Formalization: Minimax Inference for Threshold Modified Treatment Policies with Continuous Treatments
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.
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Basic 67 declarations This cold Stage-2 scaffold defines the observed and full-data experiments, the modeling assumptions, model classes, rate objects, and minimax criteria used by the threshold-clamp frontier.
Threshold-clamp minimax frontier: shared formal world
This cold Stage-2 scaffold defines the observed and full-data experiments, the modeling assumptions, model classes, rate objects, and minimax criteria used by the threshold-clamp frontier. Proof obligations are split into the theorem and helper modules named in the formalization plan.
The Causalean survey found reusable i.i.d.-sample and minimax infrastructure,
but no continuous-treatment clamp world. Causalean.PO is intentionally
bypassed because its finite-regime potential-outcome carrier is at a different
abstraction from the continuum-indexed standard-Borel response process here.
def StandardBorelRegularConditionalLaw Object lem:standard-borel-regular-conditional-law in the paper ↗
Kallenberg (2002), Foundations of Modern Probability, second edition, Theorem 6.3 (conditional distribution), doi:10.1007/978-1-4757-4015-8.
Definition (Lean source)
def
StandardBorelRegularConditionalLaw :
Sort 0
given by
∀ (S T : Type*) [MeasurableSpace S] [StandardBorelSpace S] [MeasurableSpace T] [StandardBorelSpace T] (μ : Measure (S × T)),
IsProbabilityMeasure μ → ∃ K : Kernel S T, (∀ s, IsProbabilityMeasure (K s)) ∧ ∀ (B : Set S) (C : Set T), MeasurableSet B → MeasurableSet C → μ (B ×ˢ C)
= ∫⁻ s in B, K s C ∂μ.map fst
Hoeffding (1963), “Probability inequalities for sums of bounded random variables”, Theorem 2 specialized to unit ranges and two tails, doi:10.1080/01621459.1963.10500830.
Definition (Lean source)
def
HoeffdingBoundedAverage :
Sort 0
given by
∀ (m : ℕ) (Ω : Type*) [MeasurableSpace Ω] (μ : Measure Ω),
IsProbabilityMeasure μ → ∀ Z : Fin m → Ω → ℝ, (∀ i, Measurable (Z i)) → iIndepFun Z μ → (∀ i, ∀ᵐ ω ∂μ, Z i ω ∈ Icc (0 : ℝ) 1) → ∀ t : ℝ, 0 < t → μ.real {ω | |(m : ℝ)⁻¹ * ∑ i, Z i ω - (m : ℝ)⁻¹ * ∑ i, ∫ ω, Z i ω ∂μ| > t}
≤ 2 * exp (-2 * (m : ℝ) * t ^ 2)
One observed unit O = (X,A,Y) in a finite-stratum continuous-treatment model.
Definition (Lean source)
instance instMeasurableSpaceClampObs
The observation carrier inherits the product Borel structure.
Definition (Lean source)
instance instMeasurableSpaceClampObs {J : ℕ} : MeasurableSpace (ClampObs J) :=
comap (fun o : ClampObs J => (o.X, o.A, o.Y)) inferInstance
A law and its law-pinned finite-stratum nuisance functions.
The lower-threshold modified-treatment policy d_delta(a) = max a delta.
Definition (Lean source)
Conditional mass collapsed to the threshold by the clamp.
def retainedMean
The retained natural-course contribution above the threshold.
The observed-data clamp target.
Definition (Lean source)
parameters
def
clampFunctional P delta :
ℝ
given by
def observedSampleCoordinate
The designated i-th observation in the canonical finite sample.
def IidSampling
The designated observations are the canonical first n coordinate maps under the n-fold product law; they are independent and each has law P.
Definition (Lean source)
parameters
def
IidSampling P n :
Prop
given by
clause 1
IsProbabilityMeasure P.dataMeasure
clause 2
∀ᵐ o ∂P.dataMeasure, o.A ∈ Icc (0 : ℝ) 1
clause 3
∀ᵐ o ∂P.dataMeasure, o.Y ∈ Icc (0 : ℝ) 1
clause 4
def CondDensityLaw
The declared conditional density is nonnegative and gives every finite-stratum conditional treatment probability.
def StratumMass
Every stratum mass is the corresponding atom of the X marginal and is bounded below by pmin.
def PolynomialThinning
The two-sided polynomial density envelope, with no smoothness imposed on the treatment density.
The local-polynomial order: the greatest natural number strictly below beta, including the integer-order correction.
Definition (Lean source)
parameters
beta :
ℝ
def
ellOf beta :
ℕ
given by
⌈beta⌉₊ - 1
def HolderRegression
Continuous bounded regression, its conditional-expectation law tie, and the stated Taylor-remainder Hölder condition.
Definition (Lean source)
parameters
def
HolderRegression P beta L :
Prop
given by
clause 1
ContinuousOn (P.mu x) (Icc (0 : ℝ) 1)
clause 3
=ᵐ[P.dataMeasure] fun o => P.mu o.X o.A
clause 4
s ∈ Set.Icc (0 : ℝ) 1 :
t ∈ Set.Icc (0 : ℝ) 1 :
|P.mu x t
- ∑ j ∈ range (ellOf beta + 1), iteratedDerivWithin j (P.mu x) (Icc (0 : ℝ) 1) s * (t - s) ^ j / (Nat.factorial j : ℝ)|
≤ L * |t - s| ^ beta
structure ClampModel
The finite-stratum clamp model, with exactly the four core member atoms.
Definition (Lean source)
parameters
assumes
probability :
IsProbabilityMeasure P.dataMeasure
outcomeSupport :
∀ᵐ o ∂P.dataMeasure, o.Y ∈ Icc (0 : ℝ) 1
@realizes A(a.s. range [0,1]) @realizes O(treatment coordinate supported on [0,1])
condDensity :
@realizes Y(a.s. range [0,1]) @realizes O(outcome coordinate supported on [0,1])
stratumMass :
StratumMass P pmin
thinning :
PolynomialThinning P kappa cminus cplus
holder :
HolderRegression P beta L
Standing declared-space constraints for the frontier constants.
Definition (Lean source)
parameters
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
def
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha :
Prop
given by
clause 1
0 < J
clause 2
0 < beta
clause 3
0 ≤ kappa
clause 4
0 < L
clause 5
0 < cminus
clause 6
cminus ≤ kappa + 1
clause 7
kappa + 1 ≤ cplus
clause 8
0 < pmin
clause 9
pmin ≤ 1 / (J : ℝ)
clause 10
0 < deltaBar
clause 11
deltaBar < 1
clause 12
0 < alpha
clause 13
alpha < 1 / 2
theorem one_le_J
Admissibility of the regime constants forces the number of strata to be at least one.
Formal statement
hypotheses
J :
ℕ
beta :
ℝ
kappa :
ℝ
L :
ℝ
cminus :
ℝ
cplus :
ℝ
pmin :
ℝ
deltaBar :
ℝ
alpha :
ℝ
h :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
1 ≤ J
Proof (Lean source)
lemma RegimeConstants.one_le_J
(h : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) : 1 ≤ J := by
exact Nat.one_le_iff_ne_zero.2 (Nat.ne_of_gt h.1)
theorem alpha_pos
Admissibility of the regime constants forces the error level to be positive.
Formal statement
hypotheses
J :
ℕ
beta :
ℝ
kappa :
ℝ
L :
ℝ
cminus :
ℝ
cplus :
ℝ
pmin :
ℝ
deltaBar :
ℝ
alpha :
ℝ
h :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
0 < alpha
Proof (Lean source)
lemma RegimeConstants.alpha_pos
(h : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) : 0 < alpha := by
rcases h with ⟨_, _, _, _, _, _, _, _, _, _, _, halpha, _⟩
exact halpha
theorem alpha_lt_half
Admissibility of the regime constants forces the error level to be less than one half.
Formal statement
hypotheses
J :
ℕ
beta :
ℝ
kappa :
ℝ
L :
ℝ
cminus :
ℝ
cplus :
ℝ
pmin :
ℝ
deltaBar :
ℝ
alpha :
ℝ
h :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
alpha < 1 / 2
Proof (Lean source)
lemma RegimeConstants.alpha_lt_half
(h : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) : alpha < 1 / 2 := by
rcases h with ⟨_, _, _, _, _, _, _, _, _, _, _, _, halpha⟩
exact halpha
def testMultiplier
The conditional test multiplier on its declared core domain.
A deterministic threshold sequence stays in [0,deltaBar].
Definition (Lean source)
parameters
deltaBar :
ℝ
delta :
ℕ → ℝ
n :
def
ThresholdSequence deltaBar delta :
Prop
given by
delta n ∈ Icc (0 : ℝ) deltaBar
def bandwidthCrossingSet
The information-balance crossing set.
Definition (Lean source)
parameters
n :
ℕ
delta beta kappa deltaBar :
ℝ
def
given by
{h | 0 < h ∧ h ≤ 1 - deltaBar ∧ 1 ≤ (n : ℝ) * h ^ (2 * beta + 1) * (delta + h) ^ kappa}
First information-balance crossing, with the prescribed empty-set fallback.
Definition (Lean source)
parameters
n :
ℕ
delta beta kappa deltaBar :
ℝ
def
infoBandwidth n delta beta kappa deltaBar :
ℝ
given by
by classical exact if (bandwidthCrossingSet n delta beta kappa deltaBar).Nonempty then sInf (bandwidthCrossingSet n delta beta kappa deltaBar) else 1
- deltaBar
Candidate regular-plus-atom estimation frontier.
Definition (Lean source)
parameters
n :
ℕ
delta kappa h beta :
ℝ
def
clampFrontier n delta kappa h beta :
ℝ
given by
(n : ℝ) ^ (-(1 : ℝ) / 2) + delta ^ (kappa + 1) * h ^ beta
Critical threshold scale.
Definition (Lean source)
parameters
n :
ℕ
beta kappa :
ℝ
def
deltaCrit n beta kappa :
ℝ
given by
(n : ℝ) ^ (-(1 : ℝ) / (2 * (beta * kappa + 2 * beta + kappa + 1)))
Boundary-design scale.
Definition (Lean source)
parameters
n :
ℕ
beta kappa :
ℝ
def
deltaEdge n beta kappa :
ℝ
given by
(n : ℝ) ^ (-(1 : ℝ) / (2 * beta + kappa + 1))
The canonical product law of the observed sample.
def Estimator
Observed-sample point estimators.
Observed-sample confidence-set procedures.
An estimator is measurable with respect to the observed sample.
Definition (Lean source)
parameters
n J :
ℕ
est :
Estimator n J
def
ObservedMeasurableEstimator est :
Prop
given by
Measurable est
A confidence procedure has measurable ordered endpoints and returns exactly the corresponding closed interval.
Definition (Lean source)
parameters
n J :
ℕ
C :
def
ObservedMeasurableInterval C :
Prop
def intervalLength
Length of the convex hull of a real confidence set.
def intervalLengthENNReal
Extended-real interval length, used so non-integrable expected lengths are represented by ∞ rather than the junk value of the real Bochner integral.
Definition (Lean source)
def estimatorRisk
Absolute-error risk of an observed-sample estimator under one law.
Definition (Lean source)
parameters
def
estimatorRisk P n delta est :
ℝ
given by
∫ z, |est z - clampFunctional P delta| ∂iidProduct P n
Minimax absolute-error risk over the clamp model.
Definition (Lean source)
parameters
J n :
ℕ
beta kappa L cminus cplus pmin delta :
ℝ
def
observedMinimaxRisk J n beta kappa L cminus cplus pmin delta :
ℝ
given by
sInf {r : ℝ | ∃ est : Estimator n J, ObservedMeasurableEstimator est ∧ (∀ z, est z ∈ Icc (0 : ℝ) 1) ∧ r = sSup {v : ℝ | ∃ P : ClampLaw J, ClampModel P beta kappa L cminus cplus pmin ∧ v = estimatorRisk P n delta est}}
Uniform coverage of a confidence procedure over the clamp model.
Definition (Lean source)
parameters
def
UniformCoverage J n beta kappa L cminus cplus pmin delta alpha C :
Prop
Minimax worst-case expected length among uniformly honest intervals.
Definition (Lean source)
parameters
J n :
ℕ
beta kappa L cminus cplus pmin delta alpha :
ℝ
def
observedMinimaxLength J n beta kappa L cminus cplus pmin delta alpha :
ℝ≥0∞
given by
sInf {r : ℝ≥0∞ | ∃ C : ConfidenceProcedure n J, UniformCoverage J n beta kappa L cminus cplus pmin delta alpha C ∧ r = sSup {v : ℝ≥0∞ | ∃ P : ClampLaw J, ClampModel P beta kappa L cminus cplus pmin ∧ v = ∫⁻ z, intervalLengthENNReal (C z) ∂iidProduct P n}}
A full-data law packages its own standard-Borel latent carrier, so different members of a full-data model class need not share a carrier. The observed-model restriction is deliberately imposed by the class predicates below, rather than by this carrier.
Definition (Lean source)
parameters
J :
ℕ
fields
latentCarrier :
observedMargin :
ClampLaw J
@realizes P_full(full-data probability law on the member-specific carrier)
probability :
IsProbabilityMeasure fullMeasure
margin_eq :
fullMeasure.map fst = observedMargin.dataMeasure
g :
Fin J → ℝ → latentCarrier → ℝ
@realizes g_x(structural response map on [0,1]×mathcal_U)
pot :
ℝ → (ClampObs J × latentCarrier) → ℝ
@realizes Y_a(potential-outcome process a↦Y(a))
pot_jointlyMeasurable :
Measurable (uncurry pot)
instance instMeasurableSpaceLatentCarrier
the canonical inst measurable space latent carrier instance is defined for the specified J input, the specified PF input.
Definition (Lean source)
instance (PF : FullDataLaw J) : MeasurableSpace PF.latentCarrier :=
PF.latentMeasurable
instance instStandardBorelSpaceLatentCarrier
the canonical inst standard borel space latent carrier instance is defined for the specified J input, the specified PF input.
Definition (Lean source)
instance (PF : FullDataLaw J) : StandardBorelSpace PF.latentCarrier :=
PF.latentStandardBorel
instance instNonemptyLatentCarrier
the canonical inst nonempty latent carrier instance is defined for the specified J input, the specified PF input.
Definition (Lean source)
instance (PF : FullDataLaw J) : Nonempty PF.latentCarrier :=
PF.latentNonempty
def LatentResponseConsistency
Simultaneous latent-response consistency outside one common null set.
Definition (Lean source)
parameters
J :
ℕ
PF :
def
LatentResponseConsistency PF :
Prop
given by
clause 3
∀ᵐ z ∂PF.fullMeasure, (∀ a ∈ Icc (0 : ℝ) 1, PF.pot a z = PF.g z.1.X a z.2) ∧ z.1.Y
= PF.g z.1.X z.1.A z.2
def fullDataStratumMass
The actual X=x marginal mass under a full-data law. This is computed from the observed marginal measure, which FullDataLaw.margin_eq pins to the (X,A,Y) projection of the full-data measure; it deliberately does not use the auxiliary ClampLaw.px field.
Definition (Lean source)
parameters
def
fullDataStratumMass PF x :
ℝ
given by
(PF.observedMargin.dataMeasure.map (fun o => o.X)).real {x}
def LatentExchangeability
Finite-stratum conditional independence of treatment and latent response, written as the exact stratumwise product-moment factorization using the actual X-marginal mass.
Definition (Lean source)
parameters
J :
ℕ
PF :
x :
Fin J
f :
ℝ → ℝ
g :
PF.latentCarrier → ℝ
(∃ M, ∀ a, |f a| ≤ M)
(∃ M, ∀ u, |g u| ≤ M)
def
LatentExchangeability PF :
Prop
given by
fullDataStratumMass PF x * (∫ z in {z : ClampObs J × PF.latentCarrier | z.1.X = x}, f z.1.A * g z.2 ∂PF.fullMeasure)
def fullDataResponseMean
Full-data conditional response mean in a finite stratum.
Definition (Lean source)
parameters
def
fullDataResponseMean PF x a :
ℝ
given by
(fullDataStratumMass PF x)⁻¹ * ∫ z in {z : ClampObs J × PF.latentCarrier | z.1.X = x}, PF.g x a z.2 ∂PF.fullMeasure
def FullDataResponseContinuity
Continuity of the fiberwise structural response mean on the declared threshold range.
Definition (Lean source)
parameters
def
FullDataResponseContinuity PF deltaBar :
Prop
given by
def causalClampMean
The full-data clamp-policy mean.
Definition (Lean source)
parameters
def
causalClampMean PF delta :
ℝ
given by
∫ z, PF.pot (clampPolicy delta z.1.A) z ∂PF.fullMeasure
structure FullDataClampModel
The structural full-data class. Each member carries its own latent carrier; the observed margin belongs to the fixed-Hölder model, and the three causal member conditions hold on that same package.
Definition (Lean source)
parameters
assumes
observedModel :
ClampModel PF.observedMargin beta kappa L cminus cplus pmin
consistency :
LatentResponseConsistency (PF := PF)
exchangeability :
LatentExchangeability (PF := PF)
responseContinuity :
FullDataResponseContinuity PF deltaBar
def causalEstimatorRisk
Absolute risk of an observed-sample estimator for one full-data law.
Definition (Lean source)
parameters
def
causalEstimatorRisk PF n delta est :
ℝ
given by
∫ z, |est z - causalClampMean (PF := PF) (delta := delta)| ∂iidProduct PF.observedMargin n
def causalFrontierCriteria
The pair (R_{n,F}^star, L_{n,F}^star) of full-data criteria, with expected length valued in ℝ≥0∞ so infinite expectations are preserved.
Definition (Lean source)
parameters
J n :
ℕ
beta kappa L cminus cplus pmin deltaBar delta alpha :
ℝ
def
causalFrontierCriteria J n beta kappa L cminus cplus pmin deltaBar delta alpha :
ℝ × ℝ≥0∞
given by
(sInf {r : ℝ | ∃ est : Estimator n J, ObservedMeasurableEstimator est ∧ (∀ z, est z ∈ Icc (0 : ℝ) 1) ∧ r = sSup {v : ℝ | ∃ PF : FullDataLaw J, FullDataClampModel PF beta kappa L cminus cplus pmin deltaBar ∧ v = causalEstimatorRisk (PF := PF) (n := n) (delta := delta) (est := est)}}, sInf {r : ℝ≥0∞ | ∃ C : ConfidenceProcedure n J, ObservedMeasurableInterval C ∧ (∀ PF : FullDataLaw J, FullDataClampModel PF beta kappa L cminus cplus pmin deltaBar → 1 - alpha ≤ (iidProduct PF.observedMargin n).real {z | causalClampMean (PF := PF) (delta := delta) ∈ C z}) ∧ r = sSup {v : ℝ≥0∞ | ∃ PF : FullDataLaw J, FullDataClampModel PF beta kappa L cminus cplus pmin deltaBar ∧ v = ∫⁻ z, intervalLengthENNReal (C z) ∂iidProduct PF.observedMargin n}})
Declared design spaces for the continuity-only model, with no Hölder exponent or radius among its parameters.
Definition (Lean source)
parameters
J :
ℕ
kappa cminus cplus pmin deltaBar :
ℝ
def
ContDesignConstants J kappa cminus cplus pmin deltaBar :
Prop
given by
clause 1
0 < J
clause 2
0 ≤ kappa
clause 3
0 < cminus
clause 4
cminus ≤ kappa + 1
clause 5
kappa + 1 ≤ cplus
clause 6
0 < pmin
clause 7
pmin ≤ 1 / (J : ℝ)
clause 8
0 < deltaBar
clause 9
deltaBar < 1
def ContRegimeConstants
Declared spaces for the continuity-only inference regime.
Definition (Lean source)
parameters
J :
ℕ
kappa cminus cplus pmin deltaBar alpha :
ℝ
def
ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha :
Prop
given by
clause 1
ContDesignConstants J kappa cminus cplus pmin deltaBar
clause 2
0 < alpha
clause 3
alpha < 1 / 2
structure ContClampModel
The continuity-only observed model: the three shared law conditions and existence of a continuous conditional-regression version on the threshold range, with no common modulus or Hölder radius.
Definition (Lean source)
parameters
assumes
probability :
IsProbabilityMeasure P.dataMeasure
treatmentSupport :
∀ᵐ o ∂P.dataMeasure, o.A ∈ Icc (0 : ℝ) 1
outcomeSupport :
∀ᵐ o ∂P.dataMeasure, o.Y ∈ Icc (0 : ℝ) 1
condDensity :
stratumMass :
StratumMass P pmin
thinning :
PolynomialThinning P kappa cminus cplus
continuousVersion :
∃ mu : Fin J → ℝ → ℝ,
(∀ x, ContinuousOn (mu x) (Icc (0 : ℝ) deltaBar)) ∧
(P.dataMeasure[(fun o : ClampObs J => o.Y) | comap (fun o : ClampObs J => (o.X, o.A)) inferInstance] =ᵐ[P.dataMeasure] fun o => mu o.X o.A)
def contRegression
The selected continuous conditional-regression version. Its uniqueness is proved separately from positivity of the treatment density.
Definition (Lean source)
parameters
J :
ℕ
P :
ClampLaw J
kappa cminus cplus pmin deltaBar :
ℝ
hP :
ContClampModel P kappa cminus cplus pmin deltaBar
def
given by
choose hP.continuousVersion
def contClampFunctional
The continuity-only observed clamp functional.
Definition (Lean source)
parameters
J :
ℕ
P :
ClampLaw J
kappa cminus cplus pmin deltaBar :
ℝ
hP :
ContClampModel P kappa cminus cplus pmin deltaBar
delta :
ℝ
def
contClampFunctional P kappa cminus cplus pmin deltaBar hP delta :
ℝ
given by
retainedMean P delta
+ ∑ x : Fin J, P.px x * atomMass P x delta * contRegression P kappa cminus cplus pmin deltaBar hP x delta
The continuity-only regular-plus-atom frontier.
Definition (Lean source)
parameters
n :
ℕ
delta kappa :
ℝ
def
contFrontier n delta kappa :
ℝ
given by
(n : ℝ) ^ (-(1 : ℝ) / 2) + delta ^ (kappa + 1)
structure ContFullDataClampModel
Continuity-only full-data membership over the same law-specific latent carrier, differing from the fixed-Hölder class only in its observed margin.
Definition (Lean source)
parameters
assumes
observedModel :
ContClampModel PF.observedMargin kappa cminus cplus pmin deltaBar
consistency :
LatentResponseConsistency (PF := PF)
exchangeability :
LatentExchangeability (PF := PF)
responseContinuity :
FullDataResponseContinuity PF deltaBar
The common observed-law fields used by the full-data identification bridge.
Definition (Lean source)
parameters
assumes
probability :
IsProbabilityMeasure P.dataMeasure
treatmentSupport :
∀ᵐ o ∂P.dataMeasure, o.A ∈ Icc (0 : ℝ) 1
outcomeSupport :
∀ᵐ o ∂P.dataMeasure, o.Y ∈ Icc (0 : ℝ) 1
condDensity :
stratumMass :
StratumMass P pmin
thinning :
PolynomialThinning P kappa cminus cplus
theorem toBridge
Forget the fixed-Hölder regression field when proving the causal bridge. The result uses the hP condition. This is the stated conclusion.
Formal statement
hypotheses
beta :
ℝ
kappa :
ℝ
L :
ℝ
cminus :
ℝ
cplus :
ℝ
pmin :
ℝ
hP :
ClampModel P beta kappa L cminus cplus pmin
conclusion
BridgeClampModel P kappa cminus cplus pmin
Proof (Lean source)
-- @node: ClampModel.toBridge
theorem ClampModel.toBridge
(hP : ClampModel P beta kappa L cminus cplus pmin) :
BridgeClampModel P kappa cminus cplus pmin :=
⟨hP.probability, hP.treatmentSupport, hP.outcomeSupport, hP.condDensity,
hP.stratumMass, hP.thinning⟩
theorem toBridge
Forget the qualitative regression field when proving the causal bridge. The result uses the hP condition. This is the stated conclusion.
Formal statement
hypotheses
kappa :
ℝ
cminus :
ℝ
cplus :
ℝ
pmin :
ℝ
deltaBar :
ℝ
hP :
ContClampModel P kappa cminus cplus pmin deltaBar
conclusion
BridgeClampModel P kappa cminus cplus pmin
Proof (Lean source)
-- @node: ContClampModel.toBridge
theorem ContClampModel.toBridge
(hP : ContClampModel P kappa cminus cplus pmin deltaBar) :
BridgeClampModel P kappa cminus cplus pmin :=
⟨hP.probability, hP.treatmentSupport, hP.outcomeSupport, hP.condDensity,
hP.stratumMass, hP.thinning⟩
structure BridgeFullDataClampModel
The causal assumptions together with precisely the common observed-law fields needed by the measure-factorization argument.
Definition (Lean source)
parameters
assumes
observedModel :
BridgeClampModel PF.observedMargin kappa cminus cplus pmin
consistency :
LatentResponseConsistency (PF := PF)
exchangeability :
LatentExchangeability (PF := PF)
responseContinuity :
FullDataResponseContinuity PF deltaBar
theorem toBridge
The common bridge view of a fixed-Hölder full-data member. The result uses the hPF condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta :
ℝ
kappa :
ℝ
L :
ℝ
cminus :
ℝ
cplus :
ℝ
pmin :
ℝ
deltaBar :
ℝ
PF :
hPF :
FullDataClampModel PF beta kappa L cminus cplus pmin deltaBar
conclusion
BridgeFullDataClampModel PF kappa cminus cplus pmin deltaBar
Proof (Lean source)
-- @node: FullDataClampModel.toBridge
theorem FullDataClampModel.toBridge {PF : FullDataLaw J}
(hPF : FullDataClampModel PF beta kappa L cminus cplus pmin deltaBar) :
BridgeFullDataClampModel PF kappa cminus cplus pmin deltaBar :=
⟨hPF.observedModel.toBridge, hPF.consistency, hPF.exchangeability,
hPF.responseContinuity⟩
theorem toBridge
The common bridge view of a continuity-only full-data member. The result uses the hPF condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa :
ℝ
cminus :
ℝ
cplus :
ℝ
pmin :
ℝ
deltaBar :
ℝ
PF :
hPF :
ContFullDataClampModel PF kappa cminus cplus pmin deltaBar
conclusion
BridgeFullDataClampModel PF kappa cminus cplus pmin deltaBar
Proof (Lean source)
-- @node: ContFullDataClampModel.toBridge
theorem ContFullDataClampModel.toBridge {PF : FullDataLaw J}
(hPF : ContFullDataClampModel PF kappa cminus cplus pmin deltaBar) :
BridgeFullDataClampModel PF kappa cminus cplus pmin deltaBar :=
⟨hPF.observedModel.toBridge, hPF.consistency, hPF.exchangeability,
hPF.responseContinuity⟩
def contEstimatorRisk
Absolute risk for the continuity-only observed functional.
Definition (Lean source)
parameters
J :
ℕ
P :
ClampLaw J
n :
ℕ
kappa cminus cplus pmin deltaBar delta :
ℝ
hP :
ContClampModel P kappa cminus cplus pmin deltaBar
est :
Estimator n J
def
contEstimatorRisk P n kappa cminus cplus pmin deltaBar delta hP est :
ℝ
given by
∫ z, |est z
- contClampFunctional P kappa cminus cplus pmin deltaBar hP delta| ∂iidProduct P n
The four continuity-only observed and causal decision criteria, ordered as observed risk, observed length, causal risk, and causal length.
Definition (Lean source)
parameters
J n :
ℕ
kappa cminus cplus pmin deltaBar delta alpha :
ℝ
def
contFrontierCriteria J n kappa cminus cplus pmin deltaBar delta alpha :
ℝ × ℝ≥0∞ × ℝ × ℝ≥0∞
given by
(sInf {r : ℝ | ∃ est : Estimator n J, ObservedMeasurableEstimator est ∧ (∀ z, est z ∈ Icc (0 : ℝ) 1) ∧ r = sSup {v : ℝ | ∃ (P : ClampLaw J) (hP : ContClampModel P kappa cminus cplus pmin deltaBar), v = contEstimatorRisk P n kappa cminus cplus pmin deltaBar delta hP est}}, sInf {r : ℝ≥0∞ | ∃ C : ConfidenceProcedure n J, ObservedMeasurableInterval C ∧ (∀ (P : ClampLaw J) (hP : ContClampModel P kappa cminus cplus pmin deltaBar), 1 - alpha ≤ (iidProduct P n).real {z | contClampFunctional P kappa cminus cplus pmin deltaBar hP delta ∈ C z}) ∧ r = sSup {v : ℝ≥0∞ | ∃ (P : ClampLaw J) (_hP : ContClampModel P kappa cminus cplus pmin deltaBar), v = ∫⁻ z, intervalLengthENNReal (C z) ∂iidProduct P n}}, sInf {r : ℝ | ∃ est : Estimator n J, ObservedMeasurableEstimator est ∧ (∀ z, est z ∈ Icc (0 : ℝ) 1) ∧ r = sSup {v : ℝ | ∃ (PF : FullDataLaw J) (_hPF : ContFullDataClampModel PF kappa cminus cplus pmin deltaBar), v = causalEstimatorRisk (PF := PF) (n := n) (delta := delta) (est := est)}}, sInf {r : ℝ≥0∞ | ∃ C : ConfidenceProcedure n J, ObservedMeasurableInterval C ∧ (∀ (PF : FullDataLaw J), ContFullDataClampModel PF kappa cminus cplus pmin deltaBar → 1 - alpha ≤ (iidProduct PF.observedMargin n).real {z | causalClampMean (PF := PF) (delta := delta) ∈ C z}) ∧ r = sSup {v : ℝ≥0∞ | ∃ (PF : FullDataLaw J) (_hPF : ContFullDataClampModel PF kappa cminus cplus pmin deltaBar), v = ∫⁻ z, intervalLengthENNReal (C z) ∂iidProduct PF.observedMargin n}})
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.Bandwidth 9 declarations AsympSeq records the explicit eventual two-sided constant sandwich used by the paper.
Information-balance bandwidth regimes
AsympSeq records the explicit eventual two-sided constant sandwich used by
the paper. The theorem keeps the threshold sequence arbitrary.
Two positive constants eventually sandwich one nonnegative sequence by another.
Definition (Lean source)
parameters
f g :
ℕ → ℝ
def
AsympSeq f g :
Prop
Eventually the infimum in infoBandwidth is the unique positive information-balance root. The result uses the hbeta condition, the hkappa condition, the hdeltaBar condition, the hdelta condition. This is the stated conclusion.
Formal statement
hypotheses
beta kappa deltaBar :
ℝ
hbeta :
0 < beta
hkappa :
0 ≤ kappa
hdeltaBar :
0 < deltaBar ∧ deltaBar < 1
deltaSeq :
ℕ → ℝ
hdelta :
ThresholdSequence deltaBar deltaSeq
conclusion
∀ᶠ n in atTop, 0 < infoBandwidth n (deltaSeq n) beta kappa deltaBar ∧ infoBandwidth n (deltaSeq n) beta kappa deltaBar
≤ 1 - deltaBar ∧
(n : ℝ) * infoBandwidth n (deltaSeq n) beta kappa deltaBar ^ (2 * beta + 1) * (deltaSeq n + infoBandwidth n (deltaSeq n) beta kappa deltaBar) ^ kappa
= 1
Proof (Lean source)
-- @node: infoBandwidth_eventually_balance
lemma infoBandwidth_eventually_balance
(beta kappa deltaBar : ℝ) (hbeta : 0 < beta) (hkappa : 0 ≤ kappa)
(hdeltaBar : 0 < deltaBar ∧ deltaBar < 1)
(deltaSeq : ℕ → ℝ) (hdelta : ThresholdSequence deltaBar deltaSeq) :
∀ᶠ n in atTop,
0 < infoBandwidth n (deltaSeq n) beta kappa deltaBar ∧
infoBandwidth n (deltaSeq n) beta kappa deltaBar ≤ 1 - deltaBar ∧
(n : ℝ) * infoBandwidth n (deltaSeq n) beta kappa deltaBar ^
(2 * beta + 1) *
(deltaSeq n + infoBandwidth n (deltaSeq n) beta kappa deltaBar) ^ kappa = 1 := by
let H : ℝ := 1 - deltaBar
have hH : 0 < H := sub_pos.mpr hdeltaBar.2
have hp : 0 < 2 * beta + 1 := by linarith
have hK : 0 < H ^ (2 * beta + 1) * H ^ kappa :=
mul_pos (Real.rpow_pos_of_pos hH _) (Real.rpow_pos_of_pos hH _)
obtain ⟨N, hN⟩ := exists_nat_gt (1 / (H ^ (2 * beta + 1) * H ^ kappa))
filter_upwards [eventually_ge_atTop N] with n hn
have hnpos : 0 < n := by
by_contra hn0
have hnz : n = 0 := Nat.eq_zero_of_not_pos hn0
have hNz : N = 0 := Nat.le_zero.mp (hnz ▸ hn)
subst N
norm_num at hN
have : 0 < (H ^ kappa)⁻¹ * (H ^ (2 * beta + 1))⁻¹ := by positivity
linarith
let F : ℝ → ℝ := fun h => (n : ℝ) * h ^ (2 * beta + 1) *
(deltaSeq n + h) ^ kappa
have hd := hdelta n
have hFH : 1 ≤ F H := by
have hbase : H ≤ deltaSeq n + H := by linarith [hd.1]
have hrpow : H ^ kappa ≤ (deltaSeq n + H) ^ kappa :=
Real.rpow_le_rpow (le_of_lt hH) hbase hkappa
have hnK : 1 < (n : ℝ) * (H ^ (2 * beta + 1) * H ^ kappa) := by
have hnR : 1 / (H ^ (2 * beta + 1) * H ^ kappa) < (n : ℝ) :=
lt_of_lt_of_le hN (by exact_mod_cast hn)
calc
1 = (1 / (H ^ (2 * beta + 1) * H ^ kappa)) *
(H ^ (2 * beta + 1) * H ^ kappa) := by field_simp
_ < (n : ℝ) * (H ^ (2 * beta + 1) * H ^ kappa) :=
mul_lt_mul_of_pos_right hnR hK
dsimp [F]
calc
1 ≤ (n : ℝ) * (H ^ (2 * beta + 1) * H ^ kappa) := hnK.le
_ = (n : ℝ) * H ^ (2 * beta + 1) * H ^ kappa := by ring
_ ≤ (n : ℝ) * H ^ (2 * beta + 1) * (deltaSeq n + H) ^ kappa := by
gcongr
have hFcont : ContinuousOn F (Icc 0 H) := by
unfold F
by_cases hkzero : kappa = 0
· simp [hkzero]
exact continuousOn_const.mul
(continuousOn_id.rpow continuousOn_const (fun _ _ => inr hp))
· exact (continuousOn_const.mul
(continuousOn_id.rpow continuousOn_const (fun _ _ => inr hp))).mul
((continuousOn_const.add continuousOn_id).rpow continuousOn_const
(fun _ _ => inr (lt_of_le_of_ne hkappa (Ne.symm hkzero))))
have hF0 : F 0 = 0 := by
simp [F, Real.zero_rpow hp.ne']
obtain ⟨r, hrI, hr⟩ : ∃ r ∈ Icc (0 : ℝ) H, F r = 1 := by
have hone : (1 : ℝ) ∈ Icc (F 0) (F H) := by simpa [hF0] using hFH
rcases intermediate_value_Icc hH.le hFcont hone with ⟨r, hrI, hr⟩
exact ⟨r, hrI, hr⟩
have hrpos : 0 < r := by
rcases hrI with ⟨hr0, _⟩
exact lt_of_le_of_ne hr0 (fun hre => by subst r; simp [hF0] at hr)
have hFstrict : StrictMonoOn F (Icc 0 H) := by
intro a ha b hb hab
have haPow : a ^ (2 * beta + 1) < b ^ (2 * beta + 1) :=
Real.rpow_lt_rpow ha.1 hab hp
have hbpos : 0 < b := lt_of_le_of_lt ha.1 hab
have hdb : 0 < deltaSeq n + b := by linarith [hd.1]
have hfac : 0 < (n : ℝ) * (deltaSeq n + b) ^ kappa :=
mul_pos (by exact_mod_cast hnpos) (Real.rpow_pos_of_pos hdb _)
have hmono : (deltaSeq n + a) ^ kappa ≤ (deltaSeq n + b) ^ kappa := by
apply Real.rpow_le_rpow
· linarith [hd.1, ha.1]
· linarith
· exact hkappa
dsimp [F]
calc
(n : ℝ) * a ^ (2 * beta + 1) * (deltaSeq n + a) ^ kappa ≤
(n : ℝ) * a ^ (2 * beta + 1) * (deltaSeq n + b) ^ kappa := by
exact mul_le_mul_of_nonneg_left hmono
(mul_nonneg (Nat.cast_nonneg _) (Real.rpow_nonneg ha.1 _))
_ < (n : ℝ) * b ^ (2 * beta + 1) * (deltaSeq n + b) ^ kappa := by
exact mul_lt_mul_of_pos_right
(mul_lt_mul_of_pos_left haPow (Nat.cast_pos.mpr hnpos))
(Real.rpow_pos_of_pos hdb _)
have hset : bandwidthCrossingSet n (deltaSeq n) beta kappa deltaBar =
Icc r H := by
ext h
simp only [bandwidthCrossingSet, Set.mem_setOf_eq, Set.mem_Icc]
constructor
· rintro ⟨hh0, hhH, hhF⟩
refine ⟨?_, hhH⟩
by_contra hnot
have hlt : h < r := lt_of_not_ge hnot
have := hFstrict ⟨hh0.le, hhH⟩ hrI hlt
dsimp [F] at this
linarith
· rintro ⟨hrh, hhH⟩
have hh0 : 0 < h := hrpos.trans_le hrh
refine ⟨hh0, hhH, ?_⟩
have := hFstrict.monotoneOn hrI ⟨hh0.le, hhH⟩ hrh
dsimp [F] at this
linarith
have hnonempty : (bandwidthCrossingSet n (deltaSeq n) beta kappa deltaBar).Nonempty := by
rw [hset]
exact Set.nonempty_Icc.2 hrI.2
have hbw : infoBandwidth n (deltaSeq n) beta kappa deltaBar = r := by
simp [infoBandwidth, hnonempty, hset, hrI.2]
rw [hbw]
exact ⟨hrpos, hrI.2, by simpa [F] using hr⟩
The balance equation bounds its positive root above by the interior-scale closed form. The result uses the hn condition, the hd condition, the hh condition, the hp condition, the hk condition, the heq condition. This is the stated conclusion.
Formal statement
hypotheses
n d h p k :
ℝ
hn :
0 < n
hd :
0 < d
hh :
0 < h
hp :
0 < p
hk :
0 ≤ k
heq :
n * h ^ p * (d + h) ^ k = 1
conclusion
h ≤ (n * d ^ k) ^ (-1 / p)
Proof (Lean source)
-- @node: balance_root_le_interior_scale
lemma balance_root_le_interior_scale (n d h p k : ℝ)
(hn : 0 < n) (hd : 0 < d) (hh : 0 < h) (hp : 0 < p) (hk : 0 ≤ k)
(heq : n * h ^ p * (d + h) ^ k = 1) :
h ≤ (n * d ^ k) ^ (-1 / p) := by
have hX : 0 < n * d ^ k := mul_pos hn (Real.rpow_pos_of_pos hd _)
rw [← Real.rpow_le_rpow_iff hh.le (Real.rpow_nonneg hX.le _) hp]
rw [← Real.rpow_mul hX.le]
have hexp : (-1 / p) * p = -1 := by field_simp
rw [hexp, Real.rpow_neg_one]
have hdk : d ^ k ≤ (d + h) ^ k :=
Real.rpow_le_rpow hd.le (by linarith) hk
have hmul : n * h ^ p * d ^ k ≤ n * h ^ p * (d + h) ^ k :=
mul_le_mul_of_nonneg_left hdk
(mul_nonneg hn.le (Real.rpow_nonneg hh.le p))
rw [heq] at hmul
rw [inv_eq_one_div]
apply (le_div_iff₀ hX).2
calc
h ^ p * (n * d ^ k) = n * h ^ p * d ^ k := by ring
_ ≤ 1 := hmul
When the root lies below the threshold, it is also bounded below by a fixed multiple of the interior scale. The result uses the hn condition, the hd condition, the hh condition, the hp condition, the hk condition, the hhd condition, the heq condition. This is the stated conclusion.
Formal statement
hypotheses
n d h p k :
ℝ
hn :
0 < n
hd :
0 < d
hh :
0 < h
hp :
0 < p
hk :
0 ≤ k
hhd :
h ≤ d
heq :
n * h ^ p * (d + h) ^ k = 1
conclusion
2 ^ (-k / p) * (n * d ^ k) ^ (-1 / p) ≤ h
Proof (Lean source)
-- @node: interior_scale_le_balance_root
lemma interior_scale_le_balance_root (n d h p k : ℝ)
(hn : 0 < n) (hd : 0 < d) (hh : 0 < h) (hp : 0 < p) (hk : 0 ≤ k)
(hhd : h ≤ d) (heq : n * h ^ p * (d + h) ^ k = 1) :
2 ^ (-k / p) * (n * d ^ k) ^ (-1 / p) ≤ h := by
have htwo : 0 < (2 : ℝ) := by norm_num
have hX : 0 < n * d ^ k := mul_pos hn (Real.rpow_pos_of_pos hd _)
have hc : 0 < (2 : ℝ) ^ (-k / p) := Real.rpow_pos_of_pos htwo _
rw [← Real.rpow_le_rpow_iff
(mul_nonneg hc.le (Real.rpow_nonneg hX.le _)) hh.le hp]
rw [Real.mul_rpow hc.le (Real.rpow_nonneg hX.le _)]
rw [← Real.rpow_mul htwo.le, ← Real.rpow_mul hX.le]
have hcExp : (-k / p) * p = -k := by field_simp
have hXExp : (-1 / p) * p = -1 := by field_simp
rw [hcExp, hXExp, Real.rpow_neg_one]
have hadd : d + h ≤ 2 * d := by linarith
have hpowadd : (d + h) ^ k ≤ (2 * d) ^ k :=
Real.rpow_le_rpow (by linarith) hadd hk
rw [Real.mul_rpow (by norm_num : (0 : ℝ) ≤ 2) hd.le] at hpowadd
have hmul : n * h ^ p * (d + h) ^ k ≤ n * h ^ p * (2 ^ k * d ^ k) :=
mul_le_mul_of_nonneg_left hpowadd
(mul_nonneg hn.le (Real.rpow_nonneg hh.le p))
rw [heq] at hmul
rw [Real.rpow_neg (by norm_num : (0 : ℝ) ≤ 2)]
have hden : 0 < 2 ^ k * (n * d ^ k) :=
mul_pos (Real.rpow_pos_of_pos htwo _) hX
have hmain : 1 / (2 ^ k * (n * d ^ k)) ≤ h ^ p := by
apply (div_le_iff₀ hden).2
calc
1 ≤ n * h ^ p * (2 ^ k * d ^ k) := hmul
_ = h ^ p * (2 ^ k * (n * d ^ k)) := by ring
calc
(2 ^ k)⁻¹ * (n * d ^ k)⁻¹ = 1 / (2 ^ k * (n * d ^ k)) := by field_simp
_ ≤ h ^ p := hmain
theorem interior_scale_le_threshold_of_edge_le Lemma interior_scale_le_threshold_of_edge_le in the paper ↗
Above the edge scale, the interior closed-form scale is no larger than the threshold. The result uses the hn condition, the hd condition, the hp condition, the hk condition, the hed condition. This is the stated conclusion.
Formal statement
hypotheses
n d p k :
ℝ
hn :
0 < n
hd :
0 < d
hp :
0 < p
hk :
0 ≤ k
hed :
n ^ (-1 / (p + k)) ≤ d
conclusion
(n * d ^ k) ^ (-1 / p) ≤ d
Proof (Lean source)
-- @node: interior_scale_le_threshold_of_edge_le
lemma interior_scale_le_threshold_of_edge_le (n d p k : ℝ)
(hn : 0 < n) (hd : 0 < d) (hp : 0 < p) (hk : 0 ≤ k)
(hed : n ^ (-1 / (p + k)) ≤ d) :
(n * d ^ k) ^ (-1 / p) ≤ d := by
have hpk : 0 < p + k := by linarith
have hepos : 0 < n ^ (-1 / (p + k)) := Real.rpow_pos_of_pos hn _
have hnpk : n * (n ^ (-1 / (p + k))) ^ (p + k) = 1 := by
rw [← Real.rpow_mul hn.le]
have : (-1 / (p + k)) * (p + k) = -1 := by field_simp
rw [this, Real.rpow_neg_one]
field_simp
have hdpow : (n ^ (-1 / (p + k))) ^ (p + k) ≤ d ^ (p + k) :=
Real.rpow_le_rpow hepos.le hed hpk.le
have hone : 1 ≤ n * d ^ (p + k) := by
rw [← hnpk]
exact mul_le_mul_of_nonneg_left hdpow hn.le
have hX : 0 < n * d ^ k := mul_pos hn (Real.rpow_pos_of_pos hd _)
rw [← Real.rpow_le_rpow_iff (Real.rpow_nonneg hX.le _) hd.le hp]
rw [← Real.rpow_mul hX.le]
have hxexp : (-1 / p) * p = -1 := by field_simp
rw [hxexp, Real.rpow_neg_one, inv_eq_one_div]
apply (div_le_iff₀ hX).2
calc
1 ≤ n * d ^ (p + k) := hone
_ = d ^ p * (n * d ^ k) := by rw [Real.rpow_add hd]; ring
Every positive balance root is at most the edge scale. The result uses the hn condition, the hd condition, the hh condition, the hp condition, the hk condition, the heq condition. This is the stated conclusion.
Formal statement
hypotheses
n d h p k :
ℝ
hn :
0 < n
hd :
0 ≤ d
hh :
0 < h
hp :
0 < p
hk :
0 ≤ k
heq :
n * h ^ p * (d + h) ^ k = 1
conclusion
h ≤ n ^ (-1 / (p + k))
Proof (Lean source)
-- @node: balance_root_le_edge_scale
lemma balance_root_le_edge_scale (n d h p k : ℝ)
(hn : 0 < n) (hd : 0 ≤ d) (hh : 0 < h) (hp : 0 < p) (hk : 0 ≤ k)
(heq : n * h ^ p * (d + h) ^ k = 1) :
h ≤ n ^ (-1 / (p + k)) := by
have hpk : 0 < p + k := by linarith
have hpow : h ^ k ≤ (d + h) ^ k :=
Real.rpow_le_rpow hh.le (by linarith) hk
have hmul : n * h ^ p * h ^ k ≤ n * h ^ p * (d + h) ^ k :=
mul_le_mul_of_nonneg_left hpow
(mul_nonneg hn.le (Real.rpow_nonneg hh.le p))
rw [heq] at hmul
rw [← Real.rpow_le_rpow_iff hh.le (Real.rpow_nonneg hn.le _) hpk]
rw [← Real.rpow_mul hn.le]
have hexp : (-1 / (p + k)) * (p + k) = -1 := by field_simp
rw [hexp, Real.rpow_neg_one, inv_eq_one_div]
apply (le_div_iff₀ hn).2
calc
h ^ (p + k) * n = n * (h ^ p * h ^ k) := by rw [Real.rpow_add hh]; ring
_ ≤ 1 := by simpa [mul_assoc] using hmul
Under an edge-scale upper bound on the threshold, a positive balance root is bounded below by a fixed multiple of that edge scale. The result uses the hn condition, the hd condition, the hh condition, the hp condition, the hk condition, the hD condition, the heq condition, the hde condition, the hhe condition. This is the stated conclusion.
Formal statement
hypotheses
n d h p k D :
ℝ
hn :
0 < n
hd :
0 ≤ d
hh :
0 < h
hp :
0 < p
hk :
0 ≤ k
hD :
0 < D
heq :
n * h ^ p * (d + h) ^ k = 1
hde :
d ≤ D * n ^ (-1 / (p + k))
hhe :
h ≤ n ^ (-1 / (p + k))
conclusion
(D + 1) ^ (-k / p) * n ^ (-1 / (p + k)) ≤ h
Proof (Lean source)
-- @node: edge_scale_le_balance_root
lemma edge_scale_le_balance_root (n d h p k D : ℝ)
(hn : 0 < n) (hd : 0 ≤ d) (hh : 0 < h) (hp : 0 < p) (hk : 0 ≤ k)
(hD : 0 < D) (heq : n * h ^ p * (d + h) ^ k = 1)
(hde : d ≤ D * n ^ (-1 / (p + k)))
(hhe : h ≤ n ^ (-1 / (p + k))) :
(D + 1) ^ (-k / p) * n ^ (-1 / (p + k)) ≤ h := by
have hpk : 0 < p + k := by linarith
let e := n ^ (-1 / (p + k))
have hepos : 0 < e := Real.rpow_pos_of_pos hn _
have hDp : 0 < D + 1 := by linarith
have hcpos : 0 < (D + 1) ^ (-k / p) := Real.rpow_pos_of_pos hDp _
rw [← Real.rpow_le_rpow_iff (mul_nonneg hcpos.le hepos.le) hh.le hp]
rw [Real.mul_rpow hcpos.le hepos.le]
rw [← Real.rpow_mul hDp.le, ← Real.rpow_mul hn.le]
have hcExp : (-k / p) * p = -k := by field_simp
have heExp : (-1 / (p + k)) * p = -p / (p + k) := by ring
rw [hcExp, heExp]
have hsum : d + h ≤ (D + 1) * e := by dsimp [e] at *; linarith
have hpow : (d + h) ^ k ≤ ((D + 1) * e) ^ k :=
Real.rpow_le_rpow (by linarith) hsum hk
rw [Real.mul_rpow hDp.le hepos.le] at hpow
have hmul : n * h ^ p * (d + h) ^ k ≤ n * h ^ p * ((D + 1) ^ k * e ^ k) :=
mul_le_mul_of_nonneg_left hpow
(mul_nonneg hn.le (Real.rpow_nonneg hh.le p))
rw [heq] at hmul
have hne : n * e ^ (p + k) = 1 := by
dsimp [e]
rw [← Real.rpow_mul hn.le]
have : (-1 / (p + k)) * (p + k) = -1 := by field_simp
rw [this, Real.rpow_neg_one]
field_simp
rw [Real.rpow_neg hDp.le]
have hnpart : n ^ (-p / (p + k)) = e ^ p := by
dsimp [e]
rw [← Real.rpow_mul hn.le]
congr 1
ring
rw [hnpart]
have htarget : ((D + 1) ^ k)⁻¹ * e ^ p =
1 / (n * (D + 1) ^ k * e ^ k) := by
have hene : n * (e ^ p * e ^ k) = 1 := by
rw [← Real.rpow_add hepos]
exact hne
field_simp
nlinarith
rw [htarget]
have hden : 0 < n * (D + 1) ^ k * e ^ k := by positivity
apply (div_le_iff₀ hden).2
calc
1 ≤ n * h ^ p * ((D + 1) ^ k * e ^ k) := hmul
_ = h ^ p * (n * (D + 1) ^ k * e ^ k) := by ring
The crossing bandwidth has the stated balance, effective count, and the two regimes separated by deltaEdge. This is the stated conclusion.
Formal statement
hypotheses
beta kappa deltaBar :
ℝ
conclusion
∃ cBalance CBalance cFar CFar : ℝ,
0 < cBalance
cBalance ≤ CBalance
0 < cFar
cFar ≤ CFar
conclusion 1
J :
ℕ
P :
ClampLaw J
L cminus cplus pmin alpha :
ℝ
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
PolynomialThinning P kappa cminus cplus
deltaSeq :
ℕ → ℝ
ThresholdSequence deltaBar deltaSeq
hSeq :
fun n => infoBandwidth n (deltaSeq n) beta kappa deltaBar
∀ᶠ n : ℕ in atTop, cBalance
≤ (n : ℝ) * (hSeq n) ^ (2 * beta + 1) * (deltaSeq n + hSeq n) ^ kappa ∧
(n : ℝ) * (hSeq n) ^ (2 * beta + 1) * (deltaSeq n + hSeq n) ^ kappa ≤ CBalance
∀ᶠ n : ℕ in atTop, cBalance * (hSeq n) ^ (-2 * beta)
≤ (n : ℝ) * hSeq n * (deltaSeq n + hSeq n) ^ kappa ∧
(n : ℝ) * hSeq n * (deltaSeq n + hSeq n) ^ kappa ≤ CBalance * (hSeq n) ^ (-2 * beta)
conclusion 2
D :
ℝ
0 < D
∃ cNear CNear : ℝ,
0 < cNear
cNear ≤ CNear
J :
ℕ
P :
ClampLaw J
L cminus cplus pmin alpha :
ℝ
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
PolynomialThinning P kappa cminus cplus
deltaSeq :
ℕ → ℝ
ThresholdSequence deltaBar deltaSeq
hSeq :
fun n => infoBandwidth n (deltaSeq n) beta kappa deltaBar
∀ᶠ n : ℕ in atTop, cNear * (n : ℝ) ^ (-(1 : ℝ) / (2 * beta + kappa + 1))
≤ hSeq n ∧ hSeq n ≤ CNear * (n : ℝ) ^ (-(1 : ℝ) / (2 * beta + kappa + 1))
Proof (Lean source)
-- @node: lem:bandwidth-phases
lemma bandwidth_phases
(beta kappa deltaBar : ℝ) :
∃ cBalance CBalance cFar CFar : ℝ,
0 < cBalance ∧ cBalance ≤ CBalance ∧ 0 < cFar ∧ cFar ≤ CFar ∧
(∀ (J : ℕ) (P : ClampLaw J) (L cminus cplus pmin alpha : ℝ),
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha →
PolynomialThinning P kappa cminus cplus →
∀ (deltaSeq : ℕ → ℝ), ThresholdSequence deltaBar deltaSeq →
let hSeq := fun n => infoBandwidth n (deltaSeq n) beta kappa deltaBar
(∀ᶠ n : ℕ in atTop,
cBalance ≤ (n : ℝ) * (hSeq n) ^ (2 * beta + 1) *
(deltaSeq n + hSeq n) ^ kappa ∧
(n : ℝ) * (hSeq n) ^ (2 * beta + 1) *
(deltaSeq n + hSeq n) ^ kappa ≤ CBalance) ∧
(∀ᶠ n : ℕ in atTop,
cBalance * (hSeq n) ^ (-2 * beta) ≤
(n : ℝ) * hSeq n * (deltaSeq n + hSeq n) ^ kappa ∧
(n : ℝ) * hSeq n * (deltaSeq n + hSeq n) ^ kappa ≤
CBalance * (hSeq n) ^ (-2 * beta)) ∧
(Tendsto (fun n => deltaSeq n / deltaEdge n beta kappa) atTop atTop →
∀ᶠ n : ℕ in atTop,
cFar * (((n : ℝ) * (deltaSeq n) ^ kappa) ^
(-(1 : ℝ) / (2 * beta + 1))) ≤ hSeq n ∧
hSeq n ≤ CFar * (((n : ℝ) * (deltaSeq n) ^ kappa) ^
(-(1 : ℝ) / (2 * beta + 1))))) ∧
∀ D : ℝ, 0 < D →
∃ cNear CNear : ℝ, 0 < cNear ∧ cNear ≤ CNear ∧
∀ (J : ℕ) (P : ClampLaw J) (L cminus cplus pmin alpha : ℝ),
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha →
PolynomialThinning P kappa cminus cplus →
∀ (deltaSeq : ℕ → ℝ), ThresholdSequence deltaBar deltaSeq →
(∀ᶠ n : ℕ in atTop,
|deltaSeq n| ≤ D * |deltaEdge n beta kappa|) →
let hSeq := fun n => infoBandwidth n (deltaSeq n) beta kappa deltaBar
∀ᶠ n : ℕ in atTop,
cNear * (n : ℝ) ^ (-(1 : ℝ) / (2 * beta + kappa + 1)) ≤ hSeq n ∧
hSeq n ≤ CNear * (n : ℝ) ^
(-(1 : ℝ) / (2 * beta + kappa + 1)) := by
let p : ℝ := 2 * beta + 1
let rawFar : ℝ := 2 ^ (-kappa / p)
let cFar : ℝ := min 1 rawFar
have hrawFar : 0 < rawFar := Real.rpow_pos_of_pos (by norm_num) _
have hcFar : 0 < cFar := lt_min (by norm_num) hrawFar
have hcFarOne : cFar ≤ 1 := min_le_left _ _
refine ⟨1, 1, cFar, 1, by norm_num, le_rfl, hcFar, hcFarOne, ?_, ?_⟩
· intro J P L cminus cplus pmin alpha hreg hthin deltaSeq hdelta
rcases hreg with ⟨hJ, hbeta, hkappa, hL, hcminus, hcm, hcp, hpmin,
hpminJ, hdeltaBar_pos, hdeltaBar_lt, halpha, halpha_lt⟩
have hp : 0 < p := by dsimp [p]; linarith
have hrawFarOne : rawFar ≤ 1 := by
rw [show (1 : ℝ) = 2 ^ (0 : ℝ) by norm_num]
apply Real.rpow_le_rpow_of_exponent_le (by norm_num)
exact div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hkappa) hp.le
have hcFarEq : cFar = rawFar := min_eq_right hrawFarOne
dsimp
have hbalance := infoBandwidth_eventually_balance beta kappa deltaBar hbeta hkappa
⟨hdeltaBar_pos, hdeltaBar_lt⟩ deltaSeq hdelta
refine ⟨?_, ?_, ?_⟩
· filter_upwards [hbalance] with n hn
simpa using ⟨hn.2.2.ge, hn.2.2.le⟩
· filter_upwards [hbalance] with n hn
have hh := hn.1
have heq := hn.2.2
have hsplit : infoBandwidth n (deltaSeq n) beta kappa deltaBar ^
(2 * beta + 1) =
infoBandwidth n (deltaSeq n) beta kappa deltaBar ^ (2 * beta) *
infoBandwidth n (deltaSeq n) beta kappa deltaBar := by
simpa using Real.rpow_add hh (2 * beta) 1
have hneg : infoBandwidth n (deltaSeq n) beta kappa deltaBar ^ (-2 * beta) =
(infoBandwidth n (deltaSeq n) beta kappa deltaBar ^ (2 * beta))⁻¹ := by
rw [show -2 * beta = -(2 * beta) by ring, Real.rpow_neg hh.le]
rw [hsplit] at heq
rw [hneg]
have hpowpos :
0 < infoBandwidth n (deltaSeq n) beta kappa deltaBar ^ (2 * beta) :=
Real.rpow_pos_of_pos hh _
constructor <;> field_simp <;> nlinarith
· intro hfar
have hratio := (Filter.tendsto_atTop.1 hfar) 1
filter_upwards [hbalance, hratio] with n hn hratioN
have hnpos : 0 < (n : ℝ) := by
by_contra hn0
have hnzero : (n : ℝ) = 0 :=
le_antisymm (le_of_not_gt hn0) (Nat.cast_nonneg n)
rw [hnzero, zero_mul] at hn
norm_num at hn
have hedgePos : 0 < deltaEdge n beta kappa :=
Real.rpow_pos_of_pos hnpos _
have hdeltaPos : 0 < deltaSeq n := by
have : deltaEdge n beta kappa ≤ deltaSeq n := by
have := (le_div_iff₀ hedgePos).mp hratioN
simpa using this
exact hedgePos.trans_le this
have hedge : (n : ℝ) ^ (-1 / (p + kappa)) ≤ deltaSeq n := by
have hle : deltaEdge n beta kappa ≤ deltaSeq n := by
have := (le_div_iff₀ hedgePos).mp hratioN
simpa using this
simpa [deltaEdge, p, show 2 * beta + kappa + 1 =
(2 * beta + 1) + kappa by ring] using hle
let h := infoBandwidth n (deltaSeq n) beta kappa deltaBar
let w := ((n : ℝ) * (deltaSeq n) ^ kappa) ^ (-1 / p)
have hwUpper : h ≤ w := by
exact balance_root_le_interior_scale (n : ℝ) (deltaSeq n) h p kappa hnpos
hdeltaPos hn.1 hp hkappa (by simpa [h, p] using hn.2.2)
have hwDelta : w ≤ deltaSeq n := by
exact interior_scale_le_threshold_of_edge_le (n : ℝ) (deltaSeq n) p kappa
hnpos hdeltaPos hp hkappa hedge
have hwLower : rawFar * w ≤ h := by
exact interior_scale_le_balance_root (n : ℝ) (deltaSeq n) h p kappa hnpos
hdeltaPos hn.1 hp hkappa (hwUpper.trans hwDelta)
(by simpa [h, p] using hn.2.2)
simpa [h, w, hcFarEq, rawFar, p] using ⟨hwLower, hwUpper⟩
· intro D hD
let rawNear : ℝ := (D + 1) ^ (-kappa / p)
let cNear : ℝ := min 1 rawNear
have hDp : 0 < D + 1 := by linarith
have hrawNear : 0 < rawNear := Real.rpow_pos_of_pos hDp _
have hcNear : 0 < cNear := lt_min (by norm_num) hrawNear
have hcNearOne : cNear ≤ 1 := min_le_left _ _
refine ⟨cNear, 1, hcNear, hcNearOne, ?_⟩
intro J P L cminus cplus pmin alpha hreg hthin deltaSeq hdelta hnear
rcases hreg with ⟨hJ, hbeta, hkappa, hL, hcminus, hcm, hcp, hpmin,
hpminJ, hdeltaBar_pos, hdeltaBar_lt, halpha, halpha_lt⟩
have hp : 0 < p := by dsimp [p]; linarith
have hrawNearOne : rawNear ≤ 1 := by
rw [show (1 : ℝ) = (D + 1) ^ (0 : ℝ) by simp]
apply Real.rpow_le_rpow_of_exponent_le (by linarith)
exact div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hkappa) hp.le
have hcNearEq : cNear = rawNear := min_eq_right hrawNearOne
dsimp
have hbalance := infoBandwidth_eventually_balance beta kappa deltaBar hbeta hkappa
⟨hdeltaBar_pos, hdeltaBar_lt⟩ deltaSeq hdelta
filter_upwards [hbalance, hnear] with n hn hnearN
have hnpos : 0 < (n : ℝ) := by
by_contra hn0
have hnzero : (n : ℝ) = 0 := le_antisymm (le_of_not_gt hn0) (Nat.cast_nonneg n)
rw [hnzero, zero_mul] at hn
norm_num at hn
have hedgePos : 0 < deltaEdge n beta kappa := Real.rpow_pos_of_pos hnpos _
have hdeltaNonneg : 0 ≤ deltaSeq n := (hdelta n).1
have hdeltaEdge : deltaSeq n ≤ D * deltaEdge n beta kappa := by
simpa [abs_of_nonneg hdeltaNonneg, abs_of_pos hedgePos] using hnearN
let h := infoBandwidth n (deltaSeq n) beta kappa deltaBar
let e := (n : ℝ) ^ (-1 / (p + kappa))
have hedgeEq : deltaEdge n beta kappa = e := by
simp [deltaEdge, e, p, show 2 * beta + kappa + 1 =
(2 * beta + 1) + kappa by ring]
have hde : deltaSeq n ≤ D * e := by simpa [hedgeEq] using hdeltaEdge
have hhe : h ≤ e := by
exact balance_root_le_edge_scale (n : ℝ) (deltaSeq n) h p kappa hnpos
hdeltaNonneg hn.1 hp hkappa (by simpa [h, p] using hn.2.2)
have hlow : rawNear * e ≤ h := by
exact edge_scale_le_balance_root (n : ℝ) (deltaSeq n) h p kappa D hnpos
hdeltaNonneg hn.1 hp hkappa hD (by simpa [h, p] using hn.2.2) hde hhe
simpa [h, e, hcNearEq, rawNear, p, show 2 * beta + kappa + 1 =
(2 * beta + 1) + kappa by ring] using ⟨hlow, hhe⟩
theorem infoBandwidth_interior_asymp
If the threshold is asymptotically above the design edge, the information bandwidth has the interior closed-form order. Unlike bandwidth_phases, this purely analytic projection does not require an otherwise unused model law. The result uses the hreg condition, the hdelta condition, the hfar condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
deltaSeq :
ℕ → ℝ
hdelta :
ThresholdSequence deltaBar deltaSeq
conclusion
AsympSeq (fun n => infoBandwidth n (deltaSeq n) beta kappa deltaBar) (fun n => ((n : ℝ) * (deltaSeq n) ^ kappa) ^ (-(1 : ℝ) / (2 * beta + 1)))
Proof (Lean source)
-- @node: infoBandwidth_interior_asymp
lemma infoBandwidth_interior_asymp
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(deltaSeq : ℕ → ℝ) (hdelta : ThresholdSequence deltaBar deltaSeq)
(hfar : Tendsto (fun n => deltaSeq n / deltaEdge n beta kappa) atTop atTop) :
AsympSeq (fun n => infoBandwidth n (deltaSeq n) beta kappa deltaBar)
(fun n => ((n : ℝ) * (deltaSeq n) ^ kappa) ^
(-(1 : ℝ) / (2 * beta + 1))) := by
rcases hreg with ⟨hJ, hbeta, hkappa, hL, hcminus, hcm, hcp, hpmin,
hpminJ, hdeltaBar_pos, hdeltaBar_lt, halpha, halpha_lt⟩
have hbalance := infoBandwidth_eventually_balance beta kappa deltaBar hbeta hkappa
⟨hdeltaBar_pos, hdeltaBar_lt⟩ deltaSeq hdelta
let p : ℝ := 2 * beta + 1
let c : ℝ := 2 ^ (-kappa / p)
have hp : 0 < p := by dsimp [p]; linarith
have hc : 0 < c := Real.rpow_pos_of_pos (by norm_num) _
have hcOne : c ≤ 1 := by
rw [show (1 : ℝ) = 2 ^ (0 : ℝ) by norm_num]
apply Real.rpow_le_rpow_of_exponent_le (by norm_num)
exact div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hkappa) hp.le
refine ⟨c, 1, hc, hcOne, ?_⟩
have hratio := (Filter.tendsto_atTop.1 hfar) 1
filter_upwards [hbalance, hratio] with n hn hratioN
have hnpos : 0 < (n : ℝ) := by
by_contra hn0
have hnzero : (n : ℝ) = 0 :=
le_antisymm (le_of_not_gt hn0) (Nat.cast_nonneg n)
rw [hnzero, zero_mul] at hn
norm_num at hn
have hedgePos : 0 < deltaEdge n beta kappa :=
Real.rpow_pos_of_pos hnpos _
have hdeltaPos : 0 < deltaSeq n := by
have hle : deltaEdge n beta kappa ≤ deltaSeq n := by
have := (le_div_iff₀ hedgePos).mp hratioN
simpa using this
exact hedgePos.trans_le hle
have hedge : (n : ℝ) ^ (-1 / (p + kappa)) ≤ deltaSeq n := by
have hle : deltaEdge n beta kappa ≤ deltaSeq n := by
have := (le_div_iff₀ hedgePos).mp hratioN
simpa using this
simpa [deltaEdge, p, show 2 * beta + kappa + 1 =
(2 * beta + 1) + kappa by ring] using hle
let h := infoBandwidth n (deltaSeq n) beta kappa deltaBar
let w := ((n : ℝ) * (deltaSeq n) ^ kappa) ^ (-1 / p)
have hwUpper : h ≤ w := by
exact balance_root_le_interior_scale (n : ℝ) (deltaSeq n) h p kappa hnpos
hdeltaPos hn.1 hp hkappa (by simpa [h, p] using hn.2.2)
have hwDelta : w ≤ deltaSeq n := by
exact interior_scale_le_threshold_of_edge_le (n : ℝ) (deltaSeq n) p kappa
hnpos hdeltaPos hp hkappa hedge
have hwLower : c * w ≤ h := by
exact interior_scale_le_balance_root (n : ℝ) (deltaSeq n) h p kappa hnpos
hdeltaPos hn.1 hp hkappa (hwUpper.trans hwDelta)
(by simpa [h, p] using hn.2.2)
simpa [h, w, c, p] using ⟨hwLower, hwUpper⟩
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.CausalBridgeIdentification 11 declarations This file turns the normalized-stratum product law into the conditional integral identity and the pathwise decomposition used by the two causal bridge theorems.
Identification identities for the causal clamp bridge
This file turns the normalized-stratum product law into the conditional integral identity and the pathwise decomposition used by the two causal bridge theorems.
def structuralResponseExtension
A globally measurable extension of a structural response map from its declared unit-dose domain.
Definition (Lean source)
parameters
def
structuralResponseExtension PF hcons x :
ℝ × PF.latentCarrier → ℝ
given by
Function.extend (fun z : Icc (0 : ℝ) 1 × PF.latentCarrier => (z.1.1, z.2)) (fun z => PF.g x z.1.1 z.2) 0
theorem structuralResponseExtension_measurable
The structural-response extension is measurable. The result uses the hcons condition. This is the stated conclusion.
Formal statement
hypotheses
conclusion
Measurable (structuralResponseExtension PF hcons x)
Proof (Lean source)
-- @node: structuralResponseExtension_measurable
lemma structuralResponseExtension_measurable (PF : FullDataLaw J)
(hcons : LatentResponseConsistency PF) (x : Fin J) :
Measurable (structuralResponseExtension PF hcons x) := by
let e : Icc (0 : ℝ) 1 × PF.latentCarrier → ℝ × PF.latentCarrier :=
fun z => (z.1.1, z.2)
have he : MeasurableEmbedding e :=
(MeasurableEmbedding.subtype_coe measurableSet_Icc).prodMap
MeasurableEmbedding.id
exact he.measurable_extend (hcons.1 x) measurable_const
theorem structuralResponseExtension_eq
On the unit dose interval, the measurable extension equals the original structural response. The result uses the hcons condition, the ha condition. This is the stated conclusion.
Formal statement
hypotheses
conclusion
structuralResponseExtension PF hcons x (a, u) = PF.g x a u
Proof (Lean source)
-- @node: structuralResponseExtension_eq
lemma structuralResponseExtension_eq (PF : FullDataLaw J)
(hcons : LatentResponseConsistency PF) (x : Fin J) (a : ℝ)
(ha : a ∈ Icc (0 : ℝ) 1) (u : PF.latentCarrier) :
structuralResponseExtension PF hcons x (a, u) = PF.g x a u := by
let e : Icc (0 : ℝ) 1 × PF.latentCarrier → ℝ × PF.latentCarrier :=
fun z => (z.1.1, z.2)
have he : Injective e :=
((MeasurableEmbedding.subtype_coe measurableSet_Icc).prodMap
MeasurableEmbedding.id).injective
simp only [structuralResponseExtension]
exact he.extend_apply (fun z => PF.g x z.1.1 z.2) 0 (⟨a, ha⟩, u)
theorem fullDataResponseMean_setIntegral
The full-data stratum conditional mean satisfies the same design-cell integral identity as the observed regression. The result uses the hPF condition, the hkappa condition, the hcplus condition, the hpmin condition, the hB condition, the hBunit condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin deltaBar :
ℝ
PF :
hPF :
BridgeFullDataClampModel PF kappa cminus cplus pmin deltaBar
hkappa :
0 ≤ kappa
hcplus :
0 ≤ cplus
hpmin :
0 < pmin
x :
Fin J
B :
Set ℝ
hB :
hBunit :
B ⊆ Icc (0 : ℝ) 1
conclusion
(PF.observedMargin.px x)⁻¹ * (∫ z in {z : ClampObs J × PF.latentCarrier | z.1.X = x ∧ z.1.A ∈ B}, z.1.Y ∂PF.fullMeasure)
= ∫ a in B, fullDataResponseMean PF x a * PF.observedMargin.pi x a
Proof (Lean source)
-- @node: fullDataResponseMean_setIntegral
lemma fullDataResponseMean_setIntegral
(J : ℕ) (kappa cminus cplus pmin deltaBar : ℝ)
(PF : FullDataLaw J)
(hPF : BridgeFullDataClampModel PF kappa cminus cplus pmin deltaBar)
(hkappa : 0 ≤ kappa) (hcplus : 0 ≤ cplus) (hpmin : 0 < pmin)
(x : Fin J) (B : Set ℝ) (hB : MeasurableSet B)
(hBunit : B ⊆ Icc (0 : ℝ) 1) :
(PF.observedMargin.px x)⁻¹ *
(∫ z in {z : ClampObs J × PF.latentCarrier | z.1.X = x ∧ z.1.A ∈ B},
z.1.Y ∂PF.fullMeasure) =
∫ a in B, fullDataResponseMean PF x a * PF.observedMargin.pi x a := by
classical
let μU := Measure.map (fun z : ClampObs J × PF.latentCarrier => z.2)
((ofReal (PF.observedMargin.px x))⁻¹ •
PF.fullMeasure.restrict {z | z.1.X = x})
let F : ℝ × PF.latentCarrier → ℝ := fun w =>
if w.1 ∈ B then structuralResponseExtension PF hPF.consistency x w else 0
have hF : Measurable F := by
exact (structuralResponseExtension_measurable PF hPF.consistency x).piecewise
(hB.preimage measurable_fst) measurable_const
have hFbound : ∀ w, ‖F w‖ ≤ (1 : ℝ) := by
intro w
by_cases hw : w.1 ∈ B
· have hunit := hBunit hw
rw [show F w = PF.g x w.1 w.2 by
simp [F, hw, structuralResponseExtension_eq PF hPF.consistency x w.1 hunit w.2]]
rw [Real.norm_eq_abs]
exact abs_le.2 ⟨by linarith [(hPF.consistency.2.1 x w.1 w.2 hunit).1],
(hPF.consistency.2.1 x w.1 w.2 hunit).2⟩
· simp [F, hw]
have hprod := conditional_stratum_integral_prod J kappa cminus cplus
pmin PF hPF.observedModel hPF.exchangeability hkappa hcplus hpmin x F hF 1 hFbound
have hcell : MeasurableSet
{z : ClampObs J × PF.latentCarrier | z.1.X = x ∧ z.1.A ∈ B} := by
have hX : Measurable (fun z : ClampObs J × PF.latentCarrier => z.1.X) :=
(measurable_fst.comp (Measurable.of_comap_le le_rfl)).comp measurable_fst
have hA : Measurable (fun z : ClampObs J × PF.latentCarrier => z.1.A) :=
(measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))).comp measurable_fst
exact (measurableSet_eq_fun
hX measurable_const).inter (hB.preimage hA)
have hstratum : MeasurableSet
{z : ClampObs J × PF.latentCarrier | z.1.X = x} := by
have hX : Measurable (fun z : ClampObs J × PF.latentCarrier => z.1.X) :=
(measurable_fst.comp (Measurable.of_comap_le le_rfl)).comp measurable_fst
exact measurableSet_eq_fun
hX measurable_const
have hcellEq :
(∫ z in {z : ClampObs J × PF.latentCarrier | z.1.X = x ∧ z.1.A ∈ B},
z.1.Y ∂PF.fullMeasure) =
∫ z in {z : ClampObs J × PF.latentCarrier | z.1.X = x},
F (z.1.A, z.2) ∂PF.fullMeasure := by
rw [← integral_indicator hcell, ← integral_indicator hstratum]
apply integral_congr_ae
filter_upwards [hPF.consistency.2.2] with z hz
by_cases hx : z.1.X = x
· by_cases hBa : z.1.A ∈ B
· have ha : z.1.A ∈ Icc (0 : ℝ) 1 := hBunit hBa
simp only [Set.indicator_of_mem
(show z ∈ {z : ClampObs J × PF.latentCarrier | z.1.X = x ∧ z.1.A ∈ B}
from ⟨hx, hBa⟩),
Set.indicator_of_mem
(show z ∈ {z : ClampObs J × PF.latentCarrier | z.1.X = x} from hx)]
rw [show F (z.1.A, z.2) = PF.g x z.1.A z.2 by
simp [F, hBa,
structuralResponseExtension_eq PF hPF.consistency x z.1.A ha z.2],
← hx, ← hz.2]
· simp [F, hx, hBa]
· simp [F, hx]
have hsupp : ∀ᵐ a ∂conditionalTreatmentMeasure PF.observedMargin x,
a ∈ Icc (0 : ℝ) 1 := by
unfold conditionalTreatmentMeasure
exact (withDensity_absolutelyContinuous _ _).ae_le
(ae_restrict_mem measurableSet_Icc)
calc
_ = (PF.observedMargin.px x)⁻¹ *
(∫ z in {z : ClampObs J × PF.latentCarrier | z.1.X = x},
F (z.1.A, z.2) ∂PF.fullMeasure) := by rw [hcellEq]
_ = ∫ a, ∫ u, F (a, u) ∂μU
∂conditionalTreatmentMeasure PF.observedMargin x := hprod
_ = ∫ a, B.indicator (fullDataResponseMean PF x) a
∂conditionalTreatmentMeasure PF.observedMargin x := by
apply integral_congr_ae
filter_upwards [hsupp] with a ha
by_cases hBa : a ∈ B
· rw [Set.indicator_of_mem hBa]
simp only [F, if_pos hBa]
rw [integral_congr_ae (ae_of_all _ fun u =>
structuralResponseExtension_eq PF hPF.consistency x a ha u)]
exact conditional_stratum_latent_integral_eq_mean J kappa cminus
cplus pmin PF hPF.observedModel hpmin hPF.consistency x a ha
· rw [show (fun u => F (a, u)) = fun _ => 0 by
funext u
simp [F, hBa], integral_zero, Set.indicator_of_notMem hBa]
_ = ∫ a in Icc (0 : ℝ) 1,
PF.observedMargin.pi x a * B.indicator (fullDataResponseMean PF x) a :=
conditionalTreatment_integral_eq_density PF.observedMargin
hPF.observedModel x _
_ = ∫ a in B, fullDataResponseMean PF x a * PF.observedMargin.pi x a := by
rw [← integral_indicator measurableSet_Icc, ← integral_indicator hB]
apply integral_congr_ae
filter_upwards with a
by_cases hBa : a ∈ B
· have ha := hBunit hBa
simp [Set.indicator_of_mem hBa, ha, mul_comm]
· by_cases ha : a ∈ Icc (0 : ℝ) 1 <;>
simp [Set.indicator_of_notMem hBa, ha]
theorem continuous_mul_treatmentDensity_integrable
A continuous regression times the polynomially bounded treatment density is integrable on the threshold interval. The result uses the hcond condition, the hthin condition, the hkappa condition, the hcplus condition, the hdelta condition, the hmu condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
kappa cminus cplus deltaBar :
ℝ
hcond :
hthin :
PolynomialThinning P kappa cminus cplus
hkappa :
0 ≤ kappa
hcplus :
0 ≤ cplus
hdelta :
deltaBar ≤ 1
x :
Fin J
mu :
ℝ → ℝ
hmu :
ContinuousOn mu (Icc (0 : ℝ) deltaBar)
conclusion
Proof (Lean source)
-- @node: continuous_mul_treatmentDensity_integrable
lemma continuous_mul_treatmentDensity_integrable
(P : ClampLaw J) (kappa cminus cplus deltaBar : ℝ)
(hcond : CondDensityLaw P)
(hthin : PolynomialThinning P kappa cminus cplus)
(hkappa : 0 ≤ kappa) (hcplus : 0 ≤ cplus)
(hdelta : deltaBar ≤ 1) (x : Fin J) (mu : ℝ → ℝ)
(hmu : ContinuousOn mu (Icc (0 : ℝ) deltaBar)) :
IntegrableOn (fun a => mu a * P.pi x a) (Icc (0 : ℝ) deltaBar) volume := by
have hmuInt : IntegrableOn mu (Icc (0 : ℝ) deltaBar) volume :=
hmu.integrableOn_Icc
have hpiMeas : AEStronglyMeasurable (P.pi x)
(volume.restrict (Icc (0 : ℝ) deltaBar)) := by
exact (aemeasurable_restrict_of_measurable_subtype measurableSet_Icc
(hcond.1 x)).mono_measure
(Measure.restrict_mono_set volume (Set.Icc_subset_Icc_right hdelta)) |>.aestronglyMeasurable
have hpiBound : ∀ᵐ a ∂volume.restrict (Icc (0 : ℝ) deltaBar),
‖P.pi x a‖ ≤ cplus := by
filter_upwards [hcond.2.1 x |>.filter_mono
(ae_mono (Measure.restrict_mono_set volume (Set.Icc_subset_Icc_right hdelta))),
hthin x |>.filter_mono
(ae_mono (Measure.restrict_mono_set volume (Set.Icc_subset_Icc_right hdelta))),
ae_restrict_mem measurableSet_Icc] with a hnon henv ha
rw [Real.norm_eq_abs, abs_of_nonneg hnon]
exact henv.2.trans (mul_le_of_le_one_right hcplus
(Real.rpow_le_one ha.1 (ha.2.trans hdelta) hkappa))
have h := hmuInt.bdd_mul hpiMeas hpiBound
simpa only [IntegrableOn, mul_comm] using h
theorem continuous_regression_eq_of_density_integrals
Equality of all density-weighted cell integrals identifies two continuous regressions pointwise on the threshold interval. The result uses the hcond condition, the hthin condition, the hkappa condition, the hcminus condition, the hcplus condition, the hdelta condition, the hdeltaOne condition, the hcont₁ condition, the hcont₂ condition, the hint₁ condition, the hint₂ condition, the hint condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
kappa cminus cplus deltaBar :
ℝ
hcond :
hthin :
PolynomialThinning P kappa cminus cplus
hkappa :
0 ≤ kappa
hcminus :
0 < cminus
hcplus :
0 ≤ cplus
hdelta :
0 < deltaBar
hdeltaOne :
deltaBar ≤ 1
x :
Fin J
mu₁ mu₂ :
ℝ → ℝ
hcont₁ :
ContinuousOn mu₁ (Icc (0 : ℝ) deltaBar)
hcont₂ :
ContinuousOn mu₂ (Icc (0 : ℝ) deltaBar)
hint₁ :
hint₂ :
Proof (Lean source)
-- @node: continuous_regression_eq_of_density_integrals
lemma continuous_regression_eq_of_density_integrals
(P : ClampLaw J) (kappa cminus cplus deltaBar : ℝ)
(hcond : CondDensityLaw P)
(hthin : PolynomialThinning P kappa cminus cplus)
(hkappa : 0 ≤ kappa) (hcminus : 0 < cminus) (hcplus : 0 ≤ cplus)
(hdelta : 0 < deltaBar) (hdeltaOne : deltaBar ≤ 1)
(x : Fin J) (mu₁ mu₂ : ℝ → ℝ)
(hcont₁ : ContinuousOn mu₁ (Icc (0 : ℝ) deltaBar))
(hcont₂ : ContinuousOn mu₂ (Icc (0 : ℝ) deltaBar))
(hint₁ : IntegrableOn (fun a => mu₁ a * P.pi x a)
(Icc (0 : ℝ) deltaBar) volume)
(hint₂ : IntegrableOn (fun a => mu₂ a * P.pi x a)
(Icc (0 : ℝ) deltaBar) volume)
(hint : ∀ B : Set ℝ, MeasurableSet B → B ⊆ Icc (0 : ℝ) deltaBar →
(∫ a in B, mu₁ a * P.pi x a) = ∫ a in B, mu₂ a * P.pi x a) :
EqOn mu₁ mu₂ (Icc (0 : ℝ) deltaBar) := by
have haeProd : (fun a => mu₁ a * P.pi x a) =ᵐ[
volume.restrict (Icc (0 : ℝ) deltaBar)]
fun a => mu₂ a * P.pi x a := by
refine Integrable.ae_eq_of_forall_setIntegral_eq
(fun a => mu₁ a * P.pi x a) (fun a => mu₂ a * P.pi x a)
hint₁ hint₂ ?_
intro S hS _
rw [Measure.restrict_restrict hS]
exact hint (S ∩ Icc (0 : ℝ) deltaBar)
(hS.inter measurableSet_Icc) inter_subset_right
have hpiPos : ∀ᵐ a ∂volume.restrict (Icc (0 : ℝ) deltaBar),
0 < P.pi x a := by
filter_upwards [hthin x |>.filter_mono
(ae_mono (Measure.restrict_mono_set volume
(Set.Icc_subset_Icc_right hdeltaOne))),
ae_restrict_mem measurableSet_Icc,
(volume.ae_ne 0).filter_mono (ae_mono Measure.restrict_le_self)] with a henv ha ha0
have haPos : 0 < a := lt_of_le_of_ne ha.1 (Ne.symm ha0)
exact lt_of_lt_of_le (mul_pos hcminus (Real.rpow_pos_of_pos haPos kappa)) henv.1
have hae : mu₁ =ᵐ[volume.restrict (Icc (0 : ℝ) deltaBar)] mu₂ := by
filter_upwards [haeProd, hpiPos] with a heq hpos
exact (mul_right_cancel₀ (ne_of_gt hpos)) heq
exact volume.eqOn_Icc_of_ae_eq (ne_of_lt hdelta) hae hcont₁ hcont₂
theorem fullData_observed_setIntegral
Set integrals of observed-coordinate functions agree under a full-data law and its observed margin. The result uses the hS condition, the hf condition. This is the stated conclusion.
Formal statement
Proof (Lean source)
-- @node: fullData_observed_setIntegral
lemma fullData_observed_setIntegral (PF : FullDataLaw J) (S : Set (ClampObs J))
(hS : MeasurableSet S) (f : ClampObs J → ℝ) (hf : Measurable f) :
(∫ z in (fun z : ClampObs J × PF.latentCarrier => z.1) ⁻¹' S,
f z.1 ∂PF.fullMeasure) =
∫ o in S, f o ∂PF.observedMargin.dataMeasure := by
rw [← integral_indicator (hS.preimage measurable_fst), ← integral_indicator hS]
rw [← PF.margin_eq]
exact (integral_map measurable_fst.aemeasurable
(hf.indicator hS).aestronglyMeasurable).symm
theorem fullDataResponseMean_eq_mu
Full-data and fixed-Hölder observed regressions agree pointwise on the declared threshold interval. The result uses the hPF condition, the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
PF :
hPF :
FullDataClampModel PF beta kappa L cminus cplus pmin deltaBar
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
x :
Fin J
a :
ℝ
a ∈ Icc (0 : ℝ) deltaBar
conclusion
fullDataResponseMean PF x a = PF.observedMargin.mu x a
Proof (Lean source)
-- @node: fullDataResponseMean_eq_mu
lemma fullDataResponseMean_eq_mu
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(PF : FullDataLaw J)
(hPF : FullDataClampModel PF beta kappa L cminus cplus pmin deltaBar)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
∀ (x : Fin J) (a : ℝ), a ∈ Icc (0 : ℝ) deltaBar →
fullDataResponseMean PF x a = PF.observedMargin.mu x a := by
have hreg' := hreg
rcases hreg' with
⟨_, _, hkappa, _, hcminus, _, hcplusLower, hpmin, _, hdelta,
hdeltaOne, _, _⟩
have hcplus : 0 ≤ cplus := le_trans (by linarith) hcplusLower
intro x a ha
have hEq : EqOn (fullDataResponseMean PF x) (PF.observedMargin.mu x)
(Icc (0 : ℝ) deltaBar) := by
apply continuous_regression_eq_of_density_integrals PF.observedMargin kappa
cminus cplus deltaBar hPF.observedModel.condDensity
hPF.observedModel.thinning hkappa hcminus hcplus hdelta hdeltaOne.le x
· exact hPF.responseContinuity x
· exact (hPF.observedModel.holder x).1.mono
(Set.Icc_subset_Icc_right hdeltaOne.le)
· exact continuous_mul_treatmentDensity_integrable PF.observedMargin kappa
cminus cplus deltaBar hPF.observedModel.condDensity
hPF.observedModel.thinning hkappa hcplus hdeltaOne.le x _
(hPF.responseContinuity x)
· exact continuous_mul_treatmentDensity_integrable PF.observedMargin kappa
cminus cplus deltaBar hPF.observedModel.condDensity
hPF.observedModel.thinning hkappa hcplus hdeltaOne.le x _
((hPF.observedModel.holder x).1.mono
(Set.Icc_subset_Icc_right hdeltaOne.le))
· intro B hB hBdelta
have hBunit : B ⊆ Icc (0 : ℝ) 1 :=
hBdelta.trans (Set.Icc_subset_Icc_right hdeltaOne.le)
have hfull := fullDataResponseMean_setIntegral J kappa cminus
cplus pmin deltaBar PF hPF.toBridge hkappa hcplus hpmin x B hB hBunit
have hobs := holderRegression_setIntegral PF.observedMargin
hPF.observedModel hkappa hcplus hpmin x B hB hBunit
have hmargin :
(∫ z in {z : ClampObs J × PF.latentCarrier |
z.1.X = x ∧ z.1.A ∈ B}, z.1.Y ∂PF.fullMeasure) =
∫ o in {o : ClampObs J | o.X = x ∧ o.A ∈ B},
o.Y ∂PF.observedMargin.dataMeasure := by
exact fullData_observed_setIntegral PF
{o : ClampObs J | o.X = x ∧ o.A ∈ B}
(by
exact (measurableSet_eq_fun
(measurable_fst.comp (Measurable.of_comap_le le_rfl)) measurable_const).inter
(hB.preimage (measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))))) _ clampOutcome_measurable
have hpx : 0 < PF.observedMargin.px x :=
lt_of_lt_of_le hpmin (hPF.observedModel.stratumMass x).2
calc
(∫ a in B, fullDataResponseMean PF x a * PF.observedMargin.pi x a) =
(PF.observedMargin.px x)⁻¹ *
(∫ z in {z : ClampObs J × PF.latentCarrier |
z.1.X = x ∧ z.1.A ∈ B}, z.1.Y ∂PF.fullMeasure) := hfull.symm
_ = (PF.observedMargin.px x)⁻¹ *
(∫ o in {o : ClampObs J | o.X = x ∧ o.A ∈ B},
o.Y ∂PF.observedMargin.dataMeasure) := by rw [hmargin]
_ = (PF.observedMargin.px x)⁻¹ * (PF.observedMargin.px x *
∫ a in B, PF.observedMargin.mu x a * PF.observedMargin.pi x a) := by
rw [hobs]
_ = ∫ a in B, PF.observedMargin.mu x a * PF.observedMargin.pi x a := by
field_simp
exact hEq ha
theorem fullDataResponseMean_eq_contRegression
Full-data and continuity-only observed regressions agree pointwise on the declared threshold interval. The result uses the hPF condition, the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin deltaBar :
ℝ
PF :
hPF :
ContFullDataClampModel PF kappa cminus cplus pmin deltaBar
hreg :
ContDesignConstants J kappa cminus cplus pmin deltaBar
x :
Fin J
a :
ℝ
a ∈ Icc (0 : ℝ) deltaBar
conclusion
fullDataResponseMean PF x a
= contRegression PF.observedMargin kappa cminus cplus pmin deltaBar hPF.observedModel x a
Proof (Lean source)
-- @node: fullDataResponseMean_eq_contRegression
lemma fullDataResponseMean_eq_contRegression
(J : ℕ) (kappa cminus cplus pmin deltaBar : ℝ)
(PF : FullDataLaw J)
(hPF : ContFullDataClampModel PF kappa cminus cplus pmin deltaBar)
(hreg : ContDesignConstants J kappa cminus cplus pmin deltaBar) :
∀ (x : Fin J) (a : ℝ), a ∈ Icc (0 : ℝ) deltaBar →
fullDataResponseMean PF x a =
contRegression PF.observedMargin kappa cminus cplus pmin deltaBar
hPF.observedModel x a := by
rcases hreg with ⟨_, hkappa, hcminus, _, hcplusLower, hpmin, _, hdelta, hdeltaOne⟩
have hcplus : 0 ≤ cplus := le_trans (by linarith) hcplusLower
let mu := contRegression PF.observedMargin kappa cminus cplus pmin deltaBar
hPF.observedModel
have hcont : ∀ x, ContinuousOn (mu x) (Icc (0 : ℝ) deltaBar) := by
simpa [mu, contRegression] using
(Classical.choose_spec hPF.observedModel.continuousVersion).1
have hver : PF.observedMargin.dataMeasure[(fun o : ClampObs J => o.Y) |
comap (fun o : ClampObs J => (o.X, o.A)) inferInstance]
=ᵐ[PF.observedMargin.dataMeasure] fun o => mu o.X o.A := by
simpa [mu, contRegression] using
(Classical.choose_spec hPF.observedModel.continuousVersion).2
intro x a ha
apply continuous_regression_eq_of_density_integrals PF.observedMargin kappa
cminus cplus deltaBar hPF.observedModel.condDensity hPF.observedModel.thinning
hkappa hcminus hcplus hdelta hdeltaOne.le x
(fullDataResponseMean PF x) (mu x)
(hPF.responseContinuity x) (hcont x)
· exact continuous_mul_treatmentDensity_integrable PF.observedMargin kappa
cminus cplus deltaBar hPF.observedModel.condDensity hPF.observedModel.thinning
hkappa hcplus hdeltaOne.le x _ (hPF.responseContinuity x)
· exact continuous_mul_treatmentDensity_integrable PF.observedMargin kappa
cminus cplus deltaBar hPF.observedModel.condDensity hPF.observedModel.thinning
hkappa hcplus hdeltaOne.le x _ (hcont x)
· intro B hB hBdelta
have hBunit : B ⊆ Icc (0 : ℝ) 1 :=
hBdelta.trans (Set.Icc_subset_Icc_right hdeltaOne.le)
have hfull := fullDataResponseMean_setIntegral J kappa cminus cplus pmin
deltaBar PF hPF.toBridge hkappa hcplus hpmin x B hB hBunit
have hweighted := continuous_mul_treatmentDensity_integrable PF.observedMargin
kappa cminus cplus deltaBar hPF.observedModel.condDensity
hPF.observedModel.thinning hkappa hcplus hdeltaOne.le x (mu x) (hcont x)
have hmuInt : Integrable (B.indicator (mu x))
(conditionalTreatmentMeasure PF.observedMargin x) := by
unfold conditionalTreatmentMeasure
have hpiAe : AEMeasurable (fun t : ℝ => ofReal
(PF.observedMargin.pi x t))
(volume.restrict (Icc (0 : ℝ) 1)) :=
(aemeasurable_restrict_of_measurable_subtype measurableSet_Icc
(hPF.observedModel.condDensity.1 x)).ennreal_ofReal
refine (integrable_withDensity_iff_integrable_smul₀' hpiAe
(ae_of_all _ fun _ => ENNReal.ofReal_lt_top)).2 ?_
have hi := (hweighted.mono_set hBdelta).integrable_indicator hB
have hir := hi.mono_measure
(Measure.restrict_le_self (μ := volume) (s := Icc (0 : ℝ) 1))
apply hir.congr
filter_upwards [hPF.observedModel.condDensity.2.1 x] with t ht
simp [indicator, ht, mul_comm]
have hobs := regressionVersion_setIntegral PF.observedMargin
hPF.observedModel.toBridge hkappa hcplus hpmin mu hver x B hB hBunit hmuInt
have hmargin := fullData_observed_setIntegral PF
{o : ClampObs J | o.X = x ∧ o.A ∈ B}
((measurableSet_eq_fun
(measurable_fst.comp (Measurable.of_comap_le le_rfl)) measurable_const).inter
(hB.preimage (measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))))) _ clampOutcome_measurable
have hmargin' :
(∫ z in {z : ClampObs J × PF.latentCarrier |
z.1.X = x ∧ z.1.A ∈ B}, z.1.Y ∂PF.fullMeasure) =
∫ o in {o : ClampObs J | o.X = x ∧ o.A ∈ B},
o.Y ∂PF.observedMargin.dataMeasure := by
have hs : {z : ClampObs J × PF.latentCarrier |
z.1.X = x ∧ z.1.A ∈ B} =
(fun z : ClampObs J × PF.latentCarrier => z.1) ⁻¹'
{o : ClampObs J | o.X = x ∧ o.A ∈ B} := by
ext z
rfl
rw [hs]
exact hmargin
have hpx : 0 < PF.observedMargin.px x :=
lt_of_lt_of_le hpmin (hPF.observedModel.stratumMass x).2
calc
(∫ t in B, fullDataResponseMean PF x t * PF.observedMargin.pi x t) =
(PF.observedMargin.px x)⁻¹ *
(∫ z in {z : ClampObs J × PF.latentCarrier |
z.1.X = x ∧ z.1.A ∈ B}, z.1.Y ∂PF.fullMeasure) := hfull.symm
_ = (PF.observedMargin.px x)⁻¹ *
(∫ o in {o : ClampObs J | o.X = x ∧ o.A ∈ B},
o.Y ∂PF.observedMargin.dataMeasure) := by rw [hmargin']
_ = (PF.observedMargin.px x)⁻¹ * (PF.observedMargin.px x *
∫ t in B, mu x t * PF.observedMargin.pi x t) := by rw [hobs]
_ = ∫ t in B, mu x t * PF.observedMargin.pi x t := by field_simp
· exact ha
theorem causalClamp_pathwise
Simultaneous consistency gives the pathwise lower-clamp decomposition. The result uses the hcons condition, the hsupp condition, the hdelta condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
PF :
hcons :
hsupp :
∀ᵐ o ∂PF.observedMargin.dataMeasure, o.A ∈ Icc (0 : ℝ) 1
delta :
ℝ
hdelta :
delta ∈ Icc (0 : ℝ) 1
conclusion
∀ᵐ z ∂PF.fullMeasure, PF.pot (clampPolicy delta z.1.A) z
= (if delta < z.1.A then z.1.Y else 0) + (if z.1.A ≤ delta then PF.pot delta z else 0)
Proof (Lean source)
-- @node: causalClamp_pathwise
lemma causalClamp_pathwise
(PF : FullDataLaw J) (hcons : LatentResponseConsistency PF)
(hsupp : ∀ᵐ o ∂PF.observedMargin.dataMeasure, o.A ∈ Icc (0 : ℝ) 1)
(delta : ℝ) (hdelta : delta ∈ Icc (0 : ℝ) 1) :
∀ᵐ z ∂PF.fullMeasure,
PF.pot (clampPolicy delta z.1.A) z =
(if delta < z.1.A then z.1.Y else 0) +
(if z.1.A ≤ delta then PF.pot delta z else 0) := by
have hsuppFull : ∀ᵐ z ∂PF.fullMeasure, z.1.A ∈ Icc (0 : ℝ) 1 := by
rw [← PF.margin_eq] at hsupp
exact (MeasureTheory.ae_map_iff
(show AEMeasurable (fun z : ClampObs J × PF.latentCarrier => z.1) PF.fullMeasure
from measurable_fst.aemeasurable)
(measurableSet_Icc.preimage
(measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))))).mp hsupp
filter_upwards [hcons.2.2, hsuppFull] with z hz ha
by_cases hlt : delta < z.1.A
· have hclamp : clampPolicy delta z.1.A = z.1.A := by
simp [clampPolicy, max_eq_left hlt.le]
rw [if_pos hlt, if_neg (not_le.mpr hlt), hclamp, hz.1 z.1.A ha, hz.2]
ring
· have hle : z.1.A ≤ delta := le_of_not_gt hlt
have hclamp : clampPolicy delta z.1.A = delta := by
simp [clampPolicy, max_eq_right hle]
rw [if_neg hlt, if_pos hle, hclamp]
ring
theorem fullData_clampAtom_integral_eq
The expectation of the clamped atom contribution is the finite-stratum sum of atom masses times full-data response means. The result uses the hPF condition, the hkappa condition, the hcplus condition, the hpmin condition, the hdeltaBarOne condition, the hdelta condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin deltaBar :
ℝ
PF :
hPF :
BridgeFullDataClampModel PF kappa cminus cplus pmin deltaBar
hkappa :
0 ≤ kappa
hcplus :
0 ≤ cplus
hpmin :
0 < pmin
hdeltaBarOne :
deltaBar ≤ 1
delta :
ℝ
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
conclusion
(∫ z, if z.1.A ≤ delta then PF.pot delta z else 0 ∂PF.fullMeasure)
= ∑ x : Fin J, PF.observedMargin.px x * atomMass PF.observedMargin x delta * fullDataResponseMean PF x delta
Proof (Lean source)
-- @node: fullData_clampAtom_integral_eq
lemma fullData_clampAtom_integral_eq
(J : ℕ) (kappa cminus cplus pmin deltaBar : ℝ)
(PF : FullDataLaw J)
(hPF : BridgeFullDataClampModel PF kappa cminus cplus pmin deltaBar)
(hkappa : 0 ≤ kappa) (hcplus : 0 ≤ cplus) (hpmin : 0 < pmin)
(hdeltaBarOne : deltaBar ≤ 1)
(delta : ℝ) (hdelta : delta ∈ Icc (0 : ℝ) deltaBar) :
(∫ z, if z.1.A ≤ delta then PF.pot delta z else 0 ∂PF.fullMeasure) =
∑ x : Fin J, PF.observedMargin.px x *
atomMass PF.observedMargin x delta * fullDataResponseMean PF x delta := by
classical
have hdeltaUnit : delta ∈ Icc (0 : ℝ) 1 :=
⟨hdelta.1, hdelta.2.trans hdeltaBarOne⟩
let f : ClampObs J × PF.latentCarrier → ℝ := fun z =>
if z.1.A ≤ delta then PF.pot delta z else 0
have hfMeas : Measurable f := by
have hpot : Measurable (fun z : ClampObs J × PF.latentCarrier => PF.pot delta z) :=
PF.pot_jointlyMeasurable.comp (measurable_const.prodMk measurable_id)
exact hpot.piecewise
(measurableSet_Iic.preimage
((measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))).comp measurable_fst)) measurable_const
have hfInt : Integrable f PF.fullMeasure := by
let _ := PF.probability
refine Integrable.of_bound hfMeas.aestronglyMeasurable 1 ?_
filter_upwards [hPF.consistency.2.2] with z hz
by_cases hle : z.1.A ≤ delta
· simp only [f, if_pos hle, Real.norm_eq_abs]
rw [hz.1 delta hdeltaUnit]
exact abs_le.2 ⟨by linarith [(hPF.consistency.2.1 z.1.X delta z.2 hdeltaUnit).1],
(hPF.consistency.2.1 z.1.X delta z.2 hdeltaUnit).2⟩
· simp [f, hle]
have hXMeas : ∀ x : Fin J, MeasurableSet
((fun z : ClampObs J × PF.latentCarrier => z.1.X) ⁻¹' {x}) := by
intro x
exact (measurableSet_singleton x).preimage
((measurable_fst.comp (Measurable.of_comap_le le_rfl)).comp measurable_fst)
rw [show (∫ z, if z.1.A ≤ delta then PF.pot delta z else 0 ∂PF.fullMeasure) =
∫ z, f z ∂PF.fullMeasure from rfl,
integral_eq_sum_setIntegral_fiber hXMeas hfInt]
apply Finset.sum_congr rfl
intro x _
let F : ℝ × PF.latentCarrier → ℝ := fun w =>
if w.1 ≤ delta then structuralResponseExtension PF hPF.consistency x (delta, w.2) else 0
have hF : Measurable F := by
have hg : Measurable (fun w : ℝ × PF.latentCarrier =>
structuralResponseExtension PF hPF.consistency x (delta, w.2)) :=
(structuralResponseExtension_measurable PF hPF.consistency x).comp
(measurable_const.prodMk measurable_snd)
exact hg.piecewise (measurableSet_Iic.preimage measurable_fst) measurable_const
have hFbound : ∀ w, ‖F w‖ ≤ (1 : ℝ) := by
intro w
by_cases hle : w.1 ≤ delta
· simp only [F, if_pos hle]
rw [structuralResponseExtension_eq PF hPF.consistency x delta hdeltaUnit w.2,
Real.norm_eq_abs]
exact abs_le.2 ⟨by linarith [(hPF.consistency.2.1 x delta w.2 hdeltaUnit).1],
(hPF.consistency.2.1 x delta w.2 hdeltaUnit).2⟩
· simp [F, hle]
have hprod := conditional_stratum_integral_prod J kappa cminus cplus
pmin PF hPF.observedModel hPF.exchangeability hkappa hcplus hpmin x F hF 1 hFbound
have hstratum : MeasurableSet
{z : ClampObs J × PF.latentCarrier | z.1.X = x} := by
exact measurableSet_eq_fun
((measurable_fst.comp (Measurable.of_comap_le le_rfl)).comp measurable_fst)
measurable_const
have hcell : (∫ z in {z : ClampObs J × PF.latentCarrier | z.1.X = x}, f z
∂PF.fullMeasure) =
∫ z in {z : ClampObs J × PF.latentCarrier | z.1.X = x}, F (z.1.A, z.2)
∂PF.fullMeasure := by
apply setIntegral_congr_ae hstratum
exact hPF.consistency.2.2.mono (fun z hz hx => by
by_cases hle : z.1.A ≤ delta
· simp only [f, F, if_pos hle]
rw [structuralResponseExtension_eq PF hPF.consistency x delta hdeltaUnit z.2,
← hx, ← hz.1 delta hdeltaUnit]
· simp [f, F, hle])
change (∫ z in {z : ClampObs J × PF.latentCarrier | z.1.X = x}, f z
∂PF.fullMeasure) = _
rw [hcell]
have hrhs : (∫ a, ∫ u, F (a, u)
∂Measure.map (fun z : ClampObs J × PF.latentCarrier => z.2)
((ofReal (PF.observedMargin.px x))⁻¹ •
PF.fullMeasure.restrict {z | z.1.X = x})
∂conditionalTreatmentMeasure PF.observedMargin x) =
atomMass PF.observedMargin x delta * fullDataResponseMean PF x delta := by
rw [show (∫ a, ∫ u, F (a, u)
∂Measure.map (fun z : ClampObs J × PF.latentCarrier => z.2)
((ofReal (PF.observedMargin.px x))⁻¹ •
PF.fullMeasure.restrict {z | z.1.X = x})
∂conditionalTreatmentMeasure PF.observedMargin x) =
∫ a, indicator (Iic delta) (fun _ => fullDataResponseMean PF x delta) a
∂conditionalTreatmentMeasure PF.observedMargin x by
apply integral_congr_ae
filter_upwards with a
by_cases hle : a ≤ delta
· simp only [F, if_pos hle, Set.indicator_of_mem
(show a ∈ Iic delta from hle)]
rw [integral_congr_ae (ae_of_all _ fun u =>
structuralResponseExtension_eq PF hPF.consistency x delta hdeltaUnit u)]
exact conditional_stratum_latent_integral_eq_mean J kappa cminus
cplus pmin PF hPF.observedModel hpmin hPF.consistency x delta hdeltaUnit
· simp [F, hle],
conditionalTreatment_integral_eq_density PF.observedMargin hPF.observedModel x]
simp only [atomMass]
rw [← integral_indicator measurableSet_Icc]
calc
(∫ a, (Icc (0 : ℝ) 1).indicator
(fun a => PF.observedMargin.pi x a *
(Iic delta).indicator (fun _ => fullDataResponseMean PF x delta) a) a) =
∫ a, (Icc (0 : ℝ) delta).indicator
(fun a => PF.observedMargin.pi x a * fullDataResponseMean PF x delta) a := by
congr 1
funext a
by_cases ha : a ∈ Icc (0 : ℝ) delta
· have haUnit : a ∈ Icc (0 : ℝ) 1 :=
⟨ha.1, ha.2.trans hdeltaUnit.2⟩
have hIic : a ∈ Iic delta := ha.2
simp [indicator, ha, haUnit, hIic]
· by_cases haUnit : a ∈ Icc (0 : ℝ) 1
· by_cases hle : a ≤ delta
· exact elim (ha ⟨haUnit.1, hle⟩)
· have hnotIic : a ∉ Iic delta := hle
simp [indicator, ha, haUnit, hnotIic]
· simp [indicator, ha, haUnit]
_ = ∫ a in Icc (0 : ℝ) delta,
PF.observedMargin.pi x a * fullDataResponseMean PF x delta :=
integral_indicator measurableSet_Icc
_ = (∫ a in Icc (0 : ℝ) delta, PF.observedMargin.pi x a) *
fullDataResponseMean PF x delta := by rw [integral_mul_const]
have hpx : 0 < PF.observedMargin.px x :=
lt_of_lt_of_le hpmin (hPF.observedModel.stratumMass x).2
rw [hrhs] at hprod
field_simp at hprod ⊢
exact hprod
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.CausalBridgeMeasure 7 declarations This module derives the normalized stratum product law and continuity of the full-data response mean used by the causal clamp bridge.
Conditional full-data stratum measure helpers
This module derives the normalized stratum product law and continuity of the full-data response mean used by the causal clamp bridge.
theorem conditional_stratum_treatment_latent_indep Lemma conditional_stratum_treatment_latent_indep in the paper ↗
After restriction to a positive-mass stratum and normalization, treatment and the latent response coordinate are independent. The result uses the hexch condition, the hpx condition, the hmass condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
PF :
hexch :
LatentExchangeability (PF := PF)
x :
Fin J
hpx :
0 < PF.observedMargin.px x
hmass :
fullDataStratumMass PF x = PF.observedMargin.px x
Proof (Lean source)
lemma conditional_stratum_treatment_latent_indep
(J : ℕ)
(PF : FullDataLaw J)
(hexch : LatentExchangeability (PF := PF)) (x : Fin J)
(hpx : 0 < PF.observedMargin.px x)
(hmass : fullDataStratumMass PF x = PF.observedMargin.px x) :
IndepFun
(fun z : ClampObs J × PF.latentCarrier => z.1.A) (fun z => z.2)
((ofReal (PF.observedMargin.px x))⁻¹ •
PF.fullMeasure.restrict {z | z.1.X = x}) := by
letI : IsProbabilityMeasure PF.fullMeasure := PF.probability
have hX : Measurable (fun o : ClampObs J => o.X) := by
exact measurable_fst.comp (Measurable.of_comap_le le_rfl)
have hfullX : Measurable (fun z : ClampObs J × PF.latentCarrier => z.1.X) :=
hX.comp measurable_fst
have hA : Measurable (fun z : ClampObs J × PF.latentCarrier => z.1.A) := by
have hAo : Measurable (fun o : ClampObs J => o.A) := by
exact measurable_fst.comp (measurable_snd.comp (Measurable.of_comap_le le_rfl))
exact hAo.comp measurable_fst
have hU : Measurable (fun z : ClampObs J × PF.latentCarrier => z.2) := measurable_snd
rw [ProbabilityTheory.indepFun_iff_measure_inter_preimage_eq_mul]
intro s t hs ht
have he := hexch x (s.indicator (fun _ => (1 : ℝ)))
(t.indicator (fun _ => (1 : ℝ)))
have hfs : Measurable (s.indicator (fun _ => (1 : ℝ))) :=
measurable_const.indicator hs
have hgt : Measurable (t.indicator (fun _ => (1 : ℝ))) :=
measurable_const.indicator ht
specialize he hfs hgt
⟨1, fun a => by by_cases ha : a ∈ s <;> simp [indicator, ha]⟩
⟨1, fun u => by by_cases hu : u ∈ t <;> simp [indicator, hu]⟩
rw [hmass] at he
have hSx : MeasurableSet {z : ClampObs J × PF.latentCarrier | z.1.X = x} :=
hfullX (measurableSet_singleton x)
have hAs : MeasurableSet ((fun z : ClampObs J × PF.latentCarrier => z.1.A) ⁻¹' s) := hA hs
have hUt : MeasurableSet ((fun z : ClampObs J × PF.latentCarrier => z.2) ⁻¹' t) := hU ht
have hIfg : (∫ z in {z : ClampObs J × PF.latentCarrier | z.1.X = x},
s.indicator (fun _ => (1 : ℝ)) z.1.A * t.indicator (fun _ => (1 : ℝ)) z.2
∂PF.fullMeasure) =
PF.fullMeasure.real ({z : ClampObs J × PF.latentCarrier | z.1.X = x} ∩
((fun z => z.1.A) ⁻¹' s ∩ (fun z => z.2) ⁻¹' t)) := by
rw [← integral_indicator hSx]
rw [← integral_indicator_one (hSx.inter (hAs.inter hUt))]
congr 1
funext z
by_cases hx : z.1.X = x <;> by_cases ha : z.1.A ∈ s <;>
by_cases hu : z.2 ∈ t <;> simp [indicator, hx, ha, hu]
have hIf : (∫ z in {z : ClampObs J × PF.latentCarrier | z.1.X = x},
s.indicator (fun _ => (1 : ℝ)) z.1.A ∂PF.fullMeasure) =
PF.fullMeasure.real ({z : ClampObs J × PF.latentCarrier | z.1.X = x} ∩
(fun z => z.1.A) ⁻¹' s) := by
rw [← integral_indicator hSx]
rw [← integral_indicator_one (hSx.inter hAs)]
congr 1
funext z
by_cases hx : z.1.X = x <;> by_cases ha : z.1.A ∈ s <;>
simp [indicator, hx, ha]
have hIg : (∫ z in {z : ClampObs J × PF.latentCarrier | z.1.X = x},
t.indicator (fun _ => (1 : ℝ)) z.2 ∂PF.fullMeasure) =
PF.fullMeasure.real ({z : ClampObs J × PF.latentCarrier | z.1.X = x} ∩
(fun z => z.2) ⁻¹' t) := by
rw [← integral_indicator hSx]
rw [← integral_indicator_one (hSx.inter hUt)]
congr 1
funext z
by_cases hx : z.1.X = x <;> by_cases hu : z.2 ∈ t <;>
simp [indicator, hx, hu]
rw [hIfg, hIf, hIg] at he
have he' : PF.observedMargin.px x *
PF.fullMeasure.real ((fun z : ClampObs J × PF.latentCarrier => z.2) ⁻¹' t ∩
({z | z.1.X = x} ∩ (fun z => z.1.A) ⁻¹' s)) =
PF.fullMeasure.real ({z | z.1.X = x} ∩ (fun z => z.1.A) ⁻¹' s) *
PF.fullMeasure.real ((fun z => z.2) ⁻¹' t ∩ {z | z.1.X = x}) := by
convert he using 1 <;> congr 2 <;> ext z <;>
simp only [Set.mem_inter_iff, Set.mem_preimage, Set.mem_ofPred_eq] <;> tauto
have hscalar : (ofReal (PF.observedMargin.px x))⁻¹ ≠ ⊤ :=
ENNReal.inv_ne_top.mpr (by simp [ENNReal.ofReal_eq_zero, not_le.mpr hpx])
have hfinite (u : Set (ClampObs J × PF.latentCarrier)) :
(ofReal (PF.observedMargin.px x))⁻¹ *
(PF.fullMeasure.restrict {z | z.1.X = x}) u ≠ ⊤ :=
ENNReal.mul_ne_top hscalar (measure_ne_top _ _)
apply (ENNReal.toReal_eq_toReal_iff' (hfinite _)
(ENNReal.mul_ne_top (hfinite _) (hfinite _))).mp
simp only [ENNReal.toReal_mul, ENNReal.toReal_inv,
ENNReal.toReal_ofReal hpx.le]
rw [Measure.restrict_apply (hAs.inter hUt), Measure.restrict_apply hAs,
Measure.restrict_apply hUt]
simp only [Set.inter_left_comm, Set.inter_comm]
change (PF.observedMargin.px x)⁻¹ *
PF.fullMeasure.real ((fun z : ClampObs J × PF.latentCarrier => z.2) ⁻¹' t ∩
({z | z.1.X = x} ∩ (fun z => z.1.A) ⁻¹' s)) =
((PF.observedMargin.px x)⁻¹ *
PF.fullMeasure.real ({z | z.1.X = x} ∩ (fun z => z.1.A) ⁻¹' s)) *
((PF.observedMargin.px x)⁻¹ *
PF.fullMeasure.real ((fun z => z.2) ⁻¹' t ∩ {z | z.1.X = x}))
calc
_ = (PF.observedMargin.px x)⁻¹ ^ 2 *
(PF.observedMargin.px x *
PF.fullMeasure.real ((fun z : ClampObs J × PF.latentCarrier => z.2) ⁻¹' t ∩
({z | z.1.X = x} ∩ (fun z => z.1.A) ⁻¹' s))) := by
field_simp [ne_of_gt hpx]
_ = (PF.observedMargin.px x)⁻¹ ^ 2 *
(PF.fullMeasure.real ({z | z.1.X = x} ∩ (fun z => z.1.A) ⁻¹' s) *
PF.fullMeasure.real ((fun z => z.2) ⁻¹' t ∩ {z | z.1.X = x})) := by
rw [he']
_ = _ := by ring
The treatment marginal of the normalized full-data stratum is the declared conditional treatment law. The result uses the hmodel condition, the hkappa condition, the hcplus condition, the hpmin condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin :
ℝ
PF :
hmodel :
BridgeClampModel PF.observedMargin kappa cminus cplus pmin
hkappa :
0 ≤ kappa
hcplus :
0 ≤ cplus
hpmin :
0 < pmin
x :
Fin J
conclusion
Measure.map (fun z : ClampObs J × PF.latentCarrier => z.1.A) ((ofReal (PF.observedMargin.px x))⁻¹ • PF.fullMeasure.restrict {z | z.1.X = x})
= conditionalTreatmentMeasure PF.observedMargin x
Proof (Lean source)
lemma conditional_stratum_treatment_law
(J : ℕ)
(kappa cminus cplus pmin : ℝ)
(PF : FullDataLaw J)
(hmodel : BridgeClampModel PF.observedMargin kappa cminus cplus pmin)
(hkappa : 0 ≤ kappa) (hcplus : 0 ≤ cplus) (hpmin : 0 < pmin)
(x : Fin J) :
Measure.map (fun z : ClampObs J × PF.latentCarrier => z.1.A)
((ofReal (PF.observedMargin.px x))⁻¹ •
PF.fullMeasure.restrict {z | z.1.X = x}) =
conditionalTreatmentMeasure PF.observedMargin x := by
have hpx : 0 < PF.observedMargin.px x :=
lt_of_lt_of_le hpmin (hmodel.stratumMass x).2
have hfst : Measurable (fun z : ClampObs J × PF.latentCarrier => z.1) := measurable_fst
have hX : Measurable (fun o : ClampObs J => o.X) :=
measurable_fst.comp (Measurable.of_comap_le le_rfl)
have hA : Measurable (fun o : ClampObs J => o.A) :=
measurable_fst.comp (measurable_snd.comp (Measurable.of_comap_le le_rfl))
have hrestrict :
Measure.map (fun z : ClampObs J × PF.latentCarrier => z.1)
(PF.fullMeasure.restrict {z | z.1.X = x}) =
PF.observedMargin.dataMeasure.restrict {o | o.X = x} := by
let s : Set (ClampObs J) := (fun o => o.X) ⁻¹' {x}
change Measure.map (fun z : ClampObs J × PF.latentCarrier => z.1)
(PF.fullMeasure.restrict ((fun z => z.1) ⁻¹' s)) =
PF.observedMargin.dataMeasure.restrict s
calc
_ = (Measure.map (fun z : ClampObs J × PF.latentCarrier => z.1) PF.fullMeasure).restrict s :=
(Measure.restrict_map hfst (hX (measurableSet_singleton x))).symm
_ = _ := by rw [PF.margin_eq]
have hunscaled :
Measure.map (fun z : ClampObs J × PF.latentCarrier => z.1.A)
(PF.fullMeasure.restrict {z | z.1.X = x}) =
ofReal (PF.observedMargin.px x) •
conditionalTreatmentMeasure PF.observedMargin x := by
rw [← stratumTreatmentMeasure_eq PF.observedMargin hmodel hkappa hcplus hpmin x]
rw [← hrestrict, Measure.map_map hA hfst]
rfl
rw [Measure.map_smul, hunscaled, smul_smul]
have hz : ofReal (PF.observedMargin.px x) ≠ 0 := by
rw [ne_eq, ENNReal.ofReal_eq_zero]
exact not_le.mpr hpx
rw [ENNReal.inv_mul_cancel hz ENNReal.ofReal_ne_top, one_smul]
A bounded measurable function of treatment and the latent coordinate integrates by iterated integration under the normalized stratum law. The result uses the hmodel condition, the hexch condition, the hkappa condition, the hcplus condition, the hpmin condition, the hF condition, the hM condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin :
ℝ
PF :
hmodel :
BridgeClampModel PF.observedMargin kappa cminus cplus pmin
hexch :
LatentExchangeability (PF := PF)
hkappa :
0 ≤ kappa
hcplus :
0 ≤ cplus
hpmin :
0 < pmin
x :
Fin J
F :
ℝ × PF.latentCarrier → ℝ
hF :
M :
ℝ
hM :
∀ z, ‖F z‖ ≤ M
conclusion
(PF.observedMargin.px x)⁻¹ * (∫ z in {z : ClampObs J × PF.latentCarrier | z.1.X = x}, F (z.1.A, z.2) ∂PF.fullMeasure)
Proof (Lean source)
lemma conditional_stratum_integral_prod
(J : ℕ)
(kappa cminus cplus pmin : ℝ)
(PF : FullDataLaw J)
(hmodel : BridgeClampModel PF.observedMargin kappa cminus cplus pmin)
(hexch : LatentExchangeability (PF := PF))
(hkappa : 0 ≤ kappa) (hcplus : 0 ≤ cplus) (hpmin : 0 < pmin)
(x : Fin J) (F : ℝ × PF.latentCarrier → ℝ) (hF : Measurable F)
(M : ℝ) (hM : ∀ z, ‖F z‖ ≤ M) :
(PF.observedMargin.px x)⁻¹ *
(∫ z in {z : ClampObs J × PF.latentCarrier | z.1.X = x}, F (z.1.A, z.2) ∂PF.fullMeasure) =
∫ a, ∫ u, F (a, u)
∂Measure.map (fun z : ClampObs J × PF.latentCarrier => z.2)
((ofReal (PF.observedMargin.px x))⁻¹ •
PF.fullMeasure.restrict {z | z.1.X = x})
∂conditionalTreatmentMeasure PF.observedMargin x := by
let μ := (ofReal (PF.observedMargin.px x))⁻¹ •
PF.fullMeasure.restrict {z : ClampObs J × PF.latentCarrier | z.1.X = x}
have hpx : 0 < PF.observedMargin.px x :=
lt_of_lt_of_le hpmin (hmodel.stratumMass x).2
have hmass : fullDataStratumMass PF x = PF.observedMargin.px x :=
(hmodel.stratumMass x).1.symm
have hA : Measurable (fun z : ClampObs J × PF.latentCarrier => z.1.A) := by
have hAo : Measurable (fun o : ClampObs J => o.A) :=
measurable_fst.comp (measurable_snd.comp (Measurable.of_comap_le le_rfl))
exact hAo.comp measurable_fst
have hU : Measurable (fun z : ClampObs J × PF.latentCarrier => z.2) := measurable_snd
have hpair : Measurable (fun z : ClampObs J × PF.latentCarrier => (z.1.A, z.2)) :=
Measurable.prod hA hU
letI : IsProbabilityMeasure PF.fullMeasure := PF.probability
letI : IsFiniteMeasure μ := by
refine ⟨?_⟩
dsimp [μ]
apply ENNReal.mul_lt_top
· rw [ENNReal.inv_lt_top]
exact ENNReal.ofReal_pos.mpr hpx
· exact measure_lt_top _ _
have hFind : Integrable (fun z : ClampObs J × PF.latentCarrier => F (z.1.A, z.2)) μ :=
Integrable.of_bound (hF.comp hpair).aestronglyMeasurable M
(ae_of_all _ fun z => hM (z.1.A, z.2))
have hind := conditional_stratum_treatment_latent_indep J PF hexch x hpx hmass
have hjoint : Measure.map (fun z : ClampObs J × PF.latentCarrier => (z.1.A, z.2)) μ =
(Measure.map (fun z : ClampObs J × PF.latentCarrier => z.1.A) μ).prod
(Measure.map (fun z : ClampObs J × PF.latentCarrier => z.2) μ) :=
(ProbabilityTheory.indepFun_iff_map_prod_eq_prod_map_map
hA.aemeasurable hU.aemeasurable).mp hind
have hprodInt : Integrable F
((Measure.map (fun z : ClampObs J × PF.latentCarrier => z.1.A) μ).prod
(Measure.map (fun z : ClampObs J × PF.latentCarrier => z.2) μ)) := by
rw [← hjoint]
exact (integrable_map_measure hF.aestronglyMeasurable hpair.aemeasurable).2 hFind
have hAlaw : Measure.map (fun z : ClampObs J × PF.latentCarrier => z.1.A) μ =
conditionalTreatmentMeasure PF.observedMargin x := by
simpa [μ] using conditional_stratum_treatment_law J kappa cminus
cplus pmin PF hmodel hkappa hcplus hpmin x
calc
_ = ∫ z, F (z.1.A, z.2) ∂μ := by
rw [show μ = (ofReal (PF.observedMargin.px x))⁻¹ •
PF.fullMeasure.restrict {z : ClampObs J × PF.latentCarrier | z.1.X = x} from rfl,
integral_smul_measure]
simp only [ENNReal.toReal_inv, ENNReal.toReal_ofReal hpx.le, smul_eq_mul]
_ = ∫ w, F w ∂Measure.map (fun z : ClampObs J × PF.latentCarrier => (z.1.A, z.2)) μ := by
exact (integral_map hpair.aemeasurable hF.aestronglyMeasurable).symm
_ = ∫ w, F w ∂(Measure.map (fun z : ClampObs J × PF.latentCarrier => z.1.A) μ).prod
(Measure.map (fun z : ClampObs J × PF.latentCarrier => z.2) μ) := by rw [hjoint]
_ = ∫ a, ∫ u, F (a, u) ∂Measure.map
(fun z : ClampObs J × PF.latentCarrier => z.2) μ
∂Measure.map (fun z : ClampObs J × PF.latentCarrier => z.1.A) μ :=
integral_prod F hprodInt
_ = _ := by rw [hAlaw]
theorem conditional_stratum_latent_integral_eq_mean Lemma conditional_stratum_latent_integral_eq_mean in the paper ↗
The latent-coordinate integral under the normalized stratum law is the full-data response mean. The result uses the hmodel condition, the hpmin condition, the hcons condition, the ha condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin :
ℝ
PF :
hmodel :
BridgeClampModel PF.observedMargin kappa cminus cplus pmin
hpmin :
0 < pmin
hcons :
LatentResponseConsistency (PF := PF)
x :
Fin J
a :
ℝ
ha :
a ∈ Icc (0 : ℝ) 1
conclusion
(∫ u, PF.g x a u ∂Measure.map (fun z : ClampObs J × PF.latentCarrier => z.2) ((ofReal (PF.observedMargin.px x))⁻¹ • PF.fullMeasure.restrict {z | z.1.X = x}))
= fullDataResponseMean (PF := PF) (x := x) (a := a)
Proof (Lean source)
lemma conditional_stratum_latent_integral_eq_mean
(J : ℕ)
(kappa cminus cplus pmin : ℝ)
(PF : FullDataLaw J)
(hmodel : BridgeClampModel PF.observedMargin kappa cminus cplus pmin)
(hpmin : 0 < pmin)
(hcons : LatentResponseConsistency (PF := PF))
(x : Fin J) (a : ℝ) (ha : a ∈ Icc (0 : ℝ) 1) :
(∫ u, PF.g x a u
∂Measure.map (fun z : ClampObs J × PF.latentCarrier => z.2)
((ofReal (PF.observedMargin.px x))⁻¹ •
PF.fullMeasure.restrict {z | z.1.X = x})) =
fullDataResponseMean (PF := PF) (x := x) (a := a) := by
have hpx : 0 < PF.observedMargin.px x := by
exact lt_of_lt_of_le hpmin (hmodel.stratumMass x).2
have hmass : fullDataStratumMass PF x = PF.observedMargin.px x :=
(hmodel.stratumMass x).1.symm
have hg : Measurable (fun u : PF.latentCarrier => PF.g x a u) := by
exact (hcons.1 x).comp (by fun_prop : Measurable fun u : PF.latentCarrier =>
((⟨a, ha⟩ : Icc (0 : ℝ) 1), u))
rw [integral_map measurable_snd.aemeasurable hg.aestronglyMeasurable,
integral_smul_measure]
simp only [ENNReal.toReal_inv, ENNReal.toReal_ofReal hpx.le, smul_eq_mul,
fullDataResponseMean]
rw [hmass]
Conditional L1 response distance bounds the distance between full-data response means. The result uses the hmodel condition, the hpmin condition, the hcons condition, the hs condition, the ht condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin :
ℝ
PF :
hmodel :
ClampModel PF.observedMargin beta kappa L cminus cplus pmin
hpmin :
0 < pmin
hcons :
LatentResponseConsistency (PF := PF)
x :
Fin J
s t :
ℝ
hs :
s ∈ Icc (0 : ℝ) 1
ht :
t ∈ Icc (0 : ℝ) 1
conclusion
dist (fullDataResponseMean (PF := PF) (x := x) (a := t)) (fullDataResponseMean (PF := PF) (x := x) (a := s))
≤ (PF.observedMargin.px x)⁻¹ * ∫ z in {z : ClampObs J × PF.latentCarrier | z.1.X = x}, |PF.pot t z
- PF.pot s z| ∂PF.fullMeasure
Proof (Lean source)
lemma fullDataResponseMean_dist_le
(J : ℕ)
(beta kappa L cminus cplus pmin : ℝ)
(PF : FullDataLaw J)
(hmodel : ClampModel PF.observedMargin beta kappa L cminus cplus pmin)
(hpmin : 0 < pmin) (hcons : LatentResponseConsistency (PF := PF))
(x : Fin J) (s t : ℝ) (hs : s ∈ Icc (0 : ℝ) 1)
(ht : t ∈ Icc (0 : ℝ) 1) :
dist (fullDataResponseMean (PF := PF) (x := x) (a := t))
(fullDataResponseMean (PF := PF) (x := x) (a := s)) ≤
(PF.observedMargin.px x)⁻¹ *
∫ z in {z : ClampObs J × PF.latentCarrier | z.1.X = x},
|PF.pot t z - PF.pot s z| ∂PF.fullMeasure := by
letI : IsProbabilityMeasure PF.fullMeasure := PF.probability
have hpx : 0 < PF.observedMargin.px x :=
lt_of_lt_of_le hpmin (hmodel.stratumMass x).2
have hmass : fullDataStratumMass PF x = PF.observedMargin.px x :=
(hmodel.stratumMass x).1.symm
have hg (a : ℝ) (ha : a ∈ Icc (0 : ℝ) 1) :
Measurable (fun z : ClampObs J × PF.latentCarrier => PF.g x a z.2) := by
exact ((hcons.1 x).comp (by fun_prop : Measurable fun z : ClampObs J × PF.latentCarrier =>
((⟨a, ha⟩ : Icc (0 : ℝ) 1), z.2)))
have hgInt (a : ℝ) (ha : a ∈ Icc (0 : ℝ) 1) :
Integrable (fun z : ClampObs J × PF.latentCarrier => PF.g x a z.2)
(PF.fullMeasure.restrict {z | z.1.X = x}) := by
apply Integrable.of_bound (hg a ha).aestronglyMeasurable 1
exact ae_of_all _ fun z => by
rw [Real.norm_eq_abs]
exact abs_le.2 ⟨by linarith [(hcons.2.1 x a z.2 ha).1],
(hcons.2.1 x a z.2 ha).2⟩
have hpotg (a : ℝ) (ha : a ∈ Icc (0 : ℝ) 1) :
(fun z : ClampObs J × PF.latentCarrier => PF.pot a z) =ᵐ[
PF.fullMeasure.restrict {z | z.1.X = x}]
(fun z => PF.g x a z.2) := by
filter_upwards [ae_restrict_of_ae hcons.2.2,
ae_restrict_mem (by
have hXm : Measurable (fun z : ClampObs J × PF.latentCarrier => z.1.X) :=
(measurable_fst.comp (Measurable.of_comap_le le_rfl)).comp measurable_fst
exact hXm (measurableSet_singleton x))] with z hz hzx
simpa [hzx] using hz.1 a ha
have hpotInt (a : ℝ) (ha : a ∈ Icc (0 : ℝ) 1) :
Integrable (fun z : ClampObs J × PF.latentCarrier => PF.pot a z)
(PF.fullMeasure.restrict {z | z.1.X = x}) :=
(hgInt a ha).congr (hpotg a ha).symm
rw [fullDataResponseMean, fullDataResponseMean, Real.dist_eq]
rw [hmass]
calc
|(PF.observedMargin.px x)⁻¹ *
(∫ z in {z | z.1.X = x}, PF.g x t z.2 ∂PF.fullMeasure) -
(PF.observedMargin.px x)⁻¹ *
(∫ z in {z | z.1.X = x}, PF.g x s z.2 ∂PF.fullMeasure)| =
(PF.observedMargin.px x)⁻¹ *
|(∫ z in {z | z.1.X = x}, PF.g x t z.2 ∂PF.fullMeasure) -
(∫ z in {z | z.1.X = x}, PF.g x s z.2 ∂PF.fullMeasure)| := by
rw [← mul_sub, abs_mul, abs_of_nonneg (inv_nonneg.mpr hpx.le)]
_ = (PF.observedMargin.px x)⁻¹ *
|∫ z in {z | z.1.X = x},
(PF.g x t z.2 - PF.g x s z.2) ∂PF.fullMeasure| := by
rw [integral_sub (hgInt t ht) (hgInt s hs)]
_ = (PF.observedMargin.px x)⁻¹ *
|∫ z in {z | z.1.X = x},
(PF.pot t z - PF.pot s z) ∂PF.fullMeasure| := by
congr 2
apply integral_congr_ae
filter_upwards [hpotg t ht, hpotg s hs] with z htz hsz
rw [htz, hsz]
_ ≤ _ := mul_le_mul_of_nonneg_left abs_integral_le_integral_abs
(inv_nonneg.mpr hpx.le)
The full-data mean-continuity member is exactly continuity of the structural response mean on the declared threshold interval. The result uses the hcont condition. This is the stated conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma fullDataResponseMean_continuousOn
(J : ℕ) (PF : FullDataLaw J) (deltaBar : ℝ)
(hcont : FullDataResponseContinuity PF deltaBar) (x : Fin J) :
ContinuousOn (fun a => fullDataResponseMean (PF := PF) (x := x) (a := a))
(Icc (0 : ℝ) deltaBar) := by
exact hcont x
Two continuous conditional-regression versions on the threshold interval coincide there under the positive polynomial treatment-density envelope. The result uses the hP condition, the hreg condition, the hcont₁ condition, the hcont₂ condition, the hver₁ condition, the hver₂ condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
kappa cminus cplus pmin deltaBar :
ℝ
hP :
ContClampModel P kappa cminus cplus pmin deltaBar
hreg :
ContDesignConstants J kappa cminus cplus pmin deltaBar
mu₁ mu₂ :
Fin J → ℝ → ℝ
hcont₁ :
∀ x, ContinuousOn (mu₁ x) (Icc (0 : ℝ) deltaBar)
hcont₂ :
∀ x, ContinuousOn (mu₂ x) (Icc (0 : ℝ) deltaBar)
hver₁ :
=ᵐ[P.dataMeasure] fun o => mu₁ o.X o.A
hver₂ :
=ᵐ[P.dataMeasure] fun o => mu₂ o.X o.A
x :
Proof (Lean source)
-- @node: lem:continuity-regression-extension-unique
lemma cont_regression_extension_unique
(P : ClampLaw J) (kappa cminus cplus pmin deltaBar : ℝ)
(hP : ContClampModel P kappa cminus cplus pmin deltaBar)
(hreg : ContDesignConstants J kappa cminus cplus pmin deltaBar)
(mu₁ mu₂ : Fin J → ℝ → ℝ)
(hcont₁ : ∀ x, ContinuousOn (mu₁ x) (Icc (0 : ℝ) deltaBar))
(hcont₂ : ∀ x, ContinuousOn (mu₂ x) (Icc (0 : ℝ) deltaBar))
(hver₁ : P.dataMeasure[(fun o : ClampObs J => o.Y) |
comap (fun o : ClampObs J => (o.X, o.A)) inferInstance]
=ᵐ[P.dataMeasure] fun o => mu₁ o.X o.A)
(hver₂ : P.dataMeasure[(fun o : ClampObs J => o.Y) |
comap (fun o : ClampObs J => (o.X, o.A)) inferInstance]
=ᵐ[P.dataMeasure] fun o => mu₂ o.X o.A) :
∀ x, EqOn (mu₁ x) (mu₂ x) (Icc (0 : ℝ) deltaBar) := by
rcases hreg with ⟨_, hkappa, hcminus, _, hcplus, hpmin, _, hdeltaBar, hdeltaBar_lt⟩
intro x
have hversions :
(fun o : ClampObs J => mu₁ o.X o.A) =ᵐ[P.dataMeasure]
fun o => mu₂ o.X o.A := hver₁.symm.trans hver₂
have hpx : 0 < P.px x := lt_of_lt_of_le hpmin (hP.stratumMass x).2
let f₁ : ℝ → ℝ := fun a => if a ∈ Icc (0 : ℝ) deltaBar then mu₁ x a else 0
let f₂ : ℝ → ℝ := fun a => if a ∈ Icc (0 : ℝ) deltaBar then mu₂ x a else 0
have hf₁ : Measurable f₁ := by
let g : ℝ → ℝ := fun _ => 0
have hg : ContinuousOn g (Icc (0 : ℝ) deltaBar)ᶜ :=
continuous_const.continuousOn
change Measurable ((Icc (0 : ℝ) deltaBar).piecewise (mu₁ x) g)
exact (hcont₁ x).measurable_piecewise hg measurableSet_Icc
have hf₂ : Measurable f₂ := by
let g : ℝ → ℝ := fun _ => 0
have hg : ContinuousOn g (Icc (0 : ℝ) deltaBar)ᶜ :=
continuous_const.continuousOn
change Measurable ((Icc (0 : ℝ) deltaBar).piecewise (mu₂ x) g)
exact (hcont₂ x).measurable_piecewise hg measurableSet_Icc
let B : Set ℝ := {a | f₁ a ≠ f₂ a}
have hB : MeasurableSet B := (measurableSet_eq_fun hf₁ hf₂).compl
have hBthreshold : B ⊆ Icc (0 : ℝ) deltaBar := by
intro a ha
by_contra hnot
change (if a ∈ Icc (0 : ℝ) deltaBar then mu₁ x a else 0) ≠
(if a ∈ Icc (0 : ℝ) deltaBar then mu₂ x a else 0) at ha
simp only [if_neg hnot] at ha
exact ha rfl
have hBunit : B ⊆ Icc (0 : ℝ) 1 := by
intro a ha
exact ⟨(hBthreshold ha).1, (hBthreshold ha).2.trans hdeltaBar_lt.le⟩
have hbadNull : P.dataMeasure
{o : ClampObs J | o.X = x ∧ o.A ∈ B} = 0 := by
apply measure_mono_null _ (ae_iff.mp hversions)
intro o ho
have ha := hBthreshold ho.2
have hneq : mu₁ x o.A ≠ mu₂ x o.A := by
intro heq
apply ho.2
change (if o.A ∈ Icc (0 : ℝ) deltaBar then mu₁ x o.A else 0) =
(if o.A ∈ Icc (0 : ℝ) deltaBar then mu₂ x o.A else 0)
simpa only [if_pos ha] using heq
exact fun heq => hneq (by simpa [ho.1] using heq)
have hdensity := (hP.condDensity.2.2 x).2 B hB hBunit
have hleft : P.dataMeasure.real
{o : ClampObs J | o.X = x ∧ o.A ∈ B} = 0 := by
simp [Measure.real, hbadNull]
rw [hleft] at hdensity
have hintegral : (∫ a in B, P.pi x a) = 0 := by
nlinarith
have hcplus0 : 0 ≤ cplus := le_trans (by linarith) hcplus
have hpiInt : IntegrableOn (P.pi x) B volume := by
apply IntegrableOn.of_bound (lt_of_le_of_lt (measure_mono hBunit) measure_Icc_lt_top)
· exact ((aemeasurable_restrict_of_measurable_subtype measurableSet_Icc
(hP.condDensity.1 x)).mono_measure
(Measure.restrict_mono_set volume hBunit)).aestronglyMeasurable
· filter_upwards [(hP.condDensity.2.1 x).filter_mono
(ae_mono (Measure.restrict_mono_set volume hBunit)),
(hP.thinning x).filter_mono
(ae_mono (Measure.restrict_mono_set volume hBunit)),
ae_restrict_mem hB] with a hnon hthin ha
rw [Real.norm_eq_abs, abs_of_nonneg hnon]
calc
P.pi x a ≤ cplus * a ^ kappa := hthin.2
_ ≤ cplus := by
exact mul_le_of_le_one_right hcplus0
(Real.rpow_le_one (hBunit ha).1 (hBunit ha).2 hkappa)
have hnonneg : 0 ≤ᵐ[volume.restrict B] P.pi x :=
(hP.condDensity.2.1 x).filter_mono
(ae_mono (Measure.restrict_mono_set volume hBunit))
have hpizero : P.pi x =ᵐ[volume.restrict B] (fun _ : ℝ => (0 : ℝ)) :=
(setIntegral_eq_zero_iff_of_nonneg_ae hnonneg hpiInt).mp hintegral
have hnezero : ∀ᵐ a ∂volume.restrict B, a ≠ 0 :=
(volume.ae_ne 0).filter_mono (ae_mono Measure.restrict_le_self)
have hfalse : ∀ᵐ _a ∂volume.restrict B, False := by
filter_upwards [hpizero, hnezero,
(hP.thinning x).filter_mono
(ae_mono (Measure.restrict_mono_set volume hBunit)),
ae_restrict_mem hB] with a hzero ha0 hthin ha
have haPos : 0 < a := lt_of_le_of_ne (hBunit ha).1 (Ne.symm ha0)
have hpiPos : 0 < P.pi x a := lt_of_lt_of_le
(mul_pos hcminus (Real.rpow_pos_of_pos haPos kappa)) hthin.1
linarith
have hBnull : volume B = 0 := by
have hu : (volume.restrict B) univ = 0 := by
simpa using (ae_iff.mp hfalse)
simpa [Measure.restrict_apply_univ] using hu
have houtside : ∀ᵐ a ∂volume, a ∉ B := by
rw [ae_iff]
convert hBnull using 1
congr 1
ext a
simp
have hglobal : f₁ =ᵐ[volume] f₂ := by
filter_upwards [houtside] with a ha
exact not_ne_iff.mp ha
have hthreshold : mu₁ x =ᵐ[volume.restrict (Icc (0 : ℝ) deltaBar)] mu₂ x := by
filter_upwards [ae_restrict_of_ae hglobal,
ae_restrict_mem measurableSet_Icc] with a heq ha
simpa only [f₁, f₂, if_pos ha] using heq
exact volume.eqOn_Icc_of_ae_eq (ne_of_lt hdeltaBar) hthreshold
(hcont₁ x) (hcont₂ x)
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.ContinuityCriteriaLift 2 declarations Concrete transport of continuity-only decision criteria
Concrete transport of continuity-only decision criteria
theorem causalEstimatorRisk_eq_contEstimatorRisk
Identification rewrites the causal risk of every observed-sample estimator as the corresponding continuity-only observed risk. The result uses the hreg condition, the hdelta condition, the hPF condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
kappa cminus cplus pmin deltaBar delta :
ℝ
hreg :
ContDesignConstants J kappa cminus cplus pmin deltaBar
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
PF :
hPF :
ContFullDataClampModel PF kappa cminus cplus pmin deltaBar
est :
Estimator n J
conclusion
causalEstimatorRisk PF n delta est
= contEstimatorRisk PF.observedMargin n kappa cminus cplus pmin deltaBar delta hPF.observedModel est
Proof (Lean source)
lemma causalEstimatorRisk_eq_contEstimatorRisk
(J n : ℕ) (kappa cminus cplus pmin deltaBar delta : ℝ)
(hreg : ContDesignConstants J kappa cminus cplus pmin deltaBar)
(hdelta : delta ∈ Icc (0 : ℝ) deltaBar)
(PF : FullDataLaw J)
(hPF : ContFullDataClampModel PF kappa cminus cplus pmin deltaBar)
(est : Estimator n J) :
causalEstimatorRisk PF n delta est =
contEstimatorRisk PF.observedMargin n kappa cminus cplus pmin deltaBar
delta hPF.observedModel est := by
unfold causalEstimatorRisk contEstimatorRisk
rw [(continuity_causal_bridge J kappa cminus cplus pmin deltaBar PF hPF
hreg).2 delta hdelta]
theorem contFrontierCriteria_observed_causal_eq
The paper's quantile lift and identification bridge instantiate the abstract transport interface without importing the later frontier theorem. The result uses the hreg condition, the hdelta condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
kappa cminus cplus pmin deltaBar delta alpha :
ℝ
hreg :
ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
conclusion
crit :
contFrontierCriteria J n kappa cminus cplus pmin deltaBar delta alpha
crit.2.2.1 = crit.1
crit.2.2.2 = crit.2.1
Proof (Lean source)
lemma contFrontierCriteria_observed_causal_eq
(J n : ℕ) (kappa cminus cplus pmin deltaBar delta alpha : ℝ)
(hreg : ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha)
(hdelta : delta ∈ Icc (0 : ℝ) deltaBar) :
let crit := contFrontierCriteria J n kappa cminus cplus pmin
deltaBar delta alpha
crit.2.2.1 = crit.1 ∧ crit.2.2.2 = crit.2.1 := by
have hsurj : ∀ (P : ClampLaw J),
ContClampModel P kappa cminus cplus pmin deltaBar →
∃ PF : FullDataLaw J,
ContFullDataClampModel PF kappa cminus cplus pmin deltaBar ∧
PF.observedMargin = P := by
intro P hP
exact exists_fullData_cont_lift P hP hreg.1.2.2.2.2.2.1
have htarget : ∀ (PF : FullDataLaw J)
(hPF : ContFullDataClampModel PF kappa cminus cplus pmin deltaBar),
causalClampMean PF delta =
contClampFunctional PF.observedMargin kappa cminus cplus pmin deltaBar
hPF.observedModel delta := by
intro PF hPF
exact (continuity_causal_bridge J kappa cminus cplus pmin deltaBar PF hPF
hreg.1).2 delta hdelta
exact contFrontierCriteria_eq_of_surjective J n kappa cminus cplus pmin
deltaBar delta alpha hsurj htarget
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.ContinuityCriteriaTransport 1 declarations Abstract transport of continuity-only decision criteria
Abstract transport of continuity-only decision criteria
theorem contFrontierCriteria_eq_of_surjective
Surjectivity of the observed-margin map and point identification of the causal target are sufficient to identify both continuity-only causal decision criteria with their observed counterparts. The result uses the hsurj condition, the htarget condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
kappa cminus cplus pmin deltaBar delta alpha :
ℝ
hsurj :
∀ (P : ClampLaw J)
if
ContClampModel P kappa cminus cplus pmin deltaBar
then
∃ PF : FullDataLaw J,
ContFullDataClampModel PF kappa cminus cplus pmin deltaBar ∧ PF.observedMargin = P
htarget :
∀ (PF : FullDataLaw J) (hPF : ContFullDataClampModel PF kappa cminus cplus pmin deltaBar),
causalClampMean PF delta
= contClampFunctional PF.observedMargin kappa cminus cplus pmin deltaBar hPF.observedModel delta
conclusion
crit :
contFrontierCriteria J n kappa cminus cplus pmin deltaBar delta alpha
crit.2.2.1 = crit.1
crit.2.2.2 = crit.2.1
Proof (Lean source)
lemma contFrontierCriteria_eq_of_surjective
(J n : ℕ) (kappa cminus cplus pmin deltaBar delta alpha : ℝ)
(hsurj : ∀ (P : ClampLaw J),
ContClampModel P kappa cminus cplus pmin deltaBar →
∃ PF : FullDataLaw J,
ContFullDataClampModel PF kappa cminus cplus pmin deltaBar ∧
PF.observedMargin = P)
(htarget : ∀ (PF : FullDataLaw J)
(hPF : ContFullDataClampModel PF kappa cminus cplus pmin deltaBar),
causalClampMean PF delta =
contClampFunctional PF.observedMargin kappa cminus cplus pmin deltaBar
hPF.observedModel delta) :
let crit := contFrontierCriteria J n kappa cminus cplus pmin
deltaBar delta alpha
crit.2.2.1 = crit.1 ∧ crit.2.2.2 = crit.2.1 := by
dsimp only
constructor
· unfold contFrontierCriteria
apply congrArg sInf
ext r
constructor
· rintro ⟨est, hm, hrange, rfl⟩
refine ⟨est, hm, hrange, ?_⟩
apply congrArg sSup
ext v
constructor
· rintro ⟨PF, hPF, rfl⟩
refine ⟨PF.observedMargin, hPF.observedModel, ?_⟩
unfold causalEstimatorRisk contEstimatorRisk
rw [htarget PF hPF]
· rintro ⟨P, hP, rfl⟩
obtain ⟨PF, hPF, hmargin⟩ := hsurj P hP
subst P
refine ⟨PF, hPF, ?_⟩
unfold causalEstimatorRisk contEstimatorRisk
rw [htarget PF hPF]
· rintro ⟨est, hm, hrange, rfl⟩
refine ⟨est, hm, hrange, ?_⟩
apply congrArg sSup
ext v
constructor
· rintro ⟨P, hP, rfl⟩
obtain ⟨PF, hPF, hmargin⟩ := hsurj P hP
subst P
refine ⟨PF, hPF, ?_⟩
unfold causalEstimatorRisk contEstimatorRisk
rw [htarget PF hPF]
· rintro ⟨PF, hPF, rfl⟩
refine ⟨PF.observedMargin, hPF.observedModel, ?_⟩
unfold causalEstimatorRisk contEstimatorRisk
rw [htarget PF hPF]
· unfold contFrontierCriteria
apply congrArg sInf
ext r
constructor
· rintro ⟨C, hm, hcov, rfl⟩
refine ⟨C, hm, ?_, ?_⟩
· intro P hP
obtain ⟨PF, hPF, hmargin⟩ := hsurj P hP
subst P
simpa only [htarget PF hPF] using hcov PF hPF
· apply congrArg sSup
ext v
constructor
· rintro ⟨PF, hPF, rfl⟩
exact ⟨PF.observedMargin, hPF.observedModel, rfl⟩
· rintro ⟨P, hP, rfl⟩
obtain ⟨PF, hPF, hmargin⟩ := hsurj P hP
exact ⟨PF, hPF, by rw [hmargin]⟩
· rintro ⟨C, hm, hcov, rfl⟩
refine ⟨C, hm, ?_, ?_⟩
· intro PF hPF
simpa only [htarget PF hPF] using hcov PF.observedMargin hPF.observedModel
· apply congrArg sSup
ext v
constructor
· rintro ⟨P, hP, rfl⟩
obtain ⟨PF, hPF, hmargin⟩ := hsurj P hP
exact ⟨PF, hPF, by rw [hmargin]⟩
· rintro ⟨PF, hPF, rfl⟩
exact ⟨PF.observedMargin, hPF.observedModel, rfl⟩
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.ContinuityExpectedLength 3 declarations This file bounds the expected length of the two-block Hoeffding interval by its deterministic root-block radii and the polynomial threshold-mass envelope.
Expected length of the continuity-only interval
This file bounds the expected length of the two-block Hoeffding interval by its deterministic root-block radii and the polynomial threshold-mass envelope.
theorem contHoeffdingInterval_length_le_radius
The continuity-only interval has length at most twice its displayed radius, before intersection with the outcome range. This is the stated conclusion.
Formal statement
hypotheses
conclusion
intervalLength (contHoeffdingInterval B z delta alpha)
Proof (Lean source)
lemma contHoeffdingInterval_length_le_radius {J n : ℕ} (B : SplitBlocks n)
(delta alpha : ℝ) (z : Fin n → ClampObs J) :
intervalLength (contHoeffdingInterval B z delta alpha) ≤
2 * (sqrt (log (4 / alpha) / (2 * (B.I0.card : ℝ))) +
sqrt (log (4 / alpha) / (2 * (B.I1.card : ℝ))) +
(∑ x : Fin J, atomEstimate B z x delta) / 2) := by
let radius :=
sqrt (log (4 / alpha) / (2 * (B.I0.card : ℝ))) +
sqrt (log (4 / alpha) / (2 * (B.I1.card : ℝ))) +
(∑ x : Fin J, atomEstimate B z x delta) / 2
have hrad : 0 ≤ radius := by
dsimp [radius]
have hatom : 0 ≤ ∑ x : Fin J, atomEstimate B z x delta :=
sum_nonneg fun x _ => (atomEstimate_mem_Icc B z x delta).1
positivity
have hcI := clampUnit_mem_Icc
(retainedEstimate B z delta +
(1 / 2 : ℝ) * ∑ x : Fin J, atomEstimate B z x delta)
have hord : max 0 (contFallbackEstimator B z delta - radius) ≤
min 1 (contFallbackEstimator B z delta + radius) := by
apply le_min
· exact max_le (by norm_num) (by
unfold contFallbackEstimator
linarith [hcI.2])
· exact max_le (by
unfold contFallbackEstimator
linarith [hcI.1]) (by linarith)
simp only [contHoeffdingInterval, intervalLength]
rw [csSup_Icc hord, csInf_Icc hord]
apply max_le
· positivity
· have hhi := min_le_right (1 : ℝ)
(contFallbackEstimator B z delta + radius)
have hlo := le_max_right (0 : ℝ)
(contFallbackEstimator B z delta - radius)
dsimp [radius] at *
linarith
theorem contHoeffdingInterval_integral_length_le
Under a continuity-only model, the real expected interval length is bounded by the two deterministic Hoeffding radii, the atom-estimation noise, and the polynomial threshold-mass envelope. The result uses the hP condition, the hreg condition, the hdelta condition, the hcard1 condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
B :
kappa cminus cplus pmin deltaBar delta alpha :
ℝ
hP :
ContClampModel P kappa cminus cplus pmin deltaBar
hreg :
ContDesignConstants J kappa cminus cplus pmin deltaBar
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
hcard1 :
0 < B.I1.card
conclusion
(∫ z, intervalLength (contHoeffdingInterval B z delta alpha) ∂iidProduct P n)
+ (J : ℝ) * ((1 / 2 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹ + cplus * delta ^ (kappa + 1) / (kappa + 1))
Proof (Lean source)
lemma contHoeffdingInterval_integral_length_le
{J n : ℕ} (P : ClampLaw J) (B : SplitBlocks n)
(kappa cminus cplus pmin deltaBar delta alpha : ℝ)
(hP : ContClampModel P kappa cminus cplus pmin deltaBar)
(hreg : ContDesignConstants J kappa cminus cplus pmin deltaBar)
(hdelta : delta ∈ Icc (0 : ℝ) deltaBar)
(hcard1 : 0 < B.I1.card) :
(∫ z, intervalLength (contHoeffdingInterval B z delta alpha)
∂iidProduct P n) ≤
2 * sqrt (log (4 / alpha) / (2 * (B.I0.card : ℝ))) +
2 * sqrt (log (4 / alpha) / (2 * (B.I1.card : ℝ))) +
(J : ℝ) * ((1 / 2 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹ +
cplus * delta ^ (kappa + 1) / (kappa + 1)) := by
letI : IsProbabilityMeasure P.dataMeasure := hP.probability
letI : IsProbabilityMeasure (iidProduct P n) := by
unfold iidProduct
infer_instance
let f := fun z : Fin n → ClampObs J =>
intervalLength (contHoeffdingInterval B z delta alpha)
let atomTotal := fun z : Fin n → ClampObs J =>
∑ x : Fin J, atomEstimate B z x delta
let t0 := sqrt (log (4 / alpha) / (2 * (B.I0.card : ℝ)))
let t1 := sqrt (log (4 / alpha) / (2 * (B.I1.card : ℝ)))
let env := cplus * delta ^ (kappa + 1) / (kappa + 1)
have hatomMeas : Measurable atomTotal :=
Finset.measurable_fun_sum _ fun x _ => atomEstimate_measurable B x delta
have hatomInt : Integrable atomTotal (iidProduct P n) := by
refine Integrable.of_bound hatomMeas.aestronglyMeasurable J ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg]
· exact (Finset.sum_le_sum fun x _ => (atomEstimate_mem_Icc B z x delta).2).trans_eq
(by simp)
· exact sum_nonneg fun x _ => (atomEstimate_mem_Icc B z x delta).1
have hlenMeas : Measurable f := by
let center := fun z : Fin n → ClampObs J => contFallbackEstimator B z delta
let radius := fun z : Fin n → ClampObs J =>
t0 + t1 + atomTotal z / 2
let lo := fun z => max 0 (center z - radius z)
let hi := fun z => min 1 (center z + radius z)
have hc : Measurable center := by
dsimp [center]
exact clampUnit_measurable'.comp
((retainedEstimate_measurable B delta).add
(measurable_const.mul hatomMeas))
have hr : Measurable radius :=
(measurable_const.add measurable_const).add (hatomMeas.div_const 2)
have hlo : Measurable lo := measurable_const.max (hc.sub hr)
have hhi : Measurable hi := measurable_const.min (hc.add hr)
have hord : ∀ z, lo z ≤ hi z := by
intro z
have hrad : 0 ≤ radius z := by
dsimp [radius, atomTotal]
have ha : 0 ≤ ∑ x : Fin J, atomEstimate B z x delta :=
sum_nonneg fun x _ => (atomEstimate_mem_Icc B z x delta).1
positivity
have hcI := clampUnit_mem_Icc
(retainedEstimate B z delta +
(1 / 2 : ℝ) * ∑ x : Fin J, atomEstimate B z x delta)
apply le_min
· exact max_le (by norm_num) (by
dsimp [lo, hi, center]
unfold contFallbackEstimator
linarith [hcI.2])
· exact max_le (by
dsimp [lo, hi, center]
unfold contFallbackEstimator
linarith [hcI.1]) (by
dsimp [lo, hi, center]
linarith)
have heq : f = fun z => max 0 (hi z - lo z) := by
funext z
dsimp [f, lo, hi, center, radius, t0, t1, atomTotal]
simp only [contHoeffdingInterval, intervalLength]
rw [csSup_Icc (hord z), csInf_Icc (hord z)]
rw [heq]
exact measurable_const.max (hhi.sub hlo)
have hlenInt : Integrable f (iidProduct P n) := by
refine Integrable.of_bound hlenMeas.aestronglyMeasurable 1 ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (le_max_left _ _)]
let radius := t0 + t1 + atomTotal z / 2
have hrad : 0 ≤ radius := by
dsimp [radius, atomTotal, t0, t1]
have ha : 0 ≤ ∑ x : Fin J, atomEstimate B z x delta :=
sum_nonneg fun x _ => (atomEstimate_mem_Icc B z x delta).1
positivity
have hcI := clampUnit_mem_Icc
(retainedEstimate B z delta +
(1 / 2 : ℝ) * ∑ x : Fin J, atomEstimate B z x delta)
have hord : max 0 (contFallbackEstimator B z delta - radius) ≤
min 1 (contFallbackEstimator B z delta + radius) := by
apply le_min
· exact max_le (by norm_num) (by
unfold contFallbackEstimator
linarith [hcI.2])
· exact max_le (by
unfold contFallbackEstimator
linarith [hcI.1]) (by linarith)
dsimp [f, radius, t0, t1, atomTotal]
simp only [contHoeffdingInterval, intervalLength]
rw [csSup_Icc hord, csInf_Icc hord]
apply max_le
· norm_num
· have hhi' := min_le_left (1 : ℝ)
(contFallbackEstimator B z delta +
(sqrt (log (4 / alpha) / (2 * (B.I0.card : ℝ))) +
sqrt (log (4 / alpha) / (2 * (B.I1.card : ℝ))) +
(∑ x : Fin J, atomEstimate B z x delta) / 2))
have hlo' := le_max_left (0 : ℝ)
(contFallbackEstimator B z delta -
(sqrt (log (4 / alpha) / (2 * (B.I0.card : ℝ))) +
sqrt (log (4 / alpha) / (2 * (B.I1.card : ℝ))) +
(∑ x : Fin J, atomEstimate B z x delta) / 2))
linarith
have hatomIntLe : (∫ z, atomTotal z ∂iidProduct P n) ≤
(J : ℝ) * ((1 / 2 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹ + env) := by
rw [integral_finsetSum]
· calc
(∑ x : Fin J, ∫ z, atomEstimate B z x delta ∂iidProduct P n) ≤
∑ _x : Fin J,
((1 / 2 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹ + env) := by
apply Finset.sum_le_sum
intro x hx
let coeff := P.px x * atomMass P x delta
have hax : Integrable (fun z : Fin n → ClampObs J => atomEstimate B z x delta)
(iidProduct P n) := by
refine Integrable.of_bound
(atomEstimate_measurable B x delta).aestronglyMeasurable 1 ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (atomEstimate_mem_Icc B z x delta).1]
exact (atomEstimate_mem_Icc B z x delta).2
have habs : |∫ z, atomEstimate B z x delta - coeff ∂iidProduct P n| ≤
(1 / 2 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹ := by
calc
_ ≤ ∫ z, |atomEstimate B z x delta - coeff| ∂iidProduct P n :=
abs_integral_le_integral_abs
_ ≤ _ := cont_atomEstimate_l1_le P hP B hcard1 x hdelta.1
(hdelta.2.trans hreg.2.2.2.2.2.2.2.2.le)
have hcoeff :=
(cont_atomCoefficient_nonneg_le_envelope P hP hreg x hdelta).2
have hintconst :
(∫ _z : Fin n → ClampObs J, coeff ∂iidProduct P n) = coeff := by simp
have hrewrite :
(∫ z, atomEstimate B z x delta - coeff ∂iidProduct P n) =
(∫ z, atomEstimate B z x delta ∂iidProduct P n) - coeff := by
rw [integral_sub hax (integrable_const coeff), hintconst]
rw [hrewrite] at habs
dsimp [env, coeff]
linarith [le_abs_self
((∫ z, atomEstimate B z x delta ∂iidProduct P n) - coeff)]
_ = (J : ℝ) *
((1 / 2 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹ + env) := by
simp
ring
· intro x hx
refine Integrable.of_bound (atomEstimate_measurable B x delta).aestronglyMeasurable 1 ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (atomEstimate_mem_Icc B z x delta).1]
exact (atomEstimate_mem_Icc B z x delta).2
have hpoint : ∀ z, f z ≤ 2 * t0 + 2 * t1 + atomTotal z := by
intro z
dsimp [f, t0, t1, atomTotal]
convert contHoeffdingInterval_length_le_radius B delta alpha z using 1 <;> ring
have hrhsInt : Integrable (fun z => 2 * t0 + 2 * t1 + atomTotal z)
(iidProduct P n) := (integrable_const _).add hatomInt
calc
(∫ z, f z ∂iidProduct P n) ≤
∫ z, (2 * t0 + 2 * t1 + atomTotal z) ∂iidProduct P n :=
integral_mono_ae hlenInt hrhsInt (ae_of_all _ hpoint)
_ = 2 * t0 + 2 * t1 + ∫ z, atomTotal z ∂iidProduct P n := by
rw [integral_add (integrable_const _) hatomInt, integral_const]
simp
_ ≤ 2 * t0 + 2 * t1 +
(J : ℝ) * ((1 / 2 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹ + env) :=
by linarith [hatomIntLe]
_ = _ := rfl
theorem contHoeffdingInterval_lintegral_length_le
The real expected-length bound upgrades to the extended-real convention used by the continuity-only minimax criterion. The result uses the hP condition, the hreg condition, the hdelta condition, the hcard1 condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
B :
kappa cminus cplus pmin deltaBar delta alpha :
ℝ
hP :
ContClampModel P kappa cminus cplus pmin deltaBar
hreg :
ContDesignConstants J kappa cminus cplus pmin deltaBar
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
hcard1 :
0 < B.I1.card
conclusion
(∫⁻ z, intervalLengthENNReal (contHoeffdingInterval B z delta alpha) ∂iidProduct P n)
Proof (Lean source)
lemma contHoeffdingInterval_lintegral_length_le
{J n : ℕ} (P : ClampLaw J) (B : SplitBlocks n)
(kappa cminus cplus pmin deltaBar delta alpha : ℝ)
(hP : ContClampModel P kappa cminus cplus pmin deltaBar)
(hreg : ContDesignConstants J kappa cminus cplus pmin deltaBar)
(hdelta : delta ∈ Icc (0 : ℝ) deltaBar)
(hcard1 : 0 < B.I1.card) :
(∫⁻ z, intervalLengthENNReal (contHoeffdingInterval B z delta alpha)
∂iidProduct P n) ≤
ofReal
(2 * sqrt (log (4 / alpha) / (2 * (B.I0.card : ℝ))) +
2 * sqrt (log (4 / alpha) / (2 * (B.I1.card : ℝ))) +
(J : ℝ) * ((1 / 2 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹ +
cplus * delta ^ (kappa + 1) / (kappa + 1))) := by
letI : IsProbabilityMeasure P.dataMeasure := hP.probability
letI : IsProbabilityMeasure (iidProduct P n) := by
unfold iidProduct
infer_instance
let f := fun z : Fin n → ClampObs J =>
intervalLength (contHoeffdingInterval B z delta alpha)
have hfmeas : Measurable f := by
let center := fun z : Fin n → ClampObs J => contFallbackEstimator B z delta
let radius := fun z : Fin n → ClampObs J =>
sqrt (log (4 / alpha) / (2 * (B.I0.card : ℝ))) +
sqrt (log (4 / alpha) / (2 * (B.I1.card : ℝ))) +
(∑ x : Fin J, atomEstimate B z x delta) / 2
let lo := fun z => max 0 (center z - radius z)
let hi := fun z => min 1 (center z + radius z)
have hc : Measurable center := by
dsimp [center]
exact clampUnit_measurable'.comp
((retainedEstimate_measurable B delta).add
(measurable_const.mul
(Finset.measurable_fun_sum _ fun x _ => atomEstimate_measurable B x delta)))
have hr : Measurable radius :=
(measurable_const.add measurable_const).add
((Finset.measurable_fun_sum _ fun x _ =>
atomEstimate_measurable B x delta).div_const 2)
have hlo : Measurable lo := measurable_const.max (hc.sub hr)
have hhi : Measurable hi := measurable_const.min (hc.add hr)
have hord : ∀ z, lo z ≤ hi z := by
intro z
have hrad : 0 ≤ radius z := by
dsimp [radius]
have ha : 0 ≤ ∑ x : Fin J, atomEstimate B z x delta :=
sum_nonneg fun x _ => (atomEstimate_mem_Icc B z x delta).1
positivity
have hcI := clampUnit_mem_Icc
(retainedEstimate B z delta +
(1 / 2 : ℝ) * ∑ x : Fin J, atomEstimate B z x delta)
apply le_min
· exact max_le (by norm_num) (by
dsimp [lo, hi, center]
unfold contFallbackEstimator
linarith [hcI.2])
· exact max_le (by
dsimp [lo, hi, center]
unfold contFallbackEstimator
linarith [hcI.1]) (by
dsimp [lo, hi, center]
linarith)
have heq : f = fun z => max 0 (hi z - lo z) := by
funext z
dsimp [f, lo, hi, center, radius]
simp only [contHoeffdingInterval, intervalLength]
rw [csSup_Icc (hord z), csInf_Icc (hord z)]
rw [heq]
exact measurable_const.max (hhi.sub hlo)
have hfint : Integrable f (iidProduct P n) := by
refine Integrable.of_bound hfmeas.aestronglyMeasurable 1 ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (le_max_left _ _)]
let radius :=
sqrt (log (4 / alpha) / (2 * (B.I0.card : ℝ))) +
sqrt (log (4 / alpha) / (2 * (B.I1.card : ℝ))) +
(∑ x : Fin J, atomEstimate B z x delta) / 2
have hrad : 0 ≤ radius := by
dsimp [radius]
have ha : 0 ≤ ∑ x : Fin J, atomEstimate B z x delta :=
sum_nonneg fun x _ => (atomEstimate_mem_Icc B z x delta).1
positivity
have hcI := clampUnit_mem_Icc
(retainedEstimate B z delta +
(1 / 2 : ℝ) * ∑ x : Fin J, atomEstimate B z x delta)
have hord : max 0 (contFallbackEstimator B z delta - radius) ≤
min 1 (contFallbackEstimator B z delta + radius) := by
apply le_min
· exact max_le (by norm_num) (by
unfold contFallbackEstimator
linarith [hcI.2])
· exact max_le (by
unfold contFallbackEstimator
linarith [hcI.1]) (by linarith)
dsimp [f, radius]
simp only [contHoeffdingInterval, intervalLength]
rw [csSup_Icc hord, csInf_Icc hord]
apply max_le
· norm_num
· have hhi := min_le_left (1 : ℝ)
(contFallbackEstimator B z delta +
(sqrt (log (4 / alpha) / (2 * (B.I0.card : ℝ))) +
sqrt (log (4 / alpha) / (2 * (B.I1.card : ℝ))) +
(∑ x : Fin J, atomEstimate B z x delta) / 2))
have hlo := le_max_left (0 : ℝ)
(contFallbackEstimator B z delta -
(sqrt (log (4 / alpha) / (2 * (B.I0.card : ℝ))) +
sqrt (log (4 / alpha) / (2 * (B.I1.card : ℝ))) +
(∑ x : Fin J, atomEstimate B z x delta) / 2))
linarith
have hfnn : 0 ≤ᵐ[iidProduct P n] f :=
ae_of_all _ fun z => le_max_left (0 : ℝ)
(sSup (contHoeffdingInterval B z delta alpha) -
sInf (contHoeffdingInterval B z delta alpha))
simp_rw [intervalLengthENNReal]
rw [← ofReal_integral_eq_lintegral_ofReal hfint hfnn]
exact ENNReal.ofReal_le_ofReal
(contHoeffdingInterval_integral_length_le P B kappa cminus cplus pmin
deltaBar delta alpha hP hreg hdelta hcard1)
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.ContinuityLengthLower 5 declarations Two-point lower bounds for continuity-only honest confidence length
Two-point lower bounds for continuity-only honest confidence length
theorem contMinimaxLength_lower_of_two_point
A close pair in the continuity-only model lower-bounds minimax honest expected interval length. The result uses the hP0 condition, the hP1 condition, the hgap condition, the hsep condition, the hac condition, the hint condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P0 P1 :
ClampLaw J
n :
ℕ
kappa cminus cplus pmin deltaBar delta alpha gap :
ℝ
hP0 :
ContClampModel P0 kappa cminus cplus pmin deltaBar
hP1 :
ContClampModel P1 kappa cminus cplus pmin deltaBar
hgap :
0 ≤ gap
hsep :
gap
≤ |contClampFunctional P1 kappa cminus cplus pmin deltaBar hP1 delta
- contClampFunctional P0 kappa cminus cplus pmin deltaBar hP0 delta|
hac :
iidProduct P1 n ≪ iidProduct P0 n
hint :
Integrable (fun z => (((iidProduct P1 n).rnDeriv (iidProduct P0 n) z).toReal - 1) ^ 2) (iidProduct P0 n)
conclusion
≤ (contFrontierCriteria J n kappa cminus cplus pmin deltaBar delta alpha).2.1
Proof (Lean source)
lemma contMinimaxLength_lower_of_two_point
(P0 P1 : ClampLaw J) (n : ℕ)
(kappa cminus cplus pmin deltaBar delta alpha gap : ℝ)
(hP0 : ContClampModel P0 kappa cminus cplus pmin deltaBar)
(hP1 : ContClampModel P1 kappa cminus cplus pmin deltaBar)
(hgap : 0 ≤ gap)
(hsep : gap ≤ |contClampFunctional P1 kappa cminus cplus pmin deltaBar hP1 delta -
contClampFunctional P0 kappa cminus cplus pmin deltaBar hP0 delta|)
(hac : iidProduct P1 n ≪ iidProduct P0 n)
(hint : Integrable (fun z => (((iidProduct P1 n).rnDeriv
(iidProduct P0 n) z).toReal - 1) ^ 2) (iidProduct P0 n)) :
ofReal ((1 - 2 * alpha -
(1 / 2) * sqrt (productChiSq P1 P0 n)) * gap) ≤
(contFrontierCriteria J n kappa cminus cplus pmin deltaBar delta alpha).2.1 := by
let theta0 := contClampFunctional P0 kappa cminus cplus pmin deltaBar hP0 delta
let theta1 := contClampFunctional P1 kappa cminus cplus pmin deltaBar hP1 delta
let outer : Set ℝ≥0∞ := {r : ℝ≥0∞ | ∃ C : ConfidenceProcedure n J,
ObservedMeasurableInterval C ∧
(∀ (P : ClampLaw J)
(hP : ContClampModel P kappa cminus cplus pmin deltaBar),
1 - alpha ≤ (iidProduct P n).real
{z | contClampFunctional P kappa cminus cplus pmin deltaBar hP delta ∈ C z}) ∧
r = sSup {v : ℝ≥0∞ | ∃ (P : ClampLaw J)
(_hP : ContClampModel P kappa cminus cplus pmin deltaBar),
v = ∫⁻ z, intervalLengthENNReal (C z) ∂iidProduct P n}}
change _ ≤ sInf outer
apply le_sInf
intro r hr
rcases hr with ⟨C, ⟨lo, hi, hlo, hhi, hord, hC⟩, hcover, rfl⟩
let A : Set (Fin n → ClampObs J) := {z | theta0 ∈ C z}
let B : Set (Fin n → ClampObs J) := {z | theta1 ∈ C z}
have hA : MeasurableSet A := by
simp only [A, hC, mem_Icc]
exact
((measurableSet_le hlo measurable_const).inter
(measurableSet_le measurable_const hhi))
have hB : MeasurableSet B := by
simp only [B, hC, mem_Icc]
exact
((measurableSet_le hlo measurable_const).inter
(measurableSet_le measurable_const hhi))
letI : IsProbabilityMeasure P0.dataMeasure := hP0.probability
letI : IsProbabilityMeasure P1.dataMeasure := hP1.probability
letI : IsProbabilityMeasure (iidProduct P0 n) := by unfold iidProduct; infer_instance
letI : IsProbabilityMeasure (iidProduct P1 n) := by unfold iidProduct; infer_instance
have hcov0 : 1 - alpha ≤ (iidProduct P0 n).real A := by
simpa [A, theta0] using hcover P0 hP0
have hcov1 : 1 - alpha ≤ (iidProduct P1 n).real B := by
simpa [B, theta1] using hcover P1 hP1
have htv : tvDist (iidProduct P1 n) (iidProduct P0 n) ≤
(1 / 2) * sqrt (productChiSq P1 P0 n) := by
simpa [productChiSq] using
tvDist_le_half_sqrt_chiSqDiv
(iidProduct P1 n) (iidProduct P0 n) hac hint
have hcov1_at0 : 1 - alpha -
(1 / 2) * sqrt (productChiSq P1 P0 n) ≤
(iidProduct P0 n).real B := by
have hmove := measureReal_sub_le_tvDist
(μ := iidProduct P0 n) (ν := iidProduct P1 n) hB
rw [tvDist_symm] at hmove
linarith
have hinter : 1 - 2 * alpha -
(1 / 2) * sqrt (productChiSq P1 P0 n) ≤
(iidProduct P0 n).real (A ∩ B) := by
have hunion := measureReal_union_add_inter (μ := iidProduct P0 n)
(s := A) (t := B) hB (measure_ne_top _ _) (measure_ne_top _ _)
have hunionle : (iidProduct P0 n).real (A ∪ B) ≤ 1 :=
ENNReal.toReal_mono ENNReal.one_ne_top prob_le_one
linarith
have hpoint (z : Fin n → ClampObs J) :
(A ∩ B).indicator (fun _ => ofReal gap) z ≤
intervalLengthENNReal (C z) := by
by_cases hz : z ∈ A ∩ B
· rw [Set.indicator_of_mem hz]
apply ENNReal.ofReal_le_ofReal
rw [hC z]
simp only [intervalLength, csSup_Icc (hord z), csInf_Icc (hord z)]
rcases hz with ⟨hz0, hz1⟩
have hz0' : lo z ≤ theta0 ∧ theta0 ≤ hi z := by
simpa [A, theta0, hC z] using hz0
have hz1' : lo z ≤ theta1 ∧ theta1 ≤ hi z := by
simpa [B, theta1, hC z] using hz1
rcases hz0' with ⟨hlo0, hhi0⟩
rcases hz1' with ⟨hlo1, hhi1⟩
have habs : |contClampFunctional P1 kappa cminus cplus pmin deltaBar hP1 delta -
contClampFunctional P0 kappa cminus cplus pmin deltaBar hP0 delta| ≤
hi z - lo z := by
rw [abs_le]
constructor <;> linarith
exact (hsep.trans habs).trans (le_max_right _ _)
· simp [hz, intervalLengthENNReal]
have hlen : ofReal gap * (iidProduct P0 n) (A ∩ B) ≤
∫⁻ z, intervalLengthENNReal (C z) ∂iidProduct P0 n := by
calc
_ = ∫⁻ z, (A ∩ B).indicator (fun _ => ofReal gap) z
∂iidProduct P0 n := by
rw [lintegral_indicator (hA.inter hB), lintegral_const]
simp
_ ≤ _ := lintegral_mono hpoint
have hreal : ofReal (1 - 2 * alpha -
(1 / 2) * sqrt (productChiSq P1 P0 n)) ≤
(iidProduct P0 n) (A ∩ B) := by
rw [← ENNReal.ofReal_toReal (measure_ne_top _ _)]
exact ENNReal.ofReal_le_ofReal hinter
have htarget : ofReal ((1 - 2 * alpha -
(1 / 2) * sqrt (productChiSq P1 P0 n)) * gap) ≤
∫⁻ z, intervalLengthENNReal (C z) ∂iidProduct P0 n := by
rw [mul_comm, ENNReal.ofReal_mul hgap]
exact (mul_le_mul le_rfl hreal bot_le bot_le).trans hlen
apply htarget.trans
exact le_sSup ⟨P0, hP0, rfl⟩
theorem contMinimaxLength_lower_of_two_point_chi
A numerical upper bound on the product chi-square divergence gives a corresponding explicit honest-length lower bound. The result uses the hP0 condition, the hP1 condition, the hgap condition, the hsep condition, the hac condition, the hint condition, the hchi0 condition, the hchi condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P0 P1 :
ClampLaw J
n :
ℕ
kappa cminus cplus pmin deltaBar delta alpha gap chi :
ℝ
hP0 :
ContClampModel P0 kappa cminus cplus pmin deltaBar
hP1 :
ContClampModel P1 kappa cminus cplus pmin deltaBar
hgap :
0 ≤ gap
hsep :
gap
≤ |contClampFunctional P1 kappa cminus cplus pmin deltaBar hP1 delta
- contClampFunctional P0 kappa cminus cplus pmin deltaBar hP0 delta|
hac :
iidProduct P1 n ≪ iidProduct P0 n
hint :
Integrable (fun z => (((iidProduct P1 n).rnDeriv (iidProduct P0 n) z).toReal - 1) ^ 2) (iidProduct P0 n)
hchi0 :
0 ≤ chi
hchi :
productChiSq P1 P0 n ≤ chi
conclusion
≤ (contFrontierCriteria J n kappa cminus cplus pmin deltaBar delta alpha).2.1
Proof (Lean source)
lemma contMinimaxLength_lower_of_two_point_chi
(P0 P1 : ClampLaw J) (n : ℕ)
(kappa cminus cplus pmin deltaBar delta alpha gap chi : ℝ)
(hP0 : ContClampModel P0 kappa cminus cplus pmin deltaBar)
(hP1 : ContClampModel P1 kappa cminus cplus pmin deltaBar)
(hgap : 0 ≤ gap)
(hsep : gap ≤ |contClampFunctional P1 kappa cminus cplus pmin deltaBar hP1 delta -
contClampFunctional P0 kappa cminus cplus pmin deltaBar hP0 delta|)
(hac : iidProduct P1 n ≪ iidProduct P0 n)
(hint : Integrable (fun z => (((iidProduct P1 n).rnDeriv
(iidProduct P0 n) z).toReal - 1) ^ 2) (iidProduct P0 n))
(hchi0 : 0 ≤ chi) (hchi : productChiSq P1 P0 n ≤ chi) :
ofReal ((1 - 2 * alpha - (1 / 2) * sqrt chi) * gap) ≤
(contFrontierCriteria J n kappa cminus cplus pmin deltaBar delta alpha).2.1 := by
have hsqrt := Real.sqrt_le_sqrt hchi
have hreal : (1 - 2 * alpha - (1 / 2) * sqrt chi) * gap ≤
(1 - 2 * alpha -
(1 / 2) * sqrt (productChiSq P1 P0 n)) * gap :=
mul_le_mul_of_nonneg_right (by linarith) hgap
exact (ENNReal.ofReal_le_ofReal hreal).trans
(contMinimaxLength_lower_of_two_point P0 P1 n kappa cminus cplus pmin
deltaBar delta alpha gap hP0 hP1 hgap hsep hac hint)
theorem contMinimaxLength_eventually_ge_root
The continuity-only minimax honest length has the regular root-sample-size lower bound, uniformly over threshold sequences. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin deltaBar alpha :
ℝ
hreg :
ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha
conclusion
∃ c : ℝ,
0 < c
conclusion 1
deltaSeq :
ℕ → ℝ
ThresholdSequence deltaBar deltaSeq
≤ (contFrontierCriteria J n kappa cminus cplus pmin deltaBar (deltaSeq n) alpha).2.1
Proof (Lean source)
lemma contMinimaxLength_eventually_ge_root
(J : ℕ) (kappa cminus cplus pmin deltaBar alpha : ℝ)
(hreg : ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha) :
∃ c : ℝ, 0 < c ∧ ∀ deltaSeq : ℕ → ℝ,
ThresholdSequence deltaBar deltaSeq → ∀ᶠ n : ℕ in atTop,
ofReal (c * (n : ℝ) ^ (-(1 : ℝ) / 2)) ≤
(contFrontierCriteria J n kappa cminus cplus pmin deltaBar
(deltaSeq n) alpha).2.1 := by
let eta : ℝ := 1 - 2 * alpha
let tau : ℝ := eta / 16
let chi : ℝ := eta ^ 2 / 16
let floor : ℝ := eta - (1 / 2) * Real.sqrt chi
let c : ℝ := floor * tau
have heta : 0 < eta := by dsimp [eta]; linarith [hreg.2.2]
have heta1 : eta < 1 := by dsimp [eta]; linarith [hreg.2.1]
have htau : 0 < tau := div_pos heta (by norm_num)
have hchi0 : 0 ≤ chi := by dsimp [chi]; positivity
have hsqrt : Real.sqrt chi = eta / 4 := by
rw [show chi = (eta / 4) ^ 2 by dsimp [chi]; ring,
Real.sqrt_sq_eq_abs, abs_of_pos (div_pos heta (by norm_num))]
have hfloor : 0 < floor := by dsimp [floor]; rw [hsqrt]; linarith
refine ⟨c, by dsimp [c]; positivity, ?_⟩
intro deltaSeq hdelta
filter_upwards [Filter.eventually_ge_atTop 1] with n hn
have hnpos : 0 < n := by omega
have hnR : 0 < (n : ℝ) := by exact_mod_cast hnpos
let root : ℝ := (n : ℝ) ^ (-(1 : ℝ) / 2)
let eps : ℝ := tau * root
have hroot : 0 < root := Real.rpow_pos_of_pos hnR _
have heps : 0 < eps := mul_pos htau hroot
have heta_le : eta ≤ 1 := heta1.le
have hroot1 : root ≤ 1 := by
rw [show (1 : ℝ) = 1 ^ (-(1 : ℝ) / 2) by norm_num]
exact Real.rpow_le_rpow_of_nonpos (by norm_num) (by exact_mod_cast hn)
(by norm_num)
have heps_le : eps ≤ 1 / 4 := by
dsimp [eps, tau]
nlinarith
have hepsb : |eps| ≤ 1 / 2 := by rw [abs_of_pos heps]; linarith
have hepss : |eps| < 1 / 2 := by rw [abs_of_pos heps]; linarith
let P0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
let P1 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => eps)
have hw := continuityConstant_twoPoint J n kappa cminus cplus pmin deltaBar
(deltaSeq n) eps hreg.1 (hdelta n) heps.le heps_le
rcases hw with ⟨hP0, hP1, hsepEq, hchiExp⟩
have hrootSq : root ^ 2 = (n : ℝ)⁻¹ := by
dsimp [root]
rw [← Real.rpow_natCast, ← Real.rpow_mul hnR.le]
norm_num
rw [Real.rpow_neg_one]
have hepsSq : eps ^ 2 = tau ^ 2 * (n : ℝ)⁻¹ := by
dsimp [eps]
rw [mul_pow, hrootSq]
have hnE : (n : ℝ) * (4 * eps ^ 2) = eta ^ 2 / 64 := by
rw [hepsSq]
dsimp [tau]
field_simp [hnR.ne']
ring
have hxabs : |eta ^ 2 / 64| ≤ 1 := by
rw [abs_of_nonneg (by positivity)]
nlinarith [sq_nonneg eta]
have hexp : exp (eta ^ 2 / 64) ≤
1 + eta ^ 2 / 64 + (eta ^ 2 / 64) ^ 2 :=
Causalean.Stat.Concentration.exp_le_one_add_add_sq hxabs
have hpoly : eta ^ 2 / 64 + (eta ^ 2 / 64) ^ 2 ≤ chi := by
dsimp [chi]
have heta_sq : eta ^ 2 ≤ 1 := by nlinarith [sq_nonneg eta]
nlinarith [sq_nonneg (eta ^ 2 / 64)]
have hchi : productChiSq P1 P0 n ≤ chi := by
rw [hnE] at hchiExp
linarith
have hac1 := minimaxDataMeasure_ac_center J kappa hreg.1.1 hreg.1.2.1
(fun _ : Fin J × ℝ => eps) measurable_const (fun _ => hepss)
have hint1 := minimaxDataMeasure_sq_integrable_center J kappa hreg.1.1 hreg.1.2.1
(fun _ : Fin J × ℝ => eps) measurable_const (fun _ => hepss)
letI : IsProbabilityMeasure (minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => eps)) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hreg.1.1 hreg.1.2.1 _
measurable_const (fun _ => hepsb)
letI : IsProbabilityMeasure (minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => 0)) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hreg.1.1 hreg.1.2.1 _
measurable_const (fun _ => by norm_num)
have hac := pi_iid_absolutelyContinuous _ _ hac1 n
have hint := pi_iid_integrable_sq_dev _ _ hac1 hint1 n
have hlower := contMinimaxLength_lower_of_two_point_chi P0 P1 n kappa cminus
cplus pmin deltaBar (deltaSeq n) alpha eps chi hP0 hP1 heps.le
(by simpa [P0, P1] using hsepEq.le)
(by simpa [P0, P1, iidProduct, minimaxClampLaw] using hac)
(by simpa [P0, P1, iidProduct, minimaxClampLaw] using hint)
hchi0 hchi
dsimp [c, floor, eps, tau, root, eta] at hlower ⊢
simpa [mul_assoc] using hlower
theorem contMinimaxLength_eventually_ge_atom
Narrow continuous bumps give an atom-scale lower bound for continuity-only minimax honest length. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin deltaBar alpha :
ℝ
hreg :
ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha
conclusion
∃ c : ℝ,
0 < c
conclusion 1
deltaSeq :
ℕ → ℝ
ThresholdSequence deltaBar deltaSeq
≤ (contFrontierCriteria J n kappa cminus cplus pmin deltaBar (deltaSeq n) alpha).2.1
Proof (Lean source)
lemma contMinimaxLength_eventually_ge_atom
(J : ℕ) (kappa cminus cplus pmin deltaBar alpha : ℝ)
(hreg : ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha) :
∃ c : ℝ, 0 < c ∧ ∀ deltaSeq : ℕ → ℝ,
ThresholdSequence deltaBar deltaSeq → ∀ᶠ n : ℕ in atTop,
ofReal (c * (deltaSeq n) ^ (kappa + 1)) ≤
(contFrontierCriteria J n kappa cminus cplus pmin deltaBar
(deltaSeq n) alpha).2.1 := by
let eta : ℝ := 1 - 2 * alpha
let amplitude : ℝ := 1 / 4
let chi : ℝ := eta ^ 2 / 16
let floor : ℝ := eta - (1 / 2) * Real.sqrt chi
let c : ℝ := floor * amplitude
have heta : 0 < eta := by dsimp [eta]; linarith [hreg.2.2]
have heta1 : eta < 1 := by dsimp [eta]; linarith [hreg.2.1]
have hchi0 : 0 ≤ chi := by dsimp [chi]; positivity
have hsqrt : Real.sqrt chi = eta / 4 := by
rw [show chi = (eta / 4) ^ 2 by dsimp [chi]; ring,
Real.sqrt_sq_eq_abs, abs_of_pos (div_pos heta (by norm_num))]
have hfloor : 0 < floor := by dsimp [floor]; rw [hsqrt]; linarith
refine ⟨c, by dsimp [c, amplitude]; positivity, ?_⟩
intro deltaSeq hdelta
filter_upwards [Filter.eventually_ge_atTop 1] with n hn
have hnpos : 0 < n := by omega
have hnR : 0 < (n : ℝ) := by exact_mod_cast hnpos
let A : ℝ := 8 * (kappa + 1) * amplitude ^ 2 * (deltaBar + 1) ^ kappa
let K : ℝ := eta ^ 2 + 64 * A
let h : ℝ := eta ^ 2 / (K * (n : ℝ))
have hk1 : 0 ≤ kappa + 1 := by linarith [hreg.1.2.1]
have hd1 : 0 ≤ deltaBar + 1 := by
linarith [hreg.1.2.2.2.2.2.2.2.1]
have hA : 0 ≤ A := by
dsimp [A, amplitude]
exact mul_nonneg (mul_nonneg (mul_nonneg (by norm_num) hk1) (sq_nonneg _))
(Real.rpow_nonneg hd1 _)
have hK : 0 < K := by dsimp [K]; positivity
have hh : 0 < h := by dsimp [h]; positivity
have hh1 : h ≤ 1 := by
rw [div_le_one (mul_pos hK hnR)]
have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast hn
dsimp [K]
nlinarith [sq_nonneg eta]
have hdh : deltaSeq n + h ≤ deltaBar + 1 := by
linarith [(hdelta n).2]
have hpow : (deltaSeq n + h) ^ kappa ≤ (deltaBar + 1) ^ kappa :=
Real.rpow_le_rpow (add_nonneg (hdelta n).1 hh.le) hdh hreg.1.2.1
have hcoef : 0 ≤ 8 * (kappa + 1) * amplitude ^ 2 * h := by
exact mul_nonneg (mul_nonneg (mul_nonneg (by norm_num) hk1) (sq_nonneg _)) hh.le
have hcore : (n : ℝ) * (8 * (kappa + 1) * amplitude ^ 2 * h *
(deltaBar + 1) ^ kappa) = A * eta ^ 2 / K := by
dsimp [h, A]
field_simp [hK.ne', hnR.ne']
have hfrac : A * eta ^ 2 / K ≤ eta ^ 2 / 64 := by
rw [div_le_iff₀ hK]
dsimp [K]
nlinarith [sq_nonneg eta]
have hexponent : (n : ℝ) * (8 * (kappa + 1) * amplitude ^ 2 * h *
(deltaSeq n + h) ^ kappa) ≤ eta ^ 2 / 64 := by
calc
_ ≤ (n : ℝ) * (8 * (kappa + 1) * amplitude ^ 2 * h *
(deltaBar + 1) ^ kappa) := by gcongr
_ = A * eta ^ 2 / K := hcore
_ ≤ _ := hfrac
have hw := continuityBump_twoPoint J n kappa cminus cplus pmin deltaBar
(deltaSeq n) h amplitude hreg.1 (hdelta n) hh (by norm_num [amplitude])
(by norm_num [amplitude])
let q := fun a : ℝ => amplitude *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - deltaSeq n) / h)
let P0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
let P1 := minimaxClampLaw J kappa (fun p => q p.2)
rcases hw with ⟨hP0, hP1, hsep, hchiExp⟩
have hxabs : |eta ^ 2 / 64| ≤ 1 := by
rw [abs_of_nonneg (by positivity)]
nlinarith [sq_nonneg eta]
have hexp : exp (eta ^ 2 / 64) ≤
1 + eta ^ 2 / 64 + (eta ^ 2 / 64) ^ 2 :=
Causalean.Stat.Concentration.exp_le_one_add_add_sq hxabs
have hpoly : eta ^ 2 / 64 + (eta ^ 2 / 64) ^ 2 ≤ chi := by
dsimp [chi]
have heta_sq : eta ^ 2 ≤ 1 := by nlinarith [sq_nonneg eta]
nlinarith [sq_nonneg (eta ^ 2 / 64)]
have hchi : productChiSq P1 P0 n ≤ chi := by
have hbound := hchiExp.trans (Real.exp_le_exp.mpr hexponent)
linarith
have hqmeas : Measurable q := by dsimp [q]; fun_prop
have hq0 (a : ℝ) : 0 ≤ q a := by
exact mul_nonneg (by norm_num [amplitude])
(CausalSmith.Stat.DoseResponseMinimax.doseBump_nonneg _)
have hqbound (a : ℝ) : |q a| ≤ 1 / 2 := by
rw [abs_of_nonneg (hq0 a)]
have hb := CausalSmith.Stat.DoseResponseMinimax.doseBump_le_one
((a - deltaSeq n) / h)
dsimp [q, amplitude]
nlinarith
have hqstrict (p : Fin J × ℝ) : |q p.2| < 1 / 2 := by
rw [abs_of_nonneg (hq0 p.2)]
have hb := CausalSmith.Stat.DoseResponseMinimax.doseBump_le_one
((p.2 - deltaSeq n) / h)
dsimp [q, amplitude]
nlinarith
have hac1 := minimaxDataMeasure_ac_center J kappa hreg.1.1 hreg.1.2.1
(fun p : Fin J × ℝ => q p.2) (hqmeas.comp measurable_snd) hqstrict
have hint1 := minimaxDataMeasure_sq_integrable_center J kappa hreg.1.1 hreg.1.2.1
(fun p : Fin J × ℝ => q p.2) (hqmeas.comp measurable_snd) hqstrict
letI : IsProbabilityMeasure (minimaxDataMeasure J kappa
(fun p : Fin J × ℝ => q p.2)) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hreg.1.1 hreg.1.2.1 _
(hqmeas.comp measurable_snd) (fun p => hqbound p.2)
letI : IsProbabilityMeasure (minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => 0)) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hreg.1.1 hreg.1.2.1 _
measurable_const (fun _ => by norm_num)
have hac := pi_iid_absolutelyContinuous _ _ hac1 n
have hint := pi_iid_integrable_sq_dev _ _ hac1 hint1 n
have hlower := contMinimaxLength_lower_of_two_point_chi P0 P1 n kappa cminus
cplus pmin deltaBar (deltaSeq n) alpha
(amplitude * (deltaSeq n) ^ (kappa + 1)) chi hP0 hP1
(mul_nonneg (by norm_num [amplitude]) (Real.rpow_nonneg (hdelta n).1 _)) hsep
(by simpa [P0, P1, q, iidProduct, minimaxClampLaw] using hac)
(by simpa [P0, P1, q, iidProduct, minimaxClampLaw] using hint)
hchi0 hchi
dsimp [c, floor, amplitude, eta] at hlower ⊢
simpa [mul_assoc] using hlower
theorem contMinimaxLength_eventually_ge_frontier
The regular and atom experiments combine into the full continuity frontier lower bound for minimax honest expected length. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin deltaBar alpha :
ℝ
hreg :
ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha
conclusion
∃ c : ℝ,
0 < c
conclusion 1
deltaSeq :
ℕ → ℝ
ThresholdSequence deltaBar deltaSeq
≤ (contFrontierCriteria J n kappa cminus cplus pmin deltaBar (deltaSeq n) alpha).2.1
Proof (Lean source)
lemma contMinimaxLength_eventually_ge_frontier
(J : ℕ) (kappa cminus cplus pmin deltaBar alpha : ℝ)
(hreg : ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha) :
∃ c : ℝ, 0 < c ∧ ∀ deltaSeq : ℕ → ℝ,
ThresholdSequence deltaBar deltaSeq → ∀ᶠ n : ℕ in atTop,
ofReal (c * contFrontier n (deltaSeq n) kappa) ≤
(contFrontierCriteria J n kappa cminus cplus pmin deltaBar
(deltaSeq n) alpha).2.1 := by
obtain ⟨c0, hc0, h0⟩ := contMinimaxLength_eventually_ge_root J kappa
cminus cplus pmin deltaBar alpha hreg
obtain ⟨c1, hc1, h1⟩ := contMinimaxLength_eventually_ge_atom J kappa
cminus cplus pmin deltaBar alpha hreg
let c := min c0 c1 / 2
refine ⟨c, by dsimp [c]; positivity, ?_⟩
intro deltaSeq hdelta
filter_upwards [h0 deltaSeq hdelta, h1 deltaSeq hdelta] with n hn0 hn1
let root : ℝ := (n : ℝ) ^ (-(1 : ℝ) / 2)
let atom : ℝ := (deltaSeq n) ^ (kappa + 1)
have hroot : 0 ≤ root := Real.rpow_nonneg (Nat.cast_nonneg n) _
have hatom : 0 ≤ atom := Real.rpow_nonneg (hdelta n).1 _
by_cases hle : c0 * root ≤ c1 * atom
· apply (ENNReal.ofReal_le_ofReal ?_).trans hn1
dsimp [c, contFrontier, root, atom]
calc
min c0 c1 / 2 *
((n : ℝ) ^ (-(1 : ℝ) / 2) + (deltaSeq n) ^ (kappa + 1)) =
(min c0 c1 * (n : ℝ) ^ (-(1 : ℝ) / 2) +
min c0 c1 * (deltaSeq n) ^ (kappa + 1)) / 2 := by ring
_ ≤ (c0 * (n : ℝ) ^ (-(1 : ℝ) / 2) +
c1 * (deltaSeq n) ^ (kappa + 1)) / 2 := by
apply div_le_div_of_nonneg_right _ (by norm_num)
exact add_le_add
(mul_le_mul_of_nonneg_right (min_le_left _ _) hroot)
(mul_le_mul_of_nonneg_right (min_le_right _ _) hatom)
_ ≤ c1 * (deltaSeq n) ^ (kappa + 1) := by
dsimp [root, atom] at hle
linarith
· apply (ENNReal.ofReal_le_ofReal ?_).trans hn0
have hle' : c1 * atom ≤ c0 * root := le_of_not_ge hle
dsimp [c, contFrontier, root, atom]
calc
min c0 c1 / 2 *
((n : ℝ) ^ (-(1 : ℝ) / 2) + (deltaSeq n) ^ (kappa + 1)) =
(min c0 c1 * (n : ℝ) ^ (-(1 : ℝ) / 2) +
min c0 c1 * (deltaSeq n) ^ (kappa + 1)) / 2 := by ring
_ ≤ (c0 * (n : ℝ) ^ (-(1 : ℝ) / 2) +
c1 * (deltaSeq n) ^ (kappa + 1)) / 2 := by
apply div_le_div_of_nonneg_right _ (by norm_num)
exact add_le_add
(mul_le_mul_of_nonneg_right (min_le_left _ _) hroot)
(mul_le_mul_of_nonneg_right (min_le_right _ _) hatom)
_ ≤ c0 * (n : ℝ) ^ (-(1 : ℝ) / 2) := by
dsimp [root, atom] at hle'
linarith
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.ContinuityProcedureBounds 11 declarations Finite-sample bounds for the continuity-only procedure
Finite-sample bounds for the continuity-only procedure
theorem blockAverage_hoeffding
Hoeffding's inequality for an empirical average over a deterministic block of the canonical finite-product sample. The result uses the hf condition, the hf01 condition, the ht condition. This is the stated conclusion.
Formal statement
Proof (Lean source)
lemma blockAverage_hoeffding
(_HoeffdingBoundedAverage_of_gate : HoeffdingBoundedAverage)
{X : Type} [MeasurableSpace X] (P : Measure X) [IsProbabilityMeasure P]
{n : ℕ} (I : Finset (Fin n)) (f : X → ℝ) (hf : Measurable f)
(hf01 : ∀ x, f x ∈ Icc (0 : ℝ) 1) (t : ℝ) (ht : 0 < t) :
(Measure.pi (fun _ : Fin n => P)).real
{z | |blockAverage I z f - ∫ x, f x ∂P| > t} ≤
2 * exp (-2 * (I.card : ℝ) * t ^ 2) := by
classical
by_cases hI : I.card = 0
· exact (measureReal_le_one (μ := Measure.pi fun _ : Fin n => P)
(s := {z | |blockAverage I z f - ∫ x, f x ∂P| > t})).trans (by
simp [hI])
let S := Causalean.Stat.iidSample_infinitePi P
let sumFin : (Fin n → X) → ℝ := fun z => ∑ i ∈ I, f (z i)
let sumInf : (ℕ → X) → ℝ :=
Causalean.Stat.Concentration.bernoulliCount S f I.card
let A : Set ℝ := {y | |(I.card : ℝ)⁻¹ * y - ∫ x, f x ∂P| > t}
have hA : MeasurableSet A := by
exact measurableSet_lt measurable_const
(((measurable_const.mul measurable_id).sub measurable_const).abs)
have hsumFin : Measurable sumFin := by
exact Finset.measurable_fun_sum I fun i _ => hf.comp (measurable_pi_apply i)
have hsumInf : Measurable sumInf := by
dsimp [sumInf]
exact Finset.measurable_fun_sum _ fun i _ => hf.comp (S.meas i)
have hlaw : (Measure.pi (fun _ : Fin n => P)).map sumFin =
(Measure.infinitePi (fun _ : ℕ => P)).map sumInf := by
calc
_ = (Measure.pi (fun _ : Fin I.card => P)).map
(fun z => ∑ j : Fin I.card, f (z j)) :=
block_count_law_eq P I f hf
_ = _ := range_count_transport P I.card f hf
have htail := hoeffding_abs_ge S hf
(a := 0) (b := 1) (by norm_num) (ae_of_all _ hf01) I.card
(Nat.pos_of_ne_zero hI) ht.le
have hfinite : (Measure.pi (fun _ : Fin n => P)).real (sumFin ⁻¹' A) =
((Measure.pi (fun _ : Fin n => P)).map sumFin).real A := by
simpa only [measureReal_def] using congrArg toReal
(Measure.map_apply hsumFin hA).symm
have hinf : (Measure.infinitePi (fun _ : ℕ => P)).real (sumInf ⁻¹' A) =
((Measure.infinitePi (fun _ : ℕ => P)).map sumInf).real A := by
simpa only [measureReal_def] using congrArg toReal
(Measure.map_apply hsumInf hA).symm
have hle : (Measure.infinitePi (fun _ : ℕ => P)).real (sumInf ⁻¹' A) ≤
2 * exp (-2 * (I.card : ℝ) * t ^ 2) := by
refine (measureReal_mono
(μ := Measure.infinitePi (fun _ : ℕ => P))
(s₁ := sumInf ⁻¹' A)
(s₂ := {w | t ≤ |S.sampleMean f I.card w - ∫ x, f x ∂P|}) ?_).trans ?_
· intro w hw
change t < |(I.card : ℝ)⁻¹ * sumInf w - ∫ x, f x ∂P| at hw
change t ≤ |S.sampleMean f I.card w - ∫ x, f x ∂P|
simpa [S, sumInf, sampleMean,
Causalean.Stat.Concentration.bernoulliCount] using le_of_lt hw
· simpa [S, sumInf, A, sampleMean] using htail
calc
(Measure.pi (fun _ : Fin n => P)).real
{z | |blockAverage I z f - ∫ x, f x ∂P| > t} =
(Measure.pi (fun _ : Fin n => P)).real (sumFin ⁻¹' A) := by
rfl
_ = ((Measure.pi (fun _ : Fin n => P)).map sumFin).real A := hfinite
_ = ((Measure.infinitePi (fun _ : ℕ => P)).map sumInf).real A := by rw [hlaw]
_ = (Measure.infinitePi (fun _ : ℕ => P)).real (sumInf ⁻¹' A) := hinf.symm
_ ≤ _ := hle
theorem cont_iidProduct_outcomeSupport
Outcome support lifts coordinatewise to the canonical finite product for a continuity-only model. The result uses the hP condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
kappa cminus cplus pmin deltaBar :
ℝ
hP :
ContClampModel P kappa cminus cplus pmin deltaBar
Proof (Lean source)
lemma cont_iidProduct_outcomeSupport {J n : ℕ} (P : ClampLaw J)
{kappa cminus cplus pmin deltaBar : ℝ}
(hP : ContClampModel P kappa cminus cplus pmin deltaBar) :
∀ᵐ z ∂iidProduct P n, ∀ i : Fin n, (z i).Y ∈ Icc (0 : ℝ) 1 := by
let _ := hP.probability
letI : IsProbabilityMeasure (iidProduct P n) := by
unfold iidProduct
infer_instance
apply ae_all_iff.mpr
intro i
have hmap : (iidProduct P n).map (Function.eval i) = P.dataMeasure := by
simpa [iidProduct] using (Measure.pi_map_eval (fun _ : Fin n => P.dataMeasure) i)
have hs := hP.outcomeSupport
rw [← hmap] at hs
exact (ae_map_iff (μ := iidProduct P n) (f := Function.eval i)
(p := fun o : ClampObs J => o.Y ∈ Icc (0 : ℝ) 1)
(measurable_pi_apply i).aemeasurable
(measurableSet_Icc.preimage clampOutcome_measurable)).mp hs
theorem cont_retainedEstimate_l1_le
The retained-block empirical mean has root-block-size absolute risk in the continuity-only model. The result uses the hP condition, the hcard condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
kappa cminus cplus pmin deltaBar delta :
ℝ
hP :
ContClampModel P kappa cminus cplus pmin deltaBar
B :
hcard :
0 < B.I0.card
conclusion
(∫ z, |retainedEstimate B z delta - retainedMean P delta| ∂iidProduct P n)
≤ (1 / 2 : ℝ) * sqrt (B.I0.card : ℝ)⁻¹
Proof (Lean source)
lemma cont_retainedEstimate_l1_le {J n : ℕ} (P : ClampLaw J)
{kappa cminus cplus pmin deltaBar delta : ℝ}
(hP : ContClampModel P kappa cminus cplus pmin deltaBar)
(B : SplitBlocks n) (hcard : 0 < B.I0.card) :
(∫ z, |retainedEstimate B z delta - retainedMean P delta| ∂iidProduct P n) ≤
(1 / 2 : ℝ) * sqrt (B.I0.card : ℝ)⁻¹ := by
let _ := hP.probability
let f : ClampObs J → ℝ := fun o => if delta < o.A then clampUnit o.Y else 0
have hf : Measurable f := by
exact Measurable.ite
(measurableSet_lt measurable_const
(measurable_fst.comp (measurable_snd.comp (Measurable.of_comap_le le_rfl))))
(clampUnit_measurable'.comp clampOutcome_measurable) measurable_const
have hf0 : ∀ o, 0 ≤ f o := by
intro o; dsimp [f]; split_ifs
· exact (clampUnit_mem_Icc o.Y).1
· norm_num
have hf1 : ∀ o, f o ≤ 1 := by
intro o; dsimp [f]; split_ifs
· exact (clampUnit_mem_Icc o.Y).2
· norm_num
have hm : ∫ o, f o ∂P.dataMeasure = retainedMean P delta := by
calc
_ = ∫ o, (if delta < o.A then o.Y else 0) ∂P.dataMeasure := by
apply integral_congr_ae
filter_upwards [hP.outcomeSupport] with o ho
simp [f, clampUnit, ho.1, ho.2]
_ = _ := by
unfold retainedMean
apply integral_congr_ae
filter_upwards with o
by_cases ha : delta < o.A <;> simp [indicator, ha]
have heq : (∫ z, |retainedEstimate B z delta - retainedMean P delta|
∂iidProduct P n) =
∫ z, |blockAverage B.I0 z f - retainedMean P delta| ∂iidProduct P n := by
apply integral_congr_ae
filter_upwards [cont_iidProduct_outcomeSupport P hP] with z hz
congr 2
unfold retainedEstimate blockAverage
congr 1
apply Finset.sum_congr rfl
intro i hi
simp [f, clampUnit, (hz i).1, (hz i).2]
rw [heq]
exact blockAverage_l1_le P B.I0 hcard f hf hf0 hf1 (retainedMean P delta) hm
theorem cont_atomEstimate_l1_le
A stratum-threshold empirical mass has root-block-size absolute risk in the continuity-only model. The result uses the hP condition, the hcard condition, the hdelta condition, the hdelta1 condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
kappa cminus cplus pmin deltaBar delta :
ℝ
hP :
ContClampModel P kappa cminus cplus pmin deltaBar
B :
hcard :
0 < B.I1.card
x :
Fin J
hdelta :
0 ≤ delta
hdelta1 :
delta ≤ 1
conclusion
(∫ z, |atomEstimate B z x delta - P.px x * atomMass P x delta| ∂iidProduct P n)
≤ (1 / 2 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹
Proof (Lean source)
lemma cont_atomEstimate_l1_le {J n : ℕ} (P : ClampLaw J)
{kappa cminus cplus pmin deltaBar delta : ℝ}
(hP : ContClampModel P kappa cminus cplus pmin deltaBar)
(B : SplitBlocks n) (hcard : 0 < B.I1.card) (x : Fin J)
(hdelta : 0 ≤ delta) (hdelta1 : delta ≤ 1) :
(∫ z, |atomEstimate B z x delta - P.px x * atomMass P x delta|
∂iidProduct P n) ≤ (1 / 2 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹ := by
let _ := hP.probability
let f : ClampObs J → ℝ := fun o => if o.X = x ∧ o.A ≤ delta then 1 else 0
have hX : Measurable (fun o : ClampObs J => o.X) :=
measurable_fst.comp (Measurable.of_comap_le le_rfl)
have hA : Measurable (fun o : ClampObs J => o.A) :=
measurable_fst.comp (measurable_snd.comp (Measurable.of_comap_le le_rfl))
have hf : Measurable f := by
exact Measurable.ite
((measurableSet_eq_fun hX measurable_const).inter
(measurableSet_le hA measurable_const)) measurable_const measurable_const
have hf0 : ∀ o, 0 ≤ f o := by intro o; dsimp [f]; split_ifs <;> norm_num
have hf1 : ∀ o, f o ≤ 1 := by intro o; dsimp [f]; split_ifs <;> norm_num
have hm : ∫ o, f o ∂P.dataMeasure = P.px x * atomMass P x delta := by
have hset : MeasurableSet {o : ClampObs J |
o.X = x ∧ o.A ∈ Icc (0 : ℝ) delta} :=
(measurableSet_eq_fun hX measurable_const).inter
(measurableSet_Icc.preimage hA)
calc
_ = ∫ o, indicator {o : ClampObs J |
o.X = x ∧ o.A ∈ Icc (0 : ℝ) delta} (fun _ => (1 : ℝ)) o
∂P.dataMeasure := by
apply integral_congr_ae
filter_upwards [hP.treatmentSupport] with o ho
by_cases hx : o.X = x <;> by_cases ha : o.A ≤ delta <;>
simp [f, indicator, hx, ha, ho.1]
_ = P.dataMeasure.real {o : ClampObs J |
o.X = x ∧ o.A ∈ Icc (0 : ℝ) delta} := by
rw [integral_indicator hset]
simp
_ = _ := by
rw [(hP.condDensity.2.2 x).2 (Icc 0 delta) measurableSet_Icc]
· rfl
· intro a ha
exact ⟨ha.1, ha.2.trans hdelta1⟩
simpa [atomEstimate, f] using
(blockAverage_l1_le P B.I1 hcard f hf hf0 hf1
(P.px x * atomMass P x delta) hm)
theorem cont_atomCoefficient_nonneg_le_envelope
The polynomial thinning envelope controls each population atom coefficient in the continuity-only model. The result uses the hP condition, the hreg condition, the hdelta condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
kappa cminus cplus pmin deltaBar delta :
ℝ
hP :
ContClampModel P kappa cminus cplus pmin deltaBar
hreg :
ContDesignConstants J kappa cminus cplus pmin deltaBar
x :
Fin J
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
Proof (Lean source)
lemma cont_atomCoefficient_nonneg_le_envelope {J : ℕ} (P : ClampLaw J)
{kappa cminus cplus pmin deltaBar delta : ℝ}
(hP : ContClampModel P kappa cminus cplus pmin deltaBar)
(hreg : ContDesignConstants J kappa cminus cplus pmin deltaBar)
(x : Fin J) (hdelta : delta ∈ Icc (0 : ℝ) deltaBar) :
P.px x * atomMass P x delta ∈ Icc 0
(cplus * delta ^ (kappa + 1) / (kappa + 1)) := by
rcases hreg with ⟨hJ, hkappa, hcminus, hcminus_le, hcplus, hpmin,
hpmin_le, hdeltaBar, hdeltaBar1⟩
have hdelta1 : delta ≤ 1 := hdelta.2.trans hdeltaBar1.le
have hsubset : Icc (0 : ℝ) delta ⊆ Icc (0 : ℝ) 1 := fun _ ha =>
⟨ha.1, ha.2.trans hdelta1⟩
have hpiInt : Integrable (P.pi x)
(volume.restrict (Icc (0 : ℝ) 1)) := by
apply IntegrableOn.of_bound measure_Icc_lt_top
· exact (aemeasurable_restrict_of_measurable_subtype measurableSet_Icc
(hP.condDensity.1 x)).aestronglyMeasurable
· have hcplus0 : 0 ≤ cplus := by linarith
filter_upwards [ae_restrict_mem measurableSet_Icc,
hP.condDensity.2.1 x, hP.thinning x] with a ha hnon hthin
rw [Real.norm_eq_abs, abs_of_nonneg hnon]
calc
P.pi x a ≤ cplus * a ^ kappa := hthin.2
_ ≤ cplus := by
simpa only [mul_one] using mul_le_mul_of_nonneg_left
(Real.rpow_le_one ha.1 ha.2 hkappa) hcplus0
have hpiDelta : IntegrableOn (P.pi x) (Icc (0 : ℝ) delta) volume :=
hpiInt.mono_measure (Measure.restrict_mono_set volume hsubset)
have hpowInt : IntegrableOn (fun a : ℝ => a ^ kappa)
(Icc 0 delta) volume := by
rw [← intervalIntegrable_iff_integrableOn_Icc_of_le hdelta.1]
exact intervalIntegral.intervalIntegrable_rpow (inl hkappa)
have hthinDelta := (hP.thinning x).filter_mono
(ae_mono (Measure.restrict_mono_set volume hsubset))
have hpowEval : (∫ a in Icc (0 : ℝ) delta, a ^ kappa) =
delta ^ (kappa + 1) / (kappa + 1) := by
rw [integral_Icc_eq_integral_Ioc,
← intervalIntegral.integral_of_le hdelta.1,
integral_rpow (inl (by linarith : -1 < kappa))]
rw [Real.zero_rpow (by linarith : kappa + 1 ≠ 0), sub_zero]
have hmass0 : 0 ≤ atomMass P x delta := by
unfold atomMass
exact integral_nonneg_of_ae
((hP.condDensity.2.1 x).filter_mono
(ae_mono (Measure.restrict_mono_set volume hsubset)))
have hmassUpper : atomMass P x delta ≤
cplus * delta ^ (kappa + 1) / (kappa + 1) := by
unfold atomMass
calc
(∫ a in Icc (0 : ℝ) delta, P.pi x a) ≤
∫ a in Icc (0 : ℝ) delta, cplus * a ^ kappa :=
integral_mono_ae hpiDelta (hpowInt.const_mul cplus)
(hthinDelta.mono fun _ ha => ha.2)
_ = _ := by rw [integral_const_mul, hpowEval]; ring
have hXm : Measurable (fun o : ClampObs J => o.X) :=
measurable_fst.comp (Measurable.of_comap_le le_rfl)
let _ := hP.probability
letI : IsProbabilityMeasure (P.dataMeasure.map fun o => o.X) :=
Measure.isProbabilityMeasure_map hXm.aemeasurable
have hpx0 : 0 ≤ P.px x := by
rw [(hP.stratumMass x).1]
exact measureReal_nonneg
have hpx1 : P.px x ≤ 1 := by
rw [(hP.stratumMass x).1]
exact measureReal_le_one
exact ⟨mul_nonneg hpx0 hmass0,
(mul_le_mul_of_nonneg_right hpx1 hmass0).trans (by simpa using hmassUpper)⟩
theorem contFallbackEstimator_risk_le_explicit
The fixed one-half fallback has an explicit root-block plus threshold-mass absolute-risk bound. The result uses the hP condition, the hreg condition, the hdelta condition, the hcard0 condition, the hcard1 condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
B :
kappa cminus cplus pmin deltaBar delta :
ℝ
hP :
ContClampModel P kappa cminus cplus pmin deltaBar
hreg :
ContDesignConstants J kappa cminus cplus pmin deltaBar
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
hcard0 :
0 < B.I0.card
hcard1 :
0 < B.I1.card
conclusion
contEstimatorRisk P n kappa cminus cplus pmin deltaBar delta hP (contFallbackEstimator B · delta)
≤ (1 / 2 : ℝ) * sqrt (B.I0.card : ℝ)⁻¹
+ (J : ℝ) * ((1 / 4 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹ + (1 / 2 : ℝ) * (cplus * delta ^ (kappa + 1) / (kappa + 1)))
Proof (Lean source)
lemma contFallbackEstimator_risk_le_explicit {J n : ℕ} (P : ClampLaw J)
(B : SplitBlocks n) (kappa cminus cplus pmin deltaBar delta : ℝ)
(hP : ContClampModel P kappa cminus cplus pmin deltaBar)
(hreg : ContDesignConstants J kappa cminus cplus pmin deltaBar)
(hdelta : delta ∈ Icc (0 : ℝ) deltaBar)
(hcard0 : 0 < B.I0.card) (hcard1 : 0 < B.I1.card) :
contEstimatorRisk P n kappa cminus cplus pmin deltaBar delta hP
(contFallbackEstimator B · delta) ≤
(1 / 2 : ℝ) * sqrt (B.I0.card : ℝ)⁻¹ +
(J : ℝ) * ((1 / 4 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹ +
(1 / 2 : ℝ) * (cplus * delta ^ (kappa + 1) / (kappa + 1))) := by
letI : IsProbabilityMeasure P.dataMeasure := hP.probability
letI : IsProbabilityMeasure (iidProduct P n) := by unfold iidProduct; infer_instance
let target := contClampFunctional P kappa cminus cplus pmin deltaBar hP delta
let e0 := fun z : Fin n → ClampObs J =>
|retainedEstimate B z delta - retainedMean P delta|
let ex := fun x : Fin J => fun z : Fin n → ClampObs J =>
|atomEstimate B z x delta - P.px x * atomMass P x delta|
let env := cplus * delta ^ (kappa + 1) / (kappa + 1)
have ht := contClampFunctional_mem_Icc P kappa cminus cplus pmin deltaBar
delta hP hreg hdelta
have hpoint (z : Fin n → ClampObs J) :
|contFallbackEstimator B z delta - target| ≤
e0 z + ∑ x : Fin J, ((1 / 2 : ℝ) * ex x z + (1 / 2 : ℝ) * env) := by
have hclamp := abs_clampUnit_sub_le
(retainedEstimate B z delta +
(1 / 2 : ℝ) * ∑ x : Fin J, atomEstimate B z x delta) target ht
have hx (x : Fin J) :
|(1 / 2 : ℝ) * atomEstimate B z x delta -
P.px x * atomMass P x delta *
contRegression P kappa cminus cplus pmin deltaBar hP x delta| ≤
(1 / 2 : ℝ) * ex x z + (1 / 2 : ℝ) * env := by
let coeff := P.px x * atomMass P x delta
have hc := cont_atomCoefficient_nonneg_le_envelope P hP hreg x hdelta
have hm := contRegression_mem_Icc P kappa cminus cplus pmin deltaBar
hP hreg x delta hdelta
have hmhalf : |(1 / 2 : ℝ) -
contRegression P kappa cminus cplus pmin deltaBar hP x delta| ≤ 1 / 2 := by
rw [abs_le]
constructor <;> linarith [hm.1, hm.2]
calc
_ = |(1 / 2 : ℝ) *
(atomEstimate B z x delta - coeff) +
coeff * ((1 / 2 : ℝ) -
contRegression P kappa cminus cplus pmin deltaBar hP x delta)| := by
congr 1
dsimp [coeff]
ring
_ ≤ |(1 / 2 : ℝ) * (atomEstimate B z x delta - coeff)| +
|coeff * ((1 / 2 : ℝ) -
contRegression P kappa cminus cplus pmin deltaBar hP x delta)| :=
abs_add_le _ _
_ ≤ (1 / 2 : ℝ) * ex x z + (1 / 2 : ℝ) * env := by
rw [abs_mul, abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 1 / 2),
abs_mul, abs_of_nonneg hc.1]
dsimp [ex, coeff, env]
nlinarith [mul_le_mul_of_nonneg_left hmhalf hc.1,
mul_le_mul_of_nonneg_left hc.2 (by norm_num : (0 : ℝ) ≤ 1 / 2)]
calc
|contFallbackEstimator B z delta - target| ≤
|(retainedEstimate B z delta - retainedMean P delta) +
∑ x : Fin J, ((1 / 2 : ℝ) * atomEstimate B z x delta -
P.px x * atomMass P x delta *
contRegression P kappa cminus cplus pmin deltaBar hP x delta)| := by
refine hclamp.trans_eq ?_
congr 1
dsimp [target]
unfold contClampFunctional
rw [Finset.mul_sum, Finset.sum_sub_distrib]
ring
_ ≤ e0 z + ∑ x : Fin J,
|(1 / 2 : ℝ) * atomEstimate B z x delta -
P.px x * atomMass P x delta *
contRegression P kappa cminus cplus pmin deltaBar hP x delta| := by
let fsum := fun x : Fin J =>
(1 / 2 : ℝ) * atomEstimate B z x delta -
P.px x * atomMass P x delta *
contRegression P kappa cminus cplus pmin deltaBar hP x delta
have hs : |∑ x : Fin J, fsum x| ≤ ∑ x : Fin J, |fsum x| := by
simpa using Finset.abs_sum_le_sum_abs fsum univ
exact (abs_add_le _ _).trans (add_le_add_right hs _)
_ ≤ _ := by
exact add_le_add_right (Finset.sum_le_sum fun x _ => hx x) _
have he0int : Integrable e0 (iidProduct P n) := by
refine Integrable.of_bound
((retainedEstimate_measurable B delta).sub measurable_const).abs.aestronglyMeasurable 2 ?_
filter_upwards [cont_iidProduct_outcomeSupport P hP] with z hz
dsimp [e0]
rw [abs_abs]
have hr := blockAverage_mem_Icc B.I0 z
(fun o : ClampObs J => if delta < o.A then clampUnit o.Y else 0) (fun o => by
split_ifs
· exact (clampUnit_mem_Icc o.Y)
· exact ⟨by norm_num, by norm_num⟩)
have hre : retainedEstimate B z delta = blockAverage B.I0 z
(fun o : ClampObs J => if delta < o.A then clampUnit o.Y else 0) := by
unfold retainedEstimate blockAverage
congr 1
apply Finset.sum_congr rfl
intro i hi
simp [clampUnit, (hz i).1, (hz i).2]
have hm0 : |retainedMean P delta| ≤ 1 := by
let g := indicator {o : ClampObs J | delta < o.A} (fun o => o.Y)
have hgmeas : Measurable g := clampOutcome_measurable.indicator
(measurableSet_lt measurable_const
(measurable_fst.comp (measurable_snd.comp (Measurable.of_comap_le le_rfl))))
have hgint : Integrable g P.dataMeasure := by
refine Integrable.of_bound hgmeas.aestronglyMeasurable 1 ?_
filter_upwards [hP.outcomeSupport] with o ho
by_cases ha : delta < o.A <;> simp [g, indicator, ha, abs_of_nonneg ho.1, ho.2]
have hg0 : 0 ≤ᵐ[P.dataMeasure] g := by
filter_upwards [hP.outcomeSupport] with o ho
by_cases ha : delta < o.A <;> simp [g, indicator, ha, ho.1]
have hg1 : g ≤ᵐ[P.dataMeasure] fun _ => (1 : ℝ) := by
filter_upwards [hP.outcomeSupport] with o ho
by_cases ha : delta < o.A <;> simp [g, indicator, ha, ho.2]
have hmI : retainedMean P delta ∈ Icc (0 : ℝ) 1 := by
have hret : retainedMean P delta = ∫ o, g o ∂P.dataMeasure := by
unfold retainedMean
apply integral_congr_ae
filter_upwards with o
by_cases ha : delta < o.A <;> simp [g, indicator, ha]
rw [hret]
exact ⟨integral_nonneg_of_ae hg0,
(integral_mono_ae hgint (integrable_const 1) hg1).trans_eq (by simp)⟩
rw [abs_of_nonneg hmI.1]
exact hmI.2
rw [hre]
rw [abs_le]
have hmBounds := (abs_le.mp hm0)
exact ⟨by linarith [hr.1, hmBounds.2], by linarith [hr.2, hmBounds.1]⟩
have hexint (x : Fin J) : Integrable (ex x) (iidProduct P n) := by
refine Integrable.of_bound
((atomEstimate_measurable B x delta).sub measurable_const).abs.aestronglyMeasurable 2 ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (abs_nonneg _)]
change |atomEstimate B z x delta - P.px x * atomMass P x delta| ≤ 2
have ha := atomEstimate_mem_Icc B z x delta
have hc := cont_atomCoefficient_nonneg_le_envelope P hP hreg x hdelta
have hcoef1 : P.px x * atomMass P x delta ≤ 1 := by
unfold atomMass
rw [← (hP.condDensity.2.2 x).2 (Icc 0 delta) measurableSet_Icc]
· exact measureReal_le_one
· intro a ha'
exact ⟨ha'.1, ha'.2.trans
(hdelta.2.trans hreg.2.2.2.2.2.2.2.2.le)⟩
rw [abs_le]
exact ⟨by linarith [ha.1, hcoef1], by linarith [ha.2, hc.1]⟩
have hrhsInt : Integrable
(fun z => e0 z + ∑ x : Fin J,
((1 / 2 : ℝ) * ex x z + (1 / 2 : ℝ) * env)) (iidProduct P n) :=
he0int.add (integrable_finsetSum univ fun x _ =>
((hexint x).const_mul (1 / 2)).add (integrable_const _))
have hlhsInt : Integrable
(fun z => |contFallbackEstimator B z delta - target|) (iidProduct P n) := by
have hest : Measurable (fun z : Fin n → ClampObs J =>
contFallbackEstimator B z delta) := by
unfold contFallbackEstimator
exact clampUnit_measurable'.comp ((retainedEstimate_measurable B delta).add
(measurable_const.mul (Finset.measurable_fun_sum _ fun x _ =>
atomEstimate_measurable B x delta)))
refine Integrable.of_bound
((hest.sub measurable_const).abs.aestronglyMeasurable) 1 ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_abs, abs_le]
have hs := clampUnit_mem_Icc
(retainedEstimate B z delta + (1 / 2 : ℝ) *
∑ x : Fin J, atomEstimate B z x delta)
change target ∈ Icc (0 : ℝ) 1 at ht
change contFallbackEstimator B z delta - target ∈ Icc (-1 : ℝ) 1
exact ⟨by unfold contFallbackEstimator; linarith [hs.1, ht.2],
by unfold contFallbackEstimator; linarith [hs.2, ht.1]⟩
unfold contEstimatorRisk
change (∫ z, |contFallbackEstimator B z delta - target| ∂iidProduct P n) ≤ _
calc
_ ≤ ∫ z, (e0 z + ∑ x : Fin J,
((1 / 2 : ℝ) * ex x z + (1 / 2 : ℝ) * env)) ∂iidProduct P n :=
integral_mono_ae hlhsInt hrhsInt (ae_of_all _ hpoint)
_ = (∫ z, e0 z ∂iidProduct P n) + ∑ x : Fin J,
((1 / 2 : ℝ) * ∫ z, ex x z ∂iidProduct P n + (1 / 2 : ℝ) * env) := by
have hsumInt : Integrable (fun z => ∑ x : Fin J,
((1 / 2 : ℝ) * ex x z + (1 / 2 : ℝ) * env)) (iidProduct P n) :=
integrable_finsetSum univ fun x _ =>
((hexint x).const_mul (1 / 2)).add (integrable_const _)
rw [integral_add he0int hsumInt, integral_finsetSum]
· congr 1
apply Finset.sum_congr rfl
intro x hx
rw [integral_add ((hexint x).const_mul (1 / 2)) (integrable_const _),
integral_const_mul, integral_const]
simp
· intro x hx
exact (hexint x).const_mul (1 / 2) |>.add (integrable_const _)
_ ≤ (1 / 2 : ℝ) * sqrt (B.I0.card : ℝ)⁻¹ +
∑ _x : Fin J, ((1 / 4 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹ +
(1 / 2 : ℝ) * env) := by
apply add_le_add
· exact cont_retainedEstimate_l1_le P hP B hcard0
· apply Finset.sum_le_sum
intro x hx
have h := cont_atomEstimate_l1_le P hP B hcard1 x hdelta.1
(hdelta.2.trans hreg.2.2.2.2.2.2.2.2.le)
nlinarith
_ = _ := by simp [env]; ring
theorem cont_atomTotal_eq_blockAverage
Summing the stratum indicators gives the single threshold indicator. This is the stated conclusion.
Formal statement
hypotheses
conclusion
(∑ x : Fin J, atomEstimate B z x delta)
= blockAverage B.I1 z (fun o => if o.A ≤ delta then 1 else 0)
Proof (Lean source)
lemma cont_atomTotal_eq_blockAverage {J n : ℕ} (B : SplitBlocks n)
(z : Fin n → ClampObs J) (delta : ℝ) :
(∑ x : Fin J, atomEstimate B z x delta) =
blockAverage B.I1 z (fun o => if o.A ≤ delta then 1 else 0) := by
classical
unfold atomEstimate blockAverage
rw [← Finset.mul_sum]
congr 1
calc
(∑ x : Fin J, ∑ i ∈ B.I1,
if (z i).X = x ∧ (z i).A ≤ delta then (1 : ℝ) else 0) =
∑ i ∈ B.I1, ∑ x : Fin J,
if (z i).X = x ∧ (z i).A ≤ delta then (1 : ℝ) else 0 := by
rw [Finset.sum_comm]
_ = ∑ i ∈ B.I1, if (z i).A ≤ delta then (1 : ℝ) else 0 := by
apply Finset.sum_congr rfl
intro i hi
by_cases ha : (z i).A ≤ delta <;> simp [ha]
theorem cont_atomTotal_integral_eq
The population counterpart of the total empirical threshold mass. The result uses the hP condition, the hdelta condition, the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
kappa cminus cplus pmin deltaBar delta :
ℝ
hP :
ContClampModel P kappa cminus cplus pmin deltaBar
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
hreg :
ContDesignConstants J kappa cminus cplus pmin deltaBar
Proof (Lean source)
lemma cont_atomTotal_integral_eq {J : ℕ} (P : ClampLaw J)
{kappa cminus cplus pmin deltaBar delta : ℝ}
(hP : ContClampModel P kappa cminus cplus pmin deltaBar)
(hdelta : delta ∈ Icc (0 : ℝ) deltaBar)
(hreg : ContDesignConstants J kappa cminus cplus pmin deltaBar) :
(∫ o, (if o.A ≤ delta then (1 : ℝ) else 0) ∂P.dataMeasure) =
∑ x : Fin J, P.px x * atomMass P x delta := by
letI : IsProbabilityMeasure P.dataMeasure := hP.probability
have hdelta1 : delta ≤ 1 := hdelta.2.trans hreg.2.2.2.2.2.2.2.2.le
have hX : Measurable (fun o : ClampObs J => o.X) :=
measurable_fst.comp (Measurable.of_comap_le le_rfl)
have hA : Measurable (fun o : ClampObs J => o.A) :=
measurable_fst.comp (measurable_snd.comp (Measurable.of_comap_le le_rfl))
have hone (x : Fin J) :
(∫ o, (if o.X = x ∧ o.A ≤ delta then (1 : ℝ) else 0) ∂P.dataMeasure) =
P.px x * atomMass P x delta := by
have hset : MeasurableSet {o : ClampObs J |
o.X = x ∧ o.A ∈ Icc (0 : ℝ) delta} :=
(measurableSet_eq_fun hX measurable_const).inter
(measurableSet_Icc.preimage hA)
calc
_ = ∫ o, indicator {o : ClampObs J |
o.X = x ∧ o.A ∈ Icc (0 : ℝ) delta}
(fun _ => (1 : ℝ)) o ∂P.dataMeasure := by
apply integral_congr_ae
filter_upwards [hP.treatmentSupport] with o ho
by_cases hx : o.X = x <;> by_cases ha : o.A ≤ delta <;>
simp [indicator, hx, ha, ho.1]
_ = P.dataMeasure.real {o : ClampObs J |
o.X = x ∧ o.A ∈ Icc (0 : ℝ) delta} := by
rw [integral_indicator hset]
simp
_ = _ := by
rw [(hP.condDensity.2.2 x).2 (Icc 0 delta) measurableSet_Icc]
· rfl
· intro a ha
exact ⟨ha.1, ha.2.trans hdelta1⟩
calc
_ = ∫ o, ∑ x : Fin J,
(if o.X = x ∧ o.A ≤ delta then (1 : ℝ) else 0) ∂P.dataMeasure := by
apply integral_congr_ae
filter_upwards with o
by_cases ha : o.A ≤ delta <;> simp [ha]
_ = ∑ x : Fin J, ∫ o,
(if o.X = x ∧ o.A ≤ delta then (1 : ℝ) else 0) ∂P.dataMeasure := by
apply integral_finsetSum
intro x hx
have hf : Measurable (fun o : ClampObs J =>
if o.X = x ∧ o.A ≤ delta then (1 : ℝ) else 0) :=
Measurable.ite
((measurableSet_eq_fun hX measurable_const).inter
(measurableSet_le hA measurable_const)) measurable_const measurable_const
exact Integrable.of_bound hf.aestronglyMeasurable 1 (by
filter_upwards with o
split_ifs <;> norm_num)
_ = _ := Finset.sum_congr rfl fun x _ => hone x
theorem contFallbackEstimator_mem_interval_of_good
A deterministic deviation implication used by the two-block interval. The result uses the hP condition, the hreg condition, the hdelta condition, the h0 condition, the h1 condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
B :
kappa cminus cplus pmin deltaBar delta alpha t0 t1 :
ℝ
hP :
ContClampModel P kappa cminus cplus pmin deltaBar
hreg :
ContDesignConstants J kappa cminus cplus pmin deltaBar
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
h0 :
|retainedEstimate B z delta - retainedMean P delta| ≤ t0
h1 :
conclusion
contClampFunctional P kappa cminus cplus pmin deltaBar hP delta ∈ Icc (max 0 (contFallbackEstimator B z delta - (t0 + t1 + (∑ x : Fin J, atomEstimate B z x delta) / 2))) (min 1 (contFallbackEstimator B z delta + (t0 + t1 + (∑ x : Fin J, atomEstimate B z x delta) / 2)))
Proof (Lean source)
lemma contFallbackEstimator_mem_interval_of_good {J n : ℕ} (P : ClampLaw J)
(B : SplitBlocks n) (kappa cminus cplus pmin deltaBar delta alpha t0 t1 : ℝ)
(hP : ContClampModel P kappa cminus cplus pmin deltaBar)
(hreg : ContDesignConstants J kappa cminus cplus pmin deltaBar)
(hdelta : delta ∈ Icc (0 : ℝ) deltaBar)
(z : Fin n → ClampObs J)
(h0 : |retainedEstimate B z delta - retainedMean P delta| ≤ t0)
(h1 : |(∑ x : Fin J, atomEstimate B z x delta) -
∑ x : Fin J, P.px x * atomMass P x delta| ≤ t1) :
contClampFunctional P kappa cminus cplus pmin deltaBar hP delta ∈
Icc
(max 0 (contFallbackEstimator B z delta -
(t0 + t1 + (∑ x : Fin J, atomEstimate B z x delta) / 2)))
(min 1 (contFallbackEstimator B z delta +
(t0 + t1 + (∑ x : Fin J, atomEstimate B z x delta) / 2))) := by
let E := ∑ x : Fin J, atomEstimate B z x delta
let M := ∑ x : Fin J, P.px x * atomMass P x delta
let T := ∑ x : Fin J, P.px x * atomMass P x delta *
contRegression P kappa cminus cplus pmin deltaBar hP x delta
let target := contClampFunctional P kappa cminus cplus pmin deltaBar hP delta
have hE : 0 ≤ E := sum_nonneg fun x _ =>
(atomEstimate_mem_Icc B z x delta).1
have hM : 0 ≤ M := sum_nonneg fun x _ =>
(cont_atomCoefficient_nonneg_le_envelope P hP hreg x hdelta).1
have hT0 : 0 ≤ T := sum_nonneg fun x _ => mul_nonneg
(cont_atomCoefficient_nonneg_le_envelope P hP hreg x hdelta).1
(contRegression_mem_Icc P kappa cminus cplus pmin deltaBar
hP hreg x delta hdelta).1
have hTM : T ≤ M := by
apply Finset.sum_le_sum
intro x hx
have hc := cont_atomCoefficient_nonneg_le_envelope P hP hreg x hdelta
have hm := contRegression_mem_Icc P kappa cminus cplus pmin deltaBar
hP hreg x delta hdelta
simpa using mul_le_of_le_one_right hc.1 hm.2
have hEM : M - E ≤ t1 := by
have := (abs_le.mp h1).1
dsimp [E, M] at this ⊢
linarith
have ht1 : 0 ≤ t1 := (abs_nonneg _).trans h1
have hatom : |(1 / 2 : ℝ) * E - T| ≤ t1 + E / 2 := by
rw [abs_le]
constructor <;> dsimp [E, M, T] at * <;> linarith
have htI := contClampFunctional_mem_Icc P kappa cminus cplus pmin deltaBar
delta hP hreg hdelta
have hdev : |contFallbackEstimator B z delta - target| ≤ t0 + t1 + E / 2 := by
have hc := abs_clampUnit_sub_le
(retainedEstimate B z delta + (1 / 2 : ℝ) * E) target htI
calc
_ ≤ |(retainedEstimate B z delta - retainedMean P delta) +
((1 / 2 : ℝ) * E - T)| := by
refine hc.trans_eq ?_
congr 1
dsimp [target, T]
unfold contClampFunctional
ring
_ ≤ |retainedEstimate B z delta - retainedMean P delta| +
|(1 / 2 : ℝ) * E - T| := abs_add_le _ _
_ ≤ _ := by linarith
change target ∈ Icc _ _
constructor
· apply max_le htI.1
dsimp [E] at hdev ⊢
rw [abs_le] at hdev
linarith [hdev.2]
· apply le_min htI.2
dsimp [E] at hdev ⊢
rw [abs_le] at hdev
linarith [hdev.1]
theorem cont_hoeffding_radius_tail_le
The displayed square-root radius calibrates the two-sided Hoeffding tail to alpha / 2. The result uses the hm condition, the ha condition, the ha1 condition. This is the stated conclusion.
Formal statement
Proof (Lean source)
lemma cont_hoeffding_radius_tail_le (m : ℕ) (alpha : ℝ)
(hm : 0 < m) (ha : 0 < alpha) (ha1 : alpha < 1 / 2) :
2 * exp (-2 * (m : ℝ) *
(sqrt (log (4 / alpha) / (2 * (m : ℝ)))) ^ 2) ≤
alpha / 2 := by
have hmR : 0 < (m : ℝ) := by positivity
have hquot : 1 < 4 / alpha := by
rw [lt_div_iff₀ ha]
linarith
have hlog : 0 ≤ log (4 / alpha) := (Real.log_pos hquot).le
rw [Real.sq_sqrt (div_nonneg hlog (by positivity : 0 ≤ 2 * (m : ℝ)))]
have he : -2 * (m : ℝ) * (log (4 / alpha) / (2 * (m : ℝ))) =
-log (4 / alpha) := by field_simp
rw [he, Real.exp_neg, Real.exp_log (by positivity : 0 < 4 / alpha)]
field_simp
norm_num
theorem contHoeffdingInterval_coverage_model
The continuity-only interval covers its model-specific target with the advertised finite-sample probability. The result uses the HoeffdingBoundedAverage_of_gate condition, the hP condition, the hreg condition, the hdelta condition, the hcard0 condition, the hcard1 condition. This is the stated conclusion.
Formal statement
hypotheses
HoeffdingBoundedAverage_of_gate :
J n :
ℕ
P :
ClampLaw J
B :
kappa cminus cplus pmin deltaBar delta alpha :
ℝ
hP :
ContClampModel P kappa cminus cplus pmin deltaBar
hreg :
ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
hcard0 :
0 < B.I0.card
hcard1 :
0 < B.I1.card
conclusion
1 - alpha
≤ (iidProduct P n).real {z | contClampFunctional P kappa cminus cplus pmin deltaBar hP delta ∈ contHoeffdingInterval B z delta alpha}
Proof (Lean source)
lemma contHoeffdingInterval_coverage_model
(HoeffdingBoundedAverage_of_gate : HoeffdingBoundedAverage)
{J n : ℕ} (P : ClampLaw J) (B : SplitBlocks n)
(kappa cminus cplus pmin deltaBar delta alpha : ℝ)
(hP : ContClampModel P kappa cminus cplus pmin deltaBar)
(hreg : ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha)
(hdelta : delta ∈ Icc (0 : ℝ) deltaBar)
(hcard0 : 0 < B.I0.card) (hcard1 : 0 < B.I1.card) :
1 - alpha ≤ (iidProduct P n).real {z |
contClampFunctional P kappa cminus cplus pmin deltaBar hP delta ∈
contHoeffdingInterval B z delta alpha} := by
letI : IsProbabilityMeasure P.dataMeasure := hP.probability
letI : IsProbabilityMeasure (iidProduct P n) := by unfold iidProduct; infer_instance
let f0 : ClampObs J → ℝ := fun o => if delta < o.A then clampUnit o.Y else 0
let f1 : ClampObs J → ℝ := fun o => if o.A ≤ delta then 1 else 0
let t0 := sqrt (log (4 / alpha) / (2 * (B.I0.card : ℝ)))
let t1 := sqrt (log (4 / alpha) / (2 * (B.I1.card : ℝ)))
let target := contClampFunctional P kappa cminus cplus pmin deltaBar hP delta
let good : Set (Fin n → ClampObs J) := {z | target ∈ contHoeffdingInterval B z delta alpha}
let bad0 : Set (Fin n → ClampObs J) :=
{z | |blockAverage B.I0 z f0 - ∫ o, f0 o ∂P.dataMeasure| > t0}
let bad1 : Set (Fin n → ClampObs J) :=
{z | |blockAverage B.I1 z f1 - ∫ o, f1 o ∂P.dataMeasure| > t1}
have hf0 : Measurable f0 := by
exact Measurable.ite
(measurableSet_lt measurable_const
(measurable_fst.comp (measurable_snd.comp (Measurable.of_comap_le le_rfl))))
(clampUnit_measurable'.comp clampOutcome_measurable) measurable_const
have hf1 : Measurable f1 := by
exact Measurable.ite
(measurableSet_le
(measurable_fst.comp (measurable_snd.comp (Measurable.of_comap_le le_rfl)))
measurable_const) measurable_const measurable_const
have hf001 : ∀ o, f0 o ∈ Icc (0 : ℝ) 1 := by
intro o; dsimp [f0]; split_ifs
· exact clampUnit_mem_Icc _
· exact ⟨by norm_num, by norm_num⟩
have hf101 : ∀ o, f1 o ∈ Icc (0 : ℝ) 1 := by
intro o; dsimp [f1]; split_ifs <;> constructor <;> norm_num
have ht0 : 0 < t0 := by
dsimp [t0]
have hq : 1 < 4 / alpha := by
rw [lt_div_iff₀ hreg.2.1]
linarith [hreg.2.2]
exact Real.sqrt_pos.2 (div_pos (Real.log_pos hq) (by positivity))
have ht1 : 0 < t1 := by
dsimp [t1]
have hq : 1 < 4 / alpha := by
rw [lt_div_iff₀ hreg.2.1]
linarith [hreg.2.2]
exact Real.sqrt_pos.2 (div_pos (Real.log_pos hq) (by positivity))
have hb0 : (iidProduct P n).real bad0 ≤ alpha / 2 := by
exact (blockAverage_hoeffding HoeffdingBoundedAverage_of_gate P.dataMeasure
B.I0 f0 hf0 hf001 t0 ht0).trans
(cont_hoeffding_radius_tail_le B.I0.card alpha hcard0 hreg.2.1 hreg.2.2)
have hb1 : (iidProduct P n).real bad1 ≤ alpha / 2 := by
exact (blockAverage_hoeffding HoeffdingBoundedAverage_of_gate P.dataMeasure
B.I1 f1 hf1 hf101 t1 ht1).trans
(cont_hoeffding_radius_tail_le B.I1.card alpha hcard1 hreg.2.1 hreg.2.2)
have hm0 : ∫ o, f0 o ∂P.dataMeasure = retainedMean P delta := by
calc
_ = ∫ o, (if delta < o.A then o.Y else 0) ∂P.dataMeasure := by
apply integral_congr_ae
filter_upwards [hP.outcomeSupport] with o ho
simp [f0, clampUnit, ho.1, ho.2]
_ = _ := by
unfold retainedMean
apply integral_congr_ae
filter_upwards with o
by_cases ha : delta < o.A <;> simp [indicator, ha]
have hm1 : ∫ o, f1 o ∂P.dataMeasure =
∑ x : Fin J, P.px x * atomMass P x delta :=
cont_atomTotal_integral_eq P hP hdelta hreg.1
have hsub : ∀ᵐ z ∂iidProduct P n, z ∈ goodᶜ → z ∈ bad0 ∪ bad1 := by
filter_upwards [cont_iidProduct_outcomeSupport P hP] with z hz
intro hzg
by_contra hzbad
have hz0 : |blockAverage B.I0 z f0 - ∫ o, f0 o ∂P.dataMeasure| ≤ t0 :=
le_of_not_gt (fun h => hzbad (inl h))
have hz1 : |blockAverage B.I1 z f1 - ∫ o, f1 o ∂P.dataMeasure| ≤ t1 :=
le_of_not_gt (fun h => hzbad (inr h))
have hre : retainedEstimate B z delta = blockAverage B.I0 z f0 := by
unfold retainedEstimate blockAverage
congr 1
apply Finset.sum_congr rfl
intro i hi
simp [f0, clampUnit, (hz i).1, (hz i).2]
have hmem := contFallbackEstimator_mem_interval_of_good P B kappa cminus cplus
pmin deltaBar delta alpha t0 t1 hP hreg.1 hdelta z
(by simpa [hre, hm0] using hz0)
(by simpa [cont_atomTotal_eq_blockAverage, hm1] using hz1)
exact hzg hmem
have hfail : (iidProduct P n).real goodᶜ ≤ alpha := by
have hm := measure_mono_ae hsub
have hmreal : (iidProduct P n).real goodᶜ ≤ (iidProduct P n).real (bad0 ∪ bad1) :=
ENNReal.toReal_mono (measure_ne_top _ _) hm
exact hmreal.trans ((measureReal_union_le bad0 bad1).trans (by linarith))
have hgoodMeas : MeasurableSet good := by
dsimp [good, target]
simp only [contHoeffdingInterval]
have hc : Measurable (fun z : Fin n → ClampObs J =>
contFallbackEstimator B z delta) := by
unfold contFallbackEstimator
exact clampUnit_measurable'.comp ((retainedEstimate_measurable B delta).add
(measurable_const.mul (Finset.measurable_fun_sum _ fun x _ =>
atomEstimate_measurable B x delta)))
have hr : Measurable (fun z : Fin n → ClampObs J =>
sqrt (log (4 / alpha) / (2 * (B.I0.card : ℝ))) +
sqrt (log (4 / alpha) / (2 * (B.I1.card : ℝ))) +
(∑ x : Fin J, atomEstimate B z x delta) / 2) :=
(measurable_const.add measurable_const).add
((Finset.measurable_fun_sum _ fun x _ =>
atomEstimate_measurable B x delta).div_const 2)
exact (measurableSet_le (measurable_const.max (hc.sub hr)) measurable_const).inter
(measurableSet_le measurable_const (measurable_const.min (hc.add hr)))
have hadd := measureReal_add_measureReal_compl (μ := iidProduct P n) hgoodMeas
have huniv : (iidProduct P n).real univ = 1 := by simp
rw [huniv] at hadd
change 1 - alpha ≤ (iidProduct P n).real good
linarith
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.ContinuityRates 2 declarations Elementary rates for the continuity-only frontier
Elementary rates for the continuity-only frontier
theorem contFrontier_elbow_asymp
At the continuity elbow the two summands of the frontier have root-sample size order. The result uses the hkappa condition. This is the stated conclusion.
Formal statement
hypotheses
kappa :
ℝ
hkappa :
0 ≤ kappa
conclusion
AsympSeq (fun n => contFrontier n ((n : ℝ) ^ (-(1 : ℝ) / (2 * (kappa + 1)))) kappa) (fun n => (n : ℝ) ^ (-(1 : ℝ) / 2))
Proof (Lean source)
lemma contFrontier_elbow_asymp (kappa : ℝ) (hkappa : 0 ≤ kappa) :
AsympSeq
(fun n => contFrontier n
((n : ℝ) ^ (-(1 : ℝ) / (2 * (kappa + 1)))) kappa)
(fun n => (n : ℝ) ^ (-(1 : ℝ) / 2)) := by
refine ⟨1, 2, by norm_num, by norm_num, ?_⟩
filter_upwards [eventually_gt_atTop 0] with n hn
have hnR : 0 < (n : ℝ) := by exact_mod_cast hn
have hk1 : 0 < kappa + 1 := by linarith
have hp :
(((n : ℝ) ^ (-(1 : ℝ) / (2 * (kappa + 1)))) ^ (kappa + 1)) =
(n : ℝ) ^ (-(1 : ℝ) / 2) := by
rw [← Real.rpow_mul hnR.le]
congr 1
field_simp [hk1.ne']
rw [contFrontier, hp]
have hnonneg := Real.rpow_nonneg hnR.le (-(1 : ℝ) / 2)
constructor <;> nlinarith
theorem contFrontier_zero_asymp
At threshold zero the continuity-only frontier is exactly the regular root-sample-size term. The result uses the hkappa condition. This is the stated conclusion.
Formal statement
hypotheses
kappa :
ℝ
hkappa :
0 ≤ kappa
conclusion
AsympSeq (fun n => contFrontier n 0 kappa) (fun n => (n : ℝ) ^ (-(1 : ℝ) / 2))
Proof (Lean source)
lemma contFrontier_zero_asymp (kappa : ℝ) (hkappa : 0 ≤ kappa) :
AsympSeq (fun n => contFrontier n 0 kappa)
(fun n => (n : ℝ) ^ (-(1 : ℝ) / 2)) := by
refine ⟨1, 2, by norm_num, by norm_num, ?_⟩
filter_upwards with n
have hk1 : kappa + 1 ≠ 0 := by linarith
simp only [contFrontier, Real.zero_rpow hk1, add_zero, one_mul]
have hnonneg := Real.rpow_nonneg (Nat.cast_nonneg n) (-(1 : ℝ) / 2)
constructor <;> nlinarith
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.ContinuityRiskLower 4 declarations Two-point lower bounds for continuity-only risk
Two-point lower bounds for continuity-only risk
theorem contMinimaxRisk_lower_of_two_point
A close pair in the continuity-only model lower-bounds its observed minimax absolute risk. The result uses the hreg condition, the hdelta condition, the hP0 condition, the hP1 condition, the hgap0 condition, the hsep condition, the hac condition, the hint condition, the hchi0 condition, the hchi4 condition, the hchi condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P0 P1 :
ClampLaw J
n :
ℕ
kappa cminus cplus pmin deltaBar delta gap chi :
ℝ
hreg :
ContDesignConstants J kappa cminus cplus pmin deltaBar
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
hP0 :
ContClampModel P0 kappa cminus cplus pmin deltaBar
hP1 :
ContClampModel P1 kappa cminus cplus pmin deltaBar
hgap0 :
0 ≤ gap
hsep :
gap
≤ |contClampFunctional P1 kappa cminus cplus pmin deltaBar hP1 delta
- contClampFunctional P0 kappa cminus cplus pmin deltaBar hP0 delta|
hac :
iidProduct P1 n ≪ iidProduct P0 n
hint :
Integrable (fun z => (((iidProduct P1 n).rnDeriv (iidProduct P0 n) z).toReal - 1) ^ 2) (iidProduct P0 n)
hchi0 :
0 ≤ chi
hchi4 :
chi < 4
hchi :
productChiSq P1 P0 n ≤ chi
conclusion
((1 - (1 / 2) * sqrt chi) / 4) * gap
≤ (contFrontierCriteria J n kappa cminus cplus pmin deltaBar delta 0).1
Proof (Lean source)
lemma contMinimaxRisk_lower_of_two_point
(P0 P1 : ClampLaw J) (n : ℕ)
(kappa cminus cplus pmin deltaBar delta gap chi : ℝ)
(hreg : ContDesignConstants J kappa cminus cplus pmin deltaBar)
(hdelta : delta ∈ Icc (0 : ℝ) deltaBar)
(hP0 : ContClampModel P0 kappa cminus cplus pmin deltaBar)
(hP1 : ContClampModel P1 kappa cminus cplus pmin deltaBar)
(hgap0 : 0 ≤ gap)
(hsep : gap ≤ |contClampFunctional P1 kappa cminus cplus pmin deltaBar hP1 delta -
contClampFunctional P0 kappa cminus cplus pmin deltaBar hP0 delta|)
(hac : iidProduct P1 n ≪ iidProduct P0 n)
(hint : Integrable (fun z => (((iidProduct P1 n).rnDeriv
(iidProduct P0 n) z).toReal - 1) ^ 2) (iidProduct P0 n))
(hchi0 : 0 ≤ chi) (hchi4 : chi < 4)
(hchi : productChiSq P1 P0 n ≤ chi) :
((1 - (1 / 2) * sqrt chi) / 4) * gap ≤
(contFrontierCriteria J n kappa cminus cplus pmin deltaBar delta 0).1 := by
let target := fun (P : ClampLaw J)
(hP : ContClampModel P kappa cminus cplus pmin deltaBar) =>
contClampFunctional P kappa cminus cplus pmin deltaBar hP delta
let outer : Set ℝ := {r : ℝ | ∃ est : Estimator n J,
ObservedMeasurableEstimator est ∧ (∀ z, est z ∈ Icc (0 : ℝ) 1) ∧
r = sSup {v : ℝ | ∃ (P : ClampLaw J)
(hP : ContClampModel P kappa cminus cplus pmin deltaBar),
v = contEstimatorRisk P n kappa cminus cplus pmin deltaBar delta hP est}}
change _ ≤ sInf outer
have houter : outer.Nonempty := by
let est : Estimator n J := fun _ => 0
refine ⟨sSup {v : ℝ | ∃ (P : ClampLaw J)
(hP : ContClampModel P kappa cminus cplus pmin deltaBar),
v = contEstimatorRisk P n kappa cminus cplus pmin deltaBar delta hP est},
est, measurable_const, fun _ => ⟨le_rfl, by norm_num⟩, rfl⟩
apply le_csInf houter
intro r hr
rcases hr with ⟨est, hest, hestrange, rfl⟩
let risks : Set ℝ := {v : ℝ | ∃ (P : ClampLaw J)
(hP : ContClampModel P kappa cminus cplus pmin deltaBar),
v = contEstimatorRisk P n kappa cminus cplus pmin deltaBar delta hP est}
have hrisk_le (P : ClampLaw J)
(hP : ContClampModel P kappa cminus cplus pmin deltaBar) :
contEstimatorRisk P n kappa cminus cplus pmin deltaBar delta hP est ≤ 1 := by
letI : IsProbabilityMeasure P.dataMeasure := hP.probability
letI : IsProbabilityMeasure (iidProduct P n) := by unfold iidProduct; infer_instance
have ht := contClampFunctional_mem_Icc P kappa cminus cplus pmin deltaBar
delta hP hreg hdelta
unfold contEstimatorRisk
have hfint : Integrable (fun z => |est z - target P hP|) (iidProduct P n) := by
refine Integrable.of_bound ((hest.sub measurable_const).abs.aestronglyMeasurable) 1 ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_abs, abs_le]
exact ⟨by dsimp [target]; linarith [(hestrange z).1, ht.2],
by dsimp [target]; linarith [(hestrange z).2, ht.1]⟩
calc
(∫ z, |est z - target P hP| ∂iidProduct P n) ≤
∫ _z, (1 : ℝ) ∂iidProduct P n := by
apply integral_mono_ae hfint (integrable_const 1)
filter_upwards with z
rw [abs_le]
exact ⟨by dsimp [target]; linarith [(hestrange z).1, ht.2],
by dsimp [target]; linarith [(hestrange z).2, ht.1]⟩
_ = 1 := by simp
have hbdd : BddAbove risks := by
refine ⟨1, ?_⟩
rintro v ⟨P, hP, rfl⟩
exact hrisk_le P hP
have hle0 : contEstimatorRisk P0 n kappa cminus cplus pmin deltaBar delta hP0 est ≤
sSup risks := le_csSup hbdd ⟨P0, hP0, rfl⟩
have hle1 : contEstimatorRisk P1 n kappa cminus cplus pmin deltaBar delta hP1 est ≤
sSup risks := le_csSup hbdd ⟨P1, hP1, rfl⟩
letI : IsProbabilityMeasure P0.dataMeasure := hP0.probability
letI : IsProbabilityMeasure P1.dataMeasure := hP1.probability
letI : IsProbabilityMeasure (iidProduct P0 n) := by unfold iidProduct; infer_instance
letI : IsProbabilityMeasure (iidProduct P1 n) := by unfold iidProduct; infer_instance
have htest := two_point_lower_bound_of_chiSqDiv_le
(P₀ := iidProduct P1 n) (P₁ := iidProduct P0 n) hest
(θ₀ := target P1 hP1) (θ₁ := target P0 hP0)
(s := gap / 2) (c := chi) (by dsimp [target]; convert hsep using 1 <;> ring)
hac hint hchi
have hfloor : 0 < (1 - (1 / 2) * sqrt chi) / 2 := by
have hsqrt : sqrt chi < 2 := by
rw [Real.sqrt_lt' (by norm_num : (0 : ℝ) < 2)]
convert hchi4 using 1 <;> norm_num
linarith
have hint0 : Integrable (fun z => |est z - target P0 hP0|) (iidProduct P0 n) := by
refine Integrable.of_bound ((hest.sub measurable_const).abs.aestronglyMeasurable) 1 ?_
have ht := contClampFunctional_mem_Icc P0 kappa cminus cplus pmin deltaBar
delta hP0 hreg hdelta
filter_upwards with z
rw [Real.norm_eq_abs, abs_abs, abs_le]
exact ⟨by dsimp [target]; linarith [(hestrange z).1, ht.2],
by dsimp [target]; linarith [(hestrange z).2, ht.1]⟩
have hint1 : Integrable (fun z => |est z - target P1 hP1|) (iidProduct P1 n) := by
refine Integrable.of_bound ((hest.sub measurable_const).abs.aestronglyMeasurable) 1 ?_
have ht := contClampFunctional_mem_Icc P1 kappa cminus cplus pmin deltaBar
delta hP1 hreg hdelta
filter_upwards with z
rw [Real.norm_eq_abs, abs_abs, abs_le]
exact ⟨by dsimp [target]; linarith [(hestrange z).1, ht.2],
by dsimp [target]; linarith [(hestrange z).2, ht.1]⟩
have hevent0 := mul_meas_ge_le_integral_of_nonneg
(ae_of_all _ fun z => abs_nonneg (est z - target P1 hP1)) hint1 (gap / 2)
have hevent1 := mul_meas_ge_le_integral_of_nonneg
(ae_of_all _ fun z => abs_nonneg (est z - target P0 hP0)) hint0 (gap / 2)
have hmaxEvent : ((1 - (1 / 2) * sqrt chi) / 2) * (gap / 2) ≤
max (contEstimatorRisk P1 n kappa cminus cplus pmin deltaBar delta hP1 est)
(contEstimatorRisk P0 n kappa cminus cplus pmin deltaBar delta hP0 est) := by
unfold contEstimatorRisk
have hg2 : 0 ≤ gap / 2 := by positivity
calc
_ ≤ (gap / 2) * max
((iidProduct P1 n).real {z | gap / 2 ≤ |est z - target P1 hP1|})
((iidProduct P0 n).real {z | gap / 2 ≤ |est z - target P0 hP0|}) := by
rw [mul_comm]
exact mul_le_mul_of_nonneg_left htest hg2
_ = max ((gap / 2) * (iidProduct P1 n).real
{z | gap / 2 ≤ |est z - target P1 hP1|})
((gap / 2) * (iidProduct P0 n).real
{z | gap / 2 ≤ |est z - target P0 hP0|}) := by
rw [mul_max_of_nonneg _ _ hg2]
_ ≤ _ := max_le_max hevent0 hevent1
calc
((1 - (1 / 2) * sqrt chi) / 4) * gap =
((1 - (1 / 2) * sqrt chi) / 2) * (gap / 2) := by ring
_ ≤ max (contEstimatorRisk P1 n kappa cminus cplus pmin deltaBar delta hP1 est)
(contEstimatorRisk P0 n kappa cminus cplus pmin deltaBar delta hP0 est) := hmaxEvent
_ ≤ sSup risks := max_le hle1 hle0
theorem contMinimaxRisk_eventually_ge_root
The continuity-only minimax risk has the regular root-sample lower bound, uniformly over threshold sequences. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin deltaBar alpha :
ℝ
hreg :
ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha
conclusion
∃ c : ℝ,
0 < c
conclusion 1
deltaSeq :
ℕ → ℝ
ThresholdSequence deltaBar deltaSeq
∀ᶠ n : ℕ in atTop, c * (n : ℝ) ^ (-(1 : ℝ) / 2)
≤ (contFrontierCriteria J n kappa cminus cplus pmin deltaBar (deltaSeq n) alpha).1
Proof (Lean source)
lemma contMinimaxRisk_eventually_ge_root
(J : ℕ) (kappa cminus cplus pmin deltaBar alpha : ℝ)
(hreg : ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha) :
∃ c : ℝ, 0 < c ∧ ∀ deltaSeq : ℕ → ℝ,
ThresholdSequence deltaBar deltaSeq → ∀ᶠ n : ℕ in atTop,
c * (n : ℝ) ^ (-(1 : ℝ) / 2) ≤
(contFrontierCriteria J n kappa cminus cplus pmin deltaBar
(deltaSeq n) alpha).1 := by
let chi : ℝ := exp (1 / 4 : ℝ) - 1
let floor : ℝ := (1 - (1 / 2) * sqrt chi) / 4
let c : ℝ := floor / 4
have hchi0 : 0 ≤ chi := sub_nonneg.mpr (Real.one_le_exp (by norm_num))
have hchi4 : chi < 4 := by
dsimp [chi]
have he : exp (1 / 4 : ℝ) < exp 1 :=
Real.exp_lt_exp.mpr (by norm_num)
linarith [Real.exp_one_lt_three]
have hfloor : 0 < floor := by
dsimp [floor]
have hs : sqrt chi < 2 := by
rw [Real.sqrt_lt' (by norm_num : (0 : ℝ) < 2)]
convert hchi4 using 1 <;> norm_num
linarith
refine ⟨c, by dsimp [c]; positivity, ?_⟩
intro deltaSeq hdelta
filter_upwards [Filter.eventually_ge_atTop 1] with n hn
have hnpos : 0 < n := by omega
have hnR : 0 < (n : ℝ) := by exact_mod_cast hnpos
let root : ℝ := (n : ℝ) ^ (-(1 : ℝ) / 2)
let eps : ℝ := (1 / 4 : ℝ) * root
have hroot : 0 < root := Real.rpow_pos_of_pos hnR _
have heps : 0 < eps := mul_pos (by norm_num) hroot
have heps_le : eps ≤ 1 / 4 := by
have hroot1 : root ≤ 1 := by
rw [show (1 : ℝ) = 1 ^ (-(1 : ℝ) / 2) by norm_num]
exact Real.rpow_le_rpow_of_nonpos (by norm_num) (by exact_mod_cast hn)
(by norm_num)
dsimp [eps]
nlinarith
have hepsb : |eps| ≤ 1 / 2 := by rw [abs_of_pos heps]; linarith
have hepss : |eps| < 1 / 2 := by rw [abs_of_pos heps]; linarith
let P0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
let P1 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => eps)
have hw := continuityConstant_twoPoint J n kappa cminus cplus pmin deltaBar
(deltaSeq n) eps hreg.1 (hdelta n) heps.le heps_le
rcases hw with ⟨hP0, hP1, hsepEq, hchiExp⟩
have hrootSq : root ^ 2 = (n : ℝ)⁻¹ := by
dsimp [root]
rw [← Real.rpow_natCast, ← Real.rpow_mul hnR.le]
norm_num
rw [Real.rpow_neg_one]
have hepsSq : eps ^ 2 = (1 / 16 : ℝ) * (n : ℝ)⁻¹ := by
dsimp [eps]
rw [mul_pow, hrootSq]
norm_num
have hnE : (n : ℝ) * (4 * eps ^ 2) = 1 / 4 := by
rw [hepsSq]
field_simp [hnR.ne']
ring
have hchi : productChiSq P1 P0 n ≤ chi := by
dsimp [chi]
rw [hnE] at hchiExp
linarith
have hac1 := minimaxDataMeasure_ac_center J kappa hreg.1.1 hreg.1.2.1
(fun _ : Fin J × ℝ => eps) measurable_const (fun _ => hepss)
have hint1 := minimaxDataMeasure_sq_integrable_center J kappa hreg.1.1 hreg.1.2.1
(fun _ : Fin J × ℝ => eps) measurable_const (fun _ => hepss)
letI : IsProbabilityMeasure (minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => eps)) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hreg.1.1 hreg.1.2.1 _
measurable_const (fun _ => hepsb)
letI : IsProbabilityMeasure (minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => 0)) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hreg.1.1 hreg.1.2.1 _
measurable_const (fun _ => by norm_num)
have hac := pi_iid_absolutelyContinuous _ _ hac1 n
have hint := pi_iid_integrable_sq_dev _ _ hac1 hint1 n
have hlower := contMinimaxRisk_lower_of_two_point P0 P1 n kappa cminus
cplus pmin deltaBar (deltaSeq n) eps chi hreg.1 (hdelta n) hP0 hP1
heps.le (by simpa [P0, P1] using hsepEq.le)
(by simpa [P0, P1, iidProduct, minimaxClampLaw] using hac)
(by simpa [P0, P1, iidProduct, minimaxClampLaw] using hint)
hchi0 hchi4 hchi
change _ ≤ (contFrontierCriteria J n kappa cminus cplus pmin deltaBar
(deltaSeq n) 0).1 at hlower
dsimp [c, floor, eps, root] at hlower ⊢
change _ ≤ (contFrontierCriteria J n kappa cminus cplus pmin deltaBar
(deltaSeq n) 0).1
convert hlower using 1 <;> ring
theorem contMinimaxRisk_eventually_ge_atom
Arbitrarily narrow continuous bumps give an atom-scale minimax lower bound uniformly over the whole threshold range. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin deltaBar alpha :
ℝ
hreg :
ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha
conclusion
∃ c : ℝ,
0 < c
conclusion 1
deltaSeq :
ℕ → ℝ
ThresholdSequence deltaBar deltaSeq
∀ᶠ n : ℕ in atTop, c * (deltaSeq n) ^ (kappa + 1)
≤ (contFrontierCriteria J n kappa cminus cplus pmin deltaBar (deltaSeq n) alpha).1
Proof (Lean source)
lemma contMinimaxRisk_eventually_ge_atom
(J : ℕ) (kappa cminus cplus pmin deltaBar alpha : ℝ)
(hreg : ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha) :
∃ c : ℝ, 0 < c ∧ ∀ deltaSeq : ℕ → ℝ,
ThresholdSequence deltaBar deltaSeq → ∀ᶠ n : ℕ in atTop,
c * (deltaSeq n) ^ (kappa + 1) ≤
(contFrontierCriteria J n kappa cminus cplus pmin deltaBar
(deltaSeq n) alpha).1 := by
let amplitude : ℝ := 1 / 4
let chi : ℝ := exp 1 - 1
let floor : ℝ := (1 - (1 / 2) * sqrt chi) / 4
let c : ℝ := floor * amplitude
have hchi0 : 0 ≤ chi := sub_nonneg.mpr (Real.one_le_exp zero_le_one)
have hchi4 : chi < 4 := by dsimp [chi]; linarith [Real.exp_one_lt_three]
have hfloor : 0 < floor := by
dsimp [floor]
have hs : sqrt chi < 2 := by
rw [Real.sqrt_lt' (by norm_num : (0 : ℝ) < 2)]
convert hchi4 using 1 <;> norm_num
linarith
refine ⟨c, by dsimp [c, amplitude]; positivity, ?_⟩
intro deltaSeq hdelta
filter_upwards [Filter.eventually_ge_atTop 1] with n hn
have hnpos : 0 < n := by omega
have hnR : 0 < (n : ℝ) := by exact_mod_cast hnpos
let A : ℝ := 8 * (kappa + 1) * amplitude ^ 2 * (deltaBar + 1) ^ kappa
let K : ℝ := 1 + A
let h : ℝ := 1 / (K * (n : ℝ))
have hk1 : 0 ≤ kappa + 1 := by linarith [hreg.1.2.1]
have hd1 : 0 ≤ deltaBar + 1 := by
linarith [hreg.1.2.2.2.2.2.2.2.1]
have hA : 0 ≤ A := by
dsimp [A, amplitude]
exact mul_nonneg (mul_nonneg (mul_nonneg (by norm_num) hk1) (sq_nonneg _))
(Real.rpow_nonneg hd1 _)
have hK : 0 < K := by dsimp [K]; linarith
have hh : 0 < h := by dsimp [h]; positivity
have hh1 : h ≤ 1 := by
rw [show h = (K * (n : ℝ))⁻¹ by simp [h]]
rw [inv_le_one₀ (mul_pos hK hnR)]
nlinarith [show (1 : ℝ) ≤ n by exact_mod_cast hn]
have hdh : deltaSeq n + h ≤ deltaBar + 1 := by
linarith [(hdelta n).2]
have hpow : (deltaSeq n + h) ^ kappa ≤ (deltaBar + 1) ^ kappa := by
exact Real.rpow_le_rpow (add_nonneg (hdelta n).1 hh.le) hdh hreg.1.2.1
have hcoef : 0 ≤ 8 * (kappa + 1) * amplitude ^ 2 * h := by
exact mul_nonneg (mul_nonneg (mul_nonneg (by norm_num)
(by linarith [hreg.1.2.1])) (sq_nonneg _)) hh.le
have hexp : (n : ℝ) * (8 * (kappa + 1) * amplitude ^ 2 * h *
(deltaSeq n + h) ^ kappa) ≤ 1 := by
have hcore : (n : ℝ) * (8 * (kappa + 1) * amplitude ^ 2 * h *
(deltaBar + 1) ^ kappa) = A / K := by
dsimp [h, A]
field_simp [hK.ne', hnR.ne']
<;> ring
calc
_ ≤ (n : ℝ) * (8 * (kappa + 1) * amplitude ^ 2 * h *
(deltaBar + 1) ^ kappa) := by gcongr
_ = A / K := hcore
_ ≤ 1 := by rw [div_le_one hK]; dsimp [K]; linarith
have hw := continuityBump_twoPoint J n kappa cminus cplus pmin deltaBar
(deltaSeq n) h amplitude hreg.1 (hdelta n) hh (by norm_num [amplitude])
(by norm_num [amplitude])
let q := fun a : ℝ => amplitude *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - deltaSeq n) / h)
let P0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
let P1 := minimaxClampLaw J kappa (fun p => q p.2)
rcases hw with ⟨hP0, hP1, hsep, hchiExp⟩
have hchi : productChiSq P1 P0 n ≤ chi := by
dsimp [chi]
have := hchiExp.trans (Real.exp_le_exp.mpr hexp)
linarith
have hqmeas : Measurable q := by dsimp [q]; fun_prop
have hq0 (a : ℝ) : 0 ≤ q a := by
exact mul_nonneg (by norm_num [amplitude])
(CausalSmith.Stat.DoseResponseMinimax.doseBump_nonneg _)
have hqbound (a : ℝ) : |q a| ≤ 1 / 2 := by
rw [abs_of_nonneg (hq0 a)]
have hb := CausalSmith.Stat.DoseResponseMinimax.doseBump_le_one
((a - deltaSeq n) / h)
dsimp [q, amplitude]
nlinarith
have hqstrict (p : Fin J × ℝ) : |q p.2| < 1 / 2 := by
rw [abs_of_nonneg (hq0 p.2)]
have hb := CausalSmith.Stat.DoseResponseMinimax.doseBump_le_one
((p.2 - deltaSeq n) / h)
dsimp [q, amplitude]
nlinarith
have hac1 := minimaxDataMeasure_ac_center J kappa hreg.1.1 hreg.1.2.1
(fun p : Fin J × ℝ => q p.2) (hqmeas.comp measurable_snd) hqstrict
have hint1 := minimaxDataMeasure_sq_integrable_center J kappa hreg.1.1 hreg.1.2.1
(fun p : Fin J × ℝ => q p.2) (hqmeas.comp measurable_snd) hqstrict
letI : IsProbabilityMeasure (minimaxDataMeasure J kappa
(fun p : Fin J × ℝ => q p.2)) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hreg.1.1 hreg.1.2.1 _
(hqmeas.comp measurable_snd) (fun p => hqbound p.2)
letI : IsProbabilityMeasure (minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => 0)) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hreg.1.1 hreg.1.2.1 _
measurable_const (fun _ => by norm_num)
have hac := pi_iid_absolutelyContinuous _ _ hac1 n
have hint := pi_iid_integrable_sq_dev _ _ hac1 hint1 n
have hlower := contMinimaxRisk_lower_of_two_point P0 P1 n kappa cminus cplus
pmin deltaBar (deltaSeq n) (amplitude * (deltaSeq n) ^ (kappa + 1)) chi
hreg.1 (hdelta n) hP0 hP1
(mul_nonneg (by norm_num [amplitude]) (Real.rpow_nonneg (hdelta n).1 _))
hsep
(by simpa [P0, P1, q, iidProduct, minimaxClampLaw] using hac)
(by simpa [P0, P1, q, iidProduct, minimaxClampLaw] using hint)
hchi0 hchi4 hchi
change _ ≤ (contFrontierCriteria J n kappa cminus cplus pmin deltaBar
(deltaSeq n) 0).1 at hlower
dsimp [c, floor] at hlower ⊢
change _ ≤ (contFrontierCriteria J n kappa cminus cplus pmin deltaBar
(deltaSeq n) 0).1
convert hlower using 1 <;> ring
theorem contMinimaxRisk_eventually_ge_frontier
The regular and atom experiments combine into the full continuity frontier lower bound. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin deltaBar alpha :
ℝ
hreg :
ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha
conclusion
∃ c : ℝ,
0 < c
conclusion 1
deltaSeq :
ℕ → ℝ
ThresholdSequence deltaBar deltaSeq
∀ᶠ n : ℕ in atTop, c * contFrontier n (deltaSeq n) kappa
≤ (contFrontierCriteria J n kappa cminus cplus pmin deltaBar (deltaSeq n) alpha).1
Proof (Lean source)
lemma contMinimaxRisk_eventually_ge_frontier
(J : ℕ) (kappa cminus cplus pmin deltaBar alpha : ℝ)
(hreg : ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha) :
∃ c : ℝ, 0 < c ∧ ∀ deltaSeq : ℕ → ℝ,
ThresholdSequence deltaBar deltaSeq → ∀ᶠ n : ℕ in atTop,
c * contFrontier n (deltaSeq n) kappa ≤
(contFrontierCriteria J n kappa cminus cplus pmin deltaBar
(deltaSeq n) alpha).1 := by
obtain ⟨c0, hc0, h0⟩ := contMinimaxRisk_eventually_ge_root J kappa cminus
cplus pmin deltaBar alpha hreg
obtain ⟨c1, hc1, h1⟩ := contMinimaxRisk_eventually_ge_atom J kappa cminus
cplus pmin deltaBar alpha hreg
let c := min c0 c1 / 2
refine ⟨c, by dsimp [c]; positivity, ?_⟩
intro deltaSeq hdelta
filter_upwards [h0 deltaSeq hdelta, h1 deltaSeq hdelta] with n hn0 hn1
have hr0 : 0 ≤ (n : ℝ) ^ (-(1 : ℝ) / 2) :=
Real.rpow_nonneg (Nat.cast_nonneg n) _
have ha0 : 0 ≤ (deltaSeq n) ^ (kappa + 1) :=
Real.rpow_nonneg (hdelta n).1 _
dsimp [c, contFrontier]
have hmin0 : min c0 c1 ≤ c0 := min_le_left _ _
have hmin1 : min c0 c1 ≤ c1 := min_le_right _ _
nlinarith [mul_le_mul_of_nonneg_right hmin0 hr0,
mul_le_mul_of_nonneg_right hmin1 ha0]
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.ContinuityUpper 3 declarations Elementary upper bounds for the continuity-only procedure
Elementary upper bounds for the continuity-only procedure
theorem contRegression_mem_Icc
The selected continuous conditional-mean version remains in the outcome range throughout the threshold interval. The result uses the hP condition, the hreg condition, the ha condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
kappa cminus cplus pmin deltaBar :
ℝ
hP :
ContClampModel P kappa cminus cplus pmin deltaBar
hreg :
ContDesignConstants J kappa cminus cplus pmin deltaBar
x :
Fin J
a :
ℝ
ha :
a ∈ Icc (0 : ℝ) deltaBar
conclusion
contRegression P kappa cminus cplus pmin deltaBar hP x a ∈ Icc (0 : ℝ) 1
Proof (Lean source)
lemma contRegression_mem_Icc
(P : ClampLaw J) (kappa cminus cplus pmin deltaBar : ℝ)
(hP : ContClampModel P kappa cminus cplus pmin deltaBar)
(hreg : ContDesignConstants J kappa cminus cplus pmin deltaBar)
(x : Fin J) (a : ℝ) (ha : a ∈ Icc (0 : ℝ) deltaBar) :
contRegression P kappa cminus cplus pmin deltaBar hP x a ∈
Icc (0 : ℝ) 1 := by
let m : Fin J → ℝ → ℝ :=
contRegression P kappa cminus cplus pmin deltaBar hP
let ce : ClampObs J → ℝ :=
P.dataMeasure[(fun o : ClampObs J => o.Y) |
comap (fun o : ClampObs J => (o.X, o.A)) inferInstance]
have hspec := Classical.choose_spec hP.continuousVersion
have hmcont : ∀ y, ContinuousOn (m y) (Icc (0 : ℝ) deltaBar) := hspec.1
have hver : ce =ᵐ[P.dataMeasure] fun o => m o.X o.A := hspec.2
letI : IsProbabilityMeasure P.dataMeasure := hP.probability
have hYint : Integrable (fun o : ClampObs J => o.Y) P.dataMeasure := by
refine Integrable.of_bound clampOutcome_measurable.aestronglyMeasurable 1 ?_
filter_upwards [hP.outcomeSupport] with o ho
rw [Real.norm_eq_abs, abs_of_nonneg ho.1]
exact ho.2
have hce0 : 0 ≤ᵐ[P.dataMeasure] ce :=
MeasureTheory.condExp_nonneg (hP.outcomeSupport.mono fun _ ho => ho.1)
have hce1 : ce ≤ᵐ[P.dataMeasure] fun _ => (1 : ℝ) := by
have hmono := MeasureTheory.condExp_mono
(m := comap (fun o : ClampObs J => (o.X, o.A)) inferInstance)
hYint (integrable_const 1)
(hP.outcomeSupport.mono fun _ ho => ho.2)
rw [MeasureTheory.condExp_const clampDesign_measurable.comap_le (1 : ℝ)] at hmono
exact hmono
have hmrange : ∀ᵐ o ∂P.dataMeasure, m o.X o.A ∈ Icc (0 : ℝ) 1 := by
filter_upwards [hver, hce0, hce1] with o he h0 h1
rw [he] at h0 h1
exact ⟨h0, h1⟩
let mc : Fin J → ℝ → ℝ := fun y t => clampUnit (m y t)
have hmccont : ∀ y, ContinuousOn (mc y) (Icc (0 : ℝ) deltaBar) := by
intro y
unfold mc clampUnit
fun_prop
have hverc : ce =ᵐ[P.dataMeasure] fun o => mc o.X o.A := by
filter_upwards [hver, hmrange] with o he hm
rw [he]
simp [mc, clampUnit, hm.1, hm.2]
have heq := (cont_regression_extension_unique P kappa cminus cplus pmin deltaBar
hP hreg m mc hmcont hmccont hver hverc x) ha
change m x a ∈ Icc (0 : ℝ) 1
rw [heq]
exact clampUnit_mem_Icc _
theorem contClampFunctional_integral_eq
The continuity-only functional is the expectation of the observed outcome above the threshold and of the selected continuous regression below it. The result uses the hP condition, the hreg condition, the hdelta condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
kappa cminus cplus pmin deltaBar delta :
ℝ
hP :
ContClampModel P kappa cminus cplus pmin deltaBar
hreg :
ContDesignConstants J kappa cminus cplus pmin deltaBar
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
conclusion
(∫ o, if delta < o.A then o.Y else contRegression P kappa cminus cplus pmin deltaBar hP o.X delta ∂P.dataMeasure)
= contClampFunctional P kappa cminus cplus pmin deltaBar hP delta
Proof (Lean source)
lemma contClampFunctional_integral_eq
(P : ClampLaw J) (kappa cminus cplus pmin deltaBar delta : ℝ)
(hP : ContClampModel P kappa cminus cplus pmin deltaBar)
(hreg : ContDesignConstants J kappa cminus cplus pmin deltaBar)
(hdelta : delta ∈ Icc (0 : ℝ) deltaBar) :
(∫ o, if delta < o.A then o.Y else
contRegression P kappa cminus cplus pmin deltaBar hP o.X delta
∂P.dataMeasure) =
contClampFunctional P kappa cminus cplus pmin deltaBar hP delta := by
letI : IsProbabilityMeasure P.dataMeasure := hP.probability
let mu := contRegression P kappa cminus cplus pmin deltaBar hP
let atomTerm : Fin J → ClampObs J → ℝ := fun x o =>
if o.X = x ∧ o.A ≤ delta then mu x delta else 0
rcases hreg with ⟨hJ, hkappa, hcminus, hcminus_le, hcplus, hpmin,
hpmin_le, hdeltaBar, hdeltaBar_one⟩
have hreg' : ContDesignConstants J kappa cminus cplus pmin deltaBar :=
⟨hJ, hkappa, hcminus, hcminus_le, hcplus, hpmin, hpmin_le,
hdeltaBar, hdeltaBar_one⟩
have hdelta1 : delta ≤ 1 := hdelta.2.trans hdeltaBar_one.le
have hatomInt (x : Fin J) : Integrable (atomTerm x) P.dataMeasure := by
have hm : Measurable (atomTerm x) := by
dsimp [atomTerm]
have hX : Measurable (fun o : ClampObs J => o.X) :=
measurable_fst.comp (Measurable.of_comap_le le_rfl)
have hA : Measurable (fun o : ClampObs J => o.A) :=
measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))
exact Measurable.ite
((measurableSet_eq_fun hX measurable_const).inter
(measurableSet_le hA measurable_const)) measurable_const measurable_const
refine Integrable.of_bound hm.aestronglyMeasurable 1 ?_
filter_upwards with o
have hr := contRegression_mem_Icc P kappa cminus cplus pmin deltaBar
hP hreg' x delta hdelta
dsimp [atomTerm]
split_ifs
· rw [abs_of_nonneg hr.1]
exact hr.2
· simp
have hpoint : ∀ᵐ o ∂P.dataMeasure,
(if delta < o.A then o.Y else mu o.X delta) =
o.Y * indicator {o : ClampObs J | delta < o.A}
(fun _ => (1 : ℝ)) o + ∑ x : Fin J, atomTerm x o := by
filter_upwards [hP.treatmentSupport] with o ho
by_cases ha : delta < o.A
· have hn (x : Fin J) : ¬(o.X = x ∧ o.A ≤ delta) := fun hx => by linarith
simp [atomTerm, indicator, ha, hn]
· have hle : o.A ≤ delta := le_of_not_gt ha
simp [atomTerm, indicator, ha, hle]
rw [integral_congr_ae hpoint, integral_add]
· rw [integral_finsetSum univ]
· unfold contClampFunctional retainedMean
congr 1
apply Finset.sum_congr rfl
intro x hx
have hevent : (∫ o,
(if o.X = x ∧ o.A ≤ delta then (1 : ℝ) else 0) ∂P.dataMeasure) =
P.px x * atomMass P x delta := by
have hX : Measurable (fun o : ClampObs J => o.X) :=
measurable_fst.comp (Measurable.of_comap_le le_rfl)
have hA : Measurable (fun o : ClampObs J => o.A) :=
measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))
have hset : MeasurableSet {o : ClampObs J |
o.X = x ∧ o.A ∈ Icc (0 : ℝ) delta} :=
(measurableSet_eq_fun hX measurable_const).inter
(measurableSet_Icc.preimage hA)
calc
_ = ∫ o, indicator {o : ClampObs J |
o.X = x ∧ o.A ∈ Icc (0 : ℝ) delta}
(fun _ => (1 : ℝ)) o ∂P.dataMeasure := by
apply integral_congr_ae
filter_upwards [hP.treatmentSupport] with o ho
by_cases hx' : o.X = x <;> by_cases ha' : o.A ≤ delta <;>
simp [indicator, hx', ha', ho.1]
_ = P.dataMeasure.real {o : ClampObs J |
o.X = x ∧ o.A ∈ Icc (0 : ℝ) delta} := by
rw [integral_indicator hset]
simp
_ = P.px x * atomMass P x delta := by
rw [(hP.condDensity.2.2 x).2 (Icc 0 delta) measurableSet_Icc]
· rfl
· intro a ha'
exact ⟨ha'.1, ha'.2.trans hdelta1⟩
calc
(∫ o, atomTerm x o ∂P.dataMeasure) =
mu x delta * ∫ o,
(if o.X = x ∧ o.A ≤ delta then (1 : ℝ) else 0)
∂P.dataMeasure := by
rw [← integral_const_mul]
apply integral_congr_ae
filter_upwards with o
by_cases h : o.X = x ∧ o.A ≤ delta <;> simp [atomTerm, h]
_ = mu x delta * (P.px x * atomMass P x delta) := by rw [hevent]
_ = P.px x * atomMass P x delta * mu x delta := by ring
· exact fun x _ => hatomInt x
· exact Integrable.of_bound
((clampOutcome_measurable.mul
(Measurable.indicator measurable_const
(measurableSet_lt measurable_const
(measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))))))).aestronglyMeasurable 1
(by filter_upwards [hP.outcomeSupport] with o ho
by_cases ha : delta < o.A
· simp [indicator, ha, abs_of_nonneg ho.1, ho.2]
· simp [indicator, ha])
· exact integrable_finsetSum univ fun x _ => hatomInt x
theorem contClampFunctional_mem_Icc
The continuity-only clamp functional lies in the outcome range. The result uses the hP condition, the hreg condition, the hdelta condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
kappa cminus cplus pmin deltaBar delta :
ℝ
hP :
ContClampModel P kappa cminus cplus pmin deltaBar
hreg :
ContDesignConstants J kappa cminus cplus pmin deltaBar
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
conclusion
contClampFunctional P kappa cminus cplus pmin deltaBar hP delta ∈ Icc (0 : ℝ) 1
Proof (Lean source)
lemma contClampFunctional_mem_Icc
(P : ClampLaw J) (kappa cminus cplus pmin deltaBar delta : ℝ)
(hP : ContClampModel P kappa cminus cplus pmin deltaBar)
(hreg : ContDesignConstants J kappa cminus cplus pmin deltaBar)
(hdelta : delta ∈ Icc (0 : ℝ) deltaBar) :
contClampFunctional P kappa cminus cplus pmin deltaBar hP delta ∈
Icc (0 : ℝ) 1 := by
letI : IsProbabilityMeasure P.dataMeasure := hP.probability
let g : ClampObs J → ℝ := fun o => if delta < o.A then o.Y else
contRegression P kappa cminus cplus pmin deltaBar hP o.X delta
have hgmeas : Measurable g := by
dsimp [g]
exact Measurable.ite
(measurableSet_lt measurable_const
(measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl)))) clampOutcome_measurable
((measurable_of_countable (fun x : Fin J =>
contRegression P kappa cminus cplus pmin deltaBar hP x delta)).comp
(measurable_fst.comp (Measurable.of_comap_le le_rfl)))
have hg0 : 0 ≤ᵐ[P.dataMeasure] g := by
filter_upwards [hP.outcomeSupport] with o ho
dsimp [g]
split_ifs
· exact ho.1
· exact (contRegression_mem_Icc P kappa cminus cplus pmin deltaBar
hP hreg o.X delta hdelta).1
have hg1 : g ≤ᵐ[P.dataMeasure] fun _ => (1 : ℝ) := by
filter_upwards [hP.outcomeSupport] with o ho
dsimp [g]
split_ifs
· exact ho.2
· exact (contRegression_mem_Icc P kappa cminus cplus pmin deltaBar
hP hreg o.X delta hdelta).2
rw [← contClampFunctional_integral_eq P kappa cminus cplus pmin deltaBar
delta hP hreg hdelta]
constructor
· exact integral_nonneg_of_ae hg0
· calc
(∫ o, g o ∂P.dataMeasure) ≤ ∫ _o, (1 : ℝ) ∂P.dataMeasure :=
integral_mono_ae
(Integrable.of_bound hgmeas.aestronglyMeasurable 1
(hg0.and hg1 |>.mono fun o h => by
rw [Real.norm_eq_abs, abs_of_nonneg h.1]
exact h.2)) (integrable_const 1) hg1
_ = 1 := by simp
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.ContinuityWitness 3 declarations This file places the canonical Bernoulli regression laws in the qualitative continuity model without imposing a common modulus of continuity.
Continuity-only canonical witnesses
This file places the canonical Bernoulli regression laws in the qualitative continuity model without imposing a common modulus of continuity.
theorem minimaxClampModel_continuous
A bounded continuous Bernoulli regression on the canonical design law is a member of the continuity-only model. The result uses the hJ condition, the hkappa condition, the hcminus condition, the hcplus condition, the hpmin condition, the hqmeas condition, the hqbound condition, the hcont condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin deltaBar :
ℝ
hJ :
0 < J
hkappa :
0 ≤ kappa
hcminus :
cminus ≤ kappa + 1
hcplus :
kappa + 1 ≤ cplus
hpmin :
pmin ≤ 1 / (J : ℝ)
q :
ℝ → ℝ
hqmeas :
hqbound :
∀ a, |q a| ≤ 1 / 2
hcont :
ContinuousOn (fun a => 1 / 2 + q a) (Icc (0 : ℝ) deltaBar)
conclusion
ContClampModel (minimaxClampLaw J kappa (fun p => q p.2)) kappa cminus cplus pmin deltaBar
Proof (Lean source)
lemma minimaxClampModel_continuous
(J : ℕ) (kappa cminus cplus pmin deltaBar : ℝ)
(hJ : 0 < J) (hkappa : 0 ≤ kappa)
(hcminus : cminus ≤ kappa + 1) (hcplus : kappa + 1 ≤ cplus)
(hpmin : pmin ≤ 1 / (J : ℝ))
(q : ℝ → ℝ) (hqmeas : Measurable q) (hqbound : ∀ a, |q a| ≤ 1 / 2)
(hcont : ContinuousOn (fun a => 1 / 2 + q a) (Icc (0 : ℝ) deltaBar)) :
ContClampModel (minimaxClampLaw J kappa (fun p => q p.2))
kappa cminus cplus pmin deltaBar := by
let g : Fin J × ℝ → ℝ := fun p => q p.2
have hgmeas : Measurable g := hqmeas.comp measurable_snd
have hgbound : ∀ p, |g p| ≤ 1 / 2 := fun p => hqbound p.2
have hprob := minimaxDataMeasure_isProbabilityMeasure J kappa hJ hkappa g hgmeas hgbound
refine ⟨hprob, minimaxDataMeasure_ae_treatment J kappa g hgmeas hgbound,
(minimaxDataMeasure_ae_bernoulli J kappa g hgmeas).mono (fun o ho => by
rcases ho with ho | ho <;> simp [ho]), ?_, ?_, ?_, ?_⟩
· refine ⟨?_, ?_, ?_⟩
· intro x
fun_prop
· intro x
filter_upwards [ae_restrict_mem measurableSet_Icc] with a ha
exact mul_nonneg (by linarith) (Real.rpow_nonneg ha.1 _)
· intro x
constructor
· change 1 / (J : ℝ) = _
change 1 / (J : ℝ) =
(Measure.map (fun o : ClampObs J => o.X)
(minimaxDataMeasure J kappa g)).real {x}
rw [minimaxDataMeasure_map_X J kappa hJ hkappa g hgmeas hgbound]
exact (minimaxStratumMeasure_real_singleton J hJ x).symm
· intro B hB hsub
have hevent : {o : ClampObs J | o.X = x ∧ o.A ∈ B} =
(fun o => (o.X, o.A)) ⁻¹' ({x} ×ˢ B) := by ext o; simp
rw [hevent]
have hrect : MeasurableSet (({x} : Set (Fin J)) ×ˢ B) :=
(measurableSet_singleton x).prod hB
change (minimaxDataMeasure J kappa g).real
{o : ClampObs J | o.X = x ∧ o.A ∈ B} = _
rw [hevent, measureReal_def, ← Measure.map_apply measurable_obs_design_cont hrect,
minimaxDataMeasure_map_design J kappa g hgmeas hgbound]
change (minimaxDesignMeasure J kappa).real ({x} ×ˢ B) = _
rw [minimaxDesignMeasure_real_rectangle J kappa hJ hkappa x B hB hsub]
rfl
· intro x
constructor
· change 1 / (J : ℝ) = _
change 1 / (J : ℝ) =
(Measure.map (fun o : ClampObs J => o.X)
(minimaxDataMeasure J kappa g)).real {x}
rw [minimaxDataMeasure_map_X J kappa hJ hkappa g hgmeas hgbound]
exact (minimaxStratumMeasure_real_singleton J hJ x).symm
· exact hpmin
· intro x
filter_upwards [ae_restrict_mem measurableSet_Icc] with a ha
constructor
· exact mul_le_mul_of_nonneg_right hcminus (Real.rpow_nonneg ha.1 _)
· exact mul_le_mul_of_nonneg_right hcplus (Real.rpow_nonneg ha.1 _)
· refine ⟨fun _ => 1 / 2 + q, fun _ => hcont, ?_⟩
let μ := minimaxDataMeasure J kappa g
let design : ClampObs J → Fin J × ℝ := fun o => (o.X, o.A)
let m : ClampObs J → ℝ := fun o => 1 / 2 + q o.A
letI : IsProbabilityMeasure μ := hprob
have hdesign : Measurable design := measurable_obs_design_cont
have hY : Integrable (fun o : ClampObs J => o.Y) μ := by
refine Integrable.of_bound
(measurable_snd.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))).aestronglyMeasurable 1 ?_
filter_upwards [minimaxDataMeasure_ae_bernoulli J kappa g hgmeas] with o ho
rcases ho with ho | ho <;> rw [ho] <;> norm_num
have hm_meas : Measurable m := by
dsimp [m]
exact measurable_const.add (hqmeas.comp
(measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))))
have hm : Integrable m μ := by
refine Integrable.of_bound hm_meas.aestronglyMeasurable 1 ?_
filter_upwards with o
rw [Real.norm_eq_abs]
dsimp [m]
have hb := hqbound o.A
rw [abs_le] at hb
exact abs_le.2 ⟨by linarith [hb.1], by linarith [hb.2]⟩
have hm_design :
AEStronglyMeasurable[comap design inferInstance] m μ := by
have hfactor : m = (fun p : Fin J × ℝ => 1 / 2 + q p.2) ∘ design := rfl
rw [hfactor]
exact ((measurable_const.add (hqmeas.comp measurable_snd)).comp
(Measurable.of_comap_le le_rfl)).aestronglyMeasurable
have hintegral : ∀ T : Set (Fin J × ℝ), MeasurableSet T →
(∫ o in design ⁻¹' T, o.Y ∂μ) = ∫ o in design ⁻¹' T, m o ∂μ := by
intro T hT
rw [minimaxDataMeasure_integral_Y_design J kappa hJ hkappa g hgmeas hgbound T hT]
rw [← integral_indicator (hT.preimage hdesign), ← integral_indicator hT]
rw [← minimaxDataMeasure_map_design J kappa g hgmeas hgbound,
integral_map hdesign.aemeasurable]
· rfl
· exact ((measurable_const.add hgmeas).indicator hT).aestronglyMeasurable
exact condExp_eq_of_integral_preimage_eq
μ design hdesign (fun o => o.Y) m hY hm hm_design hintegral
theorem minimaxConstant_mem_cont_model
Constant shifts belong to the continuity-only model. The result uses the hreg condition, the heps condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin deltaBar eps :
ℝ
hreg :
ContDesignConstants J kappa cminus cplus pmin deltaBar
heps :
|eps| ≤ 1 / 2
conclusion
ContClampModel (minimaxClampLaw J kappa (fun _ : Fin J × ℝ => eps)) kappa cminus cplus pmin deltaBar
Proof (Lean source)
lemma minimaxConstant_mem_cont_model
(J : ℕ) (kappa cminus cplus pmin deltaBar eps : ℝ)
(hreg : ContDesignConstants J kappa cminus cplus pmin deltaBar)
(heps : |eps| ≤ 1 / 2) :
ContClampModel (minimaxClampLaw J kappa (fun _ : Fin J × ℝ => eps))
kappa cminus cplus pmin deltaBar := by
rcases hreg with ⟨hJ, hkappa, hcminus, hcminus_le, hcplus, hpmin,
hpmin_le, hdeltaBar, hdeltaBar_one⟩
apply minimaxClampModel_continuous J kappa cminus cplus pmin deltaBar hJ hkappa
hcminus_le hcplus hpmin_le (fun _ => eps) measurable_const (fun _ => heps)
fun_prop
theorem minimaxDoseBump_mem_cont_model
A fixed-height bump with an arbitrarily small positive width belongs to the same continuity-only class. In particular, the class membership carries no width-dependent Hölder radius. The result uses the hreg condition, the hh condition, the hamp condition, the hamp_le condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin deltaBar delta h amplitude :
ℝ
hreg :
ContDesignConstants J kappa cminus cplus pmin deltaBar
hh :
0 < h
hamp :
0 ≤ amplitude
hamp_le :
amplitude ≤ 1 / 4
conclusion
q :
fun a : ℝ => amplitude * CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
ContClampModel (minimaxClampLaw J kappa (fun p => q p.2)) kappa cminus cplus pmin deltaBar
Proof (Lean source)
lemma minimaxDoseBump_mem_cont_model
(J : ℕ) (kappa cminus cplus pmin deltaBar delta h amplitude : ℝ)
(hreg : ContDesignConstants J kappa cminus cplus pmin deltaBar)
(hh : 0 < h) (hamp : 0 ≤ amplitude) (hamp_le : amplitude ≤ 1 / 4) :
let q := fun a : ℝ => amplitude *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
ContClampModel (minimaxClampLaw J kappa (fun p => q p.2))
kappa cminus cplus pmin deltaBar := by
dsimp only
rcases hreg with ⟨hJ, hkappa, hcminus, hcminus_le, hcplus, hpmin,
hpmin_le, hdeltaBar, hdeltaBar_one⟩
let q := fun a : ℝ => amplitude *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
apply minimaxClampModel_continuous J kappa cminus cplus pmin deltaBar hJ hkappa
hcminus_le hcplus hpmin_le q
· dsimp [q]
fun_prop
· intro a
have hb0 := CausalSmith.Stat.DoseResponseMinimax.doseBump_nonneg
((a - delta) / h)
have hb1 := CausalSmith.Stat.DoseResponseMinimax.doseBump_le_one
((a - delta) / h)
rw [abs_of_nonneg (mul_nonneg hamp hb0)]
nlinarith
· apply Continuous.continuousOn
dsimp [q]
have hb : Continuous CausalSmith.Stat.DoseResponseMinimax.doseBump := by
unfold CausalSmith.Stat.DoseResponseMinimax.doseBump
exact (CausalSmith.Stat.DoseResponseMinimax.doseContDiffBump.contDiff
(n := ⊤)).continuous
exact continuous_const.add (continuous_const.mul
(hb.comp ((continuous_id.sub continuous_const).div_const h)))
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.ContinuityWitnessFunctional 5 declarations Functionals of the canonical continuity witnesses
Functionals of the canonical continuity witnesses
theorem contRegression_minimaxClampLaw_eq
The selected continuous regression of a canonical Bernoulli witness is the displayed Bernoulli mean throughout the threshold range. The result uses the hreg condition, the hqmeas condition, the hqbound condition, the hcont condition, the ha condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin deltaBar :
ℝ
hreg :
ContDesignConstants J kappa cminus cplus pmin deltaBar
q :
ℝ → ℝ
hqmeas :
hqbound :
∀ a, |q a| ≤ 1 / 2
hcont :
ContinuousOn (fun a => 1 / 2 + q a) (Icc (0 : ℝ) deltaBar)
x :
Fin J
a :
ℝ
ha :
a ∈ Icc (0 : ℝ) deltaBar
conclusion
P :
minimaxClampLaw J kappa (fun p => q p.2)
hP :
minimaxClampModel_continuous J kappa cminus cplus pmin deltaBar hreg.1 hreg.2.1 hreg.2.2.2.1 hreg.2.2.2.2.1 hreg.2.2.2.2.2.2.1 q hqmeas hqbound hcont
contRegression P kappa cminus cplus pmin deltaBar hP x a = 1 / 2 + q a
Proof (Lean source)
lemma contRegression_minimaxClampLaw_eq
(J : ℕ) (kappa cminus cplus pmin deltaBar : ℝ)
(hreg : ContDesignConstants J kappa cminus cplus pmin deltaBar)
(q : ℝ → ℝ) (hqmeas : Measurable q)
(hqbound : ∀ a, |q a| ≤ 1 / 2)
(hcont : ContinuousOn (fun a => 1 / 2 + q a) (Icc (0 : ℝ) deltaBar))
(x : Fin J) (a : ℝ) (ha : a ∈ Icc (0 : ℝ) deltaBar) :
let P := minimaxClampLaw J kappa (fun p => q p.2)
let hP := minimaxClampModel_continuous J kappa cminus cplus pmin deltaBar
hreg.1 hreg.2.1 hreg.2.2.2.1 hreg.2.2.2.2.1 hreg.2.2.2.2.2.2.1
q hqmeas hqbound hcont
contRegression P kappa cminus cplus pmin deltaBar hP x a = 1 / 2 + q a := by
dsimp only
let g : Fin J × ℝ → ℝ := fun p => q p.2
let P := minimaxClampLaw J kappa g
have hP := minimaxClampModel_continuous J kappa cminus cplus pmin deltaBar
hreg.1 hreg.2.1 hreg.2.2.2.1 hreg.2.2.2.2.1 hreg.2.2.2.2.2.2.1
q hqmeas hqbound hcont
have hchosen := Classical.choose_spec hP.continuousVersion
let μ := minimaxDataMeasure J kappa g
let design : ClampObs J → Fin J × ℝ := fun o => (o.X, o.A)
let m : ClampObs J → ℝ := fun o => 1 / 2 + q o.A
have hgmeas : Measurable g := hqmeas.comp measurable_snd
have hgbound : ∀ p, |g p| ≤ 1 / 2 := fun p => hqbound p.2
letI : IsProbabilityMeasure μ :=
minimaxDataMeasure_isProbabilityMeasure J kappa hreg.1 hreg.2.1
g hgmeas hgbound
have hdesign : Measurable design := measurable_minimax_design
have hY : Integrable (fun o : ClampObs J => o.Y) μ := by
refine Integrable.of_bound clampOutcome_measurable.aestronglyMeasurable 1 ?_
filter_upwards [minimaxDataMeasure_ae_bernoulli J kappa g hgmeas] with o ho
rcases ho with ho | ho <;> rw [ho] <;> norm_num
have hm_meas : Measurable m := by
exact measurable_const.add (hqmeas.comp
(measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))))
have hm : Integrable m μ := by
refine Integrable.of_bound hm_meas.aestronglyMeasurable 1 ?_
filter_upwards with o
rw [Real.norm_eq_abs]
change |1 / 2 + q o.A| ≤ 1
have hb := hqbound o.A
rw [abs_le] at hb
exact abs_le.2 ⟨by linarith [hb.1], by linarith [hb.2]⟩
have hm_design :
AEStronglyMeasurable[comap design inferInstance] m μ := by
have heq : m = (fun p : Fin J × ℝ => 1 / 2 + q p.2) ∘ design := rfl
rw [heq]
exact ((measurable_const.add (hqmeas.comp measurable_snd)).comp
(Measurable.of_comap_le le_rfl)).aestronglyMeasurable
have hint : ∀ T : Set (Fin J × ℝ), MeasurableSet T →
(∫ o in design ⁻¹' T, o.Y ∂μ) = ∫ o in design ⁻¹' T, m o ∂μ := by
intro T hT
rw [minimaxDataMeasure_integral_Y_design J kappa hreg.1 hreg.2.1
g hgmeas hgbound T hT]
rw [← integral_indicator (hT.preimage hdesign), ← integral_indicator hT]
rw [← minimaxDataMeasure_map_design J kappa g hgmeas hgbound,
integral_map hdesign.aemeasurable]
· rfl
· exact ((measurable_const.add hgmeas).indicator hT).aestronglyMeasurable
have hver :
P.dataMeasure[(fun o : ClampObs J => o.Y) |
comap (fun o : ClampObs J => (o.X, o.A)) inferInstance]
=ᵐ[P.dataMeasure] fun o => 1 / 2 + q o.A := by
exact condExp_eq_of_integral_preimage_eq
μ design hdesign (fun o => o.Y) m hY hm hm_design hint
exact (cont_regression_extension_unique P kappa cminus cplus pmin deltaBar
hP hreg _ (fun _ => 1 / 2 + q) hchosen.1 (fun _ => hcont)
hchosen.2 hver x) ha
theorem contClampFunctional_minimaxClampLaw_eq
The continuity functional of a canonical witness agrees with its explicit clamp functional on the threshold range. The result uses the hreg condition, the hdelta condition, the hqmeas condition, the hqbound condition, the hcont condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin deltaBar delta :
ℝ
hreg :
ContDesignConstants J kappa cminus cplus pmin deltaBar
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
q :
ℝ → ℝ
hqmeas :
hqbound :
∀ a, |q a| ≤ 1 / 2
hcont :
ContinuousOn (fun a => 1 / 2 + q a) (Icc (0 : ℝ) deltaBar)
conclusion
P :
minimaxClampLaw J kappa (fun p => q p.2)
hP :
minimaxClampModel_continuous J kappa cminus cplus pmin deltaBar hreg.1 hreg.2.1 hreg.2.2.2.1 hreg.2.2.2.2.1 hreg.2.2.2.2.2.2.1 q hqmeas hqbound hcont
contClampFunctional P kappa cminus cplus pmin deltaBar hP delta = clampFunctional P delta
Proof (Lean source)
lemma contClampFunctional_minimaxClampLaw_eq
(J : ℕ) (kappa cminus cplus pmin deltaBar delta : ℝ)
(hreg : ContDesignConstants J kappa cminus cplus pmin deltaBar)
(hdelta : delta ∈ Icc (0 : ℝ) deltaBar)
(q : ℝ → ℝ) (hqmeas : Measurable q)
(hqbound : ∀ a, |q a| ≤ 1 / 2)
(hcont : ContinuousOn (fun a => 1 / 2 + q a) (Icc (0 : ℝ) deltaBar)) :
let P := minimaxClampLaw J kappa (fun p => q p.2)
let hP := minimaxClampModel_continuous J kappa cminus cplus pmin deltaBar
hreg.1 hreg.2.1 hreg.2.2.2.1 hreg.2.2.2.2.1 hreg.2.2.2.2.2.2.1
q hqmeas hqbound hcont
contClampFunctional P kappa cminus cplus pmin deltaBar hP delta =
clampFunctional P delta := by
dsimp only
unfold contClampFunctional clampFunctional
congr 1
apply Finset.sum_congr rfl
intro x hx
rw [contRegression_minimaxClampLaw_eq J kappa cminus cplus pmin deltaBar
hreg q hqmeas hqbound hcont x delta hdelta]
rfl
theorem minimaxContinuityBumpSeparation
A fixed-height continuity bump changes the threshold functional by at least its atom contribution. The result uses the hJ condition, the hkappa condition, the hdelta condition, the hh condition, the hamp condition, the hamp_le condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa delta h amplitude :
ℝ
hJ :
0 < J
hkappa :
0 ≤ kappa
hdelta :
delta ∈ Icc (0 : ℝ) 1
hh :
0 < h
hamp :
0 ≤ amplitude
hamp_le :
amplitude ≤ 1 / 4
conclusion
q :
fun a : ℝ => amplitude * CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
amplitude * delta ^ (kappa + 1)
≤ clampFunctional (minimaxClampLaw J kappa (fun p => q p.2)) delta
- clampFunctional (minimaxClampLaw J kappa (fun _ => 0)) delta
Proof (Lean source)
lemma minimaxContinuityBumpSeparation
(J : ℕ) (kappa delta h amplitude : ℝ)
(hJ : 0 < J) (hkappa : 0 ≤ kappa)
(hdelta : delta ∈ Icc (0 : ℝ) 1) (hh : 0 < h)
(hamp : 0 ≤ amplitude) (hamp_le : amplitude ≤ 1 / 4) :
let q := fun a : ℝ => amplitude *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
amplitude * delta ^ (kappa + 1) ≤
clampFunctional (minimaxClampLaw J kappa (fun p => q p.2)) delta -
clampFunctional (minimaxClampLaw J kappa (fun _ => 0)) delta := by
dsimp only
let q := fun a : ℝ => amplitude *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
have hqmeas : Measurable q := by dsimp [q]; fun_prop
have hq0 (a : ℝ) : 0 ≤ q a := by
exact mul_nonneg hamp
(CausalSmith.Stat.DoseResponseMinimax.doseBump_nonneg _)
have hqbound (a : ℝ) : |q a| ≤ 1 / 2 := by
rw [abs_of_nonneg (hq0 a)]
have hb := CausalSmith.Stat.DoseResponseMinimax.doseBump_le_one
((a - delta) / h)
dsimp [q]
nlinarith
rw [minimaxClampFunctional J kappa delta hJ hkappa hdelta.1 q hqmeas hqbound,
minimaxClampFunctional J kappa delta hJ hkappa hdelta.1
(fun _ => 0) measurable_const (fun _ => by norm_num)]
haveI : IsProbabilityMeasure (minimaxTreatmentMeasure kappa) :=
minimaxTreatmentMeasure_isProbabilityMeasure kappa hkappa
have hqint : Integrable q (minimaxTreatmentMeasure kappa) := by
refine Integrable.of_bound hqmeas.aestronglyMeasurable (1 / 2) ?_
filter_upwards with a
simpa [Real.norm_eq_abs] using hqbound a
have honeint : Integrable (fun _ : ℝ => (1 / 2 : ℝ))
(minimaxTreatmentMeasure kappa) := integrable_const _
rw [show (∫ a in Ioi delta, (1 / 2 + q a) ∂minimaxTreatmentMeasure kappa) =
(∫ a in Ioi delta, (1 / 2 : ℝ) ∂minimaxTreatmentMeasure kappa) +
∫ a in Ioi delta, q a ∂minimaxTreatmentMeasure kappa by
rw [integral_add honeint.integrableOn hqint.integrableOn],
show (∫ a in Ioi delta, (1 / 2 + 0) ∂minimaxTreatmentMeasure kappa) =
∫ a in Ioi delta, (1 / 2 : ℝ) ∂minimaxTreatmentMeasure kappa by simp]
have hqdelta : q delta = amplitude := by
simp [q, hh.ne', CausalSmith.Stat.DoseResponseMinimax.doseBump_zero]
rw [hqdelta]
have hnonneg : 0 ≤ ∫ a in Ioi delta, q a ∂minimaxTreatmentMeasure kappa :=
integral_nonneg_of_ae (ae_of_all _ hq0)
nlinarith
theorem continuityBump_twoPoint
The canonical center and a fixed-height narrow bump form an admissible continuity-only two-point experiment with an explicit product chi-square budget and atom-scale functional separation. The result uses the hreg condition, the hdelta condition, the hh condition, the hamp condition, the hamp_le condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
kappa cminus cplus pmin deltaBar delta h amplitude :
ℝ
hreg :
ContDesignConstants J kappa cminus cplus pmin deltaBar
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
hh :
0 < h
hamp :
0 ≤ amplitude
hamp_le :
amplitude ≤ 1 / 4
conclusion
q :
fun a : ℝ => amplitude * CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
Q0 :
minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
Q1 :
minimaxClampLaw J kappa (fun p => q p.2)
∃ (hQ0 : ContClampModel Q0 kappa cminus cplus pmin deltaBar) (hQ1 : ContClampModel Q1 kappa cminus cplus pmin deltaBar),
conclusion 1
amplitude * delta ^ (kappa + 1)
≤ |contClampFunctional Q1 kappa cminus cplus pmin deltaBar hQ1 delta
- contClampFunctional Q0 kappa cminus cplus pmin deltaBar hQ0 delta|
conclusion 2
1 + productChiSq Q1 Q0 n
≤ exp ((n : ℝ) * (8 * (kappa + 1) * amplitude ^ 2 * h * (delta + h) ^ kappa))
Proof (Lean source)
lemma continuityBump_twoPoint
(J n : ℕ) (kappa cminus cplus pmin deltaBar delta h amplitude : ℝ)
(hreg : ContDesignConstants J kappa cminus cplus pmin deltaBar)
(hdelta : delta ∈ Icc (0 : ℝ) deltaBar)
(hh : 0 < h) (hamp : 0 ≤ amplitude) (hamp_le : amplitude ≤ 1 / 4) :
let q := fun a : ℝ => amplitude *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
let Q0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
let Q1 := minimaxClampLaw J kappa (fun p => q p.2)
∃ (hQ0 : ContClampModel Q0 kappa cminus cplus pmin deltaBar)
(hQ1 : ContClampModel Q1 kappa cminus cplus pmin deltaBar),
amplitude * delta ^ (kappa + 1) ≤
|contClampFunctional Q1 kappa cminus cplus pmin deltaBar hQ1 delta -
contClampFunctional Q0 kappa cminus cplus pmin deltaBar hQ0 delta| ∧
1 + productChiSq Q1 Q0 n ≤
exp ((n : ℝ) * (8 * (kappa + 1) * amplitude ^ 2 * h *
(delta + h) ^ kappa)) := by
dsimp only
let q := fun a : ℝ => amplitude *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
let Q0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
let Q1 := minimaxClampLaw J kappa (fun p => q p.2)
have hcont0 : ContinuousOn (fun _ : ℝ => (1 / 2 : ℝ) + 0)
(Icc (0 : ℝ) deltaBar) := by fun_prop
have hqmeas : Measurable q := by dsimp [q]; fun_prop
have hq0 (a : ℝ) : 0 ≤ q a := mul_nonneg hamp
(CausalSmith.Stat.DoseResponseMinimax.doseBump_nonneg _)
have hqbound (a : ℝ) : |q a| ≤ 1 / 2 := by
rw [abs_of_nonneg (hq0 a)]
have hb := CausalSmith.Stat.DoseResponseMinimax.doseBump_le_one
((a - delta) / h)
dsimp [q]
nlinarith
have hqstrict (p : Fin J × ℝ) : |q p.2| < 1 / 2 := by
rw [abs_of_nonneg (hq0 p.2)]
have hb := CausalSmith.Stat.DoseResponseMinimax.doseBump_le_one
((p.2 - delta) / h)
dsimp [q]
nlinarith
have hcont1 : ContinuousOn (fun a => 1 / 2 + q a)
(Icc (0 : ℝ) deltaBar) := by
apply Continuous.continuousOn
dsimp [q]
have hb : Continuous CausalSmith.Stat.DoseResponseMinimax.doseBump := by
unfold CausalSmith.Stat.DoseResponseMinimax.doseBump
exact (CausalSmith.Stat.DoseResponseMinimax.doseContDiffBump.contDiff
(n := ⊤)).continuous
exact continuous_const.add (continuous_const.mul
(hb.comp ((continuous_id.sub continuous_const).div_const h)))
let hQ0 := minimaxConstant_mem_cont_model J kappa cminus cplus pmin deltaBar 0
hreg (by norm_num)
let hQ1 := minimaxDoseBump_mem_cont_model J kappa cminus cplus pmin deltaBar
delta h amplitude hreg hh hamp hamp_le
refine ⟨hQ0, hQ1, ?_, ?_⟩
· rw [contClampFunctional_minimaxClampLaw_eq J kappa cminus cplus pmin
deltaBar delta hreg hdelta q hqmeas hqbound hcont1,
contClampFunctional_minimaxClampLaw_eq J kappa cminus cplus pmin
deltaBar delta hreg hdelta (fun _ => 0) measurable_const
(fun _ => by norm_num) hcont0]
have hsep := minimaxContinuityBumpSeparation J kappa delta h amplitude
hreg.1 hreg.2.1 ⟨hdelta.1, hdelta.2.trans hreg.2.2.2.2.2.2.2.2.le⟩
hh hamp hamp_le
rw [abs_of_nonneg (hsep.trans' (mul_nonneg hamp
(Real.rpow_nonneg hdelta.1 _)))]
exact hsep
· let I := ∫ p, 4 * ((fun p : Fin J × ℝ => q p.2) p) ^ 2
∂minimaxDesignMeasure J kappa
have hI : I ≤ 8 * (kappa + 1) * amplitude ^ 2 * h *
(delta + h) ^ kappa := by
have hi := minimaxLocal_designIntegral_le J 0 kappa delta h amplitude
hreg.1 hreg.2.1 hdelta.1 hh hamp
simpa [I, q] using hi
have hI0 : 0 ≤ I := integral_nonneg_of_ae (ae_of_all _ fun _ => by positivity)
have heq := minimaxProduct_chiSqDiv_center J n kappa hreg.1 hreg.2.1
(fun p : Fin J × ℝ => q p.2) (hqmeas.comp measurable_snd) hqstrict
change 1 + productChiSq Q1 Q0 n = (1 + I) ^ n at heq
rw [heq]
calc
(1 + I) ^ n ≤ (exp I) ^ n := pow_le_pow_left₀ (by linarith)
(by simpa [add_comm] using Real.add_one_le_exp I) n
_ = exp ((n : ℝ) * I) := by rw [Real.exp_nat_mul]
_ ≤ exp ((n : ℝ) * (8 * (kappa + 1) * amplitude ^ 2 * h *
(delta + h) ^ kappa)) := Real.exp_le_exp.mpr
(mul_le_mul_of_nonneg_left hI (Nat.cast_nonneg n))
theorem continuityConstant_twoPoint
Two constant Bernoulli regressions give the regular root-sample-size two-point experiment inside the continuity-only model. The result uses the hreg condition, the hdelta condition, the heps condition, the heps_le condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
kappa cminus cplus pmin deltaBar delta eps :
ℝ
hreg :
ContDesignConstants J kappa cminus cplus pmin deltaBar
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
heps :
0 ≤ eps
heps_le :
eps ≤ 1 / 4
conclusion
Q0 :
minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
Q1 :
minimaxClampLaw J kappa (fun _ : Fin J × ℝ => eps)
∃ (hQ0 : ContClampModel Q0 kappa cminus cplus pmin deltaBar) (hQ1 : ContClampModel Q1 kappa cminus cplus pmin deltaBar),
conclusion 1
eps
= |contClampFunctional Q1 kappa cminus cplus pmin deltaBar hQ1 delta
- contClampFunctional Q0 kappa cminus cplus pmin deltaBar hQ0 delta|
conclusion 2
1 + productChiSq Q1 Q0 n ≤ exp ((n : ℝ) * (4 * eps ^ 2))
Proof (Lean source)
lemma continuityConstant_twoPoint
(J n : ℕ) (kappa cminus cplus pmin deltaBar delta eps : ℝ)
(hreg : ContDesignConstants J kappa cminus cplus pmin deltaBar)
(hdelta : delta ∈ Icc (0 : ℝ) deltaBar)
(heps : 0 ≤ eps) (heps_le : eps ≤ 1 / 4) :
let Q0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
let Q1 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => eps)
∃ (hQ0 : ContClampModel Q0 kappa cminus cplus pmin deltaBar)
(hQ1 : ContClampModel Q1 kappa cminus cplus pmin deltaBar),
eps = |contClampFunctional Q1 kappa cminus cplus pmin deltaBar hQ1 delta -
contClampFunctional Q0 kappa cminus cplus pmin deltaBar hQ0 delta| ∧
1 + productChiSq Q1 Q0 n ≤ exp ((n : ℝ) * (4 * eps ^ 2)) := by
dsimp only
let Q0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
let Q1 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => eps)
have hepsBound : |eps| ≤ 1 / 2 := by rw [abs_of_nonneg heps]; linarith
have hepsStrict : |eps| < 1 / 2 := by
rw [abs_of_nonneg heps]
linarith
have hcont0 : ContinuousOn (fun _ : ℝ => (1 / 2 : ℝ) + 0)
(Icc (0 : ℝ) deltaBar) := by fun_prop
have hcont1 : ContinuousOn (fun _ : ℝ => (1 / 2 : ℝ) + eps)
(Icc (0 : ℝ) deltaBar) := by fun_prop
let hQ0 := minimaxConstant_mem_cont_model J kappa cminus cplus pmin deltaBar 0
hreg (by norm_num)
let hQ1 := minimaxConstant_mem_cont_model J kappa cminus cplus pmin deltaBar eps
hreg hepsBound
refine ⟨hQ0, hQ1, ?_, ?_⟩
· rw [contClampFunctional_minimaxClampLaw_eq J kappa cminus cplus pmin
deltaBar delta hreg hdelta (fun _ => eps) measurable_const
(fun _ => hepsBound) hcont1,
contClampFunctional_minimaxClampLaw_eq J kappa cminus cplus pmin
deltaBar delta hreg hdelta (fun _ => 0) measurable_const
(fun _ => by norm_num) hcont0,
minimaxGlobalSeparation J kappa delta eps hreg.1 hreg.2.1
⟨hdelta.1, hdelta.2.trans hreg.2.2.2.2.2.2.2.2.le⟩ hepsBound,
abs_of_nonneg heps]
· have heq := minimaxProduct_chiSqDiv_center J n kappa hreg.1 hreg.2.1
(fun _ : Fin J × ℝ => eps) measurable_const (fun _ => hepsStrict)
have hI : (∫ _p, 4 * eps ^ 2 ∂minimaxDesignMeasure J kappa) = 4 * eps ^ 2 := by
letI : IsProbabilityMeasure (minimaxDesignMeasure J kappa) :=
minimaxDesignMeasure_isProbabilityMeasure J kappa hreg.1 hreg.2.1
simp
change 1 + productChiSq Q1 Q0 n =
(1 + ∫ _p, 4 * eps ^ 2 ∂minimaxDesignMeasure J kappa) ^ n at heq
rw [hI] at heq
rw [heq]
calc
(1 + 4 * eps ^ 2) ^ n ≤ (exp (4 * eps ^ 2)) ^ n :=
pow_le_pow_left₀ (by positivity)
(by simpa [add_comm] using Real.add_one_le_exp (4 * eps ^ 2)) n
_ = exp ((n : ℝ) * (4 * eps ^ 2)) := by rw [Real.exp_nat_mul]
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.Design 29 declarations This module gives full definitions of the paper's reference Gram, deterministic three-way split, local count and total Gram, exact intercept weights, stabilized estimator, bias-aware interval, and realized-design modulus
Realized local-polynomial design and estimators
This module gives full definitions of the paper's reference Gram, deterministic three-way split, local count and total Gram, exact intercept weights, stabilized estimator, bias-aware interval, and realized-design modulus handle.
def monomialVec
Monomial basis (1,u,...,u^ell).
Definition (Lean source)
def refMomentMatrix
Normalized population reference moment matrix for (rho+u)^kappa.
Definition (Lean source)
parameters
ell :
ℕ
kappa rho :
ℝ
given by
fun i j
=> (∫ u in Icc (0 : ℝ) 1, monomialVec ell u i * monomialVec ell u j * (rho + u) ^ kappa) / (∫ u in Icc (0 : ℝ) 1, (rho + u) ^ kappa)
def matrixQuadratic
Quadratic form of a real square matrix.
def quadraticMin
Smallest quadratic-form value on the Euclidean unit sphere.
Uniform population-Gram constant.
Definition (Lean source)
parameters
ell :
ℕ
kappa cminus cplus :
ℝ
def
lambdaStar ell kappa cminus cplus :
ℝ
given by
(cminus / cplus) * sInf {q : ℝ | ∃ rho : ℝ, 0 ≤ rho ∧ q = quadraticMin (refMomentMatrix ell kappa rho)}
Three deterministic, pairwise-disjoint sample blocks, each of size at least floor(n/4).
def scaledDose
Rescaled location of one observed treatment relative to the moving threshold.
Definition (Lean source)
parameters
def
scaledDose delta h o :
ℝ
given by
(o.A - delta) / h
def localCount
Number of observations in stratum x and the local threshold window.
Definition (Lean source)
parameters
def
localCount B z x delta h :
ℕ
given by
by classical exact ∑ i ∈ B.I2, if (z i).X = x ∧
(z i).A ∈ Icc delta (delta + h) then 1 else 0
def localKernelWeight
The zero-one localization weight on the full finite design.
Definition (Lean source)
parameters
def
localKernelWeight B z x delta h i :
ℝ
given by
if i ∈ B.I2 ∧ (z i).X = x ∧ scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1 then 1 else 0
def localGram
Unnormalized total local Gram matrix, realized by Causalean's weighted monomial design matrix.
Definition (Lean source)
parameters
given by
theorem localGram_apply
For sample blocks, a realized sample, a stratum, a polynomial degree, a threshold and bandwidth, and two matrix coordinates, the local Gram entry equals the active-window monomial sum.
Formal statement
hypotheses
conclusion
conclusion 1
localGram B z x ell delta h r s = ∑ i ∈ B.I2, if (z i).X = x
conclusion 2
scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1 then monomialVec ell (scaledDose delta h (z i)) r * monomialVec ell (scaledDose delta h (z i)) s else 0
Proof (Lean source)
lemma localGram_apply (B : SplitBlocks n) (z : Fin n → ClampObs J)
(x : Fin J) (ell : ℕ) (delta h : ℝ) (r s : Fin (ell + 1)) :
localGram B z x ell delta h r s = ∑ i ∈ B.I2,
if (z i).X = x ∧ scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1 then
monomialVec ell (scaledDose delta h (z i)) r *
monomialVec ell (scaledDose delta h (z i)) s
else 0 := by
classical
simp only [localGram, designMatrix]
rw [show (∑ i, localKernelWeight B z x delta h i *
scaledDose delta h (z i) ^ (r : ℕ) * scaledDose delta h (z i) ^ (s : ℕ)) =
∑ i ∈ B.I2, localKernelWeight B z x delta h i *
scaledDose delta h (z i) ^ (r : ℕ) * scaledDose delta h (z i) ^ (s : ℕ) by
symm
apply Finset.sum_subset (Finset.subset_univ B.I2)
intro i _ hi
simp [localKernelWeight, hi]]
apply Finset.sum_congr rfl
intro i hi
by_cases ha : (z i).X = x ∧ scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1
· simp [localKernelWeight, hi, ha, monomialVec]
· rw [if_neg ha]
have hw : localKernelWeight B z x delta h i = 0 := by
rw [localKernelWeight, if_neg]
exact fun hactive => ha hactive.2
rw [hw]
simp
The good-design event, encoded by its load-bearing quadratic-form lower bound.
Definition (Lean source)
parameters
def
GoodGramEvent B z x ell kappa cminus cplus delta h :
Prop
given by
clause 1
0 < localCount B z x delta h
clause 2
v :
Fin (ell + 1) → ℝ
lambdaStar ell kappa cminus cplus * (localCount B z x delta h : ℝ) / 2 * (∑ j, (v j) ^ 2)
≤ matrixQuadratic (localGram B z x ell delta h) v
def interceptWeight
Exact local-polynomial intercept weight from Causalean's equivalent kernel on the good event, and zero off it.
Definition (Lean source)
parameters
def
interceptWeight B z x ell kappa cminus cplus delta h i :
ℝ
given by
by classical exact if GoodGramEvent B z x ell kappa cminus cplus delta h then equivKernelWeight ell (fun j => scaledDose delta h (z j)) (localKernelWeight B z x delta h) i else 0
Given a good design, an observation in the regression block, the required stratum match, and local-window membership, the intercept weight is the zeroth coordinate of the inverse-Gram feature vector.
Formal statement
hypotheses
J :
ℕ
n :
ℕ
B :
x :
Fin J
ell :
ℕ
kappa cminus cplus delta h :
ℝ
i :
Fin n
hgood :
GoodGramEvent B z x ell kappa cminus cplus delta h
hi :
i ∈ B.I2
hix :
(z i).X = x
hu :
scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1
conclusion
interceptWeight B z x ell kappa cminus cplus delta h i
Proof (Lean source)
lemma interceptWeight_eq_mulVec (B : SplitBlocks n) (z : Fin n → ClampObs J)
(x : Fin J) (ell : ℕ) (kappa cminus cplus delta h : ℝ) (i : Fin n)
(hgood : GoodGramEvent B z x ell kappa cminus cplus delta h)
(hi : i ∈ B.I2) (hix : (z i).X = x)
(hu : scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1) :
interceptWeight B z x ell kappa cminus cplus delta h i =
((localGram B z x ell delta h)⁻¹ *ᵥ
monomialVec ell (scaledDose delta h (z i))) 0 := by
classical
rw [interceptWeight, if_pos hgood]
simp only [equivKernelWeight, mulVec,
dotProduct, localGram, monomialVec]
simp [localKernelWeight, hi, hix, hu]
theorem interceptWeight_eq_equivKernel_of_good
On the good-design event, the intercept weight equals the equivalent-kernel weight.
Formal statement
hypotheses
J :
ℕ
n :
ℕ
B :
x :
Fin J
ell :
ℕ
kappa cminus cplus delta h :
ℝ
hgood :
GoodGramEvent B z x ell kappa cminus cplus delta h
i :
Fin n
conclusion
interceptWeight B z x ell kappa cminus cplus delta h i
Proof (Lean source)
lemma interceptWeight_eq_equivKernel_of_good
(B : SplitBlocks n) (z : Fin n → ClampObs J) (x : Fin J)
(ell : ℕ) (kappa cminus cplus delta h : ℝ)
(hgood : GoodGramEvent B z x ell kappa cminus cplus delta h) (i : Fin n) :
interceptWeight B z x ell kappa cminus cplus delta h i =
equivKernelWeight ell
(fun j => scaledDose delta h (z j)) (localKernelWeight B z x delta h) i := by
simp [interceptWeight, hgood]
theorem interceptWeight_eq_zero_of_inactive
On the good-design event, if an observation is inactive, its intercept weight is zero.
Formal statement
hypotheses
J :
ℕ
n :
ℕ
B :
x :
Fin J
ell :
ℕ
kappa cminus cplus delta h :
ℝ
i :
Fin n
hgood :
GoodGramEvent B z x ell kappa cminus cplus delta h
hi :
¬ (i ∈ B.I2 ∧ (z i).X = x ∧ scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1)
conclusion
interceptWeight B z x ell kappa cminus cplus delta h i = 0
Proof (Lean source)
lemma interceptWeight_eq_zero_of_inactive
(B : SplitBlocks n) (z : Fin n → ClampObs J) (x : Fin J)
(ell : ℕ) (kappa cminus cplus delta h : ℝ) (i : Fin n)
(hgood : GoodGramEvent B z x ell kappa cminus cplus delta h)
(hi : ¬ (i ∈ B.I2 ∧ (z i).X = x ∧
scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1)) :
interceptWeight B z x ell kappa cminus cplus delta h i = 0 := by
rw [interceptWeight_eq_equivKernel_of_good
B z x ell kappa cminus cplus delta h hgood i]
simp only [equivKernelWeight]
have hw : localKernelWeight B z x delta h i = 0 := by
rw [localKernelWeight, if_neg hi]
simp [hw]
Projection to the outcome range [0,1].
Empirical average over a deterministic finite block.
def retainedEstimate
Retained-course empirical mean on block zero.
Definition (Lean source)
parameters
def
retainedEstimate B z delta :
ℝ
given by
blockAverage B.I0 z (fun o => if delta < o.A then o.Y else 0)
def atomEstimate
Empirical threshold mass in one stratum on block one.
Definition (Lean source)
parameters
def
atomEstimate B z x delta :
ℝ
given by
blockAverage B.I1 z (fun o => if o.X = x ∧ o.A ≤ delta then 1 else 0)
def localRegressionEstimate
Stabilized local regression value, with the prescribed one-half fallback.
Definition (Lean source)
parameters
def
localRegressionEstimate B z x ell kappa cminus cplus delta h :
ℝ
given by
by classical exact if GoodGramEvent B z x ell kappa cminus cplus delta h then clampUnit (∑ i ∈ B.I2, interceptWeight B z x ell kappa cminus cplus delta h i * (z i).Y) else 1 / 2
def totalGramEstimator
Sample-split total-Gram-stabilized estimator of the clamp functional.
Definition (Lean source)
parameters
J :
ℕ
n :
ℕ
B :
ell :
ℕ
kappa cminus cplus delta h :
ℝ
def
totalGramEstimator B z ell kappa cminus cplus delta h :
ℝ
given by
clampUnit (retainedEstimate B z delta + ∑ x : Fin J, atomEstimate B z x delta * localRegressionEstimate B z x ell kappa cminus cplus delta h)
def contFallbackEstimator
Continuity-only estimator using the retained-course block and the fixed one-half regression fallback for the total empirical atom mass.
Definition (Lean source)
parameters
def
contFallbackEstimator B z delta :
ℝ
given by
def contHoeffdingInterval
Continuity-only Hoeffding interval, intersected with the outcome range.
Definition (Lean source)
parameters
def
given by
atomTotal :
∑ x : Fin J, atomEstimate B z x delta
center :
contFallbackEstimator B z delta
radius :
t0 + t1 + atomTotal / 2
def stratumRadius
The per-stratum bias-plus-noise radius, including the singular-Gram fallback.
Definition (Lean source)
parameters
def
stratumRadius B z x ell beta kappa L cminus cplus delta h tAlpha b1 :
ℝ
given by
by classical exact if GoodGramEvent B z x ell kappa cminus cplus delta h then (atomEstimate B z x delta + b1) * (L * h ^ beta * ∑ i ∈ B.I2, |interceptWeight B z x ell kappa cminus cplus delta h i| + tAlpha * sqrt (∑ i ∈ B.I2, (interceptWeight B z x ell kappa cminus cplus delta h i) ^ 2))
+ b1 else atomEstimate B z x delta
+ b1
def honestInterval
Bias-aware interval intersected with [0,1].
Definition (Lean source)
parameters
J :
ℕ
n :
ℕ
B :
ell :
ℕ
beta kappa L cminus cplus delta h alpha :
ℝ
def
given by
b0 :
tAlpha / sqrt (B.I0.card : ℝ)
center :
totalGramEstimator B z ell kappa cminus cplus delta h
b1 :
tAlpha / sqrt (B.I1.card : ℝ)
radius :
b0 + ∑ x : Fin J, stratumRadius B z x ell beta kappa L cminus cplus delta h tAlpha b1
def exactHolderBias
Exact worst-case Hölder bias of affine weights on a realized local window. The derivatives are intrinsic to [0,1], so the value is invariant under any change to an ambient extension of the regression.
Definition (Lean source)
parameters
J :
ℕ
n :
ℕ
B :
_x :
Fin J
@realizes exactHolderBias(realized Hölder-bias supremum)
ell :
ℕ
beta L delta :
ℝ
w :
Fin n → ℝ
def
exactHolderBias B z _x ell beta L delta w :
ℝ
given by
sSup {b : ℝ | ∃ f : ℝ → ℝ, ContinuousOn f (Icc (0 : ℝ) 1) ∧ (∀ a ∈ Icc (0 : ℝ) 1, f a ∈ Icc (0 : ℝ) 1) ∧ (∀ s ∈ Icc (0 : ℝ) 1, ∀ t ∈ Icc (0 : ℝ) 1, |f t - ∑ j ∈ range (ell + 1), iteratedDerivWithin j f (Icc (0 : ℝ) 1) s * (t - s) ^ j / (Nat.factorial j : ℝ)| ≤ L * |t - s| ^ beta) ∧ b = |(∑ i ∈ B.I2, w i * f (z i).A) - f delta|}
def realizedExactModulus
Conditional affine modulus for one realized design. Only observations in the local stratum-window may receive weight, and the full Hölder bias—not a coarse L h^beta sum |w_i| upper bound—is optimized.
Definition (Lean source)
parameters
J :
ℕ
n :
ℕ
@realizes realizedExactModulus(exact bias-plus-l2 infimum)
B :
x :
Fin J
ell :
ℕ
beta L delta h t :
ℝ
_ht :
0 ≤ t
def
realizedExactModulus B z x ell beta L delta h t _ht :
ℝ
given by
sInf {v : ℝ | ∃ w : Fin n → ℝ, (∀ i, i ∉ B.I2 ∨ (z i).X ≠ x ∨ scaledDose delta h (z i) ∉ Icc (0 : ℝ) 1 → w i = 0) ∧ (∀ j : Fin (ell + 1), (∑ i ∈ B.I2, w i * monomialVec ell (scaledDose delta h (z i)) j) = if j = 0 then 1 else 0) ∧ v = exactHolderBias B z x ell beta L delta w + t * sqrt (∑ i ∈ B.I2, (w i) ^ 2)}
def exactModulusHandle
The realized-design exact-modulus handle integrates the conditional sample-dependent modulus under a probability/support-pinned polynomial- thinning design law, for a positive bandwidth and a nonnegative test multiplier.
Definition (Lean source)
parameters
J :
ℕ
n :
ℕ
P :
ClampLaw J
B :
x :
Fin J
ell :
ℕ
beta kappa L cminus cplus pmin delta h t :
ℝ
_hh :
0 < h
_ht :
0 ≤ t
_hmodel :
ClampModel P beta kappa L cminus cplus pmin
def
exactModulusHandle P B x ell beta kappa L cminus cplus pmin delta h t _hh _ht _hmodel :
ℝ
given by
∫ z, realizedExactModulus B z x ell beta L delta h t _ht ∂iidProduct P n
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.Divergence 2 declarations The upper inequality uses the Causalean quarter-band lemma.
Two-sided Bernoulli divergence bridges
The upper inequality uses the Causalean quarter-band lemma. The lower inequality is the missing direction needed by the one-cell calibration and follows from the exact Bernoulli formula (or Pinsker with the exact two-atom variation).
theorem bernoulli_kl_band
On the quarter window, Bernoulli KL from 1/2+g to 1/2 is comparable to g^2 in both directions. The result uses the hg condition. This is the stated conclusion.
Formal statement
hypotheses
g :
ℝ
hg :
|g| ≤ (1 : ℝ) / 4
conclusion
conclusion 1
2 * g ^ 2
conclusion 2
≤ 4 * g ^ 2
Proof (Lean source)
lemma bernoulli_kl_band (g : ℝ) (hg : |g| ≤ (1 : ℝ) / 4) :
2 * g ^ 2 ≤
(klDiv
(bernoulliLaw (1 / 2 + g))
(bernoulliLaw (1 / 2))).toReal ∧
(klDiv
(bernoulliLaw (1 / 2 + g))
(bernoulliLaw (1 / 2))).toReal
≤ 4 * g ^ 2 := by
have hp_lo : (1 : ℝ) / 4 ≤ 1 / 2 + g := by
rw [abs_le] at hg
linarith
have hp_hi : 1 / 2 + g ≤ (3 : ℝ) / 4 := by
rw [abs_le] at hg
linarith
have hq_lo : (1 : ℝ) / 4 ≤ 1 / 2 := by norm_num
have hq_hi : (1 / 2 : ℝ) ≤ 3 / 4 := by norm_num
have hp0 : 0 ≤ 1 / 2 + g := by linarith
have hp1 : 1 / 2 + g ≤ 1 := by linarith
let μ := bernoulliLaw (1 / 2 + g)
let ν := bernoulliLaw (1 / 2)
haveI : IsProbabilityMeasure μ :=
Causalean.Mathlib.Probability.bernoulliLaw_isProbabilityMeasure
hp0 hp1
haveI : IsProbabilityMeasure ν :=
Causalean.Mathlib.Probability.bernoulliLaw_isProbabilityMeasure
(by norm_num) (by norm_num)
have hac : μ ≪ ν :=
bernoulliLaw_ac_of_reference_interior
(by norm_num) (by norm_num)
have hKLle :=
Causalean.Mathlib.Probability.bernoulliLaw_klDiv_le_four_sq_sub
hp_lo hp_hi hq_lo hq_hi
have hfinite : klDiv μ ν ≠ ⊤ :=
ne_top_of_le_ne_top ENNReal.ofReal_ne_top hKLle
have hpinsker := Causalean.Stat.pinskerBound_of_ac_of_ne_top μ ν hac hfinite
have hgap : |g| ≤ tvDist μ ν := by
have h := abs_measureReal_sub_le_tvDist
(μ := μ) (ν := ν) (measurableSet_singleton (1 : ℝ))
have hμ1 : μ.real {(1 : ℝ)} = 1 / 2 + g := by
norm_num [μ, bernoulliLaw, Measure.real, hp0]
have hν1 : ν.real {(1 : ℝ)} = 1 / 2 := by
simp [ν, bernoulliLaw, Measure.real]
rw [hμ1, hν1] at h
simpa [abs_sub_comm] using h
constructor
· have hsqrt : |g| ≤ sqrt ((klDiv μ ν).toReal / 2) :=
hgap.trans hpinsker
have hsq := (sq_le_sq₀ (abs_nonneg g)
(Real.sqrt_nonneg _)).2 hsqrt
rw [sq_abs, Real.sq_sqrt (by positivity)] at hsq
dsimp [μ, ν] at hsq ⊢
linarith
· have hboundFinite : ofReal (4 * (1 / 2 + g - 1 / 2) ^ 2) ≠ ⊤ :=
ENNReal.ofReal_ne_top
have hreal := (ENNReal.toReal_le_toReal hfinite hboundFinite).2 hKLle
rw [ENNReal.toReal_ofReal (by positivity)] at hreal
simpa [μ, ν] using hreal
theorem localized_design_kl_band
Integrating the scalar band against a nonnegative design density preserves both inequalities, yielding the localized one-observation KL order. The result uses the hpi_int condition, the hgamma_meas condition, the hkl_meas condition, the hpi condition, the hgamma condition. This is the stated conclusion.
Formal statement
hypotheses
pi gamma :
ℝ → ℝ
hpi_int :
Integrable pi (volume.restrict (Icc (0 : ℝ) 1))
hgamma_meas :
Measurable gamma
hkl_meas :
Measurable fun a
hpi :
∀ᵐ a ∂volume.restrict (Icc (0 : ℝ) 1), 0 ≤ pi a
hgamma :
∀ a ∈ Icc (0 : ℝ) 1, |gamma a| ≤ (1 : ℝ) / 4
conclusion
conclusion 1
2 * ∫ a in Icc (0 : ℝ) 1, (gamma a) ^ 2 * pi a
≤ ∫ a in Icc (0 : ℝ) 1, (klDiv (bernoulliLaw (1 / 2 + gamma a)) (bernoulliLaw (1 / 2))).toReal * pi a
conclusion 2
(∫ a in Icc (0 : ℝ) 1, (klDiv (bernoulliLaw (1 / 2 + gamma a)) (bernoulliLaw (1 / 2))).toReal * pi a)
≤ 4 * ∫ a in Icc (0 : ℝ) 1, (gamma a) ^ 2 * pi a
Proof (Lean source)
lemma localized_design_kl_band (pi gamma : ℝ → ℝ)
(hpi_int : Integrable pi (volume.restrict (Icc (0 : ℝ) 1)))
(hgamma_meas : Measurable gamma)
(hkl_meas : Measurable fun a =>
(klDiv
(bernoulliLaw (1 / 2 + gamma a))
(bernoulliLaw (1 / 2))).toReal)
(hpi : ∀ᵐ a ∂volume.restrict (Icc (0 : ℝ) 1), 0 ≤ pi a)
(hgamma : ∀ a ∈ Icc (0 : ℝ) 1, |gamma a| ≤ (1 : ℝ) / 4) :
2 * ∫ a in Icc (0 : ℝ) 1, (gamma a) ^ 2 * pi a ≤
∫ a in Icc (0 : ℝ) 1,
(klDiv
(bernoulliLaw (1 / 2 + gamma a))
(bernoulliLaw (1 / 2))).toReal * pi a ∧
(∫ a in Icc (0 : ℝ) 1,
(klDiv
(bernoulliLaw (1 / 2 + gamma a))
(bernoulliLaw (1 / 2))).toReal * pi a) ≤
4 * ∫ a in Icc (0 : ℝ) 1, (gamma a) ^ 2 * pi a := by
let μ := volume.restrict (Icc (0 : ℝ) 1)
have hmem : ∀ᵐ a ∂μ, a ∈ Icc (0 : ℝ) 1 :=
ae_restrict_mem measurableSet_Icc
have hsqInt : Integrable (fun a => gamma a ^ 2 * pi a) μ := by
apply hpi_int.bdd_mul (c := (1 : ℝ) / 16)
· fun_prop
· filter_upwards [hmem] with a ha
have hb := hgamma a ha
rw [Real.norm_eq_abs, abs_sq]
have hs := (sq_le_sq₀ (abs_nonneg (gamma a))
(by norm_num : (0 : ℝ) ≤ 1 / 4)).2 hb
rw [sq_abs] at hs
nlinarith
have hklInt : Integrable (fun a =>
(klDiv
(bernoulliLaw (1 / 2 + gamma a))
(bernoulliLaw (1 / 2))).toReal * pi a) μ := by
apply hpi_int.bdd_mul (c := (1 : ℝ) / 4)
· exact hkl_meas.aestronglyMeasurable
· filter_upwards [hmem] with a ha
have hb := (bernoulli_kl_band (gamma a) (hgamma a ha)).2
have hnonneg : 0 ≤ (klDiv
(bernoulliLaw (1 / 2 + gamma a))
(bernoulliLaw (1 / 2))).toReal := ENNReal.toReal_nonneg
rw [Real.norm_eq_abs, abs_of_nonneg hnonneg]
have hgam := hgamma a ha
have hs := (sq_le_sq₀ (abs_nonneg (gamma a))
(by norm_num : (0 : ℝ) ≤ 1 / 4)).2 hgam
rw [sq_abs] at hs
nlinarith
constructor
· rw [← integral_const_mul]
apply integral_mono_ae (hsqInt.const_mul 2) hklInt
filter_upwards [hpi, hmem] with a hpa ha
simpa [mul_assoc] using mul_le_mul_of_nonneg_right
(bernoulli_kl_band (gamma a) (hgamma a ha)).1 hpa
· rw [← integral_const_mul]
apply integral_mono_ae hklInt (hsqInt.const_mul 4)
filter_upwards [hpi, hmem] with a hpa ha
simpa [mul_assoc] using mul_le_mul_of_nonneg_right
(bernoulli_kl_band (gamma a) (hgamma a ha)).2 hpa
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.EstimatorMeasurable 11 declarations Measurability of the realized total-Gram estimator
Measurability of the realized total-Gram estimator
theorem localCount_measurable
local count is measurable for the specified J input, the specified n input, the specified B input, the specified x input, the specified delta input, the specified h input.
Formal statement
hypotheses
Proof (Lean source)
lemma localCount_measurable {J n : ℕ} (B : SplitBlocks n) (x : Fin J)
(delta h : ℝ) : Measurable fun z : Fin n → ClampObs J =>
localCount B z x delta h := by
classical
unfold localCount
refine Finset.measurable_fun_sum _ fun i _ => Measurable.ite ?_ measurable_const measurable_const
exact (measurableSet_eq_fun
(measurable_obs_X.comp (measurable_pi_apply i)) measurable_const).inter
(measurableSet_Icc.preimage
(measurable_obs_A.comp (measurable_pi_apply i)))
theorem localGram_entry_measurable
local gram entry is measurable for the specified J input, the specified n input, the specified ell input, the specified B input, the specified x input, the specified r input, the specified s input, the specified delta input, the specified h input.
Formal statement
Proof (Lean source)
lemma localGram_entry_measurable {J n ell : ℕ} (B : SplitBlocks n)
(x : Fin J) (r s : Fin (ell + 1)) (delta h : ℝ) :
Measurable fun z : Fin n → ClampObs J => localGram B z x ell delta h r s := by
classical
simp only [localGram_apply]
refine Finset.measurable_fun_sum _ fun i _ => Measurable.ite ?_ ?_ measurable_const
· exact (measurableSet_eq_fun
(measurable_obs_X.comp (measurable_pi_apply i)) measurable_const).inter
(measurableSet_Icc.preimage
(((measurable_obs_A.comp (measurable_pi_apply i)).sub measurable_const).div_const h))
· have hd : Measurable (fun z : Fin n → ClampObs J => ((z i).A - delta) / h) :=
((measurable_obs_A.comp (measurable_pi_apply i)).sub measurable_const).div_const h
exact (hd.pow_const (r : ℕ)).mul (hd.pow_const (s : ℕ))
theorem localGram_measurable
local gram is measurable for the specified J input, the specified n input, the specified ell input, the specified B input, the specified x input, the specified delta input, the specified h input.
Formal statement
hypotheses
Proof (Lean source)
lemma localGram_measurable {J n ell : ℕ} (B : SplitBlocks n)
(x : Fin J) (delta h : ℝ) :
Measurable fun z : Fin n → ClampObs J => localGram B z x ell delta h := by
exact measurable_pi_lambda _ fun r => measurable_pi_lambda _ fun s =>
localGram_entry_measurable B x r s delta h
theorem localGram_isHermitian
the stated local gram is hermitian property holds for the specified J input, the specified n input, the specified ell input, the specified B input, the specified z input, the specified x input, the specified delta input, the specified h input.
Formal statement
hypotheses
conclusion
(localGram B z x ell delta h).IsHermitian
Proof (Lean source)
lemma localGram_isHermitian {J n ell : ℕ} (B : SplitBlocks n)
(z : Fin n → ClampObs J) (x : Fin J) (delta h : ℝ) :
(localGram B z x ell delta h).IsHermitian := by
exact designMatrix_isHermitian
(fun i => scaledDose delta h (z i)) (localKernelWeight B z x delta h)
theorem localGram_inv_measurable
local gram inv is measurable for the specified J input, the specified n input, the specified ell input, the specified B input, the specified x input, the specified delta input, the specified h input.
Formal statement
hypotheses
Proof (Lean source)
lemma localGram_inv_measurable {J n ell : ℕ} (B : SplitBlocks n)
(x : Fin J) (delta h : ℝ) :
Measurable fun z : Fin n → ClampObs J =>
(localGram B z x ell delta h)⁻¹ := by
classical
simp only [Matrix.inv_def]
have hG := localGram_measurable (ell := ell) B x delta h
have hdet : Measurable fun z : Fin n → ClampObs J =>
(localGram B z x ell delta h).det :=
continuous_id.matrix_det.measurable.comp hG
have hadj : Measurable fun z : Fin n → ClampObs J =>
(localGram B z x ell delta h).adjugate :=
continuous_id.matrix_adjugate.measurable.comp hG
convert (measurable_inv.comp hdet).smul hadj using 1
funext z
rw [Ring.inverse_eq_inv]
rfl
theorem goodGramEvent_measurable
good gram event is measurable for the specified J input, the specified n input, the specified ell input, the specified B input, the specified x input, the specified kappa input, the specified cminus input, the specified cplus input, the specified delta input, the specified h input.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma goodGramEvent_measurable {J n ell : ℕ} (B : SplitBlocks n)
(x : Fin J) (kappa cminus cplus delta h : ℝ) :
MeasurableSet {z : Fin n → ClampObs J |
GoodGramEvent B z x ell kappa cminus cplus delta h} := by
classical
let a : (Fin n → ClampObs J) → ℝ := fun z =>
lambdaStar ell kappa cminus cplus * (localCount B z x delta h : ℝ) / 2
let F : (Fin n → ClampObs J) → Matrix (Fin (ell + 1)) (Fin (ell + 1)) ℝ :=
fun z => localGram B z x ell delta h - a z • 1
have ha : Measurable a := by
dsimp [a]
exact (measurable_const.mul
((measurable_of_countable (fun m : ℕ => (m : ℝ))).comp
(localCount_measurable B x delta h))).div_const 2
have hF : Measurable F := by
dsimp [F]
exact measurable_pi_lambda _ fun r => measurable_pi_lambda _ fun s => by
simp only [Matrix.sub_apply, Matrix.smul_apply]
exact (localGram_entry_measurable B x r s delta h).sub
(ha.mul measurable_const)
have hpsd : MeasurableSet {A : Matrix (Fin (ell + 1)) (Fin (ell + 1)) ℝ |
A.PosSemidef} :=
Causalean.Mathlib.Analysis.isClosed_posSemidef.measurableSet
have heq : {z : Fin n → ClampObs J |
GoodGramEvent B z x ell kappa cminus cplus delta h} =
{z | 0 < localCount B z x delta h} ∩ F ⁻¹' {A | A.PosSemidef} := by
ext z
simp only [Set.mem_setOf_eq, Set.mem_inter_iff, Set.mem_preimage]
unfold GoodGramEvent
constructor
· rintro ⟨hcount, hquad⟩
refine ⟨hcount, Matrix.PosSemidef.of_dotProduct_mulVec_nonneg ?_ ?_⟩
· dsimp [F]
exact (localGram_isHermitian B z x delta h).sub
(Matrix.isHermitian_one.smul (star_trivial (a z)))
· intro v
have hv := hquad v
dsimp [F, a]
have hid : star v ⬝ᵥ
((localGram B z x ell delta h -
(lambdaStar ell kappa cminus cplus *
(localCount B z x delta h : ℝ) / 2) • 1) *ᵥ v) =
matrixQuadratic (localGram B z x ell delta h) v -
lambdaStar ell kappa cminus cplus *
(localCount B z x delta h : ℝ) / 2 * ∑ j, v j ^ 2 := by
rw [Matrix.sub_mulVec, dotProduct_sub, Matrix.smul_mulVec,
Matrix.one_mulVec]
simp only [matrixQuadratic, dotProduct]
congr 1
· simp [mulVec, dotProduct, Finset.mul_sum, mul_assoc]
· rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro j hj
simp
ring
rw [hid]
exact sub_nonneg.mpr hv
· rintro ⟨hcount, hpsd'⟩
refine ⟨hcount, fun v => ?_⟩
have hv := hpsd'.dotProduct_mulVec_nonneg v
dsimp [F, a] at hv
have hid : star v ⬝ᵥ
((localGram B z x ell delta h -
(lambdaStar ell kappa cminus cplus *
(localCount B z x delta h : ℝ) / 2) • 1) *ᵥ v) =
matrixQuadratic (localGram B z x ell delta h) v -
lambdaStar ell kappa cminus cplus *
(localCount B z x delta h : ℝ) / 2 * ∑ j, v j ^ 2 := by
rw [Matrix.sub_mulVec, dotProduct_sub, Matrix.smul_mulVec,
Matrix.one_mulVec]
simp only [matrixQuadratic, dotProduct]
congr 1
· simp [mulVec, dotProduct, Finset.mul_sum, mul_assoc]
· rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro j hj
simp
ring
rw [hid] at hv
exact sub_nonneg.mp hv
rw [heq]
exact (measurableSet_lt measurable_const
(localCount_measurable B x delta h)).inter
(hpsd.preimage hF)
theorem interceptWeight_measurable
intercept weight is measurable for the specified J input, the specified n input, the specified ell input, the specified B input, the specified x input, the specified kappa input, the specified cminus input, the specified cplus input, the specified delta input, the specified h input, the specified i input.
Formal statement
hypotheses
conclusion
=> interceptWeight B z x ell kappa cminus cplus delta h i
Proof (Lean source)
lemma interceptWeight_measurable {J n ell : ℕ} (B : SplitBlocks n)
(x : Fin J) (kappa cminus cplus delta h : ℝ) (i : Fin n) :
Measurable fun z : Fin n → ClampObs J =>
interceptWeight B z x ell kappa cminus cplus delta h i := by
classical
unfold interceptWeight
have hs := scaledDose_measurable (J := J) i delta h
refine Measurable.ite
(goodGramEvent_measurable B x kappa cminus cplus delta h) ?_ measurable_const
have hw : Measurable fun z : Fin n → ClampObs J =>
localKernelWeight B z x delta h i := by
unfold localKernelWeight
by_cases hi : i ∈ B.I2
· simp only [hi, true_and]
exact Measurable.ite
((measurableSet_eq_fun
(measurable_obs_X.comp (measurable_pi_apply i)) measurable_const).inter
(measurableSet_Icc.preimage hs)) measurable_const measurable_const
· simp [hi]
simp only [equivKernelWeight]
refine Finset.measurable_fun_sum _ fun j _ => ?_
exact (measurable_pi_apply j |>.comp
(measurable_pi_apply 0 |>.comp
(localGram_inv_measurable (ell := ell) B x delta h))).mul
(hw.mul (hs.pow_const (j : ℕ)))
theorem retainedEstimate_measurable
retained estimate is measurable for the specified J input, the specified n input, the specified B input, the specified delta input.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma retainedEstimate_measurable {J n : ℕ} (B : SplitBlocks n) (delta : ℝ) :
Measurable fun z : Fin n → ClampObs J => retainedEstimate B z delta := by
classical
unfold retainedEstimate blockAverage
refine measurable_const.mul (Finset.measurable_fun_sum _ fun i _ => ?_)
exact Measurable.ite
(measurableSet_lt measurable_const
(measurable_obs_A.comp (measurable_pi_apply i)))
(measurable_obs_Y.comp (measurable_pi_apply i)) measurable_const
theorem atomEstimate_measurable
atom estimate is measurable for the specified J input, the specified n input, the specified B input, the specified x input, the specified delta input.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma atomEstimate_measurable {J n : ℕ} (B : SplitBlocks n)
(x : Fin J) (delta : ℝ) :
Measurable fun z : Fin n → ClampObs J => atomEstimate B z x delta := by
classical
unfold atomEstimate blockAverage
refine measurable_const.mul (Finset.measurable_fun_sum _ fun i _ =>
Measurable.ite ?_ measurable_const measurable_const)
exact (measurableSet_eq_fun
(measurable_obs_X.comp (measurable_pi_apply i)) measurable_const).inter
(measurableSet_le
(measurable_obs_A.comp (measurable_pi_apply i)) measurable_const)
theorem localRegressionEstimate_measurable
local regression estimate is measurable for the specified J input, the specified n input, the specified ell input, the specified B input, the specified x input, the specified kappa input, the specified cminus input, the specified cplus input, the specified delta input, the specified h input.
Formal statement
hypotheses
conclusion
=> localRegressionEstimate B z x ell kappa cminus cplus delta h
Proof (Lean source)
lemma localRegressionEstimate_measurable {J n ell : ℕ} (B : SplitBlocks n)
(x : Fin J) (kappa cminus cplus delta h : ℝ) :
Measurable fun z : Fin n → ClampObs J =>
localRegressionEstimate B z x ell kappa cminus cplus delta h := by
classical
unfold localRegressionEstimate
refine Measurable.ite
(goodGramEvent_measurable B x kappa cminus cplus delta h) ?_ measurable_const
apply clampUnit_measurable.comp
refine Finset.measurable_fun_sum _ fun i _ => ?_
exact (interceptWeight_measurable B x kappa cminus cplus delta h i).mul
(measurable_obs_Y.comp (measurable_pi_apply i))
theorem totalGramEstimator_measurable
total gram estimator is measurable for the specified J input, the specified n input, the specified ell input, the specified B input, the specified kappa input, the specified cminus input, the specified cplus input, the specified delta input, the specified h input.
Formal statement
hypotheses
conclusion
=> totalGramEstimator B z ell kappa cminus cplus delta h
Proof (Lean source)
theorem totalGramEstimator_measurable {J n ell : ℕ} (B : SplitBlocks n)
(kappa cminus cplus delta h : ℝ) :
Measurable fun z : Fin n → ClampObs J =>
totalGramEstimator B z ell kappa cminus cplus delta h := by
classical
unfold totalGramEstimator
apply clampUnit_measurable.comp
refine (retainedEstimate_measurable B delta).add
(Finset.measurable_fun_sum _ fun x _ => ?_)
exact (atomEstimate_measurable B x delta).mul
(localRegressionEstimate_measurable B x kappa cminus cplus delta h)
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.FixedPositive 1 declarations This module closes the fixed-threshold branch from the moving-threshold rate lemmas.
Fixed-positive-threshold frontier
This module closes the fixed-threshold branch from the moving-threshold rate lemmas.
theorem fixed_positive_frontier_asymp
If the threshold converges to a positive constant, the actual frontier has the fixed-threshold nonparametric order. The result uses the hreg condition, the hdelta0 condition, the hdelta condition, the htend condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha delta0 :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
hdelta0 :
0 < delta0
deltaSeq :
ℕ → ℝ
hdelta :
ThresholdSequence deltaBar deltaSeq
conclusion
AsympSeq (fun n => clampFrontier n (deltaSeq n) kappa (infoBandwidth n (deltaSeq n) beta kappa deltaBar) beta) (fun n => (n : ℝ) ^ (-beta / (2 * beta + 1)))
Proof (Lean source)
-- @node: fixed_positive_frontier_asymp
lemma fixed_positive_frontier_asymp
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha delta0 : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(hdelta0 : 0 < delta0) (deltaSeq : ℕ → ℝ)
(hdelta : ThresholdSequence deltaBar deltaSeq)
(htend : Tendsto deltaSeq atTop (nhds delta0)) :
AsympSeq
(fun n => clampFrontier n (deltaSeq n) kappa
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) beta)
(fun n => (n : ℝ) ^ (-beta / (2 * beta + 1))) := by
rcases hreg with ⟨hJ, hbeta, hkappa, hL, hcminus, hcm, hcp, hpmin,
hpminJ, hdeltaBar_pos, hdeltaBar_lt, halpha, halpha_lt⟩
have hreg' : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha :=
⟨hJ, hbeta, hkappa, hL, hcminus, hcm, hcp, hpmin, hpminJ,
hdeltaBar_pos, hdeltaBar_lt, halpha, halpha_lt⟩
have hratio := fixed_positive_threshold_ratio_critical_top J beta kappa L cminus
cplus pmin deltaBar alpha delta0 hreg' hdelta0 deltaSeq htend
have hsuper := supercritical_frontier_asymp J beta kappa L cminus cplus pmin
deltaBar alpha hreg' deltaSeq hdelta hratio
let a : ℝ := (beta * kappa + 2 * beta + kappa + 1) / (2 * beta + 1)
have ha : 0 < a := by dsimp [a]; positivity
have hpow : Tendsto (fun n => (deltaSeq n) ^ a) atTop (nhds (delta0 ^ a)) :=
(Real.continuousAt_rpow_const delta0 a (inr ha.le)).tendsto.comp htend
have hbase : ∀ᶠ n : ℕ in atTop, 0 < (n : ℝ) ^ (-beta / (2 * beta + 1)) := by
filter_upwards [eventually_gt_atTop 0] with n hn
exact Real.rpow_pos_of_pos (by exact_mod_cast hn) _
have hinterior : AsympSeq
(fun n => (deltaSeq n) ^ (kappa + 1) *
((n : ℝ) * (deltaSeq n) ^ kappa) ^ (-beta / (2 * beta + 1)))
(fun n => (n : ℝ) ^ (-beta / (2 * beta + 1))) := by
apply asympSeq_of_tendsto_div_pos _ _ (delta0 ^ a)
(Real.rpow_pos_of_pos hdelta0 _)
· exact hbase
· refine hpow.congr' ?_
filter_upwards [eventually_gt_atTop 0,
htend (Ioi_mem_nhds hdelta0)] with n hn hd
have hnR : 0 < (n : ℝ) := by exact_mod_cast hn
have hexact := atom_interior_scale_exact (n : ℝ) (deltaSeq n) beta kappa
hnR hd hbeta
have hflat : ((n : ℝ) * (deltaSeq n) ^ kappa) ^
(-beta / (2 * beta + 1)) =
(((n : ℝ) * (deltaSeq n) ^ kappa) ^
(-(1 : ℝ) / (2 * beta + 1))) ^ beta := by
rw [← Real.rpow_mul
(mul_nonneg hnR.le (Real.rpow_nonneg hd.le kappa))]
congr 1
ring
rw [hflat, hexact]
rw [mul_div_cancel_left₀]
exact (Real.rpow_pos_of_pos hnR _).ne'
exact asympSeq_trans hsuper hinterior
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.HonestCoverage 4 declarations Finite-sample coverage of the bias-aware interval
Finite-sample coverage of the bias-aware interval
theorem honest_blockAverage_hoeffding
Hoeffding for a deterministic block, proved directly from the Causalean finite-product theorem and hence requiring no external theorem gate. The result uses the hf condition, the hf01 condition, the ht condition. This is the stated conclusion.
Formal statement
Proof (Lean source)
lemma honest_blockAverage_hoeffding
{X : Type} [MeasurableSpace X] (P : Measure X) [IsProbabilityMeasure P]
{n : ℕ} (I : Finset (Fin n)) (f : X → ℝ) (hf : Measurable f)
(hf01 : ∀ x, f x ∈ Icc (0 : ℝ) 1) (t : ℝ) (ht : 0 < t) :
(Measure.pi (fun _ : Fin n => P)).real
{z | |blockAverage I z f - ∫ x, f x ∂P| > t} ≤
2 * exp (-2 * (I.card : ℝ) * t ^ 2) := by
classical
by_cases hI : I.card = 0
· exact (measureReal_le_one (μ := Measure.pi fun _ : Fin n => P)
(s := {z | |blockAverage I z f - ∫ x, f x ∂P| > t})).trans (by
simp [hI])
let S := Causalean.Stat.iidSample_infinitePi P
let sumFin : (Fin n → X) → ℝ := fun z => ∑ i ∈ I, f (z i)
let sumInf : (ℕ → X) → ℝ :=
Causalean.Stat.Concentration.bernoulliCount S f I.card
let A : Set ℝ := {y | |(I.card : ℝ)⁻¹ * y - ∫ x, f x ∂P| > t}
have hA : MeasurableSet A := by
exact measurableSet_lt measurable_const
(((measurable_const.mul measurable_id).sub measurable_const).abs)
have hsumFin : Measurable sumFin := by
exact Finset.measurable_fun_sum I fun i _ => hf.comp (measurable_pi_apply i)
have hsumInf : Measurable sumInf := by
dsimp [sumInf]
exact Finset.measurable_fun_sum _ fun i _ => hf.comp (S.meas i)
have hlaw : (Measure.pi (fun _ : Fin n => P)).map sumFin =
(Measure.infinitePi (fun _ : ℕ => P)).map sumInf := by
calc
_ = (Measure.pi (fun _ : Fin I.card => P)).map
(fun z => ∑ j : Fin I.card, f (z j)) := block_count_law_eq P I f hf
_ = _ := range_count_transport P I.card f hf
have htail := hoeffding_abs_ge S hf
(a := 0) (b := 1) (by norm_num) (ae_of_all _ hf01) I.card
(Nat.pos_of_ne_zero hI) ht.le
have hfinite : (Measure.pi (fun _ : Fin n => P)).real (sumFin ⁻¹' A) =
((Measure.pi (fun _ : Fin n => P)).map sumFin).real A := by
simpa only [measureReal_def] using congrArg toReal
(Measure.map_apply hsumFin hA).symm
have hinf : (Measure.infinitePi (fun _ : ℕ => P)).real (sumInf ⁻¹' A) =
((Measure.infinitePi (fun _ : ℕ => P)).map sumInf).real A := by
simpa only [measureReal_def] using congrArg toReal
(Measure.map_apply hsumInf hA).symm
have hle : (Measure.infinitePi (fun _ : ℕ => P)).real (sumInf ⁻¹' A) ≤
2 * exp (-2 * (I.card : ℝ) * t ^ 2) := by
refine (measureReal_mono
(μ := Measure.infinitePi (fun _ : ℕ => P))
(s₁ := sumInf ⁻¹' A)
(s₂ := {w | t ≤ |S.sampleMean f I.card w - ∫ x, f x ∂P|}) ?_).trans ?_
· intro w hw
change t < |(I.card : ℝ)⁻¹ * sumInf w - ∫ x, f x ∂P| at hw
change t ≤ |S.sampleMean f I.card w - ∫ x, f x ∂P|
simpa [S, sumInf, sampleMean,
Causalean.Stat.Concentration.bernoulliCount] using le_of_lt hw
· simpa [S, sumInf, A, sampleMean] using htail
calc
(Measure.pi (fun _ : Fin n => P)).real
{z | |blockAverage I z f - ∫ x, f x ∂P| > t} =
(Measure.pi (fun _ : Fin n => P)).real (sumFin ⁻¹' A) := by rfl
_ = ((Measure.pi (fun _ : Fin n => P)).map sumFin).real A := hfinite
_ = ((Measure.infinitePi (fun _ : ℕ => P)).map sumInf).real A := by rw [hlaw]
_ = (Measure.infinitePi (fun _ : ℕ => P)).real (sumInf ⁻¹' A) := hinf.symm
_ ≤ _ := hle
theorem honest_radius_tail_le
The common logarithmic radius gives each of the 2J+1 bad events ample budget under the union bound. The result uses the hJ condition, the hm condition, the ha condition, the ha1 condition. This is the stated conclusion.
Formal statement
Proof (Lean source)
lemma honest_radius_tail_le (J m : ℕ) (alpha : ℝ)
(hJ : 0 < J) (hm : 0 < m) (ha : 0 < alpha) (ha1 : alpha < 1 / 2) :
2 * exp (-2 * (m : ℝ) *
(sqrt (log (12 * (J : ℝ) / alpha) / 2) /
sqrt (m : ℝ)) ^ 2) ≤ alpha / (6 * (J : ℝ)) := by
have hJR : 0 < (J : ℝ) := by positivity
have hmR : 0 < (m : ℝ) := by positivity
have hJone : (1 : ℝ) ≤ J := by exact_mod_cast hJ
have hq : 1 < 12 * (J : ℝ) / alpha := by
rw [lt_div_iff₀ ha]
have : alpha < 6 := lt_trans ha1 (by norm_num)
nlinarith
have hlog : 0 ≤ log (12 * (J : ℝ) / alpha) := (Real.log_pos hq).le
rw [div_pow, Real.sq_sqrt (by positivity : 0 ≤ (m : ℝ)),
Real.sq_sqrt (div_nonneg hlog (by norm_num : (0 : ℝ) ≤ 2))]
have he : -2 * (m : ℝ) *
((log (12 * (J : ℝ) / alpha) / 2) / (m : ℝ)) =
-log (12 * (J : ℝ) / alpha) := by field_simp
rw [he, Real.exp_neg, Real.exp_log (by positivity : 0 < 12 * (J : ℝ) / alpha)]
field_simp
norm_num
theorem honest_weighted_tail_le
honest weighted tail satisfies the stated upper bound for the specified J input, the specified alpha input, the specified hJ input, the specified ha input, the specified ha1 input.
Formal statement
Proof (Lean source)
lemma honest_weighted_tail_le (J : ℕ) (alpha : ℝ)
(hJ : 0 < J) (ha : 0 < alpha) (ha1 : alpha < 1 / 2) :
2 * exp (-2 *
(sqrt (log (12 * (J : ℝ) / alpha) / 2)) ^ 2) ≤
alpha / (6 * (J : ℝ)) := by
have hJR : 0 < (J : ℝ) := by positivity
have hJone : (1 : ℝ) ≤ J := by exact_mod_cast hJ
have hq : 1 < 12 * (J : ℝ) / alpha := by
rw [lt_div_iff₀ ha]
have : alpha < 6 := lt_trans ha1 (by norm_num)
nlinarith
have hlog : 0 ≤ log (12 * (J : ℝ) / alpha) := (Real.log_pos hq).le
rw [Real.sq_sqrt (div_nonneg hlog (by norm_num : (0 : ℝ) ≤ 2))]
have he : -2 * (log (12 * (J : ℝ) / alpha) / 2) =
-log (12 * (J : ℝ) / alpha) := by ring
rw [he, Real.exp_neg, Real.exp_log (by positivity : 0 < 12 * (J : ℝ) / alpha)]
field_simp
norm_num
theorem honestInterval_coverage_model
Every fixed model law is covered by the bias-aware interval with probability at least 1-alpha. The result uses the hmodel condition, the hJ condition, the hbeta condition, the hL condition, the ha condition, the ha1 condition, the hn condition, the hdelta condition, the hupper condition, the hh condition, the hlambda condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
B :
beta kappa L cminus cplus pmin delta h alpha :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hJ :
0 < J
hbeta :
0 < beta
hL :
0 < L
ha :
0 < alpha
ha1 :
alpha < 1 / 2
hn :
4 ≤ n
hdelta :
0 ≤ delta
hupper :
delta + h ≤ 1
hh :
0 < h
hlambda :
0 < lambdaStar (ellOf beta) kappa cminus cplus
conclusion
1 - alpha
≤ (iidProduct P n).real {z | clampFunctional P delta ∈ honestInterval B z (ellOf beta) beta kappa L cminus cplus delta h alpha}
Proof (Lean source)
lemma honestInterval_coverage_model
{J n : ℕ} (P : ClampLaw J) (B : SplitBlocks n)
(beta kappa L cminus cplus pmin delta h alpha : ℝ)
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hJ : 0 < J) (hbeta : 0 < beta) (hL : 0 < L)
(ha : 0 < alpha) (ha1 : alpha < 1 / 2)
(hn : 4 ≤ n) (hdelta : 0 ≤ delta) (hupper : delta + h ≤ 1)
(hh : 0 < h)
(hlambda : 0 < lambdaStar (ellOf beta) kappa cminus cplus) :
1 - alpha ≤ (iidProduct P n).real {z |
clampFunctional P delta ∈
honestInterval B z (ellOf beta) beta kappa L cminus cplus delta h alpha} := by
classical
letI : IsProbabilityMeasure P.dataMeasure := hmodel.probability
letI : IsProbabilityMeasure (iidProduct P n) := by unfold iidProduct; infer_instance
let t := sqrt (log (12 * (J : ℝ) / alpha) / 2)
let b0 := t / sqrt (B.I0.card : ℝ)
let b1 := t / sqrt (B.I1.card : ℝ)
let f0 : ClampObs J → ℝ := fun o => if delta < o.A then clampUnit o.Y else 0
let f1 : Fin J → ClampObs J → ℝ := fun x o =>
if o.X = x ∧ o.A ≤ delta then 1 else 0
let bad0 : Set (Fin n → ClampObs J) :=
{z | |blockAverage B.I0 z f0 - ∫ o, f0 o ∂P.dataMeasure| > b0}
let bad1 : Fin J → Set (Fin n → ClampObs J) := fun x =>
{z | |blockAverage B.I1 z (f1 x) - ∫ o, f1 x o ∂P.dataMeasure| > b1}
let bad2 : Fin J → Set (Fin n → ClampObs J) := fun x => {z |
GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h ∧
t * sqrt (∑ i ∈ B.I2,
(interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i) ^ 2) ≤
|∑ i ∈ B.I2, interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
((z i).Y - clampRegressionExtension P ((z i).X, (z i).A))|}
let bad := bad0 ∪ (⋃ x, bad1 x) ∪ (⋃ x, bad2 x)
let good : Set (Fin n → ClampObs J) := {z |
clampFunctional P delta ∈
honestInterval B z (ellOf beta) beta kappa L cminus cplus delta h alpha}
have hcard0 : 0 < B.I0.card := by have := B.card_I0; omega
have hcard1 : 0 < B.I1.card := by have := B.card_I1; omega
have ht : 0 < t := by
dsimp [t]
have hJR : (1 : ℝ) ≤ J := by exact_mod_cast hJ
have hq : 1 < 12 * (J : ℝ) / alpha := by
rw [lt_div_iff₀ ha]
nlinarith
exact Real.sqrt_pos.2 (div_pos (Real.log_pos hq) (by norm_num))
have hb0pos : 0 < b0 := div_pos ht (Real.sqrt_pos.2 (by positivity))
have hb1pos : 0 < b1 := div_pos ht (Real.sqrt_pos.2 (by positivity))
have hf0 : Measurable f0 := by
exact Measurable.ite
(measurableSet_lt measurable_const
(measurable_fst.comp (measurable_snd.comp (Measurable.of_comap_le le_rfl))))
(clampUnit_measurable'.comp clampOutcome_measurable) measurable_const
have hf1 : ∀ x, Measurable (f1 x) := by
intro x
exact Measurable.ite
((measurableSet_eq_fun
(measurable_fst.comp (Measurable.of_comap_le le_rfl)) measurable_const).inter
(measurableSet_le
(measurable_fst.comp (measurable_snd.comp (Measurable.of_comap_le le_rfl)))
measurable_const)) measurable_const measurable_const
have hf001 : ∀ o, f0 o ∈ Icc (0 : ℝ) 1 := by
intro o; dsimp [f0]; split_ifs
· exact clampUnit_mem_Icc _
· norm_num
have hf101 : ∀ x o, f1 x o ∈ Icc (0 : ℝ) 1 := by
intro x o; dsimp [f1]; split_ifs <;> norm_num
have hbad0 : (iidProduct P n).real bad0 ≤ alpha / (6 * (J : ℝ)) := by
have hcal := honest_radius_tail_le J (B.I0.card) alpha hJ hcard0 ha ha1
exact (honest_blockAverage_hoeffding P.dataMeasure B.I0 f0 hf0 hf001 b0
hb0pos).trans (by simpa [b0, t] using hcal)
have hbad1 : ∀ x, (iidProduct P n).real (bad1 x) ≤ alpha / (6 * (J : ℝ)) := by
intro x
have hcal := honest_radius_tail_le J (B.I1.card) alpha hJ hcard1 ha ha1
exact (honest_blockAverage_hoeffding P.dataMeasure B.I1 (f1 x) (hf1 x)
(fun o => hf101 x o) b1 hb1pos).trans (by simpa [b1, t] using hcal)
have hbad2 : ∀ x, (iidProduct P n).real (bad2 x) ≤ alpha / (6 * (J : ℝ)) := by
intro x
exact (block_weighted_tail_on_good_gram P hmodel B x delta h t
⟨0, by omega⟩ hlambda ht.le).trans (by
simpa [t] using honest_weighted_tail_le J alpha hJ ha ha1)
have hm0 : ∫ o, f0 o ∂P.dataMeasure = retainedMean P delta := by
calc
_ = ∫ o, (if delta < o.A then o.Y else 0) ∂P.dataMeasure := by
apply integral_congr_ae
filter_upwards [hmodel.outcomeSupport] with o ho
simp [f0, clampUnit, ho.1, ho.2]
_ = _ := by
unfold retainedMean
apply integral_congr_ae
filter_upwards with o
by_cases hd : delta < o.A <;> simp [indicator, hd]
have hm1 : ∀ x, ∫ o, f1 x o ∂P.dataMeasure = P.px x * atomMass P x delta := by
intro x
simpa [f1] using atomEvent_integral_eq P hmodel x (by linarith)
have hsub : ∀ᵐ z ∂iidProduct P n, z ∈ goodᶜ → z ∈ bad := by
filter_upwards [iidProduct_outcomeSupport P hmodel] with z hz
intro hzg
by_contra hzbad
have hnb0 : z ∉ bad0 := fun h => hzbad (inl (inl h))
have hnb1 : ∀ x, z ∉ bad1 x := by
intro x hx
exact hzbad (inl (inr (Set.mem_iUnion.2 ⟨x, hx⟩)))
have hnb2 : ∀ x, z ∉ bad2 x := by
intro x hx
exact hzbad (inr (Set.mem_iUnion.2 ⟨x, hx⟩))
have hret : |retainedEstimate B z delta - retainedMean P delta| ≤ b0 := by
have hnb0' := hnb0
dsimp [bad0] at hnb0'
have := le_of_not_gt hnb0'
have hre : retainedEstimate B z delta = blockAverage B.I0 z f0 := by
unfold retainedEstimate blockAverage
congr 1
apply Finset.sum_congr rfl
intro i hi
simp [f0, clampUnit, (hz i).1, (hz i).2]
simpa [hre, hm0] using this
have hatom : ∀ x, |atomEstimate B z x delta - P.px x * atomMass P x delta| ≤ b1 := by
intro x
have hxnot := hnb1 x
dsimp [bad1] at hxnot
have hx := le_of_not_gt hxnot
simpa [bad1, f1, atomEstimate, hm1 x] using hx
have hlocal : ∀ x,
GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h →
|localRegressionEstimate B z x (ellOf beta) kappa cminus cplus delta h -
P.mu x delta| ≤
L * h ^ beta * ∑ i ∈ B.I2,
|interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i| +
t * sqrt (∑ i ∈ B.I2,
(interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i) ^ 2) := by
intro x hxg
have hnoise : |∑ i ∈ B.I2,
interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
((z i).Y - clampRegressionExtension P ((z i).X, (z i).A))| ≤
t * sqrt (∑ i ∈ B.I2,
(interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i) ^ 2) := by
exact le_of_not_ge (fun hge => hnb2 x ⟨hxg, hge⟩)
exact (localRegression_good_error_le_empirical P hmodel hbeta hL B z x
hdelta hh hupper hlambda hxg).trans (add_le_add_right hnoise _)
have hmem := clampFunctional_mem_honestInterval_of_deviations P hmodel
hdelta (by linarith) hh hbeta.le hL.le ht.le hb0pos.le hb1pos.le B z rfl rfl rfl
hret hatom hlocal
exact hzg hmem
have hbad : (iidProduct P n).real bad ≤ alpha := by
calc
_ ≤ (iidProduct P n).real bad0 +
(iidProduct P n).real (⋃ x, bad1 x) +
(iidProduct P n).real (⋃ x, bad2 x) := by
exact (measureReal_union_le _ _).trans (add_le_add_left (measureReal_union_le _ _) _)
_ ≤ alpha / (6 * (J : ℝ)) +
∑ x : Fin J, alpha / (6 * (J : ℝ)) +
∑ x : Fin J, alpha / (6 * (J : ℝ)) := by
gcongr
· exact (measureReal_iUnion_fintype_le bad1).trans
(Finset.sum_le_sum fun x _ => hbad1 x)
· exact (measureReal_iUnion_fintype_le bad2).trans
(Finset.sum_le_sum fun x _ => hbad2 x)
_ ≤ alpha := by
have hJR : 0 < (J : ℝ) := by positivity
have hJone : (1 : ℝ) ≤ J := by exact_mod_cast hJ
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
field_simp
nlinarith
have hfail : (iidProduct P n).real goodᶜ ≤ alpha := by
exact (ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono_ae hsub)).trans hbad
have hgoodMeas : MeasurableSet good := by
dsimp [good]
simp only [honestInterval]
let center := fun z : Fin n → ClampObs J =>
totalGramEstimator B z (ellOf beta) kappa cminus cplus delta h
let radius := fun z : Fin n → ClampObs J =>
sqrt (log (12 * (J : ℝ) / alpha) / 2) /
sqrt (B.I0.card : ℝ) +
∑ x : Fin J, stratumRadius B z x (ellOf beta) beta kappa L
cminus cplus delta h
(sqrt (log (12 * (J : ℝ) / alpha) / 2))
(sqrt (log (12 * (J : ℝ) / alpha) / 2) /
sqrt (B.I1.card : ℝ))
have hc : Measurable center := totalGramEstimator_measurable B
kappa cminus cplus delta h
have hr : Measurable radius := by
exact measurable_const.add (Finset.measurable_fun_sum _ fun x _ =>
honest_stratumRadius_measurable B x beta kappa L cminus cplus delta h
(sqrt (log (12 * (J : ℝ) / alpha) / 2))
(sqrt (log (12 * (J : ℝ) / alpha) / 2) /
sqrt (B.I1.card : ℝ)))
exact (measurableSet_le (measurable_const.max (hc.sub hr)) measurable_const).inter
(measurableSet_le measurable_const (measurable_const.min (hc.add hr)))
have hadd := measureReal_add_measureReal_compl (μ := iidProduct P n) hgoodMeas
have huniv : (iidProduct P n).real univ = 1 := by simp
rw [huniv] at hadd
change 1 - alpha ≤ (iidProduct P n).real good
linarith
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.HonestExpectedLength 9 declarations This module reduces the expected length of the clipped honest interval to its displayed sample-dependent radius.
Expected-length reductions for the bias-aware interval
This module reduces the expected length of the clipped honest interval to its displayed sample-dependent radius. It also records the exact mean of the atom-frequency estimate, the population input needed to bound that radius.
theorem integral_sqrt_le_sqrt_integral
On a probability space, integrating the square root of a nonnegative integrable function is bounded by the square root of its integral. The result uses the hf condition, the hf0 condition, the hfInt condition. This is the stated conclusion.
Formal statement
hypotheses
Proof (Lean source)
lemma integral_sqrt_le_sqrt_integral
{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
(f : Ω → ℝ) (hf : Measurable f) (hf0 : ∀ ω, 0 ≤ f ω)
(hfInt : Integrable f μ) :
(∫ ω, sqrt (f ω) ∂μ) ≤ sqrt (∫ ω, f ω ∂μ) := by
have hsMeas : AEStronglyMeasurable (fun ω => sqrt (f ω)) μ :=
hf.sqrt.aestronglyMeasurable
have hsSqInt : Integrable (fun ω => (sqrt (f ω)) ^ 2) μ := by
simpa only [Real.sq_sqrt (hf0 _)] using hfInt
have hsLp : MemLp (fun ω => sqrt (f ω)) 2 μ :=
(memLp_two_iff_integrable_sq hsMeas).mpr hsSqInt
have hcs :=
integral_abs_le_sqrt_integral_sq
μ (fun ω => sqrt (f ω)) hsLp
simpa only [Real.norm_eq_abs, abs_of_nonneg (Real.sqrt_nonneg _),
Real.sq_sqrt (hf0 _)] using hcs
theorem integral_mul_I1_I2
Statistics computed from the atom block and local-regression block factor under the canonical product law. The result uses the hprob condition, the F condition, the G condition, the hF condition, the hG condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
B :
hprob :
IsProbabilityMeasure P.dataMeasure
F :
((i : {i // i ∈ B.I1}) → ClampObs J) → ℝ
G :
((i : {i // i ∈ B.I2}) → ClampObs J) → ℝ
hF :
hG :
conclusion
(∫ z, F (finsetCoordProj B.I1 z) * G (finsetCoordProj B.I2 z) ∂iidProduct P n)
= (∫ z, F (finsetCoordProj B.I1 z) ∂iidProduct P n) * ∫ z, G (finsetCoordProj B.I2 z) ∂iidProduct P n
Proof (Lean source)
lemma integral_mul_I1_I2
{J n : ℕ} (P : ClampLaw J) (B : SplitBlocks n)
(hprob : IsProbabilityMeasure P.dataMeasure)
(F : ((i : {i // i ∈ B.I1}) → ClampObs J) → ℝ)
(G : ((i : {i // i ∈ B.I2}) → ClampObs J) → ℝ)
(hF : Measurable F) (hG : Measurable G) :
(∫ z, F (finsetCoordProj B.I1 z) *
G (finsetCoordProj B.I2 z) ∂iidProduct P n) =
(∫ z, F (finsetCoordProj B.I1 z) ∂iidProduct P n) *
∫ z, G (finsetCoordProj B.I2 z) ∂iidProduct P n := by
letI : IsProbabilityMeasure P.dataMeasure := hprob
have hi : iIndepFun
(fun (i : Fin n) (z : Fin n → ClampObs J) => z i) (iidProduct P n) := by
unfold iidProduct
exact ProbabilityTheory.iIndepFun_pi
(X := fun _ : Fin n => id) (fun _ => measurable_id.aemeasurable)
have hind := hi.indepFun_finset B.I1 B.I2 B.disjoint12
(fun i => measurable_pi_apply i)
change (∫ z, F (fun i => z i.1) * G (fun i => z i.1) ∂iidProduct P n) =
(∫ z, F (fun i => z i.1) ∂iidProduct P n) *
∫ z, G (fun i => z i.1) ∂iidProduct P n
exact hind.integral_fun_comp_mul_comp
(measurable_finsetCoordProj B.I1).aemeasurable
(measurable_finsetCoordProj B.I2).aemeasurable
hF.aestronglyMeasurable hG.aestronglyMeasurable
def honestRadius
The displayed radius of the bias-aware interval.
Definition (Lean source)
parameters
n :
ℕ
J :
ℕ
B :
beta kappa L cminus cplus delta h alpha :
ℝ
def
honestRadius B z beta kappa L cminus cplus delta h alpha :
ℝ
theorem honestRadius_nonneg
Under the regime restrictions and nonnegative bandwidth, the displayed honest radius is nonnegative. The result uses the hreg condition, the hn condition, the hh condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
B :
beta kappa L cminus cplus pmin deltaBar alpha delta h :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
hn :
4 ≤ n
hh :
0 ≤ h
conclusion
0 ≤ honestRadius B z beta kappa L cminus cplus delta h alpha
Proof (Lean source)
lemma honestRadius_nonneg
(J n : ℕ) (B : SplitBlocks n)
(beta kappa L cminus cplus pmin deltaBar alpha delta h : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(hn : 4 ≤ n) (hh : 0 ≤ h) (z : Fin n → ClampObs J) :
0 ≤ honestRadius B z beta kappa L cminus cplus delta h alpha := by
simpa only [honestRadius] using
honestInterval_radius_nonneg J n B beta kappa L cminus cplus pmin
deltaBar alpha delta h hreg hn hh z
theorem honestInterval_length_le_two_radius
Clipping the honest interval to the outcome range cannot make it longer than twice its displayed radius. The result uses the hreg condition, the hn condition, the hh condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
B :
beta kappa L cminus cplus pmin deltaBar alpha delta h :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
hn :
4 ≤ n
hh :
0 ≤ h
conclusion
≤ 2 * honestRadius B z beta kappa L cminus cplus delta h alpha
Proof (Lean source)
lemma honestInterval_length_le_two_radius
(J n : ℕ) (B : SplitBlocks n)
(beta kappa L cminus cplus pmin deltaBar alpha delta h : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(hn : 4 ≤ n) (hh : 0 ≤ h) (z : Fin n → ClampObs J) :
intervalLength
(honestInterval B z (ellOf beta) beta kappa L cminus cplus delta h alpha) ≤
2 * honestRadius B z beta kappa L cminus cplus delta h alpha := by
let center := totalGramEstimator B z (ellOf beta) kappa cminus cplus delta h
let radius := honestRadius B z beta kappa L cminus cplus delta h alpha
have hr : 0 ≤ radius := by
simpa only [radius] using honestRadius_nonneg J n B beta kappa L cminus
cplus pmin deltaBar alpha delta h hreg hn hh z
have hmin : min 1 (center + radius) ≤ center + radius := min_le_right _ _
have hmax : center - radius ≤ max 0 (center - radius) := le_max_right _ _
simp only [honestInterval, honest_intervalLength_Icc_eq]
change max 0
(min 1 (center + radius) - max 0 (center - radius)) ≤ 2 * radius
apply max_le
· positivity
· linarith
theorem honestInterval_modelSup_le
A uniform modelwise expected-length bound passes through the real supremum convention used for the concrete honest interval. The result uses the hreg condition, the hupper condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
B :
beta kappa L cminus cplus pmin deltaBar alpha delta h U :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
hupper :
∀ (P : ClampLaw J)
if
ClampModel P beta kappa L cminus cplus pmin
then
(∫ z, intervalLength (honestInterval B z (ellOf beta) beta kappa L cminus cplus delta h alpha) ∂iidProduct P n)
≤ U
conclusion
sSup {v : ℝ | ∃ P : ClampLaw J, ClampModel P beta kappa L cminus cplus pmin ∧ IidSampling P n ∧ v = ∫ z, intervalLength (honestInterval B z (ellOf beta) beta kappa L cminus cplus delta h alpha) ∂iidProduct P n}
≤ U
Proof (Lean source)
lemma honestInterval_modelSup_le
(J n : ℕ) (B : SplitBlocks n)
(beta kappa L cminus cplus pmin deltaBar alpha delta h U : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(hupper : ∀ (P : ClampLaw J),
ClampModel P beta kappa L cminus cplus pmin →
(∫ z, intervalLength
(honestInterval B z (ellOf beta) beta kappa L cminus cplus delta h alpha)
∂iidProduct P n) ≤ U) :
sSup {v : ℝ | ∃ P : ClampLaw J,
ClampModel P beta kappa L cminus cplus pmin ∧ IidSampling P n ∧
v = ∫ z, intervalLength
(honestInterval B z (ellOf beta) beta kappa L cminus cplus delta h alpha)
∂iidProduct P n} ≤ U := by
apply csSup_le
· let P0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
have hP0 : ClampModel P0 beta kappa L cminus cplus pmin :=
minimaxCenter_mem_model J beta kappa L cminus cplus pmin deltaBar alpha hreg
exact ⟨(∫ z, intervalLength
(honestInterval B z (ellOf beta) beta kappa L cminus cplus delta h alpha)
∂iidProduct P0 n), P0, hP0, clampModel_iidSampling hP0, rfl⟩
· intro v hv
rcases hv with ⟨P, hP, _hsampling, rfl⟩
exact hupper P hP
theorem integral_atomEstimate_le_envelope
The mean empirical atom frequency is bounded by its population thinning envelope plus the root-block fluctuation scale. The result uses the hmodel condition, the hreg condition, the hcard condition, the hdelta condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
B :
x :
Fin J
beta kappa L cminus cplus pmin deltaBar alpha delta :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
hcard :
0 < B.I1.card
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
conclusion
(∫ z, atomEstimate B z x delta ∂iidProduct P n)
≤ (1 / 2 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹ + cplus * delta ^ (kappa + 1) / (kappa + 1)
Proof (Lean source)
lemma integral_atomEstimate_le_envelope
{J n : ℕ} (P : ClampLaw J) (B : SplitBlocks n) (x : Fin J)
(beta kappa L cminus cplus pmin deltaBar alpha delta : ℝ)
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(hcard : 0 < B.I1.card) (hdelta : delta ∈ Icc (0 : ℝ) deltaBar) :
(∫ z, atomEstimate B z x delta ∂iidProduct P n) ≤
(1 / 2 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹ +
cplus * delta ^ (kappa + 1) / (kappa + 1) := by
letI : IsProbabilityMeasure P.dataMeasure := hmodel.probability
letI : IsProbabilityMeasure (iidProduct P n) := by
unfold iidProduct
infer_instance
let m := P.px x * atomMass P x delta
let f := fun z : Fin n → ClampObs J => atomEstimate B z x delta
have hfMeas : Measurable f := atomEstimate_measurable B x delta
have hfInt : Integrable f (iidProduct P n) := by
refine Integrable.of_bound hfMeas.aestronglyMeasurable 1 ?_
filter_upwards with z
have hz := atomEstimate_mem_Icc B z x delta
rw [Real.norm_eq_abs, abs_of_nonneg hz.1]
exact hz.2
have hd1 : delta ≤ 1 := hdelta.2.trans (by
rcases hreg with ⟨_, _, _, _, _, _, _, _, _, _, hdBar, _, _⟩
exact hdBar.le)
have hdiffInt : Integrable (fun z => |f z - m|) (iidProduct P n) := by
exact (hfInt.sub (integrable_const m)).abs
have hpoint : ∀ z, f z ≤ |f z - m| + m := by
intro z
linarith [le_abs_self (f z - m)]
have hint : (∫ z, f z ∂iidProduct P n) ≤
(∫ z, |f z - m| ∂iidProduct P n) + m := by
calc
_ ≤ ∫ z, (|f z - m| + m) ∂iidProduct P n :=
integral_mono hfInt (hdiffInt.add (integrable_const m)) hpoint
_ = _ := by rw [integral_add hdiffInt (integrable_const m)]; simp
have hfluct : (∫ z, |f z - m| ∂iidProduct P n) ≤
(1 / 2 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹ := by
simpa [f, m] using atomEstimate_l1_le P hmodel B hcard x hdelta.1 hd1
have hmean : m ≤ cplus * delta ^ (kappa + 1) / (kappa + 1) := by
have habs := abs_atomCoefficient_le_envelope J P hmodel hreg x hdelta
have hm0 := atomCoefficient_mem_Icc P hmodel x hd1 |>.1
simpa [m, abs_of_nonneg hm0] using habs
exact hint.trans (add_le_add hfluct hmean)
theorem integral_sqrt_interceptWeight_energy_le
The expected square root of the stabilized local-weight energy is bounded by the square root of the existing integrated energy envelope. The result uses the hsampling condition, the hh condition, the hlambda condition, the hp condition, the hcard condition, the hmean condition. This is the stated conclusion.
Formal statement
hypotheses
J n ell :
ℕ
P :
ClampLaw J
B :
x :
Fin J
kappa cminus cplus delta h p :
ℝ
hsampling :
IidSampling P n
hh :
0 < h
hlambda :
0 < lambdaStar ell kappa cminus cplus
hp :
0 < p
hcard :
0 < B.I2.card
hmean :
∫ o, localWindowWeight x delta h o ∂P.dataMeasure = p
conclusion
(∫ z, sqrt (∑ i ∈ B.I2, (interceptWeight B z x ell kappa cminus cplus delta h i) ^ 2) ∂iidProduct P n)
≤ sqrt (2 * (4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2) / ((B.I2.card : ℝ) * p) + 2 * (4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2) * exp (-((B.I2.card : ℝ) * p) / 20))
Proof (Lean source)
lemma integral_sqrt_interceptWeight_energy_le
{J n ell : ℕ} (P : ClampLaw J) (B : SplitBlocks n) (x : Fin J)
(kappa cminus cplus delta h p : ℝ)
(hsampling : IidSampling P n) (hh : 0 < h)
(hlambda : 0 < lambdaStar ell kappa cminus cplus)
(hp : 0 < p) (hcard : 0 < B.I2.card)
(hmean : ∫ o, localWindowWeight x delta h o ∂P.dataMeasure = p) :
(∫ z, sqrt (∑ i ∈ B.I2,
(interceptWeight B z x ell kappa cminus cplus delta h i) ^ 2)
∂iidProduct P n) ≤
sqrt
(2 * (4 * (ell + 1 : ℝ) /
lambdaStar ell kappa cminus cplus ^ 2) /
((B.I2.card : ℝ) * p) +
2 * (4 * (ell + 1 : ℝ) /
lambdaStar ell kappa cminus cplus ^ 2) *
exp (-((B.I2.card : ℝ) * p) / 20)) := by
letI : IsProbabilityMeasure P.dataMeasure := hsampling.1
letI : IsProbabilityMeasure (iidProduct P n) := by
unfold iidProduct
infer_instance
let energy : (Fin n → ClampObs J) → ℝ := fun z => ∑ i ∈ B.I2,
(interceptWeight B z x ell kappa cminus cplus delta h i) ^ 2
have heMeas : Measurable energy := by
exact Finset.measurable_fun_sum B.I2 fun i _ =>
(interceptWeight_measurable (ell := ell) B x kappa cminus cplus delta h i).pow_const 2
have he0 : ∀ z, 0 ≤ energy z := fun z => sum_nonneg fun _ _ => sq_nonneg _
have heInt : Integrable energy (iidProduct P n) := by
have hbase := localRegression_weight_energy_integrable B P x kappa cminus
cplus delta h hh hlambda
apply hbase.congr
filter_upwards with z
dsimp only [energy]
simp_rw [localRegressionDesignWeight_apply (ell := ell), ite_pow,
zero_pow (by norm_num : 2 ≠ 0)]
rw [← Finset.sum_filter]
simp
have hsqrt := integral_sqrt_le_sqrt_integral (iidProduct P n) energy heMeas he0 heInt
refine hsqrt.trans (Real.sqrt_le_sqrt ?_)
calc
(∫ z, energy z ∂iidProduct P n) =
∫ z, ∑ i,
(localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h
(designVector
(fun o : ClampObs J => (o.X, o.A)) z) i) ^ 2 ∂iidProduct P n := by
apply integral_congr_ae
filter_upwards with z
dsimp only [energy]
simp_rw [localRegressionDesignWeight_apply (ell := ell), ite_pow,
zero_pow (by norm_num : 2 ≠ 0)]
rw [← Finset.sum_filter]
simp
_ ≤ _ := localRegression_weight_energy_integral_le P hsampling B x
kappa cminus cplus delta h p hh hlambda hp hcard hmean
theorem stratumRadius_le_good_expression_add_bad
Each per-stratum radius is bounded by a good-Gram bias/noise expression plus the indicator of Gram failure. The result uses the hbeta condition, the hL condition, the hh condition, the ht condition, the hb1 condition, the hlambda condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
B :
x :
Fin J
beta kappa L cminus cplus delta h t b1 :
ℝ
hbeta :
0 ≤ beta
hL :
0 ≤ L
hh :
0 < h
ht :
0 ≤ t
hb1 :
0 ≤ b1
hlambda :
0 < lambdaStar (ellOf beta) kappa cminus cplus
conclusion
stratumRadius B z x (ellOf beta) beta kappa L cminus cplus delta h t b1
≤ (atomEstimate B z x delta + b1) * (L * h ^ beta * (2 * sqrt (ellOf beta + 1 : ℝ) / lambdaStar (ellOf beta) kappa cminus cplus) + t * sqrt (∑ i ∈ B.I2, (interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i) ^ 2))
+ b1
+ {w : Fin n → ClampObs J | ¬ GoodGramEvent B w x (ellOf beta) kappa cminus cplus delta h}.indicator (fun _ => (1 : ℝ)) z
Proof (Lean source)
lemma stratumRadius_le_good_expression_add_bad
{J n : ℕ} (B : SplitBlocks n) (z : Fin n → ClampObs J) (x : Fin J)
(beta kappa L cminus cplus delta h t b1 : ℝ)
(hbeta : 0 ≤ beta) (hL : 0 ≤ L) (hh : 0 < h)
(ht : 0 ≤ t) (hb1 : 0 ≤ b1)
(hlambda : 0 < lambdaStar (ellOf beta) kappa cminus cplus) :
stratumRadius B z x (ellOf beta) beta kappa L cminus cplus delta h t b1 ≤
(atomEstimate B z x delta + b1) *
(L * h ^ beta *
(2 * sqrt (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus) +
t * sqrt (∑ i ∈ B.I2,
(interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i) ^ 2)) +
b1 +
{w : Fin n → ClampObs J |
¬ GoodGramEvent B w x (ellOf beta) kappa cminus cplus delta h}.indicator
(fun _ => (1 : ℝ)) z := by
have hatom := atomEstimate_mem_Icc B z x delta
have hfac : 0 ≤ atomEstimate B z x delta + b1 := by linarith [hatom.1]
have hbias : 0 ≤ L * h ^ beta *
(2 * sqrt (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus) := by positivity
have hnoise : 0 ≤ t * sqrt (∑ i ∈ B.I2,
(interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i) ^ 2) :=
mul_nonneg ht (Real.sqrt_nonneg _)
by_cases hg : GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h
· rw [stratumRadius, if_pos hg, Set.indicator_of_notMem]
· simp only [add_zero]
have hinner :
L * h ^ beta * ∑ i ∈ B.I2,
|interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i| +
t * sqrt (∑ i ∈ B.I2,
(interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i) ^ 2) ≤
L * h ^ beta *
(2 * sqrt (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus) +
t * sqrt (∑ i ∈ B.I2,
(interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i) ^ 2) :=
add_le_add (mul_le_mul_of_nonneg_left
(goodGram_weight_controls B z x kappa cminus cplus delta h hh hlambda hg).2.1
(mul_nonneg hL (Real.rpow_nonneg hh.le _))) le_rfl
exact add_le_add (mul_le_mul_of_nonneg_left hinner hfac) le_rfl
· simpa using hg
· rw [stratumRadius, if_neg hg, Set.indicator_of_mem]
· nlinarith [mul_nonneg hfac (add_nonneg hbias hnoise), hatom.2]
· simpa using hg
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.HonestIntervalBasic 7 declarations Measurability and elementary length bounds for the honest interval
Measurability and elementary length bounds for the honest interval
theorem honest_intervalLength_Icc_eq
The convex-hull length of a nonempty closed interval is its nonnegative endpoint difference. This is the stated conclusion.
Formal statement
Proof (Lean source)
lemma honest_intervalLength_Icc_eq (a b : ℝ) :
intervalLength (Icc a b) = max 0 (b - a) := by
unfold intervalLength
by_cases h : a ≤ b
· simp [h]
· have h' : b < a := lt_of_not_ge h
simp [Set.Icc_eq_empty h, h'.le]
theorem honest_stratumRadius_measurable
The sample-dependent per-stratum honest radius is measurable. This is the stated conclusion.
Formal statement
hypotheses
conclusion
=> stratumRadius B z x ell beta kappa L cminus cplus delta h tAlpha b1
Proof (Lean source)
lemma honest_stratumRadius_measurable {J n ell : ℕ} (B : SplitBlocks n)
(x : Fin J) (beta kappa L cminus cplus delta h tAlpha b1 : ℝ) :
Measurable fun z : Fin n → ClampObs J =>
stratumRadius B z x ell beta kappa L cminus cplus delta h tAlpha b1 := by
classical
unfold stratumRadius
refine Measurable.ite
(goodGramEvent_measurable B x kappa cminus cplus delta h) ?_ ?_
· have habs := Finset.measurable_fun_sum B.I2 fun i _ =>
(interceptWeight_measurable (ell := ell) B x kappa cminus cplus delta h i).abs
have hsq := Finset.measurable_fun_sum B.I2 fun i _ =>
(interceptWeight_measurable (ell := ell) B x kappa cminus cplus delta h i).pow_const 2
exact (((atomEstimate_measurable B x delta).add measurable_const).mul
(((measurable_const.mul measurable_const).mul habs).add
(measurable_const.mul hsq.sqrt))).add measurable_const
· exact (atomEstimate_measurable B x delta).add measurable_const
theorem honestInterval_radius_nonneg
Once the split blocks are nonempty, the displayed honest radius is nonnegative under the regime restrictions. The result uses the hreg condition, the hn condition, the hh condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
B :
beta kappa L cminus cplus pmin deltaBar alpha delta h :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
hn :
4 ≤ n
hh :
0 ≤ h
Proof (Lean source)
lemma honestInterval_radius_nonneg
(J n : ℕ) (B : SplitBlocks n)
(beta kappa L cminus cplus pmin deltaBar alpha delta h : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(hn : 4 ≤ n) (hh : 0 ≤ h) :
∀ z : Fin n → ClampObs J,
let tAlpha := sqrt (log (12 * (J : ℝ) / alpha) / 2)
let b0 := tAlpha / sqrt (B.I0.card : ℝ)
let b1 := tAlpha / sqrt (B.I1.card : ℝ)
0 ≤ b0 + ∑ x : Fin J,
stratumRadius B z x (ellOf beta) beta kappa L cminus cplus delta h
tAlpha b1 := by
intro z
dsimp only
rcases hreg with ⟨hJ, hbeta, hkappa, hL, hcminus, hcminus_le, hcplus,
hpmin, hpmin_le, hdeltaBar, hdeltaBar1, ha, ha_half⟩
have ht : 0 ≤ sqrt (log (12 * (J : ℝ) / alpha) / 2) :=
Real.sqrt_nonneg _
have hI0 : 0 < (B.I0.card : ℝ) := by
have hc := B.card_I0
exact_mod_cast (show 0 < B.I0.card by omega)
have hI1 : 0 < (B.I1.card : ℝ) := by
have hc := B.card_I1
exact_mod_cast (show 0 < B.I1.card by omega)
have hb0 : 0 ≤ sqrt (log (12 * (J : ℝ) / alpha) / 2) /
sqrt (B.I0.card : ℝ) := div_nonneg ht (Real.sqrt_nonneg _)
have hb1 : 0 ≤ sqrt (log (12 * (J : ℝ) / alpha) / 2) /
sqrt (B.I1.card : ℝ) := div_nonneg ht (Real.sqrt_nonneg _)
apply add_nonneg hb0
apply sum_nonneg
intro x hx
unfold stratumRadius atomEstimate blockAverage
split_ifs <;> positivity
theorem honestInterval_observedMeasurable
The honest interval has measurable ordered endpoints for all sufficiently large sample sizes. The result uses the hreg condition, the hn condition, the hh condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
B :
beta kappa L cminus cplus pmin deltaBar alpha delta :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
hn :
4 ≤ n
hh :
0 ≤ infoBandwidth n delta beta kappa deltaBar
conclusion
ObservedMeasurableInterval (fun z : Fin n → ClampObs J => honestInterval B z (ellOf beta) beta kappa L cminus cplus delta (infoBandwidth n delta beta kappa deltaBar) alpha)
Proof (Lean source)
lemma honestInterval_observedMeasurable
(J n : ℕ) (B : SplitBlocks n)
(beta kappa L cminus cplus pmin deltaBar alpha delta : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(hn : 4 ≤ n)
(hh : 0 ≤ infoBandwidth n delta beta kappa deltaBar) :
ObservedMeasurableInterval (fun z : Fin n → ClampObs J =>
honestInterval B z (ellOf beta) beta kappa L cminus cplus delta
(infoBandwidth n delta beta kappa deltaBar) alpha) := by
classical
let tAlpha := sqrt (log (12 * (J : ℝ) / alpha) / 2)
let b0 := tAlpha / sqrt (B.I0.card : ℝ)
let b1 := tAlpha / sqrt (B.I1.card : ℝ)
let center := fun z : Fin n → ClampObs J =>
totalGramEstimator B z (ellOf beta) kappa cminus cplus delta
(infoBandwidth n delta beta kappa deltaBar)
let radius := fun z : Fin n → ClampObs J => b0 + ∑ x : Fin J,
stratumRadius B z x (ellOf beta) beta kappa L cminus cplus delta
(infoBandwidth n delta beta kappa deltaBar) tAlpha b1
let lo := fun z => max 0 (center z - radius z)
let hi := fun z => min 1 (center z + radius z)
have hc : Measurable center :=
totalGramEstimator_measurable B kappa cminus cplus delta _
have hr : Measurable radius := measurable_const.add
(Finset.measurable_fun_sum _ fun x _ =>
honest_stratumRadius_measurable B x beta kappa L cminus cplus delta _
tAlpha b1)
have hlo : Measurable lo := measurable_const.max (hc.sub hr)
have hhi : Measurable hi := measurable_const.min (hc.add hr)
refine ⟨lo, hi, hlo, hhi, ?_, ?_⟩
· intro z
have hcenter := clampUnit_mem_Icc
(retainedEstimate B z delta + ∑ x : Fin J,
atomEstimate B z x delta * localRegressionEstimate B z x (ellOf beta)
kappa cminus cplus delta (infoBandwidth n delta beta kappa deltaBar))
have hradius : 0 ≤ radius z := by
simpa only [radius, tAlpha, b0, b1] using
honestInterval_radius_nonneg J n B beta kappa L cminus cplus pmin
deltaBar alpha delta (infoBandwidth n delta beta kappa deltaBar)
hreg hn hh z
dsimp only [lo, hi]
apply le_min
· apply max_le
· norm_num
· dsimp only [center]
rw [show totalGramEstimator B z (ellOf beta) kappa cminus cplus delta
(infoBandwidth n delta beta kappa deltaBar) =
clampUnit (retainedEstimate B z delta + ∑ x : Fin J,
atomEstimate B z x delta * localRegressionEstimate B z x
(ellOf beta) kappa cminus cplus delta
(infoBandwidth n delta beta kappa deltaBar)) by rfl]
linarith [hcenter.2]
· apply max_le
· dsimp only [center]
rw [show totalGramEstimator B z (ellOf beta) kappa cminus cplus delta
(infoBandwidth n delta beta kappa deltaBar) =
clampUnit (retainedEstimate B z delta + ∑ x : Fin J,
atomEstimate B z x delta * localRegressionEstimate B z x
(ellOf beta) kappa cminus cplus delta
(infoBandwidth n delta beta kappa deltaBar)) by rfl]
linarith [hcenter.1]
· linarith
· intro z
rfl
theorem honestInterval_length_le_one
Intersecting the honest interval with the outcome range bounds its length by one on every sample. This is the stated conclusion.
Formal statement
hypotheses
conclusion
≤ 1
Proof (Lean source)
lemma honestInterval_length_le_one {J n : ℕ} (B : SplitBlocks n)
(beta kappa L cminus cplus delta h alpha : ℝ)
(z : Fin n → ClampObs J) :
intervalLength (honestInterval B z (ellOf beta) beta kappa L cminus cplus
delta h alpha) ≤ 1 := by
simp only [honestInterval, honest_intervalLength_Icc_eq]
apply max_le
· norm_num
· have hmin := min_le_left (1 : ℝ)
(totalGramEstimator B z (ellOf beta) kappa cminus cplus delta h +
(sqrt (log (12 * (J : ℝ) / alpha) / 2) /
sqrt (B.I0.card : ℝ) + ∑ x : Fin J,
stratumRadius B z x (ellOf beta) beta kappa L cminus cplus delta h
(sqrt (log (12 * (J : ℝ) / alpha) / 2))
(sqrt (log (12 * (J : ℝ) / alpha) / 2) /
sqrt (B.I1.card : ℝ))))
have hmax := le_max_left (0 : ℝ)
(totalGramEstimator B z (ellOf beta) kappa cminus cplus delta h -
(sqrt (log (12 * (J : ℝ) / alpha) / 2) /
sqrt (B.I0.card : ℝ) + ∑ x : Fin J,
stratumRadius B z x (ellOf beta) beta kappa L cminus cplus delta h
(sqrt (log (12 * (J : ℝ) / alpha) / 2))
(sqrt (log (12 * (J : ℝ) / alpha) / 2) /
sqrt (B.I1.card : ℝ))))
linarith
theorem honestInterval_length_measurable
The displayed interval length is measurable as a function of the observed sample. This is the stated conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma honestInterval_length_measurable {J n : ℕ} (B : SplitBlocks n)
(beta kappa L cminus cplus delta h alpha : ℝ) :
Measurable fun z : Fin n → ClampObs J =>
intervalLength (honestInterval B z (ellOf beta) beta kappa L cminus
cplus delta h alpha) := by
classical
let tAlpha := sqrt (log (12 * (J : ℝ) / alpha) / 2)
let b0 := tAlpha / sqrt (B.I0.card : ℝ)
let b1 := tAlpha / sqrt (B.I1.card : ℝ)
let center := fun z : Fin n → ClampObs J =>
totalGramEstimator B z (ellOf beta) kappa cminus cplus delta h
let radius := fun z : Fin n → ClampObs J => b0 + ∑ x : Fin J,
stratumRadius B z x (ellOf beta) beta kappa L cminus cplus delta h tAlpha b1
have hc : Measurable center :=
totalGramEstimator_measurable B kappa cminus cplus delta h
have hr : Measurable radius := measurable_const.add
(Finset.measurable_fun_sum _ fun x _ =>
honest_stratumRadius_measurable B x beta kappa L cminus cplus delta h
tAlpha b1)
have hm : Measurable fun z =>
max 0 (min 1 (center z + radius z) - max 0 (center z - radius z)) :=
measurable_const.max
((measurable_const.min (hc.add hr)).sub (measurable_const.max (hc.sub hr)))
simpa only [honestInterval, honest_intervalLength_Icc_eq, center, radius,
tAlpha, b0, b1] using hm
theorem clampProcedureWorstLength_honestInterval_le_ofReal
A real uniform expected-length upper bound upgrades to the extended-real worst-length convention. The result uses the hupper condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
B :
beta kappa L cminus cplus pmin delta h alpha U :
ℝ
hupper :
sSup {v : ℝ | ∃ P : ClampLaw J, ClampModel P beta kappa L cminus cplus pmin ∧ IidSampling P n ∧ v = ∫ z, intervalLength (honestInterval B z (ellOf beta) beta kappa L cminus cplus delta h alpha) ∂iidProduct P n}
≤ U
conclusion
clampProcedureWorstLength J n beta kappa L cminus cplus pmin (fun z => honestInterval B z (ellOf beta) beta kappa L cminus cplus delta h alpha)
≤ ofReal U
Proof (Lean source)
lemma clampProcedureWorstLength_honestInterval_le_ofReal
(J n : ℕ) (B : SplitBlocks n)
(beta kappa L cminus cplus pmin delta h alpha U : ℝ)
(hupper : sSup {v : ℝ | ∃ P : ClampLaw J,
ClampModel P beta kappa L cminus cplus pmin ∧ IidSampling P n ∧
v = ∫ z, intervalLength
(honestInterval B z (ellOf beta) beta kappa L cminus cplus delta h alpha)
∂iidProduct P n} ≤ U) :
clampProcedureWorstLength J n beta kappa L cminus cplus pmin
(fun z => honestInterval B z (ellOf beta) beta kappa L cminus cplus
delta h alpha) ≤ ofReal U := by
unfold clampProcedureWorstLength
apply sSup_le
intro v hv
rcases hv with ⟨P, hP, rfl⟩
letI : IsProbabilityMeasure P.dataMeasure := hP.probability
letI : IsProbabilityMeasure (iidProduct P n) := by unfold iidProduct; infer_instance
let f := fun z : Fin n → ClampObs J => intervalLength
(honestInterval B z (ellOf beta) beta kappa L cminus cplus delta h alpha)
have hfmeas : Measurable f := honestInterval_length_measurable B beta kappa L
cminus cplus delta h alpha
have hfnn : ∀ᵐ z ∂iidProduct P n, 0 ≤ f z :=
ae_of_all _ fun _ => le_max_left _ _
have hfint : Integrable f (iidProduct P n) := by
refine Integrable.of_bound hfmeas.aestronglyMeasurable 1 ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (le_max_left _ _)]
exact honestInterval_length_le_one B beta kappa L cminus cplus delta h alpha z
simp_rw [intervalLengthENNReal]
rw [← ofReal_integral_eq_lintegral_ofReal hfint hfnn]
apply ENNReal.ofReal_le_ofReal
have hbdd : BddAbove {v : ℝ | ∃ Q : ClampLaw J,
ClampModel Q beta kappa L cminus cplus pmin ∧ IidSampling Q n ∧
v = ∫ z, intervalLength
(honestInterval B z (ellOf beta) beta kappa L cminus cplus delta h alpha)
∂iidProduct Q n} := by
refine ⟨1, ?_⟩
rintro w ⟨Q, hQ, _hiid, rfl⟩
letI : IsProbabilityMeasure Q.dataMeasure := hQ.probability
letI : IsProbabilityMeasure (iidProduct Q n) := by unfold iidProduct; infer_instance
have hqmeas := honestInterval_length_measurable (J := J) B beta kappa L
cminus cplus delta h alpha
have hqint : Integrable (fun z : Fin n → ClampObs J => intervalLength
(honestInterval B z (ellOf beta) beta kappa L cminus cplus delta h alpha))
(iidProduct Q n) := by
refine Integrable.of_bound hqmeas.aestronglyMeasurable 1 ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (le_max_left _ _)]
exact honestInterval_length_le_one B beta kappa L cminus cplus delta h alpha z
exact (integral_mono_ae hqint (integrable_const 1)
(ae_of_all _ fun z => honestInterval_length_le_one B beta kappa L
cminus cplus delta h alpha z)).trans_eq (by simp)
have hmem : (∫ z, f z ∂iidProduct P n) ∈ {v : ℝ | ∃ Q : ClampLaw J,
ClampModel Q beta kappa L cminus cplus pmin ∧ IidSampling Q n ∧
v = ∫ z, intervalLength
(honestInterval B z (ellOf beta) beta kappa L cminus cplus delta h alpha)
∂iidProduct Q n} := ⟨P, hP, clampModel_iidSampling hP, rfl⟩
exact (le_csSup hbdd hmem).trans hupper
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.HonestLengthLower 9 declarations This file isolates the low-level expected-length converse for the fixed-Hölder clamp model.
Two-point lower bounds for honest confidence length
This file isolates the low-level expected-length converse for the fixed-Hölder clamp model. It deliberately defines the worst length of one procedure here, so that the headline module can reuse the results without an import cycle.
def clampProcedureWorstLength
Worst expected extended interval length of one procedure over the fixed-Hölder clamp model.
Definition (Lean source)
parameters
def
clampProcedureWorstLength J n beta kappa L cminus cplus pmin C :
ℝ≥0∞
given by
sSup {v : ℝ≥0∞ | ∃ P : ClampLaw J, ClampModel P beta kappa L cminus cplus pmin ∧ v = ∫⁻ z, intervalLengthENNReal (C z) ∂iidProduct P n}
theorem procedureWorstLength_lower_of_two_point
A close pair in the fixed-Hölder model lower-bounds the worst expected length of every interval procedure that is honest over that model. The result uses the hC condition, the hP0 condition, the hP1 condition, the hgap condition, the hsep condition, the hac condition, the hint condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P0 P1 :
ClampLaw J
n :
ℕ
beta kappa L cminus cplus pmin delta alpha gap :
ℝ
C :
hC :
UniformCoverage J n beta kappa L cminus cplus pmin delta alpha C
hP0 :
ClampModel P0 beta kappa L cminus cplus pmin
hP1 :
ClampModel P1 beta kappa L cminus cplus pmin
hgap :
0 ≤ gap
hsep :
gap ≤ |clampFunctional P1 delta - clampFunctional P0 delta|
hac :
iidProduct P1 n ≪ iidProduct P0 n
hint :
Integrable (fun z => (((iidProduct P1 n).rnDeriv (iidProduct P0 n) z).toReal - 1) ^ 2) (iidProduct P0 n)
conclusion
≤ clampProcedureWorstLength J n beta kappa L cminus cplus pmin C
Proof (Lean source)
lemma procedureWorstLength_lower_of_two_point
(P0 P1 : ClampLaw J) (n : ℕ)
(beta kappa L cminus cplus pmin delta alpha gap : ℝ)
(C : ConfidenceProcedure n J)
(hC : UniformCoverage J n beta kappa L cminus cplus pmin delta alpha C)
(hP0 : ClampModel P0 beta kappa L cminus cplus pmin)
(hP1 : ClampModel P1 beta kappa L cminus cplus pmin)
(hgap : 0 ≤ gap)
(hsep : gap ≤ |clampFunctional P1 delta - clampFunctional P0 delta|)
(hac : iidProduct P1 n ≪ iidProduct P0 n)
(hint : Integrable (fun z => (((iidProduct P1 n).rnDeriv
(iidProduct P0 n) z).toReal - 1) ^ 2) (iidProduct P0 n)) :
ofReal ((1 - 2 * alpha -
(1 / 2) * sqrt (productChiSq P1 P0 n)) * gap) ≤
clampProcedureWorstLength J n beta kappa L cminus cplus pmin C := by
rcases hC with ⟨⟨lo, hi, hlo, hhi, hord, hCeq⟩, hcover⟩
let theta0 := clampFunctional P0 delta
let theta1 := clampFunctional P1 delta
let A : Set (Fin n → ClampObs J) := {z | theta0 ∈ C z}
let B : Set (Fin n → ClampObs J) := {z | theta1 ∈ C z}
have hA : MeasurableSet A := by
simp only [A, hCeq, mem_Icc]
exact (measurableSet_le hlo measurable_const).inter
(measurableSet_le measurable_const hhi)
have hB : MeasurableSet B := by
simp only [B, hCeq, mem_Icc]
exact (measurableSet_le hlo measurable_const).inter
(measurableSet_le measurable_const hhi)
letI : IsProbabilityMeasure P0.dataMeasure := hP0.probability
letI : IsProbabilityMeasure P1.dataMeasure := hP1.probability
letI : IsProbabilityMeasure (iidProduct P0 n) := by unfold iidProduct; infer_instance
letI : IsProbabilityMeasure (iidProduct P1 n) := by unfold iidProduct; infer_instance
have hcov0 : 1 - alpha ≤ (iidProduct P0 n).real A := by
simpa [A, theta0] using hcover P0 hP0
have hcov1 : 1 - alpha ≤ (iidProduct P1 n).real B := by
simpa [B, theta1] using hcover P1 hP1
have htv : tvDist (iidProduct P1 n) (iidProduct P0 n) ≤
(1 / 2) * sqrt (productChiSq P1 P0 n) := by
simpa [productChiSq] using
tvDist_le_half_sqrt_chiSqDiv
(iidProduct P1 n) (iidProduct P0 n) hac hint
have hcov1_at0 : 1 - alpha -
(1 / 2) * sqrt (productChiSq P1 P0 n) ≤
(iidProduct P0 n).real B := by
have hmove := measureReal_sub_le_tvDist
(μ := iidProduct P0 n) (ν := iidProduct P1 n) hB
rw [tvDist_symm] at hmove
linarith
have hinter : 1 - 2 * alpha -
(1 / 2) * sqrt (productChiSq P1 P0 n) ≤
(iidProduct P0 n).real (A ∩ B) := by
have hunion := measureReal_union_add_inter (μ := iidProduct P0 n)
(s := A) (t := B) hB (measure_ne_top _ _) (measure_ne_top _ _)
have hunionle : (iidProduct P0 n).real (A ∪ B) ≤ 1 :=
ENNReal.toReal_mono ENNReal.one_ne_top prob_le_one
linarith
have hpoint (z : Fin n → ClampObs J) :
(A ∩ B).indicator (fun _ => ofReal gap) z ≤
intervalLengthENNReal (C z) := by
by_cases hz : z ∈ A ∩ B
· rw [Set.indicator_of_mem hz]
apply ENNReal.ofReal_le_ofReal
rw [hCeq z]
simp only [intervalLength, csSup_Icc (hord z), csInf_Icc (hord z)]
rcases hz with ⟨hz0, hz1⟩
have hz0' : lo z ≤ theta0 ∧ theta0 ≤ hi z := by
simpa [A, theta0, hCeq z] using hz0
have hz1' : lo z ≤ theta1 ∧ theta1 ≤ hi z := by
simpa [B, theta1, hCeq z] using hz1
have habs : |clampFunctional P1 delta - clampFunctional P0 delta| ≤
hi z - lo z := by
rw [abs_le]
constructor <;> linarith [hz0'.1, hz0'.2, hz1'.1, hz1'.2]
exact (hsep.trans habs).trans (le_max_right _ _)
· simp [hz, intervalLengthENNReal]
have hlen : ofReal gap * (iidProduct P0 n) (A ∩ B) ≤
∫⁻ z, intervalLengthENNReal (C z) ∂iidProduct P0 n := by
calc
_ = ∫⁻ z, (A ∩ B).indicator (fun _ => ofReal gap) z
∂iidProduct P0 n := by
rw [lintegral_indicator (hA.inter hB), lintegral_const]
simp
_ ≤ _ := lintegral_mono hpoint
have hreal : ofReal (1 - 2 * alpha -
(1 / 2) * sqrt (productChiSq P1 P0 n)) ≤
(iidProduct P0 n) (A ∩ B) := by
rw [← ENNReal.ofReal_toReal (measure_ne_top _ _)]
exact ENNReal.ofReal_le_ofReal hinter
have htarget : ofReal ((1 - 2 * alpha -
(1 / 2) * sqrt (productChiSq P1 P0 n)) * gap) ≤
∫⁻ z, intervalLengthENNReal (C z) ∂iidProduct P0 n := by
rw [mul_comm, ENNReal.ofReal_mul hgap]
exact (mul_le_mul le_rfl hreal bot_le bot_le).trans hlen
exact htarget.trans (le_sSup ⟨P0, hP0, rfl⟩)
theorem observedMinimaxLength_lower_of_two_point
The same two-point experiment lower-bounds the minimax honest expected length. The result uses the hP0 condition, the hP1 condition, the hgap condition, the hsep condition, the hac condition, the hint condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P0 P1 :
ClampLaw J
n :
ℕ
beta kappa L cminus cplus pmin delta alpha gap :
ℝ
hP0 :
ClampModel P0 beta kappa L cminus cplus pmin
hP1 :
ClampModel P1 beta kappa L cminus cplus pmin
hgap :
0 ≤ gap
hsep :
gap ≤ |clampFunctional P1 delta - clampFunctional P0 delta|
hac :
iidProduct P1 n ≪ iidProduct P0 n
hint :
Integrable (fun z => (((iidProduct P1 n).rnDeriv (iidProduct P0 n) z).toReal - 1) ^ 2) (iidProduct P0 n)
conclusion
≤ observedMinimaxLength J n beta kappa L cminus cplus pmin delta alpha
Proof (Lean source)
lemma observedMinimaxLength_lower_of_two_point
(P0 P1 : ClampLaw J) (n : ℕ)
(beta kappa L cminus cplus pmin delta alpha gap : ℝ)
(hP0 : ClampModel P0 beta kappa L cminus cplus pmin)
(hP1 : ClampModel P1 beta kappa L cminus cplus pmin)
(hgap : 0 ≤ gap)
(hsep : gap ≤ |clampFunctional P1 delta - clampFunctional P0 delta|)
(hac : iidProduct P1 n ≪ iidProduct P0 n)
(hint : Integrable (fun z => (((iidProduct P1 n).rnDeriv
(iidProduct P0 n) z).toReal - 1) ^ 2) (iidProduct P0 n)) :
ofReal ((1 - 2 * alpha -
(1 / 2) * sqrt (productChiSq P1 P0 n)) * gap) ≤
observedMinimaxLength J n beta kappa L cminus cplus pmin delta alpha := by
unfold observedMinimaxLength
apply le_sInf
intro r hr
rcases hr with ⟨C, hC, rfl⟩
exact procedureWorstLength_lower_of_two_point P0 P1 n beta kappa L cminus
cplus pmin delta alpha gap C hC hP0 hP1 hgap hsep hac hint
theorem observedMinimaxLength_lower_of_two_point_chi
A numerical product chi-square budget can replace the exact divergence in the two-point expected-length bound. The result uses the hP0 condition, the hP1 condition, the hgap condition, the hsep condition, the hac condition, the hint condition, the hchi0 condition, the hchi condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P0 P1 :
ClampLaw J
n :
ℕ
beta kappa L cminus cplus pmin delta alpha gap chi :
ℝ
hP0 :
ClampModel P0 beta kappa L cminus cplus pmin
hP1 :
ClampModel P1 beta kappa L cminus cplus pmin
hgap :
0 ≤ gap
hsep :
gap ≤ |clampFunctional P1 delta - clampFunctional P0 delta|
hac :
iidProduct P1 n ≪ iidProduct P0 n
hint :
Integrable (fun z => (((iidProduct P1 n).rnDeriv (iidProduct P0 n) z).toReal - 1) ^ 2) (iidProduct P0 n)
hchi0 :
0 ≤ chi
hchi :
productChiSq P1 P0 n ≤ chi
conclusion
≤ observedMinimaxLength J n beta kappa L cminus cplus pmin delta alpha
Proof (Lean source)
lemma observedMinimaxLength_lower_of_two_point_chi
(P0 P1 : ClampLaw J) (n : ℕ)
(beta kappa L cminus cplus pmin delta alpha gap chi : ℝ)
(hP0 : ClampModel P0 beta kappa L cminus cplus pmin)
(hP1 : ClampModel P1 beta kappa L cminus cplus pmin)
(hgap : 0 ≤ gap)
(hsep : gap ≤ |clampFunctional P1 delta - clampFunctional P0 delta|)
(hac : iidProduct P1 n ≪ iidProduct P0 n)
(hint : Integrable (fun z => (((iidProduct P1 n).rnDeriv
(iidProduct P0 n) z).toReal - 1) ^ 2) (iidProduct P0 n))
(hchi0 : 0 ≤ chi) (hchi : productChiSq P1 P0 n ≤ chi) :
ofReal ((1 - 2 * alpha - (1 / 2) * sqrt chi) * gap) ≤
observedMinimaxLength J n beta kappa L cminus cplus pmin delta alpha := by
have hsqrt := Real.sqrt_le_sqrt hchi
have hreal : (1 - 2 * alpha - (1 / 2) * sqrt chi) * gap ≤
(1 - 2 * alpha -
(1 / 2) * sqrt (productChiSq P1 P0 n)) * gap :=
mul_le_mul_of_nonneg_right (by linarith) hgap
exact (ENNReal.ofReal_le_ofReal hreal).trans
(observedMinimaxLength_lower_of_two_point P0 P1 n beta kappa L cminus
cplus pmin delta alpha gap hP0 hP1 hgap hsep hac hint)
theorem observedMinimaxLength_le_clampProcedureWorstLength
Every honest procedure's worst length dominates the minimax honest length. The result uses the hC condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
beta kappa L cminus cplus pmin delta alpha :
ℝ
C :
hC :
UniformCoverage J n beta kappa L cminus cplus pmin delta alpha C
conclusion
observedMinimaxLength J n beta kappa L cminus cplus pmin delta alpha
≤ clampProcedureWorstLength J n beta kappa L cminus cplus pmin C
Proof (Lean source)
lemma observedMinimaxLength_le_clampProcedureWorstLength
(J n : ℕ) (beta kappa L cminus cplus pmin delta alpha : ℝ)
(C : ConfidenceProcedure n J)
(hC : UniformCoverage J n beta kappa L cminus cplus pmin delta alpha C) :
observedMinimaxLength J n beta kappa L cminus cplus pmin delta alpha ≤
clampProcedureWorstLength J n beta kappa L cminus cplus pmin C := by
unfold observedMinimaxLength
apply sInf_le
exact ⟨C, hC, rfl⟩
theorem observedMinimaxLength_eventually_ge_root
The fixed-Hölder minimax honest length has the root-sample-size lower bound, uniformly over admissible threshold sequences. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
∃ c : ℝ,
0 < c
conclusion 1
deltaSeq :
ℕ → ℝ
ThresholdSequence deltaBar deltaSeq
≤ observedMinimaxLength J n beta kappa L cminus cplus pmin (deltaSeq n) alpha
Proof (Lean source)
lemma observedMinimaxLength_eventually_ge_root
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
∃ c : ℝ, 0 < c ∧ ∀ deltaSeq : ℕ → ℝ,
ThresholdSequence deltaBar deltaSeq → ∀ᶠ n : ℕ in atTop,
ofReal (c * (n : ℝ) ^ (-(1 : ℝ) / 2)) ≤
observedMinimaxLength J n beta kappa L cminus cplus pmin
(deltaSeq n) alpha := by
let eta : ℝ := 1 - 2 * alpha
let tau : ℝ := eta / 16
let chi : ℝ := eta ^ 2 / 16
let floor : ℝ := eta - (1 / 2) * Real.sqrt chi
let c : ℝ := floor * tau
have heta : 0 < eta := by dsimp [eta]; linarith [hreg.2.2.2.2.2.2.2.2.2.2.2.2]
have heta1 : eta < 1 := by dsimp [eta]; linarith [hreg.2.2.2.2.2.2.2.2.2.2.2.1]
have htau : 0 < tau := div_pos heta (by norm_num)
have hchi0 : 0 ≤ chi := by dsimp [chi]; positivity
have hsqrt : Real.sqrt chi = eta / 4 := by
rw [show chi = (eta / 4) ^ 2 by dsimp [chi]; ring,
Real.sqrt_sq_eq_abs, abs_of_pos (div_pos heta (by norm_num))]
have hfloor : 0 < floor := by dsimp [floor]; rw [hsqrt]; linarith
refine ⟨c, by dsimp [c]; positivity, ?_⟩
intro deltaSeq hdelta
filter_upwards [Filter.eventually_ge_atTop 1] with n hn
have hnpos : 0 < n := by omega
have hnR : 0 < (n : ℝ) := by exact_mod_cast hnpos
let root : ℝ := (n : ℝ) ^ (-(1 : ℝ) / 2)
let eps : ℝ := tau * root
have hroot : 0 < root := Real.rpow_pos_of_pos hnR _
have heps : 0 < eps := mul_pos htau hroot
have hroot1 : root ≤ 1 := by
rw [show (1 : ℝ) = 1 ^ (-(1 : ℝ) / 2) by norm_num]
exact Real.rpow_le_rpow_of_nonpos (by norm_num) (by exact_mod_cast hn) (by norm_num)
have heps_le : eps ≤ 1 / 4 := by
dsimp [eps, tau]
nlinarith
have hepsb : |eps| ≤ 1 / 2 := by rw [abs_of_pos heps]; linarith
have hepss : |eps| < 1 / 2 := by rw [abs_of_pos heps]; linarith
let P0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
let P1 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => eps)
have hP0 : ClampModel P0 beta kappa L cminus cplus pmin :=
minimaxCenter_mem_model J beta kappa L cminus cplus pmin deltaBar alpha hreg
have hP1 : ClampModel P1 beta kappa L cminus cplus pmin :=
minimaxConstant_mem_model J beta kappa L cminus cplus pmin deltaBar alpha
eps hreg hepsb
have hdelta1 : deltaSeq n ∈ Icc (0 : ℝ) 1 :=
⟨(hdelta n).1, (hdelta n).2.trans hreg.2.2.2.2.2.2.2.2.2.2.1.le⟩
have hsepEq := minimaxGlobalSeparation J kappa (deltaSeq n) eps hreg.1
hreg.2.2.1 hdelta1 hepsb
have hsep : eps ≤ |clampFunctional P1 (deltaSeq n) -
clampFunctional P0 (deltaSeq n)| := by
rw [show P1 = minimaxClampLaw J kappa (fun _ : Fin J × ℝ => eps) by rfl,
show P0 = minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0) by rfl,
hsepEq, abs_of_pos heps]
have hrootSq : root ^ 2 = (n : ℝ)⁻¹ := by
dsimp [root]
rw [← Real.rpow_natCast, ← Real.rpow_mul hnR.le]
norm_num
rw [Real.rpow_neg_one]
have hepsSq : eps ^ 2 = tau ^ 2 * (n : ℝ)⁻¹ := by
dsimp [eps]
rw [mul_pow, hrootSq]
letI : IsProbabilityMeasure (minimaxDesignMeasure J kappa) :=
minimaxDesignMeasure_isProbabilityMeasure J kappa hreg.1 hreg.2.2.1
have hintEq : (∫ p, 4 * ((fun _ : Fin J × ℝ => eps) p) ^ 2
∂minimaxDesignMeasure J kappa) = 4 * eps ^ 2 := by simp
have hchiEq := minimaxProduct_chiSqDiv_center J n kappa hreg.1 hreg.2.2.1
(fun _ : Fin J × ℝ => eps) measurable_const (fun _ => hepss)
rw [hintEq] at hchiEq
have hnE : (n : ℝ) * (4 * eps ^ 2) = eta ^ 2 / 64 := by
rw [hepsSq]
dsimp [tau]
field_simp [hnR.ne']
ring
have hxabs : |eta ^ 2 / 64| ≤ 1 := by
rw [abs_of_nonneg (by positivity)]
nlinarith [sq_nonneg eta]
have hexp : exp (eta ^ 2 / 64) ≤
1 + eta ^ 2 / 64 + (eta ^ 2 / 64) ^ 2 :=
Causalean.Stat.Concentration.exp_le_one_add_add_sq hxabs
have hpoly : eta ^ 2 / 64 + (eta ^ 2 / 64) ^ 2 ≤ chi := by
dsimp [chi]
have heta_sq : eta ^ 2 ≤ 1 := by nlinarith [sq_nonneg eta]
nlinarith [sq_nonneg (eta ^ 2 / 64)]
have hpow : (1 + 4 * eps ^ 2) ^ n ≤ exp (eta ^ 2 / 64) := by
calc
_ ≤ (exp (4 * eps ^ 2)) ^ n :=
pow_le_pow_left₀ (by positivity)
(by simpa [add_comm] using Real.add_one_le_exp (4 * eps ^ 2)) n
_ = exp ((n : ℝ) * (4 * eps ^ 2)) := by rw [Real.exp_nat_mul]
_ = _ := by rw [hnE]
have hchi : productChiSq P1 P0 n ≤ chi := by
change chiSqDiv
(Measure.pi (fun _ : Fin n => minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => eps)))
(Measure.pi (fun _ : Fin n => minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => 0))) ≤ chi
linarith [hchiEq, hpow, hexp, hpoly]
have hac1 := minimaxDataMeasure_ac_center J kappa hreg.1 hreg.2.2.1
(fun _ : Fin J × ℝ => eps) measurable_const (fun _ => hepss)
have hint1 := minimaxDataMeasure_sq_integrable_center J kappa hreg.1 hreg.2.2.1
(fun _ : Fin J × ℝ => eps) measurable_const (fun _ => hepss)
letI : IsProbabilityMeasure (minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => eps)) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hreg.1 hreg.2.2.1 _
measurable_const (fun _ => hepsb)
letI : IsProbabilityMeasure (minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => 0)) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hreg.1 hreg.2.2.1 _
measurable_const (fun _ => by norm_num)
have hac := pi_iid_absolutelyContinuous _ _ hac1 n
have hint := pi_iid_integrable_sq_dev _ _ hac1 hint1 n
have hlower := observedMinimaxLength_lower_of_two_point_chi P0 P1 n beta
kappa L cminus cplus pmin (deltaSeq n) alpha eps chi hP0 hP1 heps.le hsep
(by simpa [P0, P1, iidProduct, minimaxClampLaw] using hac)
(by simpa [P0, P1, iidProduct, minimaxClampLaw] using hint) hchi0 hchi
dsimp [c, floor, eps, tau, root, eta] at hlower ⊢
simpa [mul_assoc] using hlower
theorem observedMinimaxLength_eventually_ge_local
A fixed contraction of the information bandwidth gives the local fixed-Hölder lower component for minimax honest expected length. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
∃ c : ℝ,
0 < c
conclusion 1
deltaSeq :
ℕ → ℝ
ThresholdSequence deltaBar deltaSeq
∀ᶠ n : ℕ in atTop, let h := infoBandwidth n (deltaSeq n) beta kappa deltaBar ofReal (c * (deltaSeq n) ^ (kappa + 1) * h ^ beta)
≤ observedMinimaxLength J n beta kappa L cminus cplus pmin (deltaSeq n) alpha
Proof (Lean source)
lemma observedMinimaxLength_eventually_ge_local
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
∃ c : ℝ, 0 < c ∧ ∀ deltaSeq : ℕ → ℝ,
ThresholdSequence deltaBar deltaSeq → ∀ᶠ n : ℕ in atTop,
let h := infoBandwidth n (deltaSeq n) beta kappa deltaBar
ofReal (c * (deltaSeq n) ^ (kappa + 1) * h ^ beta) ≤
observedMinimaxLength J n beta kappa L cminus cplus pmin
(deltaSeq n) alpha := by
have hr := hreg
rcases hr with ⟨hJ, hbeta, hkappa, hL, hcminus, hcminus_le, hcplus,
hpmin, hpmin_le, hdeltaBar, hdeltaBar1, halpha, halpha1⟩
obtain ⟨amplitude, hamp, hamp_le, hamp_model⟩ :=
exists_minimaxBump_amplitude beta L hbeta hL
let eta : ℝ := 1 - 2 * alpha
let s : ℝ := eta ^ 2 / 64
let d0 : ℝ := 8 * (kappa + 1) * amplitude ^ 2
let p : ℝ := 2 * beta + 1
let t : ℝ := s / (d0 + 1)
let chi : ℝ := eta ^ 2 / 16
let floor : ℝ := eta - (1 / 2) * Real.sqrt chi
let c : ℝ := floor * amplitude * t ^ beta
have heta : 0 < eta := by dsimp [eta]; linarith
have heta1 : eta < 1 := by dsimp [eta]; linarith
have hs : 0 < s := by dsimp [s]; positivity
have hs1 : s ≤ 1 := by dsimp [s]; nlinarith [sq_nonneg eta]
have hd0 : 0 ≤ d0 := by dsimp [d0]; positivity
have hp : 1 < p := by dsimp [p]; linarith
have ht : 0 < t := div_pos hs (by linarith)
have ht1 : t ≤ 1 := by
dsimp [t]
rw [div_le_one (by linarith : 0 < d0 + 1)]
linarith
have htp : t ^ p ≤ t :=
Real.rpow_le_self_of_le_one ht.le ht1 hp.le
have hdsmall : d0 * t ^ p ≤ s := by
calc
d0 * t ^ p ≤ d0 * t := mul_le_mul_of_nonneg_left htp hd0
_ ≤ s := by
dsimp [t]
rw [← mul_div_assoc, div_le_iff₀ (by linarith : 0 < d0 + 1)]
nlinarith [hs.le]
have hchi0 : 0 ≤ chi := by dsimp [chi]; positivity
have hsqrt : Real.sqrt chi = eta / 4 := by
rw [show chi = (eta / 4) ^ 2 by dsimp [chi]; ring,
Real.sqrt_sq_eq_abs, abs_of_pos (div_pos heta (by norm_num))]
have hfloor : 0 < floor := by dsimp [floor]; rw [hsqrt]; linarith
have hxabs : |s| ≤ 1 := by rw [abs_of_pos hs]; exact hs1
have hexp : exp s ≤ 1 + s + s ^ 2 :=
Causalean.Stat.Concentration.exp_le_one_add_add_sq hxabs
have hpoly : s + s ^ 2 ≤ chi := by
dsimp [s, chi]
have heta_sq : eta ^ 2 ≤ 1 := by nlinarith [sq_nonneg eta]
nlinarith [sq_nonneg (eta ^ 2 / 64)]
refine ⟨c, by dsimp [c]; positivity, ?_⟩
intro deltaSeq hdelta
have hbal := infoBandwidth_eventually_balance beta kappa deltaBar hbeta hkappa
⟨hdeltaBar, hdeltaBar1⟩ deltaSeq hdelta
filter_upwards [hbal] with n hbn
dsimp only
let delta := deltaSeq n
let h := infoBandwidth n delta beta kappa deltaBar
let hsml := t * h
let q := fun a : ℝ => amplitude * hsml ^ beta *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / hsml)
let P0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
let P1 := minimaxClampLaw J kappa (fun z => q z.2)
have hh : 0 < h := hbn.1
have hh1 : h ≤ 1 := hbn.2.1.trans (by linarith)
have hhs : 0 < hsml := mul_pos ht hh
have hhs1 : hsml ≤ 1 := (mul_le_of_le_one_left hh.le ht1).trans hh1
have hd : delta ∈ Icc (0 : ℝ) 1 :=
⟨(hdelta n).1, (hdelta n).2.trans hdeltaBar1.le⟩
have hshape := hamp_model hd.1 hd.2 hhs hhs1
have hqmeas : Measurable q := by dsimp [q]; fun_prop
have hq0 (a : ℝ) : 0 ≤ q a := by
dsimp [q]
exact mul_nonneg (mul_nonneg hamp.le (Real.rpow_nonneg hhs.le _))
(CausalSmith.Stat.DoseResponseMinimax.doseBump_nonneg _)
have hqbound (a : ℝ) : |q a| ≤ 1 / 2 := by
rw [abs_of_nonneg (hq0 a)]
have hb := CausalSmith.Stat.DoseResponseMinimax.doseBump_le_one
((a - delta) / hsml)
have hpw := Real.rpow_le_one hhs.le hhs1 hbeta.le
dsimp [q]
calc
_ ≤ amplitude * hsml ^ beta := by
nlinarith [mul_nonneg hamp.le (Real.rpow_nonneg hhs.le beta)]
_ ≤ amplitude := by simpa using mul_le_mul_of_nonneg_left hpw hamp.le
_ ≤ 1 / 2 := by linarith
have hqstrict (z : Fin J × ℝ) : |q z.2| < 1 / 2 := by
rw [abs_of_nonneg (hq0 z.2)]
have hb := CausalSmith.Stat.DoseResponseMinimax.doseBump_le_one
((z.2 - delta) / hsml)
have hpw := Real.rpow_le_one hhs.le hhs1 hbeta.le
have hqamp : q z.2 ≤ amplitude := by
dsimp [q]
calc
_ ≤ amplitude * hsml ^ beta := by
nlinarith [mul_nonneg hamp.le (Real.rpow_nonneg hhs.le beta)]
_ ≤ amplitude := by simpa using mul_le_mul_of_nonneg_left hpw hamp.le
linarith [hamp_le]
have hP0 := minimaxCenter_mem_model J beta kappa L cminus cplus pmin
deltaBar alpha hreg
have hP1 : ClampModel P1 beta kappa L cminus cplus pmin := by
apply minimaxClampModel_of_taylor J beta kappa L cminus cplus pmin
hJ hbeta hkappa hcminus_le hcplus hpmin_le q hqmeas hqbound
exact hshape.1
exact hshape.2.1
exact hshape.2.2
have hsep0 := minimaxLocalSeparation J beta kappa delta hsml amplitude
hJ hbeta hkappa hd hhs hhs1 hamp.le hamp_le
have hsep : amplitude * delta ^ (kappa + 1) * hsml ^ beta ≤
|clampFunctional P1 delta - clampFunctional P0 delta| := by
have hn : 0 ≤ clampFunctional P1 delta - clampFunctional P0 delta :=
(mul_nonneg (mul_nonneg hamp.le (Real.rpow_nonneg hd.1 _))
(Real.rpow_nonneg hhs.le _)).trans (by simpa [P0, P1, q] using hsep0)
rw [abs_of_nonneg hn]
simpa [P0, P1, q] using hsep0
let I := ∫ z, 4 * ((fun z : Fin J × ℝ => q z.2) z) ^ 2
∂minimaxDesignMeasure J kappa
have hI := minimaxLocal_designIntegral_le J beta kappa delta hsml amplitude
hJ hkappa hd.1 hhs hamp.le
have hnI : (n : ℝ) * I ≤ s := by
have hw : delta + hsml ≤ delta + h := by
dsimp [hsml]
nlinarith [mul_le_of_le_one_left hh.le ht1]
have hpoww := Real.rpow_le_rpow (add_nonneg hd.1 hhs.le) hw hkappa
have hb := hbn.2.2
calc
_ ≤ (n : ℝ) * (8 * (kappa + 1) * amplitude ^ 2 *
hsml ^ (2 * beta + 1) * (delta + hsml) ^ kappa) :=
mul_le_mul_of_nonneg_left (by simpa [I, q] using hI) (Nat.cast_nonneg n)
_ ≤ d0 * t ^ p *
((n : ℝ) * h ^ (2 * beta + 1) * (delta + h) ^ kappa) := by
dsimp [d0, p, hsml]
rw [Real.mul_rpow ht.le hh.le]
ring_nf
gcongr
_ = d0 * t ^ p := by rw [hb, mul_one]
_ ≤ s := hdsmall
have hchiEq := minimaxProduct_chiSqDiv_center J n kappa hJ hkappa
(fun z : Fin J × ℝ => q z.2) (hqmeas.comp measurable_snd) hqstrict
have hpow : (1 + I) ^ n ≤ exp s := by
calc
_ ≤ (exp I) ^ n := pow_le_pow_left₀ (by positivity)
(by simpa [add_comm] using Real.add_one_le_exp I) n
_ = exp ((n : ℝ) * I) := by rw [Real.exp_nat_mul]
_ ≤ _ := Real.exp_le_exp.mpr hnI
have hchi : productChiSq P1 P0 n ≤ chi := by
change chiSqDiv _ _ ≤ chi
dsimp [P0, P1, iidProduct, minimaxClampLaw]
change 1 + chiSqDiv _ _ = (1 + I) ^ n at hchiEq
linarith [hexp, hpoly]
have hac1 : minimaxDataMeasure J kappa (fun z : Fin J × ℝ => q z.2) ≪
minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => 0) :=
minimaxDataMeasure_ac_center J kappa hJ hkappa _
(hqmeas.comp measurable_snd) hqstrict
have hint1 : Integrable
(fun o => (((minimaxDataMeasure J kappa
(fun z : Fin J × ℝ => q z.2)).rnDeriv
(minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => 0)) o).toReal - 1) ^ 2)
(minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => 0)) := by
exact minimaxDataMeasure_sq_integrable_center J kappa hJ hkappa _
(hqmeas.comp measurable_snd) hqstrict
letI : IsProbabilityMeasure
(minimaxDataMeasure J kappa (fun z : Fin J × ℝ => q z.2)) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hJ hkappa _
(hqmeas.comp measurable_snd) (fun z => (hqstrict z).le)
letI : IsProbabilityMeasure
(minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => 0)) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hJ hkappa _ measurable_const
(fun _ => by norm_num)
have hlower := observedMinimaxLength_lower_of_two_point_chi P0 P1 n beta
kappa L cminus cplus pmin delta alpha
(amplitude * delta ^ (kappa + 1) * hsml ^ beta) chi hP0 hP1
(mul_nonneg (mul_nonneg hamp.le (Real.rpow_nonneg hd.1 _))
(Real.rpow_nonneg hhs.le _)) hsep
(by simpa [P0, P1, iidProduct, minimaxClampLaw] using
pi_iid_absolutelyContinuous _ _ hac1 n)
(by simpa [P0, P1, iidProduct, minimaxClampLaw] using
pi_iid_integrable_sq_dev _ _ hac1 hint1 n)
hchi0 hchi
dsimp [c, floor, eta, hsml] at hlower ⊢
rw [Real.mul_rpow ht.le hh.le] at hlower
simpa [add_comm, mul_comm, mul_left_comm, mul_assoc] using hlower
theorem observedMinimaxLength_eventually_ge_frontier
The root and localized experiments combine into the full fixed-Hölder frontier lower bound for minimax honest expected length. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
∃ c : ℝ,
0 < c
conclusion 1
deltaSeq :
ℕ → ℝ
ThresholdSequence deltaBar deltaSeq
∀ᶠ n : ℕ in atTop, ofReal (c * clampFrontier n (deltaSeq n) kappa (infoBandwidth n (deltaSeq n) beta kappa deltaBar) beta)
≤ observedMinimaxLength J n beta kappa L cminus cplus pmin (deltaSeq n) alpha
Proof (Lean source)
lemma observedMinimaxLength_eventually_ge_frontier
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
∃ c : ℝ, 0 < c ∧ ∀ deltaSeq : ℕ → ℝ,
ThresholdSequence deltaBar deltaSeq → ∀ᶠ n : ℕ in atTop,
ofReal (c * clampFrontier n (deltaSeq n) kappa
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) beta) ≤
observedMinimaxLength J n beta kappa L cminus cplus pmin
(deltaSeq n) alpha := by
obtain ⟨c0, hc0, h0⟩ := observedMinimaxLength_eventually_ge_root J beta
kappa L cminus cplus pmin deltaBar alpha hreg
obtain ⟨c1, hc1, h1⟩ := observedMinimaxLength_eventually_ge_local J beta
kappa L cminus cplus pmin deltaBar alpha hreg
let c := min c0 c1 / 2
refine ⟨c, by dsimp [c]; positivity, ?_⟩
intro deltaSeq hdelta
have hbal := infoBandwidth_eventually_balance beta kappa deltaBar hreg.2.1
hreg.2.2.1 ⟨hreg.2.2.2.2.2.2.2.2.2.1,
hreg.2.2.2.2.2.2.2.2.2.2.1⟩ deltaSeq hdelta
filter_upwards [h0 deltaSeq hdelta, h1 deltaSeq hdelta, hbal] with n hn0 hn1 hbn
let h := infoBandwidth n (deltaSeq n) beta kappa deltaBar
let root : ℝ := (n : ℝ) ^ (-(1 : ℝ) / 2)
let localTerm : ℝ := (deltaSeq n) ^ (kappa + 1) * h ^ beta
have hroot : 0 ≤ root := Real.rpow_nonneg (Nat.cast_nonneg n) _
have hlocal : 0 ≤ localTerm := mul_nonneg
(Real.rpow_nonneg (hdelta n).1 _)
(Real.rpow_nonneg hbn.1.le _)
by_cases hle : c0 * root ≤ c1 * localTerm
· apply (ENNReal.ofReal_le_ofReal ?_).trans
(by simpa [mul_assoc] using hn1)
dsimp [c, clampFrontier, root, localTerm, h]
calc
min c0 c1 / 2 *
((n : ℝ) ^ (-(1 : ℝ) / 2) +
(deltaSeq n) ^ (kappa + 1) *
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) ^ beta) =
(min c0 c1 * (n : ℝ) ^ (-(1 : ℝ) / 2) +
min c0 c1 * ((deltaSeq n) ^ (kappa + 1) *
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) ^ beta)) / 2 := by ring
_ ≤ (c0 * (n : ℝ) ^ (-(1 : ℝ) / 2) +
c1 * ((deltaSeq n) ^ (kappa + 1) *
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) ^ beta)) / 2 := by
apply div_le_div_of_nonneg_right _ (by norm_num)
exact add_le_add
(mul_le_mul_of_nonneg_right (min_le_left _ _) hroot)
(mul_le_mul_of_nonneg_right (min_le_right _ _) hlocal)
_ ≤ c1 * ((deltaSeq n) ^ (kappa + 1) *
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) ^ beta) := by
dsimp [root, localTerm, h] at hle
linarith
· apply (ENNReal.ofReal_le_ofReal ?_).trans hn0
have hle' : c1 * localTerm ≤ c0 * root := le_of_not_ge hle
dsimp [c, clampFrontier, root, localTerm, h]
calc
min c0 c1 / 2 *
((n : ℝ) ^ (-(1 : ℝ) / 2) +
(deltaSeq n) ^ (kappa + 1) *
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) ^ beta) =
(min c0 c1 * (n : ℝ) ^ (-(1 : ℝ) / 2) +
min c0 c1 * ((deltaSeq n) ^ (kappa + 1) *
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) ^ beta)) / 2 := by ring
_ ≤ (c0 * (n : ℝ) ^ (-(1 : ℝ) / 2) +
c1 * ((deltaSeq n) ^ (kappa + 1) *
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) ^ beta)) / 2 := by
apply div_le_div_of_nonneg_right _ (by norm_num)
exact add_le_add
(mul_le_mul_of_nonneg_right (min_le_left _ _) hroot)
(mul_le_mul_of_nonneg_right (min_le_right _ _) hlocal)
_ ≤ c0 * (n : ℝ) ^ (-(1 : ℝ) / 2) := by
dsimp [root, localTerm, h] at hle'
linarith
theorem procedureWorstLength_eventually_ge_frontier
The same eventual frontier lower bound holds for the worst expected length of every uniformly honest procedure. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
∃ c : ℝ,
0 < c
conclusion 1
deltaSeq :
ℕ → ℝ
ThresholdSequence deltaBar deltaSeq
∀ᶠ n : ℕ in atTop, ∀ C : ConfidenceProcedure n J, UniformCoverage J n beta kappa L cminus cplus pmin (deltaSeq n) alpha C → ofReal (c * clampFrontier n (deltaSeq n) kappa (infoBandwidth n (deltaSeq n) beta kappa deltaBar) beta)
≤ clampProcedureWorstLength J n beta kappa L cminus cplus pmin C
Proof (Lean source)
lemma procedureWorstLength_eventually_ge_frontier
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
∃ c : ℝ, 0 < c ∧ ∀ deltaSeq : ℕ → ℝ,
ThresholdSequence deltaBar deltaSeq → ∀ᶠ n : ℕ in atTop,
∀ C : ConfidenceProcedure n J,
UniformCoverage J n beta kappa L cminus cplus pmin (deltaSeq n) alpha C →
ofReal (c * clampFrontier n (deltaSeq n) kappa
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) beta) ≤
clampProcedureWorstLength J n beta kappa L cminus cplus pmin C := by
obtain ⟨c, hc, hmin⟩ := observedMinimaxLength_eventually_ge_frontier J beta
kappa L cminus cplus pmin deltaBar alpha hreg
refine ⟨c, hc, ?_⟩
intro deltaSeq hdelta
filter_upwards [hmin deltaSeq hdelta] with n hn
intro C hC
exact hn.trans (observedMinimaxLength_le_clampProcedureWorstLength J n beta
kappa L cminus cplus pmin (deltaSeq n) alpha C hC)
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.HonestLengthUpper 3 declarations Eventual expected-length upper bound for the honest interval
Eventual expected-length upper bound for the honest interval
theorem exp_balanced_edge_eventually_le_root
Exponential decay along the smallest effective-sample polynomial implied by the balance equation dominates the root-sample scale. The result uses the hbeta condition, the hkappa condition, the hc condition. This is the stated conclusion.
Formal statement
Proof (Lean source)
lemma exp_balanced_edge_eventually_le_root
(beta kappa c : ℝ) (hbeta : 0 < beta) (hkappa : 0 ≤ kappa) (hc : 0 < c) :
∀ᶠ n : ℕ in atTop,
exp (-c * (n : ℝ) ^ (2 * beta / (2 * beta + kappa + 1))) ≤
(n : ℝ) ^ (-(1 : ℝ) / 2) := by
let a : ℝ := 2 * beta / (2 * beta + kappa + 1)
have ha : 0 < a := by dsimp [a]; positivity
have htend : Tendsto (fun n : ℕ => (n : ℝ) ^ a) atTop atTop :=
(tendsto_rpow_atTop ha).comp tendsto_natCast_atTop_atTop
have ho :=
(isLittleO_exp_neg_mul_rpow_atTop hc (-(1 : ℝ) / (2 * a))).eventuallyLE
have hoc := htend.eventually ho
filter_upwards [hoc, eventually_ge_atTop 1] with n hn hn1
rw [Real.norm_eq_abs, Real.norm_eq_abs,
abs_of_pos (Real.exp_pos _), abs_of_pos (Real.rpow_pos_of_pos (by positivity) _)] at hn
have hnpos : 0 < (n : ℝ) := by positivity
calc
exp (-c * (n : ℝ) ^ (2 * beta / (2 * beta + kappa + 1))) =
exp (-c * (n : ℝ) ^ a) := by rfl
_ ≤ ((n : ℝ) ^ a) ^ (-(1 : ℝ) / (2 * a)) := hn
_ = (n : ℝ) ^ (-(1 : ℝ) / 2) := by
rw [← Real.rpow_mul hnpos.le]
congr 1
field_simp
theorem balancedEffectiveSample_ge_edge
The balance equation forces the effective local sample size to grow at least at the edge-regime polynomial rate. The result uses the hn condition, the hbeta condition, the hkappa condition, the hdelta condition, the hh condition, the hbalance condition. This is the stated conclusion.
Formal statement
hypotheses
n :
ℕ
beta kappa delta h :
ℝ
hn :
0 < n
hbeta :
0 < beta
hkappa :
0 ≤ kappa
hdelta :
0 ≤ delta
hh :
0 < h
hbalance :
(n : ℝ) * h ^ (2 * beta + 1) * (delta + h) ^ kappa = 1
conclusion
(n : ℝ) ^ (2 * beta / (2 * beta + kappa + 1)) ≤ (n : ℝ) * h * (delta + h) ^ kappa
Proof (Lean source)
lemma balancedEffectiveSample_ge_edge
{n : ℕ} {beta kappa delta h : ℝ}
(hn : 0 < n) (hbeta : 0 < beta) (hkappa : 0 ≤ kappa)
(hdelta : 0 ≤ delta) (hh : 0 < h)
(hbalance : (n : ℝ) * h ^ (2 * beta + 1) * (delta + h) ^ kappa = 1) :
(n : ℝ) ^ (2 * beta / (2 * beta + kappa + 1)) ≤
(n : ℝ) * h * (delta + h) ^ kappa := by
have hnR : 0 < (n : ℝ) := by exact_mod_cast hn
let p : ℝ := 2 * beta + 1
have hp : 0 < p := by dsimp [p]; linarith
have hedge := balance_root_le_edge_scale (n : ℝ) delta h p kappa
hnR hdelta hh hp hkappa (by simpa [p] using hbalance)
have heff := effectiveSampleSize_eq_rpow hh hbalance
rw [heff]
have hpw := Real.rpow_le_rpow_of_nonpos hh hedge (by linarith : -2 * beta ≤ 0)
calc
(n : ℝ) ^ (2 * beta / (2 * beta + kappa + 1)) =
((n : ℝ) ^ (-1 / (p + kappa))) ^ (-2 * beta) := by
rw [← Real.rpow_mul hnR.le]
congr 1
dsimp [p]
field_simp
<;> ring
_ ≤ h ^ (-2 * beta) := hpw
theorem honestInterval_worstLength_eventually_le_frontier
Uniform eventual expected-length upper bound for the concrete bias-aware interval, including the supremum over all laws in the admissible regime, is controlled by the frontier rate.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
∃ C : ℝ,
0 < C
conclusion 1
SplitBlocks n, ∀ᶠ n in atTop, sSup {v : ℝ | ∃ P : ClampLaw J, ClampModel P beta kappa L cminus cplus pmin ∧ IidSampling P n ∧ v = ∫ z, intervalLength (honestInterval (Bseq n) z (ellOf beta) beta kappa L cminus cplus (deltaSeq n) (infoBandwidth n (deltaSeq n) beta kappa deltaBar) alpha) ∂iidProduct P n}
≤ C * clampFrontier n (deltaSeq n) kappa (infoBandwidth n (deltaSeq n) beta kappa deltaBar) beta
Proof (Lean source)
lemma honestInterval_worstLength_eventually_le_frontier
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
∃ C : ℝ, 0 < C ∧
∀ (deltaSeq : ℕ → ℝ), ThresholdSequence deltaBar deltaSeq →
∀ Bseq : ∀ n, SplitBlocks n,
∀ᶠ n in atTop,
sSup {v : ℝ | ∃ P : ClampLaw J,
ClampModel P beta kappa L cminus cplus pmin ∧ IidSampling P n ∧
v = ∫ z, intervalLength
(honestInterval (Bseq n) z (ellOf beta) beta kappa L cminus cplus
(deltaSeq n) (infoBandwidth n (deltaSeq n) beta kappa deltaBar) alpha)
∂iidProduct P n} ≤
C * clampFrontier n (deltaSeq n) kappa
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) beta := by
rcases total_gram_stabilization J beta kappa L cminus cplus pmin
deltaBar alpha hreg with ⟨Ct, ct, hCt, hct, hlambda, hstab⟩
have hreg' := hreg
rcases hreg' with
⟨hJ, hbeta, hkappa, hL, hcminus, hcminus_le, hcplus,
hpmin, hpmin_le, hdeltaBar, hdeltaBar1, halpha, halpha1⟩
let t : ℝ := sqrt (log (12 * (J : ℝ) / alpha) / 2)
let energyC : ℝ := sqrt
(336 * (4 * (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus ^ 2) /
(pmin * cminus / (2 * (2 : ℝ) ^ kappa)))
let biasC : ℝ := L * (2 * sqrt (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus)
let localC : ℝ := biasC + t * energyC
let atomK : ℝ := cplus / (kappa + 1)
let rootCoeff : ℝ := (3 / 2 : ℝ) + 3 * t
let radiusC : ℝ := rootCoeff * localC + atomK * localC + 3 * t + Ct
let C : ℝ := 2 * (3 * t + (J : ℝ) * radiusC) + 1
have ht : 0 ≤ t := Real.sqrt_nonneg _
have henergyC : 0 ≤ energyC := Real.sqrt_nonneg _
have hbiasC : 0 ≤ biasC := by dsimp [biasC]; positivity
have hlocalC : 0 ≤ localC := by dsimp [localC]; positivity
have hatomK : 0 ≤ atomK := by
dsimp [atomK]
exact div_nonneg (by linarith) (by linarith)
have hrootCoeff : 0 ≤ rootCoeff := by dsimp [rootCoeff]; positivity
have hradiusC : 0 ≤ radiusC := by dsimp [radiusC]; positivity
have hC : 0 < C := by dsimp [C]; positivity
refine ⟨C, hC, ?_⟩
intro deltaSeq hdelta Bseq
have hbal := infoBandwidth_eventually_balance beta kappa deltaBar hbeta hkappa
⟨hdeltaBar, hdeltaBar1⟩ deltaSeq hdelta
have hdecay := exp_balanced_edge_eventually_le_root beta kappa ct hbeta hkappa hct
filter_upwards [hbal, hdecay, eventually_ge_atTop 8] with n hbn hdecayN hn
let delta := deltaSeq n
let h := infoBandwidth n delta beta kappa deltaBar
let r0 := (n : ℝ) ^ (-(1 : ℝ) / 2)
let d0 := delta ^ (kappa + 1)
let a0 := d0 * h ^ beta
let b0 := t / sqrt ((Bseq n).I0.card : ℝ)
let b1 := t / sqrt ((Bseq n).I1.card : ℝ)
have hnpos : 0 < n := by omega
have hh : 0 < h := by simpa [h, delta] using hbn.1
have hhbar : h ≤ 1 - deltaBar := by simpa [h, delta] using hbn.2.1
have hh1 : h ≤ 1 := hhbar.trans (by linarith)
have hbalance : (n : ℝ) * h ^ (2 * beta + 1) * (delta + h) ^ kappa = 1 := by
simpa [h, delta] using hbn.2.2
have hdeltaN : delta ∈ Icc (0 : ℝ) deltaBar := hdelta n
have hupper : delta + h ≤ 1 := by linarith [hdeltaN.2, hhbar]
have hr0 : 0 ≤ r0 := Real.rpow_nonneg (Nat.cast_nonneg n) _
have hd0 : 0 ≤ d0 := Real.rpow_nonneg hdeltaN.1 _
have hhbeta : 0 ≤ h ^ beta := Real.rpow_nonneg hh.le _
have hhbeta1 : h ^ beta ≤ 1 := by
simpa using Real.rpow_le_one hh.le hh1 hbeta.le
have ha0 : 0 ≤ a0 := mul_nonneg hd0 hhbeta
have hsqrt0 : sqrt ((Bseq n).I0.card : ℝ)⁻¹ ≤ 3 * r0 := by
apply splitBlock_invSqrt_le_root (by exact_mod_cast hnpos)
have hc := (Bseq n).card_I0
omega
have hsqrt1 : sqrt ((Bseq n).I1.card : ℝ)⁻¹ ≤ 3 * r0 := by
apply splitBlock_invSqrt_le_root (by exact_mod_cast hnpos)
have hc := (Bseq n).card_I1
omega
have hb0 : b0 ≤ 3 * t * r0 := by
calc
b0 = t * sqrt ((Bseq n).I0.card : ℝ)⁻¹ := by
simp [b0, div_eq_mul_inv, Real.sqrt_inv]
_ ≤ t * (3 * r0) := mul_le_mul_of_nonneg_left hsqrt0 ht
_ = 3 * t * r0 := by ring
have hb1 : b1 ≤ 3 * t * r0 := by
calc
b1 = t * sqrt ((Bseq n).I1.card : ℝ)⁻¹ := by
simp [b1, div_eq_mul_inv, Real.sqrt_inv]
_ ≤ t * (3 * r0) := mul_le_mul_of_nonneg_left hsqrt1 ht
_ = 3 * t * r0 := by ring
have hb10 : 0 ≤ b1 := by dsimp [b1]; positivity
have heffLower := balancedEffectiveSample_ge_edge hnpos hbeta hkappa
hdeltaN.1 hh hbalance
have htailExp : Ct * exp (-ct * (n : ℝ) * h * (delta + h) ^ kappa) ≤
Ct * r0 := by
apply mul_le_mul_of_nonneg_left _ hCt.le
calc
exp (-ct * (n : ℝ) * h * (delta + h) ^ kappa) =
exp (-ct * ((n : ℝ) * h * (delta + h) ^ kappa)) := by
congr 1
ring
_ ≤ exp (-ct * (n : ℝ) ^
(2 * beta / (2 * beta + kappa + 1))) := by
apply Real.exp_le_exp.mpr
nlinarith
_ ≤ r0 := by simpa [r0] using hdecayN
apply honestInterval_modelSup_le J n (Bseq n) beta kappa L cminus cplus
pmin deltaBar alpha delta h (C * clampFrontier n delta kappa h beta) hreg
intro P hP
letI : IsProbabilityMeasure P.dataMeasure := hP.probability
letI : IsProbabilityMeasure (iidProduct P n) := by unfold iidProduct; infer_instance
have hsampling : IidSampling P n := clampModel_iidSampling hP
have htail (x : Fin J) :
(iidProduct P n).real {z |
¬ GoodGramEvent (Bseq n) z x (ellOf beta) kappa cminus cplus delta h} ≤
Ct * r0 := by
exact (hstab P hP n hsampling (Bseq n) delta hdeltaN x).1 |>.trans htailExp
have hradBound (x : Fin J) :
(∫ z, stratumRadius (Bseq n) z x (ellOf beta) beta kappa L cminus cplus
delta h t b1 ∂iidProduct P n) ≤ radiusC * (r0 + a0) := by
have hfac := integral_stratumRadius_le_factored P (Bseq n) x beta kappa L
cminus cplus delta h t b1 hJ hP.probability hbeta.le hL.le hh ht hb10 hlambda
have hatom := integral_atomEstimate_le_envelope P (Bseq n) x beta kappa L
cminus cplus pmin deltaBar alpha delta hP hreg (by
have hc := (Bseq n).card_I1
omega) hdeltaN
have henergy := integral_sqrt_interceptWeight_energy_balanced_le
(ell := ellOf beta) P hP hsampling (Bseq n) x hn hbeta hkappa hcminus
(by linarith [hcplus] : 0 ≤ cplus) hpmin hdeltaN.1 hh hupper hlambda hbalance
have hfirst :
(∫ z, atomEstimate (Bseq n) z x delta ∂iidProduct P n) + b1 ≤
rootCoeff * r0 + atomK * d0 := by
calc
_ ≤ ((1 / 2 : ℝ) * sqrt ((Bseq n).I1.card : ℝ)⁻¹ +
cplus * d0 / (kappa + 1)) + 3 * t * r0 :=
add_le_add hatom hb1
_ = ((1 / 2 : ℝ) * sqrt ((Bseq n).I1.card : ℝ)⁻¹ +
atomK * d0) + 3 * t * r0 := by
dsimp [atomK]
ring
_ ≤ rootCoeff * r0 + atomK * d0 := by
dsimp [rootCoeff]
nlinarith [hsqrt1]
have hsecond :
L * h ^ beta * (2 * sqrt (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus) +
t * (∫ z, sqrt (∑ i ∈ (Bseq n).I2,
interceptWeight (Bseq n) z x (ellOf beta) kappa cminus cplus delta h i ^ 2)
∂iidProduct P n) ≤ localC * h ^ beta := by
calc
_ ≤ biasC * h ^ beta + t * (energyC * h ^ beta) := by
apply add_le_add
· dsimp [biasC]; ring_nf; exact le_rfl
· exact mul_le_mul_of_nonneg_left henergy ht
_ = localC * h ^ beta := by dsimp [localC]; ring
have hprod :
((∫ z, atomEstimate (Bseq n) z x delta ∂iidProduct P n) + b1) *
(L * h ^ beta * (2 * sqrt (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus) +
t * (∫ z, sqrt (∑ i ∈ (Bseq n).I2,
interceptWeight (Bseq n) z x (ellOf beta) kappa cminus cplus delta h i ^ 2)
∂iidProduct P n)) ≤
rootCoeff * localC * r0 + atomK * localC * a0 := by
have hfirst0 : 0 ≤ (∫ z, atomEstimate (Bseq n) z x delta
∂iidProduct P n) + b1 := by
apply add_nonneg
· exact integral_nonneg fun _ => (atomEstimate_mem_Icc (Bseq n) _ x delta).1
· exact hb10
have hsecond0 : 0 ≤
L * h ^ beta * (2 * sqrt (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus) +
t * (∫ z, sqrt (∑ i ∈ (Bseq n).I2,
interceptWeight (Bseq n) z x (ellOf beta) kappa cminus cplus delta h i ^ 2)
∂iidProduct P n) := by positivity
calc
_ ≤ (rootCoeff * r0 + atomK * d0) * (localC * h ^ beta) :=
mul_le_mul hfirst hsecond hsecond0
(add_nonneg (mul_nonneg hrootCoeff hr0) (mul_nonneg hatomK hd0))
_ = rootCoeff * localC * (r0 * h ^ beta) + atomK * localC * a0 := by
dsimp [a0]
ring
_ ≤ rootCoeff * localC * r0 + atomK * localC * a0 := by
exact add_le_add
(mul_le_mul_of_nonneg_left
(mul_le_of_le_one_right hr0 hhbeta1) (mul_nonneg hrootCoeff hlocalC))
le_rfl
calc
_ ≤ _ := hfac
_ ≤ (rootCoeff * localC * r0 + atomK * localC * a0) +
3 * t * r0 + Ct * r0 := by
exact add_le_add (add_le_add hprod hb1) (htail x)
_ ≤ radiusC * (r0 + a0) := by
dsimp [radiusC]
have hJunk : 0 ≤ atomK * localC := mul_nonneg hatomK hlocalC
have hmain0 : 0 ≤ rootCoeff * localC + 3 * t + Ct := by positivity
nlinarith [mul_nonneg hJunk hr0,
mul_nonneg hmain0 ha0]
have hradInt (x : Fin J) := stratumRadius_integrable P (Bseq n) x beta kappa L
cminus cplus delta h t b1 hP.probability hbeta.le hL.le hh ht hb10 hlambda
have hsumInt : Integrable (fun z : Fin n → ClampObs J => ∑ x : Fin J,
stratumRadius (Bseq n) z x (ellOf beta) beta kappa L cminus cplus
delta h t b1) (iidProduct P n) :=
integrable_finsetSum univ fun x _ => hradInt x
have hradiusInt : Integrable (fun z : Fin n → ClampObs J =>
honestRadius (Bseq n) z beta kappa L cminus cplus delta h alpha)
(iidProduct P n) := by
simpa [honestRadius, t, b0, b1] using (integrable_const b0).add hsumInt
have hlenInt : Integrable (fun z : Fin n → ClampObs J => intervalLength
(honestInterval (Bseq n) z (ellOf beta) beta kappa L cminus cplus
delta h alpha)) (iidProduct P n) := by
refine Integrable.of_bound
(honestInterval_length_measurable (Bseq n) beta kappa L cminus cplus
delta h alpha |>.aestronglyMeasurable) 1 ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (le_max_left _ _)]
exact honestInterval_length_le_one (Bseq n) beta kappa L cminus cplus delta h alpha z
have hlenRad :
(∫ z, intervalLength (honestInterval (Bseq n) z (ellOf beta) beta kappa L
cminus cplus delta h alpha) ∂iidProduct P n) ≤
∫ z, 2 * honestRadius (Bseq n) z beta kappa L cminus cplus delta h alpha
∂iidProduct P n :=
integral_mono hlenInt (hradiusInt.const_mul 2) fun z =>
honestInterval_length_le_two_radius J n (Bseq n) beta kappa L cminus cplus
pmin deltaBar alpha delta h hreg (by omega) hh.le z
calc
_ ≤ ∫ z, 2 * honestRadius (Bseq n) z beta kappa L cminus cplus delta h alpha
∂iidProduct P n := hlenRad
_ = 2 * (b0 + ∑ x : Fin J, ∫ z, stratumRadius (Bseq n) z x
(ellOf beta) beta kappa L cminus cplus delta h t b1 ∂iidProduct P n) := by
rw [integral_const_mul]
change 2 * (∫ z, b0 + ∑ x : Fin J, stratumRadius (Bseq n) z x
(ellOf beta) beta kappa L cminus cplus delta h t b1 ∂iidProduct P n) = _
rw [integral_add (integrable_const _) hsumInt, integral_const,
integral_finset_sum univ (fun x _ => hradInt x)]
simp
_ ≤ 2 * (3 * t * r0 + (J : ℝ) * (radiusC * (r0 + a0))) := by
apply mul_le_mul_of_nonneg_left _ (by norm_num)
calc
b0 + ∑ x : Fin J, ∫ z, stratumRadius (Bseq n) z x
(ellOf beta) beta kappa L cminus cplus delta h t b1 ∂iidProduct P n ≤
3 * t * r0 + ∑ _x : Fin J, radiusC * (r0 + a0) := by
exact add_le_add hb0 (Finset.sum_le_sum fun x _ => hradBound x)
_ = 3 * t * r0 + (J : ℝ) * (radiusC * (r0 + a0)) := by
simp
_ ≤ C * (r0 + a0) := by
dsimp [C]
have hrsum : 0 ≤ r0 + a0 := add_nonneg hr0 ha0
-- … truncated; follow the source link for the rest …
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.HonestMixedTerm 5 declarations Disjoint-block factorization for the honest-radius mixed term
Disjoint-block factorization for the honest-radius mixed term
theorem interceptWeight_congr_I2
Local-polynomial intercept weights only depend on observations in the local-regression block. The result uses the hz condition, the hi condition. This is the stated conclusion.
Formal statement
hypotheses
J n ell :
ℕ
B :
x :
Fin J
kappa cminus cplus delta h :
ℝ
hz :
∀ i ∈ B.I2, z' i = z i
i :
Fin n
hi :
i ∈ B.I2
conclusion
interceptWeight B z' x ell kappa cminus cplus delta h i
= interceptWeight B z x ell kappa cminus cplus delta h i
Proof (Lean source)
lemma interceptWeight_congr_I2 {J n ell : ℕ} (B : SplitBlocks n)
(z z' : Fin n → ClampObs J) (x : Fin J)
(kappa cminus cplus delta h : ℝ)
(hz : ∀ i ∈ B.I2, z' i = z i) (i : Fin n) (hi : i ∈ B.I2) :
interceptWeight B z' x ell kappa cminus cplus delta h i =
interceptWeight B z x ell kappa cminus cplus delta h i := by
classical
have hcount : localCount B z' x delta h = localCount B z x delta h := by
unfold localCount
apply Finset.sum_congr rfl
intro j hj
rw [hz j hj]
have hkernel : localKernelWeight B z' x delta h =
localKernelWeight B z x delta h := by
funext j
by_cases hj : j ∈ B.I2
· simp [localKernelWeight, hj, hz j hj]
· simp [localKernelWeight, hj]
have hgram : localGram B z' x ell delta h = localGram B z x ell delta h := by
ext a b
simp only [localGram, designMatrix]
apply Finset.sum_congr rfl
intro j hj
by_cases hj2 : j ∈ B.I2
· rw [hz j hj2, hkernel]
· have hk' : localKernelWeight B z' x delta h j = 0 := by
simp [localKernelWeight, hj2]
have hk : localKernelWeight B z x delta h j = 0 := by
simp [localKernelWeight, hj2]
rw [hk', hk]
simp
have hgood : GoodGramEvent B z' x ell kappa cminus cplus delta h ↔
GoodGramEvent B z x ell kappa cminus cplus delta h := by
unfold GoodGramEvent
rw [hcount, hgram]
have hdesign := hgram
change designMatrix ell
(fun j => scaledDose delta h (z' j)) (localKernelWeight B z' x delta h) =
designMatrix ell
(fun j => scaledDose delta h (z j)) (localKernelWeight B z x delta h) at hdesign
unfold interceptWeight
by_cases hg : GoodGramEvent B z x ell kappa cminus cplus delta h
· rw [if_pos (hgood.mpr hg), if_pos hg]
unfold equivKernelWeight
rw [hdesign, hkernel]
apply Finset.sum_congr rfl
intro k hk
congr 3
exact congrArg (scaledDose delta h) (hz i hi)
· rw [if_neg (fun hg' => hg (hgood.mp hg')), if_neg hg]
theorem integral_atomEstimate_mul_sqrt_weightEnergy
Exact factorization of the atom-frequency/weight-energy mixed moment across the disjoint estimation blocks. The result uses the hJ condition, the hprob condition. This is the stated conclusion.
Formal statement
hypotheses
J n ell :
ℕ
P :
ClampLaw J
B :
x :
Fin J
kappa cminus cplus delta h :
ℝ
hJ :
0 < J
hprob :
IsProbabilityMeasure P.dataMeasure
conclusion
(∫ z, atomEstimate B z x delta * sqrt (∑ i ∈ B.I2, (interceptWeight B z x ell kappa cminus cplus delta h i) ^ 2) ∂iidProduct P n)
= (∫ z, atomEstimate B z x delta ∂iidProduct P n) * ∫ z, sqrt (∑ i ∈ B.I2, (interceptWeight B z x ell kappa cminus cplus delta h i) ^ 2) ∂iidProduct P n
Proof (Lean source)
lemma integral_atomEstimate_mul_sqrt_weightEnergy
{J n ell : ℕ} (P : ClampLaw J) (B : SplitBlocks n) (x : Fin J)
(kappa cminus cplus delta h : ℝ) (hJ : 0 < J)
(hprob : IsProbabilityMeasure P.dataMeasure) :
(∫ z, atomEstimate B z x delta *
sqrt (∑ i ∈ B.I2,
(interceptWeight B z x ell kappa cminus cplus delta h i) ^ 2)
∂iidProduct P n) =
(∫ z, atomEstimate B z x delta ∂iidProduct P n) *
∫ z, sqrt (∑ i ∈ B.I2,
(interceptWeight B z x ell kappa cminus cplus delta h i) ^ 2)
∂iidProduct P n := by
classical
let o0 : ClampObs J := ⟨⟨0, hJ⟩, 0, 0⟩
let extendBlock (I : Finset (Fin n))
(u : (i : {i // i ∈ I}) → ClampObs J) : Fin n → ClampObs J :=
fun i => if hi : i ∈ I then u ⟨i, hi⟩ else o0
have hextMeas (I : Finset (Fin n)) : Measurable (extendBlock I) := by
apply measurable_pi_lambda
intro i
by_cases hi : i ∈ I
· simpa [extendBlock, hi] using
(measurable_pi_apply (⟨i, hi⟩ : {i // i ∈ I}))
· simp only [extendBlock, hi, dite_false]
exact measurable_const
let F : ((i : {i // i ∈ B.I1}) → ClampObs J) → ℝ := fun u =>
atomEstimate B (extendBlock B.I1 u) x delta
let G : ((i : {i // i ∈ B.I2}) → ClampObs J) → ℝ := fun u =>
sqrt (∑ i ∈ B.I2,
(interceptWeight B (extendBlock B.I2 u) x ell kappa cminus cplus delta h i) ^ 2)
have hF : Measurable F := by
exact (atomEstimate_measurable B x delta).comp (hextMeas B.I1)
have hG : Measurable G := by
apply Measurable.sqrt
exact Finset.measurable_fun_sum B.I2 fun i _ =>
((interceptWeight_measurable B x kappa cminus cplus delta h i).comp
(hextMeas B.I2)).pow_const 2
have hFproj (z : Fin n → ClampObs J) :
F (finsetCoordProj B.I1 z) = atomEstimate B z x delta := by
unfold F atomEstimate blockAverage
congr 1
apply Finset.sum_congr rfl
intro i hi
simp [extendBlock, finsetCoordProj, hi]
have hGproj (z : Fin n → ClampObs J) :
G (finsetCoordProj B.I2 z) =
sqrt (∑ i ∈ B.I2,
(interceptWeight B z x ell kappa cminus cplus delta h i) ^ 2) := by
unfold G
congr 1
apply Finset.sum_congr rfl
intro i hi
congr 1
exact (interceptWeight_congr_I2 (ell := ell) B
(extendBlock B.I2 (finsetCoordProj B.I2 z)) z x
kappa cminus cplus delta h
(by
intro j hj
dsimp only [extendBlock]
rw [dif_pos hj]
rfl) i hi).symm
have hfactor := integral_mul_I1_I2 P B hprob F G hF hG
calc
_ = ∫ z, F (finsetCoordProj B.I1 z) *
G (finsetCoordProj B.I2 z) ∂iidProduct P n := by
apply integral_congr_ae
filter_upwards with z
rw [hFproj, hGproj]
_ = (∫ z, F (finsetCoordProj B.I1 z) ∂iidProduct P n) *
∫ z, G (finsetCoordProj B.I2 z) ∂iidProduct P n := hfactor
_ = _ := by
congr 1
· apply integral_congr_ae
filter_upwards with z
rw [hFproj]
· apply integral_congr_ae
filter_upwards with z
rw [hGproj]
At the information-balanced bandwidth, the expected square-root weight energy has the same h^beta scale as the local stochastic error. The result uses the hmodel condition, the hsampling condition, the hn condition, the hbeta condition, the hkappa condition, the hcminus condition, the hcplus condition, the hpmin condition, the hdelta condition, the hh condition, the hupper condition, the hlambda condition, the hbalance condition. This is the stated conclusion.
Formal statement
hypotheses
J n ell :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta h :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hsampling :
IidSampling P n
B :
x :
Fin J
hn :
8 ≤ n
hbeta :
0 < beta
hkappa :
0 ≤ kappa
hcminus :
0 < cminus
hcplus :
0 ≤ cplus
hpmin :
0 < pmin
hdelta :
0 ≤ delta
hh :
0 < h
hupper :
delta + h ≤ 1
hlambda :
0 < lambdaStar ell kappa cminus cplus
hbalance :
(n : ℝ) * h ^ (2 * beta + 1) * (delta + h) ^ kappa = 1
conclusion
(∫ z, sqrt (∑ i ∈ B.I2, (interceptWeight B z x ell kappa cminus cplus delta h i) ^ 2) ∂iidProduct P n)
≤ sqrt (336 * (4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2) / (pmin * cminus / (2 * (2 : ℝ) ^ kappa))) * h ^ beta
Proof (Lean source)
lemma integral_sqrt_interceptWeight_energy_balanced_le
{J n ell : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin delta h : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hsampling : IidSampling P n) (B : SplitBlocks n) (x : Fin J)
(hn : 8 ≤ n) (hbeta : 0 < beta) (hkappa : 0 ≤ kappa)
(hcminus : 0 < cminus) (hcplus : 0 ≤ cplus) (hpmin : 0 < pmin)
(hdelta : 0 ≤ delta) (hh : 0 < h) (hupper : delta + h ≤ 1)
(hlambda : 0 < lambdaStar ell kappa cminus cplus)
(hbalance : (n : ℝ) * h ^ (2 * beta + 1) *
(delta + h) ^ kappa = 1) :
(∫ z, sqrt (∑ i ∈ B.I2,
(interceptWeight B z x ell kappa cminus cplus delta h i) ^ 2)
∂iidProduct P n) ≤
sqrt
(336 * (4 * (ell + 1 : ℝ) /
lambdaStar ell kappa cminus cplus ^ 2) /
(pmin * cminus / (2 * (2 : ℝ) ^ kappa))) * h ^ beta := by
let p := ∫ o, localWindowWeight x delta h o ∂P.dataMeasure
let D := 4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2
let K := pmin * cminus / (2 * (2 : ℝ) ^ kappa)
have hK : 0 < K := by dsimp [K]; positivity
have hpLower : K * h * (delta + h) ^ kappa ≤ p := by
have hl := localWindowWeight_integral_lower P hmodel hkappa hcminus hcplus
hpmin x hdelta hh hupper
have hdh : 0 < delta + h := by linarith
have hrpow : ((delta + h) / 2) ^ kappa =
(delta + h) ^ kappa / (2 : ℝ) ^ kappa := by
rw [Real.div_rpow hdh.le (by norm_num : (0 : ℝ) ≤ 2)]
rw [hrpow] at hl
dsimp [K, p]
convert hl using 1 <;> ring
have hp0 : 0 < p := lt_of_lt_of_le (by
exact mul_pos (mul_pos hK hh)
(Real.rpow_pos_of_pos (by linarith) _)) hpLower
have hcardNat : 0 < B.I2.card := lt_of_lt_of_le (by omega) B.card_I2
have hcard : (n : ℝ) / 8 ≤ (B.I2.card : ℝ) := by
have hqcard := B.card_I2
have hncard : n ≤ 8 * B.I2.card := by omega
have hncardR : (n : ℝ) ≤ 8 * (B.I2.card : ℝ) := by exact_mod_cast hncard
linarith
have hraw := integral_sqrt_interceptWeight_energy_le P B x kappa cminus
cplus delta h p hsampling hh hlambda hp0 hcardNat rfl
have hD : 0 ≤ D := by dsimp [D]; positivity
have hrate := balancedEffectiveSample_sqrt_le (D := D) (K := K)
(p := p) (m := (B.I2.card : ℝ)) (Nat.zero_lt_of_lt hn) hh hD hK
hcard hpLower hbalance
dsimp [D, K] at hrate
exact hraw.trans hrate
theorem integral_stratumRadius_le_factored
Integrated version of the good/bad pointwise radius decomposition. The mixed stochastic term factors exactly by sample splitting. The result uses the hJ condition, the hprob condition, the hbeta condition, the hL condition, the hh condition, the ht condition, the hb1 condition, the hlambda condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
B :
x :
Fin J
beta kappa L cminus cplus delta h t b1 :
ℝ
hJ :
0 < J
hprob :
IsProbabilityMeasure P.dataMeasure
hbeta :
0 ≤ beta
hL :
0 ≤ L
hh :
0 < h
ht :
0 ≤ t
hb1 :
0 ≤ b1
hlambda :
0 < lambdaStar (ellOf beta) kappa cminus cplus
conclusion
(∫ z, stratumRadius B z x (ellOf beta) beta kappa L cminus cplus delta h t b1 ∂iidProduct P n)
≤ ((∫ z, atomEstimate B z x delta ∂iidProduct P n) + b1) * (L * h ^ beta * (2 * sqrt (ellOf beta + 1 : ℝ) / lambdaStar (ellOf beta) kappa cminus cplus) + t * (∫ z, sqrt (∑ i ∈ B.I2, (interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i) ^ 2) ∂iidProduct P n))
+ b1
Proof (Lean source)
lemma integral_stratumRadius_le_factored
{J n : ℕ} (P : ClampLaw J) (B : SplitBlocks n) (x : Fin J)
(beta kappa L cminus cplus delta h t b1 : ℝ)
(hJ : 0 < J) (hprob : IsProbabilityMeasure P.dataMeasure)
(hbeta : 0 ≤ beta) (hL : 0 ≤ L) (hh : 0 < h)
(ht : 0 ≤ t) (hb1 : 0 ≤ b1)
(hlambda : 0 < lambdaStar (ellOf beta) kappa cminus cplus) :
(∫ z, stratumRadius B z x (ellOf beta) beta kappa L cminus cplus
delta h t b1 ∂iidProduct P n) ≤
((∫ z, atomEstimate B z x delta ∂iidProduct P n) + b1) *
(L * h ^ beta *
(2 * sqrt (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus) +
t * (∫ z, sqrt (∑ i ∈ B.I2,
(interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i) ^ 2)
∂iidProduct P n)) +
b1 + (iidProduct P n).real {z |
¬ GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h} := by
letI : IsProbabilityMeasure P.dataMeasure := hprob
letI : IsProbabilityMeasure (iidProduct P n) := by unfold iidProduct; infer_instance
let energy : (Fin n → ClampObs J) → ℝ := fun z => sqrt (∑ i ∈ B.I2,
(interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i) ^ 2)
let bad : Set (Fin n → ClampObs J) := {z |
¬ GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h}
let bias := L * h ^ beta * (2 * sqrt (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus)
let rhs : (Fin n → ClampObs J) → ℝ := fun z =>
(atomEstimate B z x delta + b1) * (bias + t * energy z) + b1 +
bad.indicator (fun _ => (1 : ℝ)) z
have henergyMeas : Measurable energy := by
exact (Finset.measurable_fun_sum B.I2 fun i _ =>
(interceptWeight_measurable B x kappa cminus cplus delta h i).pow_const 2).sqrt
have henergyBound : ∀ z, energy z ≤ sqrt
(4 * (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus ^ 2) := by
intro z
apply Real.sqrt_le_sqrt
have he := localRegression_weight_energy_le (ell := ellOf beta) B z x
kappa cminus cplus delta h hh hlambda
simpa only [localRegressionDesignWeight_apply, ite_pow,
zero_pow (by norm_num : 2 ≠ 0), ← Finset.sum_filter,
Finset.filter_mem_eq_inter, Finset.univ_inter] using he
have henergyInt : Integrable energy (iidProduct P n) := by
refine Integrable.of_bound henergyMeas.aestronglyMeasurable
(sqrt (4 * (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus ^ 2)) ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (Real.sqrt_nonneg _)]
exact henergyBound z
have hatomInt : Integrable (fun z : Fin n → ClampObs J =>
atomEstimate B z x delta) (iidProduct P n) := by
refine Integrable.of_bound (atomEstimate_measurable B x delta).aestronglyMeasurable 1 ?_
filter_upwards with z
have hz := atomEstimate_mem_Icc B z x delta
rw [Real.norm_eq_abs, abs_of_nonneg hz.1]
exact hz.2
have hbadMeas : MeasurableSet bad :=
(goodGramEvent_measurable B x kappa cminus cplus delta h).compl
have hbadInt : Integrable (bad.indicator (fun _ => (1 : ℝ))) (iidProduct P n) :=
(integrable_const 1).indicator hbadMeas
have hrhsInt : Integrable rhs (iidProduct P n) := by
dsimp [rhs]
have hleft := hatomInt.add (integrable_const b1)
have hright := (integrable_const bias).add (henergyInt.const_mul t)
exact (hleft.mul_bdd hright.aestronglyMeasurable (by
filter_upwards with z
rw [Real.norm_eq_abs]
have hz := atomEstimate_mem_Icc B z x delta
change |bias + t * energy z| ≤ _
rw [abs_of_nonneg (by positivity : 0 ≤ bias + t * energy z)]
exact add_le_add le_rfl (mul_le_mul_of_nonneg_left (henergyBound z) ht))).add
(integrable_const b1) |>.add hbadInt
have hradMeas : Measurable (fun z : Fin n → ClampObs J =>
stratumRadius B z x (ellOf beta) beta kappa L cminus cplus delta h t b1) :=
honest_stratumRadius_measurable B x beta kappa L cminus cplus delta h t b1
have hpoint (z : Fin n → ClampObs J) :
stratumRadius B z x (ellOf beta) beta kappa L cminus cplus delta h t b1 ≤ rhs z :=
stratumRadius_le_good_expression_add_bad B z x beta kappa L cminus cplus
delta h t b1 hbeta hL hh ht hb1 hlambda
have hrad0 (z : Fin n → ClampObs J) : 0 ≤
stratumRadius B z x (ellOf beta) beta kappa L cminus cplus delta h t b1 := by
unfold stratumRadius atomEstimate blockAverage
split_ifs <;> positivity
have hradInt : Integrable (fun z : Fin n → ClampObs J =>
stratumRadius B z x (ellOf beta) beta kappa L cminus cplus delta h t b1)
(iidProduct P n) := hrhsInt.mono' hradMeas.aestronglyMeasurable (by
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (hrad0 z)]
exact hpoint z)
have hmixInt : Integrable (fun z : Fin n → ClampObs J =>
atomEstimate B z x delta * energy z) (iidProduct P n) :=
henergyInt.bdd_mul (atomEstimate_measurable B x delta).aestronglyMeasurable (by
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (atomEstimate_mem_Icc B z x delta).1]
exact (atomEstimate_mem_Icc B z x delta).2)
have hfactor := integral_atomEstimate_mul_sqrt_weightEnergy (ell := ellOf beta)
P B x kappa cminus
cplus delta h hJ hprob
change (∫ z, atomEstimate B z x delta * energy z ∂iidProduct P n) = _ at hfactor
have hprodIntegral :
(∫ z, (atomEstimate B z x delta + b1) * (bias + t * energy z)
∂iidProduct P n) =
((∫ z, atomEstimate B z x delta ∂iidProduct P n) + b1) *
(bias + t * ∫ z, energy z ∂iidProduct P n) := by
have hA : Integrable (fun z : Fin n → ClampObs J =>
bias * atomEstimate B z x delta) (iidProduct P n) :=
hatomInt.const_mul bias
have hB : Integrable (fun z : Fin n → ClampObs J =>
t * (atomEstimate B z x delta * energy z)) (iidProduct P n) :=
hmixInt.const_mul t
have hC : Integrable (fun _z : Fin n → ClampObs J => b1 * bias)
(iidProduct P n) := integrable_const _
have hD : Integrable (fun z : Fin n → ClampObs J => b1 * t * energy z)
(iidProduct P n) := by
simpa only [mul_assoc] using henergyInt.const_mul (b1 * t)
rw [show (fun z : Fin n → ClampObs J =>
(atomEstimate B z x delta + b1) * (bias + t * energy z)) =
fun z => bias * atomEstimate B z x delta +
t * (atomEstimate B z x delta * energy z) +
b1 * bias + b1 * t * energy z by funext z; ring]
have hABCD := integral_add ((hA.add hB).add hC) hD
have hABC := integral_add (hA.add hB) hC
have hAB := integral_add hA hB
calc
_ = (∫ z, bias * atomEstimate B z x delta +
t * (atomEstimate B z x delta * energy z) + b1 * bias
∂iidProduct P n) +
∫ z, b1 * t * energy z ∂iidProduct P n := by
simpa only [Pi.add_apply] using hABCD
_ = ((∫ z, bias * atomEstimate B z x delta +
t * (atomEstimate B z x delta * energy z) ∂iidProduct P n) +
∫ _z, b1 * bias ∂iidProduct P n) +
∫ z, b1 * t * energy z ∂iidProduct P n := by
rw [show (∫ z, bias * atomEstimate B z x delta +
t * (atomEstimate B z x delta * energy z) + b1 * bias
∂iidProduct P n) = _ by simpa only [Pi.add_apply] using hABC]
_ = (((∫ z, bias * atomEstimate B z x delta ∂iidProduct P n) +
∫ z, t * (atomEstimate B z x delta * energy z) ∂iidProduct P n) +
∫ _z, b1 * bias ∂iidProduct P n) +
∫ z, b1 * t * energy z ∂iidProduct P n := by
rw [show (∫ z, bias * atomEstimate B z x delta +
t * (atomEstimate B z x delta * energy z) ∂iidProduct P n) = _ by
simpa only [Pi.add_apply] using hAB]
_ = _ := by
rw [integral_const_mul, integral_const_mul, integral_const,
integral_const_mul, hfactor]
simp
ring
calc
_ ≤ ∫ z, rhs z ∂iidProduct P n := integral_mono hradInt hrhsInt hpoint
_ = _ := by
dsimp [rhs]
have hprodInt : Integrable (fun z : Fin n → ClampObs J =>
(atomEstimate B z x delta + b1) * (bias + t * energy z))
(iidProduct P n) :=
(hatomInt.add (integrable_const b1)).mul_bdd
((integrable_const bias).add (henergyInt.const_mul t)).aestronglyMeasurable
(by
filter_upwards with z
rw [Real.norm_eq_abs]
change |bias + t * energy z| ≤ _
rw [abs_of_nonneg (by positivity : 0 ≤ bias + t * energy z)]
exact add_le_add le_rfl
(mul_le_mul_of_nonneg_left (henergyBound z) ht))
calc
(∫ z, (atomEstimate B z x delta + b1) * (bias + t * energy z) + b1 +
bad.indicator (fun _ => (1 : ℝ)) z ∂iidProduct P n) =
(∫ z, (atomEstimate B z x delta + b1) * (bias + t * energy z) + b1
∂iidProduct P n) + ∫ z, bad.indicator (fun _ => (1 : ℝ)) z
∂iidProduct P n := integral_add
(hprodInt.add (integrable_const b1)) hbadInt
_ = ((∫ z, (atomEstimate B z x delta + b1) * (bias + t * energy z)
∂iidProduct P n) + ∫ _z, b1 ∂iidProduct P n) +
∫ z, bad.indicator (fun _ => (1 : ℝ)) z ∂iidProduct P n := by
rw [integral_add hprodInt (integrable_const b1)]
_ = _ := by
rw [hprodIntegral, integral_const, integral_indicator hbadMeas, integral_const]
simp [measureReal_def, energy, bad, bias]
theorem stratumRadius_integrable
The displayed per-stratum radius is integrable under every product law. This is the bookkeeping input needed to integrate the finite sum of radii. The result uses the hprob condition, the hbeta condition, the hL condition, the hh condition, the ht condition, the hb1 condition, the hlambda condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
B :
x :
Fin J
beta kappa L cminus cplus delta h t b1 :
ℝ
hprob :
IsProbabilityMeasure P.dataMeasure
hbeta :
0 ≤ beta
hL :
0 ≤ L
hh :
0 < h
ht :
0 ≤ t
hb1 :
0 ≤ b1
hlambda :
0 < lambdaStar (ellOf beta) kappa cminus cplus
conclusion
Integrable (fun z : Fin n → ClampObs J => stratumRadius B z x (ellOf beta) beta kappa L cminus cplus delta h t b1) (iidProduct P n)
Proof (Lean source)
lemma stratumRadius_integrable
{J n : ℕ} (P : ClampLaw J) (B : SplitBlocks n) (x : Fin J)
(beta kappa L cminus cplus delta h t b1 : ℝ)
(hprob : IsProbabilityMeasure P.dataMeasure)
(hbeta : 0 ≤ beta) (hL : 0 ≤ L) (hh : 0 < h)
(ht : 0 ≤ t) (hb1 : 0 ≤ b1)
(hlambda : 0 < lambdaStar (ellOf beta) kappa cminus cplus) :
Integrable (fun z : Fin n → ClampObs J =>
stratumRadius B z x (ellOf beta) beta kappa L cminus cplus
delta h t b1) (iidProduct P n) := by
letI : IsProbabilityMeasure P.dataMeasure := hprob
letI : IsProbabilityMeasure (iidProduct P n) := by unfold iidProduct; infer_instance
let energy : (Fin n → ClampObs J) → ℝ := fun z => sqrt (∑ i ∈ B.I2,
(interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i) ^ 2)
have henergyMeas : Measurable energy := by
exact (Finset.measurable_fun_sum B.I2 fun i _ =>
(interceptWeight_measurable B x kappa cminus cplus delta h i).pow_const 2).sqrt
have henergyBound : ∀ z, energy z ≤ sqrt
(4 * (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus ^ 2) := by
intro z
apply Real.sqrt_le_sqrt
have he := localRegression_weight_energy_le (ell := ellOf beta) B z x
kappa cminus cplus delta h hh hlambda
simpa only [localRegressionDesignWeight_apply, ite_pow,
zero_pow (by norm_num : 2 ≠ 0), ← Finset.sum_filter,
Finset.filter_mem_eq_inter, Finset.univ_inter] using he
have henergyInt : Integrable energy (iidProduct P n) := by
refine Integrable.of_bound henergyMeas.aestronglyMeasurable
(sqrt (4 * (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus ^ 2)) ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (Real.sqrt_nonneg _)]
exact henergyBound z
have hatomInt : Integrable (fun z : Fin n → ClampObs J =>
atomEstimate B z x delta) (iidProduct P n) := by
refine Integrable.of_bound
(atomEstimate_measurable B x delta).aestronglyMeasurable 1 ?_
filter_upwards with z
have hz := atomEstimate_mem_Icc B z x delta
rw [Real.norm_eq_abs, abs_of_nonneg hz.1]
exact hz.2
let bad : Set (Fin n → ClampObs J) := {z |
¬ GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h}
let bias := L * h ^ beta * (2 * sqrt (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus)
let rhs : (Fin n → ClampObs J) → ℝ := fun z =>
(atomEstimate B z x delta + b1) * (bias + t * energy z) + b1 +
bad.indicator (fun _ => (1 : ℝ)) z
have hrhsInt : Integrable rhs (iidProduct P n) := by
have hbadMeas : MeasurableSet bad :=
(goodGramEvent_measurable B x kappa cminus cplus delta h).compl
dsimp [rhs]
exact (((hatomInt.add (integrable_const b1)).mul_bdd
((integrable_const bias).add (henergyInt.const_mul t)).aestronglyMeasurable
(by
filter_upwards with z
rw [Real.norm_eq_abs]
change |bias + t * energy z| ≤ _
rw [abs_of_nonneg (by positivity : 0 ≤ bias + t * energy z)]
exact add_le_add le_rfl (mul_le_mul_of_nonneg_left (henergyBound z) ht)))
|>.add (integrable_const b1)).add ((integrable_const 1).indicator hbadMeas)
have hradMeas := honest_stratumRadius_measurable (ell := ellOf beta)
B x beta kappa L cminus cplus
delta h t b1
refine hrhsInt.mono' hradMeas.aestronglyMeasurable ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (by
unfold stratumRadius atomEstimate blockAverage
split_ifs <;> positivity)]
dsimp [rhs, bias, energy, bad]
exact stratumRadius_le_good_expression_add_bad B z x beta kappa L cminus cplus
delta h t b1 hbeta hL hh ht hb1 hlambda
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.HonestPointwise 4 declarations Deterministic error event for the bias-aware interval
Deterministic error event for the bias-aware interval
theorem localRegression_holderBias_le_empirical
The intermediate, realized-weight form of the Hölder bias bound. This is the form used by the honest interval, before replacing the realized l1 weight norm by its deterministic good-Gram upper bound. The result uses the hmodel condition, the hbeta condition, the hL condition, the hdelta condition, the hh condition, the hupper condition, the hlambda condition, the hgood condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta h :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hbeta :
0 < beta
hL :
0 < L
B :
x :
Fin J
hdelta :
0 ≤ delta
hh :
0 < h
hupper :
delta + h ≤ 1
hlambda :
0 < lambdaStar (ellOf beta) kappa cminus cplus
hgood :
GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h
conclusion
|(∑ i ∈ B.I2, interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i * clampRegressionExtension P ((z i).X, (z i).A))
- P.mu x delta|
≤ L * h ^ beta * ∑ i ∈ B.I2, |interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i|
Proof (Lean source)
lemma localRegression_holderBias_le_empirical {J n : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin delta h : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hbeta : 0 < beta) (hL : 0 < L)
(B : SplitBlocks n) (z : Fin n → ClampObs J) (x : Fin J)
(hdelta : 0 ≤ delta) (hh : 0 < h) (hupper : delta + h ≤ 1)
(hlambda : 0 < lambdaStar (ellOf beta) kappa cminus cplus)
(hgood : GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h) :
|(∑ i ∈ B.I2, interceptWeight B z x (ellOf beta) kappa cminus cplus
delta h i * clampRegressionExtension P ((z i).X, (z i).A)) -
P.mu x delta| ≤
L * h ^ beta * ∑ i ∈ B.I2,
|interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i| := by
classical
let ell := ellOf beta
let w : Fin n → ℝ := fun i =>
interceptWeight B z x ell kappa cminus cplus delta h i
let T : ℝ → ℝ := fun a => ∑ j ∈ range (ell + 1),
(iteratedDerivWithin j (P.mu x) (Icc (0 : ℝ) 1) delta /
(Nat.factorial j : ℝ)) * (a - delta) ^ j
have hdeltaI : delta ∈ Icc (0 : ℝ) 1 := ⟨hdelta, by linarith⟩
have hrepro := (goodGram_weight_controls B z x kappa cminus cplus delta h
hh hlambda hgood).1
have hpoly : ∑ i ∈ B.I2, w i * T (z i).A = P.mu x delta := by
have hp := honest_weightedPolynomial_reproduce B.I2 w (fun i => (z i).A)
(fun j => iteratedDerivWithin j (P.mu x) (Icc (0 : ℝ) 1) delta /
(Nat.factorial j : ℝ)) delta h hh hrepro
simpa [T, ell, w] using hp
have hrewrite :
(∑ i ∈ B.I2, w i * clampRegressionExtension P ((z i).X, (z i).A)) -
P.mu x delta =
∑ i ∈ B.I2, w i * (P.mu x (z i).A - T (z i).A) := by
rw [← hpoly, ← Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro i hi
by_cases ha : (z i).X = x ∧ scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1
· have hu := ha.2
have hAlo : delta ≤ (z i).A := by
have hs := (div_nonneg_iff.mp hu.1).resolve_right
(fun hn => (not_lt_of_ge hn.2 hh))
exact sub_nonneg.mp hs.1
have hAhi : (z i).A ≤ delta + h := by
have hs := (div_le_iff₀ hh).mp hu.2
linarith
have hAI : (z i).A ∈ Icc (0 : ℝ) 1 :=
⟨hdelta.trans hAlo, hAhi.trans hupper⟩
rw [show (z i).X = x from ha.1, clampRegressionExtension_eq P x hAI]
simp only [w]
ring
· have hw0 : w i = 0 := by
dsimp [w, ell]
exact interceptWeight_eq_zero_of_inactive B z x (ellOf beta)
kappa cminus cplus delta h i hgood (fun hi => ha ⟨hi.2.1, hi.2.2⟩)
simp [hw0]
rw [hrewrite]
calc
|∑ i ∈ B.I2, w i * (P.mu x (z i).A - T (z i).A)| ≤
∑ i ∈ B.I2, |w i| * |P.mu x (z i).A - T (z i).A| := by
exact (Finset.abs_sum_le_sum_abs _ _).trans_eq (by
apply Finset.sum_congr rfl
intro i hi
rw [abs_mul])
_ ≤ ∑ i ∈ B.I2, |w i| * (L * h ^ beta) := by
apply Finset.sum_le_sum
intro i hi
by_cases ha : (z i).X = x ∧ scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1
· apply mul_le_mul_of_nonneg_left _ (abs_nonneg _)
have hu := ha.2
have hAlo : delta ≤ (z i).A := by
have hs := (div_nonneg_iff.mp hu.1).resolve_right
(fun hn => (not_lt_of_ge hn.2 hh))
exact sub_nonneg.mp hs.1
have hAhi : (z i).A ≤ delta + h := by
have hs := (div_le_iff₀ hh).mp hu.2
linarith
have hAI : (z i).A ∈ Icc (0 : ℝ) 1 :=
⟨hdelta.trans hAlo, hAhi.trans hupper⟩
have hrem := (hmodel.holder x).2.2.2 delta hdeltaI (z i).A hAI
have habs : |(z i).A - delta| ≤ h := by
rw [abs_of_nonneg (sub_nonneg.mpr hAlo)]
linarith
have hpw := Real.rpow_le_rpow (abs_nonneg _) habs hbeta.le
have hrem' : |P.mu x (z i).A - T (z i).A| ≤
L * |(z i).A - delta| ^ beta := by
simpa [T, ell, div_eq_mul_inv, mul_comm, mul_left_comm, mul_assoc] using hrem
exact hrem'.trans (mul_le_mul_of_nonneg_left hpw hL.le)
· have hw0 : w i = 0 := by
dsimp [w, ell]
exact interceptWeight_eq_zero_of_inactive B z x (ellOf beta)
kappa cminus cplus delta h i hgood (fun hi => ha ⟨hi.2.1, hi.2.2⟩)
simp [hw0]
_ = L * h ^ beta * ∑ i ∈ B.I2, |w i| := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro i hi
ring
theorem localRegression_good_error_le_empirical
Good-Gram local error decomposition retaining the empirical Hölder radius. The result uses the hmodel condition, the hbeta condition, the hL condition, the hdelta condition, the hh condition, the hupper condition, the hlambda condition, the hgood condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta h :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hbeta :
0 < beta
hL :
0 < L
B :
x :
Fin J
hdelta :
0 ≤ delta
hh :
0 < h
hupper :
delta + h ≤ 1
hlambda :
0 < lambdaStar (ellOf beta) kappa cminus cplus
hgood :
GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h
conclusion
|localRegressionEstimate B z x (ellOf beta) kappa cminus cplus delta h - P.mu x delta|
≤ L * h ^ beta * ∑ i ∈ B.I2, |interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i|
+ |∑ i ∈ B.I2, interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i * ((z i).Y - clampRegressionExtension P ((z i).X, (z i).A))|
Proof (Lean source)
lemma localRegression_good_error_le_empirical {J n : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin delta h : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hbeta : 0 < beta) (hL : 0 < L)
(B : SplitBlocks n) (z : Fin n → ClampObs J) (x : Fin J)
(hdelta : 0 ≤ delta) (hh : 0 < h) (hupper : delta + h ≤ 1)
(hlambda : 0 < lambdaStar (ellOf beta) kappa cminus cplus)
(hgood : GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h) :
|localRegressionEstimate B z x (ellOf beta) kappa cminus cplus delta h -
P.mu x delta| ≤
L * h ^ beta * ∑ i ∈ B.I2,
|interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i| +
|∑ i ∈ B.I2,
interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
((z i).Y -
clampRegressionExtension P ((z i).X, (z i).A))| := by
have hmu := (hmodel.holder x).2.1 delta ⟨hdelta, by linarith⟩
rw [localRegressionEstimate, if_pos hgood]
change |clampUnit (∑ i ∈ B.I2,
interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
(z i).Y) - P.mu x delta| ≤ _
refine (abs_clampUnit_sub_le _ _ hmu).trans ?_
have hbias := localRegression_holderBias_le_empirical P hmodel hbeta hL B z x
hdelta hh hupper hlambda hgood
have htri := abs_add_le
(∑ i ∈ B.I2, interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
((z i).Y - clampRegressionExtension P ((z i).X, (z i).A)))
((∑ i ∈ B.I2, interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
clampRegressionExtension P ((z i).X, (z i).A)) - P.mu x delta)
have hid :
(∑ i ∈ B.I2, interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
(z i).Y) - P.mu x delta =
(∑ i ∈ B.I2, interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
((z i).Y - clampRegressionExtension P ((z i).X, (z i).A))) +
((∑ i ∈ B.I2, interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
clampRegressionExtension P ((z i).X, (z i).A)) - P.mu x delta) := by
simp_rw [mul_sub, Finset.sum_sub_distrib]
ring
rw [hid]
exact htri.trans (by linarith)
theorem totalGramEstimator_error_le_honestRadius
Retained-mean, atom-mass, and good-Gram local-regression deviations imply that the total estimator lies within the displayed honest radius. The result uses the hmodel condition, the hdelta condition, the hdelta1 condition, the hh condition, the hbeta condition, the hL condition, the htAlpha condition, the hb0 condition, the hb1 condition, the hret condition, the hatom condition, the hlocal condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta h tAlpha b0 b1 :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hdelta :
0 ≤ delta
hdelta1 :
delta ≤ 1
hh :
0 < h
hbeta :
0 ≤ beta
hL :
0 ≤ L
htAlpha :
0 ≤ tAlpha
hb0 :
0 ≤ b0
hb1 :
0 ≤ b1
B :
hret :
|retainedEstimate B z delta - retainedMean P delta| ≤ b0
hatom :
hlocal :
∀ x : Fin J
if
GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h
then
|localRegressionEstimate B z x (ellOf beta) kappa cminus cplus delta h - P.mu x delta|
≤ L * h ^ beta * ∑ i ∈ B.I2, |interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i|
conclusion
≤ b0
Proof (Lean source)
lemma totalGramEstimator_error_le_honestRadius
{J n : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin delta h tAlpha b0 b1 : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hdelta : 0 ≤ delta) (hdelta1 : delta ≤ 1) (hh : 0 < h)
(hbeta : 0 ≤ beta) (hL : 0 ≤ L) (htAlpha : 0 ≤ tAlpha)
(hb0 : 0 ≤ b0) (hb1 : 0 ≤ b1)
(B : SplitBlocks n) (z : Fin n → ClampObs J)
(hret : |retainedEstimate B z delta - retainedMean P delta| ≤ b0)
(hatom : ∀ x : Fin J,
|atomEstimate B z x delta - P.px x * atomMass P x delta| ≤ b1)
(hlocal : ∀ x : Fin J,
GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h →
|localRegressionEstimate B z x (ellOf beta) kappa cminus cplus delta h -
P.mu x delta| ≤
L * h ^ beta * ∑ i ∈ B.I2,
|interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i| +
tAlpha * sqrt (∑ i ∈ B.I2,
(interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i) ^ 2)) :
|totalGramEstimator B z (ellOf beta) kappa cminus cplus delta h -
clampFunctional P delta| ≤
b0 + ∑ x : Fin J,
stratumRadius B z x (ellOf beta) beta kappa L cminus cplus delta h
tAlpha b1 := by
let center := retainedEstimate B z delta + ∑ x : Fin J,
atomEstimate B z x delta * localRegressionEstimate B z x (ellOf beta)
kappa cminus cplus delta h
have htarget := clampFunctional_mem_Icc P hmodel hdelta hdelta1
have hclamp := abs_clampUnit_sub_le center (clampFunctional P delta) htarget
have hbase : |totalGramEstimator B z (ellOf beta) kappa cminus cplus delta h -
clampFunctional P delta| ≤
|retainedEstimate B z delta - retainedMean P delta| +
∑ x : Fin J, |atomEstimate B z x delta *
localRegressionEstimate B z x (ellOf beta) kappa cminus cplus delta h -
P.px x * atomMass P x delta * P.mu x delta| := by
calc
_ ≤ |center - clampFunctional P delta| := by
simpa [totalGramEstimator, center] using hclamp
_ = |(retainedEstimate B z delta - retainedMean P delta) +
∑ x : Fin J, (atomEstimate B z x delta *
localRegressionEstimate B z x (ellOf beta) kappa cminus cplus delta h -
P.px x * atomMass P x delta * P.mu x delta)| := by
congr 1
simp only [clampFunctional, Finset.sum_sub_distrib]
ring
_ ≤ |retainedEstimate B z delta - retainedMean P delta| +
|∑ x : Fin J, (atomEstimate B z x delta *
localRegressionEstimate B z x (ellOf beta) kappa cminus cplus delta h -
P.px x * atomMass P x delta * P.mu x delta)| := abs_add_le _ _
_ ≤ _ := by
simpa [add_comm] using add_le_add_left
(Finset.abs_sum_le_sum_abs
(fun x : Fin J => atomEstimate B z x delta *
localRegressionEstimate B z x (ellOf beta) kappa cminus cplus delta h -
P.px x * atomMass P x delta * P.mu x delta) univ)
|retainedEstimate B z delta - retainedMean P delta|
refine hbase.trans (add_le_add hret ?_)
apply Finset.sum_le_sum
intro x hx
let a := P.px x * atomMass P x delta
let ah := atomEstimate B z x delta
have haI := atomCoefficient_mem_Icc P hmodel x hdelta1
have hahI := atomEstimate_mem_Icc B z x delta
have ha0 : 0 ≤ a := by simpa [a] using haI.1
have hah0 : 0 ≤ ah := by simpa [ah] using hahI.1
have haUpper : a ≤ ah + b1 := by
have habs := hatom x
dsimp [a, ah] at habs ⊢
rw [abs_le] at habs
linarith
by_cases hg : GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h
· rw [stratumRadius, if_pos hg]
let R := L * h ^ beta * ∑ i ∈ B.I2,
|interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i| +
tAlpha * sqrt (∑ i ∈ B.I2,
(interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i) ^ 2)
have hR0 : 0 ≤ R := by
dsimp [R]
have hsumAbs : 0 ≤ ∑ i ∈ B.I2,
|interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i| :=
sum_nonneg fun _ _ => abs_nonneg _
have hsumSq : 0 ≤ ∑ i ∈ B.I2,
(interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i) ^ 2 :=
sum_nonneg fun _ _ => sq_nonneg _
exact add_nonneg
(mul_nonneg (mul_nonneg hL (Real.rpow_nonneg hh.le _)) hsumAbs)
(mul_nonneg htAlpha (Real.sqrt_nonneg _))
have hmhatI := localRegressionEstimate_mem_Icc (ell := ellOf beta)
B z x kappa cminus cplus delta h
have hmuI := (hmodel.holder x).2.1 delta ⟨hdelta, hdelta1⟩
have hprod : |ah * localRegressionEstimate B z x (ellOf beta) kappa
cminus cplus delta h - a * P.mu x delta| ≤ b1 + (ah + b1) * R := by
have hid : ah * localRegressionEstimate B z x (ellOf beta) kappa cminus
cplus delta h - a * P.mu x delta =
(ah - a) * localRegressionEstimate B z x (ellOf beta) kappa cminus
cplus delta h + a * (localRegressionEstimate B z x (ellOf beta)
kappa cminus cplus delta h - P.mu x delta) := by ring
rw [hid]
calc
_ ≤ |ah - a| * |localRegressionEstimate B z x (ellOf beta) kappa
cminus cplus delta h| +
a * |localRegressionEstimate B z x (ellOf beta) kappa cminus
cplus delta h - P.mu x delta| := by
simpa [abs_mul, abs_of_nonneg ha0] using abs_add_le
((ah - a) * localRegressionEstimate B z x (ellOf beta) kappa
cminus cplus delta h)
(a * (localRegressionEstimate B z x (ellOf beta) kappa cminus
cplus delta h - P.mu x delta))
_ ≤ b1 + a * R := by
apply add_le_add
· exact mul_le_of_le_one_right (abs_nonneg _) (by
rw [abs_of_nonneg hmhatI.1]
exact hmhatI.2) |>.trans (hatom x)
· exact mul_le_mul_of_nonneg_left (hlocal x hg) ha0
_ ≤ b1 + (ah + b1) * R := by
simpa [add_comm] using
add_le_add_left (mul_le_mul_of_nonneg_right haUpper hR0) b1
simpa [a, ah, R, add_comm] using hprod
· rw [stratumRadius, if_neg hg]
have hlocalI := localRegressionEstimate_mem_Icc (ell := ellOf beta)
B z x kappa cminus cplus delta h
have hmuI := (hmodel.holder x).2.1 delta ⟨hdelta, hdelta1⟩
have herr : |localRegressionEstimate B z x (ellOf beta) kappa cminus cplus
delta h - P.mu x delta| ≤ 1 := by
rw [abs_le]
constructor <;> linarith [hlocalI.1, hlocalI.2, hmuI.1, hmuI.2]
let u := ah * localRegressionEstimate B z x (ellOf beta) kappa cminus
cplus delta h
let v := a * P.mu x delta
have hu0 : 0 ≤ u := mul_nonneg hah0 hlocalI.1
have hv0 : 0 ≤ v := mul_nonneg ha0 hmuI.1
have huu : u ≤ ah := by dsimp [u]; nlinarith [hlocalI.2]
have hvv : v ≤ a := by dsimp [v]; nlinarith [hmuI.2]
have huv : |u - v| ≤ max ah a := by
rw [abs_le]
constructor
· linarith [hu0, hvv, le_max_right ah a]
· linarith [hv0, huu, le_max_left ah a]
have hmax : max ah a ≤ ah + b1 := max_le (by linarith) haUpper
simpa [u, v, a, ah] using huv.trans hmax
theorem clampFunctional_mem_honestInterval_of_deviations
The deterministic deviations used in the coverage proof put the target inside the clipped bias-aware interval. The result uses the hmodel condition, the hdelta condition, the hdelta1 condition, the hh condition, the hbeta condition, the hL condition, the htAlpha condition, the hb0 condition, the hb1 condition, the htdef condition, the hb0def condition, the hb1def condition, the hret condition, the hatom condition, the hlocal condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta h alpha tAlpha b0 b1 :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hdelta :
0 ≤ delta
hdelta1 :
delta ≤ 1
hh :
0 < h
hbeta :
0 ≤ beta
hL :
0 ≤ L
htAlpha :
0 ≤ tAlpha
hb0 :
0 ≤ b0
hb1 :
0 ≤ b1
B :
hb0def :
b0 = tAlpha / sqrt (B.I0.card : ℝ)
hb1def :
b1 = tAlpha / sqrt (B.I1.card : ℝ)
hret :
|retainedEstimate B z delta - retainedMean P delta| ≤ b0
hatom :
hlocal :
∀ x : Fin J
if
GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h
then
|localRegressionEstimate B z x (ellOf beta) kappa cminus cplus delta h - P.mu x delta|
≤ L * h ^ beta * ∑ i ∈ B.I2, |interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i|
conclusion
Proof (Lean source)
lemma clampFunctional_mem_honestInterval_of_deviations
{J n : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin delta h alpha tAlpha b0 b1 : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hdelta : 0 ≤ delta) (hdelta1 : delta ≤ 1) (hh : 0 < h)
(hbeta : 0 ≤ beta) (hL : 0 ≤ L) (htAlpha : 0 ≤ tAlpha)
(hb0 : 0 ≤ b0) (hb1 : 0 ≤ b1)
(B : SplitBlocks n) (z : Fin n → ClampObs J)
(htdef : tAlpha = sqrt (log (12 * (J : ℝ) / alpha) / 2))
(hb0def : b0 = tAlpha / sqrt (B.I0.card : ℝ))
(hb1def : b1 = tAlpha / sqrt (B.I1.card : ℝ))
(hret : |retainedEstimate B z delta - retainedMean P delta| ≤ b0)
(hatom : ∀ x : Fin J,
|atomEstimate B z x delta - P.px x * atomMass P x delta| ≤ b1)
(hlocal : ∀ x : Fin J,
GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h →
|localRegressionEstimate B z x (ellOf beta) kappa cminus cplus delta h -
P.mu x delta| ≤
L * h ^ beta * ∑ i ∈ B.I2,
|interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i| +
tAlpha * sqrt (∑ i ∈ B.I2,
(interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i) ^ 2)) :
clampFunctional P delta ∈
honestInterval B z (ellOf beta) beta kappa L cminus cplus delta h alpha := by
let radius := b0 + ∑ x : Fin J,
stratumRadius B z x (ellOf beta) beta kappa L cminus cplus delta h tAlpha b1
have hdev := totalGramEstimator_error_le_honestRadius P hmodel hdelta hdelta1 hh
hbeta hL htAlpha hb0 hb1 B z hret hatom hlocal
have htI := clampFunctional_mem_Icc P hmodel hdelta hdelta1
have hr0 : 0 ≤ radius := by
dsimp [radius]
apply add_nonneg hb0
apply sum_nonneg
intro x hx
unfold stratumRadius atomEstimate blockAverage
split_ifs <;> positivity
simp only [honestInterval]
rw [← htdef, ← hb0def, ← hb1def]
change clampFunctional P delta ∈ Icc
(max 0 (totalGramEstimator B z (ellOf beta) kappa cminus cplus delta h - radius))
(min 1 (totalGramEstimator B z (ellOf beta) kappa cminus cplus delta h + radius))
constructor
· apply max_le htI.1
dsimp [radius] at hdev ⊢
rw [abs_le] at hdev
linarith
· apply le_min htI.2
dsimp [radius] at hdev ⊢
rw [abs_le] at hdev
linarith
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.LocalWindowGram 15 declarations This module identifies the fixed-stratum treatment marginal with its declared density law and transports integrals through that measure identity.
Local-window population-law bridge
This module identifies the fixed-stratum treatment marginal with its declared density law and transports integrals through that measure identity.
def localWindowWeight
Unit local-window weight in a fixed stratum.
Definition (Lean source)
parameters
def
localWindowWeight x delta h o :
ℝ
given by
if o.X = x ∧ scaledDose delta h o ∈ Icc (0 : ℝ) 1 then 1 else 0
def localWindowFeature
Monomial feature clipped outside the local window, so its global envelope is one while its weighted Gram agrees with the total local Gram.
Definition (Lean source)
parameters
def
localWindowFeature delta h j o :
ℝ
given by
theorem measurable_localWindowWeight
the stated measurable local window weight property holds for the specified J input, the specified x input, the specified delta input, the specified h input.
Formal statement
Proof (Lean source)
lemma measurable_localWindowWeight {J : ℕ} (x : Fin J) (delta h : ℝ) :
Measurable (localWindowWeight x delta h) := by
have hu : Measurable (fun o : ClampObs J => scaledDose delta h o) :=
(measurable_clampObs_A.sub measurable_const).div_const _
apply Measurable.ite
· exact (measurable_clampObs_X (J := J) (measurableSet_singleton x)).inter
(hu measurableSet_Icc)
· exact measurable_const
· exact measurable_const
theorem measurable_localWindowFeature
the stated measurable local window feature property holds for the specified J input, the specified ell input, the specified delta input, the specified h input, the specified j input.
Formal statement
hypotheses
conclusion
Measurable (localWindowFeature (J := J) delta h j)
Proof (Lean source)
lemma measurable_localWindowFeature {J ell : ℕ} (delta h : ℝ)
(j : Fin (ell + 1)) : Measurable (localWindowFeature (J := J) delta h j) := by
have hu : Measurable (fun o : ClampObs J => scaledDose delta h o) :=
(measurable_clampObs_A.sub measurable_const).div_const _
apply Measurable.ite
· exact hu measurableSet_Icc
· exact hu.pow_const _
· exact measurable_const
theorem localWindowWeight_mem_unit
For a stratum, threshold, and bandwidth, the local-window weight lies between zero and one for every observation.
Formal statement
hypotheses
conclusion
localWindowWeight x delta h o ∈ Icc (0 : ℝ) 1
Proof (Lean source)
lemma localWindowWeight_mem_unit {J : ℕ} (x : Fin J) (delta h : ℝ) :
∀ o : ClampObs J, localWindowWeight x delta h o ∈ Icc (0 : ℝ) 1 := by
intro o
by_cases ho : o.X = x ∧ 0 ≤ scaledDose delta h o ∧ scaledDose delta h o ≤ 1
· simp [localWindowWeight, Set.mem_Icc, ho]
· simp [localWindowWeight, Set.mem_Icc, ho]
theorem abs_localWindowFeature_le_one
For a polynomial degree, threshold, bandwidth, and basis coordinate, the local-window feature has absolute value at most one for every observation.
Formal statement
hypotheses
conclusion
|localWindowFeature (J := J) delta h j o| ≤ 1
Proof (Lean source)
lemma abs_localWindowFeature_le_one {J ell : ℕ} (delta h : ℝ)
(j : Fin (ell + 1)) :
∀ o : ClampObs J, |localWindowFeature (J := J) delta h j o| ≤ 1 := by
intro o
by_cases hu : scaledDose delta h o ∈ Icc (0 : ℝ) 1
· simp only [localWindowFeature, if_pos hu, monomialVec]
rw [abs_of_nonneg (pow_nonneg hu.1 _)]
exact pow_le_one₀ hu.1 hu.2
· simp [localWindowFeature, hu]
theorem stratumTreatmentMeasure_eq
The treatment marginal inside a fixed stratum is its stratum mass times the declared conditional treatment measure. The result uses the hmodel condition, the hkappa condition, the hcplus condition, the hpmin condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
kappa cminus cplus pmin :
ℝ
hmodel :
BridgeClampModel P kappa cminus cplus pmin
hkappa :
0 ≤ kappa
hcplus :
0 ≤ cplus
hpmin :
0 < pmin
x :
Fin J
conclusion
= ofReal (P.px x) • conditionalTreatmentMeasure P x
Proof (Lean source)
-- @node: stratumTreatmentMeasure_eq
lemma stratumTreatmentMeasure_eq {J : ℕ} (P : ClampLaw J)
{kappa cminus cplus pmin : ℝ}
(hmodel : BridgeClampModel P kappa cminus cplus pmin)
(hkappa : 0 ≤ kappa) (hcplus : 0 ≤ cplus) (hpmin : 0 < pmin)
(x : Fin J) :
(P.dataMeasure.restrict {o : ClampObs J | o.X = x}).map (fun o => o.A) =
ofReal (P.px x) • conditionalTreatmentMeasure P x := by
letI : IsProbabilityMeasure P.dataMeasure := hmodel.probability
have hpx : 0 ≤ P.px x := le_trans hpmin.le (hmodel.stratumMass x).2
have hpiInt : Integrable (P.pi x)
(volume.restrict (Icc (0 : ℝ) 1)) := by
apply IntegrableOn.of_bound measure_Icc_lt_top
· exact (aemeasurable_restrict_of_measurable_subtype measurableSet_Icc
(hmodel.condDensity.1 x)).aestronglyMeasurable
· filter_upwards [ae_restrict_mem measurableSet_Icc,
hmodel.condDensity.2.1 x, hmodel.thinning x] with a ha hnon hthin
rw [Real.norm_eq_abs, abs_of_nonneg hnon]
calc
P.pi x a ≤ cplus * a ^ kappa := hthin.2
_ ≤ cplus := by
simpa using mul_le_mul_of_nonneg_left
(Real.rpow_le_one ha.1 ha.2 hkappa) hcplus
let μ := (P.dataMeasure.restrict {o : ClampObs J | o.X = x}).map (fun o => o.A)
let ν := ofReal (P.px x) • conditionalTreatmentMeasure P x
have hcondFinite : IsFiniteMeasure (conditionalTreatmentMeasure P x) := by
refine ⟨?_⟩
simp only [conditionalTreatmentMeasure]
rw [withDensity_apply _ MeasurableSet.univ]
simp only [Measure.restrict_univ]
exact lt_of_le_of_lt (lintegral_mono fun a => Real.ofReal_le_enorm _)
hpiInt.hasFiniteIntegral
letI : IsFiniteMeasure (conditionalTreatmentMeasure P x) := hcondFinite
have hνFinite : IsFiniteMeasure ν := by
refine ⟨?_⟩
dsimp [ν]
change ofReal (P.px x) *
(conditionalTreatmentMeasure P x) univ < ⊤
exact ENNReal.mul_lt_top ENNReal.ofReal_lt_top (measure_lt_top _ _)
letI : IsFiniteMeasure ν := hνFinite
have hμae : ∀ᵐ a ∂μ, a ∈ Icc (0 : ℝ) 1 := by
rw [ae_map_iff measurable_clampObs_A.aemeasurable]
· exact ae_restrict_of_ae hmodel.treatmentSupport
· exact measurableSet_Icc
have hνae : ∀ᵐ a ∂ν, a ∈ Icc (0 : ℝ) 1 := by
dsimp [ν, conditionalTreatmentMeasure]
exact Measure.smul_absolutelyContinuous.ae_le
((withDensity_absolutelyContinuous _ _).ae_le
(ae_restrict_mem measurableSet_Icc))
change μ = ν
rw [← Measure.restrict_eq_self_of_ae_mem hμae,
← Measure.restrict_eq_self_of_ae_mem hνae]
apply (Measure.restrict_congr_meas measurableSet_Icc).2
intro t ht htm
apply (ENNReal.toReal_eq_toReal_iff'
(measure_ne_top μ t) (measure_ne_top ν t)).mp
change μ.real t = ν.real t
have hsource : μ.real t =
P.dataMeasure.real {o : ClampObs J | o.X = x ∧ o.A ∈ t} := by
simp only [μ, measureReal_def]
rw [Measure.map_apply measurable_clampObs_A htm,
Measure.restrict_apply (measurable_clampObs_A htm)]
congr 2
ext o
simp [and_comm]
have htarget : (conditionalTreatmentMeasure P x).real t =
∫ a in t, P.pi x a := by
have hpiT : Integrable (P.pi x) (volume.restrict t) :=
hpiInt.mono_measure (Measure.restrict_mono_set volume ht)
simp only [conditionalTreatmentMeasure, measureReal_def]
rw [withDensity_apply _ htm]
rw [Measure.restrict_restrict htm,
inter_eq_self_of_subset_left ht]
rw [← ofReal_integral_eq_lintegral_ofReal hpiT]
· rw [ENNReal.toReal_ofReal]
exact integral_nonneg_of_ae ((hmodel.condDensity.2.1 x).filter_mono
(ae_mono (Measure.restrict_mono_set volume ht)))
· exact (hmodel.condDensity.2.1 x).filter_mono
(ae_mono (Measure.restrict_mono_set volume ht))
rw [hsource]
have hlaw := (hmodel.condDensity.2.2 x).2 t htm ht
rw [hlaw, show ν.real t = P.px x * (conditionalTreatmentMeasure P x).real t by
simp [ν, measureReal_def, hpx]]
rw [htarget]
theorem stratumTreatment_integral_eq
Integration form of stratumTreatmentMeasure_eq. The result uses the hmodel condition, the hkappa condition, the hcplus condition, the hpmin condition, the hf condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
kappa cminus cplus pmin :
ℝ
hmodel :
BridgeClampModel P kappa cminus cplus pmin
hkappa :
0 ≤ kappa
hcplus :
0 ≤ cplus
hpmin :
0 < pmin
x :
Fin J
f :
ℝ → ℝ
hf :
Proof (Lean source)
-- @node: stratumTreatment_integral_eq
lemma stratumTreatment_integral_eq {J : ℕ} (P : ClampLaw J)
{kappa cminus cplus pmin : ℝ}
(hmodel : BridgeClampModel P kappa cminus cplus pmin)
(hkappa : 0 ≤ kappa) (hcplus : 0 ≤ cplus) (hpmin : 0 < pmin)
(x : Fin J) (f : ℝ → ℝ)
(hf : Integrable f (conditionalTreatmentMeasure P x)) :
(∫ o, indicator {o : ClampObs J | o.X = x} (fun o => f o.A) o
∂P.dataMeasure) =
P.px x * ∫ a, f a ∂conditionalTreatmentMeasure P x := by
have hX : MeasurableSet {o : ClampObs J | o.X = x} :=
measurable_clampObs_X (measurableSet_singleton x)
have hpx : 0 ≤ P.px x := le_trans hpmin.le (hmodel.stratumMass x).2
rw [integral_indicator hX]
rw [← integral_map measurable_clampObs_A.aemeasurable]
· rw [stratumTreatmentMeasure_eq P hmodel hkappa hcplus hpmin x,
integral_smul_measure, ENNReal.toReal_ofReal hpx]
rfl
· rw [stratumTreatmentMeasure_eq P hmodel hkappa hcplus hpmin x]
exact (hf.smul_measure ENNReal.ofReal_ne_top).aestronglyMeasurable
theorem conditionalTreatment_integral_eq_density
Integration against the declared conditional treatment measure is integration against its real density on [0,1]. The result uses the hmodel condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
kappa cminus cplus pmin :
ℝ
hmodel :
BridgeClampModel P kappa cminus cplus pmin
x :
Fin J
f :
ℝ → ℝ
conclusion
(∫ a, f a ∂conditionalTreatmentMeasure P x) = ∫ a in Icc (0 : ℝ) 1, P.pi x a * f a
Proof (Lean source)
-- @node: conditionalTreatment_integral_eq_density
lemma conditionalTreatment_integral_eq_density {J : ℕ} (P : ClampLaw J)
{kappa cminus cplus pmin : ℝ}
(hmodel : BridgeClampModel P kappa cminus cplus pmin)
(x : Fin J) (f : ℝ → ℝ) :
(∫ a, f a ∂conditionalTreatmentMeasure P x) =
∫ a in Icc (0 : ℝ) 1, P.pi x a * f a := by
have hpiMeas : AEMeasurable (fun a : ℝ => ofReal (P.pi x a))
(volume.restrict (Icc (0 : ℝ) 1)) :=
(aemeasurable_restrict_of_measurable_subtype measurableSet_Icc
(hmodel.condDensity.1 x)).ennreal_ofReal
rw [conditionalTreatmentMeasure,
integral_withDensity_eq_integral_toReal_smul₀ hpiMeas]
· apply integral_congr_ae
filter_upwards [hmodel.condDensity.2.1 x] with a ha
simp [ENNReal.toReal_ofReal ha, smul_eq_mul, mul_comm]
· exact ae_of_all _ (fun _ => ENNReal.ofReal_lt_top)
theorem scaledDose_mem_Icc_iff
Positive bandwidth identifies the scaled unit window with the original treatment interval. The result uses the hh condition. This is the stated conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
-- @node: scaledDose_mem_Icc_iff
lemma scaledDose_mem_Icc_iff {J : ℕ} {delta h : ℝ} (hh : 0 < h)
(o : ClampObs J) :
scaledDose delta h o ∈ Icc (0 : ℝ) 1 ↔
o.A ∈ Icc delta (delta + h) := by
simp only [scaledDose, Set.mem_Icc]
constructor
· intro hu
constructor
· exact sub_nonneg.mp (((div_nonneg_iff).mp hu.1 |>.resolve_right
(fun hneg => (not_lt_of_ge hneg.2 hh))).1)
· have := (div_le_iff₀ hh).mp hu.2
linarith
· intro ha
constructor
· exact div_nonneg (sub_nonneg.mpr ha.1) hh.le
· exact (div_le_iff₀ hh).mpr (by linarith)
The population local-window mass is the stratum mass times the declared density integral over that window. The result uses the hmodel condition, the hh condition, the hwindow condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta h :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
x :
Fin J
hh :
0 < h
conclusion
(∫ o, localWindowWeight x delta h o ∂P.dataMeasure)
= P.px x * ∫ a in Icc delta (delta + h), P.pi x a
Proof (Lean source)
-- @node: localWindowWeight_integral_eq
lemma localWindowWeight_integral_eq {J : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin delta h : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(x : Fin J) (hh : 0 < h)
(hwindow : Icc delta (delta + h) ⊆ Icc (0 : ℝ) 1) :
(∫ o, localWindowWeight x delta h o ∂P.dataMeasure) =
P.px x * ∫ a in Icc delta (delta + h), P.pi x a := by
let E : Set (ClampObs J) := {o | o.X = x ∧ o.A ∈ Icc delta (delta + h)}
have hE : MeasurableSet E :=
(measurable_clampObs_X (measurableSet_singleton x)).inter
(measurable_clampObs_A measurableSet_Icc)
have hfun : localWindowWeight x delta h = indicator E (fun _ => (1 : ℝ)) := by
funext o
by_cases ho : o.X = x ∧ o.A ∈ Icc delta (delta + h)
· rw [Set.indicator_of_mem (show o ∈ E from ho)]
simp [localWindowWeight, ho.1, (scaledDose_mem_Icc_iff hh o).2 ho.2]
· rw [Set.indicator_of_notMem (show o ∉ E from ho)]
have hscaled : ¬(o.X = x ∧ scaledDose delta h o ∈ Icc (0 : ℝ) 1) :=
fun hz => ho ⟨hz.1, (scaledDose_mem_Icc_iff hh o).1 hz.2⟩
change (if o.X = x ∧ scaledDose delta h o ∈ Icc (0 : ℝ) 1
then 1 else 0) = 0
exact if_neg hscaled
rw [hfun, integral_indicator hE, setIntegral_const, smul_eq_mul, mul_one]
exact (hmodel.condDensity.2.2 x).2 (Icc delta (delta + h))
measurableSet_Icc hwindow
A positive interval captures a fixed positive fraction of the power mass at its upper endpoint. The result uses the hdelta condition, the hh condition, the hkappa condition. This is the stated conclusion.
Formal statement
hypotheses
delta h kappa :
ℝ
hdelta :
0 ≤ delta
hh :
0 < h
hkappa :
0 ≤ kappa
conclusion
h / 2 * ((delta + h) / 2) ^ kappa ≤ ∫ a in delta..delta + h, a ^ kappa
Proof (Lean source)
-- @node: integral_rpow_window_lower
lemma integral_rpow_window_lower {delta h kappa : ℝ}
(hdelta : 0 ≤ delta) (hh : 0 < h) (hkappa : 0 ≤ kappa) :
h / 2 * ((delta + h) / 2) ^ kappa ≤
∫ a in delta..delta + h, a ^ kappa := by
let m := delta + h / 2
have hdm : delta ≤ m := by dsimp [m]; linarith
have hmb : m ≤ delta + h := by dsimp [m]; linarith
have hm0 : 0 ≤ m := by dsimp [m]; linarith
have hb2 : (delta + h) / 2 ≤ m := by dsimp [m]; linarith
have hconst :
(∫ _a in m..delta + h, ((delta + h) / 2) ^ kappa) ≤
∫ a in m..delta + h, a ^ kappa := by
apply intervalIntegral.integral_mono_on hmb
· exact intervalIntegrable_const
· exact intervalIntegral.intervalIntegrable_rpow (inl hkappa)
· intro a ha
exact Real.rpow_le_rpow (by positivity) (hb2.trans ha.1) hkappa
have hsub : (∫ a in m..delta + h, a ^ kappa) ≤
∫ a in delta..delta + h, a ^ kappa := by
apply intervalIntegral.integral_mono_interval hdm hmb le_rfl
· filter_upwards [ae_restrict_mem measurableSet_Ioc] with a ha
exact Real.rpow_nonneg (hdelta.trans ha.1.le) _
· apply Continuous.intervalIntegrable
fun_prop
calc
h / 2 * ((delta + h) / 2) ^ kappa =
∫ _a in m..delta + h, ((delta + h) / 2) ^ kappa := by
rw [intervalIntegral.integral_const]
dsimp [m]
ring
_ ≤ ∫ a in m..delta + h, a ^ kappa := hconst
_ ≤ ∫ a in delta..delta + h, a ^ kappa := hsub
theorem localWindowWeight_integral_lower
The local stratum-window probability has the uniform polynomial lower bound obtained by integrating over the upper half of the window. The result uses the hmodel condition, the hkappa condition, the hcminus condition, the hcplus condition, the hpmin condition, the hdelta condition, the hh condition, the hupper condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta h :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hkappa :
0 ≤ kappa
hcminus :
0 < cminus
hcplus :
0 ≤ cplus
hpmin :
0 < pmin
x :
Fin J
hdelta :
0 ≤ delta
hh :
0 < h
hupper :
delta + h ≤ 1
conclusion
pmin * (cminus * ((delta + h) / 2) ^ kappa * (h / 2))
≤ ∫ o, localWindowWeight x delta h o ∂P.dataMeasure
Proof (Lean source)
-- @node: localWindowWeight_integral_lower
lemma localWindowWeight_integral_lower {J : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin delta h : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hkappa : 0 ≤ kappa) (hcminus : 0 < cminus) (hcplus : 0 ≤ cplus)
(hpmin : 0 < pmin) (x : Fin J) (hdelta : 0 ≤ delta) (hh : 0 < h)
(hupper : delta + h ≤ 1) :
pmin * (cminus * ((delta + h) / 2) ^ kappa * (h / 2)) ≤
∫ o, localWindowWeight x delta h o ∂P.dataMeasure := by
let T : Set ℝ := Icc delta (delta + h)
have hT : T ⊆ Icc (0 : ℝ) 1 := by
intro a ha
exact ⟨hdelta.trans ha.1, ha.2.trans hupper⟩
have hpiInt : Integrable (P.pi x)
(volume.restrict (Icc (0 : ℝ) 1)) := by
apply IntegrableOn.of_bound measure_Icc_lt_top
· exact (aemeasurable_restrict_of_measurable_subtype measurableSet_Icc
(hmodel.condDensity.1 x)).aestronglyMeasurable
· filter_upwards [ae_restrict_mem measurableSet_Icc,
hmodel.condDensity.2.1 x, hmodel.thinning x] with a ha hnon hthin
rw [Real.norm_eq_abs, abs_of_nonneg hnon]
calc
P.pi x a ≤ cplus * a ^ kappa := hthin.2
_ ≤ cplus := by
simpa using mul_le_mul_of_nonneg_left
(Real.rpow_le_one ha.1 ha.2 hkappa) hcplus
have hpiT : IntegrableOn (P.pi x) T volume :=
hpiInt.mono_measure (Measure.restrict_mono_set volume hT)
have hthinT : ∀ᵐ a ∂volume.restrict T, cminus * a ^ kappa ≤ P.pi x a :=
((hmodel.thinning x).filter_mono
(ae_mono (Measure.restrict_mono_set volume hT))).mono
(fun _ ha => ha.1)
have hpowT : IntegrableOn (fun a : ℝ => a ^ kappa) T volume := by
dsimp [T]
rw [← intervalIntegrable_iff_integrableOn_Icc_of_le (by linarith)]
exact intervalIntegral.intervalIntegrable_rpow (inl hkappa)
have hlower : cminus * (h / 2 * ((delta + h) / 2) ^ kappa) ≤
∫ a in T, P.pi x a := by
calc
cminus * (h / 2 * ((delta + h) / 2) ^ kappa) ≤
cminus * (∫ a in delta..delta + h, a ^ kappa) := by
exact mul_le_mul_of_nonneg_left
(integral_rpow_window_lower hdelta hh hkappa) hcminus.le
_ = ∫ a in T, cminus * a ^ kappa := by
dsimp [T]
rw [integral_Icc_eq_integral_Ioc, ← intervalIntegral.integral_of_le (by linarith),
intervalIntegral.integral_const_mul]
_ ≤ ∫ a in T, P.pi x a :=
integral_mono_ae (hpowT.const_mul cminus) hpiT hthinT
rw [localWindowWeight_integral_eq P hmodel x hh hT]
have hpx : pmin ≤ P.px x := (hmodel.stratumMass x).2
have hfactor : 0 ≤ cminus * ((delta + h) / 2) ^ kappa * (h / 2) := by
positivity
calc
pmin * (cminus * ((delta + h) / 2) ^ kappa * (h / 2)) ≤
P.px x * (cminus * ((delta + h) / 2) ^ kappa * (h / 2)) :=
mul_le_mul_of_nonneg_right hpx hfactor
_ ≤ P.px x * ∫ a in T, P.pi x a := by
have hlower' : cminus * ((delta + h) / 2) ^ kappa * (h / 2) ≤
∫ a in T, P.pi x a := by
nlinarith [hlower]
exact mul_le_mul_of_nonneg_left hlower' (hpmin.le.trans hpx)
A local-window Gram entry is the corresponding density-weighted monomial integral over the original treatment window. The result uses the hmodel condition, the hkappa condition, the hcplus condition, the hpmin condition, the hh condition, the hwindow condition. This is the stated conclusion.
Formal statement
hypotheses
J ell :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta h :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hkappa :
0 ≤ kappa
hcplus :
0 ≤ cplus
hpmin :
0 < pmin
x :
Fin J
hh :
0 < h
j k :
Fin (ell + 1)
conclusion
(∫ o, localWindowWeight x delta h o * localWindowFeature delta h j o * localWindowFeature delta h k o ∂P.dataMeasure)
= P.px x * ∫ a in Icc delta (delta + h), P.pi x a * monomialVec ell ((a - delta) / h) j * monomialVec ell ((a - delta) / h) k
Proof (Lean source)
-- @node: localWindowGramEntry_integral_eq
lemma localWindowGramEntry_integral_eq {J ell : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin delta h : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hkappa : 0 ≤ kappa) (hcplus : 0 ≤ cplus) (hpmin : 0 < pmin)
(x : Fin J) (hh : 0 < h)
(hwindow : Icc delta (delta + h) ⊆ Icc (0 : ℝ) 1)
(j k : Fin (ell + 1)) :
(∫ o, localWindowWeight x delta h o *
localWindowFeature delta h j o * localWindowFeature delta h k o
∂P.dataMeasure) =
P.px x * ∫ a in Icc delta (delta + h),
P.pi x a * monomialVec ell ((a - delta) / h) j *
monomialVec ell ((a - delta) / h) k := by
have hpiInt : Integrable (P.pi x)
(volume.restrict (Icc (0 : ℝ) 1)) := by
apply IntegrableOn.of_bound measure_Icc_lt_top
· exact (aemeasurable_restrict_of_measurable_subtype measurableSet_Icc
(hmodel.condDensity.1 x)).aestronglyMeasurable
· filter_upwards [ae_restrict_mem measurableSet_Icc,
hmodel.condDensity.2.1 x, hmodel.thinning x] with a ha hnon hthin
rw [Real.norm_eq_abs, abs_of_nonneg hnon]
calc
P.pi x a ≤ cplus * a ^ kappa := hthin.2
_ ≤ cplus := by
simpa using mul_le_mul_of_nonneg_left
(Real.rpow_le_one ha.1 ha.2 hkappa) hcplus
let f : ℝ → ℝ := fun a =>
if (a - delta) / h ∈ Icc (0 : ℝ) 1 then
monomialVec ell ((a - delta) / h) j *
monomialVec ell ((a - delta) / h) k else 0
have hfmeas : Measurable f := by
dsimp [f]
apply Measurable.ite
· exact (((measurable_id.sub measurable_const).div_const _)
measurableSet_Icc)
· exact (((measurable_id.sub measurable_const).div_const _).pow_const _).mul
(((measurable_id.sub measurable_const).div_const _).pow_const _)
· exact measurable_const
have hfbound : ∀ a, |f a| ≤ 1 := by
intro a
by_cases ha : (a - delta) / h ∈ Icc (0 : ℝ) 1
· simp only [f, if_pos ha, monomialVec, abs_mul, abs_pow]
have habs : |(a - delta) / h| ≤ 1 := by
rw [abs_of_nonneg ha.1]
exact ha.2
calc
|(a - delta) / h| ^ (j : ℕ) * |(a - delta) / h| ^ (k : ℕ) ≤
1 * 1 := mul_le_mul (pow_le_one₀ (abs_nonneg _) habs)
(pow_le_one₀ (abs_nonneg _) habs)
(pow_nonneg (abs_nonneg _) _) (by norm_num)
_ = 1 := by norm_num
· simp only [f, if_neg ha, abs_zero]
norm_num
have hcondFinite : IsFiniteMeasure (conditionalTreatmentMeasure P x) := by
refine ⟨?_⟩
simp only [conditionalTreatmentMeasure]
rw [withDensity_apply _ MeasurableSet.univ]
simp only [Measure.restrict_univ]
exact lt_of_le_of_lt (lintegral_mono fun a => Real.ofReal_le_enorm _)
hpiInt.hasFiniteIntegral
letI : IsFiniteMeasure (conditionalTreatmentMeasure P x) := hcondFinite
have hfint : Integrable f (conditionalTreatmentMeasure P x) :=
Integrable.of_bound hfmeas.aestronglyMeasurable 1
(Filter.Eventually.of_forall fun a => by simpa [Real.norm_eq_abs] using hfbound a)
have hfun : (fun o => localWindowWeight x delta h o *
localWindowFeature delta h j o * localWindowFeature delta h k o) =
indicator {o : ClampObs J | o.X = x} (fun o => f o.A) := by
funext o
by_cases hx : o.X = x
· by_cases hu : scaledDose delta h o ∈ Icc (0 : ℝ) 1
· rw [show localWindowWeight x delta h o = 1 by
rw [localWindowWeight, if_pos ⟨hx, hu⟩],
show localWindowFeature delta h j o = monomialVec ell
(scaledDose delta h o) j by
rw [localWindowFeature, if_pos hu],
show localWindowFeature delta h k o = monomialVec ell
(scaledDose delta h o) k by
rw [localWindowFeature, if_pos hu],
Set.indicator_of_mem (show o ∈ {o : ClampObs J | o.X = x} from hx)]
change (1 * monomialVec ell ((o.A - delta) / h) j *
monomialVec ell ((o.A - delta) / h) k) = f o.A
have hu' : (o.A - delta) / h ∈ Icc (0 : ℝ) 1 := by
simpa [scaledDose] using hu
dsimp [f]
rw [if_pos hu']
ring
· rw [show localWindowWeight x delta h o = 0 by
rw [localWindowWeight, if_neg (fun H => hu H.2)],
zero_mul, zero_mul,
Set.indicator_of_mem (show o ∈ {o : ClampObs J | o.X = x} from hx)]
have hu' : ¬((o.A - delta) / h ∈ Icc (0 : ℝ) 1) := by
simpa [scaledDose] using hu
dsimp [f]
rw [if_neg hu']
· simp [localWindowWeight, hx]
rw [hfun]
rw [stratumTreatment_integral_eq P hmodel.toBridge hkappa hcplus hpmin x f hfint,
conditionalTreatment_integral_eq_density P hmodel.toBridge x f]
congr 1
rw [← MeasureTheory.integral_indicator
(measurableSet_Icc : MeasurableSet (Icc (0 : ℝ) 1)),
← MeasureTheory.integral_indicator
(measurableSet_Icc : MeasurableSet (Icc delta (delta + h)))]
apply integral_congr_ae
refine Filter.Eventually.of_forall ?_
intro a
by_cases hw : a ∈ Icc delta (delta + h)
· have hu := (scaledDose_mem_Icc_iff (J := J) hh ⟨x, a, 0⟩).2 hw
have hu' : (a - delta) / h ∈ Icc (0 : ℝ) 1 := by
simpa [scaledDose] using hu
have ha01 : a ∈ Icc (0 : ℝ) 1 := hwindow hw
rw [Set.indicator_of_mem ha01, Set.indicator_of_mem hw]
dsimp [f]
rw [if_pos hu']
ring
· have hu : ¬((a - delta) / h ∈ Icc (0 : ℝ) 1) := by
intro hu
exact hw ((scaledDose_mem_Icc_iff (J := J) hh ⟨x, a, 0⟩).1 hu)
rw [Set.indicator_of_notMem hw]
by_cases ha01 : a ∈ Icc (0 : ℝ) 1
· rw [Set.indicator_of_mem ha01]
dsimp [f]
rw [if_neg hu, mul_zero]
· rw [Set.indicator_of_notMem ha01]
The population local-window Gram dominates lambdaStar times its own local mass, using only the two-sided polynomial density envelope. The result uses the hmodel condition, the hkappa condition, the hcminus condition, the hcplus condition, the hpmin condition, the hdelta condition, the hh condition, the hupper condition. This is the stated conclusion.
Formal statement
hypotheses
J ell :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta h :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hkappa :
0 ≤ kappa
hcminus :
0 < cminus
hcplus :
0 < cplus
hpmin :
0 < pmin
x :
Fin J
hdelta :
0 ≤ delta
hh :
0 < h
hupper :
delta + h ≤ 1
v :
Fin (ell + 1) → ℝ
conclusion
lambdaStar ell kappa cminus cplus * (∫ o, localWindowWeight x delta h o ∂P.dataMeasure) * (∑ j, (v j) ^ 2)
≤ ∑ j, ∑ k, v j * v k * ∫ o, localWindowWeight x delta h o * localWindowFeature (J := J) delta h j o * localWindowFeature (J := J) delta h k o ∂P.dataMeasure
Proof (Lean source)
-- @node: localWindow_populationGram_coercive
lemma localWindow_populationGram_coercive {J ell : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin delta h : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hkappa : 0 ≤ kappa) (hcminus : 0 < cminus) (hcplus : 0 < cplus)
(hpmin : 0 < pmin) (x : Fin J) (hdelta : 0 ≤ delta) (hh : 0 < h)
(hupper : delta + h ≤ 1) (v : Fin (ell + 1) → ℝ) :
lambdaStar ell kappa cminus cplus *
(∫ o, localWindowWeight x delta h o ∂P.dataMeasure) *
(∑ j, (v j) ^ 2) ≤
∑ j, ∑ k, v j * v k *
∫ o, localWindowWeight x delta h o *
localWindowFeature (J := J) delta h j o *
localWindowFeature (J := J) delta h k o ∂P.dataMeasure := by
let T : Set ℝ := Icc delta (delta + h)
let poly : ℝ → ℝ := fun a => ∑ j, v j * ((a - delta) / h) ^ (j : ℕ)
have hT : T ⊆ Icc (0 : ℝ) 1 := by
intro a ha
exact ⟨hdelta.trans ha.1, ha.2.trans hupper⟩
have hpx : 0 ≤ P.px x := hpmin.le.trans (hmodel.stratumMass x).2
have hpiInt : Integrable (P.pi x)
(volume.restrict (Icc (0 : ℝ) 1)) := by
apply IntegrableOn.of_bound measure_Icc_lt_top
· exact (aemeasurable_restrict_of_measurable_subtype measurableSet_Icc
(hmodel.condDensity.1 x)).aestronglyMeasurable
· filter_upwards [ae_restrict_mem measurableSet_Icc,
hmodel.condDensity.2.1 x, hmodel.thinning x] with a ha hnon hthin
rw [Real.norm_eq_abs, abs_of_nonneg hnon]
calc
P.pi x a ≤ cplus * a ^ kappa := hthin.2
_ ≤ cplus := by
simpa using mul_le_mul_of_nonneg_left
(Real.rpow_le_one ha.1 ha.2 hkappa) hcplus.le
have hpiT : IntegrableOn (P.pi x) T volume :=
hpiInt.mono_measure (Measure.restrict_mono_set volume hT)
have hpowT : IntegrableOn (fun a : ℝ => a ^ kappa) T volume := by
dsimp [T]
rw [← intervalIntegrable_iff_integrableOn_Icc_of_le (by linarith)]
exact intervalIntegral.intervalIntegrable_rpow (inl hkappa)
have hmassUpper :
(∫ o, localWindowWeight x delta h o ∂P.dataMeasure) ≤
P.px x * (cplus * ∫ a in T, a ^ kappa) := by
rw [localWindowWeight_integral_eq P hmodel x hh hT]
apply mul_le_mul_of_nonneg_left _ hpx
calc
(∫ a in T, P.pi x a) ≤ ∫ a in T, cplus * a ^ kappa := by
apply integral_mono_ae hpiT (hpowT.const_mul cplus)
exact ((hmodel.thinning x).filter_mono
(ae_mono (Measure.restrict_mono_set volume hT))).mono (fun _ ha => ha.2)
_ = cplus * ∫ a in T, a ^ kappa := by rw [integral_const_mul]
have hpolyMeas : AEStronglyMeasurable poly (volume.restrict T) := by
apply Measurable.aestronglyMeasurable
dsimp [poly]
fun_prop
have hpolyBound : ∀ᵐ a ∂volume.restrict T,
‖poly a ^ 2‖ ≤ (∑ j, |v j|) ^ 2 := by
filter_upwards [ae_restrict_mem measurableSet_Icc] with a ha
have hu : (a - delta) / h ∈ Icc (0 : ℝ) 1 := by
constructor
· exact div_nonneg (sub_nonneg.mpr ha.1) hh.le
· exact (div_le_iff₀ hh).2 (by linarith [ha.2])
have habs : |poly a| ≤ ∑ j, |v j| := by
dsimp [poly]
calc
|∑ j, v j * ((a - delta) / h) ^ (j : ℕ)| ≤
∑ j, |v j * ((a - delta) / h) ^ (j : ℕ)| :=
Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ j, |v j| := by
apply Finset.sum_le_sum
intro j hj
rw [abs_mul]
exact mul_le_of_le_one_right (abs_nonneg _) (by
rw [abs_pow, abs_of_nonneg hu.1]
exact pow_le_one₀ hu.1 hu.2)
have hsumNonneg : 0 ≤ ∑ j, |v j| := sum_nonneg fun _ _ => abs_nonneg _
rw [Real.norm_eq_abs, abs_sq, sq_le_sq]
simpa [abs_of_nonneg hsumNonneg] using habs
have hpolyPi : IntegrableOn (fun a => poly a ^ 2 * P.pi x a) T volume :=
hpiT.bdd_mul (hpolyMeas.pow 2) hpolyBound
have hquadIdentity :
(∑ j, ∑ k, v j * v k *
∫ o, localWindowWeight x delta h o *
localWindowFeature (J := J) delta h j o *
localWindowFeature (J := J) delta h k o ∂P.dataMeasure) =
P.px x * ∫ a in T, poly a ^ 2 * P.pi x a := by
simp_rw [localWindowGramEntry_integral_eq P hmodel hkappa hcplus.le hpmin x hh hT]
simp_rw [show ∀ (j k : Fin (ell + 1)) (I : ℝ),
v j * v k * (P.px x * I) = P.px x * (v j * v k * I) by
intros; ring]
simp_rw [← Finset.mul_sum]
congr 1
have hterm (j k : Fin (ell + 1)) : IntegrableOn
(fun a => v j * v k * (P.pi x a * monomialVec ell ((a - delta) / h) j *
monomialVec ell ((a - delta) / h) k)) T volume := by
have hb : AEStronglyMeasurable (fun a => v j * v k *
(monomialVec ell ((a - delta) / h) j *
monomialVec ell ((a - delta) / h) k)) (volume.restrict T) := by
simp only [monomialVec]
fun_prop
have hbound : ∀ᵐ a ∂volume.restrict T,
‖v j * v k * (monomialVec ell ((a - delta) / h) j *
monomialVec ell ((a - delta) / h) k)‖ ≤ |v j * v k| := by
filter_upwards [ae_restrict_mem measurableSet_Icc] with a ha
have hu : (a - delta) / h ∈ Icc (0 : ℝ) 1 := by
constructor
· exact div_nonneg (sub_nonneg.mpr ha.1) hh.le
· exact (div_le_iff₀ hh).2 (by linarith [ha.2])
rw [Real.norm_eq_abs]
calc
|v j * v k * (monomialVec ell ((a - delta) / h) j *
monomialVec ell ((a - delta) / h) k)| ≤ |v j * v k| * 1 := by
simp only [monomialVec, abs_mul, abs_pow, abs_of_nonneg hu.1]
gcongr
have hj : ((a - delta) / h) ^ (j : ℕ) ≤ 1 := pow_le_one₀ hu.1 hu.2
have hk : ((a - delta) / h) ^ (k : ℕ) ≤ 1 := pow_le_one₀ hu.1 hu.2
simpa using mul_le_mul hj hk (pow_nonneg hu.1 _) (by norm_num : (0 : ℝ) ≤ 1)
_ = |v j * v k| := mul_one _
have hi := hpiT.mul_bdd hb hbound
apply hi.congr
filter_upwards with a
ring
rw [show (∫ a in T, poly a ^ 2 * P.pi x a) =
∫ a in T, ∑ j, ∑ k, v j * v k *
(P.pi x a * monomialVec ell ((a - delta) / h) j *
monomialVec ell ((a - delta) / h) k) by
apply integral_congr_ae
filter_upwards with a
dsimp [poly, monomialVec]
simp only [pow_two, Finset.sum_mul, Finset.mul_sum]
apply Finset.sum_congr rfl
intro j hj
apply Finset.sum_congr rfl
intro k hk
ring]
simp_rw [← integral_const_mul]
have hinner (j : Fin (ell + 1)) : IntegrableOn
(fun a => ∑ k, v j * v k *
(P.pi x a * monomialVec ell ((a - delta) / h) j *
monomialVec ell ((a - delta) / h) k)) T volume :=
integrable_finsetSum univ (fun k _ => hterm j k)
rw [integral_finsetSum univ (fun j _ => hinner j)]
apply Finset.sum_congr rfl
intro j hj
rw [integral_finsetSum univ (fun k _ => hterm j k)]
have hthinLower :
P.px x * (cminus * ∫ a in T, poly a ^ 2 * a ^ kappa) ≤
P.px x * ∫ a in T, poly a ^ 2 * P.pi x a := by
apply mul_le_mul_of_nonneg_left _ hpx
rw [← integral_const_mul]
apply integral_mono_ae
· simpa [mul_assoc] using
(hpowT.bdd_mul (hpolyMeas.pow 2) hpolyBound).const_mul cminus
· exact hpolyPi
· filter_upwards [ae_restrict_mem measurableSet_Icc,
(hmodel.thinning x).filter_mono
(ae_mono (Measure.restrict_mono_set volume hT))] with a ha hthin
simpa [mul_assoc, mul_left_comm, mul_comm] using
mul_le_mul_of_nonneg_left hthin.1 (sq_nonneg (poly a))
have hscaled :=
CausalSmith.Stat.LmtpThresholdAtomFrontier.lambdaStar_scaled_power_window_coercive
ell hkappa hcminus hcplus hdelta hh v
have hscaledT :
lambdaStar ell kappa cminus cplus * cplus * (∫ a in T, a ^ kappa) *
(∑ j, (v j) ^ 2) ≤
cminus * ∫ a in T, poly a ^ 2 * a ^ kappa := by
dsimp [T, poly]
rw [integral_Icc_eq_integral_Ioc, integral_Icc_eq_integral_Ioc,
← intervalIntegral.integral_of_le (by linarith),
← intervalIntegral.integral_of_le (by linarith)]
simpa [mul_comm] using hscaled
rw [hquadIdentity]
calc
lambdaStar ell kappa cminus cplus *
(∫ o, localWindowWeight x delta h o ∂P.dataMeasure) *
(∑ j, (v j) ^ 2) ≤
P.px x * (lambdaStar ell kappa cminus cplus * cplus *
(∫ a in T, a ^ kappa) * (∑ j, (v j) ^ 2)) := by
have hlam : 0 ≤ lambdaStar ell kappa cminus cplus :=
(lambdaStar_pos ell hkappa hcminus hcplus).le
calc
_ ≤ lambdaStar ell kappa cminus cplus *
(P.px x * (cplus * ∫ a in T, a ^ kappa)) *
(∑ j, (v j) ^ 2) := by
gcongr
_ = _ := by ring
_ ≤ P.px x * (cminus * ∫ a in T, poly a ^ 2 * a ^ kappa) :=
mul_le_mul_of_nonneg_left hscaledT hpx
_ ≤ P.px x * ∫ a in T, poly a ^ 2 * P.pi x a := hthinLower
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.MinimaxDivergence 10 declarations Chi-squared control for the canonical minimax witnesses
Chi-squared control for the canonical minimax witnesses
def minimaxMarkKernel
The signed Bernoulli mark kernel before translating its marks to {0,1}.
Definition (Lean source)
parameters
given by
clause 1
toFun := fun p => twoPointMean (1 / 2) (g p)
clause 2
measurable' := (measurable_twoPointMean (1 / 2)).comp hg
theorem minimaxMarkKernel_isMarkov
the stated minimax mark kernel is markov property holds for the specified J input, the specified g input, the specified hgm input, the specified hg input.
Formal statement
hypotheses
conclusion
IsMarkovKernel (minimaxMarkKernel g hgm)
Proof (Lean source)
lemma minimaxMarkKernel_isMarkov {J : ℕ} {g : Fin J × ℝ → ℝ}
(hgm : Measurable g) (hg : ∀ p, |g p| ≤ 1 / 2) :
IsMarkovKernel (minimaxMarkKernel g hgm) := by
refine ⟨fun p => ?_⟩
exact twoPointMean_isProbabilityMeasure
(by norm_num) (hg p)
def minimaxObsEquiv
Retaining the design and translating the centered mark is a measurable equivalence with the paper's observation carrier.
Definition (Lean source)
parameters
J :
ℕ
given by
clause 1
toEquiv := { toFun := fun p => ClampObs.mk p.1.1 p.1.2 (p.2 + 1 / 2) invFun := fun o => ((o.X, o.A), o.Y - 1 / 2) left_inv := by intro p; rcases p with ⟨⟨x, a⟩, y⟩; simp right_inv := by intro o; rcases o with ⟨x, a, y⟩; simp }
clause 2
measurable_toFun := by rw [measurable_comap_iff] fun_prop
clause 3
measurable_invFun := by let hall : Measurable (fun o : ClampObs J => (o.X, o.A, o.Y)) := Measurable.of_comap_le le_rfl exact ((measurable_fst.comp hall).prodMk (measurable_fst.comp (measurable_snd.comp hall))).prodMk ((measurable_snd.comp (measurable_snd.comp hall)).sub measurable_const)
theorem minimaxDataMeasure_eq_map_attachKernel
The bind construction of the witness law is precisely the attached mark kernel, transported through minimaxObsEquiv. The result uses the hJ condition, the hkappa condition, the hg condition, the hgb condition. This is the stated conclusion.
Formal statement
hypotheses
conclusion
minimaxDataMeasure J kappa g
Proof (Lean source)
lemma minimaxDataMeasure_eq_map_attachKernel {J : ℕ} {kappa : ℝ}
{g : Fin J × ℝ → ℝ} (hJ : 0 < J) (hkappa : 0 ≤ kappa)
(hg : Measurable g) (hgb : ∀ p, |g p| ≤ 1 / 2) :
minimaxDataMeasure J kappa g =
(attachKernel (minimaxDesignMeasure J kappa)
(minimaxMarkKernel g hg)).map (minimaxObsEquiv J) := by
letI : IsProbabilityMeasure (minimaxDesignMeasure J kappa) :=
minimaxDesignMeasure_isProbabilityMeasure J kappa hJ hkappa
letI : IsMarkovKernel (minimaxMarkKernel g hg) :=
minimaxMarkKernel_isMarkov hg hgb
ext S hS
have hpre : MeasurableSet ((minimaxObsEquiv J) ⁻¹' S) :=
hS.preimage (minimaxObsEquiv J).measurable
rw [Measure.map_apply (minimaxObsEquiv J).measurable hS]
unfold minimaxDataMeasure
rw [Measure.bind_apply hS (measurable_minimaxOutcomeKernel g hg).aemeasurable]
rw [attachKernel, Measure.compProd_apply hpre]
apply lintegral_congr
intro p
change (Measure.map (fun y : ℝ => ClampObs.mk p.1 p.2 (y + 1 / 2))
(twoPointMean (1 / 2) (g p))) S = _
rw [Measure.map_apply (by rw [measurable_comap_iff]; fun_prop) hS]
rfl
theorem minimaxDataMeasure_chiSqDiv_center
Exact one-observation chi-squared divergence for a common-design centered Bernoulli perturbation. The result uses the hJ condition, the hkappa condition, the hg condition, the hgb condition. This is the stated conclusion.
Formal statement
hypotheses
conclusion
= ∫ p, 4 * (g p) ^ 2 ∂minimaxDesignMeasure J kappa
Proof (Lean source)
lemma minimaxDataMeasure_chiSqDiv_center (J : ℕ) (kappa : ℝ)
(hJ : 0 < J) (hkappa : 0 ≤ kappa) (g : Fin J × ℝ → ℝ)
(hg : Measurable g) (hgb : ∀ p, |g p| < 1 / 2) :
chiSqDiv (minimaxDataMeasure J kappa g)
(minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => (0 : ℝ))) =
∫ p, 4 * (g p) ^ 2 ∂minimaxDesignMeasure J kappa := by
letI : IsProbabilityMeasure (minimaxDesignMeasure J kappa) :=
minimaxDesignMeasure_isProbabilityMeasure J kappa hJ hkappa
letI : IsMarkovKernel (minimaxMarkKernel g hg) :=
minimaxMarkKernel_isMarkov hg (fun p => (hgb p).le)
letI : IsMarkovKernel
(minimaxMarkKernel (fun _ : Fin J × ℝ => (0 : ℝ)) measurable_const) :=
minimaxMarkKernel_isMarkov (J := J) measurable_const (fun _ => by norm_num)
rw [minimaxDataMeasure_eq_map_attachKernel hJ hkappa hg (fun p => (hgb p).le),
minimaxDataMeasure_eq_map_attachKernel hJ hkappa measurable_const (fun _ => by norm_num),
chiSqDiv_map_measurableEquiv]
have hatt := one_add_chiSqDiv_attachKernel
(minimaxDesignMeasure J kappa) (minimaxMarkKernel g hg)
(minimaxMarkKernel (fun _ : Fin J × ℝ => (0 : ℝ)) measurable_const)
(fun p => twoPointMean_ac_center (hgb p))
(minimax_attach_sq_integrable hJ hkappa hg hgb)
have hatt' : 1 + chiSqDiv
(attachKernel (minimaxDesignMeasure J kappa)
(minimaxMarkKernel g hg))
(attachKernel (minimaxDesignMeasure J kappa)
(minimaxMarkKernel (fun _ : Fin J × ℝ => (0 : ℝ)) measurable_const)) =
∫ p, (1 + 4 * (g p) ^ 2) ∂minimaxDesignMeasure J kappa := by
calc
_ = ∫ p, (1 + chiSqDiv
((minimaxMarkKernel g hg) p)
((minimaxMarkKernel (fun _ : Fin J × ℝ => (0 : ℝ)) measurable_const) p))
∂minimaxDesignMeasure J kappa := hatt
_ = _ := by
apply integral_congr_ae
filter_upwards with p
congr 1
exact chiSqDiv_twoPointMean_centerHalf (hgb p)
rw [integral_add (integrable_const 1)
(Integrable.of_bound (by fun_prop) 1 (by
filter_upwards with p
rw [Real.norm_eq_abs, abs_of_nonneg (by positivity : 0 ≤ 4 * g p ^ 2)]
rcases abs_lt.mp (hgb p) with ⟨hl, hr⟩
nlinarith [sq_nonneg (g p - 1 / 2), sq_nonneg (g p + 1 / 2)]))] at hatt'
have hmass : (minimaxDesignMeasure J kappa).real univ = 1 := by simp
simp only [integral_const, hmass, one_smul] at hatt'
linarith
theorem minimaxDataMeasure_ac_center
the stated minimax data measure ac center property holds for the specified J input, the specified kappa input, the specified hJ input, the specified hkappa input, the specified g input, the specified hg input, the specified hgb input.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma minimaxDataMeasure_ac_center (J : ℕ) (kappa : ℝ)
(hJ : 0 < J) (hkappa : 0 ≤ kappa) (g : Fin J × ℝ → ℝ)
(hg : Measurable g) (hgb : ∀ p, |g p| < 1 / 2) :
minimaxDataMeasure J kappa g ≪
minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => (0 : ℝ)) := by
letI : IsProbabilityMeasure (minimaxDesignMeasure J kappa) :=
minimaxDesignMeasure_isProbabilityMeasure J kappa hJ hkappa
letI : IsMarkovKernel (minimaxMarkKernel g hg) :=
minimaxMarkKernel_isMarkov hg (fun p => (hgb p).le)
letI : IsMarkovKernel
(minimaxMarkKernel (fun _ : Fin J × ℝ => (0 : ℝ)) measurable_const) :=
minimaxMarkKernel_isMarkov (J := J) measurable_const (fun _ => by norm_num)
rw [minimaxDataMeasure_eq_map_attachKernel hJ hkappa hg (fun p => (hgb p).le),
minimaxDataMeasure_eq_map_attachKernel hJ hkappa measurable_const (fun _ => by norm_num)]
exact (Measure.AbsolutelyContinuous.compProd_right
(ae_of_all _ fun p => twoPointMean_ac_center (hgb p))).map
(minimaxObsEquiv J).measurable
theorem minimaxDataMeasure_sq_integrable_center
the stated minimax data measure sq integrable center property holds for the specified J input, the specified kappa input, the specified hJ input, the specified hkappa input, the specified g input, the specified hg input, the specified hgb input.
Formal statement
hypotheses
conclusion
Integrable (fun o => (((minimaxDataMeasure J kappa g).rnDeriv (minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => (0 : ℝ))) o).toReal - 1) ^ 2) (minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => (0 : ℝ)))
Proof (Lean source)
lemma minimaxDataMeasure_sq_integrable_center (J : ℕ) (kappa : ℝ)
(hJ : 0 < J) (hkappa : 0 ≤ kappa) (g : Fin J × ℝ → ℝ)
(hg : Measurable g) (hgb : ∀ p, |g p| < 1 / 2) :
Integrable (fun o => (((minimaxDataMeasure J kappa g).rnDeriv
(minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => (0 : ℝ))) o).toReal - 1) ^ 2)
(minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => (0 : ℝ))) := by
letI : IsProbabilityMeasure (minimaxDesignMeasure J kappa) :=
minimaxDesignMeasure_isProbabilityMeasure J kappa hJ hkappa
letI : IsMarkovKernel (minimaxMarkKernel g hg) :=
minimaxMarkKernel_isMarkov hg (fun p => (hgb p).le)
letI : IsMarkovKernel
(minimaxMarkKernel (fun _ : Fin J × ℝ => (0 : ℝ)) measurable_const) :=
minimaxMarkKernel_isMarkov (J := J) measurable_const (fun _ => by norm_num)
let mg := attachKernel (minimaxDesignMeasure J kappa)
(minimaxMarkKernel g hg)
let m0 := attachKernel (minimaxDesignMeasure J kappa)
(minimaxMarkKernel (fun _ : Fin J × ℝ => (0 : ℝ)) measurable_const)
rw [minimaxDataMeasure_eq_map_attachKernel hJ hkappa hg (fun p => (hgb p).le),
minimaxDataMeasure_eq_map_attachKernel hJ hkappa measurable_const (fun _ => by norm_num)]
apply (integrable_map_equiv (minimaxObsEquiv J) _).2
have hi := minimax_attach_sq_integrable hJ hkappa hg hgb
refine hi.congr ?_
have hrn := (minimaxObsEquiv J).measurableEmbedding.rnDeriv_map mg m0
have hjoint :=
rnDeriv_compProd_right_of_forall_ac
(μ := minimaxDesignMeasure J kappa) (κ := minimaxMarkKernel g hg)
(η := minimaxMarkKernel (fun _ : Fin J × ℝ => (0 : ℝ)) measurable_const)
(ae_of_all _ fun p => twoPointMean_ac_center (hgb p))
filter_upwards [hrn, hjoint] with p hp hj
simp only [Function.comp_apply]
have hj' : (mg.rnDeriv m0 p) =
(minimaxMarkKernel g hg).rnDeriv
(minimaxMarkKernel (fun _ : Fin J × ℝ => (0 : ℝ)) measurable_const)
p.1 p.2 := by
simpa [mg, m0, attachKernel] using hj
rw [hp, hj']
theorem minimaxProduct_chiSqDiv_center
Exact iid tensorization of the canonical common-design experiment. The result uses the hJ condition, the hkappa condition, the hg condition, the hgb condition. This is the stated conclusion.
Formal statement
hypotheses
conclusion
1
+ chiSqDiv (Measure.pi (fun _ : Fin n => minimaxDataMeasure J kappa g)) (Measure.pi (fun _ : Fin n => minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => (0 : ℝ))))
= (1 + ∫ p, 4 * (g p) ^ 2 ∂minimaxDesignMeasure J kappa) ^ n
Proof (Lean source)
lemma minimaxProduct_chiSqDiv_center (J n : ℕ) (kappa : ℝ)
(hJ : 0 < J) (hkappa : 0 ≤ kappa) (g : Fin J × ℝ → ℝ)
(hg : Measurable g) (hgb : ∀ p, |g p| < 1 / 2) :
1 + chiSqDiv
(Measure.pi (fun _ : Fin n => minimaxDataMeasure J kappa g))
(Measure.pi (fun _ : Fin n =>
minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => (0 : ℝ)))) =
(1 + ∫ p, 4 * (g p) ^ 2 ∂minimaxDesignMeasure J kappa) ^ n := by
letI : IsProbabilityMeasure (minimaxDataMeasure J kappa g) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hJ hkappa g hg
(fun p => (hgb p).le)
letI : IsProbabilityMeasure
(minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => (0 : ℝ))) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hJ hkappa _ measurable_const
(fun _ => by norm_num)
rw [one_add_chiSqDiv_pi_iid_general
(minimaxDataMeasure J kappa g)
(minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => (0 : ℝ)))
(minimaxDataMeasure_ac_center J kappa hJ hkappa g hg hgb)
(minimaxDataMeasure_sq_integrable_center J kappa hJ hkappa g hg hgb) n,
minimaxDataMeasure_chiSqDiv_center J kappa hJ hkappa g hg hgb]
theorem minimaxLocal_designIntegral_le
The squared local bump has the information-balance design integral order. The result uses the hJ condition, the hkappa condition, the hdelta condition, the hh condition, the hamp condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa delta h amplitude :
ℝ
hJ :
0 < J
hkappa :
0 ≤ kappa
hdelta :
0 ≤ delta
hh :
0 < h
hamp :
0 ≤ amplitude
conclusion
q :
fun a : ℝ
=> amplitude * h ^ beta * CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
(∫ p, 4 * (q p.2) ^ 2 ∂minimaxDesignMeasure J kappa)
≤ 8 * (kappa + 1) * amplitude ^ 2 * h ^ (2 * beta + 1) * (delta + h) ^ kappa
Proof (Lean source)
lemma minimaxLocal_designIntegral_le
(J : ℕ) (beta kappa delta h amplitude : ℝ)
(hJ : 0 < J) (hkappa : 0 ≤ kappa) (hdelta : 0 ≤ delta)
(hh : 0 < h) (hamp : 0 ≤ amplitude) :
let q := fun a : ℝ => amplitude * h ^ beta *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
(∫ p, 4 * (q p.2) ^ 2 ∂minimaxDesignMeasure J kappa) ≤
8 * (kappa + 1) * amplitude ^ 2 * h ^ (2 * beta + 1) *
(delta + h) ^ kappa := by
dsimp only
let q := fun a : ℝ => amplitude * h ^ beta *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
let S : Set ℝ := Icc (delta - h) (delta + h)
have hqmeas : Measurable q := by dsimp [q]; fun_prop
have hq0 (a : ℝ) : 0 ≤ q a := by
dsimp [q]
exact mul_nonneg (mul_nonneg hamp (Real.rpow_nonneg hh.le _))
(CausalSmith.Stat.DoseResponseMinimax.doseBump_nonneg _)
have hqle (a : ℝ) : q a ≤ amplitude * h ^ beta := by
dsimp [q]
have hb := CausalSmith.Stat.DoseResponseMinimax.doseBump_le_one ((a - delta) / h)
nlinarith [mul_nonneg hamp (Real.rpow_nonneg hh.le beta)]
have hqzero (a : ℝ) (ha : a ∉ S) : q a = 0 := by
have hu : (a - delta) / h ∉ Icc (-1 : ℝ) 1 := by
intro hu
have hul := (le_div_iff₀ hh).mp hu.1
have hur := (div_le_iff₀ hh).mp hu.2
exact ha ⟨by linarith [hul], by linarith [hur]⟩
have habs : 1 ≤ |(a - delta) / h| := by
rw [Set.mem_Icc, not_and_or, not_le, not_le] at hu
rcases hu with hu | hu
· rw [abs_of_neg (by linarith : (a - delta) / h < 0)]; linarith
· exact le_trans (by linarith : 1 ≤ (a - delta) / h) (le_abs_self _)
simp [q, CausalSmith.Stat.DoseResponseMinimax.doseBump_eq_zero_of_one_le_abs habs]
letI : IsProbabilityMeasure (minimaxDesignMeasure J kappa) :=
minimaxDesignMeasure_isProbabilityMeasure J kappa hJ hkappa
letI : Nonempty (Fin J) := Fin.pos_iff_nonempty.mp hJ
letI : IsProbabilityMeasure (minimaxStratumMeasure J) := by
unfold minimaxStratumMeasure
infer_instance
letI : IsProbabilityMeasure (minimaxTreatmentMeasure kappa) :=
minimaxTreatmentMeasure_isProbabilityMeasure kappa hkappa
have hint : Integrable (fun p : Fin J × ℝ => 4 * (q p.2) ^ 2)
(minimaxDesignMeasure J kappa) := by
refine Integrable.of_bound
(((hqmeas.comp measurable_snd).pow_const 2).const_mul 4).aestronglyMeasurable
(4 * (amplitude * h ^ beta) ^ 2) ?_
filter_upwards with p
rw [Real.norm_eq_abs, abs_of_nonneg (by positivity : 0 ≤ 4 * q p.2 ^ 2)]
nlinarith [(sq_le_sq₀ (hq0 p.2)
(mul_nonneg hamp (Real.rpow_nonneg hh.le _))).2 (hqle p.2)]
rw [minimaxDesignMeasure, integral_prod _ hint]
simp only [integral_const]
rw [show (minimaxStratumMeasure J).real univ = 1 by simp]
simp only [one_smul]
rw [minimaxTreatmentMeasure,
integral_withDensity_eq_integral_toReal_smul
(by fun_prop : Measurable fun a : ℝ =>
ofReal ((kappa + 1) * a ^ kappa)) (by simp)]
simp only [smul_eq_mul]
let T : Set ℝ := Icc (0 : ℝ) 1
have hdensity (a : ℝ) (ha : a ∈ T) : 0 ≤ (kappa + 1) * a ^ kappa :=
mul_nonneg (by linarith) (Real.rpow_nonneg ha.1 _)
let K : ℝ := 4 * (amplitude * h ^ beta) ^ 2 *
((kappa + 1) * (delta + h) ^ kappa)
have hK0 : 0 ≤ K := by dsimp [K]; positivity
have hpoint : ∀ a, T.indicator
(fun a => 4 * q a ^ 2 * ((kappa + 1) * a ^ kappa)) a ≤
S.indicator (fun _ => K) a := by
intro a
by_cases haT : a ∈ T
· simp only [Set.indicator_of_mem haT]
by_cases haS : a ∈ S
· simp only [Set.indicator_of_mem haS]
have harpow : a ^ kappa ≤ (delta + h) ^ kappa :=
Real.rpow_le_rpow haT.1 haS.2 hkappa
have hsquare : q a ^ 2 ≤ (amplitude * h ^ beta) ^ 2 :=
(sq_le_sq₀ (hq0 a)
(mul_nonneg hamp (Real.rpow_nonneg hh.le _))).2 (hqle a)
dsimp [K]
calc
4 * q a ^ 2 * ((kappa + 1) * a ^ kappa) ≤
4 * (amplitude * h ^ beta) ^ 2 * ((kappa + 1) * a ^ kappa) := by
exact mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left hsquare (by norm_num)) (hdensity a haT)
_ ≤ 4 * (amplitude * h ^ beta) ^ 2 *
((kappa + 1) * (delta + h) ^ kappa) := by gcongr
· rw [hqzero a haS]
simp [haS]
· by_cases haS : a ∈ S <;> simp [haT, haS, hK0]
have hmono : (∫ a, T.indicator
(fun a => 4 * q a ^ 2 * ((kappa + 1) * a ^ kappa)) a) ≤
∫ a, S.indicator (fun _ => K) a := by
apply integral_mono_ae
· apply IntegrableOn.integrable_indicator (s := T) (f := fun a =>
4 * q a ^ 2 * ((kappa + 1) * a ^ kappa)) (hs := measurableSet_Icc)
refine IntegrableOn.of_bound measure_Icc_lt_top
(((hqmeas.pow_const 2).const_mul 4).mul
(measurable_const.mul (measurable_id.pow_const kappa))).aestronglyMeasurable
(4 * (amplitude * h ^ beta) ^ 2 * (kappa + 1)) ?_
filter_upwards [ae_restrict_mem measurableSet_Icc] with a ha
rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg (by positivity) (hdensity a ha))]
have haq : q a ^ 2 ≤ (amplitude * h ^ beta) ^ 2 :=
(sq_le_sq₀ (hq0 a)
(mul_nonneg hamp (Real.rpow_nonneg hh.le _))).2 (hqle a)
have har : a ^ kappa ≤ 1 := by
simpa using Real.rpow_le_one ha.1 ha.2 hkappa
calc
4 * q a ^ 2 * ((kappa + 1) * a ^ kappa) ≤
4 * (amplitude * h ^ beta) ^ 2 * ((kappa + 1) * a ^ kappa) := by
exact mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left haq (by norm_num)) (hdensity a ha)
_ ≤ 4 * (amplitude * h ^ beta) ^ 2 * (kappa + 1) := by
have hdle : (kappa + 1) * a ^ kappa ≤ kappa + 1 := by
simpa using mul_le_mul_of_nonneg_left har (by linarith : 0 ≤ kappa + 1)
exact mul_le_mul_of_nonneg_left hdle (by positivity)
· apply IntegrableOn.integrable_indicator (s := S) (f := fun _ : ℝ => K)
(hs := measurableSet_Icc)
refine IntegrableOn.of_bound measure_Icc_lt_top aestronglyMeasurable_const K ?_
filter_upwards with a
rw [Real.norm_eq_abs, abs_of_nonneg hK0]
· exact Filter.Eventually.of_forall hpoint
have hrewrite : (∫ a in Icc (0 : ℝ) 1,
(ofReal ((kappa + 1) * a ^ kappa)).toReal * (4 * q a ^ 2)) =
∫ a, T.indicator (fun a => 4 * q a ^ 2 * ((kappa + 1) * a ^ kappa)) a := by
rw [← integral_indicator measurableSet_Icc]
apply integral_congr_ae
filter_upwards with a
by_cases ha : a ∈ T
· simp [T, ha, ENNReal.toReal_ofReal (hdensity a ha)]
ring
· simp [T, ha]
rw [hrewrite]
calc
_ ≤ ∫ a, S.indicator (fun _ => K) a := hmono
_ = K * (2 * h) := by simp [S, measureReal_def, hh.le]; ring
_ = 8 * (kappa + 1) * amplitude ^ 2 * h ^ (2 * beta + 1) *
(delta + h) ^ kappa := by
dsimp [K]
rw [show h ^ (2 * beta + 1) = h ^ beta * h ^ beta * h by
rw [show 2 * beta + 1 = (beta + beta) + 1 by ring,
Real.rpow_add hh, Real.rpow_add hh, Real.rpow_one]]
ring
theorem minimaxConstant_productChiSq_le_exp_four
the stated minimax constant product chi sq bound exp four property holds for the specified J input, the specified n input, the specified kappa input, the specified hJ input, the specified hkappa input, the specified hn input.
Formal statement
hypotheses
J n :
ℕ
kappa :
ℝ
hJ :
0 < J
hkappa :
0 ≤ kappa
hn :
5 ≤ n
conclusion
eps :
(n : ℝ) ^ (-(1 : ℝ) / 2)
chiSqDiv (Measure.pi (fun _ : Fin n => minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => eps))) (Measure.pi (fun _ : Fin n => minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => 0)))
≤ exp 4
Proof (Lean source)
lemma minimaxConstant_productChiSq_le_exp_four
(J n : ℕ) (kappa : ℝ) (hJ : 0 < J) (hkappa : 0 ≤ kappa)
(hn : 5 ≤ n) :
let eps := (n : ℝ) ^ (-(1 : ℝ) / 2)
chiSqDiv
(Measure.pi (fun _ : Fin n => minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => eps)))
(Measure.pi (fun _ : Fin n => minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => 0))) ≤ exp 4 := by
dsimp only
let eps : ℝ := (n : ℝ) ^ (-(1 : ℝ) / 2)
have hnpos : 0 < n := by omega
have hnR : 0 < (n : ℝ) := by exact_mod_cast hnpos
have heps0 : 0 < eps := Real.rpow_pos_of_pos hnR _
have hepslt : |eps| < 1 / 2 := by
rw [abs_of_pos heps0]
have hn4 : (4 : ℝ) < n := by exact_mod_cast hn
calc
eps < (4 : ℝ) ^ (-(1 : ℝ) / 2) := by
dsimp [eps]
exact Real.rpow_lt_rpow_of_neg (by norm_num) hn4 (by norm_num)
_ = 1 / 2 := by norm_num
have hepsSq : eps ^ 2 = (n : ℝ)⁻¹ := by
dsimp [eps]
rw [← Real.rpow_natCast, ← Real.rpow_mul hnR.le]
norm_num
rw [Real.rpow_neg_one]
have hneps : (n : ℝ) * eps ^ 2 = 1 := by
rw [hepsSq]
exact mul_inv_cancel₀ hnR.ne'
letI : IsProbabilityMeasure (minimaxDesignMeasure J kappa) :=
minimaxDesignMeasure_isProbabilityMeasure J kappa hJ hkappa
have hint : (∫ p, 4 * ((fun _ : Fin J × ℝ => eps) p) ^ 2
∂minimaxDesignMeasure J kappa) = 4 * eps ^ 2 := by simp
have hchiEq := minimaxProduct_chiSqDiv_center J n kappa hJ hkappa
(fun _ : Fin J × ℝ => eps) measurable_const (fun _ => hepslt)
rw [hint] at hchiEq
have hbase : 1 + 4 * eps ^ 2 ≤ exp (4 * eps ^ 2) := by
simpa [add_comm] using Real.add_one_le_exp (4 * eps ^ 2)
have hpow : (1 + 4 * eps ^ 2) ^ n ≤ exp 4 := by
calc
_ ≤ (exp (4 * eps ^ 2)) ^ n :=
pow_le_pow_left₀ (by positivity) hbase n
_ = exp ((n : ℝ) * (4 * eps ^ 2)) := by
rw [Real.exp_nat_mul]
_ = exp 4 := by rw [show (n : ℝ) * (4 * eps ^ 2) = 4 by nlinarith]
change chiSqDiv
(Measure.pi (fun _ : Fin n => minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => eps)))
(Measure.pi (fun _ : Fin n => minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => 0))) ≤ exp 4
linarith [hchiEq, hpow]
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.MinimaxFunctional 6 declarations Clamp functional on the canonical witness family
Clamp functional on the canonical witness family
theorem minimaxAtomMass
the stated minimax atom mass property holds for the specified J input, the specified kappa input, the specified delta input, the specified hkappa input, the specified hdelta input, the specified q input, the specified x input.
Formal statement
hypotheses
conclusion
atomMass (minimaxClampLaw J kappa (fun p => q p.2)) x delta = delta ^ (kappa + 1)
Proof (Lean source)
lemma minimaxAtomMass (J : ℕ) (kappa delta : ℝ) (hkappa : 0 ≤ kappa)
(hdelta : 0 ≤ delta) (q : ℝ → ℝ) (x : Fin J) :
atomMass (minimaxClampLaw J kappa (fun p => q p.2)) x delta =
delta ^ (kappa + 1) := by
unfold atomMass minimaxClampLaw
rw [integral_Icc_eq_integral_Ioc, ← intervalIntegral.integral_of_le hdelta,
intervalIntegral.integral_const_mul,
integral_rpow (inl (by linarith : -1 < kappa))]
rw [Real.zero_rpow (by linarith : kappa + 1 ≠ 0)]
have hk1 : kappa + 1 ≠ 0 := by linarith
field_simp [hk1]
ring
theorem minimaxRetainedMean
the stated minimax retained mean property holds for the specified J input, the specified kappa input, the specified delta input, the specified hJ input, the specified hkappa input, the specified q input, the specified hqmeas input, the specified hqbound input.
Formal statement
hypotheses
J :
ℕ
kappa delta :
ℝ
hJ :
0 < J
hkappa :
0 ≤ kappa
q :
ℝ → ℝ
hqmeas :
hqbound :
∀ a, |q a| ≤ 1 / 2
conclusion
retainedMean (minimaxClampLaw J kappa (fun p => q p.2)) delta
= ∫ a in Ioi delta, (1 / 2 + q a) ∂minimaxTreatmentMeasure kappa
Proof (Lean source)
lemma minimaxRetainedMean (J : ℕ) (kappa delta : ℝ)
(hJ : 0 < J) (hkappa : 0 ≤ kappa)
(q : ℝ → ℝ) (hqmeas : Measurable q) (hqbound : ∀ a, |q a| ≤ 1 / 2) :
retainedMean (minimaxClampLaw J kappa (fun p => q p.2)) delta =
∫ a in Set.Ioi delta, (1 / 2 + q a) ∂minimaxTreatmentMeasure kappa := by
let g : Fin J × ℝ → ℝ := fun p => q p.2
have hgmeas : Measurable g := hqmeas.comp measurable_snd
have hgbound : ∀ p, |g p| ≤ 1 / 2 := fun p => hqbound p.2
let T : Set (Fin J × ℝ) := (Set.univ : Set (Fin J)) ×ˢ Set.Ioi delta
have hT : MeasurableSet T := MeasurableSet.univ.prod measurableSet_Ioi
have hreg := minimaxDataMeasure_integral_Y_design J kappa hJ hkappa g hgmeas
hgbound T hT
haveI : Nonempty (Fin J) := Fin.pos_iff_nonempty.mp hJ
haveI : IsProbabilityMeasure (minimaxStratumMeasure J) := by
unfold minimaxStratumMeasure
infer_instance
haveI : IsProbabilityMeasure (minimaxTreatmentMeasure kappa) :=
minimaxTreatmentMeasure_isProbabilityMeasure kappa hkappa
have hprod := setIntegral_prod_mul
(μ := minimaxStratumMeasure J) (ν := minimaxTreatmentMeasure kappa)
(fun _ : Fin J => (1 : ℝ)) (fun a => 1 / 2 + q a)
(Set.univ : Set (Fin J)) (Set.Ioi delta)
rw [show retainedMean (minimaxClampLaw J kappa (fun p => q p.2)) delta =
∫ o in (fun o : ClampObs J => (o.X, o.A)) ⁻¹' T, o.Y
∂minimaxDataMeasure J kappa g by
unfold retainedMean
change (∫ o, o.Y * indicator {o : ClampObs J | delta < o.A}
(fun _ => (1 : ℝ)) o ∂minimaxDataMeasure J kappa g) = _
have hd : Measurable (fun o : ClampObs J => (o.X, o.A)) := by
let hall : Measurable (fun o : ClampObs J => (o.X, o.A, o.Y)) :=
Measurable.of_comap_le le_rfl
exact (measurable_fst.comp hall).prodMk
(measurable_fst.comp (measurable_snd.comp hall))
calc
(∫ o, o.Y * indicator {o : ClampObs J | delta < o.A}
(fun _ => (1 : ℝ)) o ∂minimaxDataMeasure J kappa g) =
∫ o, ((fun o : ClampObs J => (o.X, o.A)) ⁻¹' T).indicator
(fun o => o.Y) o ∂minimaxDataMeasure J kappa g := by
apply integral_congr_ae
filter_upwards with o
simp [T, indicator, g]
_ = _ := integral_indicator (hT.preimage hd)]
rw [hreg]
change (∫ p in T, (1 / 2 + q p.2) ∂minimaxDesignMeasure J kappa) = _
rw [minimaxDesignMeasure]
calc
(∫ p in T, (1 / 2 + q p.2)
∂(minimaxStratumMeasure J).prod (minimaxTreatmentMeasure kappa)) =
(∫ _x in (Set.univ : Set (Fin J)), (1 : ℝ) ∂minimaxStratumMeasure J) *
∫ a in Set.Ioi delta, (1 / 2 + q a) ∂minimaxTreatmentMeasure kappa := by
simpa [T] using hprod
_ = _ := by simp
theorem minimaxClampFunctional
the stated minimax clamp functional property holds for the specified J input, the specified kappa input, the specified delta input, the specified hJ input, the specified hkappa input, the specified hdelta input, the specified q input, the specified hqmeas input, the specified hqbound input.
Formal statement
hypotheses
J :
ℕ
kappa delta :
ℝ
hJ :
0 < J
hkappa :
0 ≤ kappa
hdelta :
0 ≤ delta
q :
ℝ → ℝ
hqmeas :
hqbound :
∀ a, |q a| ≤ 1 / 2
conclusion
clampFunctional (minimaxClampLaw J kappa (fun p => q p.2)) delta
= (∫ a in Ioi delta, (1 / 2 + q a) ∂minimaxTreatmentMeasure kappa)
+ delta ^ (kappa + 1) * (1 / 2 + q delta)
Proof (Lean source)
lemma minimaxClampFunctional (J : ℕ) (kappa delta : ℝ)
(hJ : 0 < J) (hkappa : 0 ≤ kappa) (hdelta : 0 ≤ delta)
(q : ℝ → ℝ) (hqmeas : Measurable q) (hqbound : ∀ a, |q a| ≤ 1 / 2) :
clampFunctional (minimaxClampLaw J kappa (fun p => q p.2)) delta =
(∫ a in Ioi delta, (1 / 2 + q a) ∂minimaxTreatmentMeasure kappa) +
delta ^ (kappa + 1) * (1 / 2 + q delta) := by
rw [clampFunctional, minimaxRetainedMean J kappa delta hJ hkappa q hqmeas hqbound]
simp_rw [minimaxAtomMass J kappa delta hkappa hdelta q]
simp [minimaxClampLaw]
have hJR : (J : ℝ) ≠ 0 := by exact_mod_cast hJ.ne'
field_simp [hJR]
theorem minimaxTreatmentMeasure_real_Ioi
the stated minimax treatment measure real ioi property holds for the specified kappa input, the specified delta input, the specified hkappa input, the specified hdelta input.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma minimaxTreatmentMeasure_real_Ioi (kappa delta : ℝ)
(hkappa : 0 ≤ kappa) (hdelta : delta ∈ Icc (0 : ℝ) 1) :
(minimaxTreatmentMeasure kappa).real (Ioi delta) =
1 - delta ^ (kappa + 1) := by
letI : IsProbabilityMeasure (minimaxTreatmentMeasure kappa) :=
minimaxTreatmentMeasure_isProbabilityMeasure kappa hkappa
have hicc : (minimaxTreatmentMeasure kappa).real (Icc (0 : ℝ) delta) =
delta ^ (kappa + 1) := by
rw [minimaxTreatmentMeasure_real kappa hkappa _ measurableSet_Icc
(fun a ha => ⟨ha.1, ha.2.trans hdelta.2⟩)]
rw [integral_Icc_eq_integral_Ioc, ← intervalIntegral.integral_of_le hdelta.1,
intervalIntegral.integral_const_mul,
integral_rpow (inl (by linarith : -1 < kappa))]
rw [Real.zero_rpow (by linarith : kappa + 1 ≠ 0)]
have hk1 : kappa + 1 ≠ 0 := by linarith
field_simp [hk1]
ring
have hiic : (minimaxTreatmentMeasure kappa).real (Iic delta) =
delta ^ (kappa + 1) := by
rw [← hicc]
apply measureReal_congr
filter_upwards [minimaxTreatmentMeasure_ae_mem_Icc kappa] with a ha
apply propext
change a ≤ delta ↔ 0 ≤ a ∧ a ≤ delta
constructor
· intro had
exact ⟨ha.1, had⟩
· intro had
exact had.2
rw [← Set.compl_Iic, measureReal_compl measurableSet_Iic, hiic]
simp
theorem minimaxGlobalSeparation
the stated minimax global separation property holds for the specified J input, the specified kappa input, the specified delta input, the specified eps input, the specified hJ input, the specified hkappa input, the specified hdelta input, the specified heps input.
Formal statement
hypotheses
J :
ℕ
kappa delta eps :
ℝ
hJ :
0 < J
hkappa :
0 ≤ kappa
hdelta :
delta ∈ Icc (0 : ℝ) 1
heps :
|eps| ≤ 1 / 2
conclusion
clampFunctional (minimaxClampLaw J kappa (fun _ => eps)) delta
- clampFunctional (minimaxClampLaw J kappa (fun _ => 0)) delta
= eps
Proof (Lean source)
lemma minimaxGlobalSeparation (J : ℕ) (kappa delta eps : ℝ)
(hJ : 0 < J) (hkappa : 0 ≤ kappa) (hdelta : delta ∈ Icc (0 : ℝ) 1)
(heps : |eps| ≤ 1 / 2) :
clampFunctional (minimaxClampLaw J kappa (fun _ => eps)) delta -
clampFunctional (minimaxClampLaw J kappa (fun _ => 0)) delta = eps := by
rw [minimaxClampFunctional J kappa delta hJ hkappa hdelta.1
(fun _ => eps) measurable_const (fun _ => heps),
minimaxClampFunctional J kappa delta hJ hkappa hdelta.1
(fun _ => 0) measurable_const (fun _ => by norm_num)]
have hconst (c : ℝ) :
(∫ _a in Ioi delta, c ∂minimaxTreatmentMeasure kappa) =
c * (minimaxTreatmentMeasure kappa).real (Ioi delta) := by
simp [mul_comm]
rw [show (∫ a in Ioi delta, (1 / 2 + eps) ∂minimaxTreatmentMeasure kappa) =
(1 / 2 + eps) * (minimaxTreatmentMeasure kappa).real (Ioi delta) by
exact hconst _,
show (∫ a in Ioi delta, (1 / 2 + 0) ∂minimaxTreatmentMeasure kappa) =
(1 / 2) * (minimaxTreatmentMeasure kappa).real (Ioi delta) by
simpa using hconst (1 / 2)]
rw [minimaxTreatmentMeasure_real_Ioi kappa delta hkappa hdelta]
ring
theorem minimaxLocalSeparation
the stated minimax local separation property holds for the specified J input, the specified beta input, the specified kappa input, the specified delta input, the specified h input, the specified amplitude input, the specified hJ input, the specified hbeta input, the specified hkappa input, the specified hdelta input, the specified hh input, the specified hh1 input, the specified hamp input, the specified hamp_le input.
Formal statement
hypotheses
J :
ℕ
beta kappa delta h amplitude :
ℝ
hJ :
0 < J
hbeta :
0 < beta
hkappa :
0 ≤ kappa
hdelta :
delta ∈ Icc (0 : ℝ) 1
hh :
0 < h
hh1 :
h ≤ 1
hamp :
0 ≤ amplitude
hamp_le :
amplitude ≤ 1 / 4
conclusion
q :
fun a : ℝ
=> amplitude * h ^ beta * CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
amplitude * delta ^ (kappa + 1) * h ^ beta
≤ clampFunctional (minimaxClampLaw J kappa (fun p => q p.2)) delta
- clampFunctional (minimaxClampLaw J kappa (fun _ => 0)) delta
Proof (Lean source)
lemma minimaxLocalSeparation (J : ℕ) (beta kappa delta h amplitude : ℝ)
(hJ : 0 < J) (hbeta : 0 < beta) (hkappa : 0 ≤ kappa)
(hdelta : delta ∈ Icc (0 : ℝ) 1) (hh : 0 < h)
(hh1 : h ≤ 1) (hamp : 0 ≤ amplitude) (hamp_le : amplitude ≤ 1 / 4) :
let q := fun a : ℝ => amplitude * h ^ beta *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
amplitude * delta ^ (kappa + 1) * h ^ beta ≤
clampFunctional (minimaxClampLaw J kappa (fun p => q p.2)) delta -
clampFunctional (minimaxClampLaw J kappa (fun _ => 0)) delta := by
dsimp only
let q := fun a : ℝ => amplitude * h ^ beta *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
have hqmeas : Measurable q := by
dsimp [q]
fun_prop
have hqnonneg : ∀ a, 0 ≤ q a := by
intro a
dsimp [q]
exact mul_nonneg (mul_nonneg hamp (Real.rpow_nonneg hh.le _))
(CausalSmith.Stat.DoseResponseMinimax.doseBump_nonneg _)
have hqbound : ∀ a, |q a| ≤ 1 / 2 := by
intro a
rw [abs_of_nonneg (hqnonneg a)]
have hb := CausalSmith.Stat.DoseResponseMinimax.doseBump_le_one ((a - delta) / h)
calc
q a ≤ amplitude * h ^ beta := by
dsimp [q]
nlinarith [mul_nonneg hamp (Real.rpow_nonneg hh.le beta)]
_ ≤ 1 / 2 := by
have hp : h ^ beta ≤ 1 := Real.rpow_le_one hh.le hh1 hbeta.le
calc
amplitude * h ^ beta ≤ amplitude * 1 := mul_le_mul_of_nonneg_left hp hamp
_ ≤ 1 / 2 := by linarith
rw [minimaxClampFunctional J kappa delta hJ hkappa hdelta.1 q hqmeas hqbound,
minimaxClampFunctional J kappa delta hJ hkappa hdelta.1
(fun _ => 0) measurable_const (fun _ => by norm_num)]
haveI : IsProbabilityMeasure (minimaxTreatmentMeasure kappa) :=
minimaxTreatmentMeasure_isProbabilityMeasure kappa hkappa
have hqint : Integrable q (minimaxTreatmentMeasure kappa) := by
refine Integrable.of_bound hqmeas.aestronglyMeasurable (1 / 2) ?_
filter_upwards with a
simpa [Real.norm_eq_abs] using hqbound a
have honeint : Integrable (fun _ : ℝ => (1 / 2 : ℝ))
(minimaxTreatmentMeasure kappa) := integrable_const _
rw [show (∫ a in Ioi delta, (1 / 2 + q a) ∂minimaxTreatmentMeasure kappa) =
(∫ a in Ioi delta, (1 / 2 : ℝ) ∂minimaxTreatmentMeasure kappa) +
∫ a in Ioi delta, q a ∂minimaxTreatmentMeasure kappa by
rw [integral_add honeint.integrableOn hqint.integrableOn],
show (∫ a in Ioi delta, (1 / 2 + 0) ∂minimaxTreatmentMeasure kappa) =
∫ a in Ioi delta, (1 / 2 : ℝ) ∂minimaxTreatmentMeasure kappa by simp]
have hqdelta : q delta = amplitude * h ^ beta := by
simp [q, hh.ne', CausalSmith.Stat.DoseResponseMinimax.doseBump_zero]
rw [hqdelta]
have hnonnegint : 0 ≤ ∫ a in Ioi delta, q a ∂minimaxTreatmentMeasure kappa :=
integral_nonneg_of_ae (ae_of_all _ hqnonneg)
nlinarith
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.MinimaxHolder 2 declarations Taylor certification for the localized minimax bump
Taylor certification for the localized minimax bump
theorem taylorWithin_remainder_of_holder
A global smoothness and top-derivative Hölder bound imply the exact within-interval Taylor remainder convention used by HolderRegression. The result uses the hbeta condition, the hL condition, the hf condition, the htop condition. This is the stated conclusion.
Formal statement
hypotheses
f :
ℝ → ℝ
beta L :
ℝ
hbeta :
0 < beta
hL :
0 ≤ L
hf :
ContDiff ℝ (ellOf beta) f
htop :
∀ x ∈ Icc (0 : ℝ) 1,
∀ y ∈ Icc (0 : ℝ) 1,
|iteratedDeriv (ellOf beta) f x - iteratedDeriv (ellOf beta) f y|
≤ L * |x - y| ^ (beta - (ellOf beta : ℝ))
s ∈ Set.Icc (0 : ℝ) 1 :
t ∈ Set.Icc (0 : ℝ) 1 :
conclusion
|f t
- ∑ j ∈ range (ellOf beta + 1), iteratedDerivWithin j f (Icc (0 : ℝ) 1) s * (t - s) ^ j / (Nat.factorial j : ℝ)|
≤ L * |t - s| ^ beta
Proof (Lean source)
lemma taylorWithin_remainder_of_holder
{f : ℝ → ℝ} {beta L : ℝ} (hbeta : 0 < beta) (hL : 0 ≤ L)
(hf : ContDiff ℝ (ellOf beta) f)
(htop : ∀ x ∈ Icc (0 : ℝ) 1, ∀ y ∈ Icc (0 : ℝ) 1,
|iteratedDeriv (ellOf beta) f x - iteratedDeriv (ellOf beta) f y| ≤
L * |x - y| ^ (beta - (ellOf beta : ℝ))) :
∀ s ∈ Icc (0 : ℝ) 1, ∀ t ∈ Icc (0 : ℝ) 1,
|f t - ∑ j ∈ range (ellOf beta + 1),
iteratedDerivWithin j f (Icc (0 : ℝ) 1) s *
(t - s) ^ j / (Nat.factorial j : ℝ)| ≤ L * |t - s| ^ beta := by
intro s hs t ht
have hrem := holder_taylor_remainder
(f := f) (M := L) (β := beta) (lo := 0) (hi := 1) (t := s) (a := t)
hbeta hL hs ht (by simpa [ellOf_eq_holderDerivOrder] using hf)
(by simpa [ellOf_eq_holderDerivOrder] using htop)
have hpoly :
Causalean.Stat.Nonparametric.taylorPoly (ellOf beta) f s t =
∑ j ∈ range (ellOf beta + 1),
iteratedDerivWithin j f (Icc (0 : ℝ) 1) s *
(t - s) ^ j / (Nat.factorial j : ℝ) := by
unfold Causalean.Stat.Nonparametric.taylorPoly
apply Finset.sum_congr rfl
intro j hj
have hjle : j ≤ ellOf beta := Nat.lt_succ_iff.mp (Finset.mem_range.mp hj)
have hwithin : iteratedDerivWithin j f (Icc (0 : ℝ) 1) s =
iteratedDeriv j f s := by
exact iteratedDerivWithin_eq_iteratedDeriv (uniqueDiffOn_Icc (by norm_num))
((hf.of_le (WithTop.coe_le_coe.mpr (ENat.coe_le_coe.mpr hjle))).contDiffAt) hs
rw [hwithin]
ring
rw [← hpoly]
calc
|f t - Causalean.Stat.Nonparametric.taylorPoly (ellOf beta) f s t| ≤
L / (Nat.factorial (ellOf beta) : ℝ) * |t - s| ^ beta := by
simpa [ellOf_eq_holderDerivOrder] using hrem
_ ≤ L * |t - s| ^ beta := by
have hfacNat : 1 ≤ Nat.factorial (ellOf beta) :=
Nat.succ_le_iff.mpr (Nat.factorial_pos _)
have hfac : (1 : ℝ) ≤ Nat.factorial (ellOf beta) := by exact_mod_cast hfacNat
have hdiv : L / (Nat.factorial (ellOf beta) : ℝ) ≤ L := by
exact div_le_self hL hfac
exact mul_le_mul_of_nonneg_right hdiv (Real.rpow_nonneg (abs_nonneg _) _)
theorem exists_minimaxBump_amplitude
Fixed amplitude and the localized smooth bump satisfy the paper's exact Taylor-remainder Hölder member uniformly over all admissible centers and bandwidths. The result uses the hbeta condition, the hL condition. This is the stated conclusion.
Formal statement
hypotheses
beta L :
ℝ
hbeta :
0 < beta
hL :
0 < L
conclusion
∃ amplitude : ℝ,
0 < amplitude
amplitude ≤ 1 / 4
conclusion 1
delta h :
ℝ
0 ≤ delta
delta ≤ 1
0 < h
h ≤ 1
q :
fun a : ℝ
=> amplitude * h ^ beta * CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
ContinuousOn (fun a => 1 / 2 + q a) (Icc (0 : ℝ) 1)
a ∈ Set.Icc (0 : ℝ) 1 :
1 / 2 + q a ∈ Icc (0 : ℝ) 1
s ∈ Set.Icc (0 : ℝ) 1 :
t ∈ Set.Icc (0 : ℝ) 1 :
|(1 / 2 + q t)
- ∑ j ∈ range (ellOf beta + 1), iteratedDerivWithin j (fun a => 1 / 2 + q a) (Icc (0 : ℝ) 1) s * (t - s) ^ j / (Nat.factorial j : ℝ)|
≤ L * |t - s| ^ beta
Proof (Lean source)
lemma exists_minimaxBump_amplitude (beta L : ℝ) (hbeta : 0 < beta) (hL : 0 < L) :
∃ amplitude : ℝ, 0 < amplitude ∧ amplitude ≤ 1 / 4 ∧
∀ {delta h : ℝ}, 0 ≤ delta → delta ≤ 1 → 0 < h → h ≤ 1 →
let q := fun a : ℝ => amplitude * h ^ beta *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
ContinuousOn (fun a => 1 / 2 + q a) (Icc (0 : ℝ) 1) ∧
(∀ a ∈ Icc (0 : ℝ) 1, 1 / 2 + q a ∈ Icc (0 : ℝ) 1) ∧
∀ s ∈ Icc (0 : ℝ) 1, ∀ t ∈ Icc (0 : ℝ) 1,
|(1 / 2 + q t) - ∑ j ∈ range (ellOf beta + 1),
iteratedDerivWithin j (fun a => 1 / 2 + q a) (Icc (0 : ℝ) 1) s *
(t - s) ^ j / (Nat.factorial j : ℝ)| ≤ L * |t - s| ^ beta := by
let M := min L 1
have hM : 0 < M := lt_min hL zero_lt_one
obtain ⟨amplitude, hamp, hampM, hball⟩ :=
CausalSmith.Stat.DoseResponseMinimax.doseBump_holder_gate beta M 0 1 hbeta hM
refine ⟨amplitude, hamp, hampM.trans ?_, ?_⟩
· exact (div_le_div_of_nonneg_right (min_le_right L 1) (by norm_num)).trans_eq (by norm_num)
intro delta h hdelta hdelta1 hh hh1
dsimp only
let q := fun a : ℝ => amplitude * h ^ beta *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
have hqball := hball (ζ := (1 : ℝ)) (h := h) (inr rfl) hh hh1
have hsmoothTop : ContDiff ℝ (⊤ : ℕ∞) q := by
dsimp [q]
exact contDiff_const.mul
(CausalSmith.Stat.DoseResponseMinimax.doseContDiffBump.contDiff.comp
((contDiff_id.sub contDiff_const).div_const h))
have hsmooth : ContDiff ℝ (ellOf beta) (fun a => 1 / 2 + q a) :=
contDiff_const.add (hsmoothTop.of_le
(WithTop.coe_le_coe.mpr
(show (ellOf beta : ℕ∞) ≤ (⊤ : ℕ∞) from le_top)))
have hcont : ContinuousOn (fun a => 1 / 2 + q a) (Icc (0 : ℝ) 1) :=
hsmooth.continuous.continuousOn
have hqnonneg (a : ℝ) : 0 ≤ q a := by
dsimp [q]
exact mul_nonneg (mul_nonneg hamp.le (Real.rpow_nonneg hh.le _))
(CausalSmith.Stat.DoseResponseMinimax.doseBump_nonneg _)
have hqle (a : ℝ) : q a ≤ 1 / 4 := by
have hb := CausalSmith.Stat.DoseResponseMinimax.doseBump_le_one ((a - delta) / h)
have hhpow : h ^ beta ≤ 1 := Real.rpow_le_one hh.le hh1 hbeta.le
have ha4 : amplitude ≤ 1 / 4 := hampM.trans
((div_le_div_of_nonneg_right (min_le_right L 1) (by norm_num)).trans_eq (by norm_num))
calc
q a ≤ amplitude * h ^ beta * 1 :=
mul_le_mul_of_nonneg_left hb (mul_nonneg hamp.le (Real.rpow_nonneg hh.le _))
_ ≤ amplitude * 1 := by
simpa only [mul_one] using mul_le_mul_of_nonneg_left hhpow hamp.le
_ ≤ 1 / 4 := by simpa using ha4
refine ⟨hcont, ?_, ?_⟩
· intro a ha
exact ⟨by linarith [hqnonneg a], by linarith [hqle a]⟩
apply taylorWithin_remainder_of_holder hbeta hL.le hsmooth
intro x hx y hy
have hxd : x - delta ∈ Icc (-1 : ℝ) 1 := by
constructor <;> linarith [hx.1, hx.2]
have hyd : y - delta ∈ Icc (-1 : ℝ) 1 := by
constructor <;> linarith [hy.1, hy.2]
have htop := hqball.2.2 (x - delta)
(by simpa [CausalSmith.Stat.DoseResponseMinimax.doseWindow] using hxd)
(y - delta)
(by simpa [CausalSmith.Stat.DoseResponseMinimax.doseWindow] using hyd)
have htranslate (j : ℕ) (a : ℝ) :
iteratedDeriv j q a =
iteratedDeriv j
(fun u : ℝ => amplitude * h ^ beta *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((u - 0) / h))
(a - delta) := by
let f0 := fun u : ℝ => amplitude * h ^ beta *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((u - 0) / h)
have hfun : q = fun a => f0 (a - delta) := by
funext a
simp [q, f0]
change iteratedDeriv j q a = iteratedDeriv j f0 (a - delta)
rw [hfun]
exact congrFun (iteratedDeriv_comp_sub_const (n := j) (f := f0) (s := delta)) a
have hdist : |(x - delta) - (y - delta)| = |x - y| := by ring_nf
have htopq :
|iteratedDeriv (ellOf beta) q x - iteratedDeriv (ellOf beta) q y| ≤
M * |x - y| ^ (beta - (ellOf beta : ℝ)) := by
rw [htranslate (ellOf beta) x, htranslate (ellOf beta) y]
simpa [ellOf, hdist] using htop
have hML : M ≤ L := min_le_left _ _
have htopL := htopq.trans (mul_le_mul_of_nonneg_right hML
(Real.rpow_nonneg (abs_nonneg _) _))
by_cases hell : ellOf beta = 0
· simpa [hell, iteratedDeriv_zero] using htopL
· have hellpos : 0 < ellOf beta := Nat.pos_of_ne_zero hell
rw [iteratedDeriv_const_add hellpos (1 / 2 : ℝ),
iteratedDeriv_const_add hellpos (1 / 2 : ℝ)]
exact htopL
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.MinimaxLaw 13 declarations The centered two-point law on {-1/2,1/2} is translated to a genuine Bernoulli outcome on {0,1}.
Bernoulli laws over the canonical minimax design
The centered two-point law on {-1/2,1/2} is translated to a genuine
Bernoulli outcome on {0,1}. Keeping this construction as an iterated kernel
makes the common design marginal and regression identities transparent.
def minimaxOutcomeKernel
The Bernoulli observation kernel with centered conditional mean g.
theorem measurable_minimaxOutcomeKernel
the stated measurable minimax outcome kernel property holds for the specified J input, the specified g input, the specified hg input.
Formal statement
Proof (Lean source)
lemma measurable_minimaxOutcomeKernel {J : ℕ} (g : Fin J × ℝ → ℝ)
(hg : Measurable g) : Measurable (minimaxOutcomeKernel g) := by
classical
refine Measure.measurable_of_measurable_coe _ ?_
intro S hS
unfold minimaxOutcomeKernel twoPointMean
simp_rw [Measure.map_apply (measurable_clampObs_mk _) hS]
simp only [Measure.add_apply, Measure.smul_apply, Measure.dirac_apply, smul_eq_mul]
have hplus : Measurable (fun p : Fin J × ℝ =>
((fun y : ℝ => ClampObs.mk p.1 p.2 (y + 1 / 2)) ⁻¹' S).indicator
(1 : ℝ → ℝ≥0∞) (1 / 2)) := by
rw [show (fun p : Fin J × ℝ =>
((fun y : ℝ => ClampObs.mk p.1 p.2 (y + 1 / 2)) ⁻¹' S).indicator
(1 : ℝ → ℝ≥0∞) (1 / 2)) =
fun p => if ClampObs.mk p.1 p.2 1 ∈ S then 1 else 0 by
funext p
simp [Set.indicator_apply]
norm_num]
exact Measurable.ite (hS.preimage (by rw [measurable_comap_iff]; fun_prop))
measurable_const measurable_const
have hminus : Measurable (fun p : Fin J × ℝ =>
((fun y : ℝ => ClampObs.mk p.1 p.2 (y + 1 / 2)) ⁻¹' S).indicator
(1 : ℝ → ℝ≥0∞) (-1 / 2)) := by
rw [show (fun p : Fin J × ℝ =>
((fun y : ℝ => ClampObs.mk p.1 p.2 (y + 1 / 2)) ⁻¹' S).indicator
(1 : ℝ → ℝ≥0∞) (-1 / 2)) =
fun p => if ClampObs.mk p.1 p.2 0 ∈ S then 1 else 0 by
funext p
simp [Set.indicator_apply]
norm_num]
exact Measurable.ite (hS.preimage (by rw [measurable_comap_iff]; fun_prop))
measurable_const measurable_const
exact ((ENNReal.measurable_ofReal.comp
(measurable_const.add (hg.div_const _) |>.div_const _)).mul hplus).add
((ENNReal.measurable_ofReal.comp
(measurable_const.sub (hg.div_const _) |>.div_const _)).mul (by
simpa only [neg_div] using hminus))
def minimaxDataMeasure
Joint observation law induced by the canonical design and centered mean g.
Definition (Lean source)
parameters
given by
def minimaxClampLaw
Canonical law package associated with a centered Bernoulli regression.
Definition (Lean source)
parameters
def
given by
clause 1
dataMeasure := minimaxDataMeasure J kappa g
clause 2
px := fun _ => 1 / (J : ℝ)
clause 3
pi := fun _ a => (kappa + 1) * a ^ kappa
clause 4
mu := fun x a => 1 / 2 + g (x, a)
theorem minimaxOutcomeKernel_isProbabilityMeasure
minimax outcome kernel is a probability measure for the specified J input, the specified g input, the specified hg input, the specified p input.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma minimaxOutcomeKernel_isProbabilityMeasure {J : ℕ}
{g : Fin J × ℝ → ℝ} (hg : ∀ p, |g p| ≤ 1 / 2) (p : Fin J × ℝ) :
IsProbabilityMeasure (minimaxOutcomeKernel g p) := by
letI : IsProbabilityMeasure
(twoPointMean (1 / 2) (g p)) :=
twoPointMean_isProbabilityMeasure (by norm_num) (hg p)
exact Measure.isProbabilityMeasure_map (measurable_clampObs_mk p).aemeasurable
theorem minimaxDataMeasure_isProbabilityMeasure
minimax data measure is a probability measure for the specified J input, the specified kappa input, the specified hJ input, the specified hkappa input, the specified g input, the specified hgmeas input, the specified hg input.
Formal statement
hypotheses
conclusion
IsProbabilityMeasure (minimaxDataMeasure J kappa g)
Proof (Lean source)
lemma minimaxDataMeasure_isProbabilityMeasure (J : ℕ) (kappa : ℝ)
(hJ : 0 < J) (hkappa : 0 ≤ kappa) (g : Fin J × ℝ → ℝ)
(hgmeas : Measurable g) (hg : ∀ p, |g p| ≤ 1 / 2) :
IsProbabilityMeasure (minimaxDataMeasure J kappa g) := by
letI : IsProbabilityMeasure (minimaxDesignMeasure J kappa) :=
minimaxDesignMeasure_isProbabilityMeasure J kappa hJ hkappa
exact isProbabilityMeasure_bind (measurable_minimaxOutcomeKernel g hgmeas).aemeasurable
(Filter.Eventually.of_forall (minimaxOutcomeKernel_isProbabilityMeasure hg))
theorem minimaxOutcomeKernel_map_design
the stated minimax outcome kernel map design property holds for the specified J input, the specified g input, the specified hg input, the specified p input.
Formal statement
hypotheses
conclusion
(minimaxOutcomeKernel g p).map (fun o => (o.X, o.A)) = Measure.dirac p
Proof (Lean source)
lemma minimaxOutcomeKernel_map_design {J : ℕ}
{g : Fin J × ℝ → ℝ} (hg : ∀ p, |g p| ≤ 1 / 2) (p : Fin J × ℝ) :
(minimaxOutcomeKernel g p).map (fun o => (o.X, o.A)) = Measure.dirac p := by
letI : IsProbabilityMeasure
(twoPointMean (1 / 2) (g p)) :=
twoPointMean_isProbabilityMeasure (by norm_num) (hg p)
unfold minimaxOutcomeKernel
rw [Measure.map_map measurable_clampObs_design (measurable_clampObs_mk p)]
have hcomp : ((fun o : ClampObs J => (o.X, o.A)) ∘
fun y : ℝ => ClampObs.mk p.1 p.2 (y + 1 / 2)) = fun _ : ℝ => p := by
funext y
rfl
rw [hcomp]
rw [Measure.map_const _ p, measure_univ, one_smul]
theorem minimaxDataMeasure_map_design
the stated minimax data measure map design property holds for the specified J input, the specified kappa input, the specified g input, the specified hgmeas input, the specified hg input.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma minimaxDataMeasure_map_design (J : ℕ) (kappa : ℝ)
(g : Fin J × ℝ → ℝ) (hgmeas : Measurable g)
(hg : ∀ p, |g p| ≤ 1 / 2) :
(minimaxDataMeasure J kappa g).map (fun o => (o.X, o.A)) =
minimaxDesignMeasure J kappa := by
ext S hS
rw [Measure.map_apply measurable_clampObs_design hS]
unfold minimaxDataMeasure
rw [Measure.bind_apply (hS.preimage measurable_clampObs_design)
(measurable_minimaxOutcomeKernel g hgmeas).aemeasurable]
have hfiber (p : Fin J × ℝ) :
(minimaxOutcomeKernel g p) ((fun o : ClampObs J => (o.X, o.A)) ⁻¹' S) =
S.indicator (fun _ => (1 : ℝ≥0∞)) p := by
rw [← Measure.map_apply measurable_clampObs_design hS,
minimaxOutcomeKernel_map_design hg p]
by_cases hp : p ∈ S <;> simp [Measure.dirac_apply, indicator, hp]
simp_rw [hfiber]
rw [lintegral_indicator hS]
simp [Pi.one_apply]
theorem minimaxDataMeasure_map_X
the stated minimax data measure map x property holds for the specified J input, the specified kappa input, the specified hJ input, the specified hkappa input, the specified g input, the specified hgmeas input, the specified hg input.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma minimaxDataMeasure_map_X (J : ℕ) (kappa : ℝ)
(hJ : 0 < J) (hkappa : 0 ≤ kappa)
(g : Fin J × ℝ → ℝ) (hgmeas : Measurable g)
(hg : ∀ p, |g p| ≤ 1 / 2) :
(minimaxDataMeasure J kappa g).map (fun o => o.X) = minimaxStratumMeasure J := by
letI : IsProbabilityMeasure (minimaxTreatmentMeasure kappa) :=
minimaxTreatmentMeasure_isProbabilityMeasure kappa hkappa
calc
(minimaxDataMeasure J kappa g).map (fun o => o.X) =
((minimaxDataMeasure J kappa g).map (fun o => (o.X, o.A))).map fst := by
rw [Measure.map_map measurable_fst measurable_clampObs_design]
rfl
_ = (minimaxDesignMeasure J kappa).map fst := by
rw [minimaxDataMeasure_map_design J kappa g hgmeas hg]
_ = minimaxStratumMeasure J := by
rw [minimaxDesignMeasure, Measure.map_fst_prod, measure_univ, one_smul]
theorem minimaxTreatmentMeasure_ae_mem_Icc
minimax treatment measure almost everywhere lies in the stated closed interval for the specified kappa input.
Formal statement
Proof (Lean source)
lemma minimaxTreatmentMeasure_ae_mem_Icc (kappa : ℝ) :
∀ᵐ a ∂minimaxTreatmentMeasure kappa, a ∈ Icc (0 : ℝ) 1 := by
rw [ae_iff]
unfold minimaxTreatmentMeasure
change ((volume.restrict (Icc (0 : ℝ) 1)).withDensity
(fun a => ofReal ((kappa + 1) * a ^ kappa)))
(Icc (0 : ℝ) 1)ᶜ = 0
rw [withDensity_apply _ measurableSet_Icc.compl]
apply setLIntegral_measure_zero
rw [Measure.restrict_apply measurableSet_Icc.compl]
simp
theorem minimaxDataMeasure_ae_treatment
the stated minimax data measure almost everywhere treatment property holds for the specified J input, the specified kappa input, the specified g input, the specified hgmeas input, the specified hg input.
Formal statement
hypotheses
conclusion
∀ᵐ o ∂minimaxDataMeasure J kappa g, o.A ∈ Icc (0 : ℝ) 1
Proof (Lean source)
lemma minimaxDataMeasure_ae_treatment (J : ℕ) (kappa : ℝ)
(g : Fin J × ℝ → ℝ) (hgmeas : Measurable g)
(hg : ∀ p, |g p| ≤ 1 / 2) :
∀ᵐ o ∂minimaxDataMeasure J kappa g, o.A ∈ Icc (0 : ℝ) 1 := by
letI : SFinite (minimaxTreatmentMeasure kappa) := by
unfold minimaxTreatmentMeasure
infer_instance
have hs : MeasurableSet {p : Fin J × ℝ | p.2 ∈ Icc (0 : ℝ) 1} :=
measurableSet_Icc.preimage measurable_snd
rw [← ae_map_iff measurable_clampObs_design.aemeasurable hs,
minimaxDataMeasure_map_design J kappa g hgmeas hg]
rw [ae_iff]
have hset : {p : Fin J × ℝ | ¬p.2 ∈ Icc (0 : ℝ) 1} =
(Set.univ : Set (Fin J)) ×ˢ (Icc (0 : ℝ) 1)ᶜ := by ext p; simp
rw [hset, minimaxDesignMeasure, Measure.prod_prod]
rw [show minimaxTreatmentMeasure kappa (Icc (0 : ℝ) 1)ᶜ = 0 by
exact ae_iff.mp (minimaxTreatmentMeasure_ae_mem_Icc kappa)]
simp
theorem minimaxOutcomeKernel_ae_support
the stated minimax outcome kernel almost everywhere support property holds for the specified J input, the specified g input, the specified p input.
Formal statement
Proof (Lean source)
lemma minimaxOutcomeKernel_ae_support {J : ℕ}
(g : Fin J × ℝ → ℝ) (p : Fin J × ℝ) :
∀ᵐ o ∂minimaxOutcomeKernel g p,
o.X = p.1 ∧ o.A = p.2 ∧ (o.Y = 0 ∨ o.Y = 1) := by
unfold minimaxOutcomeKernel
have hs : MeasurableSet {o : ClampObs J |
o.X = p.1 ∧ o.A = p.2 ∧ (o.Y = 0 ∨ o.Y = 1)} := by
have hd : MeasurableSet {o : ClampObs J | (o.X, o.A) = p} :=
(measurableSet_singleton p).preimage measurable_clampObs_design
have hy : MeasurableSet {o : ClampObs J | o.Y = 0 ∨ o.Y = 1} :=
((measurableSet_singleton (0 : ℝ)).preimage measurable_clampObs_Y).union
((measurableSet_singleton (1 : ℝ)).preimage measurable_clampObs_Y)
convert hd.inter hy using 1 <;> ext o <;> simp [Prod.ext_iff, and_assoc]
rw [ae_map_iff (measurable_clampObs_mk p).aemeasurable hs]
rw [ae_iff]
unfold twoPointMean
rw [Measure.add_apply, Measure.smul_apply, Measure.smul_apply]
simp
norm_num
theorem minimaxDataMeasure_ae_bernoulli
the stated minimax data measure almost everywhere bernoulli property holds for the specified J input, the specified kappa input, the specified g input, the specified hgmeas input.
Formal statement
hypotheses
conclusion
∀ᵐ o ∂minimaxDataMeasure J kappa g, o.Y = 0 ∨ o.Y = 1
Proof (Lean source)
lemma minimaxDataMeasure_ae_bernoulli (J : ℕ) (kappa : ℝ)
(g : Fin J × ℝ → ℝ) (hgmeas : Measurable g) :
∀ᵐ o ∂minimaxDataMeasure J kappa g, o.Y = 0 ∨ o.Y = 1 := by
rw [ae_iff]
unfold minimaxDataMeasure
have hbad : MeasurableSet {o : ClampObs J | ¬(o.Y = 0 ∨ o.Y = 1)} := by
exact (((measurableSet_singleton (0 : ℝ)).preimage measurable_clampObs_Y).union
((measurableSet_singleton (1 : ℝ)).preimage measurable_clampObs_Y)).compl
rw [Measure.bind_apply hbad (measurable_minimaxOutcomeKernel g hgmeas).aemeasurable]
apply lintegral_eq_zero_of_ae_eq_zero
filter_upwards [] with p
exact ae_iff.mp ((minimaxOutcomeKernel_ae_support g p).mono (fun o ho => ho.2.2))
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.MinimaxMembership 3 declarations Model membership of canonical Bernoulli witnesses
Model membership of canonical Bernoulli witnesses
theorem minimaxClampModel_of_taylor
the stated minimax clamp model of taylor property holds for the specified J input, the specified beta input, the specified kappa input, the specified L input, the specified cminus input, the specified cplus input, the specified pmin input, the specified hJ input, the specified hbeta input, the specified hkappa input, the specified hcminus input, the specified hcplus input, the specified hpmin input, the specified q input, the specified hqmeas input, the specified hqbound input, the specified hcont input, the specified hrange input, the specified htaylor input.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin :
ℝ
hJ :
0 < J
hbeta :
0 < beta
hkappa :
0 ≤ kappa
hcminus :
cminus ≤ kappa + 1
hcplus :
kappa + 1 ≤ cplus
hpmin :
pmin ≤ 1 / (J : ℝ)
q :
ℝ → ℝ
hqmeas :
hqbound :
∀ a, |q a| ≤ 1 / 2
hcont :
ContinuousOn (fun a => 1 / 2 + q a) (Icc (0 : ℝ) 1)
conclusion
ClampModel (minimaxClampLaw J kappa (fun p => q p.2)) beta kappa L cminus cplus pmin
Proof (Lean source)
lemma minimaxClampModel_of_taylor
(J : ℕ) (beta kappa L cminus cplus pmin : ℝ)
(hJ : 0 < J) (hbeta : 0 < beta) (hkappa : 0 ≤ kappa)
(hcminus : cminus ≤ kappa + 1) (hcplus : kappa + 1 ≤ cplus)
(hpmin : pmin ≤ 1 / (J : ℝ))
(q : ℝ → ℝ) (hqmeas : Measurable q) (hqbound : ∀ a, |q a| ≤ 1 / 2)
(hcont : ContinuousOn (fun a => 1 / 2 + q a) (Icc (0 : ℝ) 1))
(hrange : ∀ a ∈ Icc (0 : ℝ) 1, 1 / 2 + q a ∈ Icc (0 : ℝ) 1)
(htaylor : ∀ s ∈ Icc (0 : ℝ) 1, ∀ t ∈ Icc (0 : ℝ) 1,
|(1 / 2 + q t) - ∑ j ∈ range (ellOf beta + 1),
iteratedDerivWithin j (fun a => 1 / 2 + q a) (Icc (0 : ℝ) 1) s *
(t - s) ^ j / (Nat.factorial j : ℝ)| ≤ L * |t - s| ^ beta) :
ClampModel (minimaxClampLaw J kappa (fun p => q p.2))
beta kappa L cminus cplus pmin := by
let g : Fin J × ℝ → ℝ := fun p => q p.2
have hgmeas : Measurable g := hqmeas.comp measurable_snd
have hgbound : ∀ p, |g p| ≤ 1 / 2 := fun p => hqbound p.2
have hprob := minimaxDataMeasure_isProbabilityMeasure J kappa hJ hkappa g hgmeas hgbound
refine ⟨hprob, minimaxDataMeasure_ae_treatment J kappa g hgmeas hgbound,
(minimaxDataMeasure_ae_bernoulli J kappa g hgmeas).mono (fun o ho => by
rcases ho with ho | ho <;> simp [ho]), ?_, ?_, ?_, ?_⟩
· refine ⟨?_, ?_, ?_⟩
· intro x
fun_prop
· intro x
filter_upwards [ae_restrict_mem measurableSet_Icc] with a ha
exact mul_nonneg (by linarith) (Real.rpow_nonneg ha.1 _)
· intro x
constructor
· change 1 / (J : ℝ) = _
change 1 / (J : ℝ) =
(Measure.map (fun o : ClampObs J => o.X)
(minimaxDataMeasure J kappa g)).real {x}
rw [minimaxDataMeasure_map_X J kappa hJ hkappa g hgmeas hgbound]
exact (minimaxStratumMeasure_real_singleton J hJ x).symm
· intro B hB hsub
have hevent : {o : ClampObs J | o.X = x ∧ o.A ∈ B} =
(fun o => (o.X, o.A)) ⁻¹' ({x} ×ˢ B) := by ext o; simp
rw [hevent]
have hrect : MeasurableSet (({x} : Set (Fin J)) ×ˢ B) :=
(measurableSet_singleton x).prod hB
change (minimaxDataMeasure J kappa g).real
{o : ClampObs J | o.X = x ∧ o.A ∈ B} = _
rw [hevent, measureReal_def, ← Measure.map_apply measurable_obs_design hrect,
minimaxDataMeasure_map_design J kappa g hgmeas hgbound]
change (minimaxDesignMeasure J kappa).real ({x} ×ˢ B) = _
rw [minimaxDesignMeasure_real_rectangle J kappa hJ hkappa x B hB hsub]
rfl
· intro x
constructor
· change 1 / (J : ℝ) = _
change 1 / (J : ℝ) =
(Measure.map (fun o : ClampObs J => o.X)
(minimaxDataMeasure J kappa g)).real {x}
rw [minimaxDataMeasure_map_X J kappa hJ hkappa g hgmeas hgbound]
exact (minimaxStratumMeasure_real_singleton J hJ x).symm
· exact hpmin
· intro x
filter_upwards [ae_restrict_mem measurableSet_Icc] with a ha
constructor
· exact mul_le_mul_of_nonneg_right hcminus (Real.rpow_nonneg ha.1 _)
· exact mul_le_mul_of_nonneg_right hcplus (Real.rpow_nonneg ha.1 _)
· intro x
refine ⟨hcont, hrange, ?_, htaylor⟩
let μ := minimaxDataMeasure J kappa g
let design : ClampObs J → Fin J × ℝ := fun o => (o.X, o.A)
let m : ClampObs J → ℝ := fun o => 1 / 2 + q o.A
letI : IsProbabilityMeasure μ := hprob
have hdesign : Measurable design := measurable_obs_design
have hY : Integrable (fun o : ClampObs J => o.Y) μ := by
refine Integrable.of_bound
(measurable_snd.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))).aestronglyMeasurable 1 ?_
filter_upwards [minimaxDataMeasure_ae_bernoulli J kappa g hgmeas] with o ho
rcases ho with ho | ho <;> rw [ho] <;> norm_num
have hm_meas : Measurable m := by
dsimp [m]
exact measurable_const.add (hqmeas.comp
(measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))))
have hm : Integrable m μ := by
refine Integrable.of_bound hm_meas.aestronglyMeasurable 1 ?_
filter_upwards with o
rw [Real.norm_eq_abs]
change |1 / 2 + q o.A| ≤ 1
have hb := hqbound o.A
rw [abs_le] at hb
exact abs_le.2 ⟨by linarith [hb.1], by linarith [hb.2]⟩
have hm_design :
AEStronglyMeasurable[comap design inferInstance] m μ := by
have hfactor : m = (fun p : Fin J × ℝ => 1 / 2 + q p.2) ∘ design := rfl
rw [hfactor]
exact ((measurable_const.add (hqmeas.comp measurable_snd)).comp
(Measurable.of_comap_le le_rfl)).aestronglyMeasurable
have hintegral : ∀ T : Set (Fin J × ℝ), MeasurableSet T →
(∫ o in design ⁻¹' T, o.Y ∂μ) = ∫ o in design ⁻¹' T, m o ∂μ := by
intro T hT
rw [minimaxDataMeasure_integral_Y_design J kappa hJ hkappa g hgmeas hgbound T hT]
rw [← integral_indicator (hT.preimage hdesign), ← integral_indicator hT]
have hmap := minimaxDataMeasure_map_design J kappa g hgmeas hgbound
rw [← hmap, integral_map hdesign.aemeasurable]
· rfl
· exact ((measurable_const.add hgmeas).indicator hT).aestronglyMeasurable
exact condExp_eq_of_integral_preimage_eq
μ design hdesign (fun o => o.Y) m hY hm hm_design hintegral
theorem minimaxCenter_mem_model
The unperturbed one-half Bernoulli regression is a member of every admissible clamp model and supplies model-class nonemptiness. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
Proof (Lean source)
lemma minimaxCenter_mem_model
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
ClampModel (minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0))
beta kappa L cminus cplus pmin := by
rcases hreg with
⟨hJ, hbeta, hkappa, hL, hcminus, hcminus_le, hcplus,
hpmin, hpmin_le, hdeltaBar, hdeltaBar1, halpha, halpha1⟩
apply minimaxClampModel_of_taylor J beta kappa L cminus cplus pmin
hJ hbeta hkappa hcminus_le hcplus hpmin_le (fun _ => 0)
· exact measurable_const
· intro a
norm_num
· fun_prop
· intro a ha
norm_num
· intro s hs t ht
simp_rw [iteratedDerivWithin_const]
rw [Finset.sum_eq_single 0]
· simp
exact mul_nonneg hL.le (Real.rpow_nonneg (abs_nonneg _) _)
· intro b hb hb0
simp [hb0]
· simp
theorem minimaxConstant_mem_model
the stated minimax constant membership model property holds for the specified J input, the specified beta input, the specified kappa input, the specified L input, the specified cminus input, the specified cplus input, the specified pmin input, the specified deltaBar input, the specified alpha input, the specified eps input, the specified hreg input, the specified heps input.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha eps :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
heps :
|eps| ≤ 1 / 2
conclusion
Proof (Lean source)
lemma minimaxConstant_mem_model
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha eps : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(heps : |eps| ≤ 1 / 2) :
ClampModel (minimaxClampLaw J kappa (fun _ : Fin J × ℝ => eps))
beta kappa L cminus cplus pmin := by
rcases hreg with
⟨hJ, hbeta, hkappa, hL, hcminus, hcminus_le, hcplus,
hpmin, hpmin_le, hdeltaBar, hdeltaBar1, halpha, halpha1⟩
apply minimaxClampModel_of_taylor J beta kappa L cminus cplus pmin
hJ hbeta hkappa hcminus_le hcplus hpmin_le (fun _ => eps)
· exact measurable_const
· intro a
exact heps
· fun_prop
· intro a ha
rw [abs_le] at heps
constructor <;> linarith
· intro s hs t ht
simp_rw [iteratedDerivWithin_const]
rw [Finset.sum_eq_single 0]
· simp
exact mul_nonneg hL.le (Real.rpow_nonneg (abs_nonneg _) _)
· intro b hb hb0
simp [hb0]
· simp
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.MinimaxRegression 1 declarations Regression identities for the canonical Bernoulli witnesses
Regression identities for the canonical Bernoulli witnesses
theorem minimaxDataMeasure_integral_Y_design
Integrating the outcome over a design event integrates the advertised Bernoulli mean over the common design law. The result uses the hJ condition, the hkappa condition, the hgmeas condition, the hg condition, the hT condition. This is the stated conclusion.
Formal statement
Proof (Lean source)
lemma minimaxDataMeasure_integral_Y_design
(J : ℕ) (kappa : ℝ) (hJ : 0 < J) (hkappa : 0 ≤ kappa)
(g : Fin J × ℝ → ℝ) (hgmeas : Measurable g)
(hg : ∀ p, |g p| ≤ 1 / 2) (T : Set (Fin J × ℝ)) (hT : MeasurableSet T) :
(∫ o in (fun o : ClampObs J => (o.X, o.A)) ⁻¹' T, o.Y
∂minimaxDataMeasure J kappa g) =
∫ p in T, (1 / 2 + g p) ∂minimaxDesignMeasure J kappa := by
classical
let P := minimaxDataMeasure J kappa g
let D := minimaxDesignMeasure J kappa
let E : Set (ClampObs J) := (fun o => (o.X, o.A)) ⁻¹' T
let F : ClampObs J → ℝ := E.indicator (fun o => o.Y)
haveI : IsProbabilityMeasure P :=
minimaxDataMeasure_isProbabilityMeasure J kappa hJ hkappa g hgmeas hg
have hFmeas : Measurable F :=
measurable_obs_Y.indicator (hT.preimage measurable_obs_design)
have hFint : Integrable F P := by
refine integrable_of_measurable_ae_bound hFmeas 1 ?_
filter_upwards [minimaxDataMeasure_ae_bernoulli J kappa g hgmeas] with o ho
rcases ho with ho | ho <;>
by_cases hmem : o ∈ E <;> simp [F, indicator, hmem, ho]
have hbind := integral_bind_map
(m := D)
(κ := fun p => twoPointMean (1 / 2) (g p))
(g := fun p y => ClampObs.mk p.1 p.2 (y + 1 / 2))
(f := F) (fun p => measurable_obs_mk p)
(measurable_minimaxOutcomeKernel g hgmeas) (by
change Integrable F (minimaxDataMeasure J kappa g)
exact hFint)
have hfiber (p : Fin J × ℝ) :
(∫ y, F (ClampObs.mk p.1 p.2 (y + 1 / 2))
∂Causalean.Mathlib.Probability.twoPointMean (1 / 2) (g p)) =
T.indicator (fun p => 1 / 2 + g p) p := by
by_cases hp : p ∈ T
· rw [Set.indicator_of_mem hp]
have hFE : ∀ y : ℝ,
F (ClampObs.mk p.1 p.2 (y + 1 / 2)) = y + 1 / 2 := by
intro y
simp [F, E, indicator, hp]
simp_rw [hFE]
rw [twoPointMean_integral (by norm_num) (hg p)]
ring
· simp only [indicator, hp, if_false]
have hzero : ∀ y : ℝ, F (ClampObs.mk p.1 p.2 (y + 1 / 2)) = 0 := by
intro y
simp [F, E, indicator, hp]
simp_rw [hzero]
simp
rw [show (∫ o in E, o.Y ∂P) = ∫ o, F o ∂P by
rw [integral_indicator (hT.preimage measurable_obs_design)]]
change (∫ o, F o ∂P) = _
rw [show (∫ o, F o ∂P) =
∫ p, ∫ y, F (ClampObs.mk p.1 p.2 (y + 1 / 2))
∂Causalean.Mathlib.Probability.twoPointMean (1 / 2) (g p) ∂D by
change (∫ o, F o ∂(minimaxDesignMeasure J kappa).bind
(fun p => (twoPointMean (1 / 2) (g p)).map
(fun y => ClampObs.mk p.1 p.2 (y + 1 / 2)))) = _
exact hbind]
rw [← integral_indicator hT]
apply integral_congr_ae
filter_upwards with p
rw [hfiber]
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.MinimaxWitness 9 declarations The witness design uses uniform finite strata and the normalized polynomial density (κ + 1) a^κ on [0,1].
Canonical design for the clamp minimax witnesses
The witness design uses uniform finite strata and the normalized polynomial
density (κ + 1) a^κ on [0,1]. Outcome tilts are added in later lemmas.
def minimaxStratumMeasure
Uniform probability law on the finite stratum space.
def minimaxTreatmentMeasure
The normalized polynomial treatment law on [0,1].
theorem minimaxTreatmentDensity_integral
the stated minimax treatment density integral property holds for the specified kappa input, the specified hkappa input.
Formal statement
hypotheses
kappa :
ℝ
hkappa :
0 ≤ kappa
conclusion
(∫ a in Icc (0 : ℝ) 1, (kappa + 1) * a ^ kappa) = 1
Proof (Lean source)
lemma minimaxTreatmentDensity_integral (kappa : ℝ) (hkappa : 0 ≤ kappa) :
(∫ a in Icc (0 : ℝ) 1, (kappa + 1) * a ^ kappa) = 1 := by
rw [integral_Icc_eq_integral_Ioc, ← intervalIntegral.integral_of_le (by norm_num),
intervalIntegral.integral_const_mul, integral_rpow (inl (by linarith : -1 < kappa))]
rw [Real.one_rpow, Real.zero_rpow (by linarith : kappa + 1 ≠ 0)]
field_simp
norm_num
theorem minimaxTreatmentMeasure_isProbabilityMeasure
minimax treatment measure is a probability measure for the specified kappa input, the specified hkappa input.
Formal statement
hypotheses
kappa :
ℝ
hkappa :
0 ≤ kappa
conclusion
Proof (Lean source)
lemma minimaxTreatmentMeasure_isProbabilityMeasure (kappa : ℝ) (hkappa : 0 ≤ kappa) :
IsProbabilityMeasure (minimaxTreatmentMeasure kappa) := by
rw [isProbabilityMeasure_iff]
rw [minimaxTreatmentMeasure, withDensity_apply _ MeasurableSet.univ,
Measure.restrict_univ]
have hfint : Integrable (fun a : ℝ => (kappa + 1) * a ^ kappa)
(volume.restrict (Icc (0 : ℝ) 1)) := by
change IntegrableOn (fun a : ℝ => (kappa + 1) * a ^ kappa)
(Icc (0 : ℝ) 1) volume
rw [← intervalIntegrable_iff_integrableOn_Icc_of_le (by norm_num)]
exact (intervalIntegral.intervalIntegrable_rpow (inl hkappa)).const_mul (kappa + 1)
have hfnn : 0 ≤ᵐ[volume.restrict (Icc (0 : ℝ) 1)]
(fun a : ℝ => (kappa + 1) * a ^ kappa) := by
filter_upwards [ae_restrict_mem measurableSet_Icc] with a ha
exact mul_nonneg (by linarith) (Real.rpow_nonneg ha.1 _)
rw [← ofReal_integral_eq_lintegral_ofReal hfint hfnn,
minimaxTreatmentDensity_integral kappa hkappa]
simp
def minimaxDesignMeasure
The common (X,A) law of every minimax witness.
Definition (Lean source)
parameters
J :
ℕ
kappa :
ℝ
given by
theorem minimaxDesignMeasure_isProbabilityMeasure
minimax design measure is a probability measure for the specified J input, the specified kappa input, the specified hJ input, the specified hkappa input.
Formal statement
hypotheses
J :
ℕ
kappa :
ℝ
hJ :
0 < J
hkappa :
0 ≤ kappa
conclusion
IsProbabilityMeasure (minimaxDesignMeasure J kappa)
Proof (Lean source)
lemma minimaxDesignMeasure_isProbabilityMeasure (J : ℕ) (kappa : ℝ)
(hJ : 0 < J) (hkappa : 0 ≤ kappa) :
IsProbabilityMeasure (minimaxDesignMeasure J kappa) := by
letI : Nonempty (Fin J) := Fin.pos_iff_nonempty.mp hJ
letI : IsProbabilityMeasure (minimaxTreatmentMeasure kappa) :=
minimaxTreatmentMeasure_isProbabilityMeasure kappa hkappa
unfold minimaxDesignMeasure minimaxStratumMeasure
infer_instance
theorem minimaxStratumMeasure_real_singleton
the stated minimax stratum measure real singleton property holds for the specified J input, the specified hJ input, the specified x input.
Formal statement
Proof (Lean source)
lemma minimaxStratumMeasure_real_singleton (J : ℕ) (hJ : 0 < J) (x : Fin J) :
(minimaxStratumMeasure J).real {x} = 1 / (J : ℝ) := by
classical
rw [measureReal_def, minimaxStratumMeasure, ProbabilityTheory.uniformOn_univ]
simp [measureReal_def, ENNReal.toReal_div, hJ.ne']
theorem minimaxTreatmentMeasure_real
the stated minimax treatment measure real property holds for the specified kappa input, the specified hkappa input, the specified B input, the specified hB input, the specified hsub input.
Formal statement
hypotheses
conclusion
(minimaxTreatmentMeasure kappa).real B = ∫ a in B, (kappa + 1) * a ^ kappa
Proof (Lean source)
lemma minimaxTreatmentMeasure_real (kappa : ℝ) (hkappa : 0 ≤ kappa)
(B : Set ℝ) (hB : MeasurableSet B) (hsub : B ⊆ Icc (0 : ℝ) 1) :
(minimaxTreatmentMeasure kappa).real B =
∫ a in B, (kappa + 1) * a ^ kappa := by
let f : ℝ → ℝ := fun a => (kappa + 1) * a ^ kappa
have hfint : Integrable f (volume.restrict (Icc (0 : ℝ) 1)) := by
change IntegrableOn f (Icc (0 : ℝ) 1) volume
rw [← intervalIntegrable_iff_integrableOn_Icc_of_le (by norm_num)]
exact (intervalIntegral.intervalIntegrable_rpow (inl hkappa)).const_mul (kappa + 1)
have hfB : Integrable f (volume.restrict B) := by
exact hfint.mono_measure (Measure.restrict_mono hsub le_rfl)
have hfnn : 0 ≤ᵐ[volume.restrict B] f := by
filter_upwards [ae_restrict_mem hB] with a ha
exact mul_nonneg (by linarith) (Real.rpow_nonneg (hsub ha).1 _)
have hrestrict :
(volume.restrict (Icc (0 : ℝ) 1)).restrict B = volume.restrict B := by
rw [Measure.restrict_restrict hB]
congr 1
exact Set.inter_eq_left.mpr hsub
rw [measureReal_def, minimaxTreatmentMeasure, withDensity_apply _ hB]
have he := ofReal_integral_eq_lintegral_ofReal hfB hfnn
change (∫⁻ a, ofReal ((kappa + 1) * a ^ kappa)
∂(volume.restrict (Icc (0 : ℝ) 1)).restrict B).toReal = _
rw [hrestrict]
rw [← he]
rw [ENNReal.toReal_ofReal (integral_nonneg_of_ae hfnn)]
theorem minimaxDesignMeasure_real_rectangle
the stated minimax design measure real rectangle property holds for the specified J input, the specified kappa input, the specified hJ input, the specified hkappa input, the specified x input, the specified B input, the specified hB input, the specified hsub input.
Formal statement
hypotheses
conclusion
(minimaxDesignMeasure J kappa).real ({x} ×ˢ B)
= (1 / (J : ℝ)) * ∫ a in B, (kappa + 1) * a ^ kappa
Proof (Lean source)
lemma minimaxDesignMeasure_real_rectangle (J : ℕ) (kappa : ℝ)
(hJ : 0 < J) (hkappa : 0 ≤ kappa) (x : Fin J)
(B : Set ℝ) (hB : MeasurableSet B) (hsub : B ⊆ Icc (0 : ℝ) 1) :
(minimaxDesignMeasure J kappa).real ({x} ×ˢ B) =
(1 / (J : ℝ)) * ∫ a in B, (kappa + 1) * a ^ kappa := by
letI : Nonempty (Fin J) := Fin.pos_iff_nonempty.mp hJ
letI : IsProbabilityMeasure (minimaxTreatmentMeasure kappa) :=
minimaxTreatmentMeasure_isProbabilityMeasure kappa hkappa
rw [minimaxDesignMeasure, measureReal_prod_prod,
minimaxStratumMeasure_real_singleton J hJ x,
minimaxTreatmentMeasure_real kappa hkappa B hB hsub]
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.PhaseRates 28 declarations The statement spells out every limiting regime and retains the almost-sure zero-threshold reduction to the ordinary bounded-mean estimator.
Four-regime threshold phase diagram
The statement spells out every limiting regime and retains the almost-sure zero-threshold reduction to the ordinary bounded-mean estimator.
theorem treatment_strictly_positive_ae
Under the conditional-density model, the continuous treatment is strictly positive almost surely; the boundary singleton has no mass in any stratum. The result uses the hmodel condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
conclusion
∀ᵐ o ∂P.dataMeasure, 0 < o.A
Proof (Lean source)
-- @node: treatment_strictly_positive_ae
lemma treatment_strictly_positive_ae
(J : ℕ) (P : ClampLaw J) (beta kappa L cminus cplus pmin : ℝ)
(hmodel : ClampModel P beta kappa L cminus cplus pmin) :
∀ᵐ o ∂P.dataMeasure, 0 < o.A := by
letI := hmodel.probability
have hstratum_zero : ∀ x : Fin J,
P.dataMeasure {o : ClampObs J | o.X = x ∧ o.A ∈ ({0} : Set ℝ)} = 0 := by
intro x
have hreal := (hmodel.condDensity.2.2 x).2 ({0} : Set ℝ) (measurableSet_singleton 0)
(by
intro a ha
have : a = 0 := by simpa using ha
subst a
norm_num)
rw [show (∫ a in ({0} : Set ℝ), P.pi x a) = 0 by simp, mul_zero,
measureReal_def] at hreal
rcases (ENNReal.toReal_eq_zero_iff _).mp hreal with hz | htop
· exact hz
· exact (measure_ne_top P.dataMeasure _ htop).elim
have hA_zero : P.dataMeasure {o : ClampObs J | o.A = 0} = 0 := by
rw [show {o : ClampObs J | o.A = 0} =
⋃ x : Fin J, {o : ClampObs J | o.X = x ∧ o.A ∈ ({0} : Set ℝ)} by
ext o
simp]
exact measure_iUnion_null hstratum_zero
have hne : ∀ᵐ o ∂P.dataMeasure, o.A ≠ 0 := by
rw [ae_iff]
simpa only [not_not] using hA_zero
filter_upwards [hmodel.treatmentSupport, hne] with o ho hne
exact lt_of_le_of_ne ho.1 (Ne.symm hne)
theorem zero_threshold_functional
At threshold zero the observed clamp functional is the ordinary outcome mean, including the boundary point because the treatment law has no atom there. The result uses the hmodel condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
conclusion
clampFunctional P 0 = ∫ o, o.Y ∂P.dataMeasure
Proof (Lean source)
-- @node: zero_threshold_functional
lemma zero_threshold_functional
(J : ℕ) (P : ClampLaw J) (beta kappa L cminus cplus pmin : ℝ)
(hmodel : ClampModel P beta kappa L cminus cplus pmin) :
clampFunctional P 0 = ∫ o, o.Y ∂P.dataMeasure := by
have hretained : retainedMean P 0 = ∫ o, o.Y ∂P.dataMeasure := by
apply integral_congr_ae
filter_upwards [treatment_strictly_positive_ae J P beta kappa L cminus cplus pmin
hmodel] with o ho
simp [ho]
simp [clampFunctional, hretained, atomMass]
theorem zero_threshold_totalGramEstimator
At threshold zero every empirical atom term vanishes almost surely, so the total-Gram estimator is exactly the clamped block-zero outcome mean. The result uses the hmodel condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
B :
beta kappa L cminus cplus pmin deltaBar :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
Proof (Lean source)
-- @node: zero_threshold_totalGramEstimator
lemma zero_threshold_totalGramEstimator
(J n : ℕ) (P : ClampLaw J) (B : SplitBlocks n)
(beta kappa L cminus cplus pmin deltaBar : ℝ)
(hmodel : ClampModel P beta kappa L cminus cplus pmin) :
let h := infoBandwidth n 0 beta kappa deltaBar
(fun z : Fin n → ClampObs J =>
totalGramEstimator B z (ellOf beta) kappa cminus cplus 0 h) =ᵐ[iidProduct P n]
(fun z => clampUnit (blockAverage B.I0 z (fun o => o.Y))) := by
letI := hmodel.probability
have hcoord : ∀ i : Fin n, ∀ᵐ z ∂iidProduct P n, 0 < (z i).A := by
intro i
have hmap : (iidProduct P n).map (Function.eval i) = P.dataMeasure := by
simpa [iidProduct] using
(Measure.pi_map_eval (fun _ : Fin n => P.dataMeasure) i)
have hpos := treatment_strictly_positive_ae J P beta kappa L cminus cplus pmin
hmodel
rw [← hmap] at hpos
have hAmeas : Measurable (fun o : ClampObs J => o.A) := by
have ht : Measurable (fun o : ClampObs J => (o.X, o.A, o.Y)) := by
rw [measurable_iff_comap_le]
rfl
exact (measurable_fst.comp measurable_snd).comp ht
exact (ae_map_iff (μ := iidProduct P n) (f := Function.eval i)
(p := fun o : ClampObs J => 0 < o.A)
(measurable_pi_apply i).aemeasurable
(measurableSet_Ioi.preimage hAmeas)).mp hpos
have hall : ∀ᵐ z ∂iidProduct P n, ∀ i : Fin n, 0 < (z i).A :=
ae_all_iff.mpr hcoord
dsimp
filter_upwards [hall] with z hz
have hatom : ∀ x : Fin J, atomEstimate B z x 0 = 0 := by
intro x
simp [atomEstimate, blockAverage, fun i => not_le_of_gt (hz i)]
simp [totalGramEstimator, retainedEstimate, blockAverage, hatom, hz]
theorem zero_threshold_atom_and_frontier
At the identity threshold the density-created atom vanishes, and hence the candidate frontier has exactly its root-sample-size summand. The result uses the hkappa condition. This is the stated conclusion.
Formal statement
hypotheses
conclusion
conclusion 2
clampFrontier n 0 kappa (infoBandwidth n 0 beta kappa deltaBar) beta
= (n : ℝ) ^ (-(1 : ℝ) / 2)
Proof (Lean source)
-- @node: zero_threshold_atom_and_frontier
lemma zero_threshold_atom_and_frontier
(J n : ℕ) (P : ClampLaw J) (beta kappa deltaBar : ℝ)
(hkappa : 0 ≤ kappa) :
(∀ x : Fin J, atomMass P x 0 = 0) ∧
clampFrontier n 0 kappa (infoBandwidth n 0 beta kappa deltaBar) beta =
(n : ℝ) ^ (-(1 : ℝ) / 2) := by
constructor
· intro x
simp [atomMass]
· simp [clampFrontier, Real.zero_rpow (by linarith : kappa + 1 ≠ 0)]
theorem critical_scale_div_edge_scale_tendsto_top
The critical threshold is asymptotically separated above the design-edge scale under the declared exponent restrictions. The result uses the hreg condition. This is the stated conclusion.
Formal statement
Proof (Lean source)
-- @node: critical_scale_div_edge_scale_tendsto_top
lemma critical_scale_div_edge_scale_tendsto_top
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
Tendsto (fun n => deltaCrit n beta kappa / deltaEdge n beta kappa)
atTop atTop := by
rcases hreg with ⟨hJ, hbeta, hkappa, hL, hcminus, hcm, hcp, hpmin,
hpminJ, hdeltaBar_pos, hdeltaBar_lt, halpha, halpha_lt⟩
let D : ℝ := beta * kappa + 2 * beta + kappa + 1
let E : ℝ := 2 * beta + kappa + 1
have hD : 0 < D := by
dsimp [D]
nlinarith
have hE : 0 < E := by
dsimp [E]
linarith
have hgap : 0 < 1 / E - 1 / (2 * D) := by
rw [sub_pos, div_lt_div_iff₀ (by positivity : 0 < (2 : ℝ) * D) hE]
dsimp [D, E]
nlinarith
refine ((tendsto_rpow_atTop hgap).comp
(tendsto_natCast_atTop_atTop (R := ℝ))).congr' ?_
filter_upwards [eventually_gt_atTop 0] with n hn
have hn' : 0 < (n : ℝ) := by exact_mod_cast hn
symm
calc
deltaCrit n beta kappa / deltaEdge n beta kappa =
(n : ℝ) ^ (-1 / (2 * D)) / (n : ℝ) ^ (-1 / E) := by
simp [deltaCrit, deltaEdge, D, E]
_ = (n : ℝ) ^ ((-1 / (2 * D)) - (-1 / E)) :=
(Real.rpow_sub hn' _ _).symm
_ = (n : ℝ) ^ (1 / E - 1 / (2 * D)) := by
congr 1
ring
theorem edge_atom_scale_isLittleO_root
At the design edge, the largest boundary-band atom term is negligible relative to the root-sample-size scale. The result uses the hbeta condition, the hkappa condition. This is the stated conclusion.
Formal statement
Proof (Lean source)
-- @node: edge_atom_scale_isLittleO_root
lemma edge_atom_scale_isLittleO_root
(beta kappa : ℝ) (hbeta : 0 < beta) (hkappa : 0 ≤ kappa) :
Tendsto (fun n =>
(deltaEdge n beta kappa) ^ (beta + kappa + 1) /
(n : ℝ) ^ (-(1 : ℝ) / 2)) atTop (nhds 0) := by
let E : ℝ := 2 * beta + kappa + 1
let gap : ℝ := (beta + kappa + 1) / E - 1 / 2
have hE : 0 < E := by dsimp [E]; linarith
have hgap : 0 < gap := by
dsimp [gap, E]
rw [sub_pos, div_lt_div_iff₀ (by norm_num : (0 : ℝ) < 2) hE]
linarith
refine ((tendsto_rpow_neg_atTop hgap).comp
(tendsto_natCast_atTop_atTop (R := ℝ))).congr' ?_
filter_upwards [eventually_gt_atTop 0] with n hn
have hnR : 0 < (n : ℝ) := by exact_mod_cast hn
symm
rw [deltaEdge, ← Real.rpow_mul hnR.le]
rw [← Real.rpow_sub hnR]
congr 1
dsimp [gap, E]
ring
In the interior-bandwidth regime, the atom contribution has the exact power-law form used to compare thresholds with the critical scale. The result uses the hn condition, the hd condition, the hbeta condition. This is the stated conclusion.
Formal statement
hypotheses
n d beta kappa :
ℝ
hn :
0 < n
hd :
0 < d
hbeta :
0 < beta
conclusion
d ^ (kappa + 1) * ((n * d ^ kappa) ^ (-(1 : ℝ) / (2 * beta + 1))) ^ beta
= n ^ (-beta / (2 * beta + 1)) * d ^ ((beta * kappa + 2 * beta + kappa + 1) / (2 * beta + 1))
Proof (Lean source)
-- @node: atom_interior_scale_exact
lemma atom_interior_scale_exact (n d beta kappa : ℝ) (hn : 0 < n) (hd : 0 < d)
(hbeta : 0 < beta) :
d ^ (kappa + 1) *
((n * d ^ kappa) ^ (-(1 : ℝ) / (2 * beta + 1))) ^ beta =
n ^ (-beta / (2 * beta + 1)) *
d ^ ((beta * kappa + 2 * beta + kappa + 1) /
(2 * beta + 1)) := by
have hp : 0 < 2 * beta + 1 := by linarith
rw [← Real.rpow_mul (mul_nonneg hn.le (Real.rpow_nonneg hd.le kappa))]
rw [Real.mul_rpow hn.le (Real.rpow_nonneg hd.le kappa)]
rw [← Real.rpow_mul hd.le]
calc
_ = n ^ ((-(1 : ℝ) / (2 * beta + 1)) * beta) *
(d ^ (kappa + 1) *
d ^ (kappa * ((-(1 : ℝ) / (2 * beta + 1)) * beta))) := by ring
_ = n ^ ((-(1 : ℝ) / (2 * beta + 1)) * beta) *
d ^ ((kappa + 1) +
kappa * ((-(1 : ℝ) / (2 * beta + 1)) * beta)) := by
exact congrArg
(fun t : ℝ => n ^ ((-(1 : ℝ) / (2 * beta + 1)) * beta) * t)
(Real.rpow_add hd _ _).symm
_ = n ^ (-beta / (2 * beta + 1)) *
d ^ ((beta * kappa + 2 * beta + kappa + 1) /
(2 * beta + 1)) := by
congr 1 <;> field_simp <;> ring
theorem critical_atom_interior_scale_exact
At the critical threshold, the interior-form atom contribution is exactly the root-sample-size scale. The result uses the hn condition, the hbeta condition, the hkappa condition. This is the stated conclusion.
Formal statement
Proof (Lean source)
-- @node: critical_atom_interior_scale_exact
lemma critical_atom_interior_scale_exact
(n : ℕ) (beta kappa : ℝ) (hn : 0 < n) (hbeta : 0 < beta)
(hkappa : 0 ≤ kappa) :
(deltaCrit n beta kappa) ^ (kappa + 1) *
(((n : ℝ) * (deltaCrit n beta kappa) ^ kappa) ^
(-(1 : ℝ) / (2 * beta + 1))) ^ beta =
(n : ℝ) ^ (-(1 : ℝ) / 2) := by
have hnR : 0 < (n : ℝ) := by exact_mod_cast hn
let D : ℝ := beta * kappa + 2 * beta + kappa + 1
have hD : 0 < D := by dsimp [D]; nlinarith
have hp : 0 < 2 * beta + 1 := by linarith
have hc : 0 < deltaCrit n beta kappa := Real.rpow_pos_of_pos hnR _
rw [← Real.rpow_mul
(mul_nonneg hnR.le (Real.rpow_nonneg hc.le kappa))]
rw [Real.mul_rpow hnR.le (Real.rpow_nonneg hc.le kappa)]
rw [← Real.rpow_mul hc.le]
calc
_ = (n : ℝ) ^ ((-(1 : ℝ) / (2 * beta + 1)) * beta) *
((deltaCrit n beta kappa) ^ (kappa + 1) *
(deltaCrit n beta kappa) ^
(kappa * ((-(1 : ℝ) / (2 * beta + 1)) * beta))) := by ring
_ = (n : ℝ) ^ ((-(1 : ℝ) / (2 * beta + 1)) * beta) *
(deltaCrit n beta kappa) ^
((kappa + 1) + kappa * ((-(1 : ℝ) / (2 * beta + 1)) * beta)) := by
exact congrArg
(fun t : ℝ =>
(n : ℝ) ^ ((-(1 : ℝ) / (2 * beta + 1)) * beta) * t)
(Real.rpow_add hc _ _).symm
_ = (n : ℝ) ^ ((-(1 : ℝ) / (2 * beta + 1)) * beta) *
(n : ℝ) ^ ((-(1 : ℝ) / (2 * D)) *
((kappa + 1) + kappa * ((-(1 : ℝ) / (2 * beta + 1)) * beta))) := by
rw [deltaCrit, ← Real.rpow_mul hnR.le]
_ = (n : ℝ) ^ (-(1 : ℝ) / 2) := by
rw [← Real.rpow_add hnR]
congr 1
dsimp [D]
field_simp
ring
theorem atom_interior_div_root_exact
The interior-form atom contribution, normalized by the root-sample-size scale, is exactly a positive power of the threshold-to-critical-scale ratio. The result uses the hn condition, the hd condition, the hbeta condition, the hkappa condition. This is the stated conclusion.
Formal statement
hypotheses
n :
ℕ
d beta kappa :
ℝ
hn :
0 < n
hd :
0 < d
hbeta :
0 < beta
hkappa :
0 ≤ kappa
conclusion
(d ^ (kappa + 1) * (((n : ℝ) * d ^ kappa) ^ (-(1 : ℝ) / (2 * beta + 1))) ^ beta) / (n : ℝ) ^ (-(1 : ℝ) / 2)
= (d / deltaCrit n beta kappa) ^ ((beta * kappa + 2 * beta + kappa + 1) / (2 * beta + 1))
Proof (Lean source)
-- @node: atom_interior_div_root_exact
lemma atom_interior_div_root_exact
(n : ℕ) (d beta kappa : ℝ) (hn : 0 < n) (hd : 0 < d)
(hbeta : 0 < beta) (hkappa : 0 ≤ kappa) :
(d ^ (kappa + 1) *
(((n : ℝ) * d ^ kappa) ^ (-(1 : ℝ) / (2 * beta + 1))) ^ beta) /
(n : ℝ) ^ (-(1 : ℝ) / 2) =
(d / deltaCrit n beta kappa) ^
((beta * kappa + 2 * beta + kappa + 1) / (2 * beta + 1)) := by
have hnR : 0 < (n : ℝ) := by exact_mod_cast hn
let D : ℝ := beta * kappa + 2 * beta + kappa + 1
let p : ℝ := 2 * beta + 1
have hD : 0 < D := by dsimp [D]; nlinarith
have hp : 0 < p := by dsimp [p]; linarith
have hc : 0 < deltaCrit n beta kappa := Real.rpow_pos_of_pos hnR _
rw [atom_interior_scale_exact (n : ℝ) d beta kappa hnR hd hbeta]
have hleft :
((n : ℝ) ^ (-beta / p) * d ^ (D / p)) /
(n : ℝ) ^ (-(1 : ℝ) / 2) =
(n : ℝ) ^ (1 / (2 * p)) * d ^ (D / p) := by
calc
((n : ℝ) ^ (-beta / p) * d ^ (D / p)) /
(n : ℝ) ^ (-(1 : ℝ) / 2) =
((n : ℝ) ^ (-beta / p) /
(n : ℝ) ^ (-(1 : ℝ) / 2)) * d ^ (D / p) := by ring
_ = (n : ℝ) ^ ((-beta / p) - (-(1 : ℝ) / 2)) * d ^ (D / p) := by
rw [Real.rpow_sub hnR]
_ = (n : ℝ) ^ (1 / (2 * p)) * d ^ (D / p) := by
congr 2
dsimp [p]
field_simp
ring
rw [show (beta * kappa + 2 * beta + kappa + 1) / (2 * beta + 1) =
D / p by rfl]
rw [hleft]
rw [Real.div_rpow hd.le hc.le]
have hcrit : (deltaCrit n beta kappa) ^ (D / p) =
(n : ℝ) ^ (-1 / (2 * p)) := by
rw [deltaCrit, ← Real.rpow_mul hnR.le]
congr 1
dsimp [D, p]
field_simp
rw [hcrit]
calc
(n : ℝ) ^ (1 / (2 * p)) * d ^ (D / p) =
d ^ (D / p) * (n : ℝ) ^ (1 / (2 * p)) := mul_comm _ _
_ = d ^ (D / p) * ((n : ℝ) ^ (-1 / (2 * p)))⁻¹ := by
congr 1
rw [← Real.rpow_neg hnR.le]
congr 1
ring
_ = d ^ (D / p) / (n : ℝ) ^ (-1 / (2 * p)) := by
simp only [div_eq_mul_inv]
theorem atom_interior_normalized_tendsto_critical
If the threshold-to-critical-scale ratio converges to a positive constant, the normalized interior atom contribution converges to the corresponding positive power of that constant. The result uses the hbeta condition, the hkappa condition, the hc0 condition, the hratio condition. This is the stated conclusion.
Formal statement
hypotheses
beta kappa c0 :
ℝ
hbeta :
0 < beta
hkappa :
0 ≤ kappa
hc0 :
0 < c0
deltaSeq :
ℕ → ℝ
Proof (Lean source)
-- @node: atom_interior_normalized_tendsto_critical
lemma atom_interior_normalized_tendsto_critical
(beta kappa c0 : ℝ) (hbeta : 0 < beta) (hkappa : 0 ≤ kappa)
(hc0 : 0 < c0) (deltaSeq : ℕ → ℝ)
(hratio : Tendsto (fun n => deltaSeq n / deltaCrit n beta kappa)
atTop (nhds c0)) :
Tendsto (fun n =>
((deltaSeq n) ^ (kappa + 1) *
(((n : ℝ) * (deltaSeq n) ^ kappa) ^
(-(1 : ℝ) / (2 * beta + 1))) ^ beta) /
(n : ℝ) ^ (-(1 : ℝ) / 2)) atTop
(nhds (c0 ^ ((beta * kappa + 2 * beta + kappa + 1) /
(2 * beta + 1)))) := by
let a : ℝ := (beta * kappa + 2 * beta + kappa + 1) / (2 * beta + 1)
have ha : 0 < a := by
dsimp [a]
positivity
have hpow := (Real.continuousAt_rpow_const c0 a (inr ha.le)).tendsto.comp hratio
refine hpow.congr' ?_
have hratioPos : ∀ᶠ n in atTop,
0 < deltaSeq n / deltaCrit n beta kappa :=
hratio (Ioi_mem_nhds hc0)
filter_upwards [eventually_gt_atTop 0, hratioPos] with n hn hnratio
have hnR : 0 < (n : ℝ) := by exact_mod_cast hn
have hcrit : 0 < deltaCrit n beta kappa := Real.rpow_pos_of_pos hnR _
have hdelta : 0 < deltaSeq n :=
((div_pos_iff.mp hnratio).resolve_right fun hneg =>
(not_lt_of_ge hcrit.le) hneg.2).1
symm
exact atom_interior_div_root_exact n (deltaSeq n) beta kappa hn hdelta hbeta hkappa
theorem asympSeq_of_tendsto_div_pos
A positive finite limit for a ratio gives an eventual two-sided constant sandwich of its numerator by its denominator. The result uses the hc condition, the hg condition, the hfg condition. This is the stated conclusion.
Formal statement
Proof (Lean source)
-- @node: asympSeq_of_tendsto_div_pos
lemma asympSeq_of_tendsto_div_pos (f g : ℕ → ℝ) (c : ℝ) (hc : 0 < c)
(hg : ∀ᶠ n in atTop, 0 < g n)
(hfg : Tendsto (fun n => f n / g n) atTop (nhds c)) :
AsympSeq f g := by
refine ⟨c / 2, 2 * c, by linarith, by linarith, ?_⟩
have hratio : ∀ᶠ n in atTop, f n / g n ∈ Ioo (c / 2) (2 * c) :=
hfg (Ioo_mem_nhds (by linarith) (by linarith))
filter_upwards [hg, hratio] with n hgn hn
constructor
· exact ((le_div_iff₀ hgn).mp hn.1.le)
· exact ((div_le_iff₀ hgn).mp hn.2.le)
theorem critical_interior_atom_asymp
In the critical regime, the exact interior-form atom term has root-sample-size order. The result uses the hbeta condition, the hkappa condition, the hc0 condition, the hratio condition. This is the stated conclusion.
Formal statement
hypotheses
beta kappa c0 :
ℝ
hbeta :
0 < beta
hkappa :
0 ≤ kappa
hc0 :
0 < c0
deltaSeq :
ℕ → ℝ
conclusion
AsympSeq (fun n => (deltaSeq n) ^ (kappa + 1) * (((n : ℝ) * (deltaSeq n) ^ kappa) ^ (-(1 : ℝ) / (2 * beta + 1))) ^ beta) (fun n => (n : ℝ) ^ (-(1 : ℝ) / 2))
Proof (Lean source)
-- @node: critical_interior_atom_asymp
lemma critical_interior_atom_asymp
(beta kappa c0 : ℝ) (hbeta : 0 < beta) (hkappa : 0 ≤ kappa)
(hc0 : 0 < c0) (deltaSeq : ℕ → ℝ)
(hratio : Tendsto (fun n => deltaSeq n / deltaCrit n beta kappa)
atTop (nhds c0)) :
AsympSeq (fun n =>
(deltaSeq n) ^ (kappa + 1) *
(((n : ℝ) * (deltaSeq n) ^ kappa) ^
(-(1 : ℝ) / (2 * beta + 1))) ^ beta)
(fun n => (n : ℝ) ^ (-(1 : ℝ) / 2)) := by
let a : ℝ := (beta * kappa + 2 * beta + kappa + 1) / (2 * beta + 1)
have ha : 0 < a := by dsimp [a]; positivity
have hlimit := atom_interior_normalized_tendsto_critical beta kappa c0 hbeta
hkappa hc0 deltaSeq hratio
have hpow : 0 < c0 ^ a := Real.rpow_pos_of_pos hc0 _
apply asympSeq_of_tendsto_div_pos _ _ (c0 ^ a) hpow
· filter_upwards [eventually_gt_atTop 0] with n hn
exact Real.rpow_pos_of_pos (by exact_mod_cast hn) _
· simpa [a] using hlimit
theorem atom_interior_normalized_tendsto_top
If the threshold is asymptotically far above the critical scale, the interior atom contribution dominates the root-sample-size term. The result uses the hbeta condition, the hkappa condition, the hratio condition. This is the stated conclusion.
Formal statement
hypotheses
beta kappa :
ℝ
hbeta :
0 < beta
hkappa :
0 ≤ kappa
deltaSeq :
ℕ → ℝ
Proof (Lean source)
-- @node: atom_interior_normalized_tendsto_top
lemma atom_interior_normalized_tendsto_top
(beta kappa : ℝ) (hbeta : 0 < beta) (hkappa : 0 ≤ kappa)
(deltaSeq : ℕ → ℝ)
(hratio : Tendsto (fun n => deltaSeq n / deltaCrit n beta kappa)
atTop atTop) :
Tendsto (fun n =>
((deltaSeq n) ^ (kappa + 1) *
(((n : ℝ) * (deltaSeq n) ^ kappa) ^
(-(1 : ℝ) / (2 * beta + 1))) ^ beta) /
(n : ℝ) ^ (-(1 : ℝ) / 2)) atTop atTop := by
let a : ℝ := (beta * kappa + 2 * beta + kappa + 1) / (2 * beta + 1)
have ha : 0 < a := by
dsimp [a]
positivity
have hpow := (tendsto_rpow_atTop ha).comp hratio
refine hpow.congr' ?_
have hratioOne : ∀ᶠ n in atTop,
1 ≤ deltaSeq n / deltaCrit n beta kappa :=
(Filter.tendsto_atTop.1 hratio) 1
filter_upwards [eventually_gt_atTop 0, hratioOne] with n hn hnratioOne
have hnR : 0 < (n : ℝ) := by exact_mod_cast hn
have hcrit : 0 < deltaCrit n beta kappa := Real.rpow_pos_of_pos hnR _
have hnratio : 0 < deltaSeq n / deltaCrit n beta kappa := by linarith
have hdelta : 0 < deltaSeq n :=
((div_pos_iff.mp hnratio).resolve_right fun hneg =>
(not_lt_of_ge hcrit.le) hneg.2).1
symm
exact atom_interior_div_root_exact n (deltaSeq n) beta kappa hn hdelta hbeta hkappa
theorem supercritical_root_le_interior_atom
In the supercritical regime, the root-sample-size contribution is eventually bounded by the exact interior-form atom contribution. The result uses the hbeta condition, the hkappa condition, the hratio condition. This is the stated conclusion.
Formal statement
hypotheses
beta kappa :
ℝ
hbeta :
0 < beta
hkappa :
0 ≤ kappa
deltaSeq :
ℕ → ℝ
conclusion
∀ᶠ n : ℕ in atTop, (n : ℝ) ^ (-(1 : ℝ) / 2)
≤ (deltaSeq n) ^ (kappa + 1) * (((n : ℝ) * (deltaSeq n) ^ kappa) ^ (-(1 : ℝ) / (2 * beta + 1))) ^ beta
Proof (Lean source)
-- @node: supercritical_root_le_interior_atom
lemma supercritical_root_le_interior_atom
(beta kappa : ℝ) (hbeta : 0 < beta) (hkappa : 0 ≤ kappa)
(deltaSeq : ℕ → ℝ)
(hratio : Tendsto (fun n => deltaSeq n / deltaCrit n beta kappa)
atTop atTop) :
∀ᶠ n : ℕ in atTop,
(n : ℝ) ^ (-(1 : ℝ) / 2) ≤
(deltaSeq n) ^ (kappa + 1) *
(((n : ℝ) * (deltaSeq n) ^ kappa) ^
(-(1 : ℝ) / (2 * beta + 1))) ^ beta := by
have htop := atom_interior_normalized_tendsto_top beta kappa hbeta hkappa
deltaSeq hratio
have hratioOne : ∀ᶠ n in atTop, 1 ≤
((deltaSeq n) ^ (kappa + 1) *
(((n : ℝ) * (deltaSeq n) ^ kappa) ^
(-(1 : ℝ) / (2 * beta + 1))) ^ beta) /
(n : ℝ) ^ (-(1 : ℝ) / 2) :=
(Filter.tendsto_atTop.1 htop) 1
filter_upwards [eventually_gt_atTop 0, hratioOne] with n hn hratioN
have hroot : 0 < (n : ℝ) ^ (-(1 : ℝ) / 2) :=
Real.rpow_pos_of_pos (by exact_mod_cast hn) _
simpa using (le_div_iff₀ hroot).mp hratioN
theorem critical_ratio_tendsto_edge_ratio_top
A threshold asymptotic to a positive multiple of the critical scale lies asymptotically above the design-edge scale. The result uses the hreg condition, the hc0 condition, the hratio condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha c0 :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
hc0 :
0 < c0
deltaSeq :
ℕ → ℝ
Proof (Lean source)
-- @node: critical_ratio_tendsto_edge_ratio_top
lemma critical_ratio_tendsto_edge_ratio_top
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha c0 : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(hc0 : 0 < c0) (deltaSeq : ℕ → ℝ)
(hratio : Tendsto (fun n => deltaSeq n / deltaCrit n beta kappa)
atTop (nhds c0)) :
Tendsto (fun n => deltaSeq n / deltaEdge n beta kappa) atTop atTop := by
have hscale := critical_scale_div_edge_scale_tendsto_top J beta kappa L cminus
cplus pmin deltaBar alpha hreg
have hprod := Filter.Tendsto.pos_mul_atTop hc0 hratio hscale
refine hprod.congr' ?_
filter_upwards [eventually_gt_atTop 0] with n hn
have hnR : 0 < (n : ℝ) := by exact_mod_cast hn
have hc : deltaCrit n beta kappa ≠ 0 :=
(Real.rpow_pos_of_pos hnR _).ne'
have he : deltaEdge n beta kappa ≠ 0 :=
(Real.rpow_pos_of_pos hnR _).ne'
field_simp
theorem supercritical_ratio_tendsto_edge_ratio_top
A threshold asymptotically larger than the critical scale also lies asymptotically above the smaller design-edge scale. The result uses the hreg condition, the hratio condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
deltaSeq :
ℕ → ℝ
Proof (Lean source)
-- @node: supercritical_ratio_tendsto_edge_ratio_top
lemma supercritical_ratio_tendsto_edge_ratio_top
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(deltaSeq : ℕ → ℝ)
(hratio : Tendsto (fun n => deltaSeq n / deltaCrit n beta kappa)
atTop atTop) :
Tendsto (fun n => deltaSeq n / deltaEdge n beta kappa) atTop atTop := by
have hscale := critical_scale_div_edge_scale_tendsto_top J beta kappa L cminus
cplus pmin deltaBar alpha hreg
have hprod := Filter.Tendsto.atTop_mul_atTop₀ hratio hscale
refine hprod.congr' ?_
filter_upwards [eventually_gt_atTop 0] with n hn
have hnR : 0 < (n : ℝ) := by exact_mod_cast hn
have hc : deltaCrit n beta kappa ≠ 0 :=
(Real.rpow_pos_of_pos hnR _).ne'
have he : deltaEdge n beta kappa ≠ 0 :=
(Real.rpow_pos_of_pos hnR _).ne'
field_simp
theorem fixed_positive_threshold_ratio_critical_top
A threshold converging to a positive fixed value is asymptotically above the vanishing critical scale. The result uses the hreg condition, the hdelta0 condition, the hdelta condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha delta0 :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
hdelta0 :
0 < delta0
deltaSeq :
ℕ → ℝ
Proof (Lean source)
-- @node: fixed_positive_threshold_ratio_critical_top
lemma fixed_positive_threshold_ratio_critical_top
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha delta0 : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(hdelta0 : 0 < delta0) (deltaSeq : ℕ → ℝ)
(hdelta : Tendsto deltaSeq atTop (nhds delta0)) :
Tendsto (fun n => deltaSeq n / deltaCrit n beta kappa) atTop atTop := by
rcases hreg with ⟨hJ, hbeta, hkappa, hL, hcminus, hcm, hcp, hpmin,
hpminJ, hdeltaBar_pos, hdeltaBar_lt, halpha, halpha_lt⟩
let D : ℝ := beta * kappa + 2 * beta + kappa + 1
have hD : 0 < D := by dsimp [D]; nlinarith
have ha : 0 < 1 / (2 * D) := by positivity
have hpow : Tendsto (fun n : ℕ => (n : ℝ) ^ (1 / (2 * D))) atTop atTop :=
(tendsto_rpow_atTop ha).comp (tendsto_natCast_atTop_atTop (R := ℝ))
have hprod := Filter.Tendsto.pos_mul_atTop hdelta0 hdelta hpow
refine hprod.congr' ?_
filter_upwards [eventually_gt_atTop 0] with n hn
have hnR : 0 < (n : ℝ) := by exact_mod_cast hn
rw [deltaCrit]
apply (eq_div_iff (Real.rpow_pos_of_pos hnR _).ne').2
rw [mul_assoc, ← Real.rpow_add hnR]
have hexp : 1 / (2 * D) +
(-1 / (2 * (beta * kappa + 2 * beta + kappa + 1))) = 0 := by
dsimp [D]
ring
rw [hexp]
simp
theorem zero_threshold_stabilizedWorstRisk_asymp
At the identity threshold, the concrete stabilized estimator inherits the root-sample-size risk sandwich from the minimax theorem. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
Bseq :
∀ n
conclusion
SplitBlocks n, AsympSeq (fun n => stabilizedWorstRisk J n (Bseq n) beta kappa L cminus cplus pmin deltaBar 0) (fun n => (n : ℝ) ^ (-(1 : ℝ) / 2))
Proof (Lean source)
-- @node: zero_threshold_stabilizedWorstRisk_asymp
lemma zero_threshold_stabilizedWorstRisk_asymp
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
∀ Bseq : ∀ n, SplitBlocks n,
AsympSeq
(fun n => stabilizedWorstRisk J n (Bseq n) beta kappa L cminus cplus
pmin deltaBar 0)
(fun n => (n : ℝ) ^ (-(1 : ℝ) / 2)) := by
rcases hreg with ⟨hJ, hbeta, hkappa, hL, hcminus, hcm, hcp, hpmin,
hpminJ, hdeltaBar_pos, hdeltaBar_lt, halpha, halpha_lt⟩
have hreg' : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha :=
⟨hJ, hbeta, hkappa, hL, hcminus, hcm, hcp, hpmin, hpminJ,
hdeltaBar_pos, hdeltaBar_lt, halpha, halpha_lt⟩
rcases clamp_minimax_risk J beta kappa L cminus cplus pmin deltaBar alpha hreg' with
⟨c, C, amplitude, hc, hcC, hamp, hamp_le, hrisk⟩
have hzero : ThresholdSequence deltaBar (fun _ => 0) := by
intro n
exact ⟨le_rfl, hdeltaBar_pos.le⟩
intro Bseq
refine ⟨c, C, hc, hcC.le, ?_⟩
filter_upwards [hrisk (fun _ => 0) hzero Bseq] with n hn
dsimp at hn
have hfront :
clampFrontier n 0 kappa (infoBandwidth n 0 beta kappa deltaBar) beta =
(n : ℝ) ^ (-(1 : ℝ) / 2) := by
simp [clampFrontier, Real.zero_rpow (by linarith : kappa + 1 ≠ 0)]
rw [hfront] at hn
exact ⟨hn.1.trans hn.2.1, hn.2.2.1⟩
theorem asymp_add_reference_left
Adding a nonnegative reference term preserves an asymptotic comparison when the other summand already has that reference order. The result uses the hfg condition, the hg condition. This is the stated conclusion.
Formal statement
Proof (Lean source)
-- @node: asymp_add_reference_left
lemma asymp_add_reference_left (f g : ℕ → ℝ)
(hfg : AsympSeq f g) (hg : ∀ᶠ n in atTop, 0 ≤ g n) :
AsympSeq (fun n => g n + f n) g := by
rcases hfg with ⟨c, C, hc, hcC, hfg⟩
refine ⟨c, C + 1, hc, by linarith, ?_⟩
filter_upwards [hfg, hg] with n hn hgn
constructor
· linarith [hn.1]
· nlinarith [hn.2]
theorem asymp_add_eventually_le_left
If a nonnegative summand is eventually no larger than a sequence already of the target order, adding it preserves that asymptotic order. The result uses the hfs condition, the hg condition. This is the stated conclusion.
Formal statement
Proof (Lean source)
-- @node: asymp_add_eventually_le_left
lemma asymp_add_eventually_le_left (f g s : ℕ → ℝ)
(hfs : AsympSeq f s) (hg : ∀ᶠ n in atTop, 0 ≤ g n ∧ g n ≤ f n) :
AsympSeq (fun n => g n + f n) s := by
rcases hfs with ⟨c, C, hc, hcC, hfs⟩
refine ⟨c, 2 * C, hc, by linarith, ?_⟩
filter_upwards [hfs, hg] with n hn hgn
constructor
· linarith [hn.1, hgn.1]
· nlinarith [hn.2, hgn.2]
theorem asympSeq_rpow
Raising eventually nonnegative comparable sequences to a fixed positive real power preserves their asymptotic comparison. The result uses the ha condition, the hfg condition, the hf condition, the hg condition. This is the stated conclusion.
Formal statement
Proof (Lean source)
-- @node: asympSeq_rpow
lemma asympSeq_rpow (f g : ℕ → ℝ) (a : ℝ) (ha : 0 < a)
(hfg : AsympSeq f g) (hf : ∀ᶠ n in atTop, 0 ≤ f n)
(hg : ∀ᶠ n in atTop, 0 ≤ g n) :
AsympSeq (fun n => (f n) ^ a) (fun n => (g n) ^ a) := by
rcases hfg with ⟨c, C, hc, hcC, hfg⟩
refine ⟨c ^ a, C ^ a, Real.rpow_pos_of_pos hc _,
Real.rpow_le_rpow hc.le hcC ha.le, ?_⟩
filter_upwards [hfg, hf, hg] with n hn hfn hgn
constructor
· rw [← Real.mul_rpow hc.le hgn]
exact Real.rpow_le_rpow (mul_nonneg hc.le hgn) hn.1 ha.le
· rw [← Real.mul_rpow (hc.le.trans hcC) hgn]
exact Real.rpow_le_rpow hfn hn.2 ha.le
theorem asympSeq_mul_nonnegative_left
Multiplication by a common eventually nonnegative factor preserves an asymptotic comparison. The result uses the hfg condition, the hq condition. This is the stated conclusion.
Formal statement
Proof (Lean source)
-- @node: asympSeq_mul_nonnegative_left
lemma asympSeq_mul_nonnegative_left (q f g : ℕ → ℝ)
(hfg : AsympSeq f g) (hq : ∀ᶠ n in atTop, 0 ≤ q n) :
AsympSeq (fun n => q n * f n) (fun n => q n * g n) := by
rcases hfg with ⟨c, C, hc, hcC, hfg⟩
refine ⟨c, C, hc, hcC, ?_⟩
filter_upwards [hfg, hq] with n hn hqn
constructor
· calc
c * (q n * g n) = q n * (c * g n) := by ring
_ ≤ q n * f n := mul_le_mul_of_nonneg_left hn.1 hqn
· calc
q n * f n ≤ q n * (C * g n) :=
mul_le_mul_of_nonneg_left hn.2 hqn
_ = C * (q n * g n) := by ring
theorem asympSeq_trans
Asymptotic comparison is transitive. The result uses the hfg condition, the hgs condition. This is the stated conclusion.
Formal statement
Proof (Lean source)
-- @node: asympSeq_trans
lemma asympSeq_trans {f g s : ℕ → ℝ} (hfg : AsympSeq f g)
(hgs : AsympSeq g s) : AsympSeq f s := by
rcases hfg with ⟨c₁, C₁, hc₁, hcC₁, hfg⟩
rcases hgs with ⟨c₂, C₂, hc₂, hcC₂, hgs⟩
refine ⟨c₁ * c₂, C₁ * C₂, mul_pos hc₁ hc₂, ?_, ?_⟩
· exact mul_le_mul hcC₁ hcC₂ hc₂.le (hc₁.le.trans hcC₁)
· filter_upwards [hfg, hgs] with n hn hns
constructor
· calc
(c₁ * c₂) * s n = c₁ * (c₂ * s n) := by ring
_ ≤ c₁ * g n := mul_le_mul_of_nonneg_left hns.1 hc₁.le
_ ≤ f n := hn.1
· calc
f n ≤ C₁ * g n := hn.2
_ ≤ C₁ * (C₂ * s n) :=
mul_le_mul_of_nonneg_left hns.2 (hc₁.le.trans hcC₁)
_ = (C₁ * C₂) * s n := by ring
theorem atomSeq_asymp_interior_of_edge_ratio_top
In the above-edge regime, replacing the implicit information bandwidth by its interior closed form preserves the order of the atom contribution. The result uses the hreg condition, the hdelta condition, the hfar condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
deltaSeq :
ℕ → ℝ
hdelta :
ThresholdSequence deltaBar deltaSeq
conclusion
AsympSeq (fun n => (deltaSeq n) ^ (kappa + 1) * (infoBandwidth n (deltaSeq n) beta kappa deltaBar) ^ beta) (fun n => (deltaSeq n) ^ (kappa + 1) * (((n : ℝ) * (deltaSeq n) ^ kappa) ^ (-(1 : ℝ) / (2 * beta + 1))) ^ beta)
Proof (Lean source)
-- @node: atomSeq_asymp_interior_of_edge_ratio_top
lemma atomSeq_asymp_interior_of_edge_ratio_top
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(deltaSeq : ℕ → ℝ) (hdelta : ThresholdSequence deltaBar deltaSeq)
(hfar : Tendsto (fun n => deltaSeq n / deltaEdge n beta kappa)
atTop atTop) :
AsympSeq
(fun n => (deltaSeq n) ^ (kappa + 1) *
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) ^ beta)
(fun n => (deltaSeq n) ^ (kappa + 1) *
(((n : ℝ) * (deltaSeq n) ^ kappa) ^
(-(1 : ℝ) / (2 * beta + 1))) ^ beta) := by
rcases hreg with ⟨hJ, hbeta, hkappa, hL, hcminus, hcm, hcp, hpmin,
hpminJ, hdeltaBar_pos, hdeltaBar_lt, halpha, halpha_lt⟩
have hreg' : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha :=
⟨hJ, hbeta, hkappa, hL, hcminus, hcm, hcp, hpmin, hpminJ,
hdeltaBar_pos, hdeltaBar_lt, halpha, halpha_lt⟩
have hbw := infoBandwidth_interior_asymp J beta kappa L cminus cplus pmin
deltaBar alpha hreg' deltaSeq hdelta hfar
apply asympSeq_mul_nonnegative_left
· apply asympSeq_rpow _ _ beta hbeta hbw
· exact (infoBandwidth_eventually_balance beta kappa deltaBar hbeta hkappa
⟨hdeltaBar_pos, hdeltaBar_lt⟩ deltaSeq hdelta).mono
(fun _ hn => hn.1.le)
· exact Filter.Eventually.of_forall fun n => Real.rpow_nonneg
(mul_nonneg (Nat.cast_nonneg n)
(Real.rpow_nonneg (hdelta n).1 kappa)) _
· exact Filter.Eventually.of_forall fun n =>
Real.rpow_nonneg (hdelta n).1 (kappa + 1)
theorem critical_atom_and_frontier_asymp
At a positive finite critical ratio, the actual implicit-bandwidth atom term and the resulting frontier both have root-sample-size order. The result uses the hreg condition, the hc0 condition, the hdelta condition, the hratio condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha c0 :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
hc0 :
0 < c0
deltaSeq :
ℕ → ℝ
hdelta :
ThresholdSequence deltaBar deltaSeq
conclusion
atomSeq :
fun n
=> (deltaSeq n) ^ (kappa + 1) * (infoBandwidth n (deltaSeq n) beta kappa deltaBar) ^ beta
rSeq :
fun n
=> clampFrontier n (deltaSeq n) kappa (infoBandwidth n (deltaSeq n) beta kappa deltaBar) beta
Proof (Lean source)
-- @node: critical_atom_and_frontier_asymp
lemma critical_atom_and_frontier_asymp
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha c0 : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(hc0 : 0 < c0) (deltaSeq : ℕ → ℝ)
(hdelta : ThresholdSequence deltaBar deltaSeq)
(hratio : Tendsto (fun n => deltaSeq n / deltaCrit n beta kappa)
atTop (nhds c0)) :
let atomSeq := fun n => (deltaSeq n) ^ (kappa + 1) *
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) ^ beta
let rSeq := fun n => clampFrontier n (deltaSeq n) kappa
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) beta
AsympSeq atomSeq (fun n => (n : ℝ) ^ (-(1 : ℝ) / 2)) ∧
AsympSeq rSeq (fun n => (n : ℝ) ^ (-(1 : ℝ) / 2)) := by
rcases hreg with ⟨hJ, hbeta, hkappa, hL, hcminus, hcm, hcp, hpmin,
hpminJ, hdeltaBar_pos, hdeltaBar_lt, halpha, halpha_lt⟩
have hreg' : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha :=
⟨hJ, hbeta, hkappa, hL, hcminus, hcm, hcp, hpmin, hpminJ,
hdeltaBar_pos, hdeltaBar_lt, halpha, halpha_lt⟩
have hedge := critical_ratio_tendsto_edge_ratio_top J beta kappa L cminus
cplus pmin deltaBar alpha c0 hreg' hc0 deltaSeq hratio
have hactual := atomSeq_asymp_interior_of_edge_ratio_top J beta kappa L cminus
cplus pmin deltaBar alpha hreg' deltaSeq hdelta hedge
have hinterior := critical_interior_atom_asymp beta kappa c0 hbeta hkappa
hc0 deltaSeq hratio
have hatom := asympSeq_trans hactual hinterior
dsimp
refine ⟨hatom, ?_⟩
simpa [clampFrontier] using asymp_add_reference_left _ _ hatom
(Filter.Eventually.of_forall fun n => Real.rpow_nonneg (Nat.cast_nonneg n) _)
theorem critical_interior_frontier_asymp
At a positive finite critical ratio, adding the regular root term to the interior-form atom term preserves root-sample-size order. The result uses the hbeta condition, the hkappa condition, the hc0 condition, the hratio condition. This is the stated conclusion.
Formal statement
hypotheses
beta kappa c0 :
ℝ
hbeta :
0 < beta
hkappa :
0 ≤ kappa
hc0 :
0 < c0
deltaSeq :
ℕ → ℝ
conclusion
AsympSeq (fun n => (n : ℝ) ^ (-(1 : ℝ) / 2) + (deltaSeq n) ^ (kappa + 1) * (((n : ℝ) * (deltaSeq n) ^ kappa) ^ (-(1 : ℝ) / (2 * beta + 1))) ^ beta) (fun n => (n : ℝ) ^ (-(1 : ℝ) / 2))
Proof (Lean source)
-- @node: critical_interior_frontier_asymp
lemma critical_interior_frontier_asymp
(beta kappa c0 : ℝ) (hbeta : 0 < beta) (hkappa : 0 ≤ kappa)
(hc0 : 0 < c0) (deltaSeq : ℕ → ℝ)
(hratio : Tendsto (fun n => deltaSeq n / deltaCrit n beta kappa)
atTop (nhds c0)) :
AsympSeq (fun n =>
(n : ℝ) ^ (-(1 : ℝ) / 2) +
(deltaSeq n) ^ (kappa + 1) *
(((n : ℝ) * (deltaSeq n) ^ kappa) ^
(-(1 : ℝ) / (2 * beta + 1))) ^ beta)
(fun n => (n : ℝ) ^ (-(1 : ℝ) / 2)) := by
apply asymp_add_reference_left
· exact critical_interior_atom_asymp beta kappa c0 hbeta hkappa hc0
deltaSeq hratio
· filter_upwards [eventually_gt_atTop 0] with n hn
exact (Real.rpow_pos_of_pos (by exact_mod_cast hn) _).le
theorem supercritical_frontier_asymp
In the supercritical moving-threshold regime, the actual implicit-bandwidth frontier is comparable to the interior atom scale, since that scale dominates the root-sample-size summand. The result uses the hreg condition, the hdelta condition, the hratio condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
deltaSeq :
ℕ → ℝ
hdelta :
ThresholdSequence deltaBar deltaSeq
conclusion
AsympSeq (fun n => clampFrontier n (deltaSeq n) kappa (infoBandwidth n (deltaSeq n) beta kappa deltaBar) beta) (fun n => (deltaSeq n) ^ (kappa + 1) * ((n : ℝ) * (deltaSeq n) ^ kappa) ^ (-beta / (2 * beta + 1)))
Proof (Lean source)
-- @node: supercritical_frontier_asymp
lemma supercritical_frontier_asymp
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(deltaSeq : ℕ → ℝ) (hdelta : ThresholdSequence deltaBar deltaSeq)
(hratio : Tendsto (fun n => deltaSeq n / deltaCrit n beta kappa)
atTop atTop) :
AsympSeq
(fun n => clampFrontier n (deltaSeq n) kappa
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) beta)
(fun n => (deltaSeq n) ^ (kappa + 1) *
((n : ℝ) * (deltaSeq n) ^ kappa) ^
(-beta / (2 * beta + 1))) := by
rcases hreg with ⟨hJ, hbeta, hkappa, hL, hcminus, hcm, hcp, hpmin,
hpminJ, hdeltaBar_pos, hdeltaBar_lt, halpha, halpha_lt⟩
have hreg' : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha :=
⟨hJ, hbeta, hkappa, hL, hcminus, hcm, hcp, hpmin, hpminJ,
hdeltaBar_pos, hdeltaBar_lt, halpha, halpha_lt⟩
let interior : ℕ → ℝ := fun n =>
(deltaSeq n) ^ (kappa + 1) *
(((n : ℝ) * (deltaSeq n) ^ kappa) ^
(-(1 : ℝ) / (2 * beta + 1))) ^ beta
let atom : ℕ → ℝ := fun n =>
(deltaSeq n) ^ (kappa + 1) *
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) ^ beta
have hedge := supercritical_ratio_tendsto_edge_ratio_top J beta kappa L cminus
cplus pmin deltaBar alpha hreg' deltaSeq hratio
have ha := atomSeq_asymp_interior_of_edge_ratio_top J beta kappa L cminus
cplus pmin deltaBar alpha hreg' deltaSeq hdelta hedge
have hroot := supercritical_root_le_interior_atom beta kappa hbeta hkappa
deltaSeq hratio
rcases ha with ⟨c, C, hc, hcC, ha⟩
refine ⟨c, C + 1, hc, by linarith, ?_⟩
have hratioOne : ∀ᶠ n in atTop,
1 ≤ deltaSeq n / deltaCrit n beta kappa :=
(Filter.tendsto_atTop.1 hratio) 1
filter_upwards [ha, hroot, eventually_gt_atTop 0, hratioOne] with n han hr hn hrat
have hnR : 0 < (n : ℝ) := by exact_mod_cast hn
have hcrit : 0 < deltaCrit n beta kappa := Real.rpow_pos_of_pos hnR _
have hd : 0 < deltaSeq n := by
have : 0 < deltaSeq n / deltaCrit n beta kappa := by linarith
rcases div_pos_iff.mp this with h | h
· exact h.1
· exact (not_lt_of_ge hcrit.le h.2).elim
have hinterior_nonneg : 0 ≤ interior n := by
dsimp [interior]
exact mul_nonneg (Real.rpow_nonneg hd.le _)
(Real.rpow_nonneg
(Real.rpow_nonneg
(mul_nonneg hnR.le (Real.rpow_nonneg hd.le kappa)) _ ) beta)
have hatom_nonneg : 0 ≤ atom n := by
exact (mul_nonneg hc.le hinterior_nonneg).trans han.1
have htarget : interior n =
(deltaSeq n) ^ (kappa + 1) *
((n : ℝ) * (deltaSeq n) ^ kappa) ^
(-beta / (2 * beta + 1)) := by
dsimp [interior]
rw [← Real.rpow_mul
(mul_nonneg hnR.le (Real.rpow_nonneg hd.le kappa))]
congr 2
ring
rw [← htarget]
change c * interior n ≤
(n : ℝ) ^ (-(1 : ℝ) / 2) + atom n ∧
(n : ℝ) ^ (-(1 : ℝ) / 2) + atom n ≤ (C + 1) * interior n
constructor
· exact han.1.trans (le_add_of_nonneg_left
(Real.rpow_nonneg (Nat.cast_nonneg n) _))
· calc
(n : ℝ) ^ (-(1 : ℝ) / 2) + atom n ≤ interior n + C * interior n :=
add_le_add hr han.2
_ = (C + 1) * interior n := by ring
theorem regular_frontier_asymp
Below the critical scale, the actual implicit-bandwidth frontier has root-sample-size order. The result uses the hreg condition, the hdelta condition, the hratio condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
deltaSeq :
ℕ → ℝ
hdelta :
ThresholdSequence deltaBar deltaSeq
conclusion
AsympSeq (fun n => clampFrontier n (deltaSeq n) kappa (infoBandwidth n (deltaSeq n) beta kappa deltaBar) beta) (fun n => (n : ℝ) ^ (-(1 : ℝ) / 2))
Proof (Lean source)
-- @node: regular_frontier_asymp
lemma regular_frontier_asymp
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(deltaSeq : ℕ → ℝ) (hdelta : ThresholdSequence deltaBar deltaSeq)
(hratio : Tendsto (fun n => deltaSeq n / deltaCrit n beta kappa)
atTop (nhds 0)) :
AsympSeq (fun n => clampFrontier n (deltaSeq n) kappa
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) beta)
(fun n => (n : ℝ) ^ (-(1 : ℝ) / 2)) := by
rcases hreg with ⟨hJ, hbeta, hkappa, hL, hcminus, hcm, hcp, hpmin,
hpminJ, hdeltaBar_pos, hdeltaBar_lt, halpha, halpha_lt⟩
have hbalance := infoBandwidth_eventually_balance beta kappa deltaBar hbeta
hkappa ⟨hdeltaBar_pos, hdeltaBar_lt⟩ deltaSeq hdelta
have hsmall : ∀ᶠ n in atTop, deltaSeq n / deltaCrit n beta kappa < 1 :=
hratio (Iio_mem_nhds (by norm_num))
refine ⟨1, 2, by norm_num, by norm_num, ?_⟩
filter_upwards [hbalance, hsmall, eventually_gt_atTop 0] with n hb hs hn
have hnR : 0 < (n : ℝ) := by exact_mod_cast hn
have hroot : 0 < (n : ℝ) ^ (-(1 : ℝ) / 2) := Real.rpow_pos_of_pos hnR _
have hatom : (deltaSeq n) ^ (kappa + 1) *
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) ^ beta ≤
(n : ℝ) ^ (-(1 : ℝ) / 2) := by
rcases eq_or_lt_of_le (hdelta n).1 with hd | hd
· simp [← hd, Real.zero_rpow (by linarith : kappa + 1 ≠ 0), hroot.le]
· let w := ((n : ℝ) * (deltaSeq n) ^ kappa) ^
(-(1 : ℝ) / (2 * beta + 1))
have hbw : infoBandwidth n (deltaSeq n) beta kappa deltaBar ≤ w := by
exact balance_root_le_interior_scale (n : ℝ) (deltaSeq n)
(infoBandwidth n (deltaSeq n) beta kappa deltaBar)
(2 * beta + 1) kappa hnR hd hb.1 (by linarith) hkappa
(by simpa [w] using hb.2.2)
have hatomInterior : (deltaSeq n) ^ (kappa + 1) *
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) ^ beta ≤
(deltaSeq n) ^ (kappa + 1) * w ^ beta := by gcongr; exact hb.1.le
have hratioNonneg : 0 ≤ deltaSeq n / deltaCrit n beta kappa :=
div_nonneg hd.le (Real.rpow_pos_of_pos hnR _).le
have hpow : (deltaSeq n / deltaCrit n beta kappa) ^
((beta * kappa + 2 * beta + kappa + 1) / (2 * beta + 1)) ≤ 1 := by
simpa using Real.rpow_le_rpow hratioNonneg hs.le (show 0 ≤
(beta * kappa + 2 * beta + kappa + 1) / (2 * beta + 1) by positivity)
have hinterior : (deltaSeq n) ^ (kappa + 1) * w ^ beta ≤
(n : ℝ) ^ (-(1 : ℝ) / 2) := by
apply (div_le_one hroot).mp
rw [atom_interior_div_root_exact n (deltaSeq n) beta kappa hn hd hbeta hkappa]
exact hpow
exact hatomInterior.trans hinterior
simp only [clampFrontier]
have hatom_nonneg : 0 ≤ (deltaSeq n) ^ (kappa + 1) *
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) ^ beta :=
mul_nonneg (Real.rpow_nonneg (hdelta n).1 _) (Real.rpow_nonneg hb.1.le _)
constructor <;> nlinarith
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.Pushforward 4 declarations The conditional treatment law is represented by its density against Lebesgue measure.
Clamp pushforward and exposure support
The conditional treatment law is represented by its density against Lebesgue measure. The clamp pushforward is stated as an equality of measures, including the Dirac mass at the threshold and the two-sided atom-mass bound.
def conditionalTreatmentMeasure
Conditional natural-treatment measure in stratum x.
def IsConditionalSupportPoint
Pointwise support expressed by positivity of every open neighborhood.
The threshold clamp produces the retained continuous law plus a Dirac atom, whose mass is sandwiched by the integrated polynomial envelope. The result uses the hmodel condition, the hreg condition, the hdelta condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
x :
Fin J
delta :
ℝ
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
Proof (Lean source)
-- @node: prop:pushforward-setup
lemma clamp_pushforward_decomposition
(P : ClampLaw J) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(x : Fin J) (delta : ℝ) (hdelta : delta ∈ Icc (0 : ℝ) deltaBar) :
(conditionalTreatmentMeasure P x).map (clampPolicy delta) =
(conditionalTreatmentMeasure P x).restrict (Ioc delta 1) +
ofReal (atomMass P x delta) • Measure.dirac delta ∧
cminus * delta ^ (kappa + 1) / (kappa + 1) ≤ atomMass P x delta ∧
atomMass P x delta ≤ cplus * delta ^ (kappa + 1) / (kappa + 1) := by
rcases hreg with ⟨hJ, hbeta, hkappa, hL, hcminus, hcm, hcp, hpmin,
hpminJ, hdeltaBar_pos, hdeltaBar_lt, halpha, halpha_lt⟩
have hdelta0 : 0 ≤ delta := hdelta.1
have hdelta1 : delta ≤ 1 := hdelta.2.trans hdeltaBar_lt.le
let μ : Measure ℝ := conditionalTreatmentMeasure P x
have hclamp : Measurable (clampPolicy delta) := by
exact measurable_id.max measurable_const
have hmap : μ.map (clampPolicy delta) =
μ.restrict (Ioi delta) + μ (Iic delta) • Measure.dirac delta := by
calc
μ.map (clampPolicy delta) =
(μ.restrict (Iic delta) + μ.restrict (Iic delta)ᶜ).map
(clampPolicy delta) := by
rw [Measure.restrict_add_restrict_compl measurableSet_Iic]
_ = (μ.restrict (Iic delta)).map (clampPolicy delta) +
(μ.restrict (Iic delta)ᶜ).map (clampPolicy delta) := by
rw [Measure.map_add _ _ hclamp]
_ = (μ.restrict (Iic delta)).map (fun _ => delta) +
(μ.restrict (Iic delta)ᶜ).map id := by
congr 1
· apply Measure.map_congr
filter_upwards [ae_restrict_mem measurableSet_Iic] with a ha
have ha' : a ≤ delta := ha
exact max_eq_right ha'
· apply Measure.map_congr
filter_upwards [ae_restrict_mem measurableSet_Iic.compl] with a ha
have hda : delta < a := by simpa using ha
simpa [clampPolicy, max_eq_left hda.le]
_ = μ (Iic delta) • Measure.dirac delta + μ.restrict (Ioi delta) := by
simp [Measure.map_const]
_ = μ.restrict (Ioi delta) + μ (Iic delta) • Measure.dirac delta :=
add_comm _ _
have hupper : μ.restrict (Ioi delta) = μ.restrict (Ioc delta 1) := by
dsimp [μ, conditionalTreatmentMeasure]
rw [restrict_withDensity measurableSet_Ioi, restrict_withDensity measurableSet_Ioc,
Measure.restrict_restrict measurableSet_Ioi,
Measure.restrict_restrict measurableSet_Ioc]
congr 2
ext a
simp only [Set.mem_inter_iff, Set.mem_Ioi, Set.mem_Icc, Set.mem_Ioc]
constructor
· rintro ⟨hda, ha0, ha1⟩
exact ⟨⟨hda, ha1⟩, ha0, ha1⟩
· rintro ⟨⟨hda, ha1⟩, ha0, _⟩
exact ⟨hda, ha0, ha1⟩
have hpiInt : Integrable (P.pi x)
(volume.restrict (Icc (0 : ℝ) 1)) := by
apply IntegrableOn.of_bound measure_Icc_lt_top
· exact (aemeasurable_restrict_of_measurable_subtype measurableSet_Icc
(hmodel.condDensity.1 x)).aestronglyMeasurable
· filter_upwards [ae_restrict_mem measurableSet_Icc,
hmodel.condDensity.2.1 x, hmodel.thinning x] with a ha hnon hthin
rw [Real.norm_eq_abs, abs_of_nonneg hnon]
calc
P.pi x a ≤ cplus * a ^ kappa := hthin.2
_ ≤ cplus * 1 := by
have hcplus : 0 ≤ cplus := by linarith
gcongr
exact Real.rpow_le_one ha.1 ha.2 hkappa
_ = cplus := mul_one _
have hmass : μ (Iic delta) = ofReal (atomMass P x delta) := by
dsimp [μ, conditionalTreatmentMeasure]
rw [withDensity_apply _ measurableSet_Iic,
← ofReal_integral_eq_lintegral_ofReal hpiInt.restrict]
· congr 2
rw [Measure.restrict_restrict measurableSet_Iic]
congr 1
ext a
simp only [Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc]
constructor
· rintro ⟨had, ha0, ha1⟩
exact ⟨ha0, had⟩
· rintro ⟨ha0, had⟩
exact ⟨had, ha0, had.trans hdelta1⟩
· exact (hmodel.condDensity.2.1 x).filter_mono ae_restrict_le
refine ⟨?_, ?_⟩
· simpa [μ, hupper, hmass] using hmap
have hsubset : Icc (0 : ℝ) delta ⊆ Icc (0 : ℝ) 1 := by
intro a ha
exact ⟨ha.1, ha.2.trans hdelta1⟩
have hpiDelta : IntegrableOn (P.pi x) (Icc (0 : ℝ) delta) volume :=
hpiInt.mono_measure (Measure.restrict_mono_set volume hsubset)
have hpowInt : IntegrableOn (fun a : ℝ => a ^ kappa) (Icc 0 delta) volume := by
rw [← intervalIntegrable_iff_integrableOn_Icc_of_le hdelta0]
exact intervalIntegral.intervalIntegrable_rpow (inl hkappa)
have hthinDelta := (hmodel.thinning x).filter_mono
(ae_mono (Measure.restrict_mono_set volume hsubset))
have hpowEval : (∫ a in Icc (0 : ℝ) delta, a ^ kappa) =
delta ^ (kappa + 1) / (kappa + 1) := by
rw [integral_Icc_eq_integral_Ioc, ← intervalIntegral.integral_of_le hdelta0,
integral_rpow (inl (by linarith : -1 < kappa))]
rw [Real.zero_rpow (by linarith : kappa + 1 ≠ 0), sub_zero]
constructor
· rw [atomMass]
calc
cminus * delta ^ (kappa + 1) / (kappa + 1) =
cminus * ∫ a in Icc (0 : ℝ) delta, a ^ kappa := by rw [hpowEval]; ring
_ = ∫ a in Icc (0 : ℝ) delta, cminus * a ^ kappa :=
by rw [integral_const_mul]
_ ≤ ∫ a in Icc (0 : ℝ) delta, P.pi x a :=
integral_mono_ae (hpowInt.const_mul cminus) hpiDelta
(hthinDelta.mono fun _ ha => ha.1)
· rw [atomMass]
calc
(∫ a in Icc (0 : ℝ) delta, P.pi x a) ≤
∫ a in Icc (0 : ℝ) delta, cplus * a ^ kappa :=
integral_mono_ae hpiDelta (hpowInt.const_mul cplus)
(hthinDelta.mono fun _ ha => ha.2)
_ = cplus * ∫ a in Icc (0 : ℝ) delta, a ^ kappa := by rw [integral_const_mul]
_ = cplus * delta ^ (kappa + 1) / (kappa + 1) := by rw [hpowEval]; ring
Every clamped treatment value is almost surely a support point of the natural conditional treatment law, including the identity case at zero. The result uses the hmodel condition, the hreg condition, the hdelta condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
x :
Fin J
delta :
ℝ
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
conclusion
∀ᵐ a ∂conditionalTreatmentMeasure P x, IsConditionalSupportPoint (conditionalTreatmentMeasure P x) (clampPolicy delta a)
Proof (Lean source)
-- @node: prop:policy-support
lemma clamp_policy_support
(P : ClampLaw J) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(x : Fin J) (delta : ℝ) (hdelta : delta ∈ Icc (0 : ℝ) deltaBar) :
∀ᵐ a ∂conditionalTreatmentMeasure P x,
IsConditionalSupportPoint (conditionalTreatmentMeasure P x)
(clampPolicy delta a) := by
rcases hreg with ⟨hJ, hbeta, hkappa, hL, hcminus, hcm, hcp, hpmin,
hpminJ, hdeltaBar_pos, hdeltaBar_lt, halpha, halpha_lt⟩
let μ₀ : Measure ℝ := volume.restrict (Icc (0 : ℝ) 1)
let f : ℝ → ENNReal := fun a => ofReal (P.pi x a)
have hf : AEMeasurable f μ₀ := by
apply AEMeasurable.ennreal_ofReal
exact aemeasurable_restrict_of_measurable_subtype measurableSet_Icc
(hmodel.condDensity.1 x)
have hμ : conditionalTreatmentMeasure P x = μ₀.withDensity f := rfl
have hinterior : ∀ᵐ a ∂μ₀, a ∈ Ioo (0 : ℝ) 1 := by
rw [ae_iff, Measure.restrict_apply₀]
· have hset : {a : ℝ | ¬a ∈ Ioo (0 : ℝ) 1} ∩ Icc 0 1 = {0, 1} := by
ext a
simp only [Set.mem_inter_iff, Set.mem_ofPred_eq, Set.mem_Ioo, Set.mem_Icc,
Set.mem_insert_iff, Set.mem_singleton_iff]
constructor
· rintro ⟨hnot, ha0, ha1⟩
by_cases hzero : a = 0
· exact Or.inl hzero
· right
have hpos : 0 < a := lt_of_le_of_ne ha0 (Ne.symm hzero)
exact le_antisymm ha1 (le_of_not_gt fun hlt => hnot ⟨hpos, hlt⟩)
· rintro (rfl | rfl) <;> simp
rw [hset]
rw [Set.insert_eq]
exact measure_union_null Real.volume_singleton Real.volume_singleton
· exact measurableSet_Ioo.compl.nullMeasurableSet
have hfpos : ∀ᵐ a ∂μ₀, f a ≠ 0 := by
filter_upwards [hinterior, hmodel.thinning x] with a ha hthin
rw [ne_eq, ENNReal.ofReal_eq_zero]
exact not_le.mpr (lt_of_lt_of_le
(mul_pos hcminus (Real.rpow_pos_of_pos ha.1 kappa)) hthin.1)
have hsourceInterior : ∀ᵐ a ∂conditionalTreatmentMeasure P x,
a ∈ Ioo (0 : ℝ) 1 := by
rw [hμ]
exact (withDensity_absolutelyContinuous μ₀ f).ae_le hinterior
filter_upwards [hsourceInterior] with a ha
have hdelta01 : delta ∈ Icc (0 : ℝ) 1 :=
⟨hdelta.1, le_trans hdelta.2 (le_of_lt hdeltaBar_lt)⟩
have hy : clampPolicy delta a ∈ Ioo (0 : ℝ) 1 := by
constructor
· exact lt_of_lt_of_le ha.1 (le_max_left _ _)
· exact max_lt ha.2 (lt_of_le_of_lt hdelta.2 hdeltaBar_lt)
intro U hU hyU
have hμ₀U : 0 < μ₀ U := by
rcases (Metric.isOpen_iff.mp hU) _ hyU with ⟨ε, hε, hball⟩
let r := min ε (min (clampPolicy delta a) (1 - clampPolicy delta a))
have hr : 0 < r := by
dsimp [r]
exact lt_min hε (lt_min hy.1 (sub_pos.mpr hy.2))
have hsub : ball (clampPolicy delta a) r ⊆ U ∩ Icc (0 : ℝ) 1 := by
intro z hz
have hz' : |z - clampPolicy delta a| < r := by
simpa [Real.dist_eq] using hz
constructor
· exact hball (Metric.ball_subset_ball (min_le_left _ _) hz)
· constructor
· have : -(clampPolicy delta a) < z - clampPolicy delta a :=
lt_of_le_of_lt (neg_le_neg (min_le_right ε _ |>.trans (min_le_left _ _)))
(abs_lt.mp hz').1
linarith
· have : z - clampPolicy delta a < 1 - clampPolicy delta a :=
lt_of_lt_of_le (abs_lt.mp hz').2
(min_le_right ε _ |>.trans (min_le_right _ _))
linarith
rw [Measure.restrict_apply hU.measurableSet]
refine lt_of_lt_of_le ?_ (measure_mono hsub)
rw [Real.ball_eq_Ioo, Real.volume_Ioo, ENNReal.ofReal_pos]
linarith
rw [hμ]
rw [pos_iff_ne_zero] at hμ₀U ⊢
intro hzero
apply hμ₀U
rw [← (withDensity_apply_eq_zero' hf).mp hzero]
apply measure_congr
filter_upwards [hfpos] with z hz
apply propext
constructor
· exact fun hzU => ⟨hz, hzU⟩
· exact fun hzU => hzU.2
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.RegressionVersion 11 declarations A measurable global version of the clamp regression
A measurable global version of the clamp regression
theorem clampDesign_measurable
clamp design is measurable for the specified J input.
Formal statement
Proof (Lean source)
lemma clampDesign_measurable {J : ℕ} :
Measurable (fun o : ClampObs J => (o.X, o.A)) := by
let h : Measurable (fun o : ClampObs J => (o.X, o.A, o.Y)) :=
Measurable.of_comap_le le_rfl
exact (measurable_fst.comp h).prodMk
(measurable_fst.comp (measurable_snd.comp h))
theorem clampOutcome_measurable
clamp outcome is measurable for the specified J input.
Formal statement
Proof (Lean source)
lemma clampOutcome_measurable {J : ℕ} : Measurable (fun o : ClampObs J => o.Y) := by
exact measurable_snd.comp (measurable_snd.comp (Measurable.of_comap_le le_rfl))
def clampRegressionExtension
Extend the model's regression from its declared dose interval by zero.
theorem clampRegressionExtension_measurable
clamp regression extension is measurable for the specified J input, the specified P input, the specified beta input, the specified L input, the specified hholder input.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma clampRegressionExtension_measurable {J : ℕ} (P : ClampLaw J)
{beta L : ℝ} (hholder : HolderRegression P beta L) :
Measurable (clampRegressionExtension P) := by
apply measurable_from_prod_countable_right
intro x
exact measurable_mu_extension P hholder x
theorem clampRegressionExtension_eq
clamp regression extension satisfies the stated identity for the specified J input, the specified P input, the specified x input, the specified a input, the specified ha input.
Formal statement
hypotheses
conclusion
clampRegressionExtension P (x, a) = P.mu x a
Proof (Lean source)
lemma clampRegressionExtension_eq {J : ℕ} (P : ClampLaw J)
(x : Fin J) {a : ℝ} (ha : a ∈ Icc (0 : ℝ) 1) :
clampRegressionExtension P (x, a) = P.mu x a := by
simp [clampRegressionExtension, ha]
theorem clampRegressionExtension_mem_Icc
clamp regression extension lies in the stated closed interval for the specified J input, the specified P input, the specified beta input, the specified L input, the specified hholder input, the specified d input.
Formal statement
hypotheses
conclusion
clampRegressionExtension P d ∈ Icc (0 : ℝ) 1
Proof (Lean source)
lemma clampRegressionExtension_mem_Icc {J : ℕ} (P : ClampLaw J)
{beta L : ℝ} (hholder : HolderRegression P beta L) (d : Fin J × ℝ) :
clampRegressionExtension P d ∈ Icc (0 : ℝ) 1 := by
by_cases ha : d.2 ∈ Icc (0 : ℝ) 1
· simpa [clampRegressionExtension, ha] using (hholder d.1).2.1 d.2 ha
· simp [clampRegressionExtension, ha]
theorem clampOutcome_integrable
clamp outcome is integrable for the specified J input, the specified P input, the specified beta input, the specified kappa input, the specified L input, the specified cminus input, the specified cplus input, the specified pmin input, the specified hmodel input.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
conclusion
Integrable (fun o : ClampObs J => o.Y) P.dataMeasure
Proof (Lean source)
lemma clampOutcome_integrable {J : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin) :
Integrable (fun o : ClampObs J => o.Y) P.dataMeasure := by
let _ := hmodel.probability
refine Integrable.of_bound clampOutcome_measurable.aestronglyMeasurable 1 ?_
filter_upwards [hmodel.outcomeSupport] with o ho
rw [Real.norm_eq_abs]
exact abs_le.2 ⟨by linarith [ho.1], ho.2⟩
theorem clampRegressionExtension_ae_eq_mu
the stated clamp regression extension almost everywhere identity mu property holds for the specified J input, the specified P input, the specified beta input, the specified kappa input, the specified L input, the specified cminus input, the specified cplus input, the specified pmin input, the specified hmodel input.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
conclusion
(fun o : ClampObs J => clampRegressionExtension P (o.X, o.A))
=ᵐ[P.dataMeasure] fun o => P.mu o.X o.A
Proof (Lean source)
lemma clampRegressionExtension_ae_eq_mu {J : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin) :
(fun o : ClampObs J => clampRegressionExtension P (o.X, o.A)) =ᵐ[P.dataMeasure]
fun o => P.mu o.X o.A := by
filter_upwards [hmodel.treatmentSupport] with o ho
exact clampRegressionExtension_eq P o.X ho
theorem holderRegression_setIntegral
The conditional-expectation regression tie implies the cellwise set-integral identity used by the causal bridge. The result uses the hmodel condition, the hkappa condition, the hcplus condition, the hpmin condition, the hB condition, the hBIcc condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hkappa :
0 ≤ kappa
hcplus :
0 ≤ cplus
hpmin :
0 < pmin
x :
Fin J
B :
Set ℝ
hB :
hBIcc :
B ⊆ Icc (0 : ℝ) 1
conclusion
(∫ o in {o : ClampObs J | o.X = x ∧ o.A ∈ B}, o.Y ∂P.dataMeasure)
= P.px x * ∫ a in B, P.mu x a * P.pi x a
Proof (Lean source)
lemma holderRegression_setIntegral {J : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hkappa : 0 ≤ kappa) (hcplus : 0 ≤ cplus) (hpmin : 0 < pmin)
(x : Fin J) (B : Set ℝ) (hB : MeasurableSet B)
(hBIcc : B ⊆ Icc (0 : ℝ) 1) :
(∫ o in {o : ClampObs J | o.X = x ∧ o.A ∈ B}, o.Y ∂P.dataMeasure) =
P.px x * ∫ a in B, P.mu x a * P.pi x a := by
let _ := hmodel.probability
let design : ClampObs J → Fin J × ℝ := fun o => (o.X, o.A)
let E : Set (ClampObs J) := {o | o.X = x ∧ o.A ∈ B}
have hdesign : Measurable design := clampDesign_measurable
have hE : MeasurableSet E :=
(measurableSet_eq_fun
(measurable_fst.comp (Measurable.of_comap_le le_rfl)) measurable_const).inter
(hB.preimage
(measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))))
have hEcond : MeasurableSet[comap design inferInstance] E := by
have hrect : MeasurableSet (({x} : Set (Fin J)) ×ˢ B) :=
(measurableSet_singleton x).prod hB
have hpre : E = design ⁻¹' (({x} : Set (Fin J)) ×ˢ B) := by
ext o
simp [E, design]
rw [hpre]
exact hrect.preimage (Measurable.of_comap_le le_rfl)
have hY := clampOutcome_integrable P hmodel
have hcond := (hmodel.holder x).2.2.1
have hf := conditionalRegression_integrable P hmodel
hkappa hcplus x B hB
have htreat := stratumTreatment_integral_eq P hmodel.toBridge
hkappa hcplus hpmin x (B.indicator (P.mu x)) hf
rw [conditionalTreatment_integral_eq_density P hmodel.toBridge x] at htreat
calc
(∫ o in {o : ClampObs J | o.X = x ∧ o.A ∈ B}, o.Y ∂P.dataMeasure) =
∫ o in E, o.Y ∂P.dataMeasure := by rfl
_ = ∫ o in E,
P.dataMeasure[(fun o : ClampObs J => o.Y) |
comap design inferInstance] o ∂P.dataMeasure :=
(setIntegral_condExp hdesign.comap_le hY hEcond).symm
_ = ∫ o in E, P.mu o.X o.A ∂P.dataMeasure := by
exact setIntegral_congr_ae hE (hcond.mono fun o ho _ => ho)
_ = ∫ o, indicator {o : ClampObs J | o.X = x}
(fun o => B.indicator (P.mu x) o.A) o ∂P.dataMeasure := by
rw [← integral_indicator hE]
apply integral_congr_ae
filter_upwards with o
by_cases hx : o.X = x <;> by_cases ha : o.A ∈ B <;>
simp [E, indicator, hx, ha]
_ = P.px x * ∫ a in Icc (0 : ℝ) 1,
P.pi x a * B.indicator (P.mu x) a := htreat
_ = P.px x * ∫ a in B, P.mu x a * P.pi x a := by
congr 1
rw [← integral_indicator measurableSet_Icc, ← integral_indicator hB]
apply integral_congr_ae
filter_upwards with a
by_cases ha : a ∈ B
· have haI := hBIcc ha
simp [Set.indicator_of_mem ha, haI, mul_comm]
· by_cases haI : a ∈ Icc (0 : ℝ) 1 <;>
simp [Set.indicator_of_notMem ha, haI]
theorem regressionVersion_setIntegral
Any integrable conditional-regression version satisfies the same cellwise set-integral identity; this is the smoothness-free form used by the continuity-only causal bridge. The result uses the hmodel condition, the hkappa condition, the hcplus condition, the hpmin condition, the hver condition, the hB condition, the hBIcc condition, the hmuInt condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
kappa cminus cplus pmin :
ℝ
hmodel :
BridgeClampModel P kappa cminus cplus pmin
hkappa :
0 ≤ kappa
hcplus :
0 ≤ cplus
hpmin :
0 < pmin
mu :
Fin J → ℝ → ℝ
hver :
=ᵐ[P.dataMeasure] fun o => mu o.X o.A
x :
Fin J
B :
Set ℝ
hB :
hBIcc :
B ⊆ Icc (0 : ℝ) 1
hmuInt :
Integrable (B.indicator (mu x)) (conditionalTreatmentMeasure P x)
conclusion
(∫ o in {o : ClampObs J | o.X = x ∧ o.A ∈ B}, o.Y ∂P.dataMeasure)
= P.px x * ∫ a in B, mu x a * P.pi x a
Proof (Lean source)
-- @node: regressionVersion_setIntegral
lemma regressionVersion_setIntegral {J : ℕ} (P : ClampLaw J)
{kappa cminus cplus pmin : ℝ}
(hmodel : BridgeClampModel P kappa cminus cplus pmin)
(hkappa : 0 ≤ kappa) (hcplus : 0 ≤ cplus) (hpmin : 0 < pmin)
(mu : Fin J → ℝ → ℝ)
(hver : P.dataMeasure[(fun o : ClampObs J => o.Y) |
comap (fun o : ClampObs J => (o.X, o.A)) inferInstance]
=ᵐ[P.dataMeasure] fun o => mu o.X o.A)
(x : Fin J) (B : Set ℝ) (hB : MeasurableSet B)
(hBIcc : B ⊆ Icc (0 : ℝ) 1)
(hmuInt : Integrable (B.indicator (mu x)) (conditionalTreatmentMeasure P x)) :
(∫ o in {o : ClampObs J | o.X = x ∧ o.A ∈ B}, o.Y ∂P.dataMeasure) =
P.px x * ∫ a in B, mu x a * P.pi x a := by
let _ := hmodel.probability
let design : ClampObs J → Fin J × ℝ := fun o => (o.X, o.A)
let E : Set (ClampObs J) := {o | o.X = x ∧ o.A ∈ B}
have hdesign : Measurable design := clampDesign_measurable
have hE : MeasurableSet E :=
(measurableSet_eq_fun
(measurable_fst.comp (Measurable.of_comap_le le_rfl)) measurable_const).inter
(hB.preimage (measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))))
have hEcond : MeasurableSet[comap design inferInstance] E := by
rw [show E = design ⁻¹' (({x} : Set (Fin J)) ×ˢ B) by ext o; simp [E, design]]
exact ((measurableSet_singleton x).prod hB).preimage (Measurable.of_comap_le le_rfl)
have hY : Integrable (fun o : ClampObs J => o.Y) P.dataMeasure := by
refine Integrable.of_bound clampOutcome_measurable.aestronglyMeasurable 1 ?_
filter_upwards [hmodel.outcomeSupport] with o ho
simpa [Real.norm_eq_abs] using (abs_le.2 ⟨by linarith [ho.1], ho.2⟩)
have htreat := stratumTreatment_integral_eq P hmodel hkappa hcplus hpmin
x (B.indicator (mu x)) hmuInt
rw [conditionalTreatment_integral_eq_density P hmodel x] at htreat
calc
(∫ o in {o : ClampObs J | o.X = x ∧ o.A ∈ B}, o.Y ∂P.dataMeasure) =
∫ o in E, o.Y ∂P.dataMeasure := by rfl
_ = ∫ o in E, P.dataMeasure[(fun o : ClampObs J => o.Y) |
comap design inferInstance] o ∂P.dataMeasure :=
(setIntegral_condExp hdesign.comap_le hY hEcond).symm
_ = ∫ o in E, mu o.X o.A ∂P.dataMeasure :=
setIntegral_congr_ae hE (hver.mono fun o ho _ => ho)
_ = ∫ o, indicator {o : ClampObs J | o.X = x}
(fun o => B.indicator (mu x) o.A) o ∂P.dataMeasure := by
rw [← integral_indicator hE]
apply integral_congr_ae
filter_upwards with o
by_cases hx : o.X = x <;> by_cases ha : o.A ∈ B <;>
simp [E, indicator, hx, ha]
_ = P.px x * ∫ a in Icc (0 : ℝ) 1,
P.pi x a * B.indicator (mu x) a := htreat
_ = P.px x * ∫ a in B, mu x a * P.pi x a := by
congr 1
rw [← integral_indicator measurableSet_Icc, ← integral_indicator hB]
apply integral_congr_ae
filter_upwards with a
by_cases ha : a ∈ B
· have haI := hBIcc ha
simp [Set.indicator_of_mem ha, haI, mul_comm]
· by_cases haI : a ∈ Icc (0 : ℝ) 1 <;>
simp [Set.indicator_of_notMem ha, haI]
theorem clampRegression_condExp
the stated clamp regression cond exp property holds for the specified J input, the specified P input, the specified beta input, the specified kappa input, the specified L input, the specified cminus input, the specified cplus input, the specified pmin input, the specified hmodel input, the specified x₀ input.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
x₀ :
Fin J
conclusion
=ᵐ[P.dataMeasure] clampRegressionExtension P ∘ fun o => (o.X, o.A)
Proof (Lean source)
theorem clampRegression_condExp {J : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(x₀ : Fin J) :
P.dataMeasure[(fun o : ClampObs J => o.Y) |
comap (fun o : ClampObs J => (o.X, o.A)) inferInstance] =ᵐ[P.dataMeasure]
clampRegressionExtension P ∘ fun o => (o.X, o.A) := by
have hraw := (hmodel.holder x₀).2.2.1
exact hraw.trans <| (clampRegressionExtension_ae_eq_mu P hmodel).symm
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.SampleBlocks 5 declarations These build-inline bridges restrict the canonical finite product sample to an arbitrary deterministic block, reindex that block by Fin I.card, and connect the resulting coordinate sum to the range-indexed Causalean count
Finite-product transport to deterministic sample blocks
These build-inline bridges restrict the canonical finite product sample to an
arbitrary deterministic block, reindex that block by Fin I.card, and connect
the resulting coordinate sum to the range-indexed Causalean count API through
the canonical infinite-product sample. No ambient i.i.d. stream is assumed.
Membership in the observed clamp model supplies the canonical finite-product sampling certificate at every horizon. Independence and marginal laws follow from the product measure rather than from an ambient sample stream. The result uses the hP condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
ClampLaw J
beta kappa L cminus cplus pmin :
ℝ
hP :
ClampModel P beta kappa L cminus cplus pmin
conclusion
IidSampling P n
Proof (Lean source)
lemma clampModel_iidSampling {J n : ℕ} {P : ClampLaw J}
{beta kappa L cminus cplus pmin : ℝ}
(hP : ClampModel P beta kappa L cminus cplus pmin) : IidSampling P n := by
let _ : IsProbabilityMeasure P.dataMeasure := hP.probability
refine ⟨hP.probability, hP.treatmentSupport, hP.outcomeSupport, ?_, ?_⟩
· change iIndepFun (fun i z => z i) (Measure.pi fun _ : Fin n => P.dataMeasure)
exact iIndepFun_pi
(μ := fun _ : Fin n => P.dataMeasure)
(X := fun _ : Fin n => id) (fun _ => measurable_id.aemeasurable)
· intro i
change (Measure.pi (fun _ : Fin n => P.dataMeasure)).map (fun z => z i) =
P.dataMeasure
exact (MeasureTheory.measurePreserving_eval
(fun _ : Fin n => P.dataMeasure) i).map_eq
theorem contClampModel_iidSampling
Continuity-only model membership supplies the same canonical finite-product sampling certificate. The result uses the hP condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
ClampLaw J
kappa cminus cplus pmin deltaBar :
ℝ
hP :
ContClampModel P kappa cminus cplus pmin deltaBar
conclusion
IidSampling P n
Proof (Lean source)
lemma contClampModel_iidSampling {J n : ℕ} {P : ClampLaw J}
{kappa cminus cplus pmin deltaBar : ℝ}
(hP : ContClampModel P kappa cminus cplus pmin deltaBar) : IidSampling P n := by
let _ : IsProbabilityMeasure P.dataMeasure := hP.probability
refine ⟨hP.probability, hP.treatmentSupport, hP.outcomeSupport, ?_, ?_⟩
· change iIndepFun (fun i z => z i) (Measure.pi fun _ : Fin n => P.dataMeasure)
exact iIndepFun_pi
(μ := fun _ : Fin n => P.dataMeasure)
(X := fun _ : Fin n => id) (fun _ => measurable_id.aemeasurable)
· intro i
change (Measure.pi (fun _ : Fin n => P.dataMeasure)).map (fun z => z i) =
P.dataMeasure
exact (MeasureTheory.measurePreserving_eval
(fun _ : Fin n => P.dataMeasure) i).map_eq
theorem block_reindexed_law_eq
After increasing-order reindexing, the tuple retained by a deterministic block has the ordinary product law on Fin I.card. This is the stated conclusion.
Formal statement
hypotheses
Proof (Lean source)
lemma block_reindexed_law_eq {X : Type*} [MeasurableSpace X]
(P : Measure X) [IsProbabilityMeasure P] {n : ℕ} (I : Finset (Fin n)) :
(Measure.pi (fun _ : Fin n => P)).map
(fun z => fun j : Fin I.card => z ((I.orderIsoOfFin rfl) j).1) =
Measure.pi (fun _ : Fin I.card => P) := by
let e : Fin I.card ↪ Fin n :=
⟨fun j => ((I.orderIsoOfFin rfl) j).1,
fun _ _ h => (I.orderIsoOfFin rfl).injective (Subtype.ext h)⟩
have hindep : iIndepFun (fun j : Fin I.card => fun z : Fin n → X => z (e j))
(Measure.pi fun _ : Fin n => P) := by
exact (iIndepFun_pi
(μ := fun _ : Fin n => P)
(X := fun _ : Fin n => id) (fun _ => measurable_id.aemeasurable)).precomp e.injective
have hmap := hindep.map_fun_eq_pi_map
(fun j => (measurable_pi_apply (e j)).aemeasurable)
calc
(Measure.pi (fun _ : Fin n => P)).map
(fun z => fun j : Fin I.card => z ((I.orderIsoOfFin rfl) j).1) =
Measure.pi (fun j : Fin I.card =>
(Measure.pi (fun _ : Fin n => P)).map (fun z => z (e j))) := by
simpa [e] using hmap
_ = Measure.pi (fun _ : Fin I.card => P) := by
congr 1
funext j
exact (MeasureTheory.measurePreserving_eval
(fun _ : Fin n => P) (e j)).map_eq
theorem block_count_law_eq
The law of a statistic summed over an arbitrary deterministic block is the law of the same statistic summed over Fin I.card product coordinates. The result uses the hf condition. This is the stated conclusion.
Formal statement
hypotheses
X :
Type*
P :
n :
ℕ
f :
X → ℝ
hf :
Proof (Lean source)
lemma block_count_law_eq {X : Type*} [MeasurableSpace X]
(P : Measure X) [IsProbabilityMeasure P] {n : ℕ} (I : Finset (Fin n))
(f : X → ℝ) (hf : Measurable f) :
(Measure.pi (fun _ : Fin n => P)).map
(fun z => ∑ i ∈ I, f (z i)) =
(Measure.pi (fun _ : Fin I.card => P)).map
(fun z => ∑ j : Fin I.card, f (z j)) := by
let restrictBlock : (Fin n → X) → (Fin I.card → X) :=
fun z j => z ((I.orderIsoOfFin rfl) j).1
let blockSum : (Fin I.card → X) → ℝ := fun z => ∑ j, f (z j)
have hrestrict : Measurable restrictBlock := by
exact measurable_pi_lambda _ fun j => measurable_pi_apply _
have hsum : Measurable blockSum := by
exact Finset.measurable_fun_sum _ fun j _ => hf.comp (measurable_pi_apply j)
have hfun : (fun z : Fin n → X => ∑ i ∈ I, f (z i)) =
blockSum ∘ restrictBlock := by
funext z
dsimp [blockSum, restrictBlock]
rw [← Finset.sum_attach]
exact (Equiv.sum_comp (I.orderIsoOfFin rfl).toEquiv
(fun i : {i : Fin n // i ∈ I} => f (z i.1))).symm
rw [hfun, ← MeasureTheory.Measure.map_map hsum hrestrict,
block_reindexed_law_eq P I]
theorem range_count_transport
A range-indexed count on the canonical infinite-product stream has the same law as the corresponding sum on the finite product Fin m → X. The result uses the hf condition. This is the stated conclusion.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma range_count_transport {X : Type*} [MeasurableSpace X]
(P : Measure X) [IsProbabilityMeasure P] (m : ℕ)
(f : X → ℝ) (hf : Measurable f) :
(Measure.pi (fun _ : Fin m => P)).map
(fun z => ∑ j : Fin m, f (z j)) =
(Measure.infinitePi (fun _ : ℕ => P)).map
(bernoulliCount
(Causalean.Stat.iidSample_infinitePi P) f m) := by
let S := Causalean.Stat.iidSample_infinitePi P
let first : (ℕ → X) → (Fin m → X) := fun w j => S.Z j w
let total : (Fin m → X) → ℝ := fun z => ∑ j, f (z j)
have hfirst : Measurable first :=
Causalean.Stat.iidSample_finN_measurable S m
have htotal : Measurable total := by
exact Finset.measurable_fun_sum _ fun j _ => hf.comp (measurable_pi_apply j)
have hfirstLaw :
(Measure.infinitePi (fun _ : ℕ => P)).map first =
Measure.pi (fun _ : Fin m => P) := by
exact Causalean.Stat.iidSample_finN_pushforward S m
have hcount :
bernoulliCount S f m = total ∘ first := by
funext w
change (∑ i ∈ range m, f (S.Z i w)) =
∑ j : Fin m, f (S.Z j w)
exact (Fin.sum_univ_eq_sum_range (fun i => f (S.Z i w)) m).symm
rw [← hfirstLaw, MeasureTheory.Measure.map_map htotal hfirst, ← hcount]
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.ShiftedPowerCoercivity 4 declarations This module isolates the reusable compactification step behind the population Gram lower bound for polynomially thinned local designs.
Uniform coercivity for shifted-power moment matrices
This module isolates the reusable compactification step behind the population Gram lower bound for polynomially thinned local designs.
Normalized monomial moment matrices weighted by (ρ + u)^κ are uniformly coercive over all nonnegative shifts ρ when κ is nonnegative. The result uses the hkappa condition. This is the stated conclusion.
Formal statement
hypotheses
ell :
ℕ
kappa :
ℝ
hkappa :
0 ≤ kappa
conclusion
∃ c : ℝ,
0 < c
conclusion 1
c * (∑ i, (z i) ^ 2) ≤ matrixQuadratic (refMomentMatrix ell kappa rho) z
Proof (Lean source)
theorem uniform_shifted_power_moment_coercivity
(ell : ℕ) {kappa : ℝ} (hkappa : 0 ≤ kappa) :
∃ c : ℝ, 0 < c ∧ ∀ rho : ℝ, 0 ≤ rho →
∀ z : Fin (ell + 1) → ℝ,
c * (∑ i, (z i) ^ 2) ≤
matrixQuadratic (refMomentMatrix ell kappa rho) z := by
obtain ⟨c, hcpos, hcoercive⟩ := powerPolynomialEnergy_coercive ell hkappa
refine ⟨c, hcpos, ?_⟩
intro rho hrho z
let P : ℝ → ℝ := fun u => ∑ i, z i * u ^ (i : ℕ)
let D : ℝ := ∫ u in (0 : ℝ)..1, (rho + u) ^ kappa
let N : ℝ := ∫ u in (0 : ℝ)..1, (P u) ^ 2 * (rho + u) ^ kappa
let E : ℝ := powerPolynomialEnergy ell kappa z
let R : ℝ := (rho + 1) ^ kappa
have hrho1 : 0 < rho + 1 := by linarith
have hRpos : 0 < R := Real.rpow_pos_of_pos hrho1 kappa
have hDpos : 0 < D := by
dsimp [D]
apply intervalIntegral.integral_pos (by norm_num)
· fun_prop
· intro x hx
exact Real.rpow_nonneg (by linarith [hx.1]) _
· refine ⟨1, by norm_num, ?_⟩
exact Real.rpow_pos_of_pos hrho1 kappa
have hDle : D ≤ R := by
dsimp [D, R]
have hmono := intervalIntegral.integral_mono_on
(show (0 : ℝ) ≤ 1 by norm_num)
(show IntervalIntegrable (fun u : ℝ => (rho + u) ^ kappa) volume 0 1 by
apply Continuous.intervalIntegrable
fun_prop)
(show IntervalIntegrable (fun _u : ℝ => (rho + 1) ^ kappa) volume 0 1 by
apply Continuous.intervalIntegrable
fun_prop)
(fun u hu => Real.rpow_le_rpow (by linarith [hu.1]) (by linarith [hu.2]) hkappa)
simpa using hmono
have hEN : R * E ≤ N := by
dsimp [N, E, R]
rw [powerPolynomialEnergy_eq_integral ell hkappa]
rw [← intervalIntegral.integral_const_mul]
apply intervalIntegral.integral_mono_on (by norm_num)
· apply Continuous.intervalIntegrable
fun_prop
· apply Continuous.intervalIntegrable
fun_prop
· intro u hu
have hu0 : 0 ≤ u := hu.1
have hbase : u * (rho + 1) ≤ rho + u := by
nlinarith [hrho, hu.2]
have hpow := Real.rpow_le_rpow (mul_nonneg hu0 hrho1.le) hbase hkappa
rw [Real.mul_rpow hu0 hrho1.le] at hpow
have hsq : 0 ≤ (∑ i, z i * u ^ (i : ℕ)) ^ 2 := sq_nonneg _
nlinarith [mul_le_mul_of_nonneg_left hpow hsq]
have hE_nonneg : 0 ≤ E := by
exact (mul_nonneg hcpos.le (sum_nonneg fun _ _ => sq_nonneg _)).trans
(hcoercive z)
have hED : D * E ≤ N :=
(mul_le_mul_of_nonneg_right hDle hE_nonneg).trans hEN
calc
c * ∑ i, (z i) ^ 2 ≤ E := hcoercive z
_ ≤ N / D := (le_div_iff₀ hDpos).2 (by simpa [mul_comm] using hED)
_ = matrixQuadratic (refMomentMatrix ell kappa rho) z := by
rw [matrixQuadratic_refMomentMatrix_eq ell hkappa rho]
The uniform population-Gram constant is positive under positive envelope constants and a nonnegative thinning exponent. The result uses the hkappa condition, the hcminus condition, the hcplus condition. This is the stated conclusion.
Formal statement
hypotheses
ell :
ℕ
kappa cminus cplus :
ℝ
hkappa :
0 ≤ kappa
hcminus :
0 < cminus
hcplus :
0 < cplus
conclusion
0 < lambdaStar ell kappa cminus cplus
Proof (Lean source)
theorem lambdaStar_pos
(ell : ℕ) {kappa cminus cplus : ℝ}
(hkappa : 0 ≤ kappa) (hcminus : 0 < cminus) (hcplus : 0 < cplus) :
0 < lambdaStar ell kappa cminus cplus := by
obtain ⟨c, hcpos, hcoercive⟩ :=
uniform_shifted_power_moment_coercivity ell hkappa
have hqmin (rho : ℝ) (hrho : 0 ≤ rho) :
c ≤ quadraticMin (refMomentMatrix ell kappa rho) := by
unfold quadraticMin
apply le_csInf
· let e : Fin (ell + 1) := 0
let zunit : Fin (ell + 1) → ℝ := single e 1
refine ⟨matrixQuadratic (refMomentMatrix ell kappa rho) zunit,
zunit, ?_, rfl⟩
classical
simp [zunit, e, Pi.single_apply]
· intro q hq
rcases hq with ⟨z, hzunit, rfl⟩
have h := hcoercive rho hrho z
rw [hzunit, mul_one] at h
exact h
have hset_nonempty :
{q : ℝ | ∃ rho : ℝ, 0 ≤ rho ∧
q = quadraticMin (refMomentMatrix ell kappa rho)}.Nonempty := by
refine ⟨quadraticMin (refMomentMatrix ell kappa 0), 0, le_rfl, rfl⟩
have hsinf : c ≤ sInf {q : ℝ | ∃ rho : ℝ, 0 ≤ rho ∧
q = quadraticMin (refMomentMatrix ell kappa rho)} := by
apply le_csInf hset_nonempty
intro q hq
rcases hq with ⟨rho, hrho, rfl⟩
exact hqmin rho hrho
unfold lambdaStar
exact mul_pos (div_pos hcminus hcplus) (lt_of_lt_of_le hcpos hsinf)
theorem lambdaStar_refMomentMatrix_coercive
lambdaStar gives the advertised coercivity for every normalized shifted power moment matrix. The result uses the hkappa condition, the hcminus condition, the hcplus condition, the hrho condition. This is the stated conclusion.
Formal statement
hypotheses
ell :
ℕ
kappa cminus cplus :
ℝ
hkappa :
0 ≤ kappa
hcminus :
0 < cminus
hcplus :
0 < cplus
rho :
ℝ
hrho :
0 ≤ rho
v :
Fin (ell + 1) → ℝ
conclusion
lambdaStar ell kappa cminus cplus * (∑ i, (v i) ^ 2)
≤ (cminus / cplus) * matrixQuadratic (refMomentMatrix ell kappa rho) v
Proof (Lean source)
-- @node: lambdaStar_refMomentMatrix_coercive
theorem lambdaStar_refMomentMatrix_coercive
(ell : ℕ) {kappa cminus cplus : ℝ}
(hkappa : 0 ≤ kappa) (hcminus : 0 < cminus) (hcplus : 0 < cplus)
(rho : ℝ) (hrho : 0 ≤ rho) (v : Fin (ell + 1) → ℝ) :
lambdaStar ell kappa cminus cplus * (∑ i, (v i) ^ 2) ≤
(cminus / cplus) * matrixQuadratic (refMomentMatrix ell kappa rho) v := by
classical
obtain ⟨c, hc, hcoercive⟩ :=
uniform_shifted_power_moment_coercivity ell hkappa
let Q := {q : ℝ | ∃ z : Fin (ell + 1) → ℝ,
(∑ i, (z i) ^ 2) = 1 ∧
q = matrixQuadratic (refMomentMatrix ell kappa rho) z}
have hQne : Q.Nonempty := by
let e : Fin (ell + 1) := 0
let z : Fin (ell + 1) → ℝ := single e 1
refine ⟨matrixQuadratic (refMomentMatrix ell kappa rho) z, z, ?_, rfl⟩
simp [z, e, Pi.single_apply]
have hQbelow : BddBelow Q := by
refine ⟨c, ?_⟩
intro q hq
rcases hq with ⟨z, hz, rfl⟩
simpa [hz] using hcoercive rho hrho z
have hsinf_le : sInf {q : ℝ | ∃ r : ℝ, 0 ≤ r ∧
q = quadraticMin (refMomentMatrix ell kappa r)} ≤
quadraticMin (refMomentMatrix ell kappa rho) := by
apply csInf_le
· refine ⟨c, ?_⟩
intro q hq
rcases hq with ⟨r, hr, rfl⟩
unfold quadraticMin
exact le_csInf (by
let e : Fin (ell + 1) := 0
let z : Fin (ell + 1) → ℝ := single e 1
refine ⟨matrixQuadratic (refMomentMatrix ell kappa r) z, z, ?_, rfl⟩
simp [z, e, Pi.single_apply]) (by
intro q hq
rcases hq with ⟨z, hz, rfl⟩
simpa [hz] using hcoercive r hr z)
· exact ⟨rho, hrho, rfl⟩
have hqmin : quadraticMin (refMomentMatrix ell kappa rho) *
(∑ i, (v i) ^ 2) ≤ matrixQuadratic (refMomentMatrix ell kappa rho) v := by
let S : ℝ := ∑ i, (v i) ^ 2
by_cases hS : S = 0
· have hv : v = 0 := by
funext i
have hi : (v i) ^ 2 = 0 := le_antisymm
(hS ▸ Finset.single_le_sum (fun j _ => sq_nonneg (v j))
(Finset.mem_univ i)) (sq_nonneg _)
exact sq_eq_zero_iff.mp hi
simp [S, hS, hv, matrixQuadratic]
· have hSpos : 0 < S := lt_of_le_of_ne
(sum_nonneg fun _ _ => sq_nonneg _) (Ne.symm hS)
let r := Real.sqrt S
let z : Fin (ell + 1) → ℝ := r⁻¹ • v
have hr : 0 < r := Real.sqrt_pos.2 hSpos
have hr_sq0 : r ^ 2 = S := Real.sq_sqrt hSpos.le
have hz : ∑ i, (z i) ^ 2 = 1 := by
change ∑ i, (r⁻¹ * v i) ^ 2 = 1
calc
∑ i, (r⁻¹ * v i) ^ 2 = r⁻¹ ^ 2 * ∑ i, (v i) ^ 2 := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro i hi
ring
_ = r⁻¹ ^ 2 * r ^ 2 := by rw [show (∑ i, (v i) ^ 2) = S from rfl, ← hr_sq0]
_ = 1 := by field_simp
have hmin : quadraticMin (refMomentMatrix ell kappa rho) ≤
matrixQuadratic (refMomentMatrix ell kappa rho) z := by
unfold quadraticMin
exact csInf_le hQbelow ⟨z, hz, rfl⟩
have hscale : matrixQuadratic (refMomentMatrix ell kappa rho) z =
r⁻¹ ^ 2 * matrixQuadratic (refMomentMatrix ell kappa rho) v := by
simp [z, matrixQuadratic, Finset.mul_sum]
ring_nf
rw [hscale] at hmin
rw [show (∑ i, (v i) ^ 2) = r ^ 2 by exact hr_sq0.symm]
calc
quadraticMin (refMomentMatrix ell kappa rho) * r ^ 2 ≤
(r⁻¹ ^ 2 * matrixQuadratic (refMomentMatrix ell kappa rho) v) * r ^ 2 :=
mul_le_mul_of_nonneg_right hmin (sq_nonneg r)
_ = _ := by field_simp
unfold lambdaStar
have hcoef : 0 ≤ cminus / cplus := div_nonneg hcminus.le hcplus.le
have hsum : 0 ≤ ∑ i, (v i) ^ 2 :=
sum_nonneg fun _ _ => sq_nonneg _
calc
cminus / cplus * sInf {q : ℝ | ∃ r : ℝ, 0 ≤ r ∧
q = quadraticMin (refMomentMatrix ell kappa r)} *
(∑ i, (v i) ^ 2) ≤
cminus / cplus * quadraticMin (refMomentMatrix ell kappa rho) *
(∑ i, (v i) ^ 2) := by
exact mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left hsinf_le hcoef) hsum
_ ≤ cminus / cplus * matrixQuadratic (refMomentMatrix ell kappa rho) v := by
rw [mul_assoc]
exact mul_le_mul_of_nonneg_left hqmin (div_pos hcminus hcplus).le
The normalized lambdaStar coercivity, transported from the unit interval to a positive local window. The result uses the hkappa condition, the hcminus condition, the hcplus condition, the hdelta condition, the hh condition. This is the stated conclusion.
Formal statement
hypotheses
ell :
ℕ
kappa cminus cplus delta h :
ℝ
hkappa :
0 ≤ kappa
hcminus :
0 < cminus
hcplus :
0 < cplus
hdelta :
0 ≤ delta
hh :
0 < h
v :
Fin (ell + 1) → ℝ
conclusion
lambdaStar ell kappa cminus cplus * cplus * (∫ a in delta..delta + h, a ^ kappa) * (∑ i, (v i) ^ 2)
≤ cminus * ∫ a in delta..delta
+ h, (∑ i, v i * ((a - delta) / h) ^ (i : ℕ)) ^ 2 * a ^ kappa
Proof (Lean source)
-- @node: lambdaStar_scaled_power_window_coercive
theorem lambdaStar_scaled_power_window_coercive
(ell : ℕ) {kappa cminus cplus delta h : ℝ}
(hkappa : 0 ≤ kappa) (hcminus : 0 < cminus) (hcplus : 0 < cplus)
(hdelta : 0 ≤ delta) (hh : 0 < h) (v : Fin (ell + 1) → ℝ) :
lambdaStar ell kappa cminus cplus * cplus *
(∫ a in delta..delta + h, a ^ kappa) * (∑ i, (v i) ^ 2) ≤
cminus * ∫ a in delta..delta + h,
(∑ i, v i * ((a - delta) / h) ^ (i : ℕ)) ^ 2 * a ^ kappa := by
let rho := delta / h
let D := ∫ u in (0 : ℝ)..1, (rho + u) ^ kappa
let N := ∫ u in (0 : ℝ)..1,
(∑ i, v i * u ^ (i : ℕ)) ^ 2 * (rho + u) ^ kappa
have hrho : 0 ≤ rho := div_nonneg hdelta hh.le
have hD : 0 < D := by
dsimp [D]
apply intervalIntegral.integral_pos (by norm_num)
· fun_prop
· intro u hu
exact Real.rpow_nonneg (by linarith [hu.1]) _
· refine ⟨1, by norm_num, ?_⟩
exact Real.rpow_pos_of_pos (by dsimp [rho]; positivity) _
have href := lambdaStar_refMomentMatrix_coercive ell hkappa hcminus hcplus
rho hrho v
rw [matrixQuadratic_refMomentMatrix_eq ell hkappa rho] at href
change lambdaStar ell kappa cminus cplus * (∑ i, (v i) ^ 2) ≤
(cminus / cplus) * (N / D) at href
have hunit :
lambdaStar ell kappa cminus cplus * cplus * D * (∑ i, (v i) ^ 2) ≤
cminus * N := by
have hcD : 0 < cplus * D := mul_pos hcplus hD
calc
lambdaStar ell kappa cminus cplus * cplus * D * (∑ i, (v i) ^ 2) =
(lambdaStar ell kappa cminus cplus * (∑ i, (v i) ^ 2)) *
(cplus * D) := by ring
_ ≤ ((cminus / cplus) * (N / D)) * (cplus * D) :=
mul_le_mul_of_nonneg_right href hcD.le
_ = cminus * N := by field_simp [hcplus.ne', hD.ne']
have hscale (F : ℝ → ℝ) :
(∫ a in delta..delta + h, F a) = h * ∫ u in (0 : ℝ)..1, F (delta + h * u) := by
rw [intervalIntegral.mul_integral_comp_add_mul]
congr 1 <;> ring
have hpow (u : ℝ) (hu : u ∈ Icc (0 : ℝ) 1) :
(delta + h * u) ^ kappa = h ^ kappa * (rho + u) ^ kappa := by
have hr : delta + h * u = h * (rho + u) := by
dsimp [rho]
field_simp [hh.ne']
rw [hr, Real.mul_rpow hh.le (add_nonneg hrho hu.1)]
have hden :
(∫ a in delta..delta + h, a ^ kappa) = h * h ^ kappa * D := by
rw [hscale]
change h * (∫ u in (0 : ℝ)..1, (delta + h * u) ^ kappa) = _
rw [show (∫ u in (0 : ℝ)..1, (delta + h * u) ^ kappa) =
∫ u in (0 : ℝ)..1, h ^ kappa * (rho + u) ^ kappa by
apply intervalIntegral.integral_congr
intro u hu
rw [Set.uIcc_of_le (by norm_num)] at hu
exact hpow u hu]
rw [intervalIntegral.integral_const_mul]
simp only [D]
ring
have hnum :
(∫ a in delta..delta + h,
(∑ i, v i * ((a - delta) / h) ^ (i : ℕ)) ^ 2 * a ^ kappa) =
h * h ^ kappa * N := by
rw [hscale]
change h * (∫ u in (0 : ℝ)..1,
(∑ i, v i * ((delta + h * u - delta) / h) ^ (i : ℕ)) ^ 2 *
(delta + h * u) ^ kappa) = _
rw [show (∫ u in (0 : ℝ)..1,
(∑ i, v i * ((delta + h * u - delta) / h) ^ (i : ℕ)) ^ 2 *
(delta + h * u) ^ kappa) =
∫ u in (0 : ℝ)..1,
h ^ kappa * ((∑ i, v i * u ^ (i : ℕ)) ^ 2 * (rho + u) ^ kappa) by
apply intervalIntegral.integral_congr
intro u hu
rw [Set.uIcc_of_le (by norm_num)] at hu
dsimp only
rw [show (delta + h * u - delta) / h = u by
field_simp [hh.ne'] <;> ring,
hpow u hu]
ring]
rw [intervalIntegral.integral_const_mul]
simp only [N]
ring
rw [hden, hnum]
have hhpow : 0 ≤ h * h ^ kappa :=
mul_nonneg hh.le (Real.rpow_nonneg hh.le _)
calc
lambdaStar ell kappa cminus cplus * cplus * (h * h ^ kappa * D) *
(∑ i, (v i) ^ 2) =
(h * h ^ kappa) *
(lambdaStar ell kappa cminus cplus * cplus * D *
(∑ i, (v i) ^ 2)) := by ring
_ ≤ (h * h ^ kappa) * (cminus * N) :=
mul_le_mul_of_nonneg_left hunit hhpow
_ = cminus * (h * h ^ kappa * N) := by ring
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.SurjectivityLift 3 declarations Quantile lifts of observed clamp laws
Quantile lifts of observed clamp laws
theorem exists_fullData_quantile_lift
A measurable bounded conditional-regression version can be realized by a single independent uniform latent variable while preserving the observed law. This uses a probability law, bounded outcomes, positive stratum masses, a measurable bounded regression, the conditional-mean identity, and stratumwise continuity; such a full-data lift exists.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
deltaBar :
ℝ
hprob :
IsProbabilityMeasure P.dataMeasure
hout :
∀ᵐ o ∂P.dataMeasure, o.Y ∈ Icc (0 : ℝ) 1
m :
Fin J × ℝ → ℝ
hm :
hm_unit :
∀ d, m d ∈ Icc (0 : ℝ) 1
hcond :
=ᵐ[P.dataMeasure] fun o => m (o.X, o.A)
hm_cont :
conclusion
∃ PF : FullDataLaw J,
conclusion 1
PF.observedMargin = P
conclusion 2
conclusion 3
conclusion 4
FullDataResponseContinuity PF deltaBar
Proof (Lean source)
lemma exists_fullData_quantile_lift
{J : ℕ} (P : ClampLaw J) (deltaBar : ℝ)
(hprob : IsProbabilityMeasure P.dataMeasure)
(hout : ∀ᵐ o ∂P.dataMeasure, o.Y ∈ Icc (0 : ℝ) 1)
(hmass_pos : ∀ x : Fin J,
0 < (P.dataMeasure.map (fun o => o.X)).real {x})
(m : Fin J × ℝ → ℝ) (hm : Measurable m)
(hm_unit : ∀ d, m d ∈ Icc (0 : ℝ) 1)
(hcond : P.dataMeasure[(fun o : ClampObs J => o.Y) |
comap (fun o : ClampObs J => (o.X, o.A)) inferInstance]
=ᵐ[P.dataMeasure] fun o => m (o.X, o.A))
(hm_cont : ∀ x : Fin J,
ContinuousOn (fun a => m (x, a)) (Icc (0 : ℝ) deltaBar)) :
∃ PF : FullDataLaw J,
PF.observedMargin = P ∧ LatentResponseConsistency PF ∧
LatentExchangeability PF ∧ FullDataResponseContinuity PF deltaBar := by
let _ := hprob
let D : ClampObs J → Fin J × ℝ := fun o => (o.X, o.A)
let Y : ClampObs J → ℝ := fun o => o.Y
have hD : Measurable D := clampDesign_measurable
have hY : Measurable Y := clampOutcome_measurable
let ν : Measure (Fin J × ℝ) := P.dataMeasure.map D
letI : IsProbabilityMeasure ν := Measure.isProbabilityMeasure_map hD.aemeasurable
let κraw : Kernel (Fin J × ℝ) ℝ :=
ProbabilityTheory.condDistrib Y D P.dataMeasure
letI : IsMarkovKernel κraw := inferInstance
let κ0 : Kernel (Fin J × ℝ) ℝ := κraw.map clampUnit
letI : IsMarkovKernel κ0 :=
ProbabilityTheory.Kernel.IsMarkovKernel.map κraw measurable_clampUnit
let fiberMean : Fin J × ℝ → ℝ := fun d => ∫ y, y ∂κ0 d
have hκraw_κ0 : (κraw : (Fin J × ℝ) → Measure ℝ) =ᵐ[ν] κ0 := by
have hclamp : (fun o : ClampObs J => clampUnit (Y o)) =ᵐ[P.dataMeasure] Y := by
filter_upwards [hout] with o ho
simp only [Y]
unfold clampUnit
simp [ho.1, ho.2]
have hc : (ProbabilityTheory.condDistrib (clampUnit ∘ Y) D P.dataMeasure :
Kernel (Fin J × ℝ) ℝ) =ᵐ[ν]
(ProbabilityTheory.condDistrib Y D P.dataMeasure).map clampUnit :=
ProbabilityTheory.condDistrib_comp (μ := P.dataMeasure)
(X := D) hY.aemeasurable measurable_clampUnit
have heq : ProbabilityTheory.condDistrib (clampUnit ∘ Y) D P.dataMeasure = κraw :=
ProbabilityTheory.condDistrib_congr_left hclamp
rw [heq] at hc
simpa only [ν, κ0, κraw] using hc
have hfiber_meas : Measurable fiberMean := by
exact stronglyMeasurable_id.integral_kernel.measurable
have hfiber_ae : fiberMean =ᵐ[ν] m := by
have hce := ProbabilityTheory.condExp_ae_eq_integral_condDistrib
hD hY.aemeasurable stronglyMeasurable_id
(Integrable.of_bound hY.aestronglyMeasurable 1 <| hout.mono fun o ho => by
change |o.Y| ≤ 1
exact abs_le.2 ⟨by linarith [ho.1], ho.2⟩)
have hraw : (fun o => ∫ y, y ∂κraw (D o)) =ᵐ[P.dataMeasure]
fun o => m (D o) := hce.symm.trans hcond
have hzero : (fun d => ∫ y, y ∂κraw d) =ᵐ[ν] fiberMean :=
hκraw_κ0.mono fun d hd => by simp only [fiberMean, hd]
have hpull : (fun o => fiberMean (D o)) =ᵐ[P.dataMeasure] fun o => m (D o) :=
(hzero.comp_tendsto (Measure.tendsto_ae_map hD.aemeasurable)).symm.trans hraw
exact (MeasureTheory.ae_map_iff hD.aemeasurable
(measurableSet_eq_fun hfiber_meas hm)).mpr hpull
let bad : Set (Fin J × ℝ) := {d | fiberMean d ≠ m d}
have hbad : MeasurableSet bad := (measurableSet_eq_fun hfiber_meas hm).compl
let κdirac : Kernel (Fin J × ℝ) ℝ :=
ProbabilityTheory.Kernel.deterministic m hm
letI : IsMarkovKernel κdirac := inferInstance
let κ : Kernel (Fin J × ℝ) ℝ :=
ProbabilityTheory.Kernel.piecewise hbad κdirac κ0
letI : IsMarkovKernel κ := inferInstance
have hκ_ae : (κ : (Fin J × ℝ) → Measure ℝ) =ᵐ[ν] κ0 := by
filter_upwards [hfiber_ae] with d hd
simp [κ, bad, ProbabilityTheory.Kernel.piecewise_apply, hd]
have hκ_support : SupportedOnUnitInterval κ := by
intro d
by_cases hd : d ∈ bad
· rw [ProbabilityTheory.Kernel.piecewise_apply, if_pos hd]
rw [ProbabilityTheory.Kernel.deterministic_apply]
simp [hm_unit d]
· rw [ProbabilityTheory.Kernel.piecewise_apply, if_neg hd]
simp only [κ, κ0]
rw [ProbabilityTheory.Kernel.map_apply' κraw measurable_clampUnit d measurableSet_Icc]
have hpre : clampUnit ⁻¹' Icc (0 : ℝ) 1 = univ := by
ext y
simp only [Set.mem_preimage, Set.mem_Icc, Set.mem_univ, iff_true]
exact clampUnit_in_unit y
rw [hpre]
haveI : IsProbabilityMeasure (κraw d) := inferInstance
exact measure_univ
have hκ_mean : ∀ d, (∫ y, y ∂κ d) = m d := by
intro d
by_cases hd : d ∈ bad
· rw [ProbabilityTheory.Kernel.integral_piecewise, if_pos hd]
simp [κ, κdirac]
· rw [ProbabilityTheory.Kernel.integral_piecewise, if_neg hd]
exact not_ne_iff.mp hd
let q : (Fin J × ℝ) → unitInterval → ℝ := fun d u => kernelUnitQuantile κ d u
have hq : Measurable (Function.uncurry q) := by
exact measurable_subtype_coe.comp (measurable_kernelUnitQuantile κ)
have hqlaw : ∀ d, Measure.map (q d) volume = κ d := by
intro d
exact map_kernelUnitQuantile κ hκ_support d
let base : Measure ((Fin J × ℝ) × unitInterval) := ν.prod volume
letI : IsProbabilityMeasure base := inferInstance
let H : ((Fin J × ℝ) × unitInterval) → (Fin J × ℝ) × ℝ :=
fun p => (p.1, q p.1 p.2)
have hH : Measurable H := measurable_fst.prodMk hq
have hbase_joint : Measure.map H base = ν.compProd κ := by
exact map_prod_realization_eq_compProd ν volume κ q hq hqlaw
have hκraw_joint : ν.compProd κraw = Measure.map (fun o => (D o, Y o)) P.dataMeasure :=
ProbabilityTheory.compProd_map_condDistrib hY.aemeasurable
have hκraw_κ : (κraw : (Fin J × ℝ) → Measure ℝ) =ᵐ[ν] κ :=
hκraw_κ0.trans hκ_ae.symm
have hjoint : Measure.map H base = Measure.map (fun o => (D o, Y o)) P.dataMeasure := by
rw [hbase_joint, Measure.compProd_congr hκraw_κ.symm, hκraw_joint]
let obsMk : (Fin J × ℝ) × ℝ → ClampObs J :=
fun p => ClampObs.mk p.1.1 p.1.2 p.2
have hobsMk : Measurable obsMk := measurable_clampObs_of_design_outcome
have hobs_joint : Measure.map obsMk (Measure.map H base) = P.dataMeasure := by
rw [hjoint, Measure.map_map hobsMk (hD.prodMk hY)]
have hid : obsMk ∘ (fun o : ClampObs J => (D o, Y o)) = id := by
funext o
rfl
rw [hid, Measure.map_id]
let F : ((Fin J × ℝ) × unitInterval) → ClampObs J × unitInterval :=
fun p => (obsMk (H p), p.2)
have hF : Measurable F := (hobsMk.comp hH).prodMk measurable_snd
have hmargin : (Measure.map F base).map fst = P.dataMeasure := by
rw [Measure.map_map measurable_fst hF]
have hcomp : fst ∘ F = obsMk ∘ H := by funext p; rfl
rw [hcomp, ← Measure.map_map hobsMk hH, hobs_joint]
let potfun : ℝ → (ClampObs J × unitInterval) → ℝ :=
fun a z => q (z.1.X, a) z.2
have hpotfun : Measurable (Function.uncurry potfun) := by
change Measurable (Function.uncurry q ∘
fun p : ℝ × (ClampObs J × unitInterval) => ((p.2.1.X, p.1), p.2.2))
have hXobs : Measurable (fun o : ClampObs J => o.X) :=
measurable_fst.comp (Measurable.of_comap_le le_rfl)
have hX : Measurable (fun p : ℝ × (ClampObs J × unitInterval) => p.2.1.X) :=
hXobs.comp (measurable_fst.comp measurable_snd)
have ha : Measurable (fun p : ℝ × (ClampObs J × unitInterval) => p.1) :=
measurable_fst
have hu : Measurable (fun p : ℝ × (ClampObs J × unitInterval) => p.2.2) :=
measurable_snd.comp measurable_snd
exact hq.comp ((hX.prodMk ha).prodMk hu)
have hgmeas : ∀ x : Fin J,
Measurable (fun z : Icc (0 : ℝ) 1 × unitInterval => q (x, z.1) z.2) := by
intro x
change Measurable (Function.uncurry q ∘
fun z : Icc (0 : ℝ) 1 × unitInterval => ((x, (z.1 : ℝ)), z.2))
exact hq.comp <| ((measurable_const.prodMk
(measurable_subtype_coe.comp measurable_fst)).prodMk measurable_snd)
have hactual : ∀ᵐ z ∂Measure.map F base,
z.1.Y = q (z.1.X, z.1.A) z.2 := by
have hleft : Measurable (fun z : ClampObs J × unitInterval => z.1.Y) :=
clampOutcome_measurable.comp measurable_fst
have hright : Measurable (fun z : ClampObs J × unitInterval =>
q (z.1.X, z.1.A) z.2) := by
change Measurable (Function.uncurry q ∘ fun z : ClampObs J × unitInterval =>
((z.1.X, z.1.A), z.2))
exact hq.comp ((clampDesign_measurable.comp measurable_fst).prodMk measurable_snd)
rw [MeasureTheory.ae_map_iff hF.aemeasurable (measurableSet_eq_fun hleft hright)]
exact Filter.Eventually.of_forall fun p => rfl
let PF : FullDataLaw J := {
latentCarrier := unitInterval
observedMargin := P
fullMeasure := Measure.map F base
probability := Measure.isProbabilityMeasure_map hF.aemeasurable
margin_eq := hmargin
g := fun x a u => q (x, a) u
pot := potfun
pot_jointlyMeasurable := hpotfun
}
refine ⟨PF, rfl, ?_, ?_, ?_⟩
· refine ⟨?_, ?_, ?_⟩
· exact hgmeas
· intro x a u ha
exact (kernelUnitQuantile κ (x, a) u).property
· filter_upwards [hactual] with z hz
exact ⟨fun a _ => rfl, hz⟩
· intro x f g hf hg _ _
let sx : Set (Fin J × ℝ) := {d | d.1 = x}
let sz : Set (ClampObs J × unitInterval) := {z | z.1.X = x}
have hsx : MeasurableSet sx := measurable_fst (measurableSet_singleton x)
have hsz : MeasurableSet sz :=
((measurable_fst.comp (Measurable.of_comap_le le_rfl)).comp measurable_fst)
(measurableSet_singleton x)
let Af : ℝ := ∫ d, sx.indicator (fun d => f d.2) d ∂ν
let Gu : ℝ := ∫ u, g u ∂volume
have hfi : Measurable (sx.indicator (fun d => f d.2)) :=
(hf.comp measurable_snd).indicator hsx
have hone : Measurable (sx.indicator (fun _ => (1 : ℝ))) :=
measurable_const.indicator hsx
have hjoint : (∫ z in sz, f z.1.A * g z.2 ∂Measure.map F base) = Af * Gu := by
rw [← integral_indicator hsz]
have hi : Measurable (sz.indicator (fun z => f z.1.A * g z.2)) :=
((hf.comp ((measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))).comp measurable_fst)).mul
(hg.comp measurable_snd)).indicator hsz
rw [integral_map hF.aemeasurable hi.aestronglyMeasurable]
change (∫ p : (Fin J × ℝ) × unitInterval,
sz.indicator (fun z => f z.1.A * g z.2) (F p) ∂base) = _
have heq : (fun p : (Fin J × ℝ) × unitInterval =>
sz.indicator (fun z => f z.1.A * g z.2) (F p)) =
fun p => sx.indicator (fun d => f d.2) p.1 * g p.2 := by
funext p
by_cases hp : p.1.1 = x <;> simp [sx, sz, F, H, obsMk, indicator, hp]
rw [heq, show base = ν.prod volume from rfl, integral_prod_mul]
have hfint : (∫ z in sz, f z.1.A ∂Measure.map F base) = Af := by
rw [← integral_indicator hsz]
have hi : Measurable (sz.indicator (fun z => f z.1.A)) :=
(hf.comp ((measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))).comp measurable_fst)).indicator hsz
rw [integral_map hF.aemeasurable hi.aestronglyMeasurable]
change (∫ p : (Fin J × ℝ) × unitInterval,
sz.indicator (fun z => f z.1.A) (F p) ∂base) = _
have heq : (fun p : (Fin J × ℝ) × unitInterval =>
sz.indicator (fun z => f z.1.A) (F p)) =
fun p => sx.indicator (fun d => f d.2) p.1 * (1 : ℝ) := by
funext p
by_cases hp : p.1.1 = x <;> simp [sx, sz, F, H, obsMk, indicator, hp]
rw [heq, show base = ν.prod volume from rfl]
calc
_ = (∫ d, sx.indicator (fun d => f d.2) d ∂ν) *
∫ _u : unitInterval, (1 : ℝ) ∂volume := by
exact integral_prod_mul
(μ := ν) (ν := volume)
(sx.indicator (fun d => f d.2)) (fun _ : unitInterval => (1 : ℝ))
_ = Af := by simp [Af]
have hgint : (∫ z in sz, g z.2 ∂Measure.map F base) =
ν.real sx * Gu := by
rw [← integral_indicator hsz]
have hi : Measurable (sz.indicator (fun z => g z.2)) :=
(hg.comp measurable_snd).indicator hsz
rw [integral_map hF.aemeasurable hi.aestronglyMeasurable]
change (∫ p : (Fin J × ℝ) × unitInterval,
sz.indicator (fun z => g z.2) (F p) ∂base) = _
have heq : (fun p : (Fin J × ℝ) × unitInterval =>
sz.indicator (fun z => g z.2) (F p)) =
fun p => sx.indicator (fun _ => (1 : ℝ)) p.1 * g p.2 := by
funext p
by_cases hp : p.1.1 = x <;> simp [sx, sz, F, H, obsMk, indicator, hp]
rw [heq, show base = ν.prod volume from rfl, integral_prod_mul]
have honeint : (∫ d, sx.indicator (fun _ => (1 : ℝ)) d ∂ν) =
ν.real sx := by
rw [show (fun _ : Fin J × ℝ => (1 : ℝ)) = 1 from rfl]
-- … truncated; follow the source link for the rest …
theorem exists_fullData_holder_lift
Fixed-Hölder observed laws have a quantile full-data lift. The result uses the hP condition, the hJ condition, the hpmin condition, the hdelta condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin deltaBar :
ℝ
hP :
ClampModel P beta kappa L cminus cplus pmin
hJ :
0 < J
hpmin :
0 < pmin
hdelta :
deltaBar ≤ 1
conclusion
∃ PF : FullDataLaw J,
conclusion 1
FullDataClampModel PF beta kappa L cminus cplus pmin deltaBar
conclusion 2
PF.observedMargin = P
Proof (Lean source)
lemma exists_fullData_holder_lift
{J : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin deltaBar : ℝ}
(hP : ClampModel P beta kappa L cminus cplus pmin)
(hJ : 0 < J) (hpmin : 0 < pmin) (hdelta : deltaBar ≤ 1) :
∃ PF : FullDataLaw J,
FullDataClampModel PF beta kappa L cminus cplus pmin deltaBar ∧
PF.observedMargin = P := by
let m : Fin J × ℝ → ℝ := clampRegressionExtension P
have hm : Measurable m := clampRegressionExtension_measurable P hP.holder
have hm_unit : ∀ d, m d ∈ Icc (0 : ℝ) 1 :=
clampRegressionExtension_mem_Icc P hP.holder
have hcond : P.dataMeasure[(fun o : ClampObs J => o.Y) |
comap (fun o : ClampObs J => (o.X, o.A)) inferInstance]
=ᵐ[P.dataMeasure] fun o => m (o.X, o.A) :=
(hP.holder ⟨0, hJ⟩).2.2.1.trans
(clampRegressionExtension_ae_eq_mu P hP).symm
have hm_cont : ∀ x : Fin J,
ContinuousOn (fun a => m (x, a)) (Icc (0 : ℝ) deltaBar) := by
intro x
apply (hP.holder x).1.mono (Set.Icc_subset_Icc le_rfl hdelta) |>.congr
intro a ha
unfold m
exact clampRegressionExtension_eq P x ⟨ha.1, ha.2.trans hdelta⟩
have hmass_pos : ∀ x : Fin J,
0 < (P.dataMeasure.map (fun o => o.X)).real {x} := by
intro x
rw [← (hP.stratumMass x).1]
exact hpmin.trans_le (hP.stratumMass x).2
obtain ⟨PF, hmargin, hcons, hexch, hcont⟩ :=
exists_fullData_quantile_lift P deltaBar hP.probability hP.outcomeSupport
hmass_pos m hm hm_unit hcond hm_cont
refine ⟨PF, ?_, hmargin⟩
exact ⟨hmargin ▸ hP, hcons, hexch, hcont⟩
theorem exists_fullData_cont_lift
Continuity-only observed laws have the same quantile full-data lift. The result uses the hP condition, the hpmin condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
kappa cminus cplus pmin deltaBar :
ℝ
hP :
ContClampModel P kappa cminus cplus pmin deltaBar
hpmin :
0 < pmin
conclusion
∃ PF : FullDataLaw J,
conclusion 1
ContFullDataClampModel PF kappa cminus cplus pmin deltaBar
conclusion 2
PF.observedMargin = P
Proof (Lean source)
lemma exists_fullData_cont_lift
{J : ℕ} (P : ClampLaw J)
{kappa cminus cplus pmin deltaBar : ℝ}
(hP : ContClampModel P kappa cminus cplus pmin deltaBar)
(hpmin : 0 < pmin) :
∃ PF : FullDataLaw J,
ContFullDataClampModel PF kappa cminus cplus pmin deltaBar ∧
PF.observedMargin = P := by
let _ := hP.probability
let D : ClampObs J → Fin J × ℝ := fun o => (o.X, o.A)
let Y : ClampObs J → ℝ := fun o => o.Y
have hD : Measurable D := clampDesign_measurable
have hY : Measurable Y := clampOutcome_measurable
let ν : Measure (Fin J × ℝ) := P.dataMeasure.map D
let κraw : Kernel (Fin J × ℝ) ℝ :=
ProbabilityTheory.condDistrib Y D P.dataMeasure
letI : IsMarkovKernel κraw := inferInstance
let κ0 : Kernel (Fin J × ℝ) ℝ := κraw.map clampUnit
let r : Fin J × ℝ → ℝ := fun d => ∫ y, y ∂κ0 d
have hr_meas : Measurable r := stronglyMeasurable_id.integral_kernel.measurable
have hr_unit : ∀ d, r d ∈ Icc (0 : ℝ) 1 := by
intro d
have hi : Integrable clampUnit (κraw d) := by
refine Integrable.of_bound measurable_clampUnit.aestronglyMeasurable 1 ?_
exact Filter.Eventually.of_forall fun y => by
rw [Real.norm_eq_abs, abs_of_nonneg (clampUnit_in_unit y).1]
exact (clampUnit_in_unit y).2
have hr : r d = ∫ y, clampUnit y ∂κraw d := by
simp only [r, κ0]
rw [ProbabilityTheory.Kernel.map_apply κraw measurable_clampUnit d]
simpa only [id_eq] using
(integral_map measurable_clampUnit.aemeasurable
stronglyMeasurable_id.aestronglyMeasurable)
rw [hr]
constructor
· exact integral_nonneg fun y => (clampUnit_in_unit y).1
· calc
_ ≤ ∫ _y : ℝ, (1 : ℝ) ∂κraw d :=
integral_mono hi (integrable_const 1) fun y => (clampUnit_in_unit y).2
_ = 1 := by simp
let mu := contRegression P kappa cminus cplus pmin deltaBar hP
have hmu_cont : ∀ x, ContinuousOn (mu x) (Icc (0 : ℝ) deltaBar) :=
(Classical.choose_spec hP.continuousVersion).1
have hmu_ce : P.dataMeasure[(fun o : ClampObs J => o.Y) |
comap (fun o : ClampObs J => (o.X, o.A)) inferInstance]
=ᵐ[P.dataMeasure] fun o => mu o.X o.A :=
(Classical.choose_spec hP.continuousVersion).2
let c : Fin J × ℝ → ℝ := fun d =>
if d.2 ∈ Icc (0 : ℝ) deltaBar then clampUnit (mu d.1 d.2) else 0
have hc_meas : Measurable c := by
apply measurable_from_prod_countable_right
intro x
have hcOn : ContinuousOn (fun a => clampUnit (mu x a))
(Icc (0 : ℝ) deltaBar) :=
continuous_clampUnit'.comp_continuousOn (hmu_cont x)
have hzOn : ContinuousOn (fun _a : ℝ => (0 : ℝ))
(Icc (0 : ℝ) deltaBar)ᶜ := continuous_const.continuousOn
convert hcOn.measurable_piecewise hzOn measurableSet_Icc using 1
funext a
by_cases ha : a ∈ Icc (0 : ℝ) deltaBar <;>
simp_all [c, Set.piecewise]
let E : Set (Fin J × ℝ) := {d | d.2 ∈ Icc (0 : ℝ) deltaBar}
have hE : MeasurableSet E := measurable_snd measurableSet_Icc
let m : Fin J × ℝ → ℝ := E.piecewise c r
have hm : Measurable m := hc_meas.piecewise hE hr_meas
have hm_unit : ∀ d, m d ∈ Icc (0 : ℝ) 1 := by
intro d
by_cases hd : d ∈ E
· rw [show m d = if d ∈ E then c d else r d by rfl, if_pos hd]
have hd' : d.2 ∈ Icc (0 : ℝ) deltaBar := hd
rw [show c d = if d.2 ∈ Icc (0 : ℝ) deltaBar then
clampUnit (mu d.1 d.2) else 0 by rfl, if_pos hd']
exact clampUnit_in_unit _
· rw [show m d = if d ∈ E then c d else r d by rfl, if_neg hd]
exact hr_unit d
have hraw_κ0 : (κraw : (Fin J × ℝ) → Measure ℝ) =ᵐ[ν] κ0 := by
have hclamp : (fun o : ClampObs J => clampUnit (Y o)) =ᵐ[P.dataMeasure] Y := by
filter_upwards [hP.outcomeSupport] with o ho
simp only [Y]
unfold clampUnit
simp [ho.1, ho.2]
have hc' : (ProbabilityTheory.condDistrib (clampUnit ∘ Y) D P.dataMeasure :
Kernel (Fin J × ℝ) ℝ) =ᵐ[ν]
(ProbabilityTheory.condDistrib Y D P.dataMeasure).map clampUnit :=
ProbabilityTheory.condDistrib_comp (μ := P.dataMeasure)
(X := D) hY.aemeasurable measurable_clampUnit
have heq : ProbabilityTheory.condDistrib (clampUnit ∘ Y) D P.dataMeasure = κraw :=
ProbabilityTheory.condDistrib_congr_left hclamp
rw [heq] at hc'
simpa only [ν, κ0, κraw] using hc'
have hce := ProbabilityTheory.condExp_ae_eq_integral_condDistrib
hD hY.aemeasurable stronglyMeasurable_id
(Integrable.of_bound hY.aestronglyMeasurable 1 <|
hP.outcomeSupport.mono fun o ho => by
change |o.Y| ≤ 1
exact abs_le.2 ⟨by linarith [ho.1], ho.2⟩)
have hraw_mu : (fun o => ∫ y, y ∂κraw (D o)) =ᵐ[P.dataMeasure]
fun o => mu o.X o.A := hce.symm.trans hmu_ce
have hr_mu : (fun o => r (D o)) =ᵐ[P.dataMeasure] fun o => mu o.X o.A := by
have hmeans : (fun d => ∫ y, y ∂κraw d) =ᵐ[ν] r :=
hraw_κ0.mono fun d hd => by simp only [r, hd]
have hpull := hmeans.comp_tendsto (Measure.tendsto_ae_map hD.aemeasurable)
exact hpull.symm.trans hraw_mu
have hcond : P.dataMeasure[(fun o : ClampObs J => o.Y) |
comap (fun o : ClampObs J => (o.X, o.A)) inferInstance]
=ᵐ[P.dataMeasure] fun o => m (o.X, o.A) := by
filter_upwards [hmu_ce, hr_mu] with o hceo hro
rw [hceo]
by_cases ho : D o ∈ E
· rw [show m (o.X, o.A) = if D o ∈ E then c (D o) else r (D o) by rfl,
if_pos ho]
rw [show c (D o) = if D o ∈ E then clampUnit (mu o.X o.A) else 0 by rfl,
if_pos ho, ← hro]
rw [show clampUnit (r (D o)) = r (D o) by
unfold clampUnit
rw [max_eq_right (hr_unit (D o)).1, min_eq_right (hr_unit (D o)).2]]
· rw [show m (o.X, o.A) = if D o ∈ E then c (D o) else r (D o) by rfl,
if_neg ho]
exact hro.symm
have hm_cont : ∀ x : Fin J,
ContinuousOn (fun a => m (x, a)) (Icc (0 : ℝ) deltaBar) := by
intro x
have hclamp : ContinuousOn (fun a => clampUnit (mu x a))
(Icc (0 : ℝ) deltaBar) :=
continuous_clampUnit'.comp_continuousOn (hmu_cont x)
apply hclamp.congr
intro a ha
change m (x, a) = clampUnit (mu x a)
rw [show m (x, a) = if (x, a) ∈ E then c (x, a) else r (x, a) by rfl,
if_pos (show (x, a) ∈ E by exact ha)]
rw [show c (x, a) = if a ∈ Icc (0 : ℝ) deltaBar then
clampUnit (mu x a) else 0 by rfl, if_pos ha]
have hmass_pos : ∀ x : Fin J,
0 < (P.dataMeasure.map (fun o => o.X)).real {x} := by
intro x
rw [← (hP.stratumMass x).1]
exact hpmin.trans_le (hP.stratumMass x).2
obtain ⟨PF, hmargin, hcons, hexch, hcont⟩ :=
exists_fullData_quantile_lift P deltaBar hP.probability hP.outcomeSupport
hmass_pos m hm hm_unit hcond hm_cont
refine ⟨PF, ?_, hmargin⟩
exact ⟨hmargin ▸ hP, hcons, hexch, hcont⟩
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.TotalGram 6 declarations This file isolates the note's main realized-design bottleneck.
Uniform total-Gram stabilization
This file isolates the note's main realized-design bottleneck. Constants are quantified before laws, sample sizes, thresholds, blocks, and strata.
If the reference eigenvalue is positive and the realized Gram matrix is good, the intercept weights reproduce every polynomial basis coordinate.
Formal statement
hypotheses
J n ell :
ℕ
B :
x :
Fin J
kappa cminus cplus delta h :
ℝ
hlambda :
0 < lambdaStar ell kappa cminus cplus
hgood :
GoodGramEvent B z x ell kappa cminus cplus delta h
j :
Fin (ell + 1)
conclusion
(∑ i ∈ B.I2, interceptWeight B z x ell kappa cminus cplus delta h i * (scaledDose delta h (z i)) ^ (j : ℕ))
= if j = 0 then 1 else 0
Proof (Lean source)
lemma goodGram_reproduction {J n ell : ℕ}
(B : SplitBlocks n) (z : Fin n → ClampObs J) (x : Fin J)
(kappa cminus cplus delta h : ℝ)
(hlambda : 0 < lambdaStar ell kappa cminus cplus)
(hgood : GoodGramEvent B z x ell kappa cminus cplus delta h) :
∀ j : Fin (ell + 1),
(∑ i ∈ B.I2,
interceptWeight B z x ell kappa cminus cplus delta h i *
(scaledDose delta h (z i)) ^ (j : ℕ)) =
if j = 0 then 1 else 0 := by
classical
let A := localGram B z x ell delta h
let N : ℝ := localCount B z x delta h
let a := lambdaStar ell kappa cminus cplus * N / 2
have hN : 0 < N := by
dsimp [N]
exact_mod_cast hgood.1
have ha : 0 < a := by dsimp [a]; positivity
have hunit : IsUnit A := coerciveMatrix_isUnit ha hgood.2
have hdet : IsUnit A.det := (Matrix.isUnit_iff_isUnit_det A).mp hunit
let U : Fin n → ℝ := fun i => scaledDose delta h (z i)
let W : Fin n → ℝ := localKernelWeight B z x delta h
have hdet' : IsUnit
(designMatrix ell U W).det := by
simpa [A, U, W, localGram] using hdet
intro j
have hrep := equivKernelWeight_reproduces hdet'
(j : ℕ) (Nat.le_of_lt_succ j.isLt)
calc
(∑ i ∈ B.I2,
interceptWeight B z x ell kappa cminus cplus delta h i * U i ^ (j : ℕ)) =
∑ i ∈ B.I2,
equivKernelWeight ell U W i * U i ^ (j : ℕ) := by
apply Finset.sum_congr rfl
intro i hi
rw [interceptWeight_eq_equivKernel_of_good
B z x ell kappa cminus cplus delta h hgood i]
_ = ∑ i,
equivKernelWeight ell U W i * U i ^ (j : ℕ) := by
apply Finset.sum_subset (Finset.subset_univ B.I2)
intro i _ hi
have hw : W i = 0 := by
simp [W, localKernelWeight, hi]
simp [equivKernelWeight, hw]
_ = if j = 0 then 1 else 0 := by
simpa [U] using hrep
Summing over the canonical enumeration of a finite set agrees with summing directly over that set.
Formal statement
On a good Gram event, every active intercept weight has the inverse-count scale dictated by coercivity. The result uses the hlambda condition, the hgood condition. This is the stated conclusion.
Formal statement
hypotheses
J n ell :
ℕ
B :
x :
Fin J
kappa cminus cplus delta h :
ℝ
hlambda :
0 < lambdaStar ell kappa cminus cplus
hgood :
GoodGramEvent B z x ell kappa cminus cplus delta h
i :
Fin n
conclusion
|interceptWeight B z x ell kappa cminus cplus delta h i|
≤ (lambdaStar ell kappa cminus cplus * (localCount B z x delta h : ℝ) / 2)⁻¹ * sqrt (ell + 1 : ℝ)
Proof (Lean source)
-- @node: goodGram_interceptWeight_pointwise
lemma goodGram_interceptWeight_pointwise {J n ell : ℕ}
(B : SplitBlocks n) (z : Fin n → ClampObs J) (x : Fin J)
(kappa cminus cplus delta h : ℝ)
(hlambda : 0 < lambdaStar ell kappa cminus cplus)
(hgood : GoodGramEvent B z x ell kappa cminus cplus delta h) (i : Fin n) :
|interceptWeight B z x ell kappa cminus cplus delta h i| ≤
(lambdaStar ell kappa cminus cplus * (localCount B z x delta h : ℝ) / 2)⁻¹ *
sqrt (ell + 1 : ℝ) := by
classical
by_cases hi : i ∈ B.I2 ∧ (z i).X = x ∧
scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1
· let A := localGram B z x ell delta h
let a := lambdaStar ell kappa cminus cplus * (localCount B z x delta h : ℝ) / 2
let s := monomialVec ell (scaledDose delta h (z i))
have hN : 0 < (localCount B z x delta h : ℝ) := by exact_mod_cast hgood.1
have ha : 0 < a := by dsimp [a]; positivity
have hinv := coerciveMatrix_inv_mulVec_euclidean_le ha hgood.2 s
have hcoord : |A⁻¹.mulVec s 0| ≤ sqrt (∑ k, (A⁻¹.mulVec s k) ^ 2) := by
rw [← Real.sqrt_sq_eq_abs]
exact Real.sqrt_le_sqrt (Finset.single_le_sum
(fun j _ => sq_nonneg (A⁻¹.mulVec s j)) (Finset.mem_univ 0))
have hs : sqrt (∑ k, (s k) ^ 2) ≤ sqrt (ell + 1 : ℝ) := by
apply Real.sqrt_le_sqrt
calc
∑ k, (s k) ^ 2 ≤ ∑ _k : Fin (ell + 1), (1 : ℝ) := by
apply Finset.sum_le_sum
intro k hk
dsimp [s, monomialVec]
have hu := hi.2.2
have hp0 : 0 ≤ scaledDose delta h (z i) ^ (k : ℕ) := pow_nonneg hu.1 _
have hp1 : scaledDose delta h (z i) ^ (k : ℕ) ≤ 1 := pow_le_one₀ hu.1 hu.2
nlinarith
_ = ell + 1 := by simp
rw [interceptWeight_eq_mulVec B z x ell kappa cminus cplus delta h i
hgood hi.1 hi.2.1 hi.2.2]
simpa only [A, a, s] using
hcoord.trans (hinv.trans
(mul_le_mul_of_nonneg_left hs (inv_nonneg.mpr ha.le)))
· rw [interceptWeight_eq_zero_of_inactive
B z x ell kappa cminus cplus delta h i hgood hi, abs_zero]
have hN : 0 < (localCount B z x delta h : ℝ) := by exact_mod_cast hgood.1
positivity
On a good Gram event, the exact weights reproduce every basis monomial and obey explicit count-normalized l1 and squared-l2 bounds. The result uses the hh condition, the hlambda condition, the hgood condition. This is the stated conclusion.
Formal statement
hypotheses
J n ell :
ℕ
B :
x :
Fin J
kappa cminus cplus delta h :
ℝ
hh :
0 < h
hlambda :
0 < lambdaStar ell kappa cminus cplus
hgood :
GoodGramEvent B z x ell kappa cminus cplus delta h
conclusion
conclusion 1
j :
Fin (ell + 1)
(∑ i ∈ B.I2, interceptWeight B z x ell kappa cminus cplus delta h i * (scaledDose delta h (z i)) ^ (j : ℕ))
= if j = 0 then 1 else 0
conclusion 2
(∑ i ∈ B.I2, |interceptWeight B z x ell kappa cminus cplus delta h i|)
≤ 2 * sqrt (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus
conclusion 3
(∑ i ∈ B.I2, (interceptWeight B z x ell kappa cminus cplus delta h i) ^ 2)
≤ (4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2) / (localCount B z x delta h : ℝ)
Proof (Lean source)
-- @node: goodGram_weight_controls
lemma goodGram_weight_controls {J n ell : ℕ}
(B : SplitBlocks n) (z : Fin n → ClampObs J) (x : Fin J)
(kappa cminus cplus delta h : ℝ)
(hh : 0 < h)
(hlambda : 0 < lambdaStar ell kappa cminus cplus)
(hgood : GoodGramEvent B z x ell kappa cminus cplus delta h) :
(∀ j : Fin (ell + 1),
(∑ i ∈ B.I2,
interceptWeight B z x ell kappa cminus cplus delta h i *
(scaledDose delta h (z i)) ^ (j : ℕ)) =
if j = 0 then 1 else 0) ∧
(∑ i ∈ B.I2,
|interceptWeight B z x ell kappa cminus cplus delta h i|) ≤
2 * sqrt (ell + 1 : ℝ) /
lambdaStar ell kappa cminus cplus ∧
(∑ i ∈ B.I2,
(interceptWeight B z x ell kappa cminus cplus delta h i) ^ 2) ≤
(4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2) /
(localCount B z x delta h : ℝ) := by
classical
let lam := lambdaStar ell kappa cminus cplus
let N := localCount B z x delta h
have hNnat : 0 < N := hgood.1
have hN : (0 : ℝ) < N := by exact_mod_cast hNnat
have hpoint (i : Fin n) :
|interceptWeight B z x ell kappa cminus cplus delta h i| ≤
if i ∈ B.I2 ∧ (z i).X = x ∧
scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1 then
(lam * (N : ℝ) / 2)⁻¹ * sqrt (ell + 1 : ℝ)
else 0 := by
by_cases hi : i ∈ B.I2 ∧ (z i).X = x ∧
scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1
· rw [if_pos hi]
exact goodGram_interceptWeight_pointwise B z x kappa cminus cplus delta h
hlambda hgood i
· rw [if_neg hi, interceptWeight_eq_zero_of_inactive
B z x ell kappa cminus cplus delta h i hgood hi, abs_zero]
have hcount :
(∑ i ∈ B.I2, if (z i).X = x ∧
scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1 then (1 : ℝ) else 0) = N := by
simpa only [N] using local_active_count_eq B z x delta h hh
refine ⟨goodGram_reproduction B z x kappa cminus cplus delta h hlambda hgood, ?_, ?_⟩
· calc
(∑ i ∈ B.I2,
|interceptWeight B z x ell kappa cminus cplus delta h i|) ≤
∑ i ∈ B.I2, if (z i).X = x ∧
scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1 then
(lam * (N : ℝ) / 2)⁻¹ * sqrt (ell + 1 : ℝ) else 0 := by
apply Finset.sum_le_sum
intro i hi
simpa [hi] using hpoint i
_ = N * ((lam * (N : ℝ) / 2)⁻¹ * sqrt (ell + 1 : ℝ)) := by
rw [← hcount]
simp_rw [Finset.sum_mul]
apply Finset.sum_congr rfl
intro i hi
by_cases ha : (z i).X = x ∧
scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1 <;> simp [ha]
_ = 2 * sqrt (ell + 1 : ℝ) / lam := by
field_simp [ne_of_gt hlambda, ne_of_gt hN]
<;> ring
· have hsqpoint (i : Fin n) :
(interceptWeight B z x ell kappa cminus cplus delta h i) ^ 2 ≤
if i ∈ B.I2 ∧ (z i).X = x ∧
scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1 then
((lam * (N : ℝ) / 2)⁻¹ * sqrt (ell + 1 : ℝ)) ^ 2
else 0 := by
have habs := hpoint i
by_cases hi : i ∈ B.I2 ∧ (z i).X = x ∧
scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1
· rw [if_pos hi] at habs ⊢
rw [← sq_abs]
gcongr
· rw [if_neg hi] at habs ⊢
have hz : interceptWeight B z x ell kappa cminus cplus delta h i = 0 := by
exact interceptWeight_eq_zero_of_inactive
B z x ell kappa cminus cplus delta h i hgood hi
simp [hz]
calc
(∑ i ∈ B.I2,
(interceptWeight B z x ell kappa cminus cplus delta h i) ^ 2) ≤
∑ i ∈ B.I2, if (z i).X = x ∧
scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1 then
((lam * (N : ℝ) / 2)⁻¹ * sqrt (ell + 1 : ℝ)) ^ 2 else 0 := by
apply Finset.sum_le_sum
intro i hi
simpa [hi] using hsqpoint i
_ = N * ((lam * (N : ℝ) / 2)⁻¹ * sqrt (ell + 1 : ℝ)) ^ 2 := by
rw [← hcount]
simp_rw [Finset.sum_mul]
apply Finset.sum_congr rfl
intro i hi
by_cases ha : (z i).X = x ∧
scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1 <;> simp [ha]
_ = (4 * (ell + 1 : ℝ) / lam ^ 2) / N := by
simp only [mul_pow]
rw [Real.sq_sqrt (by positivity : (0 : ℝ) ≤ ell + 1)]
field_simp [ne_of_gt hlambda, ne_of_gt hN]
<;> ring
theorem totalGram_lambdaStar_pos
The population constant used by total-Gram stabilization is positive under the standing regime constraints. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
0 < lambdaStar (ellOf beta) kappa cminus cplus
Proof (Lean source)
-- @node: totalGram_lambdaStar_pos
lemma totalGram_lambdaStar_pos
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
0 < lambdaStar (ellOf beta) kappa cminus cplus := by
simp only [RegimeConstants] at hreg
rcases hreg with
⟨_hJ, _hbeta, hkappa, _hL, hcminus, _hcminus_le, hkappa_le_cplus,
_hpmin, _hpmin_le, _hdeltaBar, _hdeltaBar_lt, _halpha, _halpha_lt⟩
exact lambdaStar_pos (ellOf beta) hkappa hcminus (by linarith)
Uniform positivity, singular-design tail, exact reproduction, and l1/l2 weight control follow from admissible regime constants. The constants are uniform over laws, samples, thresholds, and strata.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
∃ C c : ℝ,
conclusion 1
P :
ClampLaw J
ClampModel P beta kappa L cminus cplus pmin
n :
ℕ
IidSampling P n
B :
delta :
ℝ
delta ∈ Icc (0 : ℝ) deltaBar
x :
Fin J
h :
infoBandwidth n delta beta kappa deltaBar
≤ C * exp (-c * (n : ℝ) * h * (delta + h) ^ kappa)
GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h
(∑ i ∈ B.I2, interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i * (scaledDose delta h (z i)) ^ (j : ℕ))
= if j = 0 then 1 else 0
(∑ i ∈ B.I2, |interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i|) ≤ C
(∑ i ∈ B.I2, (interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i) ^ 2)
≤ C / (localCount B z x delta h : ℝ)
Proof (Lean source)
-- @node: lem:total-gram
lemma total_gram_stabilization
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
∃ C c : ℝ, 0 < C ∧ 0 < c ∧
0 < lambdaStar (ellOf beta) kappa cminus cplus ∧
∀ (P : ClampLaw J), ClampModel P beta kappa L cminus cplus pmin →
∀ (n : ℕ), IidSampling P n →
∀ (B : SplitBlocks n) (delta : ℝ), delta ∈ Icc (0 : ℝ) deltaBar →
∀ x : Fin J,
let h := infoBandwidth n delta beta kappa deltaBar
(iidProduct P n).real {z |
¬ GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h} ≤
C * exp (-c * (n : ℝ) * h * (delta + h) ^ kappa) ∧
∀ z : Fin n → ClampObs J,
GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h →
(∀ j : Fin (ellOf beta + 1),
(∑ i ∈ B.I2,
interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
(scaledDose delta h (z i)) ^ (j : ℕ)) =
if j = 0 then 1 else 0) ∧
(∑ i ∈ B.I2,
|interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i|) ≤ C ∧
(∑ i ∈ B.I2,
(interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i) ^ 2) ≤
C / (localCount B z x delta h : ℝ) := by
have _hlambda : 0 < lambdaStar (ellOf beta) kappa cminus cplus :=
totalGram_lambdaStar_pos J beta kappa L cminus cplus pmin deltaBar alpha hreg
classical
simp only [RegimeConstants] at hreg
rcases hreg with
⟨hJ, hbeta, hkappa, hL, hcminus, hcminus_le, hcplus_lower,
hpmin, hpmin_le, hdeltaBar, hdeltaBar_lt, halpha, halpha_lt⟩
let ell := ellOf beta
let lam := lambdaStar ell kappa cminus cplus
let d : ℝ := ell + 1
let r := localizedGramRate (Fin (ell + 1)) lam 1
let K := pmin * cminus / (2 * (2 : ℝ) ^ kappa)
let c := r * K / 8
let A := 2 * (1 + d * d)
let C := A * exp (8 * c) + 2 * sqrt d / lam + 4 * d / lam ^ 2 + 1
have hlam : 0 < lam := by simpa [ell, lam] using _hlambda
have hd : 0 < d := by dsimp [d]; positivity
have hr : 0 < r := by
dsimp [r]
unfold localizedGramRate
apply lt_min (by norm_num)
positivity
have hK : 0 < K := by dsimp [K]; positivity
have hc : 0 < c := by dsimp [c]; positivity
have hA : 1 ≤ A := by dsimp [A, d]; nlinarith [sq_nonneg (ell + 1 : ℝ)]
have hC : 0 < C := by dsimp [C]; positivity
have hCtail : A * exp (8 * c) ≤ C := by
dsimp [C]
have h1 : 0 ≤ 2 * sqrt d / lam := by positivity
have h2 : 0 ≤ 4 * d / lam ^ 2 := by positivity
linarith
have hCA : A ≤ C := by
calc
A = A * 1 := by ring
_ ≤ A * exp (8 * c) :=
mul_le_mul_of_nonneg_left (Real.one_le_exp (by positivity)) (by linarith [hA])
_ ≤ C := hCtail
have hCl1 : 2 * sqrt d / lam ≤ C := by
dsimp [C]
have h0 : 0 ≤ A * exp (8 * c) := by positivity
have h2 : 0 ≤ 4 * d / lam ^ 2 := by positivity
linarith
have hCl2 : 4 * d / lam ^ 2 ≤ C := by
dsimp [C]
have h0 : 0 ≤ A * exp (8 * c) := by positivity
have h1 : 0 ≤ 2 * sqrt d / lam := by positivity
linarith
refine ⟨C, c, hC, hc, _hlambda, ?_⟩
intro P hmodel n hsampling B delta hdelta x
dsimp only
let h := infoBandwidth n delta beta kappa deltaBar
have hhgeom := infoBandwidth_pos_le (n := n) hbeta hkappa hdelta
⟨hdeltaBar, hdeltaBar_lt⟩
have hh : 0 < h := by simpa [h] using hhgeom.1
have hupper : delta + h ≤ 1 := by
have := hhgeom.2
linarith [hdelta.2]
let q := localWindowWeight x delta h
let phi : Fin (ell + 1) → ClampObs J → ℝ := localWindowFeature delta h
let p := ∫ o, q o ∂P.dataMeasure
let F : (Fin n → ClampObs J) → (Fin B.I2.card → ClampObs J) :=
fun z j => z ((B.I2.orderIsoOfFin rfl) j)
haveI : IsProbabilityMeasure P.dataMeasure := hmodel.probability
have hp0 : 0 ≤ p := by
dsimp [p, q]
exact integral_nonneg_of_ae (ae_of_all _ fun o => (localWindowWeight_mem_unit x delta h o).1)
have hpLower : K * h * (delta + h) ^ kappa ≤ p := by
have hl := localWindowWeight_integral_lower P hmodel hkappa hcminus
(by linarith : 0 ≤ cplus) hpmin x hdelta.1 hh hupper
have hwpos : 0 < delta + h := by linarith [hdelta.1]
have htwo : (0 : ℝ) < 2 := by norm_num
have hrpow : ((delta + h) / 2) ^ kappa =
(delta + h) ^ kappa / (2 : ℝ) ^ kappa := by
rw [Real.div_rpow hwpos.le htwo.le]
rw [hrpow] at hl
calc
K * h * (delta + h) ^ kappa =
pmin * (cminus * ((delta + h) ^ kappa / (2 : ℝ) ^ kappa) * (h / 2)) := by
dsimp [K]
ring
_ ≤ p := by simpa [p, q] using hl
have hqmeas : Measurable q := measurable_localWindowWeight x delta h
have hphimeas : ∀ j, Measurable (phi j) := measurable_localWindowFeature delta h
have hq : ∀ᵐ o ∂P.dataMeasure, 0 ≤ q o ∧ q o ≤ 1 :=
ae_of_all _ (localWindowWeight_mem_unit x delta h)
have hphi : ∀ j, ∀ᵐ o ∂P.dataMeasure, |phi j o| ≤ 1 :=
fun j => ae_of_all _ (abs_localWindowFeature_le_one delta h j)
have hcoercive : ∀ v : Fin (ell + 1) → ℝ,
lam * p * (∑ j, (v j) ^ 2) ≤
∑ j, ∑ k, v j * v k * ∫ o, q o * phi j o * phi k o ∂P.dataMeasure := by
intro v
simpa [ell, lam, p, q, phi] using
localWindow_populationGram_coercive P hmodel hkappa hcminus
(by linarith : 0 < cplus) hpmin x hdelta.1 hh hupper v
letI : IsProbabilityMeasure (iidProduct P n) := by
unfold iidProduct
infer_instance
have hsum (f : Fin n → ℝ) :
(∑ j : Fin B.I2.card, f ((B.I2.orderIsoOfFin rfl) j)) = ∑ i ∈ B.I2, f i := by
calc
_ = ∑ i : B.I2, f i := Equiv.sum_comp (B.I2.orderIsoOfFin rfl).toEquiv
(fun i : B.I2 => f i)
_ = _ := (Finset.sum_subtype B.I2 (fun _ => Iff.rfl) f).symm
have hgood (z : Fin n → ClampObs J) :
LocalizedGramGood lam q phi (F z) ↔
GoodGramEvent B z x ell kappa cminus cplus delta h := by
simp only [LocalizedGramGood, GoodGramEvent]
have hcount : localCount q (F z) =
(localCount B z x delta h : ℝ) := by
unfold localCount
change (∑ j : Fin B.I2.card, q (z ((B.I2.orderIsoOfFin rfl) j))) = _
rw [show (∑ j : Fin B.I2.card, q (z ((B.I2.orderIsoOfFin rfl) j))) =
∑ i ∈ B.I2, q (z i) by exact hsum (fun i => q (z i))]
calc
(∑ i ∈ B.I2, q (z i)) =
∑ i ∈ B.I2, if (z i).X = x ∧
scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1 then 1 else 0 := by
apply Finset.sum_congr rfl
intro i hi
simp [q, localWindowWeight]
_ = (localCount B z x delta h : ℝ) := local_active_count_eq B z x delta h hh
have hentry (j k : Fin (ell + 1)) :
localGramEntry q phi (F z) j k =
localGram B z x ell delta h j k := by
unfold localGramEntry
change (∑ a : Fin B.I2.card, q (z ((B.I2.orderIsoOfFin rfl) a)) *
phi j (z ((B.I2.orderIsoOfFin rfl) a)) *
phi k (z ((B.I2.orderIsoOfFin rfl) a))) = _
rw [show (∑ a : Fin B.I2.card, q (z ((B.I2.orderIsoOfFin rfl) a)) *
phi j (z ((B.I2.orderIsoOfFin rfl) a)) *
phi k (z ((B.I2.orderIsoOfFin rfl) a))) =
∑ i ∈ B.I2, q (z i) * phi j (z i) * phi k (z i) by
exact hsum (fun i => q (z i) * phi j (z i) * phi k (z i))]
rw [localGram_apply]
apply Finset.sum_congr rfl
intro i hi
by_cases ha : (z i).X = x ∧ scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1
· simp [q, phi, localWindowWeight, localWindowFeature, monomialVec, ha]
· by_cases hu : scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1
· have hx : (z i).X ≠ x := fun hx => ha ⟨hx, hu⟩
simp [q, phi, localWindowWeight, localWindowFeature, monomialVec, ha, hx, hu]
· simp [q, phi, localWindowWeight, localWindowFeature, monomialVec, ha, hu]
have hquad (v : Fin (ell + 1) → ℝ) :
localGramQuadratic q phi (F z) v =
matrixQuadratic (localGram B z x ell delta h) v := by
unfold localGramQuadratic matrixQuadratic
simp_rw [hentry]
apply Finset.sum_congr rfl
intro j hj
apply Finset.sum_congr rfl
intro k hk
ring
rw [hcount]
constructor
· rintro ⟨hN, hg⟩
refine ⟨by exact_mod_cast hN, ?_⟩
intro v
have hv := hg v
rw [hquad] at hv
dsimp [lam, ell] at hv ⊢
ring_nf at hv ⊢
exact hv
· rintro ⟨hN, hg⟩
refine ⟨by exact_mod_cast hN, ?_⟩
intro v
have hv := hg v
rw [← hquad] at hv
dsimp [lam, ell] at hv ⊢
ring_nf at hv ⊢
exact hv
have hprob : (iidProduct P n).real {z |
¬ GoodGramEvent B z x ell kappa cminus cplus delta h} ≤ 1 := measureReal_le_one
by_cases hlarge : 8 ≤ n
· have hN : 0 < B.I2.card := lt_of_lt_of_le (by omega) B.card_I2
have hp : 0 < p := lt_of_lt_of_le
(mul_pos (mul_pos hK hh) (Real.rpow_pos_of_pos (by linarith [hdelta.1]) kappa))
hpLower
have hlocal := localized_empiricalGram_coercive_of_pos
P.dataMeasure q phi hqmeas hphimeas hN hp hlam (by norm_num)
hq hphi rfl hcoercive
have htail : (iidProduct P n).real {z |
¬ GoodGramEvent B z x ell kappa cminus cplus delta h} ≤
A * exp (-(B.I2.card : ℝ) * p * r) := by
let E : Set (Fin B.I2.card → ClampObs J) := {omega |
¬ LocalizedGramGood lam q phi omega}
have hset : {z : Fin n → ClampObs J |
¬ GoodGramEvent B z x ell kappa cminus cplus delta h} = F ⁻¹' E := by
ext z
simp only [Set.mem_setOf_eq, Set.mem_preimage]
exact not_congr (hgood z).symm
have hle : (iidProduct P n) (F ⁻¹' E) ≤
((iidProduct P n).map F) E := Measure.le_map_apply (by fun_prop) E
have hreal : (iidProduct P n).real (F ⁻¹' E) ≤
((iidProduct P n).map F).real E := by
exact ENNReal.toReal_mono (measure_ne_top _ _) hle
rw [hset]
rw [iid_block_product_law P hsampling B.I2] at hreal
exact hreal.trans (by simpa [E, A, d, r] using hlocal)
have hcard : (n : ℝ) / 8 ≤ B.I2.card := by
have hqcard := B.card_I2
have hncard : n ≤ 8 * B.I2.card := by omega
have hncardR : (n : ℝ) ≤ 8 * (B.I2.card : ℝ) := by exact_mod_cast hncard
linarith
have hexp : c * (n : ℝ) * h * (delta + h) ^ kappa ≤
(B.I2.card : ℝ) * p * r := by
have ht : 0 ≤ h * (delta + h) ^ kappa :=
mul_nonneg hh.le (Real.rpow_nonneg (by linarith [hdelta.1]) kappa)
have hcn : c * (n : ℝ) ≤ r * K * (B.I2.card : ℝ) := by
dsimp [c]
nlinarith [mul_pos hr hK]
calc
c * (n : ℝ) * h * (delta + h) ^ kappa =
(c * (n : ℝ)) * (h * (delta + h) ^ kappa) := by ring
_ ≤ (r * K * (B.I2.card : ℝ)) *
(h * (delta + h) ^ kappa) := mul_le_mul_of_nonneg_right hcn ht
_ = (B.I2.card : ℝ) * (K * h * (delta + h) ^ kappa) * r := by ring
_ ≤ (B.I2.card : ℝ) * p * r := by
exact mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left hpLower (Nat.cast_nonneg _)) hr.le
refine ⟨htail.trans ?_, ?_⟩
· simpa [h] using mul_le_mul hCA (Real.exp_le_exp.mpr (neg_le_neg hexp))
(Real.exp_pos _).le (by linarith [hA])
-- … truncated; follow the source link for the rest …
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.UpperEmpirical 4 declarations
theorem blockAverage_l1_le
Expected absolute error of a bounded empirical average over an arbitrary nonempty deterministic block of a finite product sample. The result uses the hI condition, the hf condition, the hf0 condition, the hf1 condition, the hm condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
hI :
0 < I.card
f :
ClampObs J → ℝ
hf :
hf0 :
∀ o, 0 ≤ f o
hf1 :
∀ o, f o ≤ 1
m :
ℝ
hm :
∫ o, f o ∂P.dataMeasure = m
IsProbabilityMeasure P.dataMeasure
conclusion
(∫ z, |blockAverage I z f - m| ∂iidProduct P n) ≤ (1 / 2 : ℝ) * sqrt (I.card : ℝ)⁻¹
Proof (Lean source)
lemma blockAverage_l1_le {J n : ℕ} (P : ClampLaw J)
(I : Finset (Fin n)) (hI : 0 < I.card)
(f : ClampObs J → ℝ) (hf : Measurable f)
(hf0 : ∀ o, 0 ≤ f o) (hf1 : ∀ o, f o ≤ 1)
(m : ℝ) (hm : ∫ o, f o ∂P.dataMeasure = m)
[IsProbabilityMeasure P.dataMeasure] :
(∫ z, |blockAverage I z f - m| ∂iidProduct P n) ≤
(1 / 2 : ℝ) * sqrt (I.card : ℝ)⁻¹ := by
letI : IsProbabilityMeasure (iidProduct P n) := by
unfold iidProduct
infer_instance
letI : MeasurableSpace Unit := ⊥
let design : ClampObs J → Unit := fun _ => ()
let mD : Unit → ℝ := fun _ => m
let w : (Fin n → Unit) → Fin n → ℝ := fun _ i =>
if i ∈ I then (I.card : ℝ)⁻¹ else 0
have hd : Measurable design := measurable_const
have hmD : Measurable mD := measurable_const
have hcond :
P.dataMeasure[f | comap design inferInstance] =ᵐ[P.dataMeasure]
mD ∘ design := by
have hspace : comap design inferInstance = ⊥ := by
ext s
rw [MeasurableSpace.measurableSet_comap,
MeasurableSpace.measurableSet_bot_iff]
constructor
· rintro ⟨t, ht, rfl⟩
rcases MeasurableSpace.measurableSet_bot_iff.mp ht with rfl | rfl <;> simp
· rintro (rfl | rfl)
· exact ⟨∅, MeasurableSet.empty, by simp⟩
· exact ⟨Set.univ, MeasurableSet.univ, by simp⟩
rw [hspace, condExp_bot]
filter_upwards with o
simp [mD, hm]
have hw : Measurable w := by
apply measurable_pi_lambda
intro i
by_cases hi : i ∈ I <;> simp [w, hi]
have henergy (z : Fin n → ClampObs J) :
(∑ i, w (designVector design z) i ^ 2) =
(I.card : ℝ)⁻¹ := by
classical
simp only [w]
have hcard : (I.card : ℝ) ≠ 0 := by exact_mod_cast hI.ne'
simp [hcard]
field_simp
have henergyInt : Integrable (fun z : Fin n → ClampObs J =>
∑ i, w (designVector design z) i ^ 2)
(iidProduct P n) := by
convert integrable_const (μ := iidProduct P n) (I.card : ℝ)⁻¹ using 1
funext z
exact henergy z
have hcenter (o : ClampObs J) : |f o - m| ≤ 1 := by
have hm0 : 0 ≤ m := by rw [← hm]; exact integral_nonneg hf0
have hm1 : m ≤ 1 := by
have hfint : Integrable f P.dataMeasure := Integrable.of_bound hf.aestronglyMeasurable 1
(ae_of_all _ fun o => by
rw [Real.norm_eq_abs, abs_of_nonneg (hf0 o)]
exact hf1 o)
rw [← hm]
calc
(∫ o, f o ∂P.dataMeasure) ≤ ∫ _o, (1 : ℝ) ∂P.dataMeasure :=
integral_mono_ae hfint (integrable_const 1) (ae_of_all _ hf1)
_ = 1 := by simp
rw [abs_le]
constructor <;> linarith [hf0 o, hf1 o]
let S := Causalean.Mathlib.Probability.weightedCenteredSum design f mD w
have hSmeas : Measurable S := by
dsimp [S]
unfold Causalean.Mathlib.Probability.weightedCenteredSum
exact Finset.measurable_fun_sum _ fun i _ =>
((measurable_pi_apply i).comp
(hw.comp (measurable_designVector design hd))).mul
((hf.comp (measurable_pi_apply i)).sub
(hmD.comp (hd.comp (measurable_pi_apply i))))
have hSbound (z : Fin n → ClampObs J) : |S z| ≤ 1 := by
dsimp [S]
unfold Causalean.Mathlib.Probability.weightedCenteredSum
calc
_ ≤ ∑ i, |w (designVector design z) i *
(f (z i) - mD (design (z i)))| := Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ i, if i ∈ I then (I.card : ℝ)⁻¹ else 0 := by
apply Finset.sum_le_sum
intro i hi
by_cases hiI : i ∈ I
· simp only [w, mD, hiI, if_true, abs_mul, abs_inv]
rw [abs_of_nonneg (Nat.cast_nonneg I.card)]
simpa using mul_le_mul_of_nonneg_left (hcenter (z i))
(inv_nonneg.mpr (Nat.cast_nonneg I.card))
· simp [w, hiI]
_ = ∑ i ∈ I, (I.card : ℝ)⁻¹ := by simp
_ = 1 := by
rw [Finset.sum_const, nsmul_eq_mul]
have hc : (I.card : ℝ) ≠ 0 := by exact_mod_cast hI.ne'
field_simp
have hSsq : Integrable (fun z => (S z) ^ 2) (iidProduct P n) := by
refine Integrable.of_bound (hSmeas.pow_const 2).aestronglyMeasurable 1 ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
rw [← sq_abs]
nlinarith [hSbound z, abs_nonneg (S z)]
have hbase := product_weighted_centered_l1_le
P.dataMeasure design hd f hf hf0 hf1 mD hmD hcond w hw henergyInt hSsq
have hSid (z : Fin n → ClampObs J) : S z = blockAverage I z f - m := by
unfold S Causalean.Mathlib.Probability.weightedCenteredSum blockAverage
simp only [w, mD]
rw [show (∑ x, (if x ∈ I then (I.card : ℝ)⁻¹ else 0) * (f (z x) - m)) =
∑ i ∈ I, (I.card : ℝ)⁻¹ * (f (z i) - m) by simp]
have hsum : ∑ i ∈ I, (I.card : ℝ)⁻¹ * (f (z i) - m) =
(I.card : ℝ)⁻¹ * ∑ i ∈ I, f (z i) - m := by
rw [Finset.mul_sum]
simp_rw [mul_sub]
rw [Finset.sum_sub_distrib, Finset.sum_const, nsmul_eq_mul]
have hc : (I.card : ℝ) ≠ 0 := by exact_mod_cast hI.ne'
field_simp
exact hsum
rw [show (∫ z, |blockAverage I z f - m| ∂iidProduct P n) =
∫ z, |S z| ∂iidProduct P n by
apply integral_congr_ae
filter_upwards with z
rw [hSid]]
have henergyIntegral : (∫ z : Fin n → ClampObs J,
∑ i, w (designVector design z) i ^ 2
∂iidProduct P n) = (I.card : ℝ)⁻¹ := by
calc
_ = ∫ _z : Fin n → ClampObs J, (I.card : ℝ)⁻¹ ∂iidProduct P n :=
integral_congr_ae (ae_of_all _ henergy)
_ = _ := by simp
rw [show Measure.pi (fun _ : Fin n => P.dataMeasure) = iidProduct P n by rfl,
henergyIntegral] at hbase
simpa only [S] using hbase
theorem retainedEstimate_l1_le
retained estimate absolute-error satisfies the stated upper bound for the specified J input, the specified n input, the specified P input, the specified beta input, the specified kappa input, the specified L input, the specified cminus input, the specified cplus input, the specified pmin input, the specified delta input, the specified hmodel input, the specified B input, the specified hcard input.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
B :
hcard :
0 < B.I0.card
conclusion
(∫ z, |retainedEstimate B z delta - retainedMean P delta| ∂iidProduct P n)
≤ (1 / 2 : ℝ) * sqrt (B.I0.card : ℝ)⁻¹
Proof (Lean source)
lemma retainedEstimate_l1_le {J n : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin delta : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(B : SplitBlocks n) (hcard : 0 < B.I0.card) :
(∫ z, |retainedEstimate B z delta - retainedMean P delta| ∂iidProduct P n) ≤
(1 / 2 : ℝ) * sqrt (B.I0.card : ℝ)⁻¹ := by
let _ := hmodel.probability
let f : ClampObs J → ℝ := fun o => if delta < o.A then clampUnit o.Y else 0
have hf : Measurable f := by
have hA : Measurable (fun o : ClampObs J => o.A) :=
measurable_fst.comp (measurable_snd.comp (Measurable.of_comap_le le_rfl))
apply Measurable.ite
· exact measurableSet_lt measurable_const hA
· exact clampUnit_measurable'.comp clampOutcome_measurable
· exact measurable_const
have hf0 : ∀ o, 0 ≤ f o := by
intro o
dsimp [f]
split_ifs
· exact (clampUnit_mem_Icc o.Y).1
· norm_num
have hf1 : ∀ o, f o ≤ 1 := by
intro o
dsimp [f]
split_ifs
· exact (clampUnit_mem_Icc o.Y).2
· norm_num
have hm : ∫ o, f o ∂P.dataMeasure = retainedMean P delta := by
calc
_ = ∫ o, (if delta < o.A then o.Y else 0) ∂P.dataMeasure := by
apply integral_congr_ae
filter_upwards [hmodel.outcomeSupport] with o ho
simp [f, clampUnit, ho.1, ho.2]
_ = _ := by
unfold retainedMean
apply integral_congr_ae
filter_upwards with o
by_cases ha : delta < o.A <;> simp [indicator, ha]
have hb := blockAverage_l1_le P B.I0 hcard f hf hf0 hf1
(retainedMean P delta) hm
calc
(∫ z, |retainedEstimate B z delta - retainedMean P delta| ∂iidProduct P n) =
∫ z, |blockAverage B.I0 z f - retainedMean P delta| ∂iidProduct P n := by
apply integral_congr_ae
filter_upwards [iidProduct_outcomeSupport P hmodel] with z hz
congr 2
unfold retainedEstimate blockAverage
congr 1
apply Finset.sum_congr rfl
intro i hi
simp [f, clampUnit, (hz i).1, (hz i).2]
_ ≤ _ := hb
theorem atomEvent_integral_eq
atom event integral satisfies the stated identity for the specified J input, the specified P input, the specified beta input, the specified kappa input, the specified L input, the specified cminus input, the specified cplus input, the specified pmin input, the specified delta input, the specified hmodel input, the specified x input, the specified hdelta1 input.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
x :
Fin J
hdelta1 :
delta ≤ 1
conclusion
(∫ o, (if o.X = x ∧ o.A ≤ delta then (1 : ℝ) else 0) ∂P.dataMeasure)
= P.px x * atomMass P x delta
Proof (Lean source)
lemma atomEvent_integral_eq {J : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin delta : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(x : Fin J) (hdelta1 : delta ≤ 1) :
(∫ o, (if o.X = x ∧ o.A ≤ delta then (1 : ℝ) else 0) ∂P.dataMeasure) =
P.px x * atomMass P x delta := by
let _ := hmodel.probability
have hX : Measurable (fun o : ClampObs J => o.X) :=
measurable_fst.comp (Measurable.of_comap_le le_rfl)
have hA : Measurable (fun o : ClampObs J => o.A) :=
measurable_fst.comp (measurable_snd.comp (Measurable.of_comap_le le_rfl))
have hset : MeasurableSet {o : ClampObs J |
o.X = x ∧ o.A ∈ Icc (0 : ℝ) delta} :=
(measurableSet_eq_fun hX measurable_const).inter (measurableSet_Icc.preimage hA)
calc
_ = ∫ o, indicator {o : ClampObs J |
o.X = x ∧ o.A ∈ Icc (0 : ℝ) delta} (fun _ => (1 : ℝ)) o
∂P.dataMeasure := by
apply integral_congr_ae
filter_upwards [hmodel.treatmentSupport] with o ho
by_cases hx : o.X = x <;> by_cases ha : o.A ≤ delta <;>
simp [indicator, hx, ha, ho.1]
_ = P.dataMeasure.real {o : ClampObs J |
o.X = x ∧ o.A ∈ Icc (0 : ℝ) delta} := by
rw [integral_indicator hset]
simp
_ = P.px x * atomMass P x delta := by
rw [(hmodel.condDensity.2.2 x).2 (Icc 0 delta) measurableSet_Icc]
· rfl
· intro a ha
exact ⟨ha.1, ha.2.trans hdelta1⟩
theorem atomEstimate_l1_le
atom estimate absolute-error satisfies the stated upper bound for the specified J input, the specified n input, the specified P input, the specified beta input, the specified kappa input, the specified L input, the specified cminus input, the specified cplus input, the specified pmin input, the specified delta input, the specified hmodel input, the specified B input, the specified hcard input, the specified x input, the specified hdelta input, the specified hdelta1 input.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
B :
hcard :
0 < B.I1.card
x :
Fin J
hdelta :
0 ≤ delta
hdelta1 :
delta ≤ 1
conclusion
(∫ z, |atomEstimate B z x delta - P.px x * atomMass P x delta| ∂iidProduct P n)
≤ (1 / 2 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹
Proof (Lean source)
lemma atomEstimate_l1_le {J n : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin delta : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(B : SplitBlocks n) (hcard : 0 < B.I1.card) (x : Fin J)
(hdelta : 0 ≤ delta) (hdelta1 : delta ≤ 1) :
(∫ z, |atomEstimate B z x delta - P.px x * atomMass P x delta|
∂iidProduct P n) ≤ (1 / 2 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹ := by
let _ := hmodel.probability
let f : ClampObs J → ℝ := fun o => if o.X = x ∧ o.A ≤ delta then 1 else 0
have hX : Measurable (fun o : ClampObs J => o.X) :=
measurable_fst.comp (Measurable.of_comap_le le_rfl)
have hA : Measurable (fun o : ClampObs J => o.A) :=
measurable_fst.comp (measurable_snd.comp (Measurable.of_comap_le le_rfl))
have hf : Measurable f := by
apply Measurable.ite
· exact (measurableSet_eq_fun hX measurable_const).inter
(measurableSet_le hA measurable_const)
· exact measurable_const
· exact measurable_const
have hf0 : ∀ o, 0 ≤ f o := by intro o; dsimp [f]; split_ifs <;> norm_num
have hf1 : ∀ o, f o ≤ 1 := by intro o; dsimp [f]; split_ifs <;> norm_num
have hset : MeasurableSet {o : ClampObs J |
o.X = x ∧ o.A ∈ Icc (0 : ℝ) delta} :=
(measurableSet_eq_fun hX measurable_const).inter (measurableSet_Icc.preimage hA)
have hm : ∫ o, f o ∂P.dataMeasure = P.px x * atomMass P x delta := by
calc
_ = ∫ o, indicator {o : ClampObs J |
o.X = x ∧ o.A ∈ Icc (0 : ℝ) delta} (fun _ => (1 : ℝ)) o
∂P.dataMeasure := by
apply integral_congr_ae
filter_upwards [hmodel.treatmentSupport] with o ho
by_cases hx : o.X = x <;> by_cases ha : o.A ≤ delta <;>
simp [f, indicator, hx, ha, ho.1]
_ = P.dataMeasure.real {o : ClampObs J |
o.X = x ∧ o.A ∈ Icc (0 : ℝ) delta} := by
rw [integral_indicator hset]
simp
_ = P.px x * atomMass P x delta := by
rw [(hmodel.condDensity.2.2 x).2 (Icc 0 delta) measurableSet_Icc]
· rfl
· intro a ha
exact ⟨ha.1, ha.2.trans hdelta1⟩
have hb := blockAverage_l1_le P B.I1 hcard f hf hf0 hf1
(P.px x * atomMass P x delta) hm
simpa [atomEstimate, f] using hb
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.UpperNoise 27 declarations Design-measurable weights for the local-regression noise term
Design-measurable weights for the local-regression noise term
theorem clampUnit_measurable'
clamp unit is measurable.
Formal statement
conclusion
Proof (Lean source)
theorem clampUnit_mem_Icc
clamp unit lies in the stated closed interval for the specified t input.
Proof (Lean source)
lemma clampUnit_mem_Icc (t : ℝ) : clampUnit t ∈ Icc (0 : ℝ) 1 := by
unfold clampUnit
constructor <;> simp
theorem abs_clampUnit_sub_le
abs clamp unit sub satisfies the stated upper bound for the specified t input, the specified q input, the specified hq input.
Proof (Lean source)
lemma abs_clampUnit_sub_le (t q : ℝ) (hq : q ∈ Icc (0 : ℝ) 1) :
|clampUnit t - q| ≤ |t - q| := by
have hcq : clampUnit q = q := by simp [clampUnit, hq.1, hq.2]
calc
|clampUnit t - q| = |clampUnit t - clampUnit q| := by rw [hcq]
|min 1 (max 0 t) - min 1 (max 0 q)| ≤
max |(1 : ℝ) - 1| |max 0 t - max 0 q| :=
abs_min_sub_min_le_max _ _ _ _
_ ≤ |t - q| := by
apply max_le
· simp
· exact (abs_max_sub_max_le_max (0 : ℝ) t 0 q).trans (by simp)
theorem clampOutcome_ae_eq
clamp outcome almost everywhere satisfies the stated identity for the specified J input, the specified P input, the specified beta input, the specified kappa input, the specified L input, the specified cminus input, the specified cplus input, the specified pmin input, the specified hmodel input.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
Proof (Lean source)
lemma clampOutcome_ae_eq {J : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin) :
(fun o : ClampObs J => clampUnit o.Y) =ᵐ[P.dataMeasure] fun o => o.Y := by
filter_upwards [hmodel.outcomeSupport] with o ho
simp [clampUnit, ho.1, ho.2]
theorem iidProduct_outcomeSupport
the stated i.i.d. product outcome support property holds for the specified J input, the specified n input, the specified P input, the specified beta input, the specified kappa input, the specified L input, the specified cminus input, the specified cplus input, the specified pmin input, the specified hmodel input.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
Proof (Lean source)
lemma iidProduct_outcomeSupport {J n : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin) :
∀ᵐ z ∂iidProduct P n, ∀ i : Fin n, (z i).Y ∈ Icc (0 : ℝ) 1 := by
let _ := hmodel.probability
letI : IsProbabilityMeasure (iidProduct P n) := by
unfold iidProduct
infer_instance
apply ae_all_iff.mpr
intro i
have hmap : (iidProduct P n).map (Function.eval i) = P.dataMeasure := by
simpa [iidProduct] using
(Measure.pi_map_eval (fun _ : Fin n => P.dataMeasure) i)
have hs := hmodel.outcomeSupport
rw [← hmap] at hs
have hYmeas : Measurable (fun o : ClampObs J => o.Y) := clampOutcome_measurable
exact (ae_map_iff (μ := iidProduct P n) (f := Function.eval i)
(p := fun o : ClampObs J => o.Y ∈ Icc (0 : ℝ) 1)
(measurable_pi_apply i).aemeasurable
(measurableSet_Icc.preimage hYmeas)).mp hs
theorem clampOutcome_condExp
the stated clamp outcome cond exp property holds for the specified J input, the specified P input, the specified beta input, the specified kappa input, the specified L input, the specified cminus input, the specified cplus input, the specified pmin input, the specified hmodel input, the specified x₀ input.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
x₀ :
Fin J
conclusion
P.dataMeasure[(fun o : ClampObs J => clampUnit o.Y) | comap (fun o : ClampObs J => (o.X, o.A)) inferInstance]
=ᵐ[P.dataMeasure] clampRegressionExtension P ∘ fun o => (o.X, o.A)
Proof (Lean source)
lemma clampOutcome_condExp {J : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin) (x₀ : Fin J) :
P.dataMeasure[(fun o : ClampObs J => clampUnit o.Y) |
comap (fun o : ClampObs J => (o.X, o.A)) inferInstance] =ᵐ[P.dataMeasure]
clampRegressionExtension P ∘ fun o => (o.X, o.A) := by
exact (condExp_congr_ae (clampOutcome_ae_eq P hmodel)).trans
(clampRegression_condExp P hmodel x₀)
def clampDesignLift
A canonical observation with the specified design and dummy outcome.
theorem clampDesignLift_measurable
clamp design lift is measurable for the specified J input.
Formal statement
Proof (Lean source)
lemma clampDesignLift_measurable {J : ℕ} : Measurable (clampDesignLift (J := J)) := by
rw [measurable_iff_comap_le]
change comap clampDesignLift
(comap (fun o : ClampObs J => (o.X, o.A, o.Y)) inferInstance) ≤ _
rw [MeasurableSpace.comap_comp]
simpa [Function.comp_def, clampDesignLift] using
(measurable_fst.prodMk (measurable_snd.prodMk measurable_const)).comap_le
def localRegressionDesignWeight
The realized local-polynomial weight regarded as a function only of the full design vector.
Definition (Lean source)
parameters
def
localRegressionDesignWeight B x kappa cminus cplus delta h d i :
ℝ
given by
if i ∈ B.I2 then interceptWeight B (fun j => clampDesignLift (d j)) x ell kappa cminus cplus delta h i else 0
theorem localRegressionDesignWeight_measurable
local regression design weight is measurable for the specified J input, the specified n input, the specified ell input, the specified B input, the specified x input, the specified kappa input, the specified cminus input, the specified cplus input, the specified delta input, the specified h input.
Formal statement
hypotheses
conclusion
Measurable (localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h)
Proof (Lean source)
lemma localRegressionDesignWeight_measurable {J n ell : ℕ} (B : SplitBlocks n)
(x : Fin J) (kappa cminus cplus delta h : ℝ) :
Measurable (localRegressionDesignWeight (ell := ell) B x
kappa cminus cplus delta h) := by
classical
apply measurable_pi_lambda
intro i
by_cases hi : i ∈ B.I2
· simp only [localRegressionDesignWeight, hi, if_true]
apply (interceptWeight_measurable B x kappa cminus cplus delta h i).comp
exact measurable_pi_lambda _ fun j =>
clampDesignLift_measurable.comp (measurable_pi_apply j)
· simp [localRegressionDesignWeight, hi]
theorem localRegressionDesignWeight_apply
the stated local regression design weight apply property holds for the specified J input, the specified n input, the specified ell input, the specified B input, the specified z input, the specified x input, the specified kappa input, the specified cminus input, the specified cplus input, the specified delta input, the specified h input, the specified i input.
Formal statement
hypotheses
conclusion
localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h (designVector (fun o : ClampObs J => (o.X, o.A)) z) i
= if i ∈ B.I2 then interceptWeight B z x ell kappa cminus cplus delta h i else 0
Proof (Lean source)
lemma localRegressionDesignWeight_apply {J n ell : ℕ} (B : SplitBlocks n)
(z : Fin n → ClampObs J) (x : Fin J) (kappa cminus cplus delta h : ℝ)
(i : Fin n) :
localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h
(designVector (fun o : ClampObs J => (o.X, o.A)) z) i =
if i ∈ B.I2 then interceptWeight B z x ell kappa cminus cplus delta h i else 0 := by
classical
by_cases hi : i ∈ B.I2
· simp only [localRegressionDesignWeight, hi, if_true,
designVector]
unfold interceptWeight GoodGramEvent localCount localGram scaledDose
rfl
· simp [localRegressionDesignWeight, hi]
theorem weightedCenteredSum_localRegression
the stated weighted centered sum local regression property holds for the specified J input, the specified n input, the specified ell input, the specified B input, the specified P input, the specified z input, the specified x input, the specified kappa input, the specified cminus input, the specified cplus input, the specified delta input, the specified h input.
Formal statement
hypotheses
conclusion
weightedCenteredSum (fun o : ClampObs J => (o.X, o.A)) (fun o => o.Y) (clampRegressionExtension P) (localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h) z
= ∑ i ∈ B.I2, interceptWeight B z x ell kappa cminus cplus delta h i * ((z i).Y - clampRegressionExtension P ((z i).X, (z i).A))
Proof (Lean source)
lemma weightedCenteredSum_localRegression {J n ell : ℕ} (B : SplitBlocks n)
(P : ClampLaw J) (z : Fin n → ClampObs J) (x : Fin J)
(kappa cminus cplus delta h : ℝ) :
weightedCenteredSum
(fun o : ClampObs J => (o.X, o.A)) (fun o => o.Y)
(clampRegressionExtension P)
(localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h) z =
∑ i ∈ B.I2, interceptWeight B z x ell kappa cminus cplus delta h i *
((z i).Y - clampRegressionExtension P ((z i).X, (z i).A)) := by
classical
unfold weightedCenteredSum
simp_rw [localRegressionDesignWeight_apply (ell := ell)]
simp only [ite_mul, zero_mul]
rw [← Finset.sum_filter]
simp only [Finset.filter_mem_eq_inter]
rw [Finset.univ_inter]
theorem weightedCenteredSum_localRegression_clamp
the stated weighted centered sum local regression clamp property holds for the specified J input, the specified n input, the specified ell input, the specified B input, the specified P input, the specified z input, the specified x input, the specified kappa input, the specified cminus input, the specified cplus input, the specified delta input, the specified h input.
Formal statement
hypotheses
conclusion
weightedCenteredSum (fun o : ClampObs J => (o.X, o.A)) (fun o => clampUnit o.Y) (clampRegressionExtension P) (localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h) z
= ∑ i ∈ B.I2, interceptWeight B z x ell kappa cminus cplus delta h i * (clampUnit (z i).Y - clampRegressionExtension P ((z i).X, (z i).A))
Proof (Lean source)
lemma weightedCenteredSum_localRegression_clamp {J n ell : ℕ} (B : SplitBlocks n)
(P : ClampLaw J) (z : Fin n → ClampObs J) (x : Fin J)
(kappa cminus cplus delta h : ℝ) :
weightedCenteredSum
(fun o : ClampObs J => (o.X, o.A)) (fun o => clampUnit o.Y)
(clampRegressionExtension P)
(localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h) z =
∑ i ∈ B.I2, interceptWeight B z x ell kappa cminus cplus delta h i *
(clampUnit (z i).Y - clampRegressionExtension P ((z i).X, (z i).A)) := by
classical
unfold weightedCenteredSum
simp_rw [localRegressionDesignWeight_apply (ell := ell)]
simp only [ite_mul, zero_mul]
rw [← Finset.sum_filter]
simp only [Finset.filter_mem_eq_inter]
rw [Finset.univ_inter]
theorem localRegression_weight_energy_le
local regression weight energy satisfies the stated upper bound for the specified J input, the specified n input, the specified ell input, the specified B input, the specified z input, the specified x input, the specified kappa input, the specified cminus input, the specified cplus input, the specified delta input, the specified h input, the specified hh input, the specified hlambda input.
Formal statement
hypotheses
J n ell :
ℕ
B :
x :
Fin J
kappa cminus cplus delta h :
ℝ
hh :
0 < h
hlambda :
0 < lambdaStar ell kappa cminus cplus
conclusion
∑ i, (localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h (designVector (fun o : ClampObs J => (o.X, o.A)) z) i) ^ 2
≤ 4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2
Proof (Lean source)
lemma localRegression_weight_energy_le {J n ell : ℕ} (B : SplitBlocks n)
(z : Fin n → ClampObs J) (x : Fin J) (kappa cminus cplus delta h : ℝ)
(hh : 0 < h) (hlambda : 0 < lambdaStar ell kappa cminus cplus) :
∑ i, (localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h
(designVector
(fun o : ClampObs J => (o.X, o.A)) z) i) ^ 2 ≤
4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2 := by
classical
simp_rw [localRegressionDesignWeight_apply (ell := ell), ite_pow, zero_pow (by norm_num : 2 ≠ 0)]
rw [← Finset.sum_filter]
simp only [Finset.filter_mem_eq_inter, Finset.univ_inter]
by_cases hg : GoodGramEvent B z x ell kappa cminus cplus delta h
· have hc := (goodGram_weight_controls B z x kappa cminus cplus delta h
hh hlambda hg).2.2
have hN : (1 : ℝ) ≤ localCount B z x delta h := by exact_mod_cast hg.1
calc
_ ≤ (4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2) /
(localCount B z x delta h : ℝ) := hc
_ ≤ 4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2 := by
exact div_le_self (by positivity) hN
· have hwzero (i : Fin n) :
interceptWeight B z x ell kappa cminus cplus delta h i = 0 := by
unfold interceptWeight
simp [hg]
simp [hwzero]
positivity
theorem localRegression_weight_energy_integrable
local regression weight energy is integrable for the specified J input, the specified n input, the specified ell input, the specified B input, the specified P input, the specified x input, the specified kappa input, the specified cminus input, the specified cplus input, the specified delta input, the specified h input, the specified hh input, the specified hlambda input.
Formal statement
hypotheses
J n ell :
ℕ
B :
P :
ClampLaw J
IsProbabilityMeasure P.dataMeasure
x :
Fin J
kappa cminus cplus delta h :
ℝ
hh :
0 < h
hlambda :
0 < lambdaStar ell kappa cminus cplus
conclusion
Integrable (fun z : Fin n → ClampObs J => ∑ i, (localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h (designVector (fun o : ClampObs J => (o.X, o.A)) z) i) ^ 2) (iidProduct P n)
Proof (Lean source)
lemma localRegression_weight_energy_integrable {J n ell : ℕ} (B : SplitBlocks n)
(P : ClampLaw J) [IsProbabilityMeasure P.dataMeasure]
(x : Fin J) (kappa cminus cplus delta h : ℝ)
(hh : 0 < h) (hlambda : 0 < lambdaStar ell kappa cminus cplus) :
Integrable (fun z : Fin n → ClampObs J =>
∑ i, (localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h
(designVector
(fun o : ClampObs J => (o.X, o.A)) z) i) ^ 2) (iidProduct P n) := by
letI : IsProbabilityMeasure (iidProduct P n) := by
unfold iidProduct
infer_instance
let w := localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h
have hmeas : Measurable (fun z : Fin n → ClampObs J =>
∑ i, (w (designVector
(fun o : ClampObs J => (o.X, o.A)) z) i) ^ 2) := by
have hw : Measurable w := localRegressionDesignWeight_measurable B x
kappa cminus cplus delta h
have hd := measurable_designVector
(N := n) (fun o : ClampObs J => (o.X, o.A)) clampDesign_measurable
exact Finset.measurable_fun_sum _ fun i _ =>
((measurable_pi_apply i).comp (hw.comp hd)).pow_const 2
apply Integrable.of_bound hmeas.aestronglyMeasurable
(4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2)
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (sum_nonneg fun _ _ => sq_nonneg _)]
exact localRegression_weight_energy_le B z x kappa cminus cplus delta h hh hlambda
theorem finiteProduct_bernoulliCount_lower_tail
the stated finite product bernoulli count lower tail property holds for the specified N input, the specified X input, the specified P input, the specified q input, the specified hqmeas input, the specified hq01 input, the specified hN input, the specified p input, the specified hp input, the specified hmean input.
Formal statement
hypotheses
N :
ℕ
X :
Type*
P :
q :
X → ℝ
hqmeas :
hq01 :
∀ x, q x = 0 ∨ q x = 1
hN :
0 < N
p :
ℝ
hp :
0 < p
hmean :
∫ x, q x ∂P = p
Proof (Lean source)
lemma finiteProduct_bernoulliCount_lower_tail
{N : ℕ} {X : Type*} [MeasurableSpace X]
(P : Measure X) [IsProbabilityMeasure P] (q : X → ℝ)
(hqmeas : Measurable q) (hq01 : ∀ x, q x = 0 ∨ q x = 1)
(hN : 0 < N) (p : ℝ) (hp : 0 < p) (hmean : ∫ x, q x ∂P = p) :
(Measure.pi (fun _ : Fin N => P)).real
{z | ∑ i, q (z i) ≤ (N : ℝ) * p / 2} ≤
2 * exp (-((N : ℝ) * p) / 20) := by
letI : IsProbabilityMeasure (Measure.pi (fun _ : Fin N => P)) := inferInstance
have hqIcc : ∀ x, q x ∈ Icc (0 : ℝ) 1 := by
intro x
rcases hq01 x with hx | hx <;> simp [hx]
have hqint : Integrable q P := by
refine Integrable.of_bound hqmeas.aestronglyMeasurable 1 ?_
filter_upwards with x
rcases hq01 x with hx | hx <;> simp [hx]
have hp_le : p ≤ 1 := by
calc
p = ∫ x, q x ∂P := hmean.symm
_ ≤ ∫ _x, (1 : ℝ) ∂P := integral_mono_ae hqint (integrable_const 1)
(ae_of_all _ fun x => (hqIcc x).2)
_ = 1 := by simp
have henvelope : ∀ᵐ x ∂P, |q x - ∫ y, q y ∂P| ≤ 1 := by
filter_upwards with x
rw [hmean]
exact abs_le.2 ⟨by linarith [(hqIcc x).1], by linarith [(hqIcc x).2]⟩
have hvar : ∫ x, (q x - ∫ y, q y ∂P) ^ 2 ∂P ≤ p := by
rw [hmean]
have hq2int : Integrable (fun x => q x ^ 2) P := by
exact Integrable.of_bound (hqmeas.pow_const 2).aestronglyMeasurable 1
(ae_of_all _ fun x => by rcases hq01 x with hx | hx <;> simp [hx])
calc
∫ x, (q x - p) ^ 2 ∂P =
(∫ x, q x ^ 2 ∂P) - 2 * p * (∫ x, q x ∂P) + p ^ 2 := by
rw [show (fun x => (q x - p) ^ 2) =
fun x => q x ^ 2 - 2 * p * q x + p ^ 2 by funext x; ring]
integral_linearity
rw [integral_const]
simp
_ = p - p ^ 2 := by
rw [show (∫ x, q x ^ 2 ∂P) = p by
rw [← hmean]
apply integral_congr_ae
filter_upwards with x
rcases hq01 x with hx | hx <;> simp [hx]]
rw [hmean]
ring
_ ≤ p := by nlinarith [sq_nonneg p]
let g : Unit → X → ℝ := fun _ => q
let b : Unit → ℝ := fun _ => 1
let sigma2 : Unit → ℝ := fun _ => p
let eta : Unit → ℝ := fun _ => (N : ℝ) * p / 2
have hbern := iid_sum_bernstein_union_bound
P g (fun _ => hqmeas) (fun _ => hqint) b sigma2 eta
(fun _ => by simp [b]) (fun _ => hp.le) (fun _ => by dsimp [eta]; positivity)
hN (fun _ => by simpa [g] using henvelope) (fun _ => by simpa [g] using hvar)
calc
_ ≤ (Measure.pi (fun _ : Fin N => P)).real
{z | ∃ a : Unit, eta a ≤
|(∑ i, g a (z i)) - (N : ℝ) * ∫ x, g a x ∂P|} := by
refine measureReal_mono ?_ (measure_ne_top _ _)
intro z hz
refine ⟨(), ?_⟩
simp only [eta, g, hmean]
change (∑ i, q (z i)) ≤ (N : ℝ) * p / 2 at hz
have hdiff : (∑ i, q (z i)) - (N : ℝ) * p ≤ 0 := by
have hNp : 0 ≤ (N : ℝ) * p := by positivity
linarith
rw [abs_of_nonpos hdiff]
linarith
_ ≤ ∑ a : Unit, 2 * exp
(-(eta a) ^ 2 / (2 * (2 * (N : ℝ) * sigma2 a + b a * eta a))) := hbern
_ = 2 * exp (-((N : ℝ) * p) / 20) := by
simp only [Fintype.sum_unique, eta, sigma2, b]
have hNp : 0 < (N : ℝ) * p := mul_pos (by exact_mod_cast hN) hp
congr 2
field_simp
ring
theorem iid_block_product_law'
the stated i.i.d. block product law property holds for the specified J input, the specified n input, the specified P input, the specified hsampling input, the specified I input.
Formal statement
hypotheses
conclusion
= Measure.pi (fun _ : Fin I.card => P.dataMeasure)
Proof (Lean source)
lemma iid_block_product_law' {J n : ℕ} (P : ClampLaw J)
(hsampling : IidSampling P n) (I : Finset (Fin n)) :
(iidProduct P n).map (fun z => fun j : Fin I.card =>
z ((I.orderIsoOfFin rfl) j)) =
Measure.pi (fun _ : Fin I.card => P.dataMeasure) := by
let _ : IsProbabilityMeasure P.dataMeasure := hsampling.1
simpa [iidProduct] using block_reindexed_law_eq P.dataMeasure I
theorem localCount_lower_tail
the stated local count lower tail property holds for the specified J input, the specified n input, the specified P input, the specified hsampling input, the specified B input, the specified x input, the specified delta input, the specified h input, the specified p input, the specified hh input, the specified hp input, the specified hmean input, the specified hcard input.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
hsampling :
IidSampling P n
B :
x :
Fin J
delta h p :
ℝ
hh :
0 < h
hp :
0 < p
hmean :
∫ o, localWindowWeight x delta h o ∂P.dataMeasure = p
hcard :
0 < B.I2.card
conclusion
(iidProduct P n).real {z | (localCount B z x delta h : ℝ) ≤ (B.I2.card : ℝ) * p / 2}
≤ 2 * exp (-((B.I2.card : ℝ) * p) / 20)
Proof (Lean source)
lemma localCount_lower_tail {J n : ℕ} (P : ClampLaw J)
(hsampling : IidSampling P n) (B : SplitBlocks n) (x : Fin J)
(delta h p : ℝ) (hh : 0 < h) (hp : 0 < p)
(hmean : ∫ o, localWindowWeight x delta h o ∂P.dataMeasure = p)
(hcard : 0 < B.I2.card) :
(iidProduct P n).real {z |
(localCount B z x delta h : ℝ) ≤ (B.I2.card : ℝ) * p / 2} ≤
2 * exp (-((B.I2.card : ℝ) * p) / 20) := by
letI : IsProbabilityMeasure P.dataMeasure := hsampling.1
let F : (Fin n → ClampObs J) → (Fin B.I2.card → ClampObs J) :=
fun z j => z ((B.I2.orderIsoOfFin rfl) j)
let q := localWindowWeight x delta h
have hF : Measurable F := measurable_pi_lambda _ fun j => measurable_pi_apply _
have hqmeas : Measurable q := measurable_localWindowWeight x delta h
have hq01 : ∀ o, q o = 0 ∨ q o = 1 := by
intro o
unfold q localWindowWeight
split_ifs <;> simp
let E : Set (Fin B.I2.card → ClampObs J) :=
{w | ∑ j, q (w j) ≤ (B.I2.card : ℝ) * p / 2}
have hE : MeasurableSet E := measurableSet_le
(Finset.measurable_fun_sum _ fun j _ => hqmeas.comp (measurable_pi_apply j))
measurable_const
have hsum (z : Fin n → ClampObs J) :
(∑ j, q (F z j)) = (localCount B z x delta h : ℝ) := by
change (∑ j : Fin B.I2.card, q (z ((B.I2.orderIsoOfFin rfl) j))) = _
rw [show (∑ j : Fin B.I2.card, q (z ((B.I2.orderIsoOfFin rfl) j))) =
∑ i ∈ B.I2, q (z i) by
calc
_ = ∑ i : B.I2, q (z i) := Equiv.sum_comp
(B.I2.orderIsoOfFin rfl).toEquiv (fun i : B.I2 => q (z i))
_ = _ := (Finset.sum_subtype B.I2 (fun _ => Iff.rfl) (fun i => q (z i))).symm]
unfold q localWindowWeight localCount
simp_rw [scaledDose_mem_Icc_iff hh]
norm_cast
have hset : {z : Fin n → ClampObs J |
(localCount B z x delta h : ℝ) ≤ (B.I2.card : ℝ) * p / 2} = F ⁻¹' E := by
ext z
simp only [Set.mem_setOf_eq, Set.mem_preimage]
change (localCount B z x delta h : ℝ) ≤ _ ↔ (∑ j, q (F z j)) ≤ _
rw [hsum]
rw [hset, measureReal_def, ← Measure.map_apply hF hE,
iid_block_product_law' P hsampling B.I2]
exact finiteProduct_bernoulliCount_lower_tail P.dataMeasure q hqmeas hq01
hcard p hp hmean
theorem localRegression_weight_energy_integral_le
local regression weight energy integral satisfies the stated upper bound for the specified J input, the specified n input, the specified ell input, the specified P input, the specified hsampling input, the specified B input, the specified x input, the specified kappa input, the specified cminus input, the specified cplus input, the specified delta input, the specified h input, the specified p input, the specified hh input, the specified hlambda input, the specified hp input, the specified hcard input, the specified hmean input.
Formal statement
hypotheses
J n ell :
ℕ
P :
ClampLaw J
hsampling :
IidSampling P n
B :
x :
Fin J
kappa cminus cplus delta h p :
ℝ
hh :
0 < h
hlambda :
0 < lambdaStar ell kappa cminus cplus
hp :
0 < p
hcard :
0 < B.I2.card
hmean :
∫ o, localWindowWeight x delta h o ∂P.dataMeasure = p
conclusion
(∫ z, ∑ i, (localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h (designVector (fun o : ClampObs J => (o.X, o.A)) z) i) ^ 2 ∂iidProduct P n)
≤ 2 * (4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2) / ((B.I2.card : ℝ) * p)
+ 2 * (4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2) * exp (-((B.I2.card : ℝ) * p) / 20)
Proof (Lean source)
lemma localRegression_weight_energy_integral_le {J n ell : ℕ}
(P : ClampLaw J) (hsampling : IidSampling P n) (B : SplitBlocks n)
(x : Fin J) (kappa cminus cplus delta h p : ℝ)
(hh : 0 < h) (hlambda : 0 < lambdaStar ell kappa cminus cplus)
(hp : 0 < p) (hcard : 0 < B.I2.card)
(hmean : ∫ o, localWindowWeight x delta h o ∂P.dataMeasure = p) :
(∫ z, ∑ i,
(localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h
(designVector
(fun o : ClampObs J => (o.X, o.A)) z) i) ^ 2 ∂iidProduct P n) ≤
2 * (4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2) /
((B.I2.card : ℝ) * p) +
2 * (4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2) *
exp (-((B.I2.card : ℝ) * p) / 20) := by
letI : IsProbabilityMeasure P.dataMeasure := hsampling.1
letI : IsProbabilityMeasure (iidProduct P n) := by
unfold iidProduct
infer_instance
let K := 4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2
let mp := (B.I2.card : ℝ) * p
let energy : (Fin n → ClampObs J) → ℝ := fun z => ∑ i,
(localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h
(designVector
(fun o : ClampObs J => (o.X, o.A)) z) i) ^ 2
let low : Set (Fin n → ClampObs J) :=
{z | (localCount B z x delta h : ℝ) ≤ mp / 2}
have hK : 0 ≤ K := by dsimp [K]; positivity
have hmp : 0 < mp := by dsimp [mp]; positivity
have henergyInt : Integrable energy (iidProduct P n) := by
exact localRegression_weight_energy_integrable B P x kappa cminus cplus delta h
hh hlambda
have hlow : MeasurableSet low := by
exact measurableSet_le
((measurable_of_countable (fun m : ℕ => (m : ℝ))).comp
(localCount_measurable B x delta h)) measurable_const
have hpoint (z : Fin n → ClampObs J) :
energy z ≤ 2 * K / mp + K * low.indicator (fun _ => (1 : ℝ)) z := by
by_cases hzlow : z ∈ low
· have he := localRegression_weight_energy_le B z x kappa cminus cplus
delta h hh hlambda
simp only [Set.indicator_of_mem hzlow, mul_one]
have hterm : 0 ≤ 2 * K / mp := by positivity
exact he.trans (by dsimp [energy, K] at he ⊢; linarith)
· simp only [Set.indicator_of_notMem hzlow, mul_zero, add_zero]
by_cases hg : GoodGramEvent B z x ell kappa cminus cplus delta h
· have he := (goodGram_weight_controls B z x kappa cminus cplus delta h
hh hlambda hg).2.2
have hN : mp / 2 < (localCount B z x delta h : ℝ) := by
exact lt_of_not_ge hzlow
have hNpos : 0 < (localCount B z x delta h : ℝ) := by exact_mod_cast hg.1
calc
energy z = ∑ i ∈ B.I2,
(interceptWeight B z x ell kappa cminus cplus delta h i) ^ 2 := by
dsimp [energy]
simp_rw [localRegressionDesignWeight_apply (ell := ell), ite_pow,
zero_pow (by norm_num : 2 ≠ 0)]
rw [← Finset.sum_filter]
simp
_ ≤ K / (localCount B z x delta h : ℝ) := by simpa [K] using he
_ ≤ 2 * K / mp := by
rw [div_le_iff₀ hNpos, div_mul_eq_mul_div, le_div_iff₀ hmp]
have hmul := mul_le_mul_of_nonneg_left hN.le hK
nlinarith
· have hwzero (i : Fin n) :
localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h
(designVector
(fun o : ClampObs J => (o.X, o.A)) z) i = 0 := by
rw [localRegressionDesignWeight_apply (ell := ell)]
split_ifs
· unfold interceptWeight
simp [hg]
· rfl
simp [energy, hwzero]
positivity
have hrhsInt : Integrable (fun z => 2 * K / mp +
K * low.indicator (fun _ => (1 : ℝ)) z) (iidProduct P n) :=
(integrable_const _).add ((integrable_const 1).indicator hlow |>.const_mul K)
calc
∫ z, energy z ∂iidProduct P n ≤
∫ z, (2 * K / mp + K * low.indicator (fun _ => (1 : ℝ)) z)
∂iidProduct P n := integral_mono henergyInt hrhsInt hpoint
_ = 2 * K / mp + K * (iidProduct P n).real low := by
rw [integral_add, integral_const, integral_const_mul]
· rw [integral_indicator hlow, integral_const]
simp [measureReal_def, hlow]
· exact integrable_const _
· exact (integrable_const 1).indicator hlow |>.const_mul K
_ ≤ 2 * K / mp + K * (2 * exp (-mp / 20)) := by
gcongr
simpa [low, mp] using
localCount_lower_tail P hsampling B x delta h p hh hp hmean hcard
_ = 2 * K / mp + 2 * K * exp (-mp / 20) := by ring
_ = _ := by simp [K, mp]
theorem localRegression_centered_l1_le
local regression centered absolute-error satisfies the stated upper bound for the specified J input, the specified n input, the specified ell input, the specified P input, the specified beta input, the specified kappa input, the specified L input, the specified cminus input, the specified cplus input, the specified pmin input, the specified hmodel input, the specified hsampling input, the specified B input, the specified x input, the specified delta input, the specified h input, the specified p input, the specified hh input, the specified hlambda input, the specified hkappa input, the specified hcplus input, the specified hpmin input, the specified hp input, the specified hcard input, the specified hmean input.
Formal statement
hypotheses
J n ell :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hsampling :
IidSampling P n
B :
x :
Fin J
delta h p :
ℝ
hh :
0 < h
hlambda :
0 < lambdaStar ell kappa cminus cplus
hkappa :
0 ≤ kappa
hcplus :
0 ≤ cplus
hpmin :
0 < pmin
hp :
0 < p
hcard :
0 < B.I2.card
hmean :
∫ o, localWindowWeight x delta h o ∂P.dataMeasure = p
conclusion
(∫ z, |∑ i ∈ B.I2, interceptWeight B z x ell kappa cminus cplus delta h i * (clampUnit (z i).Y - clampRegressionExtension P ((z i).X, (z i).A))| ∂iidProduct P n)
≤ (1 / 2 : ℝ) * sqrt (2 * (4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2) / ((B.I2.card : ℝ) * p) + 2 * (4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2) * exp (-((B.I2.card : ℝ) * p) / 20))
Proof (Lean source)
lemma localRegression_centered_l1_le {J n ell : ℕ}
(P : ClampLaw J) {beta kappa L cminus cplus pmin : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hsampling : IidSampling P n) (B : SplitBlocks n) (x : Fin J)
(delta h p : ℝ) (hh : 0 < h)
(hlambda : 0 < lambdaStar ell kappa cminus cplus)
(hkappa : 0 ≤ kappa) (hcplus : 0 ≤ cplus) (hpmin : 0 < pmin)
(hp : 0 < p) (hcard : 0 < B.I2.card)
(hmean : ∫ o, localWindowWeight x delta h o ∂P.dataMeasure = p) :
(∫ z, |∑ i ∈ B.I2,
interceptWeight B z x ell kappa cminus cplus delta h i *
(clampUnit (z i).Y - clampRegressionExtension P ((z i).X, (z i).A))|
∂iidProduct P n) ≤
(1 / 2 : ℝ) * sqrt
(2 * (4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2) /
((B.I2.card : ℝ) * p) +
2 * (4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2) *
exp (-((B.I2.card : ℝ) * p) / 20)) := by
letI : IsProbabilityMeasure P.dataMeasure := hmodel.probability
letI : IsProbabilityMeasure (iidProduct P n) := by
unfold iidProduct
infer_instance
let design : ClampObs J → Fin J × ℝ := fun o => (o.X, o.A)
let Y : ClampObs J → ℝ := fun o => clampUnit o.Y
let w := localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h
let S := weightedCenteredSum design Y
(clampRegressionExtension P) w
let energy : (Fin n → ClampObs J) → ℝ := fun z => ∑ i,
(w (designVector design z) i) ^ 2
have hw : Measurable w := localRegressionDesignWeight_measurable B x
kappa cminus cplus delta h
have hd : Measurable design := clampDesign_measurable
have hY : Measurable Y := clampUnit_measurable'.comp clampOutcome_measurable
have hmD := clampRegressionExtension_measurable P hmodel.holder
have henergyInt : Integrable energy (iidProduct P n) := by
simpa [energy, w, design] using
localRegression_weight_energy_integrable B P x kappa cminus cplus delta h hh hlambda
have hSmeas : Measurable S := by
dsimp [S]
unfold weightedCenteredSum
exact Finset.measurable_fun_sum _ fun i _ =>
((measurable_pi_apply i).comp
(hw.comp (measurable_designVector design hd))).mul
((hY.comp (measurable_pi_apply i)).sub
(hmD.comp (hd.comp (measurable_pi_apply i))))
have hSsq : Integrable (fun z => (S z) ^ 2) (iidProduct P n) := by
refine Integrable.mono (henergyInt.const_mul (n : ℝ))
(hSmeas.pow_const 2).aestronglyMeasurable ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _), Real.norm_eq_abs,
abs_of_nonneg (mul_nonneg (Nat.cast_nonneg _) (sum_nonneg fun _ _ => sq_nonneg _))]
have hcs := Finset.sum_mul_sq_le_sq_mul_sq univ
(fun i => w (designVector design z) i)
(fun i => Y (z i) - clampRegressionExtension P (design (z i)))
have hr : ∑ i, (Y (z i) - clampRegressionExtension P (design (z i))) ^ 2 ≤
(n : ℝ) := by
calc
_ ≤ ∑ _i : Fin n, (1 : ℝ) := by
apply Finset.sum_le_sum
intro i hi
have hYi := clampUnit_mem_Icc (z i).Y
have hmi := clampRegressionExtension_mem_Icc P hmodel.holder (design (z i))
have hY0 : 0 ≤ Y (z i) := by simpa [Y] using hYi.1
have hY1 : Y (z i) ≤ 1 := by simpa [Y] using hYi.2
have hm0 : 0 ≤ clampRegressionExtension P (design (z i)) := hmi.1
have hm1 : clampRegressionExtension P (design (z i)) ≤ 1 := hmi.2
have hlo : -(1 : ℝ) ≤ Y (z i) - clampRegressionExtension P (design (z i)) := by
linarith
have hhi : Y (z i) - clampRegressionExtension P (design (z i)) ≤ 1 := by
linarith
nlinarith [mul_nonneg (sub_nonneg.mpr hlo) (sub_nonneg.mpr hhi)]
_ = n := by simp
calc
S z ^ 2 ≤ energy z * ∑ i,
(Y (z i) - clampRegressionExtension P (design (z i))) ^ 2 := hcs
_ ≤ energy z * n := mul_le_mul_of_nonneg_left hr
(sum_nonneg fun _ _ => sq_nonneg _)
_ = n * energy z := mul_comm _ _
have hbase := product_weighted_centered_l1_le
P.dataMeasure design hd Y hY (fun o => (clampUnit_mem_Icc o.Y).1)
(fun o => (clampUnit_mem_Icc o.Y).2) (clampRegressionExtension P) hmD
(clampOutcome_condExp P hmodel x) w hw henergyInt hSsq
rw [show (∫ z, |∑ i ∈ B.I2,
interceptWeight B z x ell kappa cminus cplus delta h i *
(clampUnit (z i).Y - clampRegressionExtension P ((z i).X, (z i).A))|
∂iidProduct P n) = ∫ z, |S z| ∂iidProduct P n by
apply integral_congr_ae
filter_upwards with z
congr 1
simpa [S, design, Y, w] using
(weightedCenteredSum_localRegression_clamp (ell := ell) B P z x
kappa cminus cplus delta h).symm]
refine hbase.trans (mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt ?_) (by norm_num))
exact localRegression_weight_energy_integral_le P hsampling B x
kappa cminus cplus delta h p hh hlambda hp hcard hmean
theorem effectiveSampleSize_eq_rpow
The information-balance identity expressed as the reciprocal effective sample size used by the local-regression variance bound. The result uses the hh condition, the hb condition. This is the stated conclusion.
Formal statement
hypotheses
n :
ℕ
beta kappa delta h :
ℝ
hh :
0 < h
hb :
(n : ℝ) * h ^ (2 * beta + 1) * (delta + h) ^ kappa = 1
conclusion
(n : ℝ) * h * (delta + h) ^ kappa = h ^ (-2 * beta)
Proof (Lean source)
lemma effectiveSampleSize_eq_rpow {n : ℕ} {beta kappa delta h : ℝ}
(hh : 0 < h)
(hb : (n : ℝ) * h ^ (2 * beta + 1) * (delta + h) ^ kappa = 1) :
(n : ℝ) * h * (delta + h) ^ kappa = h ^ (-2 * beta) := by
have hp := Real.rpow_add hh (2 * beta) 1
rw [Real.rpow_one] at hp
have hnhe : (n : ℝ) * (h ^ (2 * beta) * h) * (delta + h) ^ kappa = 1 := by
rw [← hp]
exact hb
have hpow : 0 < h ^ (2 * beta) := Real.rpow_pos_of_pos hh _
rw [show -2 * beta = -(2 * beta) by ring, Real.rpow_neg hh.le,
inv_eq_one_div]
apply (eq_div_iff hpow.ne').2
nlinarith
theorem exp_neg_scaled_le_inv
the stated exp neg scaled bound inv property holds for the specified t input, the specified ht input.
Formal statement
Proof (Lean source)
lemma exp_neg_scaled_le_inv {t : ℝ} (ht : 0 < t) :
exp (-t / 20) ≤ 20 / t := by
have hu : 0 < t / 20 := div_pos ht (by norm_num)
have he : t / 20 ≤ exp (t / 20) :=
le_trans (le_add_of_nonneg_right zero_le_one) (Real.add_one_le_exp (t / 20))
have hi : exp (-(t / 20)) ≤ (t / 20)⁻¹ := by
rw [Real.exp_neg]
exact (inv_le_inv₀ (Real.exp_pos _) hu).2 he
rw [inv_div] at hi
simpa [div_eq_mul_inv] using hi
theorem balancedEffectiveSample_sqrt_le
A deterministic conversion of the count-weighted noise expression to the balanced bandwidth rate. The constants include the split-block factor eight and the elementary exponential bound exp (-t/20) ≤ 20/t. The result uses the hn condition, the hh condition, the hD condition, the hK condition, the hm condition, the hp condition, the hb condition. This is the stated conclusion.
Formal statement
hypotheses
n :
ℕ
beta kappa delta h p m D K :
ℝ
hn :
0 < n
hh :
0 < h
hD :
0 ≤ D
hK :
0 < K
hm :
(n : ℝ) / 8 ≤ m
hp :
K * h * (delta + h) ^ kappa ≤ p
hb :
(n : ℝ) * h ^ (2 * beta + 1) * (delta + h) ^ kappa = 1
Proof (Lean source)
lemma balancedEffectiveSample_sqrt_le {n : ℕ}
{beta kappa delta h p m D K : ℝ}
(hn : 0 < n) (hh : 0 < h) (hD : 0 ≤ D) (hK : 0 < K)
(hm : (n : ℝ) / 8 ≤ m)
(hp : K * h * (delta + h) ^ kappa ≤ p)
(hb : (n : ℝ) * h ^ (2 * beta + 1) * (delta + h) ^ kappa = 1) :
sqrt (2 * D / (m * p) + 2 * D * exp (-(m * p) / 20)) ≤
sqrt (336 * D / K) * h ^ beta := by
have hnR : 0 < (n : ℝ) := by exact_mod_cast hn
have heff := effectiveSampleSize_eq_rpow hh hb
have hbase : 0 < K * h * (delta + h) ^ kappa := by
have heffpos : 0 < (n : ℝ) * h * (delta + h) ^ kappa := by
rw [heff]
exact Real.rpow_pos_of_pos hh _
have htail : 0 < (delta + h) ^ kappa := by
have heffpos' : 0 < (delta + h) ^ kappa * ((n : ℝ) * h) := by
simpa [mul_comm] using heffpos
exact pos_of_mul_pos_left heffpos' (mul_nonneg hnR.le hh.le)
positivity
have hp0 : 0 < p := hbase.trans_le hp
have hm0 : 0 < m := lt_of_lt_of_le (div_pos hnR (by norm_num)) hm
have ht0 : 0 < m * p := mul_pos hm0 hp0
have htLower : (K / 8) * h ^ (-2 * beta) ≤ m * p := by
calc
(K / 8) * h ^ (-2 * beta) =
((n : ℝ) / 8) * (K * h * (delta + h) ^ kappa) := by
rw [← heff]
ring
_ ≤ m * p := mul_le_mul hm hp (by positivity) (by positivity)
have hinv : (m * p)⁻¹ ≤ (8 / K) * h ^ (2 * beta) := by
have hi := (inv_le_inv₀ ht0
(by positivity : 0 < (K / 8) * h ^ (-2 * beta))).2 htLower
rw [show -2 * beta = -(2 * beta) by ring, Real.rpow_neg hh.le] at hi
calc
(m * p)⁻¹ ≤ ((K / 8) * (h ^ (2 * beta))⁻¹)⁻¹ := hi
_ = (8 / K) * h ^ (2 * beta) := by field_simp
have hexp : exp (-(m * p) / 20) ≤ 20 * (m * p)⁻¹ := by
have he := exp_neg_scaled_le_inv ht0
simpa [div_eq_mul_inv] using he
have hinside : 2 * D / (m * p) + 2 * D * exp (-(m * p) / 20) ≤
(336 * D / K) * h ^ (2 * beta) := by
rw [div_eq_mul_inv]
calc
2 * D * (m * p)⁻¹ + 2 * D * exp (-(m * p) / 20) ≤
2 * D * (m * p)⁻¹ + 2 * D * (20 * (m * p)⁻¹) := by gcongr
_ = 42 * D * (m * p)⁻¹ := by ring
_ ≤ 42 * D * ((8 / K) * h ^ (2 * beta)) := by gcongr
_ = (336 * D / K) * h ^ (2 * beta) := by ring
calc
sqrt (2 * D / (m * p) + 2 * D * exp (-(m * p) / 20)) ≤
sqrt ((336 * D / K) * h ^ (2 * beta)) := Real.sqrt_le_sqrt hinside
_ = sqrt (336 * D / K) * sqrt (h ^ (2 * beta)) := by
rw [Real.sqrt_mul (by positivity : 0 ≤ 336 * D / K)]
_ = sqrt (336 * D / K) * h ^ beta := by
congr 1
rw [show 2 * beta = beta * 2 by ring, Real.rpow_mul hh.le, Real.rpow_two]
exact (Real.sqrt_sq_eq_abs (h ^ beta)).trans
(abs_of_nonneg (Real.rpow_nonneg hh.le _))
theorem localRegression_holderBias_le
On a good realized Gram event, the conditional mean part of the local polynomial estimator has the stated Hölder bias. The result uses the hmodel condition, the hbeta condition, the hL condition, the hdelta condition, the hh condition, the hupper condition, the hlambda condition, the hgood condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta h :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hbeta :
0 < beta
hL :
0 < L
B :
x :
Fin J
hdelta :
0 ≤ delta
hh :
0 < h
hupper :
delta + h ≤ 1
hlambda :
0 < lambdaStar (ellOf beta) kappa cminus cplus
hgood :
GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h
conclusion
|(∑ i ∈ B.I2, interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i * clampRegressionExtension P ((z i).X, (z i).A))
- P.mu x delta|
Proof (Lean source)
lemma localRegression_holderBias_le {J n : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin delta h : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hbeta : 0 < beta) (hL : 0 < L)
(B : SplitBlocks n) (z : Fin n → ClampObs J) (x : Fin J)
(hdelta : 0 ≤ delta) (hh : 0 < h) (hupper : delta + h ≤ 1)
(hlambda : 0 < lambdaStar (ellOf beta) kappa cminus cplus)
(hgood : GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h) :
|(∑ i ∈ B.I2, interceptWeight B z x (ellOf beta) kappa cminus cplus
delta h i * clampRegressionExtension P ((z i).X, (z i).A)) -
P.mu x delta| ≤
(2 * sqrt (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus) * L * h ^ beta := by
classical
let ell := ellOf beta
let w : Fin n → ℝ := fun i =>
interceptWeight B z x ell kappa cminus cplus delta h i
let T : ℝ → ℝ := fun a => ∑ j ∈ range (ell + 1),
(iteratedDerivWithin j (P.mu x) (Icc (0 : ℝ) 1) delta /
(Nat.factorial j : ℝ)) * (a - delta) ^ j
have hdeltaI : delta ∈ Icc (0 : ℝ) 1 := ⟨hdelta, by linarith⟩
have hrepro := (goodGram_weight_controls B z x kappa cminus cplus delta h
hh hlambda hgood).1
have hpoly : ∑ i ∈ B.I2, w i * T (z i).A = P.mu x delta := by
have hp := weightedPolynomial_reproduce B.I2 w (fun i => (z i).A)
(fun j => iteratedDerivWithin j (P.mu x) (Icc (0 : ℝ) 1) delta /
(Nat.factorial j : ℝ)) delta h hh hrepro
simpa [T, ell, w] using hp
have hrewrite :
(∑ i ∈ B.I2, w i * clampRegressionExtension P ((z i).X, (z i).A)) -
P.mu x delta =
∑ i ∈ B.I2, w i * (P.mu x (z i).A - T (z i).A) := by
rw [← hpoly, ← Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro i hi
by_cases ha : (z i).X = x ∧ scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1
· have hu := ha.2
have hAlo : delta ≤ (z i).A := by
have := (div_nonneg_iff.mp hu.1).resolve_right
(fun hn => (not_lt_of_ge hn.2 hh))
exact sub_nonneg.mp this.1
have hAhi : (z i).A ≤ delta + h := by
have := (div_le_iff₀ hh).mp hu.2
linarith
have hAI : (z i).A ∈ Icc (0 : ℝ) 1 :=
⟨hdelta.trans hAlo, hAhi.trans hupper⟩
rw [show (z i).X = x from ha.1, clampRegressionExtension_eq P x hAI]
simp only [w]
ring
· have hw0 : w i = 0 := by
dsimp [w, ell]
exact interceptWeight_eq_zero_of_inactive B z x (ellOf beta)
kappa cminus cplus delta h i hgood (fun hi => ha ⟨hi.2.1, hi.2.2⟩)
simp [hw0]
rw [hrewrite]
calc
|∑ i ∈ B.I2, w i * (P.mu x (z i).A - T (z i).A)| ≤
∑ i ∈ B.I2, |w i| * |P.mu x (z i).A - T (z i).A| := by
exact (Finset.abs_sum_le_sum_abs _ _).trans_eq (by
apply Finset.sum_congr rfl
intro i hi
rw [abs_mul])
_ ≤ ∑ i ∈ B.I2, |w i| * (L * h ^ beta) := by
apply Finset.sum_le_sum
intro i hi
by_cases ha : (z i).X = x ∧ scaledDose delta h (z i) ∈ Icc (0 : ℝ) 1
· apply mul_le_mul_of_nonneg_left _ (abs_nonneg _)
have hu := ha.2
have hAlo : delta ≤ (z i).A := by
have hs := (div_nonneg_iff.mp hu.1).resolve_right
(fun hn => (not_lt_of_ge hn.2 hh))
exact sub_nonneg.mp hs.1
have hAhi : (z i).A ≤ delta + h := by
have hs := (div_le_iff₀ hh).mp hu.2
linarith
have hAI : (z i).A ∈ Icc (0 : ℝ) 1 :=
⟨hdelta.trans hAlo, hAhi.trans hupper⟩
have hrem := (hmodel.holder x).2.2.2 delta hdeltaI (z i).A hAI
have habs : |(z i).A - delta| ≤ h := by
rw [abs_of_nonneg (sub_nonneg.mpr hAlo)]
linarith
have hpw := Real.rpow_le_rpow (abs_nonneg _) habs hbeta.le
have hrem' : |P.mu x (z i).A - T (z i).A| ≤
L * |(z i).A - delta| ^ beta := by
simpa [T, ell, div_eq_mul_inv, mul_comm, mul_left_comm, mul_assoc] using hrem
exact hrem'.trans (mul_le_mul_of_nonneg_left hpw hL.le)
· have hw0 : w i = 0 := by
dsimp [w, ell]
exact interceptWeight_eq_zero_of_inactive B z x (ellOf beta)
kappa cminus cplus delta h i hgood (fun hi => ha ⟨hi.2.1, hi.2.2⟩)
simp [hw0]
_ = (∑ i ∈ B.I2, |w i|) * (L * h ^ beta) := by
rw [Finset.sum_mul]
_ ≤ (2 * sqrt (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus) * L * h ^ beta := by
have hl1 := (goodGram_weight_controls B z x kappa cminus cplus delta h
hh hlambda hgood).2.1
have hfac : 0 ≤ L * h ^ beta := mul_nonneg hL.le (Real.rpow_nonneg hh.le _)
simpa [w, ell, mul_assoc] using mul_le_mul_of_nonneg_right hl1 hfac
theorem localRegression_good_error_le
Pointwise good-event decomposition of local-regression error into its centered stochastic sum and deterministic Hölder bias. The result uses the hmodel condition, the hbeta condition, the hL condition, the hdelta condition, the hh condition, the hupper condition, the hlambda condition, the hgood condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta h :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hbeta :
0 < beta
hL :
0 < L
B :
x :
Fin J
hdelta :
0 ≤ delta
hh :
0 < h
hupper :
delta + h ≤ 1
hlambda :
0 < lambdaStar (ellOf beta) kappa cminus cplus
hgood :
GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h
conclusion
|clampUnit (∑ i ∈ B.I2, interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i * clampUnit (z i).Y)
- P.mu x delta|
≤ |∑ i ∈ B.I2, interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i * (clampUnit (z i).Y - clampRegressionExtension P ((z i).X, (z i).A))|
Proof (Lean source)
lemma localRegression_good_error_le {J n : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin delta h : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hbeta : 0 < beta) (hL : 0 < L)
(B : SplitBlocks n) (z : Fin n → ClampObs J) (x : Fin J)
(hdelta : 0 ≤ delta) (hh : 0 < h) (hupper : delta + h ≤ 1)
(hlambda : 0 < lambdaStar (ellOf beta) kappa cminus cplus)
(hgood : GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h) :
|clampUnit (∑ i ∈ B.I2,
interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
clampUnit (z i).Y) - P.mu x delta| ≤
|∑ i ∈ B.I2,
interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
(clampUnit (z i).Y -
clampRegressionExtension P ((z i).X, (z i).A))| +
(2 * sqrt (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus) * L * h ^ beta := by
have hmu := (hmodel.holder x).2.1 delta ⟨hdelta, by linarith⟩
refine (abs_clampUnit_sub_le _ _ hmu).trans ?_
have htri := abs_add_le
(∑ i ∈ B.I2,
interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
(clampUnit (z i).Y -
clampRegressionExtension P ((z i).X, (z i).A)))
((∑ i ∈ B.I2,
interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
clampRegressionExtension P ((z i).X, (z i).A)) - P.mu x delta)
have hid :
(∑ i ∈ B.I2,
interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
clampUnit (z i).Y) - P.mu x delta =
(∑ i ∈ B.I2,
interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
(clampUnit (z i).Y -
clampRegressionExtension P ((z i).X, (z i).A))) +
((∑ i ∈ B.I2,
interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
clampRegressionExtension P ((z i).X, (z i).A)) - P.mu x delta) := by
simp_rw [mul_sub, Finset.sum_sub_distrib]
ring
rw [hid]
exact htri.trans (add_le_add_right
(localRegression_holderBias_le P hmodel hbeta hL B z x hdelta hh hupper
hlambda hgood) _)
theorem localRegression_centered_balanced_l1_le
The centered local-regression noise at the information-balanced bandwidth is uniformly of order h^beta. The result uses the hmodel condition, the hsampling condition, the hn condition, the hbeta condition, the hkappa condition, the hcminus condition, the hcplus condition, the hpmin condition, the hdelta condition, the hh condition, the hupper condition, the hlambda condition, the hbalance condition. This is the stated conclusion.
Formal statement
hypotheses
J n ell :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta h :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hsampling :
IidSampling P n
B :
x :
Fin J
hn :
8 ≤ n
hbeta :
0 < beta
hkappa :
0 ≤ kappa
hcminus :
0 < cminus
hcplus :
0 ≤ cplus
hpmin :
0 < pmin
hdelta :
0 ≤ delta
hh :
0 < h
hupper :
delta + h ≤ 1
hlambda :
0 < lambdaStar ell kappa cminus cplus
hbalance :
(n : ℝ) * h ^ (2 * beta + 1) * (delta + h) ^ kappa = 1
conclusion
(∫ z, |∑ i ∈ B.I2, interceptWeight B z x ell kappa cminus cplus delta h i * (clampUnit (z i).Y - clampRegressionExtension P ((z i).X, (z i).A))| ∂iidProduct P n)
≤ (1 / 2) * sqrt (336 * (4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2) / (pmin * cminus / (2 * (2 : ℝ) ^ kappa))) * h ^ beta
Proof (Lean source)
lemma localRegression_centered_balanced_l1_le {J n ell : ℕ}
(P : ClampLaw J) {beta kappa L cminus cplus pmin delta h : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hsampling : IidSampling P n) (B : SplitBlocks n) (x : Fin J)
(hn : 8 ≤ n) (hbeta : 0 < beta) (hkappa : 0 ≤ kappa)
(hcminus : 0 < cminus) (hcplus : 0 ≤ cplus) (hpmin : 0 < pmin)
(hdelta : 0 ≤ delta) (hh : 0 < h) (hupper : delta + h ≤ 1)
(hlambda : 0 < lambdaStar ell kappa cminus cplus)
(hbalance : (n : ℝ) * h ^ (2 * beta + 1) *
(delta + h) ^ kappa = 1) :
(∫ z, |∑ i ∈ B.I2,
interceptWeight B z x ell kappa cminus cplus delta h i *
(clampUnit (z i).Y -
clampRegressionExtension P ((z i).X, (z i).A))| ∂iidProduct P n) ≤
(1 / 2) * sqrt
(336 * (4 * (ell + 1 : ℝ) /
lambdaStar ell kappa cminus cplus ^ 2) /
(pmin * cminus / (2 * (2 : ℝ) ^ kappa))) * h ^ beta := by
let p := ∫ o, localWindowWeight x delta h o ∂P.dataMeasure
let D := 4 * (ell + 1 : ℝ) / lambdaStar ell kappa cminus cplus ^ 2
let K := pmin * cminus / (2 * (2 : ℝ) ^ kappa)
have hK : 0 < K := by dsimp [K]; positivity
have hpLower : K * h * (delta + h) ^ kappa ≤ p := by
have hl := localWindowWeight_integral_lower P hmodel hkappa hcminus hcplus
hpmin x hdelta hh hupper
have hdh : 0 < delta + h := by linarith
have hrpow : ((delta + h) / 2) ^ kappa =
(delta + h) ^ kappa / (2 : ℝ) ^ kappa := by
rw [Real.div_rpow hdh.le (by norm_num : (0 : ℝ) ≤ 2)]
rw [hrpow] at hl
dsimp [K, p]
convert hl using 1 <;> ring
have hp0 : 0 < p := lt_of_lt_of_le (by
exact mul_pos (mul_pos hK hh)
(Real.rpow_pos_of_pos (by linarith) _)) hpLower
have hcardNat : 0 < B.I2.card := lt_of_lt_of_le (by omega) B.card_I2
have hcard : (n : ℝ) / 8 ≤ (B.I2.card : ℝ) := by
have hqcard := B.card_I2
have hncard : n ≤ 8 * B.I2.card := by omega
have hncardR : (n : ℝ) ≤ 8 * (B.I2.card : ℝ) := by exact_mod_cast hncard
linarith
have hraw := localRegression_centered_l1_le P hmodel hsampling B x delta h p
hh hlambda hkappa hcplus hpmin hp0 hcardNat rfl
have hD : 0 ≤ D := by dsimp [D]; positivity
have hrate := balancedEffectiveSample_sqrt_le (D := D) (K := K)
(p := p) (m := (B.I2.card : ℝ)) (Nat.zero_lt_of_lt hn) hh hD hK
hcard hpLower hbalance
dsimp [D, K] at hrate
exact hraw.trans ((mul_le_mul_of_nonneg_left hrate
(by norm_num : (0 : ℝ) ≤ 1 / 2)).trans_eq (by ring))
theorem localRegression_risk_le_noise_bias_bad
Integrated Good/bad decomposition for one stratum's stabilized local regression. The only exceptional contribution is the real probability of a bad Gram event. The result uses the hmodel condition, the hbeta condition, the hL condition, the hdelta condition, the hh condition, the hupper condition, the hlambda condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta h :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hbeta :
0 < beta
hL :
0 < L
B :
x :
Fin J
hdelta :
0 ≤ delta
hh :
0 < h
hupper :
delta + h ≤ 1
hlambda :
0 < lambdaStar (ellOf beta) kappa cminus cplus
conclusion
(∫ z, |localRegressionEstimate B z x (ellOf beta) kappa cminus cplus delta h - P.mu x delta| ∂iidProduct P n)
≤ (∫ z, |∑ i ∈ B.I2, interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i * (clampUnit (z i).Y - clampRegressionExtension P ((z i).X, (z i).A))| ∂iidProduct P n)
Proof (Lean source)
lemma localRegression_risk_le_noise_bias_bad {J n : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin delta h : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hbeta : 0 < beta) (hL : 0 < L)
(B : SplitBlocks n) (x : Fin J)
(hdelta : 0 ≤ delta) (hh : 0 < h) (hupper : delta + h ≤ 1)
(hlambda : 0 < lambdaStar (ellOf beta) kappa cminus cplus) :
(∫ z, |localRegressionEstimate B z x (ellOf beta) kappa cminus cplus
delta h - P.mu x delta| ∂iidProduct P n) ≤
(∫ z, |∑ i ∈ B.I2,
interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
(clampUnit (z i).Y -
clampRegressionExtension P ((z i).X, (z i).A))| ∂iidProduct P n) +
(2 * sqrt (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus) * L * h ^ beta +
(iidProduct P n).real {z |
¬ GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h} := by
let _ := hmodel.probability
letI : IsProbabilityMeasure (iidProduct P n) := by
unfold iidProduct
infer_instance
let bad : Set (Fin n → ClampObs J) := {z |
¬ GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h}
let noise : (Fin n → ClampObs J) → ℝ := fun z => |∑ i ∈ B.I2,
interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
(clampUnit (z i).Y - clampRegressionExtension P ((z i).X, (z i).A))|
let bias := (2 * sqrt (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus) * L * h ^ beta
have hbad : MeasurableSet bad := by
exact (goodGramEvent_measurable B x kappa cminus cplus delta h).compl
have hnoiseMeas : Measurable noise := by
dsimp [noise]
apply Measurable.abs
refine Finset.measurable_fun_sum _ fun i _ => ?_
exact (interceptWeight_measurable B x kappa cminus cplus delta h i).mul
(((clampUnit_measurable'.comp
(clampOutcome_measurable.comp (measurable_pi_apply i))).sub
((clampRegressionExtension_measurable P hmodel.holder).comp
((clampDesign_measurable.comp (measurable_pi_apply i))))))
have hnoiseInt : Integrable noise (iidProduct P n) := by
let C := 2 * sqrt (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus
refine Integrable.of_bound hnoiseMeas.aestronglyMeasurable C ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (abs_nonneg _)]
by_cases hg : GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h
· calc
noise z ≤ ∑ i ∈ B.I2,
|interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i| := by
dsimp [noise]
calc
_ ≤ ∑ i ∈ B.I2,
|interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
(clampUnit (z i).Y -
clampRegressionExtension P ((z i).X, (z i).A))| :=
Finset.abs_sum_le_sum_abs _ _
_ ≤ _ := by
apply Finset.sum_le_sum
intro i hi
rw [abs_mul]
have hy := clampUnit_mem_Icc (z i).Y
have hm := clampRegressionExtension_mem_Icc P hmodel.holder ((z i).X, (z i).A)
have hr : |clampUnit (z i).Y -
clampRegressionExtension P ((z i).X, (z i).A)| ≤ 1 := by
rw [abs_le]
constructor <;> linarith [hy.1, hy.2, hm.1, hm.2]
simpa using mul_le_of_le_one_right (abs_nonneg _) hr
_ ≤ C := (goodGram_weight_controls B z x kappa cminus cplus delta h
hh hlambda hg).2.1
· have hw (i : Fin n) : interceptWeight B z x (ellOf beta) kappa cminus cplus
delta h i = 0 := by simp [interceptWeight, hg]
simp [noise, hw, C]
positivity
have herrMeas : Measurable fun z : Fin n → ClampObs J =>
|localRegressionEstimate B z x (ellOf beta) kappa cminus cplus delta h -
P.mu x delta| :=
((localRegressionEstimate_measurable B x kappa cminus cplus delta h).sub
measurable_const).abs
have herrInt : Integrable (fun z : Fin n → ClampObs J =>
|localRegressionEstimate B z x (ellOf beta) kappa cminus cplus delta h -
P.mu x delta|) (iidProduct P n) := by
refine Integrable.of_bound herrMeas.aestronglyMeasurable 1 ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (abs_nonneg _)]
have he : localRegressionEstimate B z x (ellOf beta) kappa cminus cplus
delta h ∈ Icc (0 : ℝ) 1 := by
unfold localRegressionEstimate
split_ifs
· exact clampUnit_mem_Icc _
· norm_num
have hm := (hmodel.holder x).2.1 delta ⟨hdelta, by linarith⟩
rw [abs_le]
constructor <;> linarith [he.1, he.2, hm.1, hm.2]
have hbias0 : 0 ≤ bias := by dsimp [bias]; positivity
have hrhsInt : Integrable (fun z => noise z + bias + bad.indicator (fun _ => (1 : ℝ)) z)
(iidProduct P n) := (hnoiseInt.add (integrable_const bias)).add
(Integrable.indicator (integrable_const 1) hbad)
have hae : ∀ᵐ z ∂iidProduct P n,
|localRegressionEstimate B z x (ellOf beta) kappa cminus cplus delta h -
P.mu x delta| ≤ noise z + bias + bad.indicator (fun _ => (1 : ℝ)) z := by
filter_upwards [iidProduct_outcomeSupport P hmodel] with z hz
by_cases hg : GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h
· have heqY (i : Fin n) : (z i).Y = clampUnit (z i).Y := by
simp [clampUnit, (hz i).1, (hz i).2]
simp only [localRegressionEstimate, hg, if_true]
have hb := localRegression_good_error_le P hmodel hbeta hL B z x
hdelta hh hupper hlambda hg
have hnotbad : z ∉ bad := by simpa [bad] using hg
rw [Set.indicator_of_notMem hnotbad, add_zero]
have hsumY : (∑ i ∈ B.I2,
interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i * (z i).Y) =
∑ i ∈ B.I2,
interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
clampUnit (z i).Y := by
apply Finset.sum_congr rfl
intro i hi
exact congrArg
(fun y => interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i * y)
(heqY i)
rw [hsumY]
change |clampUnit (∑ i ∈ B.I2,
interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
clampUnit (z i).Y) - P.mu x delta| ≤ noise z + bias
exact hb
· have he : |(1 / 2 : ℝ) - P.mu x delta| ≤ 1 := by
have hm := (hmodel.holder x).2.1 delta ⟨hdelta, by linarith⟩
rw [abs_le]
constructor <;> linarith [hm.1, hm.2]
have hn0 : 0 ≤ noise z := abs_nonneg _
have hmem : z ∈ bad := by simpa [bad] using hg
rw [Set.indicator_of_mem hmem]
simp only [localRegressionEstimate, hg, if_false]
exact he.trans (by linarith [hn0, hbias0])
have hint := integral_mono_ae herrInt hrhsInt hae
have hnbInt : Integrable (fun z => noise z + bias) (iidProduct P n) :=
hnoiseInt.add (integrable_const bias)
have hindInt : Integrable (bad.indicator (fun _ => (1 : ℝ))) (iidProduct P n) :=
Integrable.indicator (integrable_const 1) hbad
have houter :
(∫ z, noise z + bias + bad.indicator (fun _ => (1 : ℝ)) z ∂iidProduct P n) =
(∫ z, noise z + bias ∂iidProduct P n) +
∫ z, bad.indicator (fun _ => (1 : ℝ)) z ∂iidProduct P n := by
simpa only [Pi.add_apply] using integral_add hnbInt hindInt
have hinner : (∫ z, noise z + bias ∂iidProduct P n) =
(∫ z, noise z ∂iidProduct P n) + ∫ _z, bias ∂iidProduct P n := by
simpa only [Pi.add_apply] using integral_add hnoiseInt (integrable_const bias)
have hindval : (∫ z, bad.indicator (fun _ => (1 : ℝ)) z ∂iidProduct P n) =
(iidProduct P n).real bad := by
rw [integral_indicator hbad, integral_const]
simp [measureReal_def]
rw [houter, hinner, integral_const, hindval] at hint
simpa [noise, bias, bad, measureReal_def, add_assoc] using hint
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.UpperTotal 12 declarations
theorem clampFunctional_integral_eq
clamp functional integral satisfies the stated identity for the specified J input, the specified P input, the specified beta input, the specified kappa input, the specified L input, the specified cminus input, the specified cplus input, the specified pmin input, the specified delta input, the specified hmodel input, the specified hdelta input, the specified hdelta1 input.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hdelta :
0 ≤ delta
hdelta1 :
delta ≤ 1
conclusion
(∫ o, if delta < o.A then o.Y else P.mu o.X delta ∂P.dataMeasure)
= clampFunctional P delta
Proof (Lean source)
lemma clampFunctional_integral_eq {J : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin delta : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hdelta : 0 ≤ delta) (hdelta1 : delta ≤ 1) :
(∫ o, if delta < o.A then o.Y else P.mu o.X delta ∂P.dataMeasure) =
clampFunctional P delta := by
let _ := hmodel.probability
let atomTerm : Fin J → ClampObs J → ℝ := fun x o =>
if o.X = x ∧ o.A ≤ delta then P.mu x delta else 0
have hatomInt (x : Fin J) : Integrable (atomTerm x) P.dataMeasure := by
have hX : Measurable (fun o : ClampObs J => o.X) :=
measurable_fst.comp (Measurable.of_comap_le le_rfl)
have hA : Measurable (fun o : ClampObs J => o.A) :=
measurable_fst.comp (measurable_snd.comp (Measurable.of_comap_le le_rfl))
have hm : Measurable (atomTerm x) := by
dsimp [atomTerm]
apply Measurable.ite
· exact (measurableSet_eq_fun hX measurable_const).inter
(measurableSet_le hA measurable_const)
· exact measurable_const
· exact measurable_const
refine Integrable.of_bound hm.aestronglyMeasurable 1 ?_
filter_upwards with o
have hmu := (hmodel.holder x).2.1 delta
⟨hdelta, hdelta1⟩
dsimp [atomTerm]
split_ifs
· rw [abs_of_nonneg hmu.1]
exact hmu.2
· simp
have hpoint : ∀ᵐ o ∂P.dataMeasure,
(if delta < o.A then o.Y else P.mu o.X delta) =
o.Y * indicator {o : ClampObs J | delta < o.A} (fun _ => (1 : ℝ)) o +
∑ x : Fin J, atomTerm x o := by
filter_upwards [hmodel.treatmentSupport] with o ho
by_cases ha : delta < o.A
· have hnone (x : Fin J) : ¬(o.X = x ∧ o.A ≤ delta) := fun h => by linarith
simp [atomTerm, indicator, ha, hnone]
· have hle : o.A ≤ delta := le_of_not_gt ha
simp [atomTerm, indicator, ha, hle]
rw [integral_congr_ae hpoint]
rw [integral_add]
· rw [integral_finsetSum univ]
· unfold clampFunctional retainedMean
congr 1
apply Finset.sum_congr rfl
intro x hx
have hevent := atomEvent_integral_eq P hmodel x hdelta1
calc
(∫ o, atomTerm x o ∂P.dataMeasure) =
P.mu x delta * ∫ o,
(if o.X = x ∧ o.A ≤ delta then (1 : ℝ) else 0) ∂P.dataMeasure := by
rw [← integral_const_mul]
apply integral_congr_ae
filter_upwards with o
by_cases h : o.X = x ∧ o.A ≤ delta <;> simp [atomTerm, h]
_ = P.mu x delta * (P.px x * atomMass P x delta) := by rw [hevent]
_ = P.px x * atomMass P x delta * P.mu x delta := by ring
· exact fun x _ => hatomInt x
· exact Integrable.of_bound
((clampOutcome_measurable.mul
(Measurable.indicator measurable_const
(measurableSet_lt measurable_const
(measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))))))).aestronglyMeasurable 1
(by filter_upwards [hmodel.outcomeSupport] with o ho
by_cases ha : delta < o.A
· simp [indicator, ha, abs_of_nonneg ho.1, ho.2]
· simp [indicator, ha])
· exact integrable_finsetSum univ fun x _ => hatomInt x
theorem clampFunctional_mem_Icc
clamp functional lies in the stated closed interval for the specified J input, the specified P input, the specified beta input, the specified kappa input, the specified L input, the specified cminus input, the specified cplus input, the specified pmin input, the specified delta input, the specified hmodel input, the specified hdelta input, the specified hdelta1 input.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hdelta :
0 ≤ delta
hdelta1 :
delta ≤ 1
conclusion
clampFunctional P delta ∈ Icc (0 : ℝ) 1
Proof (Lean source)
lemma clampFunctional_mem_Icc {J : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin delta : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hdelta : 0 ≤ delta) (hdelta1 : delta ≤ 1) :
clampFunctional P delta ∈ Icc (0 : ℝ) 1 := by
let _ := hmodel.probability
let g : ClampObs J → ℝ := fun o => if delta < o.A then o.Y else P.mu o.X delta
have hmuMeas : Measurable (fun o : ClampObs J => P.mu o.X delta) := by
exact (measurable_of_countable (fun x : Fin J => P.mu x delta)).comp
(measurable_fst.comp (Measurable.of_comap_le le_rfl))
have hgmeas : Measurable g := by
dsimp [g]
exact Measurable.ite
(measurableSet_lt measurable_const
(measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))))
clampOutcome_measurable
hmuMeas
have hg0 : ∀ᵐ o ∂P.dataMeasure, 0 ≤ g o := by
filter_upwards [hmodel.outcomeSupport] with o ho
dsimp [g]
split_ifs
· exact ho.1
· exact ((hmodel.holder _).2.1 delta ⟨hdelta, hdelta1⟩).1
have hg1 : ∀ᵐ o ∂P.dataMeasure, g o ≤ 1 := by
filter_upwards [hmodel.outcomeSupport] with o ho
dsimp [g]
split_ifs
· exact ho.2
· exact ((hmodel.holder _).2.1 delta ⟨hdelta, hdelta1⟩).2
rw [← clampFunctional_integral_eq P hmodel hdelta hdelta1]
constructor
· exact integral_nonneg_of_ae hg0
· calc
(∫ o, g o ∂P.dataMeasure) ≤ ∫ _o, (1 : ℝ) ∂P.dataMeasure :=
integral_mono_ae
(Integrable.of_bound
hgmeas.aestronglyMeasurable 1
(hg0.and hg1 |>.mono fun o h => by
rw [Real.norm_eq_abs, abs_of_nonneg h.1]; exact h.2))
(integrable_const 1) hg1
_ = 1 := by simp
theorem localRegressionEstimate_mem_Icc
local regression estimate lies in the stated closed interval for the specified J input, the specified n input, the specified ell input, the specified B input, the specified z input, the specified x input, the specified kappa input, the specified cminus input, the specified cplus input, the specified delta input, the specified h input.
Formal statement
hypotheses
conclusion
localRegressionEstimate B z x ell kappa cminus cplus delta h ∈ Icc (0 : ℝ) 1
Proof (Lean source)
lemma localRegressionEstimate_mem_Icc {J n ell : ℕ}
(B : SplitBlocks n) (z : Fin n → ClampObs J) (x : Fin J)
(kappa cminus cplus delta h : ℝ) :
localRegressionEstimate B z x ell kappa cminus cplus delta h ∈
Icc (0 : ℝ) 1 := by
unfold localRegressionEstimate
split_ifs
· exact clampUnit_mem_Icc _
· norm_num
theorem blockAverage_mem_Icc
block average lies in the stated closed interval for the specified X input, the specified n input, the specified I input, the specified z input, the specified f input, the specified hf input.
Formal statement
Proof (Lean source)
lemma blockAverage_mem_Icc {X : Type*} {n : ℕ} (I : Finset (Fin n))
(z : Fin n → X) (f : X → ℝ) (hf : ∀ u, f u ∈ Icc (0 : ℝ) 1) :
blockAverage I z f ∈ Icc (0 : ℝ) 1 := by
by_cases hI : I.card = 0
· simp [blockAverage, hI]
have hcard : 0 < (I.card : ℝ) := by exact_mod_cast Nat.pos_of_ne_zero hI
have hsum0 : 0 ≤ ∑ i ∈ I, f (z i) :=
sum_nonneg fun i _ => (hf (z i)).1
have hsum1 : (∑ i ∈ I, f (z i)) ≤ (I.card : ℝ) := by
calc
(∑ i ∈ I, f (z i)) ≤ ∑ _i ∈ I, (1 : ℝ) :=
Finset.sum_le_sum fun i _ => (hf (z i)).2
_ = (I.card : ℝ) := by simp
unfold blockAverage
constructor
· positivity
· calc
(I.card : ℝ)⁻¹ * ∑ i ∈ I, f (z i) ≤
(I.card : ℝ)⁻¹ * (I.card : ℝ) :=
mul_le_mul_of_nonneg_left hsum1 (inv_nonneg.mpr hcard.le)
_ = 1 := inv_mul_cancel₀ (ne_of_gt hcard)
theorem atomEstimate_mem_Icc
atom estimate lies in the stated closed interval for the specified J input, the specified n input, the specified B input, the specified z input, the specified x input, the specified delta input.
Formal statement
hypotheses
conclusion
atomEstimate B z x delta ∈ Icc (0 : ℝ) 1
Proof (Lean source)
lemma atomEstimate_mem_Icc {J n : ℕ} (B : SplitBlocks n)
(z : Fin n → ClampObs J) (x : Fin J) (delta : ℝ) :
atomEstimate B z x delta ∈ Icc (0 : ℝ) 1 := by
apply blockAverage_mem_Icc
intro o
split_ifs <;> norm_num
theorem splitBlock_invSqrt_le_root
the stated split block inv sqrt bound root property holds for the specified n input, the specified m input, the specified hn input, the specified hm input.
Formal statement
hypotheses
n m :
ℕ
hn :
0 < n
hm :
n ≤ 8 * m
conclusion
sqrt (m : ℝ)⁻¹ ≤ 3 * (n : ℝ) ^ (-(1 : ℝ) / 2)
Proof (Lean source)
lemma splitBlock_invSqrt_le_root {n m : ℕ} (hn : 0 < n) (hm : n ≤ 8 * m) :
sqrt (m : ℝ)⁻¹ ≤ 3 * (n : ℝ) ^ (-(1 : ℝ) / 2) := by
have hnR : 0 < (n : ℝ) := by exact_mod_cast hn
have hmR : 0 < (m : ℝ) := by
have : 0 < m := by omega
exact_mod_cast this
have hInv : (m : ℝ)⁻¹ ≤ 9 * (n : ℝ)⁻¹ := by
have hR : (n : ℝ) ≤ 8 * (m : ℝ) := by exact_mod_cast hm
rw [show 9 * (n : ℝ)⁻¹ = 9 / (n : ℝ) by ring]
rw [le_div_iff₀ hnR]
rw [inv_mul_le_iff₀ hmR]
nlinarith
have hl0 : 0 ≤ sqrt (m : ℝ)⁻¹ := Real.sqrt_nonneg _
have hr0 : 0 ≤ 3 * (n : ℝ) ^ (-(1 : ℝ) / 2) := by positivity
rw [← sq_le_sq₀ hl0 hr0]
rw [Real.sq_sqrt (inv_nonneg.mpr (Nat.cast_nonneg m))]
calc
(m : ℝ)⁻¹ ≤ 9 * (n : ℝ)⁻¹ := hInv
_ = (3 * (n : ℝ) ^ (-(1 : ℝ) / 2)) ^ 2 := by
rw [mul_pow]
norm_num
rw [← Real.rpow_natCast]
rw [← Real.rpow_mul hnR.le]
norm_num
rw [Real.rpow_neg_one]
theorem balancedGramTail_le
balanced gram tail satisfies the stated upper bound for the specified n input, the specified beta input, the specified kappa input, the specified delta input, the specified h input, the specified C input, the specified c input, the specified hn input, the specified hbeta input, the specified hh input, the specified hh1 input, the specified hC input, the specified hc input, the specified hbalance input.
Formal statement
hypotheses
n :
ℕ
beta kappa delta h C c :
ℝ
hn :
0 < n
hbeta :
0 < beta
hh :
0 < h
hh1 :
h ≤ 1
hC :
0 ≤ C
hc :
0 < c
hbalance :
(n : ℝ) * h ^ (2 * beta + 1) * (delta + h) ^ kappa = 1
conclusion
C * exp (-c * (n : ℝ) * h * (delta + h) ^ kappa) ≤ (C / c) * h ^ beta
Proof (Lean source)
lemma balancedGramTail_le {n : ℕ} {beta kappa delta h C c : ℝ}
(hn : 0 < n) (hbeta : 0 < beta) (hh : 0 < h) (hh1 : h ≤ 1)
(hC : 0 ≤ C) (hc : 0 < c)
(hbalance : (n : ℝ) * h ^ (2 * beta + 1) * (delta + h) ^ kappa = 1) :
C * exp (-c * (n : ℝ) * h * (delta + h) ^ kappa) ≤
(C / c) * h ^ beta := by
have heff := effectiveSampleSize_eq_rpow hh hbalance
let e : ℝ := (n : ℝ) * h * (delta + h) ^ kappa
have hepos : 0 < e := by
rw [show e = h ^ (-2 * beta) by simpa [e] using heff]
exact Real.rpow_pos_of_pos hh _
let t : ℝ := 20 * c * e
have ht : 0 < t := by dsimp [t]; positivity
have hexp := exp_neg_scaled_le_inv ht
have harg : -t / 20 = -c * e := by dsimp [t]; ring
rw [harg] at hexp
have hinv : 20 / t = (1 / c) * h ^ (2 * beta) := by
have heq : e = h ^ (-2 * beta) := by simpa [e] using heff
change 20 / (20 * c * e) = _
rw [heq]
rw [show -2 * beta = -(2 * beta) by ring, Real.rpow_neg hh.le]
have hp : 0 < h ^ (2 * beta) := Real.rpow_pos_of_pos hh _
field_simp [hc.ne', hp.ne']
have hpow : h ^ (2 * beta) ≤ h ^ beta :=
Real.rpow_le_rpow_of_exponent_ge hh hh1 (by linarith)
calc
C * exp (-c * (n : ℝ) * h * (delta + h) ^ kappa) =
C * exp (-c * e) := by
congr 2
dsimp [e]
ring
_ ≤ C * (20 / t) := mul_le_mul_of_nonneg_left hexp hC
_ = (C / c) * h ^ (2 * beta) := by rw [hinv]; ring
_ ≤ (C / c) * h ^ beta := by
exact mul_le_mul_of_nonneg_left hpow (div_nonneg hC hc.le)
theorem atomCoefficient_mem_Icc
atom coefficient lies in the stated closed interval for the specified J input, the specified P input, the specified beta input, the specified kappa input, the specified L input, the specified cminus input, the specified cplus input, the specified pmin input, the specified delta input, the specified hmodel input, the specified x input, the specified hdelta1 input.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
x :
Fin J
hdelta1 :
delta ≤ 1
Proof (Lean source)
lemma atomCoefficient_mem_Icc {J : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin delta : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(x : Fin J) (hdelta1 : delta ≤ 1) :
P.px x * atomMass P x delta ∈ Icc (0 : ℝ) 1 := by
let _ := hmodel.probability
let f : ClampObs J → ℝ := fun o =>
if o.X = x ∧ o.A ≤ delta then 1 else 0
have hX : Measurable (fun o : ClampObs J => o.X) :=
measurable_fst.comp (Measurable.of_comap_le le_rfl)
have hA : Measurable (fun o : ClampObs J => o.A) :=
measurable_fst.comp (measurable_snd.comp (Measurable.of_comap_le le_rfl))
have hf : Measurable f := by
dsimp [f]
exact Measurable.ite
((measurableSet_eq_fun hX measurable_const).inter
(measurableSet_le hA measurable_const)) measurable_const measurable_const
have hf0 : ∀ o, 0 ≤ f o := by intro o; dsimp [f]; split_ifs <;> norm_num
have hf1 : ∀ o, f o ≤ 1 := by intro o; dsimp [f]; split_ifs <;> norm_num
have hfint : Integrable f P.dataMeasure := by
refine Integrable.of_bound hf.aestronglyMeasurable 1 ?_
filter_upwards with o
dsimp [f]
split_ifs <;> norm_num
have hint : ∫ o, f o ∂P.dataMeasure = P.px x * atomMass P x delta := by
simpa [f] using atomEvent_integral_eq P hmodel x hdelta1
rw [← hint]
constructor
· exact integral_nonneg_of_ae (ae_of_all _ hf0)
· calc
(∫ o, f o ∂P.dataMeasure) ≤ ∫ _o, (1 : ℝ) ∂P.dataMeasure :=
integral_mono_ae hfint (integrable_const 1) (ae_of_all _ hf1)
_ = 1 := by simp
theorem abs_atomCoefficient_le_envelope
the stated abs atom coefficient bound envelope property holds for the specified J input, the specified P input, the specified beta input, the specified kappa input, the specified L input, the specified cminus input, the specified cplus input, the specified pmin input, the specified deltaBar input, the specified alpha input, the specified delta input, the specified hmodel input, the specified hreg input, the specified x input, the specified hdelta input.
Formal statement
hypotheses
J :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin deltaBar alpha delta :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
x :
Fin J
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
conclusion
|P.px x * atomMass P x delta| ≤ cplus * delta ^ (kappa + 1) / (kappa + 1)
Proof (Lean source)
lemma abs_atomCoefficient_le_envelope
(J : ℕ) (P : ClampLaw J)
{beta kappa L cminus cplus pmin deltaBar alpha delta : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(x : Fin J) (hdelta : delta ∈ Icc (0 : ℝ) deltaBar) :
|P.px x * atomMass P x delta| ≤
cplus * delta ^ (kappa + 1) / (kappa + 1) := by
have hreg' := hreg
rcases hreg' with
⟨hJ, hbeta, hkappa, hL, hcminus, hcminus_le, hcplus,
hpmin, hpmin_le, hdeltaBar, hdeltaBar1, halpha, halpha1⟩
have hdelta1 : delta ≤ 1 := hdelta.2.trans hdeltaBar1.le
have hcoef := atomCoefficient_mem_Icc P hmodel x hdelta1
have hpush := clamp_pushforward_decomposition P beta kappa L cminus cplus
pmin deltaBar alpha hmodel hreg x delta hdelta
have hmass0 : 0 ≤ atomMass P x delta := by
have hlower0 : 0 ≤ cminus * delta ^ (kappa + 1) / (kappa + 1) :=
div_nonneg (mul_nonneg hcminus.le (Real.rpow_nonneg hdelta.1 _)) (by linarith)
exact hlower0.trans hpush.2.1
have hX : Measurable (fun o : ClampObs J => o.X) :=
measurable_fst.comp (Measurable.of_comap_le le_rfl)
let _ := hmodel.probability
letI : IsProbabilityMeasure (P.dataMeasure.map (fun o => o.X)) :=
Measure.isProbabilityMeasure_map hX.aemeasurable
have hpx0 : 0 ≤ P.px x := by
rw [(hmodel.stratumMass x).1]
exact measureReal_nonneg
have hpx1 : P.px x ≤ 1 := by
rw [(hmodel.stratumMass x).1]
exact measureReal_le_one
rw [abs_of_nonneg hcoef.1]
calc
P.px x * atomMass P x delta ≤ 1 * atomMass P x delta :=
mul_le_mul_of_nonneg_right hpx1 hmass0
_ ≤ _ := by simpa using hpush.2.2
theorem totalGramEstimator_error_le
Deterministic decomposition of the total estimator into the retained-block error, the atom-mass empirical errors, and the local-regression errors. The result uses the hmodel condition, the hdelta condition, the hdelta1 condition. This is the stated conclusion.
Formal statement
hypotheses
J n ell :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta h :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hdelta :
0 ≤ delta
hdelta1 :
delta ≤ 1
B :
conclusion
|totalGramEstimator B z ell kappa cminus cplus delta h - clampFunctional P delta|
≤ |retainedEstimate B z delta
- retainedMean P delta|
+ ∑ x : Fin J, (|atomEstimate B z x delta - P.px x * atomMass P x delta| + |P.px x * atomMass P x delta| * |localRegressionEstimate B z x ell kappa cminus cplus delta h - P.mu x delta|)
Proof (Lean source)
lemma totalGramEstimator_error_le {J n ell : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin delta h : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hdelta : 0 ≤ delta) (hdelta1 : delta ≤ 1)
(B : SplitBlocks n) (z : Fin n → ClampObs J) :
|totalGramEstimator B z ell kappa cminus cplus delta h -
clampFunctional P delta| ≤
|retainedEstimate B z delta - retainedMean P delta| +
∑ x : Fin J,
(|atomEstimate B z x delta - P.px x * atomMass P x delta| +
|P.px x * atomMass P x delta| *
|localRegressionEstimate B z x ell kappa cminus cplus delta h -
P.mu x delta|) := by
let center := retainedEstimate B z delta + ∑ x : Fin J,
atomEstimate B z x delta *
localRegressionEstimate B z x ell kappa cminus cplus delta h
have htarget := clampFunctional_mem_Icc P hmodel hdelta hdelta1
have hclamp := abs_clampUnit_sub_le center (clampFunctional P delta) htarget
have hprod (x : Fin J) :
|atomEstimate B z x delta *
localRegressionEstimate B z x ell kappa cminus cplus delta h -
P.px x * atomMass P x delta * P.mu x delta| ≤
|atomEstimate B z x delta - P.px x * atomMass P x delta| +
|P.px x * atomMass P x delta| *
|localRegressionEstimate B z x ell kappa cminus cplus delta h -
P.mu x delta| := by
let ah := atomEstimate B z x delta
let a := P.px x * atomMass P x delta
let mh := localRegressionEstimate B z x ell kappa cminus cplus delta h
let m := P.mu x delta
have hmh := localRegressionEstimate_mem_Icc (ell := ell)
B z x kappa cminus cplus delta h
have hid : ah * mh - a * m = (ah - a) * mh + a * (mh - m) := by ring
rw [hid]
calc
|(ah - a) * mh + a * (mh - m)| ≤
|(ah - a) * mh| + |a * (mh - m)| := abs_add_le _ _
_ = |ah - a| * |mh| + |a| * |mh - m| := by rw [abs_mul, abs_mul]
_ ≤ |ah - a| + |a| * |mh - m| := by
apply add_le_add_left
apply mul_le_of_le_one_right (abs_nonneg _)
rw [abs_of_nonneg hmh.1]
exact hmh.2
have hsum :
|∑ x : Fin J,
(atomEstimate B z x delta *
localRegressionEstimate B z x ell kappa cminus cplus delta h -
P.px x * atomMass P x delta * P.mu x delta)| ≤
∑ x : Fin J,
(|atomEstimate B z x delta - P.px x * atomMass P x delta| +
|P.px x * atomMass P x delta| *
|localRegressionEstimate B z x ell kappa cminus cplus delta h -
P.mu x delta|) := by
exact (Finset.abs_sum_le_sum_abs _ _).trans
(Finset.sum_le_sum fun x _ => hprod x)
calc
|totalGramEstimator B z ell kappa cminus cplus delta h -
clampFunctional P delta| ≤ |center - clampFunctional P delta| := by
simpa [totalGramEstimator, center] using hclamp
_ = |(retainedEstimate B z delta - retainedMean P delta) +
∑ x : Fin J,
(atomEstimate B z x delta *
localRegressionEstimate B z x ell kappa cminus cplus delta h -
P.px x * atomMass P x delta * P.mu x delta)| := by
congr 1
simp only [clampFunctional, Finset.sum_sub_distrib]
ring
_ ≤ |retainedEstimate B z delta - retainedMean P delta| +
|∑ x : Fin J,
(atomEstimate B z x delta *
localRegressionEstimate B z x ell kappa cminus cplus delta h -
P.px x * atomMass P x delta * P.mu x delta)| := abs_add_le _ _
_ ≤ _ := add_le_add_right hsum _
theorem totalGramEstimator_risk_decomposition
Integrated form of totalGramEstimator_error_le, with every finite-sample term exposed for the concentration and empirical-process bounds. The result uses the hmodel condition, the hdelta condition, the hdelta1 condition. This is the stated conclusion.
Formal statement
hypotheses
J n ell :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin delta h :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hdelta :
0 ≤ delta
hdelta1 :
delta ≤ 1
B :
conclusion
(∫ z, |totalGramEstimator B z ell kappa cminus cplus delta h - clampFunctional P delta| ∂iidProduct P n)
≤ (∫ z, |retainedEstimate B z delta - retainedMean P delta| ∂iidProduct P n)
+ ∑ x : Fin J, ((∫ z, |atomEstimate B z x delta - P.px x * atomMass P x delta| ∂iidProduct P n) + |P.px x * atomMass P x delta| * ∫ z, |localRegressionEstimate B z x ell kappa cminus cplus delta h - P.mu x delta| ∂iidProduct P n)
Proof (Lean source)
lemma totalGramEstimator_risk_decomposition {J n ell : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin delta h : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hdelta : 0 ≤ delta) (hdelta1 : delta ≤ 1)
(B : SplitBlocks n) :
(∫ z, |totalGramEstimator B z ell kappa cminus cplus delta h -
clampFunctional P delta| ∂iidProduct P n) ≤
(∫ z, |retainedEstimate B z delta - retainedMean P delta|
∂iidProduct P n) +
∑ x : Fin J,
((∫ z, |atomEstimate B z x delta - P.px x * atomMass P x delta|
∂iidProduct P n) +
|P.px x * atomMass P x delta| *
∫ z, |localRegressionEstimate B z x ell kappa cminus cplus delta h -
P.mu x delta| ∂iidProduct P n) := by
let _ := hmodel.probability
letI : IsProbabilityMeasure (iidProduct P n) := by
unfold iidProduct
infer_instance
let eret : (Fin n → ClampObs J) → ℝ := fun z =>
|retainedEstimate B z delta - retainedMean P delta|
let eatom : Fin J → (Fin n → ClampObs J) → ℝ := fun x z =>
|atomEstimate B z x delta - P.px x * atomMass P x delta|
let elocal : Fin J → (Fin n → ClampObs J) → ℝ := fun x z =>
|localRegressionEstimate B z x ell kappa cminus cplus delta h - P.mu x delta|
let lhs : (Fin n → ClampObs J) → ℝ := fun z =>
|totalGramEstimator B z ell kappa cminus cplus delta h - clampFunctional P delta|
have hretBase : Integrable (fun z : Fin n → ClampObs J =>
retainedEstimate B z delta) (iidProduct P n) := by
refine Integrable.of_bound
(retainedEstimate_measurable B delta).aestronglyMeasurable 1 ?_
filter_upwards [iidProduct_outcomeSupport P hmodel] with z hz
have hrange : retainedEstimate B z delta ∈ Icc (0 : ℝ) 1 := by
have hi := blockAverage_mem_Icc B.I0 (fun i : Fin n => i)
(fun i => if delta < (z i).A then (z i).Y else 0) (by
intro i
by_cases ha : delta < (z i).A
· simpa [ha] using hz i
· simp [ha])
simpa [retainedEstimate, blockAverage] using hi
rw [Real.norm_eq_abs, abs_of_nonneg hrange.1]
exact hrange.2
have hretInt : Integrable eret (iidProduct P n) := by
exact (hretBase.sub (integrable_const (retainedMean P delta))).abs
have hatomInt (x : Fin J) : Integrable (eatom x) (iidProduct P n) := by
apply Integrable.abs
apply Integrable.sub
· refine Integrable.of_bound
(atomEstimate_measurable B x delta).aestronglyMeasurable 1 ?_
filter_upwards with z
have hr := atomEstimate_mem_Icc B z x delta
rw [Real.norm_eq_abs, abs_of_nonneg hr.1]
exact hr.2
· exact integrable_const _
have hlocalInt (x : Fin J) : Integrable (elocal x) (iidProduct P n) := by
apply Integrable.abs
apply Integrable.sub
· refine Integrable.of_bound
(localRegressionEstimate_measurable B x kappa cminus cplus delta h).aestronglyMeasurable
1 ?_
filter_upwards with z
have hr := localRegressionEstimate_mem_Icc (ell := ell)
B z x kappa cminus cplus delta h
rw [Real.norm_eq_abs, abs_of_nonneg hr.1]
exact hr.2
· exact integrable_const _
have hlhsInt : Integrable lhs (iidProduct P n) := by
apply Integrable.abs
exact ((Integrable.of_bound
(totalGramEstimator_measurable B kappa cminus cplus delta h).aestronglyMeasurable
1 (by
filter_upwards with z
have hr := clampUnit_mem_Icc
(retainedEstimate B z delta + ∑ x : Fin J,
atomEstimate B z x delta *
localRegressionEstimate B z x ell kappa cminus cplus delta h)
rw [show totalGramEstimator B z ell kappa cminus cplus delta h =
clampUnit (retainedEstimate B z delta + ∑ x : Fin J,
atomEstimate B z x delta *
localRegressionEstimate B z x ell kappa cminus cplus delta h) by rfl]
rw [Real.norm_eq_abs, abs_of_nonneg hr.1]
exact hr.2)).sub (integrable_const _))
have hrhsInt : Integrable (fun z => eret z + ∑ x : Fin J,
(eatom x z + |P.px x * atomMass P x delta| * elocal x z))
(iidProduct P n) :=
hretInt.add (integrable_finsetSum univ fun x _ =>
(hatomInt x).add ((hlocalInt x).const_mul _))
have hmono : (∫ z, lhs z ∂iidProduct P n) ≤
∫ z, (eret z + ∑ x : Fin J,
(eatom x z + |P.px x * atomMass P x delta| * elocal x z))
∂iidProduct P n := by
exact integral_mono_ae hlhsInt hrhsInt
(ae_of_all _ fun z => totalGramEstimator_error_le P hmodel hdelta hdelta1 B z)
calc
(∫ z, |totalGramEstimator B z ell kappa cminus cplus delta h -
clampFunctional P delta| ∂iidProduct P n) =
∫ z, lhs z ∂iidProduct P n := rfl
_ ≤ _ := hmono
_ = (∫ z, eret z ∂iidProduct P n) + ∫ z, ∑ x : Fin J,
(eatom x z + |P.px x * atomMass P x delta| * elocal x z)
∂iidProduct P n := integral_add hretInt
(integrable_finsetSum univ fun x _ =>
(hatomInt x).add ((hlocalInt x).const_mul _))
_ = (∫ z, eret z ∂iidProduct P n) + ∑ x : Fin J,
∫ z, (eatom x z + |P.px x * atomMass P x delta| * elocal x z)
∂iidProduct P n := by
rw [integral_finsetSum univ]
exact fun x _ => (hatomInt x).add ((hlocalInt x).const_mul _)
_ = _ := by
congr 1
apply Finset.sum_congr rfl
intro x hx
rw [integral_add (hatomInt x) ((hlocalInt x).const_mul _), integral_const_mul]
theorem totalGramEstimator_risk_le_explicit
Explicit finite-sample risk bound at an information-balanced bandwidth, before the elementary eventual-rate simplifications. The result uses the hmodel condition, the hreg condition, the hsampling condition, the hn condition, the hdelta condition, the hh condition, the hupper condition, the hbalance condition, the hlambda condition, the htail condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin deltaBar alpha delta h Ctail ctail :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
hsampling :
IidSampling P n
B :
hn :
8 ≤ n
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
hh :
0 < h
hupper :
delta + h ≤ 1
hbalance :
(n : ℝ) * h ^ (2 * beta + 1) * (delta + h) ^ kappa = 1
hlambda :
0 < lambdaStar (ellOf beta) kappa cminus cplus
conclusion
(∫ z, |totalGramEstimator B z (ellOf beta) kappa cminus cplus delta h - clampFunctional P delta| ∂iidProduct P n)
≤ (1 / 2 : ℝ) * sqrt (B.I0.card : ℝ)⁻¹
+ (J : ℝ) * ((1 / 2 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹ + (cplus * delta ^ (kappa + 1) / (kappa + 1)) * ((1 / 2 : ℝ) * sqrt (336 * (4 * (ellOf beta + 1 : ℝ) / lambdaStar (ellOf beta) kappa cminus cplus ^ 2) / (pmin * cminus / (2 * (2 : ℝ) ^ kappa))) * h ^ beta + (2 * sqrt (ellOf beta + 1 : ℝ) / lambdaStar (ellOf beta) kappa cminus cplus) * L * h ^ beta + Ctail * exp (-ctail * (n : ℝ) * h * (delta + h) ^ kappa)))
Proof (Lean source)
lemma totalGramEstimator_risk_le_explicit
{J n : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin deltaBar alpha delta h Ctail ctail : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(hsampling : IidSampling P n) (B : SplitBlocks n)
(hn : 8 ≤ n) (hdelta : delta ∈ Icc (0 : ℝ) deltaBar)
(hh : 0 < h) (hupper : delta + h ≤ 1)
(hbalance : (n : ℝ) * h ^ (2 * beta + 1) * (delta + h) ^ kappa = 1)
(hlambda : 0 < lambdaStar (ellOf beta) kappa cminus cplus)
(htail : ∀ x : Fin J,
(iidProduct P n).real {z |
¬ GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h} ≤
Ctail * exp (-ctail * (n : ℝ) * h * (delta + h) ^ kappa)) :
(∫ z, |totalGramEstimator B z (ellOf beta) kappa cminus cplus delta h -
clampFunctional P delta| ∂iidProduct P n) ≤
(1 / 2 : ℝ) * sqrt (B.I0.card : ℝ)⁻¹ +
(J : ℝ) * ((1 / 2 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹ +
(cplus * delta ^ (kappa + 1) / (kappa + 1)) *
((1 / 2 : ℝ) * sqrt
(336 * (4 * (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus ^ 2) /
(pmin * cminus / (2 * (2 : ℝ) ^ kappa))) * h ^ beta +
(2 * sqrt (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus) * L * h ^ beta +
Ctail * exp (-ctail * (n : ℝ) * h *
(delta + h) ^ kappa))) := by
rcases hreg with
⟨hJ, hbeta, hkappa, hL, hcminus, hcminus_le, hcplus,
hpmin, hpmin_le, hdeltaBar, hdeltaBar1, halpha, halpha1⟩
have hregFull : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha :=
⟨hJ, hbeta, hkappa, hL, hcminus, hcminus_le, hcplus,
hpmin, hpmin_le, hdeltaBar, hdeltaBar1, halpha, halpha1⟩
have hdelta1 : delta ≤ 1 := hdelta.2.trans hdeltaBar1.le
have hcard0 : 0 < B.I0.card := lt_of_lt_of_le (by omega) B.card_I0
have hcard1 : 0 < B.I1.card := lt_of_lt_of_le (by omega) B.card_I1
let noiseC : ℝ := (1 / 2 : ℝ) * sqrt
(336 * (4 * (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus ^ 2) /
(pmin * cminus / (2 * (2 : ℝ) ^ kappa)))
let biasC : ℝ := 2 * sqrt (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus
let tail : ℝ := Ctail * exp
(-ctail * (n : ℝ) * h * (delta + h) ^ kappa)
let localBound : ℝ := noiseC * h ^ beta + biasC * L * h ^ beta + tail
let atomBound : ℝ := cplus * delta ^ (kappa + 1) / (kappa + 1)
have hnoise (x : Fin J) :
(∫ z, |∑ i ∈ B.I2,
interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
(clampUnit (z i).Y -
clampRegressionExtension P ((z i).X, (z i).A))| ∂iidProduct P n) ≤
noiseC * h ^ beta := by
simpa [noiseC] using localRegression_centered_balanced_l1_le
(ell := ellOf beta) P hmodel hsampling B x hn hbeta hkappa hcminus
(by linarith) hpmin hdelta.1 hh hupper hlambda hbalance
have hlocal (x : Fin J) :
(∫ z, |localRegressionEstimate B z x (ellOf beta) kappa cminus cplus
delta h - P.mu x delta| ∂iidProduct P n) ≤ localBound := by
have hd := localRegression_risk_le_noise_bias_bad P hmodel hbeta hL B x
hdelta.1 hh hupper hlambda
calc
_ ≤
(∫ z, |∑ i ∈ B.I2,
interceptWeight B z x (ellOf beta) kappa cminus cplus delta h i *
(clampUnit (z i).Y -
clampRegressionExtension P ((z i).X, (z i).A))| ∂iidProduct P n) +
biasC * L * h ^ beta +
(iidProduct P n).real {z |
¬ GoodGramEvent B z x (ellOf beta) kappa cminus cplus delta h} := by
simpa [biasC] using hd
_ ≤ noiseC * h ^ beta + biasC * L * h ^ beta + tail := by
exact add_le_add (add_le_add (hnoise x) le_rfl) (htail x)
_ = localBound := rfl
have hatomBound0 : 0 ≤ atomBound := by
dsimp [atomBound]
exact div_nonneg (mul_nonneg (by linarith) (Real.rpow_nonneg hdelta.1 _)) (by linarith)
have hdec := totalGramEstimator_risk_decomposition (ell := ellOf beta) (h := h)
P hmodel hdelta.1 hdelta1 B
calc
_ ≤
(∫ z, |retainedEstimate B z delta - retainedMean P delta| ∂iidProduct P n) +
∑ x : Fin J,
((∫ z, |atomEstimate B z x delta - P.px x * atomMass P x delta|
∂iidProduct P n) +
|P.px x * atomMass P x delta| *
∫ z, |localRegressionEstimate B z x (ellOf beta) kappa cminus cplus
delta h - P.mu x delta| ∂iidProduct P n) := hdec
_ ≤ (1 / 2 : ℝ) * sqrt (B.I0.card : ℝ)⁻¹ +
∑ _x : Fin J,
((1 / 2 : ℝ) * sqrt (B.I1.card : ℝ)⁻¹ +
atomBound * localBound) := by
apply add_le_add
· exact retainedEstimate_l1_le P hmodel B hcard0
· apply Finset.sum_le_sum
intro x hx
apply add_le_add
· exact atomEstimate_l1_le P hmodel B hcard1 x hdelta.1 hdelta1
· exact mul_le_mul
(by simpa [atomBound] using
abs_atomCoefficient_le_envelope J P hmodel hregFull x hdelta)
(hlocal x) (integral_nonneg fun _ => abs_nonneg _) hatomBound0
_ = _ := by
simp [atomBound, localBound, noiseC, biasC, tail]
ring
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.Helpers.WeightedConcentration 7 declarations This module instantiates Causalean's finite-product conditional Hoeffding theorem.
Random-design weighted concentration on the good-Gram event
This module instantiates Causalean's finite-product conditional Hoeffding theorem. The local-polynomial weights are stabilized off the good-Gram event so their realized energy is everywhere positive; the resulting tail is then restricted back to the good event used by the atom-fallback interval.
theorem good_gram_positive_energy
On the good-Gram event the exact equivalent-kernel weights have strictly positive realized squared energy. The result uses the hlambda condition, the hgood condition. This is the stated conclusion.
Formal statement
hypotheses
J n ell :
ℕ
B :
x :
Fin J
kappa cminus cplus delta h :
ℝ
hlambda :
0 < lambdaStar ell kappa cminus cplus
hgood :
GoodGramEvent B z x ell kappa cminus cplus delta h
conclusion
0 < ∑ i, (localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h (designVector (fun o : ClampObs J => (o.X, o.A)) z) i) ^ 2
Proof (Lean source)
lemma good_gram_positive_energy {J n ell : ℕ}
(B : SplitBlocks n) (z : Fin n → ClampObs J) (x : Fin J)
(kappa cminus cplus delta h : ℝ)
(hlambda : 0 < lambdaStar ell kappa cminus cplus)
(hgood : GoodGramEvent B z x ell kappa cminus cplus delta h) :
0 < ∑ i, (localRegressionDesignWeight (ell := ell) B x
kappa cminus cplus delta h
(designVector
(fun o : ClampObs J => (o.X, o.A)) z) i) ^ 2 := by
classical
have hrep := goodGram_reproduction B z x kappa cminus cplus delta h
hlambda hgood (0 : Fin (ell + 1))
simp only [Fin.isValue, if_pos] at hrep
simp_rw [localRegressionDesignWeight_apply (ell := ell), ite_pow, zero_pow
(by norm_num : 2 ≠ 0)]
rw [← Finset.sum_filter]
simp only [Finset.filter_mem_eq_inter, Finset.univ_inter]
by_contra hnot
have hle : (∑ i ∈ B.I2,
(interceptWeight B z x ell kappa cminus cplus delta h i) ^ 2) ≤ 0 :=
le_of_not_gt hnot
have hsqzero (i : Fin n) (hi : i ∈ B.I2) :
(interceptWeight B z x ell kappa cminus cplus delta h i) ^ 2 = 0 := by
apply le_antisymm
· exact hle.trans' (Finset.single_le_sum
(fun j _ => sq_nonneg
(interceptWeight B z x ell kappa cminus cplus delta h j)) hi)
· exact sq_nonneg _
have hwzero (i : Fin n) (hi : i ∈ B.I2) :
interceptWeight B z x ell kappa cminus cplus delta h i = 0 :=
sq_eq_zero_iff.mp (hsqzero i hi)
rw [Finset.sum_eq_zero (fun i hi => by simp [hwzero i hi])] at hrep
norm_num at hrep
def stabilizedWeight
Complete-design-measurable local-polynomial weights, replaced off the good-Gram event by one fixed unit coordinate.
Definition (Lean source)
parameters
given by
by classical exact fun d i
=> if GoodGramEvent B (fun j => clampDesignLift (d j)) x ell kappa cminus cplus delta h then localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h d i else if i = i₀ then 1 else 0
theorem stabilizedWeight_measurable
stabilized weight is measurable for the specified J input, the specified n input, the specified ell input, the specified B input, the specified x input, the specified kappa input, the specified cminus input, the specified cplus input, the specified delta input, the specified h input, the specified i₀ input.
Formal statement
hypotheses
conclusion
Measurable (stabilizedWeight (ell := ell) B x kappa cminus cplus delta h i₀)
Proof (Lean source)
lemma stabilizedWeight_measurable {J n ell : ℕ} (B : SplitBlocks n)
(x : Fin J) (kappa cminus cplus delta h : ℝ) (i₀ : Fin n) :
Measurable (stabilizedWeight (ell := ell) B x
kappa cminus cplus delta h i₀) := by
classical
apply measurable_pi_lambda
intro i
refine Measurable.ite ?_ ?_ measurable_const
· apply (goodGramEvent_measurable B x kappa cminus cplus delta h).preimage
exact measurable_pi_lambda _ fun j =>
clampDesignLift_measurable.comp (measurable_pi_apply j)
· exact (measurable_pi_apply i).comp
(localRegressionDesignWeight_measurable (ell := ell) B x
kappa cminus cplus delta h)
theorem stabilizedWeight_eq_on_good
the stated stabilized weight identity on good property holds for the specified J input, the specified n input, the specified ell input, the specified B input, the specified z input, the specified x input, the specified kappa input, the specified cminus input, the specified cplus input, the specified delta input, the specified h input, the specified i₀ input, the specified hgood input.
Formal statement
hypotheses
J n ell :
ℕ
B :
x :
Fin J
kappa cminus cplus delta h :
ℝ
i₀ :
Fin n
hgood :
GoodGramEvent B z x ell kappa cminus cplus delta h
conclusion
stabilizedWeight (ell := ell) B x kappa cminus cplus delta h i₀ (designVector (fun o : ClampObs J => (o.X, o.A)) z)
= localRegressionDesignWeight (ell := ell) B x kappa cminus cplus delta h (designVector (fun o : ClampObs J => (o.X, o.A)) z)
Proof (Lean source)
lemma stabilizedWeight_eq_on_good {J n ell : ℕ} (B : SplitBlocks n)
(z : Fin n → ClampObs J) (x : Fin J)
(kappa cminus cplus delta h : ℝ) (i₀ : Fin n)
(hgood : GoodGramEvent B z x ell kappa cminus cplus delta h) :
stabilizedWeight (ell := ell) B x kappa cminus cplus delta h i₀
(designVector
(fun o : ClampObs J => (o.X, o.A)) z) =
localRegressionDesignWeight (ell := ell) B x
kappa cminus cplus delta h
(designVector
(fun o : ClampObs J => (o.X, o.A)) z) := by
funext i
have hgood' : GoodGramEvent B
(fun j => clampDesignLift
(designVector
(fun o : ClampObs J => (o.X, o.A)) z j)) x ell
kappa cminus cplus delta h := by
change GoodGramEvent B z x ell kappa cminus cplus delta h
exact hgood
simp [stabilizedWeight, hgood']
theorem stabilizedWeight_positive_energy
the stated stabilized weight positive energy property holds for the specified J input, the specified n input, the specified ell input, the specified B input, the specified z input, the specified x input, the specified kappa input, the specified cminus input, the specified cplus input, the specified delta input, the specified h input, the specified i₀ input, the specified hlambda input.
Formal statement
hypotheses
J n ell :
ℕ
B :
x :
Fin J
kappa cminus cplus delta h :
ℝ
i₀ :
Fin n
hlambda :
0 < lambdaStar ell kappa cminus cplus
conclusion
0 < realizedWeightEnergy (stabilizedWeight (ell := ell) B x kappa cminus cplus delta h i₀) (designVector (fun o : ClampObs J => (o.X, o.A)) z)
Proof (Lean source)
lemma stabilizedWeight_positive_energy {J n ell : ℕ} (B : SplitBlocks n)
(z : Fin n → ClampObs J) (x : Fin J)
(kappa cminus cplus delta h : ℝ) (i₀ : Fin n)
(hlambda : 0 < lambdaStar ell kappa cminus cplus) :
0 < Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.realizedWeightEnergy
(stabilizedWeight (ell := ell) B x kappa cminus cplus delta h i₀)
(designVector
(fun o : ClampObs J => (o.X, o.A)) z) := by
classical
unfold Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.realizedWeightEnergy
by_cases hgood : GoodGramEvent B z x ell kappa cminus cplus delta h
· rw [stabilizedWeight_eq_on_good B z x kappa cminus cplus delta h i₀ hgood]
exact good_gram_positive_energy B z x kappa cminus cplus delta h hlambda hgood
· have hone : stabilizedWeight (ell := ell) B x kappa cminus cplus delta h i₀
(designVector
(fun o : ClampObs J => (o.X, o.A)) z) i₀ = 1 := by
have hgood' : ¬ GoodGramEvent B
(fun j => clampDesignLift
(designVector
(fun o : ClampObs J => (o.X, o.A)) z j)) x ell
kappa cminus cplus delta h := by
change ¬ GoodGramEvent B z x ell kappa cminus cplus delta h
exact hgood
simp [stabilizedWeight, hgood']
have hsingle : (0 : ℝ) <
(stabilizedWeight (ell := ell) B x kappa cminus cplus delta h i₀
(designVector
(fun o : ClampObs J => (o.X, o.A)) z) i₀) ^ 2 := by
rw [hone]
norm_num
exact lt_of_lt_of_le hsingle
(Finset.single_le_sum (fun i _ => sq_nonneg
(stabilizedWeight (ell := ell) B x kappa cminus cplus delta h i₀
(designVector
(fun o : ClampObs J => (o.X, o.A)) z) i)) (Finset.mem_univ i₀))
theorem stabilized_weighted_tail
The finite-product random-design Hoeffding tail for the stabilized weights. This is the direct Causalean theorem at the paper's observed-sample abstraction. The result uses the hmodel condition, the hlambda condition, the ht condition. This is the stated conclusion.
Formal statement
hypotheses
J n ell :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
B :
x :
Fin J
delta h t :
ℝ
i₀ :
Fin n
hlambda :
0 < lambdaStar ell kappa cminus cplus
ht :
0 ≤ t
conclusion
(iidProduct P n).real {z | t * sqrt (realizedWeightEnergy (stabilizedWeight (ell := ell) B x kappa cminus cplus delta h i₀) (designVector (fun o : ClampObs J => (o.X, o.A)) z)) ≤ |weightedCenteredSum (fun o : ClampObs J => (o.X, o.A)) (fun o => o.Y) (clampRegressionExtension P) (stabilizedWeight (ell := ell) B x kappa cminus cplus delta h i₀) z|}
≤ 2 * exp (-2 * t ^ 2)
Proof (Lean source)
lemma stabilized_weighted_tail {J n ell : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(B : SplitBlocks n) (x : Fin J) (delta h t : ℝ) (i₀ : Fin n)
(hlambda : 0 < lambdaStar ell kappa cminus cplus) (ht : 0 ≤ t) :
(iidProduct P n).real {z |
t * sqrt
(realizedWeightEnergy
(stabilizedWeight (ell := ell) B x kappa cminus cplus delta h i₀)
(designVector
(fun o : ClampObs J => (o.X, o.A)) z)) ≤
|weightedCenteredSum
(fun o : ClampObs J => (o.X, o.A)) (fun o => o.Y)
(clampRegressionExtension P)
(stabilizedWeight (ell := ell) B x kappa cminus cplus delta h i₀) z|} ≤
2 * exp (-2 * t ^ 2) := by
letI : IsProbabilityMeasure P.dataMeasure := hmodel.probability
apply product_weighted_centered_tail_le
P.dataMeasure (fun o : ClampObs J => (o.X, o.A)) clampDesign_measurable
(fun o : ClampObs J => o.Y) clampOutcome_measurable
(hmodel.outcomeSupport.mono fun _ ho => ho.1)
(hmodel.outcomeSupport.mono fun _ ho => ho.2)
(clampRegressionExtension P) (clampRegressionExtension_measurable P hmodel.holder)
(clampRegression_condExp P hmodel x)
(stabilizedWeight (ell := ell) B x kappa cminus cplus delta h i₀)
(stabilizedWeight_measurable B x kappa cminus cplus delta h i₀)
(by
filter_upwards with z
exact stabilizedWeight_positive_energy B z x kappa cminus cplus delta h i₀ hlambda)
ht
theorem block_weighted_tail_on_good_gram
Restricting the stabilized tail back to the good-Gram event recovers the paper's exact local-polynomial weighted residual sum. The result uses the hmodel condition, the hlambda condition, the ht condition. This is the stated conclusion.
Formal statement
hypotheses
J n ell :
ℕ
P :
ClampLaw J
beta kappa L cminus cplus pmin :
ℝ
hmodel :
ClampModel P beta kappa L cminus cplus pmin
B :
x :
Fin J
delta h t :
ℝ
i₀ :
Fin n
hlambda :
0 < lambdaStar ell kappa cminus cplus
ht :
0 ≤ t
conclusion
(iidProduct P n).real {z | GoodGramEvent B z x ell kappa cminus cplus delta h ∧ t * sqrt (∑ i ∈ B.I2, (interceptWeight B z x ell kappa cminus cplus delta h i) ^ 2) ≤ |∑ i ∈ B.I2, interceptWeight B z x ell kappa cminus cplus delta h i * ((z i).Y - clampRegressionExtension P ((z i).X, (z i).A))|}
≤ 2 * exp (-2 * t ^ 2)
Proof (Lean source)
lemma block_weighted_tail_on_good_gram {J n ell : ℕ} (P : ClampLaw J)
{beta kappa L cminus cplus pmin : ℝ}
(hmodel : ClampModel P beta kappa L cminus cplus pmin)
(B : SplitBlocks n) (x : Fin J) (delta h t : ℝ) (i₀ : Fin n)
(hlambda : 0 < lambdaStar ell kappa cminus cplus) (ht : 0 ≤ t) :
(iidProduct P n).real {z |
GoodGramEvent B z x ell kappa cminus cplus delta h ∧
t * sqrt (∑ i ∈ B.I2,
(interceptWeight B z x ell kappa cminus cplus delta h i) ^ 2) ≤
|∑ i ∈ B.I2,
interceptWeight B z x ell kappa cminus cplus delta h i *
((z i).Y - clampRegressionExtension P ((z i).X, (z i).A))|} ≤
2 * exp (-2 * t ^ 2) := by
letI : IsProbabilityMeasure P.dataMeasure := hmodel.probability
letI : IsProbabilityMeasure (iidProduct P n) := by
unfold iidProduct
infer_instance
have htail := stabilized_weighted_tail P hmodel B x delta h t i₀ hlambda ht
refine (measureReal_mono (h₂ := measure_ne_top _ _) ?_).trans htail
intro z hz
rcases hz with ⟨hgood, hbound⟩
change t * sqrt
(realizedWeightEnergy
(stabilizedWeight (ell := ell) B x kappa cminus cplus delta h i₀)
(designVector
(fun o : ClampObs J => (o.X, o.A)) z)) ≤
|weightedCenteredSum
(fun o : ClampObs J => (o.X, o.A)) (fun o => o.Y)
(clampRegressionExtension P)
(stabilizedWeight (ell := ell) B x kappa cminus cplus delta h i₀) z|
have hw := stabilizedWeight_eq_on_good B z x kappa cminus cplus delta h i₀ hgood
have hcentered :
weightedCenteredSum
(fun o : ClampObs J => (o.X, o.A)) (fun o => o.Y)
(clampRegressionExtension P)
(stabilizedWeight (ell := ell) B x kappa cminus cplus delta h i₀) z =
weightedCenteredSum
(fun o : ClampObs J => (o.X, o.A)) (fun o => o.Y)
(clampRegressionExtension P)
(localRegressionDesignWeight (ell := ell) B x
kappa cminus cplus delta h) z := by
unfold weightedCenteredSum
rw [hw]
rw [hcentered, weightedCenteredSum_localRegression]
unfold realizedWeightEnergy
rw [hw]
simp_rw [localRegressionDesignWeight_apply (ell := ell), ite_pow, zero_pow
(by norm_num : 2 ≠ 0)]
rw [← Finset.sum_filter]
simpa only [Finset.filter_mem_eq_inter, Finset.univ_inter] using hbound
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.OpenQuestions 6 declarations The declaration is a never-proved proposition, not a theorem.
Open question: sharp confidence-length constant
The declaration is a never-proved proposition, not a theorem. It asks for one sharp constant over every threshold regime and attainment by intervals whose radius is built from the realized-design exact-modulus handle.
def worstExactModulus
Worst-case integrated exact-modulus radius over the model and strata.
Definition (Lean source)
parameters
def
worstExactModulus J n B beta kappa L cminus cplus pmin deltaBar delta alpha :
ℝ
given by
sSup {v : ℝ | ∃ (P : ClampLaw J) (x : Fin J), ∃ hmodel : ClampModel P beta kappa L cminus cplus pmin, ∃ hh : 0 < infoBandwidth n delta beta kappa deltaBar, v = exactModulusHandle P B x (ellOf beta) beta kappa L cminus cplus pmin delta (infoBandwidth n delta beta kappa deltaBar) (sqrt (log (12 * (J : ℝ) / alpha) / 2)) hh (Real.sqrt_nonneg _) hmodel}
def worstRealizedExactModulus
Worst conditional exact modulus on the realized design.
Definition (Lean source)
parameters
J n :
ℕ
B :
beta kappa L _cminus _cplus deltaBar delta alpha :
ℝ
def
worstRealizedExactModulus J n B beta kappa L _cminus _cplus deltaBar delta alpha z :
ℝ
given by
sSup {v : ℝ | ∃ x : Fin J, ∃ hh : 0 < infoBandwidth n delta beta kappa deltaBar, v = realizedExactModulus B z x (ellOf beta) beta L delta (infoBandwidth n delta beta kappa deltaBar) (sqrt (log (12 * (J : ℝ) / alpha) / 2)) (Real.sqrt_nonneg _)}
def exactModulusInterval
Conditional-modulus interval centered at the realized total-Gram estimate.
Definition (Lean source)
parameters
n :
ℕ
J :
ℕ
B :
beta kappa L cminus cplus _pmin deltaBar delta alpha :
ℝ
def
exactModulusInterval B beta kappa L cminus cplus _pmin deltaBar delta alpha :
given by
fun z
=> let center := totalGramEstimator B z (ellOf beta) kappa cminus cplus delta (infoBandwidth n delta beta kappa deltaBar) let radius := worstRealizedExactModulus J n B beta kappa L cminus cplus deltaBar delta alpha z Icc (max 0 (center - radius)) (min 1 (center + radius))
def HasThresholdRegimeLimit
The threshold sequence approaches one of the finite or infinite phase regimes; oscillating sequences with no ratio limit are intentionally excluded.
def LeastFavorableMixedExperimentMatch
The conditional modulus interval matches both the integrated realized modulus and the minimax length of the least-favorable mixed regular-plus-local experiment.
Definition (Lean source)
parameters
def
LeastFavorableMixedExperimentMatch J beta kappa L cminus cplus pmin deltaBar alpha deltaSeq Bseq :
Prop
given by
clause 1
∃ D amplitude : ℝ,
0 < D
0 < amplitude
amplitude ≤ 1 / 4
∃ P0 Preg Ploc : ℕ → ClampLaw J,
∀ᶠ n in atTop, ClampModel (P0 n) beta kappa L cminus cplus pmin ∧ ClampModel (Preg n) beta kappa L cminus cplus pmin ∧ ClampModel (Ploc n) beta kappa L cminus cplus pmin ∧ IidSampling (P0 n) n ∧ IidSampling (Preg n) n ∧ IidSampling (Ploc n) n ∧ GlobalBernoulliShift (P0 n) (Preg n) ((n : ℝ) ^ (-(1 : ℝ) / 2)) ∧ LocalizedBernoulliPerturbation (P0 n) (Ploc n) beta (deltaSeq n) (infoBandwidth n (deltaSeq n) beta kappa deltaBar) amplitude ∧ productChiSq (Preg n) (P0 n) n
≤ D ∧ productChiSq (Ploc n) (P0 n) n ≤ D
Tendsto (fun n => (|clampFunctional (Preg n) (deltaSeq n) - clampFunctional (P0 n) (deltaSeq n)| + |clampFunctional (Ploc n) (deltaSeq n) - clampFunctional (P0 n) (deltaSeq n)|) / (2 * worstExactModulus J n (Bseq n) beta kappa L cminus cplus pmin deltaBar (deltaSeq n) alpha)) atTop (nhds 1)
clause 2
Tendsto (fun n => confidenceWorstLength J n beta kappa L cminus cplus pmin (exactModulusInterval (Bseq n) beta kappa L cminus cplus pmin deltaBar (deltaSeq n) alpha) / ofReal (2 * worstExactModulus J n (Bseq n) beta kappa L cminus cplus pmin deltaBar (deltaSeq n) alpha)) atTop (nhds 1)
clause 3
Tendsto (fun n => ofReal (2 * worstExactModulus J n (Bseq n) beta kappa L cminus cplus pmin deltaBar (deltaSeq n) alpha) / observedMinimaxLength J n beta kappa L cminus cplus pmin (deltaSeq n) alpha) atTop (nhds 1)
Does a realized-design exact-modulus interval attain one sharp asymptotic constant for L_n^star / r_n across regular, critical, atom-dominated, and fixed-threshold limits while retaining the stated model class?
Definition (Lean source)
parameters
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
def
SharpConfidenceLengthConstantQuestion :
Prop
given by
∃ c : ℝ,
0 < c
clause 1
Tendsto (fun n => observedMinimaxLength J n beta kappa L cminus cplus pmin (deltaSeq n) alpha / ofReal (clampFrontier n (deltaSeq n) kappa (infoBandwidth n (deltaSeq n) beta kappa deltaBar) beta)) atTop (nhds (ofReal c))
∃ Bseq : ∀ n,
SplitBlocks n, (∀ᶠ n in atTop, ∀ P : ClampLaw J, ClampModel P beta kappa L cminus cplus pmin → 1 - alpha ≤ (iidProduct P n).real {z | clampFunctional P (deltaSeq n) ∈ exactModulusInterval (Bseq n) beta kappa L cminus cplus pmin deltaBar (deltaSeq n) alpha z})
Tendsto (fun n => confidenceWorstLength J n beta kappa L cminus cplus pmin (exactModulusInterval (Bseq n) beta kappa L cminus cplus pmin deltaBar (deltaSeq n) alpha) / ofReal (clampFrontier n (deltaSeq n) kappa (infoBandwidth n (deltaSeq n) beta kappa deltaBar) beta)) atTop (nhds (ofReal c))
LeastFavorableMixedExperimentMatch J beta kappa L cminus cplus pmin deltaBar alpha deltaSeq Bseq
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.TCausalBridge 2 declarations The theorem includes the stratumwise product identity, pointwise identification, support, simultaneous pathwise clamp decomposition, and target equality.
Full-data to observed-data clamp bridge
The theorem includes the stratumwise product identity, pointwise identification, support, simultaneous pathwise clamp decomposition, and target equality.
Latent randomization and structural-mean continuity identify the observed continuous regression on the declared threshold range and hence identify the clamp mean. Joint measurability of the potential-outcome process is part of FullDataLaw, and all remaining causal conditions are supplied by the bundled full-data model membership. The result uses the hPF condition, the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
PF :
hPF :
FullDataClampModel PF beta kappa L cminus cplus pmin deltaBar
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
conclusion 1
(PF.observedMargin.px x)⁻¹ * (∫ z in {z : ClampObs J × PF.latentCarrier | z.1.X = x ∧ z.1.A ∈ B}, z.1.Y ∂PF.fullMeasure)
= ∫ a in B, fullDataResponseMean PF x a * PF.observedMargin.pi x a
conclusion 2
fullDataResponseMean PF x a = PF.observedMargin.mu x a
conclusion 3
delta :
ℝ
delta ∈ Icc (0 : ℝ) deltaBar
x :
Fin J
∀ᵐ a ∂conditionalTreatmentMeasure PF.observedMargin x, IsConditionalSupportPoint (conditionalTreatmentMeasure PF.observedMargin x) (clampPolicy delta a)
∀ᵐ z ∂PF.fullMeasure, PF.pot (clampPolicy delta z.1.A) z
= (if delta < z.1.A then z.1.Y else 0) + (if z.1.A ≤ delta then PF.pot delta z else 0)
causalClampMean PF delta
= retainedMean PF.observedMargin delta
causalClampMean PF delta = clampFunctional PF.observedMargin delta
Proof (Lean source)
-- @node: prop:causal-bridge
theorem causal_bridge
(J : ℕ)
(beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(PF : FullDataLaw J)
(hPF : FullDataClampModel PF beta kappa L cminus cplus pmin deltaBar)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
(∀ (x : Fin J) (B : Set ℝ), MeasurableSet B → B ⊆ Icc (0 : ℝ) 1 →
(PF.observedMargin.px x)⁻¹ *
(∫ z in
{z : ClampObs J × PF.latentCarrier | z.1.X = x ∧ z.1.A ∈ B},
z.1.Y ∂PF.fullMeasure) =
∫ a in B, fullDataResponseMean PF x a * PF.observedMargin.pi x a) ∧
(∀ (x : Fin J) (a : ℝ), a ∈ Icc (0 : ℝ) deltaBar →
fullDataResponseMean PF x a = PF.observedMargin.mu x a) ∧
∀ delta : ℝ, delta ∈ Icc (0 : ℝ) deltaBar →
(∀ x : Fin J, ∀ᵐ a ∂conditionalTreatmentMeasure PF.observedMargin x,
IsConditionalSupportPoint (conditionalTreatmentMeasure PF.observedMargin x)
(clampPolicy delta a)) ∧
(∀ᵐ z ∂PF.fullMeasure,
PF.pot (clampPolicy delta z.1.A) z =
(if delta < z.1.A then z.1.Y else 0) +
(if z.1.A ≤ delta then PF.pot delta z else 0)) ∧
causalClampMean PF delta =
retainedMean PF.observedMargin delta +
∑ x : Fin J, PF.observedMargin.px x *
atomMass PF.observedMargin x delta * PF.observedMargin.mu x delta ∧
causalClampMean PF delta = clampFunctional PF.observedMargin delta := by
have hreg' := hreg
rcases hreg' with
⟨_, _, hkappa, _, _, _, hcplusLower, hpmin, _, _, hdeltaBarOne, _, _⟩
have hcplus : 0 ≤ cplus := le_trans (by linarith) hcplusLower
have hmean := fullDataResponseMean_eq_mu J beta kappa L cminus cplus pmin
deltaBar alpha PF hPF hreg
refine ⟨?_, hmean, ?_⟩
· intro x B hB hBunit
exact fullDataResponseMean_setIntegral J kappa cminus cplus pmin deltaBar
PF hPF.toBridge hkappa hcplus hpmin x B hB hBunit
· intro delta hdelta
have hdeltaUnit : delta ∈ Icc (0 : ℝ) 1 := by
exact ⟨hdelta.1, hdelta.2.trans hdeltaBarOne.le⟩
have hsupport : ∀ x : Fin J, ∀ᵐ a ∂conditionalTreatmentMeasure PF.observedMargin x,
IsConditionalSupportPoint (conditionalTreatmentMeasure PF.observedMargin x)
(clampPolicy delta a) := fun x =>
clamp_policy_support PF.observedMargin beta kappa L cminus cplus pmin
deltaBar alpha hPF.observedModel hreg x delta hdelta
have hpath := causalClamp_pathwise PF hPF.consistency
hPF.observedModel.treatmentSupport delta hdeltaUnit
refine ⟨hsupport, hpath, ?_⟩
have hYFull : ∀ᵐ z ∂PF.fullMeasure, z.1.Y ∈ Icc (0 : ℝ) 1 := by
have hsupp := hPF.observedModel.outcomeSupport
rw [← PF.margin_eq] at hsupp
exact (MeasureTheory.ae_map_iff measurable_fst.aemeasurable
(measurableSet_Icc.preimage clampOutcome_measurable)).mp
hsupp
let fRet : ClampObs J × PF.latentCarrier → ℝ := fun z =>
if delta < z.1.A then z.1.Y else 0
let fAtom : ClampObs J × PF.latentCarrier → ℝ := fun z =>
if z.1.A ≤ delta then PF.pot delta z else 0
have hfRetMeas : Measurable fRet := by
exact (clampOutcome_measurable.comp measurable_fst).piecewise
(measurableSet_Ioi.preimage
((measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))).comp measurable_fst)) measurable_const
have hfAtomMeas : Measurable fAtom := by
have hp : Measurable (fun z : ClampObs J × PF.latentCarrier => PF.pot delta z) :=
PF.pot_jointlyMeasurable.comp (measurable_const.prodMk measurable_id)
exact hp.piecewise
(measurableSet_Iic.preimage
((measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))).comp measurable_fst)) measurable_const
let _ := PF.probability
have hfRetInt : Integrable fRet PF.fullMeasure := by
refine Integrable.of_bound hfRetMeas.aestronglyMeasurable 1 ?_
filter_upwards [hYFull] with z hY
by_cases hlt : delta < z.1.A
· simp only [fRet, if_pos hlt, Real.norm_eq_abs]
exact abs_le.2 ⟨by linarith [hY.1], hY.2⟩
· simp [fRet, hlt]
have hfAtomInt : Integrable fAtom PF.fullMeasure := by
refine Integrable.of_bound hfAtomMeas.aestronglyMeasurable 1 ?_
filter_upwards [hPF.consistency.2.2] with z hz
by_cases hle : z.1.A ≤ delta
· simp only [fAtom, if_pos hle, Real.norm_eq_abs]
rw [hz.1 delta hdeltaUnit]
exact abs_le.2 ⟨by
linarith [(hPF.consistency.2.1 z.1.X delta z.2 hdeltaUnit).1],
(hPF.consistency.2.1 z.1.X delta z.2 hdeltaUnit).2⟩
· simp [fAtom, hle]
have hret : (∫ z, fRet z ∂PF.fullMeasure) =
retainedMean PF.observedMargin delta := by
have hm : Measurable (fun o : ClampObs J =>
if delta < o.A then o.Y else 0) := by
exact clampOutcome_measurable.piecewise
(measurableSet_Ioi.preimage
(measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl)))) measurable_const
have h := fullData_observed_setIntegral PF univ MeasurableSet.univ
(fun o : ClampObs J => if delta < o.A then o.Y else 0) hm
simpa [fRet, retainedMean, indicator] using h
have hatom := fullData_clampAtom_integral_eq J kappa cminus cplus
pmin deltaBar PF hPF.toBridge hkappa hcplus hpmin hdeltaBarOne.le delta hdelta
have hpsi : causalClampMean PF delta =
retainedMean PF.observedMargin delta +
∑ x : Fin J, PF.observedMargin.px x *
atomMass PF.observedMargin x delta * fullDataResponseMean PF x delta := by
calc
causalClampMean PF delta = ∫ z, fRet z + fAtom z ∂PF.fullMeasure := by
apply integral_congr_ae
exact hpath.mono (fun z hz => by simpa [causalClampMean, fRet, fAtom] using hz)
_ = (∫ z, fRet z ∂PF.fullMeasure) + ∫ z, fAtom z ∂PF.fullMeasure := by
rw [integral_add hfRetInt hfAtomInt]
_ = _ := by rw [hret, hatom]
have htarget : causalClampMean PF delta =
retainedMean PF.observedMargin delta +
∑ x : Fin J, PF.observedMargin.px x *
atomMass PF.observedMargin x delta * PF.observedMargin.mu x delta := by
rw [hpsi]
congr 1
apply Finset.sum_congr rfl
intro x _
rw [hmean x delta hdelta]
exact ⟨htarget, by simpa [clampFunctional] using htarget⟩
The same latent-response argument identifies the qualitative continuous regression version and the continuity-only clamp functional, without imposing a fixed-Hölder observed model or a shared latent carrier. The result uses the hPF condition, the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin deltaBar :
ℝ
PF :
hPF :
ContFullDataClampModel PF kappa cminus cplus pmin deltaBar
hreg :
ContDesignConstants J kappa cminus cplus pmin deltaBar
conclusion
conclusion 1
fullDataResponseMean PF x a
= contRegression PF.observedMargin kappa cminus cplus pmin deltaBar hPF.observedModel x a
conclusion 2
delta :
ℝ
delta ∈ Icc (0 : ℝ) deltaBar
causalClampMean PF delta
= contClampFunctional PF.observedMargin kappa cminus cplus pmin deltaBar hPF.observedModel delta
Proof (Lean source)
-- @node: prop:continuity-causal-bridge
theorem continuity_causal_bridge
(J : ℕ) (kappa cminus cplus pmin deltaBar : ℝ)
(PF : FullDataLaw J)
(hPF : ContFullDataClampModel PF kappa cminus cplus pmin deltaBar)
(hreg : ContDesignConstants J kappa cminus cplus pmin deltaBar) :
(∀ (x : Fin J) (a : ℝ), a ∈ Icc (0 : ℝ) deltaBar →
fullDataResponseMean PF x a =
contRegression PF.observedMargin kappa cminus cplus pmin deltaBar
hPF.observedModel x a) ∧
∀ delta : ℝ, delta ∈ Icc (0 : ℝ) deltaBar →
causalClampMean PF delta =
contClampFunctional PF.observedMargin kappa cminus cplus pmin deltaBar
hPF.observedModel delta := by
have hreg' := hreg
rcases hreg' with
⟨_, hkappa, _, _, hcplusLower, hpmin, _, _, hdeltaBarOne⟩
have hcplus : 0 ≤ cplus := le_trans (by linarith) hcplusLower
have hmean := fullDataResponseMean_eq_contRegression J kappa cminus cplus
pmin deltaBar PF hPF hreg
refine ⟨hmean, ?_⟩
intro delta hdelta
have hdeltaUnit : delta ∈ Icc (0 : ℝ) 1 :=
⟨hdelta.1, hdelta.2.trans hdeltaBarOne.le⟩
have hpath := causalClamp_pathwise PF hPF.consistency
hPF.observedModel.treatmentSupport delta hdeltaUnit
have hYFull : ∀ᵐ z ∂PF.fullMeasure, z.1.Y ∈ Icc (0 : ℝ) 1 := by
have hsupp := hPF.observedModel.outcomeSupport
rw [← PF.margin_eq] at hsupp
exact (MeasureTheory.ae_map_iff measurable_fst.aemeasurable
(measurableSet_Icc.preimage clampOutcome_measurable)).mp hsupp
let fRet : ClampObs J × PF.latentCarrier → ℝ := fun z =>
if delta < z.1.A then z.1.Y else 0
let fAtom : ClampObs J × PF.latentCarrier → ℝ := fun z =>
if z.1.A ≤ delta then PF.pot delta z else 0
have hfRetMeas : Measurable fRet := by
exact (clampOutcome_measurable.comp measurable_fst).piecewise
(measurableSet_Ioi.preimage
((measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))).comp measurable_fst)) measurable_const
have hfAtomMeas : Measurable fAtom := by
have hp : Measurable (fun z : ClampObs J × PF.latentCarrier => PF.pot delta z) :=
PF.pot_jointlyMeasurable.comp (measurable_const.prodMk measurable_id)
exact hp.piecewise
(measurableSet_Iic.preimage
((measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl))).comp measurable_fst)) measurable_const
let _ := PF.probability
have hfRetInt : Integrable fRet PF.fullMeasure := by
refine Integrable.of_bound hfRetMeas.aestronglyMeasurable 1 ?_
filter_upwards [hYFull] with z hY
by_cases hlt : delta < z.1.A
· simp only [fRet, if_pos hlt, Real.norm_eq_abs]
exact abs_le.2 ⟨by linarith [hY.1], hY.2⟩
· simp [fRet, hlt]
have hfAtomInt : Integrable fAtom PF.fullMeasure := by
refine Integrable.of_bound hfAtomMeas.aestronglyMeasurable 1 ?_
filter_upwards [hPF.consistency.2.2] with z hz
by_cases hle : z.1.A ≤ delta
· simp only [fAtom, if_pos hle, Real.norm_eq_abs]
rw [hz.1 delta hdeltaUnit]
exact abs_le.2 ⟨by
linarith [(hPF.consistency.2.1 z.1.X delta z.2 hdeltaUnit).1],
(hPF.consistency.2.1 z.1.X delta z.2 hdeltaUnit).2⟩
· simp [fAtom, hle]
have hret : (∫ z, fRet z ∂PF.fullMeasure) =
retainedMean PF.observedMargin delta := by
have hm : Measurable (fun o : ClampObs J =>
if delta < o.A then o.Y else 0) := by
exact clampOutcome_measurable.piecewise
(measurableSet_Ioi.preimage
(measurable_fst.comp (measurable_snd.comp
(Measurable.of_comap_le le_rfl)))) measurable_const
have h := fullData_observed_setIntegral PF univ MeasurableSet.univ
(fun o : ClampObs J => if delta < o.A then o.Y else 0) hm
simpa [fRet, retainedMean, indicator] using h
have hatom := fullData_clampAtom_integral_eq J kappa cminus cplus pmin
deltaBar PF hPF.toBridge hkappa hcplus hpmin hdeltaBarOne.le delta hdelta
have hpsi : causalClampMean PF delta =
retainedMean PF.observedMargin delta +
∑ x : Fin J, PF.observedMargin.px x *
atomMass PF.observedMargin x delta * fullDataResponseMean PF x delta := by
calc
causalClampMean PF delta = ∫ z, fRet z + fAtom z ∂PF.fullMeasure := by
apply integral_congr_ae
exact hpath.mono (fun z hz => by
simpa [causalClampMean, fRet, fAtom] using hz)
_ = (∫ z, fRet z ∂PF.fullMeasure) + ∫ z, fAtom z ∂PF.fullMeasure := by
rw [integral_add hfRetInt hfAtomInt]
_ = _ := by rw [hret, hatom]
rw [hpsi]
unfold contClampFunctional
congr 1
apply Finset.sum_congr rfl
intro x _
rw [hmean x delta hdelta]
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.TCausalFrontierLift 18 declarations All procedures remain functions only of the observed sample.
Lift of the observed frontier to the full-data causal class
All procedures remain functions only of the observed sample. The quantified
class is the full-data class, and the criteria are the two components of
causalFrontierCriteria.
theorem standardBorelRegularConditionalLaw_mathlib
Mathlib's standard-Borel disintegration kernel discharges the paper's regular-conditional-law gate. This is the stated conclusion.
Formal statement
conclusion
Proof (Lean source)
-- @node: standardBorelRegularConditionalLaw_mathlib
lemma standardBorelRegularConditionalLaw_mathlib :
StandardBorelRegularConditionalLaw := by
intro S T _ _ _ _ μ hμ
let _ := hμ
let _ : Nonempty T :=
⟨(choice (nonempty_of_isProbabilityMeasure μ)).2⟩
let K : Kernel S T := μ.condKernel
refine ⟨K, fun _ => inferInstance, ?_⟩
intro B C hB hC
have h := Measure.setLIntegral_condKernel
(ρ := μ) (f := fun _ : S × T => (1 : ℝ≥0∞)) measurable_const hB hC
simpa [K, Measure.fst] using h.symm
Conditional on the disclosed regular-conditional-law gate, every observed model law admits a structural full-data lift and the observed and causal finite-sample criteria agree. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
StandardBorelRegularConditionalLaw_of_gate :
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
conclusion 1
P :
ClampLaw J
ClampModel P beta kappa L cminus cplus pmin
∃ PF : FullDataLaw J,
FullDataClampModel PF beta kappa L cminus cplus pmin deltaBar
PF.observedMargin = P
conclusion 2
(causalFrontierCriteria J n beta kappa L cminus cplus pmin deltaBar delta alpha).1
= observedMinimaxRisk J n beta kappa L cminus cplus pmin delta
(causalFrontierCriteria J n beta kappa L cminus cplus pmin deltaBar delta alpha).2
= observedMinimaxLength J n beta kappa L cminus cplus pmin delta alpha
Proof (Lean source)
-- @node: prop:observed-margin-surjectivity
theorem observed_margin_surjectivity
(StandardBorelRegularConditionalLaw_of_gate :
StandardBorelRegularConditionalLaw)
(J : ℕ)
(beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
(∀ (P : ClampLaw J), ClampModel P beta kappa L cminus cplus pmin →
∃ PF : FullDataLaw J,
FullDataClampModel PF beta kappa L cminus cplus pmin deltaBar ∧
PF.observedMargin = P) ∧
∀ (n : ℕ) (delta : ℝ), delta ∈ Icc (0 : ℝ) deltaBar →
(causalFrontierCriteria J n beta kappa L cminus cplus pmin deltaBar
delta alpha).1 =
observedMinimaxRisk J n beta kappa L cminus cplus pmin delta ∧
(causalFrontierCriteria J n beta kappa L cminus cplus pmin deltaBar
delta alpha).2 =
observedMinimaxLength J n beta kappa L cminus cplus pmin delta alpha := by
have hpmin : 0 < pmin := hreg.2.2.2.2.2.2.2.1
have hJ : 0 < J := hreg.1
have hdeltaBar : deltaBar ≤ 1 :=
(hreg.2.2.2.2.2.2.2.2.2.2.1).le
have hsurj : ∀ (P : ClampLaw J), ClampModel P beta kappa L cminus cplus pmin →
∃ PF : FullDataLaw J,
FullDataClampModel PF beta kappa L cminus cplus pmin deltaBar ∧
PF.observedMargin = P := by
intro P hP
exact exists_fullData_holder_lift P hP hJ hpmin hdeltaBar
refine ⟨hsurj, ?_⟩
intro n delta hdelta
have htarget (PF : FullDataLaw J)
(hPF : FullDataClampModel PF beta kappa L cminus cplus pmin deltaBar) :
causalClampMean PF delta = clampFunctional PF.observedMargin delta :=
((causal_bridge J beta kappa L cminus cplus pmin deltaBar alpha PF hPF hreg).2.2
delta hdelta).2.2.2
constructor
· unfold causalFrontierCriteria observedMinimaxRisk
apply congrArg sInf
ext r
constructor
· rintro ⟨est, hm, hrange, rfl⟩
refine ⟨est, hm, hrange, ?_⟩
apply congrArg sSup
ext v
constructor
· rintro ⟨PF, hPF, rfl⟩
refine ⟨PF.observedMargin, hPF.observedModel, ?_⟩
unfold causalEstimatorRisk estimatorRisk
rw [htarget PF hPF]
· rintro ⟨P, hP, rfl⟩
obtain ⟨PF, hPF, hmargin⟩ := hsurj P hP
refine ⟨PF, hPF, ?_⟩
unfold causalEstimatorRisk estimatorRisk
rw [htarget PF hPF, hmargin]
· rintro ⟨est, hm, hrange, rfl⟩
refine ⟨est, hm, hrange, ?_⟩
apply congrArg sSup
ext v
constructor
· rintro ⟨P, hP, rfl⟩
obtain ⟨PF, hPF, hmargin⟩ := hsurj P hP
refine ⟨PF, hPF, ?_⟩
unfold causalEstimatorRisk estimatorRisk
rw [htarget PF hPF, hmargin]
· rintro ⟨PF, hPF, rfl⟩
refine ⟨PF.observedMargin, hPF.observedModel, ?_⟩
unfold causalEstimatorRisk estimatorRisk
rw [htarget PF hPF]
· unfold causalFrontierCriteria observedMinimaxLength UniformCoverage
apply congrArg sInf
ext r
constructor
· rintro ⟨C, hm, hcov, rfl⟩
refine ⟨C, ⟨hm, ?_⟩, ?_⟩
· intro P hP
obtain ⟨PF, hPF, hmargin⟩ := hsurj P hP
simpa only [hmargin, htarget PF hPF] using hcov PF hPF
· apply congrArg sSup
ext v
constructor
· rintro ⟨PF, hPF, rfl⟩
exact ⟨PF.observedMargin, hPF.observedModel, rfl⟩
· rintro ⟨P, hP, rfl⟩
obtain ⟨PF, hPF, hmargin⟩ := hsurj P hP
exact ⟨PF, hPF, by rw [hmargin]⟩
· rintro ⟨C, ⟨hm, hcov⟩, rfl⟩
refine ⟨C, hm, ?_, ?_⟩
· intro PF hPF
simpa only [htarget PF hPF] using hcov PF.observedMargin hPF.observedModel
· apply congrArg sSup
ext v
constructor
· rintro ⟨P, hP, rfl⟩
obtain ⟨PF, hPF, hmargin⟩ := hsurj P hP
exact ⟨PF, hPF, by rw [hmargin]⟩
· rintro ⟨PF, hPF, rfl⟩
exact ⟨PF.observedMargin, hPF.observedModel, rfl⟩
Conditional on the same disclosed gate, the observed-margin map is onto for the continuity-only class and its observed and causal criteria agree. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
StandardBorelRegularConditionalLaw_of_gate :
J :
ℕ
kappa cminus cplus pmin deltaBar alpha :
ℝ
hreg :
ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha
conclusion
conclusion 1
P :
ClampLaw J
ContClampModel P kappa cminus cplus pmin deltaBar
∃ PF : FullDataLaw J,
ContFullDataClampModel PF kappa cminus cplus pmin deltaBar
PF.observedMargin = P
conclusion 2
n :
ℕ
delta :
ℝ
delta ∈ Icc (0 : ℝ) deltaBar
crit :
contFrontierCriteria J n kappa cminus cplus pmin deltaBar delta alpha
crit.2.2.1 = crit.1
crit.2.2.2 = crit.2.1
Proof (Lean source)
-- @node: prop:continuity-observed-margin-surjectivity
theorem continuity_observed_margin_surjectivity
(StandardBorelRegularConditionalLaw_of_gate :
StandardBorelRegularConditionalLaw)
(J : ℕ)
(kappa cminus cplus pmin deltaBar alpha : ℝ)
(hreg : ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha) :
(∀ (P : ClampLaw J), ContClampModel P kappa cminus cplus pmin deltaBar →
∃ PF : FullDataLaw J,
ContFullDataClampModel PF kappa cminus cplus pmin deltaBar ∧
PF.observedMargin = P) ∧
∀ (n : ℕ) (delta : ℝ), delta ∈ Icc (0 : ℝ) deltaBar →
let crit := contFrontierCriteria J n kappa cminus cplus pmin
deltaBar delta alpha
crit.2.2.1 = crit.1 ∧ crit.2.2.2 = crit.2.1 := by
rcases hreg.1 with ⟨hJ, hkappa, hcminus, hcminus_le, hcplus,
hpmin, hpmin_le, hdeltaBar, hdeltaBar_one⟩
have hdesign : ContDesignConstants J kappa cminus cplus pmin deltaBar :=
⟨hJ, hkappa, hcminus, hcminus_le, hcplus, hpmin, hpmin_le,
hdeltaBar, hdeltaBar_one⟩
have hsurj : ∀ (P : ClampLaw J),
ContClampModel P kappa cminus cplus pmin deltaBar →
∃ PF : FullDataLaw J,
ContFullDataClampModel PF kappa cminus cplus pmin deltaBar ∧
PF.observedMargin = P := by
intro P hP
exact exists_fullData_cont_lift P hP hpmin
refine ⟨hsurj, ?_⟩
intro n delta hdelta
have htarget (PF : FullDataLaw J)
(hPF : ContFullDataClampModel PF kappa cminus cplus pmin deltaBar) :
causalClampMean PF delta =
contClampFunctional PF.observedMargin kappa cminus cplus pmin deltaBar
hPF.observedModel delta :=
(continuity_causal_bridge J kappa cminus cplus pmin deltaBar PF hPF hdesign).2
delta hdelta
dsimp only
constructor
· unfold contFrontierCriteria
apply congrArg sInf
ext r
constructor
· rintro ⟨est, hm, hrange, rfl⟩
refine ⟨est, hm, hrange, ?_⟩
apply congrArg sSup
ext v
constructor
· rintro ⟨PF, hPF, rfl⟩
refine ⟨PF.observedMargin, hPF.observedModel, ?_⟩
unfold causalEstimatorRisk contEstimatorRisk
rw [htarget PF hPF]
· rintro ⟨P, hP, rfl⟩
obtain ⟨PF, hPF, hmargin⟩ := hsurj P hP
subst P
refine ⟨PF, hPF, ?_⟩
unfold causalEstimatorRisk contEstimatorRisk
rw [htarget PF hPF]
· rintro ⟨est, hm, hrange, rfl⟩
refine ⟨est, hm, hrange, ?_⟩
apply congrArg sSup
ext v
constructor
· rintro ⟨P, hP, rfl⟩
obtain ⟨PF, hPF, hmargin⟩ := hsurj P hP
subst P
refine ⟨PF, hPF, ?_⟩
unfold causalEstimatorRisk contEstimatorRisk
rw [htarget PF hPF]
· rintro ⟨PF, hPF, rfl⟩
refine ⟨PF.observedMargin, hPF.observedModel, ?_⟩
unfold causalEstimatorRisk contEstimatorRisk
rw [htarget PF hPF]
· unfold contFrontierCriteria
apply congrArg sInf
ext r
constructor
· rintro ⟨C, hm, hcov, rfl⟩
refine ⟨C, hm, ?_, ?_⟩
· intro P hP
obtain ⟨PF, hPF, hmargin⟩ := hsurj P hP
subst P
simpa only [htarget PF hPF] using hcov PF hPF
· apply congrArg sSup
ext v
constructor
· rintro ⟨PF, hPF, rfl⟩
exact ⟨PF.observedMargin, hPF.observedModel, rfl⟩
· rintro ⟨P, hP, rfl⟩
obtain ⟨PF, hPF, hmargin⟩ := hsurj P hP
exact ⟨PF, hPF, by rw [hmargin]⟩
· rintro ⟨C, hm, hcov, rfl⟩
refine ⟨C, hm, ?_, ?_⟩
· intro PF hPF
simpa only [htarget PF hPF] using hcov PF.observedMargin hPF.observedModel
· apply congrArg sSup
ext v
constructor
· rintro ⟨P, hP, rfl⟩
obtain ⟨PF, hPF, hmargin⟩ := hsurj P hP
exact ⟨PF, hPF, by rw [hmargin]⟩
· rintro ⟨PF, hPF, rfl⟩
exact ⟨PF.observedMargin, hPF.observedModel, rfl⟩
Worst-case causal risk of the same observed-sample total-Gram estimator.
Definition (Lean source)
parameters
def
causalStabilizedWorstRisk J n B beta kappa L cminus cplus pmin deltaBar delta :
ℝ
given by
sSup {v : ℝ | ∃ PF : FullDataLaw J, FullDataClampModel PF beta kappa L cminus cplus pmin deltaBar ∧ v = causalEstimatorRisk (PF := PF) (n := n) (delta := delta) (est := fun z => totalGramEstimator B z (ellOf beta) kappa cminus cplus delta (infoBandwidth n delta beta kappa deltaBar))}
Worst-case full-data coverage of the same observed-sample interval.
Definition (Lean source)
parameters
def
causalStabilizedCoverage J n B beta kappa L cminus cplus pmin deltaBar delta alpha :
ℝ
given by
sInf {v : ℝ | ∃ PF : FullDataLaw J, FullDataClampModel PF beta kappa L cminus cplus pmin deltaBar ∧ v = (iidProduct PF.observedMargin n).real {z | causalClampMean (PF := PF) (delta := delta) ∈ stabilizedInterval B beta kappa L cminus cplus deltaBar delta alpha z}}
Worst-case full-data expected length of that interval, as an extended expectation so a non-integrable length contributes ∞.
Definition (Lean source)
parameters
def
causalStabilizedWorstLength J n B beta kappa L cminus cplus pmin deltaBar delta alpha :
ℝ≥0∞
given by
sSup {v : ℝ≥0∞ | ∃ PF : FullDataLaw J, FullDataClampModel PF beta kappa L cminus cplus pmin deltaBar ∧ v = ∫⁻ z, intervalLengthENNReal (stabilizedInterval B beta kappa L cminus cplus deltaBar delta alpha z) ∂iidProduct PF.observedMargin n}
The convex-hull length of a closed real interval is its nonnegative endpoint difference, including the empty-interval case. This is the stated conclusion.
Formal statement
Proof (Lean source)
-- @node: intervalLength_Icc_eq
lemma intervalLength_Icc_eq (a b : ℝ) :
intervalLength (Icc a b) = max 0 (b - a) := by
unfold intervalLength
by_cases h : a ≤ b
· simp [h]
· have h' : b < a := lt_of_not_ge h
simp [Set.Icc_eq_empty h, h'.le]
theorem stratumRadius_measurable
The sample-dependent per-stratum radius is measurable. This is the stated conclusion.
Formal statement
hypotheses
conclusion
=> stratumRadius B z x ell beta kappa L cminus cplus delta h tAlpha b1
Proof (Lean source)
-- @node: stratumRadius_measurable
lemma stratumRadius_measurable {J n ell : ℕ} (B : SplitBlocks n) (x : Fin J)
(beta kappa L cminus cplus delta h tAlpha b1 : ℝ) :
Measurable fun z : Fin n → ClampObs J =>
stratumRadius B z x ell beta kappa L cminus cplus delta h tAlpha b1 := by
classical
unfold stratumRadius
refine Measurable.ite (goodGramEvent_measurable B x kappa cminus cplus delta h) ?_ ?_
· have habs := Finset.measurable_fun_sum B.I2 fun i _ =>
(interceptWeight_measurable (ell := ell) B x kappa cminus cplus delta h i).abs
have hsq := Finset.measurable_fun_sum B.I2 fun i _ =>
(interceptWeight_measurable (ell := ell) B x kappa cminus cplus delta h i).pow_const 2
exact (((atomEstimate_measurable B x delta).add measurable_const).mul
(((measurable_const.mul measurable_const).mul habs).add
(measurable_const.mul hsq.sqrt))).add measurable_const
· exact (atomEstimate_measurable B x delta).add measurable_const
theorem stabilizedInterval_radius_nonneg
Once all three blocks are nonempty, the total displayed confidence radius is nonnegative under the declared parameter restrictions. The result uses the hreg condition, the hn condition, the hh condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
B :
beta kappa L cminus cplus pmin deltaBar alpha delta h :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
hn :
4 ≤ n
hh :
0 ≤ h
Proof (Lean source)
-- @node: stabilizedInterval_radius_nonneg
lemma stabilizedInterval_radius_nonneg
(J n : ℕ) (B : SplitBlocks n)
(beta kappa L cminus cplus pmin deltaBar alpha delta h : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(hn : 4 ≤ n) (hh : 0 ≤ h) :
∀ z : Fin n → ClampObs J,
let tAlpha := sqrt (log (12 * (J : ℝ) / alpha) / 2)
let b0 := tAlpha / sqrt (B.I0.card : ℝ)
let b1 := tAlpha / sqrt (B.I1.card : ℝ)
0 ≤ b0 + ∑ x : Fin J,
stratumRadius B z x (ellOf beta) beta kappa L cminus cplus delta h tAlpha b1 := by
intro z
dsimp only
rcases hreg with ⟨hJ, hbeta, hkappa, hL, hcminus, hcminus_le, hcplus,
hpmin, hpmin_le, hdeltaBar, hdeltaBar1, ha, ha_half⟩
have ht : 0 ≤ sqrt (log (12 * (J : ℝ) / alpha) / 2) :=
Real.sqrt_nonneg _
have hI0 : 0 < (B.I0.card : ℝ) := by
have hc := B.card_I0
exact_mod_cast (show 0 < B.I0.card by omega)
have hI1 : 0 < (B.I1.card : ℝ) := by
have hc := B.card_I1
exact_mod_cast (show 0 < B.I1.card by omega)
have hb0 : 0 ≤ sqrt (log (12 * (J : ℝ) / alpha) / 2) /
sqrt (B.I0.card : ℝ) := div_nonneg ht (Real.sqrt_nonneg _)
have hb1 : 0 ≤ sqrt (log (12 * (J : ℝ) / alpha) / 2) /
sqrt (B.I1.card : ℝ) := div_nonneg ht (Real.sqrt_nonneg _)
apply add_nonneg hb0
apply sum_nonneg
intro x hx
unfold stratumRadius atomEstimate blockAverage
split_ifs <;> positivity
theorem stabilizedInterval_length_measurable
Length of the displayed bias-aware interval is measurable as a function of the observed sample. This is the stated conclusion.
Formal statement
hypotheses
conclusion
=> intervalLength (stabilizedInterval B beta kappa L cminus cplus deltaBar delta alpha z)
Proof (Lean source)
-- @node: stabilizedInterval_length_measurable
lemma stabilizedInterval_length_measurable (J n : ℕ) (B : SplitBlocks n)
(beta kappa L cminus cplus deltaBar delta alpha : ℝ) :
Measurable fun z : Fin n → ClampObs J =>
intervalLength (stabilizedInterval B beta kappa L cminus cplus deltaBar delta alpha z) := by
classical
let tAlpha := sqrt (log (12 * (J : ℝ) / alpha) / 2)
let b0 := tAlpha / sqrt (B.I0.card : ℝ)
let b1 := tAlpha / sqrt (B.I1.card : ℝ)
let center := fun z : Fin n → ClampObs J =>
totalGramEstimator B z (ellOf beta) kappa cminus cplus delta
(infoBandwidth n delta beta kappa deltaBar)
let radius := fun z : Fin n → ClampObs J => b0 + ∑ x : Fin J,
stratumRadius B z x (ellOf beta) beta kappa L cminus cplus delta
(infoBandwidth n delta beta kappa deltaBar) tAlpha b1
have hc : Measurable center := totalGramEstimator_measurable B kappa cminus cplus delta _
have hr : Measurable radius := measurable_const.add
(Finset.measurable_fun_sum _ fun x _ =>
stratumRadius_measurable B x beta kappa L cminus cplus delta _ tAlpha b1)
have hlo : Measurable fun z => max 0 (center z - radius z) :=
measurable_const.max (hc.sub hr)
have hhi : Measurable fun z => min 1 (center z + radius z) :=
measurable_const.min (hc.add hr)
have hd : Measurable fun z =>
min 1 (center z + radius z) - max 0 (center z - radius z) := hhi.sub hlo
have hm : Measurable fun z =>
max 0 (min 1 (center z + radius z) - max 0 (center z - radius z)) :=
measurable_const.max hd
simpa only [stabilizedInterval, honestInterval, intervalLength_Icc_eq,
center, radius, tAlpha, b0, b1] using hm
theorem stabilizedInterval_observedMeasurable
For sufficiently large samples the displayed endpoints are ordered and measurable, hence the bias-aware construction is an admissible observed confidence procedure. The result uses the hreg condition, the hn condition, the hh condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
B :
beta kappa L cminus cplus pmin deltaBar alpha delta :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
hn :
4 ≤ n
hh :
0 ≤ infoBandwidth n delta beta kappa deltaBar
conclusion
ObservedMeasurableInterval (stabilizedInterval (J := J) B beta kappa L cminus cplus deltaBar delta alpha)
Proof (Lean source)
-- @node: stabilizedInterval_observedMeasurable
lemma stabilizedInterval_observedMeasurable
(J n : ℕ) (B : SplitBlocks n)
(beta kappa L cminus cplus pmin deltaBar alpha delta : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(hn : 4 ≤ n)
(hh : 0 ≤ infoBandwidth n delta beta kappa deltaBar) :
ObservedMeasurableInterval
(stabilizedInterval (J := J) B beta kappa L cminus cplus deltaBar delta alpha) := by
classical
let tAlpha := sqrt (log (12 * (J : ℝ) / alpha) / 2)
let b0 := tAlpha / sqrt (B.I0.card : ℝ)
let b1 := tAlpha / sqrt (B.I1.card : ℝ)
let center := fun z : Fin n → ClampObs J =>
totalGramEstimator B z (ellOf beta) kappa cminus cplus delta
(infoBandwidth n delta beta kappa deltaBar)
let radius := fun z : Fin n → ClampObs J => b0 + ∑ x : Fin J,
stratumRadius B z x (ellOf beta) beta kappa L cminus cplus delta
(infoBandwidth n delta beta kappa deltaBar) tAlpha b1
let lo := fun z => max 0 (center z - radius z)
let hi := fun z => min 1 (center z + radius z)
have hc : Measurable center := totalGramEstimator_measurable B kappa cminus cplus delta _
have hr : Measurable radius := measurable_const.add
(Finset.measurable_fun_sum _ fun x _ =>
stratumRadius_measurable B x beta kappa L cminus cplus delta _ tAlpha b1)
have hlo : Measurable lo := measurable_const.max (hc.sub hr)
have hhi : Measurable hi := measurable_const.min (hc.add hr)
refine ⟨lo, hi, hlo, hhi, ?_, ?_⟩
· intro z
have hcenter := clampUnit_mem_Icc
(retainedEstimate B z delta + ∑ x : Fin J,
atomEstimate B z x delta * localRegressionEstimate B z x (ellOf beta)
kappa cminus cplus delta (infoBandwidth n delta beta kappa deltaBar))
have hradius : 0 ≤ radius z := by
simpa only [radius, tAlpha, b0, b1] using
stabilizedInterval_radius_nonneg J n B beta kappa L cminus cplus pmin
deltaBar alpha delta (infoBandwidth n delta beta kappa deltaBar) hreg hn hh z
dsimp only [lo, hi]
apply le_min
· apply max_le
· norm_num
· dsimp only [center]
rw [show totalGramEstimator B z (ellOf beta) kappa cminus cplus delta
(infoBandwidth n delta beta kappa deltaBar) =
clampUnit (retainedEstimate B z delta + ∑ x : Fin J,
atomEstimate B z x delta * localRegressionEstimate B z x (ellOf beta)
kappa cminus cplus delta
(infoBandwidth n delta beta kappa deltaBar)) by rfl]
linarith [hcenter.2]
· apply max_le
· dsimp only [center]
rw [show totalGramEstimator B z (ellOf beta) kappa cminus cplus delta
(infoBandwidth n delta beta kappa deltaBar) =
clampUnit (retainedEstimate B z delta + ∑ x : Fin J,
atomEstimate B z x delta * localRegressionEstimate B z x (ellOf beta)
kappa cminus cplus delta
(infoBandwidth n delta beta kappa deltaBar)) by rfl]
linarith [hcenter.1]
· linarith
· intro z
rfl
theorem stabilizedInterval_length_le_one
Intersecting the bias-aware interval with the outcome range bounds its convex-hull length by one on every sample. This is the stated conclusion.
Formal statement
hypotheses
conclusion
intervalLength (stabilizedInterval B beta kappa L cminus cplus deltaBar delta alpha z) ≤ 1
Proof (Lean source)
-- @node: stabilizedInterval_length_le_one
lemma stabilizedInterval_length_le_one (J n : ℕ) (B : SplitBlocks n)
(beta kappa L cminus cplus deltaBar delta alpha : ℝ)
(z : Fin n → ClampObs J) :
intervalLength (stabilizedInterval B beta kappa L cminus cplus deltaBar delta alpha z) ≤ 1 := by
simp only [stabilizedInterval, honestInterval, intervalLength_Icc_eq]
apply max_le
· norm_num
· have hmin := min_le_left (1 : ℝ)
(totalGramEstimator B z (ellOf beta) kappa cminus cplus delta
(infoBandwidth n delta beta kappa deltaBar) +
(sqrt (log (12 * (J : ℝ) / alpha) / 2) /
sqrt (B.I0.card : ℝ) + ∑ x : Fin J,
stratumRadius B z x (ellOf beta) beta kappa L cminus cplus delta
(infoBandwidth n delta beta kappa deltaBar)
(sqrt (log (12 * (J : ℝ) / alpha) / 2))
(sqrt (log (12 * (J : ℝ) / alpha) / 2) /
sqrt (B.I1.card : ℝ))))
have hmax := le_max_left (0 : ℝ)
(totalGramEstimator B z (ellOf beta) kappa cminus cplus delta
(infoBandwidth n delta beta kappa deltaBar) -
(sqrt (log (12 * (J : ℝ) / alpha) / 2) /
sqrt (B.I0.card : ℝ) + ∑ x : Fin J,
stratumRadius B z x (ellOf beta) beta kappa L cminus cplus delta
(infoBandwidth n delta beta kappa deltaBar)
(sqrt (log (12 * (J : ℝ) / alpha) / 2))
(sqrt (log (12 * (J : ℝ) / alpha) / 2) /
sqrt (B.I1.card : ℝ))))
linarith
theorem confidenceWorstLength_stabilizedInterval_le_ofReal
A real expected-length bound for the concrete interval upgrades to the extended expected-length convention used by the minimax criteria. The result uses the hupper condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
B :
beta kappa L cminus cplus pmin deltaBar delta alpha U :
ℝ
hupper :
stabilizedWorstLength J n B beta kappa L cminus cplus pmin deltaBar delta alpha ≤ U
conclusion
confidenceWorstLength J n beta kappa L cminus cplus pmin (stabilizedInterval B beta kappa L cminus cplus deltaBar delta alpha)
≤ ofReal U
Proof (Lean source)
-- @node: confidenceWorstLength_stabilizedInterval_le_ofReal
lemma confidenceWorstLength_stabilizedInterval_le_ofReal
(J n : ℕ) (B : SplitBlocks n)
(beta kappa L cminus cplus pmin deltaBar delta alpha U : ℝ)
(hupper : stabilizedWorstLength J n B beta kappa L cminus cplus pmin
deltaBar delta alpha ≤ U) :
confidenceWorstLength J n beta kappa L cminus cplus pmin
(stabilizedInterval B beta kappa L cminus cplus deltaBar delta alpha) ≤
ofReal U := by
unfold stabilizedWorstLength at hupper
unfold confidenceWorstLength
apply sSup_le
intro v hv
rcases hv with ⟨P, hP, rfl⟩
letI : IsProbabilityMeasure P.dataMeasure := hP.probability
letI : IsProbabilityMeasure (iidProduct P n) := by
unfold iidProduct
infer_instance
let f := fun z : Fin n → ClampObs J =>
intervalLength (stabilizedInterval B beta kappa L cminus cplus deltaBar delta alpha z)
have hfmeas : Measurable f :=
stabilizedInterval_length_measurable J n B beta kappa L cminus cplus deltaBar delta alpha
have hfnn : ∀ᵐ z ∂iidProduct P n, 0 ≤ f z := ae_of_all _ fun _ => le_max_left _ _
have hfint : Integrable f (iidProduct P n) := by
refine Integrable.of_bound hfmeas.aestronglyMeasurable 1 ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (le_max_left _ _)]
exact stabilizedInterval_length_le_one J n B beta kappa L cminus cplus
deltaBar delta alpha z
simp_rw [intervalLengthENNReal]
rw [← ofReal_integral_eq_lintegral_ofReal hfint hfnn]
apply ENNReal.ofReal_le_ofReal
have hbdd : BddAbove {v : ℝ | ∃ Q : ClampLaw J,
ClampModel Q beta kappa L cminus cplus pmin ∧ IidSampling Q n ∧
v = ∫ z, intervalLength
(stabilizedInterval B beta kappa L cminus cplus deltaBar delta alpha z)
∂iidProduct Q n} := by
refine ⟨1, ?_⟩
rintro w ⟨Q, hQ, _hiid, rfl⟩
letI : IsProbabilityMeasure Q.dataMeasure := hQ.probability
letI : IsProbabilityMeasure (iidProduct Q n) := by
unfold iidProduct
infer_instance
have hqmeas := stabilizedInterval_length_measurable J n B beta kappa L cminus
cplus deltaBar delta alpha
have hqint : Integrable (fun z : Fin n → ClampObs J =>
intervalLength (stabilizedInterval B beta kappa L cminus cplus deltaBar delta alpha z))
(iidProduct Q n) := by
refine Integrable.of_bound hqmeas.aestronglyMeasurable 1 ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_of_nonneg (le_max_left _ _)]
exact stabilizedInterval_length_le_one J n B beta kappa L cminus cplus
deltaBar delta alpha z
have hmono := integral_mono_ae hqint (integrable_const 1)
(ae_of_all _ fun z => stabilizedInterval_length_le_one J n B beta kappa L
cminus cplus deltaBar delta alpha z)
simpa using hmono
have hmem : (∫ z, f z ∂iidProduct P n) ∈ {v : ℝ | ∃ Q : ClampLaw J,
ClampModel Q beta kappa L cminus cplus pmin ∧ IidSampling Q n ∧
v = ∫ z, intervalLength
(stabilizedInterval B beta kappa L cminus cplus deltaBar delta alpha z)
∂iidProduct Q n} := ⟨P, hP, clampModel_iidSampling hP, rfl⟩
exact (le_csSup hbdd hmem).trans hupper
theorem observedMinimaxLength_le_confidenceWorstLength
Any uniformly honest concrete interval upper-bounds the observed minimax expected length by its own worst-case length. The result uses the hC condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
beta kappa L cminus cplus pmin delta alpha :
ℝ
C :
hC :
UniformCoverage J n beta kappa L cminus cplus pmin delta alpha C
conclusion
observedMinimaxLength J n beta kappa L cminus cplus pmin delta alpha
≤ confidenceWorstLength J n beta kappa L cminus cplus pmin C
Proof (Lean source)
-- @node: observedMinimaxLength_le_confidenceWorstLength
lemma observedMinimaxLength_le_confidenceWorstLength
(J n : ℕ) (beta kappa L cminus cplus pmin delta alpha : ℝ)
(C : ConfidenceProcedure n J)
(hC : UniformCoverage J n beta kappa L cminus cplus pmin delta alpha C) :
observedMinimaxLength J n beta kappa L cminus cplus pmin delta alpha ≤
confidenceWorstLength J n beta kappa L cminus cplus pmin C := by
unfold observedMinimaxLength confidenceWorstLength
apply sInf_le
exact ⟨C, hC, rfl⟩
theorem causalStabilizedWorstRisk_eq_observed
Surjectivity and identification transport the concrete estimator's worst-case risk from the observed class to the full-data class. The result uses the hreg condition, the hdelta condition. This is the stated conclusion.
Formal statement
hypotheses
StandardBorelRegularConditionalLaw_of_gate :
J n :
ℕ
B :
beta kappa L cminus cplus pmin deltaBar alpha delta :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
conclusion
causalStabilizedWorstRisk J n B beta kappa L cminus cplus pmin deltaBar delta
= stabilizedWorstRisk J n B beta kappa L cminus cplus pmin deltaBar delta
Proof (Lean source)
-- @node: causalStabilizedWorstRisk_eq_observed
lemma causalStabilizedWorstRisk_eq_observed
(StandardBorelRegularConditionalLaw_of_gate :
StandardBorelRegularConditionalLaw)
(J n : ℕ) (B : SplitBlocks n)
(beta kappa L cminus cplus pmin deltaBar alpha delta : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(hdelta : delta ∈ Icc (0 : ℝ) deltaBar) :
causalStabilizedWorstRisk J n B beta kappa L cminus cplus pmin deltaBar delta =
stabilizedWorstRisk J n B beta kappa L cminus cplus pmin deltaBar delta := by
have hsurj := (observed_margin_surjectivity
StandardBorelRegularConditionalLaw_of_gate J beta kappa L cminus cplus pmin
deltaBar alpha hreg).1
unfold causalStabilizedWorstRisk stabilizedWorstRisk
congr 1
ext v
constructor
· rintro ⟨PF, hPF, rfl⟩
have hbridge := (causal_bridge J beta kappa L cminus cplus pmin deltaBar alpha
PF hPF hreg).2.2 delta hdelta
refine ⟨PF.observedMargin, hPF.observedModel,
clampModel_iidSampling hPF.observedModel, ?_⟩
simp only [causalEstimatorRisk, estimatorRisk, stabilizedEstimatorRisk,
hbridge.2.2.2]
· rintro ⟨P, hP, _hiid, rfl⟩
rcases hsurj P hP with ⟨PF, hPF, rfl⟩
have hbridge := (causal_bridge J beta kappa L cminus cplus pmin deltaBar alpha
PF hPF hreg).2.2 delta hdelta
refine ⟨PF, hPF, ?_⟩
simp only [causalEstimatorRisk, estimatorRisk, stabilizedEstimatorRisk,
hbridge.2.2.2]
theorem causalStabilizedCoverage_eq_observed
Surjectivity and identification transport coverage of the concrete bias-aware interval. The result uses the hreg condition, the hdelta condition. This is the stated conclusion.
Formal statement
hypotheses
StandardBorelRegularConditionalLaw_of_gate :
J n :
ℕ
B :
beta kappa L cminus cplus pmin deltaBar alpha delta :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
conclusion
causalStabilizedCoverage J n B beta kappa L cminus cplus pmin deltaBar delta alpha
= stabilizedCoverage J n B beta kappa L cminus cplus pmin deltaBar delta alpha
Proof (Lean source)
-- @node: causalStabilizedCoverage_eq_observed
lemma causalStabilizedCoverage_eq_observed
(StandardBorelRegularConditionalLaw_of_gate :
StandardBorelRegularConditionalLaw)
(J n : ℕ) (B : SplitBlocks n)
(beta kappa L cminus cplus pmin deltaBar alpha delta : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(hdelta : delta ∈ Icc (0 : ℝ) deltaBar) :
causalStabilizedCoverage J n B beta kappa L cminus cplus pmin deltaBar delta alpha =
stabilizedCoverage J n B beta kappa L cminus cplus pmin deltaBar delta alpha := by
have hsurj := (observed_margin_surjectivity
StandardBorelRegularConditionalLaw_of_gate J beta kappa L cminus cplus pmin
deltaBar alpha hreg).1
unfold causalStabilizedCoverage stabilizedCoverage
congr 1
ext v
constructor
· rintro ⟨PF, hPF, rfl⟩
have hbridge := (causal_bridge J beta kappa L cminus cplus pmin deltaBar alpha
PF hPF hreg).2.2 delta hdelta
refine ⟨PF.observedMargin, hPF.observedModel,
clampModel_iidSampling hPF.observedModel, ?_⟩
congr 2
ext z
simp only [hbridge.2.2.2]
· rintro ⟨P, hP, _hiid, rfl⟩
rcases hsurj P hP with ⟨PF, hPF, rfl⟩
have hbridge := (causal_bridge J beta kappa L cminus cplus pmin deltaBar alpha
PF hPF hreg).2.2 delta hdelta
refine ⟨PF, hPF, ?_⟩
congr 2
ext z
simp only [hbridge.2.2.2]
theorem causalStabilizedWorstLength_eq_confidenceWorstLength
Expected interval length depends only on the observed margin, so surjectivity transports its worst case without any target calculation. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
StandardBorelRegularConditionalLaw_of_gate :
J n :
ℕ
B :
beta kappa L cminus cplus pmin deltaBar alpha delta :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
causalStabilizedWorstLength J n B beta kappa L cminus cplus pmin deltaBar delta alpha
= confidenceWorstLength J n beta kappa L cminus cplus pmin (stabilizedInterval B beta kappa L cminus cplus deltaBar delta alpha)
Proof (Lean source)
-- @node: causalStabilizedWorstLength_eq_confidenceWorstLength
lemma causalStabilizedWorstLength_eq_confidenceWorstLength
(StandardBorelRegularConditionalLaw_of_gate :
StandardBorelRegularConditionalLaw)
(J n : ℕ) (B : SplitBlocks n)
(beta kappa L cminus cplus pmin deltaBar alpha delta : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
causalStabilizedWorstLength J n B beta kappa L cminus cplus pmin deltaBar delta alpha =
confidenceWorstLength J n beta kappa L cminus cplus pmin
(stabilizedInterval B beta kappa L cminus cplus deltaBar delta alpha) := by
have hsurj := (observed_margin_surjectivity
StandardBorelRegularConditionalLaw_of_gate J beta kappa L cminus cplus pmin
deltaBar alpha hreg).1
unfold causalStabilizedWorstLength confidenceWorstLength
congr 1
ext v
constructor
· rintro ⟨PF, hPF, rfl⟩
exact ⟨PF.observedMargin, hPF.observedModel, rfl⟩
· rintro ⟨P, hP, rfl⟩
rcases hsurj P hP with ⟨PF, hPF, rfl⟩
exact ⟨PF, hPF, rfl⟩
Identification transports both observed brackets to the full-data class; the same observed estimator and interval attain both causal frontiers. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
∃ c C : ℝ,
0 < c
c < C
conclusion 1
SplitBlocks n, ∀ᶠ n in atTop, let delta := deltaSeq n let h := infoBandwidth n delta beta kappa deltaBar let r := clampFrontier n delta kappa h beta let crit := causalFrontierCriteria J n beta kappa L cminus cplus pmin deltaBar delta alpha ObservedMeasurableEstimator (n := n) (J := J) (fun z : Fin n → ClampObs J => totalGramEstimator (Bseq n) z (ellOf beta) kappa cminus cplus delta h) ∧ ObservedMeasurableInterval (n := n) (J := J) (stabilizedInterval (J := J) (Bseq n) beta kappa L cminus cplus deltaBar delta alpha) ∧ c * r
≤ crit.1 ∧
crit.1
≤ causalStabilizedWorstRisk J n (Bseq n) beta kappa L cminus cplus pmin deltaBar delta ∧
causalStabilizedWorstRisk J n (Bseq n) beta kappa L cminus cplus pmin deltaBar delta
≤ C * r ∧
1 - alpha
≤ causalStabilizedCoverage J n (Bseq n) beta kappa L cminus cplus pmin deltaBar delta alpha ∧
causalStabilizedWorstLength J n (Bseq n) beta kappa L cminus cplus pmin deltaBar delta alpha
≤ ofReal (C * r) ∧
ofReal (c * r) ≤ crit.2 ∧
crit.2 ≤ ofReal (C * r)
Proof (Lean source)
-- @node: thm:causal-frontier-lift
theorem causal_frontier_lift
(J : ℕ)
(beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
∃ c C : ℝ, 0 < c ∧ c < C ∧
∀ (deltaSeq : ℕ → ℝ), ThresholdSequence deltaBar deltaSeq →
∀ Bseq : ∀ n, SplitBlocks n,
∀ᶠ n in atTop,
let delta := deltaSeq n
let h := infoBandwidth n delta beta kappa deltaBar
let r := clampFrontier n delta kappa h beta
let crit := causalFrontierCriteria J n beta kappa L cminus cplus
pmin deltaBar delta alpha
ObservedMeasurableEstimator (n := n) (J := J)
(fun z : Fin n → ClampObs J =>
totalGramEstimator (Bseq n) z (ellOf beta) kappa cminus cplus delta h) ∧
ObservedMeasurableInterval (n := n) (J := J)
(stabilizedInterval (J := J) (Bseq n) beta kappa L cminus cplus deltaBar
delta alpha) ∧
c * r ≤ crit.1 ∧
crit.1 ≤ causalStabilizedWorstRisk J n (Bseq n) beta kappa L
cminus cplus pmin deltaBar delta ∧
causalStabilizedWorstRisk J n (Bseq n) beta kappa L
cminus cplus pmin deltaBar delta ≤ C * r ∧
1 - alpha ≤ causalStabilizedCoverage J n (Bseq n) beta kappa L
cminus cplus pmin deltaBar delta alpha ∧
causalStabilizedWorstLength J n (Bseq n) beta kappa L
cminus cplus pmin deltaBar delta alpha ≤ ofReal (C * r) ∧
ofReal (c * r) ≤ crit.2 ∧
crit.2 ≤ ofReal (C * r) := by
rcases hreg with ⟨hJ, hbeta, hkappa, hL, hcminus, hcminus_le, hcplus,
hpmin, hpmin_le, hdeltaBar, hdeltaBar1, halpha, halpha_half⟩
have hreg' : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha :=
⟨hJ, hbeta, hkappa, hL, hcminus, hcminus_le, hcplus, hpmin, hpmin_le,
hdeltaBar, hdeltaBar1, halpha, halpha_half⟩
have hgate : StandardBorelRegularConditionalLaw.{0, 0} := by
exact standardBorelRegularConditionalLaw_mathlib.{0, 0}
rcases clamp_minimax_risk J beta kappa L cminus cplus pmin deltaBar alpha hreg' with
⟨cRisk, CRisk, amplitude, hcRisk, hcRiskC, hamp, hamp_le, hRisk⟩
rcases honest_coverage_and_length J beta kappa L cminus cplus pmin deltaBar alpha
hreg' with ⟨cLen, CLen, hcLen, hcLenC, hLength⟩
let c := min cRisk cLen
let C := max CRisk CLen
have hc : 0 < c := lt_min hcRisk hcLen
have hcC : c < C :=
lt_of_le_of_lt (min_le_left _ _) (hcRiskC.trans_le (le_max_left _ _))
refine ⟨c, C, hc, hcC, ?_⟩
intro deltaSeq hdeltaSeq Bseq
have hBand := infoBandwidth_eventually_balance beta kappa deltaBar hbeta hkappa
⟨hdeltaBar, hdeltaBar1⟩ deltaSeq hdeltaSeq
have hRisk' := hRisk deltaSeq hdeltaSeq Bseq
have hLength' := hLength deltaSeq hdeltaSeq Bseq
filter_upwards [hRisk', hLength', hBand, eventually_ge_atTop 4] with
n hnRisk hnLength hnBand hn4
dsimp only at hnRisk hnLength ⊢
let delta := deltaSeq n
let h := infoBandwidth n delta beta kappa deltaBar
let r := clampFrontier n delta kappa h beta
have hdelta : delta ∈ Icc (0 : ℝ) deltaBar := hdeltaSeq n
have hupper : delta + infoBandwidth n delta beta kappa deltaBar ≤ 1 := by
dsimp [delta]
linarith [hdelta.2, hnBand.2.1]
have hr : 0 ≤ r := by
dsimp [r, clampFrontier, h, delta]
exact add_nonneg (Real.rpow_nonneg (Nat.cast_nonneg n) _)
(mul_nonneg (Real.rpow_nonneg hdelta.1 _)
(Real.rpow_nonneg hnBand.1.le _))
have hcRisk' : c ≤ cRisk := min_le_left _ _
have hcLen' : c ≤ cLen := min_le_right _ _
have hCRisk : CRisk ≤ C := le_max_left _ _
have hCLen : CLen ≤ C := le_max_right _ _
have hcrit := (observed_margin_surjectivity
hgate J beta kappa L cminus cplus pmin
deltaBar alpha hreg').2 n delta hdelta
have hriskEq := causalStabilizedWorstRisk_eq_observed
hgate J n (Bseq n) beta kappa L cminus
cplus pmin deltaBar alpha delta hreg' hdelta
have hcovEq := causalStabilizedCoverage_eq_observed
hgate J n (Bseq n) beta kappa L cminus
cplus pmin deltaBar alpha delta hreg' hdelta
have hmeasEst : ObservedMeasurableEstimator
(fun z : Fin n → ClampObs J =>
totalGramEstimator (Bseq n) z (ellOf beta) kappa cminus cplus delta h) := by
exact totalGramEstimator_measurable (Bseq n) kappa cminus cplus delta h
have hmeasInt := stabilizedInterval_observedMeasurable J n (Bseq n) beta kappa L
cminus cplus pmin deltaBar alpha delta hreg' hn4 hnBand.1.le
have huniform := stabilizedInterval_uniformCoverage J n (Bseq n) beta kappa L
cminus cplus pmin deltaBar delta alpha hreg' hn4 hdelta hnBand.1
hupper
have hlenExt := confidenceWorstLength_stabilizedInterval_le_ofReal J n (Bseq n)
beta kappa L cminus cplus pmin deltaBar delta alpha (CLen * r)
hnLength.2.2.1
have hlenC : confidenceWorstLength J n beta kappa L cminus cplus pmin
(stabilizedInterval (Bseq n) beta kappa L cminus cplus deltaBar delta alpha) ≤
ofReal (C * r) :=
hlenExt.trans (ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_right hCLen hr))
have hminUpper := observedMinimaxLength_le_confidenceWorstLength J n beta kappa L
cminus cplus pmin delta alpha _ huniform
refine ⟨hmeasEst, hmeasInt, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩
· rw [hcrit.1]
exact (mul_le_mul_of_nonneg_right hcRisk' hr).trans hnRisk.1
· rw [hcrit.1, hriskEq]
exact hnRisk.2.1
· rw [hriskEq]
exact hnRisk.2.2.1.trans (mul_le_mul_of_nonneg_right hCRisk hr)
· rw [hcovEq]
exact hnLength.1
· rw [causalStabilizedWorstLength_eq_confidenceWorstLength
hgate J n (Bseq n) beta kappa L cminus
cplus pmin deltaBar alpha delta hreg']
exact hlenC
· rw [hcrit.2]
exact (ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_right hcLen' hr)).trans
hnLength.2.2.2.1
· rw [hcrit.2]
exact hminUpper.trans hlenC
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.THonestLength 24 declarations The statement gives finite-sample eventual uniform coverage, expected-length control for the concrete atom-fallback interval, and a converse for every uniformly honest confidence procedure.
Uniformly honest confidence length frontier
The statement gives finite-sample eventual uniform coverage, expected-length control for the concrete atom-fallback interval, and a converse for every uniformly honest confidence procedure.
The paper's confidence procedure at a fixed sample size.
Definition (Lean source)
parameters
def
given by
fun z
=> honestInterval B z (ellOf beta) beta kappa L cminus cplus delta (infoBandwidth n delta beta kappa deltaBar) alpha
Worst-case coverage of the concrete interval over i.i.d. model laws.
Definition (Lean source)
parameters
def
stabilizedCoverage J n B beta kappa L cminus cplus pmin deltaBar delta alpha :
ℝ
given by
sInf {v : ℝ | ∃ P : ClampLaw J, ClampModel P beta kappa L cminus cplus pmin ∧ IidSampling P n ∧ v = (iidProduct P n).real {z | clampFunctional P delta ∈ stabilizedInterval B beta kappa L cminus cplus deltaBar delta alpha z}}
Worst-case expected length of the concrete interval.
Definition (Lean source)
parameters
def
stabilizedWorstLength J n B beta kappa L cminus cplus pmin deltaBar delta alpha :
ℝ
given by
sSup {v : ℝ | ∃ P : ClampLaw J, ClampModel P beta kappa L cminus cplus pmin ∧ IidSampling P n ∧ v = ∫ z, intervalLength (stabilizedInterval B beta kappa L cminus cplus deltaBar delta alpha z) ∂iidProduct P n}
Worst-case expected length of any given confidence procedure.
Definition (Lean source)
parameters
def
confidenceWorstLength J n beta kappa L cminus cplus pmin C :
ℝ≥0∞
given by
sSup {v : ℝ≥0∞ | ∃ P : ClampLaw J, ClampModel P beta kappa L cminus cplus pmin ∧ v = ∫⁻ z, intervalLengthENNReal (C z) ∂iidProduct P n}
theorem stabilizedCoverage_ge
The modelwise finite-sample coverage inequality passes through the worst-model infimum defining the paper's concrete coverage criterion. The result uses the hreg condition, the hn condition, the hdelta condition, the hh condition, the hupper condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
B :
beta kappa L cminus cplus pmin deltaBar delta alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
hn :
4 ≤ n
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
hh :
0 < infoBandwidth n delta beta kappa deltaBar
hupper :
delta + infoBandwidth n delta beta kappa deltaBar ≤ 1
conclusion
1 - alpha ≤ stabilizedCoverage J n B beta kappa L cminus cplus pmin deltaBar delta alpha
Proof (Lean source)
lemma stabilizedCoverage_ge
(J n : ℕ) (B : SplitBlocks n)
(beta kappa L cminus cplus pmin deltaBar delta alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(hn : 4 ≤ n) (hdelta : delta ∈ Icc (0 : ℝ) deltaBar)
(hh : 0 < infoBandwidth n delta beta kappa deltaBar)
(hupper : delta + infoBandwidth n delta beta kappa deltaBar ≤ 1) :
1 - alpha ≤ stabilizedCoverage J n B beta kappa L cminus cplus pmin
deltaBar delta alpha := by
have hreg' := hreg
rcases hreg' with ⟨hJ, hbeta, hkappa, hL, hcminus, hcminusK, hcplus,
hpmin, hpminJ, hdbar, hdbar1, ha, ha1⟩
unfold stabilizedCoverage
apply le_csInf
· let P0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
have hP0 : ClampModel P0 beta kappa L cminus cplus pmin :=
minimaxCenter_mem_model J beta kappa L cminus cplus pmin deltaBar alpha hreg
exact ⟨(iidProduct P0 n).real {z | clampFunctional P0 delta ∈
stabilizedInterval B beta kappa L cminus cplus deltaBar delta alpha z},
P0, hP0, clampModel_iidSampling hP0, rfl⟩
· intro v hv
rcases hv with ⟨P, hP, _hiid, rfl⟩
simpa [stabilizedInterval] using
honestInterval_coverage_model P B beta kappa L cminus cplus pmin delta
(infoBandwidth n delta beta kappa deltaBar) alpha hP hJ hbeta hL ha ha1
hn hdelta.1 hupper hh
(totalGram_lambdaStar_pos J beta kappa L cminus cplus pmin deltaBar alpha hreg)
The concrete stabilized interval is an admissible uniformly covering procedure at every sample size where the bandwidth lies inside the design support. The result uses the hreg condition, the hn condition, the hdelta condition, the hh condition, the hupper condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
B :
beta kappa L cminus cplus pmin deltaBar delta alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
hn :
4 ≤ n
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
hh :
0 < infoBandwidth n delta beta kappa deltaBar
hupper :
delta + infoBandwidth n delta beta kappa deltaBar ≤ 1
conclusion
UniformCoverage J n beta kappa L cminus cplus pmin delta alpha (stabilizedInterval B beta kappa L cminus cplus deltaBar delta alpha)
Proof (Lean source)
lemma stabilizedInterval_uniformCoverage
(J n : ℕ) (B : SplitBlocks n)
(beta kappa L cminus cplus pmin deltaBar delta alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(hn : 4 ≤ n) (hdelta : delta ∈ Icc (0 : ℝ) deltaBar)
(hh : 0 < infoBandwidth n delta beta kappa deltaBar)
(hupper : delta + infoBandwidth n delta beta kappa deltaBar ≤ 1) :
UniformCoverage J n beta kappa L cminus cplus pmin delta alpha
(stabilizedInterval B beta kappa L cminus cplus deltaBar delta alpha) := by
constructor
· exact honestInterval_observedMeasurable J n B beta kappa L cminus cplus
pmin deltaBar alpha delta hreg hn hh.le
· intro P hP
have hreg' := hreg
rcases hreg' with ⟨hJ, hbeta, hkappa, hL, hcminus, hcminusK, hcplus,
hpmin, hpminJ, hdbar, hdbar1, ha, ha1⟩
simpa [stabilizedInterval] using
honestInterval_coverage_model P B beta kappa L cminus cplus pmin delta
(infoBandwidth n delta beta kappa deltaBar) alpha hP hJ hbeta hL ha ha1
hn hdelta.1 hupper hh
(totalGram_lambdaStar_pos J beta kappa L cminus cplus pmin deltaBar alpha hreg)
The bias-aware interval is uniformly honest and rate-optimal in worst-case expected length, including singular-Gram samples via its atom fallback. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
∃ c C : ℝ,
0 < c
c < C
conclusion 1
SplitBlocks n, ∀ᶠ n in atTop, let delta := deltaSeq n let h := infoBandwidth n delta beta kappa deltaBar let r := clampFrontier n delta kappa h beta 1
- alpha
≤ stabilizedCoverage J n (Bseq n) beta kappa L cminus cplus pmin deltaBar delta alpha ∧
c * r
≤ stabilizedWorstLength J n (Bseq n) beta kappa L cminus cplus pmin deltaBar delta alpha ∧
stabilizedWorstLength J n (Bseq n) beta kappa L cminus cplus pmin deltaBar delta alpha
≤ C * r ∧
ofReal (c * r)
≤ observedMinimaxLength J n beta kappa L cminus cplus pmin delta alpha ∧
∀ Cn : ConfidenceProcedure n J,
≤ confidenceWorstLength J n beta kappa L cminus cplus pmin Cn
Proof (Lean source)
-- @node: thm:honest-length
theorem honest_coverage_and_length
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
∃ c C : ℝ, 0 < c ∧ c < C ∧
∀ (deltaSeq : ℕ → ℝ), ThresholdSequence deltaBar deltaSeq →
∀ Bseq : ∀ n, SplitBlocks n,
∀ᶠ n in atTop,
let delta := deltaSeq n
let h := infoBandwidth n delta beta kappa deltaBar
let r := clampFrontier n delta kappa h beta
1 - alpha ≤ stabilizedCoverage J n (Bseq n) beta kappa L cminus cplus
pmin deltaBar delta alpha ∧
c * r ≤ stabilizedWorstLength J n (Bseq n) beta kappa L cminus cplus
pmin deltaBar delta alpha ∧
stabilizedWorstLength J n (Bseq n) beta kappa L cminus cplus
pmin deltaBar delta alpha ≤ C * r ∧
ofReal (c * r) ≤ observedMinimaxLength J n beta kappa L cminus cplus
pmin delta alpha ∧
∀ Cn : ConfidenceProcedure n J,
UniformCoverage J n beta kappa L cminus cplus pmin delta alpha Cn →
ofReal (c * r) ≤
confidenceWorstLength J n beta kappa L cminus cplus pmin Cn := by
obtain ⟨Cu, hCu, huRaw⟩ := honestInterval_worstLength_eventually_le_frontier
J beta kappa L cminus cplus pmin deltaBar alpha hreg
have hu : ∀ (deltaSeq : ℕ → ℝ), ThresholdSequence deltaBar deltaSeq →
∀ Bseq : ∀ n, SplitBlocks n, ∀ᶠ n in atTop,
stabilizedWorstLength J n (Bseq n) beta kappa L cminus cplus pmin
deltaBar (deltaSeq n) alpha ≤ Cu * clampFrontier n (deltaSeq n) kappa
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) beta := by
intro deltaSeq hdelta Bseq
simpa [stabilizedWorstLength, stabilizedInterval] using huRaw deltaSeq hdelta Bseq
obtain ⟨co, hco, ho⟩ := observedMinimaxLength_eventually_ge_frontier J beta
kappa L cminus cplus pmin deltaBar alpha hreg
obtain ⟨cp, hcp, hp⟩ := procedureWorstLength_eventually_ge_frontier J beta
kappa L cminus cplus pmin deltaBar alpha hreg
let c := min co cp / 2
let C := Cu + c + 1
have hc : 0 < c := by dsimp [c]; positivity
have hcC : c < C := by dsimp [C]; linarith
refine ⟨c, C, hc, hcC, ?_⟩
intro deltaSeq hdelta Bseq
have hbal := infoBandwidth_eventually_balance beta kappa deltaBar hreg.2.1
hreg.2.2.1 ⟨hreg.2.2.2.2.2.2.2.2.2.1,
hreg.2.2.2.2.2.2.2.2.2.2.1⟩ deltaSeq hdelta
filter_upwards [hu deltaSeq hdelta Bseq, ho deltaSeq hdelta,
hp deltaSeq hdelta, hbal, eventually_ge_atTop 8] with n hun hon hpn hbn hn
let delta := deltaSeq n
let h := infoBandwidth n delta beta kappa deltaBar
let r := clampFrontier n delta kappa h beta
have hdeltaN : delta ∈ Icc (0 : ℝ) deltaBar := hdelta n
have hh : 0 < h := by simpa [h, delta] using hbn.1
have hupper : delta + h ≤ 1 := by
have := hbn.2.1
dsimp [h, delta]
linarith [hdeltaN.2]
have hrpos : 0 < r := by
dsimp [r, clampFrontier]
exact add_pos_of_pos_of_nonneg
(Real.rpow_pos_of_pos (by exact_mod_cast (show 0 < n by omega)) _)
(mul_nonneg (Real.rpow_nonneg hdeltaN.1 _) (Real.rpow_nonneg hh.le _))
have hr : 0 ≤ r := hrpos.le
have hcov : 1 - alpha ≤ stabilizedCoverage J n (Bseq n) beta kappa L
cminus cplus pmin deltaBar delta alpha :=
stabilizedCoverage_ge J n (Bseq n) beta kappa L cminus cplus pmin deltaBar
delta alpha hreg (by omega) hdeltaN (by simpa [h]) (by simpa [h] using hupper)
have hUC : UniformCoverage J n beta kappa L cminus cplus pmin delta alpha
(stabilizedInterval (Bseq n) beta kappa L cminus cplus deltaBar delta alpha) :=
stabilizedInterval_uniformCoverage J n (Bseq n) beta kappa L cminus cplus pmin
deltaBar delta alpha hreg (by omega) hdeltaN (by simpa [h])
(by simpa [h] using hupper)
have hc_co : c ≤ co := by
dsimp [c]
have hm := min_le_left co cp
nlinarith [hco]
have hc_cp : c ≤ cp := by
dsimp [c]
have hm := min_le_right co cp
nlinarith [hcp]
have hobs : ofReal (c * r) ≤ observedMinimaxLength J n beta kappa L
cminus cplus pmin delta alpha := by
exact (ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_right hc_co hr)).trans
(by simpa [r, h, delta] using hon)
have hconcreteENN : ofReal (c * r) ≤ ofReal
(stabilizedWorstLength J n (Bseq n) beta kappa L cminus cplus pmin
deltaBar delta alpha) := by
calc
_ ≤ ofReal (cp * r) := ENNReal.ofReal_le_ofReal
(mul_le_mul_of_nonneg_right hc_cp hr)
_ ≤ clampProcedureWorstLength J n beta kappa L cminus cplus pmin
(stabilizedInterval (Bseq n) beta kappa L cminus cplus deltaBar delta alpha) := by
simpa [r, h, delta] using hpn _ hUC
_ ≤ _ := by
apply clampProcedureWorstLength_honestInterval_le_ofReal
rfl
have hswpos : 0 < stabilizedWorstLength J n (Bseq n) beta kappa L
cminus cplus pmin deltaBar delta alpha := by
apply ENNReal.ofReal_pos.mp
exact (ENNReal.ofReal_pos.2 (mul_pos hc hrpos)).trans_le hconcreteENN
have hconcrete : c * r ≤ stabilizedWorstLength J n (Bseq n) beta kappa L
cminus cplus pmin deltaBar delta alpha :=
(ENNReal.ofReal_le_ofReal_iff hswpos.le).mp hconcreteENN
have huC : stabilizedWorstLength J n (Bseq n) beta kappa L cminus cplus pmin
deltaBar delta alpha ≤ C * r := by
have hCuC : Cu ≤ C := by dsimp [C]; linarith
have hun' : stabilizedWorstLength J n (Bseq n) beta kappa L cminus cplus pmin
deltaBar delta alpha ≤ Cu * r := by simpa [r, h, delta] using hun
exact hun'.trans (mul_le_mul_of_nonneg_right hCuC hr)
refine ⟨hcov, hconcrete, huC, hobs, ?_⟩
intro Cn hCn
have hpc := hpn Cn hCn
exact (ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_right hc_cp hr)).trans
(by simpa [confidenceWorstLength, clampProcedureWorstLength, r, h, delta] using hpc)
Worst-case continuity-only risk of the fixed one-half fallback estimator.
Definition (Lean source)
parameters
def
contStabilizedWorstRisk J n B kappa cminus cplus pmin deltaBar delta :
ℝ
given by
sSup {v : ℝ | ∃ (P : ClampLaw J) (hP : ContClampModel P kappa cminus cplus pmin deltaBar), IidSampling P n ∧ v = contEstimatorRisk P n kappa cminus cplus pmin deltaBar delta hP (contFallbackEstimator B · delta)}
Worst-case coverage of the continuity-only Hoeffding interval.
Definition (Lean source)
parameters
def
contStabilizedCoverage J n B kappa cminus cplus pmin deltaBar delta alpha :
ℝ
given by
sInf {v : ℝ | ∃ (P : ClampLaw J) (hP : ContClampModel P kappa cminus cplus pmin deltaBar), IidSampling P n ∧ v = (iidProduct P n).real {z | contClampFunctional P kappa cminus cplus pmin deltaBar hP delta ∈ contHoeffdingInterval B z delta alpha}}
Worst-case extended expected length of the continuity-only interval.
Definition (Lean source)
parameters
def
contStabilizedWorstLength J n B kappa cminus cplus pmin deltaBar delta alpha :
ℝ≥0∞
given by
sSup {v : ℝ≥0∞ | ∃ (P : ClampLaw J) (_hP : ContClampModel P kappa cminus cplus pmin deltaBar), IidSampling P n ∧ v = ∫⁻ z, intervalLengthENNReal (contHoeffdingInterval B z delta alpha) ∂iidProduct P n}
Worst-case causal risk of the same continuity-only observed-sample fallback estimator, over law-specific latent carriers.
Definition (Lean source)
parameters
def
contCausalStabilizedWorstRisk J n B kappa cminus cplus pmin deltaBar delta :
ℝ
given by
sSup {v : ℝ | ∃ (PF : FullDataLaw J) (_hPF : ContFullDataClampModel PF kappa cminus cplus pmin deltaBar), v = causalEstimatorRisk PF n delta (contFallbackEstimator B · delta)}
Worst-case causal coverage of the same continuity-only interval, over law-specific latent carriers.
Definition (Lean source)
parameters
def
contCausalStabilizedCoverage J n B kappa cminus cplus pmin deltaBar delta alpha :
ℝ
given by
sInf {v : ℝ | ∃ (PF : FullDataLaw J) (_hPF : ContFullDataClampModel PF kappa cminus cplus pmin deltaBar), v = (iidProduct PF.observedMargin n).real {z | causalClampMean PF delta ∈ contHoeffdingInterval B z delta alpha}}
Worst-case causal expected length of the same continuity-only interval, over law-specific latent carriers.
Definition (Lean source)
parameters
def
contCausalStabilizedWorstLength J n B kappa cminus cplus pmin deltaBar delta alpha :
ℝ≥0∞
given by
sSup {v : ℝ≥0∞ | ∃ (PF : FullDataLaw J) (_hPF : ContFullDataClampModel PF kappa cminus cplus pmin deltaBar), v = ∫⁻ z, intervalLengthENNReal (contHoeffdingInterval B z delta alpha) ∂iidProduct PF.observedMargin n}
theorem contFallbackEstimator_admissible
The fixed-fallback estimator is measurable and remains in the outcome range. This is the stated conclusion.
Formal statement
hypotheses
conclusion
conclusion 1
Proof (Lean source)
lemma contFallbackEstimator_admissible {J n : ℕ} (B : SplitBlocks n)
(delta : ℝ) :
ObservedMeasurableEstimator (fun z : Fin n → ClampObs J =>
contFallbackEstimator B z delta) ∧
∀ z : Fin n → ClampObs J,
contFallbackEstimator B z delta ∈ Icc (0 : ℝ) 1 := by
constructor
· unfold ObservedMeasurableEstimator contFallbackEstimator
apply clampUnit_measurable'.comp
exact (retainedEstimate_measurable B delta).add
(measurable_const.mul (Finset.measurable_fun_sum _ fun x _ =>
atomEstimate_measurable B x delta))
· intro z
exact clampUnit_mem_Icc _
theorem contCausalStabilizedWorstRisk_eq_observed
Surjectivity and identification equate the concrete observed and causal worst-case risks. The result uses the hreg condition, the hdelta condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
B :
kappa cminus cplus pmin deltaBar delta alpha :
ℝ
hreg :
ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
conclusion
contCausalStabilizedWorstRisk J n B kappa cminus cplus pmin deltaBar delta
= contStabilizedWorstRisk J n B kappa cminus cplus pmin deltaBar delta
Proof (Lean source)
lemma contCausalStabilizedWorstRisk_eq_observed
(J n : ℕ) (B : SplitBlocks n)
(kappa cminus cplus pmin deltaBar delta alpha : ℝ)
(hreg : ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha)
(hdelta : delta ∈ Icc (0 : ℝ) deltaBar) :
contCausalStabilizedWorstRisk J n B kappa cminus cplus pmin deltaBar delta =
contStabilizedWorstRisk J n B kappa cminus cplus pmin deltaBar delta := by
unfold contCausalStabilizedWorstRisk contStabilizedWorstRisk
apply congrArg sSup
ext v
constructor
· rintro ⟨PF, hPF, rfl⟩
refine ⟨PF.observedMargin, hPF.observedModel,
contClampModel_iidSampling hPF.observedModel, ?_⟩
exact causalEstimatorRisk_eq_contEstimatorRisk J n kappa cminus cplus pmin
deltaBar delta hreg.1 hdelta PF hPF _
· rintro ⟨P, hP, _hiid, rfl⟩
obtain ⟨PF, hPF, hmargin⟩ := exists_fullData_cont_lift P hP
hreg.1.2.2.2.2.2.1
subst P
refine ⟨PF, hPF, ?_⟩
exact (causalEstimatorRisk_eq_contEstimatorRisk J n kappa cminus cplus pmin
deltaBar delta hreg.1 hdelta PF hPF _).symm
theorem contCausalStabilizedCoverage_eq_observed
Surjectivity and identification equate the concrete observed and causal worst-case coverages. The result uses the hreg condition, the hdelta condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
B :
kappa cminus cplus pmin deltaBar delta alpha :
ℝ
hreg :
ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
conclusion
contCausalStabilizedCoverage J n B kappa cminus cplus pmin deltaBar delta alpha
= contStabilizedCoverage J n B kappa cminus cplus pmin deltaBar delta alpha
Proof (Lean source)
lemma contCausalStabilizedCoverage_eq_observed
(J n : ℕ) (B : SplitBlocks n)
(kappa cminus cplus pmin deltaBar delta alpha : ℝ)
(hreg : ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha)
(hdelta : delta ∈ Icc (0 : ℝ) deltaBar) :
contCausalStabilizedCoverage J n B kappa cminus cplus pmin deltaBar delta alpha =
contStabilizedCoverage J n B kappa cminus cplus pmin deltaBar delta alpha := by
have htarget (PF : FullDataLaw J)
(hPF : ContFullDataClampModel PF kappa cminus cplus pmin deltaBar) :
causalClampMean PF delta = contClampFunctional PF.observedMargin kappa
cminus cplus pmin deltaBar hPF.observedModel delta :=
(continuity_causal_bridge J kappa cminus cplus pmin deltaBar PF hPF hreg.1).2
delta hdelta
unfold contCausalStabilizedCoverage contStabilizedCoverage
apply congrArg sInf
ext v
constructor
· rintro ⟨PF, hPF, rfl⟩
refine ⟨PF.observedMargin, hPF.observedModel,
contClampModel_iidSampling hPF.observedModel, ?_⟩
rw [htarget PF hPF]
· rintro ⟨P, hP, _hiid, rfl⟩
obtain ⟨PF, hPF, hmargin⟩ := exists_fullData_cont_lift P hP
hreg.1.2.2.2.2.2.1
subst P
refine ⟨PF, hPF, ?_⟩
rw [htarget PF hPF]
theorem contCausalStabilizedWorstLength_eq_observed
Surjectivity equates the concrete observed and causal worst-case expected lengths. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
B :
kappa cminus cplus pmin deltaBar delta alpha :
ℝ
hreg :
ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha
conclusion
contCausalStabilizedWorstLength J n B kappa cminus cplus pmin deltaBar delta alpha
= contStabilizedWorstLength J n B kappa cminus cplus pmin deltaBar delta alpha
Proof (Lean source)
lemma contCausalStabilizedWorstLength_eq_observed
(J n : ℕ) (B : SplitBlocks n)
(kappa cminus cplus pmin deltaBar delta alpha : ℝ)
(hreg : ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha) :
contCausalStabilizedWorstLength J n B kappa cminus cplus pmin deltaBar delta alpha =
contStabilizedWorstLength J n B kappa cminus cplus pmin deltaBar delta alpha := by
unfold contCausalStabilizedWorstLength contStabilizedWorstLength
apply congrArg sSup
ext v
constructor
· rintro ⟨PF, hPF, rfl⟩
exact ⟨PF.observedMargin, hPF.observedModel,
contClampModel_iidSampling hPF.observedModel, rfl⟩
· rintro ⟨P, hP, _hiid, rfl⟩
obtain ⟨PF, hPF, hmargin⟩ := exists_fullData_cont_lift P hP
hreg.1.2.2.2.2.2.1
exact ⟨PF, hPF, by rw [hmargin]⟩
theorem contFrontierCriteria_risk_le_stabilized
The fixed fallback is an admissible competitor for the observed minimax risk criterion. The result uses the hreg condition, the hdelta condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
B :
kappa cminus cplus pmin deltaBar delta alpha :
ℝ
hreg :
ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
conclusion
(contFrontierCriteria J n kappa cminus cplus pmin deltaBar delta alpha).1
≤ contStabilizedWorstRisk J n B kappa cminus cplus pmin deltaBar delta
Proof (Lean source)
lemma contFrontierCriteria_risk_le_stabilized
(J n : ℕ) (B : SplitBlocks n)
(kappa cminus cplus pmin deltaBar delta alpha : ℝ)
(hreg : ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha)
(hdelta : delta ∈ Icc (0 : ℝ) deltaBar) :
(contFrontierCriteria J n kappa cminus cplus pmin deltaBar delta alpha).1 ≤
contStabilizedWorstRisk J n B kappa cminus cplus pmin deltaBar delta := by
simp only [contFrontierCriteria, fst]
unfold contStabilizedWorstRisk
let outer : Set ℝ := {r : ℝ | ∃ est : Estimator n J,
ObservedMeasurableEstimator est ∧ (∀ z, est z ∈ Icc (0 : ℝ) 1) ∧
r = sSup {v : ℝ | ∃ (P : ClampLaw J)
(hP : ContClampModel P kappa cminus cplus pmin deltaBar),
v = contEstimatorRisk P n kappa cminus cplus pmin deltaBar delta hP est}}
change sInf outer ≤ _
have houterBelow : BddBelow outer := by
refine ⟨0, ?_⟩
rintro r ⟨est, hmeas, hrange, rfl⟩
let risks : Set ℝ := {v : ℝ | ∃ (P : ClampLaw J)
(hP : ContClampModel P kappa cminus cplus pmin deltaBar),
v = contEstimatorRisk P n kappa cminus cplus pmin deltaBar delta hP est}
have hbdd : BddAbove risks := by
refine ⟨1, ?_⟩
rintro v ⟨P, hP, rfl⟩
letI : IsProbabilityMeasure P.dataMeasure := hP.probability
letI : IsProbabilityMeasure (iidProduct P n) := by unfold iidProduct; infer_instance
let target := contClampFunctional P kappa cminus cplus pmin deltaBar hP delta
have ht := contClampFunctional_mem_Icc P kappa cminus cplus pmin deltaBar
delta hP hreg.1 hdelta
have hpoint (z : Fin n → ClampObs J) : |est z - target| ≤ 1 := by
rw [abs_le]
dsimp [target]
constructor <;> linarith [(hrange z).1, (hrange z).2, ht.1, ht.2]
unfold contEstimatorRisk
change (∫ z, |est z - target| ∂iidProduct P n) ≤ 1
have hmono := integral_mono_ae
(Integrable.of_bound
((hmeas.sub (measurable_const : Measurable
(fun _ : Fin n → ClampObs J => target))).abs.aestronglyMeasurable) 1
(ae_of_all _ fun z => by
rw [Real.norm_eq_abs, abs_abs]
exact hpoint z))
(integrable_const (μ := iidProduct P n) 1) (ae_of_all _ hpoint)
simpa using hmono
let P0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
have hP0 : ContClampModel P0 kappa cminus cplus pmin deltaBar :=
minimaxClampModel_continuous J kappa cminus cplus pmin deltaBar
hreg.1.1 hreg.1.2.1 hreg.1.2.2.2.1 hreg.1.2.2.2.2.1
hreg.1.2.2.2.2.2.2.1 (fun _ => 0) measurable_const
(fun _ => by norm_num) continuous_const.continuousOn
have hnon : 0 ≤ contEstimatorRisk P0 n kappa cminus cplus pmin deltaBar
delta hP0 est := by
unfold contEstimatorRisk
exact integral_nonneg_of_ae (ae_of_all _ fun z => abs_nonneg _)
exact hnon.trans (le_csSup hbdd ⟨P0, hP0, rfl⟩)
apply csInf_le houterBelow
refine ⟨(fun z => contFallbackEstimator B z delta),
(contFallbackEstimator_admissible B delta).1,
(contFallbackEstimator_admissible B delta).2, ?_⟩
apply congrArg sSup
ext v
constructor
· rintro ⟨P, hP, _hiid, hv⟩
exact ⟨P, hP, hv⟩
· rintro ⟨P, hP, hv⟩
exact ⟨P, hP, contClampModel_iidSampling hP, hv⟩
theorem contStabilizedCoverage_ge
The concrete continuity interval has uniform finite-sample coverage once both deterministic blocks are nonempty. The result uses the HoeffdingBoundedAverage_of_gate condition, the hreg condition, the hdelta condition, the hcard0 condition, the hcard1 condition. This is the stated conclusion.
Formal statement
hypotheses
HoeffdingBoundedAverage_of_gate :
J n :
ℕ
B :
kappa cminus cplus pmin deltaBar delta alpha :
ℝ
hreg :
ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha
hdelta :
delta ∈ Icc (0 : ℝ) deltaBar
hcard0 :
0 < B.I0.card
hcard1 :
0 < B.I1.card
conclusion
1 - alpha ≤ contStabilizedCoverage J n B kappa cminus cplus pmin deltaBar delta alpha
Proof (Lean source)
lemma contStabilizedCoverage_ge
(HoeffdingBoundedAverage_of_gate : HoeffdingBoundedAverage)
(J n : ℕ) (B : SplitBlocks n)
(kappa cminus cplus pmin deltaBar delta alpha : ℝ)
(hreg : ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha)
(hdelta : delta ∈ Icc (0 : ℝ) deltaBar)
(hcard0 : 0 < B.I0.card) (hcard1 : 0 < B.I1.card) :
1 - alpha ≤ contStabilizedCoverage J n B kappa cminus cplus pmin
deltaBar delta alpha := by
unfold contStabilizedCoverage
apply le_csInf
· let P0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
have hP0 : ContClampModel P0 kappa cminus cplus pmin deltaBar :=
minimaxClampModel_continuous J kappa cminus cplus pmin deltaBar
hreg.1.1 hreg.1.2.1 hreg.1.2.2.2.1 hreg.1.2.2.2.2.1
hreg.1.2.2.2.2.2.2.1 (fun _ => 0) measurable_const
(fun _ => by norm_num) continuous_const.continuousOn
exact ⟨(iidProduct P0 n).real {z |
contClampFunctional P0 kappa cminus cplus pmin deltaBar hP0 delta ∈
contHoeffdingInterval B z delta alpha}, P0, hP0,
contClampModel_iidSampling hP0, rfl⟩
· intro v hv
rcases hv with ⟨P, hP, _hiid, rfl⟩
exact contHoeffdingInterval_coverage_model HoeffdingBoundedAverage_of_gate
P B kappa cminus cplus pmin deltaBar delta alpha hP hreg hdelta hcard0 hcard1
theorem contStabilizedWorstRisk_eventually_le_frontier
The concrete fixed-fallback worst-case risk attains the continuity rate. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin deltaBar alpha :
ℝ
hreg :
ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha
conclusion
∃ C : ℝ,
0 < C
conclusion 1
SplitBlocks n, ∀ᶠ n : ℕ in atTop, contStabilizedWorstRisk J n (Bseq n) kappa cminus cplus pmin deltaBar (deltaSeq n)
≤ C * contFrontier n (deltaSeq n) kappa
Proof (Lean source)
lemma contStabilizedWorstRisk_eventually_le_frontier
(J : ℕ) (kappa cminus cplus pmin deltaBar alpha : ℝ)
(hreg : ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha) :
∃ C : ℝ, 0 < C ∧ ∀ deltaSeq : ℕ → ℝ,
ThresholdSequence deltaBar deltaSeq → ∀ Bseq : ∀ n, SplitBlocks n,
∀ᶠ n : ℕ in atTop,
contStabilizedWorstRisk J n (Bseq n) kappa cminus cplus pmin deltaBar
(deltaSeq n) ≤ C * contFrontier n (deltaSeq n) kappa := by
let C : ℝ := 3 * (1 + (J : ℝ)) * (1 + cplus / (kappa + 1))
have hk1 : 0 < kappa + 1 := by linarith [hreg.1.2.1]
have hcplus0 : 0 < cplus := hk1.trans_le hreg.1.2.2.2.2.1
have hC : 0 < C := by dsimp [C]; positivity
refine ⟨C, hC, ?_⟩
intro deltaSeq hdelta Bseq
filter_upwards [eventually_ge_atTop 8] with n hn
have hnpos : 0 < n := by omega
have hcard0 : 0 < (Bseq n).I0.card := by
have := (Bseq n).card_I0
omega
have hcard1 : 0 < (Bseq n).I1.card := by
have := (Bseq n).card_I1
omega
let root : ℝ := (n : ℝ) ^ (-(1 : ℝ) / 2)
let atom : ℝ := (deltaSeq n) ^ (kappa + 1)
have hroot : 0 ≤ root := Real.rpow_nonneg (Nat.cast_nonneg n) _
have hatom : 0 ≤ atom := Real.rpow_nonneg (hdelta n).1 _
have hsqrt0 : sqrt ((Bseq n).I0.card : ℝ)⁻¹ ≤ 3 * root := by
apply splitBlock_invSqrt_le_root hnpos
have := (Bseq n).card_I0
omega
have hsqrt1 : sqrt ((Bseq n).I1.card : ℝ)⁻¹ ≤ 3 * root := by
apply splitBlock_invSqrt_le_root hnpos
have := (Bseq n).card_I1
omega
apply csSup_le
· let P0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
have hP0 : ContClampModel P0 kappa cminus cplus pmin deltaBar :=
minimaxClampModel_continuous J kappa cminus cplus pmin deltaBar
hreg.1.1 hreg.1.2.1 hreg.1.2.2.2.1 hreg.1.2.2.2.2.1
hreg.1.2.2.2.2.2.2.1 (fun _ => 0) measurable_const
(fun _ => by norm_num) continuous_const.continuousOn
exact ⟨contEstimatorRisk P0 n kappa cminus cplus pmin deltaBar
(deltaSeq n) hP0 (contFallbackEstimator (Bseq n) · (deltaSeq n)),
P0, hP0, contClampModel_iidSampling hP0, rfl⟩
· intro v hv
rcases hv with ⟨P, hP, _hiid, rfl⟩
have he := contFallbackEstimator_risk_le_explicit P (Bseq n) kappa cminus
cplus pmin deltaBar (deltaSeq n) hP hreg.1 (hdelta n) hcard0 hcard1
have hcoef : (3 / 2 : ℝ) + (J : ℝ) * (3 / 4) ≤ C := by
dsimp [C]
have hJ : 0 ≤ (J : ℝ) := Nat.cast_nonneg J
have hratio : 0 ≤ cplus / (kappa + 1) := by positivity
nlinarith
have hatomcoef : (J : ℝ) * ((1 / 2 : ℝ) * (cplus / (kappa + 1))) ≤ C := by
dsimp [C]
have hJ : 0 ≤ (J : ℝ) := Nat.cast_nonneg J
have hratio : 0 ≤ cplus / (kappa + 1) := by positivity
nlinarith
calc
_ ≤ (1 / 2 : ℝ) * (3 * root) +
(J : ℝ) * ((1 / 4 : ℝ) * (3 * root) +
(1 / 2 : ℝ) * (cplus * atom / (kappa + 1))) := by
exact he.trans (add_le_add
(mul_le_mul_of_nonneg_left hsqrt0 (by norm_num))
(mul_le_mul_of_nonneg_left
(add_le_add
(mul_le_mul_of_nonneg_left hsqrt1 (by norm_num)) le_rfl)
(Nat.cast_nonneg J)))
_ = ((3 / 2 : ℝ) + (J : ℝ) * (3 / 4)) * root +
((J : ℝ) * ((1 / 2 : ℝ) * (cplus / (kappa + 1)))) * atom := by ring
_ ≤ C * root + C * atom := add_le_add
(mul_le_mul_of_nonneg_right hcoef hroot)
(mul_le_mul_of_nonneg_right hatomcoef hatom)
_ = C * contFrontier n (deltaSeq n) kappa := by
dsimp [root, atom]
unfold contFrontier
ring
theorem contStabilizedWorstLength_eventually_le_frontier
The extended expected length of the concrete interval attains the same continuity frontier. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin deltaBar alpha :
ℝ
hreg :
ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha
conclusion
∃ C : ℝ,
0 < C
conclusion 1
SplitBlocks n, ∀ᶠ n : ℕ in atTop, contStabilizedWorstLength J n (Bseq n) kappa cminus cplus pmin deltaBar (deltaSeq n) alpha
≤ ofReal (C * contFrontier n (deltaSeq n) kappa)
Proof (Lean source)
lemma contStabilizedWorstLength_eventually_le_frontier
(J : ℕ) (kappa cminus cplus pmin deltaBar alpha : ℝ)
(hreg : ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha) :
∃ C : ℝ, 0 < C ∧ ∀ deltaSeq : ℕ → ℝ,
ThresholdSequence deltaBar deltaSeq → ∀ Bseq : ∀ n, SplitBlocks n,
∀ᶠ n : ℕ in atTop,
contStabilizedWorstLength J n (Bseq n) kappa cminus cplus pmin deltaBar
(deltaSeq n) alpha ≤ ofReal
(C * contFrontier n (deltaSeq n) kappa) := by
let K := sqrt (log (4 / alpha) / 2)
let C : ℝ := 1 + 12 * K + (J : ℝ) * (3 / 2 + cplus / (kappa + 1))
have hk1 : 0 < kappa + 1 := by linarith [hreg.1.2.1]
have hcplus0 : 0 < cplus := hk1.trans_le hreg.1.2.2.2.2.1
have hquot : 1 < 4 / alpha := by
rw [lt_div_iff₀ hreg.2.1]
linarith [hreg.2.2]
have hloghalf : 0 ≤ log (4 / alpha) / 2 :=
div_nonneg (Real.log_pos hquot).le (by norm_num)
have hK : 0 ≤ K := Real.sqrt_nonneg _
have hC : 0 < C := by
dsimp [C]
have hJ : 0 ≤ (J : ℝ) := Nat.cast_nonneg J
have hratio : 0 ≤ cplus / (kappa + 1) := by positivity
nlinarith
refine ⟨C, hC, ?_⟩
intro deltaSeq hdelta Bseq
filter_upwards [eventually_ge_atTop 8] with n hn
have hnpos : 0 < n := by omega
have hcard0 : 0 < (Bseq n).I0.card := by
have := (Bseq n).card_I0
omega
have hcard1 : 0 < (Bseq n).I1.card := by
have := (Bseq n).card_I1
omega
let root : ℝ := (n : ℝ) ^ (-(1 : ℝ) / 2)
let atom : ℝ := (deltaSeq n) ^ (kappa + 1)
have hroot : 0 ≤ root := Real.rpow_nonneg (Nat.cast_nonneg n) _
have hatom : 0 ≤ atom := Real.rpow_nonneg (hdelta n).1 _
have hsqrt0 : sqrt ((Bseq n).I0.card : ℝ)⁻¹ ≤ 3 * root := by
apply splitBlock_invSqrt_le_root hnpos
have := (Bseq n).card_I0
omega
have hsqrt1 : sqrt ((Bseq n).I1.card : ℝ)⁻¹ ≤ 3 * root := by
apply splitBlock_invSqrt_le_root hnpos
have := (Bseq n).card_I1
omega
have ht0 : sqrt (log (4 / alpha) /
(2 * ((Bseq n).I0.card : ℝ))) ≤ K * (3 * root) := by
rw [show log (4 / alpha) / (2 * ((Bseq n).I0.card : ℝ)) =
(log (4 / alpha) / 2) * (((Bseq n).I0.card : ℝ)⁻¹) by
field_simp]
rw [Real.sqrt_mul hloghalf]
exact mul_le_mul_of_nonneg_left hsqrt0 hK
have ht1 : sqrt (log (4 / alpha) /
(2 * ((Bseq n).I1.card : ℝ))) ≤ K * (3 * root) := by
rw [show log (4 / alpha) / (2 * ((Bseq n).I1.card : ℝ)) =
(log (4 / alpha) / 2) * (((Bseq n).I1.card : ℝ)⁻¹) by
field_simp]
rw [Real.sqrt_mul hloghalf]
exact mul_le_mul_of_nonneg_left hsqrt1 hK
unfold contStabilizedWorstLength
apply sSup_le
intro v hv
rcases hv with ⟨P, hP, _hiid, rfl⟩
have he := contHoeffdingInterval_lintegral_length_le P (Bseq n) kappa cminus
cplus pmin deltaBar (deltaSeq n) alpha hP hreg.1 (hdelta n) hcard1
apply he.trans
apply ENNReal.ofReal_le_ofReal
have hrootcoef : 12 * K + (J : ℝ) * (3 / 2) ≤ C := by
dsimp [C]
have hJ : 0 ≤ (J : ℝ) := Nat.cast_nonneg J
have hratio : 0 ≤ cplus / (kappa + 1) := by positivity
nlinarith
have hatomcoef : (J : ℝ) * (cplus / (kappa + 1)) ≤ C := by
dsimp [C]
have hJ : 0 ≤ (J : ℝ) := Nat.cast_nonneg J
have hratio : 0 ≤ cplus / (kappa + 1) := by positivity
nlinarith
calc
_ ≤ 2 * (K * (3 * root)) + 2 * (K * (3 * root)) +
(J : ℝ) * ((1 / 2 : ℝ) * (3 * root) +
cplus * atom / (kappa + 1)) := by
exact add_le_add (add_le_add
(mul_le_mul_of_nonneg_left ht0 (by norm_num))
(mul_le_mul_of_nonneg_left ht1 (by norm_num)))
(mul_le_mul_of_nonneg_left
(add_le_add
(mul_le_mul_of_nonneg_left hsqrt1 (by norm_num)) le_rfl)
(Nat.cast_nonneg J))
_ = (12 * K + (J : ℝ) * (3 / 2)) * root +
((J : ℝ) * (cplus / (kappa + 1))) * atom := by ring
_ ≤ C * root + C * atom := add_le_add
(mul_le_mul_of_nonneg_right hrootcoef hroot)
(mul_le_mul_of_nonneg_right hatomcoef hatom)
_ = C * contFrontier n (deltaSeq n) kappa := by
dsimp [root, atom]
unfold contFrontier
ring
theorem contFrontierCriteria_fixed_positive_not_tendsto_zero
At a threshold converging to a positive interior point, neither the observed nor causal continuity-only minimax risk can converge to zero. The result uses the hreg condition, the hdelta condition, the hdelta0 condition, the htend condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin deltaBar alpha :
ℝ
hreg :
ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha
deltaSeq :
ℕ → ℝ
delta0 :
ℝ
hdelta :
ThresholdSequence deltaBar deltaSeq
hdelta0 :
delta0 ∈ Ioc (0 : ℝ) deltaBar
conclusion
conclusion 1
¬ Tendsto (fun n => (contFrontierCriteria J n kappa cminus cplus pmin deltaBar (deltaSeq n) alpha).1) atTop (nhds 0)
conclusion 2
¬ Tendsto (fun n => (contFrontierCriteria J n kappa cminus cplus pmin deltaBar (deltaSeq n) alpha).2.2.1) atTop (nhds 0)
Proof (Lean source)
lemma contFrontierCriteria_fixed_positive_not_tendsto_zero
(J : ℕ) (kappa cminus cplus pmin deltaBar alpha : ℝ)
(hreg : ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha)
(deltaSeq : ℕ → ℝ) (delta0 : ℝ)
(hdelta : ThresholdSequence deltaBar deltaSeq)
(hdelta0 : delta0 ∈ Ioc (0 : ℝ) deltaBar)
(htend : Tendsto deltaSeq atTop (nhds delta0)) :
¬ Tendsto
(fun n => (contFrontierCriteria J n kappa cminus cplus pmin deltaBar
(deltaSeq n) alpha).1) atTop (nhds 0) ∧
¬ Tendsto
(fun n => (contFrontierCriteria J n kappa cminus cplus pmin deltaBar
(deltaSeq n) alpha).2.2.1) atTop (nhds 0) := by
obtain ⟨c, hc, hlower⟩ := contMinimaxRisk_eventually_ge_frontier J kappa
cminus cplus pmin deltaBar alpha hreg
have hk1 : 0 < kappa + 1 := by linarith [hreg.1.2.1]
have hpow : Tendsto (fun n => c * (deltaSeq n) ^ (kappa + 1)) atTop
(nhds (c * delta0 ^ (kappa + 1))) := by
exact tendsto_const_nhds.mul
((Real.continuousAt_rpow_const delta0 (kappa + 1) (inr hk1.le)).tendsto.comp htend)
have hle : ∀ᶠ n : ℕ in atTop,
c * (deltaSeq n) ^ (kappa + 1) ≤
(contFrontierCriteria J n kappa cminus cplus pmin deltaBar
(deltaSeq n) alpha).1 := by
filter_upwards [hlower deltaSeq hdelta] with n hn
have hr : 0 ≤ (n : ℝ) ^ (-(1 : ℝ) / 2) :=
Real.rpow_nonneg (Nat.cast_nonneg n) _
calc
c * (deltaSeq n) ^ (kappa + 1) ≤
c * contFrontier n (deltaSeq n) kappa := by
unfold contFrontier
nlinarith
_ ≤ _ := hn
have hobs : ¬ Tendsto
(fun n => (contFrontierCriteria J n kappa cminus cplus pmin deltaBar
(deltaSeq n) alpha).1) atTop (nhds 0) := by
intro hz
have hlim : c * delta0 ^ (kappa + 1) ≤ 0 :=
le_of_tendsto_of_tendsto hpow hz hle
have : 0 < c * delta0 ^ (kappa + 1) :=
mul_pos hc (Real.rpow_pos_of_pos hdelta0.1 _)
linarith
refine ⟨hobs, ?_⟩
intro hcausal
have heq : ∀ n,
(contFrontierCriteria J n kappa cminus cplus pmin deltaBar
(deltaSeq n) alpha).2.2.1 =
(contFrontierCriteria J n kappa cminus cplus pmin deltaBar
(deltaSeq n) alpha).1 := by
intro n
exact (contFrontierCriteria_observed_causal_eq J n kappa cminus cplus pmin
deltaBar (deltaSeq n) alpha hreg (hdelta n)).1
apply hobs
exact hcausal.congr' (Filter.Eventually.of_forall heq)
theorem contFrontierLength_fixed_positive_not_tendsto_zero
A threshold converging to a positive interior point also prevents the observed minimax honest length from converging to zero. The result uses the hreg condition, the hdelta condition, the hdelta0 condition, the htend condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
kappa cminus cplus pmin deltaBar alpha :
ℝ
hreg :
ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha
deltaSeq :
ℕ → ℝ
delta0 :
ℝ
hdelta :
ThresholdSequence deltaBar deltaSeq
hdelta0 :
delta0 ∈ Ioc (0 : ℝ) deltaBar
conclusion
¬ Tendsto (fun n => (contFrontierCriteria J n kappa cminus cplus pmin deltaBar (deltaSeq n) alpha).2.1) atTop (nhds 0)
Proof (Lean source)
lemma contFrontierLength_fixed_positive_not_tendsto_zero
(J : ℕ) (kappa cminus cplus pmin deltaBar alpha : ℝ)
(hreg : ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha)
(deltaSeq : ℕ → ℝ) (delta0 : ℝ)
(hdelta : ThresholdSequence deltaBar deltaSeq)
(hdelta0 : delta0 ∈ Ioc (0 : ℝ) deltaBar)
(htend : Tendsto deltaSeq atTop (nhds delta0)) :
¬ Tendsto
(fun n => (contFrontierCriteria J n kappa cminus cplus pmin deltaBar
(deltaSeq n) alpha).2.1) atTop (nhds 0) := by
obtain ⟨c, hc, hlower⟩ := contMinimaxLength_eventually_ge_frontier J kappa
cminus cplus pmin deltaBar alpha hreg
have hk1 : 0 < kappa + 1 := by linarith [hreg.1.2.1]
have hpowR : Tendsto (fun n => c * (deltaSeq n) ^ (kappa + 1)) atTop
(nhds (c * delta0 ^ (kappa + 1))) :=
tendsto_const_nhds.mul
((Real.continuousAt_rpow_const delta0 (kappa + 1) (inr hk1.le)).tendsto.comp htend)
have hpow : Tendsto (fun n => ofReal
(c * (deltaSeq n) ^ (kappa + 1))) atTop
(nhds (ofReal (c * delta0 ^ (kappa + 1)))) :=
ENNReal.continuous_ofReal.continuousAt.tendsto.comp hpowR
have hle : ∀ᶠ n : ℕ in atTop,
ofReal (c * (deltaSeq n) ^ (kappa + 1)) ≤
(contFrontierCriteria J n kappa cminus cplus pmin deltaBar
(deltaSeq n) alpha).2.1 := by
filter_upwards [hlower deltaSeq hdelta] with n hn
apply (ENNReal.ofReal_le_ofReal ?_).trans hn
have hr := Real.rpow_nonneg (Nat.cast_nonneg n) (-(1 : ℝ) / 2)
unfold contFrontier
nlinarith
intro hz
have hlim : ofReal (c * delta0 ^ (kappa + 1)) ≤ 0 :=
le_of_tendsto_of_tendsto hpow hz hle
have hp : 0 < c * delta0 ^ (kappa + 1) :=
mul_pos hc (Real.rpow_pos_of_pos hdelta0.1 _)
have : (0 : ℝ≥0∞) < ofReal (c * delta0 ^ (kappa + 1)) :=
ENNReal.ofReal_pos.mpr hp
exact (not_lt_of_ge hlim) this
Conditional on the disclosed Hoeffding gate, the fixed-fallback estimator and interval attain the continuity-only frontier, with identical observed and causal decision criteria. The result uses the HoeffdingBoundedAverage_of_gate condition, the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
HoeffdingBoundedAverage_of_gate :
J :
ℕ
kappa cminus cplus pmin deltaBar alpha :
ℝ
hreg :
ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha
conclusion
∃ c C : ℝ,
0 < c
c < C
conclusion 1
SplitBlocks n, ∀ᶠ n in atTop, let delta := deltaSeq n let s := contFrontier n delta kappa let crit := contFrontierCriteria J n kappa cminus cplus pmin deltaBar delta alpha c * s
≤ crit.1 ∧
crit.1 ≤ contStabilizedWorstRisk J n (Bseq n) kappa cminus cplus pmin deltaBar delta ∧
contStabilizedWorstRisk J n (Bseq n) kappa cminus cplus pmin deltaBar delta ≤ C * s ∧
1 - alpha
≤ contStabilizedCoverage J n (Bseq n) kappa cminus cplus pmin deltaBar delta alpha ∧
contStabilizedWorstLength J n (Bseq n) kappa cminus cplus pmin deltaBar delta alpha
≤ ofReal (C * s) ∧
ofReal (c * s) ≤ crit.2.1 ∧
crit.2.2.1 = crit.1 ∧
crit.2.2.2 = crit.2.1 ∧
crit.2.2.1
≤ contCausalStabilizedWorstRisk J n (Bseq n) kappa cminus cplus pmin deltaBar delta ∧
contCausalStabilizedWorstRisk J n (Bseq n) kappa cminus cplus pmin deltaBar delta ≤ C * s ∧
1 - alpha
≤ contCausalStabilizedCoverage J n (Bseq n) kappa cminus cplus pmin deltaBar delta alpha ∧
contCausalStabilizedWorstLength J n (Bseq n) kappa cminus cplus pmin deltaBar delta alpha
≤ ofReal (C * s)
conclusion 2
AsympSeq (fun n => contFrontier n ((n : ℝ) ^ (-(1 : ℝ) / (2 * (kappa + 1)))) kappa) (fun n => (n : ℝ) ^ (-(1 : ℝ) / 2))
conclusion 3
AsympSeq (fun n => contFrontier n 0 kappa) (fun n => (n : ℝ) ^ (-(1 : ℝ) / 2))
conclusion 4
¬ Tendsto (fun n => (contFrontierCriteria J n kappa cminus cplus pmin deltaBar (deltaSeq n) alpha).1) atTop (nhds 0)
¬ Tendsto (fun n => (contFrontierCriteria J n kappa cminus cplus pmin deltaBar (deltaSeq n) alpha).2.1) atTop (nhds 0)
Proof (Lean source)
-- @node: thm:continuity-only-frontier
theorem continuity_only_frontier
(HoeffdingBoundedAverage_of_gate : HoeffdingBoundedAverage)
(J : ℕ) (kappa cminus cplus pmin deltaBar alpha : ℝ)
(hreg : ContRegimeConstants J kappa cminus cplus pmin deltaBar alpha) :
∃ c C : ℝ, 0 < c ∧ c < C ∧
(∀ (deltaSeq : ℕ → ℝ), ThresholdSequence deltaBar deltaSeq →
∀ Bseq : ∀ n, SplitBlocks n,
∀ᶠ n in atTop,
let delta := deltaSeq n
let s := contFrontier n delta kappa
let crit := contFrontierCriteria J n kappa cminus cplus pmin
deltaBar delta alpha
c * s ≤ crit.1 ∧
crit.1 ≤ contStabilizedWorstRisk J n (Bseq n) kappa cminus cplus pmin
deltaBar delta ∧
contStabilizedWorstRisk J n (Bseq n) kappa cminus cplus pmin
deltaBar delta ≤ C * s ∧
1 - alpha ≤ contStabilizedCoverage J n (Bseq n) kappa cminus cplus pmin
deltaBar delta alpha ∧
contStabilizedWorstLength J n (Bseq n) kappa cminus cplus pmin
deltaBar delta alpha ≤ ofReal (C * s) ∧
ofReal (c * s) ≤ crit.2.1 ∧
crit.2.2.1 = crit.1 ∧ crit.2.2.2 = crit.2.1 ∧
crit.2.2.1 ≤ contCausalStabilizedWorstRisk J n (Bseq n) kappa cminus
cplus pmin deltaBar delta ∧
contCausalStabilizedWorstRisk J n (Bseq n) kappa cminus cplus pmin
deltaBar delta ≤ C * s ∧
1 - alpha ≤ contCausalStabilizedCoverage J n (Bseq n) kappa cminus
cplus pmin deltaBar delta alpha ∧
contCausalStabilizedWorstLength J n (Bseq n) kappa cminus cplus pmin
deltaBar delta alpha ≤ ofReal (C * s)) ∧
AsympSeq
(fun n => contFrontier n
((n : ℝ) ^ (-(1 : ℝ) / (2 * (kappa + 1)))) kappa)
(fun n => (n : ℝ) ^ (-(1 : ℝ) / 2)) ∧
AsympSeq (fun n => contFrontier n 0 kappa)
(fun n => (n : ℝ) ^ (-(1 : ℝ) / 2)) ∧
∀ (deltaSeq : ℕ → ℝ) (delta0 : ℝ),
ThresholdSequence deltaBar deltaSeq →
delta0 ∈ Ioc (0 : ℝ) deltaBar →
Tendsto deltaSeq atTop (nhds delta0) →
¬ Tendsto
(fun n => (contFrontierCriteria J n kappa cminus cplus pmin deltaBar
(deltaSeq n) alpha).1) atTop (nhds 0) ∧
¬ Tendsto
(fun n => (contFrontierCriteria J n kappa cminus cplus pmin deltaBar
(deltaSeq n) alpha).2.1) atTop (nhds 0) := by
obtain ⟨cr, hcr, hRiskLower⟩ := contMinimaxRisk_eventually_ge_frontier J kappa
cminus cplus pmin deltaBar alpha hreg
obtain ⟨cl, hcl, hLengthLower⟩ := contMinimaxLength_eventually_ge_frontier J kappa
cminus cplus pmin deltaBar alpha hreg
obtain ⟨CR, hCR, hRiskUpper⟩ :=
contStabilizedWorstRisk_eventually_le_frontier J kappa cminus cplus pmin
deltaBar alpha hreg
obtain ⟨CL, hCL, hLengthUpper⟩ :=
contStabilizedWorstLength_eventually_le_frontier J kappa cminus cplus pmin
deltaBar alpha hreg
let c := min cr cl
let C := max CR CL + c + 1
have hc : 0 < c := by dsimp [c]; positivity
have hcC : c < C := by
dsimp [C]
have : 0 < max CR CL := lt_max_of_lt_left hCR
linarith
have hCRle : CR ≤ C := by
dsimp [C]
have := le_max_left CR CL
linarith
have hCLle : CL ≤ C := by
dsimp [C]
have := le_max_right CR CL
linarith
refine ⟨c, C, hc, hcC, ?_,
contFrontier_elbow_asymp kappa hreg.1.2.1,
contFrontier_zero_asymp kappa hreg.1.2.1, ?_⟩
· intro deltaSeq hdelta Bseq
filter_upwards [hRiskLower deltaSeq hdelta,
hLengthLower deltaSeq hdelta,
hRiskUpper deltaSeq hdelta Bseq,
hLengthUpper deltaSeq hdelta Bseq,
eventually_ge_atTop 8] with n hnRL hnLL hnRU hnLU hn8
let delta := deltaSeq n
let s := contFrontier n delta kappa
let crit := contFrontierCriteria J n kappa cminus cplus pmin
deltaBar delta alpha
have hdeltaN : delta ∈ Icc (0 : ℝ) deltaBar := hdelta n
have hs : 0 ≤ s := by
dsimp [s, delta]
unfold contFrontier
exact add_nonneg (Real.rpow_nonneg (Nat.cast_nonneg n) _)
(Real.rpow_nonneg (hdelta n).1 _)
have hcard0 : 0 < (Bseq n).I0.card := by
have := (Bseq n).card_I0
omega
have hcard1 : 0 < (Bseq n).I1.card := by
have := (Bseq n).card_I1
omega
have hcritEq := contFrontierCriteria_observed_causal_eq J n kappa cminus
cplus pmin deltaBar delta alpha hreg hdeltaN
have hRiskEq := contCausalStabilizedWorstRisk_eq_observed J n (Bseq n)
kappa cminus cplus pmin deltaBar delta alpha hreg hdeltaN
have hCovEq := contCausalStabilizedCoverage_eq_observed J n (Bseq n)
kappa cminus cplus pmin deltaBar delta alpha hreg hdeltaN
have hLenEq := contCausalStabilizedWorstLength_eq_observed J n (Bseq n)
kappa cminus cplus pmin deltaBar delta alpha hreg
have hcov := contStabilizedCoverage_ge HoeffdingBoundedAverage_of_gate J n
(Bseq n) kappa cminus cplus pmin deltaBar delta alpha hreg hdeltaN
hcard0 hcard1
have hlowRisk : c * s ≤ crit.1 := by
exact (mul_le_mul_of_nonneg_right (min_le_left cr cl) hs).trans
(by simpa [c, s, crit, delta] using hnRL)
have hlowLen : ofReal (c * s) ≤ crit.2.1 := by
exact (ENNReal.ofReal_le_ofReal
(mul_le_mul_of_nonneg_right (min_le_right cr cl) hs)).trans
(by simpa [c, s, crit, delta] using hnLL)
have hupRisk : contStabilizedWorstRisk J n (Bseq n) kappa cminus cplus
pmin deltaBar delta ≤ C * s := by
calc
_ ≤ CR * s := by simpa [s, delta] using hnRU
_ ≤ C * s := mul_le_mul_of_nonneg_right hCRle hs
have hupLen : contStabilizedWorstLength J n (Bseq n) kappa cminus cplus
pmin deltaBar delta alpha ≤ ofReal (C * s) := by
calc
_ ≤ ofReal (CL * s) := by simpa [s, delta] using hnLU
_ ≤ ofReal (C * s) :=
ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_right hCLle hs)
refine ⟨hlowRisk,
contFrontierCriteria_risk_le_stabilized J n (Bseq n) kappa cminus
cplus pmin deltaBar delta alpha hreg hdeltaN,
hupRisk, hcov, hupLen, hlowLen, hcritEq.1, hcritEq.2, ?_, ?_, ?_, ?_⟩
· rw [hcritEq.1, hRiskEq]
exact contFrontierCriteria_risk_le_stabilized J n (Bseq n) kappa cminus
cplus pmin deltaBar delta alpha hreg hdeltaN
· rw [hRiskEq]
exact hupRisk
· rw [hCovEq]
exact hcov
· rw [hLenEq]
exact hupLen
· intro deltaSeq delta0 hdelta hdelta0 htend
exact ⟨(contFrontierCriteria_fixed_positive_not_tendsto_zero J kappa
cminus cplus pmin deltaBar alpha hreg deltaSeq delta0 hdelta hdelta0 htend).1,
contFrontierLength_fixed_positive_not_tendsto_zero J kappa cminus cplus
pmin deltaBar alpha hreg deltaSeq delta0 hdelta hdelta0 htend⟩
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.TMinimaxRisk 18 declarations The upper bound is attached to the concrete total-Gram estimator.
Matched minimax absolute-risk frontier
The upper bound is attached to the concrete total-Gram estimator. The lower bound contains both same-class experiments and their product-divergence bounds.
def stabilizedEstimatorRisk
Risk of the paper's concrete stabilized estimator.
Definition (Lean source)
parameters
def
stabilizedEstimatorRisk P n B beta kappa cminus cplus deltaBar delta :
ℝ
given by
estimatorRisk P n delta (fun z => totalGramEstimator B z (ellOf beta) kappa cminus cplus delta (infoBandwidth n delta beta kappa deltaBar))
Worst-case risk of the concrete estimator over i.i.d. laws in the model.
Definition (Lean source)
parameters
def
stabilizedWorstRisk J n B beta kappa L cminus cplus pmin deltaBar delta :
ℝ
given by
sSup {v : ℝ | ∃ P : ClampLaw J, ClampModel P beta kappa L cminus cplus pmin ∧ IidSampling P n ∧ v = stabilizedEstimatorRisk P n B beta kappa cminus cplus deltaBar delta}
Product chi-squared divergence used by the same-class witnesses.
Definition (Lean source)
parameters
def
productChiSq P Q n :
ℝ
given by
The product law from P is absolutely continuous with respect to the product law from Q, and its squared Radon--Nikodym deviation is integrable for the sample size n; together these conditions define well-posed real product chi-squared divergence.
Definition (Lean source)
parameters
def
ProperProductChiSq P Q n :
Prop
given by
clause 1
iidProduct P n ≪ iidProduct Q n
clause 2
Integrable (fun z => (((iidProduct P n).rnDeriv (iidProduct Q n) z).toReal - 1) ^ 2) (iidProduct Q n)
The outcome in a lower-bound experiment is genuinely Bernoulli.
Definition (Lean source)
parameters
J :
ℕ
P :
ClampLaw J
def
BernoulliOutcomeLaw P :
Prop
given by
∀ᵐ o ∂P.dataMeasure, o.Y = 0 ∨ o.Y = 1
Uniform eventual upper bound for the concrete stabilized estimator, including the supremum over all model laws. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
∃ C : ℝ,
0 < C
conclusion 1
SplitBlocks n, ∀ᶠ n in atTop, stabilizedWorstRisk J n (Bseq n) beta kappa L cminus cplus pmin deltaBar (deltaSeq n)
≤ C * clampFrontier n (deltaSeq n) kappa (infoBandwidth n (deltaSeq n) beta kappa deltaBar) beta
Proof (Lean source)
lemma stabilizedWorstRisk_eventually_le_frontier
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
∃ C : ℝ, 0 < C ∧
∀ (deltaSeq : ℕ → ℝ), ThresholdSequence deltaBar deltaSeq →
∀ Bseq : ∀ n, SplitBlocks n,
∀ᶠ n in atTop,
stabilizedWorstRisk J n (Bseq n) beta kappa L cminus cplus pmin
deltaBar (deltaSeq n) ≤
C * clampFrontier n (deltaSeq n) kappa
(infoBandwidth n (deltaSeq n) beta kappa deltaBar) beta := by
rcases total_gram_stabilization J beta kappa L cminus cplus pmin
deltaBar alpha hreg with ⟨Ct, ct, hCt, hct, hlambda, hstab⟩
have hreg' := hreg
rcases hreg' with
⟨hJ, hbeta, hkappa, hL, hcminus, hcminus_le, hcplus,
hpmin, hpmin_le, hdeltaBar, hdeltaBar1, halpha, halpha1⟩
let noiseC : ℝ := (1 / 2 : ℝ) * sqrt
(336 * (4 * (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus ^ 2) /
(pmin * cminus / (2 * (2 : ℝ) ^ kappa)))
let biasC : ℝ := 2 * sqrt (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus
let localK : ℝ := noiseC + biasC * L + Ct / ct
let atomK : ℝ := cplus / (kappa + 1)
let C : ℝ := 2 * (1 + (J : ℝ)) * (1 + atomK * localK)
have hnoiseC : 0 ≤ noiseC := by dsimp [noiseC]; positivity
have hbiasC : 0 ≤ biasC := by dsimp [biasC]; positivity
have hlocalK : 0 ≤ localK := by dsimp [localK]; positivity
have hatomK : 0 ≤ atomK := by
dsimp [atomK]
exact div_nonneg (by linarith) (by linarith)
have hC : 0 < C := by dsimp [C]; positivity
refine ⟨C, hC, ?_⟩
intro deltaSeq hdelta Bseq
have hbal := infoBandwidth_eventually_balance beta kappa deltaBar hbeta hkappa
⟨hdeltaBar, hdeltaBar1⟩ deltaSeq hdelta
filter_upwards [hbal, eventually_ge_atTop 8] with n hbn hn
let delta := deltaSeq n
let h := infoBandwidth n delta beta kappa deltaBar
let r0 := (n : ℝ) ^ (-(1 : ℝ) / 2)
let a0 := delta ^ (kappa + 1) * h ^ beta
have hnpos : 0 < n := by omega
have hh : 0 < h := by simpa [h, delta] using hbn.1
have hhbar : h ≤ 1 - deltaBar := by simpa [h, delta] using hbn.2.1
have hh1 : h ≤ 1 := hhbar.trans (by linarith)
have hbalance : (n : ℝ) * h ^ (2 * beta + 1) * (delta + h) ^ kappa = 1 := by
simpa [h, delta] using hbn.2.2
have hdeltaN : delta ∈ Icc (0 : ℝ) deltaBar := hdelta n
have hupper : delta + h ≤ 1 := by linarith [hdeltaN.2, hhbar]
have hr0 : 0 ≤ r0 := Real.rpow_nonneg (Nat.cast_nonneg n) _
have ha0 : 0 ≤ a0 := mul_nonneg
(Real.rpow_nonneg hdeltaN.1 _) (Real.rpow_nonneg hh.le _)
have hsqrt0 : sqrt ((Bseq n).I0.card : ℝ)⁻¹ ≤ 3 * r0 := by
have hm : n ≤ 8 * (Bseq n).I0.card := by
have := (Bseq n).card_I0
omega
exact splitBlock_invSqrt_le_root hnpos hm
have hsqrt1 : sqrt ((Bseq n).I1.card : ℝ)⁻¹ ≤ 3 * r0 := by
have hm : n ≤ 8 * (Bseq n).I1.card := by
have := (Bseq n).card_I1
omega
exact splitBlock_invSqrt_le_root hnpos hm
apply csSup_le
· let P0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
have hP0 : ClampModel P0 beta kappa L cminus cplus pmin := by
exact minimaxCenter_mem_model J beta kappa L cminus cplus pmin deltaBar alpha hreg
exact ⟨stabilizedEstimatorRisk P0 n (Bseq n) beta kappa cminus cplus
deltaBar delta, P0, hP0, clampModel_iidSampling hP0, rfl⟩
· intro v hv
rcases hv with ⟨P, hP, hsampling, rfl⟩
have htail (x : Fin J) :
(iidProduct P n).real {z |
¬ GoodGramEvent (Bseq n) z x (ellOf beta) kappa cminus cplus delta h} ≤
Ct * exp (-ct * (n : ℝ) * h * (delta + h) ^ kappa) := by
simpa [h, delta] using (hstab P hP n hsampling (Bseq n) delta hdeltaN x).1
have hexp : Ct * exp (-ct * (n : ℝ) * h * (delta + h) ^ kappa) ≤
(Ct / ct) * h ^ beta :=
balancedGramTail_le hnpos hbeta hh hh1 hCt.le hct hbalance
have hexpl := totalGramEstimator_risk_le_explicit P hP hreg hsampling
(Bseq n) hn hdeltaN hh hupper hbalance hlambda htail
have hlocal :
(1 / 2 : ℝ) * sqrt
(336 * (4 * (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus ^ 2) /
(pmin * cminus / (2 * (2 : ℝ) ^ kappa))) * h ^ beta +
(2 * sqrt (ellOf beta + 1 : ℝ) /
lambdaStar (ellOf beta) kappa cminus cplus) * L * h ^ beta +
Ct * exp (-ct * (n : ℝ) * h * (delta + h) ^ kappa) ≤
localK * h ^ beta := by
calc
_ ≤ noiseC * h ^ beta + biasC * L * h ^ beta + (Ct / ct) * h ^ beta := by
exact add_le_add (add_le_add le_rfl le_rfl) hexp
_ = localK * h ^ beta := by simp [noiseC, biasC, localK]; ring
have hraw : stabilizedEstimatorRisk P n (Bseq n) beta kappa cminus cplus
deltaBar delta ≤ 2 * r0 + (J : ℝ) * (2 * r0 + atomK * localK * a0) := by
unfold stabilizedEstimatorRisk estimatorRisk
have hatomExpr0 : 0 ≤ cplus * delta ^ (kappa + 1) / (kappa + 1) :=
div_nonneg (mul_nonneg (by linarith) (Real.rpow_nonneg hdeltaN.1 _)) (by linarith)
calc
_ ≤ _ := hexpl
_ ≤ 2 * r0 + (J : ℝ) *
(2 * r0 + (cplus * delta ^ (kappa + 1) / (kappa + 1)) *
(localK * h ^ beta)) := by
apply add_le_add
· nlinarith [hsqrt0]
· apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg J)
apply add_le_add
· nlinarith [hsqrt1]
· exact mul_le_mul_of_nonneg_left hlocal hatomExpr0
_ = 2 * r0 + (J : ℝ) * (2 * r0 + atomK * localK * a0) := by
dsimp [atomK, a0]
ring
calc
stabilizedEstimatorRisk P n (Bseq n) beta kappa cminus cplus deltaBar delta ≤
2 * r0 + (J : ℝ) * (2 * r0 + atomK * localK * a0) := hraw
_ ≤ C * (r0 + a0) := by
have hq : 0 ≤ atomK * localK := mul_nonneg hatomK hlocalK
have hJ0 : 0 ≤ (J : ℝ) := Nat.cast_nonneg J
have hqa : 0 ≤ (atomK * localK) * a0 := mul_nonneg hq ha0
have hcoeff : (J : ℝ) ≤ 2 * (1 + (J : ℝ)) := by
have hJ1 : (J : ℝ) ≤ 1 + (J : ℝ) := le_add_of_nonneg_left zero_le_one
have hsum0 : 0 ≤ 1 + (J : ℝ) := by positivity
exact hJ1.trans (by linarith)
change 2 * r0 + (J : ℝ) * (2 * r0 + atomK * localK * a0) ≤
(2 * (1 + (J : ℝ)) * (1 + atomK * localK)) * (r0 + a0)
calc
_ = 2 * (1 + (J : ℝ)) * r0 +
(J : ℝ) * ((atomK * localK) * a0) := by ring
_ ≤ 2 * (1 + (J : ℝ)) * r0 +
(2 * (1 + (J : ℝ))) * ((atomK * localK) * a0) := by
apply add_le_add_right
exact mul_le_mul_of_nonneg_right hcoeff hqa
_ ≤ (2 * (1 + (J : ℝ)) * (1 + atomK * localK)) *
(r0 + a0) := by
have hfac : 0 ≤ 2 * (1 + (J : ℝ)) := by positivity
have hrest : 0 ≤
(2 * (1 + (J : ℝ))) * a0 +
(2 * (1 + (J : ℝ))) * ((atomK * localK) * r0) :=
add_nonneg (mul_nonneg hfac ha0)
(mul_nonneg hfac (mul_nonneg hq hr0))
calc
_ ≤ 2 * (1 + (J : ℝ)) * r0 +
2 * (1 + (J : ℝ)) * (atomK * localK * a0) +
(2 * (1 + (J : ℝ)) * a0 +
2 * (1 + (J : ℝ)) * (atomK * localK * r0)) :=
le_add_of_nonneg_right hrest
_ = _ := by ring
_ = C * clampFrontier n delta kappa h beta := by
simp [clampFrontier, r0, a0]
A global Bernoulli regression mean shift of the advertised size.
Definition (Lean source)
parameters
def
GlobalBernoulliShift P Q eps :
Prop
given by
clause 1
clause 2
clause 3
theorem minimaxConstant_globalShift
the stated minimax constant global shift property holds for the specified J input, the specified kappa input, the specified eps input, the specified heps input.
Formal statement
hypotheses
J :
ℕ
kappa eps :
ℝ
heps :
|eps| ≤ 1 / 2
conclusion
GlobalBernoulliShift (minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)) (minimaxClampLaw J kappa (fun _ : Fin J × ℝ => eps)) eps
Proof (Lean source)
lemma minimaxConstant_globalShift (J : ℕ) (kappa eps : ℝ)
(heps : |eps| ≤ 1 / 2) :
GlobalBernoulliShift
(minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0))
(minimaxClampLaw J kappa (fun _ : Fin J × ℝ => eps)) eps := by
have hmap0 := minimaxDataMeasure_map_design J kappa
(fun _ : Fin J × ℝ => 0) measurable_const (fun _ => by norm_num)
have hmap1 := minimaxDataMeasure_map_design J kappa
(fun _ : Fin J × ℝ => eps) measurable_const (fun _ => heps)
have hshared : SharedClampDesign
(minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0))
(minimaxClampLaw J kappa (fun _ : Fin J × ℝ => eps)) := by
refine ⟨?_, rfl, rfl⟩
change (minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => 0)).map
(fun o => (o.X, o.A)) =
(minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => eps)).map
(fun o => (o.X, o.A))
rw [hmap0, hmap1]
refine ⟨hshared,
minimaxDataMeasure_ae_bernoulli J kappa
(fun _ : Fin J × ℝ => 0) measurable_const,
minimaxDataMeasure_ae_bernoulli J kappa
(fun _ : Fin J × ℝ => eps) measurable_const, ?_⟩
intro x a ha
simp [minimaxClampLaw]
theorem globalBernoulli_witness_eventually
The canonical constant Bernoulli shift supplies the global root-sample witness, uniformly over every admissible threshold sequence. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
deltaSeq :
ℕ → ℝ
ThresholdSequence deltaBar deltaSeq
conclusion
∀ᶠ n in atTop, ∃ P0 P1 : ClampLaw J, ClampModel P0 beta kappa L cminus cplus pmin ∧ ClampModel P1 beta kappa L cminus cplus pmin ∧ IidSampling P0 n ∧ IidSampling P1 n ∧ GlobalBernoulliShift P0 P1 ((n : ℝ) ^ (-(1 : ℝ) / 2)) ∧ (1 / 2 : ℝ) * (n : ℝ) ^ (-(1 : ℝ) / 2)
≤ |clampFunctional P1 (deltaSeq n) - clampFunctional P0 (deltaSeq n)| ∧
ProperProductChiSq P1 P0 n ∧
productChiSq P1 P0 n ≤ exp 4
Proof (Lean source)
lemma globalBernoulli_witness_eventually
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
∀ (deltaSeq : ℕ → ℝ), ThresholdSequence deltaBar deltaSeq →
∀ᶠ n in atTop, ∃ P0 P1 : ClampLaw J,
ClampModel P0 beta kappa L cminus cplus pmin ∧
ClampModel P1 beta kappa L cminus cplus pmin ∧
IidSampling P0 n ∧ IidSampling P1 n ∧
GlobalBernoulliShift P0 P1 ((n : ℝ) ^ (-(1 : ℝ) / 2)) ∧
(1 / 2 : ℝ) * (n : ℝ) ^ (-(1 : ℝ) / 2) ≤
|clampFunctional P1 (deltaSeq n) - clampFunctional P0 (deltaSeq n)| ∧
ProperProductChiSq P1 P0 n ∧
productChiSq P1 P0 n ≤ exp 4 := by
intro deltaSeq hdelta
have hreg' := hreg
rcases hreg' with
⟨hJ, hbeta, hkappa, hL, hcminus, hcminus_le, hcplus,
hpmin, hpmin_le, hdeltaBar, hdeltaBar1, halpha, halpha1⟩
filter_upwards [eventually_ge_atTop 5] with n hn
have hnpos : 0 < n := by omega
have hnR : 0 < (n : ℝ) := by exact_mod_cast hnpos
let eps : ℝ := (n : ℝ) ^ (-(1 : ℝ) / 2)
have heps0 : 0 < eps := Real.rpow_pos_of_pos hnR _
have hepshalf : |eps| ≤ 1 / 2 := by
rw [abs_of_pos heps0]
have hn4Nat : 4 ≤ n := by omega
have hn4 : (4 : ℝ) ≤ n := by exact_mod_cast hn4Nat
calc
eps ≤ (4 : ℝ) ^ (-(1 : ℝ) / 2) := by
dsimp [eps]
exact Real.rpow_le_rpow_of_nonpos (by norm_num) hn4 (by norm_num)
_ = 1 / 2 := by norm_num
have hepslt : |eps| < 1 / 2 := by
rw [abs_of_pos heps0]
have hn4 : (4 : ℝ) < n := by exact_mod_cast hn
calc
eps < (4 : ℝ) ^ (-(1 : ℝ) / 2) := by
dsimp [eps]
exact Real.rpow_lt_rpow_of_neg (by norm_num) hn4 (by norm_num)
_ = 1 / 2 := by norm_num
let P0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
let P1 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => eps)
have hP0 : ClampModel P0 beta kappa L cminus cplus pmin := by
exact minimaxCenter_mem_model J beta kappa L cminus cplus pmin deltaBar alpha hreg
have hP1 : ClampModel P1 beta kappa L cminus cplus pmin := by
exact minimaxConstant_mem_model J beta kappa L cminus cplus pmin deltaBar alpha
eps hreg hepshalf
have hshift : GlobalBernoulliShift P0 P1 eps := by
exact minimaxConstant_globalShift J kappa eps hepshalf
have hdelta1 : deltaSeq n ∈ Icc (0 : ℝ) 1 :=
⟨(hdelta n).1, (hdelta n).2.trans hdeltaBar1.le⟩
have hsep := minimaxGlobalSeparation J kappa (deltaSeq n) eps hJ hkappa
hdelta1 hepshalf
have hsepAbs : |clampFunctional P1 (deltaSeq n) -
clampFunctional P0 (deltaSeq n)| = eps := by
rw [show P1 = minimaxClampLaw J kappa (fun _ : Fin J × ℝ => eps) by rfl,
show P0 = minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0) by rfl,
hsep, abs_of_pos heps0]
have hchi : productChiSq P1 P0 n ≤ exp 4 := by
change chiSqDiv
(Measure.pi (fun _ : Fin n => minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => eps)))
(Measure.pi (fun _ : Fin n => minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => 0))) ≤ exp 4
simpa [eps] using minimaxConstant_productChiSq_le_exp_four J n kappa hJ hkappa hn
have hac1 := minimaxDataMeasure_ac_center J kappa hJ hkappa
(fun _ : Fin J × ℝ => eps) measurable_const (fun _ => hepslt)
have hint1 := minimaxDataMeasure_sq_integrable_center J kappa hJ hkappa
(fun _ : Fin J × ℝ => eps) measurable_const (fun _ => hepslt)
letI : IsProbabilityMeasure (minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => eps)) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hJ hkappa _ measurable_const
(fun _ => hepshalf)
letI : IsProbabilityMeasure (minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => 0)) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hJ hkappa _ measurable_const
(fun _ => by norm_num)
have hac : iidProduct P1 n ≪ iidProduct P0 n := by
simpa [P0, P1, iidProduct, minimaxClampLaw] using
Causalean.Stat.pi_iid_absolutelyContinuous _ _ hac1 n
have hint : Integrable (fun z => (((iidProduct P1 n).rnDeriv
(iidProduct P0 n) z).toReal - 1) ^ 2) (iidProduct P0 n) := by
simpa [P0, P1, iidProduct, minimaxClampLaw] using
pi_iid_integrable_sq_dev _ _ hac1 hint1 n
refine ⟨P0, P1, hP0, hP1, clampModel_iidSampling hP0,
clampModel_iidSampling hP1, ?_, ?_, ⟨hac, hint⟩, hchi⟩
· simpa [eps] using hshift
· rw [hsepAbs]
change (1 / 2 : ℝ) * eps ≤ eps
nlinarith [heps0]
A common-design Bernoulli perturbation localized to a width-h window at the threshold, with fixed small amplitude times h^beta and a continuous bump supported on [-1,1].
Definition (Lean source)
parameters
def
LocalizedBernoulliPerturbation P Q beta delta h amplitude :
Prop
given by
clause 1
clause 2
clause 3
clause 4
0 < amplitude
clause 5
amplitude ≤ 1 / 4
clause 6
∃ bump : ℝ → ℝ,
Continuous bump
bump 0 = 1
u :
0 ≤ bump u
u :
u ∉ Icc (-1 : ℝ) 1
bump u = 0
u ∈ Set.Icc (-1 : ℝ) 1 :
|bump u| ≤ 1
theorem minimaxBump_localizedPerturbation
the stated minimax bump localized perturbation property holds for the specified J input, the specified beta input, the specified kappa input, the specified delta input, the specified h input, the specified amplitude input, the specified hbeta input, the specified hh input, the specified hh1 input, the specified hamp input, the specified hamp_le input.
Formal statement
hypotheses
J :
ℕ
beta kappa delta h amplitude :
ℝ
hbeta :
0 < beta
hh :
0 < h
hh1 :
h ≤ 1
hamp :
0 < amplitude
hamp_le :
amplitude ≤ 1 / 4
conclusion
q :
fun a : ℝ
=> amplitude * h ^ beta * CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
LocalizedBernoulliPerturbation (minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)) (minimaxClampLaw J kappa (fun p => q p.2)) beta delta h amplitude
Proof (Lean source)
lemma minimaxBump_localizedPerturbation
(J : ℕ) (beta kappa delta h amplitude : ℝ)
(hbeta : 0 < beta) (hh : 0 < h) (hh1 : h ≤ 1)
(hamp : 0 < amplitude) (hamp_le : amplitude ≤ 1 / 4) :
let q := fun a : ℝ => amplitude * h ^ beta *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
LocalizedBernoulliPerturbation
(minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0))
(minimaxClampLaw J kappa (fun p => q p.2)) beta delta h amplitude := by
dsimp only
let q := fun a : ℝ => amplitude * h ^ beta *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
have hqmeas : Measurable q := by dsimp [q]; fun_prop
have hq0 (a : ℝ) : 0 ≤ q a := by
dsimp [q]
exact mul_nonneg (mul_nonneg hamp.le (Real.rpow_nonneg hh.le _))
(CausalSmith.Stat.DoseResponseMinimax.doseBump_nonneg _)
have hqle (a : ℝ) : q a ≤ 1 / 4 := by
have hb := CausalSmith.Stat.DoseResponseMinimax.doseBump_le_one ((a - delta) / h)
have hhpow : h ^ beta ≤ 1 := Real.rpow_le_one hh.le hh1 hbeta.le
dsimp [q]
calc
amplitude * h ^ beta *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h) ≤
amplitude * h ^ beta * 1 := by gcongr
_ ≤ amplitude * 1 := by
simpa using mul_le_mul_of_nonneg_left hhpow hamp.le
_ ≤ 1 / 4 := by simpa using hamp_le
have hqbound (a : ℝ) : |q a| ≤ 1 / 2 := by
rw [abs_of_nonneg (hq0 a)]
linarith [hqle a]
have hmap0 := minimaxDataMeasure_map_design J kappa
(fun _ : Fin J × ℝ => 0) measurable_const (fun _ => by norm_num)
have hmap1 := minimaxDataMeasure_map_design J kappa
(fun p : Fin J × ℝ => q p.2) (hqmeas.comp measurable_snd)
(fun p => hqbound p.2)
have hshared : SharedClampDesign
(minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0))
(minimaxClampLaw J kappa (fun p => q p.2)) := by
refine ⟨?_, rfl, rfl⟩
change (minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => 0)).map
(fun o => (o.X, o.A)) =
(minimaxDataMeasure J kappa (fun p : Fin J × ℝ => q p.2)).map
(fun o => (o.X, o.A))
rw [hmap0, hmap1]
refine ⟨hshared,
minimaxDataMeasure_ae_bernoulli J kappa
(fun _ : Fin J × ℝ => 0) measurable_const,
minimaxDataMeasure_ae_bernoulli J kappa
(fun p : Fin J × ℝ => q p.2) (hqmeas.comp measurable_snd),
hamp, hamp_le, ?_⟩
refine ⟨CausalSmith.Stat.DoseResponseMinimax.doseBump, ?_,
CausalSmith.Stat.DoseResponseMinimax.doseBump_zero,
CausalSmith.Stat.DoseResponseMinimax.doseBump_nonneg, ?_,
fun u hu => CausalSmith.Stat.DoseResponseMinimax.doseBump_abs_le_one u, ?_⟩
· exact (CausalSmith.Stat.DoseResponseMinimax.doseContDiffBump.contDiff
(n := ⊤)).continuous
· intro u hu
apply CausalSmith.Stat.DoseResponseMinimax.doseBump_eq_zero_of_one_le_abs
rw [Set.mem_Icc, not_and_or, not_le, not_le] at hu
rcases hu with hu | hu
· rw [abs_of_neg (by linarith)]; linarith
· exact hu.le.trans (le_abs_self u)
· intro x a ha
simp [minimaxClampLaw, q]
theorem localizedBernoulli_witness_eventually
The fixed smooth bump supplies the localized information-bandwidth witness with a product chi-squared bound uniform in the threshold sequence and sample size. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
∃ amplitude D : ℝ,
0 < amplitude
amplitude ≤ 1 / 4
0 < D
conclusion 1
deltaSeq :
ℕ → ℝ
ThresholdSequence deltaBar deltaSeq
∀ᶠ n in atTop, let delta := deltaSeq n let h := infoBandwidth n delta beta kappa deltaBar ∃ Q0 Q1 : ClampLaw J, ClampModel Q0 beta kappa L cminus cplus pmin ∧ ClampModel Q1 beta kappa L cminus cplus pmin ∧ IidSampling Q0 n ∧ IidSampling Q1 n ∧ LocalizedBernoulliPerturbation Q0 Q1 beta delta h amplitude ∧ amplitude * delta ^ (kappa + 1) * h ^ beta
≤ |clampFunctional Q1 delta - clampFunctional Q0 delta| ∧
ProperProductChiSq Q1 Q0 n ∧
productChiSq Q1 Q0 n ≤ D
Proof (Lean source)
lemma localizedBernoulli_witness_eventually
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
∃ amplitude D : ℝ, 0 < amplitude ∧ amplitude ≤ 1 / 4 ∧ 0 < D ∧
∀ (deltaSeq : ℕ → ℝ), ThresholdSequence deltaBar deltaSeq →
∀ᶠ n in atTop,
let delta := deltaSeq n
let h := infoBandwidth n delta beta kappa deltaBar
∃ Q0 Q1 : ClampLaw J,
ClampModel Q0 beta kappa L cminus cplus pmin ∧
ClampModel Q1 beta kappa L cminus cplus pmin ∧
IidSampling Q0 n ∧ IidSampling Q1 n ∧
LocalizedBernoulliPerturbation Q0 Q1 beta delta h amplitude ∧
amplitude * delta ^ (kappa + 1) * h ^ beta ≤
|clampFunctional Q1 delta - clampFunctional Q0 delta| ∧
ProperProductChiSq Q1 Q0 n ∧
productChiSq Q1 Q0 n ≤ D := by
have hreg' := hreg
rcases hreg' with
⟨hJ, hbeta, hkappa, hL, hcminus, hcminus_le, hcplus,
hpmin, hpmin_le, hdeltaBar, hdeltaBar1, halpha, halpha1⟩
obtain ⟨amplitude, hamp, hamp_le, hamp_model⟩ :=
exists_minimaxBump_amplitude beta L hbeta hL
let d0 : ℝ := 8 * (kappa + 1) * amplitude ^ 2
let D : ℝ := Real.exp d0
have hD : 0 < D := Real.exp_pos _
refine ⟨amplitude, D, hamp, hamp_le, hD, ?_⟩
intro deltaSeq hdelta
have hbal := infoBandwidth_eventually_balance beta kappa deltaBar hbeta hkappa
⟨hdeltaBar, hdeltaBar1⟩ deltaSeq hdelta
filter_upwards [hbal] with n hbn
dsimp only
let delta := deltaSeq n
let h := infoBandwidth n delta beta kappa deltaBar
let q := fun a : ℝ => amplitude * h ^ beta *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / h)
let Q0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
let Q1 := minimaxClampLaw J kappa (fun p => q p.2)
have hh : 0 < h := by simpa [h, delta] using hbn.1
have hhbar : h ≤ 1 - deltaBar := by simpa [h, delta] using hbn.2.1
have hh1 : h ≤ 1 := hhbar.trans (by linarith)
have hbalance : (n : ℝ) * h ^ (2 * beta + 1) * (delta + h) ^ kappa = 1 := by
simpa [h, delta] using hbn.2.2
have hdeltaN : delta ∈ Icc (0 : ℝ) deltaBar := hdelta n
have hdelta1 : delta ∈ Icc (0 : ℝ) 1 :=
⟨hdeltaN.1, hdeltaN.2.trans hdeltaBar1.le⟩
have hshape := hamp_model hdelta1.1 hdelta1.2 hh hh1
have hqmeas : Measurable q := by dsimp [q]; fun_prop
have hq0 (a : ℝ) : 0 ≤ q a := by
dsimp [q]
exact mul_nonneg (mul_nonneg hamp.le (Real.rpow_nonneg hh.le _))
(CausalSmith.Stat.DoseResponseMinimax.doseBump_nonneg _)
have hqle (a : ℝ) : q a ≤ 1 / 4 := by
have hb := CausalSmith.Stat.DoseResponseMinimax.doseBump_le_one ((a - delta) / h)
have hhpow : h ^ beta ≤ 1 := Real.rpow_le_one hh.le hh1 hbeta.le
dsimp [q]
calc
_ ≤ amplitude * h ^ beta * 1 := by gcongr
_ ≤ amplitude * 1 := by
simpa using mul_le_mul_of_nonneg_left hhpow hamp.le
_ ≤ 1 / 4 := by simpa using hamp_le
have hqbound (a : ℝ) : |q a| ≤ 1 / 2 := by
rw [abs_of_nonneg (hq0 a)]
linarith [hqle a]
have hqstrict (p : Fin J × ℝ) : |q p.2| < 1 / 2 := by
rw [abs_of_nonneg (hq0 p.2)]
linarith [hqle p.2]
have hQ0 : ClampModel Q0 beta kappa L cminus cplus pmin := by
exact minimaxCenter_mem_model J beta kappa L cminus cplus pmin deltaBar alpha hreg
have hQ1 : ClampModel Q1 beta kappa L cminus cplus pmin := by
apply minimaxClampModel_of_taylor J beta kappa L cminus cplus pmin
hJ hbeta hkappa hcminus_le hcplus hpmin_le q hqmeas hqbound
exact hshape.1
exact hshape.2.1
exact hshape.2.2
have hpert : LocalizedBernoulliPerturbation Q0 Q1 beta delta h amplitude := by
simpa [Q0, Q1, q] using minimaxBump_localizedPerturbation
J beta kappa delta h amplitude hbeta hh hh1 hamp hamp_le
have hsep0 := minimaxLocalSeparation J beta kappa delta h amplitude
hJ hbeta hkappa hdelta1 hh hh1 hamp.le hamp_le
have hsep : amplitude * delta ^ (kappa + 1) * h ^ beta ≤
|clampFunctional Q1 delta - clampFunctional Q0 delta| := by
have hgap0 : 0 ≤ clampFunctional Q1 delta - clampFunctional Q0 delta :=
le_trans (mul_nonneg
(mul_nonneg hamp.le (Real.rpow_nonneg hdelta1.1 _))
(Real.rpow_nonneg hh.le _)) (by simpa [Q0, Q1, q] using hsep0)
rw [abs_of_nonneg hgap0]
simpa [Q0, Q1, q] using hsep0
let I : ℝ := ∫ p, 4 * ((fun p : Fin J × ℝ => q p.2) p) ^ 2
∂minimaxDesignMeasure J kappa
have hIle : I ≤ 8 * (kappa + 1) * amplitude ^ 2 *
h ^ (2 * beta + 1) * (delta + h) ^ kappa := by
simpa [I, q] using minimaxLocal_designIntegral_le J beta kappa delta h amplitude
hJ hkappa hdelta1.1 hh hamp.le
have hnI : (n : ℝ) * I ≤ d0 := by
dsimp [d0]
calc
(n : ℝ) * I ≤ (n : ℝ) *
(8 * (kappa + 1) * amplitude ^ 2 * h ^ (2 * beta + 1) *
(delta + h) ^ kappa) := mul_le_mul_of_nonneg_left hIle (Nat.cast_nonneg n)
_ = 8 * (kappa + 1) * amplitude ^ 2 := by
calc
_ = (8 * (kappa + 1) * amplitude ^ 2) *
((n : ℝ) * h ^ (2 * beta + 1) * (delta + h) ^ kappa) := by ring
_ = _ := by rw [hbalance, mul_one]
have hchiEq := minimaxProduct_chiSqDiv_center J n kappa hJ hkappa
(fun p : Fin J × ℝ => q p.2) (hqmeas.comp measurable_snd) hqstrict
change 1 + chiSqDiv
(Measure.pi (fun _ : Fin n => minimaxDataMeasure J kappa
(fun p : Fin J × ℝ => q p.2)))
(Measure.pi (fun _ : Fin n => minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => 0))) = (1 + I) ^ n at hchiEq
have hI0 : 0 ≤ I := integral_nonneg_of_ae (ae_of_all _ fun p => by positivity)
have hpow : (1 + I) ^ n ≤ D := by
calc
(1 + I) ^ n ≤ (Real.exp I) ^ n :=
pow_le_pow_left₀ (by linarith) (by simpa [add_comm] using Real.add_one_le_exp I) n
_ = Real.exp ((n : ℝ) * I) := by rw [Real.exp_nat_mul]
_ ≤ Real.exp d0 := Real.exp_le_exp.mpr hnI
_ = D := rfl
have hchi : productChiSq Q1 Q0 n ≤ D := by
change chiSqDiv
(Measure.pi (fun _ : Fin n => minimaxDataMeasure J kappa
(fun p : Fin J × ℝ => q p.2)))
(Measure.pi (fun _ : Fin n => minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => 0))) ≤ D
linarith [hchiEq, hpow]
have hac1 := minimaxDataMeasure_ac_center J kappa hJ hkappa
(fun p : Fin J × ℝ => q p.2) (hqmeas.comp measurable_snd) hqstrict
have hint1 := minimaxDataMeasure_sq_integrable_center J kappa hJ hkappa
(fun p : Fin J × ℝ => q p.2) (hqmeas.comp measurable_snd) hqstrict
letI : IsProbabilityMeasure (minimaxDataMeasure J kappa
(fun p : Fin J × ℝ => q p.2)) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hJ hkappa _
(hqmeas.comp measurable_snd) (fun p => (hqstrict p).le)
letI : IsProbabilityMeasure (minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => 0)) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hJ hkappa _ measurable_const
(fun _ => by norm_num)
have hac : iidProduct Q1 n ≪ iidProduct Q0 n := by
simpa [Q0, Q1, iidProduct, minimaxClampLaw] using
Causalean.Stat.pi_iid_absolutelyContinuous _ _ hac1 n
have hint : Integrable (fun z => (((iidProduct Q1 n).rnDeriv
(iidProduct Q0 n) z).toReal - 1) ^ 2) (iidProduct Q0 n) := by
simpa [Q0, Q1, iidProduct, minimaxClampLaw] using
pi_iid_integrable_sq_dev _ _ hac1 hint1 n
exact ⟨Q0, Q1, hQ0, hQ1, clampModel_iidSampling hQ0,
clampModel_iidSampling hQ1, hpert, hsep, ⟨hac, hint⟩, hchi⟩
theorem observedMinimaxRisk_lower_of_two_point
A close pair with a chi-squared budget below four gives an absolute-risk lower bound for the observed minimax problem. The result uses the hP0 condition, the hP1 condition, the hdelta0 condition, the hdelta1 condition, the hgap0 condition, the hsep condition, the hac condition, the hint condition, the hchi0 condition, the hchi4 condition, the hchi condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
P0 P1 :
ClampLaw J
n :
ℕ
beta kappa L cminus cplus pmin delta gap chi :
ℝ
hP0 :
ClampModel P0 beta kappa L cminus cplus pmin
hP1 :
ClampModel P1 beta kappa L cminus cplus pmin
hdelta0 :
0 ≤ delta
hdelta1 :
delta ≤ 1
hgap0 :
0 ≤ gap
hsep :
gap ≤ |clampFunctional P1 delta - clampFunctional P0 delta|
hac :
iidProduct P1 n ≪ iidProduct P0 n
hint :
Integrable (fun z => (((iidProduct P1 n).rnDeriv (iidProduct P0 n) z).toReal - 1) ^ 2) (iidProduct P0 n)
hchi0 :
0 ≤ chi
hchi4 :
chi < 4
hchi :
productChiSq P1 P0 n ≤ chi
conclusion
((1 - (1 / 2) * sqrt chi) / 4) * gap
≤ observedMinimaxRisk J n beta kappa L cminus cplus pmin delta
Proof (Lean source)
lemma observedMinimaxRisk_lower_of_two_point
(P0 P1 : ClampLaw J) (n : ℕ)
(beta kappa L cminus cplus pmin delta gap chi : ℝ)
(hP0 : ClampModel P0 beta kappa L cminus cplus pmin)
(hP1 : ClampModel P1 beta kappa L cminus cplus pmin)
(hdelta0 : 0 ≤ delta) (hdelta1 : delta ≤ 1) (hgap0 : 0 ≤ gap)
(hsep : gap ≤ |clampFunctional P1 delta - clampFunctional P0 delta|)
(hac : iidProduct P1 n ≪ iidProduct P0 n)
(hint : Integrable (fun z => (((iidProduct P1 n).rnDeriv
(iidProduct P0 n) z).toReal - 1) ^ 2) (iidProduct P0 n))
(hchi0 : 0 ≤ chi) (hchi4 : chi < 4)
(hchi : productChiSq P1 P0 n ≤ chi) :
((1 - (1 / 2) * sqrt chi) / 4) * gap ≤
observedMinimaxRisk J n beta kappa L cminus cplus pmin delta := by
let outer : Set ℝ := {r : ℝ | ∃ est : Estimator n J,
ObservedMeasurableEstimator est ∧ (∀ z, est z ∈ Icc (0 : ℝ) 1) ∧
r = sSup {v : ℝ | ∃ P : ClampLaw J,
ClampModel P beta kappa L cminus cplus pmin ∧
v = estimatorRisk P n delta est}}
change _ ≤ sInf outer
have houter : outer.Nonempty := by
let est : Estimator n J := fun _ => 0
refine ⟨sSup {v : ℝ | ∃ P : ClampLaw J,
ClampModel P beta kappa L cminus cplus pmin ∧
v = estimatorRisk P n delta est}, est, measurable_const, ?_, rfl⟩
intro z
exact ⟨le_rfl, by norm_num⟩
apply le_csInf houter
intro r hr
rcases hr with ⟨est, hest, hestrange, rfl⟩
let risks : Set ℝ := {v : ℝ | ∃ P : ClampLaw J,
ClampModel P beta kappa L cminus cplus pmin ∧
v = estimatorRisk P n delta est}
have hrisk_le (P : ClampLaw J)
(hP : ClampModel P beta kappa L cminus cplus pmin) :
estimatorRisk P n delta est ≤ 1 := by
letI : IsProbabilityMeasure P.dataMeasure := hP.probability
letI : IsProbabilityMeasure (iidProduct P n) := by
unfold iidProduct
infer_instance
have htheta := clampFunctional_mem_Icc P hP hdelta0 hdelta1
unfold estimatorRisk
have hfint : Integrable (fun z => |est z - clampFunctional P delta|)
(iidProduct P n) := by
refine Integrable.of_bound ((hest.sub measurable_const).abs.aestronglyMeasurable) 1 ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_abs, abs_le]
constructor <;> linarith [(hestrange z).1, (hestrange z).2,
htheta.1, htheta.2]
calc
(∫ z, |est z - clampFunctional P delta| ∂iidProduct P n) ≤
∫ _z, (1 : ℝ) ∂iidProduct P n := by
apply integral_mono_ae hfint (integrable_const 1)
exact ae_of_all _ fun z => by
rw [abs_le]
constructor <;> linarith [(hestrange z).1, (hestrange z).2,
htheta.1, htheta.2]
(∫ _z, (1 : ℝ) ∂iidProduct P n) = 1 := by simp
have hbdd : BddAbove risks := by
refine ⟨1, ?_⟩
intro v hv
rcases hv with ⟨P, hP, rfl⟩
exact hrisk_le P hP
have hle0 : estimatorRisk P0 n delta est ≤ sSup risks := by
apply le_csSup hbdd
exact ⟨P0, hP0, rfl⟩
have hle1 : estimatorRisk P1 n delta est ≤ sSup risks := by
apply le_csSup hbdd
exact ⟨P1, hP1, rfl⟩
letI : IsProbabilityMeasure P0.dataMeasure := hP0.probability
letI : IsProbabilityMeasure P1.dataMeasure := hP1.probability
letI : IsProbabilityMeasure (iidProduct P0 n) := by
unfold iidProduct
infer_instance
letI : IsProbabilityMeasure (iidProduct P1 n) := by
unfold iidProduct
infer_instance
have htest := two_point_lower_bound_of_chiSqDiv_le
(P₀ := iidProduct P1 n) (P₁ := iidProduct P0 n) hest
(θ₀ := clampFunctional P1 delta) (θ₁ := clampFunctional P0 delta)
(s := gap / 2) (c := chi) (by convert hsep using 1 <;> ring) hac hint hchi
have hfloor : 0 < (1 - (1 / 2) * sqrt chi) / 2 := by
have hsqrt : sqrt chi < 2 := by
rw [Real.sqrt_lt' (by norm_num : (0 : ℝ) < 2)]
convert hchi4 using 1 <;> norm_num
linarith
have hint0 : Integrable (fun z => |est z - clampFunctional P0 delta|)
(iidProduct P0 n) := by
refine Integrable.of_bound ((hest.sub measurable_const).abs.aestronglyMeasurable) 1 ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_abs]
have ht := clampFunctional_mem_Icc P0 hP0 hdelta0 hdelta1
rw [abs_le]
constructor <;> linarith [(hestrange z).1, (hestrange z).2, ht.1, ht.2]
have hint1 : Integrable (fun z => |est z - clampFunctional P1 delta|)
(iidProduct P1 n) := by
refine Integrable.of_bound ((hest.sub measurable_const).abs.aestronglyMeasurable) 1 ?_
filter_upwards with z
rw [Real.norm_eq_abs, abs_abs]
have ht := clampFunctional_mem_Icc P1 hP1 hdelta0 hdelta1
rw [abs_le]
constructor <;> linarith [(hestrange z).1, (hestrange z).2, ht.1, ht.2]
have hevent0 := mul_meas_ge_le_integral_of_nonneg
(ae_of_all _ fun z => abs_nonneg (est z - clampFunctional P1 delta)) hint1 (gap / 2)
have hevent1 := mul_meas_ge_le_integral_of_nonneg
(ae_of_all _ fun z => abs_nonneg (est z - clampFunctional P0 delta)) hint0 (gap / 2)
have hmaxEvent : ((1 - (1 / 2) * sqrt chi) / 2) * (gap / 2) ≤
max (estimatorRisk P1 n delta est) (estimatorRisk P0 n delta est) := by
unfold estimatorRisk
have hg2 : 0 ≤ gap / 2 := by positivity
calc
_ ≤ (gap / 2) * max
((iidProduct P1 n).real {z | gap / 2 ≤
|est z - clampFunctional P1 delta|})
((iidProduct P0 n).real {z | gap / 2 ≤
|est z - clampFunctional P0 delta|}) := by
rw [mul_comm]
exact mul_le_mul_of_nonneg_left htest hg2
_ = max ((gap / 2) * (iidProduct P1 n).real {z | gap / 2 ≤
|est z - clampFunctional P1 delta|})
((gap / 2) * (iidProduct P0 n).real {z | gap / 2 ≤
|est z - clampFunctional P0 delta|}) := by
rw [mul_max_of_nonneg _ _ hg2]
_ ≤ _ := max_le_max hevent0 hevent1
calc
((1 - (1 / 2) * sqrt chi) / 4) * gap =
((1 - (1 / 2) * sqrt chi) / 2) * (gap / 2) := by ring
_ ≤ max (estimatorRisk P1 n delta est) (estimatorRisk P0 n delta est) := hmaxEvent
_ ≤ sSup risks := max_le hle1 hle0
theorem observedMinimaxRisk_eventually_ge_root
A quarter-strength constant Bernoulli shift gives the parametric component of the minimax lower bound while keeping chi-squared below four. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
∃ c : ℝ,
0 < c
conclusion 1
deltaSeq :
ℕ → ℝ
ThresholdSequence deltaBar deltaSeq
∀ᶠ n : ℕ in atTop, c * (n : ℝ) ^ (-(1 : ℝ) / 2)
≤ observedMinimaxRisk J n beta kappa L cminus cplus pmin (deltaSeq n)
Proof (Lean source)
lemma observedMinimaxRisk_eventually_ge_root
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
∃ c : ℝ, 0 < c ∧ ∀ (deltaSeq : ℕ → ℝ),
ThresholdSequence deltaBar deltaSeq →
∀ᶠ n : ℕ in atTop, c * (n : ℝ) ^ (-(1 : ℝ) / 2) ≤
observedMinimaxRisk J n beta kappa L cminus cplus pmin (deltaSeq n) := by
rcases hreg with ⟨hJ, hbeta, hkappa, hL, hcminus, hcminus_le, hcplus,
hpmin, hpmin_le, hdeltaBar, hdeltaBar1, halpha, halpha1⟩
let chi : ℝ := exp (1 / 4 : ℝ) - 1
let floor : ℝ := (1 - (1 / 2) * sqrt chi) / 4
let c : ℝ := floor / 4
have hchi0 : 0 ≤ chi := by
dsimp [chi]
exact sub_nonneg.mpr (Real.one_le_exp (by norm_num))
have hchi4 : chi < 4 := by
dsimp [chi]
have he : exp (1 / 4 : ℝ) < exp 1 :=
Real.exp_lt_exp.mpr (by norm_num)
linarith [Real.exp_one_lt_three]
have hfloor : 0 < floor := by
dsimp [floor]
have hs : sqrt chi < 2 := by
rw [Real.sqrt_lt' (by norm_num : (0 : ℝ) < 2)]
convert hchi4 using 1 <;> norm_num
linarith
refine ⟨c, by dsimp [c]; positivity, ?_⟩
intro deltaSeq hdelta
filter_upwards [eventually_ge_atTop 1] with n hn
have hnpos : 0 < n := by omega
have hnR : 0 < (n : ℝ) := by exact_mod_cast hnpos
let root : ℝ := (n : ℝ) ^ (-(1 : ℝ) / 2)
let eps : ℝ := (1 / 4 : ℝ) * root
have hroot : 0 < root := Real.rpow_pos_of_pos hnR _
have heps : 0 < eps := mul_pos (by norm_num) hroot
have hepsq : eps ≤ 1 / 4 := by
have hroot1 : root ≤ 1 := by
rw [show (1 : ℝ) = 1 ^ (-(1 : ℝ) / 2) by norm_num]
exact Real.rpow_le_rpow_of_nonpos (by norm_num) (by exact_mod_cast hn) (by norm_num)
dsimp [eps]
nlinarith
have hepsb : |eps| ≤ 1 / 2 := by rw [abs_of_pos heps]; linarith
let P0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
let P1 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => eps)
have hP0 : ClampModel P0 beta kappa L cminus cplus pmin :=
minimaxCenter_mem_model J beta kappa L cminus cplus pmin deltaBar alpha
⟨hJ, hbeta, hkappa, hL, hcminus, hcminus_le, hcplus,
hpmin, hpmin_le, hdeltaBar, hdeltaBar1, halpha, halpha1⟩
have hP1 : ClampModel P1 beta kappa L cminus cplus pmin :=
minimaxConstant_mem_model J beta kappa L cminus cplus pmin deltaBar alpha eps
⟨hJ, hbeta, hkappa, hL, hcminus, hcminus_le, hcplus,
hpmin, hpmin_le, hdeltaBar, hdeltaBar1, halpha, halpha1⟩ hepsb
have hdelta1 : deltaSeq n ∈ Icc (0 : ℝ) 1 :=
⟨(hdelta n).1, (hdelta n).2.trans hdeltaBar1.le⟩
have hsepEq := minimaxGlobalSeparation J kappa (deltaSeq n) eps hJ hkappa
hdelta1 hepsb
have hsep : eps ≤ |clampFunctional P1 (deltaSeq n) -
clampFunctional P0 (deltaSeq n)| := by
rw [show P1 = minimaxClampLaw J kappa (fun _ : Fin J × ℝ => eps) by rfl,
show P0 = minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0) by rfl,
hsepEq, abs_of_pos heps]
have hrootSq : root ^ 2 = (n : ℝ)⁻¹ := by
dsimp [root]
rw [← Real.rpow_natCast, ← Real.rpow_mul hnR.le]
norm_num
rw [Real.rpow_neg_one]
have hepsSq : eps ^ 2 = (1 / 16 : ℝ) * (n : ℝ)⁻¹ := by
dsimp [eps]
rw [mul_pow, hrootSq]
norm_num
letI : IsProbabilityMeasure (minimaxDesignMeasure J kappa) :=
minimaxDesignMeasure_isProbabilityMeasure J kappa hJ hkappa
have hintEq : (∫ p, 4 * ((fun _ : Fin J × ℝ => eps) p) ^ 2
∂minimaxDesignMeasure J kappa) = 4 * eps ^ 2 := by simp
have hchiEq := minimaxProduct_chiSqDiv_center J n kappa hJ hkappa
(fun _ : Fin J × ℝ => eps) measurable_const (fun _ => by
rw [abs_of_pos heps]; linarith [hepsq])
rw [hintEq] at hchiEq
have hnE : (n : ℝ) * (4 * eps ^ 2) = 1 / 4 := by
rw [hepsSq]
field_simp [hnR.ne']
ring
have hpow : (1 + 4 * eps ^ 2) ^ n ≤ exp (1 / 4 : ℝ) := by
calc
_ ≤ (exp (4 * eps ^ 2)) ^ n :=
pow_le_pow_left₀ (by positivity)
(by simpa [add_comm] using Real.add_one_le_exp (4 * eps ^ 2)) n
_ = exp ((n : ℝ) * (4 * eps ^ 2)) := by rw [Real.exp_nat_mul]
_ = _ := by rw [hnE]
have hchi : productChiSq P1 P0 n ≤ chi := by
change chiSqDiv
(Measure.pi (fun _ : Fin n => minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => eps)))
(Measure.pi (fun _ : Fin n => minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => 0))) ≤ chi
dsimp [chi]
linarith [hchiEq, hpow]
have hac1 := minimaxDataMeasure_ac_center J kappa hJ hkappa
(fun _ : Fin J × ℝ => eps) measurable_const (fun _ => by
rw [abs_of_pos heps]; linarith [hepsq])
have hint1 := minimaxDataMeasure_sq_integrable_center J kappa hJ hkappa
(fun _ : Fin J × ℝ => eps) measurable_const (fun _ => by
rw [abs_of_pos heps]; linarith [hepsq])
letI : IsProbabilityMeasure (minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => eps)) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hJ hkappa _ measurable_const
(fun _ => hepsb)
letI : IsProbabilityMeasure (minimaxDataMeasure J kappa
(fun _ : Fin J × ℝ => 0)) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hJ hkappa _ measurable_const
(fun _ => by norm_num)
have hac := pi_iid_absolutelyContinuous _ _ hac1 n
have hint := pi_iid_integrable_sq_dev _ _ hac1 hint1 n
have hlower := observedMinimaxRisk_lower_of_two_point P0 P1 n beta kappa L
cminus cplus pmin (deltaSeq n) eps chi hP0 hP1 hdelta1.1 hdelta1.2
heps.le hsep (by simpa [P0, P1, iidProduct, minimaxClampLaw] using hac)
(by simpa [P0, P1, iidProduct, minimaxClampLaw] using hint) hchi0 hchi4 hchi
dsimp [c, floor, eps, root] at hlower ⊢
convert hlower using 1 <;> ring
theorem observedMinimaxRisk_eventually_ge_local
A fixed contraction of the information bandwidth makes the localized experiment close while preserving the local frontier order. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
∃ c : ℝ,
0 < c
conclusion 1
deltaSeq :
ℕ → ℝ
ThresholdSequence deltaBar deltaSeq
∀ᶠ n : ℕ in atTop, let h := infoBandwidth n (deltaSeq n) beta kappa deltaBar c * (deltaSeq n) ^ (kappa + 1) * h ^ beta
≤ observedMinimaxRisk J n beta kappa L cminus cplus pmin (deltaSeq n)
Proof (Lean source)
lemma observedMinimaxRisk_eventually_ge_local
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
∃ c : ℝ, 0 < c ∧ ∀ deltaSeq : ℕ → ℝ,
ThresholdSequence deltaBar deltaSeq → ∀ᶠ n : ℕ in atTop,
let h := infoBandwidth n (deltaSeq n) beta kappa deltaBar
c * (deltaSeq n) ^ (kappa + 1) * h ^ beta ≤
observedMinimaxRisk J n beta kappa L cminus cplus pmin (deltaSeq n) := by
have hr := hreg
rcases hr with ⟨hJ, hbeta, hkappa, hL, hcminus, hcminus_le, hcplus,
hpmin, hpmin_le, hdeltaBar, hdeltaBar1, halpha, halpha1⟩
obtain ⟨amplitude, hamp, hamp_le, hamp_model⟩ :=
exists_minimaxBump_amplitude beta L hbeta hL
let d0 : ℝ := 8 * (kappa + 1) * amplitude ^ 2
let p : ℝ := 2 * beta + 1
let t : ℝ := exp (-(d0 + 1) / p)
let chi : ℝ := exp 1 - 1
let floor : ℝ := (1 - (1 / 2) * sqrt chi) / 4
let c : ℝ := floor * amplitude * t ^ beta
have hd0 : 0 ≤ d0 := by dsimp [d0]; positivity
have hp : 0 < p := by dsimp [p]; linarith
have ht : 0 < t := Real.exp_pos _
have ht1 : t ≤ 1 := by
dsimp [t]
rw [Real.exp_le_one_iff]
exact div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr (by linarith)) hp.le
have htp : t ^ p = exp (-(d0 + 1)) := by
dsimp [t]
rw [← Real.exp_mul]
congr 1
field_simp [hp.ne']
have hdsmall : d0 * t ^ p ≤ 1 := by
rw [htp]
have hdexp : d0 ≤ exp d0 := by
exact (Real.add_one_le_exp d0).trans' (by linarith)
calc
d0 * exp (-(d0 + 1)) ≤ exp d0 * exp (-(d0 + 1)) :=
mul_le_mul_of_nonneg_right hdexp (Real.exp_pos _).le
_ = exp (-1) := by rw [← Real.exp_add]; congr 1; ring
_ ≤ 1 := Real.exp_le_one_iff.mpr (by norm_num)
have hchi0 : 0 ≤ chi := by dsimp [chi]; exact sub_nonneg.mpr (Real.one_le_exp zero_le_one)
have hchi4 : chi < 4 := by dsimp [chi]; linarith [Real.exp_one_lt_three]
have hfloor : 0 < floor := by
dsimp [floor]
have hs : sqrt chi < 2 := by
rw [Real.sqrt_lt' (by norm_num : (0 : ℝ) < 2)]
convert hchi4 using 1 <;> norm_num
linarith
refine ⟨c, by dsimp [c]; positivity, ?_⟩
intro deltaSeq hdelta
have hbal := infoBandwidth_eventually_balance beta kappa deltaBar hbeta hkappa
⟨hdeltaBar, hdeltaBar1⟩ deltaSeq hdelta
filter_upwards [hbal] with n hbn
dsimp only
let delta := deltaSeq n
let h := infoBandwidth n delta beta kappa deltaBar
let hs := t * h
let q := fun a : ℝ => amplitude * hs ^ beta *
CausalSmith.Stat.DoseResponseMinimax.doseBump ((a - delta) / hs)
let P0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
let P1 := minimaxClampLaw J kappa (fun z => q z.2)
have hh : 0 < h := hbn.1
have hh1 : h ≤ 1 := hbn.2.1.trans (by linarith)
have hhs : 0 < hs := mul_pos ht hh
have hhs1 : hs ≤ 1 := (mul_le_of_le_one_left hh.le ht1).trans hh1
have hd : delta ∈ Icc (0 : ℝ) 1 :=
⟨(hdelta n).1, (hdelta n).2.trans hdeltaBar1.le⟩
have hshape := hamp_model hd.1 hd.2 hhs hhs1
have hqmeas : Measurable q := by dsimp [q]; fun_prop
have hq0 (a : ℝ) : 0 ≤ q a := by
dsimp [q]
exact mul_nonneg (mul_nonneg hamp.le (Real.rpow_nonneg hhs.le _))
(CausalSmith.Stat.DoseResponseMinimax.doseBump_nonneg _)
have hqbound (a : ℝ) : |q a| ≤ 1 / 2 := by
rw [abs_of_nonneg (hq0 a)]
have hb := CausalSmith.Stat.DoseResponseMinimax.doseBump_le_one ((a-delta)/hs)
have hpw := Real.rpow_le_one hhs.le hhs1 hbeta.le
dsimp [q]
calc
_ ≤ amplitude * hs ^ beta := by nlinarith [mul_nonneg hamp.le (Real.rpow_nonneg hhs.le beta)]
_ ≤ amplitude := by simpa using mul_le_mul_of_nonneg_left hpw hamp.le
_ ≤ 1 / 2 := by linarith
have hqstrict (z : Fin J × ℝ) : |q z.2| < 1 / 2 := by
rw [abs_of_nonneg (hq0 z.2)]
have hb := CausalSmith.Stat.DoseResponseMinimax.doseBump_le_one ((z.2-delta)/hs)
have hpw := Real.rpow_le_one hhs.le hhs1 hbeta.le
have : q z.2 ≤ amplitude := by
dsimp [q]
calc
_ ≤ amplitude * hs ^ beta := by
nlinarith [mul_nonneg hamp.le (Real.rpow_nonneg hhs.le beta)]
_ ≤ amplitude := by simpa using mul_le_mul_of_nonneg_left hpw hamp.le
linarith [hamp_le]
have hP0 := minimaxCenter_mem_model J beta kappa L cminus cplus pmin deltaBar alpha hreg
have hP1 : ClampModel P1 beta kappa L cminus cplus pmin := by
apply minimaxClampModel_of_taylor J beta kappa L cminus cplus pmin
hJ hbeta hkappa hcminus_le hcplus hpmin_le q hqmeas hqbound
exact hshape.1
exact hshape.2.1
exact hshape.2.2
have hsep0 := minimaxLocalSeparation J beta kappa delta hs amplitude
hJ hbeta hkappa hd hhs hhs1 hamp.le hamp_le
have hsep : amplitude * delta ^ (kappa + 1) * hs ^ beta ≤
|clampFunctional P1 delta - clampFunctional P0 delta| := by
have hn : 0 ≤ clampFunctional P1 delta - clampFunctional P0 delta :=
(mul_nonneg (mul_nonneg hamp.le (Real.rpow_nonneg hd.1 _))
(Real.rpow_nonneg hhs.le _)).trans (by simpa [P0, P1, q] using hsep0)
rw [abs_of_nonneg hn]
simpa [P0, P1, q] using hsep0
let I := ∫ z, 4 * ((fun z : Fin J × ℝ => q z.2) z)^2 ∂minimaxDesignMeasure J kappa
have hI := minimaxLocal_designIntegral_le J beta kappa delta hs amplitude
hJ hkappa hd.1 hhs hamp.le
have hnI : (n : ℝ) * I ≤ 1 := by
have hw : delta + hs ≤ delta + h := by dsimp [hs]; nlinarith [mul_le_of_le_one_left hh.le ht1]
have hpoww := Real.rpow_le_rpow (add_nonneg hd.1 hhs.le) hw hkappa
have hb := hbn.2.2
calc
_ ≤ (n : ℝ) * (8*(kappa+1)*amplitude^2*hs^(2*beta+1)*(delta+hs)^kappa) :=
mul_le_mul_of_nonneg_left (by simpa [I, q] using hI) (Nat.cast_nonneg n)
_ ≤ d0 * t ^ p * ((n:ℝ)*h^(2*beta+1)*(delta+h)^kappa) := by
dsimp [d0, p, hs]
rw [Real.mul_rpow ht.le hh.le]
ring_nf
gcongr
_ = d0 * t ^ p := by rw [hb, mul_one]
_ ≤ 1 := hdsmall
have hchiEq := minimaxProduct_chiSqDiv_center J n kappa hJ hkappa
(fun z : Fin J × ℝ => q z.2) (hqmeas.comp measurable_snd) hqstrict
have hpow : (1 + I)^n ≤ exp 1 := by
calc
_ ≤ (exp I)^n := pow_le_pow_left₀ (by positivity)
(by simpa [add_comm] using Real.add_one_le_exp I) n
_ = exp ((n:ℝ)*I) := by rw [Real.exp_nat_mul]
_ ≤ _ := Real.exp_le_exp.mpr hnI
have hchi : productChiSq P1 P0 n ≤ chi := by
change chiSqDiv _ _ ≤ chi
dsimp [chi, P0, P1, iidProduct, minimaxClampLaw]
change chiSqDiv _ _ ≤ exp 1 - 1
change 1 + chiSqDiv _ _ = (1+I)^n at hchiEq
linarith
have hac1 : minimaxDataMeasure J kappa (fun z : Fin J × ℝ => q z.2) ≪
minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => 0) :=
minimaxDataMeasure_ac_center J kappa hJ hkappa _
(hqmeas.comp measurable_snd) hqstrict
have hint1 : Integrable
(fun o => (((minimaxDataMeasure J kappa
(fun z : Fin J × ℝ => q z.2)).rnDeriv
(minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => 0)) o).toReal - 1) ^ 2)
(minimaxDataMeasure J kappa (fun _ : Fin J × ℝ => 0)) := by
exact minimaxDataMeasure_sq_integrable_center J kappa hJ hkappa _
(hqmeas.comp measurable_snd) hqstrict
letI : IsProbabilityMeasure (minimaxDataMeasure J kappa (fun z : Fin J×ℝ => q z.2)) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hJ hkappa _
(hqmeas.comp measurable_snd) (fun z => hqstrict z |>.le)
letI : IsProbabilityMeasure (minimaxDataMeasure J kappa (fun _ : Fin J×ℝ => 0)) :=
minimaxDataMeasure_isProbabilityMeasure J kappa hJ hkappa _ measurable_const (fun _ => by norm_num)
have hl := observedMinimaxRisk_lower_of_two_point P0 P1 n beta kappa L cminus cplus
pmin delta (amplitude*delta^(kappa+1)*hs^beta) chi hP0 hP1 hd.1 hd.2
(mul_nonneg (mul_nonneg hamp.le (Real.rpow_nonneg hd.1 _))
(Real.rpow_nonneg hhs.le _)) hsep
(by simpa [P0,P1,iidProduct,minimaxClampLaw] using
pi_iid_absolutelyContinuous _ _ hac1 n)
(by simpa [P0,P1,iidProduct,minimaxClampLaw] using
pi_iid_integrable_sq_dev _ _ hac1 hint1 n)
hchi0 hchi4 hchi
dsimp [c, floor, hs]
rw [Real.mul_rpow ht.le hh.le] at hl
convert hl using 1 <;> ring
theorem observedMinimaxRisk_le_stabilizedWorstRisk
The concrete stabilized estimator is an admissible competitor in the observed minimax problem. The result uses the hreg condition, the hdelta condition. This is the stated conclusion.
Formal statement
hypotheses
J n :
ℕ
B :
beta kappa L cminus cplus pmin deltaBar alpha delta :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
hdelta :
delta ∈ Icc (0 : ℝ) 1
conclusion
observedMinimaxRisk J n beta kappa L cminus cplus pmin delta
≤ stabilizedWorstRisk J n B beta kappa L cminus cplus pmin deltaBar delta
Proof (Lean source)
lemma observedMinimaxRisk_le_stabilizedWorstRisk
(J n : ℕ) (B : SplitBlocks n)
(beta kappa L cminus cplus pmin deltaBar alpha delta : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha)
(hdelta : delta ∈ Icc (0 : ℝ) 1) :
observedMinimaxRisk J n beta kappa L cminus cplus pmin delta ≤
stabilizedWorstRisk J n B beta kappa L cminus cplus pmin deltaBar delta := by
let est : Estimator n J := fun z =>
totalGramEstimator B z (ellOf beta) kappa cminus cplus delta
(infoBandwidth n delta beta kappa deltaBar)
let outer : Set ℝ := {r : ℝ | ∃ e : Estimator n J,
ObservedMeasurableEstimator e ∧ (∀ z, e z ∈ Icc (0 : ℝ) 1) ∧
r = sSup {v : ℝ | ∃ P : ClampLaw J,
ClampModel P beta kappa L cminus cplus pmin ∧
v = estimatorRisk P n delta e}}
change sInf outer ≤ _
have houterBelow : BddBelow outer := by
refine ⟨0, ?_⟩
intro r hr
rcases hr with ⟨e, hemeas, herange, rfl⟩
let risks : Set ℝ := {v : ℝ | ∃ P : ClampLaw J,
ClampModel P beta kappa L cminus cplus pmin ∧
v = estimatorRisk P n delta e}
have hrisk_le (P : ClampLaw J)
(hP : ClampModel P beta kappa L cminus cplus pmin) :
estimatorRisk P n delta e ≤ 1 := by
letI : IsProbabilityMeasure P.dataMeasure := hP.probability
letI : IsProbabilityMeasure (iidProduct P n) := by
unfold iidProduct
infer_instance
have htheta := clampFunctional_mem_Icc P hP hdelta.1 hdelta.2
unfold estimatorRisk
calc
(∫ z, |e z - clampFunctional P delta| ∂iidProduct P n) ≤
∫ _z, (1 : ℝ) ∂iidProduct P n := by
apply integral_mono_ae
· exact Integrable.of_bound
((hemeas.sub measurable_const).abs.aestronglyMeasurable) 1
(ae_of_all _ fun z => by
rw [Real.norm_eq_abs, abs_abs, abs_le]
constructor <;> linarith [(herange z).1, (herange z).2,
htheta.1, htheta.2])
· exact integrable_const 1
· exact ae_of_all _ fun z => by
rw [abs_le]
constructor <;> linarith [(herange z).1, (herange z).2,
htheta.1, htheta.2]
_ = 1 := by simp
have hbdd : BddAbove risks := by
refine ⟨1, ?_⟩
rintro v ⟨P, hP, rfl⟩
exact hrisk_le P hP
let P0 := minimaxClampLaw J kappa (fun _ : Fin J × ℝ => 0)
have hP0 : ClampModel P0 beta kappa L cminus cplus pmin :=
minimaxCenter_mem_model J beta kappa L cminus cplus pmin deltaBar alpha hreg
have hrisk0 : 0 ≤ estimatorRisk P0 n delta e := by
unfold estimatorRisk
exact integral_nonneg_of_ae (ae_of_all _ fun z => abs_nonneg _)
exact hrisk0.trans (le_csSup hbdd ⟨P0, hP0, rfl⟩)
apply csInf_le houterBelow
refine ⟨est, ?_, ?_, ?_⟩
· exact totalGramEstimator_measurable B kappa cminus cplus delta _
· intro z
exact clampUnit_mem_Icc _
· unfold stabilizedWorstRisk
congr 1
ext v
constructor
· rintro ⟨P, hP, hsample, hv⟩
refine ⟨P, hP, ?_⟩
simpa [stabilizedEstimatorRisk, est] using hv
· rintro ⟨P, hP, hv⟩
refine ⟨P, hP, clampModel_iidSampling hP, ?_⟩
simpa [stabilizedEstimatorRisk, est] using hv
The minimax risk and concrete estimator risk are both comparable to the regular-plus-atom frontier, with explicit global and localized witnesses. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
∃ c C amplitude : ℝ,
0 < c
c < C
0 < amplitude
amplitude ≤ 1 / 4
conclusion 1
SplitBlocks n, ∀ᶠ n in atTop, let delta := deltaSeq n let h := infoBandwidth n delta beta kappa deltaBar let r := clampFrontier n delta kappa h beta c * r
≤ observedMinimaxRisk J n beta kappa L cminus cplus pmin delta ∧
observedMinimaxRisk J n beta kappa L cminus cplus pmin delta
≤ stabilizedWorstRisk J n (Bseq n) beta kappa L cminus cplus pmin deltaBar delta ∧
stabilizedWorstRisk J n (Bseq n) beta kappa L cminus cplus pmin deltaBar delta ≤ C * r ∧
(∃ P0 P1 : ClampLaw J, ClampModel P0 beta kappa L cminus cplus pmin ∧ ClampModel P1 beta kappa L cminus cplus pmin ∧ IidSampling P0 n ∧ IidSampling P1 n ∧ GlobalBernoulliShift P0 P1 ((n : ℝ) ^ (-(1 : ℝ) / 2)) ∧ c * (n : ℝ) ^ (-(1 : ℝ) / 2) ≤ |clampFunctional P1 delta - clampFunctional P0 delta| ∧ ProperProductChiSq P1 P0 n ∧ productChiSq P1 P0 n ≤ C) ∧
(∃ Q0 Q1 : ClampLaw J, ClampModel Q0 beta kappa L cminus cplus pmin ∧ ClampModel Q1 beta kappa L cminus cplus pmin ∧ IidSampling Q0 n ∧ IidSampling Q1 n ∧ LocalizedBernoulliPerturbation Q0 Q1 beta delta h amplitude ∧ c * delta ^ (kappa + 1) * h ^ beta ≤ |clampFunctional Q1 delta - clampFunctional Q0 delta| ∧ ProperProductChiSq Q1 Q0 n ∧ productChiSq Q1 Q0 n ≤ C)
Proof (Lean source)
-- @node: thm:minimax-risk
theorem clamp_minimax_risk
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
∃ c C amplitude : ℝ, 0 < c ∧ c < C ∧
0 < amplitude ∧ amplitude ≤ 1 / 4 ∧
∀ (deltaSeq : ℕ → ℝ), ThresholdSequence deltaBar deltaSeq →
∀ Bseq : ∀ n, SplitBlocks n,
∀ᶠ n in atTop,
let delta := deltaSeq n
let h := infoBandwidth n delta beta kappa deltaBar
let r := clampFrontier n delta kappa h beta
c * r ≤ observedMinimaxRisk J n beta kappa L cminus cplus pmin delta ∧
observedMinimaxRisk J n beta kappa L cminus cplus pmin delta ≤
stabilizedWorstRisk J n (Bseq n) beta kappa L cminus cplus pmin
deltaBar delta ∧
stabilizedWorstRisk J n (Bseq n) beta kappa L cminus cplus pmin
deltaBar delta ≤ C * r ∧
(∃ P0 P1 : ClampLaw J,
ClampModel P0 beta kappa L cminus cplus pmin ∧
ClampModel P1 beta kappa L cminus cplus pmin ∧
IidSampling P0 n ∧ IidSampling P1 n ∧
GlobalBernoulliShift P0 P1 ((n : ℝ) ^ (-(1 : ℝ) / 2)) ∧
c * (n : ℝ) ^ (-(1 : ℝ) / 2) ≤
|clampFunctional P1 delta - clampFunctional P0 delta| ∧
ProperProductChiSq P1 P0 n ∧
productChiSq P1 P0 n ≤ C) ∧
(∃ Q0 Q1 : ClampLaw J,
ClampModel Q0 beta kappa L cminus cplus pmin ∧
ClampModel Q1 beta kappa L cminus cplus pmin ∧
IidSampling Q0 n ∧ IidSampling Q1 n ∧
LocalizedBernoulliPerturbation Q0 Q1 beta delta h amplitude ∧
c * delta ^ (kappa + 1) * h ^ beta ≤
|clampFunctional Q1 delta - clampFunctional Q0 delta| ∧
ProperProductChiSq Q1 Q0 n ∧
productChiSq Q1 Q0 n ≤ C) := by
have hreg' := hreg
rcases hreg' with ⟨hJ, hbeta, hkappa, hL, hcminus, hcminus_le, hcplus,
hpmin, hpmin_le, hdeltaBar, hdeltaBar1, halpha, halpha1⟩
rcases stabilizedWorstRisk_eventually_le_frontier J beta kappa L cminus cplus
pmin deltaBar alpha hreg with ⟨Cupper, hCupper, hupper⟩
rcases observedMinimaxRisk_eventually_ge_root J beta kappa L cminus cplus
pmin deltaBar alpha hreg with ⟨croot, hcroot, hroot⟩
rcases observedMinimaxRisk_eventually_ge_local J beta kappa L cminus cplus
pmin deltaBar alpha hreg with ⟨clocal, hclocal, hlocal⟩
rcases localizedBernoulli_witness_eventually J beta kappa L cminus cplus
pmin deltaBar alpha hreg with ⟨amplitude, D, hamp, hamp_le, hD, hwitnessLocal⟩
let core : ℝ := min croot (min clocal (min (1 / 2 : ℝ) amplitude))
let c : ℝ := core / 2
let C : ℝ := Cupper + D + exp 4 + c + 1
have hcore : 0 < core := by
dsimp [core]
exact lt_min hcroot (lt_min hclocal (lt_min (by norm_num) hamp))
have hc : 0 < c := div_pos hcore (by norm_num)
have h2c_root : 2 * c ≤ croot := by
dsimp [c, core]
nlinarith [min_le_left croot (min clocal (min (1 / 2 : ℝ) amplitude))]
have h2c_local : 2 * c ≤ clocal := by
dsimp [c, core]
have hm := min_le_right croot (min clocal (min (1 / 2 : ℝ) amplitude))
have hm' := min_le_left clocal (min (1 / 2 : ℝ) amplitude)
linarith
have hc_half : c ≤ 1 / 2 := by
dsimp [c, core]
have hm := min_le_right croot (min clocal (min (1 / 2 : ℝ) amplitude))
have hm' := min_le_right clocal (min (1 / 2 : ℝ) amplitude)
have hm'' := min_le_left (1 / 2 : ℝ) amplitude
linarith
have hc_amp : c ≤ amplitude := by
dsimp [c, core]
have hm := min_le_right croot (min clocal (min (1 / 2 : ℝ) amplitude))
have hm' := min_le_right clocal (min (1 / 2 : ℝ) amplitude)
have hm'' := min_le_right (1 / 2 : ℝ) amplitude
linarith
have hC : 0 < C := by dsimp [C]; positivity
have hcC : c < C := by
dsimp [C]
nlinarith [hCupper, hD, Real.exp_pos 4]
have hCupperC : Cupper ≤ C := by
dsimp [C]
nlinarith [hD, Real.exp_pos 4, hc]
have hDC : D ≤ C := by
dsimp [C]
nlinarith [hCupper, Real.exp_pos 4, hc]
have hexpC : exp 4 ≤ C := by
dsimp [C]
nlinarith [hCupper, hD, hc]
refine ⟨c, C, amplitude, hc, hcC, hamp, hamp_le, ?_⟩
intro deltaSeq hdelta Bseq
have heu := hupper deltaSeq hdelta Bseq
have her := hroot deltaSeq hdelta
have hel := hlocal deltaSeq hdelta
have heg := globalBernoulli_witness_eventually J beta kappa L cminus cplus
pmin deltaBar alpha hreg deltaSeq hdelta
have hew := hwitnessLocal deltaSeq hdelta
have heb := infoBandwidth_eventually_balance beta kappa deltaBar hbeta hkappa
⟨hdeltaBar, hdeltaBar1⟩ deltaSeq hdelta
filter_upwards [heu, her, hel, heg, hew, heb] with n hun hrn hln hgn hwn hbn
dsimp only
let delta := deltaSeq n
let h := infoBandwidth n delta beta kappa deltaBar
let r0 : ℝ := (n : ℝ) ^ (-(1 : ℝ) / 2)
let a0 : ℝ := delta ^ (kappa + 1) * h ^ beta
have hdelta1 : delta ∈ Icc (0 : ℝ) 1 :=
⟨(hdelta n).1, (hdelta n).2.trans hdeltaBar1.le⟩
have hr0 : 0 ≤ r0 := Real.rpow_nonneg (Nat.cast_nonneg n) _
have ha0 : 0 ≤ a0 := mul_nonneg
(Real.rpow_nonneg (hdelta n).1 _)
(Real.rpow_nonneg (by simpa [h, delta] using hbn.1.le) _)
have hlower : c * (r0 + a0) ≤
observedMinimaxRisk J n beta kappa L cminus cplus pmin delta := by
have hprod0 : 0 ≤ (croot - 2 * c) * r0 :=
mul_nonneg (sub_nonneg.mpr h2c_root) hr0
have hprod1 : 0 ≤ (clocal - 2 * c) * a0 :=
mul_nonneg (sub_nonneg.mpr h2c_local) ha0
change croot * r0 ≤
observedMinimaxRisk J n beta kappa L cminus cplus pmin delta at hrn
have hln' : clocal * a0 ≤
observedMinimaxRisk J n beta kappa L cminus cplus pmin delta := by
dsimp at hln
convert hln using 1 <;> simp [a0, delta, h] <;> ring
nlinarith
have hmiddle := observedMinimaxRisk_le_stabilizedWorstRisk J n (Bseq n)
beta kappa L cminus cplus pmin deltaBar alpha delta hreg hdelta1
have hr : 0 ≤ clampFrontier n delta kappa h beta := by
exact add_nonneg hr0 ha0
have hupper' : stabilizedWorstRisk J n (Bseq n) beta kappa L cminus cplus
pmin deltaBar delta ≤ C * clampFrontier n delta kappa h beta :=
hun.trans (mul_le_mul_of_nonneg_right hCupperC hr)
rcases hgn with
⟨P0, P1, hP0, hP1, hiid0, hiid1, hshift, hsepG, hproperG, hchiG⟩
rcases hwn with
⟨Q0, Q1, hQ0, hQ1, hqiid0, hqiid1, hpert, hsepL, hproperL, hchiL⟩
have hsepG' : c * r0 ≤
|clampFunctional P1 delta - clampFunctional P0 delta| :=
(mul_le_mul_of_nonneg_right hc_half hr0).trans (by simpa [r0, delta] using hsepG)
have hsepL' : c * a0 ≤
|clampFunctional Q1 delta - clampFunctional Q0 delta| := by
have hampsep : amplitude * a0 ≤
|clampFunctional Q1 delta - clampFunctional Q0 delta| := by
convert hsepL using 1 <;> simp [a0, delta, h] <;> ring
exact (mul_le_mul_of_nonneg_right hc_amp ha0).trans hampsep
refine ⟨?_, hmiddle, hupper', ?_, ?_⟩
· simpa [clampFrontier, r0, a0, delta, h] using hlower
· exact ⟨P0, P1, hP0, hP1, hiid0, hiid1,
by simpa [r0] using hshift, hsepG', hproperG, hchiG.trans hexpC⟩
· refine ⟨Q0, Q1, hQ0, hQ1, hqiid0, hqiid1,
by simpa [delta, h] using hpert, ?_, hproperL, hchiL.trans hDC⟩
convert hsepL' using 1 <;> simp [a0, delta, h] <;> ring
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.TOneCellCalibration 12 declarations The concrete one-stratum design has density 2a, a triangular width-h Bernoulli regression perturbation, its exact integrated KL, and target separation.
One-cell early-kill calibration
The concrete one-stratum design has density 2a, a triangular width-h
Bernoulli regression perturbation, its exact integrated KL, and target separation.
def oneCellDensity
The concrete one-cell assignment density.
Definition (Lean source)
parameters
a :
ℝ
def
oneCellDensity a :
ℝ
given by
2 * a
def oneCellTent
Unit triangular localization around the moving threshold.
Definition (Lean source)
def oneCellRegression0
Null Bernoulli regression.
Definition (Lean source)
parameters
_a :
ℝ
def
oneCellRegression0 _a :
ℝ
given by
1 / 2
def oneCellRegression1
Width-h, height-h localized Bernoulli alternative for beta one.
Definition (Lean source)
parameters
delta h a :
ℝ
def
oneCellRegression1 delta h a :
ℝ
given by
1 / 2 + h * oneCellTent delta h a
def oneCellProductKL
Product KL of the concrete localized Bernoulli alternatives.
Definition (Lean source)
parameters
n :
ℕ
delta h :
ℝ
def
oneCellProductKL n delta h :
ℝ
given by
(n : ℝ) * ∫ a in Icc (0 : ℝ) 1, (klDiv (bernoulliLaw (oneCellRegression1 delta h a)) (bernoulliLaw (oneCellRegression0 a))).toReal * oneCellDensity a
def oneCellTargetSeparation
Total clamp-target separation of the concrete one-cell pair: the retained- course integral plus the collapsed-atom contribution.
Definition (Lean source)
parameters
delta h :
ℝ
def
oneCellTargetSeparation delta h :
ℝ
given by
+ (∫ a in Icc (0 : ℝ) delta, oneCellDensity a) * (oneCellRegression1 delta h delta - oneCellRegression0 delta)
On an interior right-hand localization window, the one-cell target separation is the atom contribution plus the exact retained-course integral. The result uses the hdelta condition, the hh condition, the hwindow condition. This is the stated conclusion.
Formal statement
hypotheses
delta h :
ℝ
hdelta :
0 ≤ delta
hh :
0 < h
hwindow :
delta + h ≤ 1
conclusion
oneCellTargetSeparation delta h = delta ^ 2 * h + delta * h ^ 2 + h ^ 3 / 3
Proof (Lean source)
-- @node: oneCellTargetSeparation_eq
lemma oneCellTargetSeparation_eq (delta h : ℝ) (hdelta : 0 ≤ delta)
(hh : 0 < h) (hwindow : delta + h ≤ 1) :
oneCellTargetSeparation delta h =
delta ^ 2 * h + delta * h ^ 2 + h ^ 3 / 3 := by
have hsmall : Ioc delta (delta + h) ⊆ Ioc delta 1 := by
intro a ha
exact ⟨ha.1, ha.2.trans hwindow⟩
have hcut : (∫ a in Ioc delta 1,
h * max 0 (1 - |a - delta| / h) * (2 * a)) =
∫ a in Ioc delta (delta + h),
h * max 0 (1 - |a - delta| / h) * (2 * a) := by
apply setIntegral_eq_of_subset_of_forall_sdiff_eq_zero measurableSet_Ioc hsmall
intro a ha
have had : delta ≤ a := ha.1.1.le
have hha : h ≤ a - delta := by
have : delta + h ≤ a :=
le_of_not_gt (fun halt => ha.2 ⟨ha.1.1, halt.le⟩)
linarith
have : 1 - |a - delta| / h ≤ 0 := by
rw [abs_of_nonneg (sub_nonneg.mpr had)]
apply sub_nonpos.mpr
exact (le_div_iff₀ hh).2 (by simpa using hha)
simp [max_eq_left this]
have hpoly : (∫ a in Ioc delta (delta + h),
h * max 0 (1 - |a - delta| / h) * (2 * a)) =
∫ a in Ioc delta (delta + h),
(2 * (delta + h) * a - 2 * a ^ 2) := by
apply integral_congr_ae
filter_upwards [ae_restrict_mem measurableSet_Ioc] with a ha
have had : 0 ≤ a - delta := sub_nonneg.mpr ha.1.le
have hratio : |a - delta| / h ≤ 1 := by
rw [abs_of_nonneg had]
exact (div_le_one hh).2 (by linarith [ha.2])
rw [max_eq_right (sub_nonneg.mpr hratio), abs_of_nonneg had]
field_simp
ring
have hint : (∫ a in Ioc delta (delta + h),
(2 * (delta + h) * a - 2 * a ^ 2)) =
delta * h ^ 2 + h ^ 3 / 3 := by
have hf : IntervalIntegrable
(fun a : ℝ => 2 * (delta + h) * a) volume delta (delta + h) :=
intervalIntegral.intervalIntegrable_id.const_mul _
have hg : IntervalIntegrable (fun a : ℝ => 2 * a ^ 2)
volume delta (delta + h) :=
(intervalIntegral.intervalIntegrable_pow 2).const_mul _
rw [← intervalIntegral.integral_of_le (by linarith : delta ≤ delta + h),
intervalIntegral.integral_sub hf hg, intervalIntegral.integral_const_mul,
integral_id, intervalIntegral.integral_const_mul, integral_pow]
ring
have hmass : (∫ a in Icc (0 : ℝ) delta, 2 * a) = delta ^ 2 := by
rw [integral_Icc_eq_integral_Ioc,
← intervalIntegral.integral_of_le hdelta,
intervalIntegral.integral_const_mul, integral_id]
ring
simp [oneCellTargetSeparation, oneCellRegression1, oneCellRegression0,
oneCellTent, oneCellDensity]
rw [hcut, hpoly, hint, hmass]
ring
The squared triangular perturbation has an exact design-weighted mass on an interior localization window. The result uses the hh condition, the hleft condition, the hright condition. This is the stated conclusion.
Formal statement
hypotheses
delta h :
ℝ
hh :
0 < h
hleft :
h ≤ delta
hright :
delta + h ≤ 1
conclusion
= (4 / 3 : ℝ) * delta * h ^ 3
Proof (Lean source)
-- @node: oneCellTentSqIntegral_eq
lemma oneCellTentSqIntegral_eq (delta h : ℝ) (hh : 0 < h)
(hleft : h ≤ delta) (hright : delta + h ≤ 1) :
(∫ a in Icc (0 : ℝ) 1,
(h * oneCellTent delta h a) ^ 2 * oneCellDensity a) =
(4 / 3 : ℝ) * delta * h ^ 3 := by
have hdelta : 0 ≤ delta := hh.le.trans hleft
have hdh : 0 ≤ delta - h := sub_nonneg.mpr hleft
have hsub : Ioc (delta - h) (delta + h) ⊆ Icc (0 : ℝ) 1 := by
intro a ha
exact ⟨hdh.trans ha.1.le, ha.2.trans hright⟩
have hcut : (∫ a in Icc (0 : ℝ) 1,
(h * oneCellTent delta h a) ^ 2 * oneCellDensity a) =
∫ a in Ioc (delta - h) (delta + h),
(h * oneCellTent delta h a) ^ 2 * oneCellDensity a := by
apply setIntegral_eq_of_subset_of_forall_sdiff_eq_zero measurableSet_Icc hsub
intro a ha
have hout : h ≤ |a - delta| := by
by_cases had : a ≤ delta
· rw [abs_of_nonpos (sub_nonpos.mpr had)]
have : a ≤ delta - h := le_of_not_gt fun halt => ha.2 ⟨halt, by linarith⟩
linarith
· rw [abs_of_pos (sub_pos.mpr (lt_of_not_ge had))]
have : delta + h ≤ a := le_of_not_gt fun halt => ha.2 ⟨by linarith, halt.le⟩
linarith
have hzero : 1 - |a - delta| / h ≤ 0 := by
exact sub_nonpos.mpr ((le_div_iff₀ hh).2 (by simpa using hout))
simp [oneCellTent, max_eq_left hzero]
rw [hcut]
rw [← intervalIntegral.integral_of_le (by linarith : delta - h ≤ delta + h)]
rw [← intervalIntegral.integral_add_adjacent_intervals
(b := delta)]
· have hleftEq : (∫ a in delta - h..delta,
(h * oneCellTent delta h a) ^ 2 * oneCellDensity a) =
∫ a in delta - h..delta, 2 * a * (a - delta + h) ^ 2 := by
apply intervalIntegral.integral_congr
intro a ha
have habounds : a ∈ Icc (delta - h) delta := by
simpa [uIcc_of_le (by linarith : delta - h ≤ delta)] using ha
have had : a ≤ delta := habounds.2
have hratio : |a - delta| / h ≤ 1 := by
rw [abs_of_nonpos (sub_nonpos.mpr had)]
exact (div_le_one hh).2 (by linarith [habounds.1])
simp only [oneCellTent, oneCellDensity, max_eq_right (sub_nonneg.mpr hratio)]
rw [abs_of_nonpos (sub_nonpos.mpr had)]
field_simp
ring
have hrightEq : (∫ a in delta..delta + h,
(h * oneCellTent delta h a) ^ 2 * oneCellDensity a) =
∫ a in delta..delta + h, 2 * a * (delta + h - a) ^ 2 := by
apply intervalIntegral.integral_congr
intro a ha
have habounds : a ∈ Icc delta (delta + h) := by
simpa [uIcc_of_le (by linarith : delta ≤ delta + h)] using ha
have had : delta ≤ a := habounds.1
have hratio : |a - delta| / h ≤ 1 := by
rw [abs_of_nonneg (sub_nonneg.mpr had)]
exact (div_le_one hh).2 (by linarith [habounds.2])
simp only [oneCellTent, oneCellDensity, max_eq_right (sub_nonneg.mpr hratio)]
rw [abs_of_nonneg (sub_nonneg.mpr had)]
field_simp
ring
rw [hleftEq, hrightEq]
simp_rw [show (fun a : ℝ => 2 * a * (a - delta + h) ^ 2) =
fun a => 2 * a ^ 3 + (4 * (h - delta)) * a ^ 2 +
(2 * (h - delta) ^ 2) * a by funext a; ring]
simp_rw [show (fun a : ℝ => 2 * a * (delta + h - a) ^ 2) =
fun a => 2 * a ^ 3 - (4 * (delta + h)) * a ^ 2 +
(2 * (delta + h) ^ 2) * a by funext a; ring]
have hpolyInt (a b A B C : ℝ) :
(∫ x in a..b, A * x ^ 3 + B * x ^ 2 + C * x) =
A * ((b ^ 4 - a ^ 4) / 4) +
B * ((b ^ 3 - a ^ 3) / 3) + C * ((b ^ 2 - a ^ 2) / 2) := by
have h3 := (intervalIntegral.intervalIntegrable_pow 3
(μ := volume) (a := a) (b := b)).const_mul A
have h2 := (intervalIntegral.intervalIntegrable_pow 2
(μ := volume) (a := a) (b := b)).const_mul B
have h1 := (intervalIntegral.intervalIntegrable_id
(μ := volume) (a := a) (b := b)).const_mul C
rw [intervalIntegral.integral_add (h3.add h2) h1,
intervalIntegral.integral_add h3 h2,
intervalIntegral.integral_const_mul, integral_pow,
intervalIntegral.integral_const_mul, integral_pow,
intervalIntegral.integral_const_mul, integral_id]
norm_num
rw [hpolyInt]
have hrightPoly : (∫ a in delta..delta + h,
2 * a ^ 3 - 4 * (delta + h) * a ^ 2 + 2 * (delta + h) ^ 2 * a) =
2 * (((delta + h) ^ 4 - delta ^ 4) / 4) +
(-4 * (delta + h)) * (((delta + h) ^ 3 - delta ^ 3) / 3) +
(2 * (delta + h) ^ 2) * (((delta + h) ^ 2 - delta ^ 2) / 2) := by
calc
_ = ∫ a in delta..delta + h,
2 * a ^ 3 + (-4 * (delta + h)) * a ^ 2 +
(2 * (delta + h) ^ 2) * a := by
apply intervalIntegral.integral_congr
intro a ha
ring
_ = _ := hpolyInt delta (delta + h) 2 (-4 * (delta + h))
(2 * (delta + h) ^ 2)
rw [hrightPoly]
ring
· exact (by
unfold oneCellTent oneCellDensity
fun_prop : Continuous (fun a : ℝ =>
(h * oneCellTent delta h a) ^ 2 * oneCellDensity a)).intervalIntegrable _ _
· exact (by
unfold oneCellTent oneCellDensity
fun_prop : Continuous (fun a : ℝ =>
(h * oneCellTent delta h a) ^ 2 * oneCellDensity a)).intervalIntegrable _ _
theorem oneCellBandwidth_far
Above the one-cell edge scale, the balance bandwidth is comparable to the interior bandwidth and its localization window is eventually interior. The result uses the hdeltaBar condition, the hdelta condition, the hfar condition. This is the stated conclusion.
Formal statement
hypotheses
deltaBar :
ℝ
hdeltaBar :
0 < deltaBar ∧ deltaBar < 1
deltaSeq :
ℕ → ℝ
hdelta :
ThresholdSequence deltaBar deltaSeq
conclusion
hSeq :
fun n => infoBandwidth n (deltaSeq n) 1 1 deltaBar
Proof (Lean source)
-- @node: oneCellBandwidth_far
lemma oneCellBandwidth_far (deltaBar : ℝ)
(hdeltaBar : 0 < deltaBar ∧ deltaBar < 1)
(deltaSeq : ℕ → ℝ) (hdelta : ThresholdSequence deltaBar deltaSeq)
(hfar : Tendsto (fun n => deltaSeq n / (n : ℝ) ^ (-(1 : ℝ) / 4))
atTop atTop) :
let hSeq := fun n => infoBandwidth n (deltaSeq n) 1 1 deltaBar
AsympSeq hSeq (fun n => ((n : ℝ) * deltaSeq n) ^ (-(1 : ℝ) / 3)) ∧
∀ᶠ n in atTop, 0 < hSeq n ∧ hSeq n ≤ deltaSeq n ∧
deltaSeq n + hSeq n ≤ 1 ∧ hSeq n ≤ 1 / 4 := by
dsimp
have hbalance := infoBandwidth_eventually_balance 1 1 deltaBar (by norm_num)
(by norm_num) hdeltaBar deltaSeq hdelta
let c : ℝ := 2 ^ (-(1 : ℝ) / 3)
have hc : 0 < c := Real.rpow_pos_of_pos (by norm_num) _
have hcOne : c ≤ 1 := by
rw [show (1 : ℝ) = 2 ^ (0 : ℝ) by norm_num]
apply Real.rpow_le_rpow_of_exponent_le (by norm_num)
norm_num
have hratio := (Filter.tendsto_atTop.1 hfar) 1
have hevent : ∀ᶠ n in atTop,
0 < infoBandwidth n (deltaSeq n) 1 1 deltaBar ∧
infoBandwidth n (deltaSeq n) 1 1 deltaBar ≤ deltaSeq n ∧
deltaSeq n + infoBandwidth n (deltaSeq n) 1 1 deltaBar ≤ 1 ∧
infoBandwidth n (deltaSeq n) 1 1 deltaBar ≤ 1 / 4 ∧
c * (((n : ℝ) * deltaSeq n) ^ (-(1 : ℝ) / 3)) ≤
infoBandwidth n (deltaSeq n) 1 1 deltaBar ∧
infoBandwidth n (deltaSeq n) 1 1 deltaBar ≤
((n : ℝ) * deltaSeq n) ^ (-(1 : ℝ) / 3) := by
filter_upwards [hbalance, hratio, eventually_ge_atTop 257] with n hn hratioN hn257
have hnpos : 0 < (n : ℝ) := by
by_contra hn0
have hnzero : (n : ℝ) = 0 := le_antisymm (le_of_not_gt hn0) (Nat.cast_nonneg n)
rw [hnzero, zero_mul] at hn
norm_num at hn
have hedgePos : 0 < (n : ℝ) ^ (-(1 : ℝ) / 4) :=
Real.rpow_pos_of_pos hnpos _
have hdeltaPos : 0 < deltaSeq n := by
have hle : (n : ℝ) ^ (-(1 : ℝ) / 4) ≤ deltaSeq n := by
simpa using (le_div_iff₀ hedgePos).mp hratioN
exact hedgePos.trans_le hle
let h := infoBandwidth n (deltaSeq n) 1 1 deltaBar
let w := ((n : ℝ) * deltaSeq n) ^ (-(1 : ℝ) / 3)
have hedge : (n : ℝ) ^ (-1 / ((3 : ℝ) + 1)) ≤ deltaSeq n := by
have hbase : (n : ℝ) ^ (-(1 : ℝ) / 4) ≤ deltaSeq n := by
simpa using (le_div_iff₀ hedgePos).mp hratioN
convert hbase using 1 <;> norm_num
have hwUpper : h ≤ w := by
simpa [w, Real.rpow_one] using
balance_root_le_interior_scale (n : ℝ) (deltaSeq n) h 3 1 hnpos
hdeltaPos hn.1 (by norm_num) (by norm_num)
(by norm_num [h, Real.rpow_natCast] at hn ⊢; simpa using hn.2.2)
have hwDelta : w ≤ deltaSeq n := by
simpa [w, Real.rpow_one] using
interior_scale_le_threshold_of_edge_le (n : ℝ) (deltaSeq n) 3 1
hnpos hdeltaPos (by norm_num) (by norm_num) hedge
have hwLower : c * w ≤ h := by
simpa [c, w, Real.rpow_one] using
interior_scale_le_balance_root (n : ℝ) (deltaSeq n) h 3 1 hnpos
hdeltaPos hn.1 (by norm_num) (by norm_num) (hwUpper.trans hwDelta)
(by norm_num [h, Real.rpow_natCast] at hn ⊢; simpa using hn.2.2)
have hwindow : deltaSeq n + h ≤ 1 :=
(add_le_add (hdelta n).2 hn.2.1).trans_eq (by ring)
have hhquarter : h ≤ 1 / 4 := by
by_contra hnot
have hhlarge : (1 : ℝ) / 4 < h := lt_of_not_ge hnot
have hEq : (n : ℝ) * h ^ 3 * (deltaSeq n + h) = 1 := by
norm_num [h, Real.rpow_natCast, Real.rpow_one] at hn ⊢
simpa using hn.2.2
have hstrict : 1 < (n : ℝ) * h ^ 3 * h := by
calc
1 = (256 : ℝ) * ((1 : ℝ) / 4) ^ 3 * ((1 : ℝ) / 4) := by norm_num
_ < (n : ℝ) * h ^ 3 * h := by
gcongr
· exact_mod_cast hn257
have : (n : ℝ) * h ^ 3 * h ≤ (n : ℝ) * h ^ 3 * (deltaSeq n + h) := by
gcongr
linarith [(hdelta n).1]
linarith
refine ⟨hn.1, hwUpper.trans hwDelta, hwindow, hhquarter, hwLower, hwUpper⟩
constructor
· exact ⟨c, 1, hc, hcOne, hevent.mono fun n hn =>
⟨hn.2.2.2.2.1, by simpa using hn.2.2.2.2.2⟩⟩
· exact hevent.mono fun n hn => ⟨hn.1, hn.2.1, hn.2.2.1, hn.2.2.2.1⟩
On an interior quarter-height window, the concrete product KL is trapped between fixed multiples of n * delta * h^3. The result uses the hh condition, the hleft condition, the hright condition, the hquarter condition. This is the stated conclusion.
Formal statement
hypotheses
n :
ℕ
delta h :
ℝ
hh :
0 < h
hleft :
h ≤ delta
hright :
delta + h ≤ 1
hquarter :
h ≤ 1 / 4
conclusion
conclusion 1
(8 / 3 : ℝ) * (n : ℝ) * delta * h ^ 3 ≤ oneCellProductKL n delta h
conclusion 2
oneCellProductKL n delta h ≤ (16 / 3 : ℝ) * (n : ℝ) * delta * h ^ 3
Proof (Lean source)
-- @node: oneCellProductKL_bounds
lemma oneCellProductKL_bounds (n : ℕ) (delta h : ℝ) (hh : 0 < h)
(hleft : h ≤ delta) (hright : delta + h ≤ 1) (hquarter : h ≤ 1 / 4) :
(8 / 3 : ℝ) * (n : ℝ) * delta * h ^ 3 ≤ oneCellProductKL n delta h ∧
oneCellProductKL n delta h ≤ (16 / 3 : ℝ) * (n : ℝ) * delta * h ^ 3 := by
let gamma : ℝ → ℝ := fun a => h * oneCellTent delta h a
have htent : ∀ a, oneCellTent delta h a ∈ Icc (0 : ℝ) 1 := by
intro a
constructor
· exact le_max_left _ _
· apply max_le (by norm_num)
have : 0 ≤ |a - delta| / h := div_nonneg (abs_nonneg _) hh.le
linarith
have hgamma0 : ∀ a, 0 ≤ gamma a := fun a => mul_nonneg hh.le (htent a).1
have hgammah : ∀ a, gamma a ≤ h := fun a => by
dsimp [gamma]
simpa using mul_le_mul_of_nonneg_left (htent a).2 hh.le
have hgamma_meas : Measurable gamma := by
dsimp [gamma]
unfold oneCellTent
fun_prop
have hkl_meas : Measurable fun a =>
(klDiv
(bernoulliLaw (1 / 2 + gamma a))
(bernoulliLaw (1 / 2))).toReal := by
have hform : (fun a =>
(klDiv
(bernoulliLaw (1 / 2 + gamma a))
(bernoulliLaw (1 / 2))).toReal) =
fun a => (1 / 2 + gamma a) * log ((1 / 2 + gamma a) / (1 / 2)) +
(1 - (1 / 2 + gamma a)) *
log ((1 - (1 / 2 + gamma a)) / (1 - (1 / 2))) := by
funext a
apply bernoulliLaw_klDiv_toReal
· linarith [hgamma0 a]
· linarith [hgammah a]
· norm_num
· norm_num
rw [hform]
fun_prop
have hpi_int : Integrable oneCellDensity
(volume.restrict (Icc (0 : ℝ) 1)) := by
apply IntegrableOn.of_bound (C := 2) measure_Icc_lt_top
· unfold oneCellDensity
fun_prop
· filter_upwards [ae_restrict_mem measurableSet_Icc] with a ha
change |2 * a| ≤ 2
rw [abs_of_nonneg (mul_nonneg (by norm_num) ha.1)]
nlinarith [ha.2]
have hband := localized_design_kl_band oneCellDensity gamma hpi_int
hgamma_meas hkl_meas
(by filter_upwards [ae_restrict_mem measurableSet_Icc] with a ha
exact mul_nonneg (by norm_num) ha.1)
(fun a ha => by
rw [abs_of_nonneg (hgamma0 a)]
exact (hgammah a).trans hquarter)
have hsquare := oneCellTentSqIntegral_eq delta h hh hleft hright
have hn0 : 0 ≤ (n : ℝ) := Nat.cast_nonneg n
constructor
· rw [oneCellProductKL]
change (8 / 3 : ℝ) * (n : ℝ) * delta * h ^ 3 ≤
(n : ℝ) * ∫ a in Icc (0 : ℝ) 1,
(klDiv
(bernoulliLaw (1 / 2 + gamma a))
(bernoulliLaw (1 / 2))).toReal *
oneCellDensity a
calc
(8 / 3 : ℝ) * (n : ℝ) * delta * h ^ 3 =
(n : ℝ) * (2 * ((4 / 3 : ℝ) * delta * h ^ 3)) := by ring
_ = (n : ℝ) * (2 * ∫ a in Icc (0 : ℝ) 1,
(gamma a) ^ 2 * oneCellDensity a) := by rw [hsquare]
_ ≤ _ := mul_le_mul_of_nonneg_left hband.1 hn0
· rw [oneCellProductKL]
change (n : ℝ) * ∫ a in Icc (0 : ℝ) 1,
(klDiv
(bernoulliLaw (1 / 2 + gamma a))
(bernoulliLaw (1 / 2))).toReal *
oneCellDensity a ≤ (16 / 3 : ℝ) * (n : ℝ) * delta * h ^ 3
calc
_ ≤ (n : ℝ) * (4 * ∫ a in Icc (0 : ℝ) 1,
(gamma a) ^ 2 * oneCellDensity a) :=
mul_le_mul_of_nonneg_left hband.2 hn0
_ = (16 / 3 : ℝ) * (n : ℝ) * delta * h ^ 3 := by rw [hsquare]; ring
The normalized interior one-cell atom scale is the 5/3 power of the threshold measured in critical-scale units. The result uses the hn condition, the hdelta condition. This is the stated conclusion.
Formal statement
hypotheses
n :
ℕ
delta :
ℝ
hn :
0 < n
hdelta :
0 < delta
conclusion
(delta ^ 2 * ((n : ℝ) * delta) ^ (-(1 : ℝ) / 3)) / (n : ℝ) ^ (-(1 : ℝ) / 2)
= (delta / (n : ℝ) ^ (-(1 : ℝ) / 10)) ^ ((5 : ℝ) / 3)
Proof (Lean source)
-- @node: oneCellNormalizedScale_eq
lemma oneCellNormalizedScale_eq (n : ℕ) (delta : ℝ) (hn : 0 < n)
(hdelta : 0 < delta) :
(delta ^ 2 * ((n : ℝ) * delta) ^ (-(1 : ℝ) / 3)) /
(n : ℝ) ^ (-(1 : ℝ) / 2) =
(delta / (n : ℝ) ^ (-(1 : ℝ) / 10)) ^ ((5 : ℝ) / 3) := by
have hnR : 0 < (n : ℝ) := by exact_mod_cast hn
have hcrit : 0 < (n : ℝ) ^ (-(1 : ℝ) / 10) := Real.rpow_pos_of_pos hnR _
rw [Real.div_rpow hdelta.le hcrit.le, Real.mul_rpow hnR.le hdelta.le]
rw [show delta ^ 2 * ((n : ℝ) ^ (-(1 : ℝ) / 3) *
delta ^ (-(1 : ℝ) / 3)) =
(n : ℝ) ^ (-(1 : ℝ) / 3) *
(delta ^ 2 * delta ^ (-(1 : ℝ) / 3)) by ring]
rw [show delta ^ 2 = delta ^ (2 : ℝ) by norm_num [Real.rpow_natCast]]
rw [← Real.rpow_add hdelta, ← Real.rpow_mul hnR.le]
field_simp
norm_num
rw [show (n : ℝ) ^ (-(1 / 3 : ℝ)) * delta ^ ((5 : ℝ) / 3) *
(n : ℝ) ^ (-(1 / 6 : ℝ)) =
delta ^ ((5 : ℝ) / 3) * ((n : ℝ) ^ (-(1 / 3 : ℝ)) *
(n : ℝ) ^ (-(1 / 6 : ℝ))) by ring]
rw [← Real.rpow_add hnR]
norm_num
ring
For beta=kappa=one, the bandwidth, KL, separation, and transition exponent are exactly the one-cell calibration stated in the note. The result uses the hdeltaBar condition, the hdelta condition, the hfar condition. This is the stated conclusion.
Formal statement
hypotheses
deltaBar :
ℝ
hdeltaBar :
0 < deltaBar ∧ deltaBar < 1
deltaSeq :
ℕ → ℝ
hdelta :
ThresholdSequence deltaBar deltaSeq
conclusion
hSeq :
fun n => infoBandwidth n (deltaSeq n) 1 1 deltaBar
AsympSeq hSeq (fun n => ((n : ℝ) * deltaSeq n) ^ (-(1 : ℝ) / 3))
AsympSeq (fun n => oneCellProductKL n (deltaSeq n) (hSeq n)) (fun n => (n : ℝ) * deltaSeq n * (hSeq n) ^ 3)
AsympSeq (fun n => oneCellTargetSeparation (deltaSeq n) (hSeq n)) (fun n => (deltaSeq n) ^ 2 * hSeq n)
AsympSeq deltaSeq (fun n => (n : ℝ) ^ (-(1 : ℝ) / 10))
↔ AsympSeq (fun n => oneCellTargetSeparation (deltaSeq n) (hSeq n)) (fun n => (n : ℝ) ^ (-(1 : ℝ) / 2))
Proof (Lean source)
-- @node: prop:one-cell-calibration
theorem one_cell_calibration (deltaBar : ℝ)
(hdeltaBar : 0 < deltaBar ∧ deltaBar < 1)
(deltaSeq : ℕ → ℝ) (hdelta : ThresholdSequence deltaBar deltaSeq)
(hfar : Tendsto (fun n => deltaSeq n / (n : ℝ) ^ (-(1 : ℝ) / 4))
atTop atTop) :
let hSeq := fun n => infoBandwidth n (deltaSeq n) 1 1 deltaBar
AsympSeq hSeq (fun n => ((n : ℝ) * deltaSeq n) ^ (-(1 : ℝ) / 3)) ∧
AsympSeq (fun n => oneCellProductKL n (deltaSeq n) (hSeq n))
(fun n => (n : ℝ) * deltaSeq n * (hSeq n) ^ 3) ∧
AsympSeq (fun n => oneCellTargetSeparation (deltaSeq n) (hSeq n))
(fun n => (deltaSeq n) ^ 2 * hSeq n) ∧
(AsympSeq deltaSeq (fun n => (n : ℝ) ^ (-(1 : ℝ) / 10)) ↔
AsympSeq (fun n => oneCellTargetSeparation (deltaSeq n) (hSeq n))
(fun n => (n : ℝ) ^ (-(1 : ℝ) / 2))) := by
dsimp
have hb := oneCellBandwidth_far deltaBar hdeltaBar deltaSeq hdelta hfar
have hbw := hb.1
have hwin := hb.2
refine ⟨hbw, ?_, ?_, ?_⟩
· refine ⟨8 / 3, 16 / 3, by norm_num, by norm_num, ?_⟩
filter_upwards [hwin] with n hn
simpa [mul_assoc] using oneCellProductKL_bounds n (deltaSeq n)
(infoBandwidth n (deltaSeq n) 1 1 deltaBar) hn.1 hn.2.1 hn.2.2.1 hn.2.2.2
· refine ⟨1, 7 / 3, by norm_num, by norm_num, ?_⟩
filter_upwards [hwin] with n hn
have hdelta0 : 0 ≤ deltaSeq n := hn.1.le.trans hn.2.1
have heq := oneCellTargetSeparation_eq (deltaSeq n)
(infoBandwidth n (deltaSeq n) 1 1 deltaBar) hdelta0 hn.1 hn.2.2.1
rw [heq]
constructor
· nlinarith [sq_nonneg (deltaSeq n),
sq_nonneg (infoBandwidth n (deltaSeq n) 1 1 deltaBar)]
· have hh0 : 0 ≤ infoBandwidth n (deltaSeq n) 1 1 deltaBar := hn.1.le
have hh_le : infoBandwidth n (deltaSeq n) 1 1 deltaBar ≤ deltaSeq n := hn.2.1
nlinarith [mul_nonneg (sq_nonneg (infoBandwidth n (deltaSeq n) 1 1 deltaBar))
(sub_nonneg.mpr hh_le),
mul_nonneg (sq_nonneg (deltaSeq n)) (sub_nonneg.mpr hh_le)]
· rcases hbw with ⟨ch, Ch, hch, hchCh, hbwEv⟩
have hscale : ∀ᶠ n in atTop,
ch * (deltaSeq n ^ 2 *
(((n : ℝ) * deltaSeq n) ^ (-(1 : ℝ) / 3))) ≤
oneCellTargetSeparation (deltaSeq n)
(infoBandwidth n (deltaSeq n) 1 1 deltaBar) ∧
oneCellTargetSeparation (deltaSeq n)
(infoBandwidth n (deltaSeq n) 1 1 deltaBar) ≤
((7 / 3 : ℝ) * Ch) * (deltaSeq n ^ 2 *
(((n : ℝ) * deltaSeq n) ^ (-(1 : ℝ) / 3))) := by
filter_upwards [hwin, hbwEv] with n hn hbn
let h := infoBandwidth n (deltaSeq n) 1 1 deltaBar
let w := ((n : ℝ) * deltaSeq n) ^ (-(1 : ℝ) / 3)
have hd0 : 0 ≤ deltaSeq n := hn.1.le.trans hn.2.1
have heq := oneCellTargetSeparation_eq (deltaSeq n) h hd0 hn.1 hn.2.2.1
have htargetLo : deltaSeq n ^ 2 * h ≤
oneCellTargetSeparation (deltaSeq n) h := by
rw [heq]
nlinarith [sq_nonneg (deltaSeq n), sq_nonneg h]
have htargetHi : oneCellTargetSeparation (deltaSeq n) h ≤
(7 / 3 : ℝ) * (deltaSeq n ^ 2 * h) := by
rw [heq]
have hh0 : 0 ≤ h := hn.1.le
have hh_le : h ≤ deltaSeq n := hn.2.1
nlinarith [mul_nonneg (sq_nonneg h) (sub_nonneg.mpr hh_le),
mul_nonneg (sq_nonneg (deltaSeq n)) (sub_nonneg.mpr hh_le)]
have hsq0 : 0 ≤ deltaSeq n ^ 2 := sq_nonneg _
constructor
· calc
ch * (deltaSeq n ^ 2 * w) = deltaSeq n ^ 2 * (ch * w) := by ring
_ ≤ deltaSeq n ^ 2 * h :=
mul_le_mul_of_nonneg_left hbn.1 hsq0
_ ≤ _ := htargetLo
· calc
_ ≤ (7 / 3 : ℝ) * (deltaSeq n ^ 2 * h) := htargetHi
_ ≤ (7 / 3 : ℝ) * (deltaSeq n ^ 2 * (Ch * w)) := by
gcongr
exact hbn.2
_ = ((7 / 3 : ℝ) * Ch) * (deltaSeq n ^ 2 * w) := by ring
constructor
· rintro ⟨cd, Cd, hcd, hcdC, hdEv⟩
let cOut : ℝ := ch * cd ^ ((5 : ℝ) / 3)
let COut : ℝ := ((7 / 3 : ℝ) * Ch) * Cd ^ ((5 : ℝ) / 3)
have hCd : 0 < Cd := hcd.trans_le hcdC
have hCh : 0 < Ch := hch.trans_le hchCh
have hcOut : 0 < cOut := mul_pos hch (Real.rpow_pos_of_pos hcd _)
have hCOut : 0 < COut := by
dsimp [COut]
exact mul_pos (mul_pos (by norm_num) hCh) (Real.rpow_pos_of_pos hCd _)
have hcC : cOut ≤ COut := by
dsimp [cOut, COut]
have : ch ≤ (7 / 3 : ℝ) * Ch := by nlinarith
gcongr
refine ⟨cOut, COut, hcOut, hcC, ?_⟩
filter_upwards [hscale, hdEv, hwin, eventually_ge_atTop 1] with n hs hd hwn hn1
have hnPos : 0 < n := lt_of_lt_of_le Nat.zero_lt_one hn1
have hnR : 0 < (n : ℝ) := by exact_mod_cast hnPos
have hdPos : 0 < deltaSeq n := hwn.1.trans_le hwn.2.1
have hcrit : 0 < (n : ℝ) ^ (-(1 : ℝ) / 10) := Real.rpow_pos_of_pos hnR _
have hroot : 0 < (n : ℝ) ^ (-(1 : ℝ) / 2) := Real.rpow_pos_of_pos hnR _
let ratio := deltaSeq n / (n : ℝ) ^ (-(1 : ℝ) / 10)
let atom := deltaSeq n ^ 2 * (((n : ℝ) * deltaSeq n) ^ (-(1 : ℝ) / 3))
have hratioLo : cd ≤ ratio := by
exact (le_div_iff₀ hcrit).2 (by simpa [ratio, mul_comm] using hd.1)
have hratioHi : ratio ≤ Cd := by
exact (div_le_iff₀ hcrit).2 (by simpa [ratio, mul_comm] using hd.2)
have hnorm : atom / (n : ℝ) ^ (-(1 : ℝ) / 2) = ratio ^ ((5 : ℝ) / 3) := by
simpa [atom, ratio] using oneCellNormalizedScale_eq n (deltaSeq n) hnPos hdPos
have hatom : atom = ratio ^ ((5 : ℝ) / 3) *
(n : ℝ) ^ (-(1 : ℝ) / 2) := (div_eq_iff hroot.ne').mp hnorm
have hpowLo := Real.rpow_le_rpow hcd.le hratioLo (by norm_num : (0 : ℝ) ≤ 5 / 3)
have hpowHi := Real.rpow_le_rpow (le_trans hcd.le hratioLo) hratioHi
(by norm_num : (0 : ℝ) ≤ 5 / 3)
constructor
· calc
cOut * (n : ℝ) ^ (-(1 : ℝ) / 2) =
ch * (cd ^ ((5 : ℝ) / 3) * (n : ℝ) ^ (-(1 : ℝ) / 2)) := by
simp [cOut]; ring
_ ≤ ch * atom := by rw [hatom]; gcongr
_ ≤ _ := hs.1
· calc
_ ≤ ((7 / 3 : ℝ) * Ch) * atom := hs.2
_ ≤ ((7 / 3 : ℝ) * Ch) *
(Cd ^ ((5 : ℝ) / 3) * (n : ℝ) ^ (-(1 : ℝ) / 2)) := by
rw [hatom]
gcongr
_ = COut * (n : ℝ) ^ (-(1 : ℝ) / 2) := by simp [COut]; ring
· rintro ⟨cs, Cs, hcs, hcsC, hsEv⟩
let K : ℝ := (7 / 3 : ℝ) * Ch
have hCh : 0 < Ch := hch.trans_le hchCh
have hK : 0 < K := mul_pos (by norm_num) hCh
let lo : ℝ := (cs / K) ^ ((3 : ℝ) / 5)
let up : ℝ := (Cs / ch) ^ ((3 : ℝ) / 5)
have hCs : 0 < Cs := hcs.trans_le hcsC
have hlo : 0 < lo := Real.rpow_pos_of_pos (div_pos hcs hK) _
have hup : 0 < up := Real.rpow_pos_of_pos (div_pos hCs hch) _
refine ⟨lo, lo + up, hlo, le_add_of_nonneg_right hup.le, ?_⟩
filter_upwards [hscale, hsEv, hwin, eventually_ge_atTop 1] with n hscaleN hsN hwn hn1
have hnPos : 0 < n := lt_of_lt_of_le Nat.zero_lt_one hn1
have hnR : 0 < (n : ℝ) := by exact_mod_cast hnPos
have hdPos : 0 < deltaSeq n := hwn.1.trans_le hwn.2.1
have hcrit : 0 < (n : ℝ) ^ (-(1 : ℝ) / 10) := Real.rpow_pos_of_pos hnR _
have hroot : 0 < (n : ℝ) ^ (-(1 : ℝ) / 2) := Real.rpow_pos_of_pos hnR _
let ratio := deltaSeq n / (n : ℝ) ^ (-(1 : ℝ) / 10)
let atom := deltaSeq n ^ 2 * (((n : ℝ) * deltaSeq n) ^ (-(1 : ℝ) / 3))
have hratio : 0 < ratio := div_pos hdPos hcrit
have hnorm : atom / (n : ℝ) ^ (-(1 : ℝ) / 2) = ratio ^ ((5 : ℝ) / 3) := by
simpa [atom, ratio] using oneCellNormalizedScale_eq n (deltaSeq n) hnPos hdPos
have hatom : atom = ratio ^ ((5 : ℝ) / 3) *
(n : ℝ) ^ (-(1 : ℝ) / 2) := (div_eq_iff hroot.ne').mp hnorm
have hpLo : cs / K ≤ ratio ^ ((5 : ℝ) / 3) := by
apply (div_le_iff₀ hK).2
have hmul := hsN.1.trans (by simpa [K] using hscaleN.2)
change cs * (n : ℝ) ^ (-(1 : ℝ) / 2) ≤ K * atom at hmul
rw [hatom] at hmul
nlinarith
have hpHi : ratio ^ ((5 : ℝ) / 3) ≤ Cs / ch := by
apply (le_div_iff₀ hch).2
have hmul := hscaleN.1.trans hsN.2
change ch * atom ≤ Cs * (n : ℝ) ^ (-(1 : ℝ) / 2) at hmul
rw [hatom] at hmul
nlinarith
have hloRatio : lo ≤ ratio := by
have hp := Real.rpow_le_rpow (div_nonneg hcs.le hK.le) hpLo
(by norm_num : (0 : ℝ) ≤ 3 / 5)
have hcollapse : (ratio ^ ((5 : ℝ) / 3)) ^ ((3 : ℝ) / 5) = ratio := by
rw [← Real.rpow_mul hratio.le]
norm_num
simpa [lo, hcollapse] using hp
have hratioUp : ratio ≤ up := by
have hp := Real.rpow_le_rpow (Real.rpow_nonneg hratio.le _) hpHi
(by norm_num : (0 : ℝ) ≤ 3 / 5)
have hcollapse : (ratio ^ ((5 : ℝ) / 3)) ^ ((3 : ℝ) / 5) = ratio := by
rw [← Real.rpow_mul hratio.le]
norm_num
simpa [up, hcollapse] using hp
constructor
· exact (le_div_iff₀ hcrit).mp (by simpa [ratio, mul_comm] using hloRatio)
· have hdu : deltaSeq n ≤ up * (n : ℝ) ^ (-(1 : ℝ) / 10) :=
(div_le_iff₀ hcrit).mp (by simpa [ratio] using hratioUp)
exact hdu.trans (mul_le_mul_of_nonneg_right
(le_add_of_nonneg_left hlo.le) hcrit.le)
CausalSmith.Stat.STAT_LmtpThresholdAtomFrontier_Research.TPhaseDiagram 3 declarations This module assembles the rate comparisons and the zero-threshold reduction.
Four-regime threshold phase diagram
This module assembles the rate comparisons and the zero-threshold reduction.
theorem zero_threshold_stabilizedCoverage_eventually
At the identity threshold, the stabilized interval is eventually uniformly honest over the bounded-outcome model. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
Bseq :
∀ n
conclusion
SplitBlocks n, ∀ᶠ n in atTop, 1 - alpha
≤ stabilizedCoverage J n (Bseq n) beta kappa L cminus cplus pmin deltaBar 0 alpha
Proof (Lean source)
lemma zero_threshold_stabilizedCoverage_eventually
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
∀ Bseq : ∀ n, SplitBlocks n,
∀ᶠ n in atTop,
1 - alpha ≤ stabilizedCoverage J n (Bseq n) beta kappa L cminus cplus
pmin deltaBar 0 alpha := by
rcases honest_coverage_and_length J beta kappa L cminus cplus pmin deltaBar
alpha hreg with ⟨c, C, hc, hcC, hlength⟩
have hzero : ThresholdSequence deltaBar (fun _ => 0) := by
intro n
exact ⟨le_rfl, hreg.2.2.2.2.2.2.2.2.2.1.le⟩
intro Bseq
filter_upwards [hlength (fun _ => 0) hzero Bseq] with n hn
exact hn.1
theorem zero_threshold_stabilizedWorstLength_asymp
At the identity threshold, the bias-aware interval's worst-case expected length inherits the root-sample-size sandwich from the honest-length theorem. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
Bseq :
∀ n
conclusion
SplitBlocks n, AsympSeq (fun n => stabilizedWorstLength J n (Bseq n) beta kappa L cminus cplus pmin deltaBar 0 alpha) (fun n => (n : ℝ) ^ (-(1 : ℝ) / 2))
Proof (Lean source)
-- @node: zero_threshold_stabilizedWorstLength_asymp
lemma zero_threshold_stabilizedWorstLength_asymp
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
∀ Bseq : ∀ n, SplitBlocks n,
AsympSeq
(fun n => stabilizedWorstLength J n (Bseq n) beta kappa L cminus cplus
pmin deltaBar 0 alpha)
(fun n => (n : ℝ) ^ (-(1 : ℝ) / 2)) := by
rcases honest_coverage_and_length J beta kappa L cminus cplus pmin deltaBar
alpha hreg with ⟨c, C, hc, hcC, hlength⟩
have hzero : ThresholdSequence deltaBar (fun _ => 0) := by
intro n
exact ⟨le_rfl, hreg.2.2.2.2.2.2.2.2.2.1.le⟩
intro Bseq
refine ⟨c, C, hc, hcC.le, ?_⟩
filter_upwards [hlength (fun _ => 0) hzero Bseq] with n hn
dsimp at hn
have hfront :
clampFrontier n 0 kappa (infoBandwidth n 0 beta kappa deltaBar) beta =
(n : ℝ) ^ (-(1 : ℝ) / 2) := by
simp [clampFrontier,
Real.zero_rpow (by linarith [hreg.2.2.1] : kappa + 1 ≠ 0)]
rw [hfront] at hn
exact ⟨hn.2.1, hn.2.2.1⟩
The critical scale is separated from the design-edge scale; below, at, and above it the frontier has the four rates stated in the paper. The result uses the hreg condition. This is the stated conclusion.
Formal statement
hypotheses
J :
ℕ
beta kappa L cminus cplus pmin deltaBar alpha :
ℝ
hreg :
RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha
conclusion
conclusion 2
deltaSeq :
ℕ → ℝ
ThresholdSequence deltaBar deltaSeq
hSeq :
fun n => infoBandwidth n (deltaSeq n) beta kappa deltaBar
atomSeq :
fun n => (deltaSeq n) ^ (kappa + 1) * (hSeq n) ^ beta
rSeq :
fun n => clampFrontier n (deltaSeq n) kappa (hSeq n) beta
AsympSeq rSeq (fun n => (n : ℝ) ^ (-(1 : ℝ) / 2))
AsympSeq rSeq (fun n => (deltaSeq n) ^ (kappa + 1) * ((n : ℝ) * (deltaSeq n) ^ kappa) ^ (-beta / (2 * beta + 1)))
AsympSeq rSeq (fun n => (n : ℝ) ^ (-beta / (2 * beta + 1)))
clampFunctional P 0 = ∫ o, o.Y ∂P.dataMeasure
clampFrontier n 0 kappa (infoBandwidth n 0 beta kappa deltaBar) beta
= (n : ℝ) ^ (-(1 : ℝ) / 2)
n :
ℕ
P :
ClampLaw J
B :
ClampModel P beta kappa L cminus cplus pmin
h :
infoBandwidth n 0 beta kappa deltaBar
Bseq :
∀ n
SplitBlocks n, (∀ᶠ n in atTop, 1 - alpha ≤ stabilizedCoverage J n (Bseq n) beta kappa L cminus cplus pmin deltaBar 0 alpha)
AsympSeq (fun n => stabilizedWorstRisk J n (Bseq n) beta kappa L cminus cplus pmin deltaBar 0) (fun n => (n : ℝ) ^ (-(1 : ℝ) / 2))
AsympSeq (fun n => stabilizedWorstLength J n (Bseq n) beta kappa L cminus cplus pmin deltaBar 0 alpha) (fun n => (n : ℝ) ^ (-(1 : ℝ) / 2))
Proof (Lean source)
-- @node: thm:phase-diagram
theorem clamp_phase_diagram
(J : ℕ) (beta kappa L cminus cplus pmin deltaBar alpha : ℝ)
(hreg : RegimeConstants J beta kappa L cminus cplus pmin deltaBar alpha) :
Tendsto (fun n => deltaCrit n beta kappa / deltaEdge n beta kappa)
atTop atTop ∧
∀ (deltaSeq : ℕ → ℝ), ThresholdSequence deltaBar deltaSeq →
let hSeq := fun n => infoBandwidth n (deltaSeq n) beta kappa deltaBar
let atomSeq := fun n =>
(deltaSeq n) ^ (kappa + 1) * (hSeq n) ^ beta
let rSeq := fun n => clampFrontier n (deltaSeq n) kappa (hSeq n) beta
(Tendsto (fun n => deltaSeq n / deltaCrit n beta kappa) atTop (nhds 0) →
AsympSeq rSeq (fun n => (n : ℝ) ^ (-(1 : ℝ) / 2))) ∧
(∀ c0 : ℝ, 0 < c0 →
Tendsto (fun n => deltaSeq n / deltaCrit n beta kappa) atTop (nhds c0) →
AsympSeq atomSeq (fun n => (n : ℝ) ^ (-(1 : ℝ) / 2)) ∧
AsympSeq rSeq (fun n => (n : ℝ) ^ (-(1 : ℝ) / 2))) ∧
(Tendsto (fun n => deltaSeq n / deltaCrit n beta kappa) atTop atTop →
Tendsto deltaSeq atTop (nhds 0) →
AsympSeq rSeq (fun n =>
(deltaSeq n) ^ (kappa + 1) *
((n : ℝ) * (deltaSeq n) ^ kappa) ^
(-beta / (2 * beta + 1)))) ∧
(∀ delta0 : ℝ, delta0 ∈ Ioc (0 : ℝ) deltaBar →
Tendsto deltaSeq atTop (nhds delta0) →
AsympSeq rSeq (fun n => (n : ℝ) ^ (-beta / (2 * beta + 1)))) ∧
(∀ (n : ℕ) (P : ClampLaw J),
ClampModel P beta kappa L cminus cplus pmin →
(∀ x : Fin J, atomMass P x 0 = 0) ∧
clampFunctional P 0 = ∫ o, o.Y ∂P.dataMeasure ∧
clampFrontier n 0 kappa (infoBandwidth n 0 beta kappa deltaBar) beta =
(n : ℝ) ^ (-(1 : ℝ) / 2)) ∧
(∀ (n : ℕ) (P : ClampLaw J) (B : SplitBlocks n),
ClampModel P beta kappa L cminus cplus pmin →
let h := infoBandwidth n 0 beta kappa deltaBar
(fun z : Fin n → ClampObs J =>
totalGramEstimator B z (ellOf beta) kappa cminus cplus 0 h) =ᵐ[iidProduct P n]
(fun z => clampUnit (blockAverage B.I0 z (fun o => o.Y)))) ∧
∀ Bseq : ∀ n, SplitBlocks n,
(∀ᶠ n in atTop,
1 - alpha ≤ stabilizedCoverage J n (Bseq n) beta kappa L cminus cplus
pmin deltaBar 0 alpha) ∧
AsympSeq
(fun n => stabilizedWorstRisk J n (Bseq n) beta kappa L cminus cplus
pmin deltaBar 0)
(fun n => (n : ℝ) ^ (-(1 : ℝ) / 2)) ∧
AsympSeq
(fun n => stabilizedWorstLength J n (Bseq n) beta kappa L cminus cplus
pmin deltaBar 0 alpha)
(fun n => (n : ℝ) ^ (-(1 : ℝ) / 2)) := by
refine ⟨critical_scale_div_edge_scale_tendsto_top J beta kappa L cminus cplus
pmin deltaBar alpha hreg, ?_⟩
intro deltaSeq hdelta
dsimp
refine ⟨regular_frontier_asymp J beta kappa L cminus cplus pmin deltaBar
alpha hreg deltaSeq hdelta, ?_, ?_, ?_, ?_, ?_, ?_⟩
· intro c0 hc0 hratio
exact critical_atom_and_frontier_asymp J beta kappa L cminus cplus pmin
deltaBar alpha c0 hreg hc0 deltaSeq hdelta hratio
· intro hratio _hzero
exact supercritical_frontier_asymp J beta kappa L cminus cplus pmin
deltaBar alpha hreg deltaSeq hdelta hratio
· intro delta0 hdelta0 htend
exact fixed_positive_frontier_asymp J beta kappa L cminus cplus pmin
deltaBar alpha delta0 hreg hdelta0.1 deltaSeq hdelta htend
· intro n P hmodel
refine ⟨(zero_threshold_atom_and_frontier J n P beta kappa deltaBar
hreg.2.2.1).1, zero_threshold_functional J P beta kappa L cminus cplus
pmin hmodel, ?_⟩
exact (zero_threshold_atom_and_frontier J n P beta kappa deltaBar
hreg.2.2.1).2
· intro n P B hmodel
exact zero_threshold_totalGramEstimator J n P B beta kappa L cminus cplus
pmin deltaBar hmodel
· intro Bseq
exact ⟨zero_threshold_stabilizedCoverage_eventually J beta kappa L cminus
cplus pmin deltaBar alpha hreg Bseq,
zero_threshold_stabilizedWorstRisk_asymp J beta kappa L cminus cplus
pmin deltaBar alpha hreg Bseq,
zero_threshold_stabilizedWorstLength_asymp J beta kappa L cminus cplus
pmin deltaBar alpha hreg Bseq⟩