Formalization: Treatment Effect Estimation with Missing Outcomes and Many Covariate Categories
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 (15d86b3) — statements may have moved since the audit.
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Basic 74 declarations Finite full-data and observed experiments for the rare-arrival MAR frontier.
Finite full-data and observed experiments for the rare-arrival MAR frontier. The finite-table encoding is deliberate: the available POSystem and MinimaxATE models have a different observational alphabet. The generic minimax value is reused.
For the specified inputs and assumptions, the mathematical structure records the components specified by this declaration.
Definition (Lean source)
structure FullRecord (d : ℕ) where
X : Fin d -- @realizes \(X\)(baseline category); @realizes \(d\)(alphabet size)
A : Bool -- @realizes \(A\)(binary randomized arm)
S0 : Bool -- @realizes \(S(a)\)(control surrogate)
S1 : Bool -- @realizes \(S(a)\)(treated surrogate)
Y0 : Bool -- @realizes \(Y(a)\)(control outcome)
Y1 : Bool -- @realizes \(Y(a)\)(treated outcome)
R : Bool -- @realizes \(R\)(arrival flag)
deriving DecidableEq
instance instDecidableEqFullRecord
For the specified inputs and assumptions, the mathematical structure records the components specified by this declaration.
Definition (Lean source)
deriving DecidableEq
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
instance instMeasurableSpaceFullRecord
the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
instance (d : ℕ) : MeasurableSpace (FullRecord d) := ⊤
For the specified inputs and assumptions, the mathematical structure records the components specified by this declaration.
Definition (Lean source)
structure ObsRecord (d : ℕ) where
X : Fin d
A : Bool
S : Bool
R : Bool
RY : Bool
deriving DecidableEq, Fintype
instance instDecidableEqObsRecord
For the specified inputs and assumptions, the mathematical structure records the components specified by this declaration.
Definition (Lean source)
deriving DecidableEq, Fintype
instance instFintypeObsRecord
For the specified inputs and assumptions, the mathematical structure records the components specified by this declaration.
Definition (Lean source)
deriving DecidableEq, Fintype
instance instMeasurableSpaceObsRecord
the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
instance (d : ℕ) : MeasurableSpace (ObsRecord d) := ⊤
def obs
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def Cell
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
inCell r j :
Prop
given by
clause 1
r.A = j.1
clause 2
r.X = j.2.1
clause 3
r.S = j.2.2
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
d :
ℕ
def
FullLaw d :
Type
given by
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def arrivedCell
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def cellProb
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def arrivalProb
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
arrivalProb P j :
ℝ
given by
if 0 < cellProb P j then arrivedCell P j / cellProb P j else 0
def cellMean
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
cellMean P j :
ℝ
given by
if 0 < arrivedCell P j then P.1.real {r | inCell r j ∧ r.R = true ∧ r.Y = true} / arrivedCell P j else 0
def cellContribution
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
cellContribution P j :
ℝ
given by
if 0 < arrivedCell P j then cellProb P j * cellMean P j else 0
def armSign
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def IIDSampling
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
IIDSampling n μ O P :
Prop
def RandomizedIndependence
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
d :
ℕ
P :
FullLaw d
def
RandomizedIndependence P :
Prop
given by
IndepFun FullRecord.A (fun r : FullRecord d => (r.X, r.S0, r.S1, r.Y0, r.Y1)) P.1
def BalancedRandomization
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def SurrogateConsistency
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
d :
ℕ
P :
FullLaw d
def
SurrogateConsistency P :
Prop
given by
∀ᵐ r : FullRecord d ∂(P.1), r.S = if r.A then r.S1 else r.S0
def OutcomeConsistency
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
d :
ℕ
P :
FullLaw d
def
OutcomeConsistency P :
Prop
given by
∀ᵐ r : FullRecord d ∂(P.1), r.Y = if r.A then r.Y1 else r.Y0
def ArrivalMAR
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def OccupiedCellArrival
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def EffectiveSizeScope
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n :
ℕ
q :
ℝ
def
EffectiveSizeScope n q :
Prop
given by
clause 1
1 ≤ n
clause 2
0 < q
def effectiveSize
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n :
ℕ
q :
ℝ
def
effectiveSize n q :
ℝ
given by
n * q
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
structure UnrestrictedArrivalModelClass
For the specified inputs and assumptions, the mathematical structure records the components specified by this declaration.
Definition (Lean source)
parameters
assumes
n_pos :
1 ≤ n
d_pos :
1 ≤ d
q_pos :
0 < q
q_le_one :
q ≤ 1
randomized :
balanced :
surrogate :
outcome :
mar :
arrival :
structure RareArrivalModelClass
For the specified inputs and assumptions, the mathematical structure records the components specified by this declaration.
Definition (Lean source)
parameters
assumes
iid :
randomized :
balanced :
surrogate :
outcome :
mar :
arrival :
slice :
RareArrivalSlice n q
n_pos :
1 ≤ n
d_pos :
1 ≤ d
q_pos :
0 < q
q_le_one :
q ≤ 1
def observedExperiment
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def ate
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
d :
ℕ
P :
FullLaw d
def
ate P :
ℝ
given by
∫ r : FullRecord d, ((if r.Y1 then (1 : ℝ) else 0) - (if r.Y0 then (1 : ℝ) else 0)) ∂(P.1)
def cellFunctional
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
d :
ℕ
P :
FullLaw d
def
cellFunctional P :
ℝ
given by
2 * ∑ j : Cell d, armSign j.1 * cellContribution P j
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n d :
ℕ
q :
ℝ
def
frontierRate n d q :
ℝ
given by
def BoundedRealKernel
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
Z :
Type*
def
given by
{K : Kernel Z ℝ // IsMarkovKernel K ∧ ∀ o, K o (Icc (-1 : ℝ) 1) = 1}
def Estimator
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def ofMap
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
given by
⟨Kernel.deterministic f hf.1, inferInstance, by intro s rw [Kernel.deterministic_apply, Measure.dirac_apply] rw [Set.indicator_of_mem (hf.2 s)] rfl⟩
def toBoundedKernel
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
theorem toBoundedKernel_val
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
def squaredRisk
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def minimaxRisk
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n d :
ℕ
q :
ℝ
def
minimaxRisk n d q :
ℝ
given by
minimaxValueReal (E := Estimator n d) (Θ := {P : FullLaw d // RareArrivalModelClass n d q P}) (fun T P => squaredRisk T P.1)
def EndpointPair
the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def
EndpointPair :
Type
given by
{p : ℝ × ℝ // -1 ≤ p.1 ∧ p.1 ≤ p.2 ∧ p.2 ≤ 1}
the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def
ConnectedInterval :
Type
given by
instance instMeasurableSpaceConnectedInterval
the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def lo
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Definition (Lean source)
def hi
The interval has right endpoint equal to its second coordinate.
Definition (Lean source)
theorem lower
The interval has left endpoint at least minus one.
Formal statement
Proof (Lean source)
lemma ConnectedInterval.lower (I : ConnectedInterval) : -1 ≤ I.lo := I.2.1
theorem ordered
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma ConnectedInterval.ordered (I : ConnectedInterval) : I.lo ≤ I.hi := I.2.2.1
theorem upper
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma ConnectedInterval.upper (I : ConnectedInterval) : I.hi ≤ 1 := I.2.2.2
theorem measurable_connectedInterval_lo
the stated mathematical conclusion holds.
Formal statement
conclusion
Measurable ConnectedInterval.lo
Proof (Lean source)
@[fun_prop] lemma measurable_connectedInterval_lo : Measurable ConnectedInterval.lo :=
measurable_fst.comp measurable_subtype_coe
theorem measurable_connectedInterval_hi
the stated mathematical conclusion holds.
Formal statement
conclusion
Measurable ConnectedInterval.hi
Proof (Lean source)
@[fun_prop] lemma measurable_connectedInterval_hi : Measurable ConnectedInterval.hi :=
measurable_snd.comp measurable_subtype_coe
def closedLeft
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def closedRight
The connected interval is closed at its right endpoint.
Definition (Lean source)
The interval contains the point exactly when both endpoint tests hold.
Definition (Lean source)
parameters
t :
ℝ
def
intervalContains I t :
Prop
given by
clause 1
if I.closedLeft then I.lo ≤ t else I.lo < t
clause 2
if I.closedRight then t ≤ I.hi else t < I.hi
def IntervalProcedure
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n d :
ℕ
def
IntervalProcedure n d :
Type
given by
def ofMap
A measurable interval-valued map defines the corresponding deterministic interval procedure.
Definition (Lean source)
parameters
def
given by
⟨Kernel.deterministic f hf, inferInstance⟩
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
HonestInterval n d q α T :
Prop
given by
1 - α ≤ ∫ s, (T.1 s).real {I | intervalContains I (ate P)} ∂(sampleLaw n P)
def intervalRisk
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
intervalRisk T P :
ℝ
given by
∫ s, (∫ I : ConnectedInterval, I.hi - I.lo ∂(T.1 s)) ∂(sampleLaw n P)
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n d :
ℕ
q α :
ℝ
def
intervalLengthRisk n d q α :
ℝ
given by
minimaxValueReal (E := {T : IntervalProcedure n d // HonestInterval n d q α T}) (Θ := {P : FullLaw d // RareArrivalModelClass n d q P}) (fun T P => intervalRisk T.1 P.1)
def unrestrictedMinimaxRisk
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n d :
ℕ
q :
ℝ
def
unrestrictedMinimaxRisk n d q :
ℝ
given by
minimaxValueReal (E := Estimator n d) (Θ := {P : FullLaw d // UnrestrictedArrivalModelClass n d q P}) (fun T P => squaredRisk T P.1)
def ZengRecord
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
d :
ℕ
def
ZengLaw d :
Type
given by
def zengCategory
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def zengArm
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def zengMean
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def zengDiscreteClass
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
d :
ℕ
ε :
ℝ
given by
{P | 0 < ε ∧ ε < 1 / 2 ∧ 1 ≤ d ∧ ∀ x : Fin d, 0 < zengCategory P x → ε * zengCategory P x ≤ zengArm P x true ∧ zengArm P x true ≤ (1 - ε) * zengCategory P x}
def zengATE
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
d :
ℕ
P :
ZengLaw d
def
zengATE P :
ℝ
given by
∑ x : Fin d, if h : 0 < zengCategory P x then zengCategory P x * (zengMean P x true h - zengMean P x false h) else 0
def ReciprocalBestApproximation
the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def
ReciprocalBestApproximation :
Sort 0
given by
∀ K : ℕ
if
2 ≤ K
then
bestUniformApproxError (fun z : ℝ => z⁻¹) 1 ((K : ℝ) ^ 2) K
= (((K : ℝ) ^ 2 - 1) / (2 * (K : ℝ) ^ 2)) * (((K : ℝ) - 1) / ((K : ℝ) + 1)) ^ K
the stated mathematical conclusion holds.
Formal statement
conclusion
Proof (Lean source)
-- keep: discharges cited node lem:reciprocal-best-approximation; stable staging theorem, relink to promoted Causalean module after study
-- @node: lem:reciprocal-best-approximation
lemma reciprocalBestApproximation_proved : ReciprocalBestApproximation := by
intro K hK
exact bestUniformApproxError_reciprocal_one_sq K hK
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Activation 14 declarations Explicit finite activated laws and the two prior-mixture experiments.
Explicit finite activated laws and the two prior-mixture experiments.
def lowerDegree
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def lowerEndpoint
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n :
ℕ
q :
ℝ
def
lowerEndpoint n q :
ℝ
given by
(lowerDegree n q : ℝ) ^ 2
theorem lower_activation_probability_le_one
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
q * lowerEndpoint n q ≤ 1
Proof (Lean source)
-- @node: lower_activation_probability_le_one
lemma lower_activation_probability_le_one (n : ℕ) (q : ℝ)
(hq : 0 ≤ q) (hslice : RareArrivalSlice n q) :
q * lowerEndpoint n q ≤ 1 := by
have hN : 0 ≤ effectiveSize n q := by
unfold effectiveSize
positivity
have hlog : 1 ≤ logScale n q := by
unfold logScale
have harg : exp 1 ≤ exp 1 + effectiveSize n q := by linarith
have hpos : 0 < exp (1 : ℝ) := Real.exp_pos _
have := Real.log_le_log hpos harg
simpa using this
have hceil : (lowerDegree n q : ℝ) ≤ 2 * logScale n q := by
have h := Nat.ceil_lt_add_one (show 0 ≤ logScale n q by linarith)
unfold lowerDegree
exact le_of_lt (lt_of_lt_of_le h (by linarith))
have hdegree : 0 ≤ (lowerDegree n q : ℝ) := by positivity
have hsq : lowerEndpoint n q ≤ 4 * (logScale n q) ^ 2 := by
unfold lowerEndpoint
nlinarith [sq_nonneg ((lowerDegree n q : ℝ) - 2 * logScale n q)]
have hslice' : q * (logScale n q) ^ 2 ≤ 1 / 64 := hslice
nlinarith
def rareMass
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def rareCount
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def FiniteReciprocalPrior
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
H :
ℝ
prior :
Measure ℝ
def
FiniteReciprocalPrior H prior :
Prop
given by
clause 1
IsProbabilityMeasure prior
def baselineMass
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def lowerCellZ
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
lowerCellZ η n d q z x :
ℝ
given by
if x.val < rareCount η n d q then z x else lowerEndpoint n q
def bernWeight
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def lowerFullRecord
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def activatedLaw
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
given by
∑ x : Fin d, ∑ a : Bool, ∑ y₁ : Bool, ∑ arrival : Bool, ofReal (baselineMass η n d q x / 2 * bernWeight (if x.val < rareCount η n d q then (z x)⁻¹ else 0) y₁ * bernWeight (q * lowerCellZ η n d q z x) arrival) • Measure.dirac (lowerFullRecord x a y₁ arrival)
def augmentedOutcomeWeight
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
augmentedOutcomeWeight η n d q z x a flag arrival one :
ℝ
given by
if !flag then if !arrival && !one then 1 else 0 else if one then if arrival && a && x.val < rareCount η n d q then 1 / lowerEndpoint n q else 0 else if arrival then if a && x.val < rareCount η n d q then (z x - 1) / lowerEndpoint n q else lowerCellZ η n d q z x / lowerEndpoint n q else 1
- lowerCellZ η n d q z x / lowerEndpoint n q
def augmentedOneRecord
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
given by
∑ x : Fin d, ∑ a : Bool, ∑ flag : Bool, ∑ arrival : Bool, ∑ one : Bool, ofReal (baselineMass η n d q x / 2 * bernWeight (q * lowerEndpoint n q) flag * augmentedOutcomeWeight η n d q z x a flag arrival one) • Measure.dirac (flag, ⟨x, a, false, arrival, one⟩)
def activationHandle
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
η :
ℝ
@realizes \(\eta\)(universal constant selected before n,d,q)
_hη :
0 < η
@realizes \(\eta\)(positive universal constant) @realizes \(b_0\)(activation-scope positive numerator)
n d :
ℕ
q :
ℝ
_hn :
1 ≤ n
@realizes \(b_0\)(activation-scope positive sample size)
_hd :
1 ≤ d
_hq_pos :
0 < q
@realizes \(b_0\)(activation-scope positive arrival floor)
_hq_le_one :
q ≤ 1
_hregime :
RareArrivalSlice n q
prior0 :
Measure ℝ
@realizes \(\Pi_0\)(first prior carrier)
prior1 :
Measure ℝ
@realizes \(\Pi_1\)(second prior carrier)
_hprior0 :
FiniteReciprocalPrior (lowerEndpoint n q) prior0
@realizes \(\Pi_0\)(finite probability supported on [1,H])
_hprior1 :
FiniteReciprocalPrior (lowerEndpoint n q) prior1
@realizes \(\Pi_1\)(finite probability supported on [1,H])
_hmoments :
∀ v : ℕ
if
v ≤ lowerDegree n q
then
(∫ z, z ^ v ∂prior0) = ∫ z, z ^ v ∂prior1
_hseparation :
1 / 12 ≤ |(∫ z, z⁻¹ ∂prior1) - (∫ z, z⁻¹ ∂prior0)|
def
given by
((Measure.pi (fun _ : Fin d => prior0)).bind (fun z => if hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q) then Measure.pi (fun _ : Fin n => augmentedOneRecord η n d q z hz) else 0), (Measure.pi (fun _ : Fin d => prior1)).bind (fun z => if hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q) then Measure.pi (fun _ : Fin n => augmentedOneRecord η n d q z hz) else 0))
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.BiasVariance 11 declarations The bounded auxiliary risk from marked moments and pilot selection.
The bounded auxiliary risk from marked moments and pilot selection.
theorem mixedCountEstimator_eq_zero_of_effectiveSize_le_one
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
mixedCountEstimator n d q sample = 0
Proof (Lean source)
-- @node: mixedCountEstimator_eq_zero_of_effectiveSize_le_one
lemma mixedCountEstimator_eq_zero_of_effectiveSize_le_one
(n d : ℕ) (q : ℝ) (hsmall : effectiveSize n q ≤ 1)
(sample : Fin n → ObsRecord d) :
mixedCountEstimator n d q sample = 0 := by
simp [mixedCountEstimator, auxiliaryMixedValue, hsmall]
theorem riskEnvelope_eq_one_of_effectiveSize_le_one
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: riskEnvelope_eq_one_of_effectiveSize_le_one
lemma riskEnvelope_eq_one_of_effectiveSize_le_one
(n d : ℕ) (q : ℝ) (hsmall : effectiveSize n q ≤ 1) :
riskEnvelope n d q = 1 := by
simp [riskEnvelope, hsmall]
theorem mixedCountEstimator_risk_eq_sq_of_effectiveSize_le_one
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ sample, (mixedCountEstimator n d q sample - cellFunctional P) ^ 2 ∂(sampleLaw n P))
= (cellFunctional P) ^ 2
Proof (Lean source)
-- @node: mixedCountEstimator_risk_eq_sq_of_effectiveSize_le_one
lemma mixedCountEstimator_risk_eq_sq_of_effectiveSize_le_one
(n d : ℕ) (q : ℝ) (P : FullLaw d)
(hsmall : effectiveSize n q ≤ 1) :
(∫ sample, (mixedCountEstimator n d q sample - cellFunctional P) ^ 2
∂(sampleLaw n P)) = (cellFunctional P) ^ 2 := by
let : IsProbabilityMeasure P.1 := P.2
have : IsProbabilityMeasure (P.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop : AEMeasurable obs P.1)
have : IsProbabilityMeasure (sampleLaw n P) := by
unfold sampleLaw
infer_instance
simp only [mixedCountEstimator_eq_zero_of_effectiveSize_le_one n d q hsmall,
zero_sub, neg_sq]
simp
theorem unrestricted_auxiliary_risk_envelope_small
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
P :
FullLaw d
hP :
UnrestrictedArrivalModelClass n d q P
hsmall :
effectiveSize n q ≤ 1
conclusion
(∫ sample, (mixedCountEstimator n d q sample - cellFunctional P) ^ 2 ∂(sampleLaw n P))
≤ riskEnvelope n d q
Proof (Lean source)
-- @node: unrestricted_auxiliary_risk_envelope_small
lemma unrestricted_auxiliary_risk_envelope_small (n d : ℕ) (q : ℝ)
(P : FullLaw d) (hP : UnrestrictedArrivalModelClass n d q P)
(hsmall : effectiveSize n q ≤ 1) :
(∫ sample, (mixedCountEstimator n d q sample - cellFunctional P) ^ 2
∂(sampleLaw n P)) ≤ riskEnvelope n d q := by
rw [mixedCountEstimator_risk_eq_sq_of_effectiveSize_le_one n d q P hsmall,
riskEnvelope_eq_one_of_effectiveSize_le_one n d q hsmall]
have htarget := cellFunctional_mem_Icc P hP
nlinarith [htarget.1, htarget.2]
theorem unrestricted_mixedCountEstimator_risk_le_four
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ sample, (mixedCountEstimator n d q sample - cellFunctional P) ^ 2 ∂(sampleLaw n P)) ≤ 4
Proof (Lean source)
-- @node: unrestricted_mixedCountEstimator_risk_le_four
lemma unrestricted_mixedCountEstimator_risk_le_four (n d : ℕ) (q : ℝ)
(P : FullLaw d) (hP : UnrestrictedArrivalModelClass n d q P) :
(∫ sample, (mixedCountEstimator n d q sample - cellFunctional P) ^ 2
∂(sampleLaw n P)) ≤ 4 := by
let : IsProbabilityMeasure P.1 := P.2
have : IsProbabilityMeasure (P.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop : AEMeasurable obs P.1)
have : IsProbabilityMeasure (sampleLaw n P) := by
unfold sampleLaw
infer_instance
calc
_ ≤ ∫ _sample : Fin n → ObsRecord d, (4 : ℝ) ∂sampleLaw n P := by
apply integral_mono (Integrable.of_finite) (integrable_const _)
intro sample
exact sq_sub_le_four_of_mem_Icc
((mixed_count_estimator_regular n d q).2 sample)
(cellFunctional_mem_Icc P hP)
_ = 4 := by simp
theorem unrestricted_auxiliary_risk_envelope_active
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
P :
FullLaw d
hP :
UnrestrictedArrivalModelClass n d q P
hb :
needleBranch n d q
hlarge :
¬ effectiveSize n q ≤ 1
conclusion
(∫ sample, (mixedCountEstimator n d q sample - cellFunctional P) ^ 2 ∂sampleLaw n P)
≤ riskEnvelope n d q
Proof (Lean source)
-- @node: unrestricted_auxiliary_risk_envelope_active
lemma unrestricted_auxiliary_risk_envelope_active (n d : ℕ) (q : ℝ)
(P : FullLaw d) (hP : UnrestrictedArrivalModelClass n d q P)
(hb : needleBranch n d q) (hlarge : ¬ effectiveSize n q ≤ 1) :
(∫ sample, (mixedCountEstimator n d q sample - cellFunctional P) ^ 2
∂sampleLaw n P) ≤ riskEnvelope n d q := by
have ht := deterministicRisk_mixedCountEstimator_le_streamRisk_add_tail n d q P
rw [unrestricted_cell_identification P hP] at ht
rw [← integral_sq_poissonAuxiliaryMixedValue_eq_streams] at ht
have hi := integral_sq_poissonAuxiliaryMixedValue_le_active_budget n d q P hP hb hlarge
simp only [riskEnvelope, if_neg hlarge, not_true_eq_false, hb, if_false]
apply le_min (unrestricted_mixedCountEstimator_risk_le_four n d q P hP)
simp only [deterministicRisk, unrestricted_cell_identification P hP] at ht
exact ht.trans (add_le_add hi le_rfl)
theorem abs_integral_poissonRawMixedValue_bias_le_ratio
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
|(∫ s, poissonRawMixedValue n d q s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
- cellFunctional P|
≤ 8 * d / (exp 1 * streamEffectiveSize n q)
Proof (Lean source)
-- @node: abs_integral_poissonRawMixedValue_bias_le_ratio
lemma abs_integral_poissonRawMixedValue_bias_le_ratio
(n d : ℕ) (q : ℝ) (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) (hb : ¬ needleBranch n d q) :
|(∫ s, poissonRawMixedValue n d q s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) - cellFunctional P| ≤
8 * d / (exp 1 * streamEffectiveSize n q) := by
classical
have hn : 0 < n := lt_of_lt_of_le Nat.zero_lt_one hP.n_pos
have hj (j : Cell d) : Integrable (fun s ↦ armSign j.1 * poissonMemberWeight n j s *
poissonSelectedCellEstimate n d q j s)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
simpa only [mul_assoc] using
((memLp_poissonMemberWeight_mul_selectedCellEstimate n d q P j).integrable
(by norm_num)).const_mul (armSign j.1)
have hc (j : Cell d) : cellContribution P j = cellProb P j * cellMean P j := by
by_cases ha : 0 < arrivedCell P j
· simp [cellContribution, ha]
· simp [cellContribution, cellMean, ha]
change |(∫ s, 2 * ∑ j : Cell d, armSign j.1 * poissonMemberWeight n j s *
poissonSelectedCellEstimate n d q j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) - cellFunctional P| ≤ _
rw [integral_const_mul, integral_finsetSum _ (fun j _ ↦ hj j)]
simp_rw [mul_assoc (armSign _) (poissonMemberWeight _ _ _) _, integral_const_mul,
integral_poissonMemberWeight_mul_selectedCellEstimate n d q P _ hn]
rw [cellFunctional]
simp_rw [hc, ← mul_assoc]
rw [← mul_sub, ← Finset.sum_sub_distrib]
have heq : (∑ j : Cell d, (
armSign j.1 * cellProb P j *
(∫ s, poissonSelectedCellEstimate n d q j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) -
armSign j.1 * cellProb P j * cellMean P j)) =
∑ j : Cell d, armSign j.1 * cellProb P j *
((∫ s, poissonSelectedCellEstimate n d q j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) - cellMean P j) := by
apply Finset.sum_congr rfl
intro j _
ring
rw [heq]
simpa only [poissonSelectedCellEstimate, hb, false_and, if_false] using
abs_ratioBranch_aggregate_bias_le n d q P hP
theorem variance_poissonRawMixedValue_le_ratio_budget
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
≤ 4 * (4 / streamEffectiveSize n q + 1 / streamSize n)
Proof (Lean source)
-- @node: variance_poissonRawMixedValue_le_ratio_budget
lemma variance_poissonRawMixedValue_le_ratio_budget
(n d : ℕ) (q : ℝ) (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) (hb : ¬ needleBranch n d q) :
variance (poissonRawMixedValue n d q)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
4 * (4 / streamEffectiveSize n q + 1 / streamSize n) := by
rw [variance_poissonRawMixedValue_eq_four_sum]
simp only [poissonSelectedCellEstimate, hb, false_and, if_false]
exact mul_le_mul_of_nonneg_left
(sum_variance_poissonMemberWeight_mul_ratioBranch_le n d q P hP) (by norm_num)
theorem integral_sq_poissonAuxiliaryMixedValue_le_ratio_budget
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
P :
FullLaw d
hP :
UnrestrictedArrivalModelClass n d q P
hb :
¬ needleBranch n d q
hlarge :
¬ effectiveSize n q ≤ 1
conclusion
(∫ s, (poissonAuxiliaryMixedValue n d q s - cellFunctional P) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
≤ (8 * d / (exp 1 * streamEffectiveSize n q)) ^ 2
+ 4 * (4 / streamEffectiveSize n q + 1 / streamSize n)
Proof (Lean source)
-- @node: integral_sq_poissonAuxiliaryMixedValue_le_ratio_budget
lemma integral_sq_poissonAuxiliaryMixedValue_le_ratio_budget
(n d : ℕ) (q : ℝ) (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) (hb : ¬ needleBranch n d q)
(hlarge : ¬ effectiveSize n q ≤ 1) :
(∫ s, (poissonAuxiliaryMixedValue n d q s - cellFunctional P) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
(8 * d / (exp 1 * streamEffectiveSize n q)) ^ 2 +
4 * (4 / streamEffectiveSize n q + 1 / streamSize n) := by
have hbias := abs_integral_poissonRawMixedValue_bias_le_ratio n d q P hP hb
have hs := mul_self_le_mul_self (abs_nonneg _) hbias
have hs' : ((∫ s, poissonRawMixedValue n d q s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) -
cellFunctional P) ^ 2 ≤ (8 * d / (exp 1 * streamEffectiveSize n q)) ^ 2 := by
simpa only [← sq, sq_abs] using hs
have hv := variance_poissonRawMixedValue_le_ratio_budget n d q P hP hb
refine (integral_sq_poissonAuxiliaryMixedValue_le_raw n d q P hP hlarge).trans ?_
rw [integral_sq_poissonRawMixedValue_error_eq]
exact (add_le_add hv hs').trans_eq (add_comm _ _)
theorem unrestricted_auxiliary_risk_envelope_ratio
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
P :
FullLaw d
hP :
UnrestrictedArrivalModelClass n d q P
hb :
¬ needleBranch n d q
hlarge :
¬ effectiveSize n q ≤ 1
conclusion
(∫ sample, (mixedCountEstimator n d q sample - cellFunctional P) ^ 2 ∂sampleLaw n P)
≤ riskEnvelope n d q
Proof (Lean source)
-- @node: unrestricted_auxiliary_risk_envelope_ratio
lemma unrestricted_auxiliary_risk_envelope_ratio (n d : ℕ) (q : ℝ)
(P : FullLaw d) (hP : UnrestrictedArrivalModelClass n d q P)
(hb : ¬ needleBranch n d q) (hlarge : ¬ effectiveSize n q ≤ 1) :
(∫ sample, (mixedCountEstimator n d q sample - cellFunctional P) ^ 2
∂sampleLaw n P) ≤ riskEnvelope n d q := by
have ht := deterministicRisk_mixedCountEstimator_le_streamRisk_add_tail n d q P
rw [unrestricted_cell_identification P hP] at ht
rw [← integral_sq_poissonAuxiliaryMixedValue_eq_streams] at ht
have hi := integral_sq_poissonAuxiliaryMixedValue_le_ratio_budget n d q P hP hb hlarge
simp only [riskEnvelope, if_neg hlarge, hb, not_false_eq_true, if_true]
apply le_min (unrestricted_mixedCountEstimator_risk_le_four n d q P hP)
simp only [deterministicRisk, unrestricted_cell_identification P hP] at ht
exact ht.trans (add_le_add hi le_rfl)
theorem unrestricted_auxiliary_risk_envelope
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ sample, (mixedCountEstimator n d q sample - cellFunctional P) ^ 2 ∂(sampleLaw n P))
≤ riskEnvelope n d q
Proof (Lean source)
lemma unrestricted_auxiliary_risk_envelope (n d : ℕ) (q : ℝ)
(P : FullLaw d) (hP : UnrestrictedArrivalModelClass n d q P) :
(∫ sample, (mixedCountEstimator n d q sample - cellFunctional P) ^ 2
∂(sampleLaw n P)) ≤ riskEnvelope n d q := by
by_cases hsmall : effectiveSize n q ≤ 1
· exact unrestricted_auxiliary_risk_envelope_small n d q P hP hsmall
· by_cases hcap : riskEnvelope n d q = 4
· rw [hcap]
exact unrestricted_mixedCountEstimator_risk_le_four n d q P hP
· by_cases hb : needleBranch n d q
· exact unrestricted_auxiliary_risk_envelope_active n d q P hP hb hsmall
· exact unrestricted_auxiliary_risk_envelope_ratio n d q P hP hb hsmall
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.CellIntensity 4 declarations Effective cell intensities inside the ambient marked Poisson sample.
Effective cell intensities inside the ambient marked Poisson sample.
def arrivedStreamIntensity
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
arrivedStreamIntensity n P pool j :
ℝ≥0
given by
theorem coe_arrivedStreamIntensity
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(arrivedStreamIntensity n P pool j : ℝ) = streamSize n * arrivedCell P j
Proof (Lean source)
lemma coe_arrivedStreamIntensity (n : ℕ) {d : ℕ}
(P : FullLaw d) (pool : Fin 3) (j : Cell d) :
(arrivedStreamIntensity n P pool j : ℝ) =
streamSize n * arrivedCell P j := by
unfold arrivedStreamIntensity
change ((n : ℝ) / 2) *
((markedObsLaw P) (streamEvent pool j true false)).toReal = _
rw [← measureReal_def, markedObsLaw_streamEvent_real,
map_obsStreamEvent_arrived_real]
unfold streamSize
ring
theorem streamEffectiveSize_mul_cellProb_le_arrivedStreamIntensity
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
streamEffectiveSize n q * cellProb P j ≤ (arrivedStreamIntensity n P pool j : ℝ)
Proof (Lean source)
lemma streamEffectiveSize_mul_cellProb_le_arrivedStreamIntensity
(n d : ℕ) (q : ℝ) (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) (pool : Fin 3) (j : Cell d) :
streamEffectiveSize n q * cellProb P j ≤
(arrivedStreamIntensity n P pool j : ℝ) := by
rw [coe_arrivedStreamIntensity]
unfold streamEffectiveSize
have hm : 0 ≤ streamSize n := by
unfold streamSize
positivity
by_cases hj : 0 < cellProb P j
· simpa only [mul_assoc] using
mul_le_mul_of_nonneg_left (hP.arrival j hj) hm
· have hj' : cellProb P j = 0 := le_antisymm (le_of_not_gt hj) measureReal_nonneg
rw [hj', mul_zero]
exact mul_nonneg hm measureReal_nonneg
theorem integral_sq_markedPoissonFactorialBranch_le_exp_of_light
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, (poissonFactorialBranch n d q j s) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
lemma integral_sq_markedPoissonFactorialBranch_le_exp_of_light
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d)
(hbranch : needleBranch n d q)
(hlight : streamSize n * arrivedCell P j ≤ needleRadius n q) :
(∫ s, (poissonFactorialBranch n d q j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
exp (logScale n q / 8) := by
apply integral_sq_poissonFactorialBranch_le_exp_effective
n d q j (markedObsLaw P) ((n : ℝ≥0) / 2) hbranch
change (arrivedStreamIntensity n P 2 j : ℝ) ≤ needleRadius n q
rw [coe_arrivedStreamIntensity]
exact hlight
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.ChebyshevNeedle 1 declarations Chebyshev needle bounds used by the light-cell analysis.
Chebyshev needle bounds used by the light-cell analysis.
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
k :
ℕ
x :
ℝ
hx :
x ∈ Icc (-1) 1
conclusion
conclusion 1
0 ≤ 1 - (Chebyshev.T ℝ (k : ℤ)).eval x
conclusion 2
1 - (Chebyshev.T ℝ (k : ℤ)).eval x ≤ (k : ℝ)^2 * (1 - x)
Proof (Lean source)
lemma cheb_secant (k : ℕ) (x : ℝ) (hx : x ∈ Icc (-1) 1) :
0 ≤ 1 - (Chebyshev.T ℝ (k : ℤ)).eval x ∧
1 - (Chebyshev.T ℝ (k : ℤ)).eval x ≤ (k : ℝ)^2 * (1 - x) := by
have hT := Chebyshev.abs_eval_T_real_le_one (k : ℤ)
(show |x| ≤ 1 by exact abs_le.mpr ⟨hx.1, hx.2⟩)
have hderiv (y : ℝ) (hy : y ∈ Ico x 1) :
|(derivative (Chebyshev.T ℝ (k : ℤ))).eval y| ≤ (k : ℝ)^2 := by
have hyy : |y| ≤ 1 := abs_le.mpr ⟨le_trans hx.1 hy.1, le_of_lt hy.2⟩
have hh := Chebyshev.abs_iterate_derivative_T_real_le (k : ℤ) 1 hyy
simpa [Chebyshev.derivative_T_eval_one] using hh
have hmv := norm_image_sub_le_of_norm_deriv_le_segment'
(f := fun y : ℝ => (Chebyshev.T ℝ (k : ℤ)).eval y)
(f' := fun y : ℝ => (derivative (Chebyshev.T ℝ (k : ℤ))).eval y)
(a := x) (b := 1) (C := (k : ℝ)^2)
(fun y _ => (Chebyshev.T ℝ (k : ℤ)).hasDerivWithinAt y (Icc x 1))
(by intro y hy; simpa only [Real.norm_eq_abs] using hderiv y hy)
(1 : ℝ) (show (1 : ℝ) ∈ Icc x 1 from ⟨hx.2, le_refl _⟩)
constructor
· have := (abs_le.mp hT).2
linarith
· rw [Chebyshev.T_eval_one] at hmv
have := (abs_le.mp (by simpa only [Real.norm_eq_abs] using hmv)).2
linarith
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.CoefficientOneNorm 11 declarations Coefficient one-norm transport lemmas for the rescaled Chebyshev needle.
Coefficient one-norm transport lemmas for the rescaled Chebyshev needle.
theorem polynomialCoeffOneNorm_nonneg
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma polynomialCoeffOneNorm_nonneg (p : Polynomial ℝ) :
0 ≤ polynomialCoeffOneNorm p := by
change 0 ≤ p.sum (fun _ c => |c|)
exact Finset.sum_nonneg fun _ _ => abs_nonneg _
theorem sum_abs_coeff_le_polynomialCoeffOneNorm
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma sum_abs_coeff_le_polynomialCoeffOneNorm (p : Polynomial ℝ) (s : Finset ℕ) :
(∑ i ∈ s, |p.coeff i|) ≤ polynomialCoeffOneNorm p := by
classical
change (∑ i ∈ s, |p.coeff i|) ≤ ∑ i ∈ p.support, |p.coeff i|
have heq : (∑ i ∈ s ∩ p.support, |p.coeff i|) =
∑ i ∈ s, |p.coeff i| := by
apply Finset.sum_subset Finset.inter_subset_left
intro i his hi
have hnmem : i ∉ p.support := by
intro hiSupport
exact hi (Finset.mem_inter.mpr ⟨his, hiSupport⟩)
simp only [mem_support_iff, ne_eq, not_not] at hnmem
simp [hnmem]
rw [← heq]
exact Finset.sum_le_sum_of_subset_of_nonneg inter_subset_right
(fun i _ _ => abs_nonneg (p.coeff i))
theorem polynomialCoeffOneNorm_add_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
p q :
conclusion
Proof (Lean source)
lemma polynomialCoeffOneNorm_add_le (p q : Polynomial ℝ) :
polynomialCoeffOneNorm (p + q) ≤
polynomialCoeffOneNorm p + polynomialCoeffOneNorm q := by
let m := max p.natDegree q.natDegree + 1
rw [coefficientOneNorm_range (p + q)
(lt_of_le_of_lt (natDegree_add_le _ _) (lt_succ_self _)),
coefficientOneNorm_range p
(lt_of_le_of_lt (le_max_left _ _) (lt_succ_self _)),
coefficientOneNorm_range q
(lt_of_le_of_lt (le_max_right _ _) (lt_succ_self _)),
← Finset.sum_add_distrib]
exact Finset.sum_le_sum fun i _ => by
simpa only [coeff_add] using abs_add_le (p.coeff i) (q.coeff i)
theorem polynomialCoeffOneNorm_sub_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
p q :
conclusion
Proof (Lean source)
lemma polynomialCoeffOneNorm_sub_le (p q : Polynomial ℝ) :
polynomialCoeffOneNorm (p - q) ≤
polynomialCoeffOneNorm p + polynomialCoeffOneNorm q := by
let m := max p.natDegree q.natDegree + 1
rw [coefficientOneNorm_range (p - q)
(lt_of_le_of_lt (natDegree_sub_le _ _) (lt_succ_self _)),
coefficientOneNorm_range p
(lt_of_le_of_lt (le_max_left _ _) (lt_succ_self _)),
coefficientOneNorm_range q
(lt_of_le_of_lt (le_max_right _ _) (lt_succ_self _)),
← Finset.sum_add_distrib]
exact Finset.sum_le_sum fun i _ => by
simpa only [coeff_sub] using abs_sub (p.coeff i) (q.coeff i)
theorem polynomialCoeffOneNorm_C
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma polynomialCoeffOneNorm_C (c : ℝ) :
polynomialCoeffOneNorm (C c) = |c| := by
by_cases hc : c = 0
· simp [hc, polynomialCoeffOneNorm,
polyCoeffL1]
· rw [coefficientOneNorm_range (C c) (m := 1) (by simp)]
simp
theorem polynomialCoeffOneNorm_sum_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
polynomialCoeffOneNorm (∑ i ∈ s, f i) ≤ ∑ i ∈ s, polynomialCoeffOneNorm (f i)
Proof (Lean source)
lemma polynomialCoeffOneNorm_sum_le {ι : Type*} (s : Finset ι)
(f : ι → Polynomial ℝ) :
polynomialCoeffOneNorm (∑ i ∈ s, f i) ≤
∑ i ∈ s, polynomialCoeffOneNorm (f i) := by
classical
induction s using Finset.induction_on with
| empty => simp [polynomialCoeffOneNorm,
polyCoeffL1]
| @insert i s hi ih =>
simp only [Finset.sum_insert hi]
exact (polynomialCoeffOneNorm_add_le _ _).trans (add_le_add_right ih _)
theorem polynomialCoeffOneNorm_pow_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma polynomialCoeffOneNorm_pow_le (p : Polynomial ℝ) (m : ℕ) :
polynomialCoeffOneNorm (p ^ m) ≤ polynomialCoeffOneNorm p ^ m := by
induction m with
| zero =>
rw [pow_zero, pow_zero, show (1 : Polynomial ℝ) = C 1 by simp,
polynomialCoeffOneNorm_C]
norm_num
| succ m ih =>
rw [pow_succ, pow_succ]
exact (polynomialCoeffOneNorm_mul_le _ _).trans
(mul_le_mul_of_nonneg_right ih (polynomialCoeffOneNorm_nonneg p))
theorem polynomialCoeffOneNorm_comp_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
p q :
hq :
1 ≤ polynomialCoeffOneNorm q
conclusion
polynomialCoeffOneNorm (p.comp q)
≤ polynomialCoeffOneNorm p * polynomialCoeffOneNorm q ^ p.natDegree
Proof (Lean source)
lemma polynomialCoeffOneNorm_comp_le (p q : Polynomial ℝ)
(hq : 1 ≤ polynomialCoeffOneNorm q) :
polynomialCoeffOneNorm (p.comp q) ≤
polynomialCoeffOneNorm p * polynomialCoeffOneNorm q ^ p.natDegree := by
classical
rw [Polynomial.comp_eq_sum_left]
calc
polynomialCoeffOneNorm
(p.sum fun e a => C a * q ^ e) ≤
∑ e ∈ p.support, polynomialCoeffOneNorm (C (p.coeff e) * q ^ e) := by
simpa only [Polynomial.sum] using
polynomialCoeffOneNorm_sum_le p.support
(fun e => C (p.coeff e) * q ^ e)
_ ≤ ∑ e ∈ p.support, |p.coeff e| * polynomialCoeffOneNorm q ^ e := by
apply Finset.sum_le_sum
intro e he
calc
polynomialCoeffOneNorm (C (p.coeff e) * q ^ e) ≤
polynomialCoeffOneNorm (C (p.coeff e)) *
polynomialCoeffOneNorm (q ^ e) :=
polynomialCoeffOneNorm_mul_le _ _
_ = |p.coeff e| * polynomialCoeffOneNorm (q ^ e) := by
rw [polynomialCoeffOneNorm_C]
_ ≤ |p.coeff e| * polynomialCoeffOneNorm q ^ e := by
gcongr
exact polynomialCoeffOneNorm_pow_le q e
_ ≤ ∑ e ∈ p.support,
|p.coeff e| * polynomialCoeffOneNorm q ^ p.natDegree := by
apply Finset.sum_le_sum
intro e he
gcongr
exact le_natDegree_of_ne_zero (mem_support_iff.mp he)
_ = polynomialCoeffOneNorm p *
polynomialCoeffOneNorm q ^ p.natDegree := by
change (∑ e ∈ p.support,
|p.coeff e| * polynomialCoeffOneNorm q ^ p.natDegree) =
(∑ e ∈ p.support, |p.coeff e|) *
polynomialCoeffOneNorm q ^ p.natDegree
rw [Finset.sum_mul]
theorem polynomialCoeffOneNorm_divX_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma polynomialCoeffOneNorm_divX_le (p : Polynomial ℝ) :
polynomialCoeffOneNorm p.divX ≤ polynomialCoeffOneNorm p := by
by_cases hp : p = 0
· simp [hp, polynomialCoeffOneNorm,
polyCoeffL1]
· rw [coefficientOneNorm_range p.divX
(lt_of_le_of_lt natDegree_divX_le (lt_succ_self _)),
coefficientOneNorm_range p (lt_succ_self _)]
rw [Finset.sum_range_succ, Finset.sum_range_succ']
simp only [coeff_divX]
have htop : p.coeff (p.natDegree + 1) = 0 :=
coeff_eq_zero_of_natDegree_lt (lt_succ_self _)
simp only [htop, abs_zero, add_zero]
exact le_add_of_nonneg_right (abs_nonneg (p.coeff 0))
theorem polynomialCoeffOneNorm_one_sub_two_X
the stated mathematical conclusion holds.
Formal statement
conclusion
polynomialCoeffOneNorm (C 1 - C 2 * X : Polynomial ℝ) = 3
Proof (Lean source)
lemma polynomialCoeffOneNorm_one_sub_two_X :
polynomialCoeffOneNorm (C 1 - C 2 * X : Polynomial ℝ) = 3 := by
rw [coefficientOneNorm_range (C 1 - C 2 * X : Polynomial ℝ) (m := 2) (by
exact lt_of_le_of_lt (natDegree_sub_le _ _) (by norm_num))]
norm_num [Finset.sum_range_succ, coeff_sub, coeff_one, coeff_C_mul, coeff_X]
theorem C_mul_divX_comp_scale
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma C_mul_divX_comp_scale (p : Polynomial ℝ) (B : ℝ) :
C B * (p.divX.comp (C B * X)) = (p.comp (C B * X)).divX := by
ext i
simp only [coeff_C_mul, coeff_divX, comp_C_mul_X_coeff]
rw [pow_succ]
ring
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.CoefficientOverlap 10 declarations The exact coefficient-overlap expression controlling the marked factorial second moment.
The exact coefficient-overlap expression controlling the marked factorial second moment.
def factorialOverlap
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def needleCoefficientOverlap
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n d :
ℕ
q :
ℝ
lam :
ℝ≥0
def
needleCoefficientOverlap n d q lam :
ℝ
given by
∑ v ∈ Icc 1 (needleDegree n q - 1), ∑ w ∈ Icc 1 (needleDegree n q - 1), |needleCoeff n d q v * needleCoeff n d q w| * factorialOverlap lam v w
theorem needle_scaled_polynomial_oneNorm_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
≤ (8 : ℝ) ^ needleDegree n q
Proof (Lean source)
lemma needle_scaled_polynomial_oneNorm_le (n d : ℕ) (q : ℝ)
(hbranch : needleBranch n d q) :
polynomialCoeffOneNorm
((1 - chebNeedle n d q).comp (C (needleRadius n q) * X)) ≤
(8 : ℝ) ^ needleDegree n q := by
let k := needleDegree n q
let B := needleRadius n q
let T := T ℝ (k : ℤ)
let p : Polynomial ℝ :=
1 - T.comp (C 1 - C (2 / B) * X)
let r : Polynomial ℝ :=
1 - T.comp (C 1 - C 2 * X)
have hell : 128 ≤ logScale n q := hbranch.1
have hB : 0 < B := by dsimp [B, needleRadius]; positivity
have hk : 2 ≤ k := by
dsimp [k, needleDegree]
exact Nat.le_floor (by linarith)
have hpcomp : p.comp (C B * X) = r := by
dsimp [p, r]
rw [Polynomial.sub_comp, Polynomial.one_comp, Polynomial.comp_assoc]
congr 2
rw [Polynomial.sub_comp, Polynomial.C_comp, Polynomial.mul_comp,
Polynomial.C_comp, Polynomial.X_comp]
ext i
simp only [Polynomial.coeff_sub, Polynomial.coeff_C,
Polynomial.coeff_C_mul, Polynomial.coeff_X]
split_ifs <;> field_simp [hB.ne']
have hscaled :
(chebNeedle n d q).comp (C B * X) =
C (1 / (2 * (k : ℝ) ^ 2)) * r.divX := by
rw [chebNeedle, if_pos hbranch]
dsimp [k, B, p] at hpcomp ⊢
rw [Polynomial.mul_comp, Polynomial.C_comp]
have hshift := C_mul_divX_comp_scale p B
rw [hpcomp] at hshift
calc
C (B / (2 * (k : ℝ) ^ 2)) *
p.divX.comp (C B * X) =
C (1 / (2 * (k : ℝ) ^ 2)) *
(C B * p.divX.comp (C B * X)) := by
rw [← mul_assoc]
congr 1
rw [← Polynomial.C_mul]
congr 1
field_simp [hB.ne']
_ = _ := by rw [hshift]
have hrnorm : polynomialCoeffOneNorm r ≤
1 + (1 + sqrt 2) ^ k * 3 ^ k := by
dsimp [r, T]
calc
polynomialCoeffOneNorm
(1 - (T ℝ (k : ℤ)).comp
(C 1 - C 2 * X)) ≤
1 + polynomialCoeffOneNorm
((T ℝ (k : ℤ)).comp
(C 1 - C 2 * X)) := by
calc
_ ≤ polynomialCoeffOneNorm (1 : Polynomial ℝ) +
polynomialCoeffOneNorm
((T ℝ (k : ℤ)).comp
(C 1 - C 2 * X)) :=
polynomialCoeffOneNorm_sub_le _ _
_ = _ := by
rw [show (1 : Polynomial ℝ) = C 1 by simp,
polynomialCoeffOneNorm_C]
norm_num
_ ≤ 1 + (polynomialCoeffOneNorm (T ℝ (k : ℤ)) *
polynomialCoeffOneNorm
(C 1 - C 2 * X) ^
(T ℝ (k : ℤ)).natDegree) := by
gcongr
apply polynomialCoeffOneNorm_comp_le
rw [polynomialCoeffOneNorm_one_sub_two_X]
norm_num
_ = 1 + polynomialCoeffOneNorm
(T ℝ (k : ℤ)) * 3 ^ k := by
rw [polynomialCoeffOneNorm_one_sub_two_X,
Polynomial.Chebyshev.natDegree_T]
simp
_ ≤ _ := by
gcongr
exact polynomialCoeffOneNorm_chebyshev_le k
rw [Polynomial.sub_comp, Polynomial.one_comp, hscaled]
calc
polynomialCoeffOneNorm
(1 - C (1 / (2 * (k : ℝ) ^ 2)) * r.divX) ≤
1 + (1 / (2 * (k : ℝ) ^ 2)) * polynomialCoeffOneNorm r := by
calc
_ ≤ polynomialCoeffOneNorm 1 +
polynomialCoeffOneNorm
(C (1 / (2 * (k : ℝ) ^ 2)) * r.divX) :=
polynomialCoeffOneNorm_sub_le _ _
_ ≤ 1 + (1 / (2 * (k : ℝ) ^ 2)) *
polynomialCoeffOneNorm r.divX := by
rw [show (1 : Polynomial ℝ) = C 1 by simp,
polynomialCoeffOneNorm_C, abs_one]
gcongr
calc
polynomialCoeffOneNorm
(C (1 / (2 * (k : ℝ) ^ 2)) * r.divX) ≤
polynomialCoeffOneNorm
(C (1 / (2 * (k : ℝ) ^ 2))) *
polynomialCoeffOneNorm r.divX :=
polynomialCoeffOneNorm_mul_le _ _
_ = _ := by
rw [polynomialCoeffOneNorm_C, abs_of_nonneg]
positivity
_ ≤ _ := by
gcongr
exact polynomialCoeffOneNorm_divX_le r
_ ≤ (8 : ℝ) ^ k := by
have hsqrt : sqrt 2 ≤ (3 : ℝ) / 2 := by
rw [Real.sqrt_le_iff]
constructor <;> norm_num
have hbase : 3 * (1 + sqrt 2) ≤ 8 := by linarith
have hr8 : polynomialCoeffOneNorm r ≤ 1 + (8 : ℝ) ^ k := by
calc
_ ≤ 1 + (1 + sqrt 2) ^ k * 3 ^ k := hrnorm
_ = 1 + (3 * (1 + sqrt 2)) ^ k := by rw [mul_pow]; ring
_ ≤ 1 + 8 ^ k := by gcongr
have hkreal : (2 : ℝ) ≤ k := by exact_mod_cast hk
have hc0 : 0 ≤ 1 / (2 * (k : ℝ) ^ 2) := by positivity
have hc8 : 1 / (2 * (k : ℝ) ^ 2) ≤ (1 : ℝ) / 8 := by
apply (div_le_iff₀ (by positivity : (0 : ℝ) < 2 * (k : ℝ) ^ 2)).2
nlinarith
have hpow : (64 : ℝ) ≤ 8 ^ k := by
calc
(64 : ℝ) = 8 ^ 2 := by norm_num
_ ≤ 8 ^ k := pow_le_pow_right₀ (by norm_num) hk
nlinarith [mul_le_mul_of_nonneg_left hr8 hc0]
theorem needle_scaled_coefficient_oneNorm_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
≤ (8 : ℝ) ^ needleDegree n q
Proof (Lean source)
lemma needle_scaled_coefficient_oneNorm_le (n d : ℕ) (q : ℝ)
(hbranch : needleBranch n d q) :
(∑ v ∈ Icc 1 (needleDegree n q - 1),
|needleCoeff n d q v * needleRadius n q ^ v|) ≤
(8 : ℝ) ^ needleDegree n q := by
let B := needleRadius n q
let R := (1 - chebNeedle n d q).comp (C B * X)
calc
(∑ v ∈ Icc 1 (needleDegree n q - 1),
|needleCoeff n d q v * needleRadius n q ^ v|) =
∑ v ∈ Icc 1 (needleDegree n q - 1), |R.coeff v| := by
apply Finset.sum_congr rfl
intro v hv
dsimp [R, B]
rw [Polynomial.comp_C_mul_X_coeff]
simp only [needleCoeff, if_pos hbranch]
_ ≤ polynomialCoeffOneNorm R :=
sum_abs_coeff_le_polynomialCoeffOneNorm R _
_ ≤ (8 : ℝ) ^ needleDegree n q :=
needle_scaled_polynomial_oneNorm_le n d q hbranch
def NeedleCoefficientOverlapBound
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
NeedleCoefficientOverlapBound n d q :
Prop
given by
theorem needleCoefficientOverlap_bound
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma needleCoefficientOverlap_bound (n d : ℕ) (q : ℝ) :
NeedleCoefficientOverlapBound n d q := by
intro hbranch lam hlam
exact (needleCoefficientOverlap_le_exp_mul_sq n d q lam hbranch hlam
((8 : ℝ) ^ needleDegree n q)
(needle_scaled_coefficient_oneNorm_le n d q hbranch)).trans
(needle_overlap_exponential_budget n d q hbranch)
theorem weightedFactorial_le_eventCount_descFactorial
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
weightedFactorial A B v s ≤ (eventCount s B).descFactorial v
Proof (Lean source)
lemma weightedFactorial_le_eventCount_descFactorial
{X : Type*} [MeasurableSpace X] (A B : Set X) (hAB : A ⊆ B)
(v : ℕ) (hv : 1 ≤ v) (s : FiniteSample X) :
weightedFactorial A B v s ≤ (eventCount s B).descFactorial v := by
have hcount : eventCount s A ≤ eventCount s B := eventCount_mono s hAB
cases v with
| zero => omega
| succ v =>
unfold weightedFactorial
simp only [Nat.add_sub_cancel]
by_cases hB : eventCount s B = 0
· have hA : eventCount s A = 0 := by omega
simp [hA, hB]
· have hpos : 0 < eventCount s B := Nat.pos_of_ne_zero hB
rw [show eventCount s B = (eventCount s B - 1) + 1 by omega,
Nat.succ_descFactorial_succ]
push_cast
gcongr
exact_mod_cast (show eventCount s A ≤ eventCount s B - 1 + 1 by omega)
theorem integral_mul_weightedFactorial_le_effective
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, weightedFactorial A B v s * weightedFactorial A B w s ∂finitePoissonSampleLaw Q lam)
≤ factorialOverlap (lam * (Q B).toNNReal) v w
Proof (Lean source)
lemma integral_mul_weightedFactorial_le_effective
{X : Type*} [MeasurableSpace X] [Fintype X] [MeasurableSingletonClass X]
[DecidableEq X] (Q : Measure X) [IsProbabilityMeasure Q]
(lam : ℝ≥0) (A B : Set X) (hA : MeasurableSet A) (hB : MeasurableSet B)
(hAB : A ⊆ B) (v w : ℕ) (hv : 1 ≤ v) (hw : 1 ≤ w) :
(∫ s, weightedFactorial A B v s * weightedFactorial A B w s
∂finitePoissonSampleLaw Q lam) ≤
factorialOverlap (lam * (Q B).toNNReal) v w := by
let t : ℝ≥0 := lam * (Q B).toNNReal
have hWeighted := integrable_mul_weightedFactorial Q lam A B A B
hA hB hA hB hAB hAB v w hv hw
have hPoisson : Integrable (fun N : ℕ =>
(N.descFactorial v : ℝ) * (N.descFactorial w : ℝ))
(poissonMeasure t) := by
have hsum := integrable_finsetSum (range (min v w + 1))
(fun r _ =>
(poisson_descFactorial_integrable
t (v + w - r)).const_mul
((v.choose r : ℝ) * (w.choose r : ℝ) * (Nat.factorial r : ℝ)))
refine hsum.congr (Filter.Eventually.of_forall fun N => ?_)
change (∑ r ∈ range (min v w + 1),
(v.choose r : ℝ) * (w.choose r : ℝ) * (Nat.factorial r : ℝ) *
(N.descFactorial (v + w - r) : ℝ)) =
(N.descFactorial v : ℝ) * (N.descFactorial w : ℝ)
rw [descFactorial_mul N v w]
have hmap : Measure.map (fun s : FiniteSample X => eventCount s B)
(finitePoissonSampleLaw Q lam) = poissonMeasure t := by
exact finitePoissonSampleLaw_map_eventCount Q lam B hB
have hCount : Integrable (fun s : FiniteSample X =>
((eventCount s B).descFactorial v : ℝ) *
((eventCount s B).descFactorial w : ℝ))
(finitePoissonSampleLaw Q lam) := by
rw [← hmap] at hPoisson
exact hPoisson.comp_aemeasurable (measurable_eventCount B hB).aemeasurable
calc
_ ≤ ∫ s, ((eventCount s B).descFactorial v : ℝ) *
((eventCount s B).descFactorial w : ℝ)
∂finitePoissonSampleLaw Q lam := by
apply integral_mono hWeighted hCount
intro s
exact mul_le_mul
(weightedFactorial_le_eventCount_descFactorial A B hAB v hv s)
(weightedFactorial_le_eventCount_descFactorial A B hAB w hw s)
(weightedFactorial_nonneg_le_countFactorial A B hAB w hw s).1
(by positivity)
_ = factorialOverlap t v w := by
unfold factorialOverlap
have hmixed :=
poisson_descFactorial_mixed t v w
rw [← hmixed, ← hmap]
have hMap : Integrable (fun N : ℕ =>
(N.descFactorial v : ℝ) * (N.descFactorial w : ℝ))
(Measure.map (fun s : FiniteSample X => eventCount s B)
(finitePoissonSampleLaw Q lam)) := by
rw [hmap]
exact hPoisson
exact (integral_map (measurable_eventCount B hB).aemeasurable
hMap.aestronglyMeasurable).symm
theorem integral_sq_poissonFactorialBranch_le_effective_overlap
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, (poissonFactorialBranch n d q j s) ^ 2 ∂finitePoissonSampleLaw Q lam)
Proof (Lean source)
lemma integral_sq_poissonFactorialBranch_le_effective_overlap (n d : ℕ) (q : ℝ)
(j : Cell d) (Q : Measure (ObsRecord d × Fin 3)) [IsProbabilityMeasure Q]
(lam : ℝ≥0) :
(∫ s, (poissonFactorialBranch n d q j s) ^ 2
∂finitePoissonSampleLaw Q lam) ≤
needleCoefficientOverlap n d q
(lam * (Q (streamEvent 2 j true false)).toNNReal) := by
by_cases hb : needleBranch n d q
· let S := Icc 1 (needleDegree n q - 1)
let A := streamEvent 2 j true true
let B := streamEvent 2 j true false
have hA : MeasurableSet A := Set.Finite.measurableSet (Set.toFinite _)
have hB : MeasurableSet B := Set.Finite.measurableSet (Set.toFinite _)
have hAB : A ⊆ B := streamEvent_ones_subset_arrived 2 j
have hv (v : ℕ) (hvS : v ∈ S) : 1 ≤ v := (Finset.mem_Icc.mp hvS).1
have hint (v w : ℕ) (hvS : v ∈ S) (hwS : w ∈ S) :
Integrable (fun s : FiniteSample (ObsRecord d × Fin 3) =>
(needleCoeff n d q v * weightedFactorial A B v s) *
(needleCoeff n d q w * weightedFactorial A B w s))
(finitePoissonSampleLaw Q lam) := by
simpa only [mul_assoc, mul_left_comm, mul_comm] using
(integrable_mul_weightedFactorial Q lam A B A B hA hB hA hB hAB hAB
v w (hv v hvS) (hv w hwS)).const_mul
(needleCoeff n d q v * needleCoeff n d q w)
simp only [poissonFactorialBranch, hb, if_true, pow_two]
change (∫ s, (∑ v ∈ S, needleCoeff n d q v * weightedFactorial A B v s) *
(∑ w ∈ S, needleCoeff n d q w * weightedFactorial A B w s)
∂finitePoissonSampleLaw Q lam) ≤
needleCoefficientOverlap n d q (lam * (Q B).toNNReal)
simp_rw [Finset.sum_mul, Finset.mul_sum]
rw [integral_finsetSum]
· apply Finset.sum_le_sum
intro v hvS
rw [integral_finsetSum]
· apply Finset.sum_le_sum
intro w hwS
have hfun : (fun s : FiniteSample (ObsRecord d × Fin 3) =>
(needleCoeff n d q v * weightedFactorial A B v s) *
(needleCoeff n d q w * weightedFactorial A B w s)) =
(fun s => (needleCoeff n d q v * needleCoeff n d q w) *
(weightedFactorial A B v s * weightedFactorial A B w s)) := by
funext s
ring
rw [hfun, integral_const_mul]
change needleCoeff n d q v * needleCoeff n d q w *
(∫ s, weightedFactorial A B v s * weightedFactorial A B w s
∂finitePoissonSampleLaw Q lam) ≤
|needleCoeff n d q v * needleCoeff n d q w| *
factorialOverlap (lam * (Q B).toNNReal) v w
have hcross := integral_mul_weightedFactorial_le_effective Q lam A B
hA hB hAB v w (hv v hvS) (hv w hwS)
have hcross0 : 0 ≤ (∫ s, weightedFactorial A B v s *
weightedFactorial A B w s ∂finitePoissonSampleLaw Q lam) :=
integral_nonneg fun s => mul_nonneg
(weightedFactorial_nonneg_le_countFactorial A B hAB v (hv v hvS) s).1
(weightedFactorial_nonneg_le_countFactorial A B hAB w (hv w hwS) s).1
calc
_ ≤ |needleCoeff n d q v * needleCoeff n d q w| *
(∫ s, weightedFactorial A B v s * weightedFactorial A B w s
∂finitePoissonSampleLaw Q lam) := by
gcongr
exact le_abs_self _
_ ≤ _ := mul_le_mul_of_nonneg_left hcross (abs_nonneg _)
· intro w hwS
exact hint v w hvS hwS
· intro v hvS
exact integrable_finsetSum S fun w hwS => hint v w hvS hwS
· simp [poissonFactorialBranch, needleCoefficientOverlap, needleCoeff, hb]
theorem integral_sq_poissonFactorialBranch_le_exp_effective
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, (poissonFactorialBranch n d q j s) ^ 2 ∂finitePoissonSampleLaw Q lam)
Proof (Lean source)
lemma integral_sq_poissonFactorialBranch_le_exp_effective
(n d : ℕ) (q : ℝ) (j : Cell d)
(Q : Measure (ObsRecord d × Fin 3)) [IsProbabilityMeasure Q]
(lam : ℝ≥0) (hbranch : needleBranch n d q)
(hlam : ((lam * (Q (streamEvent 2 j true false)).toNNReal : ℝ≥0) : ℝ) ≤
needleRadius n q) :
(∫ s, (poissonFactorialBranch n d q j s) ^ 2
∂finitePoissonSampleLaw Q lam) ≤ exp (logScale n q / 8) :=
(integral_sq_poissonFactorialBranch_le_effective_overlap n d q j Q lam).trans
(needleCoefficientOverlap_bound n d q hbranch _ hlam)
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.CountBridge 37 declarations Exact count bridges from the mixed-stream encoding to finite Poisson samples.
Exact count bridges from the mixed-stream encoding to finite Poisson samples.
def streamEvent
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def obsStreamEvent
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def markedObsLaw
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def uniformPoolMass
the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
theorem uniformPoolMass_sum
the stated mathematical conclusion holds.
Formal statement
conclusion
∑ i : Fin 3, uniformPoolMass i = 1
Proof (Lean source)
theorem labelLaw_uniformPoolMass
the stated mathematical conclusion holds.
Formal statement
conclusion
Proof (Lean source)
lemma labelLaw_uniformPoolMass :
labelLaw uniformPoolMass uniformPoolMass_sum =
(uniformOfFintype (Fin 3)).toMeasure := by
apply Measure.ext_of_singleton
intro i
rw [PMF.toMeasure_apply_singleton _ _ (MeasurableSet.singleton _)]
simp [labelLaw, uniformPoolMass, PMF.ofFintype_apply]
theorem streamEvent_eq_prod
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
streamEvent pool j arrived ones = obsStreamEvent j arrived ones ×ˢ {pool}
Proof (Lean source)
lemma streamEvent_eq_prod {d : ℕ} (pool : Fin 3) (j : Cell d)
(arrived ones : Bool) :
streamEvent pool j arrived ones = obsStreamEvent j arrived ones ×ˢ {pool} := by
ext z
simp only [streamEvent, obsStreamEvent, Set.mem_ofPred_eq, Set.mem_prod,
Set.mem_singleton_iff]
tauto
theorem markedObsLaw_streamEvent_real
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma markedObsLaw_streamEvent_real {d : ℕ} (P : FullLaw d) (pool : Fin 3)
(j : Cell d) (arrived ones : Bool) :
(markedObsLaw P).real (streamEvent pool j arrived ones) =
(P.1.map obs).real (obsStreamEvent j arrived ones) / 3 := by
rw [streamEvent_eq_prod]
rw [markedObsLaw, measureReal_def, Measure.prod_prod,
PMF.toMeasure_apply_singleton _ _ (MeasurableSet.singleton _),
PMF.uniformOfFintype_apply]
norm_num [ENNReal.toReal_mul, ENNReal.toReal_inv, div_eq_mul_inv]
rfl
theorem markedObsLaw_isProbabilityMeasure
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma markedObsLaw_isProbabilityMeasure {d : ℕ} (P : FullLaw d) :
IsProbabilityMeasure (markedObsLaw P) := by
letI : IsProbabilityMeasure P.1 := P.2
letI : IsProbabilityMeasure (P.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
letI : IsProbabilityMeasure (uniformOfFintype (Fin 3)).toMeasure :=
isProbabilityMeasure _
unfold markedObsLaw
infer_instance
theorem map_obs_isProbabilityMeasure
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma map_obs_isProbabilityMeasure {d : ℕ} (P : FullLaw d) :
IsProbabilityMeasure (P.1.map obs) := by
letI : IsProbabilityMeasure P.1 := P.2
exact Measure.isProbabilityMeasure_map (by fun_prop)
theorem map_unshuffle_markedObsLaw
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma map_unshuffle_markedObsLaw {d : ℕ} (P : FullLaw d) (lam : ℝ≥0) :
Measure.map unshuffle (finitePoissonSampleLaw (markedObsLaw P) lam) =
Measure.pi (fun _ : Fin 3 =>
finitePoissonSampleLaw (P.1.map obs) (lam * (1 / 3))) := by
letI : IsProbabilityMeasure P.1 := P.2
calc
Measure.map unshuffle (finitePoissonSampleLaw (markedObsLaw P) lam) =
labeledStreamLaw (P.1.map obs) uniformPoolMass uniformPoolMass_sum lam := by
simp only [labeledStreamLaw, markedObsLaw, labelLaw_uniformPoolMass]
_ = Measure.pi (fun i : Fin 3 =>
finitePoissonSampleLaw (P.1.map obs) (lam * uniformPoolMass i)) :=
labeledStreamLaw_eq_independent (P.1.map obs)
uniformPoolMass uniformPoolMass_sum lam
_ = _ := by simp [uniformPoolMass]
theorem map_obsStreamEvent_membership_real
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma map_obsStreamEvent_membership_real {d : ℕ} (P : FullLaw d) (j : Cell d) :
(P.1.map obs).real (obsStreamEvent j false false) = cellProb P j := by
rw [measureReal_def, Measure.map_apply (by fun_prop)
(Set.Finite.measurableSet (Set.toFinite _))]
unfold cellProb
congr 2
ext r
simp [obsStreamEvent, obs, inCell]
theorem map_obsStreamEvent_arrived_real
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma map_obsStreamEvent_arrived_real {d : ℕ} (P : FullLaw d) (j : Cell d) :
(P.1.map obs).real (obsStreamEvent j true false) = arrivedCell P j := by
rw [measureReal_def, Measure.map_apply (by fun_prop)
(Set.Finite.measurableSet (Set.toFinite _))]
unfold arrivedCell
congr 2
ext r
simp [obsStreamEvent, obs, inCell]
tauto
theorem map_obsStreamEvent_arrivedOne_real
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma map_obsStreamEvent_arrivedOne_real {d : ℕ} (P : FullLaw d) (j : Cell d) :
(P.1.map obs).real (obsStreamEvent j true true) =
P.1.real {r | inCell r j ∧ r.R = true ∧ r.Y = true} := by
rw [measureReal_def, Measure.map_apply (by fun_prop)
(Set.Finite.measurableSet (Set.toFinite _)), measureReal_def]
congr 2
ext r
simp [obsStreamEvent, obs, inCell]
tauto
theorem streamCount_eq_eventCount
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
streamCount records assign pool j arrived ones
= eventCount (fixedSizeEmbed k (fun i ↦ (records i, assign i))) (streamEvent pool j arrived ones)
Proof (Lean source)
lemma streamCount_eq_eventCount {k d : ℕ} (records : Fin k → ObsRecord d)
(assign : Fin k → Fin 3) (pool : Fin 3) (j : Cell d)
(arrived ones : Bool) :
streamCount records assign pool j arrived ones =
eventCount (fixedSizeEmbed k (fun i ↦ (records i, assign i)))
(streamEvent pool j arrived ones) := by
classical
simp [streamCount, eventCount, streamEvent, FiniteSample.count,
FiniteSample.points, fixedSizeEmbed]
apply congrArg card
grind
theorem streamEvent_ones_subset_arrived
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma streamEvent_ones_subset_arrived {d : ℕ} (pool : Fin 3) (j : Cell d) :
streamEvent pool j true true ⊆ streamEvent pool j true false := by
intro z hz
rcases hz with ⟨hp, ha, hx, hs, hr, _hy⟩
exact ⟨hp, ha, hx, hs, hr, by simp⟩
theorem memberCount_eq_eventCount
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
memberCount records assign j
Proof (Lean source)
lemma memberCount_eq_eventCount {k d : ℕ} (records : Fin k → ObsRecord d)
(assign : Fin k → Fin 3) (j : Cell d) :
memberCount records assign j =
eventCount (fixedSizeEmbed k (fun i ↦ (records i, assign i)))
(streamEvent 0 j false false) := by
exact streamCount_eq_eventCount records assign 0 j false false
theorem pilotCount_eq_eventCount
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
pilotCount records assign j
Proof (Lean source)
lemma pilotCount_eq_eventCount {k d : ℕ} (records : Fin k → ObsRecord d)
(assign : Fin k → Fin 3) (j : Cell d) :
pilotCount records assign j =
eventCount (fixedSizeEmbed k (fun i ↦ (records i, assign i)))
(streamEvent 1 j true false) := by
exact streamCount_eq_eventCount records assign 1 j true false
theorem arrivedCount_eq_eventCount
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
arrivedCount records assign j
Proof (Lean source)
lemma arrivedCount_eq_eventCount {k d : ℕ} (records : Fin k → ObsRecord d)
(assign : Fin k → Fin 3) (j : Cell d) :
arrivedCount records assign j =
eventCount (fixedSizeEmbed k (fun i ↦ (records i, assign i)))
(streamEvent 2 j true false) := by
exact streamCount_eq_eventCount records assign 2 j true false
theorem arrivedOneCount_eq_eventCount
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
arrivedOneCount records assign j
Proof (Lean source)
lemma arrivedOneCount_eq_eventCount {k d : ℕ} (records : Fin k → ObsRecord d)
(assign : Fin k → Fin 3) (j : Cell d) :
arrivedOneCount records assign j =
eventCount (fixedSizeEmbed k (fun i ↦ (records i, assign i)))
(streamEvent 2 j true true) := by
exact streamCount_eq_eventCount records assign 2 j true true
theorem arrivedOne_fallingFactorial_eq_weightedFactorial
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(arrivedOneCount records assign j : ℝ) * fallingFactorial (arrivedCount records assign j - 1) (v - 1)
= weightedFactorial (streamEvent 2 j true true) (streamEvent 2 j true false) v (fixedSizeEmbed k (fun i ↦ (records i, assign i)))
Proof (Lean source)
lemma arrivedOne_fallingFactorial_eq_weightedFactorial {k d v : ℕ}
(records : Fin k → ObsRecord d) (assign : Fin k → Fin 3) (j : Cell d) :
(arrivedOneCount records assign j : ℝ) *
fallingFactorial (arrivedCount records assign j - 1) (v - 1) =
weightedFactorial (streamEvent 2 j true true) (streamEvent 2 j true false) v
(fixedSizeEmbed k (fun i ↦ (records i, assign i))) := by
rw [arrivedOneCount, arrivedCount,
streamCount_eq_eventCount, streamCount_eq_eventCount]
rfl
theorem factorialBranch_eq_weightedFactorialSum
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
factorialBranch n q records assign j
= if needleBranch n d q then ∑ v ∈ Icc 1 (needleDegree n q - 1), needleCoeff n d q v * weightedFactorial (streamEvent 2 j true true) (streamEvent 2 j true false) v (fixedSizeEmbed k (fun i ↦ (records i, assign i))) else 0
Proof (Lean source)
lemma factorialBranch_eq_weightedFactorialSum {k d : ℕ} (n : ℕ) (q : ℝ)
(records : Fin k → ObsRecord d) (assign : Fin k → Fin 3) (j : Cell d) :
factorialBranch n q records assign j =
if needleBranch n d q then
∑ v ∈ Icc 1 (needleDegree n q - 1),
needleCoeff n d q v *
weightedFactorial (streamEvent 2 j true true)
(streamEvent 2 j true false) v
(fixedSizeEmbed k (fun i ↦ (records i, assign i)))
else 0 := by
unfold factorialBranch
split_ifs with hbranch
· apply Finset.sum_congr rfl
intro v hv
rw [← arrivedOne_fallingFactorial_eq_weightedFactorial
(records := records) (assign := assign) (j := j) (v := v)]
ring
· rfl
def poissonFactorialBranch
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
poissonFactorialBranch n d q j s :
ℝ
given by
if needleBranch n d q then ∑ v ∈ Icc 1 (needleDegree n q - 1), needleCoeff n d q v * weightedFactorial (streamEvent 2 j true true) (streamEvent 2 j true false) v s else 0
def poissonRatioBranch
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
poissonRatioBranch j s :
ℝ
given by
if 0 < eventCount s (streamEvent 2 j true false) then eventCount s (streamEvent 2 j true true) / eventCount s (streamEvent 2 j true false) else 0
def poissonSelectedCellEstimate
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
theorem ratioBranch_eq_poissonRatioBranch
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
ratioBranch records assign j
= poissonRatioBranch j (fixedSizeEmbed k (fun i ↦ (records i, assign i)))
Proof (Lean source)
lemma ratioBranch_eq_poissonRatioBranch {k d : ℕ}
(records : Fin k → ObsRecord d) (assign : Fin k → Fin 3) (j : Cell d) :
ratioBranch records assign j = poissonRatioBranch j
(fixedSizeEmbed k (fun i ↦ (records i, assign i))) := by
rw [ratioBranch, poissonRatioBranch, arrivedCount_eq_eventCount,
arrivedOneCount_eq_eventCount]
theorem selectedCellEstimate_eq_poissonSelectedCellEstimate
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
selectedCellEstimate n q records assign j
= poissonSelectedCellEstimate n d q j (fixedSizeEmbed k (fun i ↦ (records i, assign i)))
Proof (Lean source)
lemma selectedCellEstimate_eq_poissonSelectedCellEstimate {k d : ℕ}
(n : ℕ) (q : ℝ) (records : Fin k → ObsRecord d)
(assign : Fin k → Fin 3) (j : Cell d) :
selectedCellEstimate n q records assign j = poissonSelectedCellEstimate n d q j
(fixedSizeEmbed k (fun i ↦ (records i, assign i))) := by
unfold selectedCellEstimate poissonSelectedCellEstimate
rw [pilotCount_eq_eventCount, ratioBranch_eq_poissonRatioBranch]
split_ifs with hbranch
· rw [factorialBranch_eq_weightedFactorialSum]
simp only [poissonFactorialBranch, hbranch.1, if_true]
· rfl
theorem eventCount_mono
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
eventCount s A ≤ eventCount s B
Proof (Lean source)
lemma eventCount_mono {X : Type*} [MeasurableSpace X]
(s : FiniteSample X) {A B : Set X} (hAB : A ⊆ B) :
eventCount s A ≤ eventCount s B := by
classical
unfold eventCount
apply Finset.card_le_card
intro i hi
simp only [mem_filter, Finset.mem_univ, true_and] at hi ⊢
exact hAB hi
theorem poissonRatioBranch_mem_unitInterval
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
poissonRatioBranch j s ∈ Icc (0 : ℝ) 1
Proof (Lean source)
lemma poissonRatioBranch_mem_unitInterval {d : ℕ} (j : Cell d)
(s : FiniteSample (ObsRecord d × Fin 3)) :
poissonRatioBranch j s ∈ Icc (0 : ℝ) 1 := by
have hsub := streamEvent_ones_subset_arrived 2 j
have hle := eventCount_mono s hsub
unfold poissonRatioBranch
split_ifs with hpos
· constructor
· positivity
· exact (div_le_one (by exact_mod_cast hpos)).2 (by exact_mod_cast hle)
· simp
theorem measurable_poissonRatioBranch
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma measurable_poissonRatioBranch {d : ℕ} (j : Cell d) :
Measurable (poissonRatioBranch j) := by
let A := streamEvent 2 j true true
let B := streamEvent 2 j true false
have hA : MeasurableSet A := Set.Finite.measurableSet (Set.toFinite _)
have hB : MeasurableSet B := Set.Finite.measurableSet (Set.toFinite _)
unfold poissonRatioBranch
apply Measurable.ite
· exact (measurable_eventCount B hB) measurableSet_Ioi
· exact ((measurable_of_countable (fun z : ℕ => (z : ℝ))).comp
(measurable_eventCount A hA)).div
((measurable_of_countable (fun z : ℕ => (z : ℝ))).comp
(measurable_eventCount B hB))
· exact measurable_const
theorem integrable_poissonRatioBranch_sq
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Integrable (fun s => (poissonRatioBranch j s) ^ 2) (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
lemma integrable_poissonRatioBranch_sq (n d : ℕ) (P : FullLaw d) (j : Cell d) :
Integrable (fun s => (poissonRatioBranch j s) ^ 2)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
apply Integrable.of_bound ((measurable_poissonRatioBranch j).pow_const 2).aestronglyMeasurable 1
filter_upwards [] with s
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
rcases poissonRatioBranch_mem_unitInterval j s with ⟨hzero, hone⟩
nlinarith [mul_nonneg hzero (sub_nonneg.mpr hone)]
theorem map_finitePoissonSampleLaw_finiteSampleMap
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
X Y :
Q :
f :
X → Y
hf :
lam :
ℝ≥0
conclusion
Measure.map (finiteSampleMap f) (finitePoissonSampleLaw Q lam)
= (letI : IsProbabilityMeasure (Measure.map f Q) := Measure.isProbabilityMeasure_map hf.aemeasurable finitePoissonSampleLaw (Measure.map f Q) lam)
Proof (Lean source)
lemma map_finitePoissonSampleLaw_finiteSampleMap
{X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
(Q : Measure X) [IsProbabilityMeasure Q]
(f : X → Y) (hf : Measurable f) (lam : ℝ≥0) :
Measure.map (finiteSampleMap f) (finitePoissonSampleLaw Q lam) =
(letI : IsProbabilityMeasure (Measure.map f Q) :=
Measure.isProbabilityMeasure_map hf.aemeasurable
finitePoissonSampleLaw (Measure.map f Q) lam) := by
letI : IsProbabilityMeasure (Measure.map f Q) :=
Measure.isProbabilityMeasure_map hf.aemeasurable
let F := finiteSampleMap f
have hF : Measurable F := measurable_finiteSampleMap f hf
apply Measure.ext_of_iUnion_eq_univ
(show (⋃ n : ℕ, FiniteSample.count ⁻¹' ({n} : Set ℕ)) = univ by ext s; simp)
intro n
rw [Measure.restrict_map hF
(measurable_finiteSample_count (MeasurableSet.singleton n))]
have hpre : F ⁻¹' (FiniteSample.count ⁻¹' ({n} : Set ℕ)) =
FiniteSample.count ⁻¹' ({n} : Set ℕ) := by ext s; rfl
rw [hpre, finitePoissonSampleLaw_restrict_count_eq,
finitePoissonSampleLaw_restrict_count_eq, Measure.map_smul,
Measure.map_map hF (measurable_fixedSizeEmbed n)]
have hfun : F ∘ fixedSizeEmbed n =
fixedSizeEmbed n ∘ (fun x : Fin n → X => fun i => f (x i)) := by
funext x
exact finiteSampleMap_fixedSizeEmbed f n x
rw [hfun]
congr 1
let G : (Fin n → X) → (Fin n → Y) := fun x i => f (x i)
have hG : Measurable G :=
measurable_pi_lambda _ fun i => hf.comp (measurable_pi_apply i)
calc
Measure.map (fixedSizeEmbed n ∘ G) (Measure.pi fun _ : Fin n => Q) =
Measure.map (fixedSizeEmbed n) (Measure.map G (Measure.pi fun _ : Fin n => Q)) :=
(Measure.map_map (measurable_fixedSizeEmbed n) hG).symm
_ = Measure.map (fixedSizeEmbed n)
(Measure.pi fun _ : Fin n => Measure.map f Q) := by
rw [Measure.pi_map_pi (fun _ => hf.aemeasurable)]
theorem finitePoissonSampleLaw_map_eventCount
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Measure.map (fun s : FiniteSample X => eventCount s B) (finitePoissonSampleLaw Q intensity)
= poissonMeasure (intensity * (Q B).toNNReal)
Proof (Lean source)
lemma finitePoissonSampleLaw_map_eventCount
{X : Type*} [MeasurableSpace X] [Fintype X] [MeasurableSingletonClass X]
[DecidableEq X] (Q : Measure X) [IsProbabilityMeasure Q]
(intensity : ℝ≥0) (B : Set X) (hB : MeasurableSet B) :
Measure.map (fun s : FiniteSample X => eventCount s B)
(finitePoissonSampleLaw Q intensity) =
poissonMeasure (intensity * (Q B).toNNReal) := by
classical
let f : X → Bool := fun x => decide (x ∈ B)
have hf : Measurable f := measurable_of_finite _
let QB := Measure.map f Q
letI : IsProbabilityMeasure QB :=
Measure.isProbabilityMeasure_map hf.aemeasurable
let H : FiniteSample Bool → (Bool → ℕ) :=
fun s => finiteSampleHistogram
s.points
have hpoint (s : FiniteSample X) : eventCount s B = H (finiteSampleMap f s) true := by
rcases s with ⟨n, x⟩
simp only [eventCount, H,
finiteSampleHistogram,
finiteSampleMap, FiniteSample.count, FiniteSample.points, f]
rw [Fintype.card_subtype]
change (Finset.univ.filter (fun i : Fin n => x i ∈ B)).card =
(Finset.univ.filter (fun i : Fin n => decide (x i ∈ B) = true)).card
apply congrArg Finset.card
ext i
simp
have hmass : QB {true} = Q B := by
rw [Measure.map_apply hf (MeasurableSet.singleton true)]
congr 1
ext x
simp [f]
have hH : Measurable H :=
Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.measurable_finiteSampleHistogram
have hEval : Measurable (Function.eval true : (Bool → ℕ) → ℕ) :=
measurable_pi_apply true
calc
Measure.map (fun s : FiniteSample X => eventCount s B)
(finitePoissonSampleLaw Q intensity) =
Measure.map (Function.eval true)
(Measure.map H
(Measure.map (finiteSampleMap f) (finitePoissonSampleLaw Q intensity))) := by
rw [Measure.map_map hEval hH,
Measure.map_map (hEval.comp hH) (measurable_finiteSampleMap f hf)]
· congr 1
funext s
exact hpoint s
_ = Measure.map (Function.eval true)
(independentPoissonCountLaw
QB intensity) := by
rw [map_finitePoissonSampleLaw_finiteSampleMap Q f hf intensity,
finitePoissonSampleLaw_map_histogram]
_ = poissonMeasure (intensity * (QB {true}).toNNReal) := by
unfold independentPoissonCountLaw
rw [Measure.pi_map_eval]
simp
_ = _ := by rw [hmass]
theorem integral_poissonFactorialBranch
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, poissonFactorialBranch n d q j s ∂finitePoissonSampleLaw Q lam)
= if needleBranch n d q then ∑ v ∈ Icc 1 (needleDegree n q - 1), needleCoeff n d q v * (lam : ℝ) ^ v * (Q (streamEvent 2 j true true)).toReal * (Q (streamEvent 2 j true false)).toReal ^ (v - 1) else 0
Proof (Lean source)
lemma integral_poissonFactorialBranch (n d : ℕ) (q : ℝ) (j : Cell d)
(Q : Measure (ObsRecord d × Fin 3)) [IsProbabilityMeasure Q] (lam : ℝ≥0) :
(∫ s, poissonFactorialBranch n d q j s ∂finitePoissonSampleLaw Q lam) =
if needleBranch n d q then
∑ v ∈ Icc 1 (needleDegree n q - 1),
needleCoeff n d q v * (lam : ℝ) ^ v *
(Q (streamEvent 2 j true true)).toReal *
(Q (streamEvent 2 j true false)).toReal ^ (v - 1)
else 0 := by
classical
have hOne : MeasurableSet (streamEvent 2 j true true) :=
Set.Finite.measurableSet (Set.toFinite _)
have hArrived : MeasurableSet (streamEvent 2 j true false) :=
Set.Finite.measurableSet (Set.toFinite _)
have hsub := streamEvent_ones_subset_arrived 2 j
by_cases hbranch : needleBranch n d q
· simp only [poissonFactorialBranch, if_pos hbranch]
rw [integral_finsetSum]
· apply Finset.sum_congr rfl
intro v hv
have hv1 : 1 ≤ v := (Finset.mem_Icc.mp hv).1
rw [integral_const_mul,
finitePoisson_nestedEvent_factorialMoment Q lam
(streamEvent 2 j true true) (streamEvent 2 j true false)
hOne hArrived hsub v hv1]
ring
· intro v hv
have hv1 : 1 ≤ v := (Finset.mem_Icc.mp hv).1
exact (integrable_weightedFactorial Q lam
(streamEvent 2 j true true) (streamEvent 2 j true false)
hOne hArrived hsub v hv1).const_mul _
· simp [poissonFactorialBranch, hbranch]
theorem integrable_sq_weightedFactorial
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Integrable (fun s : FiniteSample X => (weightedFactorial A B v s) ^ 2) (finitePoissonSampleLaw Q lam)
Proof (Lean source)
lemma integrable_sq_weightedFactorial
{X : Type*} [MeasurableSpace X] (Q : Measure X) [IsProbabilityMeasure Q]
(lam : ℝ≥0) (A B : Set X) (hA : MeasurableSet A) (hB : MeasurableSet B)
(hAB : A ⊆ B) (v : ℕ) (hv : 1 ≤ v) :
Integrable (fun s : FiniteSample X => (weightedFactorial A B v s) ^ 2)
(finitePoissonSampleLaw Q lam) := by
have hPoisson : Integrable (fun N : ℕ => (N.descFactorial v : ℝ) ^ 2)
(poissonMeasure lam) := by
have hsum := integrable_finsetSum (range (min v v + 1))
(fun r _ =>
(poisson_descFactorial_integrable
lam (v + v - r)).const_mul
((v.choose r : ℝ) * (v.choose r : ℝ) * (Nat.factorial r : ℝ)))
refine hsum.congr (Filter.Eventually.of_forall fun N => ?_)
change (∑ r ∈ range (min v v + 1),
(v.choose r : ℝ) * (v.choose r : ℝ) * (Nat.factorial r : ℝ) *
(N.descFactorial (v + v - r) : ℝ)) = (N.descFactorial v : ℝ) ^ 2
rw [pow_two,
descFactorial_mul N v v]
have hCount : Integrable
(fun s : FiniteSample X => (s.count.descFactorial v : ℝ) ^ 2)
(finitePoissonSampleLaw Q lam) := by
rw [← finitePoissonSampleLaw_map_count Q lam] at hPoisson
exact hPoisson.comp_aemeasurable measurable_finiteSample_count.aemeasurable
have hmeas : AEStronglyMeasurable
(fun s : FiniteSample X => (weightedFactorial A B v s) ^ 2)
(finitePoissonSampleLaw Q lam) :=
((measurable_weightedFactorial A B hA hB v).pow_const 2).aestronglyMeasurable
apply integrable_of_le_of_le hmeas
(Filter.Eventually.of_forall fun _ => sq_nonneg _)
_ (integrable_zero _ _ _) hCount
filter_upwards [] with s
have hs := weightedFactorial_nonneg_le_countFactorial A B hAB v hv s
nlinarith
theorem integrable_mul_weightedFactorial
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
X :
Type*
Q :
lam :
ℝ≥0
A B C D :
Set X
hA :
hB :
hC :
hD :
hAB :
A ⊆ B
hCD :
C ⊆ D
v w :
ℕ
hv :
1 ≤ v
hw :
1 ≤ w
conclusion
Integrable (fun s : FiniteSample X => weightedFactorial A B v s * weightedFactorial C D w s) (finitePoissonSampleLaw Q lam)
Proof (Lean source)
lemma integrable_mul_weightedFactorial
{X : Type*} [MeasurableSpace X] (Q : Measure X) [IsProbabilityMeasure Q]
(lam : ℝ≥0) (A B C D : Set X)
(hA : MeasurableSet A) (hB : MeasurableSet B)
(hC : MeasurableSet C) (hD : MeasurableSet D)
(hAB : A ⊆ B) (hCD : C ⊆ D) (v w : ℕ) (hv : 1 ≤ v) (hw : 1 ≤ w) :
Integrable (fun s : FiniteSample X =>
weightedFactorial A B v s * weightedFactorial C D w s)
(finitePoissonSampleLaw Q lam) := by
have hPoisson : Integrable (fun N : ℕ =>
(N.descFactorial v : ℝ) * (N.descFactorial w : ℝ))
(poissonMeasure lam) := by
have hsum := integrable_finsetSum (range (min v w + 1))
(fun r _ =>
(poisson_descFactorial_integrable
lam (v + w - r)).const_mul
((v.choose r : ℝ) * (w.choose r : ℝ) * (Nat.factorial r : ℝ)))
refine hsum.congr (Filter.Eventually.of_forall fun N => ?_)
change (∑ r ∈ range (min v w + 1),
(v.choose r : ℝ) * (w.choose r : ℝ) * (Nat.factorial r : ℝ) *
(N.descFactorial (v + w - r) : ℝ)) =
(N.descFactorial v : ℝ) * (N.descFactorial w : ℝ)
rw [descFactorial_mul N v w]
have hCount : Integrable (fun s : FiniteSample X =>
(s.count.descFactorial v : ℝ) * (s.count.descFactorial w : ℝ))
(finitePoissonSampleLaw Q lam) := by
rw [← finitePoissonSampleLaw_map_count Q lam] at hPoisson
exact hPoisson.comp_aemeasurable measurable_finiteSample_count.aemeasurable
have hmeas : AEStronglyMeasurable (fun s : FiniteSample X =>
weightedFactorial A B v s * weightedFactorial C D w s)
(finitePoissonSampleLaw Q lam) :=
((measurable_weightedFactorial A B hA hB v).mul
(measurable_weightedFactorial C D hC hD w)).aestronglyMeasurable
apply integrable_of_le_of_le hmeas
(Filter.Eventually.of_forall fun s => mul_nonneg
(weightedFactorial_nonneg_le_countFactorial A B hAB v hv s).1
(weightedFactorial_nonneg_le_countFactorial C D hCD w hw s).1)
_ (integrable_zero _ _ _) hCount
filter_upwards [] with s
exact mul_le_mul
(weightedFactorial_nonneg_le_countFactorial A B hAB v hv s).2
(weightedFactorial_nonneg_le_countFactorial C D hCD w hw s).2
(weightedFactorial_nonneg_le_countFactorial C D hCD w hw s).1
(by positivity)
theorem integral_markedPoissonFactorialBranch
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, poissonFactorialBranch n d q j s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
= if needleBranch n d q then ∑ v ∈ Icc 1 (needleDegree n q - 1), needleCoeff n d q v * ((((n : ℝ≥0) / 2 : ℝ≥0) : ℝ) ^ v) * (P.1.real {r | inCell r j ∧ r.R = true ∧ r.Y = true} / 3) * (arrivedCell P j / 3) ^ (v - 1) else 0
Proof (Lean source)
lemma integral_markedPoissonFactorialBranch (n d : ℕ) (q : ℝ) (P : FullLaw d)
(j : Cell d) :
(∫ s, poissonFactorialBranch n d q j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) =
if needleBranch n d q then
∑ v ∈ Icc 1 (needleDegree n q - 1),
needleCoeff n d q v * ((((n : ℝ≥0) / 2 : ℝ≥0) : ℝ) ^ v) *
(P.1.real {r | inCell r j ∧ r.R = true ∧ r.Y = true} / 3) *
(arrivedCell P j / 3) ^ (v - 1)
else 0 := by
rw [integral_poissonFactorialBranch]
congr 1
apply Finset.sum_congr rfl
intro v hv
change needleCoeff n d q v * ((((n : ℝ≥0) / 2 : ℝ≥0) : ℝ) ^ v) *
(markedObsLaw P).real (streamEvent 2 j true true) *
((markedObsLaw P).real (streamEvent 2 j true false)) ^ (v - 1) = _
rw [markedObsLaw_streamEvent_real, markedObsLaw_streamEvent_real,
map_obsStreamEvent_arrivedOne_real, map_obsStreamEvent_arrived_real]
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.CrossCellIndependence 10 declarations Disjoint cell histograms in the marked Poisson experiment are independent.
Disjoint cell histograms in the marked Poisson experiment are independent. This supplies the cross-cell independence required after roadmap (21).
theorem eventCount_eq_sum_singleton_histogram Lemma eventCount_eq_sum_singleton_histogram in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
eventCount s B
= ∑ x ∈ Finset.univ.filter (fun x ↦ x ∈ B), finiteSampleHistogram s.points x
Proof (Lean source)
-- @node: eventCount_eq_sum_singleton_histogram
lemma eventCount_eq_sum_singleton_histogram {X : Type*} [MeasurableSpace X]
[Fintype X] [DecidableEq X]
(s : FiniteSample X) (B : Set X) :
eventCount s B = ∑ x ∈ Finset.univ.filter (fun x ↦ x ∈ B),
finiteSampleHistogram s.points x := by
classical
have h := Finset.sum_card_fiberwise_eq_card_filter
(Finset.univ : Finset (Fin s.count)) (Finset.univ.filter (fun x ↦ x ∈ B)) s.points
simp only [mem_filter, Finset.mem_univ, true_and] at h
rw [eventCount, ← h]
apply Finset.sum_congr rfl
intro x hx
simp [finiteSampleHistogram, Fintype.card_subtype]
def markedCellAlphabet
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def poissonCellHistogram
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
theorem disjoint_markedCellAlphabet
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
d :
ℕ
Cell d
hij :
i ≠ j
conclusion
Proof (Lean source)
-- @node: disjoint_markedCellAlphabet
lemma disjoint_markedCellAlphabet {d : ℕ} {i j : Cell d} (hij : i ≠ j) :
Disjoint (markedCellAlphabet i) (markedCellAlphabet j) := by
classical
apply Finset.disjoint_left.mpr
intro z hi hj
simp only [markedCellAlphabet, mem_filter, Finset.mem_univ, true_and] at hi hj
apply hij
apply Prod.ext
· exact hi.1.symm.trans hj.1
· exact Prod.ext (hi.2.1.symm.trans hj.2.1) (hi.2.2.symm.trans hj.2.2)
theorem indepFun_poissonCellHistogram
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
IndepFun (poissonCellHistogram i) (poissonCellHistogram j) (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: indepFun_poissonCellHistogram
lemma indepFun_poissonCellHistogram (n d : ℕ) (P : FullLaw d)
{i j : Cell d} (hij : i ≠ j) :
IndepFun (poissonCellHistogram i) (poissonCellHistogram j)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
classical
let H := fun s : FiniteSample (ObsRecord d × Fin 3) ↦ finiteSampleHistogram s.points
have hH : Measurable H := by
apply measurable_pi_lambda
intro z
have he : (fun s : FiniteSample (ObsRecord d × Fin 3) ↦ H s z) =
(fun s ↦ eventCount s {z}) := by
funext s
simp [H, finiteSampleHistogram, eventCount, Fintype.card_subtype]
rw [he]
exact measurable_eventCount _ (MeasurableSet.singleton _)
have hind := (iIndepFun_pi (μ := fun z : ObsRecord d × Fin 3 ↦
poissonMeasure (((n : ℝ≥0) / 2) * ((markedObsLaw P) {z}).toNNReal))
(X := fun _ ↦ id) (fun _ ↦ measurable_id.aemeasurable)).indepFun_finset
(markedCellAlphabet i) (markedCellAlphabet j) (disjoint_markedCellAlphabet hij)
(fun _ ↦ measurable_pi_apply _)
change IndepFun (fun h (z : markedCellAlphabet i) ↦ h z)
(fun h (z : markedCellAlphabet j) ↦ h z)
(independentPoissonCountLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) at hind
rw [← finitePoissonSampleLaw_map_histogram (markedObsLaw P) ((n : ℝ≥0) / 2)] at hind
rw [indepFun_iff_measure_inter_preimage_eq_mul] at hind ⊢
intro a b ha hb
have hi : Measurable (fun h : (ObsRecord d × Fin 3) → ℕ ↦
fun z : markedCellAlphabet i ↦ h z) := by fun_prop
have hj : Measurable (fun h : (ObsRecord d × Fin 3) → ℕ ↦
fun z : markedCellAlphabet j ↦ h z) := by fun_prop
have h := hind a b ha hb
rw [Measure.map_apply hH ((ha.preimage hi).inter (hb.preimage hj)),
Measure.map_apply hH (ha.preimage hi), Measure.map_apply hH (hb.preimage hj)] at h
exact h
def cellHistogramEventCount
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
cellHistogramEventCount j pool arrived ones h :
ℕ
given by
∑ z : markedCellAlphabet j, if z.val ∈ streamEvent pool j arrived ones then h z else 0
theorem cellHistogramEventCount_poissonCellHistogram
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
cellHistogramEventCount j pool arrived ones (poissonCellHistogram j s)
= eventCount s (streamEvent pool j arrived ones)
Proof (Lean source)
-- @node: cellHistogramEventCount_poissonCellHistogram
lemma cellHistogramEventCount_poissonCellHistogram {d : ℕ} (j : Cell d)
(pool : Fin 3) (arrived ones : Bool) (s : FiniteSample (ObsRecord d × Fin 3)) :
cellHistogramEventCount j pool arrived ones (poissonCellHistogram j s) =
eventCount s (streamEvent pool j arrived ones) := by
classical
rw [eventCount_eq_sum_singleton_histogram]
unfold cellHistogramEventCount poissonCellHistogram
rw [Finset.sum_coe_sort (markedCellAlphabet j) (fun z ↦
if z ∈ streamEvent pool j arrived ones then finiteSampleHistogram s.points z else 0)]
unfold markedCellAlphabet
simp only [Finset.sum_filter]
apply Finset.sum_congr rfl
intro z hz
by_cases he : z ∈ streamEvent pool j arrived ones
· have hc : z ∈ markedCellAlphabet j := by
simp only [markedCellAlphabet, mem_filter, Finset.mem_univ, true_and]
exact ⟨he.2.1, he.2.2.1, he.2.2.2.1⟩
simp [he, he.2.1, he.2.2.1, he.2.2.2.1]
· simp [he]
def histogramWeightedSelectedEstimate
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
histogramWeightedSelectedEstimate n d q j h :
ℝ
given by
M :
C' :
C :
U :
(M : ℝ) / streamSize n * (if needleBranch n d q ∧ (C' : ℝ) ≤ needleRadius n q / 4 then (if needleBranch n d q then ∑ v ∈ Icc 1 (needleDegree n q - 1), needleCoeff n d q v * ((U : ℝ) * ((C - 1).descFactorial (v - 1) : ℝ)) else 0) else if 0 < C then (U : ℝ) / C else 0)
theorem histogramWeightedSelectedEstimate_poissonCellHistogram
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
histogramWeightedSelectedEstimate n d q j (poissonCellHistogram j s)
= poissonMemberWeight n j s * poissonSelectedCellEstimate n d q j s
Proof (Lean source)
-- @node: histogramWeightedSelectedEstimate_poissonCellHistogram
lemma histogramWeightedSelectedEstimate_poissonCellHistogram (n d : ℕ) (q : ℝ)
(j : Cell d) (s : FiniteSample (ObsRecord d × Fin 3)) :
histogramWeightedSelectedEstimate n d q j (poissonCellHistogram j s) =
poissonMemberWeight n j s * poissonSelectedCellEstimate n d q j s := by
simp only [histogramWeightedSelectedEstimate,
cellHistogramEventCount_poissonCellHistogram, poissonMemberWeight,
poissonSelectedCellEstimate, poissonFactorialBranch, poissonRatioBranch, weightedFactorial]
theorem indepFun_memberWeight_mul_selectedCellEstimate
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
IndepFun (fun s ↦ poissonMemberWeight n i s * poissonSelectedCellEstimate n d q i s) (fun s ↦ poissonMemberWeight n j s * poissonSelectedCellEstimate n d q j s) (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: indepFun_memberWeight_mul_selectedCellEstimate
lemma indepFun_memberWeight_mul_selectedCellEstimate (n d : ℕ) (q : ℝ) (P : FullLaw d)
{i j : Cell d} (hij : i ≠ j) :
IndepFun (fun s ↦ poissonMemberWeight n i s * poissonSelectedCellEstimate n d q i s)
(fun s ↦ poissonMemberWeight n j s * poissonSelectedCellEstimate n d q j s)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
have h := (indepFun_poissonCellHistogram n d P hij).comp
(measurable_of_countable (histogramWeightedSelectedEstimate n d q i))
(measurable_of_countable (histogramWeightedSelectedEstimate n d q j))
simpa only [Function.comp_def, histogramWeightedSelectedEstimate_poissonCellHistogram] using h
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.EnvelopeRate 1 declarations Uniform comparison of the explicit envelope with the frontier rate.
Uniform comparison of the explicit envelope with the frontier rate.
theorem riskEnvelope_le_frontierRate
the stated mathematical conclusion holds.
Formal statement
conclusion
∃ C : ℝ,
0 < C
Proof (Lean source)
lemma riskEnvelope_le_frontierRate :
∃ C : ℝ, 0 < C ∧ ∀ (n d : ℕ) (q : ℝ),
1 ≤ n → 1 ≤ d → 0 < q → q ≤ 1 →
riskEnvelope n d q ≤ C * frontierRate n d q := by
refine ⟨10 ^ 30, by norm_num, ?_⟩
intro n d q hn hd hq hq1
let N := effectiveSize n q
let m := streamSize n
let N₀ := streamEffectiveSize n q
let ell := logScale n q
let k := needleDegree n q
let B := needleRadius n q
let gamma := log 2 - 1 / 2
change (if N ≤ 1 then 1 else if ¬ needleBranch n d q then
min 4 (((8 * d / (exp 1 * N₀)) ^ 2) +
4 * (4 / N₀ + 1 / m) + 4 * exp (-(n : ℝ) * gamma))
else
let b := 2 * (4 * d * B / (N₀ * (k : ℝ) ^ 2) + 3 * exp (-16 * ell))
let v := 8 * (4 / N₀ + 1 / m) +
64 * d * (exp (ell / 8) + 1) *
(B ^ 2 / N₀ ^ 2 + B / (m * N₀)) +
32 * exp (-32 * ell) * (1 + 1 / m)
min 4 (b ^ 2 + v + 4 * exp (-(n : ℝ) * gamma))) ≤
10 ^ 30 * min 1 (N⁻¹ + ((d : ℝ) / (N * ell)) ^ 2)
have hnR : 0 < (n : ℝ) := by exact_mod_cast hn
have hdR : 0 < (d : ℝ) := by exact_mod_cast hd
have hNpos : 0 < N := by
dsimp [N, effectiveSize]
positivity
have hNn : N ≤ (n : ℝ) := by
dsimp [N, effectiveSize]
nlinarith [mul_nonneg hnR.le hq.le]
have hm : m = (n : ℝ) / 6 := rfl
have hN₀ : N₀ = N / 6 := by
dsimp [N₀, streamEffectiveSize, streamSize, N, effectiveSize]
ring
have hell : ell = log (exp 1 + N) := rfl
have hR0 : 0 ≤ N⁻¹ + ((d : ℝ) / (N * ell)) ^ 2 :=
add_nonneg (inv_nonneg.mpr hNpos.le) (sq_nonneg _)
by_cases hsmall : N ≤ 1
· rw [if_pos hsmall]
have hInv : 1 ≤ N⁻¹ := (one_le_inv₀ hNpos).2 hsmall
have hR1 : 1 ≤ N⁻¹ + ((d : ℝ) / (N * ell)) ^ 2 := by
nlinarith [sq_nonneg ((d : ℝ) / (N * ell))]
rw [min_eq_left hR1]
norm_num
· rw [if_neg hsmall]
have hN1 : 1 < N := lt_of_not_ge hsmall
obtain ⟨hell1, hexpell, hexpneg⟩ := logscale_basic N ell hN1 hell
have hellpos : 0 < ell := lt_trans (by norm_num) hell1
have htail := poisson_tail_rate n N hn hNpos hNn
have hmpos : 0 < m := by rw [hm]; positivity
have hN₀pos : 0 < N₀ := by rw [hN₀]; positivity
have hN₀inv : 1 / N₀ = 6 * N⁻¹ := by
rw [hN₀]
field_simp
have hminv : 1 / m ≤ 6 * N⁻¹ := by
have hmN : N / 6 ≤ m := by rw [hm]; linarith
calc
1 / m ≤ 1 / (N / 6) := one_div_le_one_div_of_le (by positivity) hmN
_ = 6 * N⁻¹ := by field_simp
by_cases hbranch : needleBranch n d q
· rw [if_neg (not_not_intro hbranch)]
rcases hbranch with ⟨hell128, hdsqrt⟩
have hB : B = 256 * ell := rfl
have hxfloor : 1 ≤ ell / 64 := by linarith
have hk : ell / 128 ≤ (k : ℝ) := by
have hf := half_le_natFloor hxfloor
dsimp [k, needleDegree]
convert hf using 1 <;> ring
have hkpos : 0 < (k : ℝ) := lt_of_lt_of_le (by positivity) hk
have hkSq : (ell / 128) ^ 2 ≤ (k : ℝ) ^ 2 :=
pow_le_pow_left₀ (by positivity) hk 2
obtain ⟨hgrowth, hell_d⟩ :=
needle_growth N ell d hN1 hell128 hexpell hdsqrt hdR.le
have hexp16 : exp (-16 * ell) ≤ N⁻¹ := by
exact (Real.exp_le_exp.mpr (by nlinarith)).trans hexpneg
have hexp32 : exp (-32 * ell) ≤ N⁻¹ := by
exact (Real.exp_le_exp.mpr (by nlinarith)).trans hexpneg
have hinv_ratio : N⁻¹ ≤ 2 * (d : ℝ) / (N * ell) := by
rw [inv_eq_one_div]
apply (div_le_div_iff₀ hNpos (mul_pos hNpos hellpos)).2
have hp : 0 ≤ N * (2 * (d : ℝ) - ell) :=
mul_nonneg hNpos.le (sub_nonneg.mpr hell_d)
nlinarith
have hmain : 4 * (d : ℝ) * B / (N₀ * (k : ℝ) ^ 2) ≤
100663296 * ((d : ℝ) / (N * ell)) := by
rw [hB, hN₀]
calc
4 * (d : ℝ) * (256 * ell) / (N / 6 * (k : ℝ) ^ 2) =
6144 * (d : ℝ) * ell / (N * (k : ℝ) ^ 2) := by field_simp; ring
_ ≤ 6144 * (d : ℝ) * ell / (N * (ell / 128) ^ 2) := by
have hnumer : 0 ≤ 6144 * (d : ℝ) * ell := by positivity
have hden := mul_le_mul_of_nonneg_left hkSq hNpos.le
exact div_le_div_of_nonneg_left hnumer
(mul_pos hNpos (sq_pos_of_pos (by positivity : 0 < ell / 128))) hden
_ = 100663296 * ((d : ℝ) / (N * ell)) := by field_simp; ring
have hbias_inside :
4 * (d : ℝ) * B / (N₀ * (k : ℝ) ^ 2) +
3 * exp (-16 * ell) ≤
100663302 * ((d : ℝ) / (N * ell)) := by
have htail_ratio : 3 * exp (-16 * ell) ≤
6 * ((d : ℝ) / (N * ell)) := by
calc
3 * exp (-16 * ell) ≤ 3 * N⁻¹ :=
mul_le_mul_of_nonneg_left hexp16 (by norm_num)
_ ≤ 3 * (2 * (d : ℝ) / (N * ell)) :=
mul_le_mul_of_nonneg_left hinv_ratio (by norm_num)
_ = 6 * ((d : ℝ) / (N * ell)) := by ring
linarith
have hbias_sq :
(2 * (4 * (d : ℝ) * B / (N₀ * (k : ℝ) ^ 2) +
3 * exp (-16 * ell))) ^ 2 ≤
40532401478172816 * ((d : ℝ) / (N * ell)) ^ 2 := by
have hins0 : 0 ≤ 4 * (d : ℝ) * B / (N₀ * (k : ℝ) ^ 2) +
3 * exp (-16 * ell) := by rw [hB]; positivity
have hs := pow_le_pow_left₀ hins0 hbias_inside 2
calc
(2 * (4 * (d : ℝ) * B / (N₀ * (k : ℝ) ^ 2) +
3 * exp (-16 * ell))) ^ 2 ≤
(2 * (100663302 * ((d : ℝ) / (N * ell)))) ^ 2 := by
exact pow_le_pow_left₀ (mul_nonneg (by norm_num) hins0)
(mul_le_mul_of_nonneg_left hbias_inside (by norm_num)) 2
_ = 40532401478172816 * ((d : ℝ) / (N * ell)) ^ 2 := by ring
have hN₀m : N₀ ≤ m := by
rw [hN₀, hm]
linarith
have hB1 : 1 ≤ B := by rw [hB]; nlinarith
have hBterms : B ^ 2 / N₀ ^ 2 + B / (m * N₀) ≤
4718592 * ell ^ 2 / N ^ 2 := by
have hden : N₀ ^ 2 ≤ m * N₀ := by
rw [sq]
exact mul_le_mul_of_nonneg_right hN₀m hN₀pos.le
have hsecond : B / (m * N₀) ≤ B ^ 2 / N₀ ^ 2 := by
calc
B / (m * N₀) ≤ B / N₀ ^ 2 :=
div_le_div_of_nonneg_left (zero_le_one.trans hB1) (sq_pos_of_pos hN₀pos) hden
_ ≤ B ^ 2 / N₀ ^ 2 :=
div_le_div_of_nonneg_right (by nlinarith [sq_nonneg (B - 1)])
(sq_nonneg N₀)
calc
B ^ 2 / N₀ ^ 2 + B / (m * N₀) ≤ 2 * (B ^ 2 / N₀ ^ 2) := by linarith
_ = 4718592 * ell ^ 2 / N ^ 2 := by rw [hB, hN₀]; field_simp; ring
have hscale : (d : ℝ) * (exp (ell / 8) + 1) * ell ^ 2 / N ^ 2 ≤
262144 * ((d : ℝ) / (N * ell)) ^ 2 := by
calc
(d : ℝ) * (exp (ell / 8) + 1) * ell ^ 2 / N ^ 2 =
(d : ℝ) * ((exp (ell / 8) + 1) * ell ^ 4) /
(N ^ 2 * ell ^ 2) := by field_simp
_ ≤ (d : ℝ) * (262144 * d) / (N ^ 2 * ell ^ 2) := by
exact div_le_div_of_nonneg_right
(mul_le_mul_of_nonneg_left hgrowth hdR.le) (by positivity)
_ = 262144 * ((d : ℝ) / (N * ell)) ^ 2 := by field_simp
have hcomplex :
64 * (d : ℝ) * (exp (ell / 8) + 1) *
(B ^ 2 / N₀ ^ 2 + B / (m * N₀)) ≤
79164837199872 * ((d : ℝ) / (N * ell)) ^ 2 := by
calc
64 * (d : ℝ) * (exp (ell / 8) + 1) *
(B ^ 2 / N₀ ^ 2 + B / (m * N₀)) ≤
64 * (d : ℝ) * (exp (ell / 8) + 1) *
(4718592 * ell ^ 2 / N ^ 2) :=
mul_le_mul_of_nonneg_left hBterms (by positivity)
_ = 301989888 *
((d : ℝ) * (exp (ell / 8) + 1) * ell ^ 2 / N ^ 2) := by ring
_ ≤ 301989888 * (262144 * ((d : ℝ) / (N * ell)) ^ 2) :=
mul_le_mul_of_nonneg_left hscale (by norm_num)
_ = 79164837199872 * ((d : ℝ) / (N * ell)) ^ 2 := by ring
have hbase : 8 * (4 / N₀ + 1 / m) ≤ 240 * N⁻¹ := by
have hfour : 4 / N₀ = 24 * N⁻¹ := by
rw [show 4 / N₀ = 4 * (1 / N₀) by ring, hN₀inv]
ring
rw [hfour]
nlinarith
have hinv1 : N⁻¹ ≤ 1 := (inv_le_one₀ hNpos).2 hN1.le
have hmInv6 : 1 / m ≤ 6 := hminv.trans (by nlinarith)
have hvtail : 32 * exp (-32 * ell) * (1 + 1 / m) ≤
224 * N⁻¹ := by
calc
32 * exp (-32 * ell) * (1 + 1 / m) ≤
32 * N⁻¹ * 7 :=
mul_le_mul (mul_le_mul_of_nonneg_left hexp32 (by norm_num))
(by linarith) (by positivity) (by positivity)
_ = 224 * N⁻¹ := by ring
refine min_four_le_const_min_one ?_ hR0 (by norm_num : (4 : ℝ) ≤ 10 ^ 30)
have htail' : 4 * exp (-(n : ℝ) * gamma) ≤ 32 * N⁻¹ := by
dsimp [gamma]
nlinarith
calc
(2 * (4 * (d : ℝ) * B / (N₀ * (k : ℝ) ^ 2) +
3 * exp (-16 * ell))) ^ 2 +
(8 * (4 / N₀ + 1 / m) +
64 * (d : ℝ) * (exp (ell / 8) + 1) *
(B ^ 2 / N₀ ^ 2 + B / (m * N₀)) +
32 * exp (-32 * ell) * (1 + 1 / m)) +
4 * exp (-(n : ℝ) * gamma) ≤
(40532401478172816 + 79164837199872) *
((d : ℝ) / (N * ell)) ^ 2 + 496 * N⁻¹ := by
linarith
_ ≤ 10 ^ 30 * (N⁻¹ + ((d : ℝ) / (N * ell)) ^ 2) := by
nlinarith [inv_nonneg.mpr hNpos.le,
sq_nonneg ((d : ℝ) / (N * ell))]
· rw [if_pos hbranch]
have hfrac0 : 0 ≤ 8 * (d : ℝ) / (exp 1 * N₀) := by positivity
have hfrac : 8 * (d : ℝ) / (exp 1 * N₀) ≤
48 * (d : ℝ) / N := by
calc
8 * (d : ℝ) / (exp 1 * N₀) ≤ 8 * (d : ℝ) / N₀ := by
apply (div_le_div_iff₀ (mul_pos (Real.exp_pos 1) hN₀pos) hN₀pos).2
have he := Real.one_le_exp (show (0 : ℝ) ≤ 1 by norm_num)
nlinarith [mul_nonneg (show (0 : ℝ) ≤ 8 * d by positivity)
(sub_nonneg.mpr he)]
_ = 48 * (d : ℝ) / N := by rw [hN₀]; field_simp; norm_num
have hfrac_sq : (8 * (d : ℝ) / (exp 1 * N₀)) ^ 2 ≤
2304 * ((d : ℝ) / N) ^ 2 := by
have := pow_le_pow_left₀ hfrac0 hfrac 2
calc
(8 * (d : ℝ) / (exp 1 * N₀)) ^ 2 ≤
(48 * (d : ℝ) / N) ^ 2 := this
_ = 2304 * ((d : ℝ) / N) ^ 2 := by ring
have hnbranch : ell < 128 ∨ (d : ℝ) < sqrt N := by
simp only [needleBranch, not_and_or, not_le] at hbranch
exact hbranch
have hdratio : ((d : ℝ) / N) ^ 2 ≤
N⁻¹ + 16384 * ((d : ℝ) / (N * ell)) ^ 2 := by
rcases hnbranch with hell128 | hdsqrt
· have heq : (d : ℝ) / N = ell * ((d : ℝ) / (N * ell)) := by
field_simp
rw [heq]
have hs := sq_nonneg ((d : ℝ) / (N * ell))
have hellsq : ell ^ 2 ≤ 16384 := by nlinarith [sq_nonneg (128 - ell)]
calc
(ell * ((d : ℝ) / (N * ell))) ^ 2 =
ell ^ 2 * ((d : ℝ) / (N * ell)) ^ 2 := by ring
_ ≤ 16384 * ((d : ℝ) / (N * ell)) ^ 2 :=
mul_le_mul_of_nonneg_right hellsq hs
_ ≤ N⁻¹ + 16384 * ((d : ℝ) / (N * ell)) ^ 2 :=
le_add_of_nonneg_left (inv_nonneg.mpr hNpos.le)
· have hsqrt : (sqrt N) ^ 2 = N := Real.sq_sqrt hNpos.le
have hdsq : (d : ℝ) ^ 2 ≤ N := by
nlinarith [sq_nonneg (sqrt N - d)]
have hratioN : ((d : ℝ) / N) ^ 2 ≤ N⁻¹ := by
calc
((d : ℝ) / N) ^ 2 = (d : ℝ) ^ 2 / N ^ 2 := by ring
_ ≤ N / N ^ 2 := div_le_div_of_nonneg_right hdsq (sq_nonneg N)
_ = N⁻¹ := by field_simp
exact hratioN.trans (le_add_of_nonneg_right (by positivity))
have hvar : 4 * (4 / N₀ + 1 / m) ≤ 120 * N⁻¹ := by
have hfour : 4 / N₀ = 24 * N⁻¹ := by
-- … truncated; follow the source link for the rest …
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.FactorialMean 4 declarations The finite coefficient expansion and exact Chebyshev mean of the marked factorial branch, including zero-arrival cells, in roadmap (8).
The finite coefficient expansion and exact Chebyshev mean of the marked factorial branch, including zero-arrival cells, in roadmap (8).
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: chebNeedle_eval_zero
lemma chebNeedle_eval_zero (n d : ℕ) (q : ℝ) :
(chebNeedle n d q).eval 0 = 1 := by
by_cases hb : needleBranch n d q
· have hB : 0 < needleRadius n q := by
unfold needleRadius
have := hb.1
positivity
have hk : 0 < needleDegree n q := by
unfold needleDegree
exact Nat.floor_pos.mpr (by linarith [hb.1])
let p : Polynomial ℝ := 1 - (Chebyshev.T ℝ (needleDegree n q : ℤ)).comp
(C 1 - C (2 / needleRadius n q) * X)
have hc : p.coeff 1 = (2 / needleRadius n q) * (needleDegree n q : ℝ)^2 := by
have hder : p.coeff 1 = (derivative p).eval 0 := by
rw [← coeff_zero_eq_eval_zero, coeff_derivative]
norm_num
rw [hder]
simp [p, derivative_sub, derivative_comp, Chebyshev.derivative_T_eval_one]
have hform : (chebNeedle n d q).eval 0 =
(needleRadius n q / (2 * (needleDegree n q : ℝ)^2)) * p.coeff 1 := by
simp [chebNeedle, hb, p, ← coeff_zero_eq_eval_zero, coeff_divX]
rw [hform, hc]
have hk0 : (needleDegree n q : ℝ) ≠ 0 := by exact_mod_cast hk.ne'
field_simp
· simp [chebNeedle, hb]
theorem natDegree_one_sub_chebNeedle_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: natDegree_one_sub_chebNeedle_le
lemma natDegree_one_sub_chebNeedle_le (n d : ℕ) (q : ℝ)
(hb : needleBranch n d q) :
(1 - chebNeedle n d q).natDegree ≤ needleDegree n q - 1 := by
let a : Polynomial ℝ := C 1 - C (2 / needleRadius n q) * X
have ha : a.natDegree ≤ 1 := by
apply (natDegree_sub_le _ _).trans
apply max_le (by simp)
exact (natDegree_C_mul_le _ _).trans (by simp)
have hp : (1 - (Chebyshev.T ℝ (needleDegree n q : ℤ)).comp a).natDegree ≤
needleDegree n q := by
apply (natDegree_sub_le _ _).trans
apply max_le
· simp
· apply (natDegree_comp_le).trans
rw [Chebyshev.natDegree_T]
simp only [Int.natAbs_natCast]
exact (Nat.mul_le_mul_left _ ha).trans_eq (Nat.mul_one _)
have hQ : (chebNeedle n d q).natDegree ≤ needleDegree n q - 1 := by
rw [chebNeedle, if_pos hb]
apply (natDegree_C_mul_le _ _).trans
rw [natDegree_divX_eq_natDegree_tsub_one]
exact Nat.sub_le_sub_right hp 1
exact (natDegree_sub_le _ _).trans (max_le (by simp) hQ)
theorem sum_needleCoeff_mul_pow
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
= 1 - (chebNeedle n d q).eval t
Proof (Lean source)
-- @node: sum_needleCoeff_mul_pow
lemma sum_needleCoeff_mul_pow (n d : ℕ) (q t : ℝ)
(hb : needleBranch n d q) :
(∑ v ∈ Icc 1 (needleDegree n q - 1), needleCoeff n d q v * t ^ v) =
1 - (chebNeedle n d q).eval t := by
classical
let p := 1 - chebNeedle n d q
have hp0 : p.coeff 0 = 0 := by
simp [p, coeff_zero_eq_eval_zero, chebNeedle_eval_zero]
have hk : 0 < needleDegree n q := by
unfold needleDegree
exact Nat.floor_pos.mpr (by linarith [hb.1])
have hdeg : p.natDegree < needleDegree n q :=
lt_of_le_of_lt (natDegree_one_sub_chebNeedle_le n d q hb) (sub_lt hk (by decide))
have heval := Polynomial.eval_eq_sum_range' hdeg t
rw [Nat.range_eq_Icc_zero_sub_one _ hk.ne'] at heval
have hsum : (∑ v ∈ Icc 0 (needleDegree n q - 1), p.coeff v * t ^ v) =
∑ v ∈ Icc 1 (needleDegree n q - 1), p.coeff v * t ^ v := by
symm
apply Finset.sum_subset
· intro v hv
simp only [Finset.mem_Icc] at hv ⊢
omega
· intro v hv hn
have hv0 : v = 0 := by
simp only [Finset.mem_Icc] at hv hn
omega
simp [hv0, hp0]
rw [hsum] at heval
simp only [p, eval_sub, eval_one] at heval
simpa only [needleCoeff, if_pos hb] using heval.symm
theorem integral_markedPoissonFactorialBranch_eq_needle
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, poissonFactorialBranch n d q j s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: integral_markedPoissonFactorialBranch_eq_needle
lemma integral_markedPoissonFactorialBranch_eq_needle
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d) :
(∫ s, poissonFactorialBranch n d q j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) =
cellMean P j * (1 - (chebNeedle n d q).eval
(streamSize n * arrivedCell P j)) := by
classical
let : IsProbabilityMeasure P.1 := P.2
by_cases hb : needleBranch n d q
· rw [integral_markedPoissonFactorialBranch, if_pos hb,
← sum_needleCoeff_mul_pow n d q _ hb, Finset.mul_sum]
apply Finset.sum_congr rfl
intro v hv
have hvpos : 1 ≤ v := (Finset.mem_Icc.mp hv).1
have hpow : v - 1 + 1 = v := by omega
by_cases hj : 0 < arrivedCell P j
· rw [cellMean, if_pos hj]
simp only [NNReal.coe_div, NNReal.coe_natCast, NNReal.coe_ofNat]
rw [show streamSize n * arrivedCell P j =
((n : ℝ) / 2) * (arrivedCell P j / 3) by unfold streamSize; ring, mul_pow]
have ha0 := hj.ne'
have ha : (arrivedCell P j / 3) ^ v =
(arrivedCell P j / 3) ^ (v - 1) * (arrivedCell P j / 3) := by
conv_lhs => rw [← hpow, pow_succ]
rw [ha]
field_simp
· have hz : arrivedCell P j = 0 :=
le_antisymm (le_of_not_gt hj) measureReal_nonneg
have hone : P.1.real {r | inCell r j ∧ r.R = true ∧ r.Y = true} = 0 := by
apply le_antisymm _ measureReal_nonneg
calc
_ ≤ arrivedCell P j := measureReal_mono
(fun r hr ↦ ⟨hr.1, hr.2.1⟩) (measure_ne_top P.1 _)
_ = 0 := hz
simp [cellMean, hj, hone]
· simp [integral_markedPoissonFactorialBranch, chebNeedle, hb]
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.FullBias 5 declarations Assembly of the light and heavy needle biases and the complementary pilot term into the signed bias bound in roadmap (20).
Assembly of the light and heavy needle biases and the complementary pilot term into the signed bias bound in roadmap (20).
theorem abs_sum_pilot_needle_bias_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
|2 * ∑ j : Cell d, armSign j.1 * cellProb P j * cellMean P j * (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real {s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} * (chebNeedle n d q).eval (streamSize n * arrivedCell P j)|
≤ 8 * d * needleRadius n q / (streamEffectiveSize n q * (needleDegree n q : ℝ)^2)
Proof (Lean source)
-- @node: abs_sum_pilot_needle_bias_le
lemma abs_sum_pilot_needle_bias_le
(n d : ℕ) (q : ℝ) (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) (hb : needleBranch n d q) :
|2 * ∑ j : Cell d, armSign j.1 * cellProb P j * cellMean P j *
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} *
(chebNeedle n d q).eval (streamSize n * arrivedCell P j)| ≤
8 * d * needleRadius n q /
(streamEffectiveSize n q * (needleDegree n q : ℝ)^2) +
4 * exp (-16 * logScale n q) := by
classical
let f : Cell d → ℝ := fun j ↦ armSign j.1 * cellProb P j * cellMean P j *
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} *
(chebNeedle n d q).eval (streamSize n * arrivedCell P j)
have hsplit : (∑ j : Cell d, f j) =
(∑ j ∈ (Finset.univ : Finset (Cell d)).filter
(fun j ↦ streamSize n * arrivedCell P j ≤ needleRadius n q), f j) +
(∑ j ∈ (Finset.univ : Finset (Cell d)).filter
(fun j ↦ needleRadius n q < streamSize n * arrivedCell P j), f j) := by
rw [Finset.sum_filter, Finset.sum_filter, ← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro j _
by_cases hj : streamSize n * arrivedCell P j ≤ needleRadius n q
· simp [hj, not_lt.mpr hj]
· simp [hj, lt_of_not_ge hj]
have hl := abs_sum_light_pilot_needle_bias_le n d q P hP hb
(fun j ↦ (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4})
(fun _ ↦ ⟨measureReal_nonneg, measureReal_le_one⟩)
have hh := abs_sum_heavy_pilot_needle_bias_le n d q P hb
change |2 * ∑ j : Cell d, f j| ≤ _
rw [hsplit, mul_add]
exact (abs_add_le _ _).trans (add_le_add hl hh)
theorem abs_sum_markedPoissonSelectedCellEstimate_bias_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
|2 * ∑ j : Cell d, armSign j.1 * cellProb P j * ((∫ s, poissonSelectedCellEstimate n d q j s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) - cellMean P j)|
≤ 2 * (4 * d * needleRadius n q / (streamEffectiveSize n q * (needleDegree n q : ℝ)^2) + 3 * exp (-16 * logScale n q))
Proof (Lean source)
-- @node: abs_sum_markedPoissonSelectedCellEstimate_bias_le
lemma abs_sum_markedPoissonSelectedCellEstimate_bias_le
(n d : ℕ) (q : ℝ) (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) (hb : needleBranch n d q) :
|2 * ∑ j : Cell d, armSign j.1 * cellProb P j *
((∫ s, poissonSelectedCellEstimate n d q j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) - cellMean P j)| ≤
2 * (4 * d * needleRadius n q /
(streamEffectiveSize n q * (needleDegree n q : ℝ)^2) +
3 * exp (-16 * logScale n q)) := by
classical
let f : Cell d → ℝ := fun j ↦ armSign j.1 * cellProb P j * cellMean P j *
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} *
(chebNeedle n d q).eval (streamSize n * arrivedCell P j)
let g : Cell d → ℝ := fun j ↦ armSign j.1 * cellProb P j * cellMean P j *
(1 - (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4}) *
exp (-(streamSize n * arrivedCell P j))
have heq : (2 * ∑ j : Cell d, armSign j.1 * cellProb P j *
((∫ s, poissonSelectedCellEstimate n d q j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) - cellMean P j)) =
-(2 * ∑ j : Cell d, f j) - (2 * ∑ j : Cell d, g j) := by
have hj (j : Cell d) : armSign j.1 * cellProb P j *
((∫ s, poissonSelectedCellEstimate n d q j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) - cellMean P j) =
-f j - g j := by
rw [markedPoissonSelectedCellEstimate_bias n d q P j hb]
dsimp [f, g]
ring
simp_rw [hj]
rw [Finset.sum_sub_distrib, Finset.sum_neg_distrib]
ring
have hf := abs_sum_pilot_needle_bias_le n d q P hP hb
have hg := abs_sum_pilot_heavy_exp_bias_le n d q P hb
rw [heq]
calc
_ ≤ |2 * ∑ j : Cell d, f j| + |2 * ∑ j : Cell d, g j| := by
simpa only [abs_neg] using abs_sub (-(2 * ∑ j : Cell d, f j))
(2 * ∑ j : Cell d, g j)
_ ≤ (8 * d * needleRadius n q /
(streamEffectiveSize n q * (needleDegree n q : ℝ)^2) +
4 * exp (-16 * logScale n q)) +
2 * exp (-16 * logScale n q) := add_le_add hf hg
_ = _ := by ring
theorem memLp_poissonMemberWeight_mul_selectedCellEstimate
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
MemLp (fun s ↦ poissonMemberWeight n j s * poissonSelectedCellEstimate n d q j s) 2 (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: memLp_poissonMemberWeight_mul_selectedCellEstimate
lemma memLp_poissonMemberWeight_mul_selectedCellEstimate
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d) :
MemLp (fun s ↦ poissonMemberWeight n j s * poissonSelectedCellEstimate n d q j s) 2
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
have h := (memLp_poissonMemberWeight_mul_ratioBranch n d P j).add
(memLp_poissonMemberWeight_mul_selectedCorrection n d q P j)
convert h using 1
funext s
simp only [Pi.add_apply]
ring
theorem integral_poissonMemberWeight_mul_selectedCellEstimate
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, poissonMemberWeight n j s * poissonSelectedCellEstimate n d q j s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
= cellProb P j * (∫ s, poissonSelectedCellEstimate n d q j s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: integral_poissonMemberWeight_mul_selectedCellEstimate
lemma integral_poissonMemberWeight_mul_selectedCellEstimate
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d) (hn : 0 < n) :
(∫ s, poissonMemberWeight n j s * poissonSelectedCellEstimate n d q j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) =
cellProb P j * (∫ s, poissonSelectedCellEstimate n d q j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
have hD := (memLp_poissonRatioBranch n d P j).integrable (by norm_num)
have hC := (memLp_poissonSelectedCellEstimate_sub_ratio n d q P j).integrable
(by norm_num)
have hWD := (memLp_poissonMemberWeight_mul_ratioBranch n d P j).integrable
(by norm_num)
have hWC := (memLp_poissonMemberWeight_mul_selectedCorrection n d q P j).integrable
(by norm_num)
have hmeanD := (indepFun_poissonMemberWeight_ratioBranch n d P j).integral_fun_mul_eq_mul_integral
(measurable_poissonMemberWeight n d j).aestronglyMeasurable
(measurable_poissonRatioBranch j).aestronglyMeasurable
have hiC := indepFun_poissonMemberWeight_selectedCorrection n d q P j
have hmeanC := hiC.integral_fun_mul_eq_mul_integral
(measurable_poissonMemberWeight n d j).aestronglyMeasurable
((measurable_poissonSelectedCellEstimate n d q j).sub
(measurable_poissonRatioBranch j)).aestronglyMeasurable
have heq : (fun s ↦ poissonMemberWeight n j s * poissonSelectedCellEstimate n d q j s) =
(fun s ↦ poissonMemberWeight n j s * poissonRatioBranch j s +
poissonMemberWeight n j s *
(poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) := by
funext s
ring
have hGeq : (fun s ↦ poissonRatioBranch j s +
(poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) =
poissonSelectedCellEstimate n d q j := by
funext s
ring
rw [heq, integral_add hWD hWC, hmeanD, hmeanC,
integral_poissonMemberWeight n d P j hn]
rw [← mul_add, ← integral_add hD hC, hGeq]
theorem abs_integral_raw_poissonSelected_estimator_bias_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
|(∫ s, 2 * ∑ j : Cell d, armSign j.1 * poissonMemberWeight n j s * poissonSelectedCellEstimate n d q j s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
- cellFunctional P|
≤ 2 * (4 * d * needleRadius n q / (streamEffectiveSize n q * (needleDegree n q : ℝ)^2) + 3 * exp (-16 * logScale n q))
Proof (Lean source)
-- @node: abs_integral_raw_poissonSelected_estimator_bias_le
lemma abs_integral_raw_poissonSelected_estimator_bias_le
(n d : ℕ) (q : ℝ) (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) (hb : needleBranch n d q) :
|(∫ s, 2 * ∑ j : Cell d, armSign j.1 * poissonMemberWeight n j s *
poissonSelectedCellEstimate n d q j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) - cellFunctional P| ≤
2 * (4 * d * needleRadius n q /
(streamEffectiveSize n q * (needleDegree n q : ℝ)^2) +
3 * exp (-16 * logScale n q)) := by
classical
have hn : 0 < n := lt_of_lt_of_le Nat.zero_lt_one hP.n_pos
have hj (j : Cell d) : Integrable (fun s ↦ armSign j.1 * poissonMemberWeight n j s *
poissonSelectedCellEstimate n d q j s)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
simpa only [mul_assoc] using
((memLp_poissonMemberWeight_mul_selectedCellEstimate n d q P j).integrable
(by norm_num)).const_mul (armSign j.1)
have hc (j : Cell d) : cellContribution P j = cellProb P j * cellMean P j := by
by_cases ha : 0 < arrivedCell P j
· simp [cellContribution, ha]
· simp [cellContribution, cellMean, ha]
rw [integral_const_mul, integral_finsetSum _ (fun j _ ↦ hj j)]
simp_rw [mul_assoc (armSign _) (poissonMemberWeight _ _ _) _, integral_const_mul,
integral_poissonMemberWeight_mul_selectedCellEstimate n d q P _ hn]
rw [cellFunctional]
simp_rw [hc, ← mul_assoc]
rw [← mul_sub, ← Finset.sum_sub_distrib]
have heq : (∑ j : Cell d, (
armSign j.1 * cellProb P j *
(∫ s, poissonSelectedCellEstimate n d q j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) -
armSign j.1 * cellProb P j * cellMean P j)) =
∑ j : Cell d, armSign j.1 * cellProb P j *
((∫ s, poissonSelectedCellEstimate n d q j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) - cellMean P j) := by
apply Finset.sum_congr rfl
intro j _
ring
rw [heq]
exact abs_sum_markedPoissonSelectedCellEstimate_bias_le n d q P hP hb
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.FullFactorialMoment 5 declarations Full-range factorial second moments and the false-light product in (10)--(13).
Full-range factorial second moments and the false-light product in (10)--(13).
theorem factorialOverlap_le_intensity_add_orders_pow Lemma factorialOverlap_le_intensity_add_orders_pow in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: factorialOverlap_le_intensity_add_orders_pow
lemma factorialOverlap_le_intensity_add_orders_pow (lam : ℝ≥0) (v w : ℕ) :
factorialOverlap lam v w ≤ ((lam : ℝ) + v + w) ^ (v + w) := by
have hv0 : (0 : ℝ) ≤ v := by positivity
have hw0 : (0 : ℝ) ≤ w := by positivity
have ht : 0 ≤ (lam : ℝ) := lam.coe_nonneg
calc
_ ≤ ∑ r ∈ range (v + 1),
(v.choose r : ℝ) * (w : ℝ) ^ r * (lam : ℝ) ^ (v - r) * (lam : ℝ) ^ w := by
unfold factorialOverlap
apply (Finset.sum_le_sum fun r hr ↦ ?_).trans
(Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_mono (by omega : min v w + 1 ≤ v + 1))
(fun r _ _ ↦ by positivity))
have hrv : r ≤ v := by have := Finset.mem_range.mp hr; omega
have hc : (w.choose r : ℝ) * (Nat.factorial r : ℝ) ≤ (w : ℝ) ^ r := by
have h := Nat.choose_le_pow_div (α := ℝ) r w
exact (le_div_iff₀ (by positivity : (0 : ℝ) < Nat.factorial r)).mp h
have he : v + w - r = (v - r) + w := by omega
rw [he, pow_add]
calc
_ = (v.choose r : ℝ) * ((w.choose r : ℝ) * Nat.factorial r) *
(lam : ℝ) ^ (v - r) * (lam : ℝ) ^ w := by ring
_ ≤ _ := by gcongr
_ = ((w : ℝ) + lam) ^ v * (lam : ℝ) ^ w := by
rw [← Finset.sum_mul, add_pow]
congr 1
apply Finset.sum_congr rfl
intro r hr
ring
_ ≤ ((lam : ℝ) + v + w) ^ v * ((lam : ℝ) + v + w) ^ w := by
apply mul_le_mul (pow_le_pow_left₀ (by positivity) (by linarith) _)
(pow_le_pow_left₀ ht (by linarith) _) (by positivity) (by positivity)
_ = _ := (pow_add _ _ _).symm
theorem needleCoefficientOverlap_le_full_growth
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
needleCoefficientOverlap n d q lam
≤ (64 : ℝ) ^ needleDegree n q * max 1 (((lam + 2 * needleDegree n q) / needleRadius n q) ^ (2 * needleDegree n q))
Proof (Lean source)
-- @node: needleCoefficientOverlap_le_full_growth
lemma needleCoefficientOverlap_le_full_growth (n d : ℕ) (q : ℝ) (lam : ℝ≥0)
(hb : needleBranch n d q) :
needleCoefficientOverlap n d q lam ≤ (64 : ℝ) ^ needleDegree n q *
max 1 (((lam + 2 * needleDegree n q) / needleRadius n q) ^
(2 * needleDegree n q)) := by
let k := needleDegree n q
let B := needleRadius n q
let S := Icc 1 (k - 1)
let x : ℝ := ((lam : ℝ) + 2 * k) / B
let M : ℝ := max 1 (x ^ (2 * k))
let a : ℕ → ℝ := fun v ↦ |needleCoeff n d q v * B ^ v|
have hB : 0 < B := by dsimp [B, needleRadius]; linarith [hb.1]
have hx : 0 ≤ x := by dsimp [x]; positivity
have hM : 0 ≤ M := le_trans (by norm_num) (le_max_left _ _)
have hpow (r : ℕ) (hr : r ≤ 2 * k) : x ^ r ≤ M := by
by_cases hx1 : x ≤ 1
· exact (pow_le_one₀ hx hx1).trans (le_max_left _ _)
· exact (pow_le_pow_right₀ (le_of_not_ge hx1) hr).trans (le_max_right _ _)
have hterm (v w : ℕ) (hv : v ∈ S) (hw : w ∈ S) :
|needleCoeff n d q v * needleCoeff n d q w| * factorialOverlap lam v w ≤
a v * a w * M := by
have hvk : v ≤ k := by have := (Finset.mem_Icc.mp hv).2; dsimp [S] at hv; omega
have hwk : w ≤ k := by have := (Finset.mem_Icc.mp hw).2; dsimp [S] at hw; omega
have horders : (v : ℝ) + w ≤ 2 * k := by exact_mod_cast (show v + w ≤ 2 * k by omega)
have ho : factorialOverlap lam v w ≤ ((lam : ℝ) + 2 * k) ^ (v + w) := by
apply (factorialOverlap_le_intensity_add_orders_pow lam v w).trans
apply pow_le_pow_left₀ (by positivity)
linarith
calc
_ ≤ |needleCoeff n d q v * needleCoeff n d q w| *
(((lam : ℝ) + 2 * k) ^ (v + w)) := by gcongr
_ = a v * a w * x ^ (v + w) := by
dsimp [a, x]
rw [div_pow, pow_add B, abs_mul, abs_mul, abs_mul,
abs_pow, abs_pow, abs_of_pos hB]
field_simp
_ ≤ _ := mul_le_mul_of_nonneg_left (hpow _ (by omega)) (by positivity)
change (∑ v ∈ S, ∑ w ∈ S,
|needleCoeff n d q v * needleCoeff n d q w| * factorialOverlap lam v w) ≤ _
calc
_ ≤ ∑ v ∈ S, ∑ w ∈ S, a v * a w * M := by
apply Finset.sum_le_sum
intro v hv
apply Finset.sum_le_sum
intro w hw
exact hterm v w hv hw
_ = (∑ v ∈ S, a v) ^ 2 * M := by
simp_rw [← Finset.sum_mul, ← Finset.mul_sum]
rw [← Finset.sum_mul]
ring
_ ≤ ((8 : ℝ) ^ k) ^ 2 * M := by
apply mul_le_mul_of_nonneg_right _ hM
apply pow_le_pow_left₀ (Finset.sum_nonneg fun _ _ ↦ abs_nonneg _)
exact needle_scaled_coefficient_oneNorm_le n d q hb
_ = _ := by
change ((8 : ℝ) ^ k) ^ 2 * M = (64 : ℝ) ^ k * M
rw [pow_right_comm]
norm_num
theorem integral_sq_markedPoissonFactorialBranch_le_full_growth
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, (poissonFactorialBranch n d q j s) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
≤ (64 : ℝ) ^ needleDegree n q * max 1 (((streamSize n * arrivedCell P j + 2 * needleDegree n q) / needleRadius n q) ^ (2 * needleDegree n q))
Proof (Lean source)
-- @node: integral_sq_markedPoissonFactorialBranch_le_full_growth
lemma integral_sq_markedPoissonFactorialBranch_le_full_growth
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d)
(hb : needleBranch n d q) :
(∫ s, (poissonFactorialBranch n d q j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
(64 : ℝ) ^ needleDegree n q *
max 1 (((streamSize n * arrivedCell P j + 2 * needleDegree n q) /
needleRadius n q) ^ (2 * needleDegree n q)) := by
have h := (integral_sq_poissonFactorialBranch_le_effective_overlap n d q j
(markedObsLaw P) ((n : ℝ≥0) / 2)).trans
(needleCoefficientOverlap_le_full_growth n d q
(arrivedStreamIntensity n P 2 j) hb)
simpa only [coe_arrivedStreamIntensity] using h
theorem markedPoisson_pilot_light_mul_factorial_secondMoment_le_intensity
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real {s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} * (∫ s, (poissonFactorialBranch n d q j s) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: markedPoisson_pilot_light_mul_factorial_secondMoment_le_intensity
lemma markedPoisson_pilot_light_mul_factorial_secondMoment_le_intensity
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d)
(hb : needleBranch n d q)
(hh : needleRadius n q < streamSize n * arrivedCell P j) :
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} *
(∫ s, (poissonFactorialBranch n d q j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
exp (-(streamSize n * arrivedCell P j) / 8) := by
have hl : 0 < logScale n q := by linarith [hb.1]
have hk : (needleDegree n q : ℝ) ≤ logScale n q / 64 :=
Nat.floor_le (by positivity)
have hx : 1 ≤ (streamSize n * arrivedCell P j + 2 * needleDegree n q) /
needleRadius n q := by
apply (le_div_iff₀ (by dsimp [needleRadius]; positivity)).2
have hk0 : (0 : ℝ) ≤ needleDegree n q := by positivity
linarith
have hp := one_le_pow₀ hx (n := 2 * needleDegree n q)
calc
_ ≤ exp (-(streamSize n * arrivedCell P j) / 4) *
((64 : ℝ) ^ needleDegree n q *
max 1 (((streamSize n * arrivedCell P j + 2 * needleDegree n q) /
needleRadius n q) ^ (2 * needleDegree n q))) := by
apply mul_le_mul
(markedPoisson_pilot_light_probability_le_of_heavy n d q P j hb hh)
(integral_sq_markedPoissonFactorialBranch_le_full_growth n d q P j hb)
(integral_nonneg fun _ ↦ sq_nonneg _) (Real.exp_pos _).le
_ ≤ _ := by
rw [max_eq_right hp, ← mul_assoc]
exact heavy_factorial_tail_growth_le (logScale n q)
(streamSize n * arrivedCell P j) (needleDegree n q) hl hk hh.le
theorem markedPoisson_pilot_light_mul_factorial_secondMoment_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real {s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} * (∫ s, (poissonFactorialBranch n d q j s) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: markedPoisson_pilot_light_mul_factorial_secondMoment_le
lemma markedPoisson_pilot_light_mul_factorial_secondMoment_le
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d)
(hb : needleBranch n d q)
(hh : needleRadius n q < streamSize n * arrivedCell P j) :
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} *
(∫ s, (poissonFactorialBranch n d q j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
exp (-32 * logScale n q) := by
apply (markedPoisson_pilot_light_mul_factorial_secondMoment_le_intensity
n d q P j hb hh).trans
apply Real.exp_le_exp.mpr
change 256 * logScale n q < streamSize n * arrivedCell P j at hh
linarith
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.FullVarianceBudget 4 declarations Full correction second-moment budget and weighted selected-cell variance comparison for roadmap (21)--(24).
Full correction second-moment budget and weighted selected-cell variance comparison for roadmap (21)--(24). Cross-cell independence is left to the signed raw-estimator assembly.
theorem sum_integral_sq_memberWeight_mul_selectedCorrection_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
8 * (∑ j : Cell d, ∫ s, (poissonMemberWeight n j s * (poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
≤ 64 * d * (exp (logScale n q / 8) + 1) * (needleRadius n q ^ 2 / streamEffectiveSize n q ^ 2 + needleRadius n q / (streamSize n * streamEffectiveSize n q))
Proof (Lean source)
-- @node: sum_integral_sq_memberWeight_mul_selectedCorrection_le
lemma sum_integral_sq_memberWeight_mul_selectedCorrection_le
(n d : ℕ) (q : ℝ) (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) (hb : needleBranch n d q) :
8 * (∑ j : Cell d, ∫ s, (poissonMemberWeight n j s *
(poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
64 * d * (exp (logScale n q / 8) + 1) *
(needleRadius n q ^ 2 / streamEffectiveSize n q ^ 2 +
needleRadius n q / (streamSize n * streamEffectiveSize n q)) +
32 * exp (-32 * logScale n q) * (1 + 1 / streamSize n) := by
classical
let f : Cell d → ℝ := fun j ↦ ∫ s, (poissonMemberWeight n j s *
(poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)
have hsplit : (∑ j : Cell d, f j) =
(∑ j ∈ (Finset.univ : Finset (Cell d)).filter
(fun j ↦ streamSize n * arrivedCell P j ≤ needleRadius n q), f j) +
(∑ j ∈ (Finset.univ : Finset (Cell d)).filter
(fun j ↦ needleRadius n q < streamSize n * arrivedCell P j), f j) := by
rw [Finset.sum_filter, Finset.sum_filter, ← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro j _
by_cases hj : streamSize n * arrivedCell P j ≤ needleRadius n q
· simp [hj, not_lt.mpr hj]
· simp [hj, lt_of_not_ge hj]
change 8 * (∑ j : Cell d, f j) ≤ _
rw [hsplit, mul_add]
exact add_le_add
(sum_light_integral_sq_poissonMemberWeight_mul_selectedCorrection_le n d q P hP hb)
(sum_heavy_integral_sq_memberWeight_mul_selectedCorrection_le n d q P
(lt_of_lt_of_le Nat.zero_lt_one hP.n_pos) hb)
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Proof (Lean source)
-- @node: selected_variance_add_le_twice
lemma selected_variance_add_le_twice {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsFiniteMeasure μ] {X Y : Ω → ℝ}
(hX : MemLp X 2 μ) (hY : MemLp Y 2 μ) :
variance (fun s ↦ X s + Y s) μ ≤ 2 * variance X μ + 2 * variance Y μ := by
have hsub := variance_nonneg (X := X - Y) (μ := μ)
rw [variance_sub hX hY] at hsub
rw [variance_fun_add hX hY]
linarith
theorem variance_memberWeight_mul_selectedCellEstimate_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
variance (fun s ↦ poissonMemberWeight n j s * poissonSelectedCellEstimate n d q j s) (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
≤ 2 * variance (fun s ↦ poissonMemberWeight n j s * poissonRatioBranch j s) (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
+ 2 * (∫ s, (poissonMemberWeight n j s * (poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: variance_memberWeight_mul_selectedCellEstimate_le
lemma variance_memberWeight_mul_selectedCellEstimate_le
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d) :
variance (fun s ↦ poissonMemberWeight n j s *
poissonSelectedCellEstimate n d q j s)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
2 * variance (fun s ↦ poissonMemberWeight n j s * poissonRatioBranch j s)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) +
2 * (∫ s, (poissonMemberWeight n j s *
(poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
have hD := memLp_poissonMemberWeight_mul_ratioBranch n d P j
have hC := memLp_poissonMemberWeight_mul_selectedCorrection n d q P j
have heq : (fun s ↦ poissonMemberWeight n j s * poissonSelectedCellEstimate n d q j s) =
(fun s ↦ poissonMemberWeight n j s * poissonRatioBranch j s +
poissonMemberWeight n j s *
(poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) := by
funext s
ring
rw [heq]
exact (selected_variance_add_le_twice _ hD hC).trans
(add_le_add le_rfl (mul_le_mul_of_nonneg_left
(variance_le_expectation_sq hC.aestronglyMeasurable) (by norm_num : (0 : ℝ) ≤ 2)))
theorem four_sum_variance_memberWeight_mul_selectedCellEstimate_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
4 * (∑ j : Cell d, variance (fun s ↦ poissonMemberWeight n j s * poissonSelectedCellEstimate n d q j s) (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)))
≤ 8 * (4 / streamEffectiveSize n q + 1 / streamSize n)
+ 64 * d * (exp (logScale n q / 8) + 1) * (needleRadius n q ^ 2 / streamEffectiveSize n q ^ 2 + needleRadius n q / (streamSize n * streamEffectiveSize n q))
Proof (Lean source)
-- @node: four_sum_variance_memberWeight_mul_selectedCellEstimate_le
lemma four_sum_variance_memberWeight_mul_selectedCellEstimate_le
(n d : ℕ) (q : ℝ) (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) (hb : needleBranch n d q) :
4 * (∑ j : Cell d, variance (fun s ↦ poissonMemberWeight n j s *
poissonSelectedCellEstimate n d q j s)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))) ≤
8 * (4 / streamEffectiveSize n q + 1 / streamSize n) +
64 * d * (exp (logScale n q / 8) + 1) *
(needleRadius n q ^ 2 / streamEffectiveSize n q ^ 2 +
needleRadius n q / (streamSize n * streamEffectiveSize n q)) +
32 * exp (-32 * logScale n q) * (1 + 1 / streamSize n) := by
have hsum := Finset.sum_le_sum (s := (Finset.univ : Finset (Cell d)))
(fun j _ ↦ variance_memberWeight_mul_selectedCellEstimate_le n d q P j)
rw [Finset.sum_add_distrib, ← Finset.mul_sum, ← Finset.mul_sum] at hsum
have h := mul_le_mul_of_nonneg_left hsum (by norm_num : (0 : ℝ) ≤ 4)
have hratio := sum_variance_poissonMemberWeight_mul_ratioBranch_le n d q P hP
have hcorr := sum_integral_sq_memberWeight_mul_selectedCorrection_le n d q P hP hb
linarith
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.HeavyBias 4 declarations Heavy needle bias via the exact factorial mean and Cauchy--Schwarz, as required by roadmap (18)--(20).
Heavy needle bias via the exact factorial mean and Cauchy--Schwarz, as required by roadmap (18)--(20).
theorem sq_integral_markedPoissonFactorialBranch_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, poissonFactorialBranch n d q j s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ^ 2
≤ (∫ s, (poissonFactorialBranch n d q j s) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: sq_integral_markedPoissonFactorialBranch_le
lemma sq_integral_markedPoissonFactorialBranch_le
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d) :
(∫ s, poissonFactorialBranch n d q j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ^ 2 ≤
(∫ s, (poissonFactorialBranch n d q j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
have hv := variance_nonneg (poissonFactorialBranch n d q j)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
rw [variance_eq_sub (memLp_poissonFactorialBranch n d q P j)] at hv
exact sub_nonneg.mp hv
theorem markedPoisson_pilot_light_mul_abs_factorial_mean_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real {s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} * |∫ s, poissonFactorialBranch n d q j s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)|
Proof (Lean source)
-- @node: markedPoisson_pilot_light_mul_abs_factorial_mean_le
lemma markedPoisson_pilot_light_mul_abs_factorial_mean_le
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d)
(hb : needleBranch n d q)
(hh : needleRadius n q < streamSize n * arrivedCell P j) :
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} *
|∫ s, poissonFactorialBranch n d q j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)| ≤
exp (-(streamSize n * arrivedCell P j) / 16) := by
let p := (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4}
let a := ∫ s, poissonFactorialBranch n d q j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)
let b := ∫ s, (poissonFactorialBranch n d q j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)
have hp0 : 0 ≤ p := measureReal_nonneg
have hp1 : p ≤ 1 := measureReal_le_one
have ha : a ^ 2 ≤ b := sq_integral_markedPoissonFactorialBranch_le n d q P j
have hprod : p * b ≤ exp (-(streamSize n * arrivedCell P j) / 8) :=
markedPoisson_pilot_light_mul_factorial_secondMoment_le_intensity n d q P j hb hh
have hs : (p * |a|) ^ 2 ≤ exp (-(streamSize n * arrivedCell P j) / 8) := by
calc
_ = p * (p * a ^ 2) := by rw [mul_pow, sq_abs]; ring
_ ≤ p * a ^ 2 := by
have h := mul_le_mul_of_nonneg_right hp1 (mul_nonneg hp0 (sq_nonneg a))
simpa only [one_mul] using h
_ ≤ p * b := mul_le_mul_of_nonneg_left ha hp0
_ ≤ _ := hprod
have he : (exp (-(streamSize n * arrivedCell P j) / 16)) ^ 2 =
exp (-(streamSize n * arrivedCell P j) / 8) := by
rw [← Real.exp_nat_mul]
congr 1
norm_num
ring
change p * |a| ≤ _
nlinarith [Real.exp_pos (-(streamSize n * arrivedCell P j) / 16)]
theorem markedPoisson_heavy_needle_bias_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
cellMean P j * (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real {s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} * |(chebNeedle n d q).eval (streamSize n * arrivedCell P j)|
Proof (Lean source)
-- @node: markedPoisson_heavy_needle_bias_le
lemma markedPoisson_heavy_needle_bias_le
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d)
(hb : needleBranch n d q)
(hh : needleRadius n q < streamSize n * arrivedCell P j) :
cellMean P j *
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} *
|(chebNeedle n d q).eval (streamSize n * arrivedCell P j)| ≤
2 * exp (-(streamSize n * arrivedCell P j) / 16) := by
let p := (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4}
let a := ∫ s, poissonFactorialBranch n d q j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)
have hp0 : 0 ≤ p := measureReal_nonneg
have hmu := cellMean_mem_Icc P j
have hid : cellMean P j * (chebNeedle n d q).eval
(streamSize n * arrivedCell P j) = cellMean P j - a := by
dsimp [a]
rw [integral_markedPoissonFactorialBranch_eq_needle n d q P j]
ring
have ht0 : 0 ≤ streamSize n * arrivedCell P j := by
have : 0 ≤ arrivedCell P j := measureReal_nonneg
unfold streamSize
positivity
have hp : p ≤ exp (-(streamSize n * arrivedCell P j) / 16) := by
apply (markedPoisson_pilot_light_probability_le_of_heavy n d q P j hb hh).trans
apply Real.exp_le_exp.mpr
linarith
have ha := markedPoisson_pilot_light_mul_abs_factorial_mean_le n d q P j hb hh
change cellMean P j * p * _ ≤ _
calc
_ = p * |cellMean P j - a| := by
rw [← hid, abs_mul, abs_of_nonneg hmu.1]
ring
_ ≤ p * (cellMean P j + |a|) := by
apply mul_le_mul_of_nonneg_left _ hp0
exact (abs_sub _ _).trans_eq (by rw [abs_of_nonneg hmu.1])
_ ≤ p + p * |a| := by nlinarith [mul_le_mul_of_nonneg_left hmu.2 hp0]
_ ≤ _ := by change p * |a| ≤ _ at ha; linarith
theorem abs_sum_heavy_pilot_needle_bias_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
|2 * ∑ j ∈ (Finset.univ : Finset (Cell d)).filter (fun j ↦ needleRadius n q < streamSize n * arrivedCell P j), armSign j.1 * cellProb P j * cellMean P j * (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real {s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} * (chebNeedle n d q).eval (streamSize n * arrivedCell P j)|
Proof (Lean source)
-- @node: abs_sum_heavy_pilot_needle_bias_le
lemma abs_sum_heavy_pilot_needle_bias_le
(n d : ℕ) (q : ℝ) (P : FullLaw d) (hb : needleBranch n d q) :
|2 * ∑ j ∈ (Finset.univ : Finset (Cell d)).filter
(fun j ↦ needleRadius n q < streamSize n * arrivedCell P j),
armSign j.1 * cellProb P j * cellMean P j *
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} *
(chebNeedle n d q).eval (streamSize n * arrivedCell P j)| ≤
4 * exp (-16 * logScale n q) := by
classical
have hj (j : Cell d) (hh : needleRadius n q < streamSize n * arrivedCell P j) :
|armSign j.1 * cellProb P j * cellMean P j *
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} *
(chebNeedle n d q).eval (streamSize n * arrivedCell P j)| ≤
cellProb P j * (2 * exp (-16 * logScale n q)) := by
have hp : 0 ≤ cellProb P j := measureReal_nonneg
have hmu := cellMean_mem_Icc P j
have hsign : |armSign j.1| = 1 := by cases j.1 <;> simp [armSign]
rw [abs_mul, abs_mul, abs_mul, abs_mul, hsign, one_mul,
abs_of_nonneg hp, abs_of_nonneg hmu.1, abs_of_nonneg measureReal_nonneg]
calc
_ = cellProb P j * (cellMean P j *
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} *
|(chebNeedle n d q).eval (streamSize n * arrivedCell P j)|) := by ring
_ ≤ cellProb P j * (2 * exp (-(streamSize n * arrivedCell P j) / 16)) :=
mul_le_mul_of_nonneg_left (markedPoisson_heavy_needle_bias_le n d q P j hb hh) hp
_ ≤ _ := by
gcongr
change 256 * logScale n q < streamSize n * arrivedCell P j at hh
linarith
rw [abs_mul, abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 2)]
calc
_ ≤ 2 * ∑ j ∈ (Finset.univ : Finset (Cell d)).filter
(fun j ↦ needleRadius n q < streamSize n * arrivedCell P j),
|armSign j.1 * cellProb P j * cellMean P j *
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} *
(chebNeedle n d q).eval (streamSize n * arrivedCell P j)| := by
gcongr
exact abs_sum_le_sum_abs _ _
_ ≤ 2 * ∑ j : Cell d, cellProb P j * (2 * exp (-16 * logScale n q)) := by
apply mul_le_mul_of_nonneg_left _ (by norm_num)
rw [Finset.sum_filter]
apply Finset.sum_le_sum
intro j _
split_ifs with hh
· exact hj j hh
· exact mul_nonneg measureReal_nonneg (by positivity)
_ = _ := by rw [← Finset.sum_mul, sum_cellProb_eq_one]; ring
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.HeavyCorrection 4 declarations Heavy-cell correction second moments, retaining the random membership weights, as required by roadmap (24).
Heavy-cell correction second moments, retaining the random membership weights, as required by roadmap (24).
theorem integral_sq_selectedCorrection_eq_pilot_mul
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, (poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
= (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real {s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} * (∫ s, (poissonFactorialBranch n d q j s - poissonRatioBranch j s) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: integral_sq_selectedCorrection_eq_pilot_mul
lemma integral_sq_selectedCorrection_eq_pilot_mul
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d)
(hb : needleBranch n d q) :
(∫ s, (poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) =
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} *
(∫ s, (poissonFactorialBranch n d q j s - poissonRatioBranch j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
classical
let H : FiniteSample (ObsRecord d) → ℝ := fun s ↦
if needleBranch n d q then
∑ v ∈ Icc 1 (needleDegree n q - 1), needleCoeff n d q v *
weightedFactorial (obsStreamEvent j true true) (obsStreamEvent j true false) v s
else 0
let D : FiniteSample (ObsRecord d) → ℝ := fun s ↦
if 0 < eventCount s (obsStreamEvent j true false) then
eventCount s (obsStreamEvent j true true) /
eventCount s (obsStreamEvent j true false) else 0
have hH : Measurable H := by
simp only [H, if_pos hb]
apply Finset.measurable_sum
intro v hv
exact (measurable_weightedFactorial _ _
(Set.Finite.measurableSet (Set.toFinite _))
(Set.Finite.measurableSet (Set.toFinite _)) v).const_mul _
have hc (a o : Bool) : Measurable (fun s : FiniteSample (ObsRecord d) ↦
(eventCount s (obsStreamEvent j a o) : ℝ)) :=
(measurable_of_countable (fun k : ℕ ↦ (k : ℝ))).comp
(measurable_eventCount _ (Set.Finite.measurableSet (Set.toFinite _)))
have hD : Measurable D := by
apply Measurable.ite
· exact (measurable_eventCount _ (Set.Finite.measurableSet (Set.toFinite _)))
measurableSet_Ioi
· exact (hc true true).div (hc true false)
· fun_prop
have hDs (s : FiniteSample (ObsRecord d × Fin 3)) :
D (unshuffle s 2) = poissonRatioBranch j s := by
simp only [D, poissonRatioBranch, eventCount_unshuffle]
have hHs (s : FiniteSample (ObsRecord d × Fin 3)) :
H (unshuffle s 2) = poissonFactorialBranch n d q j s := by
simp only [H, poissonFactorialBranch, weightedFactorial, eventCount_unshuffle]
have hgate := integral_pilot_gated_estimationStatistic n d q P j
(fun s ↦ (H s - D s) ^ 2) ((hH.sub hD).pow_const 2)
simp only [hHs, hDs] at hgate
rw [← hgate]
apply integral_congr_ae
filter_upwards [] with s
simp only [poissonSelectedCellEstimate, hb, true_and]
split_ifs <;> simp
theorem integral_sq_selectedCorrection_le_of_heavy
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, (poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: integral_sq_selectedCorrection_le_of_heavy
lemma integral_sq_selectedCorrection_le_of_heavy
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d)
(hb : needleBranch n d q)
(hh : needleRadius n q < streamSize n * arrivedCell P j) :
(∫ s, (poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
4 * exp (-32 * logScale n q) := by
have hH := (memLp_two_iff_integrable_sq
(measurable_poissonFactorialBranch n d q j).aestronglyMeasurable).1
(memLp_poissonFactorialBranch n d q P j)
have hdiff := (memLp_two_iff_integrable_sq
((measurable_poissonFactorialBranch n d q j).sub
(measurable_poissonRatioBranch j)).aestronglyMeasurable).1
((memLp_poissonFactorialBranch n d q P j).sub
(memLp_poissonRatioBranch n d P j))
have hmoment :
(∫ s, (poissonFactorialBranch n d q j s - poissonRatioBranch j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
2 * ((∫ s, (poissonFactorialBranch n d q j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) + 1) := by
calc
_ ≤ ∫ s, 2 * ((poissonFactorialBranch n d q j s) ^ 2 + 1)
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2) := by
apply integral_mono hdiff ((hH.add (integrable_const 1)).const_mul 2)
intro s
dsimp only [Pi.add_apply]
have hD := poissonRatioBranch_mem_unitInterval j s
have hDs : (poissonRatioBranch j s) ^ 2 ≤ 1 := by nlinarith [hD.1, hD.2]
have hsq := add_sq_le (a := poissonFactorialBranch n d q j s)
(b := -poissonRatioBranch j s)
simp only [neg_sq, ← sub_eq_add_neg] at hsq
exact hsq.trans (by linarith)
_ = _ := by rw [integral_const_mul, integral_add hH (integrable_const 1)]; simp
rw [integral_sq_selectedCorrection_eq_pilot_mul n d q P j hb]
have hp := markedPoisson_pilot_light_probability_le_exp_of_heavy n d q P j hb hh
have hprod := markedPoisson_pilot_light_mul_factorial_secondMoment_le n d q P j hb hh
have h := mul_le_mul_of_nonneg_left hmoment
(show 0 ≤ (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4}
from measureReal_nonneg)
nlinarith
theorem integral_sq_memberWeight_mul_selectedCorrection_le_of_heavy
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, (poissonMemberWeight n j s * (poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
≤ 4 * exp (-32 * logScale n q) * (∫ s, (poissonMemberWeight n j s) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: integral_sq_memberWeight_mul_selectedCorrection_le_of_heavy
lemma integral_sq_memberWeight_mul_selectedCorrection_le_of_heavy
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d) (hn : 0 < n)
(hb : needleBranch n d q)
(hh : needleRadius n q < streamSize n * arrivedCell P j) :
(∫ s, (poissonMemberWeight n j s *
(poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
4 * exp (-32 * logScale n q) *
(∫ s, (poissonMemberWeight n j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
rw [integral_sq_poissonMemberWeight_mul_selectedCorrection n d q P j hn,
integral_sq_poissonMemberWeight n d P j hn]
have h := mul_le_mul_of_nonneg_left
(integral_sq_selectedCorrection_le_of_heavy n d q P j hb hh)
(show 0 ≤ (cellProb P j) ^ 2 + cellProb P j / streamSize n from
add_nonneg (sq_nonneg _) (div_nonneg measureReal_nonneg (by
unfold streamSize; positivity)))
simpa only [mul_comm] using h
theorem sum_heavy_integral_sq_memberWeight_mul_selectedCorrection_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
8 * (∑ j ∈ (Finset.univ : Finset (Cell d)).filter (fun j ↦ needleRadius n q < streamSize n * arrivedCell P j), ∫ s, (poissonMemberWeight n j s * (poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: sum_heavy_integral_sq_memberWeight_mul_selectedCorrection_le
lemma sum_heavy_integral_sq_memberWeight_mul_selectedCorrection_le
(n d : ℕ) (q : ℝ) (P : FullLaw d) (hn : 0 < n)
(hb : needleBranch n d q) :
8 * (∑ j ∈ (Finset.univ : Finset (Cell d)).filter
(fun j ↦ needleRadius n q < streamSize n * arrivedCell P j),
∫ s, (poissonMemberWeight n j s *
(poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
32 * exp (-32 * logScale n q) * (1 + 1 / streamSize n) := by
classical
calc
_ ≤ 8 * ∑ j : Cell d, 4 * exp (-32 * logScale n q) *
(∫ s, (poissonMemberWeight n j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
rw [Finset.sum_filter]
apply mul_le_mul_of_nonneg_left _ (by norm_num)
apply Finset.sum_le_sum
intro j _
split_ifs with hh
· exact integral_sq_memberWeight_mul_selectedCorrection_le_of_heavy n d q P j hn hb hh
· exact mul_nonneg (by positivity) (integral_nonneg fun _ ↦ sq_nonneg _)
_ = 32 * exp (-32 * logScale n q) *
(∑ j : Cell d, ∫ s, (poissonMemberWeight n j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
rw [← Finset.mul_sum]
ring
_ ≤ _ := mul_le_mul_of_nonneg_left
(sum_integral_sq_poissonMemberWeight_le n d P hn) (by positivity)
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.HeavyFactorialBudget 4 declarations Numerical heavy-cell exponential budgets for the false-light product (13).
Numerical heavy-cell exponential budgets for the false-light product (13). These isolate the analytic step after the factorial second-moment bound (10).
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
ell t :
ℝ
k :
ℕ
hell :
0 < ell
hk :
(k : ℝ) ≤ ell / 64
ht :
256 * ell ≤ t
conclusion
6 * (k : ℝ) + 2 * (k : ℝ) * ((t + 2 * k) / (256 * ell)) ≤ t / 8
Proof (Lean source)
-- @node: heavy_factorial_exponent_budget
lemma heavy_factorial_exponent_budget (ell t : ℝ) (k : ℕ)
(hell : 0 < ell) (hk : (k : ℝ) ≤ ell / 64) (ht : 256 * ell ≤ t) :
6 * (k : ℝ) + 2 * (k : ℝ) * ((t + 2 * k) / (256 * ell)) ≤ t / 8 := by
have hk0 : (0 : ℝ) ≤ k := by positivity
have ht0 : 0 ≤ t := by linarith
have hkt : (k : ℝ) ≤ t / 16384 := by linarith
have hratio : 2 * (k : ℝ) / (256 * ell) ≤ 1 / 8192 := by
apply (div_le_iff₀ (by positivity : (0 : ℝ) < 256 * ell)).2
linarith
have hprod := mul_le_mul_of_nonneg_right hratio
(show 0 ≤ t + 2 * (k : ℝ) by positivity)
have heq : 2 * (k : ℝ) * ((t + 2 * k) / (256 * ell)) =
(2 * (k : ℝ) / (256 * ell)) * (t + 2 * k) := by ring
rw [heq]
linarith
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: heavy_factorial_tail_growth_le
lemma heavy_factorial_tail_growth_le (ell t : ℝ) (k : ℕ)
(hell : 0 < ell) (hk : (k : ℝ) ≤ ell / 64) (ht : 256 * ell ≤ t) :
exp (-t / 4) * (64 : ℝ) ^ k *
((t + 2 * k) / (256 * ell)) ^ (2 * k) ≤ exp (-t / 8) := by
let x := (t + 2 * k) / (256 * ell)
have ht0 : 0 ≤ t := by linarith
have hx0 : 0 ≤ x := by dsimp [x]; positivity
have hx : x ≤ exp x := by linarith [Real.add_one_le_exp x]
have htwo : (2 : ℝ) ≤ exp 1 := by
have h := Real.add_one_le_exp (1 : ℝ)
norm_num at h
exact h
have hcoeff : (64 : ℝ) ^ k ≤ exp (6 * (k : ℝ)) := by
calc
(64 : ℝ) ^ k = ((2 : ℝ) ^ 6) ^ k := by norm_num
_ = (2 : ℝ) ^ (6 * k) := by rw [← pow_mul]
_ ≤ (exp 1) ^ (6 * k) := by gcongr
_ = exp (6 * (k : ℝ)) := by
rw [← Real.exp_nat_mul]
congr 1
norm_num
have hgrowth : x ^ (2 * k) ≤ exp (2 * (k : ℝ) * x) := by
calc
_ ≤ (exp x) ^ (2 * k) := by gcongr
_ = _ := by rw [← Real.exp_nat_mul]; congr 1; push_cast; ring
calc
_ ≤ exp (-t / 4) * exp (6 * (k : ℝ)) *
exp (2 * (k : ℝ) * x) := by gcongr
_ = exp (-t / 4 + 6 * (k : ℝ) + 2 * (k : ℝ) * x) := by
rw [← Real.exp_add, ← Real.exp_add]
_ ≤ exp (-t / 8) := by
apply Real.exp_le_exp.mpr
have h := heavy_factorial_exponent_budget ell t k hell hk ht
change 6 * (k : ℝ) + 2 * (k : ℝ) * x ≤ t / 8 at h
linarith
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: heavy_factorial_tail_max_growth_le
lemma heavy_factorial_tail_max_growth_le (ell t : ℝ) (k : ℕ)
(hell : 0 < ell) (hk : (k : ℝ) ≤ ell / 64) (ht : 256 * ell ≤ t) :
exp (-t / 4) * ((64 : ℝ) ^ k *
max 1 (((t + 2 * k) / (256 * ell)) ^ (2 * k))) ≤
exp (-32 * ell) := by
have hx : 1 ≤ (t + 2 * (k : ℝ)) / (256 * ell) := by
apply (le_div_iff₀ (by positivity : (0 : ℝ) < 256 * ell)).2
have hk0 : (0 : ℝ) ≤ k := by positivity
linarith
have hpow : 1 ≤ ((t + 2 * (k : ℝ)) / (256 * ell)) ^ (2 * k) :=
one_le_pow₀ hx
rw [max_eq_right hpow, ← mul_assoc]
apply (heavy_factorial_tail_growth_le ell t k hell hk ht).trans
apply Real.exp_le_exp.mpr
linarith
theorem markedPoisson_pilot_light_mul_factorial_growth_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real {s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} * ((64 : ℝ) ^ needleDegree n q * max 1 (((streamSize n * arrivedCell P j + 2 * needleDegree n q) / needleRadius n q) ^ (2 * needleDegree n q)))
Proof (Lean source)
-- @node: markedPoisson_pilot_light_mul_factorial_growth_le
lemma markedPoisson_pilot_light_mul_factorial_growth_le
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d)
(hb : needleBranch n d q)
(hh : needleRadius n q < streamSize n * arrivedCell P j) :
(finitePoissonSampleLaw
(markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount
s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} *
((64 : ℝ) ^ needleDegree n q *
max 1 (((streamSize n * arrivedCell P j + 2 * needleDegree n q) /
needleRadius n q) ^ (2 * needleDegree n q))) ≤
exp (-32 * logScale n q) := by
have hl : 0 < logScale n q := by linarith [hb.1]
have hk : (needleDegree n q : ℝ) ≤ logScale n q / 64 :=
Nat.floor_le (by positivity)
calc
_ ≤ exp (-(streamSize n * arrivedCell P j) / 4) *
((64 : ℝ) ^ needleDegree n q *
max 1 (((streamSize n * arrivedCell P j + 2 * needleDegree n q) /
needleRadius n q) ^ (2 * needleDegree n q))) := by
apply mul_le_mul_of_nonneg_right
(markedPoisson_pilot_light_probability_le_of_heavy n d q P j hb hh)
positivity
_ ≤ _ := heavy_factorial_tail_max_growth_le (logScale n q)
(streamSize n * arrivedCell P j) (needleDegree n q) hl hk hh.le
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.IdealEstimator 9 declarations The ideal estimator evaluated on an uncapped marked Poisson sample.
The ideal estimator evaluated on an uncapped marked Poisson sample.
def poissonMemberWeight
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
poissonMemberWeight n j s :
ℝ
given by
def poissonRawMixedValue
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
poissonRawMixedValue n d q s :
ℝ
given by
def poissonAuxiliaryMixedValue
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
poissonAuxiliaryMixedValue n d q s :
ℝ
given by
if effectiveSize n q ≤ 1 then 0 else clip (poissonRawMixedValue n d q s)
theorem memberWeight_eq_poissonMemberWeight
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
memberWeight n records assign j
= poissonMemberWeight n j (fixedSizeEmbed k (fun i ↦ (records i, assign i)))
Proof (Lean source)
lemma memberWeight_eq_poissonMemberWeight {k d : ℕ} (n : ℕ)
(records : Fin k → ObsRecord d) (assign : Fin k → Fin 3) (j : Cell d) :
memberWeight n records assign j = poissonMemberWeight n j
(fixedSizeEmbed k (fun i ↦ (records i, assign i))) := by
rw [memberWeight, poissonMemberWeight, memberCount_eq_eventCount]
theorem auxiliaryMixedValue_eq_poissonAuxiliaryMixedValue
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
auxiliaryMixedValue n q records assign
= poissonAuxiliaryMixedValue n d q (fixedSizeEmbed k (fun i ↦ (records i, assign i)))
Proof (Lean source)
lemma auxiliaryMixedValue_eq_poissonAuxiliaryMixedValue {k d : ℕ}
(n : ℕ) (q : ℝ) (records : Fin k → ObsRecord d)
(assign : Fin k → Fin 3) :
auxiliaryMixedValue n q records assign =
poissonAuxiliaryMixedValue n d q
(fixedSizeEmbed k (fun i ↦ (records i, assign i))) := by
unfold auxiliaryMixedValue poissonAuxiliaryMixedValue poissonRawMixedValue
split_ifs
· rfl
· congr 2
apply Finset.sum_congr rfl
intro j hj
rw [memberWeight_eq_poissonMemberWeight,
selectedCellEstimate_eq_poissonSelectedCellEstimate]
theorem measurable_poissonFactorialBranch
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
@[fun_prop] lemma measurable_poissonFactorialBranch (n d : ℕ) (q : ℝ)
(j : Cell d) : Measurable (poissonFactorialBranch n d q j) := by
unfold poissonFactorialBranch
split_ifs
· apply Finset.measurable_sum
intro v hv
exact (measurable_weightedFactorial
(streamEvent 2 j true true) (streamEvent 2 j true false)
(Set.Finite.measurableSet (Set.toFinite _))
(Set.Finite.measurableSet (Set.toFinite _)) v).const_mul _
· fun_prop
theorem measurable_poissonSelectedCellEstimate
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
@[fun_prop] lemma measurable_poissonSelectedCellEstimate (n d : ℕ) (q : ℝ)
(j : Cell d) : Measurable (poissonSelectedCellEstimate n d q j) := by
unfold poissonSelectedCellEstimate
apply Measurable.ite
· by_cases hbranch : needleBranch n d q
· simp only [hbranch, true_and]
have hc : Measurable (fun s : FiniteSample (ObsRecord d × Fin 3) =>
(eventCount s (streamEvent 1 j true false) : ℝ)) :=
(measurable_of_countable (fun z : ℕ => (z : ℝ))).comp
(measurable_eventCount _ (Set.Finite.measurableSet (Set.toFinite _)))
exact hc measurableSet_Iic
· simp [hbranch]
· exact measurable_poissonFactorialBranch n d q j
· exact measurable_poissonRatioBranch j
theorem measurable_poissonAuxiliaryMixedValue
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
@[fun_prop] lemma measurable_poissonAuxiliaryMixedValue (n d : ℕ) (q : ℝ) :
Measurable (poissonAuxiliaryMixedValue n d q) := by
unfold poissonAuxiliaryMixedValue poissonRawMixedValue poissonMemberWeight clip
split_ifs
· fun_prop
· apply Measurable.max measurable_const
apply Measurable.min measurable_const
apply Measurable.const_mul
apply Finset.measurable_sum
intro j hj
have hmember : Measurable (fun s : FiniteSample (ObsRecord d × Fin 3) =>
(eventCount s (streamEvent 0 j false false) : ℝ) / streamSize n) := by
apply Measurable.div_const
exact (measurable_of_countable (fun z : ℕ => (z : ℝ))).comp
(measurable_eventCount _ (Set.Finite.measurableSet (Set.toFinite _)))
exact (hmember.const_mul _).mul
(measurable_poissonSelectedCellEstimate n d q j)
theorem poissonAuxiliaryMixedValue_mem_Icc
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
poissonAuxiliaryMixedValue n d q s ∈ Icc (-1 : ℝ) 1
Proof (Lean source)
lemma poissonAuxiliaryMixedValue_mem_Icc (n d : ℕ) (q : ℝ)
(s : FiniteSample (ObsRecord d × Fin 3)) :
poissonAuxiliaryMixedValue n d q s ∈ Icc (-1 : ℝ) 1 := by
unfold poissonAuxiliaryMixedValue clip
split_ifs
· norm_num
· exact ⟨le_max_left _ _, (max_le_iff.mpr ⟨by norm_num, min_le_left _ _⟩)⟩
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.IdealRiskTransfer 1 declarations Transfer of ideal risk between a marked Poisson sample and independent streams.
Transfer of ideal risk between a marked Poisson sample and independent streams.
theorem integral_sq_poissonAuxiliaryMixedValue_eq_streams
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, (poissonAuxiliaryMixedValue n d q s - θ) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
= ∫ s, (streamAuxiliaryMixedValue n d q s - θ) ^ 2 ∂Measure.pi (fun _ : Fin 3 ↦ finitePoissonSampleLaw (P.1.map obs) (((n : ℝ≥0) / 2) * (1 / 3)))
Proof (Lean source)
lemma integral_sq_poissonAuxiliaryMixedValue_eq_streams
(n d : ℕ) (q θ : ℝ) (P : FullLaw d) :
(∫ s, (poissonAuxiliaryMixedValue n d q s - θ) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) =
∫ s, (streamAuxiliaryMixedValue n d q s - θ) ^ 2
∂Measure.pi (fun _ : Fin 3 ↦
finitePoissonSampleLaw (P.1.map obs)
(((n : ℝ≥0) / 2) * (1 / 3))) := by
letI : IsProbabilityMeasure P.1 := P.2
letI : IsProbabilityMeasure (P.1.map obs) := map_obs_isProbabilityMeasure P
rw [← map_unshuffle_markedObsLaw P ((n : ℝ≥0) / 2)]
let f : (Fin 3 → FiniteSample (ObsRecord d)) → ℝ := fun s ↦
(streamAuxiliaryMixedValue n d q s - θ) ^ 2
have hf : Measurable f :=
((measurable_streamAuxiliaryMixedValue n d q).sub measurable_const).pow_const 2
change _ = ∫ s, f s ∂Measure.map unshuffle
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
rw [integral_map measurable_unshuffle.aemeasurable hf.aestronglyMeasurable]
apply integral_congr_ae
filter_upwards [] with s
dsimp [f]
rw [streamAuxiliaryMixedValue_unshuffle]
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.MembershipMoments 9 declarations Exact random membership-weight moments in roadmap (3), (21), and (24).
Exact random membership-weight moments in roadmap (3), (21), and (24).
def membershipStreamIntensity
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
membershipStreamIntensity n P j :
ℝ≥0
given by
theorem coe_membershipStreamIntensity
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(membershipStreamIntensity n P j : ℝ) = streamSize n * cellProb P j
Proof (Lean source)
lemma coe_membershipStreamIntensity (n : ℕ) {d : ℕ}
(P : FullLaw d) (j : Cell d) :
(membershipStreamIntensity n P j : ℝ) = streamSize n * cellProb P j := by
unfold membershipStreamIntensity
change ((n : ℝ) / 2) *
((markedObsLaw P) (streamEvent 0 j false false)).toReal = _
rw [← measureReal_def, markedObsLaw_streamEvent_real,
map_obsStreamEvent_membership_real]
unfold streamSize
ring
theorem map_membershipCount_eq_poisson
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Measure.map (fun s ↦ eventCount s (streamEvent 0 j false false)) (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
= poissonMeasure (membershipStreamIntensity n P j)
Proof (Lean source)
-- @node: map_membershipCount_eq_poisson
lemma map_membershipCount_eq_poisson (n d : ℕ) (P : FullLaw d) (j : Cell d) :
Measure.map (fun s ↦ eventCount s (streamEvent 0 j false false))
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) =
poissonMeasure (membershipStreamIntensity n P j) := by
classical
exact finitePoissonSampleLaw_map_eventCount (markedObsLaw P) ((n : ℝ≥0) / 2)
(streamEvent 0 j false false) (Set.Finite.measurableSet (Set.toFinite _))
theorem memLp_poissonMemberWeight
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: memLp_poissonMemberWeight
lemma memLp_poissonMemberWeight (n d : ℕ) (P : FullLaw d) (j : Cell d) :
MemLp (poissonMemberWeight n j) 2
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
classical
have h := poisson_natCast_memLp_two
(membershipStreamIntensity n P j)
rw [← map_membershipCount_eq_poisson n d P j] at h
have hc := h.comp_of_map
(measurable_eventCount (streamEvent 0 j false false)
(Set.Finite.measurableSet (Set.toFinite _))).aemeasurable
change MemLp (fun s ↦ (eventCount s (streamEvent 0 j false false) : ℝ) /
streamSize n) 2 _
simpa [Function.comp_def, div_eq_mul_inv] using
hc.mul_const (streamSize n)⁻¹
theorem integral_poissonMemberWeight
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, poissonMemberWeight n j s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
= cellProb P j
Proof (Lean source)
-- @node: integral_poissonMemberWeight
lemma integral_poissonMemberWeight (n d : ℕ) (P : FullLaw d) (j : Cell d)
(hn : 0 < n) :
(∫ s, poissonMemberWeight n j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) = cellProb P j := by
classical
have hm : streamSize n ≠ 0 := by
unfold streamSize
exact div_ne_zero (by exact_mod_cast hn.ne') (by norm_num)
unfold poissonMemberWeight
rw [integral_div]
rw [← integral_map
(measurable_eventCount (streamEvent 0 j false false)
(Set.Finite.measurableSet (Set.toFinite _))).aemeasurable
(measurable_of_countable (fun k : ℕ ↦ (k : ℝ))).aestronglyMeasurable,
map_membershipCount_eq_poisson, poisson_count_first_moment,
coe_membershipStreamIntensity]
exact mul_div_cancel_left₀ _ hm
theorem integral_sq_poissonMemberWeight
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, (poissonMemberWeight n j s) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
= (cellProb P j) ^ 2 + cellProb P j / streamSize n
Proof (Lean source)
-- @node: integral_sq_poissonMemberWeight
lemma integral_sq_poissonMemberWeight (n d : ℕ) (P : FullLaw d) (j : Cell d)
(hn : 0 < n) :
(∫ s, (poissonMemberWeight n j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) =
(cellProb P j) ^ 2 + cellProb P j / streamSize n := by
classical
have hm : streamSize n ≠ 0 := by
unfold streamSize
exact div_ne_zero (by exact_mod_cast hn.ne') (by norm_num)
simp only [poissonMemberWeight, div_pow]
rw [integral_div]
rw [← integral_map
(measurable_eventCount (streamEvent 0 j false false)
(Set.Finite.measurableSet (Set.toFinite _))).aemeasurable
(measurable_of_countable (fun k : ℕ ↦ (k : ℝ) ^ 2)).aestronglyMeasurable,
map_membershipCount_eq_poisson, poisson_count_second_moment,
coe_membershipStreamIntensity]
field_simp
theorem variance_poissonMemberWeight
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
= cellProb P j / streamSize n
Proof (Lean source)
-- @node: variance_poissonMemberWeight
lemma variance_poissonMemberWeight (n d : ℕ) (P : FullLaw d) (j : Cell d)
(hn : 0 < n) :
variance (poissonMemberWeight n j)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) =
cellProb P j / streamSize n := by
rw [variance_eq_sub (memLp_poissonMemberWeight n d P j)]
change (∫ s, (poissonMemberWeight n j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) -
(∫ s, poissonMemberWeight n j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ^ 2 = _
rw [integral_sq_poissonMemberWeight n d P j hn,
integral_poissonMemberWeight n d P j hn]
ring
theorem sum_integral_sq_poissonMemberWeight_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∑ j : Cell d, ∫ s, (poissonMemberWeight n j s) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
≤ 1 + 1 / streamSize n
Proof (Lean source)
-- @node: sum_integral_sq_poissonMemberWeight_le
lemma sum_integral_sq_poissonMemberWeight_le (n d : ℕ) (P : FullLaw d)
(hn : 0 < n) :
(∑ j : Cell d, ∫ s, (poissonMemberWeight n j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
1 + 1 / streamSize n := by
have hp (j : Cell d) : cellProb P j ≤ 1 := by
calc
cellProb P j ≤ ∑ i : Cell d, cellProb P i :=
Finset.single_le_sum
(fun i _ ↦ (show 0 ≤ cellProb P i from measureReal_nonneg)) (Finset.mem_univ j)
_ = 1 := sum_cellProb_eq_one P
calc
_ = ∑ j : Cell d, ((cellProb P j) ^ 2 + cellProb P j / streamSize n) := by
apply Finset.sum_congr rfl
intro j hj
exact integral_sq_poissonMemberWeight n d P j hn
_ ≤ ∑ j : Cell d, (cellProb P j + cellProb P j / streamSize n) := by
apply Finset.sum_le_sum
intro j hj
have hnonneg : 0 ≤ cellProb P j := measureReal_nonneg
nlinarith [hp j]
_ = _ := by
rw [Finset.sum_add_distrib, ← Finset.sum_div, sum_cellProb_eq_one]
theorem sum_membershipVariance_mul_ratioSecondMoment_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∑ j : Cell d, variance (poissonMemberWeight n j) (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) * (∫ s, (poissonRatioBranch j s) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)))
≤ 1 / streamSize n
Proof (Lean source)
-- @node: sum_membershipVariance_mul_ratioSecondMoment_le
lemma sum_membershipVariance_mul_ratioSecondMoment_le
(n d : ℕ) (P : FullLaw d) (hn : 0 < n) :
(∑ j : Cell d,
variance (poissonMemberWeight n j)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) *
(∫ s, (poissonRatioBranch j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))) ≤
1 / streamSize n := by
have hm : 0 < streamSize n := by
unfold streamSize
exact div_pos (by exact_mod_cast hn) (by norm_num)
calc
_ ≤ ∑ j : Cell d, cellProb P j / streamSize n := by
apply Finset.sum_le_sum
intro j hj
rw [variance_poissonMemberWeight n d P j hn]
have hsecond : (∫ s, (poissonRatioBranch j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤ 1 := by
calc
_ ≤ ∫ _s : FiniteSample (ObsRecord d × Fin 3), (1 : ℝ)
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2) := by
apply integral_mono (integrable_poissonRatioBranch_sq n d P j)
(integrable_const _)
intro s
rcases poissonRatioBranch_mem_unitInterval j s with ⟨hs0, hs1⟩
nlinarith
_ = 1 := by simp
simpa using mul_le_mul_of_nonneg_left hsecond
(div_nonneg (show 0 ≤ cellProb P j from measureReal_nonneg) hm.le)
_ = _ := by rw [← Finset.sum_div, sum_cellProb_eq_one]
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.MembershipRatio 6 declarations Independent membership and estimation streams give the weighted ratio variance identity and its aggregate bound in roadmap (21).
Independent membership and estimation streams give the weighted ratio variance identity and its aggregate bound in roadmap (21).
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: measurable_poissonMemberWeight
@[fun_prop] lemma measurable_poissonMemberWeight (n d : ℕ) (j : Cell d) :
Measurable (poissonMemberWeight n j) := by
unfold poissonMemberWeight
exact ((measurable_of_countable (fun k : ℕ ↦ (k : ℝ))).comp
(measurable_eventCount _ (Set.Finite.measurableSet (Set.toFinite _)))).div_const _
theorem memLp_poissonRatioBranch
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: memLp_poissonRatioBranch
lemma memLp_poissonRatioBranch (n d : ℕ) (P : FullLaw d) (j : Cell d) :
MemLp (poissonRatioBranch j) 2
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
apply (memLp_two_iff_integrable_sq
(measurable_poissonRatioBranch j).aestronglyMeasurable).2
exact integrable_poissonRatioBranch_sq n d P j
theorem indepFun_poissonMemberWeight_ratioBranch
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
IndepFun (poissonMemberWeight n j) (poissonRatioBranch j) (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: indepFun_poissonMemberWeight_ratioBranch
lemma indepFun_poissonMemberWeight_ratioBranch (n d : ℕ) (P : FullLaw d)
(j : Cell d) :
IndepFun (poissonMemberWeight n j) (poissonRatioBranch j)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
classical
let W : FiniteSample (ObsRecord d) → ℝ := fun s ↦
eventCount s (obsStreamEvent j false false) / streamSize n
let D : FiniteSample (ObsRecord d) → ℝ := fun s ↦
if 0 < eventCount s (obsStreamEvent j true false) then
eventCount s (obsStreamEvent j true true) /
eventCount s (obsStreamEvent j true false) else 0
have hcount (a o : Bool) : Measurable (fun s : FiniteSample (ObsRecord d) ↦
(eventCount s (obsStreamEvent j a o) : ℝ)) :=
(measurable_of_countable (fun k : ℕ ↦ (k : ℝ))).comp
(measurable_eventCount _ (Set.Finite.measurableSet (Set.toFinite _)))
have hW : Measurable W := (hcount false false).div_const _
have hD : Measurable D := by
apply Measurable.ite
· exact (measurable_eventCount _ (Set.Finite.measurableSet (Set.toFinite _)))
measurableSet_Ioi
· exact (hcount true true).div (hcount true false)
· fun_prop
have hcoords := (iIndepFun_pi (μ := fun _ : Fin 3 ↦
finitePoissonSampleLaw (P.1.map obs) (((n : ℝ≥0) / 2) * (1 / 3)))
(X := fun _ ↦ id) (fun _ ↦ measurable_id.aemeasurable)).indepFun
(show (0 : Fin 3) ≠ 2 by decide)
have hind := hcoords.comp hW hD
rw [← map_unshuffle_markedObsLaw P ((n : ℝ≥0) / 2)] at hind
have hWs (s : FiniteSample (ObsRecord d × Fin 3)) :
W (unshuffle s 0) = poissonMemberWeight n j s := by
simp only [W, poissonMemberWeight, eventCount_unshuffle]
have hDs (s : FiniteSample (ObsRecord d × Fin 3)) :
D (unshuffle s 2) = poissonRatioBranch j s := by
simp only [D, poissonRatioBranch, eventCount_unshuffle]
rw [indepFun_iff_measure_inter_preimage_eq_mul] at hind ⊢
intro a b ha hb
have h := hind a b ha hb
simp only [id_eq] at h
rw [Measure.map_apply measurable_unshuffle
((ha.preimage (hW.comp (measurable_pi_apply 0))).inter
(hb.preimage (hD.comp (measurable_pi_apply 2)))),
Measure.map_apply measurable_unshuffle (ha.preimage (hW.comp (measurable_pi_apply 0))),
Measure.map_apply measurable_unshuffle (hb.preimage (hD.comp (measurable_pi_apply 2)))] at h
simpa only [Set.preimage_inter, Set.preimage_preimage, Function.comp_def, hWs, hDs] using h
theorem memLp_poissonMemberWeight_mul_ratioBranch
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
MemLp (fun s ↦ poissonMemberWeight n j s * poissonRatioBranch j s) 2 (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: memLp_poissonMemberWeight_mul_ratioBranch
lemma memLp_poissonMemberWeight_mul_ratioBranch (n d : ℕ) (P : FullLaw d)
(j : Cell d) :
MemLp (fun s ↦ poissonMemberWeight n j s * poissonRatioBranch j s) 2
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
apply (memLp_poissonMemberWeight n d P j).of_le_mul
((measurable_poissonMemberWeight n d j).mul
(measurable_poissonRatioBranch j)).aestronglyMeasurable (c := 1)
filter_upwards [] with s
simp only [Pi.mul_apply, norm_mul, one_mul]
apply mul_le_of_le_one_right (norm_nonneg _)
rcases poissonRatioBranch_mem_unitInterval j s with ⟨h0, h1⟩
simpa [Real.norm_eq_abs, abs_of_nonneg h0] using h1
theorem variance_poissonMemberWeight_mul_ratioBranch
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
variance (fun s ↦ poissonMemberWeight n j s * poissonRatioBranch j s) (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
= (cellProb P j) ^ 2 * variance (poissonRatioBranch j) (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
+ (cellProb P j / streamSize n) * (∫ s, (poissonRatioBranch j s) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: variance_poissonMemberWeight_mul_ratioBranch
lemma variance_poissonMemberWeight_mul_ratioBranch (n d : ℕ) (P : FullLaw d)
(j : Cell d) (hn : 0 < n) :
variance (fun s ↦ poissonMemberWeight n j s * poissonRatioBranch j s)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) =
(cellProb P j) ^ 2 * variance (poissonRatioBranch j)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) +
(cellProb P j / streamSize n) *
(∫ s, (poissonRatioBranch j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
have hi := indepFun_poissonMemberWeight_ratioBranch n d P j
have hmean := hi.integral_fun_mul_eq_mul_integral
(measurable_poissonMemberWeight n d j).aestronglyMeasurable
(measurable_poissonRatioBranch j).aestronglyMeasurable
have hsecond := (hi.comp (measurable_id.pow_const 2) (measurable_id.pow_const 2)).integral_fun_mul_eq_mul_integral
((measurable_poissonMemberWeight n d j).pow_const 2).aestronglyMeasurable
((measurable_poissonRatioBranch j).pow_const 2).aestronglyMeasurable
simp only [Function.comp_def, id_eq] at hsecond
rw [variance_eq_sub (memLp_poissonMemberWeight_mul_ratioBranch n d P j),
variance_eq_sub (memLp_poissonRatioBranch n d P j)]
simp only [Pi.pow_apply]
simp only [mul_pow]
rw [hmean, hsecond, integral_poissonMemberWeight n d P j hn,
integral_sq_poissonMemberWeight n d P j hn]
ring
theorem sum_variance_poissonMemberWeight_mul_ratioBranch_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∑ j : Cell d, variance (fun s ↦ poissonMemberWeight n j s * poissonRatioBranch j s) (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)))
≤ 4 / streamEffectiveSize n q + 1 / streamSize n
Proof (Lean source)
-- @node: sum_variance_poissonMemberWeight_mul_ratioBranch_le
lemma sum_variance_poissonMemberWeight_mul_ratioBranch_le
(n d : ℕ) (q : ℝ) (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) :
(∑ j : Cell d, variance
(fun s ↦ poissonMemberWeight n j s * poissonRatioBranch j s)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))) ≤
4 / streamEffectiveSize n q + 1 / streamSize n := by
have hn : 0 < n := lt_of_lt_of_le Nat.zero_lt_one hP.n_pos
simp_rw [variance_poissonMemberWeight_mul_ratioBranch n d P _ hn]
rw [Finset.sum_add_distrib]
apply add_le_add (sum_cellProb_sq_mul_ratioVariance_le n d q P hP)
simpa only [variance_poissonMemberWeight n d P _ hn] using
sum_membershipVariance_mul_ratioSecondMoment_le n d P hn
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.PilotBias 4 declarations Pilot independence and the exact selected-cell bias in roadmap (16).
Pilot independence and the exact selected-cell bias in roadmap (16).
theorem indepFun_pilotCount_estimationStatistic
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
IndepFun (fun s ↦ eventCount s (streamEvent 1 j true false)) (fun s ↦ F (unshuffle s 2)) (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: indepFun_pilotCount_estimationStatistic
lemma indepFun_pilotCount_estimationStatistic (n d : ℕ) (P : FullLaw d)
(j : Cell d) (F : FiniteSample (ObsRecord d) → ℝ) (hF : Measurable F) :
IndepFun (fun s ↦ eventCount s (streamEvent 1 j true false))
(fun s ↦ F (unshuffle s 2))
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
classical
let C : FiniteSample (ObsRecord d) → ℕ := fun s ↦
eventCount s (obsStreamEvent j true false)
have hC : Measurable C :=
measurable_eventCount _ (Set.Finite.measurableSet (Set.toFinite _))
have hcoords := (iIndepFun_pi (μ := fun _ : Fin 3 ↦
finitePoissonSampleLaw (P.1.map obs) (((n : ℝ≥0) / 2) * (1 / 3)))
(X := fun _ ↦ id) (fun _ ↦ measurable_id.aemeasurable)).indepFun
(show (1 : Fin 3) ≠ 2 by decide)
have hind := hcoords.comp hC hF
rw [← map_unshuffle_markedObsLaw P ((n : ℝ≥0) / 2)] at hind
rw [indepFun_iff_measure_inter_preimage_eq_mul] at hind ⊢
intro a b ha hb
have h := hind a b ha hb
simp only [id_eq] at h
rw [Measure.map_apply measurable_unshuffle
((ha.preimage (hC.comp (measurable_pi_apply 1))).inter
(hb.preimage (hF.comp (measurable_pi_apply 2)))),
Measure.map_apply measurable_unshuffle (ha.preimage (hC.comp (measurable_pi_apply 1))),
Measure.map_apply measurable_unshuffle (hb.preimage (hF.comp (measurable_pi_apply 2)))] at h
simpa only [Set.preimage_inter, Set.preimage_preimage, Function.comp_def,
C, eventCount_unshuffle] using h
theorem integral_pilot_gated_estimationStatistic
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, (if (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4 then F (unshuffle s 2) else 0) ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
= (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real {s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} * (∫ s, F (unshuffle s 2) ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: integral_pilot_gated_estimationStatistic
lemma integral_pilot_gated_estimationStatistic (n d : ℕ) (q : ℝ)
(P : FullLaw d) (j : Cell d) (F : FiniteSample (ObsRecord d) → ℝ)
(hF : Measurable F) :
(∫ s, (if (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4
then F (unshuffle s 2) else 0)
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) =
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} *
(∫ s, F (unshuffle s 2)
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
classical
let g : ℕ → ℝ := fun c ↦ if (c : ℝ) ≤ needleRadius n q / 4 then 1 else 0
have hg : Measurable g := measurable_of_countable _
have hi := (indepFun_pilotCount_estimationStatistic n d P j F hF).comp hg measurable_id
have hcount := measurable_eventCount (streamEvent 1 j true false)
(Set.Finite.measurableSet (Set.toFinite _))
have hstat : Measurable (fun s : FiniteSample (ObsRecord d × Fin 3) ↦ F (unshuffle s 2)) :=
hF.comp ((measurable_pi_apply (2 : Fin 3)).comp measurable_unshuffle)
have h := hi.integral_fun_mul_eq_mul_integral
(hg.comp hcount).aestronglyMeasurable hstat.aestronglyMeasurable
simp only [Function.comp_def, id_eq, g, ite_mul, one_mul, zero_mul] at h
rw [h]
congr 1
have hs : MeasurableSet {s : FiniteSample (ObsRecord d × Fin 3) |
(eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} :=
((measurable_of_countable (fun c : ℕ ↦ (c : ℝ))).comp hcount) measurableSet_Iic
simpa only [indicator, Set.mem_ofPred_eq, Pi.one_apply] using
(integral_indicator_one (μ := finitePoissonSampleLaw (markedObsLaw P)
((n : ℝ≥0) / 2)) hs)
theorem integral_poissonSelectedCellEstimate_eq_pilot_mixture
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, poissonSelectedCellEstimate n d q j s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
= (∫ s, poissonRatioBranch j s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
+ (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real {s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} * ((∫ s, poissonFactorialBranch n d q j s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) - (∫ s, poissonRatioBranch j s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)))
Proof (Lean source)
-- @node: integral_poissonSelectedCellEstimate_eq_pilot_mixture
lemma integral_poissonSelectedCellEstimate_eq_pilot_mixture
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d)
(hb : needleBranch n d q) :
(∫ s, poissonSelectedCellEstimate n d q j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) =
(∫ s, poissonRatioBranch j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) +
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} *
((∫ s, poissonFactorialBranch n d q j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) -
(∫ s, poissonRatioBranch j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))) := by
classical
let H : FiniteSample (ObsRecord d) → ℝ := fun s ↦
if needleBranch n d q then
∑ v ∈ Icc 1 (needleDegree n q - 1), needleCoeff n d q v *
weightedFactorial (obsStreamEvent j true true) (obsStreamEvent j true false) v s
else 0
let D : FiniteSample (ObsRecord d) → ℝ := fun s ↦
if 0 < eventCount s (obsStreamEvent j true false) then
eventCount s (obsStreamEvent j true true) /
eventCount s (obsStreamEvent j true false) else 0
have hH : Measurable H := by
simp only [H, if_pos hb]
apply Finset.measurable_sum
intro v hv
exact (measurable_weightedFactorial _ _
(Set.Finite.measurableSet (Set.toFinite _))
(Set.Finite.measurableSet (Set.toFinite _)) v).const_mul _
have hc (a o : Bool) : Measurable (fun s : FiniteSample (ObsRecord d) ↦
(eventCount s (obsStreamEvent j a o) : ℝ)) :=
(measurable_of_countable (fun k : ℕ ↦ (k : ℝ))).comp
(measurable_eventCount _ (Set.Finite.measurableSet (Set.toFinite _)))
have hD : Measurable D := by
apply Measurable.ite
· exact (measurable_eventCount _ (Set.Finite.measurableSet (Set.toFinite _)))
measurableSet_Ioi
· exact (hc true true).div (hc true false)
· fun_prop
have hDs (s : FiniteSample (ObsRecord d × Fin 3)) :
D (unshuffle s 2) = poissonRatioBranch j s := by
simp only [D, poissonRatioBranch, eventCount_unshuffle]
have hHs (s : FiniteSample (ObsRecord d × Fin 3)) :
H (unshuffle s 2) = poissonFactorialBranch n d q j s := by
simp only [H, poissonFactorialBranch, weightedFactorial, eventCount_unshuffle]
have hgate := integral_pilot_gated_estimationStatistic n d q P j
(fun s ↦ H s - D s) (hH.sub hD)
simp only [hHs, hDs] at hgate
have hDint := (memLp_poissonRatioBranch n d P j).integrable (by norm_num)
have hHint := (memLp_poissonFactorialBranch n d q P j).integrable (by norm_num)
have hCint := (memLp_poissonSelectedCellEstimate_sub_ratio n d q P j).integrable
(by norm_num)
have hdecomp : (fun s ↦ poissonSelectedCellEstimate n d q j s) =
(fun s ↦ poissonRatioBranch j s +
(poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) := by
funext s
ring
rw [hdecomp, integral_add hDint hCint]
have hcorr : (fun s ↦ poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s) =
(fun s ↦ if (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4
then poissonFactorialBranch n d q j s - poissonRatioBranch j s else 0) := by
funext s
simp only [poissonSelectedCellEstimate, hb, true_and]
split_ifs <;> simp
rw [hcorr, hgate, integral_sub hHint hDint]
theorem markedPoissonSelectedCellEstimate_bias
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, poissonSelectedCellEstimate n d q j s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
- cellMean P j
= -cellMean P j * ((finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real {s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} * (chebNeedle n d q).eval (streamSize n * arrivedCell P j) + (1 - (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real {s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4}) * exp (-(streamSize n * arrivedCell P j)))
Proof (Lean source)
-- @node: markedPoissonSelectedCellEstimate_bias
lemma markedPoissonSelectedCellEstimate_bias
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d)
(hb : needleBranch n d q) :
(∫ s, poissonSelectedCellEstimate n d q j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) - cellMean P j =
-cellMean P j *
((finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} *
(chebNeedle n d q).eval (streamSize n * arrivedCell P j) +
(1 - (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4}) *
exp (-(streamSize n * arrivedCell P j))) := by
rw [integral_poissonSelectedCellEstimate_eq_pilot_mixture n d q P j hb,
integral_markedPoissonFactorialBranch_eq_needle, integral_markedPoissonRatioBranch]
rw [neg_mul]
ring
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.PilotBiasBounds 3 declarations Complementary pilot probabilities and the signed false-heavy bias contribution in roadmap (19)--(20), together with the heavy-cell pilot bound for (24).
Complementary pilot probabilities and the signed false-heavy bias contribution in roadmap (19)--(20), together with the heavy-cell pilot bound for (24).
theorem markedPoisson_pilot_heavy_probability_eq_one_sub_light
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real {s | needleRadius n q / 4 < (eventCount s (streamEvent 1 j true false) : ℝ)}
= 1
- (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real {s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4}
Proof (Lean source)
-- @node: markedPoisson_pilot_heavy_probability_eq_one_sub_light
lemma markedPoisson_pilot_heavy_probability_eq_one_sub_light
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d) :
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | needleRadius n q / 4 < (eventCount s (streamEvent 1 j true false) : ℝ)} =
1 - (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} := by
have hs : MeasurableSet {s : FiniteSample (ObsRecord d × Fin 3) |
(eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} :=
((measurable_of_countable (fun c : ℕ ↦ (c : ℝ))).comp
(measurable_eventCount _ (Set.Finite.measurableSet (Set.toFinite _))))
measurableSet_Iic
have heq : {s : FiniteSample (ObsRecord d × Fin 3) |
needleRadius n q / 4 < (eventCount s (streamEvent 1 j true false) : ℝ)} =
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4}ᶜ := by
ext s
simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_le]
rw [heq, measureReal_compl hs, probReal_univ]
theorem abs_sum_pilot_heavy_exp_bias_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
|2 * ∑ j : Cell d, armSign j.1 * cellProb P j * cellMean P j * (1 - (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real {s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4}) * exp (-(streamSize n * arrivedCell P j))|
Proof (Lean source)
-- @node: abs_sum_pilot_heavy_exp_bias_le
lemma abs_sum_pilot_heavy_exp_bias_le
(n d : ℕ) (q : ℝ) (P : FullLaw d) (hb : needleBranch n d q) :
|2 * ∑ j : Cell d, armSign j.1 * cellProb P j * cellMean P j *
(1 - (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4}) *
exp (-(streamSize n * arrivedCell P j))| ≤
2 * exp (-16 * logScale n q) := by
classical
have hj (j : Cell d) :
|armSign j.1 * cellProb P j * cellMean P j *
(1 - (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4}) *
exp (-(streamSize n * arrivedCell P j))| ≤
cellProb P j * exp (-16 * logScale n q) := by
rw [← markedPoisson_pilot_heavy_probability_eq_one_sub_light]
have hp : 0 ≤ cellProb P j := measureReal_nonneg
have hmu := cellMean_mem_Icc P j
have hsign : |armSign j.1| = 1 := by cases j.1 <;> simp [armSign]
rw [abs_mul, abs_mul, abs_mul, abs_mul, hsign, one_mul,
abs_of_nonneg hp, abs_of_nonneg hmu.1,
abs_of_nonneg measureReal_nonneg, abs_of_pos (Real.exp_pos _)]
calc
_ = cellProb P j * cellMean P j *
((finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | needleRadius n q / 4 < (eventCount s (streamEvent 1 j true false) : ℝ)} *
exp (-(streamSize n * arrivedCell P j))) := by ring
_ ≤ cellProb P j * 1 * exp (-16 * logScale n q) := by
gcongr
· exact hmu.2
· exact markedPoisson_pilot_heavy_mul_exp_le n d q P j hb
_ = _ := by ring
rw [abs_mul, abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 2)]
calc
_ ≤ 2 * ∑ j : Cell d, |armSign j.1 * cellProb P j * cellMean P j *
(1 - (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4}) *
exp (-(streamSize n * arrivedCell P j))| := by
gcongr
exact abs_sum_le_sum_abs _ _
_ ≤ 2 * ∑ j : Cell d, cellProb P j * exp (-16 * logScale n q) := by
gcongr with j
exact hj j
_ = _ := by rw [← Finset.sum_mul, sum_cellProb_eq_one, one_mul]
theorem markedPoisson_pilot_light_probability_le_exp_of_heavy
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real {s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4}
Proof (Lean source)
-- @node: markedPoisson_pilot_light_probability_le_exp_of_heavy
lemma markedPoisson_pilot_light_probability_le_exp_of_heavy
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d)
(hb : needleBranch n d q)
(hh : needleRadius n q < streamSize n * arrivedCell P j) :
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} ≤
exp (-32 * logScale n q) := by
apply (markedPoisson_pilot_light_probability_le_of_heavy n d q P j hb hh).trans
apply Real.exp_le_exp.mpr
have hl := hb.1
change 256 * logScale n q < streamSize n * arrivedCell P j at hh
linarith
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.PilotCorrection 15 declarations Square integrability, independent membership weighting, and the light-cell correction bounds in roadmap (22)--(23).
Square integrability, independent membership weighting, and the light-cell correction bounds in roadmap (22)--(23).
theorem memLp_poissonFactorialBranch
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: memLp_poissonFactorialBranch
lemma memLp_poissonFactorialBranch (n d : ℕ) (q : ℝ) (P : FullLaw d)
(j : Cell d) :
MemLp (poissonFactorialBranch n d q j) 2
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
classical
unfold poissonFactorialBranch
split_ifs
· apply memLp_finsetSum
intro v hv
have hA : MeasurableSet (streamEvent 2 j true true) :=
Set.Finite.measurableSet (Set.toFinite _)
have hB : MeasurableSet (streamEvent 2 j true false) :=
Set.Finite.measurableSet (Set.toFinite _)
have hmem := (memLp_two_iff_integrable_sq
(measurable_weightedFactorial _ _ hA hB v).aestronglyMeasurable).2
(integrable_sq_weightedFactorial (markedObsLaw P) ((n : ℝ≥0) / 2)
_ _ hA hB (streamEvent_ones_subset_arrived 2 j) v
(Finset.mem_Icc.mp hv).1)
exact hmem.const_mul _
· simp
theorem poissonSelectedCellEstimate_eq_ratio_add_correction
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
poissonSelectedCellEstimate n d q j s
= poissonRatioBranch j s
+ (if needleBranch n d q ∧ eventCount s (streamEvent 1 j true false) ≤ needleRadius n q / 4 then poissonFactorialBranch n d q j s - poissonRatioBranch j s else 0)
Proof (Lean source)
-- @node: poissonSelectedCellEstimate_eq_ratio_add_correction
lemma poissonSelectedCellEstimate_eq_ratio_add_correction (n d : ℕ) (q : ℝ)
(j : Cell d) (s : FiniteSample (ObsRecord d × Fin 3)) :
poissonSelectedCellEstimate n d q j s = poissonRatioBranch j s +
(if needleBranch n d q ∧
eventCount s (streamEvent 1 j true false) ≤ needleRadius n q / 4
then poissonFactorialBranch n d q j s - poissonRatioBranch j s else 0) := by
unfold poissonSelectedCellEstimate
split_ifs <;> ring
theorem norm_poissonSelectedCellEstimate_sub_ratio_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
‖poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s‖
≤ ‖poissonFactorialBranch n d q j s - poissonRatioBranch j s‖
Proof (Lean source)
-- @node: norm_poissonSelectedCellEstimate_sub_ratio_le
lemma norm_poissonSelectedCellEstimate_sub_ratio_le (n d : ℕ) (q : ℝ)
(j : Cell d) (s : FiniteSample (ObsRecord d × Fin 3)) :
‖poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s‖ ≤
‖poissonFactorialBranch n d q j s - poissonRatioBranch j s‖ := by
unfold poissonSelectedCellEstimate
split_ifs
· exact le_rfl
· simp only [sub_self, norm_zero]
exact norm_nonneg _
theorem memLp_poissonSelectedCellEstimate_sub_ratio
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
MemLp (fun s ↦ poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s) 2 (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: memLp_poissonSelectedCellEstimate_sub_ratio
lemma memLp_poissonSelectedCellEstimate_sub_ratio (n d : ℕ) (q : ℝ)
(P : FullLaw d) (j : Cell d) :
MemLp (fun s ↦ poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s) 2
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
apply ((memLp_poissonFactorialBranch n d q P j).sub
(memLp_poissonRatioBranch n d P j)).of_le
· exact ((measurable_poissonSelectedCellEstimate n d q j).sub
(measurable_poissonRatioBranch j)).aestronglyMeasurable
· filter_upwards [] with s
exact norm_poissonSelectedCellEstimate_sub_ratio_le n d q j s
theorem integral_sq_poissonSelectedCellEstimate_sub_ratio_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, (poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
≤ 2 * ((∫ s, (poissonFactorialBranch n d q j s) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) + 1)
Proof (Lean source)
-- @node: integral_sq_poissonSelectedCellEstimate_sub_ratio_le
lemma integral_sq_poissonSelectedCellEstimate_sub_ratio_le
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d) :
(∫ s, (poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
2 * ((∫ s, (poissonFactorialBranch n d q j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) + 1) := by
have hH := (memLp_two_iff_integrable_sq
(measurable_poissonFactorialBranch n d q j).aestronglyMeasurable).1
(memLp_poissonFactorialBranch n d q P j)
have hC := (memLp_two_iff_integrable_sq
((measurable_poissonSelectedCellEstimate n d q j).sub
(measurable_poissonRatioBranch j)).aestronglyMeasurable).1
(memLp_poissonSelectedCellEstimate_sub_ratio n d q P j)
calc
_ ≤ ∫ s, 2 * ((poissonFactorialBranch n d q j s) ^ 2 + 1)
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2) := by
apply integral_mono hC ((hH.add (integrable_const 1)).const_mul 2)
intro s
have hD := poissonRatioBranch_mem_unitInterval j s
have hDs : (poissonRatioBranch j s) ^ 2 ≤ 1 := by
nlinarith [hD.1, hD.2]
dsimp only [Pi.sub_apply, Pi.add_apply]
unfold poissonSelectedCellEstimate
split_ifs
· have hsq := add_sq_le (a := poissonFactorialBranch n d q j s)
(b := -poissonRatioBranch j s)
simp only [neg_sq, ← sub_eq_add_neg] at hsq
exact hsq.trans (by linarith)
· simp only [sub_self, zero_pow (by decide : 2 ≠ 0)]
positivity
_ = _ := by rw [integral_const_mul, integral_add hH (integrable_const 1)]; simp
theorem integral_sq_poissonSelectedCellEstimate_sub_ratio_le_of_light
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, (poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: integral_sq_poissonSelectedCellEstimate_sub_ratio_le_of_light
lemma integral_sq_poissonSelectedCellEstimate_sub_ratio_le_of_light
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d)
(hbranch : needleBranch n d q)
(hlight : streamSize n * arrivedCell P j ≤ needleRadius n q) :
(∫ s, (poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
2 * (exp (logScale n q / 8) + 1) := by
apply (integral_sq_poissonSelectedCellEstimate_sub_ratio_le n d q P j).trans
gcongr
exact integral_sq_markedPoissonFactorialBranch_le_exp_of_light
n d q P j hbranch hlight
theorem indepFun_poissonMemberWeight_pilot_estimation
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
IndepFun (poissonMemberWeight n j) (fun s ↦ F (unshuffle s 1, unshuffle s 2)) (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: indepFun_poissonMemberWeight_pilot_estimation
lemma indepFun_poissonMemberWeight_pilot_estimation (n d : ℕ) (P : FullLaw d)
(j : Cell d)
(F : FiniteSample (ObsRecord d) × FiniteSample (ObsRecord d) → ℝ)
(hF : Measurable F) :
IndepFun (poissonMemberWeight n j)
(fun s ↦ F (unshuffle s 1, unshuffle s 2))
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
classical
let W : FiniteSample (ObsRecord d) → ℝ := fun s ↦
eventCount s (obsStreamEvent j false false) / streamSize n
have hW : Measurable W :=
((measurable_of_countable (fun k : ℕ ↦ (k : ℝ))).comp
(measurable_eventCount _ (Set.Finite.measurableSet (Set.toFinite _)))).div_const _
have hcoords := (iIndepFun_pi (μ := fun _ : Fin 3 ↦
finitePoissonSampleLaw (P.1.map obs) (((n : ℝ≥0) / 2) * (1 / 3)))
(X := fun _ ↦ id) (fun _ ↦ measurable_id.aemeasurable)).indepFun_prodMk
(fun _ ↦ measurable_pi_apply _) 1 2 0 (by decide) (by decide)
have hind := hcoords.symm.comp hW hF
rw [← map_unshuffle_markedObsLaw P ((n : ℝ≥0) / 2)] at hind
have hWs (s : FiniteSample (ObsRecord d × Fin 3)) :
W (unshuffle s 0) = poissonMemberWeight n j s := by
simp only [W, poissonMemberWeight, eventCount_unshuffle]
rw [indepFun_iff_measure_inter_preimage_eq_mul] at hind ⊢
intro a b ha hb
have h := hind a b ha hb
simp only [id_eq] at h
have hpair : Measurable (fun s : Fin 3 → FiniteSample (ObsRecord d) ↦
(s 1, s 2)) := (measurable_pi_apply 1).prodMk (measurable_pi_apply 2)
rw [Measure.map_apply measurable_unshuffle
((ha.preimage (hW.comp (measurable_pi_apply 0))).inter (hb.preimage (hF.comp hpair))),
Measure.map_apply measurable_unshuffle (ha.preimage (hW.comp (measurable_pi_apply 0))),
Measure.map_apply measurable_unshuffle (hb.preimage (hF.comp hpair))] at h
simpa only [Set.preimage_inter, Set.preimage_preimage, Function.comp_def, hWs] using h
theorem indepFun_poissonMemberWeight_selectedCorrection
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
IndepFun (poissonMemberWeight n j) (fun s ↦ poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s) (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: indepFun_poissonMemberWeight_selectedCorrection
lemma indepFun_poissonMemberWeight_selectedCorrection (n d : ℕ) (q : ℝ)
(P : FullLaw d) (j : Cell d) :
IndepFun (poissonMemberWeight n j)
(fun s ↦ poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
classical
let H : FiniteSample (ObsRecord d) → ℝ := fun s ↦
if needleBranch n d q then
∑ v ∈ Icc 1 (needleDegree n q - 1), needleCoeff n d q v *
weightedFactorial (obsStreamEvent j true true) (obsStreamEvent j true false) v s
else 0
let D : FiniteSample (ObsRecord d) → ℝ := fun s ↦
if 0 < eventCount s (obsStreamEvent j true false) then
eventCount s (obsStreamEvent j true true) /
eventCount s (obsStreamEvent j true false) else 0
have hH : Measurable H := by
dsimp [H]
split_ifs
· apply Finset.measurable_sum
intro v hv
exact (measurable_weightedFactorial _ _
(Set.Finite.measurableSet (Set.toFinite _))
(Set.Finite.measurableSet (Set.toFinite _)) v).const_mul _
· fun_prop
have hc (a o : Bool) : Measurable (fun s : FiniteSample (ObsRecord d) ↦
(eventCount s (obsStreamEvent j a o) : ℝ)) :=
(measurable_of_countable (fun k : ℕ ↦ (k : ℝ))).comp
(measurable_eventCount _ (Set.Finite.measurableSet (Set.toFinite _)))
have hD : Measurable D := by
apply Measurable.ite
· exact (measurable_eventCount _ (Set.Finite.measurableSet (Set.toFinite _)))
measurableSet_Ioi
· exact (hc true true).div (hc true false)
· fun_prop
let F : FiniteSample (ObsRecord d) × FiniteSample (ObsRecord d) → ℝ := fun z ↦
(if needleBranch n d q ∧
eventCount z.1 (obsStreamEvent j true false) ≤ needleRadius n q / 4
then H z.2 else D z.2) - D z.2
have hF : Measurable F := by
apply Measurable.sub _ (hD.comp measurable_snd)
apply Measurable.ite
· by_cases hb : needleBranch n d q
· simp only [hb, true_and]
exact ((hc true false).comp measurable_fst) measurableSet_Iic
· simp [hb]
· exact hH.comp measurable_snd
· exact hD.comp measurable_snd
have hDs (s : FiniteSample (ObsRecord d × Fin 3)) :
D (unshuffle s 2) = poissonRatioBranch j s := by
simp only [D, poissonRatioBranch, eventCount_unshuffle]
have hHs (s : FiniteSample (ObsRecord d × Fin 3)) :
H (unshuffle s 2) = poissonFactorialBranch n d q j s := by
simp only [H, poissonFactorialBranch, weightedFactorial, eventCount_unshuffle]
have hFs : (fun s ↦ F (unshuffle s 1, unshuffle s 2)) =
(fun s ↦ poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s) := by
funext s
simp only [F, eventCount_unshuffle, hDs, hHs, poissonSelectedCellEstimate]
rw [← hFs]
exact indepFun_poissonMemberWeight_pilot_estimation n d P j F hF
theorem memLp_poissonMemberWeight_mul_selectedCorrection
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
MemLp (fun s ↦ poissonMemberWeight n j s * (poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) 2 (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: memLp_poissonMemberWeight_mul_selectedCorrection
lemma memLp_poissonMemberWeight_mul_selectedCorrection
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d) :
MemLp (fun s ↦ poissonMemberWeight n j s *
(poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) 2
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
have hW := (memLp_two_iff_integrable_sq
(measurable_poissonMemberWeight n d j).aestronglyMeasurable).1
(memLp_poissonMemberWeight n d P j)
have hC := (memLp_two_iff_integrable_sq
((measurable_poissonSelectedCellEstimate n d q j).sub
(measurable_poissonRatioBranch j)).aestronglyMeasurable).1
(memLp_poissonSelectedCellEstimate_sub_ratio n d q P j)
have hi := (indepFun_poissonMemberWeight_selectedCorrection n d q P j).comp
(measurable_id.pow_const 2) (measurable_id.pow_const 2)
have h := hi.integrable_mul hW hC
change Integrable (fun s ↦ (poissonMemberWeight n j s) ^ 2 *
(poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s) ^ 2) _ at h
apply (memLp_two_iff_integrable_sq
((measurable_poissonMemberWeight n d j).mul
((measurable_poissonSelectedCellEstimate n d q j).sub
(measurable_poissonRatioBranch j))).aestronglyMeasurable).2
simpa only [Pi.mul_apply, Pi.sub_apply, mul_pow] using h
theorem integral_sq_poissonMemberWeight_mul_selectedCorrection
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, (poissonMemberWeight n j s * (poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
= ((cellProb P j) ^ 2 + cellProb P j / streamSize n) * (∫ s, (poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: integral_sq_poissonMemberWeight_mul_selectedCorrection
lemma integral_sq_poissonMemberWeight_mul_selectedCorrection
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d) (hn : 0 < n) :
(∫ s, (poissonMemberWeight n j s *
(poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) =
((cellProb P j) ^ 2 + cellProb P j / streamSize n) *
(∫ s, (poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
have hi := (indepFun_poissonMemberWeight_selectedCorrection n d q P j).comp
(measurable_id.pow_const 2) (measurable_id.pow_const 2)
have h := hi.integral_fun_mul_eq_mul_integral
((measurable_poissonMemberWeight n d j).pow_const 2).aestronglyMeasurable
(((measurable_poissonSelectedCellEstimate n d q j).sub
(measurable_poissonRatioBranch j)).pow_const 2).aestronglyMeasurable
simp only [Function.comp_def, id_eq] at h
simp_rw [mul_pow]
rw [h, integral_sq_poissonMemberWeight n d P j hn]
theorem integral_sq_poissonMemberWeight_mul_selectedCorrection_le_of_light
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, (poissonMemberWeight n j s * (poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: integral_sq_poissonMemberWeight_mul_selectedCorrection_le_of_light
lemma integral_sq_poissonMemberWeight_mul_selectedCorrection_le_of_light
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d) (hn : 0 < n)
(hbranch : needleBranch n d q)
(hlight : streamSize n * arrivedCell P j ≤ needleRadius n q) :
(∫ s, (poissonMemberWeight n j s *
(poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
2 * ((cellProb P j) ^ 2 + cellProb P j / streamSize n) *
(exp (logScale n q / 8) + 1) := by
rw [integral_sq_poissonMemberWeight_mul_selectedCorrection n d q P j hn]
calc
_ ≤ ((cellProb P j) ^ 2 + cellProb P j / streamSize n) *
(2 * (exp (logScale n q / 8) + 1)) :=
mul_le_mul_of_nonneg_left
(integral_sq_poissonSelectedCellEstimate_sub_ratio_le_of_light
n d q P j hbranch hlight)
(add_nonneg (sq_nonneg _) (div_nonneg measureReal_nonneg (by
unfold streamSize; positivity)))
_ = _ := by ring
theorem sum_light_integral_sq_poissonMemberWeight_mul_selectedCorrection_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
8 * (∑ j ∈ (Finset.univ : Finset (Cell d)).filter (fun j ↦ streamSize n * arrivedCell P j ≤ needleRadius n q), ∫ s, (poissonMemberWeight n j s * (poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
≤ 64 * d * (exp (logScale n q / 8) + 1) * (needleRadius n q ^ 2 / streamEffectiveSize n q ^ 2 + needleRadius n q / (streamSize n * streamEffectiveSize n q))
Proof (Lean source)
-- @node: sum_light_integral_sq_poissonMemberWeight_mul_selectedCorrection_le
lemma sum_light_integral_sq_poissonMemberWeight_mul_selectedCorrection_le
(n d : ℕ) (q : ℝ) (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P)
(hbranch : needleBranch n d q) :
8 * (∑ j ∈ (Finset.univ : Finset (Cell d)).filter
(fun j ↦ streamSize n * arrivedCell P j ≤ needleRadius n q),
∫ s, (poissonMemberWeight n j s *
(poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
64 * d * (exp (logScale n q / 8) + 1) *
(needleRadius n q ^ 2 / streamEffectiveSize n q ^ 2 +
needleRadius n q / (streamSize n * streamEffectiveSize n q)) := by
classical
have hn : 0 < n := lt_of_lt_of_le Nat.zero_lt_one hP.n_pos
have hm : 0 < streamSize n := by unfold streamSize; positivity
have hN : 0 < streamEffectiveSize n q := by
unfold streamEffectiveSize
exact mul_pos hm hP.q_pos
have hB : 0 ≤ needleRadius n q := by
unfold needleRadius
linarith [hbranch.1]
let c : ℝ := 2 * (needleRadius n q ^ 2 / streamEffectiveSize n q ^ 2 +
needleRadius n q / (streamSize n * streamEffectiveSize n q)) *
(exp (logScale n q / 8) + 1)
have hc : 0 ≤ c := by dsimp [c]; positivity
have hj (j : Cell d) (hlight : streamSize n * arrivedCell P j ≤ needleRadius n q) :
(∫ s, (poissonMemberWeight n j s *
(poissonSelectedCellEstimate n d q j s - poissonRatioBranch j s)) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤ c := by
have hp0 : 0 ≤ cellProb P j := measureReal_nonneg
have hp : cellProb P j ≤ needleRadius n q / streamEffectiveSize n q := by
apply (le_div_iff₀ hN).2
have hi := streamEffectiveSize_mul_cellProb_le_arrivedStreamIntensity
n d q P hP 2 j
rw [coe_arrivedStreamIntensity] at hi
simpa only [mul_comm] using hi.trans hlight
have hmass : (cellProb P j) ^ 2 + cellProb P j / streamSize n ≤
needleRadius n q ^ 2 / streamEffectiveSize n q ^ 2 +
needleRadius n q / (streamSize n * streamEffectiveSize n q) := by
calc
_ ≤ (needleRadius n q / streamEffectiveSize n q) ^ 2 +
(needleRadius n q / streamEffectiveSize n q) / streamSize n := by
gcongr
_ = _ := by rw [div_pow]; field_simp
exact (integral_sq_poissonMemberWeight_mul_selectedCorrection_le_of_light
n d q P j hn hbranch hlight).trans
(mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left hmass (by norm_num)) (by positivity))
calc
_ ≤ 8 * ∑ _j : Cell d, c := by
rw [Finset.sum_filter]
apply mul_le_mul_of_nonneg_left _ (by norm_num)
apply Finset.sum_le_sum
intro j hjmem
split_ifs with hlight
· exact hj j hlight
· exact hc
_ = _ := by simp [c]; ring
theorem cheb_needle_intensity_abs_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
t * |(chebNeedle n d q).eval t| ≤ needleRadius n q / (needleDegree n q : ℝ)^2
Proof (Lean source)
-- @node: cheb_needle_intensity_abs_le
lemma cheb_needle_intensity_abs_le (n d : ℕ) (q t : ℝ)
(hbranch : needleBranch n d q) (ht : t ∈ Icc 0 (needleRadius n q)) :
t * |(chebNeedle n d q).eval t| ≤
needleRadius n q / (needleDegree n q : ℝ)^2 := by
have hB : 0 < needleRadius n q := by
unfold needleRadius
linarith [hbranch.1]
have hk : 0 < needleDegree n q := by
unfold needleDegree
apply Nat.floor_pos.mpr
linarith [hbranch.1]
have hk2 : 0 < (needleDegree n q : ℝ)^2 := by positivity
let p : Polynomial ℝ := 1 -
(Chebyshev.T ℝ (needleDegree n q : ℤ)).comp
(C 1 - C (2 / needleRadius n q) * X)
have hp0 : p.coeff 0 = 0 := by
simp [p, coeff_zero_eq_eval_zero, Chebyshev.T_eval_one]
have hdiv : p.divX.eval t * t = p.eval t := by
have h := congrArg (eval t) (divX_mul_X_add p)
simpa [hp0] using h
have hprod : (chebNeedle n d q).eval t * t =
(needleRadius n q / (2 * (needleDegree n q : ℝ)^2)) *
(1 - (Chebyshev.T ℝ (needleDegree n q : ℤ)).eval
(1 - 2 * t / needleRadius n q)) := by
simp only [chebNeedle, if_pos hbranch, eval_mul, eval_C]
rw [mul_assoc, hdiv]
simp only [p, eval_sub, eval_one, eval_comp, eval_mul, eval_C, eval_X]
congr 3
ring
have hx : |1 - 2 * t / needleRadius n q| ≤ 1 := by
apply abs_le.mpr
have hratio : 0 ≤ 2 * t / needleRadius n q :=
div_nonneg (mul_nonneg (by norm_num) ht.1) hB.le
have hu : 2 * t / needleRadius n q ≤ 2 :=
(div_le_iff₀ hB).2 (by linarith [ht.2])
constructor <;> linarith
have hT := Chebyshev.abs_eval_T_real_le_one (needleDegree n q : ℤ) hx
have hab : |1 - (Chebyshev.T ℝ (needleDegree n q : ℤ)).eval
(1 - 2 * t / needleRadius n q)| ≤ 2 := by
apply abs_le.mpr
constructor <;> linarith [(abs_le.mp hT).1, (abs_le.mp hT).2]
calc
_ = |(chebNeedle n d q).eval t * t| := by
rw [abs_mul, abs_of_nonneg ht.1, mul_comm]
_ = (needleRadius n q / (2 * (needleDegree n q : ℝ)^2)) *
|1 - (Chebyshev.T ℝ (needleDegree n q : ℤ)).eval
(1 - 2 * t / needleRadius n q)| := by
rw [hprod, abs_mul, abs_of_nonneg (by positivity)]
_ ≤ (needleRadius n q / (2 * (needleDegree n q : ℝ)^2)) * 2 :=
mul_le_mul_of_nonneg_left hab (by positivity)
_ = _ := by field_simp
theorem cellProb_mul_cellMean_abs_needle_le_of_light
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
P :
FullLaw d
hP :
UnrestrictedArrivalModelClass n d q P
j :
Cell d
hbranch :
needleBranch n d q
hlight :
conclusion
≤ needleRadius n q / (streamEffectiveSize n q * (needleDegree n q : ℝ)^2)
Proof (Lean source)
-- @node: cellProb_mul_cellMean_abs_needle_le_of_light
lemma cellProb_mul_cellMean_abs_needle_le_of_light
(n d : ℕ) (q : ℝ) (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) (j : Cell d)
(hbranch : needleBranch n d q)
(hlight : streamSize n * arrivedCell P j ≤ needleRadius n q) :
cellProb P j * cellMean P j *
|(chebNeedle n d q).eval (streamSize n * arrivedCell P j)| ≤
needleRadius n q /
(streamEffectiveSize n q * (needleDegree n q : ℝ)^2) := by
have hm : 0 < streamSize n := by
unfold streamSize
have hn : 0 < (n : ℝ) := by
exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hP.n_pos)
positivity
have hN : 0 < streamEffectiveSize n q := by
unfold streamEffectiveSize
exact mul_pos hm hP.q_pos
have ht : 0 ≤ streamSize n * arrivedCell P j :=
mul_nonneg hm.le measureReal_nonneg
have hi := streamEffectiveSize_mul_cellProb_le_arrivedStreamIntensity
n d q P hP 2 j
rw [coe_arrivedStreamIntensity] at hi
have hneedle := cheb_needle_intensity_abs_le n d q
(streamSize n * arrivedCell P j) hbranch ⟨ht, hlight⟩
have hmu := cellMean_mem_Icc P j
have hp : 0 ≤ cellProb P j := measureReal_nonneg
have hmass : streamEffectiveSize n q * (cellProb P j * cellMean P j) ≤
streamSize n * arrivedCell P j := by
calc
_ ≤ streamEffectiveSize n q * (cellProb P j * 1) := by gcongr; exact hmu.2
_ ≤ _ := by simpa using hi
have h := (mul_le_mul_of_nonneg_right hmass
(abs_nonneg ((chebNeedle n d q).eval (streamSize n * arrivedCell P j)))).trans
hneedle
have hdiv : cellProb P j * cellMean P j *
|(chebNeedle n d q).eval (streamSize n * arrivedCell P j)| ≤
(needleRadius n q / (needleDegree n q : ℝ)^2) / streamEffectiveSize n q :=
(le_div_iff₀ hN).2 (by nlinarith [h])
simpa only [div_div, mul_comm] using hdiv
theorem abs_sum_light_pilot_needle_bias_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
P :
FullLaw d
hP :
UnrestrictedArrivalModelClass n d q P
hbranch :
needleBranch n d q
pilot :
Cell d → ℝ
hpilot :
∀ j, pilot j ∈ Icc (0 : ℝ) 1
conclusion
|2 * ∑ j ∈ (Finset.univ : Finset (Cell d)).filter (fun j ↦ streamSize n * arrivedCell P j ≤ needleRadius n q), armSign j.1 * cellProb P j * cellMean P j * pilot j * (chebNeedle n d q).eval (streamSize n * arrivedCell P j)|
≤ 8 * d * needleRadius n q / (streamEffectiveSize n q * (needleDegree n q : ℝ)^2)
Proof (Lean source)
-- @node: abs_sum_light_pilot_needle_bias_le
lemma abs_sum_light_pilot_needle_bias_le
(n d : ℕ) (q : ℝ) (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P)
(hbranch : needleBranch n d q) (pilot : Cell d → ℝ)
(hpilot : ∀ j, pilot j ∈ Icc (0 : ℝ) 1) :
|2 * ∑ j ∈ (Finset.univ : Finset (Cell d)).filter
(fun j ↦ streamSize n * arrivedCell P j ≤ needleRadius n q),
armSign j.1 * cellProb P j * cellMean P j * pilot j *
(chebNeedle n d q).eval (streamSize n * arrivedCell P j)| ≤
8 * d * needleRadius n q /
(streamEffectiveSize n q * (needleDegree n q : ℝ)^2) := by
classical
let c := needleRadius n q /
(streamEffectiveSize n q * (needleDegree n q : ℝ)^2)
have hc : 0 ≤ c := by
dsimp [c, needleRadius, streamEffectiveSize, streamSize]
have := hbranch.1
have := hP.q_pos
positivity
have hj (j : Cell d)
(hlight : streamSize n * arrivedCell P j ≤ needleRadius n q) :
|armSign j.1 * cellProb P j * cellMean P j * pilot j *
(chebNeedle n d q).eval (streamSize n * arrivedCell P j)| ≤ c := by
have hsign : |armSign j.1| = 1 := by cases j.1 <;> simp [armSign]
have hp : 0 ≤ cellProb P j := measureReal_nonneg
have hmu := cellMean_mem_Icc P j
have hpm : 0 ≤ cellProb P j * cellMean P j := mul_nonneg hp hmu.1
rw [abs_mul, abs_mul, abs_mul, abs_mul, hsign, one_mul,
abs_of_nonneg hp,
abs_of_nonneg (cellMean_mem_Icc P j).1, abs_of_nonneg (hpilot j).1]
calc
_ ≤ cellProb P j * cellMean P j * 1 *
|(chebNeedle n d q).eval (streamSize n * arrivedCell P j)| := by
gcongr
exact (hpilot j).2
_ ≤ c := by
simpa only [mul_one] using
cellProb_mul_cellMean_abs_needle_le_of_light n d q P hP j hbranch hlight
rw [abs_mul, abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 2)]
calc
_ ≤ 2 * ∑ j ∈ (Finset.univ : Finset (Cell d)).filter
(fun j ↦ streamSize n * arrivedCell P j ≤ needleRadius n q),
|armSign j.1 * cellProb P j * cellMean P j * pilot j *
(chebNeedle n d q).eval (streamSize n * arrivedCell P j)| := by
gcongr
exact abs_sum_le_sum_abs _ _
_ ≤ 2 * ∑ _j : Cell d, c := by
rw [Finset.sum_filter]
apply mul_le_mul_of_nonneg_left _ (by norm_num)
apply Finset.sum_le_sum
intro j _
split_ifs with hlight
· exact hj j hlight
· exact hc
_ = _ := by simp [c]; ring
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.PilotTailBounds 3 declarations Pilot-stream Poisson tail terms in roadmap (12) and (19).
Pilot-stream Poisson tail terms in roadmap (12) and (19).
theorem markedPoisson_pilot_light_probability_le_of_heavy
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real {s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4}
Proof (Lean source)
-- @node: markedPoisson_pilot_light_probability_le_of_heavy
lemma markedPoisson_pilot_light_probability_le_of_heavy
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d)
(hbranch : needleBranch n d q)
(hheavy : needleRadius n q < streamSize n * arrivedCell P j) :
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | (eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} ≤
exp (-(streamSize n * arrivedCell P j) / 4) := by
have hB : 0 ≤ needleRadius n q / 4 := by
unfold needleRadius
have := hbranch.1
positivity
have hk : (⌊needleRadius n q / 4⌋₊ : ℝ) <
(arrivedStreamIntensity n P 1 j : ℝ) / 4 := by
rw [coe_arrivedStreamIntensity]
exact (Nat.floor_le hB).trans_lt (by linarith)
have htail := poisson_le_cutoff_of_cutoff_lt_quarter
(arrivedStreamIntensity n P 1 j) ⌊needleRadius n q / 4⌋₊ hk
have hmap := finitePoissonSampleLaw_map_eventCount (markedObsLaw P)
((n : ℝ≥0) / 2) (streamEvent 1 j true false)
(Set.Finite.measurableSet (Set.toFinite _))
simp only [arrivedStreamIntensity] at htail
rw [← hmap, Measure.map_apply
(measurable_eventCount _ (Set.Finite.measurableSet (Set.toFinite _)))
((Set.to_countable _).measurableSet)] at htail
have hset : {s : FiniteSample (ObsRecord d × Fin 3) |
(eventCount s (streamEvent 1 j true false) : ℝ) ≤ needleRadius n q / 4} =
(fun s ↦ eventCount s (streamEvent 1 j true false)) ⁻¹'
{w : ℕ | w ≤ ⌊needleRadius n q / 4⌋₊} := by
ext s
exact (Nat.le_floor_iff hB).symm
rw [hset, measureReal_def]
have hreal := ENNReal.toReal_mono (by finiteness) htail
change _ ≤ (ofReal
(exp (-(arrivedStreamIntensity n P 1 j : ℝ) / 4))).toReal at hreal
simpa only [ENNReal.toReal_ofReal (Real.exp_nonneg _),
coe_arrivedStreamIntensity] using hreal
theorem poisson_pilot_heavy_mul_exp_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: poisson_pilot_heavy_mul_exp_le
lemma poisson_pilot_heavy_mul_exp_le (n d : ℕ) (q : ℝ) (lam : ℝ≥0)
(hbranch : needleBranch n d q) :
(poissonMeasure lam).real {w : ℕ | needleRadius n q / 4 < (w : ℝ)} *
exp (-(lam : ℝ)) ≤ exp (-16 * logScale n q) := by
have hB : 0 ≤ needleRadius n q / 4 := by
unfold needleRadius
have := hbranch.1
positivity
have hset : {w : ℕ | needleRadius n q / 4 < (w : ℝ)} =
{w : ℕ | ⌊needleRadius n q / 4⌋₊ < w} := by
ext w
exact (Nat.floor_lt hB).symm
rw [hset]
have ht := poisson_upper_tail_mul_exp_le lam ⌊needleRadius n q / 4⌋₊
(r := 1) (by norm_num)
simp only [neg_mul, one_mul] at ht
apply ht.trans
apply Real.exp_le_exp.mpr
have hf := Nat.lt_floor_add_one (needleRadius n q / 4)
have hl := Real.log_two_gt_d9
have hk : 0 ≤ (⌊needleRadius n q / 4⌋₊ : ℝ) := Nat.cast_nonneg _
have hlog : logScale n q ≥ 128 := hbranch.1
norm_num only at ht ⊢
change 256 * logScale n q / 4 < (⌊needleRadius n q / 4⌋₊ : ℝ) + 1 at hf
nlinarith
theorem markedPoisson_pilot_heavy_mul_exp_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real {s | needleRadius n q / 4 < (eventCount s (streamEvent 1 j true false) : ℝ)} * exp (-(streamSize n * arrivedCell P j))
Proof (Lean source)
-- @node: markedPoisson_pilot_heavy_mul_exp_le
lemma markedPoisson_pilot_heavy_mul_exp_le
(n d : ℕ) (q : ℝ) (P : FullLaw d) (j : Cell d)
(hbranch : needleBranch n d q) :
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | needleRadius n q / 4 < (eventCount s (streamEvent 1 j true false) : ℝ)} *
exp (-(streamSize n * arrivedCell P j)) ≤
exp (-16 * logScale n q) := by
have hmap := finitePoissonSampleLaw_map_eventCount (markedObsLaw P)
((n : ℝ≥0) / 2) (streamEvent 1 j true false)
(Set.Finite.measurableSet (Set.toFinite _))
have hprob :
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)).real
{s | needleRadius n q / 4 < (eventCount s (streamEvent 1 j true false) : ℝ)} =
(poissonMeasure (arrivedStreamIntensity n P 1 j)).real
{w : ℕ | needleRadius n q / 4 < (w : ℝ)} := by
simp only [arrivedStreamIntensity]
rw [← hmap]
simp only [measureReal_def]
rw [Measure.map_apply
(measurable_eventCount _ (Set.Finite.measurableSet (Set.toFinite _)))
((Set.to_countable _).measurableSet)]
rfl
rw [hprob]
simpa only [coe_arrivedStreamIntensity] using
poisson_pilot_heavy_mul_exp_le n d q (arrivedStreamIntensity n P 1 j) hbranch
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.PoissonThinning 1 declarations Poisson cap and thinning bookkeeping for the three streams.
Poisson cap and thinning bookkeeping for the three streams.
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma poisson_half_prefix_tail (n : ℕ) :
(poissonMeasure ((n : ℝ≥0) / 2)).real {k : ℕ | n < k} ≤
exp (-(n : ℝ) * (log 2 - 1 / 2)) := by
have h := poisson_upper_tail_mul_exp_le
((n : ℝ≥0) / 2) n (r := 1) (by norm_num)
have hexp : 0 < exp (-((n : ℝ) / 2)) := Real.exp_pos _
apply le_of_mul_le_mul_right _ hexp
calc
(poissonMeasure ((n : ℝ≥0) / 2)).real {k : ℕ | n < k} *
exp (-((n : ℝ) / 2)) ≤ exp (-log 2 * (n : ℝ)) := by
convert h using 1 <;> norm_num [NNReal.coe_div]
_ = exp (-(n : ℝ) * (log 2 - 1 / 2)) *
exp (-((n : ℝ) / 2)) := by
rw [← Real.exp_add]
congr 1
ring
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.PrefixIdealTransfer 3 declarations Transfer from the finite prefix estimator to the independent ideal streams.
Transfer from the finite prefix estimator to the independent ideal streams.
theorem streamAuxiliaryMixedValue_labeledPrefix
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
streamAuxiliaryMixedValue n d q (labeledPrefix sample k hk assign)
= auxiliaryMixedValue n q (fun i ↦ sample (castLE hk i)) assign
Proof (Lean source)
lemma streamAuxiliaryMixedValue_labeledPrefix {n d k : ℕ}
(q : ℝ) (sample : Fin n → ObsRecord d) (hk : k ≤ n)
(assign : Fin k → Fin 3) :
streamAuxiliaryMixedValue n d q (labeledPrefix sample k hk assign) =
auxiliaryMixedValue n q (fun i ↦ sample (castLE hk i)) assign := by
unfold labeledPrefix
rw [streamAuxiliaryMixedValue_unshuffle]
rw [auxiliaryMixedValue_eq_poissonAuxiliaryMixedValue]
congr 1
theorem mixedCountEstimator_eq_fixedStatistic
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
mixedCountEstimator n d q sample
= fixedStatistic uniformPoolMass uniformPoolMass_sum ((n : ℝ≥0) / 2) sample (streamAuxiliaryMixedValue n d q) 0
Proof (Lean source)
lemma mixedCountEstimator_eq_fixedStatistic (n d : ℕ) (q : ℝ)
(sample : Fin n → ObsRecord d) :
mixedCountEstimator n d q sample =
fixedStatistic uniformPoolMass uniformPoolMass_sum ((n : ℝ≥0) / 2)
sample (streamAuxiliaryMixedValue n d q) 0 := by
rw [fixedStatistic_eq_finite_sum]
simp only [mul_zero, add_zero]
unfold mixedCountEstimator
rw [← Fin.sum_univ_eq_sum_range]
apply Fintype.sum_congr
intro k
have hkn : k.val ≤ n := le_of_lt_succ k.isLt
simp only [dif_pos hkn]
congr 1
simp only [uniformPoolMass, streamAuxiliaryMixedValue_labeledPrefix]
rw [Finset.sum_div]
apply Finset.sum_congr rfl
intro assign ha
simp [uniformPoolMass, div_eq_mul_inv]
ring
theorem deterministicRisk_mixedCountEstimator_le_streamRisk_add_tail
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
deterministicRisk (mixedCountEstimator n d q) P
≤ (∫ s, (streamAuxiliaryMixedValue n d q s - ate P) ^ 2 ∂Measure.pi (fun _ : Fin 3 ↦ finitePoissonSampleLaw (P.1.map obs) (((n : ℝ≥0) / 2) * (1 / 3))))
Proof (Lean source)
lemma deterministicRisk_mixedCountEstimator_le_streamRisk_add_tail
(n d : ℕ) (q : ℝ) (P : FullLaw d) :
deterministicRisk (mixedCountEstimator n d q) P ≤
(∫ s, (streamAuxiliaryMixedValue n d q s - ate P) ^ 2
∂Measure.pi (fun _ : Fin 3 ↦
finitePoissonSampleLaw (P.1.map obs)
(((n : ℝ≥0) / 2) * (1 / 3)))) +
4 * exp (-(n : ℝ) * (log 2 - 1 / 2)) := by
letI : IsProbabilityMeasure P.1 := P.2
letI : IsProbabilityMeasure (P.1.map obs) := map_obs_isProbabilityMeasure P
have htransfer := fixedRisk_le_independentRisk_add_tail
(P.1.map obs) uniformPoolMass uniformPoolMass_sum ((n : ℝ≥0) / 2) n
(measurable_streamAuxiliaryMixedValue n d q)
(a := (-1 : ℝ)) (b := 1) (theta := ate P) (zOver := 0)
(streamAuxiliaryMixedValue_mem_Icc n d q)
(ate_mem_Icc_of_fullLaw P) (by norm_num)
have hfun : (fun sample : Fin n → ObsRecord d ↦
fixedStatistic uniformPoolMass uniformPoolMass_sum ((n : ℝ≥0) / 2)
sample (streamAuxiliaryMixedValue n d q) 0) =
mixedCountEstimator n d q := by
funext sample
exact (mixedCountEstimator_eq_fixedStatistic n d q sample).symm
rw [hfun] at htransfer
have htransfer' : deterministicRisk (mixedCountEstimator n d q) P ≤
(∫ s, (streamAuxiliaryMixedValue n d q s - ate P) ^ 2
∂Measure.pi (fun _ : Fin 3 ↦
finitePoissonSampleLaw (P.1.map obs)
(((n : ℝ≥0) / 2) * (1 / 3)))) +
4 * (poissonMeasure ((n : ℝ≥0) / 2)).real (Ioi n) := by
norm_num at htransfer
simpa [deterministicRisk, sqRisk, fixedPoolLaw, sampleLaw, uniformPoolMass]
using htransfer
refine htransfer'.trans ?_
gcongr
exact poisson_half_prefix_tail n
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.PrefixTransfer 3 declarations Conditional averaging contracts squared risk.
Conditional averaging contracts squared risk.
theorem finite_weighted_square_with_remainder Lemma finite_weighted_square_with_remainder in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
((∑ i ∈ s, w i * a i) - c) ^ 2 ≤ (1 - ∑ i ∈ s, w i) * c ^ 2 + ∑ i ∈ s, w i * (a i - c) ^ 2
Proof (Lean source)
lemma finite_weighted_square_with_remainder {ι : Type*} (s : Finset ι)
(w a : ι → ℝ) (c : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hmass : ∑ i ∈ s, w i ≤ 1) :
((∑ i ∈ s, w i * a i) - c) ^ 2 ≤
(1 - ∑ i ∈ s, w i) * c ^ 2 +
∑ i ∈ s, w i * (a i - c) ^ 2 := by
have hc : ConvexOn ℝ univ (fun x : ℝ => x ^ 2) :=
(show Even (2 : ℕ) by decide).convexOn_pow
have hj := hc.map_add_sum_le hw
(show (1 - ∑ i ∈ s, w i) + ∑ i ∈ s, w i = 1 by ring)
(by simp : ∀ i ∈ s, a i - c ∈ univ)
(sub_nonneg.mpr hmass) (by simp : -c ∈ univ)
have he : (∑ i ∈ s, w i * a i) - c =
(1 - ∑ i ∈ s, w i) * (-c) + ∑ i ∈ s, w i * (a i - c) := by
simp_rw [mul_sub]
rw [Finset.sum_sub_distrib, ← Finset.sum_mul]
ring
simpa only [smul_eq_mul, Function.comp_apply, neg_sq, he] using hj
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
((∑ i : ι, f i) / (Fintype.card ι : ℝ) - c)^2
≤ (∑ i : ι, (f i - c)^2) / (Fintype.card ι : ℝ)
Proof (Lean source)
lemma finite_uniform_square {ι : Type*} [Fintype ι] [Nonempty ι]
(f : ι → ℝ) (c : ℝ) :
((∑ i : ι, f i) / (Fintype.card ι : ℝ) - c)^2 ≤
(∑ i : ι, (f i - c)^2) / (Fintype.card ι : ℝ) := by
have h := sum_div_card_sq_le_sum_sq_div_card
(s := (Finset.univ : Finset ι)) (f := fun i => f i - c)
have hc : (Fintype.card ι : ℝ) ≠ 0 := by positivity
have he : (∑ i : ι, (f i - c)) / (Fintype.card ι : ℝ) =
(∑ i : ι, f i) / (Fintype.card ι : ℝ) - c := by
simp only [Finset.sum_sub_distrib, Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
field_simp
rw [← he]
simpa using h
theorem mixed_count_derandomization
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma mixed_count_derandomization (n d : ℕ) (q : ℝ) (P : FullLaw d) :
deterministicRisk (mixedCountEstimator n d q) P ≤
auxiliaryMixedRisk n q P := by
classical
let w : ℕ → ℝ := fun k =>
(poissonMeasure ((n : ℝ≥0) / 2)).real ({k} : Set ℕ)
let a : (Fin n → ObsRecord d) → ℕ → ℝ := fun sample k =>
if h : k ≤ n then
(∑ assign : Fin k → Fin 3,
auxiliaryMixedValue n q (fun i => sample (castLE h i)) assign) /
(3 : ℝ) ^ k
else 0
have hw (k : ℕ) : 0 ≤ w k := measureReal_nonneg
have hmass : (∑ k ∈ range (n + 1), w k) ≤ 1 := by
have h := MeasureTheory.sum_measureReal_le_measureReal_univ
(μ := poissonMeasure ((n : ℝ≥0) / 2))
(s := range (n + 1)) (t := fun k : ℕ => ({k} : Set ℕ))
have hm : ∀ k ∈ range (n + 1), MeasurableSet ({k} : Set ℕ) :=
fun _ _ => MeasurableSet.singleton _
have hd : (↑(range (n + 1)) : Set ℕ).PairwiseDisjoint
(fun k : ℕ => ({k} : Set ℕ)) := by
intro i hi j hj hij
exact Set.disjoint_singleton.mpr hij
simpa [w] using h hm hd
have hpoint (sample : Fin n → ObsRecord d) :
(mixedCountEstimator n d q sample - ate P) ^ 2 ≤
(∑ k ∈ range (n + 1),
if h : k ≤ n then w k *
((∑ assign : Fin k → Fin 3,
(auxiliaryMixedValue n q (fun i => sample (castLE h i)) assign -
ate P) ^ 2) / (3 : ℝ) ^ k)
else 0) +
(1 - ∑ k ∈ range (n + 1), w k) * (ate P) ^ 2 := by
have houter := finite_weighted_square_with_remainder
(range (n + 1)) w (a sample) (ate P)
(fun k _ => hw k) hmass
have hinner (k : ℕ) (h : k ≤ n) :
(a sample k - ate P) ^ 2 ≤
(∑ assign : Fin k → Fin 3,
(auxiliaryMixedValue n q (fun i => sample (castLE h i)) assign -
ate P) ^ 2) / (3 : ℝ) ^ k := by
simpa [a, h] using finite_uniform_square
(fun assign : Fin k → Fin 3 =>
auxiliaryMixedValue n q (fun i => sample (castLE h i)) assign)
(ate P)
have hsum :
(∑ k ∈ range (n + 1), w k * (a sample k - ate P) ^ 2) ≤
(∑ k ∈ range (n + 1),
if h : k ≤ n then w k *
((∑ assign : Fin k → Fin 3,
(auxiliaryMixedValue n q (fun i => sample (castLE h i)) assign -
ate P) ^ 2) / (3 : ℝ) ^ k)
else 0) := by
apply Finset.sum_le_sum
intro k hk
have h : k ≤ n := by simpa using hk
simpa only [a, dif_pos h] using
mul_le_mul_of_nonneg_left (hinner k h) (hw k)
have he : mixedCountEstimator n d q sample =
∑ k ∈ range (n + 1), w k * a sample k := by
simp [mixedCountEstimator, w, a]
rw [he]
linarith
letI : IsProbabilityMeasure P.1 := P.2
haveI : IsProbabilityMeasure (P.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop : AEMeasurable obs P.1)
haveI : IsProbabilityMeasure (sampleLaw n P) := by
unfold sampleLaw
infer_instance
change (∫ sample, (mixedCountEstimator n d q sample - ate P) ^ 2
∂(sampleLaw n P)) ≤
∫ sample, (∑ k ∈ range (n + 1),
if h : k ≤ n then w k *
((∑ assign : Fin k → Fin 3,
(auxiliaryMixedValue n q (fun i => sample (castLE h i)) assign -
ate P) ^ 2) / (3 : ℝ) ^ k)
else 0) +
(1 - ∑ k ∈ range (n + 1), w k) * (ate P) ^ 2
∂(sampleLaw n P)
apply integral_mono_of_nonneg
· exact Filter.Eventually.of_forall (fun sample => sq_nonneg _)
· exact Integrable.of_finite
· exact Filter.Eventually.of_forall hpoint
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.RatioAggregate 3 declarations Aggregate deterministic bounds for the empirical-ratio branch.
Aggregate deterministic bounds for the empirical-ratio branch.
theorem cellMean_mem_Icc
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma cellMean_mem_Icc (P : FullLaw d) (j : Cell d) :
cellMean P j ∈ Icc (0 : ℝ) 1 := by
letI : IsProbabilityMeasure P.1 := P.2
by_cases hpos : 0 < arrivedCell P j
· rw [cellMean, if_pos hpos]
constructor
· exact div_nonneg measureReal_nonneg hpos.le
· apply (div_le_one hpos).2
exact measureReal_mono (fun r hr ↦ ⟨hr.1, hr.2.1⟩)
(measure_ne_top P.1 _)
· simp [cellMean, hpos]
theorem cell_exp_envelope
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma cell_exp_envelope {eps t z : ℝ} (heps : 0 < eps) (ht : 0 < t)
(hz : 0 ≤ z) :
z * exp (-eps * t * z) ≤ 1 / (eps * exp 1 * t) := by
have hbase : exp 1 * (eps * t * z) ≤ exp (eps * t * z) :=
Real.exp_one_mul_le_exp
have hmul := mul_le_mul_of_nonneg_right hbase
(Real.exp_pos (-(eps * t * z))).le
have hprod : exp 1 * (eps * t * z) * exp (-(eps * t * z)) ≤ 1 := by
calc
_ ≤ exp (eps * t * z) * exp (-(eps * t * z)) := hmul
_ = 1 := by rw [← Real.exp_add]; ring_nf; simp
apply (le_div_iff₀ (by positivity : 0 < eps * exp 1 * t)).2
calc
z * exp (-eps * t * z) * (eps * exp 1 * t) =
exp 1 * (eps * t * z) * exp (-(eps * t * z)) := by ring
_ ≤ 1 := hprod
theorem abs_ratioBranch_aggregate_bias_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
abs (2 * ∑ j : Cell d, armSign j.1 * cellProb P j * ((∫ s, poissonRatioBranch j s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) - cellMean P j))
≤ 8 * d / (exp 1 * streamEffectiveSize n q)
Proof (Lean source)
lemma abs_ratioBranch_aggregate_bias_le (n d : ℕ) (q : ℝ)
(P : FullLaw d) (hP : UnrestrictedArrivalModelClass n d q P) :
abs (2 * ∑ j : Cell d, armSign j.1 * cellProb P j *
((∫ s, poissonRatioBranch j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) -
cellMean P j)) ≤
8 * d / (exp 1 * streamEffectiveSize n q) := by
have hm : 0 < streamSize n := by
unfold streamSize
have hn : 0 < (n : ℝ) := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hP.n_pos)
positivity
have hq : 0 < q := hP.q_pos
have hN0 : 0 < streamEffectiveSize n q := by
unfold streamEffectiveSize
positivity
have hcell (j : Cell d) :
abs (armSign j.1 * cellProb P j *
((∫ s, poissonRatioBranch j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) -
cellMean P j)) ≤
1 / (exp 1 * streamEffectiveSize n q) := by
rw [markedPoissonRatioBranch_bias]
have hmu := cellMean_mem_Icc P j
have hp : 0 ≤ cellProb P j := measureReal_nonneg
have harr : q * cellProb P j ≤ arrivedCell P j := by
by_cases hpj : 0 < cellProb P j
· exact hP.arrival j hpj
· have hp0 : cellProb P j = 0 := le_antisymm (le_of_not_gt hpj) hp
rw [hp0, mul_zero]
exact measureReal_nonneg
have hexp : exp (-streamSize n * arrivedCell P j) ≤
exp (-q * streamSize n * cellProb P j) := by
apply Real.exp_le_exp.mpr
have := mul_le_mul_of_nonneg_left harr hm.le
nlinarith
have hfirst :
abs (armSign j.1 * cellProb P j *
(-cellMean P j * exp (-streamSize n * arrivedCell P j))) ≤
cellProb P j * exp (-q * streamSize n * cellProb P j) := by
rw [abs_mul, abs_mul, abs_mul]
have hsign : abs (armSign j.1) = 1 := by cases j.1 <;> simp [armSign]
rw [hsign, one_mul]
rw [abs_of_nonneg hp, abs_neg, abs_of_nonneg hmu.1,
abs_of_pos (Real.exp_pos _)]
calc
cellProb P j * (cellMean P j * exp (-streamSize n * arrivedCell P j)) ≤
cellProb P j * (1 * exp (-streamSize n * arrivedCell P j)) := by
gcongr
exact hmu.2
_ ≤ cellProb P j * exp (-q * streamSize n * cellProb P j) := by
gcongr
simpa using hexp
refine hfirst.trans ?_
have he := cell_exp_envelope hq hm hp
simpa [streamEffectiveSize, mul_comm, mul_left_comm, mul_assoc] using he
calc
abs (2 * ∑ j : Cell d, armSign j.1 * cellProb P j *
((∫ s, poissonRatioBranch j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) -
cellMean P j)) =
2 * abs (∑ j : Cell d, armSign j.1 * cellProb P j *
((∫ s, poissonRatioBranch j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) -
cellMean P j)) := by rw [abs_mul, abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 2)]
_ ≤ 2 * ∑ j : Cell d, abs (armSign j.1 * cellProb P j *
((∫ s, poissonRatioBranch j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) -
cellMean P j)) := by
gcongr
exact abs_sum_le_sum_abs _ _
_ ≤ 2 * ∑ _j : Cell d,
1 / (exp 1 * streamEffectiveSize n q) := by
gcongr with j
exact hcell j
_ = 8 * d / (exp 1 * streamEffectiveSize n q) := by
simp [div_eq_mul_inv]
ring
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.RatioMean 6 declarations The exact, zero-safe ratio-branch mean in the marked Poisson experiment.
The exact, zero-safe ratio-branch mean in the marked Poisson experiment.
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
eventCount A s
= (eventCount s A : ℝ)
Proof (Lean source)
-- @node: successFraction_eventCount_eq_cast
lemma successFraction_eventCount_eq_cast {X : Type*} [MeasurableSpace X]
(A : Set X) (s : FiniteSample X) :
eventCount A s =
(eventCount s A : ℝ) := by
classical
simp only [eventCount,
iidEventCount,
eventCount, Finset.card_filter, Nat.cast_sum, Nat.cast_ite,
Nat.cast_one, Nat.cast_zero]
theorem poissonRatioBranch_eq_successFraction
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: poissonRatioBranch_eq_successFraction
lemma poissonRatioBranch_eq_successFraction {d : ℕ} (j : Cell d)
(s : FiniteSample (ObsRecord d × Fin 3)) :
poissonRatioBranch j s =
successFraction
(streamEvent 2 j true true) (streamEvent 2 j true false) s := by
classical
simp only [poissonRatioBranch,
successFraction,
successFraction_eventCount_eq_cast, Nat.cast_eq_zero]
by_cases h : eventCount s (streamEvent 2 j true false) = 0
· simp [h]
· simp [h, Nat.pos_of_ne_zero h]
theorem integral_markedPoissonRatioBranch_of_pos
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, poissonRatioBranch j s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: integral_markedPoissonRatioBranch_of_pos
lemma integral_markedPoissonRatioBranch_of_pos (n d : ℕ) (P : FullLaw d)
(j : Cell d) (hpos : 0 < arrivedCell P j) :
(∫ s, poissonRatioBranch j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) =
cellMean P j * (1 - exp (-streamSize n * arrivedCell P j)) := by
letI := markedObsLaw_isProbabilityMeasure P
simp_rw [poissonRatioBranch_eq_successFraction]
rw [finitePoisson_successFraction_mean
(markedObsLaw P) ((n : ℝ≥0) / 2)
(Set.Finite.measurableSet (Set.toFinite _))
(Set.Finite.measurableSet (Set.toFinite _))
(streamEvent_ones_subset_arrived 2 j)
(by rw [markedObsLaw_streamEvent_real, map_obsStreamEvent_arrived_real]; positivity)]
rw [markedObsLaw_streamEvent_real, markedObsLaw_streamEvent_real,
map_obsStreamEvent_arrivedOne_real, map_obsStreamEvent_arrived_real]
simp only [cellMean, if_pos hpos, NNReal.coe_div, NNReal.coe_natCast,
NNReal.coe_ofNat]
congr 1
· ring
· congr 2
unfold streamSize
ring
theorem integral_markedPoissonRatioBranch_of_zero
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, poissonRatioBranch j s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) = 0
Proof (Lean source)
-- @node: integral_markedPoissonRatioBranch_of_zero
lemma integral_markedPoissonRatioBranch_of_zero (n d : ℕ) (P : FullLaw d)
(j : Cell d) (hzero : arrivedCell P j = 0) :
(∫ s, poissonRatioBranch j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) = 0 := by
have hmass : (markedObsLaw P) (streamEvent 2 j true false) = 0 := by
apply (measureReal_eq_zero_iff).mp
rw [markedObsLaw_streamEvent_real, map_obsStreamEvent_arrived_real, hzero,
zero_div]
simp_rw [poissonRatioBranch_eq_successFraction]
exact finitePoisson_successFraction_mean_of_mass_zero
(markedObsLaw P) ((n : ℝ≥0) / 2)
(Set.Finite.measurableSet (Set.toFinite _)) hmass
theorem integral_markedPoissonRatioBranch
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, poissonRatioBranch j s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: integral_markedPoissonRatioBranch
lemma integral_markedPoissonRatioBranch (n d : ℕ) (P : FullLaw d)
(j : Cell d) :
(∫ s, poissonRatioBranch j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) =
cellMean P j * (1 - exp (-streamSize n * arrivedCell P j)) := by
by_cases hpos : 0 < arrivedCell P j
· exact integral_markedPoissonRatioBranch_of_pos n d P j hpos
· have hzero : arrivedCell P j = 0 :=
le_antisymm (le_of_not_gt hpos) measureReal_nonneg
rw [integral_markedPoissonRatioBranch_of_zero n d P j hzero]
simp [cellMean, hpos]
theorem markedPoissonRatioBranch_bias
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, poissonRatioBranch j s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
- cellMean P j
Proof (Lean source)
-- @node: markedPoissonRatioBranch_bias
lemma markedPoissonRatioBranch_bias (n d : ℕ) (P : FullLaw d) (j : Cell d) :
(∫ s, poissonRatioBranch j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) - cellMean P j =
-cellMean P j * exp (-streamSize n * arrivedCell P j) := by
rw [integral_markedPoissonRatioBranch]
ring
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.RatioSecondMoment 5 declarations Second moments for the zero-safe marked Poisson ratio branch.
Second moments for the zero-safe marked Poisson ratio branch.
theorem iid_successFraction_count_fibre_centered_sq_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ x : Fin n → X, (successFraction A B (fixedSizeEmbed n x) - P.real A / P.real B) ^ 2 ∂(Measure.pi (fun _ : Fin n ↦ P)).restrict {x | iidEventCount B x = (k : ℝ)})
Proof (Lean source)
lemma iid_successFraction_count_fibre_centered_sq_le
{X : Type*} [MeasurableSpace X] [Fintype X] [MeasurableSingletonClass X]
(P : Measure X) [IsProbabilityMeasure P] (n k : ℕ) (hk : 0 < k)
{A B : Set X} (hA : MeasurableSet A) (hB : MeasurableSet B)
(hAB : A ⊆ B) (hPB : 0 < P.real B) :
(∫ x : Fin n → X,
(successFraction A B (fixedSizeEmbed n x) - P.real A / P.real B) ^ 2
∂(Measure.pi (fun _ : Fin n ↦ P)).restrict
{x | iidEventCount B x = (k : ℝ)}) ≤
(k : ℝ)⁻¹ *
(Measure.pi (fun _ : Fin n ↦ P)).real
{x | iidEventCount B x = (k : ℝ)} := by
classical
let T := Finset.univ.filter (fun U : Finset (Fin n) ↦ U.card = k)
let R := fun U : Finset (Fin n) ↦ ratioPattern B U
have hset : {x : Fin n → X | iidEventCount B x = (k : ℝ)} =
⋃ U ∈ T, R U := by
ext x
simp only [Set.mem_ofPred_eq, Set.mem_iUnion]
constructor
· intro hx
let U := Finset.univ.filter (fun i ↦ x i ∈ B)
refine ⟨U, ?_, ?_⟩
· simp only [T, mem_filter, Finset.mem_univ, true_and]
exact_mod_cast (iidEventCount_eq_card B x).symm.trans hx
· exact (mem_ratioPattern_iff B U x).mpr rfl
· rintro ⟨U, hU, hx⟩
have hcard : U.card = k := (Finset.mem_filter.mp hU).2
rw [iidEventCount_eq_card B x, (mem_ratioPattern_iff B U x).mp hx, hcard]
have hm : ∀ U ∈ T, MeasurableSet (R U) :=
fun U _ ↦ measurable_ratioPattern hB U
have hd : PairwiseDisjoint (↑T) R :=
fun U _ V _ hUV ↦ disjoint_ratioPatterns B U V hUV
have hkR : (k : ℝ) ≠ 0 := by exact_mod_cast hk.ne'
have hPBR : P.real B ≠ 0 := ne_of_gt hPB
let f : (Fin n → X) → ℝ := fun x ↦
(successFraction A B (fixedSizeEmbed n x) - P.real A / P.real B) ^ 2
have hf : Integrable f (Measure.pi (fun _ : Fin n ↦ P)) := by
apply Integrable.of_bound
(((measurable_successFraction hA hB).comp
(measurable_fixedSizeEmbed n) |>.sub measurable_const).pow_const 2).aestronglyMeasurable 1
filter_upwards [] with x
rcases successFraction_bounds hAB (fixedSizeEmbed n x) with ⟨hs0, hs1⟩
have hp0 : 0 ≤ P.real A / P.real B := div_nonneg measureReal_nonneg hPB.le
have hp1 : P.real A / P.real B ≤ 1 := (div_le_one hPB).2 (measureReal_mono hAB)
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
change (successFraction A B (fixedSizeEmbed n x) - P.real A / P.real B) ^ 2 ≤ 1
nlinarith
rw [hset, integral_biUnion_finset T hm hd (fun _ _ ↦ hf.integrableOn),
measureReal_biUnion_finset hd hm]
rw [Finset.mul_sum]
apply Finset.sum_le_sum
intro U hU
have hcard : U.card = k := (Finset.mem_filter.mp hU).2
have heq : f =ᵐ[(Measure.pi (fun _ : Fin n ↦ P)).restrict (R U)]
fun x ↦
(P.real B * iidEventCount A x - P.real A * (k : ℝ)) ^ 2 /
((P.real B) ^ 2 * (k : ℝ) ^ 2) := by
filter_upwards [ae_restrict_mem (hm U hU)] with x hx
have hxB : iidEventCount B x = (k : ℝ) := by
rw [iidEventCount_eq_card B x, (mem_ratioPattern_iff B U x).mp hx, hcard]
simp only [f, successFraction, eventCount_fixedSizeEmbed, hxB, hkR, if_false]
field_simp
rw [integral_congr_ae heq, integral_div]
have hpat := iid_nested_pattern_centered_sq_le P U hA hB hAB
rw [hcard] at hpat
apply (div_le_iff₀ (mul_pos (sq_pos_of_pos hPB) (sq_pos_of_pos (by exact_mod_cast hk)))).2
calc
(∫ x, (P.real B * iidEventCount A x - P.real A * (k : ℝ)) ^ 2
∂(Measure.pi (fun _ : Fin n ↦ P)).restrict (R U)) ≤
(k : ℝ) * (P.real B) ^ 2 *
(Measure.pi (fun _ : Fin n ↦ P)).real (R U) := hpat
_ = ((k : ℝ)⁻¹ * (Measure.pi (fun _ : Fin n ↦ P)).real (R U)) *
((P.real B) ^ 2 * (k : ℝ) ^ 2) := by field_simp
theorem iid_successFraction_centered_sq_le_inverseCount
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ x : Fin n → X, (successFraction A B (fixedSizeEmbed n x) - P.real A / P.real B) ^ 2 ∂Measure.pi (fun _ : Fin n ↦ P))
≤ ∫ x : Fin n → X, (if 0 < iidEventCount B x then (iidEventCount B x)⁻¹ else 1) ∂Measure.pi (fun _ : Fin n ↦ P)
Proof (Lean source)
lemma iid_successFraction_centered_sq_le_inverseCount
{X : Type*} [MeasurableSpace X] [Fintype X] [MeasurableSingletonClass X]
(P : Measure X) [IsProbabilityMeasure P] (n : ℕ)
{A B : Set X} (hA : MeasurableSet A) (hB : MeasurableSet B)
(hAB : A ⊆ B) (hPB : 0 < P.real B) :
(∫ x : Fin n → X,
(successFraction A B (fixedSizeEmbed n x) - P.real A / P.real B) ^ 2
∂Measure.pi (fun _ : Fin n ↦ P)) ≤
∫ x : Fin n → X,
(if 0 < iidEventCount B x then (iidEventCount B x)⁻¹ else 1)
∂Measure.pi (fun _ : Fin n ↦ P) := by
classical
let μ : Measure (Fin n → X) := Measure.pi (fun _ : Fin n ↦ P)
let F : ℕ → Set (Fin n → X) := fun k ↦ {x | iidEventCount B x = (k : ℝ)}
let T := range (n + 1)
let f : (Fin n → X) → ℝ := fun x ↦
(successFraction A B (fixedSizeEmbed n x) - P.real A / P.real B) ^ 2
let g : (Fin n → X) → ℝ := fun x ↦
if 0 < iidEventCount B x then (iidEventCount B x)⁻¹ else 1
have hm : ∀ k ∈ T, MeasurableSet (F k) := fun k _ ↦
(measurable_iidEventCount hB n) (measurableSet_singleton _)
have hd : PairwiseDisjoint (↑T) F := by
intro k _ l _ hkl
apply Set.disjoint_left.mpr
intro x hx hy
apply hkl
exact_mod_cast (hx.symm.trans hy : (k : ℝ) = (l : ℝ))
have hcover : (⋃ k ∈ T, F k) = univ := by
apply Set.eq_univ_of_forall
intro x
let U := Finset.univ.filter (fun i : Fin n ↦ x i ∈ B)
have hU : U.card ≤ n :=
(Finset.card_le_card (filter_subset _ _)).trans (by simp)
exact Set.mem_iUnion.mpr ⟨U.card, Set.mem_iUnion.mpr
⟨Finset.mem_range.mpr (by omega), iidEventCount_eq_card B x⟩⟩
have hf : Integrable f μ := Integrable.of_finite
have hg : Integrable g μ := Integrable.of_finite
change (∫ x, f x ∂μ) ≤ ∫ x, g x ∂μ
rw [← Measure.restrict_univ (μ := μ), ← hcover,
integral_biUnion_finset T hm hd (fun _ _ ↦ hf.integrableOn),
integral_biUnion_finset T hm hd (fun _ _ ↦ hg.integrableOn)]
apply Finset.sum_le_sum
intro k hkT
by_cases hk0 : k = 0
· subst k
have hpoint : ∀ x ∈ F 0, f x ≤ g x := by
intro x hx
have hp0 : 0 ≤ P.real A / P.real B := div_nonneg measureReal_nonneg hPB.le
have hp1 : P.real A / P.real B ≤ 1 := (div_le_one hPB).2 (measureReal_mono hAB)
have hx0 : iidEventCount B x = 0 := by simpa [F] using hx
have hs0 : successFraction A B (fixedSizeEmbed n x) = 0 := by
simp [successFraction, eventCount_fixedSizeEmbed, hx0]
simp only [f, g, hs0, hx0, lt_self_iff_false, if_false, zero_sub]
nlinarith
apply integral_mono_ae hf.restrict hg.restrict
filter_upwards [ae_restrict_mem (hm 0 hkT)] with x hx
exact hpoint x hx
· have hkpos : 0 < k := Nat.pos_of_ne_zero hk0
have hleft := iid_successFraction_count_fibre_centered_sq_le
P n k hkpos hA hB hAB hPB
have hright : (∫ x, g x ∂μ.restrict (F k)) =
(k : ℝ)⁻¹ * μ.real (F k) := by
have heq : g =ᵐ[μ.restrict (F k)] (fun _ ↦ (k : ℝ)⁻¹) := by
filter_upwards [ae_restrict_mem (hm k hkT)] with x hx
have hxk : iidEventCount B x = (k : ℝ) := by simpa [F] using hx
simp [g, hxk, hkpos]
rw [integral_congr_ae heq, integral_const]
rw [measureReal_restrict_apply_univ]
simp [smul_eq_mul, mul_comm]
change (∫ x, f x ∂μ.restrict (F k)) ≤ _
rw [hright]
exact hleft
theorem poisson_zeroSafeInverse_expectation_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
lambda :
ℝ≥0
conclusion
(∫ k : ℕ, (if 0 < k then ((k : ℝ)⁻¹) else 1) ∂poissonMeasure lambda)
≤ 4 / (1 + (lambda : ℝ))
Proof (Lean source)
lemma poisson_zeroSafeInverse_expectation_le (lambda : ℝ≥0) :
(∫ k : ℕ, (if 0 < k then ((k : ℝ)⁻¹) else 1)
∂poissonMeasure lambda) ≤ 4 / (1 + (lambda : ℝ)) := by
let g : ℕ → ℝ := fun k ↦ if 0 < k then ((k : ℝ)⁻¹) else 1
have hg : Integrable g (poissonMeasure lambda) := by
apply Integrable.of_bound (measurable_of_countable _).aestronglyMeasurable 1
filter_upwards with k
rw [Real.norm_eq_abs, abs_of_nonneg]
· by_cases hk : 0 < k
· simp only [g, hk, if_true]
apply (inv_le_one₀ (by positivity : 0 < (k : ℝ))).2
exact_mod_cast hk
· simp [g, hk]
· simp only [g]
split_ifs
· positivity
· positivity
by_cases hlambda : (lambda : ℝ) ≤ 1
· calc
(∫ k : ℕ, (if 0 < k then ((k : ℝ)⁻¹) else 1)
∂poissonMeasure lambda) = ∫ k, g k ∂poissonMeasure lambda := rfl
_ ≤ ∫ _k : ℕ, (1 : ℝ) ∂poissonMeasure lambda := by
apply integral_mono hg (integrable_const 1)
intro k
by_cases hk : 0 < k
· simp only [g, hk, if_true]
apply (inv_le_one₀ (by positivity : 0 < (k : ℝ))).2
exact_mod_cast hk
· simp [g, hk]
_ = 1 := by simp
_ ≤ 4 / (1 + (lambda : ℝ)) := by
apply (le_div_iff₀ (by positivity : 0 < 1 + (lambda : ℝ))).2
linarith
· have hlambda_pos : 0 < (lambda : ℝ) := by
have : 1 < (lambda : ℝ) := lt_of_not_ge hlambda
linarith
let r : ℕ → ℝ := fun k ↦ (((k + 1 : ℕ) : ℝ))⁻¹
have hr : Integrable r (poissonMeasure lambda) :=
(shifted_reciprocal_memLp_two lambda).integrable
(by norm_num)
have hpoint (k : ℕ) : g k ≤ 2 * r k := by
by_cases hk : k = 0
· subst k
simp [g, r]
· have hkpos : 0 < k := Nat.pos_of_ne_zero hk
simp only [g, hkpos, if_true, r]
rw [← one_div, show 2 * (((k + 1 : ℕ) : ℝ))⁻¹ =
2 / (((k + 1 : ℕ) : ℝ)) by rw [div_eq_mul_inv]]
apply (div_le_div_iff₀ (by positivity : 0 < (k : ℝ))
(by positivity : 0 < ((k + 1 : ℕ) : ℝ))).2
norm_num
exact_mod_cast (show k + 1 ≤ 2 * k by omega)
have hshift :=
Causalean.Mathlib.Probability.Poisson.shifted_reciprocal_first_moment lambda
have hr_eq : (∫ k : ℕ, r k ∂poissonMeasure lambda) =
(1 - exp (-(lambda : ℝ))) / (lambda : ℝ) := by
apply (eq_div_iff hlambda_pos.ne').2
simpa [r, mul_comm] using hshift
calc
(∫ k : ℕ, (if 0 < k then ((k : ℝ)⁻¹) else 1)
∂poissonMeasure lambda) = ∫ k, g k ∂poissonMeasure lambda := rfl
_ ≤ ∫ k, 2 * r k ∂poissonMeasure lambda := by
exact integral_mono hg (hr.const_mul 2) hpoint
_ = 2 * ((1 - exp (-(lambda : ℝ))) / (lambda : ℝ)) := by
rw [integral_const_mul, hr_eq]
_ ≤ 2 / (lambda : ℝ) := by
have hexp : 0 ≤ exp (-(lambda : ℝ)) := (Real.exp_pos _).le
rw [← mul_div_assoc]
apply div_le_div_of_nonneg_right _ hlambda_pos.le
linarith
_ ≤ 4 / (1 + (lambda : ℝ)) := by
apply (div_le_div_iff₀ hlambda_pos (by positivity : 0 < 1 + (lambda : ℝ))).2
have : 1 ≤ (lambda : ℝ) := le_of_not_ge hlambda
nlinarith
theorem finitePoisson_successFraction_centered_sq_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s : FiniteSample X, (successFraction A B s - P.real A / P.real B) ^ 2 ∂finitePoissonSampleLaw P lambda)
≤ 4 / (1 + (lambda : ℝ) * P.real B)
Proof (Lean source)
lemma finitePoisson_successFraction_centered_sq_le
{X : Type*} [MeasurableSpace X] [Fintype X] [MeasurableSingletonClass X]
[DecidableEq X] (P : Measure X) [IsProbabilityMeasure P]
(lambda : ℝ≥0) {A B : Set X}
(hA : MeasurableSet A) (hB : MeasurableSet B)
(hAB : A ⊆ B) (hPB : 0 < P.real B) :
(∫ s : FiniteSample X,
(successFraction A B s - P.real A / P.real B) ^ 2
∂finitePoissonSampleLaw P lambda) ≤
4 / (1 + (lambda : ℝ) * P.real B) := by
classical
let target : ℝ := P.real A / P.real B
let f : FiniteSample X → ℝ := fun s ↦ (successFraction A B s - target) ^ 2
let g : FiniteSample X → ℝ := fun s ↦
if 0 < Causalean.Stat.FiniteRaoBlackwell.Poisson.FinitePartition.NestedEventMoment.eventCount s B
then ((Causalean.Stat.FiniteRaoBlackwell.Poisson.FinitePartition.NestedEventMoment.eventCount s B : ℝ))⁻¹
else 1
have hf : Measurable f := by
exact ((measurable_successFraction hA hB).sub measurable_const).pow_const 2
have hg : Measurable g := by
have hh : Measurable (fun k : ℕ ↦ if 0 < k then ((k : ℝ)⁻¹) else 1) :=
measurable_of_countable _
exact hh.comp
(measurable_eventCount B hB)
have hfb (s : FiniteSample X) : ‖f s‖ ≤ 1 := by
rcases successFraction_bounds hAB s with ⟨hs0, hs1⟩
have ht0 : 0 ≤ target := div_nonneg measureReal_nonneg hPB.le
have ht1 : target ≤ 1 := (div_le_one hPB).2 (measureReal_mono hAB)
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
dsimp [f]
nlinarith
have hgb (s : FiniteSample X) : ‖g s‖ ≤ 1 := by
rw [Real.norm_eq_abs, abs_of_nonneg]
· by_cases hk : 0 <
Causalean.Stat.FiniteRaoBlackwell.Poisson.FinitePartition.NestedEventMoment.eventCount s B
· simp only [g, hk, if_true]
apply (inv_le_one₀ (by exact_mod_cast hk : (0 : ℝ) < _)).2
exact_mod_cast hk
· simp [g, hk]
· simp only [g]
split_ifs <;> positivity
rw [show (fun s : FiniteSample X ↦
(successFraction A B s - P.real A / P.real B) ^ 2) = f by rfl]
rw [integral_finitePoissonSampleLaw_eq_integral_iid P lambda f hf 1 hfb]
calc
(∫ n : ℕ, ∫ x : Fin n → X, f (fixedSizeEmbed n x)
∂Measure.pi (fun _ : Fin n ↦ P) ∂poissonMeasure lambda) ≤
∫ n : ℕ, ∫ x : Fin n → X, g (fixedSizeEmbed n x)
∂Measure.pi (fun _ : Fin n ↦ P) ∂poissonMeasure lambda := by
apply integral_mono
· exact Integrable.of_bound (measurable_of_countable _).aestronglyMeasurable 1
(Filter.Eventually.of_forall fun n ↦ by
simpa using norm_integral_le_of_norm_le_const
(μ := Measure.pi (fun _ : Fin n ↦ P))
(f := fun x : Fin n → X ↦ f (fixedSizeEmbed n x))
(Filter.Eventually.of_forall fun x ↦ hfb _))
· exact Integrable.of_bound (measurable_of_countable _).aestronglyMeasurable 1
(Filter.Eventually.of_forall fun n ↦ by
simpa using norm_integral_le_of_norm_le_const
(μ := Measure.pi (fun _ : Fin n ↦ P))
(f := fun x : Fin n → X ↦ g (fixedSizeEmbed n x))
(Filter.Eventually.of_forall fun x ↦ hgb _))
· intro n
calc
(∫ x : Fin n → X, f (fixedSizeEmbed n x)
∂Measure.pi (fun _ : Fin n ↦ P)) ≤
∫ x : Fin n → X,
(if 0 < iidEventCount B x then (iidEventCount B x)⁻¹ else 1)
∂Measure.pi (fun _ : Fin n ↦ P) := by
simpa [f, target] using
iid_successFraction_centered_sq_le_inverseCount P n hA hB hAB hPB
_ = ∫ x : Fin n → X, g (fixedSizeEmbed n x)
∂Measure.pi (fun _ : Fin n ↦ P) := by
apply integral_congr_ae
filter_upwards with x
have hc := successFraction_eventCount_eq_cast B (fixedSizeEmbed n x)
rw [eventCount_fixedSizeEmbed]
at hc
by_cases hk : 0 <
Causalean.Stat.FiniteRaoBlackwell.Poisson.FinitePartition.NestedEventMoment.eventCount
(fixedSizeEmbed n x) B
· have hkr : 0 < iidEventCount B x := by
rw [hc]
exact_mod_cast hk
simp [g, hk, hkr, hc]
· have hk0 :
Causalean.Stat.FiniteRaoBlackwell.Poisson.FinitePartition.NestedEventMoment.eventCount
(fixedSizeEmbed n x) B = 0 := Nat.eq_zero_of_not_pos hk
have hreal0 : iidEventCount B x = 0 := by simp [hc, hk0]
simp [g, hk, hreal0]
_ = ∫ s : FiniteSample X, g s ∂finitePoissonSampleLaw P lambda := by
rw [integral_finitePoissonSampleLaw_eq_integral_iid P lambda g hg 1 hgb]
_ = ∫ k : ℕ, (if 0 < k then ((k : ℝ)⁻¹) else 1)
∂poissonMeasure (lambda * (P B).toNNReal) := by
have hmap := finitePoissonSampleLaw_map_eventCount P lambda B hB
let h : ℕ → ℝ := fun k ↦ if 0 < k then ((k : ℝ)⁻¹) else 1
have hh : Measurable h := measurable_of_countable _
rw [← integral_map
(measurable_eventCount B hB).aemeasurable
hh.aestronglyMeasurable, hmap]
_ ≤ 4 / (1 + ((lambda * (P B).toNNReal : ℝ≥0) : ℝ)) :=
poisson_zeroSafeInverse_expectation_le _
_ = 4 / (1 + (lambda : ℝ) * P.real B) := by
have hmass : ((P B).toNNReal : ℝ) = (P B).toReal := by
calc
((P B).toNNReal : ℝ) =
(((P B).toNNReal : ℝ≥0∞).toReal) := (ENNReal.coe_toReal _).symm
_ = (P B).toReal := congrArg toReal
(ENNReal.coe_toNNReal (measure_ne_top P B))
simp [NNReal.coe_mul, Measure.real, hmass]
theorem integral_sq_markedPoissonRatioBranch_sub_cellMean_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, (poissonRatioBranch j s - cellMean P j) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
≤ 4 / (1 + streamSize n * arrivedCell P j)
Proof (Lean source)
lemma integral_sq_markedPoissonRatioBranch_sub_cellMean_le
(n d : ℕ) (P : FullLaw d) (j : Cell d)
(hpos : 0 < arrivedCell P j) :
(∫ s, (poissonRatioBranch j s - cellMean P j) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
4 / (1 + streamSize n * arrivedCell P j) := by
letI := markedObsLaw_isProbabilityMeasure P
have h := finitePoisson_successFraction_centered_sq_le
(markedObsLaw P) ((n : ℝ≥0) / 2)
(Set.Finite.measurableSet (Set.toFinite (streamEvent 2 j true true)))
(Set.Finite.measurableSet (Set.toFinite (streamEvent 2 j true false)))
(streamEvent_ones_subset_arrived 2 j)
(by rw [markedObsLaw_streamEvent_real, map_obsStreamEvent_arrived_real]; positivity)
simp_rw [poissonRatioBranch_eq_successFraction]
rw [cellMean, if_pos hpos]
rw [markedObsLaw_streamEvent_real, markedObsLaw_streamEvent_real,
map_obsStreamEvent_arrivedOne_real, map_obsStreamEvent_arrived_real] at h
have hratio :
P.1.real {r | inCell r j ∧ r.R = true ∧ r.Y = true} /
arrivedCell P j =
(P.1.real {r | inCell r j ∧ r.R = true ∧ r.Y = true} / 3) /
(arrivedCell P j / 3) := by
field_simp [ne_of_gt hpos]
rw [hratio]
convert h using 1
unfold streamSize
push_cast
ring
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.RatioVariance 6 declarations Null-cell completion and variance bounds for roadmap (15).
Null-cell completion and variance bounds for roadmap (15).
theorem integral_sq_markedPoissonRatioBranch_of_zero
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, (poissonRatioBranch j s) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
= 0
Proof (Lean source)
-- @node: integral_sq_markedPoissonRatioBranch_of_zero
lemma integral_sq_markedPoissonRatioBranch_of_zero (n d : ℕ) (P : FullLaw d)
(j : Cell d) (hzero : arrivedCell P j = 0) :
(∫ s, (poissonRatioBranch j s) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) = 0 := by
have hint : Integrable (poissonRatioBranch j)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
apply Integrable.of_bound (measurable_poissonRatioBranch j).aestronglyMeasurable 1
filter_upwards [] with s
rcases poissonRatioBranch_mem_unitInterval j s with ⟨hs0, hs1⟩
simpa [Real.norm_eq_abs, abs_of_nonneg hs0] using hs1
apply le_antisymm _ (integral_nonneg fun s ↦ sq_nonneg _)
calc
_ ≤ ∫ s, poissonRatioBranch j s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2) := by
apply integral_mono (integrable_poissonRatioBranch_sq n d P j) hint
intro s
rcases poissonRatioBranch_mem_unitInterval j s with ⟨hs0, hs1⟩
nlinarith
_ = 0 := integral_markedPoissonRatioBranch_of_zero n d P j hzero
theorem markedPoissonRatioBranch_centered_secondMoment_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, (poissonRatioBranch j s - cellMean P j) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
≤ 4 / (1 + streamSize n * arrivedCell P j)
Proof (Lean source)
-- @node: markedPoissonRatioBranch_centered_secondMoment_le
lemma markedPoissonRatioBranch_centered_secondMoment_le
(n d : ℕ) (P : FullLaw d) (j : Cell d) :
(∫ s, (poissonRatioBranch j s - cellMean P j) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
4 / (1 + streamSize n * arrivedCell P j) := by
by_cases hpos : 0 < arrivedCell P j
· exact integral_sq_markedPoissonRatioBranch_sub_cellMean_le n d P j hpos
· have hzero : arrivedCell P j = 0 :=
le_antisymm (le_of_not_gt hpos) measureReal_nonneg
simp only [cellMean, hpos, if_false, sub_zero]
rw [integral_sq_markedPoissonRatioBranch_of_zero n d P j hzero, hzero]
norm_num
theorem markedPoissonRatioBranch_variance_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
≤ 4 / (1 + streamSize n * arrivedCell P j)
Proof (Lean source)
-- @node: markedPoissonRatioBranch_variance_le
lemma markedPoissonRatioBranch_variance_le
(n d : ℕ) (P : FullLaw d) (j : Cell d) :
variance (poissonRatioBranch j)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
4 / (1 + streamSize n * arrivedCell P j) := by
have hm := (measurable_poissonRatioBranch j).aestronglyMeasurable
(μ := finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
calc
_ = variance (fun s ↦ poissonRatioBranch j s - cellMean P j)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) :=
(variance_sub_const hm (cellMean P j)).symm
_ ≤ ∫ s, (poissonRatioBranch j s - cellMean P j) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2) :=
variance_le_expectation_sq (hm.sub aestronglyMeasurable_const)
_ ≤ _ := markedPoissonRatioBranch_centered_secondMoment_le n d P j
theorem sum_cellProb_eq_one
Given the probability law, the cell probabilities sum to one.
Formal statement
Proof (Lean source)
-- @node: sum_cellProb_eq_one
lemma sum_cellProb_eq_one (P : FullLaw d) :
(∑ j : Cell d, cellProb P j) = 1 := by
classical
letI : IsProbabilityMeasure P.1 := P.2
have h := sum_measureReal_preimage_singleton (μ := P.1)
(Finset.univ : Finset (Cell d))
(f := fun r : FullRecord d ↦ (r.A, r.X, r.S)) (by intro; simp)
have heq (j : Cell d) : {r : FullRecord d | inCell r j} =
(fun r : FullRecord d ↦ (r.A, r.X, r.S)) ⁻¹' {j} := by
ext r
simp [inCell, Prod.ext_iff]
simpa [cellProb, heq] using h
theorem cellProb_sq_div_one_add_intensity_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(cellProb P j) ^ 2 / (1 + streamSize n * arrivedCell P j)
≤ cellProb P j / streamEffectiveSize n q
Proof (Lean source)
-- @node: cellProb_sq_div_one_add_intensity_le
lemma cellProb_sq_div_one_add_intensity_le
(n d : ℕ) (q : ℝ) (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) (j : Cell d) :
(cellProb P j) ^ 2 / (1 + streamSize n * arrivedCell P j) ≤
cellProb P j / streamEffectiveSize n q := by
have hm : 0 < streamSize n := by
unfold streamSize
have hn : 0 < (n : ℝ) := by exact_mod_cast hP.n_pos
positivity
have hN : 0 < streamEffectiveSize n q := by
unfold streamEffectiveSize
exact mul_pos hm hP.q_pos
have hp : 0 ≤ cellProb P j := measureReal_nonneg
have ht : 0 ≤ streamSize n * arrivedCell P j := mul_nonneg hm.le measureReal_nonneg
have hlo := streamEffectiveSize_mul_cellProb_le_arrivedStreamIntensity
n d q P hP 2 j
rw [coe_arrivedStreamIntensity] at hlo
apply (div_le_div_iff₀ (by positivity : 0 < 1 + streamSize n * arrivedCell P j) hN).2
have hmul := mul_le_mul_of_nonneg_left hlo hp
nlinarith
theorem sum_cellProb_sq_mul_ratioVariance_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∑ j : Cell d, (cellProb P j) ^ 2 * variance (poissonRatioBranch j) (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)))
≤ 4 / streamEffectiveSize n q
Proof (Lean source)
-- @node: sum_cellProb_sq_mul_ratioVariance_le
lemma sum_cellProb_sq_mul_ratioVariance_le
(n d : ℕ) (q : ℝ) (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) :
(∑ j : Cell d, (cellProb P j) ^ 2 * variance (poissonRatioBranch j)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))) ≤
4 / streamEffectiveSize n q := by
calc
_ ≤ ∑ j : Cell d, 4 * (cellProb P j / streamEffectiveSize n q) := by
apply Finset.sum_le_sum
intro j hj
calc
_ ≤ (cellProb P j) ^ 2 * (4 / (1 + streamSize n * arrivedCell P j)) :=
mul_le_mul_of_nonneg_left (markedPoissonRatioBranch_variance_le n d P j)
(sq_nonneg _)
_ = 4 * ((cellProb P j) ^ 2 / (1 + streamSize n * arrivedCell P j)) := by ring
_ ≤ _ := mul_le_mul_of_nonneg_left
(cellProb_sq_div_one_add_intensity_le n d q P hP j) (by norm_num)
_ = _ := by
rw [← Finset.mul_sum, ← Finset.sum_div, sum_cellProb_eq_one]
ring
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.RawRiskAssembly 5 declarations Square integrability, bias--variance decomposition, and clipping assemble ideal squared risk from the actual raw-estimator variance and the proved bias budget.
Square integrability, bias--variance decomposition, and clipping assemble ideal squared risk from the actual raw-estimator variance and the proved bias budget. Cross-cell independence is still needed to bound that variance.
theorem memLp_poissonRawMixedValue
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: memLp_poissonRawMixedValue
lemma memLp_poissonRawMixedValue (n d : ℕ) (q : ℝ) (P : FullLaw d) :
MemLp (poissonRawMixedValue n d q) 2
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
have hj (j : Cell d) :=
(memLp_poissonMemberWeight_mul_selectedCellEstimate n d q P j).const_mul
(armSign j.1)
have hs := memLp_finsetSum (Finset.univ : Finset (Cell d)) (fun j _ ↦ hj j)
change MemLp (fun s ↦ 2 * ∑ j : Cell d, armSign j.1 *
poissonMemberWeight n j s * poissonSelectedCellEstimate n d q j s) 2 _
simpa only [mul_assoc] using hs.const_mul (2 : ℝ)
theorem integral_sq_error_eq_variance_add_bias_sq Lemma integral_sq_error_eq_variance_add_bias_sq in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, (X s - θ) ^ 2 ∂μ) = variance X μ + ((∫ s, X s ∂μ) - θ) ^ 2
Proof (Lean source)
-- @node: integral_sq_error_eq_variance_add_bias_sq
lemma integral_sq_error_eq_variance_add_bias_sq {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] {X : Ω → ℝ}
(hX : MemLp X 2 μ) (θ : ℝ) :
(∫ s, (X s - θ) ^ 2 ∂μ) = variance X μ + ((∫ s, X s ∂μ) - θ) ^ 2 := by
have hsub : MemLp (fun s ↦ X s - θ) 2 μ := hX.sub (memLp_const θ)
have hv := variance_eq_sub hsub
rw [variance_sub_const hX.aestronglyMeasurable θ] at hv
have hm : (∫ s, X s - θ ∂μ) = (∫ s, X s ∂μ) - θ := by
rw [integral_sub (hX.integrable (by norm_num)) (integrable_const θ)]
simp
simp only [Pi.pow_apply] at hv
rw [hm] at hv
linarith
theorem integral_sq_poissonRawMixedValue_error_eq
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, (poissonRawMixedValue n d q s - θ) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
+ ((∫ s, poissonRawMixedValue n d q s ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) - θ) ^ 2
Proof (Lean source)
-- @node: integral_sq_poissonRawMixedValue_error_eq
lemma integral_sq_poissonRawMixedValue_error_eq (n d : ℕ) (q θ : ℝ) (P : FullLaw d) :
(∫ s, (poissonRawMixedValue n d q s - θ) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) =
variance (poissonRawMixedValue n d q)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) +
((∫ s, poissonRawMixedValue n d q s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) - θ) ^ 2 := by
exact integral_sq_error_eq_variance_add_bias_sq _ (memLp_poissonRawMixedValue n d q P) θ
theorem integral_sq_poissonAuxiliaryMixedValue_le_raw
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
P :
FullLaw d
hP :
UnrestrictedArrivalModelClass n d q P
hlarge :
¬ effectiveSize n q ≤ 1
conclusion
(∫ s, (poissonAuxiliaryMixedValue n d q s - cellFunctional P) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
≤ ∫ s, (poissonRawMixedValue n d q s - cellFunctional P) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)
Proof (Lean source)
-- @node: integral_sq_poissonAuxiliaryMixedValue_le_raw
lemma integral_sq_poissonAuxiliaryMixedValue_le_raw (n d : ℕ) (q : ℝ) (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) (hlarge : ¬ effectiveSize n q ≤ 1) :
(∫ s, (poissonAuxiliaryMixedValue n d q s - cellFunctional P) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
∫ s, (poissonRawMixedValue n d q s - cellFunctional P) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2) := by
have haux : MemLp (poissonAuxiliaryMixedValue n d q) 2
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
apply memLp_of_bounded (a := -1) (b := 1) _
(measurable_poissonAuxiliaryMixedValue n d q).aestronglyMeasurable
filter_upwards [] with s
exact poissonAuxiliaryMixedValue_mem_Icc n d q s
apply integral_mono
(haux.sub (memLp_const (cellFunctional P))).integrable_sq
((memLp_poissonRawMixedValue n d q P).sub
(memLp_const (cellFunctional P))).integrable_sq
intro s
change (poissonAuxiliaryMixedValue n d q s - cellFunctional P) ^ 2 ≤
(poissonRawMixedValue n d q s - cellFunctional P) ^ 2
simp only [poissonAuxiliaryMixedValue, if_neg hlarge]
exact clip_sq_sub_le _ _ (cellFunctional_mem_Icc P hP)
theorem integral_sq_poissonAuxiliaryMixedValue_le_variance_add_active_bias_sq
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
P :
FullLaw d
hP :
UnrestrictedArrivalModelClass n d q P
hb :
needleBranch n d q
hlarge :
¬ effectiveSize n q ≤ 1
conclusion
(∫ s, (poissonAuxiliaryMixedValue n d q s - cellFunctional P) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
+ (2 * (4 * d * needleRadius n q / (streamEffectiveSize n q * (needleDegree n q : ℝ)^2) + 3 * exp (-16 * logScale n q))) ^ 2
Proof (Lean source)
-- @node: integral_sq_poissonAuxiliaryMixedValue_le_variance_add_active_bias_sq
lemma integral_sq_poissonAuxiliaryMixedValue_le_variance_add_active_bias_sq
(n d : ℕ) (q : ℝ) (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) (hb : needleBranch n d q)
(hlarge : ¬ effectiveSize n q ≤ 1) :
(∫ s, (poissonAuxiliaryMixedValue n d q s - cellFunctional P) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
variance (poissonRawMixedValue n d q)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) +
(2 * (4 * d * needleRadius n q /
(streamEffectiveSize n q * (needleDegree n q : ℝ)^2) +
3 * exp (-16 * logScale n q))) ^ 2 := by
have h := abs_integral_raw_poissonSelected_estimator_bias_le n d q P hP hb
change |(∫ s, poissonRawMixedValue n d q s
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) - cellFunctional P| ≤ _ at h
refine (integral_sq_poissonAuxiliaryMixedValue_le_raw n d q P hP hlarge).trans ?_
rw [integral_sq_poissonRawMixedValue_error_eq]
apply add_le_add_right
have hs := mul_self_le_mul_self (abs_nonneg _) h
simpa only [← sq, sq_abs] using hs
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.RiskGeometry 4 declarations Bounded-target and clipping geometry used by the risk certificate.
Bounded-target and clipping geometry used by the risk certificate.
theorem ate_mem_Icc_of_fullLaw
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Proof (Lean source)
lemma ate_mem_Icc_of_fullLaw {d : ℕ} (P : FullLaw d) :
ate P ∈ Icc (-1 : ℝ) 1 := by
rw [ate_eq_potentialOutcomeMass]
letI : IsProbabilityMeasure P.1 := P.2
have h0 : (0 : ℝ) ≤ P.1.real {r | r.Y0 = true} := measureReal_nonneg
have h1 : (0 : ℝ) ≤ P.1.real {r | r.Y1 = true} := measureReal_nonneg
have h0' : P.1.real {r | r.Y0 = true} ≤ 1 := measureReal_le_one
have h1' : P.1.real {r | r.Y1 = true} ≤ 1 := measureReal_le_one
constructor <;> linarith
theorem cellFunctional_mem_Icc
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
cellFunctional P ∈ Icc (-1 : ℝ) 1
Proof (Lean source)
lemma cellFunctional_mem_Icc {n d : ℕ} {q : ℝ} (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) :
cellFunctional P ∈ Icc (-1 : ℝ) 1 := by
rw [← unrestricted_cell_identification P hP]
exact ate_mem_Icc_of_fullLaw P
theorem clip_sq_sub_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
u theta :
ℝ
htheta :
theta ∈ Icc (-1 : ℝ) 1
conclusion
(clip u - theta) ^ 2 ≤ (u - theta) ^ 2
Proof (Lean source)
lemma clip_sq_sub_le (u theta : ℝ) (htheta : theta ∈ Icc (-1 : ℝ) 1) :
(clip u - theta) ^ 2 ≤ (u - theta) ^ 2 := by
rcases htheta with ⟨htheta0, htheta1⟩
unfold clip
by_cases hu0 : u < -1
· have hmax : max (-1) (min 1 u) = -1 := by
rw [max_eq_left]
exact (min_le_right 1 u).trans (le_of_lt hu0)
rw [hmax]
nlinarith [sq_nonneg (u - theta), sq_nonneg (-1 - theta)]
· have hu0' : -1 ≤ u := le_of_not_gt hu0
by_cases hu1 : 1 < u
· have hmax : max (-1) (min 1 u) = 1 := by
rw [min_eq_left (le_of_lt hu1), max_eq_right]
norm_num
rw [hmax]
nlinarith [sq_nonneg (u - theta), sq_nonneg (1 - theta)]
· have hu1' : u ≤ 1 := le_of_not_gt hu1
rw [min_eq_right hu1', max_eq_right hu0']
theorem sq_sub_le_four_of_mem_Icc
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma sq_sub_le_four_of_mem_Icc {u theta : ℝ}
(hu : u ∈ Icc (-1 : ℝ) 1)
(htheta : theta ∈ Icc (-1 : ℝ) 1) :
(u - theta) ^ 2 ≤ 4 := by
rcases hu with ⟨hu0, hu1⟩
rcases htheta with ⟨htheta0, htheta1⟩
nlinarith [sq_nonneg (u - theta + 2), sq_nonneg (2 - (u - theta))]
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.SignedVariance 3 declarations Cross-cell independence assembles the signed raw variance and the active ideal risk bound in roadmap (20)--(24), retaining random membership weights.
Cross-cell independence assembles the signed raw variance and the active ideal risk bound in roadmap (20)--(24), retaining random membership weights.
theorem variance_poissonRawMixedValue_eq_four_sum
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
= 4 * ∑ j : Cell d, variance (fun s ↦ poissonMemberWeight n j s * poissonSelectedCellEstimate n d q j s) (finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
Proof (Lean source)
-- @node: variance_poissonRawMixedValue_eq_four_sum
lemma variance_poissonRawMixedValue_eq_four_sum (n d : ℕ) (q : ℝ) (P : FullLaw d) :
variance (poissonRawMixedValue n d q)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) =
4 * ∑ j : Cell d, variance (fun s ↦ poissonMemberWeight n j s *
poissonSelectedCellEstimate n d q j s)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) := by
classical
let μ := finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)
let X (j : Cell d) := fun s ↦ poissonMemberWeight n j s *
poissonSelectedCellEstimate n d q j s
let Z (j : Cell d) := fun s ↦ armSign j.1 * X j s
have hZ (j : Cell d) : MemLp (Z j) 2 μ :=
(memLp_poissonMemberWeight_mul_selectedCellEstimate n d q P j).const_mul _
have hi : Pairwise (↑(Finset.univ : Finset (Cell d)))
(fun i j ↦ IndepFun (Z i) (Z j) μ) := by
intro i _ j _ hij
exact (indepFun_memberWeight_mul_selectedCellEstimate n d q P hij).comp
(measurable_const.mul measurable_id) (measurable_const.mul measurable_id)
have hv := IndepFun.variance_sum (fun j _ ↦ hZ j) hi
have hsign (j : Cell d) : armSign j.1 ^ 2 = 1 := by
cases j.1 <;> norm_num [armSign]
change variance (fun s ↦ 2 * ∑ j : Cell d, armSign j.1 *
poissonMemberWeight n j s * poissonSelectedCellEstimate n d q j s) μ = _
have heq : (fun s ↦ ∑ j : Cell d, armSign j.1 *
poissonMemberWeight n j s * poissonSelectedCellEstimate n d q j s) =
∑ j : Cell d, Z j := by
funext s
simp only [Finset.sum_apply, Z, X, mul_assoc]
rw [variance_const_mul, heq]
rw [hv]
norm_num only [show (2 : ℝ) ^ 2 = 4 by norm_num]
congr 1
apply Finset.sum_congr rfl
intro j _
change variance (fun s ↦ armSign j.1 * X j s) μ = _
rw [variance_const_mul, hsign, one_mul]
theorem variance_poissonRawMixedValue_le_active_budget
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
≤ 8 * (4 / streamEffectiveSize n q + 1 / streamSize n)
+ 64 * d * (exp (logScale n q / 8) + 1) * (needleRadius n q ^ 2 / streamEffectiveSize n q ^ 2 + needleRadius n q / (streamSize n * streamEffectiveSize n q))
Proof (Lean source)
-- @node: variance_poissonRawMixedValue_le_active_budget
lemma variance_poissonRawMixedValue_le_active_budget (n d : ℕ) (q : ℝ) (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) (hb : needleBranch n d q) :
variance (poissonRawMixedValue n d q)
(finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
8 * (4 / streamEffectiveSize n q + 1 / streamSize n) +
64 * d * (exp (logScale n q / 8) + 1) *
(needleRadius n q ^ 2 / streamEffectiveSize n q ^ 2 +
needleRadius n q / (streamSize n * streamEffectiveSize n q)) +
32 * exp (-32 * logScale n q) * (1 + 1 / streamSize n) := by
rw [variance_poissonRawMixedValue_eq_four_sum]
exact four_sum_variance_memberWeight_mul_selectedCellEstimate_le n d q P hP hb
theorem integral_sq_poissonAuxiliaryMixedValue_le_active_budget
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
P :
FullLaw d
hP :
UnrestrictedArrivalModelClass n d q P
hb :
needleBranch n d q
hlarge :
¬ effectiveSize n q ≤ 1
conclusion
(∫ s, (poissonAuxiliaryMixedValue n d q s - cellFunctional P) ^ 2 ∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2))
≤ (2 * (4 * d * needleRadius n q / (streamEffectiveSize n q * (needleDegree n q : ℝ)^2) + 3 * exp (-16 * logScale n q))) ^ 2
+ (8 * (4 / streamEffectiveSize n q + 1 / streamSize n) + 64 * d * (exp (logScale n q / 8) + 1) * (needleRadius n q ^ 2 / streamEffectiveSize n q ^ 2 + needleRadius n q / (streamSize n * streamEffectiveSize n q)) + 32 * exp (-32 * logScale n q) * (1 + 1 / streamSize n))
Proof (Lean source)
-- @node: integral_sq_poissonAuxiliaryMixedValue_le_active_budget
lemma integral_sq_poissonAuxiliaryMixedValue_le_active_budget
(n d : ℕ) (q : ℝ) (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) (hb : needleBranch n d q)
(hlarge : ¬ effectiveSize n q ≤ 1) :
(∫ s, (poissonAuxiliaryMixedValue n d q s - cellFunctional P) ^ 2
∂finitePoissonSampleLaw (markedObsLaw P) ((n : ℝ≥0) / 2)) ≤
(2 * (4 * d * needleRadius n q /
(streamEffectiveSize n q * (needleDegree n q : ℝ)^2) +
3 * exp (-16 * logScale n q))) ^ 2 +
(8 * (4 / streamEffectiveSize n q + 1 / streamSize n) +
64 * d * (exp (logScale n q / 8) + 1) *
(needleRadius n q ^ 2 / streamEffectiveSize n q ^ 2 +
needleRadius n q / (streamSize n * streamEffectiveSize n q)) +
32 * exp (-32 * logScale n q) * (1 + 1 / streamSize n)) := by
have hr := integral_sq_poissonAuxiliaryMixedValue_le_variance_add_active_bias_sq
n d q P hP hb hlarge
have hv := variance_poissonRawMixedValue_le_active_budget n d q P hP hb
exact hr.trans (by linarith)
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Certificate.StreamBridge 12 declarations Ordered-stream representation of the ideal marked Poisson estimator.
Ordered-stream representation of the ideal marked Poisson estimator.
theorem eventCount_unshuffle
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
eventCount (unshuffle s pool) (obsStreamEvent j arrived ones)
= eventCount s (streamEvent pool j arrived ones)
Proof (Lean source)
lemma eventCount_unshuffle (s : FiniteSample (ObsRecord d × Fin 3)) (pool : Fin 3)
(j : Cell d) (arrived ones : Bool) :
eventCount (unshuffle s pool) (obsStreamEvent j arrived ones) =
eventCount s (streamEvent pool j arrived ones) := by
classical
rcases s with ⟨k, z⟩
unfold eventCount unshuffle
simp only [FiniteSample.count, FiniteSample.points]
let w : Fin k → Fin 3 := fun i => (z i).2
let y : Fin k → ObsRecord d := fun i => (z i).1
let e (i : Fin (wordHistogram w pool)) : Fin k :=
wordUnshuffleEquiv w ⟨pool, i⟩
apply Finset.card_bij (fun i _ => e i)
· intro i hi
change i ∈ Finset.univ.filter (fun i =>
gatherWord (fun k => (z k).2) (fun k => (z k).1) pool i ∈
obsStreamEvent j arrived ones) at hi
change e i ∈ Finset.univ.filter (fun i => z i ∈
streamEvent pool j arrived ones)
rw [mem_filter] at hi ⊢
rcases hi with ⟨_, hi⟩
refine ⟨Finset.mem_univ _, ?_⟩
have hlabel := (wordFiberEquiv w pool i).property
change w (e i) = pool at hlabel
simpa [e, w, y, gatherWord, obsStreamEvent, streamEvent, hlabel] using hi
· intro i hi i' hi' he
have hsigma : (⟨pool, i⟩ : Σ a, Fin (wordHistogram w a)) = ⟨pool, i'⟩ :=
(wordUnshuffleEquiv w).injective he
cases hsigma
rfl
· intro i hi
change i ∈ Finset.univ.filter (fun i => z i ∈
streamEvent pool j arrived ones) at hi
rw [mem_filter] at hi
rcases hi with ⟨_, hi⟩
have hipool : w i = pool := by simpa [w, streamEvent] using hi.1
let u : {a : Fin k // w a = pool} := ⟨i, hipool⟩
let r : Fin (wordHistogram w pool) := (wordFiberEquiv w pool).symm u
refine ⟨r, ?_, ?_⟩
· change r ∈ Finset.univ.filter (fun r =>
gatherWord (fun k => (z k).2) (fun k => (z k).1) pool r ∈
obsStreamEvent j arrived ones)
rw [mem_filter]
refine ⟨Finset.mem_univ _, ?_⟩
have heq : e r = i := by
change ((wordFiberEquiv w pool) r).1 = i
exact congrArg val ((wordFiberEquiv w pool).apply_symm_apply u)
simpa [e, w, y, gatherWord, obsStreamEvent, streamEvent, heq] using hi.2
· change ((wordFiberEquiv w pool) r).1 = i
exact congrArg val ((wordFiberEquiv w pool).apply_symm_apply u)
def streamFactorialBranch
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
streamFactorialBranch n d q j s :
ℝ
given by
if needleBranch n d q then ∑ v ∈ Icc 1 (needleDegree n q - 1), needleCoeff n d q v * weightedFactorial (obsStreamEvent j true true) (obsStreamEvent j true false) v (s 2) else 0
def streamRatioBranch
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
streamRatioBranch j s :
ℝ
given by
if 0 < eventCount (s 2) (obsStreamEvent j true false) then eventCount (s 2) (obsStreamEvent j true true) / eventCount (s 2) (obsStreamEvent j true false) else 0
def streamSelectedCellEstimate
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def streamAuxiliaryMixedValue
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
streamAuxiliaryMixedValue n d q s :
ℝ
given by
if effectiveSize n q
≤ 1 then 0 else clip (2 * ∑ j : Cell d, armSign j.1 * (eventCount (s 0) (obsStreamEvent j false false) / streamSize n) * streamSelectedCellEstimate n d q j s)
theorem streamAuxiliaryMixedValue_unshuffle
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma streamAuxiliaryMixedValue_unshuffle (n d : ℕ) (q : ℝ)
(s : FiniteSample (ObsRecord d × Fin 3)) :
streamAuxiliaryMixedValue n d q (unshuffle s) =
poissonAuxiliaryMixedValue n d q s := by
unfold streamAuxiliaryMixedValue poissonAuxiliaryMixedValue
split_ifs
· rfl
· congr 2
unfold poissonMemberWeight
apply Finset.sum_congr rfl
intro j hj
rw [eventCount_unshuffle]
congr 2
unfold streamSelectedCellEstimate poissonSelectedCellEstimate
rw [eventCount_unshuffle]
split_ifs
· unfold streamFactorialBranch poissonFactorialBranch
split_ifs
· apply Finset.sum_congr rfl
intro v hv
unfold weightedFactorial
rw [eventCount_unshuffle, eventCount_unshuffle]
· rfl
· unfold streamRatioBranch poissonRatioBranch
rw [eventCount_unshuffle, eventCount_unshuffle]
theorem measurable_streamEventCount
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Measurable (fun s : Fin 3 → FiniteSample (ObsRecord d) ↦ eventCount (s pool) (obsStreamEvent j arrived ones))
Proof (Lean source)
lemma measurable_streamEventCount {d : ℕ} (pool : Fin 3)
(j : Cell d) (arrived ones : Bool) :
Measurable (fun s : Fin 3 → FiniteSample (ObsRecord d) ↦
eventCount (s pool) (obsStreamEvent j arrived ones)) := by
exact (measurable_eventCount _
(Set.Finite.measurableSet (Set.toFinite _))).comp
(measurable_pi_apply pool)
theorem measurable_streamFactorialBranch
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
@[fun_prop] lemma measurable_streamFactorialBranch (n d : ℕ) (q : ℝ)
(j : Cell d) : Measurable (streamFactorialBranch n d q j) := by
unfold streamFactorialBranch
split_ifs
· apply Finset.measurable_sum
intro v hv
exact (measurable_weightedFactorial
(obsStreamEvent j true true) (obsStreamEvent j true false)
(Set.Finite.measurableSet (Set.toFinite _))
(Set.Finite.measurableSet (Set.toFinite _)) v |>.comp
(measurable_pi_apply 2)).const_mul _
· fun_prop
theorem measurable_streamRatioBranch
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
@[fun_prop] lemma measurable_streamRatioBranch {d : ℕ} (j : Cell d) :
Measurable (streamRatioBranch j) := by
have hnum : Measurable (fun s : Fin 3 → FiniteSample (ObsRecord d) ↦
(eventCount (s 2) (obsStreamEvent j true true) : ℝ)) :=
(measurable_of_countable (fun z : ℕ ↦ (z : ℝ))).comp
(measurable_streamEventCount 2 j true true)
have hden : Measurable (fun s : Fin 3 → FiniteSample (ObsRecord d) ↦
(eventCount (s 2) (obsStreamEvent j true false) : ℝ)) :=
(measurable_of_countable (fun z : ℕ ↦ (z : ℝ))).comp
(measurable_streamEventCount 2 j true false)
unfold streamRatioBranch
apply Measurable.ite
· exact (measurable_streamEventCount 2 j true false) measurableSet_Ioi
· exact hnum.div hden
· fun_prop
theorem measurable_streamSelectedCellEstimate
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
@[fun_prop] lemma measurable_streamSelectedCellEstimate (n d : ℕ) (q : ℝ)
(j : Cell d) : Measurable (streamSelectedCellEstimate n d q j) := by
unfold streamSelectedCellEstimate
apply Measurable.ite
· by_cases hbranch : needleBranch n d q
· simp only [hbranch, true_and]
have hc : Measurable (fun s : Fin 3 → FiniteSample (ObsRecord d) ↦
(eventCount (s 1) (obsStreamEvent j true false) : ℝ)) :=
(measurable_of_countable (fun z : ℕ ↦ (z : ℝ))).comp
(measurable_streamEventCount 1 j true false)
exact hc measurableSet_Iic
· simp [hbranch]
· exact measurable_streamFactorialBranch n d q j
· exact measurable_streamRatioBranch j
theorem measurable_streamAuxiliaryMixedValue
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
@[fun_prop] lemma measurable_streamAuxiliaryMixedValue (n d : ℕ) (q : ℝ) :
Measurable (streamAuxiliaryMixedValue n d q) := by
unfold streamAuxiliaryMixedValue clip
split_ifs
· fun_prop
· apply Measurable.max measurable_const
apply Measurable.min measurable_const
apply Measurable.const_mul
apply Finset.measurable_sum
intro j hj
have hmember : Measurable
(fun s : Fin 3 → FiniteSample (ObsRecord d) ↦
(eventCount (s 0) (obsStreamEvent j false false) : ℝ) /
streamSize n) := by
apply Measurable.div_const
exact (measurable_of_countable (fun z : ℕ ↦ (z : ℝ))).comp
(measurable_streamEventCount 0 j false false)
exact (hmember.const_mul _).mul
(measurable_streamSelectedCellEstimate n d q j)
theorem streamAuxiliaryMixedValue_mem_Icc
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
streamAuxiliaryMixedValue n d q s ∈ Icc (-1 : ℝ) 1
Proof (Lean source)
lemma streamAuxiliaryMixedValue_mem_Icc (n d : ℕ) (q : ℝ)
(s : Fin 3 → FiniteSample (ObsRecord d)) :
streamAuxiliaryMixedValue n d q s ∈ Icc (-1 : ℝ) 1 := by
unfold streamAuxiliaryMixedValue clip
split_ifs
· norm_num
· exact ⟨le_max_left _ _,
(max_le_iff.mpr ⟨by norm_num, min_le_left _ _⟩)⟩
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.CompleteArrivalFamily 15 declarations A normalized one-cell complete-arrival Bernoulli family.
A normalized one-cell complete-arrival Bernoulli family.
def completeRecord
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def completeMeasure
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
given by
∑ a : Bool, ∑ y : Bool, ofReal (completeBernWeight (1/2) a * completeBernWeight u y) • Measure.dirac (completeRecord x a y)
theorem completeMeasure_univ
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma completeMeasure_univ {d : ℕ} (x : Fin d) (u : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) : completeMeasure x u univ = 1 := by
simp [completeMeasure, completeBernWeight]
have hu' : 0 ≤ 1 - u := by linarith [hu.2]
norm_num at ⊢
have hhalf : ofReal (1 / 2 : ℝ) = (2 : ℝ≥0∞)⁻¹ := by
rw [ENNReal.ofReal_div_of_pos (by norm_num)]
norm_num
rw [hhalf]
calc
_ = (2⁻¹ + 2⁻¹ : ℝ≥0∞) *
(ofReal u + ofReal (1 - u)) := by ring
_ = 1 := by rw [← ENNReal.ofReal_add hu.1 hu', ENNReal.inv_two_add_inv_two]; norm_num
def completeFamilyLaw
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
theorem completeFamilyLaw_ate
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma completeFamilyLaw_ate {d : ℕ} (x : Fin d) (u : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) : ate (completeFamilyLaw x u hu) = u := by
let f : FullRecord d → ℝ := fun r =>
(if r.Y1 then 1 else 0) - (if r.Y0 then 1 else 0)
have hInt (a y : Bool) : Integrable f
(ofReal (completeBernWeight (1/2) a * completeBernWeight u y) •
Measure.dirac (completeRecord x a y)) := by
exact (integrable_dirac (by simp)).smul_measure ENNReal.ofReal_ne_top
unfold ate completeFamilyLaw completeMeasure
change (∫ r, f r ∂(∑ a : Bool, ∑ y : Bool,
ofReal (completeBernWeight (1/2) a * completeBernWeight u y) •
Measure.dirac (completeRecord x a y))) = u
have hinner (a : Bool) :
(∫ r, f r ∂(∑ y : Bool,
ofReal (completeBernWeight (1/2) a * completeBernWeight u y) •
Measure.dirac (completeRecord x a y))) =
∑ y : Bool, ∫ r, f r ∂(ofReal
(completeBernWeight (1/2) a * completeBernWeight u y) •
Measure.dirac (completeRecord x a y)) := by
exact integral_finsetSum_measure (by intro y hy; exact hInt a y)
have hOuter : ∀ a : Bool, Integrable f (∑ y : Bool,
ofReal (completeBernWeight (1/2) a * completeBernWeight u y) •
Measure.dirac (completeRecord x a y)) := by
intro a
apply (integrable_finsetSum_measure).2
intro y hy
exact hInt a y
rw [integral_finsetSum_measure (by intro a ha; exact hOuter a)]
simp_rw [hinner]
simp [completeRecord, completeBernWeight, integral_smul_measure, integral_dirac]
have hu' : 0 ≤ 1 - u := by linarith [hu.2]
have hu2 : 0 ≤ (1 - 2⁻¹ : ℝ) * u := mul_nonneg (by norm_num) hu.1
simp [f, ENNReal.toReal_ofReal hu.1, ENNReal.toReal_ofReal hu']
rw [ENNReal.toReal_ofReal hu2]
ring
theorem completeFamilyLaw_obs_ac
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
((completeFamilyLaw x u hu).1.map obs) ≪ ((completeFamilyLaw x (1 / 2) (by norm_num : (0 : ℝ) ≤ 1 / 2 ∧ (1 / 2 : ℝ) ≤ 1)).1.map obs)
Proof (Lean source)
lemma completeFamilyLaw_obs_ac {d : ℕ} (x : Fin d) (u : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) :
((completeFamilyLaw x u hu).1.map obs) ≪
((completeFamilyLaw x (1 / 2) (by norm_num :
(0 : ℝ) ≤ 1 / 2 ∧ (1 / 2 : ℝ) ≤ 1)).1.map obs) := by
intro s hs
rw [Measure.map_apply (by fun_prop) (by simp)] at hs ⊢
simp [completeFamilyLaw, completeMeasure, completeBernWeight,
completeRecord] at hs ⊢
norm_num at hs
tauto
theorem completeFamilyLaw_obs_atom
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
o.S = false ∧
o.R
= true then (1 / 2 : ℝ) * (if o.A then completeBernWeight u o.RY else if o.RY then 0 else 1) else 0
Proof (Lean source)
lemma completeFamilyLaw_obs_atom {d : ℕ} (x : Fin d) (u : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) (o : ObsRecord d) :
((completeFamilyLaw x u hu).1.map obs).real {o} =
if o.X = x ∧ o.S = false ∧ o.R = true then
(1 / 2 : ℝ) *
(if o.A then completeBernWeight u o.RY else if o.RY then 0 else 1)
else 0 := by
have h1 : 0 ≤ u * (1 / 2 : ℝ) := mul_nonneg hu.1 (by norm_num)
have h2 : 0 ≤ (1 / 2 : ℝ) + u * (-1 / 2) := by nlinarith [hu.2]
rw [measureReal_def, Measure.map_apply (by fun_prop) (by simp)]
rcases o with ⟨X, A, S, R, RY⟩
by_cases hx : X = x
· subst X
cases A <;> cases S <;> cases R <;> cases RY <;>
simp [completeFamilyLaw, completeMeasure, completeBernWeight, completeRecord,
obs, FullRecord.Y, FullRecord.S, indicator, hu.1, sub_nonneg.mpr hu.2,
ENNReal.toReal_add, ENNReal.toReal_ofReal, h1, h2] <;>
try ring
rw [ENNReal.toReal_ofReal h1, ENNReal.toReal_ofReal h2]
ring
· have hneq : x ≠ X := Ne.symm hx
cases A <;> cases S <;> cases R <;> cases RY <;>
simp [completeFamilyLaw, completeMeasure, completeBernWeight, completeRecord,
obs, FullRecord.Y, FullRecord.S, indicator, hx, hneq]
theorem completeFamilyLaw_obs_chi
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
chiSqDiv ((completeFamilyLaw x u hu).1.map obs) ((completeFamilyLaw x (1 / 2) (by norm_num)).1.map obs)
= 2 * (u - 1 / 2) ^ 2
Proof (Lean source)
lemma completeFamilyLaw_obs_chi {d : ℕ} (x : Fin d) (u : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) :
chiSqDiv ((completeFamilyLaw x u hu).1.map obs)
((completeFamilyLaw x (1 / 2) (by norm_num)).1.map obs) =
2 * (u - 1 / 2) ^ 2 := by
let P₁ := completeFamilyLaw x u hu
let P₀ := completeFamilyLaw x (1 / 2)
(by norm_num : (0 : ℝ) ≤ 1 / 2 ∧ (1 / 2 : ℝ) ≤ 1)
letI : IsProbabilityMeasure P₁.1 := P₁.2
letI : IsProbabilityMeasure P₀.1 := P₀.2
let μ := P₁.1.map obs
let ν := P₀.1.map obs
letI : IsProbabilityMeasure μ := Measure.isProbabilityMeasure_map (by fun_prop)
letI : IsProbabilityMeasure ν := Measure.isProbabilityMeasure_map (by fun_prop)
have hac := completeFamilyLaw_obs_ac x u hu
have h := finite_one_add_chiSqDiv μ ν hac
dsimp [μ, ν, P₁, P₀] at h
simp_rw [completeFamilyLaw_obs_atom x u hu,
completeFamilyLaw_obs_atom x (1 / 2)
(by norm_num : (0 : ℝ) ≤ 1 / 2 ∧ (1 / 2 : ℝ) ≤ 1)] at h
let a : ObsRecord d := ⟨x, false, false, true, false⟩
let b : ObsRecord d := ⟨x, true, false, true, false⟩
let c : ObsRecord d := ⟨x, true, false, true, true⟩
let f : ObsRecord d → ℝ := fun o =>
(if o.X = x ∧ o.S = false ∧ o.R = true then
(1 / 2 : ℝ) * (if o.A then completeBernWeight u o.RY else if o.RY then 0 else 1)
else 0) ^ 2 /
(if o.X = x ∧ o.S = false ∧ o.R = true then
(1 / 2 : ℝ) *
(if o.A then completeBernWeight (1 / 2) o.RY else if o.RY then 0 else 1)
else 0)
have hsum : (∑ o : ObsRecord d, f o) =
∑ o ∈ ({a, b, c} : Finset (ObsRecord d)), f o := by
symm
apply Finset.sum_subset (Finset.subset_univ _)
intro o ho hn
rcases o with ⟨X, A, S, R, RY⟩
by_cases hx : X = x
· subst X
cases A <;> cases S <;> cases R <;> cases RY <;>
simp [f, a, b, c] at hn ⊢
· simp [f, hx]
have hvalue : (∑ o : ObsRecord d, f o) =
1 + 2 * (u - 1 / 2) ^ 2 := by
rw [hsum]
simp [f, a, b, c, completeBernWeight]
ring
change 1 + chiSqDiv
((completeFamilyLaw x u hu).1.map obs)
((completeFamilyLaw x (1 / 2) (by norm_num)).1.map obs) =
∑ o : ObsRecord d, f o at h
linarith [hvalue]
theorem completeFamilyLaw_balanced
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
BalancedRandomization (completeFamilyLaw x u hu)
Proof (Lean source)
lemma completeFamilyLaw_balanced {d : ℕ} (x : Fin d) (u : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) :
BalancedRandomization (completeFamilyLaw x u hu) := by
rw [BalancedRandomization, Measure.real]
simp [completeFamilyLaw, completeMeasure, completeBernWeight, completeRecord]
have hu' : 0 ≤ 1 - u := by linarith [hu.2]
have hw : ofReal u + ofReal (1 - u) = 1 := by
rw [← ENNReal.ofReal_add hu.1 hu']
norm_num
rw [← mul_add, hw]
norm_num
theorem completeFamilyLaw_consistency
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
SurrogateConsistency (completeFamilyLaw x u hu)
conclusion 2
OutcomeConsistency (completeFamilyLaw x u hu)
Proof (Lean source)
lemma completeFamilyLaw_consistency {d : ℕ} (x : Fin d) (u : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) :
SurrogateConsistency (completeFamilyLaw x u hu) ∧
OutcomeConsistency (completeFamilyLaw x u hu) := by
constructor
· filter_upwards [] with r
rfl
· filter_upwards [] with r
rfl
theorem completeFamilyLaw_arrivedCell
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma completeFamilyLaw_arrivedCell {d : ℕ} (x : Fin d) (u : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) (j : Cell d) :
arrivedCell (completeFamilyLaw x u hu) j =
cellProb (completeFamilyLaw x u hu) j := by
simp [arrivedCell, cellProb, Measure.real, completeFamilyLaw, completeMeasure,
completeBernWeight, completeRecord, inCell, indicator]
theorem completeFamilyLaw_mar
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
ArrivalMAR (completeFamilyLaw x u hu)
Proof (Lean source)
lemma completeFamilyLaw_mar {d : ℕ} (x : Fin d) (u : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) : ArrivalMAR (completeFamilyLaw x u hu) := by
intro j y r
cases r <;>
simp [Measure.real, completeFamilyLaw, completeMeasure,
completeBernWeight, completeRecord, cellProb, inCell, indicator] <;> ring
theorem completeFamilyLaw_randomized
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
RandomizedIndependence (completeFamilyLaw x u hu)
Proof (Lean source)
lemma completeFamilyLaw_randomized {d : ℕ} (x : Fin d) (u : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) :
RandomizedIndependence (completeFamilyLaw x u hu) := by
rw [RandomizedIndependence, indepFun_iff_measure_inter_preimage_eq_mul]
intro s t _ _
have hu' : 0 ≤ 1 - u := by linarith [hu.2]
have hw (v : ℝ) (hv : 0 ≤ v) :
ofReal ((1 - 2⁻¹ : ℝ) * v) = (2⁻¹ : ℝ≥0∞) * ofReal v := by
norm_num [ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 1 / 2)]
norm_num [ENNReal.ofReal_div_of_pos]
have hsum : ofReal u + ofReal (1 - u) = 1 := by
rw [← ENNReal.ofReal_add hu.1 hu']
norm_num
have htwo : (2⁻¹ : ℝ≥0∞) * 2 = 1 := by
rw [mul_two]
exact ENNReal.inv_two_add_inv_two
simp [completeFamilyLaw, completeMeasure, completeRecord, completeBernWeight,
indicator, hw u hu.1, hw (1-u) hu']
by_cases hsT : true ∈ s <;> by_cases hsF : false ∈ s <;>
by_cases htT : (x, false, false, false, true) ∈ t <;>
by_cases htF : (x, false, false, false, false) ∈ t <;>
simp [hsT, hsF, htT, htF, Set.mem_inter_iff] <;>
simp only [← mul_add, hsum, mul_one] <;>
simp only [← two_mul, ← mul_assoc, htwo, one_mul] <;>
rw [mul_comm (2 : ℝ≥0∞) (2⁻¹ : ℝ≥0∞), htwo] <;>
simp <;> simpa only [mul_comm] using htwo.symm
theorem completeFamilyLaw_in_unrestricted_class_of_randomized
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
hn :
1 ≤ n
hd :
1 ≤ d
x :
Fin d
u :
ℝ
hu :
0 ≤ u ∧ u ≤ 1
hR :
RandomizedIndependence (completeFamilyLaw x u hu)
conclusion
UnrestrictedArrivalModelClass n d 1 (completeFamilyLaw x u hu)
Proof (Lean source)
lemma completeFamilyLaw_in_unrestricted_class_of_randomized {n d : ℕ}
(hn : 1 ≤ n) (hd : 1 ≤ d) (x : Fin d) (u : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1)
(hR : RandomizedIndependence (completeFamilyLaw x u hu)) :
UnrestrictedArrivalModelClass n d 1 (completeFamilyLaw x u hu) := by
refine ⟨hn, hd, by norm_num, by norm_num, hR,
completeFamilyLaw_balanced x u hu,
(completeFamilyLaw_consistency x u hu).1,
(completeFamilyLaw_consistency x u hu).2,
completeFamilyLaw_mar x u hu, ?_⟩
intro j hj
rw [completeFamilyLaw_arrivedCell]
simp
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.CompleteArrivalTesting 2 declarations A quantitative testing floor for the complete-arrival two-point experiment.
A quantitative testing floor for the complete-arrival two-point experiment.
the stated mathematical conclusion holds.
Formal statement
conclusion
exp (1 / 8 : ℝ) < 2
Proof (Lean source)
lemma complete_arrival_exp_eighth_lt_two : exp (1 / 8 : ℝ) < 2 := by
have hsq : exp (1 / 2 : ℝ) * exp (1 / 2 : ℝ) = exp 1 := by
rw [← Real.exp_add]
norm_num
have hmon : exp (1 / 8 : ℝ) ≤ exp (1 / 2 : ℝ) := by
exact Real.exp_le_exp.mpr (by norm_num)
have hpos : 0 < exp (1 / 2 : ℝ) := Real.exp_pos _
nlinarith [Real.exp_one_lt_three]
theorem complete_arrival_product_testing_half Lemma complete_arrival_product_testing_half in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Ω :
n :
ℕ
hn :
1 ≤ n
P₁ P₀ :
hac :
P₁ ≪ P₀
hint :
Integrable (fun x => ((P₁.rnDeriv P₀ x).toReal - 1) ^ 2) P₀
hchi :
chiSqDiv P₁ P₀ ≤ (1 / 8 : ℝ) / n
Proof (Lean source)
lemma complete_arrival_product_testing_half {Ω : Type}
[MeasurableSpace Ω] [MeasurableSingletonClass Ω]
(n : ℕ) (hn : 1 ≤ n) (P₁ P₀ : Measure Ω)
[IsProbabilityMeasure P₁] [IsProbabilityMeasure P₀]
(hac : P₁ ≪ P₀)
(hint : Integrable (fun x => ((P₁.rnDeriv P₀ x).toReal - 1) ^ 2) P₀)
(hchi : chiSqDiv P₁ P₀ ≤ (1 / 8 : ℝ) / n) :
∀ A : Set (Fin n → Ω), MeasurableSet A →
(Measure.pi (fun _ : Fin n => P₁)).real Aᶜ +
(Measure.pi (fun _ : Fin n => P₀)).real A ≥ 1 / 2 := by
intro A hA
let Q₁ : Measure (Fin n → Ω) := Measure.pi (fun _ => P₁)
let Q₀ : Measure (Fin n → Ω) := Measure.pi (fun _ => P₀)
have hprod_ac : Q₁ ≪ Q₀ :=
Causalean.Mathlib.Probability.ProductAbsolutelyContinuous.pi_iid_absolutelyContinuous
P₁ P₀ hac n
have hprod_int : Integrable
(fun x => ((Q₁.rnDeriv Q₀ x).toReal - 1) ^ 2) Q₀ :=
Causalean.Stat.pi_iid_integrable_sq_dev P₁ P₀ hac hint n
have hchi_non : 0 ≤ chiSqDiv P₁ P₀ :=
Causalean.Stat.chiSqDiv_nonneg
have hnpos : (0 : ℝ) < n := by exact_mod_cast hn
have hmul : (n : ℝ) * chiSqDiv P₁ P₀ ≤ 1 / 8 := by
have h := mul_le_mul_of_nonneg_left hchi (le_of_lt hnpos)
field_simp [hnpos.ne'] at h
linarith
have hprod_chi : chiSqDiv Q₁ Q₀ ≤
exp (1 / 8 : ℝ) - 1 := by
have hident := one_add_chiSqDiv_pi_iid_general
P₁ P₀ hac hint n
change 1 + chiSqDiv Q₁ Q₀ =
(1 + chiSqDiv P₁ P₀) ^ n at hident
have hpow : (1 + chiSqDiv P₁ P₀) ^ n ≤
exp ((n : ℝ) * chiSqDiv P₁ P₀) := by
calc
_ ≤ (exp (chiSqDiv P₁ P₀)) ^ n :=
pow_le_pow_left₀ (by linarith)
(by linarith [Real.add_one_le_exp (chiSqDiv P₁ P₀)]) n
_ = _ := by rw [← Real.exp_nat_mul]
linarith [hpow, Real.exp_le_exp.mpr hmul]
letI : IsProbabilityMeasure Q₁ := by dsimp [Q₁]; infer_instance
letI : IsProbabilityMeasure Q₀ := by dsimp [Q₀]; infer_instance
have htv := Causalean.Stat.tvDist_le_half_sqrt_chiSqDiv
Q₁ Q₀ hprod_ac hprod_int
have hchi_lt : chiSqDiv Q₁ Q₀ < 1 := by
linarith [complete_arrival_exp_eighth_lt_two]
have htv_le : tvDist Q₁ Q₀ ≤ 1 / 2 := by
have hsqrt := Real.sqrt_le_sqrt (le_of_lt hchi_lt)
have h := htv.trans (mul_le_mul_of_nonneg_left
(by simpa only [Real.sqrt_one] using hsqrt) (by norm_num : (0 : ℝ) ≤ 1 / 2))
simpa using h
have htest := one_sub_tvDist_le_test
(μ := Q₁) (ν := Q₀) hA.compl
simpa only [compl_compl] using (le_trans (by linarith : (1 / 2 : ℝ) ≤
1 - tvDist Q₁ Q₀) htest)
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Estimator 26 declarations The finite-work mixed-count estimator and its deterministic risk envelope.
The finite-work mixed-count estimator and its deterministic risk envelope.
def streamSize
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n :
ℕ
def
streamSize n :
ℝ
given by
(n : ℝ) / 6
def streamEffectiveSize
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def needleRadius
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n d :
ℕ
q :
ℝ
def
needleBranch n d q :
Prop
def chebNeedle
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n d :
ℕ
q :
ℝ
def
given by
if needleBranch n d q then C (needleRadius n q / (2 * (needleDegree n q : ℝ) ^ 2)) * (1 - (T ℝ (needleDegree n q : ℤ)).comp (C 1 - C (2 / needleRadius n q) * X)).divX else 1
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n d :
ℕ
q :
ℝ
v :
ℕ
def
needleCoeff n d q v :
ℝ
given by
if needleBranch n d q then (1 - chebNeedle n d q).coeff v else 0
def fallingFactorial
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
z v :
ℕ
def
fallingFactorial z v :
ℝ
given by
(z.descFactorial v : ℝ)
def streamCount
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
streamCount records assign pool j arrived ones :
ℕ
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def pilotCount
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def arrivedCount
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
memberWeight n records assign j :
ℝ
given by
memberCount records assign j / streamSize n
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
ratioBranch records assign j :
ℝ
given by
if 0 < arrivedCount records assign j then arrivedOneCount records assign j / arrivedCount records assign j else 0
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
factorialBranch n q records assign j :
ℝ
given by
if needleBranch n d q then ∑ v ∈ Icc 1 (needleDegree n q - 1), needleCoeff n d q v * arrivedOneCount records assign j * fallingFactorial (arrivedCount records assign j - 1) (v - 1) else 0
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
selectedCellEstimate n q records assign j :
ℝ
given by
if needleBranch n d q ∧
pilotCount records assign j
def clip
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def auxiliaryMixedValue
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
auxiliaryMixedValue n q records assign :
ℝ
given by
if effectiveSize n q
≤ 1 then 0 else clip (2 * ∑ j : Cell d, armSign j.1 * memberWeight n records assign j * selectedCellEstimate n q records assign j)
def mixedCountEstimator
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
mixedCountEstimator n d q sample :
ℝ
given by
∑ k ∈ range (n + 1), if h : k
≤ n then (poissonMeasure ((n : ℝ≥0) / 2)).real ({k} : Set ℕ) * ((∑ assign : Fin k → Fin 3, auxiliaryMixedValue n q (fun i => sample (castLE h i)) assign) / (3 : ℝ) ^ k) else 0
theorem mixed_count_estimator_regular Assumption ass:filler:39:prop:fixed-d-effective-sample-reduction:27:a_fixed_d_estimator_regular in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: mixed_count_estimator_regular
lemma mixed_count_estimator_regular (n d : ℕ) (q : ℝ) :
Measurable (mixedCountEstimator n d q) ∧
∀ s, mixedCountEstimator n d q s ∈ Icc (-1 : ℝ) 1 := by
constructor
· exact measurable_of_finite _
· intro s
have hclip (k : ℕ) (records : Fin k → ObsRecord d)
(assign : Fin k → Fin 3) :
-1 ≤ auxiliaryMixedValue n q records assign ∧
auxiliaryMixedValue n q records assign ≤ 1 := by
unfold auxiliaryMixedValue clip
split_ifs <;> constructor <;> simp [le_max_iff, max_le_iff, min_le_iff]
have hweight (k : ℕ) :
0 ≤ (poissonMeasure ((n : ℝ≥0) / 2)).real ({k} : Set ℕ) :=
measureReal_nonneg
have hmass :
(∑ k ∈ range (n + 1),
(poissonMeasure ((n : ℝ≥0) / 2)).real ({k} : Set ℕ)) ≤ 1 := by
have h := MeasureTheory.sum_measureReal_le_measureReal_univ
(μ := poissonMeasure ((n : ℝ≥0) / 2))
(s := range (n + 1)) (t := fun k : ℕ => ({k} : Set ℕ))
have hm : ∀ k ∈ range (n + 1), MeasurableSet ({k} : Set ℕ) :=
fun _ _ => MeasurableSet.singleton _
have hd : (↑(range (n + 1)) : Set ℕ).PairwiseDisjoint
(fun k : ℕ => ({k} : Set ℕ)) := by
intro i hi j hj hij
exact Set.disjoint_singleton.mpr hij
simpa using h hm hd
have havg (k : ℕ) (h : k ≤ n) :
-1 ≤ (∑ assign : Fin k → Fin 3,
auxiliaryMixedValue n q (fun i => s (castLE h i)) assign) /
(3 : ℝ) ^ k ∧
(∑ assign : Fin k → Fin 3,
auxiliaryMixedValue n q (fun i => s (castLE h i)) assign) /
(3 : ℝ) ^ k ≤ 1 := by
have hp : (0 : ℝ) < 3 ^ k := by positivity
have hc : (card (Fin k → Fin 3) : ℝ) = 3 ^ k := by simp
constructor
· apply (le_div_iff₀ hp).2
rw [neg_mul, one_mul, ← hc]
simpa using Finset.sum_le_sum (s := univ)
(fun a _ => (hclip k _ a).1)
· apply (div_le_iff₀ hp).2
rw [one_mul, ← hc]
simpa using Finset.sum_le_sum (s := univ)
(fun a _ => (hclip k _ a).2)
have hlow :
-(∑ k ∈ range (n + 1),
(poissonMeasure ((n : ℝ≥0) / 2)).real ({k} : Set ℕ)) ≤
mixedCountEstimator n d q s := by
unfold mixedCountEstimator
rw [← Finset.sum_neg_distrib]
apply Finset.sum_le_sum
intro k hk
have h : k ≤ n := by simpa using hk
simp only [dif_pos h]
simpa only [mul_neg, mul_one] using
mul_le_mul_of_nonneg_left (havg k h).1 (hweight k)
have hupp : mixedCountEstimator n d q s ≤
∑ k ∈ range (n + 1),
(poissonMeasure ((n : ℝ≥0) / 2)).real ({k} : Set ℕ) := by
unfold mixedCountEstimator
apply Finset.sum_le_sum
intro k hk
have h : k ≤ n := by simpa using hk
simp only [dif_pos h]
simpa only [mul_one] using
mul_le_mul_of_nonneg_left (havg k h).2 (hweight k)
exact ⟨by linarith, hupp.trans hmass⟩
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
auxiliaryMixedRisk n q P :
ℝ
given by
∫ sample, (∑ k ∈ range (n + 1), if h : k ≤ n then (poissonMeasure ((n : ℝ≥0) / 2)).real ({k} : Set ℕ) * ((∑ assign : Fin k → Fin 3, (auxiliaryMixedValue n q (fun i => sample (castLE h i)) assign - ate P) ^ 2) / (3 : ℝ) ^ k) else 0)
+ (1 - ∑ k ∈ range (n + 1), (poissonMeasure ((n : ℝ≥0) / 2)).real ({k} : Set ℕ)) * (ate P) ^ 2 ∂(sampleLaw n P)
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n d :
ℕ
q :
ℝ
def
riskEnvelope n d q :
ℝ
given by
N :
effectiveSize n q
m :
N₀ :
ell :
logScale n q
k :
needleDegree n q
B :
needleRadius n q
γ :
log 2 - 1 / 2
if N
≤ 1 then 1 else if ¬ needleBranch n d q then min 4 (((8 * d / (exp 1 * N₀)) ^ 2) + 4 * (4 / N₀ + 1 / m) + 4 * exp (-(n : ℝ) * γ)) else let b := 2 * (4 * d * B / (N₀ * (k : ℝ) ^ 2) + 3 * exp (-16 * ell)) let v := 8 * (4 / N₀ + 1 / m)
+ 64 * d * (exp (ell / 8) + 1) * (B ^ 2 / N₀ ^ 2 + B / (m * N₀))
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
riskEnvelopeEndpoints n d q α sample :
ℝ × ℝ
given by
(max (-1) (mixedCountEstimator n d q sample - sqrt (riskEnvelope n d q / α)), min 1 (mixedCountEstimator n d q sample + sqrt (riskEnvelope n d q / α)))
def riskEnvelopeInterval
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
given by
Icc (riskEnvelopeEndpoints n d q α sample).1 (riskEnvelopeEndpoints n d q α sample).2
def completeArrivalHTEstimator
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Identification 7 declarations Finite-cell identification under randomization, consistency, and MAR.
Finite-cell identification under randomization, consistency, and MAR.
theorem cellContribution_eq_cellOutcomeMass
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma cellContribution_eq_cellOutcomeMass {n d : ℕ} {q : ℝ}
(P : FullLaw d) (hP : UnrestrictedArrivalModelClass n d q P)
(j : Cell d) :
cellContribution P j = P.1.real {r | inCell r j ∧ r.Y = true} := by
classical
letI : IsProbabilityMeasure P.1 := P.2
by_cases hp : 0 < cellProb P j
· have ha : 0 < arrivedCell P j :=
lt_of_lt_of_le (mul_pos hP.q_pos hp) (hP.arrival j hp)
have hm := hP.mar j true true
have hmass : cellContribution P j =
cellProb P j *
(P.1.real {r | inCell r j ∧ r.R = true ∧ r.Y = true} /
arrivedCell P j) := by
simp [cellContribution, cellMean, ha]
rw [hmass]
have : arrivedCell P j ≠ 0 := ne_of_gt ha
calc
cellProb P j *
(P.1.real {r | inCell r j ∧ r.R = true ∧ r.Y = true} /
arrivedCell P j) =
(P.1.real {r | inCell r j ∧ r.R = true ∧ r.Y = true} *
cellProb P j) / arrivedCell P j := by ring
_ = P.1.real {r | inCell r j ∧ r.Y = true} := by
rw [hm]
have ha' : P.1.real {r | inCell r j ∧ r.R = true} ≠ 0 := by
simpa only [arrivedCell] using this
exact mul_div_cancel_left₀ _ ha'
· have hzero : cellProb P j = 0 :=
le_antisymm (le_of_not_gt hp) (measureReal_nonneg)
have hsub : P.1.real {r | inCell r j ∧ r.Y = true} ≤ cellProb P j := by
exact measureReal_mono (by intro r hr; exact hr.1)
have hy : P.1.real {r | inCell r j ∧ r.Y = true} = 0 :=
le_antisymm (hsub.trans_eq hzero) measureReal_nonneg
have ha : arrivedCell P j = 0 :=
le_antisymm ((measureReal_mono (by intro r hr; exact hr.1)).trans_eq hzero)
measureReal_nonneg
simp [cellContribution, ha, hy]
theorem sum_cellOutcomeMass_eq_armOutcomeMass
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma sum_cellOutcomeMass_eq_armOutcomeMass {d : ℕ} (P : FullLaw d) (a : Bool) :
(∑ x : Fin d, ∑ s : Bool,
P.1.real {r | inCell r (a, x, s) ∧ r.Y = true}) =
P.1.real {r | r.A = a ∧ r.Y = true} := by
classical
letI : IsProbabilityMeasure P.1 := P.2
let μ := P.1.restrict {r | r.A = a ∧ r.Y = true}
have hμ : IsFiniteMeasure μ := inferInstance
have hsum := sum_measureReal_preimage_singleton (μ := μ)
(Finset.univ : Finset (Fin d × Bool))
(f := fun r : FullRecord d => (r.X, r.S)) (by intro; simp)
calc
(∑ x : Fin d, ∑ s : Bool,
P.1.real {r | inCell r (a, x, s) ∧ r.Y = true}) =
∑ j : Fin d × Bool, μ.real ((fun r : FullRecord d => (r.X, r.S)) ⁻¹' {j}) := by
rw [← Finset.sum_product']
apply Finset.sum_congr rfl
intro x hx
simp only [μ, measureReal_def, Measure.restrict_apply]
have hs : {r : FullRecord d | inCell r (a, x.1, x.2) ∧ r.Y = true} =
((fun r : FullRecord d => (r.X, r.S)) ⁻¹' {x}) ∩
{r | r.A = a ∧ r.Y = true} := by
rcases x with ⟨x, s⟩
ext r
simp only [Set.mem_ofPred_eq, Set.mem_inter_iff, Set.mem_preimage,
Set.mem_singleton_iff, Prod.mk.injEq, inCell]
tauto
rw [hs]
rw [Measure.restrict_apply (by simp)]
_ = μ.real ((fun r : FullRecord d => (r.X, r.S)) ⁻¹' (Finset.univ : Finset (Fin d × Bool))) := hsum
_ = P.1.real {r | r.A = a ∧ r.Y = true} := by
simp [μ, measureReal_def, Measure.restrict_apply]
theorem randomized_armOutcomeMass
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma randomized_armOutcomeMass {d : ℕ} (P : FullLaw d)
(hR : RandomizedIndependence P) (hB : BalancedRandomization P)
(a : Bool) :
2 * P.1.real {r | r.A = a ∧ r.Y = true} =
P.1.real {r | (if a then r.Y1 else r.Y0) = true} := by
classical
letI : IsProbabilityMeasure P.1 := P.2
let y : FullRecord d → Bool := fun r => if a then r.Y1 else r.Y0
have hInd : IndepFun (fun r : FullRecord d => r.A) y P.1 := by
have h := hR.comp (φ := id)
(ψ := fun z : Fin d × Bool × Bool × Bool × Bool => if a then z.2.2.2.2 else z.2.2.2.1)
(by fun_prop) (by fun_prop)
simpa [y, Function.comp_def] using h
have hm := hInd.measure_inter_preimage_eq_mul ({a} : Set Bool) ({true} : Set Bool)
(by simp) (by simp)
change P.1 {r | r.A = a ∧ y r = true} =
P.1 {r | r.A = a} * P.1 {r | y r = true} at hm
have hmreal : P.1.real {r | r.A = a ∧ y r = true} =
P.1.real {r | r.A = a} * P.1.real {r | y r = true} := by
simpa only [measureReal_def, ENNReal.toReal_mul] using
congrArg ENNReal.toReal hm
have ha : P.1.real {r | r.A = a} = 1 / 2 := by
cases a with
| true => exact hB
| false =>
have hcomp : {r : FullRecord d | r.A = false} =
{r : FullRecord d | r.A = true}ᶜ := by ext r; cases r.A <;> simp
rw [hcomp, measureReal_compl (by simp), hB]
norm_num
rw [show {r : FullRecord d | r.A = a ∧ r.Y = true} =
{r | r.A = a ∧ y r = true} from by
ext r
simp only [Set.mem_setOf_eq]
constructor
· rintro ⟨h, hy⟩
simpa [y, FullRecord.Y, h] using hy
· rintro ⟨h, hy⟩
exact ⟨h, by simpa [y, FullRecord.Y, h] using hy⟩]
rw [hmreal, ha]
simp [y]
theorem cellFunctional_eq_armOutcomeMass
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma cellFunctional_eq_armOutcomeMass {n d : ℕ} {q : ℝ}
(P : FullLaw d) (hP : UnrestrictedArrivalModelClass n d q P) :
cellFunctional P =
2 * (P.1.real {r | r.A = true ∧ r.Y = true} -
P.1.real {r | r.A = false ∧ r.Y = true}) := by
classical
unfold cellFunctional
rw [Fintype.sum_prod_type]
simp_rw [cellContribution_eq_cellOutcomeMass P hP]
simp [Fintype.univ_bool, armSign, Fintype.sum_prod_type]
have ht := sum_cellOutcomeMass_eq_armOutcomeMass P true
have hf := sum_cellOutcomeMass_eq_armOutcomeMass P false
simp [Fintype.univ_bool] at ht hf
rw [ht, hf]
ring
theorem ate_eq_potentialOutcomeMass
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma ate_eq_potentialOutcomeMass {d : ℕ} (P : FullLaw d) :
ate P = P.1.real {r | r.Y1 = true} - P.1.real {r | r.Y0 = true} := by
classical
letI : IsProbabilityMeasure P.1 := P.2
have hv (a : Bool) :
(fun r : FullRecord d => if (if a then r.Y1 else r.Y0) then (1 : ℝ) else 0) =
({r : FullRecord d | (if a then r.Y1 else r.Y0) = true}.indicator 1) := by
funext r
rcases r with ⟨x, a', s₀, s₁, y₀, y₁, arrival⟩
cases a <;> cases y₀ <;> cases y₁ <;> simp [indicator]
have hi (a : Bool) :
Integrable (fun r : FullRecord d =>
if (if a then r.Y1 else r.Y0) then (1 : ℝ) else 0) P.1 := by
rw [hv]
exact (integrable_const (1 : ℝ)).indicator (by simp)
have he (a : Bool) :
(∫ r : FullRecord d,
(if (if a then r.Y1 else r.Y0) then (1 : ℝ) else 0) ∂P.1) =
P.1.real {r | (if a then r.Y1 else r.Y0) = true} := by
rw [hv]
exact integral_indicator_one (by simp)
unfold ate
rw [integral_sub (by simpa using hi true) (by simpa using hi false)]
simpa using congrArg₂ (· - ·) (he true) (he false)
theorem unrestricted_cell_identification
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
ate P = cellFunctional P
Proof (Lean source)
lemma unrestricted_cell_identification {n d : ℕ} {q : ℝ} (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) :
ate P = cellFunctional P := by
rw [ate_eq_potentialOutcomeMass P, cellFunctional_eq_armOutcomeMass P hP]
have ht := randomized_armOutcomeMass P hP.randomized hP.balanced true
have hf := randomized_armOutcomeMass P hP.randomized hP.balanced false
have ht' : 2 * P.1.real {r | r.A = true ∧ r.Y = true} =
P.1.real {r | r.Y1 = true} := by simpa using ht
have hf' : 2 * P.1.real {r | r.A = false ∧ r.Y = true} =
P.1.real {r | r.Y0 = true} := by simpa using hf
linear_combination -ht' + hf'
theorem cell_identification_background
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
ate P = cellFunctional P
Proof (Lean source)
lemma cell_identification_background {n d : ℕ} {q : ℝ} (P : FullLaw d)
(hP : RareArrivalModelClass n d q P) :
ate P = cellFunctional P := by
apply unrestricted_cell_identification P
exact
{ n_pos := hP.n_pos
d_pos := hP.d_pos
q_pos := hP.q_pos
q_le_one := hP.q_le_one
randomized := hP.randomized
balanced := hP.balanced
surrogate := hP.surrogate
outcome := hP.outcome
mar := hP.mar
arrival := hP.arrival }
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.IntervalConcentration 7 declarations Variance and Chebyshev concentration of the latent rare-cell target in equations (8) and (12) of the connected-interval lower bound.
Variance and Chebyshev concentration of the latent rare-cell target in equations (8) and (12) of the connected-interval lower bound. The connection of this target to the activated full-law ATE is supplied separately.
theorem finiteReciprocalPrior_reciprocal_ae_unit Lemma finiteReciprocalPrior_reciprocal_ae_unit in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
∀ᵐ z ∂π, z⁻¹ ∈ Icc (0 : ℝ) 1
Proof (Lean source)
-- @node: finiteReciprocalPrior_reciprocal_ae_unit
lemma finiteReciprocalPrior_reciprocal_ae_unit {H : ℝ} {π : Measure ℝ}
(hπ : FiniteReciprocalPrior H π) : ∀ᵐ z ∂π, z⁻¹ ∈ Icc (0 : ℝ) 1 := by
letI := hπ.1
obtain ⟨s, hs, hsupport⟩ := hπ.2
have hae : ∀ᵐ z ∂π, z ∈ (s : Set ℝ) :=
(ae_mem_iff_measure_eq s.measurableSet.nullMeasurableSet).2 (by simpa using hs)
filter_upwards [hae] with z hz
have hlow := (hsupport z hz).1
exact ⟨inv_nonneg.mpr (by linarith), inv_le_one_of_one_le₀ hlow⟩
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
theorem intervalPriorTarget_integral
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ z, intervalPriorTarget J b z ∂Measure.pi (fun _ : Fin J => π))
= (J : ℝ) * b * ∫ z, z⁻¹ ∂π
Proof (Lean source)
-- @node: intervalPriorTarget_integral
lemma intervalPriorTarget_integral {H : ℝ} {π : Measure ℝ}
(hπ : FiniteReciprocalPrior H π) (J : ℕ) (b : ℝ) :
(∫ z, intervalPriorTarget J b z ∂Measure.pi (fun _ : Fin J => π)) =
(J : ℝ) * b * ∫ z, z⁻¹ ∂π := by
letI := hπ.1
have hf : Integrable (fun z : ℝ => z⁻¹) π := finiteReciprocalPrior_integrable hπ _
unfold intervalPriorTarget
rw [integral_const_mul, integral_finsetSum]
· simp_rw [integral_comp_eval (μ := fun _ : Fin J => π) hf.aestronglyMeasurable]
simp only [Finset.sum_const, Finset.card_fin, nsmul_eq_mul]
ring
· intro i hi
exact integrable_comp_eval (μ := fun _ : Fin J => π) (i := i) hf
theorem intervalPriorTarget_variance_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: intervalPriorTarget_variance_le
lemma intervalPriorTarget_variance_le {H : ℝ} {π : Measure ℝ}
(hπ : FiniteReciprocalPrior H π) (J : ℕ) (b : ℝ) :
variance (intervalPriorTarget J b) (Measure.pi (fun _ : Fin J => π)) ≤
(J : ℝ) * b ^ 2 / 4 := by
letI := hπ.1
have hbound := finiteReciprocalPrior_reciprocal_ae_unit hπ
have hf : MemLp (fun z : ℝ => z⁻¹) 2 π :=
memLp_of_bounded hbound (by fun_prop) 2
have hvar : variance (fun z : ℝ => z⁻¹) π ≤ 1 / 4 := by
have h := variance_le_sq_of_bounded hbound (show AEMeasurable (fun z : ℝ => z⁻¹) π by fun_prop)
norm_num at h ⊢
exact h
unfold intervalPriorTarget
rw [variance_const_mul]
have heq : (fun z : Fin J → ℝ => ∑ i, (z i)⁻¹) =
∑ i : Fin J, fun z : Fin J → ℝ => (z i)⁻¹ := by
funext z
simp only [Finset.sum_apply]
rw [heq, variance_sum_pi (fun _ => hf)]
simp only [Finset.sum_const, Finset.card_fin, nsmul_eq_mul]
nlinarith [mul_le_mul_of_nonneg_left hvar (show 0 ≤ (J : ℝ) * b ^ 2 by positivity)]
theorem intervalPriorTarget_chebyshev
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(Measure.pi (fun _ : Fin J => π)).real {z | w < |intervalPriorTarget J b z - (J : ℝ) * b * ∫ z, z⁻¹ ∂π|}
≤ ((J : ℝ) * b ^ 2 / 4) / w ^ 2
Proof (Lean source)
-- @node: intervalPriorTarget_chebyshev
lemma intervalPriorTarget_chebyshev {H : ℝ} {π : Measure ℝ}
(hπ : FiniteReciprocalPrior H π) (J : ℕ) (b w : ℝ) (hw : 0 < w) :
(Measure.pi (fun _ : Fin J => π)).real
{z | w < |intervalPriorTarget J b z - (J : ℝ) * b * ∫ z, z⁻¹ ∂π|} ≤
((J : ℝ) * b ^ 2 / 4) / w ^ 2 := by
letI := hπ.1
have hf : MemLp (fun z : ℝ => z⁻¹) 2 π :=
memLp_of_bounded (finiteReciprocalPrior_reciprocal_ae_unit hπ) (by fun_prop) 2
have hsum : MemLp (intervalPriorTarget J b) 2 (Measure.pi (fun _ : Fin J => π)) := by
have heq : intervalPriorTarget J b =
fun z : Fin J → ℝ => b * (∑ i : Fin J, fun z : Fin J → ℝ => (z i)⁻¹) z := by
funext z
simp [intervalPriorTarget, Finset.sum_apply]
rw [heq]
exact (memLp_finsetSum' univ fun i hi =>
hf.comp_measurePreserving (measurePreserving_eval (fun _ : Fin J => π) i)).const_mul b
have htail := meas_ge_le_variance_div_sq hsum hw
rw [intervalPriorTarget_integral hπ J b] at htail
have hmono : (Measure.pi (fun _ : Fin J => π))
{z | w < |intervalPriorTarget J b z - (J : ℝ) * b * ∫ z, z⁻¹ ∂π|} ≤
ofReal (((J : ℝ) * b ^ 2 / 4) / w ^ 2) := by
have hsubset :
{z : Fin J → ℝ | w < |intervalPriorTarget J b z - (J : ℝ) * b * ∫ z, z⁻¹ ∂π|} ⊆
{z | w ≤ |intervalPriorTarget J b z - (J : ℝ) * b * ∫ z, z⁻¹ ∂π|} :=
by
intro z hz
simp only [Set.mem_setOf_eq] at hz ⊢
exact le_of_lt hz
refine (measure_mono hsubset).trans (htail.trans ?_)
exact ENNReal.ofReal_le_ofReal
(div_le_div_of_nonneg_right (intervalPriorTarget_variance_le hπ J b) (sq_nonneg w))
have hreal := ENNReal.toReal_mono (by simp) hmono
simpa only [Measure.real, ENNReal.toReal_ofReal (by positivity :
0 ≤ ((J : ℝ) * b ^ 2 / 4) / w ^ 2)] using hreal
theorem intervalPriorTarget_concentration_budget
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
H :
ℝ
π :
Measure ℝ
hπ :
J M :
ℕ
b Δ u :
ℝ
hb :
0 < b
hΔ :
0 < Δ
hu :
0 < u
hM :
1 ≤ M
hgap :
(J : ℝ) * b / 12 ≤ Δ
hJ :
18432 * (M : ℝ) ^ 2 / u ≤ (J : ℝ)
conclusion
(Measure.pi (fun _ : Fin J => π)).real {z | Δ / (8 * M) < |intervalPriorTarget J b z - (J : ℝ) * b * ∫ z, z⁻¹ ∂π|}
≤ u / 8
Proof (Lean source)
-- @node: intervalPriorTarget_concentration_budget
lemma intervalPriorTarget_concentration_budget {H : ℝ} {π : Measure ℝ}
(hπ : FiniteReciprocalPrior H π) (J M : ℕ) (b Δ u : ℝ)
(hb : 0 < b) (hΔ : 0 < Δ) (hu : 0 < u) (hM : 1 ≤ M)
(hgap : (J : ℝ) * b / 12 ≤ Δ)
(hJ : 18432 * (M : ℝ) ^ 2 / u ≤ (J : ℝ)) :
(Measure.pi (fun _ : Fin J => π)).real
{z | Δ / (8 * M) <
|intervalPriorTarget J b z - (J : ℝ) * b * ∫ z, z⁻¹ ∂π|} ≤ u / 8 := by
have hMr : 0 < (M : ℝ) := by exact_mod_cast (by omega : 0 < M)
have hJr : 0 < (J : ℝ) := lt_of_lt_of_le (by positivity) hJ
have hJu : 18432 * (M : ℝ) ^ 2 ≤ (J : ℝ) * u := (div_le_iff₀ hu).mp hJ
apply (intervalPriorTarget_chebyshev hπ J b (Δ / (8 * M)) (by positivity)).trans
apply (div_le_iff₀ (by positivity : 0 < (Δ / (8 * M)) ^ 2)).mpr
have hsq : ((J : ℝ) * b) ^ 2 ≤ 144 * Δ ^ 2 := by
have hlinear : (J : ℝ) * b ≤ 12 * Δ := by linarith
have hs := mul_self_le_mul_self (by positivity : 0 ≤ (J : ℝ) * b) hlinear
nlinarith
have hprod := mul_le_mul_of_nonneg_left hsq hu.le
have hcount := mul_le_mul_of_nonneg_right hJu (sq_nonneg b)
field_simp
nlinarith
theorem intervalInterpolatedPrior_grid_concentration
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
H :
ℝ
π₀ π₁ :
Measure ℝ
h₀ :
h₁ :
J M :
ℕ
b u :
ℝ
hb :
0 < b
hu :
0 < u
hM :
1 ≤ M
hgap :
1 / 12 ≤ |(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀|
hJ :
18432 * (M : ℝ) ^ 2 / u ≤ (J : ℝ)
j :
Fin (M + 1)
conclusion
(Measure.pi (fun _ : Fin J => intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))).real {z | ((J : ℝ) * b * |(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀|) / (8 * M) < |intervalPriorTarget J b z - ((J : ℝ) * b * (∫ z, z⁻¹ ∂π₀) + (j : ℝ) * ((J : ℝ) * b * ((∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀) / M))|}
≤ u / 8
Proof (Lean source)
-- @node: intervalInterpolatedPrior_grid_concentration
lemma intervalInterpolatedPrior_grid_concentration {H : ℝ} {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior H π₀) (h₁ : FiniteReciprocalPrior H π₁)
(J M : ℕ) (b u : ℝ) (hb : 0 < b) (hu : 0 < u) (hM : 1 ≤ M)
(hgap : 1 / 12 ≤ |(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀|)
(hJ : 18432 * (M : ℝ) ^ 2 / u ≤ (J : ℝ)) :
∀ j : Fin (M + 1),
(Measure.pi (fun _ : Fin J =>
intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))).real
{z | ((J : ℝ) * b * |(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀|) / (8 * M) <
|intervalPriorTarget J b z -
((J : ℝ) * b * (∫ z, z⁻¹ ∂π₀) +
(j : ℝ) * ((J : ℝ) * b *
((∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀) / M))|} ≤ u / 8 := by
have hMr : 0 < (M : ℝ) := by exact_mod_cast (by omega : 0 < M)
have hJr : 0 < (J : ℝ) := lt_of_lt_of_le (by positivity) hJ
have hdiff : 0 < |(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀| :=
lt_of_lt_of_le (by norm_num) hgap
intro j
have ht : (j : ℝ) / M ∈ Icc (0 : ℝ) 1 := by
refine ⟨by positivity, (div_le_one hMr).mpr ?_⟩
exact_mod_cast (by omega : j.val ≤ M)
have hprior := finiteReciprocalPrior_interpolate h₀ h₁ ht
have hsegment : (J : ℝ) * b / 12 ≤
(J : ℝ) * b * |(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀| := by
nlinarith [mul_le_mul_of_nonneg_left hgap (show 0 ≤ (J : ℝ) * b by positivity)]
have htail := intervalPriorTarget_concentration_budget hprior J M b
((J : ℝ) * b * |(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀|) u hb
(by positivity) hu hM hsegment hJ
have hmean : (J : ℝ) * b *
(∫ z, z⁻¹ ∂intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M)) =
(J : ℝ) * b * (∫ z, z⁻¹ ∂π₀) +
(j : ℝ) * ((J : ℝ) * b *
((∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀) / M) := by
rw [intervalInterpolatedPrior_integral h₀ h₁ ht]
ring
simpa only [hmean] using htail
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.IntervalNeighborhood 7 declarations Neighborhood coverage transfer and expected length for the many-prior interval argument.
Neighborhood coverage transfer and expected length for the many-prior interval argument.
theorem connectedInterval_neighborhood_iff
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∃ x : ℝ, I.lo ≤ x ∧ x ≤ I.hi ∧ |x - t| ≤ w) ↔ t ∈ Icc (I.lo - w) (I.hi + w)
Proof (Lean source)
-- @node: connectedInterval_neighborhood_iff
lemma connectedInterval_neighborhood_iff (I : ConnectedInterval) (t w : ℝ)
(hw : 0 ≤ w) :
(∃ x : ℝ, I.lo ≤ x ∧ x ≤ I.hi ∧ |x - t| ≤ w) ↔
t ∈ Icc (I.lo - w) (I.hi + w) := by
constructor
· rintro ⟨x, hxlo, hxhi, hx⟩
rcases abs_le.mp hx with ⟨hx0, hx1⟩
constructor <;> linarith
· rintro ⟨ht0, ht1⟩
refine ⟨max I.lo (t - w), le_max_left _ _, ?_, ?_⟩
· exact max_le I.ordered (by linarith)
· rw [abs_le]
constructor
· have := le_max_right I.lo (t - w)
linarith
· have : max I.lo (t - w) ≤ t + w := max_le (by linarith) (by linarith)
linarith
theorem connectedInterval_measurableSet_neighborhood
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
θ w :
ℝ
hw :
0 ≤ w
conclusion
MeasurableSet {I : ConnectedInterval | ∃ x : ℝ, I.lo ≤ x ∧ x ≤ I.hi ∧ |x - θ| ≤ w}
Proof (Lean source)
-- @node: connectedInterval_measurableSet_neighborhood
lemma connectedInterval_measurableSet_neighborhood (θ w : ℝ) (hw : 0 ≤ w) :
MeasurableSet {I : ConnectedInterval | ∃ x : ℝ,
I.lo ≤ x ∧ x ≤ I.hi ∧ |x - θ| ≤ w} := by
have hF : Measurable (fun I : ConnectedInterval =>
(I.lo, I.hi, I.closedLeft, I.closedRight)) := by
exact (measurable_fst.comp measurable_subtype_coe).prodMk
((measurable_snd.comp measurable_subtype_coe).prodMk
(measurable_const.prodMk measurable_const))
have hlo : Measurable (fun I : ConnectedInterval => I.lo - w) := by
first | fun_prop | exact (measurable_connectedInterval_lo).sub_const w
have hhi : Measurable (fun I : ConnectedInterval => I.hi + w) := by
first | fun_prop | exact (measurable_connectedInterval_hi).add_const w
have heq : {I : ConnectedInterval | ∃ x : ℝ,
I.lo ≤ x ∧ x ≤ I.hi ∧ |x - θ| ≤ w} =
{I | I.lo - w ≤ θ} ∩ {I | θ ≤ I.hi + w} := by
ext I
exact connectedInterval_neighborhood_iff I θ w hw
rw [heq]
exact (measurableSet_le hlo measurable_const).inter
(measurableSet_le measurable_const hhi)
theorem connectedInterval_joint_neighborhood_coverage
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
μ :
θ w u ε :
ℝ
hcoverage :
u ≤ μ.real {z | intervalContains z.2 z.1}
hconcentration :
μ.real {z | w < |z.1 - θ|} ≤ ε
conclusion
u - ε ≤ μ.real {z | ∃ x : ℝ, z.2.lo ≤ x ∧ x ≤ z.2.hi ∧ |x - θ| ≤ w}
Proof (Lean source)
-- @node: connectedInterval_joint_neighborhood_coverage
lemma connectedInterval_joint_neighborhood_coverage
(μ : Measure (ℝ × ConnectedInterval)) [IsProbabilityMeasure μ]
(θ w u ε : ℝ)
(hcoverage : u ≤ μ.real {z | intervalContains z.2 z.1})
(hconcentration : μ.real {z | w < |z.1 - θ|} ≤ ε) :
u - ε ≤ μ.real {z | ∃ x : ℝ,
z.2.lo ≤ x ∧ x ≤ z.2.hi ∧ |x - θ| ≤ w} := by
have hsubset : {z : ℝ × ConnectedInterval | intervalContains z.2 z.1} ⊆
{z | ∃ x : ℝ, z.2.lo ≤ x ∧ x ≤ z.2.hi ∧ |x - θ| ≤ w} ∪
{z | w < |z.1 - θ|} := by
intro z hz
by_cases hnear : |z.1 - θ| ≤ w
· left
have hlo : z.2.lo ≤ z.1 := by
have h := hz.1
change (if z.2.closedLeft then z.2.lo ≤ z.1 else z.2.lo < z.1) at h
split at h
· exact h
· exact h.le
have hhi : z.1 ≤ z.2.hi := by
have h := hz.2
change (if z.2.closedRight then z.1 ≤ z.2.hi else z.1 < z.2.hi) at h
split at h
· exact h
· exact h.le
exact ⟨z.1, hlo, hhi, hnear⟩
· exact inr (lt_of_not_ge hnear)
have hmono := measureReal_mono (μ := μ) hsubset
have hunion := measureReal_union_le (μ := μ)
{z | ∃ x : ℝ, z.2.lo ≤ x ∧ x ≤ z.2.hi ∧ |x - θ| ≤ w}
{z | w < |z.1 - θ|}
linarith
theorem connectedInterval_prior_neighborhood_coverage
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
μ :
θ w u ε :
ℝ
hw :
0 ≤ w
hcoverage :
u ≤ μ.real {z | intervalContains z.2 z.1}
hconcentration :
μ.real {z | w < |z.1 - θ|} ≤ ε
Proof (Lean source)
-- @node: connectedInterval_prior_neighborhood_coverage
lemma connectedInterval_prior_neighborhood_coverage
(μ : Measure (ℝ × ConnectedInterval)) [IsProbabilityMeasure μ]
(θ w u ε : ℝ) (hw : 0 ≤ w)
(hcoverage : u ≤ μ.real {z | intervalContains z.2 z.1})
(hconcentration : μ.real {z | w < |z.1 - θ|} ≤ ε) :
u - ε ≤ (μ.map snd).real {I | ∃ x : ℝ,
I.lo ≤ x ∧ x ≤ I.hi ∧ |x - θ| ≤ w} := by
rw [map_measureReal_apply measurable_snd
(connectedInterval_measurableSet_neighborhood θ w hw)]
exact connectedInterval_joint_neighborhood_coverage μ θ w u ε
hcoverage hconcentration
theorem connectedInterval_expected_neighborhood_length_of_tv
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
M :
ℕ
ν :
∀ j, IsProbabilityMeasure (ν j)
θ δ w p ε :
ℝ
hδ :
0 < δ
hw :
0 ≤ w
hhit :
htv :
∀ j, tvDist ν₀ (ν j) ≤ ε
conclusion
δ * (((M + 1 : ℕ) : ℝ) * (p - ε) - 1) - 2 * w ≤ ∫ I : ConnectedInterval, I.hi - I.lo ∂ν₀
Proof (Lean source)
-- @node: connectedInterval_expected_neighborhood_length_of_tv
lemma connectedInterval_expected_neighborhood_length_of_tv
(ν₀ : Measure ConnectedInterval) [IsProbabilityMeasure ν₀]
(M : ℕ) (ν : Fin (M + 1) → Measure ConnectedInterval)
[∀ j, IsProbabilityMeasure (ν j)] (θ δ w p ε : ℝ)
(hδ : 0 < δ) (hw : 0 ≤ w)
(hhit : ∀ j : Fin (M + 1), p ≤ (ν j).real
{I | ∃ x : ℝ, I.lo ≤ x ∧ x ≤ I.hi ∧
|x - (θ + (j : ℝ) * δ)| ≤ w})
(htv : ∀ j, tvDist ν₀ (ν j) ≤ ε) :
δ * (((M + 1 : ℕ) : ℝ) * (p - ε) - 1) - 2 * w ≤
∫ I : ConnectedInterval, I.hi - I.lo ∂ν₀ := by
have hF : Measurable (fun I : ConnectedInterval =>
(I.lo, I.hi, I.closedLeft, I.closedRight)) := by
exact (measurable_fst.comp measurable_subtype_coe).prodMk
((measurable_snd.comp measurable_subtype_coe).prodMk
(measurable_const.prodMk measurable_const))
have hlo : Measurable (fun I : ConnectedInterval => I.lo - w) := by
first | fun_prop | exact (measurable_connectedInterval_lo).sub_const w
have hhi : Measurable (fun I : ConnectedInterval => I.hi + w) := by
first | fun_prop | exact (measurable_connectedInterval_hi).add_const w
let L : ConnectedInterval → ℝ := fun I => I.lo - w
let U : ConnectedInterval → ℝ := fun I => I.hi + w
have hcoverage : ∀ j : Fin (M + 1), p - ε ≤ ν₀.real
(coverageEvent L U
(gridPoint θ δ j)) := by
intro j
have heq : {I : ConnectedInterval | ∃ x : ℝ,
I.lo ≤ x ∧ x ≤ I.hi ∧ |x - (θ + (j : ℝ) * δ)| ≤ w} =
coverageEvent L U
(gridPoint θ δ j) := by
ext I
exact connectedInterval_neighborhood_iff I _ w hw
have hh := hhit j
rw [heq] at hh
have hgap := measureReal_sub_le_tvDist (μ := ν₀) (ν := ν j)
(A := coverageEvent L U
(gridPoint θ δ j))
(measurableSet_coverageEvent
hlo hhi)
linarith [htv j]
have hbound :=
expected_interval_length_lower_of_grid
ν₀ M θ δ (p - ε) hlo hhi hδ hcoverage
have hlen : Integrable (fun I : ConnectedInterval => I.hi - I.lo) ν₀ :=
Integrable.of_bound
((measurable_connectedInterval_hi).sub
(measurable_connectedInterval_lo)).aestronglyMeasurable 2
(Filter.Eventually.of_forall (fun I => by
rw [Real.norm_eq_abs, abs_of_nonneg (sub_nonneg.mpr I.ordered)]
linarith [I.lower, I.upper]))
have hexpand : ∀ I : ConnectedInterval,
intervalLength (I.lo - w) (I.hi + w) =
(I.hi - I.lo) + 2 * w := by
intro I
dsimp [intervalLength, L, U]
rw [max_eq_right (by have h := I.2.2.1; linarith)]
ring
simp_rw [hexpand] at hbound
have hlenw : Integrable (fun I : ConnectedInterval => (I.hi - I.lo) + 2 * w) ν₀ :=
hlen.add (integrable_const (2 * w))
rw [← ofReal_integral_eq_lintegral_ofReal hlenw
(Filter.Eventually.of_forall (fun I => by
change 0 ≤ I.hi - I.lo + 2 * w
linarith [I.ordered]))] at hbound
have hreal := (ENNReal.ofReal_le_ofReal_iff
(integral_nonneg (fun I => by
change 0 ≤ I.hi - I.lo + 2 * w
linarith [I.ordered]))).mp hbound
have heq : (∫ I : ConnectedInterval, (I.hi - I.lo) + 2 * w ∂ν₀) =
(∫ I : ConnectedInterval, I.hi - I.lo ∂ν₀) + 2 * w := by
integral_linearity
simp
rw [heq] at hreal
linarith
theorem connectedInterval_many_prior_length_floor
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
M :
ℕ
ν :
∀ j, IsProbabilityMeasure (ν j)
θ Δ u :
ℝ
hΔ :
0 < Δ
hu :
0 < u
hM :
8 / u ≤ (M : ℝ)
hhit :
htv :
∀ j, tvDist ν₀ (ν j) ≤ u / 16
conclusion
(21 * u / 32) * Δ ≤ ∫ I : ConnectedInterval, I.hi - I.lo ∂ν₀
Proof (Lean source)
-- @node: connectedInterval_many_prior_length_floor
lemma connectedInterval_many_prior_length_floor
(ν₀ : Measure ConnectedInterval) [IsProbabilityMeasure ν₀]
(M : ℕ) (ν : Fin (M + 1) → Measure ConnectedInterval)
[∀ j, IsProbabilityMeasure (ν j)] (θ Δ u : ℝ)
(hΔ : 0 < Δ) (hu : 0 < u) (hM : 8 / u ≤ (M : ℝ))
(hhit : ∀ j : Fin (M + 1), 7 * u / 8 ≤ (ν j).real
{I | ∃ x : ℝ, I.lo ≤ x ∧ x ≤ I.hi ∧
|x - (θ + (j : ℝ) * (Δ / M))| ≤ Δ / (8 * M)})
(htv : ∀ j, tvDist ν₀ (ν j) ≤ u / 16) :
(21 * u / 32) * Δ ≤ ∫ I : ConnectedInterval, I.hi - I.lo ∂ν₀ := by
have hMr : 0 < (M : ℝ) := lt_of_lt_of_le (by positivity) hM
have hMu : 8 ≤ (M : ℝ) * u := (div_le_iff₀ hu).mp hM
have hbound := connectedInterval_expected_neighborhood_length_of_tv
ν₀ M ν θ (Δ / M) (Δ / (8 * M)) (7 * u / 8) (u / 16)
(by positivity) (by positivity) hhit htv
apply le_trans ?_ hbound
push_cast
rw [show Δ / (M : ℝ) * (((M : ℝ) + 1) * (7 * u / 8 - u / 16) - 1) -
2 * (Δ / (8 * M)) =
(Δ * (((M : ℝ) + 1) * (7 * u / 8 - u / 16) - 1) - Δ / 4) / M by ring]
apply (le_div_iff₀ hMr).mpr
have harith : (21 * u / 32) * Δ * M ≤
Δ * (((M : ℝ) + 1) * (7 * u / 8 - u / 16) - 1) - Δ / 4 := by
nlinarith [mul_nonneg hΔ.le hu.le]
exact harith
theorem connectedInterval_many_prior_length_of_joint_coverage
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
M :
ℕ
μ :
∀ j, IsProbabilityMeasure (μ j)
θ Δ u :
ℝ
hΔ :
0 < Δ
hu :
0 < u
hM :
8 / u ≤ (M : ℝ)
hcoverage :
∀ j, u ≤ (μ j).real {z | intervalContains z.2 z.1}
hconcentration :
conclusion
Proof (Lean source)
-- @node: connectedInterval_many_prior_length_of_joint_coverage
lemma connectedInterval_many_prior_length_of_joint_coverage
(M : ℕ) (μ : Fin (M + 1) → Measure (ℝ × ConnectedInterval))
[∀ j, IsProbabilityMeasure (μ j)] (θ Δ u : ℝ)
(hΔ : 0 < Δ) (hu : 0 < u) (hM : 8 / u ≤ (M : ℝ))
(hcoverage : ∀ j, u ≤ (μ j).real {z | intervalContains z.2 z.1})
(hconcentration : ∀ j : Fin (M + 1),
(μ j).real {z | Δ / (8 * M) < |z.1 - (θ + (j : ℝ) * (Δ / M))|} ≤ u / 8)
(htv : ∀ j, tvDist
((μ 0).map snd) ((μ j).map snd) ≤ u / 16) :
(21 * u / 32) * Δ ≤
∫ I : ConnectedInterval, I.hi - I.lo ∂((μ 0).map snd) := by
have hMr : 0 < (M : ℝ) := lt_of_lt_of_le (by positivity) hM
let ν : Fin (M + 1) → Measure ConnectedInterval := fun j => (μ j).map snd
have (j : Fin (M + 1)) : IsProbabilityMeasure (ν j) := by
dsimp [ν]
exact Measure.isProbabilityMeasure_map measurable_snd.aemeasurable
apply connectedInterval_many_prior_length_floor (ν 0) M ν θ Δ u hΔ hu hM
· intro j
have h := connectedInterval_prior_neighborhood_coverage (μ j)
(θ + (j : ℝ) * (Δ / M)) (Δ / (8 * M)) u (u / 8)
(by positivity) (hcoverage j) (hconcentration j)
convert h using 1; ring
· exact htv
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.IntervalPriors 7 declarations The finite interpolated reciprocal priors in equations (5) and (7) of the connected-interval lower bound.
The finite interpolated reciprocal priors in equations (5) and (7) of the connected-interval lower bound.
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: finiteReciprocalPrior_integrable
lemma finiteReciprocalPrior_integrable {H : ℝ} {π : Measure ℝ}
(hπ : FiniteReciprocalPrior H π) (f : ℝ → ℝ) : Integrable f π := by
let := hπ.1
obtain ⟨s, hs, _⟩ := hπ.2
have hae : ∀ᵐ z ∂π, z ∈ (s : Set ℝ) :=
(ae_mem_iff_measure_eq s.measurableSet.nullMeasurableSet).2 (by simpa using hs)
have hres : π.restrict (s : Set ℝ) = π := Measure.restrict_eq_self_of_ae_mem hae
have hi : IntegrableOn f (s : Set ℝ) π := IntegrableOn.of_finite s.finite_toSet
simpa only [IntegrableOn, hres] using hi
theorem finiteReciprocalPrior_interpolate
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
FiniteReciprocalPrior H (intervalInterpolatedPrior π₀ π₁ t)
Proof (Lean source)
-- @node: finiteReciprocalPrior_interpolate
lemma finiteReciprocalPrior_interpolate {H : ℝ} {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior H π₀) (h₁ : FiniteReciprocalPrior H π₁)
{t : ℝ} (ht : t ∈ Icc (0 : ℝ) 1) :
FiniteReciprocalPrior H (intervalInterpolatedPrior π₀ π₁ t) := by
classical
let := h₀.1
let := h₁.1
have hweights : ofReal (1 - t) + ofReal t = 1 := by
rw [← ENNReal.ofReal_add (by linarith [ht.2]) ht.1]
norm_num
have hp : IsProbabilityMeasure (intervalInterpolatedPrior π₀ π₁ t) := by
constructor
simp only [intervalInterpolatedPrior, Measure.add_apply, Measure.smul_apply,
measure_univ, smul_eq_mul, mul_one]
exact hweights
refine ⟨hp, ?_⟩
obtain ⟨s₀, hs₀, hmem₀⟩ := h₀.2
obtain ⟨s₁, hs₁, hmem₁⟩ := h₁.2
have hmass₀ : π₀ (↑(s₀ ∪ s₁) : Set ℝ) = 1 := by
apply le_antisymm (by simpa using (measure_mono (subset_univ _) :
π₀ (↑(s₀ ∪ s₁) : Set ℝ) ≤ π₀ univ))
rw [← hs₀]
exact measure_mono (by simp only [Finset.coe_union]; exact subset_union_left)
have hmass₁ : π₁ (↑(s₀ ∪ s₁) : Set ℝ) = 1 := by
apply le_antisymm (by simpa using (measure_mono (subset_univ _) :
π₁ (↑(s₀ ∪ s₁) : Set ℝ) ≤ π₁ univ))
rw [← hs₁]
exact measure_mono (by simp only [Finset.coe_union]; exact subset_union_right)
refine ⟨s₀ ∪ s₁, ?_, ?_⟩
· simp only [intervalInterpolatedPrior, Measure.add_apply, Measure.smul_apply,
hmass₀, hmass₁, smul_eq_mul, mul_one]
exact hweights
· intro z hz
rcases Finset.mem_union.mp hz with hz | hz
· exact hmem₀ z hz
· exact hmem₁ z hz
theorem intervalInterpolatedPrior_integral
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ z, f z ∂intervalInterpolatedPrior π₀ π₁ t)
= (1 - t) * (∫ z, f z ∂π₀) + t * (∫ z, f z ∂π₁)
Proof (Lean source)
-- @node: intervalInterpolatedPrior_integral
lemma intervalInterpolatedPrior_integral {H : ℝ} {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior H π₀) (h₁ : FiniteReciprocalPrior H π₁)
{t : ℝ} (ht : t ∈ Icc (0 : ℝ) 1) (f : ℝ → ℝ) :
(∫ z, f z ∂intervalInterpolatedPrior π₀ π₁ t) =
(1 - t) * (∫ z, f z ∂π₀) + t * (∫ z, f z ∂π₁) := by
rw [intervalInterpolatedPrior, integral_add_measure
((finiteReciprocalPrior_integrable h₀ f).smul_measure (by simp))
((finiteReciprocalPrior_integrable h₁ f).smul_measure (by simp)),
integral_smul_measure, integral_smul_measure]
simp only [ENNReal.toReal_ofReal (show 0 ≤ 1 - t by linarith [ht.2]),
ENNReal.toReal_ofReal ht.1, smul_eq_mul]
theorem intervalInterpolatedPrior_moments
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
H :
ℝ
π₀ π₁ :
Measure ℝ
h₀ :
h₁ :
K :
ℕ
hm :
∀ v : ℕ
if
v ≤ K
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
t :
ℝ
ht :
t ∈ Icc (0 : ℝ) 1
v :
ℕ
hv :
v ≤ K
conclusion
(∫ z, z ^ v ∂intervalInterpolatedPrior π₀ π₁ t) = ∫ z, z ^ v ∂π₀
Proof (Lean source)
-- @node: intervalInterpolatedPrior_moments
lemma intervalInterpolatedPrior_moments {H : ℝ} {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior H π₀) (h₁ : FiniteReciprocalPrior H π₁)
(K : ℕ) (hm : ∀ v : ℕ, v ≤ K → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
{t : ℝ} (ht : t ∈ Icc (0 : ℝ) 1) (v : ℕ) (hv : v ≤ K) :
(∫ z, z ^ v ∂intervalInterpolatedPrior π₀ π₁ t) = ∫ z, z ^ v ∂π₀ := by
rw [intervalInterpolatedPrior_integral h₀ h₁ ht, ← hm v hv]
ring
theorem intervalInterpolatedPrior_grid
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
H :
ℝ
π₀ π₁ :
Measure ℝ
h₀ :
h₁ :
K :
ℕ
hm :
∀ v : ℕ
if
v ≤ K
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
M :
ℕ
hM :
1 ≤ M
h :
Fin (M + 1)
conclusion
conclusion 1
FiniteReciprocalPrior H (intervalInterpolatedPrior π₀ π₁ ((h : ℝ) / M))
conclusion 2
(∫ z, z ^ v ∂intervalInterpolatedPrior π₀ π₁ ((h : ℝ) / M))
= ∫ z, z ^ v ∂intervalInterpolatedPrior π₀ π₁ ((h' : ℝ) / M)
conclusion 3
(∫ z, z⁻¹ ∂intervalInterpolatedPrior π₀ π₁ ((h : ℝ) / M))
= (∫ z, z⁻¹ ∂π₀) + (h : ℝ) * (((∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀) / M)
Proof (Lean source)
-- @node: intervalInterpolatedPrior_grid
lemma intervalInterpolatedPrior_grid {H : ℝ} {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior H π₀) (h₁ : FiniteReciprocalPrior H π₁)
(K : ℕ) (hm : ∀ v : ℕ, v ≤ K → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
(M : ℕ) (hM : 1 ≤ M) :
∀ h : Fin (M + 1),
FiniteReciprocalPrior H (intervalInterpolatedPrior π₀ π₁ ((h : ℝ) / M)) ∧
(∀ h' : Fin (M + 1), ∀ v : ℕ, v ≤ K →
(∫ z, z ^ v ∂intervalInterpolatedPrior π₀ π₁ ((h : ℝ) / M)) =
∫ z, z ^ v ∂intervalInterpolatedPrior π₀ π₁ ((h' : ℝ) / M)) ∧
(∫ z, z⁻¹ ∂intervalInterpolatedPrior π₀ π₁ ((h : ℝ) / M)) =
(∫ z, z⁻¹ ∂π₀) + (h : ℝ) *
(((∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀) / M) := by
have hMpos : (0 : ℝ) < M := by exact_mod_cast (by omega : 0 < M)
have ht (h : Fin (M + 1)) : (h : ℝ) / M ∈ Icc (0 : ℝ) 1 := by
refine ⟨by positivity, (div_le_one hMpos).mpr ?_⟩
exact_mod_cast (by omega : h.val ≤ M)
intro h
refine ⟨finiteReciprocalPrior_interpolate h₀ h₁ (ht h), ?_, ?_⟩
· intro h' v hv
rw [intervalInterpolatedPrior_moments h₀ h₁ K hm (ht h) v hv,
intervalInterpolatedPrior_moments h₀ h₁ K hm (ht h') v hv]
· rw [intervalInterpolatedPrior_integral h₀ h₁ (ht h)]
ring
theorem intervalInterpolatedPrior_reciprocal_spacing
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
|(∫ z, z⁻¹ ∂intervalInterpolatedPrior π₀ π₁ (((h : ℝ) + 1) / M))
- (∫ z, z⁻¹ ∂intervalInterpolatedPrior π₀ π₁ ((h : ℝ) / M))|
= |(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀| / M
Proof (Lean source)
-- @node: intervalInterpolatedPrior_reciprocal_spacing
lemma intervalInterpolatedPrior_reciprocal_spacing {H : ℝ} {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior H π₀) (h₁ : FiniteReciprocalPrior H π₁)
(M h : ℕ) (hh : h < M) :
|(∫ z, z⁻¹ ∂intervalInterpolatedPrior π₀ π₁ (((h : ℝ) + 1) / M)) -
(∫ z, z⁻¹ ∂intervalInterpolatedPrior π₀ π₁ ((h : ℝ) / M))| =
|(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀| / M := by
have hM : (0 : ℝ) < M := by exact_mod_cast (by omega : 0 < M)
have ht : (h : ℝ) / M ∈ Icc (0 : ℝ) 1 := by
refine ⟨by positivity, (div_le_one hM).mpr ?_⟩
exact_mod_cast (le_of_lt hh)
have ht' : ((h : ℝ) + 1) / M ∈ Icc (0 : ℝ) 1 := by
refine ⟨by positivity, (div_le_one hM).mpr ?_⟩
exact_mod_cast (by omega : h + 1 ≤ M)
rw [intervalInterpolatedPrior_integral h₀ h₁ ht',
intervalInterpolatedPrior_integral h₀ h₁ ht]
have heq (a b : ℝ) :
(1 - ((h : ℝ) + 1) / M) * a + (((h : ℝ) + 1) / M) * b -
((1 - (h : ℝ) / M) * a + ((h : ℝ) / M) * b) = (b - a) / M := by
ring
rw [heq, abs_div, abs_of_pos hM]
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.IntervalRegimes 4 declarations The bounded-effective-size and small-rare-count cases in equations (18)--(19).
The bounded-effective-size and small-rare-count cases in equations (18)--(19).
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: interval_many_scale_sqrt
lemma interval_many_scale_sqrt (n d : ℕ) (q : ℝ)
(hN : 0 < effectiveSize n q) (hell : 0 < logScale n q) :
sqrt (min 1 (((d : ℝ) / (effectiveSize n q * logScale n q)) ^ 2)) =
min 1 ((d : ℝ) / (effectiveSize n q * logScale n q)) := by
let x := (d : ℝ) / (effectiveSize n q * logScale n q)
have hx : 0 ≤ x := by dsimp [x]; positivity
change sqrt (min 1 (x ^ 2)) = min 1 x
by_cases hx1 : x ≤ 1
· rw [min_eq_right (by nlinarith : x ^ 2 ≤ 1), Real.sqrt_sq hx,
min_eq_right hx1]
· rw [min_eq_left (by nlinarith : 1 ≤ x ^ 2), Real.sqrt_one,
min_eq_left (le_of_not_ge hx1)]
theorem interval_bounded_size_parametric_scale Lemma interval_bounded_size_parametric_scale in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: interval_bounded_size_parametric_scale
lemma interval_bounded_size_parametric_scale (N T : ℝ)
(hN : 0 < N) (hNT : N ≤ T) :
min 1 (sqrt T)⁻¹ ≤ min 1 (sqrt N)⁻¹ := by
apply min_le_min_left
exact (inv_le_inv₀ (Real.sqrt_pos.mpr (hN.trans_le hNT))
(Real.sqrt_pos.mpr hN)).mpr (Real.sqrt_le_sqrt hNT)
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d D :
ℕ
q :
ℝ
hd :
1 ≤ d
hN :
1 ≤ effectiveSize n q
hell :
1 ≤ logScale n q
hcapacity :
(D : ℝ) ≤ (2 * rareMass η n q)⁻¹
hsmall :
rareCount η n d q < D
conclusion
Proof (Lean source)
-- @node: interval_small_rare_count_scale
lemma interval_small_rare_count_scale (η : ℝ) (n d D : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hN : 1 ≤ effectiveSize n q) (hell : 1 ≤ logScale n q)
(hcapacity : (D : ℝ) ≤ (2 * rareMass η n q)⁻¹)
(hsmall : rareCount η n d q < D) :
min 1 ((d : ℝ) / (effectiveSize n q * logScale n q)) ≤
(D : ℝ) * min 1 (sqrt (effectiveSize n q))⁻¹ := by
have hs : 1 ≤ sqrt (effectiveSize n q) := Real.one_le_sqrt.mpr hN
have hinv : (sqrt (effectiveSize n q))⁻¹ ≤ 1 := by
simpa using (inv_le_inv₀ (by linarith : 0 < sqrt (effectiveSize n q))
(by norm_num : (0 : ℝ) < 1)).mpr hs
rw [min_eq_right hinv]
exact interval_small_dimension_scale n d D q hN hell
(rareCount_small_dimension η n d D q hd hcapacity hsmall)
theorem interval_many_scale_of_large_regime Lemma interval_many_scale_of_large_regime in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η T c₀ c₁ :
ℝ
D :
ℕ
hT :
1 ≤ T
hD :
1 ≤ D
hc₀ :
0 < c₀
hc₁ :
0 < c₁
n d :
ℕ
q L :
ℝ
hd :
1 ≤ d
hN :
0 < effectiveSize n q
hell :
1 ≤ logScale n q
hparam :
hcapacity :
T ≤ effectiveSize n q → (D : ℝ) ≤ (2 * rareMass η n q)⁻¹
hlarge :
T
≤ effectiveSize n q → D ≤ rareCount η n d q → c₁ * min 1 ((d : ℝ) / (effectiveSize n q * logScale n q)) ≤ L
Proof (Lean source)
-- @node: interval_many_scale_of_large_regime
lemma interval_many_scale_of_large_regime (η T c₀ c₁ : ℝ) (D : ℕ)
(hT : 1 ≤ T) (hD : 1 ≤ D) (hc₀ : 0 < c₀) (hc₁ : 0 < c₁)
(n d : ℕ) (q L : ℝ) (hd : 1 ≤ d)
(hN : 0 < effectiveSize n q) (hell : 1 ≤ logScale n q)
(hparam : c₀ * min 1 (sqrt (effectiveSize n q))⁻¹ ≤ L)
(hcapacity : T ≤ effectiveSize n q → (D : ℝ) ≤ (2 * rareMass η n q)⁻¹)
(hlarge : T ≤ effectiveSize n q → D ≤ rareCount η n d q →
c₁ * min 1 ((d : ℝ) / (effectiveSize n q * logScale n q)) ≤ L) :
min c₁ (min (c₀ * min 1 (sqrt T)⁻¹) (c₀ / D)) *
min 1 ((d : ℝ) / (effectiveSize n q * logScale n q)) ≤ L := by
let c := min c₁ (min (c₀ * min 1 (sqrt T)⁻¹) (c₀ / D))
let b := min 1 ((d : ℝ) / (effectiveSize n q * logScale n q))
have hb : 0 ≤ b := by dsimp [b]; positivity
change c * b ≤ L
by_cases hNT : T ≤ effectiveSize n q
· by_cases hJ : D ≤ rareCount η n d q
· exact (mul_le_mul_of_nonneg_right (min_le_left _ _) hb).trans (hlarge hNT hJ)
· have hscale := interval_small_rare_count_scale η n d D q hd
(hT.trans hNT) hell (hcapacity hNT) (lt_of_not_ge hJ)
have hc : c ≤ c₀ / D := (min_le_right _ _).trans (min_le_right _ _)
have hDpos : (0 : ℝ) < D := by exact_mod_cast (lt_of_lt_of_le (by decide : 0 < 1) hD)
have hcpos : 0 ≤ c := by dsimp [c]; positivity
have hmul := mul_le_mul_of_nonneg_left hscale hcpos
have hcoef : c * (D : ℝ) ≤ c₀ := (le_div_iff₀ hDpos).mp hc
have hmul' := mul_le_mul_of_nonneg_right hcoef
(show 0 ≤ min 1 (sqrt (effectiveSize n q))⁻¹ by positivity)
have hfinish : c * ((D : ℝ) * min 1 (sqrt (effectiveSize n q))⁻¹) ≤ L := by
calc
c * ((D : ℝ) * min 1 (sqrt (effectiveSize n q))⁻¹) ≤
c₀ * min 1 (sqrt (effectiveSize n q))⁻¹ := by
simpa [mul_assoc] using hmul'
_ ≤ L := hparam
exact hmul.trans hfinish
· have hc : c ≤ c₀ * min 1 (sqrt T)⁻¹ :=
(min_le_right _ _).trans (min_le_left _ _)
have hcpos : 0 ≤ c := by dsimp [c]; positivity
have hb1 : b ≤ 1 := min_le_left _ _
have hscale := interval_bounded_size_parametric_scale
(effectiveSize n q) T hN (le_of_not_ge hNT)
calc
c * b ≤ c := by nlinarith
_ ≤ c₀ * min 1 (sqrt T)⁻¹ := hc
_ ≤ c₀ * min 1 (sqrt (effectiveSize n q))⁻¹ :=
mul_le_mul_of_nonneg_left hscale hc₀.le
_ ≤ L := hparam
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.IntervalScales 5 declarations Rare-cell mass and the complementary small-dimension scale in equations (17)--(18) of the connected-interval frontier proof.
Rare-cell mass and the complementary small-dimension scale in equations (17)--(18) of the connected-interval frontier proof.
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: rareCount_mass_le_half
lemma rareCount_mass_le_half (η : ℝ) (n d : ℕ) (q : ℝ)
(hb : 0 < rareMass η n q) :
(rareCount η n d q : ℝ) * rareMass η n q ≤ 1 / 2 := by
have hcount : rareCount η n d q ≤ floor ((2 * rareMass η n q)⁻¹) :=
min_le_right _ _
have hfloor := Nat.floor_le (show 0 ≤ (2 * rareMass η n q)⁻¹ by positivity)
have hle : (rareCount η n d q : ℝ) ≤ (2 * rareMass η n q)⁻¹ :=
(Nat.cast_le.mpr hcount).trans hfloor
have hmul := mul_le_mul_of_nonneg_right hle hb.le
have hid : (2 * rareMass η n q)⁻¹ * rareMass η n q = 1 / 2 := by
field_simp
linarith [hid]
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: rareCount_mass_lower
lemma rareCount_mass_lower (η : ℝ) (n d : ℕ) (q : ℝ)
(hd : 2 ≤ d) (hb : 0 < rareMass η n q) (hbsmall : rareMass η n q ≤ 1 / 4) :
min ((d : ℝ) * rareMass η n q / 2) (1 / 4) ≤
(rareCount η n d q : ℝ) * rareMass η n q := by
by_cases hbranch : d - 1 ≤ floor ((2 * rareMass η n q)⁻¹)
· rw [rareCount, min_eq_left hbranch]
have hdim : (d : ℝ) / 2 ≤ ((d - 1 : ℕ) : ℝ) := by
rw [Nat.cast_sub (by omega : 1 ≤ d)]
have : (2 : ℝ) ≤ d := by exact_mod_cast hd
push_cast
linarith
apply (min_le_left _ _).trans
nlinarith
· rw [rareCount, min_eq_right (le_of_not_ge hbranch)]
have hf := Nat.sub_one_lt_floor ((2 * rareMass η n q)⁻¹)
have hmul := mul_lt_mul_of_pos_right hf hb
have hid : (2 * rareMass η n q)⁻¹ * rareMass η n q = 1 / 2 := by
field_simp
apply (min_le_right _ _).trans
nlinarith
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η Δ :
ℝ
n d :
ℕ
q :
ℝ
hη :
0 < η
hηsmall :
η ≤ 1 / 2
hd :
2 ≤ d
hN :
0 < effectiveSize n q
hell :
0 < logScale n q
hbsmall :
rareMass η n q ≤ 1 / 4
hseparation :
(rareCount η n d q : ℝ) * rareMass η n q / 12 ≤ Δ
conclusion
Proof (Lean source)
-- @node: interval_prior_segment_width_lower
lemma interval_prior_segment_width_lower (η Δ : ℝ) (n d : ℕ) (q : ℝ)
(hη : 0 < η) (hηsmall : η ≤ 1 / 2)
(hd : 2 ≤ d) (hN : 0 < effectiveSize n q) (hell : 0 < logScale n q)
(hbsmall : rareMass η n q ≤ 1 / 4)
(hseparation : (rareCount η n d q : ℝ) * rareMass η n q / 12 ≤ Δ) :
(η / 24) * min 1 ((d : ℝ) / (effectiveSize n q * logScale n q)) ≤ Δ := by
have hb : 0 < rareMass η n q := by
unfold rareMass
positivity
have hmass := rareCount_mass_lower η n d q hd hb hbsmall
have hscale : (η / 24) * min 1 ((d : ℝ) /
(effectiveSize n q * logScale n q)) ≤
min ((d : ℝ) * rareMass η n q / 2) (1 / 4) / 12 := by
rw [← min_div_div_right (by norm_num : (0 : ℝ) ≤ 12)]
apply le_min
· have h := mul_le_mul_of_nonneg_left
(min_le_right (1 : ℝ) ((d : ℝ) / (effectiveSize n q * logScale n q)))
(show 0 ≤ η / 24 by positivity)
convert h using 1
dsimp [rareMass]
ring
· have h := mul_le_mul_of_nonneg_left
(min_le_left (1 : ℝ) ((d : ℝ) / (effectiveSize n q * logScale n q)))
(show 0 ≤ η / 24 by positivity)
linarith
exact hscale.trans ((div_le_div_of_nonneg_right hmass (by norm_num)).trans hseparation)
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: rareCount_small_dimension
lemma rareCount_small_dimension (η : ℝ) (n d D : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hcapacity : (D : ℝ) ≤ (2 * rareMass η n q)⁻¹)
(hsmall : rareCount η n d q < D) : d ≤ D := by
have hfloor : D ≤ floor ((2 * rareMass η n q)⁻¹) := by
exact (Nat.le_floor_iff (by
exact (show (0 : ℝ) ≤ D by positivity).trans hcapacity)).mpr hcapacity
have hbranch : d - 1 < D := by
unfold rareCount at hsmall
omega
omega
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
≤ (D : ℝ) * (sqrt (effectiveSize n q))⁻¹
Proof (Lean source)
-- @node: interval_small_dimension_scale
lemma interval_small_dimension_scale (n d D : ℕ) (q : ℝ)
(hN : 1 ≤ effectiveSize n q) (hell : 1 ≤ logScale n q) (hd : d ≤ D) :
min 1 ((d : ℝ) / (effectiveSize n q * logScale n q)) ≤
(D : ℝ) * (sqrt (effectiveSize n q))⁻¹ := by
have hNpos : 0 < effectiveSize n q := by linarith
have hspos : 0 < sqrt (effectiveSize n q) := Real.sqrt_pos.mpr hNpos
have hs : sqrt (effectiveSize n q) ≤ effectiveSize n q := by
apply Real.sqrt_le_iff.mpr
refine ⟨hNpos.le, ?_⟩
nlinarith
have hden : sqrt (effectiveSize n q) ≤ effectiveSize n q * logScale n q := by
nlinarith
have hdim : (d : ℝ) ≤ D := by exact_mod_cast hd
apply (min_le_right _ _).trans
rw [← div_eq_mul_inv]
apply (div_le_div_iff₀ (by positivity) hspos).mpr
have hmul := mul_le_mul_of_nonneg_right hdim hspos.le
have hmul' := mul_le_mul_of_nonneg_left hden (show (0 : ℝ) ≤ D by positivity)
exact hmul.trans hmul'
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.IntervalThresholds 5 declarations Uniform effective-size thresholds for equations (9)--(10) of the connected-interval proof.
Uniform effective-size thresholds for equations (9)--(10) of the connected-interval proof. These are numerical estimates, preceding the activated likelihood and model-mixture transfer.
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
D n :
ℕ
q :
ℝ
hη :
0 < η
hN :
0 < effectiveSize n q
hell :
1 ≤ logScale n q
hlarge :
2 * η * D ≤ effectiveSize n q
conclusion
(D : ℝ) ≤ (2 * rareMass η n q)⁻¹
Proof (Lean source)
-- @node: interval_capacity_of_large_size
lemma interval_capacity_of_large_size (η : ℝ) (D n : ℕ) (q : ℝ)
(hη : 0 < η) (hN : 0 < effectiveSize n q)
(hell : 1 ≤ logScale n q) (hlarge : 2 * η * D ≤ effectiveSize n q) :
(D : ℝ) ≤ (2 * rareMass η n q)⁻¹ := by
have hb : 0 < 2 * rareMass η n q := by unfold rareMass; positivity
rw [inv_eq_one_div]
apply (le_div_iff₀ hb).mpr
rw [show (D : ℝ) * (2 * rareMass η n q) =
(2 * η * D) / (effectiveSize n q * logScale n q) by unfold rareMass; ring]
apply (div_le_iff₀ (by positivity : 0 < effectiveSize n q * logScale n q)).mpr
nlinarith [mul_le_mul_of_nonneg_left hell hN.le]
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
-- @node: interval_activation_tail_le_exp
lemma interval_activation_tail_le_exp (n d : ℕ) (q : ℝ)
(hN : 0 < effectiveSize n q) (hell : 1 ≤ logScale n q) :
(rareCount (exp (-32)) n d q : ℝ) *
(exp 1 * exp (-32)) ^ (lowerDegree n q + 1) ≤
exp (32 - 29 * logScale n q) := by
let N := effectiveSize n q
let L := logScale n q
have hNL : 0 < N * L := by dsimp [N, L]; positivity
have hcap : (rareCount (exp (-32)) n d q : ℝ) ≤
N * L / (2 * exp (-32)) := by
have h := rareCount_mass_le_half (exp (-32)) n d q
(show 0 < rareMass (exp (-32)) n q by unfold rareMass; positivity)
apply (le_div_iff₀ (by positivity : 0 < 2 * exp (-32))).mpr
dsimp [rareMass] at h
have h' := (div_le_iff₀ hNL).mp (show
(rareCount (exp (-32)) n d q : ℝ) * exp (-32) / (N * L) ≤ 1 / 2 by
simpa [N, L, mul_div_assoc] using h)
nlinarith
have hexpL : exp L = exp 1 + N := by
dsimp [L, N, logScale]
rw [Real.exp_log (by positivity)]
have hNexp : N ≤ exp L := by rw [hexpL]; linarith [Real.exp_pos (1 : ℝ)]
have hLexp : L ≤ exp L := by linarith [Real.add_one_le_exp L]
have hprod : N * L ≤ exp (2 * L) := by
rw [show 2 * L = L + L by ring, Real.exp_add]
exact mul_le_mul hNexp hLexp (by dsimp [L]; linarith) (Real.exp_nonneg _)
have hdegree : L ≤ (lowerDegree n q + 1 : ℕ) := by
have h := Nat.le_ceil (logScale n q)
dsimp [L]
unfold lowerDegree
push_cast
linarith
have hpow : (exp 1 * exp (-32)) ^ (lowerDegree n q + 1) ≤
exp (-31 * L) := by
rw [← Real.exp_add, show (1 : ℝ) + -32 = -31 by norm_num,
← Real.exp_nat_mul]
apply Real.exp_le_exp.mpr
nlinarith
calc
_ ≤ (N * L / (2 * exp (-32))) * exp (-31 * L) :=
mul_le_mul hcap hpow (by positivity) (by positivity)
_ ≤ (exp (2 * L) / exp (-32)) * exp (-31 * L) := by
apply mul_le_mul_of_nonneg_right _ (Real.exp_nonneg _)
exact (div_le_div_of_nonneg_left hNL.le (Real.exp_pos (-32)) (by
linarith [Real.exp_pos (-32)])).trans
(div_le_div_of_nonneg_right hprod (Real.exp_nonneg (-32)))
_ = exp (32 - 29 * logScale n q) := by
rw [← Real.exp_sub, ← Real.exp_add]
congr 1
dsimp [L]
ring
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
≤ u / 16
Proof (Lean source)
-- @node: interval_activation_tail_budget
lemma interval_activation_tail_budget (u : ℝ) (n d : ℕ) (q : ℝ)
(hu : 0 < u) (hN : 0 < effectiveSize n q) (hell : 1 ≤ logScale n q)
(hlarge : exp ((32 - log (u / 16)) / 29) ≤ effectiveSize n q) :
(rareCount (exp (-32)) n d q : ℝ) *
(exp 1 * exp (-32)) ^ (lowerDegree n q + 1) ≤ u / 16 := by
have hlog := Real.log_le_log (Real.exp_pos ((32 - log (u / 16)) / 29))
(hlarge.trans (show effectiveSize n q ≤ exp 1 + effectiveSize n q by
linarith [Real.exp_pos (1 : ℝ)]))
rw [Real.log_exp] at hlog
change (32 - log (u / 16)) / 29 ≤ logScale n q at hlog
apply (interval_activation_tail_le_exp n d q hN hell).trans
calc
exp (32 - 29 * logScale n q) ≤ exp (log (u / 16)) := by
apply Real.exp_le_exp.mpr
linarith
_ = u / 16 := Real.exp_log (by positivity)
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
u :
ℝ
D :
ℕ
hu :
0 < u
conclusion
∃ T : ℝ,
1 ≤ T
conclusion 1
Proof (Lean source)
-- @node: interval_activation_thresholds
lemma interval_activation_thresholds (u : ℝ) (D : ℕ) (hu : 0 < u) :
∃ T : ℝ, 1 ≤ T ∧ ∀ (n d : ℕ) (q : ℝ), 0 < effectiveSize n q →
T ≤ effectiveSize n q →
(D : ℝ) ≤ (2 * rareMass (exp (-32)) n q)⁻¹ ∧
(rareCount (exp (-32)) n d q : ℝ) *
(exp 1 * exp (-32)) ^ (lowerDegree n q + 1) ≤ u / 16 := by
let T := max 1 (max (2 * exp (-32) * D)
(exp ((32 - log (u / 16)) / 29)))
refine ⟨T, le_max_left _ _, ?_⟩
intro n d q hN hlarge
have hell : 1 ≤ logScale n q := by
unfold logScale
have h := Real.log_le_log (Real.exp_pos (1 : ℝ))
(show exp 1 ≤ exp 1 + effectiveSize n q by linarith)
simpa using h
have hcap : 2 * exp (-32) * D ≤ effectiveSize n q :=
(le_max_left _ _).trans ((le_max_right _ _).trans hlarge)
have htail : exp ((32 - log (u / 16)) / 29) ≤ effectiveSize n q :=
(le_max_right _ _).trans ((le_max_right _ _).trans hlarge)
exact ⟨interval_capacity_of_large_size _ D n q (Real.exp_pos _) hN hell hcap,
interval_activation_tail_budget u n d q hu hN hell htail⟩
theorem interval_activation_grid_thresholds Lemma interval_activation_grid_thresholds in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: interval_activation_grid_thresholds
lemma interval_activation_grid_thresholds (u : ℝ) (hu : 0 < u) :
let M := ceil (8 / u) + 1
let D := ceil (18432 * (M : ℝ) ^ 2 / u) + 1
1 ≤ M ∧ (M : ℝ)⁻¹ ≤ u / 8 ∧ 1 ≤ D ∧
18432 * (M : ℝ) ^ 2 / u ≤ (D : ℝ) := by
dsimp only
have hM : 1 ≤ ceil (8 / u) + 1 := by omega
have hMr : 0 < ((ceil (8 / u) + 1 : ℕ) : ℝ) := by exact_mod_cast hM
have hceil := Nat.le_ceil (8 / u)
have hgrid : 8 ≤ u * ((ceil (8 / u) + 1 : ℕ) : ℝ) := by
have h := (div_le_iff₀ hu).mp hceil
push_cast
nlinarith
refine ⟨hM, ?_, by omega, ?_⟩
· rw [inv_eq_one_div]
apply (div_le_iff₀ hMr).mpr
nlinarith
· have h := Nat.le_ceil (18432 * (((ceil (8 / u) + 1 : ℕ) : ℝ)) ^ 2 / u)
push_cast at *
linarith
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.LargeAlphabet.CollisionProbability 11 declarations Actual iid treated-cell collision probabilities for the large-alphabet experiment.
Actual iid treated-cell collision probabilities for the large-alphabet experiment.
theorem largeAlphabetLaw_treated_cell_mass
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: largeAlphabetLaw_treated_cell_mass
lemma largeAlphabetLaw_treated_cell_mass {d : ℕ} (q : ℝ) (ξ : Fin d → Bool)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (x : Fin d) :
((largeAlphabetLaw q ξ hd hq hq1).1.map obs).real
{o | o.A = true ∧ o.X = x} = 1 / (2 * (d : ℝ)) := by
classical
rw [measureReal_def, Measure.map_apply (by fun_prop) (by measurability)]
change (largeAlphabetLaw q ξ hd hq hq1).1.real _ = _
rw [largeAlphabetLaw_real_event]
simp [obs, largeAlphabetRecord, completeBernWeight,
Finset.sum_ite_irrel, Finset.sum_ite_eq']
ring
theorem largeAlphabet_iid_pair_collision
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
-- @node: largeAlphabet_iid_pair_collision
lemma largeAlphabet_iid_pair_collision {n d : ℕ}
(μ : Measure (ObsRecord d)) [IsProbabilityMeasure μ] (hd : 1 ≤ d)
(hm : ∀ x : Fin d, μ.real {o | o.A = true ∧ o.X = x} = 1 / (2 * (d : ℝ)))
(i j : Fin n) (hij : i ≠ j) :
(Measure.pi (fun _ : Fin n => μ)).real
{s | (s i).A = true ∧ (s j).A = true ∧ (s i).X = (s j).X} ≤
1 / (4 * (d : ℝ)) := by
classical
let Q := Measure.pi (fun _ : Fin n => μ)
let C (x : Fin d) : Set (ObsRecord d) := {o | o.A = true ∧ o.X = x}
have hind : iIndepFun (fun k : Fin n => fun s : Fin n → ObsRecord d => s k) Q :=
iIndepFun_pi (fun _ => aemeasurable_id)
have hcoord (k : Fin n) (x : Fin d) : Q.real {s | s k ∈ C x} =
1 / (2 * (d : ℝ)) := by
have hp := (measurePreserving_eval (fun _ : Fin n => μ) k).map_eq
have heq := congrArg (fun ν : Measure (ObsRecord d) => ν.real (C x)) hp
rw [measureReal_def, Measure.map_apply (by fun_prop) (by measurability)] at heq
exact heq.trans (hm x)
have hp (x : Fin d) : Q.real {s | s i ∈ C x ∧ s j ∈ C x} =
(1 / (2 * (d : ℝ))) ^ 2 := by
have heq := (hind.indepFun hij).measure_inter_preimage_eq_mul (C x) (C x)
(by measurability) (by measurability)
have hr := congrArg toReal heq
rw [ENNReal.toReal_mul] at hr
change Q.real {s | s i ∈ C x ∧ s j ∈ C x} =
Q.real {s | s i ∈ C x} * Q.real {s | s j ∈ C x} at hr
rw [hcoord, hcoord] at hr
simpa [pow_two] using hr
have hcover : {s : Fin n → ObsRecord d | (s i).A = true ∧ (s j).A = true ∧ (s i).X = (s j).X} ⊆
⋃ x : Fin d, {s | s i ∈ C x ∧ s j ∈ C x} := by
intro s hs
exact mem_iUnion.mpr ⟨(s i).X, ⟨⟨hs.1, rfl⟩, ⟨hs.2.1, hs.2.2.symm⟩⟩⟩
calc
Q.real _ ≤ Q.real (⋃ x : Fin d, {s | s i ∈ C x ∧ s j ∈ C x}) :=
measureReal_mono hcover (measure_ne_top Q _)
_ ≤ ∑ x : Fin d, Q.real {s | s i ∈ C x ∧ s j ∈ C x} :=
measureReal_iUnion_fintype_le _
_ = 1 / (4 * (d : ℝ)) := by
simp_rw [hp]
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
have hdpos : 0 < (d : ℝ) := by exact_mod_cast hd
field_simp
<;> ring
theorem largeAlphabetComparisonLaw_treated_cell_mass
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: largeAlphabetComparisonLaw_treated_cell_mass
lemma largeAlphabetComparisonLaw_treated_cell_mass (d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (x : Fin d) :
((largeAlphabetComparisonLaw d q hd hq hq1).1.map obs).real
{o | o.A = true ∧ o.X = x} = 1 / (2 * (d : ℝ)) := by
classical
rw [measureReal_def, Measure.map_apply (by fun_prop) (by measurability)]
change (largeAlphabetComparisonLaw d q hd hq hq1).1.real _ = _
rw [largeAlphabetComparisonLaw_real_event]
simp [obs, largeAlphabetComparisonWeight, completeBernWeight,
largeAlphabetComparisonArrival, Finset.sum_ite_irrel, Finset.sum_ite_eq']
ring
theorem largeAlphabetLaw_pair_collision
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: largeAlphabetLaw_pair_collision
lemma largeAlphabetLaw_pair_collision {n d : ℕ} (q : ℝ) (ξ : Fin d → Bool)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (i j : Fin n) (hij : i ≠ j) :
(sampleLaw n (largeAlphabetLaw q ξ hd hq hq1)).real
{s | (s i).A = true ∧ (s j).A = true ∧ (s i).X = (s j).X} ≤
1 / (4 * (d : ℝ)) := by
let := (largeAlphabetLaw q ξ hd hq hq1).2
let : IsProbabilityMeasure ((largeAlphabetLaw q ξ hd hq hq1).1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
exact largeAlphabet_iid_pair_collision _ hd
(largeAlphabetLaw_treated_cell_mass q ξ hd hq hq1) i j hij
theorem largeAlphabetComparisonLaw_pair_collision
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(sampleLaw n (largeAlphabetComparisonLaw d q hd hq hq1)).real {s | (s i).A = true ∧ (s j).A = true ∧ (s i).X = (s j).X}
≤ 1 / (4 * (d : ℝ))
Proof (Lean source)
-- @node: largeAlphabetComparisonLaw_pair_collision
lemma largeAlphabetComparisonLaw_pair_collision {n d : ℕ} (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (i j : Fin n) (hij : i ≠ j) :
(sampleLaw n (largeAlphabetComparisonLaw d q hd hq hq1)).real
{s | (s i).A = true ∧ (s j).A = true ∧ (s i).X = (s j).X} ≤
1 / (4 * (d : ℝ)) := by
let := (largeAlphabetComparisonLaw d q hd hq hq1).2
let : IsProbabilityMeasure ((largeAlphabetComparisonLaw d q hd hq hq1).1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
exact largeAlphabet_iid_pair_collision _ hd
(largeAlphabetComparisonLaw_treated_cell_mass d q hd hq hq1) i j hij
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
hn :
1 ≤ n
hd :
1024 * n ^ 2 ≤ d
conclusion
(n.choose 2 : ℝ) * (1 / (4 * (d : ℝ))) ≤ 1 / 8192
Proof (Lean source)
-- @node: largeAlphabet_collision_budget
lemma largeAlphabet_collision_budget (n d : ℕ) (hn : 1 ≤ n)
(hd : 1024 * n ^ 2 ≤ d) :
(n.choose 2 : ℝ) * (1 / (4 * (d : ℝ))) ≤ 1 / 8192 := by
have hnat : 2 * n.choose 2 ≤ n ^ 2 := by
rw [Nat.choose_two_right]
have hm := Nat.mul_le_mul_left n (sub_le n 1)
simp only [pow_two]
omega
have hc : 2 * (n.choose 2 : ℝ) ≤ (n : ℝ) ^ 2 := by exact_mod_cast hnat
have hdreal : 1024 * (n : ℝ) ^ 2 ≤ d := by exact_mod_cast hd
have hnreal : 1 ≤ (n : ℝ) := by exact_mod_cast hn
have hdpos : 0 < (d : ℝ) := by nlinarith
rw [mul_one_div, div_le_iff₀ (by positivity : 0 < 4 * (d : ℝ))]
nlinarith
theorem largeAlphabetLaw_collision_probability
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
≤ 1 / 8192
Proof (Lean source)
-- @node: largeAlphabetLaw_collision_probability
lemma largeAlphabetLaw_collision_probability (n d : ℕ) (q : ℝ) (ξ : Fin d → Bool)
(hn : 1 ≤ n) (hd : 1024 * n ^ 2 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) :
(sampleLaw n (largeAlphabetLaw q ξ (by nlinarith) hq hq1)).real
{s | ¬ noRepeatedTreatedCell s} ≤ 1 / 8192 := by
have hdpos : 1 ≤ d := by nlinarith
let := (largeAlphabetLaw q ξ hdpos hq hq1).2
let : IsProbabilityMeasure ((largeAlphabetLaw q ξ hdpos hq hq1).1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
let : IsProbabilityMeasure (sampleLaw n (largeAlphabetLaw q ξ hdpos hq hq1)) := by
unfold sampleLaw
infer_instance
exact (largeAlphabet_collision_union_bound _ (1 / (4 * (d : ℝ)))
(fun i j hij => largeAlphabetLaw_pair_collision q ξ hdpos hq hq1 i j hij.ne)).trans
(largeAlphabet_collision_budget n d hn hd)
theorem largeAlphabetComparisonLaw_collision_probability
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1024 * n ^ 2 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
conclusion
(sampleLaw n (largeAlphabetComparisonLaw d q (by nlinarith) hq hq1)).real {s | ¬ noRepeatedTreatedCell s}
≤ 1 / 8192
Proof (Lean source)
-- @node: largeAlphabetComparisonLaw_collision_probability
lemma largeAlphabetComparisonLaw_collision_probability (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1024 * n ^ 2 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) :
(sampleLaw n (largeAlphabetComparisonLaw d q (by nlinarith) hq hq1)).real
{s | ¬ noRepeatedTreatedCell s} ≤ 1 / 8192 := by
have hdpos : 1 ≤ d := by nlinarith
let := (largeAlphabetComparisonLaw d q hdpos hq hq1).2
let : IsProbabilityMeasure ((largeAlphabetComparisonLaw d q hdpos hq hq1).1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
let : IsProbabilityMeasure (sampleLaw n (largeAlphabetComparisonLaw d q hdpos hq hq1)) := by
unfold sampleLaw
infer_instance
exact (largeAlphabet_collision_union_bound _ (1 / (4 * (d : ℝ)))
(fun i j hij => largeAlphabetComparisonLaw_pair_collision q hdpos hq hq1 i j hij.ne)).trans
(largeAlphabet_collision_budget n d hn hd)
theorem largeAlphabetSampleMixture_real_event
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(largeAlphabetSampleMixture n d q hd hq hq1).real E
= ∫ ξ, (sampleLaw n (largeAlphabetLaw q ξ hd hq hq1)).real E ∂binaryCellPrior d
Proof (Lean source)
-- @node: largeAlphabetSampleMixture_real_event
lemma largeAlphabetSampleMixture_real_event (n d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (E : Set (Fin n → ObsRecord d)) :
(largeAlphabetSampleMixture n d q hd hq hq1).real E =
∫ ξ, (sampleLaw n (largeAlphabetLaw q ξ hd hq hq1)).real E
∂binaryCellPrior d := by
classical
let μ : Measure Bool := (1 / 2 : ℝ≥0∞) • Measure.dirac false +
(1 / 2 : ℝ≥0∞) • Measure.dirac true
let : IsProbabilityMeasure μ := fairBinary_probability
let : IsProbabilityMeasure (binaryCellPrior d) := inferInstanceAs
(IsProbabilityMeasure (Measure.pi (fun _ : Fin d => μ)))
let : IsProbabilityMeasure (largeAlphabetSampleMixture n d q hd hq hq1) :=
largeAlphabetSampleMixture_probability n d q hd hq hq1
rw [probability_measureReal_eq_tsum_singletons _ E (by measurability),
tsum_fintype]
have hterm (s : Fin n → ObsRecord d) :
E.indicator (fun s => (largeAlphabetSampleMixture n d q hd hq hq1).real {s}) s =
∫ ξ, E.indicator (fun s => (sampleLaw n (largeAlphabetLaw q ξ hd hq hq1)).real {s}) s
∂binaryCellPrior d := by
by_cases hs : s ∈ E
· simp only [Set.indicator_of_mem hs, largeAlphabetSampleMixture_real_singleton]
· simp [Set.indicator_of_notMem hs]
simp_rw [hterm]
rw [← integral_finsetSum _ (fun _ _ => Integrable.of_finite)]
apply integral_congr_ae
filter_upwards [] with ξ
let := (largeAlphabetLaw q ξ hd hq hq1).2
let : IsProbabilityMeasure ((largeAlphabetLaw q ξ hd hq hq1).1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
let : IsProbabilityMeasure (sampleLaw n (largeAlphabetLaw q ξ hd hq hq1)) := by
unfold sampleLaw
infer_instance
rw [probability_measureReal_eq_tsum_singletons _ E (by measurability),
tsum_fintype]
theorem largeAlphabetSampleMixture_collision_probability
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1024 * n ^ 2 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
conclusion
≤ 1 / 8192
Proof (Lean source)
-- @node: largeAlphabetSampleMixture_collision_probability
lemma largeAlphabetSampleMixture_collision_probability (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1024 * n ^ 2 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) :
(largeAlphabetSampleMixture n d q (by nlinarith) hq hq1).real
{s | ¬ noRepeatedTreatedCell s} ≤ 1 / 8192 := by
let μ : Measure Bool := (1 / 2 : ℝ≥0∞) • Measure.dirac false +
(1 / 2 : ℝ≥0∞) • Measure.dirac true
let : IsProbabilityMeasure μ := fairBinary_probability
let : IsProbabilityMeasure (binaryCellPrior d) := inferInstanceAs
(IsProbabilityMeasure (Measure.pi (fun _ : Fin d => μ)))
rw [largeAlphabetSampleMixture_real_event]
calc
_ ≤ ∫ _ξ : Fin d → Bool, (1 / 8192 : ℝ) ∂binaryCellPrior d :=
integral_mono Integrable.of_finite Integrable.of_finite
(fun ξ => largeAlphabetLaw_collision_probability n d q ξ hn hd hq hq1)
_ = 1 / 8192 := by simp
theorem largeAlphabet_actual_collision_testing
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1024 * n ^ 2 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
κ :
hT :
conclusion
1 - (1 / 8192 : ℝ)
≤ (largeAlphabetSampleMixture n d q (by nlinarith) hq hq1 ⊗ₘ κ.1).real T
+ (sampleLaw n (largeAlphabetComparisonLaw d q (by nlinarith) hq hq1) ⊗ₘ κ.1).real Tᶜ
Proof (Lean source)
-- @node: largeAlphabet_actual_collision_testing
lemma largeAlphabet_actual_collision_testing (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1024 * n ^ 2 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1)
(κ : BoundedKernel (Fin n → ObsRecord d))
(T : Set ((Fin n → ObsRecord d) × ℝ)) (hT : MeasurableSet T) :
1 - (1 / 8192 : ℝ) ≤
(largeAlphabetSampleMixture n d q (by nlinarith) hq hq1 ⊗ₘ κ.1).real T +
(sampleLaw n (largeAlphabetComparisonLaw d q (by nlinarith) hq hq1) ⊗ₘ κ.1).real Tᶜ := by
have hdpos : 1 ≤ d := by nlinarith
let := (largeAlphabetComparisonLaw d q hdpos hq hq1).2
let : IsProbabilityMeasure ((largeAlphabetComparisonLaw d q hdpos hq hq1).1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
let : IsProbabilityMeasure (sampleLaw n (largeAlphabetComparisonLaw d q hdpos hq hq1)) := by
unfold sampleLaw
infer_instance
let : IsProbabilityMeasure (largeAlphabetSampleMixture n d q hdpos hq hq1) :=
largeAlphabetSampleMixture_probability n d q hdpos hq hq1
exact largeAlphabet_collision_testing _ _
(largeAlphabetSampleMixture_matching n d q hdpos hq hq1)
(largeAlphabetSampleMixture_collision_probability n d q hn hd hq hq1)
(largeAlphabetComparisonLaw_collision_probability n d q hn hd hq hq1) κ T hT
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.LargeAlphabet.CollisionTV 4 declarations The event on which every treated baseline category occurs at most once.
The event on which every treated baseline category occurs at most once.
def noRepeatedTreatedCell
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
theorem largeAlphabet_collision_union_bound
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
Q :
b :
ℝ
conclusion
Q.real {s | ¬ noRepeatedTreatedCell s} ≤ (n.choose 2 : ℝ) * b
Proof (Lean source)
-- @node: largeAlphabet_collision_union_bound
lemma largeAlphabet_collision_union_bound {n d : ℕ}
(Q : Measure (Fin n → ObsRecord d)) [IsProbabilityMeasure Q]
(b : ℝ)
(hpair : ∀ i j : Fin n, i < j →
Q.real {s | (s i).A = true ∧ (s j).A = true ∧ (s i).X = (s j).X} ≤ b) :
Q.real {s | ¬ noRepeatedTreatedCell s} ≤ (n.choose 2 : ℝ) * b := by
classical
let pairs := (Finset.univ ×ˢ (Finset.univ : Finset (Fin n))).filter
(fun ij => ij.1 < ij.2)
let events (ij : Fin n × Fin n) : Set (Fin n → ObsRecord d) :=
{s | (s ij.1).A = true ∧ (s ij.2).A = true ∧ (s ij.1).X = (s ij.2).X}
have hcover : {s | ¬ noRepeatedTreatedCell s} ⊆ ⋃ ij ∈ pairs, events ij := by
intro s hs
simp only [Set.mem_ofPred_eq, noRepeatedTreatedCell, not_forall,
not_not] at hs
obtain ⟨i, j, hij, hi, hj, hx⟩ := hs
rcases lt_or_gt_of_ne hij with hlt | hgt
· exact Set.mem_iUnion.mpr ⟨(i, j), Set.mem_iUnion.mpr
⟨by simp [pairs, hlt], ⟨hi, hj, hx⟩⟩⟩
· exact Set.mem_iUnion.mpr ⟨(j, i), Set.mem_iUnion.mpr
⟨by simp [pairs, hgt], ⟨hj, hi, hx.symm⟩⟩⟩
calc
Q.real {s | ¬ noRepeatedTreatedCell s} ≤ Q.real (⋃ ij ∈ pairs, events ij) :=
measureReal_mono hcover (measure_ne_top Q _)
_ ≤ ∑ ij ∈ pairs, Q.real (events ij) := measureReal_biUnion_finset_le pairs events
_ ≤ ∑ _ij ∈ pairs, b := by
apply Finset.sum_le_sum
intro ij hij
exact hpair ij.1 ij.2 (by simpa [pairs] using hij)
_ = (n.choose 2 : ℝ) * b := by
simp only [Finset.sum_const, nsmul_eq_mul]
congr 1
simpa [pairs] using congrArg (fun k : ℕ => (k : ℝ))
(Finset.card_product_filter_lt (s := (Finset.univ : Finset (Fin n))))
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: largeAlphabet_tv_le_collision_mass
lemma largeAlphabet_tv_le_collision_mass
{Z : Type*} [Finite Z] [MeasurableSpace Z] [MeasurableSingletonClass Z]
(Q₀ Q₁ : Measure Z) [IsProbabilityMeasure Q₀] [IsProbabilityMeasure Q₁]
(E : Set Z) (hmatch : ∀ z ∈ E, Q₀.real {z} = Q₁.real {z}) :
tvDist Q₀ Q₁ ≤ (Q₀.real Eᶜ + Q₁.real Eᶜ) / 2 := by
classical
let _ := ofFinite Z
have hmass (Q : Measure Z) [IsProbabilityMeasure Q] :
Q.real Eᶜ = ∑ z : Z, Eᶜ.indicator (fun z => Q.real {z}) z := by
simpa only [tsum_fintype] using
probability_measureReal_eq_tsum_singletons Q Eᶜ (by measurability)
have hsum : (∑ z : Z, |Q₀.real {z} - Q₁.real {z}|) ≤
Q₀.real Eᶜ + Q₁.real Eᶜ := by
rw [hmass Q₀, hmass Q₁, ← Finset.sum_add_distrib]
apply Finset.sum_le_sum
intro z _
by_cases hz : z ∈ E
· simp [hmatch z hz, indicator, hz]
· simp only [Set.mem_compl_iff, hz, not_false_eq_true, Set.indicator_of_mem]
apply abs_le.mpr
constructor <;> linarith [measureReal_nonneg (μ := Q₀) (s := {z}),
measureReal_nonneg (μ := Q₁) (s := {z})]
have htv := tvDist_le_half_tsum_singleton_abs Q₀ Q₁
rw [tsum_fintype] at htv
exact htv.trans (by linarith)
theorem largeAlphabet_collision_testing
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
Q₀ Q₁ :
hmatch :
∀ s
then
Q₀.real {s} = Q₁.real {s}
hcollision₀ :
Q₀.real {s | ¬ noRepeatedTreatedCell s} ≤ 1 / 8192
hcollision₁ :
Q₁.real {s | ¬ noRepeatedTreatedCell s} ≤ 1 / 8192
κ :
hT :
Proof (Lean source)
-- @node: largeAlphabet_collision_testing
lemma largeAlphabet_collision_testing
{n d : ℕ} (Q₀ Q₁ : Measure (Fin n → ObsRecord d))
[IsProbabilityMeasure Q₀] [IsProbabilityMeasure Q₁]
(hmatch : ∀ s, noRepeatedTreatedCell s → Q₀.real {s} = Q₁.real {s})
(hcollision₀ : Q₀.real {s | ¬ noRepeatedTreatedCell s} ≤ 1 / 8192)
(hcollision₁ : Q₁.real {s | ¬ noRepeatedTreatedCell s} ≤ 1 / 8192)
(κ : BoundedKernel (Fin n → ObsRecord d))
(T : Set ((Fin n → ObsRecord d) × ℝ)) (hT : MeasurableSet T) :
1 - (1 / 8192 : ℝ) ≤ (Q₀ ⊗ₘ κ.1).real T + (Q₁ ⊗ₘ κ.1).real Tᶜ := by
let : IsMarkovKernel κ.1 := κ.2.1
have htv := largeAlphabet_tv_le_collision_mass Q₀ Q₁
{s | noRepeatedTreatedCell s} hmatch
have hbudget : tvDist Q₀ Q₁ ≤ 1 / 8192 := by
change tvDist Q₀ Q₁ ≤
(Q₀.real {s | ¬ noRepeatedTreatedCell s} +
Q₁.real {s | ¬ noRepeatedTreatedCell s}) / 2 at htv
linarith
exact (sub_le_sub_left hbudget 1).trans
((randomized_kernel_tv_comparison Q₀ Q₁ κ.1).2.2.1 T hT)
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.LargeAlphabet.ComparisonLaw 23 declarations The comparison member with independent treated outcome and arrival draws.
The comparison member with independent treated outcome and arrival draws. This is the law P_* in equations (3) and (5) of the large-alphabet roadmap.
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
q :
ℝ
hq :
0 < q
hq1 :
q ≤ 1
conclusion
conclusion 1
q / (1 + q) ∈ Icc (0 : ℝ) 1
conclusion 2
q ≤ (1 + q) / 2
conclusion 3
(1 + q) / 2 ∈ Icc (0 : ℝ) 1
Proof (Lean source)
-- @node: largeAlphabetComparison_parameters
lemma largeAlphabetComparison_parameters (q : ℝ) (hq : 0 < q) (hq1 : q ≤ 1) :
q / (1 + q) ∈ Icc (0 : ℝ) 1 ∧
q ≤ (1 + q) / 2 ∧ (1 + q) / 2 ∈ Icc (0 : ℝ) 1 := by
have hp : 0 < 1 + q := by linarith
refine ⟨⟨by positivity, (div_le_iff₀ hp).mpr (by linarith)⟩,
by linarith, by constructor <;> linarith⟩
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
q :
ℝ
a :
def
largeAlphabetComparisonArrival q a :
ℝ
given by
if a then (1 + q) / 2 else 1
theorem largeAlphabetComparisonArrival_bounds
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
q ≤ largeAlphabetComparisonArrival q a
conclusion 2
largeAlphabetComparisonArrival q a ∈ Icc (0 : ℝ) 1
Proof (Lean source)
lemma largeAlphabetComparisonArrival_bounds (q : ℝ) (hq : 0 < q)
(hq1 : q ≤ 1) (a : Bool) :
q ≤ largeAlphabetComparisonArrival q a ∧
largeAlphabetComparisonArrival q a ∈ Icc (0 : ℝ) 1 := by
have hp := largeAlphabetComparison_parameters q hq hq1
cases a
· simp only [largeAlphabetComparisonArrival, Bool.false_eq_true, if_false]
exact ⟨hq1, by norm_num⟩
· exact ⟨hp.2.1, hp.2.2⟩
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
largeAlphabetComparisonWeight d q a y r :
ℝ
given by
(1 / (2 * (d : ℝ))) * completeBernWeight (q / (1 + q)) y * completeBernWeight (largeAlphabetComparisonArrival q a) r
theorem largeAlphabetComparisonWeight_nonneg
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
0 ≤ largeAlphabetComparisonWeight d q a y r
Proof (Lean source)
lemma largeAlphabetComparisonWeight_nonneg (d : ℕ) (q : ℝ)
(hq : 0 < q) (hq1 : q ≤ 1) (a y r : Bool) :
0 ≤ largeAlphabetComparisonWeight d q a y r := by
have hy := (largeAlphabetComparison_parameters q hq hq1).1
have hr := (largeAlphabetComparisonArrival_bounds q hq hq1 a).2
have hb (p : ℝ) (hp : p ∈ Icc (0 : ℝ) 1) (b : Bool) :
0 ≤ completeBernWeight p b := by
cases b <;> simp [completeBernWeight] <;> linarith [hp.1, hp.2]
exact mul_nonneg (mul_nonneg (by positivity) (hb _ hy y)) (hb _ hr r)
def largeAlphabetComparisonMeasure
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
d :
ℕ
q :
ℝ
def
given by
∑ x : Fin d, ∑ a : Bool, ∑ y : Bool, ∑ r : Bool, ofReal (largeAlphabetComparisonWeight d q a y r) • Measure.dirac (⟨x, a, false, false, false, y, r⟩ : FullRecord d)
theorem largeAlphabetComparisonWeight_sum_arrival
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
= ofReal ((1 / (2 * (d : ℝ))) * completeBernWeight (q / (1 + q)) y)
Proof (Lean source)
lemma largeAlphabetComparisonWeight_sum_arrival (d : ℕ) (q : ℝ)
(hq : 0 < q) (hq1 : q ≤ 1) (a y : Bool) :
(∑ r : Bool, ofReal (largeAlphabetComparisonWeight d q a y r)) =
ofReal ((1 / (2 * (d : ℝ))) * completeBernWeight (q / (1 + q)) y) := by
rw [Fintype.sum_bool, ← ENNReal.ofReal_add
(largeAlphabetComparisonWeight_nonneg d q hq hq1 a y true)
(largeAlphabetComparisonWeight_nonneg d q hq hq1 a y false)]
congr 1
simp only [largeAlphabetComparisonWeight, completeBernWeight, if_true,
Bool.false_eq_true, if_false]
ring
theorem largeAlphabetComparisonMeasure_univ
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
conclusion
largeAlphabetComparisonMeasure d q univ = 1
Proof (Lean source)
lemma largeAlphabetComparisonMeasure_univ (d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) :
largeAlphabetComparisonMeasure d q univ = 1 := by
have hdpos : 0 < (d : ℝ) := by exact_mod_cast hd
have hp := (largeAlphabetComparison_parameters q hq hq1).1
have hsum : (∑ y : Bool,
ofReal ((1 / (2 * (d : ℝ))) * completeBernWeight (q / (1 + q)) y)) =
ofReal (1 / (2 * (d : ℝ))) := by
simp only [Fintype.sum_bool, completeBernWeight, if_true, Bool.false_eq_true, if_false]
rw [← ENNReal.ofReal_add (mul_nonneg (by positivity) hp.1)
(mul_nonneg (by positivity) (sub_nonneg.mpr hp.2))]
congr 1
ring
simp only [largeAlphabetComparisonMeasure, Measure.finsetSum_apply, Measure.smul_apply,
Measure.dirac_apply_of_mem (mem_univ _), smul_eq_mul, mul_one,
largeAlphabetComparisonWeight_sum_arrival d q hq hq1, hsum]
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_bool, Fintype.card_fin,
nsmul_eq_mul]
rw [← ENNReal.ofReal_natCast, ← ENNReal.ofReal_natCast,
← ENNReal.ofReal_mul (by positivity), ← ENNReal.ofReal_mul (by positivity)]
have heq : (d : ℝ) * (2 * (1 / (2 * (d : ℝ)))) = 1 := by field_simp
norm_num only [Nat.cast_ofNat] at ⊢
rw [heq]
norm_num
def largeAlphabetComparisonLaw
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
def
given by
⟨largeAlphabetComparisonMeasure d q, ⟨largeAlphabetComparisonMeasure_univ d q hd hq hq1⟩⟩
theorem largeAlphabetComparisonLaw_real_event
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(largeAlphabetComparisonLaw d q hd hq hq1).1.real E
= by classical exact ∑ x : Fin d, ∑ a : Bool, ∑ y : Bool, ∑ r : Bool, if (⟨x, a, false, false, false, y, r⟩ : FullRecord d) ∈ E then largeAlphabetComparisonWeight d q a y r else 0
Proof (Lean source)
-- @node: largeAlphabetComparisonLaw_real_event
lemma largeAlphabetComparisonLaw_real_event (d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (E : Set (FullRecord d)) :
(largeAlphabetComparisonLaw d q hd hq hq1).1.real E = by
classical
exact ∑ x : Fin d, ∑ a : Bool, ∑ y : Bool, ∑ r : Bool,
if (⟨x, a, false, false, false, y, r⟩ : FullRecord d) ∈ E then
largeAlphabetComparisonWeight d q a y r else 0 := by
classical
simp only [largeAlphabetComparisonLaw, largeAlphabetComparisonMeasure, Measure.real,
Measure.finsetSum_apply, Measure.smul_apply, smul_eq_mul, Measure.dirac_apply,
Set.indicator_apply, Pi.one_apply, mul_ite, mul_one, mul_zero]
have hfinite (x : Fin d) (a y r : Bool) :
(if (⟨x, a, false, false, false, y, r⟩ : FullRecord d) ∈ E then
ofReal (largeAlphabetComparisonWeight d q a y r) else 0) ≠ ⊤ := by
split_ifs <;> simp
rw [ENNReal.toReal_sum (by simp only [ENNReal.sum_ne_top]; intros; apply hfinite)]
apply Finset.sum_congr rfl
intro x _
rw [ENNReal.toReal_sum (by simp only [ENNReal.sum_ne_top]; intros; apply hfinite)]
apply Finset.sum_congr rfl
intro a _
rw [ENNReal.toReal_sum (by simp only [ENNReal.sum_ne_top]; intros; apply hfinite)]
apply Finset.sum_congr rfl
intro y _
rw [ENNReal.toReal_sum (by intros; apply hfinite)]
apply Finset.sum_congr rfl
intro r _
split_ifs
· exact ENNReal.toReal_ofReal (largeAlphabetComparisonWeight_nonneg d q hq hq1 a y r)
· rfl
theorem largeAlphabetComparisonLaw_integral
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ w, f w ∂(largeAlphabetComparisonLaw d q hd hq hq1).1)
Proof (Lean source)
-- @node: largeAlphabetComparisonLaw_integral
lemma largeAlphabetComparisonLaw_integral (d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (f : FullRecord d → ℝ) :
(∫ w, f w ∂(largeAlphabetComparisonLaw d q hd hq hq1).1) =
∑ x : Fin d, ∑ a : Bool, ∑ y : Bool, ∑ r : Bool,
largeAlphabetComparisonWeight d q a y r *
f ⟨x, a, false, false, false, y, r⟩ := by
have hi (x : Fin d) (a y r : Bool) : Integrable f
(ofReal (largeAlphabetComparisonWeight d q a y r) •
Measure.dirac (⟨x, a, false, false, false, y, r⟩ : FullRecord d)) := by
exact (integrable_dirac (by simp)).smul_measure ENNReal.ofReal_ne_top
have hir (x : Fin d) (a y : Bool) : Integrable f
(∑ r : Bool, ofReal (largeAlphabetComparisonWeight d q a y r) •
Measure.dirac (⟨x, a, false, false, false, y, r⟩ : FullRecord d)) := by
apply integrable_finsetSum_measure.mpr
intro r _
exact hi x a y r
have hiy (x : Fin d) (a : Bool) : Integrable f
(∑ y : Bool, ∑ r : Bool, ofReal (largeAlphabetComparisonWeight d q a y r) •
Measure.dirac (⟨x, a, false, false, false, y, r⟩ : FullRecord d)) := by
apply integrable_finsetSum_measure.mpr
intro y _
exact hir x a y
have hia (x : Fin d) : Integrable f
(∑ a : Bool, ∑ y : Bool, ∑ r : Bool,
ofReal (largeAlphabetComparisonWeight d q a y r) •
Measure.dirac (⟨x, a, false, false, false, y, r⟩ : FullRecord d)) := by
apply integrable_finsetSum_measure.mpr
intro a _
exact hiy x a
change (∫ w, f w ∂largeAlphabetComparisonMeasure d q) = _
rw [largeAlphabetComparisonMeasure, integral_finsetSum_measure (by intro x _; exact hia x)]
apply Finset.sum_congr rfl
intro x _
rw [integral_finsetSum_measure (by intro a _; exact hiy x a)]
apply Finset.sum_congr rfl
intro a _
rw [integral_finsetSum_measure (by intro y _; exact hir x a y)]
apply Finset.sum_congr rfl
intro y _
rw [integral_finsetSum_measure (by intro r _; exact hi x a y r)]
apply Finset.sum_congr rfl
intro r _
rw [integral_smul_measure, integral_dirac,
ENNReal.toReal_ofReal (largeAlphabetComparisonWeight_nonneg d q hq hq1 a y r)]
rfl
theorem largeAlphabetComparisonLaw_ate
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
conclusion
ate (largeAlphabetComparisonLaw d q hd hq hq1) = q / (1 + q)
Proof (Lean source)
-- @node: largeAlphabetComparisonLaw_ate
lemma largeAlphabetComparisonLaw_ate (d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) :
ate (largeAlphabetComparisonLaw d q hd hq hq1) = q / (1 + q) := by
have hdpos : 0 < (d : ℝ) := by exact_mod_cast hd
rw [ate, largeAlphabetComparisonLaw_integral]
simp [largeAlphabetComparisonWeight, completeBernWeight, largeAlphabetComparisonArrival]
field_simp
<;> ring
theorem largeAlphabetComparisonLaw_observed_atom
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
-- @node: largeAlphabetComparisonLaw_observed_atom
lemma largeAlphabetComparisonLaw_observed_atom (d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (o : ObsRecord d) :
((largeAlphabetComparisonLaw d q hd hq hq1).1.map obs).real {o} =
if o.S = false then (1 / (2 * (d : ℝ))) *
(if o.A then
if o.R then (if o.RY then q / 2 else 1 / 2)
else (if o.RY then 0 else (1 - q) / 2)
else if o.R = true ∧ o.RY = false then 1 else 0)
else 0 := by
classical
have hp : 1 + q ≠ 0 := by linarith
rw [measureReal_def, Measure.map_apply (by fun_prop) (by simp)]
change (largeAlphabetComparisonLaw d q hd hq hq1).1.real (obs ⁻¹' {o}) = _
rw [largeAlphabetComparisonLaw_real_event]
rcases o with ⟨x, a, s, r, ry⟩
cases a <;> cases s <;> cases r <;> cases ry <;>
simp [obs, FullRecord.S, FullRecord.Y, largeAlphabetComparisonWeight,
completeBernWeight, largeAlphabetComparisonArrival, ite_and,
Finset.sum_ite_irrel, Finset.sum_ite_eq'] <;>
field_simp <;> ring
theorem largeAlphabetComparisonLaw_cellProb
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
cellProb (largeAlphabetComparisonLaw d q hd hq hq1) (a, x, s)
= if s then 0 else 1 / (2 * (d : ℝ))
Proof (Lean source)
-- @node: largeAlphabetComparisonLaw_cellProb
lemma largeAlphabetComparisonLaw_cellProb (d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (a : Bool) (x : Fin d) (s : Bool) :
cellProb (largeAlphabetComparisonLaw d q hd hq hq1) (a, x, s) =
if s then 0 else 1 / (2 * (d : ℝ)) := by
classical
rw [cellProb, largeAlphabetComparisonLaw_real_event]
cases a <;> cases s <;>
simp [inCell, FullRecord.S, largeAlphabetComparisonWeight,
completeBernWeight, largeAlphabetComparisonArrival,
Finset.sum_ite_irrel, Finset.sum_ite_eq'] <;> ring
theorem largeAlphabetComparisonLaw_arrivedCell
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
arrivedCell (largeAlphabetComparisonLaw d q hd hq hq1) (a, x, s)
= largeAlphabetComparisonArrival q a * cellProb (largeAlphabetComparisonLaw d q hd hq hq1) (a, x, s)
Proof (Lean source)
-- @node: largeAlphabetComparisonLaw_arrivedCell
lemma largeAlphabetComparisonLaw_arrivedCell (d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (a : Bool) (x : Fin d) (s : Bool) :
arrivedCell (largeAlphabetComparisonLaw d q hd hq hq1) (a, x, s) =
largeAlphabetComparisonArrival q a *
cellProb (largeAlphabetComparisonLaw d q hd hq hq1) (a, x, s) := by
classical
rw [arrivedCell, largeAlphabetComparisonLaw_real_event, largeAlphabetComparisonLaw_cellProb]
cases a <;> cases s <;>
simp [inCell, FullRecord.S, largeAlphabetComparisonWeight,
completeBernWeight, largeAlphabetComparisonArrival,
Finset.sum_ite_irrel, Finset.sum_ite_eq'] <;> ring
theorem largeAlphabetComparisonLaw_mar
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
conclusion
ArrivalMAR (largeAlphabetComparisonLaw d q hd hq hq1)
Proof (Lean source)
-- @node: largeAlphabetComparisonLaw_mar
lemma largeAlphabetComparisonLaw_mar (d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) :
ArrivalMAR (largeAlphabetComparisonLaw d q hd hq hq1) := by
classical
intro j y r
rcases j with ⟨a, x, s⟩
rw [largeAlphabetComparisonLaw_cellProb]
simp_rw [largeAlphabetComparisonLaw_real_event]
cases a <;> cases s <;> cases y <;> cases r <;>
simp [inCell, FullRecord.S, FullRecord.Y, largeAlphabetComparisonWeight,
completeBernWeight, largeAlphabetComparisonArrival, ite_and,
Finset.sum_ite_irrel, Finset.sum_ite_eq']
all_goals first | (left; ring) | ring
theorem largeAlphabetComparisonLaw_occupiedCellArrival
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
conclusion
OccupiedCellArrival q (largeAlphabetComparisonLaw d q hd hq hq1)
Proof (Lean source)
lemma largeAlphabetComparisonLaw_occupiedCellArrival (d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) :
OccupiedCellArrival q (largeAlphabetComparisonLaw d q hd hq hq1) := by
intro j hj
rcases j with ⟨a, x, s⟩
rw [largeAlphabetComparisonLaw_arrivedCell]
exact mul_le_mul_of_nonneg_right
(largeAlphabetComparisonArrival_bounds q hq hq1 a).1 hj.le
theorem largeAlphabetComparisonLaw_balanced
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
conclusion
BalancedRandomization (largeAlphabetComparisonLaw d q hd hq hq1)
Proof (Lean source)
lemma largeAlphabetComparisonLaw_balanced (d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) :
BalancedRandomization (largeAlphabetComparisonLaw d q hd hq hq1) := by
classical
have hdpos : 0 < (d : ℝ) := by exact_mod_cast hd
rw [BalancedRandomization, largeAlphabetComparisonLaw_real_event]
calc
_ = ∑ _x : Fin d, 1 / (2 * (d : ℝ)) := by
apply Finset.sum_congr rfl
intro x _
simp [largeAlphabetComparisonWeight, completeBernWeight, largeAlphabetComparisonArrival]
ring
_ = 1 / 2 := by simp; field_simp
theorem largeAlphabetComparisonLaw_arm_event_factor
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
conclusion
(largeAlphabetComparisonLaw d q hd hq hq1).1.real {w | w.A ∈ s ∧ (w.X, w.S0, w.S1, w.Y0, w.Y1) ∈ t}
= (by classical exact ((if true ∈ s then (1 : ℝ) else 0) + (if false ∈ s then (1 : ℝ) else 0)) / 2) * (largeAlphabetComparisonLaw d q hd hq hq1).1.real {w | (w.X, w.S0, w.S1, w.Y0, w.Y1) ∈ t}
Proof (Lean source)
-- @node: largeAlphabetComparisonLaw_arm_event_factor
lemma largeAlphabetComparisonLaw_arm_event_factor (d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (s : Set Bool)
(t : Set (Fin d × Bool × Bool × Bool × Bool)) :
(largeAlphabetComparisonLaw d q hd hq hq1).1.real
{w | w.A ∈ s ∧ (w.X, w.S0, w.S1, w.Y0, w.Y1) ∈ t} =
(by classical exact ((if true ∈ s then (1 : ℝ) else 0) +
(if false ∈ s then (1 : ℝ) else 0)) / 2) *
(largeAlphabetComparisonLaw d q hd hq hq1).1.real
{w | (w.X, w.S0, w.S1, w.Y0, w.Y1) ∈ t} := by
classical
rw [largeAlphabetComparisonLaw_real_event, largeAlphabetComparisonLaw_real_event,
Finset.mul_sum]
apply Finset.sum_congr rfl
intro x _
by_cases hsT : true ∈ s <;> by_cases hsF : false ∈ s <;>
by_cases htT : (x, false, false, false, true) ∈ t <;>
by_cases htF : (x, false, false, false, false) ∈ t <;>
simp [largeAlphabetComparisonWeight, completeBernWeight,
largeAlphabetComparisonArrival, hsT, hsF, htT, htF] <;> ring
theorem largeAlphabetComparisonLaw_arm_mass
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma largeAlphabetComparisonLaw_arm_mass (d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (s : Set Bool) :
(largeAlphabetComparisonLaw d q hd hq hq1).1.real {w | w.A ∈ s} =
(by classical exact ((if true ∈ s then (1 : ℝ) else 0) +
(if false ∈ s then (1 : ℝ) else 0)) / 2) := by
let := (largeAlphabetComparisonLaw d q hd hq hq1).2
simpa using largeAlphabetComparisonLaw_arm_event_factor d q hd hq hq1 s univ
theorem largeAlphabetComparisonLaw_randomized
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
conclusion
RandomizedIndependence (largeAlphabetComparisonLaw d q hd hq hq1)
Proof (Lean source)
lemma largeAlphabetComparisonLaw_randomized (d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) :
RandomizedIndependence (largeAlphabetComparisonLaw d q hd hq hq1) := by
let := (largeAlphabetComparisonLaw d q hd hq hq1).2
rw [RandomizedIndependence, indepFun_iff_measure_inter_preimage_eq_mul]
intro s t _ _
apply (ENNReal.toReal_eq_toReal_iff' (measure_ne_top _ _) (by finiteness)).mp
rw [ENNReal.toReal_mul]
change (largeAlphabetComparisonLaw d q hd hq hq1).1.real
{w | w.A ∈ s ∧ (w.X, w.S0, w.S1, w.Y0, w.Y1) ∈ t} =
(largeAlphabetComparisonLaw d q hd hq hq1).1.real {w | w.A ∈ s} *
(largeAlphabetComparisonLaw d q hd hq hq1).1.real
{w | (w.X, w.S0, w.S1, w.Y0, w.Y1) ∈ t}
rw [largeAlphabetComparisonLaw_arm_event_factor, largeAlphabetComparisonLaw_arm_mass]
theorem largeAlphabetComparisonLaw_model
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
conclusion
UnrestrictedArrivalModelClass n d q (largeAlphabetComparisonLaw d q hd hq hq1)
Proof (Lean source)
-- @node: largeAlphabetComparisonLaw_model
lemma largeAlphabetComparisonLaw_model (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) :
UnrestrictedArrivalModelClass n d q (largeAlphabetComparisonLaw d q hd hq hq1) := by
refine ⟨hn, hd, hq, hq1, largeAlphabetComparisonLaw_randomized d q hd hq hq1,
largeAlphabetComparisonLaw_balanced d q hd hq hq1, ?_, ?_,
largeAlphabetComparisonLaw_mar d q hd hq hq1,
largeAlphabetComparisonLaw_occupiedCellArrival d q hd hq hq1⟩
all_goals filter_upwards [] with w; rfl
theorem largeAlphabet_one_visit_matching
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
-- @node: largeAlphabet_one_visit_matching
lemma largeAlphabet_one_visit_matching (d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (o : ObsRecord d) :
(∫ b : Bool,
((largeAlphabetLaw q (fun _ : Fin d => b) hd hq hq1).1.map obs).real {o}
∂((1 / 2 : ℝ≥0∞) • Measure.dirac false +
(1 / 2 : ℝ≥0∞) • Measure.dirac true)) =
((largeAlphabetComparisonLaw d q hd hq hq1).1.map obs).real {o} := by
rw [fairBinary_integral]
simp_rw [largeAlphabetLaw_observed_atom, largeAlphabetComparisonLaw_observed_atom]
rcases o with ⟨x, a, s, r, ry⟩
cases a <;> cases s <;> cases r <;> cases ry <;>
simp [largeAlphabetArrival, completeBernWeight] <;> ring
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.LargeAlphabet.FuzzyPrior 5 declarations The fair product prior on binary cell outcome parameters for the large-alphabet converse.
The fair product prior on binary cell outcome parameters for the large-alphabet converse.
def binaryCellPrior
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def cellPriorTarget
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: fairBinary_integral
lemma fairBinary_integral (f : Bool → ℝ) :
(∫ b, f b ∂((1 / 2 : ℝ≥0∞) • Measure.dirac false +
(1 / 2 : ℝ≥0∞) • Measure.dirac true)) = (f false + f true) / 2 := by
rw [integral_add_measure
((integrable_dirac (by simp)).smul_measure (by norm_num))
((integrable_dirac (by simp)).smul_measure (by norm_num)),
integral_smul_measure, integral_smul_measure]
simp
ring
the stated mathematical conclusion holds.
Formal statement
conclusion
Proof (Lean source)
-- @node: fairBinary_probability
lemma fairBinary_probability :
IsProbabilityMeasure ((1 / 2 : ℝ≥0∞) • Measure.dirac false +
(1 / 2 : ℝ≥0∞) • Measure.dirac true) := by
constructor
norm_num
exact ENNReal.inv_two_add_inv_two
theorem binaryCellPrior_target_concentration Lemma binaryCellPrior_target_concentration in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
1 ≤ d
h :
ℝ
hh :
0 < h
conclusion
Proof (Lean source)
lemma binaryCellPrior_target_concentration (d : ℕ) (hd : 1 ≤ d) (h : ℝ) (hh : 0 < h) :
(binaryCellPrior d).real {ξ | h < |cellPriorTarget ξ - 1 / 2|} ≤
1 / (4 * (d : ℝ) * h ^ 2) := by
let μ : Measure Bool := (1 / 2 : ℝ≥0∞) • Measure.dirac false +
(1 / 2 : ℝ≥0∞) • Measure.dirac true
let : IsProbabilityMeasure μ := fairBinary_probability
let g : Bool → ℝ := fun b => if b then 1 else 0
have hg : MemLp g 2 μ := MemLp.of_discrete
have hmean : (∫ b, g b ∂μ) = 1 / 2 := by
rw [show μ = _ from rfl, fairBinary_integral]
norm_num [g]
have hvar : variance g μ = 1 / 4 := by
rw [variance_eq_integral hg.aemeasurable, hmean]
rw [show μ = _ from rfl, fairBinary_integral]
norm_num [g]
have hdpos : 0 < (d : ℝ) := by exact_mod_cast hd
have hcheb := iid_sum_chebyshev
(N := d) μ g hg (mul_pos hdpos hh)
rw [hmean, hvar] at hcheb
have hsub : {ξ | h < |cellPriorTarget ξ - 1 / 2|} ⊆
{ξ : Fin d → Bool | (d : ℝ) * h ≤ |(∑ x, g (ξ x)) - d * (1 / 2)|} := by
intro ξ hξ
have heq : (∑ x, g (ξ x)) - d * (1 / 2) =
(d : ℝ) * (cellPriorTarget ξ - 1 / 2) := by
dsimp [cellPriorTarget, g]
field_simp
change (d : ℝ) * h ≤ |(∑ x, g (ξ x)) - d * (1 / 2)|
rw [heq, abs_mul, abs_of_pos hdpos]
exact le_of_lt (mul_lt_mul_of_pos_left hξ hdpos)
have hprob := (measure_mono hsub).trans hcheb
have hreal := ENNReal.toReal_mono (by simp) hprob
rw [ENNReal.toReal_ofReal (by positivity)] at hreal
change (binaryCellPrior d).real {ξ | h < |cellPriorTarget ξ - 1 / 2|} ≤ _ at hreal
have halg : (d : ℝ) * (1 / 4) / ((d : ℝ) * h) ^ 2 =
1 / (4 * (d : ℝ) * h ^ 2) := by
field_simp
simpa only [halg] using hreal
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.LargeAlphabet.MemberLaw 20 declarations The deterministic binary-cell members of the large-alphabet lower-bound experiment.
The deterministic binary-cell members of the large-alphabet lower-bound experiment.
def largeAlphabetRecord
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def largeAlphabetMeasure
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
given by
∑ x : Fin d, ∑ a : Bool, ∑ r : Bool, ofReal ((1 / (2 * (d : ℝ))) * completeBernWeight (largeAlphabetArrival q ξ x a) r) • Measure.dirac (largeAlphabetRecord ξ x a r)
theorem largeAlphabetArrival_bounds
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
q ≤ largeAlphabetArrival q ξ x a
conclusion 2
largeAlphabetArrival q ξ x a ∈ Icc (0 : ℝ) 1
Proof (Lean source)
-- @node: largeAlphabetArrival_bounds
lemma largeAlphabetArrival_bounds {d : ℕ} (q : ℝ) (ξ : Fin d → Bool)
(hq : 0 < q) (hq1 : q ≤ 1) (x : Fin d) (a : Bool) :
q ≤ largeAlphabetArrival q ξ x a ∧
largeAlphabetArrival q ξ x a ∈ Icc (0 : ℝ) 1 := by
unfold largeAlphabetArrival
split_ifs <;> refine ⟨by linarith, ?_, ?_⟩ <;> linarith
theorem largeAlphabet_weight_nonneg
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
0 ≤ (1 / (2 * (d : ℝ))) * completeBernWeight (largeAlphabetArrival q ξ x a) r
Proof (Lean source)
-- @node: largeAlphabet_weight_nonneg
lemma largeAlphabet_weight_nonneg {d : ℕ} (q : ℝ) (ξ : Fin d → Bool)
(hq : 0 < q) (hq1 : q ≤ 1) (x : Fin d) (a r : Bool) :
0 ≤ (1 / (2 * (d : ℝ))) * completeBernWeight (largeAlphabetArrival q ξ x a) r := by
have hb := (largeAlphabetArrival_bounds q ξ hq hq1 x a).2
apply mul_nonneg (by positivity)
cases r <;> simp only [completeBernWeight, Bool.false_eq_true, if_false, if_true]
· linarith [hb.2]
· exact hb.1
theorem largeAlphabetMeasure_univ
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
largeAlphabetMeasure q ξ univ = 1
Proof (Lean source)
-- @node: largeAlphabetMeasure_univ
lemma largeAlphabetMeasure_univ {d : ℕ} (q : ℝ) (ξ : Fin d → Bool)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) :
largeAlphabetMeasure q ξ univ = 1 := by
have hdpos : 0 < (d : ℝ) := by exact_mod_cast hd
have hsum (x : Fin d) (a : Bool) :
(∑ r : Bool, ofReal ((1 / (2 * (d : ℝ))) *
completeBernWeight (largeAlphabetArrival q ξ x a) r)) =
ofReal (1 / (2 * (d : ℝ))) := by
simp only [Fintype.sum_bool, completeBernWeight, if_true, Bool.false_eq_true, if_false]
rw [← ENNReal.ofReal_add
(by simpa [completeBernWeight] using largeAlphabet_weight_nonneg q ξ hq hq1 x a true)
(by simpa [completeBernWeight] using largeAlphabet_weight_nonneg q ξ hq hq1 x a false)]
congr 1
ring
simp only [largeAlphabetMeasure, Measure.finsetSum_apply, Measure.smul_apply,
Measure.dirac_apply_of_mem (mem_univ _), smul_eq_mul, mul_one, hsum]
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_bool, Fintype.card_fin,
nsmul_eq_mul]
rw [← ENNReal.ofReal_natCast, ← ENNReal.ofReal_natCast,
← ENNReal.ofReal_mul (by positivity), ← ENNReal.ofReal_mul (by positivity)]
have heq : (d : ℝ) * (2 * (1 / (2 * (d : ℝ)))) = 1 := by field_simp
norm_num only [Nat.cast_ofNat] at ⊢
rw [heq]
norm_num
def largeAlphabetLaw
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
theorem largeAlphabetLaw_real_event
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(largeAlphabetLaw q ξ hd hq hq1).1.real E
= by classical exact ∑ x : Fin d, ∑ a : Bool, ∑ r : Bool, if largeAlphabetRecord ξ x a r ∈ E then (1 / (2 * (d : ℝ))) * completeBernWeight (largeAlphabetArrival q ξ x a) r else 0
Proof (Lean source)
-- @node: largeAlphabetLaw_real_event
lemma largeAlphabetLaw_real_event {d : ℕ} (q : ℝ) (ξ : Fin d → Bool)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (E : Set (FullRecord d)) :
(largeAlphabetLaw q ξ hd hq hq1).1.real E = by
classical
exact ∑ x : Fin d, ∑ a : Bool, ∑ r : Bool,
if largeAlphabetRecord ξ x a r ∈ E then
(1 / (2 * (d : ℝ))) * completeBernWeight (largeAlphabetArrival q ξ x a) r
else 0 := by
classical
simp only [largeAlphabetLaw, largeAlphabetMeasure, Measure.real,
Measure.finsetSum_apply, Measure.smul_apply, smul_eq_mul, Measure.dirac_apply,
Set.indicator_apply, Pi.one_apply, mul_ite, mul_one, mul_zero]
rw [ENNReal.toReal_sum (by
simp only [ENNReal.sum_ne_top]
intros
split_ifs <;> simp only [ENNReal.ofReal_ne_top, ne_eq, ENNReal.zero_ne_top, not_false_eq_true])]
apply Finset.sum_congr rfl
intro x _
rw [ENNReal.toReal_sum (by
simp only [ENNReal.sum_ne_top]
intros
split_ifs <;> simp only [ENNReal.ofReal_ne_top, ne_eq, ENNReal.zero_ne_top, not_false_eq_true])]
apply Finset.sum_congr rfl
intro a _
rw [ENNReal.toReal_sum (by
intros
split_ifs <;> simp only [ENNReal.ofReal_ne_top, ne_eq, ENNReal.zero_ne_top, not_false_eq_true])]
apply Finset.sum_congr rfl
intro r _
split_ifs
· exact ENNReal.toReal_ofReal (largeAlphabet_weight_nonneg q ξ hq hq1 x a r)
· rfl
theorem largeAlphabetLaw_cellProb
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: largeAlphabetLaw_cellProb
lemma largeAlphabetLaw_cellProb {d : ℕ} (q : ℝ) (ξ : Fin d → Bool)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (a : Bool) (x : Fin d) (s : Bool) :
cellProb (largeAlphabetLaw q ξ hd hq hq1) (a, x, s) =
if s then 0 else 1 / (2 * (d : ℝ)) := by
classical
rw [cellProb, largeAlphabetLaw_real_event]
cases a <;> cases s <;>
simp [inCell, largeAlphabetRecord, FullRecord.S, completeBernWeight,
Finset.sum_ite_irrel, Finset.sum_ite_eq'] <;> ring
theorem largeAlphabetLaw_arrivedCell
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
arrivedCell (largeAlphabetLaw q ξ hd hq hq1) (a, x, s)
= largeAlphabetArrival q ξ x a * cellProb (largeAlphabetLaw q ξ hd hq hq1) (a, x, s)
Proof (Lean source)
-- @node: largeAlphabetLaw_arrivedCell
lemma largeAlphabetLaw_arrivedCell {d : ℕ} (q : ℝ) (ξ : Fin d → Bool)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (a : Bool) (x : Fin d) (s : Bool) :
arrivedCell (largeAlphabetLaw q ξ hd hq hq1) (a, x, s) =
largeAlphabetArrival q ξ x a * cellProb (largeAlphabetLaw q ξ hd hq hq1) (a, x, s) := by
classical
rw [arrivedCell, largeAlphabetLaw_real_event, largeAlphabetLaw_cellProb]
cases a <;> cases s <;>
simp [inCell, largeAlphabetRecord, FullRecord.S, completeBernWeight,
Finset.sum_ite_irrel, Finset.sum_ite_eq'] <;> ring
theorem largeAlphabetLaw_mar
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
ArrivalMAR (largeAlphabetLaw q ξ hd hq hq1)
Proof (Lean source)
-- @node: largeAlphabetLaw_mar
lemma largeAlphabetLaw_mar {d : ℕ} (q : ℝ) (ξ : Fin d → Bool)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) :
ArrivalMAR (largeAlphabetLaw q ξ hd hq hq1) := by
classical
intro j y r
rcases j with ⟨a, x, s⟩
rw [largeAlphabetLaw_cellProb]
simp_rw [largeAlphabetLaw_real_event]
cases a <;> cases s <;> cases y <;> cases r <;>
simp [inCell, largeAlphabetRecord, FullRecord.S, FullRecord.Y, completeBernWeight,
Finset.sum_ite_irrel, Finset.sum_ite_eq', ite_and]
all_goals first | (left; ring) | (split_ifs <;> ring) | ring
theorem largeAlphabetLaw_occupiedCellArrival
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
OccupiedCellArrival q (largeAlphabetLaw q ξ hd hq hq1)
Proof (Lean source)
-- @node: largeAlphabetLaw_occupiedCellArrival
lemma largeAlphabetLaw_occupiedCellArrival {d : ℕ} (q : ℝ) (ξ : Fin d → Bool)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) :
OccupiedCellArrival q (largeAlphabetLaw q ξ hd hq hq1) := by
intro j hj
rcases j with ⟨a, x, s⟩
rw [largeAlphabetLaw_arrivedCell]
exact mul_le_mul_of_nonneg_right
(largeAlphabetArrival_bounds q ξ hq hq1 x a).1 hj.le
theorem largeAlphabetLaw_balanced
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
BalancedRandomization (largeAlphabetLaw q ξ hd hq hq1)
Proof (Lean source)
-- @node: largeAlphabetLaw_balanced
lemma largeAlphabetLaw_balanced {d : ℕ} (q : ℝ) (ξ : Fin d → Bool)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) :
BalancedRandomization (largeAlphabetLaw q ξ hd hq hq1) := by
classical
have hdpos : 0 < (d : ℝ) := by exact_mod_cast hd
rw [BalancedRandomization, largeAlphabetLaw_real_event]
calc
_ = ∑ _x : Fin d, 1 / (2 * (d : ℝ)) := by
apply Finset.sum_congr rfl
intro x _
simp [largeAlphabetRecord, completeBernWeight]
ring
_ = 1 / 2 := by simp; field_simp
theorem largeAlphabetLaw_arm_event_factor
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
conclusion
(largeAlphabetLaw q ξ hd hq hq1).1.real {w | w.A ∈ s ∧ (w.X, w.S0, w.S1, w.Y0, w.Y1) ∈ t}
= (by classical exact ((if true ∈ s then (1 : ℝ) else 0) + (if false ∈ s then (1 : ℝ) else 0)) / 2) * (largeAlphabetLaw q ξ hd hq hq1).1.real {w | (w.X, w.S0, w.S1, w.Y0, w.Y1) ∈ t}
Proof (Lean source)
-- @node: largeAlphabetLaw_arm_event_factor
lemma largeAlphabetLaw_arm_event_factor {d : ℕ} (q : ℝ) (ξ : Fin d → Bool)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (s : Set Bool)
(t : Set (Fin d × Bool × Bool × Bool × Bool)) :
(largeAlphabetLaw q ξ hd hq hq1).1.real
{w | w.A ∈ s ∧ (w.X, w.S0, w.S1, w.Y0, w.Y1) ∈ t} =
(by classical exact ((if true ∈ s then (1 : ℝ) else 0) +
(if false ∈ s then (1 : ℝ) else 0)) / 2) *
(largeAlphabetLaw q ξ hd hq hq1).1.real
{w | (w.X, w.S0, w.S1, w.Y0, w.Y1) ∈ t} := by
classical
rw [largeAlphabetLaw_real_event, largeAlphabetLaw_real_event, Finset.mul_sum]
apply Finset.sum_congr rfl
intro x _
by_cases hsT : true ∈ s <;> by_cases hsF : false ∈ s <;>
by_cases ht : (x, false, false, false, ξ x) ∈ t <;>
simp [largeAlphabetRecord, completeBernWeight, hsT, hsF, ht] <;> ring
theorem largeAlphabetLaw_arm_mass
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: largeAlphabetLaw_arm_mass
lemma largeAlphabetLaw_arm_mass {d : ℕ} (q : ℝ) (ξ : Fin d → Bool)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (s : Set Bool) :
(largeAlphabetLaw q ξ hd hq hq1).1.real {w | w.A ∈ s} =
(by classical exact ((if true ∈ s then (1 : ℝ) else 0) +
(if false ∈ s then (1 : ℝ) else 0)) / 2) := by
let := (largeAlphabetLaw q ξ hd hq hq1).2
simpa using largeAlphabetLaw_arm_event_factor q ξ hd hq hq1 s univ
theorem largeAlphabetLaw_randomized
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
RandomizedIndependence (largeAlphabetLaw q ξ hd hq hq1)
Proof (Lean source)
-- @node: largeAlphabetLaw_randomized
lemma largeAlphabetLaw_randomized {d : ℕ} (q : ℝ) (ξ : Fin d → Bool)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) :
RandomizedIndependence (largeAlphabetLaw q ξ hd hq hq1) := by
let := (largeAlphabetLaw q ξ hd hq hq1).2
rw [RandomizedIndependence, indepFun_iff_measure_inter_preimage_eq_mul]
intro s t _ _
apply (ENNReal.toReal_eq_toReal_iff' (measure_ne_top _ _) (by finiteness)).mp
rw [ENNReal.toReal_mul]
change (largeAlphabetLaw q ξ hd hq hq1).1.real
{w | w.A ∈ s ∧ (w.X, w.S0, w.S1, w.Y0, w.Y1) ∈ t} =
(largeAlphabetLaw q ξ hd hq hq1).1.real {w | w.A ∈ s} *
(largeAlphabetLaw q ξ hd hq hq1).1.real
{w | (w.X, w.S0, w.S1, w.Y0, w.Y1) ∈ t}
rw [largeAlphabetLaw_arm_event_factor, largeAlphabetLaw_arm_mass]
theorem largeAlphabetLaw_model
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
UnrestrictedArrivalModelClass n d q (largeAlphabetLaw q ξ hd hq hq1)
Proof (Lean source)
-- @node: largeAlphabetLaw_model
lemma largeAlphabetLaw_model {n d : ℕ} (q : ℝ) (ξ : Fin d → Bool)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) :
UnrestrictedArrivalModelClass n d q (largeAlphabetLaw q ξ hd hq hq1) := by
refine ⟨hn, hd, hq, hq1, largeAlphabetLaw_randomized q ξ hd hq hq1,
largeAlphabetLaw_balanced q ξ hd hq hq1, ?_, ?_,
largeAlphabetLaw_mar q ξ hd hq hq1,
largeAlphabetLaw_occupiedCellArrival q ξ hd hq hq1⟩
all_goals filter_upwards [] with w; rfl
theorem largeAlphabetLaw_integral
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ w, f w ∂(largeAlphabetLaw q ξ hd hq hq1).1)
= ∑ x : Fin d, ∑ a : Bool, ∑ r : Bool, (1 / (2 * (d : ℝ))) * completeBernWeight (largeAlphabetArrival q ξ x a) r * f (largeAlphabetRecord ξ x a r)
Proof (Lean source)
-- @node: largeAlphabetLaw_integral
lemma largeAlphabetLaw_integral {d : ℕ} (q : ℝ) (ξ : Fin d → Bool)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (f : FullRecord d → ℝ) :
(∫ w, f w ∂(largeAlphabetLaw q ξ hd hq hq1).1) =
∑ x : Fin d, ∑ a : Bool, ∑ r : Bool,
(1 / (2 * (d : ℝ))) * completeBernWeight (largeAlphabetArrival q ξ x a) r *
f (largeAlphabetRecord ξ x a r) := by
have hi (x : Fin d) (a r : Bool) : Integrable f
(ofReal ((1 / (2 * (d : ℝ))) *
completeBernWeight (largeAlphabetArrival q ξ x a) r) •
Measure.dirac (largeAlphabetRecord ξ x a r)) := by
exact (integrable_dirac (by simp)).smul_measure ENNReal.ofReal_ne_top
have hir (x : Fin d) (a : Bool) : Integrable f
(∑ r : Bool, ofReal ((1 / (2 * (d : ℝ))) *
completeBernWeight (largeAlphabetArrival q ξ x a) r) •
Measure.dirac (largeAlphabetRecord ξ x a r)) := by
apply integrable_finsetSum_measure.mpr
intro r _
exact hi x a r
have hia (x : Fin d) : Integrable f
(∑ a : Bool, ∑ r : Bool, ofReal ((1 / (2 * (d : ℝ))) *
completeBernWeight (largeAlphabetArrival q ξ x a) r) •
Measure.dirac (largeAlphabetRecord ξ x a r)) := by
apply integrable_finsetSum_measure.mpr
intro a _
exact hir x a
change (∫ w, f w ∂largeAlphabetMeasure q ξ) = _
rw [largeAlphabetMeasure, integral_finsetSum_measure (by intro x _; exact hia x)]
apply Finset.sum_congr rfl
intro x _
rw [integral_finsetSum_measure (by intro a _; exact hir x a)]
apply Finset.sum_congr rfl
intro a _
rw [integral_finsetSum_measure (by intro r _; exact hi x a r)]
apply Finset.sum_congr rfl
intro r _
rw [integral_smul_measure, integral_dirac]
rw [ENNReal.toReal_ofReal (largeAlphabet_weight_nonneg q ξ hq hq1 x a r)]
rfl
theorem largeAlphabetLaw_ate
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
ate (largeAlphabetLaw q ξ hd hq hq1) = cellPriorTarget ξ
Proof (Lean source)
-- @node: largeAlphabetLaw_ate
lemma largeAlphabetLaw_ate {d : ℕ} (q : ℝ) (ξ : Fin d → Bool)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) :
ate (largeAlphabetLaw q ξ hd hq hq1) = cellPriorTarget ξ := by
rw [ate, largeAlphabetLaw_integral]
unfold cellPriorTarget
rw [Finset.sum_div]
apply Finset.sum_congr rfl
intro x _
cases hx : ξ x <;>
simp [largeAlphabetRecord, completeBernWeight, largeAlphabetArrival, hx] <;> ring
theorem largeAlphabetLaw_observed_atom
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
o.RY
= (o.R && (o.A && ξ o.X)) then (1 / (2 * (d : ℝ))) * completeBernWeight (largeAlphabetArrival q ξ o.X o.A) o.R else 0
Proof (Lean source)
-- @node: largeAlphabetLaw_observed_atom
lemma largeAlphabetLaw_observed_atom {d : ℕ} (q : ℝ) (ξ : Fin d → Bool)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (o : ObsRecord d) :
((largeAlphabetLaw q ξ hd hq hq1).1.map obs).real {o} =
if o.S = false ∧ o.RY = (o.R && (o.A && ξ o.X)) then
(1 / (2 * (d : ℝ))) * completeBernWeight (largeAlphabetArrival q ξ o.X o.A) o.R
else 0 := by
classical
rw [measureReal_def, Measure.map_apply (by fun_prop) (by simp)]
change (largeAlphabetLaw q ξ hd hq hq1).1.real (obs ⁻¹' {o}) = _
rw [largeAlphabetLaw_real_event]
rcases o with ⟨x, a, s, r, ry⟩
cases a <;> cases s <;> cases r <;> cases ry <;>
simp [obs, largeAlphabetRecord, FullRecord.S, FullRecord.Y, completeBernWeight,
ite_and, Finset.sum_ite_irrel, Finset.sum_ite_eq']
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.LargeAlphabet.RiskAveraging 5 declarations Prior averaging of the common randomized test and member squared risks.
Prior averaging of the common randomized test and member squared risks.
theorem largeAlphabetSampleMixture_integral
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, f s ∂largeAlphabetSampleMixture n d q hd hq hq1)
= ∫ ξ, (∫ s, f s ∂sampleLaw n (largeAlphabetLaw q ξ hd hq hq1)) ∂binaryCellPrior d
Proof (Lean source)
-- @node: largeAlphabetSampleMixture_integral
lemma largeAlphabetSampleMixture_integral (n d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1)
(f : (Fin n → ObsRecord d) → ℝ) :
(∫ s, f s ∂largeAlphabetSampleMixture n d q hd hq hq1) =
∫ ξ, (∫ s, f s ∂sampleLaw n (largeAlphabetLaw q ξ hd hq hq1))
∂binaryCellPrior d := by
classical
let μ : Measure Bool := (1 / 2 : ℝ≥0∞) • Measure.dirac false +
(1 / 2 : ℝ≥0∞) • Measure.dirac true
let : IsProbabilityMeasure μ := fairBinary_probability
let : IsProbabilityMeasure (binaryCellPrior d) := inferInstanceAs
(IsProbabilityMeasure (Measure.pi (fun _ : Fin d => μ)))
let : IsProbabilityMeasure (largeAlphabetSampleMixture n d q hd hq hq1) :=
largeAlphabetSampleMixture_probability n d q hd hq hq1
rw [integral_fintype Integrable.of_finite]
simp only [smul_eq_mul]
simp_rw [largeAlphabetSampleMixture_real_singleton]
simp_rw [← integral_mul_const]
rw [← integral_finsetSum _ (fun _ _ => Integrable.of_finite)]
apply integral_congr_ae
filter_upwards [] with ξ
let := (largeAlphabetLaw q ξ hd hq hq1).2
let : IsProbabilityMeasure ((largeAlphabetLaw q ξ hd hq hq1).1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
let : IsProbabilityMeasure (sampleLaw n (largeAlphabetLaw q ξ hd hq hq1)) := by
unfold sampleLaw
infer_instance
exact (integral_fintype Integrable.of_finite).symm
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: largeAlphabet_joint_event_integral
lemma largeAlphabet_joint_event_integral
{Z : Type*} [MeasurableSpace Z] (Q : Measure Z) [IsProbabilityMeasure Q]
(κ : BoundedKernel Z) (E : Set (Z × ℝ)) (hE : MeasurableSet E) :
(Q ⊗ₘ κ.1).real E =
∫ s, ∫ w, E.indicator (fun _ => (1 : ℝ)) (s, w) ∂κ.1 s ∂Q := by
let : IsMarkovKernel κ.1 := κ.2.1
have hi : Integrable (E.indicator (fun _ => (1 : ℝ))) (Q ⊗ₘ κ.1) :=
(integrable_const 1).indicator hE
rw [← Measure.integral_compProd hi, integral_indicator_const 1 hE]
simp
theorem largeAlphabetSampleMixture_joint_event
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
κ :
hE :
conclusion
(largeAlphabetSampleMixture n d q hd hq hq1 ⊗ₘ κ.1).real E
= ∫ ξ, (sampleLaw n (largeAlphabetLaw q ξ hd hq hq1) ⊗ₘ κ.1).real E ∂binaryCellPrior d
Proof (Lean source)
-- @node: largeAlphabetSampleMixture_joint_event
lemma largeAlphabetSampleMixture_joint_event (n d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1)
(κ : BoundedKernel (Fin n → ObsRecord d))
(E : Set ((Fin n → ObsRecord d) × ℝ)) (hE : MeasurableSet E) :
(largeAlphabetSampleMixture n d q hd hq hq1 ⊗ₘ κ.1).real E =
∫ ξ, (sampleLaw n (largeAlphabetLaw q ξ hd hq hq1) ⊗ₘ κ.1).real E
∂binaryCellPrior d := by
let : IsProbabilityMeasure (largeAlphabetSampleMixture n d q hd hq hq1) :=
largeAlphabetSampleMixture_probability n d q hd hq hq1
rw [largeAlphabet_joint_event_integral _ κ E hE,
largeAlphabetSampleMixture_integral]
apply integral_congr_ae
filter_upwards [] with ξ
let := (largeAlphabetLaw q ξ hd hq hq1).2
let : IsProbabilityMeasure ((largeAlphabetLaw q ξ hd hq hq1).1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
let : IsProbabilityMeasure (sampleLaw n (largeAlphabetLaw q ξ hd hq hq1)) := by
unfold sampleLaw
infer_instance
exact (largeAlphabet_joint_event_integral _ κ E hE).symm
theorem largeAlphabet_prior_test_risk
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(h ^ 2 / 16) * (largeAlphabetSampleMixture n d q hd hq hq1 ⊗ₘ T.1).real {z | z.2 ≤ a - h / 2}
≤ (∫ ξ, squaredRisk T (largeAlphabetLaw q ξ hd hq hq1) ∂binaryCellPrior d)
Proof (Lean source)
-- @node: largeAlphabet_prior_test_risk
lemma largeAlphabet_prior_test_risk (n d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1)
(T : Estimator n d) (a h : ℝ) (hh : 0 ≤ h) :
(h ^ 2 / 16) *
(largeAlphabetSampleMixture n d q hd hq hq1 ⊗ₘ T.1).real
{z | z.2 ≤ a - h / 2} ≤
(∫ ξ, squaredRisk T (largeAlphabetLaw q ξ hd hq hq1) ∂binaryCellPrior d) +
(h ^ 2 / 16) * (binaryCellPrior d).real
{ξ | h / 4 < |cellPriorTarget ξ - a|} := by
classical
let μ : Measure Bool := (1 / 2 : ℝ≥0∞) • Measure.dirac false +
(1 / 2 : ℝ≥0∞) • Measure.dirac true
let : IsProbabilityMeasure μ := fairBinary_probability
let : IsProbabilityMeasure (binaryCellPrior d) := inferInstanceAs
(IsProbabilityMeasure (Measure.pi (fun _ : Fin d => μ)))
rw [largeAlphabetSampleMixture_joint_event n d q hd hq hq1 T _ (by measurability),
← integral_const_mul]
have hp (ξ : Fin d → Bool) := by
let := (largeAlphabetLaw q ξ hd hq hq1).2
let : IsProbabilityMeasure ((largeAlphabetLaw q ξ hd hq hq1).1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
let : IsProbabilityMeasure (sampleLaw n (largeAlphabetLaw q ξ hd hq hq1)) := by
unfold sampleLaw
infer_instance
exact largeAlphabet_member_test_risk
(sampleLaw n (largeAlphabetLaw q ξ hd hq hq1)) T a h
(ate (largeAlphabetLaw q ξ hd hq hq1)) hh
have havg := integral_mono (μ := binaryCellPrior d)
Integrable.of_finite Integrable.of_finite hp
simp_rw [largeAlphabetLaw_ate] at havg
rw [integral_add Integrable.of_finite Integrable.of_finite] at havg
have hpenalty :
(∫ ξ, (if h / 4 < |cellPriorTarget ξ - a| then h ^ 2 / 16 else 0)
∂binaryCellPrior d) =
(h ^ 2 / 16) * (binaryCellPrior d).real
{ξ | h / 4 < |cellPriorTarget ξ - a|} := by
change (∫ ξ, {ξ | h / 4 < |cellPriorTarget ξ - a|}.indicator
(fun _ => h ^ 2 / 16) ξ ∂binaryCellPrior d) = _
rw [integral_indicator_const _ (by measurability), smul_eq_mul, mul_comm]
rw [hpenalty] at havg
simpa only [squaredRisk, largeAlphabetLaw_ate] using havg
theorem largeAlphabet_prior_squaredRisk_le_worst
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ ξ, squaredRisk T (largeAlphabetLaw q ξ hd hq hq1) ∂binaryCellPrior d)
≤ worstCaseRiskReal (fun (T : Estimator n d) (P : {P : FullLaw d // UnrestrictedArrivalModelClass n d q P}) => squaredRisk T P.1) T
Proof (Lean source)
-- @node: largeAlphabet_prior_squaredRisk_le_worst
lemma largeAlphabet_prior_squaredRisk_le_worst (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1)
(T : Estimator n d) :
(∫ ξ, squaredRisk T (largeAlphabetLaw q ξ hd hq hq1) ∂binaryCellPrior d) ≤
worstCaseRiskReal
(fun (T : Estimator n d) (P : {P : FullLaw d //
UnrestrictedArrivalModelClass n d q P}) => squaredRisk T P.1) T := by
let μ : Measure Bool := (1 / 2 : ℝ≥0∞) • Measure.dirac false +
(1 / 2 : ℝ≥0∞) • Measure.dirac true
let : IsProbabilityMeasure μ := fairBinary_probability
let : IsProbabilityMeasure (binaryCellPrior d) := inferInstanceAs
(IsProbabilityMeasure (Measure.pi (fun _ : Fin d => μ)))
have hbdd : BddAbove (range (fun P : {P : FullLaw d //
UnrestrictedArrivalModelClass n d q P} => squaredRisk T P.1)) := by
refine ⟨4, ?_⟩
rintro _ ⟨P, rfl⟩
exact (unrestricted_squaredRisk_bounds T P.1).2
have hle (ξ : Fin d → Bool) := le_worstCaseRisk
(risk := fun (T : Estimator n d) (P : {P : FullLaw d //
UnrestrictedArrivalModelClass n d q P}) => squaredRisk T P.1)
(e := T) hbdd ⟨_, largeAlphabetLaw_model q ξ hn hd hq hq1⟩
simpa using integral_mono (μ := binaryCellPrior d)
Integrable.of_finite (integrable_const _) hle
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.LargeAlphabet.RiskTesting 6 declarations Squared-loss consequences of the common randomized threshold test.
Squared-loss consequences of the common randomized threshold test.
theorem largeAlphabet_joint_loss_integrable Lemma largeAlphabet_joint_loss_integrable in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: largeAlphabet_joint_loss_integrable
lemma largeAlphabet_joint_loss_integrable
{Z : Type*} [Finite Z] [MeasurableSpace Z] [MeasurableSingletonClass Z]
(Q : Measure Z) [IsProbabilityMeasure Q] (κ : BoundedKernel Z) (t : ℝ) :
Integrable (fun z : Z × ℝ => (z.2 - t) ^ 2) (Q ⊗ₘ κ.1) := by
let : IsMarkovKernel κ.1 := κ.2.1
apply (Measure.integrable_compProd_iff (by fun_prop)).mpr
refine ⟨?_, Integrable.of_finite⟩
filter_upwards [] with o
have hb : ∀ᵐ w ∂(κ.1 o), w ∈ Icc (-1 : ℝ) 1 :=
(ae_mem_iff_measure_eq (μ := κ.1 o) measurableSet_Icc.nullMeasurableSet).2
(by simpa only [measure_univ] using κ.2.2 o)
have hm : AEStronglyMeasurable (fun w : ℝ => w) (κ.1 o) := by fun_prop
exact ((memLp_of_bounded hb hm 2).sub (memLp_const t)).integrable_sq
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: largeAlphabet_kernel_event_loss
lemma largeAlphabet_kernel_event_loss
{Z : Type*} [Finite Z] [MeasurableSpace Z] [MeasurableSingletonClass Z]
(Q : Measure Z) [IsProbabilityMeasure Q] (κ : BoundedKernel Z)
(t b : ℝ) (E : Set (Z × ℝ)) (hE : MeasurableSet E)
(hdom : ∀ z ∈ E, b ≤ (z.2 - t) ^ 2) :
b * (Q ⊗ₘ κ.1).real E ≤ ∫ o, ∫ w, (w - t) ^ 2 ∂κ.1 o ∂Q := by
let : IsMarkovKernel κ.1 := κ.2.1
have hi := largeAlphabet_joint_loss_integrable Q κ t
have hmono : ∫ z, E.indicator (fun _ => b) z ∂(Q ⊗ₘ κ.1) ≤
∫ z : Z × ℝ, (z.2 - t) ^ 2 ∂(Q ⊗ₘ κ.1) := by
apply integral_mono ((integrable_const b).indicator hE) hi
intro z
by_cases hz : z ∈ E
· simpa only [indicator_of_mem hz] using hdom z hz
· simp only [indicator_of_notMem hz]
exact sq_nonneg _
rw [integral_indicator_const b hE, smul_eq_mul, mul_comm,
Measure.integral_compProd hi] at hmono
exact hmono
theorem largeAlphabet_good_target_test_loss Lemma largeAlphabet_good_target_test_loss in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
a h t w :
ℝ
hh :
0 ≤ h
ht :
|t - a| ≤ h / 4
hw :
w ≤ a - h / 2
conclusion
h ^ 2 / 16 ≤ (w - t) ^ 2
Proof (Lean source)
-- @node: largeAlphabet_good_target_test_loss
lemma largeAlphabet_good_target_test_loss (a h t w : ℝ) (hh : 0 ≤ h)
(ht : |t - a| ≤ h / 4) (hw : w ≤ a - h / 2) :
h ^ 2 / 16 ≤ (w - t) ^ 2 := by
have htlow := (abs_le.mp ht).1
have herr : h / 4 ≤ t - w := by linarith
nlinarith [sq_nonneg (t - w - h / 4)]
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
a h w :
ℝ
hh :
0 ≤ h
hw :
a - h / 2 < w
conclusion
h ^ 2 / 16 ≤ (w - (a - h)) ^ 2
Proof (Lean source)
-- @node: largeAlphabet_comparison_test_loss
lemma largeAlphabet_comparison_test_loss (a h w : ℝ) (hh : 0 ≤ h)
(hw : a - h / 2 < w) :
h ^ 2 / 16 ≤ (w - (a - h)) ^ 2 := by
have herr : h / 2 ≤ w - (a - h) := by linarith
nlinarith [sq_nonneg (w - (a - h) - h / 2)]
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Z :
Q :
κ :
a h t :
ℝ
hh :
0 ≤ h
conclusion
(h ^ 2 / 16) * (Q ⊗ₘ κ.1).real {z | z.2 ≤ a - h / 2}
≤ (∫ o, ∫ w, (w - t) ^ 2 ∂κ.1 o ∂Q) + (if h / 4 < |t - a| then h ^ 2 / 16 else 0)
Proof (Lean source)
-- @node: largeAlphabet_member_test_risk
lemma largeAlphabet_member_test_risk
{Z : Type*} [Finite Z] [MeasurableSpace Z] [MeasurableSingletonClass Z]
(Q : Measure Z) [IsProbabilityMeasure Q] (κ : BoundedKernel Z)
(a h t : ℝ) (hh : 0 ≤ h) :
(h ^ 2 / 16) * (Q ⊗ₘ κ.1).real {z | z.2 ≤ a - h / 2} ≤
(∫ o, ∫ w, (w - t) ^ 2 ∂κ.1 o ∂Q) +
(if h / 4 < |t - a| then h ^ 2 / 16 else 0) := by
let : IsMarkovKernel κ.1 := κ.2.1
by_cases ht : h / 4 < |t - a|
· rw [if_pos ht]
have hp : (Q ⊗ₘ κ.1).real {z | z.2 ≤ a - h / 2} ≤ 1 :=
measureReal_le_one
have hm := mul_le_mul_of_nonneg_left hp (by positivity : 0 ≤ h ^ 2 / 16)
have hr : 0 ≤ ∫ o, ∫ w, (w - t) ^ 2 ∂κ.1 o ∂Q :=
integral_nonneg (fun _ => integral_nonneg (fun _ => sq_nonneg _))
linarith
· rw [if_neg ht, add_zero]
apply largeAlphabet_kernel_event_loss Q κ t (h ^ 2 / 16) _
(by measurability)
intro z hz
exact largeAlphabet_good_target_test_loss a h t z.2 hh (le_of_not_gt ht) hz
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Z :
Q :
κ :
a h :
ℝ
hh :
0 ≤ h
conclusion
(h ^ 2 / 16) * (Q ⊗ₘ κ.1).real {z | ¬ z.2 ≤ a - h / 2}
≤ ∫ o, ∫ w, (w - (a - h)) ^ 2 ∂κ.1 o ∂Q
Proof (Lean source)
-- @node: largeAlphabet_comparison_test_risk
lemma largeAlphabet_comparison_test_risk
{Z : Type*} [Finite Z] [MeasurableSpace Z] [MeasurableSingletonClass Z]
(Q : Measure Z) [IsProbabilityMeasure Q] (κ : BoundedKernel Z)
(a h : ℝ) (hh : 0 ≤ h) :
(h ^ 2 / 16) * (Q ⊗ₘ κ.1).real {z | ¬ z.2 ≤ a - h / 2} ≤
∫ o, ∫ w, (w - (a - h)) ^ 2 ∂κ.1 o ∂Q := by
apply largeAlphabet_kernel_event_loss Q κ (a - h) (h ^ 2 / 16) _
(by measurability)
intro z hz
exact largeAlphabet_comparison_test_loss a h z.2 hh (lt_of_not_ge hz)
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.LargeAlphabet.SampleMatching 10 declarations Independence of distinct treated prior coordinates and sample-atom matching.
Independence of distinct treated prior coordinates and sample-atom matching.
theorem binaryCellPrior_integral_distinct_prod Lemma binaryCellPrior_integral_distinct_prod in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: binaryCellPrior_integral_distinct_prod
lemma binaryCellPrior_integral_distinct_prod {ι : Type*} [Fintype ι]
{d : ℕ} (x : ι → Fin d) (hx : Injective x) (f : ι → Bool → ℝ) :
(∫ ξ, ∏ i, f i (ξ (x i)) ∂binaryCellPrior d) =
∏ i, (f i false + f i true) / 2 := by
let μ : Measure Bool := (1 / 2 : ℝ≥0∞) • Measure.dirac false +
(1 / 2 : ℝ≥0∞) • Measure.dirac true
let : IsProbabilityMeasure μ := fairBinary_probability
have hi : iIndepFun (fun j : Fin d => fun ξ : Fin d → Bool => ξ j)
(binaryCellPrior d) := iIndepFun_pi (fun _ => aemeasurable_id)
have hprod := (hi.precomp hx).integral_fun_prod_comp
(f := f) (fun i => (measurable_pi_apply (x i)).aemeasurable)
(fun _ => by fun_prop)
rw [hprod]
apply Finset.prod_congr rfl
intro i _
change (∫ ξ, f i (ξ (x i)) ∂Measure.pi (fun _ : Fin d => μ)) = _
have hp := measurePreserving_eval (fun _ : Fin d => μ) (x i)
rw [← integral_map hp.measurable.aemeasurable (by fun_prop), hp.map_eq]
exact fairBinary_integral (f i)
theorem binaryCellPrior_integral_partial_distinct_prod Lemma binaryCellPrior_integral_partial_distinct_prod in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: binaryCellPrior_integral_partial_distinct_prod
lemma binaryCellPrior_integral_partial_distinct_prod {ι : Type*} [Fintype ι]
{d : ℕ} (p : ι → Prop) (x : ι → Fin d) (f : ι → Bool → ℝ)
(hx : Injective (fun i : {i // p i} => x i))
(hc : ∀ i, ¬ p i → f i true = f i false) :
(∫ ξ, ∏ i, f i (ξ (x i)) ∂binaryCellPrior d) =
∏ i, (f i false + f i true) / 2 := by
classical
let μ : Measure Bool := (1 / 2 : ℝ≥0∞) • Measure.dirac false +
(1 / 2 : ℝ≥0∞) • Measure.dirac true
let : IsProbabilityMeasure μ := fairBinary_probability
let : IsProbabilityMeasure (binaryCellPrior d) := inferInstanceAs
(IsProbabilityMeasure (Measure.pi (fun _ : Fin d => μ)))
have hconst (ξ : Fin d → Bool) (i : {i // ¬ p i}) :
f i (ξ (x i)) = f i false := by
cases ξ (x i)
· rfl
· exact hc i i.property
have heq : (fun ξ : Fin d → Bool => ∏ i, f i (ξ (x i))) =
(fun ξ => (∏ i : {i // p i}, f i (ξ (x i))) *
(∏ i : {i // ¬ p i}, f i false)) := by
funext ξ
rw [← Fintype.prod_subtype_mul_prod_subtype p]
simp_rw [hconst]
rw [heq, integral_mul_const,
binaryCellPrior_integral_distinct_prod (fun i : {i // p i} => x i) hx]
rw [← Fintype.prod_subtype_mul_prod_subtype p]
congr 1
apply Finset.prod_congr rfl
intro i _
rw [hc i i.property]
ring
theorem largeAlphabetLaw_atom_coordinate
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: largeAlphabetLaw_atom_coordinate
lemma largeAlphabetLaw_atom_coordinate {d : ℕ} (q : ℝ) (ξ : Fin d → Bool)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (o : ObsRecord d) :
((largeAlphabetLaw q ξ hd hq hq1).1.map obs).real {o} =
((largeAlphabetLaw q (fun _ : Fin d => ξ o.X) hd hq hq1).1.map obs).real {o} := by
simp only [largeAlphabetLaw_observed_atom, largeAlphabetArrival]
theorem largeAlphabetLaw_control_atom_constant
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: largeAlphabetLaw_control_atom_constant
lemma largeAlphabetLaw_control_atom_constant {d : ℕ} (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (o : ObsRecord d)
(ha : o.A ≠ true) :
((largeAlphabetLaw q (fun _ : Fin d => true) hd hq hq1).1.map obs).real {o} =
((largeAlphabetLaw q (fun _ : Fin d => false) hd hq hq1).1.map obs).real {o} := by
have hf : o.A = false := Bool.eq_false_iff.mpr ha
simp only [largeAlphabetLaw_observed_atom, largeAlphabetArrival, hf,
Bool.false_and, Bool.and_false, Bool.false_eq_true, if_false]
theorem largeAlphabet_sample_atom_prod
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: largeAlphabet_sample_atom_prod
lemma largeAlphabet_sample_atom_prod {n d : ℕ} (P : FullLaw d)
(s : Fin n → ObsRecord d) :
(sampleLaw n P).real {s} = ∏ i, (P.1.map obs).real {s i} := by
let := P.2
rw [sampleLaw, Measure.real, Measure.pi_singleton, ENNReal.toReal_prod]
rfl
theorem largeAlphabet_sample_atom_matching
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
hs :
conclusion
(∫ ξ, (sampleLaw n (largeAlphabetLaw q ξ hd hq hq1)).real {s} ∂binaryCellPrior d)
= (sampleLaw n (largeAlphabetComparisonLaw d q hd hq hq1)).real {s}
Proof (Lean source)
-- @node: largeAlphabet_sample_atom_matching
lemma largeAlphabet_sample_atom_matching {n d : ℕ} (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (s : Fin n → ObsRecord d)
(hs : noRepeatedTreatedCell s) :
(∫ ξ, (sampleLaw n (largeAlphabetLaw q ξ hd hq hq1)).real {s}
∂binaryCellPrior d) =
(sampleLaw n (largeAlphabetComparisonLaw d q hd hq hq1)).real {s} := by
classical
simp_rw [largeAlphabet_sample_atom_prod, largeAlphabetLaw_atom_coordinate]
rw [binaryCellPrior_integral_partial_distinct_prod
(fun i => (s i).A = true) (fun i => (s i).X)
(fun i b => ((largeAlphabetLaw q (fun _ : Fin d => b) hd hq hq1).1.map obs).real {s i})]
· apply Finset.prod_congr rfl
intro i _
simpa only [fairBinary_integral] using
largeAlphabet_one_visit_matching d q hd hq hq1 (s i)
· intro i j hij
apply Subtype.ext
by_contra hne
exact hs i j hne i.property j.property hij
· intro i hi
exact largeAlphabetLaw_control_atom_constant q hd hq hq1 (s i) hi
def largeAlphabetSampleMixture
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
theorem largeAlphabetSampleMixture_real_singleton
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(largeAlphabetSampleMixture n d q hd hq hq1).real {s}
= ∫ ξ, (sampleLaw n (largeAlphabetLaw q ξ hd hq hq1)).real {s} ∂binaryCellPrior d
Proof (Lean source)
lemma largeAlphabetSampleMixture_real_singleton (n d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (s : Fin n → ObsRecord d) :
(largeAlphabetSampleMixture n d q hd hq hq1).real {s} =
∫ ξ, (sampleLaw n (largeAlphabetLaw q ξ hd hq hq1)).real {s}
∂binaryCellPrior d := by
classical
unfold largeAlphabetSampleMixture Measure.real
simp only [Measure.finsetSum_apply, Measure.smul_apply, smul_eq_mul,
Measure.dirac_apply, Set.indicator_apply, Set.mem_singleton_iff, Pi.one_apply]
simp only [mul_ite, mul_one, mul_zero, Finset.sum_ite_eq', Finset.mem_univ, if_true]
exact ENNReal.toReal_ofReal (integral_nonneg (fun _ => measureReal_nonneg))
theorem largeAlphabetSampleMixture_probability
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
conclusion
IsProbabilityMeasure (largeAlphabetSampleMixture n d q hd hq hq1)
Proof (Lean source)
lemma largeAlphabetSampleMixture_probability (n d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) :
IsProbabilityMeasure (largeAlphabetSampleMixture n d q hd hq hq1) := by
classical
let μ : Measure Bool := (1 / 2 : ℝ≥0∞) • Measure.dirac false +
(1 / 2 : ℝ≥0∞) • Measure.dirac true
let : IsProbabilityMeasure μ := fairBinary_probability
let : IsProbabilityMeasure (binaryCellPrior d) := inferInstanceAs
(IsProbabilityMeasure (Measure.pi (fun _ : Fin d => μ)))
have hsum : (∑ s : Fin n → ObsRecord d,
∫ ξ, (sampleLaw n (largeAlphabetLaw q ξ hd hq hq1)).real {s}
∂binaryCellPrior d) = 1 := by
rw [← integral_finsetSum _ (fun _ _ => Integrable.of_finite)]
have hnorm (ξ : Fin d → Bool) :
(∑ s : Fin n → ObsRecord d,
(sampleLaw n (largeAlphabetLaw q ξ hd hq hq1)).real {s}) = 1 := by
let := (largeAlphabetLaw q ξ hd hq hq1).2
let : IsProbabilityMeasure ((largeAlphabetLaw q ξ hd hq hq1).1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
let : IsProbabilityMeasure (sampleLaw n (largeAlphabetLaw q ξ hd hq hq1)) := by
unfold sampleLaw
infer_instance
rw [sum_measureReal_singleton]
simp
simp_rw [hnorm]
simp
constructor
unfold largeAlphabetSampleMixture
simp only [Measure.finsetSum_apply, Measure.smul_apply, smul_eq_mul,
Measure.dirac_apply_of_mem (Set.mem_univ _), mul_one]
rw [← ENNReal.ofReal_sum_of_nonneg
(fun _ _ => integral_nonneg (fun _ => measureReal_nonneg)), hsum]
simp
theorem largeAlphabetSampleMixture_matching
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
hs :
conclusion
(largeAlphabetSampleMixture n d q hd hq hq1).real {s}
= (sampleLaw n (largeAlphabetComparisonLaw d q hd hq hq1)).real {s}
Proof (Lean source)
-- @node: largeAlphabetSampleMixture_matching
lemma largeAlphabetSampleMixture_matching (n d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) (s : Fin n → ObsRecord d)
(hs : noRepeatedTreatedCell s) :
(largeAlphabetSampleMixture n d q hd hq hq1).real {s} =
(sampleLaw n (largeAlphabetComparisonLaw d q hd hq hq1)).real {s} := by
rw [largeAlphabetSampleMixture_real_singleton]
exact largeAlphabet_sample_atom_matching q hd hq hq1 s hs
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Lower.ActivationCalibration 7 declarations Ceiling-degree calibration of the actual activated cutoff.
Ceiling-degree calibration of the actual activated cutoff. The rounding allowance changes only the numerical effective-size threshold in (9).
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: activation_mean_le_factorial_order
lemma activation_mean_le_factorial_order (η : ℝ) (n : ℕ) (q : ℝ)
(hη : 0 ≤ η) (hN : 0 < effectiveSize n q) (hell : 1 ≤ logScale n q) :
(n : ℝ) * (rareMass η n q * (q * lowerEndpoint n q)) ≤
(2 * η) * (lowerDegree n q + 1 : ℕ) := by
have hn0 : 0 < (n : ℝ) := by
have h : 0 < (n : ℝ) * q := hN
exact (mul_pos_iff.mp h).elim (fun h => h.1)
(fun h => False.elim (not_lt_of_ge (Nat.cast_nonneg n) h.1))
have hq0 : 0 < q := (mul_pos_iff.mp hN).elim (fun h => h.2)
(fun h => False.elim (not_lt_of_ge (Nat.cast_nonneg n) h.1))
have hL : 0 < logScale n q := by linarith
have hK : (lowerDegree n q : ℝ) ≤ 2 * logScale n q := by
have h := Nat.ceil_lt_add_one hL.le
simpa only [lowerDegree] using h.le.trans (by linarith)
have hKr : 0 ≤ (lowerDegree n q : ℝ) := by positivity
have hsq : (lowerDegree n q : ℝ) ^ 2 ≤
2 * logScale n q * (lowerDegree n q + 1 : ℕ) := by
push_cast
nlinarith [mul_le_mul_of_nonneg_right hK hKr]
have heq : (n : ℝ) * (rareMass η n q * (q * lowerEndpoint n q)) =
η * (lowerDegree n q : ℝ) ^ 2 / logScale n q := by
unfold rareMass lowerEndpoint effectiveSize at *
field_simp [ne_of_gt hn0, ne_of_gt hq0]
rw [heq]
apply (div_le_iff₀ hL).mpr
nlinarith [mul_le_mul_of_nonneg_left hsq hη]
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
((n : ℝ) * (rareMass (exp (-32)) n q * (q * lowerEndpoint n q))) ^ (lowerDegree n q + 1) / ((lowerDegree n q + 1).factorial : ℝ)
Proof (Lean source)
-- @node: activation_factorial_ratio_le_exp
lemma activation_factorial_ratio_le_exp (n : ℕ) (q : ℝ)
(hN : 0 < effectiveSize n q) (hell : 1 ≤ logScale n q) :
((n : ℝ) * (rareMass (exp (-32)) n q * (q * lowerEndpoint n q))) ^
(lowerDegree n q + 1) / ((lowerDegree n q + 1).factorial : ℝ) ≤
exp (-30 * logScale n q) := by
let r := lowerDegree n q + 1
have hmean := activation_mean_le_factorial_order (exp (-32)) n q
(Real.exp_nonneg _) hN hell
have hdegree : logScale n q ≤ (r : ℝ) := by
have h := Nat.le_ceil (logScale n q)
dsimp [r, lowerDegree]
push_cast
linarith
have hcoef : 2 * exp (-32) ≤ exp (-31) := by
rw [show (-31 : ℝ) = 1 + -32 by norm_num, Real.exp_add]
have h := Real.add_one_le_exp (1 : ℝ)
nlinarith [Real.exp_pos (-32)]
have hnonneg : 0 ≤ (n : ℝ) *
(rareMass (exp (-32)) n q * (q * lowerEndpoint n q)) := by
unfold rareMass lowerEndpoint effectiveSize at *
have : 0 < (n : ℝ) * q := hN
have hn : 0 < (n : ℝ) := by
by_contra h
have : n = 0 := by exact_mod_cast (le_antisymm (le_of_not_gt h) (Nat.cast_nonneg n))
simp [this] at hN
have hq : 0 < q := ((mul_pos_iff.mp hN).resolve_right (by rintro ⟨h, _⟩; linarith)).2
positivity
calc
_ ≤ ((exp (-31)) * (r : ℝ)) ^ r / (r.factorial : ℝ) := by
apply div_le_div_of_nonneg_right _ (by positivity)
exact pow_le_pow_left₀ hnonneg
(hmean.trans (mul_le_mul_of_nonneg_right hcoef (by positivity))) r
_ = exp (-31) ^ r * ((r : ℝ) ^ r / (r.factorial : ℝ)) := by ring
_ ≤ exp (-31) ^ r * exp (r : ℝ) :=
mul_le_mul_of_nonneg_left (Real.pow_div_factorial_le_exp r (by positivity) r)
(by positivity)
_ = exp (-30 * (r : ℝ)) := by
rw [← Real.exp_nat_mul, ← Real.exp_add]
congr 1
ring
_ ≤ exp (-30 * logScale n q) := Real.exp_le_exp.mpr (by linarith)
theorem activatedAugmentedMixture_cutoff_le_exp
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
conclusion
(activatedAugmentedMixture (exp (-32)) n d q π).real {s | ¬ ∀ x : Fin d, x.val < rareCount (exp (-32)) n d q → (∑ i ∈ Finset.univ.filter (fun i => (s i).2.X = x), if (s i).1 then 1 else 0) ≤ lowerDegree n q}
Proof (Lean source)
-- @node: activatedAugmentedMixture_cutoff_le_exp
lemma activatedAugmentedMixture_cutoff_le_exp (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hq : 0 < q) (hslice : RareArrivalSlice n q)
{π : Measure ℝ} (hπ : FiniteReciprocalPrior (lowerEndpoint n q) π) :
(activatedAugmentedMixture (exp (-32)) n d q π).real
{s | ¬ ∀ x : Fin d, x.val < rareCount (exp (-32)) n d q →
(∑ i ∈ Finset.univ.filter (fun i => (s i).2.X = x),
if (s i).1 then 1 else 0) ≤ lowerDegree n q} ≤
exp (32 - 28 * logScale n q) := by
have hN : 0 < effectiveSize n q := by unfold effectiveSize; positivity
have hell : 1 ≤ logScale n q := by
unfold logScale
have h := Real.log_le_log (Real.exp_pos (1 : ℝ))
(show exp 1 ≤ exp 1 + effectiveSize n q by linarith)
simpa using h
have hb : 0 < rareMass (exp (-32)) n q := by unfold rareMass; positivity
have hcap : (rareCount (exp (-32)) n d q : ℝ) ≤
effectiveSize n q * logScale n q / (2 * exp (-32)) := by
have h := rareCount_mass_le_half (exp (-32)) n d q hb
apply (le_div_iff₀ (by positivity : 0 < 2 * exp (-32))).mpr
dsimp [rareMass] at h
have h' := (div_le_iff₀ (by positivity :
0 < effectiveSize n q * logScale n q)).mp (show
(rareCount (exp (-32)) n d q : ℝ) * exp (-32) /
(effectiveSize n q * logScale n q) ≤ 1 / 2 by
simpa [mul_div_assoc] using h)
nlinarith
have hNexp : effectiveSize n q ≤ exp (logScale n q) := by
rw [logScale, Real.exp_log (by positivity)]
linarith [Real.exp_pos (1 : ℝ)]
have hLexp : logScale n q ≤ exp (logScale n q) := by
linarith [Real.add_one_le_exp (logScale n q)]
have hprod : effectiveSize n q * logScale n q ≤ exp (2 * logScale n q) := by
rw [show 2 * logScale n q = logScale n q + logScale n q by ring, Real.exp_add]
exact mul_le_mul hNexp hLexp (by linarith) (Real.exp_nonneg _)
apply (activatedAugmentedMixture_factorial_cutoff_bound (exp (-32)) n d
(lowerDegree n q) q hn hd hb hq hslice hπ).trans
calc
_ ≤ (effectiveSize n q * logScale n q / (2 * exp (-32))) *
exp (-30 * logScale n q) :=
mul_le_mul hcap (activation_factorial_ratio_le_exp n q hN hell)
(by unfold lowerEndpoint; positivity)
(div_nonneg (mul_nonneg hN.le (by linarith)) (by positivity))
_ ≤ (exp (2 * logScale n q) / exp (-32)) *
exp (-30 * logScale n q) := by
apply mul_le_mul_of_nonneg_right _ (Real.exp_nonneg _)
exact (div_le_div_of_nonneg_left (mul_nonneg hN.le (by linarith)) (Real.exp_pos (-32))
(by linarith [Real.exp_pos (-32)])).trans
(div_le_div_of_nonneg_right hprod (Real.exp_nonneg (-32)))
_ = _ := by
rw [← Real.exp_sub, ← Real.exp_add]
congr 1
ring
theorem activatedAugmentedMixture_cutoff_budget
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
u :
ℝ
n d :
ℕ
q :
ℝ
hu :
0 < u
hn :
1 ≤ n
hd :
1 ≤ d
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
hlarge :
conclusion
(activatedAugmentedMixture (exp (-32)) n d q π).real {s | ¬ ∀ x : Fin d, x.val < rareCount (exp (-32)) n d q → (∑ i ∈ Finset.univ.filter (fun i => (s i).2.X = x), if (s i).1 then 1 else 0) ≤ lowerDegree n q}
≤ u / 16
Proof (Lean source)
-- @node: activatedAugmentedMixture_cutoff_budget
lemma activatedAugmentedMixture_cutoff_budget (u : ℝ) (n d : ℕ) (q : ℝ)
(hu : 0 < u) (hn : 1 ≤ n) (hd : 1 ≤ d) (hq : 0 < q)
(hslice : RareArrivalSlice n q)
{π : Measure ℝ} (hπ : FiniteReciprocalPrior (lowerEndpoint n q) π)
(hlarge : exp ((32 - log (u / 16)) / 28) ≤ effectiveSize n q) :
(activatedAugmentedMixture (exp (-32)) n d q π).real
{s | ¬ ∀ x : Fin d, x.val < rareCount (exp (-32)) n d q →
(∑ i ∈ Finset.univ.filter (fun i => (s i).2.X = x),
if (s i).1 then 1 else 0) ≤ lowerDegree n q} ≤ u / 16 := by
have hlog := Real.log_le_log
(Real.exp_pos ((32 - log (u / 16)) / 28))
(hlarge.trans (show effectiveSize n q ≤ exp 1 + effectiveSize n q by
linarith [Real.exp_pos (1 : ℝ)]))
rw [Real.log_exp] at hlog
change (32 - log (u / 16)) / 28 ≤ logScale n q at hlog
apply (activatedAugmentedMixture_cutoff_le_exp n d q hn hd hq hslice hπ).trans
calc
exp (32 - 28 * logScale n q) ≤ exp (log (u / 16)) :=
Real.exp_le_exp.mpr (by linarith)
_ = u / 16 := Real.exp_log (by positivity)
theorem activatedAugmentedMixture_interval_minimax_calibrated
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d M :
ℕ
q α :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hq :
0 < q
hslice :
RareArrivalSlice n q
π₀ π₁ :
Measure ℝ
h₀ :
FiniteReciprocalPrior (lowerEndpoint n q) π₀
h₁ :
FiniteReciprocalPrior (lowerEndpoint n q) π₁
hm :
∀ v : ℕ
if
v ≤ lowerDegree n q
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
hgap :
1 / 12 ≤ (∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀
hα :
0 ≤ α
hα1 :
α < 1
hM :
1 ≤ M
hgrid :
8 / (1 - α) ≤ (M : ℝ)
hlarge :
conclusion
(21 * (1 - α) / 32) * ((rareCount (exp (-32)) n d q : ℝ) * rareMass (exp (-32)) n q * ((∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀))
≤ intervalLengthRisk n d q α
Proof (Lean source)
-- @node: activatedAugmentedMixture_interval_minimax_calibrated
lemma activatedAugmentedMixture_interval_minimax_calibrated
(n d M : ℕ) (q α : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hq : 0 < q) (hslice : RareArrivalSlice n q)
{π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior (lowerEndpoint n q) π₀)
(h₁ : FiniteReciprocalPrior (lowerEndpoint n q) π₁)
(hm : ∀ v : ℕ, v ≤ lowerDegree n q → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
(hgap : 1 / 12 ≤ (∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀)
(hα : 0 ≤ α) (hα1 : α < 1) (hM : 1 ≤ M)
(hgrid : 8 / (1 - α) ≤ (M : ℝ))
(hJ : 18432 * (M : ℝ) ^ 2 / (1 - α) ≤
(rareCount (exp (-32)) n d q : ℝ))
(hlarge : exp ((32 - log ((1 - α) / 16)) / 28) ≤ effectiveSize n q) :
(21 * (1 - α) / 32) * ((rareCount (exp (-32)) n d q : ℝ) *
rareMass (exp (-32)) n q *
((∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀)) ≤ intervalLengthRisk n d q α := by
have hN : 0 < effectiveSize n q := by unfold effectiveSize; positivity
have hell : 1 ≤ logScale n q := by
unfold logScale
have h := Real.log_le_log (Real.exp_pos (1 : ℝ))
(show exp 1 ≤ exp 1 + effectiveSize n q by linarith)
simpa using h
have hb : 0 < rareMass (exp (-32)) n q := by unfold rareMass; positivity
apply activatedAugmentedMixture_interval_minimax_of_cutoff (exp (-32)) n d M q α
hn hd hb hq hslice h₀ h₁ (lowerDegree n q) hm hgap hα hα1 hM hgrid hJ
exact activatedAugmentedMixture_cutoff_budget (1 - α) n d q (by linarith)
hn hd hq hslice (finiteReciprocalPrior_interpolate h₀ h₁ (by simp)) hlarge
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
∃ π₀ π₁ : Measure ℝ,
FiniteReciprocalPrior (lowerEndpoint n q) π₀
FiniteReciprocalPrior (lowerEndpoint n q) π₁
conclusion 2
1 / 12 ≤ (∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀
Proof (Lean source)
-- @node: interval_ordered_reciprocal_priors
lemma interval_ordered_reciprocal_priors
(n : ℕ) (q : ℝ) (hK : 2 ≤ lowerDegree n q) :
∃ π₀ π₁ : Measure ℝ,
FiniteReciprocalPrior (lowerEndpoint n q) π₀ ∧
FiniteReciprocalPrior (lowerEndpoint n q) π₁ ∧
(∀ v : ℕ, v ≤ lowerDegree n q → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁) ∧
1 / 12 ≤ (∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀ := by
obtain ⟨π₀, π₁, hp₀, hp₁, ⟨s₀, s₁, hs₀, hs₁, hm₀, hm₁⟩, hm, hgap⟩ :=
moment_matched_reciprocal_priors (lowerDegree n q) hK
have h₀ : FiniteReciprocalPrior (lowerEndpoint n q) π₀ :=
⟨hp₀, s₀, hs₀, hm₀⟩
have h₁ : FiniteReciprocalPrior (lowerEndpoint n q) π₁ :=
⟨hp₁, s₁, hs₁, hm₁⟩
by_cases horder : (∫ z, z⁻¹ ∂π₀) ≤ ∫ z, z⁻¹ ∂π₁
· refine ⟨π₀, π₁, h₀, h₁, hm, ?_⟩
simpa only [abs_of_nonneg (sub_nonneg.mpr horder)] using hgap
· refine ⟨π₁, π₀, h₁, h₀, fun v hv => (hm v hv).symm, ?_⟩
rw [abs_of_nonpos (by linarith)] at hgap
linarith
theorem interval_activated_large_regime
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
α :
ℝ
hα :
0 ≤ α
hα1 :
α < 1
conclusion
∃ (T : ℝ) (D : ℕ) (cAct : ℝ),
1 ≤ T
1 ≤ D
0 < cAct
conclusion 1
Proof (Lean source)
-- @node: interval_activated_large_regime
lemma interval_activated_large_regime
(α : ℝ) (hα : 0 ≤ α) (hα1 : α < 1) :
∃ (T : ℝ) (D : ℕ) (cAct : ℝ), 1 ≤ T ∧ 1 ≤ D ∧ 0 < cAct ∧
∀ (n d : ℕ) (q : ℝ), 1 ≤ n → 1 ≤ d → 0 < q → q ≤ 1 →
RareArrivalSlice n q → T ≤ effectiveSize n q →
(D : ℝ) ≤ (2 * rareMass (exp (-32)) n q)⁻¹ ∧
(D ≤ rareCount (exp (-32)) n d q →
cAct * min 1 ((d : ℝ) / (effectiveSize n q * logScale n q)) ≤
intervalLengthRisk n d q α) := by
let u := 1 - α
let η := exp (-32)
let M := ceil (8 / u) + 1
let D := ceil (18432 * (M : ℝ) ^ 2 / u) + 1
have hu : 0 < u := by dsimp [u]; linarith
obtain ⟨hM, hgridInv, hD, hDreal⟩ := interval_activation_grid_thresholds u hu
have hD' : 1 ≤ D := hD
have hη : 0 < η := Real.exp_pos _
have hηsmall : η ≤ 1 / 2 := by
have h := Real.add_one_le_exp (32 : ℝ)
dsimp [η]
rw [Real.exp_neg, inv_eq_one_div]
apply (div_le_iff₀ (Real.exp_pos 32)).mpr
linarith
let T := max 1 (max (exp 2) (max (4 * η)
(max (2 * η * D) (exp ((32 - log (u / 16)) / 28)))))
let cAct := (21 * u / 32) * (η / 24)
refine ⟨T, D, cAct, le_max_left _ _, hD, by dsimp [cAct]; positivity, ?_⟩
intro n d q hn hd hq _hq1 hslice hlarge
have hN : 0 < effectiveSize n q := by unfold effectiveSize; positivity
have hell : 1 ≤ logScale n q := by
unfold logScale
have h := Real.log_le_log (Real.exp_pos (1 : ℝ))
(show exp 1 ≤ exp 1 + effectiveSize n q by linarith)
simpa using h
have hTexp : exp 2 ≤ effectiveSize n q :=
(le_max_left _ _).trans ((le_max_right _ _).trans hlarge)
have hTmass : 4 * η ≤ effectiveSize n q :=
(le_max_left _ _).trans ((le_max_right _ _).trans
((le_max_right _ _).trans hlarge))
have hTcap : 2 * η * D ≤ effectiveSize n q :=
(le_max_left _ _).trans ((le_max_right _ _).trans
((le_max_right _ _).trans ((le_max_right _ _).trans hlarge)))
have hTtail : exp ((32 - log (u / 16)) / 28) ≤ effectiveSize n q :=
(le_max_right _ _).trans ((le_max_right _ _).trans
((le_max_right _ _).trans ((le_max_right _ _).trans hlarge)))
refine ⟨interval_capacity_of_large_size η D n q hη hN hell hTcap, ?_⟩
intro hJ
change D ≤ rareCount η n d q at hJ
have hK : 2 ≤ lowerDegree n q := by
have hlog := Real.log_le_log (Real.exp_pos (2 : ℝ))
(hTexp.trans (show effectiveSize n q ≤ exp 1 + effectiveSize n q by
linarith [Real.exp_pos (1 : ℝ)]))
rw [Real.log_exp] at hlog
have hceil := Nat.le_ceil (logScale n q)
have : (2 : ℝ) ≤ (lowerDegree n q : ℝ) := hlog.trans hceil
exact_mod_cast this
obtain ⟨π₀, π₁, h₀, h₁, hm, hgap⟩ :=
interval_ordered_reciprocal_priors n q hK
have hJreal : 18432 * (M : ℝ) ^ 2 / (1 - α) ≤ (rareCount η n d q : ℝ) :=
hDreal.trans (Nat.cast_le.mpr hJ)
have hgrid : 8 / (1 - α) ≤ (M : ℝ) := by
have h := Nat.le_ceil (8 / u)
dsimp [M]
push_cast
dsimp [u] at h
linarith
have hlength := activatedAugmentedMixture_interval_minimax_calibrated n d M q α
hn hd hq hslice h₀ h₁ hm hgap hα hα1 hM hgrid hJreal hTtail
have hbsmall : rareMass η n q ≤ 1 / 4 := by
unfold rareMass
apply (div_le_iff₀ (by positivity : 0 < effectiveSize n q * logScale n q)).mpr
nlinarith [mul_le_mul_of_nonneg_left hell hN.le]
have hd2 : 2 ≤ d := by
have hc : rareCount η n d q ≤ d - 1 := min_le_left _ _
omega
have hsep : (rareCount η n d q : ℝ) * rareMass η n q / 12 ≤
(rareCount η n d q : ℝ) * rareMass η n q *
((∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀) := by
have hb : 0 < rareMass η n q := by unfold rareMass; positivity
nlinarith [mul_le_mul_of_nonneg_left hgap
(mul_nonneg (Nat.cast_nonneg (rareCount η n d q)) hb.le)]
have hwidth := interval_prior_segment_width_lower η _ n d q hη hηsmall hd2 hN
(by linarith) hbsmall hsep
have hfinal : cAct * min 1 ((d : ℝ) / (effectiveSize n q * logScale n q)) ≤
(21 * (1 - α) / 32) * ((rareCount η n d q : ℝ) * rareMass η n q *
((∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀)) := by
simpa only [cAct, u, mul_assoc] using
mul_le_mul_of_nonneg_left hwidth (show 0 ≤ 21 * (1 - α) / 32 by positivity)
exact hfinal.trans hlength
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Lower.ActivationCoverage 11 declarations Uniform honesty transfers to neighborhood hits in the actual activated prior mixture.
Uniform honesty transfers to neighborhood hits in the actual activated prior mixture. The target concentration budget is kept separate from coverage, so equations (13)--(16) and the minimax reduction are assembled without assuming averaged honesty. The actual activation-cutoff tail remains a separate input to the lower-bound assembly.
theorem intervalContains_subset_neighborhood
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
t θ w :
ℝ
hnear :
|t - θ| ≤ w
conclusion
{I : ConnectedInterval | intervalContains I t} ⊆ {I | ∃ x : ℝ, I.lo ≤ x ∧ x ≤ I.hi ∧ |x - θ| ≤ w}
Proof (Lean source)
-- @node: intervalContains_subset_neighborhood
lemma intervalContains_subset_neighborhood (t θ w : ℝ) (hnear : |t - θ| ≤ w) :
{I : ConnectedInterval | intervalContains I t} ⊆
{I | ∃ x : ℝ, I.lo ≤ x ∧ x ≤ I.hi ∧ |x - θ| ≤ w} := by
intro I hI
have hlo : I.lo ≤ t := by
have h := hI.1
change (if I.closedLeft then I.lo ≤ t else I.lo < t) at h
split at h
· exact h
· exact h.le
have hhi : t ≤ I.hi := by
have h := hI.2
change (if I.closedRight then t ≤ I.hi else t < I.hi) at h
split at h
· exact h
· exact h.le
exact ⟨t, hlo, hhi, hnear⟩
theorem intervalProcedure_real_event
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
T :
μ :
A :
hA :
Proof (Lean source)
-- @node: intervalProcedure_real_event
lemma intervalProcedure_real_event {n d : ℕ} (T : IntervalProcedure n d)
(μ : Measure (Fin n → ObsRecord d)) [IsProbabilityMeasure μ]
(A : Set ConnectedInterval) (hA : MeasurableSet A) :
(T.1 ∘ₘ μ).real A = ∫ s, (T.1 s).real A ∂μ := by
let := T.2
rw [measureReal_def, Measure.bind_apply hA T.1.aemeasurable, ←
integral_toReal (T.1.measurable_coe hA).aemeasurable
(Filter.Eventually.of_forall fun s =>
(lt_top_iff_ne_top).2 (measure_ne_top (T.1 s) A))]
rfl
theorem honestInterval_neighborhood_coverage
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q α :
ℝ
T :
hT :
HonestInterval n d q α T
hα :
0 ≤ α
P :
FullLaw d
hP :
RareArrivalModelClass n d q P
θ w :
ℝ
conclusion
1 - α - (if w < |ate P - θ| then 1 else 0)
≤ ∫ s, (T.1 s).real {I | ∃ x : ℝ, I.lo ≤ x ∧ x ≤ I.hi ∧ |x - θ| ≤ w} ∂sampleLaw n P
Proof (Lean source)
-- @node: honestInterval_neighborhood_coverage
lemma honestInterval_neighborhood_coverage {n d : ℕ} {q α : ℝ}
(T : IntervalProcedure n d) (hT : HonestInterval n d q α T)
(hα : 0 ≤ α) (P : FullLaw d) (hP : RareArrivalModelClass n d q P)
(θ w : ℝ) :
1 - α - (if w < |ate P - θ| then 1 else 0) ≤
∫ s, (T.1 s).real {I | ∃ x : ℝ,
I.lo ≤ x ∧ x ≤ I.hi ∧ |x - θ| ≤ w} ∂sampleLaw n P := by
classical
let := T.2
let := P.2
let : IsProbabilityMeasure (sampleLaw n P) := by
unfold sampleLaw
let : IsProbabilityMeasure (P.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
infer_instance
by_cases hbad : w < |ate P - θ|
· simp only [if_pos hbad]
have hnonneg : 0 ≤ ∫ s, (T.1 s).real {I | ∃ x : ℝ,
I.lo ≤ x ∧ x ≤ I.hi ∧ |x - θ| ≤ w} ∂sampleLaw n P :=
integral_nonneg (fun _ => measureReal_nonneg)
linarith
· simp only [if_neg hbad, sub_zero]
apply (hT P hP).trans
apply integral_mono Integrable.of_finite Integrable.of_finite
intro s
exact measureReal_mono (intervalContains_subset_neighborhood (ate P) θ w
(le_of_not_gt hbad)) (measure_ne_top (T.1 s) _)
theorem intervalPrior_initial_coordinates_map Lemma intervalPrior_initial_coordinates_map in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: intervalPrior_initial_coordinates_map
lemma intervalPrior_initial_coordinates_map {π : Measure ℝ} [IsProbabilityMeasure π]
(d J : ℕ) (hJ : J ≤ d) :
(Measure.pi (fun _ : Fin d => π)).map
(fun z : Fin d → ℝ => fun x : Fin J => z (x.castLE hJ)) =
Measure.pi (fun _ : Fin J => π) := by
have hind : iIndepFun (fun x : Fin d => fun z : Fin d → ℝ => z x)
(Measure.pi (fun _ : Fin d => π)) :=
iIndepFun_pi (X := fun _ (z : ℝ) => z) (fun _ => measurable_id.aemeasurable)
have hrestrict := hind.precomp (Fin.castLE_injective hJ)
rw [hrestrict.map_fun_eq_pi_map (fun x => (measurable_pi_apply _).aemeasurable)]
simp_rw [(measurePreserving_eval (fun _ : Fin d => π) _).map_eq]
theorem intervalPrior_full_alphabet_concentration_budget Lemma intervalPrior_full_alphabet_concentration_budget in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
H :
ℝ
π :
Measure ℝ
hπ :
d J M :
ℕ
hJd :
J ≤ d
b Δ u :
ℝ
hb :
0 < b
hΔ :
0 < Δ
hu :
0 < u
hM :
1 ≤ M
hgap :
(J : ℝ) * b / 12 ≤ Δ
hJ :
18432 * (M : ℝ) ^ 2 / u ≤ (J : ℝ)
Proof (Lean source)
-- @node: intervalPrior_full_alphabet_concentration_budget
lemma intervalPrior_full_alphabet_concentration_budget {H : ℝ} {π : Measure ℝ}
(hπ : FiniteReciprocalPrior H π) (d J M : ℕ) (hJd : J ≤ d)
(b Δ u : ℝ) (hb : 0 < b) (hΔ : 0 < Δ) (hu : 0 < u) (hM : 1 ≤ M)
(hgap : (J : ℝ) * b / 12 ≤ Δ)
(hJ : 18432 * (M : ℝ) ^ 2 / u ≤ (J : ℝ)) :
(Measure.pi (fun _ : Fin d => π)).real
{z | Δ / (8 * M) < |b * (∑ x : Fin d,
if x.val < J then (z x)⁻¹ else 0) - (J : ℝ) * b * ∫ z, z⁻¹ ∂π|} ≤ u / 8 := by
let := hπ.1
let restrict : (Fin d → ℝ) → (Fin J → ℝ) :=
fun z x => z (x.castLE hJd)
have hr : Measurable restrict := by fun_prop
have hevent : MeasurableSet {z : Fin J → ℝ | Δ / (8 * M) <
|intervalPriorTarget J b z - (J : ℝ) * b * ∫ z, z⁻¹ ∂π|} := by
have hf : Measurable (intervalPriorTarget J b) := by
unfold intervalPriorTarget
fun_prop
simpa only [Function.comp_def, Real.norm_eq_abs] using
(measurableSet_lt measurable_const (measurable_norm.comp (hf.sub_const _)))
have hmap := intervalPrior_initial_coordinates_map d J hJd (π := π)
have hprob := map_measureReal_apply (μ := Measure.pi (fun _ : Fin d => π)) hr hevent
rw [hmap] at hprob
have htail := intervalPriorTarget_concentration_budget hπ J M b Δ u hb hΔ hu hM hgap hJ
rw [hprob] at htail
simpa only [Set.preimage_ofPred_eq, intervalPriorTarget,
activation_sum_initial_cells d J hJd, restrict] using htail
theorem activatedAugmentedMixture_neighborhood_coverage
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q α θ w ε :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
T :
hT :
HonestInterval n d q α T
hα :
0 ≤ α
conclusion
1 - α - ε
≤ ∫ s, (T.1 s).real {I | ∃ x : ℝ, I.lo ≤ x ∧ x ≤ I.hi ∧ |x - θ| ≤ w} ∂((activatedAugmentedMixture η n d q π).map (fun s i => (s i).2))
Proof (Lean source)
-- @node: activatedAugmentedMixture_neighborhood_coverage
lemma activatedAugmentedMixture_neighborhood_coverage
(η : ℝ) (n d : ℕ) (q α θ w ε : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π : Measure ℝ}
(hπ : FiniteReciprocalPrior (lowerEndpoint n q) π)
(T : IntervalProcedure n d) (hT : HonestInterval n d q α T) (hα : 0 ≤ α)
(htail : (Measure.pi (fun _ : Fin d => π)).real
{z | w < |rareMass η n q * (∑ x : Fin d,
if x.val < rareCount η n d q then (z x)⁻¹ else 0) - θ|} ≤ ε) :
1 - α - ε ≤
∫ s, (T.1 s).real {I | ∃ x : ℝ,
I.lo ≤ x ∧ x ≤ I.hi ∧ |x - θ| ≤ w}
∂((activatedAugmentedMixture η n d q π).map (fun s i => (s i).2)) := by
classical
let := hπ.1
let μ := Measure.pi (fun _ : Fin d => π)
let target : (Fin d → ℝ) → ℝ := fun z => rareMass η n q *
∑ x : Fin d, if x.val < rareCount η n d q then (z x)⁻¹ else 0
let bad : Set (Fin d → ℝ) := {z | w < |target z - θ|}
have hmeas : Measurable target := by
dsimp [target]
apply Measurable.const_mul
apply Finset.measurable_sum
intro x hx
by_cases h : x.val < rareCount η n d q <;> simp only [h, if_true, if_false]
· fun_prop
· fun_prop
have hbad : MeasurableSet bad := by
simpa only [bad, Function.comp_def, Real.norm_eq_abs] using
(measurableSet_lt measurable_const (measurable_norm.comp (hmeas.sub_const θ)))
rw [activatedAugmentedMixture_projected_integral η n d q hn hd hb hq hslice hπ]
have hle : ∀ᵐ z ∂μ, 1 - α - bad.indicator (fun _ => (1 : ℝ)) z ≤
(if hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q) then
∫ s, (T.1 s).real {I | ∃ x : ℝ,
I.lo ≤ x ∧ x ≤ I.hi ∧ |x - θ| ≤ w}
∂sampleLaw n (activatedFullLaw η n d q z hd hb hq hslice hz)
else 0) := by
filter_upwards [finiteReciprocalPrior_pi_ae_support hπ d] with z hz
rw [dif_pos hz]
have htarget : ate (activatedFullLaw η n d q z hd hb hq hslice hz) = target z := by
change (∫ r : FullRecord d,
((if r.Y1 then (1 : ℝ) else 0) - (if r.Y0 then (1 : ℝ) else 0))
∂activatedLaw η n d q z hz) = target z
exact activatedLaw_target_integral η n d q z hb hq hslice hz
have h := honestInterval_neighborhood_coverage T hT hα
(activatedFullLaw η n d q z hd hb hq hslice hz)
(activatedFullLaw_model η n d q z hd hb hq hslice hz hn) θ w
simpa only [htarget, Set.indicator_apply, Set.mem_ofPred_eq, bad, Pi.one_apply]
using h
have hbound := integral_mono_ae (finiteReciprocalPrior_pi_integrable hπ d _)
(finiteReciprocalPrior_pi_integrable hπ d _) hle
have heq : (∫ z, 1 - α - bad.indicator (fun _ => (1 : ℝ)) z ∂μ) =
1 - α - μ.real bad := by
rw [integral_sub (integrable_const _) (finiteReciprocalPrior_pi_integrable hπ d _),
integral_const, integral_indicator_const 1 hbad]
simp [μ]
rw [heq] at hbound
exact (sub_le_sub_left htail (1 - α)).trans hbound
theorem activatedAugmentedMixture_neighborhood_coverage_budget
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d M :
ℕ
q α Δ :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
T :
hT :
HonestInterval n d q α T
hα :
0 ≤ α
hα1 :
α < 1
hΔ :
0 < Δ
hM :
1 ≤ M
hgap :
(rareCount η n d q : ℝ) * rareMass η n q / 12 ≤ Δ
hJ :
18432 * (M : ℝ) ^ 2 / (1 - α) ≤ (rareCount η n d q : ℝ)
conclusion
7 * (1 - α) / 8
≤ (T.1 ∘ₘ ((activatedAugmentedMixture η n d q π).map (fun s i => (s i).2))).real {I | ∃ x : ℝ, I.lo ≤ x ∧ x ≤ I.hi ∧ |x - ((rareCount η n d q : ℝ) * rareMass η n q * ∫ z, z⁻¹ ∂π)| ≤ Δ / (8 * M)}
Proof (Lean source)
-- @node: activatedAugmentedMixture_neighborhood_coverage_budget
lemma activatedAugmentedMixture_neighborhood_coverage_budget
(η : ℝ) (n d M : ℕ) (q α Δ : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π : Measure ℝ}
(hπ : FiniteReciprocalPrior (lowerEndpoint n q) π)
(T : IntervalProcedure n d) (hT : HonestInterval n d q α T)
(hα : 0 ≤ α) (hα1 : α < 1) (hΔ : 0 < Δ) (hM : 1 ≤ M)
(hgap : (rareCount η n d q : ℝ) * rareMass η n q / 12 ≤ Δ)
(hJ : 18432 * (M : ℝ) ^ 2 / (1 - α) ≤ (rareCount η n d q : ℝ)) :
7 * (1 - α) / 8 ≤
(T.1 ∘ₘ ((activatedAugmentedMixture η n d q π).map
(fun s i => (s i).2))).real {I | ∃ x : ℝ,
I.lo ≤ x ∧ x ≤ I.hi ∧
|x - ((rareCount η n d q : ℝ) * rareMass η n q * ∫ z, z⁻¹ ∂π)| ≤
Δ / (8 * M)} := by
let := activatedAugmentedMixture_isProbabilityMeasure η n d q hn hd hb hq hslice hπ
have hf : Measurable (fun s : Fin n → Bool × ObsRecord d => fun i => (s i).2) := by
fun_prop
let := Measure.isProbabilityMeasure_map hf.aemeasurable
(μ := activatedAugmentedMixture η n d q π)
have hMr : 0 < (M : ℝ) := by exact_mod_cast (by omega : 0 < M)
rw [intervalProcedure_real_event T _ _
(connectedInterval_measurableSet_neighborhood _ _ (by positivity))]
have htail := intervalPrior_full_alphabet_concentration_budget hπ d
(rareCount η n d q) M ((min_le_left _ _).trans (sub_le d 1))
(rareMass η n q) Δ (1 - α) hb hΔ (by linarith) hM hgap hJ
have h := activatedAugmentedMixture_neighborhood_coverage η n d q α
((rareCount η n d q : ℝ) * rareMass η n q * ∫ z, z⁻¹ ∂π)
(Δ / (8 * M)) ((1 - α) / 8) hn hd hb hq hslice hπ T hT hα htail
convert h using 1
ring
theorem activatedAugmentedMixture_reported_length_le_worst
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
T :
conclusion
(∫ I : ConnectedInterval, I.hi - I.lo ∂(T.1 ∘ₘ ((activatedAugmentedMixture η n d q π).map (fun s i => (s i).2))))
≤ worstCaseRiskReal (fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) => intervalRisk T P.1) ()
Proof (Lean source)
-- @node: activatedAugmentedMixture_reported_length_le_worst
lemma activatedAugmentedMixture_reported_length_le_worst
(η : ℝ) (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π : Measure ℝ}
(hπ : FiniteReciprocalPrior (lowerEndpoint n q) π)
(T : IntervalProcedure n d) :
(∫ I : ConnectedInterval, I.hi - I.lo ∂(T.1 ∘ₘ
((activatedAugmentedMixture η n d q π).map (fun s i => (s i).2)))) ≤
worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) =>
intervalRisk T P.1) () := by
let := T.2
let := activatedAugmentedMixture_isProbabilityMeasure η n d q hn hd hb hq hslice hπ
have hf : Measurable (fun s : Fin n → Bool × ObsRecord d => fun i => (s i).2) := by
fun_prop
let := Measure.isProbabilityMeasure_map hf.aemeasurable
(μ := activatedAugmentedMixture η n d q π)
have hF : Measurable (fun I : ConnectedInterval =>
(I.lo, I.hi, I.closedLeft, I.closedRight)) := by
exact (measurable_fst.comp measurable_subtype_coe).prodMk
((measurable_snd.comp measurable_subtype_coe).prodMk
(measurable_const.prodMk measurable_const))
have hlen : Measurable (fun I : ConnectedInterval => I.hi - I.lo) := by
first
| fun_prop
| exact (measurable_connectedInterval_hi).sub
(measurable_connectedInterval_lo)
have hint : Integrable (fun I : ConnectedInterval => I.hi - I.lo)
(T.1 ∘ₘ ((activatedAugmentedMixture η n d q π).map (fun s i => (s i).2))) :=
Integrable.of_bound hlen.aestronglyMeasurable 2
(Eventually.of_forall fun I => by
rw [Real.norm_eq_abs, abs_of_nonneg (sub_nonneg.mpr I.ordered)]
linarith [I.lower, I.upper])
rw [integral_bind T.1.measurable hint]
exact activatedAugmentedMixture_intervalRisk_le_worst η n d q hn hd hb hq hslice hπ T
theorem activatedAugmentedMixture_grid_length_le_worst
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d M :
ℕ
q α :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π₀ π₁ :
Measure ℝ
h₀ :
FiniteReciprocalPrior (lowerEndpoint n q) π₀
h₁ :
FiniteReciprocalPrior (lowerEndpoint n q) π₁
K :
ℕ
hm :
∀ v : ℕ
if
v ≤ K
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
hgap :
1 / 12 ≤ (∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀
hα :
0 ≤ α
hα1 :
α < 1
hM :
1 ≤ M
hgrid :
8 / (1 - α) ≤ (M : ℝ)
hJ :
18432 * (M : ℝ) ^ 2 / (1 - α) ≤ (rareCount η n d q : ℝ)
htv :
∀ j : Fin (M + 1),
tvDist ((activatedAugmentedMixture η n d q (intervalInterpolatedPrior π₀ π₁ ((0 : ℝ) / M))).map (fun s i => (s i).2)) ((activatedAugmentedMixture η n d q (intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))).map (fun s i => (s i).2))
≤ (1 - α) / 16
T :
hT :
HonestInterval n d q α T
conclusion
(21 * (1 - α) / 32) * ((rareCount η n d q : ℝ) * rareMass η n q * ((∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀))
≤ worstCaseRiskReal (fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) => intervalRisk T P.1) ()
Proof (Lean source)
-- @node: activatedAugmentedMixture_grid_length_le_worst
lemma activatedAugmentedMixture_grid_length_le_worst
(η : ℝ) (n d M : ℕ) (q α : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior (lowerEndpoint n q) π₀)
(h₁ : FiniteReciprocalPrior (lowerEndpoint n q) π₁)
(K : ℕ) (hm : ∀ v : ℕ, v ≤ K → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
(hgap : 1 / 12 ≤ (∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀)
(hα : 0 ≤ α) (hα1 : α < 1) (hM : 1 ≤ M)
(hgrid : 8 / (1 - α) ≤ (M : ℝ))
(hJ : 18432 * (M : ℝ) ^ 2 / (1 - α) ≤ (rareCount η n d q : ℝ))
(htv : ∀ j : Fin (M + 1), tvDist
((activatedAugmentedMixture η n d q
(intervalInterpolatedPrior π₀ π₁ ((0 : ℝ) / M))).map (fun s i => (s i).2))
((activatedAugmentedMixture η n d q
(intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))).map (fun s i => (s i).2)) ≤
(1 - α) / 16)
(T : IntervalProcedure n d) (hT : HonestInterval n d q α T) :
(21 * (1 - α) / 32) * ((rareCount η n d q : ℝ) * rareMass η n q *
((∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀)) ≤
worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) =>
intervalRisk T P.1) () := by
let J : ℝ := rareCount η n d q
let b := rareMass η n q
let Δ := J * b * ((∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀)
let θ := J * b * ∫ z, z⁻¹ ∂π₀
let Q : Fin (M + 1) → Measure (Fin n → ObsRecord d) := fun j =>
(activatedAugmentedMixture η n d q
(intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))).map (fun s i => (s i).2)
let ν : Fin (M + 1) → Measure ConnectedInterval := fun j => T.1 ∘ₘ Q j
have hpriors := intervalInterpolatedPrior_grid h₀ h₁ K hm M hM
let := T.2
have hQ (j : Fin (M + 1)) : IsProbabilityMeasure (Q j) := by
let := activatedAugmentedMixture_isProbabilityMeasure η n d q hn hd hb hq hslice
(hpriors j).1
apply Measure.isProbabilityMeasure_map
have hf : Measurable (fun s : Fin n → Bool × ObsRecord d => fun i => (s i).2) := by
fun_prop
exact hf.aemeasurable
have hν (j : Fin (M + 1)) : IsProbabilityMeasure (ν j) := by
dsimp [ν]
infer_instance
have hu : 0 < 1 - α := by linarith
have hMr : 0 < (M : ℝ) := by exact_mod_cast (by omega : 0 < M)
have hJpos : 0 < J := lt_of_lt_of_le (by positivity) hJ
have hΔ : 0 < Δ := by dsimp [Δ, b]; positivity
have hΔgap : J * b / 12 ≤ Δ := by
dsimp [Δ]
nlinarith [mul_le_mul_of_nonneg_left hgap (mul_nonneg hJpos.le hb.le)]
have hlength := connectedInterval_many_prior_length_floor (ν 0) M ν θ Δ (1 - α)
hΔ hu hgrid (fun j => ?_) (fun j => ?_)
· exact hlength.trans (activatedAugmentedMixture_reported_length_le_worst η n d q
hn hd hb hq hslice (hpriors 0).1 T)
· have h := activatedAugmentedMixture_neighborhood_coverage_budget η n d M q α Δ
hn hd hb hq hslice (hpriors j).1 T hT hα hα1 hΔ hM hΔgap hJ
have hcenter : J * b * (∫ z, z⁻¹ ∂intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M)) =
θ + (j : ℝ) * (Δ / M) := by
rw [(hpriors j).2.2]
dsimp [θ, Δ]
ring
change 7 * (1 - α) / 8 ≤ (ν j).real {I | ∃ x : ℝ,
I.lo ≤ x ∧ x ≤ I.hi ∧
|x - (J * b * ∫ z, z⁻¹ ∂intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))| ≤
Δ / (8 * M)} at h
rw [hcenter] at h
exact h
· exact (tvDist_bind_le (Q 0) (Q j) T.1).trans (by simpa [Q] using htv j)
theorem honestInterval_nonempty
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q α :
ℝ
hα :
0 ≤ α
conclusion
Nonempty {T : IntervalProcedure n d // HonestInterval n d q α T}
Proof (Lean source)
-- @node: honestInterval_nonempty
lemma honestInterval_nonempty (n d : ℕ) (q α : ℝ) (hα : 0 ≤ α) :
Nonempty {T : IntervalProcedure n d // HonestInterval n d q α T} := by
let I : ConnectedInterval := ⟨(-1, 1), by norm_num, by norm_num, by norm_num⟩
let T : IntervalProcedure n d := ⟨Kernel.const _ (Measure.dirac I), inferInstance⟩
refine ⟨⟨T, ?_⟩⟩
intro P hP
let := P.2
let := Measure.isProbabilityMeasure_map (show Measurable obs by fun_prop).aemeasurable
(μ := P.1)
letI : IsProbabilityMeasure (sampleLaw n P) := by unfold sampleLaw; infer_instance
have hmem : I ∈ {I | intervalContains I (ate P)} := by
exact ate_mem_unit_interval P
have heq (s : Fin n → ObsRecord d) :
(T.1 s).real {I | intervalContains I (ate P)} = 1 := by
rw [Kernel.const_apply, Measure.real, Measure.dirac_apply_of_mem hmem]
simp
simp_rw [heq]
simp only [integral_const, probReal_univ, smul_eq_mul, one_mul]
linarith
theorem activatedAugmentedMixture_interval_minimax_of_cutoff
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d M :
ℕ
q α :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π₀ π₁ :
Measure ℝ
h₀ :
FiniteReciprocalPrior (lowerEndpoint n q) π₀
h₁ :
FiniteReciprocalPrior (lowerEndpoint n q) π₁
K :
ℕ
hm :
∀ v : ℕ
if
v ≤ K
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
hgap :
1 / 12 ≤ (∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀
hα :
0 ≤ α
hα1 :
α < 1
hM :
1 ≤ M
hgrid :
8 / (1 - α) ≤ (M : ℝ)
hJ :
18432 * (M : ℝ) ^ 2 / (1 - α) ≤ (rareCount η n d q : ℝ)
hbad :
(activatedAugmentedMixture η n d q (intervalInterpolatedPrior π₀ π₁ ((0 : ℝ) / M))).real {s | ¬ ∀ x : Fin d, x.val < rareCount η n d q → (∑ i ∈ Finset.univ.filter (fun i => (s i).2.X = x), if (s i).1 then 1 else 0) ≤ K}
≤ (1 - α) / 16
conclusion
(21 * (1 - α) / 32) * ((rareCount η n d q : ℝ) * rareMass η n q * ((∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀))
≤ intervalLengthRisk n d q α
Proof (Lean source)
-- @node: activatedAugmentedMixture_interval_minimax_of_cutoff
lemma activatedAugmentedMixture_interval_minimax_of_cutoff
(η : ℝ) (n d M : ℕ) (q α : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior (lowerEndpoint n q) π₀)
(h₁ : FiniteReciprocalPrior (lowerEndpoint n q) π₁)
(K : ℕ) (hm : ∀ v : ℕ, v ≤ K → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
(hgap : 1 / 12 ≤ (∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀)
(hα : 0 ≤ α) (hα1 : α < 1) (hM : 1 ≤ M)
(hgrid : 8 / (1 - α) ≤ (M : ℝ))
(hJ : 18432 * (M : ℝ) ^ 2 / (1 - α) ≤ (rareCount η n d q : ℝ))
(hbad : (activatedAugmentedMixture η n d q
(intervalInterpolatedPrior π₀ π₁ ((0 : ℝ) / M))).real
{s | ¬ ∀ x : Fin d, x.val < rareCount η n d q →
(∑ i ∈ Finset.univ.filter (fun i => (s i).2.X = x),
if (s i).1 then 1 else 0) ≤ K} ≤ (1 - α) / 16) :
(21 * (1 - α) / 32) * ((rareCount η n d q : ℝ) * rareMass η n q *
((∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀)) ≤ intervalLengthRisk n d q α := by
have htv (j : Fin (M + 1)) :=
activatedAugmentedMixture_projected_grid_tv_le_bad_event η n d q hn hd hb hq hslice
h₀ h₁ K hm M hM 0 j
have htvbudget (j : Fin (M + 1)) : tvDist
((activatedAugmentedMixture η n d q
(intervalInterpolatedPrior π₀ π₁ ((0 : ℝ) / M))).map (fun s i => (s i).2))
((activatedAugmentedMixture η n d q
(intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))).map (fun s i => (s i).2)) ≤
(1 - α) / 16 := by
simpa only [Fin.val_zero, Nat.cast_zero] using (htv j).trans
(by simpa only [Fin.val_zero, Nat.cast_zero] using hbad)
letI := honestInterval_nonempty n d q α hα
unfold intervalLengthRisk
apply le_minimaxValue
intro T
exact activatedAugmentedMixture_grid_length_le_worst η n d M q α hn hd hb hq hslice
h₀ h₁ K hm hgap hα hα1 hM hgrid hJ htvbudget T.1 T.2
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Lower.ActivationIntervalTransfer 4 declarations Averaging randomized interval lengths in the actual activated experiment and comparing the prior average with the model-class worst-case length.
Averaging randomized interval lengths in the actual activated experiment and comparing the prior average with the model-class worst-case length.
theorem activatedAugmentedMixture_integral
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
conclusion
(∫ s, f s ∂activatedAugmentedMixture η n d q π)
= ∫ z, (∫ s, f s ∂activatedAugmentedSample η n d q z) ∂Measure.pi (fun _ : Fin d => π)
Proof (Lean source)
-- @node: activatedAugmentedMixture_integral
lemma activatedAugmentedMixture_integral (η : ℝ) (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π : Measure ℝ} (hπ : FiniteReciprocalPrior (lowerEndpoint n q) π)
(f : (Fin n → Bool × ObsRecord d) → ℝ) :
(∫ s, f s ∂activatedAugmentedMixture η n d q π) =
∫ z, (∫ s, f s ∂activatedAugmentedSample η n d q z)
∂Measure.pi (fun _ : Fin d => π) := by
classical
let := activatedAugmentedMixture_isProbabilityMeasure η n d q hn hd hb hq hslice hπ
rw [integral_fintype Integrable.of_finite]
simp_rw [activatedAugmentedMixture_real_singleton, smul_eq_mul,
← integral_mul_const]
rw [← integral_finsetSum _ (fun s _ => finiteReciprocalPrior_pi_integrable hπ d _)]
apply integral_congr_ae
filter_upwards [finiteReciprocalPrior_pi_ae_support hπ d] with z hz
rw [activatedAugmentedSample, dif_pos hz]
let := augmentedOneRecord_isProbabilityMeasure η n d q z hn hd hb hq hslice hz
simpa only [smul_eq_mul] using (integral_fintype (f := f)
(μ := Measure.pi (fun _ : Fin n => augmentedOneRecord η n d q z hz))
Integrable.of_finite).symm
theorem activatedAugmentedMixture_projected_integral
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
conclusion
(∫ s, f s ∂((activatedAugmentedMixture η n d q π).map (fun s i => (s i).2)))
= ∫ z, (if hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q) then ∫ s, f s ∂sampleLaw n (activatedFullLaw η n d q z hd hb hq hslice hz) else 0) ∂Measure.pi (fun _ : Fin d => π)
Proof (Lean source)
-- @node: activatedAugmentedMixture_projected_integral
lemma activatedAugmentedMixture_projected_integral (η : ℝ) (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π : Measure ℝ}
(hπ : FiniteReciprocalPrior (lowerEndpoint n q) π)
(f : (Fin n → ObsRecord d) → ℝ) :
(∫ s, f s ∂((activatedAugmentedMixture η n d q π).map
(fun s i => (s i).2))) =
∫ z, (if hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q) then
∫ s, f s ∂sampleLaw n (activatedFullLaw η n d q z hd hb hq hslice hz)
else 0) ∂Measure.pi (fun _ : Fin d => π) := by
have hf : Measurable (fun s : Fin n → Bool × ObsRecord d => fun i => (s i).2) := by
fun_prop
rw [integral_map hf.aemeasurable (by fun_prop), activatedAugmentedMixture_integral η n d q hn hd hb hq hslice hπ]
apply integral_congr_ae
filter_upwards [finiteReciprocalPrior_pi_ae_support hπ d] with z hz
rw [dif_pos hz, activatedAugmentedSample, dif_pos hz]
rw [← activated_sample_projection η n d q z hn hd hb hq hslice hz,
integral_map hf.aemeasurable (by fun_prop)]
theorem activatedAugmentedMixture_intervalRisk_average
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
T :
conclusion
(∫ s, (∫ I : ConnectedInterval, I.hi - I.lo ∂T.1 s) ∂((activatedAugmentedMixture η n d q π).map (fun s i => (s i).2)))
= ∫ z, (if hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q) then intervalRisk T (activatedFullLaw η n d q z hd hb hq hslice hz) else 0) ∂Measure.pi (fun _ : Fin d => π)
Proof (Lean source)
-- @node: activatedAugmentedMixture_intervalRisk_average
lemma activatedAugmentedMixture_intervalRisk_average (η : ℝ) (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π : Measure ℝ}
(hπ : FiniteReciprocalPrior (lowerEndpoint n q) π)
(T : IntervalProcedure n d) :
(∫ s, (∫ I : ConnectedInterval, I.hi - I.lo ∂T.1 s)
∂((activatedAugmentedMixture η n d q π).map (fun s i => (s i).2))) =
∫ z, (if hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q) then
intervalRisk T (activatedFullLaw η n d q z hd hb hq hslice hz)
else 0) ∂Measure.pi (fun _ : Fin d => π) := by
exact activatedAugmentedMixture_projected_integral η n d q hn hd hb hq hslice hπ _
theorem activatedAugmentedMixture_intervalRisk_le_worst
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
T :
conclusion
(∫ s, (∫ I : ConnectedInterval, I.hi - I.lo ∂T.1 s) ∂((activatedAugmentedMixture η n d q π).map (fun s i => (s i).2)))
≤ worstCaseRiskReal (fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) => intervalRisk T P.1) ()
Proof (Lean source)
-- @node: activatedAugmentedMixture_intervalRisk_le_worst
lemma activatedAugmentedMixture_intervalRisk_le_worst (η : ℝ) (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π : Measure ℝ}
(hπ : FiniteReciprocalPrior (lowerEndpoint n q) π)
(T : IntervalProcedure n d) :
(∫ s, (∫ I : ConnectedInterval, I.hi - I.lo ∂T.1 s)
∂((activatedAugmentedMixture η n d q π).map (fun s i => (s i).2))) ≤
worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) =>
intervalRisk T P.1) () := by
let := hπ.1
rw [activatedAugmentedMixture_intervalRisk_average η n d q hn hd hb hq hslice hπ T]
have hbdd : BddAbove (range (fun P : {P : FullLaw d // RareArrivalModelClass n d q P} =>
intervalRisk T P.1)) := by
refine ⟨2, ?_⟩
rintro _ ⟨P, rfl⟩
exact (intervalRisk_bounds T P.1).2
have hle : ∀ᵐ z ∂Measure.pi (fun _ : Fin d => π),
(if hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q) then
intervalRisk T (activatedFullLaw η n d q z hd hb hq hslice hz) else 0) ≤
worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) =>
intervalRisk T P.1) () := by
filter_upwards [finiteReciprocalPrior_pi_ae_support hπ d] with z hz
rw [dif_pos hz]
exact le_worstCaseRisk
(risk := fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) =>
intervalRisk T P.1) (e := ()) hbdd ⟨_, activatedFullLaw_model η n d q z hd hb hq hslice hz hn⟩
simpa using integral_mono_ae (finiteReciprocalPrior_pi_integrable hπ d _)
(integrable_const _) hle
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Lower.ActivationLikelihood 4 declarations Projection from the activated experiment to every observed record.
Projection from the activated experiment to every observed record.
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∑ flag : Bool, bernWeight (q * lowerEndpoint n q) flag * augmentedOutcomeWeight η n d q z x a flag arrival one)
= ∑ y₁ : Bool, bernWeight (if x.val < rareCount η n d q then (z x)⁻¹ else 0) y₁ * bernWeight (q * lowerCellZ η n d q z x) arrival * (if (arrival && (a && y₁)) = one then (1 : ℝ) else 0)
Proof (Lean source)
lemma activation_atom_weight (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin d → ℝ) (x : Fin d) (a arrival one : Bool)
(hz : 1 ≤ z x ∧ z x ≤ lowerEndpoint n q) :
(∑ flag : Bool, bernWeight (q * lowerEndpoint n q) flag *
augmentedOutcomeWeight η n d q z x a flag arrival one) =
∑ y₁ : Bool,
bernWeight (if x.val < rareCount η n d q then (z x)⁻¹ else 0) y₁ *
bernWeight (q * lowerCellZ η n d q z x) arrival *
(if (arrival && (a && y₁)) = one then (1 : ℝ) else 0) := by
have hH : lowerEndpoint n q ≠ 0 := by linarith
have hz0 : z x ≠ 0 := by linarith
by_cases hr : x.val < rareCount η n d q
· cases a <;> cases arrival <;> cases one <;>
simp [bernWeight, augmentedOutcomeWeight, lowerCellZ, hr] <;>
field_simp <;> ring
· cases a <;> cases arrival <;> cases one <;>
simp [bernWeight, augmentedOutcomeWeight, lowerCellZ, hr] <;>
field_simp <;> simp
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
z :
Fin d → ℝ
x :
Fin d
a flag arrival one :
hz :
1 ≤ z x ∧ z x ≤ lowerEndpoint n q
conclusion
0 ≤ augmentedOutcomeWeight η n d q z x a flag arrival one
Proof (Lean source)
lemma augmentedOutcomeWeight_nonneg (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin d → ℝ) (x : Fin d) (a flag arrival one : Bool)
(hz : 1 ≤ z x ∧ z x ≤ lowerEndpoint n q) :
0 ≤ augmentedOutcomeWeight η n d q z x a flag arrival one := by
have hH : 0 < lowerEndpoint n q := by linarith [hz.1, hz.2]
have hcell : 0 ≤ lowerCellZ η n d q z x ∧
lowerCellZ η n d q z x ≤ lowerEndpoint n q := by
unfold lowerCellZ
split_ifs <;> constructor <;> linarith [hz.1, hz.2]
unfold augmentedOutcomeWeight
split_ifs <;> try positivity
all_goals try exact div_nonneg (by linarith [hz.1]) (le_of_lt hH)
all_goals apply sub_nonneg.mpr
all_goals exact (div_le_one hH).2 hcell.2
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma ofReal_signed_mul_bool_sum (b : ℝ) (w : Bool → ℝ)
(hw : ∀ flag, 0 ≤ w flag) :
(∑ flag : Bool, ofReal (b * w flag)) =
ofReal (b * ∑ flag : Bool, w flag) := by
simp_rw [ENNReal.ofReal_mul' (hw _)]
rw [← Finset.mul_sum]
rw [ENNReal.ofReal_mul' (Finset.sum_nonneg (fun flag _ => hw flag))]
rw [ENNReal.ofReal_sum_of_nonneg (fun flag _ => hw flag)]
theorem activated_projection
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
z :
Fin d → ℝ
_hn :
1 ≤ n
_hd :
1 ≤ d
hq :
0 < q
hslice :
RareArrivalSlice n q
hz :
∀ x, 1 ≤ z x ∧ z x ≤ lowerEndpoint n q
conclusion
Proof (Lean source)
lemma activated_projection (η : ℝ) (n d : ℕ) (q : ℝ) (z : Fin d → ℝ)
(_hn : 1 ≤ n) (_hd : 1 ≤ d) (hq : 0 < q)
(hslice : RareArrivalSlice n q)
(hz : ∀ x, 1 ≤ z x ∧ z x ≤ lowerEndpoint n q)
-- @realizes \(Z\)(latent reciprocal variable in [1,H])
:
(augmentedOneRecord η n d q z (by simpa [Set.mem_Icc] using hz)).map Prod.snd =
(activatedLaw η n d q z (by simpa [Set.mem_Icc] using hz)).map obs := by
classical
have hH : q * lowerEndpoint n q ≤ 1 :=
lower_activation_probability_le_one n q (le_of_lt hq) hslice
have hHpos : 0 ≤ lowerEndpoint n q := by
unfold lowerEndpoint
positivity
have hcell (x : Fin d) : 0 ≤ lowerCellZ η n d q z x ∧
lowerCellZ η n d q z x ≤ lowerEndpoint n q := by
unfold lowerCellZ
split_ifs <;> constructor <;> linarith [(hz x).1, (hz x).2]
have hprob (x : Fin d) : 0 ≤ q * lowerCellZ η n d q z x ∧
q * lowerCellZ η n d q z x ≤ 1 := by
constructor
· exact mul_nonneg (le_of_lt hq) (hcell x).1
· exact (mul_le_mul_of_nonneg_left (hcell x).2 (le_of_lt hq)).trans hH
have hbw (p : ℝ) (hp : 0 ≤ p ∧ p ≤ 1) (v : Bool) : 0 ≤ bernWeight p v := by
cases v <;> simp [bernWeight] <;> linarith [hp.1, hp.2]
have hyprob (x : Fin d) :
0 ≤ (if x.val < rareCount η n d q then (z x)⁻¹ else 0) ∧
(if x.val < rareCount η n d q then (z x)⁻¹ else 0) ≤ 1 := by
split_ifs
· constructor
· exact inv_nonneg.mpr (le_trans (by norm_num) (hz x).1)
· exact inv_le_one₀ (by linarith [(hz x).1]) |>.mpr (hz x).1
· constructor <;> norm_num
have hcoeff (x : Fin d) (a arrival one : Bool) :
(∑ flag : Bool, ofReal
(baselineMass η n d q x / 2 * bernWeight (q * lowerEndpoint n q) flag *
augmentedOutcomeWeight η n d q z x a flag arrival one)) =
∑ y₁ : Bool, ofReal
(baselineMass η n d q x / 2 *
(bernWeight (if x.val < rareCount η n d q then (z x)⁻¹ else 0) y₁ *
bernWeight (q * lowerCellZ η n d q z x) arrival *
(if (arrival && (a && y₁)) = one then (1 : ℝ) else 0))) := by
let b := baselineMass η n d q x / 2
have hleft (flag : Bool) :
0 ≤ bernWeight (q * lowerEndpoint n q) flag *
augmentedOutcomeWeight η n d q z x a flag arrival one :=
mul_nonneg (hbw _ ⟨mul_nonneg (le_of_lt hq) hHpos, hH⟩ flag)
(augmentedOutcomeWeight_nonneg η n d q z x a flag arrival one (hz x))
have hright (y₁ : Bool) :
0 ≤ bernWeight (if x.val < rareCount η n d q then (z x)⁻¹ else 0) y₁ *
bernWeight (q * lowerCellZ η n d q z x) arrival *
(if (arrival && (a && y₁)) = one then (1 : ℝ) else 0) := by
apply mul_nonneg
· exact mul_nonneg (hbw _ (hyprob x) y₁) (hbw _ (hprob x) arrival)
· split_ifs <;> norm_num
simp_rw [show ∀ flag : Bool,
baselineMass η n d q x / 2 * bernWeight (q * lowerEndpoint n q) flag *
augmentedOutcomeWeight η n d q z x a flag arrival one =
b * (bernWeight (q * lowerEndpoint n q) flag *
augmentedOutcomeWeight η n d q z x a flag arrival one) from
fun _ => by simp [b, mul_assoc]]
rw [ofReal_signed_mul_bool_sum b _ hleft]
rw [activation_atom_weight η n d q z x a arrival one (hz x)]
rw [← ofReal_signed_mul_bool_sum b _ hright]
have hsnd : Measurable (Prod.snd : Bool × ObsRecord d → ObsRecord d) := measurable_snd
have hobs : Measurable (obs : FullRecord d → ObsRecord d) := by fun_prop
simp only [augmentedOneRecord, activatedLaw]
rw [Measure.map_finset_sum' hsnd.aemeasurable]
rw [Measure.map_finset_sum' hobs.aemeasurable]
simp_rw [Measure.map_finset_sum' hsnd.aemeasurable]
simp_rw [Measure.map_finset_sum' hobs.aemeasurable]
simp only [Measure.map_smul, Measure.map_dirac, obs, lowerFullRecord,
FullRecord.S, FullRecord.Y]
apply Finset.sum_congr rfl
intro x _
apply Finset.sum_congr rfl
intro a _
rw [Finset.sum_comm]
conv_rhs => rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro arrival _
rw [Finset.sum_comm]
simp_rw [← Finset.sum_smul]
simp_rw [hcoeff x a arrival]
simp_rw [Finset.sum_smul]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro y₁ _
cases a <;> cases arrival <;> cases y₁ <;>
simp [mul_assoc]
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Lower.ActivationMixture 8 declarations Construct the augmented prior mixture from its averaged atoms, with normalization and event probabilities derived from the supported experiment.
Construct the augmented prior mixture from its averaged atoms, with normalization and event probabilities derived from the supported experiment.
theorem finiteReciprocalPrior_pi_integrable Lemma finiteReciprocalPrior_pi_integrable in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Integrable f (Measure.pi (fun _ : Fin J => π))
Proof (Lean source)
-- @node: finiteReciprocalPrior_pi_integrable
lemma finiteReciprocalPrior_pi_integrable {H : ℝ} {π : Measure ℝ}
(hπ : FiniteReciprocalPrior H π) (J : ℕ) (f : (Fin J → ℝ) → ℝ) :
Integrable f (Measure.pi (fun _ : Fin J => π)) := by
let := hπ.1
obtain ⟨s, hs, _⟩ := hπ.2
have hae : ∀ᵐ z ∂π, z ∈ (s : Set ℝ) :=
(ae_mem_iff_measure_eq s.measurableSet.nullMeasurableSet).2 (by simpa using hs)
have hprod : ∀ᵐ z ∂Measure.pi (fun _ : Fin J => π), ∀ x, z x ∈ (s : Set ℝ) := by
rw [ae_all_iff]
intro x
exact Measure.tendsto_eval_ae_ae.eventually hae
have hfinite : {z : Fin J → ℝ | ∀ x, z x ∈ (s : Set ℝ)}.Finite :=
Set.Finite.pi' (fun _ => s.finite_toSet)
have hres : (Measure.pi (fun _ : Fin J => π)).restrict
{z : Fin J → ℝ | ∀ x, z x ∈ (s : Set ℝ)} =
Measure.pi (fun _ : Fin J => π) :=
Measure.restrict_eq_self_of_ae_mem hprod
have hi : IntegrableOn f {z : Fin J → ℝ | ∀ x, z x ∈ (s : Set ℝ)}
(Measure.pi (fun _ : Fin J => π)) := IntegrableOn.of_finite hfinite
simpa only [IntegrableOn, hres] using hi
def activatedAugmentedMixture
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
theorem activatedAugmentedMixture_real_singleton
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(activatedAugmentedMixture η n d q π).real {s}
Proof (Lean source)
-- @node: activatedAugmentedMixture_real_singleton
lemma activatedAugmentedMixture_real_singleton (η : ℝ) (n d : ℕ) (q : ℝ)
(π : Measure ℝ) (s : Fin n → Bool × ObsRecord d) :
(activatedAugmentedMixture η n d q π).real {s} =
∫ z, (activatedAugmentedSample η n d q z).real {s}
∂Measure.pi (fun _ : Fin d => π) := by
classical
unfold activatedAugmentedMixture Measure.real
simp only [Measure.finsetSum_apply, Measure.smul_apply, smul_eq_mul,
Measure.dirac_apply, Set.indicator_apply, Set.mem_singleton_iff, Pi.one_apply]
simp only [mul_ite, mul_one, mul_zero, Finset.sum_ite_eq', Finset.mem_univ, if_true]
exact ENNReal.toReal_ofReal (integral_nonneg (fun _ => measureReal_nonneg))
theorem activatedAugmentedMixture_isProbabilityMeasure
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
conclusion
IsProbabilityMeasure (activatedAugmentedMixture η n d q π)
Proof (Lean source)
-- @node: activatedAugmentedMixture_isProbabilityMeasure
lemma activatedAugmentedMixture_isProbabilityMeasure (η : ℝ) (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π : Measure ℝ}
(hπ : FiniteReciprocalPrior (lowerEndpoint n q) π) :
IsProbabilityMeasure (activatedAugmentedMixture η n d q π) := by
classical
let := hπ.1
have hsum : (∑ s : Fin n → Bool × ObsRecord d,
∫ z, (activatedAugmentedSample η n d q z).real {s}
∂Measure.pi (fun _ : Fin d => π)) = 1 := by
rw [← integral_finsetSum _ (fun s _ => finiteReciprocalPrior_pi_integrable hπ d _)]
calc
(∫ z, ∑ s : Fin n → Bool × ObsRecord d,
(activatedAugmentedSample η n d q z).real {s}
∂Measure.pi (fun _ : Fin d => π)) = ∫ _z, (1 : ℝ)
∂Measure.pi (fun _ : Fin d => π) := by
apply integral_congr_ae
filter_upwards [finiteReciprocalPrior_pi_ae_support hπ d] with z hz
rw [activatedAugmentedSample, dif_pos hz]
let := augmentedOneRecord_isProbabilityMeasure η n d q z hn hd hb hq hslice hz
rw [sum_measureReal_singleton]
simp
_ = 1 := by simp
constructor
unfold activatedAugmentedMixture
simp only [Measure.finsetSum_apply, Measure.smul_apply, smul_eq_mul,
Measure.dirac_apply_of_mem (Set.mem_univ _), mul_one]
rw [← ENNReal.ofReal_sum_of_nonneg
(fun s _ => integral_nonneg (fun _ => measureReal_nonneg)), hsum]
simp
theorem activatedAugmentedMixture_real_event
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
conclusion
(activatedAugmentedMixture η n d q π).real A
Proof (Lean source)
-- @node: activatedAugmentedMixture_real_event
lemma activatedAugmentedMixture_real_event (η : ℝ) (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π : Measure ℝ}
(hπ : FiniteReciprocalPrior (lowerEndpoint n q) π)
(A : Set (Fin n → Bool × ObsRecord d)) :
(activatedAugmentedMixture η n d q π).real A =
∫ z, (activatedAugmentedSample η n d q z).real A
∂Measure.pi (fun _ : Fin d => π) := by
classical
let := activatedAugmentedMixture_isProbabilityMeasure η n d q hn hd hb hq hslice hπ
rw [← Set.coe_toFinset A, ← sum_measureReal_singleton]
simp_rw [activatedAugmentedMixture_real_singleton]
rw [← integral_finsetSum _ (fun s _ => finiteReciprocalPrior_pi_integrable hπ d _)]
apply integral_congr_ae
filter_upwards [finiteReciprocalPrior_pi_ae_support hπ d] with z hz
rw [activatedAugmentedSample, dif_pos hz]
let := augmentedOneRecord_isProbabilityMeasure η n d q z hn hd hb hq hslice hz
exact sum_measureReal_singleton _
theorem activatedAugmentedMixture_grid_tv_le_bad_event
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π₀ π₁ :
Measure ℝ
h₀ :
FiniteReciprocalPrior (lowerEndpoint n q) π₀
h₁ :
FiniteReciprocalPrior (lowerEndpoint n q) π₁
K :
ℕ
hm :
∀ v : ℕ
if
v ≤ K
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
M :
ℕ
hM :
1 ≤ M
j j' :
Fin (M + 1)
conclusion
tvDist (activatedAugmentedMixture η n d q (intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))) (activatedAugmentedMixture η n d q (intervalInterpolatedPrior π₀ π₁ ((j' : ℝ) / M)))
≤ (activatedAugmentedMixture η n d q (intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))).real {s | ¬ ∀ x : Fin d, x.val < rareCount η n d q → (∑ i ∈ Finset.univ.filter (fun i => (s i).2.X = x), if (s i).1 then 1 else 0) ≤ K}
Proof (Lean source)
-- @node: activatedAugmentedMixture_grid_tv_le_bad_event
lemma activatedAugmentedMixture_grid_tv_le_bad_event (η : ℝ) (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior (lowerEndpoint n q) π₀)
(h₁ : FiniteReciprocalPrior (lowerEndpoint n q) π₁)
(K : ℕ) (hm : ∀ v : ℕ, v ≤ K → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
(M : ℕ) (hM : 1 ≤ M) (j j' : Fin (M + 1)) :
tvDist
(activatedAugmentedMixture η n d q
(intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M)))
(activatedAugmentedMixture η n d q
(intervalInterpolatedPrior π₀ π₁ ((j' : ℝ) / M))) ≤
(activatedAugmentedMixture η n d q
(intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))).real
{s | ¬ ∀ x : Fin d, x.val < rareCount η n d q →
(∑ i ∈ Finset.univ.filter (fun i => (s i).2.X = x),
if (s i).1 then 1 else 0) ≤ K} := by
have hgrid := intervalInterpolatedPrior_grid h₀ h₁ K hm M hM
let := activatedAugmentedMixture_isProbabilityMeasure η n d q hn hd hb hq hslice
(hgrid j).1
let := activatedAugmentedMixture_isProbabilityMeasure η n d q hn hd hb hq hslice
(hgrid j').1
exact activatedAugmentedSample_grid_tv_le_bad_event η n d q hn hd hb hq hslice
h₀ h₁ K hm M hM j j' _ _
(activatedAugmentedMixture_real_singleton η n d q _)
(activatedAugmentedMixture_real_singleton η n d q _)
theorem activatedAugmentedMixture_projected_real_event
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
conclusion
= ∫ z, (if hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q) then (sampleLaw n (activatedFullLaw η n d q z hd hb hq hslice hz)).real A else 0) ∂Measure.pi (fun _ : Fin d => π)
Proof (Lean source)
-- @node: activatedAugmentedMixture_projected_real_event
lemma activatedAugmentedMixture_projected_real_event (η : ℝ) (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π : Measure ℝ}
(hπ : FiniteReciprocalPrior (lowerEndpoint n q) π)
(A : Set (Fin n → ObsRecord d)) :
((activatedAugmentedMixture η n d q π).map (fun s i => (s i).2)).real A =
∫ z, (if hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q) then
(sampleLaw n (activatedFullLaw η n d q z hd hb hq hslice hz)).real A
else 0) ∂Measure.pi (fun _ : Fin d => π) := by
have hf : Measurable (fun s : Fin n → Bool × ObsRecord d => fun i => (s i).2) := by
fun_prop
have hA : MeasurableSet A := A.toFinite.measurableSet
rw [map_measureReal_apply hf hA,
activatedAugmentedMixture_real_event η n d q hn hd hb hq hslice hπ]
apply integral_congr_ae
filter_upwards [finiteReciprocalPrior_pi_ae_support hπ d] with z hz
rw [dif_pos hz, activatedAugmentedSample, dif_pos hz]
rw [← activated_sample_projection η n d q z hn hd hb hq hslice hz,
map_measureReal_apply hf hA]
theorem activatedAugmentedMixture_projected_grid_tv_le_bad_event
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π₀ π₁ :
Measure ℝ
h₀ :
FiniteReciprocalPrior (lowerEndpoint n q) π₀
h₁ :
FiniteReciprocalPrior (lowerEndpoint n q) π₁
K :
ℕ
hm :
∀ v : ℕ
if
v ≤ K
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
M :
ℕ
hM :
1 ≤ M
j j' :
Fin (M + 1)
conclusion
tvDist ((activatedAugmentedMixture η n d q (intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))).map (fun s i => (s i).2)) ((activatedAugmentedMixture η n d q (intervalInterpolatedPrior π₀ π₁ ((j' : ℝ) / M))).map (fun s i => (s i).2))
≤ (activatedAugmentedMixture η n d q (intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))).real {s | ¬ ∀ x : Fin d, x.val < rareCount η n d q → (∑ i ∈ Finset.univ.filter (fun i => (s i).2.X = x), if (s i).1 then 1 else 0) ≤ K}
Proof (Lean source)
-- @node: activatedAugmentedMixture_projected_grid_tv_le_bad_event
lemma activatedAugmentedMixture_projected_grid_tv_le_bad_event
(η : ℝ) (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior (lowerEndpoint n q) π₀)
(h₁ : FiniteReciprocalPrior (lowerEndpoint n q) π₁)
(K : ℕ) (hm : ∀ v : ℕ, v ≤ K → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
(M : ℕ) (hM : 1 ≤ M) (j j' : Fin (M + 1)) :
tvDist
((activatedAugmentedMixture η n d q
(intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))).map (fun s i => (s i).2))
((activatedAugmentedMixture η n d q
(intervalInterpolatedPrior π₀ π₁ ((j' : ℝ) / M))).map (fun s i => (s i).2)) ≤
(activatedAugmentedMixture η n d q
(intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))).real
{s | ¬ ∀ x : Fin d, x.val < rareCount η n d q →
(∑ i ∈ Finset.univ.filter (fun i => (s i).2.X = x),
if (s i).1 then 1 else 0) ≤ K} := by
have hgrid := intervalInterpolatedPrior_grid h₀ h₁ K hm M hM
let := activatedAugmentedMixture_isProbabilityMeasure η n d q hn hd hb hq hslice
(hgrid j).1
let := activatedAugmentedMixture_isProbabilityMeasure η n d q hn hd hb hq hslice
(hgrid j').1
have hf : Measurable (fun s : Fin n → Bool × ObsRecord d => fun i => (s i).2) := by
fun_prop
apply activation_projected_tv_le_bad_event _ _ _ _ _ hf
intro s hs
rw [activatedAugmentedMixture_real_singleton, activatedAugmentedMixture_real_singleton]
exact activatedAugmentedSample_grid_singleton_eq η n d q hn hd hb hq hslice
h₀ h₁ K hm M hM j j' s hs
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Lower.ActivationModel 11 declarations Cell probabilities and MAR factorization for the activated full-data laws.
Cell probabilities and MAR factorization for the activated full-data laws. These construction identities supply the model obligations used when averaging coverage over the finite reciprocal priors.
theorem activatedFullLaw_real_event
Given the event, the stated activated-law mass identity holds.
Formal statement
hypotheses
η :
ℝ
n :
ℕ
d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
E :
Set (FullRecord d)
conclusion
(activatedFullLaw η n d q z hd hb hq hslice hz).1.real E
= by classical exact ∑ x : Fin d, ∑ a : Bool, ∑ y : Bool, ∑ r : Bool, if lowerFullRecord x a y r ∈ E then baselineMass η n d q x / 2 * bernWeight (if x.val < rareCount η n d q then (z x)⁻¹ else 0) y * bernWeight (q * lowerCellZ η n d q z x) r else 0
Proof (Lean source)
-- @node: activatedFullLaw_real_event
lemma activatedFullLaw_real_event (E : Set (FullRecord d)) :
(activatedFullLaw η n d q z hd hb hq hslice hz).1.real E = by
classical
exact ∑ x : Fin d, ∑ a : Bool, ∑ y : Bool, ∑ r : Bool,
if lowerFullRecord x a y r ∈ E then
baselineMass η n d q x / 2 *
bernWeight (if x.val < rareCount η n d q then (z x)⁻¹ else 0) y *
bernWeight (q * lowerCellZ η n d q z x) r else 0 := by
classical
simp only [activatedFullLaw, activatedLaw, Measure.real, Measure.finsetSum_apply,
Measure.smul_apply, smul_eq_mul, Measure.dirac_apply, Set.indicator_apply,
Pi.one_apply]
simp only [mul_ite, mul_one, mul_zero]
rw [ENNReal.toReal_sum (by simp [ENNReal.sum_ne_top, ite_eq_iff])]
apply Finset.sum_congr rfl
intro x _
rw [ENNReal.toReal_sum (by simp [ENNReal.sum_ne_top, ite_eq_iff])]
apply Finset.sum_congr rfl
intro a _
rw [ENNReal.toReal_sum (by simp [ENNReal.sum_ne_top, ite_eq_iff])]
apply Finset.sum_congr rfl
intro y _
rw [ENNReal.toReal_sum (by simp [ENNReal.sum_ne_top, ite_eq_iff])]
apply Finset.sum_congr rfl
intro r _
by_cases he : lowerFullRecord x a y r ∈ E
· simp only [if_pos he]
exact ENNReal.toReal_ofReal
(activatedLaw_weight_nonneg η n d q z hb hq hslice hz x y r)
· simp only [if_neg he, ENNReal.toReal_zero]
theorem activatedFullLaw_cellProb
Given an arm, cell, and surrogate value, the activated cell probability has the displayed form.
Formal statement
hypotheses
η :
ℝ
n :
ℕ
d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
a :
x :
Fin d
s :
conclusion
cellProb (activatedFullLaw η n d q z hd hb hq hslice hz) (a, x, s)
= if s then 0 else baselineMass η n d q x / 2
Proof (Lean source)
-- @node: activatedFullLaw_cellProb
lemma activatedFullLaw_cellProb (a : Bool) (x : Fin d) (s : Bool) :
cellProb (activatedFullLaw η n d q z hd hb hq hslice hz) (a, x, s) =
if s then 0 else baselineMass η n d q x / 2 := by
classical
rw [cellProb, activatedFullLaw_real_event η n d q z hd hb hq hslice hz]
cases a <;> cases s <;>
simp [inCell, lowerFullRecord, FullRecord.S, bernWeight,
Finset.sum_ite_irrel, Finset.sum_ite_eq']
all_goals split_ifs <;> ring
theorem activatedFullLaw_arrivedCell
Given an arm, cell, and surrogate value, the activated arrived-cell mass has the displayed form.
Formal statement
hypotheses
η :
ℝ
n :
ℕ
d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
a :
x :
Fin d
s :
conclusion
arrivedCell (activatedFullLaw η n d q z hd hb hq hslice hz) (a, x, s)
= q * lowerCellZ η n d q z x * cellProb (activatedFullLaw η n d q z hd hb hq hslice hz) (a, x, s)
Proof (Lean source)
-- @node: activatedFullLaw_arrivedCell
lemma activatedFullLaw_arrivedCell (a : Bool) (x : Fin d) (s : Bool) :
arrivedCell (activatedFullLaw η n d q z hd hb hq hslice hz) (a, x, s) =
q * lowerCellZ η n d q z x *
cellProb (activatedFullLaw η n d q z hd hb hq hslice hz) (a, x, s) := by
classical
rw [arrivedCell, activatedFullLaw_real_event η n d q z hd hb hq hslice hz,
activatedFullLaw_cellProb η n d q z hd hb hq hslice hz]
cases a <;> cases s <;>
simp [inCell, lowerFullRecord, FullRecord.S, bernWeight,
Finset.sum_ite_irrel, Finset.sum_ite_eq']
all_goals split_ifs <;> ring
theorem activatedFullLaw_occupiedCellArrival
The activated law satisfies occupied-cell arrival.
Formal statement
hypotheses
η :
ℝ
n :
ℕ
d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
conclusion
OccupiedCellArrival q (activatedFullLaw η n d q z hd hb hq hslice hz)
Proof (Lean source)
-- @node: activatedFullLaw_occupiedCellArrival
lemma activatedFullLaw_occupiedCellArrival :
OccupiedCellArrival q (activatedFullLaw η n d q z hd hb hq hslice hz) := by
intro j hj
rcases j with ⟨a, x, s⟩
rw [activatedFullLaw_arrivedCell η n d q z hd hb hq hslice hz]
have hcell : 1 ≤ lowerCellZ η n d q z x := by
unfold lowerCellZ
split_ifs
· exact (hz x).1
· exact lowerEndpoint_one_le n q hq.le
have hfactor : q ≤ q * lowerCellZ η n d q z x := by nlinarith
exact mul_le_mul_of_nonneg_right hfactor hj.le
theorem activatedFullLaw_mar
The activated law satisfies arrival MAR.
Formal statement
hypotheses
η :
ℝ
n :
ℕ
d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
conclusion
ArrivalMAR (activatedFullLaw η n d q z hd hb hq hslice hz)
Proof (Lean source)
-- @node: activatedFullLaw_mar
lemma activatedFullLaw_mar :
ArrivalMAR (activatedFullLaw η n d q z hd hb hq hslice hz) := by
classical
intro j y r
rcases j with ⟨a, x, s⟩
rw [activatedFullLaw_cellProb η n d q z hd hb hq hslice hz]
simp_rw [activatedFullLaw_real_event η n d q z hd hb hq hslice hz]
cases a <;> cases s <;> cases y <;> cases r <;>
simp [inCell, lowerFullRecord, FullRecord.S, FullRecord.Y, bernWeight, Finset.sum_add_distrib,
Finset.sum_ite_irrel, Finset.sum_ite_eq']
all_goals split_ifs <;> ring
all_goals simp only [true_or]
theorem activatedFullLaw_consistency
The activated law satisfies surrogate and outcome consistency.
Formal statement
hypotheses
η :
ℝ
n :
ℕ
d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
conclusion
conclusion 1
SurrogateConsistency (activatedFullLaw η n d q z hd hb hq hslice hz)
conclusion 2
OutcomeConsistency (activatedFullLaw η n d q z hd hb hq hslice hz)
Proof (Lean source)
-- @node: activatedFullLaw_consistency
lemma activatedFullLaw_consistency :
SurrogateConsistency (activatedFullLaw η n d q z hd hb hq hslice hz) ∧
OutcomeConsistency (activatedFullLaw η n d q z hd hb hq hslice hz) := by
constructor <;> filter_upwards [] with r <;> rfl
theorem activatedFullLaw_balanced
The activated law has balanced treatment assignment.
Formal statement
hypotheses
η :
ℝ
n :
ℕ
d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
conclusion
BalancedRandomization (activatedFullLaw η n d q z hd hb hq hslice hz)
Proof (Lean source)
-- @node: activatedFullLaw_balanced
lemma activatedFullLaw_balanced :
BalancedRandomization (activatedFullLaw η n d q z hd hb hq hslice hz) := by
classical
rw [BalancedRandomization, activatedFullLaw_real_event η n d q z hd hb hq hslice hz]
calc
_ = ∑ x : Fin d, baselineMass η n d q x / 2 := by
apply Finset.sum_congr rfl
intro x _
by_cases hr : x.val < rareCount η n d q <;>
simp [lowerFullRecord, bernWeight, hr] <;> ring
_ = 1 / 2 := by rw [← Finset.sum_div, baselineMass_sum η n d q hd]
theorem activatedFullLaw_arm_event_factor
For arm and latent events, the activated-law joint mass factors.
Formal statement
hypotheses
η :
ℝ
n :
ℕ
d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
conclusion
(activatedFullLaw η n d q z hd hb hq hslice hz).1.real {r | r.A ∈ s ∧ (r.X, r.S0, r.S1, r.Y0, r.Y1) ∈ t}
= (by classical exact ((if true ∈ s then (1 : ℝ) else 0) + (if false ∈ s then (1 : ℝ) else 0)) / 2) * (activatedFullLaw η n d q z hd hb hq hslice hz).1.real {r | (r.X, r.S0, r.S1, r.Y0, r.Y1) ∈ t}
Proof (Lean source)
-- @node: activatedFullLaw_arm_event_factor
lemma activatedFullLaw_arm_event_factor (s : Set Bool)
(t : Set (Fin d × Bool × Bool × Bool × Bool)) :
(activatedFullLaw η n d q z hd hb hq hslice hz).1.real
{r | r.A ∈ s ∧ (r.X, r.S0, r.S1, r.Y0, r.Y1) ∈ t} =
(by classical exact ((if true ∈ s then (1 : ℝ) else 0) +
(if false ∈ s then (1 : ℝ) else 0)) / 2) *
(activatedFullLaw η n d q z hd hb hq hslice hz).1.real
{r | (r.X, r.S0, r.S1, r.Y0, r.Y1) ∈ t} := by
classical
rw [activatedFullLaw_real_event η n d q z hd hb hq hslice hz,
activatedFullLaw_real_event η n d q z hd hb hq hslice hz, Finset.mul_sum]
apply Finset.sum_congr rfl
intro x _
by_cases hsT : true ∈ s <;> by_cases hsF : false ∈ s <;>
by_cases htT : (x, false, false, false, true) ∈ t <;>
by_cases htF : (x, false, false, false, false) ∈ t <;>
simp [lowerFullRecord, bernWeight, hsT, hsF, htT, htF]
all_goals split_ifs <;> ring
theorem activatedFullLaw_arm_mass
For an arm event, the activated-law arm mass has the displayed value.
Formal statement
hypotheses
η :
ℝ
n :
ℕ
d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
conclusion
(activatedFullLaw η n d q z hd hb hq hslice hz).1.real {r | r.A ∈ s}
Proof (Lean source)
-- @node: activatedFullLaw_arm_mass
lemma activatedFullLaw_arm_mass (s : Set Bool) :
(activatedFullLaw η n d q z hd hb hq hslice hz).1.real {r | r.A ∈ s} =
(by classical exact ((if true ∈ s then (1 : ℝ) else 0) +
(if false ∈ s then (1 : ℝ) else 0)) / 2) := by
let := (activatedFullLaw η n d q z hd hb hq hslice hz).2
simpa using activatedFullLaw_arm_event_factor η n d q z hd hb hq hslice hz s univ
theorem activatedFullLaw_randomized
The activated law has randomized treatment independent of latent variables.
Formal statement
hypotheses
η :
ℝ
n :
ℕ
d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
conclusion
RandomizedIndependence (activatedFullLaw η n d q z hd hb hq hslice hz)
Proof (Lean source)
-- @node: activatedFullLaw_randomized
lemma activatedFullLaw_randomized :
RandomizedIndependence (activatedFullLaw η n d q z hd hb hq hslice hz) := by
let := (activatedFullLaw η n d q z hd hb hq hslice hz).2
rw [RandomizedIndependence, indepFun_iff_measure_inter_preimage_eq_mul]
intro s t _ _
apply (ENNReal.toReal_eq_toReal_iff' (measure_ne_top _ _) (by finiteness)).mp
rw [ENNReal.toReal_mul]
change (activatedFullLaw η n d q z hd hb hq hslice hz).1.real
{r | r.A ∈ s ∧ (r.X, r.S0, r.S1, r.Y0, r.Y1) ∈ t} =
(activatedFullLaw η n d q z hd hb hq hslice hz).1.real {r | r.A ∈ s} *
(activatedFullLaw η n d q z hd hb hq hslice hz).1.real
{r | (r.X, r.S0, r.S1, r.Y0, r.Y1) ∈ t}
rw [activatedFullLaw_arm_event_factor η n d q z hd hb hq hslice hz,
activatedFullLaw_arm_mass η n d q z hd hb hq hslice hz]
theorem activatedFullLaw_model
Given a positive sample size, the activated law belongs to the rare-arrival model.
Formal statement
hypotheses
η :
ℝ
n :
ℕ
d :
ℕ
q :
ℝ
hd :
1 ≤ d
hq :
0 < q
hn :
1 ≤ n
conclusion
RareArrivalModelClass n d q (activatedFullLaw η n d q z hd hb hq hslice hz)
Proof (Lean source)
-- @node: activatedFullLaw_model
lemma activatedFullLaw_model (hn : 1 ≤ n) :
RareArrivalModelClass n d q
(activatedFullLaw η n d q z hd hb hq hslice hz) := by
have hq1 : q ≤ 1 := by
have hH := lowerEndpoint_one_le n q hq.le
have hactivation := lower_activation_probability_le_one n q hq.le hslice
nlinarith
exact ⟨sampleLaw_iid n _,
activatedFullLaw_randomized η n d q z hd hb hq hslice hz,
activatedFullLaw_balanced η n d q z hd hb hq hslice hz,
(activatedFullLaw_consistency η n d q z hd hb hq hslice hz).1,
(activatedFullLaw_consistency η n d q z hd hb hq hslice hz).2,
activatedFullLaw_mar η n d q z hd hb hq hslice hz,
activatedFullLaw_occupiedCellArrival η n d q z hd hb hq hslice hz,
hslice, hn, hd, hq, hq1⟩
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Lower.ActivationPolynomial 9 declarations The conditional rare-cell likelihood in equation (6) is polynomial, with degree at most the number of revealed activations.
The conditional rare-cell likelihood in equation (6) is polynomial, with degree at most the number of revealed activations. Matching moments therefore identifies its prior averages below the activation cutoff.
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
H :
ℝ
a flag arrival one :
def
given by
if !flag then if !arrival && !one then 1 else 0 else if one then if arrival && a then C H⁻¹ else 0 else if arrival then if a then C H⁻¹ * (X - 1) else C H⁻¹ * X else 1
- C H⁻¹ * X
theorem activationRareOutcomePolynomial_eval
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(activationRareOutcomePolynomial (lowerEndpoint n q) a flag arrival one).eval (z x)
= augmentedOutcomeWeight η n d q z x a flag arrival one
Proof (Lean source)
-- @node: activationRareOutcomePolynomial_eval
lemma activationRareOutcomePolynomial_eval (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin d → ℝ) (x : Fin d) (a flag arrival one : Bool)
(hx : x.val < rareCount η n d q) :
(activationRareOutcomePolynomial (lowerEndpoint n q) a flag arrival one).eval (z x) =
augmentedOutcomeWeight η n d q z x a flag arrival one := by
cases a <;> cases flag <;> cases arrival <;> cases one <;>
simp only [activationRareOutcomePolynomial, augmentedOutcomeWeight, lowerCellZ, hx,
Bool.not_false, Bool.not_true, Bool.false_eq_true,
Bool.and_false, Bool.and_true, Bool.false_and, Bool.true_and,
if_true, if_false, eval_zero, eval_one, eval_C, eval_mul, eval_sub, eval_X]
<;> simp [div_eq_mul_inv, mul_comm]
theorem activationRareOutcomePolynomial_natDegree
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
H :
ℝ
a flag arrival one :
conclusion
(activationRareOutcomePolynomial H a flag arrival one).natDegree ≤ if flag then 1 else 0
Proof (Lean source)
-- @node: activationRareOutcomePolynomial_natDegree
lemma activationRareOutcomePolynomial_natDegree (H : ℝ)
(a flag arrival one : Bool) :
(activationRareOutcomePolynomial H a flag arrival one).natDegree ≤
if flag then 1 else 0 := by
have hmul : (C H⁻¹ * X : Polynomial ℝ).natDegree ≤ 1 := by
simpa using (natDegree_mul_le (p := (C H⁻¹ : Polynomial ℝ)) (q := X))
have hsub : (X - 1 : Polynomial ℝ).natDegree ≤ 1 := by
simpa using (natDegree_sub_le (X : Polynomial ℝ) 1)
have hmulsub : (C H⁻¹ * (X - 1) : Polynomial ℝ).natDegree ≤ 1 := by
exact (natDegree_mul_le (p := C H⁻¹) (q := X - 1)).trans (by simpa using hsub)
have hlast : (1 - C H⁻¹ * X : Polynomial ℝ).natDegree ≤ 1 :=
(natDegree_sub_le _ _).trans (max_le (by simp) hmul)
cases a <;> cases flag <;> cases arrival <;> cases one <;>
simp [activationRareOutcomePolynomial] <;> assumption
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
given by
∏ i ∈ s, activationRareOutcomePolynomial H (a i) (flag i) (arrival i) (one i)
theorem activationRareCellPolynomial_natDegree
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(activationRareCellPolynomial s H a flag arrival one).natDegree
≤ ∑ i ∈ s, if flag i then 1 else 0
Proof (Lean source)
-- @node: activationRareCellPolynomial_natDegree
lemma activationRareCellPolynomial_natDegree {ι : Type*} (s : Finset ι)
(H : ℝ) (a flag arrival one : ι → Bool) :
(activationRareCellPolynomial s H a flag arrival one).natDegree ≤
∑ i ∈ s, if flag i then 1 else 0 := by
apply (natDegree_prod_le s _).trans
exact Finset.sum_le_sum (fun i _ =>
activationRareOutcomePolynomial_natDegree H (a i) (flag i) (arrival i) (one i))
theorem activationRareCellPolynomial_eval
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
ι :
Type*
s :
Finset ι
η :
ℝ
n d :
ℕ
q :
ℝ
z :
Fin d → ℝ
x :
Fin d
a flag arrival one :
ι → Bool
hx :
x.val < rareCount η n d q
conclusion
(activationRareCellPolynomial s (lowerEndpoint n q) a flag arrival one).eval (z x)
= ∏ i ∈ s, augmentedOutcomeWeight η n d q z x (a i) (flag i) (arrival i) (one i)
Proof (Lean source)
-- @node: activationRareCellPolynomial_eval
lemma activationRareCellPolynomial_eval {ι : Type*} (s : Finset ι)
(η : ℝ) (n d : ℕ) (q : ℝ) (z : Fin d → ℝ) (x : Fin d)
(a flag arrival one : ι → Bool) (hx : x.val < rareCount η n d q) :
(activationRareCellPolynomial s (lowerEndpoint n q) a flag arrival one).eval (z x) =
∏ i ∈ s, augmentedOutcomeWeight η n d q z x (a i) (flag i) (arrival i) (one i) := by
simp only [activationRareCellPolynomial, eval_prod]
exact Finset.prod_congr rfl (fun i _ =>
activationRareOutcomePolynomial_eval η n d q z x _ _ _ _ hx)
theorem finiteReciprocalPrior_polynomial_integral_eq Lemma finiteReciprocalPrior_polynomial_integral_eq in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
H :
ℝ
π₀ π₁ :
Measure ℝ
h₀ :
h₁ :
K :
ℕ
hm :
∀ v : ℕ
if
v ≤ K
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
p :
hp :
p.natDegree ≤ K
conclusion
(∫ z, p.eval z ∂π₀) = ∫ z, p.eval z ∂π₁
Proof (Lean source)
-- @node: finiteReciprocalPrior_polynomial_integral_eq
lemma finiteReciprocalPrior_polynomial_integral_eq {H : ℝ} {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior H π₀) (h₁ : FiniteReciprocalPrior H π₁)
(K : ℕ) (hm : ∀ v : ℕ, v ≤ K → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
(p : Polynomial ℝ) (hp : p.natDegree ≤ K) :
(∫ z, p.eval z ∂π₀) = ∫ z, p.eval z ∂π₁ := by
simp_rw [eval_eq_sum_range]
rw [integral_finsetSum _ (fun i _ => finiteReciprocalPrior_integrable h₀ _),
integral_finsetSum _ (fun i _ => finiteReciprocalPrior_integrable h₁ _)]
apply Finset.sum_congr rfl
intro i hi
rw [integral_const_mul, integral_const_mul, hm i]
exact (le_of_lt_succ (Finset.mem_range.mp hi)).trans hp
theorem activationRareCellPolynomial_integral_eq
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
ι :
Type*
s :
Finset ι
H :
ℝ
π₀ π₁ :
Measure ℝ
h₀ :
h₁ :
K :
ℕ
hm :
∀ v : ℕ
if
v ≤ K
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
a flag arrival one :
ι → Bool
hcount :
(∑ i ∈ s, if flag i then 1 else 0) ≤ K
conclusion
(∫ z, (activationRareCellPolynomial s H a flag arrival one).eval z ∂π₀)
= ∫ z, (activationRareCellPolynomial s H a flag arrival one).eval z ∂π₁
Proof (Lean source)
-- @node: activationRareCellPolynomial_integral_eq
lemma activationRareCellPolynomial_integral_eq {ι : Type*} (s : Finset ι)
{H : ℝ} {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior H π₀) (h₁ : FiniteReciprocalPrior H π₁)
(K : ℕ) (hm : ∀ v : ℕ, v ≤ K → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
(a flag arrival one : ι → Bool)
(hcount : (∑ i ∈ s, if flag i then 1 else 0) ≤ K) :
(∫ z, (activationRareCellPolynomial s H a flag arrival one).eval z ∂π₀) =
∫ z, (activationRareCellPolynomial s H a flag arrival one).eval z ∂π₁ := by
exact finiteReciprocalPrior_polynomial_integral_eq h₀ h₁ K hm _
((activationRareCellPolynomial_natDegree s H a flag arrival one).trans hcount)
theorem activationRareCellPolynomial_grid_integral_eq
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
ι :
Type*
s :
Finset ι
H :
ℝ
π₀ π₁ :
Measure ℝ
h₀ :
h₁ :
K :
ℕ
hm :
∀ v : ℕ
if
v ≤ K
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
M :
ℕ
hM :
1 ≤ M
j j' :
Fin (M + 1)
a flag arrival one :
ι → Bool
hcount :
(∑ i ∈ s, if flag i then 1 else 0) ≤ K
conclusion
(∫ z, (activationRareCellPolynomial s H a flag arrival one).eval z ∂intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))
= ∫ z, (activationRareCellPolynomial s H a flag arrival one).eval z ∂intervalInterpolatedPrior π₀ π₁ ((j' : ℝ) / M)
Proof (Lean source)
-- @node: activationRareCellPolynomial_grid_integral_eq
lemma activationRareCellPolynomial_grid_integral_eq {ι : Type*} (s : Finset ι)
{H : ℝ} {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior H π₀) (h₁ : FiniteReciprocalPrior H π₁)
(K : ℕ) (hm : ∀ v : ℕ, v ≤ K → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
(M : ℕ) (hM : 1 ≤ M) (j j' : Fin (M + 1))
(a flag arrival one : ι → Bool)
(hcount : (∑ i ∈ s, if flag i then 1 else 0) ≤ K) :
(∫ z, (activationRareCellPolynomial s H a flag arrival one).eval z
∂intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M)) =
∫ z, (activationRareCellPolynomial s H a flag arrival one).eval z
∂intervalInterpolatedPrior π₀ π₁ ((j' : ℝ) / M) := by
have hgrid := intervalInterpolatedPrior_grid h₀ h₁ K hm M hM
exact activationRareCellPolynomial_integral_eq s (hgrid j).1 (hgrid j').1 K
((hgrid j).2.1 j') a flag arrival one hcount
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Lower.ActivationPriorTarget 10 declarations Extend independent rare-cell priors to the full alphabet and identify the actual activated ATE with the reciprocal target in equations (7)--(12).
Extend independent rare-cell priors to the full alphabet and identify the actual activated ATE with the reciprocal target in equations (7)--(12).
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: lowerEndpoint_one_le
lemma lowerEndpoint_one_le (n : ℕ) (q : ℝ) (hq : 0 ≤ q) :
1 ≤ lowerEndpoint n q := by
have hN : 0 ≤ effectiveSize n q := by unfold effectiveSize; positivity
have hlog : 1 ≤ logScale n q := by
unfold logScale
have h := Real.log_le_log (Real.exp_pos (1 : ℝ))
(show exp 1 ≤ exp 1 + effectiveSize n q by linarith)
simpa using h
have hceil : (1 : ℝ) ≤ lowerDegree n q :=
hlog.trans (Nat.le_ceil (logScale n q))
unfold lowerEndpoint
nlinarith
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
activationExtendRare η n d q z x :
ℝ
given by
if h : x.val < rareCount η n d q then z ⟨x.val, h⟩ else lowerEndpoint n q
theorem activationExtendRare_eval
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: activationExtendRare_eval
lemma activationExtendRare_eval (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin (rareCount η n d q) → ℝ) (x : Fin (rareCount η n d q)) :
activationExtendRare η n d q z
(x.castLE ((min_le_left _ _).trans (sub_le d 1))) = z x := by
change (if h : x.val < rareCount η n d q then z ⟨x.val, h⟩ else
lowerEndpoint n q) = z x
rw [dif_pos x.isLt]
theorem activationExtendRare_support
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
activationExtendRare η n d q z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)
Proof (Lean source)
-- @node: activationExtendRare_support
lemma activationExtendRare_support (η : ℝ) (n d : ℕ) (q : ℝ) (hq : 0 ≤ q)
(z : Fin (rareCount η n d q) → ℝ)
(hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)) :
∀ x, activationExtendRare η n d q z x ∈ Icc (1 : ℝ) (lowerEndpoint n q) := by
intro x
unfold activationExtendRare
split_ifs with h
· exact hz ⟨x.val, h⟩
· exact ⟨lowerEndpoint_one_le n q hq, le_rfl⟩
theorem finiteReciprocalPrior_pi_ae_support Lemma finiteReciprocalPrior_pi_ae_support in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
∀ᵐ z ∂Measure.pi (fun _ : Fin J => π), ∀ x, z x ∈ Icc (1 : ℝ) H
Proof (Lean source)
-- @node: finiteReciprocalPrior_pi_ae_support
lemma finiteReciprocalPrior_pi_ae_support {H : ℝ} {π : Measure ℝ}
(hπ : FiniteReciprocalPrior H π) (J : ℕ) :
∀ᵐ z ∂Measure.pi (fun _ : Fin J => π), ∀ x, z x ∈ Icc (1 : ℝ) H := by
let := hπ.1
obtain ⟨s, hs, hsupport⟩ := hπ.2
have hae : ∀ᵐ z ∂π, z ∈ (s : Set ℝ) :=
(ae_mem_iff_measure_eq s.measurableSet.nullMeasurableSet).2 (by simpa using hs)
have hinterval : ∀ᵐ z ∂π, z ∈ Icc (1 : ℝ) H := by
filter_upwards [hae] with z hz
exact hsupport z hz
rw [ae_all_iff]
intro x
exact Measure.tendsto_eval_ae_ae.eventually hinterval
def activatedRarePriorATE
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
activatedRarePriorATE η n d q hd hb hq hslice z :
ℝ
given by
if hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q) then ate (activatedFullLaw η n d q (activationExtendRare η n d q z) hd hb hq hslice (activationExtendRare_support η n d q hq.le z hz)) else 0
theorem activatedRarePriorATE_ae_eq
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
conclusion
activatedRarePriorATE η n d q hd hb hq hslice
=ᵐ[ Measure.pi (fun _ : Fin (rareCount η n d q) => π)] intervalPriorTarget (rareCount η n d q) (rareMass η n q)
Proof (Lean source)
-- @node: activatedRarePriorATE_ae_eq
lemma activatedRarePriorATE_ae_eq (η : ℝ) (n d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hb : 0 < rareMass η n q) (hq : 0 < q)
(hslice : RareArrivalSlice n q) {π : Measure ℝ}
(hπ : FiniteReciprocalPrior (lowerEndpoint n q) π) :
activatedRarePriorATE η n d q hd hb hq hslice =ᵐ[
Measure.pi (fun _ : Fin (rareCount η n d q) => π)]
intervalPriorTarget (rareCount η n d q) (rareMass η n q) := by
filter_upwards [finiteReciprocalPrior_pi_ae_support hπ (rareCount η n d q)] with z hz
rw [activatedRarePriorATE, dif_pos hz, activatedFullLaw_ate]
congr 1
funext x
exact activationExtendRare_eval η n d q z x
theorem activatedRarePriorATE_integral
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
conclusion
(∫ z, activatedRarePriorATE η n d q hd hb hq hslice z ∂Measure.pi (fun _ : Fin (rareCount η n d q) => π))
= (rareCount η n d q : ℝ) * rareMass η n q * ∫ z, z⁻¹ ∂π
Proof (Lean source)
-- @node: activatedRarePriorATE_integral
lemma activatedRarePriorATE_integral (η : ℝ) (n d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hb : 0 < rareMass η n q) (hq : 0 < q)
(hslice : RareArrivalSlice n q) {π : Measure ℝ}
(hπ : FiniteReciprocalPrior (lowerEndpoint n q) π) :
(∫ z, activatedRarePriorATE η n d q hd hb hq hslice z
∂Measure.pi (fun _ : Fin (rareCount η n d q) => π)) =
(rareCount η n d q : ℝ) * rareMass η n q * ∫ z, z⁻¹ ∂π := by
rw [integral_congr_ae (activatedRarePriorATE_ae_eq η n d q hd hb hq hslice hπ)]
exact intervalPriorTarget_integral hπ _ _
theorem activatedRarePriorATE_variance_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
conclusion
variance (activatedRarePriorATE η n d q hd hb hq hslice) (Measure.pi (fun _ : Fin (rareCount η n d q) => π))
≤ (rareCount η n d q : ℝ) * (rareMass η n q) ^ 2 / 4
Proof (Lean source)
-- @node: activatedRarePriorATE_variance_le
lemma activatedRarePriorATE_variance_le (η : ℝ) (n d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hb : 0 < rareMass η n q) (hq : 0 < q)
(hslice : RareArrivalSlice n q) {π : Measure ℝ}
(hπ : FiniteReciprocalPrior (lowerEndpoint n q) π) :
variance (activatedRarePriorATE η n d q hd hb hq hslice)
(Measure.pi (fun _ : Fin (rareCount η n d q) => π)) ≤
(rareCount η n d q : ℝ) * (rareMass η n q) ^ 2 / 4 := by
rw [variance_congr (activatedRarePriorATE_ae_eq η n d q hd hb hq hslice hπ)]
exact intervalPriorTarget_variance_le hπ _ _
theorem activatedRarePriorATE_grid_concentration
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d M :
ℕ
q u :
ℝ
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π₀ π₁ :
Measure ℝ
h₀ :
FiniteReciprocalPrior (lowerEndpoint n q) π₀
h₁ :
FiniteReciprocalPrior (lowerEndpoint n q) π₁
hu :
0 < u
hM :
1 ≤ M
hgap :
1 / 12 ≤ |(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀|
hJ :
18432 * (M : ℝ) ^ 2 / u ≤ (rareCount η n d q : ℝ)
j :
Fin (M + 1)
conclusion
(Measure.pi (fun _ : Fin (rareCount η n d q) => intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))).real {z | ((rareCount η n d q : ℝ) * rareMass η n q * |(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀|) / (8 * M) < |activatedRarePriorATE η n d q hd hb hq hslice z - ((rareCount η n d q : ℝ) * rareMass η n q * (∫ z, z⁻¹ ∂π₀) + (j : ℝ) * ((rareCount η n d q : ℝ) * rareMass η n q * ((∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀) / M))|}
≤ u / 8
Proof (Lean source)
-- @node: activatedRarePriorATE_grid_concentration
lemma activatedRarePriorATE_grid_concentration (η : ℝ) (n d M : ℕ) (q u : ℝ)
(hd : 1 ≤ d) (hb : 0 < rareMass η n q) (hq : 0 < q)
(hslice : RareArrivalSlice n q) {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior (lowerEndpoint n q) π₀)
(h₁ : FiniteReciprocalPrior (lowerEndpoint n q) π₁)
(hu : 0 < u) (hM : 1 ≤ M)
(hgap : 1 / 12 ≤ |(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀|)
(hJ : 18432 * (M : ℝ) ^ 2 / u ≤ (rareCount η n d q : ℝ)) :
∀ j : Fin (M + 1),
(Measure.pi (fun _ : Fin (rareCount η n d q) =>
intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))).real
{z | ((rareCount η n d q : ℝ) * rareMass η n q *
|(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀|) / (8 * M) <
|activatedRarePriorATE η n d q hd hb hq hslice z -
((rareCount η n d q : ℝ) * rareMass η n q * (∫ z, z⁻¹ ∂π₀) +
(j : ℝ) * ((rareCount η n d q : ℝ) * rareMass η n q *
((∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀) / M))|} ≤ u / 8 := by
intro j
have hMr : 0 < (M : ℝ) := by exact_mod_cast (by omega : 0 < M)
have ht : (j : ℝ) / M ∈ Icc (0 : ℝ) 1 := by
refine ⟨by positivity, (div_le_one hMr).mpr ?_⟩
exact_mod_cast (by omega : j.val ≤ M)
have hae := activatedRarePriorATE_ae_eq η n d q hd hb hq hslice
(finiteReciprocalPrior_interpolate h₀ h₁ ht)
have hevent :
{z | ((rareCount η n d q : ℝ) * rareMass η n q *
|(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀|) / (8 * M) <
|activatedRarePriorATE η n d q hd hb hq hslice z -
((rareCount η n d q : ℝ) * rareMass η n q * (∫ z, z⁻¹ ∂π₀) +
(j : ℝ) * ((rareCount η n d q : ℝ) * rareMass η n q *
((∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀) / M))|} =ᵐ[
Measure.pi (fun _ : Fin (rareCount η n d q) =>
intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))]
{z | ((rareCount η n d q : ℝ) * rareMass η n q *
|(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀|) / (8 * M) <
|intervalPriorTarget (rareCount η n d q) (rareMass η n q) z -
((rareCount η n d q : ℝ) * rareMass η n q * (∫ z, z⁻¹ ∂π₀) +
(j : ℝ) * ((rareCount η n d q : ℝ) * rareMass η n q *
((∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀) / M))|} := by
filter_upwards [hae] with z hz
change (_ < |activatedRarePriorATE η n d q hd hb hq hslice z - _|) =
(_ < |intervalPriorTarget (rareCount η n d q) (rareMass η n q) z - _|)
rw [hz]
unfold Measure.real
rw [measure_congr hevent]
exact intervalInterpolatedPrior_grid_concentration h₀ h₁ (rareCount η n d q)
M (rareMass η n q) u hb hu hM hgap hJ j
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Lower.ActivationProbability 7 declarations Probability certificates and fixed-sample projection for the activated experiment used in equation (6) of the connected-interval lower bound.
Probability certificates and fixed-sample projection for the activated experiment used in equation (6) of the connected-interval lower bound.
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
= 1
Proof (Lean source)
-- @node: augmentedOutcomeWeight_sum
lemma augmentedOutcomeWeight_sum (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin d → ℝ) (x : Fin d) (a flag : Bool)
(hz : 1 ≤ z x ∧ z x ≤ lowerEndpoint n q) :
(∑ arrival : Bool, ∑ one : Bool,
augmentedOutcomeWeight η n d q z x a flag arrival one) = 1 := by
have hH : lowerEndpoint n q ≠ 0 := by linarith [hz.1, hz.2]
simp only [Fintype.sum_bool]
by_cases hr : x.val < rareCount η n d q
· cases a <;> cases flag <;>
simp [augmentedOutcomeWeight, lowerCellZ, hr]
all_goals field_simp; ring
· cases a <;> cases flag <;>
simp [augmentedOutcomeWeight, lowerCellZ, hr, hH]
theorem augmentedOneRecord_univ
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
z :
Fin d → ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
hz :
∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)
conclusion
augmentedOneRecord η n d q z hz univ = 1
Proof (Lean source)
-- @node: augmentedOneRecord_univ
lemma augmentedOneRecord_univ (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin d → ℝ) (hn : 1 ≤ n) (hd : 1 ≤ d)
(hb : 0 < rareMass η n q) (hq : 0 < q)
(hslice : RareArrivalSlice n q)
(hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)) :
augmentedOneRecord η n d q z hz univ = 1 := by
have hobs : Measurable (obs : FullRecord d → ObsRecord d) := by fun_prop
have hp := activated_projection η n d q z hn hd hq hslice hz
have hm := congrArg (fun μ : Measure (ObsRecord d) => μ univ) hp
simpa only [Measure.map_apply measurable_snd MeasurableSet.univ,
Measure.map_apply hobs MeasurableSet.univ,
Set.preimage_univ, activatedLaw_univ η n d q z hd hb hq hslice hz] using hm
theorem augmentedOneRecord_isProbabilityMeasure
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
z :
Fin d → ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
hz :
∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)
conclusion
IsProbabilityMeasure (augmentedOneRecord η n d q z hz)
Proof (Lean source)
-- @node: augmentedOneRecord_isProbabilityMeasure
lemma augmentedOneRecord_isProbabilityMeasure (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin d → ℝ) (hn : 1 ≤ n) (hd : 1 ≤ d)
(hb : 0 < rareMass η n q) (hq : 0 < q)
(hslice : RareArrivalSlice n q)
(hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)) :
IsProbabilityMeasure (augmentedOneRecord η n d q z hz) :=
⟨augmentedOneRecord_univ η n d q z hn hd hb hq hslice hz⟩
theorem activated_sample_projection
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
z :
Fin d → ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
hz :
∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)
conclusion
= sampleLaw n (activatedFullLaw η n d q z hd hb hq hslice hz)
Proof (Lean source)
-- @node: activated_sample_projection
lemma activated_sample_projection (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin d → ℝ) (hn : 1 ≤ n) (hd : 1 ≤ d)
(hb : 0 < rareMass η n q) (hq : 0 < q)
(hslice : RareArrivalSlice n q)
(hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)) :
(Measure.pi (fun _ : Fin n => augmentedOneRecord η n d q z hz)).map
(fun s i => (s i).2) =
sampleLaw n (activatedFullLaw η n d q z hd hb hq hslice hz) := by
let := augmentedOneRecord_isProbabilityMeasure η n d q z hn hd hb hq hslice hz
rw [Measure.pi_map_pi (fun _ => measurable_snd.aemeasurable)]
simp_rw [activated_projection η n d q z hn hd hq hslice hz]
rfl
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∑ arrival : Bool, ∑ one : Bool, ofReal (baselineMass η n d q x / 2 * bernWeight (q * lowerEndpoint n q) flag * augmentedOutcomeWeight η n d q z x a flag arrival one))
Proof (Lean source)
-- @node: augmentedOneRecord_marginal_weight
lemma augmentedOneRecord_marginal_weight (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin d → ℝ) (x : Fin d) (a flag : Bool)
(hz : 1 ≤ z x ∧ z x ≤ lowerEndpoint n q) :
(∑ arrival : Bool, ∑ one : Bool,
ofReal (baselineMass η n d q x / 2 *
bernWeight (q * lowerEndpoint n q) flag *
augmentedOutcomeWeight η n d q z x a flag arrival one)) =
ofReal (baselineMass η n d q x / 2 *
bernWeight (q * lowerEndpoint n q) flag) := by
let b := baselineMass η n d q x / 2 * bernWeight (q * lowerEndpoint n q) flag
change (∑ arrival : Bool, ∑ one : Bool,
ofReal (b * augmentedOutcomeWeight η n d q z x a flag arrival one)) = _
simp_rw [ofReal_signed_mul_bool_sum b _ (fun one =>
augmentedOutcomeWeight_nonneg η n d q z x a flag _ one hz)]
rw [ofReal_signed_mul_bool_sum b _ (fun arrival =>
Finset.sum_nonneg (fun one _ =>
augmentedOutcomeWeight_nonneg η n d q z x a flag arrival one hz))]
rw [augmentedOutcomeWeight_sum η n d q z x a flag hz, mul_one]
theorem augmentedOneRecord_membership_activation_map Lemma augmentedOneRecord_membership_activation_map in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(augmentedOneRecord η n d q z hz).map (fun r => (r.2.X, r.2.A, r.1))
= ∑ x : Fin d, ∑ a : Bool, ∑ flag : Bool, ofReal (baselineMass η n d q x / 2 * bernWeight (q * lowerEndpoint n q) flag) • Measure.dirac (x, a, flag)
Proof (Lean source)
-- @node: augmentedOneRecord_membership_activation_map
lemma augmentedOneRecord_membership_activation_map (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin d → ℝ)
(hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)) :
(augmentedOneRecord η n d q z hz).map
(fun r => (r.2.X, r.2.A, r.1)) =
∑ x : Fin d, ∑ a : Bool, ∑ flag : Bool,
ofReal (baselineMass η n d q x / 2 *
bernWeight (q * lowerEndpoint n q) flag) • Measure.dirac (x, a, flag) := by
classical
have hf : Measurable (fun r : Bool × ObsRecord d => (r.2.X, r.2.A, r.1)) := by
fun_prop
unfold augmentedOneRecord
simp_rw [Measure.map_finset_sum' hf.aemeasurable, Measure.map_smul,
Measure.map_dirac]
apply Finset.sum_congr rfl
intro x _
apply Finset.sum_congr rfl
intro a _
apply Finset.sum_congr rfl
intro flag _
simp_rw [← Finset.sum_smul]
rw [augmentedOneRecord_marginal_weight η n d q z x a flag (hz x)]
theorem activated_sample_membership_activation_map
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
z :
Fin d → ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
hz :
∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)
conclusion
(Measure.pi (fun _ : Fin n => augmentedOneRecord η n d q z hz)).map (fun s i => ((s i).2.X, (s i).2.A, (s i).1))
= Measure.pi (fun _ : Fin n => ∑ x : Fin d, ∑ a : Bool, ∑ flag : Bool, ofReal (baselineMass η n d q x / 2 * bernWeight (q * lowerEndpoint n q) flag) • Measure.dirac (x, a, flag))
Proof (Lean source)
-- @node: activated_sample_membership_activation_map
lemma activated_sample_membership_activation_map (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin d → ℝ) (hn : 1 ≤ n) (hd : 1 ≤ d)
(hb : 0 < rareMass η n q) (hq : 0 < q)
(hslice : RareArrivalSlice n q)
(hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)) :
(Measure.pi (fun _ : Fin n => augmentedOneRecord η n d q z hz)).map
(fun s i => ((s i).2.X, (s i).2.A, (s i).1)) =
Measure.pi (fun _ : Fin n =>
∑ x : Fin d, ∑ a : Bool, ∑ flag : Bool,
ofReal (baselineMass η n d q x / 2 *
bernWeight (q * lowerEndpoint n q) flag) • Measure.dirac (x, a, flag)) := by
let := augmentedOneRecord_isProbabilityMeasure η n d q z hn hd hb hq hslice hz
have hf : Measurable (fun r : Bool × ObsRecord d => (r.2.X, r.2.A, r.1)) := by
fun_prop
rw [Measure.pi_map_pi (fun _ => hf.aemeasurable)]
simp_rw [augmentedOneRecord_membership_activation_map η n d q z hz]
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Lower.ActivationProduct 6 declarations Joint rare-cell likelihood cancellation under independent moment-matched priors, the exact-cancellation step preceding the tail bound in equation (6).
Joint rare-cell likelihood cancellation under independent moment-matched priors, the exact-cancellation step preceding the tail bound in equation (6).
theorem finiteReciprocalPrior_product_polynomial_integral_eq Lemma finiteReciprocalPrior_product_polynomial_integral_eq in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
J :
ℕ
H :
ℝ
π₀ π₁ :
Measure ℝ
h₀ :
h₁ :
K :
ℕ
hm :
∀ v : ℕ
if
v ≤ K
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
p :
Fin J → Polynomial ℝ
hp :
∀ x, (p x).natDegree ≤ K
Proof (Lean source)
-- @node: finiteReciprocalPrior_product_polynomial_integral_eq
lemma finiteReciprocalPrior_product_polynomial_integral_eq
{J : ℕ} {H : ℝ} {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior H π₀) (h₁ : FiniteReciprocalPrior H π₁)
(K : ℕ) (hm : ∀ v : ℕ, v ≤ K → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
(p : Fin J → Polynomial ℝ) (hp : ∀ x, (p x).natDegree ≤ K) :
(∫ z, ∏ x, (p x).eval (z x) ∂Measure.pi (fun _ : Fin J => π₀)) =
∫ z, ∏ x, (p x).eval (z x) ∂Measure.pi (fun _ : Fin J => π₁) := by
let := h₀.1
let := h₁.1
rw [integral_fintype_prod_eq_prod (fun x t => (p x).eval t),
integral_fintype_prod_eq_prod (fun x t => (p x).eval t)]
exact Finset.prod_congr rfl (fun x _ =>
finiteReciprocalPrior_polynomial_integral_eq h₀ h₁ K hm (p x) (hp x))
theorem activationRareRecords_factorization
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∏ i ∈ s, (activationRareOutcomePolynomial H (a i) (flag i) (arrival i) (one i)).eval (z (cell i)))
= ∏ x : Fin J, (activationRareCellPolynomial (s.filter (fun i => cell i = x)) H a flag arrival one).eval (z x)
Proof (Lean source)
-- @node: activationRareRecords_factorization
lemma activationRareRecords_factorization {ι : Type*} (s : Finset ι)
{J : ℕ} (cell : ι → Fin J) (H : ℝ)
(a flag arrival one : ι → Bool) (z : Fin J → ℝ) :
(∏ i ∈ s, (activationRareOutcomePolynomial H (a i) (flag i)
(arrival i) (one i)).eval (z (cell i))) =
∏ x : Fin J, (activationRareCellPolynomial (s.filter (fun i => cell i = x))
H a flag arrival one).eval (z x) := by
classical
rw [← Finset.prod_fiberwise s cell]
apply Finset.prod_congr rfl
intro x _
simp only [activationRareCellPolynomial, eval_prod]
apply Finset.prod_congr rfl
intro i hi
rw [(Finset.mem_filter.mp hi).2]
theorem activationRareRecords_integral_eq
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
ι :
Type*
s :
Finset ι
J :
ℕ
cell :
ι → Fin J
H :
ℝ
π₀ π₁ :
Measure ℝ
h₀ :
h₁ :
K :
ℕ
hm :
∀ v : ℕ
if
v ≤ K
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
a flag arrival one :
ι → Bool
hcount :
∀ x, (∑ i ∈ s.filter (fun i => cell i = x), if flag i then 1 else 0) ≤ K
conclusion
(∫ z, ∏ i ∈ s, (activationRareOutcomePolynomial H (a i) (flag i) (arrival i) (one i)).eval (z (cell i)) ∂Measure.pi (fun _ : Fin J => π₀))
= ∫ z, ∏ i ∈ s, (activationRareOutcomePolynomial H (a i) (flag i) (arrival i) (one i)).eval (z (cell i)) ∂Measure.pi (fun _ : Fin J => π₁)
Proof (Lean source)
-- @node: activationRareRecords_integral_eq
lemma activationRareRecords_integral_eq {ι : Type*} (s : Finset ι)
{J : ℕ} (cell : ι → Fin J) {H : ℝ} {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior H π₀) (h₁ : FiniteReciprocalPrior H π₁)
(K : ℕ) (hm : ∀ v : ℕ, v ≤ K → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
(a flag arrival one : ι → Bool)
(hcount : ∀ x, (∑ i ∈ s.filter (fun i => cell i = x), if flag i then 1 else 0) ≤ K) :
(∫ z, ∏ i ∈ s, (activationRareOutcomePolynomial H (a i) (flag i)
(arrival i) (one i)).eval (z (cell i)) ∂Measure.pi (fun _ : Fin J => π₀)) =
∫ z, ∏ i ∈ s, (activationRareOutcomePolynomial H (a i) (flag i)
(arrival i) (one i)).eval (z (cell i)) ∂Measure.pi (fun _ : Fin J => π₁) := by
classical
simp_rw [activationRareRecords_factorization]
apply finiteReciprocalPrior_product_polynomial_integral_eq h₀ h₁ K hm
intro x
exact (activationRareCellPolynomial_natDegree _ H a flag arrival one).trans (hcount x)
theorem activationRareRecords_grid_integral_eq
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
ι :
Type*
s :
Finset ι
J :
ℕ
cell :
ι → Fin J
H :
ℝ
π₀ π₁ :
Measure ℝ
h₀ :
h₁ :
K :
ℕ
hm :
∀ v : ℕ
if
v ≤ K
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
M :
ℕ
hM :
1 ≤ M
j j' :
Fin (M + 1)
a flag arrival one :
ι → Bool
hcount :
∀ x, (∑ i ∈ s.filter (fun i => cell i = x), if flag i then 1 else 0) ≤ K
conclusion
(∫ z, ∏ i ∈ s, (activationRareOutcomePolynomial H (a i) (flag i) (arrival i) (one i)).eval (z (cell i)) ∂Measure.pi (fun _ : Fin J => intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M)))
= ∫ z, ∏ i ∈ s, (activationRareOutcomePolynomial H (a i) (flag i) (arrival i) (one i)).eval (z (cell i)) ∂Measure.pi (fun _ : Fin J => intervalInterpolatedPrior π₀ π₁ ((j' : ℝ) / M))
Proof (Lean source)
-- @node: activationRareRecords_grid_integral_eq
lemma activationRareRecords_grid_integral_eq {ι : Type*} (s : Finset ι)
{J : ℕ} (cell : ι → Fin J) {H : ℝ} {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior H π₀) (h₁ : FiniteReciprocalPrior H π₁)
(K : ℕ) (hm : ∀ v : ℕ, v ≤ K → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
(M : ℕ) (hM : 1 ≤ M) (j j' : Fin (M + 1))
(a flag arrival one : ι → Bool)
(hcount : ∀ x, (∑ i ∈ s.filter (fun i => cell i = x), if flag i then 1 else 0) ≤ K) :
(∫ z, ∏ i ∈ s, (activationRareOutcomePolynomial H (a i) (flag i)
(arrival i) (one i)).eval (z (cell i))
∂Measure.pi (fun _ : Fin J => intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))) =
∫ z, ∏ i ∈ s, (activationRareOutcomePolynomial H (a i) (flag i)
(arrival i) (one i)).eval (z (cell i))
∂Measure.pi (fun _ : Fin J => intervalInterpolatedPrior π₀ π₁ ((j' : ℝ) / M)) := by
have hgrid := intervalInterpolatedPrior_grid h₀ h₁ K hm M hM
exact activationRareRecords_integral_eq s cell (hgrid j).1 (hgrid j').1 K
((hgrid j).2.1 j') a flag arrival one hcount
theorem activationRareRecords_augmented_eq
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∏ i ∈ s, (activationRareOutcomePolynomial (lowerEndpoint n q) (a i) (flag i) (arrival i) (one i)).eval (z (cell i)))
= ∏ i ∈ s, augmentedOutcomeWeight η n d q (activationExtendRare η n d q z) ((cell i).castLE ((min_le_left _ _).trans (sub_le d 1))) (a i) (flag i) (arrival i) (one i)
Proof (Lean source)
-- @node: activationRareRecords_augmented_eq
lemma activationRareRecords_augmented_eq {ι : Type*} (s : Finset ι)
(η : ℝ) (n d : ℕ) (q : ℝ)
(cell : ι → Fin (rareCount η n d q))
(a flag arrival one : ι → Bool) (z : Fin (rareCount η n d q) → ℝ) :
(∏ i ∈ s, (activationRareOutcomePolynomial (lowerEndpoint n q)
(a i) (flag i) (arrival i) (one i)).eval (z (cell i))) =
∏ i ∈ s, augmentedOutcomeWeight η n d q (activationExtendRare η n d q z)
((cell i).castLE ((min_le_left _ _).trans (sub_le d 1)))
(a i) (flag i) (arrival i) (one i) := by
apply Finset.prod_congr rfl
intro i _
have heval := activationRareOutcomePolynomial_eval η n d q
(activationExtendRare η n d q z)
((cell i).castLE ((min_le_left _ _).trans (sub_le d 1)))
(a i) (flag i) (arrival i) (one i) (cell i).isLt
rw [activationExtendRare_eval] at heval
exact heval
theorem activationRareRecords_augmented_grid_integral_eq
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
ι :
Type*
s :
Finset ι
η :
ℝ
n d :
ℕ
q :
ℝ
π₀ π₁ :
Measure ℝ
h₀ :
FiniteReciprocalPrior (lowerEndpoint n q) π₀
h₁ :
FiniteReciprocalPrior (lowerEndpoint n q) π₁
K :
ℕ
hm :
∀ v : ℕ
if
v ≤ K
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
M :
ℕ
hM :
1 ≤ M
j j' :
Fin (M + 1)
a flag arrival one :
ι → Bool
hcount :
∀ x, (∑ i ∈ s.filter (fun i => cell i = x), if flag i then 1 else 0) ≤ K
conclusion
(∫ z, ∏ i ∈ s, augmentedOutcomeWeight η n d q (activationExtendRare η n d q z) ((cell i).castLE ((min_le_left _ _).trans (sub_le d 1))) (a i) (flag i) (arrival i) (one i) ∂Measure.pi (fun _ : Fin (rareCount η n d q) => intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M)))
= ∫ z, ∏ i ∈ s, augmentedOutcomeWeight η n d q (activationExtendRare η n d q z) ((cell i).castLE ((min_le_left _ _).trans (sub_le d 1))) (a i) (flag i) (arrival i) (one i) ∂Measure.pi (fun _ : Fin (rareCount η n d q) => intervalInterpolatedPrior π₀ π₁ ((j' : ℝ) / M))
Proof (Lean source)
-- @node: activationRareRecords_augmented_grid_integral_eq
lemma activationRareRecords_augmented_grid_integral_eq {ι : Type*} (s : Finset ι)
(η : ℝ) (n d : ℕ) (q : ℝ)
(cell : ι → Fin (rareCount η n d q)) {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior (lowerEndpoint n q) π₀)
(h₁ : FiniteReciprocalPrior (lowerEndpoint n q) π₁)
(K : ℕ) (hm : ∀ v : ℕ, v ≤ K → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
(M : ℕ) (hM : 1 ≤ M) (j j' : Fin (M + 1))
(a flag arrival one : ι → Bool)
(hcount : ∀ x, (∑ i ∈ s.filter (fun i => cell i = x), if flag i then 1 else 0) ≤ K) :
(∫ z, ∏ i ∈ s, augmentedOutcomeWeight η n d q (activationExtendRare η n d q z)
((cell i).castLE ((min_le_left _ _).trans (sub_le d 1)))
(a i) (flag i) (arrival i) (one i)
∂Measure.pi (fun _ : Fin (rareCount η n d q) =>
intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))) =
∫ z, ∏ i ∈ s, augmentedOutcomeWeight η n d q (activationExtendRare η n d q z)
((cell i).castLE ((min_le_left _ _).trans (sub_le d 1)))
(a i) (flag i) (arrival i) (one i)
∂Measure.pi (fun _ : Fin (rareCount η n d q) =>
intervalInterpolatedPrior π₀ π₁ ((j' : ℝ) / M)) := by
simp_rw [← activationRareRecords_augmented_eq]
exact activationRareRecords_grid_integral_eq s cell h₀ h₁ K hm M hM j j'
a flag arrival one hcount
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Lower.ActivationRiskTransfer 7 declarations Prior averaging and centered-risk transfer for randomized estimators in equations (14)--(16) of the activated point-risk lower bound.
Prior averaging and centered-risk transfer for randomized estimators in equations (14)--(16) of the activated point-risk lower bound.
theorem activated_deterministic_centered_risk_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s, (f s - θ) ^ 2 ∂sampleLaw n P) ≤ 2 * deterministicRisk f P + 2 * (ate P - θ) ^ 2
Proof (Lean source)
-- @node: activated_deterministic_centered_risk_le
lemma activated_deterministic_centered_risk_le {n d : ℕ}
(f : (Fin n → ObsRecord d) → ℝ) (P : FullLaw d) (θ : ℝ) :
(∫ s, (f s - θ) ^ 2 ∂sampleLaw n P) ≤
2 * deterministicRisk f P + 2 * (ate P - θ) ^ 2 := by
let := P.2
let := Measure.isProbabilityMeasure_map (show Measurable obs by fun_prop).aemeasurable
(μ := P.1)
letI : IsProbabilityMeasure (sampleLaw n P) := by unfold sampleLaw; infer_instance
have h := activated_centered_risk_le (sampleLaw n P) f (fun _ => ate P) θ
Integrable.of_finite Integrable.of_finite Integrable.of_finite
simpa [deterministicRisk] using h
theorem activated_full_prior_target_second_moment_le Lemma activated_full_prior_target_second_moment_le in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
-- @node: activated_full_prior_target_second_moment_le
lemma activated_full_prior_target_second_moment_le
(η : ℝ) (n d : ℕ) (q : ℝ) {π : Measure ℝ}
(hπ : FiniteReciprocalPrior (lowerEndpoint n q) π) :
(∫ z, (rareMass η n q * (∑ x : Fin d,
if x.val < rareCount η n d q then (z x)⁻¹ else 0) -
(rareCount η n d q : ℝ) * rareMass η n q * (∫ z, z⁻¹ ∂π)) ^ 2
∂Measure.pi (fun _ : Fin d => π)) ≤
(rareCount η n d q : ℝ) * (rareMass η n q) ^ 2 / 4 := by
let := hπ.1
let J := rareCount η n d q
have hJ : J ≤ d := (min_le_left _ _).trans (sub_le d 1)
let restrict : (Fin d → ℝ) → (Fin J → ℝ) := fun z x => z (x.castLE hJ)
have hr : Measurable restrict := by fun_prop
have hmap := intervalPrior_initial_coordinates_map d J hJ (π := π)
have hm : Measurable (intervalPriorTarget J (rareMass η n q)) := by
unfold intervalPriorTarget
apply Measurable.const_mul
apply Finset.measurable_sum
intro i hi
fun_prop
have hv := intervalPriorTarget_variance_le hπ J (rareMass η n q)
rw [variance_eq_integral hm.aemeasurable,
intervalPriorTarget_integral hπ] at hv
rw [← hmap, integral_map hr.aemeasurable ((hm.sub_const _).pow_const 2).aestronglyMeasurable] at hv
simpa only [intervalPriorTarget, activation_sum_initial_cells d J hJ, restrict, J]
using hv
theorem activated_prior_squaredRisk_le_worst
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
T :
Estimator n d
conclusion
(∫ z, (if hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q) then squaredRisk T (activatedFullLaw η n d q z hd hb hq hslice hz) else 0) ∂Measure.pi (fun _ : Fin d => π))
≤ worstCaseRiskReal (fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) => squaredRisk T P.1) ()
Proof (Lean source)
-- @node: activated_prior_squaredRisk_le_worst
lemma activated_prior_squaredRisk_le_worst (η : ℝ) (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π : Measure ℝ}
(hπ : FiniteReciprocalPrior (lowerEndpoint n q) π) (T : Estimator n d) :
(∫ z, (if hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q) then
squaredRisk T (activatedFullLaw η n d q z hd hb hq hslice hz) else 0)
∂Measure.pi (fun _ : Fin d => π)) ≤
worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) =>
squaredRisk T P.1) () := by
let := hπ.1
have hbdd : BddAbove (range (fun P : {P : FullLaw d // RareArrivalModelClass n d q P} =>
squaredRisk T P.1)) := by
refine ⟨4, ?_⟩
rintro _ ⟨P, rfl⟩
exact (unrestricted_squaredRisk_bounds T P.1).2
have hle : ∀ᵐ z ∂Measure.pi (fun _ : Fin d => π),
(if hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q) then
squaredRisk T (activatedFullLaw η n d q z hd hb hq hslice hz) else 0) ≤
worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) =>
squaredRisk T P.1) () := by
filter_upwards [finiteReciprocalPrior_pi_ae_support hπ d] with z hz
rw [dif_pos hz]
exact le_worstCaseRisk
(risk := fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) =>
squaredRisk T P.1) (e := ()) hbdd
⟨_, activatedFullLaw_model η n d q z hd hb hq hslice hz hn⟩
simpa using integral_mono_ae (finiteReciprocalPrior_pi_integrable hπ d _)
(integrable_const _) hle
theorem activatedAugmentedMixture_centered_risk_le_worst Lemma activatedAugmentedMixture_centered_risk_le_worst in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
T :
Estimator n d
conclusion
(∫ s, (kernelMean T.toBoundedKernel.1 clip s - (rareCount η n d q : ℝ) * rareMass η n q * (∫ z, z⁻¹ ∂π)) ^ 2 ∂((activatedAugmentedMixture η n d q π).map (fun s i => (s i).2)))
≤ 2 * worstCaseRiskReal (fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) => squaredRisk T P.1) ()
+ (rareCount η n d q : ℝ) * (rareMass η n q) ^ 2 / 2
Proof (Lean source)
lemma activatedAugmentedMixture_centered_risk_le_worst
(η : ℝ) (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π : Measure ℝ}
(hπ : FiniteReciprocalPrior (lowerEndpoint n q) π) (T : Estimator n d) :
(∫ s, (kernelMean T.toBoundedKernel.1 clip s -
(rareCount η n d q : ℝ) * rareMass η n q * (∫ z, z⁻¹ ∂π)) ^ 2
∂((activatedAugmentedMixture η n d q π).map (fun s i => (s i).2))) ≤
2 * worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) =>
squaredRisk T P.1) () +
(rareCount η n d q : ℝ) * (rareMass η n q) ^ 2 / 2 := by
classical
let := hπ.1
let f := kernelMean T.toBoundedKernel.1 clip
let θ := (rareCount η n d q : ℝ) * rareMass η n q * (∫ z, z⁻¹ ∂π)
let r := fun z : Fin d → ℝ => if hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)
then squaredRisk T (activatedFullLaw η n d q z hd hb hq hslice hz) else 0
let v := fun z : Fin d → ℝ => (rareMass η n q * (∑ x : Fin d,
if x.val < rareCount η n d q then (z x)⁻¹ else 0) - θ) ^ 2
rw [activatedAugmentedMixture_projected_integral η n d q hn hd hb hq hslice hπ]
have hpoint : ∀ᵐ z ∂Measure.pi (fun _ : Fin d => π),
(if hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q) then
∫ s, (f s - θ) ^ 2 ∂sampleLaw n
(activatedFullLaw η n d q z hd hb hq hslice hz) else 0) ≤ 2 * r z + 2 * v z := by
filter_upwards [finiteReciprocalPrior_pi_ae_support hπ d] with z hz
simp only [r, dif_pos hz]
let P := activatedFullLaw η n d q z hd hb hq hslice hz
have h := activated_deterministic_centered_risk_le f P θ
have hj := complete_arrival_kernel_mean_risk_le T P
have ht : ate P = rareMass η n q * (∑ x : Fin d,
if x.val < rareCount η n d q then (z x)⁻¹ else 0) := by
exact activatedLaw_target_integral η n d q z hb hq hslice hz
dsimp only [v]
rw [← ht]
linarith
have havg := integral_mono_ae (finiteReciprocalPrior_pi_integrable hπ d _)
(finiteReciprocalPrior_pi_integrable hπ d _) hpoint
rw [integral_add ((finiteReciprocalPrior_pi_integrable hπ d r).const_mul 2)
((finiteReciprocalPrior_pi_integrable hπ d v).const_mul 2),
integral_const_mul, integral_const_mul] at havg
have hr := activated_prior_squaredRisk_le_worst η n d q hn hd hb hq hslice hπ T
have hv := activated_full_prior_target_second_moment_le η n d q hπ
change (∫ z, v z ∂Measure.pi (fun _ : Fin d => π)) ≤ _ at hv
change (∫ z, r z ∂Measure.pi (fun _ : Fin d => π)) ≤ _ at hr
dsimp only [f, θ] at havg
linarith
theorem activatedAugmentedMixture_squaredRisk_lower
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π₀ π₁ :
Measure ℝ
h₀ :
FiniteReciprocalPrior (lowerEndpoint n q) π₀
h₁ :
FiniteReciprocalPrior (lowerEndpoint n q) π₁
hgap :
1 / 12 ≤ |(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀|
hJ :
2048 ≤ rareCount η n d q
htv :
tvDist ((activatedAugmentedMixture η n d q π₀).map (fun s i => (s i).2)) ((activatedAugmentedMixture η n d q π₁).map (fun s i => (s i).2))
≤ 1 / 8
T :
Estimator n d
conclusion
(1 / 4096) * ((rareCount η n d q : ℝ) * rareMass η n q) ^ 2
≤ worstCaseRiskReal (fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) => squaredRisk T P.1) ()
Proof (Lean source)
-- @node: activatedAugmentedMixture_squaredRisk_lower
lemma activatedAugmentedMixture_squaredRisk_lower
(η : ℝ) (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior (lowerEndpoint n q) π₀)
(h₁ : FiniteReciprocalPrior (lowerEndpoint n q) π₁)
(hgap : 1 / 12 ≤ |(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀|)
(hJ : 2048 ≤ rareCount η n d q)
(htv : tvDist
((activatedAugmentedMixture η n d q π₀).map (fun s i => (s i).2))
((activatedAugmentedMixture η n d q π₁).map (fun s i => (s i).2)) ≤ 1 / 8)
(T : Estimator n d) :
(1 / 4096) * ((rareCount η n d q : ℝ) * rareMass η n q) ^ 2 ≤
worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) =>
squaredRisk T P.1) () := by
let := T.toBoundedKernel.2.1
let := activatedAugmentedMixture_isProbabilityMeasure η n d q hn hd hb hq hslice h₀
let := activatedAugmentedMixture_isProbabilityMeasure η n d q hn hd hb hq hslice h₁
have hf : Measurable (fun s : Fin n → Bool × ObsRecord d => fun i => (s i).2) := by
fun_prop
let := Measure.isProbabilityMeasure_map hf.aemeasurable
(μ := activatedAugmentedMixture η n d q π₀)
let := Measure.isProbabilityMeasure_map hf.aemeasurable
(μ := activatedAugmentedMixture η n d q π₁)
have hclip : Measurable clip := by unfold clip; fun_prop
have hm := Causalean.Stat.measurable_kernelMean T.toBoundedKernel.1 hclip
have hsep : (rareCount η n d q : ℝ) * rareMass η n q / 12 ≤
|(rareCount η n d q : ℝ) * rareMass η n q * (∫ z, z⁻¹ ∂π₁) -
(rareCount η n d q : ℝ) * rareMass η n q * (∫ z, z⁻¹ ∂π₀)| := by
rw [← mul_sub, abs_mul, abs_of_nonneg (by positivity)]
nlinarith [mul_le_mul_of_nonneg_left hgap
(show 0 ≤ (rareCount η n d q : ℝ) * rareMass η n q by positivity)]
exact activated_fuzzy_risk_lower_of_center_comparison _ _ _ hm _ _ _ _ _
hJ hb.le hsep htv Integrable.of_finite Integrable.of_finite
(activatedAugmentedMixture_centered_risk_le_worst η n d q hn hd hb hq hslice h₀ T)
(activatedAugmentedMixture_centered_risk_le_worst η n d q hn hd hb hq hslice h₁ T)
theorem activatedAugmentedMixture_minimaxRisk_lower
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π₀ π₁ :
Measure ℝ
h₀ :
FiniteReciprocalPrior (lowerEndpoint n q) π₀
h₁ :
FiniteReciprocalPrior (lowerEndpoint n q) π₁
hgap :
1 / 12 ≤ |(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀|
hJ :
2048 ≤ rareCount η n d q
htv :
tvDist ((activatedAugmentedMixture η n d q π₀).map (fun s i => (s i).2)) ((activatedAugmentedMixture η n d q π₁).map (fun s i => (s i).2))
≤ 1 / 8
conclusion
(1 / 4096) * ((rareCount η n d q : ℝ) * rareMass η n q) ^ 2 ≤ minimaxRisk n d q
Proof (Lean source)
-- @node: activatedAugmentedMixture_minimaxRisk_lower
lemma activatedAugmentedMixture_minimaxRisk_lower
(η : ℝ) (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior (lowerEndpoint n q) π₀)
(h₁ : FiniteReciprocalPrior (lowerEndpoint n q) π₁)
(hgap : 1 / 12 ≤ |(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀|)
(hJ : 2048 ≤ rareCount η n d q)
(htv : tvDist
((activatedAugmentedMixture η n d q π₀).map (fun s i => (s i).2))
((activatedAugmentedMixture η n d q π₁).map (fun s i => (s i).2)) ≤ 1 / 8) :
(1 / 4096) * ((rareCount η n d q : ℝ) * rareMass η n q) ^ 2 ≤ minimaxRisk n d q := by
let T₀ : Estimator n d :=
Estimator.ofMap (fun _ => 0) ⟨measurable_const, by intro s; norm_num⟩
letI : Nonempty (Estimator n d) := ⟨T₀⟩
unfold minimaxRisk
apply le_minimaxValue
intro T
exact activatedAugmentedMixture_squaredRisk_lower η n d q hn hd hb hq hslice
h₀ h₁ hgap hJ htv T
theorem activatedAugmentedMixture_minimaxRisk_calibrated
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hq :
0 < q
hslice :
RareArrivalSlice n q
π₀ π₁ :
Measure ℝ
h₀ :
FiniteReciprocalPrior (lowerEndpoint n q) π₀
h₁ :
FiniteReciprocalPrior (lowerEndpoint n q) π₁
hm :
∀ v : ℕ
if
v ≤ lowerDegree n q
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
hgap :
1 / 12 ≤ |(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀|
hlarge :
conclusion
≤ minimaxRisk n d q
Proof (Lean source)
-- @node: activatedAugmentedMixture_minimaxRisk_calibrated
lemma activatedAugmentedMixture_minimaxRisk_calibrated
(n d : ℕ) (q : ℝ) (hn : 1 ≤ n) (hd : 1 ≤ d)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior (lowerEndpoint n q) π₀)
(h₁ : FiniteReciprocalPrior (lowerEndpoint n q) π₁)
(hm : ∀ v : ℕ, v ≤ lowerDegree n q → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
(hgap : 1 / 12 ≤ |(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀|)
(hJ : 2048 ≤ rareCount (exp (-32)) n d q)
(hlarge : exp ((32 - log ((2 : ℝ) / 16)) / 28) ≤ effectiveSize n q) :
(1 / 4096) * ((rareCount (exp (-32)) n d q : ℝ) *
rareMass (exp (-32)) n q) ^ 2 ≤ minimaxRisk n d q := by
have hN : 0 < effectiveSize n q := by unfold effectiveSize; positivity
have hell : 0 < logScale n q := by
unfold logScale
apply Real.log_pos
linarith [Real.add_one_le_exp (1 : ℝ)]
have hb : 0 < rareMass (exp (-32)) n q := by
unfold rareMass
positivity
let := activatedAugmentedMixture_isProbabilityMeasure (exp (-32)) n d q
hn hd hb hq hslice h₀
let := activatedAugmentedMixture_isProbabilityMeasure (exp (-32)) n d q
hn hd hb hq hslice h₁
have htv := activation_projected_tv_le_bad_event
(activatedAugmentedMixture (exp (-32)) n d q π₀)
(activatedAugmentedMixture (exp (-32)) n d q π₁)
{s | ∀ x : Fin d, x.val < rareCount (exp (-32)) n d q →
(∑ i ∈ Finset.univ.filter (fun i => (s i).2.X = x),
if (s i).1 then 1 else 0) ≤ lowerDegree n q}
(by
intro s hs
rw [activatedAugmentedMixture_real_singleton, activatedAugmentedMixture_real_singleton]
exact activatedAugmentedSample_prior_singleton_eq (exp (-32)) n d q
hn hd hb hq hslice h₀ h₁ (lowerDegree n q) hm s hs)
(fun s i => (s i).2) (by fun_prop)
have htail := activatedAugmentedMixture_cutoff_budget 2 n d q (by norm_num)
hn hd hq hslice h₀ hlarge
apply activatedAugmentedMixture_minimaxRisk_lower (exp (-32)) n d q
hn hd hb hq hslice h₀ h₁ hgap hJ
exact htv.trans (by simpa only [Set.compl_setOf, show (2 : ℝ) / 16 = 1 / 8 by norm_num] using htail)
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Lower.ActivationSampleLikelihood 7 declarations Full augmented sample likelihood cancellation below the activation cutoff in equation (6).
Full augmented sample likelihood cancellation below the activation cutoff in equation (6). Reservoir and zero-mass cells contribute constant polynomials; only rare cells consume the matched-moment budget.
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
given by
if x.val < rareCount η n d q then activationRareOutcomePolynomial (lowerEndpoint n q) a flag arrival one else C (augmentedOutcomeWeight η n d q (fun _ => 0) x a flag arrival one)
theorem activationOutcomePolynomial_eval
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(activationOutcomePolynomial η n d q x a flag arrival one).eval (z x)
= augmentedOutcomeWeight η n d q z x a flag arrival one
Proof (Lean source)
-- @node: activationOutcomePolynomial_eval
lemma activationOutcomePolynomial_eval (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin d → ℝ) (x : Fin d) (a flag arrival one : Bool) :
(activationOutcomePolynomial η n d q x a flag arrival one).eval (z x) =
augmentedOutcomeWeight η n d q z x a flag arrival one := by
by_cases hx : x.val < rareCount η n d q
· simp only [activationOutcomePolynomial, hx, if_true]
exact activationRareOutcomePolynomial_eval η n d q z x a flag arrival one hx
· cases a <;> cases flag <;> cases arrival <;> cases one <;>
simp [activationOutcomePolynomial, augmentedOutcomeWeight, lowerCellZ, hx]
theorem activationOutcomePolynomial_natDegree
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(activationOutcomePolynomial η n d q x a flag arrival one).natDegree
≤ if x.val < rareCount η n d q then (if flag then 1 else 0) else 0
Proof (Lean source)
-- @node: activationOutcomePolynomial_natDegree
lemma activationOutcomePolynomial_natDegree (η : ℝ) (n d : ℕ) (q : ℝ)
(x : Fin d) (a flag arrival one : Bool) :
(activationOutcomePolynomial η n d q x a flag arrival one).natDegree ≤
if x.val < rareCount η n d q then (if flag then 1 else 0) else 0 := by
by_cases hx : x.val < rareCount η n d q
· simp only [activationOutcomePolynomial, hx, if_true]
exact activationRareOutcomePolynomial_natDegree _ a flag arrival one
· simp [activationOutcomePolynomial, hx]
theorem activationSampleOutcome_factorization
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
ι :
Type*
s :
Finset ι
η :
ℝ
n d :
ℕ
q :
ℝ
cell :
ι → Fin d
a flag arrival one :
ι → Bool
z :
Fin d → ℝ
conclusion
(∏ i ∈ s, augmentedOutcomeWeight η n d q z (cell i) (a i) (flag i) (arrival i) (one i))
= ∏ x : Fin d, (∏ i ∈ s.filter (fun i => cell i = x), activationOutcomePolynomial η n d q x (a i) (flag i) (arrival i) (one i)).eval (z x)
Proof (Lean source)
-- @node: activationSampleOutcome_factorization
lemma activationSampleOutcome_factorization {ι : Type*} (s : Finset ι)
(η : ℝ) (n d : ℕ) (q : ℝ) (cell : ι → Fin d)
(a flag arrival one : ι → Bool) (z : Fin d → ℝ) :
(∏ i ∈ s, augmentedOutcomeWeight η n d q z (cell i)
(a i) (flag i) (arrival i) (one i)) =
∏ x : Fin d, (∏ i ∈ s.filter (fun i => cell i = x),
activationOutcomePolynomial η n d q x (a i) (flag i)
(arrival i) (one i)).eval (z x) := by
classical
rw [← Finset.prod_fiberwise s cell]
apply Finset.prod_congr rfl
intro x _
rw [eval_prod]
apply Finset.prod_congr rfl
intro i hi
rw [(Finset.mem_filter.mp hi).2]
exact (activationOutcomePolynomial_eval η n d q z x _ _ _ _).symm
theorem activationSampleOutcome_integral_eq Lemma activationSampleOutcome_integral_eq in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
ι :
Type*
s :
Finset ι
η :
ℝ
n d :
ℕ
q :
ℝ
cell :
ι → Fin d
π₀ π₁ :
Measure ℝ
h₀ :
FiniteReciprocalPrior (lowerEndpoint n q) π₀
h₁ :
FiniteReciprocalPrior (lowerEndpoint n q) π₁
K :
ℕ
hm :
∀ v : ℕ
if
v ≤ K
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
a flag arrival one :
ι → Bool
conclusion
(∫ z, ∏ i ∈ s, augmentedOutcomeWeight η n d q z (cell i) (a i) (flag i) (arrival i) (one i) ∂Measure.pi (fun _ : Fin d => π₀))
= ∫ z, ∏ i ∈ s, augmentedOutcomeWeight η n d q z (cell i) (a i) (flag i) (arrival i) (one i) ∂Measure.pi (fun _ : Fin d => π₁)
Proof (Lean source)
-- @node: activationSampleOutcome_integral_eq
lemma activationSampleOutcome_integral_eq {ι : Type*} (s : Finset ι)
(η : ℝ) (n d : ℕ) (q : ℝ) (cell : ι → Fin d)
{π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior (lowerEndpoint n q) π₀)
(h₁ : FiniteReciprocalPrior (lowerEndpoint n q) π₁)
(K : ℕ) (hm : ∀ v : ℕ, v ≤ K → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
(a flag arrival one : ι → Bool)
(hcount : ∀ x : Fin d, x.val < rareCount η n d q →
(∑ i ∈ s.filter (fun i => cell i = x), if flag i then 1 else 0) ≤ K) :
(∫ z, ∏ i ∈ s, augmentedOutcomeWeight η n d q z (cell i)
(a i) (flag i) (arrival i) (one i) ∂Measure.pi (fun _ : Fin d => π₀)) =
∫ z, ∏ i ∈ s, augmentedOutcomeWeight η n d q z (cell i)
(a i) (flag i) (arrival i) (one i) ∂Measure.pi (fun _ : Fin d => π₁) := by
classical
simp_rw [activationSampleOutcome_factorization]
apply finiteReciprocalPrior_product_polynomial_integral_eq h₀ h₁ K hm
intro x
apply (natDegree_prod_le _ _).trans
by_cases hx : x.val < rareCount η n d q
· exact (Finset.sum_le_sum (fun i _ => by
simpa only [hx, if_true] using
activationOutcomePolynomial_natDegree η n d q x (a i) (flag i)
(arrival i) (one i))).trans (hcount x hx)
· have hzero : (∑ i ∈ s.filter (fun i => cell i = x),
(activationOutcomePolynomial η n d q x (a i) (flag i)
(arrival i) (one i)).natDegree) = 0 := by
apply Finset.sum_eq_zero
intro i _
simp [activationOutcomePolynomial, hx]
rw [hzero]
exact zero_le K
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
ι :
Type*
s :
Finset ι
η :
ℝ
n d :
ℕ
q :
ℝ
cell :
ι → Fin d
π₀ π₁ :
Measure ℝ
h₀ :
FiniteReciprocalPrior (lowerEndpoint n q) π₀
h₁ :
FiniteReciprocalPrior (lowerEndpoint n q) π₁
K :
ℕ
hm :
∀ v : ℕ
if
v ≤ K
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
a flag arrival one :
ι → Bool
conclusion
(∫ z, ∏ i ∈ s, (baselineMass η n d q (cell i) / 2 * bernWeight (q * lowerEndpoint n q) (flag i) * augmentedOutcomeWeight η n d q z (cell i) (a i) (flag i) (arrival i) (one i)) ∂Measure.pi (fun _ : Fin d => π₀))
= ∫ z, ∏ i ∈ s, (baselineMass η n d q (cell i) / 2 * bernWeight (q * lowerEndpoint n q) (flag i) * augmentedOutcomeWeight η n d q z (cell i) (a i) (flag i) (arrival i) (one i)) ∂Measure.pi (fun _ : Fin d => π₁)
Proof (Lean source)
-- @node: activationSampleWeight_integral_eq
lemma activationSampleWeight_integral_eq {ι : Type*} (s : Finset ι)
(η : ℝ) (n d : ℕ) (q : ℝ) (cell : ι → Fin d)
{π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior (lowerEndpoint n q) π₀)
(h₁ : FiniteReciprocalPrior (lowerEndpoint n q) π₁)
(K : ℕ) (hm : ∀ v : ℕ, v ≤ K → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
(a flag arrival one : ι → Bool)
(hcount : ∀ x : Fin d, x.val < rareCount η n d q →
(∑ i ∈ s.filter (fun i => cell i = x), if flag i then 1 else 0) ≤ K) :
(∫ z, ∏ i ∈ s, (baselineMass η n d q (cell i) / 2 *
bernWeight (q * lowerEndpoint n q) (flag i) *
augmentedOutcomeWeight η n d q z (cell i) (a i) (flag i) (arrival i) (one i))
∂Measure.pi (fun _ : Fin d => π₀)) =
∫ z, ∏ i ∈ s, (baselineMass η n d q (cell i) / 2 *
bernWeight (q * lowerEndpoint n q) (flag i) *
augmentedOutcomeWeight η n d q z (cell i) (a i) (flag i) (arrival i) (one i))
∂Measure.pi (fun _ : Fin d => π₁) := by
simp_rw [Finset.prod_mul_distrib]
rw [integral_const_mul, integral_const_mul]
rw [activationSampleOutcome_integral_eq s η n d q cell h₀ h₁ K hm
a flag arrival one hcount]
theorem activationSampleWeight_grid_integral_eq
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
ι :
Type*
s :
Finset ι
η :
ℝ
n d :
ℕ
q :
ℝ
cell :
ι → Fin d
π₀ π₁ :
Measure ℝ
h₀ :
FiniteReciprocalPrior (lowerEndpoint n q) π₀
h₁ :
FiniteReciprocalPrior (lowerEndpoint n q) π₁
K :
ℕ
hm :
∀ v : ℕ
if
v ≤ K
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
M :
ℕ
hM :
1 ≤ M
j j' :
Fin (M + 1)
a flag arrival one :
ι → Bool
conclusion
(∫ z, ∏ i ∈ s, (baselineMass η n d q (cell i) / 2 * bernWeight (q * lowerEndpoint n q) (flag i) * augmentedOutcomeWeight η n d q z (cell i) (a i) (flag i) (arrival i) (one i)) ∂Measure.pi (fun _ : Fin d => intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M)))
= ∫ z, ∏ i ∈ s, (baselineMass η n d q (cell i) / 2 * bernWeight (q * lowerEndpoint n q) (flag i) * augmentedOutcomeWeight η n d q z (cell i) (a i) (flag i) (arrival i) (one i)) ∂Measure.pi (fun _ : Fin d => intervalInterpolatedPrior π₀ π₁ ((j' : ℝ) / M))
Proof (Lean source)
-- @node: activationSampleWeight_grid_integral_eq
lemma activationSampleWeight_grid_integral_eq {ι : Type*} (s : Finset ι)
(η : ℝ) (n d : ℕ) (q : ℝ) (cell : ι → Fin d)
{π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior (lowerEndpoint n q) π₀)
(h₁ : FiniteReciprocalPrior (lowerEndpoint n q) π₁)
(K : ℕ) (hm : ∀ v : ℕ, v ≤ K → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
(M : ℕ) (hM : 1 ≤ M) (j j' : Fin (M + 1))
(a flag arrival one : ι → Bool)
(hcount : ∀ x : Fin d, x.val < rareCount η n d q →
(∑ i ∈ s.filter (fun i => cell i = x), if flag i then 1 else 0) ≤ K) :
(∫ z, ∏ i ∈ s, (baselineMass η n d q (cell i) / 2 *
bernWeight (q * lowerEndpoint n q) (flag i) *
augmentedOutcomeWeight η n d q z (cell i) (a i) (flag i) (arrival i) (one i))
∂Measure.pi (fun _ : Fin d => intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))) =
∫ z, ∏ i ∈ s, (baselineMass η n d q (cell i) / 2 *
bernWeight (q * lowerEndpoint n q) (flag i) *
augmentedOutcomeWeight η n d q z (cell i) (a i) (flag i) (arrival i) (one i))
∂Measure.pi (fun _ : Fin d => intervalInterpolatedPrior π₀ π₁ ((j' : ℝ) / M)) := by
have hgrid := intervalInterpolatedPrior_grid h₀ h₁ K hm M hM
exact activationSampleWeight_integral_eq s η n d q cell (hgrid j).1 (hgrid j').1 K
((hgrid j).2.1 j') a flag arrival one hcount
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Lower.ActivationSampleMass 12 declarations Atomic probabilities of the actual augmented fixed-horizon experiment and matched-prior cancellation on the activation cutoff event in equation (6).
Atomic probabilities of the actual augmented fixed-horizon experiment and matched-prior cancellation on the activation cutoff event in equation (6).
theorem augmentedOneRecord_singleton
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
augmentedOneRecord η n d q z hz {r}
= if r.2.S then 0 else ofReal (baselineMass η n d q r.2.X / 2 * bernWeight (q * lowerEndpoint n q) r.1 * augmentedOutcomeWeight η n d q z r.2.X r.2.A r.1 r.2.R r.2.RY)
Proof (Lean source)
-- @node: augmentedOneRecord_singleton
lemma augmentedOneRecord_singleton (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin d → ℝ) (hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q))
(r : Bool × ObsRecord d) :
augmentedOneRecord η n d q z hz {r} =
if r.2.S then 0 else
ofReal (baselineMass η n d q r.2.X / 2 *
bernWeight (q * lowerEndpoint n q) r.1 *
augmentedOutcomeWeight η n d q z r.2.X r.2.A r.1 r.2.R r.2.RY) := by
classical
rcases r with ⟨flag, x, a, s, arrival, one⟩
simp only [augmentedOneRecord, Measure.finsetSum_apply, Measure.smul_apply,
smul_eq_mul, Measure.dirac_apply, Set.indicator_apply, Set.mem_singleton_iff,
Pi.one_apply, Prod.mk.injEq, ObsRecord.mk.injEq]
simp only [mul_ite, mul_one, mul_zero, ite_and]
cases s <;> simp only [Bool.false_eq_true, ↓reduceIte,
Finset.sum_ite_irrel, Finset.sum_ite_eq', Finset.mem_univ,
Finset.sum_const_zero]
theorem augmentedOneRecord_weight_nonneg
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
z :
Fin d → ℝ
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
hz :
∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)
conclusion
0
≤ baselineMass η n d q r.2.X / 2 * bernWeight (q * lowerEndpoint n q) r.1 * augmentedOutcomeWeight η n d q z r.2.X r.2.A r.1 r.2.R r.2.RY
Proof (Lean source)
-- @node: augmentedOneRecord_weight_nonneg
lemma augmentedOneRecord_weight_nonneg (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin d → ℝ) (hb : 0 < rareMass η n q) (hq : 0 < q)
(hslice : RareArrivalSlice n q)
(hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q))
(r : Bool × ObsRecord d) :
0 ≤ baselineMass η n d q r.2.X / 2 *
bernWeight (q * lowerEndpoint n q) r.1 *
augmentedOutcomeWeight η n d q z r.2.X r.2.A r.1 r.2.R r.2.RY := by
have hH : 0 ≤ lowerEndpoint n q := by unfold lowerEndpoint; positivity
have hp : 0 ≤ q * lowerEndpoint n q ∧ q * lowerEndpoint n q ≤ 1 :=
⟨mul_nonneg hq.le hH, lower_activation_probability_le_one n q hq.le hslice⟩
have hbern : 0 ≤ bernWeight (q * lowerEndpoint n q) r.1 := by
cases r.1 <;> simp [bernWeight] <;> linarith [hp.1, hp.2]
exact mul_nonneg
(mul_nonneg (div_nonneg (baselineMass_nonneg η n d q hb r.2.X) (by norm_num))
hbern)
(augmentedOutcomeWeight_nonneg η n d q z _ _ _ _ _ (hz r.2.X))
theorem augmentedOneRecord_real_singleton
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
z :
Fin d → ℝ
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
hz :
∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)
conclusion
(augmentedOneRecord η n d q z hz).real {r}
= if r.2.S then 0 else baselineMass η n d q r.2.X / 2 * bernWeight (q * lowerEndpoint n q) r.1 * augmentedOutcomeWeight η n d q z r.2.X r.2.A r.1 r.2.R r.2.RY
Proof (Lean source)
-- @node: augmentedOneRecord_real_singleton
lemma augmentedOneRecord_real_singleton (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin d → ℝ) (hb : 0 < rareMass η n q) (hq : 0 < q)
(hslice : RareArrivalSlice n q)
(hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q))
(r : Bool × ObsRecord d) :
(augmentedOneRecord η n d q z hz).real {r} =
if r.2.S then 0 else
baselineMass η n d q r.2.X / 2 * bernWeight (q * lowerEndpoint n q) r.1 *
augmentedOutcomeWeight η n d q z r.2.X r.2.A r.1 r.2.R r.2.RY := by
rw [Measure.real, augmentedOneRecord_singleton]
split_ifs
· simp
· exact ENNReal.toReal_ofReal
(augmentedOneRecord_weight_nonneg η n d q z hb hq hslice hz r)
theorem activated_sample_real_singleton
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
z :
Fin d → ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
hz :
∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)
conclusion
= ∏ i, if (s i).2.S then 0 else baselineMass η n d q (s i).2.X / 2 * bernWeight (q * lowerEndpoint n q) (s i).1 * augmentedOutcomeWeight η n d q z (s i).2.X (s i).2.A (s i).1 (s i).2.R (s i).2.RY
Proof (Lean source)
-- @node: activated_sample_real_singleton
lemma activated_sample_real_singleton (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin d → ℝ) (hn : 1 ≤ n) (hd : 1 ≤ d)
(hb : 0 < rareMass η n q) (hq : 0 < q) (hslice : RareArrivalSlice n q)
(hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q))
(s : Fin n → Bool × ObsRecord d) :
(Measure.pi (fun _ : Fin n => augmentedOneRecord η n d q z hz)).real {s} =
∏ i, if (s i).2.S then 0 else
baselineMass η n d q (s i).2.X / 2 *
bernWeight (q * lowerEndpoint n q) (s i).1 *
augmentedOutcomeWeight η n d q z (s i).2.X (s i).2.A (s i).1
(s i).2.R (s i).2.RY := by
let := augmentedOneRecord_isProbabilityMeasure η n d q z hn hd hb hq hslice hz
rw [Measure.real, Measure.pi_singleton, ENNReal.toReal_prod]
exact Finset.prod_congr rfl (fun i _ =>
augmentedOneRecord_real_singleton η n d q z hb hq hslice hz (s i))
def activatedAugmentedSample
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
given by
if hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q) then Measure.pi (fun _ : Fin n => augmentedOneRecord η n d q z hz) else 0
theorem activatedAugmentedSample_prior_singleton
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
conclusion
= if ∀ i, (s i).2.S = false then ∫ z, ∏ i, (baselineMass η n d q (s i).2.X / 2 * bernWeight (q * lowerEndpoint n q) (s i).1 * augmentedOutcomeWeight η n d q z (s i).2.X (s i).2.A (s i).1 (s i).2.R (s i).2.RY) ∂Measure.pi (fun _ : Fin d => π) else 0
Proof (Lean source)
-- @node: activatedAugmentedSample_prior_singleton
lemma activatedAugmentedSample_prior_singleton (η : ℝ) (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q)
{π : Measure ℝ} (hπ : FiniteReciprocalPrior (lowerEndpoint n q) π)
(s : Fin n → Bool × ObsRecord d) :
(∫ z, (activatedAugmentedSample η n d q z).real {s}
∂Measure.pi (fun _ : Fin d => π)) =
if ∀ i, (s i).2.S = false then
∫ z, ∏ i, (baselineMass η n d q (s i).2.X / 2 *
bernWeight (q * lowerEndpoint n q) (s i).1 *
augmentedOutcomeWeight η n d q z (s i).2.X (s i).2.A (s i).1
(s i).2.R (s i).2.RY) ∂Measure.pi (fun _ : Fin d => π)
else 0 := by
classical
by_cases hs : ∀ i, (s i).2.S = false
· rw [if_pos hs]
apply integral_congr_ae
filter_upwards [finiteReciprocalPrior_pi_ae_support hπ d] with z hz
rw [activatedAugmentedSample, dif_pos hz,
activated_sample_real_singleton η n d q z hn hd hb hq hslice hz]
exact Finset.prod_congr rfl (fun i _ => by simp only [hs i, Bool.false_eq_true, if_false])
· rw [if_neg hs]
calc
(∫ z, (activatedAugmentedSample η n d q z).real {s}
∂Measure.pi (fun _ : Fin d => π)) = ∫ _z, (0 : ℝ)
∂Measure.pi (fun _ : Fin d => π) := by
apply integral_congr_ae
filter_upwards [finiteReciprocalPrior_pi_ae_support hπ d] with z hz
rw [activatedAugmentedSample, dif_pos hz,
activated_sample_real_singleton η n d q z hn hd hb hq hslice hz]
obtain ⟨i, hi⟩ := not_forall.mp hs
apply Finset.prod_eq_zero (Finset.mem_univ i)
have hi' : (s i).2.S = true := by cases h : (s i).2.S <;> simp_all
simp only [hi', if_true]
_ = 0 := by simp
theorem activatedAugmentedSample_prior_singleton_eq
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π₀ π₁ :
Measure ℝ
h₀ :
FiniteReciprocalPrior (lowerEndpoint n q) π₀
h₁ :
FiniteReciprocalPrior (lowerEndpoint n q) π₁
K :
ℕ
hm :
∀ v : ℕ
if
v ≤ K
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
conclusion
Proof (Lean source)
-- @node: activatedAugmentedSample_prior_singleton_eq
lemma activatedAugmentedSample_prior_singleton_eq (η : ℝ) (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q)
{π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior (lowerEndpoint n q) π₀)
(h₁ : FiniteReciprocalPrior (lowerEndpoint n q) π₁)
(K : ℕ) (hm : ∀ v : ℕ, v ≤ K → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
(s : Fin n → Bool × ObsRecord d)
(hcount : ∀ x : Fin d, x.val < rareCount η n d q →
(∑ i ∈ Finset.univ.filter (fun i => (s i).2.X = x),
if (s i).1 then 1 else 0) ≤ K) :
(∫ z, (activatedAugmentedSample η n d q z).real {s}
∂Measure.pi (fun _ : Fin d => π₀)) =
∫ z, (activatedAugmentedSample η n d q z).real {s}
∂Measure.pi (fun _ : Fin d => π₁) := by
rw [activatedAugmentedSample_prior_singleton η n d q hn hd hb hq hslice h₀,
activatedAugmentedSample_prior_singleton η n d q hn hd hb hq hslice h₁]
split_ifs
· exact activationSampleWeight_integral_eq univ η n d q
(fun i => (s i).2.X) h₀ h₁ K hm
(fun i => (s i).2.A) (fun i => (s i).1)
(fun i => (s i).2.R) (fun i => (s i).2.RY) hcount
· rfl
theorem activatedAugmentedSample_grid_singleton_eq
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π₀ π₁ :
Measure ℝ
h₀ :
FiniteReciprocalPrior (lowerEndpoint n q) π₀
h₁ :
FiniteReciprocalPrior (lowerEndpoint n q) π₁
K :
ℕ
hm :
∀ v : ℕ
if
v ≤ K
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
M :
ℕ
hM :
1 ≤ M
j j' :
Fin (M + 1)
conclusion
(∫ z, (activatedAugmentedSample η n d q z).real {s} ∂Measure.pi (fun _ : Fin d => intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M)))
= ∫ z, (activatedAugmentedSample η n d q z).real {s} ∂Measure.pi (fun _ : Fin d => intervalInterpolatedPrior π₀ π₁ ((j' : ℝ) / M))
Proof (Lean source)
-- @node: activatedAugmentedSample_grid_singleton_eq
lemma activatedAugmentedSample_grid_singleton_eq (η : ℝ) (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q)
{π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior (lowerEndpoint n q) π₀)
(h₁ : FiniteReciprocalPrior (lowerEndpoint n q) π₁)
(K : ℕ) (hm : ∀ v : ℕ, v ≤ K → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
(M : ℕ) (hM : 1 ≤ M) (j j' : Fin (M + 1))
(s : Fin n → Bool × ObsRecord d)
(hcount : ∀ x : Fin d, x.val < rareCount η n d q →
(∑ i ∈ Finset.univ.filter (fun i => (s i).2.X = x),
if (s i).1 then 1 else 0) ≤ K) :
(∫ z, (activatedAugmentedSample η n d q z).real {s}
∂Measure.pi (fun _ : Fin d => intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))) =
∫ z, (activatedAugmentedSample η n d q z).real {s}
∂Measure.pi (fun _ : Fin d => intervalInterpolatedPrior π₀ π₁ ((j' : ℝ) / M)) := by
have hgrid := intervalInterpolatedPrior_grid h₀ h₁ K hm M hM
exact activatedAugmentedSample_prior_singleton_eq η n d q hn hd hb hq hslice
(hgrid j).1 (hgrid j').1 K ((hgrid j).2.1 j') s hcount
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
μ.real (A ∩ G) = ν.real (A ∩ G)
Proof (Lean source)
-- @node: activation_good_event_real_eq
lemma activation_good_event_real_eq {Ω : Type*} [Fintype Ω]
[MeasurableSpace Ω] [MeasurableSingletonClass Ω]
(μ ν : Measure Ω) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
(G A : Set Ω) (hatom : ∀ s ∈ G, μ.real {s} = ν.real {s}) :
μ.real (A ∩ G) = ν.real (A ∩ G) := by
classical
rw [← Set.coe_toFinset (A ∩ G), ← sum_measureReal_singleton,
← sum_measureReal_singleton]
exact Finset.sum_congr rfl (fun s hs => hatom s (Set.mem_toFinset.mp hs).2)
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: activation_tv_le_bad_event
lemma activation_tv_le_bad_event {Ω : Type*} [Fintype Ω]
[MeasurableSpace Ω] [MeasurableSingletonClass Ω]
(μ ν : Measure Ω) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
(G : Set Ω) (hatom : ∀ s ∈ G, μ.real {s} = ν.real {s}) :
tvDist μ ν ≤ μ.real Gᶜ := by
classical
have hG : MeasurableSet G := G.toFinite.measurableSet
have hgood : μ.real G = ν.real G := by
simpa using activation_good_event_real_eq μ ν G Set.univ hatom
have hbad : μ.real Gᶜ = ν.real Gᶜ := by
rw [measureReal_compl hG, measureReal_compl hG, probReal_univ,
probReal_univ, hgood]
unfold tvDist
apply ciSup_le
rintro ⟨A, hA⟩
have hsplit (ρ : Measure Ω) [IsProbabilityMeasure ρ] :
ρ.real A = ρ.real (A ∩ G) + ρ.real (A ∩ Gᶜ) := by
rw [← measureReal_union (Set.disjoint_left.mpr (by
intro s hs ht
exact ht.2 hs.2)) (hA.inter hG.compl)]
congr 1
exact (Set.inter_union_compl A G).symm
rw [hsplit μ, hsplit ν, activation_good_event_real_eq μ ν G A hatom]
have hμ := measureReal_mono (μ := μ) (Set.inter_subset_right : A ∩ Gᶜ ⊆ Gᶜ)
have hν := measureReal_mono (μ := ν) (Set.inter_subset_right : A ∩ Gᶜ ⊆ Gᶜ)
rw [← hbad] at hν
have hμ0 : 0 ≤ μ.real (A ∩ Gᶜ) := measureReal_nonneg
have hν0 : 0 ≤ ν.real (A ∩ Gᶜ) := measureReal_nonneg
rw [abs_le]
constructor <;> linarith
theorem activation_projected_tv_le_bad_event Lemma activation_projected_tv_le_bad_event in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: activation_projected_tv_le_bad_event
lemma activation_projected_tv_le_bad_event {Ω Ξ : Type*} [Fintype Ω]
[MeasurableSpace Ω] [MeasurableSingletonClass Ω] [MeasurableSpace Ξ]
(μ ν : Measure Ω) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
(G : Set Ω) (hatom : ∀ s ∈ G, μ.real {s} = ν.real {s})
(f : Ω → Ξ) (hf : Measurable f) :
tvDist (μ.map f) (ν.map f) ≤ μ.real Gᶜ := by
have htv := activation_tv_le_bad_event μ ν G hatom
unfold tvDist
apply ciSup_le
rintro ⟨A, hA⟩
rw [map_measureReal_apply hf hA, map_measureReal_apply hf hA]
exact (abs_measureReal_sub_le_tvDist (hf hA)).trans htv
theorem activatedAugmentedSample_grid_tv_le_bad_event
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π₀ π₁ :
Measure ℝ
h₀ :
FiniteReciprocalPrior (lowerEndpoint n q) π₀
h₁ :
FiniteReciprocalPrior (lowerEndpoint n q) π₁
K :
ℕ
hm :
∀ v : ℕ
if
v ≤ K
then
(∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁
M :
ℕ
hM :
1 ≤ M
j j' :
Fin (M + 1)
Q Q' :
hQ :
∀ s,
Q.real {s}
= ∫ z, (activatedAugmentedSample η n d q z).real {s} ∂Measure.pi (fun _ : Fin d => intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M))
hQ' :
∀ s,
Q'.real {s}
= ∫ z, (activatedAugmentedSample η n d q z).real {s} ∂Measure.pi (fun _ : Fin d => intervalInterpolatedPrior π₀ π₁ ((j' : ℝ) / M))
Proof (Lean source)
-- @node: activatedAugmentedSample_grid_tv_le_bad_event
lemma activatedAugmentedSample_grid_tv_le_bad_event (η : ℝ) (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q)
{π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior (lowerEndpoint n q) π₀)
(h₁ : FiniteReciprocalPrior (lowerEndpoint n q) π₁)
(K : ℕ) (hm : ∀ v : ℕ, v ≤ K → (∫ z, z ^ v ∂π₀) = ∫ z, z ^ v ∂π₁)
(M : ℕ) (hM : 1 ≤ M) (j j' : Fin (M + 1))
(Q Q' : Measure (Fin n → Bool × ObsRecord d))
[IsProbabilityMeasure Q] [IsProbabilityMeasure Q']
(hQ : ∀ s, Q.real {s} = ∫ z, (activatedAugmentedSample η n d q z).real {s}
∂Measure.pi (fun _ : Fin d => intervalInterpolatedPrior π₀ π₁ ((j : ℝ) / M)))
(hQ' : ∀ s, Q'.real {s} = ∫ z, (activatedAugmentedSample η n d q z).real {s}
∂Measure.pi (fun _ : Fin d => intervalInterpolatedPrior π₀ π₁ ((j' : ℝ) / M))) :
tvDist Q Q' ≤ Q.real
{s | ¬ ∀ x : Fin d, x.val < rareCount η n d q →
(∑ i ∈ Finset.univ.filter (fun i => (s i).2.X = x),
if (s i).1 then 1 else 0) ≤ K} := by
apply activation_tv_le_bad_event
intro s hs
rw [hQ s, hQ' s]
exact activatedAugmentedSample_grid_singleton_eq η n d q hn hd hb hq hslice
h₀ h₁ K hm M hM j j' s hs
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Lower.ActivationTail 11 declarations Reduction of the actual activation-cutoff probability to a finite union under the parameter-independent iid membership, arm and flag distribution.
Reduction of the actual activation-cutoff probability to a finite union under the parameter-independent iid membership, arm and flag distribution. These identities isolate the binomial tail calculation in equation (6).
theorem activated_sample_membership_activation_event
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
z :
Fin d → ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
hz :
∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)
conclusion
(Measure.pi (fun _ : Fin n => augmentedOneRecord η n d q z hz)).real {s | (fun i => ((s i).2.X, (s i).2.A, (s i).1)) ∈ A}
= (Measure.pi (fun _ : Fin n => ∑ x : Fin d, ∑ a : Bool, ∑ flag : Bool, ofReal (baselineMass η n d q x / 2 * bernWeight (q * lowerEndpoint n q) flag) • Measure.dirac (x, a, flag))).real A
Proof (Lean source)
-- @node: activated_sample_membership_activation_event
lemma activated_sample_membership_activation_event (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin d → ℝ) (hn : 1 ≤ n) (hd : 1 ≤ d)
(hb : 0 < rareMass η n q) (hq : 0 < q) (hslice : RareArrivalSlice n q)
(hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q))
(A : Set (Fin n → Fin d × Bool × Bool)) :
(Measure.pi (fun _ : Fin n => augmentedOneRecord η n d q z hz)).real
{s | (fun i => ((s i).2.X, (s i).2.A, (s i).1)) ∈ A} =
(Measure.pi (fun _ : Fin n =>
∑ x : Fin d, ∑ a : Bool, ∑ flag : Bool,
ofReal (baselineMass η n d q x / 2 *
bernWeight (q * lowerEndpoint n q) flag) • Measure.dirac (x, a, flag))).real A := by
have hf : Measurable (fun s : Fin n → Bool × ObsRecord d =>
fun i => ((s i).2.X, (s i).2.A, (s i).1)) := by fun_prop
rw [← activated_sample_membership_activation_map η n d q z hn hd hb hq hslice hz,
map_measureReal_apply hf (MeasurableSet.of_discrete)]
rfl
theorem activatedAugmentedMixture_membership_activation_event
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
conclusion
(activatedAugmentedMixture η n d q π).real {s | (fun i => ((s i).2.X, (s i).2.A, (s i).1)) ∈ A}
= (Measure.pi (fun _ : Fin n => ∑ x : Fin d, ∑ a : Bool, ∑ flag : Bool, ofReal (baselineMass η n d q x / 2 * bernWeight (q * lowerEndpoint n q) flag) • Measure.dirac (x, a, flag))).real A
Proof (Lean source)
-- @node: activatedAugmentedMixture_membership_activation_event
lemma activatedAugmentedMixture_membership_activation_event (η : ℝ) (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π : Measure ℝ}
(hπ : FiniteReciprocalPrior (lowerEndpoint n q) π)
(A : Set (Fin n → Fin d × Bool × Bool)) :
(activatedAugmentedMixture η n d q π).real
{s | (fun i => ((s i).2.X, (s i).2.A, (s i).1)) ∈ A} =
(Measure.pi (fun _ : Fin n =>
∑ x : Fin d, ∑ a : Bool, ∑ flag : Bool,
ofReal (baselineMass η n d q x / 2 *
bernWeight (q * lowerEndpoint n q) flag) • Measure.dirac (x, a, flag))).real A := by
let := hπ.1
rw [activatedAugmentedMixture_real_event η n d q hn hd hb hq hslice hπ]
calc
_ = ∫ _z : Fin d → ℝ,
(Measure.pi (fun _ : Fin n =>
∑ x : Fin d, ∑ a : Bool, ∑ flag : Bool,
ofReal (baselineMass η n d q x / 2 *
bernWeight (q * lowerEndpoint n q) flag) • Measure.dirac (x, a, flag))).real A
∂Measure.pi (fun _ : Fin d => π) := by
apply integral_congr_ae
filter_upwards [finiteReciprocalPrior_pi_ae_support hπ d] with z hz
rw [activatedAugmentedSample, dif_pos hz]
exact activated_sample_membership_activation_event η n d q z hn hd hb hq hslice hz A
_ = _ := by simp
theorem activatedAugmentedMixture_cutoff_probability
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d K :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
conclusion
(activatedAugmentedMixture η n d q π).real {s | ¬ ∀ x : Fin d, x.val < rareCount η n d q → (∑ i ∈ Finset.univ.filter (fun i => (s i).2.X = x), if (s i).1 then 1 else 0) ≤ K}
= (Measure.pi (fun _ : Fin n => ∑ x : Fin d, ∑ a : Bool, ∑ flag : Bool, ofReal (baselineMass η n d q x / 2 * bernWeight (q * lowerEndpoint n q) flag) • Measure.dirac (x, a, flag))).real {s | ¬ ∀ x : Fin d, x.val < rareCount η n d q → (∑ i ∈ Finset.univ.filter (fun i => (s i).1 = x), if (s i).2.2 then 1 else 0) ≤ K}
Proof (Lean source)
-- @node: activatedAugmentedMixture_cutoff_probability
lemma activatedAugmentedMixture_cutoff_probability (η : ℝ) (n d K : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π : Measure ℝ}
(hπ : FiniteReciprocalPrior (lowerEndpoint n q) π) :
(activatedAugmentedMixture η n d q π).real
{s | ¬ ∀ x : Fin d, x.val < rareCount η n d q →
(∑ i ∈ Finset.univ.filter (fun i => (s i).2.X = x),
if (s i).1 then 1 else 0) ≤ K} =
(Measure.pi (fun _ : Fin n =>
∑ x : Fin d, ∑ a : Bool, ∑ flag : Bool,
ofReal (baselineMass η n d q x / 2 *
bernWeight (q * lowerEndpoint n q) flag) • Measure.dirac (x, a, flag))).real
{s | ¬ ∀ x : Fin d, x.val < rareCount η n d q →
(∑ i ∈ Finset.univ.filter (fun i => (s i).1 = x),
if (s i).2.2 then 1 else 0) ≤ K} := by
exact activatedAugmentedMixture_membership_activation_event η n d q hn hd hb hq hslice hπ
{s | ¬ ∀ x : Fin d, x.val < rareCount η n d q →
(∑ i ∈ Finset.univ.filter (fun i => (s i).1 = x),
if (s i).2.2 then 1 else 0) ≤ K}
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d K :
ℕ
q ε :
ℝ
htail :
∀ x : Fin d
if
x.val < rareCount η n d q
then
μ.real {s | K < (∑ i ∈ Finset.univ.filter (fun i => (s i).1 = x), if (s i).2.2 then 1 else 0)}
≤ ε
conclusion
μ.real {s | ¬ ∀ x : Fin d, x.val < rareCount η n d q → (∑ i ∈ Finset.univ.filter (fun i => (s i).1 = x), if (s i).2.2 then 1 else 0) ≤ K}
≤ (rareCount η n d q : ℝ) * ε
Proof (Lean source)
-- @node: activation_cutoff_union_bound
lemma activation_cutoff_union_bound (η : ℝ) (n d K : ℕ) (q ε : ℝ)
(μ : Measure (Fin n → Fin d × Bool × Bool))
(htail : ∀ x : Fin d, x.val < rareCount η n d q →
μ.real {s | K < (∑ i ∈ Finset.univ.filter (fun i => (s i).1 = x),
if (s i).2.2 then 1 else 0)} ≤ ε) :
μ.real {s | ¬ ∀ x : Fin d, x.val < rareCount η n d q →
(∑ i ∈ Finset.univ.filter (fun i => (s i).1 = x),
if (s i).2.2 then 1 else 0) ≤ K} ≤ (rareCount η n d q : ℝ) * ε := by
classical
have hJ : rareCount η n d q ≤ d :=
(min_le_left _ _).trans (sub_le d 1)
let cell : Fin (rareCount η n d q) → Fin d := fun x => ⟨x.val, lt_of_lt_of_le x.isLt hJ⟩
let E (x : Fin (rareCount η n d q)) : Set (Fin n → Fin d × Bool × Bool) :=
{s | K < (∑ i ∈ Finset.univ.filter (fun i => (s i).1 = cell x),
if (s i).2.2 then 1 else 0)}
have heq : {s | ¬ ∀ x : Fin d, x.val < rareCount η n d q →
(∑ i ∈ Finset.univ.filter (fun i => (s i).1 = x),
if (s i).2.2 then 1 else 0) ≤ K} = ⋃ x, E x := by
ext s
simp only [Set.mem_ofPred_eq, not_forall, not_le, Set.mem_iUnion]
constructor
· rintro ⟨x, hx, hcount⟩
exact ⟨⟨x.val, hx⟩, hcount⟩
· rintro ⟨x, hx⟩
exact ⟨cell x, x.isLt, hx⟩
rw [heq]
calc
_ ≤ ∑ x, μ.real (E x) := measureReal_iUnion_fintype_le E
_ ≤ ∑ _x : Fin (rareCount η n d q), ε :=
Finset.sum_le_sum (fun x _ => htail (cell x) x.isLt)
_ = _ := by simp
theorem activation_event_descFactorial_eq_weighted Lemma activation_event_descFactorial_eq_weighted in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
((eventCount (fixedSizeEmbed n s) A).descFactorial (K + 1) : ℝ)
= weightedFactorial A A (K + 1) (fixedSizeEmbed n s)
Proof (Lean source)
-- @node: activation_event_descFactorial_eq_weighted
lemma activation_event_descFactorial_eq_weighted {X : Type*} [MeasurableSpace X]
(A : Set X) (n K : ℕ) (s : Fin n → X) :
((eventCount
(fixedSizeEmbed n s) A).descFactorial
(K + 1) : ℝ) =
weightedFactorial
A A (K + 1)
(fixedSizeEmbed n s) := by
unfold weightedFactorial
simp only [Nat.add_sub_cancel]
generalize eventCount
(fixedSizeEmbed n s) A = c
cases c with
| zero => simp
| succ c =>
rw [Nat.succ_descFactorial_succ]
simp only [Nat.add_sub_cancel, Nat.cast_mul]
theorem activation_event_descFactorial_integral Lemma activation_event_descFactorial_integral in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ s : Fin n → X, ((eventCount (fixedSizeEmbed n s) A).descFactorial (K + 1) : ℝ) ∂Measure.pi (fun _ : Fin n => P))
= (n.descFactorial (K + 1) : ℝ) * P.real A ^ (K + 1)
Proof (Lean source)
-- @node: activation_event_descFactorial_integral
lemma activation_event_descFactorial_integral {X : Type*} [MeasurableSpace X]
(P : Measure X) [IsProbabilityMeasure P] (A : Set X) (hA : MeasurableSet A)
(n K : ℕ) :
(∫ s : Fin n → X,
((eventCount
(fixedSizeEmbed n s) A).descFactorial
(K + 1) : ℝ) ∂Measure.pi (fun _ : Fin n => P)) =
(n.descFactorial (K + 1) : ℝ) * P.real A ^ (K + 1) := by
simp_rw [activation_event_descFactorial_eq_weighted]
rw [integral_weightedFactorial_fixedCount
P A A hA hA (Subset.rfl) (K + 1) n (by omega)]
simp only [Nat.add_sub_cancel, Measure.real, pow_succ]
ring
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
≤ ((n : ℝ) * P.real A) ^ (K + 1) / ((K + 1).factorial : ℝ)
Proof (Lean source)
-- @node: activation_event_factorial_tail
lemma activation_event_factorial_tail {X : Type*} [MeasurableSpace X] [MeasurableSingletonClass X] [Finite X]
(P : Measure X) [IsProbabilityMeasure P] (A : Set X) (hA : MeasurableSet A)
(n K : ℕ) :
(Measure.pi (fun _ : Fin n => P)).real
{s | K < eventCount
(fixedSizeEmbed n s) A} ≤
((n : ℝ) * P.real A) ^ (K + 1) / ((K + 1).factorial : ℝ) := by
let μ := Measure.pi (fun _ : Fin n => P)
let f (s : Fin n → X) : ℝ :=
((eventCount
(fixedSizeEmbed n s) A).descFactorial
(K + 1) : ℝ)
have hsubset : {s | K < eventCount
(fixedSizeEmbed n s) A} ⊆
{s | ((K + 1).factorial : ℝ) ≤ f s} := by
intro s hs
have h := Nat.descFactorial_le (K + 1) (succ_le_of_lt hs)
rw [Nat.descFactorial_self] at h
change ((K + 1).factorial : ℝ) ≤
((eventCount
(fixedSizeEmbed n s) A).descFactorial
(K + 1) : ℝ)
exact_mod_cast h
have hmarkov := mul_meas_ge_le_integral_of_nonneg
(μ := μ) (Filter.Eventually.of_forall fun s => (Nat.cast_nonneg _ : 0 ≤ f s))
(Integrable.of_finite : Integrable f μ) ((K + 1).factorial : ℝ)
have hmono := measureReal_mono (μ := μ) hsubset
have hfac : (0 : ℝ) < ((K + 1).factorial : ℝ) := by positivity
apply (le_div_iff₀ hfac).mpr
have hmoment := activation_event_descFactorial_integral P A hA n K
have hpow : (n.descFactorial (K + 1) : ℝ) ≤ (n : ℝ) ^ (K + 1) := by
exact_mod_cast Nat.descFactorial_le_pow n (K + 1)
have hbound : (∫ s, f s ∂μ) ≤ ((n : ℝ) * P.real A) ^ (K + 1) := by
rw [show (∫ s, f s ∂μ) = (n.descFactorial (K + 1) : ℝ) * P.real A ^ (K + 1)
from hmoment, mul_pow]
exact mul_le_mul_of_nonneg_right hpow (by positivity)
exact (by nlinarith : _ ≤ ∫ s, f s ∂μ).trans hbound
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: activation_count_eq_eventCount
lemma activation_count_eq_eventCount (n d : ℕ)
(x : Fin d) (s : Fin n → Fin d × Bool × Bool) :
(∑ i ∈ Finset.univ.filter (fun i => (s i).1 = x),
if (s i).2.2 then 1 else 0) =
eventCount
(fixedSizeEmbed n s)
{r | r.1 = x ∧ r.2.2 = true} := by
classical
unfold eventCount
fixedSizeEmbed
FiniteSample.count
FiniteSample.points
simp only [Finset.sum_boole, Finset.filter_filter, Set.mem_ofPred_eq, Nat.cast_id]
theorem activation_membership_flag_probability
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∑ y : Fin d, ∑ a : Bool, ∑ flag : Bool, ofReal (baselineMass η n d q y / 2 * bernWeight (q * lowerEndpoint n q) flag) • Measure.dirac (y, a, flag)).real {r | r.1 = x ∧ r.2.2 = true}
= baselineMass η n d q x * (q * lowerEndpoint n q)
Proof (Lean source)
-- @node: activation_membership_flag_probability
lemma activation_membership_flag_probability (η : ℝ) (n d : ℕ) (q : ℝ)
(hb : 0 < rareMass η n q) (hq : 0 < q) (hslice : RareArrivalSlice n q)
(x : Fin d) :
(∑ y : Fin d, ∑ a : Bool, ∑ flag : Bool,
ofReal (baselineMass η n d q y / 2 *
bernWeight (q * lowerEndpoint n q) flag) • Measure.dirac (y, a, flag)).real
{r | r.1 = x ∧ r.2.2 = true} =
baselineMass η n d q x * (q * lowerEndpoint n q) := by
classical
have hb0 := baselineMass_nonneg η n d q hb x
have hH : 0 ≤ q * lowerEndpoint n q := by
unfold lowerEndpoint
positivity
have hweight : 0 ≤ baselineMass η n d q x / 2 * (q * lowerEndpoint n q) :=
mul_nonneg (div_nonneg hb0 (by norm_num)) hH
have hevent : MeasurableSet {r : Fin d × Bool × Bool | r.1 = x ∧ r.2.2 = true} :=
MeasurableSet.of_discrete
simp only [Measure.real]
simp [bernWeight, indicator]
rw [Finset.sum_add_distrib]
simp only [Finset.sum_ite_eq', Finset.mem_univ, if_true, ENNReal.toReal_add ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top,
ENNReal.toReal_ofReal hweight]
ring
theorem activation_membership_flag_isProbabilityMeasure
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
conclusion
IsProbabilityMeasure (∑ x : Fin d, ∑ a : Bool, ∑ flag : Bool, ofReal (baselineMass η n d q x / 2 * bernWeight (q * lowerEndpoint n q) flag) • Measure.dirac (x, a, flag))
Proof (Lean source)
-- @node: activation_membership_flag_isProbabilityMeasure
lemma activation_membership_flag_isProbabilityMeasure (η : ℝ) (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) :
IsProbabilityMeasure
(∑ x : Fin d, ∑ a : Bool, ∑ flag : Bool,
ofReal (baselineMass η n d q x / 2 *
bernWeight (q * lowerEndpoint n q) flag) • Measure.dirac (x, a, flag)) := by
have hz : ∀ x : Fin d, (fun _ : Fin d => (1 : ℝ)) x ∈ Icc (1 : ℝ) (lowerEndpoint n q) :=
fun _ => ⟨le_rfl, lowerEndpoint_one_le n q hq.le⟩
let := augmentedOneRecord_isProbabilityMeasure η n d q (fun _ => 1)
hn hd hb hq hslice hz
rw [← augmentedOneRecord_membership_activation_map η n d q (fun _ => 1) hz]
exact Measure.isProbabilityMeasure_map (by fun_prop)
theorem activatedAugmentedMixture_factorial_cutoff_bound
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d K :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π :
Measure ℝ
hπ :
FiniteReciprocalPrior (lowerEndpoint n q) π
conclusion
(activatedAugmentedMixture η n d q π).real {s | ¬ ∀ x : Fin d, x.val < rareCount η n d q → (∑ i ∈ Finset.univ.filter (fun i => (s i).2.X = x), if (s i).1 then 1 else 0) ≤ K}
≤ (rareCount η n d q : ℝ) * (((n : ℝ) * (rareMass η n q * (q * lowerEndpoint n q))) ^ (K + 1) / ((K + 1).factorial : ℝ))
Proof (Lean source)
-- @node: activatedAugmentedMixture_factorial_cutoff_bound
lemma activatedAugmentedMixture_factorial_cutoff_bound (η : ℝ) (n d K : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hb : 0 < rareMass η n q)
(hq : 0 < q) (hslice : RareArrivalSlice n q) {π : Measure ℝ}
(hπ : FiniteReciprocalPrior (lowerEndpoint n q) π) :
(activatedAugmentedMixture η n d q π).real
{s | ¬ ∀ x : Fin d, x.val < rareCount η n d q →
(∑ i ∈ Finset.univ.filter (fun i => (s i).2.X = x),
if (s i).1 then 1 else 0) ≤ K} ≤
(rareCount η n d q : ℝ) *
(((n : ℝ) * (rareMass η n q * (q * lowerEndpoint n q))) ^ (K + 1) /
((K + 1).factorial : ℝ)) := by
let P : Measure (Fin d × Bool × Bool) :=
∑ x : Fin d, ∑ a : Bool, ∑ flag : Bool,
ofReal (baselineMass η n d q x / 2 *
bernWeight (q * lowerEndpoint n q) flag) • Measure.dirac (x, a, flag)
let : IsProbabilityMeasure P :=
activation_membership_flag_isProbabilityMeasure η n d q hn hd hb hq hslice
rw [activatedAugmentedMixture_cutoff_probability η n d K q hn hd hb hq hslice hπ]
apply activation_cutoff_union_bound
intro x hx
simp_rw [activation_count_eq_eventCount]
have htail := activation_event_factorial_tail P
{r | r.1 = x ∧ r.2.2 = true} (MeasurableSet.of_discrete) n K
have hprob : P.real {r | r.1 = x ∧ r.2.2 = true} =
rareMass η n q * (q * lowerEndpoint n q) := by
rw [activation_membership_flag_probability η n d q hb hq hslice x]
simp [baselineMass, hx]
simpa only [hprob] using htail
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Lower.ActivationTarget 8 declarations The activated full-data target is the rare-cell reciprocal sum used in equations (7)--(8) of the connected-interval lower bound.
The activated full-data target is the rare-cell reciprocal sum used in equations (7)--(8) of the connected-interval lower bound.
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: activation_sum_initial_cells
lemma activation_sum_initial_cells (d J : ℕ) (hJ : J ≤ d) (f : Fin d → ℝ) :
(∑ x : Fin d, if x.val < J then f x else 0) =
∑ x : Fin J, f (x.castLE hJ) := by
classical
rw [← Finset.sum_filter]
symm
apply Finset.sum_bij (fun x _ => x.castLE hJ)
· intro x hx
simp [x.isLt]
· intro x hx y hy hxy
apply Fin.ext
exact congrArg (fun t : Fin d => t.val) hxy
· intro y hy
have hyl : y.val < J := (Finset.mem_filter.mp hy).2
exact ⟨⟨y.val, hyl⟩, Finset.mem_univ _, Fin.ext rfl⟩
· intro x hx
rfl
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: baselineMass_sum
lemma baselineMass_sum (η : ℝ) (n d : ℕ) (q : ℝ) (hd : 1 ≤ d) :
(∑ x : Fin d, baselineMass η n d q x) = 1 := by
classical
have hJ : rareCount η n d q < d := by
have h := min_le_left (d - 1) (floor ((2 * rareMass η n q)⁻¹))
change rareCount η n d q ≤ d - 1 at h
omega
let reservoir : Fin d := ⟨rareCount η n d q, hJ⟩
have hsplit (x : Fin d) : baselineMass η n d q x =
(if x.val < rareCount η n d q then rareMass η n q else 0) +
(if x = reservoir then 1 - rareCount η n d q * rareMass η n q else 0) := by
have heq : x = reservoir ↔ x.val = rareCount η n d q := by
simp [reservoir, Fin.ext_iff]
by_cases hl : x.val < rareCount η n d q
· have hn : x.val ≠ rareCount η n d q := by omega
simp [baselineMass, hl, heq, hn]
· simp [baselineMass, hl, heq]
simp_rw [hsplit]
rw [Finset.sum_add_distrib, activation_sum_initial_cells d _ (le_of_lt hJ)]
simp
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
0 ≤ baselineMass η n d q x
Proof (Lean source)
-- @node: baselineMass_nonneg
lemma baselineMass_nonneg (η : ℝ) (n d : ℕ) (q : ℝ)
(hb : 0 < rareMass η n q) (x : Fin d) :
0 ≤ baselineMass η n d q x := by
have hmass := rareCount_mass_le_half η n d q hb
unfold baselineMass
split_ifs <;> linarith
theorem activatedLaw_weight_nonneg
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
z :
Fin d → ℝ
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
hz :
∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)
x :
Fin d
y arrival :
conclusion
0
≤ baselineMass η n d q x / 2 * bernWeight (if x.val < rareCount η n d q then (z x)⁻¹ else 0) y * bernWeight (q * lowerCellZ η n d q z x) arrival
Proof (Lean source)
-- @node: activatedLaw_weight_nonneg
lemma activatedLaw_weight_nonneg (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin d → ℝ) (hb : 0 < rareMass η n q) (hq : 0 < q)
(hslice : RareArrivalSlice n q)
(hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q))
(x : Fin d) (y arrival : Bool) :
0 ≤ baselineMass η n d q x / 2 *
bernWeight (if x.val < rareCount η n d q then (z x)⁻¹ else 0) y *
bernWeight (q * lowerCellZ η n d q z x) arrival := by
have hH : 0 ≤ lowerEndpoint n q := by unfold lowerEndpoint; positivity
have hcell : 0 ≤ lowerCellZ η n d q z x ∧
lowerCellZ η n d q z x ≤ lowerEndpoint n q := by
unfold lowerCellZ
split_ifs <;> constructor <;> linarith [(hz x).1, (hz x).2]
have hprob : 0 ≤ q * lowerCellZ η n d q z x ∧
q * lowerCellZ η n d q z x ≤ 1 :=
⟨mul_nonneg hq.le hcell.1,
(mul_le_mul_of_nonneg_left hcell.2 hq.le).trans
(lower_activation_probability_le_one n q hq.le hslice)⟩
have hy : 0 ≤ (if x.val < rareCount η n d q then (z x)⁻¹ else 0) ∧
(if x.val < rareCount η n d q then (z x)⁻¹ else 0) ≤ 1 := by
split_ifs
· exact ⟨inv_nonneg.mpr (by linarith [(hz x).1]),
inv_le_one_of_one_le₀ (hz x).1⟩
· norm_num
have hbern (p : ℝ) (hp : 0 ≤ p ∧ p ≤ 1) (v : Bool) :
0 ≤ bernWeight p v := by
cases v <;> simp [bernWeight] <;> linarith [hp.1, hp.2]
exact mul_nonneg
(mul_nonneg (div_nonneg (baselineMass_nonneg η n d q hb x) (by norm_num))
(hbern _ hy y)) (hbern _ hprob arrival)
theorem activatedLaw_target_integral
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
z :
Fin d → ℝ
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
hz :
∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)
conclusion
(∫ r : FullRecord d, ((if r.Y1 then (1 : ℝ) else 0) - (if r.Y0 then (1 : ℝ) else 0)) ∂activatedLaw η n d q z hz)
Proof (Lean source)
-- @node: activatedLaw_target_integral
lemma activatedLaw_target_integral (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin d → ℝ) (hb : 0 < rareMass η n q) (hq : 0 < q)
(hslice : RareArrivalSlice n q)
(hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)) :
(∫ r : FullRecord d,
((if r.Y1 then (1 : ℝ) else 0) - (if r.Y0 then (1 : ℝ) else 0))
∂activatedLaw η n d q z hz) =
rareMass η n q * ∑ x : Fin d,
if x.val < rareCount η n d q then (z x)⁻¹ else 0 := by
let f : FullRecord d → ℝ := fun r =>
(if r.Y1 then 1 else 0) - (if r.Y0 then 1 else 0)
let μ (x : Fin d) (a y arrival : Bool) : Measure (FullRecord d) :=
ofReal (baselineMass η n d q x / 2 *
bernWeight (if x.val < rareCount η n d q then (z x)⁻¹ else 0) y *
bernWeight (q * lowerCellZ η n d q z x) arrival) •
Measure.dirac (lowerFullRecord x a y arrival)
have hInt (x : Fin d) (a y arrival : Bool) : Integrable f (μ x a y arrival) :=
(integrable_dirac (by simp)).smul_measure ENNReal.ofReal_ne_top
have hIntY (x : Fin d) (a y : Bool) :
Integrable f (∑ arrival, μ x a y arrival) :=
integrable_finsetSum_measure.mpr (fun arrival _ => hInt x a y arrival)
have hIntA (x : Fin d) (a : Bool) :
Integrable f (∑ y, ∑ arrival, μ x a y arrival) :=
integrable_finsetSum_measure.mpr (fun y _ => hIntY x a y)
have hIntX (x : Fin d) :
Integrable f (∑ a, ∑ y, ∑ arrival, μ x a y arrival) :=
integrable_finsetSum_measure.mpr (fun a _ => hIntA x a)
change (∫ r, f r ∂(∑ x, ∑ a, ∑ y, ∑ arrival, μ x a y arrival)) = _
rw [integral_finsetSum_measure (fun x _ => hIntX x)]
simp_rw [integral_finsetSum_measure (fun a _ => hIntA _ a),
integral_finsetSum_measure (fun y _ => hIntY _ _ y),
integral_finsetSum_measure (fun arrival _ => hInt _ _ _ arrival)]
simp only [μ, integral_smul_measure, integral_dirac, smul_eq_mul]
simp_rw [ENNReal.toReal_ofReal (activatedLaw_weight_nonneg η n d q z hb hq hslice hz _ _ _)]
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro x _
by_cases hr : x.val < rareCount η n d q
· simp [f, lowerFullRecord, bernWeight, hr, baselineMass]
ring
· simp [f, lowerFullRecord, bernWeight, hr]
theorem activatedLaw_univ
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
z :
Fin d → ℝ
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
hz :
∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)
conclusion
activatedLaw η n d q z hz univ = 1
Proof (Lean source)
-- @node: activatedLaw_univ
lemma activatedLaw_univ (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin d → ℝ) (hd : 1 ≤ d) (hb : 0 < rareMass η n q) (hq : 0 < q)
(hslice : RareArrivalSlice n q)
(hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)) :
activatedLaw η n d q z hz univ = 1 := by
classical
let w (x : Fin d) (y arrival : Bool) : ℝ :=
baselineMass η n d q x / 2 *
bernWeight (if x.val < rareCount η n d q then (z x)⁻¹ else 0) y *
bernWeight (q * lowerCellZ η n d q z x) arrival
have hw (x : Fin d) (y arrival : Bool) : 0 ≤ w x y arrival :=
activatedLaw_weight_nonneg η n d q z hb hq hslice hz x y arrival
have hcell (x : Fin d) :
(∑ y : Bool, ∑ arrival : Bool, ofReal (w x y arrival)) =
ofReal (baselineMass η n d q x / 2) := by
simp_rw [← ENNReal.ofReal_sum_of_nonneg (fun arrival _ => hw x _ arrival)]
rw [← ENNReal.ofReal_sum_of_nonneg (fun y _ =>
Finset.sum_nonneg (fun arrival _ => hw x y arrival))]
congr 1
by_cases hr : x.val < rareCount η n d q <;>
simp [w, bernWeight, hr] <;> ring
simp only [activatedLaw, Measure.finsetSum_apply, Measure.smul_apply,
Measure.dirac_apply_of_mem (Set.mem_univ _), smul_eq_mul, mul_one]
change (∑ x : Fin d, ∑ _a : Bool, ∑ y : Bool, ∑ arrival : Bool,
ofReal (w x y arrival)) = 1
simp_rw [hcell]
have harm (x : Fin d) :
(∑ _a : Bool, ofReal (baselineMass η n d q x / 2)) =
ofReal (baselineMass η n d q x) := by
rw [← ENNReal.ofReal_sum_of_nonneg (fun _ _ =>
div_nonneg (baselineMass_nonneg η n d q hb x) (by norm_num))]
congr 1
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_bool, nsmul_eq_mul, Nat.cast_ofNat]
ring
simp_rw [harm]
rw [← ENNReal.ofReal_sum_of_nonneg (fun x _ => baselineMass_nonneg η n d q hb x),
baselineMass_sum η n d q hd]
norm_num
def activatedFullLaw
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
η :
ℝ
n d :
ℕ
q :
ℝ
z :
Fin d → ℝ
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
hz :
∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)
def
given by
⟨activatedLaw η n d q z hz, ⟨activatedLaw_univ η n d q z hd hb hq hslice hz⟩⟩
theorem activatedFullLaw_ate
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
z :
Fin d → ℝ
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
hz :
∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)
conclusion
ate (activatedFullLaw η n d q z hd hb hq hslice hz)
= intervalPriorTarget (rareCount η n d q) (rareMass η n q) (fun x => z (x.castLE (show rareCount η n d q ≤ d from (min_le_left _ _).trans (sub_le d 1))))
Proof (Lean source)
-- @node: activatedFullLaw_ate
lemma activatedFullLaw_ate (η : ℝ) (n d : ℕ) (q : ℝ)
(z : Fin d → ℝ) (hd : 1 ≤ d) (hb : 0 < rareMass η n q) (hq : 0 < q)
(hslice : RareArrivalSlice n q)
(hz : ∀ x, z x ∈ Icc (1 : ℝ) (lowerEndpoint n q)) :
ate (activatedFullLaw η n d q z hd hb hq hslice hz) =
intervalPriorTarget (rareCount η n d q) (rareMass η n q)
(fun x => z (x.castLE (show rareCount η n d q ≤ d from
(min_le_left _ _).trans (sub_le d 1)))) := by
change (∫ r : FullRecord d,
((if r.Y1 then (1 : ℝ) else 0) - (if r.Y0 then (1 : ℝ) else 0))
∂activatedLaw η n d q z hz) = _
rw [activatedLaw_target_integral η n d q z hb hq hslice hz,
activation_sum_initial_cells d (rareCount η n d q) (show rareCount η n d q ≤ d from
(min_le_left _ _).trans (sub_le d 1)) (fun x => (z x)⁻¹)]
rfl
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Lower.FuzzyRisk 7 declarations Centered squared-risk comparison and variance absorption for equations (13)--(16) of the activated point-risk lower bound.
Centered squared-risk comparison and variance absorption for equations (13)--(16) of the activated point-risk lower bound.
theorem activatedRarePriorATE_mean_separation
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
η :
ℝ
n d :
ℕ
q :
ℝ
hd :
1 ≤ d
hb :
0 < rareMass η n q
hq :
0 < q
hslice :
RareArrivalSlice n q
π₀ π₁ :
Measure ℝ
h₀ :
FiniteReciprocalPrior (lowerEndpoint n q) π₀
h₁ :
FiniteReciprocalPrior (lowerEndpoint n q) π₁
hgap :
1 / 12 ≤ |(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀|
conclusion
(rareCount η n d q : ℝ) * rareMass η n q / 12
≤ |(∫ z, activatedRarePriorATE η n d q hd hb hq hslice z ∂Measure.pi (fun _ : Fin (rareCount η n d q) => π₁))
- (∫ z, activatedRarePriorATE η n d q hd hb hq hslice z ∂Measure.pi (fun _ : Fin (rareCount η n d q) => π₀))|
Proof (Lean source)
-- @node: activatedRarePriorATE_mean_separation
lemma activatedRarePriorATE_mean_separation (η : ℝ) (n d : ℕ) (q : ℝ)
(hd : 1 ≤ d) (hb : 0 < rareMass η n q) (hq : 0 < q)
(hslice : RareArrivalSlice n q) {π₀ π₁ : Measure ℝ}
(h₀ : FiniteReciprocalPrior (lowerEndpoint n q) π₀)
(h₁ : FiniteReciprocalPrior (lowerEndpoint n q) π₁)
(hgap : 1 / 12 ≤ |(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀|) :
(rareCount η n d q : ℝ) * rareMass η n q / 12 ≤
|(∫ z, activatedRarePriorATE η n d q hd hb hq hslice z
∂Measure.pi (fun _ : Fin (rareCount η n d q) => π₁)) -
(∫ z, activatedRarePriorATE η n d q hd hb hq hslice z
∂Measure.pi (fun _ : Fin (rareCount η n d q) => π₀))| := by
rw [activatedRarePriorATE_integral η n d q hd hb hq hslice h₁,
activatedRarePriorATE_integral η n d q hd hb hq hslice h₀,
← mul_sub, abs_mul, abs_of_nonneg
(mul_nonneg (Nat.cast_nonneg _) hb.le)]
have h := mul_le_mul_of_nonneg_left hgap
(mul_nonneg (Nat.cast_nonneg (rareCount η n d q)) hb.le)
nlinarith [h]
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
Measure Ω
T τ :
Ω → ℝ
θ :
ℝ
herror :
Integrable (fun ω => (T ω - τ ω) ^ 2) μ
htarget :
Integrable (fun ω => (τ ω - θ) ^ 2) μ
hcenter :
Integrable (fun ω => (T ω - θ) ^ 2) μ
conclusion
(∫ ω, (T ω - θ) ^ 2 ∂μ) ≤ 2 * (∫ ω, (T ω - τ ω) ^ 2 ∂μ) + 2 * (∫ ω, (τ ω - θ) ^ 2 ∂μ)
Proof (Lean source)
-- @node: activated_centered_risk_le
lemma activated_centered_risk_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) (T τ : Ω → ℝ) (θ : ℝ)
(herror : Integrable (fun ω => (T ω - τ ω) ^ 2) μ)
(htarget : Integrable (fun ω => (τ ω - θ) ^ 2) μ)
(hcenter : Integrable (fun ω => (T ω - θ) ^ 2) μ) :
(∫ ω, (T ω - θ) ^ 2 ∂μ) ≤
2 * (∫ ω, (T ω - τ ω) ^ 2 ∂μ) +
2 * (∫ ω, (τ ω - θ) ^ 2 ∂μ) := by
calc
_ ≤ ∫ ω, 2 * (T ω - τ ω) ^ 2 + 2 * (τ ω - θ) ^ 2 ∂μ := by
apply integral_mono hcenter ((herror.const_mul 2).add (htarget.const_mul 2))
intro ω
have h := add_sq_le (a := T ω - τ ω) (b := τ ω - θ)
change (T ω - θ) ^ 2 ≤ 2 * (T ω - τ ω) ^ 2 + 2 * (τ ω - θ) ^ 2
nlinarith [h]
_ = _ := by rw [integral_add (herror.const_mul 2) (htarget.const_mul 2),
integral_const_mul, integral_const_mul]
theorem activated_error_probability_mul_sq_le_risk Lemma activated_error_probability_mul_sq_le_risk in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
μ.real {ω | s ≤ |T ω - θ|} * s ^ 2 ≤ ∫ ω, (T ω - θ) ^ 2 ∂μ
Proof (Lean source)
-- @node: activated_error_probability_mul_sq_le_risk
lemma activated_error_probability_mul_sq_le_risk {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (T : Ω → ℝ)
(hT : Measurable T) (θ s : ℝ) (hs : 0 ≤ s)
(hint : Integrable (fun ω => (T ω - θ) ^ 2) μ) :
μ.real {ω | s ≤ |T ω - θ|} * s ^ 2 ≤ ∫ ω, (T ω - θ) ^ 2 ∂μ := by
classical
let E := {ω | s ≤ |T ω - θ|}
have hE : MeasurableSet E := by
dsimp [E]
have hm : Measurable (fun ω => |T ω - θ|) := by fun_prop
exact measurableSet_le measurable_const hm
have hmono := integral_mono ((integrable_const (s ^ 2)).indicator hE) hint
(fun ω => by
by_cases hω : ω ∈ E
· rw [indicator_of_mem hω]
have hmiss : s ≤ |T ω - θ| := hω
simpa only [sq_abs] using (sq_le_sq₀ hs (abs_nonneg _)).2 hmiss
· rw [indicator_of_notMem hω]
exact sq_nonneg _)
rw [integral_indicator_const _ hE] at hmono
simpa only [smul_eq_mul] using hmono
theorem activated_center_testing_risk_lower Lemma activated_center_testing_risk_lower in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Ω :
Type*
Q₀ Q₁ :
T :
Ω → ℝ
hT :
θ₀ θ₁ :
ℝ
h₀ :
Integrable (fun ω => (T ω - θ₀) ^ 2) Q₀
h₁ :
Integrable (fun ω => (T ω - θ₁) ^ 2) Q₁
Proof (Lean source)
-- @node: activated_center_testing_risk_lower
lemma activated_center_testing_risk_lower {Ω : Type*} [MeasurableSpace Ω]
(Q₀ Q₁ : Measure Ω) [IsProbabilityMeasure Q₀] [IsProbabilityMeasure Q₁]
(T : Ω → ℝ) (hT : Measurable T) (θ₀ θ₁ : ℝ)
(h₀ : Integrable (fun ω => (T ω - θ₀) ^ 2) Q₀)
(h₁ : Integrable (fun ω => (T ω - θ₁) ^ 2) Q₁) :
|θ₁ - θ₀| ^ 2 / 8 * (1 - tvDist Q₀ Q₁) ≤
max (∫ ω, (T ω - θ₀) ^ 2 ∂Q₀) (∫ ω, (T ω - θ₁) ^ 2 ∂Q₁) := by
let s := |θ₁ - θ₀| / 2
have hs : 0 ≤ s := by dsimp [s]; positivity
have hsep : 2 * s ≤ |θ₀ - θ₁| := by dsimp [s]; rw [abs_sub_comm]; linarith
have htest := real_two_point_lower_bound
(P₀ := Q₀) (P₁ := Q₁) hT hsep
have hr₀ := activated_error_probability_mul_sq_le_risk Q₀ T hT θ₀ s hs h₀
have hr₁ := activated_error_probability_mul_sq_le_risk Q₁ T hT θ₁ s hs h₁
have hmax : max (Q₀.real {ω | s ≤ |T ω - θ₀|})
(Q₁.real {ω | s ≤ |T ω - θ₁|}) * s ^ 2 ≤
max (∫ ω, (T ω - θ₀) ^ 2 ∂Q₀) (∫ ω, (T ω - θ₁) ^ 2 ∂Q₁) := by
rw [max_mul_of_nonneg _ _ (sq_nonneg s)]
exact max_le_max hr₀ hr₁
have h := (mul_le_mul_of_nonneg_right htest (sq_nonneg s)).trans hmax
calc
_ = (1 - tvDist Q₀ Q₁) / 2 * s ^ 2 := by dsimp [s]; ring
_ ≤ _ := h
theorem activated_fuzzy_risk_variance_subtraction Lemma activated_fuzzy_risk_variance_subtraction in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Δ J b R C₀ C₁ tv :
ℝ
htv :
tv ≤ 1 / 8
htest :
Δ ^ 2 / 8 * (1 - tv) ≤ max C₀ C₁
h₀ :
C₀ ≤ 2 * R + J * b ^ 2 / 2
h₁ :
C₁ ≤ 2 * R + J * b ^ 2 / 2
conclusion
7 * Δ ^ 2 / 128 - J * b ^ 2 / 4 ≤ R
Proof (Lean source)
-- @node: activated_fuzzy_risk_variance_subtraction
lemma activated_fuzzy_risk_variance_subtraction (Δ J b R C₀ C₁ tv : ℝ)
(htv : tv ≤ 1 / 8)
(htest : Δ ^ 2 / 8 * (1 - tv) ≤ max C₀ C₁)
(h₀ : C₀ ≤ 2 * R + J * b ^ 2 / 2)
(h₁ : C₁ ≤ 2 * R + J * b ^ 2 / 2) :
7 * Δ ^ 2 / 128 - J * b ^ 2 / 4 ≤ R := by
have hupper := max_le h₀ h₁
have hbudget := mul_le_mul_of_nonneg_left (show 7 / 8 ≤ 1 - tv by linarith)
(show 0 ≤ Δ ^ 2 / 8 by positivity)
linarith [hbudget.trans (htest.trans hupper)]
theorem activated_fuzzy_risk_absorb_variance Lemma activated_fuzzy_risk_absorb_variance in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
J :
ℕ
b Δ R :
ℝ
hJ :
2048 ≤ J
hb :
0 ≤ b
hgap :
(J : ℝ) * b / 12 ≤ Δ
hrisk :
7 * Δ ^ 2 / 128 - (J : ℝ) * b ^ 2 / 4 ≤ R
conclusion
(1 / 4096) * ((J : ℝ) * b) ^ 2 ≤ R
Proof (Lean source)
-- @node: activated_fuzzy_risk_absorb_variance
lemma activated_fuzzy_risk_absorb_variance (J : ℕ) (b Δ R : ℝ)
(hJ : 2048 ≤ J) (hb : 0 ≤ b) (hgap : (J : ℝ) * b / 12 ≤ Δ)
(hrisk : 7 * Δ ^ 2 / 128 - (J : ℝ) * b ^ 2 / 4 ≤ R) :
(1 / 4096) * ((J : ℝ) * b) ^ 2 ≤ R := by
have hJreal : (2048 : ℝ) ≤ J := by exact_mod_cast hJ
have hΔ : 0 ≤ Δ := (by positivity : 0 ≤ (J : ℝ) * b / 12).trans hgap
have hsq := (sq_le_sq₀ (by positivity : 0 ≤ (J : ℝ) * b / 12) hΔ).2 hgap
have hpenalty := mul_le_mul_of_nonneg_right hJreal
(show 0 ≤ (J : ℝ) * b ^ 2 by positivity)
nlinarith
theorem activated_fuzzy_risk_lower_of_center_comparison Lemma activated_fuzzy_risk_lower_of_center_comparison in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Ω :
Type*
Q₀ Q₁ :
T :
Ω → ℝ
hT :
θ₀ θ₁ b R :
ℝ
J :
ℕ
hJ :
2048 ≤ J
hb :
0 ≤ b
hgap :
(J : ℝ) * b / 12 ≤ |θ₁ - θ₀|
htv :
tvDist Q₀ Q₁ ≤ 1 / 8
hint₀ :
Integrable (fun ω => (T ω - θ₀) ^ 2) Q₀
hint₁ :
Integrable (fun ω => (T ω - θ₁) ^ 2) Q₁
hcenter₀ :
(∫ ω, (T ω - θ₀) ^ 2 ∂Q₀) ≤ 2 * R + (J : ℝ) * b ^ 2 / 2
hcenter₁ :
(∫ ω, (T ω - θ₁) ^ 2 ∂Q₁) ≤ 2 * R + (J : ℝ) * b ^ 2 / 2
conclusion
(1 / 4096) * ((J : ℝ) * b) ^ 2 ≤ R
Proof (Lean source)
-- @node: activated_fuzzy_risk_lower_of_center_comparison
lemma activated_fuzzy_risk_lower_of_center_comparison
{Ω : Type*} [MeasurableSpace Ω]
(Q₀ Q₁ : Measure Ω) [IsProbabilityMeasure Q₀] [IsProbabilityMeasure Q₁]
(T : Ω → ℝ) (hT : Measurable T) (θ₀ θ₁ b R : ℝ) (J : ℕ)
(hJ : 2048 ≤ J) (hb : 0 ≤ b)
(hgap : (J : ℝ) * b / 12 ≤ |θ₁ - θ₀|)
(htv : tvDist Q₀ Q₁ ≤ 1 / 8)
(hint₀ : Integrable (fun ω => (T ω - θ₀) ^ 2) Q₀)
(hint₁ : Integrable (fun ω => (T ω - θ₁) ^ 2) Q₁)
(hcenter₀ : (∫ ω, (T ω - θ₀) ^ 2 ∂Q₀) ≤ 2 * R + (J : ℝ) * b ^ 2 / 2)
(hcenter₁ : (∫ ω, (T ω - θ₁) ^ 2 ∂Q₁) ≤ 2 * R + (J : ℝ) * b ^ 2 / 2) :
(1 / 4096) * ((J : ℝ) * b) ^ 2 ≤ R := by
have htest := activated_center_testing_risk_lower Q₀ Q₁ T hT θ₀ θ₁ hint₀ hint₁
have hsubtract := activated_fuzzy_risk_variance_subtraction
|θ₁ - θ₀| (J : ℝ) b R _ _ (tvDist Q₀ Q₁)
htv htest hcenter₀ hcenter₁
exact activated_fuzzy_risk_absorb_variance J b |θ₁ - θ₀| R hJ hb hgap hsubtract
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.Lower.FuzzyTesting 2 declarations Fuzzy testing comparison in the original fixed-sample experiment.
Fuzzy testing comparison in the original fixed-sample experiment.
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Proof (Lean source)
-- @node: activated_min_one_sq
lemma activated_min_one_sq (u : ℝ) (hu : 0 ≤ u) :
(min 1 u) ^ 2 = min 1 (u ^ 2) := by
by_cases h : u ≤ 1
· rw [min_eq_right h, min_eq_right (by nlinarith : u ^ 2 ≤ 1)]
· have h' : 1 ≤ u := le_of_not_ge h
rw [min_eq_left h', min_eq_left (by nlinarith : 1 ≤ u ^ 2)]
norm_num
theorem activated_fuzzy_lower
the stated mathematical conclusion holds.
Formal statement
conclusion
∃ c : ℝ,
0 < c
conclusion 1
Proof (Lean source)
lemma activated_fuzzy_lower
: ∃ c : ℝ, 0 < c ∧
∀ (n d : ℕ) (q : ℝ), 1 ≤ n → 1 ≤ d → 0 < q → q ≤ 1 →
RareArrivalSlice n q → sqrt (effectiveSize n q) ≤ d →
c * min 1 (((d : ℝ) / (effectiveSize n q * logScale n q)) ^ 2) ≤
minimaxRisk n d q := by
let η := exp (-32)
let T := max 1 (max (exp 2) (max (4 * η)
(max (2 * η * 2048) (exp ((32 - log ((2 : ℝ) / 16)) / 28)))))
let c := min (1 / (256 * T)) (min (1 / (256 * (2048 : ℝ) ^ 2)) (η ^ 2 / 16384))
have hT : 1 ≤ T := le_max_left _ _
have hη : 0 < η := Real.exp_pos _
have hc : 0 < c := by dsimp [c]; positivity
refine ⟨c, hc, ?_⟩
intro n d q hn hd hq hq1 hslice _hdim
let N := effectiveSize n q
let L := logScale n q
let u := (d : ℝ) / (N * L)
have hN : 0 < N := by dsimp [N, effectiveSize]; positivity
have hL : 1 ≤ L := by
dsimp [L, logScale]
have h := Real.log_le_log (Real.exp_pos (1 : ℝ))
(show exp 1 ≤ exp 1 + effectiveSize n q by linarith)
simpa using h
have hu : 0 ≤ u := by dsimp [u]; positivity
have hcap : 0 ≤ min 1 (u ^ 2) := le_min (by norm_num) (sq_nonneg _)
have hone := (oneCell_minimax_risk_floors q hn hd hq hq1 hslice).1
have hcT : c ≤ 1 / (256 * T) := min_le_left _ _
have hcD : c ≤ 1 / (256 * (2048 : ℝ) ^ 2) :=
(min_le_right _ _).trans (min_le_left _ _)
have hcη : c ≤ η ^ 2 / 16384 :=
(min_le_right _ _).trans (min_le_right _ _)
change c * min 1 (u ^ 2) ≤ minimaxRisk n d q
by_cases hlarge : T ≤ N
· have hN1 : 1 ≤ N := hT.trans hlarge
have hTexp : exp 2 ≤ N :=
(le_max_left _ _).trans ((le_max_right _ _).trans hlarge)
have hTmass : 4 * η ≤ N :=
(le_max_left _ _).trans ((le_max_right _ _).trans
((le_max_right _ _).trans hlarge))
have hTcapacity : 2 * η * 2048 ≤ N :=
(le_max_left _ _).trans ((le_max_right _ _).trans
((le_max_right _ _).trans ((le_max_right _ _).trans hlarge)))
have hTtail : exp ((32 - log ((2 : ℝ) / 16)) / 28) ≤ N :=
(le_max_right _ _).trans ((le_max_right _ _).trans
((le_max_right _ _).trans ((le_max_right _ _).trans hlarge)))
have hcapacity := interval_capacity_of_large_size η 2048 n q hη hN hL hTcapacity
by_cases hJ : 2048 ≤ rareCount η n d q
· have hK : 2 ≤ lowerDegree n q := by
have hlog := Real.log_le_log (Real.exp_pos (2 : ℝ))
(hTexp.trans (show N ≤ exp 1 + N by linarith [Real.exp_pos (1 : ℝ)]))
rw [Real.log_exp] at hlog
have hceil := Nat.le_ceil (logScale n q)
have : (2 : ℝ) ≤ (lowerDegree n q : ℝ) := hlog.trans hceil
exact_mod_cast this
obtain ⟨π₀, π₁, h₀, h₁, hm, hgap⟩ :=
interval_ordered_reciprocal_priors n q hK
have hgapabs : 1 / 12 ≤ |(∫ z, z⁻¹ ∂π₁) - ∫ z, z⁻¹ ∂π₀| :=
hgap.trans (le_abs_self _)
have hrisk := activatedAugmentedMixture_minimaxRisk_calibrated n d q
hn hd hq hslice h₀ h₁ hm hgapabs hJ hTtail
have hb : 0 < rareMass η n q := by unfold rareMass; positivity
have hbsmall : rareMass η n q ≤ 1 / 4 := by
unfold rareMass
apply (div_le_iff₀ (by positivity : 0 < effectiveSize n q * logScale n q)).mpr
nlinarith [mul_le_mul_of_nonneg_left hL hN.le]
have hηsmall : η ≤ 1 / 2 := by
dsimp [η]
rw [Real.exp_neg, inv_eq_one_div]
apply (div_le_iff₀ (Real.exp_pos 32)).mpr
linarith [Real.add_one_le_exp (32 : ℝ)]
have hd2 : 2 ≤ d := by
have hjd : rareCount η n d q ≤ d - 1 := min_le_left _ _
omega
have hwidth := interval_prior_segment_width_lower η
((rareCount η n d q : ℝ) * rareMass η n q / 12) n d q
hη hηsmall hd2 hN (by linarith) hbsmall le_rfl
have hmass : η / 2 * min 1 u ≤ (rareCount η n d q : ℝ) * rareMass η n q := by
change η / 24 * min 1 u ≤ _ at hwidth
linarith
have hsq := (sq_le_sq₀ (by positivity : 0 ≤ η / 2 * min 1 u)
(by positivity : 0 ≤ (rareCount η n d q : ℝ) * rareMass η n q)).2 hmass
rw [mul_pow, activated_min_one_sq u hu] at hsq
have hbound : c * min 1 (u ^ 2) ≤
(1 / 4096) * ((rareCount η n d q : ℝ) * rareMass η n q) ^ 2 := by
nlinarith [mul_le_mul_of_nonneg_right hcη hcap]
exact hbound.trans hrisk
· have hdsmall := rareCount_small_dimension η n d 2048 q hd hcapacity (by omega)
have hscale := interval_small_dimension_scale n d 2048 q hN1 hL hdsmall
change min 1 u ≤ (2048 : ℝ) * (sqrt N)⁻¹ at hscale
have hs := (sq_le_sq₀ (le_min (by norm_num) hu) (by positivity)).2 hscale
rw [activated_min_one_sq u hu, mul_pow, inv_pow, Real.sq_sqrt hN.le] at hs
have hmin : min 1 N⁻¹ = N⁻¹ := min_eq_right (inv_le_one_of_one_le₀ hN1)
change (1 / 256 : ℝ) * min 1 N⁻¹ ≤ _ at hone
rw [hmin] at hone
have hbnd := mul_le_mul_of_nonneg_left hs hc.le
have hcoef : c * (2048 : ℝ) ^ 2 ≤ 1 / 256 := by
nlinarith [mul_le_mul_of_nonneg_right hcD (by positivity : 0 ≤ (2048 : ℝ) ^ 2)]
have hbound := mul_le_mul_of_nonneg_right hcoef (inv_nonneg.mpr hN.le)
exact hbnd.trans (by simpa only [mul_assoc] using hbound.trans hone)
· have hsmall : N ≤ T := le_of_not_ge hlarge
have hrecip : T⁻¹ ≤ min 1 N⁻¹ := by
apply le_min (inv_le_one_of_one_le₀ hT)
exact (inv_le_inv₀ (show 0 < T by linarith [hT]) hN).mpr hsmall
have hc' : c ≤ (1 / 256 : ℝ) * T⁻¹ := by
convert hcT using 1
ring
have hbound := mul_le_mul_of_nonneg_left hrecip (by norm_num : (0 : ℝ) ≤ 1 / 256)
have hcap1 := mul_le_mul_of_nonneg_left (min_le_left (1 : ℝ) (u ^ 2)) hc.le
exact hcap1.trans (by simpa only [mul_one] using (hc'.trans hbound).trans hone)
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.MomentPriors 3 declarations Finite reciprocal-moment priors from the alternation dual and the cited gate.
Finite reciprocal-moment priors from the alternation dual and the cited gate.
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
K :
ℕ
hK :
3 ≤ K
conclusion
(3 / 32 : ℝ) ≤ (((K : ℝ) - 1) / ((K : ℝ) + 1)) ^ K
Proof (Lean source)
lemma reciprocal_ratio_power_lower (K : ℕ) (hK : 3 ≤ K) :
(3 / 32 : ℝ) ≤ (((K : ℝ) - 1) / ((K : ℝ) + 1)) ^ K := by
let k : ℝ := K
have hk : (3 : ℝ) ≤ k := by
change (3 : ℝ) ≤ (K : ℝ)
exact_mod_cast hK
have hk0 : 0 < k := by linarith
let x : ℝ := k⁻¹
have hx0 : 0 ≤ x := by dsimp [x]; positivity
have hx1 : x < 1 := by
dsimp [x]
have h : 1 / k < 1 := (div_lt_iff₀ hk0).2 (by linarith)
simpa only [one_div] using h
have hlog := Real.log_div_le_sum_range_add hx0 hx1 0
simp only [Finset.range_zero, Finset.sum_empty, zero_add, pow_one,
mul_zero] at hlog
have hk1 : 0 < k - 1 := by linarith
have hk2 : 0 < k + 1 := by linarith
have hr : 0 < (k - 1) / (k + 1) := div_pos hk1 hk2
have hratio : (1 + x) / (1 - x) = ((k - 1) / (k + 1))⁻¹ := by
dsimp [x]
field_simp
rw [hratio, Real.log_inv] at hlog
have hbound : k * log ((k - 1) / (k + 1)) ≥ -(9 / 4 : ℝ) := by
have hx : x = 1 / k := by simp [x]
rw [hx] at hlog
have hden : 0 < 1 - (1 / k)^2 := by
have : (1 / k : ℝ) < 1 := by simpa [x] using hx1
nlinarith [sq_nonneg (1 - 1 / k)]
have hsq : 8 * k^2 ≤ 9 * (k^2 - 1) := by nlinarith
have hfrac : 2 * k^2 / (k^2 - 1) ≤ 9 / 4 := by
apply (div_le_iff₀ (by nlinarith : 0 < k^2 - 1)).2
nlinarith
have heq : k * (2 * (1 / k) / (1 - (1 / k)^2)) =
2 * k^2 / (k^2 - 1) := by field_simp
have hlog2 : -log ((k - 1) / (k + 1)) ≤
2 * (1 / k) / (1 - (1 / k)^2) := by
have h := mul_le_mul_of_nonneg_left hlog (by norm_num : (0 : ℝ) ≤ 2)
simpa [div_eq_mul_inv, mul_assoc] using h
have h := mul_le_mul_of_nonneg_left hlog2 hk0.le
nlinarith
have hpow : log (((k - 1) / (k + 1)) ^ K) =
k * log ((k - 1) / (k + 1)) := by
rw [Real.log_pow]
have hExp : exp (-(9 / 4 : ℝ)) ≤ ((k - 1) / (k + 1)) ^ K := by
calc
_ ≤ exp (k * log ((k - 1) / (k + 1))) :=
Real.exp_monotone hbound
_ = _ := by rw [← hpow, Real.exp_log (pow_pos hr _)]
have hnum : exp (9 / 4 : ℝ) ≤ 32 / 3 := by
have h1 : exp 1 ≤ 11 / 4 :=
le_of_lt (Real.exp_one_lt_d9.trans (by norm_num))
have hq : exp (1 / 4 : ℝ) ≤ 9 / 7 := by
convert Real.exp_le_two_add_div_two_sub (x := (1 / 4 : ℝ))
(by norm_num) (by norm_num) using 1
norm_num
have he : exp (9 / 4 : ℝ) =
(exp 1)^2 * exp (1 / 4 : ℝ) := by
rw [show (9 / 4 : ℝ) = 1 + 1 + 1 / 4 by norm_num,
Real.exp_add, Real.exp_add]
ring
rw [he]
nlinarith [Real.exp_pos (1 / 4 : ℝ), Real.exp_pos 1]
have hnum2 : (3 / 32 : ℝ) ≤ exp (-(9 / 4 : ℝ)) := by
rw [Real.exp_neg]
have hh : (3 / 32 : ℝ) ≤ 1 / exp (9 / 4 : ℝ) :=
(le_div_iff₀ (Real.exp_pos _)).2 (by nlinarith)
simpa only [one_div] using hh
exact hnum2.trans hExp
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
K :
ℕ
hK :
2 ≤ K
conclusion
(1 / 12 : ℝ) ≤ 2 * bestUniformApproxError (fun z : ℝ => z⁻¹) 1 ((K : ℝ)^2) K
Proof (Lean source)
lemma reciprocal_best_error_lower (K : ℕ) (hK : 2 ≤ K) :
(1 / 12 : ℝ) ≤
2 * bestUniformApproxError (fun z : ℝ => z⁻¹) 1 ((K : ℝ)^2) K := by
rw [reciprocalBestApproximation_proved K hK]
by_cases h2 : K = 2
· subst K
norm_num
· have h3 : 3 ≤ K := by omega
have hk : (3 : ℝ) ≤ K := by exact_mod_cast h3
have hcoeff : (4 / 9 : ℝ) ≤
(((K : ℝ)^2 - 1) / (2 * (K : ℝ)^2)) := by
apply (le_div_iff₀ (by positivity : 0 < 2 * (K : ℝ)^2)).2
nlinarith
calc
(1 / 12 : ℝ) = 2 * ((4 / 9 : ℝ) * (3 / 32 : ℝ)) := by norm_num
_ ≤ 2 * ((((K : ℝ)^2 - 1) / (2 * (K : ℝ)^2)) *
(((K : ℝ) - 1) / ((K : ℝ) + 1)) ^ K) := by
gcongr
exact reciprocal_ratio_power_lower K h3
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
K :
ℕ
hK :
2 ≤ K
conclusion
∃ prior0 prior1 : Measure ℝ,
IsProbabilityMeasure prior0
IsProbabilityMeasure prior1
conclusion 1
∃ s₀ s₁ : Finset ℝ,
conclusion 2
v :
ℕ
v ≤ K
(∫ z, z ^ v ∂prior0) = ∫ z, z ^ v ∂prior1
conclusion 3
1 / 12 ≤ |(∫ z, z⁻¹ ∂prior1) - (∫ z, z⁻¹ ∂prior0)|
Proof (Lean source)
lemma moment_matched_reciprocal_priors
(K : ℕ) (hK : 2 ≤ K) :
∃ prior0 prior1 : Measure ℝ,
IsProbabilityMeasure prior0 ∧ IsProbabilityMeasure prior1 ∧
(∃ s₀ s₁ : Finset ℝ,
prior0 (s₀ : Set ℝ) = 1 ∧ prior1 (s₁ : Set ℝ) = 1 ∧
(∀ z ∈ s₀, 1 ≤ z ∧ z ≤ (K : ℝ) ^ 2) ∧
(∀ z ∈ s₁, 1 ≤ z ∧ z ≤ (K : ℝ) ^ 2)) ∧
(∀ v : ℕ, v ≤ K →
(∫ z, z ^ v ∂prior0) = ∫ z, z ^ v ∂prior1) ∧
1 / 12 ≤ |(∫ z, z⁻¹ ∂prior1) - (∫ z, z⁻¹ ∂prior0)| := by
classical
have hKreal : (2 : ℝ) ≤ K := by exact_mod_cast hK
have hrs : (1 : ℝ) < (K : ℝ)^2 := by nlinarith
have hcont : ContinuousOn (fun z : ℝ => z⁻¹) (Icc 1 ((K : ℝ)^2)) := by
apply continuousOn_id.inv₀
intro z hz
have hz1 : 1 ≤ z := hz.1
simpa only [id_eq] using (ne_of_gt (lt_of_lt_of_le zero_lt_one hz1))
obtain ⟨D⟩ := exists_finiteMomentDual hrs hcont K
let C := NormalizedFiniteSignedMomentCertificate.ofFiniteMomentDual D
let p₀ := C.orientedPrior0 (fun z : ℝ => z⁻¹)
let p₁ := C.orientedPrior1 (fun z : ℝ => z⁻¹)
have hp₀ : IsProbabilityMeasure p₀ := inferInstance
have hp₁ : IsProbabilityMeasure p₁ := inferInstance
let s : Finset ℝ := (Set.finite_range C.node).toFinset
have hs : (s : Set ℝ) = range C.node := Set.Finite.coe_toFinset _
have h0 : C.positivePrior (s : Set ℝ) = 1 := by
rw [hs]
calc
_ = C.positivePrior univ :=
(ae_mem_iff_measure_eq
(Set.finite_range C.node).measurableSet.nullMeasurableSet).1
C.jordanPriors_ae_mem_range.1
_ = 1 := measure_univ
have h1 : C.negativePrior (s : Set ℝ) = 1 := by
rw [hs]
calc
_ = C.negativePrior univ :=
(ae_mem_iff_measure_eq
(Set.finite_range C.node).measurableSet.nullMeasurableSet).1
C.jordanPriors_ae_mem_range.2
_ = 1 := measure_univ
have hs₀ : p₀ (s : Set ℝ) = 1 := by
dsimp [p₀, NormalizedFiniteSignedMomentCertificate.orientedPrior0]
split_ifs <;> assumption
have hs₁ : p₁ (s : Set ℝ) = 1 := by
dsimp [p₁, NormalizedFiniteSignedMomentCertificate.orientedPrior1]
split_ifs <;> assumption
have hmem : ∀ z ∈ s, 1 ≤ z ∧ z ≤ (K : ℝ)^2 := by
intro z hz
have hz' : z ∈ (s : Set ℝ) := hz
rw [hs] at hz'
obtain ⟨i, rfl⟩ := hz'
exact D.nodes_mem i
refine ⟨p₀, p₁, hp₀, hp₁, ⟨s, s, hs₀, hs₁, hmem, hmem⟩,
(fun v hv => C.orientedPriors_moments_eq _ v hv), ?_⟩
have hsep := C.orientedPrior_target_separation (fun z : ℝ => z⁻¹)
have htarget : |∑ i, C.weight i * (C.node i)⁻¹| =
bestUniformApproxError (fun z : ℝ => z⁻¹) 1 ((K : ℝ)^2) K :=
D.target_abs_eq
rw [htarget] at hsep
change (1 / 12 : ℝ) ≤ |(∫ z, z⁻¹ ∂p₁) - (∫ z, z⁻¹ ∂p₀)|
rw [show (∫ z, z⁻¹ ∂p₁) - (∫ z, z⁻¹ ∂p₀) =
2 * bestUniformApproxError (fun z : ℝ => z⁻¹) 1 ((K : ℝ)^2) K from hsep]
exact (reciprocal_best_error_lower K hK).trans (le_abs_self _)
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.NearComplete.CompleteArrivalTwoPoint 14 declarations Complete-arrival frontier for the finite randomized experiment.
Complete-arrival frontier for the finite randomized experiment.
theorem complete_arrival_ht_estimator_regular Lemma complete_arrival_ht_estimator_regular in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
conclusion
conclusion 1
Proof (Lean source)
lemma complete_arrival_ht_estimator_regular (n d : ℕ) :
Measurable (completeArrivalHTEstimator n d) ∧
∀ s, completeArrivalHTEstimator n d s ∈ Icc (-1 : ℝ) 1 := by
constructor
· exact measurable_of_finite _
· intro s
unfold completeArrivalHTEstimator clip
constructor
· exact le_max_left _ _
· exact (max_le_iff.mpr ⟨by norm_num, min_le_left _ _⟩)
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma clip_sq_error_le (u v : ℝ) (hv : v ∈ Icc (-1 : ℝ) 1) :
(clip u - v) ^ 2 ≤ (u - v) ^ 2 := by
rcases hv with ⟨hvlo, hvhi⟩
unfold clip
rcases le_total u 1 with hu | hu
· rw [min_eq_right hu]
rcases le_total (-1 : ℝ) u with hlu | hlu
· rw [max_eq_right hlu]
· rw [max_eq_left hlu]
nlinarith
· rw [min_eq_left hu]
rw [max_eq_right (by norm_num : (-1 : ℝ) ≤ 1)]
nlinarith
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma complete_arrival_score_variance_le {d : ℕ} (P : FullLaw d) :
variance
(fun o : ObsRecord d =>
2 * armSign o.A * (if o.RY then (1 : ℝ) else 0)) (P.1.map obs) ≤ 4 := by
let μ := P.1.map obs
letI : IsProbabilityMeasure P.1 := P.2
letI : IsProbabilityMeasure μ :=
Measure.isProbabilityMeasure_map (by fun_prop : AEMeasurable obs P.1)
have hbound : ∀ᵐ o ∂μ,
2 * armSign o.A * (if o.RY then (1 : ℝ) else 0) ∈ Icc (-2) 2 := by
filter_upwards [] with o
cases hA : o.A <;> cases hRY : o.RY <;>
simp [armSign]
have hmeas : AEMeasurable
(fun o : ObsRecord d =>
2 * armSign o.A * (if o.RY then (1 : ℝ) else 0)) μ := by
fun_prop
have h := ProbabilityTheory.variance_le_sq_of_bounded hbound hmeas
norm_num at h ⊢
exact h
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma complete_arrival_no_absent_cell {n d : ℕ} (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d 1 P) (j : Cell d) :
P.1.real {r | inCell r j ∧ r.R = false} = 0 := by
letI : IsProbabilityMeasure P.1 := P.2
have hbound : arrivedCell P j ≤ cellProb P j :=
measureReal_mono (by intro r hr; exact hr.1)
have heq : arrivedCell P j = cellProb P j := by
by_cases hp : 0 < cellProb P j
· exact le_antisymm hbound (by simpa using hP.arrival j hp)
· have hz : cellProb P j = 0 :=
le_antisymm (le_of_not_gt hp) measureReal_nonneg
exact (le_antisymm (hbound.trans_eq hz) measureReal_nonneg).trans hz.symm
have hdiff : {r : FullRecord d | inCell r j ∧ r.R = false} =
{r | inCell r j} \ {r | inCell r j ∧ r.R = true} := by
ext r
cases r.R <;> simp <;> tauto
rw [hdiff, measureReal_sdiff (by intro r hr; exact hr.1) (by simp)]
exact sub_eq_zero.mpr heq.symm
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
∀ᵐ r : FullRecord d ∂P.1, r.R = true
Proof (Lean source)
lemma complete_arrival_almost_sure {n d : ℕ} (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d 1 P) :
∀ᵐ r : FullRecord d ∂P.1, r.R = true := by
letI : IsProbabilityMeasure P.1 := P.2
have hzero : P.1.real {r : FullRecord d | r.R = false} = 0 := by
let μ := P.1.restrict {r : FullRecord d | r.R = false}
have hsum := sum_measureReal_preimage_singleton (μ := μ)
(Finset.univ : Finset (Cell d))
(f := fun r : FullRecord d => (r.A, r.X, r.S)) (by intro; simp)
have hcell (j : Cell d) :
μ.real ((fun r : FullRecord d => (r.A, r.X, r.S)) ⁻¹' {j}) = 0 := by
have hs : ((fun r : FullRecord d => (r.A, r.X, r.S)) ⁻¹' {j}) ∩
{r : FullRecord d | r.R = false} =
{r | inCell r j ∧ r.R = false} := by
ext r
rcases j with ⟨a, x, s⟩
simp [inCell, Prod.mk.injEq, and_assoc, and_left_comm, and_comm]
change (P.1.restrict {r : FullRecord d | r.R = false}).real
((fun r : FullRecord d => (r.A, r.X, r.S)) ⁻¹' {j}) = 0
rw [measureReal_def, Measure.restrict_apply (by simp)]
rw [hs]
exact congrArg toReal (by
exact (measureReal_eq_zero_iff (by simp)).mp
(complete_arrival_no_absent_cell P hP j))
simp_rw [hcell] at hsum
simpa [μ, measureReal_def, Measure.restrict_apply] using hsum.symm
have hzero' : P.1 {r : FullRecord d | r.R = false} = 0 := by
exact (measureReal_eq_zero_iff (by simp)).mp hzero
exact (ae_iff).2 (by simpa using hzero')
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma complete_arrival_score_mean {n d : ℕ} (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d 1 P) :
(∫ o : ObsRecord d,
2 * armSign o.A * (if o.R && o.RY then (1 : ℝ) else 0) ∂(P.1.map obs)) = ate P := by
letI : IsProbabilityMeasure P.1 := P.2
have hmap :
(∫ o : ObsRecord d,
2 * armSign o.A * (if o.R && o.RY then (1 : ℝ) else 0) ∂(P.1.map obs)) =
∫ r : FullRecord d,
2 * armSign r.A * (if r.R && r.Y then (1 : ℝ) else 0) ∂P.1 := by
rw [integral_map (by fun_prop) (by fun_prop)]
simp [obs, Bool.and_assoc]
rw [hmap]
have hae := complete_arrival_almost_sure P hP
have hreplace :
(∫ r : FullRecord d,
2 * armSign r.A * (if r.R && r.Y then (1 : ℝ) else 0) ∂P.1) =
∫ r : FullRecord d,
2 * armSign r.A * (if r.Y then (1 : ℝ) else 0) ∂P.1 := by
apply integral_congr_ae
filter_upwards [hae] with r hr
simp [hr]
rw [hreplace]
let E (a : Bool) : Set (FullRecord d) := {r | r.A = a ∧ r.Y = true}
have hpoint :
(fun r : FullRecord d => 2 * armSign r.A * (if r.Y then (1 : ℝ) else 0)) =
(E true).indicator (fun _ => (2 : ℝ)) -
(E false).indicator (fun _ => (2 : ℝ)) := by
funext r
cases hA : r.A <;> cases hY : r.Y <;>
simp [E, armSign, indicator, hA, hY]
rw [hpoint]
have hmeas (a : Bool) : MeasurableSet (E a) := by simp [E]
change (∫ r : FullRecord d,
(E true).indicator (fun _ => (2 : ℝ)) r -
(E false).indicator (fun _ => (2 : ℝ)) r ∂P.1) = ate P
rw [integral_sub ((integrable_const (2 : ℝ)).indicator (hmeas true))
((integrable_const (2 : ℝ)).indicator (hmeas false))]
rw [integral_indicator_const (2 : ℝ) (hmeas true),
integral_indicator_const (2 : ℝ) (hmeas false)]
rw [ate_eq_potentialOutcomeMass P]
have ht := randomized_armOutcomeMass P hP.randomized hP.balanced true
have hf := randomized_armOutcomeMass P hP.randomized hP.balanced false
simp only [E] at *
simp only [ite_true, Bool.false_eq_true, ite_false] at ht hf
linear_combination ht - hf
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Proof (Lean source)
lemma ate_mem_unit_interval {d : ℕ} (P : FullLaw d) :
ate P ∈ Icc (-1 : ℝ) 1 := by
rw [ate_eq_potentialOutcomeMass]
have h₀ : (0 : ℝ) ≤ P.1.real {r | r.Y0 = true} := measureReal_nonneg
have h₁ : (0 : ℝ) ≤ P.1.real {r | r.Y1 = true} := measureReal_nonneg
have h₀' : P.1.real {r | r.Y0 = true} ≤ 1 := by
letI : IsProbabilityMeasure P.1 := P.2
exact measureReal_le_one
have h₁' : P.1.real {r | r.Y1 = true} ≤ 1 := by
letI : IsProbabilityMeasure P.1 := P.2
exact measureReal_le_one
constructor <;> linarith
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma unrestricted_squaredRisk_bounds {n d : ℕ} (T : Estimator n d)
(P : FullLaw d) : 0 ≤ squaredRisk T P ∧ squaredRisk T P ≤ 4 := by
letI : IsProbabilityMeasure P.1 := P.2
letI : IsMarkovKernel T.toBoundedKernel.1 := T.toBoundedKernel.2.1
have hATE := ate_mem_unit_interval P
rw [Set.mem_Icc] at hATE
have hinner_nonneg (s : Fin n → ObsRecord d) :
0 ≤ ∫ t, (t - ate P) ^ 2 ∂(T.toBoundedKernel.1 s) := by
exact integral_nonneg (fun t => sq_nonneg _)
have hinner_le (s : Fin n → ObsRecord d) :
(∫ t, (t - ate P) ^ 2 ∂(T.toBoundedKernel.1 s)) ≤ 4 := by
haveI : IsProbabilityMeasure (T.toBoundedKernel.1 s) := inferInstance
have hs : ∀ᵐ t ∂(T.toBoundedKernel.1 s), t ∈ Icc (-1 : ℝ) 1 :=
(mem_ae_iff_prob_eq_one measurableSet_Icc).mpr (T.toBoundedKernel.2.2 s)
have hb : ∀ᵐ t ∂(T.toBoundedKernel.1 s), (t - ate P) ^ 2 ≤ 4 := by
filter_upwards [hs] with t ht
rw [Set.mem_Icc] at ht
nlinarith [sq_nonneg (t - ate P + 2), sq_nonneg (t - ate P - 2)]
simpa using (integral_mono_of_nonneg (μ := T.toBoundedKernel.1 s)
(Filter.Eventually.of_forall (fun t => sq_nonneg (t - ate P)))
(integrable_const 4) hb)
haveI : IsProbabilityMeasure (sampleLaw n P) := by
unfold sampleLaw
letI : IsProbabilityMeasure (P.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop : AEMeasurable obs P.1)
infer_instance
constructor
· exact integral_nonneg hinner_nonneg
· simpa [squaredRisk] using (integral_mono_of_nonneg (μ := sampleLaw n P)
(Filter.Eventually.of_forall hinner_nonneg) (integrable_const 4)
(Filter.Eventually.of_forall hinner_le))
theorem unrestricted_minimax_lower_of_two_point Lemma unrestricted_minimax_lower_of_two_point in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q c :
ℝ
P₀ P₁ :
FullLaw d
hP₀ :
UnrestrictedArrivalModelClass n d q P₀
hP₁ :
UnrestrictedArrivalModelClass n d q P₁
conclusion
c ≤ unrestrictedMinimaxRisk n d q
Proof (Lean source)
lemma unrestricted_minimax_lower_of_two_point {n d : ℕ} {q c : ℝ}
(P₀ P₁ : FullLaw d)
(hP₀ : UnrestrictedArrivalModelClass n d q P₀)
(hP₁ : UnrestrictedArrivalModelClass n d q P₁)
(htwo : ∀ T : Estimator n d,
c ≤ max (squaredRisk T P₀) (squaredRisk T P₁)) :
c ≤ unrestrictedMinimaxRisk n d q := by
let T₀ : Estimator n d :=
Estimator.ofMap (fun _ => 0) ⟨measurable_const, by intro s; norm_num⟩
letI : Nonempty (Estimator n d) := ⟨T₀⟩
let θ₀ : {P : FullLaw d // UnrestrictedArrivalModelClass n d q P} := ⟨P₀, hP₀⟩
let θ₁ : {P : FullLaw d // UnrestrictedArrivalModelClass n d q P} := ⟨P₁, hP₁⟩
unfold unrestrictedMinimaxRisk
apply le_minimaxValue_of_two_point θ₀ θ₁
· intro T
refine ⟨4, ?_⟩
rintro r ⟨θ, rfl⟩
exact (unrestricted_squaredRisk_bounds T θ.1).2
· intro T
exact htwo T
theorem complete_arrival_chisq_testing_floor Lemma complete_arrival_chisq_testing_floor in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma complete_arrival_chisq_testing_floor {n d : ℕ} (hn : 1 ≤ n)
(P₀ P₁ : FullLaw d)
(hac : (P₁.1.map obs) ≪ (P₀.1.map obs))
(hchi : chiSqDiv (P₁.1.map obs) (P₀.1.map obs) ≤
(1 / 8 : ℝ) / n) :
∀ A : Set (Fin n → ObsRecord d), MeasurableSet A →
(sampleLaw n P₁).real Aᶜ + (sampleLaw n P₀).real A ≥ 1 / 2 := by
letI : IsProbabilityMeasure P₀.1 := P₀.2
letI : IsProbabilityMeasure P₁.1 := P₁.2
letI : IsProbabilityMeasure (P₀.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
letI : IsProbabilityMeasure (P₁.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
simpa only [sampleLaw] using
complete_arrival_product_testing_half n hn
(P₁.1.map obs) (P₀.1.map obs) hac (Integrable.of_finite) hchi
theorem complete_arrival_kernel_mean_risk_le Lemma complete_arrival_kernel_mean_risk_le in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
deterministicRisk (kernelMean T.toBoundedKernel.1 clip) P ≤ squaredRisk T P
Proof (Lean source)
lemma complete_arrival_kernel_mean_risk_le {n d : ℕ} (T : Estimator n d)
(P : FullLaw d) :
deterministicRisk (Causalean.Stat.kernelMean T.toBoundedKernel.1 clip) P ≤
squaredRisk T P := by
letI : IsMarkovKernel T.toBoundedKernel.1 := T.toBoundedKernel.2.1
letI : IsProbabilityMeasure P.1 := P.2
letI : IsProbabilityMeasure (P.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
letI : IsProbabilityMeasure (sampleLaw n P) := by
unfold sampleLaw
infer_instance
have hclip_meas : Measurable clip := by
unfold clip
fun_prop
have hclip_bound : UniformlyBounded clip := by
refine ⟨1, by norm_num, ?_⟩
intro t
rw [abs_le]
simp [clip]
have hclip_abs (t : ℝ) : |clip t| ≤ 1 := by
rw [abs_le]
constructor <;> simp [clip]
let f := Causalean.Stat.kernelMean T.toBoundedKernel.1 clip
have hf : Measurable f := measurable_kernelMean T.toBoundedKernel.1 hclip_meas
have hpoint (s : Fin n → ObsRecord d) :
(f s - ate P) ^ 2 ≤ ∫ t, (t - ate P) ^ 2 ∂(T.toBoundedKernel.1 s) := by
calc
_ ≤ ∫ t, (clip t - ate P) ^ 2 ∂(T.toBoundedKernel.1 s) :=
sqLoss_kernelMean_le T.toBoundedKernel.1 hclip_meas
hclip_bound (ate P) s
_ = _ := by
apply integral_congr_ae
filter_upwards [(mem_ae_iff_prob_eq_one measurableSet_Icc).mpr (T.toBoundedKernel.2.2 s)]
with t ht
simp [clip, ht.1, ht.2]
have hfbound : ∀ s, f s ∈ Icc (-1 : ℝ) 1 := by
intro s
rw [Set.mem_Icc, ← abs_le]
exact abs_kernelMean_le T.toBoundedKernel.1 (M := 1)
hclip_abs s
have hmeas : Measurable (fun s ↦ ∫ t, (t - ate P) ^ 2 ∂(T.toBoundedKernel.1 s)) := by
exact ((measurable_id.sub measurable_const).pow_const 2).stronglyMeasurable
|>.integral_kernel.measurable
have hbound (s : Fin n → ObsRecord d) :
|(∫ t, (t - ate P) ^ 2 ∂(T.toBoundedKernel.1 s))| ≤ 4 := by
have hATE := ate_mem_unit_interval P
have hs : ∀ᵐ t ∂(T.toBoundedKernel.1 s), t ∈ Icc (-1 : ℝ) 1 :=
(mem_ae_iff_prob_eq_one measurableSet_Icc).mpr (T.toBoundedKernel.2.2 s)
have hle : ∀ᵐ t ∂(T.toBoundedKernel.1 s), (t - ate P) ^ 2 ≤ 4 := by
filter_upwards [hs] with t ht
rw [Set.mem_Icc] at ht hATE
nlinarith [sq_nonneg (t - ate P + 2), sq_nonneg (t - ate P - 2)]
have hnon : 0 ≤ ∫ t, (t - ate P) ^ 2 ∂(T.toBoundedKernel.1 s) :=
integral_nonneg (fun _ ↦ sq_nonneg _)
rw [abs_of_nonneg hnon]
simpa using integral_mono_of_nonneg
(Filter.Eventually.of_forall fun t ↦ sq_nonneg _)
(integrable_const 4) hle
unfold deterministicRisk squaredRisk
exact integral_mono_of_nonneg
(Filter.Eventually.of_forall fun _ ↦ sq_nonneg _)
(Integrable.of_bound hmeas.aestronglyMeasurable 4
(Filter.Eventually.of_forall fun s ↦ by
simpa [Real.norm_eq_abs] using hbound s))
(Filter.Eventually.of_forall hpoint)
theorem complete_arrival_two_point_decision_reduction Lemma complete_arrival_two_point_decision_reduction in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma complete_arrival_two_point_decision_reduction {n d : ℕ}
(P₀ P₁ : FullLaw d) (δ c : ℝ) (hδ : 0 < δ) (hc : 0 ≤ c)
(hgap : ate P₁ = ate P₀ + δ)
(hfloor : ∀ A : Set (Fin n → ObsRecord d), MeasurableSet A →
(sampleLaw n P₁).real Aᶜ + (sampleLaw n P₀).real A ≥ c) :
∀ T : Estimator n d,
c * δ ^ 2 / 8 ≤ max (squaredRisk T P₀) (squaredRisk T P₁) := by
intro T
let f := Causalean.Stat.kernelMean T.toBoundedKernel.1 clip
let E₀ : Set (Fin n → ObsRecord d) := {s | δ / 2 ≤ |f s - ate P₀|}
let E₁ : Set (Fin n → ObsRecord d) := {s | δ / 2 ≤ |f s - ate P₁|}
let A : Set (Fin n → ObsRecord d) :=
{s | |f s - ate P₁| ≤ |f s - ate P₀|}
letI : IsMarkovKernel T.toBoundedKernel.1 := T.toBoundedKernel.2.1
letI : IsProbabilityMeasure P₀.1 := P₀.2
letI : IsProbabilityMeasure P₁.1 := P₁.2
letI : IsProbabilityMeasure (P₀.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
letI : IsProbabilityMeasure (P₁.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
letI : IsProbabilityMeasure (sampleLaw n P₀) := by
unfold sampleLaw
infer_instance
letI : IsProbabilityMeasure (sampleLaw n P₁) := by
unfold sampleLaw
infer_instance
have hf : Measurable f := measurable_kernelMean T.toBoundedKernel.1 (by
unfold clip
fun_prop)
have hA : MeasurableSet A := by
exact measurableSet_le
(continuous_abs.measurable.comp (hf.sub measurable_const))
(continuous_abs.measurable.comp (hf.sub measurable_const))
have htri (s : Fin n → ObsRecord d) :
δ ≤ |f s - ate P₀| + |f s - ate P₁| := by
have ht := abs_sub_le (ate P₁) (f s) (ate P₀)
have hdiff : |ate P₁ - ate P₀| = δ := by
rw [hgap]
simpa using abs_of_pos hδ
calc
δ = |ate P₁ - ate P₀| := hdiff.symm
_ ≤ _ := by simpa [abs_sub_comm, add_comm] using ht
have hA_E₀ : A ⊆ E₀ := by
intro s hs
simp only [A, E₀, Set.mem_setOf_eq] at hs ⊢
linarith [htri s]
have hAc_E₁ : Aᶜ ⊆ E₁ := by
intro s hs
simp only [A, E₁, Set.mem_compl_iff, Set.mem_setOf_eq, not_le] at hs ⊢
linarith [htri s]
have htest : c ≤ (sampleLaw n P₁).real E₁ +
(sampleLaw n P₀).real E₀ := by
calc
c ≤ (sampleLaw n P₁).real Aᶜ +
(sampleLaw n P₀).real A := hfloor A hA
_ ≤ _ := add_le_add
(measureReal_mono hAc_E₁ (by finiteness))
(measureReal_mono hA_E₀ (by finiteness))
have hbound (P : FullLaw d) : ∀ s, f s ∈ Icc (-1 : ℝ) 1 := by
intro s
rw [Set.mem_Icc, ← abs_le]
exact abs_kernelMean_le T.toBoundedKernel.1 (M := 1)
(fun t => by rw [abs_le]; constructor <;> simp [clip]) s
have hint₀ : Integrable (fun s ↦ (f s - ate P₀) ^ 2) (sampleLaw n P₀) :=
mse_integrable_of_estimator_bound _ f hf (by norm_num)
(hbound P₀)
have hint₁ : Integrable (fun s ↦ (f s - ate P₁) ^ 2) (sampleLaw n P₁) :=
mse_integrable_of_estimator_bound _ f hf (by norm_num)
(hbound P₁)
have hmse₀ : (δ / 2) ^ 2 * (sampleLaw n P₀).real E₀ ≤
deterministicRisk f P₀ := by
unfold deterministicRisk E₀
have hset : {s | (δ / 2) ^ 2 ≤ (f s - ate P₀) ^ 2} =
{s | δ / 2 ≤ |f s - ate P₀|} := by
ext s
simp only [Set.mem_setOf_eq]
rw [sq_le_sq, abs_of_nonneg (show (0 : ℝ) ≤ δ / 2 by positivity)]
rw [← hset]
exact mul_meas_ge_le_integral_of_nonneg
(Filter.Eventually.of_forall fun s ↦ sq_nonneg _) hint₀ _
have hmse₁ : (δ / 2) ^ 2 * (sampleLaw n P₁).real E₁ ≤
deterministicRisk f P₁ := by
unfold deterministicRisk E₁
have hset : {s | (δ / 2) ^ 2 ≤ (f s - ate P₁) ^ 2} =
{s | δ / 2 ≤ |f s - ate P₁|} := by
ext s
simp only [Set.mem_setOf_eq]
rw [sq_le_sq, abs_of_nonneg (show (0 : ℝ) ≤ δ / 2 by positivity)]
rw [← hset]
exact mul_meas_ge_le_integral_of_nonneg
(Filter.Eventually.of_forall fun s ↦ sq_nonneg _) hint₁ _
have hmax : c / 2 ≤ max ((sampleLaw n P₀).real E₀)
((sampleLaw n P₁).real E₁) := by
nlinarith [le_max_left ((sampleLaw n P₀).real E₀)
((sampleLaw n P₁).real E₁),
le_max_right ((sampleLaw n P₀).real E₀)
((sampleLaw n P₁).real E₁)]
calc
c * δ ^ 2 / 8 ≤ (δ / 2) ^ 2 *
max ((sampleLaw n P₀).real E₀) ((sampleLaw n P₁).real E₁) := by
nlinarith [sq_nonneg δ]
_ = max ((δ / 2) ^ 2 * (sampleLaw n P₀).real E₀)
((δ / 2) ^ 2 * (sampleLaw n P₁).real E₁) := by
by_cases hle : (sampleLaw n P₀).real E₀ ≤
(sampleLaw n P₁).real E₁
· rw [max_eq_right hle, max_eq_right]
exact mul_le_mul_of_nonneg_left hle (sq_nonneg _)
· have hge := le_of_not_ge hle
rw [max_eq_left hge, max_eq_left]
exact mul_le_mul_of_nonneg_left hge (sq_nonneg _)
_ ≤ max (deterministicRisk f P₀) (deterministicRisk f P₁) :=
max_le_max hmse₀ hmse₁
_ ≤ max (squaredRisk T P₀) (squaredRisk T P₁) :=
max_le_max (complete_arrival_kernel_mean_risk_le T P₀)
(complete_arrival_kernel_mean_risk_le T P₁)
theorem complete_arrival_two_point_minimax_floor Lemma complete_arrival_two_point_minimax_floor in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
hn :
1 ≤ n
P₀ P₁ :
FullLaw d
hP₀ :
UnrestrictedArrivalModelClass n d 1 P₀
hP₁ :
UnrestrictedArrivalModelClass n d 1 P₁
δ :
ℝ
hδ :
0 < δ
hgap :
ate P₁ = ate P₀ + δ
hδsq :
δ ^ 2 = 1 / (16 * (n : ℝ))
conclusion
1 / (256 * (n : ℝ)) ≤ unrestrictedMinimaxRisk n d 1
Proof (Lean source)
lemma complete_arrival_two_point_minimax_floor {n d : ℕ} (hn : 1 ≤ n)
(P₀ P₁ : FullLaw d)
(hP₀ : UnrestrictedArrivalModelClass n d 1 P₀)
(hP₁ : UnrestrictedArrivalModelClass n d 1 P₁)
(δ : ℝ) (hδ : 0 < δ) (hgap : ate P₁ = ate P₀ + δ)
(hδsq : δ ^ 2 = 1 / (16 * (n : ℝ)))
(hac : (P₁.1.map obs) ≪ (P₀.1.map obs))
(hchi : chiSqDiv (P₁.1.map obs) (P₀.1.map obs) ≤
(1 / 8 : ℝ) / n) :
1 / (256 * (n : ℝ)) ≤ unrestrictedMinimaxRisk n d 1 := by
apply unrestricted_minimax_lower_of_two_point P₀ P₁ hP₀ hP₁
intro T
have hfloor := complete_arrival_chisq_testing_floor hn P₀ P₁ hac hchi
have hrisk := complete_arrival_two_point_decision_reduction P₀ P₁ δ (1 / 2)
hδ (by norm_num) hgap hfloor T
have hnpos : (0 : ℝ) < n := by exact_mod_cast (lt_of_lt_of_le (by omega : 0 < 1) hn)
rw [hδsq] at hrisk
convert hrisk using 1 <;> ring
theorem complete_arrival_frontier_direct
Uniformly over sample size and alphabet size, the complete-arrival minimax and estimator risks have the displayed parametric bounds.
Formal statement
hypotheses
n :
ℕ
d :
ℕ
n d :
ℕ
1 ≤ n
1 ≤ d
conclusion
conclusion 1
1 / (256 * (n : ℝ)) ≤ unrestrictedMinimaxRisk n d 1
conclusion 2
unrestrictedMinimaxRisk n d 1
≤ worstCaseRiskReal (fun (_ : Unit) (P : {P : FullLaw d // UnrestrictedArrivalModelClass n d 1 P}) => deterministicRisk (completeArrivalHTEstimator n d) P.1) ()
conclusion 3
worstCaseRiskReal (fun (_ : Unit) (P : {P : FullLaw d // UnrestrictedArrivalModelClass n d 1 P}) => deterministicRisk (completeArrivalHTEstimator n d) P.1) ()
≤ 4 / (n : ℝ)
Proof (Lean source)
theorem complete_arrival_frontier_direct :
∀ (n d : ℕ), 1 ≤ n → 1 ≤ d →
1 / (256 * (n : ℝ)) ≤ unrestrictedMinimaxRisk n d 1 ∧
unrestrictedMinimaxRisk n d 1 ≤
Causalean.Stat.worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // UnrestrictedArrivalModelClass n d 1 P}) =>
deterministicRisk (completeArrivalHTEstimator n d) P.1) () ∧
Causalean.Stat.worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // UnrestrictedArrivalModelClass n d 1 P}) =>
deterministicRisk (completeArrivalHTEstimator n d) P.1) () ≤
4 / (n : ℝ) := by
intro n d hn hd
constructor
· let x : Fin d := ⟨0, hd⟩
let δ : ℝ := 1 / (4 * sqrt n)
have hnR : (0 : ℝ) < n := by exact_mod_cast (lt_of_lt_of_le (by omega : 0 < 1) hn)
have hsqrt : 0 < sqrt (n : ℝ) := Real.sqrt_pos.2 hnR
have hδ : 0 < δ := by dsimp [δ]; positivity
have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast hn
have hsqrt1 : (1 : ℝ) ≤ sqrt n := by
rw [← Real.sqrt_one]
exact Real.sqrt_le_sqrt hn1
have hδle : δ ≤ 1 / 4 := by
dsimp [δ]
rw [div_le_iff₀ (by positivity : 0 < 4 * sqrt (n : ℝ))]
nlinarith
have hu₀ : (0 : ℝ) ≤ 1 / 2 ∧ (1 / 2 : ℝ) ≤ 1 := by norm_num
have hu₁ : (0 : ℝ) ≤ 1 / 2 + δ ∧ (1 / 2 + δ : ℝ) ≤ 1 := by
constructor <;> linarith
let P₀ := completeFamilyLaw x (1 / 2) hu₀
let P₁ := completeFamilyLaw x (1 / 2 + δ) hu₁
have hgap : ate P₁ = ate P₀ + δ := by
simp [P₀, P₁, completeFamilyLaw_ate]
have hδsq : δ ^ 2 = 1 / (16 * (n : ℝ)) := by
dsimp [δ]
calc
(1 / (4 * sqrt (n : ℝ))) ^ 2 =
1 / (16 * (sqrt (n : ℝ)) ^ 2) := by ring
_ = 1 / (16 * (n : ℝ)) := by rw [Real.sq_sqrt (le_of_lt hnR)]
have hP₀ : UnrestrictedArrivalModelClass n d 1 P₀ := by
apply completeFamilyLaw_in_unrestricted_class_of_randomized hn hd x (1 / 2) hu₀
exact completeFamilyLaw_randomized x (1 / 2) hu₀
have hP₁ : UnrestrictedArrivalModelClass n d 1 P₁ := by
apply completeFamilyLaw_in_unrestricted_class_of_randomized hn hd x (1 / 2 + δ) hu₁
exact completeFamilyLaw_randomized x (1 / 2 + δ) hu₁
have hac : (P₁.1.map obs) ≪ (P₀.1.map obs) := by
simpa [P₀, P₁] using completeFamilyLaw_obs_ac x (1 / 2 + δ) hu₁
have hchi : chiSqDiv (P₁.1.map obs) (P₀.1.map obs) ≤
(1 / 8 : ℝ) / n := by
have heq : chiSqDiv (P₁.1.map obs) (P₀.1.map obs) =
2 * δ ^ 2 := by
simpa [P₀, P₁] using
completeFamilyLaw_obs_chi x (1 / 2 + δ) hu₁
rw [heq, hδsq]
apply le_of_eq
field_simp [hnR.ne']
ring
exact complete_arrival_two_point_minimax_floor hn P₀ P₁ hP₀ hP₁ δ
hδ hgap hδsq hac hchi
constructor
· let f := completeArrivalHTEstimator n d
have hEstimator := complete_arrival_ht_estimator_regular n d
let T : Estimator n d :=
Estimator.ofMap f hEstimator
have hrisk : ∀ (T : Estimator n d)
(P : {P : FullLaw d // UnrestrictedArrivalModelClass n d 1 P}),
0 ≤ squaredRisk T P.1 := by
intro T P
exact integral_nonneg (fun s => integral_nonneg (fun y => sq_nonneg _))
have hval : unrestrictedMinimaxRisk n d 1 ≤
Causalean.Stat.worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // UnrestrictedArrivalModelClass n d 1 P}) =>
squaredRisk T P.1) () := by
exact Causalean.Stat.minimaxValue_le_worstCaseRisk_of_nonneg hrisk T
have heq :
(fun (_ : Unit) (P : {P : FullLaw d // UnrestrictedArrivalModelClass n d 1 P}) =>
squaredRisk T P.1) =
(fun (_ : Unit) P => deterministicRisk f P.1) := by
funext _ P
simp [squaredRisk, deterministicRisk, T, Estimator.ofMap, Kernel.deterministic_apply]
rw [heq] at hval
exact hval
· unfold Causalean.Stat.worstCaseRiskReal
by_cases hne : Nonempty
{P : FullLaw d // UnrestrictedArrivalModelClass n d 1 P}
· letI := hne
apply ciSup_le
intro P
let μ := P.1.1.map obs
let ξ : ObsRecord d → ℝ := fun o =>
2 * armSign o.A * (if o.R && o.RY then (1 : ℝ) else 0)
letI : IsProbabilityMeasure P.1.1 := P.1.2
letI : IsProbabilityMeasure μ :=
Measure.isProbabilityMeasure_map (by fun_prop : AEMeasurable obs P.1.1)
letI : IsProbabilityMeasure (sampleLaw n P.1) := by
unfold sampleLaw
infer_instance
have htarget : ate P.1 ∈ Icc (-1 : ℝ) 1 :=
ate_mem_unit_interval P.1
have hbound : ∀ᵐ o ∂μ, ξ o ∈ Icc (-2 : ℝ) 2 := by
filter_upwards [] with o
cases hA : o.A <;> cases hR : o.R <;> cases hRY : o.RY <;>
simp [ξ, armSign, hA, hR, hRY]
have hLp : MemLp ξ 2 μ :=
memLp_of_bounded hbound (by fun_prop) 2
have hsquare : (∫ o, (ξ o) ^ 2 ∂μ) ≤ 4 := by
have hint : Integrable (fun o => (ξ o) ^ 2) μ := hLp.integrable_sq
have h := integral_mono hint (integrable_const (4 : ℝ))
simpa using h (fun o => by
cases hA : o.A <;> cases hR : o.R <;> cases hRY : o.RY <;>
norm_num [ξ, armSign, hA, hR, hRY])
have hmean : (∫ o, ξ o ∂μ) = ate P.1 :=
complete_arrival_score_mean P.1 P.2
have hiid := Causalean.Mathlib.Probability.iid_mean_sq_le
μ hn ξ hLp
have hsumLp : MemLp
(fun s : Fin n → ObsRecord d => ∑ i : Fin n, ξ (s i)) 2
(sampleLaw n P.1) := by
simpa [sampleLaw, μ] using
(memLp_finset_sum univ (fun i _ =>
hLp.comp_measurePreserving
(measurePreserving_eval (fun _ : Fin n => μ) i)))
have hdevInt : Integrable
(fun s : Fin n → ObsRecord d =>
((n : ℝ)⁻¹ * ∑ i : Fin n, ξ (s i) - ate P.1) ^ 2)
(sampleLaw n P.1) :=
((hsumLp.const_mul _).sub (memLp_const _)).integrable_sq
have hpoint (s : Fin n → ObsRecord d) :
(completeArrivalHTEstimator n d s - ate P.1) ^ 2 ≤
((n : ℝ)⁻¹ * ∑ i : Fin n, ξ (s i) - ate P.1) ^ 2 := by
have hraw : (2 / (n : ℝ)) * ∑ i : Fin n,
armSign (s i).A *
(if (s i).R && (s i).RY then (1 : ℝ) else 0) =
(n : ℝ)⁻¹ * ∑ i : Fin n, ξ (s i) := by
simp only [ξ]
simp_rw [mul_assoc]
rw [← Finset.mul_sum]
ring
unfold completeArrivalHTEstimator
rw [hraw]
exact clip_sq_error_le _ _ htarget
have hrisk : deterministicRisk (completeArrivalHTEstimator n d) P.1 ≤
(∫ s, ((n : ℝ)⁻¹ * ∑ i : Fin n, ξ (s i) - ate P.1) ^ 2
∂(sampleLaw n P.1)) := by
unfold deterministicRisk
exact integral_mono_of_nonneg (ae_of_all _ (fun s => sq_nonneg _)) hdevInt
(ae_of_all _ hpoint)
calc
deterministicRisk (completeArrivalHTEstimator n d) P.1 ≤ _ := hrisk
_ ≤ (∫ o, (ξ o) ^ 2 ∂μ) / (n : ℝ) := by
simpa [sampleLaw, μ, hmean] using hiid
_ ≤ 4 / (n : ℝ) := by
exact div_le_div_of_nonneg_right hsquare (by positivity)
· haveI : IsEmpty {P : FullLaw d // UnrestrictedArrivalModelClass n d 1 P} :=
not_nonempty_iff.mp hne
simp
positivity
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.NearComplete.ObservedContrast 9 declarations Armwise missing-outcome deficits and the clipped observed-contrast risk envelope.
Armwise missing-outcome deficits and the clipped observed-contrast risk envelope.
def armDeficit
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
theorem sum_armCell_event_mass
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: sum_armCell_event_mass
lemma sum_armCell_event_mass {d : ℕ} (P : FullLaw d) (a : Bool)
(E : Set (FullRecord d)) :
(∑ j : Fin d × Bool, P.1.real {r | inCell r (a, j.1, j.2) ∧ r ∈ E}) =
P.1.real {r | r.A = a ∧ r ∈ E} := by
classical
let : IsProbabilityMeasure P.1 := P.2
let μ := P.1.restrict {r | r.A = a ∧ r ∈ E}
have hsum := sum_measureReal_preimage_singleton (μ := μ)
(Finset.univ : Finset (Fin d × Bool))
(f := fun r : FullRecord d => (r.X, r.S)) (by intro; simp)
calc
(∑ j : Fin d × Bool, P.1.real {r | inCell r (a, j.1, j.2) ∧ r ∈ E}) =
∑ j : Fin d × Bool, μ.real ((fun r : FullRecord d => (r.X, r.S)) ⁻¹' {j}) := by
apply Finset.sum_congr rfl
intro j hj
have hs : {r : FullRecord d | inCell r (a, j.1, j.2) ∧ r ∈ E} =
((fun r : FullRecord d => (r.X, r.S)) ⁻¹' {j}) ∩
{r | r.A = a ∧ r ∈ E} := by
rcases j with ⟨x, s⟩
ext r
simp [inCell, Prod.mk.injEq, and_assoc, and_left_comm, and_comm]
rw [hs]
simp [μ, measureReal_def, Measure.restrict_apply]
_ = μ.real ((fun r : FullRecord d => (r.X, r.S)) ⁻¹'
(Finset.univ : Finset (Fin d × Bool))) := hsum
_ = P.1.real {r | r.A = a ∧ r ∈ E} := by
simp [μ, measureReal_def, Measure.restrict_apply]
theorem absent_cell_mass_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
-- @node: absent_cell_mass_le
lemma absent_cell_mass_le {n d : ℕ} {q : ℝ} (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) (j : Cell d) :
P.1.real {r | inCell r j ∧ r.R = false} ≤ (1 - q) * cellProb P j := by
let : IsProbabilityMeasure P.1 := P.2
have harr : q * cellProb P j ≤ arrivedCell P j := by
by_cases hp : 0 < cellProb P j
· exact hP.arrival j hp
· have hz : cellProb P j = 0 :=
le_antisymm (le_of_not_gt hp) measureReal_nonneg
rw [hz, mul_zero]
exact measureReal_nonneg
have hdiff : {r : FullRecord d | inCell r j ∧ r.R = false} =
{r | inCell r j} \ {r | inCell r j ∧ r.R = true} := by
ext r
cases r.R <;> simp <;> tauto
rw [hdiff, measureReal_sdiff (by intro r hr; exact hr.1) (by simp)]
change cellProb P j - arrivedCell P j ≤ _
linarith
theorem balanced_arm_mass
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
P.1.real {r | r.A = a} = 1 / 2
Proof (Lean source)
-- @node: balanced_arm_mass
lemma balanced_arm_mass {d : ℕ} (P : FullLaw d)
(hB : BalancedRandomization P) (a : Bool) :
P.1.real {r | r.A = a} = 1 / 2 := by
let : IsProbabilityMeasure P.1 := P.2
cases a with
| true => exact hB
| false =>
have hc : {r : FullRecord d | r.A = false} = {r | r.A = true}ᶜ := by
ext r
cases r.A <;> simp
rw [hc, measureReal_compl (by simp), hB]
norm_num
theorem armDeficit_eq_missingOutcomeMass
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: armDeficit_eq_missingOutcomeMass
lemma armDeficit_eq_missingOutcomeMass {d : ℕ} (P : FullLaw d) (a : Bool) :
armDeficit P a = 2 * P.1.real {r | r.A = a ∧ r.R = false ∧ r.Y = true} := by
classical
let : IsProbabilityMeasure P.1 := P.2
unfold armDeficit
have hi : (fun r : FullRecord d =>
if r.A = a ∧ r.R = false ∧ r.Y = true then (1 : ℝ) else 0) =
{r : FullRecord d | r.A = a ∧ r.R = false ∧ r.Y = true}.indicator 1 := by
funext r
simp [indicator]
rw [hi, integral_indicator_one (by simp)]
theorem armDeficit_bounds
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
-- @node: armDeficit_bounds
lemma armDeficit_bounds {n d : ℕ} {q : ℝ} (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) (a : Bool) :
0 ≤ armDeficit P a ∧ armDeficit P a ≤ 1 - q := by
classical
let : IsProbabilityMeasure P.1 := P.2
rw [armDeficit_eq_missingOutcomeMass]
constructor
· positivity
· have hsum : (∑ j : Fin d × Bool, cellProb P (a, j.1, j.2)) = 1 / 2 := by
have h := sum_armCell_event_mass P a univ
simp only [Set.mem_univ, and_true] at h
exact h.trans (balanced_arm_mass P hP.balanced a)
have habsent : P.1.real {r | r.A = a ∧ r.R = false} ≤ (1 - q) / 2 := by
calc
P.1.real {r | r.A = a ∧ r.R = false} =
∑ j : Fin d × Bool, P.1.real {r | inCell r (a, j.1, j.2) ∧ r.R = false} :=
(sum_armCell_event_mass P a {r | r.R = false}).symm
_ ≤ ∑ j : Fin d × Bool, (1 - q) * cellProb P (a, j.1, j.2) :=
Finset.sum_le_sum (fun j _ => absent_cell_mass_le P hP _)
_ = (1 - q) / 2 := by rw [← Finset.mul_sum, hsum]; ring
have hout : P.1.real {r | r.A = a ∧ r.R = false ∧ r.Y = true} ≤
P.1.real {r | r.A = a ∧ r.R = false} :=
measureReal_mono (by intro r hr; exact ⟨hr.1, hr.2.1⟩)
linarith
theorem observed_contrast_score_mean
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: observed_contrast_score_mean
lemma observed_contrast_score_mean {n d : ℕ} {q : ℝ} (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) :
(∫ o : ObsRecord d,
2 * armSign o.A * (if o.R && o.RY then (1 : ℝ) else 0) ∂(P.1.map obs)) =
ate P - armDeficit P true + armDeficit P false := by
classical
let : IsProbabilityMeasure P.1 := P.2
rw [integral_map (by fun_prop) (by fun_prop)]
let E (a : Bool) : Set (FullRecord d) := {r | r.A = a ∧ r.Y = true}
let M (a : Bool) : Set (FullRecord d) :=
{r | r.A = a ∧ r.R = false ∧ r.Y = true}
have hp : (fun r : FullRecord d =>
2 * armSign (obs r).A * (if (obs r).R && (obs r).RY then (1 : ℝ) else 0)) =
(fun r => (E true).indicator (fun _ => (2 : ℝ)) r -
(E false).indicator (fun _ => (2 : ℝ)) r -
(M true).indicator (fun _ => (2 : ℝ)) r +
(M false).indicator (fun _ => (2 : ℝ)) r) := by
funext r
cases hA : r.A <;> cases hR : r.R <;> cases hY : r.Y <;>
simp [obs, armSign, E, M, indicator, hA, hR, hY]
rw [hp]
have hi (a : Bool) : Integrable ((E a).indicator (fun _ => (2 : ℝ))) P.1 := by
exact (integrable_const (2 : ℝ)).indicator (by simp [E])
have hm (a : Bool) : Integrable ((M a).indicator (fun _ => (2 : ℝ))) P.1 := by
exact (integrable_const (2 : ℝ)).indicator (by simp [M])
have hit := hi true
have hif := hi false
have hmt := hm true
have hmf := hm false
integral_linearity
rw [integral_indicator_const (s := E true) (2 : ℝ) (by simp [E]),
integral_indicator_const (s := E false) (2 : ℝ) (by simp [E]),
integral_indicator_const (s := M true) (2 : ℝ) (by simp [M]),
integral_indicator_const (s := M false) (2 : ℝ) (by simp [M])]
simp only [smul_eq_mul]
rw [armDeficit_eq_missingOutcomeMass, armDeficit_eq_missingOutcomeMass,
ate_eq_potentialOutcomeMass]
have ht := randomized_armOutcomeMass P hP.randomized hP.balanced true
have hf := randomized_armOutcomeMass P hP.randomized hP.balanced false
simp only [ite_true, Bool.false_eq_true, ite_false] at ht hf
dsimp only [E, M]
linear_combination ht - hf
theorem observed_contrast_score_bias_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
-- @node: observed_contrast_score_bias_le
lemma observed_contrast_score_bias_le {n d : ℕ} {q : ℝ} (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) :
((∫ o : ObsRecord d,
2 * armSign o.A * (if o.R && o.RY then (1 : ℝ) else 0) ∂(P.1.map obs)) -
ate P) ^ 2 ≤ (1 - q) ^ 2 := by
rw [observed_contrast_score_mean P hP]
have ht := armDeficit_bounds P hP true
have hf := armDeficit_bounds P hP false
have habs : |ate P - armDeficit P true + armDeficit P false - ate P| ≤ 1 - q := by
rw [abs_le]
constructor <;> linarith
have hsq := (sq_le_sq₀ (abs_nonneg _) (sub_nonneg.mpr hP.q_le_one)).mpr habs
simpa only [sq_abs] using hsq
theorem observed_contrast_risk_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
deterministicRisk (completeArrivalHTEstimator n d) P ≤ 4 / (n : ℝ) + (1 - q) ^ 2
Proof (Lean source)
-- @node: observed_contrast_risk_le
lemma observed_contrast_risk_le {n d : ℕ} {q : ℝ} (P : FullLaw d)
(hP : UnrestrictedArrivalModelClass n d q P) :
deterministicRisk (completeArrivalHTEstimator n d) P ≤
4 / (n : ℝ) + (1 - q) ^ 2 := by
classical
let : IsProbabilityMeasure P.1 := P.2
let μ := P.1.map obs
let : IsProbabilityMeasure μ := Measure.isProbabilityMeasure_map (by fun_prop)
let : IsProbabilityMeasure (sampleLaw n P) := by
unfold sampleLaw
infer_instance
let ξ : ObsRecord d → ℝ := fun o =>
2 * armSign o.A * (if o.R && o.RY then (1 : ℝ) else 0)
let V : (Fin n → ObsRecord d) → ℝ := fun s => (n : ℝ)⁻¹ * ∑ i, ξ (s i)
have hξ : MemLp ξ 2 μ := ⟨by fun_prop, eLpNorm_lt_top_of_finite⟩
have hV : MemLp V 2 (sampleLaw n P) := ⟨by fun_prop, eLpNorm_lt_top_of_finite⟩
have hmean : (∫ s, V s ∂sampleLaw n P) = ∫ o, ξ o ∂μ :=
iid_average_integral μ n hP.n_pos ξ
(hξ.integrable (by norm_num))
have hvar : variance V (sampleLaw n P) = (n : ℝ)⁻¹ * variance ξ μ :=
Causalean.Mathlib.Probability.iid_average_variance μ n ξ hξ
have hbound : ∀ᵐ o ∂μ, ξ o ∈ Icc (-2) 2 := by
filter_upwards [] with o
dsimp only [ξ]
cases o.A <;> cases o.R <;> cases o.RY <;> norm_num [armSign]
have hvarle : variance ξ μ ≤ 4 := by
have h := variance_le_sq_of_bounded hbound hξ.aemeasurable
norm_num at h
exact h
have hdecomp : (∫ s, (V s - ate P) ^ 2 ∂sampleLaw n P) =
(n : ℝ)⁻¹ * variance ξ μ + ((∫ o, ξ o ∂μ) - ate P) ^ 2 := by
have h := variance_eq_sub (hV.sub (memLp_const (ate P)))
change variance (fun s => V s - ate P) (sampleLaw n P) =
(∫ s, (V s - ate P) ^ 2 ∂sampleLaw n P) -
(∫ s, V s - ate P ∂sampleLaw n P) ^ 2 at h
rw [variance_sub_const hV.aestronglyMeasurable, hvar,
integral_sub (hV.integrable (by norm_num)) (integrable_const _),
integral_const, probReal_univ, one_smul, hmean] at h
linarith
have hclip : ∀ s, (completeArrivalHTEstimator n d s - ate P) ^ 2 ≤
(V s - ate P) ^ 2 := by
intro s
have he : completeArrivalHTEstimator n d s = clip (V s) := by
unfold completeArrivalHTEstimator V ξ
congr 1
simp only [Finset.mul_sum]
apply Finset.sum_congr rfl
intro i hi
simp only [div_eq_mul_inv]
ring
rw [he]
exact clip_sq_error_le _ _ (ate_mem_unit_interval P)
calc
deterministicRisk (completeArrivalHTEstimator n d) P ≤
∫ s, (V s - ate P) ^ 2 ∂sampleLaw n P :=
integral_mono Integrable.of_finite Integrable.of_finite hclip
_ = (n : ℝ)⁻¹ * variance ξ μ + ((∫ o, ξ o ∂μ) - ate P) ^ 2 := hdecomp
_ ≤ 4 / (n : ℝ) + (1 - q) ^ 2 := by
have hb := observed_contrast_score_bias_le P hP
change ((∫ o, ξ o ∂μ) - ate P) ^ 2 ≤ (1 - q) ^ 2 at hb
have hv := mul_le_mul_of_nonneg_left hvarle (inv_nonneg.mpr (Nat.cast_nonneg n))
simpa only [div_eq_mul_inv, mul_comm] using add_le_add hv hb
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.OneCell 28 declarations One-cell testing lower bounds inside the original rare-arrival model.
One-cell testing lower bounds inside the original rare-arrival model.
def oneCellRecord
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def oneCellMeasure
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
given by
∑ a : Bool, ∑ y : Bool, ∑ r : Bool, ofReal (completeBernWeight (1 / 2) a * completeBernWeight u y * completeBernWeight q r) • Measure.dirac (oneCellRecord x a y r)
theorem oneCellMeasure_univ
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
oneCellMeasure x u q univ = 1
Proof (Lean source)
lemma oneCellMeasure_univ {d : ℕ} (x : Fin d) (u q : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) (hq : 0 ≤ q ∧ q ≤ 1) :
oneCellMeasure x u q univ = 1 := by
have hu' : 0 ≤ 1 - u := by linarith [hu.2]
have hq' : 0 ≤ 1 - q := by linarith [hq.2]
simp [oneCellMeasure, completeBernWeight]
norm_num at ⊢
rw [ENNReal.ofReal_mul (p := (1 / 2 : ℝ) * u)
(mul_nonneg (by norm_num) hu.1),
ENNReal.ofReal_mul (p := (1 / 2 : ℝ) * u)
(mul_nonneg (by norm_num) hu.1),
ENNReal.ofReal_mul (p := (1 / 2 : ℝ) * (1 - u))
(mul_nonneg (by norm_num) hu'),
ENNReal.ofReal_mul (p := (1 / 2 : ℝ) * (1 - u))
(mul_nonneg (by norm_num) hu')]
simp only [ENNReal.ofReal_mul (p := (1 / 2 : ℝ)) (by norm_num)]
have hu_sum : ofReal u + ofReal (1 - u) = 1 := by
rw [← ENNReal.ofReal_add hu.1 hu']
norm_num
have hq_sum : ofReal q + ofReal (1 - q) = 1 := by
rw [← ENNReal.ofReal_add hq.1 hq']
norm_num
calc
_ = (ofReal (1 / 2 : ℝ) + ofReal (1 / 2 : ℝ)) *
(ofReal u + ofReal (1 - u)) *
(ofReal q + ofReal (1 - q)) := by ring
_ = 1 := by
rw [hu_sum, hq_sum]
simp only [mul_one]
rw [← ENNReal.ofReal_add (by norm_num : (0 : ℝ) ≤ 1 / 2)
(by norm_num : (0 : ℝ) ≤ 1 / 2)]
norm_num
def oneCellFamilyLaw
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
theorem oneCellFamilyLaw_ate
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
ate (oneCellFamilyLaw x u q hu hq) = u
Proof (Lean source)
lemma oneCellFamilyLaw_ate {d : ℕ} (x : Fin d) (u q : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) (hq : 0 ≤ q ∧ q ≤ 1) :
ate (oneCellFamilyLaw x u q hu hq) = u := by
let f : FullRecord d → ℝ := fun r =>
(if r.Y1 then 1 else 0) - (if r.Y0 then 1 else 0)
have hInt (a y r : Bool) : Integrable f
(ofReal (completeBernWeight (1 / 2) a *
completeBernWeight u y * completeBernWeight q r) •
Measure.dirac (oneCellRecord x a y r)) := by
exact (integrable_dirac (by simp)).smul_measure ENNReal.ofReal_ne_top
unfold ate oneCellFamilyLaw oneCellMeasure
change (∫ w, f w ∂(∑ a : Bool, ∑ y : Bool, ∑ r : Bool,
ofReal (completeBernWeight (1 / 2) a *
completeBernWeight u y * completeBernWeight q r) •
Measure.dirac (oneCellRecord x a y r))) = u
have hr (a y : Bool) :
(∫ w, f w ∂(∑ r : Bool,
ofReal (completeBernWeight (1 / 2) a *
completeBernWeight u y * completeBernWeight q r) •
Measure.dirac (oneCellRecord x a y r))) =
∑ r : Bool, ∫ w, f w ∂(ofReal
(completeBernWeight (1 / 2) a * completeBernWeight u y *
completeBernWeight q r) • Measure.dirac (oneCellRecord x a y r)) := by
exact integral_finsetSum_measure (by intro r _; exact hInt a y r)
have hy (a : Bool) :
(∫ w, f w ∂(∑ y : Bool, ∑ r : Bool,
ofReal (completeBernWeight (1 / 2) a *
completeBernWeight u y * completeBernWeight q r) •
Measure.dirac (oneCellRecord x a y r))) =
∑ y : Bool, ∫ w, f w ∂(∑ r : Bool,
ofReal (completeBernWeight (1 / 2) a *
completeBernWeight u y * completeBernWeight q r) •
Measure.dirac (oneCellRecord x a y r)) := by
exact integral_finsetSum_measure (by
intro y _
apply (integrable_finsetSum_measure).2
intro r _
exact hInt a y r)
rw [integral_finsetSum_measure]
· simp_rw [hy, hr]
simp [f, oneCellRecord, completeBernWeight, integral_smul_measure,
integral_dirac]
have hu' : 0 ≤ 1 - u := by linarith [hu.2]
have hq' : 0 ≤ 1 - q := by linarith [hq.2]
norm_num [ENNReal.toReal_ofReal, hu.1, hu', hq.1, hq']
ring
· intro a ha
apply (integrable_finsetSum_measure).2
intro y hy
apply (integrable_finsetSum_measure).2
intro r hr
exact hInt a y r
theorem oneCellFamilyLaw_consistency
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
SurrogateConsistency (oneCellFamilyLaw x u q hu hq)
conclusion 2
OutcomeConsistency (oneCellFamilyLaw x u q hu hq)
Proof (Lean source)
lemma oneCellFamilyLaw_consistency {d : ℕ} (x : Fin d) (u q : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) (hq : 0 ≤ q ∧ q ≤ 1) :
SurrogateConsistency (oneCellFamilyLaw x u q hu hq) ∧
OutcomeConsistency (oneCellFamilyLaw x u q hu hq) := by
constructor
· filter_upwards [] with r
rfl
· filter_upwards [] with r
rfl
theorem oneCellFamilyLaw_support
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
∀ᵐ w : FullRecord d ∂((oneCellFamilyLaw x u q hu hq).1), w.X
Proof (Lean source)
lemma oneCellFamilyLaw_support {d : ℕ} (x : Fin d) (u q : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) (hq : 0 ≤ q ∧ q ≤ 1) :
∀ᵐ w : FullRecord d ∂((oneCellFamilyLaw x u q hu hq).1),
w.X = x ∧ w.S0 = false ∧ w.S1 = false ∧ w.Y0 = false := by
simp [oneCellFamilyLaw, oneCellMeasure, oneCellRecord, Measure.ae_smul_measure]
theorem oneCellFamilyLaw_iid
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
IIDSampling n (sampleLaw n (oneCellFamilyLaw x u q hu hq)) (fun (i : Fin n) (s : Fin n → ObsRecord d) => s i) (oneCellFamilyLaw x u q hu hq)
Proof (Lean source)
lemma oneCellFamilyLaw_iid {n d : ℕ} (x : Fin d) (u q : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) (hq : 0 ≤ q ∧ q ≤ 1) :
IIDSampling n (sampleLaw n (oneCellFamilyLaw x u q hu hq))
(fun (i : Fin n) (s : Fin n → ObsRecord d) => s i)
(oneCellFamilyLaw x u q hu hq) :=
sampleLaw_iid n _
theorem oneCellFamilyLaw_balanced
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
BalancedRandomization (oneCellFamilyLaw x u q hu hq)
Proof (Lean source)
lemma oneCellFamilyLaw_balanced {d : ℕ} (x : Fin d) (u q : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) (hq : 0 ≤ q ∧ q ≤ 1) :
BalancedRandomization (oneCellFamilyLaw x u q hu hq) := by
have hu' : 0 ≤ 1 - u := by linarith [hu.2]
have hq' : 0 ≤ 1 - q := by linarith [hq.2]
simp [BalancedRandomization, Measure.real, oneCellFamilyLaw, oneCellMeasure,
oneCellRecord, completeBernWeight, hu.1, hu']
have hu_sum : ofReal u + ofReal (1 - u) = 1 := by
rw [← ENNReal.ofReal_add hu.1 hu']
norm_num
have hq_sum : ofReal q + ofReal (1 - q) = 1 := by
rw [← ENNReal.ofReal_add hq.1 hq']
norm_num
rw [show (2⁻¹ * ofReal u * ofReal q +
2⁻¹ * ofReal u * ofReal (1 - q) +
(2⁻¹ * ofReal (1 - u) * ofReal q +
2⁻¹ * ofReal (1 - u) * ofReal (1 - q))) =
(2⁻¹ : ℝ≥0∞) *
(ofReal u + ofReal (1 - u)) *
(ofReal q + ofReal (1 - q)) by ring,
hu_sum, hq_sum]
norm_num
theorem oneCellFamilyLaw_arrival_mass
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma oneCellFamilyLaw_arrival_mass {d : ℕ} (x : Fin d) (u q : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) (hq : 0 ≤ q ∧ q ≤ 1) :
(oneCellFamilyLaw x u q hu hq).1.real {r | r.R = true} = q := by
have hu' : 0 ≤ 1 - u := by linarith [hu.2]
simp [Measure.real, oneCellFamilyLaw, oneCellMeasure, oneCellRecord,
completeBernWeight, hu.1, hu']
have hu_sum : ofReal u + ofReal (1 - u) = 1 := by
rw [← ENNReal.ofReal_add hu.1 hu']
norm_num
have hhalf : ofReal (1 / 2 : ℝ) = (2 : ℝ≥0∞)⁻¹ := by
rw [ENNReal.ofReal_div_of_pos (by norm_num)]
norm_num
norm_num only [ENNReal.ofReal_mul, hu.1, hu', hq.1] at ⊢
rw [ENNReal.ofReal_mul (p := (1 / 2 : ℝ) * u) (mul_nonneg (by norm_num) hu.1),
ENNReal.ofReal_mul (p := (1 / 2 : ℝ) * (1 - u)) (mul_nonneg (by norm_num) hu'),
ENNReal.ofReal_mul (p := (1 / 2 : ℝ)) (by norm_num)]
rw [hhalf]
rw [ENNReal.ofReal_mul (p := (1 / 2 : ℝ)) (by norm_num), hhalf]
calc
_ = (((2 : ℝ≥0∞)⁻¹ + 2⁻¹) *
(ofReal u + ofReal (1 - u)) * ofReal q).toReal := by
congr 1
ring
_ = q := by rw [hu_sum, ENNReal.inv_two_add_inv_two]; simp [hq.1]
theorem oneCellFamilyLaw_randomized
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
RandomizedIndependence (oneCellFamilyLaw x u q hu hq)
Proof (Lean source)
lemma oneCellFamilyLaw_randomized {d : ℕ} (x : Fin d) (u q : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) (hq : 0 ≤ q ∧ q ≤ 1) :
RandomizedIndependence (oneCellFamilyLaw x u q hu hq) := by
rw [RandomizedIndependence, indepFun_iff_measure_inter_preimage_eq_mul]
intro s t _ _
have hu' : 0 ≤ 1 - u := by linarith [hu.2]
have hq' : 0 ≤ 1 - q := by linarith [hq.2]
have hsum_u : ofReal u + ofReal (1 - u) = 1 := by
rw [← ENNReal.ofReal_add hu.1 hu']
norm_num
have hsum_q : ofReal q + ofReal (1 - q) = 1 := by
rw [← ENNReal.ofReal_add hq.1 hq']
norm_num
have hhalf : (1 - 2⁻¹ : ℝ) = 2⁻¹ := by norm_num
have hfactor (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) :
ofReal (2⁻¹ * a * b) =
(2⁻¹ : ℝ≥0∞) * ofReal a * ofReal b := by
rw [ENNReal.ofReal_mul (mul_nonneg (by norm_num) ha),
ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 2⁻¹)]
norm_num [ENNReal.ofReal_div_of_pos]
have htwo : (2⁻¹ : ℝ≥0∞) * 2 = 1 := by
rw [mul_two]
exact ENNReal.inv_two_add_inv_two
have hdouble (z : ℝ≥0∞) : (2⁻¹ : ℝ≥0∞) * (z + z) = z := by
rw [← two_mul, ← mul_assoc, htwo]
simp
simp [oneCellFamilyLaw, oneCellMeasure, oneCellRecord, completeBernWeight,
indicator]
rw [hhalf]
simp only [hfactor u q hu.1 hq.1, hfactor u (1-q) hu.1 hq',
hfactor (1-u) q hu' hq.1, hfactor (1-u) (1-q) hu' hq']
by_cases hsT : true ∈ s <;> by_cases hsF : false ∈ s <;>
by_cases htT : (x, false, false, false, true) ∈ t <;>
by_cases htF : (x, false, false, false, false) ∈ t <;>
simp [hsT, hsF, htT, htF, Set.mem_inter_iff, ← mul_add, hsum_u, hsum_q] <;>
norm_num [ENNReal.inv_two_add_inv_two] <;>
simp [pow_two, mul_assoc, mul_comm, mul_left_comm, htwo, hdouble]
all_goals
calc
_ = (2⁻¹ : ℝ≥0∞) * ofReal u := by ring
_ = (2⁻¹ : ℝ≥0∞) * (2⁻¹ * (ofReal u + ofReal u)) := by
rw [hdouble]
_ = _ := by ring
theorem oneCellFamilyLaw_mar
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
ArrivalMAR (oneCellFamilyLaw x u q hu hq)
Proof (Lean source)
lemma oneCellFamilyLaw_mar {d : ℕ} (x : Fin d) (u q : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) (hq : 0 ≤ q ∧ q ≤ 1) :
ArrivalMAR (oneCellFamilyLaw x u q hu hq) := by
have hu' : 0 ≤ 1 - u := sub_nonneg.mpr hu.2
have hq' : 0 ≤ 1 - q := sub_nonneg.mpr hq.2
have h₀ : 0 ≤ u * (1 / 2) + u * q * (-1 / 2) := by
nlinarith [mul_nonneg hu.1 hq']
have h₁ : 0 ≤ 1 / 2 + u * (-1 / 2) + u * q * (1 / 2) + q * (-1 / 2) := by
nlinarith [mul_nonneg hu' hq']
have h₂ : 0 ≤ u * q * (1 / 2) := by
nlinarith [mul_nonneg hu.1 hq.1]
have h₃ : 0 ≤ u * q * (-1 / 2) + q * (1 / 2) := by
nlinarith [mul_nonneg hu' hq.1]
have h₄ : 0 ≤ (1 / 2 : ℝ) * u * q :=
mul_nonneg (mul_nonneg (by norm_num) hu.1) hq.1
have h₅ : 0 ≤ (1 / 2 : ℝ) * u * (1 - q) :=
mul_nonneg (mul_nonneg (by norm_num) hu.1) hq'
have h₆ : 0 ≤ (1 / 2 : ℝ) * (1 - u) * q :=
mul_nonneg (mul_nonneg (by norm_num) hu') hq.1
have h₇ : 0 ≤ (1 / 2 : ℝ) * (1 - u) * (1 - q) :=
mul_nonneg (mul_nonneg (by norm_num) hu') hq'
intro j y r
cases j with
| mk a xs =>
cases xs with
| mk x' s =>
cases a <;> cases s <;> cases y <;> cases r <;>
by_cases hx : x = x'
all_goals simp [Measure.real, oneCellFamilyLaw, oneCellMeasure,
completeBernWeight, oneCellRecord, FullRecord.Y, FullRecord.S, cellProb,
inCell, indicator, hx, hu.1, hu', hq.1, hq', h₀, h₁, h₂, h₃,
h₄, h₅, h₆, h₇, ENNReal.toReal_add, ENNReal.toReal_ofReal,
ENNReal.toReal_mul]
all_goals
repeat rw [ENNReal.toReal_add (by finiteness) (by finiteness)]
try simp [ENNReal.toReal_mul, ENNReal.toReal_ofReal, hu.1, hu', hq.1, hq']
ring
theorem oneCellFamilyLaw_arrivedCell
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
arrivedCell (oneCellFamilyLaw x u q hu hq) j
= q * cellProb (oneCellFamilyLaw x u q hu hq) j
Proof (Lean source)
lemma oneCellFamilyLaw_arrivedCell {d : ℕ} (x : Fin d) (u q : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) (hq : 0 ≤ q ∧ q ≤ 1) (j : Cell d) :
arrivedCell (oneCellFamilyLaw x u q hu hq) j =
q * cellProb (oneCellFamilyLaw x u q hu hq) j := by
have hu' : 0 ≤ 1 - u := sub_nonneg.mpr hu.2
have hq' : 0 ≤ 1 - q := sub_nonneg.mpr hq.2
have h₀ : 0 ≤ u * (1 / 2) + u * q * (-1 / 2) := by
nlinarith [mul_nonneg hu.1 hq']
have h₁ : 0 ≤ 1 / 2 + u * (-1 / 2) + u * q * (1 / 2) + q * (-1 / 2) := by
nlinarith [mul_nonneg hu' hq']
have h₂ : 0 ≤ u * q * (1 / 2) := by
nlinarith [mul_nonneg hu.1 hq.1]
have h₃ : 0 ≤ u * q * (-1 / 2) + q * (1 / 2) := by
nlinarith [mul_nonneg hu' hq.1]
have h₄ : 0 ≤ (1 / 2 : ℝ) * u * q :=
mul_nonneg (mul_nonneg (by norm_num) hu.1) hq.1
have h₅ : 0 ≤ (1 / 2 : ℝ) * u * (1 - q) :=
mul_nonneg (mul_nonneg (by norm_num) hu.1) hq'
have h₆ : 0 ≤ (1 / 2 : ℝ) * (1 - u) * q :=
mul_nonneg (mul_nonneg (by norm_num) hu') hq.1
have h₇ : 0 ≤ (1 / 2 : ℝ) * (1 - u) * (1 - q) :=
mul_nonneg (mul_nonneg (by norm_num) hu') hq'
cases j with
| mk a xs =>
cases xs with
| mk x' s =>
cases a <;> cases s <;>
by_cases hx : x = x'
all_goals simp [arrivedCell, cellProb, Measure.real,
oneCellFamilyLaw, oneCellMeasure, completeBernWeight, oneCellRecord,
FullRecord.Y, FullRecord.S, inCell, indicator, hx, hu.1, hu', hq.1,
hq', h₀, h₁, h₂, h₃, h₄, h₅, h₆, h₇, ENNReal.toReal_add,
ENNReal.toReal_ofReal, ENNReal.toReal_mul]
all_goals try rw [ENNReal.toReal_ofReal h₂]
all_goals try rw [ENNReal.toReal_ofReal h₃]
all_goals try rw [ENNReal.toReal_ofReal h₀]
all_goals try rw [ENNReal.toReal_ofReal h₁]
all_goals try ring
all_goals
repeat rw [ENNReal.toReal_add (by finiteness) (by finiteness)]
try simp [ENNReal.toReal_mul, ENNReal.toReal_ofReal, hu.1, hu', hq.1, hq']
ring
all_goals
rw [ENNReal.toReal_ofReal h₂, ENNReal.toReal_ofReal h₃,
ENNReal.toReal_ofReal h₀, ENNReal.toReal_ofReal h₁]
ring
theorem oneCellFamilyLaw_model
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
x :
Fin d
u q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hu :
0 ≤ u ∧ u ≤ 1
hq :
0 < q
hq1 :
q ≤ 1
hslice :
RareArrivalSlice n q
conclusion
RareArrivalModelClass n d q (oneCellFamilyLaw x u q hu ⟨hq.le, hq1⟩)
Proof (Lean source)
lemma oneCellFamilyLaw_model {n d : ℕ} (x : Fin d) (u q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hu : 0 ≤ u ∧ u ≤ 1)
(hq : 0 < q) (hq1 : q ≤ 1) (hslice : RareArrivalSlice n q) :
RareArrivalModelClass n d q (oneCellFamilyLaw x u q hu ⟨hq.le, hq1⟩) := by
refine ⟨oneCellFamilyLaw_iid (n := n) x u q hu ⟨hq.le, hq1⟩,
oneCellFamilyLaw_randomized x u q hu ⟨hq.le, hq1⟩,
oneCellFamilyLaw_balanced x u q hu ⟨hq.le, hq1⟩,
(oneCellFamilyLaw_consistency x u q hu ⟨hq.le, hq1⟩).1,
(oneCellFamilyLaw_consistency x u q hu ⟨hq.le, hq1⟩).2,
oneCellFamilyLaw_mar x u q hu ⟨hq.le, hq1⟩, ?_, hslice,
hn, hd, hq, hq1⟩
intro j hj
rw [oneCellFamilyLaw_arrivedCell]
theorem oneCellFamilyLaw_obs_ac
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
((oneCellFamilyLaw x u q hu hq).1.map obs) ≪ ((oneCellFamilyLaw x (1 / 2) q (by norm_num) hq).1.map obs)
Proof (Lean source)
lemma oneCellFamilyLaw_obs_ac {d : ℕ} (x : Fin d) (u q : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) (hq : 0 ≤ q ∧ q ≤ 1) :
((oneCellFamilyLaw x u q hu hq).1.map obs) ≪
((oneCellFamilyLaw x (1 / 2) q (by norm_num) hq).1.map obs) := by
apply Measure.AbsolutelyContinuous.map
· let μ : Bool × Bool × Bool → Measure (FullRecord d) := fun t =>
ofReal (completeBernWeight (1 / 2) t.1 *
completeBernWeight u t.2.1 * completeBernWeight q t.2.2) •
Measure.dirac (oneCellRecord x t.1 t.2.1 t.2.2)
let ν : Bool × Bool × Bool → Measure (FullRecord d) := fun t =>
ofReal (completeBernWeight (1 / 2) t.1 *
completeBernWeight (1 / 2) t.2.1 * completeBernWeight q t.2.2) •
Measure.dirac (oneCellRecord x t.1 t.2.1 t.2.2)
have hatom (t : Bool × Bool × Bool) : μ t ≪ ν t := by
rcases t with ⟨a, y, r⟩
by_cases hν : ofReal (completeBernWeight (1 / 2) a *
completeBernWeight (1 / 2) y * completeBernWeight q r) = 0
· have hμ : ofReal (completeBernWeight (1 / 2) a *
completeBernWeight u y * completeBernWeight q r) = 0 := by
cases a <;> cases y <;> cases r <;>
simp [completeBernWeight, ENNReal.ofReal_eq_zero, hq.1, hq.2] at hν ⊢ <;>
nlinarith
change (ofReal (completeBernWeight (1 / 2) a *
completeBernWeight u y * completeBernWeight q r) •
Measure.dirac (oneCellRecord x a y r)) ≪
ofReal (completeBernWeight (1 / 2) a *
completeBernWeight (1 / 2) y * completeBernWeight q r) •
Measure.dirac (oneCellRecord x a y r)
rw [hμ, hν]
· exact Measure.smul_absolutelyContinuous.trans
(Measure.AbsolutelyContinuous.rfl.smul_right hν)
have hfin (s : Finset (Bool × Bool × Bool)) :
(∑ t ∈ s, μ t) ≪ ∑ t ∈ s, ν t := by
classical
induction s using Finset.induction_on with
| empty => simp
| @insert t s ht ih =>
simp only [Finset.sum_insert ht]
exact (hatom t).add ih
have hsum : (∑ t, μ t) ≪ ∑ t, ν t := by
simpa using hfin univ
simpa [oneCellFamilyLaw, oneCellMeasure, μ, ν, Fintype.sum_prod_type] using hsum
· fun_prop
theorem oneCellFamilyLaw_obs_atom
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
o.S
= false then if o.R then (1 / 2 : ℝ) * q * (if o.A then completeBernWeight u o.RY else if o.RY then 0 else 1) else if o.RY then 0 else (1 / 2 : ℝ) * (1 - q) else 0
Proof (Lean source)
lemma oneCellFamilyLaw_obs_atom {d : ℕ} (x : Fin d) (u q : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) (hq : 0 ≤ q ∧ q ≤ 1) (o : ObsRecord d) :
((oneCellFamilyLaw x u q hu hq).1.map obs).real {o} =
if o.X = x ∧ o.S = false then
if o.R then
(1 / 2 : ℝ) * q *
(if o.A then completeBernWeight u o.RY else if o.RY then 0 else 1)
else if o.RY then 0 else (1 / 2 : ℝ) * (1 - q)
else 0 := by
have hu' : 0 ≤ 1 - u := sub_nonneg.mpr hu.2
have hq' : 0 ≤ 1 - q := sub_nonneg.mpr hq.2
have h₀ : 0 ≤ u * (1 / 2) + u * q * (-1 / 2) := by
nlinarith [mul_nonneg hu.1 hq']
have h₁ : 0 ≤ 1 / 2 + u * (-1 / 2) + u * q * (1 / 2) + q * (-1 / 2) := by
nlinarith [mul_nonneg hu' hq']
have h₂ : 0 ≤ u * q * (1 / 2) := by
nlinarith [mul_nonneg hu.1 hq.1]
have h₃ : 0 ≤ u * q * (-1 / 2) + q * (1 / 2) := by
nlinarith [mul_nonneg hu' hq.1]
rw [measureReal_def, Measure.map_apply (by fun_prop) (by simp)]
rcases o with ⟨X, A, S, R, RY⟩
by_cases hx : X = x
· subst X
cases A <;> cases S <;> cases R <;> cases RY <;>
simp [oneCellFamilyLaw, oneCellMeasure, completeBernWeight, oneCellRecord,
obs, FullRecord.Y, FullRecord.S, indicator, hu.1, hu', hq.1, hq',
ENNReal.toReal_add, ENNReal.toReal_ofReal, h₀, h₁, h₂, h₃,
ENNReal.toReal_mul] <;>
try ring
all_goals try { rw [ENNReal.toReal_ofReal h₀, ENNReal.toReal_ofReal h₁]; ring }
all_goals try { rw [ENNReal.toReal_ofReal h₂, ENNReal.toReal_ofReal h₃]; ring }
all_goals
rw [ENNReal.toReal_add (by finiteness) (by finiteness),
ENNReal.toReal_mul, ENNReal.toReal_mul,
ENNReal.toReal_mul, ENNReal.toReal_mul]
simp [ENNReal.toReal_ofReal, hu.1, hu', hq']
ring
· have hneq : x ≠ X := Ne.symm hx
cases A <;> cases S <;> cases R <;> cases RY <;>
simp [oneCellFamilyLaw, oneCellMeasure, completeBernWeight, oneCellRecord,
obs, FullRecord.Y, FullRecord.S, indicator, hx, hneq]
theorem oneCellFamilyLaw_obs_chi
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
chiSqDiv ((oneCellFamilyLaw x u q hu hq).1.map obs) ((oneCellFamilyLaw x (1 / 2) q (by norm_num) hq).1.map obs)
= 2 * q * (u - 1 / 2) ^ 2
Proof (Lean source)
lemma oneCellFamilyLaw_obs_chi {d : ℕ} (x : Fin d) (u q : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) (hq : 0 ≤ q ∧ q ≤ 1) :
chiSqDiv ((oneCellFamilyLaw x u q hu hq).1.map obs)
((oneCellFamilyLaw x (1 / 2) q (by norm_num) hq).1.map obs) =
2 * q * (u - 1 / 2) ^ 2 := by
let P₁ := oneCellFamilyLaw x u q hu hq
let P₀ := oneCellFamilyLaw x (1 / 2) q
(by norm_num : (0 : ℝ) ≤ 1 / 2 ∧ (1 / 2 : ℝ) ≤ 1) hq
letI : IsProbabilityMeasure P₁.1 := P₁.2
letI : IsProbabilityMeasure P₀.1 := P₀.2
let μ := P₁.1.map obs
let ν := P₀.1.map obs
letI : IsProbabilityMeasure μ := Measure.isProbabilityMeasure_map (by fun_prop)
letI : IsProbabilityMeasure ν := Measure.isProbabilityMeasure_map (by fun_prop)
have hac := oneCellFamilyLaw_obs_ac x u q hu hq
have h := finite_one_add_chiSqDiv μ ν hac
dsimp [μ, ν, P₁, P₀] at h
simp_rw [oneCellFamilyLaw_obs_atom x u q hu hq,
oneCellFamilyLaw_obs_atom x (1 / 2) q
(by norm_num : (0 : ℝ) ≤ 1 / 2 ∧ (1 / 2 : ℝ) ≤ 1) hq] at h
let f : ObsRecord d → ℝ := fun o =>
(if o.X = x ∧ o.S = false then
if o.R then
(1 / 2 : ℝ) * q *
(if o.A then completeBernWeight u o.RY else if o.RY then 0 else 1)
else if o.RY then 0 else (1 / 2 : ℝ) * (1 - q)
else 0) ^ 2 /
(if o.X = x ∧ o.S = false then
if o.R then
(1 / 2 : ℝ) * q *
(if o.A then completeBernWeight (1 / 2) o.RY else if o.RY then 0 else 1)
else if o.RY then 0 else (1 / 2 : ℝ) * (1 - q)
else 0)
let a : ObsRecord d := ⟨x, false, false, false, false⟩
let b : ObsRecord d := ⟨x, true, false, false, false⟩
let c : ObsRecord d := ⟨x, false, false, true, false⟩
let e₀ : ObsRecord d := ⟨x, true, false, true, false⟩
let e₁ : ObsRecord d := ⟨x, true, false, true, true⟩
have hsum : (∑ o : ObsRecord d, f o) =
∑ o ∈ ({a, b, c, e₀, e₁} : Finset (ObsRecord d)), f o := by
symm
apply Finset.sum_subset (Finset.subset_univ _)
intro o ho hn
rcases o with ⟨X, A, S, R, RY⟩
by_cases hx : X = x
· subst X
cases A <;> cases S <;> cases R <;> cases RY <;>
simp [f, a, b, c, e₀, e₁] at hn ⊢
· simp [f, hx]
have hvalue : (∑ o : ObsRecord d, f o) =
1 + 2 * q * (u - 1 / 2) ^ 2 := by
classical
rw [hsum]
rcases eq_or_ne q 0 with rfl | hqne
· norm_num [f, a, b, c, e₀, e₁]
· rcases eq_or_ne (1 - q) 0 with hqone | hqone
· have hqeq : q = 1 := by linarith
subst q
norm_num [f, a, b, c, e₀, e₁, completeBernWeight]
field_simp
ring
· simp [f, a, b, c, e₀, e₁, completeBernWeight, hqne, hqone]
field_simp
ring
change 1 + chiSqDiv
((oneCellFamilyLaw x u q hu hq).1.map obs)
((oneCellFamilyLaw x (1 / 2) q (by norm_num) hq).1.map obs) =
∑ o : ObsRecord d, f o at h
linarith [hvalue]
def oneCellStep
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n :
ℕ
q :
ℝ
def
oneCellStep n q :
ℝ
given by
theorem oneCellStep_pos
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma oneCellStep_pos (n : ℕ) (q : ℝ) (hn : 1 ≤ n) (hq : 0 < q) :
0 < oneCellStep n q := by
have hN : 0 < effectiveSize n q := by
unfold effectiveSize
positivity
unfold oneCellStep
exact lt_min (by norm_num) (by positivity)
theorem oneCellStep_le_quarter
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
theorem oneCellStep_mem
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n :
ℕ
q :
ℝ
hn :
1 ≤ n
hq :
0 < q
Proof (Lean source)
lemma oneCellStep_mem (n : ℕ) (q : ℝ) (hn : 1 ≤ n) (hq : 0 < q) :
0 ≤ 1 / 2 + oneCellStep n q ∧ 1 / 2 + oneCellStep n q ≤ 1 := by
have hp := oneCellStep_pos n q hn hq
have hle := oneCellStep_le_quarter n q
constructor <;> linarith
theorem oneCellFamilyLaw_obs_chi_step
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
chiSqDiv ((oneCellFamilyLaw x (1 / 2 + oneCellStep n q) q (oneCellStep_mem n q hn hq) ⟨hq.le, hq1⟩).1.map obs) ((oneCellFamilyLaw x (1 / 2) q (by norm_num) ⟨hq.le, hq1⟩).1.map obs)
≤ (1 / 8 : ℝ) / n
Proof (Lean source)
lemma oneCellFamilyLaw_obs_chi_step {d : ℕ} (x : Fin d) (n : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hq : 0 < q) (hq1 : q ≤ 1) :
chiSqDiv
((oneCellFamilyLaw x (1 / 2 + oneCellStep n q) q
(oneCellStep_mem n q hn hq) ⟨hq.le, hq1⟩).1.map obs)
((oneCellFamilyLaw x (1 / 2) q (by norm_num) ⟨hq.le, hq1⟩).1.map obs) ≤
(1 / 8 : ℝ) / n := by
rw [oneCellFamilyLaw_obs_chi]
have hN : 0 < effectiveSize n q := by
unfold effectiveSize
positivity
have hsqrt : 0 < sqrt (effectiveSize n q) := Real.sqrt_pos.2 hN
have hle : oneCellStep n q ≤ 1 / (4 * sqrt (effectiveSize n q)) :=
min_le_right _ _
have hp := oneCellStep_pos n q hn hq
have hsquare : (oneCellStep n q) ^ 2 ≤
(1 / (4 * sqrt (effectiveSize n q))) ^ 2 := by
nlinarith
have hsqrt_sq : (sqrt (effectiveSize n q)) ^ 2 = effectiveSize n q :=
Real.sq_sqrt hN.le
have hbudget : 16 * effectiveSize n q * (oneCellStep n q) ^ 2 ≤ 1 := by
rw [div_pow] at hsquare
field_simp [hsqrt.ne'] at hsquare
nlinarith
have hnR : (0 : ℝ) < n := by positivity
have heq : (1 / 2 + oneCellStep n q - 1 / 2) ^ 2 =
(oneCellStep n q) ^ 2 := by ring
rw [heq, le_div_iff₀ hnR]
unfold effectiveSize at hbudget
nlinarith
theorem oneCell_product_testing_half
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(sampleLaw n (oneCellFamilyLaw x (1 / 2 + oneCellStep n q) q (oneCellStep_mem n q hn hq) ⟨hq.le, hq1⟩)).real Aᶜ
+ (sampleLaw n (oneCellFamilyLaw x (1 / 2) q (by norm_num) ⟨hq.le, hq1⟩)).real A
≥ 1 / 2
Proof (Lean source)
lemma oneCell_product_testing_half {d : ℕ} (x : Fin d) (n : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hq : 0 < q) (hq1 : q ≤ 1) :
∀ A : Set (Fin n → ObsRecord d), MeasurableSet A →
(sampleLaw n (oneCellFamilyLaw x (1 / 2 + oneCellStep n q) q
(oneCellStep_mem n q hn hq) ⟨hq.le, hq1⟩)).real Aᶜ +
(sampleLaw n (oneCellFamilyLaw x (1 / 2) q (by norm_num)
⟨hq.le, hq1⟩)).real A ≥ 1 / 2 := by
let L₁ := oneCellFamilyLaw x (1 / 2 + oneCellStep n q) q
(oneCellStep_mem n q hn hq) ⟨hq.le, hq1⟩
let L₀ := oneCellFamilyLaw x (1 / 2) q (by norm_num) ⟨hq.le, hq1⟩
letI : IsProbabilityMeasure L₁.1 := L₁.2
letI : IsProbabilityMeasure L₀.1 := L₀.2
let P₁ := L₁.1.map obs
let P₀ := L₀.1.map obs
letI : IsProbabilityMeasure P₁ := Measure.isProbabilityMeasure_map (by fun_prop)
letI : IsProbabilityMeasure P₀ := Measure.isProbabilityMeasure_map (by fun_prop)
have hac : P₁ ≪ P₀ := oneCellFamilyLaw_obs_ac x
(1 / 2 + oneCellStep n q) q (oneCellStep_mem n q hn hq) ⟨hq.le, hq1⟩
have hint : Integrable (fun z => ((P₁.rnDeriv P₀ z).toReal - 1) ^ 2) P₀ :=
Integrable.of_finite
have hchi : chiSqDiv P₁ P₀ ≤ (1 / 8 : ℝ) / n :=
oneCellFamilyLaw_obs_chi_step x n q hn hq hq1
simpa [sampleLaw, P₁, P₀, L₁, L₀] using
(complete_arrival_product_testing_half n hn P₁ P₀ hac hint hchi)
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n :
ℕ
q :
ℝ
hn :
1 ≤ n
hq :
0 < q
conclusion
Proof (Lean source)
lemma oneCellStep_sq_lower (n : ℕ) (q : ℝ) (hn : 1 ≤ n) (hq : 0 < q) :
(1 / 16 : ℝ) * min 1 (effectiveSize n q)⁻¹ ≤ (oneCellStep n q) ^ 2 := by
have hN : 0 < effectiveSize n q := by
unfold effectiveSize
positivity
have hsqrt : 0 < sqrt (effectiveSize n q) := Real.sqrt_pos.2 hN
have hsqrt_sq : (sqrt (effectiveSize n q)) ^ 2 = effectiveSize n q :=
Real.sq_sqrt hN.le
unfold oneCellStep
by_cases h : (1 / 4 : ℝ) ≤ 1 / (4 * sqrt (effectiveSize n q))
· rw [min_eq_left h]
have hmin : min 1 (effectiveSize n q)⁻¹ ≤ 1 := min_le_left _ _
nlinarith
· rw [min_eq_right (le_of_not_ge h)]
have hmin : min 1 (effectiveSize n q)⁻¹ ≤ (effectiveSize n q)⁻¹ :=
min_le_right _ _
have heq : (1 / (4 * sqrt (effectiveSize n q))) ^ 2 =
(1 / 16 : ℝ) * (effectiveSize n q)⁻¹ := by
rw [div_pow, mul_pow, hsqrt_sq]
field_simp
norm_num
rw [heq]
exact mul_le_mul_of_nonneg_left hmin (by norm_num)
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma oneCell_symmetric_testing_pair (δ : ℝ) (hδ : 0 < δ) (hδle : δ ≤ 1 / 2) :
∃ u₀ ∈ Icc (1 / 4 : ℝ) (3 / 4 : ℝ),
∃ u₁ ∈ Icc (1 / 4 : ℝ) (3 / 4 : ℝ),
u₀ ≠ u₁ ∧ u₁ - u₀ = δ := by
refine ⟨1 / 2 - δ / 2, ?_, 1 / 2 + δ / 2, ?_, ?_, ?_⟩
· constructor <;> linarith
· constructor <;> linarith
· linarith
· ring
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma oneCell_linear_grid (M : ℕ) (hM : 0 < M) (w : ℝ)
(hw : 0 < w) (hwle : w ≤ 1 / 2) :
∃ grid : Fin (M + 1) → ℝ,
Injective grid ∧
(∀ i, grid i ∈ Icc (1 / 4 : ℝ) (3 / 4 : ℝ)) := by
have hMr : (0 : ℝ) < M := by exact_mod_cast hM
refine ⟨fun i => 1 / 2 - w / 2 + w * (i : ℝ) / M, ?_, ?_⟩
· intro i j hij
have hcast : (i : ℝ) = (j : ℝ) := by
have hdiv : w * (i : ℝ) / M = w * (j : ℝ) / M := by linarith
have hmul : w * (i : ℝ) = w * (j : ℝ) :=
(div_left_inj' (ne_of_gt hMr)).mp hdiv
exact mul_left_cancel₀ (ne_of_gt hw) hmul
exact Fin.ext (by exact_mod_cast hcast)
· intro i
have hi : (i : ℝ) ≤ M := by
exact_mod_cast (Nat.lt_succ_iff.mp i.isLt)
have hit : 0 ≤ w * (i : ℝ) / M := by positivity
have hit' : w * (i : ℝ) / M ≤ w := by
apply (div_le_iff₀ hMr).2
exact mul_le_mul_of_nonneg_left hi hw.le
constructor <;> dsimp <;> linarith
theorem oneCellFamilyLaw_noSurrogate
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma oneCellFamilyLaw_noSurrogate {d : ℕ} (x : Fin d) (u q : ℝ)
(hu : 0 ≤ u ∧ u ≤ 1) (hq : 0 ≤ q ∧ q ≤ 1) :
∀ᵐ r : FullRecord d ∂((oneCellFamilyLaw x u q hu hq).1),
r.S0 = false ∧ r.S1 = false := by
filter_upwards [oneCellFamilyLaw_support x u q hu hq] with r hr
exact ⟨hr.2.1, hr.2.2.1⟩
def noSurrogateRisk
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n d :
ℕ
q :
ℝ
def
noSurrogateRisk n d q :
ℝ
given by
minimaxValueReal (E := Estimator n d) (Θ := {P : FullLaw d // RareArrivalModelClass n d q P ∧ ∀ᵐ r : FullRecord d ∂(P.1), r.S0 = false ∧ r.S1 = false}) (fun T P => squaredRisk T P.1)
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.RandomizedKernel 30 declarations Barycenters, clipping and uniform-seed representations of all bounded scalar kernels, and total-variation comparison of joint input/action experiments.
Barycenters, clipping and uniform-seed representations of all bounded scalar kernels, and total-variation comparison of joint input/action experiments. Common independent seeds and deterministic projections are constant-kernel and deterministic-kernel specializations.
def BoundedKernel
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def barycenter
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
barycenter κ o :
ℝ
given by
∫ h, h ∂(κ.1 o)
def kernelRisk
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
kernelRisk Q θ κ param :
ℝ
given by
∫ o, (∫ h, (h - θ param) ^ 2 ∂(κ.1 o)) ∂(Q param)
def mapRisk
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
mapRisk Q θ f param :
ℝ
given by
∫ o, (f o - θ param) ^ 2 ∂(Q param)
def realKernelRisk
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
realKernelRisk Q θ κ param :
ℝ≥0∞
given by
∫⁻ o, (∫⁻ h, ofReal ((h - θ param) ^ 2) ∂(κ o)) ∂(Q param)
def realKernelClip
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def UnitSeed
the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def
given by
Icc (0 : ℝ) 1
def uniformSeedLaw
the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
theorem kernel_barycenter_variance_identity Lemma kernel_barycenter_variance_identity in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ h, (h - c) ^ 2 ∂(κ.1 o))
= (barycenter κ o - c) ^ 2 + ∫ h, (h - barycenter κ o) ^ 2 ∂(κ.1 o)
Proof (Lean source)
-- @node: kernel_barycenter_variance_identity
lemma kernel_barycenter_variance_identity {Z : Type*} [MeasurableSpace Z]
(κ : BoundedKernel Z) (o : Z) (c : ℝ) :
(∫ h, (h - c) ^ 2 ∂(κ.1 o)) = (barycenter κ o - c) ^ 2 +
∫ h, (h - barycenter κ o) ^ 2 ∂(κ.1 o) := by
let : IsMarkovKernel κ.1 := κ.2.1
have hbounded : ∀ᵐ h ∂(κ.1 o), h ∈ Icc (-1 : ℝ) 1 :=
(ae_mem_iff_measure_eq (μ := κ.1 o) measurableSet_Icc.nullMeasurableSet).2
(by simpa only [measure_univ] using κ.2.2 o)
have hmeas : AEStronglyMeasurable (fun h : ℝ => h) (κ.1 o) := by fun_prop
have hLp : MemLp (fun h : ℝ => h) 2 (κ.1 o) :=
memLp_of_bounded hbounded hmeas 2
have hshiftLp : MemLp (fun h : ℝ => h - c) 2 (κ.1 o) :=
hLp.sub (memLp_const c)
have hmean : (∫ h, h - c ∂(κ.1 o)) = barycenter κ o - c := by
rw [integral_sub (hLp.integrable (by norm_num)) (integrable_const c)]
simp [barycenter]
have hvar := variance_eq_sub hshiftLp
rw [variance_sub_const hmeas c, variance_eq_integral hLp.aemeasurable] at hvar
change (∫ h, (h - ∫ x, x ∂(κ.1 o)) ^ 2 ∂(κ.1 o)) =
(∫ h, (h - c) ^ 2 ∂(κ.1 o)) - (∫ h, h - c ∂(κ.1 o)) ^ 2 at hvar
rw [hmean] at hvar
dsimp only [barycenter] at *
linarith
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
barycenter κ o ∈ Icc (-1 : ℝ) 1
Proof (Lean source)
-- @node: kernel_barycenter_mem
lemma kernel_barycenter_mem {Z : Type*} [MeasurableSpace Z]
(κ : BoundedKernel Z) (o : Z) : barycenter κ o ∈ Icc (-1 : ℝ) 1 := by
let : IsMarkovKernel κ.1 := κ.2.1
have hb : ∀ᵐ h ∂(κ.1 o), h ∈ Icc (-1 : ℝ) 1 :=
(ae_mem_iff_measure_eq (μ := κ.1 o) measurableSet_Icc.nullMeasurableSet).2
(by simpa only [measure_univ] using κ.2.2 o)
have hi : Integrable (fun h : ℝ => h) (κ.1 o) := by
apply Integrable.of_mem_Icc (-1) 1
· fun_prop
· exact hb
constructor
· have h := integral_mono_ae (integrable_const (-1 : ℝ)) hi
(hb.mono fun _ hh => hh.1)
simpa [barycenter] using h
· have h := integral_mono_ae hi (integrable_const (1 : ℝ))
(hb.mono fun _ hh => hh.2)
simpa [barycenter] using h
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(barycenter κ o - c) ^ 2 ≤ ∫ h, (h - c) ^ 2 ∂(κ.1 o)
Proof (Lean source)
-- @node: kernel_barycenter_sq_le
lemma kernel_barycenter_sq_le {Z : Type*} [MeasurableSpace Z]
(κ : BoundedKernel Z) (o : Z) (c : ℝ) :
(barycenter κ o - c) ^ 2 ≤ ∫ h, (h - c) ^ 2 ∂(κ.1 o) := by
rw [kernel_barycenter_variance_identity]
exact le_add_of_nonneg_right (integral_nonneg fun _ => sq_nonneg _)
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: kernel_barycenter_risk_le
lemma kernel_barycenter_risk_le
{Z Λ : Type*} [Finite Z] [MeasurableSpace Z] [MeasurableSingletonClass Z]
(Q : Λ → Measure Z) (hQ : ∀ param, IsProbabilityMeasure (Q param))
(θ : Λ → ℝ) (κ : BoundedKernel Z) (param : Λ) :
mapRisk Q θ (barycenter κ) param ≤ kernelRisk Q θ κ param := by
let : IsProbabilityMeasure (Q param) := hQ param
exact integral_mono Integrable.of_finite Integrable.of_finite
(fun o => kernel_barycenter_sq_le κ o (θ param))
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Z Λ :
Q :
Λ → Measure Z
hQ :
∀ param, IsProbabilityMeasure (Q param)
θ :
Λ → ℝ
hθ :
∀ param, θ param ∈ Icc (-1 : ℝ) 1
κ :
param :
Λ
conclusion
kernelRisk Q θ κ param ∈ Icc (0 : ℝ) 4
Proof (Lean source)
-- @node: bounded_kernelRisk_bounds
lemma bounded_kernelRisk_bounds
{Z Λ : Type*} [Finite Z] [MeasurableSpace Z] [MeasurableSingletonClass Z]
(Q : Λ → Measure Z) (hQ : ∀ param, IsProbabilityMeasure (Q param))
(θ : Λ → ℝ) (hθ : ∀ param, θ param ∈ Icc (-1 : ℝ) 1)
(κ : BoundedKernel Z) (param : Λ) : kernelRisk Q θ κ param ∈ Icc (0 : ℝ) 4 := by
let : IsProbabilityMeasure (Q param) := hQ param
let : IsMarkovKernel κ.1 := κ.2.1
have hinner (o : Z) : (∫ h, (h - θ param) ^ 2 ∂(κ.1 o)) ≤ 4 := by
have hb : ∀ᵐ h ∂(κ.1 o), h ∈ Icc (-1 : ℝ) 1 :=
(ae_mem_iff_measure_eq (μ := κ.1 o) measurableSet_Icc.nullMeasurableSet).2
(by simpa only [measure_univ] using κ.2.2 o)
have hs : ∀ᵐ h ∂(κ.1 o), (h - θ param) ^ 2 ∈ Icc (0 : ℝ) 4 := by
filter_upwards [hb] with h hh
refine ⟨sq_nonneg _, ?_⟩
obtain ⟨hl, hu⟩ := hh
obtain ⟨cl, cu⟩ := hθ param
nlinarith [sq_nonneg (h - θ param + 2), sq_nonneg (h - θ param - 2),
mul_nonneg (by linarith : 0 ≤ 2 - (h - θ param))
(by linarith : 0 ≤ 2 + (h - θ param))]
have hi : Integrable (fun h => (h - θ param) ^ 2) (κ.1 o) := by
apply Integrable.of_mem_Icc 0 4
· fun_prop
· exact hs
have h := integral_mono_ae hi (integrable_const (4 : ℝ))
(hs.mono fun _ hh => hh.2)
simpa using h
refine ⟨integral_nonneg (fun _ => integral_nonneg (fun _ => sq_nonneg _)), ?_⟩
have h := integral_mono (μ := Q param) Integrable.of_finite
(integrable_const (4 : ℝ)) hinner
simpa [kernelRisk] using h
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
given by
⟨Kernel.deterministic (fun o => (f o).1) (by fun_prop), inferInstance, by intro o rw [Kernel.deterministic_apply, Measure.dirac_apply] rw [Set.indicator_of_mem (f o).2] rfl⟩
theorem boundedMapKernel_risk
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: boundedMapKernel_risk
lemma boundedMapKernel_risk {Z Λ : Type*} [MeasurableSpace Z]
(Q : Λ → Measure Z) (θ : Λ → ℝ)
(f : Z → Icc (-1 : ℝ) 1) (hf : Measurable f) (param : Λ) :
kernelRisk Q θ (boundedMapKernel f hf) param =
mapRisk Q θ (fun o => (f o).1) param := by
simp [kernelRisk, mapRisk, boundedMapKernel, Kernel.deterministic_apply]
theorem bounded_kernel_minimax_eq_deterministic Lemma bounded_kernel_minimax_eq_deterministic in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Z Λ :
Q :
Λ → Measure Z
hQ :
∀ param, IsProbabilityMeasure (Q param)
θ :
Λ → ℝ
hθ :
∀ param, θ param ∈ Icc (-1 : ℝ) 1
conclusion
minimaxValueReal (kernelRisk Q θ)
= minimaxValueReal (fun (f : Z → Icc (-1 : ℝ) 1) param => mapRisk Q θ (fun o => (f o).1) param)
Proof (Lean source)
-- @node: bounded_kernel_minimax_eq_deterministic
lemma bounded_kernel_minimax_eq_deterministic
{Z Λ : Type*} [Finite Z] [MeasurableSpace Z] [MeasurableSingletonClass Z]
(Q : Λ → Measure Z) (hQ : ∀ param, IsProbabilityMeasure (Q param))
(θ : Λ → ℝ) (hθ : ∀ param, θ param ∈ Icc (-1 : ℝ) 1) :
minimaxValueReal (kernelRisk Q θ) =
minimaxValueReal
(fun (f : Z → Icc (-1 : ℝ) 1) param => mapRisk Q θ (fun o => (f o).1) param) := by
let D : (Z → Icc (-1 : ℝ) 1) → Λ → ℝ :=
fun f param => mapRisk Q θ (fun o => (f o).1) param
let z : Z → Icc (-1 : ℝ) 1 := fun _ => ⟨0, by norm_num⟩
let : Nonempty (Z → Icc (-1 : ℝ) 1) := ⟨z⟩
let : Nonempty (BoundedKernel Z) := ⟨boundedMapKernel z (by fun_prop)⟩
have hknonneg : ∀ κ param, 0 ≤ kernelRisk Q θ κ param :=
fun κ param => (bounded_kernelRisk_bounds Q hQ θ hθ κ param).1
have hdnonneg : ∀ f param, 0 ≤ D f param :=
fun _ _ => integral_nonneg (fun _ => sq_nonneg _)
have hkbdd : ∀ κ, BddAbove (range (kernelRisk Q θ κ)) := by
intro κ
refine ⟨4, ?_⟩
rintro _ ⟨param, rfl⟩
exact (bounded_kernelRisk_bounds Q hQ θ hθ κ param).2
apply le_antisymm
· apply minimaxValue_le_minimaxValue
(bddBelow_range_worstCaseRisk hknonneg)
intro f
have hf : Measurable f := by fun_prop
refine ⟨boundedMapKernel f hf, le_of_eq ?_⟩
unfold worstCaseRiskReal
simp_rw [boundedMapKernel_risk]
· apply minimaxValue_le_minimaxValue
(bddBelow_range_worstCaseRisk hdnonneg)
intro κ
let f : Z → Icc (-1 : ℝ) 1 := fun o => ⟨barycenter κ o, kernel_barycenter_mem κ o⟩
refine ⟨f, ?_⟩
exact worstCaseRisk_mono_class_of_nonneg
(risk := fun (_ : Unit) param => D f param)
(risk' := fun (_ : Unit) param => kernelRisk Q θ κ param) (e := ())
id (hkbdd κ) (hknonneg κ) (kernel_barycenter_risk_le Q hQ θ κ)
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: realKernelClip_sq_le
lemma realKernelClip_sq_le (h c : ℝ) (hc : c ∈ Icc (-1 : ℝ) 1) :
(realKernelClip h - c) ^ 2 ≤ (h - c) ^ 2 := by
obtain ⟨hcl, hcu⟩ := hc
unfold realKernelClip
rcases le_total h 1 with hu | hu
· rw [min_eq_right hu]
rcases le_total (-1 : ℝ) h with hl | hl
· rw [max_eq_right hl]
· rw [max_eq_left hl]
nlinarith
· rw [min_eq_left hu, max_eq_right (by norm_num : (-1 : ℝ) ≤ 1)]
nlinarith
the stated mathematical conclusion holds.
Formal statement
conclusion
Proof (Lean source)
-- @node: measurable_realKernelClip
@[fun_prop] lemma measurable_realKernelClip : Measurable realKernelClip := by
unfold realKernelClip
fun_prop
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: realKernelClip_risk_le
lemma realKernelClip_risk_le {Z Λ : Type*} [MeasurableSpace Z]
(Q : Λ → Measure Z) (θ : Λ → ℝ)
(hθ : ∀ param, θ param ∈ Icc (-1 : ℝ) 1)
(κ : Kernel Z ℝ) (param : Λ) :
realKernelRisk Q θ (κ.map realKernelClip) param ≤ realKernelRisk Q θ κ param := by
apply lintegral_mono
intro o
dsimp only
rw [Kernel.map_apply κ measurable_realKernelClip,
lintegral_map (by fun_prop) measurable_realKernelClip]
apply lintegral_mono
intro h
exact ENNReal.ofReal_le_ofReal (realKernelClip_sq_le h (θ param) (hθ param))
the stated mathematical conclusion holds.
Formal statement
conclusion
Proof (Lean source)
-- @node: uniformSeedLaw_probability
lemma uniformSeedLaw_probability : IsProbabilityMeasure uniformSeedLaw := by
constructor
rw [uniformSeedLaw, Measure.comap_apply val val_injective
(fun s hs => (MeasurableEmbedding.subtype_coe measurableSet_Icc).measurableSet_image.mpr hs)
_ MeasurableSet.univ]
rw [image_univ, Subtype.range_coe]
change volume (Icc (0 : ℝ) 1) = 1
simp only [Real.volume_Icc, sub_zero, ENNReal.ofReal_one]
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
∃ κ : BoundedKernel Z, ∀ o, κ.1 o = uniformSeedLaw.map (fun v => (f (o, v)).1)
Proof (Lean source)
-- @node: boundedKernel_of_uniformSeed
lemma boundedKernel_of_uniformSeed
{Z : Type*} [Finite Z] [MeasurableSpace Z] [MeasurableSingletonClass Z]
(f : Z × UnitSeed → Icc (-1 : ℝ) 1) (hf : Measurable f) :
∃ κ : BoundedKernel Z, ∀ o,
κ.1 o = uniformSeedLaw.map (fun v => (f (o, v)).1) := by
letI : IsProbabilityMeasure uniformSeedLaw := uniformSeedLaw_probability
have hm (o : Z) : Measurable (fun v => (f (o, v)).1) := by fun_prop
let κ : Kernel Z ℝ := ⟨fun o => uniformSeedLaw.map (fun v => (f (o, v)).1),
measurable_of_finite _⟩
have hk : IsMarkovKernel κ := ⟨fun o => Measure.isProbabilityMeasure_map (hm o).aemeasurable⟩
refine ⟨⟨κ, hk, ?_⟩, fun _ => rfl⟩
intro o
change uniformSeedLaw.map (fun v => (f (o, v)).1) (Icc (-1 : ℝ) 1) = 1
rw [Measure.map_apply (hm o) measurableSet_Icc]
have hpre : (fun v => (f (o, v)).1) ⁻¹' Icc (-1 : ℝ) 1 = univ := by
ext v
simp only [mem_preimage, mem_univ, iff_true]
exact (f (o, v)).2
rw [hpre, measure_univ]
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: realKernelClip_mem
lemma realKernelClip_mem (h : ℝ) : realKernelClip h ∈ Icc (-1 : ℝ) 1 := by
exact ⟨le_max_left _ _, max_le (by norm_num) (min_le_left _ _)⟩
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
-- @node: realKernelClip_eq_self
lemma realKernelClip_eq_self {h : ℝ} (hh : h ∈ Icc (-1 : ℝ) 1) :
realKernelClip h = h := by
exact (congrArg (max (-1)) (min_eq_right hh.2)).trans (max_eq_right hh.1)
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(κ.1 o).map realKernelClip = κ.1 o
Proof (Lean source)
-- @node: boundedKernel_map_clip
lemma boundedKernel_map_clip {Z : Type*} [MeasurableSpace Z]
(κ : BoundedKernel Z) (o : Z) : (κ.1 o).map realKernelClip = κ.1 o := by
letI : IsMarkovKernel κ.1 := κ.2.1
have hb : ∀ᵐ h ∂(κ.1 o), h ∈ Icc (-1 : ℝ) 1 :=
(ae_mem_iff_measure_eq (μ := κ.1 o) measurableSet_Icc.nullMeasurableSet).2
(by simpa only [measure_univ] using κ.2.2 o)
calc
(κ.1 o).map realKernelClip = (κ.1 o).map id :=
Measure.map_congr (hb.mono fun _ hh => realKernelClip_eq_self hh)
_ = κ.1 o := Measure.map_id
theorem boundedKernel_uniformSeed_representation Lemma boundedKernel_uniformSeed_representation in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Z :
κ :
conclusion
∃ f : Z × UnitSeed → Icc (-1 : ℝ) 1,
conclusion 1
o :
κ.1 o = uniformSeedLaw.map (fun v => (f (o, v)).1)
Proof (Lean source)
-- @node: boundedKernel_uniformSeed_representation
lemma boundedKernel_uniformSeed_representation
{Z : Type*} [Finite Z] [MeasurableSpace Z] [MeasurableSingletonClass Z]
(κ : BoundedKernel Z) :
∃ f : Z × UnitSeed → Icc (-1 : ℝ) 1,
Measurable f ∧ ∀ o, κ.1 o = uniformSeedLaw.map (fun v => (f (o, v)).1) := by
letI : IsMarkovKernel κ.1 := κ.2.1
obtain ⟨g, hg, hmap⟩ := κ.1.exists_measurable_map_eq_unitInterval
let f : Z × UnitSeed → Icc (-1 : ℝ) 1 :=
fun p => ⟨realKernelClip (g p.1 p.2), realKernelClip_mem _⟩
have hf : Measurable f := by fun_prop
refine ⟨f, hf, fun o => ?_⟩
have hgo : Measurable (g o) := by fun_prop
change κ.1 o = (volume : Measure UnitSeed).map (realKernelClip ∘ g o)
rw [← Measure.map_map measurable_realKernelClip hgo, hmap o,
boundedKernel_map_clip]
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
given by
by letI : IsMarkovKernel κ := hk refine ⟨κ.map realKernelClip, Kernel.IsMarkovKernel.map κ measurable_realKernelClip, fun o
=> ?_⟩ rw [Kernel.map_apply κ measurable_realKernelClip, Measure.map_apply measurable_realKernelClip measurableSet_Icc] have hpre : realKernelClip ⁻¹' Icc (-1 : ℝ) 1 = univ := by ext h simp only [mem_preimage, mem_univ, iff_true] exact realKernelClip_mem h rw [hpre, measure_univ]
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: bounded_realKernelRisk_eq_ofReal
lemma bounded_realKernelRisk_eq_ofReal
{Z Λ : Type*} [Finite Z] [MeasurableSpace Z] [MeasurableSingletonClass Z]
(Q : Λ → Measure Z) (hQ : ∀ param, IsProbabilityMeasure (Q param))
(θ : Λ → ℝ) (κ : BoundedKernel Z) (param : Λ) :
realKernelRisk Q θ κ.1 param = ofReal (kernelRisk Q θ κ param) := by
letI : IsProbabilityMeasure (Q param) := hQ param
letI : IsMarkovKernel κ.1 := κ.2.1
have hi (o : Z) : Integrable (fun h => (h - θ param) ^ 2) (κ.1 o) := by
have hb : ∀ᵐ h ∂(κ.1 o), h ∈ Icc (-1 : ℝ) 1 :=
(ae_mem_iff_measure_eq (μ := κ.1 o) measurableSet_Icc.nullMeasurableSet).2
(by simpa only [measure_univ] using κ.2.2 o)
have hm : AEStronglyMeasurable (fun h : ℝ => h) (κ.1 o) := by fun_prop
exact ((memLp_of_bounded hb hm 2).sub (memLp_const (θ param))).integrable_sq
unfold realKernelRisk kernelRisk
rw [ofReal_integral_eq_lintegral_ofReal Integrable.of_finite
(Filter.Eventually.of_forall fun o => integral_nonneg fun h => sq_nonneg _)]
apply lintegral_congr
intro o
exact (ofReal_integral_eq_lintegral_ofReal (hi o)
(Filter.Eventually.of_forall fun h => sq_nonneg (h - θ param))).symm
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Z Λ :
Q :
Λ → Measure Z
hQ :
∀ param, IsProbabilityMeasure (Q param)
θ :
Λ → ℝ
hθ :
∀ param, θ param ∈ Icc (-1 : ℝ) 1
conclusion
= minimaxValueReal (kernelRisk Q θ)
Proof (Lean source)
-- @node: real_kernel_minimax_eq_bounded
lemma real_kernel_minimax_eq_bounded
{Z Λ : Type*} [Finite Z] [MeasurableSpace Z] [MeasurableSingletonClass Z]
(Q : Λ → Measure Z) (hQ : ∀ param, IsProbabilityMeasure (Q param))
(θ : Λ → ℝ) (hθ : ∀ param, θ param ∈ Icc (-1 : ℝ) 1) :
(⨅ κ : {κ : Kernel Z ℝ // IsMarkovKernel κ},
⨆ param : Λ, realKernelRisk Q θ κ.1 param).toReal =
minimaxValueReal (kernelRisk Q θ) := by
have heq : (⨅ κ : {κ : Kernel Z ℝ // IsMarkovKernel κ},
⨆ param : Λ, realKernelRisk Q θ κ.1 param) =
minimaxValueENNRealOfReal (kernelRisk Q θ) := by
apply le_antisymm
· apply le_iInf
intro κ
refine (iInf_le _ ⟨κ.1, κ.2.1⟩).trans ?_
apply iSup_le
intro param
rw [bounded_realKernelRisk_eq_ofReal Q hQ]
exact le_iSup (fun param => ofReal (kernelRisk Q θ κ param)) param
· apply le_iInf
intro κ
let b := clippedRealKernel κ.1 κ.2
change (⨅ b : BoundedKernel Z, ⨆ param, ofReal (kernelRisk Q θ b param)) ≤ _
refine (iInf_le _ b).trans ?_
apply iSup_le
intro param
rw [← bounded_realKernelRisk_eq_ofReal Q hQ]
exact (realKernelClip_risk_le Q θ hθ κ.1 param).trans
(le_iSup (realKernelRisk Q θ κ.1) param)
rw [heq]
apply minimaxValueENNRealOfReal_toReal_of_nonneg_of_bddAbove
· exact fun κ param => (bounded_kernelRisk_bounds Q hQ θ hθ κ param).1
· intro κ
refine ⟨4, ?_⟩
rintro _ ⟨param, rfl⟩
exact (bounded_kernelRisk_bounds Q hQ θ hθ κ param).2
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Z Λ :
Q :
Λ → Measure Z
hQ :
∀ param, IsProbabilityMeasure (Q param)
θ :
Λ → ℝ
hθ :
∀ param, θ param ∈ Icc (-1 : ℝ) 1
conclusion
conclusion 2
κ :
param :
Λ
conclusion 3
minimaxValueReal (kernelRisk Q θ)
= minimaxValueReal (fun (f : Z → Icc (-1 : ℝ) 1) param => mapRisk Q θ (fun o => (f o).1) param)
conclusion 4
conclusion 5
= minimaxValueReal (kernelRisk Q θ)
conclusion 6
conclusion 7
κ :
∃ f : Z × UnitSeed → Icc (-1 : ℝ) 1,
o :
κ.1 o = uniformSeedLaw.map (fun v => (f (o, v)).1)
conclusion 8
f :
Z × UnitSeed → Icc (-1 : ℝ) 1
∃ κ : BoundedKernel Z, ∀ o, κ.1 o = uniformSeedLaw.map (fun v => (f (o, v)).1)
Proof (Lean source)
-- @node: lem:randomized-kernel-barycenter
lemma randomized_kernel_barycenter
{Z Λ : Type*} [Finite Z] [MeasurableSpace Z] [MeasurableSingletonClass Z]
(Q : Λ → Measure Z) (hQ : ∀ param, IsProbabilityMeasure (Q param))
(θ : Λ → ℝ) (hθ : ∀ param, θ param ∈ Icc (-1 : ℝ) 1) :
(∀ κ : BoundedKernel Z, ∀ o, barycenter κ o ∈ Icc (-1 : ℝ) 1) ∧
(∀ (κ : BoundedKernel Z) (param : Λ),
mapRisk Q θ (barycenter κ) param ≤ kernelRisk Q θ κ param) ∧
minimaxValueReal (kernelRisk Q θ) =
minimaxValueReal
(fun (f : Z → Icc (-1 : ℝ) 1) param => mapRisk Q θ (fun o => (f o).1) param) ∧
(∀ (κ : Kernel Z ℝ), IsMarkovKernel κ → ∀ param,
realKernelRisk Q θ (κ.map realKernelClip) param ≤ realKernelRisk Q θ κ param) ∧
(⨅ κ : {κ : Kernel Z ℝ // IsMarkovKernel κ},
⨆ param : Λ, realKernelRisk Q θ κ.1 param).toReal =
minimaxValueReal (kernelRisk Q θ) ∧
IsProbabilityMeasure uniformSeedLaw ∧
(∀ κ : BoundedKernel Z, ∃ f : Z × UnitSeed → Icc (-1 : ℝ) 1,
Measurable f ∧ ∀ o, κ.1 o = uniformSeedLaw.map (fun v => (f (o, v)).1)) ∧
(∀ f : Z × UnitSeed → Icc (-1 : ℝ) 1, Measurable f →
∃ κ : BoundedKernel Z, ∀ o,
κ.1 o = uniformSeedLaw.map (fun v => (f (o, v)).1)) := by
refine ⟨kernel_barycenter_mem, kernel_barycenter_risk_le Q hQ θ,
bounded_kernel_minimax_eq_deterministic Q hQ θ hθ, ?_⟩
refine ⟨fun κ _ param => realKernelClip_risk_le Q θ hθ κ param, ?_,
uniformSeedLaw_probability, ?_, ?_⟩
· exact real_kernel_minimax_eq_bounded Q hQ θ hθ
· exact boundedKernel_uniformSeed_representation
· exact boundedKernel_of_uniformSeed
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Proof (Lean source)
-- @node: lem:randomized-kernel-tv-comparison
lemma randomized_kernel_tv_comparison
{Z H : Type*} [Finite Z] [MeasurableSpace Z] [MeasurableSingletonClass Z]
[MeasurableSpace H] (Q₀ Q₁ : Measure Z)
[IsProbabilityMeasure Q₀] [IsProbabilityMeasure Q₁]
(κ : Kernel Z H) [IsMarkovKernel κ] :
tvDist (κ ∘ₘ Q₀) (κ ∘ₘ Q₁) ≤
tvDist (Q₀ ⊗ₘ κ) (Q₁ ⊗ₘ κ) ∧
tvDist (Q₀ ⊗ₘ κ) (Q₁ ⊗ₘ κ) = tvDist Q₀ Q₁ ∧
(∀ T : Set (Z × H), MeasurableSet T →
1 - tvDist Q₀ Q₁ ≤ (Q₀ ⊗ₘ κ).real T + (Q₁ ⊗ₘ κ).real Tᶜ) ∧
(∀ T : Set H, MeasurableSet T →
1 - tvDist Q₀ Q₁ ≤ (κ ∘ₘ Q₀).real T + (κ ∘ₘ Q₁).real Tᶜ) := by
exact ⟨Causalean.Stat.tvDist_bind_le_compProd Q₀ Q₁ κ,
tvDist_compProd_eq Q₀ Q₁ κ,
one_sub_tvDist_le_compProd_test Q₀ Q₁ κ,
one_sub_tvDist_le_bind_test Q₀ Q₁ κ⟩
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.SyntheticCompletion 39 declarations Law-independent fair-coin completion of a discrete observational record.
Law-independent fair-coin completion of a discrete observational record.
def categoryLaw
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def propensity
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
propensity P x :
ℝ
given by
def outcomeMean
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def parameters
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
d :
ℕ
P :
ZengLaw d
def
given by
clause 1
μ := categoryLaw P
clause 2
μ_prob := by letI : IsProbabilityMeasure P.1 := P.2 exact Measure.isProbabilityMeasure_map (by fun_prop)
clause 3
e := propensity P
clause 4
q₀ := fun x => outcomeMean P x false
clause 5
q₁ := fun x => outcomeMean P x true
clause 6
he := by fun_prop
clause 7
hq₀ := by fun_prop
clause 8
hq₁ := by fun_prop
clause 9
he01 := by letI : IsProbabilityMeasure P.1 := P.2 intro x rw [Set.mem_Icc] by_cases hx : 0 < zengCategory P x · simp only [propensity, if_pos hx] constructor · exact div_nonneg measureReal_nonneg (le_of_lt hx) · apply (div_le_one hx).2 exact measureReal_mono (by intro r hr; exact hr.1) (measure_ne_top P.1 _) · simp [propensity, hx]
clause 10
hq₀01 := by letI : IsProbabilityMeasure P.1 := P.2 intro x rw [Set.mem_Icc] by_cases hb : 0 < zengArm P x false · simp only [outcomeMean, if_pos hb] constructor · exact div_nonneg measureReal_nonneg (le_of_lt hb) · apply (div_le_one hb).2 exact measureReal_mono (by intro r hr; exact ⟨hr.1, hr.2.1⟩) (measure_ne_top P.1 _) · simp [outcomeMean, hb]
clause 11
hq₁01 := by letI : IsProbabilityMeasure P.1 := P.2 intro x rw [Set.mem_Icc] by_cases hb : 0 < zengArm P x true · simp only [outcomeMean, if_pos hb] constructor · exact div_nonneg measureReal_nonneg (le_of_lt hb) · apply (div_le_one hb).2 exact measureReal_mono (by intro r hr; exact ⟨hr.1, hr.2.1⟩) (measure_ne_top P.1 _) · simp [outcomeMean, hb]
def fairCoin
the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
instance instIsProbabilityMeasureBoolFairCoin
the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
instance : IsProbabilityMeasure fairCoin := by
refine ⟨?_⟩
simp [fairCoin, ENNReal.inv_two_add_inv_two]
def toFullRecord
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def fullMeasure
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
theorem fullMeasure_probability
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma fullMeasure_probability {d : ℕ} (P : ZengLaw d) :
IsProbabilityMeasure (fullMeasure P) := by
letI : IsProbabilityMeasure (parameters P).fullLaw :=
(parameters P).fullLaw_probability
unfold fullMeasure
exact Measure.isProbabilityMeasure_map (by fun_prop)
def fullLaw
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
theorem arm_sum
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma arm_sum {d : ℕ} (P : ZengLaw d) (x : Fin d) :
zengArm P x false + zengArm P x true = zengCategory P x := by
letI : IsProbabilityMeasure P.1 := P.2
rw [zengArm, zengArm, zengCategory, ← measureReal_union]
· congr 1
ext r
cases h : r.2.1 <;> simp [h]
· rw [Set.disjoint_left]
simp
· measurability
theorem arm_pos
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
0 < zengArm P x b
Proof (Lean source)
lemma arm_pos {d : ℕ} {q : ℝ} (P : ZengLaw d)
(hP : P ∈ zengDiscreteClass d q) (x : Fin d)
(hx : 0 < zengCategory P x) (b : Bool) : 0 < zengArm P x b := by
rcases hP with ⟨hq, hqhalf, hd, hbounds⟩
cases b with
| true => exact lt_of_lt_of_le (mul_pos hq hx) (hbounds x hx).1
| false =>
have hu := (hbounds x hx).2
have hs := arm_sum P x
have : q < 1 := by linarith
nlinarith
theorem categoryLaw_singleton_real
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma categoryLaw_singleton_real {d : ℕ} (P : ZengLaw d) (x : Fin d) :
(categoryLaw P).real {x} = zengCategory P x := by
letI : IsProbabilityMeasure P.1 := P.2
rw [categoryLaw, Measure.real, Measure.map_apply (by fun_prop) (by simp),
zengCategory, measureReal_def]
rfl
theorem outcomeMean_eq_zengMean
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
outcomeMean P x b = zengMean P x b hx
Proof (Lean source)
lemma outcomeMean_eq_zengMean {d : ℕ} {q : ℝ} (P : ZengLaw d)
(hP : P ∈ zengDiscreteClass d q) (x : Fin d)
(hx : 0 < zengCategory P x) (b : Bool) :
outcomeMean P x b = zengMean P x b hx := by
have hb := arm_pos P hP x hx b
simp [outcomeMean, zengMean, hb]
theorem success_real_singleton
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
theorem arm_real_split
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma arm_real_split {d : ℕ} (P : ZengLaw d) (x : Fin d) (b : Bool) :
zengArm P x b =
P.1.real (singleton (x, b, false)) +
P.1.real (singleton (x, b, true)) := by
letI : IsProbabilityMeasure P.1 := P.2
rw [zengArm, ← measureReal_union]
· congr 1
ext r
rcases r with ⟨x', b', y'⟩
change (x' = x ∧ b' = b) ↔
(x', b', y') = (x, b, false) ∨ (x', b', y') = (x, b, true)
cases y' <;> simp
· rw [Set.disjoint_left]
intro r hr₀ hr₁
change r = (x, b, false) at hr₀
change r = (x, b, true) at hr₁
subst r
exact Bool.noConfusion (congrArg (fun z => z.2.2) hr₁)
· exact MeasurableSet.singleton _
theorem selected_eq
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(parameters P).selected = P.1
Proof (Lean source)
lemma selected_eq {d : ℕ} {q : ℝ} (P : ZengLaw d)
(hP : P ∈ zengDiscreteClass d q) :
(parameters P).selected = P.1 := by
letI : IsProbabilityMeasure P.1 := P.2
letI : IsProbabilityMeasure (parameters P).selected :=
(parameters P).selected_probability
apply Measure.ext_of_measureReal_singleton
rintro ⟨x, b, y⟩
have hcell := (parameters P).selected_cell ({x} : Set (Fin d)) (by simp) b y
have hset : {z : SelectedCoord (Fin d) | z.1 ∈ ({x} : Set (Fin d)) ∧
z.2.1 = b ∧ z.2.2 = y} = {(x, b, y)} := by
ext z
rcases z with ⟨x', b', y'⟩
simp
rw [hset] at hcell
rw [MeasureTheory.lintegral_singleton] at hcell
have hcellReal := congrArg toReal hcell
rw [Measure.real, hcellReal, ENNReal.toReal_mul, ENNReal.toReal_ofReal]
· change cellMass (propensity P x) (outcomeMean P x false)
(outcomeMean P x true) b y * ((categoryLaw P) {x}).toReal = _
change _ = P.1.real (singleton (x, b, y))
rw [show ((categoryLaw P) {x}).toReal = zengCategory P x by
simpa [Measure.real] using categoryLaw_singleton_real P x]
by_cases hx : 0 < zengCategory P x
· have hbpos := arm_pos P hP x hx b
have hcatne : zengCategory P x ≠ 0 := ne_of_gt hx
have hbne : zengArm P x b ≠ 0 := ne_of_gt hbpos
rw [show propensity P x = zengArm P x true / zengCategory P x by
simp [propensity, hx]]
rw [show outcomeMean P x false = zengMean P x false hx from
outcomeMean_eq_zengMean P hP x hx false]
rw [show outcomeMean P x true = zengMean P x true hx from
outcomeMean_eq_zengMean P hP x hx true]
have hsfalse := arm_real_split P x false
have hstrue := arm_real_split P x true
have hcat := arm_sum P x
cases b <;> cases y <;>
simp [cellMass, bitMass, zengMean, success_real_singleton] <;>
field_simp <;>
first
| (rw [← hcat, hsfalse, hstrue]; ring)
| (rw [← hcat, hsfalse]; ring)
| (rw [hstrue]; ring)
| (rw [hsfalse]; ring)
| ring
· have hcat0 : zengCategory P x = 0 :=
le_antisymm (le_of_not_gt hx) measureReal_nonneg
have hatom : P.1.real (singleton (x, b, y)) = 0 := by
have hsub : singleton (x, b, y) ⊆ {r | r.1 = x} := by
intro r hr
change r = (x, b, y) at hr
subst r
rfl
apply le_antisymm _ measureReal_nonneg
apply le_trans (measureReal_mono hsub (measure_ne_top P.1 _))
simpa [zengCategory] using le_of_eq hcat0
simp [hcat0, hatom]
· unfold cellMass
apply mul_nonneg
· exact bitMass_nonneg _ ((parameters P).he01 x) _
· split
· exact bitMass_nonneg _ ((parameters P).hq₁01 x) _
· exact bitMass_nonneg _ ((parameters P).hq₀01 x) _
theorem bind_map_eq_prod_map
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
μ.bind (fun x => ν.map (f x)) = (μ.prod ν).map (fun z => f z.1 z.2)
Proof (Lean source)
lemma bind_map_eq_prod_map {α β γ : Type*}
[MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ]
[MeasurableSingletonClass α] [Countable α]
(μ : Measure α) (ν : Measure β) [SFinite ν]
(f : α → β → γ) (hf : Measurable (fun z : α × β => f z.1 z.2)) :
μ.bind (fun x => ν.map (f x)) =
(μ.prod ν).map (fun z => f z.1 z.2) := by
ext s hs
rw [Measure.bind_apply hs (measurable_of_countable _).aemeasurable,
Measure.map_apply hf hs,
Measure.prod_apply (hs.preimage hf)]
apply lintegral_congr
intro x
have hfx : Measurable (f x) :=
hf.comp (measurable_const.prodMk measurable_id)
rw [Measure.map_apply hfx hs]
rfl
theorem ofReal_bernoulli_marginal
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma ofReal_bernoulli_marginal (p q : ℝ)
(hp : p ∈ Icc (0 : ℝ) 1) (hq : q ∈ Icc (0 : ℝ) 1)
(y : Bool) :
ofReal (p * bitMass q y) +
ofReal ((1 - p) * bitMass q y) =
ofReal (bitMass q y) := by
have hbp : 0 ≤ bitMass p false := bitMass_nonneg p hp false
have hpt : 0 ≤ bitMass p true := bitMass_nonneg p hp true
have hbq : 0 ≤ bitMass q y := bitMass_nonneg q hq y
rw [← ENNReal.ofReal_add (mul_nonneg hp.1 hbq)
(mul_nonneg (sub_nonneg.mpr hp.2) hbq)]
congr 1
cases y <;> simp [bitMass] <;> ring
theorem remainingPairSetMass_selected
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma remainingPairSetMass_selected (q₀ q₁ : ℝ)
(hq₀ : q₀ ∈ Icc (0 : ℝ) 1) (hq₁ : q₁ ∈ Icc (0 : ℝ) 1)
(a y : Bool) :
remainingPairSetMass q₀ q₁ {v | (if a then v.2 else v.1) = y} =
ofReal (bitMass (if a then q₁ else q₀) y) := by
cases a with
| false =>
simpa [remainingPairSetMass, bitMass, mul_comm] using
(ofReal_bernoulli_marginal q₁ q₀ hq₁ hq₀ y)
| true =>
simpa [remainingPairSetMass, bitMass, add_comm] using
(ofReal_bernoulli_marginal q₀ q₁ hq₀ hq₁ y)
theorem latent_cell_mass
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(parameters P).fullLaw {z | z.1 = x ∧ z.2.1 = b ∧ (if a then z.2.2.2 else z.2.2.1) = y}
= ofReal (bitMass (propensity P x) b * bitMass (if a then outcomeMean P x true else outcomeMean P x false) y) * (categoryLaw P) {x}
Proof (Lean source)
lemma latent_cell_mass {d : ℕ} (P : ZengLaw d)
(x : Fin d) (b a y : Bool) :
(parameters P).fullLaw
{z | z.1 = x ∧ z.2.1 = b ∧
(if a then z.2.2.2 else z.2.2.1) = y} =
ofReal
(bitMass (propensity P x) b *
bitMass (if a then outcomeMean P x true else outcomeMean P x false) y) *
(categoryLaw P) {x} := by
let t : Set (Bool × Bool) :=
{v | (if a then v.2 else v.1) = y}
have h := (parameters P).conditional_independence_finite
({x} : Set (Fin d)) (by simp) ({b} : Set Bool) t
have hset : {z : FullCoord (Fin d) |
z.1 ∈ ({x} : Set (Fin d)) ∧ z.2.1 ∈ ({b} : Set Bool) ∧ z.2.2 ∈ t} =
{z | z.1 = x ∧ z.2.1 = b ∧
(if a then z.2.2.2 else z.2.2.1) = y} := by
ext z
simp [t]
rw [hset] at h
rw [h, MeasureTheory.lintegral_singleton]
have he := (parameters P).he01 x
have hq₀ := (parameters P).hq₀01 x
have hq₁ := (parameters P).hq₁01 x
rw [show t = {v | (if a then v.2 else v.1) = y} from rfl,
remainingPairSetMass_selected _ _ hq₀ hq₁]
have hb : 0 ≤ bitMass (propensity P x) b :=
bitMass_nonneg _ he b
rw [show firstMarkSetMass ((parameters P).e x) {b} =
ofReal (bitMass (propensity P x) b) by
change firstMarkSetMass (propensity P x) {b} = _
cases b <;> simp [firstMarkSetMass, bitMass]]
rw [← ENNReal.ofReal_mul hb]
rfl
theorem full_joint_mass
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
= (parameters P).fullLaw {z | z.1 = x ∧ z.2.1 = (if r then a else !a) ∧ (if a then z.2.2.2 else z.2.2.1) = y} * fairCoin {a}
Proof (Lean source)
lemma full_joint_mass {d : ℕ} (P : ZengLaw d)
(a : Bool) (x : Fin d) (r y : Bool) :
(fullMeasure P)
{w | inCell w (a, x, false) ∧ w.R = r ∧ w.Y = y} =
(parameters P).fullLaw
{z | z.1 = x ∧ z.2.1 = (if r then a else !a) ∧
(if a then z.2.2.2 else z.2.2.1) = y} *
fairCoin {a} := by
let E : Set (FullCoord (Fin d)) :=
{z | z.1 = x ∧ z.2.1 = (if r then a else !a) ∧
(if a then z.2.2.2 else z.2.2.1) = y}
have hpre : toFullRecord ⁻¹'
{w | inCell w (a, x, false) ∧ w.R = r ∧ w.Y = y} =
E ×ˢ ({a} : Set Bool) := by
ext z
rcases z with ⟨⟨x', b, y₀, y₁⟩, a'⟩
cases a <;> cases a' <;> cases b <;> cases r <;>
cases y₀ <;> cases y₁ <;> cases y <;>
simp [E, toFullRecord, inCell, FullRecord.S, FullRecord.Y]
rw [fullMeasure, Measure.map_apply (by fun_prop) (by simp), hpre,
Measure.prod_prod]
theorem full_joint_mass_real
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
= (1 / 2 : ℝ) * zengCategory P x * bitMass (propensity P x) (if r then a else !a) * bitMass (if a then outcomeMean P x true else outcomeMean P x false) y
Proof (Lean source)
lemma full_joint_mass_real {d : ℕ} (P : ZengLaw d)
(a : Bool) (x : Fin d) (r y : Bool) :
(fullMeasure P).real
{w | inCell w (a, x, false) ∧ w.R = r ∧ w.Y = y} =
(1 / 2 : ℝ) * zengCategory P x *
bitMass (propensity P x) (if r then a else !a) *
bitMass (if a then outcomeMean P x true else outcomeMean P x false) y := by
rw [Measure.real, full_joint_mass, latent_cell_mass]
rw [ENNReal.toReal_mul, ENNReal.toReal_mul, ENNReal.toReal_ofReal]
· rw [show ((categoryLaw P) {x}).toReal = zengCategory P x by
simpa [Measure.real] using categoryLaw_singleton_real P x]
have hcoin : (fairCoin {a}).toReal = (1 / 2 : ℝ) := by
cases a <;> simp [fairCoin]
rw [hcoin]
ring
· exact mul_nonneg
(bitMass_nonneg _ ((parameters P).he01 x) _)
(bitMass_nonneg _ (by
split
· exact (parameters P).hq₁01 x
· exact (parameters P).hq₀01 x) _)
theorem measureReal_split_bool
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma measureReal_split_bool {α : Type*} [MeasurableSpace α]
(μ : Measure α) [IsFiniteMeasure μ] (E : Set α) (hE : MeasurableSet E)
(f : α → Bool) (hf : Measurable f) :
μ.real E = ∑ b : Bool, μ.real {w | w ∈ E ∧ f w = b} := by
rw [Fintype.sum_bool, ← measureReal_union]
· congr 1
ext w
cases h : f w <;> simp [h]
· rw [Set.disjoint_left]
intro a ha hb
simp only [Set.mem_setOf_eq] at ha hb
simp [ha.2] at hb
· exact hE.inter (measurableSet_eq_fun hf measurable_const)
theorem full_cellProb_false
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
cellProb (fullLaw P) (a, x, false) = (1 / 2 : ℝ) * zengCategory P x
Proof (Lean source)
lemma full_cellProb_false {d : ℕ} (P : ZengLaw d) (a : Bool) (x : Fin d) :
cellProb (fullLaw P) (a, x, false) =
(1 / 2 : ℝ) * zengCategory P x := by
letI : IsProbabilityMeasure (fullMeasure P) := fullMeasure_probability P
unfold cellProb
change (fullMeasure P).real {r | inCell r (a, x, false)} = _
rw [measureReal_split_bool (fullMeasure P)
{w | inCell w (a, x, false)} (by simp) FullRecord.R (by fun_prop)]
simp only [Fintype.sum_bool, Set.mem_setOf_eq, and_assoc]
rw [measureReal_split_bool (fullMeasure P)
{w | inCell w (a, x, false) ∧ w.R = false} (by simp)
FullRecord.Y (by fun_prop),
measureReal_split_bool (fullMeasure P)
{w | inCell w (a, x, false) ∧ w.R = true} (by simp)
FullRecord.Y (by fun_prop)]
simp only [Fintype.sum_bool, Set.mem_setOf_eq, and_assoc]
simp_rw [full_joint_mass_real]
cases a <;> simp [bitMass] <;> ring
theorem full_arrived_false
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
arrivedCell (fullLaw P) (a, x, false)
= (1 / 2 : ℝ) * zengCategory P x * bitMass (propensity P x) a
Proof (Lean source)
lemma full_arrived_false {d : ℕ} (P : ZengLaw d) (a : Bool) (x : Fin d) :
arrivedCell (fullLaw P) (a, x, false) =
(1 / 2 : ℝ) * zengCategory P x * bitMass (propensity P x) a := by
letI : IsProbabilityMeasure (fullMeasure P) := fullMeasure_probability P
unfold arrivedCell
change (fullMeasure P).real {r | inCell r (a, x, false) ∧ r.R = true} = _
rw [measureReal_split_bool (fullMeasure P)
{w | inCell w (a, x, false) ∧ w.R = true} (by simp)
FullRecord.Y (by fun_prop)]
simp only [Fintype.sum_bool, Set.mem_setOf_eq, and_assoc]
simp_rw [full_joint_mass_real]
cases a <;> simp [bitMass] <;> ring
theorem full_cellR_false
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
= (1 / 2 : ℝ) * zengCategory P x * bitMass (propensity P x) (if r then a else !a)
Proof (Lean source)
lemma full_cellR_false {d : ℕ} (P : ZengLaw d)
(a : Bool) (x : Fin d) (r : Bool) :
(fullMeasure P).real {w | inCell w (a, x, false) ∧ w.R = r} =
(1 / 2 : ℝ) * zengCategory P x *
bitMass (propensity P x) (if r then a else !a) := by
letI : IsProbabilityMeasure (fullMeasure P) := fullMeasure_probability P
rw [measureReal_split_bool (fullMeasure P)
{w | inCell w (a, x, false) ∧ w.R = r} (by simp)
FullRecord.Y (by fun_prop)]
simp only [Fintype.sum_bool, Set.mem_setOf_eq, and_assoc]
simp_rw [full_joint_mass_real]
cases a <;> cases r <;> simp [bitMass] <;> ring
theorem full_cellY_false
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
= (1 / 2 : ℝ) * zengCategory P x * bitMass (if a then outcomeMean P x true else outcomeMean P x false) y
Proof (Lean source)
lemma full_cellY_false {d : ℕ} (P : ZengLaw d)
(a : Bool) (x : Fin d) (y : Bool) :
(fullMeasure P).real {w | inCell w (a, x, false) ∧ w.Y = y} =
(1 / 2 : ℝ) * zengCategory P x *
bitMass (if a then outcomeMean P x true else outcomeMean P x false) y := by
letI : IsProbabilityMeasure (fullMeasure P) := fullMeasure_probability P
rw [measureReal_split_bool (fullMeasure P)
{w | inCell w (a, x, false) ∧ w.Y = y} (by simp)
FullRecord.R (by fun_prop)]
simp only [Fintype.sum_bool, Set.mem_setOf_eq, and_assoc]
have hreorder (r : Bool) :
(fullMeasure P).real
{w | inCell w (a, x, false) ∧ w.Y = y ∧ w.R = r} =
(fullMeasure P).real
{w | inCell w (a, x, false) ∧ w.R = r ∧ w.Y = y} := by
congr 1
ext w
simp [and_left_comm, and_comm]
rw [hreorder false, hreorder true, full_joint_mass_real, full_joint_mass_real]
cases a <;> cases y <;>
simp [Fintype.sum_bool, bitMass] <;> try ring_nf
theorem indepFun_map
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma indepFun_map {Ω X Y Z : Type*} [MeasurableSpace Ω]
[MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace Z]
{μ : Measure Ω} {φ : Ω → X} {f : X → Y} {g : X → Z}
(hφ : Measurable φ) (hf : Measurable f) (hg : Measurable g)
(h : IndepFun (f ∘ φ) (g ∘ φ) μ) : IndepFun f g (μ.map φ) := by
rw [indepFun_iff_measure_inter_preimage_eq_mul] at h ⊢
intro s t hs ht
rw [Measure.map_apply hφ ((hs.preimage hf).inter (ht.preimage hg)),
Measure.map_apply hφ (hs.preimage hf), Measure.map_apply hφ (ht.preimage hg)]
exact h s t hs ht
theorem fullLaw_randomized
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma fullLaw_randomized {d : ℕ} (P : ZengLaw d) :
RandomizedIndependence (fullLaw P) := by
let L := (parameters P).fullLaw
letI : IsProbabilityMeasure L := (parameters P).fullLaw_probability
have hsource : IndepFun (fun z : FullCoord (Fin d) × Bool => z.2)
(fun z => z.1) (L.prod fairCoin) :=
(indepFun_prod (μ := L) (ν := fairCoin) measurable_id measurable_id).symm
apply indepFun_map (φ := toFullRecord) (by fun_prop) (by fun_prop) (by fun_prop)
convert hsource.comp measurable_id
(measurable_of_finite (fun z : FullCoord (Fin d) =>
(z.1, false, false, z.2.2.1, z.2.2.2))) using 1 <;> rfl
theorem fullLaw_balanced
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma fullLaw_balanced {d : ℕ} (P : ZengLaw d) :
BalancedRandomization (fullLaw P) := by
letI : IsProbabilityMeasure (parameters P).fullLaw :=
(parameters P).fullLaw_probability
rw [BalancedRandomization]
change (fullMeasure P).real {r : FullRecord d | r.A = true} = 1 / 2
rw [Measure.real, fullMeasure,
Measure.map_apply (by fun_prop) (by simp)]
have hpre : toFullRecord ⁻¹' {r : FullRecord d | r.A = true} =
univ ×ˢ ({true} : Set Bool) := by
ext z
simp [toFullRecord]
rw [hpre, Measure.prod_prod]
simp [fairCoin]
theorem fullLaw_surrogate
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma fullLaw_surrogate {d : ℕ} (P : ZengLaw d) :
SurrogateConsistency (fullLaw P) := by
filter_upwards [] with r
rfl
theorem fullLaw_outcome
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma fullLaw_outcome {d : ℕ} (P : ZengLaw d) :
OutcomeConsistency (fullLaw P) := by
filter_upwards [] with r
rfl
theorem fullMeasure_real_of_surrogate_true
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(fullMeasure P).real E = 0
Proof (Lean source)
lemma fullMeasure_real_of_surrogate_true {d : ℕ} (P : ZengLaw d)
(E : Set (FullRecord d)) (hE : MeasurableSet E)
(htrue : ∀ w, w ∈ E → w.S = true) : (fullMeasure P).real E = 0 := by
rw [Measure.real, fullMeasure, Measure.map_apply (by fun_prop) hE]
have hpre : toFullRecord ⁻¹' E = ∅ := by
ext z
constructor
· intro hz
have h := htrue (toFullRecord z) hz
have hf : (false : Bool) = true := by
change (if z.2 then false else false) = true at h
simpa using h
exact Bool.noConfusion hf
· simp
rw [hpre]
simp
theorem fullLaw_mar
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma fullLaw_mar {d : ℕ} (P : ZengLaw d) : ArrivalMAR (fullLaw P) := by
rintro ⟨a, x, s⟩ y r
cases s with
| false =>
change (fullMeasure P).real
{w | inCell w (a, x, false) ∧ w.R = r ∧ w.Y = y} *
cellProb (fullLaw P) (a, x, false) =
(fullMeasure P).real {w | inCell w (a, x, false) ∧ w.R = r} *
(fullMeasure P).real {w | inCell w (a, x, false) ∧ w.Y = y}
rw [full_joint_mass_real, full_cellProb_false, full_cellR_false,
full_cellY_false]
ring
| true =>
have hzero (E : Set (FullRecord d)) (hE : MeasurableSet E)
(hsub : ∀ w, w ∈ E → inCell w (a, x, true)) :
(fullMeasure P).real E = 0 :=
fullMeasure_real_of_surrogate_true P E hE
(fun w hw => (hsub w hw).2.2)
change (fullMeasure P).real
{w | inCell w (a, x, true) ∧ w.R = r ∧ w.Y = y} *
cellProb (fullLaw P) (a, x, true) =
(fullMeasure P).real {w | inCell w (a, x, true) ∧ w.R = r} *
(fullMeasure P).real {w | inCell w (a, x, true) ∧ w.Y = y}
rw [hzero _ (by simp) (by intro w hw; exact hw.1),
hzero _ (by simp) (by intro w hw; exact hw.1),
hzero _ (by simp) (by intro w hw; exact hw.1)]
simp
theorem fullLaw_arrival
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
OccupiedCellArrival q (fullLaw P)
Proof (Lean source)
lemma fullLaw_arrival {d : ℕ} {q : ℝ} (P : ZengLaw d)
(hP : P ∈ zengDiscreteClass d q) : OccupiedCellArrival q (fullLaw P) := by
rintro ⟨a, x, s⟩ hcell
cases s with
| true =>
have hz : cellProb (fullLaw P) (a, x, true) = 0 := by
unfold cellProb
change (fullMeasure P).real {w | inCell w (a, x, true)} = 0
exact fullMeasure_real_of_surrogate_true P _ (by simp)
(by intro w hw; exact hw.2.2)
rw [hz] at hcell
exact (lt_irrefl 0 hcell).elim
| false =>
rw [full_cellProb_false, full_arrived_false]
rw [full_cellProb_false] at hcell
have hx : 0 < zengCategory P x := by nlinarith
have hb := hP.2.2.2 x hx
have hprop : propensity P x = zengArm P x true / zengCategory P x := by
simp [propensity, hx]
rw [hprop]
cases a with
| false =>
simp only [bitMass, Bool.false_eq_true, if_false]
have hdiv : q ≤ 1 - zengArm P x true / zengCategory P x := by
have hu : zengArm P x true / zengCategory P x ≤ 1 - q :=
(div_le_iff₀ hx).2 (by simpa [mul_comm] using hb.2)
linarith
simpa [mul_comm, mul_left_comm, mul_assoc] using
mul_le_mul_of_nonneg_left hdiv (le_of_lt hcell)
| true =>
simp only [bitMass, Bool.true_eq, if_true]
have hdiv : q ≤ zengArm P x true / zengCategory P x :=
(le_div_iff₀ hx).2 (by simpa [mul_comm] using hb.1)
simpa [mul_comm, mul_left_comm, mul_assoc] using
mul_le_mul_of_nonneg_left hdiv (le_of_lt hcell)
theorem full_cellContribution_false
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
cellContribution (fullLaw P) (a, x, false)
Proof (Lean source)
lemma full_cellContribution_false {d : ℕ} {q : ℝ} (P : ZengLaw d)
(hP : P ∈ zengDiscreteClass d q) (a : Bool) (x : Fin d) :
cellContribution (fullLaw P) (a, x, false) =
(1 / 2 : ℝ) * zengCategory P x *
(if a then outcomeMean P x true else outcomeMean P x false) := by
by_cases hx : 0 < zengCategory P x
· have harr : 0 < arrivedCell (fullLaw P) (a, x, false) := by
rw [full_arrived_false]
have hb := hP.2.2.2 x hx
have hq := hP.1
have hbit : 0 < bitMass (propensity P x) a := by
rw [show propensity P x = zengArm P x true / zengCategory P x by
simp [propensity, hx]]
cases a with
| false =>
simp only [bitMass, Bool.false_eq_true, if_false]
have hu : zengArm P x true / zengCategory P x < 1 := by
apply (div_lt_one hx).2
nlinarith [hb.2]
linarith
| true =>
simp only [bitMass, Bool.true_eq, if_true]
exact div_pos (lt_of_lt_of_le (mul_pos hq hx) hb.1) hx
positivity
unfold cellContribution cellMean
rw [if_pos harr, if_pos harr, full_cellProb_false]
change (1 / 2 : ℝ) * zengCategory P x *
((fullMeasure P).real
{r | inCell r (a, x, false) ∧ r.R = true ∧ r.Y = true} /
arrivedCell (fullLaw P) (a, x, false)) = _
rw [full_joint_mass_real, full_arrived_false]
have hmass : (1 / 2 : ℝ) * zengCategory P x *
bitMass (propensity P x) a ≠ 0 := by
rw [← full_arrived_false]
exact ne_of_gt harr
have hcatne : zengCategory P x ≠ 0 := ne_of_gt hx
have hbitne : bitMass (propensity P x) a ≠ 0 := by
intro hz
apply hmass
rw [hz]
ring
cases a <;> simp [bitMass] at hbitne ⊢ <;>
field_simp [hcatne, hbitne] <;> simp
· have hx0 : zengCategory P x = 0 :=
le_antisymm (le_of_not_gt hx) measureReal_nonneg
have harr : arrivedCell (fullLaw P) (a, x, false) = 0 := by
rw [full_arrived_false, hx0]
ring
simp [cellContribution, harr, hx0]
theorem full_cellContribution_true
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
cellContribution (fullLaw P) (a, x, true) = 0
Proof (Lean source)
lemma full_cellContribution_true {d : ℕ} (P : ZengLaw d)
(a : Bool) (x : Fin d) : cellContribution (fullLaw P) (a, x, true) = 0 := by
have harr : arrivedCell (fullLaw P) (a, x, true) = 0 := by
unfold arrivedCell
change (fullMeasure P).real {w | inCell w (a, x, true) ∧ w.R = true} = 0
exact fullMeasure_real_of_surrogate_true P _ (by simp)
(by intro w hw; exact hw.1.2.2)
simp [cellContribution, harr]
theorem fullLaw_ate
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
ate (fullLaw P) = zengATE P
Proof (Lean source)
lemma fullLaw_ate {n d : ℕ} {q : ℝ} (hn : 1 ≤ n)
(hq : 0 < q) (hqhalf : q < 1 / 2) (P : ZengLaw d)
(hP : P ∈ zengDiscreteClass d q) : ate (fullLaw P) = zengATE P := by
have hmodel : UnrestrictedArrivalModelClass n d q (fullLaw P) :=
{ n_pos := hn
d_pos := hP.2.2.1
q_pos := hq
q_le_one := by linarith
randomized := fullLaw_randomized P
balanced := fullLaw_balanced P
surrogate := fullLaw_surrogate P
outcome := fullLaw_outcome P
mar := fullLaw_mar P
arrival := fullLaw_arrival P hP }
rw [unrestricted_cell_identification (fullLaw P) hmodel]
unfold cellFunctional zengATE
simp only [Fintype.sum_prod_type, Fintype.sum_bool,
full_cellContribution_false P hP, full_cellContribution_true P,
armSign, Bool.false_eq_true, if_true, if_false, zero_mul, mul_zero,
add_zero, zero_add, one_mul, neg_one_mul]
rw [← Finset.sum_add_distrib, Finset.mul_sum]
apply Finset.sum_congr rfl
intro x hxmem
by_cases hx : 0 < zengCategory P x
· rw [dif_pos hx, outcomeMean_eq_zengMean P hP x hx false,
outcomeMean_eq_zengMean P hP x hx true]
ring
· have hx0 : zengCategory P x = 0 :=
le_antisymm (le_of_not_gt hx) measureReal_nonneg
simp [hx, hx0]
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.SyntheticKernel 16 declarations Law-independent fair-coin completion of a discrete observational record.
Law-independent fair-coin completion of a discrete observational record.
def syntheticRecord
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def syntheticOneRecord
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
d :
ℕ
r :
given by
(1 / 2 : ℝ≥0∞) • Measure.dirac (syntheticRecord r false)
+ (1 / 2 : ℝ≥0∞) • Measure.dirac (syntheticRecord r true)
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
d :
ℕ
r :
conclusion
= Measure.map (syntheticRecord r) ((1 / 2 : ℝ≥0∞) • Measure.dirac false + (1 / 2 : ℝ≥0∞) • Measure.dirac true)
Proof (Lean source)
lemma syntheticOneRecord_eq_coin_map {d : ℕ} (r : ZengRecord d) :
syntheticOneRecord r =
Measure.map (syntheticRecord r)
((1 / 2 : ℝ≥0∞) • Measure.dirac false +
(1 / 2 : ℝ≥0∞) • Measure.dirac true) := by
rw [Measure.map_add _ _ (by fun_prop)]
simp [syntheticOneRecord, Measure.map_smul]
theorem observedLaw_eq_bind
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma SyntheticCompletion.observedLaw_eq_bind {d : ℕ} {q : ℝ} (P : ZengLaw d)
(hP : P ∈ zengDiscreteClass d q) :
(fullMeasure P).map obs =
P.1.bind (fun r : ZengRecord d => syntheticOneRecord r) := by
let L := (parameters P).fullLaw
haveI : IsProbabilityMeasure L := (parameters P).fullLaw_probability
have hobs (z : FullCoord (Fin d) × Bool) :
obs (toFullRecord z) = syntheticRecord (selectMark z.1) z.2 := by
rcases z with ⟨⟨x, b, y₀, y₁⟩, a⟩
cases a <;> cases b <;> cases y₀ <;> cases y₁ <;> rfl
have hpair :
(L.prod fairCoin).map
(fun z : FullCoord (Fin d) × Bool => (selectMark z.1, z.2)) =
P.1.prod fairCoin := by
change Measure.map (map selectMark id) (L.prod fairCoin) = _
rw [← Measure.map_prod_map (f := selectMark) (g := id) L fairCoin
(by fun_prop) measurable_id]
rw [(parameters P).selectMark_map, selected_eq P hP]
simp
calc
(fullMeasure P).map obs =
(L.prod fairCoin).map
(fun z => syntheticRecord (selectMark z.1) z.2) := by
rw [fullMeasure, Measure.map_map (by fun_prop) (by fun_prop)]
congr 1
funext z
exact hobs z
_ = (P.1.prod fairCoin).map (fun z => syntheticRecord z.1 z.2) := by
rw [← hpair, Measure.map_map (by fun_prop) (by fun_prop)]
rfl
_ = P.1.bind (fun r => fairCoin.map (syntheticRecord r)) := by
exact (bind_map_eq_prod_map P.1 fairCoin syntheticRecord (by fun_prop)).symm
_ = P.1.bind (fun r : ZengRecord d => syntheticOneRecord r) := by
apply Measure.bind_congr_right
filter_upwards [] with r
simpa [fairCoin] using (syntheticOneRecord_eq_coin_map r).symm
def syntheticProductLaw
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma syntheticProductLaw_eq_coin_map {n d : ℕ}
(s : Fin n → ZengRecord d) :
syntheticProductLaw s =
Measure.map (fun u : Fin n → Bool => fun i => syntheticRecord (s i) (u i))
(Measure.pi (fun _ : Fin n =>
(1 / 2 : ℝ≥0∞) • Measure.dirac false +
(1 / 2 : ℝ≥0∞) • Measure.dirac true)) := by
let coin : Measure Bool :=
(1 / 2 : ℝ≥0∞) • Measure.dirac false +
(1 / 2 : ℝ≥0∞) • Measure.dirac true
haveI : IsProbabilityMeasure coin := by
refine ⟨?_⟩
simp [coin, ENNReal.inv_two_add_inv_two]
letI (i : Fin n) : IsProbabilityMeasure (coin.map (syntheticRecord (s i))) :=
Measure.isProbabilityMeasure_map (by fun_prop)
change syntheticProductLaw s =
(Measure.pi (fun _ : Fin n => coin)).map
(fun u i => syntheticRecord (s i) (u i))
rw [Measure.pi_map_pi (fun i => by fun_prop)]
unfold syntheticProductLaw
congr 1
funext i
exact syntheticOneRecord_eq_coin_map (s i)
theorem syntheticSample_bind_eq_pi
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
P :
ZengLaw d
conclusion
= Measure.pi (fun _ : Fin n => P.1.bind syntheticOneRecord)
Proof (Lean source)
lemma syntheticSample_bind_eq_pi {n d : ℕ} (P : ZengLaw d) :
(zengSampleLaw n P).bind syntheticProductLaw =
Measure.pi (fun _ : Fin n => P.1.bind syntheticOneRecord) := by
let coinPi := Measure.pi (fun _ : Fin n => SyntheticCompletion.fairCoin)
haveI : IsProbabilityMeasure P.1 := P.2
haveI : IsProbabilityMeasure coinPi := by
dsimp [coinPi]
infer_instance
have hkernel : syntheticProductLaw =
fun s : Fin n → ZengRecord d =>
coinPi.map (fun u i => syntheticRecord (s i) (u i)) := by
funext s
simpa [coinPi, SyntheticCompletion.fairCoin] using
syntheticProductLaw_eq_coin_map s
rw [hkernel]
rw [SyntheticCompletion.bind_map_eq_prod_map _ _ _ (measurable_of_finite _)]
let e := MeasurableEquiv.arrowProdEquivProdArrow (ZengRecord d) Bool (Fin n)
have hpair :
((zengSampleLaw n P).prod coinPi).map e.symm =
Measure.pi (fun _ : Fin n => P.1.prod SyntheticCompletion.fairCoin) := by
exact (measurePreserving_arrowProdEquivProdArrow
(ZengRecord d) Bool (Fin n) (fun _ => P.1)
(fun _ => SyntheticCompletion.fairCoin)).symm.map_eq
calc
((zengSampleLaw n P).prod coinPi).map
(fun z i => syntheticRecord (z.1 i) (z.2 i)) =
(((zengSampleLaw n P).prod coinPi).map e.symm).map
(fun w i => syntheticRecord (w i).1 (w i).2) := by
rw [Measure.map_map (by fun_prop) e.symm.measurable]
rfl
_ = (Measure.pi (fun _ : Fin n =>
P.1.prod SyntheticCompletion.fairCoin)).map
(fun w i => syntheticRecord (w i).1 (w i).2) := by rw [hpair]
_ = Measure.pi (fun _ : Fin n =>
(P.1.prod SyntheticCompletion.fairCoin).map
(fun z => syntheticRecord z.1 z.2)) := by
exact Measure.pi_map_pi (μ := fun _ : Fin n =>
P.1.prod SyntheticCompletion.fairCoin)
(f := fun _ z => syntheticRecord z.1 z.2)
(fun _ => (measurable_of_finite _).aemeasurable)
_ = Measure.pi (fun _ : Fin n => P.1.bind syntheticOneRecord) := by
congr 1
funext i
rw [← SyntheticCompletion.bind_map_eq_prod_map P.1
SyntheticCompletion.fairCoin syntheticRecord (measurable_of_finite _)]
apply Measure.bind_congr_right
filter_upwards [] with r
simpa [SyntheticCompletion.fairCoin] using
(syntheticOneRecord_eq_coin_map r).symm
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma syntheticOneRecord_probability {d : ℕ} (r : ZengRecord d) :
IsProbabilityMeasure (syntheticOneRecord r) := by
refine ⟨?_⟩
simp [syntheticOneRecord]
exact ENNReal.inv_two_add_inv_two
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma syntheticProductLaw_probability {n d : ℕ}
(s : Fin n → ZengRecord d) : IsProbabilityMeasure (syntheticProductLaw s) := by
unfold syntheticProductLaw
letI (i : Fin n) : IsProbabilityMeasure (syntheticOneRecord (s i)) :=
syntheticOneRecord_probability (s i)
infer_instance
def syntheticProductKernel
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
instance syntheticProductKernel_isMarkovKernel
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
instance syntheticProductKernel_isMarkovKernel (n d : ℕ) :
IsMarkovKernel (syntheticProductKernel n d) where
isProbabilityMeasure s := syntheticProductLaw_probability s
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n d :
ℕ
T :
Estimator n d
def
given by
by let K := syntheticProductKernel n d letI : IsMarkovKernel T.toBoundedKernel.1 := T.toBoundedKernel.2.1 refine ⟨T.toBoundedKernel.1 ∘ₖ K, inferInstance, ?_⟩ intro s rw [Kernel.comp_apply' _ _ _ (by simp)] simp only [T.toBoundedKernel.2.2] simp
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
thetaPoly n d ε s :
ℝ
given by
(∑ u : Fin n → Bool, mixedCountEstimator n d ε (fun i => syntheticRecord (s i) (u i))) / (2 : ℝ) ^ n
theorem sampleLaw_iid
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma sampleLaw_iid (n : ℕ) {d : ℕ} (P : FullLaw d) :
IIDSampling n (sampleLaw n P)
(fun (i : Fin n) (s : Fin n → ObsRecord d) => s i) P := by
let μ : Measure (ObsRecord d) := P.1.map obs
letI : IsProbabilityMeasure P.1 := P.2
letI : IsProbabilityMeasure μ :=
Measure.isProbabilityMeasure_map (by fun_prop : AEMeasurable obs P.1)
refine ⟨?_, ?_, ?_⟩
· intro i
fun_prop
· simpa [sampleLaw, μ] using
(iIndepFun_pi (μ := fun _ : Fin n => μ)
(X := fun _ : Fin n => id) (by intro i; fun_prop))
· intro i
simpa [sampleLaw, μ] using
(measurePreserving_eval (fun _ : Fin n => μ) i).map_eq
theorem synthetic_randomization_kernel
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
∃ Pstar : FullLaw d,
conclusion 1
UnrestrictedArrivalModelClass n d q Pstar
conclusion 2
conclusion 3
conclusion 4
ate Pstar = zengATE P
Proof (Lean source)
lemma synthetic_randomization_kernel {n d : ℕ} {q : ℝ}
(hn : 1 ≤ n) (hq : 0 < q) (hqhalf : q < 1 / 2)
(P : ZengLaw d) (hP : P ∈ zengDiscreteClass d q) :
∃ Pstar : FullLaw d,
UnrestrictedArrivalModelClass n d q Pstar ∧
IIDSampling n (sampleLaw n Pstar)
(fun (i : Fin n) (s : Fin n → ObsRecord d) => s i) Pstar ∧
(zengSampleLaw n P).bind syntheticProductLaw = sampleLaw n Pstar ∧
ate Pstar = zengATE P := by
refine ⟨SyntheticCompletion.fullLaw P, ?_, sampleLaw_iid n _, ?_, ?_⟩
· exact
{ n_pos := hn
d_pos := hP.2.2.1
q_pos := hq
q_le_one := by linarith
randomized := SyntheticCompletion.fullLaw_randomized P
balanced := SyntheticCompletion.fullLaw_balanced P
surrogate := SyntheticCompletion.fullLaw_surrogate P
outcome := SyntheticCompletion.fullLaw_outcome P
mar := SyntheticCompletion.fullLaw_mar P
arrival := SyntheticCompletion.fullLaw_arrival P hP }
· rw [sampleLaw, syntheticSample_bind_eq_pi]
congr 1
funext i
exact (SyntheticCompletion.observedLaw_eq_bind P hP).symm
· exact SyntheticCompletion.fullLaw_ate hn hq hqhalf P hP
theorem syntheticPullbackEstimator_risk_eq
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
zengSquaredRisk (syntheticPullbackEstimator T) P = squaredRisk T Pstar
Proof (Lean source)
lemma syntheticPullbackEstimator_risk_eq {n d : ℕ}
(T : Estimator n d) (P : ZengLaw d) (Pstar : FullLaw d)
(hsample : (zengSampleLaw n P).bind syntheticProductLaw = sampleLaw n Pstar)
(htarget : ate Pstar = zengATE P) :
zengSquaredRisk (syntheticPullbackEstimator T) P =
squaredRisk T Pstar := by
letI : IsProbabilityMeasure Pstar.1 := Pstar.2
letI : IsProbabilityMeasure (Pstar.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
letI : IsProbabilityMeasure (sampleLaw n Pstar) := by
unfold sampleLaw
infer_instance
have hF : Integrable
(fun s : Fin n → ObsRecord d =>
∫ t, (t - ate Pstar) ^ 2 ∂T.toBoundedKernel.1 s)
(sampleLaw n Pstar) := Integrable.of_finite
unfold zengSquaredRisk squaredRisk
rw [← htarget]
rw [← hsample]
have hmeas : Measurable (syntheticProductLaw (n := n) (d := d)) :=
(syntheticProductKernel n d).measurable
have hFbind : Integrable
(fun s : Fin n → ObsRecord d =>
∫ t, (t - ate Pstar) ^ 2 ∂T.toBoundedKernel.1 s)
((zengSampleLaw n P).bind syntheticProductLaw) := by
rw [hsample]
exact hF
rw [integral_bind hmeas hFbind]
congr 1
funext s
letI : IsMarkovKernel T.toBoundedKernel.1 := T.toBoundedKernel.2.1
letI : IsProbabilityMeasure (syntheticProductLaw s) :=
syntheticProductLaw_probability s
letI : IsProbabilityMeasure (T.toBoundedKernel.1 ∘ₘ syntheticProductLaw s) := by
infer_instance
have hsupp : (T.toBoundedKernel.1 ∘ₘ syntheticProductLaw s) (Icc (-1 : ℝ) 1) = 1 := by
rw [Measure.bind_apply measurableSet_Icc T.toBoundedKernel.1.measurable.aemeasurable]
simp only [T.toBoundedKernel.2.2]
simp
have hlossInt : Integrable (fun t : ℝ => (t - ate Pstar) ^ 2)
(T.toBoundedKernel.1 ∘ₘ syntheticProductLaw s) := by
refine (integrable_const ((1 + |ate Pstar|) ^ 2)).mono'
((measurable_id.sub measurable_const).pow_const 2).aestronglyMeasurable ?_
filter_upwards [(mem_ae_iff_prob_eq_one measurableSet_Icc).2 hsupp] with t ht
rcases ht with ⟨htlo, hthi⟩
have htAbs : |t| ≤ 1 := abs_le.mpr ⟨htlo, hthi⟩
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _), sq_le_sq]
have hnonneg : 0 ≤ (1 : ℝ) + |ate Pstar| := by positivity
rw [abs_of_nonneg hnonneg]
calc
|t - ate Pstar| ≤ |t| + |ate Pstar| := abs_sub _ _
_ ≤ 1 + |ate Pstar| := add_le_add_left htAbs _
simpa [syntheticPullbackEstimator, syntheticProductKernel,
Kernel.comp_apply, Kernel.ofFunOfCountable] using
(ProbabilityTheory.Kernel.integral_comp
(κ := Kernel.ofFunOfCountable syntheticProductLaw) (η := T.toBoundedKernel.1)
(a := s) (f := fun t : ℝ => (t - ate Pstar) ^ 2) hlossInt)
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.ZengLower.BoundedRisk 12 declarations Bounded randomized decision class for the fixed-positivity comparison.
Bounded randomized decision class for the fixed-positivity comparison.
def ZengEstimator
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n d :
ℕ
def
ZengEstimator n d :
Type
given by
def zengSampleLaw
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def zengSquaredRisk
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
zengSquaredRisk T P :
ℝ
given by
∫ s, (∫ t, (t - zengATE P) ^ 2 ∂(T.1 s)) ∂(zengSampleLaw n P)
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
zengDeterministicRisk f P :
ℝ
given by
∫ s, (f s - zengATE P) ^ 2 ∂(zengSampleLaw n P)
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def zengZeroControlClass
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def zengZeroControlRisk
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n d :
ℕ
ε :
ℝ
def
zengZeroControlRisk n d ε :
ℝ
given by
minimaxValueReal (E := ZengEstimator n d) (Θ := {P : ZengLaw d // P ∈ zengZeroControlClass d ε}) (fun T P => zengSquaredRisk T P.1)
def zengMinimaxRisk
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n d :
ℕ
ε :
ℝ
def
zengMinimaxRisk n d ε :
ℝ
given by
minimaxValueReal (E := ZengEstimator n d) (Θ := {P : ZengLaw d // P ∈ zengDiscreteClass d ε}) (fun T P => zengSquaredRisk T P.1)
theorem zeng_category_sum
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma zeng_category_sum {d : ℕ} (P : ZengLaw d) :
∑ x : Fin d, zengCategory P x = 1 := by
letI : IsProbabilityMeasure P.1 := P.2
have h := MeasureTheory.sum_measureReal_preimage_singleton
(μ := P.1) (s := univ) (f := fun r : ZengRecord d => r.1)
(by intro x hx; exact MeasurableSet.of_discrete)
calc
∑ x : Fin d, zengCategory P x =
∑ x : Fin d, P.1.real ((fun r : ZengRecord d => r.1) ⁻¹' {x}) := by
congr 1
_ = 1 := by simpa using h
theorem zeng_mean_mem_Icc
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma zeng_mean_mem_Icc {d : ℕ} (P : ZengLaw d) (x : Fin d)
(b : Bool) (h : 0 < zengCategory P x) :
zengMean P x b h ∈ Icc (0 : ℝ) 1 := by
letI : IsProbabilityMeasure P.1 := P.2
have hnum : 0 ≤ P.1.real {r | r.1 = x ∧ r.2.1 = b ∧ r.2.2 = true} :=
measureReal_nonneg
have hle : P.1.real {r | r.1 = x ∧ r.2.1 = b ∧ r.2.2 = true} ≤
zengArm P x b := by
apply measureReal_mono (by
intro r hr
exact ⟨hr.1, hr.2.1⟩) (by finiteness)
have hden : 0 ≤ zengArm P x b := measureReal_nonneg
rcases hden.eq_or_lt with hzero | hpos
· have hnumzero := le_antisymm (hzero ▸ hle) hnum
simp [zengMean, ← hzero, hnumzero, Set.mem_Icc]
· simp only [zengMean, Set.mem_Icc]
constructor
· exact div_nonneg hnum (le_of_lt hpos)
· exact (div_le_iff₀ hpos).mpr (by simpa using hle)
theorem zeng_ate_mem_Icc
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Proof (Lean source)
lemma zeng_ate_mem_Icc {d : ℕ} (P : ZengLaw d) :
zengATE P ∈ Icc (-1 : ℝ) 1 := by
letI : IsProbabilityMeasure P.1 := P.2
have hterm (x : Fin d) :
-zengCategory P x ≤
(if h : 0 < zengCategory P x then
zengCategory P x * (zengMean P x true h - zengMean P x false h) else 0) ∧
(if h : 0 < zengCategory P x then
zengCategory P x * (zengMean P x true h - zengMean P x false h) else 0) ≤
zengCategory P x := by
by_cases hx : 0 < zengCategory P x
· simp only [dif_pos hx]
have htrue := zeng_mean_mem_Icc P x true hx
have hfalse := zeng_mean_mem_Icc P x false hx
rw [Set.mem_Icc] at htrue hfalse
constructor
· have hd : -1 ≤ zengMean P x true hx - zengMean P x false hx := by
linarith [htrue.1, hfalse.2]
calc
-zengCategory P x = zengCategory P x * (-1) := by ring
_ ≤ _ := mul_le_mul_of_nonneg_left hd (le_of_lt hx)
· have hd : zengMean P x true hx - zengMean P x false hx ≤ 1 := by
linarith [htrue.2, hfalse.1]
calc
_ ≤ zengCategory P x * 1 := mul_le_mul_of_nonneg_left hd (le_of_lt hx)
_ = zengCategory P x := by ring
· simp only [dif_neg hx]
have hc : 0 ≤ zengCategory P x := measureReal_nonneg
constructor <;> linarith
have hsum := zeng_category_sum P
rw [Set.mem_Icc]
constructor
· calc
(-1 : ℝ) = ∑ x : Fin d, -zengCategory P x := by
simp [Finset.sum_neg_distrib, hsum]
_ ≤ zengATE P := by
simp only [zengATE]
exact Finset.sum_le_sum (fun x _ => (hterm x).1)
· calc
zengATE P ≤ ∑ x : Fin d, zengCategory P x := by
simp only [zengATE]
exact Finset.sum_le_sum (fun x _ => (hterm x).2)
_ = 1 := hsum
theorem zeng_squaredRisk_bounds
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma zeng_squaredRisk_bounds {n d : ℕ} (ε : ℝ) (T : ZengEstimator n d)
(P : ZengLaw d) (hP : P ∈ zengDiscreteClass d ε) :
0 ≤ zengSquaredRisk T P ∧ zengSquaredRisk T P ≤ 4 := by
letI : IsProbabilityMeasure P.1 := P.2
letI : IsMarkovKernel T.1 := T.2.1
have hATE := zeng_ate_mem_Icc P
rw [Set.mem_Icc] at hATE
have hinner_nonneg (s : Fin n → ZengRecord d) :
0 ≤ ∫ t, (t - zengATE P) ^ 2 ∂(T.1 s) := by
exact integral_nonneg (fun t => sq_nonneg _)
have hinner_le (s : Fin n → ZengRecord d) :
(∫ t, (t - zengATE P) ^ 2 ∂(T.1 s)) ≤ 4 := by
haveI : IsProbabilityMeasure (T.1 s) := inferInstance
have hs : ∀ᵐ t ∂(T.1 s), t ∈ Icc (-1 : ℝ) 1 :=
(mem_ae_iff_prob_eq_one measurableSet_Icc).mpr (T.2.2 s)
have hb : ∀ᵐ t ∂(T.1 s), (t - zengATE P) ^ 2 ≤ 4 := by
filter_upwards [hs] with t ht
rw [Set.mem_Icc] at ht
nlinarith [sq_nonneg (t - zengATE P + 2), sq_nonneg (t - zengATE P - 2)]
simpa using (integral_mono_of_nonneg (μ := T.1 s)
(Filter.Eventually.of_forall (fun t => sq_nonneg (t - zengATE P)))
(integrable_const 4) hb)
haveI : IsProbabilityMeasure (zengSampleLaw n P) := by
unfold zengSampleLaw
infer_instance
constructor
· exact integral_nonneg hinner_nonneg
· simpa [zengSquaredRisk] using (integral_mono_of_nonneg (μ := zengSampleLaw n P)
(Filter.Eventually.of_forall hinner_nonneg) (integrable_const 4)
(Filter.Eventually.of_forall hinner_le))
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.ZengLower.Concentration 1 declarations Chebyshev concentration and conditioning for iid relaxed rare-cell priors.
Chebyshev concentration and conditioning for iid relaxed rare-cell priors.
def iidRareGood
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
iidRareGood ν massRadius targetRadius z :
Prop
given by
clause 1
|(∑ x, (z x).1) - d * ∫ t, t.1 ∂ν| < massRadius
clause 2
|(∑ x, targetFunctional (z x)) - d * ∫ t, targetFunctional t ∂ν| < targetRadius
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.ZengLower.ConditionedPredictiveTV 2 declarations Total-variation cost of conditioning the relaxed iid prior.
Total-variation cost of conditioning the relaxed iid prior.
def relaxedCountKernel
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
d :
ℕ
a0 :
ℝ
lam :
ℝ≥0
def
relaxedCountKernel d a0 lam :
given by
(Kernel.const (Fin d → MarkedParam) (relaxedDummyCountLaw d a0 lam)).prod (coordinatewiseFiniteKernel (markedPoissonKernel (lam : ℝ)) d)
instance relaxedCountKernel_isMarkov
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
noncomputable instance relaxedCountKernel_isMarkov
(d : ℕ) (a0 : ℝ) (lam : ℝ≥0) :
IsMarkovKernel (relaxedCountKernel d a0 lam) := by
unfold relaxedCountKernel
infer_instance
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.ZengLower.DimensionPadding 13 declarations Zero-mass padding for the finite Zeng experiment.
Zero-mass padding for the finite Zeng experiment.
def padZengRecord
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def padZengLaw
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
given by
⟨P.1.map (padZengRecord h), by letI : IsProbabilityMeasure P.1 := P.2 exact Measure.isProbabilityMeasure_map (measurable_of_finite _).aemeasurable⟩
theorem padZengLaw_category_castLE
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma padZengLaw_category_castLE {r d : ℕ} (h : r ≤ d)
(P : ZengLaw r) (x : Fin r) :
zengCategory (padZengLaw h P) (castLE h x) = zengCategory P x := by
rw [zengCategory, zengCategory, Measure.real, padZengLaw,
Measure.map_apply (measurable_of_finite _) MeasurableSet.of_discrete]
apply congrArg toReal
congr 1
ext z
exact Fin.castLE_injective h |>.eq_iff
theorem padZengLaw_arm_castLE
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma padZengLaw_arm_castLE {r d : ℕ} (h : r ≤ d)
(P : ZengLaw r) (x : Fin r) (a : Bool) :
zengArm (padZengLaw h P) (castLE h x) a = zengArm P x a := by
rw [zengArm, zengArm, Measure.real, padZengLaw,
Measure.map_apply (measurable_of_finite _) MeasurableSet.of_discrete]
apply congrArg toReal
congr 1
ext z
simp [padZengRecord, Fin.castLE_injective h |>.eq_iff]
theorem padZengLaw_success_castLE
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma padZengLaw_success_castLE {r d : ℕ} (h : r ≤ d)
(P : ZengLaw r) (x : Fin r) (a : Bool) :
(padZengLaw h P).1.real
{z | z.1 = castLE h x ∧ z.2.1 = a ∧ z.2.2 = true} =
P.1.real {z | z.1 = x ∧ z.2.1 = a ∧ z.2.2 = true} := by
rw [Measure.real, padZengLaw,
Measure.map_apply (measurable_of_finite _) MeasurableSet.of_discrete]
apply congrArg toReal
congr 1
ext z
simp [padZengRecord, Fin.castLE_injective h |>.eq_iff]
theorem padZengLaw_category_new
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
zengCategory (padZengLaw h P) x = 0
Proof (Lean source)
lemma padZengLaw_category_new {r d : ℕ} (h : r ≤ d)
(P : ZengLaw r) (x : Fin d) (hx : r ≤ x) :
zengCategory (padZengLaw h P) x = 0 := by
rw [zengCategory, Measure.real, padZengLaw,
Measure.map_apply (measurable_of_finite _) MeasurableSet.of_discrete]
have hpre : padZengRecord h ⁻¹' {z : ZengRecord d | z.1 = x} = ∅ := by
ext z
constructor
· intro hz
have hv := congrArg (fun w : Fin d => (w : ℕ)) hz
simp [padZengRecord] at hv
omega
· simp
rw [hpre]
simp
theorem padZengLaw_mem_class
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
padZengLaw h P ∈ zengDiscreteClass d q
Proof (Lean source)
lemma padZengLaw_mem_class {r d : ℕ} {q : ℝ} (h : r ≤ d)
(P : ZengLaw r) (hP : P ∈ zengDiscreteClass r q) :
padZengLaw h P ∈ zengDiscreteClass d q := by
refine ⟨hP.1, hP.2.1, hP.2.2.1.trans h, ?_⟩
intro x hx
by_cases hxold : (x : ℕ) < r
· let x₀ : Fin r := ⟨x, hxold⟩
have he : castLE h x₀ = x := Fin.ext (by rfl)
rw [← he, padZengLaw_category_castLE] at hx
rw [← he, padZengLaw_category_castLE, padZengLaw_arm_castLE]
exact hP.2.2.2 x₀ hx
· rw [padZengLaw_category_new h P x (Nat.le_of_not_gt hxold)] at hx
linarith
theorem padZengLaw_ate
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma padZengLaw_ate {r d : ℕ} (h : r ≤ d) (P : ZengLaw r) :
zengATE (padZengLaw h P) = zengATE P := by
unfold zengATE
rw [← sum_fin_pad h (fun x => if hx : 0 < zengCategory P x then
zengCategory P x * (zengMean P x true hx - zengMean P x false hx) else 0)]
apply Fintype.sum_congr
intro x
by_cases hxold : (x : ℕ) < r
· let x₀ : Fin r := ⟨x, hxold⟩
have he : castLE h x₀ = x := Fin.ext (by rfl)
simp only [hxold, dite_true]
have hterm :
(if hp : 0 < zengCategory (padZengLaw h P) (castLE h x₀) then
zengCategory (padZengLaw h P) (castLE h x₀) *
(zengMean (padZengLaw h P) (castLE h x₀) true hp -
zengMean (padZengLaw h P) (castLE h x₀) false hp) else 0) =
if hp : 0 < zengCategory P x₀ then
zengCategory P x₀ *
(zengMean P x₀ true hp - zengMean P x₀ false hp) else 0 := by
by_cases hp : 0 < zengCategory P x₀
· have hppad : 0 < zengCategory (padZengLaw h P) (castLE h x₀) := by
rwa [padZengLaw_category_castLE]
rw [dif_pos hp, dif_pos hppad, padZengLaw_category_castLE]
unfold zengMean
rw [padZengLaw_arm_castLE, padZengLaw_arm_castLE,
padZengLaw_success_castLE, padZengLaw_success_castLE]
· have hppad : ¬0 < zengCategory (padZengLaw h P) (castLE h x₀) := by
rwa [padZengLaw_category_castLE]
rw [dif_neg hp, dif_neg hppad]
simpa only [he] using hterm
· simp only [hxold, dite_false]
have hz := padZengLaw_category_new h P x (Nat.le_of_not_gt hxold)
rw [dif_neg (by rw [hz]; linarith)]
def padZengSample
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
theorem padZengLaw_sample_map
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma padZengLaw_sample_map {n r d : ℕ} (h : r ≤ d) (P : ZengLaw r) :
(zengSampleLaw n P).map (padZengSample h) =
zengSampleLaw n (padZengLaw h P) := by
unfold zengSampleLaw padZengSample
letI : IsProbabilityMeasure P.1 := P.2
letI : IsProbabilityMeasure ((padZengLaw h P).1) := (padZengLaw h P).2
rw [Measure.pi_map_pi (fun _ => (measurable_of_finite _).aemeasurable)]
congr 1
def pullbackZengEstimator
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
given by
by letI : IsMarkovKernel T.1 := T.2.1 let K := T.1.comap (padZengSample h) (measurable_of_finite _) have hK : IsMarkovKernel K := Kernel.IsMarkovKernel.comap T.1 (measurable_of_finite _) refine ⟨K, hK, ?_⟩ intro s rw [Kernel.comap_apply] exact T.2.2 _
theorem zengSquaredRisk_pullback_padding
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma zengSquaredRisk_pullback_padding {n r d : ℕ} (h : r ≤ d)
(T : ZengEstimator n d) (P : ZengLaw r) :
zengSquaredRisk (pullbackZengEstimator h T) P =
zengSquaredRisk T (padZengLaw h P) := by
unfold zengSquaredRisk
rw [padZengLaw_ate]
let g : (Fin n → ZengRecord d) → ℝ := fun s =>
∫ t, (t - zengATE P) ^ 2 ∂(T.1 s)
have hg : Measurable g := by fun_prop
have hi := MeasureTheory.integral_map
(μ := zengSampleLaw n P)
(measurable_of_finite (padZengSample h)).aemeasurable
hg.aestronglyMeasurable
rw [padZengLaw_sample_map h P] at hi
simpa [g, pullbackZengEstimator, Kernel.comap_apply] using hi.symm
theorem zengMinimaxRisk_mono_dimension
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n r d :
ℕ
hr :
1 ≤ r
h :
r ≤ d
q :
ℝ
conclusion
zengMinimaxRisk n r q ≤ zengMinimaxRisk n d q
Proof (Lean source)
lemma zengMinimaxRisk_mono_dimension {n r d : ℕ} (hr : 1 ≤ r) (h : r ≤ d)
(q : ℝ) : zengMinimaxRisk n r q ≤ zengMinimaxRisk n d q := by
let T₀ : ZengEstimator n d :=
⟨Kernel.const _ (Measure.dirac 0), inferInstance, by intro s; simp [Kernel.const_apply]⟩
letI : Nonempty (ZengEstimator n d) := ⟨T₀⟩
unfold zengMinimaxRisk
apply minimaxValue_le_minimaxValue
(bddBelow_range_worstCaseRisk fun _ _ => by
unfold zengSquaredRisk
positivity)
intro T
refine ⟨pullbackZengEstimator h T, ?_⟩
change worstCaseRiskReal
(fun (_ : Unit) (P : {P : ZengLaw r // P ∈ zengDiscreteClass r q}) =>
zengSquaredRisk (pullbackZengEstimator h T) P.1) () ≤
worstCaseRiskReal
(fun (_ : Unit) (P : {P : ZengLaw d // P ∈ zengDiscreteClass d q}) =>
zengSquaredRisk T P.1) ()
apply worstCaseRisk_mono_class_of_nonneg
(fun P : {P : ZengLaw r // P ∈ zengDiscreteClass r q} =>
⟨padZengLaw h P.1, padZengLaw_mem_class h P.1 P.2⟩)
· refine ⟨4, ?_⟩
rintro y ⟨Q, rfl⟩
exact (zeng_squaredRisk_bounds q T Q.1 Q.2).2
· intro Q
exact (zeng_squaredRisk_bounds q T Q.1 Q.2).1
· rintro ⟨P, hP⟩
exact (zengSquaredRisk_pullback_padding h T P).le
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.ZengLower.Experiment 3 declarations Finite zero-control observational laws for the cited fixed-positivity experiment.
Finite zero-control observational laws for the cited fixed-positivity experiment.
def zengBernWeight
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def zengFiniteTable
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
d :
ℕ
p π μ :
Fin d → ℝ
def
given by
∑ x : Fin d, ∑ b : Bool, ∑ y : Bool, ofReal (p x * zengBernWeight (π x) b * (if b then zengBernWeight (μ x) y else if y then 0 else 1)) • Measure.dirac (x, b, y)
theorem zeng_finiteTable_zero_control
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
d :
ℕ
ε :
ℝ
hd :
1 ≤ d
hε :
0 < ε ∧ ε < 1 / 2
p π μ :
Fin d → ℝ
hp :
∀ x, 0 ≤ p x
hsum :
∑ x : Fin d, p x = 1
hπ :
∀ x
if
0 < p x
then
ε ≤ π x ∧ π x ≤ 1 - ε
hμ :
∀ x, 0 ≤ μ x ∧ μ x ≤ 1
conclusion
∃ P : ZengLaw d,
conclusion 1
P.1 = zengFiniteTable d p π μ
conclusion 2
P ∈ zengZeroControlClass d ε
Proof (Lean source)
lemma zeng_finiteTable_zero_control (d : ℕ) (ε : ℝ)
(hd : 1 ≤ d) (hε : 0 < ε ∧ ε < 1 / 2)
(p π μ : Fin d → ℝ)
(hp : ∀ x, 0 ≤ p x) (hsum : ∑ x : Fin d, p x = 1)
(hπ : ∀ x, 0 < p x → ε ≤ π x ∧ π x ≤ 1 - ε)
(hμ : ∀ x, 0 ≤ μ x ∧ μ x ≤ 1) :
∃ P : ZengLaw d, P.1 = zengFiniteTable d p π μ ∧
P ∈ zengZeroControlClass d ε ∧
zengATE P = ∑ x : Fin d, p x * μ x := by
have hπ01 (x : Fin d) (hx : 0 < p x) : 0 ≤ π x ∧ π x ≤ 1 := by
constructor
· exact le_trans (le_of_lt hε.1) (hπ x hx).1
· exact le_trans (hπ x hx).2 (by linarith [hε.1])
let M := zengFiniteTable d p π μ
have hM_univ : M univ = 1 := by
rw [zengFiniteTable_univ d p π μ hp hπ01 hμ, hsum]
norm_num
let P : ZengLaw d := ⟨M, ⟨hM_univ⟩⟩
have hcat (x : Fin d) : zengCategory P x = p x := by
exact zengFiniteTable_category_real d p π μ hp hπ01 hμ x
have harm_true (x : Fin d) : zengArm P x true = p x * π x := by
exact zengFiniteTable_true_arm_real d p π μ hp
(fun x hx => (hπ01 x hx).1) hμ x
have hnum_true (x : Fin d) :
P.1.real {r | r.1 = x ∧ r.2.1 = true ∧ r.2.2 = true} =
p x * π x * μ x := by
exact zengFiniteTable_true_outcome_real d p π μ hp
(fun x hx => (hπ01 x hx).1) (fun x => (hμ x).1) x
have hnum_false (x : Fin d) :
P.1.real {r | r.1 = x ∧ r.2.1 = false ∧ r.2.2 = true} = 0 := by
exact zengFiniteTable_false_outcome_real d p π μ x
have hmean_true (x : Fin d) (hx : 0 < zengCategory P x) :
zengMean P x true hx = μ x := by
have hpx : 0 < p x := by simpa [hcat x] using hx
have hπx : 0 < π x := lt_of_lt_of_le hε.1 (hπ x hpx).1
rw [zengMean, hnum_true x, harm_true x]
field_simp
have hmean_false (x : Fin d) (hx : 0 < zengCategory P x) :
zengMean P x false hx = 0 := by
simp [zengMean, hnum_false x]
refine ⟨P, rfl, ?_, ?_⟩
· constructor
· refine ⟨hε.1, hε.2, hd, ?_⟩
intro x hx
have hpx : 0 < p x := by simpa [hcat x] using hx
rw [hcat x, harm_true x]
constructor
· simpa [mul_comm] using
(mul_le_mul_of_nonneg_left (hπ x hpx).1 (hp x))
· simpa [mul_comm] using
(mul_le_mul_of_nonneg_left (hπ x hpx).2 (hp x))
· intro x hx
exact hmean_false x hx
· unfold zengATE
apply Finset.sum_congr rfl
intro x hx
by_cases hpx : 0 < p x
· rw [dif_pos (by simpa [hcat x] using hpx)]
simp only [hmean_true x, hmean_false x, sub_zero]
rw [hcat x]
· rw [dif_neg (by simpa [hcat x] using hpx)]
have hpx0 : p x = 0 := le_antisymm (le_of_not_gt hpx) (hp x)
simp [hpx0]
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.ZengLower.FixedPoissonBridge 1 declarations Exact prefix transfer and the remaining many-cell lower-bound obligation.
Exact prefix transfer and the remaining many-cell lower-bound obligation.
instance zengFiniteTable_isFiniteMeasure
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
noncomputable instance zengFiniteTable_isFiniteMeasure
(d : ℕ) (p π μ : Fin d → ℝ) :
IsFiniteMeasure (zengFiniteTable d p π μ) := by
apply IsFiniteMeasure.mk
simp [zengFiniteTable]
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.ZengLower.Main 1 declarations The cited fixed-positivity lower bound and its paper-local proof.
The cited fixed-positivity lower bound and its paper-local proof.
the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def
ZengTheoremTwoLower :
Sort 0
given by
∀ ε : ℝ,
0 < ε → ε < 1 / 2 → ∃ (cε c₀ : ℝ) (n₀ : ℕ), 0 < cε ∧ 0 < c₀ ∧ ∀ (n d : ℕ), n₀
≤ n → 1 ≤ d → (d : ℝ) ≤ c₀ * n * log n → cε * ((n : ℝ)⁻¹ + ((d : ℝ) / ((n : ℝ) * log n)) ^ 2) ≤ zengZeroControlRisk n d ε
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.ZengLower.ManyCellComposition 2 declarations Paper moment certificates composed with conditioned fuzzy testing.
Paper moment certificates composed with conditioned fuzzy testing.
def swap
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
given by
clause 1
prior0 := M.prior1
clause 2
prior1 := M.prior0
clause 3
probability0 := M.probability1
clause 4
probability1 := M.probability0
clause 5
degree_pos := M.degree_pos
clause 6
degree_calibration := M.degree_calibration
clause 7
finite_support := by obtain ⟨s0, s1, hs0, hs1, h0, h1⟩ := M.finite_support exact ⟨s1, s0, hs1, hs0, h1, h0⟩
clause 8
matched_moments := by intro i j k hijk exact (M.matched_moments i j k hijk).symm
clause 9
mean_p_eq := M.mean_p_eq.symm
clause 10
mean_p_le := by rw [← M.mean_p_eq] exact M.mean_p_le
clause 11
separated_pmu := by simpa [abs_sub_comm] using M.separated_pmu
theorem gap_or_swap_gap
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
c₄ / ((n : ℝ) * log n)
≤ (∫ t, targetFunctional t ∂M.prior1)
- ∫ t, targetFunctional t ∂M.prior0 ∨ c₄ / ((n : ℝ) * log n)
≤ (∫ t, targetFunctional t ∂M.swap.prior1)
- ∫ t, targetFunctional t ∂M.swap.prior0
Proof (Lean source)
lemma ZengMomentPriorPair.gap_or_swap_gap
{ε c₁ c₂ c₄ : ℝ} {n d K : ℕ}
(M : ZengMomentPriorPair ε c₁ c₂ c₄ n d K) :
c₄ / ((n : ℝ) * log n) ≤
(∫ t, targetFunctional t ∂M.prior1) -
∫ t, targetFunctional t ∂M.prior0 ∨
c₄ / ((n : ℝ) * log n) ≤
(∫ t, targetFunctional t ∂M.swap.prior1) -
∫ t, targetFunctional t ∂M.swap.prior0 := by
let a := (∫ t, targetFunctional t ∂M.prior1) -
∫ t, targetFunctional t ∂M.prior0
have hsep : c₄ / ((n : ℝ) * log n) ≤ |a| := by
simpa [a, targetFunctional] using M.separated_pmu
by_cases ha : 0 ≤ a
· left
simpa [abs_of_nonneg ha, a] using hsep
· right
have hneg : a < 0 := lt_of_not_ge ha
simpa [ZengMomentPriorPair.swap, a, abs_of_neg hneg] using hsep
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.ZengLower.MomentCertificate 1 declarations Quantitative reciprocal moment certificate required by the Zeng lower bound.
Quantitative reciprocal moment certificate required by the Zeng lower bound.
structure ZengMomentPriorPair
For the specified inputs and assumptions, the mathematical structure records the components specified by this declaration.
Definition (Lean source)
parameters
ε c₁ c₂ c₄ :
ℝ
n d K :
ℕ
fields
prior0 :
Measure (ℝ × ℝ × ℝ)
prior1 :
Measure (ℝ × ℝ × ℝ)
probability0 :
IsProbabilityMeasure prior0
probability1 :
IsProbabilityMeasure prior1
degree_pos :
1 ≤ K
finite_support :
matched_moments :
∀ i j k : ℕ
if
i + j + k ≤ 3 * K
then
(∫ z, z.1 ^ i * (z.1 * z.2.1) ^ j * (z.1 * z.2.1 * z.2.2) ^ k ∂prior0)
= ∫ z, z.1 ^ i * (z.1 * z.2.1) ^ j * (z.1 * z.2.1 * z.2.2) ^ k ∂prior1
mean_p_eq :
(∫ z, z.1 ∂prior0) = ∫ z, z.1 ∂prior1
mean_p_le :
(∫ z, z.1 ∂prior0) ≤ ((d : ℝ))⁻¹
separated_pmu :
c₄ / ((n : ℝ) * log n) ≤ |(∫ z, z.1 * z.2.2 ∂prior1) - (∫ z, z.1 * z.2.2 ∂prior0)|
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.ZengLower.MomentCertificateProof 1 declarations
theorem exists_triplePriors
Given a paper-local moment-prior pair, a generic triple-prior certificate exists at the same scale and target gap.
Formal statement
hypotheses
conclusion
Proof (Lean source)
theorem ZengMomentPriorPair.exists_triplePriors
{ε c₁ c₂ c₄ : ℝ} {n d K : ℕ}
(M : ZengMomentPriorPair ε c₁ c₂ c₄ n d K) :
Nonempty (TriplePriors K (c₁ * log n / n) ε
(c₄ / ((n : ℝ) * log n))) := by
letI : IsProbabilityMeasure M.prior0 := M.probability0
letI : IsProbabilityMeasure M.prior1 := M.probability1
let B : Set MarkedParam := {z | 0 ≤ z.1 ∧
z.1 ≤ c₁ * log n / n ∧ ε ≤ z.2.1 ∧ z.2.1 ≤ 1 - ε ∧
0 ≤ z.2.2 ∧ z.2.2 ≤ 1}
obtain ⟨s0, s1, hs0, hs1, h0, h1⟩ := M.finite_support
have hsup0 : M.prior0 B = 1 := by
apply le_antisymm prob_le_one
rw [← hs0]
apply measure_mono
intro z hz
exact h0 z hz
have hsup1 : M.prior1 B = 1 := by
apply le_antisymm prob_le_one
rw [← hs1]
apply measure_mono
intro z hz
exact h1 z hz
exact ⟨{
positive_degree := M.degree_pos
ν₀ := M.prior0
ν₁ := M.prior1
probability₀ := M.probability0
probability₁ := M.probability1
finite₀ := ⟨s0, hs0⟩
finite₁ := ⟨s1, hs1⟩
supported₀ := hsup0
supported₁ := hsup1
mixed_match := by
intro i j k hijk
simpa [mixedMonomial] using M.matched_moments i j k hijk
mean_p_match := M.mean_p_eq
target_gap := by
rw [abs_sub_comm]
simpa [targetFunctional] using M.separated_pmu }⟩
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.ZengLower.MomentGapCap 1 declarations
def weaken_separation
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
ε c₁ c₂ c₄ c₄' :
ℝ
n d K :
ℕ
M :
ZengMomentPriorPair ε c₁ c₂ c₄ n d K
hden :
0 ≤ (n : ℝ) * log n
hc₄' :
c₄' ≤ c₄
def
given by
by refine { M with separated_pmu := ?_ } exact (div_le_div_of_nonneg_right hc₄' hden).trans M.separated_pmu
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.ZengLower.Parametric 20 declarations A two-point zero-control submodel for the parametric term.
A two-point zero-control submodel for the parametric term.
def zengOneCellMass
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
theorem zengOneCellMass_nonneg
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma zengOneCellMass_nonneg {d : ℕ} (x₀ x : Fin d) :
0 ≤ zengOneCellMass x₀ x := by
unfold zengOneCellMass
split <;> norm_num
theorem zengOneCellMass_sum
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma zengOneCellMass_sum {d : ℕ} (x₀ : Fin d) :
∑ x : Fin d, zengOneCellMass x₀ x = 1 := by
simp [zengOneCellMass]
theorem zeng_parametric_exists
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
d :
ℕ
ε u :
ℝ
hd :
1 ≤ d
hε :
0 < ε ∧ ε < 1 / 2
hu :
0 ≤ u ∧ u ≤ 1
conclusion
∃ P : ZengLaw d,
conclusion 1
P.1 = zengFiniteTable d (zengOneCellMass ⟨0, hd⟩) (fun _ => 1 / 2) (fun _ => u)
conclusion 2
P ∈ zengZeroControlClass d ε
conclusion 3
zengATE P = u
Proof (Lean source)
lemma zeng_parametric_exists (d : ℕ) (ε u : ℝ) (hd : 1 ≤ d)
(hε : 0 < ε ∧ ε < 1 / 2) (hu : 0 ≤ u ∧ u ≤ 1) :
∃ P : ZengLaw d,
P.1 = zengFiniteTable d (zengOneCellMass ⟨0, hd⟩) (fun _ => 1 / 2)
(fun _ => u) ∧
P ∈ zengZeroControlClass d ε ∧ zengATE P = u := by
obtain ⟨P, hmeasure, hclass, hATE⟩ := zeng_finiteTable_zero_control d ε hd hε
(zengOneCellMass ⟨0, hd⟩) (fun _ => 1 / 2) (fun _ => u)
(zengOneCellMass_nonneg ⟨0, hd⟩) (zengOneCellMass_sum ⟨0, hd⟩)
(by intro x hx; constructor <;> linarith [hε.2]) (fun _ => hu)
refine ⟨P, hmeasure, hclass, ?_⟩
rw [hATE]
simp [zengOneCellMass]
def zengParametricLaw
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
d :
ℕ
ε u :
ℝ
hd :
1 ≤ d
hε :
0 < ε ∧ ε < 1 / 2
hu :
0 ≤ u ∧ u ≤ 1
def
given by
choose (zeng_parametric_exists d ε u hd hε hu)
theorem zengParametricLaw_measure
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
d :
ℕ
ε u :
ℝ
hd :
1 ≤ d
hε :
0 < ε ∧ ε < 1 / 2
hu :
0 ≤ u ∧ u ≤ 1
conclusion
(zengParametricLaw d ε u hd hε hu).1
= zengFiniteTable d (zengOneCellMass ⟨0, hd⟩) (fun _ => 1 / 2) (fun _ => u)
Proof (Lean source)
lemma zengParametricLaw_measure (d : ℕ) (ε u : ℝ) (hd : 1 ≤ d)
(hε : 0 < ε ∧ ε < 1 / 2) (hu : 0 ≤ u ∧ u ≤ 1) :
(zengParametricLaw d ε u hd hε hu).1 =
zengFiniteTable d (zengOneCellMass ⟨0, hd⟩) (fun _ => 1 / 2)
(fun _ => u) :=
(Classical.choose_spec (zeng_parametric_exists d ε u hd hε hu)).1
theorem zengParametricLaw_mem
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
d :
ℕ
ε u :
ℝ
hd :
1 ≤ d
hε :
0 < ε ∧ ε < 1 / 2
hu :
0 ≤ u ∧ u ≤ 1
conclusion
zengParametricLaw d ε u hd hε hu ∈ zengZeroControlClass d ε
Proof (Lean source)
lemma zengParametricLaw_mem (d : ℕ) (ε u : ℝ) (hd : 1 ≤ d)
(hε : 0 < ε ∧ ε < 1 / 2) (hu : 0 ≤ u ∧ u ≤ 1) :
zengParametricLaw d ε u hd hε hu ∈ zengZeroControlClass d ε :=
by
unfold zengParametricLaw
exact (Classical.choose_spec (zeng_parametric_exists d ε u hd hε hu)).2.1
theorem zengParametricLaw_ate
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
d :
ℕ
ε u :
ℝ
hd :
1 ≤ d
hε :
0 < ε ∧ ε < 1 / 2
hu :
0 ≤ u ∧ u ≤ 1
conclusion
zengATE (zengParametricLaw d ε u hd hε hu) = u
Proof (Lean source)
lemma zengParametricLaw_ate (d : ℕ) (ε u : ℝ) (hd : 1 ≤ d)
(hε : 0 < ε ∧ ε < 1 / 2) (hu : 0 ≤ u ∧ u ≤ 1) :
zengATE (zengParametricLaw d ε u hd hε hu) = u :=
by
unfold zengParametricLaw
exact (Classical.choose_spec (zeng_parametric_exists d ε u hd hε hu)).2.2
theorem zengParametricLaw_singleton
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(zengParametricLaw d ε u hd hε hu).1.real {r}
= if r.1 = (⟨0, hd⟩ : Fin d) then if r.2.1 then (if r.2.2 then u / 2 else (1 - u) / 2) else (if r.2.2 then 0 else 1 / 2) else 0
Proof (Lean source)
lemma zengParametricLaw_singleton (d : ℕ) (ε u : ℝ) (hd : 1 ≤ d)
(hε : 0 < ε ∧ ε < 1 / 2) (hu : 0 ≤ u ∧ u ≤ 1)
(r : ZengRecord d) :
(zengParametricLaw d ε u hd hε hu).1.real {r} =
if r.1 = (⟨0, hd⟩ : Fin d) then
if r.2.1 then (if r.2.2 then u / 2 else (1 - u) / 2)
else (if r.2.2 then 0 else 1 / 2)
else 0 := by
rcases r with ⟨x, b, y⟩
rw [zengParametricLaw_measure]
simp [zengFiniteTable, Measure.real, indicator, zengOneCellMass]
rw [Finset.sum_eq_single x]
· cases b
· cases y
· by_cases hx : x = (⟨0, hd⟩ : Fin d)
· simp [hx, zengBernWeight]
norm_num [ENNReal.toReal_ofReal]
· simp [hx]
· norm_num [zengBernWeight, ENNReal.toReal_ofReal]
· cases y
· by_cases hx : x = (⟨0, hd⟩ : Fin d)
· have hnon : 0 ≤ 1 - u := sub_nonneg.mpr hu.2
simp [hx, zengBernWeight, ENNReal.toReal_ofReal hnon]
ring
· simp [hx]
· by_cases hx : x = (⟨0, hd⟩ : Fin d)
· simp [hx, zengBernWeight, ENNReal.toReal_ofReal hu.1]
ring
· simp [hx]
· intro x' hx hne
simp [hne]
· simp
theorem zengParametricLaw_ac_half
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
d :
ℕ
ε u :
ℝ
hd :
1 ≤ d
hε :
0 < ε ∧ ε < 1 / 2
hu :
0 ≤ u ∧ u ≤ 1
conclusion
(zengParametricLaw d ε u hd hε hu).1 ≪ (zengParametricLaw d ε (1 / 2) hd hε (by norm_num)).1
Proof (Lean source)
lemma zengParametricLaw_ac_half (d : ℕ) (ε u : ℝ) (hd : 1 ≤ d)
(hε : 0 < ε ∧ ε < 1 / 2) (hu : 0 ≤ u ∧ u ≤ 1) :
(zengParametricLaw d ε u hd hε hu).1 ≪
(zengParametricLaw d ε (1 / 2) hd hε (by norm_num)).1 := by
intro s hzero
rw [zengParametricLaw_measure] at hzero ⊢
simp [zengFiniteTable, zengOneCellMass, zengBernWeight] at hzero ⊢
intro i
by_cases hi : i = (⟨0, hd⟩ : Fin d)
· subst i
have hsupport := hzero (⟨0, hd⟩ : Fin d)
norm_num at hsupport
simp [hsupport]
· simp [hi]
theorem zengParametricLaw_chiSq_half
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
d :
ℕ
ε u :
ℝ
hd :
1 ≤ d
hε :
0 < ε ∧ ε < 1 / 2
hu :
0 ≤ u ∧ u ≤ 1
conclusion
1
+ chiSqDiv (zengParametricLaw d ε u hd hε hu).1 (zengParametricLaw d ε (1 / 2) hd hε (by norm_num)).1
= 1 + 2 * (u - 1 / 2) ^ 2
Proof (Lean source)
lemma zengParametricLaw_chiSq_half (d : ℕ) (ε u : ℝ) (hd : 1 ≤ d)
(hε : 0 < ε ∧ ε < 1 / 2) (hu : 0 ≤ u ∧ u ≤ 1) :
1 + chiSqDiv
(zengParametricLaw d ε u hd hε hu).1
(zengParametricLaw d ε (1 / 2) hd hε (by norm_num)).1 =
1 + 2 * (u - 1 / 2) ^ 2 := by
let P₁ := zengParametricLaw d ε u hd hε hu
let P₀ := zengParametricLaw d ε (1 / 2) hd hε (by norm_num)
letI : IsProbabilityMeasure P₁.1 := P₁.2
letI : IsProbabilityMeasure P₀.1 := P₀.2
change 1 + chiSqDiv P₁.1 P₀.1 = _
rw [finite_one_add_chiSqDiv _ _
(zengParametricLaw_ac_half d ε u hd hε hu)]
simp_rw [zengParametricLaw_singleton]
simp only [Fintype.sum_prod_type]
simp
rw [Finset.sum_eq_single (⟨0, hd⟩ : Fin d)]
· norm_num
ring
· intro x hx hne
simp [hne]
· simp
def zengParametricStep
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
theorem zengParametricStep_pos
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma zengParametricStep_pos (n : ℕ) (hn : 1 ≤ n) :
0 < zengParametricStep n := by
unfold zengParametricStep
positivity
theorem zengParametricStep_le_quarter
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma zengParametricStep_le_quarter (n : ℕ) (hn : 1 ≤ n) :
zengParametricStep n ≤ 1 / 4 := by
have hnR : (1 : ℝ) ≤ n := by exact_mod_cast hn
have hsqrt : (1 : ℝ) ≤ sqrt n := by
rw [← Real.sqrt_one]
exact Real.sqrt_le_sqrt hnR
unfold zengParametricStep
rw [div_le_iff₀ (by positivity : (0 : ℝ) < 4 * sqrt n)]
nlinarith
theorem zengParametricPerturbation_mem
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n :
ℕ
hn :
1 ≤ n
conclusion
conclusion 1
0 ≤ (1 / 2 + zengParametricStep n)
conclusion 2
(1 / 2 + zengParametricStep n) ≤ 1
Proof (Lean source)
lemma zengParametricPerturbation_mem (n : ℕ) (hn : 1 ≤ n) :
0 ≤ (1 / 2 + zengParametricStep n) ∧
(1 / 2 + zengParametricStep n) ≤ 1 := by
have hp := zengParametricStep_pos n hn
have hq := zengParametricStep_le_quarter n hn
constructor <;> linarith
theorem zengParametricPerturbation_chiSq
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
ε :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hε :
0 < ε ∧ ε < 1 / 2
conclusion
chiSqDiv (zengParametricLaw d ε (1 / 2 + zengParametricStep n) hd hε (zengParametricPerturbation_mem n hn)).1 (zengParametricLaw d ε (1 / 2) hd hε (by norm_num)).1
≤ (1 / 8 : ℝ) / n
Proof (Lean source)
lemma zengParametricPerturbation_chiSq (n d : ℕ) (ε : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hε : 0 < ε ∧ ε < 1 / 2) :
chiSqDiv
(zengParametricLaw d ε (1 / 2 + zengParametricStep n) hd hε
(zengParametricPerturbation_mem n hn)).1
(zengParametricLaw d ε (1 / 2) hd hε (by norm_num)).1 ≤
(1 / 8 : ℝ) / n := by
have hEq := zengParametricLaw_chiSq_half d ε
(1 / 2 + zengParametricStep n) hd hε
(zengParametricPerturbation_mem n hn)
have hchi : chiSqDiv
(zengParametricLaw d ε (1 / 2 + zengParametricStep n) hd hε
(zengParametricPerturbation_mem n hn)).1
(zengParametricLaw d ε (1 / 2) hd hε (by norm_num)).1 =
2 * (zengParametricStep n) ^ 2 := by
linarith
rw [hchi]
have hnR : (0 : ℝ) < n := by positivity
have hsqrt : sqrt (n : ℝ) ≠ 0 := by positivity
have hsqrt_sq : (sqrt (n : ℝ)) ^ 2 = n := by
rw [Real.sq_sqrt (le_of_lt hnR)]
unfold zengParametricStep
rw [div_pow]
field_simp
nlinarith
def ZengParametricDecisionReduction
the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n d :
ℕ
ε c :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hε :
0 < ε ∧ ε < 1 / 2
0 < c
(∀ A : Set (Fin n → ZengRecord d), MeasurableSet A → (zengSampleLaw n (zengParametricLaw d ε (1 / 2 + zengParametricStep n) hd hε (zengParametricPerturbation_mem n hn))).real Aᶜ + (zengSampleLaw n (zengParametricLaw d ε (1 / 2) hd hε (by norm_num))).real A ≥ c)
T :
ZengEstimator n d
def
ZengParametricDecisionReduction :
Prop
given by
c * (zengParametricStep n) ^ 2 / 8
≤ max (zengSquaredRisk T (zengParametricLaw d ε (1 / 2) hd hε (by norm_num))) (zengSquaredRisk T (zengParametricLaw d ε (1 / 2 + zengParametricStep n) hd hε (zengParametricPerturbation_mem n hn)))
theorem zengParametricDecisionReduction_proved
the stated mathematical conclusion holds.
Formal statement
conclusion
Proof (Lean source)
theorem zengParametricDecisionReduction_proved : ZengParametricDecisionReduction := by
intro n d ε c hn hd hε hc htest T
let P₀ := zengParametricLaw d ε (1 / 2) hd hε (by norm_num)
let P₁ := zengParametricLaw d ε (1 / 2 + zengParametricStep n) hd hε
(zengParametricPerturbation_mem n hn)
let f := kernelMean T.1 zengClip
let δ := zengParametricStep n
let θ₀ : ℝ := 1 / 2
let θ₁ : ℝ := 1 / 2 + δ
let E₀ : Set (Fin n → ZengRecord d) := {s | δ / 2 ≤ |f s - θ₀|}
let E₁ : Set (Fin n → ZengRecord d) := {s | δ / 2 ≤ |f s - θ₁|}
let A : Set (Fin n → ZengRecord d) := {s | |f s - θ₁| ≤ |f s - θ₀|}
letI : IsMarkovKernel T.1 := T.2.1
letI : IsProbabilityMeasure P₀.1 := P₀.2
letI : IsProbabilityMeasure P₁.1 := P₁.2
letI : IsProbabilityMeasure (zengSampleLaw n P₀) := by
unfold zengSampleLaw
infer_instance
letI : IsProbabilityMeasure (zengSampleLaw n P₁) := by
unfold zengSampleLaw
infer_instance
have hδ : 0 < δ := zengParametricStep_pos n hn
have hf : Measurable f := Causalean.Stat.measurable_kernelMean T.1 zengClip_measurable
have hA : MeasurableSet A := by
exact measurableSet_le
(continuous_abs.measurable.comp (hf.sub measurable_const))
(continuous_abs.measurable.comp (hf.sub measurable_const))
have hA_E₀ : A ⊆ E₀ := by
intro s hs
simp only [A, E₀, Set.mem_setOf_eq] at hs ⊢
have htri : δ ≤ |f s - θ₀| + |f s - θ₁| := by
have := abs_sub_le θ₁ (f s) θ₀
dsimp [θ₀, θ₁, δ] at this ⊢
norm_num at this
rw [abs_of_pos (zengParametricStep_pos n hn)] at this
simpa [abs_sub_comm, add_comm] using this
linarith
have hAc_E₁ : Aᶜ ⊆ E₁ := by
intro s hs
simp only [A, E₁, Set.mem_compl_iff, Set.mem_setOf_eq, not_le] at hs ⊢
have htri : δ ≤ |f s - θ₀| + |f s - θ₁| := by
have := abs_sub_le θ₁ (f s) θ₀
dsimp [θ₀, θ₁, δ] at this ⊢
norm_num at this
rw [abs_of_pos (zengParametricStep_pos n hn)] at this
simpa [abs_sub_comm, add_comm] using this
linarith
have htest' : c ≤ (zengSampleLaw n P₁).real E₁ +
(zengSampleLaw n P₀).real E₀ := by
calc
c ≤ (zengSampleLaw n P₁).real Aᶜ +
(zengSampleLaw n P₀).real A := htest A hA
_ ≤ _ := add_le_add
(measureReal_mono hAc_E₁ (by finiteness))
(measureReal_mono hA_E₀ (by finiteness))
have hfbound : ∀ s, f s ∈ Icc (-1 : ℝ) 1 := by
intro s
rw [Set.mem_Icc, ← abs_le]
exact abs_kernelMean_le T.1 (M := 1)
zengClip_abs_le s
have hint₀ : Integrable (fun s ↦ (f s - θ₀) ^ 2) (zengSampleLaw n P₀) :=
mse_integrable_of_estimator_bound _ f hf (by norm_num) hfbound
have hint₁ : Integrable (fun s ↦ (f s - θ₁) ^ 2) (zengSampleLaw n P₁) :=
mse_integrable_of_estimator_bound _ f hf (by norm_num) hfbound
have hmse₀ : (δ / 2) ^ 2 * (zengSampleLaw n P₀).real E₀ ≤
zengDeterministicRisk f P₀ := by
unfold zengDeterministicRisk E₀
rw [zengParametricLaw_ate]
have hset : {s | (δ / 2) ^ 2 ≤ (f s - θ₀) ^ 2} =
{s | δ / 2 ≤ |f s - θ₀|} := by
ext s
simp only [Set.mem_setOf_eq]
rw [sq_le_sq, abs_of_nonneg (show (0 : ℝ) ≤ δ / 2 by positivity)]
rw [← hset]
exact mul_meas_ge_le_integral_of_nonneg
(Filter.Eventually.of_forall fun s ↦ sq_nonneg _) hint₀ _
have hmse₁ : (δ / 2) ^ 2 * (zengSampleLaw n P₁).real E₁ ≤
zengDeterministicRisk f P₁ := by
unfold zengDeterministicRisk E₁
rw [zengParametricLaw_ate]
have hset : {s | (δ / 2) ^ 2 ≤ (f s - θ₁) ^ 2} =
{s | δ / 2 ≤ |f s - θ₁|} := by
ext s
simp only [Set.mem_setOf_eq]
rw [sq_le_sq, abs_of_nonneg (show (0 : ℝ) ≤ δ / 2 by positivity)]
rw [← hset]
exact mul_meas_ge_le_integral_of_nonneg
(Filter.Eventually.of_forall fun s ↦ sq_nonneg _) hint₁ _
have hmax : c / 2 ≤ max ((zengSampleLaw n P₀).real E₀)
((zengSampleLaw n P₁).real E₁) := by
nlinarith [le_max_left ((zengSampleLaw n P₀).real E₀)
((zengSampleLaw n P₁).real E₁),
le_max_right ((zengSampleLaw n P₀).real E₀)
((zengSampleLaw n P₁).real E₁)]
calc
c * δ ^ 2 / 8 ≤ (δ / 2) ^ 2 *
max ((zengSampleLaw n P₀).real E₀)
((zengSampleLaw n P₁).real E₁) := by nlinarith [sq_nonneg δ]
_ = max ((δ / 2) ^ 2 * (zengSampleLaw n P₀).real E₀)
((δ / 2) ^ 2 * (zengSampleLaw n P₁).real E₁) := by
by_cases hle : (zengSampleLaw n P₀).real E₀ ≤
(zengSampleLaw n P₁).real E₁
· rw [max_eq_right hle, max_eq_right]
exact mul_le_mul_of_nonneg_left hle (sq_nonneg _)
· have hge := le_of_not_ge hle
rw [max_eq_left hge, max_eq_left]
exact mul_le_mul_of_nonneg_left hge (sq_nonneg _)
_ ≤ max (zengDeterministicRisk f P₀) (zengDeterministicRisk f P₁) :=
max_le_max hmse₀ hmse₁
_ ≤ max (zengSquaredRisk T P₀) (zengSquaredRisk T P₁) :=
max_le_max (zeng_kernelMean_risk_le T P₀) (zeng_kernelMean_risk_le T P₁)
theorem zeng_parametric_lower_of_decisionReduction
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
hdecision :
conclusion
∃ c : ℝ,
0 < c
Proof (Lean source)
theorem zeng_parametric_lower_of_decisionReduction
(hdecision : ZengParametricDecisionReduction) :
∃ c : ℝ, 0 < c ∧
∀ (n d : ℕ) (ε : ℝ), 1 ≤ n → 1 ≤ d →
0 < ε ∧ ε < 1 / 2 →
c / n ≤ zengZeroControlRisk n d ε := by
obtain ⟨ctest, hctest, htest⟩ :=
Causalean.Stat.le_cam_two_point_chisq.2 (1 / 8) (by norm_num)
refine ⟨ctest / 128, by positivity, ?_⟩
intro n d ε hn hd hε
let P₁ := zengParametricLaw d ε (1 / 2 + zengParametricStep n) hd hε
(zengParametricPerturbation_mem n hn)
let P₀ := zengParametricLaw d ε (1 / 2) hd hε (by norm_num)
letI : IsProbabilityMeasure P₁.1 := P₁.2
letI : IsProbabilityMeasure P₀.1 := P₀.2
have hfloor : ∀ A : Set (Fin n → ZengRecord d), MeasurableSet A →
(zengSampleLaw n P₁).real Aᶜ + (zengSampleLaw n P₀).real A ≥ ctest := by
intro A hA
apply htest P₁.1 P₀.1
(zengParametricLaw_ac_half d ε
(1 / 2 + zengParametricStep n) hd hε
(zengParametricPerturbation_mem n hn))
(Integrable.of_finite) n (by omega)
(zengParametricPerturbation_chiSq n d ε hn hd hε) A hA
have htwo : ∀ T : ZengEstimator n d,
ctest * (zengParametricStep n) ^ 2 / 8 ≤
max (zengSquaredRisk T P₀) (zengSquaredRisk T P₁) := by
exact hdecision n d ε ctest hn hd hε hctest hfloor
let θ₀ : {P : ZengLaw d // P ∈ zengZeroControlClass d ε} :=
⟨P₀, zengParametricLaw_mem d ε (1 / 2) hd hε (by norm_num)⟩
let θ₁ : {P : ZengLaw d // P ∈ zengZeroControlClass d ε} :=
⟨P₁, zengParametricLaw_mem d ε
(1 / 2 + zengParametricStep n) hd hε
(zengParametricPerturbation_mem n hn)⟩
let T₀ : ZengEstimator n d :=
⟨Kernel.const _ (Measure.dirac 0), inferInstance, by
intro s
simp [Kernel.const_apply]⟩
letI : Nonempty (ZengEstimator n d) := ⟨T₀⟩
have hmini : ctest * (zengParametricStep n) ^ 2 / 8 ≤
zengZeroControlRisk n d ε := by
apply le_minimaxValue_of_two_point θ₀ θ₁
· intro T
refine ⟨4, ?_⟩
rintro r ⟨θ, rfl⟩
exact (zeng_squaredRisk_bounds ε T θ.1 θ.2.1).2
· intro T
exact htwo T
apply le_trans ?_ hmini
have hnR : (0 : ℝ) < n := by positivity
have hsqrt : sqrt (n : ℝ) ≠ 0 := by positivity
have hsqrt_sq : (sqrt (n : ℝ)) ^ 2 = n :=
Real.sq_sqrt (le_of_lt hnR)
unfold zengParametricStep
rw [div_pow]
field_simp
nlinarith
theorem zeng_parametric_lower
the stated mathematical conclusion holds.
Formal statement
conclusion
∃ c : ℝ,
0 < c
Proof (Lean source)
theorem zeng_parametric_lower :
∃ c : ℝ, 0 < c ∧
∀ (n d : ℕ) (ε : ℝ), 1 ≤ n → 1 ≤ d →
0 < ε ∧ ε < 1 / 2 →
c / n ≤ zengZeroControlRisk n d ε :=
zeng_parametric_lower_of_decisionReduction zengParametricDecisionReduction_proved
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.ZengLower.PredictiveProductBridge 7 declarations Product-prior factorization for the current marked-Poisson experiment.
Product-prior factorization for the current marked-Poisson experiment.
def coordinatewiseFiniteKernel
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
given by
clause 2
| d + 1
=> let lastK := K.comap (fun z : Fin (d + 1) → Θ => z (last d)) (measurable_pi_apply _) let initK := (coordinatewiseFiniteKernel K d).comap (fun z : Fin (d + 1) → Θ => Fin.init z) (by exact measurable_pi_iff.2 fun i => measurable_pi_apply i.castSucc) (lastK.prod initK).map (piFinSuccAbove (fun _ : Fin (d + 1) => X) (last d)).symm
instance coordinatewiseFiniteKernel_isMarkov
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
noncomputable instance coordinatewiseFiniteKernel_isMarkov
{Θ X : Type*} [MeasurableSpace Θ] [MeasurableSpace X]
(K : Kernel Θ X) [IsMarkovKernel K] (d : ℕ) :
IsMarkovKernel (coordinatewiseFiniteKernel K d) := by
induction d with
| zero =>
unfold coordinatewiseFiniteKernel
infer_instance
| succ d ih =>
letI : IsMarkovKernel (coordinatewiseFiniteKernel K d) := ih
unfold coordinatewiseFiniteKernel
exact Kernel.IsMarkovKernel.map _
(piFinSuccAbove
(fun _ : Fin (d + 1) => X) (last d)).symm.measurable
theorem measurable_poissonMeasure_rate_local
the stated mathematical conclusion holds.
Formal statement
conclusion
Proof (Lean source)
lemma measurable_poissonMeasure_rate_local :
Measurable (fun r : NNReal => poissonMeasure r) := by
apply Measure.measurable_of_measurable_coe
intro s hs
simp only [poissonMeasure, Measure.sum_apply _ hs]
apply Measurable.tsum
intro k
simp only [Measure.smul_apply, Measure.dirac_apply' _ hs, smul_eq_mul]
fun_prop
theorem measurable_markedPoissonLaw_local
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma measurable_markedPoissonLaw_local (n : ℝ) :
Measurable (markedPoissonLaw n) := by
have h₁ : Measurable (fun t : MarkedParam =>
(⟨poissonMeasure (nnRate (n * t.1 * t.2.1 * t.2.2)), inferInstance⟩ :
ProbabilityMeasure ℕ)) :=
Measurable.subtype_mk (measurable_poissonMeasure_rate_local.comp (by
unfold nnRate
fun_prop))
have h₂ : Measurable (fun t : MarkedParam =>
(⟨poissonMeasure (nnRate (n * t.1 * t.2.1 * (1 - t.2.2))), inferInstance⟩ :
ProbabilityMeasure ℕ)) :=
Measurable.subtype_mk (measurable_poissonMeasure_rate_local.comp (by
unfold nnRate
fun_prop))
have h₃ : Measurable (fun t : MarkedParam =>
(⟨poissonMeasure (nnRate (n * t.1 * (1 - t.2.1))), inferInstance⟩ :
ProbabilityMeasure ℕ)) :=
Measurable.subtype_mk (measurable_poissonMeasure_rate_local.comp (by
unfold nnRate
fun_prop))
have h₁₂ : Measurable (fun t : MarkedParam =>
(⟨(poissonMeasure (nnRate (n * t.1 * t.2.1 * t.2.2))).prod
(poissonMeasure (nnRate (n * t.1 * t.2.1 * (1 - t.2.2)))), inferInstance⟩ :
ProbabilityMeasure (ℕ × ℕ))) :=
Measurable.subtype_mk
(ProbabilityMeasure.measurable_fun_prod.comp (h₁.prodMk h₂))
exact ProbabilityMeasure.measurable_fun_prod.comp (h₁₂.prodMk h₃)
def markedPoissonKernel
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
n :
ℝ
def
given by
clause 1
toFun := markedPoissonLaw n
clause 2
measurable' := measurable_markedPoissonLaw_local n
instance markedPoissonKernel_isMarkov
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
noncomputable instance markedPoissonKernel_isMarkov (n : ℝ) :
IsMarkovKernel (markedPoissonKernel n) where
isProbabilityMeasure t := by
change IsProbabilityMeasure (markedPoissonLaw n t)
unfold markedPoissonLaw
infer_instance
instance relaxedDummyCountLaw_isProbability
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
noncomputable instance relaxedDummyCountLaw_isProbability
(d : ℕ) (a0 : ℝ) (lam : ℝ≥0) :
IsProbabilityMeasure (relaxedDummyCountLaw d a0 lam) := by
unfold relaxedDummyCountLaw
infer_instance
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.ZengLower.RelaxedExactTransfer 2 declarations Normalization and fixed-sample risk transfer for relaxed Zeng intensities.
Normalization and fixed-sample risk transfer for relaxed Zeng intensities.
def normalizedRelaxedZengLaw
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
instance normalizedRelaxedZengLaw_isProbabilityMeasure
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
noncomputable instance normalizedRelaxedZengLaw_isProbabilityMeasure
{d : ℕ} (a0 : ℝ) (z : Fin d → MarkedParam)
(P₀ : Measure (RelaxedZengRecord d)) [IsProbabilityMeasure P₀] :
IsProbabilityMeasure (normalizedRelaxedZengLaw a0 z P₀).1 :=
(normalizedRelaxedZengLaw a0 z P₀).2
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.ZengLower.RelaxedFuzzy 1 declarations Fuzzy testing lower bounds for conditioned iid relaxed rare-cell priors.
Fuzzy testing lower bounds for conditioned iid relaxed rare-cell priors.
def iidRareTargetSum
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
d :
ℕ
z :
Fin d → MarkedParam
def
iidRareTargetSum z :
ℝ
given by
∑ x, targetFunctional (z x)
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.ZengLower.RelaxedPoissonBridge 2 declarations Finite-intensity Poisson representation of the relaxed Zeng table.
Finite-intensity Poisson representation of the relaxed Zeng table.
instance relaxedZengTable_isFiniteMeasure
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
noncomputable instance relaxedZengTable_isFiniteMeasure
(d : ℕ) (a0 : ℝ) (z : Fin d → MarkedParam) :
IsFiniteMeasure (relaxedZengTable d a0 z) := by
apply IsFiniteMeasure.mk
simp [relaxedZengTable]
def relaxedDummyCountLaw
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
d :
ℕ
a0 :
ℝ
lam :
ℝ≥0
def
given by
((poissonMeasure 0).prod (poissonMeasure (lam * (ofReal ((1 - d * a0) / 2)).toNNReal))).prod (poissonMeasure (lam * (ofReal ((1 - d * a0) / 2)).toNNReal))
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.ZengLower.RelaxedPrefixReconstruction 8 declarations This module proves that ordering independent atomless marks and then reading a fixed prefix gives exactly the same observation law as reading the prefix of the uniformly ordered unmarked Poisson sample reconstructed from
Marked and unmarked prefix reconstruction
This module proves that ordering independent atomless marks and then reading a fixed prefix gives exactly the same observation law as reading the prefix of the uniformly ordered unmarked Poisson sample reconstructed from split counts.
def unmarkedPrefixObservations
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
theorem measurable_unmarkedPrefixObservations
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
X :
Type u_1
x₀ :
X
n :
ℕ
conclusion
Proof (Lean source)
@[fun_prop] lemma measurable_unmarkedPrefixObservations (x₀ : X) (n : ℕ) :
Measurable (unmarkedPrefixObservations x₀ n : FiniteSample X → Fin n → X) := by
unfold unmarkedPrefixObservations
apply measurable_pi_lambda
intro k t ht
rw [MeasurableSpace.measurableSet_iInf]
intro m
change MeasurableSet ((fun x : Fin m → X =>
(if h : n ≤ m then fun k => x (castLE h k) else fun _ => x₀) k) ⁻¹' t)
by_cases h : n ≤ m
· simp only [dif_pos h]
exact ht.preimage (measurable_pi_apply (castLE h k))
· simp only [dif_neg h]
exact measurable_const ht
def relaxedZengClip
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
theorem measurable_relaxedZengClip
the stated mathematical conclusion holds.
Formal statement
conclusion
Proof (Lean source)
@[fun_prop] lemma measurable_relaxedZengClip : Measurable relaxedZengClip := by
unfold relaxedZengClip
fun_prop
def unmarkedPrefixDecisionKernel
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
given by
T.1 ∘ₖ Kernel.deterministic (unmarkedPrefixObservations x₀ n) (measurable_unmarkedPrefixObservations x₀ n)
instance unmarkedPrefixDecisionKernel_isMarkov
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
instance unmarkedPrefixDecisionKernel_isMarkov {n d : ℕ}
(T : ZengEstimator n (d + 1)) (x₀ : RelaxedZengRecord d) :
IsMarkovKernel (unmarkedPrefixDecisionKernel T x₀) := by
letI : IsMarkovKernel T.1 := T.2.1
unfold unmarkedPrefixDecisionKernel
infer_instance
def relaxedCountPrefixEstimator
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
given by
theorem measurable_relaxedCountPrefixEstimator
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
@[fun_prop] theorem measurable_relaxedCountPrefixEstimator {n d : ℕ}
(T : ZengEstimator n (d + 1)) (x₀ : RelaxedZengRecord d) :
Measurable (relaxedCountPrefixEstimator T x₀) := by
exact measurable_kernelMean _ measurable_relaxedZengClip
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.ZengLower.RelaxedSplitReconstruction 5 declarations This module reconstructs the full relaxed histogram and then a uniformly ordered raw Poisson sample from the split dummy/rare three-count statistic.
Split-count sufficiency for the relaxed Zeng experiment
This module reconstructs the full relaxed histogram and then a uniformly ordered raw Poisson sample from the split dummy/rare three-count statistic. The reconstruction is parameter-free and remains exact when the dummy cell has both treated-failure and control-failure mass.
def unpackRelaxedThreeCounts
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
def unsplitRelaxedCounts
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
theorem measurable_unsplitRelaxedCounts
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
@[fun_prop]
lemma measurable_unsplitRelaxedCounts {d : ℕ} :
Measurable (unsplitRelaxedCounts (d := d)) := by
fun_prop
def relaxedSplitReconstructionKernel
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
d :
ℕ
def
relaxedSplitReconstructionKernel d :
instance relaxedSplitReconstructionKernel_isMarkov
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
noncomputable instance relaxedSplitReconstructionKernel_isMarkov (d : ℕ) :
IsMarkovKernel (relaxedSplitReconstructionKernel d) := by
unfold relaxedSplitReconstructionKernel
infer_instance
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.Helpers.ZengLower.RelaxedTable 3 declarations Relaxed finite-intensity tables for the many-cell lower bound.
Relaxed finite-intensity tables for the many-cell lower bound.
def RelaxedZengRecord
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def relaxedEmbedZengRecord
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
def relaxedZengTable
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.PFixedDEffectiveSampleReduction 2 declarations PFixedDEffectiveSampleReduction for the finite rare-arrival experiment.
PFixedDEffectiveSampleReduction for the finite rare-arrival experiment.
theorem fixed_d_frontier_rate_theta
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
d :
ℕ
n :
ℕ → ℕ
q :
ℕ → ℝ
hn :
∀ t, 1 ≤ n t
hq :
∀ t, 0 < q t
hN :
conclusion
Proof (Lean source)
lemma fixed_d_frontier_rate_theta
(d : ℕ) (n : ℕ → ℕ) (q : ℕ → ℝ)
(hn : ∀ t, 1 ≤ n t) (hq : ∀ t, 0 < q t)
(hN : Tendsto (fun t => effectiveSize (n t) (q t)) atTop atTop) :
(fun t => frontierRate (n t) d (q t)) =Θ[atTop]
(fun t => (effectiveSize (n t) (q t))⁻¹) := by
constructor
· rw [Asymptotics.isBigO_iff]
refine ⟨2, ?_⟩
filter_upwards [hN.eventually_ge_atTop (max 2 ((d : ℝ) ^ 2))] with t ht
have hNpos : 0 < effectiveSize (n t) (q t) := by
dsimp [effectiveSize]
exact mul_pos (by exact_mod_cast hn t) (hq t)
have hL : 1 ≤ logScale (n t) (q t) := by
change 1 ≤ log (exp 1 + effectiveSize (n t) (q t))
rw [Real.le_log_iff_exp_le (by positivity)]
linarith
have hdN : (d : ℝ) ^ 2 ≤ effectiveSize (n t) (q t) :=
le_trans (le_max_right _ _) ht
have hden : 0 < effectiveSize (n t) (q t) * logScale (n t) (q t) :=
mul_pos hNpos (by linarith)
have hb : ((d : ℝ) / (effectiveSize (n t) (q t) * logScale (n t) (q t))) ^ 2 ≤
(effectiveSize (n t) (q t))⁻¹ := by
rw [div_pow]
apply (div_le_iff₀ (pow_pos hden 2)).2
have hL2 : 1 ≤ (logScale (n t) (q t)) ^ 2 := by nlinarith
have heq : (effectiveSize (n t) (q t))⁻¹ *
(effectiveSize (n t) (q t) * logScale (n t) (q t)) ^ 2 =
effectiveSize (n t) (q t) * (logScale (n t) (q t)) ^ 2 := by
field_simp [ne_of_gt hNpos]
rw [heq]
nlinarith [mul_nonneg (le_of_lt hNpos) (sub_nonneg.mpr hL2)]
have ha : 0 ≤ (effectiveSize (n t) (q t))⁻¹ := inv_nonneg.mpr (le_of_lt hNpos)
have hr : 0 ≤ frontierRate (n t) d (q t) := by
unfold frontierRate
exact le_min (by norm_num) (add_nonneg ha (sq_nonneg _))
rw [Real.norm_eq_abs, Real.norm_eq_abs, abs_of_nonneg hr, abs_of_nonneg ha]
unfold frontierRate
exact (min_le_right _ _).trans (by linarith)
· rw [Asymptotics.isBigO_iff]
refine ⟨1, ?_⟩
filter_upwards [hN.eventually_ge_atTop (max 2 ((d : ℝ) ^ 2))] with t ht
have hNpos : 0 < effectiveSize (n t) (q t) := by
dsimp [effectiveSize]
exact mul_pos (by exact_mod_cast hn t) (hq t)
have ha : 0 ≤ (effectiveSize (n t) (q t))⁻¹ := inv_nonneg.mpr (le_of_lt hNpos)
have ha1 : (effectiveSize (n t) (q t))⁻¹ ≤ 1 := by
apply (inv_le_one₀ hNpos).2
linarith [le_trans (le_max_left (2 : ℝ) ((d : ℝ) ^ 2)) ht]
have hr : 0 ≤ frontierRate (n t) d (q t) := by
unfold frontierRate
exact le_min (by norm_num) (add_nonneg ha (sq_nonneg _))
rw [Real.norm_eq_abs, Real.norm_eq_abs, abs_of_nonneg ha, abs_of_nonneg hr]
simp only [one_mul]
unfold frontierRate
exact le_min ha1 (le_add_of_nonneg_right (sq_nonneg _))
theorem fixed_d_effective_sample_reduction
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
1 ≤ d
n :
ℕ → ℕ
q :
ℕ → ℝ
hn :
∀ t, 1 ≤ n t
hq :
∀ t, 0 < q t ∧ q t ≤ 1
hslice :
∀ t, RareArrivalSlice (n t) (q t)
hN :
conclusion
conclusion 1
conclusion 2
conclusion 3
(fun t => noSurrogateRisk (n t) d (q t))
=Θ[atTop] (fun t => (effectiveSize (n t) (q t))⁻¹)
Proof (Lean source)
theorem fixed_d_effective_sample_reduction
(d : ℕ) (hd : 1 ≤ d) (n : ℕ → ℕ) (q : ℕ → ℝ)
(hn : ∀ t, 1 ≤ n t) (hq : ∀ t, 0 < q t ∧ q t ≤ 1)
(hslice : ∀ t, RareArrivalSlice (n t) (q t))
(hN : Tendsto (fun t => effectiveSize (n t) (q t)) atTop atTop) :
(fun t => frontierRate (n t) d (q t)) =Θ[atTop]
(fun t => (effectiveSize (n t) (q t))⁻¹) ∧
(fun t => minimaxRisk (n t) d (q t)) =Θ[atTop]
(fun t => (effectiveSize (n t) (q t))⁻¹) ∧
(fun t => noSurrogateRisk (n t) d (q t)) =Θ[atTop]
(fun t => (effectiveSize (n t) (q t))⁻¹) := by
refine ⟨fixed_d_frontier_rate_theta d n q hn (fun t => (hq t).1) hN, ?_, ?_⟩
· obtain ⟨c, C, hc, hcC, hbounds⟩ :=
matched_minimax_frontier
have hθ : (fun t => minimaxRisk (n t) d (q t)) =Θ[atTop]
(fun t => frontierRate (n t) d (q t)) := by
constructor
· rw [Asymptotics.isBigO_iff]
refine ⟨C, ?_⟩
filter_upwards with t
obtain ⟨hl, hu⟩ := hbounds (n t) d (q t) (hn t) hd (hq t).1
(hq t).2 (hslice t)
have hr : 0 ≤ frontierRate (n t) d (q t) := by
unfold frontierRate
exact le_min (by norm_num) (add_nonneg
(inv_nonneg.mpr (le_of_lt (mul_pos
(by exact_mod_cast hn t) (hq t).1))) (sq_nonneg _))
have hm : 0 ≤ minimaxRisk (n t) d (q t) :=
le_trans (mul_nonneg (le_of_lt hc) hr) hl
rw [Real.norm_eq_abs, Real.norm_eq_abs, abs_of_nonneg hm, abs_of_nonneg hr]
exact hu
· rw [Asymptotics.isBigO_iff]
refine ⟨c⁻¹, ?_⟩
filter_upwards with t
obtain ⟨hl, hu⟩ := hbounds (n t) d (q t) (hn t) hd (hq t).1
(hq t).2 (hslice t)
have hr : 0 ≤ frontierRate (n t) d (q t) := by
unfold frontierRate
exact le_min (by norm_num) (add_nonneg
(inv_nonneg.mpr (le_of_lt (mul_pos
(by exact_mod_cast hn t) (hq t).1))) (sq_nonneg _))
have hm : 0 ≤ minimaxRisk (n t) d (q t) :=
le_trans (mul_nonneg (le_of_lt hc) hr) hl
rw [Real.norm_eq_abs, Real.norm_eq_abs, abs_of_nonneg hr, abs_of_nonneg hm]
calc
frontierRate (n t) d (q t) ≤ minimaxRisk (n t) d (q t) / c :=
(le_div_iff₀ hc).2 (by simpa [mul_comm] using hl)
_ = c⁻¹ * minimaxRisk (n t) d (q t) := by ring
exact hθ.trans (fixed_d_frontier_rate_theta d n q hn
(fun t => (hq t).1) hN)
· let c : ℝ := 1 / 256
have hc : 0 < c := by norm_num [c]
obtain ⟨C, hC, hupp⟩ := mixed_count_upper
have hUpper (t : ℕ) :
noSurrogateRisk (n t) d (q t) ≤ C * frontierRate (n t) d (q t) := by
let f := mixedCountEstimator (n t) d (q t)
have hEstimator := mixed_count_estimator_regular (n t) d (q t)
let T : Estimator (n t) d :=
Estimator.ofMap f hEstimator
have hrisk : ∀ (T : Estimator (n t) d)
(P : {P : FullLaw d // RareArrivalModelClass (n t) d (q t) P ∧
∀ᵐ r : FullRecord d ∂(P.1), r.S0 = false ∧ r.S1 = false}),
0 ≤ squaredRisk T P.1 := by
intro T P
exact integral_nonneg (fun s => integral_nonneg (fun y => sq_nonneg _))
have hval : noSurrogateRisk (n t) d (q t) ≤
Causalean.Stat.worstCaseRiskReal
(fun (T : Estimator (n t) d)
(P : {P : FullLaw d // RareArrivalModelClass (n t) d (q t) P ∧
∀ᵐ r : FullRecord d ∂(P.1), r.S0 = false ∧ r.S1 = false}) =>
squaredRisk T P.1) T :=
minimaxValue_le_worstCaseRisk_of_nonneg hrisk T
refine hval.trans ?_
unfold Causalean.Stat.worstCaseRiskReal
by_cases hne : Nonempty {P : FullLaw d // RareArrivalModelClass (n t) d (q t) P ∧
∀ᵐ r : FullRecord d ∂(P.1), r.S0 = false ∧ r.S1 = false}
· letI := hne
apply ciSup_le
intro P
have hdet : squaredRisk T P.1 = deterministicRisk f P.1 := by
simp [squaredRisk, deterministicRisk, T, Estimator.ofMap, Kernel.deterministic_apply]
change squaredRisk T P.1 ≤ C * frontierRate (n t) d (q t)
rw [hdet]
exact (hupp (n t) d (q t) P.1 P.2.1).1.trans
(hupp (n t) d (q t) P.1 P.2.1).2.1
· haveI : IsEmpty {P : FullLaw d // RareArrivalModelClass (n t) d (q t) P ∧
∀ᵐ r : FullRecord d ∂(P.1), r.S0 = false ∧ r.S1 = false} :=
not_nonempty_iff.mp hne
have hNpos : 0 < effectiveSize (n t) (q t) := by
unfold effectiveSize
exact mul_pos (by exact_mod_cast hn t) (hq t).1
have hr : 0 ≤ frontierRate (n t) d (q t) := by
unfold frontierRate
exact le_min (by norm_num) (add_nonneg
(inv_nonneg.mpr (le_of_lt hNpos)) (sq_nonneg _))
simpa using (mul_nonneg (le_of_lt hC) hr)
have hLower (t : ℕ) :
c * min 1 (effectiveSize (n t) (q t))⁻¹ ≤
noSurrogateRisk (n t) d (q t) := by
simpa [c] using
(oneCell_minimax_risk_floors (q t) (hn t) hd (hq t).1
(hq t).2 (hslice t)).2
constructor
· have hO : (fun t => noSurrogateRisk (n t) d (q t)) =O[atTop]
(fun t => frontierRate (n t) d (q t)) := by
rw [Asymptotics.isBigO_iff]
refine ⟨C, ?_⟩
filter_upwards with t
have hNpos : 0 < effectiveSize (n t) (q t) := by
unfold effectiveSize
exact mul_pos (by exact_mod_cast hn t) (hq t).1
have hr : 0 ≤ frontierRate (n t) d (q t) := by
unfold frontierRate
exact le_min (by norm_num) (add_nonneg
(inv_nonneg.mpr (le_of_lt hNpos)) (sq_nonneg _))
have hm : 0 ≤ noSurrogateRisk (n t) d (q t) := by
exact le_trans (mul_nonneg (le_of_lt hc)
(le_min (by norm_num) (inv_nonneg.mpr (le_of_lt hNpos)))) (hLower t)
rw [Real.norm_eq_abs, Real.norm_eq_abs, abs_of_nonneg hm, abs_of_nonneg hr]
exact hUpper t
exact hO.trans (fixed_d_frontier_rate_theta d n q hn
(fun t => (hq t).1) hN).1
· rw [Asymptotics.isBigO_iff]
refine ⟨c⁻¹, ?_⟩
filter_upwards [hN.eventually_ge_atTop 1] with t ht
have hNpos : 0 < effectiveSize (n t) (q t) := by
unfold effectiveSize
exact mul_pos (by exact_mod_cast hn t) (hq t).1
have ha : 0 ≤ (effectiveSize (n t) (q t))⁻¹ :=
inv_nonneg.mpr (le_of_lt hNpos)
have ha1 : (effectiveSize (n t) (q t))⁻¹ ≤ 1 :=
(inv_le_one₀ hNpos).2 ht
have hl : c * (effectiveSize (n t) (q t))⁻¹ ≤
noSurrogateRisk (n t) d (q t) := by
simpa [min_eq_right ha1] using hLower t
have hm : 0 ≤ noSurrogateRisk (n t) d (q t) :=
le_trans (mul_nonneg (le_of_lt hc) ha) hl
rw [Real.norm_eq_abs, Real.norm_eq_abs, abs_of_nonneg ha, abs_of_nonneg hm]
calc
(effectiveSize (n t) (q t))⁻¹ ≤
noSurrogateRisk (n t) d (q t) / c :=
(le_div_iff₀ hc).2 (by simpa [mul_comm] using hl)
_ = c⁻¹ * noSurrogateRisk (n t) d (q t) := by ring
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.PPhaseBoundaries 5 declarations PPhaseBoundaries for the finite rare-arrival experiment.
PPhaseBoundaries for the finite rare-arrival experiment.
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma phase_min_rate_tendsto (a b : ℕ → ℝ)
(ha : ∀ t, 0 ≤ a t) (hb : ∀ t, 0 ≤ b t)
(ha0 : Tendsto a atTop (nhds 0)) :
Tendsto (fun t => min 1 (a t + b t)) atTop (nhds 0) ↔
Tendsto b atTop (nhds 0) := by
constructor
· intro h
have hsmall : ∀ᶠ t in atTop, min 1 (a t + b t) < 1 :=
h.eventually (Iio_mem_nhds (by norm_num : (0 : ℝ) < 1))
have hsum : Tendsto (fun t => a t + b t) atTop (nhds 0) := by
apply h.congr'
filter_upwards [hsmall] with t ht
have hab : a t + b t < 1 := by
by_contra hnot
have hge : 1 ≤ a t + b t := le_of_not_gt hnot
simp [min_eq_left hge] at ht
exact min_eq_right (le_of_lt hab)
exact squeeze_zero hb (fun t => le_add_of_nonneg_left (ha t)) hsum
· intro hb0
have hsum : Tendsto (fun t => a t + b t) atTop (nhds 0) := by
convert ha0.add hb0 using 1 <;> simp
simpa using ((tendsto_const_nhds : Tendsto (fun _ : ℕ => (1 : ℝ)) atTop (nhds 1)).min hsum)
theorem phase_frontier_rate_tendsto
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ → ℕ
q :
ℕ → ℝ
hn :
∀ t, 1 ≤ n t
hq :
∀ t, 0 < q t
hN :
conclusion
Proof (Lean source)
lemma phase_frontier_rate_tendsto
(n d : ℕ → ℕ) (q : ℕ → ℝ)
(hn : ∀ t, 1 ≤ n t) (hq : ∀ t, 0 < q t)
(hN : Tendsto (fun t => effectiveSize (n t) (q t)) atTop atTop) :
Tendsto (fun t => frontierRate (n t) (d t) (q t)) atTop (nhds 0) ↔
(fun t => (d t : ℝ)) =o[atTop]
(fun t => effectiveSize (n t) (q t) * logScale (n t) (q t)) := by
let a : ℕ → ℝ := fun t => (effectiveSize (n t) (q t))⁻¹
let x : ℕ → ℝ := fun t => (d t : ℝ) /
(effectiveSize (n t) (q t) * logScale (n t) (q t))
have hNpos (t : ℕ) : 0 < effectiveSize (n t) (q t) := by
unfold effectiveSize
exact mul_pos (by exact_mod_cast hn t) (hq t)
have hL (t : ℕ) : 0 < logScale (n t) (q t) := by
change 0 < log (exp 1 + effectiveSize (n t) (q t))
apply Real.log_pos
have : 1 < exp (1 : ℝ) := Real.one_lt_exp_iff.mpr (by norm_num)
linarith [hNpos t]
have hx (t : ℕ) : 0 ≤ x t := div_nonneg (Nat.cast_nonneg _) (le_of_lt (mul_pos (hNpos t) (hL t)))
have ha0 : Tendsto a atTop (nhds 0) := tendsto_inv_atTop_zero.comp hN
have hsq : Tendsto (fun t => (x t) ^ 2) atTop (nhds 0) ↔
Tendsto x atTop (nhds 0) := by
constructor
· intro h
have hs := Real.continuous_sqrt.tendsto (0 : ℝ) |>.comp h
have hs' : Tendsto (fun t => sqrt ((x t) ^ 2)) atTop (nhds 0) := by
convert hs using 1 <;> simp [Function.comp_def]
exact hs'.congr' (Eventually.of_forall (fun t => by
simp [Real.sqrt_sq_eq_abs, abs_of_nonneg (hx t)]))
· intro h
convert h.pow 2 using 1 <;> norm_num
have hlittle : (fun t => (d t : ℝ)) =o[atTop]
(fun t => effectiveSize (n t) (q t) * logScale (n t) (q t)) ↔
Tendsto x atTop (nhds 0) := by
have hden : ∀ t, effectiveSize (n t) (q t) * logScale (n t) (q t) = 0 →
(d t : ℝ) = 0 := by
intro t ht
exact False.elim ((ne_of_gt (mul_pos (hNpos t) (hL t))) ht)
exact Asymptotics.isLittleO_iff_tendsto hden
change Tendsto (fun t => min 1 (a t + (x t) ^ 2)) atTop (nhds 0) ↔ _
rw [phase_min_rate_tendsto a (fun t => (x t) ^ 2)
(fun t => inv_nonneg.mpr (le_of_lt (hNpos t)))
(fun t => sq_nonneg _) ha0, hsq, hlittle]
theorem phase_rate_lower_bigO
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ → ℕ
q :
ℕ → ℝ
hn :
∀ t, 1 ≤ n t
hq :
∀ t, 0 < q t
hN :
conclusion
(fun t => (effectiveSize (n t) (q t))⁻¹)
=O[atTop] (fun t => frontierRate (n t) (d t) (q t))
Proof (Lean source)
lemma phase_rate_lower_bigO
(n d : ℕ → ℕ) (q : ℕ → ℝ)
(hn : ∀ t, 1 ≤ n t) (hq : ∀ t, 0 < q t)
(hN : Tendsto (fun t => effectiveSize (n t) (q t)) atTop atTop) :
(fun t => (effectiveSize (n t) (q t))⁻¹) =O[atTop]
(fun t => frontierRate (n t) (d t) (q t)) := by
rw [Asymptotics.isBigO_iff]
refine ⟨1, ?_⟩
filter_upwards [hN.eventually_ge_atTop 1] with t ht
have hNpos : 0 < effectiveSize (n t) (q t) := by
unfold effectiveSize
exact mul_pos (by exact_mod_cast hn t) (hq t)
have ha : 0 ≤ (effectiveSize (n t) (q t))⁻¹ := inv_nonneg.mpr (le_of_lt hNpos)
have ha1 : (effectiveSize (n t) (q t))⁻¹ ≤ 1 := (inv_le_one₀ hNpos).2 ht
have hr : 0 ≤ frontierRate (n t) (d t) (q t) := by
unfold frontierRate
exact le_min (by norm_num) (add_nonneg ha (sq_nonneg _))
rw [Real.norm_eq_abs, Real.norm_eq_abs, abs_of_nonneg ha, abs_of_nonneg hr]
simp only [one_mul]
unfold frontierRate
exact le_min ha1 (le_add_of_nonneg_right (sq_nonneg _))
theorem phase_dimension_rate_bigO
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ → ℕ
q :
ℕ → ℝ
hn :
∀ t, 1 ≤ n t
hq :
∀ t, 0 < q t
hN :
conclusion
(fun t => frontierRate (n t) (d t) (q t))
=O[atTop] (fun t => (effectiveSize (n t) (q t))⁻¹) ↔ (fun t => (d t : ℝ)) =O[atTop] (fun t => sqrt (effectiveSize (n t) (q t)) * logScale (n t) (q t))
Proof (Lean source)
lemma phase_dimension_rate_bigO
(n d : ℕ → ℕ) (q : ℕ → ℝ)
(hn : ∀ t, 1 ≤ n t) (hq : ∀ t, 0 < q t)
(hN : Tendsto (fun t => effectiveSize (n t) (q t)) atTop atTop) :
(fun t => frontierRate (n t) (d t) (q t)) =O[atTop]
(fun t => (effectiveSize (n t) (q t))⁻¹) ↔
(fun t => (d t : ℝ)) =O[atTop]
(fun t => sqrt (effectiveSize (n t) (q t)) * logScale (n t) (q t)) := by
let N : ℕ → ℝ := fun t => effectiveSize (n t) (q t)
let L : ℕ → ℝ := fun t => logScale (n t) (q t)
let B : ℕ → ℝ := fun t => sqrt (N t) * L t
have hNpos (t : ℕ) : 0 < N t := by
dsimp [N, effectiveSize]
exact mul_pos (by exact_mod_cast hn t) (hq t)
have hL (t : ℕ) : 0 < L t := by
change 0 < log (exp 1 + N t)
apply Real.log_pos
have : 1 < exp (1 : ℝ) := Real.one_lt_exp_iff.mpr (by norm_num)
linarith [hNpos t]
have hB (t : ℕ) : 0 < B t := mul_pos (Real.sqrt_pos.2 (hNpos t)) (hL t)
have hden (t : ℕ) : 0 < N t * L t := mul_pos (hNpos t) (hL t)
have hident (t : ℕ) : (N t)⁻¹ * (N t * L t)^2 = (B t)^2 := by
dsimp [B]
simp only [mul_pow, Real.sq_sqrt (le_of_lt (hNpos t))]
field_simp [ne_of_gt (hNpos t)]
have ha (t : ℕ) : 0 ≤ (N t)⁻¹ := inv_nonneg.mpr (le_of_lt (hNpos t))
have hr (t : ℕ) : 0 ≤ frontierRate (n t) (d t) (q t) := by
unfold frontierRate
exact le_min (by norm_num) (add_nonneg (ha t) (sq_nonneg _))
constructor
· intro hO
obtain ⟨c, hc⟩ := Asymptotics.isBigO_iff.mp hO
have hrate0 : Tendsto (fun t => frontierRate (n t) (d t) (q t)) atTop (nhds 0) := by
have hca : Tendsto (fun t => c * (N t)⁻¹) atTop (nhds 0) := by
convert (tendsto_const_nhds : Tendsto (fun _ : ℕ => c) atTop (nhds c)).mul
(tendsto_inv_atTop_zero.comp hN) using 1 <;> simp [N]
apply squeeze_zero' (Eventually.of_forall hr) ?_ hca
filter_upwards [hc] with t ht
rw [Real.norm_eq_abs, Real.norm_eq_abs, abs_of_nonneg (hr t),
abs_of_nonneg (ha t)] at ht
exact ht
have hsmall : ∀ᶠ t in atTop, frontierRate (n t) (d t) (q t) < 1 :=
hrate0.eventually (Iio_mem_nhds (by norm_num : (0 : ℝ) < 1))
rw [Asymptotics.isBigO_iff]
refine ⟨max 1 c, ?_⟩
filter_upwards [hc, hsmall] with t hct hst
have hsum : (N t)⁻¹ + ((d t : ℝ) / (N t * L t)) ^ 2 < 1 := by
by_contra hnot
have hge : 1 ≤ (N t)⁻¹ + ((d t : ℝ) / (N t * L t)) ^ 2 :=
le_of_not_gt hnot
simp [frontierRate, N, L, min_eq_left hge] at hst
have hrate : frontierRate (n t) (d t) (q t) =
(N t)⁻¹ + ((d t : ℝ) / (N t * L t)) ^ 2 := by
simp [frontierRate, N, L, min_eq_right (le_of_lt hsum)]
rw [Real.norm_eq_abs, Real.norm_eq_abs, abs_of_nonneg (hr t),
abs_of_nonneg (ha t), hrate] at hct
have hb : ((d t : ℝ) / (N t * L t)) ^ 2 ≤ c * (N t)⁻¹ := by
nlinarith [ha t]
have hsq : (d t : ℝ)^2 ≤ c * (B t)^2 := by
rw [div_pow] at hb
have hp := (mul_le_mul_of_nonneg_right hb (le_of_lt (pow_pos (hden t) 2)))
rw [div_mul_cancel₀ _ (ne_of_gt (pow_pos (hden t) 2)), mul_assoc, hident] at hp
nlinarith
have hK : 0 ≤ (max 1 c) * B t := mul_nonneg (by positivity) (le_of_lt (hB t))
have hcm : c ≤ (max 1 c)^2 := by
have h1 : 1 ≤ max 1 c := le_max_left _ _
have hc' : c ≤ max 1 c := le_max_right _ _
nlinarith
have hdim : (d t : ℝ) ≤ max 1 c * B t := by
have hB2 : 0 ≤ (B t)^2 := sq_nonneg _
nlinarith [mul_nonneg (sub_nonneg.mpr hcm) hB2]
change ‖(d t : ℝ)‖ ≤ max 1 c * ‖B t‖
have hd0 : (0 : ℝ) ≤ (d t : ℝ) := Nat.cast_nonneg _
rw [Real.norm_eq_abs, Real.norm_eq_abs, abs_of_nonneg hd0,
abs_of_nonneg (le_of_lt (hB t))]
exact hdim
· intro hO
obtain ⟨c, hc⟩ := Asymptotics.isBigO_iff.mp hO
rw [Asymptotics.isBigO_iff]
refine ⟨1 + c^2, ?_⟩
filter_upwards [hc] with t hct
rw [Real.norm_eq_abs, Real.norm_eq_abs,
abs_of_nonneg (Nat.cast_nonneg _), abs_of_nonneg (le_of_lt (hB t))] at hct
have hsq : (d t : ℝ)^2 ≤ c^2 * (B t)^2 := by
have hcB : 0 ≤ c * B t := le_trans (Nat.cast_nonneg _) hct
nlinarith
have hb : ((d t : ℝ) / (N t * L t)) ^ 2 ≤ c^2 * (N t)⁻¹ := by
rw [div_pow]
apply (div_le_iff₀ (pow_pos (hden t) 2)).2
rw [mul_assoc, hident]
exact hsq
have hrate : frontierRate (n t) (d t) (q t) ≤ (1 + c^2) * (N t)⁻¹ := by
unfold frontierRate
calc
min 1 ((N t)⁻¹ + ((d t : ℝ) / (N t * L t)) ^ 2) ≤
(N t)⁻¹ + ((d t : ℝ) / (N t * L t)) ^ 2 := min_le_right _ _
_ ≤ (1 + c^2) * (N t)⁻¹ := by nlinarith
change ‖frontierRate (n t) (d t) (q t)‖ ≤ (1 + c^2) * ‖(N t)⁻¹‖
simpa only [Real.norm_eq_abs, abs_of_nonneg (hr t), abs_of_nonneg (ha t)]
using hrate
theorem phase_boundaries
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ → ℕ
q :
ℕ → ℝ
hn :
∀ t, 1 ≤ n t
hd :
∀ t, 1 ≤ d t
hq :
∀ t, 0 < q t ∧ q t ≤ 1
hslice :
∀ t, RareArrivalSlice (n t) (q t)
hN :
conclusion
conclusion 1
conclusion 2
(fun t => minimaxRisk (n t) (d t) (q t))
=Θ[atTop] (fun t => (effectiveSize (n t) (q t))⁻¹) ↔ (fun t => (d t : ℝ)) =O[atTop] (fun t => sqrt (effectiveSize (n t) (q t)) * logScale (n t) (q t))
Proof (Lean source)
theorem phase_boundaries (n d : ℕ → ℕ) (q : ℕ → ℝ)
(hn : ∀ t, 1 ≤ n t) (hd : ∀ t, 1 ≤ d t)
(hq : ∀ t, 0 < q t ∧ q t ≤ 1)
(hslice : ∀ t, RareArrivalSlice (n t) (q t))
(hN : Tendsto (fun t => effectiveSize (n t) (q t)) atTop atTop) :
(Tendsto (fun t => minimaxRisk (n t) (d t) (q t)) atTop (nhds 0) ↔
(fun t => (d t : ℝ)) =o[atTop]
(fun t => effectiveSize (n t) (q t) * logScale (n t) (q t))) ∧
((fun t => minimaxRisk (n t) (d t) (q t)) =Θ[atTop]
(fun t => (effectiveSize (n t) (q t))⁻¹) ↔
(fun t => (d t : ℝ)) =O[atTop]
(fun t => sqrt (effectiveSize (n t) (q t)) * logScale (n t) (q t))) := by
constructor
· obtain ⟨c, C, hc, hcC, hbounds⟩ := matched_minimax_frontier
have hC : 0 ≤ C := le_trans (le_of_lt hc) hcC
have hr (t : ℕ) : 0 ≤ frontierRate (n t) (d t) (q t) := by
unfold frontierRate
have hNpos : 0 < effectiveSize (n t) (q t) := by
unfold effectiveSize
exact mul_pos (by exact_mod_cast hn t) (hq t).1
exact le_min (by norm_num) (add_nonneg
(inv_nonneg.mpr (le_of_lt hNpos)) (sq_nonneg _))
have hboth (t : ℕ) :
c * frontierRate (n t) (d t) (q t) ≤ minimaxRisk (n t) (d t) (q t) ∧
minimaxRisk (n t) (d t) (q t) ≤ C * frontierRate (n t) (d t) (q t) :=
hbounds (n t) (d t) (q t) (hn t) (hd t) (hq t).1 (hq t).2 (hslice t)
have hiff : Tendsto (fun t => minimaxRisk (n t) (d t) (q t)) atTop (nhds 0) ↔
Tendsto (fun t => frontierRate (n t) (d t) (q t)) atTop (nhds 0) := by
constructor
· intro h
apply squeeze_zero hr (fun t => ?_)
(by convert (tendsto_const_nhds : Tendsto (fun _ : ℕ => c⁻¹) atTop (nhds c⁻¹)).mul h using 1 <;> simp)
have hle := (hboth t).1
have hle' : frontierRate (n t) (d t) (q t) ≤
minimaxRisk (n t) (d t) (q t) / c :=
(le_div_iff₀ hc).2 (by simpa [mul_comm] using hle)
simpa [div_eq_inv_mul] using hle'
· intro h
apply squeeze_zero
(fun t => le_trans (mul_nonneg (le_of_lt hc) (hr t)) (hboth t).1)
(fun t => (hboth t).2)
convert (tendsto_const_nhds : Tendsto (fun _ : ℕ => C) atTop (nhds C)).mul h using 1 <;> simp
exact hiff.trans (phase_frontier_rate_tendsto n d q hn (fun t => (hq t).1) hN)
· obtain ⟨c, C, hc, hcC, hbounds⟩ := matched_minimax_frontier
have hC : 0 ≤ C := le_trans (le_of_lt hc) hcC
have hr (t : ℕ) : 0 ≤ frontierRate (n t) (d t) (q t) := by
unfold frontierRate
have hNpos : 0 < effectiveSize (n t) (q t) := by
unfold effectiveSize
exact mul_pos (by exact_mod_cast hn t) (hq t).1
exact le_min (by norm_num) (add_nonneg
(inv_nonneg.mpr (le_of_lt hNpos)) (sq_nonneg _))
have hboth (t : ℕ) :
c * frontierRate (n t) (d t) (q t) ≤ minimaxRisk (n t) (d t) (q t) ∧
minimaxRisk (n t) (d t) (q t) ≤ C * frontierRate (n t) (d t) (q t) :=
hbounds (n t) (d t) (q t) (hn t) (hd t) (hq t).1 (hq t).2 (hslice t)
have hm (t : ℕ) : 0 ≤ minimaxRisk (n t) (d t) (q t) :=
le_trans (mul_nonneg (le_of_lt hc) (hr t)) (hboth t).1
have hRiskRate : (fun t => minimaxRisk (n t) (d t) (q t)) =Θ[atTop]
(fun t => frontierRate (n t) (d t) (q t)) := by
constructor
· rw [Asymptotics.isBigO_iff]
refine ⟨C, ?_⟩
filter_upwards with t
rw [Real.norm_eq_abs, Real.norm_eq_abs,
abs_of_nonneg (hm t), abs_of_nonneg (hr t)]
exact (hboth t).2
· rw [Asymptotics.isBigO_iff]
refine ⟨c⁻¹, ?_⟩
filter_upwards with t
rw [Real.norm_eq_abs, Real.norm_eq_abs,
abs_of_nonneg (hr t), abs_of_nonneg (hm t)]
have hle : frontierRate (n t) (d t) (q t) ≤
minimaxRisk (n t) (d t) (q t) / c :=
(le_div_iff₀ hc).2 (by simpa [mul_comm] using (hboth t).1)
simpa [div_eq_inv_mul] using hle
have hbase := phase_rate_lower_bigO n d q hn (fun t => (hq t).1) hN
have hdim := phase_dimension_rate_bigO n d q hn (fun t => (hq t).1) hN
constructor
· intro htheta
exact hdim.mp (hRiskRate.2.trans htheta.1)
· intro hO
exact ⟨hRiskRate.1.trans (hdim.mpr hO), hbase.trans hRiskRate.2⟩
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.PUnrestrictedCompleteArrivalPhaseBoundary 1 declarations PUnrestrictedCompleteArrivalPhaseBoundary for the finite rare-arrival experiment.
PUnrestrictedCompleteArrivalPhaseBoundary for the finite rare-arrival experiment.
theorem unrestricted_complete_arrival_phase_boundary
the stated mathematical conclusion holds.
Formal statement
conclusion
∃ c C : ℝ,
0 < c
c ≤ C
conclusion 1
d :
ℕ → ℕ
(∀ n, 1 ≤ d n)
∀ᶠ n : ℕ in atTop, c / (n : ℝ)
≤ unrestrictedMinimaxRisk n (d n) 1 ∧ unrestrictedMinimaxRisk n (d n) 1 ≤ C / (n : ℝ)
Proof (Lean source)
theorem unrestricted_complete_arrival_phase_boundary :
∃ c C : ℝ, 0 < c ∧ c ≤ C ∧
∀ d : ℕ → ℕ, (∀ n, 1 ≤ d n) →
∀ᶠ n : ℕ in atTop,
c / (n : ℝ) ≤ unrestrictedMinimaxRisk n (d n) 1 ∧
unrestrictedMinimaxRisk n (d n) 1 ≤ C / (n : ℝ) := by
refine ⟨1 / 256, 4, by norm_num, by norm_num, ?_⟩
intro d hd
have hdeficit : ∀ᶠ n : ℕ in atTop,
1 - (1 : ℝ) ≤ (0 : ℝ) / sqrt (n : ℝ) := by simp
have hb := unrestricted_near_complete_phase_boundary 0 (by norm_num) d hd
(fun _ => 1) (by intro n; norm_num) hdeficit
filter_upwards [hb] with n hn
simpa [div_eq_mul_inv, mul_comm, mul_assoc] using hn
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.PUnrestrictedFixedQPhaseBoundaries 3 declarations PUnrestrictedFixedQPhaseBoundaries for the finite rare-arrival experiment.
PUnrestrictedFixedQPhaseBoundaries for the finite rare-arrival experiment.
theorem fixedQRate_eq_frontierRate_one
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma fixedQRate_eq_frontierRate_one (n d : ℕ) :
fixedQRate n d = frontierRate n d 1 := by
simp [fixedQRate, frontierRate, effectiveSize, logScale]
theorem fixedQRate_tendsto_iff_dimension_littleO
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma fixedQRate_tendsto_iff_dimension_littleO (d : ℕ → ℕ) :
Tendsto (fun n => fixedQRate n (d n)) atTop (nhds 0) ↔
(fun n => (d n : ℝ)) =o[atTop]
(fun n => (n : ℝ) * log (exp 1 + n)) := by
let a : ℕ → ℝ := fun n => (n : ℝ)⁻¹
let x : ℕ → ℝ := fun n => (d n : ℝ) /
((n : ℝ) * log (exp 1 + n))
have hlog (n : ℕ) : 0 < log (exp 1 + n) := by
apply Real.log_pos
have : 1 < exp (1 : ℝ) := Real.one_lt_exp_iff.mpr (by norm_num)
linarith [Nat.cast_nonneg (α := ℝ) n]
have hx (n : ℕ) : 0 ≤ x n := by
dsimp [x]
exact div_nonneg (Nat.cast_nonneg _) (mul_nonneg (Nat.cast_nonneg _) (le_of_lt (hlog n)))
have ha0 : Tendsto a atTop (nhds 0) := by
simpa only [a, Function.comp_def] using (tendsto_inv_atTop_zero.comp
(tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop))
have hsq : Tendsto (fun n => (x n) ^ 2) atTop (nhds 0) ↔
Tendsto x atTop (nhds 0) := by
constructor
· intro h
have hs := Real.continuous_sqrt.tendsto (0 : ℝ) |>.comp h
have hs' : Tendsto (fun n => sqrt ((x n) ^ 2)) atTop (nhds 0) := by
convert hs using 1 <;> simp [Function.comp_def]
exact hs'.congr' (Eventually.of_forall (fun n => by
simp [Real.sqrt_sq_eq_abs, abs_of_nonneg (hx n)]))
· intro h
convert h.pow 2 using 1 <;> norm_num
have hlittle : (fun n => (d n : ℝ)) =o[atTop]
(fun n => (n : ℝ) * log (exp 1 + n)) ↔
Tendsto x atTop (nhds 0) := by
have hden : ∀ᶠ (n : ℕ) in (atTop : Filter ℕ),
(n : ℝ) * log (exp 1 + n) = 0 → (d n : ℝ) = 0 := by
filter_upwards [eventually_ge_atTop 1] with n hn hzero
have hn' : (0 : ℝ) < n := by exact_mod_cast hn
exact False.elim ((ne_of_gt (mul_pos hn' (hlog n))) hzero)
exact Asymptotics.isLittleO_iff_tendsto' hden
change Tendsto (fun n => min 1 (a n + (x n) ^ 2)) atTop (nhds 0) ↔ _
rw [phase_min_rate_tendsto a (fun n => (x n) ^ 2)
(fun n => inv_nonneg.mpr (Nat.cast_nonneg _))
(fun n => sq_nonneg _) ha0, hsq, hlittle]
theorem unrestricted_fixed_q_phase_boundaries
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
d :
ℕ → ℕ
(∀ n, 1 ≤ d n)
conclusion 2
∃ c : ℝ,
0 < c
M :
ℝ
0 < M
∃ C : ℝ,
c ≤ C
∀ᶠ n in (atTop : Filter ℕ), c * ((n : ℕ) : ℝ)⁻¹
≤ unrestrictedMinimaxRisk n (d n) q ∧
unrestrictedMinimaxRisk n (d n) q ≤ C * ((n : ℕ) : ℝ)⁻¹
conclusion 3
d :
ℕ → ℕ
(∀ n, 1 ≤ d n)
(∃ c C : ℝ, 0 < c ∧ c ≤ C ∧ ∀ᶠ n in (atTop : Filter ℕ), c * ((n : ℕ) : ℝ)⁻¹ ≤ unrestrictedMinimaxRisk n (d n) q ∧ unrestrictedMinimaxRisk n (d n) q ≤ C * ((n : ℕ) : ℝ)⁻¹)
Proof (Lean source)
-- @node: prop:unrestricted-fixed-q-phase-boundaries
theorem unrestricted_fixed_q_phase_boundaries
(hZeng_of_gate : ZengTheoremTwoLower)
(q : ℝ) (hq : 0 < q ∧ q < 1 / 2) :
(∀ (d : ℕ → ℕ), (∀ n, 1 ≤ d n) →
(Tendsto (fun n => unrestrictedMinimaxRisk n (d n) q) atTop (nhds 0) ↔
(fun n => (d n : ℝ)) =o[atTop]
(fun n => (n : ℝ) * log (exp 1 + n)))) ∧
(∃ c : ℝ, 0 < c ∧
∀ M : ℝ, 0 < M →
∃ C : ℝ, c ≤ C ∧
∀ (d : ℕ → ℕ), (∀ n, 1 ≤ d n) →
(∀ᶠ n in (atTop : Filter ℕ),
(d n : ℝ) ≤ M * sqrt n * log (exp 1 + n)) →
(∀ᶠ n in (atTop : Filter ℕ),
c * ((n : ℕ) : ℝ)⁻¹ ≤ unrestrictedMinimaxRisk n (d n) q ∧
unrestrictedMinimaxRisk n (d n) q ≤ C * ((n : ℕ) : ℝ)⁻¹)) ∧
(∀ (d : ℕ → ℕ), (∀ n, 1 ≤ d n) →
(∃ c C : ℝ, 0 < c ∧ c ≤ C ∧
∀ᶠ n in (atTop : Filter ℕ),
c * ((n : ℕ) : ℝ)⁻¹ ≤ unrestrictedMinimaxRisk n (d n) q ∧
unrestrictedMinimaxRisk n (d n) q ≤ C * ((n : ℕ) : ℝ)⁻¹) →
(fun n => (d n : ℝ)) =O[atTop]
(fun n => sqrt n * log (exp 1 + n))) := by
obtain ⟨a, B, ha, haB, hfrontier⟩ :=
unrestricted_fixed_q_minimax_frontier hZeng_of_gate q hq.1 hq.2
have hB : 0 ≤ B := le_trans (le_of_lt ha) haB
refine ⟨?_, ?_, ?_⟩
· intro d hd
have hrate : Tendsto (fun n => unrestrictedMinimaxRisk n (d n) q) atTop
(nhds 0) ↔ Tendsto (fun n => fixedQRate n (d n)) atTop (nhds 0) := by
have hr (n : ℕ) : 0 ≤ fixedQRate n (d n) := by
unfold fixedQRate
exact le_min (by norm_num) (add_nonneg (inv_nonneg.mpr (Nat.cast_nonneg _))
(sq_nonneg _))
have hbound : ∀ᶠ n in (atTop : Filter ℕ),
a * fixedQRate n (d n) ≤ unrestrictedMinimaxRisk n (d n) q ∧
unrestrictedMinimaxRisk n (d n) q ≤ B * fixedQRate n (d n) := by
filter_upwards [eventually_ge_atTop 1] with n hn
obtain ⟨hlo, hhi, _, _⟩ := hfrontier n (d n) hn (hd n)
exact ⟨hlo, hhi⟩
constructor
· intro h
have hupper : Tendsto (fun n => a⁻¹ * unrestrictedMinimaxRisk n (d n) q)
atTop (nhds 0) := by
convert tendsto_const_nhds.mul h using 1 <;> simp
apply squeeze_zero' (Eventually.of_forall hr) ?_ hupper
filter_upwards [hbound] with n hn
calc
fixedQRate n (d n) ≤ unrestrictedMinimaxRisk n (d n) q / a :=
(le_div_iff₀ ha).2 (by simpa [mul_comm] using hn.1)
_ = a⁻¹ * unrestrictedMinimaxRisk n (d n) q := by ring
· intro h
have hupper : Tendsto (fun n => B * fixedQRate n (d n))
atTop (nhds 0) := by
convert tendsto_const_nhds.mul h using 1 <;> simp
apply squeeze_zero' ?_ (hbound.mono fun n hn => hn.2) hupper
filter_upwards [hbound] with n hn
exact (mul_nonneg (le_of_lt ha) (hr n)).trans hn.1
exact hrate.trans (fixedQRate_tendsto_iff_dimension_littleO d)
· refine ⟨a, ha, ?_⟩
intro M hM
refine ⟨B * (1 + M ^ 2), ?_, ?_⟩
· nlinarith [sq_nonneg M, mul_nonneg hB (sq_nonneg M)]
· intro d hd hdim
filter_upwards [eventually_ge_atTop 1, hdim] with n hn hdn
obtain ⟨hlo, hhi, _, _⟩ := hfrontier n (d n) hn (hd n)
have hnpos : (0 : ℝ) < n := by exact_mod_cast hn
have hlog : 0 < log (exp 1 + n) := by
apply Real.log_pos
have he : 1 < exp (1 : ℝ) := Real.one_lt_exp_iff.mpr (by norm_num)
linarith
have hden : 0 < (n : ℝ) * log (exp 1 + n) :=
mul_pos hnpos hlog
have hroot : (sqrt n) ^ 2 = (n : ℝ) :=
Real.sq_sqrt (le_of_lt hnpos)
have hdim_sq : (d n : ℝ)^2 ≤
M^2 * ((n : ℝ) * log (exp 1 + n)^2) := by
have hright : 0 ≤ M * sqrt n * log (exp 1 + n) := by positivity
calc
(d n : ℝ)^2 ≤ (M * sqrt n * log (exp 1 + n))^2 := by
nlinarith [mul_nonneg (sub_nonneg.mpr hdn)
(add_nonneg hright (Nat.cast_nonneg (d n)))]
_ = M^2 * ((n : ℝ) * log (exp 1 + n)^2) := by
rw [mul_pow, mul_pow, hroot]
ring
have hsq : ((d n : ℝ) / ((n : ℝ) * log (exp 1 + n))) ^ 2 ≤
M^2 * (n : ℝ)⁻¹ := by
rw [div_pow]
apply (div_le_iff₀ (pow_pos hden 2)).2
have heq : (n : ℝ)⁻¹ * ((n : ℝ) * log (exp 1 + n)) ^ 2 =
(n : ℝ) * log (exp 1 + n)^2 := by
field_simp [ne_of_gt hnpos]
rw [mul_assoc, heq]
exact hdim_sq
have hinv : (n : ℝ)⁻¹ ≤ 1 := (inv_le_one₀ hnpos).2 (by exact_mod_cast hn)
have hrate_low : (n : ℝ)⁻¹ ≤ fixedQRate n (d n) := by
unfold fixedQRate
exact le_min hinv (le_add_of_nonneg_right (sq_nonneg _))
have hrate_up : fixedQRate n (d n) ≤
(1 + M^2) * (n : ℝ)⁻¹ := by
unfold fixedQRate
calc
min 1 ((n : ℝ)⁻¹ + _) ≤ (n : ℝ)⁻¹ + _ := min_le_right _ _
_ ≤ (1 + M^2) * (n : ℝ)⁻¹ := by nlinarith
exact ⟨(mul_le_mul_of_nonneg_left hrate_low (le_of_lt ha)).trans hlo,
hhi.trans (by
calc
B * fixedQRate n (d n) ≤ B * ((1 + M^2) * (n : ℝ)⁻¹) :=
mul_le_mul_of_nonneg_left hrate_up hB
_ = (B * (1 + M^2)) * (n : ℝ)⁻¹ := by ring)⟩
· intro d hd hparam
obtain ⟨c, C, hc, hcC, hbound⟩ := hparam
have hC : 0 ≤ C := le_trans (le_of_lt hc) hcC
have hratebound : ∀ᶠ n in (atTop : Filter ℕ),
a * fixedQRate n (d n) ≤ unrestrictedMinimaxRisk n (d n) q ∧
unrestrictedMinimaxRisk n (d n) q ≤ C * (n : ℝ)⁻¹ := by
filter_upwards [eventually_ge_atTop 1, hbound] with n hn hbc
exact ⟨(hfrontier n (d n) hn (hd n)).1, hbc.2⟩
have hrate0 : Tendsto (fun n => fixedQRate n (d n)) atTop (nhds 0) := by
have hupper : Tendsto (fun n : ℕ => (C / a) * (n : ℝ)⁻¹)
atTop (nhds 0) := by
have hinv : Tendsto (fun n : ℕ => (n : ℝ)⁻¹) atTop (nhds 0) := by
simpa only [Function.comp_def] using (tendsto_inv_atTop_zero.comp
(tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n : ℝ))
atTop atTop))
simpa only [mul_zero] using
(tendsto_const_nhds : Tendsto (fun _ : ℕ => C / a) atTop (nhds (C / a))).mul hinv
apply squeeze_zero' ?_ ?_ hupper
· filter_upwards [eventually_ge_atTop 1] with n hn
unfold fixedQRate
exact le_min (by norm_num) (add_nonneg (inv_nonneg.mpr (Nat.cast_nonneg _))
(sq_nonneg _))
· filter_upwards [hratebound] with n hn
calc
fixedQRate n (d n) ≤ unrestrictedMinimaxRisk n (d n) q / a :=
(le_div_iff₀ ha).2 (by simpa [mul_comm] using hn.1)
_ ≤ (C / a) * (n : ℝ)⁻¹ := by
exact div_le_div_of_nonneg_right hn.2 (le_of_lt ha) |>.trans_eq (by ring)
have hsmall : ∀ᶠ n in (atTop : Filter ℕ), fixedQRate n (d n) < 1 :=
hrate0.eventually (Iio_mem_nhds (by norm_num : (0 : ℝ) < 1))
let K : ℝ := max 1 (C / a)
have hK : 0 ≤ K := le_trans (by norm_num) (le_max_left _ _)
have hK2 : C / a ≤ K^2 := by
have h1 : 1 ≤ K := le_max_left _ _
have h2 : C / a ≤ K := le_max_right _ _
nlinarith
rw [Asymptotics.isBigO_iff]
refine ⟨K, ?_⟩
filter_upwards [eventually_ge_atTop 1, hratebound, hsmall] with n hn hb hs
have hnpos : (0 : ℝ) < n := by exact_mod_cast hn
have hlog : 0 < log (exp 1 + n) := by
apply Real.log_pos
have he : 1 < exp (1 : ℝ) := Real.one_lt_exp_iff.mpr (by norm_num)
linarith
have hden : 0 < (n : ℝ) * log (exp 1 + n) :=
mul_pos hnpos hlog
have hsum : (n : ℝ)⁻¹ +
((d n : ℝ) / ((n : ℝ) * log (exp 1 + n)))^2 < 1 := by
by_contra hnot
have hge : 1 ≤ (n : ℝ)⁻¹ +
((d n : ℝ) / ((n : ℝ) * log (exp 1 + n)))^2 :=
le_of_not_gt hnot
simp [fixedQRate, min_eq_left hge] at hs
have hsq : ((d n : ℝ) / ((n : ℝ) * log (exp 1 + n)))^2 ≤
(C / a) * (n : ℝ)⁻¹ := by
have hr : fixedQRate n (d n) = (n : ℝ)⁻¹ +
((d n : ℝ) / ((n : ℝ) * log (exp 1 + n)))^2 := by
simp [fixedQRate, min_eq_right (le_of_lt hsum)]
have hu : fixedQRate n (d n) ≤ (C / a) * (n : ℝ)⁻¹ := by
calc
fixedQRate n (d n) ≤ unrestrictedMinimaxRisk n (d n) q / a :=
(le_div_iff₀ ha).2 (by simpa [mul_comm] using hb.1)
_ ≤ (C / a) * (n : ℝ)⁻¹ := by
calc
unrestrictedMinimaxRisk n (d n) q / a ≤
(C * (n : ℝ)⁻¹) / a :=
div_le_div_of_nonneg_right hb.2 (le_of_lt ha)
_ = (C / a) * (n : ℝ)⁻¹ := by ring
nlinarith [inv_nonneg.mpr (le_of_lt hnpos)]
have hdsq : (d n : ℝ)^2 ≤
(C / a) * ((n : ℝ) * log (exp 1 + n)^2) := by
rw [div_pow] at hsq
have hp := mul_le_mul_of_nonneg_right hsq (le_of_lt (pow_pos hden 2))
have heq : (n : ℝ)⁻¹ * ((n : ℝ) * log (exp 1 + n))^2 =
(n : ℝ) * log (exp 1 + n)^2 := by
field_simp [ne_of_gt hnpos]
rw [div_mul_cancel₀ _ (ne_of_gt (pow_pos hden 2)), mul_assoc, heq] at hp
exact hp
have hroot : (sqrt n)^2 = (n : ℝ) := Real.sq_sqrt (le_of_lt hnpos)
have hdim : (d n : ℝ) ≤ K * (sqrt n * log (exp 1 + n)) := by
have hprod : 0 ≤ (n : ℝ) * log (exp 1 + n)^2 := by positivity
have hkdsq : (d n : ℝ)^2 ≤
K^2 * ((n : ℝ) * log (exp 1 + n)^2) := by
nlinarith [mul_nonneg (sub_nonneg.mpr hK2) hprod]
have hright : 0 ≤ K * (sqrt n * log (exp 1 + n)) := by positivity
have hright_sq : (K * (sqrt n * log (exp 1 + n)))^2 =
K^2 * ((n : ℝ) * log (exp 1 + n)^2) := by
rw [mul_pow, mul_pow, hroot]
rw [← hright_sq] at hkdsq
nlinarith [Nat.cast_nonneg (α := ℝ) (d n)]
change ‖(d n : ℝ)‖ ≤ K * ‖Real.sqrt n * log (exp 1 + n)‖
rw [Real.norm_eq_abs, Real.norm_eq_abs,
abs_of_nonneg (Nat.cast_nonneg (α := ℝ) (d n)),
abs_of_nonneg (mul_nonneg (Real.sqrt_nonneg _) (le_of_lt hlog))]
exact hdim
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.PUnrestrictedNearCompletePhaseBoundary 1 declarations A sufficient near-complete window for parametric point risk uniformly in dimension.
A sufficient near-complete window for parametric point risk uniformly in dimension.
theorem unrestricted_near_complete_phase_boundary
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
M :
ℝ
hM :
0 ≤ M
d :
ℕ → ℕ
hd :
∀ n, 1 ≤ d n
q :
ℕ → ℝ
hq :
∀ n, 0 < q n ∧ q n ≤ 1
conclusion
∀ᶠ n : ℕ in atTop, 1 / (256 * (n : ℝ))
≤ unrestrictedMinimaxRisk n (d n) (q n) ∧
unrestrictedMinimaxRisk n (d n) (q n) ≤ (4 + M ^ 2) / (n : ℝ)
Proof (Lean source)
theorem unrestricted_near_complete_phase_boundary
(M : ℝ) (hM : 0 ≤ M) (d : ℕ → ℕ) (hd : ∀ n, 1 ≤ d n)
(q : ℕ → ℝ) (hq : ∀ n, 0 < q n ∧ q n ≤ 1)
(hdeficit : ∀ᶠ n : ℕ in atTop, 1 - q n ≤ M / sqrt (n : ℝ)) :
∀ᶠ n : ℕ in atTop,
1 / (256 * (n : ℝ)) ≤ unrestrictedMinimaxRisk n (d n) (q n) ∧
unrestrictedMinimaxRisk n (d n) (q n) ≤ (4 + M ^ 2) / (n : ℝ) := by
obtain ⟨C, hC, hmix, henvelope⟩ := unrestricted_near_complete_arrival_envelope
filter_upwards [hdeficit, eventually_ge_atTop (1 : ℕ)] with n hdef hn
have hnR : (0 : ℝ) < n := by exact_mod_cast hn
obtain ⟨hlower, hupper, hHT⟩ := henvelope n (d n) (q n) hn (hd n)
(hq n).1 (hq n).2
refine ⟨hlower, ?_⟩
have hupper' : unrestrictedMinimaxRisk n (d n) (q n) ≤
4 / (n : ℝ) + (1 - q n) ^ 2 :=
hupper.trans ((min_le_right _ _).trans (min_le_right _ _))
have hsqrt : 0 < sqrt (n : ℝ) := Real.sqrt_pos.2 hnR
have hscaled : (1 - q n) * sqrt (n : ℝ) ≤ M :=
(le_div_iff₀ hsqrt).1 hdef
have hnonneg : 0 ≤ (1 - q n) * sqrt (n : ℝ) :=
mul_nonneg (sub_nonneg.2 (hq n).2) hsqrt.le
have hsquare : ((1 - q n) * sqrt (n : ℝ)) ^ 2 ≤ M ^ 2 := by
nlinarith
have hbudget : (1 - q n) ^ 2 ≤ M ^ 2 / (n : ℝ) := by
apply (le_div_iff₀ hnR).2
simpa only [mul_pow, Real.sq_sqrt hnR.le] using hsquare
calc
unrestrictedMinimaxRisk n (d n) (q n) ≤
4 / (n : ℝ) + (1 - q n) ^ 2 := hupper'
_ ≤ 4 / (n : ℝ) + M ^ 2 / (n : ℝ) := add_le_add_right hbudget _
_ = (4 + M ^ 2) / (n : ℝ) := by rw [add_div]
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.PUnrestrictedUniformNearCompleteElbow 1 declarations The near-complete elbow is necessary and sufficient for parametric point risk uniformly over every alphabet size.
The near-complete elbow is necessary and sufficient for parametric point risk uniformly over every alphabet size. Supremum bounds use an upper bound and witnesses above every strictly smaller threshold.
theorem unrestricted_uniform_near_complete_elbow
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
q :
ℕ → ℝ
hq :
∀ n, 0 < q n ∧ q n ≤ 1
conclusion
conclusion 1
n :
ℕ
1 ≤ n
d :
ℕ
1 ≤ d
(n : ℝ) * unrestrictedMinimaxRisk n d (q n) ≤ 4 + n * (1 - q n) ^ 2
b :
ℝ
b < (1 / 2048 : ℝ) * (1 + n * (1 - q n) ^ 2)
∃ d : ℕ,
1 ≤ d
b < (n : ℝ) * unrestrictedMinimaxRisk n d (q n)
conclusion 2
(∃ (M : ℝ) (n₀ : ℕ), 0 ≤ M ∧ ∀ n, n₀ ≤ n → 1 - q n ≤ M / sqrt (n : ℝ))
↔ (∃ (D : ℝ) (n₀ : ℕ), ∀ n, n₀ ≤ n → ∀ d, 1 ≤ d → unrestrictedMinimaxRisk n d (q n) ≤ D / n)
conclusion 4
∃ᶠ n : ℕ in atTop, C < (n : ℝ) * unrestrictedMinimaxRisk n (1024 * n ^ 2) (q n)
Proof (Lean source)
theorem unrestricted_uniform_near_complete_elbow
(q : ℕ → ℝ) (hq : ∀ n, 0 < q n ∧ q n ≤ 1) :
(∀ n : ℕ, 1 ≤ n →
(∀ d : ℕ, 1 ≤ d →
(n : ℝ) * unrestrictedMinimaxRisk n d (q n) ≤ 4 + n * (1 - q n) ^ 2) ∧
(∀ b : ℝ, b < (1 / 2048 : ℝ) * (1 + n * (1 - q n) ^ 2) →
∃ d : ℕ, 1 ≤ d ∧
b < (n : ℝ) * unrestrictedMinimaxRisk n d (q n))) ∧
((∃ (M : ℝ) (n₀ : ℕ), 0 ≤ M ∧ ∀ n, n₀ ≤ n →
1 - q n ≤ M / sqrt (n : ℝ)) ↔
(∃ (D : ℝ) (n₀ : ℕ), ∀ n, n₀ ≤ n → ∀ d, 1 ≤ d →
unrestrictedMinimaxRisk n d (q n) ≤ D / n)) ∧
(∀ n d : ℕ, 1 ≤ n → 1 ≤ d →
1 / (256 * (n : ℝ)) ≤ unrestrictedMinimaxRisk n d (q n)) ∧
(¬ BddAbove (range (fun n : ℕ => sqrt (n : ℝ) * (1 - q n))) →
∀ C : ℝ, ∃ᶠ n : ℕ in atTop,
C < (n : ℝ) * unrestrictedMinimaxRisk n (1024 * n ^ 2) (q n)) := by
obtain ⟨C₀, hC₀, hmix, henv⟩ := unrestricted_near_complete_arrival_envelope
have upper (n d : ℕ) (hn : 1 ≤ n) (hd : 1 ≤ d) :
unrestrictedMinimaxRisk n d (q n) ≤ 4 / (n : ℝ) + (1 - q n) ^ 2 := by
exact (henv n d (q n) hn hd (hq n).1 (hq n).2).2.1.trans
((min_le_right _ _).trans (min_le_right _ _))
have large (n : ℕ) (hn : 1 ≤ n) :
(1 / 2048 : ℝ) * (1 + n * (1 - q n) ^ 2) ≤
(n : ℝ) * unrestrictedMinimaxRisk n (1024 * n ^ 2) (q n) := by
have hnR : (0 : ℝ) < n := by exact_mod_cast hn
have hl := (unrestricted_large_alphabet_deficit_lower n (1024 * n ^ 2)
(q n) hn le_rfl (hq n).1 (hq n).2).2.2.1
have hm := mul_le_mul_of_nonneg_left hl hnR.le
have heq : (n : ℝ) * ((1 / 2048 : ℝ) * ((n : ℝ)⁻¹ + (1 - q n) ^ 2)) =
(1 / 2048 : ℝ) * (1 + n * (1 - q n) ^ 2) := by
field_simp
rwa [heq] at hm
have alphabet (n : ℕ) (hn : 1 ≤ n) : 1 ≤ 1024 * n ^ 2 := by
have : 1 ≤ n ^ 2 := one_le_pow₀ hn
omega
refine ⟨?_, ?_, ?_, ?_⟩
· intro n hn
have hnR : (0 : ℝ) < n := by exact_mod_cast hn
constructor
· intro d hd
have h := mul_le_mul_of_nonneg_left (upper n d hn hd) hnR.le
have heq : (n : ℝ) * (4 / (n : ℝ) + (1 - q n) ^ 2) =
4 + n * (1 - q n) ^ 2 := by
field_simp
rwa [heq] at h
· intro b hb
exact ⟨1024 * n ^ 2, alphabet n hn, hb.trans_le (large n hn)⟩
· constructor
· rintro ⟨M, n₀, hM, hdef⟩
refine ⟨4 + M ^ 2, max n₀ 1, ?_⟩
intro n hn d hd
have hn1 : 1 ≤ n := (le_max_right n₀ 1).trans hn
have hnR : (0 : ℝ) < n := by exact_mod_cast hn1
have hsqrt : 0 < sqrt (n : ℝ) := Real.sqrt_pos.2 hnR
have hscaled := (le_div_iff₀ hsqrt).1
(hdef n ((le_max_left n₀ 1).trans hn))
have hnon : 0 ≤ (1 - q n) * sqrt (n : ℝ) :=
mul_nonneg (sub_nonneg.2 (hq n).2) hsqrt.le
have hsquare : ((1 - q n) * sqrt (n : ℝ)) ^ 2 ≤ M ^ 2 := by
nlinarith
have hbudget : (1 - q n) ^ 2 ≤ M ^ 2 / (n : ℝ) := by
apply (le_div_iff₀ hnR).2
simpa only [mul_pow, Real.sq_sqrt hnR.le] using hsquare
calc
unrestrictedMinimaxRisk n d (q n) ≤ 4 / (n : ℝ) + (1 - q n) ^ 2 :=
upper n d hn1 hd
_ ≤ 4 / (n : ℝ) + M ^ 2 / (n : ℝ) := add_le_add_right hbudget _
_ = (4 + M ^ 2) / (n : ℝ) := by rw [add_div]
· rintro ⟨D, n₀, hD⟩
refine ⟨Real.sqrt (max (2048 * D) 0), max n₀ 1, Real.sqrt_nonneg _, ?_⟩
intro n hn
have hn1 : 1 ≤ n := (le_max_right n₀ 1).trans hn
have hnR : (0 : ℝ) < n := by exact_mod_cast hn1
have hsqrt : 0 < sqrt (n : ℝ) := Real.sqrt_pos.2 hnR
have hbound := mul_le_mul_of_nonneg_left
(hD n ((le_max_left n₀ 1).trans hn) _ (alphabet n hn1)) hnR.le
have hcancel : (n : ℝ) * (D / (n : ℝ)) = D := by
field_simp
rw [hcancel] at hbound
have hl := (large n hn1).trans hbound
have hsq : (sqrt (n : ℝ) * (1 - q n)) ^ 2 = n * (1 - q n) ^ 2 := by
rw [mul_pow, Real.sq_sqrt hnR.le]
have hM : (sqrt (max (2048 * D) 0)) ^ 2 = max (2048 * D) 0 :=
Real.sq_sqrt (le_max_right _ _)
have hscaled : (1 - q n) * sqrt (n : ℝ) ≤
sqrt (max (2048 * D) 0) := by
have := le_max_left (2048 * D) (0 : ℝ)
have := Real.sqrt_nonneg (max (2048 * D) 0)
nlinarith [sq_nonneg (sqrt (n : ℝ) * (1 - q n) -
sqrt (max (2048 * D) 0))]
exact (le_div_iff₀ hsqrt).2 hscaled
· intro n d hn hd
exact (henv n d (q n) hn hd (hq n).1 (hq n).2).1
· intro hunbounded C
by_contra hnot
have hevent : ∀ᶠ n : ℕ in atTop,
(n : ℝ) * unrestrictedMinimaxRisk n (1024 * n ^ 2) (q n) ≤ C := by
simpa only [Frequently, not_not, not_lt] using hnot
apply hunbounded
apply Filter.IsBoundedUnder.bddAbove_range
apply Filter.isBoundedUnder_of_eventually_le
(a := sqrt (max (2048 * C) 0))
filter_upwards [hevent, eventually_ge_atTop (1 : ℕ)] with n hbound hn
have hnR : (0 : ℝ) ≤ n := Nat.cast_nonneg n
have hl := (large n hn).trans hbound
have hsq : (sqrt (n : ℝ) * (1 - q n)) ^ 2 = n * (1 - q n) ^ 2 := by
rw [mul_pow, Real.sq_sqrt hnR]
have hM : (sqrt (max (2048 * C) 0)) ^ 2 = max (2048 * C) 0 :=
Real.sq_sqrt (le_max_right _ _)
have := le_max_left (2048 * C) (0 : ℝ)
have := Real.sqrt_nonneg (max (2048 * C) 0)
nlinarith [sq_nonneg (sqrt (n : ℝ) * (1 - q n) -
sqrt (max (2048 * C) 0))]
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.TConnectedIntervalFrontier 20 declarations TConnectedIntervalFrontier for the finite rare-arrival experiment.
TConnectedIntervalFrontier for the finite rare-arrival experiment.
theorem riskEnvelopeEndpoints_length_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(riskEnvelopeEndpoints n d q α s).2 - (riskEnvelopeEndpoints n d q α s).1
≤ 2 * sqrt (riskEnvelope n d q / α)
Proof (Lean source)
lemma riskEnvelopeEndpoints_length_le (n d : ℕ) (q α : ℝ)
(s : Fin n → ObsRecord d) :
(riskEnvelopeEndpoints n d q α s).2 -
(riskEnvelopeEndpoints n d q α s).1 ≤
2 * sqrt (riskEnvelope n d q / α) := by
unfold riskEnvelopeEndpoints
have hlo := le_max_right (-1 : ℝ)
(mixedCountEstimator n d q s - sqrt (riskEnvelope n d q / α))
have hhi := min_le_right (1 : ℝ)
(mixedCountEstimator n d q s + sqrt (riskEnvelope n d q / α))
linarith
theorem riskEnvelopeInterval_contains_of_error_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma riskEnvelopeInterval_contains_of_error_le (n d : ℕ) (q α : ℝ)
(P : FullLaw d) (s : Fin n → ObsRecord d)
(herror : |mixedCountEstimator n d q s - ate P| ≤
sqrt (riskEnvelope n d q / α)) :
ate P ∈ riskEnvelopeInterval n d q α s := by
have htarget := ate_mem_unit_interval P
rcases abs_le.mp herror with ⟨hlo, hhi⟩
simp only [riskEnvelopeInterval, riskEnvelopeEndpoints, Set.mem_Icc] at *
constructor
· exact max_le htarget.1 (by linarith)
· exact le_min htarget.2 (by linarith)
theorem riskEnvelopeInterval_contains_of_sq_error_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q α :
ℝ
P :
FullLaw d
hratio :
0 ≤ riskEnvelope n d q / α
herror :
(mixedCountEstimator n d q s - ate P) ^ 2 ≤ riskEnvelope n d q / α
conclusion
ate P ∈ riskEnvelopeInterval n d q α s
Proof (Lean source)
lemma riskEnvelopeInterval_contains_of_sq_error_le (n d : ℕ) (q α : ℝ)
(P : FullLaw d) (s : Fin n → ObsRecord d)
(hratio : 0 ≤ riskEnvelope n d q / α)
(herror : (mixedCountEstimator n d q s - ate P) ^ 2 ≤
riskEnvelope n d q / α) :
ate P ∈ riskEnvelopeInterval n d q α s := by
apply riskEnvelopeInterval_contains_of_error_le
have hsqrt : (sqrt (riskEnvelope n d q / α)) ^ 2 =
riskEnvelope n d q / α := Real.sq_sqrt hratio
have hnonneg := Real.sqrt_nonneg (riskEnvelope n d q / α)
nlinarith [abs_nonneg (mixedCountEstimator n d q s - ate P),
sq_abs (mixedCountEstimator n d q s - ate P)]
theorem riskEnvelopeInterval_coverage_of_positive_envelope
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
1 - α ≤ (sampleLaw n P).real {s | ate P ∈ riskEnvelopeInterval n d q α s}
Proof (Lean source)
lemma riskEnvelopeInterval_coverage_of_positive_envelope (n d : ℕ) (q α : ℝ)
(P : FullLaw d) (hα : 0 < α) (henv : 0 < riskEnvelope n d q)
(hrisk : deterministicRisk (mixedCountEstimator n d q) P ≤
riskEnvelope n d q) :
1 - α ≤ (sampleLaw n P).real
{s | ate P ∈ riskEnvelopeInterval n d q α s} := by
let μ := sampleLaw n P
let f : (Fin n → ObsRecord d) → ℝ :=
fun s => (mixedCountEstimator n d q s - ate P) ^ 2
let t := riskEnvelope n d q / α
letI : IsProbabilityMeasure P.1 := P.2
letI : IsProbabilityMeasure μ := by
dsimp [μ, sampleLaw]
letI : IsProbabilityMeasure (P.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
infer_instance
have ht : 0 < t := div_pos henv hα
have hmarkov : t * μ.real {s | t ≤ f s} ≤ riskEnvelope n d q := by
calc
t * μ.real {s | t ≤ f s} ≤ ∫ s, f s ∂μ :=
mul_meas_ge_le_integral_of_nonneg
(Filter.Eventually.of_forall fun s => sq_nonneg _) Integrable.of_finite t
_ ≤ riskEnvelope n d q := hrisk
let good : Set (Fin n → ObsRecord d) :=
{s | ate P ∈ riskEnvelopeInterval n d q α s}
have hsubset : goodᶜ ⊆ {s | t ≤ f s} := by
intro s hnotgood
by_contra hlt
have hsq : f s ≤ t := le_of_lt (lt_of_not_ge hlt)
have hmem := riskEnvelopeInterval_contains_of_sq_error_le n d q α P s
(le_of_lt ht) (by simpa [f, t] using hsq)
exact hnotgood hmem
have hbad : μ.real goodᶜ ≤ α := by
have hmono := measureReal_mono (μ := μ) hsubset (measure_ne_top μ _)
have hmul := mul_le_mul_of_nonneg_left hmono (le_of_lt ht)
have hαne : α ≠ 0 := ne_of_gt hα
have henvne : riskEnvelope n d q ≠ 0 := ne_of_gt henv
dsimp [t] at hmul hmarkov
have hprod : (riskEnvelope n d q / α) * α = riskEnvelope n d q := by
field_simp
nlinarith
have hgood : MeasurableSet good := by
classical
have heq : ((fun s : Fin n → ObsRecord d => decide (s ∈ good)) ⁻¹' {true}) =
good := by
ext s
simp
rw [← heq]
exact (measurable_of_finite
(fun s : Fin n → ObsRecord d => decide (s ∈ good)))
(measurableSet_singleton true)
have htotal : μ.real goodᶜ = 1 - μ.real good := by
simpa using (measureReal_compl (μ := μ) hgood)
dsimp [μ, good] at *
linarith
theorem riskEnvelopeInterval_coverage
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
1 - α ≤ (sampleLaw n P).real {s | ate P ∈ riskEnvelopeInterval n d q α s}
Proof (Lean source)
lemma riskEnvelopeInterval_coverage (n d : ℕ) (q α : ℝ)
(P : FullLaw d) (hα : 0 < α)
(hrisk : deterministicRisk (mixedCountEstimator n d q) P ≤
riskEnvelope n d q) :
1 - α ≤ (sampleLaw n P).real
{s | ate P ∈ riskEnvelopeInterval n d q α s} := by
have henv_nonneg : 0 ≤ riskEnvelope n d q := by
have hrisk_nonneg : 0 ≤ deterministicRisk (mixedCountEstimator n d q) P := by
unfold deterministicRisk
exact integral_nonneg (fun s => sq_nonneg _)
exact hrisk_nonneg.trans hrisk
rcases eq_or_lt_of_le henv_nonneg with hzero | hpos
· let μ := sampleLaw n P
let f : (Fin n → ObsRecord d) → ℝ :=
fun s => (mixedCountEstimator n d q s - ate P) ^ 2
letI : IsProbabilityMeasure P.1 := P.2
letI : IsProbabilityMeasure μ := by
dsimp [μ, sampleLaw]
letI : IsProbabilityMeasure (P.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
infer_instance
have hzero_integral : (∫ s, f s ∂μ) = 0 := by
have hle : (∫ s, f s ∂μ) ≤ 0 := by
simpa [deterministicRisk, μ, f, ← hzero] using hrisk
exact le_antisymm hle (integral_nonneg (fun s => sq_nonneg _))
have hae : ∀ᵐ s ∂μ, f s = 0 :=
(integral_eq_zero_iff_of_nonneg
(fun s => sq_nonneg (mixedCountEstimator n d q s - ate P))
Integrable.of_finite).mp hzero_integral
let good : Set (Fin n → ObsRecord d) :=
{s | ate P ∈ riskEnvelopeInterval n d q α s}
have hgood : MeasurableSet good := by
classical
have heq : ((fun s : Fin n → ObsRecord d => decide (s ∈ good)) ⁻¹' {true}) =
good := by
ext s
simp
rw [← heq]
exact (measurable_of_finite
(fun s : Fin n → ObsRecord d => decide (s ∈ good)))
(measurableSet_singleton true)
have hgoodae : ∀ᵐ s ∂μ, s ∈ good := by
filter_upwards [hae] with s hs
apply riskEnvelopeInterval_contains_of_sq_error_le n d q α P s
· simp [← hzero]
· simpa [f, ← hzero] using le_of_eq hs
have hprob : μ good = 1 := (mem_ae_iff_prob_eq_one hgood).mp hgoodae
have hreal : μ.real good = 1 := by
simp [measureReal_def, hprob]
change 1 - α ≤ μ.real good
rw [hreal]
linarith
· exact riskEnvelopeInterval_coverage_of_positive_envelope n d q α P hα hpos hrisk
theorem riskEnvelopeEndpoints_risk_le
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
deterministicIntervalRisk (riskEnvelopeEndpoints n d q α) P
≤ 2 * sqrt (riskEnvelope n d q / α)
Proof (Lean source)
lemma riskEnvelopeEndpoints_risk_le (n d : ℕ) (q α : ℝ)
(P : FullLaw d) :
deterministicIntervalRisk (riskEnvelopeEndpoints n d q α) P ≤
2 * sqrt (riskEnvelope n d q / α) := by
letI : IsProbabilityMeasure P.1 := P.2
letI : IsProbabilityMeasure (sampleLaw n P) := by
unfold sampleLaw
letI : IsProbabilityMeasure (P.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop : AEMeasurable obs P.1)
infer_instance
unfold deterministicIntervalRisk
have hint : Integrable
(fun s : Fin n → ObsRecord d =>
(riskEnvelopeEndpoints n d q α s).2 -
(riskEnvelopeEndpoints n d q α s).1) (sampleLaw n P) :=
Integrable.of_finite
have h := integral_mono hint (integrable_const _)
(fun s => riskEnvelopeEndpoints_length_le n d q α s)
simpa using h
theorem riskEnvelope_interval_upper
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
α :
ℝ
hα :
0 < α
conclusion
∃ C : ℝ,
0 < C
conclusion 1
deterministicIntervalRisk (riskEnvelopeEndpoints n d q α) P
≤ C * sqrt (frontierRate n d q)
Proof (Lean source)
lemma riskEnvelope_interval_upper (α : ℝ) (hα : 0 < α) :
∃ C : ℝ, 0 < C ∧
∀ (n d : ℕ) (q : ℝ) (P : FullLaw d),
RareArrivalModelClass n d q P →
deterministicIntervalRisk (riskEnvelopeEndpoints n d q α) P ≤
C * sqrt (frontierRate n d q) := by
obtain ⟨C₀, hC₀, hupper⟩ := mixed_count_upper
refine ⟨2 * sqrt (C₀ / α), by positivity, ?_⟩
intro n d q P hP
have henv : riskEnvelope n d q ≤ C₀ * frontierRate n d q :=
(hupper n d q P hP).2.1
calc
deterministicIntervalRisk (riskEnvelopeEndpoints n d q α) P ≤
2 * sqrt (riskEnvelope n d q / α) :=
riskEnvelopeEndpoints_risk_le n d q α P
_ ≤ 2 * sqrt (C₀ * frontierRate n d q / α) := by
gcongr
_ = 2 * sqrt (C₀ / α) * sqrt (frontierRate n d q) := by
rw [show C₀ * frontierRate n d q / α =
(C₀ / α) * frontierRate n d q by ring]
rw [Real.sqrt_mul (by positivity : 0 ≤ C₀ / α)]
ring
theorem riskEnvelope_interval_worst_upper
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
α :
ℝ
hα :
0 < α
conclusion
∃ C : ℝ,
0 < C
conclusion 1
n d :
ℕ
q :
ℝ
1 ≤ n
1 ≤ d
0 < q
worstCaseRiskReal (fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) => deterministicIntervalRisk (riskEnvelopeEndpoints n d q α) P.1) ()
≤ C * sqrt (frontierRate n d q)
Proof (Lean source)
lemma riskEnvelope_interval_worst_upper (α : ℝ) (hα : 0 < α) :
∃ C : ℝ, 0 < C ∧
∀ (n d : ℕ) (q : ℝ), 1 ≤ n → 1 ≤ d → 0 < q →
Causalean.Stat.worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) =>
deterministicIntervalRisk (riskEnvelopeEndpoints n d q α) P.1) () ≤
C * sqrt (frontierRate n d q) := by
obtain ⟨C, hC, hbound⟩ := riskEnvelope_interval_upper α hα
refine ⟨C, hC, ?_⟩
intro n d q hn hd hq
unfold Causalean.Stat.worstCaseRiskReal
by_cases hne : Nonempty {P : FullLaw d // RareArrivalModelClass n d q P}
· letI := hne
exact ciSup_le (fun P => hbound n d q P.1 P.2)
· haveI : IsEmpty {P : FullLaw d // RareArrivalModelClass n d q P} :=
not_nonempty_iff.mp hne
simp
have hN : 0 < effectiveSize n q := by
unfold effectiveSize
positivity
have hrate : 0 ≤ frontierRate n d q := by
unfold frontierRate
exact le_min (by norm_num) (by positivity)
positivity
def riskEnvelope_connectedInterval
For the specified inputs and assumptions, the object defined here is the quantity, rule, or mathematical structure specified by this declaration.
Definition (Lean source)
parameters
def
given by
by let e := riskEnvelopeEndpoints n d q α s have hc := (mixed_count_estimator_regular n d q).2 s have hr : 0
≤ sqrt (riskEnvelope n d q / α) := Real.sqrt_nonneg _ have hlo : -1 ≤ e.1 := by dsimp [e, riskEnvelopeEndpoints] exact le_max_left _ _ have hhi : e.2 ≤ 1 := by dsimp [e, riskEnvelopeEndpoints] exact min_le_left _ _ have horder : e.1 ≤ e.2 := by dsimp [e, riskEnvelopeEndpoints] apply le_min · exact max_le (by norm_num) (by linarith [hc.2]) · exact max_le (by linarith [hc.1]) (by linarith) exact ⟨(e.1, e.2), hlo, horder, hhi⟩
theorem riskEnvelope_connectedInterval_contains
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
intervalContains (riskEnvelope_connectedInterval n d q α s) t
↔ t ∈ riskEnvelopeInterval n d q α s
Proof (Lean source)
lemma riskEnvelope_connectedInterval_contains (n d : ℕ) (q α t : ℝ)
(s : Fin n → ObsRecord d) :
intervalContains (riskEnvelope_connectedInterval n d q α s) t ↔
t ∈ riskEnvelopeInterval n d q α s := by
simp [intervalContains, riskEnvelopeInterval, riskEnvelope_connectedInterval]
theorem riskEnvelope_connectedInterval_length
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(riskEnvelope_connectedInterval n d q α s).hi
- (riskEnvelope_connectedInterval n d q α s).lo
= (riskEnvelopeEndpoints n d q α s).2 - (riskEnvelopeEndpoints n d q α s).1
Proof (Lean source)
lemma riskEnvelope_connectedInterval_length (n d : ℕ) (q α : ℝ)
(s : Fin n → ObsRecord d) :
(riskEnvelope_connectedInterval n d q α s).hi -
(riskEnvelope_connectedInterval n d q α s).lo =
(riskEnvelopeEndpoints n d q α s).2 -
(riskEnvelopeEndpoints n d q α s).1 := rfl
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q α :
ℝ
hα :
0 < α
conclusion
HonestInterval n d q α ⟨Kernel.deterministic (riskEnvelope_connectedInterval n d q α) (measurable_of_finite _), inferInstance⟩
Proof (Lean source)
lemma riskEnvelope_honestInterval (n d : ℕ) (q α : ℝ) (hα : 0 < α) :
HonestInterval n d q α
⟨Kernel.deterministic (riskEnvelope_connectedInterval n d q α)
(measurable_of_finite _), inferInstance⟩ := by
intro P hP
obtain ⟨C, hC, hupper⟩ := mixed_count_upper
have hrisk := hupper n d q P hP
have hcoverage := riskEnvelopeInterval_coverage n d q α P hα hrisk.1
let f := riskEnvelope_connectedInterval n d q α
let μ := sampleLaw n P
let good : Set (Fin n → ObsRecord d) :=
{s | ate P ∈ riskEnvelopeInterval n d q α s}
have hgood : MeasurableSet good := by
exact MeasurableSet.of_discrete
have heq (s : Fin n → ObsRecord d) :
(Kernel.deterministic f (measurable_of_finite _) s).real
{I | intervalContains I (ate P)} =
good.indicator (fun _ => (1 : ℝ)) s := by
have hset : MeasurableSet
{I : ConnectedInterval | intervalContains I (ate P)} := by
unfold intervalContains
have hlo : Measurable (fun I : ConnectedInterval => I.lo) :=
measurable_connectedInterval_lo
have hhi : Measurable (fun I : ConnectedInterval => I.hi) :=
measurable_connectedInterval_hi
have hleft : Measurable (fun I : ConnectedInterval => I.closedLeft) :=
measurable_const
have hright : Measurable (fun I : ConnectedInterval => I.closedRight) :=
measurable_const
have hseteq :
{I : ConnectedInterval |
(if I.closedLeft = true then I.lo ≤ ate P else I.lo < ate P) ∧
(if I.closedRight = true then ate P ≤ I.hi else ate P < I.hi)} =
(({I | I.closedLeft = true} ∩ {I | I.lo ≤ ate P}) ∪
({I | I.closedLeft = false} ∩ {I | I.lo < ate P})) ∩
(({I | I.closedRight = true} ∩ {I | ate P ≤ I.hi}) ∪
({I | I.closedRight = false} ∩ {I | ate P < I.hi})) := by
ext I
by_cases hl : I.closedLeft = true <;>
by_cases hr : I.closedRight = true <;> simp [hl, hr]
rw [hseteq]
measurability
rw [Kernel.deterministic_apply, Measure.real, Measure.dirac_apply' _ hset]
by_cases hmem : ate P ∈ riskEnvelopeInterval n d q α s <;>
simp [good, f, indicator, riskEnvelope_connectedInterval_contains, hmem]
change 1 - α ≤ ∫ s,
(Kernel.deterministic f (measurable_of_finite _) s).real
{I | intervalContains I (ate P)} ∂μ
simp_rw [heq]
rw [integral_indicator_const _ hgood]
simpa [μ, good] using hcoverage
theorem intervalLengthRisk_le_riskEnvelope_worst
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q α :
ℝ
hα :
0 < α
conclusion
intervalLengthRisk n d q α
≤ worstCaseRiskReal (fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) => deterministicIntervalRisk (riskEnvelopeEndpoints n d q α) P.1) ()
Proof (Lean source)
lemma intervalLengthRisk_le_riskEnvelope_worst (n d : ℕ) (q α : ℝ)
(hα : 0 < α) :
intervalLengthRisk n d q α ≤
worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) =>
deterministicIntervalRisk (riskEnvelopeEndpoints n d q α) P.1) () := by
let f := riskEnvelope_connectedInterval n d q α
let T : IntervalProcedure n d :=
IntervalProcedure.ofMap f (measurable_of_finite _)
let H : {T : IntervalProcedure n d // HonestInterval n d q α T} :=
⟨T, riskEnvelope_honestInterval n d q α hα⟩
have hrisk : ∀ (T : {T : IntervalProcedure n d // HonestInterval n d q α T})
(P : {P : FullLaw d // RareArrivalModelClass n d q P}),
0 ≤ intervalRisk T.1 P.1 := by
intro T P
exact integral_nonneg (fun s => integral_nonneg (fun I => sub_nonneg.mpr I.ordered))
have hval := minimaxValue_le_worstCaseRisk_of_nonneg hrisk H
change intervalLengthRisk n d q α ≤
worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) =>
intervalRisk H.1 P.1) () at hval
have heq :
(fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) =>
intervalRisk H.1 P.1) =
(fun (_ : Unit) P =>
deterministicIntervalRisk (riskEnvelopeEndpoints n d q α) P.1) := by
funext _ P
unfold intervalRisk deterministicIntervalRisk
congr 1
funext s
change (∫ I : ConnectedInterval, I.hi - I.lo ∂(Kernel.deterministic f
(measurable_of_finite _) s)) =
(riskEnvelopeEndpoints n d q α s).2 -
(riskEnvelopeEndpoints n d q α s).1
have hlo : Measurable (fun I : ConnectedInterval => I.lo) :=
measurable_connectedInterval_lo
have hhi : Measurable (fun I : ConnectedInterval => I.hi) :=
measurable_connectedInterval_hi
have hmeas : StronglyMeasurable
(fun I : ConnectedInterval => I.hi - I.lo) :=
(hhi.sub hlo).stronglyMeasurable
rw [Kernel.deterministic_apply, integral_dirac' _ _ hmeas]
exact riskEnvelope_connectedInterval_length n d q α s
rw [heq] at hval
exact hval
theorem connectedInterval_grid_neighborhood_length
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
s :
Finset ℕ
θ δ w :
ℝ
hδ :
0 ≤ δ
hw :
0 ≤ w
hmeet :
∀ i ∈ s, ∃ x : ℝ, I.lo ≤ x ∧ x ≤ I.hi ∧ |x - (θ + (i : ℝ) * δ)| ≤ w
conclusion
((s.card : ℝ) - 1) * δ - 2 * w ≤ I.hi - I.lo
Proof (Lean source)
-- @node: connectedInterval_grid_neighborhood_length
lemma connectedInterval_grid_neighborhood_length (I : ConnectedInterval)
(s : Finset ℕ) (θ δ w : ℝ) (hδ : 0 ≤ δ) (hw : 0 ≤ w)
(hmeet : ∀ i ∈ s, ∃ x : ℝ,
I.lo ≤ x ∧ x ≤ I.hi ∧ |x - (θ + (i : ℝ) * δ)| ≤ w) :
((s.card : ℝ) - 1) * δ - 2 * w ≤ I.hi - I.lo := by
classical
obtain hs | hs := s.eq_empty_or_nonempty
· subst s
simp only [Finset.card_empty, Nat.cast_zero]
have hlen := I.ordered
nlinarith
· have hsub : s ⊆ Icc (s.min' hs) (s.max' hs) := by
intro i hi
exact Finset.mem_Icc.mpr ⟨s.min'_le i hi, s.le_max' i hi⟩
have hcard := Finset.card_le_card hsub
rw [Nat.card_Icc] at hcard
have hminmax := s.min'_le_max' hs
have hspan : s.card + s.min' hs ≤ s.max' hs + 1 := by omega
have hspanR : (s.card : ℝ) - 1 ≤
(s.max' hs : ℝ) - (s.min' hs : ℝ) := by
exact_mod_cast (show (s.card : ℤ) - 1 ≤
(s.max' hs : ℤ) - (s.min' hs : ℤ) by omega)
obtain ⟨x, hxlo, hxhi, hx⟩ := hmeet (s.min' hs) (s.min'_mem hs)
obtain ⟨y, hylo, hyhi, hy⟩ := hmeet (s.max' hs) (s.max'_mem hs)
have hx' := (abs_le.mp hx).2
have hy' := (abs_le.mp hy).1
have hmul := mul_le_mul_of_nonneg_right hspanR hδ
nlinarith
theorem connectedInterval_expected_grid_neighborhood_length
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
s :
θ δ w :
ℝ
hδ :
0 ≤ δ
hw :
0 ≤ w
hcount :
Integrable (fun I => (s I).card : ConnectedInterval → ℝ) ν
hmeet :
∀ I i
if
i ∈ s I
then
∃ x : ℝ, I.lo ≤ x ∧ x ≤ I.hi ∧ |x - (θ + (i : ℝ) * δ)| ≤ w
conclusion
((∫ I, ((s I).card : ℝ) ∂ν) - 1) * δ - 2 * w ≤ ∫ I, I.hi - I.lo ∂ν
Proof (Lean source)
-- @node: connectedInterval_expected_grid_neighborhood_length
lemma connectedInterval_expected_grid_neighborhood_length
(ν : Measure ConnectedInterval) [IsProbabilityMeasure ν]
(s : ConnectedInterval → Finset ℕ) (θ δ w : ℝ)
(hδ : 0 ≤ δ) (hw : 0 ≤ w)
(hcount : Integrable (fun I => (s I).card : ConnectedInterval → ℝ) ν)
(hmeet : ∀ I i, i ∈ s I → ∃ x : ℝ,
I.lo ≤ x ∧ x ≤ I.hi ∧ |x - (θ + (i : ℝ) * δ)| ≤ w) :
((∫ I, ((s I).card : ℝ) ∂ν) - 1) * δ - 2 * w ≤
∫ I, I.hi - I.lo ∂ν := by
have hmeas : Measurable (fun I : ConnectedInterval => I.hi - I.lo) := by
fun_prop
have hlen : Integrable (fun I : ConnectedInterval => I.hi - I.lo) ν :=
Integrable.of_bound hmeas.aestronglyMeasurable 2
(Filter.Eventually.of_forall (fun I => by
rw [Real.norm_eq_abs, abs_of_nonneg (sub_nonneg.mpr I.ordered)]
linarith [I.lower, I.upper]))
have hleft : Integrable (fun I =>
(((s I).card : ℝ) - 1) * δ - 2 * w) ν :=
((hcount.sub (integrable_const 1)).mul_const δ).sub (integrable_const (2 * w))
have hbound := integral_mono hleft hlen (fun I =>
connectedInterval_grid_neighborhood_length I (s I) θ δ w hδ hw (hmeet I))
have heq : (∫ I, (((s I).card : ℝ) - 1) * δ - 2 * w ∂ν) =
((∫ I, ((s I).card : ℝ) ∂ν) - 1) * δ - 2 * w := by
integral_linearity
simp
rw [heq] at hbound
exact hbound
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Proof (Lean source)
lemma frontier_min_sum_le_twice_max (a b : ℝ) :
min 1 (a + b) ≤ 2 * max (min 1 a) (min 1 b) := by
by_cases ha1 : a ≤ 1
· by_cases hb1 : b ≤ 1
· simp only [min_eq_right ha1, min_eq_right hb1]
by_cases hab : a ≤ b
· rw [max_eq_right hab]
exact (min_le_right _ _).trans (by linarith)
· rw [max_eq_left (le_of_not_ge hab)]
exact (min_le_right _ _).trans (by linarith)
· have hb1' : 1 ≤ b := le_of_not_ge hb1
rw [min_eq_left hb1', max_eq_right (by simpa [min_eq_right ha1] using ha1)]
exact le_trans (min_le_left _ _) (by norm_num)
· have ha1' : 1 ≤ a := le_of_not_ge ha1
rw [min_eq_left ha1']
exact le_trans (min_le_left _ _) (by
have hmax : 1 ≤ max (1 : ℝ) (min 1 b) := le_max_left _ _
linarith)
theorem frontierRate_le_two_lower_scales
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
conclusion
frontierRate n d q
≤ 2 * max (min 1 (effectiveSize n q)⁻¹) (min 1 (((d : ℝ) / (effectiveSize n q * logScale n q)) ^ 2))
Proof (Lean source)
lemma frontierRate_le_two_lower_scales (n d : ℕ) (q : ℝ) :
frontierRate n d q ≤ 2 * max
(min 1 (effectiveSize n q)⁻¹)
(min 1 (((d : ℝ) / (effectiveSize n q * logScale n q)) ^ 2)) := by
unfold frontierRate
exact frontier_min_sum_le_twice_max _ _
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Proof (Lean source)
-- @node: frontier_sqrt_min_one
lemma frontier_sqrt_min_one (x : ℝ) :
sqrt (min 1 x) = min 1 (sqrt x) := by
by_cases hx : x ≤ 1
· rw [min_eq_right hx, min_eq_right (Real.sqrt_le_one.mpr hx)]
· have hx' : 1 ≤ x := le_of_not_ge hx
have hs : 1 ≤ sqrt x := by
simpa using Real.sqrt_le_sqrt hx'
rw [min_eq_left hx', min_eq_left hs, Real.sqrt_one]
theorem frontier_interval_lower_of_two_scales
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(c / 2) * sqrt (frontierRate n d q) ≤ L
Proof (Lean source)
-- @node: frontier_interval_lower_of_two_scales
lemma frontier_interval_lower_of_two_scales (n d : ℕ) (q c L : ℝ)
(hc : 0 ≤ c)
(hparam : c * sqrt (min 1 (effectiveSize n q)⁻¹) ≤ L)
(hmany : c * sqrt
(min 1 (((d : ℝ) / (effectiveSize n q * logScale n q)) ^ 2)) ≤ L)
(hN : 0 ≤ effectiveSize n q) :
(c / 2) * sqrt (frontierRate n d q) ≤ L := by
let A := min 1 (effectiveSize n q)⁻¹
let B := min 1 (((d : ℝ) / (effectiveSize n q * logScale n q)) ^ 2)
have hA : 0 ≤ A := le_min (by norm_num) (inv_nonneg.mpr hN)
have hmax : 0 ≤ max A B := hA.trans (le_max_left _ _)
have hboth : c * sqrt (max A B) ≤ L := by
rcases le_total A B with h | h
· simpa [max_eq_right h, B] using hmany
· simpa [max_eq_left h, A] using hparam
have hrate := frontierRate_le_two_lower_scales n d q
have hs : sqrt (frontierRate n d q) ≤ 2 * sqrt (max A B) := by
apply Real.sqrt_le_iff.mpr
refine ⟨by positivity, ?_⟩
have hsq := Real.sq_sqrt hmax
change frontierRate n d q ≤ 2 * max A B at hrate
nlinarith
have hmul := mul_le_mul_of_nonneg_left hs hc
linarith
theorem connected_interval_frontier
At every admissible miscoverage level, connected randomized intervals attain matching frontier bounds.
Formal statement
hypotheses
α :
ℝ
α :
ℝ
0 < α
α < 1
conclusion
∃ c C : ℝ,
0 < c
c ≤ C
conclusion 1
intervalLengthRisk n d q α
≤ worstCaseRiskReal (fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) => deterministicIntervalRisk (riskEnvelopeEndpoints n d q α) P.1) ()
worstCaseRiskReal (fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) => deterministicIntervalRisk (riskEnvelopeEndpoints n d q α) P.1) ()
≤ C * sqrt (frontierRate n d q)
P :
FullLaw d
RareArrivalModelClass n d q P
1 - α ≤ (sampleLaw n P).real {s | ate P ∈ riskEnvelopeInterval n d q α s}
Proof (Lean source)
theorem connected_interval_frontier :
∀ α : ℝ, 0 < α → α < 1 → -- @realizes \(\alpha\)(miscoverage level)
-- @realizes \(\underline c_\alpha\)(positive interval lower constant)
∃ c C : ℝ, 0 < c ∧ c ≤ C ∧ -- @realizes \(\overline C_\alpha\)(interval upper constant)
∀ (n d : ℕ) (q : ℝ), 1 ≤ n → 1 ≤ d → 0 < q → q ≤ 1 →
RareArrivalSlice n q →
c * sqrt (frontierRate n d q) ≤ intervalLengthRisk n d q α ∧
intervalLengthRisk n d q α ≤
worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) =>
deterministicIntervalRisk (riskEnvelopeEndpoints n d q α) P.1) () ∧
worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) =>
deterministicIntervalRisk (riskEnvelopeEndpoints n d q α) P.1) () ≤
C * sqrt (frontierRate n d q) ∧
(∀ P : FullLaw d, RareArrivalModelClass n d q P →
1 - α ≤ (sampleLaw n P).real
{s | ate P ∈ riskEnvelopeInterval n d q α s}) := by
intro α hα hα1
obtain ⟨C, hC, hUpper⟩ := riskEnvelope_interval_worst_upper α hα
obtain ⟨c₀, hc₀, hLower⟩ :
∃ c₀ : ℝ, 0 < c₀ ∧
∀ (n d : ℕ) (q : ℝ), 1 ≤ n → 1 ≤ d → 0 < q → q ≤ 1 →
RareArrivalSlice n q →
c₀ * sqrt (frontierRate n d q) ≤ intervalLengthRisk n d q α := by
obtain ⟨c₁, hc₁, hMany⟩ :
∃ c₁ : ℝ, 0 < c₁ ∧
∀ (n d : ℕ) (q : ℝ), 1 ≤ n → 1 ≤ d → 0 < q → q ≤ 1 →
RareArrivalSlice n q →
c₁ * sqrt (min 1
(((d : ℝ) / (effectiveSize n q * logScale n q)) ^ 2)) ≤
intervalLengthRisk n d q α := by
let η := exp (-32)
obtain ⟨T, D, cAct, hT, hD, hcAct, hActivated⟩ :
∃ (T : ℝ) (D : ℕ) (cAct : ℝ), 1 ≤ T ∧ 1 ≤ D ∧ 0 < cAct ∧
∀ (n d : ℕ) (q : ℝ), 1 ≤ n → 1 ≤ d → 0 < q → q ≤ 1 →
RareArrivalSlice n q → T ≤ effectiveSize n q →
(D : ℝ) ≤ (2 * rareMass η n q)⁻¹ ∧
(D ≤ rareCount η n d q →
cAct * min 1 ((d : ℝ) /
(effectiveSize n q * logScale n q)) ≤ intervalLengthRisk n d q α) := by
exact interval_activated_large_regime α hα.le hα1
let cP := 3 * (1 - α) ^ 2 / 512
have hcP : 0 < cP := by dsimp [cP]; positivity
refine ⟨min cAct (min (cP * min 1 (sqrt T)⁻¹) (cP / D)),
by positivity, ?_⟩
intro n d q hn hd hq hq1 hslice
have hN : 0 < effectiveSize n q := by unfold effectiveSize; positivity
have hell : 1 ≤ logScale n q := by
unfold logScale
have h := Real.log_le_log (Real.exp_pos (1 : ℝ))
(show exp 1 ≤ exp 1 + effectiveSize n q by linarith)
simpa using h
obtain ⟨M, grid, hM, hinj, hgrid, -, hfloor⟩ :=
oneCell_interval_minimax_floor q α hn hd hq hq1 hslice hα hα1
rw [interval_many_scale_sqrt n d q hN (by linarith)]
apply interval_many_scale_of_large_regime η T cP cAct D hT hD hcP hcAct
n d q (intervalLengthRisk n d q α) hd hN hell hfloor
· intro hNT
exact (hActivated n d q hn hd hq hq1 hslice hNT).1
· intro hNT hJ
exact (hActivated n d q hn hd hq hq1 hslice hNT).2 hJ
let cParam := 3 * (1 - α) ^ 2 / 512
have hcParam : 0 < cParam := by dsimp [cParam]; positivity
refine ⟨min cParam c₁ / 2, by positivity, ?_⟩
intro n d q hn hd hq hq1 hslice
obtain ⟨M, grid, hM, hinj, hgrid, -, hfloor⟩ :=
oneCell_interval_minimax_floor q α hn hd hq hq1 hslice hα hα1
apply frontier_interval_lower_of_two_scales n d q (min cParam c₁) _
(le_of_lt (lt_min hcParam hc₁))
· rw [frontier_sqrt_min_one, Real.sqrt_inv]
exact (mul_le_mul_of_nonneg_right (min_le_left cParam c₁)
(le_min (by norm_num) (by positivity))).trans hfloor
· exact (mul_le_mul_of_nonneg_right (min_le_right cParam c₁)
(Real.sqrt_nonneg _)).trans (hMany n d q hn hd hq hq1 hslice)
· unfold effectiveSize
positivity
refine ⟨min c₀ C, C, lt_min hc₀ hC, min_le_right _ _, ?_⟩
intro n d q hn hd hq hq1 hslice
refine ⟨?_, intervalLengthRisk_le_riskEnvelope_worst n d q α hα,
hUpper n d q hn hd hq, ?_⟩
· exact (mul_le_mul_of_nonneg_right (min_le_left c₀ C)
(Real.sqrt_nonneg _)).trans (hLower n d q hn hd hq hq1 hslice)
· intro P hP
obtain ⟨C', hC', hMixed⟩ := mixed_count_upper
have hrisk := hMixed n d q P hP
exact riskEnvelopeInterval_coverage n d q α P hα hrisk.1
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.TFullExperimentLower 2 declarations TFullExperimentLower for the finite rare-arrival experiment.
TFullExperimentLower for the finite rare-arrival experiment.
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma frontier_min_sum_le (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) :
min 1 (a + b) ≤ min 1 a + min 1 b := by
rcases le_total a 1 with ha1 | ha1
· rcases le_total b 1 with hb1 | hb1
· simpa [min_eq_right ha1, min_eq_right hb1] using
(min_le_right (1 : ℝ) (a + b))
· rw [min_eq_right ha1, min_eq_left hb1]
have : 0 ≤ a := ha
exact (min_le_left _ _).trans (by linarith)
· rw [min_eq_left ha1]
have : 0 ≤ min 1 b := le_min (by norm_num) hb
exact (min_le_left _ _).trans (by linarith)
theorem full_experiment_lower
the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
theorem full_experiment_lower :
∃ c : ℝ, 0 < c ∧ -- @realizes \(\underline c\)(universal lower constant)
∀ (n d : ℕ) (q : ℝ), 1 ≤ n → 1 ≤ d → 0 < q → q ≤ 1 →
RareArrivalSlice n q → c * frontierRate n d q ≤ minimaxRisk n d q := by
obtain ⟨c₀, cInterval, hc₀, -, hOne⟩ := one_cell_testing_family
obtain ⟨c₁, hc₁, hMany⟩ := activated_fuzzy_lower
refine ⟨min c₀ c₁ / 4, by positivity, ?_⟩
intro n d q hn hd hq hq1 hslice
obtain ⟨_, hBern, hPoint, _⟩ := hOne n d q hn hd hq hq1 hslice
obtain ⟨_, _, _, _, _, hRiskOne⟩ := hPoint
let N := effectiveSize n q
let L := logScale n q
let a := N⁻¹
let b := ((d : ℝ) / (N * L)) ^ 2
have hN : 0 < N := by
dsimp [N, effectiveSize]
positivity
have hL : 1 ≤ L := by
change 1 ≤ log (exp 1 + N)
rw [Real.le_log_iff_exp_le (by positivity)]
linarith
have ha : 0 ≤ a := by dsimp [a]; positivity
have hb : 0 ≤ b := sq_nonneg _
have hOne' : c₀ * min 1 a ≤ minimaxRisk n d q := hRiskOne.2
have hMany' : sqrt N ≤ d → c₁ * min 1 b ≤ minimaxRisk n d q := by
intro h
exact hMany n d q hn hd hq hq1 hslice h
have hr : frontierRate n d q = min 1 (a + b) := rfl
rw [hr]
by_cases hdim : sqrt N ≤ d
· have hbRisk := hMany' hdim
have hmin := frontier_min_sum_le a b ha hb
have hmax : min 1 a ≤ max (min 1 a) (min 1 b) := le_max_left _ _
have hmax' : min 1 b ≤ max (min 1 a) (min 1 b) := le_max_right _ _
have hc : 0 < min c₀ c₁ := lt_min hc₀ hc₁
have h₀ : min c₀ c₁ ≤ c₀ := min_le_left _ _
have h₁ : min c₀ c₁ ≤ c₁ := min_le_right _ _
have hma : 0 ≤ min 1 a := le_min (by norm_num) ha
have hmb : 0 ≤ min 1 b := le_min (by norm_num) hb
have hca := mul_nonneg (sub_nonneg.mpr h₀) hma
have hcb := mul_nonneg (sub_nonneg.mpr h₁) hmb
have hbound := mul_le_mul_of_nonneg_left hmin (by positivity : 0 ≤ min c₀ c₁ / 4)
nlinarith
· have hdim' : (d : ℝ) ≤ sqrt N := le_of_not_ge hdim
have hN0 : 0 ≤ N := le_of_lt hN
have hsqrt := Real.sq_sqrt hN0
have hden : 0 < N * L := mul_pos hN (by linarith)
have hb_le : b ≤ a := by
dsimp [a, b]
have hd0 : 0 ≤ (d : ℝ) := Nat.cast_nonneg _
have hs0 : 0 ≤ sqrt N := Real.sqrt_nonneg _
have hsq : ((d : ℝ) : ℝ) ^ 2 ≤ N := by nlinarith
have hNL : 0 < N * L := hden
rw [div_pow]
apply (div_le_iff₀ (pow_pos hNL 2)).2
have hL2 : 1 ≤ L ^ 2 := by nlinarith
have heq : N⁻¹ * (N * L) ^ 2 = N * L ^ 2 := by
field_simp [ne_of_gt hN]
rw [heq]
nlinarith [mul_nonneg hN0 (sub_nonneg.mpr hL2)]
have hminb : min 1 b ≤ min 1 a := min_le_min_left 1 hb_le
have hmin := frontier_min_sum_le a b ha hb
have h₀ : min c₀ c₁ ≤ c₀ := min_le_left _ _
have hma : 0 ≤ min 1 a := le_min (by norm_num) ha
have hca := mul_nonneg (sub_nonneg.mpr h₀) hma
have hbound := mul_le_mul_of_nonneg_left hmin (by positivity : 0 ≤ min c₀ c₁ / 4)
have hcb := mul_le_mul_of_nonneg_left hminb (by positivity : 0 ≤ min c₀ c₁ / 4)
nlinarith
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.TMatchedMinimaxFrontier 1 declarations TMatchedMinimaxFrontier for the finite rare-arrival experiment.
TMatchedMinimaxFrontier for the finite rare-arrival experiment.
theorem matched_minimax_frontier
the stated mathematical conclusion holds.
Formal statement
conclusion
∃ c C : ℝ,
0 < c
c ≤ C
conclusion 1
c * frontierRate n d q ≤ minimaxRisk n d q
minimaxRisk n d q ≤ C * frontierRate n d q
Proof (Lean source)
theorem matched_minimax_frontier :
-- @realizes \(\underline c\)(lower constant)
∃ c C : ℝ, 0 < c ∧ c ≤ C ∧ -- @realizes \(\overline C\)(upper constant)
∀ (n d : ℕ) (q : ℝ), 1 ≤ n → 1 ≤ d → 0 < q → q ≤ 1 →
RareArrivalSlice n q →
c * frontierRate n d q ≤ minimaxRisk n d q ∧
minimaxRisk n d q ≤ C * frontierRate n d q := by
obtain ⟨c, hc, hlow⟩ := full_experiment_lower
obtain ⟨C₀, hC₀, hupp⟩ := mixed_count_upper
let C := max c C₀
refine ⟨c, C, hc, le_max_left _ _, ?_⟩
intro n d q hn hd hq hq1 hslice
constructor
· exact hlow n d q hn hd hq hq1 hslice
· let f := mixedCountEstimator n d q
have hEstimator := mixed_count_estimator_regular n d q
let T : Estimator n d :=
Estimator.ofMap f hEstimator
have hrisk : ∀ (T : Estimator n d)
(P : {P : FullLaw d // RareArrivalModelClass n d q P}),
0 ≤ squaredRisk T P.1 := by
intro T P
exact integral_nonneg (fun s => integral_nonneg (fun t => sq_nonneg _))
have hbound : ∀ (P : {P : FullLaw d // RareArrivalModelClass n d q P}),
squaredRisk T P.1 ≤ C * frontierRate n d q := by
intro P
have h := hupp n d q P.1 P.2
have hdet : squaredRisk T P.1 = deterministicRisk f P.1 := by
simp [squaredRisk, deterministicRisk, T, Estimator.ofMap, Kernel.deterministic_apply]
rw [hdet]
have hN : 0 < effectiveSize n q := by
unfold effectiveSize
positivity
have hrate : 0 ≤ frontierRate n d q := by
unfold frontierRate
exact le_min (by norm_num) (by positivity)
exact (h.1.trans h.2.1).trans
(mul_le_mul_of_nonneg_right (le_max_right c C₀) hrate)
have hval : minimaxRisk n d q ≤ Causalean.Stat.worstCaseRiskReal
(fun (T : Estimator n d) (P : {P : FullLaw d // RareArrivalModelClass n d q P}) =>
squaredRisk T P.1) T := by
exact minimaxValue_le_worstCaseRisk_of_nonneg hrisk T
refine hval.trans ?_
unfold Causalean.Stat.worstCaseRiskReal
by_cases hne : Nonempty {P : FullLaw d // RareArrivalModelClass n d q P}
· letI := hne
exact ciSup_le hbound
· haveI : IsEmpty {P : FullLaw d // RareArrivalModelClass n d q P} := not_nonempty_iff.mp hne
simpa using (show (0 : ℝ) ≤ C * frontierRate n d q from by
have hC : 0 ≤ C := le_trans (le_of_lt hC₀) (le_max_right c C₀)
have hN : 0 < effectiveSize n q := by
unfold effectiveSize
positivity
have hrate : 0 ≤ frontierRate n d q := by
unfold frontierRate
exact le_min (by norm_num) (by positivity)
exact mul_nonneg hC hrate)
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.TMixedCountUpper 1 declarations TMixedCountUpper for the finite rare-arrival experiment.
TMixedCountUpper for the finite rare-arrival experiment.
theorem mixed_count_upper
the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
theorem mixed_count_upper :
∃ C : ℝ, 0 < C ∧ -- @realizes \(\overline C\)(universal upper constant)
∀ (n d : ℕ) (q : ℝ) (P : FullLaw d),
RareArrivalModelClass n d q P →
deterministicRisk (mixedCountEstimator n d q) P ≤ riskEnvelope n d q ∧
riskEnvelope n d q ≤ C * frontierRate n d q ∧
deterministicRisk (mixedCountEstimator n d q) P ≤ auxiliaryMixedRisk n q P := by
obtain ⟨C, hC, hcertificate⟩ := uniform_mixed_count_certificate
refine ⟨C, hC, ?_⟩
intro n d q P hP
obtain ⟨hrisk, hrate⟩ := hcertificate n d q P hP
exact ⟨hrisk, hrate, mixed_count_derandomization n d q P⟩
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.TOneCellTestingFamily 3 declarations Assembly of the one-cell testing family and its honest-interval consequence.
Assembly of the one-cell testing family and its honest-interval consequence.
theorem intervalRisk_bounds
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma intervalRisk_bounds {n d : ℕ} (T : IntervalProcedure n d) (P : FullLaw d) :
0 ≤ intervalRisk T P ∧ intervalRisk T P ≤ 2 := by
letI : IsProbabilityMeasure P.1 := P.2
letI : IsProbabilityMeasure (P.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
letI : IsProbabilityMeasure (sampleLaw n P) := by unfold sampleLaw; infer_instance
letI : IsMarkovKernel T.1 := T.2
have hinner (s : Fin n → ObsRecord d) :
0 ≤ ∫ I : ConnectedInterval, I.hi - I.lo ∂(T.1 s) ∧
(∫ I : ConnectedInterval, I.hi - I.lo ∂(T.1 s)) ≤ 2 := by
constructor
· exact integral_nonneg (fun I => sub_nonneg.mpr I.ordered)
· have hb : ∀ I : ConnectedInterval, I.hi - I.lo ≤ 2 := by
intro I
linarith [I.lower, I.upper]
simpa using integral_mono_of_nonneg (μ := T.1 s)
(Filter.Eventually.of_forall (fun I => sub_nonneg.mpr I.ordered))
(integrable_const (2 : ℝ)) (Filter.Eventually.of_forall hb)
unfold intervalRisk
constructor
· exact integral_nonneg (fun s => (hinner s).1)
· simpa using integral_mono_of_nonneg (μ := sampleLaw n P)
(Filter.Eventually.of_forall (fun s => (hinner s).1))
(integrable_const (2 : ℝ))
(Filter.Eventually.of_forall (fun s => (hinner s).2))
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q α :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
hslice :
RareArrivalSlice n q
hα :
0 < α
hα1 :
α < 1
conclusion
conclusion 2
I :
HonestInterval n d q α I
≤ ⨆ i, intervalRisk I (oneCellFamilyLaw (⟨0, hd⟩ : Fin d) (max 0 (min 1 (grid i))) q ⟨le_max_left _ _, max_le (by norm_num) (min_le_left _ _)⟩ ⟨hq.le, hq1⟩)
conclusion 3
≤ intervalLengthRisk n d q α
Proof (Lean source)
lemma oneCell_interval_minimax_floor {n d : ℕ} (q α : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1)
(hslice : RareArrivalSlice n q) (hα : 0 < α) (hα1 : α < 1) :
∃ (M : ℕ) (grid : Fin (M + 1) → ℝ),
0 < M ∧ Injective grid ∧
(∀ i, grid i ∈ Icc (1 / 4 : ℝ) (3 / 4 : ℝ)) ∧
(∀ (I : IntervalProcedure n d), HonestInterval n d q α I →
(3 * (1 - α) ^ 2 / 512) * min 1 (sqrt (effectiveSize n q))⁻¹ ≤
⨆ i, intervalRisk I
(oneCellFamilyLaw (⟨0, hd⟩ : Fin d) (max 0 (min 1 (grid i))) q
⟨le_max_left _ _, max_le (by norm_num) (min_le_left _ _)⟩
⟨hq.le, hq1⟩)) ∧
(3 * (1 - α) ^ 2 / 512) *
min 1 (sqrt (effectiveSize n q))⁻¹ ≤ intervalLengthRisk n d q α := by
let η := (1 - α) / 8
have hη : 0 < η := by dsimp [η]; linarith
have hη1 : η ≤ 1 := by dsimp [η]; linarith
obtain ⟨M, hMlarge⟩ := exists_nat_ge (8 / (1 - α) : ℝ)
have hM : 0 < M := by
have : (0 : ℝ) < M := lt_of_lt_of_le (by positivity) hMlarge
exact_mod_cast this
have hMr : (0 : ℝ) < M := by exact_mod_cast hM
have hN : 0 < effectiveSize n q := by unfold effectiveSize; positivity
have hsqrt : 0 < sqrt (effectiveSize n q) := Real.sqrt_pos.2 hN
let w := min (1 / 4 : ℝ) (η / (4 * sqrt (effectiveSize n q)))
have hw : 0 < w := by dsimp [w]; exact lt_min (by norm_num) (by positivity)
have hwle : w ≤ 1 / 4 := by dsimp [w]; exact min_le_left _ _
have hwbudget : 2 * effectiveSize n q * w ^ 2 ≤ η ^ 2 / 8 := by
have hle : w ≤ η / (4 * sqrt (effectiveSize n q)) := by
dsimp [w]; exact min_le_right _ _
have hsquare : w ^ 2 ≤ (η / (4 * sqrt (effectiveSize n q))) ^ 2 := by
nlinarith
have hsqrt_sq : (sqrt (effectiveSize n q)) ^ 2 = effectiveSize n q :=
Real.sq_sqrt hN.le
rw [div_pow, mul_pow, hsqrt_sq] at hsquare
field_simp [ne_of_gt hN] at hsquare ⊢
nlinarith
let δ := w / M
let grid : Fin (M + 1) → ℝ := fun j => 1 / 2 + (j : ℝ) * δ
have hδ : 0 < δ := by dsimp [δ]; positivity
have hgrid_inj : Injective grid := by
intro i j hij
have hcast : (i : ℝ) = (j : ℝ) := by
dsimp [grid] at hij
have := mul_right_cancel₀ (ne_of_gt hδ) (sub_eq_zero.mp (by linarith :
(i : ℝ) * δ - (j : ℝ) * δ = 0))
exact this
exact Fin.ext (by exact_mod_cast hcast)
have hgrid_mem : ∀ i, grid i ∈ Icc (1 / 4 : ℝ) (3 / 4 : ℝ) := by
intro i
have hi : (i : ℝ) ≤ M := by exact_mod_cast (Nat.lt_succ_iff.mp i.isLt)
have hiz : 0 ≤ (i : ℝ) := by positivity
have hterm0 : 0 ≤ (i : ℝ) * δ := mul_nonneg hiz hδ.le
have hterm1 : (i : ℝ) * δ ≤ w := by
rw [show (i : ℝ) * δ = ((i : ℝ) * w) / M by dsimp [δ]; ring]
apply (div_le_iff₀ hMr).2
simpa [mul_comm] using mul_le_mul_of_nonneg_right hi hw.le
dsimp [grid]
constructor <;> linarith
refine ⟨M, grid, hM, hgrid_inj, hgrid_mem, ?_⟩
let x : Fin d := ⟨0, hd⟩
let P₀ := oneCellFamilyLaw x (1 / 2) q (by norm_num) ⟨hq.le, hq1⟩
have hP₀ : RareArrivalModelClass n d q P₀ :=
oneCellFamilyLaw_model x (1 / 2) q hn hd (by norm_num) hq hq1 hslice
have hwscale : (η / 4) * min 1 (sqrt (effectiveSize n q))⁻¹ ≤ w := by
dsimp [w]
by_cases hc : (1 / 4 : ℝ) ≤ η / (4 * sqrt (effectiveSize n q))
· rw [min_eq_left hc]
have hm : min 1 (sqrt (effectiveSize n q))⁻¹ ≤ 1 := min_le_left _ _
have : η / 4 ≤ 1 / 4 := by linarith
nlinarith [mul_nonneg hη.le (show 0 ≤ min 1 (sqrt (effectiveSize n q))⁻¹ by
exact le_min (by norm_num) (by positivity))]
· rw [min_eq_right (le_of_not_ge hc)]
have hm : min 1 (sqrt (effectiveSize n q))⁻¹ ≤
(sqrt (effectiveSize n q))⁻¹ := min_le_right _ _
calc
η / 4 * min 1 (sqrt (effectiveSize n q))⁻¹ ≤
η / 4 * (sqrt (effectiveSize n q))⁻¹ :=
mul_le_mul_of_nonneg_left hm (by positivity)
_ = η / (4 * sqrt (effectiveSize n q)) := by ring
have hproc : ∀ H : {T : IntervalProcedure n d // HonestInterval n d q α T},
(3 * (1 - α) / 4) * w ≤ intervalRisk H.1 P₀ := by
intro H
have hgrid_unit (j : Fin (M + 1)) : 0 ≤ grid j ∧ grid j ≤ 1 := by
rcases hgrid_mem j with ⟨hj0, hj1⟩
constructor <;> linarith
let P : Fin (M + 1) → FullLaw d := fun j =>
oneCellFamilyLaw x (grid j) q (hgrid_unit j) ⟨hq.le, hq1⟩
have hP (j : Fin (M + 1)) : RareArrivalModelClass n d q (P j) :=
oneCellFamilyLaw_model x (grid j) q hn hd (hgrid_unit j) hq hq1 hslice
let μ₀ := H.1.1 ∘ₘ sampleLaw n P₀
let μ : Fin (M + 1) → Measure ConnectedInterval := fun j =>
H.1.1 ∘ₘ sampleLaw n (P j)
letI : IsMarkovKernel H.1.1 := H.1.2
letI : IsProbabilityMeasure P₀.1 := P₀.2
letI : IsProbabilityMeasure (P₀.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
letI : IsProbabilityMeasure (sampleLaw n P₀) := by unfold sampleLaw; infer_instance
letI : IsProbabilityMeasure μ₀ := by dsimp [μ₀]; infer_instance
letI (j : Fin (M + 1)) : IsProbabilityMeasure (P j).1 := (P j).2
letI (j : Fin (M + 1)) : IsProbabilityMeasure ((P j).1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
letI (j : Fin (M + 1)) : IsProbabilityMeasure (sampleLaw n (P j)) := by
unfold sampleLaw; infer_instance
letI (j : Fin (M + 1)) : IsProbabilityMeasure (μ j) := by dsimp [μ]; infer_instance
have hm := connectedInterval_endpoint_measurable
have hhonest : ∀ j,
1 - α ≤ (μ j).real
(coverageEvent
(fun I : ConnectedInterval => I.lo) (fun I => I.hi)
(gridPoint (M := M)
(1 / 2) δ j)) := by
intro j
have hpoint : gridPoint
(M := M) (1 / 2) δ j = grid j := by
simp [gridPoint, grid, mul_comm]
rw [hpoint]
have hsub : {I : ConnectedInterval | intervalContains I (grid j)} ⊆
coverageEvent
(fun I : ConnectedInterval => I.lo) (fun I => I.hi) (grid j) := by
intro I hI
cases hl : I.closedLeft <;> cases hr : I.closedRight <;>
simp [intervalContains, coverageEvent,
hl, hr] at hI ⊢ <;> constructor <;> linarith
rw [measureReal_bind_eq_integral (sampleLaw n (P j)) H.1.1 _
(measurableSet_coverageEvent
hm.1 hm.2)]
calc
1 - α ≤ ∫ s, (H.1.1 s).real {I | intervalContains I (ate (P j))}
∂(sampleLaw n (P j)) := H.2 (P j) (hP j)
_ = ∫ s, (H.1.1 s).real {I | intervalContains I (grid j)}
∂(sampleLaw n (P j)) := by rw [oneCellFamilyLaw_ate]
_ ≤ ∫ s, (H.1.1 s).real
(coverageEvent
(fun I : ConnectedInterval => I.lo) (fun I => I.hi) (grid j))
∂(sampleLaw n (P j)) := by
apply integral_mono_of_nonneg
· exact Filter.Eventually.of_forall (fun _ => measureReal_nonneg)
· exact Integrable.of_bound
(H.1.1.measurable_coe
(measurableSet_coverageEvent
hm.1 hm.2)).ennreal_toReal.aestronglyMeasurable 1
(Filter.Eventually.of_forall fun s => by
rw [Real.norm_eq_abs, abs_of_nonneg measureReal_nonneg]
exact measureReal_le_one)
· exact Filter.Eventually.of_forall (fun _ => measureReal_mono hsub)
have htv : ∀ j, tvDist μ₀ (μ j) ≤ η := by
intro j
have hj : (j : ℝ) * δ ≤ w := by
have hi : (j : ℝ) ≤ M := by exact_mod_cast (Nat.lt_succ_iff.mp j.isLt)
rw [show (j : ℝ) * δ = ((j : ℝ) * w) / M by dsimp [δ]; ring]
apply (div_le_iff₀ hMr).2
simpa [mul_comm] using mul_le_mul_of_nonneg_right hi hw.le
have hz0 : 0 ≤ (j : ℝ) * δ := by positivity
have hz1 : (j : ℝ) * δ ≤ 1 / 2 := hj.trans (by linarith)
have hb : 2 * effectiveSize n q * ((j : ℝ) * δ) ^ 2 ≤ η ^ 2 / 8 := by
have hsq : ((j : ℝ) * δ) ^ 2 ≤ w ^ 2 := by nlinarith
exact (mul_le_mul_of_nonneg_left hsq (by positivity)).trans hwbudget
exact (tvDist_bind_le (sampleLaw n P₀) (sampleLaw n (P j)) H.1.1).trans
(by
dsimp [P, P₀, grid]
simpa [δ, mul_comm] using
oneCell_product_tv_le x q ((j : ℝ) * δ) η hn hq hq1 hz0 hz1 hη.le hη1 hb)
have hgeneric :=
honest_interval_length_lower_of_grid_tv_positive
μ₀ M μ (1 / 2) δ α η hm.1 hm.2 hδ hα hα1 hMlarge (le_rfl)
hhonest htv
have hwidth : (M : ℝ) * δ = w := by
dsimp [δ]
field_simp [ne_of_gt hMr]
have hrisk0 := (intervalRisk_bounds H.1 P₀).1
have hreal : (3 * (1 - α) / 4) * ((M : ℝ) * δ) ≤ intervalRisk H.1 P₀ := by
rw [← ENNReal.ofReal_le_ofReal_iff hrisk0]
rw [intervalRisk_bind_lintegral]
simpa [μ₀, intervalLength,
max_eq_right] using hgeneric.2.1
simpa [hwidth] using hreal
let Iall : ConnectedInterval := ⟨(-1, 1), by norm_num, by norm_num, by norm_num⟩
let Tall : IntervalProcedure n d :=
⟨Kernel.const _ (Measure.dirac Iall), inferInstance⟩
have hTall : HonestInterval n d q α Tall := by
intro P hP
letI : IsProbabilityMeasure P.1 := P.2
letI : IsProbabilityMeasure (P.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
letI : IsProbabilityMeasure (sampleLaw n P) := by unfold sampleLaw; infer_instance
rcases ate_mem_unit_interval P with ⟨hATE0, hATE1⟩
have heq (s : Fin n → ObsRecord d) :
(Tall.1 s).real {I | intervalContains I (ate P)} = 1 := by
rw [Kernel.const_apply, Measure.real,
Measure.dirac_apply' _ (measurableSet_intervalContains (ate P))]
have hmem : Iall ∈ {I : ConnectedInterval | intervalContains I (ate P)} :=
⟨hATE0, hATE1⟩
rw [Set.indicator_of_mem hmem]
norm_num
simp_rw [heq]
simp
linarith
let HAll : {T : IntervalProcedure n d // HonestInterval n d q α T} := ⟨Tall, hTall⟩
letI : Nonempty {T : IntervalProcedure n d // HonestInterval n d q α T} := ⟨HAll⟩
let θ₀ : {P : FullLaw d // RareArrivalModelClass n d q P} := ⟨P₀, hP₀⟩
unfold intervalLengthRisk
have hmin : (3 * (1 - α) / 4) * w ≤
minimaxValueReal
(fun (T : {T : IntervalProcedure n d // HonestInterval n d q α T})
(P : {P : FullLaw d // RareArrivalModelClass n d q P}) =>
intervalRisk T.1 P.1) :=
le_minimaxValue (fun H =>
(hproc H).trans (le_worstCaseRisk
(risk := fun
(T : {T : IntervalProcedure n d // HonestInterval n d q α T})
(P : {P : FullLaw d // RareArrivalModelClass n d q P}) =>
intervalRisk T.1 P.1)
(by
show BddAbove (range (fun
P : {P : FullLaw d // RareArrivalModelClass n d q P} => intervalRisk H.1 P.1))
refine ⟨2, ?_⟩
rintro _ ⟨P, rfl⟩
exact (intervalRisk_bounds H.1 P).2)
θ₀))
have hscale : (3 * (1 - α) ^ 2 / 512) *
min 1 (sqrt (effectiveSize n q))⁻¹ ≤ (3 * (1 - α) / 4) * w := by
have hcoeff : 0 ≤ 3 * (1 - α) / 4 := by positivity
have h := mul_le_mul_of_nonneg_left hwscale hcoeff
dsimp [η] at h
nlinarith [mul_nonneg (sq_nonneg (1 - α))
(show 0 ≤ min 1 (sqrt (effectiveSize n q))⁻¹ by
exact le_min (by norm_num) (by positivity))]
constructor
· intro I hI
apply le_trans (hscale.trans (hproc ⟨I, hI⟩))
refine le_ciSup_of_le ?_ (0 : Fin (M + 1)) ?_
· refine ⟨2, ?_⟩
rintro _ ⟨i, rfl⟩
exact (intervalRisk_bounds I _).2
· norm_num [grid, P₀, x]
· exact hscale.trans hmin
theorem one_cell_testing_family
the stated mathematical conclusion holds.
Formal statement
conclusion
∃ (c : ℝ) (cInterval : ℝ → ℝ),
0 < c
conclusion 1
α :
ℝ
0 < α
α < 1
0 < cInterval α
conclusion 2
∃ bernoulliFamily : ℝ → FullLaw d,
u ∈ Set.Icc (1 / 4 : ℝ) (3 / 4 : ℝ) :
RareArrivalModelClass n d q (bernoulliFamily u)
∀ᵐ r : FullRecord d ∂((bernoulliFamily u).1), r.X
ate (bernoulliFamily u) = u
u₀ ≠ u₁
T :
Estimator n d
c * min 1 (effectiveSize n q)⁻¹
≤ max (squaredRisk T (bernoulliFamily u₀)) (squaredRisk T (bernoulliFamily u₁))
α :
ℝ
0 < α
α < 1
∃ (M : ℕ) (grid : Fin (M + 1) → ℝ),
0 < M
Injective grid
i :
grid i ∈ Icc (1 / 4 : ℝ) (3 / 4 : ℝ)
I :
HonestInterval n d q α I
≤ ⨆ i, intervalRisk I (bernoulliFamily (grid i))
Proof (Lean source)
lemma one_cell_testing_family :
∃ (c : ℝ) (cInterval : ℝ → ℝ),
0 < c ∧ (∀ α : ℝ, 0 < α → α < 1 → 0 < cInterval α) ∧
∀ (n d : ℕ) (q : ℝ) (hn : 1 ≤ n) (hd : 1 ≤ d),
0 < q → q ≤ 1 → RareArrivalSlice n q →
∃ bernoulliFamily : ℝ → FullLaw d,
(∀ u ∈ Icc (1 / 4 : ℝ) (3 / 4 : ℝ),
RareArrivalModelClass n d q (bernoulliFamily u) ∧
(∀ᵐ r : FullRecord d ∂((bernoulliFamily u).1),
r.X = (⟨0, hd⟩ : Fin d) ∧
r.S0 = false ∧ r.S1 = false ∧ r.Y0 = false) ∧
(bernoulliFamily u).1.real {r | r.R = true} = q ∧
ate (bernoulliFamily u) = u) ∧
(∃ u₀ ∈ Icc (1 / 4 : ℝ) (3 / 4 : ℝ),
∃ u₁ ∈ Icc (1 / 4 : ℝ) (3 / 4 : ℝ),
u₀ ≠ u₁ ∧
(∀ T : Estimator n d,
c * min 1 (effectiveSize n q)⁻¹ ≤
max (squaredRisk T (bernoulliFamily u₀))
(squaredRisk T (bernoulliFamily u₁))) ∧
c * min 1 (effectiveSize n q)⁻¹ ≤ minimaxRisk n d q) ∧
∀ α : ℝ, 0 < α → α < 1 →
∃ (M : ℕ) (grid : Fin (M + 1) → ℝ),
0 < M ∧ Injective grid ∧
(∀ i, grid i ∈ Icc (1 / 4 : ℝ) (3 / 4 : ℝ)) ∧
(∀ (I : IntervalProcedure n d), HonestInterval n d q α I →
cInterval α * min 1 (sqrt (effectiveSize n q))⁻¹ ≤
⨆ i, intervalRisk I (bernoulliFamily (grid i))) ∧
cInterval α * min 1 (sqrt (effectiveSize n q))⁻¹ ≤
intervalLengthRisk n d q α := by
refine ⟨1 / 256, fun α => 3 * (1 - α) ^ 2 / 512, by norm_num, ?_, ?_⟩
· intro α hα hα1
positivity
· intro n d q hn hd hq hq1 hslice
let x : Fin d := ⟨0, hd⟩
let clamp : ℝ → ℝ := fun u => max 0 (min 1 u)
have hclamp_mem (u : ℝ) : 0 ≤ clamp u ∧ clamp u ≤ 1 := by
dsimp [clamp]
exact ⟨le_max_left _ _, max_le (by norm_num) (min_le_left _ _)⟩
let family : ℝ → FullLaw d := fun u =>
oneCellFamilyLaw x (clamp u) q (hclamp_mem u) ⟨hq.le, hq1⟩
refine ⟨family, ?_, ?_, ?_⟩
· intro u hu
have hu0 : 0 ≤ u := by linarith [hu.1]
have hu1 : u ≤ 1 := by linarith [hu.2]
have hclamp : clamp u = u := by simp [clamp, min_eq_right hu1, max_eq_right hu0]
simpa [family, hclamp] using
(⟨oneCellFamilyLaw_model x u q hn hd ⟨hu0, hu1⟩ hq hq1 hslice,
oneCellFamilyLaw_support x u q ⟨hu0, hu1⟩ ⟨hq.le, hq1⟩,
oneCellFamilyLaw_arrival_mass x u q ⟨hu0, hu1⟩ ⟨hq.le, hq1⟩,
oneCellFamilyLaw_ate x u q ⟨hu0, hu1⟩ ⟨hq.le, hq1⟩⟩ :
RareArrivalModelClass n d q
(oneCellFamilyLaw x u q ⟨hu0, hu1⟩ ⟨hq.le, hq1⟩) ∧
(∀ᵐ r : FullRecord d ∂((oneCellFamilyLaw x u q ⟨hu0, hu1⟩
⟨hq.le, hq1⟩).1),
r.X = x ∧ r.S0 = false ∧ r.S1 = false ∧ r.Y0 = false) ∧
(oneCellFamilyLaw x u q ⟨hu0, hu1⟩ ⟨hq.le, hq1⟩).1.real
{r | r.R = true} = q ∧
ate (oneCellFamilyLaw x u q ⟨hu0, hu1⟩ ⟨hq.le, hq1⟩) = u)
· have hrisk := oneCell_minimax_risk_floors q hn hd hq hq1 hslice
have hstep := oneCellStep_pos n q hn hq
have hstep_le := oneCellStep_le_quarter n q
have hu₁ : (1 / 2 : ℝ) + oneCellStep n q ∈ Icc (1 / 4) (3 / 4) := by
constructor <;> linarith
refine ⟨1 / 2, by norm_num, 1 / 2 + oneCellStep n q, hu₁,
by linarith, ?_, hrisk.1⟩
have hclamp₀ : clamp (1 / 2) = 1 / 2 := by norm_num [clamp]
have hclamp₁ : clamp (1 / 2 + oneCellStep n q) = 1 / 2 + oneCellStep n q := by
dsimp [clamp]
rw [min_eq_right (by linarith : (1 / 2 : ℝ) + oneCellStep n q ≤ 1),
max_eq_right (by linarith : (0 : ℝ) ≤ 1 / 2 + oneCellStep n q)]
simpa only [family, hclamp₀, hclamp₁] using oneCell_two_point_risk_floor x q hn hq hq1
· intro α hα hα1
simpa [family, clamp, x] using
oneCell_interval_minimax_floor q α hn hd hq hq1 hslice hα hα1
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.TOneCellTestingRisk 2 declarations Squared-risk consequences of the one-cell observed-law testing family.
Squared-risk consequences of the one-cell observed-law testing family.
theorem oneCell_two_point_risk_floor
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(1 / 256 : ℝ) * min 1 (effectiveSize n q)⁻¹
≤ max (squaredRisk T (oneCellFamilyLaw x (1 / 2) q (by norm_num) ⟨hq.le, hq1⟩)) (squaredRisk T (oneCellFamilyLaw x (1 / 2 + oneCellStep n q) q (oneCellStep_mem n q hn hq) ⟨hq.le, hq1⟩))
Proof (Lean source)
lemma oneCell_two_point_risk_floor {n d : ℕ} (x : Fin d) (q : ℝ)
(hn : 1 ≤ n) (hq : 0 < q) (hq1 : q ≤ 1) :
∀ T : Estimator n d,
(1 / 256 : ℝ) * min 1 (effectiveSize n q)⁻¹ ≤
max
(squaredRisk T
(oneCellFamilyLaw x (1 / 2) q (by norm_num) ⟨hq.le, hq1⟩))
(squaredRisk T
(oneCellFamilyLaw x (1 / 2 + oneCellStep n q) q
(oneCellStep_mem n q hn hq) ⟨hq.le, hq1⟩)) := by
let P₀ := oneCellFamilyLaw x (1 / 2) q (by norm_num) ⟨hq.le, hq1⟩
let P₁ := oneCellFamilyLaw x (1 / 2 + oneCellStep n q) q
(oneCellStep_mem n q hn hq) ⟨hq.le, hq1⟩
have hgap : ate P₁ = ate P₀ + oneCellStep n q := by
simp [P₀, P₁, oneCellFamilyLaw_ate]
have hrisk := complete_arrival_two_point_decision_reduction P₀ P₁
(oneCellStep n q) (1 / 2) (oneCellStep_pos n q hn hq) (by norm_num)
hgap (oneCell_product_testing_half x n q hn hq hq1)
intro T
have hstep := oneCellStep_sq_lower n q hn hq
calc
(1 / 256 : ℝ) * min 1 (effectiveSize n q)⁻¹ ≤
(1 / 16) * (oneCellStep n q) ^ 2 := by nlinarith
_ = (1 / 2) * (oneCellStep n q) ^ 2 / 8 := by ring
_ ≤ max (squaredRisk T P₀) (squaredRisk T P₁) := hrisk T
theorem oneCell_minimax_risk_floors
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
conclusion 2
Proof (Lean source)
lemma oneCell_minimax_risk_floors {n d : ℕ} (q : ℝ)
(hn : 1 ≤ n) (hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1)
(hslice : RareArrivalSlice n q) :
(1 / 256 : ℝ) * min 1 (effectiveSize n q)⁻¹ ≤ minimaxRisk n d q ∧
(1 / 256 : ℝ) * min 1 (effectiveSize n q)⁻¹ ≤ noSurrogateRisk n d q := by
let x : Fin d := ⟨0, hd⟩
let P₀ := oneCellFamilyLaw x (1 / 2) q (by norm_num) ⟨hq.le, hq1⟩
let P₁ := oneCellFamilyLaw x (1 / 2 + oneCellStep n q) q
(oneCellStep_mem n q hn hq) ⟨hq.le, hq1⟩
have hP₀ : RareArrivalModelClass n d q P₀ :=
oneCellFamilyLaw_model x (1 / 2) q hn hd (by norm_num) hq hq1 hslice
have hP₁ : RareArrivalModelClass n d q P₁ :=
oneCellFamilyLaw_model x (1 / 2 + oneCellStep n q) q hn hd
(oneCellStep_mem n q hn hq) hq hq1 hslice
have hNS₀ : ∀ᵐ r : FullRecord d ∂P₀.1, r.S0 = false ∧ r.S1 = false :=
oneCellFamilyLaw_noSurrogate x (1 / 2) q (by norm_num) ⟨hq.le, hq1⟩
have hNS₁ : ∀ᵐ r : FullRecord d ∂P₁.1, r.S0 = false ∧ r.S1 = false :=
oneCellFamilyLaw_noSurrogate x (1 / 2 + oneCellStep n q) q
(oneCellStep_mem n q hn hq) ⟨hq.le, hq1⟩
have htwo : ∀ T : Estimator n d,
(1 / 256 : ℝ) * min 1 (effectiveSize n q)⁻¹ ≤
max (squaredRisk T P₀) (squaredRisk T P₁) := by
simpa [P₀, P₁] using oneCell_two_point_risk_floor x q hn hq hq1
let T₀ : Estimator n d :=
Estimator.ofMap (fun _ => 0) ⟨measurable_const, by intro s; norm_num⟩
letI : Nonempty (Estimator n d) := ⟨T₀⟩
constructor
· let θ₀ : {P : FullLaw d // RareArrivalModelClass n d q P} := ⟨P₀, hP₀⟩
let θ₁ : {P : FullLaw d // RareArrivalModelClass n d q P} := ⟨P₁, hP₁⟩
unfold minimaxRisk
apply le_minimaxValue_of_two_point θ₀ θ₁
· intro T
refine ⟨4, ?_⟩
rintro r ⟨θ, rfl⟩
exact (unrestricted_squaredRisk_bounds T θ.1).2
· exact htwo
· let θ₀ : {P : FullLaw d // RareArrivalModelClass n d q P ∧
∀ᵐ r : FullRecord d ∂P.1, r.S0 = false ∧ r.S1 = false} :=
⟨P₀, hP₀, hNS₀⟩
let θ₁ : {P : FullLaw d // RareArrivalModelClass n d q P ∧
∀ᵐ r : FullRecord d ∂P.1, r.S0 = false ∧ r.S1 = false} :=
⟨P₁, hP₁, hNS₁⟩
unfold noSurrogateRisk
apply le_minimaxValue_of_two_point θ₀ θ₁
· intro T
refine ⟨4, ?_⟩
rintro r ⟨θ, rfl⟩
exact (unrestricted_squaredRisk_bounds T θ.1).2
· exact htwo
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.TUniformMixedCountCertificate 1 declarations TUniformMixedCountCertificate for the finite rare-arrival experiment.
TUniformMixedCountCertificate for the finite rare-arrival experiment.
theorem uniform_mixed_count_certificate
the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
theorem uniform_mixed_count_certificate :
∃ C : ℝ, 0 < C ∧ -- @realizes \(\overline C\)(universal upper constant)
∀ (n d : ℕ) (q : ℝ) (P : FullLaw d),
RareArrivalModelClass n d q P →
deterministicRisk (mixedCountEstimator n d q) P ≤ riskEnvelope n d q ∧
riskEnvelope n d q ≤ C * frontierRate n d q := by
obtain ⟨C, hC, hcertificate⟩ := unrestricted_uniform_mixed_count_certificate
refine ⟨C, hC, ?_⟩
intro n d q P hP
have hPplus : UnrestrictedArrivalModelClass n d q P :=
{ n_pos := hP.n_pos
d_pos := hP.d_pos
q_pos := hP.q_pos
q_le_one := hP.q_le_one
randomized := hP.randomized
balanced := hP.balanced
surrogate := hP.surrogate
outcome := hP.outcome
mar := hP.mar
arrival := hP.arrival }
obtain ⟨hrisk, hrate⟩ := hcertificate n d q P hPplus hP.iid
have hident := unrestricted_cell_identification P hPplus
rw [← hident] at hrisk
exact ⟨hrisk, hrate⟩
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.TUnrestrictedCompleteArrivalFrontier 1 declarations The complete-arrival endpoint of the all-kernel near-complete envelope.
The complete-arrival endpoint of the all-kernel near-complete envelope.
theorem unrestricted_complete_arrival_frontier
Uniformly over sample size and alphabet size, the unrestricted complete-arrival frontier has parametric order.
Formal statement
hypotheses
n :
ℕ
d :
ℕ
n d :
ℕ
1 ≤ n
1 ≤ d
conclusion
conclusion 1
1 / (256 * (n : ℝ)) ≤ unrestrictedMinimaxRisk n d 1
conclusion 2
unrestrictedMinimaxRisk n d 1
≤ worstCaseRiskReal (fun (_ : Unit) (P : {P : FullLaw d // UnrestrictedArrivalModelClass n d 1 P}) => deterministicRisk (completeArrivalHTEstimator n d) P.1) ()
conclusion 3
worstCaseRiskReal (fun (_ : Unit) (P : {P : FullLaw d // UnrestrictedArrivalModelClass n d 1 P}) => deterministicRisk (completeArrivalHTEstimator n d) P.1) ()
≤ 4 / (n : ℝ)
Proof (Lean source)
theorem unrestricted_complete_arrival_frontier :
∀ (n d : ℕ), 1 ≤ n → 1 ≤ d →
1 / (256 * (n : ℝ)) ≤ unrestrictedMinimaxRisk n d 1 ∧
unrestrictedMinimaxRisk n d 1 ≤
worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // UnrestrictedArrivalModelClass n d 1 P}) =>
deterministicRisk (completeArrivalHTEstimator n d) P.1) () ∧
worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // UnrestrictedArrivalModelClass n d 1 P}) =>
deterministicRisk (completeArrivalHTEstimator n d) P.1) () ≤
4 / (n : ℝ) := by
intro n d hn hd
obtain ⟨C, hC, hmixed, hnear⟩ := unrestricted_near_complete_arrival_envelope
obtain ⟨hlo, _, hhi⟩ := hnear n d 1 hn hd (by norm_num) (by norm_num)
exact ⟨hlo, (complete_arrival_frontier_direct n d hn hd).2.1, by simpa using hhi⟩
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.TUnrestrictedFixedQMinimaxFrontier 6 declarations TUnrestrictedFixedQMinimaxFrontier for the finite rare-arrival experiment.
TUnrestrictedFixedQMinimaxFrontier for the finite rare-arrival experiment.
theorem fixedQRate_le_frontierRate
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hq :
0 < q
hq1 :
q ≤ 1
conclusion
fixedQRate n d ≤ frontierRate n d q
Proof (Lean source)
lemma fixedQRate_le_frontierRate (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hq : 0 < q) (hq1 : q ≤ 1) :
fixedQRate n d ≤ frontierRate n d q := by
have hn0 : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n)
have hN : 0 < effectiveSize n q := mul_pos hn0 hq
have hNle : effectiveSize n q ≤ n := by
unfold effectiveSize
nlinarith
have hlogpos : 0 < logScale n q := by
unfold logScale
apply Real.log_pos
have he : 1 < exp 1 := by
nlinarith [Real.add_one_le_exp (1 : ℝ)]
linarith
have hlogle : logScale n q ≤ log (exp 1 + n) := by
unfold logScale
exact Real.log_le_log (by positivity) (by linarith)
have hden : 0 < effectiveSize n q * logScale n q :=
mul_pos hN hlogpos
have hden' : effectiveSize n q * logScale n q ≤
(n : ℝ) * log (exp 1 + n) := by
apply mul_le_mul hNle hlogle hlogpos.le (by positivity)
have hfirst : (n : ℝ)⁻¹ ≤ (effectiveSize n q)⁻¹ :=
by simpa [one_div] using one_div_le_one_div_of_le hN hNle
have hsecond :
((d : ℝ) / ((n : ℝ) * log (exp 1 + n))) ^ 2 ≤
((d : ℝ) / (effectiveSize n q * logScale n q)) ^ 2 := by
have hdiv := div_le_div_of_nonneg_left (Nat.cast_nonneg d) hden hden'
have hdenBig : 0 < (n : ℝ) * log (exp 1 + n) :=
lt_of_lt_of_le hden hden'
have hdiv0 : 0 ≤ (d : ℝ) / ((n : ℝ) * log (exp 1 + n)) :=
div_nonneg (Nat.cast_nonneg d) hdenBig.le
nlinarith
unfold fixedQRate frontierRate
exact min_le_min_left 1 (add_le_add hfirst hsecond)
theorem frontierRate_le_fixedQRate_mul
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hq :
0 < q
hq1 :
q ≤ 1
conclusion
Proof (Lean source)
lemma frontierRate_le_fixedQRate_mul (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hq : 0 < q) (hq1 : q ≤ 1) :
frontierRate n d q ≤ ((1 - log q) / q) ^ 2 * fixedQRate n d := by
let N := effectiveSize n q
let L := logScale n q
let L₁ := log (exp 1 + n)
let K := (1 - log q) / q
have hn0 : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n)
have hN : 0 < N := mul_pos hn0 hq
have hL : 1 ≤ L := by
dsimp [L, logScale]
rw [Real.le_log_iff_exp_le (by positivity)]
dsimp [N, effectiveSize] at *
linarith
have hL₁ : 0 < L₁ := by
dsimp [L₁]
apply Real.log_pos
nlinarith [Real.add_one_le_exp (1 : ℝ)]
have hlogq : log q ≤ 0 := Real.log_nonpos hq.le hq1
have hlogbound : L₁ ≤ (1 - log q) * L := by
have harg : exp 1 + (n : ℝ) ≤ (exp 1 + N) / q := by
apply (le_div_iff₀ hq).2
dsimp [N, effectiveSize]
nlinarith [Real.exp_pos (1 : ℝ)]
have hbase : L₁ ≤ L - log q := by
calc
L₁ ≤ log ((exp 1 + N) / q) :=
Real.log_le_log (by positivity) harg
_ = L - log q := by
rw [Real.log_div (by positivity) (ne_of_gt hq)]
rfl
have hprod := mul_nonneg (neg_nonneg.mpr hlogq) (sub_nonneg.mpr hL)
nlinarith
have hK : 1 ≤ K := by
dsimp [K]
apply (le_div_iff₀ hq).2
linarith
have hKpos : 0 < K := lt_of_lt_of_le (by norm_num) hK
have hD : 0 < N * L := mul_pos hN (by linarith)
have hD₁ : 0 < (n : ℝ) * L₁ := mul_pos hn0 hL₁
have hDen : (n : ℝ) * L₁ ≤ K * (N * L) := by
calc
(n : ℝ) * L₁ ≤ (n : ℝ) * ((1 - log q) * L) :=
mul_le_mul_of_nonneg_left hlogbound hn0.le
_ = K * (N * L) := by
dsimp [K, N, effectiveSize]
field_simp
have hfirst : N⁻¹ ≤ K ^ 2 * (n : ℝ)⁻¹ := by
have hqK : q * K = 1 - log q := by
dsimp [K]
field_simp
have hK2 : 1 ≤ q * K ^ 2 := by
nlinarith [mul_nonneg (sub_nonneg.mpr hK) (sub_nonneg.mpr hlogq)]
have hN_eq : N = (n : ℝ) * q := rfl
have h : 1 / N ≤ K ^ 2 / (n : ℝ) := by
apply (div_le_div_iff₀ hN hn0).2
rw [hN_eq]
nlinarith [mul_nonneg hn0.le (sub_nonneg.mpr hK2)]
simpa [one_div, div_eq_mul_inv] using h
have hsecond : ((d : ℝ) / (N * L)) ^ 2 ≤
K ^ 2 * ((d : ℝ) / ((n : ℝ) * L₁)) ^ 2 := by
have hdiv : (d : ℝ) / (N * L) ≤ K * ((d : ℝ) / ((n : ℝ) * L₁)) := by
have h : (d : ℝ) / (N * L) ≤ (K * d) / ((n : ℝ) * L₁) := by
apply (div_le_div_iff₀ hD hD₁).2
have hd0 : 0 ≤ (d : ℝ) := Nat.cast_nonneg _
nlinarith [mul_nonneg hd0 (sub_nonneg.mpr hDen)]
convert h using 1 <;> ring
have hleft : 0 ≤ (d : ℝ) / (N * L) := div_nonneg (Nat.cast_nonneg _) hD.le
nlinarith [sq_nonneg (K * ((d : ℝ) / ((n : ℝ) * L₁)) -
((d : ℝ) / (N * L)))]
have hr0 : 0 ≤ (n : ℝ)⁻¹ + ((d : ℝ) / ((n : ℝ) * L₁)) ^ 2 := by positivity
have hsum : N⁻¹ + ((d : ℝ) / (N * L)) ^ 2 ≤
K ^ 2 * ((n : ℝ)⁻¹ + ((d : ℝ) / ((n : ℝ) * L₁)) ^ 2) := by
nlinarith [hfirst, hsecond]
dsimp [frontierRate, fixedQRate, N, L, L₁, K] at *
calc
min 1 ((effectiveSize n q)⁻¹ +
((d : ℝ) / (effectiveSize n q * logScale n q)) ^ 2) ≤
min 1 (K ^ 2 * ((n : ℝ)⁻¹ + ((d : ℝ) /
((n : ℝ) * log (exp 1 + n))) ^ 2)) :=
min_le_min_left 1 hsum
_ ≤ K ^ 2 * min 1 ((n : ℝ)⁻¹ + ((d : ℝ) /
((n : ℝ) * log (exp 1 + n))) ^ 2) := by
rcases le_total ((n : ℝ)⁻¹ + ((d : ℝ) /
((n : ℝ) * log (exp 1 + n))) ^ 2) 1 with h | h
· rw [min_eq_right h]
exact (min_le_right _ _)
· rw [min_eq_left h]
exact (min_le_left _ _).trans (by nlinarith [sq_nonneg K])
theorem unrestricted_fixed_q_frontier_upper
the stated mathematical conclusion holds.
Formal statement
conclusion
∃ C : ℝ,
0 < C
conclusion 1
n d :
ℕ
q :
ℝ
1 ≤ n
1 ≤ d
0 < q
q < 1 / 2
unrestrictedMinimaxRisk n d q ≤ C * frontierRate n d q
Proof (Lean source)
lemma unrestricted_fixed_q_frontier_upper :
∃ C : ℝ, 0 < C ∧ ∀ (n d : ℕ) (q : ℝ),
1 ≤ n → 1 ≤ d → 0 < q → q < 1 / 2 →
unrestrictedMinimaxRisk n d q ≤ C * frontierRate n d q := by
obtain ⟨C, hC, hupper⟩ := unrestricted_mixed_count_upper
refine ⟨C, hC, ?_⟩
intro n d q hn hd hq hqhalf
let f := mixedCountEstimator n d q
have hf := mixed_count_estimator_regular n d q
let T : Estimator n d :=
Estimator.ofMap f hf
have hrisk : ∀ (T : Estimator n d)
(P : {P : FullLaw d // UnrestrictedArrivalModelClass n d q P}),
0 ≤ squaredRisk T P.1 := by
intro T P
exact integral_nonneg (fun s => integral_nonneg (fun y => sq_nonneg _))
have hval : unrestrictedMinimaxRisk n d q ≤
Causalean.Stat.worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // UnrestrictedArrivalModelClass n d q P}) =>
deterministicRisk f P.1) () := by
have h := minimaxValue_le_worstCaseRisk_of_nonneg hrisk T
calc
unrestrictedMinimaxRisk n d q ≤
Causalean.Stat.worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // UnrestrictedArrivalModelClass n d q P}) =>
squaredRisk T P.1) () := by
simpa only [unrestrictedMinimaxRisk,
Causalean.Stat.worstCaseRiskReal] using h
_ = _ := by
congr 1
funext _ P
simp [squaredRisk, deterministicRisk, T, Estimator.ofMap, Kernel.deterministic_apply]
refine le_trans hval ?_
unfold Causalean.Stat.worstCaseRiskReal
by_cases hne : Nonempty
{P : FullLaw d // UnrestrictedArrivalModelClass n d q P}
· letI := hne
apply ciSup_le
intro P
obtain ⟨hriskP, hrateP, _⟩ := hupper n d q P.1 P.2 (sampleLaw_iid n P.1)
exact hriskP.trans hrateP
· haveI : IsEmpty {P : FullLaw d // UnrestrictedArrivalModelClass n d q P} :=
not_nonempty_iff.mp hne
simp
have hrate : 0 ≤ frontierRate n d q := by
have hN : 0 < effectiveSize n q := by
unfold effectiveSize
exact mul_pos (by exact_mod_cast hn) hq
have hexp : (1 : ℝ) < exp 1 := by
nlinarith [Real.add_one_le_exp (1 : ℝ)]
have hlog : 0 < logScale n q := by
unfold logScale
exact Real.log_pos (by linarith)
unfold frontierRate
exact le_min (by norm_num) (by positivity)
exact mul_nonneg hC.le hrate
theorem unrestricted_fixed_q_polynomial_upper
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
q :
ℝ
hq :
0 < q
hqhalf :
q < 1 / 2
conclusion
∃ C : ℝ,
0 < C
Proof (Lean source)
lemma unrestricted_fixed_q_polynomial_upper (q : ℝ)
(hq : 0 < q) (hqhalf : q < 1 / 2) :
∃ C : ℝ, 0 < C ∧ ∀ (n d : ℕ), 1 ≤ n → 1 ≤ d →
unrestrictedMinimaxRisk n d q ≤ C * fixedQRate n d := by
obtain ⟨C, hC, hupper⟩ := unrestricted_fixed_q_frontier_upper
let K := ((1 - log q) / q) ^ 2
have hK : 0 < K := by
dsimp [K]
have hlog : log q ≤ 0 :=
Real.log_nonpos hq.le (by linarith)
exact pow_pos (div_pos (by linarith) hq) _
refine ⟨C * K, mul_pos hC hK, ?_⟩
intro n d hn hd
calc
unrestrictedMinimaxRisk n d q ≤ C * frontierRate n d q :=
hupper n d q hn hd hq hqhalf
_ ≤ C * (K * fixedQRate n d) :=
mul_le_mul_of_nonneg_left
(frontierRate_le_fixedQRate_mul n d q hn hq (by linarith)) hC.le
_ = (C * K) * fixedQRate n d := by ring
theorem zengMinimaxRisk_le_unrestrictedMinimaxRisk
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hq :
0 < q
hqhalf :
q < 1 / 2
conclusion
zengMinimaxRisk n d q ≤ unrestrictedMinimaxRisk n d q
Proof (Lean source)
lemma zengMinimaxRisk_le_unrestrictedMinimaxRisk (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hq : 0 < q) (hqhalf : q < 1 / 2) :
zengMinimaxRisk n d q ≤ unrestrictedMinimaxRisk n d q := by
let φ : {P : ZengLaw d // P ∈ zengDiscreteClass d q} →
{P : FullLaw d // UnrestrictedArrivalModelClass n d q P} := fun P =>
⟨(synthetic_randomization_kernel hn hq hqhalf P.1 P.2).choose,
(synthetic_randomization_kernel hn hq hqhalf P.1 P.2).choose_spec.1⟩
let T₀ : Estimator n d :=
Estimator.ofMap (fun _ => 0) ⟨measurable_const, by intro s; norm_num⟩
letI : Nonempty (Estimator n d) := ⟨T₀⟩
have hr : ∀ (T : ZengEstimator n d)
(P : {P : ZengLaw d // P ∈ zengDiscreteClass d q}),
0 ≤ zengSquaredRisk T P.1 := by
intro T P
exact (zeng_squaredRisk_bounds q T P.1 P.2).1
unfold zengMinimaxRisk unrestrictedMinimaxRisk
apply minimaxValue_le_minimaxValue
(bddBelow_range_worstCaseRisk hr)
intro T
refine ⟨syntheticPullbackEstimator T, ?_⟩
change worstCaseRiskReal
(fun (_ : Unit) (P : {P : ZengLaw d // P ∈ zengDiscreteClass d q}) =>
zengSquaredRisk (syntheticPullbackEstimator T) P.1) () ≤
worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // UnrestrictedArrivalModelClass n d q P}) =>
squaredRisk T P.1) ()
apply worstCaseRisk_mono_class_of_nonneg φ
· refine ⟨4, ?_⟩
rintro r ⟨P, rfl⟩
exact (unrestricted_squaredRisk_bounds T P.1).2
· intro P
exact (unrestricted_squaredRisk_bounds T P.1).1
· intro P
have hspec :=
(synthetic_randomization_kernel hn hq hqhalf P.1 P.2).choose_spec
exact le_of_eq (syntheticPullbackEstimator_risk_eq T P.1 (φ P).1
hspec.2.2.1 hspec.2.2.2)
theorem unrestricted_fixed_q_minimax_frontier
Given the cited lower-bound gate, the unrestricted fixed-positivity minimax risk matches the frontier.
Formal statement
hypotheses
conclusion
∃ c C : ℝ,
0 < c
c ≤ C
conclusion 1
n d :
ℕ
1 ≤ n
1 ≤ d
c * fixedQRate n d ≤ unrestrictedMinimaxRisk n d q
unrestrictedMinimaxRisk n d q ≤ C * fixedQRate n d
c * frontierRate n d q ≤ unrestrictedMinimaxRisk n d q
unrestrictedMinimaxRisk n d q ≤ C * frontierRate n d q
Proof (Lean source)
theorem unrestricted_fixed_q_minimax_frontier
(hZeng_of_gate : ZengTheoremTwoLower) :
∀ q : ℝ, 0 < q → q < 1 / 2 →
-- @realizes \(\underline c(q)\)(fixed-q lower constant)
∃ c C : ℝ, 0 < c ∧ c ≤ C ∧ -- @realizes \(\overline C(q)\)(fixed-q upper constant)
∀ (n d : ℕ), 1 ≤ n → 1 ≤ d →
c * fixedQRate n d ≤ unrestrictedMinimaxRisk n d q ∧
unrestrictedMinimaxRisk n d q ≤ C * fixedQRate n d ∧
c * frontierRate n d q ≤ unrestrictedMinimaxRisk n d q ∧
unrestrictedMinimaxRisk n d q ≤ C * frontierRate n d q := by
intro q hq hqhalf
obtain ⟨C₁, hC₁, hupperFront⟩ := unrestricted_fixed_q_frontier_upper
obtain ⟨C₂, hC₂, hupperPoly⟩ :=
unrestricted_fixed_q_polynomial_upper q hq hqhalf
have hLower : ∃ c : ℝ, 0 < c ∧
∀ (n d : ℕ), 1 ≤ n → 1 ≤ d →
c * frontierRate n d q ≤ unrestrictedMinimaxRisk n d q := by
obtain ⟨_, _, _, hZfixed⟩ :=
zeng_fixed_positivity_polynomial_frontier hZeng_of_gate
obtain ⟨cε, _, hcε, _, hzlower⟩ := hZfixed q hq hqhalf
let K := ((1 - log q) / q) ^ 2
have hK : 0 < K := by
dsimp [K]
have hlog : log q ≤ 0 :=
Real.log_nonpos hq.le (by linarith)
exact pow_pos (div_pos (by linarith) hq) _
refine ⟨cε / K, div_pos hcε hK, ?_⟩
intro n d hn hd
calc
(cε / K) * frontierRate n d q ≤
(cε / K) * (K * fixedQRate n d) := by
exact mul_le_mul_of_nonneg_left
(frontierRate_le_fixedQRate_mul n d q hn hq (by linarith))
(div_nonneg hcε.le hK.le)
_ = cε * fixedQRate n d := by field_simp
_ ≤ zengMinimaxRisk n d q := (hzlower n d hn hd).1
_ ≤ unrestrictedMinimaxRisk n d q :=
zengMinimaxRisk_le_unrestrictedMinimaxRisk n d q hn hq hqhalf
obtain ⟨c, hc, hlower⟩ := hLower
let C := max C₁ C₂ + c
have hC : 0 < C := by
dsimp [C]
linarith [le_max_left C₁ C₂]
refine ⟨c, C, hc, by dsimp [C]; linarith [le_max_left C₁ C₂], ?_⟩
intro n d hn hd
have hfixed : 0 ≤ fixedQRate n d := by
unfold fixedQRate
exact le_min (by norm_num) (by positivity)
have hfront : 0 ≤ frontierRate n d q := by
unfold frontierRate
have hN : 0 ≤ effectiveSize n q := by
unfold effectiveSize
positivity
exact le_min (by norm_num)
(add_nonneg (inv_nonneg.mpr hN) (sq_nonneg _))
have hC₁ : C₁ ≤ C := by dsimp [C]; linarith [le_max_left C₁ C₂]
have hC₂ : C₂ ≤ C := by dsimp [C]; linarith [le_max_right C₁ C₂]
refine ⟨?_, ?_, hlower n d hn hd, ?_⟩
· exact (mul_le_mul_of_nonneg_left
(fixedQRate_le_frontierRate n d q hn hq (by linarith)) hc.le).trans
(hlower n d hn hd)
· exact (hupperPoly n d hn hd).trans
(mul_le_mul_of_nonneg_right hC₂ hfixed)
· exact (hupperFront n d q hn hd hq hqhalf).trans
(mul_le_mul_of_nonneg_right hC₁ hfront)
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.TUnrestrictedLargeAlphabetDeficitLower 5 declarations The uniform large-alphabet point-risk lower bound for all bounded estimator kernels.
The uniform large-alphabet point-risk lower bound for all bounded estimator kernels.
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
q :
ℝ
hq :
0 < q
hq1 :
q ≤ 1
conclusion
conclusion 1
1 / 2 - q / (1 + q) = (1 - q) / (2 * (1 + q))
conclusion 2
(1 - q) / 4 ≤ 1 / 2 - q / (1 + q)
conclusion 3
1 / 2 - q / (1 + q) ≤ (1 - q) / 2
Proof (Lean source)
-- @node: largeAlphabet_center_separation
lemma largeAlphabet_center_separation (q : ℝ) (hq : 0 < q) (hq1 : q ≤ 1) :
1 / 2 - q / (1 + q) = (1 - q) / (2 * (1 + q)) ∧
(1 - q) / 4 ≤ 1 / 2 - q / (1 + q) ∧
1 / 2 - q / (1 + q) ≤ (1 - q) / 2 := by
have hp : 0 < 1 + q := by linarith
have heq : 1 / 2 - q / (1 + q) = (1 - q) / (2 * (1 + q)) := by
field_simp
ring
refine ⟨heq, ?_, ?_⟩
· rw [heq]
apply (le_div_iff₀ (by positivity : 0 < 2 * (1 + q))).mpr
nlinarith
· rw [heq]
apply (div_le_iff₀ (by positivity : 0 < 2 * (1 + q))).mpr
nlinarith
theorem largeAlphabet_prior_bad_probability Lemma largeAlphabet_prior_bad_probability in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q h :
ℝ
hn :
1 ≤ n
hd :
1024 * n ^ 2 ≤ d
hq :
q ≤ 1
hregime :
(n : ℝ)⁻¹ ≤ (1 - q) ^ 2
hsep :
(1 - q) / 4 ≤ h
conclusion
Proof (Lean source)
-- @node: largeAlphabet_prior_bad_probability
lemma largeAlphabet_prior_bad_probability (n d : ℕ) (q h : ℝ)
(hn : 1 ≤ n) (hd : 1024 * n ^ 2 ≤ d) (hq : q ≤ 1)
(hregime : (n : ℝ)⁻¹ ≤ (1 - q) ^ 2)
(hsep : (1 - q) / 4 ≤ h) :
(binaryCellPrior d).real {ξ | h / 4 < |cellPriorTarget ξ - 1 / 2|} ≤
1 / 16 := by
have hnpos : 0 < (n : ℝ) := by exact_mod_cast hn
have hnreal : (1 : ℝ) ≤ n := by exact_mod_cast hn
have hdreal : 1024 * (n : ℝ) ^ 2 ≤ d := by exact_mod_cast hd
have hdpos : 0 < (d : ℝ) := by nlinarith
have hdNat : 1 ≤ d := by exact_mod_cast (show (1 : ℝ) ≤ d by nlinarith)
have hdelta : 0 < 1 - q := by
have hinv : 0 < (n : ℝ)⁻¹ := inv_pos.mpr hnpos
nlinarith
have hh : 0 < h := by linarith
have hsquare : (1 - q) ^ 2 ≤ 16 * h ^ 2 := by nlinarith
have hndelta : 1 ≤ (n : ℝ) * (1 - q) ^ 2 := by
have hm := mul_le_mul_of_nonneg_left hregime hnpos.le
simpa [ne_of_gt hnpos] using hm
have hbudget : 64 ≤ (d : ℝ) * h ^ 2 := by
have h₁ := mul_le_mul_of_nonneg_left hsquare (by positivity : 0 ≤ (n : ℝ) ^ 2)
have h₂ := mul_le_mul_of_nonneg_right hdreal (sq_nonneg h)
have h₃ := mul_le_mul_of_nonneg_left hndelta hnpos.le
nlinarith
refine (binaryCellPrior_target_concentration d hdNat (h / 4) (by positivity)).trans ?_
apply (div_le_iff₀ (by positivity : 0 < 4 * (d : ℝ) * (h / 4) ^ 2)).mpr
nlinarith
theorem largeAlphabet_center_prior_concentration Lemma largeAlphabet_center_prior_concentration in the paper ↗
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1024 * n ^ 2 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
hregime :
(n : ℝ)⁻¹ ≤ (1 - q) ^ 2
conclusion
≤ 1 / 16
Proof (Lean source)
-- @node: largeAlphabet_center_prior_concentration
lemma largeAlphabet_center_prior_concentration (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hd : 1024 * n ^ 2 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1)
(hregime : (n : ℝ)⁻¹ ≤ (1 - q) ^ 2) :
(binaryCellPrior d).real
{ξ | (1 / 2 - q / (1 + q)) / 4 < |cellPriorTarget ξ - 1 / 2|} ≤
1 / 16 := by
exact largeAlphabet_prior_bad_probability n d q (1 / 2 - q / (1 + q))
hn hd hq1 hregime (largeAlphabet_center_separation q hq hq1).2.1
theorem largeAlphabet_deficit_minimax_lower
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1024 * n ^ 2 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
hregime :
(n : ℝ)⁻¹ ≤ (1 - q) ^ 2
conclusion
(1 - q) ^ 2 / 1024 ≤ unrestrictedMinimaxRisk n d q
Proof (Lean source)
-- @node: largeAlphabet_deficit_minimax_lower
lemma largeAlphabet_deficit_minimax_lower
(n d : ℕ) (q : ℝ) (hn : 1 ≤ n) (hd : 1024 * n ^ 2 ≤ d)
(hq : 0 < q) (hq1 : q ≤ 1)
(hregime : (n : ℝ)⁻¹ ≤ (1 - q) ^ 2) :
(1 - q) ^ 2 / 1024 ≤ unrestrictedMinimaxRisk n d q := by
have hdpos : 1 ≤ d := by nlinarith
let T₀ : Estimator n d :=
Estimator.ofMap (fun _ => 0) ⟨measurable_const, by intro s; norm_num⟩
let : Nonempty (Estimator n d) := ⟨T₀⟩
unfold unrestrictedMinimaxRisk
apply Causalean.Stat.le_minimaxValue
intro T
let h : ℝ := 1 / 2 - q / (1 + q)
have hsep := (largeAlphabet_center_separation q hq hq1).2.1
have hh : 0 ≤ h := by dsimp [h]; linarith
let P := largeAlphabetComparisonLaw d q hdpos hq hq1
let := P.2
let : IsProbabilityMeasure (P.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
let : IsProbabilityMeasure (sampleLaw n P) := by
unfold sampleLaw
infer_instance
let E : Set ((Fin n → ObsRecord d) × ℝ) := {z | z.2 ≤ 1 / 2 - h / 2}
have hE : MeasurableSet E := by measurability
have havg := largeAlphabet_prior_test_risk n d q hdpos hq hq1 T (1 / 2) h hh
have hcomp := largeAlphabet_comparison_test_risk (sampleLaw n P) T (1 / 2) h hh
have htarget : 1 / 2 - h = ate P := by
rw [show ate P = q / (1 + q) from
largeAlphabetComparisonLaw_ate d q hdpos hq hq1]
dsimp [h]
ring
rw [htarget] at hcomp
change (h ^ 2 / 16) * (sampleLaw n P ⊗ₘ T.1).real Eᶜ ≤ squaredRisk T P at hcomp
have htest := largeAlphabet_actual_collision_testing n d q hn hd hq hq1 T E hE
have hbad := largeAlphabet_center_prior_concentration n d q hn hd hq hq1 hregime
have hscaled := mul_le_mul_of_nonneg_left htest
(by positivity : 0 ≤ h ^ 2 / 16)
have hpenalty := mul_le_mul_of_nonneg_left hbad
(by positivity : 0 ≤ h ^ 2 / 16)
have hbayes : h ^ 2 / 32 ≤
(∫ ξ, squaredRisk T (largeAlphabetLaw q ξ hdpos hq hq1) ∂binaryCellPrior d) +
squaredRisk T P := by
change (h ^ 2 / 16) * (1 - (1 / 8192 : ℝ)) ≤
(h ^ 2 / 16) *
((largeAlphabetSampleMixture n d q hdpos hq hq1 ⊗ₘ T.1).real E +
(sampleLaw n P ⊗ₘ T.1).real Eᶜ) at hscaled
change (h ^ 2 / 16) * (binaryCellPrior d).real
{ξ | h / 4 < |cellPriorTarget ξ - 1 / 2|} ≤ (h ^ 2 / 16) * (1 / 16) at hpenalty
change (h ^ 2 / 16) *
(largeAlphabetSampleMixture n d q hdpos hq hq1 ⊗ₘ T.1).real E ≤ _ at havg
nlinarith only [hscaled, havg, hcomp, hpenalty, sq_nonneg h]
have hbdd : BddAbove (range (fun P : {P : FullLaw d //
UnrestrictedArrivalModelClass n d q P} => squaredRisk T P.1)) := by
refine ⟨4, ?_⟩
rintro _ ⟨P, rfl⟩
exact (unrestricted_squaredRisk_bounds T P.1).2
have hb₀ := largeAlphabet_prior_squaredRisk_le_worst n d q hn hdpos hq hq1 T
have hb₁ := le_worstCaseRisk
(risk := fun (T : Estimator n d) (P : {P : FullLaw d //
UnrestrictedArrivalModelClass n d q P}) => squaredRisk T P.1)
(e := T) hbdd ⟨P, largeAlphabetComparisonLaw_model n d q hn hdpos hq hq1⟩
have hsquare : (1 - q) ^ 2 ≤ 16 * h ^ 2 := by
change (1 - q) / 4 ≤ h at hsep
nlinarith [sq_nonneg (h - (1 - q) / 4)]
nlinarith only [hbayes, hb₀, hb₁, hsquare]
theorem unrestricted_large_alphabet_deficit_lower
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1024 * n ^ 2 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
conclusion
conclusion 1
max (1 / (256 * (n : ℝ))) ((1 - q) ^ 2 / 1024) ≤ unrestrictedMinimaxRisk n d q
conclusion 2
unrestrictedMinimaxRisk n d q ≤ 4 / (n : ℝ) + (1 - q) ^ 2
conclusion 3
(1 / 2048 : ℝ) * ((n : ℝ)⁻¹ + (1 - q) ^ 2) ≤ unrestrictedMinimaxRisk n d q
conclusion 4
unrestrictedMinimaxRisk n d q ≤ 4 * ((n : ℝ)⁻¹ + (1 - q) ^ 2)
Proof (Lean source)
theorem unrestricted_large_alphabet_deficit_lower
(n d : ℕ) (q : ℝ) (hn : 1 ≤ n) (hd : 1024 * n ^ 2 ≤ d)
(hq : 0 < q) (hq1 : q ≤ 1) :
max (1 / (256 * (n : ℝ))) ((1 - q) ^ 2 / 1024) ≤
unrestrictedMinimaxRisk n d q ∧
unrestrictedMinimaxRisk n d q ≤ 4 / (n : ℝ) + (1 - q) ^ 2 ∧
(1 / 2048 : ℝ) * ((n : ℝ)⁻¹ + (1 - q) ^ 2) ≤ unrestrictedMinimaxRisk n d q ∧
unrestrictedMinimaxRisk n d q ≤ 4 * ((n : ℝ)⁻¹ + (1 - q) ^ 2) := by
have hdpos : 1 ≤ d := by nlinarith
obtain ⟨C, hC, _, henvelope⟩ := unrestricted_near_complete_arrival_envelope
obtain ⟨hfloor, hupper, _⟩ := henvelope n d q hn hdpos hq hq1
have hupper' : unrestrictedMinimaxRisk n d q ≤ 4 / (n : ℝ) + (1 - q) ^ 2 :=
hupper.trans ((min_le_right _ _).trans (min_le_right _ _))
have hdeficit : (1 - q) ^ 2 / 1024 ≤ unrestrictedMinimaxRisk n d q := by
by_cases hsmall : (1 - q) ^ 2 ≤ (n : ℝ)⁻¹
· refine le_trans ?_ hfloor
have hnpos : 0 < (n : ℝ) := by exact_mod_cast hn
have heq : 1 / (256 * (n : ℝ)) = (n : ℝ)⁻¹ / 256 := by
field_simp
rw [heq]
nlinarith [inv_nonneg.mpr hnpos.le, sq_nonneg (1 - q)]
· have hregime : (n : ℝ)⁻¹ ≤ (1 - q) ^ 2 := (lt_of_not_ge hsmall).le
exact largeAlphabet_deficit_minimax_lower n d q hn hd hq hq1 hregime
have hmax := max_le hfloor hdeficit
refine ⟨hmax, hupper', ?_, ?_⟩
· have hnpos : 0 < (n : ℝ) := by exact_mod_cast hn
have heq : 1 / (256 * (n : ℝ)) = (n : ℝ)⁻¹ / 256 := by field_simp
rw [heq] at hfloor
linarith [inv_nonneg.mpr hnpos.le]
· have heq : 4 / (n : ℝ) = 4 * (n : ℝ)⁻¹ := by rw [div_eq_mul_inv]
rw [heq] at hupper'
nlinarith [sq_nonneg (1 - q)]
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.TUnrestrictedMixedCountUpper 1 declarations TUnrestrictedMixedCountUpper for the finite rare-arrival experiment.
TUnrestrictedMixedCountUpper for the finite rare-arrival experiment.
theorem unrestricted_mixed_count_upper
the stated mathematical conclusion holds.
Formal statement
conclusion
∃ C : ℝ,
0 < C
conclusion 1
Proof (Lean source)
theorem unrestricted_mixed_count_upper :
∃ C : ℝ, 0 < C ∧ -- @realizes \(\overline C\)(universal upper constant)
∀ (n d : ℕ) (q : ℝ) (P : FullLaw d),
UnrestrictedArrivalModelClass n d q P →
IIDSampling n (sampleLaw n P)
(fun (i : Fin n) (s : Fin n → ObsRecord d) => s i) P →
deterministicRisk (mixedCountEstimator n d q) P ≤ riskEnvelope n d q ∧
riskEnvelope n d q ≤ C * frontierRate n d q ∧
deterministicRisk (mixedCountEstimator n d q) P ≤ auxiliaryMixedRisk n q P := by
obtain ⟨C, hC, hcertificate⟩ := unrestricted_uniform_mixed_count_certificate
refine ⟨C, hC, ?_⟩
intro n d q P hP hiid
obtain ⟨hrisk, hrate⟩ := hcertificate n d q P hP hiid
have hident := unrestricted_cell_identification P hP
rw [← hident] at hrisk
exact ⟨hrisk, hrate, mixed_count_derandomization n d q P⟩
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.TUnrestrictedNearCompleteArrivalEnvelope 4 declarations The all-kernel point-risk envelope at every arrival floor.
The all-kernel point-risk envelope at every arrival floor. The same universal constant also certifies the mixed-count upper theorem. No rare-arrival slice is used.
theorem unrestricted_arrival_floor_mono
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
UnrestrictedArrivalModelClass n d q P
Proof (Lean source)
-- @node: unrestricted_arrival_floor_mono
lemma unrestricted_arrival_floor_mono {n d : ℕ} {q q' : ℝ} {P : FullLaw d}
(hP : UnrestrictedArrivalModelClass n d q' P) (hq : 0 < q) (hqq' : q ≤ q') :
UnrestrictedArrivalModelClass n d q P := by
refine ⟨hP.n_pos, hP.d_pos, hq, hqq'.trans hP.q_le_one,
hP.randomized, hP.balanced, hP.surrogate, hP.outcome, hP.mar, ?_⟩
intro j hj
exact (mul_le_mul_of_nonneg_right hqq' measureReal_nonneg).trans (hP.arrival j hj)
theorem unrestricted_deterministic_upper
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q b :
ℝ
hf :
Measurable f ∧ ∀ s, f s ∈ Icc (-1 : ℝ) 1
hb :
0 ≤ b
hbound :
∀ P : FullLaw d
if
UnrestrictedArrivalModelClass n d q P
then
deterministicRisk f P ≤ b
conclusion
conclusion 1
unrestrictedMinimaxRisk n d q ≤ b
conclusion 2
worstCaseRiskReal (fun (_ : Unit) (P : {P : FullLaw d // UnrestrictedArrivalModelClass n d q P}) => deterministicRisk f P.1) ()
≤ b
Proof (Lean source)
-- @node: unrestricted_deterministic_upper
lemma unrestricted_deterministic_upper {n d : ℕ} {q b : ℝ}
(f : (Fin n → ObsRecord d) → ℝ)
(hf : Measurable f ∧ ∀ s, f s ∈ Icc (-1 : ℝ) 1) (hb : 0 ≤ b)
(hbound : ∀ P : FullLaw d, UnrestrictedArrivalModelClass n d q P →
deterministicRisk f P ≤ b) :
unrestrictedMinimaxRisk n d q ≤ b ∧
Causalean.Stat.worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // UnrestrictedArrivalModelClass n d q P}) =>
deterministicRisk f P.1) () ≤ b := by
have hw : Causalean.Stat.worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // UnrestrictedArrivalModelClass n d q P}) =>
deterministicRisk f P.1) () ≤ b := by
unfold Causalean.Stat.worstCaseRiskReal
by_cases hne : Nonempty {P : FullLaw d // UnrestrictedArrivalModelClass n d q P}
· letI := hne
exact ciSup_le (fun P => hbound P.1 P.2)
· haveI := not_nonempty_iff.mp hne
simpa using hb
refine ⟨le_trans ?_ hw, hw⟩
let T : Estimator n d := Estimator.ofMap f hf
have h := minimaxValue_le_worstCaseRisk_of_nonneg
(risk := fun (T : Estimator n d)
(P : {P : FullLaw d // UnrestrictedArrivalModelClass n d q P}) => squaredRisk T P.1)
(fun T P => (unrestricted_squaredRisk_bounds T P.1).1) T
simpa [unrestrictedMinimaxRisk, Causalean.Stat.worstCaseRiskReal,
squaredRisk, deterministicRisk, T, Estimator.ofMap, Kernel.deterministic_apply] using h
theorem unrestricted_complete_submodel_floor
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hd :
1 ≤ d
hq :
0 < q
hq1 :
q ≤ 1
conclusion
1 / (256 * (n : ℝ)) ≤ unrestrictedMinimaxRisk n d q
Proof (Lean source)
-- @node: unrestricted_complete_submodel_floor
lemma unrestricted_complete_submodel_floor {n d : ℕ} {q : ℝ}
(hn : 1 ≤ n) (hd : 1 ≤ d) (hq : 0 < q) (hq1 : q ≤ 1) :
1 / (256 * (n : ℝ)) ≤ unrestrictedMinimaxRisk n d q := by
have hfloor := (complete_arrival_frontier_direct n d hn hd).1
refine hfloor.trans ?_
let T₀ : Estimator n d :=
Estimator.ofMap (fun _ => 0) ⟨measurable_const, by intro s; norm_num⟩
letI : Nonempty (Estimator n d) := ⟨T₀⟩
apply minimaxValue_mono_class_of_nonneg
(risk := fun (T : Estimator n d)
(P : {P : FullLaw d // UnrestrictedArrivalModelClass n d 1 P}) => squaredRisk T P.1)
(risk' := fun (T : Estimator n d)
(P : {P : FullLaw d // UnrestrictedArrivalModelClass n d q P}) => squaredRisk T P.1)
(fun P => ⟨P.1, unrestricted_arrival_floor_mono P.2 hq hq1⟩)
· exact fun T P => (unrestricted_squaredRisk_bounds T P.1).1
· exact fun T P => (unrestricted_squaredRisk_bounds T P.1).1
· intro T
refine ⟨4, ?_⟩
rintro _ ⟨P, rfl⟩
exact (unrestricted_squaredRisk_bounds T P.1).2
· exact fun _ _ => le_rfl
theorem unrestricted_near_complete_arrival_envelope
the stated mathematical conclusion holds.
Formal statement
conclusion
∃ C : ℝ,
0 < C
conclusion 1
conclusion 2
n d :
ℕ
q :
ℝ
1 ≤ n
1 ≤ d
0 < q
q ≤ 1
1 / (256 * (n : ℝ)) ≤ unrestrictedMinimaxRisk n d q
unrestrictedMinimaxRisk n d q
worstCaseRiskReal (fun (_ : Unit) (P : {P : FullLaw d // UnrestrictedArrivalModelClass n d q P}) => deterministicRisk (completeArrivalHTEstimator n d) P.1) ()
≤ 4 / (n : ℝ) + (1 - q) ^ 2
Proof (Lean source)
theorem unrestricted_near_complete_arrival_envelope :
∃ C : ℝ, 0 < C ∧ -- @realizes \(\overline C\)(shared universal upper constant)
(∀ (n d : ℕ) (q : ℝ) (P : FullLaw d),
UnrestrictedArrivalModelClass n d q P →
IIDSampling n (sampleLaw n P)
(fun (i : Fin n) (s : Fin n → ObsRecord d) => s i) P →
deterministicRisk (mixedCountEstimator n d q) P ≤ riskEnvelope n d q ∧
riskEnvelope n d q ≤ C * frontierRate n d q ∧
deterministicRisk (mixedCountEstimator n d q) P ≤ auxiliaryMixedRisk n q P) ∧
∀ (n d : ℕ) (q : ℝ), 1 ≤ n → 1 ≤ d → 0 < q → q ≤ 1 →
1 / (256 * (n : ℝ)) ≤ unrestrictedMinimaxRisk n d q ∧
unrestrictedMinimaxRisk n d q ≤
min 1 (min (C * frontierRate n d q) (4 / (n : ℝ) + (1 - q) ^ 2)) ∧
worstCaseRiskReal
(fun (_ : Unit) (P : {P : FullLaw d // UnrestrictedArrivalModelClass n d q P}) =>
deterministicRisk (completeArrivalHTEstimator n d) P.1) () ≤
4 / (n : ℝ) + (1 - q) ^ 2 := by
obtain ⟨C, hC, hmixed⟩ := unrestricted_mixed_count_upper
refine ⟨C, hC, hmixed, ?_⟩
intro n d q hn hd hq hq1
have hzero : unrestrictedMinimaxRisk n d q ≤ 1 := by
apply (unrestricted_deterministic_upper (fun _ => 0)
⟨measurable_const, by intro s; norm_num⟩ (by norm_num) ?_).1
intro P hP
letI : IsProbabilityMeasure P.1 := P.2
letI : IsProbabilityMeasure (P.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop)
letI : IsProbabilityMeasure (sampleLaw n P) := by unfold sampleLaw; infer_instance
have ht := ate_mem_unit_interval P
have hs : (ate P) ^ 2 ≤ 1 := by nlinarith [ht.1, ht.2]
simpa [deterministicRisk] using hs
have hrate : 0 ≤ C * frontierRate n d q := by
apply mul_nonneg hC.le
unfold frontierRate effectiveSize
exact le_min (by norm_num) (add_nonneg (inv_nonneg.mpr (mul_nonneg (Nat.cast_nonneg _) hq.le))
(sq_nonneg _))
have hmix : unrestrictedMinimaxRisk n d q ≤ C * frontierRate n d q := by
apply (unrestricted_deterministic_upper (mixedCountEstimator n d q)
(mixed_count_estimator_regular n d q) hrate ?_).1
intro P hP
obtain ⟨hrisk, henv, _⟩ := hmixed n d q P hP (sampleLaw_iid n P)
exact hrisk.trans henv
have hht := unrestricted_deterministic_upper (q := q) (completeArrivalHTEstimator n d)
(complete_arrival_ht_estimator_regular n d)
(add_nonneg (div_nonneg (by norm_num) (Nat.cast_nonneg _)) (sq_nonneg _))
(fun P hP => observed_contrast_risk_le P hP)
exact ⟨unrestricted_complete_submodel_floor hn hd hq hq1,
le_min hzero (le_min hmix hht.1), hht.2⟩
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.TUnrestrictedUniformMixedCountCertificate 1 declarations TUnrestrictedUniformMixedCountCertificate for the finite rare-arrival experiment.
TUnrestrictedUniformMixedCountCertificate for the finite rare-arrival experiment.
theorem unrestricted_uniform_mixed_count_certificate
the stated mathematical conclusion holds.
Formal statement
conclusion
∃ C : ℝ,
0 < C
conclusion 1
(∫ s, (mixedCountEstimator n d q s - cellFunctional P) ^ 2 ∂(sampleLaw n P))
≤ riskEnvelope n d q
riskEnvelope n d q ≤ C * frontierRate n d q
Proof (Lean source)
theorem unrestricted_uniform_mixed_count_certificate :
∃ C : ℝ, 0 < C ∧ -- @realizes \(\overline C\)(universal upper constant)
∀ (n d : ℕ) (q : ℝ) (P : FullLaw d),
UnrestrictedArrivalModelClass n d q P →
IIDSampling n (sampleLaw n P)
(fun (i : Fin n) (s : Fin n → ObsRecord d) => s i) P →
(∫ s, (mixedCountEstimator n d q s - cellFunctional P) ^ 2
∂(sampleLaw n P)) ≤ riskEnvelope n d q ∧
riskEnvelope n d q ≤ C * frontierRate n d q := by
obtain ⟨C, hC, hrate⟩ := riskEnvelope_le_frontierRate
refine ⟨C, hC, ?_⟩
intro n d q P hP _
exact ⟨unrestricted_auxiliary_risk_envelope n d q P hP,
hrate n d q hP.n_pos hP.d_pos hP.q_pos hP.q_le_one⟩
CausalSmith.Stat.STAT_MarRareqLogfrontier_Research.TZengFixedPositivityPolynomialFrontier 13 declarations TZengFixedPositivityPolynomialFrontier for the finite rare-arrival experiment.
TZengFixedPositivityPolynomialFrontier for the finite rare-arrival experiment.
theorem zeng_zero_control_risk_le_minimax
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma zeng_zero_control_risk_le_minimax (n d : ℕ) (ε : ℝ) :
zengZeroControlRisk n d ε ≤ zengMinimaxRisk n d ε := by
let T₀ : ZengEstimator n d :=
⟨Kernel.const _ (Measure.dirac 0), inferInstance, by
intro s
simp [Kernel.const_apply]⟩
haveI : Nonempty (ZengEstimator n d) := ⟨T₀⟩
let φ : {P : ZengLaw d // P ∈ zengZeroControlClass d ε} →
{P : ZengLaw d // P ∈ zengDiscreteClass d ε} :=
fun P => ⟨P.1, P.2.1⟩
unfold zengZeroControlRisk zengMinimaxRisk
apply minimaxValue_mono_class_of_nonneg φ
· intro T P
exact (zeng_squaredRisk_bounds ε T P.1 P.2.1).1
· intro T P
exact (zeng_squaredRisk_bounds ε T P.1 P.2).1
· intro T
refine ⟨4, ?_⟩
rintro y ⟨P, rfl⟩
exact (zeng_squaredRisk_bounds ε T P.1 P.2).2
· intro T P
exact le_refl _
theorem zeng_parametric_minimax_lower
the stated mathematical conclusion holds.
Formal statement
conclusion
∃ c : ℝ,
0 < c
Proof (Lean source)
lemma zeng_parametric_minimax_lower :
∃ c : ℝ, 0 < c ∧
∀ (n d : ℕ) (ε : ℝ), 1 ≤ n → 1 ≤ d →
0 < ε → ε < 1 / 2 →
c / n ≤ zengMinimaxRisk n d ε := by
obtain ⟨c, hc, hparam⟩ := zeng_parametric_lower
refine ⟨c, hc, ?_⟩
intro n d ε hn hd hε0 hε1
exact (hparam n d ε hn hd ⟨hε0, hε1⟩).trans
(zeng_zero_control_risk_le_minimax n d ε)
theorem thetaPoly_pointwise_jensen
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(thetaPoly n d ε s - θ) ^ 2
≤ (∑ u : Fin n → Bool, (mixedCountEstimator n d ε (fun i => syntheticRecord (s i) (u i)) - θ) ^ 2) / (2 : ℝ) ^ n
Proof (Lean source)
lemma thetaPoly_pointwise_jensen (n d : ℕ) (ε θ : ℝ)
(s : Fin n → ZengRecord d) :
(thetaPoly n d ε s - θ) ^ 2 ≤
(∑ u : Fin n → Bool,
(mixedCountEstimator n d ε
(fun i => syntheticRecord (s i) (u i)) - θ) ^ 2) / (2 : ℝ) ^ n := by
let f : (Fin n → Bool) → ℝ := fun u =>
mixedCountEstimator n d ε (fun i => syntheticRecord (s i) (u i)) - θ
have hcard : (Finset.univ : Finset (Fin n → Bool)).card = 2 ^ n := by
simp
have hcardR : (0 : ℝ) < (2 : ℝ) ^ n := by positivity
have hsum : (∑ u : Fin n → Bool, f u) =
(2 : ℝ) ^ n * (thetaPoly n d ε s - θ) := by
simp only [f, Finset.sum_sub_distrib]
simp [thetaPoly, hcard]
field_simp
have hcs := sq_sum_le_card_mul_sum_sq (s := (Finset.univ : Finset (Fin n → Bool)))
(f := f)
rw [hcard, Nat.cast_pow, Nat.cast_ofNat, hsum] at hcs
change (thetaPoly n d ε s - θ) ^ 2 ≤
(∑ u : Fin n → Bool, f u ^ 2) / (2 : ℝ) ^ n
apply (le_div_iff₀ hcardR).2
nlinarith [sq_nonneg ((2 : ℝ) ^ n)]
theorem thetaPoly_regular
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
Proof (Lean source)
lemma thetaPoly_regular (n d : ℕ) (ε : ℝ) :
Measurable (thetaPoly n d ε) ∧
∀ s, thetaPoly n d ε s ∈ Icc (-1 : ℝ) 1 := by
constructor
· exact measurable_of_finite _
· intro s
have hcard : (Finset.univ : Finset (Fin n → Bool)).card = 2 ^ n := by
simp
have hcardR : (0 : ℝ) < (2 : ℝ) ^ n := by positivity
have hlo : -((2 : ℝ) ^ n) ≤
∑ u : Fin n → Bool,
mixedCountEstimator n d ε (fun i => syntheticRecord (s i) (u i)) := by
calc
-((2 : ℝ) ^ n) =
∑ _u : Fin n → Bool, (-1 : ℝ) := by simp [hcard]
_ ≤ _ := Finset.sum_le_sum (fun u _ => (mixed_count_estimator_regular n d ε).2 _ |>.1)
have hhi : (∑ u : Fin n → Bool,
mixedCountEstimator n d ε (fun i => syntheticRecord (s i) (u i))) ≤
(2 : ℝ) ^ n := by
calc
_ ≤ ∑ _u : Fin n → Bool, (1 : ℝ) :=
Finset.sum_le_sum (fun u _ => (mixed_count_estimator_regular n d ε).2 _ |>.2)
_ = (2 : ℝ) ^ n := by simp [hcard]
change -1 ≤ (∑ u : Fin n → Bool,
mixedCountEstimator n d ε (fun i => syntheticRecord (s i) (u i))) /
(2 : ℝ) ^ n ∧
(∑ u : Fin n → Bool,
mixedCountEstimator n d ε (fun i => syntheticRecord (s i) (u i))) /
(2 : ℝ) ^ n ≤ 1
constructor
· exact (le_div_iff₀ hcardR).2 (by nlinarith)
· exact (div_le_iff₀ hcardR).2 (by nlinarith)
theorem thetaPoly_risk_le_synthetic_average
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
zengDeterministicRisk (thetaPoly n d ε) P
≤ ∫ s, (∑ u : Fin n → Bool, (mixedCountEstimator n d ε (fun i => syntheticRecord (s i) (u i)) - zengATE P) ^ 2) / (2 : ℝ) ^ n ∂(zengSampleLaw n P)
Proof (Lean source)
lemma thetaPoly_risk_le_synthetic_average (n d : ℕ) (ε : ℝ)
(P : ZengLaw d) :
zengDeterministicRisk (thetaPoly n d ε) P ≤
∫ s, (∑ u : Fin n → Bool,
(mixedCountEstimator n d ε
(fun i => syntheticRecord (s i) (u i)) - zengATE P) ^ 2) /
(2 : ℝ) ^ n ∂(zengSampleLaw n P) := by
letI : IsProbabilityMeasure P.1 := P.2
letI : IsProbabilityMeasure (zengSampleLaw n P) := by
unfold zengSampleLaw
infer_instance
unfold zengDeterministicRisk
have hint : Integrable
(fun s : Fin n → ZengRecord d =>
(∑ u : Fin n → Bool,
(mixedCountEstimator n d ε
(fun i => syntheticRecord (s i) (u i)) - zengATE P) ^ 2) /
(2 : ℝ) ^ n) (zengSampleLaw n P) := Integrable.of_finite
exact integral_mono_of_nonneg
(Filter.Eventually.of_forall (fun s => sq_nonneg _)) hint
(Filter.Eventually.of_forall (thetaPoly_pointwise_jensen n d ε (zengATE P)))
theorem zengMinimaxRisk_le_thetaPoly_worst
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
ε :
ℝ
conclusion
zengMinimaxRisk n d ε
≤ worstCaseRiskReal (fun (_ : Unit) (P : {P : ZengLaw d // P ∈ zengDiscreteClass d ε}) => zengDeterministicRisk (thetaPoly n d ε) P.1) ()
Proof (Lean source)
lemma zengMinimaxRisk_le_thetaPoly_worst (n d : ℕ) (ε : ℝ) :
zengMinimaxRisk n d ε ≤
worstCaseRiskReal
(fun (_ : Unit) (P : {P : ZengLaw d // P ∈ zengDiscreteClass d ε}) =>
zengDeterministicRisk (thetaPoly n d ε) P.1) () := by
have hregular := thetaPoly_regular n d ε
let T : ZengEstimator n d :=
⟨Kernel.deterministic (thetaPoly n d ε) hregular.1,
inferInstance, by
intro s
rw [Kernel.deterministic_apply, Measure.dirac_apply]
rw [Set.indicator_of_mem (hregular.2 s)]
rfl⟩
have hrisk : ∀ (T : ZengEstimator n d)
(P : {P : ZengLaw d // P ∈ zengDiscreteClass d ε}),
0 ≤ zengSquaredRisk T P.1 := by
intro T P
exact integral_nonneg (fun s => integral_nonneg (fun y => sq_nonneg _))
have hval : zengMinimaxRisk n d ε ≤
worstCaseRiskReal
(fun (_ : Unit) (P : {P : ZengLaw d // P ∈ zengDiscreteClass d ε}) =>
zengSquaredRisk T P.1) () := by
exact Causalean.Stat.minimaxValue_le_worstCaseRisk_of_nonneg hrisk T
have heq :
(fun (_ : Unit) (P : {P : ZengLaw d // P ∈ zengDiscreteClass d ε}) =>
zengSquaredRisk T P.1) =
(fun (_ : Unit) P => zengDeterministicRisk (thetaPoly n d ε) P.1) := by
funext _ P
simp [zengSquaredRisk, zengDeterministicRisk, T, Kernel.deterministic_apply]
rw [heq] at hval
exact hval
theorem synthetic_coin_average_integral
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
(∫ o, F o ∂(syntheticProductLaw s))
Proof (Lean source)
lemma synthetic_coin_average_integral (n d : ℕ)
(s : Fin n → ZengRecord d) (F : (Fin n → ObsRecord d) → ℝ) :
(∫ o, F o ∂(syntheticProductLaw s)) =
(∑ u : Fin n → Bool,
F (fun i => syntheticRecord (s i) (u i))) / (2 : ℝ) ^ n := by
let coinPi : Measure (Fin n → Bool) :=
Measure.pi (fun _ : Fin n => SyntheticCompletion.fairCoin)
letI : IsProbabilityMeasure coinPi := by
dsimp [coinPi]
infer_instance
have hmap : syntheticProductLaw s =
coinPi.map (fun u i => syntheticRecord (s i) (u i)) := by
simpa [coinPi, SyntheticCompletion.fairCoin] using
syntheticProductLaw_eq_coin_map s
rw [hmap, integral_map (measurable_of_finite _).aemeasurable
(measurable_of_finite _).aestronglyMeasurable]
rw [integral_fintype (Integrable.of_finite)]
have hcoin (u : Fin n → Bool) : coinPi.real {u} = 1 / (2 : ℝ) ^ n := by
have hsingle (b : Bool) : (SyntheticCompletion.fairCoin {b}).toReal =
(1 / 2 : ℝ) := by
cases b <;> simp [SyntheticCompletion.fairCoin]
change ((Measure.pi (fun _ : Fin n => SyntheticCompletion.fairCoin)) {u}).toReal = _
rw [Measure.pi_singleton]
rw [ENNReal.toReal_prod]
simp_rw [hsingle]
simp [Finset.prod_const]
simp_rw [hcoin]
simp only [smul_eq_mul, ← Finset.mul_sum, div_eq_mul_inv, one_mul]
ring
theorem thetaPoly_risk_le_completion_risk
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
conclusion
zengDeterministicRisk (thetaPoly n d ε) P
≤ deterministicRisk (mixedCountEstimator n d ε) Pstar
Proof (Lean source)
lemma thetaPoly_risk_le_completion_risk {n d : ℕ} {ε : ℝ}
(P : ZengLaw d) (Pstar : FullLaw d)
(hsample : (zengSampleLaw n P).bind syntheticProductLaw = sampleLaw n Pstar)
(htarget : ate Pstar = zengATE P) :
zengDeterministicRisk (thetaPoly n d ε) P ≤
deterministicRisk (mixedCountEstimator n d ε) Pstar := by
let F : (Fin n → ObsRecord d) → ℝ := fun o =>
(mixedCountEstimator n d ε o - zengATE P) ^ 2
have hmeas : Measurable (syntheticProductLaw (n := n) (d := d)) :=
measurable_of_finite _
have hFbind : Integrable F ((zengSampleLaw n P).bind syntheticProductLaw) := by
rw [hsample]
letI : IsProbabilityMeasure Pstar.1 := Pstar.2
letI : IsProbabilityMeasure (Pstar.1.map obs) :=
Measure.isProbabilityMeasure_map (by fun_prop : AEMeasurable obs Pstar.1)
unfold sampleLaw
exact Integrable.of_finite
have hinner (s : Fin n → ZengRecord d) :
(∫ o, F o ∂syntheticProductLaw s) =
(∑ u : Fin n → Bool,
(mixedCountEstimator n d ε
(fun i => syntheticRecord (s i) (u i)) - zengATE P) ^ 2) /
(2 : ℝ) ^ n := by
exact synthetic_coin_average_integral n d s F
calc
zengDeterministicRisk (thetaPoly n d ε) P ≤
∫ s, (∑ u : Fin n → Bool,
(mixedCountEstimator n d ε
(fun i => syntheticRecord (s i) (u i)) - zengATE P) ^ 2) /
(2 : ℝ) ^ n ∂(zengSampleLaw n P) :=
thetaPoly_risk_le_synthetic_average n d ε P
_ = ∫ o, F o ∂((zengSampleLaw n P).bind syntheticProductLaw) := by
rw [integral_bind hmeas hFbind]
simp_rw [hinner]
_ = deterministicRisk (mixedCountEstimator n d ε) Pstar := by
rw [hsample]
simpa [F, deterministicRisk, htarget]
theorem thetaPoly_uniform_upper
the stated mathematical conclusion holds.
Formal statement
conclusion
∃ C : ℝ,
0 < C
conclusion 1
n d :
ℕ
ε :
ℝ
1 ≤ n
1 ≤ d
0 < ε
ε < 1 / 2
worstCaseRiskReal (fun (_ : Unit) (P : {P : ZengLaw d // P ∈ zengDiscreteClass d ε}) => zengDeterministicRisk (thetaPoly n d ε) P.1) ()
≤ C * frontierRate n d ε
Proof (Lean source)
lemma thetaPoly_uniform_upper :
∃ C : ℝ, 0 < C ∧
∀ (n d : ℕ) (ε : ℝ), 1 ≤ n → 1 ≤ d → 0 < ε → ε < 1 / 2 →
Causalean.Stat.worstCaseRiskReal
(fun (_ : Unit) (P : {P : ZengLaw d // P ∈ zengDiscreteClass d ε}) =>
zengDeterministicRisk (thetaPoly n d ε) P.1) () ≤
C * frontierRate n d ε := by
obtain ⟨C, hC, hupper⟩ := unrestricted_mixed_count_upper
refine ⟨C, hC, ?_⟩
intro n d ε hn hd hε hεhalf
unfold Causalean.Stat.worstCaseRiskReal
by_cases hne : Nonempty {P : ZengLaw d // P ∈ zengDiscreteClass d ε}
· letI := hne
apply ciSup_le
intro P
obtain ⟨Pstar, hclass, hiid, hsample, htarget⟩ :=
synthetic_randomization_kernel hn hε hεhalf P.1 P.2
have hrisk := thetaPoly_risk_le_completion_risk (ε := ε) P.1 Pstar hsample htarget
have hbound := hupper n d ε Pstar hclass hiid
nlinarith [hrisk, hbound.1, hbound.2.1]
· haveI : IsEmpty {P : ZengLaw d // P ∈ zengDiscreteClass d ε} :=
not_nonempty_iff.mp hne
simp
have hrate : 0 ≤ frontierRate n d ε := by
unfold frontierRate
have hN : 0 < effectiveSize n ε := by
unfold effectiveSize
positivity
positivity
exact mul_nonneg hC.le hrate
theorem zeng_frontierRate_le_fixedQRate_mul
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
n d :
ℕ
q :
ℝ
hn :
1 ≤ n
hq :
0 < q
hq1 :
q ≤ 1
conclusion
Proof (Lean source)
lemma zeng_frontierRate_le_fixedQRate_mul (n d : ℕ) (q : ℝ)
(hn : 1 ≤ n) (hq : 0 < q) (hq1 : q ≤ 1) :
frontierRate n d q ≤ ((1 - log q) / q) ^ 2 * fixedQRate n d := by
let N := effectiveSize n q
let L := logScale n q
let L₁ := log (exp 1 + n)
let K := (1 - log q) / q
have hn0 : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n)
have hN : 0 < N := mul_pos hn0 hq
have hL : 1 ≤ L := by
dsimp [L, logScale]
rw [Real.le_log_iff_exp_le (by positivity)]
dsimp [N, effectiveSize] at *
linarith
have hL₁ : 0 < L₁ := by
dsimp [L₁]
apply Real.log_pos
nlinarith [Real.add_one_le_exp (1 : ℝ)]
have hlogq : log q ≤ 0 := Real.log_nonpos hq.le hq1
have hlogbound : L₁ ≤ (1 - log q) * L := by
have harg : exp 1 + (n : ℝ) ≤ (exp 1 + N) / q := by
apply (le_div_iff₀ hq).2
dsimp [N, effectiveSize]
nlinarith [Real.exp_pos (1 : ℝ)]
have hbase : L₁ ≤ L - log q := by
calc
L₁ ≤ log ((exp 1 + N) / q) :=
Real.log_le_log (by positivity) harg
_ = L - log q := by
rw [Real.log_div (by positivity) (ne_of_gt hq)]
rfl
have hprod := mul_nonneg (neg_nonneg.mpr hlogq) (sub_nonneg.mpr hL)
nlinarith
have hK : 1 ≤ K := by
dsimp [K]
apply (le_div_iff₀ hq).2
linarith
have hKpos : 0 < K := lt_of_lt_of_le (by norm_num) hK
have hD : 0 < N * L := mul_pos hN (by linarith)
have hD₁ : 0 < (n : ℝ) * L₁ := mul_pos hn0 hL₁
have hDen : (n : ℝ) * L₁ ≤ K * (N * L) := by
calc
(n : ℝ) * L₁ ≤ (n : ℝ) * ((1 - log q) * L) :=
mul_le_mul_of_nonneg_left hlogbound hn0.le
_ = K * (N * L) := by
dsimp [K, N, effectiveSize]
field_simp
have hfirst : N⁻¹ ≤ K ^ 2 * (n : ℝ)⁻¹ := by
have hqK : q * K = 1 - log q := by
dsimp [K]
field_simp
have hK2 : 1 ≤ q * K ^ 2 := by
nlinarith [mul_nonneg (sub_nonneg.mpr hK) (sub_nonneg.mpr hlogq)]
have hN_eq : N = (n : ℝ) * q := rfl
have h : 1 / N ≤ K ^ 2 / (n : ℝ) := by
apply (div_le_div_iff₀ hN hn0).2
rw [hN_eq]
nlinarith [mul_nonneg hn0.le (sub_nonneg.mpr hK2)]
simpa [one_div, div_eq_mul_inv] using h
have hsecond : ((d : ℝ) / (N * L)) ^ 2 ≤
K ^ 2 * ((d : ℝ) / ((n : ℝ) * L₁)) ^ 2 := by
have hdiv : (d : ℝ) / (N * L) ≤ K * ((d : ℝ) / ((n : ℝ) * L₁)) := by
have h : (d : ℝ) / (N * L) ≤ (K * d) / ((n : ℝ) * L₁) := by
apply (div_le_div_iff₀ hD hD₁).2
have hd0 : 0 ≤ (d : ℝ) := Nat.cast_nonneg _
nlinarith [mul_nonneg hd0 (sub_nonneg.mpr hDen)]
convert h using 1 <;> ring
have hleft : 0 ≤ (d : ℝ) / (N * L) := div_nonneg (Nat.cast_nonneg _) hD.le
nlinarith [sq_nonneg (K * ((d : ℝ) / ((n : ℝ) * L₁)) -
((d : ℝ) / (N * L)))]
have hr0 : 0 ≤ (n : ℝ)⁻¹ + ((d : ℝ) / ((n : ℝ) * L₁)) ^ 2 := by positivity
have hsum : N⁻¹ + ((d : ℝ) / (N * L)) ^ 2 ≤
K ^ 2 * ((n : ℝ)⁻¹ + ((d : ℝ) / ((n : ℝ) * L₁)) ^ 2) := by
nlinarith [hfirst, hsecond]
dsimp [frontierRate, fixedQRate, N, L, L₁, K] at *
calc
min 1 ((effectiveSize n q)⁻¹ +
((d : ℝ) / (effectiveSize n q * logScale n q)) ^ 2) ≤
min 1 (K ^ 2 * ((n : ℝ)⁻¹ + ((d : ℝ) /
((n : ℝ) * log (exp 1 + n))) ^ 2)) :=
min_le_min_left 1 hsum
_ ≤ K ^ 2 * min 1 ((n : ℝ)⁻¹ + ((d : ℝ) /
((n : ℝ) * log (exp 1 + n))) ^ 2) := by
rcases le_total ((n : ℝ)⁻¹ + ((d : ℝ) /
((n : ℝ) * log (exp 1 + n))) ^ 2) 1 with h | h
· rw [min_eq_right h]
exact (min_le_right _ _)
· rw [min_eq_left h]
exact (min_le_left _ _).trans (by nlinarith [sq_nonneg K])
theorem zeng_minimax_upper_fixedQ
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
ε :
ℝ
hε :
0 < ε
hεhalf :
ε < 1 / 2
conclusion
∃ Cε : ℝ,
0 < Cε
Proof (Lean source)
lemma zeng_minimax_upper_fixedQ (ε : ℝ)
(hε : 0 < ε) (hεhalf : ε < 1 / 2) :
∃ Cε : ℝ, 0 < Cε ∧ ∀ (n d : ℕ), 1 ≤ n → 1 ≤ d →
zengMinimaxRisk n d ε ≤ Cε * fixedQRate n d := by
obtain ⟨C, hC, hupper⟩ := thetaPoly_uniform_upper
let K := ((1 - log ε) / ε) ^ 2
have hlog : log ε ≤ 0 :=
Real.log_nonpos hε.le (by linarith)
have hK : 0 < K := by
dsimp [K]
exact pow_pos (div_pos (by linarith) hε) _
refine ⟨C * K, mul_pos hC hK, ?_⟩
intro n d hn hd
calc
zengMinimaxRisk n d ε ≤
worstCaseRiskReal
(fun (_ : Unit) (P : {P : ZengLaw d // P ∈ zengDiscreteClass d ε}) =>
zengDeterministicRisk (thetaPoly n d ε) P.1) () :=
zengMinimaxRisk_le_thetaPoly_worst n d ε
_ ≤ C * frontierRate n d ε :=
hupper n d ε hn hd hε hεhalf
_ ≤ C * (K * fixedQRate n d) :=
mul_le_mul_of_nonneg_left
(zeng_frontierRate_le_fixedQRate_mul n d ε hn hε (by linarith)) hC.le
_ = (C * K) * fixedQRate n d := by ring
theorem zeng_minimax_lower_fixedQ
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma zeng_minimax_lower_fixedQ (hZeng_of_gate : ZengTheoremTwoLower)
(ε : ℝ) (hε : 0 < ε) (hεhalf : ε < 1 / 2) :
∃ c : ℝ, 0 < c ∧ ∀ (n d : ℕ), 1 ≤ n → 1 ≤ d →
c * fixedQRate n d ≤ zengMinimaxRisk n d ε := by
obtain ⟨c₀, cdim, n₀, hc₀, hcdim, hgate⟩ :=
hZeng_of_gate ε hε hεhalf
obtain ⟨cp, hcp, hparam⟩ := zeng_parametric_minimax_lower
obtain ⟨N₁, hN₁⟩ := exists_nat_ge (2 / cdim)
let N := max (max n₀ 3) N₁
have hNpos : 0 < N := by
dsimp [N]
omega
let csat := c₀ * (cdim / 2) ^ 2
have hcsat : 0 < csat := by dsimp [csat]; positivity
let c := min (cp / N) (min c₀ csat)
have hc : 0 < c := by
dsimp [c]
exact lt_min (div_pos hcp (by exact_mod_cast hNpos)) (lt_min hc₀ hcsat)
refine ⟨c, hc, ?_⟩
intro n d hn hd
have hrate_nonneg : 0 ≤ fixedQRate n d := by
unfold fixedQRate
exact le_min (by norm_num) (by positivity)
have hrate_one : fixedQRate n d ≤ 1 := by
unfold fixedQRate
exact min_le_left _ _
by_cases hnlarge : N ≤ n
· have hn₀ : n₀ ≤ n := le_trans (le_max_left _ _) (le_trans (le_max_left _ _) hnlarge)
have hn3 : 3 ≤ n := le_trans (le_max_right n₀ 3) (le_trans (le_max_left _ _) hnlarge)
have hnposR : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
have hlog : 1 < Real.log n := by
rw [Real.lt_log_iff_exp_lt hnposR]
have hn3R : (3 : ℝ) ≤ n := by exact_mod_cast hn3
exact Real.exp_one_lt_d9.trans (by nlinarith)
have hlogpos : 0 < Real.log n := lt_trans zero_lt_one hlog
have hden : 0 < (n : ℝ) * Real.log n := mul_pos hnposR hlogpos
have hN₁n : N₁ ≤ n := le_trans (le_max_right (max n₀ 3) N₁) hnlarge
have htwo : 2 ≤ cdim * n * Real.log n := by
have hbase : 2 ≤ cdim * n := by
have hcN : 2 / cdim ≤ (n : ℝ) := hN₁.trans (by exact_mod_cast hN₁n)
exact (div_le_iff₀ hcdim).mp hcN |>.trans_eq (by ring)
nlinarith [mul_nonneg (show 0 ≤ cdim * n by positivity) (sub_nonneg.mpr hlog.le)]
by_cases hdim : (d : ℝ) ≤ cdim * n * Real.log n
· have hpublished := (hgate n d hn₀ hd hdim).trans
(zeng_zero_control_risk_le_minimax n d ε)
have hlog_le : Real.log n ≤ Real.log (exp 1 + n) := by
exact Real.log_le_log hnposR (by nlinarith [Real.exp_pos 1])
have hden' : 0 < (n : ℝ) * Real.log (exp 1 + n) := by
apply mul_pos hnposR
apply Real.log_pos
have hnR : (1 : ℝ) ≤ n := by exact_mod_cast hn
nlinarith [Real.exp_pos 1]
have hfrac : (d : ℝ) / ((n : ℝ) * Real.log (exp 1 + n)) ≤
(d : ℝ) / ((n : ℝ) * Real.log n) := by
exact div_le_div_of_nonneg_left (Nat.cast_nonneg _) hden
(mul_le_mul_of_nonneg_left hlog_le hnposR.le)
have hraw : fixedQRate n d ≤ (n : ℝ)⁻¹ +
((d : ℝ) / ((n : ℝ) * Real.log n)) ^ 2 := by
unfold fixedQRate
calc
min 1 ((n : ℝ)⁻¹ + ((d : ℝ) /
((n : ℝ) * Real.log (exp 1 + n))) ^ 2) ≤
(n : ℝ)⁻¹ + ((d : ℝ) /
((n : ℝ) * Real.log (exp 1 + n))) ^ 2 := min_le_right _ _
_ ≤ _ := by
have hf₀ : 0 ≤ (d : ℝ) /
((n : ℝ) * Real.log (exp 1 + n)) := by positivity
have hf₁ : 0 ≤ (d : ℝ) /
((n : ℝ) * Real.log n) := by positivity
nlinarith [mul_nonneg (sub_nonneg.mpr hfrac) (add_nonneg hf₀ hf₁)]
have hc_le : c ≤ c₀ := (min_le_right _ _).trans (min_le_left _ _)
calc
c * fixedQRate n d ≤ c * ((n : ℝ)⁻¹ +
((d : ℝ) / ((n : ℝ) * Real.log n)) ^ 2) :=
mul_le_mul_of_nonneg_left hraw hc.le
_ ≤ c₀ * ((n : ℝ)⁻¹ +
((d : ℝ) / ((n : ℝ) * Real.log n)) ^ 2) :=
mul_le_mul_of_nonneg_right hc_le (by positivity)
_ ≤ zengMinimaxRisk n d ε := hpublished
· let r := ⌊cdim * n * Real.log n⌋₊
have hrpos : 1 ≤ r := by
dsimp [r]
exact Nat.floor_pos.mpr (le_trans (by norm_num) htwo)
have hrleA : (r : ℝ) ≤ cdim * n * Real.log n := by
dsimp [r]
exact Nat.floor_le (by positivity)
have hrle : r ≤ d := by
exact_mod_cast hrleA.trans (le_of_not_ge hdim)
have hrhalf : cdim / 2 * ((n : ℝ) * Real.log n) ≤ r := by
have hfloor := Nat.sub_one_lt_floor (cdim * n * Real.log n)
dsimp [r]
have hhalf : cdim / 2 * ((n : ℝ) * Real.log n) ≤
cdim * n * Real.log n - 1 := by nlinarith
exact hhalf.trans hfloor.le
have hpublished := (hgate n r hn₀ hrpos hrleA).trans
(zeng_zero_control_risk_le_minimax n r ε)
have hmono := zengMinimaxRisk_mono_dimension (n := n) hrpos hrle ε
have hratio : cdim / 2 ≤ (r : ℝ) /
((n : ℝ) * Real.log n) := (le_div_iff₀ hden).2 (by simpa using hrhalf)
have hsatlower : csat ≤ c₀ * ((n : ℝ)⁻¹ +
((r : ℝ) / ((n : ℝ) * Real.log n)) ^ 2) := by
dsimp [csat]
have hc0nonneg := hc₀.le
have hsq : (cdim / 2) ^ 2 ≤
((r : ℝ) / ((n : ℝ) * Real.log n)) ^ 2 := by nlinarith
nlinarith [mul_nonneg hc0nonneg (inv_nonneg.mpr hnposR.le)]
have hc_le_sat : c ≤ csat := (min_le_right _ _).trans (min_le_right _ _)
calc
c * fixedQRate n d ≤ c := by nlinarith
_ ≤ csat := hc_le_sat
_ ≤ c₀ * ((n : ℝ)⁻¹ +
((r : ℝ) / ((n : ℝ) * Real.log n)) ^ 2) := hsatlower
_ ≤ zengZeroControlRisk n r ε := hgate n r hn₀ hrpos hrleA
_ ≤ zengMinimaxRisk n r ε := zeng_zero_control_risk_le_minimax n r ε
_ ≤ zengMinimaxRisk n d ε := hmono
· have hnN : n ≤ N := by omega
have hcpN : cp / N ≤ cp / n := by
exact div_le_div_of_nonneg_left hcp.le (by exact_mod_cast hn) (by exact_mod_cast hnN)
have hc_le : c ≤ cp / N := min_le_left _ _
calc
c * fixedQRate n d ≤ c := by nlinarith
_ ≤ cp / N := hc_le
_ ≤ cp / n := hcpN
_ ≤ zengMinimaxRisk n d ε := hparam n d ε hn hd hε hεhalf
theorem zeng_fixed_positivity_polynomial_frontier
Given the specified inputs and assumptions, the stated mathematical conclusion holds.
Formal statement
hypotheses
hZeng_of_gate :
conclusion
∃ C_univ : ℝ,
0 < C_univ
conclusion 1
n d :
ℕ
ε :
ℝ
1 ≤ n
1 ≤ d
0 < ε
ε < 1 / 2
worstCaseRiskReal (fun (_ : Unit) (P : {P : ZengLaw d // P ∈ zengDiscreteClass d ε}) => zengDeterministicRisk (thetaPoly n d ε) P.1) ()
≤ C_univ * frontierRate n d ε
conclusion 2
ε :
ℝ
0 < ε
ε < 1 / 2
∃ cε Cε : ℝ,
0 < cε
cε ≤ Cε
n d :
ℕ
1 ≤ n
1 ≤ d
cε * fixedQRate n d ≤ zengMinimaxRisk n d ε
zengMinimaxRisk n d ε ≤ Cε * fixedQRate n d
Proof (Lean source)
theorem zeng_fixed_positivity_polynomial_frontier
(hZeng_of_gate : ZengTheoremTwoLower) :
∃ C_univ : ℝ, 0 < C_univ ∧
(∀ (n d : ℕ) (ε : ℝ), 1 ≤ n → 1 ≤ d → 0 < ε → ε < 1 / 2 →
worstCaseRiskReal
(fun (_ : Unit) (P : {P : ZengLaw d // P ∈ zengDiscreteClass d ε}) =>
zengDeterministicRisk (thetaPoly n d ε) P.1) () ≤
C_univ * frontierRate n d ε) ∧
(∀ ε : ℝ, 0 < ε → ε < 1 / 2 →
∃ cε Cε : ℝ, 0 < cε ∧ cε ≤ Cε ∧
∀ (n d : ℕ), 1 ≤ n → 1 ≤ d →
cε * fixedQRate n d ≤ zengMinimaxRisk n d ε ∧
zengMinimaxRisk n d ε ≤ Cε * fixedQRate n d) := by
obtain ⟨C, hC, hupper⟩ := thetaPoly_uniform_upper
refine ⟨C, hC, hupper, ?_⟩
intro ε hε hεhalf
obtain ⟨Cu, hCu, hupperFixed⟩ :=
zeng_minimax_upper_fixedQ ε hε hεhalf
obtain ⟨c, hc, hlower⟩ :=
zeng_minimax_lower_fixedQ hZeng_of_gate ε hε hεhalf
let Cε := max Cu c
refine ⟨c, Cε, hc, le_max_right _ _, ?_⟩
intro n d hn hd
have hrate : 0 ≤ fixedQRate n d := by
unfold fixedQRate
exact le_min (by norm_num) (by positivity)
refine ⟨hlower n d hn hd, ?_⟩
exact (hupperFixed n d hn hd).trans
(mul_le_mul_of_nonneg_right (le_max_left Cu c) hrate)