α x y z m))))
(ρ.prod (Measure.pi (fun _ : Fin m => threeScoreBaseLaw x y z))) ≤
(1 - 2 * α) / 2 := by
let u := threeScoreCalibratedU α x y z m
have htv := threeScorePerturbedLaw_commonTrial_tv_bound
ρ x y z u hxy hyz hx hy hz m
have hpow := threeScoreCalibratedU_pow_bound α hα x y z hxy hyz m hm
have hβ : 0 < 1 - 2 * α := by linarith [hα.2]
calc
_ ≤ (1 / 2 : ℝ) * sqrt
((1 + 3 * u ^ 2 * threeScoreGapSq x y z) ^ m - 1) := by
simpa [u, threeScoreGapSq] using htv
_ ≤ (1 / 2 : ℝ) * sqrt ((1 - 2 * α) ^ 2) := by
apply mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt ?_) (by norm_num)
linarith
_ = (1 - 2 * α) / 2 := by
rw [Real.sqrt_sq_eq_abs, abs_of_pos hβ]
ring
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerScoreLogCellIdentification 6 declarations Identification of the active released cell in the score-log full laws.
Identification of the active released cell in the score-log full laws.
theorem scoreLogTripleBaseline_releasedLaw_cell
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
r :
x y z :
hx :
g x = r
hy :
g y = r
hz :
g z = r
t :
ℝ
hs0 :
0 ≤ t / (((x : ℝ) + y + z) / 3)
hs1 :
t / (((x : ℝ) + y + z) / 3) ≤ 1
conclusion
q :
((x : ℝ) + y + z) / 3
releasedLaw (scoreLogTripleBaselineFullLaw g x y z (t / q))
= scoreLogReleasedCellLaw r q t (1 - q)
Proof (Lean source)
lemma scoreLogTripleBaseline_releasedLaw_cell
{ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(r : LabelSpace J) (x y z : ScoreSpace ε)
(hx : g x = r) (hy : g y = r) (hz : g z = r)
(t : ℝ)
(hs0 : 0 ≤ t / (((x : ℝ) + y + z) / 3))
(hs1 : t / (((x : ℝ) + y + z) / 3) ≤ 1) :
let q := ((x : ℝ) + y + z) / 3
releasedLaw
(scoreLogTripleBaselineFullLaw g x y z (t / q)) =
scoreLogReleasedCellLaw r q t (1 - q) := by
dsimp only
let q : ℝ := ((x : ℝ) + y + z) / 3
have ex0 : 0 ≤ (x : ℝ) := le_trans hOverlap.1.le x.property.1
have ey0 : 0 ≤ (y : ℝ) := le_trans hOverlap.1.le y.property.1
have ez0 : 0 ≤ (z : ℝ) := le_trans hOverlap.1.le z.property.1
have ex1 : (x : ℝ) ≤ 1 := by linarith [x.property.2, hOverlap.1]
have ey1 : (y : ℝ) ≤ 1 := by linarith [y.property.2, hOverlap.1]
have ez1 : (z : ℝ) ≤ 1 := by linarith [z.property.2, hOverlap.1]
have hq0 : 0 ≤ q := by dsimp [q]; positivity
have hq1 : q ≤ 1 := by dsimp [q]; linarith
let T : Measure (Observation J) :=
(trialOutcomeLaw (t / q)).map (fun v => (r, true, v))
let F : Measure (Observation J) := Measure.dirac (r, false, trialZeroOutcome)
have htCoeff :
ofReal (1 / 3 : ℝ) * ofReal (x : ℝ) +
ofReal (1 / 3 : ℝ) * ofReal (y : ℝ) +
ofReal (1 / 3 : ℝ) * ofReal (z : ℝ) =
ofReal q := by
rw [← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 1 / 3),
← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 1 / 3),
← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 1 / 3),
← ENNReal.ofReal_add (mul_nonneg (by norm_num) ex0)
(mul_nonneg (by norm_num) ey0),
← ENNReal.ofReal_add
(add_nonneg (mul_nonneg (by norm_num) ex0)
(mul_nonneg (by norm_num) ey0))
(mul_nonneg (by norm_num) ez0)]
congr 1
dsimp [q]
ring
have hcCoeff :
ofReal (1 / 3 : ℝ) * ofReal (1 - (x : ℝ)) +
ofReal (1 / 3 : ℝ) * ofReal (1 - (y : ℝ)) +
ofReal (1 / 3 : ℝ) * ofReal (1 - (z : ℝ)) =
ofReal (1 - q) := by
rw [← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 1 / 3),
← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 1 / 3),
← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 1 / 3),
← ENNReal.ofReal_add (mul_nonneg (by norm_num) (sub_nonneg.mpr ex1))
(mul_nonneg (by norm_num) (sub_nonneg.mpr ey1)),
← ENNReal.ofReal_add
(add_nonneg (mul_nonneg (by norm_num) (sub_nonneg.mpr ex1))
(mul_nonneg (by norm_num) (sub_nonneg.mpr ey1)))
(mul_nonneg (by norm_num) (sub_nonneg.mpr ez1))]
congr 1
dsimp [q]
ring
rw [scoreLogTripleBaselineFullLaw_releasedLaw,
scoreLogAtomFullLaw_releasedLaw g x (t / q) hs0 hs1,
scoreLogAtomFullLaw_releasedLaw g y (t / q) hs0 hs1,
scoreLogAtomFullLaw_releasedLaw g z (t / q) hs0 hs1,
hx, hy, hz]
unfold scoreLogReleasedCellLaw
have htCoeff' :
ofReal (1 / 3 : ℝ) * ofReal (x : ℝ) +
ofReal (1 / 3 : ℝ) * ofReal (y : ℝ) +
ofReal (1 / 3 : ℝ) * ofReal (z : ℝ) =
ofReal (((x : ℝ) + y + z) / 3) := by
simpa [q] using htCoeff
have hcCoeff' :
ofReal (1 / 3 : ℝ) * ofReal (1 - (x : ℝ)) +
ofReal (1 / 3 : ℝ) * ofReal (1 - (y : ℝ)) +
ofReal (1 / 3 : ℝ) * ofReal (1 - (z : ℝ)) =
ofReal (1 - (((x : ℝ) + y + z) / 3)) := by
simpa [q] using hcCoeff
rw [← htCoeff', ← hcCoeff']
module
theorem scoreLogTriplePerturbed_releasedLaw_cell
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
r :
x y z :
hxy :
x < y
hyz :
y < z
hx :
g x = r
hy :
g y = r
hz :
g z = r
t u :
ℝ
hs0 :
0 ≤ t / (((x : ℝ) + y + z) / 3)
hs1 :
t / (((x : ℝ) + y + z) / 3) ≤ 1
hu0 :
0 ≤ u
hu :
u * ((z : ℝ) - x) ≤ 1 / 3
conclusion
q :
((x : ℝ) + y + z) / 3
releasedLaw (scoreLogTriplePerturbedFullLaw g x y z (t / q) u)
= scoreLogReleasedCellLaw r q t (1 - q)
Proof (Lean source)
lemma scoreLogTriplePerturbed_releasedLaw_cell
{ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(r : LabelSpace J) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z)
(hx : g x = r) (hy : g y = r) (hz : g z = r)
(t u : ℝ)
(hs0 : 0 ≤ t / (((x : ℝ) + y + z) / 3))
(hs1 : t / (((x : ℝ) + y + z) / 3) ≤ 1)
(hu0 : 0 ≤ u) (hu : u * ((z : ℝ) - x) ≤ 1 / 3) :
let q := ((x : ℝ) + y + z) / 3
releasedLaw
(scoreLogTriplePerturbedFullLaw g x y z (t / q) u) =
scoreLogReleasedCellLaw r q t (1 - q) := by
dsimp only
rw [← scoreLogTriple_releasedLaw_eq g hOverlap r x y z hxy hyz
hx hy hz (t / (((x : ℝ) + y + z) / 3)) hs0 hs1 u hu0 hu]
exact scoreLogTripleBaseline_releasedLaw_cell
g hOverlap r x y z hx hy hz t hs0 hs1
theorem scoreLogTripleBaseline_armCellMass_true
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
armCellMass (scoreLogTripleBaseline x y z) g true r = ((x : ℝ) + y + z) / 3
Proof (Lean source)
lemma scoreLogTripleBaseline_armCellMass_true
{ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (r : LabelSpace J)
(x y z : ScoreSpace ε)
(hx : g x = r) (hy : g y = r) (hz : g z = r) :
armCellMass (scoreLogTripleBaseline x y z) g true r =
((x : ℝ) + y + z) / 3 := by
classical
unfold armCellMass scoreLogTripleBaseline
rw [Measure.restrict_add, Measure.restrict_add,
Measure.restrict_smul, Measure.restrict_smul, Measure.restrict_smul,
restrict_dirac, restrict_dirac, restrict_dirac]
simp only [cell, hx, hy, hz, Set.mem_ofPred_eq, if_pos]
rw [integral_add_measure, integral_add_measure,
integral_smul_measure, integral_smul_measure, integral_smul_measure]
simp [armProb]
ring
all_goals
first
| exact Integrable.add_measure
(Integrable.smul_measure (integrable_dirac (by simp [armProb])) (by simp))
(Integrable.smul_measure (integrable_dirac (by simp [armProb])) (by simp))
| exact Integrable.smul_measure (integrable_dirac (by simp [armProb])) (by simp)
theorem scoreLogTripleBaseline_armCellWeightLaw_true
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hg :
r :
x y z :
hx :
g x = r
hy :
g y = r
hz :
g z = r
hq :
0 < ((x : ℝ) + y + z) / 3
conclusion
hm :
by rw [scoreLogTripleBaseline_armCellMass_true g r x y z hx hy hz] exact hq (: 0 < armCellMass (scoreLogTripleBaseline x y z) g true r)
armCellWeightLaw (scoreLogTripleBaseline x y z) (scoreLogTripleBaseline_isProbabilityMeasure x y z) g hg true r hm
= threeScoreInverseLaw (x : ℝ) y z (1 / 3) (1 / 3) (1 / 3)
Proof (Lean source)
lemma scoreLogTripleBaseline_armCellWeightLaw_true
{ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hg : Measurable g)
(r : LabelSpace J) (x y z : ScoreSpace ε)
(hx : g x = r) (hy : g y = r) (hz : g z = r)
(hq : 0 < ((x : ℝ) + y + z) / 3) :
let hm : 0 < armCellMass (scoreLogTripleBaseline x y z) g true r := by
rw [scoreLogTripleBaseline_armCellMass_true g r x y z hx hy hz]
exact hq
armCellWeightLaw (scoreLogTripleBaseline x y z)
(scoreLogTripleBaseline_isProbabilityMeasure x y z)
g hg true r hm =
threeScoreInverseLaw (x : ℝ) y z (1 / 3) (1 / 3) (1 / 3) := by
classical
dsimp only
unfold armCellWeightLaw armCellScoreLaw
rw [scoreLogTripleBaseline_armCellMass_true g r x y z hx hy hz]
unfold scoreLogTripleBaseline
rw [
withDensity_add_measure, withDensity_add_measure,
withDensity_smul_measure, withDensity_smul_measure,
withDensity_smul_measure, dirac_withDensity, dirac_withDensity,
dirac_withDensity]
simp only [Measure.restrict_add, Measure.restrict_smul, restrict_dirac,
cell, hx, hy, hz, Set.mem_ofPred_eq, if_pos, armProb]
repeat' rw [smul_smul]
have hmap : Measurable (fun e : ScoreSpace ε => ((e : ℝ))⁻¹) := by
simpa [armProb] using (measurable_inverseArmProb (ε := ε) true)
rw [Measure.map_smul,
Measure.map_add _ _ hmap,
Measure.map_add _ _ hmap,
Measure.map_smul, Measure.map_smul, Measure.map_smul,
Measure.map_dirac' hmap, Measure.map_dirac' hmap,
Measure.map_dirac' hmap]
unfold threeScoreInverseLaw
have hden :
(x : ℝ) * (1 / 3) + (y : ℝ) * (1 / 3) + (z : ℝ) * (1 / 3) =
((x : ℝ) + y + z) / 3 := by ring
have hcoeff (v : ℝ) :
(ofReal (((x : ℝ) + y + z) / 3))⁻¹ *
(ofReal (1 / 3 : ℝ) * ofReal v) =
ofReal
(v * (1 / 3) /
((x : ℝ) * (1 / 3) + (y : ℝ) * (1 / 3) + (z : ℝ) * (1 / 3))) := by
rw [hden, ENNReal.ofReal_div_of_pos hq]
have hn :
ofReal (1 / 3 : ℝ) * ofReal v =
ofReal (v * (1 / 3)) := by
rw [← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 1 / 3)]
congr 1
ring
rw [hn]
simp only [div_eq_mul_inv, mul_comm]
rw [smul_add, smul_add]
repeat' rw [smul_smul]
rw [hcoeff (x : ℝ), hcoeff (y : ℝ), hcoeff (z : ℝ)]
simp only [one_div]
theorem scoreLogTriplePerturbed_armCellMass_true
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
r :
x y z :
hxy :
x < y
hyz :
y < z
hx :
g x = r
hy :
g y = r
hz :
g z = r
u :
ℝ
hu0 :
0 ≤ u
hu :
u * ((z : ℝ) - x) ≤ 1 / 3
conclusion
armCellMass (scoreLogTriplePerturbed x y z u) g true r = ((x : ℝ) + y + z) / 3
Proof (Lean source)
lemma scoreLogTriplePerturbed_armCellMass_true
{ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (r : LabelSpace J)
(x y z : ScoreSpace ε) (hxy : x < y) (hyz : y < z)
(hx : g x = r) (hy : g y = r) (hz : g z = r)
(u : ℝ) (hu0 : 0 ≤ u) (hu : u * ((z : ℝ) - x) ≤ 1 / 3) :
armCellMass (scoreLogTriplePerturbed x y z u) g true r =
((x : ℝ) + y + z) / 3 := by
classical
have hw := scoreLogTriple_weights_nonneg x y z hxy hyz u hu0 hu
unfold armCellMass scoreLogTriplePerturbed
rw [Measure.restrict_add, Measure.restrict_add,
Measure.restrict_smul, Measure.restrict_smul, Measure.restrict_smul,
restrict_dirac, restrict_dirac, restrict_dirac]
simp only [cell, hx, hy, hz, Set.mem_ofPred_eq, if_pos]
rw [integral_add_measure, integral_add_measure,
integral_smul_measure, integral_smul_measure, integral_smul_measure]
rw [ENNReal.toReal_ofReal hw.1, ENNReal.toReal_ofReal hw.2.1,
ENNReal.toReal_ofReal hw.2.2]
simp [armProb]
have hm := (scoreLogTriple_weight_identities x y z hxy hyz u).2
linarith
all_goals
first
| exact Integrable.add_measure
(Integrable.smul_measure (integrable_dirac (by simp [armProb])) (by simp))
(Integrable.smul_measure (integrable_dirac (by simp [armProb])) (by simp))
| exact Integrable.smul_measure (integrable_dirac (by simp [armProb])) (by simp)
theorem scoreLogTriplePerturbed_armCellWeightLaw_true
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hg :
r :
x y z :
hxy :
x < y
hyz :
y < z
hx :
g x = r
hy :
g y = r
hz :
g z = r
u :
ℝ
hu0 :
0 ≤ u
hu :
u * ((z : ℝ) - x) ≤ 1 / 3
hq :
0 < ((x : ℝ) + y + z) / 3
conclusion
wx :
1 / 3 + u * ((z : ℝ) - y)
wy :
1 / 3 - u * ((z : ℝ) - x)
wz :
1 / 3 + u * ((y : ℝ) - x)
hm :
by rw [scoreLogTriplePerturbed_armCellMass_true g r x y z hxy hyz hx hy hz u hu0 hu] exact hq (: 0 < armCellMass (scoreLogTriplePerturbed x y z u) g true r)
armCellWeightLaw (scoreLogTriplePerturbed x y z u) (scoreLogTriplePerturbed_isProbabilityMeasure x y z hxy hyz u hu0 hu) g hg true r hm
= threeScoreInverseLaw (x : ℝ) y z wx wy wz
Proof (Lean source)
lemma scoreLogTriplePerturbed_armCellWeightLaw_true
{ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hg : Measurable g)
(r : LabelSpace J) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z)
(hx : g x = r) (hy : g y = r) (hz : g z = r)
(u : ℝ) (hu0 : 0 ≤ u) (hu : u * ((z : ℝ) - x) ≤ 1 / 3)
(hq : 0 < ((x : ℝ) + y + z) / 3) :
let wx := 1 / 3 + u * ((z : ℝ) - y)
let wy := 1 / 3 - u * ((z : ℝ) - x)
let wz := 1 / 3 + u * ((y : ℝ) - x)
let hm : 0 < armCellMass (scoreLogTriplePerturbed x y z u) g true r := by
rw [scoreLogTriplePerturbed_armCellMass_true g r x y z hxy hyz hx hy hz u hu0 hu]
exact hq
armCellWeightLaw (scoreLogTriplePerturbed x y z u)
(scoreLogTriplePerturbed_isProbabilityMeasure x y z hxy hyz u hu0 hu)
g hg true r hm =
threeScoreInverseLaw (x : ℝ) y z wx wy wz := by
classical
dsimp only
have hw := scoreLogTriple_weights_nonneg x y z hxy hyz u hu0 hu
unfold armCellWeightLaw armCellScoreLaw
rw [scoreLogTriplePerturbed_armCellMass_true g r x y z hxy hyz hx hy hz u hu0 hu]
unfold scoreLogTriplePerturbed
rw [withDensity_add_measure, withDensity_add_measure,
withDensity_smul_measure, withDensity_smul_measure,
withDensity_smul_measure, dirac_withDensity, dirac_withDensity,
dirac_withDensity]
simp only [Measure.restrict_add, Measure.restrict_smul, restrict_dirac,
cell, hx, hy, hz, Set.mem_ofPred_eq, if_pos, armProb]
repeat' rw [smul_smul]
have hmap : Measurable (fun e : ScoreSpace ε => ((e : ℝ))⁻¹) := by
simpa [armProb] using (measurable_inverseArmProb (ε := ε) true)
rw [Measure.map_smul, Measure.map_add _ _ hmap,
Measure.map_add _ _ hmap, Measure.map_smul, Measure.map_smul,
Measure.map_smul, Measure.map_dirac' hmap, Measure.map_dirac' hmap,
Measure.map_dirac' hmap]
unfold threeScoreInverseLaw
have hden :
(x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) +
(y : ℝ) * (1 / 3 - u * ((z : ℝ) - x)) +
(z : ℝ) * (1 / 3 + u * ((y : ℝ) - x)) =
((x : ℝ) + y + z) / 3 := by
calc
_ = (x : ℝ) / 3 + (y : ℝ) / 3 + (z : ℝ) / 3 :=
(scoreLogTriple_weight_identities x y z hxy hyz u).2
_ = ((x : ℝ) + y + z) / 3 := by ring
have hcoeff (v w : ℝ) (hw0 : 0 ≤ w) :
(ofReal (((x : ℝ) + y + z) / 3))⁻¹ *
(ofReal w * ofReal v) =
ofReal
(v * w /
((x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) +
(y : ℝ) * (1 / 3 - u * ((z : ℝ) - x)) +
(z : ℝ) * (1 / 3 + u * ((y : ℝ) - x)))) := by
rw [hden, ENNReal.ofReal_div_of_pos hq]
have hn : ofReal w * ofReal v =
ofReal (v * w) := by
rw [← ENNReal.ofReal_mul hw0]
congr 1
ring
rw [hn]
simp only [div_eq_mul_inv, mul_comm]
rw [smul_add, smul_add]
repeat' rw [smul_smul]
rw [hcoeff (x : ℝ) _ hw.1, hcoeff (y : ℝ) _ hw.2.1,
hcoeff (z : ℝ) _ hw.2.2]
simp only [one_div]
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerScoreLogCertificate 2 declarations Honest excess-risk transfer from two endpoint-attaining external laws.
Honest excess-risk transfer from two endpoint-attaining external laws.
theorem externalExcessRisk_lower_of_twoPoint
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
hα :
0 < α ∧ α < 1 / 2
Qj :
hTrial :
JointTrialIID g Qj
hLog :
ExternalLogIID g Qj
hInd :
H0 H1 :
Measure (ScoreSpace ε)
hPH0 :
(P0, H0) ∈ ExternalLaws g
hPH1 :
(P1, H1) ∈ ExternalLaws g
W0 Delta :
ℝ
hW0 :
htheta :
ate P0 ≤ ate P1
hsep :
ate P1 - ate P0 = W0 + Delta
hDelta :
0 < Delta
htv :
tvDist (Qj P0 H0) (Qj P1 H1) ≤ (1 - 2 * α) / 2
conclusion
Delta * ((1 - 2 * α) / 2) ≤ externalExcessRisk g Qj α
Proof (Lean source)
lemma externalExcessRisk_lower_of_twoPoint
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J)
(α : ℝ) (hα : 0 < α ∧ α < 1 / 2)
(Qj : Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : JointTrialIID g Qj) (hLog : ExternalLogIID g Qj)
(hInd : ExternalLogIndependent g Qj)
(P0 P1 : Measure (FullRow ε J))
(H0 H1 : Measure (ScoreSpace ε))
(hPH0 : (P0, H0) ∈ ExternalLaws g)
(hPH1 : (P1, H1) ∈ ExternalLaws g)
(W0 Delta : ℝ)
(hW0 : sharpATELength H0 g (releasedLaw P0)
(externalLawCellMasses g (P0, H0) hPH0) = W0)
(htheta : ate P0 ≤ ate P1)
(hsep : ate P1 - ate P0 = W0 + Delta)
(hDelta : 0 < Delta)
(htv : tvDist (Qj P0 H0) (Qj P1 H1) ≤
(1 - 2 * α) / 2) :
Delta * ((1 - 2 * α) / 2) ≤ externalExcessRisk g Qj α := by
have hbase (C : ExternalIntervalProcedure ε J n m)
(hC : C ∈ externalHonestProcedures g Qj α) :
Delta * ((1 - 2 * α) / 2) ≤
sSup {u : ℝ | ∃ PH, ∃ hPH : PH ∈ ExternalLaws g,
u = ∫ x, max 0 (C.length x -
sharpATELength PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH))
∂(externalExperiment n m PH.1 PH.2)} := by
letI : IsProbabilityMeasure (Qj P0 H0) :=
externalLaw_jointExperiment_probability g Qj hTrial hLog hInd
(P0, H0) hPH0
letI : IsProbabilityMeasure (Qj P1 H1) :=
externalLaw_jointExperiment_probability g Qj hTrial hLog hInd
(P1, H1) hPH1
have hc0 := hC (P0, H0) hPH0
have hc1 := hC (P1, H1) hPH1
have hQ0 := jointTrialLog_eq_externalExperiment g Qj hTrial hLog hInd
(P0, H0) hPH0
have hQ1 := jointTrialLog_eq_externalExperiment g Qj hTrial hLog hInd
(P1, H1) hPH1
change 1 - α ≤ (externalExperiment n m P0 H0).real
{x | C.contains x (ate P0)} at hc0
change 1 - α ≤ (externalExperiment n m P1 H1).real
{x | C.contains x (ate P1)} at hc1
rw [← hQ0] at hc0
rw [← hQ1] at hc1
have hexcess := externalInterval_twoPoint_expectedExcessLength_of_tv
C (Qj P0 H0) (Qj P1 H1) (ate P0) (ate P1) W0 Delta α
htheta hsep hDelta hc0 hc1 htv
rw [← hW0] at hexcess
rw [hQ0] at hexcess
let U : Set ℝ := {u | ∃ PH, ∃ hPH : PH ∈ ExternalLaws g,
u = ∫ x, max 0 (C.length x -
sharpATELength PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH))
∂(externalExperiment n m PH.1 PH.2)}
have hUbd : BddAbove U := by
refine ⟨2, ?_⟩
rintro u ⟨PH, hPH', rfl⟩
simpa only [jointTrialLog_eq_externalExperiment g Qj hTrial hLog hInd PH hPH']
using externalExcessIntegral_le_two g Qj hTrial hLog hInd C PH hPH'
exact hexcess.trans (le_csSup hUbd ⟨(P0, H0), hPH0, rfl⟩)
unfold externalExcessRisk
apply le_csInf
· refine ⟨sSup {u : ℝ | ∃ PH, ∃ hPH : PH ∈ ExternalLaws g,
u = ∫ x, max 0 ((externalFullInterval ε J n m).length x -
sharpATELength PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH))
∂(externalExperiment n m PH.1 PH.2)}, ?_⟩
exact ⟨externalFullInterval ε J n m,
externalFullInterval_honest g α hα.1.le Qj hTrial hLog hInd, rfl⟩
· rintro v ⟨C, hC, rfl⟩
exact hbase C hC
theorem externalExcessRisk_log_liminf_of_eventual_uniform
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
Qj :
∀ n m
α beta C :
ℝ
hLower :
≤ externalExcessRisk g (Qj n m) α
hUpper :
∀ (n m : ℕ)
conclusion
beta
≤ liminf (fun m : ℕ => sInf {v : ℝ | ∃ n : ℕ, 0 < n ∧ v = sqrt (m : ℝ) * externalExcessRisk g (Qj n m) α}) atTop
Proof (Lean source)
lemma externalExcessRisk_log_liminf_of_eventual_uniform
{ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J)
(Qj : ∀ n m,
Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(α beta C : ℝ)
(hLower : ∀ᶠ m : ℕ in atTop, ∀ n : ℕ, 0 < n →
beta * (sqrt (m : ℝ))⁻¹ ≤
externalExcessRisk g (Qj n m) α)
(hUpper : ∀ (n m : ℕ), 0 < n → 0 < m →
externalNormalizer n m * externalExcessRisk g (Qj n m) α ≤ C) :
beta ≤ liminf (fun m : ℕ =>
sInf {v : ℝ | ∃ n : ℕ, 0 < n ∧
v = sqrt (m : ℝ) * externalExcessRisk g (Qj n m) α}) atTop := by
exact logCertificate_liminf_infimum_of_normalized_upper beta C
(fun n m => externalExcessRisk g (Qj n m) α) hLower hUpper
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerScoreLogExperiment 1 declarations Testing control for the actual trial-plus-score-log experiments.
Testing control for the actual trial-plus-score-log experiments.
theorem scoreLogCalibrated_jointExperiment_eventually_tv_bound
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
r :
x y z :
hxy :
x < y
hyz :
y < z
hx :
g x = r
hy :
g y = r
hz :
g z = r
t α :
ℝ
hα :
0 < α ∧ α < 1 / 2
hs0 :
0 ≤ t / (((x : ℝ) + y + z) / 3)
hs1 :
t / (((x : ℝ) + y + z) / 3) ≤ 1
Qj :
∀ n m
hTrial :
∀ n m, JointTrialIID g (Qj n m)
hLog :
∀ n m, ExternalLogIID g (Qj n m)
hInd :
∀ n m, ExternalLogIndependent g (Qj n m)
conclusion
∀ᶠ m : ℕ in atTop, ∀ n : ℕ, let u := threeScoreCalibratedU α x y z m let P₀ := scoreLogTripleBaselineFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)) let H₀ := scoreLogTripleBaseline x y z let P₁ := scoreLogTriplePerturbedFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)) u let H₁ := scoreLogTriplePerturbed x y z u tvDist (Qj n m P₀ H₀) (Qj n m P₁ H₁)
≤ (1 - 2 * α) / 2
Proof (Lean source)
lemma scoreLogCalibrated_jointExperiment_eventually_tv_bound
{ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(hg : Measurable g) (r : LabelSpace J) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z)
(hx : g x = r) (hy : g y = r) (hz : g z = r)
(t α : ℝ) (hα : 0 < α ∧ α < 1 / 2)
(hs0 : 0 ≤ t / (((x : ℝ) + y + z) / 3))
(hs1 : t / (((x : ℝ) + y + z) / 3) ≤ 1)
(Qj : ∀ n m,
Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : ∀ n m, JointTrialIID g (Qj n m))
(hLog : ∀ n m, ExternalLogIID g (Qj n m))
(hInd : ∀ n m, ExternalLogIndependent g (Qj n m)) :
∀ᶠ m : ℕ in atTop, ∀ n : ℕ,
let u := threeScoreCalibratedU α x y z m
let P₀ := scoreLogTripleBaselineFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3))
let H₀ := scoreLogTripleBaseline x y z
let P₁ := scoreLogTriplePerturbedFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)) u
let H₁ := scoreLogTriplePerturbed x y z u
tvDist (Qj n m P₀ H₀) (Qj n m P₁ H₁) ≤
(1 - 2 * α) / 2 := by
filter_upwards [threeScoreCalibratedU_eventually_weights_pos α x y z,
eventually_gt_atTop (0 : ℕ)] with m hw hm
intro n
dsimp only
let u := threeScoreCalibratedU α x y z m
let P₀ := scoreLogTripleBaselineFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3))
let H₀ := scoreLogTripleBaseline x y z
let P₁ := scoreLogTriplePerturbedFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)) u
let H₁ := scoreLogTriplePerturbed x y z u
have hu0 : 0 ≤ u :=
(threeScoreCalibratedU_pos α hα x y z hxy hyz m hm).le
have hu : u * ((z : ℝ) - x) ≤ 1 / 3 := by
dsimp [u] at hw ⊢
linarith [hw.2.1]
have hpair := scoreLogTriple_externalPair g hOverlap hg r x y z hxy hyz
hx hy hz (t / (((x : ℝ) + y + z) / 3)) hs0 hs1 u hu0 hu
have hPH₀ : (P₀, H₀) ∈ ExternalLaws g := by
simpa [P₀, H₀] using hpair.1
have hPH₁ : (P₁, H₁) ∈ ExternalLaws g := by
simpa [P₁, H₁] using hpair.2.1
have hrel : releasedLaw P₀ = releasedLaw P₁ := by
simpa [P₀, P₁] using hpair.2.2
have hQ (P : Measure (FullRow ε J)) (H : Measure (ScoreSpace ε))
(hPH : (P, H) ∈ ExternalLaws g) :
Qj n m P H =
(Measure.pi (fun _ : Fin n => releasedLaw P)).prod
(Measure.pi (fun _ : Fin m => H)) := by
rw [hInd n m (P, H) hPH, hTrial n m (P, H) hPH,
hLog n m (P, H) hPH]
rw [hQ P₀ H₀ hPH₀, hQ P₁ H₁ hPH₁]
let ρ : Measure (Fin n → Observation J) :=
Measure.pi (fun _ : Fin n => releasedLaw P₀)
have hPrel : IsProbabilityMeasure (releasedLaw P₀) := by
exact @Measure.isProbabilityMeasure_map _ _ _ _ P₀ hPH₀.2.2.probabilityP _
(by unfold releasedRecord label arm observed; fun_prop :
Measurable (releasedRecord (ε := ε) (J := J))).aemeasurable
letI : IsProbabilityMeasure (releasedLaw P₀) := hPrel
letI : IsProbabilityMeasure ρ := by dsimp [ρ]; infer_instance
have htv := threeScoreCalibratedU_commonTrial_tv_bound ρ α hα x y z
hxy hyz m hm hw.1 hw.2.1 hw.2.2
rw [← hrel]
rw [tvDist_symm]
simpa [ρ, H₀, H₁, u, scoreLogTripleBaseline,
scoreLogTriplePerturbed, threeScoreBaseLaw, threeScorePerturbedLaw] using htv
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerScoreLogFinal 7 declarations Final specialization of the calibrated score-log lower certificate.
Final specialization of the calibrated score-log lower certificate.
theorem scoreLogTripleBaseline_sharpLength
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
r :
x y z :
hxy :
x < y
hyz :
y < z
hx :
g x = r
hy :
g y = r
hz :
g z = r
t :
ℝ
ht₁ :
(x : ℝ) * (1 / 3) < t
ht₂ :
t < min ((x : ℝ) * (1 / 3) + (y : ℝ) * (1 / 3)) ((z : ℝ) * (1 / 3))
hMass :
CompatibleCellMasses (scoreLogTripleBaseline x y z) g (releasedLaw (scoreLogTripleBaselineFullLaw g x y z (t / (((x : ℝ) + y + z) / 3))))
conclusion
sharpATELength (scoreLogTripleBaseline x y z) g (releasedLaw (scoreLogTripleBaselineFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)))) hMass
= threeScoreUpperEndpoint (x : ℝ) y (1 / 3) t - threeScoreLowerEndpoint (z : ℝ) t
Proof (Lean source)
lemma scoreLogTripleBaseline_sharpLength
{ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(r : LabelSpace J) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z)
(hx : g x = r) (hy : g y = r) (hz : g z = r)
(t : ℝ)
(ht₁ : (x : ℝ) * (1 / 3) < t)
(ht₂ : t < min ((x : ℝ) * (1 / 3) + (y : ℝ) * (1 / 3))
((z : ℝ) * (1 / 3)))
(hMass : CompatibleCellMasses (scoreLogTripleBaseline x y z) g
(releasedLaw (scoreLogTripleBaselineFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3))))) :
sharpATELength (scoreLogTripleBaseline x y z) g
(releasedLaw (scoreLogTripleBaselineFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)))) hMass =
threeScoreUpperEndpoint (x : ℝ) y (1 / 3) t -
threeScoreLowerEndpoint (z : ℝ) t := by
let q : ℝ := ((x : ℝ) + y + z) / 3
have hx0 : 0 < (x : ℝ) := lt_of_lt_of_le hOverlap.1 x.property.1
have hy0 : 0 < (y : ℝ) := lt_trans hx0 hxy
have hz0 : 0 < (z : ℝ) := lt_trans hy0 hyz
have hq : 0 < q := by dsimp [q]; linarith
have ht0 : 0 < t := lt_trans (mul_pos hx0 (by norm_num)) ht₁
have htq : t < q := by
have hleft := lt_of_lt_of_le ht₂ (min_le_left _ _)
dsimp [q]
nlinarith [mul_pos hz0 (by norm_num : (0 : ℝ) < 1 / 3)]
have hs0 : 0 ≤ t / q := (div_pos ht0 hq).le
have hs1 : t / q ≤ 1 := (div_lt_one hq).2 htq |>.le
have hq1 : q < 1 := by
have hx1 : (x : ℝ) < 1 := lt_of_le_of_lt x.property.2 (by linarith [hOverlap.1])
have hy1 : (y : ℝ) < 1 := lt_of_le_of_lt y.property.2 (by linarith [hOverlap.1])
have hz1 : (z : ℝ) < 1 := lt_of_le_of_lt z.property.2 (by linarith [hOverlap.1])
dsimp [q]
linarith
have hrel := scoreLogTripleBaseline_releasedLaw_cell
g hOverlap r x y z hx hy hz t hs0 hs1
have hc := muLower_muUpper_false_eq_zero_of_scoreLogReleasedCellLaw
(scoreLogTripleBaseline x y z) g
(releasedLaw (scoreLogTripleBaselineFullLaw g x y z (t / q))) hMass
r q t (1 - q) hrel hq.le (sub_pos.mpr hq1) hs0 hs1 (by ring)
unfold sharpATELength
rw [scoreLogTripleBaseline_muUpper_true g hOverlap r x y z hxy hyz
hx hy hz t ht₁ ht₂ hMass,
scoreLogTripleBaseline_muLower_true g hOverlap r x y z hxy hyz
hx hy hz t ht₁ ht₂ hMass,
hc.1, hc.2]
ring
theorem scoreLogTriplePerturbed_sharpLength
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
r :
x y z :
hxy :
x < y
hyz :
y < z
hx :
g x = r
hy :
g y = r
hz :
g z = r
t u :
ℝ
hu0 :
0 ≤ u
hu :
u * ((z : ℝ) - x) ≤ 1 / 3
ht₁ :
(x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) < t
ht₂ :
t < min ((x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) + (y : ℝ) * (1 / 3 - u * ((z : ℝ) - x))) ((z : ℝ) * (1 / 3 + u * ((y : ℝ) - x)))
hMass :
CompatibleCellMasses (scoreLogTriplePerturbed x y z u) g (releasedLaw (scoreLogTriplePerturbedFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)) u))
conclusion
sharpATELength (scoreLogTriplePerturbed x y z u) g (releasedLaw (scoreLogTriplePerturbedFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)) u)) hMass
= threeScoreUpperEndpoint (x : ℝ) y (1 / 3 + u * ((z : ℝ) - y)) t
- threeScoreLowerEndpoint (z : ℝ) t
Proof (Lean source)
lemma scoreLogTriplePerturbed_sharpLength
{ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(r : LabelSpace J) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z)
(hx : g x = r) (hy : g y = r) (hz : g z = r)
(t u : ℝ) (hu0 : 0 ≤ u) (hu : u * ((z : ℝ) - x) ≤ 1 / 3)
(ht₁ : (x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) < t)
(ht₂ : t < min
((x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) +
(y : ℝ) * (1 / 3 - u * ((z : ℝ) - x)))
((z : ℝ) * (1 / 3 + u * ((y : ℝ) - x))))
(hMass : CompatibleCellMasses (scoreLogTriplePerturbed x y z u) g
(releasedLaw (scoreLogTriplePerturbedFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)) u))) :
sharpATELength (scoreLogTriplePerturbed x y z u) g
(releasedLaw (scoreLogTriplePerturbedFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)) u)) hMass =
threeScoreUpperEndpoint (x : ℝ) y
(1 / 3 + u * ((z : ℝ) - y)) t -
threeScoreLowerEndpoint (z : ℝ) t := by
let q : ℝ := ((x : ℝ) + y + z) / 3
let wx : ℝ := 1 / 3 + u * ((z : ℝ) - y)
let wy : ℝ := 1 / 3 - u * ((z : ℝ) - x)
let wz : ℝ := 1 / 3 + u * ((y : ℝ) - x)
have hw := scoreLogTriple_weights_nonneg x y z hxy hyz u hu0 hu
have hmoment := (scoreLogTriple_weight_identities x y z hxy hyz u).2
have hx0 : 0 < (x : ℝ) := lt_of_lt_of_le hOverlap.1 x.property.1
have hy0 : 0 < (y : ℝ) := lt_trans hx0 hxy
have hz0 : 0 < (z : ℝ) := lt_trans hy0 hyz
have hq : 0 < q := by dsimp [q]; linarith
have ht0 : 0 < t := by
have : 0 ≤ (x : ℝ) * wx := mul_nonneg hx0.le (by simpa [wx] using hw.1)
exact lt_of_le_of_lt this (by simpa [wx] using ht₁)
have htq : t < q := by
have hleft := lt_of_lt_of_le ht₂ (min_le_left _ _)
have hwz : 0 ≤ wz := by simpa [wz] using hw.2.2
have hmom : (x : ℝ) * wx + (y : ℝ) * wy + (z : ℝ) * wz = q := by
dsimp [wx, wy, wz, q]
calc
_ = (x : ℝ) / 3 + (y : ℝ) / 3 + (z : ℝ) / 3 := hmoment
_ = ((x : ℝ) + y + z) / 3 := by ring
have hleft' : t < (x : ℝ) * wx + (y : ℝ) * wy := by simpa [wx, wy] using hleft
nlinarith [mul_nonneg hz0.le hwz]
have hs0 : 0 ≤ t / q := (div_pos ht0 hq).le
have hs1 : t / q ≤ 1 := (div_lt_one hq).2 htq |>.le
have hq1 : q < 1 := by
have hx1 : (x : ℝ) < 1 := lt_of_le_of_lt x.property.2 (by linarith [hOverlap.1])
have hy1 : (y : ℝ) < 1 := lt_of_le_of_lt y.property.2 (by linarith [hOverlap.1])
have hz1 : (z : ℝ) < 1 := lt_of_le_of_lt z.property.2 (by linarith [hOverlap.1])
dsimp [q]
linarith
have hrel := scoreLogTriplePerturbed_releasedLaw_cell
g hOverlap r x y z hxy hyz hx hy hz t u hs0 hs1 hu0 hu
have hc := muLower_muUpper_false_eq_zero_of_scoreLogReleasedCellLaw
(scoreLogTriplePerturbed x y z u) g
(releasedLaw (scoreLogTriplePerturbedFullLaw g x y z (t / q) u)) hMass
r q t (1 - q) hrel hq.le (sub_pos.mpr hq1) hs0 hs1 (by ring)
unfold sharpATELength
rw [scoreLogTriplePerturbed_muUpper_true g hOverlap r x y z hxy hyz
hx hy hz t u hu0 hu ht₁ ht₂ hMass,
scoreLogTriplePerturbed_muLower_true g hOverlap r x y z hxy hyz
hx hy hz t u hu0 hu ht₁ ht₂ hMass,
hc.1, hc.2]
ring
theorem scoreLogTriple_endpoint_ate_separation
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
r :
x y z :
hxy :
x < y
hyz :
y < z
hx :
g x = r
hy :
g y = r
hz :
g z = r
t u :
ℝ
hu0 :
0 ≤ u
hu :
u * ((z : ℝ) - x) ≤ 1 / 3
hb₁ :
(x : ℝ) * (1 / 3) < t
hb₂ :
t < min ((x : ℝ) * (1 / 3) + (y : ℝ) * (1 / 3)) ((z : ℝ) * (1 / 3))
hp₁ :
(x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) < t
hp₂ :
t < min ((x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) + (y : ℝ) * (1 / 3 - u * ((z : ℝ) - x))) ((z : ℝ) * (1 / 3 + u * ((y : ℝ) - x)))
hMass₀ :
CompatibleCellMasses (scoreLogTripleBaseline x y z) g (releasedLaw (scoreLogTripleBaselineFullLaw g x y z (t / (((x : ℝ) + y + z) / 3))))
hMass₁ :
CompatibleCellMasses (scoreLogTriplePerturbed x y z u) g (releasedLaw (scoreLogTriplePerturbedFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)) u))
conclusion
ate (endpointFullLaw (scoreLogTriplePerturbed x y z u) g (releasedLaw (scoreLogTriplePerturbedFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)) u)) hMass₁ hOverlap false true)
- ate (endpointFullLaw (scoreLogTripleBaseline x y z) g (releasedLaw (scoreLogTripleBaselineFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)))) hMass₀ hOverlap true false)
= sharpATELength (scoreLogTripleBaseline x y z) g (releasedLaw (scoreLogTripleBaselineFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)))) hMass₀
+ u * ((z : ℝ) - y) * (1 - (x : ℝ) / y)
Proof (Lean source)
lemma scoreLogTriple_endpoint_ate_separation
{ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(r : LabelSpace J) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z)
(hx : g x = r) (hy : g y = r) (hz : g z = r)
(t u : ℝ) (hu0 : 0 ≤ u) (hu : u * ((z : ℝ) - x) ≤ 1 / 3)
(hb₁ : (x : ℝ) * (1 / 3) < t)
(hb₂ : t < min ((x : ℝ) * (1 / 3) + (y : ℝ) * (1 / 3))
((z : ℝ) * (1 / 3)))
(hp₁ : (x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) < t)
(hp₂ : t < min
((x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) +
(y : ℝ) * (1 / 3 - u * ((z : ℝ) - x)))
((z : ℝ) * (1 / 3 + u * ((y : ℝ) - x))))
(hMass₀ : CompatibleCellMasses (scoreLogTripleBaseline x y z) g
(releasedLaw (scoreLogTripleBaselineFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)))))
(hMass₁ : CompatibleCellMasses (scoreLogTriplePerturbed x y z u) g
(releasedLaw (scoreLogTriplePerturbedFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)) u))) :
ate (endpointFullLaw (scoreLogTriplePerturbed x y z u) g
(releasedLaw (scoreLogTriplePerturbedFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)) u)) hMass₁ hOverlap false true) -
ate (endpointFullLaw (scoreLogTripleBaseline x y z) g
(releasedLaw (scoreLogTripleBaselineFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)))) hMass₀ hOverlap true false) =
sharpATELength (scoreLogTripleBaseline x y z) g
(releasedLaw (scoreLogTripleBaselineFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)))) hMass₀ +
u * ((z : ℝ) - y) * (1 - (x : ℝ) / y) := by
have hA₀ := scoreLogTripleBaseline_lowerEndpoint_ate
g hOverlap r x y z hxy hyz hx hy hz t hb₁ hb₂ hMass₀
have hA₁ := scoreLogTriplePerturbed_upperEndpoint_ate
g hOverlap r x y z hxy hyz hx hy hz t u hu0 hu hp₁ hp₂ hMass₁
have hW₁ := scoreLogTriplePerturbed_sharpLength
g hOverlap r x y z hxy hyz hx hy hz t u hu0 hu hp₁ hp₂ hMass₁
have hshift := scoreLogTriple_sharpATELength_perturbation
g hOverlap r x y z hxy hyz hx hy hz t u hu0 hu hb₁ hb₂ hp₁ hp₂
hMass₀ hMass₁
rw [hA₁, hA₀, ← hW₁]
linarith
theorem externalExperiment_tv_eq_of_releasedLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
Qj :
hTrial :
JointTrialIID g Qj
hLog :
ExternalLogIID g Qj
hInd :
H0 H1 :
Measure (ScoreSpace ε)
hPH0 :
(P0, H0) ∈ ExternalLaws g
hPH0' :
(P0', H0) ∈ ExternalLaws g
hPH1 :
(P1, H1) ∈ ExternalLaws g
hPH1' :
(P1', H1) ∈ ExternalLaws g
hrel0 :
releasedLaw P0' = releasedLaw P0
hrel1 :
releasedLaw P1' = releasedLaw P1
Proof (Lean source)
lemma externalExperiment_tv_eq_of_releasedLaw
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J)
(Qj : Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : JointTrialIID g Qj) (hLog : ExternalLogIID g Qj)
(hInd : ExternalLogIndependent g Qj)
(P0 P0' P1 P1' : Measure (FullRow ε J))
(H0 H1 : Measure (ScoreSpace ε))
(hPH0 : (P0, H0) ∈ ExternalLaws g)
(hPH0' : (P0', H0) ∈ ExternalLaws g)
(hPH1 : (P1, H1) ∈ ExternalLaws g)
(hPH1' : (P1', H1) ∈ ExternalLaws g)
(hrel0 : releasedLaw P0' = releasedLaw P0)
(hrel1 : releasedLaw P1' = releasedLaw P1) :
tvDist (Qj P0' H0) (Qj P1' H1) =
tvDist (Qj P0 H0) (Qj P1 H1) := by
have hQ (P : Measure (FullRow ε J)) (H : Measure (ScoreSpace ε))
(hPH : (P, H) ∈ ExternalLaws g) :
Qj P H = (Measure.pi (fun _ : Fin n => releasedLaw P)).prod
(Measure.pi (fun _ : Fin m => H)) := by
rw [hInd (P, H) hPH, hTrial (P, H) hPH, hLog (P, H) hPH]
rw [hQ P0' H0 hPH0', hQ P1' H1 hPH1', hQ P0 H0 hPH0,
hQ P1 H1 hPH1, hrel0, hrel1]
def threeScoreFixedLogCertificate
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
threeScoreFixedLogCertificate α x y z :
ℝ
given by
(1 - 2 * α) * sqrt (log (1 + (1 - 2 * α) ^ 2)) / (2 * sqrt 3) * ((((z : ℝ) - y) * (1 - (x : ℝ) / y)) / sqrt (threeScoreGapSq x y z))
theorem scoreLogFixedTriple_eventually_externalExcessRisk_lower
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
r :
x y z :
hxy :
x < y
hyz :
y < z
hx :
g x = r
hy :
g y = r
hz :
g z = r
α :
ℝ
hα :
0 < α ∧ α < 1 / 2
Qj :
∀ n m
hTrial :
∀ n m, JointTrialIID g (Qj n m)
hLog :
∀ n m, ExternalLogIID g (Qj n m)
hInd :
∀ n m, ExternalLogIndependent g (Qj n m)
conclusion
≤ externalExcessRisk g (Qj n m) α
Proof (Lean source)
lemma scoreLogFixedTriple_eventually_externalExcessRisk_lower
{ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(hg : Measurable g) (r : LabelSpace J) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z)
(hx : g x = r) (hy : g y = r) (hz : g z = r)
(α : ℝ) (hα : 0 < α ∧ α < 1 / 2)
(Qj : ∀ n m,
Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : ∀ n m, JointTrialIID g (Qj n m))
(hLog : ∀ n m, ExternalLogIID g (Qj n m))
(hInd : ∀ n m, ExternalLogIndependent g (Qj n m)) :
∀ᶠ m : ℕ in atTop, ∀ n : ℕ, 0 < n →
threeScoreFixedLogCertificate α x y z *
(sqrt (m : ℝ))⁻¹ ≤
externalExcessRisk g (Qj n m) α := by
let t := threeScoreThreshold x y z
let q : ℝ := ((x : ℝ) + y + z) / 3
have hx0 : 0 < (x : ℝ) := lt_of_lt_of_le hOverlap.1 x.property.1
have hb := threeScoreThreshold_baseline x y z hx0 hxy hyz
dsimp only at hb
have hs0 : 0 ≤ t / q := by simpa [t, q] using hb.2.2.1.le
have hs1 : t / q ≤ 1 := by simpa [t, q] using hb.2.2.2.le
have htv := scoreLogCalibrated_jointExperiment_eventually_tv_bound
g hOverlap hg r x y z hxy hyz hx hy hz t α hα hs0 hs1
Qj hTrial hLog hInd
filter_upwards [threeScoreCalibratedU_eventually_thresholds
α hα x y z hx0 hxy hyz, htv, eventually_gt_atTop (0 : ℕ)]
with m hth htv_m hm
intro n hn
dsimp only at hth
let u := threeScoreCalibratedU α x y z m
let Pbase := scoreLogTripleBaselineFullLaw g x y z (t / q)
let Hbase := scoreLogTripleBaseline x y z
let Ppert := scoreLogTriplePerturbedFullLaw g x y z (t / q) u
let Hpert := scoreLogTriplePerturbed x y z u
have hpair := scoreLogTriple_externalPair g hOverlap hg r x y z hxy hyz
hx hy hz (t / q) hs0 hs1 u hth.2.2.2.1 hth.2.2.2.2.1
have hPHbase : (Pbase, Hbase) ∈ ExternalLaws g := by
simpa [Pbase, Hbase, q] using hpair.1
have hPHpert : (Ppert, Hpert) ∈ ExternalLaws g := by
simpa [Ppert, Hpert, q] using hpair.2.1
let hMass0 := externalLawCellMasses g (Pbase, Hbase) hPHbase
let hMass1 := externalLawCellMasses g (Ppert, Hpert) hPHpert
let E0 := endpointFullLaw Hbase g (releasedLaw Pbase) hMass0 hOverlap true false
let E1 := endpointFullLaw Hpert g (releasedLaw Ppert) hMass1 hOverlap false true
have hE0 : (E0, Hbase) ∈ ExternalLaws g := by
exact endpointFullLaw_mem_externalLaws Hbase g (releasedLaw Pbase)
hMass0 hOverlap true false
have hE1 : (E1, Hpert) ∈ ExternalLaws g := by
exact endpointFullLaw_mem_externalLaws Hpert g (releasedLaw Ppert)
hMass1 hOverlap false true
have hrel0 : releasedLaw E0 = releasedLaw Pbase := by
exact endpointFullLaw_releasedLaw Hbase g (releasedLaw Pbase)
hMass0 hOverlap true false
have hrel1 : releasedLaw E1 = releasedLaw Ppert := by
exact endpointFullLaw_releasedLaw Hpert g (releasedLaw Ppert)
hMass1 hOverlap false true
have htvEndpoint : tvDist
(Qj n m E0 Hbase) (Qj n m E1 Hpert) ≤ (1 - 2 * α) / 2 := by
rw [externalExperiment_tv_eq_of_releasedLaw g (Qj n m)
(hTrial n m) (hLog n m) (hInd n m)
Pbase E0 Ppert E1 Hbase Hpert hPHbase hE0 hPHpert hE1 hrel0 hrel1]
simpa [Pbase, Hbase, Ppert, Hpert, u, t, q] using htv_m n
have hW0 := scoreLogTripleBaseline_sharpLength
g hOverlap r x y z hxy hyz hx hy hz t hb.1 hb.2.1 hMass0
have hsep := scoreLogTriple_endpoint_ate_separation
g hOverlap r x y z hxy hyz hx hy hz t u hth.2.2.2.1
hth.2.2.2.2.1 hb.1 hb.2.1 hth.2.2.2.2.2.1 hth.2.2.2.2.2.2
hMass0 hMass1
let Delta : ℝ := u * ((z : ℝ) - y) * (1 - (x : ℝ) / y)
have hu0 : 0 < u := threeScoreCalibratedU_pos α hα x y z hxy hyz m
hm
have hy0 : 0 < (y : ℝ) := lt_trans hx0 hxy
have hDelta : 0 < Delta := by
dsimp [Delta]
exact mul_pos (mul_pos hu0 (sub_pos.mpr hyz))
(sub_pos.mpr ((div_lt_one hy0).2 hxy))
have hW0nonneg : 0 ≤ sharpATELength Hbase g (releasedLaw Pbase) hMass0 :=
sharpATELength_nonneg_of_overlap Hbase g (releasedLaw Pbase)
hMass0 hOverlap
have htheta : ate E0 ≤ ate E1 := by
have : ate E1 - ate E0 =
sharpATELength Hbase g (releasedLaw Pbase) hMass0 + Delta := by
simpa [E0, E1, Pbase, Hbase, Ppert, Hpert, hMass0, hMass1,
t, q, u, Delta] using hsep
linarith
have hSharpEq :
sharpATELength Hbase g (releasedLaw E0)
(externalLawCellMasses g (E0, Hbase) hE0) =
sharpATELength Hbase g (releasedLaw Pbase) hMass0 := by
simpa only [hrel0]
have hrisk := externalExcessRisk_lower_of_twoPoint g α hα (Qj n m)
(hTrial n m) (hLog n m) (hInd n m) E0 E1 Hbase Hpert hE0 hE1
(sharpATELength Hbase g (releasedLaw Pbase) hMass0) Delta
hSharpEq htheta (by
simpa [E0, E1, Pbase, Hbase, Ppert, Hpert, hMass0, hMass1,
t, q, u, Delta] using hsep) hDelta htvEndpoint
have hcoef : threeScoreFixedLogCertificate α x y z *
(sqrt (m : ℝ))⁻¹ = Delta * ((1 - 2 * α) / 2) := by
dsimp [threeScoreFixedLogCertificate, Delta, u, threeScoreCalibratedU]
simp only [div_eq_mul_inv, mul_inv]
ring
rw [hcoef]
exact hrisk
theorem scoreLog_logCertificate_liminf
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
α :
ℝ
hα :
0 < α ∧ α < 1 / 2
Qj :
∀ n m
hTrial :
∀ n m, JointTrialIID g (Qj n m)
hLog :
∀ n m, ExternalLogIID g (Qj n m)
hInd :
∀ n m, ExternalLogIndependent g (Qj n m)
C :
ℝ
hUpper :
∀ n m : ℕ
conclusion
logCertificate g α
≤ liminf (fun m : ℕ => sInf {v : ℝ | ∃ n : ℕ, 0 < n ∧ v = sqrt (m : ℝ) * externalExcessRisk g (Qj n m) α}) atTop
Proof (Lean source)
lemma scoreLog_logCertificate_liminf
{ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(hg : Measurable g) (α : ℝ) (hα : 0 < α ∧ α < 1 / 2)
(Qj : ∀ n m,
Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : ∀ n m, JointTrialIID g (Qj n m))
(hLog : ∀ n m, ExternalLogIID g (Qj n m))
(hInd : ∀ n m, ExternalLogIndependent g (Qj n m))
(C : ℝ)
(hUpper : ∀ n m : ℕ, 0 < n → 0 < m →
externalNormalizer n m * externalExcessRisk g (Qj n m) α ≤ C) :
logCertificate g α ≤
liminf (fun m : ℕ => sInf {v : ℝ | ∃ n : ℕ, 0 < n ∧
v = sqrt (m : ℝ) * externalExcessRisk g (Qj n m) α}) atTop := by
let K : ℝ := (1 - 2 * α) *
sqrt (log (1 + (1 - 2 * α) ^ 2)) / (2 * sqrt 3)
have hβ : 0 < 1 - 2 * α := by linarith [hα.2]
have hlog : 0 < log (1 + (1 - 2 * α) ^ 2) :=
Real.log_pos (by nlinarith [sq_pos_of_pos hβ])
have hK : 0 < K := by dsimp [K]; positivity
apply le_of_forall_lt
intro c hc
have hcK : c / K < fiberTripleSeparation g := by
apply (div_lt_iff₀ hK).2
simpa [logCertificate, K, mul_comm] using hc
let T : Set ℝ := {v | ∃ (r : LabelSpace J)
(e₁ e₂ e₃ : ScoreSpace ε),
e₁ < e₂ ∧ e₂ < e₃ ∧ g e₁ = r ∧ g e₂ = r ∧ g e₃ = r ∧
v = (((e₃ : ℝ) - e₂) * (1 - (e₁ : ℝ) / e₂)) /
sqrt (((e₃ : ℝ) - e₂) ^ 2 +
((e₃ : ℝ) - e₁) ^ 2 + ((e₂ : ℝ) - e₁) ^ 2)}
have hTne : T.Nonempty := by
obtain ⟨r, x, y, z, hxy, hyz, hx, hy, hz⟩ :=
fiberTriple_exists g hOverlap
exact ⟨_, r, x, y, z, hxy, hyz, hx, hy, hz, rfl⟩
have hcSup : c / K < sSup T := by
simpa [fiberTripleSeparation, T] using hcK
obtain ⟨v, hvT, hcv⟩ := exists_lt_of_lt_csSup hTne hcSup
rcases hvT with ⟨r, x, y, z, hxy, hyz, hx, hy, hz, rfl⟩
have hfixed := scoreLogFixedTriple_eventually_externalExcessRisk_lower
g hOverlap hg r x y z hxy hyz hx hy hz α hα Qj hTrial hLog hInd
have hlim := externalExcessRisk_log_liminf_of_eventual_uniform
g Qj α (threeScoreFixedLogCertificate α x y z) C hfixed hUpper
have hcoeff : K *
((((z : ℝ) - y) * (1 - (x : ℝ) / y)) /
sqrt (((z : ℝ) - y) ^ 2 + ((z : ℝ) - x) ^ 2 +
((y : ℝ) - x) ^ 2)) =
threeScoreFixedLogCertificate α x y z := by
simp only [threeScoreFixedLogCertificate, threeScoreGapSq, K]
have hcFixed : c < threeScoreFixedLogCertificate α x y z := by
have := (div_lt_iff₀ hK).1 hcv
rw [mul_comm, hcoeff] at this
exact this
exact hcFixed.trans_le hlim
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerScoreLogFullLaw 32 declarations Full causal laws generated by a score atom for the score-log lower pair.
Full causal laws generated by a score atom for the score-log lower pair.
def scoreLogArmLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
def scoreLogAtomRow
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
theorem scoreLogAtomRow_measurable
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma scoreLogAtomRow_measurable {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (e : ScoreSpace ε) :
Measurable (fun ya : OutcomeSpace × ArmSpace =>
scoreLogAtomRow g e ya.1 ya.2) := by
have hcond : MeasurableSet {ya : OutcomeSpace × ArmSpace | ya.2 = true} :=
measurableSet_eq_fun measurable_snd measurable_const
have hobs : Measurable (fun ya : OutcomeSpace × ArmSpace =>
if ya.2 then ya.1 else trialZeroOutcome) :=
Measurable.ite hcond measurable_fst measurable_const
unfold scoreLogAtomRow
fun_prop (disch := assumption)
theorem scoreLogArmLaw_isProbabilityMeasure
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma scoreLogArmLaw_isProbabilityMeasure {ε : ℝ}
(e : ScoreSpace ε) (hOverlap : Overlap ε) :
IsProbabilityMeasure (scoreLogArmLaw e) := by
have he0 : 0 ≤ (e : ℝ) := le_trans hOverlap.1.le e.property.1
have he1 : (e : ℝ) ≤ 1 := by linarith [e.property.2, hOverlap.1]
rw [isProbabilityMeasure_iff]
unfold scoreLogArmLaw
rw [Measure.add_apply, Measure.smul_apply, Measure.smul_apply]
simp only [Measure.dirac_apply, Set.mem_univ, Set.indicator_of_mem,
Pi.one_apply, smul_eq_mul, mul_one]
rw [← ENNReal.ofReal_add he0 (sub_nonneg.mpr he1)]
norm_num
theorem scoreLogArmLaw_true_mass
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma scoreLogArmLaw_true_mass {ε : ℝ}
(e : ScoreSpace ε) (hOverlap : Overlap ε) :
(scoreLogArmLaw e).real {true} = (e : ℝ) := by
have he0 : 0 ≤ (e : ℝ) := le_trans hOverlap.1.le e.property.1
unfold scoreLogArmLaw
rw [Measure.real_def, Measure.add_apply, Measure.smul_apply, Measure.smul_apply]
simp [he0]
def scoreLogAtomFullLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
given by
theorem scoreLogAtomFullLaw_isProbabilityMeasure
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma scoreLogAtomFullLaw_isProbabilityMeasure {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(e : ScoreSpace ε) (s : ℝ) (hs0 : 0 ≤ s) (hs1 : s ≤ 1) :
IsProbabilityMeasure (scoreLogAtomFullLaw g e s) := by
letI : IsProbabilityMeasure (trialOutcomeLaw s) :=
trialOutcomeLaw_isProbabilityMeasure s hs0 hs1
letI : IsProbabilityMeasure (scoreLogArmLaw e) :=
scoreLogArmLaw_isProbabilityMeasure e hOverlap
unfold scoreLogAtomFullLaw
exact Measure.isProbabilityMeasure_map
(scoreLogAtomRow_measurable g e).aemeasurable
theorem scoreLogAtomFullLaw_scoreMarginal
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
ScoreMarginal (scoreLogAtomFullLaw g e s) (Measure.dirac e)
Proof (Lean source)
lemma scoreLogAtomFullLaw_scoreMarginal {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(e : ScoreSpace ε) (s : ℝ) (hs0 : 0 ≤ s) (hs1 : s ≤ 1) :
ScoreMarginal (scoreLogAtomFullLaw g e s) (Measure.dirac e) := by
letI : IsProbabilityMeasure (trialOutcomeLaw s) :=
trialOutcomeLaw_isProbabilityMeasure s hs0 hs1
letI : IsProbabilityMeasure (scoreLogArmLaw e) :=
scoreLogArmLaw_isProbabilityMeasure e hOverlap
have hrow := scoreLogAtomRow_measurable g e
unfold ScoreMarginal scoreLogAtomFullLaw
rw [Measure.map_map (by unfold score; fun_prop) hrow]
change ((trialOutcomeLaw s).prod (scoreLogArmLaw e)).map (fun _ => e) = _
rw [Measure.map_const]
simp
theorem scoreLogAtomFullLaw_consistency
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Consistency (scoreLogAtomFullLaw g e s)
Proof (Lean source)
lemma scoreLogAtomFullLaw_consistency {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (e : ScoreSpace ε) (s : ℝ) :
Consistency (scoreLogAtomFullLaw g e s) := by
have hrow := scoreLogAtomRow_measurable g e
have hcond : MeasurableSet {ω : FullRow ε J | arm ω = true} :=
measurableSet_eq_fun (by unfold arm; fun_prop) measurable_const
have hpot : Measurable (fun ω : FullRow ε J =>
if arm ω then outcome1 ω else outcome0 ω) :=
Measurable.ite hcond (by unfold outcome1; fun_prop)
(by unfold outcome0; fun_prop)
have hset : MeasurableSet {ω : FullRow ε J |
observed ω = if arm ω then outcome1 ω else outcome0 ω} :=
measurableSet_eq_fun (by unfold observed; fun_prop) hpot
unfold Consistency scoreLogAtomFullLaw
rw [ae_map_iff hrow.aemeasurable hset]
filter_upwards [] with ya
rfl
theorem scoreLogAtomFullLaw_deterministicRelease
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
DeterministicRelease g (scoreLogAtomFullLaw g e s)
Proof (Lean source)
lemma scoreLogAtomFullLaw_deterministicRelease {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (e : ScoreSpace ε) (s : ℝ)
(hg : Measurable g) :
DeterministicRelease g (scoreLogAtomFullLaw g e s) := by
have hrow := scoreLogAtomRow_measurable g e
have hset : MeasurableSet {ω : FullRow ε J | label ω = g (score ω)} :=
measurableSet_eq_fun (by unfold label; fun_prop)
(hg.comp (by unfold score; fun_prop))
unfold DeterministicRelease scoreLogAtomFullLaw
rw [ae_map_iff hrow.aemeasurable hset]
filter_upwards [] with ya
rfl
theorem scoreLogAtomFullLaw_randomizedAssignment
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma scoreLogAtomFullLaw_randomizedAssignment {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(e : ScoreSpace ε) (s : ℝ) (hs0 : 0 ≤ s) (hs1 : s ≤ 1) :
RandomizedAssignment (scoreLogAtomFullLaw g e s) := by
intro B hB
let μ : Measure OutcomeSpace := trialOutcomeLaw s
let ν : Measure ArmSpace := scoreLogArmLaw e
let f : OutcomeSpace × ArmSpace → FullRow ε J :=
fun ya => scoreLogAtomRow g e ya.1 ya.2
have hf : Measurable f := scoreLogAtomRow_measurable g e
let U : Set OutcomeSpace := {y | (e, trialZeroOutcome, y) ∈ B}
have hU : MeasurableSet U := by
exact hB.preimage (by fun_prop : Measurable
(fun y : OutcomeSpace => (e, trialZeroOutcome, y)))
let S : Set (FullRow ε J) :=
{ω | arm ω = true ∧ latentTriple ω ∈ B}
let T : Set (FullRow ε J) := {ω | latentTriple ω ∈ B}
have hlatent : Measurable (latentTriple (ε := ε) (J := J)) := by
unfold latentTriple score outcome0 outcome1
fun_prop
have hS : MeasurableSet S :=
(measurableSet_eq_fun (by unfold arm; fun_prop) measurable_const).inter
(hB.preimage hlatent)
have hT : MeasurableSet T := hB.preimage hlatent
have hpreS : f ⁻¹' S = U ×ˢ {true} := by
ext ya
cases ha : ya.2 <;> simp [f, S, U, scoreLogAtomRow, arm,
latentTriple, score, outcome0, outcome1, ha]
have hpreT : f ⁻¹' T = U ×ˢ univ := by
ext ya
simp [f, T, U, scoreLogAtomRow, latentTriple, score, outcome0, outcome1]
have hνprob : IsProbabilityMeasure ν :=
scoreLogArmLaw_isProbabilityMeasure e hOverlap
have hνuniv : ν.real univ = 1 := isProbabilityMeasure_iff_real.mp hνprob
change (scoreLogAtomFullLaw g e s).real S =
∫ ω in T, (score ω : ℝ) ∂scoreLogAtomFullLaw g e s
have hleft : (scoreLogAtomFullLaw g e s).real S = μ.real U * (e : ℝ) := by
unfold scoreLogAtomFullLaw
rw [MeasureTheory.map_measureReal_apply hf hS, hpreS,
measureReal_prod_prod]
rw [show ν.real {true} = (e : ℝ) from scoreLogArmLaw_true_mass e hOverlap]
have hright : (∫ ω in T, (score ω : ℝ)
∂scoreLogAtomFullLaw g e s) = μ.real U * (e : ℝ) := by
unfold scoreLogAtomFullLaw
rw [MeasureTheory.setIntegral_map hT (by unfold score; fun_prop)
hf.aemeasurable]
rw [hpreT]
change ∫ _ in U ×ˢ univ, (e : ℝ) ∂(μ.prod ν) = _
rw [setIntegral_const, measureReal_prod_prod, hνuniv]
simp only [smul_eq_mul, mul_one]
exact hleft.trans hright.symm
theorem randomizedAssignment_smul_of_ne_top
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
RandomizedAssignment (c • P)
Proof (Lean source)
lemma randomizedAssignment_smul_of_ne_top {ε : ℝ} {J : ℕ}
(P : Measure (FullRow ε J)) (hP : RandomizedAssignment P)
(c : ENNReal) (hc : c ≠ ∞) : RandomizedAssignment (c • P) := by
intro B hB
have hS : MeasurableSet {ω : FullRow ε J |
arm ω = true ∧ (score ω, outcome0 ω, outcome1 ω) ∈ B} := by
exact (measurableSet_eq_fun (by unfold arm; fun_prop) measurable_const).inter
(hB.preimage (by unfold score outcome0 outcome1; fun_prop))
rw [Measure.real_def]
change ((c • P) {ω | arm ω = true ∧
(score ω, outcome0 ω, outcome1 ω) ∈ B}).toReal =
∫ ω, (score ω : ℝ) ∂(c • P).restrict
{ω | (score ω, outcome0 ω, outcome1 ω) ∈ B}
rw [Measure.smul_apply, Measure.restrict_smul, integral_smul_measure]
simp only [smul_eq_mul, ENNReal.toReal_mul]
have hr := hP B hB
rw [Measure.real_def] at hr
rw [hr]
theorem score_integrable_finite
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Integrable (fun ω => (score ω : ℝ)) P
Proof (Lean source)
lemma score_integrable_finite {ε : ℝ} {J : ℕ}
(P : Measure (FullRow ε J)) [IsFiniteMeasure P] :
Integrable (fun ω => (score ω : ℝ)) P := by
apply Integrable.of_bound (by unfold score; fun_prop) (|ε| + |1 - ε| + 1)
filter_upwards [] with ω
change |((score ω : ScoreSpace ε) : ℝ)| ≤ |ε| + |1 - ε| + 1
rcases (score ω).property with ⟨he₁, he₂⟩
have h₁ := neg_le_abs ε
have h₂ := le_abs_self (1 - ε)
have h₃ := neg_le_abs (1 - ε)
have h₄ := le_abs_self ε
exact abs_le.mpr ⟨by linarith, by linarith⟩
theorem randomizedAssignment_add
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma randomizedAssignment_add {ε : ℝ} {J : ℕ}
(P Q : Measure (FullRow ε J)) [IsFiniteMeasure P] [IsFiniteMeasure Q]
(hP : RandomizedAssignment P) (hQ : RandomizedAssignment Q) :
RandomizedAssignment (P + Q) := by
intro B hB
let T : Set (FullRow ε J) :=
{ω | (score ω, outcome0 ω, outcome1 ω) ∈ B}
have hT : MeasurableSet T :=
hB.preimage (by unfold score outcome0 outcome1; fun_prop)
have hS : MeasurableSet {ω : FullRow ε J |
arm ω = true ∧ (score ω, outcome0 ω, outcome1 ω) ∈ B} := by
exact (measurableSet_eq_fun (by unfold arm; fun_prop) measurable_const).inter hT
change (P + Q).real _ = ∫ ω in T, (score ω : ℝ) ∂(P + Q)
rw [Measure.real_def, Measure.add_apply _ _ _,
ENNReal.toReal_add (measure_ne_top P _) (measure_ne_top Q _)]
rw [Measure.restrict_add,
integral_add_measure (score_integrable_finite P).integrableOn
(score_integrable_finite Q).integrableOn]
change P.real _ + Q.real _ = _
rw [hP B hB, hQ B hB]
theorem scoreLogTripleFullLaw_randomizedAssignment
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
x y z :
s :
ℝ
hs0 :
0 ≤ s
hs1 :
s ≤ 1
w₁ w₂ w₃ :
ℝ
conclusion
RandomizedAssignment (ofReal w₁ • scoreLogAtomFullLaw g x s + ofReal w₂ • scoreLogAtomFullLaw g y s + ofReal w₃ • scoreLogAtomFullLaw g z s)
Proof (Lean source)
lemma scoreLogTripleFullLaw_randomizedAssignment {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(x y z : ScoreSpace ε) (s : ℝ) (hs0 : 0 ≤ s) (hs1 : s ≤ 1)
(w₁ w₂ w₃ : ℝ) :
RandomizedAssignment
(ofReal w₁ • scoreLogAtomFullLaw g x s +
ofReal w₂ • scoreLogAtomFullLaw g y s +
ofReal w₃ • scoreLogAtomFullLaw g z s) := by
letI : IsProbabilityMeasure (scoreLogAtomFullLaw g x s) :=
scoreLogAtomFullLaw_isProbabilityMeasure g hOverlap x s hs0 hs1
letI : IsProbabilityMeasure (scoreLogAtomFullLaw g y s) :=
scoreLogAtomFullLaw_isProbabilityMeasure g hOverlap y s hs0 hs1
letI : IsProbabilityMeasure (scoreLogAtomFullLaw g z s) :=
scoreLogAtomFullLaw_isProbabilityMeasure g hOverlap z s hs0 hs1
have hx := randomizedAssignment_smul_of_ne_top _
(scoreLogAtomFullLaw_randomizedAssignment g hOverlap x s hs0 hs1)
(ofReal w₁) (by simp)
have hy := randomizedAssignment_smul_of_ne_top _
(scoreLogAtomFullLaw_randomizedAssignment g hOverlap y s hs0 hs1)
(ofReal w₂) (by simp)
have hz := randomizedAssignment_smul_of_ne_top _
(scoreLogAtomFullLaw_randomizedAssignment g hOverlap z s hs0 hs1)
(ofReal w₃) (by simp)
letI : IsFiniteMeasure
(ofReal w₁ • scoreLogAtomFullLaw g x s) := ⟨by simp⟩
letI : IsFiniteMeasure
(ofReal w₂ • scoreLogAtomFullLaw g y s) := ⟨by simp⟩
letI : IsFiniteMeasure
(ofReal w₃ • scoreLogAtomFullLaw g z s) := ⟨by simp⟩
exact randomizedAssignment_add _ _ (randomizedAssignment_add _ _ hx hy) hz
def scoreLogTripleBaselineFullLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
given by
ofReal (1 / 3 : ℝ) • scoreLogAtomFullLaw g x s
+ ofReal (1 / 3 : ℝ) • scoreLogAtomFullLaw g y s
+ ofReal (1 / 3 : ℝ) • scoreLogAtomFullLaw g z s
def scoreLogTriplePerturbedFullLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
given by
ofReal (1 / 3 + u * ((z : ℝ) - y)) • scoreLogAtomFullLaw g x s
+ ofReal (1 / 3 - u * ((z : ℝ) - x)) • scoreLogAtomFullLaw g y s
+ ofReal (1 / 3 + u * ((y : ℝ) - x)) • scoreLogAtomFullLaw g z s
theorem scoreLogTripleBaselineFullLaw_isProbabilityMeasure
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
x y z :
s :
ℝ
hs0 :
0 ≤ s
hs1 :
s ≤ 1
conclusion
IsProbabilityMeasure (scoreLogTripleBaselineFullLaw g x y z s)
Proof (Lean source)
lemma scoreLogTripleBaselineFullLaw_isProbabilityMeasure {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(x y z : ScoreSpace ε) (s : ℝ) (hs0 : 0 ≤ s) (hs1 : s ≤ 1) :
IsProbabilityMeasure (scoreLogTripleBaselineFullLaw g x y z s) := by
letI : IsProbabilityMeasure (scoreLogAtomFullLaw g x s) :=
scoreLogAtomFullLaw_isProbabilityMeasure g hOverlap x s hs0 hs1
letI : IsProbabilityMeasure (scoreLogAtomFullLaw g y s) :=
scoreLogAtomFullLaw_isProbabilityMeasure g hOverlap y s hs0 hs1
letI : IsProbabilityMeasure (scoreLogAtomFullLaw g z s) :=
scoreLogAtomFullLaw_isProbabilityMeasure g hOverlap z s hs0 hs1
rw [isProbabilityMeasure_iff]
unfold scoreLogTripleBaselineFullLaw
rw [Measure.add_apply, Measure.add_apply]
simp only [Measure.smul_apply, measure_univ, smul_eq_mul, mul_one]
rw [← ENNReal.ofReal_add (by norm_num : (0 : ℝ) ≤ 1 / 3) (by norm_num),
← ENNReal.ofReal_add (by norm_num : (0 : ℝ) ≤ 1 / 3 + 1 / 3) (by norm_num)]
norm_num
theorem scoreLogTriplePerturbedFullLaw_isProbabilityMeasure
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
x y z :
hxy :
x < y
hyz :
y < z
s :
ℝ
hs0 :
0 ≤ s
hs1 :
s ≤ 1
u :
ℝ
hu0 :
0 ≤ u
hu :
u * ((z : ℝ) - x) ≤ 1 / 3
conclusion
IsProbabilityMeasure (scoreLogTriplePerturbedFullLaw g x y z s u)
Proof (Lean source)
lemma scoreLogTriplePerturbedFullLaw_isProbabilityMeasure {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(x y z : ScoreSpace ε) (hxy : x < y) (hyz : y < z)
(s : ℝ) (hs0 : 0 ≤ s) (hs1 : s ≤ 1)
(u : ℝ) (hu0 : 0 ≤ u) (hu : u * ((z : ℝ) - x) ≤ 1 / 3) :
IsProbabilityMeasure (scoreLogTriplePerturbedFullLaw g x y z s u) := by
letI : IsProbabilityMeasure (scoreLogAtomFullLaw g x s) :=
scoreLogAtomFullLaw_isProbabilityMeasure g hOverlap x s hs0 hs1
letI : IsProbabilityMeasure (scoreLogAtomFullLaw g y s) :=
scoreLogAtomFullLaw_isProbabilityMeasure g hOverlap y s hs0 hs1
letI : IsProbabilityMeasure (scoreLogAtomFullLaw g z s) :=
scoreLogAtomFullLaw_isProbabilityMeasure g hOverlap z s hs0 hs1
have hw := scoreLogTriple_weights_nonneg x y z hxy hyz u hu0 hu
have hsum := (scoreLogTriple_weight_identities x y z hxy hyz u).1
rw [isProbabilityMeasure_iff]
unfold scoreLogTriplePerturbedFullLaw
rw [Measure.add_apply, Measure.add_apply]
simp only [Measure.smul_apply, measure_univ, smul_eq_mul, mul_one]
rw [← ENNReal.ofReal_add hw.1 hw.2.1,
← ENNReal.ofReal_add (add_nonneg hw.1 hw.2.1) hw.2.2, hsum]
norm_num
theorem scoreLogTripleBaselineFullLaw_scoreMarginal
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
x y z :
s :
ℝ
hs0 :
0 ≤ s
hs1 :
s ≤ 1
conclusion
Proof (Lean source)
lemma scoreLogTripleBaselineFullLaw_scoreMarginal {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(x y z : ScoreSpace ε) (s : ℝ) (hs0 : 0 ≤ s) (hs1 : s ≤ 1) :
ScoreMarginal (scoreLogTripleBaselineFullLaw g x y z s)
(scoreLogTripleBaseline x y z) := by
unfold ScoreMarginal scoreLogTripleBaselineFullLaw scoreLogTripleBaseline
rw [Measure.map_add _ _ (by unfold score; fun_prop),
Measure.map_add _ _ (by unfold score; fun_prop),
Measure.map_smul, Measure.map_smul, Measure.map_smul]
rw [scoreLogAtomFullLaw_scoreMarginal g hOverlap x s hs0 hs1,
scoreLogAtomFullLaw_scoreMarginal g hOverlap y s hs0 hs1,
scoreLogAtomFullLaw_scoreMarginal g hOverlap z s hs0 hs1]
theorem scoreLogTriplePerturbedFullLaw_scoreMarginal
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
x y z :
s :
ℝ
hs0 :
0 ≤ s
hs1 :
s ≤ 1
u :
ℝ
conclusion
Proof (Lean source)
lemma scoreLogTriplePerturbedFullLaw_scoreMarginal {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(x y z : ScoreSpace ε) (s : ℝ) (hs0 : 0 ≤ s) (hs1 : s ≤ 1)
(u : ℝ) :
ScoreMarginal (scoreLogTriplePerturbedFullLaw g x y z s u)
(scoreLogTriplePerturbed x y z u) := by
unfold ScoreMarginal scoreLogTriplePerturbedFullLaw scoreLogTriplePerturbed
rw [Measure.map_add _ _ (by unfold score; fun_prop),
Measure.map_add _ _ (by unfold score; fun_prop),
Measure.map_smul, Measure.map_smul, Measure.map_smul]
rw [scoreLogAtomFullLaw_scoreMarginal g hOverlap x s hs0 hs1,
scoreLogAtomFullLaw_scoreMarginal g hOverlap y s hs0 hs1,
scoreLogAtomFullLaw_scoreMarginal g hOverlap z s hs0 hs1]
theorem scoreLogTripleBaselineFullLaw_consistency
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Consistency (scoreLogTripleBaselineFullLaw g x y z s)
Proof (Lean source)
lemma scoreLogTripleBaselineFullLaw_consistency {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (x y z : ScoreSpace ε) (s : ℝ) :
Consistency (scoreLogTripleBaselineFullLaw g x y z s) := by
unfold Consistency scoreLogTripleBaselineFullLaw
rw [ae_add_measure_iff, ae_add_measure_iff]
exact ⟨⟨Measure.ae_smul_measure (scoreLogAtomFullLaw_consistency g x s) _,
Measure.ae_smul_measure (scoreLogAtomFullLaw_consistency g y s) _⟩,
Measure.ae_smul_measure (scoreLogAtomFullLaw_consistency g z s) _⟩
theorem scoreLogTriplePerturbedFullLaw_consistency
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Consistency (scoreLogTriplePerturbedFullLaw g x y z s u)
Proof (Lean source)
lemma scoreLogTriplePerturbedFullLaw_consistency {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (x y z : ScoreSpace ε) (s u : ℝ) :
Consistency (scoreLogTriplePerturbedFullLaw g x y z s u) := by
unfold Consistency scoreLogTriplePerturbedFullLaw
rw [ae_add_measure_iff, ae_add_measure_iff]
exact ⟨⟨Measure.ae_smul_measure (scoreLogAtomFullLaw_consistency g x s) _,
Measure.ae_smul_measure (scoreLogAtomFullLaw_consistency g y s) _⟩,
Measure.ae_smul_measure (scoreLogAtomFullLaw_consistency g z s) _⟩
theorem scoreLogTripleBaselineFullLaw_deterministicRelease
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
DeterministicRelease g (scoreLogTripleBaselineFullLaw g x y z s)
Proof (Lean source)
lemma scoreLogTripleBaselineFullLaw_deterministicRelease {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hg : Measurable g)
(x y z : ScoreSpace ε) (s : ℝ) :
DeterministicRelease g (scoreLogTripleBaselineFullLaw g x y z s) := by
unfold DeterministicRelease scoreLogTripleBaselineFullLaw
rw [ae_add_measure_iff, ae_add_measure_iff]
exact ⟨⟨Measure.ae_smul_measure
(scoreLogAtomFullLaw_deterministicRelease g x s hg) _,
Measure.ae_smul_measure (scoreLogAtomFullLaw_deterministicRelease g y s hg) _⟩,
Measure.ae_smul_measure (scoreLogAtomFullLaw_deterministicRelease g z s hg) _⟩
theorem scoreLogTriplePerturbedFullLaw_deterministicRelease
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
DeterministicRelease g (scoreLogTriplePerturbedFullLaw g x y z s u)
Proof (Lean source)
lemma scoreLogTriplePerturbedFullLaw_deterministicRelease {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hg : Measurable g)
(x y z : ScoreSpace ε) (s u : ℝ) :
DeterministicRelease g (scoreLogTriplePerturbedFullLaw g x y z s u) := by
unfold DeterministicRelease scoreLogTriplePerturbedFullLaw
rw [ae_add_measure_iff, ae_add_measure_iff]
exact ⟨⟨Measure.ae_smul_measure
(scoreLogAtomFullLaw_deterministicRelease g x s hg) _,
Measure.ae_smul_measure (scoreLogAtomFullLaw_deterministicRelease g y s hg) _⟩,
Measure.ae_smul_measure (scoreLogAtomFullLaw_deterministicRelease g z s hg) _⟩
theorem scoreLogTripleBaseline_externalScoreLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
x y z :
s :
ℝ
hs0 :
0 ≤ s
hs1 :
s ≤ 1
conclusion
ExternalScoreLaw g (scoreLogTripleBaselineFullLaw g x y z s) (scoreLogTripleBaseline x y z) where probabilityP
Proof (Lean source)
lemma scoreLogTripleBaseline_externalScoreLaw {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(hg : Measurable g) (x y z : ScoreSpace ε)
(s : ℝ) (hs0 : 0 ≤ s) (hs1 : s ≤ 1) :
ExternalScoreLaw g (scoreLogTripleBaselineFullLaw g x y z s)
(scoreLogTripleBaseline x y z) where
probabilityP :=
scoreLogTripleBaselineFullLaw_isProbabilityMeasure g hOverlap x y z s hs0 hs1
probabilityH := scoreLogTripleBaseline_isProbabilityMeasure x y z
overlap := hOverlap
measurableRelease := hg
scoreMarginal :=
scoreLogTripleBaselineFullLaw_scoreMarginal g hOverlap x y z s hs0 hs1
randomizedAssignment := scoreLogTripleFullLaw_randomizedAssignment
g hOverlap x y z s hs0 hs1 (1 / 3) (1 / 3) (1 / 3)
consistency := scoreLogTripleBaselineFullLaw_consistency g x y z s
deterministicRelease :=
scoreLogTripleBaselineFullLaw_deterministicRelease g hg x y z s
theorem scoreLogTriplePerturbed_externalScoreLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
x y z :
hxy :
x < y
hyz :
y < z
s :
ℝ
hs0 :
0 ≤ s
hs1 :
s ≤ 1
u :
ℝ
hu0 :
0 ≤ u
hu :
u * ((z : ℝ) - x) ≤ 1 / 3
conclusion
ExternalScoreLaw g (scoreLogTriplePerturbedFullLaw g x y z s u) (scoreLogTriplePerturbed x y z u) where probabilityP
Proof (Lean source)
lemma scoreLogTriplePerturbed_externalScoreLaw {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(hg : Measurable g) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z)
(s : ℝ) (hs0 : 0 ≤ s) (hs1 : s ≤ 1)
(u : ℝ) (hu0 : 0 ≤ u) (hu : u * ((z : ℝ) - x) ≤ 1 / 3) :
ExternalScoreLaw g (scoreLogTriplePerturbedFullLaw g x y z s u)
(scoreLogTriplePerturbed x y z u) where
probabilityP := scoreLogTriplePerturbedFullLaw_isProbabilityMeasure
g hOverlap x y z hxy hyz s hs0 hs1 u hu0 hu
probabilityH :=
scoreLogTriplePerturbed_isProbabilityMeasure x y z hxy hyz u hu0 hu
overlap := hOverlap
measurableRelease := hg
scoreMarginal :=
scoreLogTriplePerturbedFullLaw_scoreMarginal g hOverlap x y z s hs0 hs1 u
randomizedAssignment := scoreLogTripleFullLaw_randomizedAssignment
g hOverlap x y z s hs0 hs1
(1 / 3 + u * ((z : ℝ) - y))
(1 / 3 - u * ((z : ℝ) - x))
(1 / 3 + u * ((y : ℝ) - x))
consistency := scoreLogTriplePerturbedFullLaw_consistency g x y z s u
deterministicRelease :=
scoreLogTriplePerturbedFullLaw_deterministicRelease g hg x y z s u
theorem scoreLogAtomFullLaw_releasedLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
releasedLaw (scoreLogAtomFullLaw g e s)
Proof (Lean source)
lemma scoreLogAtomFullLaw_releasedLaw {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (e : ScoreSpace ε)
(s : ℝ) (hs0 : 0 ≤ s) (hs1 : s ≤ 1) :
releasedLaw (scoreLogAtomFullLaw g e s) =
ofReal (e : ℝ) •
(trialOutcomeLaw s).map (fun y => (g e, true, y)) +
ofReal (1 - (e : ℝ)) •
Measure.dirac (g e, false, trialZeroOutcome) := by
letI : IsProbabilityMeasure (trialOutcomeLaw s) :=
trialOutcomeLaw_isProbabilityMeasure s hs0 hs1
have hrow := scoreLogAtomRow_measurable g e
have hread : Measurable (fun ya : OutcomeSpace × ArmSpace =>
(g e, ya.2, if ya.2 then ya.1 else trialZeroOutcome)) := by
have hcond : MeasurableSet {ya : OutcomeSpace × ArmSpace | ya.2 = true} :=
measurableSet_eq_fun measurable_snd measurable_const
have hobs : Measurable (fun ya : OutcomeSpace × ArmSpace =>
if ya.2 then ya.1 else trialZeroOutcome) :=
Measurable.ite hcond measurable_fst measurable_const
fun_prop (disch := assumption)
unfold releasedLaw scoreLogAtomFullLaw
rw [Measure.map_map (by unfold releasedRecord label arm observed; fun_prop) hrow]
change ((trialOutcomeLaw s).prod (scoreLogArmLaw e)).map _ = _
unfold scoreLogArmLaw
rw [Measure.prod_add _, Measure.prod_smul_right, Measure.prod_smul_right]
change Measure.map (fun ya : OutcomeSpace × ArmSpace =>
(g e, ya.2, if ya.2 then ya.1 else trialZeroOutcome)) _ = _
rw [Measure.map_add _ _ hread, Measure.map_smul, Measure.map_smul,
Measure.prod_dirac, Measure.prod_dirac,
Measure.map_map hread (by fun_prop), Measure.map_map hread (by fun_prop)]
congr 1
rw [show ((fun ya : OutcomeSpace × ArmSpace =>
(g e, ya.2, if ya.2 then ya.1 else trialZeroOutcome)) ∘
fun y => (y, false)) =
(fun _ : OutcomeSpace => (g e, false, trialZeroOutcome)) by
funext y
rfl]
rw [Measure.map_const]
simp
theorem scoreLogTripleBaselineFullLaw_releasedLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
releasedLaw (scoreLogTripleBaselineFullLaw g x y z s)
Proof (Lean source)
lemma scoreLogTripleBaselineFullLaw_releasedLaw {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (x y z : ScoreSpace ε)
(s : ℝ) :
releasedLaw (scoreLogTripleBaselineFullLaw g x y z s) =
ofReal (1 / 3 : ℝ) • releasedLaw (scoreLogAtomFullLaw g x s) +
ofReal (1 / 3 : ℝ) • releasedLaw (scoreLogAtomFullLaw g y s) +
ofReal (1 / 3 : ℝ) • releasedLaw (scoreLogAtomFullLaw g z s) := by
unfold releasedLaw scoreLogTripleBaselineFullLaw
rw [Measure.map_add _ _ (by unfold releasedRecord label arm observed; fun_prop),
Measure.map_add _ _ (by unfold releasedRecord label arm observed; fun_prop),
Measure.map_smul, Measure.map_smul, Measure.map_smul]
theorem scoreLogTriplePerturbedFullLaw_releasedLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
releasedLaw (scoreLogTriplePerturbedFullLaw g x y z s u)
Proof (Lean source)
lemma scoreLogTriplePerturbedFullLaw_releasedLaw {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (x y z : ScoreSpace ε)
(s u : ℝ) :
releasedLaw (scoreLogTriplePerturbedFullLaw g x y z s u) =
ofReal (1 / 3 + u * ((z : ℝ) - y)) •
releasedLaw (scoreLogAtomFullLaw g x s) +
ofReal (1 / 3 - u * ((z : ℝ) - x)) •
releasedLaw (scoreLogAtomFullLaw g y s) +
ofReal (1 / 3 + u * ((y : ℝ) - x)) •
releasedLaw (scoreLogAtomFullLaw g z s) := by
unfold releasedLaw scoreLogTriplePerturbedFullLaw
rw [Measure.map_add _ _ (by unfold releasedRecord label arm observed; fun_prop),
Measure.map_add _ _ (by unfold releasedRecord label arm observed; fun_prop),
Measure.map_smul, Measure.map_smul, Measure.map_smul]
theorem scoreLogTriple_releasedLaw_eq
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
r :
x y z :
hxy :
x < y
hyz :
y < z
hx :
g x = r
hy :
g y = r
hz :
g z = r
s :
ℝ
hs0 :
0 ≤ s
hs1 :
s ≤ 1
u :
ℝ
hu0 :
0 ≤ u
hu :
u * ((z : ℝ) - x) ≤ 1 / 3
conclusion
releasedLaw (scoreLogTripleBaselineFullLaw g x y z s)
= releasedLaw (scoreLogTriplePerturbedFullLaw g x y z s u)
Proof (Lean source)
lemma scoreLogTriple_releasedLaw_eq {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(r : LabelSpace J) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z)
(hx : g x = r) (hy : g y = r) (hz : g z = r)
(s : ℝ) (hs0 : 0 ≤ s) (hs1 : s ≤ 1)
(u : ℝ) (hu0 : 0 ≤ u) (hu : u * ((z : ℝ) - x) ≤ 1 / 3) :
releasedLaw (scoreLogTripleBaselineFullLaw g x y z s) =
releasedLaw (scoreLogTriplePerturbedFullLaw g x y z s u) := by
have ex0 : 0 ≤ (x : ℝ) := le_trans hOverlap.1.le x.property.1
have ey0 : 0 ≤ (y : ℝ) := le_trans hOverlap.1.le y.property.1
have ez0 : 0 ≤ (z : ℝ) := le_trans hOverlap.1.le z.property.1
have ex1 : (x : ℝ) ≤ 1 := by linarith [x.property.2, hOverlap.1]
have ey1 : (y : ℝ) ≤ 1 := by linarith [y.property.2, hOverlap.1]
have ez1 : (z : ℝ) ≤ 1 := by linarith [z.property.2, hOverlap.1]
have hw := scoreLogTriple_weights_nonneg x y z hxy hyz u hu0 hu
have hid := scoreLogTriple_weight_identities x y z hxy hyz u
let w₁ : ℝ := 1 / 3 + u * ((z : ℝ) - y)
let w₂ : ℝ := 1 / 3 - u * ((z : ℝ) - x)
let w₃ : ℝ := 1 / 3 + u * ((y : ℝ) - x)
have hw' : 0 ≤ w₁ ∧ 0 ≤ w₂ ∧ 0 ≤ w₃ := by simpa [w₁, w₂, w₃] using hw
have htReal : (1 / 3) * (x : ℝ) + (1 / 3) * (y : ℝ) +
(1 / 3) * (z : ℝ) = w₁ * (x : ℝ) + w₂ * (y : ℝ) + w₃ * (z : ℝ) := by
dsimp [w₁, w₂, w₃]
nlinarith [hid.2]
have hcReal : (1 / 3) * (1 - (x : ℝ)) + (1 / 3) * (1 - (y : ℝ)) +
(1 / 3) * (1 - (z : ℝ)) =
w₁ * (1 - (x : ℝ)) + w₂ * (1 - (y : ℝ)) + w₃ * (1 - (z : ℝ)) := by
dsimp [w₁, w₂, w₃]
nlinarith [hid.1, hid.2]
have ht := ennreal_three_weighted_eq
(x : ℝ) y z (1 / 3) (1 / 3) (1 / 3) w₁ w₂ w₃
ex0 ey0 ez0 (by norm_num) (by norm_num) (by norm_num)
hw'.1 hw'.2.1 hw'.2.2 htReal
have hc := ennreal_three_weighted_eq
(1 - (x : ℝ)) (1 - (y : ℝ)) (1 - (z : ℝ))
(1 / 3) (1 / 3) (1 / 3) w₁ w₂ w₃
(sub_nonneg.mpr ex1) (sub_nonneg.mpr ey1) (sub_nonneg.mpr ez1)
(by norm_num) (by norm_num) (by norm_num) hw'.1 hw'.2.1 hw'.2.2 hcReal
let T : Measure (Observation J) :=
(trialOutcomeLaw s).map (fun y => (r, true, y))
let F : Measure (Observation J) := Measure.dirac (r, false, trialZeroOutcome)
have hb : releasedLaw (scoreLogTripleBaselineFullLaw g x y z s) =
(ofReal (1 / 3 : ℝ) * ofReal (x : ℝ) +
ofReal (1 / 3 : ℝ) * ofReal (y : ℝ) +
ofReal (1 / 3 : ℝ) * ofReal (z : ℝ)) • T +
(ofReal (1 / 3 : ℝ) * ofReal (1 - (x : ℝ)) +
ofReal (1 / 3 : ℝ) * ofReal (1 - (y : ℝ)) +
ofReal (1 / 3 : ℝ) * ofReal (1 - (z : ℝ))) • F := by
rw [scoreLogTripleBaselineFullLaw_releasedLaw,
scoreLogAtomFullLaw_releasedLaw g x s hs0 hs1,
scoreLogAtomFullLaw_releasedLaw g y s hs0 hs1,
scoreLogAtomFullLaw_releasedLaw g z s hs0 hs1, hx, hy, hz]
dsimp [T, F]
module
have hp : releasedLaw (scoreLogTriplePerturbedFullLaw g x y z s u) =
(ofReal w₁ * ofReal (x : ℝ) +
ofReal w₂ * ofReal (y : ℝ) +
ofReal w₃ * ofReal (z : ℝ)) • T +
(ofReal w₁ * ofReal (1 - (x : ℝ)) +
ofReal w₂ * ofReal (1 - (y : ℝ)) +
ofReal w₃ * ofReal (1 - (z : ℝ))) • F := by
rw [scoreLogTriplePerturbedFullLaw_releasedLaw,
scoreLogAtomFullLaw_releasedLaw g x s hs0 hs1,
scoreLogAtomFullLaw_releasedLaw g y s hs0 hs1,
scoreLogAtomFullLaw_releasedLaw g z s hs0 hs1, hx, hy, hz]
dsimp [w₁, w₂, w₃, T, F]
module
rw [hb, hp, ht, hc]
theorem scoreLogTriple_externalPair
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
r :
x y z :
hxy :
x < y
hyz :
y < z
hx :
g x = r
hy :
g y = r
hz :
g z = r
s :
ℝ
hs0 :
0 ≤ s
hs1 :
s ≤ 1
u :
ℝ
hu0 :
0 ≤ u
hu :
u * ((z : ℝ) - x) ≤ 1 / 3
conclusion
P₀ :
scoreLogTripleBaselineFullLaw g x y z s
H₀ :
scoreLogTripleBaseline x y z
P₁ :
scoreLogTriplePerturbedFullLaw g x y z s u
H₁ :
scoreLogTriplePerturbed x y z u
Proof (Lean source)
lemma scoreLogTriple_externalPair {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(hg : Measurable g) (r : LabelSpace J) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z)
(hx : g x = r) (hy : g y = r) (hz : g z = r)
(s : ℝ) (hs0 : 0 ≤ s) (hs1 : s ≤ 1)
(u : ℝ) (hu0 : 0 ≤ u) (hu : u * ((z : ℝ) - x) ≤ 1 / 3) :
let P₀ := scoreLogTripleBaselineFullLaw g x y z s
let H₀ := scoreLogTripleBaseline x y z
let P₁ := scoreLogTriplePerturbedFullLaw g x y z s u
let H₁ := scoreLogTriplePerturbed x y z u
(P₀, H₀) ∈ ExternalLaws g ∧ (P₁, H₁) ∈ ExternalLaws g ∧
releasedLaw P₀ = releasedLaw P₁ := by
dsimp
refine ⟨?_, ?_, scoreLogTriple_releasedLaw_eq g hOverlap r x y z hxy hyz
hx hy hz s hs0 hs1 u hu0 hu⟩
· exact ⟨scoreLogTripleBaselineFullLaw_isProbabilityMeasure
g hOverlap x y z s hs0 hs1,
scoreLogTripleBaseline_isProbabilityMeasure x y z,
scoreLogTripleBaseline_externalScoreLaw g hOverlap hg x y z s hs0 hs1⟩
· exact ⟨scoreLogTriplePerturbedFullLaw_isProbabilityMeasure
g hOverlap x y z hxy hyz s hs0 hs1 u hu0 hu,
scoreLogTriplePerturbed_isProbabilityMeasure x y z hxy hyz u hu0 hu,
scoreLogTriplePerturbed_externalScoreLaw g hOverlap hg x y z hxy hyz
s hs0 hs1 u hu0 hu⟩
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerScoreLogHonestLength 2 declarations Honest two-point lower bounds for excess interval length.
Honest two-point lower bounds for excess interval length.
theorem externalInterval_twoPoint_expectedExcessLength
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
epsilon :
ℝ
J n m :
ℕ
C :
ExternalIntervalProcedure epsilon J n m
Q0 Q1 :
theta0 theta1 W0 Delta alpha :
ℝ
_htheta :
theta0 ≤ theta1
hsep :
theta1 - theta0 = W0 + Delta
hDelta :
0 ≤ Delta
hcover0 :
1 - alpha ≤ Q0.real {x | C.contains x theta0}
hcover1 :
1 - alpha ≤ Q1.real {x | C.contains x theta1}
Proof (Lean source)
lemma externalInterval_twoPoint_expectedExcessLength {epsilon : ℝ} {J n m : ℕ}
(C : ExternalIntervalProcedure epsilon J n m)
(Q0 Q1 : Measure (ExternalSample epsilon J n m))
[IsProbabilityMeasure Q0] [IsProbabilityMeasure Q1]
(theta0 theta1 W0 Delta alpha : ℝ)
(_htheta : theta0 ≤ theta1)
(hsep : theta1 - theta0 = W0 + Delta)
(hDelta : 0 ≤ Delta)
(hcover0 : 1 - alpha ≤ Q0.real {x | C.contains x theta0})
(hcover1 : 1 - alpha ≤ Q1.real {x | C.contains x theta1}) :
Delta * (1 - 2 * alpha - tvDist Q0 Q1) ≤
∫ x, max 0 (C.length x - W0) ∂Q0 := by
let A0 : Set (ExternalSample epsilon J n m) := {x | C.contains x theta0}
let A1 : Set (ExternalSample epsilon J n m) := {x | C.contains x theta1}
let I := A0 ∩ A1
have hA (theta : ℝ) : MeasurableSet {x | C.contains x theta} := by
change MeasurableSet {x | C.lo x ≤ theta ∧ theta ≤ C.hi x}
exact (measurableSet_le C.measurable_lo measurable_const).inter
(measurableSet_le measurable_const C.measurable_hi)
have hA0 : MeasurableSet A0 := hA theta0
have hA1 : MeasurableSet A1 := hA theta1
have hI : MeasurableSet I := hA0.inter hA1
have htransfer :
1 - alpha - tvDist Q0 Q1 ≤ Q0.real A1 := by
have htv := measureReal_sub_le_tvDist
(μ := Q0) (ν := Q1) hA1
dsimp [A1] at htv
linarith
have hmass :
1 - 2 * alpha - tvDist Q0 Q1 ≤ Q0.real I := by
have hu : Q0.real (A0 ∪ A1) ≤ 1 := measureReal_le_one
have hsum := measureReal_union_add_inter (μ := Q0) (s := A0) (t := A1)
hA1 (measure_ne_top Q0 A0) (measure_ne_top Q0 A1)
change Q0.real (A0 ∪ A1) + Q0.real I =
Q0.real A0 + Q0.real A1 at hsum
linarith
have hlengthInt : Integrable C.length Q0 := by
have hlengthMeas : Measurable C.length :=
C.measurable_hi.sub C.measurable_lo
apply Integrable.of_bound hlengthMeas.aestronglyMeasurable 2
filter_upwards with x
dsimp [ExternalIntervalProcedure.length]
rw [abs_of_nonneg (sub_nonneg.mpr (C.ordered x))]
linarith [C.lower_bound x, C.upper_bound x]
have hexcessInt : Integrable (fun x => max 0 (C.length x - W0)) Q0 := by
have hsub : Integrable (fun x => C.length x - W0) Q0 := by
fun_prop
simpa only [max_comm] using hsub.pos_part
have hindInt : Integrable (I.indicator (fun _ => Delta)) Q0 :=
(integrable_const _).indicator hI
have hpoint (x : ExternalSample epsilon J n m) :
I.indicator (fun _ => Delta) x ≤ max 0 (C.length x - W0) := by
by_cases hx : x ∈ I
· rw [Set.indicator_of_mem hx]
have hx0 : C.lo x ≤ theta0 ∧ theta0 ≤ C.hi x := hx.1
have hx1 : C.lo x ≤ theta1 ∧ theta1 ≤ C.hi x := hx.2
apply le_max_of_le_right
dsimp [ExternalIntervalProcedure.length]
linarith
· rw [Set.indicator_of_notMem hx]
exact le_max_left _ _
have hmono := integral_mono hindInt hexcessInt hpoint
rw [integral_indicator_const Delta hI] at hmono
simp only [smul_eq_mul] at hmono
rw [mul_comm (Q0.real I) Delta] at hmono
exact (mul_le_mul_of_nonneg_left hmass hDelta).trans hmono
theorem externalInterval_twoPoint_expectedExcessLength_of_tv
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
epsilon :
ℝ
J n m :
ℕ
C :
ExternalIntervalProcedure epsilon J n m
Q0 Q1 :
theta0 theta1 W0 Delta alpha :
ℝ
htheta :
theta0 ≤ theta1
hsep :
theta1 - theta0 = W0 + Delta
hDelta :
0 < Delta
hcover0 :
1 - alpha ≤ Q0.real {x | C.contains x theta0}
hcover1 :
1 - alpha ≤ Q1.real {x | C.contains x theta1}
htv :
tvDist Q0 Q1 ≤ (1 - 2 * alpha) / 2
conclusion
Delta * ((1 - 2 * alpha) / 2) ≤ ∫ x, max 0 (C.length x - W0) ∂Q0
Proof (Lean source)
lemma externalInterval_twoPoint_expectedExcessLength_of_tv {epsilon : ℝ}
{J n m : ℕ}
(C : ExternalIntervalProcedure epsilon J n m)
(Q0 Q1 : Measure (ExternalSample epsilon J n m))
[IsProbabilityMeasure Q0] [IsProbabilityMeasure Q1]
(theta0 theta1 W0 Delta alpha : ℝ)
(htheta : theta0 ≤ theta1)
(hsep : theta1 - theta0 = W0 + Delta)
(hDelta : 0 < Delta)
(hcover0 : 1 - alpha ≤ Q0.real {x | C.contains x theta0})
(hcover1 : 1 - alpha ≤ Q1.real {x | C.contains x theta1})
(htv : tvDist Q0 Q1 ≤ (1 - 2 * alpha) / 2) :
Delta * ((1 - 2 * alpha) / 2) ≤
∫ x, max 0 (C.length x - W0) ∂Q0 := by
have hexact := externalInterval_twoPoint_expectedExcessLength
C Q0 Q1 theta0 theta1 W0 Delta alpha htheta hsep hDelta.le
hcover0 hcover1
have hcoefficient :
(1 - 2 * alpha) / 2 ≤
1 - 2 * alpha - tvDist Q0 Q1 := by
linarith
exact (mul_le_mul_of_nonneg_left hcoefficient hDelta.le).trans hexact
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerScoreLogInverseQuantile 1 declarations Exact lower quantiles for the ordered three-atom inverse-score law.
Exact lower quantiles for the ordered three-atom inverse-score law.
theorem threeScoreInverseLaw_quantile
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
x y z w₁ w₂ w₃ u :
ℝ
hx :
0 < x
hxy :
x < y
hyz :
y < z
hw₁ :
0 ≤ w₁
hw₂ :
0 ≤ w₂
hw₃ :
0 ≤ w₃
hq :
0 < x * w₁ + y * w₂ + z * w₃
hu0 :
0 < u
hu1 :
u < 1
conclusion
quantile (threeScoreInverseLaw x y z w₁ w₂ w₃) u
= if u ≤ z * w₃ / (x * w₁ + y * w₂ + z * w₃) then 1 / z else if u ≤ z * w₃ / (x * w₁ + y * w₂ + z * w₃)
+ y * w₂ / (x * w₁ + y * w₂ + z * w₃) then 1 / y else 1 / x
Proof (Lean source)
lemma threeScoreInverseLaw_quantile
(x y z w₁ w₂ w₃ u : ℝ)
(hx : 0 < x) (hxy : x < y) (hyz : y < z)
(hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃)
(hq : 0 < x * w₁ + y * w₂ + z * w₃)
(hu0 : 0 < u) (hu1 : u < 1) :
quantile (threeScoreInverseLaw x y z w₁ w₂ w₃) u =
if u ≤ z * w₃ / (x * w₁ + y * w₂ + z * w₃) then 1 / z
else if u ≤ z * w₃ / (x * w₁ + y * w₂ + z * w₃) +
y * w₂ / (x * w₁ + y * w₂ + z * w₃) then 1 / y
else 1 / x := by
have hy : 0 < y := lt_trans hx hxy
have hz : 0 < z := lt_trans hy hyz
have hAB : 1 / z < 1 / y := one_div_lt_one_div_of_lt hy hyz
have hBC : 1 / y < 1 / x := one_div_lt_one_div_of_lt hx hxy
have ha : 0 ≤ x * w₁ / (x * w₁ + y * w₂ + z * w₃) :=
div_nonneg (mul_nonneg hx.le hw₁) hq.le
have hb : 0 ≤ y * w₂ / (x * w₁ + y * w₂ + z * w₃) :=
div_nonneg (mul_nonneg hy.le hw₂) hq.le
have hc : 0 ≤ z * w₃ / (x * w₁ + y * w₂ + z * w₃) :=
div_nonneg (mul_nonneg hz.le hw₃) hq.le
have hsum :
x * w₁ / (x * w₁ + y * w₂ + z * w₃) +
y * w₂ / (x * w₁ + y * w₂ + z * w₃) +
z * w₃ / (x * w₁ + y * w₂ + z * w₃) = 1 := by
field_simp
unfold threeScoreInverseLaw
exact orderedThreeAtomLaw_quantile (1 / z) (1 / y) (1 / x)
(x * w₁ / (x * w₁ + y * w₂ + z * w₃))
(y * w₂ / (x * w₁ + y * w₂ + z * w₃))
(z * w₃ / (x * w₁ + y * w₂ + z * w₃)) u
hAB hBC ha hb hc hsum hu0 hu1
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerScoreLogLimit 3 declarations Transfer of eventual uniform log certificates through an infimum and liminf.
Transfer of eventual uniform log certificates through an infimum and liminf.
theorem logCertificate_liminf_infimum_of_eventual_uniform
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma logCertificate_liminf_infimum_of_eventual_uniform
(beta C : ℝ) (R : ℕ → ℕ → ℝ)
(hLower : ∀ᶠ m : ℕ in atTop, ∀ n : ℕ, 0 < n →
beta * (sqrt (m : ℝ))⁻¹ ≤ R n m)
(hUpperWitness : ∀ᶠ m : ℕ in atTop, ∃ n : ℕ, 0 < n ∧
sqrt (m : ℝ) * R n m ≤ C) :
beta ≤ liminf (fun m : ℕ =>
sInf {v : ℝ | ∃ n : ℕ, 0 < n ∧
v = sqrt (m : ℝ) * R n m}) atTop := by
let F : ℕ → ℝ := fun m =>
sInf {v : ℝ | ∃ n : ℕ, 0 < n ∧
v = sqrt (m : ℝ) * R n m}
have hFLower : ∀ᶠ m : ℕ in atTop, beta ≤ F m := by
filter_upwards [hLower, eventually_gt_atTop (0 : ℕ)] with m hlower hm
have hsqrt : 0 < sqrt (m : ℝ) :=
Real.sqrt_pos.2 (Nat.cast_pos.mpr hm)
apply le_csInf
· exact ⟨Real.sqrt (m : ℝ) * R 1 m, 1, by omega, rfl⟩
· rintro v ⟨n, hn, rfl⟩
have h := mul_le_mul_of_nonneg_left (hlower n hn) hsqrt.le
field_simp at h
simpa [mul_comm] using h
have hFUpper : ∀ᶠ m : ℕ in atTop, F m ≤ C := by
filter_upwards [hLower, hUpperWitness, eventually_gt_atTop (0 : ℕ)] with
m hlower ⟨n, hn, hupper⟩ hm
have hsqrt : 0 < sqrt (m : ℝ) :=
Real.sqrt_pos.2 (Nat.cast_pos.mpr hm)
have hsetBelow : BddBelow {v : ℝ | ∃ n' : ℕ, 0 < n' ∧
v = sqrt (m : ℝ) * R n' m} := by
refine ⟨beta, ?_⟩
rintro v ⟨n', hn', rfl⟩
have h := mul_le_mul_of_nonneg_left (hlower n' hn') hsqrt.le
field_simp at h
simpa [mul_comm] using h
exact (csInf_le hsetBelow ⟨n, hn, rfl⟩).trans hupper
have hCobounded : atTop.IsCoboundedUnder (· ≥ ·) F :=
(isBoundedUnder_of_eventually_le hFUpper).isCoboundedUnder_ge
exact le_liminf_of_le hCobounded hFLower
theorem externalNormalizer_self
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma externalNormalizer_self (m : ℕ) (hm : 0 < m) :
externalNormalizer m m = sqrt (m : ℝ) / 2 := by
have hsqrt : 0 < sqrt (m : ℝ) :=
Real.sqrt_pos.2 (Nat.cast_pos.mpr hm)
unfold externalNormalizer
field_simp
ring
theorem logCertificate_liminf_infimum_of_normalized_upper
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma logCertificate_liminf_infimum_of_normalized_upper
(beta C : ℝ) (R : ℕ → ℕ → ℝ)
(hLower : ∀ᶠ m : ℕ in atTop, ∀ n : ℕ, 0 < n →
beta * (sqrt (m : ℝ))⁻¹ ≤ R n m)
(hUpper : ∀ (n m : ℕ), 0 < n → 0 < m →
externalNormalizer n m * R n m ≤ C) :
beta ≤ liminf (fun m : ℕ =>
sInf {v : ℝ | ∃ n : ℕ, 0 < n ∧
v = sqrt (m : ℝ) * R n m}) atTop := by
apply logCertificate_liminf_infimum_of_eventual_uniform beta (2 * C) R hLower
filter_upwards [eventually_gt_atTop (0 : ℕ)] with m hm
refine ⟨m, hm, ?_⟩
have h := mul_le_mul_of_nonneg_left (hUpper m m hm hm) (by norm_num : (0 : ℝ) ≤ 2)
rw [externalNormalizer_self m hm] at h
nlinarith
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerScoreLogPair 6 declarations An explicit moment-matched three-score pair for the score-log lower bound.
An explicit moment-matched three-score pair for the score-log lower bound.
def scoreLogTripleBaseline
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
x y z :
def
def scoreLogTriplePerturbed
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
theorem scoreLogTriple_weights_nonneg
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
conclusion 1
0 ≤ 1 / 3 + u * ((z : ℝ) - y)
conclusion 2
0 ≤ 1 / 3 - u * ((z : ℝ) - x)
conclusion 3
0 ≤ 1 / 3 + u * ((y : ℝ) - x)
Proof (Lean source)
lemma scoreLogTriple_weights_nonneg {ε : ℝ}
(x y z : ScoreSpace ε) (hxy : x < y) (hyz : y < z)
(u : ℝ) (hu0 : 0 ≤ u) (hu : u * ((z : ℝ) - x) ≤ 1 / 3) :
0 ≤ 1 / 3 + u * ((z : ℝ) - y) ∧
0 ≤ 1 / 3 - u * ((z : ℝ) - x) ∧
0 ≤ 1 / 3 + u * ((y : ℝ) - x) := by
have hzy : 0 ≤ (z : ℝ) - y := sub_nonneg.mpr (le_of_lt hyz)
have hyx : 0 ≤ (y : ℝ) - x := sub_nonneg.mpr (le_of_lt hxy)
constructor
· positivity
constructor
· linarith
· positivity
theorem scoreLogTriple_weight_identities
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
conclusion 1
(1 / 3 + u * ((z : ℝ) - y)) + (1 / 3 - u * ((z : ℝ) - x)) + (1 / 3 + u * ((y : ℝ) - x))
= 1
conclusion 2
(x : ℝ) * (1 / 3 + u * ((z : ℝ) - y))
+ (y : ℝ) * (1 / 3 - u * ((z : ℝ) - x))
+ (z : ℝ) * (1 / 3 + u * ((y : ℝ) - x))
= (x : ℝ) / 3 + (y : ℝ) / 3 + (z : ℝ) / 3
Proof (Lean source)
lemma scoreLogTriple_weight_identities {ε : ℝ}
(x y z : ScoreSpace ε) (hxy : x < y) (hyz : y < z) (u : ℝ) :
(1 / 3 + u * ((z : ℝ) - y)) +
(1 / 3 - u * ((z : ℝ) - x)) +
(1 / 3 + u * ((y : ℝ) - x)) = 1 ∧
(x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) +
(y : ℝ) * (1 / 3 - u * ((z : ℝ) - x)) +
(z : ℝ) * (1 / 3 + u * ((y : ℝ) - x)) =
(x : ℝ) / 3 + (y : ℝ) / 3 + (z : ℝ) / 3 := by
have h := fiberTriple_perturbation_identities (x : ℝ) y z hxy hyz
constructor <;> nlinarith [h.1, h.2.1]
theorem scoreLogTripleBaseline_isProbabilityMeasure
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
x y z :
conclusion
Proof (Lean source)
lemma scoreLogTripleBaseline_isProbabilityMeasure {ε : ℝ}
(x y z : ScoreSpace ε) :
IsProbabilityMeasure (scoreLogTripleBaseline x y z) := by
rw [isProbabilityMeasure_iff]
unfold scoreLogTripleBaseline
rw [Measure.add_apply, Measure.add_apply]
simp only [Measure.smul_apply, Measure.dirac_apply, Set.mem_univ,
Set.indicator_of_mem, Pi.one_apply, smul_eq_mul, mul_one]
rw [← ENNReal.ofReal_add (by norm_num : (0 : ℝ) ≤ 1 / 3) (by norm_num),
← ENNReal.ofReal_add (by norm_num : (0 : ℝ) ≤ 1 / 3 + 1 / 3) (by norm_num)]
norm_num
theorem scoreLogTriplePerturbed_isProbabilityMeasure
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
IsProbabilityMeasure (scoreLogTriplePerturbed x y z u)
Proof (Lean source)
lemma scoreLogTriplePerturbed_isProbabilityMeasure {ε : ℝ}
(x y z : ScoreSpace ε) (hxy : x < y) (hyz : y < z)
(u : ℝ) (hu0 : 0 ≤ u) (hu : u * ((z : ℝ) - x) ≤ 1 / 3) :
IsProbabilityMeasure (scoreLogTriplePerturbed x y z u) := by
have hw := scoreLogTriple_weights_nonneg x y z hxy hyz u hu0 hu
have hsum := (scoreLogTriple_weight_identities x y z hxy hyz u).1
rw [isProbabilityMeasure_iff]
unfold scoreLogTriplePerturbed
rw [Measure.add_apply, Measure.add_apply]
simp only [Measure.smul_apply, Measure.dirac_apply, Set.mem_univ,
Set.indicator_of_mem, Pi.one_apply, smul_eq_mul, mul_one]
rw [← ENNReal.ofReal_add hw.1 hw.2.1, ← ENNReal.ofReal_add (add_nonneg hw.1 hw.2.1) hw.2.2]
rw [hsum]
norm_num
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerScoreLogQuantileBridge 2 declarations A measure-theoretic realization of the finite three-score coupling problem.
A measure-theoretic realization of the finite three-score coupling problem.
def threeScoreOutcomeLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
def threeScoreInverseLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerScoreLogQuantileExact 2 declarations Exact quantiles of the normalized Bernoulli margin in the three-score problem.
Exact quantiles of the normalized Bernoulli margin in the three-score problem.
theorem threeScoreOutcomeLaw_isProbabilityMeasure
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
q t :
ℝ
h0 :
0 ≤ t / q
h1 :
t / q ≤ 1
conclusion
Proof (Lean source)
lemma threeScoreOutcomeLaw_isProbabilityMeasure (q t : ℝ)
(h0 : 0 ≤ t / q) (h1 : t / q ≤ 1) :
IsProbabilityMeasure (threeScoreOutcomeLaw q t) := by
rw [isProbabilityMeasure_iff]
unfold threeScoreOutcomeLaw
rw [Measure.add_apply]
simp only [Measure.smul_apply, Measure.dirac_apply, Set.mem_univ,
Set.indicator_of_mem, Pi.one_apply, smul_eq_mul, mul_one]
rw [← ENNReal.ofReal_add h0 (sub_nonneg.mpr h1)]
norm_num
theorem threeScoreOutcomeLaw_quantile
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
q t u :
ℝ
h0 :
0 ≤ t / q
h1 :
t / q ≤ 1
hu0 :
0 < u
hu1 :
u < 1
conclusion
quantile (threeScoreOutcomeLaw q t) u = if u ≤ 1 - t / q then 0 else 1
Proof (Lean source)
lemma threeScoreOutcomeLaw_quantile
(q t u : ℝ) (h0 : 0 ≤ t / q) (h1 : t / q ≤ 1)
(hu0 : 0 < u) (hu1 : u < 1) :
quantile (threeScoreOutcomeLaw q t) u =
if u ≤ 1 - t / q then 0 else 1 := by
letI := threeScoreOutcomeLaw_isProbabilityMeasure q t h0 h1
split_ifs with hu
· apply le_antisymm
· apply quantile_le_of_le_cdf hu0
have hc := threeScoreOutcomeLaw_cdf_between q t 0 h0 h1 (le_refl 0) (by norm_num)
simpa [hc] using hu
· by_contra hn
have hneg : quantile (threeScoreOutcomeLaw q t) u < 0 := lt_of_not_ge hn
have hle := (quantile_le_iff hu0 hu1).mp (le_refl (quantile (threeScoreOutcomeLaw q t) u))
rw [threeScoreOutcomeLaw_cdf_of_lt_zero q t _ h0 h1 hneg] at hle
linarith
· apply le_antisymm
· apply quantile_le_of_le_cdf hu0
rw [threeScoreOutcomeLaw_cdf_one q t h0 h1]
exact hu1.le
· by_contra hn
have hlt : quantile (threeScoreOutcomeLaw q t) u < 1 := lt_of_not_ge hn
let r := (quantile (threeScoreOutcomeLaw q t) u + 1) / 2
have hqr : quantile (threeScoreOutcomeLaw q t) u ≤ r := by
dsimp [r]
linarith
have hr0 : 0 ≤ r := by
have hq0 : 0 ≤ quantile (threeScoreOutcomeLaw q t) u := by
by_contra hn0
have hqneg : quantile (threeScoreOutcomeLaw q t) u < 0 := lt_of_not_ge hn0
have hle := (quantile_le_iff hu0 hu1).mp
(le_refl (quantile (threeScoreOutcomeLaw q t) u))
rw [threeScoreOutcomeLaw_cdf_of_lt_zero q t _ h0 h1 hqneg] at hle
linarith
dsimp [r]
linarith
have hr1 : r < 1 := by dsimp [r]; linarith
have hle := (quantile_le_iff hu0 hu1).mp hqr
rw [threeScoreOutcomeLaw_cdf_between q t r h0 h1 hr0 hr1] at hle
exact hu hle
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerScoreLogQuantileIntegrals 2 declarations Exact integrals of the Bernoulli and inverse-score quantile steps.
Exact integrals of the Bernoulli and inverse-score quantile steps.
theorem threeScore_quantileProduct_upper
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
x y z w₁ w₂ w₃ t :
ℝ
hx :
0 < x
hxy :
x < y
hyz :
y < z
hw₁ :
0 ≤ w₁
hw₂ :
0 ≤ w₂
hw₃ :
0 ≤ w₃
hsum :
w₁ + w₂ + w₃ = 1
ht₁ :
x * w₁ < t
ht₂ :
t < min (x * w₁ + y * w₂) (z * w₃)
conclusion
q :
x * w₁ + y * w₂ + z * w₃
q * (∫ u in Ioo (0 : ℝ) 1, quantile (threeScoreOutcomeLaw q t) u * quantile (threeScoreInverseLaw x y z w₁ w₂ w₃) u)
= threeScoreUpperEndpoint x y w₁ t
Proof (Lean source)
lemma threeScore_quantileProduct_upper
(x y z w₁ w₂ w₃ t : ℝ)
(hx : 0 < x) (hxy : x < y) (hyz : y < z)
(hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃)
(hsum : w₁ + w₂ + w₃ = 1)
(ht₁ : x * w₁ < t)
(ht₂ : t < min (x * w₁ + y * w₂) (z * w₃)) :
let q := x * w₁ + y * w₂ + z * w₃
q * (∫ u in Ioo (0 : ℝ) 1,
quantile (threeScoreOutcomeLaw q t) u *
quantile (threeScoreInverseLaw x y z w₁ w₂ w₃) u) =
threeScoreUpperEndpoint x y w₁ t := by
dsimp only
let q := x * w₁ + y * w₂ + z * w₃
change q * (∫ u in Ioo (0 : ℝ) 1,
quantile (threeScoreOutcomeLaw q t) u *
quantile (threeScoreInverseLaw x y z w₁ w₂ w₃) u) =
threeScoreUpperEndpoint x y w₁ t
have hy : 0 < y := lt_trans hx hxy
have hz : 0 < z := lt_trans hy hyz
have hq : 0 < q := by
dsimp [q]
have hone : 0 < w₁ + w₂ + w₃ := by rw [hsum]; norm_num
nlinarith [mul_nonneg hx.le hw₁, mul_nonneg hy.le hw₂, mul_nonneg hz.le hw₃]
have hs0 : 0 ≤ t / q := by
have ht0 : 0 < t := lt_of_le_of_lt (mul_nonneg hx.le hw₁) ht₁
positivity
have hs1 : t / q ≤ 1 := by
have htq : t < q := by
dsimp [q]
have := lt_of_lt_of_le ht₂ (min_le_left _ _)
nlinarith [mul_nonneg hz.le hw₃]
exact (div_lt_one hq).2 htq |>.le
have ha0 : 0 ≤ x * w₁ / q := div_nonneg (mul_nonneg hx.le hw₁) hq.le
have ha1 : x * w₁ / q ≤ 1 := by
apply (div_le_one hq).2
dsimp [q]
nlinarith [mul_nonneg hy.le hw₂, mul_nonneg hz.le hw₃]
have hrs : 1 - t / q ≤ 1 - x * w₁ / q := by
apply sub_le_sub_left
exact (div_lt_div_iff_of_pos_right hq).2 ht₁ |>.le
have heq : ∀ᵐ u ∂volume.restrict (Ioo (0 : ℝ) 1),
quantile (threeScoreOutcomeLaw q t) u *
quantile (threeScoreInverseLaw x y z w₁ w₂ w₃) u =
if u ≤ 1 - t / q then 0
else if u ≤ 1 - x * w₁ / q then 1 / y else 1 / x := by
filter_upwards [ae_restrict_mem measurableSet_Ioo] with u hu
rw [threeScoreOutcomeLaw_quantile q t u hs0 hs1 hu.1 hu.2,
threeScoreInverseLaw_quantile x y z w₁ w₂ w₃ u hx hxy hyz
hw₁ hw₂ hw₃ hq hu.1 hu.2]
rw [show x * w₁ + y * w₂ + z * w₃ = q by rfl]
have hthreshold :
z * w₃ / q + y * w₂ / q = 1 - x * w₁ / q := by
calc
z * w₃ / q + y * w₂ / q = (z * w₃ + y * w₂) / q := by ring
_ = (q - x * w₁) / q := by
congr 1
dsimp [q]
ring
_ = 1 - x * w₁ / q := by field_simp [ne_of_gt hq]
rw [hthreshold]
by_cases hu0 : u ≤ 1 - t / q
· simp [hu0]
· have htFirstTwo : t / q < (x * w₁ + y * w₂) / q := by
exact (div_lt_div_iff_of_pos_right hq).2
(lt_of_lt_of_le ht₂ (min_le_left _ _))
have htotal :
(x * w₁ + y * w₂) / q = 1 - z * w₃ / q := by
calc
(x * w₁ + y * w₂) / q = (q - z * w₃) / q := by
congr 1
dsimp [q]
ring
_ = 1 - z * w₃ / q := by field_simp [ne_of_gt hq]
have hthird : z * w₃ / q < u := by linarith
have hnthird : ¬u ≤ z * w₃ / q := not_le.mpr hthird
simp [hu0, hnthird]
rw [integral_congr_ae heq,
upper_step_integral (1 - t / q) (1 - x * w₁ / q) (1 / y) (1 / x)
(sub_nonneg.mpr hs1) hrs (sub_le_self 1 ha0)]
unfold threeScoreUpperEndpoint
field_simp [ne_of_gt hq, ne_of_gt hx, ne_of_gt hy]
ring
theorem threeScore_quantileProduct_lower
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
x y z w₁ w₂ w₃ t :
ℝ
hx :
0 < x
hxy :
x < y
hyz :
y < z
hw₁ :
0 ≤ w₁
hw₂ :
0 ≤ w₂
hw₃ :
0 ≤ w₃
hsum :
w₁ + w₂ + w₃ = 1
ht₁ :
x * w₁ < t
ht₂ :
t < min (x * w₁ + y * w₂) (z * w₃)
conclusion
q :
x * w₁ + y * w₂ + z * w₃
q * (∫ u in Ioo (0 : ℝ) 1, quantile (threeScoreOutcomeLaw q t) u * quantile (threeScoreInverseLaw x y z w₁ w₂ w₃) (1 - u))
= threeScoreLowerEndpoint z t
Proof (Lean source)
lemma threeScore_quantileProduct_lower
(x y z w₁ w₂ w₃ t : ℝ)
(hx : 0 < x) (hxy : x < y) (hyz : y < z)
(hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃)
(hsum : w₁ + w₂ + w₃ = 1)
(ht₁ : x * w₁ < t)
(ht₂ : t < min (x * w₁ + y * w₂) (z * w₃)) :
let q := x * w₁ + y * w₂ + z * w₃
q * (∫ u in Ioo (0 : ℝ) 1,
quantile (threeScoreOutcomeLaw q t) u *
quantile (threeScoreInverseLaw x y z w₁ w₂ w₃) (1 - u)) =
threeScoreLowerEndpoint z t := by
dsimp only
let q := x * w₁ + y * w₂ + z * w₃
change q * (∫ u in Ioo (0 : ℝ) 1,
quantile (threeScoreOutcomeLaw q t) u *
quantile (threeScoreInverseLaw x y z w₁ w₂ w₃) (1 - u)) =
threeScoreLowerEndpoint z t
have hy : 0 < y := lt_trans hx hxy
have hz : 0 < z := lt_trans hy hyz
have hq : 0 < q := by
dsimp [q]
have hone : 0 < w₁ + w₂ + w₃ := by rw [hsum]; norm_num
nlinarith [mul_nonneg hx.le hw₁, mul_nonneg hy.le hw₂, mul_nonneg hz.le hw₃]
have hs0 : 0 ≤ t / q := by
have ht0 : 0 < t := lt_of_le_of_lt (mul_nonneg hx.le hw₁) ht₁
positivity
have hs1 : t / q ≤ 1 := by
have htq : t < q := by
dsimp [q]
have := lt_of_lt_of_le ht₂ (min_le_left _ _)
nlinarith [mul_nonneg hz.le hw₃]
exact (div_lt_one hq).2 htq |>.le
have heq : ∀ᵐ u ∂volume.restrict (Ioo (0 : ℝ) 1),
quantile (threeScoreOutcomeLaw q t) u *
quantile (threeScoreInverseLaw x y z w₁ w₂ w₃) (1 - u) =
if u ≤ 1 - t / q then 0 else 1 / z := by
filter_upwards [ae_restrict_mem measurableSet_Ioo] with u hu
have huv0 : 0 < 1 - u := by linarith [hu.2]
have huv1 : 1 - u < 1 := by linarith [hu.1]
rw [threeScoreOutcomeLaw_quantile q t u hs0 hs1 hu.1 hu.2,
threeScoreInverseLaw_quantile x y z w₁ w₂ w₃ (1 - u) hx hxy hyz
hw₁ hw₂ hw₃ hq huv0 huv1]
rw [show x * w₁ + y * w₂ + z * w₃ = q by rfl]
by_cases hu0 : u ≤ 1 - t / q
· simp [hu0]
· have htThird : t / q < z * w₃ / q := by
exact (div_lt_div_iff_of_pos_right hq).2
(lt_of_lt_of_le ht₂ (min_le_right _ _))
have hfirst : 1 - u ≤ z * w₃ / q := by linarith
simp [hu0, hfirst]
rw [integral_congr_ae heq, lower_reflected_step_integral (t / q) (1 / z) hs0 hs1]
unfold threeScoreLowerEndpoint
field_simp [ne_of_gt hq, ne_of_gt hz]
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerScoreLogSharp 3 declarations Explicit three-score couplings for the score-log sharp endpoints.
Explicit three-score couplings for the score-log sharp endpoints.
def threeScoreUpperEndpoint
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
x y w₁ t :
ℝ
def
threeScoreUpperEndpoint x y w₁ t :
ℝ
given by
w₁ + (t - x * w₁) / y
def threeScoreLowerEndpoint
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
z t :
ℝ
def
threeScoreLowerEndpoint z t :
ℝ
given by
t / z
theorem threeScoreUpperEndpoint_perturbation
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
x y z t u :
ℝ
hx :
0 < x
hxy :
x < y
conclusion
threeScoreUpperEndpoint x y (1 / 3 + u * (z - y)) t
- threeScoreUpperEndpoint x y (1 / 3) t
= u * (z - y) * (1 - x / y)
Proof (Lean source)
lemma threeScoreUpperEndpoint_perturbation
(x y z t u : ℝ) (hx : 0 < x) (hxy : x < y) :
threeScoreUpperEndpoint x y (1 / 3 + u * (z - y)) t -
threeScoreUpperEndpoint x y (1 / 3) t =
u * (z - y) * (1 - x / y) := by
unfold threeScoreUpperEndpoint
exact fiberTriple_upperEndpoint_shift x y z t u (ne_of_gt (lt_trans hx hxy))
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerScoreLogSharpATE 6 declarations Concrete sharp endpoint identities for the three-score score-log laws.
Concrete sharp endpoint identities for the three-score score-log laws.
theorem muLower_muUpper_false_eq_zero_of_scoreLogReleasedCellLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
r :
q t c :
ℝ
hrel :
Prel = scoreLogReleasedCellLaw r q t c
hq0 :
0 ≤ q
hc :
0 < c
hs0 :
0 ≤ t / q
hs1 :
t / q ≤ 1
hsum :
q + c = 1
Proof (Lean source)
lemma muLower_muUpper_false_eq_zero_of_scoreLogReleasedCellLaw
{ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J))
(hMass : CompatibleCellMasses H g Prel)
(r : LabelSpace J) (q t c : ℝ)
(hrel : Prel = scoreLogReleasedCellLaw r q t c)
(hq0 : 0 ≤ q) (hc : 0 < c) (hs0 : 0 ≤ t / q) (hs1 : t / q ≤ 1)
(hsum : q + c = 1) :
muLower H g Prel hMass false = 0 ∧ muUpper H g Prel hMass false = 0 := by
have hmass (r' : LabelSpace J) :
armCellMass H g false r' = if r' = r then c else 0 := by
rw [← hMass.2.2.2 false r', hrel]
exact scoreLogReleasedCellLaw_control_mass_all r r' q t c hc.le hs0 hs1
have hout :
armCellOutcomeLaw Prel hMass.2.1 false r
(by rw [hMass.2.2.2 false r, hmass r, if_pos rfl]; exact hc) =
Measure.dirac 0 := by
unfold armCellOutcomeLaw
rw [hrel]
have hout' := scoreLogReleasedCellLaw_armCellOutcomeLaw_false
r q t c hq0 hc hs0 hs1 hsum
unfold armCellOutcomeLaw at hout'
exact hout'
constructor
· unfold muLower
rw [Finset.sum_eq_single r]
· split_ifs with hpos
· rw [hmass r, if_pos rfl, hout,
quantile_dirac_zero_product _ (fun u => 1 - u), mul_zero]
· exfalso
apply hpos
rw [hmass r, if_pos rfl]
exact hc
· intro r' _ hr'
split_ifs with hpos
· rw [hmass r', if_neg hr'] at hpos
exact (lt_irrefl 0 hpos).elim
· rfl
· simp
· unfold muUpper
rw [Finset.sum_eq_single r]
· split_ifs with hpos
· rw [hmass r, if_pos rfl, hout,
quantile_dirac_zero_product _ (fun u => u), mul_zero]
· exfalso
apply hpos
rw [hmass r, if_pos rfl]
exact hc
· intro r' _ hr'
split_ifs with hpos
· rw [hmass r', if_neg hr'] at hpos
exact (lt_irrefl 0 hpos).elim
· rfl
· simp
theorem scoreLogTripleBaseline_muUpper_true
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
r :
x y z :
hxy :
x < y
hyz :
y < z
hx :
g x = r
hy :
g y = r
hz :
g z = r
t :
ℝ
ht₁ :
(x : ℝ) * (1 / 3) < t
ht₂ :
t < min ((x : ℝ) * (1 / 3) + (y : ℝ) * (1 / 3)) ((z : ℝ) * (1 / 3))
hMass :
CompatibleCellMasses (scoreLogTripleBaseline x y z) g (releasedLaw (scoreLogTripleBaselineFullLaw g x y z (t / (((x : ℝ) + y + z) / 3))))
conclusion
muUpper (scoreLogTripleBaseline x y z) g (releasedLaw (scoreLogTripleBaselineFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)))) hMass true
= threeScoreUpperEndpoint (x : ℝ) y (1 / 3) t
Proof (Lean source)
lemma scoreLogTripleBaseline_muUpper_true
{ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(r : LabelSpace J) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z)
(hx : g x = r) (hy : g y = r) (hz : g z = r)
(t : ℝ)
(ht₁ : (x : ℝ) * (1 / 3) < t)
(ht₂ : t < min ((x : ℝ) * (1 / 3) + (y : ℝ) * (1 / 3))
((z : ℝ) * (1 / 3)))
(hMass : CompatibleCellMasses (scoreLogTripleBaseline x y z) g
(releasedLaw (scoreLogTripleBaselineFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3))))) :
muUpper (scoreLogTripleBaseline x y z) g
(releasedLaw (scoreLogTripleBaselineFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)))) hMass true =
threeScoreUpperEndpoint (x : ℝ) y (1 / 3) t := by
let q : ℝ := ((x : ℝ) + y + z) / 3
have hx0 : 0 < (x : ℝ) := lt_of_lt_of_le hOverlap.1 x.property.1
have hy0 : 0 < (y : ℝ) := lt_trans hx0 hxy
have hz0 : 0 < (z : ℝ) := lt_trans hy0 hyz
have hq : 0 < q := by dsimp [q]; linarith
have ht0 : 0 < t := lt_trans (mul_pos hx0 (by norm_num)) ht₁
have htq : t < q := by
have hleft := lt_of_lt_of_le ht₂ (min_le_left _ _)
dsimp [q]
nlinarith [mul_pos hz0 (by norm_num : (0 : ℝ) < 1 / 3)]
have hs0 : 0 ≤ t / q := (div_pos ht0 hq).le
have hs1 : t / q ≤ 1 := (div_lt_one hq).2 htq |>.le
have hrel := scoreLogTripleBaseline_releasedLaw_cell
g hOverlap r x y z hx hy hz t hs0 hs1
have hout :
armCellOutcomeLaw
(releasedLaw (scoreLogTripleBaselineFullLaw g x y z (t / q)))
hMass.2.1 true r
(by
rw [hMass.2.2.2 true r,
scoreLogTripleBaseline_armCellMass_true g r x y z hx hy hz]
exact hq) =
threeScoreOutcomeLaw q t := by
unfold armCellOutcomeLaw
rw [hrel]
have hout' := scoreLogReleasedCellLaw_armCellOutcomeLaw r q t (1 - q)
hq (sub_nonneg.mpr (by
dsimp [q]
have hx1 : (x : ℝ) ≤ 1 := by linarith [x.property.2, hOverlap.1]
have hy1 : (y : ℝ) ≤ 1 := by linarith [y.property.2, hOverlap.1]
have hz1 : (z : ℝ) ≤ 1 := by linarith [z.property.2, hOverlap.1]
linarith)) hs0 hs1 (by ring)
unfold armCellOutcomeLaw at hout'
exact hout'
unfold muUpper
rw [Finset.sum_eq_single r]
· split_ifs with hpos
· rw [scoreLogTripleBaseline_armCellMass_true g r x y z hx hy hz]
rw [hout]
rw [scoreLogTripleBaseline_armCellWeightLaw_true g hMass.2.2.1 r
x y z hx hy hz hq]
rw [intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1),
integral_Ioc_eq_integral_Ioo]
have hupper := threeScore_quantileProduct_upper
(x : ℝ) y z (1 / 3) (1 / 3) (1 / 3) t hx0 hxy hyz
(by norm_num) (by norm_num) (by norm_num) (by norm_num) ht₁ ht₂
have hqeq :
(x : ℝ) * (1 / 3) + (y : ℝ) * (1 / 3) + (z : ℝ) * (1 / 3) = q := by
dsimp [q]
ring
rw [hqeq] at hupper
exact hupper
· exfalso
apply hpos
rw [scoreLogTripleBaseline_armCellMass_true g r x y z hx hy hz]
exact hq
· intro r' _ hr'
split_ifs with hpos
· have hzero := scoreLogTripleBaseline_armCellMass_true_all
g r r' x y z hx hy hz
rw [if_neg hr'] at hzero
rw [hzero] at hpos
exact (lt_irrefl 0 hpos).elim
· rfl
· simp
theorem scoreLogTriplePerturbed_muUpper_true
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
r :
x y z :
hxy :
x < y
hyz :
y < z
hx :
g x = r
hy :
g y = r
hz :
g z = r
t u :
ℝ
hu0 :
0 ≤ u
hu :
u * ((z : ℝ) - x) ≤ 1 / 3
ht₁ :
(x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) < t
ht₂ :
t < min ((x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) + (y : ℝ) * (1 / 3 - u * ((z : ℝ) - x))) ((z : ℝ) * (1 / 3 + u * ((y : ℝ) - x)))
hMass :
CompatibleCellMasses (scoreLogTriplePerturbed x y z u) g (releasedLaw (scoreLogTriplePerturbedFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)) u))
conclusion
muUpper (scoreLogTriplePerturbed x y z u) g (releasedLaw (scoreLogTriplePerturbedFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)) u)) hMass true
= threeScoreUpperEndpoint (x : ℝ) y (1 / 3 + u * ((z : ℝ) - y)) t
Proof (Lean source)
lemma scoreLogTriplePerturbed_muUpper_true
{ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(r : LabelSpace J) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z)
(hx : g x = r) (hy : g y = r) (hz : g z = r)
(t u : ℝ) (hu0 : 0 ≤ u) (hu : u * ((z : ℝ) - x) ≤ 1 / 3)
(ht₁ : (x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) < t)
(ht₂ : t < min
((x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) +
(y : ℝ) * (1 / 3 - u * ((z : ℝ) - x)))
((z : ℝ) * (1 / 3 + u * ((y : ℝ) - x))))
(hMass : CompatibleCellMasses (scoreLogTriplePerturbed x y z u) g
(releasedLaw (scoreLogTriplePerturbedFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)) u))) :
muUpper (scoreLogTriplePerturbed x y z u) g
(releasedLaw (scoreLogTriplePerturbedFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)) u)) hMass true =
threeScoreUpperEndpoint (x : ℝ) y
(1 / 3 + u * ((z : ℝ) - y)) t := by
let wx : ℝ := 1 / 3 + u * ((z : ℝ) - y)
let wy : ℝ := 1 / 3 - u * ((z : ℝ) - x)
let wz : ℝ := 1 / 3 + u * ((y : ℝ) - x)
let q : ℝ := ((x : ℝ) + y + z) / 3
have hw := scoreLogTriple_weights_nonneg x y z hxy hyz u hu0 hu
have hsum := (scoreLogTriple_weight_identities x y z hxy hyz u).1
have hmoment := (scoreLogTriple_weight_identities x y z hxy hyz u).2
have hx0 : 0 < (x : ℝ) := lt_of_lt_of_le hOverlap.1 x.property.1
have hy0 : 0 < (y : ℝ) := lt_trans hx0 hxy
have hz0 : 0 < (z : ℝ) := lt_trans hy0 hyz
have hq : 0 < q := by dsimp [q]; linarith
have ht0 : 0 < t := by
have : 0 ≤ (x : ℝ) * wx := mul_nonneg hx0.le (by simpa [wx] using hw.1)
exact lt_of_le_of_lt this (by simpa [wx] using ht₁)
have htq : t < q := by
have hleft := lt_of_lt_of_le ht₂ (min_le_left _ _)
have hwz : 0 ≤ wz := by simpa [wz] using hw.2.2
have hmom : (x : ℝ) * wx + (y : ℝ) * wy + (z : ℝ) * wz = q := by
dsimp [wx, wy, wz, q]
calc
_ = (x : ℝ) / 3 + (y : ℝ) / 3 + (z : ℝ) / 3 := hmoment
_ = ((x : ℝ) + y + z) / 3 := by ring
have hleft' : t < (x : ℝ) * wx + (y : ℝ) * wy := by
simpa [wx, wy] using hleft
nlinarith [mul_nonneg hz0.le hwz]
have hs0 : 0 ≤ t / q := (div_pos ht0 hq).le
have hs1 : t / q ≤ 1 := (div_lt_one hq).2 htq |>.le
have hrel := scoreLogTriplePerturbed_releasedLaw_cell
g hOverlap r x y z hxy hyz hx hy hz t u hs0 hs1 hu0 hu
have hout :
armCellOutcomeLaw
(releasedLaw (scoreLogTriplePerturbedFullLaw g x y z (t / q) u))
hMass.2.1 true r
(by
rw [hMass.2.2.2 true r,
scoreLogTriplePerturbed_armCellMass_true g r x y z hxy hyz
hx hy hz u hu0 hu]
exact hq) =
threeScoreOutcomeLaw q t := by
unfold armCellOutcomeLaw
rw [hrel]
have hout' := scoreLogReleasedCellLaw_armCellOutcomeLaw r q t (1 - q)
hq (sub_nonneg.mpr (by
dsimp [q]
have hx1 : (x : ℝ) ≤ 1 := by linarith [x.property.2, hOverlap.1]
have hy1 : (y : ℝ) ≤ 1 := by linarith [y.property.2, hOverlap.1]
have hz1 : (z : ℝ) ≤ 1 := by linarith [z.property.2, hOverlap.1]
linarith)) hs0 hs1 (by ring)
unfold armCellOutcomeLaw at hout'
exact hout'
unfold muUpper
rw [Finset.sum_eq_single r]
· split_ifs with hpos
· rw [scoreLogTriplePerturbed_armCellMass_true g r x y z hxy hyz
hx hy hz u hu0 hu]
rw [hout]
rw [scoreLogTriplePerturbed_armCellWeightLaw_true g hMass.2.2.1 r
x y z hxy hyz hx hy hz u hu0 hu hq]
rw [intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1),
integral_Ioc_eq_integral_Ioo]
have hupper := threeScore_quantileProduct_upper
(x : ℝ) y z wx wy wz t hx0 hxy hyz
(by simpa [wx] using hw.1) (by simpa [wy] using hw.2.1)
(by simpa [wz] using hw.2.2) (by simpa [wx, wy, wz] using hsum)
(by simpa [wx] using ht₁) (by simpa [wx, wy, wz] using ht₂)
have hqeq : (x : ℝ) * wx + (y : ℝ) * wy + (z : ℝ) * wz = q := by
dsimp [wx, wy, wz, q]
calc
_ = (x : ℝ) / 3 + (y : ℝ) / 3 + (z : ℝ) / 3 := hmoment
_ = ((x : ℝ) + y + z) / 3 := by ring
rw [hqeq] at hupper
exact hupper
· exfalso
apply hpos
rw [scoreLogTriplePerturbed_armCellMass_true g r x y z hxy hyz
hx hy hz u hu0 hu]
exact hq
· intro r' _ hr'
split_ifs with hpos
· have hzero := scoreLogTriplePerturbed_armCellMass_true_all
g r r' x y z hxy hyz hx hy hz u hu0 hu
rw [if_neg hr'] at hzero
rw [hzero] at hpos
exact (lt_irrefl 0 hpos).elim
· rfl
· simp
theorem scoreLogTripleBaseline_muLower_true
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
r :
x y z :
hxy :
x < y
hyz :
y < z
hx :
g x = r
hy :
g y = r
hz :
g z = r
t :
ℝ
ht₁ :
(x : ℝ) * (1 / 3) < t
ht₂ :
t < min ((x : ℝ) * (1 / 3) + (y : ℝ) * (1 / 3)) ((z : ℝ) * (1 / 3))
hMass :
CompatibleCellMasses (scoreLogTripleBaseline x y z) g (releasedLaw (scoreLogTripleBaselineFullLaw g x y z (t / (((x : ℝ) + y + z) / 3))))
conclusion
muLower (scoreLogTripleBaseline x y z) g (releasedLaw (scoreLogTripleBaselineFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)))) hMass true
= threeScoreLowerEndpoint (z : ℝ) t
Proof (Lean source)
lemma scoreLogTripleBaseline_muLower_true
{ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(r : LabelSpace J) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z)
(hx : g x = r) (hy : g y = r) (hz : g z = r)
(t : ℝ)
(ht₁ : (x : ℝ) * (1 / 3) < t)
(ht₂ : t < min ((x : ℝ) * (1 / 3) + (y : ℝ) * (1 / 3))
((z : ℝ) * (1 / 3)))
(hMass : CompatibleCellMasses (scoreLogTripleBaseline x y z) g
(releasedLaw (scoreLogTripleBaselineFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3))))) :
muLower (scoreLogTripleBaseline x y z) g
(releasedLaw (scoreLogTripleBaselineFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)))) hMass true =
threeScoreLowerEndpoint (z : ℝ) t := by
let q : ℝ := ((x : ℝ) + y + z) / 3
have hx0 : 0 < (x : ℝ) := lt_of_lt_of_le hOverlap.1 x.property.1
have hy0 : 0 < (y : ℝ) := lt_trans hx0 hxy
have hz0 : 0 < (z : ℝ) := lt_trans hy0 hyz
have hq : 0 < q := by dsimp [q]; linarith
have ht0 : 0 < t := lt_trans (mul_pos hx0 (by norm_num)) ht₁
have htq : t < q := by
have hleft := lt_of_lt_of_le ht₂ (min_le_left _ _)
dsimp [q]
nlinarith [mul_pos hz0 (by norm_num : (0 : ℝ) < 1 / 3)]
have hs0 : 0 ≤ t / q := (div_pos ht0 hq).le
have hs1 : t / q ≤ 1 := (div_lt_one hq).2 htq |>.le
have hrel := scoreLogTripleBaseline_releasedLaw_cell
g hOverlap r x y z hx hy hz t hs0 hs1
have hout :
armCellOutcomeLaw
(releasedLaw (scoreLogTripleBaselineFullLaw g x y z (t / q)))
hMass.2.1 true r
(by
rw [hMass.2.2.2 true r,
scoreLogTripleBaseline_armCellMass_true g r x y z hx hy hz]
exact hq) =
threeScoreOutcomeLaw q t := by
unfold armCellOutcomeLaw
rw [hrel]
have hout' := scoreLogReleasedCellLaw_armCellOutcomeLaw r q t (1 - q)
hq (sub_nonneg.mpr (by
dsimp [q]
have hx1 : (x : ℝ) ≤ 1 := by linarith [x.property.2, hOverlap.1]
have hy1 : (y : ℝ) ≤ 1 := by linarith [y.property.2, hOverlap.1]
have hz1 : (z : ℝ) ≤ 1 := by linarith [z.property.2, hOverlap.1]
linarith)) hs0 hs1 (by ring)
unfold armCellOutcomeLaw at hout'
exact hout'
unfold muLower
rw [Finset.sum_eq_single r]
· split_ifs with hpos
· rw [scoreLogTripleBaseline_armCellMass_true g r x y z hx hy hz]
rw [hout]
rw [scoreLogTripleBaseline_armCellWeightLaw_true g hMass.2.2.1 r
x y z hx hy hz hq]
rw [intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1),
integral_Ioc_eq_integral_Ioo]
have hlower := threeScore_quantileProduct_lower
(x : ℝ) y z (1 / 3) (1 / 3) (1 / 3) t hx0 hxy hyz
(by norm_num) (by norm_num) (by norm_num) (by norm_num) ht₁ ht₂
have hqeq :
(x : ℝ) * (1 / 3) + (y : ℝ) * (1 / 3) + (z : ℝ) * (1 / 3) = q := by
dsimp [q]
ring
rw [hqeq] at hlower
exact hlower
· exfalso
apply hpos
rw [scoreLogTripleBaseline_armCellMass_true g r x y z hx hy hz]
exact hq
· intro r' _ hr'
split_ifs with hpos
· have hzero := scoreLogTripleBaseline_armCellMass_true_all
g r r' x y z hx hy hz
rw [if_neg hr'] at hzero
rw [hzero] at hpos
exact (lt_irrefl 0 hpos).elim
· rfl
· simp
theorem scoreLogTriplePerturbed_muLower_true
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
r :
x y z :
hxy :
x < y
hyz :
y < z
hx :
g x = r
hy :
g y = r
hz :
g z = r
t u :
ℝ
hu0 :
0 ≤ u
hu :
u * ((z : ℝ) - x) ≤ 1 / 3
ht₁ :
(x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) < t
ht₂ :
t < min ((x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) + (y : ℝ) * (1 / 3 - u * ((z : ℝ) - x))) ((z : ℝ) * (1 / 3 + u * ((y : ℝ) - x)))
hMass :
CompatibleCellMasses (scoreLogTriplePerturbed x y z u) g (releasedLaw (scoreLogTriplePerturbedFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)) u))
conclusion
muLower (scoreLogTriplePerturbed x y z u) g (releasedLaw (scoreLogTriplePerturbedFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)) u)) hMass true
= threeScoreLowerEndpoint (z : ℝ) t
Proof (Lean source)
lemma scoreLogTriplePerturbed_muLower_true
{ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(r : LabelSpace J) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z)
(hx : g x = r) (hy : g y = r) (hz : g z = r)
(t u : ℝ) (hu0 : 0 ≤ u) (hu : u * ((z : ℝ) - x) ≤ 1 / 3)
(ht₁ : (x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) < t)
(ht₂ : t < min
((x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) +
(y : ℝ) * (1 / 3 - u * ((z : ℝ) - x)))
((z : ℝ) * (1 / 3 + u * ((y : ℝ) - x))))
(hMass : CompatibleCellMasses (scoreLogTriplePerturbed x y z u) g
(releasedLaw (scoreLogTriplePerturbedFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)) u))) :
muLower (scoreLogTriplePerturbed x y z u) g
(releasedLaw (scoreLogTriplePerturbedFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)) u)) hMass true =
threeScoreLowerEndpoint (z : ℝ) t := by
let wx : ℝ := 1 / 3 + u * ((z : ℝ) - y)
let wy : ℝ := 1 / 3 - u * ((z : ℝ) - x)
let wz : ℝ := 1 / 3 + u * ((y : ℝ) - x)
let q : ℝ := ((x : ℝ) + y + z) / 3
have hw := scoreLogTriple_weights_nonneg x y z hxy hyz u hu0 hu
have hsum := (scoreLogTriple_weight_identities x y z hxy hyz u).1
have hmoment := (scoreLogTriple_weight_identities x y z hxy hyz u).2
have hx0 : 0 < (x : ℝ) := lt_of_lt_of_le hOverlap.1 x.property.1
have hy0 : 0 < (y : ℝ) := lt_trans hx0 hxy
have hz0 : 0 < (z : ℝ) := lt_trans hy0 hyz
have hq : 0 < q := by dsimp [q]; linarith
have ht0 : 0 < t := by
have : 0 ≤ (x : ℝ) * wx := mul_nonneg hx0.le (by simpa [wx] using hw.1)
exact lt_of_le_of_lt this (by simpa [wx] using ht₁)
have htq : t < q := by
have hleft := lt_of_lt_of_le ht₂ (min_le_left _ _)
have hwz : 0 ≤ wz := by simpa [wz] using hw.2.2
have hmom : (x : ℝ) * wx + (y : ℝ) * wy + (z : ℝ) * wz = q := by
dsimp [wx, wy, wz, q]
calc
_ = (x : ℝ) / 3 + (y : ℝ) / 3 + (z : ℝ) / 3 := hmoment
_ = ((x : ℝ) + y + z) / 3 := by ring
have hleft' : t < (x : ℝ) * wx + (y : ℝ) * wy := by
simpa [wx, wy] using hleft
nlinarith [mul_nonneg hz0.le hwz]
have hs0 : 0 ≤ t / q := (div_pos ht0 hq).le
have hs1 : t / q ≤ 1 := (div_lt_one hq).2 htq |>.le
have hrel := scoreLogTriplePerturbed_releasedLaw_cell
g hOverlap r x y z hxy hyz hx hy hz t u hs0 hs1 hu0 hu
have hout :
armCellOutcomeLaw
(releasedLaw (scoreLogTriplePerturbedFullLaw g x y z (t / q) u))
hMass.2.1 true r
(by
rw [hMass.2.2.2 true r,
scoreLogTriplePerturbed_armCellMass_true g r x y z hxy hyz
hx hy hz u hu0 hu]
exact hq) =
threeScoreOutcomeLaw q t := by
unfold armCellOutcomeLaw
rw [hrel]
have hout' := scoreLogReleasedCellLaw_armCellOutcomeLaw r q t (1 - q)
hq (sub_nonneg.mpr (by
dsimp [q]
have hx1 : (x : ℝ) ≤ 1 := by linarith [x.property.2, hOverlap.1]
have hy1 : (y : ℝ) ≤ 1 := by linarith [y.property.2, hOverlap.1]
have hz1 : (z : ℝ) ≤ 1 := by linarith [z.property.2, hOverlap.1]
linarith)) hs0 hs1 (by ring)
unfold armCellOutcomeLaw at hout'
exact hout'
unfold muLower
rw [Finset.sum_eq_single r]
· split_ifs with hpos
· rw [scoreLogTriplePerturbed_armCellMass_true g r x y z hxy hyz
hx hy hz u hu0 hu]
rw [hout]
rw [scoreLogTriplePerturbed_armCellWeightLaw_true g hMass.2.2.1 r
x y z hxy hyz hx hy hz u hu0 hu hq]
rw [intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1),
integral_Ioc_eq_integral_Ioo]
have hlower := threeScore_quantileProduct_lower
(x : ℝ) y z wx wy wz t hx0 hxy hyz
(by simpa [wx] using hw.1) (by simpa [wy] using hw.2.1)
(by simpa [wz] using hw.2.2) (by simpa [wx, wy, wz] using hsum)
(by simpa [wx] using ht₁) (by simpa [wx, wy, wz] using ht₂)
have hqeq : (x : ℝ) * wx + (y : ℝ) * wy + (z : ℝ) * wz = q := by
dsimp [wx, wy, wz, q]
calc
_ = (x : ℝ) / 3 + (y : ℝ) / 3 + (z : ℝ) / 3 := hmoment
_ = ((x : ℝ) + y + z) / 3 := by ring
rw [hqeq] at hlower
exact hlower
· exfalso
apply hpos
rw [scoreLogTriplePerturbed_armCellMass_true g r x y z hxy hyz
hx hy hz u hu0 hu]
exact hq
· intro r' _ hr'
split_ifs with hpos
· have hzero := scoreLogTriplePerturbed_armCellMass_true_all
g r r' x y z hxy hyz hx hy hz u hu0 hu
rw [if_neg hr'] at hzero
rw [hzero] at hpos
exact (lt_irrefl 0 hpos).elim
· rfl
· simp
theorem scoreLogTriple_sharpATELength_perturbation
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
r :
x y z :
hxy :
x < y
hyz :
y < z
hx :
g x = r
hy :
g y = r
hz :
g z = r
t u :
ℝ
hu0 :
0 ≤ u
hu :
u * ((z : ℝ) - x) ≤ 1 / 3
hb₁ :
(x : ℝ) * (1 / 3) < t
hb₂ :
t < min ((x : ℝ) * (1 / 3) + (y : ℝ) * (1 / 3)) ((z : ℝ) * (1 / 3))
hp₁ :
(x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) < t
hp₂ :
t < min ((x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) + (y : ℝ) * (1 / 3 - u * ((z : ℝ) - x))) ((z : ℝ) * (1 / 3 + u * ((y : ℝ) - x)))
hMass₀ :
CompatibleCellMasses (scoreLogTripleBaseline x y z) g (releasedLaw (scoreLogTripleBaselineFullLaw g x y z (t / (((x : ℝ) + y + z) / 3))))
hMass₁ :
CompatibleCellMasses (scoreLogTriplePerturbed x y z u) g (releasedLaw (scoreLogTriplePerturbedFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)) u))
conclusion
sharpATELength (scoreLogTriplePerturbed x y z u) g (releasedLaw (scoreLogTriplePerturbedFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)) u)) hMass₁
- sharpATELength (scoreLogTripleBaseline x y z) g (releasedLaw (scoreLogTripleBaselineFullLaw g x y z (t / (((x : ℝ) + y + z) / 3)))) hMass₀
= u * ((z : ℝ) - y) * (1 - (x : ℝ) / y)
Proof (Lean source)
lemma scoreLogTriple_sharpATELength_perturbation
{ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(r : LabelSpace J) (x y z : ScoreSpace ε)
(hxy : x < y) (hyz : y < z)
(hx : g x = r) (hy : g y = r) (hz : g z = r)
(t u : ℝ) (hu0 : 0 ≤ u) (hu : u * ((z : ℝ) - x) ≤ 1 / 3)
(hb₁ : (x : ℝ) * (1 / 3) < t)
(hb₂ : t < min ((x : ℝ) * (1 / 3) + (y : ℝ) * (1 / 3))
((z : ℝ) * (1 / 3)))
(hp₁ : (x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) < t)
(hp₂ : t < min
((x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) +
(y : ℝ) * (1 / 3 - u * ((z : ℝ) - x)))
((z : ℝ) * (1 / 3 + u * ((y : ℝ) - x))))
(hMass₀ : CompatibleCellMasses (scoreLogTripleBaseline x y z) g
(releasedLaw (scoreLogTripleBaselineFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)))))
(hMass₁ : CompatibleCellMasses (scoreLogTriplePerturbed x y z u) g
(releasedLaw (scoreLogTriplePerturbedFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)) u))) :
sharpATELength (scoreLogTriplePerturbed x y z u) g
(releasedLaw (scoreLogTriplePerturbedFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)) u)) hMass₁ -
sharpATELength (scoreLogTripleBaseline x y z) g
(releasedLaw (scoreLogTripleBaselineFullLaw g x y z
(t / (((x : ℝ) + y + z) / 3)))) hMass₀ =
u * ((z : ℝ) - y) * (1 - (x : ℝ) / y) := by
let q : ℝ := ((x : ℝ) + y + z) / 3
have hx0 : 0 < (x : ℝ) := lt_of_lt_of_le hOverlap.1 x.property.1
have hy0 : 0 < (y : ℝ) := lt_trans hx0 hxy
have hz0 : 0 < (z : ℝ) := lt_trans hy0 hyz
have hq : 0 < q := by dsimp [q]; linarith
have ht0 : 0 < t := lt_trans (mul_pos hx0 (by norm_num)) hb₁
have htq : t < q := by
have hleft := lt_of_lt_of_le hb₂ (min_le_left _ _)
dsimp [q]
nlinarith [mul_pos hz0 (by norm_num : (0 : ℝ) < 1 / 3)]
have hs0 : 0 ≤ t / q := (div_pos ht0 hq).le
have hs1 : t / q ≤ 1 := (div_lt_one hq).2 htq |>.le
have hq1 : q < 1 := by
have hx1 : (x : ℝ) < 1 := lt_of_le_of_lt x.property.2 (by linarith [hOverlap.1])
have hy1 : (y : ℝ) < 1 := lt_of_le_of_lt y.property.2 (by linarith [hOverlap.1])
have hz1 : (z : ℝ) < 1 := lt_of_le_of_lt z.property.2 (by linarith [hOverlap.1])
dsimp [q]
linarith
have hrel₀ := scoreLogTripleBaseline_releasedLaw_cell
g hOverlap r x y z hx hy hz t hs0 hs1
have hrel₁ := scoreLogTriplePerturbed_releasedLaw_cell
g hOverlap r x y z hxy hyz hx hy hz t u hs0 hs1 hu0 hu
have hc₀ := muLower_muUpper_false_eq_zero_of_scoreLogReleasedCellLaw
(scoreLogTripleBaseline x y z) g
(releasedLaw (scoreLogTripleBaselineFullLaw g x y z (t / q))) hMass₀
r q t (1 - q) hrel₀ hq.le (sub_pos.mpr hq1) hs0 hs1 (by ring)
have hc₁ := muLower_muUpper_false_eq_zero_of_scoreLogReleasedCellLaw
(scoreLogTriplePerturbed x y z u) g
(releasedLaw (scoreLogTriplePerturbedFullLaw g x y z (t / q) u)) hMass₁
r q t (1 - q) hrel₁ hq.le (sub_pos.mpr hq1) hs0 hs1 (by ring)
unfold sharpATELength
rw [scoreLogTriplePerturbed_muUpper_true g hOverlap r x y z hxy hyz
hx hy hz t u hu0 hu hp₁ hp₂ hMass₁,
scoreLogTriplePerturbed_muLower_true g hOverlap r x y z hxy hyz
hx hy hz t u hu0 hu hp₁ hp₂ hMass₁,
scoreLogTripleBaseline_muUpper_true g hOverlap r x y z hxy hyz
hx hy hz t hb₁ hb₂ hMass₀,
scoreLogTripleBaseline_muLower_true g hOverlap r x y z hxy hyz
hx hy hz t hb₁ hb₂ hMass₀,
hc₁.1, hc₁.2, hc₀.1, hc₀.2]
have hshift := threeScoreUpperEndpoint_perturbation
(x : ℝ) y z t u hx0 hxy
linarith
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerScoreLogTesting 8 declarations Testing estimates for the three-score external-log perturbation.
Testing estimates for the three-score external-log perturbation.
def threeScoreBaseLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
x y z :
def
def threeScorePerturbedLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
theorem threeScoreBaseLaw_isProbabilityMeasure
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma threeScoreBaseLaw_isProbabilityMeasure {ε : ℝ} (x y z : ScoreSpace ε) :
IsProbabilityMeasure (threeScoreBaseLaw x y z) := by
rw [isProbabilityMeasure_iff]
unfold threeScoreBaseLaw
rw [Measure.add_apply, Measure.add_apply, Measure.smul_apply,
Measure.smul_apply, Measure.smul_apply]
simp only [Measure.dirac_apply, Set.indicator_of_mem, Set.mem_univ,
Pi.one_apply, smul_eq_mul, mul_one]
rw [← ENNReal.ofReal_add (by norm_num : (0 : ℝ) ≤ 1 / 3)
(by norm_num : (0 : ℝ) ≤ 1 / 3),
← ENNReal.ofReal_add (by norm_num : (0 : ℝ) ≤ 1 / 3 + 1 / 3)
(by norm_num : (0 : ℝ) ≤ 1 / 3)]
norm_num
theorem threeScorePerturbedLaw_isProbabilityMeasure
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
x y z :
u :
ℝ
hx :
0 < 1 / 3 + u * ((z : ℝ) - y)
hy :
0 < 1 / 3 - u * ((z : ℝ) - x)
hz :
0 < 1 / 3 + u * ((y : ℝ) - x)
conclusion
IsProbabilityMeasure (threeScorePerturbedLaw x y z u)
Proof (Lean source)
lemma threeScorePerturbedLaw_isProbabilityMeasure {ε : ℝ}
(x y z : ScoreSpace ε) (u : ℝ)
(hx : 0 < 1 / 3 + u * ((z : ℝ) - y))
(hy : 0 < 1 / 3 - u * ((z : ℝ) - x))
(hz : 0 < 1 / 3 + u * ((y : ℝ) - x)) :
IsProbabilityMeasure (threeScorePerturbedLaw x y z u) := by
rw [isProbabilityMeasure_iff]
unfold threeScorePerturbedLaw
rw [Measure.add_apply, Measure.add_apply, Measure.smul_apply,
Measure.smul_apply, Measure.smul_apply]
simp only [Measure.dirac_apply, Set.indicator_of_mem, Set.mem_univ,
Pi.one_apply, smul_eq_mul, mul_one]
rw [← ENNReal.ofReal_add hx.le hy.le,
← ENNReal.ofReal_add (add_nonneg hx.le hy.le) hz.le]
rw [show (1 / 3 + u * ((z : ℝ) - y)) +
(1 / 3 - u * ((z : ℝ) - x)) +
(1 / 3 + u * ((y : ℝ) - x)) = 1 by ring,
ENNReal.ofReal_one]
theorem threeScorePerturbedLaw_ac_integrable
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
conclusion 1
threeScorePerturbedLaw x y z u ≪ threeScoreBaseLaw x y z
conclusion 2
Integrable (fun e => (((threeScorePerturbedLaw x y z u).rnDeriv (threeScoreBaseLaw x y z) e).toReal - 1) ^ 2) (threeScoreBaseLaw x y z)
Proof (Lean source)
lemma threeScorePerturbedLaw_ac_integrable {ε : ℝ}
(x y z : ScoreSpace ε) (u : ℝ) (hxy : x < y) (hyz : y < z) :
threeScorePerturbedLaw x y z u ≪ threeScoreBaseLaw x y z ∧
Integrable (fun e =>
(((threeScorePerturbedLaw x y z u).rnDeriv
(threeScoreBaseLaw x y z) e).toReal - 1) ^ 2)
(threeScoreBaseLaw x y z) := by
let wx : ℝ := 1 / 3 + u * ((z : ℝ) - y)
let wy : ℝ := 1 / 3 - u * ((z : ℝ) - x)
let wz : ℝ := 1 / 3 + u * ((y : ℝ) - x)
let d : ScoreSpace ε → ENNReal := fun e =>
if e = x then ofReal (3 * wx)
else if e = y then ofReal (3 * wy)
else if e = z then ofReal (3 * wz)
else 0
have hxy_ne : x ≠ y := ne_of_lt hxy
have hxz_ne : x ≠ z := ne_of_lt (lt_trans hxy hyz)
have hyz_ne : y ≠ z := ne_of_lt hyz
have hd : Measurable d := by
dsimp [d]
exact Measurable.ite (measurableSet_singleton x) measurable_const
(Measurable.ite (measurableSet_singleton y) measurable_const
(Measurable.ite (measurableSet_singleton z) measurable_const measurable_const))
have hscale (w : ℝ) :
ofReal (1 / 3 : ℝ) * ofReal (3 * w) = ofReal w := by
rw [← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 1 / 3)]
exact congrArg ofReal (by ring)
have hwd : threeScorePerturbedLaw x y z u =
(threeScoreBaseLaw x y z).withDensity d := by
unfold threeScorePerturbedLaw threeScoreBaseLaw
rw [withDensity_add_measure, withDensity_add_measure,
withDensity_smul_measure, withDensity_smul_measure,
withDensity_smul_measure, dirac_withDensity, dirac_withDensity,
dirac_withDensity]
dsimp only [d]
simp only [if_pos, if_neg hxy_ne.symm, if_neg hxz_ne.symm,
if_neg hxy_ne, if_neg hyz_ne.symm, if_neg hxz_ne, if_neg hyz_ne]
repeat' rw [smul_smul]
rw [hscale wx, hscale wy, hscale wz]
constructor
· rw [hwd]
exact withDensity_absolutelyContinuous _ _
· unfold threeScoreBaseLaw
apply Integrable.add_measure
· apply Integrable.add_measure
· exact Integrable.smul_measure (integrable_dirac (by simp [enorm])) (by simp)
· exact Integrable.smul_measure (integrable_dirac (by simp [enorm])) (by simp)
· exact Integrable.smul_measure (integrable_dirac (by simp [enorm])) (by simp)
theorem threeScorePerturbedLaw_chisq
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
x y z :
u :
ℝ
hxy :
x < y
hyz :
y < z
hx :
0 < 1 / 3 + u * ((z : ℝ) - y)
hy :
0 < 1 / 3 - u * ((z : ℝ) - x)
hz :
0 < 1 / 3 + u * ((y : ℝ) - x)
conclusion
= 3 * u ^ 2 * (((z : ℝ) - y) ^ 2 + ((z : ℝ) - x) ^ 2 + ((y : ℝ) - x) ^ 2)
Proof (Lean source)
lemma threeScorePerturbedLaw_chisq {ε : ℝ}
(x y z : ScoreSpace ε) (u : ℝ) (hxy : x < y) (hyz : y < z)
(hx : 0 < 1 / 3 + u * ((z : ℝ) - y))
(hy : 0 < 1 / 3 - u * ((z : ℝ) - x))
(hz : 0 < 1 / 3 + u * ((y : ℝ) - x)) :
chiSqDiv (threeScorePerturbedLaw x y z u)
(threeScoreBaseLaw x y z) =
3 * u ^ 2 * (((z : ℝ) - y) ^ 2 + ((z : ℝ) - x) ^ 2 +
((y : ℝ) - x) ^ 2) := by
let wx : ℝ := 1 / 3 + u * ((z : ℝ) - y)
let wy : ℝ := 1 / 3 - u * ((z : ℝ) - x)
let wz : ℝ := 1 / 3 + u * ((y : ℝ) - x)
let d : ScoreSpace ε → ENNReal := fun e =>
if e = x then ofReal (3 * wx)
else if e = y then ofReal (3 * wy)
else if e = z then ofReal (3 * wz)
else 0
have hxy_ne : x ≠ y := ne_of_lt hxy
have hxz_ne : x ≠ z := ne_of_lt (lt_trans hxy hyz)
have hyz_ne : y ≠ z := ne_of_lt hyz
have hd : Measurable d := by
dsimp [d]
exact Measurable.ite (measurableSet_singleton x) measurable_const
(Measurable.ite (measurableSet_singleton y) measurable_const
(Measurable.ite (measurableSet_singleton z) measurable_const measurable_const))
have hscale (w : ℝ) :
ofReal (1 / 3 : ℝ) * ofReal (3 * w) = ofReal w := by
rw [← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 1 / 3)]
exact congrArg ofReal (by ring)
have hwd : threeScorePerturbedLaw x y z u =
(threeScoreBaseLaw x y z).withDensity d := by
unfold threeScorePerturbedLaw threeScoreBaseLaw
rw [withDensity_add_measure, withDensity_add_measure,
withDensity_smul_measure, withDensity_smul_measure,
withDensity_smul_measure, dirac_withDensity, dirac_withDensity,
dirac_withDensity]
dsimp only [d]
simp only [if_pos, if_neg hxy_ne.symm, if_neg hxz_ne.symm,
if_neg hxy_ne, if_neg hyz_ne.symm, if_neg hxz_ne, if_neg hyz_ne]
repeat' rw [smul_smul]
rw [hscale wx, hscale wy, hscale wz]
letI : IsProbabilityMeasure (threeScoreBaseLaw x y z) :=
threeScoreBaseLaw_isProbabilityMeasure x y z
have hrn : (threeScorePerturbedLaw x y z u).rnDeriv
(threeScoreBaseLaw x y z) =ᵐ[threeScoreBaseLaw x y z] d := by
rw [hwd]
exact Measure.rnDeriv_withDensity _ hd
rw [chiSqDiv]
trans ∫ e, ((d e).toReal - 1) ^ 2 ∂(threeScoreBaseLaw x y z)
· exact integral_congr_ae <| hrn.mono fun e he => by
simpa only using congrArg (fun r : ENNReal => (r.toReal - 1) ^ 2) he
unfold threeScoreBaseLaw
rw [integral_add_measure]
· rw [integral_add_measure]
· rw [integral_smul_measure, integral_smul_measure, integral_smul_measure]
simp only [integral_dirac]
dsimp [d, wx, wy, wz]
simp only [if_pos, if_neg hxy_ne.symm, if_neg hxz_ne.symm,
if_neg hxy_ne, if_neg hyz_ne.symm, if_neg hxz_ne, if_neg hyz_ne]
rw [ENNReal.toReal_ofReal (mul_nonneg (by norm_num) hx.le),
ENNReal.toReal_ofReal (mul_nonneg (by norm_num) hy.le),
ENNReal.toReal_ofReal (mul_nonneg (by norm_num) hz.le)]
norm_num
ring
· exact Integrable.smul_measure (integrable_dirac (by simp [enorm])) (by simp)
· exact Integrable.smul_measure (integrable_dirac (by simp [enorm])) (by simp)
· apply Integrable.add_measure
· exact Integrable.smul_measure (integrable_dirac (by simp [enorm])) (by simp)
· exact Integrable.smul_measure (integrable_dirac (by simp [enorm])) (by simp)
· exact Integrable.smul_measure (integrable_dirac (by simp [enorm])) (by simp)
theorem threeScorePerturbedLaw_product_tv_bound
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
x y z :
u :
ℝ
hxy :
x < y
hyz :
y < z
hx :
0 < 1 / 3 + u * ((z : ℝ) - y)
hy :
0 < 1 / 3 - u * ((z : ℝ) - x)
hz :
0 < 1 / 3 + u * ((y : ℝ) - x)
m :
ℕ
conclusion
tvDist (Measure.pi (fun _ : Fin m => threeScorePerturbedLaw x y z u)) (Measure.pi (fun _ : Fin m => threeScoreBaseLaw x y z))
≤ (1 / 2 : ℝ) * sqrt ((1 + 3 * u ^ 2 * (((z : ℝ) - y) ^ 2 + ((z : ℝ) - x) ^ 2 + ((y : ℝ) - x) ^ 2)) ^ m - 1)
Proof (Lean source)
lemma threeScorePerturbedLaw_product_tv_bound {ε : ℝ}
(x y z : ScoreSpace ε) (u : ℝ) (hxy : x < y) (hyz : y < z)
(hx : 0 < 1 / 3 + u * ((z : ℝ) - y))
(hy : 0 < 1 / 3 - u * ((z : ℝ) - x))
(hz : 0 < 1 / 3 + u * ((y : ℝ) - x)) (m : ℕ) :
tvDist
(Measure.pi (fun _ : Fin m => threeScorePerturbedLaw x y z u))
(Measure.pi (fun _ : Fin m => threeScoreBaseLaw x y z)) ≤
(1 / 2 : ℝ) * sqrt
((1 + 3 * u ^ 2 * (((z : ℝ) - y) ^ 2 + ((z : ℝ) - x) ^ 2 +
((y : ℝ) - x) ^ 2)) ^ m - 1) := by
letI : IsProbabilityMeasure (threeScorePerturbedLaw x y z u) :=
threeScorePerturbedLaw_isProbabilityMeasure x y z u hx hy hz
letI : IsProbabilityMeasure (threeScoreBaseLaw x y z) :=
threeScoreBaseLaw_isProbabilityMeasure x y z
have h := threeScorePerturbedLaw_ac_integrable x y z u hxy hyz
simpa [threeScorePerturbedLaw_chisq x y z u hxy hyz hx hy hz] using
trialProduct_tv_le_of_chisq (threeScorePerturbedLaw x y z u)
(threeScoreBaseLaw x y z) h.1 h.2 m
theorem threeScorePerturbedLaw_commonTrial_tv_bound
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
T :
Type*
ρ :
x y z :
u :
ℝ
hxy :
x < y
hyz :
y < z
hx :
0 < 1 / 3 + u * ((z : ℝ) - y)
hy :
0 < 1 / 3 - u * ((z : ℝ) - x)
hz :
0 < 1 / 3 + u * ((y : ℝ) - x)
m :
ℕ
conclusion
tvDist (ρ.prod (Measure.pi (fun _ : Fin m => threeScorePerturbedLaw x y z u))) (ρ.prod (Measure.pi (fun _ : Fin m => threeScoreBaseLaw x y z)))
≤ (1 / 2 : ℝ) * sqrt ((1 + 3 * u ^ 2 * (((z : ℝ) - y) ^ 2 + ((z : ℝ) - x) ^ 2 + ((y : ℝ) - x) ^ 2)) ^ m - 1)
Proof (Lean source)
lemma threeScorePerturbedLaw_commonTrial_tv_bound {ε : ℝ} {T : Type*}
[MeasurableSpace T] (ρ : Measure T) [IsProbabilityMeasure ρ]
(x y z : ScoreSpace ε) (u : ℝ) (hxy : x < y) (hyz : y < z)
(hx : 0 < 1 / 3 + u * ((z : ℝ) - y))
(hy : 0 < 1 / 3 - u * ((z : ℝ) - x))
(hz : 0 < 1 / 3 + u * ((y : ℝ) - x)) (m : ℕ) :
tvDist
(ρ.prod (Measure.pi (fun _ : Fin m => threeScorePerturbedLaw x y z u)))
(ρ.prod (Measure.pi (fun _ : Fin m => threeScoreBaseLaw x y z))) ≤
(1 / 2 : ℝ) * sqrt
((1 + 3 * u ^ 2 * (((z : ℝ) - y) ^ 2 + ((z : ℝ) - x) ^ 2 +
((y : ℝ) - x) ^ 2)) ^ m - 1) := by
letI : IsProbabilityMeasure (threeScorePerturbedLaw x y z u) :=
threeScorePerturbedLaw_isProbabilityMeasure x y z u hx hy hz
letI : IsProbabilityMeasure (threeScoreBaseLaw x y z) :=
threeScoreBaseLaw_isProbabilityMeasure x y z
have hprod := tvDist_prod_le_add
ρ ρ
(Measure.pi (fun _ : Fin m => threeScorePerturbedLaw x y z u))
(Measure.pi (fun _ : Fin m => threeScoreBaseLaw x y z))
have hself : tvDist ρ ρ = 0 := by
unfold tvDist
simp
calc
_ ≤ tvDist ρ ρ +
tvDist
(Measure.pi (fun _ : Fin m => threeScorePerturbedLaw x y z u))
(Measure.pi (fun _ : Fin m => threeScoreBaseLaw x y z)) := hprod
_ = tvDist
(Measure.pi (fun _ : Fin m => threeScorePerturbedLaw x y z u))
(Measure.pi (fun _ : Fin m => threeScoreBaseLaw x y z)) := by rw [hself, zero_add]
_ ≤ _ := threeScorePerturbedLaw_product_tv_bound
x y z u hxy hyz hx hy hz m
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerScoreLogThresholds 3 declarations Explicit strict filling thresholds for the calibrated three-score pair.
Explicit strict filling thresholds for the calibrated three-score pair.
def threeScoreThreshold
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
x y z :
def
threeScoreThreshold x y z :
ℝ
given by
(((x : ℝ) / 3) + min (((x : ℝ) + y) / 3) ((z : ℝ) / 3)) / 2
theorem threeScoreThreshold_baseline
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
t :
threeScoreThreshold x y z
q :
((x : ℝ) + y + z) / 3
(x : ℝ) * (1 / 3) < t
t < min ((x : ℝ) * (1 / 3) + (y : ℝ) * (1 / 3)) ((z : ℝ) * (1 / 3))
0 < t / q
t / q < 1
Proof (Lean source)
lemma threeScoreThreshold_baseline {ε : ℝ} (x y z : ScoreSpace ε)
(hx0 : 0 < (x : ℝ)) (hxy : x < y) (hyz : y < z) :
let t := threeScoreThreshold x y z
let q := ((x : ℝ) + y + z) / 3
(x : ℝ) * (1 / 3) < t ∧
t < min ((x : ℝ) * (1 / 3) + (y : ℝ) * (1 / 3))
((z : ℝ) * (1 / 3)) ∧
0 < t / q ∧ t / q < 1 := by
dsimp only
have hxyR : (x : ℝ) < (y : ℝ) := hxy
have hyzR : (y : ℝ) < (z : ℝ) := hyz
have hy0 : 0 < (y : ℝ) := lt_trans hx0 hxy
have hz0 : 0 < (z : ℝ) := lt_trans hy0 hyz
have hlower : (x : ℝ) / 3 <
min (((x : ℝ) + y) / 3) ((z : ℝ) / 3) := by
rw [lt_min_iff]
constructor <;> linarith
have htLower : (x : ℝ) / 3 < threeScoreThreshold x y z := by
unfold threeScoreThreshold
linarith
have htUpper : threeScoreThreshold x y z <
min (((x : ℝ) + y) / 3) ((z : ℝ) / 3) := by
unfold threeScoreThreshold
linarith
have hq : 0 < ((x : ℝ) + y + z) / 3 := by linarith
have ht0 : 0 < threeScoreThreshold x y z := lt_trans (by positivity) htLower
have htq : threeScoreThreshold x y z < ((x : ℝ) + y + z) / 3 := by
have hmin := min_le_left (((x : ℝ) + y) / 3) ((z : ℝ) / 3)
linarith
refine ⟨?_, ?_, div_pos ht0 hq, (div_lt_one hq).2 htq⟩
· convert htLower using 1 <;> norm_num [div_eq_mul_inv]
· convert htUpper using 1 <;> norm_num [div_eq_mul_inv] <;> ring
theorem threeScoreCalibratedU_eventually_thresholds
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
∀ᶠ m : ℕ in atTop, let u := threeScoreCalibratedU α x y z m let t := threeScoreThreshold x y z 0 < 1 / 3
+ u * ((z : ℝ) - y) ∧ 0 < 1 / 3
- u * ((z : ℝ) - x) ∧ 0 < 1 / 3
+ u * ((y : ℝ) - x) ∧ 0
≤ u ∧
u * ((z : ℝ) - x) ≤ 1 / 3 ∧
(x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) < t ∧
t < min ((x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) + (y : ℝ) * (1 / 3 - u * ((z : ℝ) - x))) ((z : ℝ) * (1 / 3 + u * ((y : ℝ) - x)))
Proof (Lean source)
lemma threeScoreCalibratedU_eventually_thresholds {ε : ℝ}
(α : ℝ) (hα : 0 < α ∧ α < 1 / 2)
(x y z : ScoreSpace ε) (hx0 : 0 < (x : ℝ))
(hxy : x < y) (hyz : y < z) :
∀ᶠ m : ℕ in atTop,
let u := threeScoreCalibratedU α x y z m
let t := threeScoreThreshold x y z
0 < 1 / 3 + u * ((z : ℝ) - y) ∧
0 < 1 / 3 - u * ((z : ℝ) - x) ∧
0 < 1 / 3 + u * ((y : ℝ) - x) ∧
0 ≤ u ∧ u * ((z : ℝ) - x) ≤ 1 / 3 ∧
(x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) < t ∧
t < min
((x : ℝ) * (1 / 3 + u * ((z : ℝ) - y)) +
(y : ℝ) * (1 / 3 - u * ((z : ℝ) - x)))
((z : ℝ) * (1 / 3 + u * ((y : ℝ) - x))) := by
have hbase := threeScoreThreshold_baseline x y z hx0 hxy hyz
dsimp only at hbase
have hu := threeScoreCalibratedU_tendsto_zero α x y z
have hxlim : Tendsto (fun m : ℕ =>
(x : ℝ) * (1 / 3 + threeScoreCalibratedU α x y z m *
((z : ℝ) - y))) atTop (nhds ((x : ℝ) * (1 / 3))) := by
convert tendsto_const_nhds.mul
(tendsto_const_nhds.add (hu.mul_const ((z : ℝ) - y))) using 1 <;> simp
have hxylim : Tendsto (fun m : ℕ =>
(x : ℝ) * (1 / 3 + threeScoreCalibratedU α x y z m *
((z : ℝ) - y)) +
(y : ℝ) * (1 / 3 - threeScoreCalibratedU α x y z m *
((z : ℝ) - x))) atTop
(nhds ((x : ℝ) * (1 / 3) + (y : ℝ) * (1 / 3))) := by
convert
(tendsto_const_nhds.mul
(tendsto_const_nhds.add (hu.mul_const ((z : ℝ) - y)))).add
(tendsto_const_nhds.mul
(tendsto_const_nhds.sub (hu.mul_const ((z : ℝ) - x)))) using 1 <;> simp
have hzlim : Tendsto (fun m : ℕ =>
(z : ℝ) * (1 / 3 + threeScoreCalibratedU α x y z m *
((y : ℝ) - x))) atTop (nhds ((z : ℝ) * (1 / 3))) := by
convert tendsto_const_nhds.mul
(tendsto_const_nhds.add (hu.mul_const ((y : ℝ) - x))) using 1 <;> simp
filter_upwards
[threeScoreCalibratedU_eventually_weights_pos α x y z,
eventually_gt_atTop (0 : ℕ),
hxlim.eventually (Iio_mem_nhds hbase.1),
hxylim.eventually (Ioi_mem_nhds (lt_of_lt_of_le hbase.2.1
(min_le_left _ _))),
hzlim.eventually (Ioi_mem_nhds (lt_of_lt_of_le hbase.2.1
(min_le_right _ _)))]
with m hweights hm hfirst hfirstTwo hthird
dsimp only
have hu0 := (threeScoreCalibratedU_pos α hα x y z hxy hyz m hm).le
refine ⟨hweights.1, hweights.2.1, hweights.2.2, hu0, ?_, hfirst, ?_⟩
· linarith [hweights.2.1]
· exact lt_min hfirstTwo hthird
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerTrialPair 19 declarations Center-score Bernoulli trial submodels for the lower certificate.
Center-score Bernoulli trial submodels for the lower certificate.
def trialCenterScore
For the specified mathematical inputs, this definition introduces the corresponding object.
def trialZeroOutcome
This definition introduces the corresponding mathematical object.
Definition (Lean source)
def
given by
⟨0, by norm_num⟩
def trialOneOutcome
This definition introduces the corresponding mathematical object.
Definition (Lean source)
def
given by
⟨1, by norm_num⟩
def trialCenterRow
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
(trialCenterScore hOverlap, trialZeroOutcome, y, a, if a then y else trialZeroOutcome, g (trialCenterScore hOverlap))
theorem trialCenterRow_measurable
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma trialCenterRow_measurable {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε) :
Measurable (fun ya : OutcomeSpace × ArmSpace =>
trialCenterRow g hOverlap ya.1 ya.2) := by
have hcond : MeasurableSet {ya : OutcomeSpace × ArmSpace | ya.2 = true} := by
exact measurableSet_eq_fun measurable_snd measurable_const
have hobs : Measurable (fun ya : OutcomeSpace × ArmSpace =>
if ya.2 then ya.1 else trialZeroOutcome) :=
Measurable.ite hcond measurable_fst measurable_const
unfold trialCenterRow
fun_prop (disch := assumption)
def trialOutcomeLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
p :
ℝ
def
given by
ofReal p • Measure.dirac trialOneOutcome
+ ofReal (1 - p) • Measure.dirac trialZeroOutcome
theorem trialOutcomeLaw_isProbabilityMeasure
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma trialOutcomeLaw_isProbabilityMeasure (p : ℝ)
(hp0 : 0 ≤ p) (hp1 : p ≤ 1) :
IsProbabilityMeasure (trialOutcomeLaw p) := by
rw [isProbabilityMeasure_iff]
unfold trialOutcomeLaw
rw [Measure.add_apply, Measure.smul_apply, Measure.smul_apply]
simp only [Measure.dirac_apply, Set.indicator_of_mem, Set.mem_univ,
Pi.one_apply, smul_eq_mul, mul_one]
rw [← ENNReal.ofReal_add hp0 (sub_nonneg.mpr hp1)]
norm_num
def trialArmLaw
This definition introduces the corresponding mathematical object.
theorem trialArmLaw_isProbabilityMeasure
This result establishes the stated mathematical conclusion.
Formal statement
conclusion
Proof (Lean source)
lemma trialArmLaw_isProbabilityMeasure : IsProbabilityMeasure trialArmLaw := by
rw [isProbabilityMeasure_iff]
unfold trialArmLaw
rw [Measure.add_apply, Measure.smul_apply, Measure.smul_apply]
simpa only [Measure.dirac_apply, Set.indicator_of_mem, Set.mem_univ,
Pi.one_apply, smul_eq_mul, mul_one, one_div] using ENNReal.inv_two_add_inv_two
theorem trialArmLaw_true_mass
This result establishes the stated mathematical conclusion.
Formal statement
conclusion
trialArmLaw.real {true} = 1 / 2
Proof (Lean source)
lemma trialArmLaw_true_mass : trialArmLaw.real {true} = 1 / 2 := by
unfold trialArmLaw
rw [Measure.real_def, Measure.add_apply, Measure.smul_apply, Measure.smul_apply]
norm_num
def trialCenterFullLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
given by
theorem trialCenterFullLaw_isProbabilityMeasure
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
IsProbabilityMeasure (trialCenterFullLaw g hOverlap p)
Proof (Lean source)
lemma trialCenterFullLaw_isProbabilityMeasure {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) :
IsProbabilityMeasure (trialCenterFullLaw g hOverlap p) := by
letI : IsProbabilityMeasure (trialOutcomeLaw p) :=
trialOutcomeLaw_isProbabilityMeasure p hp0 hp1
letI : IsProbabilityMeasure trialArmLaw := trialArmLaw_isProbabilityMeasure
unfold trialCenterFullLaw
exact Measure.isProbabilityMeasure_map (trialCenterRow_measurable g hOverlap).aemeasurable
theorem trialCenterFullLaw_scoreMarginal
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma trialCenterFullLaw_scoreMarginal {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) :
ScoreMarginal (trialCenterFullLaw g hOverlap p)
(Measure.dirac (trialCenterScore hOverlap)) := by
letI : IsProbabilityMeasure (trialOutcomeLaw p) :=
trialOutcomeLaw_isProbabilityMeasure p hp0 hp1
letI : IsProbabilityMeasure trialArmLaw := trialArmLaw_isProbabilityMeasure
have hrow := trialCenterRow_measurable g hOverlap
unfold ScoreMarginal trialCenterFullLaw
rw [Measure.map_map (by unfold score; fun_prop) hrow]
change ((trialOutcomeLaw p).prod trialArmLaw).map
(fun _ => trialCenterScore hOverlap) = _
rw [Measure.map_const]
simp
theorem trialCenterFullLaw_consistency
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Consistency (trialCenterFullLaw g hOverlap p)
Proof (Lean source)
lemma trialCenterFullLaw_consistency {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε) (p : ℝ) :
Consistency (trialCenterFullLaw g hOverlap p) := by
have hrow := trialCenterRow_measurable g hOverlap
have hcond : MeasurableSet {ω : FullRow ε J | arm ω = true} :=
measurableSet_eq_fun (by unfold arm; fun_prop) measurable_const
have hpot : Measurable (fun ω : FullRow ε J =>
if arm ω then outcome1 ω else outcome0 ω) :=
Measurable.ite hcond (by unfold outcome1; fun_prop)
(by unfold outcome0; fun_prop)
have hset : MeasurableSet {ω : FullRow ε J |
observed ω = if arm ω then outcome1 ω else outcome0 ω} := by
exact measurableSet_eq_fun (by unfold observed; fun_prop)
hpot
unfold Consistency trialCenterFullLaw
rw [ae_map_iff hrow.aemeasurable hset]
filter_upwards [] with ya
rfl
theorem trialCenterFullLaw_deterministicRelease
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
DeterministicRelease g (trialCenterFullLaw g hOverlap p)
Proof (Lean source)
lemma trialCenterFullLaw_deterministicRelease {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε) (p : ℝ)
(hg : Measurable g) :
DeterministicRelease g (trialCenterFullLaw g hOverlap p) := by
have hrow := trialCenterRow_measurable g hOverlap
have hset : MeasurableSet {ω : FullRow ε J | label ω = g (score ω)} := by
exact measurableSet_eq_fun (by unfold label; fun_prop)
(hg.comp (by unfold score; fun_prop))
unfold DeterministicRelease trialCenterFullLaw
rw [ae_map_iff hrow.aemeasurable hset]
filter_upwards [] with ya
rfl
theorem trialOutcomeLaw_mean
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
p :
ℝ
hp0 :
0 ≤ p
hp1 :
p ≤ 1
conclusion
(∫ y : OutcomeSpace, (y : ℝ) ∂(trialOutcomeLaw p)) = p
Proof (Lean source)
lemma trialOutcomeLaw_mean (p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) :
(∫ y : OutcomeSpace, (y : ℝ) ∂(trialOutcomeLaw p)) = p := by
unfold trialOutcomeLaw
rw [integral_add_measure]
· rw [integral_smul_measure, integral_smul_measure]
simp [trialOneOutcome, trialZeroOutcome, hp0, sub_nonneg.mpr hp1, smul_eq_mul]
· exact Integrable.smul_measure (μ := Measure.dirac trialOneOutcome)
(c := ofReal p)
(integrable_dirac (f := fun y : OutcomeSpace => (y : ℝ))
(a := trialOneOutcome) (by simp [enorm])) (by simp)
· exact Integrable.smul_measure (μ := Measure.dirac trialZeroOutcome)
(c := ofReal (1 - p))
(integrable_dirac (f := fun y : OutcomeSpace => (y : ℝ))
(a := trialZeroOutcome) (by simp [enorm])) (by simp)
theorem trialCenterFullLaw_ate
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
ate (trialCenterFullLaw g hOverlap p) = p
Proof (Lean source)
lemma trialCenterFullLaw_ate {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) :
ate (trialCenterFullLaw g hOverlap p) = p := by
letI : IsProbabilityMeasure (trialOutcomeLaw p) :=
trialOutcomeLaw_isProbabilityMeasure p hp0 hp1
letI : IsProbabilityMeasure trialArmLaw := trialArmLaw_isProbabilityMeasure
have hrow := trialCenterRow_measurable g hOverlap
calc
ate (trialCenterFullLaw g hOverlap p) =
∫ ya : OutcomeSpace × ArmSpace, (ya.1 : ℝ)
∂((trialOutcomeLaw p).prod trialArmLaw) := by
unfold ate trialCenterFullLaw
rw [integral_map hrow.aemeasurable (by
unfold outcome0 outcome1
fun_prop)]
simp [trialCenterRow, outcome0, outcome1, trialZeroOutcome]
_ = ∫ y : OutcomeSpace, (y : ℝ)
∂(((trialOutcomeLaw p).prod trialArmLaw).map fst) := by
rw [integral_map measurable_fst.aemeasurable (by fun_prop)]
_ = ∫ y : OutcomeSpace, (y : ℝ) ∂(trialOutcomeLaw p) := by
rw [Measure.map_fst_prod]
rw [show trialArmLaw univ = 1 from isProbabilityMeasure_iff.mp inferInstance]
simp
_ = p := trialOutcomeLaw_mean p hp0 hp1
theorem trialCenterFullLaw_randomizedAssignment
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
RandomizedAssignment (trialCenterFullLaw g hOverlap p)
Proof (Lean source)
lemma trialCenterFullLaw_randomizedAssignment {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) :
RandomizedAssignment (trialCenterFullLaw g hOverlap p) := by
intro B hB
let μ : Measure OutcomeSpace := trialOutcomeLaw p
let ν : Measure ArmSpace := trialArmLaw
let f : OutcomeSpace × ArmSpace → FullRow ε J :=
fun ya => trialCenterRow g hOverlap ya.1 ya.2
have hf : Measurable f := trialCenterRow_measurable g hOverlap
let U : Set OutcomeSpace :=
{y | (trialCenterScore hOverlap, trialZeroOutcome, y) ∈ B}
have hU : MeasurableSet U := by
have htriple : Measurable (fun y : OutcomeSpace =>
(trialCenterScore hOverlap, trialZeroOutcome, y)) := by fun_prop
exact hB.preimage htriple
let S : Set (FullRow ε J) :=
{ω | arm ω = true ∧ latentTriple ω ∈ B}
let T : Set (FullRow ε J) := {ω | latentTriple ω ∈ B}
have hlatent : Measurable (latentTriple (ε := ε) (J := J)) := by
unfold latentTriple score outcome0 outcome1
fun_prop
have hS : MeasurableSet S :=
(measurableSet_eq_fun (by unfold arm; fun_prop) measurable_const).inter
(hB.preimage hlatent)
have hT : MeasurableSet T := hB.preimage hlatent
have hpreS : f ⁻¹' S = U ×ˢ {true} := by
ext ya
cases ha : ya.2 <;> simp [f, S, U, trialCenterRow, arm,
latentTriple, score, outcome0, outcome1, ha]
have hpreT : f ⁻¹' T = U ×ˢ univ := by
ext ya
simp [f, T, U, trialCenterRow, latentTriple, score, outcome0, outcome1]
have hνprob : IsProbabilityMeasure ν := trialArmLaw_isProbabilityMeasure
have hνuniv : ν.real univ = 1 := isProbabilityMeasure_iff_real.mp hνprob
change (trialCenterFullLaw g hOverlap p).real S =
∫ ω in T, (score ω : ℝ) ∂(trialCenterFullLaw g hOverlap p)
have hleft : (trialCenterFullLaw g hOverlap p).real S = μ.real U / 2 := by
unfold trialCenterFullLaw
rw [MeasureTheory.map_measureReal_apply hf hS, hpreS,
measureReal_prod_prod]
rw [show ν.real {true} = 1 / 2 from trialArmLaw_true_mass]
ring
have hright : (∫ ω in T, (score ω : ℝ)
∂(trialCenterFullLaw g hOverlap p)) = μ.real U / 2 := by
unfold trialCenterFullLaw
rw [MeasureTheory.setIntegral_map hT (by unfold score; fun_prop)
hf.aemeasurable]
rw [hpreT]
change ∫ _ in U ×ˢ univ, (1 / 2 : ℝ) ∂(μ.prod ν) = _
rw [setIntegral_const, measureReal_prod_prod, hνuniv]
simp only [smul_eq_mul, mul_one]
ring
exact hleft.trans hright.symm
theorem trialCenterFullLaw_externalScoreLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
ExternalScoreLaw g (trialCenterFullLaw g hOverlap p) (Measure.dirac (trialCenterScore hOverlap)) where probabilityP
Proof (Lean source)
lemma trialCenterFullLaw_externalScoreLaw {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(hg : Measurable g) (p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) :
ExternalScoreLaw g (trialCenterFullLaw g hOverlap p)
(Measure.dirac (trialCenterScore hOverlap)) where
probabilityP := trialCenterFullLaw_isProbabilityMeasure g hOverlap p hp0 hp1
probabilityH := inferInstance
overlap := hOverlap
measurableRelease := hg
scoreMarginal := trialCenterFullLaw_scoreMarginal g hOverlap p hp0 hp1
randomizedAssignment := trialCenterFullLaw_randomizedAssignment g hOverlap p hp0 hp1
consistency := trialCenterFullLaw_consistency g hOverlap p
deterministicRelease := trialCenterFullLaw_deterministicRelease g hOverlap p hg
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerTrialSharp 8 declarations Point identification of the Bernoulli trial submodels at a Dirac score.
Point identification of the Bernoulli trial submodels at a Dirac score.
theorem trialCenter_armProb
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
armProb a (trialCenterScore hOverlap) = 1 / 2
Proof (Lean source)
lemma trialCenter_armProb {ε : ℝ} (hOverlap : Overlap ε) (a : ArmSpace) :
armProb a (trialCenterScore hOverlap) = 1 / 2 := by
cases a <;> simp [armProb, trialCenterScore] <;> ring
theorem trialCenter_armCellMass
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
armCellMass (Measure.dirac (trialCenterScore hOverlap)) g a r
= if g (trialCenterScore hOverlap) = r then 1 / 2 else 0
Proof (Lean source)
lemma trialCenter_armCellMass {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(a : ArmSpace) (r : LabelSpace J) :
armCellMass (Measure.dirac (trialCenterScore hOverlap)) g a r =
if g (trialCenterScore hOverlap) = r then 1 / 2 else 0 := by
classical
unfold armCellMass
rw [setIntegral_dirac]
simp [cell, trialCenter_armProb]
theorem trialCenter_armCellScoreLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
a :
r :
hq :
0 < armCellMass (Measure.dirac (trialCenterScore hOverlap)) g a r
conclusion
= Measure.dirac (trialCenterScore hOverlap)
Proof (Lean source)
lemma trialCenter_armCellScoreLaw {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(hg : Measurable g) (a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass (Measure.dirac (trialCenterScore hOverlap)) g a r) :
armCellScoreLaw (Measure.dirac (trialCenterScore hOverlap)) inferInstance
g hg a r hq = Measure.dirac (trialCenterScore hOverlap) := by
classical
have hr : g (trialCenterScore hOverlap) = r := by
by_contra h
rw [trialCenter_armCellMass, if_neg h] at hq
exact (lt_irrefl 0) hq
unfold armCellScoreLaw
rw [trialCenter_armCellMass, if_pos hr,
dirac_withDensity, Measure.restrict_smul, restrict_dirac]
simp [cell, hr, trialCenter_armProb, smul_smul]
rw [ENNReal.mul_inv_cancel (by norm_num : (2 : ℝ≥0∞) ≠ 0)
(by norm_num : (2 : ℝ≥0∞) ≠ ⊤), one_smul]
theorem trialCenter_armCellWeightLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
a :
r :
hq :
0 < armCellMass (Measure.dirac (trialCenterScore hOverlap)) g a r
conclusion
= Measure.dirac (2 : ℝ)
Proof (Lean source)
lemma trialCenter_armCellWeightLaw {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(hg : Measurable g) (a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass (Measure.dirac (trialCenterScore hOverlap)) g a r) :
armCellWeightLaw (Measure.dirac (trialCenterScore hOverlap)) inferInstance
g hg a r hq = Measure.dirac (2 : ℝ) := by
unfold armCellWeightLaw
rw [trialCenter_armCellScoreLaw g hOverlap hg a r hq]
rw [Measure.map_dirac' (measurable_inverseArmProb a)]
simp [trialCenter_armProb]
theorem quantile_dirac_interior
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma quantile_dirac_interior (c u : ℝ) (hu0 : 0 < u) (hu1 : u < 1) :
quantile (Measure.dirac c) u = c := by
apply le_antisymm
· apply quantile_le_of_le_cdf hu0
rw [ProbabilityTheory.cdf_eq_real]
simpa [Measure.real] using hu1.le
· have hq := le_cdf_quantile (μ := Measure.dirac c) hu1
rw [ProbabilityTheory.cdf_eq_real] at hq
by_contra h
have hlt : quantile (Measure.dirac c) u < c := lt_of_not_ge h
have hc : c ∉ Iic (quantile (Measure.dirac c) u) := not_le.mpr hlt
simp [Measure.real, hc] at hq
exact (not_le_of_gt hu0) hq
theorem quantileProduct_endpoints_dirac
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma quantileProduct_endpoints_dirac (Y : Measure ℝ) (c : ℝ) :
(∫ u in (0 : ℝ)..1,
quantile Y u * quantile (Measure.dirac c) (1 - u)) =
(∫ u in (0 : ℝ)..1,
quantile Y u * quantile (Measure.dirac c) u) := by
apply intervalIntegral.integral_congr_Ioo_of_le (by norm_num)
intro u hu
dsimp
rw [quantile_dirac_interior c (1 - u) (by linarith [hu.2]) (by linarith [hu.1]),
quantile_dirac_interior c u hu.1 hu.2]
theorem trialCenter_muLower_eq_muUpper
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses (Measure.dirac (trialCenterScore hOverlap)) g Prel
a :
conclusion
muLower (Measure.dirac (trialCenterScore hOverlap)) g Prel hMass a
= muUpper (Measure.dirac (trialCenterScore hOverlap)) g Prel hMass a
Proof (Lean source)
lemma trialCenter_muLower_eq_muUpper {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(Prel : Measure (Observation J))
(hMass : CompatibleCellMasses (Measure.dirac (trialCenterScore hOverlap)) g Prel)
(a : ArmSpace) :
muLower (Measure.dirac (trialCenterScore hOverlap)) g Prel hMass a =
muUpper (Measure.dirac (trialCenterScore hOverlap)) g Prel hMass a := by
unfold muLower muUpper
apply Finset.sum_congr rfl
intro r _
split_ifs with hq
· rw [trialCenter_armCellWeightLaw g hOverlap hMass.2.2.1 a r hq]
rw [quantileProduct_endpoints_dirac]
· rfl
theorem trialCenter_sharpATELength_eq_zero
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses (Measure.dirac (trialCenterScore hOverlap)) g Prel
conclusion
sharpATELength (Measure.dirac (trialCenterScore hOverlap)) g Prel hMass = 0
Proof (Lean source)
lemma trialCenter_sharpATELength_eq_zero {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(Prel : Measure (Observation J))
(hMass : CompatibleCellMasses (Measure.dirac (trialCenterScore hOverlap)) g Prel) :
sharpATELength (Measure.dirac (trialCenterScore hOverlap)) g Prel hMass = 0 := by
unfold sharpATELength
rw [trialCenter_muLower_eq_muUpper g hOverlap Prel hMass true,
trialCenter_muLower_eq_muUpper g hOverlap Prel hMass false]
ring
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ExternalLowerTrialTesting 18 declarations Testing estimates for the Bernoulli trial pair.
Testing estimates for the Bernoulli trial pair.
theorem trialReadout_tv_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
S T :
μ ν :
f :
S → T
hf :
Proof (Lean source)
lemma trialReadout_tv_le {S T : Type*} [MeasurableSpace S] [MeasurableSpace T]
(μ ν : Measure S) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
(f : S → T) (hf : Measurable f) :
tvDist (μ.map f) (ν.map f) ≤ tvDist μ ν := by
unfold tvDist
apply ciSup_le
intro A
simp only [Measure.real_def, Measure.map_apply hf A.property]
exact abs_measureReal_sub_le_tvDist (hf A.property)
theorem trialCenter_releasedLaw_map
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
releasedLaw (trialCenterFullLaw g hOverlap p)
= ((trialOutcomeLaw p).prod trialArmLaw).map (fun ya : OutcomeSpace × ArmSpace => (g (trialCenterScore hOverlap), ya.2, if ya.2 then ya.1 else trialZeroOutcome))
Proof (Lean source)
lemma trialCenter_releasedLaw_map {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε) (p : ℝ) :
releasedLaw (trialCenterFullLaw g hOverlap p) =
((trialOutcomeLaw p).prod trialArmLaw).map
(fun ya : OutcomeSpace × ArmSpace =>
(g (trialCenterScore hOverlap), ya.2,
if ya.2 then ya.1 else trialZeroOutcome)) := by
unfold releasedLaw trialCenterFullLaw
rw [Measure.map_map (by unfold releasedRecord label arm observed; fun_prop)
(trialCenterRow_measurable g hOverlap)]
rfl
theorem trialCenter_productReleased_tv_le_latent
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
p q :
ℝ
hp0 :
0 ≤ p
hp1 :
p ≤ 1
hq0 :
0 ≤ q
hq1 :
q ≤ 1
n :
ℕ
conclusion
tvDist (Measure.pi (fun _ : Fin n => releasedLaw (trialCenterFullLaw g hOverlap p))) (Measure.pi (fun _ : Fin n => releasedLaw (trialCenterFullLaw g hOverlap q)))
≤ tvDist (Measure.pi (fun _ : Fin n => (trialOutcomeLaw p).prod trialArmLaw)) (Measure.pi (fun _ : Fin n => (trialOutcomeLaw q).prod trialArmLaw))
Proof (Lean source)
lemma trialCenter_productReleased_tv_le_latent {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(p q : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1)
(hq0 : 0 ≤ q) (hq1 : q ≤ 1) (n : ℕ) :
tvDist
(Measure.pi (fun _ : Fin n => releasedLaw (trialCenterFullLaw g hOverlap p)))
(Measure.pi (fun _ : Fin n => releasedLaw (trialCenterFullLaw g hOverlap q))) ≤
tvDist
(Measure.pi (fun _ : Fin n => (trialOutcomeLaw p).prod trialArmLaw))
(Measure.pi (fun _ : Fin n => (trialOutcomeLaw q).prod trialArmLaw)) := by
let f : OutcomeSpace × ArmSpace → Observation J := fun ya =>
(g (trialCenterScore hOverlap), ya.2,
if ya.2 then ya.1 else trialZeroOutcome)
have hf : Measurable f := by
dsimp [f]
have hcond : MeasurableSet {ya : OutcomeSpace × ArmSpace | ya.2 = true} :=
measurableSet_eq_fun measurable_snd measurable_const
have hobs : Measurable (fun ya : OutcomeSpace × ArmSpace =>
if ya.2 then ya.1 else trialZeroOutcome) :=
Measurable.ite hcond measurable_fst measurable_const
fun_prop (disch := assumption)
have hp : IsProbabilityMeasure ((trialOutcomeLaw p).prod trialArmLaw) := by
letI : IsProbabilityMeasure (trialOutcomeLaw p) :=
trialOutcomeLaw_isProbabilityMeasure p hp0 hp1
letI : IsProbabilityMeasure trialArmLaw := trialArmLaw_isProbabilityMeasure
infer_instance
have hq : IsProbabilityMeasure ((trialOutcomeLaw q).prod trialArmLaw) := by
letI : IsProbabilityMeasure (trialOutcomeLaw q) :=
trialOutcomeLaw_isProbabilityMeasure q hq0 hq1
letI : IsProbabilityMeasure trialArmLaw := trialArmLaw_isProbabilityMeasure
infer_instance
letI := hp
letI := hq
have hmap (r : ℝ) (hr0 : 0 ≤ r) (hr1 : r ≤ 1) :
Measure.pi (fun _ : Fin n => releasedLaw (trialCenterFullLaw g hOverlap r)) =
(Measure.pi (fun _ : Fin n => (trialOutcomeLaw r).prod trialArmLaw)).map
(fun z i => f (z i)) := by
letI : IsProbabilityMeasure (trialOutcomeLaw r) :=
trialOutcomeLaw_isProbabilityMeasure r hr0 hr1
letI : IsProbabilityMeasure trialArmLaw := trialArmLaw_isProbabilityMeasure
rw [Measure.pi_map_pi (fun _ => hf.aemeasurable)]
simp only [trialCenter_releasedLaw_map]
rfl
rw [hmap p hp0 hp1, hmap q hq0 hq1]
exact trialReadout_tv_le _ _ _ (measurable_pi_lambda _ fun _ => hf.comp (measurable_pi_apply _))
theorem trialOutcomeLaw_withDensity_center
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
p q :
ℝ
hq0 :
0 < q
hq1 :
q < 1
conclusion
= (trialOutcomeLaw q).withDensity (fun y : OutcomeSpace => if y = trialOneOutcome then ofReal (p / q) else ofReal ((1 - p) / (1 - q)))
Proof (Lean source)
lemma trialOutcomeLaw_withDensity_center (p q : ℝ) (hq0 : 0 < q) (hq1 : q < 1) :
trialOutcomeLaw p = (trialOutcomeLaw q).withDensity (fun y : OutcomeSpace =>
if y = trialOneOutcome then ofReal (p / q)
else ofReal ((1 - p) / (1 - q))) := by
let f : OutcomeSpace → ENNReal := fun y =>
if y = trialOneOutcome then ofReal (p / q)
else ofReal ((1 - p) / (1 - q))
have hqne : ofReal q ≠ 0 := by
intro h
have hle := ENNReal.ofReal_eq_zero.mp h
linarith
have h1qne : ofReal (1 - q) ≠ 0 := by
intro h
have hle := ENNReal.ofReal_eq_zero.mp h
linarith
change trialOutcomeLaw p = (trialOutcomeLaw q).withDensity f
ext s hs
rw [withDensity_apply _ hs, ← lintegral_indicator hs f]
unfold trialOutcomeLaw
dsimp [f]
rw [lintegral_add_measure, lintegral_smul_measure, lintegral_smul_measure]
simp only [lintegral_dirac]
have h10 : trialZeroOutcome ≠ trialOneOutcome := by
norm_num [trialZeroOutcome, trialOneOutcome]
by_cases h1 : trialOneOutcome ∈ s
· by_cases h0 : trialZeroOutcome ∈ s
· simp [h1, h0, ENNReal.ofReal_div_of_pos hq0,
ENNReal.ofReal_div_of_pos (sub_pos.mpr hq1),
ENNReal.mul_div_cancel hqne ENNReal.ofReal_ne_top,
ENNReal.mul_div_cancel h1qne ENNReal.ofReal_ne_top,
h10]
· simp [h1, h0, ENNReal.ofReal_div_of_pos hq0,
ENNReal.mul_div_cancel hqne ENNReal.ofReal_ne_top,
h10]
· by_cases h0 : trialZeroOutcome ∈ s
· simp [h1, h0, ENNReal.ofReal_div_of_pos (sub_pos.mpr hq1),
ENNReal.mul_div_cancel h1qne ENNReal.ofReal_ne_top,
h10]
· simp [h1, h0, f, h10]
theorem trialOutcomeLaw_chisq_center
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
u :
ℝ
hu :
|u| < 1 / 2
conclusion
= 4 * u ^ 2
Proof (Lean source)
lemma trialOutcomeLaw_chisq_center (u : ℝ) (hu : |u| < 1 / 2) :
chiSqDiv (trialOutcomeLaw (1 / 2 + u))
(trialOutcomeLaw (1 / 2)) = 4 * u ^ 2 := by
let p : ℝ := 1 / 2 + u
let q : ℝ := 1 / 2
have hub := abs_lt.mp hu
have hp0 : 0 ≤ p := by dsimp [p]; linarith
have hp1 : p ≤ 1 := by dsimp [p]; linarith
have hq0 : 0 < q := by norm_num [q]
have hq1 : q < 1 := by norm_num [q]
letI : IsProbabilityMeasure (trialOutcomeLaw q) :=
trialOutcomeLaw_isProbabilityMeasure q hq0.le hq1.le
let f : OutcomeSpace → ENNReal := fun y =>
if y = trialOneOutcome then ofReal (p / q)
else ofReal ((1 - p) / (1 - q))
have hf : Measurable f := by
dsimp [f]
exact Measurable.ite (measurableSet_singleton trialOneOutcome)
measurable_const measurable_const
have hwd : trialOutcomeLaw p = (trialOutcomeLaw q).withDensity f :=
trialOutcomeLaw_withDensity_center p q hq0 hq1
have hrn : (trialOutcomeLaw p).rnDeriv (trialOutcomeLaw q) =ᵐ[trialOutcomeLaw q] f := by
rw [hwd]
exact Measure.rnDeriv_withDensity _ hf
change chiSqDiv (trialOutcomeLaw p) (trialOutcomeLaw q) = 4 * u ^ 2
rw [chiSqDiv]
trans ∫ y, ((f y).toReal - 1) ^ 2 ∂(trialOutcomeLaw q)
· exact integral_congr_ae <| hrn.mono fun y hy => by
dsimp
rw [hy]
unfold trialOutcomeLaw
rw [integral_add_measure]
· rw [integral_smul_measure, integral_smul_measure]
simp [f, trialOneOutcome, trialZeroOutcome,
ENNReal.toReal_ofReal (div_nonneg hp0 hq0.le),
ENNReal.toReal_ofReal
(div_nonneg (sub_nonneg.mpr hp1) (sub_nonneg.mpr hq1.le))]
dsimp [p, q]
norm_num [q]
ring
· exact Integrable.smul_measure (integrable_dirac (by simp [enorm])) (by simp)
· exact Integrable.smul_measure (integrable_dirac (by simp [enorm])) (by simp)
theorem trialOutcomeLaw_ac_integrable_center
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
u :
ℝ
conclusion
conclusion 1
trialOutcomeLaw (1 / 2 + u) ≪ trialOutcomeLaw (1 / 2)
conclusion 2
Integrable (fun y : OutcomeSpace => (((trialOutcomeLaw (1 / 2 + u)).rnDeriv (trialOutcomeLaw (1 / 2)) y).toReal - 1) ^ 2) (trialOutcomeLaw (1 / 2))
Proof (Lean source)
lemma trialOutcomeLaw_ac_integrable_center (u : ℝ) :
trialOutcomeLaw (1 / 2 + u) ≪ trialOutcomeLaw (1 / 2) ∧
Integrable (fun y : OutcomeSpace =>
(((trialOutcomeLaw (1 / 2 + u)).rnDeriv (trialOutcomeLaw (1 / 2)) y).toReal - 1) ^ 2)
(trialOutcomeLaw (1 / 2)) := by
constructor
· rw [trialOutcomeLaw_withDensity_center (1 / 2 + u) (1 / 2)
(by norm_num) (by norm_num)]
exact withDensity_absolutelyContinuous _ _
· unfold trialOutcomeLaw
apply Integrable.add_measure
· exact Integrable.smul_measure (integrable_dirac (by simp [enorm])) (by simp)
· exact Integrable.smul_measure (integrable_dirac (by simp [enorm])) (by simp)
theorem trial_ancillary_integrable
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
S T :
μ ν :
Measure S
ρ :
Measure T
hac :
μ ≪ ν
hint :
Integrable (fun x => ((μ.rnDeriv ν x).toReal - 1) ^ 2) ν
conclusion
Integrable (fun z => (((μ.prod ρ).rnDeriv (ν.prod ρ) z).toReal - 1) ^ 2) (ν.prod ρ)
Proof (Lean source)
lemma trial_ancillary_integrable {S T : Type*} [MeasurableSpace S] [MeasurableSpace T]
(μ ν : Measure S) (ρ : Measure T)
[IsProbabilityMeasure μ] [IsProbabilityMeasure ν] [IsProbabilityMeasure ρ]
(hac : μ ≪ ν)
(hint : Integrable (fun x => ((μ.rnDeriv ν x).toReal - 1) ^ 2) ν) :
Integrable (fun z => (((μ.prod ρ).rnDeriv (ν.prod ρ) z).toReal - 1) ^ 2)
(ν.prod ρ) := by
have hrn := rnDeriv_prod_eq
μ ν ρ ρ hac (Measure.AbsolutelyContinuous.refl ρ)
have hself : ∀ᵐ z ∂ν.prod ρ, ρ.rnDeriv ρ z.2 = 1 := by
have hs : ∀ᵐ y ∂ρ, ρ.rnDeriv ρ y = 1 := by
filter_upwards [ρ.rnDeriv_self] with y hy
exact hy
have hmap : ∀ᵐ y ∂Measure.map snd (ν.prod ρ), ρ.rnDeriv ρ y = 1 := by
simpa [MeasurePreserving.map_eq (measurePreserving_snd (μ := ν) (ν := ρ))]
using hs
exact (ae_map_iff (measurePreserving_snd (μ := ν) (ν := ρ)).aemeasurable
(measurableSet_eq_fun (Measure.measurable_rnDeriv ρ ρ) measurable_const)).mp hmap
have heq : (fun z => (((μ.prod ρ).rnDeriv (ν.prod ρ) z).toReal - 1) ^ 2)
=ᵐ[ν.prod ρ] (fun z => ((μ.rnDeriv ν z.1).toReal - 1) ^ 2) := by
filter_upwards [hrn, hself] with z hz hs
simp [hz, hs]
exact (hint.comp_fst ρ).congr heq.symm
theorem trialProduct_tv_le_of_chisq
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
Proof (Lean source)
lemma trialProduct_tv_le_of_chisq {S : Type*} [MeasurableSpace S]
(μ ν : Measure S) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
(hac : μ ≪ ν)
(hint : Integrable (fun x => ((μ.rnDeriv ν x).toReal - 1) ^ 2) ν)
(n : ℕ) :
tvDist (Measure.pi (fun _ : Fin n => μ))
(Measure.pi (fun _ : Fin n => ν)) ≤
(1 / 2 : ℝ) * sqrt ((1 + chiSqDiv μ ν) ^ n - 1) := by
have hacn : Measure.pi (fun _ : Fin n => μ) ≪
Measure.pi (fun _ : Fin n => ν) :=
pi_iid_absolutelyContinuous μ ν hac n
have hintn := pi_iid_integrable_sq_dev μ ν hac hint n
have htv := tvDist_le_half_sqrt_chiSqDiv
(Measure.pi (fun _ : Fin n => μ))
(Measure.pi (fun _ : Fin n => ν)) hacn hintn
rw [show chiSqDiv
(Measure.pi (fun _ : Fin n => μ))
(Measure.pi (fun _ : Fin n => ν)) =
(1 + chiSqDiv μ ν) ^ n - 1 by
linarith [one_add_chiSqDiv_pi_iid_general μ ν hac hint n]] at htv
exact htv
theorem trialCenter_latent_chisq_center
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
u :
ℝ
hu :
|u| < 1 / 2
conclusion
chiSqDiv ((trialOutcomeLaw (1 / 2 + u)).prod trialArmLaw) ((trialOutcomeLaw (1 / 2)).prod trialArmLaw)
= 4 * u ^ 2
Proof (Lean source)
lemma trialCenter_latent_chisq_center (u : ℝ) (hu : |u| < 1 / 2) :
chiSqDiv
((trialOutcomeLaw (1 / 2 + u)).prod trialArmLaw)
((trialOutcomeLaw (1 / 2)).prod trialArmLaw) = 4 * u ^ 2 := by
have hub := abs_lt.mp hu
letI : IsProbabilityMeasure (trialOutcomeLaw (1 / 2 + u)) :=
trialOutcomeLaw_isProbabilityMeasure _ (by linarith) (by linarith)
letI : IsProbabilityMeasure (trialOutcomeLaw (1 / 2)) :=
trialOutcomeLaw_isProbabilityMeasure _ (by norm_num) (by norm_num)
letI : IsProbabilityMeasure trialArmLaw := trialArmLaw_isProbabilityMeasure
have h := trialOutcomeLaw_ac_integrable_center u
rw [chiSqDiv_prod_ancillary _ _ trialArmLaw h.1 h.2]
exact trialOutcomeLaw_chisq_center u hu
theorem trialCenter_productReleased_tv_bound
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
tvDist (Measure.pi (fun _ : Fin n => releasedLaw (trialCenterFullLaw g hOverlap (1 / 2 + u)))) (Measure.pi (fun _ : Fin n => releasedLaw (trialCenterFullLaw g hOverlap (1 / 2))))
≤ (1 / 2 : ℝ) * sqrt ((1 + 4 * u ^ 2) ^ n - 1)
Proof (Lean source)
lemma trialCenter_productReleased_tv_bound {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(u : ℝ) (hu : |u| < 1 / 2) (n : ℕ) :
tvDist
(Measure.pi (fun _ : Fin n =>
releasedLaw (trialCenterFullLaw g hOverlap (1 / 2 + u))))
(Measure.pi (fun _ : Fin n =>
releasedLaw (trialCenterFullLaw g hOverlap (1 / 2)))) ≤
(1 / 2 : ℝ) * sqrt ((1 + 4 * u ^ 2) ^ n - 1) := by
have hub := abs_lt.mp hu
let μ := (trialOutcomeLaw (1 / 2 + u)).prod trialArmLaw
let ν := (trialOutcomeLaw (1 / 2)).prod trialArmLaw
letI : IsProbabilityMeasure (trialOutcomeLaw (1 / 2 + u)) :=
trialOutcomeLaw_isProbabilityMeasure _ (by linarith) (by linarith)
letI : IsProbabilityMeasure (trialOutcomeLaw (1 / 2)) :=
trialOutcomeLaw_isProbabilityMeasure _ (by norm_num) (by norm_num)
letI : IsProbabilityMeasure trialArmLaw := trialArmLaw_isProbabilityMeasure
have h := trialOutcomeLaw_ac_integrable_center u
have hac : μ ≪ ν := h.1.prod (Measure.AbsolutelyContinuous.refl trialArmLaw)
have hint : Integrable (fun z => ((μ.rnDeriv ν z).toReal - 1) ^ 2) ν :=
trial_ancillary_integrable _ _ trialArmLaw h.1 h.2
calc
_ ≤ tvDist
(Measure.pi (fun _ : Fin n => μ))
(Measure.pi (fun _ : Fin n => ν)) :=
trialCenter_productReleased_tv_le_latent g hOverlap
(1 / 2 + u) (1 / 2) (by linarith) (by linarith)
(by norm_num) (by norm_num) n
_ ≤ (1 / 2 : ℝ) * sqrt ((1 + chiSqDiv μ ν) ^ n - 1) :=
trialProduct_tv_le_of_chisq μ ν hac hint n
_ = _ := by rw [trialCenter_latent_chisq_center u hu]
theorem trial_commonLog_tv_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Proof (Lean source)
lemma trial_commonLog_tv_le {S T : Type*} [MeasurableSpace S] [MeasurableSpace T]
(μ ν : Measure S) (ρ : Measure T)
[IsProbabilityMeasure μ] [IsProbabilityMeasure ν] [IsProbabilityMeasure ρ] :
tvDist (μ.prod ρ) (ν.prod ρ) ≤
tvDist μ ν := by
have h := Causalean.Stat.Minimax.MomentMatchedMixture.tvDist_prod_le_add μ ν ρ ρ
have hself : tvDist ρ ρ = 0 := by
unfold tvDist
simp
simpa [hself] using h
theorem trialCenter_jointExperiment_tv_bound
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
u :
ℝ
hu :
|u| < 1 / 2
Qj :
hTrial :
JointTrialIID g Qj
hLog :
ExternalLogIID g Qj
hInd :
conclusion
tvDist (Qj (trialCenterFullLaw g hOverlap (1 / 2 + u)) (Measure.dirac (trialCenterScore hOverlap))) (Qj (trialCenterFullLaw g hOverlap (1 / 2)) (Measure.dirac (trialCenterScore hOverlap)))
≤ (1 / 2 : ℝ) * sqrt ((1 + 4 * u ^ 2) ^ n - 1)
Proof (Lean source)
lemma trialCenter_jointExperiment_tv_bound {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(hg : Measurable g) (u : ℝ) (hu : |u| < 1 / 2)
(Qj : Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : JointTrialIID g Qj) (hLog : ExternalLogIID g Qj)
(hInd : ExternalLogIndependent g Qj) :
tvDist
(Qj (trialCenterFullLaw g hOverlap (1 / 2 + u))
(Measure.dirac (trialCenterScore hOverlap)))
(Qj (trialCenterFullLaw g hOverlap (1 / 2))
(Measure.dirac (trialCenterScore hOverlap))) ≤
(1 / 2 : ℝ) * sqrt ((1 + 4 * u ^ 2) ^ n - 1) := by
have hub := abs_lt.mp hu
let H : Measure (ScoreSpace ε) := Measure.dirac (trialCenterScore hOverlap)
let P1 := trialCenterFullLaw g hOverlap (1 / 2 + u)
let P0 := trialCenterFullLaw g hOverlap (1 / 2)
have hPH1 : (P1, H) ∈ ExternalLaws g :=
⟨trialCenterFullLaw_isProbabilityMeasure g hOverlap _ (by linarith) (by linarith),
inferInstance,
trialCenterFullLaw_externalScoreLaw g hOverlap hg _ (by linarith) (by linarith)⟩
have hPH0 : (P0, H) ∈ ExternalLaws g :=
⟨trialCenterFullLaw_isProbabilityMeasure g hOverlap _ (by norm_num) (by norm_num),
inferInstance,
trialCenterFullLaw_externalScoreLaw g hOverlap hg _ (by norm_num) (by norm_num)⟩
have hQ (P : Measure (FullRow ε J)) (hPH : (P, H) ∈ ExternalLaws g) :
Qj P H = (Measure.pi (fun _ : Fin n => releasedLaw P)).prod
(Measure.pi (fun _ : Fin m => H)) := by
rw [hInd (P, H) hPH, hTrial (P, H) hPH, hLog (P, H) hPH]
rw [hQ P1 hPH1, hQ P0 hPH0]
letI : IsProbabilityMeasure (releasedLaw P1) := by
exact @Measure.isProbabilityMeasure_map _ _ _ _ P1 hPH1.2.2.probabilityP _
(by unfold releasedRecord label arm observed; fun_prop :
Measurable (releasedRecord (ε := ε) (J := J))).aemeasurable
letI : IsProbabilityMeasure (releasedLaw P0) := by
exact @Measure.isProbabilityMeasure_map _ _ _ _ P0 hPH0.2.2.probabilityP _
(by unfold releasedRecord label arm observed; fun_prop :
Measurable (releasedRecord (ε := ε) (J := J))).aemeasurable
letI : IsProbabilityMeasure H := inferInstance
calc
_ ≤ tvDist
(Measure.pi (fun _ : Fin n => releasedLaw P1))
(Measure.pi (fun _ : Fin n => releasedLaw P0)) :=
trial_commonLog_tv_le _ _ (Measure.pi (fun _ : Fin m => H))
_ ≤ _ := trialCenter_productReleased_tv_bound g hOverlap u hu n
theorem externalInterval_twoPoint_expectedLength
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
C :
ExternalIntervalProcedure ε J n m
Q0 Q1 :
θ0 θ1 α :
ℝ
hθ :
θ0 ≤ θ1
hcover0 :
1 - α ≤ Q0.real {x | C.contains x θ0}
hcover1 :
1 - α ≤ Q1.real {x | C.contains x θ1}
conclusion
(θ1 - θ0) * (1 - 2 * α - tvDist Q0 Q1) ≤ ∫ x, C.length x ∂Q0
Proof (Lean source)
lemma externalInterval_twoPoint_expectedLength {ε : ℝ} {J n m : ℕ}
(C : ExternalIntervalProcedure ε J n m)
(Q0 Q1 : Measure (ExternalSample ε J n m))
[IsProbabilityMeasure Q0] [IsProbabilityMeasure Q1]
(θ0 θ1 α : ℝ) (hθ : θ0 ≤ θ1)
(hcover0 : 1 - α ≤ Q0.real {x | C.contains x θ0})
(hcover1 : 1 - α ≤ Q1.real {x | C.contains x θ1}) :
(θ1 - θ0) * (1 - 2 * α - tvDist Q0 Q1) ≤
∫ x, C.length x ∂Q0 := by
let A0 : Set (ExternalSample ε J n m) := {x | C.contains x θ0}
let A1 : Set (ExternalSample ε J n m) := {x | C.contains x θ1}
let I := A0 ∩ A1
have hA (θ : ℝ) : MeasurableSet {x | C.contains x θ} := by
change MeasurableSet {x | C.lo x ≤ θ ∧ θ ≤ C.hi x}
exact (measurableSet_le C.measurable_lo measurable_const).inter
(measurableSet_le measurable_const C.measurable_hi)
have hA0 : MeasurableSet A0 := hA θ0
have hA1 : MeasurableSet A1 := hA θ1
have hI : MeasurableSet I := hA0.inter hA1
have htransfer : 1 - α - tvDist Q0 Q1 ≤ Q0.real A1 := by
have htv := measureReal_sub_le_tvDist
(μ := Q0) (ν := Q1) hA1
dsimp [A1] at htv
linarith
have hmass : 1 - 2 * α - tvDist Q0 Q1 ≤ Q0.real I := by
have hu : Q0.real (A0 ∪ A1) ≤ 1 := measureReal_le_one
have hsum := measureReal_union_add_inter (μ := Q0) (s := A0) (t := A1)
hA1 (measure_ne_top Q0 A0) (measure_ne_top Q0 A1)
change Q0.real (A0 ∪ A1) + Q0.real I = Q0.real A0 + Q0.real A1 at hsum
linarith
have hlenMeas : Measurable C.length := C.measurable_hi.sub C.measurable_lo
have hlenNonneg (x : ExternalSample ε J n m) : 0 ≤ C.length x :=
sub_nonneg.mpr (C.ordered x)
have hlenBound (x : ExternalSample ε J n m) : C.length x ≤ 2 := by
dsimp [ExternalIntervalProcedure.length]
linarith [C.lower_bound x, C.upper_bound x]
have hlenInt : Integrable C.length Q0 := by
apply Integrable.of_bound hlenMeas.aestronglyMeasurable 2
filter_upwards with x
rw [Real.norm_eq_abs, abs_of_nonneg (hlenNonneg x)]
exact hlenBound x
have hindInt : Integrable (I.indicator (fun _ => θ1 - θ0)) Q0 :=
(integrable_const _).indicator hI
have hpoint (x : ExternalSample ε J n m) :
I.indicator (fun _ => θ1 - θ0) x ≤ C.length x := by
by_cases hx : x ∈ I
· rw [Set.indicator_of_mem hx]
have hx0 : C.lo x ≤ θ0 ∧ θ0 ≤ C.hi x := hx.1
have hx1 : C.lo x ≤ θ1 ∧ θ1 ≤ C.hi x := hx.2
dsimp [ExternalIntervalProcedure.length]
linarith
· rw [Set.indicator_of_notMem hx]
exact hlenNonneg x
have hmono := integral_mono hindInt hlenInt hpoint
rw [integral_indicator_const (θ1 - θ0) hI] at hmono
simp only [smul_eq_mul] at hmono
rw [mul_comm (Q0.real I) (θ1 - θ0)] at hmono
exact (mul_le_mul_of_nonneg_left hmass (sub_nonneg.mpr hθ)).trans hmono
theorem trial_calibrated_perturbation
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma trial_calibrated_perturbation (α : ℝ) (hα : 0 < α ∧ α < 1 / 2)
(n : ℕ) (hn : 0 < n) :
let β := 1 - 2 * α
let u := sqrt (log (1 + β ^ 2) / (n : ℝ)) / 2
0 < u ∧ |u| < 1 / 2 ∧
(1 + 4 * u ^ 2) ^ n ≤ 1 + β ^ 2 := by
dsimp
let β : ℝ := 1 - 2 * α
let x : ℝ := log (1 + β ^ 2)
let u : ℝ := sqrt (x / (n : ℝ)) / 2
have hβ0 : 0 < β := by dsimp [β]; linarith [hα.2]
have hβ1 : β < 1 := by dsimp [β]; linarith [hα.1]
have hβsq : β ^ 2 < 1 := by nlinarith
have hx0 : 0 < x := by
dsimp [x]
exact Real.log_pos (by nlinarith [sq_pos_of_pos hβ0])
have hxle : x ≤ β ^ 2 := by
dsimp [x]
nlinarith [Real.log_le_sub_one_of_pos (show 0 < 1 + β ^ 2 by positivity)]
have hn0 : (0 : ℝ) < n := by exact_mod_cast hn
have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast hn
have hq0 : 0 < x / (n : ℝ) := div_pos hx0 hn0
have hq1 : x / (n : ℝ) < 1 := (div_lt_one hn0).2 (by linarith)
have hu0 : 0 < u := by dsimp [u]; positivity
have hu1 : |u| < 1 / 2 := by
rw [abs_of_pos hu0]
dsimp [u]
have hs : sqrt (x / (n : ℝ)) < 1 := by
rw [Real.sqrt_lt' (by norm_num : (0 : ℝ) < 1)]
simpa using hq1
linarith
have h4u : 4 * u ^ 2 = x / (n : ℝ) := by
dsimp [u]
calc
4 * (sqrt (x / (n : ℝ)) / 2) ^ 2 =
(sqrt (x / (n : ℝ))) ^ 2 := by ring
_ = x / (n : ℝ) := Real.sq_sqrt hq0.le
have hpow : (1 + 4 * u ^ 2) ^ n ≤ 1 + β ^ 2 := by
rw [h4u]
have h := Real.one_sub_div_pow_le_exp_neg (n := n) (t := -x)
(by linarith : -x ≤ (n : ℝ))
have hexp : exp x = 1 + β ^ 2 := Real.exp_log (by positivity)
simpa only [neg_div, sub_neg_eq_add, neg_neg, hexp] using h
exact ⟨hu0, hu1, hpow⟩
theorem trial_calibrated_coefficient
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma trial_calibrated_coefficient (α : ℝ) (n : ℕ) :
let β := 1 - 2 * α
let u := sqrt (log (1 + β ^ 2) / (n : ℝ)) / 2
β * u / 2 = trialCertificate α * (sqrt (n : ℝ))⁻¹ := by
dsimp [trialCertificate]
rw [Real.sqrt_div (Real.log_nonneg (by nlinarith [sq_nonneg (1 - 2 * α)]))
(n : ℝ)]
ring
theorem trial_calibrated_honest_expectedLength
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
C :
ExternalIntervalProcedure ε J n m
Q0 Q1 :
α :
ℝ
hα :
0 < α ∧ α < 1 / 2
hn :
0 < n
u :
ℝ
hcover0 :
1 - α ≤ Q0.real {x | C.contains x (1 / 2)}
hcover1 :
1 - α ≤ Q1.real {x | C.contains x (1 / 2 + u)}
conclusion
trialCertificate α * (sqrt (n : ℝ))⁻¹ ≤ ∫ x, C.length x ∂Q0
Proof (Lean source)
lemma trial_calibrated_honest_expectedLength {ε : ℝ} {J n m : ℕ}
(C : ExternalIntervalProcedure ε J n m)
(Q0 Q1 : Measure (ExternalSample ε J n m))
[IsProbabilityMeasure Q0] [IsProbabilityMeasure Q1]
(α : ℝ) (hα : 0 < α ∧ α < 1 / 2) (hn : 0 < n)
(u : ℝ)
(hu : u = sqrt (log (1 + (1 - 2 * α) ^ 2) / (n : ℝ)) / 2)
(hcover0 : 1 - α ≤ Q0.real {x | C.contains x (1 / 2)})
(hcover1 : 1 - α ≤ Q1.real {x | C.contains x (1 / 2 + u)})
(htv : tvDist Q0 Q1 ≤
(1 / 2 : ℝ) * sqrt ((1 + 4 * u ^ 2) ^ n - 1)) :
trialCertificate α * (sqrt (n : ℝ))⁻¹ ≤
∫ x, C.length x ∂Q0 := by
have hcal := trial_calibrated_perturbation α hα n hn
dsimp at hcal
rw [← hu] at hcal
rcases hcal with ⟨hu0, _, hpow⟩
let β : ℝ := 1 - 2 * α
have hβ0 : 0 < β := by dsimp [β]; linarith [hα.2]
have htvhalf : tvDist Q0 Q1 ≤ β / 2 := by
calc
_ ≤ (1 / 2 : ℝ) * sqrt ((1 + 4 * u ^ 2) ^ n - 1) := htv
_ ≤ (1 / 2 : ℝ) * sqrt (β ^ 2) := by
apply mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt ?_) (by norm_num)
dsimp [β] at hpow ⊢
linarith
_ = β / 2 := by rw [Real.sqrt_sq_eq_abs, abs_of_pos hβ0]; ring
have hlen := externalInterval_twoPoint_expectedLength C Q0 Q1
(1 / 2) (1 / 2 + u) α (by linarith) hcover0 hcover1
have hcoef : β * u / 2 ≤
u * (1 - 2 * α - tvDist Q0 Q1) := by
dsimp [β] at htvhalf ⊢
nlinarith [mul_nonneg (le_of_lt hu0)
(show 0 ≤ 1 - 2 * α - 2 * tvDist Q0 Q1 by linarith)]
have hident := trial_calibrated_coefficient α n
dsimp at hident
rw [← hu] at hident
rw [← hident]
have hlen' : u * (1 - 2 * α - tvDist Q0 Q1) ≤
∫ x, C.length x ∂Q0 := by
convert hlen using 1 <;> ring
exact hcoef.trans hlen'
theorem externalLaw_jointExperiment_probability
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
Qj :
hTrial :
JointTrialIID g Qj
hLog :
ExternalLogIID g Qj
hInd :
PH :
hPH :
PH ∈ ExternalLaws g
conclusion
IsProbabilityMeasure (Qj PH.1 PH.2)
Proof (Lean source)
lemma externalLaw_jointExperiment_probability {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J)
(Qj : Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : JointTrialIID g Qj) (hLog : ExternalLogIID g Qj)
(hInd : ExternalLogIndependent g Qj)
(PH : Measure (FullRow ε J) × Measure (ScoreSpace ε))
(hPH : PH ∈ ExternalLaws g) : IsProbabilityMeasure (Qj PH.1 PH.2) := by
have hrel : IsProbabilityMeasure (releasedLaw PH.1) := by
exact @Measure.isProbabilityMeasure_map _ _ _ _ PH.1 hPH.2.2.probabilityP _
(by unfold releasedRecord label arm observed; fun_prop :
Measurable (releasedRecord (ε := ε) (J := J))).aemeasurable
letI : IsProbabilityMeasure (releasedLaw PH.1) := hrel
letI : IsProbabilityMeasure PH.2 := hPH.2.1
rw [hInd PH hPH, hTrial PH hPH, hLog PH hPH]
infer_instance
theorem trialCenter_honest_expectedLength
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
hg :
α :
ℝ
hα :
0 < α ∧ α < 1 / 2
hn :
0 < n
Qj :
hTrial :
JointTrialIID g Qj
hLog :
ExternalLogIID g Qj
hInd :
C :
ExternalIntervalProcedure ε J n m
hC :
C ∈ externalHonestProcedures g Qj α
conclusion
trialCertificate α * (sqrt (n : ℝ))⁻¹
≤ ∫ x, C.length x ∂(Qj (trialCenterFullLaw g hOverlap (1 / 2)) (Measure.dirac (trialCenterScore hOverlap)))
Proof (Lean source)
lemma trialCenter_honest_expectedLength {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hOverlap : Overlap ε)
(hg : Measurable g) (α : ℝ) (hα : 0 < α ∧ α < 1 / 2) (hn : 0 < n)
(Qj : Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : JointTrialIID g Qj) (hLog : ExternalLogIID g Qj)
(hInd : ExternalLogIndependent g Qj)
(C : ExternalIntervalProcedure ε J n m)
(hC : C ∈ externalHonestProcedures g Qj α) :
trialCertificate α * (sqrt (n : ℝ))⁻¹ ≤
∫ x, C.length x ∂(Qj (trialCenterFullLaw g hOverlap (1 / 2))
(Measure.dirac (trialCenterScore hOverlap))) := by
let u : ℝ := sqrt (log (1 + (1 - 2 * α) ^ 2) / (n : ℝ)) / 2
have hcal := trial_calibrated_perturbation α hα n hn
dsimp at hcal
change 0 < u ∧ |u| < 1 / 2 ∧ _ at hcal
have hu : |u| < 1 / 2 := hcal.2.1
let H : Measure (ScoreSpace ε) := Measure.dirac (trialCenterScore hOverlap)
let P0 := trialCenterFullLaw g hOverlap (1 / 2)
let P1 := trialCenterFullLaw g hOverlap (1 / 2 + u)
have hub := abs_lt.mp hu
have hPH0 : (P0, H) ∈ ExternalLaws g :=
⟨trialCenterFullLaw_isProbabilityMeasure g hOverlap _ (by norm_num) (by norm_num),
inferInstance,
trialCenterFullLaw_externalScoreLaw g hOverlap hg _ (by norm_num) (by norm_num)⟩
have hPH1 : (P1, H) ∈ ExternalLaws g :=
⟨trialCenterFullLaw_isProbabilityMeasure g hOverlap _ (by linarith) (by linarith),
inferInstance,
trialCenterFullLaw_externalScoreLaw g hOverlap hg _ (by linarith) (by linarith)⟩
let Q0 := Qj P0 H
let Q1 := Qj P1 H
letI : IsProbabilityMeasure Q0 :=
externalLaw_jointExperiment_probability g Qj hTrial hLog hInd (P0, H) hPH0
letI : IsProbabilityMeasure Q1 :=
externalLaw_jointExperiment_probability g Qj hTrial hLog hInd (P1, H) hPH1
have hQ0 : Q0 = externalExperiment n m P0 H := by
dsimp [Q0, externalExperiment]
rw [hInd (P0, H) hPH0, hTrial (P0, H) hPH0,
hLog (P0, H) hPH0]
have hQ1 : Q1 = externalExperiment n m P1 H := by
dsimp [Q1, externalExperiment]
rw [hInd (P1, H) hPH1, hTrial (P1, H) hPH1,
hLog (P1, H) hPH1]
have hc0 := hC (P0, H) hPH0
have hc1 := hC (P1, H) hPH1
change 1 - α ≤ (externalExperiment n m P0 H).real
{x | C.contains x (ate P0)} at hc0
change 1 - α ≤ (externalExperiment n m P1 H).real
{x | C.contains x (ate P1)} at hc1
rw [← hQ0] at hc0
rw [← hQ1] at hc1
rw [trialCenterFullLaw_ate g hOverlap _ (by norm_num) (by norm_num)] at hc0
rw [trialCenterFullLaw_ate g hOverlap _ (by linarith) (by linarith)] at hc1
have htv : tvDist Q0 Q1 ≤
(1 / 2 : ℝ) * sqrt ((1 + 4 * u ^ 2) ^ n - 1) := by
rw [tvDist_symm]
exact trialCenter_jointExperiment_tv_bound g hOverlap hg u hu Qj hTrial hLog hInd
exact trial_calibrated_honest_expectedLength C Q0 Q1 α hα hn u rfl
hc0 hc1 htv
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.FixedLaw 1 declarations Sharp potential-outcome means and ATE interval at a fixed released law.
Sharp potential-outcome means and ATE interval at a fixed released law.
theorem fixedReleasedLaw_sharp
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hOverlap :
Overlap ε
hg :
hcomp :
(CompatibleLaws H g Prel).Nonempty
conclusion
conclusion 1
a :
{v : ℝ | ∃ P ∈ CompatibleLaws H g Prel, v = ∫ ω, ((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ) ∂P}
= Icc (muLower H g Prel (compatibleCellMasses_of_nonempty H g Prel hcomp) a) (muUpper H g Prel (compatibleCellMasses_of_nonempty H g Prel hcomp) a)
conclusion 2
{v : ℝ | ∃ P ∈ CompatibleLaws H g Prel, v = ate P}
= sharpATESet H g Prel (compatibleCellMasses_of_nonempty H g Prel hcomp)
conclusion 3
∃ Pminus ∈ CompatibleLaws H g Prel, ∃ Pplus ∈ CompatibleLaws H g Prel,
ate Pminus
ate Pplus
Proof (Lean source)
lemma fixedReleasedLaw_sharp {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J))
(hOverlap : Overlap ε) (hg : Measurable g)
(hcomp : (CompatibleLaws H g Prel).Nonempty) :
(∀ a : ArmSpace,
{v : ℝ | ∃ P ∈ CompatibleLaws H g Prel,
v = ∫ ω, ((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ) ∂P} =
Icc
(muLower H g Prel (compatibleCellMasses_of_nonempty H g Prel hcomp) a)
(muUpper H g Prel (compatibleCellMasses_of_nonempty H g Prel hcomp) a)) ∧
{v : ℝ | ∃ P ∈ CompatibleLaws H g Prel, v = ate P} =
sharpATESet H g Prel (compatibleCellMasses_of_nonempty H g Prel hcomp) ∧
(∃ Pminus ∈ CompatibleLaws H g Prel,
∃ Pplus ∈ CompatibleLaws H g Prel,
ate Pminus =
muLower H g Prel (compatibleCellMasses_of_nonempty H g Prel hcomp) true -
muUpper H g Prel (compatibleCellMasses_of_nonempty H g Prel hcomp) false ∧
ate Pplus =
muUpper H g Prel (compatibleCellMasses_of_nonempty H g Prel hcomp) true -
muLower H g Prel (compatibleCellMasses_of_nonempty H g Prel hcomp) false) := by
let hMass := compatibleCellMasses_of_nonempty H g Prel hcomp
let Plo := endpointFullLaw H g Prel hMass hOverlap false false
let Phi := endpointFullLaw H g Prel hMass hOverlap true true
let Pminus := endpointFullLaw H g Prel hMass hOverlap true false
let Pplus := endpointFullLaw H g Prel hMass hOverlap false true
have hlo : CompatibleCausalLaw H g Prel Plo :=
endpointFullLaw_compatibleCausalLaw H g Prel hMass hOverlap false false
have hhi : CompatibleCausalLaw H g Prel Phi :=
endpointFullLaw_compatibleCausalLaw H g Prel hMass hOverlap true true
have hminus : CompatibleCausalLaw H g Prel Pminus :=
endpointFullLaw_compatibleCausalLaw H g Prel hMass hOverlap true false
have hplus : CompatibleCausalLaw H g Prel Pplus :=
endpointFullLaw_compatibleCausalLaw H g Prel hMass hOverlap false true
have hloMem : Plo ∈ CompatibleLaws H g Prel := ⟨hlo.probability, hlo⟩
have hhiMem : Phi ∈ CompatibleLaws H g Prel := ⟨hhi.probability, hhi⟩
have hminusMem : Pminus ∈ CompatibleLaws H g Prel :=
⟨hminus.probability, hminus⟩
have hplusMem : Pplus ∈ CompatibleLaws H g Prel :=
⟨hplus.probability, hplus⟩
have hminusAte : ate Pminus =
muLower H g Prel hMass true - muUpper H g Prel hMass false := by
letI : IsProbabilityMeasure Pminus := hminus.probability
rw [ate_eq_potentialOutcomeMeans_sub]
have h1 := endpointFullLaw_armMean H g Prel hMass hOverlap true false true
have h0 := endpointFullLaw_armMean H g Prel hMass hOverlap true false false
simpa [Pminus] using congrArg₂ (· - ·) h1 h0
have hplusAte : ate Pplus =
muUpper H g Prel hMass true - muLower H g Prel hMass false := by
letI : IsProbabilityMeasure Pplus := hplus.probability
rw [ate_eq_potentialOutcomeMeans_sub]
have h1 := endpointFullLaw_armMean H g Prel hMass hOverlap false true true
have h0 := endpointFullLaw_armMean H g Prel hMass hOverlap false true false
simpa [Pplus] using congrArg₂ (· - ·) h1 h0
refine ⟨?_, ?_, ?_⟩
· intro a
ext v
constructor
· rintro ⟨P, hP, rfl⟩
exact compatible_armMean_mem_endpoints H g Prel P hP.2 hMass hOverlap a
· intro hv
rcases Icc_subset_segment hv with ⟨c, d, hc, hd, hcd, hv⟩
let cE : ENNReal := ENNReal.ofReal c
let dE : ENNReal := ENNReal.ofReal d
let Pmix := compatibleMixture cE dE Plo Phi
have hsumE : cE + dE = 1 := by
dsimp [cE, dE]
rw [← ENNReal.ofReal_add hc hd, hcd]
norm_num
have hmix := compatibleCausalLaw_mixture H g Prel Plo Phi hlo hhi
cE dE ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top hsumE
refine ⟨Pmix, ⟨hmix.probability, hmix⟩, ?_⟩
letI : IsProbabilityMeasure Plo := hlo.probability
letI : IsProbabilityMeasure Phi := hhi.probability
have hint := integral_compatibleMixture Plo Phi cE dE
ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top
(fun ω => ((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ))
(by
cases a
· simp only [Bool.false_eq_true, ↓reduceIte]
unfold outcome0
fun_prop
· simp only [↓reduceIte]
unfold outcome1
fun_prop) 1 (by
intro ω
cases a
· simp only [Bool.false_eq_true, ↓reduceIte]
rw [Real.norm_eq_abs, abs_of_nonneg (outcome0 ω).property.1]
exact (outcome0 ω).property.2
· simp only [↓reduceIte]
rw [Real.norm_eq_abs, abs_of_nonneg (outcome1 ω).property.1]
exact (outcome1 ω).property.2)
have hL := endpointFullLaw_armMean H g Prel hMass hOverlap false false a
have hU := endpointFullLaw_armMean H g Prel hMass hOverlap true true a
change v = ∫ ω, ((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ)
∂Pmix
rw [hint, hL, hU]
dsimp [cE, dE]
rw [ENNReal.toReal_ofReal hc, ENNReal.toReal_ofReal hd]
simpa [smul_eq_mul] using hv.symm
· ext v
constructor
· rintro ⟨P, hP, rfl⟩
exact compatibleLaw_ate_mem_sharpATESet H g Prel hOverlap hcomp P hP
· intro hv
unfold sharpATESet at hv
rcases Icc_subset_segment hv with ⟨c, d, hc, hd, hcd, hv⟩
let cE : ENNReal := ENNReal.ofReal c
let dE : ENNReal := ENNReal.ofReal d
let Pmix := compatibleMixture cE dE Pminus Pplus
have hsumE : cE + dE = 1 := by
dsimp [cE, dE]
rw [← ENNReal.ofReal_add hc hd, hcd]
norm_num
have hmix := compatibleCausalLaw_mixture H g Prel Pminus Pplus
hminus hplus cE dE ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top hsumE
refine ⟨Pmix, ⟨hmix.probability, hmix⟩, ?_⟩
letI : IsProbabilityMeasure Pminus := hminus.probability
letI : IsProbabilityMeasure Pplus := hplus.probability
have hint := integral_compatibleMixture Pminus Pplus cE dE
ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top
(fun ω => (outcome1 ω : ℝ) - (outcome0 ω : ℝ))
(by unfold outcome0 outcome1; fun_prop) 1 (by
intro ω
rw [Real.norm_eq_abs]
exact abs_le.mpr ⟨by
linarith [(outcome0 ω).property.2, (outcome1 ω).property.1], by
linarith [(outcome1 ω).property.2, (outcome0 ω).property.1]⟩)
change v = ate Pmix
rw [show ate Pmix = _ from hint]
change v = cE.toReal * ate Pminus + dE.toReal * ate Pplus
rw [hminusAte, hplusAte]
dsimp [cE, dE]
rw [ENNReal.toReal_ofReal hc, ENNReal.toReal_ofReal hd]
simpa [smul_eq_mul] using hv.symm
· exact ⟨Pminus, hminusMem, Pplus, hplusMem, hminusAte, hplusAte⟩
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.FixedLawAttainment 4 declarations This file isolates the measure-theoretic reconstruction step needed to combine two potential-outcome laws that share the same score marginal.
Common-marginal gluing needed for fixed-law endpoint attainment
This file isolates the measure-theoretic reconstruction step needed to combine two potential-outcome laws that share the same score marginal. The gluing is obtained by disintegrating both laws over their first coordinate and taking the conditional product of the resulting kernels.
def glueAlongFst
For the specified mathematical inputs, this definition introduces the corresponding object.
theorem glueAlongFst_isProbabilityMeasure
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
IsProbabilityMeasure (glueAlongFst ρ₀ ρ₁)
Proof (Lean source)
lemma glueAlongFst_isProbabilityMeasure {α β γ : Type*}
[MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ]
[CountablyGenerated α]
[StandardBorelSpace β] [Nonempty β]
[StandardBorelSpace γ] [Nonempty γ]
(ρ₀ : Measure (α × β)) (ρ₁ : Measure (α × γ))
[IsProbabilityMeasure ρ₀] [IsProbabilityMeasure ρ₁] :
IsProbabilityMeasure (glueAlongFst ρ₀ ρ₁) := by
unfold glueAlongFst
infer_instance
theorem glueAlongFst_map_left
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(glueAlongFst ρ₀ ρ₁).map (fun p => (p.1, p.2.1)) = ρ₀
Proof (Lean source)
lemma glueAlongFst_map_left {α β γ : Type*}
[MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ]
[CountablyGenerated α]
[StandardBorelSpace β] [Nonempty β]
[StandardBorelSpace γ] [Nonempty γ]
(ρ₀ : Measure (α × β)) (ρ₁ : Measure (α × γ))
[IsFiniteMeasure ρ₀] [IsFiniteMeasure ρ₁] :
(glueAlongFst ρ₀ ρ₁).map (fun p => (p.1, p.2.1)) = ρ₀ := by
rw [glueAlongFst]
change (ρ₀.fst ⊗ₘ (ρ₀.condKernel ×ₖ ρ₁.condKernel)).map
(map id fst) = ρ₀
rw [← Measure.compProd_map measurable_fst]
rw [← Kernel.fst_eq, Kernel.fst_prod]
exact ρ₀.disintegrate ρ₀.condKernel
theorem glueAlongFst_map_right
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(glueAlongFst ρ₀ ρ₁).map (fun p => (p.1, p.2.2)) = ρ₁
Proof (Lean source)
lemma glueAlongFst_map_right {α β γ : Type*}
[MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ]
[CountablyGenerated α]
[StandardBorelSpace β] [Nonempty β]
[StandardBorelSpace γ] [Nonempty γ]
(ρ₀ : Measure (α × β)) (ρ₁ : Measure (α × γ))
[IsFiniteMeasure ρ₀] [IsFiniteMeasure ρ₁]
(hfst : ρ₀.fst = ρ₁.fst) :
(glueAlongFst ρ₀ ρ₁).map (fun p => (p.1, p.2.2)) = ρ₁ := by
rw [glueAlongFst]
change (ρ₀.fst ⊗ₘ (ρ₀.condKernel ×ₖ ρ₁.condKernel)).map
(map id snd) = ρ₁
rw [← Measure.compProd_map measurable_snd]
rw [← Kernel.snd_eq, Kernel.snd_prod, hfst]
exact ρ₁.disintegrate ρ₁.condKernel
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.FixedLawCellCoupling 6 declarations Assigned-cell mass identity for a compatible fixed released law.
Assigned-cell mass identity for a compatible fixed released law.
This identifies the mass used to normalize an assigned release-cell coupling with the score-induced arm-cell mass.
theorem compatible_assignedScoreLaw_eq_tilt
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hP :
CompatibleCausalLaw H g Prel P
hOverlap :
Overlap ε
a :
Proof (Lean source)
lemma compatible_assignedScoreLaw_eq_tilt {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (P : Measure (FullRow ε J))
(hP : CompatibleCausalLaw H g Prel P) (hOverlap : Overlap ε)
(a : ArmSpace) :
(P.restrict {ω | arm ω = a}).map score =
H.withDensity (fun e => ofReal (armProb a e)) := by
letI : IsProbabilityMeasure P := hP.probability
have hX : Measurable (latentTriple (ε := ε) (J := J)) := by
unfold latentTriple score outcome0 outcome1
fun_prop
have hscore : Measurable (score (ε := ε) (J := J)) := by
unfold score
fun_prop
have hf : Measurable (fun e : ScoreSpace ε =>
ofReal (armProb a e)) := by
cases a <;> simp [armProb] <;> fun_prop
calc
(P.restrict {ω | arm ω = a}).map score =
(assignedLatentLaw P a).map fst := by
unfold assignedLatentLaw
rw [Measure.map_map measurable_fst hX]
rfl
_ = (propensityTiltedLatentLaw P a).map fst :=
congrArg (fun μ : Measure (ScoreSpace ε × OutcomeSpace × OutcomeSpace) =>
μ.map fst)
(assignedLatentLaw_eq_propensityTiltedLatentLaw P hOverlap
hP.randomizedAssignment a)
_ = H.withDensity (fun e => ofReal (armProb a e)) := by
unfold propensityTiltedLatentLaw
rw [map_fst_withDensity_of_fst _ _ hf]
rw [Measure.map_map measurable_fst hX]
change (P.map score).withDensity _ = _
rw [hP.scoreMarginal]
theorem compatible_assignedCellWeightLaw_eq
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hP :
CompatibleCausalLaw H g Prel P
hOverlap :
Overlap ε
a :
r :
Proof (Lean source)
lemma compatible_assignedCellWeightLaw_eq {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (P : Measure (FullRow ε J))
(hP : CompatibleCausalLaw H g Prel P) (hOverlap : Overlap ε)
(a : ArmSpace) (r : LabelSpace J) :
(P.restrict {ω | arm ω = a ∧ g (score ω) = r}).map
(fun ω => (armProb a (score ω))⁻¹) =
((H.withDensity (fun e => ofReal (armProb a e))).restrict
(cell g r)).map (fun e => (armProb a e)⁻¹) := by
let w : ScoreSpace ε → ℝ := fun e => (armProb a e)⁻¹
have hw : Measurable w := by
cases a <;> simp [w, armProb] <;> fun_prop
have hscore : Measurable (score (ε := ε) (J := J)) := by
unfold score
fun_prop
have hcell : MeasurableSet (cell g r) :=
measurableSet_eq_fun hP.measurableRelease measurable_const
have hS : {ω : FullRow ε J | arm ω = a ∧ g (score ω) = r} =
{ω | arm ω = a} ∩ score ⁻¹' cell g r := by
ext ω
simp [cell]
have hA : MeasurableSet {ω : FullRow ε J | arm ω = a} :=
measurableSet_eq_fun (by unfold arm; fun_prop) measurable_const
have hRestrict :
(P.restrict {ω | arm ω = a}).restrict (score ⁻¹' cell g r) =
P.restrict {ω | arm ω = a ∧ g (score ω) = r} := by
rw [Measure.restrict_restrict' hA, hS, Set.inter_comm]
calc
(P.restrict {ω | arm ω = a ∧ g (score ω) = r}).map
(fun ω => (armProb a (score ω))⁻¹) =
((P.restrict {ω | arm ω = a ∧ g (score ω) = r}).map score).map w := by
rw [Measure.map_map hw hscore]
rfl
_ = (((P.restrict {ω | arm ω = a}).map score).restrict (cell g r)).map w := by
rw [Measure.restrict_map hscore hcell, hRestrict]
_ = ((H.withDensity (fun e => ofReal (armProb a e))).restrict
(cell g r)).map (fun e => (armProb a e)⁻¹) := by
rw [compatible_assignedScoreLaw_eq_tilt H g Prel P hP hOverlap a]
theorem compatibleArmCellCoupling_map_snd
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hP :
CompatibleCausalLaw H g Prel P
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
r :
hq :
0 < armCellMass H g a r
conclusion
= armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq
Proof (Lean source)
lemma compatibleArmCellCoupling_map_snd {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (P : Measure (FullRow ε J))
(hP : CompatibleCausalLaw H g Prel P)
(hMass : CompatibleCellMasses H g Prel) (hOverlap : Overlap ε)
(a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) :
(compatibleArmCellCoupling H g P a r hq).map snd =
armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq := by
unfold compatibleArmCellCoupling armCellWeightLaw armCellScoreLaw
rw [Measure.map_smul, Measure.map_map (by fun_prop) (by
cases a <;> simp [armProb, outcome0, outcome1, score] <;> fun_prop),
Measure.map_smul]
congr 1
exact compatible_assignedCellWeightLaw_eq H g Prel P hP hOverlap a r
theorem compatibleArmCellCoupling_map_fst
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hP :
CompatibleCausalLaw H g Prel P
hMass :
CompatibleCellMasses H g Prel
a :
r :
hq :
0 < armCellMass H g a r
conclusion
= armCellOutcomeLaw Prel hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq)
Proof (Lean source)
lemma compatibleArmCellCoupling_map_fst {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (P : Measure (FullRow ε J))
(hP : CompatibleCausalLaw H g Prel P)
(hMass : CompatibleCellMasses H g Prel)
(a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) :
(compatibleArmCellCoupling H g P a r hq).map fst =
armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq) := by
have hpair : Measurable (fun ω : FullRow ε J =>
(((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ),
(armProb a (score ω))⁻¹)) := by
cases a <;> simp [armProb, outcome0, outcome1, score] <;> fun_prop
unfold compatibleArmCellCoupling armCellOutcomeLaw
rw [Measure.map_smul, Measure.map_map (by fun_prop) hpair]
letI : IsProbabilityMeasure Prel := hMass.2.1
have hTmeasure : Prel {z | z.1 = r ∧ z.2.1 = a} =
ofReal (armCellMass H g a r) := by
rw [← hMass.2.2.2 a r]
exact (ENNReal.ofReal_toReal (measure_ne_top Prel _)).symm
rw [hTmeasure]
congr 1
ext B hB
have hY : Measurable (fun ω : FullRow ε J =>
((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ)) := by
cases a <;> simp [outcome0, outcome1] <;> fun_prop
have hobs : Measurable (fun z : Observation J => (z.2.2 : ℝ)) := by fun_prop
have hrecord : Measurable (releasedRecord (ε := ε) (J := J)) := by
unfold releasedRecord label arm observed
fun_prop
have hT : MeasurableSet {z : Observation J | z.1 = r ∧ z.2.1 = a} :=
(measurableSet_eq_fun measurable_fst measurable_const).inter
(measurableSet_eq_fun (measurable_fst.comp measurable_snd) measurable_const)
change ((P.restrict {ω | arm ω = a ∧ g (score ω) = r}).map
(fun ω : FullRow ε J =>
((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ))) B = _
rw [Measure.map_apply hY hB, Measure.restrict_apply (hB.preimage hY)]
rw [Measure.map_apply hobs hB, Measure.restrict_apply (hB.preimage hobs)]
rw [← hP.releasedLaw]
unfold releasedLaw
rw [Measure.map_apply hrecord ((hB.preimage hobs).inter hT)]
apply measure_congr
filter_upwards [hP.consistency, hP.deterministicRelease] with ω hcons hlabel
change ((((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ) ∈ B ∧
arm ω = a ∧ g (score ω) = r) =
((observed ω : ℝ) ∈ B ∧ label ω = r ∧ arm ω = a))
apply propext
by_cases ha : arm ω = a
· rw [ha] at hcons
simp [ha, hcons, hlabel, and_comm]
· simp [ha]
theorem compatible_assignedArmCell_measureReal
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hP :
CompatibleCausalLaw H g Prel P
hMass :
CompatibleCellMasses H g Prel
a :
r :
conclusion
P.real {ω | arm ω = a ∧ g (score ω) = r} = armCellMass H g a r
Proof (Lean source)
lemma compatible_assignedArmCell_measureReal {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (P : Measure (FullRow ε J))
(hP : CompatibleCausalLaw H g Prel P)
(hMass : CompatibleCellMasses H g Prel)
(a : ArmSpace) (r : LabelSpace J) :
P.real {ω | arm ω = a ∧ g (score ω) = r} = armCellMass H g a r := by
have hrecord : Measurable (releasedRecord (ε := ε) (J := J)) := by
unfold releasedRecord label arm observed
fun_prop
have hcell : MeasurableSet {z : Observation J | z.1 = r ∧ z.2.1 = a} :=
(measurableSet_eq_fun measurable_fst measurable_const).inter
(measurableSet_eq_fun (measurable_fst.comp measurable_snd) measurable_const)
rw [← hMass.2.2.2 a r, ← hP.releasedLaw]
unfold releasedLaw
rw [MeasureTheory.map_measureReal_apply hrecord hcell]
apply measureReal_congr
filter_upwards [hP.deterministicRelease] with ω hlabel
change (arm ω = a ∧ g (score ω) = r) = (label ω = r ∧ arm ω = a)
apply propext
simp [hlabel, and_comm]
theorem compatibleArmCellCoupling_isCoupling
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hP :
CompatibleCausalLaw H g Prel P
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
r :
hq :
0 < armCellMass H g a r
conclusion
IsCoupling (compatibleArmCellCoupling H g P a r hq) (armCellOutcomeLaw Prel hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq)) (armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq)
Proof (Lean source)
lemma compatibleArmCellCoupling_isCoupling {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (P : Measure (FullRow ε J))
(hP : CompatibleCausalLaw H g Prel P)
(hMass : CompatibleCellMasses H g Prel) (hOverlap : Overlap ε)
(a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) :
IsCoupling (compatibleArmCellCoupling H g P a r hq)
(armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq))
(armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) := by
letI : IsProbabilityMeasure P := hP.probability
let S : Set (FullRow ε J) := {ω | arm ω = a ∧ g (score ω) = r}
have hSreal : P.real S = armCellMass H g a r := by
simpa [S] using compatible_assignedArmCell_measureReal H g Prel P hP hMass a r
have hSmeasure : P S = ofReal (armCellMass H g a r) := by
rw [← hSreal]
exact (ENNReal.ofReal_toReal (measure_ne_top P S)).symm
have hSzero : P S ≠ 0 := by
rw [hSmeasure]
exact ne_of_gt (ENNReal.ofReal_pos.mpr hq)
let μ : Measure (FullRow ε J) := P.restrict S
letI : IsFiniteMeasure μ := inferInstance
letI : NeZero μ := by
constructor
intro hμ
have : P S = 0 := by
rw [← Measure.restrict_apply_univ S, show P.restrict S = μ from rfl, hμ]
simp
exact hSzero this
have hprob : IsProbabilityMeasure
((ofReal (armCellMass H g a r))⁻¹ • μ) := by
rw [← hSmeasure, ← Measure.restrict_apply_univ S]
infer_instance
have hpair : Measurable (fun ω : FullRow ε J =>
(((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ),
(armProb a (score ω))⁻¹)) := by
cases a <;> simp [armProb, outcome0, outcome1, score] <;> fun_prop
refine ⟨?_, ?_, ?_⟩
· unfold compatibleArmCellCoupling
rw [← Measure.map_smul]
change IsProbabilityMeasure
(((ofReal (armCellMass H g a r))⁻¹ • μ).map _)
exact @Measure.isProbabilityMeasure_map _ _ _ _ _ hprob _ hpair.aemeasurable
· exact compatibleArmCellCoupling_map_fst H g Prel P hP hMass a r hq
· exact compatibleArmCellCoupling_map_snd H g Prel P hP hMass hOverlap a r hq
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.FixedLawConvexity 3 declarations Convex closure of compatible fixed released laws.
Convex closure of compatible fixed released laws.
def compatibleMixture
For the specified mathematical inputs, this definition introduces the corresponding object.
theorem compatibleCausalLaw_mixture
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hP :
CompatibleCausalLaw H g Prel P
hQ :
CompatibleCausalLaw H g Prel Q
c d :
hc :
c ≠ ∞
hd :
d ≠ ∞
hsum :
c + d = 1
conclusion
CompatibleCausalLaw H g Prel (compatibleMixture c d P Q)
Proof (Lean source)
lemma compatibleCausalLaw_mixture {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (P Q : Measure (FullRow ε J))
(hP : CompatibleCausalLaw H g Prel P)
(hQ : CompatibleCausalLaw H g Prel Q)
(c d : ENNReal) (hc : c ≠ ∞) (hd : d ≠ ∞) (hsum : c + d = 1) :
CompatibleCausalLaw H g Prel (compatibleMixture c d P Q) := by
letI : IsProbabilityMeasure P := hP.probability
letI : IsProbabilityMeasure Q := hQ.probability
have hprob : IsProbabilityMeasure (compatibleMixture c d P Q) := by
rw [isProbabilityMeasure_iff]
unfold compatibleMixture
rw [Measure.add_apply, Measure.smul_apply, Measure.smul_apply]
simp only [measure_univ, smul_eq_mul, mul_one]
exact hsum
refine
{ probability := hprob
measurableRelease := hP.measurableRelease
scoreMarginal := ?_
randomizedAssignment := ?_
consistency := ?_
deterministicRelease := ?_
releasedLaw := ?_ }
· unfold ScoreMarginal compatibleMixture
rw [Measure.map_add _ _ (by unfold score; fun_prop),
Measure.map_smul, Measure.map_smul, hP.scoreMarginal, hQ.scoreMarginal,
← add_smul, hsum, one_smul]
· intro B hB
let S : Set (FullRow ε J) :=
{ω | arm ω = true ∧ (score ω, outcome0 ω, outcome1 ω) ∈ B}
let T : Set (FullRow ε J) :=
{ω | (score ω, outcome0 ω, outcome1 ω) ∈ B}
have hSintP := mixtureScore_integrable c hc P
have hSintQ := mixtureScore_integrable d hd Q
change (compatibleMixture c d P Q).real S =
∫ ω in T, (score ω : ℝ) ∂compatibleMixture c d P Q
unfold compatibleMixture
rw [mixture_real_apply c d hc hd P Q S]
rw [Measure.restrict_add,
integral_add_measure hSintP.integrableOn hSintQ.integrableOn,
Measure.restrict_smul, Measure.restrict_smul,
integral_smul_measure, integral_smul_measure]
rw [hP.randomizedAssignment B hB, hQ.randomizedAssignment B hB]
simp only [smul_eq_mul]
rfl
· unfold Consistency compatibleMixture
rw [ae_add_measure_iff]
exact ⟨Measure.smul_absolutelyContinuous.ae_le hP.consistency,
Measure.smul_absolutelyContinuous.ae_le hQ.consistency⟩
· unfold DeterministicRelease compatibleMixture
rw [ae_add_measure_iff]
exact ⟨Measure.smul_absolutelyContinuous.ae_le hP.deterministicRelease,
Measure.smul_absolutelyContinuous.ae_le hQ.deterministicRelease⟩
· unfold ReleasedLaw releasedLaw compatibleMixture
rw [Measure.map_add _ _ (by
unfold releasedRecord label arm observed
fun_prop), Measure.map_smul, Measure.map_smul,
show P.map releasedRecord = Prel from hP.releasedLaw,
show Q.map releasedRecord = Prel from hQ.releasedLaw,
← add_smul, hsum, one_smul]
theorem integral_compatibleMixture
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(∫ ω, f ω ∂compatibleMixture c d P Q) = c.toReal * ∫ ω, f ω ∂P + d.toReal * ∫ ω, f ω ∂Q
Proof (Lean source)
lemma integral_compatibleMixture {ε : ℝ} {J : ℕ}
(P Q : Measure (FullRow ε J)) [IsProbabilityMeasure P]
[IsProbabilityMeasure Q] (c d : ENNReal) (hc : c ≠ ∞) (hd : d ≠ ∞)
(f : FullRow ε J → ℝ) (hf : Measurable f) (K : ℝ)
(hK : ∀ ω, ‖f ω‖ ≤ K) :
(∫ ω, f ω ∂compatibleMixture c d P Q) =
c.toReal * ∫ ω, f ω ∂P + d.toReal * ∫ ω, f ω ∂Q := by
have hPc : Integrable f (c • P) := by
letI : IsFiniteMeasure (c • P) := ⟨by
rw [Measure.smul_apply]
exact ENNReal.mul_lt_top (lt_top_iff_ne_top.mpr hc) (measure_lt_top P univ)⟩
exact Integrable.of_bound hf.aestronglyMeasurable K
(Filter.Eventually.of_forall hK)
have hQd : Integrable f (d • Q) := by
letI : IsFiniteMeasure (d • Q) := ⟨by
rw [Measure.smul_apply]
exact ENNReal.mul_lt_top (lt_top_iff_ne_top.mpr hd) (measure_lt_top Q univ)⟩
exact Integrable.of_bound hf.aestronglyMeasurable K
(Filter.Eventually.of_forall hK)
unfold compatibleMixture
rw [integral_add_measure hPc hQd, integral_smul_measure,
integral_smul_measure]
simp only [smul_eq_mul]
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.FixedLawCoverage 4 declarations Membership of a compatible law's ATE in the population interval.
Membership of a compatible law's ATE in the population interval.
theorem compatible_armMean_eq_sum_cellCouplings
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hP :
CompatibleCausalLaw H g Prel P
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
conclusion
(∫ ω, ((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ) ∂P)
= ∑ r : LabelSpace J, if hq : 0 < armCellMass H g a r then armCellMass H g a r * ∫ p, p.1 * p.2 ∂(compatibleArmCellCoupling H g P a r hq) else 0
Proof (Lean source)
lemma compatible_armMean_eq_sum_cellCouplings {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (P : Measure (FullRow ε J))
(hP : CompatibleCausalLaw H g Prel P)
(hMass : CompatibleCellMasses H g Prel) (hOverlap : Overlap ε)
(a : ArmSpace) :
(∫ ω, ((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ) ∂P) =
∑ r : LabelSpace J,
if hq : 0 < armCellMass H g a r then
armCellMass H g a r *
∫ p, p.1 * p.2 ∂(compatibleArmCellCoupling H g P a r hq)
else 0 := by
letI : IsProbabilityMeasure P := hP.probability
rw [potentialOutcomeMean_eq_inversePropensity P hOverlap hP.randomizedAssignment a]
rw [assignedProduct_integral_eq_sum_cells P g hMass.2.2.1 hOverlap a]
apply Finset.sum_congr rfl
intro r _
by_cases hq : 0 < armCellMass H g a r
· simp only [hq, dif_pos]
exact assignedProduct_cell_eq_mass_mul_coupling H g P a r hq
· simp only [hq, dif_neg]
exact assignedProduct_cell_eq_zero H g Prel P hP hMass a r hq
theorem compatible_armMean_mem_endpoints
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hP :
CompatibleCausalLaw H g Prel P
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
conclusion
(∫ ω, ((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ) ∂P) ∈ Icc (muLower H g Prel hMass a) (muUpper H g Prel hMass a)
Proof (Lean source)
lemma compatible_armMean_mem_endpoints {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (P : Measure (FullRow ε J))
(hP : CompatibleCausalLaw H g Prel P)
(hMass : CompatibleCellMasses H g Prel) (hOverlap : Overlap ε)
(a : ArmSpace) :
(∫ ω, ((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ) ∂P) ∈
Icc (muLower H g Prel hMass a) (muUpper H g Prel hMass a) := by
let π : ∀ r : LabelSpace J, 0 < armCellMass H g a r → Measure (ℝ × ℝ) :=
fun r hq => compatibleArmCellCoupling H g P a r hq
apply armMean_mem_endpoints_of_cellCouplings H g Prel hMass a π
· intro r hq
exact compatibleArmCellCoupling_isCoupling H g Prel P hP hMass hOverlap a r hq
· intro r hq
exact (compatibleArmCell_moments H g Prel P hP hMass hOverlap a r hq).1
· intro r hq
exact (compatibleArmCell_moments H g Prel P hP hMass hOverlap a r hq).2
· exact compatible_armMean_eq_sum_cellCouplings H g Prel P hP hMass hOverlap a
theorem compatible_ate_mem_sharpATESet
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hP :
CompatibleCausalLaw H g Prel P
hOverlap :
Overlap ε
conclusion
ate P ∈ sharpATESet H g Prel (compatibleCellMasses_of_compatibleCausalLaw H g Prel P hP)
Proof (Lean source)
lemma compatible_ate_mem_sharpATESet {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (P : Measure (FullRow ε J))
(hP : CompatibleCausalLaw H g Prel P)
(hOverlap : Overlap ε) :
ate P ∈ sharpATESet H g Prel
(compatibleCellMasses_of_compatibleCausalLaw H g Prel P hP) := by
let hMass := compatibleCellMasses_of_compatibleCausalLaw H g Prel P hP
letI : IsProbabilityMeasure P := hP.probability
apply ate_mem_sharpATESet_of_armMean_bounds H g Prel hMass P
· intro a
exact (compatible_armMean_mem_endpoints H g Prel P hP hMass hOverlap a).1
· intro a
exact (compatible_armMean_mem_endpoints H g Prel P hP hMass hOverlap a).2
theorem compatibleLaw_ate_mem_sharpATESet
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hOverlap :
Overlap ε
hcomp :
(CompatibleLaws H g Prel).Nonempty
hP :
P ∈ CompatibleLaws H g Prel
conclusion
ate P ∈ sharpATESet H g Prel (compatibleCellMasses_of_nonempty H g Prel hcomp)
Proof (Lean source)
lemma compatibleLaw_ate_mem_sharpATESet {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J))
(hOverlap : Overlap ε)
(hcomp : (CompatibleLaws H g Prel).Nonempty)
(P : Measure (FullRow ε J))
(hP : P ∈ CompatibleLaws H g Prel) :
ate P ∈ sharpATESet H g Prel
(compatibleCellMasses_of_nonempty H g Prel hcomp) := by
simpa using compatible_ate_mem_sharpATESet H g Prel P hP.2 hOverlap
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.FixedLawEndpointAssembly 21 declarations Assembly of endpoint-attaining cell laws into bounded arm laws.
Assembly of endpoint-attaining cell laws into bounded arm laws.
theorem endpointCellScoreOutcomeLaw_snd
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
r :
hq :
0 < armCellMass H g a r
upper :
conclusion
(endpointCellScoreOutcomeLaw H g Prel hMass hOverlap a r hq upper).snd
= armCellOutcomeLaw Prel hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq)
Proof (Lean source)
lemma endpointCellScoreOutcomeLaw_snd {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) (upper : Bool) :
(endpointCellScoreOutcomeLaw H g Prel hMass hOverlap a r hq upper).snd =
armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq) := by
let Y := armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq)
let W := armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq
let π : Measure (ℝ × ℝ) :=
if upper then Causalean.Stat.comonotoneCoupling Y W
else Causalean.Stat.countermonotoneCoupling Y W
have hπ : IsCoupling π Y W := by
simpa [π, Y, W] using
endpointCoupling_isCoupling H g Prel hMass hOverlap a r hq upper
unfold Measure.snd
calc
(endpointCellScoreOutcomeLaw H g Prel hMass hOverlap a r hq upper).map
snd =
((endpointCellScoreOutcomeLaw H g Prel hMass hOverlap a r hq upper).map
(fun p => (p.2, (armProb a p.1)⁻¹))).map fst := by
rw [Measure.map_map measurable_fst (by
cases a <;> simp [armProb] <;> fun_prop)]
rfl
_ = π.map fst := by
rw [endpointCellScoreOutcomeLaw_map_outcome_weight
H g Prel hMass hOverlap a r hq upper]
_ = Y := hπ.map_fst
theorem endpointCellScoreOutcomeLaw_outcome_support
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
r :
hq :
0 < armCellMass H g a r
upper :
conclusion
endpointCellScoreOutcomeLaw H g Prel hMass hOverlap a r hq upper {p | p.2 ∉ Icc (0 : ℝ) 1}
= 0
Proof (Lean source)
lemma endpointCellScoreOutcomeLaw_outcome_support {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) (upper : Bool) :
endpointCellScoreOutcomeLaw H g Prel hMass hOverlap a r hq upper
{p | p.2 ∉ Icc (0 : ℝ) 1} = 0 := by
let M := endpointCellScoreOutcomeLaw H g Prel hMass hOverlap a r hq upper
have hs : MeasurableSet ({y : ℝ | y ∉ Icc (0 : ℝ) 1}) :=
measurableSet_Icc.compl
have hsnd : M.snd = armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq) :=
endpointCellScoreOutcomeLaw_snd H g Prel hMass hOverlap a r hq upper
have hset : {p : ScoreSpace ε × ℝ | p.2 ∉ Icc (0 : ℝ) 1} =
snd ⁻¹' {y : ℝ | y ∉ Icc (0 : ℝ) 1} := rfl
rw [hset, ← Measure.map_apply measurable_snd hs, show M.map snd = M.snd from rfl,
hsnd]
unfold armCellOutcomeLaw
rw [Measure.smul_apply, Measure.map_apply (by fun_prop) hs]
have hpre : (fun z : Observation J => (z.2.2 : ℝ)) ⁻¹'
{y : ℝ | y ∉ Icc (0 : ℝ) 1} = ∅ := by
ext z
exact iff_false_intro (fun hz => hz z.2.2.property)
rw [hpre]
simp
def clampOutcome
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
theorem clampOutcome_measurable
This result establishes the stated mathematical conclusion.
Formal statement
conclusion
Proof (Lean source)
theorem clampOutcome_coe_of_mem
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma clampOutcome_coe_of_mem {y : ℝ} (hy : y ∈ Icc (0 : ℝ) 1) :
((clampOutcome y : OutcomeSpace) : ℝ) = y := by
simp [clampOutcome, hy.1, hy.2]
def boundedEndpointCellLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
r :
hq :
0 < armCellMass H g a r
upper :
def
given by
(endpointCellScoreOutcomeLaw H g Prel hMass hOverlap a r hq upper).map (fun p => (p.1, clampOutcome p.2))
instance boundedEndpointCellLaw_isProbabilityMeasure
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
instance boundedEndpointCellLaw_isProbabilityMeasure {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) (upper : Bool) :
IsProbabilityMeasure
(boundedEndpointCellLaw H g Prel hMass hOverlap a r hq upper) := by
unfold boundedEndpointCellLaw
letI : IsProbabilityMeasure H := hMass.1
letI : IsProbabilityMeasure Prel := hMass.2.1
letI : Nonempty (ScoreSpace ε) := ⟨⟨1 / 2, by
constructor <;> linarith [hOverlap.1, hOverlap.2]⟩⟩
letI : IsProbabilityMeasure
(endpointCellScoreOutcomeLaw H g Prel hMass hOverlap a r hq upper) :=
endpointCellScoreOutcomeLaw_isProbabilityMeasure
H g Prel hMass hOverlap a r hq upper
exact Measure.isProbabilityMeasure_map (by
exact (measurable_fst.prodMk (clampOutcome_measurable.comp measurable_snd)).aemeasurable)
theorem boundedEndpointCellLaw_fst
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
r :
hq :
0 < armCellMass H g a r
upper :
conclusion
(boundedEndpointCellLaw H g Prel hMass hOverlap a r hq upper).fst
= armCellScoreLaw H hMass.1 g hMass.2.2.1 a r hq
Proof (Lean source)
lemma boundedEndpointCellLaw_fst {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) (upper : Bool) :
(boundedEndpointCellLaw H g Prel hMass hOverlap a r hq upper).fst =
armCellScoreLaw H hMass.1 g hMass.2.2.1 a r hq := by
unfold boundedEndpointCellLaw Measure.fst
rw [Measure.map_map measurable_fst (by
exact measurable_fst.prodMk (clampOutcome_measurable.comp measurable_snd))]
change (endpointCellScoreOutcomeLaw H g Prel hMass hOverlap a r hq upper).fst = _
exact endpointCellScoreOutcomeLaw_fst H g Prel hMass hOverlap a r hq upper
theorem boundedEndpointCellLaw_map_real_outcome
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
r :
hq :
0 < armCellMass H g a r
upper :
conclusion
(boundedEndpointCellLaw H g Prel hMass hOverlap a r hq upper).map (fun p => (p.2 : ℝ))
= armCellOutcomeLaw Prel hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq)
Proof (Lean source)
lemma boundedEndpointCellLaw_map_real_outcome {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) (upper : Bool) :
(boundedEndpointCellLaw H g Prel hMass hOverlap a r hq upper).map
(fun p => (p.2 : ℝ)) =
armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq) := by
let M := endpointCellScoreOutcomeLaw H g Prel hMass hOverlap a r hq upper
have hsupport : ∀ᵐ p ∂M, p.2 ∈ Icc (0 : ℝ) 1 := by
rw [ae_iff]
exact endpointCellScoreOutcomeLaw_outcome_support
H g Prel hMass hOverlap a r hq upper
unfold boundedEndpointCellLaw
change (M.map (fun p => (p.1, clampOutcome p.2))).map
(fun p => (p.2 : ℝ)) = _
have hb : Measurable
(fun p : ScoreSpace ε × ℝ => (p.1, clampOutcome p.2)) :=
measurable_fst.prodMk (clampOutcome_measurable.comp measurable_snd)
have hc : Measurable (fun p : ScoreSpace ε × OutcomeSpace => (p.2 : ℝ)) := by
fun_prop
rw [Measure.map_map hc hb]
calc
M.map ((fun p : ScoreSpace ε × OutcomeSpace => (p.2 : ℝ)) ∘
(fun p : ScoreSpace ε × ℝ => (p.1, clampOutcome p.2))) =
M.map snd := by
apply Measure.map_congr
filter_upwards [hsupport] with p hp
exact clampOutcome_coe_of_mem hp
_ = _ := endpointCellScoreOutcomeLaw_snd
H g Prel hMass hOverlap a r hq upper
def endpointAssignedArmLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
upper :
def
given by
Measure.sum fun r : LabelSpace J
=> if hq : 0 < armCellMass H g a r then ofReal (armCellMass H g a r) • boundedEndpointCellLaw H g Prel hMass hOverlap a r hq upper else 0
theorem endpointAssignedCell_realOutcome
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
upper :
r :
conclusion
(if hq : 0 < armCellMass H g a r then ofReal (armCellMass H g a r) • boundedEndpointCellLaw H g Prel hMass hOverlap a r hq upper else 0).map (fun p => (p.2 : ℝ))
= (Prel.restrict {z | z.1 = r ∧ z.2.1 = a}).map (fun z => (z.2.2 : ℝ))
Proof (Lean source)
lemma endpointAssignedCell_realOutcome {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) (upper : Bool)
(r : LabelSpace J) :
(if hq : 0 < armCellMass H g a r then
ofReal (armCellMass H g a r) •
boundedEndpointCellLaw H g Prel hMass hOverlap a r hq upper
else 0).map (fun p => (p.2 : ℝ)) =
(Prel.restrict {z | z.1 = r ∧ z.2.1 = a}).map
(fun z => (z.2.2 : ℝ)) := by
let C : Set (Observation J) := {z | z.1 = r ∧ z.2.1 = a}
letI : IsProbabilityMeasure Prel := hMass.2.1
split
· rename_i hq
rw [Measure.map_smul,
boundedEndpointCellLaw_map_real_outcome
H g Prel hMass hOverlap a r hq upper]
unfold armCellOutcomeLaw
rw [smul_smul]
have hC : Prel C = ofReal (armCellMass H g a r) := by
calc
Prel C = ofReal (Prel.real C) :=
(ENNReal.ofReal_toReal (measure_ne_top Prel C)).symm
_ = ofReal (armCellMass H g a r) := by
rw [hMass.2.2.2 a r]
change (ofReal (armCellMass H g a r) * (Prel C)⁻¹) •
((Prel.restrict C).map (fun z => (z.2.2 : ℝ))) = _
rw [hC, ENNReal.mul_inv_cancel
(ne_of_gt (ENNReal.ofReal_pos.mpr hq)) ENNReal.ofReal_ne_top, one_smul]
· rename_i hq
rw [Measure.map_zero]
have hnonneg : 0 ≤ armCellMass H g a r := by
rw [← hMass.2.2.2 a r]
exact measureReal_nonneg
have hqzero : armCellMass H g a r = 0 :=
le_antisymm (le_of_not_gt hq) hnonneg
have hCreal : Prel.real C = 0 := by
rw [hMass.2.2.2 a r, hqzero]
have hCzero : Prel C = 0 := (measureReal_eq_zero_iff).mp hCreal
rw [Measure.restrict_zero_set hCzero, Measure.map_zero]
theorem endpointAssignedArmLaw_fst
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
upper :
conclusion
(endpointAssignedArmLaw H g Prel hMass hOverlap a upper).fst
Proof (Lean source)
lemma endpointAssignedArmLaw_fst {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) (upper : Bool) :
(endpointAssignedArmLaw H g Prel hMass hOverlap a upper).fst =
H.withDensity (fun e => ofReal (armProb a e)) := by
unfold endpointAssignedArmLaw Measure.fst
rw [Measure.map_sum (by fun_prop)]
simp_rw [endpointAssignedCell_fst H g Prel hMass hOverlap a upper]
let μ := H.withDensity (fun e => ofReal (armProb a e))
have hcell : ∀ r : LabelSpace J, MeasurableSet (cell g r) := by
intro r
exact measurableSet_eq_fun hMass.2.2.1 measurable_const
have hdisj : Pairwise (fun r s : LabelSpace J =>
Disjoint (cell g r) (cell g s)) := by
intro r s hrs
apply Set.disjoint_left.mpr
intro e her hes
exact hrs (her.symm.trans hes)
have hunion : (⋃ r : LabelSpace J, cell g r) = univ := by
ext e
simp [cell]
rw [← Measure.restrict_iUnion hdisj hcell, hunion, Measure.restrict_univ]
def endpointArmPotentialLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
upper :
def
given by
(endpointAssignedArmLaw H g Prel hMass hOverlap a upper).withDensity (fun p => (ofReal (armProb a p.1))⁻¹)
theorem endpointArmPotentialLaw_fst
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
upper :
conclusion
(endpointArmPotentialLaw H g Prel hMass hOverlap a upper).fst = H
Proof (Lean source)
lemma endpointArmPotentialLaw_fst {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) (upper : Bool) :
(endpointArmPotentialLaw H g Prel hMass hOverlap a upper).fst = H := by
let f : ScoreSpace ε → ENNReal :=
fun e => ofReal (armProb a e)
have hf : Measurable f := by
cases a <;> simp [f, armProb] <;> fun_prop
have hf0 : ∀ e : ScoreSpace ε, f e ≠ 0 := by
intro e
apply ne_of_gt
rw [ENNReal.ofReal_pos]
rcases e.property with ⟨he0, he1⟩
cases a <;> simp [f, armProb] <;> linarith [hOverlap.1, hOverlap.2]
have hft : ∀ e : ScoreSpace ε, f e ≠ ∞ := fun _ => ENNReal.ofReal_ne_top
unfold endpointArmPotentialLaw Measure.fst
change ((endpointAssignedArmLaw H g Prel hMass hOverlap a upper).withDensity
(fun p => (f p.1)⁻¹)).map fst = H
rw [map_withDensity_of_fst _ (fun e => (f e)⁻¹) hf.fun_inv]
change ((endpointAssignedArmLaw H g Prel hMass hOverlap a upper).fst).withDensity
(fun e => (f e)⁻¹) = H
rw [endpointAssignedArmLaw_fst H g Prel hMass hOverlap a upper]
change (H.withDensity f).withDensity (fun e => (f e)⁻¹) = H
exact withDensity_inv_same hf
(Filter.Eventually.of_forall hf0) (Filter.Eventually.of_forall hft)
instance endpointArmPotentialLaw_isProbabilityMeasure
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
instance endpointArmPotentialLaw_isProbabilityMeasure {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) (upper : Bool) :
IsProbabilityMeasure
(endpointArmPotentialLaw H g Prel hMass hOverlap a upper) := by
rw [isProbabilityMeasure_iff]
have hfst := endpointArmPotentialLaw_fst H g Prel hMass hOverlap a upper
calc
endpointArmPotentialLaw H g Prel hMass hOverlap a upper univ =
(endpointArmPotentialLaw H g Prel hMass hOverlap a upper).fst univ := by
unfold Measure.fst
rw [Measure.map_apply measurable_fst MeasurableSet.univ]
simp
_ = H univ := congrArg (fun μ : Measure (ScoreSpace ε) => μ univ) hfst
_ = 1 := isProbabilityMeasure_iff.mp hMass.1
theorem endpointArmPotentialLaw_retilt
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
upper :
conclusion
(endpointArmPotentialLaw H g Prel hMass hOverlap a upper).withDensity (fun p => ofReal (armProb a p.1))
= endpointAssignedArmLaw H g Prel hMass hOverlap a upper
Proof (Lean source)
lemma endpointArmPotentialLaw_retilt {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) (upper : Bool) :
(endpointArmPotentialLaw H g Prel hMass hOverlap a upper).withDensity
(fun p => ofReal (armProb a p.1)) =
endpointAssignedArmLaw H g Prel hMass hOverlap a upper := by
let f : ScoreSpace ε × OutcomeSpace → ENNReal :=
fun p => ofReal (armProb a p.1)
have hf : Measurable f := by
cases a <;> simp [f, armProb] <;> fun_prop
have hf0 : ∀ p : ScoreSpace ε × OutcomeSpace, f p ≠ 0 := by
intro p
apply ne_of_gt
rw [ENNReal.ofReal_pos]
rcases p.1.property with ⟨he0, he1⟩
cases a <;> simp [f, armProb] <;> linarith [hOverlap.1, hOverlap.2]
unfold endpointArmPotentialLaw
change ((endpointAssignedArmLaw H g Prel hMass hOverlap a upper).withDensity
(fun p => (f p)⁻¹)).withDensity f = _
rw [← withDensity_mul _ hf.fun_inv hf]
have hfun : (fun p => (f p)⁻¹) * f = 1 := by
funext p
exact ENNReal.inv_mul_cancel (hf0 p) ENNReal.ofReal_ne_top
rw [hfun, withDensity_one]
def endpointLatentLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
upper0 upper1 :
def
endpointLatentLaw H g Prel hMass hOverlap upper0 upper1 :
given by
glueAlongFst (endpointArmPotentialLaw H g Prel hMass hOverlap false upper0) (endpointArmPotentialLaw H g Prel hMass hOverlap true upper1)
instance endpointLatentLaw_isProbabilityMeasure
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
instance endpointLatentLaw_isProbabilityMeasure {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (upper0 upper1 : Bool) :
IsProbabilityMeasure
(endpointLatentLaw H g Prel hMass hOverlap upper0 upper1) := by
unfold endpointLatentLaw
letI : IsProbabilityMeasure
(endpointArmPotentialLaw H g Prel hMass hOverlap false upper0) :=
endpointArmPotentialLaw_isProbabilityMeasure
H g Prel hMass hOverlap false upper0
letI : IsProbabilityMeasure
(endpointArmPotentialLaw H g Prel hMass hOverlap true upper1) :=
endpointArmPotentialLaw_isProbabilityMeasure
H g Prel hMass hOverlap true upper1
exact glueAlongFst_isProbabilityMeasure _ _
theorem endpointLatentLaw_map_control
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
upper0 upper1 :
conclusion
(endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).map (fun p => (p.1, p.2.1))
= endpointArmPotentialLaw H g Prel hMass hOverlap false upper0
Proof (Lean source)
lemma endpointLatentLaw_map_control {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (upper0 upper1 : Bool) :
(endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).map
(fun p => (p.1, p.2.1)) =
endpointArmPotentialLaw H g Prel hMass hOverlap false upper0 := by
unfold endpointLatentLaw
exact glueAlongFst_map_left _ _
theorem endpointLatentLaw_map_treated
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
upper0 upper1 :
conclusion
(endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).map (fun p => (p.1, p.2.2))
= endpointArmPotentialLaw H g Prel hMass hOverlap true upper1
Proof (Lean source)
lemma endpointLatentLaw_map_treated {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (upper0 upper1 : Bool) :
(endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).map
(fun p => (p.1, p.2.2)) =
endpointArmPotentialLaw H g Prel hMass hOverlap true upper1 := by
unfold endpointLatentLaw
apply glueAlongFst_map_right
rw [endpointArmPotentialLaw_fst, endpointArmPotentialLaw_fst]
theorem endpointLatentLaw_fst
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
upper0 upper1 :
conclusion
(endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).fst = H
Proof (Lean source)
lemma endpointLatentLaw_fst {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (upper0 upper1 : Bool) :
(endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).fst = H := by
unfold Measure.fst
calc
(endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).map fst =
((endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).map
(fun p => (p.1, p.2.1))).map fst := by
rw [Measure.map_map measurable_fst (by fun_prop)]
rfl
_ = (endpointArmPotentialLaw H g Prel hMass hOverlap false upper0).map
fst := by
rw [endpointLatentLaw_map_control]
_ = H := by
change (endpointArmPotentialLaw H g Prel hMass hOverlap false upper0).fst = H
exact endpointArmPotentialLaw_fst H g Prel hMass hOverlap false upper0
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.FixedLawEndpointConstruction 10 declarations Reconstruction leaves for fixed-law endpoint attainment.
Reconstruction leaves for fixed-law endpoint attainment.
def liftTransformCoupling
For the specified mathematical inputs, this definition introduces the corresponding object.
instance liftTransformCoupling_isProbabilityMeasure
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
instance liftTransformCoupling_isProbabilityMeasure {α β γ : Type*}
[MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ]
[StandardBorelSpace α] [Nonempty α]
[StandardBorelSpace β] [Nonempty β]
[CountablyGenerated γ]
(G : Measure α) (w : α → γ) (hw : Measurable w)
(π : Measure (β × γ)) [IsProbabilityMeasure G]
[IsProbabilityMeasure π] :
IsProbabilityMeasure (liftTransformCoupling G w hw π) := by
unfold liftTransformCoupling
letI : IsProbabilityMeasure (π.map swap) :=
Measure.isProbabilityMeasure_map (by fun_prop)
letI : IsProbabilityMeasure (G.map (fun x => (w x, x))) :=
Measure.isProbabilityMeasure_map (by fun_prop)
letI : IsProbabilityMeasure
(glueAlongFst (π.map swap) (G.map (fun x => (w x, x)))) :=
glueAlongFst_isProbabilityMeasure _ _
exact Measure.isProbabilityMeasure_map (by fun_prop)
theorem liftTransformCoupling_fst
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
α β γ :
Type*
Nonempty α
Nonempty β
G :
Measure α
w :
α → γ
hw :
π :
Measure (β × γ)
htransform :
π.map snd = G.map w
conclusion
(liftTransformCoupling G w hw π).fst = G
Proof (Lean source)
lemma liftTransformCoupling_fst {α β γ : Type*}
[MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ]
[StandardBorelSpace α] [Nonempty α]
[StandardBorelSpace β] [Nonempty β]
[CountablyGenerated γ]
(G : Measure α) (w : α → γ) (hw : Measurable w)
(π : Measure (β × γ)) [IsProbabilityMeasure G]
[IsProbabilityMeasure π]
(htransform : π.map snd = G.map w) :
(liftTransformCoupling G w hw π).fst = G := by
let ρ0 : Measure (γ × β) := π.map Prod.swap
let ρ1 : Measure (γ × α) := G.map (fun x => (w x, x))
have hfst : ρ0.fst = ρ1.fst := by
dsimp [ρ0, ρ1]
rw [reversedCoupling_fst, transformGraph_fst G w hw, htransform]
unfold liftTransformCoupling Measure.fst
calc
(Measure.map (fun p => (p.2.2, p.2.1))
(glueAlongFst ρ0 ρ1)).map fst =
((glueAlongFst ρ0 ρ1).map
(fun p => (p.1, p.2.2))).map snd := by
rw [Measure.map_map measurable_fst (by fun_prop),
Measure.map_map measurable_snd (by fun_prop)]
rfl
_ = ρ1.map snd := by rw [glueAlongFst_map_right ρ0 ρ1 hfst]
_ = G := by
dsimp [ρ1]
rw [Measure.map_map measurable_snd (by fun_prop)]
change Measure.map id G = G
exact Measure.map_id
theorem liftTransformCoupling_map_outcome_transform
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
α β γ :
Type*
Nonempty α
Nonempty β
G :
Measure α
w :
α → γ
hw :
π :
Measure (β × γ)
htransform :
π.map snd = G.map w
conclusion
(liftTransformCoupling G w hw π).map (fun p => (p.2, w p.1)) = π
Proof (Lean source)
lemma liftTransformCoupling_map_outcome_transform {α β γ : Type*}
[MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ]
[StandardBorelSpace α] [Nonempty α]
[StandardBorelSpace β] [Nonempty β]
[CountablyGenerated γ] [MeasurableEq γ]
(G : Measure α) (w : α → γ) (hw : Measurable w)
(π : Measure (β × γ)) [IsProbabilityMeasure G]
[IsProbabilityMeasure π]
(htransform : π.map snd = G.map w) :
(liftTransformCoupling G w hw π).map (fun p => (p.2, w p.1)) = π := by
let ρ0 : Measure (γ × β) := π.map Prod.swap
let ρ1 : Measure (γ × α) := G.map (fun x => (w x, x))
let L : Measure (γ × (β × α)) := glueAlongFst ρ0 ρ1
have hfst : ρ0.fst = ρ1.fst := by
dsimp [ρ0, ρ1]
rw [reversedCoupling_fst, transformGraph_fst G w hw, htransform]
have hright : L.map (fun p => (p.1, p.2.2)) = ρ1 := by
exact glueAlongFst_map_right ρ0 ρ1 hfst
let S : Set (γ × α) := {q | q.1 = w q.2}
have hS : MeasurableSet S :=
measurableSet_eq_fun measurable_fst (hw.comp measurable_snd)
have hSc : MeasurableSet {q : γ × α | q ∉ S} := hS.compl
have hgraphMap : Measurable (fun x : α => (w x, x)) := by fun_prop
have hrightMap : Measurable
(fun p : γ × (β × α) => (p.1, p.2.2)) := by fun_prop
have hgraphρ1 : ∀ᵐ q ∂ρ1, q ∈ S := by
rw [ae_iff]
dsimp [ρ1]
rw [Measure.map_apply hgraphMap hSc]
have hpre : (fun x : α => (w x, x)) ⁻¹' {q | q ∉ S} = ∅ := by
ext x
simp [S]
rw [hpre, measure_empty]
have hgraphLS : ∀ᵐ p ∂L, (p.1, p.2.2) ∈ S := by
rw [← hright] at hgraphρ1
rw [ae_iff] at hgraphρ1 ⊢
rw [Measure.map_apply hrightMap hSc] at hgraphρ1
exact hgraphρ1
have hgraphL : ∀ᵐ p ∂L, p.1 = w p.2.2 := by
filter_upwards [hgraphLS] with p hp
exact hp
have hleft : L.map (fun p => (p.1, p.2.1)) = ρ0 :=
glueAlongFst_map_left ρ0 ρ1
unfold liftTransformCoupling
rw [Measure.map_map (by fun_prop) (by fun_prop)]
calc
L.map ((fun q : α × β => (q.2, w q.1)) ∘
(fun p : γ × (β × α) => (p.2.2, p.2.1))) =
L.map (fun p => (p.2.1, p.1)) := by
apply Measure.map_congr
filter_upwards [hgraphL] with p hp
simp only [Function.comp_apply]
rw [hp]
_ = (L.map (fun p => (p.1, p.2.1))).map Prod.swap := by
rw [Measure.map_map (by fun_prop) (by fun_prop)]
rfl
_ = ρ0.map Prod.swap := by rw [hleft]
_ = π := by
dsimp [ρ0]
rw [Measure.map_map (by fun_prop) (by fun_prop)]
change Measure.map id π = π
exact Measure.map_id
theorem armCellScoreLaw_isProbabilityMeasure
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
g :
ScoreSpace ε → LabelSpace J
hg :
hOverlap :
Overlap ε
a :
r :
hq :
0 < armCellMass H g a r
conclusion
Proof (Lean source)
lemma armCellScoreLaw_isProbabilityMeasure {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace J) (hg : Measurable g)
(hOverlap : Overlap ε) (a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) :
IsProbabilityMeasure (armCellScoreLaw H inferInstance g hg a r hq) := by
have hInt : Integrable (armProb a : ScoreSpace ε → ℝ) H := by
apply Integrable.of_bound (by
cases a
· change AEStronglyMeasurable (fun e : ScoreSpace ε => 1 - (e : ℝ)) H
fun_prop
· change AEStronglyMeasurable (fun e : ScoreSpace ε => (e : ℝ)) H
fun_prop) 1
filter_upwards [] with e
rcases e.property with ⟨he0, he1⟩
have hzero : 0 ≤ (e : ℝ) := le_trans hOverlap.1.le he0
have hone : (e : ℝ) ≤ 1 := by linarith [he1, hOverlap.1]
cases a
· change |1 - (e : ℝ)| ≤ 1
rw [abs_of_nonneg (by linarith)]
linarith
· change |(e : ℝ)| ≤ 1
rw [abs_of_nonneg hzero]
exact hone
letI : IsFiniteMeasure
(H.withDensity (fun e => ofReal (armProb a e))) :=
isFiniteMeasure_withDensity_ofReal hInt.hasFiniteIntegral
rw [isProbabilityMeasure_iff]
unfold armCellScoreLaw
rw [Measure.smul_apply, Measure.restrict_apply MeasurableSet.univ,
Set.univ_inter, smul_eq_mul]
have hfinite :
(H.withDensity (fun e => ofReal (armProb a e))) (cell g r) ≠ ∞ :=
measure_ne_top _ _
have hmeasure :
(H.withDensity (fun e => ofReal (armProb a e))) (cell g r) =
ofReal (armCellMass H g a r) := by
rw [← propensityTilt_real_cell H g hg hOverlap a r]
exact (ENNReal.ofReal_toReal hfinite).symm
rw [hmeasure, ENNReal.inv_mul_cancel]
· exact ne_of_gt (ENNReal.ofReal_pos.mpr hq)
· exact ENNReal.ofReal_ne_top
def endpointCellScoreOutcomeLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
r :
hq :
0 < armCellMass H g a r
upper :
def
given by
by letI : IsProbabilityMeasure H := hMass.1 letI : IsProbabilityMeasure Prel := hMass.2.1 letI : Nonempty (ScoreSpace ε) := ⟨⟨1 / 2, by constructor <;> linarith [hOverlap.1, hOverlap.2]⟩⟩ let Y := armCellOutcomeLaw Prel hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq) let G := armCellScoreLaw H hMass.1 g hMass.2.2.1 a r hq let W := armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq letI : IsProbabilityMeasure Y := armCellOutcomeLaw_isProbabilityMeasure Prel a r (by rw [hMass.2.2.2 a r]; exact hq) letI : IsProbabilityMeasure G := armCellScoreLaw_isProbabilityMeasure H g hMass.2.2.1 hOverlap a r hq letI : IsProbabilityMeasure W := by dsimp [W, armCellWeightLaw] exact Measure.isProbabilityMeasure_map (by cases a <;> simp [armProb] <;> fun_prop) let π : Measure (ℝ × ℝ) := if upper then Causalean.Stat.comonotoneCoupling Y W else Causalean.Stat.countermonotoneCoupling Y W letI : IsProbabilityMeasure π := by dsimp [π] split_ifs · exact (isCoupling_comonotoneCoupling Y W).isProbabilityMeasure · exact (isCoupling_countermonotoneCoupling Y W).isProbabilityMeasure exact liftTransformCoupling G (fun e => (armProb a e)⁻¹) (by cases a <;> simp [armProb] <;> fun_prop) π
instance endpointCellScoreOutcomeLaw_isProbabilityMeasure
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
instance endpointCellScoreOutcomeLaw_isProbabilityMeasure {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) (upper : Bool) :
IsProbabilityMeasure
(endpointCellScoreOutcomeLaw H g Prel hMass hOverlap a r hq upper) := by
letI : IsProbabilityMeasure H := hMass.1
letI : IsProbabilityMeasure Prel := hMass.2.1
letI : Nonempty (ScoreSpace ε) := ⟨⟨1 / 2, by
constructor <;> linarith [hOverlap.1, hOverlap.2]⟩⟩
let Y := armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq)
let G := armCellScoreLaw H hMass.1 g hMass.2.2.1 a r hq
let W := armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq
let π : Measure (ℝ × ℝ) :=
if upper then Causalean.Stat.comonotoneCoupling Y W
else Causalean.Stat.countermonotoneCoupling Y W
letI : IsProbabilityMeasure Y :=
armCellOutcomeLaw_isProbabilityMeasure Prel a r
(by rw [hMass.2.2.2 a r]; exact hq)
letI : IsProbabilityMeasure G :=
armCellScoreLaw_isProbabilityMeasure H g hMass.2.2.1 hOverlap a r hq
letI : IsProbabilityMeasure W := by
dsimp [W, armCellWeightLaw]
exact Measure.isProbabilityMeasure_map (by
cases a <;> simp [armProb] <;> fun_prop)
letI : IsProbabilityMeasure π := by
dsimp [π]
split_ifs
· exact (isCoupling_comonotoneCoupling Y W).isProbabilityMeasure
· exact (isCoupling_countermonotoneCoupling Y W).isProbabilityMeasure
have hw : Measurable (fun e : ScoreSpace ε => (armProb a e)⁻¹) := by
cases a <;> simp [armProb] <;> fun_prop
change IsProbabilityMeasure (liftTransformCoupling G
(fun e => (armProb a e)⁻¹) hw π)
infer_instance
theorem endpointCoupling_isCoupling
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
r :
hq :
0 < armCellMass H g a r
upper :
conclusion
Y :
armCellOutcomeLaw Prel hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq)
W :
armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq
Proof (Lean source)
lemma endpointCoupling_isCoupling {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) (upper : Bool) :
let Y := armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq)
let W := armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq
IsCoupling
(if upper then comonotoneCoupling Y W
else countermonotoneCoupling Y W) Y W := by
dsimp
letI : IsProbabilityMeasure H := hMass.1
letI : IsProbabilityMeasure Prel := hMass.2.1
letI : IsProbabilityMeasure
(armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq)) :=
armCellOutcomeLaw_isProbabilityMeasure Prel a r
(by rw [hMass.2.2.2 a r]; exact hq)
letI : IsProbabilityMeasure
(armCellScoreLaw H hMass.1 g hMass.2.2.1 a r hq) :=
armCellScoreLaw_isProbabilityMeasure H g hMass.2.2.1 hOverlap a r hq
letI : IsProbabilityMeasure
(armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) := by
unfold armCellWeightLaw
exact Measure.isProbabilityMeasure_map (by
cases a <;> simp [armProb] <;> fun_prop)
cases upper
· exact isCoupling_countermonotoneCoupling _ _
· exact isCoupling_comonotoneCoupling _ _
theorem endpointCellScoreOutcomeLaw_fst
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
r :
hq :
0 < armCellMass H g a r
upper :
conclusion
(endpointCellScoreOutcomeLaw H g Prel hMass hOverlap a r hq upper).fst
= armCellScoreLaw H hMass.1 g hMass.2.2.1 a r hq
Proof (Lean source)
lemma endpointCellScoreOutcomeLaw_fst {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) (upper : Bool) :
(endpointCellScoreOutcomeLaw H g Prel hMass hOverlap a r hq upper).fst =
armCellScoreLaw H hMass.1 g hMass.2.2.1 a r hq := by
letI : IsProbabilityMeasure H := hMass.1
letI : IsProbabilityMeasure Prel := hMass.2.1
letI : Nonempty (ScoreSpace ε) := ⟨⟨1 / 2, by
constructor <;> linarith [hOverlap.1, hOverlap.2]⟩⟩
let Y := armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq)
let G := armCellScoreLaw H hMass.1 g hMass.2.2.1 a r hq
let W := armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq
let π : Measure (ℝ × ℝ) :=
if upper then Causalean.Stat.comonotoneCoupling Y W
else Causalean.Stat.countermonotoneCoupling Y W
letI : IsProbabilityMeasure Y :=
armCellOutcomeLaw_isProbabilityMeasure Prel a r
(by rw [hMass.2.2.2 a r]; exact hq)
letI : IsProbabilityMeasure G :=
armCellScoreLaw_isProbabilityMeasure H g hMass.2.2.1 hOverlap a r hq
letI : IsProbabilityMeasure W := by
dsimp [W, armCellWeightLaw]
exact Measure.isProbabilityMeasure_map (by
cases a <;> simp [armProb] <;> fun_prop)
have hπ : IsCoupling π Y W := by
simpa [π, Y, W] using
endpointCoupling_isCoupling H g Prel hMass hOverlap a r hq upper
letI : IsProbabilityMeasure π := hπ.isProbabilityMeasure
have hw : Measurable (fun e : ScoreSpace ε => (armProb a e)⁻¹) := by
cases a <;> simp [armProb] <;> fun_prop
have htransform : π.map snd = G.map
(fun e : ScoreSpace ε => (armProb a e)⁻¹) := by
rw [hπ.map_snd]
rfl
change (liftTransformCoupling G
(fun e : ScoreSpace ε => (armProb a e)⁻¹) hw π).fst = G
exact liftTransformCoupling_fst G _ hw π htransform
theorem endpointCellScoreOutcomeLaw_map_outcome_weight
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
r :
hq :
0 < armCellMass H g a r
upper :
conclusion
(endpointCellScoreOutcomeLaw H g Prel hMass hOverlap a r hq upper).map (fun p => (p.2, (armProb a p.1)⁻¹))
= (if upper then comonotoneCoupling (armCellOutcomeLaw Prel hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq)) (armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) else countermonotoneCoupling (armCellOutcomeLaw Prel hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq)) (armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq))
Proof (Lean source)
lemma endpointCellScoreOutcomeLaw_map_outcome_weight {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) (upper : Bool) :
(endpointCellScoreOutcomeLaw H g Prel hMass hOverlap a r hq upper).map
(fun p => (p.2, (armProb a p.1)⁻¹)) =
(if upper then
Causalean.Stat.comonotoneCoupling
(armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq))
(armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq)
else
Causalean.Stat.countermonotoneCoupling
(armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq))
(armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq)) := by
letI : IsProbabilityMeasure H := hMass.1
letI : IsProbabilityMeasure Prel := hMass.2.1
letI : Nonempty (ScoreSpace ε) := ⟨⟨1 / 2, by
constructor <;> linarith [hOverlap.1, hOverlap.2]⟩⟩
let Y := armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq)
let G := armCellScoreLaw H hMass.1 g hMass.2.2.1 a r hq
let W := armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq
let π : Measure (ℝ × ℝ) :=
if upper then Causalean.Stat.comonotoneCoupling Y W
else Causalean.Stat.countermonotoneCoupling Y W
letI : IsProbabilityMeasure Y :=
armCellOutcomeLaw_isProbabilityMeasure Prel a r
(by rw [hMass.2.2.2 a r]; exact hq)
letI : IsProbabilityMeasure G :=
armCellScoreLaw_isProbabilityMeasure H g hMass.2.2.1 hOverlap a r hq
letI : IsProbabilityMeasure W := by
dsimp [W, armCellWeightLaw]
exact Measure.isProbabilityMeasure_map (by
cases a <;> simp [armProb] <;> fun_prop)
have hπ : IsCoupling π Y W := by
simpa [π, Y, W] using
endpointCoupling_isCoupling H g Prel hMass hOverlap a r hq upper
letI : IsProbabilityMeasure π := hπ.isProbabilityMeasure
have hw : Measurable (fun e : ScoreSpace ε => (armProb a e)⁻¹) := by
cases a <;> simp [armProb] <;> fun_prop
have htransform : π.map snd = G.map
(fun e : ScoreSpace ε => (armProb a e)⁻¹) := by
rw [hπ.map_snd]
rfl
change (liftTransformCoupling G
(fun e : ScoreSpace ε => (armProb a e)⁻¹) hw π).map
(fun p => (p.2, (armProb a p.1)⁻¹)) = π
exact liftTransformCoupling_map_outcome_transform G _ hw π htransform
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.FixedLawEndpointFullRow 19 declarations Full-row realization of the fixed-law endpoint constructions.
Full-row realization of the fixed-law endpoint constructions.
def endpointFullLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
upper0 upper1 :
given by
((endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).withDensity (reconstructedAssignmentDensity false)).map (reconstructedFullRow g false)
+ ((endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).withDensity (reconstructedAssignmentDensity true)).map (reconstructedFullRow g true)
theorem endpointAssignmentMeasures_sum
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
upper0 upper1 :
conclusion
(endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).withDensity (reconstructedAssignmentDensity false)
+ (endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).withDensity (reconstructedAssignmentDensity true)
= endpointLatentLaw H g Prel hMass hOverlap upper0 upper1
Proof (Lean source)
lemma endpointAssignmentMeasures_sum {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (upper0 upper1 : Bool) :
(endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).withDensity
(reconstructedAssignmentDensity false) +
(endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).withDensity
(reconstructedAssignmentDensity true) =
endpointLatentLaw H g Prel hMass hOverlap upper0 upper1 := by
rw [← withDensity_add_left (endpointAssignmentDensity_measurable false)]
have hfun : reconstructedAssignmentDensity false +
reconstructedAssignmentDensity true =
(1 : ScoreSpace ε × (OutcomeSpace × OutcomeSpace) → ENNReal) := by
funext p
exact endpointAssignmentDensity_sum hOverlap p
rw [hfun, withDensity_one]
theorem endpointAssignedPotentialLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
upper0 upper1 :
a :
conclusion
((endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).withDensity (reconstructedAssignmentDensity a)).map (fun p => (p.1, if a then p.2.2 else p.2.1))
= endpointAssignedArmLaw H g Prel hMass hOverlap a (if a then upper1 else upper0)
Proof (Lean source)
lemma endpointAssignedPotentialLaw {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (upper0 upper1 : Bool) (a : ArmSpace) :
((endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).withDensity
(reconstructedAssignmentDensity a)).map
(fun p => (p.1, if a then p.2.2 else p.2.1)) =
endpointAssignedArmLaw H g Prel hMass hOverlap a
(if a then upper1 else upper0) := by
cases a
· simp only [Bool.false_eq_true, ↓reduceIte]
change ((endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).withDensity
((fun q : ScoreSpace ε × OutcomeSpace =>
ofReal (1 - (q.1 : ℝ))) ∘
(fun p => (p.1, p.2.1)))).map (fun p => (p.1, p.2.1)) = _
rw [endpoint_map_withDensity_of_comp _ _ (by fun_prop) _ (by fun_prop),
endpointLatentLaw_map_control H g Prel hMass hOverlap upper0 upper1]
simpa [armProb] using
endpointArmPotentialLaw_retilt H g Prel hMass hOverlap false upper0
· simp only [↓reduceIte]
change ((endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).withDensity
((fun q : ScoreSpace ε × OutcomeSpace =>
ofReal (q.1 : ℝ)) ∘
(fun p => (p.1, p.2.2)))).map (fun p => (p.1, p.2.2)) = _
rw [endpoint_map_withDensity_of_comp _ _ (by fun_prop) _ (by fun_prop),
endpointLatentLaw_map_treated H g Prel hMass hOverlap upper0 upper1]
simpa [armProb] using
endpointArmPotentialLaw_retilt H g Prel hMass hOverlap true upper1
theorem endpointFullLaw_isProbabilityMeasure
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
upper0 upper1 :
conclusion
IsProbabilityMeasure (endpointFullLaw H g Prel hMass hOverlap upper0 upper1)
Proof (Lean source)
lemma endpointFullLaw_isProbabilityMeasure {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (upper0 upper1 : Bool) :
IsProbabilityMeasure
(endpointFullLaw H g Prel hMass hOverlap upper0 upper1) := by
rw [isProbabilityMeasure_iff]
unfold endpointFullLaw
rw [Measure.add_apply,
Measure.map_apply (reconstructedFullRow_measurable g hMass.2.2.1 false)
MeasurableSet.univ,
Measure.map_apply (reconstructedFullRow_measurable g hMass.2.2.1 true)
MeasurableSet.univ]
simp only [preimage_univ]
rw [← Measure.add_apply,
endpointAssignmentMeasures_sum H g Prel hMass hOverlap upper0 upper1]
exact isProbabilityMeasure_iff.mp inferInstance
theorem endpointFullLaw_scoreMarginal
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
upper0 upper1 :
conclusion
ScoreMarginal (endpointFullLaw H g Prel hMass hOverlap upper0 upper1) H
Proof (Lean source)
lemma endpointFullLaw_scoreMarginal {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (upper0 upper1 : Bool) :
ScoreMarginal (endpointFullLaw H g Prel hMass hOverlap upper0 upper1) H := by
unfold ScoreMarginal endpointFullLaw
rw [Measure.map_add _ _ (by unfold score; fun_prop),
Measure.map_map (by unfold score; fun_prop)
(reconstructedFullRow_measurable g hMass.2.2.1 false),
Measure.map_map (by unfold score; fun_prop)
(reconstructedFullRow_measurable g hMass.2.2.1 true)]
change Measure.map fst
((endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).withDensity
(reconstructedAssignmentDensity false)) +
Measure.map fst
((endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).withDensity
(reconstructedAssignmentDensity true)) = H
rw [← Measure.map_add _ _ measurable_fst,
endpointAssignmentMeasures_sum H g Prel hMass hOverlap upper0 upper1]
exact endpointLatentLaw_fst H g Prel hMass hOverlap upper0 upper1
theorem endpointFullLaw_consistency
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
upper0 upper1 :
conclusion
Consistency (endpointFullLaw H g Prel hMass hOverlap upper0 upper1)
Proof (Lean source)
lemma endpointFullLaw_consistency {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (upper0 upper1 : Bool) :
Consistency (endpointFullLaw H g Prel hMass hOverlap upper0 upper1) := by
unfold Consistency endpointFullLaw
rw [ae_add_measure_iff]
constructor
· rw [ae_map_iff
(reconstructedFullRow_measurable g hMass.2.2.1 false).aemeasurable (by
exact measurableSet_eq_fun (by unfold observed; fun_prop)
(by
have hcond : MeasurableSet {ω : FullRow ε J | arm ω = true} :=
measurableSet_eq_fun (by unfold arm; fun_prop) measurable_const
exact Measurable.ite hcond (by unfold outcome1; fun_prop)
(by unfold outcome0; fun_prop)))]
filter_upwards [] with p
rfl
· rw [ae_map_iff
(reconstructedFullRow_measurable g hMass.2.2.1 true).aemeasurable (by
exact measurableSet_eq_fun (by unfold observed; fun_prop)
(by
have hcond : MeasurableSet {ω : FullRow ε J | arm ω = true} :=
measurableSet_eq_fun (by unfold arm; fun_prop) measurable_const
exact Measurable.ite hcond (by unfold outcome1; fun_prop)
(by unfold outcome0; fun_prop)))]
filter_upwards [] with p
rfl
theorem endpointFullLaw_deterministicRelease
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
upper0 upper1 :
conclusion
DeterministicRelease g (endpointFullLaw H g Prel hMass hOverlap upper0 upper1)
Proof (Lean source)
lemma endpointFullLaw_deterministicRelease {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (upper0 upper1 : Bool) :
DeterministicRelease g
(endpointFullLaw H g Prel hMass hOverlap upper0 upper1) := by
unfold DeterministicRelease endpointFullLaw
rw [ae_add_measure_iff]
constructor
· rw [ae_map_iff
(reconstructedFullRow_measurable g hMass.2.2.1 false).aemeasurable (by
exact measurableSet_eq_fun (by unfold label; fun_prop)
(hMass.2.2.1.comp (by unfold score; fun_prop)))]
filter_upwards [] with p
rfl
· rw [ae_map_iff
(reconstructedFullRow_measurable g hMass.2.2.1 true).aemeasurable (by
exact measurableSet_eq_fun (by unfold label; fun_prop)
(hMass.2.2.1.comp (by unfold score; fun_prop)))]
filter_upwards [] with p
rfl
theorem endpointFullLaw_randomizedAssignment
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
upper0 upper1 :
conclusion
RandomizedAssignment (endpointFullLaw H g Prel hMass hOverlap upper0 upper1)
Proof (Lean source)
lemma endpointFullLaw_randomizedAssignment {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (upper0 upper1 : Bool) :
RandomizedAssignment (endpointFullLaw H g Prel hMass hOverlap upper0 upper1) := by
intro B hB
let L := endpointLatentLaw H g Prel hMass hOverlap upper0 upper1
let M₀ := L.withDensity (reconstructedAssignmentDensity false)
let M₁ := L.withDensity (reconstructedAssignmentDensity true)
let D : Set (ScoreSpace ε × (OutcomeSpace × OutcomeSpace)) :=
{p | (p.1, p.2.1, p.2.2) ∈ B}
have hD : MeasurableSet D := hB.preimage (by fun_prop)
let S : Set (FullRow ε J) :=
{ω | arm ω = true ∧ (score ω, outcome0 ω, outcome1 ω) ∈ B}
let T : Set (FullRow ε J) :=
{ω | (score ω, outcome0 ω, outcome1 ω) ∈ B}
have hS : MeasurableSet S :=
(measurableSet_eq_fun (by unfold arm; fun_prop) measurable_const).inter
(hB.preimage (by unfold score outcome0 outcome1; fun_prop))
have hT : MeasurableSet T :=
hB.preimage (by unfold score outcome0 outcome1; fun_prop)
have hpre0S : reconstructedFullRow g false ⁻¹' S = ∅ := by
ext p
simp [S, reconstructedFullRow, arm]
have hpre1S : reconstructedFullRow g true ⁻¹' S = D := by
ext p
simp [S, D, reconstructedFullRow, arm, score, outcome0, outcome1]
have hpre0T : reconstructedFullRow g false ⁻¹' T = D := by
ext p
simp [T, D, reconstructedFullRow, score, outcome0, outcome1]
have hpre1T : reconstructedFullRow g true ⁻¹' T = D := by
ext p
simp [T, D, reconstructedFullRow, score, outcome0, outcome1]
letI : IsFiniteMeasure M₀ :=
endpointComponent_finite H g Prel hMass hOverlap upper0 upper1 false
letI : IsFiniteMeasure M₁ :=
endpointComponent_finite H g Prel hMass hOverlap upper0 upper1 true
have hmap0 : IsFiniteMeasure (M₀.map (reconstructedFullRow g false)) := ⟨by
rw [Measure.map_apply (reconstructedFullRow_measurable g hMass.2.2.1 false)
MeasurableSet.univ]
simpa using measure_lt_top M₀ Set.univ
⟩
have hmap1 : IsFiniteMeasure (M₁.map (reconstructedFullRow g true)) := ⟨by
rw [Measure.map_apply (reconstructedFullRow_measurable g hMass.2.2.1 true)
MeasurableSet.univ]
simpa using measure_lt_top M₁ Set.univ
⟩
change (endpointFullLaw H g Prel hMass hOverlap upper0 upper1).real S =
∫ ω in T, (score ω : ℝ) ∂endpointFullLaw H g Prel hMass hOverlap upper0 upper1
have hleft : (endpointFullLaw H g Prel hMass hOverlap upper0 upper1).real S =
∫ p in D, (p.1 : ℝ) ∂L := by
unfold endpointFullLaw
rw [Measure.real_def, Measure.add_apply,
Measure.map_apply (reconstructedFullRow_measurable g hMass.2.2.1 false) hS,
Measure.map_apply (reconstructedFullRow_measurable g hMass.2.2.1 true) hS,
hpre0S, hpre1S, measure_empty, zero_add]
change M₁.real D = _
exact endpointAssignmentTrue_real H g Prel hMass hOverlap upper0 upper1 D hD
rw [hleft]
unfold endpointFullLaw
letI : IsFiniteMeasure (M₀.map (reconstructedFullRow g false)) := hmap0
letI : IsFiniteMeasure (M₁.map (reconstructedFullRow g true)) := hmap1
rw [Measure.restrict_add,
integral_add_measure (endpointFullScore_integrable_finite _).integrableOn
(endpointFullScore_integrable_finite _).integrableOn,
setIntegral_map hT (by unfold score; fun_prop)
(reconstructedFullRow_measurable g hMass.2.2.1 false).aemeasurable,
setIntegral_map hT (by unfold score; fun_prop)
(reconstructedFullRow_measurable g hMass.2.2.1 true).aemeasurable,
hpre0T, hpre1T]
change (∫ p in D, (p.1 : ℝ) ∂L) =
(∫ p in D, (p.1 : ℝ) ∂M₀) + (∫ p in D, (p.1 : ℝ) ∂M₁)
rw [← integral_add_measure
(endpointLatentScore_integrable_finite M₀).integrableOn
(endpointLatentScore_integrable_finite M₁).integrableOn]
rw [← Measure.restrict_add,
endpointAssignmentMeasures_sum H g Prel hMass hOverlap upper0 upper1]
def endpointObservationReal
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
J :
ℕ
z :
def
given by
(z.1, z.2.1, (z.2.2 : ℝ))
theorem endpointAssignedArmLaw_releaseReal
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
upper :
conclusion
(endpointAssignedArmLaw H g Prel hMass hOverlap a upper).map (fun p => (g p.1, a, (p.2 : ℝ)))
= (Prel.restrict {z | z.2.1 = a}).map endpointObservationReal
Proof (Lean source)
lemma endpointAssignedArmLaw_releaseReal {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) (upper : Bool) :
(endpointAssignedArmLaw H g Prel hMass hOverlap a upper).map
(fun p => (g p.1, a, (p.2 : ℝ))) =
(Prel.restrict {z | z.2.1 = a}).map endpointObservationReal := by
let C : LabelSpace J → Set (Observation J) :=
fun r => {z | z.1 = r ∧ z.2.1 = a}
let A : Set (Observation J) := {z | z.2.1 = a}
have hC : ∀ r, MeasurableSet (C r) := by
intro r
exact (measurableSet_eq_fun (by fun_prop) measurable_const).inter
(measurableSet_eq_fun (by fun_prop) measurable_const)
have hdisj : Pairwise (fun r s => Disjoint (C r) (C s)) := by
intro r s hrs
apply Set.disjoint_left.mpr
intro z hzr hzs
exact hrs (hzr.1.symm.trans hzs.1)
have hunion : (⋃ r, C r) = A := by
ext z
simp [C, A]
have hpartition : Measure.sum (fun r => Prel.restrict (C r)) =
Prel.restrict A := by
rw [← Measure.restrict_iUnion hdisj hC, hunion]
let f : ScoreSpace ε × OutcomeSpace → LabelSpace J × (ArmSpace × ℝ) :=
fun p => (g p.1, a, (p.2 : ℝ))
have hf : Measurable f := by
have hg : Measurable g := hMass.2.2.1
dsimp only [f]
fun_prop
unfold endpointAssignedArmLaw
change (Measure.sum fun r : LabelSpace J =>
if hq : 0 < armCellMass H g a r then
ofReal (armCellMass H g a r) •
boundedEndpointCellLaw H g Prel hMass hOverlap a r hq upper
else 0).map f = _
rw [Measure.map_sum hf.aemeasurable]
have hterms : (fun r : LabelSpace J =>
(if hq : 0 < armCellMass H g a r then
ofReal (armCellMass H g a r) •
boundedEndpointCellLaw H g Prel hMass hOverlap a r hq upper
else 0).map f) =
(fun r => (Prel.restrict (C r)).map endpointObservationReal) := by
funext r
exact endpointAssignedCell_releaseReal
H g Prel hMass hOverlap a upper r
rw [hterms]
change Measure.sum (fun r => (Prel.restrict (C r)).map endpointObservationReal) =
(Prel.restrict A).map endpointObservationReal
rw [← hpartition,
Measure.map_sum endpointObservationReal_measurable.aemeasurable]
theorem endpointFullLaw_releaseReal
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
upper0 upper1 :
conclusion
(endpointFullLaw H g Prel hMass hOverlap upper0 upper1).map (fun ω => endpointObservationReal (releasedRecord ω))
= Prel.map endpointObservationReal
Proof (Lean source)
lemma endpointFullLaw_releaseReal {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (upper0 upper1 : Bool) :
(endpointFullLaw H g Prel hMass hOverlap upper0 upper1).map
(fun ω => endpointObservationReal (releasedRecord ω)) =
Prel.map endpointObservationReal := by
have hrel : Measurable
(fun ω : FullRow ε J => endpointObservationReal (releasedRecord ω)) := by
exact endpointObservationReal_measurable.comp (by
unfold releasedRecord label arm observed
fun_prop)
unfold endpointFullLaw
rw [Measure.map_add _ _ hrel,
Measure.map_map hrel
(reconstructedFullRow_measurable g hMass.2.2.1 false),
Measure.map_map hrel
(reconstructedFullRow_measurable g hMass.2.2.1 true)]
change ((endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).withDensity
(reconstructedAssignmentDensity false)).map
(fun p => (g p.1, false, (p.2.1 : ℝ))) +
((endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).withDensity
(reconstructedAssignmentDensity true)).map
(fun p => (g p.1, true, (p.2.2 : ℝ))) = _
have h0 := congrArg
(Measure.map (fun q : ScoreSpace ε × OutcomeSpace =>
(g q.1, false, (q.2 : ℝ))))
(endpointAssignedPotentialLaw H g Prel hMass hOverlap upper0 upper1 false)
have hg : Measurable g := hMass.2.2.1
rw [Measure.map_map (by fun_prop)
(by simp only [Bool.false_eq_true, ↓reduceIte]; fun_prop)] at h0
simp only [Bool.false_eq_true, ↓reduceIte] at h0
change ((endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).withDensity
(reconstructedAssignmentDensity false)).map
(fun p => (g p.1, false, (p.2.1 : ℝ))) =
(endpointAssignedArmLaw H g Prel hMass hOverlap false upper0).map
(fun q => (g q.1, false, (q.2 : ℝ))) at h0
have h1 := congrArg
(Measure.map (fun q : ScoreSpace ε × OutcomeSpace =>
(g q.1, true, (q.2 : ℝ))))
(endpointAssignedPotentialLaw H g Prel hMass hOverlap upper0 upper1 true)
rw [Measure.map_map (by fun_prop)
(by simp only [↓reduceIte]; fun_prop)] at h1
simp only [↓reduceIte] at h1
change ((endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).withDensity
(reconstructedAssignmentDensity true)).map
(fun p => (g p.1, true, (p.2.2 : ℝ))) =
(endpointAssignedArmLaw H g Prel hMass hOverlap true upper1).map
(fun q => (g q.1, true, (q.2 : ℝ))) at h1
rw [h0, h1,
endpointAssignedArmLaw_releaseReal H g Prel hMass hOverlap false upper0,
endpointAssignedArmLaw_releaseReal H g Prel hMass hOverlap true upper1,
← Measure.map_add _ _ endpointObservationReal_measurable]
congr 1
let S : Set (Observation J) := {z | z.2.1 = true}
have hS : MeasurableSet S :=
measurableSet_eq_fun (by fun_prop) measurable_const
have hfalse : {z : Observation J | z.2.1 = false} = Sᶜ := by
ext z
simp [S]
rw [hfalse, Measure.restrict_compl_add_restrict hS]
theorem endpointFullLaw_releasedLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
upper0 upper1 :
conclusion
ReleasedLaw (endpointFullLaw H g Prel hMass hOverlap upper0 upper1) Prel
Proof (Lean source)
lemma endpointFullLaw_releasedLaw {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (upper0 upper1 : Bool) :
ReleasedLaw (endpointFullLaw H g Prel hMass hOverlap upper0 upper1) Prel := by
have hcoe : MeasurableEmbedding
(fun y : OutcomeSpace => (y : ℝ)) :=
MeasurableEmbedding.subtype_coe measurableSet_Icc
have hinner : MeasurableEmbedding
(fun p : ArmSpace × OutcomeSpace => (p.1, (p.2 : ℝ))) := by
change MeasurableEmbedding
(map id (fun y : OutcomeSpace => (y : ℝ)))
exact MeasurableEmbedding.id.prodMap hcoe
have hemb : MeasurableEmbedding
(endpointObservationReal (J := J)) := by
change MeasurableEmbedding
(map id (fun p : ArmSpace × OutcomeSpace => (p.1, (p.2 : ℝ))))
exact MeasurableEmbedding.id.prodMap hinner
unfold ReleasedLaw releasedLaw
apply hemb.map_injective
rw [Measure.map_map hemb.measurable (by
unfold releasedRecord label arm observed
fun_prop)]
exact endpointFullLaw_releaseReal H g Prel hMass hOverlap upper0 upper1
theorem endpointFullLaw_compatibleCausalLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
upper0 upper1 :
conclusion
CompatibleCausalLaw H g Prel (endpointFullLaw H g Prel hMass hOverlap upper0 upper1)
Proof (Lean source)
lemma endpointFullLaw_compatibleCausalLaw {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (upper0 upper1 : Bool) :
CompatibleCausalLaw H g Prel
(endpointFullLaw H g Prel hMass hOverlap upper0 upper1) := by
exact
{ probability := endpointFullLaw_isProbabilityMeasure
H g Prel hMass hOverlap upper0 upper1
measurableRelease := hMass.2.2.1
scoreMarginal := endpointFullLaw_scoreMarginal
H g Prel hMass hOverlap upper0 upper1
randomizedAssignment := endpointFullLaw_randomizedAssignment
H g Prel hMass hOverlap upper0 upper1
consistency := endpointFullLaw_consistency
H g Prel hMass hOverlap upper0 upper1
deterministicRelease := endpointFullLaw_deterministicRelease
H g Prel hMass hOverlap upper0 upper1
releasedLaw := endpointFullLaw_releasedLaw
H g Prel hMass hOverlap upper0 upper1 }
theorem endpointFullLaw_potentialLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
upper0 upper1 :
a :
conclusion
(endpointFullLaw H g Prel hMass hOverlap upper0 upper1).map (fun ω => (score ω, if a then outcome1 ω else outcome0 ω))
= endpointArmPotentialLaw H g Prel hMass hOverlap a (if a then upper1 else upper0)
Proof (Lean source)
lemma endpointFullLaw_potentialLaw {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (upper0 upper1 : Bool) (a : ArmSpace) :
(endpointFullLaw H g Prel hMass hOverlap upper0 upper1).map
(fun ω => (score ω, if a then outcome1 ω else outcome0 ω)) =
endpointArmPotentialLaw H g Prel hMass hOverlap a
(if a then upper1 else upper0) := by
have hp : Measurable
(fun ω : FullRow ε J =>
(score ω, if a then outcome1 ω else outcome0 ω)) := by
cases a
· simp only [Bool.false_eq_true, ↓reduceIte]
have hs : Measurable (score (ε := ε) (J := J)) := by
unfold score
fun_prop
have ho : Measurable (outcome0 (ε := ε) (J := J)) := by
unfold outcome0
fun_prop
exact hs.prodMk ho
· simp only [↓reduceIte]
have hs : Measurable (score (ε := ε) (J := J)) := by
unfold score
fun_prop
have ho : Measurable (outcome1 (ε := ε) (J := J)) := by
unfold outcome1
fun_prop
exact hs.prodMk ho
unfold endpointFullLaw
rw [Measure.map_add _ _ hp,
Measure.map_map hp
(reconstructedFullRow_measurable g hMass.2.2.1 false),
Measure.map_map hp
(reconstructedFullRow_measurable g hMass.2.2.1 true)]
change ((endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).withDensity
(reconstructedAssignmentDensity false)).map
(fun p => (p.1, if a then p.2.2 else p.2.1)) +
((endpointLatentLaw H g Prel hMass hOverlap upper0 upper1).withDensity
(reconstructedAssignmentDensity true)).map
(fun p => (p.1, if a then p.2.2 else p.2.1)) = _
rw [← Measure.map_add _ _ (by
cases a <;> simp only [Bool.false_eq_true, ↓reduceIte] <;> fun_prop),
endpointAssignmentMeasures_sum H g Prel hMass hOverlap upper0 upper1]
cases a
· simp only [Bool.false_eq_true, ↓reduceIte]
exact endpointLatentLaw_map_control
H g Prel hMass hOverlap upper0 upper1
· simp only [↓reduceIte]
exact endpointLatentLaw_map_treated
H g Prel hMass hOverlap upper0 upper1
theorem boundedEndpointCell_productIntegral
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
r :
hq :
0 < armCellMass H g a r
upper :
conclusion
(∫ p, (p.2 : ℝ) * (armProb a p.1)⁻¹ ∂boundedEndpointCellLaw H g Prel hMass hOverlap a r hq upper)
= ∫ q, q.1 * q.2 ∂(if upper then comonotoneCoupling (armCellOutcomeLaw Prel hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq)) (armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) else countermonotoneCoupling (armCellOutcomeLaw Prel hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq)) (armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq))
Proof (Lean source)
lemma boundedEndpointCell_productIntegral {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) (r : LabelSpace J)
(hq : 0 < armCellMass H g a r) (upper : Bool) :
(∫ p, (p.2 : ℝ) * (armProb a p.1)⁻¹
∂boundedEndpointCellLaw H g Prel hMass hOverlap a r hq upper) =
∫ q, q.1 * q.2 ∂(if upper then
comonotoneCoupling
(armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq))
(armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq)
else
countermonotoneCoupling
(armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq))
(armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq)) := by
let M := endpointCellScoreOutcomeLaw H g Prel hMass hOverlap a r hq upper
let b : ScoreSpace ε × ℝ → ScoreSpace ε × OutcomeSpace :=
fun p => (p.1, clampOutcome p.2)
let t : ScoreSpace ε × ℝ → ℝ × ℝ :=
fun p => (p.2, (armProb a p.1)⁻¹)
have hb : Measurable b := by
dsimp only [b]
exact measurable_fst.prodMk (clampOutcome_measurable.comp measurable_snd)
have ht : Measurable t := by
dsimp only [t]
cases a <;> simp [armProb] <;> fun_prop
have hboundedProduct : Measurable
(fun p : ScoreSpace ε × OutcomeSpace =>
(p.2 : ℝ) * (armProb a p.1)⁻¹) := by
have hy : Measurable
(fun p : ScoreSpace ε × OutcomeSpace => (p.2 : ℝ)) := by
fun_prop
have hw : Measurable (fun e : ScoreSpace ε => (armProb a e)⁻¹) := by
cases a <;> simp [armProb] <;> fun_prop
exact hy.mul (hw.comp measurable_fst)
have hsupport : ∀ᵐ p ∂M, p.2 ∈ Icc (0 : ℝ) 1 := by
rw [ae_iff]
exact endpointCellScoreOutcomeLaw_outcome_support
H g Prel hMass hOverlap a r hq upper
unfold boundedEndpointCellLaw
rw [integral_map hb.aemeasurable hboundedProduct.aestronglyMeasurable]
calc
(∫ p, ((clampOutcome p.2 : OutcomeSpace) : ℝ) *
(armProb a p.1)⁻¹ ∂M) =
∫ p, p.2 * (armProb a p.1)⁻¹ ∂M := by
apply integral_congr_ae
filter_upwards [hsupport] with p hp
rw [clampOutcome_coe_of_mem hp]
_ = ∫ q, q.1 * q.2 ∂M.map t := by
rw [integral_map ht.aemeasurable (by fun_prop)]
_ = _ := by
rw [endpointCellScoreOutcomeLaw_map_outcome_weight
H g Prel hMass hOverlap a r hq upper]
theorem endpointArmPotentialMean_eq_sum_cellProducts
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
upper :
conclusion
(∫ p, (p.2 : ℝ) ∂endpointArmPotentialLaw H g Prel hMass hOverlap a upper)
= ∑ r : LabelSpace J, if hq : 0 < armCellMass H g a r then armCellMass H g a r * ∫ q, q.1 * q.2 ∂(if upper then comonotoneCoupling (armCellOutcomeLaw Prel hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq)) (armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) else countermonotoneCoupling (armCellOutcomeLaw Prel hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq)) (armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq)) else 0
Proof (Lean source)
lemma endpointArmPotentialMean_eq_sum_cellProducts {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) (upper : Bool) :
(∫ p, (p.2 : ℝ) ∂endpointArmPotentialLaw
H g Prel hMass hOverlap a upper) =
∑ r : LabelSpace J,
if hq : 0 < armCellMass H g a r then
armCellMass H g a r *
∫ q, q.1 * q.2 ∂(if upper then
comonotoneCoupling
(armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq))
(armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq)
else
countermonotoneCoupling
(armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq))
(armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq))
else 0 := by
let F : ScoreSpace ε × OutcomeSpace → ℝ :=
fun p => (p.2 : ℝ) * (armProb a p.1)⁻¹
have hF : Measurable F := by
dsimp only [F]
have hy : Measurable
(fun p : ScoreSpace ε × OutcomeSpace => (p.2 : ℝ)) := by fun_prop
have hw : Measurable (fun e : ScoreSpace ε => (armProb a e)⁻¹) := by
cases a <;> simp [armProb] <;> fun_prop
exact hy.mul (hw.comp measurable_fst)
have hcellInt : ∀ r : LabelSpace J, Integrable F
(if hq : 0 < armCellMass H g a r then
ofReal (armCellMass H g a r) •
boundedEndpointCellLaw H g Prel hMass hOverlap a r hq upper
else 0) := by
intro r
split
· rename_i hq
letI : IsFiniteMeasure
(ofReal (armCellMass H g a r) •
boundedEndpointCellLaw H g Prel hMass hOverlap a r hq upper) :=
⟨by
rw [Measure.smul_apply]
exact ENNReal.mul_lt_top ENNReal.ofReal_lt_top
(measure_lt_top _ _)⟩
apply Integrable.of_bound hF.aestronglyMeasurable (|ε⁻¹| + 1)
filter_upwards [] with p
have hp : ε ≤ armProb a p.1 := by
rcases p.1.property with ⟨hp0, hp1⟩
cases a <;> simp [armProb] <;> linarith
have hpa : 0 < armProb a p.1 := lt_of_lt_of_le hOverlap.1 hp
have hw0 : 0 ≤ (armProb a p.1)⁻¹ := (inv_nonneg.mpr hpa.le)
have hw1 : (armProb a p.1)⁻¹ ≤ ε⁻¹ :=
(inv_le_inv₀ hpa hOverlap.1).mpr hp
have hy := p.2.property
rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg hy.1 hw0)]
calc
(p.2 : ℝ) * (armProb a p.1)⁻¹ ≤
1 * (armProb a p.1)⁻¹ :=
mul_le_mul_of_nonneg_right hy.2 hw0
_ = (armProb a p.1)⁻¹ := one_mul _
_ ≤ ε⁻¹ := hw1
_ ≤ |ε⁻¹| := le_abs_self _
_ ≤ |ε⁻¹| + 1 := by linarith
· simp [F]
unfold endpointArmPotentialLaw
rw [integral_withDensity_eq_integral_toReal_smul
(by cases a <;> simp [armProb] <;> fun_prop)
(Filter.Eventually.of_forall fun p => by
rw [ENNReal.inv_lt_top]
apply ENNReal.ofReal_pos.mpr
rcases p.1.property with ⟨hp0, hp1⟩
cases a <;> simp [armProb] <;> linarith [hOverlap.1]) ]
have hconvert : (∫ p,
((ofReal (armProb a p.1))⁻¹).toReal • (p.2 : ℝ)
∂endpointAssignedArmLaw H g Prel hMass hOverlap a upper) =
∫ p, F p ∂endpointAssignedArmLaw H g Prel hMass hOverlap a upper := by
apply integral_congr_ae
filter_upwards [] with p
have hp : 0 < armProb a p.1 := by
rcases p.1.property with ⟨hp0, hp1⟩
cases a <;> simp [armProb] <;> linarith [hOverlap.1]
rw [ENNReal.toReal_inv, ENNReal.toReal_ofReal hp.le]
simp [F, mul_comm]
rw [hconvert]
unfold endpointAssignedArmLaw
rw [Measure.sum_fintype,
integral_finset_sum_measure (fun r _ => hcellInt r)]
apply Finset.sum_congr rfl
intro r _
by_cases hq : 0 < armCellMass H g a r
· simp only [hq, dif_pos]
rw [integral_smul_measure]
rw [ENNReal.toReal_ofReal hq.le]
change armCellMass H g a r *
(∫ p, (p.2 : ℝ) * (armProb a p.1)⁻¹
∂boundedEndpointCellLaw H g Prel hMass hOverlap a r hq upper) = _
rw [boundedEndpointCell_productIntegral
H g Prel hMass hOverlap a r hq upper]
· simp [hq]
theorem endpointArmPotentialMean_lower
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
Proof (Lean source)
lemma endpointArmPotentialMean_lower {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) :
(∫ p, (p.2 : ℝ) ∂endpointArmPotentialLaw
H g Prel hMass hOverlap a false) =
muLower H g Prel hMass a := by
rw [endpointArmPotentialMean_eq_sum_cellProducts
H g Prel hMass hOverlap a false]
unfold muLower
apply Finset.sum_congr rfl
intro r _
by_cases hq : 0 < armCellMass H g a r
· simp only [hq, dif_pos, Bool.false_eq_true, ↓reduceIte]
let Y := armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq)
let W := armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq
letI : IsProbabilityMeasure H := hMass.1
letI : IsProbabilityMeasure Prel := hMass.2.1
letI : IsProbabilityMeasure Y :=
armCellOutcomeLaw_isProbabilityMeasure Prel a r
(by rw [hMass.2.2.2 a r]; exact hq)
letI : IsProbabilityMeasure
(armCellScoreLaw H hMass.1 g hMass.2.2.1 a r hq) :=
armCellScoreLaw_isProbabilityMeasure H g hMass.2.2.1 hOverlap a r hq
letI : IsProbabilityMeasure W := by
dsimp [W, armCellWeightLaw]
exact Measure.isProbabilityMeasure_map (by
cases a <;> simp [armProb] <;> fun_prop)
congr 1
exact product_expectation_countermonotoneCoupling_interval Y W
· simp [hq]
theorem endpointArmPotentialMean_upper
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
Proof (Lean source)
lemma endpointArmPotentialMean_upper {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) :
(∫ p, (p.2 : ℝ) ∂endpointArmPotentialLaw
H g Prel hMass hOverlap a true) =
muUpper H g Prel hMass a := by
rw [endpointArmPotentialMean_eq_sum_cellProducts
H g Prel hMass hOverlap a true]
unfold muUpper
apply Finset.sum_congr rfl
intro r _
by_cases hq : 0 < armCellMass H g a r
· simp only [hq, dif_pos, ↓reduceIte]
let Y := armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq)
let W := armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq
letI : IsProbabilityMeasure H := hMass.1
letI : IsProbabilityMeasure Prel := hMass.2.1
letI : IsProbabilityMeasure Y :=
armCellOutcomeLaw_isProbabilityMeasure Prel a r
(by rw [hMass.2.2.2 a r]; exact hq)
letI : IsProbabilityMeasure
(armCellScoreLaw H hMass.1 g hMass.2.2.1 a r hq) :=
armCellScoreLaw_isProbabilityMeasure H g hMass.2.2.1 hOverlap a r hq
letI : IsProbabilityMeasure W := by
dsimp [W, armCellWeightLaw]
exact Measure.isProbabilityMeasure_map (by
cases a <;> simp [armProb] <;> fun_prop)
have hm := endpointArmCell_moments H g Prel hMass hOverlap a r hq
congr 1
exact product_expectation_comonotoneCoupling_interval Y W hm.1 hm.2
· simp [hq]
theorem endpointFullLaw_armMean
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
upper0 upper1 :
a :
conclusion
(∫ ω, ((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ) ∂endpointFullLaw H g Prel hMass hOverlap upper0 upper1)
Proof (Lean source)
lemma endpointFullLaw_armMean {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (upper0 upper1 : Bool) (a : ArmSpace) :
(∫ ω, ((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ)
∂endpointFullLaw H g Prel hMass hOverlap upper0 upper1) =
if (if a then upper1 else upper0) then muUpper H g Prel hMass a
else muLower H g Prel hMass a := by
have hmap := endpointFullLaw_potentialLaw
H g Prel hMass hOverlap upper0 upper1 a
have hint := congrArg
(fun μ : Measure (ScoreSpace ε × OutcomeSpace) =>
∫ p, (p.2 : ℝ) ∂μ) hmap
rw [integral_map (by
cases a
· simp only [Bool.false_eq_true, ↓reduceIte]
exact ((by unfold score; fun_prop : Measurable
(score (ε := ε) (J := J))).prodMk
(by unfold outcome0; fun_prop)).aemeasurable
· simp only [↓reduceIte]
exact ((by unfold score; fun_prop : Measurable
(score (ε := ε) (J := J))).prodMk
(by unfold outcome1; fun_prop)).aemeasurable)
(by fun_prop)] at hint
change (∫ ω, ((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ)
∂endpointFullLaw H g Prel hMass hOverlap upper0 upper1) =
∫ p, (p.2 : ℝ) ∂endpointArmPotentialLaw H g Prel hMass hOverlap a
(if a then upper1 else upper0) at hint
rw [hint]
generalize hu : (if a then upper1 else upper0) = u
cases u
· simp only [Bool.false_eq_true, ↓reduceIte]
simpa [hu] using endpointArmPotentialMean_lower H g Prel hMass hOverlap a
· simp only [↓reduceIte]
simpa [hu] using endpointArmPotentialMean_upper H g Prel hMass hOverlap a
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.FixedLawMembership 17 declarations One-sided membership lemmas for the fixed-released-law sharp ATE interval.
One-sided membership lemmas for the fixed-released-law sharp ATE interval.
This file isolates the algebraic passage from armwise potential-outcome mean bounds to membership of the ATE in the candidate sharp interval. It also records the quantile formulas needed when the arm-cell law has been exhibited as a coupling of the released outcome law and the inverse-score law.
theorem potentialOutcome_integrable
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Integrable (fun ω => ((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ)) P
Proof (Lean source)
lemma potentialOutcome_integrable {ε : ℝ} {J : ℕ}
(P : Measure (FullRow ε J)) [IsProbabilityMeasure P] (a : ArmSpace) :
Integrable
(fun ω => ((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ)) P := by
cases a with
| false =>
change Integrable (fun ω => ((outcome0 ω : OutcomeSpace) : ℝ)) P
apply Integrable.of_bound
(by unfold outcome0; fun_prop) 1
filter_upwards [] with ω
exact abs_le.mpr ⟨by linarith [((outcome0 ω).property).1],
((outcome0 ω).property).2⟩
| true =>
change Integrable (fun ω => ((outcome1 ω : OutcomeSpace) : ℝ)) P
apply Integrable.of_bound
(by unfold outcome1; fun_prop) 1
filter_upwards [] with ω
exact abs_le.mpr ⟨by linarith [((outcome1 ω).property).1],
((outcome1 ω).property).2⟩
theorem ate_eq_potentialOutcomeMeans_sub
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
ate P
= (∫ ω, ((outcome1 ω : OutcomeSpace) : ℝ) ∂P) - ∫ ω, ((outcome0 ω : OutcomeSpace) : ℝ) ∂P
Proof (Lean source)
lemma ate_eq_potentialOutcomeMeans_sub {ε : ℝ} {J : ℕ}
(P : Measure (FullRow ε J)) [IsProbabilityMeasure P] :
ate P =
(∫ ω, ((outcome1 ω : OutcomeSpace) : ℝ) ∂P) -
∫ ω, ((outcome0 ω : OutcomeSpace) : ℝ) ∂P := by
unfold ate
have h₁ : Integrable (fun ω => ((outcome1 ω : OutcomeSpace) : ℝ)) P := by
simpa using potentialOutcome_integrable P true
have h₀ : Integrable (fun ω => ((outcome0 ω : OutcomeSpace) : ℝ)) P := by
simpa using potentialOutcome_integrable P false
exact integral_sub h₁ h₀
theorem ate_mem_sharpATESet_of_armMean_bounds
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
P :
hLower :
∀ a : ArmSpace,
muLower H g Prel hMass a
≤ ∫ ω, ((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ) ∂P
hUpper :
∀ a : ArmSpace,
(∫ ω, ((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ) ∂P)
≤ muUpper H g Prel hMass a
conclusion
ate P ∈ sharpATESet H g Prel hMass
Proof (Lean source)
lemma ate_mem_sharpATESet_of_armMean_bounds {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(P : Measure (FullRow ε J)) [IsProbabilityMeasure P]
(hLower : ∀ a : ArmSpace,
muLower H g Prel hMass a ≤
∫ ω, ((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ) ∂P)
(hUpper : ∀ a : ArmSpace,
(∫ ω, ((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ) ∂P) ≤
muUpper H g Prel hMass a) :
ate P ∈ sharpATESet H g Prel hMass := by
rw [ate_eq_potentialOutcomeMeans_sub P]
unfold sharpATESet
constructor
· have h₁ := hLower true
have h₀ := hUpper false
simp at h₁ h₀
linarith
· have h₁ := hUpper true
have h₀ := hLower false
simp at h₁ h₀
linarith
def latentTriple
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
def assignedLatentLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
def propensityTiltedLatentLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
theorem randomizedAssignment_false_measureReal
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
P :
hRandom :
B :
hB :
conclusion
= ∫ x in B, (1 - (x.1 : ℝ)) ∂(P.map latentTriple)
Proof (Lean source)
lemma randomizedAssignment_false_measureReal
{ε : ℝ} {J : ℕ} (P : Measure (FullRow ε J)) [IsProbabilityMeasure P]
(hRandom : RandomizedAssignment P)
(B : Set (ScoreSpace ε × OutcomeSpace × OutcomeSpace))
(hB : MeasurableSet B) :
P.real {ω | arm ω = false ∧ latentTriple ω ∈ B} =
∫ x in B, (1 - (x.1 : ℝ)) ∂(P.map latentTriple) := by
have hX : Measurable (latentTriple (ε := ε) (J := J)) := by
unfold latentTriple score outcome0 outcome1
fun_prop
let C : Set (FullRow ε J) := latentTriple ⁻¹' B
let T : Set (FullRow ε J) := {ω | arm ω = true}
have hT : MeasurableSet T := by
dsimp [T]
exact measurableSet_eq_fun (by unfold arm; fun_prop) measurable_const
have hsplit := measureReal_inter_add_sdiff (μ := P) (s := C) (t := T) hT
have htreated : P.real (C ∩ T) =
∫ x in B, (x.1 : ℝ) ∂(P.map latentTriple) := by
rw [MeasureTheory.setIntegral_map hB (by fun_prop) hX.aemeasurable]
convert hRandom B hB using 1
· apply measureReal_congr
filter_upwards [] with ω
change (latentTriple ω ∈ B ∧ arm ω = true) =
(arm ω = true ∧ latentTriple ω ∈ B)
exact propext and_comm
· apply setIntegral_congr_set
filter_upwards [] with ω
rfl
have hC : P.real C = ∫ _x in B, (1 : ℝ) ∂(P.map latentTriple) := by
rw [MeasureTheory.setIntegral_const]
simp only [smul_eq_mul, mul_one]
change (P C).toReal = ((P.map latentTriple) B).toReal
rw [Measure.map_apply hX hB]
have hIntOne : IntegrableOn (fun _x : ScoreSpace ε × OutcomeSpace × OutcomeSpace =>
(1 : ℝ)) B (P.map latentTriple) := integrableOn_const
have hIntScore : IntegrableOn
(fun x : ScoreSpace ε × OutcomeSpace × OutcomeSpace => (x.1 : ℝ)) B
(P.map latentTriple) := by
apply Integrable.of_bound
((by fun_prop : AEStronglyMeasurable
(fun x : ScoreSpace ε × OutcomeSpace × OutcomeSpace => (x.1 : ℝ))
(P.map latentTriple)).restrict) (|ε| + |1 - ε| + 1)
filter_upwards [] with x
rcases x.1.property with ⟨hx₁, hx₂⟩
have h₁ := neg_le_abs ε
have h₂ := le_abs_self (1 - ε)
have h₃ := neg_le_abs (1 - ε)
have h₄ := le_abs_self ε
exact abs_le.mpr ⟨by linarith, by linarith⟩
rw [MeasureTheory.integral_sub hIntOne hIntScore]
rw [← hC, ← htreated]
have hfalse : C \ T = {ω | arm ω = false ∧ latentTriple ω ∈ B} := by
ext ω
cases hA : arm ω <;> simp [C, T, hA, and_comm]
rw [hfalse] at hsplit
linarith
theorem assignedLatentLaw_true_eq_propensityTiltedLatentLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma assignedLatentLaw_true_eq_propensityTiltedLatentLaw
{ε : ℝ} {J : ℕ} (P : Measure (FullRow ε J)) [IsProbabilityMeasure P]
(hOverlap : Overlap ε) (hRandom : RandomizedAssignment P) :
assignedLatentLaw P true = propensityTiltedLatentLaw P true := by
have hX : Measurable (latentTriple (ε := ε) (J := J)) := by
unfold latentTriple score outcome0 outcome1
fun_prop
ext B hB
have hInt : IntegrableOn
(fun x : ScoreSpace ε × OutcomeSpace × OutcomeSpace => (x.1 : ℝ)) B
(P.map latentTriple) := by
apply Integrable.of_bound
((by fun_prop : AEStronglyMeasurable
(fun x : ScoreSpace ε × OutcomeSpace × OutcomeSpace => (x.1 : ℝ))
(P.map latentTriple)).restrict) 1
filter_upwards [] with x
rcases x.1.property with ⟨hx₁, hx₂⟩
exact abs_le.mpr ⟨by linarith [hOverlap.1], by linarith [hOverlap.1]⟩
have hnonneg : 0 ≤ᵐ[(P.map latentTriple).restrict B]
fun x : ScoreSpace ε × OutcomeSpace × OutcomeSpace => (x.1 : ℝ) :=
ae_of_all _ fun x => le_trans hOverlap.1.le x.1.property.1
have hleft : assignedLatentLaw P true B ≠ ⊤ := by
rw [assignedLatentLaw, Measure.map_apply hX hB,
Measure.restrict_apply (hB.preimage hX)]
exact measure_ne_top P _
have hright : propensityTiltedLatentLaw P true B ≠ ⊤ := by
rw [propensityTiltedLatentLaw, withDensity_apply _ hB]
simp only [armProb, ↓reduceIte]
exact (MeasureTheory.ofReal_integral_eq_lintegral_ofReal hInt
hnonneg).symm.trans_ne (by simp)
rw [← ENNReal.toReal_eq_toReal_iff' hleft hright]
rw [assignedLatentLaw, Measure.map_apply hX hB,
Measure.restrict_apply (hB.preimage hX)]
change P.real (latentTriple ⁻¹' B ∩ {ω | arm ω = true}) = _
rw [propensityTiltedLatentLaw, withDensity_apply _ hB]
simp only [armProb, ↓reduceIte]
rw [← MeasureTheory.ofReal_integral_eq_lintegral_ofReal hInt
hnonneg, ENNReal.toReal_ofReal]
· rw [MeasureTheory.setIntegral_map hB (by fun_prop) hX.aemeasurable]
convert hRandom B hB using 1
· apply measureReal_congr
filter_upwards [] with ω
change (latentTriple ω ∈ B ∧ arm ω = true) =
(arm ω = true ∧ latentTriple ω ∈ B)
exact propext and_comm
· rfl
· exact integral_nonneg_of_ae hnonneg
theorem assignedLatentLaw_false_eq_propensityTiltedLatentLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma assignedLatentLaw_false_eq_propensityTiltedLatentLaw
{ε : ℝ} {J : ℕ} (P : Measure (FullRow ε J)) [IsProbabilityMeasure P]
(hOverlap : Overlap ε) (hRandom : RandomizedAssignment P) :
assignedLatentLaw P false = propensityTiltedLatentLaw P false := by
have hX : Measurable (latentTriple (ε := ε) (J := J)) := by
unfold latentTriple score outcome0 outcome1
fun_prop
ext B hB
have hInt : IntegrableOn
(fun x : ScoreSpace ε × OutcomeSpace × OutcomeSpace => 1 - (x.1 : ℝ)) B
(P.map latentTriple) := by
apply Integrable.of_bound
((by fun_prop : AEStronglyMeasurable
(fun x : ScoreSpace ε × OutcomeSpace × OutcomeSpace => 1 - (x.1 : ℝ))
(P.map latentTriple)).restrict) 1
filter_upwards [] with x
rcases x.1.property with ⟨hx₁, hx₂⟩
exact abs_le.mpr ⟨by linarith [hOverlap.1], by linarith [hOverlap.1]⟩
have hnonneg : 0 ≤ᵐ[(P.map latentTriple).restrict B]
fun x : ScoreSpace ε × OutcomeSpace × OutcomeSpace => 1 - (x.1 : ℝ) :=
ae_of_all _ fun x => show 0 ≤ 1 - (x.1 : ℝ) by
linarith [x.1.property.2, hOverlap.1]
have hleft : assignedLatentLaw P false B ≠ ⊤ := by
rw [assignedLatentLaw, Measure.map_apply hX hB,
Measure.restrict_apply (hB.preimage hX)]
exact measure_ne_top P _
have hright : propensityTiltedLatentLaw P false B ≠ ⊤ := by
rw [propensityTiltedLatentLaw, withDensity_apply _ hB]
simp only [armProb, Bool.false_eq_true, ↓reduceIte]
exact (MeasureTheory.ofReal_integral_eq_lintegral_ofReal hInt
hnonneg).symm.trans_ne (by simp)
rw [← ENNReal.toReal_eq_toReal_iff' hleft hright]
rw [assignedLatentLaw, Measure.map_apply hX hB,
Measure.restrict_apply (hB.preimage hX)]
change P.real (latentTriple ⁻¹' B ∩ {ω | arm ω = false}) = _
rw [propensityTiltedLatentLaw, withDensity_apply _ hB]
simp only [armProb, Bool.false_eq_true, ↓reduceIte]
rw [← MeasureTheory.ofReal_integral_eq_lintegral_ofReal hInt hnonneg,
ENNReal.toReal_ofReal]
· rw [← randomizedAssignment_false_measureReal P hRandom B hB]
apply measureReal_congr
filter_upwards [] with ω
change (latentTriple ω ∈ B ∧ arm ω = false) =
(arm ω = false ∧ latentTriple ω ∈ B)
exact propext and_comm
· exact integral_nonneg_of_ae hnonneg
theorem assignedLatentLaw_eq_propensityTiltedLatentLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
assignedLatentLaw P a = propensityTiltedLatentLaw P a
Proof (Lean source)
lemma assignedLatentLaw_eq_propensityTiltedLatentLaw
{ε : ℝ} {J : ℕ} (P : Measure (FullRow ε J)) [IsProbabilityMeasure P]
(hOverlap : Overlap ε) (hRandom : RandomizedAssignment P) (a : ArmSpace) :
assignedLatentLaw P a = propensityTiltedLatentLaw P a := by
cases a
· exact assignedLatentLaw_false_eq_propensityTiltedLatentLaw P hOverlap hRandom
· exact assignedLatentLaw_true_eq_propensityTiltedLatentLaw P hOverlap hRandom
theorem integral_inversePropensity_assignedLatentLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(∫ x, ((if a then x.2.2 else x.2.1 : OutcomeSpace) : ℝ) * (armProb a x.1)⁻¹ ∂(assignedLatentLaw P a))
= ∫ x, ((if a then x.2.2 else x.2.1 : OutcomeSpace) : ℝ) ∂(P.map latentTriple)
Proof (Lean source)
lemma integral_inversePropensity_assignedLatentLaw
{ε : ℝ} {J : ℕ} (P : Measure (FullRow ε J)) [IsProbabilityMeasure P]
(hOverlap : Overlap ε) (hRandom : RandomizedAssignment P) (a : ArmSpace) :
(∫ x, ((if a then x.2.2 else x.2.1 : OutcomeSpace) : ℝ) *
(armProb a x.1)⁻¹ ∂(assignedLatentLaw P a)) =
∫ x, ((if a then x.2.2 else x.2.1 : OutcomeSpace) : ℝ)
∂(P.map latentTriple) := by
rw [assignedLatentLaw_eq_propensityTiltedLatentLaw P hOverlap hRandom a]
unfold propensityTiltedLatentLaw
rw [integral_withDensity_eq_integral_toReal_smul
(by cases a <;> simp only [armProb, Bool.false_eq_true, ↓reduceIte] <;> fun_prop)
(ae_of_all _ fun _ => by finiteness)]
apply integral_congr_ae
filter_upwards [] with x
have hp : 0 < armProb a x.1 := by
rcases x.1.property with ⟨hx₁, hx₂⟩
cases a <;> simp [armProb] <;> linarith [hOverlap.1]
rw [ENNReal.toReal_ofReal hp.le]
simp only [smul_eq_mul]
field_simp
theorem potentialOutcomeMean_eq_inversePropensity
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(∫ ω, ((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ) ∂P)
= ∫ ω in {ω | arm ω = a}, ((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ) * (armProb a (score ω))⁻¹ ∂P
Proof (Lean source)
lemma potentialOutcomeMean_eq_inversePropensity {ε : ℝ} {J : ℕ}
(P : Measure (FullRow ε J)) [IsProbabilityMeasure P]
(hOverlap : Overlap ε) (hRandom : RandomizedAssignment P) (a : ArmSpace) :
(∫ ω, ((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ) ∂P) =
∫ ω in {ω | arm ω = a},
((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ) *
(armProb a (score ω))⁻¹ ∂P := by
have hX : Measurable (latentTriple (ε := ε) (J := J)) := by
unfold latentTriple score outcome0 outcome1
fun_prop
have h := integral_inversePropensity_assignedLatentLaw P hOverlap hRandom a
rw [assignedLatentLaw, MeasureTheory.integral_map] at h
· rw [MeasureTheory.integral_map] at h
· simpa [latentTriple] using h.symm
· exact hX.aemeasurable
· cases a with
| false =>
change AEStronglyMeasurable
(fun x : ScoreSpace ε × OutcomeSpace × OutcomeSpace =>
((x.2.1 : OutcomeSpace) : ℝ)) _
fun_prop
| true =>
change AEStronglyMeasurable
(fun x : ScoreSpace ε × OutcomeSpace × OutcomeSpace =>
((x.2.2 : OutcomeSpace) : ℝ)) _
fun_prop
· exact hX.aemeasurable
· cases a <;> simp only [armProb, Bool.false_eq_true, ↓reduceIte] <;> fun_prop
def compatibleArmCellCoupling
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
a :
r :
_hq :
0 < armCellMass H g a r
def
given by
(ofReal (armCellMass H g a r))⁻¹ • ((P.restrict {ω | arm ω = a ∧ g (score ω) = r}).map (fun ω => (((if a then outcome1 ω else outcome0 ω : OutcomeSpace) : ℝ), (armProb a (score ω))⁻¹)))
theorem product_expectation_countermonotoneCoupling_interval
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma product_expectation_countermonotoneCoupling_interval
(μ ν : Measure ℝ) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] :
(∫ p, p.1 * p.2 ∂(Causalean.Stat.countermonotoneCoupling μ ν)) =
∫ u in (0 : ℝ)..1,
quantile μ u * quantile ν (1 - u) := by
have hrest : volume.restrict (Ioc (0 : ℝ) 1) = unifOI := by
rw [unifOI, restrict_Ioo_eq_restrict_Ioc]
unfold Causalean.Stat.countermonotoneCoupling
let r : ℝ → ℝ := fun u => 1 - u
have hr : Measurable r := measurable_const.sub measurable_id
have hνmap : AEMeasurable (quantile ν)
(Causalean.Stat.unifOI.map r) := by
simpa [r, map_one_sub_unifOI] using
(aemeasurable_quantile_unifOI ν)
have hνr : AEMeasurable (fun u : ℝ => quantile ν (1 - u))
unifOI := by
simpa [r, Function.comp_def] using hνmap.comp_measurable hr
rw [MeasureTheory.integral_map]
· rw [intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1), hrest]
· exact (aemeasurable_quantile_unifOI μ).prodMk hνr
· fun_prop
theorem product_expectation_comonotoneCoupling_interval
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(∫ p, p.1 * p.2 ∂(comonotoneCoupling μ ν))
Proof (Lean source)
lemma product_expectation_comonotoneCoupling_interval
(μ ν : Measure ℝ) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
(hμ : MemLp (fun x : ℝ => x) 2 μ) (hν : MemLp (fun y : ℝ => y) 2 ν) :
(∫ p, p.1 * p.2 ∂(comonotoneCoupling μ ν)) =
∫ u in (0 : ℝ)..1,
quantile μ u * quantile ν u := by
have hrest : volume.restrict (Ioc (0 : ℝ) 1) = unifOI := by
rw [unifOI, restrict_Ioo_eq_restrict_Ioc]
rw [product_expectation_comonotoneCoupling μ ν hμ hν,
← integral_Ioc_eq_integral_Ioo,
intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1), hrest]
theorem product_expectation_mem_quantileInterval
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
π :
Measure (ℝ × ℝ)
μ ν :
Measure ℝ
hπ :
IsCoupling π μ ν
hμ :
MemLp (fun x : ℝ => x) 2 μ
hν :
MemLp (fun y : ℝ => y) 2 ν
Proof (Lean source)
lemma product_expectation_mem_quantileInterval
{π : Measure (ℝ × ℝ)} {μ ν : Measure ℝ}
(hπ : IsCoupling π μ ν)
[IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
(hμ : MemLp (fun x : ℝ => x) 2 μ) (hν : MemLp (fun y : ℝ => y) 2 ν) :
(∫ p, p.1 * p.2 ∂π) ∈
Icc
(∫ u in (0 : ℝ)..1,
quantile μ u * quantile ν (1 - u))
(∫ u in (0 : ℝ)..1,
quantile μ u * quantile ν u) := by
constructor
· rw [← product_expectation_countermonotoneCoupling_interval μ ν]
exact countermonotone_le_product_expectation hπ hμ hν
· rw [← product_expectation_comonotoneCoupling_interval μ ν hμ hν]
exact product_expectation_le_comonotone hπ hμ hν
theorem armMean_mem_endpoints_of_cellCouplings
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
a :
π :
∀ r : LabelSpace J
if
0 < armCellMass H g a r
then
Measure (ℝ × ℝ)
hπ :
∀ (r : LabelSpace J) (hq : 0 < armCellMass H g a r),
IsCoupling (π r hq) (armCellOutcomeLaw Prel hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq)) (armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq)
hY2 :
∀ (r : LabelSpace J) (hq : 0 < armCellMass H g a r),
MemLp (fun x : ℝ => x) 2 (armCellOutcomeLaw Prel hMass.2.1 a r (by rw [hMass.2.2.2 a r]; exact hq))
hW2 :
∀ (r : LabelSpace J) (hq : 0 < armCellMass H g a r),
MemLp (fun x : ℝ => x) 2 (armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq)
m :
ℝ
hm :
m
= ∑ r : LabelSpace J, if hq : 0 < armCellMass H g a r then armCellMass H g a r * ∫ p, p.1 * p.2 ∂(π r hq) else 0
Proof (Lean source)
lemma armMean_mem_endpoints_of_cellCouplings {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(a : ArmSpace)
(π : ∀ r : LabelSpace J, 0 < armCellMass H g a r → Measure (ℝ × ℝ))
(hπ : ∀ (r : LabelSpace J) (hq : 0 < armCellMass H g a r),
IsCoupling (π r hq)
(armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq))
(armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq))
(hY2 : ∀ (r : LabelSpace J) (hq : 0 < armCellMass H g a r),
MemLp (fun x : ℝ => x) 2
(armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq)))
(hW2 : ∀ (r : LabelSpace J) (hq : 0 < armCellMass H g a r),
MemLp (fun x : ℝ => x) 2
(armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq))
(m : ℝ)
(hm : m = ∑ r : LabelSpace J,
if hq : 0 < armCellMass H g a r then
armCellMass H g a r * ∫ p, p.1 * p.2 ∂(π r hq)
else 0) :
m ∈ Icc (muLower H g Prel hMass a) (muUpper H g Prel hMass a) := by
rw [hm]
constructor
· unfold muLower
apply Finset.sum_le_sum
intro r _
by_cases hq : 0 < armCellMass H g a r
· simp only [hq, dif_pos]
letI : IsProbabilityMeasure (π r hq) := (hπ r hq).isProbabilityMeasure
letI : IsProbabilityMeasure
(armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq)) := by
rw [← (hπ r hq).map_fst]
exact Measure.isProbabilityMeasure_map measurable_fst.aemeasurable
letI : IsProbabilityMeasure
(armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) := by
rw [← (hπ r hq).map_snd]
exact Measure.isProbabilityMeasure_map measurable_snd.aemeasurable
apply mul_le_mul_of_nonneg_left _ hq.le
exact (product_expectation_mem_quantileInterval (hπ r hq)
(hY2 r hq) (hW2 r hq)).1
· simp [hq]
· unfold muUpper
apply Finset.sum_le_sum
intro r _
by_cases hq : 0 < armCellMass H g a r
· simp only [hq, dif_pos]
letI : IsProbabilityMeasure (π r hq) := (hπ r hq).isProbabilityMeasure
letI : IsProbabilityMeasure
(armCellOutcomeLaw Prel hMass.2.1 a r
(by rw [hMass.2.2.2 a r]; exact hq)) := by
rw [← (hπ r hq).map_fst]
exact Measure.isProbabilityMeasure_map measurable_fst.aemeasurable
letI : IsProbabilityMeasure
(armCellWeightLaw H hMass.1 g hMass.2.2.1 a r hq) := by
rw [← (hπ r hq).map_snd]
exact Measure.isProbabilityMeasure_map measurable_snd.aemeasurable
apply mul_le_mul_of_nonneg_left _ hq.le
exact (product_expectation_mem_quantileInterval (hπ r hq)
(hY2 r hq) (hW2 r hq)).2
· simp [hq]
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.HighResolutionCost 11 declarations Paper-specific paired quantization coefficients and cell costs.
Paper-specific paired quantization coefficients and cell costs.
def highResolutionBeta
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
highResolutionBeta f s x z :
ℝ
given by
if s = 0 then densityExtension f x / (1 - z) else densityExtension f x / z
theorem highResolutionBeta_diagonal
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma highResolutionBeta_diagonal {ε : ℝ}
(hOverlap : Overlap ε) (f : ScoreSpace ε → ℝ)
(x : ℝ) (hx : x ∈ Icc ε (1 - ε)) :
pairedDiagonalWeight 2 (highResolutionBeta f) x =
densityExtension f x / (x * (1 - x)) := by
have hx0 : x ≠ 0 := ne_of_gt (hOverlap.1.trans_le hx.1)
have hx1 : 1 - x ≠ 0 := ne_of_gt (by linarith [hx.2, hOverlap.1])
unfold pairedDiagonalWeight highResolutionBeta
simp only [Fin.sum_univ_two, Fin.isValue, ↓reduceIte, one_ne_zero]
field_simp
ring
theorem highResolutionBeta_continuousOn
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
ContinuousOn (fun p : ℝ × ℝ => highResolutionBeta f s p.1 p.2) (Icc ε (1 - ε) ×ˢ Icc ε (1 - ε))
Proof (Lean source)
lemma highResolutionBeta_continuousOn {ε : ℝ}
(hOverlap : Overlap ε) (f : ScoreSpace ε → ℝ)
(hfcont : Continuous f) (s : Fin 2) :
ContinuousOn (fun p : ℝ × ℝ => highResolutionBeta f s p.1 p.2)
(Icc ε (1 - ε) ×ˢ Icc ε (1 - ε)) := by
have hnum : ContinuousOn
(fun p : ℝ × ℝ => densityExtension f p.1)
(Icc ε (1 - ε) ×ˢ Icc ε (1 - ε)) :=
(continuousOn_densityExtension hfcont).comp continuousOn_fst
(fun _ hp => hp.1)
by_cases hs : s = 0
· simp only [highResolutionBeta, hs, if_pos]
apply hnum.div (by fun_prop)
intro p hp
linarith [hp.2.2, hOverlap.1]
· simp only [highResolutionBeta, hs]
apply hnum.div (by fun_prop)
intro p hp
exact ne_of_gt (hOverlap.1.trans_le hp.2.1)
theorem highResolutionBeta_pos
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
hOverlap :
Overlap ε
f :
ScoreSpace ε → ℝ
hfpos :
∀ e, 0 < f e
s :
Fin 2
x z :
ℝ
hx :
x ∈ Icc ε (1 - ε)
hz :
z ∈ Icc ε (1 - ε)
conclusion
0 < highResolutionBeta f s x z
Proof (Lean source)
lemma highResolutionBeta_pos {ε : ℝ}
(hOverlap : Overlap ε) (f : ScoreSpace ε → ℝ)
(hfpos : ∀ e, 0 < f e) (s : Fin 2) (x z : ℝ)
(hx : x ∈ Icc ε (1 - ε)) (hz : z ∈ Icc ε (1 - ε)) :
0 < highResolutionBeta f s x z := by
have hfx : 0 < densityExtension f x := by
rw [densityExtension, dif_pos hx]
exact hfpos ⟨x, hx⟩
by_cases hs : s = 0
· rw [highResolutionBeta, if_pos hs]
exact div_pos hfx (by linarith [hz.2, hOverlap.1])
· rw [highResolutionBeta, if_neg hs]
exact div_pos hfx (hOverlap.1.trans_le hz.1)
theorem measurable_densityExtension
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma measurable_densityExtension {ε : ℝ} (f : ScoreSpace ε → ℝ)
(hf : Measurable f) : Measurable (densityExtension f) := by
have hemb : MeasurableEmbedding (Subtype.val : ScoreSpace ε → ℝ) :=
MeasurableEmbedding.subtype_coe measurableSet_Icc
have hext : Measurable
(Function.extend (Subtype.val : ScoreSpace ε → ℝ) f (fun _ => 0)) :=
hemb.measurable_extend hf measurable_const
convert hext using 1
funext x
by_cases hx : x ∈ Icc ε (1 - ε)
· rw [densityExtension, dif_pos hx]
exact (hemb.injective.extend_apply f (fun _ => 0) ⟨x, hx⟩).symm
· rw [densityExtension, dif_neg hx]
rw [Function.extend_apply' _ _ _ (by
intro hmem
rcases hmem with ⟨e, he⟩
apply hx
rw [← he]
exact e.property)]
theorem scoreBase_setIntegral_eq_realScoreCell
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(∫ e in cell g r, φ (e : ℝ) ∂((volume : Measure ℝ).comap (Subtype.val : ScoreSpace ε → ℝ)))
= ∫ x in realScoreCell g r, φ x
Proof (Lean source)
lemma scoreBase_setIntegral_eq_realScoreCell {ε : ℝ} {K : ℕ}
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(r : LabelSpace K) (φ : ℝ → ℝ) (hφ : Measurable φ) :
(∫ e in cell g r, φ (e : ℝ)
∂((volume : Measure ℝ).comap (Subtype.val : ScoreSpace ε → ℝ))) =
∫ x in realScoreCell g r, φ x := by
let μ : Measure (ScoreSpace ε) :=
(volume : Measure ℝ).comap (Subtype.val : ScoreSpace ε → ℝ)
have hcell : MeasurableSet (cell g r) := measurableSet_eq_fun hg measurable_const
have himageMeas : MeasurableSet (realScoreCell g r) :=
(MeasurableEmbedding.subtype_coe measurableSet_Icc).measurableSet_image' hcell
have hmap := MeasureTheory.setIntegral_map
(μ := μ) (g := (Subtype.val : ScoreSpace ε → ℝ))
(f := φ) (s := realScoreCell g r) himageMeas hφ.aestronglyMeasurable
measurable_subtype_coe.aemeasurable
have hmapMeasure : Measure.map (Subtype.val : ScoreSpace ε → ℝ) μ =
volume.restrict (Icc ε (1 - ε)) :=
map_comap_subtype_coe measurableSet_Icc volume
rw [hmapMeasure] at hmap
have himageSub : realScoreCell g r ⊆ Icc ε (1 - ε) := by
rintro x ⟨e, he, rfl⟩
exact e.property
have hrestrict :
(volume.restrict (Icc ε (1 - ε))).restrict (realScoreCell g r) =
volume.restrict (realScoreCell g r) :=
Measure.restrict_restrict_of_subset himageSub
have hpreimage : cell g r =
(Subtype.val : ScoreSpace ε → ℝ) ⁻¹' (realScoreCell g r) := by
ext e
constructor
· intro he
exact ⟨e, he, rfl⟩
· rintro ⟨e', he', heq⟩
simpa [Subtype.ext heq] using he'
rw [hpreimage, ← hmap]
change (∫ x, φ x
∂(volume.restrict (Icc ε (1 - ε))).restrict (realScoreCell g r)) = _
rw [hrestrict]
theorem treatedCandidate_eq_highResolutionCellCost
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε mf Mf :
ℝ
K :
ℕ
H :
Measure (ScoreSpace ε)
f :
ScoreSpace ε → ℝ
hDensity :
BoundedScoreDensity H f mf Mf
g :
ScoreSpace ε → LabelSpace K
hg :
r :
z :
ℝ
hz :
z ∈ Icc ε (1 - ε)
hOverlap :
Overlap ε
hfpos :
∀ e, 0 < f e
conclusion
= ∫ x in realScoreCell g r, highResolutionBeta f 1 x z * |x - z|
Proof (Lean source)
lemma treatedCandidate_eq_highResolutionCellCost {ε mf Mf : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) (f : ScoreSpace ε → ℝ)
(hDensity : BoundedScoreDensity H f mf Mf)
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(r : LabelSpace K) (z : ℝ) (hz : z ∈ Icc ε (1 - ε))
(hOverlap : Overlap ε) (hfpos : ∀ e, 0 < f e) :
(∫ e in cell g r, |1 - z⁻¹ * armProb true e| ∂H) =
∫ x in realScoreCell g r,
highResolutionBeta f 1 x z * |x - z| := by
let μ : Measure (ScoreSpace ε) :=
(volume : Measure ℝ).comap (Subtype.val : ScoreSpace ε → ℝ)
let _ : IsFiniteMeasure μ := rateLower_scoreBaseMeasure_isFinite ε
let φ : ℝ → ℝ := fun x => highResolutionBeta f 1 x z * |x - z|
have hfmeas : Measurable f := hDensity.densityLower.1
have hφmeas : Measurable φ := by
dsimp [φ, highResolutionBeta]
exact ((measurable_densityExtension f hfmeas).div_const z).mul
(by fun_prop)
have hcell : MeasurableSet (cell g r) := measurableSet_eq_fun hg measurable_const
have hzpos : 0 < z := hOverlap.1.trans_le hz.1
have hpoint (e : ScoreSpace ε) :
f e * |1 - z⁻¹ * armProb true e| = φ (e : ℝ) := by
have hext : densityExtension f (e : ℝ) = f e := by
rw [densityExtension, dif_pos e.property]
have hid : |1 - z⁻¹ * (e : ℝ)| = z⁻¹ * |(e : ℝ) - z| := by
rw [show 1 - z⁻¹ * (e : ℝ) = -z⁻¹ * ((e : ℝ) - z) by
field_simp
ring]
rw [abs_mul, abs_neg, abs_of_pos (inv_pos.mpr hzpos)]
simp only [armProb, ↓reduceIte, φ, highResolutionBeta, Fin.isValue,
one_ne_zero, hext]
rw [hid]
ring
calc
(∫ e in cell g r, |1 - z⁻¹ * armProb true e| ∂H) =
∫ e, (cell g r).indicator
(fun e => |1 - z⁻¹ * armProb true e|) e ∂H := by
rw [integral_indicator hcell]
_ = ∫ e, (ofReal (f e)).toReal *
(cell g r).indicator
(fun e => |1 - z⁻¹ * armProb true e|) e ∂μ := by
rw [hDensity.densityLower.2.1,
integral_withDensity_eq_integral_toReal_smul
hfmeas.ennreal_ofReal
(by simp)]
simp only [smul_eq_mul]
rfl
_ = ∫ e, (cell g r).indicator (fun e => φ (e : ℝ)) e ∂μ := by
apply integral_congr_ae
filter_upwards [] with e
by_cases he : e ∈ cell g r
· simp only [Set.indicator_of_mem he]
rw [ENNReal.toReal_ofReal (hfpos e).le, hpoint]
· simp [Set.indicator_of_notMem he]
_ = ∫ e in cell g r, φ (e : ℝ) ∂μ := by
rw [integral_indicator hcell]
_ = ∫ x in realScoreCell g r, φ x :=
scoreBase_setIntegral_eq_realScoreCell g hg r φ hφmeas
_ = ∫ x in realScoreCell g r,
highResolutionBeta f 1 x z * |x - z| := rfl
theorem treatedCellAbsoluteDeviation_eq_highResolutionInf
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε mf Mf :
ℝ
K :
ℕ
H :
Measure (ScoreSpace ε)
f :
ScoreSpace ε → ℝ
hDensity :
BoundedScoreDensity H f mf Mf
g :
ScoreSpace ε → LabelSpace K
hg :
r :
hOverlap :
Overlap ε
hfpos :
∀ e, 0 < f e
conclusion
cellAbsoluteDeviation H g true r
= sInf {v : ℝ | ∃ z ∈ Icc ε (1 - ε), v = ∫ x in realScoreCell g r, highResolutionBeta f 1 x z * |x - z|}
Proof (Lean source)
lemma treatedCellAbsoluteDeviation_eq_highResolutionInf
{ε mf Mf : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) (f : ScoreSpace ε → ℝ)
(hDensity : BoundedScoreDensity H f mf Mf)
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(r : LabelSpace K) (hOverlap : Overlap ε)
(hfpos : ∀ e, 0 < f e) :
cellAbsoluteDeviation H g true r =
sInf {v : ℝ | ∃ z ∈ Icc ε (1 - ε),
v = ∫ x in realScoreCell g r,
highResolutionBeta f 1 x z * |x - z|} := by
have hε : 0 < ε := hOverlap.1
have h1ε : 0 < 1 - ε := by linarith [hOverlap.2]
unfold cellAbsoluteDeviation
congr 1
ext v
constructor
· rintro ⟨c, hc, hvc⟩
have hcpos : 0 < c := lt_of_lt_of_le (inv_pos.mpr h1ε) hc.1
let z : ℝ := c⁻¹
have hz : z ∈ Icc ε (1 - ε) := by
constructor
· have h := (inv_le_inv₀ (inv_pos.mpr hε) hcpos).2 hc.2
simpa [z] using h
· have h := (inv_le_inv₀ hcpos (inv_pos.mpr h1ε)).2 hc.1
simpa [z] using h
refine ⟨z, hz, ?_⟩
have hzinv : z⁻¹ = c := by
dsimp [z]
exact inv_inv c
have hcandidate := treatedCandidate_eq_highResolutionCellCost
H f hDensity g hg r z hz hOverlap hfpos
rw [hzinv] at hcandidate
exact hvc.trans hcandidate
· rintro ⟨z, hz, hvz⟩
have hzpos : 0 < z := hε.trans_le hz.1
let c : ℝ := z⁻¹
have hc : c ∈ Icc ((1 - ε)⁻¹) ε⁻¹ := by
constructor
· exact (inv_le_inv₀ h1ε hzpos).2 hz.2
· exact (inv_le_inv₀ hzpos hε).2 hz.1
refine ⟨c, hc, ?_⟩
have hcandidate := treatedCandidate_eq_highResolutionCellCost
H f hDensity g hg r z hz hOverlap hfpos
exact hvz.trans hcandidate.symm
theorem controlCandidate_eq_highResolutionCellCost
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε mf Mf :
ℝ
K :
ℕ
H :
Measure (ScoreSpace ε)
f :
ScoreSpace ε → ℝ
hDensity :
BoundedScoreDensity H f mf Mf
g :
ScoreSpace ε → LabelSpace K
hg :
r :
z :
ℝ
hz :
z ∈ Icc ε (1 - ε)
hOverlap :
Overlap ε
hfpos :
∀ e, 0 < f e
conclusion
= ∫ x in realScoreCell g r, highResolutionBeta f 0 x z * |x - z|
Proof (Lean source)
lemma controlCandidate_eq_highResolutionCellCost {ε mf Mf : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) (f : ScoreSpace ε → ℝ)
(hDensity : BoundedScoreDensity H f mf Mf)
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(r : LabelSpace K) (z : ℝ) (hz : z ∈ Icc ε (1 - ε))
(hOverlap : Overlap ε) (hfpos : ∀ e, 0 < f e) :
(∫ e in cell g r, |1 - (1 - z)⁻¹ * armProb false e| ∂H) =
∫ x in realScoreCell g r,
highResolutionBeta f 0 x z * |x - z| := by
let μ : Measure (ScoreSpace ε) :=
(volume : Measure ℝ).comap (Subtype.val : ScoreSpace ε → ℝ)
let _ : IsFiniteMeasure μ := rateLower_scoreBaseMeasure_isFinite ε
let φ : ℝ → ℝ := fun x => highResolutionBeta f 0 x z * |x - z|
have hfmeas : Measurable f := hDensity.densityLower.1
have hφmeas : Measurable φ := by
dsimp [φ, highResolutionBeta]
exact ((measurable_densityExtension f hfmeas).div_const (1 - z)).mul
(by fun_prop)
have hcell : MeasurableSet (cell g r) := measurableSet_eq_fun hg measurable_const
have h1zpos : 0 < 1 - z := by linarith [hz.2, hOverlap.1]
have hpoint (e : ScoreSpace ε) :
f e * |1 - (1 - z)⁻¹ * armProb false e| = φ (e : ℝ) := by
have hext : densityExtension f (e : ℝ) = f e := by
rw [densityExtension, dif_pos e.property]
have hid : |1 - (1 - z)⁻¹ * (1 - (e : ℝ))| =
(1 - z)⁻¹ * |(e : ℝ) - z| := by
rw [show 1 - (1 - z)⁻¹ * (1 - (e : ℝ)) =
(1 - z)⁻¹ * ((e : ℝ) - z) by
field_simp
ring]
rw [abs_mul, abs_of_pos (inv_pos.mpr h1zpos)]
simp only [armProb, Bool.false_eq_true, ↓reduceIte, φ,
highResolutionBeta, Fin.isValue, hext]
rw [hid]
ring
calc
(∫ e in cell g r, |1 - (1 - z)⁻¹ * armProb false e| ∂H) =
∫ e, (cell g r).indicator
(fun e => |1 - (1 - z)⁻¹ * armProb false e|) e ∂H := by
rw [integral_indicator hcell]
_ = ∫ e, (ofReal (f e)).toReal *
(cell g r).indicator
(fun e => |1 - (1 - z)⁻¹ * armProb false e|) e ∂μ := by
rw [hDensity.densityLower.2.1,
integral_withDensity_eq_integral_toReal_smul
hfmeas.ennreal_ofReal (by simp)]
simp only [smul_eq_mul]
rfl
_ = ∫ e, (cell g r).indicator (fun e => φ (e : ℝ)) e ∂μ := by
apply integral_congr_ae
filter_upwards [] with e
by_cases he : e ∈ cell g r
· simp only [Set.indicator_of_mem he]
rw [ENNReal.toReal_ofReal (hfpos e).le, hpoint]
· simp [Set.indicator_of_notMem he]
_ = ∫ e in cell g r, φ (e : ℝ) ∂μ := by
rw [integral_indicator hcell]
_ = ∫ x in realScoreCell g r, φ x :=
scoreBase_setIntegral_eq_realScoreCell g hg r φ hφmeas
_ = ∫ x in realScoreCell g r,
highResolutionBeta f 0 x z * |x - z| := rfl
theorem controlCellAbsoluteDeviation_eq_highResolutionInf
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε mf Mf :
ℝ
K :
ℕ
H :
Measure (ScoreSpace ε)
f :
ScoreSpace ε → ℝ
hDensity :
BoundedScoreDensity H f mf Mf
g :
ScoreSpace ε → LabelSpace K
hg :
r :
hOverlap :
Overlap ε
hfpos :
∀ e, 0 < f e
conclusion
cellAbsoluteDeviation H g false r
= sInf {v : ℝ | ∃ z ∈ Icc ε (1 - ε), v = ∫ x in realScoreCell g r, highResolutionBeta f 0 x z * |x - z|}
Proof (Lean source)
lemma controlCellAbsoluteDeviation_eq_highResolutionInf
{ε mf Mf : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) (f : ScoreSpace ε → ℝ)
(hDensity : BoundedScoreDensity H f mf Mf)
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(r : LabelSpace K) (hOverlap : Overlap ε)
(hfpos : ∀ e, 0 < f e) :
cellAbsoluteDeviation H g false r =
sInf {v : ℝ | ∃ z ∈ Icc ε (1 - ε),
v = ∫ x in realScoreCell g r,
highResolutionBeta f 0 x z * |x - z|} := by
have hε : 0 < ε := hOverlap.1
have h1ε : 0 < 1 - ε := by linarith [hOverlap.2]
unfold cellAbsoluteDeviation
congr 1
ext v
constructor
· rintro ⟨c, hc, hvc⟩
have hcpos : 0 < c := lt_of_lt_of_le (inv_pos.mpr h1ε) hc.1
let z : ℝ := 1 - c⁻¹
have hcinvLower : ε ≤ c⁻¹ := by
have h := (inv_le_inv₀ (inv_pos.mpr hε) hcpos).2 hc.2
simpa using h
have hcinvUpper : c⁻¹ ≤ 1 - ε := by
have h := (inv_le_inv₀ hcpos (inv_pos.mpr h1ε)).2 hc.1
simpa using h
have hz : z ∈ Icc ε (1 - ε) := by
dsimp [z]
constructor <;> linarith
refine ⟨z, hz, ?_⟩
have hzc : (1 - z)⁻¹ = c := by
dsimp [z]
rw [sub_sub_cancel, inv_inv]
have hcandidate := controlCandidate_eq_highResolutionCellCost
H f hDensity g hg r z hz hOverlap hfpos
rw [hzc] at hcandidate
exact hvc.trans hcandidate
· rintro ⟨z, hz, hvz⟩
have h1zpos : 0 < 1 - z := by linarith [hz.2, hε]
let c : ℝ := (1 - z)⁻¹
have hc : c ∈ Icc ((1 - ε)⁻¹) ε⁻¹ := by
constructor
· exact (inv_le_inv₀ h1ε h1zpos).2 (by linarith [hz.1])
· exact (inv_le_inv₀ h1zpos hε).2 (by linarith [hz.2])
refine ⟨c, hc, ?_⟩
have hcandidate := controlCandidate_eq_highResolutionCellCost
H f hDensity g hg r z hz hOverlap hfpos
exact hvz.trans hcandidate.symm
theorem sum_cellAbsoluteDeviation_eq_highResolutionInfs
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε mf Mf :
ℝ
K :
ℕ
H :
Measure (ScoreSpace ε)
f :
ScoreSpace ε → ℝ
hDensity :
BoundedScoreDensity H f mf Mf
g :
ScoreSpace ε → LabelSpace K
hg :
r :
hOverlap :
Overlap ε
hfpos :
∀ e, 0 < f e
conclusion
= sInf {v : ℝ | ∃ z ∈ Icc ε (1 - ε), v = ∫ x in realScoreCell g r, highResolutionBeta f 1 x z * |x - z|}
+ sInf {v : ℝ | ∃ z ∈ Icc ε (1 - ε), v = ∫ x in realScoreCell g r, highResolutionBeta f 0 x z * |x - z|}
Proof (Lean source)
lemma sum_cellAbsoluteDeviation_eq_highResolutionInfs
{ε mf Mf : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) (f : ScoreSpace ε → ℝ)
(hDensity : BoundedScoreDensity H f mf Mf)
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(r : LabelSpace K) (hOverlap : Overlap ε)
(hfpos : ∀ e, 0 < f e) :
cellAbsoluteDeviation H g true r + cellAbsoluteDeviation H g false r =
sInf {v : ℝ | ∃ z ∈ Icc ε (1 - ε),
v = ∫ x in realScoreCell g r,
highResolutionBeta f 1 x z * |x - z|} +
sInf {v : ℝ | ∃ z ∈ Icc ε (1 - ε),
v = ∫ x in realScoreCell g r,
highResolutionBeta f 0 x z * |x - z|} := by
rw [treatedCellAbsoluteDeviation_eq_highResolutionInf
H f hDensity g hg r hOverlap hfpos,
controlCellAbsoluteDeviation_eq_highResolutionInf
H f hDensity g hg r hOverlap hfpos]
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.HighResolutionFiniteInfimum 4 declarations Finite assembly of independent reproduction-center infima.
Finite assembly of independent reproduction-center infima.
def independentCenterValues
For the specified mathematical inputs, this definition introduces the corresponding object.
def independentCenterTotalValues
For the specified mathematical inputs, this definition introduces the corresponding object.
theorem sInf_independentCenterTotalValues_eq_sum_of_nonempty_bddBelow
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ι Z :
Type*
Fintype ι
C :
ι → Set Z
cost :
ι → Z → ℝ
hne :
∀ i, (independentCenterValues C cost i).Nonempty
hbelow :
∀ i, BddBelow (independentCenterValues C cost i)
conclusion
Proof (Lean source)
lemma sInf_independentCenterTotalValues_eq_sum_of_nonempty_bddBelow
{ι Z : Type*} [Fintype ι]
(C : ι → Set Z) (cost : ι → Z → ℝ)
(hne : ∀ i, (independentCenterValues C cost i).Nonempty)
(hbelow : ∀ i, BddBelow (independentCenterValues C cost i)) :
sInf (independentCenterTotalValues C cost) =
∑ i, sInf (independentCenterValues C cost i) := by
classical
choose lower hlower using hbelow
have hne' := hne
choose initial hinitial using hne
choose z hzC hzcost using fun i => hinitial i
have htotalBelow : BddBelow (independentCenterTotalValues C cost) := by
refine ⟨∑ i, lower i, ?_⟩
rintro v ⟨w, hwC, rfl⟩
apply Finset.sum_le_sum
intro i hi
exact hlower i ⟨w i, hwC i, rfl⟩
have htotalNonempty :
(independentCenterTotalValues C cost).Nonempty :=
⟨∑ i, cost i (z i), z, hzC, rfl⟩
apply le_antisymm
· apply le_of_forall_pos_le_add
intro δ hδ
let η : ℝ := δ / ((Fintype.card ι : ℝ) + 1)
have hden : 0 < (Fintype.card ι : ℝ) + 1 := by positivity
have hη : 0 < η := div_pos hδ hden
have hbudget : (Fintype.card ι : ℝ) * η < δ := by
dsimp [η]
rw [show (Fintype.card ι : ℝ) *
(δ / ((Fintype.card ι : ℝ) + 1)) =
((Fintype.card ι : ℝ) * δ) /
((Fintype.card ι : ℝ) + 1) by ring]
rw [div_lt_iff₀ hden]
have hcard : (0 : ℝ) ≤ (Fintype.card ι : ℝ) := by positivity
nlinarith
have happrox (i : ι) : ∃ v ∈ independentCenterValues C cost i,
v < sInf (independentCenterValues C cost i) + η :=
Real.lt_sInf_add_pos (hne' i) hη
choose v hv hvalt using happrox
choose w hwC hvw using fun i => hv i
have hmember : (∑ i, cost i (w i)) ∈
independentCenterTotalValues C cost :=
⟨w, hwC, rfl⟩
have hinf : sInf (independentCenterTotalValues C cost) ≤
∑ i, cost i (w i) := csInf_le htotalBelow hmember
have hsum : (∑ i, cost i (w i)) ≤
∑ i, (sInf (independentCenterValues C cost i) + η) := by
apply Finset.sum_le_sum
intro i hi
rw [← hvw i]
exact (hvalt i).le
calc
sInf (independentCenterTotalValues C cost) ≤
∑ i, cost i (w i) := hinf
_ ≤ ∑ i, (sInf (independentCenterValues C cost i) + η) := hsum
_ = (∑ i, sInf (independentCenterValues C cost i)) +
(Fintype.card ι : ℝ) * η := by
rw [Finset.sum_add_distrib]
simp
_ ≤ (∑ i, sInf (independentCenterValues C cost i)) + δ := by
linarith
· apply le_csInf htotalNonempty
rintro v ⟨w, hwC, rfl⟩
apply Finset.sum_le_sum
intro i hi
exact csInf_le ⟨lower i, hlower i⟩ ⟨w i, hwC i, rfl⟩
theorem sInf_independentCenterTotalValues_Icc_eq_sum_of_bddBelow
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma sInf_independentCenterTotalValues_Icc_eq_sum_of_bddBelow
{ι : Type*} [Fintype ι] (a b : ℝ) (hab : a ≤ b)
(cost : ι → ℝ → ℝ)
(hbelow : ∀ i, BddBelow
(independentCenterValues (fun _ : ι => Icc a b) cost i)) :
sInf (independentCenterTotalValues (fun _ : ι => Icc a b) cost) =
∑ i, sInf
(independentCenterValues (fun _ : ι => Icc a b) cost i) := by
apply sInf_independentCenterTotalValues_eq_sum_of_nonempty_bddBelow
· intro i
exact ⟨cost i a, a, ⟨le_rfl, hab⟩, rfl⟩
· exact hbelow
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.HighResolutionObjective 3 declarations The fixed-release ambiguity as a finite paired quantization objective.
The fixed-release ambiguity as a finite paired quantization objective.
theorem highResolutionCellValues_nonempty_bddBelow
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
K :
ℕ
f :
ScoreSpace ε → ℝ
g :
ScoreSpace ε → LabelSpace K
hg :
hOverlap :
Overlap ε
hfpos :
∀ e, 0 < f e
i :
LabelSpace K × Fin 2
conclusion
cost :
fun i z
=> ∫ x in realScoreCell g i.1, highResolutionBeta f i.2 x z * |x
- z| (: (LabelSpace K × Fin 2) → ℝ → ℝ)
(independentCenterValues (fun _ => Icc ε (1 - ε)) cost i).Nonempty
BddBelow (independentCenterValues (fun _ => Icc ε (1 - ε)) cost i)
Proof (Lean source)
lemma highResolutionCellValues_nonempty_bddBelow
{ε : ℝ} {K : ℕ} (f : ScoreSpace ε → ℝ)
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(hOverlap : Overlap ε) (hfpos : ∀ e, 0 < f e)
(i : LabelSpace K × Fin 2) :
let cost : (LabelSpace K × Fin 2) → ℝ → ℝ := fun i z =>
∫ x in realScoreCell g i.1,
highResolutionBeta f i.2 x z * |x - z|
(independentCenterValues (fun _ => Icc ε (1 - ε)) cost i).Nonempty ∧
BddBelow (independentCenterValues
(fun _ => Icc ε (1 - ε)) cost i) := by
dsimp only
have hεle : ε ≤ 1 - ε := by linarith [hOverlap.2]
have hcellMeas : MeasurableSet (realScoreCell g i.1) :=
(MeasurableEmbedding.subtype_coe measurableSet_Icc).measurableSet_image'
(measurableSet_eq_fun hg measurable_const)
have hcostNonneg (z : ℝ) (hz : z ∈ Icc ε (1 - ε)) :
0 ≤ ∫ x in realScoreCell g i.1,
highResolutionBeta f i.2 x z * |x - z| := by
apply integral_nonneg_of_ae
filter_upwards [self_mem_ae_restrict hcellMeas] with x hx
rcases hx with ⟨e, he, rfl⟩
exact mul_nonneg
(le_of_lt (highResolutionBeta_pos hOverlap f hfpos i.2
(e : ℝ) z e.property hz))
(abs_nonneg _)
constructor
· refine ⟨_, ε, ⟨le_refl _, hεle⟩, rfl⟩
· refine ⟨0, ?_⟩
rintro v ⟨z, hz, rfl⟩
exact hcostNonneg z hz
theorem independentCenterTotalValues_eq_pairedPartitionValues
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
cost :
fun i z
=> ∫ x in realScoreCell g i.1, highResolutionBeta f i.2 x z * |x
- z| (: (LabelSpace K × Fin 2) → ℝ → ℝ)
independentCenterTotalValues (fun _ => Icc ε (1 - ε)) cost
= {v : ℝ | ∃ z : Fin K → Fin 2 → ℝ, (∀ r s, z r s ∈ Icc ε (1 - ε)) ∧ v = pairedPartitionCost 2 K (highResolutionBeta f) (realScoreCell g) z}
Proof (Lean source)
lemma independentCenterTotalValues_eq_pairedPartitionValues
{ε : ℝ} {K : ℕ} (f : ScoreSpace ε → ℝ)
(g : ScoreSpace ε → LabelSpace K) :
let cost : (LabelSpace K × Fin 2) → ℝ → ℝ := fun i z =>
∫ x in realScoreCell g i.1,
highResolutionBeta f i.2 x z * |x - z|
independentCenterTotalValues (fun _ => Icc ε (1 - ε)) cost =
{v : ℝ | ∃ z : Fin K → Fin 2 → ℝ,
(∀ r s, z r s ∈ Icc ε (1 - ε)) ∧
v = pairedPartitionCost 2 K (highResolutionBeta f)
(realScoreCell g) z} := by
dsimp only
ext v
constructor
· rintro ⟨z, hz, rfl⟩
refine ⟨fun r s => z (r, s), ?_, ?_⟩
· intro r s
exact hz (r, s)
· simp only [pairedPartitionCost, Fintype.sum_prod_type]
· rintro ⟨z, hz, rfl⟩
refine ⟨fun i => z i.1 i.2, ?_, ?_⟩
· intro i
exact hz i.1 i.2
· simp only [pairedPartitionCost, Fintype.sum_prod_type]
theorem worstCaseAmbiguity_eq_highResolutionObjective
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε mf Mf :
ℝ
K :
ℕ
H :
Measure (ScoreSpace ε)
f :
ScoreSpace ε → ℝ
hDensity :
BoundedScoreDensity H f mf Mf
g :
ScoreSpace ε → LabelSpace K
hg :
hOverlap :
Overlap ε
_hfcont :
hfpos :
∀ e, 0 < f e
conclusion
= sInf {v : ℝ | ∃ z : Fin K → Fin 2 → ℝ, (∀ r s, z r s ∈ Icc ε (1 - ε)) ∧ v = pairedPartitionCost 2 K (highResolutionBeta f) (realScoreCell g) z}
Proof (Lean source)
theorem worstCaseAmbiguity_eq_highResolutionObjective
{ε mf Mf : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) (f : ScoreSpace ε → ℝ)
(hDensity : BoundedScoreDensity H f mf Mf)
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(hOverlap : Overlap ε) (_hfcont : Continuous f)
(hfpos : ∀ e, 0 < f e) :
worstCaseAmbiguity H g =
sInf {v : ℝ | ∃ z : Fin K → Fin 2 → ℝ,
(∀ r s, z r s ∈ Icc ε (1 - ε)) ∧
v = pairedPartitionCost 2 K (highResolutionBeta f)
(realScoreCell g) z} := by
let _ : IsProbabilityMeasure H := hDensity.probability
let cost : (LabelSpace K × Fin 2) → ℝ → ℝ := fun i z =>
∫ x in realScoreCell g i.1,
highResolutionBeta f i.2 x z * |x - z|
have hlocal (i : LabelSpace K × Fin 2) :
(independentCenterValues (fun _ => Icc ε (1 - ε)) cost i).Nonempty ∧
BddBelow (independentCenterValues
(fun _ => Icc ε (1 - ε)) cost i) := by
simpa only [cost] using
highResolutionCellValues_nonempty_bddBelow
f g hg hOverlap hfpos i
have hεle : ε ≤ 1 - ε := by linarith [hOverlap.2]
have hassembly :
sInf (independentCenterTotalValues
(fun _ : LabelSpace K × Fin 2 => Icc ε (1 - ε)) cost) =
∑ i : LabelSpace K × Fin 2,
sInf (independentCenterValues
(fun _ => Icc ε (1 - ε)) cost i) :=
sInf_independentCenterTotalValues_Icc_eq_sum_of_bddBelow
ε (1 - ε) hεle cost (fun i => (hlocal i).2)
have hscalar (r : LabelSpace K) :
cellAbsoluteDeviation H g true r +
cellAbsoluteDeviation H g false r =
∑ s : Fin 2,
sInf (independentCenterValues
(fun _ : LabelSpace K × Fin 2 => Icc ε (1 - ε))
cost (r, s)) := by
rw [sum_cellAbsoluteDeviation_eq_highResolutionInfs
H f hDensity g hg r hOverlap hfpos]
rw [Fin.sum_univ_two]
change
sInf {v : ℝ | ∃ z ∈ Icc ε (1 - ε),
v = ∫ x in realScoreCell g r,
highResolutionBeta f 1 x z * |x - z|} +
sInf {v : ℝ | ∃ z ∈ Icc ε (1 - ε),
v = ∫ x in realScoreCell g r,
highResolutionBeta f 0 x z * |x - z|} =
sInf {v : ℝ | ∃ z ∈ Icc ε (1 - ε),
v = ∫ x in realScoreCell g r,
highResolutionBeta f 0 x z * |x - z|} +
sInf {v : ℝ | ∃ z ∈ Icc ε (1 - ε),
v = ∫ x in realScoreCell g r,
highResolutionBeta f 1 x z * |x - z|}
exact add_comm _ _
rw [(worstCaseAmbiguity_eq H g hOverlap hg).1]
calc
(∑ r : LabelSpace K,
(cellAbsoluteDeviation H g true r +
cellAbsoluteDeviation H g false r)) =
∑ r : LabelSpace K, ∑ s : Fin 2,
sInf (independentCenterValues
(fun _ : LabelSpace K × Fin 2 => Icc ε (1 - ε))
cost (r, s)) := by
apply Finset.sum_congr rfl
intro r hr
exact hscalar r
_ = ∑ i : LabelSpace K × Fin 2,
sInf (independentCenterValues
(fun _ => Icc ε (1 - ε)) cost i) := by
rw [Fintype.sum_prod_type]
_ = sInf (independentCenterTotalValues
(fun _ : LabelSpace K × Fin 2 => Icc ε (1 - ε)) cost) :=
hassembly.symm
_ = sInf {v : ℝ | ∃ z : Fin K → Fin 2 → ℝ,
(∀ r s, z r s ∈ Icc ε (1 - ε)) ∧
v = pairedPartitionCost 2 K (highResolutionBeta f)
(realScoreCell g) z} := by
rw [independentCenterTotalValues_eq_pairedPartitionValues f g]
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.HighResolutionOptimal 4 declarations Identification of optimal finite-label ambiguity with paired quantization.
Identification of optimal finite-label ambiguity with paired quantization.
theorem sInf_feasibleFiberInfs_eq_sInf_feasibleValues
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma sInf_feasibleFiberInfs_eq_sInf_feasibleValues
{ι : Type*} (p : ι → Prop) (A : ι → Set ℝ)
(hp : ∃ i, p i) (hne : ∀ i, p i → (A i).Nonempty)
(hbelow : BddBelow {x : ℝ | ∃ i, p i ∧ x ∈ A i}) :
sInf {v : ℝ | ∃ i, p i ∧ v = sInf (A i)} =
sInf {x : ℝ | ∃ i, p i ∧ x ∈ A i} := by
rcases hbelow with ⟨lower, hlower⟩
have hfiberBelow (i : ι) (hi : p i) : BddBelow (A i) := by
refine ⟨lower, ?_⟩
intro x hx
exact hlower ⟨i, hi, hx⟩
have hflatBelow : BddBelow {x : ℝ | ∃ i, p i ∧ x ∈ A i} :=
⟨lower, hlower⟩
have hflatNe : {x : ℝ | ∃ i, p i ∧ x ∈ A i}.Nonempty := by
rcases hp with ⟨i, hi⟩
rcases hne i hi with ⟨x, hx⟩
exact ⟨x, i, hi, hx⟩
have houterNe : {v : ℝ | ∃ i, p i ∧ v = sInf (A i)}.Nonempty := by
rcases hp with ⟨i, hi⟩
exact ⟨sInf (A i), i, hi, rfl⟩
have houterBelow :
BddBelow {v : ℝ | ∃ i, p i ∧ v = sInf (A i)} := by
refine ⟨lower, ?_⟩
rintro v ⟨i, hi, rfl⟩
apply le_csInf (hne i hi)
intro x hx
exact hlower ⟨i, hi, hx⟩
apply le_antisymm
· apply le_csInf hflatNe
rintro x ⟨i, hi, hx⟩
exact (csInf_le houterBelow ⟨i, hi, rfl⟩).trans
(csInf_le (hfiberBelow i hi) hx)
· apply le_csInf houterNe
rintro v ⟨i, hi, rfl⟩
apply le_csInf (hne i hi)
intro x hx
exact csInf_le hflatBelow ⟨i, hi, hx⟩
theorem highResolution_pairedPartitionCost_nonneg
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
K :
ℕ
f :
ScoreSpace ε → ℝ
hOverlap :
Overlap ε
hfpos :
∀ e, 0 < f e
hB :
IsIntervalPartition ε (1 - ε) K B
hz :
∀ r s, z r s ∈ Icc ε (1 - ε)
conclusion
0 ≤ pairedPartitionCost 2 K (highResolutionBeta f) B z
Proof (Lean source)
lemma highResolution_pairedPartitionCost_nonneg
{ε : ℝ} {K : ℕ} (f : ScoreSpace ε → ℝ)
(hOverlap : Overlap ε) (hfpos : ∀ e, 0 < f e)
(B : Fin K → Set ℝ) (hB : IsIntervalPartition ε (1 - ε) K B)
(z : Fin K → Fin 2 → ℝ)
(hz : ∀ r s, z r s ∈ Icc ε (1 - ε)) :
0 ≤ pairedPartitionCost 2 K (highResolutionBeta f) B z := by
unfold pairedPartitionCost
apply Finset.sum_nonneg
intro r hr
apply Finset.sum_nonneg
intro s hs
apply integral_nonneg_of_ae
filter_upwards [self_mem_ae_restrict (hB.1 r)] with x hx
have hxIcc : x ∈ Icc ε (1 - ε) := by
rw [← hB.2.2]
exact Set.mem_iUnion_of_mem r hx
exact mul_nonneg
(le_of_lt (highResolutionBeta_pos hOverlap f hfpos s x (z r s)
hxIcc (hz r s)))
(abs_nonneg _)
theorem measurableReleaseValues_eq_intervalPartitionValues
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
{v : ℝ | ∃ g : ScoreSpace ε → LabelSpace K, Measurable g ∧ ∃ z : Fin K → Fin 2 → ℝ, (∀ r s, z r s ∈ Icc ε (1 - ε)) ∧ v = pairedPartitionCost 2 K (highResolutionBeta f) (realScoreCell g) z}
= {v : ℝ | ∃ (B : Fin K → Set ℝ) (z : Fin K → Fin 2 → ℝ), IsIntervalPartition ε (1 - ε) K B ∧ (∀ r s, z r s ∈ Icc ε (1 - ε)) ∧ v = pairedPartitionCost 2 K (highResolutionBeta f) B z}
Proof (Lean source)
lemma measurableReleaseValues_eq_intervalPartitionValues
{ε : ℝ} {K : ℕ} (f : ScoreSpace ε → ℝ) :
{v : ℝ | ∃ g : ScoreSpace ε → LabelSpace K,
Measurable g ∧ ∃ z : Fin K → Fin 2 → ℝ,
(∀ r s, z r s ∈ Icc ε (1 - ε)) ∧
v = pairedPartitionCost 2 K (highResolutionBeta f)
(realScoreCell g) z} =
{v : ℝ | ∃ (B : Fin K → Set ℝ) (z : Fin K → Fin 2 → ℝ),
IsIntervalPartition ε (1 - ε) K B ∧
(∀ r s, z r s ∈ Icc ε (1 - ε)) ∧
v = pairedPartitionCost 2 K (highResolutionBeta f) B z} := by
ext v
constructor
· rintro ⟨g, hg, z, hz, rfl⟩
exact ⟨realScoreCell g, z, measurableRelease_intervalPartition g hg,
hz, rfl⟩
· rintro ⟨B, z, hB, hz, rfl⟩
let g := intervalPartitionRelease B hB
refine ⟨g, intervalPartitionRelease_measurable B hB, z, hz, ?_⟩
unfold g
have hcells : realScoreCell (intervalPartitionRelease B hB) = B := by
funext r
exact realScoreCell_intervalPartitionRelease B hB r
rw [hcells]
theorem optimalKAmbiguity_eq_optimalPairedCost
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε mf Mf :
ℝ
K :
ℕ
H :
Measure (ScoreSpace ε)
f :
ScoreSpace ε → ℝ
hDensity :
BoundedScoreDensity H f mf Mf
hOverlap :
Overlap ε
hfcont :
hfpos :
∀ e, 0 < f e
hK :
0 < K
conclusion
Proof (Lean source)
theorem optimalKAmbiguity_eq_optimalPairedCost
{ε mf Mf : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) (f : ScoreSpace ε → ℝ)
(hDensity : BoundedScoreDensity H f mf Mf)
(hOverlap : Overlap ε) (hfcont : Continuous f)
(hfpos : ∀ e, 0 < f e) (hK : 0 < K) :
optimalKAmbiguity ε K H =
optimalPairedCost ε (1 - ε) 2 K (highResolutionBeta f) := by
let A : (ScoreSpace ε → LabelSpace K) → Set ℝ := fun g =>
{v : ℝ | ∃ z : Fin K → Fin 2 → ℝ,
(∀ r s, z r s ∈ Icc ε (1 - ε)) ∧
v = pairedPartitionCost 2 K (highResolutionBeta f)
(realScoreCell g) z}
have hεle : ε ≤ 1 - ε := by linarith [hOverlap.2]
have hAne (g : ScoreSpace ε → LabelSpace K) : (A g).Nonempty := by
let z : Fin K → Fin 2 → ℝ := fun _ _ => ε
refine ⟨pairedPartitionCost 2 K (highResolutionBeta f)
(realScoreCell g) z, z, ?_, rfl⟩
intro r s
exact ⟨le_rfl, hεle⟩
have hreleaseNe : ∃ g : ScoreSpace ε → LabelSpace K, Measurable g := by
let r0 : LabelSpace K := ⟨0, hK⟩
exact ⟨fun _ => r0, measurable_const⟩
have hflatBelow : BddBelow
{v : ℝ | ∃ g : ScoreSpace ε → LabelSpace K,
Measurable g ∧ v ∈ A g} := by
refine ⟨0, ?_⟩
rintro v ⟨g, hg, z, hz, rfl⟩
exact highResolution_pairedPartitionCost_nonneg f hOverlap hfpos
(realScoreCell g) (measurableRelease_intervalPartition g hg) z hz
have hflatten :
sInf {v : ℝ | ∃ g : ScoreSpace ε → LabelSpace K,
Measurable g ∧ v = sInf (A g)} =
sInf {v : ℝ | ∃ g : ScoreSpace ε → LabelSpace K,
Measurable g ∧ v ∈ A g} :=
sInf_feasibleFiberInfs_eq_sInf_feasibleValues
(fun g : ScoreSpace ε → LabelSpace K => Measurable g) A
hreleaseNe (fun g hg => hAne g) hflatBelow
have houter :
{d : ℝ | ∃ g ∈ kLabelReleases ε K,
d = worstCaseAmbiguity H g} =
{v : ℝ | ∃ g : ScoreSpace ε → LabelSpace K,
Measurable g ∧ v = sInf (A g)} := by
ext v
constructor
· rintro ⟨g, hg, rfl⟩
change Measurable g at hg
refine ⟨g, hg, ?_⟩
exact worstCaseAmbiguity_eq_highResolutionObjective
H f hDensity g hg hOverlap hfcont hfpos
· rintro ⟨g, hg, rfl⟩
refine ⟨g, ?_, ?_⟩
· exact hg
· symm
exact worstCaseAmbiguity_eq_highResolutionObjective
H f hDensity g hg hOverlap hfcont hfpos
unfold optimalKAmbiguity optimalPairedCost
rw [houter, hflatten]
change
sInf {v : ℝ | ∃ g : ScoreSpace ε → LabelSpace K,
Measurable g ∧ ∃ z : Fin K → Fin 2 → ℝ,
(∀ r s, z r s ∈ Icc ε (1 - ε)) ∧
v = pairedPartitionCost 2 K (highResolutionBeta f)
(realScoreCell g) z} = _
rw [measurableReleaseValues_eq_intervalPartitionValues f]
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.HighResolutionOrderedCost 2 declarations Cost bound for the ordered paired-companding release.
Cost bound for the ordered paired-companding release.
theorem highResolutionOrderedRelease_worstCaseAmbiguity_le_compandingCost
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε mf Mf :
ℝ
H :
f :
ScoreSpace ε → ℝ
hDensity :
BoundedScoreDensity H f mf Mf
hOverlap :
Overlap ε
hfcont :
hfpos :
∀ e, 0 < f e
k :
ℕ
hk :
0 < k
conclusion
worstCaseAmbiguity H (highResolutionOrderedRelease hOverlap f hfcont hfpos k hk)
≤ pairedPartitionCost 2 k (highResolutionBeta f) (pairedCompandingCell ε (1 - ε) 2 k (highResolutionBeta f)) (pairedCompandingMidpoint ε (1 - ε) 2 k (highResolutionBeta f))
Proof (Lean source)
lemma highResolutionOrderedRelease_worstCaseAmbiguity_le_compandingCost
{ε mf Mf : ℝ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(f : ScoreSpace ε → ℝ) (hDensity : BoundedScoreDensity H f mf Mf)
(hOverlap : Overlap ε) (hfcont : Continuous f)
(hfpos : ∀ e, 0 < f e) (k : ℕ) (hk : 0 < k) :
worstCaseAmbiguity H
(highResolutionOrderedRelease hOverlap f hfcont hfpos k hk) ≤
pairedPartitionCost 2 k (highResolutionBeta f)
(pairedCompandingCell ε (1 - ε) 2 k (highResolutionBeta f))
(pairedCompandingMidpoint ε (1 - ε) 2 k (highResolutionBeta f)) := by
let g := highResolutionOrderedRelease hOverlap f hfcont hfpos k hk
let B := pairedCompandingCell ε (1 - ε) 2 k (highResolutionBeta f)
let z := pairedCompandingMidpoint ε (1 - ε) 2 k (highResolutionBeta f)
have hg : Measurable g :=
highResolutionOrderedRelease_measurable hOverlap f hfcont hfpos k hk
have hB : IsIntervalPartition ε (1 - ε) k B :=
highResolutionCompandingCell_partition hOverlap f hfcont hfpos k hk
have hz : ∀ r s, z r s ∈ Icc ε (1 - ε) :=
pairedCompandingMidpoint_mem_Icc ε (1 - ε)
(by linarith [hOverlap.2]) 2 k hk (highResolutionBeta f)
have hcellNonneg (r : LabelSpace k) (s : Fin 2) (w : ℝ)
(hw : w ∈ Icc ε (1 - ε)) :
0 ≤ ∫ x in realScoreCell g r,
highResolutionBeta f s x w * |x - w| := by
rw [highResolutionOrderedRelease_realScoreCell
hOverlap f hfcont hfpos k hk r]
apply integral_nonneg_of_ae
filter_upwards [ae_restrict_mem (hB.1 r)] with x hx
have hβ := highResolutionBeta_pos hOverlap f hfpos s x w
(by
rw [← hB.2.2]
exact Set.mem_iUnion_of_mem r hx) hw
exact mul_nonneg hβ.le (abs_nonneg _)
have hinfLe (r : LabelSpace k) (s : Fin 2) :
sInf {v : ℝ | ∃ w ∈ Icc ε (1 - ε),
v = ∫ x in realScoreCell g r,
highResolutionBeta f s x w * |x - w|} ≤
∫ x in realScoreCell g r,
highResolutionBeta f s x (z r s) * |x - z r s| := by
apply csInf_le
· refine ⟨0, ?_⟩
rintro v ⟨w, hw, rfl⟩
exact hcellNonneg r s w hw
· exact ⟨z r s, hz r s, rfl⟩
rw [(worstCaseAmbiguity_eq H g hOverlap hg).1]
calc
(∑ r : LabelSpace k,
(cellAbsoluteDeviation H g true r +
cellAbsoluteDeviation H g false r)) ≤
∑ r : LabelSpace k,
((∫ x in realScoreCell g r,
highResolutionBeta f 1 x (z r 1) * |x - z r 1|) +
∫ x in realScoreCell g r,
highResolutionBeta f 0 x (z r 0) * |x - z r 0|) := by
apply Finset.sum_le_sum
intro r hr
rw [sum_cellAbsoluteDeviation_eq_highResolutionInfs
H f hDensity g hg r hOverlap hfpos]
exact add_le_add (hinfLe r 1) (hinfLe r 0)
_ = pairedPartitionCost 2 k (highResolutionBeta f) B z := by
unfold pairedPartitionCost
simp only [Fin.sum_univ_two]
dsimp only [g]
simp_rw [highResolutionOrderedRelease_realScoreCell
hOverlap f hfcont hfpos k hk]
apply Finset.sum_congr rfl
intro r hr
ring
theorem optimalKAmbiguity_le_highResolutionOrderedRelease
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
optimalKAmbiguity ε k H
≤ worstCaseAmbiguity H (highResolutionOrderedRelease hOverlap f hfcont hfpos k hk)
Proof (Lean source)
lemma optimalKAmbiguity_le_highResolutionOrderedRelease
{ε : ℝ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(hOverlap : Overlap ε) (f : ScoreSpace ε → ℝ)
(hfcont : Continuous f) (hfpos : ∀ e, 0 < f e)
(k : ℕ) (hk : 0 < k) :
optimalKAmbiguity ε k H ≤
worstCaseAmbiguity H
(highResolutionOrderedRelease hOverlap f hfcont hfpos k hk) := by
let g := highResolutionOrderedRelease hOverlap f hfcont hfpos k hk
have hg : Measurable g :=
highResolutionOrderedRelease_measurable hOverlap f hfcont hfpos k hk
unfold optimalKAmbiguity
apply csInf_le
· refine ⟨0, ?_⟩
rintro d ⟨q, hq, rfl⟩
rw [(worstCaseAmbiguity_eq H q hOverlap hq).1]
exact Finset.sum_nonneg fun r hr => add_nonneg
(cellAbsoluteDeviation_nonneg H q hOverlap true r)
(cellAbsoluteDeviation_nonneg H q hOverlap false r)
· exact ⟨g, hg, rfl⟩
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.HighResolutionOrderedRelease 5 declarations The measurable ordered release induced by the paired companding cells.
The measurable ordered release induced by the paired companding cells.
theorem highResolutionCompandingCell_partition
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma highResolutionCompandingCell_partition {ε : ℝ}
(hOverlap : Overlap ε) (f : ScoreSpace ε → ℝ)
(hfcont : Continuous f) (hfpos : ∀ e, 0 < f e)
(k : ℕ) (hk : 0 < k) :
IsIntervalPartition ε (1 - ε) k
(pairedCompandingCell ε (1 - ε) 2 k (highResolutionBeta f)) := by
let boundary : ℕ → ℝ := fun j =>
pairedBoundary ε (1 - ε) 2 k j (highResolutionBeta f)
have hab : ε < 1 - ε := by linarith [hOverlap.2]
have hmono : ∀ i j, i ≤ j → j ≤ k → boundary i ≤ boundary j := by
intro i j hij hj
exact pairedBoundary_mono ε (1 - ε) hab.le 2 k i j hk hij hj
(highResolutionBeta f)
have hpart := orderedIntervalCells_partition boundary k hk hmono
have hzero : boundary 0 = ε :=
pairedBoundary_zero ε (1 - ε) hab.le 2 k (highResolutionBeta f)
have hlast : boundary k = 1 - ε :=
pairedBoundary_last ε (1 - ε) hab 2 k (by norm_num) hk
(highResolutionBeta f)
(highResolutionBeta_continuousOn hOverlap f hfcont)
(highResolutionBeta_pos hOverlap f hfpos)
rw [hzero, hlast] at hpart
exact hpart
def highResolutionOrderedRelease
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
intervalPartitionRelease (pairedCompandingCell ε (1 - ε) 2 k (highResolutionBeta f)) (highResolutionCompandingCell_partition hOverlap f hfcont hfpos k hk)
theorem highResolutionOrderedRelease_measurable
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Measurable (highResolutionOrderedRelease hOverlap f hfcont hfpos k hk)
Proof (Lean source)
lemma highResolutionOrderedRelease_measurable {ε : ℝ}
(hOverlap : Overlap ε) (f : ScoreSpace ε → ℝ)
(hfcont : Continuous f) (hfpos : ∀ e, 0 < f e)
(k : ℕ) (hk : 0 < k) :
Measurable (highResolutionOrderedRelease hOverlap f hfcont hfpos k hk) := by
unfold highResolutionOrderedRelease
exact intervalPartitionRelease_measurable _ _
theorem highResolutionOrderedRelease_realScoreCell
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
realScoreCell (highResolutionOrderedRelease hOverlap f hfcont hfpos k hk) r
= pairedCompandingCell ε (1 - ε) 2 k (highResolutionBeta f) r
Proof (Lean source)
lemma highResolutionOrderedRelease_realScoreCell {ε : ℝ}
(hOverlap : Overlap ε) (f : ScoreSpace ε → ℝ)
(hfcont : Continuous f) (hfpos : ∀ e, 0 < f e)
(k : ℕ) (hk : 0 < k) (r : LabelSpace k) :
realScoreCell
(highResolutionOrderedRelease hOverlap f hfcont hfpos k hk) r =
pairedCompandingCell ε (1 - ε) 2 k (highResolutionBeta f) r := by
unfold highResolutionOrderedRelease
exact realScoreCell_intervalPartitionRelease _ _ r
theorem highResolutionOrderedRelease_cell_ordConnected
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(cell (highResolutionOrderedRelease hOverlap f hfcont hfpos k hk) r).OrdConnected
Proof (Lean source)
lemma highResolutionOrderedRelease_cell_ordConnected {ε : ℝ}
(hOverlap : Overlap ε) (f : ScoreSpace ε → ℝ)
(hfcont : Continuous f) (hfpos : ∀ e, 0 < f e)
(k : ℕ) (hk : 0 < k) (r : LabelSpace k) :
(cell (highResolutionOrderedRelease hOverlap f hfcont hfpos k hk) r).OrdConnected := by
let B := pairedCompandingCell ε (1 - ε) 2 k (highResolutionBeta f)
let hB := highResolutionCompandingCell_partition hOverlap f hfcont hfpos k hk
have hBorder : (B r).OrdConnected := by
dsimp [B, pairedCompandingCell]
split_ifs
· exact ordConnected_Icc
· exact ordConnected_Ico
rw [ordConnected_iff]
intro x hx y hy hxy z hz
change highResolutionOrderedRelease hOverlap f hfcont hfpos k hk z = r
have hxB : (x : ℝ) ∈ B r := by
exact (intervalPartitionRelease_eq_iff B hB x r).mp hx
have hyB : (y : ℝ) ∈ B r := by
exact (intervalPartitionRelease_eq_iff B hB y r).mp hy
apply (intervalPartitionRelease_eq_iff B hB z r).mpr
rw [ordConnected_iff] at hBorder
exact hBorder (x : ℝ) hxB (y : ℝ) hyB hxy ⟨hz.1, hz.2⟩
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.HighResolutionPartitionRelease 6 declarations Equivalence between measurable finite score partitions and releases.
Equivalence between measurable finite score partitions and releases.
def intervalPartitionRelease
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
by have he : (e : ℝ) ∈ ⋃ r, B r := by rw [hB.2.2] exact e.property exact choose (Set.mem_iUnion.mp he)
theorem intervalPartitionRelease_mem
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(e : ℝ) ∈ B (intervalPartitionRelease B hB e)
Proof (Lean source)
lemma intervalPartitionRelease_mem {ε : ℝ} {K : ℕ}
(B : Fin K → Set ℝ)
(hB : IsIntervalPartition ε (1 - ε) K B)
(e : ScoreSpace ε) :
(e : ℝ) ∈ B (intervalPartitionRelease B hB e) := by
unfold intervalPartitionRelease
exact Classical.choose_spec (Set.mem_iUnion.mp (by
rw [hB.2.2]
exact e.property))
theorem intervalPartitionRelease_eq_iff
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
intervalPartitionRelease B hB e = r ↔ (e : ℝ) ∈ B r
Proof (Lean source)
lemma intervalPartitionRelease_eq_iff {ε : ℝ} {K : ℕ}
(B : Fin K → Set ℝ)
(hB : IsIntervalPartition ε (1 - ε) K B)
(e : ScoreSpace ε) (r : LabelSpace K) :
intervalPartitionRelease B hB e = r ↔ (e : ℝ) ∈ B r := by
constructor
· intro h
simpa [h] using intervalPartitionRelease_mem B hB e
· intro her
by_contra hne
have hdis := hB.2.1 (intervalPartitionRelease B hB e) r hne
exact Set.disjoint_left.mp hdis
(intervalPartitionRelease_mem B hB e) her
theorem intervalPartitionRelease_measurable
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma intervalPartitionRelease_measurable {ε : ℝ} {K : ℕ}
(B : Fin K → Set ℝ)
(hB : IsIntervalPartition ε (1 - ε) K B) :
Measurable (intervalPartitionRelease B hB) := by
apply measurable_to_countable'
intro r
have hpre : intervalPartitionRelease B hB ⁻¹' {r} =
(Subtype.val : ScoreSpace ε → ℝ) ⁻¹' B r := by
ext e
simp only [Set.mem_preimage, Set.mem_singleton_iff]
exact intervalPartitionRelease_eq_iff B hB e r
rw [hpre]
exact (hB.1 r).preimage measurable_subtype_coe
theorem realScoreCell_intervalPartitionRelease
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
realScoreCell (intervalPartitionRelease B hB) r = B r
Proof (Lean source)
lemma realScoreCell_intervalPartitionRelease {ε : ℝ} {K : ℕ}
(B : Fin K → Set ℝ)
(hB : IsIntervalPartition ε (1 - ε) K B)
(r : LabelSpace K) :
realScoreCell (intervalPartitionRelease B hB) r = B r := by
ext x
constructor
· rintro ⟨e, he, rfl⟩
exact (intervalPartitionRelease_eq_iff B hB e r).mp he
· intro hx
have hxUnion : x ∈ ⋃ j, B j := Set.mem_iUnion_of_mem r hx
have hxIcc : x ∈ Icc ε (1 - ε) := by
rw [← hB.2.2]
exact hxUnion
let e : ScoreSpace ε := ⟨x, hxIcc⟩
refine ⟨e, ?_, rfl⟩
exact (intervalPartitionRelease_eq_iff B hB e r).mpr hx
theorem measurableRelease_intervalPartition
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
IsIntervalPartition ε (1 - ε) K (realScoreCell g)
Proof (Lean source)
lemma measurableRelease_intervalPartition {ε : ℝ} {K : ℕ}
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g) :
IsIntervalPartition ε (1 - ε) K (realScoreCell g) :=
realScoreCell_intervalPartition g hg
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.Inference 11 declarations Finite-sample honest intervals for the known-score experiment.
Finite-sample honest intervals for the known-score experiment.
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
n K :
ℕ
fields
lo :
TrialSample n K → ℝ
hi :
TrialSample n K → ℝ
measurable_lo :
Measurable lo
measurable_hi :
Measurable hi
lower_bound :
∀ x, -1 ≤ lo x
ordered :
∀ x, lo x ≤ hi x
upper_bound :
∀ x, hi x ≤ 1
def contains
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
def length
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
length C x :
ℝ
given by
C.hi x - C.lo x
def equalWidthMidpoint
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
equalWidthMidpoint ε K r :
ℝ
given by
ε + ((r.val : ℝ) + 1 / 2) * (1 - 2 * ε) / K
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
midpointWeightedCenter ε n K x :
ℝ
given by
(n : ℝ)⁻¹ * ∑ i : Fin n, if (x i).2.1 then ((x i).2.2 : ℝ) / equalWidthMidpoint ε K (x i).1 else -((x i).2.2 : ℝ) / (1 - equalWidthMidpoint ε K (x i).1)
For the specified mathematical inputs, this definition introduces the corresponding object.
For the specified mathematical inputs, this definition introduces the corresponding object.
def IIDTrialSampling
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
K :
ℕ
n :
ℕ
trialLaw :
def
IIDTrialSampling trialLaw :
Prop
given by
trialLaw P = Measure.pi (fun _ : Fin n => releasedLaw P)
def honestProcedures
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
{gC | gC.1 ∈ kLabelReleases ε K ∧ ∀ P ∈ CausalLaws H gC.1, 1 - α ≤ (Measure.pi (fun _ : Fin n => releasedLaw P)).real {x | gC.2.contains x (ate P)}}
def minimaxHonestLength
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
minimaxHonestLength n K H α :
ℝ
given by
sInf {v : ℝ | ∃ gC ∈ honestProcedures n K H α, v = sSup {u : ℝ | ∃ P ∈ CausalLaws H gC.1, u = ∫ x, gC.2.length x ∂(Measure.pi (fun _ : Fin n => releasedLaw P))}}
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.InferenceMidpointBias 5 declarations Population bias of the equal-width midpoint inverse-propensity statistic.
Population bias of the equal-width midpoint inverse-propensity statistic.
def midpointWeightedRow
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
midpointWeightedRow ε K z :
ℝ
given by
if z.2.1 then (z.2.2 : ℝ) / equalWidthMidpoint ε K z.1 else -(z.2.2 : ℝ) / (1 - equalWidthMidpoint ε K z.1)
theorem midpointWeightedCenter_eq_rowMean
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma midpointWeightedCenter_eq_rowMean (ε : ℝ) (n K : ℕ)
(x : TrialSample n K) :
midpointWeightedCenter ε n K x =
(n : ℝ)⁻¹ * ∑ i : Fin n, midpointWeightedRow ε K (x i) := by
rfl
theorem score_sub_equalWidthMidpoint_abs_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
|(e : ℝ) - equalWidthMidpoint ε K (equalWidthRelease ε K hK e)| ≤ (1 - 2 * ε) / (2 * K)
Proof (Lean source)
lemma score_sub_equalWidthMidpoint_abs_le {ε : ℝ}
(hOverlap : Overlap ε) (K : ℕ) (hK : 0 < K) (e : ScoreSpace ε) :
|(e : ℝ) - equalWidthMidpoint ε K (equalWidthRelease ε K hK e)| ≤
(1 - 2 * ε) / (2 * K) := by
let r := equalWidthRelease ε K hK e
have he : e ∈ cell (equalWidthRelease ε K hK) r := by
rfl
have hb := equalWidthRelease_cell_subset_closedBin hOverlap hK r he
have hKreal : 0 < (K : ℝ) := by exact_mod_cast hK
have hw : 0 ≤ (1 - 2 * ε) / K :=
div_nonneg (by linarith [hOverlap.2]) hKreal.le
have hhalf : (1 - 2 * ε) / (2 * K) = ((1 - 2 * ε) / K) / 2 := by
field_simp
rw [abs_le]
rw [hhalf]
constructor <;>
dsimp [equalWidthMidpoint, r] at hb ⊢ <;>
ring_nf at hb ⊢ <;>
linarith
theorem equalWidthMidpoint_mem_Icc
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
equalWidthMidpoint ε K r ∈ Icc ε (1 - ε)
Proof (Lean source)
lemma equalWidthMidpoint_mem_Icc {ε : ℝ}
(hOverlap : Overlap ε) (K : ℕ) (hK : 0 < K) (r : LabelSpace K) :
equalWidthMidpoint ε K r ∈ Icc ε (1 - ε) := by
have hKreal : 0 < (K : ℝ) := by exact_mod_cast hK
have hwidth : 0 < 1 - 2 * ε := by linarith [hOverlap.2]
have hr0 : 0 ≤ (r : ℝ) := by positivity
have hrK : (r : ℝ) + (1 / 2 : ℝ) ≤ (K : ℝ) := by
have hrCast : (r : ℝ) + 1 ≤ (K : ℝ) := by
exact_mod_cast (Nat.succ_le_iff.mpr r.isLt)
linarith
constructor
· unfold equalWidthMidpoint
apply le_add_of_nonneg_right
exact div_nonneg (mul_nonneg (by linarith) hwidth.le) hKreal.le
· unfold equalWidthMidpoint
have hratio : ((r : ℝ) + (1 / 2 : ℝ)) / K ≤ 1 :=
(div_le_one hKreal).mpr hrK
calc
ε + ((r : ℕ) + (1 / 2 : ℝ)) * (1 - 2 * ε) / K =
ε + (((r : ℝ) + (1 / 2 : ℝ)) / K) * (1 - 2 * ε) := by ring
_ ≤ ε + 1 * (1 - 2 * ε) := by gcongr
_ ≤ 1 - ε := by ring_nf; linarith
theorem midpointWeightedRow_integral_sub_ate_abs_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
hOverlap :
Overlap ε
K :
ℕ
hK :
0 < K
H :
Measure (ScoreSpace ε)
hP :
P ∈ CausalLaws H (equalWidthRelease ε K hK)
conclusion
|(∫ z, midpointWeightedRow ε K z ∂releasedLaw P) - ate P|
≤ (1 - 2 * ε) / (2 * K * ε * (1 - ε))
Proof (Lean source)
lemma midpointWeightedRow_integral_sub_ate_abs_le
{ε : ℝ} (hOverlap : Overlap ε) (K : ℕ) (hK : 0 < K)
(H : Measure (ScoreSpace ε)) (P : Measure (FullRow ε K))
(hP : P ∈ CausalLaws H (equalWidthRelease ε K hK)) :
|(∫ z, midpointWeightedRow ε K z ∂releasedLaw P) - ate P| ≤
(1 - 2 * ε) / (2 * K * ε * (1 - ε)) := by
letI : IsProbabilityMeasure P := hP.1
let μ := P.map (latentTriple (ε := ε) (J := K))
have hlatent : Measurable (latentTriple (ε := ε) (J := K)) := by
unfold latentTriple score outcome0 outcome1
fun_prop
letI : IsProbabilityMeasure μ :=
Measure.isProbabilityMeasure_map hlatent.aemeasurable
let B := (1 - 2 * ε) / (2 * K * ε * (1 - ε))
have hB0 : 0 ≤ B := by
dsimp [B]
apply div_nonneg
· linarith [hOverlap.2]
· have hε1 : 0 < 1 - ε := by linarith [hOverlap.2]
have hKreal : 0 < (K : ℝ) := by exact_mod_cast hK
exact (mul_pos (mul_pos (mul_pos (by norm_num) hKreal) hOverlap.1) hε1).le
have hbiasInt : Integrable (midpointLatentBias ε K hK) μ := by
apply Integrable.of_bound
(midpointLatentBias_measurable ε K hK).aestronglyMeasurable B
filter_upwards [] with x
rw [Real.norm_eq_abs]
exact midpointLatentBias_abs_le hOverlap K hK x
rw [midpoint_populationBias_eq_integral hOverlap K hK P
hP.2.randomizedAssignment hP.2.consistency hP.2.deterministicRelease]
change |∫ x, midpointLatentBias ε K hK x ∂μ| ≤ B
calc
|∫ x, midpointLatentBias ε K hK x ∂μ| ≤
∫ x, |midpointLatentBias ε K hK x| ∂μ :=
MeasureTheory.abs_integral_le_integral_abs
_ ≤ ∫ _x, B ∂μ := by
apply integral_mono hbiasInt.abs (integrable_const B)
exact midpointLatentBias_abs_le hOverlap K hK
_ = B := by simp
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.InferenceMidpointConcentration 2 declarations Product-law concentration of the equal-width midpoint statistic.
Product-law concentration of the equal-width midpoint statistic.
theorem midpointWeightedRow_mem_Icc
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
midpointWeightedRow ε K z ∈ Icc (-ε⁻¹) ε⁻¹
Proof (Lean source)
lemma midpointWeightedRow_mem_Icc {ε : ℝ} (hOverlap : Overlap ε)
(K : ℕ) (hK : 0 < K) (z : Observation K) :
midpointWeightedRow ε K z ∈ Icc (-ε⁻¹) ε⁻¹ := by
have hm := equalWidthMidpoint_mem_Icc hOverlap K hK z.1
have hε : 0 < ε := hOverlap.1
have hm0 : 0 < equalWidthMidpoint ε K z.1 := hε.trans_le hm.1
have hm1 : 0 < 1 - equalWidthMidpoint ε K z.1 := by
linarith [hm.2, hOverlap.1]
by_cases ha : z.2.1 = true
· simp only [midpointWeightedRow, ha, ↓reduceIte]
constructor
· exact (div_nonneg z.2.2.property.1 hm0.le).trans' (neg_nonpos.mpr (inv_nonneg.mpr hε.le))
· have hinv := (inv_le_inv₀ hm0 hε).mpr hm.1
calc
(z.2.2 : ℝ) / equalWidthMidpoint ε K z.1 ≤
1 / equalWidthMidpoint ε K z.1 :=
div_le_div_of_nonneg_right z.2.2.property.2 hm0.le
_ ≤ ε⁻¹ := by simpa [one_div] using hinv
· have ha' : z.2.1 = false := Bool.eq_false_of_not_eq_true ha
simp only [midpointWeightedRow, ha', Bool.false_eq_true, ↓reduceIte]
have hmLower : ε ≤ 1 - equalWidthMidpoint ε K z.1 := by
linarith [hm.2]
have hinv := (inv_le_inv₀ hm1 hε).mpr hmLower
have hy : (z.2.2 : ℝ) / (1 - equalWidthMidpoint ε K z.1) ≤ ε⁻¹ := by
calc
(z.2.2 : ℝ) / (1 - equalWidthMidpoint ε K z.1) ≤
1 / (1 - equalWidthMidpoint ε K z.1) :=
div_le_div_of_nonneg_right z.2.2.property.2 hm1.le
_ ≤ ε⁻¹ := by simpa [one_div] using hinv
rw [neg_div]
constructor
· exact neg_le_neg hy
· exact (neg_nonpos.mpr (div_nonneg z.2.2.property.1 hm1.le)).trans
(inv_nonneg.mpr hε.le)
theorem midpointWeightedCenter_deviation_probability_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(Measure.pi (fun _ : Fin n => P)).real {x | (4 / ε) * sqrt (log (4 / α) / n) ≤ |midpointWeightedCenter ε n K x - ∫ z, midpointWeightedRow ε K z ∂P|}
≤ α
Proof (Lean source)
lemma midpointWeightedCenter_deviation_probability_le
{ε α : ℝ} (hOverlap : Overlap ε) (hα : 0 < α ∧ α < 1 / 2)
(K n : ℕ) (hK : 0 < K) (hn : 0 < n)
(P : Measure (Observation K)) [IsProbabilityMeasure P] :
(Measure.pi (fun _ : Fin n => P)).real
{x | (4 / ε) * sqrt (log (4 / α) / n) ≤
|midpointWeightedCenter ε n K x -
∫ z, midpointWeightedRow ε K z ∂P|} ≤ α := by
let S := Causalean.Stat.iidSample_infinitePi P
let ψ : (ℕ → Observation K) → (Fin n → Observation K) :=
fun ω k => S.Z k ω
let t := (4 / ε) * sqrt (log (4 / α) / n)
let A : Set (Fin n → Observation K) :=
{x | t ≤ |midpointWeightedCenter ε n K x -
∫ z, midpointWeightedRow ε K z ∂P|}
have hψ : Measurable ψ := Causalean.Stat.iidSample_finN_measurable S n
have hA : MeasurableSet A := by
dsimp [A, t]
apply measurableSet_le measurable_const
have hcenter : Measurable (midpointWeightedCenter ε n K) := by
rw [show midpointWeightedCenter ε n K = fun x : TrialSample n K =>
(n : ℝ)⁻¹ * ∑ i : Fin n, midpointWeightedRow ε K (x i) from
funext (midpointWeightedCenter_eq_rowMean ε n K)]
exact measurable_const.mul (Finset.measurable_fun_sum univ
fun i _ => (midpointWeightedRow_measurable' ε K).comp
(measurable_pi_apply i))
exact (hcenter.sub measurable_const).abs
have hpush : (Measure.infinitePi (fun _ : ℕ => P)).map ψ =
Measure.pi (fun _ : Fin n => P) :=
Causalean.Stat.iidSample_finN_pushforward S n
have hmeasure : (Measure.infinitePi (fun _ : ℕ => P)).real (ψ ⁻¹' A) =
(Measure.pi (fun _ : Fin n => P)).real A := by
simp only [measureReal_def]
rw [← hpush, Measure.map_apply hψ hA]
have hevent : ψ ⁻¹' A =
{ω | t ≤ |S.sampleMean (midpointWeightedRow ε K) n ω -
∫ z, midpointWeightedRow ε K z ∂P|} := by
ext ω
dsimp [A, ψ, t, sampleMean]
rw [midpointWeightedCenter_eq_rowMean]
rw [Fin.sum_univ_eq_sum_range
(fun i => midpointWeightedRow ε K (S.Z i ω)) n]
have ht0 : 0 ≤ t := by
dsimp [t]
exact mul_nonneg (div_nonneg (by norm_num) hOverlap.1.le)
(Real.sqrt_nonneg _)
have htail := hoeffding_abs_ge S
(midpointWeightedRow_measurable' ε K)
(a := -ε⁻¹) (b := ε⁻¹)
(by have hi : 0 < ε⁻¹ := inv_pos.mpr hOverlap.1; linarith)
(ae_of_all P (midpointWeightedRow_mem_Icc hOverlap K hK)) n hn ht0
have hrhs := midpointRadius_hoeffding_rhs_le hOverlap hα n hn
change (Measure.pi (fun _ : Fin n => P)).real A ≤ α
rw [← hmeasure, hevent]
exact htail.trans hrhs
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.InferenceMidpointProcedure 16 declarations The explicit midpoint-weighted honest interval procedure.
The explicit midpoint-weighted honest interval procedure.
theorem midpointWeightedRow_measurable''
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma midpointWeightedRow_measurable'' (ε : ℝ) (K : ℕ) :
Measurable (midpointWeightedRow ε K) := by
unfold midpointWeightedRow equalWidthMidpoint
apply Measurable.ite
· exact measurableSet_eq_fun (measurable_fst.comp measurable_snd) measurable_const
· fun_prop
· fun_prop
theorem midpointWeightedCenter_measurable
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma midpointWeightedCenter_measurable (ε : ℝ) (n K : ℕ) :
Measurable (midpointWeightedCenter ε n K) := by
rw [show midpointWeightedCenter ε n K = fun x : TrialSample n K =>
(n : ℝ)⁻¹ * ∑ i : Fin n, midpointWeightedRow ε K (x i) from
funext (midpointWeightedCenter_eq_rowMean ε n K)]
exact measurable_const.mul (Finset.measurable_fun_sum univ
fun i _ => (midpointWeightedRow_measurable'' ε K).comp (measurable_pi_apply i))
theorem clampATE_measurable
This result establishes the stated mathematical conclusion.
Formal statement
conclusion
Proof (Lean source)
theorem clampATE_mem_Icc
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma clampATE_mem_Icc (x : ℝ) : clampATE x ∈ Icc (-1 : ℝ) 1 := by
unfold clampATE
exact ⟨le_max_left _ _, max_le (by norm_num) (min_le_left _ _ )⟩
theorem clampATE_mono
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
theorem clampATE_abs_sub_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Proof (Lean source)
lemma clampATE_abs_sub_le (x y : ℝ) :
|clampATE x - clampATE y| ≤ |x - y| := by
unfold clampATE
calc
|max (-1) (min 1 x) - max (-1) (min 1 y)| =
|max (min 1 x) (-1) - max (min 1 y) (-1)| := by
rw [max_comm (-1), max_comm (-1)]
_ ≤ |min 1 x - min 1 y| := abs_max_sub_max_le_abs _ _ _
_ ≤ |x - y| := by
simpa using (abs_min_sub_min_le_max (1 : ℝ) x 1 y)
theorem midpointWeightedRadius_nonneg
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
0 ≤ midpointWeightedRadius ε α n K
Proof (Lean source)
lemma midpointWeightedRadius_nonneg {ε α : ℝ}
(hOverlap : Overlap ε) (hα : 0 < α ∧ α < 1 / 2)
(n K : ℕ) (hn : 0 < n) (hK : 0 < K) :
0 ≤ midpointWeightedRadius ε α n K := by
unfold midpointWeightedRadius
have hlog : 0 ≤ log (4 / α) := by
apply Real.log_nonneg
rw [le_div_iff₀ hα.1]
linarith [hα.2]
have hnR : 0 ≤ (n : ℝ) := by positivity
have hKreal : 0 < (K : ℝ) := by exact_mod_cast hK
have hε1 : 0 < 1 - ε := by linarith [hOverlap.2]
apply add_nonneg
· exact mul_nonneg (div_nonneg (by norm_num) hOverlap.1.le)
(Real.sqrt_nonneg _)
· apply div_nonneg
· linarith [hOverlap.2]
· exact (mul_pos (mul_pos (mul_pos (by norm_num) hKreal) hOverlap.1) hε1).le
def midpointInterval
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
clause 1
clause 2
clause 3
measurable_lo := clampATE_measurable.comp ((midpointWeightedCenter_measurable ε n K).sub measurable_const)
clause 4
measurable_hi := clampATE_measurable.comp ((midpointWeightedCenter_measurable ε n K).add measurable_const)
clause 5
lower_bound x := (clampATE_mem_Icc _).1
clause 6
ordered x := clampATE_mono (by have hr := midpointWeightedRadius_nonneg hOverlap hα n K hn hK linarith)
clause 7
upper_bound x := (clampATE_mem_Icc _).2
theorem midpointInterval_lo
Given overlap, the miscoverage range, positive sample and label sizes, and a released sample, the midpoint interval's lower endpoint is the clamped midpoint-weighted center minus its prescribed radius.
Formal statement
hypotheses
ε α :
ℝ
hOverlap :
Overlap ε
hα :
0 < α ∧ α < 1 / 2
n K :
ℕ
hn :
0 < n
hK :
0 < K
x :
TrialSample n K
conclusion
(midpointInterval hOverlap hα n K hn hK).lo x
Proof (Lean source)
@[simp] lemma midpointInterval_lo {ε α : ℝ} (hOverlap : Overlap ε)
(hα : 0 < α ∧ α < 1 / 2) (n K : ℕ) (hn : 0 < n) (hK : 0 < K)
(x : TrialSample n K) :
(midpointInterval hOverlap hα n K hn hK).lo x = clampATE
(midpointWeightedCenter ε n K x - midpointWeightedRadius ε α n K) := rfl
theorem midpointInterval_hi
Given overlap, the miscoverage range, positive sample and label sizes, and a released sample, the midpoint interval's upper endpoint is the clamped midpoint-weighted center plus its prescribed radius.
Formal statement
hypotheses
ε α :
ℝ
hOverlap :
Overlap ε
hα :
0 < α ∧ α < 1 / 2
n K :
ℕ
hn :
0 < n
hK :
0 < K
x :
TrialSample n K
conclusion
(midpointInterval hOverlap hα n K hn hK).hi x
Proof (Lean source)
@[simp] lemma midpointInterval_hi {ε α : ℝ} (hOverlap : Overlap ε)
(hα : 0 < α ∧ α < 1 / 2) (n K : ℕ) (hn : 0 < n) (hK : 0 < K)
(x : TrialSample n K) :
(midpointInterval hOverlap hα n K hn hK).hi x = clampATE
(midpointWeightedCenter ε n K x + midpointWeightedRadius ε α n K) := rfl
theorem midpointInterval_length_le_two_radius
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε α :
ℝ
hOverlap :
Overlap ε
hα :
0 < α ∧ α < 1 / 2
n K :
ℕ
hn :
0 < n
hK :
0 < K
x :
TrialSample n K
conclusion
Proof (Lean source)
lemma midpointInterval_length_le_two_radius {ε α : ℝ}
(hOverlap : Overlap ε) (hα : 0 < α ∧ α < 1 / 2)
(n K : ℕ) (hn : 0 < n) (hK : 0 < K) (x : TrialSample n K) :
(midpointInterval hOverlap hα n K hn hK).length x ≤
2 * midpointWeightedRadius ε α n K := by
unfold IntervalProcedure.length
rw [midpointInterval_lo, midpointInterval_hi]
have hord := (midpointInterval hOverlap hα n K hn hK).ordered x
calc
clampATE (midpointWeightedCenter ε n K x + midpointWeightedRadius ε α n K) -
clampATE (midpointWeightedCenter ε n K x - midpointWeightedRadius ε α n K) ≤
|clampATE (midpointWeightedCenter ε n K x + midpointWeightedRadius ε α n K) -
clampATE (midpointWeightedCenter ε n K x - midpointWeightedRadius ε α n K)| :=
le_abs_self _
_ ≤ |(midpointWeightedCenter ε n K x + midpointWeightedRadius ε α n K) -
(midpointWeightedCenter ε n K x - midpointWeightedRadius ε α n K)| :=
clampATE_abs_sub_le _ _
_ = 2 * midpointWeightedRadius ε α n K := by
rw [abs_of_nonneg]
· ring
· have := midpointWeightedRadius_nonneg hOverlap hα n K hn hK
linarith
theorem equalWidth_midpointInterval_honest
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma equalWidth_midpointInterval_honest {ε α : ℝ}
(hOverlap : Overlap ε) (hα : 0 < α ∧ α < 1 / 2)
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(n K : ℕ) (hn : 0 < n) (hK : 0 < K) :
(equalWidthRelease ε K hK, midpointInterval hOverlap hα n K hn hK) ∈
honestProcedures n K H α := by
refine ⟨equalWidthRelease_measurable ε K hK, ?_⟩
intro P hP
letI : IsProbabilityMeasure P := hP.1
let Q := releasedLaw P
have hrecord : Measurable (releasedRecord (ε := ε) (J := K)) := by
unfold releasedRecord label arm observed
fun_prop
letI : IsProbabilityMeasure Q := by
dsimp [Q, releasedLaw]
exact Measure.isProbabilityMeasure_map hrecord.aemeasurable
let μ := Measure.pi (fun _ : Fin n => Q)
letI : IsProbabilityMeasure μ := by
dsimp [μ]
infer_instance
let t := (4 / ε) * sqrt (log (4 / α) / n)
let m := ∫ z, midpointWeightedRow ε K z ∂Q
let bad : Set (TrialSample n K) :=
{x | t ≤ |midpointWeightedCenter ε n K x - m|}
have hbadMeas : MeasurableSet bad := by
dsimp [bad, t, m]
exact measurableSet_le measurable_const
((midpointWeightedCenter_measurable ε n K).sub measurable_const).abs
have hbad : μ.real bad ≤ α := by
dsimp [μ, bad, t, m, Q]
exact midpointWeightedCenter_deviation_probability_le
hOverlap hα K n hK hn (releasedLaw P)
have hgoodProb : 1 - α ≤ μ.real badᶜ := by
rw [MeasureTheory.measureReal_compl hbadMeas]
rw [MeasureTheory.probReal_univ]
linarith
apply hgoodProb.trans
refine measureReal_mono ?_ (measure_ne_top μ _)
intro x hx
have hxdev : |midpointWeightedCenter ε n K x - m| < t := by
exact not_le.mp hx
have hbias := midpointWeightedRow_integral_sub_ate_abs_le
hOverlap K hK H P hP
change |m - ate P| ≤
(1 - 2 * ε) / (2 * K * ε * (1 - ε)) at hbias
have hdist : |midpointWeightedCenter ε n K x - ate P| ≤
midpointWeightedRadius ε α n K := by
calc
|midpointWeightedCenter ε n K x - ate P| ≤
|midpointWeightedCenter ε n K x - m| + |m - ate P| :=
abs_sub_le _ _ _
_ ≤ t + (1 - 2 * ε) / (2 * K * ε * (1 - ε)) :=
add_le_add hxdev.le hbias
_ = midpointWeightedRadius ε α n K := by
rfl
have hbounds := abs_le.mp hdist
have hθ := ate_mem_Icc P
change ate P ∈ Icc
((midpointInterval hOverlap hα n K hn hK).lo x)
((midpointInterval hOverlap hα n K hn hK).hi x)
rw [midpointInterval_lo, midpointInterval_hi]
exact clampATE_interval_contains hθ (by linarith) (by linarith)
def midpointLengthConstant
For the specified mathematical inputs, this definition introduces the corresponding object.
theorem two_midpointWeightedRadius_le_rate
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
2 * midpointWeightedRadius ε α n K
≤ midpointLengthConstant ε α * ((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹)
Proof (Lean source)
lemma two_midpointWeightedRadius_le_rate {ε α : ℝ}
(hOverlap : Overlap ε) (hα : 0 < α ∧ α < 1 / 2)
(n K : ℕ) (hn : 0 < n) (hK : 0 < K) :
2 * midpointWeightedRadius ε α n K ≤
midpointLengthConstant ε α *
((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹) := by
have hlog : 0 ≤ log (4 / α) := by
apply Real.log_nonneg
rw [le_div_iff₀ hα.1]
linarith [hα.2]
have hε : 0 < ε := hOverlap.1
have hε1 : 0 < 1 - ε := by linarith [hOverlap.2]
have hKreal : 0 < (K : ℝ) := by exact_mod_cast hK
have hnreal : 0 < (n : ℝ) := by exact_mod_cast hn
let D := (1 - 2 * ε) / (ε * (1 - ε))
let A := (8 / ε) * sqrt (log (4 / α))
have hD : 0 ≤ D := by
dsimp [D]
exact div_nonneg (by linarith [hOverlap.2])
(mul_pos hε hε1).le
have hA : 0 ≤ A := by
dsimp [A]
exact mul_nonneg (div_nonneg (by norm_num) hε.le) (Real.sqrt_nonneg _)
have hKinv : 0 ≤ (K : ℝ)⁻¹ := inv_nonneg.mpr hKreal.le
have hninv : 0 ≤ (sqrt (n : ℝ))⁻¹ :=
inv_nonneg.mpr (Real.sqrt_nonneg _)
have hradius : 2 * midpointWeightedRadius ε α n K =
A * (sqrt (n : ℝ))⁻¹ + D * (K : ℝ)⁻¹ := by
unfold midpointWeightedRadius
rw [Real.sqrt_div hlog]
dsimp [A, D]
field_simp
ring
rw [hradius]
change A * (sqrt (n : ℝ))⁻¹ + D * (K : ℝ)⁻¹ ≤
(D + A) * ((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹)
nlinarith [mul_nonneg hD hninv, mul_nonneg hA hKinv]
theorem midpointInterval_expectedLength_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
≤ midpointLengthConstant ε α * ((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹)
Proof (Lean source)
lemma midpointInterval_expectedLength_le {ε α : ℝ}
(hOverlap : Overlap ε) (hα : 0 < α ∧ α < 1 / 2)
(n K : ℕ) (hn : 0 < n) (hK : 0 < K)
(Q : Measure (Observation K)) [IsProbabilityMeasure Q] :
(∫ x, (midpointInterval hOverlap hα n K hn hK).length x
∂Measure.pi (fun _ : Fin n => Q)) ≤
midpointLengthConstant ε α *
((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹) := by
let C := midpointInterval hOverlap hα n K hn hK
let μ := Measure.pi (fun _ : Fin n => Q)
letI : IsProbabilityMeasure μ := by
dsimp [μ]
infer_instance
have hlenInt : Integrable C.length μ := by
apply Integrable.of_bound
(C.measurable_hi.sub C.measurable_lo).aestronglyMeasurable 2
filter_upwards [] with x
have hlo := C.lower_bound x
have hord := C.ordered x
have hhi := C.upper_bound x
change |C.hi x - C.lo x| ≤ 2
rw [abs_of_nonneg (sub_nonneg.mpr hord)]
linarith
have hpoint (x : TrialSample n K) :
C.length x ≤ midpointLengthConstant ε α *
((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹) := by
exact (midpointInterval_length_le_two_radius hOverlap hα n K hn hK x).trans
(two_midpointWeightedRadius_le_rate hOverlap hα n K hn hK)
change (∫ x, C.length x ∂μ) ≤ _
calc
(∫ x, C.length x ∂μ) ≤ ∫ _x, midpointLengthConstant ε α *
((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹) ∂μ := by
apply integral_mono hlenInt (integrable_const _)
exact hpoint
_ = midpointLengthConstant ε α *
((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹) := by simp
theorem exists_equalWidth_midpointInterval
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
∃ CI : IntervalProcedure n K,
(equalWidthRelease ε K hK, CI) ∈ honestProcedures n K H α
conclusion 1
x :
conclusion 2
P ∈ CausalLaws H (equalWidthRelease ε K hK) :
(∫ x, CI.length x ∂(Measure.pi (fun _ : Fin n => releasedLaw P)))
≤ midpointLengthConstant ε α * ((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹)
Proof (Lean source)
lemma exists_equalWidth_midpointInterval {ε α : ℝ}
(hOverlap : Overlap ε) (hα : 0 < α ∧ α < 1 / 2)
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(n K : ℕ) (hn : 0 < n) (hK : 0 < K) :
∃ CI : IntervalProcedure n K,
(equalWidthRelease ε K hK, CI) ∈ honestProcedures n K H α ∧
(∀ x, CI.lo x = clampATE
(midpointWeightedCenter ε n K x - midpointWeightedRadius ε α n K) ∧
CI.hi x = clampATE
(midpointWeightedCenter ε n K x + midpointWeightedRadius ε α n K)) ∧
∀ P ∈ CausalLaws H (equalWidthRelease ε K hK),
(∫ x, CI.length x ∂(Measure.pi (fun _ : Fin n => releasedLaw P))) ≤
midpointLengthConstant ε α *
((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹) := by
let CI := midpointInterval hOverlap hα n K hn hK
refine ⟨CI, equalWidth_midpointInterval_honest hOverlap hα H n K hn hK,
?_, ?_⟩
· intro x
exact ⟨rfl, rfl⟩
· intro P hP
letI : IsProbabilityMeasure P := hP.1
have hrecord : Measurable (releasedRecord (ε := ε) (J := K)) := by
unfold releasedRecord label arm observed
fun_prop
letI : IsProbabilityMeasure (releasedLaw P) := by
unfold releasedLaw
exact Measure.isProbabilityMeasure_map hrecord.aemeasurable
exact midpointInterval_expectedLength_le hOverlap hα n K hn hK (releasedLaw P)
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.KLabelDefinitions 2 declarations Equal width finite label releases.
Equal width finite label releases.
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
⟨min (Nat.floor ((K : ℝ) * ((e : ℝ) - ε) / (1 - 2 * ε))) (K - 1), by omega⟩
theorem equalWidthRelease_measurable
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma equalWidthRelease_measurable (ε : ℝ) (K : ℕ) (hK : 0 < K) :
Measurable (equalWidthRelease ε K hK) := by
let f : ℕ → Fin K := fun n => ⟨min n (K - 1), by omega⟩
have hf : Measurable f := measurable_of_countable f
have hfloor : Measurable (fun e : ScoreSpace ε =>
Nat.floor ((K : ℝ) * ((e : ℝ) - ε) / (1 - 2 * ε))) :=
Measurable.nat_floor (by fun_prop)
convert hf.comp hfloor using 1
funext e
rfl
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.KLabelFiniteGeometry 9 declarations This file records the two-arm algebra used to turn cellwise reciprocal-score deviations into ordinary absolute score deviations.
Finite label reciprocal-score geometry
This file records the two-arm algebra used to turn cellwise reciprocal-score deviations into ordinary absolute score deviations.
theorem equalWidthRelease_cell_subset_closedBin
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
cell (equalWidthRelease ε K hK) r ⊆ {e : ScoreSpace ε | ε + (r : ℕ) * ((1 - 2 * ε) / K) ≤ (e : ℝ) ∧ (e : ℝ) ≤ ε + ((r : ℕ) + 1) * ((1 - 2 * ε) / K)}
Proof (Lean source)
lemma equalWidthRelease_cell_subset_closedBin {ε : ℝ} {K : ℕ}
(hOverlap : Overlap ε) (hK : 0 < K) (r : LabelSpace K) :
cell (equalWidthRelease ε K hK) r ⊆
{e : ScoreSpace ε |
ε + (r : ℕ) * ((1 - 2 * ε) / K) ≤ (e : ℝ) ∧
(e : ℝ) ≤ ε + ((r : ℕ) + 1) * ((1 - 2 * ε) / K)} := by
intro e he
let L : ℝ := 1 - 2 * ε
let t : ℝ := (K : ℝ) * ((e : ℝ) - ε) / L
have hL : 0 < L := by dsimp [L]; linarith [hOverlap.2]
have hKreal : 0 < (K : ℝ) := by exact_mod_cast hK
have ht0 : 0 ≤ t := by
dsimp [t]
exact div_nonneg
(mul_nonneg hKreal.le (sub_nonneg.mpr e.property.1)) hL.le
have hmin : min (Nat.floor t) (K - 1) = (r : ℕ) := by
have := congrArg val he
simpa [cell, equalWidthRelease, t, L] using this
have hrfloor : (r : ℕ) ≤ Nat.floor t := by omega
have hlowerT : (r : ℝ) ≤ t := by
have hcast : ((r : ℕ) : ℝ) ≤ (Nat.floor t : ℕ) := by exact_mod_cast hrfloor
exact hcast.trans (Nat.floor_le ht0)
have hlower : ε + (r : ℕ) * (L / K) ≤ (e : ℝ) := by
dsimp [t] at hlowerT
have h := (mul_le_mul_of_nonneg_right hlowerT (div_nonneg hL.le hKreal.le))
field_simp at h ⊢
nlinarith
have hupper : (e : ℝ) ≤ ε + ((r : ℕ) + 1) * (L / K) := by
by_cases hr : (r : ℕ) = K - 1
· have heUpper := e.property.2
rw [hr]
have hKsub : (K - 1 : ℕ) + 1 = K := by omega
have hKcast : ((K - 1 : ℕ) : ℝ) + 1 = (K : ℝ) := by
exact_mod_cast hKsub
rw [hKcast]
field_simp
dsimp [L]
linarith
· have hfloor : Nat.floor t = (r : ℕ) := by omega
have htUpper : t < (r : ℕ) + 1 := by
rw [← hfloor]
exact Nat.lt_floor_add_one t
dsimp [t] at htUpper
have h := (mul_lt_mul_of_pos_right htUpper (div_pos hL hKreal))
field_simp at h ⊢
nlinarith
simpa [L] using ⟨hlower, hupper⟩
def equalWidthBinMidpoint
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
⟨ε
+ ((r : ℕ) + (1 / 2 : ℝ)) * ((1 - 2 * ε) / K), by have hKreal : 0 < (K : ℝ) := by exact_mod_cast hK have hwidth : 0 < 1
- 2 * ε := by linarith [hOverlap.2] have hr0 : 0
≤ (r : ℝ) := by positivity have hrK : (r : ℝ)
+ (1 / 2 : ℝ) ≤ (K : ℝ) := by have hrCast : (r : ℝ)
+ 1
≤ (K : ℝ) := by exact_mod_cast (Nat.succ_le_iff.mpr r.isLt) linarith constructor · exact le_add_of_nonneg_right (mul_nonneg (by linarith) (div_nonneg hwidth.le hKreal.le)) · have hratio : ((r : ℝ) + (1 / 2 : ℝ)) / K ≤ 1 := (div_le_one hKreal).mpr hrK calc ε
+ ((r : ℕ) + (1 / 2 : ℝ)) * ((1 - 2 * ε) / K) = ε
+ (((r : ℝ) + (1 / 2 : ℝ)) / K) * (1 - 2 * ε) := by ring _ ≤ ε
+ 1 * (1 - 2 * ε) := by gcongr _ ≤ 1
- ε := by ring_nf; linarith ⟩
theorem equalWidthBinMidpoint_val
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(equalWidthBinMidpoint ε K hK hOverlap r : ℝ)
= ε + ((r : ℕ) + (1 / 2 : ℝ)) * ((1 - 2 * ε) / K)
Proof (Lean source)
lemma equalWidthBinMidpoint_val (ε : ℝ) (K : ℕ) (hK : 0 < K)
(hOverlap : Overlap ε) (r : LabelSpace K) :
(equalWidthBinMidpoint ε K hK hOverlap r : ℝ) =
ε + ((r : ℕ) + (1 / 2 : ℝ)) * ((1 - 2 * ε) / K) := by
rfl
theorem cellAbsoluteDeviation_le_center
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
a :
r :
c :
ℝ
hc :
c ∈ Icc ((1 - ε)⁻¹) ε⁻¹
conclusion
cellAbsoluteDeviation H g a r ≤ ∫ e in cell g r, |1 - c * armProb a e| ∂H
Proof (Lean source)
lemma cellAbsoluteDeviation_le_center {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(a : ArmSpace) (r : LabelSpace J) (c : ℝ)
(hc : c ∈ Icc ((1 - ε)⁻¹) ε⁻¹) :
cellAbsoluteDeviation H g a r ≤
∫ e in cell g r, |1 - c * armProb a e| ∂H := by
unfold cellAbsoluteDeviation
apply csInf_le
· refine ⟨0, ?_⟩
rintro v ⟨d, hd, rfl⟩
exact integral_nonneg fun _ => abs_nonneg _
· exact ⟨c, hc, rfl⟩
theorem sum_cellAbsoluteDeviation_le_commonCenterCandidates
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
+ ∫ e in cell g r, |1
Proof (Lean source)
lemma sum_cellAbsoluteDeviation_le_commonCenterCandidates
{ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(hOverlap : Overlap ε) (r : LabelSpace J) (z : ScoreSpace ε) :
cellAbsoluteDeviation H g true r + cellAbsoluteDeviation H g false r ≤
(∫ e in cell g r, |1 - (z : ℝ)⁻¹ * armProb true e| ∂H) +
∫ e in cell g r, |1 - (1 - (z : ℝ))⁻¹ * armProb false e| ∂H := by
have hz0 : 0 < (z : ℝ) := hOverlap.1.trans_le z.property.1
have hz1 : (z : ℝ) < 1 := z.property.2.trans_lt (by linarith [hOverlap.1])
have hε1 : 0 < 1 - ε := by linarith [hOverlap.1, hOverlap.2]
have hzCenter : (z : ℝ)⁻¹ ∈ Icc ((1 - ε)⁻¹) ε⁻¹ :=
⟨(inv_le_inv₀ hε1 hz0).mpr z.property.2,
(inv_le_inv₀ hz0 hOverlap.1).mpr z.property.1⟩
have hcontrolLower : ε ≤ 1 - (z : ℝ) := by linarith [z.property.2]
have hcontrolUpper : 1 - (z : ℝ) ≤ 1 - ε := by linarith [z.property.1]
have hcontrolCenter : (1 - (z : ℝ))⁻¹ ∈ Icc ((1 - ε)⁻¹) ε⁻¹ :=
⟨(inv_le_inv₀ hε1 (by linarith)).mpr hcontrolUpper,
(inv_le_inv₀ (by linarith) hOverlap.1).mpr hcontrolLower⟩
exact add_le_add
(cellAbsoluteDeviation_le_center H g true r (z : ℝ)⁻¹ hzCenter)
(cellAbsoluteDeviation_le_center H g false r
(1 - (z : ℝ))⁻¹ hcontrolCenter)
theorem sum_arm_reciprocalDeviation_eq
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
x z :
ℝ
hz0 :
0 < z
hz1 :
z < 1
conclusion
|1 - z⁻¹ * x| + |1 - (1 - z)⁻¹ * (1 - x)| = (z * (1 - z))⁻¹ * |x - z|
Proof (Lean source)
lemma sum_arm_reciprocalDeviation_eq {x z : ℝ}
(hz0 : 0 < z) (hz1 : z < 1) :
|1 - z⁻¹ * x| + |1 - (1 - z)⁻¹ * (1 - x)| =
(z * (1 - z))⁻¹ * |x - z| := by
have h1z : 0 < 1 - z := by linarith
have hzInv : 0 ≤ z⁻¹ := inv_nonneg.mpr hz0.le
have h1zInv : 0 ≤ (1 - z)⁻¹ := inv_nonneg.mpr h1z.le
have hfirst : |1 - z⁻¹ * x| = z⁻¹ * |x - z| := by
calc
|1 - z⁻¹ * x| = |z⁻¹ * (z - x)| := by
congr 1
field_simp
_ = |z⁻¹| * |z - x| := abs_mul _ _
_ = z⁻¹ * |x - z| := by rw [abs_of_nonneg hzInv, abs_sub_comm]
have hsecond : |1 - (1 - z)⁻¹ * (1 - x)| =
(1 - z)⁻¹ * |x - z| := by
calc
|1 - (1 - z)⁻¹ * (1 - x)| =
|(1 - z)⁻¹ * (x - z)| := by
congr 1
field_simp
ring
_ = |(1 - z)⁻¹| * |x - z| := abs_mul _ _
_ = (1 - z)⁻¹ * |x - z| := by rw [abs_of_nonneg h1zInv]
rw [hfirst, hsecond, ← add_mul]
congr 1
field_simp
ring
theorem sum_arm_reciprocalCoefficient_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(z * (1 - z))⁻¹ ≤ (ε * (1 - ε))⁻¹
Proof (Lean source)
lemma sum_arm_reciprocalCoefficient_le {ε z : ℝ}
(hOverlap : Overlap ε) (hz : z ∈ Icc ε (1 - ε)) :
(z * (1 - z))⁻¹ ≤ (ε * (1 - ε))⁻¹ := by
have hε1 : 0 < 1 - ε := by linarith [hOverlap.1, hOverlap.2]
have hz0 : 0 < z := hOverlap.1.trans_le hz.1
have hz1 : z < 1 := hz.2.trans_lt (by linarith [hOverlap.1])
have hprod : ε * (1 - ε) ≤ z * (1 - z) := by
have hnonneg : 0 ≤ (z - ε) * (1 - ε - z) :=
mul_nonneg (sub_nonneg.mpr hz.1) (sub_nonneg.mpr hz.2)
nlinarith
exact (inv_le_inv₀ (mul_pos hz0 (by linarith))
(mul_pos hOverlap.1 hε1)).mpr hprod
theorem sum_arm_reciprocalDeviation_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
|1 - z⁻¹ * x| + |1 - (1 - z)⁻¹ * (1 - x)| ≤ (ε * (1 - ε))⁻¹ * |x - z|
Proof (Lean source)
lemma sum_arm_reciprocalDeviation_le {ε x z : ℝ}
(hOverlap : Overlap ε) (hz : z ∈ Icc ε (1 - ε)) :
|1 - z⁻¹ * x| + |1 - (1 - z)⁻¹ * (1 - x)| ≤
(ε * (1 - ε))⁻¹ * |x - z| := by
rw [sum_arm_reciprocalDeviation_eq
(hOverlap.1.trans_le hz.1) (hz.2.trans_lt (by linarith [hOverlap.1]))]
exact mul_le_mul_of_nonneg_right
(sum_arm_reciprocalCoefficient_le hOverlap hz) (abs_nonneg _)
theorem sum_cellAbsoluteDeviation_le_centeredScoreIntegral
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
≤ (ε * (1 - ε))⁻¹ * ∫ e in cell g r, |(e : ℝ) - (z : ℝ)| ∂H
Proof (Lean source)
lemma sum_cellAbsoluteDeviation_le_centeredScoreIntegral
{ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace J) (hg : Measurable g)
(hOverlap : Overlap ε) (r : LabelSpace J) (z : ScoreSpace ε) :
cellAbsoluteDeviation H g true r + cellAbsoluteDeviation H g false r ≤
(ε * (1 - ε))⁻¹ *
∫ e in cell g r, |(e : ℝ) - (z : ℝ)| ∂H := by
have hcell : MeasurableSet (cell g r) :=
measurableSet_eq_fun hg measurable_const
have hpBounds (a : ArmSpace) (e : ScoreSpace ε) :
0 ≤ armProb a e ∧ armProb a e ≤ 1 := by
rcases e.property with ⟨he0, he1⟩
have he0' : 0 ≤ (e : ℝ) := hOverlap.1.le.trans he0
have he1' : (e : ℝ) ≤ 1 := he1.trans (by linarith [hOverlap.1])
cases a
· simp only [armProb, Bool.false_eq_true, ↓reduceIte]
exact ⟨by linarith, by linarith⟩
· simp only [armProb, ↓reduceIte]
exact ⟨he0', he1'⟩
have hcandidateInt (a : ArmSpace) (c : ℝ) :
IntegrableOn (fun e : ScoreSpace ε => |1 - c * armProb a e|)
(cell g r) H := by
apply Integrable.of_bound (by
cases a <;> simp [armProb] <;> fun_prop) (1 + |c|)
filter_upwards [] with e
calc
‖|1 - c * armProb a e|‖ = |1 - c * armProb a e| := by simp
_ ≤ |1| + |c * armProb a e| := abs_sub _ _
_ ≤ 1 + |c| := by
rw [abs_one, abs_mul]
have hmul := mul_le_of_le_one_right (abs_nonneg c)
(abs_le.mpr ⟨by linarith [(hpBounds a e).1], (hpBounds a e).2⟩)
linarith
have hscoreInt : IntegrableOn
(fun e : ScoreSpace ε => |(e : ℝ) - (z : ℝ)|) (cell g r) H := by
apply Integrable.of_bound (by fun_prop) 1
filter_upwards [] with e
have he := e.property
have hz := z.property
have hwidth : 1 - 2 * ε < 1 := by linarith [hOverlap.1]
rw [Real.norm_eq_abs, abs_abs]
exact abs_le.mpr
⟨by linarith [he.1, he.2, hz.1, hz.2],
by linarith [he.1, he.2, hz.1, hz.2]⟩
have hrightInt : IntegrableOn
(fun e : ScoreSpace ε =>
(ε * (1 - ε))⁻¹ * |(e : ℝ) - (z : ℝ)|) (cell g r) H :=
hscoreInt.const_mul _
calc
cellAbsoluteDeviation H g true r + cellAbsoluteDeviation H g false r ≤
(∫ e in cell g r, |1 - (z : ℝ)⁻¹ * armProb true e| ∂H) +
∫ e in cell g r,
|1 - (1 - (z : ℝ))⁻¹ * armProb false e| ∂H :=
sum_cellAbsoluteDeviation_le_commonCenterCandidates H g hOverlap r z
_ = ∫ e in cell g r,
(|1 - (z : ℝ)⁻¹ * armProb true e| +
|1 - (1 - (z : ℝ))⁻¹ * armProb false e|) ∂H := by
rw [integral_add (hcandidateInt true (z : ℝ)⁻¹)
(hcandidateInt false (1 - (z : ℝ))⁻¹)]
_ ≤ ∫ e in cell g r,
(ε * (1 - ε))⁻¹ * |(e : ℝ) - (z : ℝ)| ∂H := by
apply integral_mono
((hcandidateInt true (z : ℝ)⁻¹).add
(hcandidateInt false (1 - (z : ℝ))⁻¹)) hrightInt
intro e
simpa [armProb] using sum_arm_reciprocalDeviation_le
hOverlap z.property (x := (e : ℝ))
_ = (ε * (1 - ε))⁻¹ *
∫ e in cell g r, |(e : ℝ) - (z : ℝ)| ∂H := by
rw [integral_const_mul]
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.KLabelFiniteLower 3 declarations Measure geometry for finite-label lower bounds under arbitrary measurable cells.
Measure geometry for finite-label lower bounds under arbitrary measurable cells.
theorem boundedCell_absDeviation_tail_lower
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
volume.real B - 2 * t ≤ (volume.restrict B).real {x : ℝ | t < |x - z|}
Proof (Lean source)
lemma boundedCell_absDeviation_tail_lower
{a b z t : ℝ} {B : Set ℝ} (hBmeas : MeasurableSet B)
(hBsub : B ⊆ Icc a b) (ht : 0 ≤ t) :
volume.real B - 2 * t ≤
(volume.restrict B).real {x : ℝ | t < |x - z|} := by
let C : Set ℝ := Icc (z - t) (z + t)
let T : Set ℝ := B ∩ {x : ℝ | t < |x - z|}
have htailMeas : MeasurableSet {x : ℝ | t < |x - z|} := by
exact measurableSet_lt measurable_const ((measurable_id.sub measurable_const).norm)
have hTmeas : MeasurableSet T := hBmeas.inter htailMeas
have hcover : B ⊆ C ∪ T := by
intro x hx
by_cases hxt : t < |x - z|
· exact mem_union_right C ⟨hx, hxt⟩
· left
dsimp [C]
rw [mem_Icc]
have habs : |x - z| ≤ t := le_of_not_gt hxt
exact ⟨by linarith [neg_le_abs (x - z)], by linarith [le_abs_self (x - z)]⟩
have hBfinite : volume B ≠ ∞ :=
(measure_ne_top_of_subset hBsub measure_Icc_lt_top.ne)
have hTfinite : volume T ≠ ∞ :=
(measure_ne_top_of_subset inter_subset_left hBfinite)
have hCfinite : volume C ≠ ∞ := measure_Icc_lt_top.ne
have hUnionFinite : volume (C ∪ T) ≠ ∞ :=
(measure_union_lt_top hCfinite.lt_top hTfinite.lt_top).ne
have hmass : volume.real B ≤ volume.real C + volume.real T :=
(measureReal_mono hcover hUnionFinite).trans (measureReal_union_le C T)
have hCmass : volume.real C = 2 * t := by
dsimp [C]
rw [Real.volume_real_Icc_of_le (by linarith)]
ring
have hTmass : volume.real T =
(volume.restrict B).real {x : ℝ | t < |x - z|} := by
rw [Measure.real_def, Measure.real_def, Measure.restrict_apply htailMeas]
rw [inter_comm]
rw [hCmass, hTmass] at hmass
linarith
theorem sum_cellMass_sq_lower
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma sum_cellMass_sq_lower (K : ℕ) (hK : 0 < K) (m : Fin K → ℝ) :
(∑ j, m j) ^ 2 / K ≤ ∑ j, (m j) ^ 2 := by
have h := sq_sum_le_card_mul_sum_sq (s := univ) (f := m)
simp only [Finset.card_univ, Fintype.card_fin] at h
have hKreal : 0 < (K : ℝ) := by exact_mod_cast hK
apply (div_le_iff₀ hKreal).2
simpa [mul_comm] using h
theorem boundedCell_absDeviation_integral_lower
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
volume.real B ^ 2 / 4 ≤ ∫ x in B, |x - z|
Proof (Lean source)
lemma boundedCell_absDeviation_integral_lower
{a b z : ℝ} {B : Set ℝ} (hBmeas : MeasurableSet B)
(hBsub : B ⊆ Icc a b) :
volume.real B ^ 2 / 4 ≤ ∫ x in B, |x - z| := by
let μ : Measure ℝ := volume.restrict B
let q : ℝ → ℝ := fun t => μ.real {x : ℝ | t < |x - z|}
have hμfinite : μ univ ≠ ∞ := by
dsimp [μ]
rw [Measure.restrict_apply_univ]
exact measure_ne_top_of_subset hBsub measure_Icc_lt_top.ne
let _ : IsFiniteMeasure μ := ⟨hμfinite.lt_top⟩
have hInt : Integrable (fun x : ℝ => |x - z|) μ := by
apply Integrable.of_bound (by fun_prop) (|a| + |b| + |z| + 1)
filter_upwards [ae_restrict_mem hBmeas] with x hx
have hxI := hBsub hx
have hxabs : |x| ≤ |a| + |b| := by
apply abs_le.mpr
constructor
· calc
-(|a| + |b|) ≤ -|a| := by linarith [abs_nonneg b]
_ ≤ a := neg_abs_le a
_ ≤ x := hxI.1
· calc
x ≤ b := hxI.2
_ ≤ |b| := le_abs_self b
_ ≤ |a| + |b| := by linarith [abs_nonneg a]
calc
‖|x - z|‖ = |x - z| := by simp
_ ≤ |x| + |z| := abs_sub x z
_ ≤ |a| + |b| + |z| + 1 := by linarith
have hnonneg : 0 ≤ᵐ[μ] (fun x : ℝ => |x - z|) :=
Eventually.of_forall fun x => abs_nonneg _
have hqmeas : Measurable q := by
apply Measurable.ennreal_toReal
exact Antitone.measurable fun s t hst => measure_mono fun _ hx => hst.trans_lt hx
have hqnonneg : ∀ t, 0 ≤ q t := fun _ => measureReal_nonneg
have hlayer := lintegral_eq_lintegral_meas_lt μ hnonneg hInt.aemeasurable
have hqLIntegral :
(∫⁻ t in Ioi (0 : ℝ), ofReal (q t)) =
∫⁻ t in Ioi (0 : ℝ), μ {x : ℝ | t < |x - z|} := by
apply setLIntegral_congr_fun measurableSet_Ioi
intro t ht
dsimp [q]
rw [Measure.real_def, ENNReal.ofReal_toReal]
exact measure_ne_top_of_subset (subset_univ _) hμfinite
have hqIntegralFinite :
(∫⁻ t in Ioi (0 : ℝ), ofReal (q t)) < ∞ := by
rw [hqLIntegral, ← hlayer]
exact hInt.lintegral_lt_top
have hqInt : IntegrableOn q (Ioi (0 : ℝ)) := by
refine ⟨hqmeas.aestronglyMeasurable, ?_⟩
rw [hasFiniteIntegral_iff_norm]
simpa [Real.norm_eq_abs, abs_of_nonneg (hqnonneg _), ENNReal.ofReal_eq_coe_nnreal]
using hqIntegralFinite
have hlayerReal := hInt.integral_eq_integral_meas_lt hnonneg
change volume.real B ^ 2 / 4 ≤ ∫ x, |x - z| ∂μ
rw [hlayerReal]
let m : ℝ := volume.real B
have hm : 0 ≤ m := measureReal_nonneg
have hbaseInt : IntegrableOn (fun t : ℝ => m - 2 * t) (Ioc 0 (m / 2)) := by
exact ((continuous_const.sub (continuous_const.mul continuous_id)).integrableOn_Icc).mono_set
Ioc_subset_Icc_self
have htailInt : IntegrableOn q (Ioc 0 (m / 2)) :=
hqInt.mono_set Ioc_subset_Ioi_self
have hmono :
(∫ t in Ioc 0 (m / 2), m - 2 * t) ≤
∫ t in Ioc 0 (m / 2), q t := by
apply integral_mono_ae hbaseInt htailInt
filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht
dsimp [q, μ, m]
exact boundedCell_absDeviation_tail_lower hBmeas hBsub ht.1.le
have hsubset :
(∫ t in Ioc 0 (m / 2), q t) ≤ ∫ t in Ioi 0, q t := by
exact integral_mono_measure (Measure.restrict_mono_set volume Ioc_subset_Ioi_self)
(Eventually.of_forall fun t => hqnonneg t) hqInt
have hbase : (∫ t in Ioc 0 (m / 2), m - 2 * t) = m ^ 2 / 4 := by
rw [← intervalIntegral.integral_of_le (by linarith)]
rw [intervalIntegral.integral_sub]
· rw [intervalIntegral.integral_const_mul]
rw [integral_id]
simp [intervalIntegral.integral_const, smul_eq_mul]
ring
· exact continuous_const.intervalIntegrable 0 (m / 2)
· exact (continuous_const.mul continuous_id).intervalIntegrable 0 (m / 2)
dsimp [m] at hbase
rw [← hbase]
exact hmono.trans hsubset
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.KLabelRateLower 11 declarations Transfer of finite-cell Lebesgue quantization lower bounds to score-law ambiguity.
Transfer of finite-cell Lebesgue quantization lower bounds to score-law ambiguity.
theorem rateLower_scoreBaseMeasure_isFinite
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
conclusion
Proof (Lean source)
lemma rateLower_scoreBaseMeasure_isFinite (ε : ℝ) :
IsFiniteMeasure
((volume : Measure ℝ).comap (Subtype.val : ScoreSpace ε → ℝ)) := by
refine ⟨?_⟩
rw [(MeasurableEmbedding.subtype_coe measurableSet_Icc).comap_apply]
rw [image_univ, Subtype.range_coe_subtype]
simp only [Set.ofPred_mem_eq]
rw [Real.volume_Icc]
finiteness
def realScoreCell
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
theorem realScoreCell_intervalPartition
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
IsIntervalPartition ε (1 - ε) K (realScoreCell g)
Proof (Lean source)
lemma realScoreCell_intervalPartition {ε : ℝ} {K : ℕ}
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g) :
IsIntervalPartition ε (1 - ε) K (realScoreCell g) := by
have hcoe : MeasurableEmbedding (Subtype.val : ScoreSpace ε → ℝ) :=
MeasurableEmbedding.subtype_coe measurableSet_Icc
refine ⟨?_, ?_, ?_⟩
· intro r
apply hcoe.measurableSet_image'
exact measurableSet_eq_fun hg measurable_const
· intro i j hij
apply Set.disjoint_left.mpr
intro x hxi hxj
rcases hxi with ⟨ei, hei, rfl⟩
rcases hxj with ⟨ej, hej, heq⟩
have heij : ei = ej := Subtype.ext heq.symm
subst ej
exact hij (hei.symm.trans hej)
· ext x
constructor
· intro hx
rcases mem_iUnion.mp hx with ⟨r, e, he, rfl⟩
exact e.property
· intro hx
let e : ScoreSpace ε := ⟨x, hx⟩
exact mem_iUnion_of_mem (g e) ⟨e, rfl, rfl⟩
theorem scoreBase_setIntegral_abs_eq_realScoreCell
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(∫ e in cell g r, |(e : ℝ) - z| ∂((volume : Measure ℝ).comap (Subtype.val : ScoreSpace ε → ℝ)))
= ∫ x in realScoreCell g r, |x - z|
Proof (Lean source)
lemma scoreBase_setIntegral_abs_eq_realScoreCell {ε : ℝ} {K : ℕ}
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(r : LabelSpace K) (z : ℝ) :
(∫ e in cell g r, |(e : ℝ) - z|
∂((volume : Measure ℝ).comap (Subtype.val : ScoreSpace ε → ℝ))) =
∫ x in realScoreCell g r, |x - z| := by
let μ : Measure (ScoreSpace ε) :=
(volume : Measure ℝ).comap (Subtype.val : ScoreSpace ε → ℝ)
have hcell : MeasurableSet (cell g r) := measurableSet_eq_fun hg measurable_const
have himageMeas : MeasurableSet (realScoreCell g r) :=
(MeasurableEmbedding.subtype_coe measurableSet_Icc).measurableSet_image' hcell
have hmap := MeasureTheory.setIntegral_map
(μ := μ) (g := (Subtype.val : ScoreSpace ε → ℝ))
(f := fun x : ℝ => |x - z|) (s := realScoreCell g r) himageMeas
(by fun_prop) measurable_subtype_coe.aemeasurable
have hmapMeasure : Measure.map (Subtype.val : ScoreSpace ε → ℝ) μ =
volume.restrict (Icc ε (1 - ε)) :=
map_comap_subtype_coe measurableSet_Icc volume
rw [hmapMeasure] at hmap
have himageSub : realScoreCell g r ⊆ Icc ε (1 - ε) := by
rintro x ⟨e, he, rfl⟩
exact e.property
have hrestrict :
(volume.restrict (Icc ε (1 - ε))).restrict (realScoreCell g r) =
volume.restrict (realScoreCell g r) :=
Measure.restrict_restrict_of_subset himageSub
have hpreimage : cell g r =
(Subtype.val : ScoreSpace ε → ℝ) ⁻¹' (realScoreCell g r) := by
ext e
constructor
· intro he
exact ⟨e, he, rfl⟩
· rintro ⟨e', he', heq⟩
simpa [Subtype.ext heq] using he'
rw [hpreimage]
rw [← hmap]
change (∫ x, |x - z|
∂(volume.restrict (Icc ε (1 - ε))).restrict (realScoreCell g r)) = _
rw [hrestrict]
theorem setIntegral_lowerScoreDensity_lower
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
H :
Measure (ScoreSpace ε)
f :
ScoreSpace ε → ℝ
mf :
ℝ
hmf :
0 < mf
hf :
LowerScoreDensity H f mf
s :
Set (ScoreSpace ε)
hs :
φ :
ScoreSpace ε → ℝ
hφ :
hφH :
IntegrableOn φ s H
hφnonneg :
∀ e ∈ s, 0 ≤ φ e
conclusion
mf * (∫ e in s, φ e ∂((volume : Measure ℝ).comap (Subtype.val : ScoreSpace ε → ℝ)))
≤ ∫ e in s, φ e ∂H
Proof (Lean source)
lemma setIntegral_lowerScoreDensity_lower {ε : ℝ}
(H : Measure (ScoreSpace ε)) (f : ScoreSpace ε → ℝ)
(mf : ℝ) (hmf : 0 < mf)
(hf : LowerScoreDensity H f mf)
(s : Set (ScoreSpace ε)) (hs : MeasurableSet s)
(φ : ScoreSpace ε → ℝ)
(hφ : IntegrableOn φ s
((volume : Measure ℝ).comap (Subtype.val : ScoreSpace ε → ℝ)))
(hφH : IntegrableOn φ s H)
(hφnonneg : ∀ e ∈ s, 0 ≤ φ e) :
mf * (∫ e in s, φ e
∂((volume : Measure ℝ).comap (Subtype.val : ScoreSpace ε → ℝ))) ≤
∫ e in s, φ e ∂H := by
let μ : Measure (ScoreSpace ε) :=
(volume : Measure ℝ).comap (Subtype.val : ScoreSpace ε → ℝ)
let _ : IsFiniteMeasure μ := rateLower_scoreBaseMeasure_isFinite ε
have hfMeas : Measurable f := hf.densityLower.1
have hfLower : ∀ᵐ e ∂μ, mf ≤ f e := hf.densityLower.2.2
have hfNonneg : ∀ᵐ e ∂μ, 0 ≤ f e :=
hfLower.mono fun _ he => hmf.le.trans he
have hindicatorInt : Integrable (s.indicator φ) μ :=
hφ.integrable_indicator hs
have hleft : Integrable (fun e => mf * s.indicator φ e) μ :=
hindicatorInt.const_mul mf
have hright : Integrable (fun e => f e * s.indicator φ e) μ := by
have hwith : Integrable (s.indicator φ)
(μ.withDensity fun e => ofReal (f e)) := by
rw [← hf.densityLower.2.1]
exact hφH.integrable_indicator hs
have hweighted :=
(integrable_withDensity_iff_integrable_smul'
hfMeas.ennreal_ofReal (Eventually.of_forall fun _ => by finiteness)).mp hwith
apply hweighted.congr
filter_upwards [hfNonneg] with e he
simp only [smul_eq_mul, ENNReal.toReal_ofReal he]
rw [← integral_indicator hs, ← integral_indicator hs]
rw [hf.densityLower.2.1,
integral_withDensity_eq_integral_toReal_smul
hfMeas.ennreal_ofReal (Eventually.of_forall fun _ => by finiteness)]
simp only [smul_eq_mul]
calc
mf * ∫ e, s.indicator φ e ∂μ =
∫ e, mf * s.indicator φ e ∂μ := by rw [integral_const_mul]
_ ≤ ∫ e, f e * s.indicator φ e ∂μ := by
apply integral_mono_ae hleft hright
filter_upwards [hfLower] with e he
by_cases hes : e ∈ s
· simp only [Set.indicator_of_mem hes]
exact mul_le_mul_of_nonneg_right he (hφnonneg e hes)
· simp [hes]
_ = ∫ e, (ofReal (f e)).toReal * s.indicator φ e ∂μ := by
apply integral_congr_ae
filter_upwards [hfNonneg] with e he
rw [ENNReal.toReal_ofReal he]
theorem scoreCell_absDeviation_lower
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
K :
ℕ
H :
Measure (ScoreSpace ε)
f :
ScoreSpace ε → ℝ
mf :
ℝ
_hOverlap :
Overlap ε
hmf :
0 < mf
hf :
LowerScoreDensity H f mf
g :
ScoreSpace ε → LabelSpace K
hg :
r :
z :
ℝ
conclusion
mf * (volume.real (realScoreCell g r) ^ 2 / 4) ≤ ∫ e in cell g r, |(e : ℝ) - z| ∂H
Proof (Lean source)
lemma scoreCell_absDeviation_lower {ε : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) (f : ScoreSpace ε → ℝ)
(mf : ℝ) (_hOverlap : Overlap ε) (hmf : 0 < mf)
(hf : LowerScoreDensity H f mf)
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(r : LabelSpace K) (z : ℝ) :
mf * (volume.real (realScoreCell g r) ^ 2 / 4) ≤
∫ e in cell g r, |(e : ℝ) - z| ∂H := by
letI : IsProbabilityMeasure H := hf.probability
let μ : Measure (ScoreSpace ε) :=
(volume : Measure ℝ).comap (Subtype.val : ScoreSpace ε → ℝ)
let _ : IsFiniteMeasure μ := rateLower_scoreBaseMeasure_isFinite ε
have hcell : MeasurableSet (cell g r) := measurableSet_eq_fun hg measurable_const
have hInt : IntegrableOn (fun e : ScoreSpace ε => |(e : ℝ) - z|) (cell g r) μ := by
apply Integrable.of_bound (by fun_prop) (|ε| + |1 - ε| + |z| + 1)
filter_upwards [] with e
have he := e.property
rw [Real.norm_eq_abs, abs_abs]
calc
|(e : ℝ) - z| ≤ |(e : ℝ)| + |z| := abs_sub _ _
_ ≤ |ε| + |1 - ε| + |z| + 1 := by
have heabs : |(e : ℝ)| ≤ |ε| + |1 - ε| := by
apply abs_le.mpr
constructor
· calc
-(|ε| + |1 - ε|) ≤ -|ε| := by linarith [abs_nonneg (1 - ε)]
_ ≤ ε := neg_abs_le ε
_ ≤ (e : ℝ) := he.1
· calc
(e : ℝ) ≤ 1 - ε := he.2
_ ≤ |1 - ε| := le_abs_self (1 - ε)
_ ≤ |ε| + |1 - ε| := by linarith [abs_nonneg ε]
linarith
have hdensity := setIntegral_lowerScoreDensity_lower H f mf hmf hf
(cell g r) hcell (fun e => |(e : ℝ) - z|) hInt
(by
apply Integrable.of_bound (by fun_prop) (|ε| + |1 - ε| + |z| + 1)
filter_upwards [] with e
have he := e.property
rw [Real.norm_eq_abs, abs_abs]
have heabs : |(e : ℝ)| ≤ |ε| + |1 - ε| := by
apply abs_le.mpr
constructor
· calc
-(|ε| + |1 - ε|) ≤ -|ε| := by linarith [abs_nonneg (1 - ε)]
_ ≤ ε := neg_abs_le ε
_ ≤ (e : ℝ) := he.1
· calc
(e : ℝ) ≤ 1 - ε := he.2
_ ≤ |1 - ε| := le_abs_self (1 - ε)
_ ≤ |ε| + |1 - ε| := by linarith [abs_nonneg ε]
linarith [abs_sub (e : ℝ) z])
(fun _ _ => abs_nonneg _)
rw [scoreBase_setIntegral_abs_eq_realScoreCell g hg r z] at hdensity
have hgeom := boundedCell_absDeviation_integral_lower (z := z)
((realScoreCell_intervalPartition g hg).1 r)
(fun x hx => by
rcases hx with ⟨e, he, rfl⟩
exact e.property)
calc
mf * (volume.real (realScoreCell g r) ^ 2 / 4) ≤
mf * ∫ x in realScoreCell g r, |x - z| :=
mul_le_mul_of_nonneg_left hgeom hmf.le
_ ≤ ∫ e in cell g r, |(e : ℝ) - z| ∂H := hdensity
def reciprocalCenterScore
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
theorem reciprocalDeviation_eq_scoreDistance
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
|1 - c * armProb a e| = c * |(e : ℝ) - reciprocalCenterScore a c|
Proof (Lean source)
lemma reciprocalDeviation_eq_scoreDistance {ε c : ℝ}
(hOverlap : Overlap ε) (hc : c ∈ Icc ((1 - ε)⁻¹) ε⁻¹)
(a : ArmSpace) (e : ScoreSpace ε) :
|1 - c * armProb a e| = c * |(e : ℝ) - reciprocalCenterScore a c| := by
have hcpos : 0 < c :=
(inv_pos.mpr (by linarith [hOverlap.2])).trans_le hc.1
have hcne : c ≠ 0 := hcpos.ne'
cases a
· simp only [armProb, Bool.false_eq_true, ↓reduceIte, reciprocalCenterScore]
calc
|1 - c * (1 - (e : ℝ))| = |c * ((e : ℝ) - (1 - c⁻¹))| := by
congr 1
field_simp
ring
_ = |c| * |(e : ℝ) - (1 - c⁻¹)| := abs_mul _ _
_ = c * |(e : ℝ) - (1 - c⁻¹)| := by rw [abs_of_pos hcpos]
· simp only [armProb, ↓reduceIte, reciprocalCenterScore]
calc
|1 - c * (e : ℝ)| = |c * ((e : ℝ) - c⁻¹)| := by
rw [abs_eq_abs]
right
field_simp
ring
_ = |c| * |(e : ℝ) - c⁻¹| := abs_mul _ _
_ = c * |(e : ℝ) - c⁻¹| := by rw [abs_of_pos hcpos]
theorem cellAbsoluteDeviation_lower_cellLengthSq
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
K :
ℕ
H :
Measure (ScoreSpace ε)
f :
ScoreSpace ε → ℝ
mf :
ℝ
hOverlap :
Overlap ε
hmf :
0 < mf
hf :
LowerScoreDensity H f mf
g :
ScoreSpace ε → LabelSpace K
hg :
a :
r :
conclusion
mf * volume.real (realScoreCell g r) ^ 2 / (4 * (1 - ε)) ≤ cellAbsoluteDeviation H g a r
Proof (Lean source)
lemma cellAbsoluteDeviation_lower_cellLengthSq {ε : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) (f : ScoreSpace ε → ℝ)
(mf : ℝ) (hOverlap : Overlap ε) (hmf : 0 < mf)
(hf : LowerScoreDensity H f mf)
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(a : ArmSpace) (r : LabelSpace K) :
mf * volume.real (realScoreCell g r) ^ 2 / (4 * (1 - ε)) ≤
cellAbsoluteDeviation H g a r := by
have hε1 : 0 < 1 - ε := by linarith [hOverlap.2]
unfold cellAbsoluteDeviation
apply le_csInf
· let c : ℝ := ε⁻¹
refine ⟨∫ e in cell g r, |1 - c * armProb a e| ∂H, c, ?_, rfl⟩
dsimp [c]
constructor
· exact (inv_le_inv₀ hε1 hOverlap.1).mpr (by linarith [hOverlap.2])
· rfl
· rintro v ⟨c, hc, rfl⟩
have hcpos : 0 < c := (inv_pos.mpr hε1).trans_le hc.1
let z : ℝ := reciprocalCenterScore a c
have hscore := scoreCell_absDeviation_lower H f mf hOverlap hmf hf
g hg r z
have hrewrite :
(∫ e in cell g r, |1 - c * armProb a e| ∂H) =
c * ∫ e in cell g r, |(e : ℝ) - z| ∂H := by
rw [← integral_const_mul]
apply setIntegral_congr_fun (measurableSet_eq_fun hg measurable_const)
intro e he
exact reciprocalDeviation_eq_scoreDistance hOverlap hc a e
rw [hrewrite]
have hcoeff : (1 - ε)⁻¹ ≤ c := hc.1
have hdistNonneg : 0 ≤ ∫ e in cell g r, |(e : ℝ) - z| ∂H :=
integral_nonneg fun _ => abs_nonneg _
calc
mf * volume.real (realScoreCell g r) ^ 2 / (4 * (1 - ε)) =
(1 - ε)⁻¹ * (mf * (volume.real (realScoreCell g r) ^ 2 / 4)) := by
field_simp
_ ≤ (1 - ε)⁻¹ * (∫ e in cell g r, |(e : ℝ) - z| ∂H) :=
mul_le_mul_of_nonneg_left hscore (inv_nonneg.mpr hε1.le)
_ ≤ c * (∫ e in cell g r, |(e : ℝ) - z| ∂H) :=
mul_le_mul_of_nonneg_right hcoeff hdistNonneg
theorem worstCaseAmbiguity_finiteLabel_lower
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
mf :
ℝ
hOverlap :
Overlap ε
hmf :
0 < mf
K :
ℕ
hK :
0 < K
H :
f :
ScoreSpace ε → ℝ
hf :
LowerScoreDensity H f mf
g :
ScoreSpace ε → LabelSpace K
hg :
conclusion
mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε) * K) ≤ worstCaseAmbiguity H g
Proof (Lean source)
lemma worstCaseAmbiguity_finiteLabel_lower {ε : ℝ}
(mf : ℝ) (hOverlap : Overlap ε) (hmf : 0 < mf)
(K : ℕ) (hK : 0 < K)
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(f : ScoreSpace ε → ℝ) (hf : LowerScoreDensity H f mf)
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g) :
mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε) * K) ≤
worstCaseAmbiguity H g := by
let m : Fin K → ℝ := fun r => volume.real (realScoreCell g r)
have hpart := realScoreCell_intervalPartition g hg
have hsum : (∑ r, m r) = 1 - 2 * ε := by
dsimp [m]
calc
(∑ r, volume.real (realScoreCell g r)) =
volume.real (⋃ r, realScoreCell g r) :=
(measureReal_iUnion_fintype
(fun i j hij => hpart.2.1 i j hij) hpart.1
(fun r => measure_ne_top_of_subset
(fun x hx => by
rcases hx with ⟨e, he, rfl⟩
exact e.property)
measure_Icc_lt_top.ne)).symm
_ = volume.real (Icc ε (1 - ε)) := by rw [hpart.2.2]
_ = 1 - 2 * ε := by
rw [Real.volume_real_Icc_of_le (by linarith [hOverlap.2])]
ring
have hε1 : 0 < 1 - ε := by linarith [hOverlap.2]
have hqpos : 0 < mf / (1 - ε) := div_pos hmf hε1
have hsq := sum_cellMass_sq_lower K hK m
have hcells :
mf / (1 - ε) * (∑ r, (m r) ^ 2 / 2) ≤
∑ r, (cellAbsoluteDeviation H g true r +
cellAbsoluteDeviation H g false r) := by
rw [Finset.mul_sum]
apply Finset.sum_le_sum
intro r hr
have ht := cellAbsoluteDeviation_lower_cellLengthSq H f mf hOverlap
hmf hf g hg true r
have hc := cellAbsoluteDeviation_lower_cellLengthSq H f mf hOverlap
hmf hf g hg false r
dsimp [m]
calc
mf / (1 - ε) * (volume.real (realScoreCell g r) ^ 2 / 2) =
mf * volume.real (realScoreCell g r) ^ 2 / (4 * (1 - ε)) +
mf * volume.real (realScoreCell g r) ^ 2 / (4 * (1 - ε)) := by
field_simp
ring
_ ≤ cellAbsoluteDeviation H g true r +
cellAbsoluteDeviation H g false r := add_le_add ht hc
have hgeom :
mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε) * K) ≤
mf / (1 - ε) * (∑ r, (m r) ^ 2 / 2) := by
rw [← hsum]
have hhalf : (∑ r, m r) ^ 2 / (2 * K) ≤ ∑ r, (m r) ^ 2 / 2 := by
calc
(∑ r, m r) ^ 2 / (2 * K) = ((∑ r, m r) ^ 2 / K) / 2 := by
field_simp
_ ≤ (∑ r, (m r) ^ 2) / 2 :=
div_le_div_of_nonneg_right hsq (by norm_num)
_ = ∑ r, (m r) ^ 2 / 2 := by rw [Finset.sum_div]
calc
mf * (∑ r, m r) ^ 2 / (2 * (1 - ε) * K) =
mf / (1 - ε) * ((∑ r, m r) ^ 2 / (2 * K)) := by
field_simp
_ ≤ mf / (1 - ε) * (∑ r, (m r) ^ 2 / 2) :=
mul_le_mul_of_nonneg_left hhalf hqpos.le
rw [(worstCaseAmbiguity_eq H g hOverlap hg).1]
exact hgeom.trans hcells
theorem optimalKAmbiguity_finiteLabel_lower
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
mf :
ℝ
hOverlap :
Overlap ε
hmf :
0 < mf
K :
ℕ
hK :
0 < K
H :
f :
ScoreSpace ε → ℝ
hf :
LowerScoreDensity H f mf
conclusion
mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε) * K) ≤ optimalKAmbiguity ε K H
Proof (Lean source)
lemma optimalKAmbiguity_finiteLabel_lower {ε : ℝ}
(mf : ℝ) (hOverlap : Overlap ε) (hmf : 0 < mf)
(K : ℕ) (hK : 0 < K)
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(f : ScoreSpace ε → ℝ) (hf : LowerScoreDensity H f mf) :
mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε) * K) ≤
optimalKAmbiguity ε K H := by
unfold optimalKAmbiguity
apply le_csInf
· let r0 : LabelSpace K := ⟨0, hK⟩
let g0 : ScoreSpace ε → LabelSpace K := fun _ => r0
refine ⟨worstCaseAmbiguity H g0, g0, ?_, rfl⟩
exact measurable_const
· rintro d ⟨g, hg, rfl⟩
exact worstCaseAmbiguity_finiteLabel_lower mf hOverlap hmf K hK H f hf g hg
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.MinimaxHonestLengthFiniteLabel 4 declarations Finite-label identification lower bounds for honest released-trial intervals.
Finite-label identification lower bounds for honest released-trial intervals.
theorem honestProcedure_fairEndpoint_expectedLength_lower
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
K n :
ℕ
H :
g :
ScoreSpace ε → LabelSpace K
C :
hOverlap :
Overlap ε
hg :
α :
ℝ
hHonest :
(g, C) ∈ honestProcedures n K H α
conclusion
∃ P ∈ CausalLaws H g,
(1 - 2 * α) * worstCaseAmbiguity H g
≤ ∫ x, C.length x ∂(Measure.pi (fun _ : Fin n => releasedLaw P))
Proof (Lean source)
lemma honestProcedure_fairEndpoint_expectedLength_lower {ε : ℝ} {K n : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace K) (C : IntervalProcedure n K)
(hOverlap : Overlap ε) (hg : Measurable g) (α : ℝ)
(hHonest : (g, C) ∈ honestProcedures n K H α) :
∃ P ∈ CausalLaws H g,
(1 - 2 * α) * worstCaseAmbiguity H g ≤
∫ x, C.length x
∂(Measure.pi (fun _ : Fin n => releasedLaw P)) := by
rcases (worstCaseAmbiguity_eq H g hOverlap hg).2.2.2 with
⟨Pminus, Pplus, hminusProb, hplusProb, hminusComp, hplusComp,
_hminus, _hplus, hgap⟩
have hminusLaw : Pminus ∈ CausalLaws H g := by
refine ⟨hminusProb, ?_⟩
exact
{ probability := hminusComp.probability
measurableRelease := hminusComp.measurableRelease
scoreMarginal := hminusComp.scoreMarginal
randomizedAssignment := hminusComp.randomizedAssignment
consistency := hminusComp.consistency
deterministicRelease := hminusComp.deterministicRelease }
have hplusLaw : Pplus ∈ CausalLaws H g := by
refine ⟨hplusProb, ?_⟩
exact
{ probability := hplusComp.probability
measurableRelease := hplusComp.measurableRelease
scoreMarginal := hplusComp.scoreMarginal
randomizedAssignment := hplusComp.randomizedAssignment
consistency := hplusComp.consistency
deterministicRelease := hplusComp.deterministicRelease }
let Prel := halfOutcomeReleasedLaw H g
let Q := Measure.pi (fun _ : Fin n => Prel)
let _ : IsProbabilityMeasure Pminus := hminusProb
let _ : IsProbabilityMeasure (releasedLaw Pminus) :=
@Measure.isProbabilityMeasure_map _ _ _ _ Pminus hminusProb _
(by
exact (by
unfold releasedRecord label arm observed
fun_prop : Measurable (releasedRecord (ε := ε) (J := K))).aemeasurable)
let _ : IsProbabilityMeasure Prel := by
dsimp [Prel]
rw [← hminusComp.releasedLaw]
infer_instance
let _ : IsProbabilityMeasure Q := inferInstance
have hcoverMinus :
1 - α ≤ Q.real {x | C.contains x (ate Pminus)} := by
change 1 - α ≤
(Measure.pi (fun _ : Fin n => halfOutcomeReleasedLaw H g)).real
{x | C.contains x (ate Pminus)}
rw [← hminusComp.releasedLaw]
exact hHonest.2 Pminus hminusLaw
have hcoverPlus :
1 - α ≤ Q.real {x | C.contains x (ate Pplus)} := by
change 1 - α ≤
(Measure.pi (fun _ : Fin n => halfOutcomeReleasedLaw H g)).real
{x | C.contains x (ate Pplus)}
rw [← hplusComp.releasedLaw]
exact hHonest.2 Pplus hplusLaw
have horder : ate Pminus ≤ ate Pplus := by
rw [← sub_nonneg]
rw [hgap]
rw [(worstCaseAmbiguity_eq H g hOverlap hg).1]
exact Finset.sum_nonneg fun r _ => add_nonneg
(cellAbsoluteDeviation_nonneg H g hOverlap true r)
(cellAbsoluteDeviation_nonneg H g hOverlap false r)
have htest := intervalProcedure_twoPoint_expectedLength C Q Q
(ate Pminus) (ate Pplus) α horder hcoverMinus hcoverPlus
have htv : tvDist Q Q = 0 := by
simp [tvDist]
refine ⟨Pminus, hminusLaw, ?_⟩
calc
(1 - 2 * α) * worstCaseAmbiguity H g =
(ate Pplus - ate Pminus) *
(1 - 2 * α - tvDist Q Q) := by rw [hgap, htv]; ring
_ ≤ ∫ x, C.length x ∂Q := htest
_ = ∫ x, C.length x
∂(Measure.pi (fun _ : Fin n => releasedLaw Pminus)) := by
dsimp [Q, Prel]
rw [hminusComp.releasedLaw]
theorem intervalProcedure_expectedLength_le_two
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(∫ x, C.length x ∂Q) ≤ 2
Proof (Lean source)
lemma intervalProcedure_expectedLength_le_two {K n : ℕ}
(C : IntervalProcedure n K) (Q : Measure (TrialSample n K))
[IsProbabilityMeasure Q] :
(∫ x, C.length x ∂Q) ≤ 2 := by
have hlenMeas : Measurable C.length := C.measurable_hi.sub C.measurable_lo
have hlenNonneg (x : TrialSample n K) : 0 ≤ C.length x :=
sub_nonneg.mpr (C.ordered x)
have hlenBound (x : TrialSample n K) : C.length x ≤ 2 := by
dsimp [IntervalProcedure.length]
linarith [C.lower_bound x, C.upper_bound x]
have hlenInt : Integrable C.length Q := by
apply Integrable.of_bound hlenMeas.aestronglyMeasurable 2
filter_upwards with x
rw [Real.norm_eq_abs, abs_of_nonneg (hlenNonneg x)]
exact hlenBound x
have hmono := integral_mono hlenInt (integrable_const 2) hlenBound
simpa [probReal_univ] using hmono
theorem honestProcedure_worstCaseExpectedLength_lower
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
K n :
ℕ
H :
g :
ScoreSpace ε → LabelSpace K
C :
hOverlap :
Overlap ε
hg :
α :
ℝ
hHonest :
(g, C) ∈ honestProcedures n K H α
conclusion
(1 - 2 * α) * worstCaseAmbiguity H g
≤ sSup {u : ℝ | ∃ P ∈ CausalLaws H g, u = ∫ x, C.length x ∂(Measure.pi (fun _ : Fin n => releasedLaw P))}
Proof (Lean source)
lemma honestProcedure_worstCaseExpectedLength_lower {ε : ℝ} {K n : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace K) (C : IntervalProcedure n K)
(hOverlap : Overlap ε) (hg : Measurable g) (α : ℝ)
(hHonest : (g, C) ∈ honestProcedures n K H α) :
(1 - 2 * α) * worstCaseAmbiguity H g ≤
sSup {u : ℝ | ∃ P ∈ CausalLaws H g,
u = ∫ x, C.length x
∂(Measure.pi (fun _ : Fin n => releasedLaw P))} := by
let U : Set ℝ := {u : ℝ | ∃ P ∈ CausalLaws H g,
u = ∫ x, C.length x
∂(Measure.pi (fun _ : Fin n => releasedLaw P))}
obtain ⟨P, hP, hlen⟩ :=
honestProcedure_fairEndpoint_expectedLength_lower H g C hOverlap hg α hHonest
have hUbd : BddAbove U := by
refine ⟨2, ?_⟩
rintro u ⟨Q, hQ, rfl⟩
let _ : IsProbabilityMeasure Q := hQ.1
let _ : IsProbabilityMeasure (releasedLaw Q) :=
@Measure.isProbabilityMeasure_map _ _ _ _ Q hQ.1 _
(by
exact (by
unfold releasedRecord label arm observed
fun_prop : Measurable (releasedRecord (ε := ε) (J := K))).aemeasurable)
let _ : IsProbabilityMeasure
(Measure.pi (fun _ : Fin n => releasedLaw Q)) := inferInstance
exact intervalProcedure_expectedLength_le_two C _
exact hlen.trans (le_csSup hUbd ⟨P, hP, rfl⟩)
theorem honestProcedure_finiteLabel_expectedLength_lower
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
mf :
ℝ
hOverlap :
Overlap ε
hmf :
0 < mf
K :
ℕ
hK :
0 < K
H :
f :
ScoreSpace ε → ℝ
hf :
LowerScoreDensity H f mf
n :
ℕ
α :
ℝ
hα :
α ≤ 1 / 2
g :
ScoreSpace ε → LabelSpace K
C :
hg :
hHonest :
(g, C) ∈ honestProcedures n K H α
conclusion
(1 - 2 * α) * (mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε) * K))
≤ sSup {u : ℝ | ∃ P ∈ CausalLaws H g, u = ∫ x, C.length x ∂(Measure.pi (fun _ : Fin n => releasedLaw P))}
Proof (Lean source)
lemma honestProcedure_finiteLabel_expectedLength_lower {ε : ℝ}
(mf : ℝ) (hOverlap : Overlap ε) (hmf : 0 < mf)
(K : ℕ) (hK : 0 < K)
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(f : ScoreSpace ε → ℝ) (hf : LowerScoreDensity H f mf)
(n : ℕ) (α : ℝ) (hα : α ≤ 1 / 2)
(g : ScoreSpace ε → LabelSpace K) (C : IntervalProcedure n K)
(hg : Measurable g) (hHonest : (g, C) ∈ honestProcedures n K H α) :
(1 - 2 * α) *
(mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε) * K)) ≤
sSup {u : ℝ | ∃ P ∈ CausalLaws H g,
u = ∫ x, C.length x
∂(Measure.pi (fun _ : Fin n => releasedLaw P))} := by
have hamb := worstCaseAmbiguity_finiteLabel_lower mf hOverlap hmf
K hK H f hf g hg
exact (mul_le_mul_of_nonneg_left hamb (by linarith)).trans
(honestProcedure_worstCaseExpectedLength_lower H g C hOverlap hg α hHonest)
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.MinimaxHonestLengthFiniteLabelOuter 4 declarations Outer-infimum finite-label lower bounds for honest confidence intervals.
Outer-infimum finite-label lower bounds for honest confidence intervals.
def fullATEIntervalProcedure
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
n K :
ℕ
def
given by
clause 1
lo := fun _ => -1
clause 2
hi := fun _ => 1
clause 3
measurable_lo := measurable_const
clause 4
measurable_hi := measurable_const
clause 5
lower_bound := fun _ => le_rfl
clause 6
ordered := fun _ => by norm_num
clause 7
upper_bound := fun _ => le_rfl
theorem ate_mem_Icc_of_probability
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
ate P ∈ Icc (-1 : ℝ) 1
Proof (Lean source)
lemma ate_mem_Icc_of_probability {ε : ℝ} {K : ℕ}
(P : Measure (FullRow ε K)) [IsProbabilityMeasure P] :
ate P ∈ Icc (-1 : ℝ) 1 := by
let d : FullRow ε K → ℝ := fun ω => (outcome1 ω : ℝ) - (outcome0 ω : ℝ)
have hdMeas : Measurable d := by
dsimp [d]
unfold outcome1 outcome0
fun_prop
have hdBounds (ω : FullRow ε K) : -1 ≤ d ω ∧ d ω ≤ 1 := by
have h0 := (outcome0 ω).property
have h1 := (outcome1 ω).property
change 0 ≤ (outcome0 ω : ℝ) ∧ (outcome0 ω : ℝ) ≤ 1 at h0
change 0 ≤ (outcome1 ω : ℝ) ∧ (outcome1 ω : ℝ) ≤ 1 at h1
dsimp [d]
constructor <;> linarith
have hdInt : Integrable d P := by
apply Integrable.of_bound hdMeas.aestronglyMeasurable 1
filter_upwards with ω
rw [Real.norm_eq_abs]
exact abs_le.mpr (hdBounds ω)
have hlower := integral_mono (integrable_const (-1 : ℝ)) hdInt
(fun ω => (hdBounds ω).1)
have hupper := integral_mono hdInt (integrable_const (1 : ℝ))
(fun ω => (hdBounds ω).2)
constructor
· simpa [ate, d, probReal_univ] using hlower
· simpa [ate, d, probReal_univ] using hupper
theorem fullATEIntervalProcedure_honest
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(g, fullATEIntervalProcedure n K) ∈ honestProcedures n K H α
Proof (Lean source)
lemma fullATEIntervalProcedure_honest {ε : ℝ} {K n : ℕ}
(H : Measure (ScoreSpace ε))
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(α : ℝ) (hα : 0 ≤ α) :
(g, fullATEIntervalProcedure n K) ∈ honestProcedures n K H α := by
refine ⟨hg, ?_⟩
intro P hP
let _ : IsProbabilityMeasure P := hP.1
let _ : IsProbabilityMeasure (releasedLaw P) :=
@Measure.isProbabilityMeasure_map _ _ _ _ P hP.1 _
(by
exact (by
unfold releasedRecord label arm observed
fun_prop : Measurable (releasedRecord (ε := ε) (J := K))).aemeasurable)
let _ : IsProbabilityMeasure
(Measure.pi (fun _ : Fin n => releasedLaw P)) := inferInstance
have hate := ate_mem_Icc_of_probability P
have hevent :
{x | (fullATEIntervalProcedure n K).contains x (ate P)} = univ := by
ext x
simp [IntervalProcedure.contains, fullATEIntervalProcedure, hate.1, hate.2]
rw [hevent, probReal_univ]
linarith
theorem minimaxHonestLength_finiteLabel_lower
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε mf α :
ℝ
hOverlap :
Overlap ε
hα :
0 < α ∧ α < 1 / 2
hmf :
0 < mf
H :
Measure (ScoreSpace ε)
f :
ScoreSpace ε → ℝ
hDensity :
LowerScoreDensity H f mf
n K :
ℕ
_hn :
0 < n
hK :
0 < K
conclusion
(1 - 2 * α) * mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε) * K) ≤ minimaxHonestLength n K H α
Proof (Lean source)
lemma minimaxHonestLength_finiteLabel_lower {ε mf α : ℝ}
(hOverlap : Overlap ε) (hα : 0 < α ∧ α < 1 / 2)
(hmf : 0 < mf)
(H : Measure (ScoreSpace ε)) (f : ScoreSpace ε → ℝ)
[IsProbabilityMeasure H] (hDensity : LowerScoreDensity H f mf)
(n K : ℕ) (_hn : 0 < n) (hK : 0 < K) :
(1 - 2 * α) * mf * (1 - 2 * ε) ^ 2 /
(2 * (1 - ε) * K) ≤ minimaxHonestLength n K H α := by
let V : Set ℝ := {v : ℝ | ∃ gC ∈ honestProcedures n K H α,
v = sSup {u : ℝ | ∃ P ∈ CausalLaws H gC.1,
u = ∫ x, gC.2.length x
∂(Measure.pi (fun _ : Fin n => releasedLaw P))}}
have hVne : V.Nonempty := by
let r0 : LabelSpace K := ⟨0, hK⟩
let g0 : ScoreSpace ε → LabelSpace K := fun _ => r0
let C0 := fullATEIntervalProcedure n K
have hg0 : Measurable g0 := measurable_const
have hC0 : (g0, C0) ∈ honestProcedures n K H α :=
fullATEIntervalProcedure_honest H g0 hg0 α hα.1.le
exact ⟨sSup {u : ℝ | ∃ P ∈ CausalLaws H g0,
u = ∫ x, C0.length x
∂(Measure.pi (fun _ : Fin n => releasedLaw P))},
⟨(g0, C0), hC0, rfl⟩⟩
unfold minimaxHonestLength
change _ ≤ sInf V
apply le_csInf hVne
rintro v ⟨gC, hgC, rfl⟩
have hlower := honestProcedure_finiteLabel_expectedLength_lower
mf hOverlap hmf K hK H f hDensity n α hα.2.le
gC.1 gC.2 hgC.1 hgC
calc
(1 - 2 * α) * mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε) * K) =
(1 - 2 * α) *
(mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε) * K)) := by ring
_ ≤ _ := hlower
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.MinimaxHonestLengthRateAssembly 1 declarations Algebraic assembly of the finite-label and sampling lower rates.
Algebraic assembly of the finite-label and sampling lower rates.
theorem minimaxHonestLength_rate_assembly
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε α mf L :
ℝ
n K :
ℕ
hOverlap :
Overlap ε
hα :
0 < α ∧ α < 1 / 2
hmf :
0 < mf
hn :
0 < n
hK :
0 < K
hLabel :
((1 - 2 * α) * mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε))) * (K : ℝ)⁻¹ ≤ L
hSampling :
trialCertificate α * (sqrt (n : ℝ))⁻¹ ≤ L
conclusion
0 < c
c * ((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹) ≤ L
Proof (Lean source)
lemma minimaxHonestLength_rate_assembly {ε α mf L : ℝ} {n K : ℕ}
(hOverlap : Overlap ε) (hα : 0 < α ∧ α < 1 / 2)
(hmf : 0 < mf) (hn : 0 < n) (hK : 0 < K)
(hLabel :
((1 - 2 * α) * mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε))) *
(K : ℝ)⁻¹ ≤ L)
(hSampling :
trialCertificate α * (sqrt (n : ℝ))⁻¹ ≤ L) :
let A := (1 - 2 * α) * mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε))
let B := trialCertificate α
let c := min A B / 2
0 < c ∧
c * ((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹) ≤ L := by
dsimp
have hβ : 0 < 1 - 2 * α := by linarith [hα.2]
have hgap : 0 < 1 - 2 * ε := by linarith [hOverlap.2]
have hden : 0 < 1 - ε := by linarith [hOverlap.2]
have hA :
0 < (1 - 2 * α) * mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε)) := by
positivity
have hlog : 0 < log (1 + (1 - 2 * α) ^ 2) := by
exact Real.log_pos (by nlinarith [sq_pos_of_pos hβ])
have hB : 0 < trialCertificate α := by
unfold trialCertificate
positivity
have hc : 0 < min
((1 - 2 * α) * mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε)))
(trialCertificate α) / 2 := div_pos (lt_min hA hB) (by norm_num)
refine ⟨hc, ?_⟩
have hKinv : 0 ≤ (K : ℝ)⁻¹ :=
inv_nonneg.mpr (Nat.cast_pos.mpr hK).le
have hninv : 0 ≤ (sqrt (n : ℝ))⁻¹ :=
inv_nonneg.mpr (Real.sqrt_pos.2 (Nat.cast_pos.mpr hn)).le
have hLabelHalf :
(min
((1 - 2 * α) * mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε)))
(trialCertificate α) / 2) * (K : ℝ)⁻¹ ≤ L / 2 := by
calc
_ ≤ (((1 - 2 * α) * mf * (1 - 2 * ε) ^ 2 /
(2 * (1 - ε))) / 2) * (K : ℝ)⁻¹ :=
mul_le_mul_of_nonneg_right
(div_le_div_of_nonneg_right (min_le_left _ _) (by norm_num)) hKinv
_ = ((((1 - 2 * α) * mf * (1 - 2 * ε) ^ 2 /
(2 * (1 - ε))) * (K : ℝ)⁻¹) / 2) := by ring
_ ≤ L / 2 := by linarith
have hSamplingHalf :
(min
((1 - 2 * α) * mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε)))
(trialCertificate α) / 2) * (sqrt (n : ℝ))⁻¹ ≤ L / 2 := by
calc
_ ≤ (trialCertificate α / 2) * (sqrt (n : ℝ))⁻¹ :=
mul_le_mul_of_nonneg_right
(div_le_div_of_nonneg_right (min_le_right _ _) (by norm_num)) hninv
_ = (trialCertificate α * (sqrt (n : ℝ))⁻¹) / 2 := by ring
_ ≤ L / 2 := by linarith
calc
(min
((1 - 2 * α) * mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε)))
(trialCertificate α) / 2) *
((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹) =
(min
((1 - 2 * α) * mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε)))
(trialCertificate α) / 2) * (K : ℝ)⁻¹ +
(min
((1 - 2 * α) * mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε)))
(trialCertificate α) / 2) * (sqrt (n : ℝ))⁻¹ := by ring
_ ≤ L / 2 + L / 2 := add_le_add hLabelHalf hSamplingHalf
_ = L := by ring
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.MinimaxHonestLengthSamplingLower 3 declarations Fixed-score sampling lower bound for honest interval length.
Fixed-score sampling lower bound for honest interval length.
theorem honestProcedure_sampling_baseline_expectedLength_lower
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
K n :
ℕ
H :
g :
ScoreSpace ε → LabelSpace K
C :
hOverlap :
Overlap ε
hg :
α :
ℝ
hα :
0 < α ∧ α < 1 / 2
hn :
0 < n
hHonest :
(g, C) ∈ honestProcedures n K H α
conclusion
P₀ :
directSamplingFullLaw H g (1 / 2)
trialCertificate α * (sqrt (n : ℝ))⁻¹
≤ ∫ x, C.length x ∂(Measure.pi (fun _ : Fin n => releasedLaw P₀))
Proof (Lean source)
lemma honestProcedure_sampling_baseline_expectedLength_lower
{ε : ℝ} {K n : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace K) (C : IntervalProcedure n K)
(hOverlap : Overlap ε) (hg : Measurable g)
(α : ℝ) (hα : 0 < α ∧ α < 1 / 2) (hn : 0 < n)
(hHonest : (g, C) ∈ honestProcedures n K H α) :
let P₀ := directSamplingFullLaw H g (1 / 2)
trialCertificate α * (sqrt (n : ℝ))⁻¹ ≤
∫ x, C.length x
∂(Measure.pi (fun _ : Fin n => releasedLaw P₀)) := by
dsimp
let β : ℝ := 1 - 2 * α
let u : ℝ := sqrt (log (1 + β ^ 2) / (n : ℝ)) / 2
let P₀ := directSamplingFullLaw H g (1 / 2)
let P₁ := directSamplingFullLaw H g (1 / 2 + u)
let Q₀ := Measure.pi (fun _ : Fin n => releasedLaw P₀)
let Q₁ := Measure.pi (fun _ : Fin n => releasedLaw P₁)
have hcal := trial_calibrated_perturbation α hα n hn
change 0 < u ∧ |u| < 1 / 2 ∧
(1 + 4 * u ^ 2) ^ n ≤ 1 + β ^ 2 at hcal
rcases hcal with ⟨hu0, hu, hpow⟩
have hub := abs_lt.mp hu
have hP0 : P₀ ∈ CausalLaws H g := by
dsimp [P₀]
exact directSamplingFullLaw_mem_causalLaws H g hg hOverlap (1 / 2)
(by norm_num) (by norm_num)
have hP1 : P₁ ∈ CausalLaws H g := by
dsimp only [P₁]
exact directSamplingFullLaw_mem_causalLaws H g hg hOverlap _
(by linarith) (by linarith)
let _ : IsProbabilityMeasure P₀ := hP0.1
let _ : IsProbabilityMeasure P₁ := hP1.1
let _ : IsProbabilityMeasure (releasedLaw P₀) :=
@Measure.isProbabilityMeasure_map _ _ _ _ P₀ hP0.1 _
(by
exact (by
unfold releasedRecord label arm observed
fun_prop : Measurable (releasedRecord (ε := ε) (J := K))).aemeasurable)
let _ : IsProbabilityMeasure (releasedLaw P₁) :=
@Measure.isProbabilityMeasure_map _ _ _ _ P₁ hP1.1 _
(by
exact (by
unfold releasedRecord label arm observed
fun_prop : Measurable (releasedRecord (ε := ε) (J := K))).aemeasurable)
let _ : IsProbabilityMeasure Q₀ := inferInstance
let _ : IsProbabilityMeasure Q₁ := inferInstance
have hate0 : ate P₀ = 1 / 2 := by
dsimp [P₀]
exact directSamplingFullLaw_ate H g hg hOverlap (1 / 2)
(by norm_num) (by norm_num)
have hate1 : ate P₁ = 1 / 2 + u := by
dsimp [P₁]
exact directSamplingFullLaw_ate H g hg hOverlap (1 / 2 + u)
(by linarith) (by linarith)
have hcover0 : 1 - α ≤ Q₀.real {x | C.contains x (1 / 2)} := by
rw [← hate0]
exact hHonest.2 P₀ hP0
have hcover1 : 1 - α ≤ Q₁.real {x | C.contains x (1 / 2 + u)} := by
rw [← hate1]
exact hHonest.2 P₁ hP1
have htv : tvDist Q₀ Q₁ ≤
(1 / 2 : ℝ) * sqrt ((1 + 4 * u ^ 2) ^ n - 1) := by
rw [tvDist_symm]
dsimp [Q₀, Q₁, P₀, P₁]
exact directSampling_productReleased_tv_bound H g hg hOverlap u hu n
have hβ0 : 0 < β := by dsimp [β]; linarith [hα.2]
have htvhalf : tvDist Q₀ Q₁ ≤ β / 2 := by
calc
_ ≤ (1 / 2 : ℝ) * sqrt ((1 + 4 * u ^ 2) ^ n - 1) := htv
_ ≤ (1 / 2 : ℝ) * sqrt (β ^ 2) := by
apply mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt ?_) (by norm_num)
linarith
_ = β / 2 := by rw [Real.sqrt_sq_eq_abs, abs_of_pos hβ0]; ring
have hlen := intervalProcedure_twoPoint_expectedLength C Q₀ Q₁
(1 / 2) (1 / 2 + u) α (by linarith) hcover0 hcover1
have hcoef : β * u / 2 ≤
u * (1 - 2 * α - tvDist Q₀ Q₁) := by
dsimp [β] at htvhalf ⊢
nlinarith [mul_nonneg hu0.le
(show 0 ≤ 1 - 2 * α - 2 * tvDist Q₀ Q₁ by linarith)]
have hident := trial_calibrated_coefficient α n
change β * u / 2 =
trialCertificate α * (sqrt (n : ℝ))⁻¹ at hident
rw [← hident]
change β * u / 2 ≤ ∫ x, C.length x ∂Q₀
have hlen' : u * (1 - 2 * α - tvDist Q₀ Q₁) ≤
∫ x, C.length x ∂Q₀ := by
convert hlen using 1
ring
exact hcoef.trans hlen'
theorem honestProcedure_sampling_worstCaseExpectedLength_lower
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
K n :
ℕ
H :
g :
ScoreSpace ε → LabelSpace K
C :
hOverlap :
Overlap ε
hg :
α :
ℝ
hα :
0 < α ∧ α < 1 / 2
hn :
0 < n
hHonest :
(g, C) ∈ honestProcedures n K H α
conclusion
trialCertificate α * (sqrt (n : ℝ))⁻¹
≤ sSup {v : ℝ | ∃ P ∈ CausalLaws H g, v = ∫ x, C.length x ∂(Measure.pi (fun _ : Fin n => releasedLaw P))}
Proof (Lean source)
lemma honestProcedure_sampling_worstCaseExpectedLength_lower
{ε : ℝ} {K n : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace K) (C : IntervalProcedure n K)
(hOverlap : Overlap ε) (hg : Measurable g)
(α : ℝ) (hα : 0 < α ∧ α < 1 / 2) (hn : 0 < n)
(hHonest : (g, C) ∈ honestProcedures n K H α) :
trialCertificate α * (sqrt (n : ℝ))⁻¹ ≤
sSup {v : ℝ | ∃ P ∈ CausalLaws H g,
v = ∫ x, C.length x
∂(Measure.pi (fun _ : Fin n => releasedLaw P))} := by
let P₀ := directSamplingFullLaw H g (1 / 2)
have hP0 : P₀ ∈ CausalLaws H g := by
dsimp [P₀]
exact directSamplingFullLaw_mem_causalLaws H g hg hOverlap (1 / 2)
(by norm_num) (by norm_num)
have hlower := honestProcedure_sampling_baseline_expectedLength_lower
H g C hOverlap hg α hα hn hHonest
have hBdd : BddAbove {v : ℝ | ∃ P ∈ CausalLaws H g,
v = ∫ x, C.length x
∂(Measure.pi (fun _ : Fin n => releasedLaw P))} := by
refine ⟨2, ?_⟩
rintro v ⟨P, hP, rfl⟩
let _ : IsProbabilityMeasure P := hP.1
let _ : IsProbabilityMeasure (releasedLaw P) :=
@Measure.isProbabilityMeasure_map _ _ _ _ P hP.1 _
(by
exact (by
unfold releasedRecord label arm observed
fun_prop : Measurable (releasedRecord (ε := ε) (J := K))).aemeasurable)
let _ : IsProbabilityMeasure
(Measure.pi (fun _ : Fin n => releasedLaw P)) := inferInstance
exact intervalProcedure_expectedLength_le_two C _
exact hlower.trans (le_csSup hBdd ⟨P₀, hP0, rfl⟩)
theorem minimaxHonestLength_sampling_lower
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma minimaxHonestLength_sampling_lower {ε α : ℝ} {n K : ℕ}
(hOverlap : Overlap ε) (hα : 0 < α ∧ α < 1 / 2)
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(hn : 0 < n) (hK : 0 < K) :
trialCertificate α * (sqrt (n : ℝ))⁻¹ ≤
minimaxHonestLength n K H α := by
let V : Set ℝ := {v : ℝ | ∃ gC ∈ honestProcedures n K H α,
v = sSup {u : ℝ | ∃ P ∈ CausalLaws H gC.1,
u = ∫ x, gC.2.length x
∂(Measure.pi (fun _ : Fin n => releasedLaw P))}}
have hVne : V.Nonempty := by
let r0 : LabelSpace K := ⟨0, hK⟩
let g0 : ScoreSpace ε → LabelSpace K := fun _ => r0
let C0 := fullATEIntervalProcedure n K
have hg0 : Measurable g0 := measurable_const
have hC0 : (g0, C0) ∈ honestProcedures n K H α :=
fullATEIntervalProcedure_honest H g0 hg0 α hα.1.le
exact ⟨sSup {u : ℝ | ∃ P ∈ CausalLaws H g0,
u = ∫ x, C0.length x
∂(Measure.pi (fun _ : Fin n => releasedLaw P))},
⟨(g0, C0), hC0, rfl⟩⟩
unfold minimaxHonestLength
change _ ≤ sInf V
apply le_csInf hVne
rintro v ⟨gC, hgC, rfl⟩
exact honestProcedure_sampling_worstCaseExpectedLength_lower
H gC.1 gC.2 hOverlap hgC.1 α hα hn hgC
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.MinimaxHonestLengthSamplingPair 21 declarations Bernoulli sampling perturbations over an arbitrary fixed score law.
Bernoulli sampling perturbations over an arbitrary fixed score law.
def samplingArmOutcomeLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
def directSamplingLatentLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
def directSamplingFullLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
given by
((directSamplingLatentLaw H p).withDensity (reconstructedAssignmentDensity false)).map (reconstructedFullRow g false)
+ ((directSamplingLatentLaw H p).withDensity (reconstructedAssignmentDensity true)).map (reconstructedFullRow g true)
theorem directSamplingLatentLaw_isProbabilityMeasure
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma directSamplingLatentLaw_isProbabilityMeasure {ε : ℝ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) :
IsProbabilityMeasure (directSamplingLatentLaw H p) := by
let _ : IsProbabilityMeasure (trialOutcomeLaw p) :=
trialOutcomeLaw_isProbabilityMeasure p hp0 hp1
unfold directSamplingLatentLaw
infer_instance
theorem directSamplingAssignmentMeasures_sum
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
= directSamplingLatentLaw H p
Proof (Lean source)
lemma directSamplingAssignmentMeasures_sum {ε : ℝ}
(H : Measure (ScoreSpace ε)) (p : ℝ) (hOverlap : Overlap ε) :
(directSamplingLatentLaw H p).withDensity
(reconstructedAssignmentDensity false) +
(directSamplingLatentLaw H p).withDensity
(reconstructedAssignmentDensity true) =
directSamplingLatentLaw H p := by
rw [← withDensity_add_left (by
change Measurable (fun q : ScoreSpace ε × (OutcomeSpace × OutcomeSpace) =>
ofReal (1 - (q.1 : ℝ)))
fun_prop)]
have hfun : reconstructedAssignmentDensity false +
reconstructedAssignmentDensity true =
(1 : ScoreSpace ε × (OutcomeSpace × OutcomeSpace) → ENNReal) := by
funext q
exact directSamplingAssignmentDensity_sum hOverlap q
rw [hfun, withDensity_one]
theorem directSamplingFullLaw_map_latentTriple
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
K :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace K
hg :
hOverlap :
Overlap ε
p :
ℝ
conclusion
Proof (Lean source)
lemma directSamplingFullLaw_map_latentTriple {ε : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace K)
(hg : Measurable g) (hOverlap : Overlap ε) (p : ℝ) :
(directSamplingFullLaw H g p).map latentTriple =
directSamplingLatentLaw H p := by
unfold directSamplingFullLaw
rw [Measure.map_add _ _ (by unfold latentTriple score outcome0 outcome1; fun_prop),
Measure.map_map (by unfold latentTriple score outcome0 outcome1; fun_prop)
(reconstructedFullRow_measurable g hg false),
Measure.map_map (by unfold latentTriple score outcome0 outcome1; fun_prop)
(reconstructedFullRow_measurable g hg true)]
have hfun (a : ArmSpace) :
latentTriple ∘ reconstructedFullRow g a = id := by
funext q
cases a <;> rfl
rw [hfun false, hfun true, Measure.map_id, Measure.map_id]
change (directSamplingLatentLaw H p).withDensity
(reconstructedAssignmentDensity false) +
(directSamplingLatentLaw H p).withDensity
(reconstructedAssignmentDensity true) = directSamplingLatentLaw H p
exact directSamplingAssignmentMeasures_sum H p hOverlap
theorem directSamplingFullLaw_isProbabilityMeasure
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
K :
ℕ
H :
g :
ScoreSpace ε → LabelSpace K
hg :
hOverlap :
Overlap ε
p :
ℝ
hp0 :
0 ≤ p
hp1 :
p ≤ 1
conclusion
Proof (Lean source)
lemma directSamplingFullLaw_isProbabilityMeasure {ε : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(hOverlap : Overlap ε) (p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) :
IsProbabilityMeasure (directSamplingFullLaw H g p) := by
let _ : IsProbabilityMeasure (directSamplingLatentLaw H p) :=
directSamplingLatentLaw_isProbabilityMeasure H p hp0 hp1
rw [isProbabilityMeasure_iff]
unfold directSamplingFullLaw
rw [Measure.add_apply,
Measure.map_apply (reconstructedFullRow_measurable g hg false) MeasurableSet.univ,
Measure.map_apply (reconstructedFullRow_measurable g hg true) MeasurableSet.univ]
simp only [preimage_univ]
rw [← Measure.add_apply, directSamplingAssignmentMeasures_sum H p hOverlap]
exact isProbabilityMeasure_iff.mp inferInstance
theorem directSamplingFullLaw_scoreMarginal
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
K :
ℕ
H :
g :
ScoreSpace ε → LabelSpace K
hg :
hOverlap :
Overlap ε
p :
ℝ
hp0 :
0 ≤ p
hp1 :
p ≤ 1
conclusion
ScoreMarginal (directSamplingFullLaw H g p) H
Proof (Lean source)
lemma directSamplingFullLaw_scoreMarginal {ε : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(hOverlap : Overlap ε) (p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) :
ScoreMarginal (directSamplingFullLaw H g p) H := by
let _ : IsProbabilityMeasure (trialOutcomeLaw p) :=
trialOutcomeLaw_isProbabilityMeasure p hp0 hp1
unfold ScoreMarginal directSamplingFullLaw
rw [Measure.map_add _ _ (by unfold score; fun_prop),
Measure.map_map (by unfold score; fun_prop)
(reconstructedFullRow_measurable g hg false),
Measure.map_map (by unfold score; fun_prop)
(reconstructedFullRow_measurable g hg true)]
have hfun (a : ArmSpace) :
score ∘ reconstructedFullRow g a = fst := by
funext q
rfl
rw [hfun false, hfun true, ← Measure.map_add _ _ measurable_fst,
directSamplingAssignmentMeasures_sum H p hOverlap]
unfold directSamplingLatentLaw
rw [Measure.map_fst_prod]
simp
theorem directSamplingFullLaw_consistency
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Consistency (directSamplingFullLaw H g p)
Proof (Lean source)
lemma directSamplingFullLaw_consistency {ε : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace K)
(hg : Measurable g) (p : ℝ) :
Consistency (directSamplingFullLaw H g p) := by
unfold Consistency directSamplingFullLaw
rw [ae_add_measure_iff]
constructor
· rw [ae_map_iff (reconstructedFullRow_measurable g hg false).aemeasurable (by
exact measurableSet_eq_fun (by unfold observed; fun_prop)
(by
have hcond : MeasurableSet {ω : FullRow ε K | arm ω = true} :=
measurableSet_eq_fun (by unfold arm; fun_prop) measurable_const
exact Measurable.ite hcond (by unfold outcome1; fun_prop)
(by unfold outcome0; fun_prop)))]
filter_upwards [] with q
rfl
· rw [ae_map_iff (reconstructedFullRow_measurable g hg true).aemeasurable (by
exact measurableSet_eq_fun (by unfold observed; fun_prop)
(by
have hcond : MeasurableSet {ω : FullRow ε K | arm ω = true} :=
measurableSet_eq_fun (by unfold arm; fun_prop) measurable_const
exact Measurable.ite hcond (by unfold outcome1; fun_prop)
(by unfold outcome0; fun_prop)))]
filter_upwards [] with q
rfl
theorem directSamplingFullLaw_deterministicRelease
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
DeterministicRelease g (directSamplingFullLaw H g p)
Proof (Lean source)
lemma directSamplingFullLaw_deterministicRelease {ε : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace K)
(hg : Measurable g) (p : ℝ) :
DeterministicRelease g (directSamplingFullLaw H g p) := by
unfold DeterministicRelease directSamplingFullLaw
rw [ae_add_measure_iff]
constructor
· rw [ae_map_iff (reconstructedFullRow_measurable g hg false).aemeasurable (by
exact measurableSet_eq_fun (by unfold label; fun_prop)
(hg.comp (by unfold score; fun_prop)))]
filter_upwards [] with q
rfl
· rw [ae_map_iff (reconstructedFullRow_measurable g hg true).aemeasurable (by
exact measurableSet_eq_fun (by unfold label; fun_prop)
(hg.comp (by unfold score; fun_prop)))]
filter_upwards [] with q
rfl
theorem directSamplingFullLaw_randomizedAssignment
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
K :
ℕ
H :
g :
ScoreSpace ε → LabelSpace K
hg :
hOverlap :
Overlap ε
p :
ℝ
hp0 :
0 ≤ p
hp1 :
p ≤ 1
conclusion
Proof (Lean source)
lemma directSamplingFullLaw_randomizedAssignment {ε : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(hOverlap : Overlap ε) (p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) :
RandomizedAssignment (directSamplingFullLaw H g p) := by
intro B hB
let L := directSamplingLatentLaw H p
let M₀ := L.withDensity (reconstructedAssignmentDensity false)
let M₁ := L.withDensity (reconstructedAssignmentDensity true)
let D : Set (ScoreSpace ε × (OutcomeSpace × OutcomeSpace)) :=
{q | (q.1, q.2.1, q.2.2) ∈ B}
have hD : MeasurableSet D := hB.preimage (by fun_prop)
let S : Set (FullRow ε K) :=
{ω | arm ω = true ∧ (score ω, outcome0 ω, outcome1 ω) ∈ B}
let T : Set (FullRow ε K) :=
{ω | (score ω, outcome0 ω, outcome1 ω) ∈ B}
have hS : MeasurableSet S :=
(measurableSet_eq_fun (by unfold arm; fun_prop) measurable_const).inter
(hB.preimage (by unfold score outcome0 outcome1; fun_prop))
have hT : MeasurableSet T :=
hB.preimage (by unfold score outcome0 outcome1; fun_prop)
have hpre0S : reconstructedFullRow g false ⁻¹' S = ∅ := by
ext q
simp [S, reconstructedFullRow, arm]
have hpre1S : reconstructedFullRow g true ⁻¹' S = D := by
ext q
simp [S, D, reconstructedFullRow, arm, score, outcome0, outcome1]
have hpre0T : reconstructedFullRow g false ⁻¹' T = D := by
ext q
simp [T, D, reconstructedFullRow, score, outcome0, outcome1]
have hpre1T : reconstructedFullRow g true ⁻¹' T = D := by
ext q
simp [T, D, reconstructedFullRow, score, outcome0, outcome1]
let _ : IsFiniteMeasure M₀ :=
directSamplingComponent_finite H hOverlap p hp0 hp1 false
let _ : IsFiniteMeasure M₁ :=
directSamplingComponent_finite H hOverlap p hp0 hp1 true
have hmap0 : IsFiniteMeasure (M₀.map (reconstructedFullRow g false)) := ⟨by
rw [Measure.map_apply (reconstructedFullRow_measurable g hg false)
MeasurableSet.univ]
exact measure_lt_top M₀ Set.univ
⟩
have hmap1 : IsFiniteMeasure (M₁.map (reconstructedFullRow g true)) := ⟨by
rw [Measure.map_apply (reconstructedFullRow_measurable g hg true)
MeasurableSet.univ]
exact measure_lt_top M₁ Set.univ
⟩
change (directSamplingFullLaw H g p).real S =
∫ ω in T, (score ω : ℝ) ∂directSamplingFullLaw H g p
have hleft : (directSamplingFullLaw H g p).real S =
∫ q in D, (q.1 : ℝ) ∂L := by
unfold directSamplingFullLaw
rw [Measure.real_def, Measure.add_apply,
Measure.map_apply (reconstructedFullRow_measurable g hg false) hS,
Measure.map_apply (reconstructedFullRow_measurable g hg true) hS,
hpre0S, hpre1S, measure_empty, zero_add]
change M₁.real D = _
exact directSamplingAssignmentTrue_real H hOverlap p hp0 hp1 D hD
rw [hleft]
unfold directSamplingFullLaw
let _ : IsFiniteMeasure (M₀.map (reconstructedFullRow g false)) := hmap0
let _ : IsFiniteMeasure (M₁.map (reconstructedFullRow g true)) := hmap1
rw [Measure.restrict_add,
integral_add_measure (samplingFullScore_integrable_finite _).integrableOn
(samplingFullScore_integrable_finite _).integrableOn,
setIntegral_map hT (by unfold score; fun_prop)
(reconstructedFullRow_measurable g hg false).aemeasurable,
setIntegral_map hT (by unfold score; fun_prop)
(reconstructedFullRow_measurable g hg true).aemeasurable,
hpre0T, hpre1T]
change (∫ q in D, (q.1 : ℝ) ∂L) =
(∫ q in D, (q.1 : ℝ) ∂M₀) + (∫ q in D, (q.1 : ℝ) ∂M₁)
rw [← integral_add_measure
(samplingLatentScore_integrable_finite M₀).integrableOn
(samplingLatentScore_integrable_finite M₁).integrableOn]
rw [← Measure.restrict_add, directSamplingAssignmentMeasures_sum H p hOverlap]
theorem directSamplingFullLaw_ate
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
K :
ℕ
H :
g :
ScoreSpace ε → LabelSpace K
hg :
hOverlap :
Overlap ε
p :
ℝ
hp0 :
0 ≤ p
hp1 :
p ≤ 1
conclusion
ate (directSamplingFullLaw H g p) = p
Proof (Lean source)
lemma directSamplingFullLaw_ate {ε : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(hOverlap : Overlap ε) (p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) :
ate (directSamplingFullLaw H g p) = p := by
let _ : IsProbabilityMeasure (trialOutcomeLaw p) :=
trialOutcomeLaw_isProbabilityMeasure p hp0 hp1
let _ : IsProbabilityMeasure (directSamplingLatentLaw H p) :=
directSamplingLatentLaw_isProbabilityMeasure H p hp0 hp1
let _ : IsProbabilityMeasure (directSamplingFullLaw H g p) :=
directSamplingFullLaw_isProbabilityMeasure H g hg hOverlap p hp0 hp1
have hlatentInt : Integrable
(fun q : ScoreSpace ε × (OutcomeSpace × OutcomeSpace) =>
(q.2.2 : ℝ) - (q.2.1 : ℝ)) (directSamplingLatentLaw H p) := by
apply Integrable.of_bound (by fun_prop) 1
filter_upwards [] with q
rw [Real.norm_eq_abs]
have h0 := q.2.1.property
have h1 := q.2.2.property
change 0 ≤ (q.2.1 : ℝ) ∧ (q.2.1 : ℝ) ≤ 1 at h0
change 0 ≤ (q.2.2 : ℝ) ∧ (q.2.2 : ℝ) ≤ 1 at h1
exact abs_le.mpr ⟨by
linarith [h0.1, h1.2], by linarith [h0.2, h1.1]⟩
have houtInt : Integrable
(fun q : OutcomeSpace × OutcomeSpace => (q.2 : ℝ) - (q.1 : ℝ))
((Measure.dirac trialZeroOutcome).prod (trialOutcomeLaw p)) := by
apply Integrable.of_bound (by fun_prop) 1
filter_upwards [] with q
rw [Real.norm_eq_abs]
have h0 := q.1.property
have h1 := q.2.property
change 0 ≤ (q.1 : ℝ) ∧ (q.1 : ℝ) ≤ 1 at h0
change 0 ≤ (q.2 : ℝ) ∧ (q.2 : ℝ) ≤ 1 at h1
exact abs_le.mpr ⟨by
linarith [h0.1, h1.2], by linarith [h0.2, h1.1]⟩
have hlatentMeas : Measurable (latentTriple (ε := ε) (J := K)) := by
unfold latentTriple score outcome0 outcome1
fun_prop
calc
ate (directSamplingFullLaw H g p) =
∫ q : ScoreSpace ε × (OutcomeSpace × OutcomeSpace),
((q.2.2 : ℝ) - (q.2.1 : ℝ))
∂directSamplingLatentLaw H p := by
unfold ate
rw [← directSamplingFullLaw_map_latentTriple H g hg hOverlap p]
rw [integral_map hlatentMeas.aemeasurable (by fun_prop)]
rfl
_ = ∫ e : ScoreSpace ε,
(∫ q : OutcomeSpace × OutcomeSpace,
((q.2 : ℝ) - (q.1 : ℝ))
∂((Measure.dirac trialZeroOutcome).prod (trialOutcomeLaw p))) ∂H := by
unfold directSamplingLatentLaw
exact integral_prod _ hlatentInt
_ = ∫ q : OutcomeSpace × OutcomeSpace,
((q.2 : ℝ) - (q.1 : ℝ))
∂((Measure.dirac trialZeroOutcome).prod (trialOutcomeLaw p)) := by
simp [probReal_univ]
_ = ∫ y : OutcomeSpace, (y : ℝ) ∂(trialOutcomeLaw p) := by
rw [integral_prod _ houtInt]
simp [trialZeroOutcome]
_ = p := trialOutcomeLaw_mean p hp0 hp1
theorem directSamplingFullLaw_mem_causalLaws
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
K :
ℕ
H :
g :
ScoreSpace ε → LabelSpace K
hg :
hOverlap :
Overlap ε
p :
ℝ
hp0 :
0 ≤ p
hp1 :
p ≤ 1
conclusion
directSamplingFullLaw H g p ∈ CausalLaws H g
Proof (Lean source)
lemma directSamplingFullLaw_mem_causalLaws {ε : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(hOverlap : Overlap ε) (p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) :
directSamplingFullLaw H g p ∈ CausalLaws H g := by
have hprob := directSamplingFullLaw_isProbabilityMeasure
H g hg hOverlap p hp0 hp1
refine ⟨hprob, ?_⟩
exact
{ probability := hprob
measurableRelease := hg
scoreMarginal := directSamplingFullLaw_scoreMarginal
H g hg hOverlap p hp0 hp1
randomizedAssignment := directSamplingFullLaw_randomizedAssignment
H g hg hOverlap p hp0 hp1
consistency := directSamplingFullLaw_consistency H g hg p
deterministicRelease := directSamplingFullLaw_deterministicRelease H g hg p }
theorem directSamplingLatentLaw_withDensity
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(directSamplingLatentLaw H p).withDensity (reconstructedAssignmentDensity a)
= (H.withDensity (fun e => ofReal (armProb a e))).prod ((Measure.dirac trialZeroOutcome).prod (trialOutcomeLaw p))
Proof (Lean source)
lemma directSamplingLatentLaw_withDensity {ε : ℝ}
(H : Measure (ScoreSpace ε)) (p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1)
(a : ArmSpace) :
(directSamplingLatentLaw H p).withDensity
(reconstructedAssignmentDensity a) =
(H.withDensity (fun e => ofReal (armProb a e))).prod
((Measure.dirac trialZeroOutcome).prod (trialOutcomeLaw p)) := by
let _ : IsProbabilityMeasure (trialOutcomeLaw p) :=
trialOutcomeLaw_isProbabilityMeasure p hp0 hp1
unfold directSamplingLatentLaw reconstructedAssignmentDensity
symm
apply prod_withDensity_left
cases a <;> simp only [armProb, Bool.false_eq_true, ↓reduceIte] <;> fun_prop
theorem directSamplingReleasedComponent
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(((directSamplingLatentLaw H p).withDensity (reconstructedAssignmentDensity a)).map (reconstructedFullRow g a)).map releasedRecord
= ((H.withDensity (fun e => ofReal (armProb a e))).prod (samplingArmOutcomeLaw a p)).map (fun q => (g q.1, a, q.2))
Proof (Lean source)
lemma directSamplingReleasedComponent {ε : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) (a : ArmSpace) :
(((directSamplingLatentLaw H p).withDensity
(reconstructedAssignmentDensity a)).map
(reconstructedFullRow g a)).map releasedRecord =
((H.withDensity (fun e => ofReal (armProb a e))).prod
(samplingArmOutcomeLaw a p)).map
(fun q => (g q.1, a, q.2)) := by
let _ : IsProbabilityMeasure (trialOutcomeLaw p) :=
trialOutcomeLaw_isProbabilityMeasure p hp0 hp1
rw [Measure.map_map (by unfold releasedRecord label arm observed; fun_prop)
(reconstructedFullRow_measurable g hg a)]
rw [directSamplingLatentLaw_withDensity H p hp0 hp1 a]
let μ := H.withDensity (fun e => ofReal (armProb a e))
let ν := (Measure.dirac trialZeroOutcome).prod (trialOutcomeLaw p)
let s : OutcomeSpace × OutcomeSpace → OutcomeSpace :=
fun q => if a then q.2 else q.1
have hs : Measurable s := by
cases a <;> simp [s] <;> fun_prop
have hselect : ν.map s = samplingArmOutcomeLaw a p := by
cases a
· simp only [s, Bool.false_eq_true, ↓reduceIte, samplingArmOutcomeLaw]
rw [Measure.map_fst_prod]
simp
· simp only [s, ↓reduceIte, samplingArmOutcomeLaw]
rw [Measure.map_snd_prod]
simp
rw [← hselect]
have hprod : μ.prod (ν.map s) =
(μ.prod ν).map (map id s) := by
calc
μ.prod (ν.map s) = (μ.map id).prod (ν.map s) := by rw [Measure.map_id]
_ = (μ.prod ν).map (map id s) :=
Measure.map_prod_map μ ν measurable_id hs
rw [hprod, Measure.map_map (by fun_prop) (measurable_id.prodMap hs)]
congr 1
theorem directSamplingFullLaw_releasedComponents
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
releasedLaw (directSamplingFullLaw H g p)
Proof (Lean source)
lemma directSamplingFullLaw_releasedComponents {ε : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) :
releasedLaw (directSamplingFullLaw H g p) =
((H.withDensity (fun e => ofReal (armProb false e))).prod
(samplingArmOutcomeLaw false p)).map
(fun q => (g q.1, false, q.2)) +
((H.withDensity (fun e => ofReal (armProb true e))).prod
(samplingArmOutcomeLaw true p)).map
(fun q => (g q.1, true, q.2)) := by
unfold releasedLaw directSamplingFullLaw
rw [Measure.map_add _ _ (by unfold releasedRecord label arm observed; fun_prop)]
rw [directSamplingReleasedComponent H g hg p hp0 hp1 false,
directSamplingReleasedComponent H g hg p hp0 hp1 true]
def samplingDesignLaw
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
def samplingReadout
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
theorem samplingReadout_measurable
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma samplingReadout_measurable {K : ℕ} :
Measurable (samplingReadout (K := K)) := by
unfold samplingReadout
have hcond : MeasurableSet
{z : OutcomeSpace × (LabelSpace K × ArmSpace) | z.2.2 = true} :=
measurableSet_eq_fun (measurable_snd.comp measurable_snd) measurable_const
have hobs : Measurable
(fun z : OutcomeSpace × (LabelSpace K × ArmSpace) =>
if z.2.2 then z.1 else trialZeroOutcome) :=
Measurable.ite hcond measurable_fst measurable_const
fun_prop (disch := assumption)
theorem samplingDesignLaw_isProbabilityMeasure
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma samplingDesignLaw_isProbabilityMeasure {ε : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(hOverlap : Overlap ε) :
IsProbabilityMeasure (samplingDesignLaw H g) := by
have hmeas (a : ArmSpace) : Measurable
(fun e : ScoreSpace ε => ofReal (armProb a e)) := by
cases a
· change Measurable (fun e : ScoreSpace ε => ofReal (1 - (e : ℝ)))
fun_prop
· change Measurable (fun e : ScoreSpace ε => ofReal (e : ℝ))
fun_prop
have hsum :
H.withDensity (fun e => ofReal (armProb false e)) +
H.withDensity (fun e => ofReal (armProb true e)) = H := by
rw [← withDensity_add_left (hmeas false)]
have hfun :
(fun e : ScoreSpace ε => ofReal (armProb false e)) +
(fun e => ofReal (armProb true e)) = 1 := by
funext e
exact directSamplingAssignmentDensity_sum hOverlap
(e, (trialZeroOutcome, trialZeroOutcome))
rw [hfun, withDensity_one]
apply isProbabilityMeasure_iff.mpr
unfold samplingDesignLaw randomizedKeyLaw scoreAssignedKeyLaw
rw [Measure.add_apply,
Measure.map_apply (by fun_prop) MeasurableSet.univ,
Measure.map_apply (by fun_prop) MeasurableSet.univ]
simp only [preimage_univ]
rw [← Measure.add_apply, hsum]
exact isProbabilityMeasure_iff.mp inferInstance
theorem directSamplingFullLaw_releasedLaw
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
releasedLaw (directSamplingFullLaw H g p)
Proof (Lean source)
lemma directSamplingFullLaw_releasedLaw {ε : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) :
releasedLaw (directSamplingFullLaw H g p) =
((trialOutcomeLaw p).prod (samplingDesignLaw H g)).map
samplingReadout := by
let _ : IsProbabilityMeasure (trialOutcomeLaw p) :=
trialOutcomeLaw_isProbabilityMeasure p hp0 hp1
rw [directSamplingFullLaw_releasedComponents H g hg p hp0 hp1]
unfold samplingDesignLaw randomizedKeyLaw
rw [Measure.prod_add, Measure.map_add _ _ samplingReadout_measurable]
rw [← directSamplingReleasedComponent_design H g hg p hp0 hp1 false,
← directSamplingReleasedComponent_design H g hg p hp0 hp1 true]
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.MinimaxHonestLengthSamplingTV 2 declarations Product total-variation bound for the arbitrary-score sampling pair.
Product total-variation bound for the arbitrary-score sampling pair.
theorem directSampling_productReleased_tv_le_latent
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
K :
ℕ
H :
g :
ScoreSpace ε → LabelSpace K
hg :
hOverlap :
Overlap ε
p q :
ℝ
hp0 :
0 ≤ p
hp1 :
p ≤ 1
hq0 :
0 ≤ q
hq1 :
q ≤ 1
n :
ℕ
conclusion
tvDist (Measure.pi (fun _ : Fin n => releasedLaw (directSamplingFullLaw H g p))) (Measure.pi (fun _ : Fin n => releasedLaw (directSamplingFullLaw H g q)))
≤ tvDist (Measure.pi (fun _ : Fin n => (trialOutcomeLaw p).prod (samplingDesignLaw H g))) (Measure.pi (fun _ : Fin n => (trialOutcomeLaw q).prod (samplingDesignLaw H g)))
Proof (Lean source)
lemma directSampling_productReleased_tv_le_latent {ε : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(hOverlap : Overlap ε)
(p q : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1)
(hq0 : 0 ≤ q) (hq1 : q ≤ 1) (n : ℕ) :
tvDist
(Measure.pi (fun _ : Fin n =>
releasedLaw (directSamplingFullLaw H g p)))
(Measure.pi (fun _ : Fin n =>
releasedLaw (directSamplingFullLaw H g q))) ≤
tvDist
(Measure.pi (fun _ : Fin n =>
(trialOutcomeLaw p).prod (samplingDesignLaw H g)))
(Measure.pi (fun _ : Fin n =>
(trialOutcomeLaw q).prod (samplingDesignLaw H g))) := by
let f : OutcomeSpace × (LabelSpace K × ArmSpace) → Observation K :=
samplingReadout
have hf : Measurable f := samplingReadout_measurable
let _ : IsProbabilityMeasure (samplingDesignLaw H g) :=
samplingDesignLaw_isProbabilityMeasure H g hg hOverlap
let _ : IsProbabilityMeasure (trialOutcomeLaw p) :=
trialOutcomeLaw_isProbabilityMeasure p hp0 hp1
let _ : IsProbabilityMeasure (trialOutcomeLaw q) :=
trialOutcomeLaw_isProbabilityMeasure q hq0 hq1
have hmap (r : ℝ) (hr0 : 0 ≤ r) (hr1 : r ≤ 1) :
Measure.pi (fun _ : Fin n =>
releasedLaw (directSamplingFullLaw H g r)) =
(Measure.pi (fun _ : Fin n =>
(trialOutcomeLaw r).prod (samplingDesignLaw H g))).map
(fun z i => f (z i)) := by
let _ : IsProbabilityMeasure (trialOutcomeLaw r) :=
trialOutcomeLaw_isProbabilityMeasure r hr0 hr1
rw [Measure.pi_map_pi (fun _ => hf.aemeasurable)]
simp only [directSamplingFullLaw_releasedLaw H g hg r hr0 hr1]
rfl
rw [hmap p hp0 hp1, hmap q hq0 hq1]
exact trialReadout_tv_le _ _ _
(measurable_pi_lambda _ fun _ => hf.comp (measurable_pi_apply _))
theorem directSampling_productReleased_tv_bound
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
K :
ℕ
H :
g :
ScoreSpace ε → LabelSpace K
hg :
hOverlap :
Overlap ε
u :
ℝ
hu :
|u| < 1 / 2
n :
ℕ
conclusion
tvDist (Measure.pi (fun _ : Fin n => releasedLaw (directSamplingFullLaw H g (1 / 2 + u)))) (Measure.pi (fun _ : Fin n => releasedLaw (directSamplingFullLaw H g (1 / 2))))
≤ (1 / 2 : ℝ) * sqrt ((1 + 4 * u ^ 2) ^ n - 1)
Proof (Lean source)
lemma directSampling_productReleased_tv_bound {ε : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace K) (hg : Measurable g)
(hOverlap : Overlap ε)
(u : ℝ) (hu : |u| < 1 / 2) (n : ℕ) :
tvDist
(Measure.pi (fun _ : Fin n => releasedLaw
(directSamplingFullLaw H g (1 / 2 + u))))
(Measure.pi (fun _ : Fin n => releasedLaw
(directSamplingFullLaw H g (1 / 2)))) ≤
(1 / 2 : ℝ) * sqrt ((1 + 4 * u ^ 2) ^ n - 1) := by
have hub := abs_lt.mp hu
let ρ := samplingDesignLaw H g
let μ := (trialOutcomeLaw (1 / 2 + u)).prod ρ
let ν := (trialOutcomeLaw (1 / 2)).prod ρ
let _ : IsProbabilityMeasure (trialOutcomeLaw (1 / 2 + u)) :=
trialOutcomeLaw_isProbabilityMeasure _ (by linarith) (by linarith)
let _ : IsProbabilityMeasure (trialOutcomeLaw (1 / 2)) :=
trialOutcomeLaw_isProbabilityMeasure _ (by norm_num) (by norm_num)
let _ : IsProbabilityMeasure ρ :=
samplingDesignLaw_isProbabilityMeasure H g hg hOverlap
have h := trialOutcomeLaw_ac_integrable_center u
have hac : μ ≪ ν := h.1.prod (Measure.AbsolutelyContinuous.refl ρ)
have hint : Integrable
(fun z => ((μ.rnDeriv ν z).toReal - 1) ^ 2) ν :=
trial_ancillary_integrable _ _ ρ h.1 h.2
calc
_ ≤ tvDist
(Measure.pi (fun _ : Fin n => μ))
(Measure.pi (fun _ : Fin n => ν)) :=
directSampling_productReleased_tv_le_latent H g hg hOverlap
(1 / 2 + u) (1 / 2) (by linarith) (by linarith)
(by norm_num) (by norm_num) n
_ ≤ (1 / 2 : ℝ) *
sqrt ((1 + chiSqDiv μ ν) ^ n - 1) :=
trialProduct_tv_le_of_chisq μ ν hac hint n
_ = _ := by
change (1 / 2 : ℝ) * sqrt
((1 + chiSqDiv
((trialOutcomeLaw (1 / 2 + u)).prod ρ)
((trialOutcomeLaw (1 / 2)).prod ρ)) ^ n - 1) = _
rw [chiSqDiv_prod_ancillary
(trialOutcomeLaw (1 / 2 + u)) (trialOutcomeLaw (1 / 2)) ρ h.1 h.2]
rw [trialOutcomeLaw_chisq_center u hu]
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.MinimaxHonestLengthTesting 1 declarations Two-point testing lower bounds for ordinary released-trial confidence intervals.
Two-point testing lower bounds for ordinary released-trial confidence intervals.
theorem intervalProcedure_twoPoint_expectedLength
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
K n :
ℕ
C :
Q0 Q1 :
θ0 θ1 α :
ℝ
hθ :
θ0 ≤ θ1
hcover0 :
1 - α ≤ Q0.real {x | C.contains x θ0}
hcover1 :
1 - α ≤ Q1.real {x | C.contains x θ1}
conclusion
(θ1 - θ0) * (1 - 2 * α - tvDist Q0 Q1) ≤ ∫ x, C.length x ∂Q0
Proof (Lean source)
lemma intervalProcedure_twoPoint_expectedLength {K n : ℕ}
(C : IntervalProcedure n K)
(Q0 Q1 : Measure (TrialSample n K))
[IsProbabilityMeasure Q0] [IsProbabilityMeasure Q1]
(θ0 θ1 α : ℝ) (hθ : θ0 ≤ θ1)
(hcover0 : 1 - α ≤ Q0.real {x | C.contains x θ0})
(hcover1 : 1 - α ≤ Q1.real {x | C.contains x θ1}) :
(θ1 - θ0) * (1 - 2 * α - tvDist Q0 Q1) ≤
∫ x, C.length x ∂Q0 := by
let A0 : Set (TrialSample n K) := {x | C.contains x θ0}
let A1 : Set (TrialSample n K) := {x | C.contains x θ1}
let I := A0 ∩ A1
have hA (θ : ℝ) : MeasurableSet {x | C.contains x θ} := by
change MeasurableSet {x | C.lo x ≤ θ ∧ θ ≤ C.hi x}
exact (measurableSet_le C.measurable_lo measurable_const).inter
(measurableSet_le measurable_const C.measurable_hi)
have hA0 : MeasurableSet A0 := hA θ0
have hA1 : MeasurableSet A1 := hA θ1
have hI : MeasurableSet I := hA0.inter hA1
have htransfer : 1 - α - tvDist Q0 Q1 ≤ Q0.real A1 := by
have htv := measureReal_sub_le_tvDist
(μ := Q0) (ν := Q1) hA1
dsimp [A1] at htv
linarith
have hmass : 1 - 2 * α - tvDist Q0 Q1 ≤ Q0.real I := by
have hu : Q0.real (A0 ∪ A1) ≤ 1 := measureReal_le_one
have hsum := measureReal_union_add_inter (μ := Q0) (s := A0) (t := A1)
hA1 (measure_ne_top Q0 A0) (measure_ne_top Q0 A1)
change Q0.real (A0 ∪ A1) + Q0.real I = Q0.real A0 + Q0.real A1 at hsum
linarith
have hlenMeas : Measurable C.length := C.measurable_hi.sub C.measurable_lo
have hlenNonneg (x : TrialSample n K) : 0 ≤ C.length x :=
sub_nonneg.mpr (C.ordered x)
have hlenBound (x : TrialSample n K) : C.length x ≤ 2 := by
dsimp [IntervalProcedure.length]
linarith [C.lower_bound x, C.upper_bound x]
have hlenInt : Integrable C.length Q0 := by
apply Integrable.of_bound hlenMeas.aestronglyMeasurable 2
filter_upwards with x
rw [Real.norm_eq_abs, abs_of_nonneg (hlenNonneg x)]
exact hlenBound x
have hindInt : Integrable (I.indicator (fun _ => θ1 - θ0)) Q0 :=
(integrable_const _).indicator hI
have hpoint (x : TrialSample n K) :
I.indicator (fun _ => θ1 - θ0) x ≤ C.length x := by
by_cases hx : x ∈ I
· rw [Set.indicator_of_mem hx]
have hx0 : C.lo x ≤ θ0 ∧ θ0 ≤ C.hi x := hx.1
have hx1 : C.lo x ≤ θ1 ∧ θ1 ≤ C.hi x := hx.2
dsimp [IntervalProcedure.length]
linarith
· rw [Set.indicator_of_notMem hx]
exact hlenNonneg x
have hmono := integral_mono hindInt hlenInt hpoint
rw [integral_indicator_const (θ1 - θ0) hI] at hmono
simp only [smul_eq_mul] at hmono
rw [mul_comm (Q0.real I) (θ1 - θ0)] at hmono
exact (mul_le_mul_of_nonneg_left hmass (sub_nonneg.mpr hθ)).trans hmono
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.MinimaxHonestLengthUpper 1 declarations Outer-infimum upper bound from the explicit midpoint interval.
Outer-infimum upper bound from the explicit midpoint interval.
theorem minimaxHonestLength_midpoint_upper
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
minimaxHonestLength n K H α
≤ midpointLengthConstant ε α * ((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹)
Proof (Lean source)
lemma minimaxHonestLength_midpoint_upper {ε α : ℝ}
(hOverlap : Overlap ε) (hα : 0 < α ∧ α < 1 / 2)
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(n K : ℕ) (hn : 0 < n) (hK : 0 < K) :
minimaxHonestLength n K H α ≤
midpointLengthConstant ε α *
((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹) := by
let CI := midpointInterval hOverlap hα n K hn hK
let R := midpointLengthConstant ε α *
((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹)
let V : Set ℝ := {v : ℝ | ∃ gC ∈ honestProcedures n K H α,
v = sSup {u : ℝ | ∃ P ∈ CausalLaws H gC.1,
u = ∫ x, gC.2.length x
∂Measure.pi (fun _ : Fin n => releasedLaw P)}}
have hVbelow : BddBelow V := by
refine ⟨0, ?_⟩
rintro v ⟨gC, hgC, rfl⟩
exact innerExpectedLengthSup_nonneg H gC.1 gC.2
have hhonest : (equalWidthRelease ε K hK, CI) ∈
honestProcedures n K H α :=
equalWidth_midpointInterval_honest hOverlap hα H n K hn hK
let U : Set ℝ := {u : ℝ | ∃ P ∈ CausalLaws H (equalWidthRelease ε K hK),
u = ∫ x, CI.length x ∂Measure.pi (fun _ : Fin n => releasedLaw P)}
have hUne : U.Nonempty := by
obtain ⟨P, hP⟩ := equalWidth_causalLaws_nonempty hOverlap H K hK
refine ⟨∫ x, CI.length x ∂Measure.pi (fun _ : Fin n => releasedLaw P),
P, hP, rfl⟩
have hUsup : sSup U ≤ R := by
apply csSup_le hUne
rintro u ⟨P, hP, rfl⟩
letI : IsProbabilityMeasure P := hP.1
have hrecord : Measurable (releasedRecord (ε := ε) (J := K)) := by
unfold releasedRecord label arm observed
fun_prop
letI : IsProbabilityMeasure (releasedLaw P) := by
unfold releasedLaw
exact Measure.isProbabilityMeasure_map hrecord.aemeasurable
exact midpointInterval_expectedLength_le hOverlap hα n K hn hK (releasedLaw P)
unfold minimaxHonestLength
change sInf V ≤ R
calc
sInf V ≤ sSup U := by
apply csInf_le hVbelow
exact ⟨(equalWidthRelease ε K hK, CI), hhonest, rfl⟩
_ ≤ R := hUsup
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.Projection 19 declarations Joint empirical-measure projection for external score-log inference.
Joint empirical-measure projection for external score-log inference.
def ArmCellArray
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
J :
ℕ
α :
Type*
def
ArmCellArray J α :
Type u_1
given by
def empiricalTrialCells
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
def empiricalScoreCells
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
def outcomeCDFDistance
For the specified mathematical inputs, this definition introduces the corresponding object.
def scoreCDFDistance
For the specified mathematical inputs, this definition introduces the corresponding object.
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
ProjectionCompatible lam σ :
Prop
def projectedArmEndpoint
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
projectedArmEndpoint lam σ a upper :
ℝ
def rationalOutcomePoint
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
q :
ℚ
def
given by
⟨max 0 (min 1 (q : ℝ)), by constructor · exact le_max_left _ _ · exact max_le (by norm_num) (min_le_left _ _)⟩
def rationalScorePoint
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
def rationalOutcomeMeasure
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
l :
List (ℚ × ℚ)
def
given by
(l.map (fun p => ofReal (p.2 : ℝ) • Measure.dirac (rationalOutcomePoint p.1))).sum
def rationalScoreMeasure
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
(l.map (fun p => ofReal (p.2 : ℝ) • Measure.dirac (rationalScorePoint ε hOverlap p.1))).sum
def rationalArraySieve
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J :
ℕ
def
given by
by classical exact if hOverlap : Overlap ε then range (fun code : (ArmSpace → LabelSpace J → List (ℚ × ℚ)) × (ArmSpace → LabelSpace J → List (ℚ × ℚ)) => (fun a r => rationalOutcomeMeasure (code.1 a r), fun a r => rationalScoreMeasure ε hOverlap (code.2 a r))) ∩ {b | ProjectionCompatible b.1 b.2} else ∅
def projectionCandidates
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
{b | (∀ a r, (b.1 a r) univ < ⊤ ∧ (b.2 a r) univ < ⊤) ∧ ProjectionCompatible b.1 b.2 ∧ (∑ a : ArmSpace, ∑ r : LabelSpace J, outcomeCDFDistance (b.1 a r) (empiricalTrialCells x.1 a r)) ≤ 4 * J / (α * sqrt n) ∧ (∑ a : ArmSpace, ∑ r : LabelSpace J, scoreCDFDistance (b.2 a r) (empiricalScoreCells g x.2 a r)) ≤ 2 * J * (2 - 2 * ε) / (α * sqrt m)}
def sievedProjectionCandidates
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
sievedProjectionCandidates g α x :
given by
projectionCandidates g α x ∩ rationalArraySieve ε J
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma countable_rationalArraySieve (ε : ℝ) (J : ℕ) :
(rationalArraySieve ε J).Countable := by
classical
unfold rationalArraySieve
split_ifs
· exact (Set.countable_range _).mono Set.inter_subset_left
· exact Set.countable_empty
theorem countable_sievedProjectionCandidates Lemma countable_sievedProjectionCandidates in the paper ↗
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(sievedProjectionCandidates g α x).Countable
Proof (Lean source)
lemma countable_sievedProjectionCandidates {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(x : ExternalSample ε J n m) :
(sievedProjectionCandidates g α x).Countable := by
exact (countable_rationalArraySieve ε J).mono Set.inter_subset_right
def projectedATEInterval
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
Icc (projectedArmEndpoint lam σ true false - projectedArmEndpoint lam σ false true) (projectedArmEndpoint lam σ true true - projectedArmEndpoint lam σ false false)
def sievedTotalizedProjectionCI
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
by classical let B := sievedProjectionCandidates g α x let C := Icc (-1 : ℝ) 1 ∩ closure (convexHull ℝ (⋃ b ∈ B, projectedATEInterval b.1 b.2)) exact if B.Nonempty ∧
C.Nonempty then C else {0}
def externalProjectionCI
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionCandidateStability 3 declarations Candidate-to-population endpoint stability for the external projection.
Candidate-to-population endpoint stability for the external projection.
theorem projectionCandidate_armEndpoint_abs_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
hOverlap :
Overlap ε
x :
ExternalSample ε J n m
b :
lam :
σ :
ArmCellArray J (ScoreSpace ε)
hb :
b ∈ projectionCandidates g α x
hfinLam :
∀ a r, IsFiniteMeasure (lam a r)
hfinσ :
∀ a r, IsFiniteMeasure (σ a r)
hcompat :
ProjectionCompatible lam σ
a :
upper :
conclusion
|projectedArmEndpoint b.1 b.2 a upper - projectedArmEndpoint lam σ a upper|
≤ (ε⁻¹ + (ε ^ 2)⁻¹) * (4 * J / (α * sqrt n) + (∑ a : ArmSpace, ∑ r : LabelSpace J, outcomeCDFDistance (empiricalTrialCells x.1 a r) (lam a r)) + 2 * J * (2 - 2 * ε) / (α * sqrt m) + (∑ a : ArmSpace, ∑ r : LabelSpace J, scoreCDFDistance (empiricalScoreCells g x.2 a r) (σ a r)))
Proof (Lean source)
lemma projectionCandidate_armEndpoint_abs_le
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(hOverlap : Overlap ε)
(x : ExternalSample ε J n m)
(b : ArmCellArray J OutcomeSpace × ArmCellArray J (ScoreSpace ε))
(lam : ArmCellArray J OutcomeSpace)
(σ : ArmCellArray J (ScoreSpace ε))
(hb : b ∈ projectionCandidates g α x)
(hfinLam : ∀ a r, IsFiniteMeasure (lam a r))
(hfinσ : ∀ a r, IsFiniteMeasure (σ a r))
(hcompat : ProjectionCompatible lam σ)
(a : ArmSpace) (upper : Bool) :
|projectedArmEndpoint b.1 b.2 a upper -
projectedArmEndpoint lam σ a upper| ≤
(ε⁻¹ + (ε ^ 2)⁻¹) *
(4 * J / (α * sqrt n) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (empiricalTrialCells x.1 a r) (lam a r)) +
2 * J * (2 - 2 * ε) / (α * sqrt m) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (empiricalScoreCells g x.2 a r) (σ a r))) := by
have hfinB₁ : ∀ a r, IsFiniteMeasure (b.1 a r) :=
fun a r => ⟨(hb.1 a r).1⟩
have hfinB₂ : ∀ a r, IsFiniteMeasure (b.2 a r) :=
fun a r => ⟨(hb.1 a r).2⟩
have hε : 0 < ε := hOverlap.1
have hL : 0 ≤ ε⁻¹ + (ε ^ 2)⁻¹ := by positivity
have hArm : ∀ a upper,
|projectedArmEndpoint b.1 b.2 a upper -
projectedArmEndpoint lam σ a upper| ≤
(ε⁻¹ + (ε ^ 2)⁻¹) *
((∑ r : LabelSpace J,
outcomeCDFDistance (b.1 a r) (lam a r)) +
∑ r : LabelSpace J,
scoreCDFDistance (b.2 a r) (σ a r)) := by
intro a upper
exact projectedArmEndpoint_abs_le_of_compatible_cells hOverlap
b.1 lam b.2 σ hfinB₁ hfinLam hfinB₂ hfinσ hb.2.1 hcompat a upper
have hEndpoint := projectedArmEndpoint_abs_le_of_armBounds hOverlap
b.1 lam b.2 σ (ε⁻¹ + (ε ^ 2)⁻¹) hL hArm a upper
have hDistance := projectionCandidate_populationDistance_le_of_populationFinite
g α hOverlap x b lam σ hb
(fun a r => (hfinLam a r).measure_univ_lt_top)
(fun a r => (hfinσ a r).measure_univ_lt_top)
exact hEndpoint.trans (mul_le_mul_of_nonneg_left hDistance hL)
theorem projectionCandidate_populationArmEndpoint_abs_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
PH :
hPH :
PH ∈ ExternalLaws g
x :
ExternalSample ε J n m
b :
hb :
b ∈ projectionCandidates g α x
a :
upper :
conclusion
|projectedArmEndpoint b.1 b.2 a upper
- projectedArmEndpoint (trialPopulationCells (releasedLaw PH.1)) (scorePopulationCells PH.2 g) a upper|
≤ (ε⁻¹ + (ε ^ 2)⁻¹) * (4 * J / (α * sqrt n) + (∑ a : ArmSpace, ∑ r : LabelSpace J, outcomeCDFDistance (empiricalTrialCells x.1 a r) (trialPopulationCells (releasedLaw PH.1) a r)) + 2 * J * (2 - 2 * ε) / (α * sqrt m) + (∑ a : ArmSpace, ∑ r : LabelSpace J, scoreCDFDistance (empiricalScoreCells g x.2 a r) (scorePopulationCells PH.2 g a r)))
Proof (Lean source)
lemma projectionCandidate_populationArmEndpoint_abs_le
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(PH : Measure (FullRow ε J) × Measure (ScoreSpace ε))
(hPH : PH ∈ ExternalLaws g)
(x : ExternalSample ε J n m)
(b : ArmCellArray J OutcomeSpace × ArmCellArray J (ScoreSpace ε))
(hb : b ∈ projectionCandidates g α x)
(a : ArmSpace) (upper : Bool) :
|projectedArmEndpoint b.1 b.2 a upper -
projectedArmEndpoint (trialPopulationCells (releasedLaw PH.1))
(scorePopulationCells PH.2 g) a upper| ≤
(ε⁻¹ + (ε ^ 2)⁻¹) *
(4 * J / (α * sqrt n) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (empiricalTrialCells x.1 a r)
(trialPopulationCells (releasedLaw PH.1) a r)) +
2 * J * (2 - 2 * ε) / (α * sqrt m) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (empiricalScoreCells g x.2 a r)
(scorePopulationCells PH.2 g a r))) := by
haveI : IsProbabilityMeasure PH.2 := hPH.2.1
haveI : IsProbabilityMeasure (releasedLaw PH.1) := by
unfold releasedLaw
exact @Measure.isProbabilityMeasure_map _ _ _ _ PH.1 hPH.1 _
(by unfold releasedRecord label arm observed; fun_prop :
Measurable (releasedRecord (ε := ε) (J := J))).aemeasurable
have hfin := projectionPopulationCells_finite PH.2 g (releasedLaw PH.1)
hPH.2.2.overlap hPH.2.2.measurableRelease
exact projectionCandidate_armEndpoint_abs_le g α hPH.2.2.overlap x b
(trialPopulationCells (releasedLaw PH.1)) (scorePopulationCells PH.2 g)
hb hfin.1 hfin.2
(projectionCompatible_populationCells PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH) hPH.2.2.overlap) a upper
theorem projectionCandidate_populationATEInterval_subset
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
PH :
hPH :
PH ∈ ExternalLaws g
x :
ExternalSample ε J n m
b :
hb :
b ∈ projectionCandidates g α x
conclusion
δ :
(ε⁻¹ + (ε ^ 2)⁻¹) * (4 * J / (α * sqrt n) + (∑ a : ArmSpace, ∑ r : LabelSpace J, outcomeCDFDistance (empiricalTrialCells x.1 a r) (trialPopulationCells (releasedLaw PH.1) a r)) + 2 * J * (2 - 2 * ε) / (α * sqrt m) + (∑ a : ArmSpace, ∑ r : LabelSpace J, scoreCDFDistance (empiricalScoreCells g x.2 a r) (scorePopulationCells PH.2 g a r)))
projectedATEInterval b.1 b.2 ⊆ Icc ((projectedArmEndpoint (trialPopulationCells (releasedLaw PH.1)) (scorePopulationCells PH.2 g) true false - projectedArmEndpoint (trialPopulationCells (releasedLaw PH.1)) (scorePopulationCells PH.2 g) false true) - 2 * δ) ((projectedArmEndpoint (trialPopulationCells (releasedLaw PH.1)) (scorePopulationCells PH.2 g) true true - projectedArmEndpoint (trialPopulationCells (releasedLaw PH.1)) (scorePopulationCells PH.2 g) false false) + 2 * δ)
Proof (Lean source)
lemma projectionCandidate_populationATEInterval_subset
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(PH : Measure (FullRow ε J) × Measure (ScoreSpace ε))
(hPH : PH ∈ ExternalLaws g)
(x : ExternalSample ε J n m)
(b : ArmCellArray J OutcomeSpace × ArmCellArray J (ScoreSpace ε))
(hb : b ∈ projectionCandidates g α x) :
let δ := (ε⁻¹ + (ε ^ 2)⁻¹) *
(4 * J / (α * sqrt n) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (empiricalTrialCells x.1 a r)
(trialPopulationCells (releasedLaw PH.1) a r)) +
2 * J * (2 - 2 * ε) / (α * sqrt m) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (empiricalScoreCells g x.2 a r)
(scorePopulationCells PH.2 g a r)))
projectedATEInterval b.1 b.2 ⊆
Icc
((projectedArmEndpoint (trialPopulationCells (releasedLaw PH.1))
(scorePopulationCells PH.2 g) true false -
projectedArmEndpoint (trialPopulationCells (releasedLaw PH.1))
(scorePopulationCells PH.2 g) false true) - 2 * δ)
((projectedArmEndpoint (trialPopulationCells (releasedLaw PH.1))
(scorePopulationCells PH.2 g) true true -
projectedArmEndpoint (trialPopulationCells (releasedLaw PH.1))
(scorePopulationCells PH.2 g) false false) + 2 * δ) := by
dsimp
apply projectedATEInterval_subset_of_armEndpoint_bounds
intro a upper
exact projectionCandidate_populationArmEndpoint_abs_le g α PH hPH x b hb a upper
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionCellStability 4 declarations Cellwise stability of the quantile product used by the projected arm endpoint.
Cellwise stability of the quantile product used by the projected arm endpoint.
theorem projectedCellSummand_abs_le_of_pos
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
hOverlap :
Overlap ε
a :
upper :
μ μ' :
σ σ' :
Measure (ScoreSpace ε)
hcompat :
μ.real univ = σ.real univ
hcompat' :
μ'.real univ = σ'.real univ
hp :
0 < μ.real univ
hp' :
0 < μ'.real univ
conclusion
|μ.real univ * (∫ u in (0 : ℝ)..1, quantile (((μ univ)⁻¹ • μ).map (fun y : OutcomeSpace => (y : ℝ))) u * quantile (((σ univ)⁻¹ • σ).map (fun e => (armProb a e)⁻¹)) (if upper then u else 1 - u))
- μ'.real univ * (∫ u in (0 : ℝ)..1, quantile (((μ' univ)⁻¹ • μ').map (fun y : OutcomeSpace => (y : ℝ))) u * quantile (((σ' univ)⁻¹ • σ').map (fun e => (armProb a e)⁻¹)) (if upper then u else 1 - u))|
≤ (ε⁻¹ + (ε ^ 2)⁻¹) * (outcomeCDFDistance μ μ' + scoreCDFDistance σ σ')
Proof (Lean source)
lemma projectedCellSummand_abs_le_of_pos {ε : ℝ} (hOverlap : Overlap ε)
(a : ArmSpace) (upper : Bool)
(μ μ' : Measure OutcomeSpace) (σ σ' : Measure (ScoreSpace ε))
[IsFiniteMeasure μ] [IsFiniteMeasure μ']
[IsFiniteMeasure σ] [IsFiniteMeasure σ']
(hcompat : μ.real univ = σ.real univ)
(hcompat' : μ'.real univ = σ'.real univ)
(hp : 0 < μ.real univ) (hp' : 0 < μ'.real univ) :
|μ.real univ * (∫ u in (0 : ℝ)..1,
quantile
(((μ univ)⁻¹ • μ).map (fun y : OutcomeSpace => (y : ℝ))) u *
quantile
(((σ univ)⁻¹ • σ).map (fun e => (armProb a e)⁻¹))
(if upper then u else 1 - u)) -
μ'.real univ * (∫ u in (0 : ℝ)..1,
quantile
(((μ' univ)⁻¹ • μ').map (fun y : OutcomeSpace => (y : ℝ))) u *
quantile
(((σ' univ)⁻¹ • σ').map (fun e => (armProb a e)⁻¹))
(if upper then u else 1 - u))| ≤
(ε⁻¹ + (ε ^ 2)⁻¹) *
(outcomeCDFDistance μ μ' + scoreCDFDistance σ σ') := by
let Y : Measure ℝ := ((μ univ)⁻¹ • μ).map (fun y : OutcomeSpace => (y : ℝ))
let Y' : Measure ℝ := ((μ' univ)⁻¹ • μ').map (fun y : OutcomeSpace => (y : ℝ))
let W : Measure ℝ := ((σ univ)⁻¹ • σ).map (fun e => (armProb a e)⁻¹)
let W' : Measure ℝ := ((σ' univ)⁻¹ • σ').map (fun e => (armProb a e)⁻¹)
let A : ℝ := ∫ u in (0 : ℝ)..1, quantile Y u *
quantile W (if upper then u else 1 - u)
let A' : ℝ := ∫ u in (0 : ℝ)..1, quantile Y' u *
quantile W' (if upper then u else 1 - u)
have hσp : 0 < σ.real univ := hcompat ▸ hp
have hσp' : 0 < σ'.real univ := hcompat' ▸ hp'
letI : IsProbabilityMeasure Y := by
dsimp [Y]
rw [map_normalized_subtype_eq_finiteNormalize μ hp]
infer_instance
letI : IsProbabilityMeasure Y' := by
dsimp [Y']
rw [map_normalized_subtype_eq_finiteNormalize μ' hp']
infer_instance
letI : IsProbabilityMeasure W := by
dsimp [W]
rw [map_normalized_inverseArmProb_eq_finiteNormalize_map a σ hσp]
exact Measure.isProbabilityMeasure_map
(measurable_projectionInverseArmProbReal a).aemeasurable
letI : IsProbabilityMeasure W' := by
dsimp [W']
rw [map_normalized_inverseArmProb_eq_finiteNormalize_map a σ' hσp']
exact Measure.isProbabilityMeasure_map
(measurable_projectionInverseArmProbReal a).aemeasurable
have hYs : Y (Icc 0 1)ᶜ = 0 := by
dsimp [Y]
exact normalizedOutcome_support μ hp
have hYs' : Y' (Icc 0 1)ᶜ = 0 := by
dsimp [Y']
exact normalizedOutcome_support μ' hp'
have hWs : W (Icc 0 ε⁻¹)ᶜ = 0 := by
dsimp [W]
exact normalizedInverseArmProb_support hOverlap a σ hσp
have hWs' : W' (Icc 0 ε⁻¹)ᶜ = 0 := by
dsimp [W']
exact normalizedInverseArmProb_support hOverlap a σ' hσp'
have hεinv : 0 ≤ ε⁻¹ := (inv_pos.mpr hOverlap.1).le
have hε2inv : 0 ≤ (ε ^ 2)⁻¹ := by positivity
have hAb : |A| ≤ ε⁻¹ := by
dsimp [A]
simpa using abs_quantile_product_integral_le Y W (by norm_num) hεinv hYs hWs upper
have hAb' : |A'| ≤ ε⁻¹ := by
dsimp [A']
simpa using abs_quantile_product_integral_le Y' W' (by norm_num) hεinv hYs' hWs' upper
let DY : ℝ := ∫ u in (0 : ℝ)..1,
|quantile Y u - quantile Y' u|
let DW : ℝ := ∫ u in (0 : ℝ)..1,
|quantile W u - quantile W' u|
let O : ℝ := ∫ t in (0 : ℝ)..1,
|(μ {y | (y : ℝ) ≤ t}).toReal - (μ' {y | (y : ℝ) ≤ t}).toReal|
let S : ℝ := ∫ t in ε..(1 - ε),
|(σ {e | (e : ℝ) ≤ t}).toReal - (σ' {e | (e : ℝ) ≤ t}).toReal|
let d : ℝ := |μ.real univ - μ'.real univ|
have hDY : 0 ≤ DY := by
dsimp [DY]
exact intervalIntegral.integral_nonneg (by norm_num) (fun _ _ => abs_nonneg _)
have hDW : 0 ≤ DW := by
dsimp [DW]
exact intervalIntegral.integral_nonneg (by norm_num) (fun _ _ => abs_nonneg _)
have hO : 0 ≤ O := by
dsimp [O]
exact intervalIntegral.integral_nonneg (by norm_num) (fun _ _ => abs_nonneg _)
have hS : 0 ≤ S := by
dsimp [S]
exact intervalIntegral.integral_nonneg (by linarith [hOverlap.2])
(fun _ _ => abs_nonneg _)
have hd : 0 ≤ d := abs_nonneg _
have hout : min (μ.real univ) (μ'.real univ) * DY ≤ O + d := by
dsimp [DY, O, d, Y, Y']
simpa using finite_mass_subtype_quantile_transport_le (by norm_num) μ μ' hp hp'
have hscore : min (μ.real univ) (μ'.real univ) *
(∫ u in (0 : ℝ)..1,
|quantile
(((σ univ)⁻¹ • σ).map (fun e : ScoreSpace ε => (e : ℝ))) u -
quantile
(((σ' univ)⁻¹ • σ').map (fun e : ScoreSpace ε => (e : ℝ))) u|) ≤
S + (1 - 2 * ε) * d := by
have h := finite_mass_subtype_quantile_transport_le
(by linarith [hOverlap.2] : ε ≤ 1 - ε) σ σ' hσp hσp'
rw [← hcompat, ← hcompat'] at h
simpa [S, d, sub_sub, two_mul, Measure.map_smul] using h
have hpush : DW ≤ (ε ^ 2)⁻¹ *
(∫ u in (0 : ℝ)..1,
|quantile
(((σ univ)⁻¹ • σ).map (fun e : ScoreSpace ε => (e : ℝ))) u -
quantile
(((σ' univ)⁻¹ • σ').map (fun e : ScoreSpace ε => (e : ℝ))) u|) := by
dsimp [DW, W, W']
exact inverseArmProb_quantile_distance_le hOverlap a σ σ' hσp hσp'
have hwidth : 0 ≤ 1 - 2 * ε ∧ 1 - 2 * ε ≤ 1 := by
constructor <;> linarith [hOverlap.1, hOverlap.2]
have hweight : min (μ.real univ) (μ'.real univ) * DW ≤
(ε ^ 2)⁻¹ * (S + d) := by
have hmin : 0 ≤ min (μ.real univ) (μ'.real univ) :=
le_min measureReal_nonneg measureReal_nonneg
calc
_ ≤ min (μ.real univ) (μ'.real univ) *
((ε ^ 2)⁻¹ * ∫ u in (0 : ℝ)..1,
|quantile
(((σ univ)⁻¹ • σ).map (fun e : ScoreSpace ε => (e : ℝ))) u -
quantile
(((σ' univ)⁻¹ • σ').map (fun e : ScoreSpace ε => (e : ℝ))) u|) :=
mul_le_mul_of_nonneg_left hpush hmin
_ = (ε ^ 2)⁻¹ * (min (μ.real univ) (μ'.real univ) *
∫ u in (0 : ℝ)..1,
|quantile
(((σ univ)⁻¹ • σ).map (fun e : ScoreSpace ε => (e : ℝ))) u -
quantile
(((σ' univ)⁻¹ • σ').map (fun e : ScoreSpace ε => (e : ℝ))) u|) := by ring
_ ≤ (ε ^ 2)⁻¹ * (S + (1 - 2 * ε) * d) :=
mul_le_mul_of_nonneg_left hscore hε2inv
_ ≤ (ε ^ 2)⁻¹ * (S + d) := by
gcongr
nlinarith [mul_nonneg hd (sub_nonneg.mpr hwidth.2)]
have hprod : |A - A'| ≤ ε⁻¹ * DY + DW := by
dsimp [A, A', DY, DW]
simpa using quantile_product_integral_stability Y Y' W W'
(by norm_num) hεinv hYs hYs' hWs hWs' upper
have hmin : 0 ≤ min (μ.real univ) (μ'.real univ) :=
le_min measureReal_nonneg measureReal_nonneg
have hprodScaled : min (μ.real univ) (μ'.real univ) * |A - A'| ≤
ε⁻¹ * (O + d) + (ε ^ 2)⁻¹ * (S + d) := by
calc
_ ≤ min (μ.real univ) (μ'.real univ) * (ε⁻¹ * DY + DW) :=
mul_le_mul_of_nonneg_left hprod hmin
_ = ε⁻¹ * (min (μ.real univ) (μ'.real univ) * DY) +
min (μ.real univ) (μ'.real univ) * DW := by ring
_ ≤ ε⁻¹ * (O + d) + (ε ^ 2)⁻¹ * (S + d) :=
add_le_add (mul_le_mul_of_nonneg_left hout hεinv) hweight
have hscale : |μ.real univ * A - μ'.real univ * A'| ≤
min (μ.real univ) (μ'.real univ) * |A - A'| +
ε⁻¹ * |μ.real univ - μ'.real univ| :=
abs_mass_mul_sub_mass_mul_le
(p := μ.real univ) (q := μ'.real univ) (A := A) (A' := A')
(B := ε⁻¹) measureReal_nonneg measureReal_nonneg hAb hAb'
change |μ.real univ * A - μ'.real univ * A'| ≤ _
calc
_ ≤ min (μ.real univ) (μ'.real univ) * |A - A'| +
ε⁻¹ * |μ.real univ - μ'.real univ| := hscale
_ ≤ (ε⁻¹ * (O + d) + (ε ^ 2)⁻¹ * (S + d)) + ε⁻¹ * d :=
add_le_add hprodScaled le_rfl
_ ≤ (ε⁻¹ + (ε ^ 2)⁻¹) * ((d + O) + (d + S)) := by
nlinarith [mul_nonneg hεinv hS, mul_nonneg hε2inv hO,
mul_nonneg hε2inv hd]
_ = (ε⁻¹ + (ε ^ 2)⁻¹) *
(outcomeCDFDistance μ μ' + scoreCDFDistance σ σ') := by
unfold outcomeCDFDistance scoreCDFDistance
rw [← hcompat, ← hcompat']
theorem projectedCellTerm_abs_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
hOverlap :
Overlap ε
a :
upper :
μ :
σ :
Measure (ScoreSpace ε)
hcompat :
μ.real univ = σ.real univ
hp :
0 < μ.real univ
Proof (Lean source)
lemma projectedCellTerm_abs_le {ε : ℝ} (hOverlap : Overlap ε)
(a : ArmSpace) (upper : Bool)
(μ : Measure OutcomeSpace) (σ : Measure (ScoreSpace ε))
[IsFiniteMeasure μ] [IsFiniteMeasure σ]
(hcompat : μ.real univ = σ.real univ) (hp : 0 < μ.real univ) :
|μ.real univ * (∫ u in (0 : ℝ)..1,
quantile
(((μ univ)⁻¹ • μ).map (fun y : OutcomeSpace => (y : ℝ))) u *
quantile
(((σ univ)⁻¹ • σ).map (fun e => (armProb a e)⁻¹))
(if upper then u else 1 - u))| ≤ ε⁻¹ * μ.real univ := by
let Y : Measure ℝ := ((μ univ)⁻¹ • μ).map (fun y : OutcomeSpace => (y : ℝ))
let W : Measure ℝ := ((σ univ)⁻¹ • σ).map (fun e => (armProb a e)⁻¹)
have hσp : 0 < σ.real univ := hcompat ▸ hp
letI : IsProbabilityMeasure Y := by
dsimp [Y]
rw [map_normalized_subtype_eq_finiteNormalize μ hp]
infer_instance
letI : IsProbabilityMeasure W := by
dsimp [W]
rw [map_normalized_inverseArmProb_eq_finiteNormalize_map a σ hσp]
exact Measure.isProbabilityMeasure_map
(measurable_projectionInverseArmProbReal a).aemeasurable
have hY : Y (Icc 0 1)ᶜ = 0 := by
dsimp [Y]
exact normalizedOutcome_support μ hp
have hW : W (Icc 0 ε⁻¹)ᶜ = 0 := by
dsimp [W]
exact normalizedInverseArmProb_support hOverlap a σ hσp
have hI : |∫ u in (0 : ℝ)..1,
quantile Y u *
quantile W (if upper then u else 1 - u)| ≤ ε⁻¹ := by
simpa using abs_quantile_product_integral_le Y W (by norm_num)
(inv_pos.mpr hOverlap.1).le hY hW upper
rw [abs_mul, abs_of_pos hp]
have hmul := mul_le_mul_of_nonneg_left hI hp.le
dsimp only [Y, W] at hmul
simpa only [mul_comm] using hmul
theorem projectedCellSummand_abs_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
hOverlap :
Overlap ε
a :
upper :
μ μ' :
σ σ' :
Measure (ScoreSpace ε)
hcompat :
μ.real univ = σ.real univ
hcompat' :
μ'.real univ = σ'.real univ
conclusion
|(let q := μ.real univ if 0 < q then q * ∫ u in (0 : ℝ)..1, quantile (((μ univ)⁻¹ • μ).map (fun y : OutcomeSpace => (y : ℝ))) u * quantile (((σ univ)⁻¹ • σ).map (fun e => (armProb a e)⁻¹)) (if upper then u else 1 - u) else 0)
- (let q := μ'.real univ if 0 < q then q * ∫ u in (0 : ℝ)..1, quantile (((μ' univ)⁻¹ • μ').map (fun y : OutcomeSpace => (y : ℝ))) u * quantile (((σ' univ)⁻¹ • σ').map (fun e => (armProb a e)⁻¹)) (if upper then u else 1 - u) else 0)|
≤ (ε⁻¹ + (ε ^ 2)⁻¹) * (outcomeCDFDistance μ μ' + scoreCDFDistance σ σ')
Proof (Lean source)
lemma projectedCellSummand_abs_le {ε : ℝ} (hOverlap : Overlap ε)
(a : ArmSpace) (upper : Bool)
(μ μ' : Measure OutcomeSpace) (σ σ' : Measure (ScoreSpace ε))
[IsFiniteMeasure μ] [IsFiniteMeasure μ']
[IsFiniteMeasure σ] [IsFiniteMeasure σ']
(hcompat : μ.real univ = σ.real univ)
(hcompat' : μ'.real univ = σ'.real univ) :
|(let q := μ.real univ
if 0 < q then
q * ∫ u in (0 : ℝ)..1,
quantile
(((μ univ)⁻¹ • μ).map (fun y : OutcomeSpace => (y : ℝ))) u *
quantile
(((σ univ)⁻¹ • σ).map (fun e => (armProb a e)⁻¹))
(if upper then u else 1 - u)
else 0) -
(let q := μ'.real univ
if 0 < q then
q * ∫ u in (0 : ℝ)..1,
quantile
(((μ' univ)⁻¹ • μ').map (fun y : OutcomeSpace => (y : ℝ))) u *
quantile
(((σ' univ)⁻¹ • σ').map (fun e => (armProb a e)⁻¹))
(if upper then u else 1 - u)
else 0)| ≤
(ε⁻¹ + (ε ^ 2)⁻¹) *
(outcomeCDFDistance μ μ' + scoreCDFDistance σ σ') := by
have hL : 0 ≤ ε⁻¹ + (ε ^ 2)⁻¹ :=
add_nonneg (inv_pos.mpr hOverlap.1).le (by positivity)
have hout0 := outcomeCDFDistance_nonneg μ μ'
have hscore0 := scoreCDFDistance_nonneg hOverlap σ σ'
by_cases hp : 0 < μ.real univ
· by_cases hp' : 0 < μ'.real univ
· simpa [hp, hp'] using
projectedCellSummand_abs_le_of_pos hOverlap a upper μ μ' σ σ'
hcompat hcompat' hp hp'
· have hq' : μ'.real univ = 0 := le_antisymm (le_of_not_gt hp') measureReal_nonneg
simp only [hp, hp', if_true, if_false, sub_zero]
have hterm := projectedCellTerm_abs_le hOverlap a upper μ σ hcompat hp
have hmass : μ.real univ ≤ outcomeCDFDistance μ μ' := by
unfold outcomeCDFDistance
rw [hq', sub_zero, abs_of_pos hp]
have hi : 0 ≤ ∫ t in (0 : ℝ)..1,
|(μ {y | (y : ℝ) ≤ t}).toReal - (μ' {y | (y : ℝ) ≤ t}).toReal| :=
intervalIntegral.integral_nonneg (by norm_num) (fun _ _ => abs_nonneg _)
linarith
calc
_ ≤ ε⁻¹ * μ.real univ := hterm
_ ≤ (ε⁻¹ + (ε ^ 2)⁻¹) * μ.real univ := by
exact mul_le_mul_of_nonneg_right
(le_add_of_nonneg_right (by positivity : 0 ≤ (ε ^ 2)⁻¹)) hp.le
_ ≤ (ε⁻¹ + (ε ^ 2)⁻¹) *
(outcomeCDFDistance μ μ' + scoreCDFDistance σ σ') := by
gcongr
linarith
· have hq : μ.real univ = 0 := le_antisymm (le_of_not_gt hp) measureReal_nonneg
by_cases hp' : 0 < μ'.real univ
· simp only [hp, hp', if_true, if_false, zero_sub, abs_neg]
have hterm := projectedCellTerm_abs_le hOverlap a upper μ' σ' hcompat' hp'
have hmass : μ'.real univ ≤ outcomeCDFDistance μ μ' := by
unfold outcomeCDFDistance
rw [hq, zero_sub, abs_neg, abs_of_pos hp']
have hi : 0 ≤ ∫ t in (0 : ℝ)..1,
|(μ {y | (y : ℝ) ≤ t}).toReal - (μ' {y | (y : ℝ) ≤ t}).toReal| :=
intervalIntegral.integral_nonneg (by norm_num) (fun _ _ => abs_nonneg _)
linarith
calc
_ ≤ ε⁻¹ * μ'.real univ := hterm
_ ≤ (ε⁻¹ + (ε ^ 2)⁻¹) * μ'.real univ := by
exact mul_le_mul_of_nonneg_right
(le_add_of_nonneg_right (by positivity : 0 ≤ (ε ^ 2)⁻¹)) hp'.le
_ ≤ (ε⁻¹ + (ε ^ 2)⁻¹) *
(outcomeCDFDistance μ μ' + scoreCDFDistance σ σ') := by
gcongr
linarith
· simp only [hp, hp', if_true, if_false, sub_self, abs_zero]
exact mul_nonneg hL (add_nonneg hout0 hscore0)
theorem projectedArmEndpoint_abs_le_of_compatible_cells
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
hOverlap :
Overlap ε
lam lam' :
σ σ' :
ArmCellArray J (ScoreSpace ε)
hfinLam :
∀ a r, IsFiniteMeasure (lam a r)
hfinLam' :
∀ a r, IsFiniteMeasure (lam' a r)
hfinσ :
∀ a r, IsFiniteMeasure (σ a r)
hfinσ' :
∀ a r, IsFiniteMeasure (σ' a r)
hcompat :
ProjectionCompatible lam σ
hcompat' :
ProjectionCompatible lam' σ'
a :
upper :
conclusion
|projectedArmEndpoint lam σ a upper - projectedArmEndpoint lam' σ' a upper|
≤ (ε⁻¹ + (ε ^ 2)⁻¹) * ((∑ r : LabelSpace J, outcomeCDFDistance (lam a r) (lam' a r)) + ∑ r : LabelSpace J, scoreCDFDistance (σ a r) (σ' a r))
Proof (Lean source)
lemma projectedArmEndpoint_abs_le_of_compatible_cells
{ε : ℝ} {J : ℕ} (hOverlap : Overlap ε)
(lam lam' : ArmCellArray J OutcomeSpace)
(σ σ' : ArmCellArray J (ScoreSpace ε))
(hfinLam : ∀ a r, IsFiniteMeasure (lam a r))
(hfinLam' : ∀ a r, IsFiniteMeasure (lam' a r))
(hfinσ : ∀ a r, IsFiniteMeasure (σ a r))
(hfinσ' : ∀ a r, IsFiniteMeasure (σ' a r))
(hcompat : ProjectionCompatible lam σ)
(hcompat' : ProjectionCompatible lam' σ')
(a : ArmSpace) (upper : Bool) :
|projectedArmEndpoint lam σ a upper -
projectedArmEndpoint lam' σ' a upper| ≤
(ε⁻¹ + (ε ^ 2)⁻¹) *
((∑ r : LabelSpace J, outcomeCDFDistance (lam a r) (lam' a r)) +
∑ r : LabelSpace J, scoreCDFDistance (σ a r) (σ' a r)) := by
apply projectedArmEndpoint_abs_le_of_cellBounds lam lam' σ σ' a upper
(ε⁻¹ + (ε ^ 2)⁻¹)
intro r
letI : IsFiniteMeasure (lam a r) := hfinLam a r
letI : IsFiniteMeasure (lam' a r) := hfinLam' a r
letI : IsFiniteMeasure (σ a r) := hfinσ a r
letI : IsFiniteMeasure (σ' a r) := hfinσ' a r
exact projectedCellSummand_abs_le hOverlap a upper
(lam a r) (lam' a r) (σ a r) (σ' a r)
(hcompat a r) (hcompat' a r)
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionCellTransport 16 declarations Finite-measure transport, inverse-weight pushforward, and product-integral lemmas used by projected-cell stability.
Finite-measure transport, inverse-weight pushforward, and product-integral lemmas used by projected-cell stability.
def projectionInverseArmProbReal
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
a :
x :
ℝ
def
projectionInverseArmProbReal a x :
ℝ
given by
if a then x⁻¹ else (1 - x)⁻¹
theorem projectionInverseArmProbReal_coe
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
projectionInverseArmProbReal a (e : ℝ) = (armProb a e)⁻¹
Proof (Lean source)
lemma projectionInverseArmProbReal_coe {ε : ℝ} (a : ArmSpace) (e : ScoreSpace ε) :
projectionInverseArmProbReal a (e : ℝ) = (armProb a e)⁻¹ := by
cases a <;> rfl
theorem measurable_projectionInverseArmProbReal
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma measurable_projectionInverseArmProbReal (a : ArmSpace) :
Measurable (projectionInverseArmProbReal a) := by
cases a
· exact (measurable_const.sub measurable_id).inv
· exact measurable_id.inv
theorem projectionInverseArmProbReal_lipschitzOn
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
≤ (ε ^ 2)⁻¹ * |x - y|
Proof (Lean source)
lemma projectionInverseArmProbReal_lipschitzOn {ε : ℝ} (hOverlap : Overlap ε)
(a : ArmSpace) (x : ℝ) (hx : x ∈ Icc ε (1 - ε))
(y : ℝ) (hy : y ∈ Icc ε (1 - ε)) :
|projectionInverseArmProbReal a x - projectionInverseArmProbReal a y| ≤
(ε ^ 2)⁻¹ * |x - y| := by
let ex : ScoreSpace ε := ⟨x, hx⟩
let ey : ScoreSpace ε := ⟨y, hy⟩
have h := projectionInverseArmProb_lipschitz hOverlap a ex ey
cases a <;> simpa [ex, ey, projectionInverseArmProbReal, armProb] using h
theorem real_univ_map_subtype_val
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma real_univ_map_subtype_val {α : Set ℝ} (μ : Measure α) [IsFiniteMeasure μ] :
(μ.map (fun x : α => (x : ℝ))).real univ = μ.real univ := by
rw [Measure.real, Measure.real, Measure.map_apply (by fun_prop) MeasurableSet.univ]
simp
theorem map_subtype_val_support_Icc
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
a b :
ℝ
μ :
Measure (Icc a b)
conclusion
(μ.map (fun x : Icc a b => (x : ℝ))) (Icc a b)ᶜ = 0
Proof (Lean source)
lemma map_subtype_val_support_Icc {a b : ℝ} (μ : Measure (Icc a b)) :
(μ.map (fun x : Icc a b => (x : ℝ))) (Icc a b)ᶜ = 0 := by
rw [Measure.map_apply (by fun_prop) measurableSet_Icc.compl]
simp
theorem map_normalized_subtype_eq_finiteNormalize
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
((μ univ)⁻¹ • μ).map (fun x : α => (x : ℝ))
= (FiniteMeasure.normalize (⟨μ.map (fun x : α => (x : ℝ)), inferInstance⟩ : FiniteMeasure ℝ) : Measure ℝ)
Proof (Lean source)
lemma map_normalized_subtype_eq_finiteNormalize {α : Set ℝ}
(μ : Measure α) [IsFiniteMeasure μ] (hp : 0 < μ.real univ) :
((μ univ)⁻¹ • μ).map (fun x : α => (x : ℝ)) =
(FiniteMeasure.normalize
(⟨μ.map (fun x : α => (x : ℝ)), inferInstance⟩ : FiniteMeasure ℝ) :
Measure ℝ) := by
let μR : FiniteMeasure ℝ := ⟨μ.map (fun x : α => (x : ℝ)), inferInstance⟩
have hmass : (μR.mass : ℝ) = μ.real univ := by
change (μ.map (fun x : α => (x : ℝ))).real univ = μ.real univ
exact real_univ_map_subtype_val μ
have hμR : μR ≠ 0 := by
intro h
have : (μR.mass : ℝ) = 0 := by simp [h]
linarith
rw [Measure.map_smul, μR.toMeasure_normalize_eq_of_nonzero hμR]
change (μ univ)⁻¹ • μ.map (fun x : α => (x : ℝ)) =
(μR.mass⁻¹ : NNReal) • (μR : Measure ℝ)
have hmassENN : μ univ = (μR.mass : ENNReal) := by
dsimp [μR, FiniteMeasure.mass]
rw [Measure.map_apply (by fun_prop) MeasurableSet.univ]
simp
rw [hmassENN]
have hmass0 : μR.mass ≠ 0 := by
intro h
exact hμR ((FiniteMeasure.mass_zero_iff μR).mp h)
change (μR.mass : ENNReal)⁻¹ • (μR : Measure ℝ) =
((μR.mass⁻¹ : NNReal) : ENNReal) • (μR : Measure ℝ)
rw [ENNReal.coe_inv hmass0]
theorem finite_mass_subtype_quantile_transport_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
a b :
ℝ
hab :
a ≤ b
μ ν :
hp :
0 < μ.real univ
hq :
0 < ν.real univ
Proof (Lean source)
lemma finite_mass_subtype_quantile_transport_le {a b : ℝ} (hab : a ≤ b)
(μ ν : Measure (Icc a b)) [IsFiniteMeasure μ] [IsFiniteMeasure ν]
(hp : 0 < μ.real univ) (hq : 0 < ν.real univ) :
min (μ.real univ) (ν.real univ) *
(∫ u in (0 : ℝ)..1,
|quantile
(((μ univ)⁻¹ • μ).map (fun x : Icc a b => (x : ℝ))) u -
quantile
(((ν univ)⁻¹ • ν).map (fun x : Icc a b => (x : ℝ))) u|) ≤
(∫ t in a..b,
|(μ {x | (x : ℝ) ≤ t}).toReal -
(ν {x | (x : ℝ) ≤ t}).toReal|) +
(b - a) * |μ.real univ - ν.real univ| := by
let μR : FiniteMeasure ℝ := ⟨μ.map (fun x : Icc a b => (x : ℝ)), inferInstance⟩
let νR : FiniteMeasure ℝ := ⟨ν.map (fun x : Icc a b => (x : ℝ)), inferInstance⟩
have hμmass : (μR.mass : ℝ) = μ.real univ := by
change (μ.map (fun x : Icc a b => (x : ℝ))).real univ = μ.real univ
exact real_univ_map_subtype_val μ
have hνmass : (νR.mass : ℝ) = ν.real univ := by
change (ν.map (fun x : Icc a b => (x : ℝ))).real univ = ν.real univ
exact real_univ_map_subtype_val ν
have htransport :=
finite_mass_quantile_transport_le
μR νR hab (hμmass.symm ▸ hp) (hνmass.symm ▸ hq)
(map_subtype_val_support_Icc μ) (map_subtype_val_support_Icc ν)
rw [← map_normalized_subtype_eq_finiteNormalize μ hp,
← map_normalized_subtype_eq_finiteNormalize ν hq] at htransport
rw [hμmass, hνmass] at htransport
have hμcdf (t : ℝ) :
(μ.map (fun x : Icc a b => (x : ℝ))) (Iic t) =
μ {x | (x : ℝ) ≤ t} := by
rw [Measure.map_apply (by fun_prop) measurableSet_Iic]
rfl
have hνcdf (t : ℝ) :
(ν.map (fun x : Icc a b => (x : ℝ))) (Iic t) =
ν {x | (x : ℝ) ≤ t} := by
rw [Measure.map_apply (by fun_prop) measurableSet_Iic]
rfl
simpa only [μR, νR, FiniteMeasure.toMeasure_mk,
hμcdf, hνcdf] using htransport
theorem inverseArmProb_quantile_distance_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
Proof (Lean source)
lemma inverseArmProb_quantile_distance_le {ε : ℝ} (hOverlap : Overlap ε)
(a : ArmSpace) (μ ν : Measure (ScoreSpace ε))
[IsFiniteMeasure μ] [IsFiniteMeasure ν]
(hp : 0 < μ.real univ) (hq : 0 < ν.real univ) :
(∫ u in (0 : ℝ)..1,
|quantile
(((μ univ)⁻¹ • μ).map (fun e => (armProb a e)⁻¹)) u -
quantile
(((ν univ)⁻¹ • ν).map (fun e => (armProb a e)⁻¹)) u|) ≤
(ε ^ 2)⁻¹ *
∫ u in (0 : ℝ)..1,
|quantile
(((μ univ)⁻¹ • μ).map (fun e : ScoreSpace ε => (e : ℝ))) u -
quantile
(((ν univ)⁻¹ • ν).map (fun e : ScoreSpace ε => (e : ℝ))) u| := by
let μR : FiniteMeasure ℝ :=
⟨μ.map (fun e : ScoreSpace ε => (e : ℝ)), inferInstance⟩
let νR : FiniteMeasure ℝ :=
⟨ν.map (fun e : ScoreSpace ε => (e : ℝ)), inferInstance⟩
let μP : Measure ℝ := FiniteMeasure.normalize μR
let νP : Measure ℝ := FiniteMeasure.normalize νR
letI : IsProbabilityMeasure μP := by dsimp [μP]; infer_instance
letI : IsProbabilityMeasure νP := by dsimp [νP]; infer_instance
have hpush := quantile_pushforward_lipschitzOn
μP νP (by linarith [hOverlap.2] : ε ≤ 1 - ε) (by
dsimp [μP]
exact normalize_support_Icc
μR (by
rw [show (μR.mass : ℝ) = μ.real univ by
change (μ.map (fun e : ScoreSpace ε => (e : ℝ))).real univ = μ.real univ
exact real_univ_map_subtype_val μ]
exact hp)
(map_subtype_val_support_Icc μ)) (by
dsimp [νP]
exact normalize_support_Icc
νR (by
rw [show (νR.mass : ℝ) = ν.real univ by
change (ν.map (fun e : ScoreSpace ε => (e : ℝ))).real univ = ν.real univ
exact real_univ_map_subtype_val ν]
exact hq)
(map_subtype_val_support_Icc ν))
(by positivity : 0 ≤ (ε ^ 2)⁻¹) (projectionInverseArmProbReal a)
(measurable_projectionInverseArmProbReal a)
(projectionInverseArmProbReal_lipschitzOn hOverlap a)
have hμnorm := map_normalized_subtype_eq_finiteNormalize μ hp
have hνnorm := map_normalized_subtype_eq_finiteNormalize ν hq
change _ ≤ _
dsimp [μP, νP] at hpush
rw [← hμnorm, ← hνnorm] at hpush
rw [Measure.map_map (measurable_projectionInverseArmProbReal a) (by fun_prop),
Measure.map_map (measurable_projectionInverseArmProbReal a) (by fun_prop)] at hpush
have hfun : projectionInverseArmProbReal a ∘
(fun e : ScoreSpace ε => (e : ℝ)) = fun e => (armProb a e)⁻¹ := by
funext e
exact projectionInverseArmProbReal_coe a e
rw [hfun] at hpush
exact hpush
theorem quantile_product_integral_stability
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
μ μ' ν ν' :
A B :
ℝ
hA :
0 ≤ A
hB :
0 ≤ B
hμ :
μ (Icc 0 A)ᶜ = 0
hμ' :
μ' (Icc 0 A)ᶜ = 0
hν :
ν (Icc 0 B)ᶜ = 0
hν' :
ν' (Icc 0 B)ᶜ = 0
upper :
Proof (Lean source)
lemma quantile_product_integral_stability
(μ μ' ν ν' : Measure ℝ)
[IsProbabilityMeasure μ] [IsProbabilityMeasure μ']
[IsProbabilityMeasure ν] [IsProbabilityMeasure ν']
{A B : ℝ} (hA : 0 ≤ A) (hB : 0 ≤ B)
(hμ : μ (Icc 0 A)ᶜ = 0) (hμ' : μ' (Icc 0 A)ᶜ = 0)
(hν : ν (Icc 0 B)ᶜ = 0) (hν' : ν' (Icc 0 B)ᶜ = 0)
(upper : Bool) :
|(∫ u in (0 : ℝ)..1,
quantile μ u *
quantile ν (if upper then u else 1 - u)) -
∫ u in (0 : ℝ)..1,
quantile μ' u *
quantile ν' (if upper then u else 1 - u)| ≤
B * (∫ u in (0 : ℝ)..1,
|quantile μ u - quantile μ' u|) +
A * (∫ u in (0 : ℝ)..1,
|quantile ν u - quantile ν' u|) := by
let s : ℝ → ℝ := if upper then id else fun u => 1 - u
have hsmeas : Measurable s := by
cases upper <;> simp [s] <;> fun_prop
have hsmap : Causalean.Stat.unifOI.map s = unifOI := by
cases upper
· simpa [s] using map_one_sub_unifOI
· simp [s]
have hsp : MeasurePreserving s unifOI unifOI :=
⟨hsmeas, hsmap⟩
have hs_apply (u : ℝ) : s u = if upper then u else 1 - u := by
cases upper <;> rfl
have hμb := quantile_mem_Icc_ae μ hμ
have hμb' := quantile_mem_Icc_ae μ' hμ'
have hνb := hsp.quasiMeasurePreserving.tendsto_ae.eventually
(quantile_mem_Icc_ae ν hν)
have hνb' := hsp.quasiMeasurePreserving.tendsto_ae.eventually
(quantile_mem_Icc_ae ν' hν')
have hμi : Integrable (quantile μ) unifOI := by
apply (integrable_const A).mono'
· exact (aemeasurable_quantile_unifOI μ).aestronglyMeasurable
· filter_upwards [hμb] with u hu
simpa [Real.norm_eq_abs, abs_of_nonneg hu.1] using hu.2
have hμi' : Integrable (quantile μ') unifOI := by
apply (integrable_const A).mono'
· exact (aemeasurable_quantile_unifOI μ').aestronglyMeasurable
· filter_upwards [hμb'] with u hu
simpa [Real.norm_eq_abs, abs_of_nonneg hu.1] using hu.2
have hνae : AEMeasurable (fun u => quantile ν (s u))
unifOI := by
have h := aemeasurable_quantile_unifOI ν
rw [← hsmap] at h
exact h.comp_measurable hsmeas
have hνae' : AEMeasurable (fun u => quantile ν' (s u))
unifOI := by
have h := aemeasurable_quantile_unifOI ν'
rw [← hsmap] at h
exact h.comp_measurable hsmeas
have hνi : Integrable (fun u => quantile ν (s u))
unifOI := by
apply (integrable_const B).mono' hνae.aestronglyMeasurable
filter_upwards [hνb] with u hu
simpa [Real.norm_eq_abs, abs_of_nonneg hu.1] using hu.2
have hνi' : Integrable (fun u => quantile ν' (s u))
unifOI := by
apply (integrable_const B).mono' hνae'.aestronglyMeasurable
filter_upwards [hνb'] with u hu
simpa [Real.norm_eq_abs, abs_of_nonneg hu.1] using hu.2
have hprod : Integrable
(fun u => quantile μ u * quantile ν (s u))
unifOI := by
apply (integrable_const (A * B)).mono'
· exact ((aemeasurable_quantile_unifOI μ).mul hνae).aestronglyMeasurable
· filter_upwards [hμb, hνb] with u hy hw
rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg hy.1 hw.1)]
exact mul_le_mul hy.2 hw.2 hw.1 hA
have hprod' : Integrable
(fun u => quantile μ' u * quantile ν' (s u))
unifOI := by
apply (integrable_const (A * B)).mono'
· exact ((aemeasurable_quantile_unifOI μ').mul hνae').aestronglyMeasurable
· filter_upwards [hμb', hνb'] with u hy hw
rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg hy.1 hw.1)]
exact mul_le_mul hy.2 hw.2 hw.1 hA
have hdy : Integrable
(fun u => |quantile μ u - quantile μ' u|)
unifOI := (hμi.sub hμi').abs
have hdw : Integrable
(fun u => |quantile ν (s u) - quantile ν' (s u)|)
unifOI := (hνi.sub hνi').abs
have hdw_eq :
(∫ u, |quantile ν (s u) - quantile ν' (s u)|
∂Causalean.Stat.unifOI) =
∫ u, |quantile ν u - quantile ν' u|
∂Causalean.Stat.unifOI := by
let f : ℝ → ℝ := fun u =>
|quantile ν u - quantile ν' u|
have hf : AEStronglyMeasurable f (Causalean.Stat.unifOI.map s) := by
rw [hsmap]
exact ((aemeasurable_quantile_unifOI ν).sub
(aemeasurable_quantile_unifOI ν')).norm.aestronglyMeasurable
have hmap := integral_map hsmeas.aemeasurable hf
rw [hsmap] at hmap
exact hmap.symm
have hpoint : ∀ᵐ u ∂Causalean.Stat.unifOI,
|quantile μ u * quantile ν (s u) -
quantile μ' u * quantile ν' (s u)| ≤
B * |quantile μ u - quantile μ' u| +
A * |quantile ν (s u) - quantile ν' (s u)| := by
filter_upwards [hμb', hνb] with u hy' hw
calc
_ = |(quantile μ u - quantile μ' u) *
quantile ν (s u) +
quantile μ' u *
(quantile ν (s u) -
quantile ν' (s u))| := by ring_nf
_ ≤ |(quantile μ u - quantile μ' u) *
quantile ν (s u)| +
|quantile μ' u *
(quantile ν (s u) -
quantile ν' (s u))| := abs_add_le _ _
_ ≤ _ := by
rw [abs_mul, abs_mul, abs_of_nonneg hw.1, abs_of_nonneg hy'.1]
apply add_le_add
· simpa [mul_comm] using
(mul_le_mul_of_nonneg_left hw.2 (abs_nonneg
(quantile μ u - quantile μ' u)))
· exact mul_le_mul_of_nonneg_right hy'.2 (abs_nonneg _)
have hrest : volume.restrict (Ioc (0 : ℝ) 1) = unifOI := by
rw [unifOI, restrict_Ioo_eq_restrict_Ioc]
simp_rw [hs_apply] at hprod hprod' hdy hdw hdw_eq hpoint
rw [intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1), hrest,
intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1), hrest,
intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1), hrest,
intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1), hrest]
rw [← integral_sub hprod hprod']
calc
|∫ u, (quantile μ u *
quantile ν (if upper then u else 1 - u) -
quantile μ' u *
quantile ν' (if upper then u else 1 - u))
∂Causalean.Stat.unifOI| ≤
∫ u, |quantile μ u *
quantile ν (if upper then u else 1 - u) -
quantile μ' u *
quantile ν' (if upper then u else 1 - u)|
∂Causalean.Stat.unifOI := by
simpa only [Real.norm_eq_abs] using
(norm_integral_le_integral_norm
(μ := unifOI)
(f := fun u => quantile μ u *
quantile ν (if upper then u else 1 - u) -
quantile μ' u *
quantile ν' (if upper then u else 1 - u)))
_ ≤ ∫ u, (B * |quantile μ u - quantile μ' u| +
A * |quantile ν (if upper then u else 1 - u) -
quantile ν' (if upper then u else 1 - u)|)
∂Causalean.Stat.unifOI := integral_mono_ae
(hprod.sub hprod').abs
(hdy.const_mul B |>.add (hdw.const_mul A)) hpoint
_ = B * ∫ u, |quantile μ u - quantile μ' u|
∂Causalean.Stat.unifOI +
A * ∫ u, |quantile ν u - quantile ν' u|
∂Causalean.Stat.unifOI := by
rw [integral_add (hdy.const_mul B) (hdw.const_mul A),
integral_const_mul, integral_const_mul]
rw [← hdw_eq]
theorem normalizedOutcome_support
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
hp :
0 < μ.real univ
conclusion
(((μ univ)⁻¹ • μ).map (fun y : OutcomeSpace => (y : ℝ))) (Icc 0 1)ᶜ = 0
Proof (Lean source)
lemma normalizedOutcome_support (μ : Measure OutcomeSpace) [IsFiniteMeasure μ]
(hp : 0 < μ.real univ) :
(((μ univ)⁻¹ • μ).map (fun y : OutcomeSpace => (y : ℝ)))
(Icc 0 1)ᶜ = 0 := by
rw [map_normalized_subtype_eq_finiteNormalize μ hp]
apply normalize_support_Icc
(⟨μ.map (fun y : OutcomeSpace => (y : ℝ)), inferInstance⟩ : FiniteMeasure ℝ)
· rw [show (FiniteMeasure.mass
(⟨μ.map (fun y : OutcomeSpace => (y : ℝ)), inferInstance⟩ :
FiniteMeasure ℝ) : ℝ) = μ.real univ by
change (μ.map (fun y : OutcomeSpace => (y : ℝ))).real univ = μ.real univ
exact real_univ_map_subtype_val μ]
exact hp
· exact map_subtype_val_support_Icc μ
theorem measurable_inverseArmProb
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma measurable_inverseArmProb {ε : ℝ} (a : ArmSpace) :
Measurable (fun e : ScoreSpace ε => (armProb a e)⁻¹) := by
rw [show (fun e : ScoreSpace ε => (armProb a e)⁻¹) =
projectionInverseArmProbReal a ∘ (fun e : ScoreSpace ε => (e : ℝ)) by
funext e
exact (projectionInverseArmProbReal_coe a e).symm]
exact (measurable_projectionInverseArmProbReal a).comp (by fun_prop)
theorem normalizedInverseArmProb_support
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
Proof (Lean source)
lemma normalizedInverseArmProb_support {ε : ℝ} (hOverlap : Overlap ε)
(a : ArmSpace) (σ : Measure (ScoreSpace ε)) [IsFiniteMeasure σ]
(hp : 0 < σ.real univ) :
(((σ univ)⁻¹ • σ).map (fun e => (armProb a e)⁻¹))
(Icc 0 ε⁻¹)ᶜ = 0 := by
rw [Measure.map_apply (measurable_inverseArmProb a) measurableSet_Icc.compl]
have hpre : (fun e : ScoreSpace ε => (armProb a e)⁻¹) ⁻¹' (Icc 0 ε⁻¹)ᶜ = ∅ := by
ext e
have hb := projectionArmProb_bounds hOverlap a e
have hpos : 0 < armProb a e := lt_of_lt_of_le hOverlap.1 hb.1
have hinv : (armProb a e)⁻¹ ≤ ε⁻¹ := (inv_le_inv₀ hpos hOverlap.1).2 hb.1
simp only [mem_preimage, mem_compl_iff, mem_Icc, mem_empty_iff_false, iff_false]
exact fun h => h ⟨(inv_pos.mpr hpos).le, hinv⟩
rw [hpre]
simp
theorem map_normalized_inverseArmProb_eq_finiteNormalize_map
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
= (FiniteMeasure.normalize (⟨σ.map (fun e : ScoreSpace ε => (e : ℝ)), inferInstance⟩ : FiniteMeasure ℝ) : Measure ℝ).map (projectionInverseArmProbReal a)
Proof (Lean source)
lemma map_normalized_inverseArmProb_eq_finiteNormalize_map {ε : ℝ}
(a : ArmSpace) (σ : Measure (ScoreSpace ε)) [IsFiniteMeasure σ]
(hp : 0 < σ.real univ) :
((σ univ)⁻¹ • σ).map (fun e => (armProb a e)⁻¹) =
(FiniteMeasure.normalize
(⟨σ.map (fun e : ScoreSpace ε => (e : ℝ)), inferInstance⟩ :
FiniteMeasure ℝ) : Measure ℝ).map (projectionInverseArmProbReal a) := by
have hnorm := map_normalized_subtype_eq_finiteNormalize σ hp
calc
((σ univ)⁻¹ • σ).map (fun e => (armProb a e)⁻¹) =
(((σ univ)⁻¹ • σ).map (fun e : ScoreSpace ε => (e : ℝ))).map
(projectionInverseArmProbReal a) := by
rw [Measure.map_map (measurable_projectionInverseArmProbReal a) (by fun_prop)]
congr 1
funext e
exact (projectionInverseArmProbReal_coe a e).symm
_ = _ := by rw [hnorm]
theorem abs_quantile_product_integral_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
μ ν :
A B :
ℝ
hA :
0 ≤ A
hB :
0 ≤ B
hμ :
μ (Icc 0 A)ᶜ = 0
hν :
ν (Icc 0 B)ᶜ = 0
upper :
Proof (Lean source)
lemma abs_quantile_product_integral_le
(μ ν : Measure ℝ) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
{A B : ℝ} (hA : 0 ≤ A) (hB : 0 ≤ B)
(hμ : μ (Icc 0 A)ᶜ = 0) (hν : ν (Icc 0 B)ᶜ = 0)
(upper : Bool) :
|∫ u in (0 : ℝ)..1,
quantile μ u *
quantile ν (if upper then u else 1 - u)| ≤ A * B := by
let s : ℝ → ℝ := if upper then id else fun u => 1 - u
have hsmeas : Measurable s := by cases upper <;> simp [s] <;> fun_prop
have hsmap : Causalean.Stat.unifOI.map s = unifOI := by
cases upper
· simpa [s] using map_one_sub_unifOI
· simp [s]
have hsp : MeasurePreserving s unifOI unifOI :=
⟨hsmeas, hsmap⟩
have hs_apply (u : ℝ) : s u = if upper then u else 1 - u := by
cases upper <;> rfl
have hy := quantile_mem_Icc_ae μ hμ
have hw := hsp.quasiMeasurePreserving.tendsto_ae.eventually
(quantile_mem_Icc_ae ν hν)
have hwae : AEMeasurable (fun u => quantile ν (s u))
unifOI := by
have h := aemeasurable_quantile_unifOI ν
rw [← hsmap] at h
exact h.comp_measurable hsmeas
have hprod : Integrable
(fun u => quantile μ u * quantile ν (s u))
unifOI := by
apply (integrable_const (A * B)).mono'
· exact ((aemeasurable_quantile_unifOI μ).mul hwae).aestronglyMeasurable
· filter_upwards [hy, hw] with u hyu hwu
rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg hyu.1 hwu.1)]
exact mul_le_mul hyu.2 hwu.2 hwu.1 hA
have hnonneg : 0 ≤ ∫ u,
quantile μ u * quantile ν (s u)
∂Causalean.Stat.unifOI := integral_nonneg_of_ae (by
filter_upwards [hy, hw] with u hyu hwu
exact mul_nonneg hyu.1 hwu.1)
have hupper : (∫ u,
quantile μ u * quantile ν (s u)
∂Causalean.Stat.unifOI) ≤ A * B := by
calc
_ ≤ ∫ _ : ℝ, A * B ∂Causalean.Stat.unifOI := integral_mono_ae hprod
(integrable_const (A * B)) (by
filter_upwards [hy, hw] with u hyu hwu
exact mul_le_mul hyu.2 hwu.2 hwu.1 hA)
_ = A * B := by simp
have hrest : volume.restrict (Ioc (0 : ℝ) 1) = unifOI := by
rw [unifOI, restrict_Ioo_eq_restrict_Ioc]
simp_rw [hs_apply] at hnonneg hupper
rw [intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1), hrest,
abs_of_nonneg hnonneg]
exact hupper
theorem abs_mass_mul_sub_mass_mul_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
p q A A' B :
ℝ
hp :
0 ≤ p
hq :
0 ≤ q
hA :
|A| ≤ B
hA' :
|A'| ≤ B
conclusion
|p * A - q * A'| ≤ min p q * |A - A'| + B * |p - q|
Proof (Lean source)
lemma abs_mass_mul_sub_mass_mul_le {p q A A' B : ℝ}
(hp : 0 ≤ p) (hq : 0 ≤ q) (hA : |A| ≤ B) (hA' : |A'| ≤ B) :
|p * A - q * A'| ≤ min p q * |A - A'| + B * |p - q| := by
rcases le_total p q with hpq | hqp
· rw [min_eq_left hpq]
have hid : p * A - q * A' = p * (A - A') + (p - q) * A' := by ring
rw [hid]
calc
_ ≤ |p * (A - A')| + |(p - q) * A'| := abs_add_le _ _
_ = p * |A - A'| + |p - q| * |A'| := by
rw [abs_mul, abs_mul, abs_of_nonneg hp]
_ ≤ p * |A - A'| + |p - q| * B :=
add_le_add le_rfl (mul_le_mul_of_nonneg_left hA' (abs_nonneg _))
_ = _ := by ring
· rw [min_eq_right hqp]
have hid : p * A - q * A' = q * (A - A') + (p - q) * A := by ring
rw [hid]
calc
_ ≤ |q * (A - A')| + |(p - q) * A| := abs_add_le _ _
_ = q * |A - A'| + |p - q| * |A| := by
rw [abs_mul, abs_mul, abs_of_nonneg hq]
_ ≤ q * |A - A'| + |p - q| * B :=
add_le_add le_rfl (mul_le_mul_of_nonneg_left hA (abs_nonneg _))
_ = _ := by ring
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionCoverage 6 declarations Coverage of the atom-valid external projection interval.
Coverage of the atom-valid external projection interval.
theorem projectedArmEndpoint_lower_le_upper
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
hOverlap :
Overlap ε
lam :
σ :
ArmCellArray J (ScoreSpace ε)
hcompat :
ProjectionCompatible lam σ
a :
conclusion
Proof (Lean source)
lemma projectedArmEndpoint_lower_le_upper {ε : ℝ} {J : ℕ}
(hOverlap : Overlap ε)
(lam : ArmCellArray J OutcomeSpace)
(σ : ArmCellArray J (ScoreSpace ε))
(hfinite : ∀ a r, (lam a r) Set.univ < ⊤ ∧ (σ a r) Set.univ < ⊤)
(hcompat : ProjectionCompatible lam σ) (a : ArmSpace) :
projectedArmEndpoint lam σ a false ≤ projectedArmEndpoint lam σ a true := by
unfold projectedArmEndpoint
apply Finset.sum_le_sum
intro r _
letI : IsFiniteMeasure (lam a r) := ⟨(hfinite a r).1⟩
letI : IsFiniteMeasure (σ a r) := ⟨(hfinite a r).2⟩
let q := (lam a r).real Set.univ
by_cases hq : 0 < q
· dsimp [q] at hq
simp only [hq, if_pos, Bool.false_eq_true, if_false, if_true]
exact mul_le_mul_of_nonneg_left
(cell_lower_le_upper_of_overlap hOverlap a (lam a r) (σ a r) hq
(hcompat a r)) hq.le
· dsimp [q] at hq
simp [hq]
theorem projectedATEInterval_mem_closure_of_sieve_approx
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
x :
ExternalSample ε J n m
lam :
σ :
ArmCellArray J (ScoreSpace ε)
hOverlap :
Overlap ε
happrox :
∀ η : ℝ
if
0 < η
then
∃ b ∈ sievedProjectionCandidates g α x,
t :
ℝ
ht :
t ∈ projectedATEInterval lam σ
conclusion
Proof (Lean source)
lemma projectedATEInterval_mem_closure_of_sieve_approx {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(x : ExternalSample ε J n m)
(lam : ArmCellArray J OutcomeSpace)
(σ : ArmCellArray J (ScoreSpace ε))
(hOverlap : Overlap ε)
(happrox : ∀ η : ℝ, 0 < η →
∃ b ∈ sievedProjectionCandidates g α x,
∀ a upper,
|projectedArmEndpoint b.1 b.2 a upper -
projectedArmEndpoint lam σ a upper| < η)
{t : ℝ} (ht : t ∈ projectedATEInterval lam σ) :
t ∈ closure (⋃ b ∈ sievedProjectionCandidates g α x,
projectedATEInterval b.1 b.2) := by
classical
let L := projectedArmEndpoint lam σ true false -
projectedArmEndpoint lam σ false true
let U := projectedArmEndpoint lam σ true true -
projectedArmEndpoint lam σ false false
have ht' : L ≤ t ∧ t ≤ U := ht
apply Metric.mem_closure_iff.mpr
intro δ hδ
obtain ⟨b, hb, hnear⟩ := happrox (δ / 4) (by positivity)
let B := projectedArmEndpoint b.1 b.2 true false -
projectedArmEndpoint b.1 b.2 false true
let V := projectedArmEndpoint b.1 b.2 true true -
projectedArmEndpoint b.1 b.2 false false
have hfinite : ∀ a r, (b.1 a r) univ < ⊤ ∧
(b.2 a r) univ < ⊤ := hb.1.1
have hcompat : ProjectionCompatible b.1 b.2 := hb.1.2.1
have horder : B ≤ V := by
have h₁ := projectedArmEndpoint_lower_le_upper hOverlap b.1 b.2
hfinite hcompat true
have h₀ := projectedArmEndpoint_lower_le_upper hOverlap b.1 b.2
hfinite hcompat false
dsimp [B, V]
linarith
have hB : |B - L| < δ / 2 := by
have h₁ := hnear true false
have h₀ := hnear false true
dsimp [B, L]
rw [abs_lt] at h₁ h₀ ⊢
constructor <;> linarith
have hV : |V - U| < δ / 2 := by
have h₁ := hnear true true
have h₀ := hnear false false
dsimp [V, U]
rw [abs_lt] at h₁ h₀ ⊢
constructor <;> linarith
let z := min (max t B) V
have hz : z ∈ projectedATEInterval b.1 b.2 := by
change B ≤ z ∧ z ≤ V
dsimp [z]
exact ⟨le_min (le_max_right _ _) horder, min_le_right _ _⟩
have hdist : dist t z < δ := by
rw [Real.dist_eq]
by_cases hlow : t < B
· have hzeq : z = B := by
simp [z, max_eq_right hlow.le, min_eq_left horder]
rw [hzeq, abs_lt] at *
constructor <;> linarith [hB.1, hB.2, ht'.1]
· by_cases hhigh : V < t
· have hzeq : z = V := by
have hmax : max t B = t := max_eq_left (le_of_not_gt hlow)
simp [z, hmax, min_eq_right hhigh.le]
rw [hzeq, abs_lt] at *
constructor <;> linarith [hV.1, hV.2, ht'.2]
· have hzeq : z = t := by
have hmax : max t B = t := max_eq_left (le_of_not_gt hlow)
simp [z, hmax, min_eq_left (le_of_not_gt hhigh)]
rw [hzeq]
simpa using hδ
exact ⟨z, Set.mem_iUnion.mpr ⟨b,
Set.mem_iUnion.mpr ⟨hb, hz⟩⟩, hdist⟩
theorem projection_strict_population_approximants
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
hOverlap :
Overlap ε
x :
ExternalSample ε J n m
lam :
σ :
ArmCellArray J (ScoreSpace ε)
hcompat :
ProjectionCompatible lam σ
hout :
(∑ a : ArmSpace, ∑ r : LabelSpace J, outcomeCDFDistance (lam a r) (empiricalTrialCells x.1 a r)) < 4 * J / (α * sqrt n)
hscore :
(∑ a : ArmSpace, ∑ r : LabelSpace J, scoreCDFDistance (σ a r) (empiricalScoreCells g x.2 a r)) < 2 * J * (2 - 2 * ε) / (α * sqrt m)
η :
ℝ
0 < η
conclusion
∃ b ∈ sievedProjectionCandidates g α x,
Proof (Lean source)
lemma projection_strict_population_approximants {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(hOverlap : Overlap ε) (x : ExternalSample ε J n m)
(lam : ArmCellArray J OutcomeSpace)
(σ : ArmCellArray J (ScoreSpace ε))
(hfinite : ∀ a r, (lam a r) univ < ⊤ ∧ (σ a r) univ < ⊤)
(hcompat : ProjectionCompatible lam σ)
(hout : (∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (lam a r) (empiricalTrialCells x.1 a r)) <
4 * J / (α * sqrt n))
(hscore : (∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (σ a r) (empiricalScoreCells g x.2 a r)) <
2 * J * (2 - 2 * ε) / (α * sqrt m)) :
∀ η : ℝ, 0 < η →
∃ b ∈ sievedProjectionCandidates g α x,
∀ a upper,
|projectedArmEndpoint b.1 b.2 a upper -
projectedArmEndpoint lam σ a upper| < η := by
classical
intro η hη
let L : ℝ := ε⁻¹ + (ε ^ 2)⁻¹
have hLpos : 0 < L := by
dsimp [L]
have hε := hOverlap.1
positivity
have hL : 0 ≤ L := hLpos.le
let rout : ℝ := 4 * J / (α * sqrt n)
let rscore : ℝ := 2 * J * (2 - 2 * ε) / (α * sqrt m)
let sout : ℝ := ∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (lam a r) (empiricalTrialCells x.1 a r)
let sscore : ℝ := ∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (σ a r) (empiricalScoreCells g x.2 a r)
let δ := min (rout - sout) (min (rscore - sscore) (η / (L + 1))) / 2
have hδ : 0 < δ := by
dsimp [δ]
have h₁ : 0 < rout - sout := sub_pos.mpr hout
have h₂ : 0 < rscore - sscore := sub_pos.mpr hscore
exact div_pos (lt_min h₁ (lt_min h₂ (div_pos hη (by linarith))))
(by norm_num)
have houtδ : sout + δ ≤ rout := by
have hm := min_le_left (rout - sout) (min (rscore - sscore) (η / (L + 1)))
dsimp [δ]
linarith
have hscoreδ : sscore + δ ≤ rscore := by
have hm := min_le_left (rscore - sscore) (η / (L + 1))
have hm' := min_le_right (rout - sout) (min (rscore - sscore) (η / (L + 1)))
dsimp [δ]
linarith
obtain ⟨b, hb, hdist⟩ := exists_sievedProjectionCandidate_close_of_slack
hOverlap g α x lam σ hfinite hcompat δ hδ houtδ hscoreδ
refine ⟨b, hb, ?_⟩
have hLδ : L * δ < η := by
have hm := min_le_right (rscore - sscore) (η / (L + 1))
have hm' := min_le_right (rout - sout) (min (rscore - sscore) (η / (L + 1)))
have hδbound : δ ≤ η / (L + 1) := by
have hmin : 0 ≤ min (rout - sout)
(min (rscore - sscore) (η / (L + 1))) := by
dsimp [δ] at hδ
linarith
dsimp [δ]
linarith
have hden : 0 < L + 1 := by linarith
have hratio : L * (η / (L + 1)) < η := by
have hcancel : (L + 1) * (η / (L + 1)) = η := by field_simp
nlinarith [div_pos hη hden]
exact (mul_le_mul_of_nonneg_left hδbound hL).trans_lt hratio
have hfiniteB : ∀ a r, (b.1 a r) univ < ⊤ ∧
(b.2 a r) univ < ⊤ := hb.1.1
have hcompatB : ProjectionCompatible b.1 b.2 := hb.1.2.1
intro a upper
have hendpoint := projectedArmEndpoint_abs_le_of_compatible_cells hOverlap
lam b.1 σ b.2 (fun a r => ⟨(hfinite a r).1⟩)
(fun a r => ⟨(hfiniteB a r).1⟩)
(fun a r => ⟨(hfinite a r).2⟩)
(fun a r => ⟨(hfiniteB a r).2⟩)
hcompat hcompatB a upper
have hpair :
(∑ r : LabelSpace J, outcomeCDFDistance (lam a r) (b.1 a r)) +
(∑ r : LabelSpace J, scoreCDFDistance (σ a r) (b.2 a r)) < δ := by
have hnonneg (a : ArmSpace) :
0 ≤ (∑ r : LabelSpace J, outcomeCDFDistance (lam a r) (b.1 a r)) +
(∑ r : LabelSpace J, scoreCDFDistance (σ a r) (b.2 a r)) := by
apply add_nonneg
· exact Finset.sum_nonneg (fun r _ => outcomeCDFDistance_nonneg _ _)
· exact Finset.sum_nonneg (fun r _ => scoreCDFDistance_nonneg hOverlap _ _)
simp only [Finset.sum_add_distrib, Fintype.sum_bool] at hdist
cases a <;> linarith [hnonneg false, hnonneg true]
rw [abs_sub_comm]
exact hendpoint.trans_lt ((mul_lt_mul_of_pos_left hpair hLpos).trans hLδ)
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
PH :
hPH :
PH ∈ ExternalLaws g
x :
ExternalSample ε J n m
hout :
(∑ a : ArmSpace, ∑ r : LabelSpace J, outcomeCDFDistance (trialPopulationCells (releasedLaw PH.1) a r) (empiricalTrialCells x.1 a r)) < 4 * J / (α * sqrt n)
hscore :
(∑ a : ArmSpace, ∑ r : LabelSpace J, scoreCDFDistance (scorePopulationCells PH.2 g a r) (empiricalScoreCells g x.2 a r)) < 2 * J * (2 - 2 * ε) / (α * sqrt m)
conclusion
ate PH.1 ∈ sievedTotalizedProjectionCI g α x
Proof (Lean source)
lemma projection_strictBall_ate_mem_CI {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(PH : Measure (FullRow ε J) × Measure (ScoreSpace ε))
(hPH : PH ∈ ExternalLaws g)
(x : ExternalSample ε J n m)
(hout : (∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance
(trialPopulationCells (releasedLaw PH.1) a r)
(empiricalTrialCells x.1 a r)) <
4 * J / (α * sqrt n))
(hscore : (∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (scorePopulationCells PH.2 g a r)
(empiricalScoreCells g x.2 a r)) <
2 * J * (2 - 2 * ε) / (α * sqrt m)) :
ate PH.1 ∈ sievedTotalizedProjectionCI g α x := by
classical
let lam := trialPopulationCells (releasedLaw PH.1)
let σ := scorePopulationCells PH.2 g
let hMass := externalLawCellMasses g PH hPH
haveI : IsProbabilityMeasure PH.2 := hPH.2.1
haveI : IsProbabilityMeasure (releasedLaw PH.1) := by
unfold releasedLaw
exact @Measure.isProbabilityMeasure_map _ _ _ _ PH.1 hPH.1 _
(by unfold releasedRecord label arm observed; fun_prop :
Measurable (releasedRecord (ε := ε) (J := J))).aemeasurable
have hfin := projectionPopulationCells_finite PH.2 g (releasedLaw PH.1)
hPH.2.2.overlap hPH.2.2.measurableRelease
have hfinite : ∀ a r, (lam a r) univ < ⊤ ∧ (σ a r) univ < ⊤ := by
intro a r
exact ⟨(hfin.1 a r).measure_univ_lt_top,
(hfin.2 a r).measure_univ_lt_top⟩
have hcompat : ProjectionCompatible lam σ :=
projectionCompatible_populationCells PH.2 g (releasedLaw PH.1)
hMass hPH.2.2.overlap
have happrox := projection_strict_population_approximants g α
hPH.2.2.overlap x lam σ hfinite hcompat hout hscore
have hC : ate PH.1 ∈ closure
(⋃ b ∈ sievedProjectionCandidates g α x,
projectedATEInterval b.1 b.2) := by
apply projectedATEInterval_mem_closure_of_sieve_approx g α x lam σ
hPH.2.2.overlap happrox
have hLaw : CompatibleCausalLaw PH.2 g (releasedLaw PH.1) PH.1 :=
⟨hPH.1, hPH.2.2.measurableRelease, hPH.2.2.scoreMarginal,
hPH.2.2.randomizedAssignment, hPH.2.2.consistency,
hPH.2.2.deterministicRelease, rfl⟩
have hθ := compatible_ate_mem_sharpATESet PH.2 g
(releasedLaw PH.1) PH.1 hLaw hPH.2.2.overlap
change ate PH.1 ∈ Icc
(muLower PH.2 g (releasedLaw PH.1) hMass true -
muUpper PH.2 g (releasedLaw PH.1) hMass false)
(muUpper PH.2 g (releasedLaw PH.1) hMass true -
muLower PH.2 g (releasedLaw PH.1) hMass false) at hθ
dsimp [projectedATEInterval, lam, σ]
rw [projectedArmEndpoint_population_eq PH.2 g (releasedLaw PH.1)
hMass hPH.2.2.overlap true false,
projectedArmEndpoint_population_eq PH.2 g (releasedLaw PH.1)
hMass hPH.2.2.overlap false true,
projectedArmEndpoint_population_eq PH.2 g (releasedLaw PH.1)
hMass hPH.2.2.overlap true true,
projectedArmEndpoint_population_eq PH.2 g (releasedLaw PH.1)
hMass hPH.2.2.overlap false false]
simpa using hθ
have hB : (sievedProjectionCandidates g α x).Nonempty := by
obtain ⟨b, hb, _⟩ := happrox 1 (by norm_num)
exact ⟨b, hb⟩
have hclip : ate PH.1 ∈ Icc (-1 : ℝ) 1 := by
letI : IsProbabilityMeasure PH.1 := hPH.1
exact externalATE_mem_Icc PH.1
have hCHull : ate PH.1 ∈
closure (convexHull ℝ
(⋃ b ∈ sievedProjectionCandidates g α x,
projectedATEInterval b.1 b.2)) :=
closure_mono (subset_convexHull ℝ _) hC
have hne : (Icc (-1 : ℝ) 1 ∩
closure (convexHull ℝ
(⋃ b ∈ sievedProjectionCandidates g α x,
projectedATEInterval b.1 b.2))).Nonempty :=
⟨ate PH.1, hclip, hCHull⟩
unfold sievedTotalizedProjectionCI
simpa [hB, hne] using (show ate PH.1 ∈ Icc (-1 : ℝ) 1 ∩
closure (convexHull ℝ
(⋃ b ∈ sievedProjectionCandidates g α x,
projectedATEInterval b.1 b.2)) from ⟨hclip, hCHull⟩)
theorem externalProjectionIntervalProcedure_honest
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
hOverlap :
Overlap ε
hg :
hn :
0 < n
hm :
0 < m
hα :
0 < α
Qj :
hTrial :
JointTrialIID g Qj
hLog :
ExternalLogIID g Qj
hInd :
conclusion
externalProjectionIntervalProcedure g α hOverlap hg ∈ externalHonestProcedures g Qj α
Proof (Lean source)
lemma externalProjectionIntervalProcedure_honest {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(hOverlap : Overlap ε) (hg : Measurable g)
(hn : 0 < n) (hm : 0 < m) (hα : 0 < α)
(Qj : Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : JointTrialIID g Qj) (hLog : ExternalLogIID g Qj)
(hInd : ExternalLogIndependent g Qj) :
externalProjectionIntervalProcedure g α hOverlap hg ∈
externalHonestProcedures g Qj α := by
intro PH hPH
let μ := releasedLaw PH.1
let H := PH.2
let F : ExternalSample ε J n m → ℝ := fun x =>
∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (empiricalTrialCells x.1 a r)
(trialPopulationCells μ a r)
let G : ExternalSample ε J n m → ℝ := fun x =>
∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (empiricalScoreCells g x.2 a r)
(scorePopulationCells H g a r)
letI : IsProbabilityMeasure PH.1 := hPH.1
letI : IsProbabilityMeasure H := hPH.2.1
letI : IsProbabilityMeasure μ := by
unfold μ releasedLaw
exact @Measure.isProbabilityMeasure_map _ _ _ _ PH.1 hPH.1 _
(by unfold releasedRecord label arm observed; fun_prop :
Measurable (releasedRecord (ε := ε) (J := J))).aemeasurable
letI : IsProbabilityMeasure (Qj PH.1 PH.2) :=
externalLaw_jointExperiment_probability g Qj hTrial hLog hInd PH hPH
have hfin := projectionPopulationCells_finite H g μ hPH.2.2.overlap
hPH.2.2.measurableRelease
have hFTmeas : Measurable (fun x : TrialSample n J =>
∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (empiricalTrialCells x a r)
(trialPopulationCells μ a r)) := by
apply Finset.measurable_sum
intro a _
apply Finset.measurable_sum
intro r _
letI : IsFiniteMeasure (trialPopulationCells μ a r) := hfin.1 a r
simpa only [outcomeCDFDistance, abs_sub_comm] using
measurable_outcomeCDFDistance_empirical (trialPopulationCells μ a r)
(measure_lt_top _ _) a r
have hGTmeas : Measurable (fun x : Fin m → ScoreSpace ε =>
∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (empiricalScoreCells g x a r)
(scorePopulationCells H g a r)) := by
apply Finset.measurable_sum
intro a _
apply Finset.measurable_sum
intro r _
letI : IsFiniteMeasure (scorePopulationCells H g a r) := hfin.2 a r
simpa only [scoreCDFDistance, abs_sub_comm] using
measurable_scoreCDFDistance_empirical hPH.2.2.overlap g
hPH.2.2.measurableRelease (scorePopulationCells H g a r)
(measure_lt_top _ _) a r
have hFm : Measurable F := hFTmeas.comp measurable_fst
have hGm : Measurable G := hGTmeas.comp measurable_snd
have hFpi : Integrable (fun x : TrialSample n J =>
∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (empiricalTrialCells x a r)
(trialPopulationCells μ a r))
(Measure.pi fun _ : Fin n => μ) := by
exact integrable_finsetSum _ (fun a _ =>
integrable_finsetSum _ (fun r _ =>
integrable_outcomeCDFDistance_population_pi μ hn a r))
have hGpi : Integrable (fun x : Fin m → ScoreSpace ε =>
∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (empiricalScoreCells g x a r)
(scorePopulationCells H g a r))
(Measure.pi fun _ : Fin m => H) := by
exact integrable_finsetSum _ (fun a _ =>
integrable_finsetSum _ (fun r _ =>
integrable_scoreCDFDistance_population_pi hPH.2.2.overlap H g
hPH.2.2.measurableRelease hm a r))
have hF : Integrable F (Qj PH.1 PH.2) := by
have hmap : Measure.map fst (Qj PH.1 PH.2) =
Measure.pi (fun _ : Fin n => μ) := by
simpa [μ] using hTrial PH hPH
rw [← hmap] at hFpi
simpa [F, Function.comp_def] using
hFpi.comp_aemeasurable measurable_fst.aemeasurable
have hG : Integrable G (Qj PH.1 PH.2) := by
have hmap : Measure.map snd (Qj PH.1 PH.2) =
Measure.pi (fun _ : Fin m => H) := by
simpa [H] using hLog PH hPH
rw [← hmap] at hGpi
simpa [G, Function.comp_def] using
hGpi.comp_aemeasurable measurable_snd.aemeasurable
have hFmean : ∫ x, F x ∂Qj PH.1 PH.2 ≤ 2 * J / sqrt n := by
exact externalJointTrial_meanDistance_le g Qj hTrial PH hPH hn
(by simpa [μ] using hFTmeas)
(fun a r => integrable_outcomeCDFDistance_population_pi μ hn a r)
have hGmean : ∫ x, G x ∂Qj PH.1 PH.2 ≤
J * (2 - 2 * ε) / sqrt m := by
exact externalJointLog_meanDistance_le g Qj hLog PH hPH hm
(by simpa [H] using hGTmeas)
(fun a r => integrable_scoreCDFDistance_population_pi
hPH.2.2.overlap H g hPH.2.2.measurableRelease hm a r)
have hJ : 0 < J := by
letI : Nonempty (FullRow ε J) := nonempty_of_isProbabilityMeasure PH.1
obtain ⟨ω⟩ := ‹Nonempty (FullRow ε J)›
exact lt_of_le_of_lt (zero_le _) (label ω).isLt
have hprob := projection_strictBall_probability_at_paper_radii
(Qj PH.1 PH.2) F G hFm hGm hF hG
(Filter.Eventually.of_forall (fun x => Finset.sum_nonneg (fun a _ =>
Finset.sum_nonneg (fun r _ => outcomeCDFDistance_nonneg _ _))))
(Filter.Eventually.of_forall (fun x => Finset.sum_nonneg (fun a _ =>
Finset.sum_nonneg (fun r _ => scoreCDFDistance_nonneg hOverlap _ _))))
J n m hJ hn hm α ε hα hOverlap.2 hFmean hGmean
have hsubset : {x | F x < 4 * J / (α * sqrt n) ∧
G x < 2 * J * (2 - 2 * ε) / (α * sqrt m)} ⊆
{x | (externalProjectionIntervalProcedure g α hOverlap hg).contains x
(ate PH.1)} := by
intro x hx
change F x < _ ∧ G x < _ at hx
have hout : (∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (trialPopulationCells μ a r)
(empiricalTrialCells x.1 a r)) <
4 * J / (α * sqrt n) := by
convert hx.1 using 1
apply Finset.sum_congr rfl
intro a _
apply Finset.sum_congr rfl
intro r _
exact outcomeCDFDistance_symm _ _
have hscore : (∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (scorePopulationCells H g a r)
(empiricalScoreCells g x.2 a r)) <
2 * J * (2 - 2 * ε) / (α * sqrt m) := by
convert hx.2 using 1
apply Finset.sum_congr rfl
intro a _
apply Finset.sum_congr rfl
intro r _
exact scoreCDFDistance_symm _ _
have hmem := projection_strictBall_ate_mem_CI g α PH hPH x hout hscore
rw [← externalProjectionIntervalProcedure_Icc g α hOverlap hg x] at hmem
exact hmem
rw [← jointTrialLog_eq_externalExperiment g Qj hTrial hLog hInd PH hPH]
exact hprob.trans (measureReal_mono hsubset)
theorem externalExcessRisk_le_of_honest_uniform_bound
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
hOverlap :
Overlap ε
hg :
Qj :
hTrial :
JointTrialIID g Qj
hLog :
ExternalLogIID g Qj
hInd :
C :
ExternalIntervalProcedure ε J n m
hC :
C ∈ externalHonestProcedures g Qj α
R :
ℝ
hR :
0 ≤ R
hbound :
∀ PH,
∀ hPH : PH ∈ ExternalLaws g,
(∫ x, max 0 (C.length x - sharpATELength PH.2 g (releasedLaw PH.1) (externalLawCellMasses g PH hPH)) ∂(Qj PH.1 PH.2))
≤ R
conclusion
externalExcessRisk g Qj α ≤ R
Proof (Lean source)
lemma externalExcessRisk_le_of_honest_uniform_bound {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(hOverlap : Overlap ε) (hg : Measurable g)
(Qj : Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : JointTrialIID g Qj) (hLog : ExternalLogIID g Qj)
(hInd : ExternalLogIndependent g Qj)
(C : ExternalIntervalProcedure ε J n m)
(hC : C ∈ externalHonestProcedures g Qj α)
(R : ℝ) (hR : 0 ≤ R)
(hbound : ∀ PH, ∀ hPH : PH ∈ ExternalLaws g,
(∫ x, max 0 (C.length x -
sharpATELength PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH)) ∂(Qj PH.1 PH.2)) ≤ R) :
externalExcessRisk g Qj α ≤ R := by
let P : Measure (FullRow ε J) := trialCenterFullLaw g hOverlap (1 / 2)
let H : Measure (ScoreSpace ε) := Measure.dirac (trialCenterScore hOverlap)
have hPH : (P, H) ∈ ExternalLaws g :=
⟨trialCenterFullLaw_isProbabilityMeasure g hOverlap _ (by norm_num) (by norm_num),
inferInstance,
trialCenterFullLaw_externalScoreLaw g hOverlap hg _ (by norm_num) (by norm_num)⟩
let U (C : ExternalIntervalProcedure ε J n m) : Set ℝ :=
{u | ∃ PH, ∃ hPH : PH ∈ ExternalLaws g,
u = ∫ x, max 0 (C.length x -
sharpATELength PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH))
∂(externalExperiment n m PH.1 PH.2)}
have hUbd (C : ExternalIntervalProcedure ε J n m) : BddAbove (U C) := by
refine ⟨2, ?_⟩
rintro u ⟨PH, hPH', rfl⟩
simpa only [jointTrialLog_eq_externalExperiment g Qj hTrial hLog hInd PH hPH']
using externalExcessIntegral_le_two g Qj hTrial hLog hInd C PH hPH'
have hUne (C : ExternalIntervalProcedure ε J n m) : (U C).Nonempty := by
exact ⟨_, ⟨(P, H), hPH, rfl⟩⟩
have hUlower (C : ExternalIntervalProcedure ε J n m) : 0 ≤ sSup (U C) := by
obtain ⟨u, hu⟩ := hUne C
rcases hu with ⟨PH, hPH', rfl⟩
exact (integral_nonneg (fun _ => le_max_left _ _)).trans
(le_csSup (hUbd C) ⟨PH, hPH', rfl⟩)
have hRiskBd : BddBelow {v : ℝ | ∃ C ∈ externalHonestProcedures g Qj α,
v = sSup (U C)} := by
refine ⟨0, ?_⟩
rintro v ⟨C, hC', rfl⟩
exact hUlower C
have hUupper : sSup (U C) ≤ R := by
apply csSup_le (hUne C)
rintro v ⟨PH, hPH', rfl⟩
simpa only [jointTrialLog_eq_externalExperiment g Qj hTrial hLog hInd PH hPH']
using hbound PH hPH'
unfold externalExcessRisk
have hmem : sSup (U C) ∈
{v : ℝ | ∃ C ∈ externalHonestProcedures g Qj α, v = sSup (U C)} :=
⟨C, hC, rfl⟩
exact (csInf_le hRiskBd hmem).trans hUupper
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionDistanceTriangle 7 declarations Triangle bounds for the finite-measure CDF distances in the external projection.
Triangle bounds for the finite-measure CDF distances in the external projection.
theorem projectionFiniteCDF_intervalIntegrable Lemma projectionFiniteCDF_intervalIntegrable in the paper ↗
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
IntervalIntegrable (fun t : ℝ => ν.real {x | v x ≤ t}) volume a b
Proof (Lean source)
lemma projectionFiniteCDF_intervalIntegrable {X : Type*} [MeasurableSpace X]
(ν : Measure X) (hν : ν univ < ⊤) (v : X → ℝ)
{a b : ℝ} (hab : a ≤ b) :
IntervalIntegrable (fun t : ℝ => ν.real {x | v x ≤ t}) volume a b := by
have hInt : IntegrableOn (fun t : ℝ => ν.real {x | v x ≤ t})
(Icc a b) volume := by
apply Integrable.of_bound
(measurable_projectionFiniteCDF ν hν v).aestronglyMeasurable
(ν.real univ)
filter_upwards [] with t
have hle : ν {x | v x ≤ t} ≤ ν univ := measure_mono (Set.subset_univ _)
have hreal : ν.real {x | v x ≤ t} ≤ ν.real univ := by
exact ENNReal.toReal_mono (ne_of_lt hν) hle
rw [Real.norm_eq_abs, abs_of_nonneg measureReal_nonneg]
exact hreal
exact (intervalIntegrable_iff_integrableOn_Icc_of_le hab).mpr hInt
theorem projectionFiniteCDF_absDiff_intervalIntegrable Lemma projectionFiniteCDF_absDiff_intervalIntegrable in the paper ↗
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma projectionFiniteCDF_absDiff_intervalIntegrable {X : Type*} [MeasurableSpace X]
(μ ν : Measure X) (hμ : μ univ < ⊤) (hν : ν univ < ⊤)
(v : X → ℝ) {a b : ℝ} (hab : a ≤ b) :
IntervalIntegrable (fun t : ℝ =>
|(μ {x | v x ≤ t}).toReal - (ν {x | v x ≤ t}).toReal|)
volume a b := by
exact ((projectionFiniteCDF_intervalIntegrable μ hμ v hab).sub
(projectionFiniteCDF_intervalIntegrable ν hν v hab)).abs
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
a b :
ℝ
hab :
a ≤ b
F G H :
ℝ → ℝ
hFG :
IntervalIntegrable (fun t => |F t - G t|) volume a b
hGH :
IntervalIntegrable (fun t => |G t - H t|) volume a b
hFH :
IntervalIntegrable (fun t => |F t - H t|) volume a b
conclusion
(∫ t in a..b, |F t - H t|) ≤ (∫ t in a..b, |F t - G t|) + (∫ t in a..b, |G t - H t|)
Proof (Lean source)
lemma projectionIntegratedAbs_triangle {a b : ℝ} (hab : a ≤ b)
(F G H : ℝ → ℝ)
(hFG : IntervalIntegrable (fun t => |F t - G t|) volume a b)
(hGH : IntervalIntegrable (fun t => |G t - H t|) volume a b)
(hFH : IntervalIntegrable (fun t => |F t - H t|) volume a b) :
(∫ t in a..b, |F t - H t|) ≤
(∫ t in a..b, |F t - G t|) +
(∫ t in a..b, |G t - H t|) := by
rw [← intervalIntegral.integral_add hFG hGH]
apply intervalIntegral.integral_mono_on hab hFH (hFG.add hGH)
intro t _
calc
|F t - H t| = |(F t - G t) + (G t - H t)| := by congr 1; ring
_ ≤ |F t - G t| + |G t - H t| := abs_add_le _ _
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma outcomeCDFDistance_triangle
(μ ν ξ : Measure OutcomeSpace)
(hμ : μ univ < ⊤) (hν : ν univ < ⊤)
(hξ : ξ univ < ⊤) :
outcomeCDFDistance μ ξ ≤
outcomeCDFDistance μ ν + outcomeCDFDistance ν ξ := by
have hmass := abs_sub_le (μ.real univ) (ν.real univ) (ξ.real univ)
have hcdf := projectionIntegratedAbs_triangle (by norm_num : (0 : ℝ) ≤ 1)
(fun t => (μ {y | (y : ℝ) ≤ t}).toReal)
(fun t => (ν {y | (y : ℝ) ≤ t}).toReal)
(fun t => (ξ {y | (y : ℝ) ≤ t}).toReal)
(projectionFiniteCDF_absDiff_intervalIntegrable μ ν hμ hν
(fun y : OutcomeSpace => (y : ℝ)) (by norm_num))
(projectionFiniteCDF_absDiff_intervalIntegrable ν ξ hν hξ
(fun y : OutcomeSpace => (y : ℝ)) (by norm_num))
(projectionFiniteCDF_absDiff_intervalIntegrable μ ξ hμ hξ
(fun y : OutcomeSpace => (y : ℝ)) (by norm_num))
dsimp [outcomeCDFDistance]
linarith
theorem scoreCDFDistance_triangle
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma scoreCDFDistance_triangle {ε : ℝ} (hOverlap : Overlap ε)
(μ ν ξ : Measure (ScoreSpace ε))
(hμ : μ univ < ⊤) (hν : ν univ < ⊤)
(hξ : ξ univ < ⊤) :
scoreCDFDistance μ ξ ≤
scoreCDFDistance μ ν + scoreCDFDistance ν ξ := by
have hab : ε ≤ 1 - ε := by dsimp [Overlap] at hOverlap; linarith [hOverlap.2]
have hmass := abs_sub_le (μ.real univ) (ν.real univ) (ξ.real univ)
have hcdf := projectionIntegratedAbs_triangle hab
(fun t => (μ {e | (e : ℝ) ≤ t}).toReal)
(fun t => (ν {e | (e : ℝ) ≤ t}).toReal)
(fun t => (ξ {e | (e : ℝ) ≤ t}).toReal)
(projectionFiniteCDF_absDiff_intervalIntegrable μ ν hμ hν
(fun e : ScoreSpace ε => (e : ℝ)) hab)
(projectionFiniteCDF_absDiff_intervalIntegrable ν ξ hν hξ
(fun e : ScoreSpace ε => (e : ℝ)) hab)
(projectionFiniteCDF_absDiff_intervalIntegrable μ ξ hμ hξ
(fun e : ScoreSpace ε => (e : ℝ)) hab)
dsimp [scoreCDFDistance]
linarith
theorem projectionCandidate_populationDistance_le_of_finite
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
hOverlap :
Overlap ε
x :
ExternalSample ε J n m
b :
lam :
σ :
ArmCellArray J (ScoreSpace ε)
hb :
b ∈ projectionCandidates g α x
htrialEmp :
∀ a r, (empiricalTrialCells x.1 a r) univ < ⊤
hscoreEmp :
∀ a r, (empiricalScoreCells g x.2 a r) univ < ⊤
htrialPop :
∀ a r, (lam a r) univ < ⊤
hscorePop :
∀ a r, (σ a r) univ < ⊤
Proof (Lean source)
lemma projectionCandidate_populationDistance_le_of_finite
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(hOverlap : Overlap ε)
(x : ExternalSample ε J n m)
(b : ArmCellArray J OutcomeSpace × ArmCellArray J (ScoreSpace ε))
(lam : ArmCellArray J OutcomeSpace)
(σ : ArmCellArray J (ScoreSpace ε))
(hb : b ∈ projectionCandidates g α x)
(htrialEmp : ∀ a r, (empiricalTrialCells x.1 a r) univ < ⊤)
(hscoreEmp : ∀ a r, (empiricalScoreCells g x.2 a r) univ < ⊤)
(htrialPop : ∀ a r, (lam a r) univ < ⊤)
(hscorePop : ∀ a r, (σ a r) univ < ⊤) :
(∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (b.1 a r) (lam a r)) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (b.2 a r) (σ a r)) ≤
4 * J / (α * sqrt n) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (empiricalTrialCells x.1 a r) (lam a r)) +
2 * J * (2 - 2 * ε) / (α * sqrt m) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (empiricalScoreCells g x.2 a r) (σ a r)) := by
apply projectionCandidate_populationDistance_le g α x b lam σ hb
· intro a r
exact outcomeCDFDistance_triangle _ _ _ (hb.1 a r).1
(htrialEmp a r) (htrialPop a r)
· intro a r
exact scoreCDFDistance_triangle hOverlap _ _ _ (hb.1 a r).2
(hscoreEmp a r) (hscorePop a r)
theorem projectionCandidate_populationDistance_le_of_populationFinite
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
hOverlap :
Overlap ε
x :
ExternalSample ε J n m
b :
lam :
σ :
ArmCellArray J (ScoreSpace ε)
hb :
b ∈ projectionCandidates g α x
htrialPop :
∀ a r, (lam a r) univ < ⊤
hscorePop :
∀ a r, (σ a r) univ < ⊤
Proof (Lean source)
lemma projectionCandidate_populationDistance_le_of_populationFinite
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(hOverlap : Overlap ε)
(x : ExternalSample ε J n m)
(b : ArmCellArray J OutcomeSpace × ArmCellArray J (ScoreSpace ε))
(lam : ArmCellArray J OutcomeSpace)
(σ : ArmCellArray J (ScoreSpace ε))
(hb : b ∈ projectionCandidates g α x)
(htrialPop : ∀ a r, (lam a r) univ < ⊤)
(hscorePop : ∀ a r, (σ a r) univ < ⊤) :
(∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (b.1 a r) (lam a r)) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (b.2 a r) (σ a r)) ≤
4 * J / (α * sqrt n) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (empiricalTrialCells x.1 a r) (lam a r)) +
2 * J * (2 - 2 * ε) / (α * sqrt m) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (empiricalScoreCells g x.2 a r) (σ a r)) := by
exact projectionCandidate_populationDistance_le_of_finite g α hOverlap x b lam σ hb
(empiricalTrialCells_finite x.1) (empiricalScoreCells_finite g x.2)
htrialPop hscorePop
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionEmpirical 20 declarations This file records the finite-sample identities behind the trial half of the projection radius.
Empirical trial cell coordinates
This file records the finite-sample identities behind the trial half of the projection radius. A cell cumulative mass is a measurable finite average of coordinate indicators, and under the i.i.d. product law its expectation is the corresponding population cell mass.
def trialCellFrequency
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
trialCellFrequency a r B x :
ℝ
given by
by classical exact (n : ℝ)⁻¹ * ∑ i : Fin n, if (x i).1 = r ∧
(x i).2.1 = a ∧
(x i).2.2 ∈ B then (1 : ℝ) else 0
theorem empiricalTrialCells_real_apply
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma empiricalTrialCells_real_apply {J n : ℕ} (a : ArmSpace)
(r : LabelSpace J) (B : Set OutcomeSpace) (hB : MeasurableSet B)
(x : TrialSample n J) :
(empiricalTrialCells x a r).real B = trialCellFrequency a r B x := by
classical
unfold empiricalTrialCells trialCellFrequency Measure.real
rw [Measure.smul_apply, Measure.coe_finsetSum, Finset.sum_apply, smul_eq_mul,
ENNReal.toReal_mul, ENNReal.toReal_inv, ENNReal.toReal_natCast]
congr 1
rw [ENNReal.toReal_sum]
· apply Finset.sum_congr rfl
intro i _
by_cases hcell : (x i).1 = r ∧ (x i).2.1 = a
· simp only [hcell, and_self, ↓reduceIte]
rw [Measure.dirac_apply' _ hB]
by_cases hmem : (x i).2.2 ∈ B <;> simp [indicator, hmem]
· simp only [hcell, ↓reduceIte, Measure.coe_zero, Pi.zero_apply, ENNReal.toReal_zero]
have hfull : ¬((x i).1 = r ∧ (x i).2.1 = a ∧ (x i).2.2 ∈ B) := by
intro h
exact hcell ⟨h.1, h.2.1⟩
simp [hfull]
· intro i _
by_cases hcell : (x i).1 = r ∧ (x i).2.1 = a
· by_cases hmem : (x i).2.2 ∈ B <;> simp [hcell, hmem, indicator]
· simp [hcell]
theorem measurable_trialCellFrequency
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Measurable (trialCellFrequency (n := n) a r B)
Proof (Lean source)
@[fun_prop]
lemma measurable_trialCellFrequency {J n : ℕ} (a : ArmSpace)
(r : LabelSpace J) (B : Set OutcomeSpace) (hB : MeasurableSet B) :
Measurable (trialCellFrequency (n := n) a r B) := by
classical
unfold trialCellFrequency
apply Measurable.const_mul
apply Finset.measurable_sum
intro i _
have hi : Measurable (fun x : TrialSample n J => x i) := measurable_pi_apply i
apply Measurable.ite
· exact (measurableSet_eq_fun hi.fst
(measurable_const : Measurable (fun _ : TrialSample n J => r))).inter
((measurableSet_eq_fun hi.snd.fst
(measurable_const : Measurable (fun _ : TrialSample n J => a))).inter
(hB.preimage hi.snd.snd))
· fun_prop
· fun_prop
theorem measurable_empiricalTrialCells_real_apply
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
@[fun_prop]
lemma measurable_empiricalTrialCells_real_apply {J n : ℕ} (a : ArmSpace)
(r : LabelSpace J) (B : Set OutcomeSpace) (hB : MeasurableSet B) :
Measurable (fun x : TrialSample n J => (empiricalTrialCells x a r).real B) := by
rw [show (fun x : TrialSample n J => (empiricalTrialCells x a r).real B) =
trialCellFrequency a r B by
funext x
exact empiricalTrialCells_real_apply a r B hB x]
exact measurable_trialCellFrequency a r B hB
theorem measurable_empiricalTrialCellCDF
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
@[fun_prop]
lemma measurable_empiricalTrialCellCDF {J n : ℕ} (a : ArmSpace)
(r : LabelSpace J) (t : ℝ) :
Measurable (fun x : TrialSample n J =>
(empiricalTrialCells x a r).real {y | (y : ℝ) ≤ t}) := by
apply measurable_empiricalTrialCells_real_apply
exact measurableSet_le (by fun_prop) measurable_const
theorem integral_trialCellFrequency_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
J n :
ℕ
μ :
hn :
0 < n
a :
r :
B :
hB :
conclusion
∫ x, trialCellFrequency a r B x ∂Measure.pi (fun _ : Fin n => μ)
= μ.real {z | z.1 = r ∧ z.2.1 = a ∧ z.2.2 ∈ B}
Proof (Lean source)
lemma integral_trialCellFrequency_pi {J n : ℕ} (μ : Measure (Observation J))
[IsProbabilityMeasure μ] (hn : 0 < n) (a : ArmSpace) (r : LabelSpace J)
(B : Set OutcomeSpace) (hB : MeasurableSet B) :
∫ x, trialCellFrequency a r B x ∂Measure.pi (fun _ : Fin n => μ) =
μ.real {z | z.1 = r ∧ z.2.1 = a ∧ z.2.2 ∈ B} := by
classical
let C : Set (Observation J) := {z | z.1 = r ∧ z.2.1 = a ∧ z.2.2 ∈ B}
have hC : MeasurableSet C := by
change MeasurableSet {z : Observation J | z.1 = r ∧ z.2.1 = a ∧ z.2.2 ∈ B}
have hz : Measurable (fun z : Observation J => z) := measurable_id
exact (measurableSet_eq_fun hz.fst
(measurable_const : Measurable (fun _ : Observation J => r))).inter
((measurableSet_eq_fun hz.snd.fst
(measurable_const : Measurable (fun _ : Observation J => a))).inter
(hB.preimage hz.snd.snd))
have hcoord (i : Fin n) :
Integrable (fun x : TrialSample n J => if x i ∈ C then (1 : ℝ) else 0)
(Measure.pi fun _ : Fin n => μ) := by
have hm : Measurable (fun x : TrialSample n J => if x i ∈ C then (1 : ℝ) else 0) :=
Measurable.ite (hC.preimage (measurable_pi_apply i)) measurable_const measurable_const
exact Integrable.of_bound hm.aestronglyMeasurable 1 (Filter.Eventually.of_forall fun x => by
by_cases hx : x i ∈ C <;> simp [hx])
unfold trialCellFrequency
rw [MeasureTheory.integral_const_mul]
change (n : ℝ)⁻¹ *
∫ x : TrialSample n J, ∑ i : Fin n, if x i ∈ C then (1 : ℝ) else 0
∂Measure.pi (fun _ : Fin n => μ) = μ.real C
rw [MeasureTheory.integral_finsetSum univ
(fun i _ => hcoord i)]
have heach (i : Fin n) :
∫ x : TrialSample n J, (if x i ∈ C then (1 : ℝ) else 0)
∂Measure.pi (fun _ : Fin n => μ) = μ.real C := by
calc
(∫ x : TrialSample n J, (if x i ∈ C then (1 : ℝ) else 0)
∂Measure.pi (fun _ : Fin n => μ)) =
∫ x : TrialSample n J, C.indicator (fun _ => (1 : ℝ)) (x i)
∂Measure.pi (fun _ : Fin n => μ) := by
apply integral_congr_ae
filter_upwards [] with x
simp [indicator]
_ = ∫ z : Observation J, C.indicator (fun _ => (1 : ℝ)) z ∂μ := by
rw [MeasureTheory.integral_comp_eval (μ := fun _ : Fin n => μ) (i := i)]
exact (measurable_const.indicator hC).aestronglyMeasurable
_ = μ.real C := MeasureTheory.integral_indicator_one hC
simp_rw [heach]
simp [Nat.ne_of_gt hn]
theorem empiricalScoreCells_real_apply
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J m :
ℕ
g :
ScoreSpace ε → LabelSpace J
a :
r :
B :
Set (ScoreSpace ε)
hB :
x :
Fin m → ScoreSpace ε
Proof (Lean source)
lemma empiricalScoreCells_real_apply {ε : ℝ} {J m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (a : ArmSpace) (r : LabelSpace J)
(B : Set (ScoreSpace ε)) (hB : MeasurableSet B)
(x : Fin m → ScoreSpace ε) :
(empiricalScoreCells g x a r).real B =
∑ j : Fin m, {j | g (x j) = r ∧ x j ∈ B}.indicator
(fun j => toReal (ofReal (armProb a (x j)) / m)) j := by
classical
unfold empiricalScoreCells Measure.real
rw [Measure.coe_finsetSum, Finset.sum_apply, ENNReal.toReal_sum]
· apply Finset.sum_congr rfl
intro j _
by_cases hr : g (x j) = r
· simp only [hr, ↓reduceIte, Measure.smul_apply]
rw [Measure.dirac_apply' _ hB]
by_cases hmem : x j ∈ B <;> simp [indicator, hr, hmem, ENNReal.toReal_div]
· simp [hr]
· intro j _
by_cases hr : g (x j) = r
· by_cases hmem : x j ∈ B
· simp only [hr, ↓reduceIte, Measure.smul_apply]
rw [Measure.dirac_apply' _ hB]
simp only [indicator, if_pos hmem, Pi.one_apply, smul_eq_mul, mul_one]
by_cases hm : m = 0
· subst m
exact elim0 j
· exact ne_of_lt (ENNReal.div_lt_top ENNReal.ofReal_ne_top (by simp [hm]))
· simp [hr, hmem]
· simp [hr]
theorem variance_scoreCellIndicator
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
hOverlap :
Overlap ε
H :
g :
ScoreSpace ε → LabelSpace J
hg :
a :
r :
B :
Set (ScoreSpace ε)
hB :
conclusion
≤ 1 / 4
Proof (Lean source)
lemma variance_scoreCellIndicator {ε : ℝ} {J : ℕ}
(hOverlap : Overlap ε) (H : Measure (ScoreSpace ε))
[IsProbabilityMeasure H] (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (a : ArmSpace) (r : LabelSpace J)
(B : Set (ScoreSpace ε)) (hB : MeasurableSet B) :
variance
({e : ScoreSpace ε | g e = r ∧ e ∈ B}.indicator (armProb a)) H ≤ 1 / 4 := by
classical
have hcell : MeasurableSet {e : ScoreSpace ε | g e = r ∧ e ∈ B} :=
(measurableSet_eq_fun hg measurable_const).inter hB
have hp : Measurable (armProb a : ScoreSpace ε → ℝ) := by
unfold armProb
cases a <;> simp only [Bool.false_eq_true, ↓reduceIte] <;> fun_prop
have hm : Measurable ({e : ScoreSpace ε | g e = r ∧ e ∈ B}.indicator (armProb a)) :=
hp.indicator hcell
have hb : ∀ e : ScoreSpace ε,
{e : ScoreSpace ε | g e = r ∧ e ∈ B}.indicator (armProb a) e ∈
Icc (0 : ℝ) 1 := by
intro e
by_cases he : g e = r ∧ e ∈ B
· simpa [indicator, he] using
(show armProb a e ∈ Icc (0 : ℝ) 1 from
⟨le_trans (le_of_lt hOverlap.1) (projectionArmProb_bounds hOverlap a e).1,
(projectionArmProb_bounds hOverlap a e).2⟩)
· simp [indicator, he]
have hv := ProbabilityTheory.variance_le_sq_of_bounded
(μ := H) (a := 0) (b := 1) (Filter.Eventually.of_forall hb) hm.aemeasurable
norm_num at hv
exact hv
theorem variance_trialCellIndicator_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
variance (fun x : TrialSample n J => {z : Observation J | z.1 = r ∧ z.2.1 = a ∧ z.2.2 ∈ B}.indicator (fun _ => (1 : ℝ)) (x i)) (Measure.pi fun _ : Fin n => μ)
≤ 1 / 4
Proof (Lean source)
lemma variance_trialCellIndicator_pi {J n : ℕ}
(μ : Measure (Observation J)) [IsProbabilityMeasure μ]
(i : Fin n) (a : ArmSpace) (r : LabelSpace J)
(B : Set OutcomeSpace) (hB : MeasurableSet B) :
variance
(fun x : TrialSample n J =>
{z : Observation J | z.1 = r ∧ z.2.1 = a ∧ z.2.2 ∈ B}.indicator
(fun _ => (1 : ℝ)) (x i))
(Measure.pi fun _ : Fin n => μ) ≤ 1 / 4 := by
classical
let C : Set (Observation J) :=
{z | z.1 = r ∧ z.2.1 = a ∧ z.2.2 ∈ B}
have hC : MeasurableSet C := by
exact (measurableSet_eq_fun measurable_fst measurable_const).inter
((measurableSet_eq_fun (measurable_fst.comp measurable_snd) measurable_const).inter
(hB.preimage (measurable_snd.comp measurable_snd)))
have hm : Measurable (fun x : TrialSample n J =>
if x i ∈ C then (1 : ℝ) else 0) := by
exact Measurable.ite (hC.preimage (measurable_pi_apply i))
measurable_const measurable_const
have hb : ∀ x : TrialSample n J,
(if x i ∈ C then (1 : ℝ) else 0) ∈ Icc (0 : ℝ) 1 := by
intro x
by_cases hx : x i ∈ C <;> simp [hx]
have hv := ProbabilityTheory.variance_le_sq_of_bounded
(μ := Measure.pi fun _ : Fin n => μ)
(X := fun x : TrialSample n J => if x i ∈ C then (1 : ℝ) else 0)
(a := 0) (b := 1) (Filter.Eventually.of_forall hb) hm.aemeasurable
norm_num at hv
simpa [indicator, C] using hv
theorem variance_trialCellFrequency_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
J n :
ℕ
μ :
hn :
0 < n
a :
r :
B :
hB :
conclusion
≤ 1 / (4 * n)
Proof (Lean source)
lemma variance_trialCellFrequency_pi {J n : ℕ}
(μ : Measure (Observation J)) [IsProbabilityMeasure μ]
(hn : 0 < n) (a : ArmSpace) (r : LabelSpace J)
(B : Set OutcomeSpace) (hB : MeasurableSet B) :
variance (trialCellFrequency a r B)
(Measure.pi fun _ : Fin n => μ) ≤ 1 / (4 * n) := by
classical
let C : Set (Observation J) :=
{z | z.1 = r ∧ z.2.1 = a ∧ z.2.2 ∈ B}
let F : Observation J → ℝ := C.indicator (fun _ => 1)
have hC : MeasurableSet C := by
exact (measurableSet_eq_fun measurable_fst measurable_const).inter
((measurableSet_eq_fun (measurable_fst.comp measurable_snd) measurable_const).inter
(hB.preimage (measurable_snd.comp measurable_snd)))
have hF : Measurable F := measurable_const.indicator hC
have hbound : ∀ z, F z ∈ Icc (0 : ℝ) 1 := by
intro z
by_cases hz : z ∈ C <;> simp [F, hz]
have hlp : MemLp F 2 μ :=
memLp_of_bounded (Filter.Eventually.of_forall hbound) hF.aestronglyMeasurable 2
have hvar : variance F μ ≤ 1 / 4 := by
have hv := ProbabilityTheory.variance_le_sq_of_bounded
(μ := μ) (X := F) (a := 0) (b := 1)
(Filter.Eventually.of_forall hbound) hF.aemeasurable
norm_num at hv
exact hv
have hmain := iid_average_variance_fintype
(ι := Fin n) μ F hlp
have hfun : trialCellFrequency a r B =
(fun x : TrialSample n J => (n : ℝ)⁻¹ * ∑ i : Fin n, F (x i)) := by
funext x
unfold trialCellFrequency F C
congr 1
apply Finset.sum_congr rfl
intro i _
simp only [indicator]
rfl
rw [hfun]
simp only [Fintype.card_fin] at hmain
rw [hmain]
have hn0 : (0 : ℝ) ≤ (n : ℝ)⁻¹ := by positivity
calc
(n : ℝ)⁻¹ * variance F μ ≤ (n : ℝ)⁻¹ * (1 / 4) :=
mul_le_mul_of_nonneg_left hvar hn0
_ = 1 / (4 * n) := by field_simp
def scoreCellFrequency
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J m :
ℕ
g :
ScoreSpace ε → LabelSpace J
a :
r :
B :
Set (ScoreSpace ε)
x :
Fin m → ScoreSpace ε
def
scoreCellFrequency g a r B x :
ℝ
given by
theorem empiricalScoreCells_real_eq_scoreCellFrequency
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J m :
ℕ
hOverlap :
Overlap ε
g :
ScoreSpace ε → LabelSpace J
a :
r :
B :
Set (ScoreSpace ε)
hB :
x :
Fin m → ScoreSpace ε
conclusion
Proof (Lean source)
lemma empiricalScoreCells_real_eq_scoreCellFrequency {ε : ℝ} {J m : ℕ}
(hOverlap : Overlap ε)
(g : ScoreSpace ε → LabelSpace J) (a : ArmSpace) (r : LabelSpace J)
(B : Set (ScoreSpace ε)) (hB : MeasurableSet B)
(x : Fin m → ScoreSpace ε) :
(empiricalScoreCells g x a r).real B = scoreCellFrequency g a r B x := by
classical
rw [empiricalScoreCells_real_apply g a r B hB x]
unfold scoreCellFrequency
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro j _
by_cases hj : g (x j) = r ∧ x j ∈ B
· simp only [indicator]
rw [ENNReal.toReal_div, ENNReal.toReal_ofReal
(le_trans (le_of_lt hOverlap.1)
(projectionArmProb_bounds hOverlap a (x j)).1), ENNReal.toReal_natCast]
simp [hj, div_eq_mul_inv, mul_comm]
· simp [indicator, hj]
theorem measurable_scoreCellFrequency
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Measurable (scoreCellFrequency (m := m) g a r B)
Proof (Lean source)
@[fun_prop]
lemma measurable_scoreCellFrequency {ε : ℝ} {J m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (hg : Measurable g)
(a : ArmSpace) (r : LabelSpace J)
(B : Set (ScoreSpace ε)) (hB : MeasurableSet B) :
Measurable (scoreCellFrequency (m := m) g a r B) := by
have hcell : MeasurableSet {e : ScoreSpace ε | g e = r ∧ e ∈ B} :=
(measurableSet_eq_fun hg measurable_const).inter hB
have hp : Measurable (armProb a : ScoreSpace ε → ℝ) := by
unfold armProb
cases a <;> simp only [Bool.false_eq_true, ↓reduceIte] <;> fun_prop
unfold scoreCellFrequency
apply Measurable.const_mul
apply Finset.measurable_sum
intro j _
exact (hp.indicator hcell).comp (measurable_pi_apply j)
theorem integral_scoreCellFrequency_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J m :
ℕ
hOverlap :
Overlap ε
H :
g :
ScoreSpace ε → LabelSpace J
hg :
hm :
0 < m
a :
r :
B :
Set (ScoreSpace ε)
hB :
conclusion
∫ x, scoreCellFrequency g a r B x ∂Measure.pi (fun _ : Fin m => H)
Proof (Lean source)
lemma integral_scoreCellFrequency_pi {ε : ℝ} {J m : ℕ}
(hOverlap : Overlap ε) (H : Measure (ScoreSpace ε))
[IsProbabilityMeasure H] (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (hm : 0 < m) (a : ArmSpace) (r : LabelSpace J)
(B : Set (ScoreSpace ε)) (hB : MeasurableSet B) :
∫ x, scoreCellFrequency g a r B x ∂Measure.pi (fun _ : Fin m => H) =
∫ e, {e : ScoreSpace ε | g e = r ∧ e ∈ B}.indicator (armProb a) e ∂H := by
let F : ScoreSpace ε → ℝ :=
{e : ScoreSpace ε | g e = r ∧ e ∈ B}.indicator (armProb a)
have hcell : MeasurableSet {e : ScoreSpace ε | g e = r ∧ e ∈ B} :=
(measurableSet_eq_fun hg measurable_const).inter hB
have hp : Measurable (armProb a : ScoreSpace ε → ℝ) := by
unfold armProb
cases a <;> simp only [Bool.false_eq_true, ↓reduceIte] <;> fun_prop
have hF : Measurable F := hp.indicator hcell
have hbound : ∀ e, F e ∈ Icc (0 : ℝ) 1 := by
intro e
by_cases he : g e = r ∧ e ∈ B
· simpa [F, indicator, he] using
(show armProb a e ∈ Icc (0 : ℝ) 1 from
⟨le_trans (le_of_lt hOverlap.1) (projectionArmProb_bounds hOverlap a e).1,
(projectionArmProb_bounds hOverlap a e).2⟩)
· simp [F, he]
have hInt : Integrable F H :=
Integrable.of_bound hF.aestronglyMeasurable 1
(Filter.Eventually.of_forall fun e => by
have he := hbound e
simpa [Real.norm_eq_abs, abs_of_nonneg he.1] using he.2)
have hcoord (j : Fin m) :
Integrable (fun x : Fin m → ScoreSpace ε => F (x j))
(Measure.pi fun _ : Fin m => H) :=
MeasureTheory.integrable_comp_eval hInt
change (∫ x : Fin m → ScoreSpace ε,
(m : ℝ)⁻¹ * ∑ j : Fin m, F (x j)
∂Measure.pi (fun _ : Fin m => H)) = ∫ e, F e ∂H
rw [MeasureTheory.integral_const_mul,
MeasureTheory.integral_finsetSum univ (fun j _ => hcoord j)]
have heach (j : Fin m) :
∫ x : Fin m → ScoreSpace ε, F (x j)
∂Measure.pi (fun _ : Fin m => H) = ∫ e, F e ∂H := by
rw [MeasureTheory.integral_comp_eval (μ := fun _ : Fin m => H) (i := j)]
exact hF.aestronglyMeasurable
simp_rw [heach]
simp [Nat.ne_of_gt hm]
theorem integral_empiricalScoreCells_real_apply_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J m :
ℕ
hOverlap :
Overlap ε
H :
g :
ScoreSpace ε → LabelSpace J
hg :
hm :
0 < m
a :
r :
B :
Set (ScoreSpace ε)
hB :
conclusion
Proof (Lean source)
lemma integral_empiricalScoreCells_real_apply_pi {ε : ℝ} {J m : ℕ}
(hOverlap : Overlap ε) (H : Measure (ScoreSpace ε))
[IsProbabilityMeasure H] (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (hm : 0 < m) (a : ArmSpace) (r : LabelSpace J)
(B : Set (ScoreSpace ε)) (hB : MeasurableSet B) :
∫ x, (empiricalScoreCells g x a r).real B
∂Measure.pi (fun _ : Fin m => H) =
∫ e, {e : ScoreSpace ε | g e = r ∧ e ∈ B}.indicator (armProb a) e ∂H := by
simpa only [empiricalScoreCells_real_eq_scoreCellFrequency hOverlap g a r B hB] using
integral_scoreCellFrequency_pi hOverlap H g hg hm a r B hB
theorem variance_scoreCellFrequency_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J m :
ℕ
hOverlap :
Overlap ε
H :
g :
ScoreSpace ε → LabelSpace J
hg :
hm :
0 < m
a :
r :
B :
Set (ScoreSpace ε)
hB :
conclusion
≤ 1 / (4 * m)
Proof (Lean source)
lemma variance_scoreCellFrequency_pi {ε : ℝ} {J m : ℕ}
(hOverlap : Overlap ε) (H : Measure (ScoreSpace ε))
[IsProbabilityMeasure H] (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (hm : 0 < m) (a : ArmSpace) (r : LabelSpace J)
(B : Set (ScoreSpace ε)) (hB : MeasurableSet B) :
variance (scoreCellFrequency g a r B)
(Measure.pi fun _ : Fin m => H) ≤ 1 / (4 * m) := by
let F : ScoreSpace ε → ℝ :=
{e : ScoreSpace ε | g e = r ∧ e ∈ B}.indicator (armProb a)
have hcell : MeasurableSet {e : ScoreSpace ε | g e = r ∧ e ∈ B} :=
(measurableSet_eq_fun hg measurable_const).inter hB
have hp : Measurable (armProb a : ScoreSpace ε → ℝ) := by
unfold armProb
cases a <;> simp only [Bool.false_eq_true, ↓reduceIte] <;> fun_prop
have hF : Measurable F := hp.indicator hcell
have hbound : ∀ e, F e ∈ Icc (0 : ℝ) 1 := by
intro e
by_cases he : g e = r ∧ e ∈ B
· simpa [F, indicator, he] using
(show armProb a e ∈ Icc (0 : ℝ) 1 from
⟨le_trans (le_of_lt hOverlap.1) (projectionArmProb_bounds hOverlap a e).1,
(projectionArmProb_bounds hOverlap a e).2⟩)
· simp [F, he]
have hlp : MemLp F 2 H :=
memLp_of_bounded (Filter.Eventually.of_forall hbound) hF.aestronglyMeasurable 2
have hvar : variance F H ≤ 1 / 4 := by
simpa [F] using variance_scoreCellIndicator hOverlap H g hg a r B hB
have hmain := iid_average_variance_fintype
(ι := Fin m) H F hlp
change variance
(fun x : Fin m → ScoreSpace ε => (m : ℝ)⁻¹ * ∑ j : Fin m, F (x j))
(Measure.pi fun _ : Fin m => H) ≤ 1 / (4 * m)
simp only [Fintype.card_fin] at hmain
rw [hmain]
calc
(m : ℝ)⁻¹ * variance F H ≤ (m : ℝ)⁻¹ * (1 / 4) :=
mul_le_mul_of_nonneg_left hvar (by positivity)
_ = 1 / (4 * m) := by field_simp
theorem variance_empiricalScoreCells_real_apply_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J m :
ℕ
hOverlap :
Overlap ε
H :
g :
ScoreSpace ε → LabelSpace J
hg :
hm :
0 < m
a :
r :
B :
Set (ScoreSpace ε)
hB :
conclusion
variance (fun x : Fin m → ScoreSpace ε => (empiricalScoreCells g x a r).real B) (Measure.pi fun _ : Fin m => H)
≤ 1 / (4 * m)
Proof (Lean source)
lemma variance_empiricalScoreCells_real_apply_pi {ε : ℝ} {J m : ℕ}
(hOverlap : Overlap ε) (H : Measure (ScoreSpace ε))
[IsProbabilityMeasure H] (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (hm : 0 < m) (a : ArmSpace) (r : LabelSpace J)
(B : Set (ScoreSpace ε)) (hB : MeasurableSet B) :
variance
(fun x : Fin m → ScoreSpace ε => (empiricalScoreCells g x a r).real B)
(Measure.pi fun _ : Fin m => H) ≤ 1 / (4 * m) := by
simpa only [empiricalScoreCells_real_eq_scoreCellFrequency hOverlap g a r B hB] using
variance_scoreCellFrequency_pi hOverlap H g hg hm a r B hB
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
Proof (Lean source)
lemma projectionMeanAbs_le_sqrt_variance {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ)
(hX : MemLp X 2 μ) :
∫ x, |X x - ∫ y, X y ∂μ| ∂μ ≤
sqrt (variance X μ) := by
let Z : Ω → ℝ := fun x => X x - ∫ y, X y ∂μ
have hZ : MemLp Z 2 μ := hX.sub (memLp_const _)
have hOne : MemLp (fun _ : Ω => (1 : ℝ)) 2 μ := memLp_const _
have hh : (2 : ℝ).HolderConjugate 2 := by
rw [Real.holderConjugate_iff]
norm_num
have hZ' : MemLp Z (ofReal (2 : ℝ)) μ := by simpa using hZ
have hOne' : MemLp (fun _ : Ω => (1 : ℝ)) (ofReal (2 : ℝ)) μ := by
simpa using hOne
have h := integral_mul_norm_le_Lp_mul_Lq hh hZ' hOne'
simp only [norm_one, mul_one, integral_const, smul_eq_mul, mul_one,
Real.one_rpow, Real.norm_eq_abs] at h
rw [show μ.real univ = 1 from isProbabilityMeasure_iff_real.mp inferInstance,
Real.one_rpow, mul_one] at h
rw [ProbabilityTheory.variance_eq_integral hX.1.aemeasurable]
simpa [Z, sq_abs, Real.sqrt_eq_rpow, div_eq_mul_inv] using h
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma projectionMeanAbs_le_inv_two_sqrt {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ)
(N : ℕ) (hN : 0 < N) (hX : MemLp X 2 μ)
(hvar : variance X μ ≤ 1 / (4 * N)) :
∫ x, |X x - ∫ y, X y ∂μ| ∂μ ≤ 1 / (2 * sqrt N) := by
have hcs := projectionMeanAbs_le_sqrt_variance μ X hX
have hs := Real.sqrt_le_sqrt hvar
have hN' : (0 : ℝ) < N := by exact_mod_cast hN
have hsimp : sqrt (1 / (4 * (N : ℝ))) =
1 / (2 * sqrt N) := by
rw [Real.sqrt_div (by positivity), Real.sqrt_mul (by positivity)]
simp only [show sqrt (4 : ℝ) = 2 by
rw [show (4 : ℝ) = 2 ^ 2 by norm_num, Real.sqrt_sq (by norm_num)]]
simp only [Real.sqrt_one]
calc
∫ x, |X x - ∫ y, X y ∂μ| ∂μ ≤
sqrt (variance X μ) := hcs
_ ≤ sqrt (1 / (4 * N)) := hs
_ = 1 / (2 * sqrt N) := hsimp
theorem meanAbs_trialCellFrequency_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
J n :
ℕ
μ :
hn :
0 < n
a :
r :
B :
hB :
conclusion
∫ x, |trialCellFrequency a r B x
- μ.real {z | z.1 = r ∧ z.2.1 = a ∧ z.2.2 ∈ B}| ∂Measure.pi (fun _ : Fin n => μ)
≤ 1 / (2 * sqrt n)
Proof (Lean source)
lemma meanAbs_trialCellFrequency_pi {J n : ℕ}
(μ : Measure (Observation J)) [IsProbabilityMeasure μ]
(hn : 0 < n) (a : ArmSpace) (r : LabelSpace J)
(B : Set OutcomeSpace) (hB : MeasurableSet B) :
∫ x, |trialCellFrequency a r B x -
μ.real {z | z.1 = r ∧ z.2.1 = a ∧ z.2.2 ∈ B}|
∂Measure.pi (fun _ : Fin n => μ) ≤ 1 / (2 * sqrt n) := by
classical
let C : Set (Observation J) :=
{z | z.1 = r ∧ z.2.1 = a ∧ z.2.2 ∈ B}
have hC : MeasurableSet C := by
exact (measurableSet_eq_fun measurable_fst measurable_const).inter
((measurableSet_eq_fun (measurable_fst.comp measurable_snd) measurable_const).inter
(hB.preimage (measurable_snd.comp measurable_snd)))
have hcoord (i : Fin n) :
MemLp (fun x : TrialSample n J => if x i ∈ C then (1 : ℝ) else 0) 2
(Measure.pi fun _ : Fin n => μ) := by
have hm : Measurable (fun x : TrialSample n J =>
if x i ∈ C then (1 : ℝ) else 0) :=
Measurable.ite (hC.preimage (measurable_pi_apply i)) measurable_const measurable_const
have hb : ∀ x : TrialSample n J,
(if x i ∈ C then (1 : ℝ) else 0) ∈ Icc (0 : ℝ) 1 := by
intro x
by_cases hx : x i ∈ C <;> simp [hx]
exact memLp_of_bounded (Filter.Eventually.of_forall hb) hm.aestronglyMeasurable 2
have hX : MemLp (trialCellFrequency a r B) 2
(Measure.pi fun _ : Fin n => μ) := by
have hs := memLp_finsetSum univ (fun i _ => hcoord i)
have hs' := hs.const_mul (n : ℝ)⁻¹
unfold trialCellFrequency
simpa only [C, Set.mem_ofPred_eq] using hs'
rw [← integral_trialCellFrequency_pi μ hn a r B hB]
exact projectionMeanAbs_le_inv_two_sqrt
(Measure.pi fun _ : Fin n => μ) (trialCellFrequency a r B) n hn hX
(variance_trialCellFrequency_pi μ hn a r B hB)
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionEmpiricalFinite 2 declarations Finite mass of the empirical submeasures used in the projection.
Finite mass of the empirical submeasures used in the projection.
theorem empiricalTrialCells_finite
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(empiricalTrialCells x a r) univ < ⊤
Proof (Lean source)
lemma empiricalTrialCells_finite {J n : ℕ} (x : TrialSample n J)
(a : ArmSpace) (r : LabelSpace J) :
(empiricalTrialCells x a r) univ < ⊤ := by
classical
by_cases hn : n = 0
· subst n
simp [empiricalTrialCells]
unfold empiricalTrialCells
simp only [Measure.smul_apply, Measure.coe_finsetSum, Finset.sum_apply]
apply ENNReal.mul_lt_top
· simp [Nat.pos_of_ne_zero hn]
· apply ENNReal.sum_lt_top.mpr
intro i _
split_ifs <;> simp
theorem empiricalScoreCells_finite
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(empiricalScoreCells g x a r) univ < ⊤
Proof (Lean source)
lemma empiricalScoreCells_finite {ε : ℝ} {J m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (x : Fin m → ScoreSpace ε)
(a : ArmSpace) (r : LabelSpace J) :
(empiricalScoreCells g x a r) univ < ⊤ := by
classical
unfold empiricalScoreCells
simp only [Measure.coe_finsetSum, Finset.sum_apply]
apply ENNReal.sum_lt_top.mpr
intro i _
split_ifs
· simp only [Measure.smul_apply, Measure.dirac_apply' _ MeasurableSet.univ,
indicator, Set.mem_univ, if_pos, Pi.one_apply, smul_eq_mul, mul_one]
exact ENNReal.div_lt_top ENNReal.ofReal_ne_top (by
have hm : 0 < m := lt_of_le_of_lt (zero_le i.val) i.isLt
exact_mod_cast (Nat.ne_of_gt hm))
· simp
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionEmpiricalMean 4 declarations Expected absolute coordinate errors for the empirical projection.
Expected absolute coordinate errors for the empirical projection.
theorem meanAbs_scoreCellFrequency_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J m :
ℕ
hOverlap :
Overlap ε
H :
g :
ScoreSpace ε → LabelSpace J
hg :
hm :
0 < m
a :
r :
B :
Set (ScoreSpace ε)
hB :
conclusion
∫ x, |scoreCellFrequency g a r B x
- ∫ e, {e : ScoreSpace ε | g e = r ∧ e ∈ B}.indicator (armProb a) e ∂H| ∂Measure.pi (fun _ : Fin m => H)
≤ 1 / (2 * sqrt m)
Proof (Lean source)
lemma meanAbs_scoreCellFrequency_pi {ε : ℝ} {J m : ℕ}
(hOverlap : Overlap ε) (H : Measure (ScoreSpace ε))
[IsProbabilityMeasure H] (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (hm : 0 < m) (a : ArmSpace) (r : LabelSpace J)
(B : Set (ScoreSpace ε)) (hB : MeasurableSet B) :
∫ x, |scoreCellFrequency g a r B x -
∫ e, {e : ScoreSpace ε | g e = r ∧ e ∈ B}.indicator (armProb a) e ∂H|
∂Measure.pi (fun _ : Fin m => H) ≤ 1 / (2 * sqrt m) := by
let C : Set (ScoreSpace ε) := {e | g e = r ∧ e ∈ B}
let F : ScoreSpace ε → ℝ := C.indicator (armProb a)
have hC : MeasurableSet C :=
(measurableSet_eq_fun hg measurable_const).inter hB
have hp : Measurable (armProb a : ScoreSpace ε → ℝ) := by
unfold armProb
cases a <;> simp only [Bool.false_eq_true, ↓reduceIte] <;> fun_prop
have hF : Measurable F := hp.indicator hC
have hbound : ∀ e, F e ∈ Icc (0 : ℝ) 1 := by
intro e
by_cases he : e ∈ C
· simpa [F, indicator, he] using
(show armProb a e ∈ Icc (0 : ℝ) 1 from
⟨le_trans (le_of_lt hOverlap.1) (projectionArmProb_bounds hOverlap a e).1,
(projectionArmProb_bounds hOverlap a e).2⟩)
· simp [F, he]
have hlp : MemLp F 2 H :=
memLp_of_bounded (Filter.Eventually.of_forall hbound) hF.aestronglyMeasurable 2
have hcoord (j : Fin m) :
MemLp (fun x : Fin m → ScoreSpace ε => F (x j)) 2
(Measure.pi fun _ : Fin m => H) := by
exact hlp.comp_measurePreserving (measurePreserving_eval (fun _ : Fin m => H) j)
have hX : MemLp (scoreCellFrequency g a r B) 2
(Measure.pi fun _ : Fin m => H) := by
have hs := memLp_finsetSum univ (fun j _ => hcoord j)
have hs' := hs.const_mul (m : ℝ)⁻¹
unfold scoreCellFrequency
simpa only [F, C] using hs'
rw [← integral_scoreCellFrequency_pi hOverlap H g hg hm a r B hB]
exact projectionMeanAbs_le_inv_two_sqrt
(Measure.pi fun _ : Fin m => H) (scoreCellFrequency g a r B) m hm hX
(variance_scoreCellFrequency_pi hOverlap H g hg hm a r B hB)
theorem meanAbs_empiricalScoreCells_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J m :
ℕ
hOverlap :
Overlap ε
H :
g :
ScoreSpace ε → LabelSpace J
hg :
hm :
0 < m
a :
r :
B :
Set (ScoreSpace ε)
hB :
conclusion
∫ x, |(empiricalScoreCells g x a r).real B
- ∫ e, {e : ScoreSpace ε | g e = r ∧ e ∈ B}.indicator (armProb a) e ∂H| ∂Measure.pi (fun _ : Fin m => H)
≤ 1 / (2 * sqrt m)
Proof (Lean source)
lemma meanAbs_empiricalScoreCells_pi {ε : ℝ} {J m : ℕ}
(hOverlap : Overlap ε) (H : Measure (ScoreSpace ε))
[IsProbabilityMeasure H] (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (hm : 0 < m) (a : ArmSpace) (r : LabelSpace J)
(B : Set (ScoreSpace ε)) (hB : MeasurableSet B) :
∫ x, |(empiricalScoreCells g x a r).real B -
∫ e, {e : ScoreSpace ε | g e = r ∧ e ∈ B}.indicator (armProb a) e ∂H|
∂Measure.pi (fun _ : Fin m => H) ≤ 1 / (2 * sqrt m) := by
simpa only [empiricalScoreCells_real_eq_scoreCellFrequency hOverlap g a r B hB] using
meanAbs_scoreCellFrequency_pi hOverlap H g hg hm a r B hB
theorem sum_meanAbs_trialCellFrequency_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(∑ a : ArmSpace, ∑ r : LabelSpace J, ∫ x, |trialCellFrequency a r (B a r) x - μ.real {z | z.1 = r ∧ z.2.1 = a ∧ z.2.2 ∈ B a r}| ∂Measure.pi (fun _ : Fin n => μ))
≤ J / sqrt n
Proof (Lean source)
lemma sum_meanAbs_trialCellFrequency_pi {J n : ℕ}
(μ : Measure (Observation J)) [IsProbabilityMeasure μ] (hn : 0 < n)
(B : ArmSpace → LabelSpace J → Set OutcomeSpace)
(hB : ∀ a r, MeasurableSet (B a r)) :
(∑ a : ArmSpace, ∑ r : LabelSpace J,
∫ x, |trialCellFrequency a r (B a r) x -
μ.real {z | z.1 = r ∧ z.2.1 = a ∧ z.2.2 ∈ B a r}|
∂Measure.pi (fun _ : Fin n => μ)) ≤ J / sqrt n := by
calc
_ ≤ ∑ a : ArmSpace, ∑ _r : LabelSpace J,
(1 / (2 * sqrt n) : ℝ) := by
apply Finset.sum_le_sum
intro a _
apply Finset.sum_le_sum
intro r _
exact meanAbs_trialCellFrequency_pi μ hn a r (B a r) (hB a r)
_ = J / sqrt n := by
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin,
Fintype.card_bool, nsmul_eq_mul]
field_simp
ring
theorem sum_meanAbs_empiricalScoreCells_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J m :
ℕ
hOverlap :
Overlap ε
H :
g :
ScoreSpace ε → LabelSpace J
hg :
hm :
0 < m
B :
hB :
∀ a r, MeasurableSet (B a r)
conclusion
(∑ a : ArmSpace, ∑ r : LabelSpace J, ∫ x, |(empiricalScoreCells g x a r).real (B a r) - ∫ e, {e : ScoreSpace ε | g e = r ∧ e ∈ B a r}.indicator (armProb a) e ∂H| ∂Measure.pi (fun _ : Fin m => H))
≤ J / sqrt m
Proof (Lean source)
lemma sum_meanAbs_empiricalScoreCells_pi {ε : ℝ} {J m : ℕ}
(hOverlap : Overlap ε) (H : Measure (ScoreSpace ε))
[IsProbabilityMeasure H] (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (hm : 0 < m)
(B : ArmSpace → LabelSpace J → Set (ScoreSpace ε))
(hB : ∀ a r, MeasurableSet (B a r)) :
(∑ a : ArmSpace, ∑ r : LabelSpace J,
∫ x, |(empiricalScoreCells g x a r).real (B a r) -
∫ e, {e : ScoreSpace ε | g e = r ∧ e ∈ B a r}.indicator
(armProb a) e ∂H|
∂Measure.pi (fun _ : Fin m => H)) ≤ J / sqrt m := by
calc
_ ≤ ∑ a : ArmSpace, ∑ _r : LabelSpace J,
(1 / (2 * sqrt m) : ℝ) := by
apply Finset.sum_le_sum
intro a _
apply Finset.sum_le_sum
intro r _
exact meanAbs_empiricalScoreCells_pi hOverlap H g hg hm a r (B a r) (hB a r)
_ = J / sqrt m := by
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin,
Fintype.card_bool, nsmul_eq_mul]
field_simp
ring
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionEndpointAlgebra 7 declarations Endpoint algebra for the external projection candidate envelope.
Endpoint algebra for the external projection candidate envelope.
theorem projectedArmEndpoint_abs_le_of_cellBounds
Given the endpoint arrays, arm, endpoint direction, bound, and cellwise endpoint differences, this result bounds the absolute difference of the corresponding projected arm endpoints.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
lam lam' :
σ σ' :
ArmCellArray J (ScoreSpace ε)
a :
upper :
L :
ℝ
hcell :
∀ r : LabelSpace J,
|(let q := (lam a r).real univ if 0 < q then q * ∫ u in (0 : ℝ)..1, quantile (((lam a r univ)⁻¹ • (lam a r)).map (fun y : OutcomeSpace => (y : ℝ))) u * quantile (((σ a r univ)⁻¹ • (σ a r)).map (fun e => (armProb a e)⁻¹)) (if upper then u else 1 - u) else 0)
- (let q := (lam' a r).real univ if 0 < q then q * ∫ u in (0 : ℝ)..1, quantile (((lam' a r univ)⁻¹ • (lam' a r)).map (fun y : OutcomeSpace => (y : ℝ))) u * quantile (((σ' a r univ)⁻¹ • (σ' a r)).map (fun e => (armProb a e)⁻¹)) (if upper then u else 1 - u) else 0)|
≤ L * (outcomeCDFDistance (lam a r) (lam' a r) + scoreCDFDistance (σ a r) (σ' a r))
conclusion
|projectedArmEndpoint lam σ a upper - projectedArmEndpoint lam' σ' a upper|
≤ L * ((∑ r : LabelSpace J, outcomeCDFDistance (lam a r) (lam' a r)) + ∑ r : LabelSpace J, scoreCDFDistance (σ a r) (σ' a r))
Proof (Lean source)
lemma projectedArmEndpoint_abs_le_of_cellBounds
{ε : ℝ} {J : ℕ}
(lam lam' : ArmCellArray J OutcomeSpace)
(σ σ' : ArmCellArray J (ScoreSpace ε))
(a : ArmSpace) (upper : Bool) (L : ℝ)
(hcell : ∀ r : LabelSpace J,
|(let q := (lam a r).real univ
if 0 < q then
q * ∫ u in (0 : ℝ)..1,
quantile
(((lam a r univ)⁻¹ • (lam a r)).map
(fun y : OutcomeSpace => (y : ℝ))) u *
quantile
(((σ a r univ)⁻¹ • (σ a r)).map
(fun e => (armProb a e)⁻¹)) (if upper then u else 1 - u)
else 0) -
(let q := (lam' a r).real univ
if 0 < q then
q * ∫ u in (0 : ℝ)..1,
quantile
(((lam' a r univ)⁻¹ • (lam' a r)).map
(fun y : OutcomeSpace => (y : ℝ))) u *
quantile
(((σ' a r univ)⁻¹ • (σ' a r)).map
(fun e => (armProb a e)⁻¹)) (if upper then u else 1 - u)
else 0)| ≤
L * (outcomeCDFDistance (lam a r) (lam' a r) +
scoreCDFDistance (σ a r) (σ' a r))) :
|projectedArmEndpoint lam σ a upper -
projectedArmEndpoint lam' σ' a upper| ≤
L * ((∑ r : LabelSpace J,
outcomeCDFDistance (lam a r) (lam' a r)) +
∑ r : LabelSpace J,
scoreCDFDistance (σ a r) (σ' a r)) := by
unfold projectedArmEndpoint
rw [← Finset.sum_sub_distrib]
calc
_ ≤ ∑ r : LabelSpace J,
|(let q := (lam a r).real univ
if 0 < q then
q * ∫ u in (0 : ℝ)..1,
quantile
(((lam a r univ)⁻¹ • (lam a r)).map
(fun y : OutcomeSpace => (y : ℝ))) u *
quantile
(((σ a r univ)⁻¹ • (σ a r)).map
(fun e => (armProb a e)⁻¹)) (if upper then u else 1 - u)
else 0) -
(let q := (lam' a r).real univ
if 0 < q then
q * ∫ u in (0 : ℝ)..1,
quantile
(((lam' a r univ)⁻¹ • (lam' a r)).map
(fun y : OutcomeSpace => (y : ℝ))) u *
quantile
(((σ' a r univ)⁻¹ • (σ' a r)).map
(fun e => (armProb a e)⁻¹)) (if upper then u else 1 - u)
else 0)| := Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ r : LabelSpace J,
L * (outcomeCDFDistance (lam a r) (lam' a r) +
scoreCDFDistance (σ a r) (σ' a r)) :=
Finset.sum_le_sum (fun r _ => hcell r)
_ = _ := by simp only [← Finset.mul_sum, Finset.sum_add_distrib]
theorem projectionCandidate_populationDistance_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
x :
ExternalSample ε J n m
b :
lam :
σ :
ArmCellArray J (ScoreSpace ε)
hb :
b ∈ projectionCandidates g α x
htrial :
∀ a r,
outcomeCDFDistance (b.1 a r) (lam a r)
≤ outcomeCDFDistance (b.1 a r) (empiricalTrialCells x.1 a r)
+ outcomeCDFDistance (empiricalTrialCells x.1 a r) (lam a r)
hscore :
∀ a r,
scoreCDFDistance (b.2 a r) (σ a r)
≤ scoreCDFDistance (b.2 a r) (empiricalScoreCells g x.2 a r)
+ scoreCDFDistance (empiricalScoreCells g x.2 a r) (σ a r)
Proof (Lean source)
lemma projectionCandidate_populationDistance_le
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(x : ExternalSample ε J n m)
(b : ArmCellArray J OutcomeSpace × ArmCellArray J (ScoreSpace ε))
(lam : ArmCellArray J OutcomeSpace)
(σ : ArmCellArray J (ScoreSpace ε))
(hb : b ∈ projectionCandidates g α x)
(htrial : ∀ a r,
outcomeCDFDistance (b.1 a r) (lam a r) ≤
outcomeCDFDistance (b.1 a r) (empiricalTrialCells x.1 a r) +
outcomeCDFDistance (empiricalTrialCells x.1 a r) (lam a r))
(hscore : ∀ a r,
scoreCDFDistance (b.2 a r) (σ a r) ≤
scoreCDFDistance (b.2 a r) (empiricalScoreCells g x.2 a r) +
scoreCDFDistance (empiricalScoreCells g x.2 a r) (σ a r)) :
(∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (b.1 a r) (lam a r)) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (b.2 a r) (σ a r)) ≤
4 * J / (α * sqrt n) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (empiricalTrialCells x.1 a r) (lam a r)) +
2 * J * (2 - 2 * ε) / (α * sqrt m) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (empiricalScoreCells g x.2 a r) (σ a r)) := by
have hbt := hb.2.2.1
have hbs := hb.2.2.2
have ht : (∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (b.1 a r) (lam a r)) ≤
4 * J / (α * sqrt n) +
∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (empiricalTrialCells x.1 a r) (lam a r) := by
calc
_ ≤ ∑ a : ArmSpace, ∑ r : LabelSpace J,
(outcomeCDFDistance (b.1 a r) (empiricalTrialCells x.1 a r) +
outcomeCDFDistance (empiricalTrialCells x.1 a r) (lam a r)) := by
apply Finset.sum_le_sum
intro a _
apply Finset.sum_le_sum
intro r _
exact htrial a r
_ = (∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (b.1 a r) (empiricalTrialCells x.1 a r)) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (empiricalTrialCells x.1 a r) (lam a r)) := by
simp only [Finset.sum_add_distrib]
_ ≤ _ := add_le_add hbt le_rfl
have hs : (∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (b.2 a r) (σ a r)) ≤
2 * J * (2 - 2 * ε) / (α * sqrt m) +
∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (empiricalScoreCells g x.2 a r) (σ a r) := by
calc
_ ≤ ∑ a : ArmSpace, ∑ r : LabelSpace J,
(scoreCDFDistance (b.2 a r) (empiricalScoreCells g x.2 a r) +
scoreCDFDistance (empiricalScoreCells g x.2 a r) (σ a r)) := by
apply Finset.sum_le_sum
intro a _
apply Finset.sum_le_sum
intro r _
exact hscore a r
_ = (∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (b.2 a r) (empiricalScoreCells g x.2 a r)) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (empiricalScoreCells g x.2 a r) (σ a r)) := by
simp only [Finset.sum_add_distrib]
_ ≤ _ := add_le_add hbs le_rfl
linarith
theorem projectedATEInterval_subset_of_armEndpoint_bounds
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
projectedATEInterval lam σ ⊆ Icc ((projectedArmEndpoint lam' σ' true false - projectedArmEndpoint lam' σ' false true) - 2 * δ) ((projectedArmEndpoint lam' σ' true true - projectedArmEndpoint lam' σ' false false) + 2 * δ)
Proof (Lean source)
lemma projectedATEInterval_subset_of_armEndpoint_bounds
{ε : ℝ} {J : ℕ}
(lam lam' : ArmCellArray J OutcomeSpace)
(σ σ' : ArmCellArray J (ScoreSpace ε))
{δ : ℝ}
(h : ∀ a upper,
|projectedArmEndpoint lam σ a upper -
projectedArmEndpoint lam' σ' a upper| ≤ δ) :
projectedATEInterval lam σ ⊆
Icc
((projectedArmEndpoint lam' σ' true false -
projectedArmEndpoint lam' σ' false true) - 2 * δ)
((projectedArmEndpoint lam' σ' true true -
projectedArmEndpoint lam' σ' false false) + 2 * δ) := by
intro z hz
change projectedArmEndpoint lam σ true false -
projectedArmEndpoint lam σ false true ≤ z ∧
z ≤ projectedArmEndpoint lam σ true true -
projectedArmEndpoint lam σ false false at hz
have h₁ := (abs_le.mp (h true false))
have h₂ := (abs_le.mp (h false true))
have h₃ := (abs_le.mp (h true true))
have h₄ := (abs_le.mp (h false false))
change _ ≤ z ∧ z ≤ _
constructor <;> linarith
theorem externalProjectionCI_excessLength_le_of_armEndpoint_bounds Lemma externalProjectionCI_excessLength_le_of_armEndpoint_bounds in the paper ↗
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
x :
ExternalSample ε J n m
lam :
σ :
ArmCellArray J (ScoreSpace ε)
δ :
ℝ
hδ :
0 ≤ δ
hLU :
hclip :
(sievedProjectionCandidates g α x).Nonempty → (Icc (-1 : ℝ) 1 ∩ closure (convexHull ℝ (⋃ b ∈ sievedProjectionCandidates g α x, projectedATEInterval b.1 b.2))).Nonempty
hendpoint :
∀ b ∈ sievedProjectionCandidates g α x,
conclusion
∃ lo hi : ℝ,
conclusion 1
sievedTotalizedProjectionCI g α x = Icc lo hi
conclusion 2
max 0 ((hi - lo) - ((projectedArmEndpoint lam σ true true - projectedArmEndpoint lam σ false false) - (projectedArmEndpoint lam σ true false - projectedArmEndpoint lam σ false true)))
≤ 4 * δ
Proof (Lean source)
lemma externalProjectionCI_excessLength_le_of_armEndpoint_bounds
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(x : ExternalSample ε J n m)
(lam : ArmCellArray J OutcomeSpace)
(σ : ArmCellArray J (ScoreSpace ε)) {δ : ℝ}
(hδ : 0 ≤ δ)
(hLU : projectedArmEndpoint lam σ true false -
projectedArmEndpoint lam σ false true ≤
projectedArmEndpoint lam σ true true -
projectedArmEndpoint lam σ false false)
(hclip : (sievedProjectionCandidates g α x).Nonempty →
(Icc (-1 : ℝ) 1 ∩
closure (convexHull ℝ
(⋃ b ∈ sievedProjectionCandidates g α x,
projectedATEInterval b.1 b.2))).Nonempty)
(hendpoint : ∀ b ∈ sievedProjectionCandidates g α x, ∀ a upper,
|projectedArmEndpoint b.1 b.2 a upper -
projectedArmEndpoint lam σ a upper| ≤ δ) :
∃ lo hi : ℝ, sievedTotalizedProjectionCI g α x = Icc lo hi ∧
max 0 ((hi - lo) -
((projectedArmEndpoint lam σ true true -
projectedArmEndpoint lam σ false false) -
(projectedArmEndpoint lam σ true false -
projectedArmEndpoint lam σ false true))) ≤ 4 * δ := by
have hEach : ∀ b ∈ sievedProjectionCandidates g α x,
projectedATEInterval b.1 b.2 ⊆
Icc
((projectedArmEndpoint lam σ true false -
projectedArmEndpoint lam σ false true) - 2 * δ)
((projectedArmEndpoint lam σ true true -
projectedArmEndpoint lam σ false false) + 2 * δ) := by
intro b hb
exact projectedATEInterval_subset_of_armEndpoint_bounds
b.1 lam b.2 σ (hendpoint b hb)
have h := externalProjectionCI_excessLength_le_of_candidate_envelopes
g α x hLU (by linarith : 0 ≤ 2 * δ) hclip hEach
obtain ⟨lo, hi, heq, hbound⟩ := h
refine ⟨lo, hi, heq, ?_⟩
simpa only [show 2 * (2 * δ) = 4 * δ by ring] using hbound
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma outcomeCDFDistance_nonneg (μ ν : Measure OutcomeSpace) :
0 ≤ outcomeCDFDistance μ ν := by
unfold outcomeCDFDistance
apply add_nonneg (abs_nonneg _)
apply intervalIntegral.integral_nonneg (by norm_num)
intro t _
exact abs_nonneg _
theorem scoreCDFDistance_nonneg
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
0 ≤ scoreCDFDistance μ ν
Proof (Lean source)
lemma scoreCDFDistance_nonneg {ε : ℝ} (hOverlap : Overlap ε)
(μ ν : Measure (ScoreSpace ε)) :
0 ≤ scoreCDFDistance μ ν := by
unfold scoreCDFDistance
apply add_nonneg (abs_nonneg _)
apply intervalIntegral.integral_nonneg (by linarith [hOverlap.2])
intro t _
exact abs_nonneg _
theorem projectedArmEndpoint_abs_le_of_armBounds
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
hOverlap :
Overlap ε
lam lam' :
σ σ' :
ArmCellArray J (ScoreSpace ε)
L :
ℝ
hL :
0 ≤ L
hArm :
∀ a upper,
|projectedArmEndpoint lam σ a upper - projectedArmEndpoint lam' σ' a upper|
≤ L * ((∑ r : LabelSpace J, outcomeCDFDistance (lam a r) (lam' a r)) + ∑ r : LabelSpace J, scoreCDFDistance (σ a r) (σ' a r))
a :
upper :
conclusion
|projectedArmEndpoint lam σ a upper - projectedArmEndpoint lam' σ' a upper|
≤ L * ((∑ a : ArmSpace, ∑ r : LabelSpace J, outcomeCDFDistance (lam a r) (lam' a r)) + ∑ a : ArmSpace, ∑ r : LabelSpace J, scoreCDFDistance (σ a r) (σ' a r))
Proof (Lean source)
lemma projectedArmEndpoint_abs_le_of_armBounds
{ε : ℝ} {J : ℕ} (hOverlap : Overlap ε)
(lam lam' : ArmCellArray J OutcomeSpace)
(σ σ' : ArmCellArray J (ScoreSpace ε))
(L : ℝ) (hL : 0 ≤ L)
(hArm : ∀ a upper,
|projectedArmEndpoint lam σ a upper -
projectedArmEndpoint lam' σ' a upper| ≤
L * ((∑ r : LabelSpace J,
outcomeCDFDistance (lam a r) (lam' a r)) +
∑ r : LabelSpace J,
scoreCDFDistance (σ a r) (σ' a r)))
(a : ArmSpace) (upper : Bool) :
|projectedArmEndpoint lam σ a upper -
projectedArmEndpoint lam' σ' a upper| ≤
L * ((∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (lam a r) (lam' a r)) +
∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (σ a r) (σ' a r)) := by
have ht (a' : ArmSpace) :
0 ≤ ∑ r : LabelSpace J,
outcomeCDFDistance (lam a' r) (lam' a' r) :=
Finset.sum_nonneg (fun r _ => outcomeCDFDistance_nonneg _ _)
have hs (a' : ArmSpace) :
0 ≤ ∑ r : LabelSpace J,
scoreCDFDistance (σ a' r) (σ' a' r) :=
Finset.sum_nonneg (fun r _ => scoreCDFDistance_nonneg hOverlap _ _)
have htAll :
(∑ r : LabelSpace J,
outcomeCDFDistance (lam a r) (lam' a r)) ≤
∑ a' : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (lam a' r) (lam' a' r) := by
cases a <;> simp only [Fintype.sum_bool] <;> linarith [ht true, ht false]
have hsAll :
(∑ r : LabelSpace J,
scoreCDFDistance (σ a r) (σ' a r)) ≤
∑ a' : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (σ a' r) (σ' a' r) := by
cases a <;> simp only [Fintype.sum_bool] <;> linarith [hs true, hs false]
exact (hArm a upper).trans (mul_le_mul_of_nonneg_left
(add_le_add htAll hsAll) hL)
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionEndpointBounds 15 declarations Samplewise bounds and unconditional measurability for projection endpoints.
Samplewise bounds and unconditional measurability for projection endpoints.
The zero pair of cell arrays is a finite compatible reference. Candidate stability against this reference uniformly bounds every admitted arm endpoint, and hence both ATE endpoints, for each fixed external sample.
def projectionZeroReferenceRadius
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
projectionZeroReferenceRadius g α x :
ℝ
given by
(ε⁻¹ + (ε ^ 2)⁻¹) * (4 * J / (α * sqrt n) + (∑ a : ArmSpace, ∑ r : LabelSpace J, outcomeCDFDistance (empiricalTrialCells x.1 a r) 0) + 2 * J * (2 - 2 * ε) / (α * sqrt m) + (∑ a : ArmSpace, ∑ r : LabelSpace J, scoreCDFDistance (empiricalScoreCells g x.2 a r) 0))
theorem projectedArmEndpoint_zero
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
projectedArmEndpoint (J := J) (fun _ : ArmSpace => fun _ : LabelSpace J => (0 : Measure OutcomeSpace)) (fun _ : ArmSpace => fun _ : LabelSpace J => (0 : Measure (ScoreSpace ε))) a upper
= 0
Proof (Lean source)
lemma projectedArmEndpoint_zero {ε : ℝ} {J : ℕ}
(a : ArmSpace) (upper : Bool) :
projectedArmEndpoint (J := J)
(fun _ : ArmSpace => fun _ : LabelSpace J => (0 : Measure OutcomeSpace))
(fun _ : ArmSpace => fun _ : LabelSpace J =>
(0 : Measure (ScoreSpace ε))) a upper = 0 := by
simp [projectedArmEndpoint, Measure.real]
theorem projectionCandidate_armEndpoint_abs_le_zeroReference
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
hOverlap :
Overlap ε
x :
ExternalSample ε J n m
b :
hb :
b ∈ projectionCandidates g α x
a :
upper :
conclusion
|projectedArmEndpoint b.1 b.2 a upper| ≤ projectionZeroReferenceRadius g α x
Proof (Lean source)
lemma projectionCandidate_armEndpoint_abs_le_zeroReference
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(hOverlap : Overlap ε) (x : ExternalSample ε J n m)
(b : ArmCellArray J OutcomeSpace × ArmCellArray J (ScoreSpace ε))
(hb : b ∈ projectionCandidates g α x)
(a : ArmSpace) (upper : Bool) :
|projectedArmEndpoint b.1 b.2 a upper| ≤
projectionZeroReferenceRadius g α x := by
let lam : ArmCellArray J OutcomeSpace := fun _ _ => 0
let σ : ArmCellArray J (ScoreSpace ε) := fun _ _ => 0
have hfinLam : ∀ a r, IsFiniteMeasure (lam a r) := by
intro a r
dsimp [lam]
infer_instance
have hfinσ : ∀ a r, IsFiniteMeasure (σ a r) := by
intro a r
dsimp [σ]
infer_instance
have hcompat : ProjectionCompatible lam σ := by
intro a r
simp [lam, σ, Measure.real]
have h := projectionCandidate_armEndpoint_abs_le g α hOverlap x b lam σ
hb hfinLam hfinσ hcompat a upper
rw [show projectedArmEndpoint lam σ a upper = 0 by
exact projectedArmEndpoint_zero a upper] at h
simpa only [sub_zero, projectionZeroReferenceRadius, lam, σ] using h
theorem projectionCandidate_ATEEndpoints_abs_le_zeroReference
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
hOverlap :
Overlap ε
x :
ExternalSample ε J n m
b :
hb :
b ∈ projectionCandidates g α x
conclusion
conclusion 1
≤ 2 * projectionZeroReferenceRadius g α x
conclusion 2
≤ 2 * projectionZeroReferenceRadius g α x
Proof (Lean source)
lemma projectionCandidate_ATEEndpoints_abs_le_zeroReference
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(hOverlap : Overlap ε) (x : ExternalSample ε J n m)
(b : ArmCellArray J OutcomeSpace × ArmCellArray J (ScoreSpace ε))
(hb : b ∈ projectionCandidates g α x) :
|projectedArmEndpoint b.1 b.2 true false -
projectedArmEndpoint b.1 b.2 false true| ≤
2 * projectionZeroReferenceRadius g α x ∧
|projectedArmEndpoint b.1 b.2 true true -
projectedArmEndpoint b.1 b.2 false false| ≤
2 * projectionZeroReferenceRadius g α x := by
have hbound (a : ArmSpace) (upper : Bool) :=
projectionCandidate_armEndpoint_abs_le_zeroReference
g α hOverlap x b hb a upper
constructor
· exact (abs_sub _ _).trans (by linarith [hbound true false, hbound false true])
· exact (abs_sub _ _).trans (by linarith [hbound true true, hbound false false])
theorem bddBelow_projectedATEIntervalLower_active
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
hOverlap :
Overlap ε
x :
ExternalSample ε J n m
conclusion
Proof (Lean source)
lemma bddBelow_projectedATEIntervalLower_active
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(hOverlap : Overlap ε) (x : ExternalSample ε J n m) :
BddBelow (projectedATEIntervalLower ''
activeSievedProjectionCandidates g α x) := by
refine ⟨-2 * projectionZeroReferenceRadius g α x, ?_⟩
intro z hz
obtain ⟨b, hb, rfl⟩ := hz
rw [mem_activeSievedProjectionCandidates_iff] at hb
have hcand : b ∈ projectionCandidates g α x := hb.1.1
have habs := (projectionCandidate_ATEEndpoints_abs_le_zeroReference
g α hOverlap x b hcand).1
have hmem : projectedATEIntervalLower b ∈ projectedATEInterval b.1 b.2 := by
rw [projectedATEInterval_eq_Icc_endpoints]
exact ⟨le_rfl, hb.2⟩
change projectedATEIntervalLower b ∈ Icc
(projectedArmEndpoint b.1 b.2 true false -
projectedArmEndpoint b.1 b.2 false true)
(projectedArmEndpoint b.1 b.2 true true -
projectedArmEndpoint b.1 b.2 false false) at hmem
linarith [neg_le_of_abs_le habs, hmem.1]
theorem bddAbove_projectedATEIntervalUpper_active
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
hOverlap :
Overlap ε
x :
ExternalSample ε J n m
conclusion
Proof (Lean source)
lemma bddAbove_projectedATEIntervalUpper_active
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(hOverlap : Overlap ε) (x : ExternalSample ε J n m) :
BddAbove (projectedATEIntervalUpper ''
activeSievedProjectionCandidates g α x) := by
refine ⟨2 * projectionZeroReferenceRadius g α x, ?_⟩
intro z hz
obtain ⟨b, hb, rfl⟩ := hz
rw [mem_activeSievedProjectionCandidates_iff] at hb
have hcand : b ∈ projectionCandidates g α x := hb.1.1
have habs := (projectionCandidate_ATEEndpoints_abs_le_zeroReference
g α hOverlap x b hcand).2
have hmem : projectedATEIntervalUpper b ∈ projectedATEInterval b.1 b.2 := by
rw [projectedATEInterval_eq_Icc_endpoints]
exact ⟨hb.2, le_rfl⟩
change projectedATEIntervalUpper b ∈ Icc
(projectedArmEndpoint b.1 b.2 true false -
projectedArmEndpoint b.1 b.2 false true)
(projectedArmEndpoint b.1 b.2 true true -
projectedArmEndpoint b.1 b.2 false false) at hmem
linarith [le_abs_self (projectedArmEndpoint b.1 b.2 true true -
projectedArmEndpoint b.1 b.2 false false), hmem.2, habs]
theorem measurable_projectionActiveLowerInf
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Measurable (fun x : ExternalSample ε J n m => sInf (projectedATEIntervalLower '' activeSievedProjectionCandidates g α x))
Proof (Lean source)
@[fun_prop]
lemma measurable_projectionActiveLowerInf
{ε : ℝ} {J n m : ℕ}
(hOverlap : Overlap ε) (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (α : ℝ) :
Measurable (fun x : ExternalSample ε J n m =>
sInf (projectedATEIntervalLower ''
activeSievedProjectionCandidates g α x)) :=
measurable_sInf_activeSievedProjectionCandidates hOverlap g hg α
(bddBelow_projectedATEIntervalLower_active g α hOverlap)
theorem measurable_projectionActiveUpperSup
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Measurable (fun x : ExternalSample ε J n m => sSup (projectedATEIntervalUpper '' activeSievedProjectionCandidates g α x))
Proof (Lean source)
@[fun_prop]
lemma measurable_projectionActiveUpperSup
{ε : ℝ} {J n m : ℕ}
(hOverlap : Overlap ε) (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (α : ℝ) :
Measurable (fun x : ExternalSample ε J n m =>
sSup (projectedATEIntervalUpper ''
activeSievedProjectionCandidates g α x)) :=
measurable_sSup_activeSievedProjectionCandidates hOverlap g hg α
(bddAbove_projectedATEIntervalUpper_active g α hOverlap)
def externalProjectionLower
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
externalProjectionLower g α x :
ℝ
given by
by classical exact let A := activeSievedProjectionCandidates g α x let L := sInf (projectedATEIntervalLower '' A) let U := sSup (projectedATEIntervalUpper '' A) if A.Nonempty ∧
def externalProjectionUpper
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
externalProjectionUpper g α x :
ℝ
given by
by classical exact let A := activeSievedProjectionCandidates g α x let L := sInf (projectedATEIntervalLower '' A) let U := sSup (projectedATEIntervalUpper '' A) if A.Nonempty ∧
theorem measurable_externalProjectionLower
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Measurable (externalProjectionLower (n := n) (m := m) g α)
Proof (Lean source)
@[fun_prop]
lemma measurable_externalProjectionLower
{ε : ℝ} {J n m : ℕ}
(hOverlap : Overlap ε) (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (α : ℝ) :
Measurable (externalProjectionLower (n := n) (m := m) g α) := by
classical
have hL := measurable_projectionActiveLowerInf
(n := n) (m := m) hOverlap g hg α
have hU := measurable_projectionActiveUpperSup
(n := n) (m := m) hOverlap g hg α
have hN := measurableSet_activeSievedProjectionCandidates_nonempty
(n := n) (m := m) hOverlap g hg α
have hclip : MeasurableSet {x : ExternalSample ε J n m |
max (-1) (sInf (projectedATEIntervalLower ''
activeSievedProjectionCandidates g α x)) ≤
min 1 (sSup (projectedATEIntervalUpper ''
activeSievedProjectionCandidates g α x))} :=
measurableSet_le (measurable_const.max hL) (measurable_const.min hU)
change Measurable (fun x : ExternalSample ε J n m =>
if (activeSievedProjectionCandidates g α x).Nonempty ∧
max (-1) (sInf (projectedATEIntervalLower ''
activeSievedProjectionCandidates g α x)) ≤
min 1 (sSup (projectedATEIntervalUpper ''
activeSievedProjectionCandidates g α x)) then
max (-1) (sInf (projectedATEIntervalLower ''
activeSievedProjectionCandidates g α x)) else 0)
exact (measurable_const.max hL).ite (hN.inter hclip)
(measurable_const : Measurable (fun _ : ExternalSample ε J n m => (0 : ℝ)))
theorem measurable_externalProjectionUpper
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Measurable (externalProjectionUpper (n := n) (m := m) g α)
Proof (Lean source)
@[fun_prop]
lemma measurable_externalProjectionUpper
{ε : ℝ} {J n m : ℕ}
(hOverlap : Overlap ε) (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (α : ℝ) :
Measurable (externalProjectionUpper (n := n) (m := m) g α) := by
classical
have hL := measurable_projectionActiveLowerInf
(n := n) (m := m) hOverlap g hg α
have hU := measurable_projectionActiveUpperSup
(n := n) (m := m) hOverlap g hg α
have hN := measurableSet_activeSievedProjectionCandidates_nonempty
(n := n) (m := m) hOverlap g hg α
have hclip : MeasurableSet {x : ExternalSample ε J n m |
max (-1) (sInf (projectedATEIntervalLower ''
activeSievedProjectionCandidates g α x)) ≤
min 1 (sSup (projectedATEIntervalUpper ''
activeSievedProjectionCandidates g α x))} :=
measurableSet_le (measurable_const.max hL) (measurable_const.min hU)
change Measurable (fun x : ExternalSample ε J n m =>
if (activeSievedProjectionCandidates g α x).Nonempty ∧
max (-1) (sInf (projectedATEIntervalLower ''
activeSievedProjectionCandidates g α x)) ≤
min 1 (sSup (projectedATEIntervalUpper ''
activeSievedProjectionCandidates g α x)) then
min 1 (sSup (projectedATEIntervalUpper ''
activeSievedProjectionCandidates g α x)) else 0)
exact (measurable_const.min hU).ite (hN.inter hclip)
(measurable_const : Measurable (fun _ : ExternalSample ε J n m => (0 : ℝ)))
theorem externalProjectionCI_eq_selectedEndpoints
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
hOverlap :
Overlap ε
x :
ExternalSample ε J n m
conclusion
Proof (Lean source)
lemma externalProjectionCI_eq_selectedEndpoints
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(hOverlap : Overlap ε) (x : ExternalSample ε J n m) :
sievedTotalizedProjectionCI g α x = Icc
(externalProjectionLower g α x) (externalProjectionUpper g α x) := by
classical
by_cases hA : (activeSievedProjectionCandidates g α x).Nonempty
· rw [externalProjectionCI_eq_explicit_endpoints g α x hA
(bddBelow_projectedATEIntervalLower_active g α hOverlap x)
(bddAbove_projectedATEIntervalUpper_active g α hOverlap x)]
simp only [externalProjectionLower, externalProjectionUpper, hA, true_and]
split_ifs <;> simp
· have hunion :
(⋃ b ∈ sievedProjectionCandidates g α x,
projectedATEInterval b.1 b.2) = ∅ := by
ext z
constructor
· simp only [Set.mem_iUnion]
rintro ⟨b, hb, hz⟩
rw [projectedATEInterval_eq_Icc_endpoints] at hz
exact hA ⟨b, (mem_activeSievedProjectionCandidates_iff).2
⟨hb, hz.1.trans hz.2⟩⟩
· intro hz
simpa using hz
rw [sievedTotalizedProjectionCI]
simp only [hunion, convexHull_empty, closure_empty, Set.inter_empty,
Set.not_nonempty_empty, and_false, ↓reduceIte]
simp [externalProjectionLower, externalProjectionUpper, hA]
def externalProjectionIntervalProcedure
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
clause 1
lo := externalProjectionLower g α
clause 2
hi := externalProjectionUpper g α
clause 3
measurable_lo := measurable_externalProjectionLower hOverlap g hg α
clause 4
measurable_hi := measurable_externalProjectionUpper hOverlap g hg α
clause 5
lower_bound := by intro x have hCI := externalProjectionCI_eq_selectedEndpoints g α hOverlap x obtain ⟨z, hz⟩ := externalProjectionCI_nonempty g α x rw [hCI] at hz have hloMem : externalProjectionLower g α x ∈ sievedTotalizedProjectionCI g α x := by rw [hCI] exact ⟨le_rfl, hz.1.trans hz.2⟩ exact (externalProjectionCI_subset_Icc g α x hloMem).1
clause 6
ordered := by intro x have hCI := externalProjectionCI_eq_selectedEndpoints g α hOverlap x obtain ⟨z, hz⟩ := externalProjectionCI_nonempty g α x rw [hCI] at hz exact hz.1.trans hz.2
clause 7
upper_bound := by intro x have hCI := externalProjectionCI_eq_selectedEndpoints g α hOverlap x obtain ⟨z, hz⟩ := externalProjectionCI_nonempty g α x rw [hCI] at hz have hhiMem : externalProjectionUpper g α x ∈ sievedTotalizedProjectionCI g α x := by rw [hCI] exact ⟨hz.1.trans hz.2, le_rfl⟩ exact (externalProjectionCI_subset_Icc g α x hhiMem).2
theorem externalProjectionIntervalProcedure_Icc
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
hOverlap :
Overlap ε
hg :
x :
ExternalSample ε J n m
conclusion
Icc ((externalProjectionIntervalProcedure g α hOverlap hg).lo x) ((externalProjectionIntervalProcedure g α hOverlap hg).hi x)
= sievedTotalizedProjectionCI g α x
Proof (Lean source)
lemma externalProjectionIntervalProcedure_Icc
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(hOverlap : Overlap ε) (hg : Measurable g)
(x : ExternalSample ε J n m) :
Icc ((externalProjectionIntervalProcedure g α hOverlap hg).lo x)
((externalProjectionIntervalProcedure g α hOverlap hg).hi x) =
sievedTotalizedProjectionCI g α x := by
exact (externalProjectionCI_eq_selectedEndpoints g α hOverlap x).symm
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionEndpointGeometryMeasurable 11 declarations Closed convex hull geometry for unions of real intervals.
Closed convex hull geometry for unions of real intervals.
The main theorem identifies the closed convex hull of a nonempty bounded union of intervals with the interval between the infimum of its active lower endpoints and the supremum of its active upper endpoints. Empty intervals are discarded.
def activeIntervalIndices
For the specified mathematical inputs, this definition introduces the corresponding object.
def intervalFamilyUnion
For the specified mathematical inputs, this definition introduces the corresponding object.
theorem intervalFamilyUnion_eq_active
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma intervalFamilyUnion_eq_active {ι : Type*} (I : Set ι) (l u : ι → ℝ) :
intervalFamilyUnion I l u = intervalFamilyUnion (activeIntervalIndices I l u) l u := by
ext z
simp only [intervalFamilyUnion, activeIntervalIndices, Set.mem_iUnion,
Set.mem_Icc, Set.mem_sep_iff]
constructor
· rintro ⟨i, hi, hlz, hzu⟩
exact ⟨i, ⟨hi, hlz.trans hzu⟩, hlz, hzu⟩
· rintro ⟨i, ⟨hi, _⟩, hlz, hzu⟩
exact ⟨i, hi, hlz, hzu⟩
theorem closure_convexHull_intervalFamilyUnion_eq_Icc
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ι :
Type*
I :
Set ι
l u :
ι → ℝ
hne :
(activeIntervalIndices I l u).Nonempty
hbelow :
BddBelow (l '' activeIntervalIndices I l u)
habove :
BddAbove (u '' activeIntervalIndices I l u)
conclusion
Proof (Lean source)
lemma closure_convexHull_intervalFamilyUnion_eq_Icc {ι : Type*}
(I : Set ι) (l u : ι → ℝ)
(hne : (activeIntervalIndices I l u).Nonempty)
(hbelow : BddBelow (l '' activeIntervalIndices I l u))
(habove : BddAbove (u '' activeIntervalIndices I l u)) :
closure (convexHull ℝ (intervalFamilyUnion I l u)) =
Icc (sInf (l '' activeIntervalIndices I l u))
(sSup (u '' activeIntervalIndices I l u)) := by
let A := activeIntervalIndices I l u
let R := intervalFamilyUnion I l u
let L := sInf (l '' A)
let U := sSup (u '' A)
have hneL : (l '' A).Nonempty := hne.image l
have hneU : (u '' A).Nonempty := hne.image u
have hRactive : R = intervalFamilyUnion A l u :=
intervalFamilyUnion_eq_active I l u
have hRsub : R ⊆ Icc L U := by
rw [hRactive]
intro z hz
simp only [intervalFamilyUnion, Set.mem_iUnion, Set.mem_Icc] at hz
obtain ⟨i, hiA, hlz, hzu⟩ := hz
have hli : L ≤ l i := by
exact csInf_le hbelow ⟨i, hiA, rfl⟩
have hui : u i ≤ U := by
exact le_csSup habove ⟨i, hiA, rfl⟩
exact ⟨hli.trans hlz, hzu.trans hui⟩
have hforward : closure (convexHull ℝ R) ⊆ Icc L U :=
closure_minimal (convexHull_min hRsub (convex_Icc L U)) isClosed_Icc
have hLmem : L ∈ closure (convexHull ℝ R) := by
have hLclosure : L ∈ closure (l '' A) := csInf_mem_closure hneL hbelow
apply closure_mono _ hLclosure
intro z hz
obtain ⟨i, hiA, rfl⟩ := hz
apply subset_convexHull ℝ
rw [hRactive]
simp only [intervalFamilyUnion, Set.mem_iUnion, Set.mem_Icc]
exact ⟨i, hiA, le_rfl, hiA.2⟩
have hUmem : U ∈ closure (convexHull ℝ R) := by
have hUclosure : U ∈ closure (u '' A) := csSup_mem_closure hneU habove
apply closure_mono _ hUclosure
intro z hz
obtain ⟨i, hiA, rfl⟩ := hz
apply subset_convexHull ℝ
rw [hRactive]
simp only [intervalFamilyUnion, Set.mem_iUnion, Set.mem_Icc]
exact ⟨i, hiA, hiA.2, le_rfl⟩
have hLU : L ≤ U := by
obtain ⟨i, hiA⟩ := hne
exact (csInf_le hbelow ⟨i, hiA, rfl⟩).trans
(hiA.2.trans (le_csSup habove ⟨i, hiA, rfl⟩))
have hreverse : Icc L U ⊆ closure (convexHull ℝ R) := by
rw [← segment_eq_Icc hLU]
exact (convex_convexHull ℝ R).closure.segment_subset hLmem hUmem
exact Set.Subset.antisymm hforward hreverse
def activeSievedProjectionCandidates
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
activeSievedProjectionCandidates g α x :
theorem mem_activeSievedProjectionCandidates_iff
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
b ∈ activeSievedProjectionCandidates g α x ↔ b ∈ sievedProjectionCandidates g α x ∧
Proof (Lean source)
lemma mem_activeSievedProjectionCandidates_iff {ε : ℝ} {J n m : ℕ}
{g : ScoreSpace ε → LabelSpace J} {α : ℝ}
{x : ExternalSample ε J n m}
{b : ArmCellArray J OutcomeSpace × ArmCellArray J (ScoreSpace ε)} :
b ∈ activeSievedProjectionCandidates g α x ↔
b ∈ sievedProjectionCandidates g α x ∧
projectedATEIntervalLower b ≤ projectedATEIntervalUpper b := by
rfl
theorem closure_convexHull_sievedProjectionIntervals_eq_Icc
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
x :
ExternalSample ε J n m
hne :
(activeSievedProjectionCandidates g α x).Nonempty
hbelow :
habove :
conclusion
= Icc (sInf (projectedATEIntervalLower '' activeSievedProjectionCandidates g α x)) (sSup (projectedATEIntervalUpper '' activeSievedProjectionCandidates g α x))
Proof (Lean source)
lemma closure_convexHull_sievedProjectionIntervals_eq_Icc
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(x : ExternalSample ε J n m)
(hne : (activeSievedProjectionCandidates g α x).Nonempty)
(hbelow : BddBelow
(projectedATEIntervalLower '' activeSievedProjectionCandidates g α x))
(habove : BddAbove
(projectedATEIntervalUpper '' activeSievedProjectionCandidates g α x)) :
closure (convexHull ℝ
(⋃ b ∈ sievedProjectionCandidates g α x,
projectedATEInterval b.1 b.2)) =
Icc
(sInf (projectedATEIntervalLower ''
activeSievedProjectionCandidates g α x))
(sSup (projectedATEIntervalUpper ''
activeSievedProjectionCandidates g α x)) := by
have h := closure_convexHull_intervalFamilyUnion_eq_Icc
(sievedProjectionCandidates g α x)
projectedATEIntervalLower projectedATEIntervalUpper hne hbelow habove
simpa only [intervalFamilyUnion, activeSievedProjectionCandidates,
projectedATEInterval_eq_Icc_endpoints] using h
theorem externalProjectionCI_eq_explicit_endpoints
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
x :
ExternalSample ε J n m
hne :
(activeSievedProjectionCandidates g α x).Nonempty
hbelow :
habove :
conclusion
L :
U :
Proof (Lean source)
lemma externalProjectionCI_eq_explicit_endpoints
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(x : ExternalSample ε J n m)
(hne : (activeSievedProjectionCandidates g α x).Nonempty)
(hbelow : BddBelow
(projectedATEIntervalLower '' activeSievedProjectionCandidates g α x))
(habove : BddAbove
(projectedATEIntervalUpper '' activeSievedProjectionCandidates g α x)) :
let L := sInf (projectedATEIntervalLower ''
activeSievedProjectionCandidates g α x)
let U := sSup (projectedATEIntervalUpper ''
activeSievedProjectionCandidates g α x)
sievedTotalizedProjectionCI g α x =
if max (-1) L ≤ min 1 U then
Icc (max (-1) L) (min 1 U)
else {0} := by
dsimp only
have hraw := closure_convexHull_sievedProjectionIntervals_eq_Icc
g α x hne hbelow habove
have hB : (sievedProjectionCandidates g α x).Nonempty := by
obtain ⟨b, hb, _⟩ := hne
exact ⟨b, hb⟩
have hclip : Icc (-1 : ℝ) 1 ∩
Icc
(sInf (projectedATEIntervalLower ''
activeSievedProjectionCandidates g α x))
(sSup (projectedATEIntervalUpper ''
activeSievedProjectionCandidates g α x)) =
Icc
(max (-1) (sInf (projectedATEIntervalLower ''
activeSievedProjectionCandidates g α x)))
(min 1 (sSup (projectedATEIntervalUpper ''
activeSievedProjectionCandidates g α x))) := by
ext z
simp only [Set.mem_inter_iff, Set.mem_Icc]
constructor
· rintro ⟨⟨hzneg, hzone⟩, hzL, hzU⟩
exact ⟨max_le hzneg hzL, le_min hzone hzU⟩
· rintro ⟨hzL, hzU⟩
exact ⟨⟨le_trans (le_max_left _ _) hzL,
le_trans hzU (min_le_left _ _)⟩,
le_trans (le_max_right _ _) hzL,
le_trans hzU (min_le_right _ _)⟩
rw [sievedTotalizedProjectionCI]
rw [hraw, hclip]
simp only [hB, Set.nonempty_Icc, true_and]
theorem measurableSet_activeSievedProjectionCandidates_nonempty
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma measurableSet_activeSievedProjectionCandidates_nonempty
{ε : ℝ} {J n m : ℕ}
(hOverlap : Overlap ε) (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (α : ℝ) :
MeasurableSet {x : ExternalSample ε J n m |
(activeSievedProjectionCandidates g α x).Nonempty} := by
let S := {b ∈ rationalArraySieve ε J |
projectedATEIntervalLower b ≤ projectedATEIntervalUpper b}
have hS : S.Countable :=
(countable_rationalArraySieve ε J).mono (Set.sep_subset _ _)
have hmeas : MeasurableSet
(⋃ b ∈ S, {x : ExternalSample ε J n m |
b ∈ projectionCandidates g α x}) := by
exact MeasurableSet.biUnion hS fun b _ =>
measurableSet_mem_projectionCandidates hOverlap g hg α b
have heq : {x : ExternalSample ε J n m |
(activeSievedProjectionCandidates g α x).Nonempty} =
⋃ b ∈ S, {x | b ∈ projectionCandidates g α x} := by
ext x
simp only [Set.mem_ofPred_eq, Set.mem_iUnion, Set.nonempty_def,
activeSievedProjectionCandidates, activeIntervalIndices,
sievedProjectionCandidates, Set.mem_inter_iff, S]
aesop
rw [heq]
exact hmeas
theorem measurable_sInf_activeSievedProjectionCandidates
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
hOverlap :
Overlap ε
g :
ScoreSpace ε → LabelSpace J
hg :
α :
ℝ
hbelow :
∀ x : ExternalSample ε J n m,
conclusion
Measurable (fun x : ExternalSample ε J n m => sInf (projectedATEIntervalLower '' activeSievedProjectionCandidates g α x))
Proof (Lean source)
@[fun_prop]
lemma measurable_sInf_activeSievedProjectionCandidates
{ε : ℝ} {J n m : ℕ}
(hOverlap : Overlap ε) (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (α : ℝ)
(hbelow : ∀ x : ExternalSample ε J n m, BddBelow
(projectedATEIntervalLower '' activeSievedProjectionCandidates g α x)) :
Measurable (fun x : ExternalSample ε J n m =>
sInf (projectedATEIntervalLower ''
activeSievedProjectionCandidates g α x)) := by
apply measurable_of_Iio
intro q
let N : Set (ExternalSample ε J n m) :=
{x | (activeSievedProjectionCandidates g α x).Nonempty}
let E : Set (ExternalSample ε J n m) :=
{x | ∃ b ∈ sievedProjectionCandidates g α x,
projectedATEIntervalLower b ≤ projectedATEIntervalUpper b ∧
projectedATEIntervalLower b < q}
have hN : MeasurableSet N :=
measurableSet_activeSievedProjectionCandidates_nonempty hOverlap g hg α
have hE : MeasurableSet E :=
measurableSet_exists_sievedCandidate_lower_lt hOverlap g hg α q
have heq : (fun x : ExternalSample ε J n m =>
sInf (projectedATEIntervalLower ''
activeSievedProjectionCandidates g α x)) ⁻¹' Iio q =
(N ∩ E) ∪ (Nᶜ ∩ if 0 < q then univ else ∅) := by
ext x
by_cases hx : (activeSievedProjectionCandidates g α x).Nonempty
· have himage : (projectedATEIntervalLower ''
activeSievedProjectionCandidates g α x).Nonempty :=
hx.image projectedATEIntervalLower
rw [Set.mem_preimage, Set.mem_Iio,
csInf_lt_iff (hbelow x) himage]
constructor
· rintro ⟨z, ⟨b, hb, rfl⟩, hlt⟩
apply Set.mem_union_left
constructor
· exact hx
· exact ⟨b, hb.1, hb.2, hlt⟩
· intro hrhs
rcases hrhs with hmain | hfallback
· obtain ⟨b, hb, hord, hlt⟩ := hmain.2
exact ⟨projectedATEIntervalLower b,
⟨b, ⟨hb, hord⟩, rfl⟩, hlt⟩
· exact False.elim (hfallback.1 hx)
· have hempty : projectedATEIntervalLower ''
activeSievedProjectionCandidates g α x = ∅ :=
Set.image_eq_empty.mpr (Set.not_nonempty_iff_eq_empty.mp hx)
by_cases hq : 0 < q <;>
simp [Set.mem_preimage, hempty, N, E, hx, hq]
rw [heq]
exact (hN.inter hE).union (hN.compl.inter (by split_ifs <;> measurability))
theorem measurable_sSup_activeSievedProjectionCandidates
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
hOverlap :
Overlap ε
g :
ScoreSpace ε → LabelSpace J
hg :
α :
ℝ
habove :
∀ x : ExternalSample ε J n m,
conclusion
Measurable (fun x : ExternalSample ε J n m => sSup (projectedATEIntervalUpper '' activeSievedProjectionCandidates g α x))
Proof (Lean source)
@[fun_prop]
lemma measurable_sSup_activeSievedProjectionCandidates
{ε : ℝ} {J n m : ℕ}
(hOverlap : Overlap ε) (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (α : ℝ)
(habove : ∀ x : ExternalSample ε J n m, BddAbove
(projectedATEIntervalUpper '' activeSievedProjectionCandidates g α x)) :
Measurable (fun x : ExternalSample ε J n m =>
sSup (projectedATEIntervalUpper ''
activeSievedProjectionCandidates g α x)) := by
apply measurable_of_Iic
intro q
let N : Set (ExternalSample ε J n m) :=
{x | (activeSievedProjectionCandidates g α x).Nonempty}
let E : Set (ExternalSample ε J n m) :=
{x | ∃ b ∈ sievedProjectionCandidates g α x,
projectedATEIntervalLower b ≤ projectedATEIntervalUpper b ∧
q < projectedATEIntervalUpper b}
have hN : MeasurableSet N :=
measurableSet_activeSievedProjectionCandidates_nonempty hOverlap g hg α
have hE : MeasurableSet E :=
measurableSet_exists_sievedCandidate_lt_upper hOverlap g hg α q
have heq : (fun x : ExternalSample ε J n m =>
sSup (projectedATEIntervalUpper ''
activeSievedProjectionCandidates g α x)) ⁻¹' Iic q =
(N ∩ Eᶜ) ∪ (Nᶜ ∩ if 0 ≤ q then univ else ∅) := by
ext x
by_cases hx : (activeSievedProjectionCandidates g α x).Nonempty
· have himage : (projectedATEIntervalUpper ''
activeSievedProjectionCandidates g α x).Nonempty :=
hx.image projectedATEIntervalUpper
rw [Set.mem_preimage, Set.mem_Iic]
have hnot : sSup (projectedATEIntervalUpper ''
activeSievedProjectionCandidates g α x) ≤ q ↔
¬q < sSup (projectedATEIntervalUpper ''
activeSievedProjectionCandidates g α x) := not_lt.symm
rw [hnot, lt_csSup_iff (habove x) himage]
constructor
· intro hno
apply Set.mem_union_left
refine ⟨hx, ?_⟩
intro hE_mem
obtain ⟨b, hb, hord, hlt⟩ := hE_mem
exact hno ⟨projectedATEIntervalUpper b,
⟨b, ⟨hb, hord⟩, rfl⟩, hlt⟩
· intro hrhs hex
obtain ⟨z, ⟨b, hb, rfl⟩, hqz⟩ := hex
rcases hrhs with hmain | hfallback
· exact hmain.2 ⟨b, hb.1, hb.2, hqz⟩
· exact False.elim (hfallback.1 hx)
· have hempty : projectedATEIntervalUpper ''
activeSievedProjectionCandidates g α x = ∅ :=
Set.image_eq_empty.mpr (Set.not_nonempty_iff_eq_empty.mp hx)
by_cases hq : 0 ≤ q <;>
simp [Set.mem_preimage, hempty, N, E, hx, hq]
rw [heq]
exact (hN.inter hE.compl).union
(hN.compl.inter (by split_ifs <;> measurability))
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionEndpointMeasurable 12 declarations Measurability infrastructure for the endpoints of the external projection interval.
Measurability infrastructure for the endpoints of the external projection interval.
The rational sieve is countable. This file first proves that admission of each fixed sieve element is a measurable event of the external sample; these are the coordinate facts needed to form measurable countable endpoint envelopes.
theorem measurable_outcomeCDFDistance_empirical
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
@[fun_prop]
lemma measurable_outcomeCDFDistance_empirical {J n : ℕ}
(μ : Measure OutcomeSpace) (hμ : μ univ < ⊤)
(a : ArmSpace) (r : LabelSpace J) :
Measurable (fun x : TrialSample n J =>
outcomeCDFDistance μ (empiricalTrialCells x a r)) := by
have hmass : Measurable (fun x : TrialSample n J =>
|μ.real univ - (empiricalTrialCells x a r).real univ|) :=
(measurable_const.sub
(measurable_empiricalTrialCells_real_apply a r univ MeasurableSet.univ)).abs
have hjoint : Measurable (fun p : TrialSample n J × ℝ =>
|μ.real {y | (y : ℝ) ≤ p.2} -
(empiricalTrialCells p.1 a r).real {y | (y : ℝ) ≤ p.2}|) :=
(((measurable_projectionFiniteCDF μ hμ
(fun y : OutcomeSpace => (y : ℝ))).comp measurable_snd).sub
(measurable_empiricalTrialCellCDF_joint a r)).abs
have hint : Measurable (fun x : TrialSample n J =>
∫ t in (0 : ℝ)..1,
|μ.real {y | (y : ℝ) ≤ t} -
(empiricalTrialCells x a r).real {y | (y : ℝ) ≤ t}|) := by
rw [show (fun x : TrialSample n J =>
∫ t in (0 : ℝ)..1,
|μ.real {y | (y : ℝ) ≤ t} -
(empiricalTrialCells x a r).real {y | (y : ℝ) ≤ t}|) =
fun x => ∫ t,
|μ.real {y | (y : ℝ) ≤ t} -
(empiricalTrialCells x a r).real {y | (y : ℝ) ≤ t}|
∂(volume.restrict (Ioc 0 1)) by
funext x
rw [intervalIntegral.integral_of_le (by norm_num)]]
exact (hjoint.stronglyMeasurable.integral_prod_right'
(ν := volume.restrict (Ioc 0 1))).measurable
exact hmass.add hint
theorem measurable_scoreCDFDistance_empirical
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J m :
ℕ
hOverlap :
Overlap ε
g :
ScoreSpace ε → LabelSpace J
hg :
σ :
Measure (ScoreSpace ε)
hσ :
σ univ < ⊤
a :
r :
conclusion
Proof (Lean source)
@[fun_prop]
lemma measurable_scoreCDFDistance_empirical {ε : ℝ} {J m : ℕ}
(hOverlap : Overlap ε) (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (σ : Measure (ScoreSpace ε))
(hσ : σ Set.univ < ⊤) (a : ArmSpace) (r : LabelSpace J) :
Measurable (fun x : Fin m → ScoreSpace ε =>
scoreCDFDistance σ (empiricalScoreCells g x a r)) := by
have hmassEval : Measurable (fun x : Fin m → ScoreSpace ε =>
(empiricalScoreCells g x a r).real Set.univ) := by
rw [show (fun x : Fin m → ScoreSpace ε =>
(empiricalScoreCells g x a r).real Set.univ) =
scoreCellFrequency g a r Set.univ by
funext x
exact empiricalScoreCells_real_eq_scoreCellFrequency hOverlap g a r
Set.univ MeasurableSet.univ x]
exact measurable_scoreCellFrequency g hg a r Set.univ MeasurableSet.univ
have hmass : Measurable (fun x : Fin m → ScoreSpace ε =>
|σ.real Set.univ - (empiricalScoreCells g x a r).real Set.univ|) :=
(measurable_const.sub hmassEval).abs
have hjoint : Measurable (fun p : (Fin m → ScoreSpace ε) × ℝ =>
|σ.real {e | (e : ℝ) ≤ p.2} -
(empiricalScoreCells g p.1 a r).real {e | (e : ℝ) ≤ p.2}|) :=
(((measurable_projectionFiniteCDF σ hσ
(fun e : ScoreSpace ε => (e : ℝ))).comp measurable_snd).sub
(measurable_empiricalScoreCellCDF_joint hOverlap g hg a r)).abs
have hε : ε ≤ 1 - ε := by
dsimp [Overlap] at hOverlap
linarith [hOverlap.2]
have hint : Measurable (fun x : Fin m → ScoreSpace ε =>
∫ t in ε..(1 - ε),
|σ.real {e | (e : ℝ) ≤ t} -
(empiricalScoreCells g x a r).real {e | (e : ℝ) ≤ t}|) := by
rw [show (fun x : Fin m → ScoreSpace ε =>
∫ t in ε..(1 - ε),
|σ.real {e | (e : ℝ) ≤ t} -
(empiricalScoreCells g x a r).real {e | (e : ℝ) ≤ t}|) =
fun x => ∫ t,
|σ.real {e | (e : ℝ) ≤ t} -
(empiricalScoreCells g x a r).real {e | (e : ℝ) ≤ t}|
∂(volume.restrict (Ioc ε (1 - ε))) by
funext x
rw [intervalIntegral.integral_of_le hε]]
exact (hjoint.stronglyMeasurable.integral_prod_right'
(ν := volume.restrict (Ioc ε (1 - ε)))).measurable
exact hmass.add hint
def projectionTrialDistance
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
projectionTrialDistance b x :
ℝ
given by
def projectionScoreDistance
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
projectionScoreDistance g b x :
ℝ
given by
theorem measurable_projectionTrialDistance
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Measurable (projectionTrialDistance (n := n) (m := m) b)
Proof (Lean source)
@[fun_prop]
lemma measurable_projectionTrialDistance {ε : ℝ} {J n m : ℕ}
(b : ArmCellArray J OutcomeSpace × ArmCellArray J (ScoreSpace ε))
(hfin : ∀ a r, (b.1 a r) univ < ⊤) :
Measurable (projectionTrialDistance (n := n) (m := m) b) := by
unfold projectionTrialDistance
apply Finset.measurable_sum
intro a _
apply Finset.measurable_sum
intro r _
exact (measurable_outcomeCDFDistance_empirical (b.1 a r) (hfin a r) a r).comp
measurable_fst
theorem measurable_projectionScoreDistance
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
hOverlap :
Overlap ε
g :
ScoreSpace ε → LabelSpace J
hg :
b :
hfin :
∀ a r, (b.2 a r) univ < ⊤
conclusion
Measurable (projectionScoreDistance (n := n) (m := m) g b)
Proof (Lean source)
@[fun_prop]
lemma measurable_projectionScoreDistance {ε : ℝ} {J n m : ℕ}
(hOverlap : Overlap ε) (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g)
(b : ArmCellArray J OutcomeSpace × ArmCellArray J (ScoreSpace ε))
(hfin : ∀ a r, (b.2 a r) univ < ⊤) :
Measurable (projectionScoreDistance (n := n) (m := m) g b) := by
unfold projectionScoreDistance
apply Finset.measurable_sum
intro a _
apply Finset.measurable_sum
intro r _
exact (measurable_scoreCDFDistance_empirical hOverlap g hg (b.2 a r)
(hfin a r) a r).comp measurable_snd
theorem measurableSet_mem_projectionCandidates
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma measurableSet_mem_projectionCandidates {ε : ℝ} {J n m : ℕ}
(hOverlap : Overlap ε) (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (α : ℝ)
(b : ArmCellArray J OutcomeSpace × ArmCellArray J (ScoreSpace ε)) :
MeasurableSet {x : ExternalSample ε J n m | b ∈ projectionCandidates g α x} := by
classical
by_cases hfin : ∀ a r, (b.1 a r) univ < ⊤ ∧ (b.2 a r) univ < ⊤
· by_cases hcompat : ProjectionCompatible b.1 b.2
· have htrial : Measurable (projectionTrialDistance (n := n) (m := m) b) :=
measurable_projectionTrialDistance b (fun a r => (hfin a r).1)
have hscore : Measurable (projectionScoreDistance (n := n) (m := m) g b) :=
measurable_projectionScoreDistance hOverlap g hg b (fun a r => (hfin a r).2)
have heq : {x : ExternalSample ε J n m |
b ∈ projectionCandidates g α x} =
{x | projectionTrialDistance b x ≤ 4 * J / (α * sqrt n)} ∩
{x | projectionScoreDistance g b x ≤
2 * J * (2 - 2 * ε) / (α * sqrt m)} := by
ext x
simp only [projectionCandidates, Set.mem_ofPred_eq, Set.mem_inter_iff,
projectionTrialDistance, projectionScoreDistance]
constructor
· intro hx
exact hx.2.2
· intro hx
exact ⟨hfin, hcompat, hx⟩
rw [heq]
exact (measurableSet_le htrial measurable_const).inter
(measurableSet_le hscore measurable_const)
· have heq : {x : ExternalSample ε J n m |
b ∈ projectionCandidates g α x} = ∅ := by
ext x
simp only [projectionCandidates, Set.mem_ofPred_eq, Set.mem_empty_iff_false,
iff_false]
intro hx
exact hcompat hx.2.1
rw [heq]
exact MeasurableSet.empty
· have heq : {x : ExternalSample ε J n m |
b ∈ projectionCandidates g α x} = ∅ := by
ext x
simp only [projectionCandidates, Set.mem_ofPred_eq, Set.mem_empty_iff_false,
iff_false]
intro hx
exact hfin hx.1
rw [heq]
exact MeasurableSet.empty
def projectedATEIntervalLower
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
projectedATEIntervalLower b :
ℝ
given by
def projectedATEIntervalUpper
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
projectedATEIntervalUpper b :
ℝ
given by
theorem projectedATEInterval_eq_Icc_endpoints
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
projectedATEInterval b.1 b.2
Proof (Lean source)
lemma projectedATEInterval_eq_Icc_endpoints {ε : ℝ} {J : ℕ}
(b : ArmCellArray J OutcomeSpace × ArmCellArray J (ScoreSpace ε)) :
projectedATEInterval b.1 b.2 =
Icc (projectedATEIntervalLower b) (projectedATEIntervalUpper b) := by
rfl
theorem measurableSet_exists_sievedCandidate_lower_lt
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
MeasurableSet {x : ExternalSample ε J n m | ∃ b ∈ sievedProjectionCandidates g α x, projectedATEIntervalLower b ≤ projectedATEIntervalUpper b ∧ projectedATEIntervalLower b < q}
Proof (Lean source)
lemma measurableSet_exists_sievedCandidate_lower_lt {ε : ℝ} {J n m : ℕ}
(hOverlap : Overlap ε) (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (α q : ℝ) :
MeasurableSet {x : ExternalSample ε J n m |
∃ b ∈ sievedProjectionCandidates g α x,
projectedATEIntervalLower b ≤ projectedATEIntervalUpper b ∧
projectedATEIntervalLower b < q} := by
let S := {b ∈ rationalArraySieve ε J |
projectedATEIntervalLower b ≤ projectedATEIntervalUpper b ∧
projectedATEIntervalLower b < q}
have hS : S.Countable :=
(countable_rationalArraySieve ε J).mono (Set.sep_subset _ _)
have hmeas : MeasurableSet
(⋃ b ∈ S, {x : ExternalSample ε J n m |
b ∈ projectionCandidates g α x}) := by
exact MeasurableSet.biUnion hS fun b _ =>
measurableSet_mem_projectionCandidates hOverlap g hg α b
have heq : {x : ExternalSample ε J n m |
∃ b ∈ sievedProjectionCandidates g α x,
projectedATEIntervalLower b ≤ projectedATEIntervalUpper b ∧
projectedATEIntervalLower b < q} =
⋃ b ∈ S, {x | b ∈ projectionCandidates g α x} := by
ext x
simp only [Set.mem_ofPred_eq, Set.mem_iUnion, sievedProjectionCandidates,
Set.mem_inter_iff, S]
aesop
rw [heq]
exact hmeas
theorem measurableSet_exists_sievedCandidate_lt_upper
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
MeasurableSet {x : ExternalSample ε J n m | ∃ b ∈ sievedProjectionCandidates g α x, projectedATEIntervalLower b ≤ projectedATEIntervalUpper b ∧ q < projectedATEIntervalUpper b}
Proof (Lean source)
lemma measurableSet_exists_sievedCandidate_lt_upper {ε : ℝ} {J n m : ℕ}
(hOverlap : Overlap ε) (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (α q : ℝ) :
MeasurableSet {x : ExternalSample ε J n m |
∃ b ∈ sievedProjectionCandidates g α x,
projectedATEIntervalLower b ≤ projectedATEIntervalUpper b ∧
q < projectedATEIntervalUpper b} := by
let S := {b ∈ rationalArraySieve ε J |
projectedATEIntervalLower b ≤ projectedATEIntervalUpper b ∧
q < projectedATEIntervalUpper b}
have hS : S.Countable :=
(countable_rationalArraySieve ε J).mono (Set.sep_subset _ _)
have hmeas : MeasurableSet
(⋃ b ∈ S, {x : ExternalSample ε J n m |
b ∈ projectionCandidates g α x}) := by
exact MeasurableSet.biUnion hS fun b _ =>
measurableSet_mem_projectionCandidates hOverlap g hg α b
have heq : {x : ExternalSample ε J n m |
∃ b ∈ sievedProjectionCandidates g α x,
projectedATEIntervalLower b ≤ projectedATEIntervalUpper b ∧
q < projectedATEIntervalUpper b} =
⋃ b ∈ S, {x | b ∈ projectionCandidates g α x} := by
ext x
simp only [Set.mem_ofPred_eq, Set.mem_iUnion, sievedProjectionCandidates,
Set.mem_inter_iff, S]
aesop
rw [heq]
exact hmeas
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionGeometry 15 declarations Deterministic geometry of the external projection confidence set.
Deterministic geometry of the external projection confidence set.
The results here use only the clipping, convex-hull, closure, and singleton-fallback
operations in sievedTotalizedProjectionCI. They do not impose or derive statistical coverage.
theorem projectionArmProb_bounds
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
Proof (Lean source)
lemma projectionArmProb_bounds {ε : ℝ} (hOverlap : Overlap ε)
(a : ArmSpace) (e : ScoreSpace ε) :
ε ≤ armProb a e ∧ armProb a e ≤ 1 := by
rcases e.property with ⟨he₁, he₂⟩
rcases hOverlap with ⟨hε, hε'⟩
cases a <;> simp only [armProb, Bool.false_eq_true, ↓reduceIte] <;> constructor <;> linarith
theorem projectionArmProb_diff_abs
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
Proof (Lean source)
lemma projectionArmProb_diff_abs {ε : ℝ} (a : ArmSpace)
(e e' : ScoreSpace ε) :
|armProb a e - armProb a e'| = |(e : ℝ) - (e' : ℝ)| := by
cases a
· simp only [armProb, Bool.false_eq_true, ↓reduceIte]
have h : (1 - (e : ℝ)) - (1 - (e' : ℝ)) =
-((e : ℝ) - (e' : ℝ)) := by ring
rw [h, abs_neg]
· rfl
theorem projectionInverseArmProb_lipschitz
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
Proof (Lean source)
lemma projectionInverseArmProb_lipschitz {ε : ℝ} (hOverlap : Overlap ε)
(a : ArmSpace) (e e' : ScoreSpace ε) :
|(armProb a e)⁻¹ - (armProb a e')⁻¹| ≤
(ε ^ 2)⁻¹ * |(e : ℝ) - (e' : ℝ)| := by
have hp := (projectionArmProb_bounds hOverlap a e).1
have hq := (projectionArmProb_bounds hOverlap a e').1
have hε : 0 < ε := hOverlap.1
have hpp : 0 < armProb a e := lt_of_lt_of_le hε hp
have hqq : 0 < armProb a e' := lt_of_lt_of_le hε hq
rw [inv_sub_inv (ne_of_gt hpp) (ne_of_gt hqq), abs_div,
abs_mul, abs_of_pos hpp, abs_of_pos hqq]
rw [abs_sub_comm, projectionArmProb_diff_abs]
have hden : ε ^ 2 ≤ armProb a e * armProb a e' := by nlinarith
have hnum : 0 ≤ |(e : ℝ) - (e' : ℝ)| := abs_nonneg _
have hfrac : |(e : ℝ) - (e' : ℝ)| / (armProb a e * armProb a e') ≤
|(e : ℝ) - (e' : ℝ)| / ε ^ 2 :=
div_le_div_of_nonneg_left hnum (sq_pos_of_pos hε) hden
calc
|(e : ℝ) - (e' : ℝ)| / (armProb a e * armProb a e') ≤
|(e : ℝ) - (e' : ℝ)| / ε ^ 2 := hfrac
_ = (ε ^ 2)⁻¹ * |(e : ℝ) - (e' : ℝ)| := by ring
theorem projectionCost_bounds
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
Proof (Lean source)
lemma projectionCost_bounds {ε : ℝ} (hOverlap : Overlap ε)
(a : ArmSpace) (e e' : ScoreSpace ε) (y y' : OutcomeSpace) :
0 ≤ (y : ℝ) / armProb a e ∧
(y : ℝ) / armProb a e ≤ ε⁻¹ ∧
|(y : ℝ) / armProb a e - (y' : ℝ) / armProb a e'| ≤
ε⁻¹ * |(y : ℝ) - (y' : ℝ)| +
(ε ^ 2)⁻¹ * |(e : ℝ) - (e' : ℝ)| := by
have hp := (projectionArmProb_bounds hOverlap a e).1
have hq := (projectionArmProb_bounds hOverlap a e').1
have hε : 0 < ε := hOverlap.1
have hpp : 0 < armProb a e := lt_of_lt_of_le hε hp
have hqq : 0 < armProb a e' := lt_of_lt_of_le hε hq
have hpinv : (armProb a e)⁻¹ ≤ ε⁻¹ := (inv_le_inv₀ hpp hε).2 hp
have hdiff := projectionInverseArmProb_lipschitz hOverlap a e e'
refine ⟨div_nonneg y.property.1 (le_of_lt hpp), ?_, ?_⟩
· calc
(y : ℝ) / armProb a e = (y : ℝ) * (armProb a e)⁻¹ := div_eq_mul_inv _ _
_ ≤ 1 * (armProb a e)⁻¹ :=
mul_le_mul_of_nonneg_right y.property.2 (le_of_lt (inv_pos.mpr hpp))
_ ≤ ε⁻¹ := by simpa using hpinv
· have hrewrite : (y : ℝ) / armProb a e - (y' : ℝ) / armProb a e' =
((y : ℝ) - y') * (armProb a e)⁻¹ +
(y' : ℝ) * ((armProb a e)⁻¹ - (armProb a e')⁻¹) := by
simp only [div_eq_mul_inv]
ring
rw [hrewrite]
have hfirst : |((y : ℝ) - y') * (armProb a e)⁻¹| ≤
ε⁻¹ * |(y : ℝ) - y'| := by
rw [abs_mul, abs_of_pos (inv_pos.mpr hpp)]
calc
|(y : ℝ) - y'| * (armProb a e)⁻¹ ≤
|(y : ℝ) - y'| * ε⁻¹ :=
mul_le_mul_of_nonneg_left hpinv (abs_nonneg _)
_ = ε⁻¹ * |(y : ℝ) - y'| := mul_comm _ _
have hsecond : |(y' : ℝ) * ((armProb a e)⁻¹ - (armProb a e')⁻¹)| ≤
(ε ^ 2)⁻¹ * |(e : ℝ) - (e' : ℝ)| := by
rw [abs_mul, abs_of_nonneg y'.property.1]
calc
(y' : ℝ) * |(armProb a e)⁻¹ - (armProb a e')⁻¹| ≤
1 * |(armProb a e)⁻¹ - (armProb a e')⁻¹| :=
mul_le_mul_of_nonneg_right y'.property.2 (abs_nonneg _)
_ = |(armProb a e)⁻¹ - (armProb a e')⁻¹| := one_mul _
_ ≤ (ε ^ 2)⁻¹ * |(e : ℝ) - (e' : ℝ)| := hdiff
exact (abs_add_le _ _).trans (add_le_add hfirst hsecond)
theorem closure_convexHull_subset_Icc
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma closure_convexHull_subset_Icc {S : Set ℝ} {lo hi : ℝ}
(hS : S ⊆ Icc lo hi) :
closure (convexHull ℝ S) ⊆ Icc lo hi :=
closure_minimal (convexHull_min hS (convex_Icc lo hi)) isClosed_Icc
theorem clippedClosureConvexHull_subset_Icc
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma clippedClosureConvexHull_subset_Icc {S : Set ℝ} {lo hi : ℝ}
(hS : S ⊆ Icc lo hi) :
Icc (-1 : ℝ) 1 ∩ closure (convexHull ℝ S) ⊆
Icc (max (-1) lo) (min 1 hi) := by
intro z hz
have hzEnvelope := closure_convexHull_subset_Icc hS hz.2
exact ⟨max_le hz.1.1 hzEnvelope.1, le_min hz.1.2 hzEnvelope.2⟩
theorem projection_strictBalls_closure_of_dense Lemma projection_strictBalls_closure_of_dense in the paper ↗
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma projection_strictBalls_closure_of_dense {X : Type*} [TopologicalSpace X]
{D : Set X} (hD : Dense D) {f h : X → ℝ}
(hf : Continuous f) (hh : Continuous h) {rn rm : ℝ} {x : X}
(hfn : f x < rn) (hhm : h x < rm) :
x ∈ closure {y : X | y ∈ D ∧ f y ≤ rn ∧ h y ≤ rm} := by
let U : Set X := {y | f y < rn ∧ h y < rm}
have hU : IsOpen U := (isOpen_lt hf continuous_const).inter
(isOpen_lt hh continuous_const)
have hxU : x ∈ U := ⟨hfn, hhm⟩
have hsub : U ∩ D ⊆ {y : X | y ∈ D ∧ f y ≤ rn ∧ h y ≤ rm} := by
intro y hy
exact ⟨hy.2, hy.1.1.le, hy.1.2.le⟩
exact closure_mono hsub (hD.open_subset_closure_inter hU hxU)
theorem externalProjectionCI_nonempty
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(sievedTotalizedProjectionCI g α x).Nonempty
Proof (Lean source)
lemma externalProjectionCI_nonempty {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(x : ExternalSample ε J n m) :
(sievedTotalizedProjectionCI g α x).Nonempty := by
classical
rw [sievedTotalizedProjectionCI]
split_ifs with h
· exact h.2
· exact singleton_nonempty 0
theorem externalProjectionCI_subset_Icc
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
sievedTotalizedProjectionCI g α x ⊆ Icc (-1 : ℝ) 1
Proof (Lean source)
lemma externalProjectionCI_subset_Icc {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(x : ExternalSample ε J n m) :
sievedTotalizedProjectionCI g α x ⊆ Icc (-1 : ℝ) 1 := by
classical
rw [sievedTotalizedProjectionCI]
split_ifs
· exact inter_subset_left
· simp
theorem isClosed_externalProjectionCI
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
IsClosed (sievedTotalizedProjectionCI g α x)
Proof (Lean source)
lemma isClosed_externalProjectionCI {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(x : ExternalSample ε J n m) :
IsClosed (sievedTotalizedProjectionCI g α x) := by
classical
rw [sievedTotalizedProjectionCI]
split_ifs
· exact isClosed_Icc.inter isClosed_closure
· exact isClosed_singleton
theorem convex_externalProjectionCI
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Convex ℝ (sievedTotalizedProjectionCI g α x)
Proof (Lean source)
lemma convex_externalProjectionCI {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(x : ExternalSample ε J n m) :
Convex ℝ (sievedTotalizedProjectionCI g α x) := by
classical
rw [sievedTotalizedProjectionCI]
split_ifs
· exact (convex_Icc (-1 : ℝ) 1).inter (convex_convexHull ℝ _).closure
· exact convex_singleton 0
theorem externalProjectionCI_eq_Icc
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
∃ lo hi : ℝ,
conclusion 1
-1 ≤ lo
conclusion 2
lo ≤ hi
conclusion 3
hi ≤ 1
conclusion 4
sievedTotalizedProjectionCI g α x = Icc lo hi
Proof (Lean source)
lemma externalProjectionCI_eq_Icc {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(x : ExternalSample ε J n m) :
∃ lo hi : ℝ,
-1 ≤ lo ∧ lo ≤ hi ∧ hi ≤ 1 ∧
sievedTotalizedProjectionCI g α x = Icc lo hi := by
let S := sievedTotalizedProjectionCI g α x
have hne : S.Nonempty := externalProjectionCI_nonempty g α x
have hclosed : IsClosed S := isClosed_externalProjectionCI g α x
have hconvex : Convex ℝ S := convex_externalProjectionCI g α x
have hclip : S ⊆ Icc (-1 : ℝ) 1 := externalProjectionCI_subset_Icc g α x
have hbelow : BddBelow S := ⟨-1, fun z hz => (hclip hz).1⟩
have habove : BddAbove S := ⟨1, fun z hz => (hclip hz).2⟩
have hloMem : sInf S ∈ S := hclosed.csInf_mem hne hbelow
have hhiMem : sSup S ∈ S := hclosed.csSup_mem hne habove
have hconnected : IsConnected S := ⟨hne, hconvex.isPreconnected⟩
refine ⟨sInf S, sSup S, (hclip hloMem).1, ?_, (hclip hhiMem).2, ?_⟩
· exact csInf_le hbelow hhiMem
· exact eq_Icc_csInf_csSup_of_connected_bdd_closed
hconnected hbelow habove hclosed
theorem externalProjectionCI_excessLength_le_of_envelope Lemma externalProjectionCI_excessLength_le_of_envelope in the paper ↗
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
x :
ExternalSample ε J n m
L U δ :
ℝ
hδ :
0 ≤ δ
henv :
sievedTotalizedProjectionCI g α x ⊆ Icc (L - δ) (U + δ)
conclusion
∃ lo hi : ℝ,
conclusion 1
sievedTotalizedProjectionCI g α x = Icc lo hi
conclusion 2
max 0 ((hi - lo) - (U - L)) ≤ 2 * δ
Proof (Lean source)
lemma externalProjectionCI_excessLength_le_of_envelope
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(x : ExternalSample ε J n m) {L U δ : ℝ}
(hδ : 0 ≤ δ)
(henv : sievedTotalizedProjectionCI g α x ⊆ Icc (L - δ) (U + δ)) :
∃ lo hi : ℝ, sievedTotalizedProjectionCI g α x = Icc lo hi ∧
max 0 ((hi - lo) - (U - L)) ≤ 2 * δ := by
obtain ⟨lo, hi, _, hlohi, _, heq⟩ := externalProjectionCI_eq_Icc g α x
have hlo : L - δ ≤ lo := (henv (heq ▸ Set.left_mem_Icc.mpr hlohi)).1
have hhi : hi ≤ U + δ := (henv (heq ▸ Set.right_mem_Icc.mpr hlohi)).2
refine ⟨lo, hi, heq, ?_⟩
apply max_le (by linarith)
linarith
theorem externalProjectionCI_subset_envelope_of_candidates Lemma externalProjectionCI_subset_envelope_of_candidates in the paper ↗
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
x :
ExternalSample ε J n m
L U :
ℝ
hB :
(sievedProjectionCandidates g α x).Nonempty
hC :
(Icc (-1 : ℝ) 1 ∩ closure (convexHull ℝ (⋃ b ∈ sievedProjectionCandidates g α x, projectedATEInterval b.1 b.2))).Nonempty
hEach :
conclusion
Proof (Lean source)
lemma externalProjectionCI_subset_envelope_of_candidates
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(x : ExternalSample ε J n m) {L U : ℝ}
(hB : (sievedProjectionCandidates g α x).Nonempty)
(hC : (Icc (-1 : ℝ) 1 ∩
closure (convexHull ℝ
(⋃ b ∈ sievedProjectionCandidates g α x,
projectedATEInterval b.1 b.2))).Nonempty)
(hEach : ∀ b ∈ sievedProjectionCandidates g α x,
projectedATEInterval b.1 b.2 ⊆ Icc L U) :
sievedTotalizedProjectionCI g α x ⊆ Icc (max (-1) L) (min 1 U) := by
have hUnion :
(⋃ b ∈ sievedProjectionCandidates g α x,
projectedATEInterval b.1 b.2) ⊆ Icc L U := by
intro z hz
rcases Set.mem_iUnion.mp hz with ⟨b, hz⟩
rcases Set.mem_iUnion.mp hz with ⟨hb, hz⟩
exact hEach b hb hz
unfold sievedTotalizedProjectionCI
dsimp
rw [if_pos ⟨hB, hC⟩]
exact clippedClosureConvexHull_subset_Icc hUnion
theorem externalProjectionCI_excessLength_le_of_candidate_envelopes Lemma externalProjectionCI_excessLength_le_of_candidate_envelopes in the paper ↗
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
x :
ExternalSample ε J n m
L U δ :
ℝ
hLU :
L ≤ U
hδ :
0 ≤ δ
hclip :
(sievedProjectionCandidates g α x).Nonempty → (Icc (-1 : ℝ) 1 ∩ closure (convexHull ℝ (⋃ b ∈ sievedProjectionCandidates g α x, projectedATEInterval b.1 b.2))).Nonempty
hEach :
∀ b ∈ sievedProjectionCandidates g α x,
projectedATEInterval b.1 b.2 ⊆ Icc (L - δ) (U + δ)
conclusion
∃ lo hi : ℝ,
conclusion 1
sievedTotalizedProjectionCI g α x = Icc lo hi
conclusion 2
max 0 ((hi - lo) - (U - L)) ≤ 2 * δ
Proof (Lean source)
lemma externalProjectionCI_excessLength_le_of_candidate_envelopes
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(x : ExternalSample ε J n m) {L U δ : ℝ}
(hLU : L ≤ U) (hδ : 0 ≤ δ)
(hclip : (sievedProjectionCandidates g α x).Nonempty →
(Icc (-1 : ℝ) 1 ∩
closure (convexHull ℝ
(⋃ b ∈ sievedProjectionCandidates g α x,
projectedATEInterval b.1 b.2))).Nonempty)
(hEach : ∀ b ∈ sievedProjectionCandidates g α x,
projectedATEInterval b.1 b.2 ⊆ Icc (L - δ) (U + δ)) :
∃ lo hi : ℝ, sievedTotalizedProjectionCI g α x = Icc lo hi ∧
max 0 ((hi - lo) - (U - L)) ≤ 2 * δ := by
classical
by_cases hB : (sievedProjectionCandidates g α x).Nonempty
· have henv := externalProjectionCI_subset_envelope_of_candidates
g α x hB (hclip hB) hEach
have henv' : sievedTotalizedProjectionCI g α x ⊆ Icc (L - δ) (U + δ) := by
intro z hz
have hz' := henv hz
exact ⟨le_trans (le_max_right _ _) hz'.1,
le_trans hz'.2 (min_le_right _ _)⟩
exact externalProjectionCI_excessLength_le_of_envelope g α x hδ henv'
· have heq : sievedTotalizedProjectionCI g α x = {0} := by
unfold sievedTotalizedProjectionCI
simp [hB]
refine ⟨0, 0, ?_, ?_⟩
· simpa only [Set.Icc_self] using heq
· simp only [sub_self, zero_sub]
apply max_le (by linarith) (by linarith)
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionInference 9 declarations Probability bounds for the two empirical distances in the external projection.
Probability bounds for the two empirical distances in the external projection.
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
n m :
ℕ
hn :
0 < n
hm :
0 < m
conclusion
Proof (Lean source)
lemma externalNormalizer_mul_rootSum (n m : ℕ)
(hn : 0 < n) (hm : 0 < m) :
externalNormalizer n m *
((sqrt (n : ℝ))⁻¹ + (sqrt (m : ℝ))⁻¹) = 1 := by
have hsn : 0 < sqrt (n : ℝ) :=
Real.sqrt_pos.2 (Nat.cast_pos.mpr hn)
have hsm : 0 < sqrt (m : ℝ) :=
Real.sqrt_pos.2 (Nat.cast_pos.mpr hm)
unfold externalNormalizer
field_simp
theorem externalNormalizedRisk_le_of_rootSum_bound
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
externalNormalizedRisk g Qj α ≤ C
Proof (Lean source)
lemma externalNormalizedRisk_le_of_rootSum_bound
{ε : ℝ} {J n m : ℕ} (g : ScoreSpace ε → LabelSpace J)
(Qj : Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m)) (α C : ℝ)
(hn : 0 < n) (hm : 0 < m)
(hR : externalExcessRisk g Qj α ≤
C * ((sqrt (n : ℝ))⁻¹ + (sqrt (m : ℝ))⁻¹)) :
externalNormalizedRisk g Qj α ≤ C := by
have hnorm : 0 ≤ externalNormalizer n m := by
unfold externalNormalizer
positivity
unfold externalNormalizedRisk
calc
externalNormalizer n m * externalExcessRisk g Qj α ≤
externalNormalizer n m *
(C * ((sqrt (n : ℝ))⁻¹ +
(sqrt (m : ℝ))⁻¹)) :=
mul_le_mul_of_nonneg_left hR hnorm
_ = C := by
rw [mul_left_comm, externalNormalizer_mul_rootSum n m hn hm, mul_one]
theorem externalJointLaw_eq_experiment
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
Qj :
hTrial :
JointTrialIID g Qj
hLog :
ExternalLogIID g Qj
hInd :
PH :
hPH :
PH ∈ ExternalLaws g
conclusion
Qj PH.1 PH.2 = externalExperiment n m PH.1 PH.2
Proof (Lean source)
lemma externalJointLaw_eq_experiment {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J)
(Qj : Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : JointTrialIID g Qj) (hLog : ExternalLogIID g Qj)
(hInd : ExternalLogIndependent g Qj)
(PH : Measure (FullRow ε J) × Measure (ScoreSpace ε))
(hPH : PH ∈ ExternalLaws g) :
Qj PH.1 PH.2 = externalExperiment n m PH.1 PH.2 := by
rw [hInd PH hPH, hTrial PH hPH, hLog PH hPH]
rfl
theorem externalJointTrial_integral_eq_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
Qj :
hTrial :
JointTrialIID g Qj
PH :
hPH :
PH ∈ ExternalLaws g
F :
TrialSample n J → ℝ
hF :
conclusion
∫ x, F x.1 ∂Qj PH.1 PH.2 = ∫ x, F x ∂Measure.pi (fun _ : Fin n => releasedLaw PH.1)
Proof (Lean source)
lemma externalJointTrial_integral_eq_pi {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J)
(Qj : Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : JointTrialIID g Qj)
(PH : Measure (FullRow ε J) × Measure (ScoreSpace ε))
(hPH : PH ∈ ExternalLaws g)
(F : TrialSample n J → ℝ) (hF : Measurable F) :
∫ x, F x.1 ∂Qj PH.1 PH.2 =
∫ x, F x ∂Measure.pi (fun _ : Fin n => releasedLaw PH.1) := by
rw [← integral_map measurable_fst.aemeasurable hF.aestronglyMeasurable,
hTrial PH hPH]
theorem externalJointLog_integral_eq_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
Qj :
hLog :
ExternalLogIID g Qj
PH :
hPH :
PH ∈ ExternalLaws g
F :
(Fin m → ScoreSpace ε) → ℝ
hF :
conclusion
∫ x, F x.2 ∂Qj PH.1 PH.2 = ∫ x, F x ∂Measure.pi (fun _ : Fin m => PH.2)
Proof (Lean source)
lemma externalJointLog_integral_eq_pi {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J)
(Qj : Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hLog : ExternalLogIID g Qj)
(PH : Measure (FullRow ε J) × Measure (ScoreSpace ε))
(hPH : PH ∈ ExternalLaws g)
(F : (Fin m → ScoreSpace ε) → ℝ) (hF : Measurable F) :
∫ x, F x.2 ∂Qj PH.1 PH.2 =
∫ x, F x ∂Measure.pi (fun _ : Fin m => PH.2) := by
rw [← integral_map measurable_snd.aemeasurable hF.aestronglyMeasurable,
hLog PH hPH]
theorem externalJointTrial_meanDistance_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
Qj :
hTrial :
JointTrialIID g Qj
PH :
hPH :
PH ∈ ExternalLaws g
hn :
0 < n
hMeas :
Measurable (fun x : TrialSample n J => ∑ a : ArmSpace, ∑ r : LabelSpace J, outcomeCDFDistance (empiricalTrialCells x a r) (trialPopulationCells (releasedLaw PH.1) a r))
hInt :
∀ a r,
Integrable (fun x : TrialSample n J => outcomeCDFDistance (empiricalTrialCells x a r) (trialPopulationCells (releasedLaw PH.1) a r)) (Measure.pi (fun _ : Fin n => releasedLaw PH.1))
conclusion
(∫ x, (∑ a : ArmSpace, ∑ r : LabelSpace J, outcomeCDFDistance (empiricalTrialCells x.1 a r) (trialPopulationCells (releasedLaw PH.1) a r)) ∂Qj PH.1 PH.2)
≤ 2 * J / sqrt n
Proof (Lean source)
lemma externalJointTrial_meanDistance_le {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J)
(Qj : Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : JointTrialIID g Qj)
(PH : Measure (FullRow ε J) × Measure (ScoreSpace ε))
(hPH : PH ∈ ExternalLaws g) (hn : 0 < n)
(hMeas : Measurable (fun x : TrialSample n J =>
∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (empiricalTrialCells x a r)
(trialPopulationCells (releasedLaw PH.1) a r)))
(hInt : ∀ a r, Integrable (fun x : TrialSample n J =>
outcomeCDFDistance (empiricalTrialCells x a r)
(trialPopulationCells (releasedLaw PH.1) a r))
(Measure.pi (fun _ : Fin n => releasedLaw PH.1))) :
(∫ x, (∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (empiricalTrialCells x.1 a r)
(trialPopulationCells (releasedLaw PH.1) a r)) ∂Qj PH.1 PH.2) ≤
2 * J / sqrt n := by
haveI : IsProbabilityMeasure (releasedLaw PH.1) := by
unfold releasedLaw
exact @Measure.isProbabilityMeasure_map _ _ _ _ PH.1 hPH.1 _
(by unfold releasedRecord label arm observed; fun_prop :
Measurable (releasedRecord (ε := ε) (J := J))).aemeasurable
rw [externalJointTrial_integral_eq_pi g Qj hTrial PH hPH _ hMeas]
rw [integral_finsetSum]
· calc
_ = ∑ a : ArmSpace, ∑ r : LabelSpace J,
∫ x : TrialSample n J,
outcomeCDFDistance (empiricalTrialCells x a r)
(trialPopulationCells (releasedLaw PH.1) a r)
∂Measure.pi (fun _ : Fin n => releasedLaw PH.1) := by
apply Finset.sum_congr rfl
intro a _
rw [integral_finsetSum]
intro r _
exact hInt a r
_ ≤ _ := sum_meanOutcomeCDFDistance_trialPopulationCells_pi
(releasedLaw PH.1) hn
· intro a _
exact integrable_finsetSum _ (fun r _ => hInt a r)
theorem externalJointLog_meanDistance_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
Qj :
hLog :
ExternalLogIID g Qj
PH :
hPH :
PH ∈ ExternalLaws g
hm :
0 < m
hMeas :
Measurable (fun x : Fin m → ScoreSpace ε => ∑ a : ArmSpace, ∑ r : LabelSpace J, scoreCDFDistance (empiricalScoreCells g x a r) (scorePopulationCells PH.2 g a r))
hInt :
∀ a r,
Integrable (fun x : Fin m → ScoreSpace ε => scoreCDFDistance (empiricalScoreCells g x a r) (scorePopulationCells PH.2 g a r)) (Measure.pi (fun _ : Fin m => PH.2))
conclusion
(∫ x, (∑ a : ArmSpace, ∑ r : LabelSpace J, scoreCDFDistance (empiricalScoreCells g x.2 a r) (scorePopulationCells PH.2 g a r)) ∂Qj PH.1 PH.2)
≤ J * (2 - 2 * ε) / sqrt m
Proof (Lean source)
lemma externalJointLog_meanDistance_le {ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J)
(Qj : Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hLog : ExternalLogIID g Qj)
(PH : Measure (FullRow ε J) × Measure (ScoreSpace ε))
(hPH : PH ∈ ExternalLaws g) (hm : 0 < m)
(hMeas : Measurable (fun x : Fin m → ScoreSpace ε =>
∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (empiricalScoreCells g x a r)
(scorePopulationCells PH.2 g a r)))
(hInt : ∀ a r, Integrable (fun x : Fin m → ScoreSpace ε =>
scoreCDFDistance (empiricalScoreCells g x a r)
(scorePopulationCells PH.2 g a r))
(Measure.pi (fun _ : Fin m => PH.2))) :
(∫ x, (∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (empiricalScoreCells g x.2 a r)
(scorePopulationCells PH.2 g a r)) ∂Qj PH.1 PH.2) ≤
J * (2 - 2 * ε) / sqrt m := by
have : IsProbabilityMeasure PH.2 := hPH.2.1
rw [externalJointLog_integral_eq_pi g Qj hLog PH hPH _ hMeas]
rw [integral_finsetSum]
· calc
_ = ∑ a : ArmSpace, ∑ r : LabelSpace J,
∫ x : Fin m → ScoreSpace ε,
scoreCDFDistance (empiricalScoreCells g x a r)
(scorePopulationCells PH.2 g a r)
∂Measure.pi (fun _ : Fin m => PH.2) := by
apply Finset.sum_congr rfl
intro a _
rw [integral_finsetSum]
intro r _
exact hInt a r
_ ≤ _ := sum_meanScoreCDFDistance_scorePopulationCells_pi
hPH.2.2.overlap PH.2 g hPH.2.2.measurableRelease hm
· intro a _
exact integrable_finsetSum _ (fun r _ => hInt a r)
theorem projection_twoSource_strictBall_probability Lemma projection_twoSource_strictBall_probability in the paper ↗
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
Ω :
Type*
μ :
F G :
Ω → ℝ
hFm :
hGm :
hF :
Integrable F μ
hG :
Integrable G μ
hFnonneg :
∀ᵐ x ∂μ, 0 ≤ F x
hGnonneg :
∀ᵐ x ∂μ, 0 ≤ G x
α rn rm :
ℝ
hrn :
0 < rn
hrm :
0 < rm
hFmean :
∫ x, F x ∂μ ≤ α * rn / 2
hGmean :
∫ x, G x ∂μ ≤ α * rm / 2
conclusion
1 - α ≤ μ.real {x | F x < rn ∧ G x < rm}
Proof (Lean source)
lemma projection_twoSource_strictBall_probability
{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
(F G : Ω → ℝ) (hFm : Measurable F) (hGm : Measurable G)
(hF : Integrable F μ) (hG : Integrable G μ)
(hFnonneg : ∀ᵐ x ∂μ, 0 ≤ F x) (hGnonneg : ∀ᵐ x ∂μ, 0 ≤ G x)
(α rn rm : ℝ) (hrn : 0 < rn) (hrm : 0 < rm)
(hFmean : ∫ x, F x ∂μ ≤ α * rn / 2)
(hGmean : ∫ x, G x ∂μ ≤ α * rm / 2) :
1 - α ≤ μ.real {x | F x < rn ∧ G x < rm} := by
have htailF := mul_meas_ge_le_integral_of_nonneg hFnonneg hF rn
have htailG := mul_meas_ge_le_integral_of_nonneg hGnonneg hG rm
have hFsmall : μ.real {x | rn ≤ F x} ≤ α / 2 := by
calc
_ ≤ (α * rn / 2) / rn := (le_div_iff₀ hrn).2 (by nlinarith [htailF, hFmean])
_ = α / 2 := by field_simp
have hGsmall : μ.real {x | rm ≤ G x} ≤ α / 2 := by
calc
_ ≤ (α * rm / 2) / rm := (le_div_iff₀ hrm).2 (by nlinarith [htailG, hGmean])
_ = α / 2 := by field_simp
have hbad : μ.real ({x | rn ≤ F x} ∪ {x | rm ≤ G x}) ≤ α := by
calc
_ ≤ μ.real {x | rn ≤ F x} + μ.real {x | rm ≤ G x} :=
measureReal_union_le _ _
_ ≤ α := by linarith
have hevent : {x | F x < rn ∧ G x < rm} =
({x | rn ≤ F x} ∪ {x | rm ≤ G x})ᶜ := by
ext x
simp only [mem_ofPred_eq, mem_compl_iff, mem_union]
push Not
rfl
rw [hevent, probReal_compl_eq_one_sub]
· linarith
· exact (measurableSet_le measurable_const hFm).union
(measurableSet_le measurable_const hGm)
theorem projection_strictBall_probability_at_paper_radii Lemma projection_strictBall_probability_at_paper_radii in the paper ↗
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
Ω :
Type*
μ :
F G :
Ω → ℝ
hFm :
hGm :
hF :
Integrable F μ
hG :
Integrable G μ
hFnonneg :
∀ᵐ x ∂μ, 0 ≤ F x
hGnonneg :
∀ᵐ x ∂μ, 0 ≤ G x
J n m :
ℕ
hJ :
0 < J
hn :
0 < n
hm :
0 < m
α ε :
ℝ
hα :
0 < α
hε :
ε < 1 / 2
hFmean :
∫ x, F x ∂μ ≤ 2 * J / sqrt n
hGmean :
∫ x, G x ∂μ ≤ J * (2 - 2 * ε) / sqrt m
Proof (Lean source)
lemma projection_strictBall_probability_at_paper_radii
{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
(F G : Ω → ℝ) (hFm : Measurable F) (hGm : Measurable G)
(hF : Integrable F μ) (hG : Integrable G μ)
(hFnonneg : ∀ᵐ x ∂μ, 0 ≤ F x) (hGnonneg : ∀ᵐ x ∂μ, 0 ≤ G x)
(J n m : ℕ) (hJ : 0 < J) (hn : 0 < n) (hm : 0 < m)
(α ε : ℝ) (hα : 0 < α) (hε : ε < 1 / 2)
(hFmean : ∫ x, F x ∂μ ≤ 2 * J / sqrt n)
(hGmean : ∫ x, G x ∂μ ≤ J * (2 - 2 * ε) / sqrt m) :
1 - α ≤ μ.real {x |
F x < 4 * J / (α * sqrt n) ∧
G x < 2 * J * (2 - 2 * ε) / (α * sqrt m)} := by
have hsn : 0 < sqrt (n : ℝ) := Real.sqrt_pos.2 (by exact_mod_cast hn)
have hsm : 0 < sqrt (m : ℝ) := Real.sqrt_pos.2 (by exact_mod_cast hm)
have hJ' : (0 : ℝ) < J := by exact_mod_cast hJ
have hw : 0 < 2 - 2 * ε := by linarith
have hrn : 0 < 4 * J / (α * sqrt n) := by positivity
have hrm : 0 < 2 * J * (2 - 2 * ε) / (α * sqrt m) := by positivity
apply projection_twoSource_strictBall_probability μ F G hFm hGm hF hG
hFnonneg hGnonneg α _ _ hrn hrm
· convert hFmean using 1 <;> field_simp <;> norm_num
· convert hGmean using 1 <;> field_simp
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionIntegrated 14 declarations Integration of pointwise empirical CDF bounds over a compact support.
Integration of pointwise empirical CDF bounds over a compact support.
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Measurable (fun t : ℝ => ν.real {x | v x ≤ t})
Proof (Lean source)
lemma measurable_projectionFiniteCDF {X : Type*} [MeasurableSpace X]
(ν : Measure X) (hν : ν univ < ⊤) (v : X → ℝ) :
Measurable (fun t : ℝ => ν.real {x | v x ≤ t}) := by
apply Monotone.measurable
intro s t hst
apply ENNReal.toReal_mono
· exact ne_of_lt (lt_of_le_of_lt (measure_mono (by
intro x hx
exact trivial)) hν)
· exact measure_mono (by
intro x hx
exact le_trans hx hst)
theorem measurable_empiricalTrialCellCDF_joint
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma measurable_empiricalTrialCellCDF_joint {J n : ℕ}
(a : ArmSpace) (r : LabelSpace J) :
Measurable (fun p : TrialSample n J × ℝ =>
(empiricalTrialCells p.1 a r).real {y | (y : ℝ) ≤ p.2}) := by
classical
have heq : (fun p : TrialSample n J × ℝ =>
(empiricalTrialCells p.1 a r).real {y | (y : ℝ) ≤ p.2}) =
fun p => (n : ℝ)⁻¹ * ∑ i : Fin n,
if (p.1 i).1 = r ∧ (p.1 i).2.1 = a ∧
((p.1 i).2.2 : ℝ) ≤ p.2 then (1 : ℝ) else 0 := by
funext p
rw [empiricalTrialCells_real_apply a r _
(measurableSet_le (by fun_prop) measurable_const)]
rfl
rw [heq]
apply Measurable.const_mul
apply Finset.measurable_sum
intro i _
apply Measurable.ite
· have hi : Measurable (fun p : TrialSample n J × ℝ => p.1 i) :=
(measurable_pi_apply i).comp measurable_fst
simpa only [Set.ofPred_and, Set.inter_assoc, Function.comp_apply] using
((measurableSet_eq_fun hi.fst
(measurable_const : Measurable (fun _ : TrialSample n J × ℝ => r))).inter
(measurableSet_eq_fun hi.snd.fst
(measurable_const : Measurable (fun _ : TrialSample n J × ℝ => a)))).inter
(measurableSet_le (measurable_subtype_coe.comp hi.snd.snd) measurable_snd)
· exact measurable_const
· exact measurable_const
theorem measurable_trialPopulationCellCDF
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Measurable (fun t : ℝ => μ.real {z | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t})
Proof (Lean source)
lemma measurable_trialPopulationCellCDF {J : ℕ}
(μ : Measure (Observation J)) [IsProbabilityMeasure μ]
(a : ArmSpace) (r : LabelSpace J) :
Measurable (fun t : ℝ =>
μ.real {z | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t}) := by
let C : Set (Observation J) := {z | z.1 = r ∧ z.2.1 = a}
have hC : MeasurableSet C := by
exact (measurableSet_eq_fun measurable_fst measurable_const).inter
(measurableSet_eq_fun (measurable_snd.fst) measurable_const)
have hfin : (μ.restrict C) univ < ⊤ := by
simp
have h := measurable_projectionFiniteCDF (μ.restrict C) hfin
(fun z : Observation J => (z.2.2 : ℝ))
convert h using 1
funext t
unfold Measure.real
have hB : MeasurableSet {x : Observation J | (x.2.2 : ℝ) ≤ t} :=
measurableSet_le (measurable_subtype_coe.comp measurable_snd.snd) measurable_const
rw [Measure.restrict_apply hB]
have hset : {z : Observation J | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t} =
{x | (x.2.2 : ℝ) ≤ t} ∩ C := by
ext z
change (z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t) ↔
((z.2.2 : ℝ) ≤ t ∧ z.1 = r ∧ z.2.1 = a)
tauto
rw [hset]
theorem empiricalTrialCellCDF_mem_Icc
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma empiricalTrialCellCDF_mem_Icc {J n : ℕ} (hn : 0 < n)
(a : ArmSpace) (r : LabelSpace J) (x : TrialSample n J) (t : ℝ) :
(empiricalTrialCells x a r).real {y | (y : ℝ) ≤ t} ∈ Icc (0 : ℝ) 1 := by
rw [empiricalTrialCells_real_apply a r _
(measurableSet_le (by fun_prop) measurable_const)]
unfold trialCellFrequency
have hterm (i : Fin n) :
(if (x i).1 = r ∧ (x i).2.1 = a ∧ ((x i).2.2 : ℝ) ≤ t then
(1 : ℝ) else 0) ∈ Icc (0 : ℝ) 1 := by
split_ifs <;> norm_num
have hs0 : 0 ≤ ∑ i : Fin n,
(if (x i).1 = r ∧ (x i).2.1 = a ∧ ((x i).2.2 : ℝ) ≤ t then
(1 : ℝ) else 0) := Finset.sum_nonneg (fun i _ => (hterm i).1)
have hs1 : (∑ i : Fin n,
(if (x i).1 = r ∧ (x i).2.1 = a ∧ ((x i).2.2 : ℝ) ≤ t then
(1 : ℝ) else 0)) ≤ n := by
calc
_ ≤ ∑ _i : Fin n, (1 : ℝ) := Finset.sum_le_sum
(fun i _ => (hterm i).2)
_ = n := by simp
have hn' : (0 : ℝ) < n := Nat.cast_pos.mpr hn
constructor
· exact mul_nonneg (inv_nonneg.mpr (le_of_lt hn')) hs0
· exact (inv_mul_le_iff₀ hn').mpr (by simpa [mul_comm] using hs1)
theorem trialPopulationCellCDF_mem_Icc
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
μ.real {z | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t} ∈ Icc (0 : ℝ) 1
Proof (Lean source)
lemma trialPopulationCellCDF_mem_Icc {J : ℕ}
(μ : Measure (Observation J)) [IsProbabilityMeasure μ]
(a : ArmSpace) (r : LabelSpace J) (t : ℝ) :
μ.real {z | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t} ∈
Icc (0 : ℝ) 1 := by
constructor
· exact ENNReal.toReal_nonneg
· unfold Measure.real
have hm := measure_mono (μ := μ) (Set.subset_univ
{z : Observation J | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t})
have hfin : μ univ ≠ ⊤ := by simp
have := ENNReal.toReal_mono hfin hm
simpa [isProbabilityMeasure_iff_real.mp inferInstance] using this
theorem integrable_empiricalTrialCellCDF_joint
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Integrable (uncurry (fun (x : TrialSample n J) (t : ℝ) => |(empiricalTrialCells x a r).real {y | (y : ℝ) ≤ t} - μ.real {z | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t}|)) ((Measure.pi fun _ : Fin n => μ).prod (volume.restrict (Ioc 0 1)))
Proof (Lean source)
lemma integrable_empiricalTrialCellCDF_joint {J n : ℕ}
(μ : Measure (Observation J)) [IsProbabilityMeasure μ] (hn : 0 < n)
(a : ArmSpace) (r : LabelSpace J) :
Integrable (uncurry
(fun (x : TrialSample n J) (t : ℝ) =>
|(empiricalTrialCells x a r).real {y | (y : ℝ) ≤ t} -
μ.real {z | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t}|))
((Measure.pi fun _ : Fin n => μ).prod (volume.restrict (Ioc 0 1))) := by
have hm : Measurable (uncurry
(fun (x : TrialSample n J) (t : ℝ) =>
|(empiricalTrialCells x a r).real {y | (y : ℝ) ≤ t} -
μ.real {z | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t}|)) :=
(measurable_empiricalTrialCellCDF_joint a r |>.sub
((measurable_trialPopulationCellCDF μ a r).comp measurable_snd)).abs
apply Integrable.of_bound hm.aestronglyMeasurable 1
apply Filter.Eventually.of_forall
rintro ⟨x, t⟩
have hx := empiricalTrialCellCDF_mem_Icc hn a r x t
have hp := trialPopulationCellCDF_mem_Icc μ a r t
dsimp
rw [abs_abs]
apply abs_le.mpr
constructor <;> linarith [hx.1, hx.2, hp.1, hp.2]
theorem projectionExpectedIntegratedError_le Lemma projectionExpectedIntegratedError_le in the paper ↗
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
∫ x, (∫ t in a..b, F x t) ∂μ ≤ (b - a) * c
Proof (Lean source)
lemma projectionExpectedIntegratedError_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [SFinite μ] (a b c : ℝ) (hab : a ≤ b)
(F : Ω → ℝ → ℝ)
(hF : Integrable (uncurry F) (μ.prod (volume.restrict (Ioc a b))))
(hpoint : ∀ t ∈ Ioc a b, ∫ x, F x t ∂μ ≤ c) :
∫ x, (∫ t in a..b, F x t) ∂μ ≤ (b - a) * c := by
simp_rw [intervalIntegral.integral_of_le hab]
have hswap := MeasureTheory.integral_integral_swap hF
rw [hswap]
have hconst : Integrable (fun _ : ℝ => c) (volume.restrict (Ioc a b)) :=
integrable_const _
have hright : (∫ t in Ioc a b, (∫ x, F x t ∂μ)) ≤
∫ t in Ioc a b, c := by
apply integral_mono_ae
· exact hF.integral_prod_right
· exact hconst
· filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht
exact hpoint t ht
calc
(∫ t in Ioc a b, ∫ x, F x t ∂μ) ≤ ∫ t in Ioc a b, c := hright
_ = (b - a) * c := by
rw [← intervalIntegral.integral_of_le hab, intervalIntegral.integral_const]
simp
theorem meanIntegratedAbs_empiricalTrialCellCDF_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
∫ x, (∫ t in (0 : ℝ)..1, |(empiricalTrialCells x a r).real {y | (y : ℝ) ≤ t} - μ.real {z | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t}|) ∂Measure.pi (fun _ : Fin n => μ)
≤ 1 / (2 * sqrt n)
Proof (Lean source)
lemma meanIntegratedAbs_empiricalTrialCellCDF_pi {J n : ℕ}
(μ : Measure (Observation J)) [IsProbabilityMeasure μ] (hn : 0 < n)
(a : ArmSpace) (r : LabelSpace J) :
∫ x, (∫ t in (0 : ℝ)..1,
|(empiricalTrialCells x a r).real {y | (y : ℝ) ≤ t} -
μ.real {z | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t}|)
∂Measure.pi (fun _ : Fin n => μ) ≤ 1 / (2 * sqrt n) := by
have hpoint (t : ℝ) (_ht : t ∈ Ioc (0 : ℝ) 1) :
∫ x, |(empiricalTrialCells x a r).real {y | (y : ℝ) ≤ t} -
μ.real {z | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t}|
∂Measure.pi (fun _ : Fin n => μ) ≤ 1 / (2 * sqrt n) := by
have hB : MeasurableSet {y : OutcomeSpace | (y : ℝ) ≤ t} :=
measurableSet_le (by fun_prop) measurable_const
simpa only [empiricalTrialCells_real_apply a r _ hB, Set.mem_ofPred_eq] using
meanAbs_trialCellFrequency_pi μ hn a r _ hB
have h := projectionExpectedIntegratedError_le
(Measure.pi fun _ : Fin n => μ) 0 1 (1 / (2 * sqrt n))
(by norm_num) _ (integrable_empiricalTrialCellCDF_joint μ hn a r) hpoint
simpa using h
theorem measurable_empiricalScoreCellCDF_joint
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Measurable (fun p : (Fin m → ScoreSpace ε) × ℝ => (empiricalScoreCells g p.1 a r).real {e | (e : ℝ) ≤ p.2})
Proof (Lean source)
lemma measurable_empiricalScoreCellCDF_joint {ε : ℝ} {J m : ℕ}
(hOverlap : Overlap ε) (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (a : ArmSpace) (r : LabelSpace J) :
Measurable (fun p : (Fin m → ScoreSpace ε) × ℝ =>
(empiricalScoreCells g p.1 a r).real {e | (e : ℝ) ≤ p.2}) := by
classical
have heq : (fun p : (Fin m → ScoreSpace ε) × ℝ =>
(empiricalScoreCells g p.1 a r).real {e | (e : ℝ) ≤ p.2}) =
fun p => (m : ℝ)⁻¹ * ∑ j : Fin m,
if g (p.1 j) = r ∧ ((p.1 j : ScoreSpace ε) : ℝ) ≤ p.2 then
armProb a (p.1 j) else 0 := by
funext p
rw [empiricalScoreCells_real_eq_scoreCellFrequency hOverlap g a r _
(measurableSet_le (by fun_prop) measurable_const)]
unfold scoreCellFrequency
congr 1
apply Finset.sum_congr rfl
intro j _
by_cases hj : g (p.1 j) = r ∧ ((p.1 j : ScoreSpace ε) : ℝ) ≤ p.2
· simp [indicator, hj]
· simp [indicator, hj]
rw [heq]
apply Measurable.const_mul
apply Finset.measurable_sum
intro j _
have hj : Measurable (fun p : (Fin m → ScoreSpace ε) × ℝ => p.1 j) :=
(measurable_pi_apply j).comp measurable_fst
have hp : Measurable (armProb a : ScoreSpace ε → ℝ) := by
unfold armProb
cases a <;> simp only [Bool.false_eq_true, ↓reduceIte] <;> fun_prop
apply Measurable.ite
· exact (measurableSet_eq_fun (hg.comp hj) measurable_const).inter
(measurableSet_le (measurable_subtype_coe.comp hj) measurable_snd)
· exact hp.comp hj
· exact measurable_const
theorem measurable_scorePopulationCellCDF
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Measurable (fun t : ℝ => ∫ e, {e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator (armProb a) e ∂H)
Proof (Lean source)
lemma measurable_scorePopulationCellCDF {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace J) (hg : Measurable g)
(a : ArmSpace) (r : LabelSpace J) :
Measurable (fun t : ℝ =>
∫ e, {e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator
(armProb a) e ∂H) := by
have hp : Measurable (armProb a : ScoreSpace ε → ℝ) := by
unfold armProb
cases a <;> simp only [Bool.false_eq_true, ↓reduceIte] <;> fun_prop
have hF : Measurable (fun p : ℝ × ScoreSpace ε =>
{e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ p.1}.indicator
(armProb a) p.2) := by
apply Measurable.indicator
· exact hp.comp measurable_snd
· exact (measurableSet_eq_fun (hg.comp measurable_snd) measurable_const).inter
(measurableSet_le (measurable_subtype_coe.comp measurable_snd) measurable_fst)
exact (hF.stronglyMeasurable.integral_prod_right' (ν := H)).measurable
theorem integrable_empiricalScoreCellCDF_joint
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Integrable (uncurry (fun (x : Fin m → ScoreSpace ε) (t : ℝ) => |(empiricalScoreCells g x a r).real {e | (e : ℝ) ≤ t} - ∫ e, {e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator (armProb a) e ∂H|)) ((Measure.pi fun _ : Fin m => H).prod (volume.restrict (Ioc ε (1 - ε))))
Proof (Lean source)
lemma integrable_empiricalScoreCellCDF_joint {ε : ℝ} {J m : ℕ}
(hOverlap : Overlap ε) (H : Measure (ScoreSpace ε))
[IsProbabilityMeasure H] (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (hm : 0 < m) (a : ArmSpace) (r : LabelSpace J) :
Integrable (uncurry
(fun (x : Fin m → ScoreSpace ε) (t : ℝ) =>
|(empiricalScoreCells g x a r).real {e | (e : ℝ) ≤ t} -
∫ e, {e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator
(armProb a) e ∂H|))
((Measure.pi fun _ : Fin m => H).prod
(volume.restrict (Ioc ε (1 - ε)))) := by
have hmeas : Measurable (uncurry
(fun (x : Fin m → ScoreSpace ε) (t : ℝ) =>
|(empiricalScoreCells g x a r).real {e | (e : ℝ) ≤ t} -
∫ e, {e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator
(armProb a) e ∂H|)) :=
(measurable_empiricalScoreCellCDF_joint hOverlap g hg a r |>.sub
((measurable_scorePopulationCellCDF H g hg a r).comp measurable_snd)).abs
apply Integrable.of_bound hmeas.aestronglyMeasurable 2
apply Filter.Eventually.of_forall
rintro ⟨x, t⟩
have hB : MeasurableSet {e : ScoreSpace ε | (e : ℝ) ≤ t} :=
measurableSet_le (by fun_prop) measurable_const
dsimp
rw [empiricalScoreCells_real_eq_scoreCellFrequency hOverlap g a r _ hB]
have hp : ∀ e : ScoreSpace ε, 0 ≤ armProb a e ∧ armProb a e ≤ 1 := by
intro e
exact ⟨le_trans (le_of_lt hOverlap.1) (projectionArmProb_bounds hOverlap a e).1,
(projectionArmProb_bounds hOverlap a e).2⟩
have hpop : 0 ≤ ∫ e, {e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator
(armProb a) e ∂H := by
apply integral_nonneg
intro e
by_cases he : g e = r ∧ (e : ℝ) ≤ t <;> simp [indicator, he, (hp e).1]
have hpop1 : (∫ e, {e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator
(armProb a) e ∂H) ≤ 1 := by
calc
_ ≤ ∫ _e : ScoreSpace ε, (1 : ℝ) ∂H := by
apply integral_mono_of_nonneg
· exact Filter.Eventually.of_forall (fun e => by
by_cases he : g e = r ∧ (e : ℝ) ≤ t <;>
simp [indicator, he, (hp e).1])
· exact integrable_const _
· exact Filter.Eventually.of_forall (fun e => by
by_cases he : g e = r ∧ (e : ℝ) ≤ t <;>
simp [indicator, he, (hp e).2])
_ = 1 := by simp
have hsample : 0 ≤ scoreCellFrequency g a r {e | (e : ℝ) ≤ t} x ∧
scoreCellFrequency g a r {e | (e : ℝ) ≤ t} x ≤ 1 := by
unfold scoreCellFrequency
have hs (j : Fin m) :
0 ≤ {e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator
(armProb a) (x j) ∧
{e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator
(armProb a) (x j) ≤ 1 := by
by_cases he : g (x j) = r ∧ ((x j : ScoreSpace ε) : ℝ) ≤ t <;>
simp [indicator, he, (hp (x j)).1, (hp (x j)).2]
have hn : (0 : ℝ) < m := Nat.cast_pos.mpr hm
constructor
· exact mul_nonneg (inv_nonneg.mpr (le_of_lt hn))
(Finset.sum_nonneg (fun j _ => (hs j).1))
· apply (inv_mul_le_iff₀ hn).mpr
simpa using (Finset.sum_le_sum (fun j _ => (hs j).2) :
(∑ j : Fin m, {e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator
(armProb a) (x j)) ≤ ∑ _j : Fin m, (1 : ℝ))
rw [abs_abs]
apply abs_le.mpr
constructor <;> linarith [hsample.1, hsample.2, hpop, hpop1]
theorem meanIntegratedAbs_empiricalScoreCellCDF_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
∫ x, (∫ t in ε..(1 - ε), |(empiricalScoreCells g x a r).real {e | (e : ℝ) ≤ t} - ∫ e, {e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator (armProb a) e ∂H|) ∂Measure.pi (fun _ : Fin m => H)
≤ (1 - 2 * ε) / (2 * sqrt m)
Proof (Lean source)
lemma meanIntegratedAbs_empiricalScoreCellCDF_pi {ε : ℝ} {J m : ℕ}
(hOverlap : Overlap ε) (H : Measure (ScoreSpace ε))
[IsProbabilityMeasure H] (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (hm : 0 < m) (a : ArmSpace) (r : LabelSpace J) :
∫ x, (∫ t in ε..(1 - ε),
|(empiricalScoreCells g x a r).real {e | (e : ℝ) ≤ t} -
∫ e, {e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator
(armProb a) e ∂H|)
∂Measure.pi (fun _ : Fin m => H) ≤
(1 - 2 * ε) / (2 * sqrt m) := by
have hpoint (t : ℝ) (_ht : t ∈ Ioc ε (1 - ε)) :
∫ x, |(empiricalScoreCells g x a r).real {e | (e : ℝ) ≤ t} -
∫ e, {e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator
(armProb a) e ∂H|
∂Measure.pi (fun _ : Fin m => H) ≤ 1 / (2 * sqrt m) := by
have hB : MeasurableSet {e : ScoreSpace ε | (e : ℝ) ≤ t} :=
measurableSet_le (by fun_prop) measurable_const
exact meanAbs_empiricalScoreCells_pi hOverlap H g hg hm a r _ hB
have hε : ε ≤ 1 - ε := by linarith [hOverlap.2]
have h := projectionExpectedIntegratedError_le
(Measure.pi fun _ : Fin m => H) ε (1 - ε)
(1 / (2 * sqrt m)) hε _
(integrable_empiricalScoreCellCDF_joint hOverlap H g hg hm a r) hpoint
convert h using 1
ring
theorem sum_meanTrialCellCDFError_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(∑ a : ArmSpace, ∑ r : LabelSpace J, ((∫ x : TrialSample n J, |(empiricalTrialCells x a r).real univ - μ.real {z : Observation J | z.1 = r ∧ z.2.1 = a}| ∂Measure.pi (fun _ : Fin n => μ)) + (∫ x : TrialSample n J, (∫ t in (0 : ℝ)..1, |(empiricalTrialCells x a r).real {y | (y : ℝ) ≤ t} - μ.real {z : Observation J | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t}|) ∂Measure.pi (fun _ : Fin n => μ))))
≤ 2 * J / sqrt n
Proof (Lean source)
lemma sum_meanTrialCellCDFError_pi {J n : ℕ}
(μ : Measure (Observation J)) [IsProbabilityMeasure μ] (hn : 0 < n) :
(∑ a : ArmSpace, ∑ r : LabelSpace J,
((∫ x : TrialSample n J, |(empiricalTrialCells x a r).real univ -
μ.real {z : Observation J | z.1 = r ∧ z.2.1 = a}| ∂Measure.pi (fun _ : Fin n => μ)) +
(∫ x : TrialSample n J, (∫ t in (0 : ℝ)..1,
|(empiricalTrialCells x a r).real {y | (y : ℝ) ≤ t} -
μ.real {z : Observation J | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t}|)
∂Measure.pi (fun _ : Fin n => μ)))) ≤ 2 * J / sqrt n := by
have hmass : (∑ a : ArmSpace, ∑ r : LabelSpace J,
∫ x : TrialSample n J, |(empiricalTrialCells x a r).real univ -
μ.real {z : Observation J | z.1 = r ∧ z.2.1 = a}| ∂Measure.pi (fun _ : Fin n => μ)) ≤
J / sqrt n := by
simpa only [empiricalTrialCells_real_apply _ _ univ MeasurableSet.univ,
Set.mem_univ, and_true] using
sum_meanAbs_trialCellFrequency_pi μ hn
(fun _ _ => univ) (fun _ _ => MeasurableSet.univ)
have hcdf : (∑ a : ArmSpace, ∑ r : LabelSpace J,
∫ x : TrialSample n J, (∫ t in (0 : ℝ)..1,
|(empiricalTrialCells x a r).real {y | (y : ℝ) ≤ t} -
μ.real {z : Observation J | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t}|)
∂Measure.pi (fun _ : Fin n => μ)) ≤ J / sqrt n := by
calc
_ ≤ ∑ a : ArmSpace, ∑ _r : LabelSpace J,
(1 / (2 * sqrt n) : ℝ) := by
apply Finset.sum_le_sum
intro a _
apply Finset.sum_le_sum
intro r _
exact meanIntegratedAbs_empiricalTrialCellCDF_pi μ hn a r
_ = J / sqrt n := by
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin,
Fintype.card_bool, nsmul_eq_mul]
field_simp
ring
simp_rw [Finset.sum_add_distrib]
calc
_ ≤ J / sqrt n + J / sqrt n := add_le_add hmass hcdf
_ = 2 * J / sqrt n := by ring
theorem sum_meanScoreCellCDFError_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(∑ a : ArmSpace, ∑ r : LabelSpace J, ((∫ x : Fin m → ScoreSpace ε, |(empiricalScoreCells g x a r).real univ - ∫ e, {e : ScoreSpace ε | g e = r}.indicator (armProb a) e ∂H| ∂Measure.pi (fun _ : Fin m => H)) + (∫ x : Fin m → ScoreSpace ε, (∫ t in ε..(1 - ε), |(empiricalScoreCells g x a r).real {e | (e : ℝ) ≤ t} - ∫ e, {e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator (armProb a) e ∂H|) ∂Measure.pi (fun _ : Fin m => H))))
≤ J * (2 - 2 * ε) / sqrt m
Proof (Lean source)
lemma sum_meanScoreCellCDFError_pi {ε : ℝ} {J m : ℕ}
(hOverlap : Overlap ε) (H : Measure (ScoreSpace ε))
[IsProbabilityMeasure H] (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (hm : 0 < m) :
(∑ a : ArmSpace, ∑ r : LabelSpace J,
((∫ x : Fin m → ScoreSpace ε,
|(empiricalScoreCells g x a r).real univ -
∫ e, {e : ScoreSpace ε | g e = r}.indicator (armProb a) e ∂H|
∂Measure.pi (fun _ : Fin m => H)) +
(∫ x : Fin m → ScoreSpace ε, (∫ t in ε..(1 - ε),
|(empiricalScoreCells g x a r).real {e | (e : ℝ) ≤ t} -
∫ e, {e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator
(armProb a) e ∂H|)
∂Measure.pi (fun _ : Fin m => H)))) ≤
J * (2 - 2 * ε) / sqrt m := by
have hmass : (∑ a : ArmSpace, ∑ r : LabelSpace J,
∫ x : Fin m → ScoreSpace ε,
|(empiricalScoreCells g x a r).real univ -
∫ e, {e : ScoreSpace ε | g e = r}.indicator (armProb a) e ∂H|
∂Measure.pi (fun _ : Fin m => H)) ≤ J / sqrt m := by
simpa only [Set.mem_univ, and_true] using
sum_meanAbs_empiricalScoreCells_pi hOverlap H g hg hm
(fun _ _ => univ) (fun _ _ => MeasurableSet.univ)
have hcdf : (∑ a : ArmSpace, ∑ r : LabelSpace J,
∫ x : Fin m → ScoreSpace ε, (∫ t in ε..(1 - ε),
|(empiricalScoreCells g x a r).real {e | (e : ℝ) ≤ t} -
∫ e, {e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator
(armProb a) e ∂H|)
∂Measure.pi (fun _ : Fin m => H)) ≤
J * (1 - 2 * ε) / sqrt m := by
calc
_ ≤ ∑ a : ArmSpace, ∑ _r : LabelSpace J,
((1 - 2 * ε) / (2 * sqrt m) : ℝ) := by
apply Finset.sum_le_sum
intro a _
apply Finset.sum_le_sum
intro r _
exact meanIntegratedAbs_empiricalScoreCellCDF_pi hOverlap H g hg hm a r
_ = J * (1 - 2 * ε) / sqrt m := by
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin,
Fintype.card_bool, nsmul_eq_mul]
field_simp
ring
simp_rw [Finset.sum_add_distrib]
calc
_ ≤ J / sqrt m + J * (1 - 2 * ε) / sqrt m :=
add_le_add hmass hcdf
_ = J * (2 - 2 * ε) / sqrt m := by ring
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionPopulation 6 declarations Population arm-cell finite measures underlying the external projection.
Population arm-cell finite measures underlying the external projection.
def trialPopulationCells
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
theorem trialPopulationCells_apply
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
trialPopulationCells μ a r B = μ {z | z.1 = r ∧ z.2.1 = a ∧ z.2.2 ∈ B}
Proof (Lean source)
lemma trialPopulationCells_apply {J : ℕ} (μ : Measure (Observation J))
(a : ArmSpace) (r : LabelSpace J) (B : Set OutcomeSpace)
(hB : MeasurableSet B) :
trialPopulationCells μ a r B =
μ {z | z.1 = r ∧ z.2.1 = a ∧ z.2.2 ∈ B} := by
rw [trialPopulationCells, Measure.map_apply measurable_snd.snd hB,
Measure.restrict_apply (hB.preimage measurable_snd.snd)]
congr 1
ext z
simp only [Set.mem_inter_iff, Set.mem_preimage, Set.mem_ofPred_eq]
tauto
theorem trialPopulationCells_real_apply
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(trialPopulationCells μ a r).real B = μ.real {z | z.1 = r ∧ z.2.1 = a ∧ z.2.2 ∈ B}
Proof (Lean source)
lemma trialPopulationCells_real_apply {J : ℕ} (μ : Measure (Observation J))
(a : ArmSpace) (r : LabelSpace J) (B : Set OutcomeSpace)
(hB : MeasurableSet B) :
(trialPopulationCells μ a r).real B =
μ.real {z | z.1 = r ∧ z.2.1 = a ∧ z.2.2 ∈ B} := by
unfold Measure.real
rw [trialPopulationCells_apply μ a r B hB]
theorem outcomeCDFDistance_trialPopulationCells
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
- μ.real {z | z.1 = r ∧ z.2.1 = a}|
+ ∫ t in (0 : ℝ)..1, |(empiricalTrialCells x a r).real {y | (y : ℝ) ≤ t}
- μ.real {z | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t}|
Proof (Lean source)
lemma outcomeCDFDistance_trialPopulationCells {J n : ℕ}
(μ : Measure (Observation J)) (x : TrialSample n J)
(a : ArmSpace) (r : LabelSpace J) :
outcomeCDFDistance (empiricalTrialCells x a r) (trialPopulationCells μ a r) =
|(empiricalTrialCells x a r).real univ -
μ.real {z | z.1 = r ∧ z.2.1 = a}| +
∫ t in (0 : ℝ)..1,
|(empiricalTrialCells x a r).real {y | (y : ℝ) ≤ t} -
μ.real {z | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t}| := by
unfold outcomeCDFDistance
rw [trialPopulationCells_real_apply μ a r univ MeasurableSet.univ]
simp only [Set.mem_univ, and_true]
congr 1
apply intervalIntegral.integral_congr
intro t _
change |(empiricalTrialCells x a r).real {y | (y : ℝ) ≤ t} -
(trialPopulationCells μ a r).real {y | (y : ℝ) ≤ t}| = _
rw [trialPopulationCells_real_apply μ a r _
(measurableSet_le (by fun_prop) measurable_const)]
rfl
theorem meanOutcomeCDFDistance_trialPopulationCells_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
∫ x : TrialSample n J, outcomeCDFDistance (empiricalTrialCells x a r) (trialPopulationCells μ a r) ∂Measure.pi (fun _ : Fin n => μ)
≤ 1 / sqrt n
Proof (Lean source)
lemma meanOutcomeCDFDistance_trialPopulationCells_pi {J n : ℕ}
(μ : Measure (Observation J)) [IsProbabilityMeasure μ] (hn : 0 < n)
(a : ArmSpace) (r : LabelSpace J) :
∫ x : TrialSample n J,
outcomeCDFDistance (empiricalTrialCells x a r) (trialPopulationCells μ a r)
∂Measure.pi (fun _ : Fin n => μ) ≤ 1 / sqrt n := by
have hmass : Integrable (fun x : TrialSample n J =>
|(empiricalTrialCells x a r).real univ -
μ.real {z | z.1 = r ∧ z.2.1 = a}|)
(Measure.pi fun _ : Fin n => μ) := by
have hmeas : Measurable (fun x : TrialSample n J =>
|(empiricalTrialCells x a r).real univ -
μ.real {z | z.1 = r ∧ z.2.1 = a}|) := by
exact ((measurable_empiricalTrialCells_real_apply a r univ
MeasurableSet.univ).sub measurable_const).abs
apply Integrable.of_bound hmeas.aestronglyMeasurable 1
apply Filter.Eventually.of_forall
intro x
have hset : {y : OutcomeSpace | (y : ℝ) ≤ 1} = univ := by
ext y
simp [y.property.2]
have hx := empiricalTrialCellCDF_mem_Icc hn a r x 1
have hp := trialPopulationCellCDF_mem_Icc μ a r 1
rw [hset] at hx
have hpop : {z : Observation J | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ 1} =
{z | z.1 = r ∧ z.2.1 = a} := by
ext z
simp [z.2.2.property.2]
rw [hpop] at hp
dsimp
rw [abs_abs]
apply abs_le.mpr
constructor <;> linarith [hx.1, hx.2, hp.1, hp.2]
have hcdf : Integrable (fun x : TrialSample n J =>
∫ t in (0 : ℝ)..1,
|(empiricalTrialCells x a r).real {y | (y : ℝ) ≤ t} -
μ.real {z | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t}|)
(Measure.pi fun _ : Fin n => μ) := by
convert (integrable_empiricalTrialCellCDF_joint μ hn a r).integral_prod_left using 1
funext x
rw [intervalIntegral.integral_of_le (by norm_num)]
rfl
simp_rw [outcomeCDFDistance_trialPopulationCells]
rw [integral_add hmass hcdf]
have h₁ := meanAbs_trialCellFrequency_pi μ hn a r univ MeasurableSet.univ
have h₂ := meanIntegratedAbs_empiricalTrialCellCDF_pi μ hn a r
have h₁' : (∫ x : TrialSample n J,
|(empiricalTrialCells x a r).real univ -
μ.real {z | z.1 = r ∧ z.2.1 = a}| ∂Measure.pi (fun _ : Fin n => μ)) ≤
1 / (2 * sqrt n) := by
simpa only [empiricalTrialCells_real_apply a r univ MeasurableSet.univ,
Set.mem_univ, and_true] using h₁
calc
_ ≤ 1 / (2 * sqrt n) + 1 / (2 * sqrt n) := add_le_add h₁' h₂
_ = 1 / sqrt n := by ring
theorem sum_meanOutcomeCDFDistance_trialPopulationCells_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(∑ a : ArmSpace, ∑ r : LabelSpace J, ∫ x : TrialSample n J, outcomeCDFDistance (empiricalTrialCells x a r) (trialPopulationCells μ a r) ∂Measure.pi (fun _ : Fin n => μ))
≤ 2 * J / sqrt n
Proof (Lean source)
lemma sum_meanOutcomeCDFDistance_trialPopulationCells_pi {J n : ℕ}
(μ : Measure (Observation J)) [IsProbabilityMeasure μ] (hn : 0 < n) :
(∑ a : ArmSpace, ∑ r : LabelSpace J,
∫ x : TrialSample n J,
outcomeCDFDistance (empiricalTrialCells x a r) (trialPopulationCells μ a r)
∂Measure.pi (fun _ : Fin n => μ)) ≤ 2 * J / Real.sqrt n := by
calc
_ ≤ ∑ a : ArmSpace, ∑ _r : LabelSpace J, (1 / Real.sqrt n : ℝ) := by
apply Finset.sum_le_sum
intro a _
apply Finset.sum_le_sum
intro r _
exact meanOutcomeCDFDistance_trialPopulationCells_pi μ hn a r
_ = 2 * J / Real.sqrt n := by
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin,
Fintype.card_bool, nsmul_eq_mul]
ring
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionRiskEnvelope 4 declarations Population identification and deterministic excess-length bounds for the external projection interval.
Population identification and deterministic excess-length bounds for the external projection interval.
theorem projectedArmEndpoint_population_eq
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
upper :
conclusion
Proof (Lean source)
lemma projectedArmEndpoint_population_eq {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) (upper : Bool) :
projectedArmEndpoint (trialPopulationCells Prel) (scorePopulationCells H g)
a upper =
if upper then muUpper H g Prel hMass a else muLower H g Prel hMass a := by
unfold projectedArmEndpoint
cases upper with
| false =>
simp only [Bool.false_eq_true, ↓reduceIte]
unfold muLower
apply Finset.sum_congr rfl
intro r _
have hmass : (trialPopulationCells Prel a r).real univ =
armCellMass H g a r := by
rw [trialPopulationCells_real_apply Prel a r univ MeasurableSet.univ]
simpa only [Set.mem_univ, and_true] using hMass.2.2.2 a r
rw [hmass]
by_cases hq : 0 < armCellMass H g a r
· simp only [hq, dif_pos, if_pos]
rw [normalized_trialPopulationCell_eq_armCellOutcomeLaw H g Prel hMass a r hq,
normalized_scorePopulationCell_eq_armCellWeightLaw H g Prel hMass
hOverlap a r hq]
· simp [hq]
| true =>
simp only [↓reduceIte]
unfold muUpper
apply Finset.sum_congr rfl
intro r _
have hmass : (trialPopulationCells Prel a r).real univ =
armCellMass H g a r := by
rw [trialPopulationCells_real_apply Prel a r univ MeasurableSet.univ]
simpa only [Set.mem_univ, and_true] using hMass.2.2.2 a r
rw [hmass]
by_cases hq : 0 < armCellMass H g a r
· simp only [hq, dif_pos, if_pos]
rw [normalized_trialPopulationCell_eq_armCellOutcomeLaw H g Prel hMass a r hq,
normalized_scorePopulationCell_eq_armCellWeightLaw H g Prel hMass
hOverlap a r hq]
· simp [hq]
theorem projectedPopulationATESpan_eq_sharpATELength
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
conclusion
(projectedArmEndpoint (trialPopulationCells Prel) (scorePopulationCells H g) true true - projectedArmEndpoint (trialPopulationCells Prel) (scorePopulationCells H g) false false)
- (projectedArmEndpoint (trialPopulationCells Prel) (scorePopulationCells H g) true false - projectedArmEndpoint (trialPopulationCells Prel) (scorePopulationCells H g) false true)
= sharpATELength H g Prel hMass
Proof (Lean source)
lemma projectedPopulationATESpan_eq_sharpATELength {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) :
(projectedArmEndpoint (trialPopulationCells Prel) (scorePopulationCells H g)
true true -
projectedArmEndpoint (trialPopulationCells Prel) (scorePopulationCells H g)
false false) -
(projectedArmEndpoint (trialPopulationCells Prel) (scorePopulationCells H g)
true false -
projectedArmEndpoint (trialPopulationCells Prel) (scorePopulationCells H g)
false true) =
sharpATELength H g Prel hMass := by
rw [projectedArmEndpoint_population_eq H g Prel hMass hOverlap true true,
projectedArmEndpoint_population_eq H g Prel hMass hOverlap false false,
projectedArmEndpoint_population_eq H g Prel hMass hOverlap true false,
projectedArmEndpoint_population_eq H g Prel hMass hOverlap false true]
rfl
theorem externalProjectionIntervalProcedure_excessLength_le_of_endpoint_bounds
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
α :
ℝ
hOverlap :
Overlap ε
hg :
x :
ExternalSample ε J n m
δ :
ℝ
hδ :
0 ≤ δ
hSharp :
0 ≤ sharpATELength H g Prel hMass
hendpoint :
conclusion
max 0 ((externalProjectionIntervalProcedure g α hOverlap hg).length x - sharpATELength H g Prel hMass)
≤ 4 * δ
Proof (Lean source)
lemma externalProjectionIntervalProcedure_excessLength_le_of_endpoint_bounds
{ε : ℝ} {J n m : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(α : ℝ) (hOverlap : Overlap ε) (hg : Measurable g)
(x : ExternalSample ε J n m) {δ : ℝ}
(hδ : 0 ≤ δ) (hSharp : 0 ≤ sharpATELength H g Prel hMass)
(hendpoint : ∀ b ∈ sievedProjectionCandidates g α x, ∀ a upper,
|projectedArmEndpoint b.1 b.2 a upper -
projectedArmEndpoint (trialPopulationCells Prel)
(scorePopulationCells H g) a upper| ≤ δ) :
max 0 ((externalProjectionIntervalProcedure g α hOverlap hg).length x -
sharpATELength H g Prel hMass) ≤ 4 * δ := by
classical
let C := Icc (-1 : ℝ) 1 ∩
closure (convexHull ℝ
(⋃ b ∈ sievedProjectionCandidates g α x,
projectedATEInterval b.1 b.2))
by_cases hC : C.Nonempty
· have hLU :
projectedArmEndpoint (trialPopulationCells Prel) (scorePopulationCells H g)
true false -
projectedArmEndpoint (trialPopulationCells Prel) (scorePopulationCells H g)
false true ≤
projectedArmEndpoint (trialPopulationCells Prel) (scorePopulationCells H g)
true true -
projectedArmEndpoint (trialPopulationCells Prel) (scorePopulationCells H g)
false false := by
have hspan := projectedPopulationATESpan_eq_sharpATELength
H g Prel hMass hOverlap
linarith
have hbound := externalProjectionCI_excessLength_le_of_armEndpoint_bounds
g α x (trialPopulationCells Prel) (scorePopulationCells H g)
hδ hLU (fun _ => hC) hendpoint
obtain ⟨lo, hi, heq, hb⟩ := hbound
have hne := externalProjectionCI_nonempty g α x
have hlohi : lo ≤ hi := by
obtain ⟨z, hz⟩ := hne
rw [heq] at hz
exact hz.1.trans hz.2
let CI : ExternalIntervalProcedure ε J n m :=
externalProjectionIntervalProcedure g α hOverlap hg
have hproc : Icc (CI.lo x) (CI.hi x) = sievedTotalizedProjectionCI g α x :=
externalProjectionIntervalProcedure_Icc g α hOverlap hg x
have hprocord : CI.lo x ≤ CI.hi x := CI.ordered x
have hlo : CI.lo x = lo := by
apply le_antisymm
· have hz : lo ∈ Icc (CI.lo x) (CI.hi x) := by
rw [hproc, heq]
exact Set.left_mem_Icc.mpr hlohi
exact hz.1
· have hz : CI.lo x ∈ Icc lo hi := by
rw [← heq, ← hproc]
exact Set.left_mem_Icc.mpr hprocord
exact hz.1
have hhi : CI.hi x = hi := by
apply le_antisymm
· have hz : CI.hi x ∈ Icc lo hi := by
rw [← heq, ← hproc]
exact Set.right_mem_Icc.mpr hprocord
exact hz.2
· have hz : hi ∈ Icc (CI.lo x) (CI.hi x) := by
rw [hproc, heq]
exact Set.right_mem_Icc.mpr hlohi
exact hz.2
simpa [ExternalIntervalProcedure.length, CI, hlo, hhi,
projectedPopulationATESpan_eq_sharpATELength H g Prel hMass hOverlap] using hb
· have hfallback : sievedTotalizedProjectionCI g α x = {0} := by
rw [sievedTotalizedProjectionCI]
change (if (sievedProjectionCandidates g α x).Nonempty ∧ C.Nonempty
then C else {0}) = {0}
simp [hC]
let CI : ExternalIntervalProcedure ε J n m :=
externalProjectionIntervalProcedure g α hOverlap hg
have hproc : Icc (CI.lo x) (CI.hi x) = sievedTotalizedProjectionCI g α x :=
externalProjectionIntervalProcedure_Icc g α hOverlap hg x
have hlo : CI.lo x = 0 := by
have hz : CI.lo x ∈ sievedTotalizedProjectionCI g α x := by
rw [← hproc]
exact Set.left_mem_Icc.mpr (CI.ordered x)
simpa [hfallback] using hz
have hhi : CI.hi x = 0 := by
have hz : CI.hi x ∈ sievedTotalizedProjectionCI g α x := by
rw [← hproc]
exact Set.right_mem_Icc.mpr (CI.ordered x)
simpa [hfallback] using hz
simp only [ExternalIntervalProcedure.length, CI, hlo, hhi, sub_self]
rw [max_eq_left]
· exact mul_nonneg (by norm_num) hδ
· linarith
theorem externalProjectionIntervalProcedure_population_excessLength_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
PH :
hPH :
PH ∈ ExternalLaws g
x :
ExternalSample ε J n m
hα :
0 < α
hn :
0 < n
hm :
0 < m
hSharp :
conclusion
δ :
(ε⁻¹ + (ε ^ 2)⁻¹) * (4 * J / (α * sqrt n) + (∑ a : ArmSpace, ∑ r : LabelSpace J, outcomeCDFDistance (empiricalTrialCells x.1 a r) (trialPopulationCells (releasedLaw PH.1) a r)) + 2 * J * (2 - 2 * ε) / (α * sqrt m) + (∑ a : ArmSpace, ∑ r : LabelSpace J, scoreCDFDistance (empiricalScoreCells g x.2 a r) (scorePopulationCells PH.2 g a r)))
max 0 ((externalProjectionIntervalProcedure g α hPH.2.2.overlap hPH.2.2.measurableRelease).length x - sharpATELength PH.2 g (releasedLaw PH.1) (externalLawCellMasses g PH hPH))
≤ 4 * δ
Proof (Lean source)
lemma externalProjectionIntervalProcedure_population_excessLength_le
{ε : ℝ} {J n m : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(PH : Measure (FullRow ε J) × Measure (ScoreSpace ε))
(hPH : PH ∈ ExternalLaws g) (x : ExternalSample ε J n m)
(hα : 0 < α) (hn : 0 < n) (hm : 0 < m)
(hSharp : 0 ≤ sharpATELength PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH)) :
let δ := (ε⁻¹ + (ε ^ 2)⁻¹) *
(4 * J / (α * sqrt n) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (empiricalTrialCells x.1 a r)
(trialPopulationCells (releasedLaw PH.1) a r)) +
2 * J * (2 - 2 * ε) / (α * sqrt m) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (empiricalScoreCells g x.2 a r)
(scorePopulationCells PH.2 g a r)))
max 0 ((externalProjectionIntervalProcedure g α hPH.2.2.overlap
hPH.2.2.measurableRelease).length x -
sharpATELength PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH)) ≤ 4 * δ := by
dsimp only
have hOverlap := hPH.2.2.overlap
have hout : 0 ≤ ∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (empiricalTrialCells x.1 a r)
(trialPopulationCells (releasedLaw PH.1) a r) :=
Finset.sum_nonneg (fun a _ =>
Finset.sum_nonneg (fun r _ => outcomeCDFDistance_nonneg _ _))
have hscore : 0 ≤ ∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (empiricalScoreCells g x.2 a r)
(scorePopulationCells PH.2 g a r) :=
Finset.sum_nonneg (fun a _ =>
Finset.sum_nonneg (fun r _ => scoreCDFDistance_nonneg hOverlap _ _))
have htrialRadius : 0 ≤ 4 * J / (α * sqrt n) := by positivity
have hwidth : 0 ≤ 2 - 2 * ε := by linarith [hOverlap.2]
have hsqrtm : 0 < sqrt m := Real.sqrt_pos.2 (by exact_mod_cast hm)
have hlogRadius : 0 ≤ 2 * J * (2 - 2 * ε) / (α * sqrt m) := by
exact div_nonneg (mul_nonneg (mul_nonneg (by positivity) (by positivity)) hwidth)
(mul_nonneg hα.le hsqrtm.le)
have hL : 0 ≤ ε⁻¹ + (ε ^ 2)⁻¹ :=
add_nonneg (inv_nonneg.mpr hOverlap.1.le) (inv_nonneg.mpr (sq_nonneg ε))
apply externalProjectionIntervalProcedure_excessLength_le_of_endpoint_bounds
PH.2 g (releasedLaw PH.1) (externalLawCellMasses g PH hPH) α
hOverlap hPH.2.2.measurableRelease x
· exact mul_nonneg hL (by linarith)
· exact hSharp
· intro b hb a upper
exact projectionCandidate_populationArmEndpoint_abs_le
g α PH hPH x b hb.1 a upper
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionRiskExpectation 4 declarations Expected excess-length bounds obtained by integrating the deterministic projection-risk envelope.
Expected excess-length bounds obtained by integrating the deterministic projection-risk envelope.
theorem integrable_outcomeCDFDistance_population_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Integrable (fun x : TrialSample n J => outcomeCDFDistance (empiricalTrialCells x a r) (trialPopulationCells μ a r)) (Measure.pi fun _ : Fin n => μ)
Proof (Lean source)
lemma integrable_outcomeCDFDistance_population_pi {J n : ℕ}
(μ : Measure (Observation J)) [IsProbabilityMeasure μ] (hn : 0 < n)
(a : ArmSpace) (r : LabelSpace J) :
Integrable (fun x : TrialSample n J =>
outcomeCDFDistance (empiricalTrialCells x a r) (trialPopulationCells μ a r))
(Measure.pi fun _ : Fin n => μ) := by
have hmass : Integrable (fun x : TrialSample n J =>
|(empiricalTrialCells x a r).real univ -
μ.real {z | z.1 = r ∧ z.2.1 = a}|)
(Measure.pi fun _ : Fin n => μ) := by
have hmeas : Measurable (fun x : TrialSample n J =>
|(empiricalTrialCells x a r).real univ -
μ.real {z | z.1 = r ∧ z.2.1 = a}|) := by
exact ((measurable_empiricalTrialCells_real_apply a r univ
MeasurableSet.univ).sub measurable_const).abs
apply Integrable.of_bound hmeas.aestronglyMeasurable 1
apply Filter.Eventually.of_forall
intro x
have hset : {y : OutcomeSpace | (y : ℝ) ≤ 1} = univ := by
ext y
simp [y.property.2]
have hx := empiricalTrialCellCDF_mem_Icc hn a r x 1
have hp := trialPopulationCellCDF_mem_Icc μ a r 1
rw [hset] at hx
have hpop : {z : Observation J | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ 1} =
{z | z.1 = r ∧ z.2.1 = a} := by
ext z
simp [z.2.2.property.2]
rw [hpop] at hp
dsimp
rw [abs_abs]
apply abs_le.mpr
constructor <;> linarith [hx.1, hx.2, hp.1, hp.2]
have hcdf : Integrable (fun x : TrialSample n J =>
∫ t in (0 : ℝ)..1,
|(empiricalTrialCells x a r).real {y | (y : ℝ) ≤ t} -
μ.real {z | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t}|)
(Measure.pi fun _ : Fin n => μ) := by
convert (integrable_empiricalTrialCellCDF_joint μ hn a r).integral_prod_left using 1
funext x
rw [intervalIntegral.integral_of_le (by norm_num)]
rfl
rw [show (fun x : TrialSample n J =>
outcomeCDFDistance (empiricalTrialCells x a r) (trialPopulationCells μ a r)) =
fun x =>
|(empiricalTrialCells x a r).real univ -
μ.real {z | z.1 = r ∧ z.2.1 = a}| +
∫ t in (0 : ℝ)..1,
|(empiricalTrialCells x a r).real {y | (y : ℝ) ≤ t} -
μ.real {z | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ≤ t}| by
funext x
exact outcomeCDFDistance_trialPopulationCells μ x a r]
exact hmass.add hcdf
theorem integrable_scoreCDFDistance_population_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Integrable (fun x : Fin m → ScoreSpace ε => scoreCDFDistance (empiricalScoreCells g x a r) (scorePopulationCells H g a r)) (Measure.pi fun _ : Fin m => H)
Proof (Lean source)
lemma integrable_scoreCDFDistance_population_pi {ε : ℝ} {J m : ℕ}
(hOverlap : Overlap ε) (H : Measure (ScoreSpace ε))
[IsProbabilityMeasure H] (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (hm : 0 < m) (a : ArmSpace) (r : LabelSpace J) :
Integrable (fun x : Fin m → ScoreSpace ε =>
scoreCDFDistance (empiricalScoreCells g x a r) (scorePopulationCells H g a r))
(Measure.pi fun _ : Fin m => H) := by
have hmass : Integrable (fun x : Fin m → ScoreSpace ε =>
|(empiricalScoreCells g x a r).real univ -
∫ e, {e : ScoreSpace ε | g e = r}.indicator (armProb a) e ∂H|)
(Measure.pi fun _ : Fin m => H) := by
let F : ScoreSpace ε → ℝ :=
{e : ScoreSpace ε | g e = r}.indicator (armProb a)
have hC : MeasurableSet {e : ScoreSpace ε | g e = r} :=
measurableSet_eq_fun hg measurable_const
have hp : Measurable (armProb a : ScoreSpace ε → ℝ) := by
unfold armProb
cases a <;> simp only [Bool.false_eq_true, ↓reduceIte] <;> fun_prop
have hF : Measurable F := hp.indicator hC
have hbound : ∀ e, F e ∈ Icc (0 : ℝ) 1 := by
intro e
by_cases he : g e = r
· simpa [F, indicator, he] using
(show armProb a e ∈ Icc (0 : ℝ) 1 from
⟨le_trans (le_of_lt hOverlap.1) (projectionArmProb_bounds hOverlap a e).1,
(projectionArmProb_bounds hOverlap a e).2⟩)
· simp [F, he]
have hlp : MemLp F 2 H :=
memLp_of_bounded (Filter.Eventually.of_forall hbound) hF.aestronglyMeasurable 2
have hcoord (j : Fin m) :
MemLp (fun x : Fin m → ScoreSpace ε => F (x j)) 2
(Measure.pi fun _ : Fin m => H) :=
hlp.comp_measurePreserving (measurePreserving_eval (fun _ : Fin m => H) j)
have hX : MemLp (scoreCellFrequency g a r univ) 2
(Measure.pi fun _ : Fin m => H) := by
have hs := memLp_finsetSum univ (fun j _ => hcoord j)
have hs' := hs.const_mul (m : ℝ)⁻¹
unfold scoreCellFrequency
simpa only [F, Set.mem_univ, and_true] using hs'
have hXi : Integrable (scoreCellFrequency g a r univ)
(Measure.pi fun _ : Fin m => H) :=
memLp_one_iff_integrable.mp (hX.mono_exponent (by norm_num))
have hconst : Integrable (fun _ : Fin m → ScoreSpace ε =>
∫ e, {e : ScoreSpace ε | g e = r}.indicator (armProb a) e ∂H)
(Measure.pi fun _ : Fin m => H) := integrable_const _
convert (hXi.sub hconst).abs using 1
funext x
rw [empiricalScoreCells_real_eq_scoreCellFrequency hOverlap g a r
univ MeasurableSet.univ]
rfl
have hcdf : Integrable (fun x : Fin m → ScoreSpace ε =>
∫ t in ε..(1 - ε),
|(empiricalScoreCells g x a r).real {e | (e : ℝ) ≤ t} -
∫ e, {e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator
(armProb a) e ∂H|)
(Measure.pi fun _ : Fin m => H) := by
convert (integrable_empiricalScoreCellCDF_joint
hOverlap H g hg hm a r).integral_prod_left using 1
funext x
rw [intervalIntegral.integral_of_le (by linarith [hOverlap.2])]
rfl
rw [show (fun x : Fin m → ScoreSpace ε =>
scoreCDFDistance (empiricalScoreCells g x a r) (scorePopulationCells H g a r)) =
fun x =>
|(empiricalScoreCells g x a r).real univ -
∫ e, {e : ScoreSpace ε | g e = r}.indicator (armProb a) e ∂H| +
∫ t in ε..(1 - ε),
|(empiricalScoreCells g x a r).real {e | (e : ℝ) ≤ t} -
∫ e, {e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator
(armProb a) e ∂H| by
funext x
exact scoreCDFDistance_scorePopulationCells hOverlap H g hg x a r]
exact hmass.add hcdf
theorem externalProjectionIntervalProcedure_population_meanExcessLength_le_of_sharp_nonneg
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
Qj :
hTrial :
JointTrialIID g Qj
hLog :
ExternalLogIID g Qj
hInd :
PH :
hPH :
PH ∈ ExternalLaws g
hn :
0 < n
hm :
0 < m
α :
ℝ
hα :
0 < α
hSharp :
conclusion
∫ x, max 0 ((externalProjectionIntervalProcedure g α hPH.2.2.overlap hPH.2.2.measurableRelease).length x - sharpATELength PH.2 g (releasedLaw PH.1) (externalLawCellMasses g PH hPH)) ∂Qj PH.1 PH.2
Proof (Lean source)
lemma externalProjectionIntervalProcedure_population_meanExcessLength_le_of_sharp_nonneg
{ε : ℝ} {J n m : ℕ} (g : ScoreSpace ε → LabelSpace J)
(Qj : Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : JointTrialIID g Qj) (hLog : ExternalLogIID g Qj)
(hInd : ExternalLogIndependent g Qj)
(PH : Measure (FullRow ε J) × Measure (ScoreSpace ε))
(hPH : PH ∈ ExternalLaws g) (hn : 0 < n) (hm : 0 < m)
(α : ℝ) (hα : 0 < α)
(hSharp : 0 ≤ sharpATELength PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH)) :
∫ x, max 0
((externalProjectionIntervalProcedure g α hPH.2.2.overlap
hPH.2.2.measurableRelease).length x -
sharpATELength PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH)) ∂Qj PH.1 PH.2 ≤
4 * (ε⁻¹ + (ε ^ 2)⁻¹) *
(4 * J / (α * sqrt n) + 2 * J / sqrt n +
2 * J * (2 - 2 * ε) / (α * sqrt m) +
J * (2 - 2 * ε) / sqrt m) := by
let μ := releasedLaw PH.1
let H := PH.2
let F : ExternalSample ε J n m → ℝ := fun x =>
∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (empiricalTrialCells x.1 a r)
(trialPopulationCells μ a r)
let G : ExternalSample ε J n m → ℝ := fun x =>
∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (empiricalScoreCells g x.2 a r)
(scorePopulationCells H g a r)
haveI : IsProbabilityMeasure PH.1 := hPH.1
haveI : IsProbabilityMeasure H := hPH.2.1
haveI : IsProbabilityMeasure μ := by
unfold μ releasedLaw
exact @Measure.isProbabilityMeasure_map _ _ _ _ PH.1 hPH.1 _
(by unfold releasedRecord label arm observed; fun_prop :
Measurable (releasedRecord (ε := ε) (J := J))).aemeasurable
haveI : IsProbabilityMeasure (Qj PH.1 PH.2) := by
rw [externalJointLaw_eq_experiment g Qj hTrial hLog hInd PH hPH]
dsimp [externalExperiment]
infer_instance
have hfin := projectionPopulationCells_finite H g μ hPH.2.2.overlap
hPH.2.2.measurableRelease
have hFTmeas : Measurable (fun x : TrialSample n J =>
∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (empiricalTrialCells x a r)
(trialPopulationCells μ a r)) := by
apply Finset.measurable_sum
intro a _
apply Finset.measurable_sum
intro r _
letI : IsFiniteMeasure (trialPopulationCells μ a r) := hfin.1 a r
simpa only [outcomeCDFDistance, abs_sub_comm] using
measurable_outcomeCDFDistance_empirical (trialPopulationCells μ a r)
(measure_lt_top _ _) a r
have hGTmeas : Measurable (fun x : Fin m → ScoreSpace ε =>
∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (empiricalScoreCells g x a r)
(scorePopulationCells H g a r)) := by
apply Finset.measurable_sum
intro a _
apply Finset.measurable_sum
intro r _
letI : IsFiniteMeasure (scorePopulationCells H g a r) := hfin.2 a r
simpa only [scoreCDFDistance, abs_sub_comm] using
measurable_scoreCDFDistance_empirical hPH.2.2.overlap g
hPH.2.2.measurableRelease (scorePopulationCells H g a r)
(measure_lt_top _ _) a r
have hFmeas : Measurable F := by
exact hFTmeas.comp measurable_fst
have hGmeas : Measurable G := by
exact hGTmeas.comp measurable_snd
have hFpi : Integrable (fun x : TrialSample n J =>
∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (empiricalTrialCells x a r)
(trialPopulationCells μ a r))
(Measure.pi fun _ : Fin n => μ) := by
exact integrable_finsetSum _ (fun a _ =>
integrable_finsetSum _ (fun r _ =>
integrable_outcomeCDFDistance_population_pi μ hn a r))
have hGpi : Integrable (fun x : Fin m → ScoreSpace ε =>
∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (empiricalScoreCells g x a r)
(scorePopulationCells H g a r))
(Measure.pi fun _ : Fin m => H) := by
exact integrable_finsetSum _ (fun a _ =>
integrable_finsetSum _ (fun r _ =>
integrable_scoreCDFDistance_population_pi hPH.2.2.overlap H g
hPH.2.2.measurableRelease hm a r))
have hFint : Integrable F (Qj PH.1 PH.2) := by
have hmap : Measure.map fst (Qj PH.1 PH.2) =
Measure.pi (fun _ : Fin n => μ) := by
simpa [μ] using hTrial PH hPH
rw [← hmap] at hFpi
simpa [F, Function.comp_def] using
hFpi.comp_aemeasurable measurable_fst.aemeasurable
have hGint : Integrable G (Qj PH.1 PH.2) := by
have hmap : Measure.map snd (Qj PH.1 PH.2) =
Measure.pi (fun _ : Fin m => H) := by
simpa [H] using hLog PH hPH
rw [← hmap] at hGpi
simpa [G, Function.comp_def] using
hGpi.comp_aemeasurable measurable_snd.aemeasurable
have hFmean : ∫ x, F x ∂Qj PH.1 PH.2 ≤ 2 * J / sqrt n := by
exact externalJointTrial_meanDistance_le g Qj hTrial PH hPH hn
(by simpa [μ] using hFTmeas)
(fun a r => integrable_outcomeCDFDistance_population_pi μ hn a r)
have hGmean : ∫ x, G x ∂Qj PH.1 PH.2 ≤
J * (2 - 2 * ε) / sqrt m := by
exact externalJointLog_meanDistance_le g Qj hLog PH hPH hm
(by simpa [H] using hGTmeas)
(fun a r => integrable_scoreCDFDistance_population_pi hPH.2.2.overlap H g
hPH.2.2.measurableRelease hm a r)
let C : ExternalIntervalProcedure ε J n m :=
externalProjectionIntervalProcedure g α hPH.2.2.overlap
hPH.2.2.measurableRelease
let S := sharpATELength H g μ (externalLawCellMasses g PH hPH)
have hleftMeas : Measurable (fun x => max 0 (C.length x - S)) := by
exact measurable_const.max
((C.measurable_hi.sub C.measurable_lo).sub measurable_const)
have hleftInt : Integrable (fun x => max 0 (C.length x - S))
(Qj PH.1 PH.2) := by
apply Integrable.of_bound hleftMeas.aestronglyMeasurable 2
apply Filter.Eventually.of_forall
intro x
rw [Real.norm_eq_abs, abs_of_nonneg (le_max_left _ _)]
apply max_le (by norm_num)
dsimp [C, ExternalIntervalProcedure.length, S]
linarith [C.lower_bound x, C.upper_bound x, hSharp]
have hL : 0 ≤ ε⁻¹ + (ε ^ 2)⁻¹ :=
add_nonneg (inv_nonneg.mpr hPH.2.2.overlap.1.le)
(inv_nonneg.mpr (sq_nonneg ε))
let R : ExternalSample ε J n m → ℝ := fun x =>
4 * (ε⁻¹ + (ε ^ 2)⁻¹) *
(4 * J / (α * sqrt n) + F x +
2 * J * (2 - 2 * ε) / (α * sqrt m) + G x)
have hRint : Integrable R (Qj PH.1 PH.2) := by
dsimp [R]
fun_prop
have hpoint : ∀ x, max 0 (C.length x - S) ≤ R x := by
intro x
simpa [C, S, R, F, G, μ, H, mul_assoc] using
externalProjectionIntervalProcedure_population_excessLength_le
g α PH hPH x hα hn hm hSharp
calc
∫ x, max 0 (C.length x - S) ∂Qj PH.1 PH.2 ≤
∫ x, R x ∂Qj PH.1 PH.2 := integral_mono hleftInt hRint hpoint
_ = 4 * (ε⁻¹ + (ε ^ 2)⁻¹) *
(4 * J / (α * sqrt n) + (∫ x, F x ∂Qj PH.1 PH.2) +
2 * J * (2 - 2 * ε) / (α * sqrt m) +
(∫ x, G x ∂Qj PH.1 PH.2)) := by
dsimp [R]
rw [integral_const_mul]
congr 1
rw [integral_add
(f := fun a => 4 * J / (α * sqrt n) + F a +
2 * J * (2 - 2 * ε) / (α * sqrt m))
(g := G) (((integrable_const _).add hFint).add (integrable_const _)) hGint,
integral_add
(f := fun a => 4 * J / (α * sqrt n) + F a)
(g := fun _ => 2 * J * (2 - 2 * ε) / (α * sqrt m))
((integrable_const _).add hFint) (integrable_const _),
integral_add
(f := fun _ => 4 * J / (α * sqrt n)) (g := F)
(integrable_const _) hFint]
simp
_ ≤ 4 * (ε⁻¹ + (ε ^ 2)⁻¹) *
(4 * J / (α * sqrt n) + 2 * J / sqrt n +
2 * J * (2 - 2 * ε) / (α * sqrt m) +
J * (2 - 2 * ε) / sqrt m) := by
exact mul_le_mul_of_nonneg_left (by linarith) (mul_nonneg (by norm_num) hL)
theorem externalProjectionIntervalProcedure_population_meanExcessLength_le
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
g :
ScoreSpace ε → LabelSpace J
Qj :
hTrial :
JointTrialIID g Qj
hLog :
ExternalLogIID g Qj
hInd :
PH :
hPH :
PH ∈ ExternalLaws g
hn :
0 < n
hm :
0 < m
α :
ℝ
hα :
0 < α
conclusion
∫ x, max 0 ((externalProjectionIntervalProcedure g α hPH.2.2.overlap hPH.2.2.measurableRelease).length x - sharpATELength PH.2 g (releasedLaw PH.1) (externalLawCellMasses g PH hPH)) ∂Qj PH.1 PH.2
Proof (Lean source)
theorem externalProjectionIntervalProcedure_population_meanExcessLength_le
{ε : ℝ} {J n m : ℕ} (g : ScoreSpace ε → LabelSpace J)
(Qj : Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : JointTrialIID g Qj) (hLog : ExternalLogIID g Qj)
(hInd : ExternalLogIndependent g Qj)
(PH : Measure (FullRow ε J) × Measure (ScoreSpace ε))
(hPH : PH ∈ ExternalLaws g) (hn : 0 < n) (hm : 0 < m)
(α : ℝ) (hα : 0 < α) :
∫ x, max 0
((externalProjectionIntervalProcedure g α hPH.2.2.overlap
hPH.2.2.measurableRelease).length x -
sharpATELength PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH)) ∂Qj PH.1 PH.2 ≤
4 * (ε⁻¹ + (ε ^ 2)⁻¹) *
(4 * J / (α * sqrt n) + 2 * J / sqrt n +
2 * J * (2 - 2 * ε) / (α * sqrt m) +
J * (2 - 2 * ε) / sqrt m) := by
apply externalProjectionIntervalProcedure_population_meanExcessLength_le_of_sharp_nonneg
g Qj hTrial hLog hInd PH hPH hn hm α hα
exact sharpATELength_nonneg_of_overlap PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH) hPH.2.2.overlap
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionScorePopulation 7 declarations Population weighted score-cell submeasures for the external projection.
Population weighted score-cell submeasures for the external projection.
def scorePopulationCells
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
theorem scorePopulationCells_real_apply
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
hOverlap :
Overlap ε
H :
g :
ScoreSpace ε → LabelSpace J
hg :
a :
r :
B :
Set (ScoreSpace ε)
hB :
conclusion
(scorePopulationCells H g a r).real B
Proof (Lean source)
lemma scorePopulationCells_real_apply {ε : ℝ} {J : ℕ}
(hOverlap : Overlap ε) (H : Measure (ScoreSpace ε))
[IsProbabilityMeasure H] (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (a : ArmSpace) (r : LabelSpace J)
(B : Set (ScoreSpace ε)) (hB : MeasurableSet B) :
(scorePopulationCells H g a r).real B =
∫ e, {e : ScoreSpace ε | g e = r ∧ e ∈ B}.indicator (armProb a) e ∂H := by
let F : ScoreSpace ε → ℝ :=
{e : ScoreSpace ε | g e = r ∧ e ∈ B}.indicator (armProb a)
have hp : Measurable (armProb a : ScoreSpace ε → ℝ) := by
unfold armProb
cases a <;> simp only [Bool.false_eq_true, ↓reduceIte] <;> fun_prop
have hC : MeasurableSet {e : ScoreSpace ε | g e = r ∧ e ∈ B} :=
(measurableSet_eq_fun hg measurable_const).inter hB
have hF : Measurable F := hp.indicator hC
have hbound : ∀ e, F e ∈ Icc (0 : ℝ) 1 := by
intro e
by_cases he : g e = r ∧ e ∈ B
· simpa [F, indicator, he] using
(show armProb a e ∈ Icc (0 : ℝ) 1 from
⟨le_trans (le_of_lt hOverlap.1) (projectionArmProb_bounds hOverlap a e).1,
(projectionArmProb_bounds hOverlap a e).2⟩)
· simp [F, he]
have hInt : Integrable F H := by
apply Integrable.of_bound hF.aestronglyMeasurable 1
apply Filter.Eventually.of_forall
intro e
simpa [Real.norm_eq_abs, abs_of_nonneg (hbound e).1] using (hbound e).2
unfold scorePopulationCells Measure.real
rw [withDensity_apply _ hB]
have hlin : (∫⁻ e in B, ofReal (if g e = r then armProb a e else 0) ∂H) =
∫⁻ e, ofReal (F e) ∂H := by
rw [← lintegral_indicator hB]
apply lintegral_congr
intro e
by_cases he : g e = r <;> by_cases heB : e ∈ B <;>
simp [F, indicator, he, heB]
rw [hlin, ← ofReal_integral_eq_lintegral_ofReal hInt
(Filter.Eventually.of_forall (fun e => (hbound e).1))]
rw [ENNReal.toReal_ofReal (integral_nonneg (fun e => (hbound e).1))]
theorem scoreCDFDistance_scorePopulationCells
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J m :
ℕ
hOverlap :
Overlap ε
H :
g :
ScoreSpace ε → LabelSpace J
hg :
x :
Fin m → ScoreSpace ε
a :
r :
conclusion
+ ∫ t in ε..(1 - ε), |(empiricalScoreCells g x a r).real {e | (e : ℝ) ≤ t}
Proof (Lean source)
lemma scoreCDFDistance_scorePopulationCells {ε : ℝ} {J m : ℕ}
(hOverlap : Overlap ε) (H : Measure (ScoreSpace ε))
[IsProbabilityMeasure H] (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (x : Fin m → ScoreSpace ε)
(a : ArmSpace) (r : LabelSpace J) :
scoreCDFDistance (empiricalScoreCells g x a r) (scorePopulationCells H g a r) =
|(empiricalScoreCells g x a r).real univ -
∫ e, {e : ScoreSpace ε | g e = r}.indicator (armProb a) e ∂H| +
∫ t in ε..(1 - ε),
|(empiricalScoreCells g x a r).real {e | (e : ℝ) ≤ t} -
∫ e, {e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator
(armProb a) e ∂H| := by
unfold scoreCDFDistance
rw [scorePopulationCells_real_apply hOverlap H g hg a r univ MeasurableSet.univ]
simp only [Set.mem_univ, and_true]
congr 1
apply intervalIntegral.integral_congr
intro t _
change |(empiricalScoreCells g x a r).real {e | (e : ℝ) ≤ t} -
(scorePopulationCells H g a r).real {e | (e : ℝ) ≤ t}| = _
rw [show (scorePopulationCells H g a r).real {e | (e : ℝ) ≤ t} =
∫ e, {e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator
(armProb a) e ∂H from
scorePopulationCells_real_apply hOverlap H g hg a r _
(measurableSet_le (by fun_prop) measurable_const)]
theorem meanScoreCDFDistance_scorePopulationCells_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
∫ x : Fin m → ScoreSpace ε, scoreCDFDistance (empiricalScoreCells g x a r) (scorePopulationCells H g a r) ∂Measure.pi (fun _ : Fin m => H)
≤ (1 - ε) / sqrt m
Proof (Lean source)
lemma meanScoreCDFDistance_scorePopulationCells_pi {ε : ℝ} {J m : ℕ}
(hOverlap : Overlap ε) (H : Measure (ScoreSpace ε))
[IsProbabilityMeasure H] (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (hm : 0 < m) (a : ArmSpace) (r : LabelSpace J) :
∫ x : Fin m → ScoreSpace ε,
scoreCDFDistance (empiricalScoreCells g x a r) (scorePopulationCells H g a r)
∂Measure.pi (fun _ : Fin m => H) ≤
(1 - ε) / sqrt m := by
have hmass : Integrable (fun x : Fin m → ScoreSpace ε =>
|(empiricalScoreCells g x a r).real univ -
∫ e, {e : ScoreSpace ε | g e = r}.indicator (armProb a) e ∂H|)
(Measure.pi fun _ : Fin m => H) := by
let F : ScoreSpace ε → ℝ :=
{e : ScoreSpace ε | g e = r}.indicator (armProb a)
have hC : MeasurableSet {e : ScoreSpace ε | g e = r} :=
measurableSet_eq_fun hg measurable_const
have hp : Measurable (armProb a : ScoreSpace ε → ℝ) := by
unfold armProb
cases a <;> simp only [Bool.false_eq_true, ↓reduceIte] <;> fun_prop
have hF : Measurable F := hp.indicator hC
have hbound : ∀ e, F e ∈ Icc (0 : ℝ) 1 := by
intro e
by_cases he : g e = r
· simpa [F, indicator, he] using
(show armProb a e ∈ Icc (0 : ℝ) 1 from
⟨le_trans (le_of_lt hOverlap.1) (projectionArmProb_bounds hOverlap a e).1,
(projectionArmProb_bounds hOverlap a e).2⟩)
· simp [F, he]
have hlp : MemLp F 2 H :=
memLp_of_bounded (Filter.Eventually.of_forall hbound) hF.aestronglyMeasurable 2
have hcoord (j : Fin m) :
MemLp (fun x : Fin m → ScoreSpace ε => F (x j)) 2
(Measure.pi fun _ : Fin m => H) :=
hlp.comp_measurePreserving (measurePreserving_eval (fun _ : Fin m => H) j)
have hX : MemLp (scoreCellFrequency g a r univ) 2
(Measure.pi fun _ : Fin m => H) := by
have hs := memLp_finsetSum univ (fun j _ => hcoord j)
have hs' := hs.const_mul (m : ℝ)⁻¹
unfold scoreCellFrequency
simpa only [F, Set.mem_univ, and_true] using hs'
have hXi : Integrable (scoreCellFrequency g a r univ)
(Measure.pi fun _ : Fin m => H) :=
memLp_one_iff_integrable.mp (hX.mono_exponent (by norm_num))
have hconst : Integrable (fun _ : Fin m → ScoreSpace ε =>
∫ e, {e : ScoreSpace ε | g e = r}.indicator (armProb a) e ∂H)
(Measure.pi fun _ : Fin m => H) := integrable_const _
convert (hXi.sub hconst).abs using 1
funext x
rw [empiricalScoreCells_real_eq_scoreCellFrequency hOverlap g a r
univ MeasurableSet.univ]
rfl
have hcdf : Integrable (fun x : Fin m → ScoreSpace ε =>
∫ t in ε..(1 - ε),
|(empiricalScoreCells g x a r).real {e | (e : ℝ) ≤ t} -
∫ e, {e : ScoreSpace ε | g e = r ∧ (e : ℝ) ≤ t}.indicator
(armProb a) e ∂H|)
(Measure.pi fun _ : Fin m => H) := by
convert (integrable_empiricalScoreCellCDF_joint
hOverlap H g hg hm a r).integral_prod_left using 1
funext x
rw [intervalIntegral.integral_of_le (by linarith [hOverlap.2])]
rfl
simp_rw [scoreCDFDistance_scorePopulationCells hOverlap H g hg]
rw [integral_add hmass hcdf]
have h₁ := meanAbs_empiricalScoreCells_pi hOverlap H g hg hm a r
univ MeasurableSet.univ
have h₂ := meanIntegratedAbs_empiricalScoreCellCDF_pi hOverlap H g hg hm a r
have h₁' : (∫ x : Fin m → ScoreSpace ε,
|(empiricalScoreCells g x a r).real univ -
∫ e, {e : ScoreSpace ε | g e = r}.indicator (armProb a) e ∂H|
∂Measure.pi (fun _ : Fin m => H)) ≤
1 / (2 * sqrt m) := by
simpa only [Set.mem_univ, and_true] using h₁
calc
_ ≤ 1 / (2 * sqrt m) + (1 - 2 * ε) / (2 * sqrt m) :=
add_le_add h₁' h₂
_ = (1 - ε) / sqrt m := by ring
theorem sum_meanScoreCDFDistance_scorePopulationCells_pi
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(∑ a : ArmSpace, ∑ r : LabelSpace J, ∫ x : Fin m → ScoreSpace ε, scoreCDFDistance (empiricalScoreCells g x a r) (scorePopulationCells H g a r) ∂Measure.pi (fun _ : Fin m => H))
≤ J * (2 - 2 * ε) / sqrt m
Proof (Lean source)
lemma sum_meanScoreCDFDistance_scorePopulationCells_pi {ε : ℝ} {J m : ℕ}
(hOverlap : Overlap ε) (H : Measure (ScoreSpace ε))
[IsProbabilityMeasure H] (g : ScoreSpace ε → LabelSpace J)
(hg : Measurable g) (hm : 0 < m) :
(∑ a : ArmSpace, ∑ r : LabelSpace J,
∫ x : Fin m → ScoreSpace ε,
scoreCDFDistance (empiricalScoreCells g x a r) (scorePopulationCells H g a r)
∂Measure.pi (fun _ : Fin m => H)) ≤
J * (2 - 2 * ε) / Real.sqrt m := by
calc
_ ≤ ∑ a : ArmSpace, ∑ _r : LabelSpace J,
((1 - ε) / Real.sqrt m : ℝ) := by
apply Finset.sum_le_sum
intro a _
apply Finset.sum_le_sum
intro r _
exact meanScoreCDFDistance_scorePopulationCells_pi hOverlap H g hg hm a r
_ = J * (2 - 2 * ε) / Real.sqrt m := by
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin,
Fintype.card_bool, nsmul_eq_mul]
ring
theorem projectionCompatible_populationCells
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
μ :
Measure (Observation J)
h :
CompatibleCellMasses H g μ
hOverlap :
Overlap ε
conclusion
Proof (Lean source)
lemma projectionCompatible_populationCells {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(μ : Measure (Observation J)) (h : CompatibleCellMasses H g μ)
(hOverlap : Overlap ε) :
ProjectionCompatible (trialPopulationCells μ) (scorePopulationCells H g) := by
haveI : IsProbabilityMeasure H := h.1
intro a r
rw [trialPopulationCells_real_apply μ a r univ MeasurableSet.univ]
simp only [Set.mem_univ, and_true]
rw [scorePopulationCells_real_apply hOverlap H g h.2.2.1 a r univ MeasurableSet.univ]
simp only [Set.mem_univ, and_true]
rw [h.2.2.2 a r]
simp only [armCellMass, cell]
rw [← integral_indicator (measurableSet_eq_fun h.2.2.1 measurable_const)]
theorem projectionPopulationCells_finite
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
μ :
Measure (Observation J)
hOverlap :
Overlap ε
hg :
conclusion
Proof (Lean source)
lemma projectionPopulationCells_finite {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(μ : Measure (Observation J)) [IsProbabilityMeasure H]
[IsProbabilityMeasure μ] (hOverlap : Overlap ε) (hg : Measurable g) :
(∀ a r, IsFiniteMeasure (trialPopulationCells μ a r)) ∧
∀ a r, IsFiniteMeasure (scorePopulationCells H g a r) := by
constructor
· intro a r
unfold trialPopulationCells
infer_instance
· intro a r
let F : ScoreSpace ε → ℝ := fun e => if g e = r then armProb a e else 0
have hF : Measurable F := by
have hcell : MeasurableSet {e : ScoreSpace ε | g e = r} :=
measurableSet_eq_fun hg measurable_const
dsimp [F, armProb]
cases a <;> simp only [Bool.false_eq_true, ↓reduceIte] <;>
exact Measurable.ite hcell (by fun_prop) measurable_const
have hbound : ∀ e, |F e| ≤ 1 := by
intro e
by_cases he : g e = r
· have hp := projectionArmProb_bounds hOverlap a e
simpa [F, he, abs_of_nonneg (le_trans hOverlap.1.le hp.1)] using hp.2
· simp [F, he]
have hInt : Integrable F H := by
apply Integrable.of_bound hF.aestronglyMeasurable 1
exact Filter.Eventually.of_forall (fun e => by simpa [Real.norm_eq_abs] using hbound e)
unfold scorePopulationCells
exact isFiniteMeasure_withDensity_ofReal hInt.hasFiniteIntegral
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionSharpLength 3 declarations Nonnegativity of the population projection span under overlap.
Nonnegativity of the population projection span under overlap.
theorem cell_lower_le_upper_of_overlap
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
hOverlap :
Overlap ε
a :
μ :
σ :
Measure (ScoreSpace ε)
hp :
0 < μ.real univ
hmass :
μ.real univ = σ.real univ
Proof (Lean source)
lemma cell_lower_le_upper_of_overlap {ε : ℝ}
(hOverlap : Overlap ε) (a : ArmSpace)
(μ : Measure OutcomeSpace) (σ : Measure (ScoreSpace ε))
[IsFiniteMeasure μ] [IsFiniteMeasure σ]
(hp : 0 < μ.real univ) (hmass : μ.real univ = σ.real univ) :
(∫ u in (0 : ℝ)..1,
quantile
(((μ univ)⁻¹ • μ).map (fun y : OutcomeSpace => (y : ℝ))) u *
quantile
(((σ univ)⁻¹ • σ).map (fun e => (armProb a e)⁻¹)) (1 - u)) ≤
∫ u in (0 : ℝ)..1,
quantile
(((μ univ)⁻¹ • μ).map (fun y : OutcomeSpace => (y : ℝ))) u *
quantile
(((σ univ)⁻¹ • σ).map (fun e => (armProb a e)⁻¹)) u := by
let Y : Measure ℝ :=
((μ univ)⁻¹ • μ).map (fun y : OutcomeSpace => (y : ℝ))
let W : Measure ℝ :=
((σ univ)⁻¹ • σ).map (fun e => (armProb a e)⁻¹)
have hσp : 0 < σ.real univ := by rw [← hmass]; exact hp
letI : IsProbabilityMeasure Y := by
dsimp [Y]
rw [map_normalized_subtype_eq_finiteNormalize μ hp]
infer_instance
letI : IsProbabilityMeasure W := by
dsimp [W]
rw [map_normalized_inverseArmProb_eq_finiteNormalize_map a σ hσp]
exact Measure.isProbabilityMeasure_map
(measurable_projectionInverseArmProbReal a).aemeasurable
have hYs : Y (Icc 0 1)ᶜ = 0 := by
exact normalizedOutcome_support μ hp
have hWs : W (Icc 0 ε⁻¹)ᶜ = 0 := by
exact normalizedInverseArmProb_support hOverlap a σ hσp
have hY2 : MemLp (fun x : ℝ => x) 2 Y := by
apply memLp_of_bounded
· filter_upwards [show ∀ᵐ x ∂Y, x ∈ Icc (0 : ℝ) 1 by
rw [ae_iff]; exact hYs] with x hx
exact hx
· fun_prop
have hW2 : MemLp (fun x : ℝ => x) 2 W := by
apply memLp_of_bounded
· filter_upwards [show ∀ᵐ x ∂W, x ∈ Icc (0 : ℝ) ε⁻¹ by
rw [ae_iff]; exact hWs] with x hx
exact hx
· fun_prop
have hopt := countermonotone_le_product_expectation
(isCoupling_comonotoneCoupling Y W) hY2 hW2
have hrest : volume.restrict (Ioc (0 : ℝ) 1) = unifOI := by
rw [unifOI, restrict_Ioo_eq_restrict_Ioc]
have hcounter :
(∫ p, p.1 * p.2 ∂(Causalean.Stat.countermonotoneCoupling Y W)) =
∫ u in (0 : ℝ)..1,
quantile Y u * quantile W (1 - u) := by
unfold Causalean.Stat.countermonotoneCoupling
let r : ℝ → ℝ := fun u => 1 - u
have hr : Measurable r := measurable_const.sub measurable_id
have hWmap : AEMeasurable (quantile W)
(Causalean.Stat.unifOI.map r) := by
simpa [r, map_one_sub_unifOI] using
(aemeasurable_quantile_unifOI W)
have hWr : AEMeasurable (fun u : ℝ =>
quantile W (1 - u)) unifOI := by
simpa [r, Function.comp_def] using hWmap.comp_measurable hr
rw [MeasureTheory.integral_map]
· rw [intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1), hrest]
· exact (aemeasurable_quantile_unifOI Y).prodMk
hWr
· fun_prop
have hcomono :
(∫ p, p.1 * p.2 ∂(comonotoneCoupling Y W)) =
∫ u in (0 : ℝ)..1,
quantile Y u * quantile W u := by
rw [product_expectation_comonotoneCoupling Y W hY2 hW2,
← integral_Ioc_eq_integral_Ioo,
intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1), hrest]
rw [hcounter, hcomono] at hopt
simpa [Y, W] using hopt
theorem muLower_le_muUpper_of_overlap
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
a :
Proof (Lean source)
lemma muLower_le_muUpper_of_overlap {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) (a : ArmSpace) :
muLower H g Prel hMass a ≤ muUpper H g Prel hMass a := by
letI : IsProbabilityMeasure H := hMass.1
letI : IsProbabilityMeasure Prel := hMass.2.1
have hfin := projectionPopulationCells_finite H g Prel hOverlap hMass.2.2.1
have hl := projectedArmEndpoint_population_eq H g Prel hMass hOverlap a false
have hu := projectedArmEndpoint_population_eq H g Prel hMass hOverlap a true
simp only [Bool.false_eq_true, if_false] at hl
simp only [if_true] at hu
rw [← hl, ← hu]
unfold projectedArmEndpoint
apply Finset.sum_le_sum
intro r _
letI : IsFiniteMeasure (trialPopulationCells Prel a r) := hfin.1 a r
letI : IsFiniteMeasure (scorePopulationCells H g a r) := hfin.2 a r
let q := (trialPopulationCells Prel a r).real univ
by_cases hq : 0 < q
· dsimp [q] at hq
simp only [hq, if_pos]
apply mul_le_mul_of_nonneg_left
· apply cell_lower_le_upper_of_overlap hOverlap a
(trialPopulationCells Prel a r) (scorePopulationCells H g a r) hq
exact projectionCompatible_populationCells H g Prel hMass hOverlap a r
· exact le_of_lt hq
· dsimp [q] at hq
simp [hq]
theorem sharpATELength_nonneg_of_overlap
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
hMass :
CompatibleCellMasses H g Prel
hOverlap :
Overlap ε
conclusion
0 ≤ sharpATELength H g Prel hMass
Proof (Lean source)
theorem sharpATELength_nonneg_of_overlap {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) (hMass : CompatibleCellMasses H g Prel)
(hOverlap : Overlap ε) :
0 ≤ sharpATELength H g Prel hMass := by
have h₀ := muLower_le_muUpper_of_overlap H g Prel hMass hOverlap false
have h₁ := muLower_le_muUpper_of_overlap H g Prel hMass hOverlap true
unfold sharpATELength
linarith
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionSieveApproximation 1 declarations Arbitrarily close finite candidates inside strict projection balls.
Arbitrarily close finite candidates inside strict projection balls.
theorem exists_sievedProjectionCandidate_close_of_slack
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
hOverlap :
Overlap ε
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
x :
ExternalSample ε J n m
lam :
σ :
ArmCellArray J (ScoreSpace ε)
hcompat :
ProjectionCompatible lam σ
η :
ℝ
hη :
0 < η
hout :
+ η
≤ 4 * J / (α * sqrt n)
hscore :
+ η
≤ 2 * J * (2 - 2 * ε) / (α * sqrt m)
conclusion
∃ b ∈ sievedProjectionCandidates g α x,
(∑ a : ArmSpace, ∑ r : LabelSpace J, (outcomeCDFDistance (lam a r) (b.1 a r) + scoreCDFDistance (σ a r) (b.2 a r))) < η
Proof (Lean source)
lemma exists_sievedProjectionCandidate_close_of_slack
{ε : ℝ} {J n m : ℕ} (hOverlap : Overlap ε)
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(x : ExternalSample ε J n m)
(lam : ArmCellArray J OutcomeSpace)
(σ : ArmCellArray J (ScoreSpace ε))
(hfinite : ∀ a r, (lam a r) univ < ⊤ ∧ (σ a r) univ < ⊤)
(hcompat : ProjectionCompatible lam σ)
(η : ℝ) (hη : 0 < η)
(hout : (∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (lam a r) (empiricalTrialCells x.1 a r)) + η ≤
4 * J / (α * sqrt n))
(hscore : (∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (σ a r) (empiricalScoreCells g x.2 a r)) + η ≤
2 * J * (2 - 2 * ε) / (α * sqrt m)) :
∃ b ∈ sievedProjectionCandidates g α x,
(∑ a : ArmSpace, ∑ r : LabelSpace J,
(outcomeCDFDistance (lam a r) (b.1 a r) +
scoreCDFDistance (σ a r) (b.2 a r))) < η := by
classical
obtain ⟨code, hsieve, hcompatB, hdist⟩ :=
exists_rationalSieveArray_close hOverlap lam σ hfinite hcompat η hη
let b : ArmCellArray J OutcomeSpace × ArmCellArray J (ScoreSpace ε) :=
((fun a r => synchronizedRationalOutcomeMeasure (code a r)),
(fun a r => synchronizedRationalScoreMeasure ε hOverlap (code a r)))
have hbfinite : ∀ a r, (b.1 a r) univ < ⊤ ∧ (b.2 a r) univ < ⊤ := by
intro a r
exact ⟨synchronizedRationalOutcomeMeasure_finite (code a r),
synchronizedRationalScoreMeasure_finite hOverlap (code a r)⟩
have hnonnegOut : 0 ≤ ∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (lam a r) (b.1 a r) :=
Finset.sum_nonneg (fun a _ =>
Finset.sum_nonneg (fun r _ => outcomeCDFDistance_nonneg _ _))
have hnonnegScore : 0 ≤ ∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (σ a r) (b.2 a r) :=
Finset.sum_nonneg (fun a _ =>
Finset.sum_nonneg (fun r _ => scoreCDFDistance_nonneg hOverlap _ _))
have houtClose :
(∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (b.1 a r) (lam a r)) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (lam a r) (empiricalTrialCells x.1 a r)) ≤
4 * J / (α * sqrt n) := by
simp_rw [outcomeCDFDistance_symm (b.1 _ _) (lam _ _)]
have hdist' :
(∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (lam a r) (b.1 a r)) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (σ a r) (b.2 a r)) < η := by
simpa only [b, Finset.sum_add_distrib] using hdist
linarith
have hscoreClose :
(∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (b.2 a r) (σ a r)) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (σ a r) (empiricalScoreCells g x.2 a r)) ≤
2 * J * (2 - 2 * ε) / (α * sqrt m) := by
simp_rw [scoreCDFDistance_symm (b.2 _ _) (σ _ _)]
have hdist' :
(∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (lam a r) (b.1 a r)) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (σ a r) (b.2 a r)) < η := by
simpa only [b, Finset.sum_add_distrib] using hdist
linarith
refine ⟨b, ?_, ?_⟩
· exact ⟨projectionCandidate_of_strict_population_slack hOverlap g α x lam σ b
hbfinite hcompatB hfinite houtClose hscoreClose, hsieve⟩
· exact hdist
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionSieveDensity 10 declarations Exact-mass rational encodings for the external projection sieve.
Exact-mass rational encodings for the external projection sieve.
The outcome and score locations may differ, but every atom uses one shared rational weight. This supplies the synchronized finite arrays needed by an eventual density argument for the equal-mass projection candidates.
def ProjectionRationalAtom
This definition introduces the corresponding mathematical object.
Definition (Lean source)
def
ProjectionRationalAtom :
Type
given by
ℚ × ℚ × ℚ
def synchronizedRationalOutcomeMeasure
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
def synchronizedRationalScoreMeasure
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
theorem rationalOutcomeMeasure_apply_univ
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma rationalOutcomeMeasure_apply_univ (l : List (ℚ × ℚ)) :
rationalOutcomeMeasure l univ =
(l.map fun p => ofReal (p.2 : ℝ)).sum := by
induction l with
| nil => simp [rationalOutcomeMeasure]
| cons p l ih =>
change (ofReal (p.2 : ℝ) •
Measure.dirac (rationalOutcomePoint p.1)) univ +
rationalOutcomeMeasure l univ =
ofReal (p.2 : ℝ) +
(l.map fun p => ofReal (p.2 : ℝ)).sum
rw [ih]
simp
theorem rationalScoreMeasure_apply_univ
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma rationalScoreMeasure_apply_univ {ε : ℝ} (hOverlap : Overlap ε)
(l : List (ℚ × ℚ)) :
rationalScoreMeasure ε hOverlap l univ =
(l.map fun p => ofReal (p.2 : ℝ)).sum := by
induction l with
| nil => simp [rationalScoreMeasure]
| cons p l ih =>
change (ofReal (p.2 : ℝ) •
Measure.dirac (rationalScorePoint ε hOverlap p.1)) univ +
rationalScoreMeasure ε hOverlap l univ =
ofReal (p.2 : ℝ) +
(l.map fun p => ofReal (p.2 : ℝ)).sum
rw [ih]
simp
theorem synchronizedRationalMeasure_mass_eq
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
= synchronizedRationalScoreMeasure ε hOverlap l univ
Proof (Lean source)
lemma synchronizedRationalMeasure_mass_eq {ε : ℝ} (hOverlap : Overlap ε)
(l : List ProjectionRationalAtom) :
synchronizedRationalOutcomeMeasure l univ =
synchronizedRationalScoreMeasure ε hOverlap l univ := by
rw [synchronizedRationalOutcomeMeasure, synchronizedRationalScoreMeasure,
rationalOutcomeMeasure_apply_univ, rationalScoreMeasure_apply_univ]
apply congrArg sum
rw [List.map_map, List.map_map]
apply List.map_congr_left
intro p hp
rfl
theorem synchronizedRationalArray_compatible
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
ProjectionCompatible (fun a r => synchronizedRationalOutcomeMeasure (code a r)) (fun a r => synchronizedRationalScoreMeasure ε hOverlap (code a r))
Proof (Lean source)
lemma synchronizedRationalArray_compatible {ε : ℝ} {J : ℕ}
(hOverlap : Overlap ε)
(code : ArmSpace → LabelSpace J → List ProjectionRationalAtom) :
ProjectionCompatible
(fun a r => synchronizedRationalOutcomeMeasure (code a r))
(fun a r => synchronizedRationalScoreMeasure ε hOverlap (code a r)) := by
intro a r
exact congrArg toReal
(synchronizedRationalMeasure_mass_eq hOverlap (code a r))
theorem synchronizedRationalArray_mem_sieve
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
((fun a r => synchronizedRationalOutcomeMeasure (code a r)), (fun a r => synchronizedRationalScoreMeasure ε hOverlap (code a r))) ∈ rationalArraySieve ε J
Proof (Lean source)
lemma synchronizedRationalArray_mem_sieve {ε : ℝ} {J : ℕ}
(hOverlap : Overlap ε)
(code : ArmSpace → LabelSpace J → List ProjectionRationalAtom) :
((fun a r => synchronizedRationalOutcomeMeasure (code a r)),
(fun a r => synchronizedRationalScoreMeasure ε hOverlap (code a r))) ∈
rationalArraySieve ε J := by
classical
unfold rationalArraySieve
simp only [dif_pos hOverlap, Set.mem_inter_iff, Set.mem_range, Set.mem_setOf_eq]
constructor
· refine ⟨((fun a r => (code a r).map fun p => (p.1, p.2.2)),
(fun a r => (code a r).map fun p => (p.2.1, p.2.2))), ?_⟩
rfl
· exact synchronizedRationalArray_compatible hOverlap code
theorem synchronizedRationalOutcomeMeasure_finite
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma synchronizedRationalOutcomeMeasure_finite
(l : List ProjectionRationalAtom) :
synchronizedRationalOutcomeMeasure l univ < ⊤ := by
rw [synchronizedRationalOutcomeMeasure, rationalOutcomeMeasure_apply_univ]
induction l with
| nil => simp
| cons p l ih =>
simp only [List.map_cons, List.sum_cons, ENNReal.add_lt_top]
exact ⟨ENNReal.ofReal_lt_top, ih⟩
theorem synchronizedRationalScoreMeasure_finite
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
synchronizedRationalScoreMeasure ε hOverlap l univ < ⊤
Proof (Lean source)
lemma synchronizedRationalScoreMeasure_finite {ε : ℝ}
(hOverlap : Overlap ε) (l : List ProjectionRationalAtom) :
synchronizedRationalScoreMeasure ε hOverlap l univ < ⊤ := by
rw [synchronizedRationalScoreMeasure,
rationalScoreMeasure_apply_univ hOverlap]
induction l with
| nil => simp
| cons p l ih =>
simp only [List.map_cons, List.sum_cons, ENNReal.add_lt_top]
exact ⟨ENNReal.ofReal_lt_top, ih⟩
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.ProjectionSieveDensityBridge 27 declarations Bridge from synchronized real-line atomic approximation to the paper's compact-carrier projection sieve.
Bridge from synchronized real-line atomic approximation to the paper's compact-carrier projection sieve.
theorem outcomeCDFDistance_eq_realCdfDistance
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
μ ν :
conclusion
= cdfDistance 0 1 (μ.map (fun y : OutcomeSpace => (y : ℝ))) (ν.map (fun y : OutcomeSpace => (y : ℝ)))
Proof (Lean source)
lemma outcomeCDFDistance_eq_realCdfDistance (μ ν : Measure OutcomeSpace) :
outcomeCDFDistance μ ν =
cdfDistance 0 1
(μ.map (fun y : OutcomeSpace => (y : ℝ)))
(ν.map (fun y : OutcomeSpace => (y : ℝ))) := by
unfold outcomeCDFDistance cdfDistance
rw [Measure.real_def, Measure.real_def, Measure.real_def, Measure.real_def]
rw [Measure.map_apply (by fun_prop : Measurable fun y : OutcomeSpace => (y : ℝ))
MeasurableSet.univ, Measure.map_apply
(by fun_prop : Measurable fun y : OutcomeSpace => (y : ℝ)) MeasurableSet.univ]
simp only [preimage_univ]
congr 1
apply intervalIntegral.integral_congr
intro t ht
dsimp
simp only [Measure.real_def]
rw [Measure.map_apply (by fun_prop : Measurable fun y : OutcomeSpace => (y : ℝ))
measurableSet_Iic,
Measure.map_apply (by fun_prop : Measurable fun y : OutcomeSpace => (y : ℝ))
measurableSet_Iic]
rfl
theorem scoreCDFDistance_eq_realCdfDistance
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
μ ν :
Measure (ScoreSpace ε)
conclusion
scoreCDFDistance μ ν
= cdfDistance ε (1 - ε) (μ.map (fun e : ScoreSpace ε => (e : ℝ))) (ν.map (fun e : ScoreSpace ε => (e : ℝ)))
Proof (Lean source)
lemma scoreCDFDistance_eq_realCdfDistance {ε : ℝ}
(μ ν : Measure (ScoreSpace ε)) :
scoreCDFDistance μ ν =
cdfDistance ε (1 - ε)
(μ.map (fun e : ScoreSpace ε => (e : ℝ)))
(ν.map (fun e : ScoreSpace ε => (e : ℝ))) := by
unfold scoreCDFDistance cdfDistance
rw [Measure.real_def, Measure.real_def, Measure.real_def, Measure.real_def]
rw [Measure.map_apply (by fun_prop : Measurable fun e : ScoreSpace ε => (e : ℝ))
MeasurableSet.univ, Measure.map_apply
(by fun_prop : Measurable fun e : ScoreSpace ε => (e : ℝ)) MeasurableSet.univ]
simp only [preimage_univ]
congr 1
apply intervalIntegral.integral_congr
intro t ht
dsimp
simp only [Measure.real_def]
rw [Measure.map_apply (by fun_prop : Measurable fun e : ScoreSpace ε => (e : ℝ))
measurableSet_Iic,
Measure.map_apply (by fun_prop : Measurable fun e : ScoreSpace ε => (e : ℝ))
measurableSet_Iic]
rfl
def realOutcomePoint
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
x :
ℝ
def
given by
⟨max 0 (min 1 x), by constructor · exact le_max_left _ _ · exact max_le (by norm_num) (min_le_left _ _)⟩
theorem realOutcomePoint_continuous
This result establishes the stated mathematical conclusion.
Formal statement
conclusion
Proof (Lean source)
lemma realOutcomePoint_continuous : Continuous realOutcomePoint := by
unfold realOutcomePoint
exact Continuous.subtype_mk
(continuous_const.max (continuous_const.min continuous_id)) _
theorem realOutcomePoint_surjective
This result establishes the stated mathematical conclusion.
Formal statement
conclusion
Proof (Lean source)
lemma realOutcomePoint_surjective : Surjective realOutcomePoint := by
intro y
refine ⟨(y : ℝ), ?_⟩
apply Subtype.ext
simp [realOutcomePoint, y.property.1, y.property.2]
theorem rationalOutcomePoint_denseRange
This result establishes the stated mathematical conclusion.
Formal statement
conclusion
DenseRange (rationalOutcomePoint : ℚ → OutcomeSpace)
Proof (Lean source)
lemma rationalOutcomePoint_denseRange :
DenseRange (rationalOutcomePoint : ℚ → OutcomeSpace) := by
have h := DenseRange.comp realOutcomePoint_surjective.denseRange
(Rat.denseRange_cast : DenseRange ((↑) : ℚ → ℝ)) realOutcomePoint_continuous
convert h using 1
funext q
rfl
def outcomeToScore
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
⟨ε + (1 - 2 * ε) * (y : ℝ), by have hw : 0
≤ 1
- 2 * ε := by linarith [hOverlap.2] constructor · nlinarith [y.property.1] · nlinarith [y.property.2]⟩
theorem outcomeToScore_continuous
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma outcomeToScore_continuous {ε : ℝ} (hOverlap : Overlap ε) :
Continuous (outcomeToScore hOverlap) := by
unfold outcomeToScore
exact Continuous.subtype_mk (by fun_prop) _
theorem outcomeToScore_surjective
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma outcomeToScore_surjective {ε : ℝ} (hOverlap : Overlap ε) :
Surjective (outcomeToScore hOverlap) := by
intro e
have hw : 0 < 1 - 2 * ε := by linarith [hOverlap.2]
let y : OutcomeSpace := ⟨((e : ℝ) - ε) / (1 - 2 * ε), by
constructor
· exact div_nonneg (sub_nonneg.mpr e.property.1) hw.le
· apply (div_le_one hw).2
linarith [e.property.2]⟩
refine ⟨y, ?_⟩
apply Subtype.ext
change ε + (1 - 2 * ε) * (((e : ℝ) - ε) / (1 - 2 * ε)) = (e : ℝ)
have hcancel : (1 - 2 * ε) * (((e : ℝ) - ε) / (1 - 2 * ε)) =
(e : ℝ) - ε := by
exact mul_div_cancel₀ _ (ne_of_gt hw)
linarith
theorem rationalScorePoint_denseRange
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
hOverlap :
Overlap ε
conclusion
DenseRange (rationalScorePoint ε hOverlap : ℚ → ScoreSpace ε)
Proof (Lean source)
lemma rationalScorePoint_denseRange {ε : ℝ} (hOverlap : Overlap ε) :
DenseRange (rationalScorePoint ε hOverlap : ℚ → ScoreSpace ε) := by
have h := DenseRange.comp (outcomeToScore_surjective hOverlap).denseRange
rationalOutcomePoint_denseRange (outcomeToScore_continuous hOverlap)
convert h using 1
funext q
rfl
def outcomeSieveLocations
This definition introduces the corresponding mathematical object.
def scoreSieveLocations
For the specified mathematical inputs, this definition introduces the corresponding object.
theorem outcomeSieveLocations_countable
This result establishes the stated mathematical conclusion.
Formal statement
conclusion
outcomeSieveLocations.Countable
Proof (Lean source)
lemma outcomeSieveLocations_countable : outcomeSieveLocations.Countable := by
unfold outcomeSieveLocations
exact Set.countable_range _
theorem scoreSieveLocations_countable
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma scoreSieveLocations_countable {ε : ℝ} (hOverlap : Overlap ε) :
(scoreSieveLocations hOverlap).Countable := by
unfold scoreSieveLocations
exact Set.countable_range _
theorem outcomeSieveLocations_subset
This result establishes the stated mathematical conclusion.
Formal statement
conclusion
outcomeSieveLocations ⊆ Icc (0 : ℝ) 1
Proof (Lean source)
lemma outcomeSieveLocations_subset : outcomeSieveLocations ⊆ Icc (0 : ℝ) 1 := by
rintro x ⟨q, rfl⟩
exact (rationalOutcomePoint q).property
theorem scoreSieveLocations_subset
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma scoreSieveLocations_subset {ε : ℝ} (hOverlap : Overlap ε) :
scoreSieveLocations hOverlap ⊆ Icc ε (1 - ε) := by
rintro x ⟨q, rfl⟩
exact (rationalScorePoint ε hOverlap q).property
theorem outcomeSieveLocations_dense
This result establishes the stated mathematical conclusion.
Formal statement
conclusion
Proof (Lean source)
lemma outcomeSieveLocations_dense :
Dense ((Subtype.val : OutcomeSpace → ℝ) ⁻¹' outcomeSieveLocations) := by
have heq : ((Subtype.val : OutcomeSpace → ℝ) ⁻¹' outcomeSieveLocations) =
range (rationalOutcomePoint : ℚ → OutcomeSpace) := by
ext y
constructor
· rintro ⟨q, hq⟩
exact ⟨q, Subtype.ext hq⟩
· rintro ⟨q, rfl⟩
exact ⟨q, rfl⟩
rw [heq]
exact rationalOutcomePoint_denseRange
theorem scoreSieveLocations_dense
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma scoreSieveLocations_dense {ε : ℝ} (hOverlap : Overlap ε) :
Dense ((Subtype.val : ScoreSpace ε → ℝ) ⁻¹' scoreSieveLocations hOverlap) := by
have heq : ((Subtype.val : ScoreSpace ε → ℝ) ⁻¹' scoreSieveLocations hOverlap) =
range (rationalScorePoint ε hOverlap : ℚ → ScoreSpace ε) := by
ext e
constructor
· rintro ⟨q, hq⟩
exact ⟨q, Subtype.ext hq⟩
· rintro ⟨q, rfl⟩
exact ⟨q, rfl⟩
rw [heq]
exact rationalScorePoint_denseRange hOverlap
theorem exists_synchronized_sieveLocations_cell_approx
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
∃ (N : ℕ) (hN : 0 < N) (q : ℚ),
0 ≤ q
conclusion 1
= (q : ℝ)
= (q : ℝ)
cdfDistance 0 1 (μ.map (fun z : OutcomeSpace => (z : ℝ))) (equalAtomMeasure N (q : ℝ) (fun i => (x i : ℝ)))
+ cdfDistance ε (1 - ε) (ν.map (fun z : ScoreSpace ε => (z : ℝ))) (equalAtomMeasure N (q : ℝ) (fun i => (y i : ℝ))) < η
Proof (Lean source)
lemma exists_synchronized_sieveLocations_cell_approx {ε : ℝ}
(hOverlap : Overlap ε)
(μ : Measure OutcomeSpace) (ν : Measure (ScoreSpace ε))
[IsFiniteMeasure μ] [IsFiniteMeasure ν]
(hmass : μ.real univ = ν.real univ)
(η : ℝ) (hη : 0 < η) :
∃ (N : ℕ) (hN : 0 < N) (q : ℚ), 0 ≤ q ∧
∃ (x : Fin N → outcomeSieveLocations)
(y : Fin N → scoreSieveLocations hOverlap),
(equalAtomMeasure N (q : ℝ)
(fun i => (x i : ℝ))).real univ = (q : ℝ) ∧
(equalAtomMeasure N (q : ℝ)
(fun i => (y i : ℝ))).real univ = (q : ℝ) ∧
cdfDistance 0 1
(μ.map (fun z : OutcomeSpace => (z : ℝ)))
(equalAtomMeasure N (q : ℝ)
(fun i => (x i : ℝ))) +
cdfDistance ε (1 - ε)
(ν.map (fun z : ScoreSpace ε => (z : ℝ)))
(equalAtomMeasure N (q : ℝ)
(fun i => (y i : ℝ))) < η := by
let μr : Measure ℝ := μ.map (fun z : OutcomeSpace => (z : ℝ))
let νr : Measure ℝ := ν.map (fun z : ScoreSpace ε => (z : ℝ))
letI : IsFiniteMeasure μr := Measure.isFiniteMeasure_map μ _
letI : IsFiniteMeasure νr := Measure.isFiniteMeasure_map ν _
have hμ : μr (Icc (0 : ℝ) 1)ᶜ = 0 := by
rw [Measure.map_apply (by fun_prop : Measurable fun z : OutcomeSpace => (z : ℝ))
measurableSet_Icc.compl]
have he : (fun z : OutcomeSpace => (z : ℝ)) ⁻¹' (Icc (0 : ℝ) 1)ᶜ = ∅ := by
ext z
simp [z.property]
simp [he]
have hν : νr (Icc ε (1 - ε))ᶜ = 0 := by
rw [Measure.map_apply (by fun_prop : Measurable fun z : ScoreSpace ε => (z : ℝ))
measurableSet_Icc.compl]
have he : (fun z : ScoreSpace ε => (z : ℝ)) ⁻¹' (Icc ε (1 - ε))ᶜ = ∅ := by
ext z
simp [z.property]
simp [he]
have hm : μr.real univ = νr.real univ := by
simp only [μr, νr, Measure.real_def]
rw [Measure.map_apply (by fun_prop : Measurable fun z : OutcomeSpace => (z : ℝ))
MeasurableSet.univ,
Measure.map_apply (by fun_prop : Measurable fun z : ScoreSpace ε => (z : ℝ))
MeasurableSet.univ]
simpa only [preimage_univ, ← Measure.real_def] using hmass
exact exists_synchronized_rational_equalAtom_approx
μr νr 0 1 ε (1 - ε) (by norm_num) (by linarith [hOverlap.2])
hμ hν hm outcomeSieveLocations (scoreSieveLocations hOverlap)
outcomeSieveLocations_dense (scoreSieveLocations_dense hOverlap) η hη
theorem synchronizedOutcome_map_eq_equalAtom
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(synchronizedRationalOutcomeMeasure (List.ofFn fun i : Fin N => (x i, y i, q / N))).map (fun z : OutcomeSpace => (z : ℝ))
= equalAtomMeasure N (q : ℝ) (fun i => (rationalOutcomePoint (x i) : ℝ))
Proof (Lean source)
lemma synchronizedOutcome_map_eq_equalAtom {N : ℕ} (q : ℚ)
(x y : Fin N → ℚ) :
(synchronizedRationalOutcomeMeasure
(List.ofFn fun i : Fin N => (x i, y i, q / N))).map
(fun z : OutcomeSpace => (z : ℝ)) =
equalAtomMeasure N (q : ℝ)
(fun i => (rationalOutcomePoint (x i) : ℝ)) := by
unfold synchronizedRationalOutcomeMeasure rationalOutcomeMeasure
equalAtomMeasure
simp only [List.map_ofFn, List.sum_ofFn]
let f : OutcomeSpace → ℝ := fun z => (z : ℝ)
let M : Fin N → Measure OutcomeSpace := fun i =>
ofReal ((q / N : ℚ) : ℝ) • Measure.dirac (rationalOutcomePoint (x i))
change (∑ i, M i).map f = _
have hmap (s : Finset (Fin N)) :
(∑ i ∈ s, M i).map f = ∑ i ∈ s, (M i).map f := by
induction s using Finset.induction_on with
| empty => simp
| insert a s ha ih =>
simp only [Finset.sum_insert ha]
rw [Measure.map_add _ _ (by fun_prop : Measurable f), ih]
rw [show (∑ i : Fin N, M i).map f = ∑ i : Fin N, (M i).map f by
simpa using hmap univ]
apply Finset.sum_congr rfl
intro i hi
rw [Measure.map_smul, Measure.map_dirac']
· simp [f, Rat.cast_div, Rat.cast_natCast]
· fun_prop
theorem synchronizedScore_map_eq_equalAtom
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(synchronizedRationalScoreMeasure ε hOverlap (List.ofFn fun i : Fin N => (x i, y i, q / N))).map (fun z : ScoreSpace ε => (z : ℝ))
= equalAtomMeasure N (q : ℝ) (fun i => (rationalScorePoint ε hOverlap (y i) : ℝ))
Proof (Lean source)
lemma synchronizedScore_map_eq_equalAtom {ε : ℝ} (hOverlap : Overlap ε)
{N : ℕ} (q : ℚ) (x y : Fin N → ℚ) :
(synchronizedRationalScoreMeasure ε hOverlap
(List.ofFn fun i : Fin N => (x i, y i, q / N))).map
(fun z : ScoreSpace ε => (z : ℝ)) =
equalAtomMeasure N (q : ℝ)
(fun i => (rationalScorePoint ε hOverlap (y i) : ℝ)) := by
unfold synchronizedRationalScoreMeasure rationalScoreMeasure
equalAtomMeasure
simp only [List.map_ofFn, List.sum_ofFn]
let f : ScoreSpace ε → ℝ := fun z => (z : ℝ)
let M : Fin N → Measure (ScoreSpace ε) := fun i =>
ofReal ((q / N : ℚ) : ℝ) • Measure.dirac (rationalScorePoint ε hOverlap (y i))
change (∑ i, M i).map f = _
have hmap (s : Finset (Fin N)) :
(∑ i ∈ s, M i).map f = ∑ i ∈ s, (M i).map f := by
induction s using Finset.induction_on with
| empty => simp
| insert a s ha ih =>
simp only [Finset.sum_insert ha]
rw [Measure.map_add _ _ (by fun_prop : Measurable f), ih]
rw [show (∑ i : Fin N, M i).map f = ∑ i : Fin N, (M i).map f by
simpa using hmap univ]
apply Finset.sum_congr rfl
intro i hi
rw [Measure.map_smul, Measure.map_dirac']
· simp [f, Rat.cast_div, Rat.cast_natCast]
· fun_prop
theorem exists_synchronizedRationalCode_cell_approx
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
∃ code : List ProjectionRationalAtom,
+ scoreCDFDistance ν (synchronizedRationalScoreMeasure ε hOverlap code) < η
Proof (Lean source)
lemma exists_synchronizedRationalCode_cell_approx {ε : ℝ}
(hOverlap : Overlap ε)
(μ : Measure OutcomeSpace) (ν : Measure (ScoreSpace ε))
[IsFiniteMeasure μ] [IsFiniteMeasure ν]
(hmass : μ.real univ = ν.real univ)
(η : ℝ) (hη : 0 < η) :
∃ code : List ProjectionRationalAtom,
outcomeCDFDistance μ (synchronizedRationalOutcomeMeasure code) +
scoreCDFDistance ν (synchronizedRationalScoreMeasure ε hOverlap code) < η := by
classical
obtain ⟨N, hN, q, hq, x, y, hmx, hmy, hdist⟩ :=
exists_synchronized_sieveLocations_cell_approx hOverlap μ ν hmass η hη
let xq : Fin N → ℚ := fun i => choose (x i).property
let yq : Fin N → ℚ := fun i => choose (y i).property
have hx (i : Fin N) : (rationalOutcomePoint (xq i) : ℝ) = (x i : ℝ) :=
Classical.choose_spec (x i).property
have hy (i : Fin N) : (rationalScorePoint ε hOverlap (yq i) : ℝ) = (y i : ℝ) :=
Classical.choose_spec (y i).property
let code : List ProjectionRationalAtom :=
List.ofFn fun i : Fin N => (xq i, yq i, q / N)
refine ⟨code, ?_⟩
have hout : (synchronizedRationalOutcomeMeasure code).map
(fun z : OutcomeSpace => (z : ℝ)) =
equalAtomMeasure N (q : ℝ)
(fun i => (x i : ℝ)) := by
rw [show code = List.ofFn (fun i : Fin N => (xq i, yq i, q / N)) from rfl,
synchronizedOutcome_map_eq_equalAtom]
congr 1
funext i
exact hx i
have hscore : (synchronizedRationalScoreMeasure ε hOverlap code).map
(fun z : ScoreSpace ε => (z : ℝ)) =
equalAtomMeasure N (q : ℝ)
(fun i => (y i : ℝ)) := by
rw [show code = List.ofFn (fun i : Fin N => (xq i, yq i, q / N)) from rfl,
synchronizedScore_map_eq_equalAtom]
congr 1
funext i
exact hy i
rw [outcomeCDFDistance_eq_realCdfDistance,
scoreCDFDistance_eq_realCdfDistance, hout, hscore]
exact hdist
theorem exists_rationalSieveArray_close
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
hOverlap :
Overlap ε
lam :
σ :
ArmCellArray J (ScoreSpace ε)
hcompat :
ProjectionCompatible lam σ
η :
ℝ
hη :
0 < η
conclusion
let b := ((fun a r => synchronizedRationalOutcomeMeasure (code a r)), (fun a r => synchronizedRationalScoreMeasure ε hOverlap (code a r))) b ∈ rationalArraySieve ε J ∧
ProjectionCompatible b.1 b.2 ∧
(∑ a : ArmSpace, ∑ r : LabelSpace J, (outcomeCDFDistance (lam a r) (b.1 a r) + scoreCDFDistance (σ a r) (b.2 a r))) < η
Proof (Lean source)
lemma exists_rationalSieveArray_close {ε : ℝ} {J : ℕ}
(hOverlap : Overlap ε)
(lam : ArmCellArray J OutcomeSpace)
(σ : ArmCellArray J (ScoreSpace ε))
(hfinite : ∀ a r, (lam a r) univ < ⊤ ∧ (σ a r) univ < ⊤)
(hcompat : ProjectionCompatible lam σ)
(η : ℝ) (hη : 0 < η) :
∃ code : ArmSpace → LabelSpace J → List ProjectionRationalAtom,
let b := ((fun a r => synchronizedRationalOutcomeMeasure (code a r)),
(fun a r => synchronizedRationalScoreMeasure ε hOverlap (code a r)))
b ∈ rationalArraySieve ε J ∧ ProjectionCompatible b.1 b.2 ∧
(∑ a : ArmSpace, ∑ r : LabelSpace J,
(outcomeCDFDistance (lam a r) (b.1 a r) +
scoreCDFDistance (σ a r) (b.2 a r))) < η := by
classical
let δ : ℝ := η / (2 * (J + 1))
have hδ : 0 < δ := by dsimp [δ]; positivity
have hcell (a : ArmSpace) (r : LabelSpace J) :
∃ code : List ProjectionRationalAtom,
outcomeCDFDistance (lam a r) (synchronizedRationalOutcomeMeasure code) +
scoreCDFDistance (σ a r)
(synchronizedRationalScoreMeasure ε hOverlap code) < δ := by
letI : IsFiniteMeasure (lam a r) := ⟨(hfinite a r).1⟩
letI : IsFiniteMeasure (σ a r) := ⟨(hfinite a r).2⟩
exact exists_synchronizedRationalCode_cell_approx hOverlap
(lam a r) (σ a r) (hcompat a r) δ hδ
let code : ArmSpace → LabelSpace J → List ProjectionRationalAtom :=
fun a r => choose (hcell a r)
have hbound (a : ArmSpace) (r : LabelSpace J) :=
Classical.choose_spec (hcell a r)
refine ⟨code, synchronizedRationalArray_mem_sieve hOverlap code,
synchronizedRationalArray_compatible hOverlap code, ?_⟩
have hsum : (∑ a : ArmSpace, ∑ r : LabelSpace J,
(outcomeCDFDistance (lam a r) (synchronizedRationalOutcomeMeasure (code a r)) +
scoreCDFDistance (σ a r)
(synchronizedRationalScoreMeasure ε hOverlap (code a r)))) ≤
∑ a : ArmSpace, ∑ _r : LabelSpace J, δ := by
apply Finset.sum_le_sum
intro a ha
apply Finset.sum_le_sum
intro r hr
exact (hbound a r).le
have hcard : (∑ a : ArmSpace, ∑ _r : LabelSpace J, δ) =
(2 * (J : ℝ)) * δ := by
simp
ring
rw [hcard] at hsum
calc
_ ≤ (2 * (J : ℝ)) * δ := hsum
_ < η := by
dsimp [δ]
have hden : (0 : ℝ) < 2 * ((J : ℝ) + 1) := by positivity
calc
2 * (J : ℝ) * (η / (2 * ((J : ℝ) + 1))) =
(2 * (J : ℝ) * η) / (2 * ((J : ℝ) + 1)) := by ring
_ < η := (div_lt_iff₀ hden).2 (by nlinarith [hη])
theorem outcomeCDFDistance_symm
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma outcomeCDFDistance_symm (μ ν : Measure OutcomeSpace) :
outcomeCDFDistance μ ν = outcomeCDFDistance ν μ := by
unfold outcomeCDFDistance
rw [abs_sub_comm]
congr 1
apply intervalIntegral.integral_congr
intro t ht
dsimp
rw [abs_sub_comm]
theorem scoreCDFDistance_symm
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
μ ν :
Measure (ScoreSpace ε)
conclusion
scoreCDFDistance μ ν = scoreCDFDistance ν μ
Proof (Lean source)
lemma scoreCDFDistance_symm {ε : ℝ} (μ ν : Measure (ScoreSpace ε)) :
scoreCDFDistance μ ν = scoreCDFDistance ν μ := by
unfold scoreCDFDistance
rw [abs_sub_comm]
congr 1
apply intervalIntegral.integral_congr
intro t ht
dsimp
rw [abs_sub_comm]
theorem projectionCandidate_of_strict_population_slack
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J n m :
ℕ
hOverlap :
Overlap ε
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
x :
ExternalSample ε J n m
lam :
σ :
ArmCellArray J (ScoreSpace ε)
b :
hcompat :
ProjectionCompatible b.1 b.2
hcloseOut :
≤ 4 * J / (α * sqrt n)
hcloseScore :
≤ 2 * J * (2 - 2 * ε) / (α * sqrt m)
conclusion
b ∈ projectionCandidates g α x
Proof (Lean source)
lemma projectionCandidate_of_strict_population_slack
{ε : ℝ} {J n m : ℕ}
(hOverlap : Overlap ε)
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(x : ExternalSample ε J n m)
(lam : ArmCellArray J OutcomeSpace)
(σ : ArmCellArray J (ScoreSpace ε))
(b : ArmCellArray J OutcomeSpace × ArmCellArray J (ScoreSpace ε))
(hbfinite : ∀ a r, (b.1 a r) univ < ⊤ ∧ (b.2 a r) univ < ⊤)
(hcompat : ProjectionCompatible b.1 b.2)
(hlamfinite : ∀ a r, (lam a r) univ < ⊤ ∧ (σ a r) univ < ⊤)
(hcloseOut :
(∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (b.1 a r) (lam a r)) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (lam a r) (empiricalTrialCells x.1 a r)) ≤
4 * J / (α * sqrt n))
(hcloseScore :
(∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (b.2 a r) (σ a r)) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (σ a r) (empiricalScoreCells g x.2 a r)) ≤
2 * J * (2 - 2 * ε) / (α * sqrt m)) :
b ∈ projectionCandidates g α x := by
refine ⟨hbfinite, hcompat, ?_, ?_⟩
· have htri (a : ArmSpace) (r : LabelSpace J) :=
outcomeCDFDistance_triangle (b.1 a r) (lam a r)
(empiricalTrialCells x.1 a r) (hbfinite a r).1
(hlamfinite a r).1 (empiricalTrialCells_finite x.1 a r)
have hsum : (∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (b.1 a r) (empiricalTrialCells x.1 a r)) ≤
(∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (b.1 a r) (lam a r)) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
outcomeCDFDistance (lam a r) (empiricalTrialCells x.1 a r)) := by
simp_rw [← Finset.sum_add_distrib]
apply Finset.sum_le_sum
intro a ha
apply Finset.sum_le_sum
intro r hr
exact htri a r
exact hsum.trans hcloseOut
· have htri (a : ArmSpace) (r : LabelSpace J) :=
scoreCDFDistance_triangle hOverlap (b.2 a r) (σ a r)
(empiricalScoreCells g x.2 a r) (hbfinite a r).2
(hlamfinite a r).2 (empiricalScoreCells_finite g x.2 a r)
have hsum : (∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (b.2 a r) (empiricalScoreCells g x.2 a r)) ≤
(∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (b.2 a r) (σ a r)) +
(∑ a : ArmSpace, ∑ r : LabelSpace J,
scoreCDFDistance (σ a r) (empiricalScoreCells g x.2 a r)) := by
simp_rw [← Finset.sum_add_distrib]
apply Finset.sum_le_sum
intro a ha
apply Finset.sum_le_sum
intro r hr
exact htri a r
exact hsum.trans hcloseScore
theorem rationalSieve_dense
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
hOverlap :
Overlap ε
lam :
σ :
ArmCellArray J (ScoreSpace ε)
hcompat :
ProjectionCompatible lam σ
η :
ℝ
hη :
0 < η
conclusion
∃ b ∈ rationalArraySieve ε J,
conclusion 1
ProjectionCompatible b.1 b.2
conclusion 2
(∑ a : ArmSpace, ∑ r : LabelSpace J, (outcomeCDFDistance (lam a r) (b.1 a r) + scoreCDFDistance (σ a r) (b.2 a r))) < η
Proof (Lean source)
lemma rationalSieve_dense {ε : ℝ} {J : ℕ}
(hOverlap : Overlap ε)
(lam : ArmCellArray J OutcomeSpace)
(σ : ArmCellArray J (ScoreSpace ε))
(hfinite : ∀ a r, (lam a r) univ < ⊤ ∧ (σ a r) univ < ⊤)
(hcompat : ProjectionCompatible lam σ) (η : ℝ) (hη : 0 < η) :
∃ b ∈ rationalArraySieve ε J,
ProjectionCompatible b.1 b.2 ∧
(∑ a : ArmSpace, ∑ r : LabelSpace J,
(outcomeCDFDistance (lam a r) (b.1 a r) +
scoreCDFDistance (σ a r) (b.2 a r))) < η := by
obtain ⟨code, hb, hc, hd⟩ :=
exists_rationalSieveArray_close hOverlap lam σ hfinite hcompat η hη
exact ⟨_, hb, hc, hd⟩
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.Helpers.Quantization 26 declarations Paired scalar quantization with a common measurable partition.
Paired scalar quantization with a common measurable partition.
def pairedPartitionCost
For the specified mathematical inputs, this definition introduces the corresponding object.
def optimalPairedCost
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
optimalPairedCost a b S k β :
ℝ
given by
sInf {v : ℝ | ∃ (B : Fin k → Set ℝ) (z : Fin k → Fin S → ℝ), IsIntervalPartition a b k B ∧ (∀ j s, z j s ∈ Icc a b) ∧ v = pairedPartitionCost S k β B z}
def pairedDiagonalWeight
For the specified mathematical inputs, this definition introduces the corresponding object.
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
0 < pairedDiagonalWeight S β x
Proof (Lean source)
lemma pairedDiagonalWeight_pos (a b : ℝ) (S : ℕ) (hS : 0 < S)
(β : Fin S → ℝ → ℝ → ℝ)
(hpos : ∀ s x z, x ∈ Icc a b → z ∈ Icc a b → 0 < β s x z)
(x : ℝ) (hx : x ∈ Icc a b) :
0 < pairedDiagonalWeight S β x := by
unfold pairedDiagonalWeight
have : Nonempty (Fin S) := Fin.pos_iff_nonempty.mp hS
exact Finset.sum_pos (fun s _ => hpos s x x hx hx) univ_nonempty
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma pairedDiagonalWeight_continuousOn (a b : ℝ) (S : ℕ)
(β : Fin S → ℝ → ℝ → ℝ)
(hcont : ∀ s, ContinuousOn
(fun p : ℝ × ℝ => β s p.1 p.2)
(Icc a b ×ˢ Icc a b)) :
ContinuousOn (pairedDiagonalWeight S β) (Icc a b) := by
unfold pairedDiagonalWeight
apply continuousOn_finsetSum
intro s hs
exact (hcont s).comp (continuousOn_id.prodMk continuousOn_id)
(fun x hx => ⟨hx, hx⟩)
theorem pairedDiagonalWeight_uniform_bounds Lemma pairedDiagonalWeight_uniform_bounds in the paper ↗
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
a b :
ℝ
S :
ℕ
hS :
0 < S
β :
Fin S → ℝ → ℝ → ℝ
hcont :
∀ s, ContinuousOn (fun p : ℝ × ℝ => β s p.1 p.2) (Set.Icc a b ×ˢ Set.Icc a b)
hpos :
∀ s x z
if
x ∈ Set.Icc a b
and
z ∈ Set.Icc a b
then
0 < β s x z
conclusion
∃ c C : ℝ,
0 < c
Proof (Lean source)
lemma pairedDiagonalWeight_uniform_bounds (a b : ℝ)
(S : ℕ) (hS : 0 < S) (β : Fin S → ℝ → ℝ → ℝ)
(hcont : ∀ s, ContinuousOn
(fun p : ℝ × ℝ => β s p.1 p.2)
(Icc a b ×ˢ Icc a b))
(hpos : ∀ s x z, x ∈ Icc a b → z ∈ Icc a b → 0 < β s x z) :
∃ c C : ℝ, 0 < c ∧ ∀ x ∈ Icc a b,
c ≤ pairedDiagonalWeight S β x ∧ pairedDiagonalWeight S β x ≤ C := by
have hc := pairedDiagonalWeight_continuousOn a b S β hcont
obtain ⟨c, hcpos, hclower⟩ :=
isCompact_Icc.exists_forall_le' hc (fun x hx =>
pairedDiagonalWeight_pos a b S hS β hpos x hx)
obtain ⟨C, hC⟩ := isCompact_Icc.exists_bound_of_continuousOn hc
refine ⟨c, C, hcpos, ?_⟩
intro x hx
exact ⟨hclower x hx, le_trans (le_abs_self _) (by simpa using hC x hx)⟩
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
a b :
ℝ
hab :
a ≤ b
β :
ℝ → ℝ → ℝ
hcont :
ContinuousOn (fun p : ℝ × ℝ => β p.1 p.2) (Set.Icc a b ×ˢ Set.Icc a b)
hpos :
∀ x z
if
x ∈ Set.Icc a b
and
z ∈ Set.Icc a b
then
0 < β x z
conclusion
∃ c C : ℝ,
0 < c
conclusion 1
x ∈ Set.Icc a b :
z ∈ Set.Icc a b :
c ≤ β x z
β x z ≤ C
Proof (Lean source)
lemma pairedCoefficient_uniform_bounds (a b : ℝ) (hab : a ≤ b)
(β : ℝ → ℝ → ℝ)
(hcont : ContinuousOn (fun p : ℝ × ℝ => β p.1 p.2)
(Icc a b ×ˢ Icc a b))
(hpos : ∀ x z, x ∈ Icc a b → z ∈ Icc a b → 0 < β x z) :
∃ c C : ℝ, 0 < c ∧
∀ x ∈ Icc a b, ∀ z ∈ Icc a b, c ≤ β x z ∧ β x z ≤ C := by
have hcompact : IsCompact (Icc a b ×ˢ Icc a b) :=
isCompact_Icc.prod isCompact_Icc
have hne : (Icc a b ×ˢ Icc a b).Nonempty :=
⟨(a, a), ⟨⟨le_refl a, hab⟩, ⟨le_refl a, hab⟩⟩⟩
obtain ⟨c, hcpos, hclower⟩ :=
hcompact.exists_forall_le' hcont (fun p hp => hpos p.1 p.2 hp.1 hp.2)
obtain ⟨C, hC⟩ := hcompact.exists_bound_of_continuousOn hcont
refine ⟨c, C, hcpos, ?_⟩
intro x hx z hz
exact ⟨hclower (x, z) ⟨hx, hz⟩,
le_trans (le_abs_self _) (by simpa using hC (x, z) ⟨hx, hz⟩)⟩
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
a b :
ℝ
hab :
a ≤ b
S :
ℕ
hS :
0 < S
β :
Fin S → ℝ → ℝ → ℝ
hcont :
∀ s, ContinuousOn (fun p : ℝ × ℝ => β s p.1 p.2) (Set.Icc a b ×ˢ Set.Icc a b)
hpos :
∀ s x z
if
x ∈ Set.Icc a b
and
z ∈ Set.Icc a b
then
0 < β s x z
conclusion
∃ c C : ℝ,
0 < c
conclusion 1
s :
x ∈ Set.Icc a b :
z ∈ Set.Icc a b :
c ≤ β s x z
β s x z ≤ C
Proof (Lean source)
lemma pairedCoefficients_uniform_bounds (a b : ℝ) (hab : a ≤ b)
(S : ℕ) (hS : 0 < S) (β : Fin S → ℝ → ℝ → ℝ)
(hcont : ∀ s, ContinuousOn
(fun p : ℝ × ℝ => β s p.1 p.2)
(Icc a b ×ˢ Icc a b))
(hpos : ∀ s x z, x ∈ Icc a b → z ∈ Icc a b → 0 < β s x z) :
∃ c C : ℝ, 0 < c ∧
∀ s, ∀ x ∈ Icc a b, ∀ z ∈ Icc a b, c ≤ β s x z ∧ β s x z ≤ C := by
classical
have hsingle : ∀ s : Fin S, ∃ c C : ℝ, 0 < c ∧
∀ x ∈ Icc a b, ∀ z ∈ Icc a b,
c ≤ β s x z ∧ β s x z ≤ C := by
intro s
exact pairedCoefficient_uniform_bounds a b hab (β s) (hcont s) (hpos s)
choose c C hc hbound using hsingle
have hne : (Finset.univ : Finset (Fin S)).Nonempty :=
Finset.univ_nonempty_iff.mpr (Fin.pos_iff_nonempty.mp hS)
obtain ⟨smin, _, hmin⟩ := Finset.exists_min_image Finset.univ c hne
obtain ⟨smax, _, hmax⟩ := Finset.exists_max_image Finset.univ C hne
refine ⟨c smin, C smax, hc smin, ?_⟩
intro s x hx z hz
have hsmin := hmin s (Finset.mem_univ s)
have hsmax := hmax s (Finset.mem_univ s)
exact ⟨hsmin.trans (hbound s x hx z hz).1,
(hbound s x hx z hz).2.trans hsmax⟩
def pairedBoundary
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
pairedBoundary a b S k j β :
ℝ
given by
sInf {x : ℝ | x ∈ Icc a b ∧ (j : ℝ) / k * (∫ t in a..b, sqrt (pairedDiagonalWeight S β t)) ≤ ∫ t in a..x, sqrt (pairedDiagonalWeight S β t)}
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
pairedBoundary a b S k 0 β = a
Proof (Lean source)
lemma pairedBoundary_zero (a b : ℝ) (hab : a ≤ b) (S k : ℕ)
(β : Fin S → ℝ → ℝ → ℝ) :
pairedBoundary a b S k 0 β = a := by
unfold pairedBoundary
apply IsLeast.csInf_eq
constructor
· simp [hab]
· rintro x ⟨hx, _⟩
exact hx.1
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
pairedBoundary a b S k j β ∈ Icc a b
Proof (Lean source)
lemma pairedBoundary_mem_Icc (a b : ℝ) (hab : a ≤ b) (S k j : ℕ)
(hk : 0 < k) (hj : j ≤ k)
(β : Fin S → ℝ → ℝ → ℝ) :
pairedBoundary a b S k j β ∈ Icc a b := by
let I : ℝ := ∫ t in a..b, sqrt (pairedDiagonalWeight S β t)
have hI : 0 ≤ I := intervalIntegral.integral_nonneg_of_forall hab
(fun t => Real.sqrt_nonneg _)
have hjk : (j : ℝ) / k ≤ 1 := by
apply (div_le_one (by exact_mod_cast hk)).mpr
exact_mod_cast hj
have hratio : 0 ≤ (j : ℝ) / k := by positivity
have hfeasible : b ∈ {x : ℝ | x ∈ Icc a b ∧
(j : ℝ) / k * I ≤ ∫ t in a..x, sqrt (pairedDiagonalWeight S β t)} := by
refine ⟨⟨hab, le_refl b⟩, ?_⟩
simpa [I] using mul_le_of_le_one_left hI hjk
have hlower : a ≤ sInf {x : ℝ | x ∈ Icc a b ∧
(j : ℝ) / k * I ≤ ∫ t in a..x, sqrt (pairedDiagonalWeight S β t)} := by
apply le_csInf ⟨b, hfeasible⟩
rintro x ⟨hx, _⟩
exact hx.1
have hupper : sInf {x : ℝ | x ∈ Icc a b ∧
(j : ℝ) / k * I ≤ ∫ t in a..x, sqrt (pairedDiagonalWeight S β t)} ≤ b := by
apply csInf_le ⟨a, ?_⟩ hfeasible
rintro x ⟨hx, _⟩
exact hx.1
exact ⟨hlower, hupper⟩
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
pairedBoundary a b S k i β ≤ pairedBoundary a b S k j β
Proof (Lean source)
lemma pairedBoundary_mono (a b : ℝ) (hab : a ≤ b) (S k i j : ℕ)
(hk : 0 < k) (hij : i ≤ j) (hj : j ≤ k)
(β : Fin S → ℝ → ℝ → ℝ) :
pairedBoundary a b S k i β ≤ pairedBoundary a b S k j β := by
let I : ℝ := ∫ t in a..b, sqrt (pairedDiagonalWeight S β t)
have hI : 0 ≤ I := intervalIntegral.integral_nonneg_of_forall hab
(fun t => Real.sqrt_nonneg _)
have hsubset :
{x : ℝ | x ∈ Icc a b ∧ (j : ℝ) / k * I ≤
∫ t in a..x, sqrt (pairedDiagonalWeight S β t)} ⊆
{x : ℝ | x ∈ Icc a b ∧ (i : ℝ) / k * I ≤
∫ t in a..x, sqrt (pairedDiagonalWeight S β t)} := by
rintro x ⟨hx, hbound⟩
refine ⟨hx, ?_⟩
have hratio : (i : ℝ) / k ≤ (j : ℝ) / k := by
apply div_le_div_of_nonneg_right (by exact_mod_cast hij)
exact_mod_cast hk.le
exact (mul_le_mul_of_nonneg_right hratio hI).trans hbound
unfold pairedBoundary
change sInf {x : ℝ | x ∈ Icc a b ∧ (i : ℝ) / k * I ≤
∫ t in a..x, sqrt (pairedDiagonalWeight S β t)} ≤
sInf {x : ℝ | x ∈ Icc a b ∧ (j : ℝ) / k * I ≤
∫ t in a..x, sqrt (pairedDiagonalWeight S β t)}
apply le_csInf
· refine ⟨b, ?_⟩
have hjk : (j : ℝ) / k ≤ 1 :=
(div_le_one (by exact_mod_cast hk)).mpr (by exact_mod_cast hj)
exact ⟨⟨hab, le_refl b⟩, by
simpa [I] using mul_le_of_le_one_left hI hjk⟩
· intro x hx
apply csInf_le
· exact ⟨a, fun y hy => hy.1.1⟩
· exact hsubset hx
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
pairedBoundary a b S k k β = b
Proof (Lean source)
lemma pairedBoundary_last (a b : ℝ) (hab : a < b) (S k : ℕ) (hS : 0 < S)
(hk : 0 < k) (β : Fin S → ℝ → ℝ → ℝ)
(hcont : ∀ s, ContinuousOn
(fun p : ℝ × ℝ => β s p.1 p.2)
(Icc a b ×ˢ Icc a b))
(hpos : ∀ s x z, x ∈ Icc a b → z ∈ Icc a b → 0 < β s x z) :
pairedBoundary a b S k k β = b := by
let f : ℝ → ℝ := fun t => sqrt (pairedDiagonalWeight S β t)
have hf : ContinuousOn f (Icc a b) := by
exact Real.continuous_sqrt.continuousOn.comp
(pairedDiagonalWeight_continuousOn a b S β hcont)
(fun _ _ => Set.mem_univ _)
have hfp : ∀ x ∈ Icc a b, 0 < f x := by
intro x hx
exact Real.sqrt_pos.2 (pairedDiagonalWeight_pos a b S hS β hpos x hx)
have hratio : (k : ℝ) / k = 1 :=
div_self (by exact_mod_cast (Nat.ne_of_gt hk))
have hset :
{x : ℝ | x ∈ Icc a b ∧
(k : ℝ) / k * (∫ t in a..b, f t) ≤ ∫ t in a..x, f t} = {b} := by
ext x
simp only [Set.mem_ofPred_eq, Set.mem_singleton_iff]
constructor
· rintro ⟨hx, htarget⟩
by_contra hne
have hxb : x < b := lt_of_le_of_ne hx.2 hne
have hfxb : ContinuousOn f (Icc x b) :=
hf.mono (Set.Icc_subset_Icc_left hx.1)
have hposxb : 0 < ∫ t in x..b, f t := by
apply intervalIntegral.integral_pos hxb hfxb
· intro t ht
exact (hfp t ⟨hx.1.trans ht.1.le, ht.2⟩).le
· exact ⟨x, ⟨le_refl x, hxb.le⟩, hfp x hx⟩
have hfax : IntervalIntegrable f volume a x :=
(hf.mono (Set.Icc_subset_Icc_right hx.2)).intervalIntegrable_of_Icc hx.1
have hfxb' : IntervalIntegrable f volume x b :=
hfxb.intervalIntegrable_of_Icc hxb.le
have hadd := intervalIntegral.integral_add_adjacent_intervals hfax hfxb'
rw [hratio, one_mul] at htarget
linarith
· intro h
subst x
refine ⟨⟨hab.le, le_refl b⟩, ?_⟩
rw [hratio, one_mul]
unfold pairedBoundary
change sInf {x : ℝ | x ∈ Icc a b ∧
(k : ℝ) / k * (∫ t in a..b, f t) ≤ ∫ t in a..x, f t} = b
rw [hset]
simp
def pairedCompandingCell
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
if j.val + 1
= k then Icc (pairedBoundary a b S k j.val β) (pairedBoundary a b S k (j.val + 1) β) else Ico (pairedBoundary a b S k j.val β) (pairedBoundary a b S k (j.val + 1) β)
def pairedCompandingMidpoint
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
pairedCompandingMidpoint a b S k β j _s :
ℝ
given by
(pairedBoundary a b S k j.val β + pairedBoundary a b S k (j.val + 1) β) / 2
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
pairedCompandingMidpoint a b S k β j s ∈ Icc a b
Proof (Lean source)
lemma pairedCompandingMidpoint_mem_Icc (a b : ℝ) (hab : a ≤ b)
(S k : ℕ) (hk : 0 < k) (β : Fin S → ℝ → ℝ → ℝ) :
∀ j s, pairedCompandingMidpoint a b S k β j s ∈ Icc a b := by
intro j s
have hleft := pairedBoundary_mem_Icc a b hab S k j.val hk (le_of_lt j.isLt) β
have hright := pairedBoundary_mem_Icc a b hab S k (j.val + 1) hk j.isLt β
unfold pairedCompandingMidpoint
constructor <;> linarith [hleft.1, hleft.2, hright.1, hright.2]
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
a b :
ℝ
hab :
a ≤ b
conclusion
(∫ x in a..b, |x - (a + b) / 2|) = (b - a) ^ 2 / 4
Proof (Lean source)
lemma abs_deviation_integral_midpoint (a b : ℝ) (hab : a ≤ b) :
(∫ x in a..b, |x - (a + b) / 2|) = (b - a) ^ 2 / 4 := by
let m : ℝ := (a + b) / 2
have ham : a ≤ m := by dsimp [m]; linarith
have hmb : m ≤ b := by dsimp [m]; linarith
have hleft : (∫ x in a..m, |x - m|) =
m * (m - a) - (m ^ 2 - a ^ 2) / 2 := by
have heq : ∀ x ∈ uIcc a m, |x - m| = m - x := by
intro x hx
rw [Set.uIcc_of_le ham] at hx
rw [abs_of_nonpos (by linarith [hx.2])]
ring
rw [intervalIntegral.integral_congr heq]
rw [intervalIntegral.integral_sub]
· simp [intervalIntegral.integral_const, integral_id, smul_eq_mul]
ring
· exact continuous_const.intervalIntegrable a m
· exact continuous_id.intervalIntegrable a m
have hright : (∫ x in m..b, |x - m|) =
(b ^ 2 - m ^ 2) / 2 - m * (b - m) := by
have heq : ∀ x ∈ uIcc m b, |x - m| = x - m := by
intro x hx
rw [Set.uIcc_of_le hmb] at hx
exact abs_of_nonneg (by linarith [hx.1])
rw [intervalIntegral.integral_congr heq]
rw [intervalIntegral.integral_sub]
· simp [intervalIntegral.integral_const, integral_id, smul_eq_mul]
ring
· exact continuous_id.intervalIntegrable m b
· exact continuous_const.intervalIntegrable m b
have hi₁ : IntervalIntegrable (fun x : ℝ => |x - m|) volume a m :=
(by fun_prop : Continuous (fun x : ℝ => |x - m|)).intervalIntegrable a m
have hi₂ : IntervalIntegrable (fun x : ℝ => |x - m|) volume m b :=
(by fun_prop : Continuous (fun x : ℝ => |x - m|)).intervalIntegrable m b
change (∫ x in a..b, |x - m|) = _
rw [(intervalIntegral.integral_add_adjacent_intervals hi₁ hi₂).symm,
hleft, hright]
dsimp [m]
ring
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma pairedWeightedAbsDeviation_lower {S : ℕ} (γ : Fin S → ℝ)
(z : Fin S → ℝ) (x : ℝ) (hγ : ∀ s, 0 ≤ γ s) :
|(∑ s, γ s) * x - ∑ s, γ s * z s| ≤
∑ s, γ s * |x - z s| := by
have hsum : (∑ s, γ s) * x - ∑ s, γ s * z s =
∑ s, γ s * (x - z s) := by
rw [Finset.sum_mul, ← Finset.sum_sub_distrib]
congr 1
funext s
ring
rw [hsum]
calc
|∑ s, γ s * (x - z s)| ≤ ∑ s, |γ s * (x - z s)| :=
Finset.abs_sum_le_sum_abs _ _
_ = ∑ s, γ s * |x - z s| := by
apply Finset.sum_congr rfl
intro s hs
rw [abs_mul, abs_of_nonneg (hγ s)]
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma pairedBadPoint_cost_lower {S : ℕ} (β : Fin S → ℝ → ℝ → ℝ)
(z : Fin S → ℝ) (x c δ : ℝ)
(hc : 0 ≤ c) (hδ : 0 ≤ δ)
(hβ : ∀ s, c ≤ β s x (z s))
(hbad : ∃ s, δ ≤ |x - z s|) :
c * δ ≤ ∑ s : Fin S, β s x (z s) * |x - z s| := by
obtain ⟨s, hs⟩ := hbad
have hsingle : c * δ ≤ β s x (z s) * |x - z s| :=
mul_le_mul (hβ s) hs hδ (hc.trans (hβ s))
have hnonneg : ∀ t : Fin S, 0 ≤ β t x (z t) * |x - z t| := by
intro t
exact mul_nonneg (hc.trans (hβ t)) (abs_nonneg _)
exact hsingle.trans (Finset.single_le_sum
(fun t _ => hnonneg t) (Finset.mem_univ s))
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
|x - y| ≤ 2 * δ
Proof (Lean source)
lemma pairedGoodCell_diameter {S : ℕ} (hS : 0 < S)
(z : Fin S → ℝ) (x y δ : ℝ)
(hx : ∀ s, |x - z s| ≤ δ)
(hy : ∀ s, |y - z s| ≤ δ) :
|x - y| ≤ 2 * δ := by
let s : Fin S := ⟨0, hS⟩
calc
|x - y| = |(x - z s) + (z s - y)| := by ring
_ ≤ |x - z s| + |z s - y| := abs_add_le _ _
_ ≤ δ + δ := add_le_add (hx s) (by simpa [abs_sub_comm] using hy s)
_ = 2 * δ := by ring
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
Proof (Lean source)
lemma pairedBadSet_measure_cost {S : ℕ} (β : Fin S → ℝ → ℝ → ℝ)
(z : Fin S → ℝ) (G : Set ℝ) (c δ : ℝ)
(hG : MeasurableSet G) (hGfinite : volume G < ⊤)
(hc : 0 ≤ c) (hδ : 0 ≤ δ)
(hβ : ∀ x ∈ G, ∀ s, c ≤ β s x (z s))
(hbad : ∀ x ∈ G, ∃ s, δ ≤ |x - z s|)
(hInt : IntegrableOn
(fun x => ∑ s : Fin S, β s x (z s) * |x - z s|) G volume) :
c * δ * volume.real G ≤
∫ x in G, ∑ s : Fin S, β s x (z s) * |x - z s| := by
have : IsFiniteMeasure (volume.restrict G) :=
⟨by simpa using hGfinite⟩
have hmono := integral_mono_ae (integrable_const (c * δ)) hInt (by
filter_upwards [ae_restrict_mem hG] with x hx
exact pairedBadPoint_cost_lower β z x c δ hc hδ (hβ x hx) (hbad x hx))
simpa [Measure.real_def, smul_eq_mul, mul_comm] using hmono
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
S :
ℕ
β :
Fin S → ℝ → ℝ → ℝ
z :
Fin S → ℝ
G :
Set ℝ
c δ C :
ℝ
k :
ℕ
hG :
hGfinite :
volume G < ⊤
hc :
0 < c
hδ :
0 < δ
hk :
0 < k
hβ :
∀ x ∈ G, ∀ s, c ≤ β s x (z s)
hbad :
∀ x ∈ G, ∃ s, δ ≤ |x - z s|
hInt :
hcost :
(∫ x in G, ∑ s : Fin S, β s x (z s) * |x - z s|) ≤ C / k
conclusion
volume.real G ≤ C / (c * δ * k)
Proof (Lean source)
lemma pairedBadSet_measure_le {S : ℕ} (β : Fin S → ℝ → ℝ → ℝ)
(z : Fin S → ℝ) (G : Set ℝ) (c δ C : ℝ) (k : ℕ)
(hG : MeasurableSet G) (hGfinite : volume G < ⊤)
(hc : 0 < c) (hδ : 0 < δ) (hk : 0 < k)
(hβ : ∀ x ∈ G, ∀ s, c ≤ β s x (z s))
(hbad : ∀ x ∈ G, ∃ s, δ ≤ |x - z s|)
(hInt : IntegrableOn
(fun x => ∑ s : Fin S, β s x (z s) * |x - z s|) G volume)
(hcost : (∫ x in G, ∑ s : Fin S, β s x (z s) * |x - z s|) ≤ C / k) :
volume.real G ≤ C / (c * δ * k) := by
have hbound := pairedBadSet_measure_cost β z G c δ hG hGfinite
hc.le hδ.le hβ hbad hInt
have hcdk : 0 < c * δ * k := by positivity
apply (le_div_iff₀ hcdk).2
calc
volume.real G * (c * δ * k) = (c * δ * volume.real G) * k := by ring
_ ≤ (C / k) * k := mul_le_mul_of_nonneg_right (hbound.trans hcost) (by positivity)
_ = C := by field_simp
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
0 ≤ pairedPartitionCost S k β B z
Proof (Lean source)
lemma pairedPartitionCost_nonneg (a b : ℝ) (S k : ℕ)
(β : Fin S → ℝ → ℝ → ℝ) (B : Fin k → Set ℝ)
(z : Fin k → Fin S → ℝ)
(hB : IsIntervalPartition a b k B)
(hz : ∀ j s, z j s ∈ Icc a b)
(hβ : ∀ s x y, x ∈ Icc a b → y ∈ Icc a b → 0 ≤ β s x y) :
0 ≤ pairedPartitionCost S k β B z := by
unfold pairedPartitionCost
apply Finset.sum_nonneg
intro j hj
apply Finset.sum_nonneg
intro s hs
apply integral_nonneg_of_ae
filter_upwards [ae_restrict_mem (hB.1 j)] with x hx
have hxI : x ∈ Icc a b := by
rw [← hB.2.2]
exact Set.mem_iUnion_of_mem j hx
exact mul_nonneg (hβ s x (z j s) hxI (hz j s)) (abs_nonneg _)
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
0 ≤ optimalPairedCost a b S k β
Proof (Lean source)
lemma optimalPairedCost_nonneg (a b : ℝ) (hab : a ≤ b) (S k : ℕ) (hk : 0 < k)
(β : Fin S → ℝ → ℝ → ℝ)
(hβ : ∀ s x y, x ∈ Set.Icc a b → y ∈ Set.Icc a b → 0 ≤ β s x y) :
0 ≤ optimalPairedCost a b S k β := by
unfold optimalPairedCost
apply le_csInf
· let j₀ : Fin k := ⟨0, hk⟩
let B : Fin k → Set ℝ := fun j => if j = j₀ then Set.Icc a b else ∅
let z : Fin k → Fin S → ℝ := fun _ _ => a
refine ⟨pairedPartitionCost S k β B z, B, z, ?_, ?_, rfl⟩
· refine ⟨?_, ?_, ?_⟩
· intro j
dsimp [B]
split_ifs <;> measurability
· intro i j hij
by_cases hi : i = j₀
· by_cases hj : j = j₀
· exact (hij (hi.trans hj.symm)).elim
· simp [B, hi, hj]
· simp [B, hi]
· ext x
simp [B, j₀]
· intro j s
exact ⟨le_refl a, hab⟩
· rintro v ⟨B, z, hB, hz, rfl⟩
exact pairedPartitionCost_nonneg a b S k β B z hB hz hβ
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
optimalPairedCost a b S k β ≤ pairedPartitionCost S k β B z
Proof (Lean source)
lemma optimalPairedCost_le_feasible (a b : ℝ) (S k : ℕ)
(β : Fin S → ℝ → ℝ → ℝ) (B : Fin k → Set ℝ)
(z : Fin k → Fin S → ℝ)
(hB : IsIntervalPartition a b k B)
(hz : ∀ j s, z j s ∈ Icc a b)
(hβ : ∀ s x y, x ∈ Icc a b → y ∈ Icc a b → 0 ≤ β s x y) :
optimalPairedCost a b S k β ≤ pairedPartitionCost S k β B z := by
unfold optimalPairedCost
apply csInf_le
· refine ⟨0, ?_⟩
rintro v ⟨B', z', hB', hz', rfl⟩
exact pairedPartitionCost_nonneg a b S k β B' z' hB' hz' hβ
· exact ⟨B, z, hB, hz, rfl⟩
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
conclusion 1
Tendsto (fun k : ℕ => (k : ℝ) * optimalPairedCost a b S k β) atTop (nhds ((1 / 4 : ℝ) * (∫ x in a..b, sqrt (pairedDiagonalWeight S β x)) ^ 2))
conclusion 2
Tendsto (fun k : ℕ => (k : ℝ) * pairedPartitionCost S k β (pairedCompandingCell a b S k β) (pairedCompandingMidpoint a b S k β)) atTop (nhds ((1 / 4 : ℝ) * (∫ x in a..b, sqrt (pairedDiagonalWeight S β x)) ^ 2))
conclusion 3
k :
0 < k
IsIntervalPartition a b k (pairedCompandingCell a b S k β)
j s :
pairedCompandingMidpoint a b S k β j s ∈ Icc a b
Proof (Lean source)
lemma pairedQuantization_tendsto (a b : ℝ) (hab : a < b)
(S : ℕ) (hS : 0 < S) (β : Fin S → ℝ → ℝ → ℝ)
(hcont : ∀ s, ContinuousOn
(fun p : ℝ × ℝ => β s p.1 p.2)
(Icc a b ×ˢ Icc a b))
(hpos : ∀ s x z, x ∈ Icc a b → z ∈ Icc a b → 0 < β s x z) :
Tendsto (fun k : ℕ => (k : ℝ) * optimalPairedCost a b S k β)
atTop (nhds ((1 / 4 : ℝ) *
(∫ x in a..b, sqrt (pairedDiagonalWeight S β x)) ^ 2)) ∧
Tendsto (fun k : ℕ => (k : ℝ) *
pairedPartitionCost S k β
(pairedCompandingCell a b S k β)
(pairedCompandingMidpoint a b S k β))
atTop (nhds ((1 / 4 : ℝ) *
(∫ x in a..b, sqrt (pairedDiagonalWeight S β x)) ^ 2)) ∧
(∀ k, 0 < k →
IsIntervalPartition a b k (pairedCompandingCell a b S k β) ∧
∀ j s, pairedCompandingMidpoint a b S k β j s ∈ Icc a b) := by
refine ⟨?_, ?_, ?_⟩
· exact
optimal_scaled_cost_tendsto
a b hab S hS β hcont hpos
· exact
companding_scaled_cost_tendsto
a b hab S hS β hcont hpos
· intro k hk
refine ⟨?_, pairedCompandingMidpoint_mem_Icc a b hab.le S k hk β⟩
let f : ℕ → ℝ := fun j => pairedBoundary a b S k j β
have hm : ∀ i j, i ≤ j → j ≤ k → f i ≤ f j := by
intro i j hij hj
exact pairedBoundary_mono a b hab.le S k i j hk hij hj β
have hpart := orderedIntervalCells_partition f k hk hm
have hzero : f 0 = a := pairedBoundary_zero a b hab.le S k β
have hlast : f k = b := pairedBoundary_last a b hab S k hS hk β hcont hpos
rw [hzero, hlast] at hpart
change IsIntervalPartition a b k (fun j : Fin k =>
if j.val + 1 = k then
Icc (pairedBoundary a b S k j.val β)
(pairedBoundary a b S k (j.val + 1) β)
else
Ico (pairedBoundary a b S k j.val β)
(pairedBoundary a b S k (j.val + 1) β))
exact hpart
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.OpenQuestions 4 declarations Open normalized external-score limit question.
Open normalized external-score limit question.
def externalHonestProceduresOn
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
ε :
ℝ
J n m :
ℕ
X :
Qj :
α :
ℝ
def
given by
{C | ∀ PH ∈ X, 1 - α ≤ (Qj PH.1 PH.2).real {x | C.contains x (ate PH.1)}}
def externalExcessRiskOn
For the external model, subclass, sampling law, and miscoverage inputs, this definition gives the minimax expected positive excess length with honesty and the risk supremum restricted to the same subclass.
Definition (Lean source)
parameters
def
externalExcessRiskOn g X hX Qj α :
ℝ
given by
sInf {v : ℝ | ∃ C ∈ externalHonestProceduresOn X Qj α, v = sSup {u : ℝ | ∃ PH, ∃ hPH : PH ∈ X, u = ∫ x, max 0 (C.length x - sharpATELength PH.2 g (releasedLaw PH.1) (externalLawCellMasses g PH (hX hPH))) ∂(Qj PH.1 PH.2)}}
def externalNormalizedRiskOn
For the sample sizes, external model, subclass, sampling law, and miscoverage inputs, this definition normalizes the subclass-restricted excess risk by the two-sample root-rate factor.
Definition (Lean source)
parameters
def
externalNormalizedRiskOn n m g X hX Qj α :
ℝ
given by
externalNormalizer n m * externalExcessRiskOn g X hX Qj α
def ExternalSharpLimitQuestion
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
given by
"For each rho>0, determine the liminf and limsup of T_{n,m,alpha}(g) \ as n,m→∞ and n/m→rho over ExternalLaws(g), using the product \ externalExperiment and uniformly honest procedures. Decide whether the \ values coincide and compute their common sharp value if so. If the full-class \ sharp limit fails, identify an explicitly defined regular subclass of \ ExternalLaws(g), state its regularity conditions, and determine its common \ sharp limit for every rho>0. Restrict both uniform honesty and the supremum \ in expected excess length to that same subclass. Use the external projection \ handle to construct an attaining procedure or counterexample. This is an open \ request, not an assertion or a greatest-subclass claim."
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.TAllLabelAmbiguity 2 declarations Exact worst-case ATE ambiguity for arbitrary measurable release cells.
Exact worst-case ATE ambiguity for arbitrary measurable release cells.
theorem halfOutcome_endpoint_attainment
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
conclusion 1
conclusion 2
IsProbabilityMeasure Pminus
IsProbabilityMeasure Pplus
CompatibleCausalLaw H g (halfOutcomeReleasedLaw H g) Pminus
CompatibleCausalLaw H g (halfOutcomeReleasedLaw H g) Pplus
ate Pminus
- muUpper H g (halfOutcomeReleasedLaw H g) (halfOutcomeReleasedLaw_cellMasses H g hOverlap hg) false
ate Pplus
- muLower H g (halfOutcomeReleasedLaw H g) (halfOutcomeReleasedLaw_cellMasses H g hOverlap hg) false
ate Pplus - ate Pminus
= sharpATELength H g (halfOutcomeReleasedLaw H g) (halfOutcomeReleasedLaw_cellMasses H g hOverlap hg)
Proof (Lean source)
lemma halfOutcome_endpoint_attainment {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε) (hg : Measurable g) :
(∃ P : Measure (FullRow ε J), IsProbabilityMeasure P ∧
CompatibleCausalLaw H g (halfOutcomeReleasedLaw H g) P) ∧
(∃ Pminus Pplus : Measure (FullRow ε J),
IsProbabilityMeasure Pminus ∧ IsProbabilityMeasure Pplus ∧
CompatibleCausalLaw H g (halfOutcomeReleasedLaw H g) Pminus ∧
CompatibleCausalLaw H g (halfOutcomeReleasedLaw H g) Pplus ∧
ate Pminus =
muLower H g (halfOutcomeReleasedLaw H g)
(halfOutcomeReleasedLaw_cellMasses H g hOverlap hg) true -
muUpper H g (halfOutcomeReleasedLaw H g)
(halfOutcomeReleasedLaw_cellMasses H g hOverlap hg) false ∧
ate Pplus =
muUpper H g (halfOutcomeReleasedLaw H g)
(halfOutcomeReleasedLaw_cellMasses H g hOverlap hg) true -
muLower H g (halfOutcomeReleasedLaw H g)
(halfOutcomeReleasedLaw_cellMasses H g hOverlap hg) false ∧
ate Pplus - ate Pminus =
sharpATELength H g (halfOutcomeReleasedLaw H g)
(halfOutcomeReleasedLaw_cellMasses H g hOverlap hg)) := by
exact compatible_endpointGap_eq_sharpATELength H g
(halfOutcomeReleasedLaw H g)
(halfOutcomeReleasedLaw_cellMasses H g hOverlap hg) hOverlap
theorem worstCaseAmbiguity_eq
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
conclusion 1
conclusion 2
conclusion 3
= worstCaseAmbiguity H g
conclusion 4
IsProbabilityMeasure Pminus
IsProbabilityMeasure Pplus
CompatibleCausalLaw H g (halfOutcomeReleasedLaw H g) Pminus
CompatibleCausalLaw H g (halfOutcomeReleasedLaw H g) Pplus
ate Pminus
- muUpper H g (halfOutcomeReleasedLaw H g) (halfOutcomeReleasedLaw_cellMasses H g hOverlap hg) false
ate Pplus
- muLower H g (halfOutcomeReleasedLaw H g) (halfOutcomeReleasedLaw_cellMasses H g hOverlap hg) false
ate Pplus - ate Pminus = worstCaseAmbiguity H g
Proof (Lean source)
theorem worstCaseAmbiguity_eq {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε) (hg : Measurable g) :
worstCaseAmbiguity H g =
∑ r : LabelSpace J,
(cellAbsoluteDeviation H g true r + cellAbsoluteDeviation H g false r) ∧
(∃ P : Measure (FullRow ε J), IsProbabilityMeasure P ∧
CompatibleCausalLaw H g (halfOutcomeReleasedLaw H g) P) ∧
sharpATELength H g (halfOutcomeReleasedLaw H g)
(halfOutcomeReleasedLaw_cellMasses H g hOverlap hg) =
worstCaseAmbiguity H g ∧
(∃ Pminus Pplus : Measure (FullRow ε J),
IsProbabilityMeasure Pminus ∧ IsProbabilityMeasure Pplus ∧
CompatibleCausalLaw H g (halfOutcomeReleasedLaw H g) Pminus ∧
CompatibleCausalLaw H g (halfOutcomeReleasedLaw H g) Pplus ∧
ate Pminus =
muLower H g (halfOutcomeReleasedLaw H g)
(halfOutcomeReleasedLaw_cellMasses H g hOverlap hg) true -
muUpper H g (halfOutcomeReleasedLaw H g)
(halfOutcomeReleasedLaw_cellMasses H g hOverlap hg) false ∧
ate Pplus =
muUpper H g (halfOutcomeReleasedLaw H g)
(halfOutcomeReleasedLaw_cellMasses H g hOverlap hg) true -
muLower H g (halfOutcomeReleasedLaw H g)
(halfOutcomeReleasedLaw_cellMasses H g hOverlap hg) false ∧
ate Pplus - ate Pminus = worstCaseAmbiguity H g) := by
let S : ℝ := ∑ r : LabelSpace J,
(cellAbsoluteDeviation H g true r + cellAbsoluteDeviation H g false r)
have hupper : worstCaseAmbiguity H g ≤ S :=
worstCaseAmbiguity_le_sum_cellAbsoluteDeviation H g hOverlap hg
have hfair :
sharpATELength H g (halfOutcomeReleasedLaw H g)
(halfOutcomeReleasedLaw_cellMasses H g hOverlap hg) = S :=
halfOutcome_sharpATELength_eq_sum_cellAbsoluteDeviation H g hOverlap hg
rcases halfOutcome_endpoint_attainment H g hOverlap hg with
⟨hexists, hendpoint⟩
rcases hexists with ⟨P, hPprob, hPcomp⟩
have hPrandom : RandomizedCausalLaw H g P :=
{ probability := hPcomp.probability
measurableRelease := hPcomp.measurableRelease
scoreMarginal := hPcomp.scoreMarginal
randomizedAssignment := hPcomp.randomizedAssignment
consistency := hPcomp.consistency
deterministicRelease := hPcomp.deterministicRelease }
let A : Set ℝ := {d : ℝ | ∃ P : Measure (FullRow ε J),
∃ (_hprob : IsProbabilityMeasure P), ∃ hP : RandomizedCausalLaw H g P,
d = sharpATELength H g (releasedLaw P)
(compatibleCellMasses_of_randomizedCausalLaw H g P hP)}
have hAbdd : BddAbove A := by
refine ⟨S, ?_⟩
rintro d ⟨Q, hQprob, hQ, rfl⟩
exact randomizedCausalLaw_sharpATELength_le_sum_cellAbsoluteDeviation
H g hOverlap Q hQ
have hfairMem :
sharpATELength H g (halfOutcomeReleasedLaw H g)
(halfOutcomeReleasedLaw_cellMasses H g hOverlap hg) ∈ A := by
refine ⟨P, hPprob, hPrandom, ?_⟩
exact (sharpATELength_congr_releasedMeasure H g
(releasedLaw P) (halfOutcomeReleasedLaw H g) hPcomp.releasedLaw
(compatibleCellMasses_of_randomizedCausalLaw H g P hPrandom)
(halfOutcomeReleasedLaw_cellMasses H g hOverlap hg)).symm
have hlower : S ≤ worstCaseAmbiguity H g := by
rw [← hfair]
unfold worstCaseAmbiguity
exact le_csSup hAbdd hfairMem
have heq : worstCaseAmbiguity H g = S := le_antisymm hupper hlower
refine ⟨heq, ⟨P, hPprob, hPcomp⟩, hfair.trans heq.symm, ?_⟩
rcases hendpoint with
⟨Pminus, Pplus, hminusProb, hplusProb, hminusComp, hplusComp,
hminus, hplus, hgap⟩
exact ⟨Pminus, Pplus, hminusProb, hplusProb, hminusComp, hplusComp,
hminus, hplus, hgap.trans (hfair.trans heq.symm)⟩
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.TCompatibility 9 declarations Compatibility of a released law with a randomized causal population.
Compatibility of a released law with a randomized causal population.
theorem compatibility_armCellMass_sum
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
(∑ a : ArmSpace, armCellMass H g a r) = H.real (cell g r)
Proof (Lean source)
lemma compatibility_armCellMass_sum {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (r : LabelSpace J) :
(∑ a : ArmSpace, armCellMass H g a r) = H.real (cell g r) := by
have hint : IntegrableOn (fun e : ScoreSpace ε => (e : ℝ)) (cell g r) H := by
apply Integrable.of_bound (by fun_prop) (|ε| + |1 - ε| + 1)
filter_upwards [] with e
change |(e : ℝ)| ≤ |ε| + |1 - ε| + 1
rcases e.property with ⟨he₁, he₂⟩
have h₁ := neg_le_abs ε
have h₂ := le_abs_self (1 - ε)
have h₃ := neg_le_abs (1 - ε)
have h₄ := le_abs_self ε
exact abs_le.mpr ⟨by linarith, by linarith⟩
simp only [Fintype.sum_bool, armCellMass, armProb, Bool.false_eq_true,
↓reduceIte]
rw [integral_sub (integrable_const 1) hint]
simp
theorem releasedLabelMass_sum_arms
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Prel.real {z | z.1 = r} = ∑ a : ArmSpace, Prel.real {z | z.1 = r ∧ z.2.1 = a}
Proof (Lean source)
lemma releasedLabelMass_sum_arms {J : ℕ}
(Prel : Measure (Observation J)) [IsProbabilityMeasure Prel]
(r : LabelSpace J) :
Prel.real {z | z.1 = r} =
∑ a : ArmSpace, Prel.real {z | z.1 = r ∧ z.2.1 = a} := by
let S : Set (Observation J) := {z | z.1 = r}
let T : Set (Observation J) := {z | z.2.1 = true}
have hT : MeasurableSet T :=
measurableSet_eq_fun (measurable_fst.comp measurable_snd) measurable_const
have h := measureReal_inter_add_sdiff (μ := Prel) (s := S) (t := T) hT
have hST : S ∩ T = {z | z.1 = r ∧ z.2.1 = true} := by
ext z
simp [S, T]
have hSD : S \ T = {z | z.1 = r ∧ z.2.1 = false} := by
ext z
simp [S, T, Bool.not_eq_true]
rw [hST, hSD] at h
simpa [S, Fintype.sum_bool] using h.symm
theorem compatibleCellMasses_of_label_treated
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
IsProbabilityMeasure Prel
hg :
h :
∀ r : LabelSpace J,
Prel.real {z | z.1 = r} = H.real (cell g r) ∧
conclusion
CompatibleCellMasses H g Prel
Proof (Lean source)
lemma compatibleCellMasses_of_label_treated {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) [IsProbabilityMeasure H]
[IsProbabilityMeasure Prel] (hg : Measurable g)
(h : ∀ r : LabelSpace J,
Prel.real {z | z.1 = r} = H.real (cell g r) ∧
Prel.real {z | z.1 = r ∧ z.2.1 = true} = armCellMass H g true r) :
CompatibleCellMasses H g Prel := by
refine ⟨inferInstance, inferInstance, hg, ?_⟩
intro a r
cases a
· have hlabel := (h r).1
have htreated := (h r).2
have hsum := compatibility_armCellMass_sum H g r
have hreleased := releasedLabelMass_sum_arms Prel r
simp only [Fintype.sum_bool] at hsum hreleased
linarith
· exact (h r).2
theorem armCellMass_lower_overlap
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
ε * H.real (cell g r) ≤ armCellMass H g a r
Proof (Lean source)
lemma armCellMass_lower_overlap {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε)
(a : ArmSpace) (r : LabelSpace J) :
ε * H.real (cell g r) ≤ armCellMass H g a r := by
have hInt : IntegrableOn (fun e : ScoreSpace ε => armProb a e) (cell g r) H := by
apply Integrable.of_bound (by
cases a <;> simp only [armProb, Bool.false_eq_true, ↓reduceIte] <;> fun_prop) 1
filter_upwards [] with e
rcases e.property with ⟨he₁, he₂⟩
rcases hOverlap with ⟨hε, hε'⟩
change |armProb a e| ≤ 1
cases a <;> simp [armProb, abs_le] <;> constructor <;> linarith
have hMono : (∫ _ in cell g r, ε ∂H) ≤
∫ e in cell g r, armProb a e ∂H := by
apply integral_mono (integrable_const ε) hInt
intro e
rcases e.property with ⟨he₁, he₂⟩
rcases hOverlap with ⟨hε, hε'⟩
cases a <;> simp [armProb] <;> linarith
simpa [armCellMass, Measure.real_def, mul_comm] using hMono
theorem armCellMass_pos_of_cellMass_pos
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
a :
r :
hcell :
0 < H.real (cell g r)
conclusion
0 < armCellMass H g a r
Proof (Lean source)
lemma armCellMass_pos_of_cellMass_pos {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε)
(a : ArmSpace) (r : LabelSpace J)
(hcell : 0 < H.real (cell g r)) :
0 < armCellMass H g a r := by
have hbound := armCellMass_lower_overlap H g hOverlap a r
exact lt_of_lt_of_le (mul_pos hOverlap.1 hcell) hbound
theorem armCellMass_eq_zero_iff_cellMass_eq_zero
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
armCellMass H g a r = 0 ↔ H.real (cell g r) = 0
Proof (Lean source)
lemma armCellMass_eq_zero_iff_cellMass_eq_zero {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε)
(a : ArmSpace) (r : LabelSpace J) :
armCellMass H g a r = 0 ↔ H.real (cell g r) = 0 := by
have h₀ := armCellMass_lower_overlap H g hOverlap false r
have h₁ := armCellMass_lower_overlap H g hOverlap true r
have hsum := compatibility_armCellMass_sum H g r
have hmass : 0 ≤ H.real (cell g r) := measureReal_nonneg
simp only [Fintype.sum_bool] at hsum
constructor
· intro h
by_contra hne
have hp : 0 < H.real (cell g r) := lt_of_le_of_ne hmass (Ne.symm hne)
have := armCellMass_pos_of_cellMass_pos H g hOverlap a r hp
linarith
· intro h
rw [h] at h₀ h₁ hsum
cases a <;> simp_all <;> linarith
theorem armCellOutcomeLaw_isProbabilityMeasure
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
J :
ℕ
Prel :
Measure (Observation J)
IsProbabilityMeasure Prel
a :
r :
hq :
0 < Prel.real {z | z.1 = r ∧ z.2.1 = a}
conclusion
Proof (Lean source)
lemma armCellOutcomeLaw_isProbabilityMeasure {J : ℕ}
(Prel : Measure (Observation J)) [IsProbabilityMeasure Prel]
(a : ArmSpace) (r : LabelSpace J)
(hq : 0 < Prel.real {z | z.1 = r ∧ z.2.1 = a}) :
IsProbabilityMeasure (armCellOutcomeLaw Prel inferInstance a r hq) := by
let C : Set (Observation J) := {z | z.1 = r ∧ z.2.1 = a}
have hC : Prel C ≠ 0 :=
(measureReal_ne_zero_iff).mp (ne_of_gt hq)
have hcond : IsProbabilityMeasure (cond Prel C) :=
ProbabilityTheory.cond_isProbabilityMeasure hC
have hmap : Measurable (fun z : Observation J => (z.2.2 : ℝ)) := by fun_prop
unfold armCellOutcomeLaw
change IsProbabilityMeasure
((Prel C)⁻¹ • ((Prel.restrict C).map (fun z : Observation J => (z.2.2 : ℝ))))
rw [← Measure.map_smul]
exact @Measure.isProbabilityMeasure_map _ _ _ _ _ hcond _ hmap.aemeasurable
theorem armCellOutcomeLaw_apply
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
J :
ℕ
Prel :
Measure (Observation J)
IsProbabilityMeasure Prel
a :
r :
hq :
0 < Prel.real {z | z.1 = r ∧ z.2.1 = a}
B :
Set ℝ
hB :
conclusion
armCellOutcomeLaw Prel inferInstance a r hq B
= (Prel {z | z.1 = r ∧ z.2.1 = a})⁻¹ * Prel {z | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ∈ B}
Proof (Lean source)
lemma armCellOutcomeLaw_apply {J : ℕ}
(Prel : Measure (Observation J)) [IsProbabilityMeasure Prel]
(a : ArmSpace) (r : LabelSpace J)
(hq : 0 < Prel.real {z | z.1 = r ∧ z.2.1 = a})
(B : Set ℝ) (hB : MeasurableSet B) :
armCellOutcomeLaw Prel inferInstance a r hq B =
(Prel {z | z.1 = r ∧ z.2.1 = a})⁻¹ *
Prel {z | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ∈ B} := by
have hm : Measurable (fun z : Observation J => (z.2.2 : ℝ)) := by fun_prop
have hC : MeasurableSet {z : Observation J | z.1 = r ∧ z.2.1 = a} :=
(measurableSet_eq_fun measurable_fst measurable_const).inter
(measurableSet_eq_fun (measurable_fst.comp measurable_snd) measurable_const)
simp only [armCellOutcomeLaw, Measure.smul_apply]
rw [Measure.map_apply hm hB, Measure.restrict_apply (hm hB)]
have hset :
(fun z : Observation J => (z.2.2 : ℝ)) ⁻¹' B ∩
{z | z.1 = r ∧ z.2.1 = a} =
{z | z.1 = r ∧ z.2.1 = a ∧ (z.2.2 : ℝ) ∈ B} := by
ext z
simp only [Set.mem_inter_iff, Set.mem_preimage, Set.mem_ofPred_eq]
tauto
rw [hset]
simp only [smul_eq_mul]
theorem compatible_nonempty_iff
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
Prel :
Measure (Observation J)
@realizes H(probability normalization)
IsProbabilityMeasure Prel
@realizes Prel(probability normalization)
hOverlap :
Overlap ε
hg :
conclusion
↔ ∀ r : LabelSpace J, Prel.real {z | z.1 = r} = H.real (cell g r) ∧ Prel.real {z | z.1 = r ∧ z.2.1 = true} = armCellMass H g true r
Proof (Lean source)
theorem compatible_nonempty_iff {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(Prel : Measure (Observation J)) [IsProbabilityMeasure H]
-- @realizes H(probability normalization)
[IsProbabilityMeasure Prel] -- @realizes Prel(probability normalization)
(hOverlap : Overlap ε)
(hg : Measurable g) :
(∃ P : Measure (FullRow ε J), IsProbabilityMeasure P ∧
CompatibleCausalLaw H g Prel P) ↔
∀ r : LabelSpace J,
Prel.real {z | z.1 = r} = H.real (cell g r) ∧
Prel.real {z | z.1 = r ∧ z.2.1 = true} =
armCellMass H g true r := by
constructor
· rintro ⟨P, _, hP⟩ r
have hMass := compatibleCellMasses_of_compatibleCausalLaw H g Prel P hP
constructor
· rw [releasedLabelMass_sum_arms]
simp_rw [hMass.2.2.2]
exact (compatibility_armCellMass_sum H g r)
· exact hMass.2.2.2 true r
· intro h
have hMass : CompatibleCellMasses H g Prel :=
compatibleCellMasses_of_label_treated H g Prel hg h
let P := reconstructedFullLaw H g Prel hg
refine ⟨P, reconstructedFullLaw_isProbabilityMeasure H g Prel hg hOverlap, ?_⟩
exact reconstructedFullLaw_compatibleCausalLaw H g Prel hg hMass hOverlap
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.TExternalScoreRootOrder 1 declarations Root-two-sample order for the external score-log experiment.
Root-two-sample order for the external score-log experiment.
theorem externalExcessRisk_root_order
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
g :
ScoreSpace ε → LabelSpace J
α :
ℝ
hOverlap :
Overlap ε
hg :
hα :
0 < α ∧ α < 1 / 2
@realizes alpha(miscoverage range)
Qj :
∀ n m
hTrial :
∀ (n m : ℕ), JointTrialIID g (Qj n m)
hLog :
∀ (n m : ℕ), ExternalLogIID g (Qj n m)
hInd :
∀ (n m : ℕ), ExternalLogIndependent g (Qj n m)
conclusion
conclusion 1
0 < fiberTripleSeparation g
fiberTripleSeparation g ≤ 1
conclusion 2
∃ C : ℝ,
0 < C
n m :
ℕ
0 < n
0 < m
∃ CI : ExternalIntervalProcedure ε J n m,
CI ∈ externalHonestProcedures g (Qj n m) α
x :
Icc (CI.lo x) (CI.hi x) = externalProjectionCI g α x
PH :
hPH :
PH ∈ ExternalLaws g
(∫ x, max 0 (CI.length x - sharpATELength PH.2 g (releasedLaw PH.1) (externalLawCellMasses g PH hPH)) ∂(Qj n m PH.1 PH.2))
conclusion 3
logCertificate g α
≤ liminf (fun m : ℕ => sInf {v : ℝ | ∃ n : ℕ, 0 < n ∧ v = sqrt (m : ℝ) * externalExcessRisk g (Qj n m) α}) atTop
0 < logCertificate g α
conclusion 4
∃ C : ℝ,
0 < C
ρ :
ℝ
0 < ρ
Proof (Lean source)
theorem externalExcessRisk_root_order {ε : ℝ} {J : ℕ}
(g : ScoreSpace ε → LabelSpace J) (α : ℝ)
(hOverlap : Overlap ε) (hg : Measurable g)
(hα : 0 < α ∧ α < 1 / 2) -- @realizes alpha(miscoverage range)
(Qj : ∀ n m, Measure (FullRow ε J) → Measure (ScoreSpace ε) →
Measure (ExternalSample ε J n m))
(hTrial : ∀ (n m : ℕ), JointTrialIID g (Qj n m))
(hLog : ∀ (n m : ℕ), ExternalLogIID g (Qj n m))
(hInd : ∀ (n m : ℕ), ExternalLogIndependent g (Qj n m)) :
(0 < fiberTripleSeparation g ∧ fiberTripleSeparation g ≤ 1) ∧
(∃ C : ℝ, 0 < C ∧
∀ (n m : ℕ), 0 < n → 0 < m →
-- @realizes n(positive trial size) @realizes m(positive log size)
trialCertificate α * (sqrt (n : ℝ))⁻¹ ≤
externalExcessRisk g (Qj n m) α ∧
externalExcessRisk g (Qj n m) α ≤
C * ((sqrt (n : ℝ))⁻¹ + (sqrt (m : ℝ))⁻¹) ∧
∃ CI : ExternalIntervalProcedure ε J n m,
CI ∈ externalHonestProcedures g (Qj n m) α ∧
(∀ x, Icc (CI.lo x) (CI.hi x) = externalProjectionCI g α x) ∧
∀ PH, ∀ hPH : PH ∈ ExternalLaws g,
(∫ x, max 0 (CI.length x -
sharpATELength PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH)) ∂(Qj n m PH.1 PH.2)) ≤
C * ((sqrt (n : ℝ))⁻¹ + (sqrt (m : ℝ))⁻¹)) ∧
(logCertificate g α ≤
liminf (fun m : ℕ => sInf {v : ℝ | ∃ n : ℕ, 0 < n ∧
v = sqrt (m : ℝ) * externalExcessRisk g (Qj n m) α}) atTop ∧
0 < logCertificate g α) ∧
(∃ C : ℝ, 0 < C ∧ ∀ ρ : ℝ, 0 < ρ →
-- @realizes rho(positive asymptotic ratio)
max (trialCertificate α / (1 + sqrt ρ))
(logCertificate g α * sqrt ρ / (1 + sqrt ρ)) ≤
liminf (fun nm : ℕ × ℕ =>
externalNormalizedRisk g (Qj nm.1 nm.2) α) (ratioFilter ρ) ∧
liminf (fun nm : ℕ × ℕ =>
externalNormalizedRisk g (Qj nm.1 nm.2) α) (ratioFilter ρ) ≤
limsup (fun nm : ℕ × ℕ =>
externalNormalizedRisk g (Qj nm.1 nm.2) α) (ratioFilter ρ) ∧
limsup (fun nm : ℕ × ℕ =>
externalNormalizedRisk g (Qj nm.1 nm.2) α) (ratioFilter ρ) ≤ C ∧
0 < max (trialCertificate α / (1 + sqrt ρ))
(logCertificate g α * sqrt ρ / (1 + sqrt ρ))) := by
have hLower := externalExcessRisk_lower_certificates g α hOverlap hg hα Qj
hTrial hLog hInd
have hRate : ∃ C : ℝ, 0 < C ∧
∀ (n m : ℕ), 0 < n → 0 < m →
trialCertificate α * (sqrt (n : ℝ))⁻¹ ≤
externalExcessRisk g (Qj n m) α ∧
externalExcessRisk g (Qj n m) α ≤
C * ((sqrt (n : ℝ))⁻¹ + (sqrt (m : ℝ))⁻¹) ∧
∃ CI : ExternalIntervalProcedure ε J n m,
CI ∈ externalHonestProcedures g (Qj n m) α ∧
(∀ x, Icc (CI.lo x) (CI.hi x) = externalProjectionCI g α x) ∧
∀ PH, ∀ hPH : PH ∈ ExternalLaws g,
(∫ x, max 0 (CI.length x -
sharpATELength PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH)) ∂(Qj n m PH.1 PH.2)) ≤
C * ((sqrt (n : ℝ))⁻¹ + (sqrt (m : ℝ))⁻¹) := by
let L : ℝ := 4 * (ε⁻¹ + (ε ^ 2)⁻¹)
let A : ℝ := L * (4 * J / α + 2 * J)
let B : ℝ := L * (2 * J * (2 - 2 * ε) / α + J * (2 - 2 * ε))
let C : ℝ := A + B + 1
have hε : 0 < ε := hOverlap.1
have hαpos : 0 < α := hα.1
have hw : 0 < 2 - 2 * ε := by linarith [hOverlap.2]
have hL : 0 ≤ L := by dsimp [L]; positivity
have hA : 0 ≤ A := by
dsimp [A]
exact mul_nonneg hL (by positivity)
have hB : 0 ≤ B := by
dsimp [B]
exact mul_nonneg hL (by positivity)
have hC : 0 < C := by dsimp [C]; linarith
have hscale (n m : ℕ) (hn : 0 < n) (hm : 0 < m) :
4 * (ε⁻¹ + (ε ^ 2)⁻¹) *
(4 * J / (α * sqrt n) + 2 * J / sqrt n +
2 * J * (2 - 2 * ε) / (α * sqrt m) +
J * (2 - 2 * ε) / sqrt m) ≤
C * ((sqrt (n : ℝ))⁻¹ + (sqrt (m : ℝ))⁻¹) := by
have hAn : A ≤ C := by dsimp [C]; linarith
have hBm : B ≤ C := by dsimp [C]; linarith
calc
_ = A * (sqrt (n : ℝ))⁻¹ +
B * (sqrt (m : ℝ))⁻¹ := by
dsimp [A, B, L]
simp only [div_eq_mul_inv, mul_inv_rev]
ring
_ ≤ C * (sqrt (n : ℝ))⁻¹ +
C * (sqrt (m : ℝ))⁻¹ :=
add_le_add (mul_le_mul_of_nonneg_right hAn (by positivity))
(mul_le_mul_of_nonneg_right hBm (by positivity))
_ = _ := by ring
refine ⟨C, hC, ?_⟩
intro n m hn hm
let CI : ExternalIntervalProcedure ε J n m :=
externalProjectionIntervalProcedure g α hOverlap hg
have hCI : CI ∈ externalHonestProcedures g (Qj n m) α :=
externalProjectionIntervalProcedure_honest g α hOverlap hg hn hm hα.1
(Qj n m) (hTrial n m) (hLog n m) (hInd n m)
have hExp : ∀ PH, ∀ hPH : PH ∈ ExternalLaws g,
(∫ x, max 0 (CI.length x -
sharpATELength PH.2 g (releasedLaw PH.1)
(externalLawCellMasses g PH hPH)) ∂(Qj n m PH.1 PH.2)) ≤
C * ((sqrt (n : ℝ))⁻¹ + (sqrt (m : ℝ))⁻¹) := by
intro PH hPH
exact (externalProjectionIntervalProcedure_population_meanExcessLength_le
g (Qj n m) (hTrial n m) (hLog n m) (hInd n m)
PH hPH hn hm α hα.1).trans (hscale n m hn hm)
have hRisk : externalExcessRisk g (Qj n m) α ≤
C * ((sqrt (n : ℝ))⁻¹ + (sqrt (m : ℝ))⁻¹) :=
externalExcessRisk_le_of_honest_uniform_bound g α hOverlap hg
(Qj n m) (hTrial n m) (hLog n m) (hInd n m) CI hCI _
(by positivity) hExp
refine ⟨hLower.2.1 n m hn hm, hRisk, CI, hCI, ?_, hExp⟩
intro x
exact externalProjectionIntervalProcedure_Icc g α hOverlap hg x
refine ⟨hLower.1, hRate, hLower.2.2.1, ?_⟩
obtain ⟨C, hC, hRate⟩ := hRate
refine ⟨C, hC, ?_⟩
intro ρ hρ
let risk : ℕ × ℕ → ℝ := fun nm =>
externalNormalizedRisk g (Qj nm.1 nm.2) α
have hPoint : ∀ nm : ℕ × ℕ, 0 < nm.1 → 0 < nm.2 → risk nm ≤ C := by
intro nm hn hm
have hsum : 0 < (sqrt (nm.1 : ℝ))⁻¹ +
(sqrt (nm.2 : ℝ))⁻¹ := by positivity
have hb := (hRate nm.1 nm.2 hn hm).2.1
dsimp [risk, externalNormalizedRisk, externalNormalizer]
calc
((sqrt (nm.1 : ℝ))⁻¹ + (sqrt (nm.2 : ℝ))⁻¹)⁻¹ *
externalExcessRisk g (Qj nm.1 nm.2) α ≤
((sqrt (nm.1 : ℝ))⁻¹ + (sqrt (nm.2 : ℝ))⁻¹)⁻¹ *
(C * ((sqrt (nm.1 : ℝ))⁻¹ + (sqrt (nm.2 : ℝ))⁻¹)) :=
mul_le_mul_of_nonneg_left hb (by positivity)
_ = C := by field_simp
have hEvUpper : ∀ᶠ nm : ℕ × ℕ in ratioFilter ρ, risk nm ≤ C := by
filter_upwards [ratioFilter_eventually_pos ρ] with nm hnm
exact hPoint nm hnm.1 hnm.2
have hEvLower : ∀ᶠ nm : ℕ × ℕ in ratioFilter ρ, 0 ≤ risk nm := by
filter_upwards [ratioFilter_eventually_pos ρ] with nm hnm
have htrial := (hRate nm.1 nm.2 hnm.1 hnm.2).1
have hcert : 0 ≤ trialCertificate α := (externalCertificate_constants_pos g α hOverlap hα).1.le
have hrisk : 0 ≤ externalExcessRisk g (Qj nm.1 nm.2) α :=
le_trans (mul_nonneg hcert (by positivity)) htrial
exact mul_nonneg (by dsimp [externalNormalizer]; positivity) hrisk
haveI : (ratioFilter ρ).NeBot := ratioFilter_neBot ρ hρ
have hBddAbove : IsBoundedUnder (· ≤ ·) (ratioFilter ρ) risk :=
isBoundedUnder_of_eventually_le hEvUpper
have hBddBelow : IsBoundedUnder (· ≥ ·) (ratioFilter ρ) risk :=
isBoundedUnder_of_eventually_ge hEvLower
have hLimsup : limsup risk (ratioFilter ρ) ≤ C :=
limsup_le_of_le hBddBelow.isCoboundedUnder_le hEvUpper
refine ⟨(hLower.2.2.2 ρ hρ).1, ?_, hLimsup,
(hLower.2.2.2 ρ hρ).2⟩
exact liminf_le_limsup
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.TKLabelRate 3 declarations Uniform inverse-label bounds under a density floor, and an upper bound for every score law.
Uniform inverse-label bounds under a density floor, and an upper bound for every score law.
theorem worstCaseAmbiguity_nonneg
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
0 ≤ worstCaseAmbiguity H g
Proof (Lean source)
lemma worstCaseAmbiguity_nonneg {ε : ℝ} {K : ℕ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(g : ScoreSpace ε → LabelSpace K) (hOverlap : Overlap ε)
(hg : Measurable g) : 0 ≤ worstCaseAmbiguity H g := by
rw [(worstCaseAmbiguity_eq H g hOverlap hg).1]
apply Finset.sum_nonneg
intro r _
exact add_nonneg
(cellAbsoluteDeviation_nonneg H g hOverlap true r)
(cellAbsoluteDeviation_nonneg H g hOverlap false r)
theorem equalWidthRelease_worstCaseAmbiguity_le_all
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
worstCaseAmbiguity H (equalWidthRelease ε K hK) ≤ (1 - 2 * ε) / (2 * ε * (1 - ε) * K)
Proof (Lean source)
lemma equalWidthRelease_worstCaseAmbiguity_le_all {ε : ℝ} {K : ℕ}
(hOverlap : Overlap ε) (hK : 0 < K)
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H] :
worstCaseAmbiguity H (equalWidthRelease ε K hK) ≤
(1 - 2 * ε) / (2 * ε * (1 - ε) * K) := by
let g := equalWidthRelease ε K hK
let B : ℝ := (1 - 2 * ε) / (2 * K)
let C : ℝ := (ε * (1 - ε))⁻¹
have hg : Measurable g := equalWidthRelease_measurable ε K hK
have hB : 0 ≤ B := by
dsimp [B]
apply div_nonneg
· linarith [hOverlap.2]
· positivity
have hC : 0 ≤ C := by
dsimp [C]
exact inv_nonneg.mpr (mul_nonneg hOverlap.1.le (by linarith [hOverlap.2]))
have hcell (r : LabelSpace K) :
cellAbsoluteDeviation H g true r + cellAbsoluteDeviation H g false r ≤
C * B * H.real (cell g r) := by
let z := equalWidthBinMidpoint ε K hK hOverlap r
have hdist (e : ScoreSpace ε) (he : e ∈ cell g r) :
|(e : ℝ) - (z : ℝ)| ≤ B := by
have hb := equalWidthRelease_cell_subset_closedBin hOverlap hK r he
rcases hb with ⟨hl, hu⟩
have hz : (z : ℝ) = ε + ((r : ℕ) + (1 / 2 : ℝ)) *
((1 - 2 * ε) / K) :=
equalWidthBinMidpoint_val ε K hK hOverlap r
rw [hz]
dsimp [B] at ⊢
rw [abs_le]
constructor <;> ring_nf at hl hu ⊢ <;> linarith
have hint : IntegrableOn (fun e : ScoreSpace ε => |(e : ℝ) - (z : ℝ)|)
(cell g r) H := by
apply Integrable.of_bound (by fun_prop) 1
filter_upwards [] with e
have he := e.property
have hz := z.property
rw [Real.norm_eq_abs, abs_abs]
exact abs_le.mpr
⟨by linarith [he.1, he.2, hz.1, hz.2, hOverlap.1],
by linarith [he.1, he.2, hz.1, hz.2, hOverlap.1]⟩
have hbound :
(∫ e in cell g r, |(e : ℝ) - (z : ℝ)| ∂H) ≤
B * H.real (cell g r) := by
calc
(∫ e in cell g r, |(e : ℝ) - (z : ℝ)| ∂H) ≤
∫ _e in cell g r, B ∂H := by
apply setIntegral_mono_on hint (integrable_const B)
(measurableSet_eq_fun hg measurable_const)
exact hdist
_ = B * H.real (cell g r) := by simp [mul_comm]
calc
cellAbsoluteDeviation H g true r + cellAbsoluteDeviation H g false r ≤
C * ∫ e in cell g r, |(e : ℝ) - (z : ℝ)| ∂H :=
sum_cellAbsoluteDeviation_le_centeredScoreIntegral H g hg hOverlap r z
_ ≤ C * (B * H.real (cell g r)) :=
mul_le_mul_of_nonneg_left hbound hC
_ = C * B * H.real (cell g r) := by ring
rw [(worstCaseAmbiguity_eq H g hOverlap hg).1]
calc
(∑ r : LabelSpace K,
(cellAbsoluteDeviation H g true r + cellAbsoluteDeviation H g false r)) ≤
∑ r : LabelSpace K, C * B * H.real (cell g r) := by
apply Finset.sum_le_sum
intro r _
exact hcell r
_ = C * B * ∑ r : LabelSpace K, H.real (cell g r) := by
rw [← Finset.mul_sum]
_ = C * B := by
have h := MeasureTheory.sum_measureReal_preimage_singleton (μ := H)
(s := univ) (f := g)
(by intro r hr; exact measurableSet_eq_fun hg measurable_const)
have htotal : (∑ r : LabelSpace K, H.real (cell g r)) = 1 := by
simpa [cell, preimage, Set.mem_singleton_iff] using h
rw [htotal, mul_one]
_ = (1 - 2 * ε) / (2 * ε * (1 - ε) * K) := by
dsimp [B, C]
have hε : ε ≠ 0 := hOverlap.1.ne'
have hε1 : 1 - ε ≠ 0 := by linarith [hOverlap.2]
have hKr : (K : ℝ) ≠ 0 := by exact_mod_cast hK.ne'
field_simp
theorem optimalKAmbiguity_bounds
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
mf :
ℝ
hOverlap :
Overlap ε
hmf :
0 < mf
@realizes mf(positive density floor)
K :
ℕ
hK :
0 < K
conclusion
conclusion 1
mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε) * K) ≤ optimalKAmbiguity ε K H
conclusion 2
H :
Measure (ScoreSpace ε)
optimalKAmbiguity ε K H ≤ (1 - 2 * ε) / (2 * ε * (1 - ε) * K)
Measurable (equalWidthRelease ε K hK)
worstCaseAmbiguity H (equalWidthRelease ε K hK) ≤ (1 - 2 * ε) / (2 * ε * (1 - ε) * K)
Proof (Lean source)
theorem optimalKAmbiguity_bounds {ε : ℝ} (mf : ℝ)
(hOverlap : Overlap ε) (hmf : 0 < mf) -- @realizes mf(positive density floor)
(K : ℕ) (hK : 0 < K) -- @realizes K(positive label budget)
:
(∀ (H : Measure (ScoreSpace ε)) (f : ScoreSpace ε → ℝ),
LowerScoreDensity H f mf →
mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε) * K) ≤
optimalKAmbiguity ε K H) ∧
(∀ (H : Measure (ScoreSpace ε)), IsProbabilityMeasure H →
optimalKAmbiguity ε K H ≤
(1 - 2 * ε) / (2 * ε * (1 - ε) * K) ∧
Measurable (equalWidthRelease ε K hK) ∧
worstCaseAmbiguity H (equalWidthRelease ε K hK) ≤
(1 - 2 * ε) / (2 * ε * (1 - ε) * K)) := by
constructor
· intro H f hDensity
letI : IsProbabilityMeasure H := hDensity.probability
exact optimalKAmbiguity_finiteLabel_lower mf hOverlap hmf K hK H f hDensity
· intro H hH
letI : IsProbabilityMeasure H := hH
have heqMeas : Measurable (equalWidthRelease ε K hK) :=
equalWidthRelease_measurable ε K hK
have heqUpper := equalWidthRelease_worstCaseAmbiguity_le_all hOverlap hK H
have hoptimalUpper :
optimalKAmbiguity ε K H ≤
(1 - 2 * ε) / (2 * ε * (1 - ε) * K) := by
unfold optimalKAmbiguity
calc
sInf {d : ℝ | ∃ g ∈ kLabelReleases ε K,
d = worstCaseAmbiguity H g} ≤
worstCaseAmbiguity H (equalWidthRelease ε K hK) := by
apply csInf_le
· refine ⟨0, ?_⟩
rintro d ⟨g, hg, rfl⟩
exact worstCaseAmbiguity_nonneg H g hOverlap hg
· exact ⟨equalWidthRelease ε K hK, heqMeas, rfl⟩
_ ≤ _ := heqUpper
exact ⟨hoptimalUpper, heqMeas, heqUpper⟩
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.TMinimaxHonestLength 1 declarations Known-score minimax honest confidence-length order.
Known-score minimax honest confidence-length order.
theorem minimaxHonestLength_order
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
∃ C samplingConstant : ℝ,
0 < C
0 < samplingConstant
conclusion 1
samplingConstant * (sqrt (n : ℝ))⁻¹ ≤ minimaxHonestLength n K H α
conclusion 2
mf :
ℝ
_hmf :
0 < mf
∃ c : ℝ,
0 < c
H :
Measure (ScoreSpace ε)
f :
ScoreSpace ε → ℝ
_hH :
_hDensity :
LowerScoreDensity H f mf
n K :
ℕ
_hn :
0 < n
hK :
0 < K
c * ((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹) ≤ minimaxHonestLength n K H α
minimaxHonestLength n K H α ≤ C * ((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹)
∃ CI : IntervalProcedure n K,
(equalWidthRelease ε K hK, CI) ∈ honestProcedures n K H α
x :
P ∈ CausalLaws H (equalWidthRelease ε K hK) :
(∫ x, CI.length x ∂(Measure.pi (fun _ : Fin n => releasedLaw P)))
≤ C * ((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹)
Proof (Lean source)
theorem minimaxHonestLength_order (ε α : ℝ)
(hOverlap : Overlap ε)
(hα : 0 < α ∧ α < 1 / 2) -- @realizes alpha(miscoverage range)
:
∃ C samplingConstant : ℝ, 0 < C ∧ 0 < samplingConstant ∧
(∀ (H : Measure (ScoreSpace ε)) (_hH : IsProbabilityMeasure H)
(n K : ℕ) (_hn : 0 < n) (_hK : 0 < K),
samplingConstant * (sqrt (n : ℝ))⁻¹ ≤
minimaxHonestLength n K H α) ∧
(∀ (mf : ℝ) (_hmf : 0 < mf), -- @realizes mf(positive density floor)
∃ c : ℝ, 0 < c ∧
∀ (H : Measure (ScoreSpace ε)) (f : ScoreSpace ε → ℝ)
(_hH : IsProbabilityMeasure H) (_hDensity : LowerScoreDensity H f mf)
(n K : ℕ) (_hn : 0 < n) (hK : 0 < K),
-- @realizes n(positive trial sample size) @realizes K(positive label budget)
c * ((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹) ≤
minimaxHonestLength n K H α ∧
minimaxHonestLength n K H α ≤
C * ((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹) ∧
(∃ CI : IntervalProcedure n K,
(equalWidthRelease ε K hK, CI) ∈ honestProcedures n K H α ∧
(∀ x, CI.lo x = clampATE
(midpointWeightedCenter ε n K x - midpointWeightedRadius ε α n K) ∧
CI.hi x = clampATE
(midpointWeightedCenter ε n K x + midpointWeightedRadius ε α n K)) ∧
∀ P ∈ CausalLaws H (equalWidthRelease ε K hK),
(∫ x, CI.length x ∂(Measure.pi (fun _ : Fin n => releasedLaw P))) ≤
C * ((K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹))) := by
let B : ℝ := trialCertificate α
let U : ℝ := midpointLengthConstant ε α
let C : ℝ := max B U
have hβ : 0 < 1 - 2 * α := by linarith [hα.2]
have hlog : 0 < log (1 + (1 - 2 * α) ^ 2) := by
exact Real.log_pos (by nlinarith [sq_pos_of_pos hβ])
have hB : 0 < B := by
dsimp [B]
unfold trialCertificate
positivity
have hC : 0 < C := lt_of_lt_of_le hB (le_max_left B U)
refine ⟨C, B, hC, hB, ?_, ?_⟩
· intro H hH n K hn hK
let _ : IsProbabilityMeasure H := hH
simpa [B] using
(minimaxHonestLength_sampling_lower hOverlap hα H hn hK)
· intro mf hmf
let A : ℝ :=
(1 - 2 * α) * mf * (1 - 2 * ε) ^ 2 / (2 * (1 - ε))
let c : ℝ := min A B / 2
have hgap : 0 < 1 - 2 * ε := by linarith [hOverlap.2]
have hden : 0 < 1 - ε := by linarith [hOverlap.2]
have hA : 0 < A := by
dsimp [A]
positivity
have hc : 0 < c := by
dsimp [c]
exact div_pos (lt_min hA hB) (by norm_num)
refine ⟨c, hc, ?_⟩
intro H f hH hDensity n K hn hK
let _ : IsProbabilityMeasure H := hH
let R : ℝ := (K : ℝ)⁻¹ + (sqrt (n : ℝ))⁻¹
have hFinite := minimaxHonestLength_finiteLabel_lower
hOverlap hα hmf H f hDensity n K hn hK
have hLabel : A * (K : ℝ)⁻¹ ≤ minimaxHonestLength n K H α := by
have hKne : (K : ℝ) ≠ 0 := ne_of_gt (Nat.cast_pos.mpr hK)
have hdene : 1 - ε ≠ 0 := ne_of_gt hden
calc
A * (K : ℝ)⁻¹ =
(1 - 2 * α) * mf * (1 - 2 * ε) ^ 2 /
(2 * (1 - ε) * (K : ℝ)) := by
dsimp [A]
field_simp
_ ≤ minimaxHonestLength n K H α := hFinite
have hSampling := minimaxHonestLength_sampling_lower
hOverlap hα H hn hK
have hLowerAssembly := minimaxHonestLength_rate_assembly
hOverlap hα hmf hn hK hLabel hSampling
dsimp [A, B, c] at hLowerAssembly
have hLower : c * R ≤ minimaxHonestLength n K H α := by
simpa [c, A, B, R] using hLowerAssembly.2
have hUpperAll := minimaxHonestLength_midpoint_upper
hOverlap hα H n K hn hK
have hUC : U ≤ C := le_max_right B U
have hR : 0 ≤ R := by
dsimp [R]
positivity
have hUpper : minimaxHonestLength n K H α ≤ C * R := by
calc
minimaxHonestLength n K H α ≤ U * R := by
simpa [U, R] using hUpperAll
_ ≤ C * R := mul_le_mul_of_nonneg_right hUC hR
obtain ⟨CI, hCI, hformula, hCIU⟩ :=
exists_equalWidth_midpointInterval hOverlap hα H n K hn hK
refine ⟨by simpa [R] using hLower, by simpa [R] using hUpper,
CI, hCI, hformula, ?_⟩
intro P hP
calc
(∫ x, CI.length x ∂(Measure.pi (fun _ : Fin n => releasedLaw P))) ≤
U * R := by simpa [U, R] using hCIU P hP
_ ≤ C * R := mul_le_mul_of_nonneg_right hUC hR
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.TSharpHighResolutionConstant 3 declarations Sharp high-resolution constant and ordered companding releases.
Sharp high-resolution constant and ordered companding releases.
For the specified mathematical inputs, this definition introduces the corresponding object.
Definition (Lean source)
parameters
def
compandingAmbiguity H f K :
ℝ
given by
if hK : 0 < K then worstCaseAmbiguity H (compandingRelease f K hK) else 0
theorem exists_boundedScoreDensity_of_continuous
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
∃ mf Mf : ℝ, BoundedScoreDensity H f mf Mf
Proof (Lean source)
lemma exists_boundedScoreDensity_of_continuous {ε : ℝ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(f : ScoreSpace ε → ℝ)
(hOverlap : Overlap ε)
(hDensity : H = ((volume : Measure ℝ).comap
(Subtype.val : ScoreSpace ε → ℝ)).withDensity
(fun e => ofReal (f e)))
(hfcont : Continuous f) (hfpos : ∀ e, 0 < f e) :
∃ mf Mf : ℝ, BoundedScoreDensity H f mf Mf := by
have hne : (Set.univ : Set (ScoreSpace ε)).Nonempty :=
⟨⟨ε, ⟨le_refl _, by linarith [hOverlap.2]⟩⟩, Set.mem_univ _⟩
obtain ⟨mf, hmf, hmf_le⟩ :=
isCompact_univ.exists_forall_le' hfcont.continuousOn
(fun e _ => hfpos e)
obtain ⟨Mf, hMf⟩ :=
isCompact_univ.exists_bound_of_continuousOn hfcont.continuousOn
refine ⟨mf, Mf, ⟨inferInstance, ?_, ?_⟩⟩
· refine ⟨hfcont.measurable, hDensity, ?_⟩
exact Filter.Eventually.of_forall (fun e => hmf_le e (Set.mem_univ e))
· exact Filter.Eventually.of_forall (fun e =>
(le_abs_self _).trans (by simpa using hMf e (Set.mem_univ e)))
theorem tendsto_K_mul_optimalKAmbiguity
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
conclusion 1
Tendsto (fun K : ℕ => (K : ℝ) * optimalKAmbiguity ε K H) atTop (nhds ((1 / 4 : ℝ) * (∫ t in ε..(1 - ε), sqrt (densityExtension f t / (t * (1 - t)))) ^ 2))
conclusion 2
∃ gK : (K : ℕ) → (hK : 0 < K) → ScoreSpace ε → LabelSpace K,
Tendsto (fun K : ℕ => (K : ℝ) * (if hK : 0 < K then worstCaseAmbiguity H (gK K hK) else 0)) atTop (nhds ((1 / 4 : ℝ) * (∫ t in ε..(1 - ε), sqrt (densityExtension f t / (t * (1 - t)))) ^ 2))
K (hK : 0 < K) :
Measurable (gK K hK)
r :
x z :
x ∈ cell (gK K hK) r
z ∈ cell (gK K hK) r
y :
x ≤ y
y ≤ z
y ∈ cell (gK K hK) r
Proof (Lean source)
theorem tendsto_K_mul_optimalKAmbiguity {ε : ℝ}
(H : Measure (ScoreSpace ε)) [IsProbabilityMeasure H]
(f : ScoreSpace ε → ℝ)
(hDensity : H = ((volume : Measure ℝ).comap
(Subtype.val : ScoreSpace ε → ℝ)).withDensity
(fun e => ofReal (f e)))
(hOverlap : Overlap ε)
(hfcont : Continuous f) (hfpos : ∀ e, 0 < f e) :
Tendsto (fun K : ℕ => (K : ℝ) * optimalKAmbiguity ε K H)
atTop (nhds ((1 / 4 : ℝ) *
(∫ t in ε..(1 - ε),
sqrt (densityExtension f t / (t * (1 - t)))) ^ 2)) ∧
(∃ gK : (K : ℕ) → (hK : 0 < K) → ScoreSpace ε → LabelSpace K,
Tendsto (fun K : ℕ => (K : ℝ) *
(if hK : 0 < K then worstCaseAmbiguity H (gK K hK) else 0))
atTop (nhds ((1 / 4 : ℝ) *
(∫ t in ε..(1 - ε),
sqrt (densityExtension f t / (t * (1 - t)))) ^ 2)) ∧
∀ K (hK : 0 < K),
Measurable (gK K hK) ∧
∀ r : LabelSpace K, ∀ x z : ScoreSpace ε,
x ∈ cell (gK K hK) r →
z ∈ cell (gK K hK) r →
∀ y : ScoreSpace ε, x ≤ y → y ≤ z →
y ∈ cell (gK K hK) r) := by
obtain ⟨mf, Mf, hBounded⟩ :=
exists_boundedScoreDensity_of_continuous H f hOverlap hDensity hfcont hfpos
have hab : ε < 1 - ε := by linarith [hOverlap.2]
have hquant := pairedQuantization_tendsto ε (1 - ε) hab 2 (by norm_num)
(highResolutionBeta f)
(highResolutionBeta_continuousOn hOverlap f hfcont)
(highResolutionBeta_pos hOverlap f hfpos)
have hdiag :
(∫ t in ε..(1 - ε),
sqrt (pairedDiagonalWeight 2 (highResolutionBeta f) t)) =
∫ t in ε..(1 - ε),
sqrt (densityExtension f t / (t * (1 - t))) := by
apply intervalIntegral.integral_congr
intro t ht
change sqrt (pairedDiagonalWeight 2 (highResolutionBeta f) t) =
sqrt (densityExtension f t / (t * (1 - t)))
rw [highResolutionBeta_diagonal hOverlap f t]
simpa only [Set.uIcc_of_le hab.le] using ht
rw [hdiag] at hquant
have hoptEq :
(fun K : ℕ => (K : ℝ) * optimalKAmbiguity ε K H) =ᶠ[atTop]
(fun K : ℕ => (K : ℝ) *
optimalPairedCost ε (1 - ε) 2 K (highResolutionBeta f)) := by
apply (eventually_atTop.2 ⟨1, ?_⟩)
intro K hK
change (K : ℝ) * optimalKAmbiguity ε K H =
(K : ℝ) * optimalPairedCost ε (1 - ε) 2 K (highResolutionBeta f)
rw [optimalKAmbiguity_eq_optimalPairedCost
H f hBounded hOverlap hfcont hfpos (Nat.zero_lt_of_lt hK)]
have hopt :
Tendsto (fun K : ℕ => (K : ℝ) * optimalKAmbiguity ε K H)
atTop (nhds ((1 / 4 : ℝ) *
(∫ t in ε..(1 - ε),
sqrt (densityExtension f t / (t * (1 - t)))) ^ 2)) :=
hquant.1.congr' hoptEq.symm
refine ⟨hopt, ?_⟩
let gK : (K : ℕ) → (hK : 0 < K) →
ScoreSpace ε → LabelSpace K := fun K hK =>
highResolutionOrderedRelease hOverlap f hfcont hfpos K hK
refine ⟨gK, ?_, ?_⟩
· let middle : ℕ → ℝ := fun K => (K : ℝ) *
(if hK : 0 < K then worstCaseAmbiguity H (gK K hK) else 0)
let upper : ℕ → ℝ := fun K => (K : ℝ) *
pairedPartitionCost 2 K (highResolutionBeta f)
(pairedCompandingCell ε (1 - ε) 2 K (highResolutionBeta f))
(pairedCompandingMidpoint ε (1 - ε) 2 K (highResolutionBeta f))
have hlower : ∀ K : ℕ,
(K : ℝ) * optimalKAmbiguity ε K H ≤ middle K := by
intro K
by_cases hK : 0 < K
· simp only [middle, dif_pos hK]
exact mul_le_mul_of_nonneg_left
(optimalKAmbiguity_le_highResolutionOrderedRelease
H hOverlap f hfcont hfpos K hK)
(Nat.cast_nonneg K)
· have hK0 : K = 0 := by omega
subst K
simp [middle]
have hupper : ∀ K : ℕ, middle K ≤ upper K := by
intro K
by_cases hK : 0 < K
· simp only [middle, dif_pos hK, upper]
exact mul_le_mul_of_nonneg_left
(highResolutionOrderedRelease_worstCaseAmbiguity_le_compandingCost
H f hBounded hOverlap hfcont hfpos K hK)
(Nat.cast_nonneg K)
· have hK0 : K = 0 := by omega
subst K
simp [middle, upper]
exact hopt.squeeze hquant.2.1 hlower hupper
· intro K hK
refine ⟨highResolutionOrderedRelease_measurable
hOverlap f hfcont hfpos K hK, ?_⟩
intro r x z hx hz y hxy hyz
exact (highResolutionOrderedRelease_cell_ordConnected
hOverlap f hfcont hfpos K hK r).out hx hz ⟨hxy, hyz⟩
CausalSmith.PartialID.PID_UnlinkedPropensityAte_Research.TUniversalPointIdentification 9 declarations Universal point identification is equivalent to score recovery from the label.
Universal point identification is equivalent to score recovery from the label.
theorem cellAbsoluteDeviation_nonneg
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
0 ≤ cellAbsoluteDeviation H g a r
Proof (Lean source)
lemma cellAbsoluteDeviation_nonneg {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(hOverlap : Overlap ε)
(a : ArmSpace) (r : LabelSpace J) :
0 ≤ cellAbsoluteDeviation H g a r := by
unfold cellAbsoluteDeviation
apply le_csInf
· refine ⟨_, ε⁻¹, ?_, rfl⟩
rcases hOverlap with ⟨hε, hεhalf⟩
constructor
· simpa only [one_div] using
(one_div_le_one_div_of_le hε (show ε ≤ 1 - ε by linarith))
· exact le_refl _
· rintro v ⟨c, hc, rfl⟩
exact integral_nonneg (fun e => abs_nonneg _)
theorem cellAbsoluteDeviation_le_integral
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
a :
r :
c :
ℝ
hc :
c ∈ Icc ((1 - ε)⁻¹) ε⁻¹
conclusion
cellAbsoluteDeviation H g a r ≤ ∫ e in cell g r, |1 - c * armProb a e| ∂H
Proof (Lean source)
lemma cellAbsoluteDeviation_le_integral {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(a : ArmSpace) (r : LabelSpace J) (c : ℝ)
(hc : c ∈ Icc ((1 - ε)⁻¹) ε⁻¹) :
cellAbsoluteDeviation H g a r ≤
∫ e in cell g r, |1 - c * armProb a e| ∂H := by
unfold cellAbsoluteDeviation
apply csInf_le
· refine ⟨0, ?_⟩
rintro v ⟨d, hd, rfl⟩
exact integral_nonneg (fun e => abs_nonneg _)
· exact ⟨c, hc, rfl⟩
theorem cellAbsoluteDeviation_zero_of_ae
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
a :
r :
c :
ℝ
hc :
c ∈ Icc ((1 - ε)⁻¹) ε⁻¹
hae :
∀ᵐ e ∂(H.restrict (cell g r)), 1 - c * armProb a e = 0
conclusion
cellAbsoluteDeviation H g a r = 0
Proof (Lean source)
lemma cellAbsoluteDeviation_zero_of_ae {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(hOverlap : Overlap ε) (a : ArmSpace) (r : LabelSpace J)
(c : ℝ) (hc : c ∈ Icc ((1 - ε)⁻¹) ε⁻¹)
(hae : ∀ᵐ e ∂(H.restrict (cell g r)), 1 - c * armProb a e = 0) :
cellAbsoluteDeviation H g a r = 0 := by
apply le_antisymm
· calc
cellAbsoluteDeviation H g a r ≤
∫ e in cell g r, |1 - c * armProb a e| ∂H :=
cellAbsoluteDeviation_le_integral H g a r c hc
_ = 0 := by
apply integral_eq_zero_of_ae
filter_upwards [hae] with e he
simp [he]
· exact cellAbsoluteDeviation_nonneg H g hOverlap a r
theorem armProb_bounds
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
Proof (Lean source)
lemma armProb_bounds {ε : ℝ} (a : ArmSpace) (x : ScoreSpace ε) :
ε ≤ armProb a x ∧ armProb a x ≤ 1 - ε := by
rcases x.property with ⟨hxlo, hxhi⟩
cases a <;> simp [armProb] <;> constructor <;> linarith
theorem cellAbsoluteDeviation_zero_of_cellwiseAE
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
ε :
ℝ
J :
ℕ
H :
Measure (ScoreSpace ε)
g :
ScoreSpace ε → LabelSpace J
hOverlap :
Overlap ε
a :
r :
x :
hae :
∀ᵐ e ∂(H.restrict (cell g r)), e = x
conclusion
cellAbsoluteDeviation H g a r = 0
Proof (Lean source)
lemma cellAbsoluteDeviation_zero_of_cellwiseAE {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(hOverlap : Overlap ε) (a : ArmSpace) (r : LabelSpace J)
(x : ScoreSpace ε)
(hae : ∀ᵐ e ∂(H.restrict (cell g r)), e = x) :
cellAbsoluteDeviation H g a r = 0 := by
have hb := armProb_bounds a x
have hp : 0 < armProb a x := lt_of_lt_of_le hOverlap.1 hb.1
let c : ℝ := (armProb a x)⁻¹
have hc : c ∈ Icc ((1 - ε)⁻¹) ε⁻¹ := by
constructor
· simpa only [one_div, c] using
(one_div_le_one_div_of_le hp hb.2)
· simpa only [one_div, c] using
(one_div_le_one_div_of_le hOverlap.1 hb.1)
apply cellAbsoluteDeviation_zero_of_ae H g hOverlap a r c hc
filter_upwards [hae] with e he
simp [he, c, inv_mul_cancel₀ hp.ne']
theorem cellDeviationIntegral_continuousOn
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma cellDeviationIntegral_continuousOn {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε)
(r : LabelSpace J) :
ContinuousOn (fun c : ℝ =>
∫ e in cell g r, |1 - c * armProb true e| ∂H)
(Icc ((1 - ε)⁻¹) ε⁻¹) := by
let μ := H.restrict (cell g r)
let F : ℝ → ScoreSpace ε → ℝ := fun c e => |1 - c * armProb true e|
have hcont : ContinuousOn (fun c => ∫ e, F c e ∂μ)
(Icc ((1 - ε)⁻¹) ε⁻¹) := by
apply continuousOn_of_dominated (bound := fun _ => 1 + ε⁻¹)
· intro c hc
dsimp [F]
simp only [armProb, ↓reduceIte]
fun_prop
· intro c hc
filter_upwards [] with e
have hp := armProb_bounds true e
have hp0 : 0 ≤ armProb true e := le_trans (le_of_lt hOverlap.1) hp.1
have hp1 : armProb true e ≤ 1 := by linarith [hOverlap.1]
have hc0 : 0 ≤ c :=
le_trans (le_of_lt (inv_pos.mpr (by linarith [hOverlap.2] : 0 < 1 - ε)))
(Set.mem_Icc.mp hc).1
have hc1 : c ≤ ε⁻¹ := (Set.mem_Icc.mp hc).2
dsimp [F]
simp only [abs_abs]
have hprod0 : 0 ≤ c * armProb true e := mul_nonneg hc0 hp0
have hprod1 : c * armProb true e ≤ ε⁻¹ := by
calc
c * armProb true e ≤ c * 1 :=
mul_le_mul_of_nonneg_left hp1 hc0
_ = c := by ring
_ ≤ ε⁻¹ := hc1
exact abs_le.mpr ⟨by linarith, by linarith⟩
· exact integrable_const _
· filter_upwards [] with e
dsimp [F]
fun_prop
simpa only [μ, F] using hcont
theorem cellAbsoluteDeviation_minimizer
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
∃ c ∈ Icc ((1 - ε)⁻¹) ε⁻¹,
Proof (Lean source)
lemma cellAbsoluteDeviation_minimizer {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε)
(r : LabelSpace J) :
∃ c ∈ Icc ((1 - ε)⁻¹) ε⁻¹,
cellAbsoluteDeviation H g true r =
∫ e in cell g r, |1 - c * armProb true e| ∂H := by
have hεle : ε ≤ 1 - ε := by linarith [hOverlap.2]
have hIcc : (1 - ε)⁻¹ ≤ ε⁻¹ := by
simpa only [one_div] using one_div_le_one_div_of_le hOverlap.1 hεle
let s := Icc ((1 - ε)⁻¹) ε⁻¹
let f : ℝ → ℝ := fun c => ∫ e in cell g r, |1 - c * armProb true e| ∂H
have hs : s.Nonempty := ⟨(1 - ε)⁻¹, ⟨le_refl _, hIcc⟩⟩
obtain ⟨c, hc, heq⟩ :=
(isCompact_Icc.exists_sInf_image_eq hs
(cellDeviationIntegral_continuousOn H g hOverlap r))
refine ⟨c, hc, ?_⟩
have hset : {v : ℝ | ∃ c ∈ s, v = f c} = f '' s := by
ext v
simp only [Set.mem_ofPred_eq, Set.mem_image]
constructor
· rintro ⟨c, hc, rfl⟩
exact ⟨c, hc, rfl⟩
· rintro ⟨c, hc, rfl⟩
exact ⟨c, hc, rfl⟩
change sInf {v : ℝ | ∃ c ∈ s, v = f c} = f c
rw [hset]
exact heq
theorem scoreRecovery_iff_cellwiseAE
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
↔ (∀ r : LabelSpace J, 0 < H (cell g r) → ∃ x : ScoreSpace ε, ∀ᵐ e ∂(H.restrict (cell g r)), e = x)
Proof (Lean source)
lemma scoreRecovery_iff_cellwiseAE {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
(hOverlap : Overlap ε) (hg : Measurable g) :
(∃ h : LabelSpace J → ScoreSpace ε,
Measurable h ∧ ∀ᵐ e ∂H, e = h (g e)) ↔
(∀ r : LabelSpace J, 0 < H (cell g r) →
∃ x : ScoreSpace ε, ∀ᵐ e ∂(H.restrict (cell g r)), e = x) := by
classical
constructor
· rintro ⟨h, _, hae⟩ r _
refine ⟨h r, ?_⟩
filter_upwards [ae_restrict_of_ae hae,
self_mem_ae_restrict (measurableSet_eq_fun hg measurable_const)] with e he hr
exact he.trans (congrArg h hr)
· intro hs
have hx₀ : ε ≤ 1 - ε := by
rcases hOverlap with ⟨_, hε⟩
linarith
let x₀ : ScoreSpace ε := ⟨ε, ⟨le_refl _, hx₀⟩⟩
let h : LabelSpace J → ScoreSpace ε := fun r =>
if hr : 0 < H (cell g r) then choose (hs r hr) else x₀
refine ⟨h, measurable_of_finite _, ?_⟩
have hae : ∀ r : LabelSpace J,
∀ᵐ e ∂H, e ∈ cell g r → e = h r := by
intro r
by_cases hr : 0 < H (cell g r)
· have hspec := Classical.choose_spec (hs r hr)
have hspec' := (ae_restrict_iff' (μ := H)
(s := cell g r) (measurableSet_eq_fun hg measurable_const)).mp hspec
simpa [h, hr] using hspec'
· have hz : H (cell g r) = 0 := le_antisymm (le_of_not_gt hr) bot_le
have hzero : H.restrict (cell g r) = 0 := Measure.restrict_eq_zero.mpr hz
have htriv : ∀ᵐ e ∂(H.restrict (cell g r)), e = h r := by
rw [hzero]
simp
exact (ae_restrict_iff' (measurableSet_eq_fun hg measurable_const)).mp htriv
filter_upwards [Filter.eventually_all.mpr hae] with e he
exact he (g e) (by simp [cell])
theorem worstCaseAmbiguity_eq_zero_iff
Given the stated mathematical inputs and assumptions, this result establishes the stated mathematical conclusion.
Formal statement
hypotheses
conclusion
conclusion 1
conclusion 2
= 0 ↔ ∀ r : LabelSpace J, 0 < H (cell g r) → ∃ x : ScoreSpace ε, ∀ᵐ e ∂(H.restrict (cell g r)), e = x
Proof (Lean source)
theorem worstCaseAmbiguity_eq_zero_iff {ε : ℝ} {J : ℕ}
(H : Measure (ScoreSpace ε)) (g : ScoreSpace ε → LabelSpace J)
[IsProbabilityMeasure H] (hOverlap : Overlap ε) (hg : Measurable g) :
(worstCaseAmbiguity H g = 0 ↔
∃ h : LabelSpace J → ScoreSpace ε,
Measurable h ∧ ∀ᵐ e ∂H, e = h (g e)) ∧
(worstCaseAmbiguity H g = 0 ↔
∀ r : LabelSpace J, 0 < H (cell g r) →
∃ x : ScoreSpace ε,
∀ᵐ e ∂(H.restrict (cell g r)), e = x) := by
have hcell : worstCaseAmbiguity H g = 0 ↔
∀ r : LabelSpace J, 0 < H (cell g r) →
∃ x : ScoreSpace ε,
∀ᵐ e ∂(H.restrict (cell g r)), e = x := by
constructor
· intro hzero
have hsum := (worstCaseAmbiguity_eq H g hOverlap hg).1
rw [hsum] at hzero
have hterm : ∀ r : LabelSpace J,
cellAbsoluteDeviation H g true r +
cellAbsoluteDeviation H g false r = 0 := by
intro r
have hnonneg (s : LabelSpace J) :
0 ≤ cellAbsoluteDeviation H g true s +
cellAbsoluteDeviation H g false s :=
add_nonneg
(cellAbsoluteDeviation_nonneg H g hOverlap true s)
(cellAbsoluteDeviation_nonneg H g hOverlap false s)
exact (Finset.sum_eq_zero_iff_of_nonneg
(s := univ)
(f := fun s : LabelSpace J =>
cellAbsoluteDeviation H g true s +
cellAbsoluteDeviation H g false s)
(by intro s hs; exact hnonneg s)).mp hzero r (Finset.mem_univ r)
intro r hr
have hdev : cellAbsoluteDeviation H g true r = 0 := by
have hnonneg := cellAbsoluteDeviation_nonneg H g hOverlap false r
have hnonneg' := cellAbsoluteDeviation_nonneg H g hOverlap true r
linarith [hterm r]
obtain ⟨c, hc, hmin⟩ :=
cellAbsoluteDeviation_minimizer H g hOverlap r
have hint : IntegrableOn
(fun e : ScoreSpace ε => |1 - c * armProb true e|)
(cell g r) H := by
apply Integrable.of_bound (C := 1 + |c|)
(by simp only [armProb, ↓reduceIte]; fun_prop)
filter_upwards [] with e
have hp := armProb_bounds true e
have hp0 : 0 ≤ armProb true e := le_trans (le_of_lt hOverlap.1) hp.1
have hp1 : armProb true e ≤ 1 := by linarith [hOverlap.1]
change ‖|1 - c * armProb true e|‖ ≤ 1 + |c|
rw [Real.norm_eq_abs, abs_abs]
calc
|1 - c * armProb true e| ≤ |(1 : ℝ)| + |c * armProb true e| := by
simpa only [sub_zero, zero_sub, abs_neg] using
(abs_sub_le (1 : ℝ) 0 (c * armProb true e))
_ = 1 + |c| * armProb true e := by
rw [abs_mul, abs_of_nonneg hp0]
norm_num
_ ≤ 1 + |c| := by nlinarith [abs_nonneg c]
have hae : ∀ᵐ e ∂(H.restrict (cell g r)),
|1 - c * armProb true e| = 0 :=
(setIntegral_eq_zero_iff_of_nonneg_ae
(Filter.Eventually.of_forall (fun e => abs_nonneg _)) hint).mp
(hmin ▸ hdev)
obtain ⟨x, _, hx⟩ :=
Measure.exists_mem_of_measure_ne_zero_of_ae hr.ne' hae
refine ⟨x, ?_⟩
have hcpos : 0 < c :=
lt_of_lt_of_le (inv_pos.mpr (by linarith [hOverlap.2] : 0 < 1 - ε))
hc.1
filter_upwards [hae] with e he
apply Subtype.ext
have hxe : 1 - c * (x : ℝ) = 0 := by
simpa [armProb] using (abs_eq_zero.mp hx)
have hee : 1 - c * (e : ℝ) = 0 := by
simpa [armProb] using (abs_eq_zero.mp he)
have hmul : c * (e : ℝ) = c * (x : ℝ) := by linarith
exact mul_left_cancel₀ hcpos.ne' hmul
· intro hs
obtain ⟨h, _, hae⟩ :=
(scoreRecovery_iff_cellwiseAE H g hOverlap hg).2 hs
rw [(worstCaseAmbiguity_eq H g hOverlap hg).1]
apply Finset.sum_eq_zero
intro r hr
have hrAE : ∀ᵐ e ∂(H.restrict (cell g r)), e = h r := by
filter_upwards [ae_restrict_of_ae hae,
self_mem_ae_restrict (measurableSet_eq_fun hg measurable_const)]
with e he hmem
exact he.trans (congrArg h hmem)
rw [cellAbsoluteDeviation_zero_of_cellwiseAE H g hOverlap true r (h r) hrAE,
cellAbsoluteDeviation_zero_of_cellwiseAE H g hOverlap false r (h r) hrAE]
ring
exact ⟨hcell.trans (scoreRecovery_iff_cellwiseAE H g hOverlap hg).symm, hcell⟩