Formalization: Minimax Response Recovery under Global Overlap Tails
The complete Lean development behind this paper — every definition, lemma, and theorem of its module, including helpers the paper text never cites. Identifiers link within this page, into the Causalean library, or out to the official Mathlib docs. Note: this index was generated at a different commit than the paper's audited pin (d07574d) — statements may have moved since the audit.
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.DornScope 12 declarations The printed pointwise lower clause is obstructed by a known-regression singleton family.
Scope of Dorn's Theorem 1
The printed pointwise lower clause is obstructed by a known-regression singleton family. The source upper clause is kept as a cited, explicit gate.
def dornCanonicalLaw
A fixed, known-regression Gaussian experiment on Dorn's cube. For the stated inputs and conditions, the dornCanonicalLaw object being defined.
Definition (Lean source)
parameters
d :
ℕ
def
given by
(((ofReal ((2 : ℝ) ^ d))⁻¹ • volume.restrict (dornCube d)).prod ((realBernoulli (1 / 2)).prod (gaussianReal 0 ⟨1, by norm_num⟩)))
The canonical experiment has the stated uniform covariate law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
conclusion
Proof (Lean source)
lemma dornCanonicalLaw_covariate (d : ℕ) :
covariateLaw (dornCanonicalLaw d) =
(ofReal ((2 : ℝ) ^ d))⁻¹ • volume.restrict (dornCube d) := by
have hbern : IsProbabilityMeasure (realBernoulli (1 / 2)) := by
apply isProbabilityMeasure_iff.mpr
simp [realBernoulli]
rw [show (1 : ℝ) - 2⁻¹ = 1 / 2 by norm_num]
rw [ENNReal.ofReal_div_of_pos (by norm_num : (0 : ℝ) < 2)]
norm_num only [ENNReal.ofReal_one, ENNReal.ofReal_ofNat]
rw [one_div]
calc
(2 : ℝ≥0∞)⁻¹ + 2⁻¹ = (2 : ℝ≥0∞)⁻¹ * 2 := by ring
_ = 1 := ENNReal.inv_mul_cancel (by norm_num) (by norm_num)
letI : IsProbabilityMeasure (realBernoulli (1 / 2)) := hbern
letI : IsProbabilityMeasure (gaussianReal 0 ⟨1, by norm_num⟩) :=
instIsProbabilityMeasureGaussianReal 0 ⟨1, by norm_num⟩
haveI : IsProbabilityMeasure
((realBernoulli (1 / 2)).prod (gaussianReal 0 ⟨1, by norm_num⟩)) :=
Measure.prod.instIsProbabilityMeasure _ _
have huniv : ((realBernoulli (1 / 2)).prod
(gaussianReal 0 ⟨1, by norm_num⟩)) univ = 1 := measure_univ
simp only [covariateLaw, dornCanonicalLaw, Measure.map_fst_prod, huniv, one_smul]
The independent half-treated arm has constant selected propensity. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
conclusion
∀ᵐ x ∂covariateLaw (dornCanonicalLaw d), propensity (dornCanonicalLaw d) x = 1 / 2
Proof (Lean source)
lemma dornCanonicalLaw_propensity_ae (d : ℕ)
[IsProbabilityMeasure (dornCanonicalLaw d)] :
∀ᵐ x ∂covariateLaw (dornCanonicalLaw d),
propensity (dornCanonicalLaw d) x = 1 / 2 := by
let μx : Measure (Fin d → ℝ) :=
(ofReal ((2 : ℝ) ^ d))⁻¹ • volume.restrict (dornCube d)
let μa : Measure Bool := realBernoulli (1 / 2)
let μy : Measure ℝ := gaussianReal 0 ⟨1, by norm_num⟩
have hcube : IsProbabilityMeasure μx := by
apply isProbabilityMeasure_iff.mpr
have hvol : volume (dornCube d) = ofReal ((2 : ℝ) ^ d) := by
rw [dornCube, Set.pi_univ_Icc, Real.volume_Icc_pi]
simp [Finset.prod_const, ENNReal.ofReal_pow]
norm_num
simp only [μx, Measure.smul_apply, Measure.restrict_apply, MeasurableSet.univ,
Set.univ_inter, hvol, smul_eq_mul]
exact ENNReal.inv_mul_cancel (by positivity) (by simp)
letI : IsProbabilityMeasure μx := hcube
haveI : IsProbabilityMeasure μa := by
apply isProbabilityMeasure_iff.mpr
simp [μa, realBernoulli]
rw [show (1 : ℝ) - 2⁻¹ = 1 / 2 by norm_num]
rw [ENNReal.ofReal_div_of_pos (by norm_num : (0 : ℝ) < 2)]
norm_num only [ENNReal.ofReal_one, ENNReal.ofReal_ofNat]
rw [one_div]
calc
(2 : ℝ≥0∞)⁻¹ + 2⁻¹ = (2 : ℝ≥0∞)⁻¹ * 2 := by ring
_ = 1 := ENNReal.inv_mul_cancel (by norm_num) (by norm_num)
haveI : IsProbabilityMeasure μy := by
dsimp [μy]
exact instIsProbabilityMeasureGaussianReal 0 ⟨1, by norm_num⟩
have hmap : (dornCanonicalLaw d).map (fun z : Obs d => (z.1, z.2.1)) =
μx.prod μa := by
change (μx.prod (μa.prod μy)).map (map id fst) = μx.prod μa
rw [← Measure.map_prod_map μx (μa.prod μy) measurable_id measurable_fst]
simp
have hae := ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd
(μ := dornCanonicalLaw d) (X := fun z : Obs d => z.1)
(Y := fun z : Obs d => z.2.1) (by fun_prop)
(κ := Kernel.const (Fin d → ℝ) μa) (by
rw [hmap]
change μx.prod μa = covariateLaw (dornCanonicalLaw d) ⊗ₘ
Kernel.const (Fin d → ℝ) μa
rw [dornCanonicalLaw_covariate]
change μx.prod μa = μx ⊗ₘ Kernel.const (Fin d → ℝ) μa
simp)
filter_upwards [hae] with x hx
have hx' := congrArg (fun ν : Measure Bool => ν {true}) hx
change (condDistrib (fun z : Obs d => z.2.1) (fun z => z.1)
(dornCanonicalLaw d) x {true}).toReal = 1 / 2
rw [hx']
norm_num [μa, realBernoulli, Kernel.const_apply]
Both conditional response arms of the canonical experiment are standard Gaussian. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma dornCanonicalLaw_armKernel_ae (d : ℕ)
[IsProbabilityMeasure (dornCanonicalLaw d)] :
∀ᵐ x ∂covariateLaw (dornCanonicalLaw d),
∀ a : Bool, armKernel (dornCanonicalLaw d) x a =
gaussianReal 0 ⟨1, by norm_num⟩ := by
let μx : Measure (Fin d → ℝ) :=
(ofReal ((2 : ℝ) ^ d))⁻¹ • volume.restrict (dornCube d)
let μa : Measure Bool := realBernoulli (1 / 2)
let μy : Measure ℝ := gaussianReal 0 ⟨1, by norm_num⟩
haveI : IsProbabilityMeasure μa := by
apply isProbabilityMeasure_iff.mpr
simp [μa, realBernoulli]
rw [show (1 : ℝ) - 2⁻¹ = 1 / 2 by norm_num]
rw [ENNReal.ofReal_div_of_pos (by norm_num : (0 : ℝ) < 2)]
norm_num only [ENNReal.ofReal_one, ENNReal.ofReal_ofNat]
rw [one_div]
calc
(2 : ℝ≥0∞)⁻¹ + 2⁻¹ = (2 : ℝ≥0∞)⁻¹ * 2 := by ring
_ = 1 := ENNReal.inv_mul_cancel (by norm_num) (by norm_num)
haveI : IsProbabilityMeasure μy :=
instIsProbabilityMeasureGaussianReal 0 ⟨1, by norm_num⟩
have hpair : (dornCanonicalLaw d).map
(fun z : Obs d => (z.1, z.2.1)) = μx.prod μa := by
change (μx.prod (μa.prod μy)).map (map id fst) = μx.prod μa
rw [← Measure.map_prod_map μx (μa.prod μy) measurable_id measurable_fst]
simp
have hmap : (dornCanonicalLaw d).map
(fun z : Obs d => ((z.1, z.2.1), z.2.2)) = (μx.prod μa).prod μy := by
change (μx.prod (μa.prod μy)).map
(fun z => ((z.1, z.2.1), z.2.2)) = _
rw [← MeasurableEquiv.prodAssoc.map_measurableEquiv_injective.eq_iff]
rw [Measure.map_map (by fun_prop) (by fun_prop)]
have hf : (MeasurableEquiv.prodAssoc :
((Fin d → ℝ) × Bool) × ℝ ≃ᵐ (Fin d → ℝ) × Bool × ℝ) ∘
(fun z : (Fin d → ℝ) × Bool × ℝ => ((z.1, z.2.1), z.2.2)) = id := by
funext z
rcases z with ⟨x, a, y⟩
rfl
rw [hf, Measure.map_id]
exact (Measure.prodAssoc_prod (μ := μx) (ν := μa) (τ := μy)).symm
have hae := ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd
(μ := dornCanonicalLaw d) (X := fun z : Obs d => (z.1, z.2.1))
(Y := fun z : Obs d => z.2.2) (by fun_prop)
(κ := Kernel.const ((Fin d → ℝ) × Bool) μy) (by
rw [hmap]
change (μx.prod μa).prod μy =
((dornCanonicalLaw d).map (fun z : Obs d => (z.1, z.2.1))) ⊗ₘ
Kernel.const ((Fin d → ℝ) × Bool) μy
rw [hpair]
simp)
have hae' : ∀ᵐ xa ∂μx.prod μa,
armKernel (dornCanonicalLaw d) xa.1 xa.2 = μy := by
rw [← hpair]
filter_upwards [hae] with xa hxa
simpa [armKernel, Kernel.const_apply] using hxa
have hae'' := Measure.ae_ae_of_ae_prod hae'
rw [dornCanonicalLaw_covariate]
filter_upwards [hae''] with x hx
intro a
have ha : μa {a} ≠ 0 := by
cases a <;> norm_num [μa, realBernoulli]
exact (ae_iff_of_countable.mp hx) a ha
def dornPointwiseMissRisk
Pointwise miss probability, as an extended nonnegative risk. For the stated inputs and conditions, the dornPointwiseMissRisk object being defined.
Definition (Lean source)
parameters
d n :
ℕ
𝔓 :
Set (DornSourceLaw d)
x₀ :
Fin d → ℝ
c r :
ℝ
T :
{T : (Fin n → Obs d) → ℝ // Measurable T}
z :
{z : DornSourceLaw d // z ∈ 𝔓}
def
dornPointwiseMissRisk 𝔓 x₀ c r T z :
ℝ≥0∞
theorem knownRegression_minimax_miss_zero
If the regression curve is known across a family, its pointwise minimax miss probability vanishes for every positive threshold. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
𝔓 :
Set (DornSourceLaw d)
μref :
(Fin d → ℝ) → ℝ
x₀ :
Fin d → ℝ
c r :
ℝ
hc :
0 < c
hr :
0 < r
hμ :
∀ z ∈ 𝔓, ∀ x, z.2.1 x = μref x
conclusion
minimaxValueENNReal (dornPointwiseMissRisk (n := n) 𝔓 x₀ c r) = 0
Proof (Lean source)
lemma knownRegression_minimax_miss_zero {d n : ℕ}
(𝔓 : Set (DornSourceLaw d)) (μref : (Fin d → ℝ) → ℝ)
(x₀ : Fin d → ℝ) (c r : ℝ) (hc : 0 < c) (hr : 0 < r)
(hμ : ∀ z ∈ 𝔓, ∀ x, z.2.1 x = μref x) :
Causalean.Stat.minimaxValueENNReal
(dornPointwiseMissRisk (n := n) 𝔓 x₀ c r) = 0 := by
unfold Causalean.Stat.minimaxValueENNReal Causalean.Stat.worstCaseRiskENNReal
apply le_antisymm
· let T₀ : {T : (Fin n → Obs d) → ℝ // Measurable T} :=
⟨fun _ => μref x₀, measurable_const⟩
apply (iInf_le (fun T : {T : (Fin n → Obs d) → ℝ // Measurable T} =>
⨆ z : {z : DornSourceLaw d // z ∈ 𝔓},
dornPointwiseMissRisk 𝔓 x₀ c r T z) T₀).trans
apply iSup_le
intro z
have hz := hμ z.1 z.2 x₀
have hcr : ¬ c * r < 0 := not_lt.mpr (le_of_lt (mul_pos hc hr))
simp [dornPointwiseMissRisk, T₀, hz, hcr]
· exact bot_le
A constant half propensity has the global polynomial tail with coefficient 2 ^ (γ - 1) under any probability design. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
α :
Type*
μ :
γ :
ℝ
hγ :
1 < γ
t ∈ Set.Icc (0 : ℝ) 1 :
conclusion
μ.real {_x : α | (1 / 2 : ℝ) ≤ t} ≤ (2 : ℝ) ^ (γ - 1) * t ^ (γ - 1)
Proof (Lean source)
lemma constantHalf_globalTailInequality {α : Type*} [MeasurableSpace α]
(μ : Measure α) [IsProbabilityMeasure μ] (γ : ℝ) (hγ : 1 < γ) :
∀ t ∈ Icc (0 : ℝ) 1,
μ.real {_x : α | (1 / 2 : ℝ) ≤ t} ≤
(2 : ℝ) ^ (γ - 1) * t ^ (γ - 1) := by
intro t ht
by_cases h : (1 / 2 : ℝ) ≤ t
· have hpow : (1 / 2 : ℝ) ^ (γ - 1) ≤ t ^ (γ - 1) :=
Real.rpow_le_rpow (by norm_num) h (by linarith)
have hcoeff : 0 ≤ (2 : ℝ) ^ (γ - 1) := Real.rpow_nonneg (by norm_num) _
have hone : (2 : ℝ) ^ (γ - 1) * (1 / 2 : ℝ) ^ (γ - 1) = 1 := by
rw [← Real.mul_rpow (by norm_num) (by norm_num)]
norm_num
have hbound : (1 : ℝ) ≤ (2 : ℝ) ^ (γ - 1) * t ^ (γ - 1) := by
calc
(1 : ℝ) = (2 : ℝ) ^ (γ - 1) * (1 / 2 : ℝ) ^ (γ - 1) := hone.symm
_ ≤ (2 : ℝ) ^ (γ - 1) * t ^ (γ - 1) :=
mul_le_mul_of_nonneg_left hpow hcoeff
have huniv : {x : α | (1 / 2 : ℝ) ≤ t} = univ := by
ext x
change ((1 / 2 : ℝ) ≤ t) ↔ True
exact iff_true_intro h
rw [huniv]
simpa using hbound
· have hzero : {x : α | (1 / 2 : ℝ) ≤ t} = ∅ := by
ext x
change ((1 / 2 : ℝ) ≤ t) ↔ False
exact iff_false_intro h
rw [hzero]
simp only [measureReal_empty]
exact mul_nonneg (Real.rpow_nonneg (by norm_num) _)
(Real.rpow_nonneg ht.1 _)
A Euclidean ball centered in Dorn's closed cube has positive volume in the cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma dornCube_ball2_volume_pos {d : ℕ} (x₀ : Fin d → ℝ)
(hx₀ : x₀ ∈ dornCube d) {h : ℝ} (hh : 0 < h) :
0 < volume (ball2 x₀ h ∩ dornCube d) := by
have hclosure : dornCube d ⊆ closure (interior (dornCube d)) := by
intro y hy
change y ∈ closure (interior (Set.univ.pi (fun _ : Fin d => Icc (-1 : ℝ) 1)))
rw [interior_pi_set Set.finite_univ, closure_pi_set]
intro i hi
rw [closure_interior_Icc (by norm_num : (-1 : ℝ) ≠ 1)]
exact (show y ∈ Set.univ.pi (fun _ : Fin d => Icc (-1 : ℝ) 1) from hy) i hi
let U : Set (Fin d → ℝ) := {x | (∑ i, (x i - x₀ i) ^ 2) < h ^ 2}
have hUopen : IsOpen U := by
dsimp [U]
have hcont : Continuous (fun x : Fin d → ℝ => ∑ i, (x i - x₀ i) ^ 2) := by
fun_prop
exact isOpen_lt hcont continuous_const
have hxU : x₀ ∈ U := by
simp [U, sq_pos_of_pos hh]
have hnonempty : (U ∩ interior (dornCube d)).Nonempty :=
(mem_closure_iff.mp (hclosure hx₀)) U hUopen hxU
have hopen : IsOpen (U ∩ interior (dornCube d)) :=
hUopen.inter isOpen_interior
have hpos : 0 < volume (U ∩ interior (dornCube d)) :=
hopen.measure_pos volume hnonempty
exact lt_of_lt_of_le hpos (measure_mono (by
intro x hx
change x ∈ U ∧ x ∈ interior (dornCube d) at hx
change x ∈ ball2 x₀ h ∧ x ∈ dornCube d
exact ⟨le_of_lt (show (∑ i, (x i - x₀ i) ^ 2) < h ^ 2 from hx.1),
interior_subset hx.2⟩))
The known zero regression has zero spatial variance under the canonical design. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
M :
ℝ
hM :
0 ≤ M
conclusion
(∫ x, ((fun _ : Fin d → ℝ => (0 : ℝ)) x - ∫ y, (fun _ : Fin d → ℝ => (0 : ℝ)) y ∂covariateLaw (dornCanonicalLaw d)) ^ 2 ∂covariateLaw (dornCanonicalLaw d))
≤ M
Proof (Lean source)
lemma dornCanonicalLaw_zeroVariance (d : ℕ) (M : ℝ) (hM : 0 ≤ M) :
(∫ x, ((fun _ : Fin d → ℝ => (0 : ℝ)) x -
∫ y, (fun _ : Fin d → ℝ => (0 : ℝ)) y ∂covariateLaw (dornCanonicalLaw d)) ^ 2
∂covariateLaw (dornCanonicalLaw d)) ≤ M := by
simpa using hM
The zero regression belongs to every positive-radius Dorn seminorm class. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β L₀ :
ℝ
hL₀ :
0 ≤ L₀
conclusion
DornTopDerivativeSeminorm (d := d) β L₀ (fun _ => 0)
Proof (Lean source)
lemma dornZero_topDerivativeSeminorm (d : ℕ) (β L₀ : ℝ) (hL₀ : 0 ≤ L₀) :
DornTopDerivativeSeminorm (d := d) β L₀ (fun _ => 0) := by
unfold DornTopDerivativeSeminorm
constructor
· fun_prop
· intro α hα f hf x hx y hy
simp [coordJetOn]
exact mul_nonneg hL₀ (Real.rpow_nonneg (Real.sqrt_nonneg _) _)
A known-regression family has zero pointwise minimax error. In particular, the explicit uniform-design, half-treated, centered Gaussian singleton obeys Dorn's per-law conditions and common A3, yet has zero minimax error. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
1 ≤ d
β γ L₀ :
ℝ
hβ :
0 < β
hγ :
1 < γ
hL₀ :
0 < L₀
conclusion
conclusion 1
𝔓 :
Set (DornSourceLaw d)
μref :
(Fin d → ℝ) → ℝ
x₀ :
Fin d → ℝ
𝔓.Nonempty
(∀ z ∈ 𝔓, ∀ x, z.2.1 x = μref x)
n :
ℕ
c r :
ℝ
0 < c
0 < r
minimaxValueENNReal (dornPointwiseMissRisk (n := n) 𝔓 x₀ c r) = 0
conclusion 2
∃ q M C : ℝ,
3 < q
0 < M
0 < C
∃ hP : IsProbabilityMeasure (dornCanonicalLaw d),
@DornRemainingModel d β q M 1 C L₀ γ (dornCanonicalLaw d) hP (fun _ => 0) (fun _ => 0) (fun _ => 1 / 2)
DornA3OnOriginalClass ({(dornCanonicalLaw d, (fun _ => 0), (fun _ => 0), (fun _ => 1 / 2))} : Set (DornSourceLaw d))
n :
ℕ
c r :
ℝ
0 < c
0 < r
minimaxValueENNReal (dornPointwiseMissRisk (n := n) ({(dornCanonicalLaw d, (fun _ => 0), (fun _ => 0), (fun _ => 1 / 2))} : Set (DornSourceLaw d)) (fun _ => 0) c r)
= 0
Proof (Lean source)
lemma arbitraryFamily_lower_obstruction (d : ℕ) (hd : 1 ≤ d)
(β γ L₀ : ℝ) (hβ : 0 < β) (hγ : 1 < γ) (hL₀ : 0 < L₀) :
(∀ (𝔓 : Set (DornSourceLaw d)) (μref : (Fin d → ℝ) → ℝ)
(x₀ : Fin d → ℝ), 𝔓.Nonempty →
(∀ z ∈ 𝔓, ∀ x, z.2.1 x = μref x) →
∀ (n : ℕ) (c r : ℝ), 0 < c → 0 < r →
minimaxValueENNReal
(dornPointwiseMissRisk (n := n) 𝔓 x₀ c r) = 0) ∧
(∃ q M C : ℝ, 3 < q ∧ 0 < M ∧ 0 < C ∧
∃ hP : IsProbabilityMeasure (dornCanonicalLaw d),
@DornRemainingModel d β q M 1 C L₀ γ (dornCanonicalLaw d) hP
(fun _ => 0) (fun _ => 0) (fun _ => 1 / 2) ∧
DornA3OnOriginalClass
({(dornCanonicalLaw d, (fun _ => 0), (fun _ => 0),
(fun _ => 1 / 2))} : Set (DornSourceLaw d)) ∧
∀ (n : ℕ) (c r : ℝ), 0 < c → 0 < r →
minimaxValueENNReal
(dornPointwiseMissRisk (n := n)
({(dornCanonicalLaw d, (fun _ => 0), (fun _ => 0),
(fun _ => 1 / 2))} : Set (DornSourceLaw d))
(fun _ => 0) c r) = 0) := by
constructor
· intro 𝔓 μref x₀ _ hμ n c r hc hr
exact knownRegression_minimax_miss_zero 𝔓 μref x₀ c r hc hr hμ
· let P := dornCanonicalLaw d
have hP : IsProbabilityMeasure P := by
have hvol : volume (dornCube d) = ofReal ((2 : ℝ) ^ d) := by
rw [dornCube, Set.pi_univ_Icc, Real.volume_Icc_pi]
simp [Finset.prod_const, ENNReal.ofReal_pow]
norm_num
have hcube : IsProbabilityMeasure
((ofReal ((2 : ℝ) ^ d))⁻¹ • volume.restrict (dornCube d)) := by
apply isProbabilityMeasure_iff.mpr
simp only [Measure.smul_apply, Measure.restrict_apply, MeasurableSet.univ,
Set.univ_inter, hvol, smul_eq_mul]
exact ENNReal.inv_mul_cancel (by positivity) (by simp)
haveI : IsProbabilityMeasure
((ofReal ((2 : ℝ) ^ d))⁻¹ • volume.restrict (dornCube d)) := hcube
have hbern : IsProbabilityMeasure (realBernoulli (1 / 2)) := by
apply isProbabilityMeasure_iff.mpr
simp [realBernoulli]
rw [show (1 : ℝ) - 2⁻¹ = 1 / 2 by norm_num]
rw [ENNReal.ofReal_div_of_pos (by norm_num : (0 : ℝ) < 2)]
norm_num only [ENNReal.ofReal_one, ENNReal.ofReal_ofNat]
rw [one_div]
calc
(2 : ℝ≥0∞)⁻¹ + 2⁻¹ = (2 : ℝ≥0∞)⁻¹ * 2 := by ring
_ = 1 := ENNReal.inv_mul_cancel (by norm_num) (by norm_num)
haveI : IsProbabilityMeasure (realBernoulli (1 / 2)) := hbern
have hgauss : IsProbabilityMeasure (gaussianReal 0 ⟨1, by norm_num⟩) := by
exact instIsProbabilityMeasureGaussianReal 0 ⟨1, by norm_num⟩
haveI : IsProbabilityMeasure (gaussianReal 0 ⟨1, by norm_num⟩) := hgauss
have hinner : IsProbabilityMeasure
((realBernoulli (1 / 2)).prod (gaussianReal 0 ⟨1, by norm_num⟩)) := by
exact Measure.prod.instIsProbabilityMeasure _ _
haveI : IsProbabilityMeasure
((realBernoulli (1 / 2)).prod (gaussianReal 0 ⟨1, by norm_num⟩)) := hinner
dsimp [P, dornCanonicalLaw]
infer_instance
letI : IsProbabilityMeasure P := hP
let μy : Measure ℝ := gaussianReal 0 ⟨1, by norm_num⟩
haveI : IsProbabilityMeasure μy :=
instIsProbabilityMeasureGaussianReal 0 ⟨1, by norm_num⟩
have hInt : Integrable (fun y : ℝ => |y| ^ 4) μy := by
have hm := ProbabilityTheory.memLp_id_gaussianReal' (μ := (0 : ℝ))
(v := ⟨1, by norm_num⟩) 4 (by norm_num)
simpa [μy, Real.norm_eq_abs] using hm.integrable_norm_pow' (p := 4)
let M₀ : ℝ := max 1 |∫ y : ℝ, |y| ^ 4 ∂μy|
have hM₀ : 1 ≤ M₀ := le_max_left _ _
have hMomentBound : (∫ y : ℝ, |y| ^ 4 ∂μy) ≤ M₀ ^ 4 := by
have hle : (∫ y : ℝ, |y| ^ 4 ∂μy) ≤ M₀ :=
(le_abs_self _).trans (le_max_right _ _)
have hpow : M₀ ≤ M₀ ^ 4 := by
calc
M₀ = M₀ ^ 1 := by ring
_ ≤ M₀ ^ 4 := by
simpa [Real.rpow_natCast] using
(Real.rpow_le_rpow_of_exponent_le hM₀ (show (1 : ℝ) ≤ 4 by norm_num))
exact hle.trans hpow
have hWitness : ∃ q M C : ℝ, 3 < q ∧ 0 < M ∧ 0 < C ∧
DornRemainingModel d β q M 1 C L₀ γ P
(fun _ => 0) (fun _ => 0) (fun _ => 1 / 2) := by
refine ⟨4, M₀, 2 ^ (γ - 1), by norm_num, by linarith,
Real.rpow_pos_of_pos (by norm_num) _, ?_⟩
have hSelected : DornSelectedPropensity (dornCube d) P (fun _ => 1 / 2) := by
refine ⟨hP, measurable_const, ?_, ?_⟩
· intro x hx
norm_num
· filter_upwards [dornCanonicalLaw_propensity_ae d] with x hx
exact ⟨hx.symm, by norm_num, by norm_num⟩
refine ⟨?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩
· change covariateLaw P = _
simpa [P] using dornCanonicalLaw_covariate d
· exact hSelected.toA12
· filter_upwards [dornCanonicalLaw_armKernel_ae d] with x hx
rw [show armKernel P x true = μy by simpa [P, μy] using hx true]
exact ⟨by simpa [Real.rpow_natCast] using hInt, by simpa [Real.rpow_natCast] using hMomentBound⟩
· simpa [P] using dornCanonicalLaw_zeroVariance d M₀ (by linarith)
· exact dornZero_topDerivativeSeminorm d β L₀ hL₀.le
· exact dornZero_topDerivativeSeminorm d β L₀ hL₀.le
· refine ⟨?_, ?_⟩
· rcases hSelected with ⟨_, _, _, hae⟩
filter_upwards [hae] with x hx
exact hx.1
· haveI : IsProbabilityMeasure (covariateLaw P) := by
unfold covariateLaw
exact Measure.isProbabilityMeasure_map measurable_fst.aemeasurable
exact constantHalf_globalTailInequality (covariateLaw P) γ hγ
· filter_upwards [dornCanonicalLaw_armKernel_ae d] with x hx
constructor <;> simpa [P] using hx _
obtain ⟨q, M, C, hq, hM, hC, hModel⟩ := hWitness
have hA3 : DornA3OnOriginalClass
({(P, (fun _ => 0), (fun _ => 0),
(fun _ => 1 / 2))} : Set (DornSourceLaw d)) := by
change DornA3 (dornCube d)
({(P, (fun _ => 0), (fun _ => 0),
(fun _ => 1 / 2))} : Set (DornSourceLaw d))
refine ⟨Set.singleton_nonempty _, ?_, ?_, 1, 1 / 2, 1,
by norm_num, by norm_num, by norm_num, ?_⟩
· intro z hz
have hz' : z = (P, (fun _ => 0), (fun _ => 0),
(fun _ => 1 / 2)) := Set.mem_singleton_iff.mp hz
subst z
rw [covariateLaw, hModel.1]
have hvol : volume (dornCube d) = ofReal ((2 : ℝ) ^ d) := by
rw [dornCube, Set.pi_univ_Icc, Real.volume_Icc_pi]
simp [Finset.prod_const, ENNReal.ofReal_pow]
norm_num
have hcubeMeas : MeasurableSet (dornCube d) := by
unfold dornCube
exact MeasurableSet.univ_pi (fun _ => measurableSet_Icc)
rw [Measure.smul_apply, Measure.restrict_apply' hcubeMeas,
Set.inter_self, hvol]
simp only [smul_eq_mul]
exact ENNReal.inv_mul_cancel (by positivity) (by simp)
· intro z hz
have hz' : z = (P, (fun _ => 0), (fun _ => 0),
(fun _ => 1 / 2)) := Set.mem_singleton_iff.mp hz
subst z
refine ⟨hP, measurable_const, ?_, ?_⟩
· intro x hx
norm_num
· obtain ⟨_, _, hae⟩ := hModel.2.1
exact hae
· intro z hz x₀ hx₀ h hh
have hz' : z = (P, (fun _ => 0), (fun _ => 0),
(fun _ => 1 / 2)) := Set.mem_singleton_iff.mp hz
subst z
have hball : (ball2 x₀ h ∩ dornCube d).Nonempty := by
refine ⟨x₀, ?_, hx₀⟩
simp [ball2, sq_nonneg]
have hsup : sSup ((fun _ : Fin d → ℝ => (1 / 2 : ℝ)) ''
(ball2 x₀ h ∩ dornCube d)) = 1 / 2 := by
rw [hball.image_const]
simp
have hmeasure : 0 < (covariateLaw P).real (ball2 x₀ h) := by
have hvol := dornCube_ball2_volume_pos x₀ hx₀ hh.1
have hmass : 0 < (covariateLaw P) (ball2 x₀ h) := by
rw [covariateLaw, hModel.1]
have hcubeMeas : MeasurableSet (dornCube d) := by
unfold dornCube
exact MeasurableSet.univ_pi (fun _ => measurableSet_Icc)
rw [Measure.smul_apply, Measure.restrict_apply' hcubeMeas]
exact ENNReal.mul_pos (by simp) (ne_of_gt hvol)
haveI : IsProbabilityMeasure (covariateLaw P) := by
unfold covariateLaw
exact Measure.isProbabilityMeasure_map measurable_fst.aemeasurable
have hne : (covariateLaw P).real (ball2 x₀ h) ≠ 0 :=
(measureReal_ne_zero_iff (measure_ne_top (covariateLaw P) _)).mpr
(ne_of_gt hmass)
exact lt_of_le_of_ne (measureReal_nonneg) (Ne.symm hne)
simp only [hsup, one_mul]
have hevent : {x | (1 / 2 : ℝ) ≥ 1 / 2} ∩ ball2 x₀ h =
ball2 x₀ h := by simp
rw [hevent, div_self (ne_of_gt hmeasure)]
norm_num
refine ⟨q, M, C, hq, hM, hC, hP, hModel, hA3, ?_⟩
intro n c r hc hr
apply knownRegression_minimax_miss_zero
({(P, (fun _ => 0), (fun _ => 0),
(fun _ => 1 / 2))} : Set (DornSourceLaw d))
(fun _ => 0) (fun _ => 0) c r hc hr
intro z hz x
have hz' : z = (P, (fun _ => 0), (fun _ => 0),
(fun _ => 1 / 2)) := Set.mem_singleton_iff.mp hz
subst z
rfl
theorem dornTheorem1_scope
Dorn's Theorem 1(ii) gives a no-log L∞(P_X) upper rate after the source family, all common constants including gamma, and the selected propensity representatives have been fixed. The source comparison is supplied by the cited gate, including measurability of every displayed L∞(P_X) exceedance event and hence its ordinary probability. The second conclusion gives the known-regression obstruction to the printed lower clause, including its Gaussian witness. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
1 ≤ d
β q M σ C L₀ :
ℝ
hβ :
0 < β
hq :
3 < q
hM :
0 < M
hσ :
0 < σ
hC :
0 < C
hL₀ :
0 < L₀
_of_gate :
conclusion
conclusion 1
∃ T : ∀ n : ℕ
if
(Fin n → Obs d)
and
(Fin d → ℝ)
and
ℝ, (∀ (n : ℕ) (z : {z : DornSourceLaw d // z ∈ 𝔓}) (R : ℝ), MeasurableSet {ω | ofReal (R * (n : ℝ) ^ (-β / (2 * β + effectiveDimension d γ))) < eLpNorm (fun x => T n ω x - z.1.2.1 x) ⊤ (covariateLaw z.1.1)}) ∧
(∀ n x, Measurable (fun ω => T n ω x)) ∧
∀ ε : ℝ, 0 < ε
then
∃ R : ℝ,
0 < R ∧
limsup (fun n : ℕ => ⨆ (z : {z : DornSourceLaw d // z ∈ 𝔓}), (Measure.pi (fun _ : Fin n => z.1.1)).real {ω | ofReal (R * (n : ℝ) ^ (-β / (2 * β + effectiveDimension d γ))) < eLpNorm (fun x => T n ω x - z.1.2.1 x) ⊤ (covariateLaw z.1.1)}) atTop
≤ ε
conclusion 2
γ :
ℝ
1 < γ
𝔓 :
Set (DornSourceLaw d)
μref :
(Fin d → ℝ) → ℝ
x₀ :
Fin d → ℝ
𝔓.Nonempty
(∀ z ∈ 𝔓, ∀ x, z.2.1 x = μref x)
n :
ℕ
c r :
ℝ
0 < c
0 < r
minimaxValueENNReal (dornPointwiseMissRisk (n := n) 𝔓 x₀ c r) = 0
∃ q' M' C' : ℝ,
3 < q'
0 < M'
0 < C'
∃ hP : IsProbabilityMeasure (dornCanonicalLaw d),
@DornRemainingModel d β q' M' 1 C' L₀ γ (dornCanonicalLaw d) hP (fun _ => 0) (fun _ => 0) (fun _ => 1 / 2)
DornA3OnOriginalClass ({(dornCanonicalLaw d, (fun _ => 0), (fun _ => 0), (fun _ => 1 / 2))} : Set (DornSourceLaw d))
n :
ℕ
c r :
ℝ
0 < c
0 < r
minimaxValueENNReal (dornPointwiseMissRisk (n := n) ({(dornCanonicalLaw d, (fun _ => 0), (fun _ => 0), (fun _ => 1 / 2))} : Set (DornSourceLaw d)) (fun _ => 0) c r)
= 0
Proof (Lean source)
lemma dornTheorem1_scope (d : ℕ) (hd : 1 ≤ d)
(β q M σ C L₀ : ℝ) (hβ : 0 < β) (hq : 3 < q)
(hM : 0 < M) (hσ : 0 < σ) (hC : 0 < C) (hL₀ : 0 < L₀)
(_of_gate : DornSourceRateTranscription) :
(∀ (γ : ℝ) (𝔓 : Set (DornSourceLaw d)),
1 < γ → DornA12Family d β q M σ C L₀ γ 𝔓 →
DornA3OnOriginalClass 𝔓 →
∃ T : ∀ n : ℕ, (Fin n → Obs d) → (Fin d → ℝ) → ℝ,
(∀ (n : ℕ) (z : {z : DornSourceLaw d // z ∈ 𝔓}) (R : ℝ),
MeasurableSet
{ω | ofReal
(R * (n : ℝ) ^ (-β / (2 * β + effectiveDimension d γ))) <
eLpNorm (fun x => T n ω x - z.1.2.1 x) ⊤
(covariateLaw z.1.1)}) ∧
(∀ n x, Measurable (fun ω => T n ω x)) ∧
∀ ε : ℝ, 0 < ε → ∃ R : ℝ, 0 < R ∧
limsup (fun n : ℕ =>
⨆ (z : {z : DornSourceLaw d // z ∈ 𝔓}),
(Measure.pi (fun _ : Fin n => z.1.1)).real
{ω | ofReal
(R * (n : ℝ) ^ (-β / (2 * β + effectiveDimension d γ))) <
eLpNorm (fun x => T n ω x - z.1.2.1 x) ⊤
(covariateLaw z.1.1)}) atTop ≤ ε) ∧
(∀ γ : ℝ, 1 < γ →
(∀ (𝔓 : Set (DornSourceLaw d)) (μref : (Fin d → ℝ) → ℝ)
(x₀ : Fin d → ℝ), 𝔓.Nonempty →
(∀ z ∈ 𝔓, ∀ x, z.2.1 x = μref x) →
∀ (n : ℕ) (c r : ℝ), 0 < c → 0 < r →
minimaxValueENNReal
(dornPointwiseMissRisk (n := n) 𝔓 x₀ c r) = 0) ∧
(∃ q' M' C' : ℝ, 3 < q' ∧ 0 < M' ∧ 0 < C' ∧
∃ hP : IsProbabilityMeasure (dornCanonicalLaw d),
@DornRemainingModel d β q' M' 1 C' L₀ γ
(dornCanonicalLaw d) hP
(fun _ => 0) (fun _ => 0) (fun _ => 1 / 2) ∧
DornA3OnOriginalClass
({(dornCanonicalLaw d, (fun _ => 0), (fun _ => 0),
(fun _ => 1 / 2))} : Set (DornSourceLaw d)) ∧
∀ (n : ℕ) (c r : ℝ), 0 < c → 0 < r →
minimaxValueENNReal
(dornPointwiseMissRisk (n := n)
({(dornCanonicalLaw d, (fun _ => 0), (fun _ => 0),
(fun _ => 1 / 2))} : Set (DornSourceLaw d))
(fun _ => 0) c r) = 0)) := by
constructor
· intro γ 𝔓 hγ hA12 hA3
exact _of_gate d β q M σ C L₀ γ 𝔓 hd hβ hq hM hσ hC hL₀
hγ hA12 hA3
· intro γ hγ
exact arbitraryFamily_lower_obstruction d hd β γ L₀ hβ hγ hL₀
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.HolderCompletion 1 declarations The source class supplies only an intrinsic top-order seminorm.
Fixed-cube completion of Dorn's Hölder seminorm
The source class supplies only an intrinsic top-order seminorm. The completion and affine transport are performed on the closed cubes using within-cube jets.
theorem fixedCube_holder_completion
A response sup bound and Dorn's top-derivative seminorm control the full intrinsic Hölder radius after affine scaling. The constant depends on dimension and smoothness alone. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
1 ≤ d
β :
ℝ
hβ :
1 < β
conclusion
∃ K : ℝ,
0 < K
conclusion 1
M₀ L₀ :
ℝ
u :
(Fin d → ℝ) → ℝ
0 ≤ M₀
0 ≤ L₀
DornTopDerivativeSeminorm β L₀ u
(∀ x ∈ dornCube d, |u x| ≤ M₀)
HolderBall β (K * (M₀ + L₀)) (fun x : Fin d → ℝ => u (fun i => 2 * x i - 1))
Proof (Lean source)
lemma fixedCube_holder_completion (d : ℕ) (hd : 1 ≤ d) (β : ℝ) (hβ : 1 < β) :
∃ K : ℝ, 0 < K ∧
∀ (M₀ L₀ : ℝ) (u : (Fin d → ℝ) → ℝ),
0 ≤ M₀ → 0 ≤ L₀ →
DornTopDerivativeSeminorm β L₀ u →
(∀ x ∈ dornCube d, |u x| ≤ M₀) →
HolderBall β (K * (M₀ + L₀))
(fun x : Fin d → ℝ => u (fun i => 2 * x i - 1)) := by
let m : ℕ := polynomialDegree β
let s : ℝ := β - (m : ℝ)
have hs : 0 < s := by
dsimp [s, m, polynomialDegree]
have hceil : 1 ≤ ⌈β⌉₊ := Nat.one_le_ceil_iff.mpr (by linarith)
rw [Nat.cast_sub hceil, Nat.cast_one]
linarith [Nat.ceil_lt_add_one (by linarith : 0 ≤ β)]
have hs1 : s ≤ 1 := by
dsimp [s, m, polynomialDegree]
have hceil : 1 ≤ ⌈β⌉₊ := Nat.one_le_ceil_iff.mpr (by linarith)
rw [Nat.cast_sub hceil, Nat.cast_one]
linarith [Nat.le_ceil β]
obtain ⟨K₀, hK₀, hcomplete⟩ := cube_holder_completion d m s hs
obtain ⟨K₁, hK₁, haffine⟩ := cube_holder_affine_transport d m s hs1
let C : ℝ := (sqrt d) ^ s
have hC : 0 < C := by
dsimp [C]
exact Real.rpow_pos_of_pos (Real.sqrt_pos.2 (by exact_mod_cast hd)) _
let K : ℝ := K₁ * K₀ * (C + 1)
have hK : 0 < K := by dsimp [K]; positivity
refine ⟨K, hK, ?_⟩
intro M₀ L₀ u hM hL hu hbound
have hcube : cube d =
dornCube d := by
ext x
change (∀ i, x i ∈ Icc (-1 : ℝ) 1) ↔
x ∈ Set.univ.pi (fun _ => Icc (-1 : ℝ) 1)
exact ⟨fun hx => Set.mem_univ_pi.mpr hx, fun hx => Set.mem_univ_pi.mp hx⟩
have hunit : unitCube d = cube d := rfl
have htop : TopHolderOn (cube d)
m s (C * L₀) u := by
intro f x hx y hy
rw [hcube]
let α : Fin d → ℕ := fun i => coordCount f i
have hsum : (∑ i, α i) = m := sum_coordCount f
let P : ℕ → Prop := fun j =>
∀ g : Fin j → Fin d,
(∀ i, (Finset.univ.filter (fun k => g k = i)).card = α i) →
|coordJetOn (dornCube d) j u g x -
coordJetOn (dornCube d) j u g y| ≤
L₀ * (sqrt (∑ i, (x i - y i) ^ 2)) ^ s
have hP : P (∑ i, α i) := by
intro g hg
have hm : (∑ i, α i) = polynomialDegree β := hsum
have h := hu.2 α hm g hg x (hcube ▸ hx) y (hcube ▸ hy)
simpa only [s, m] using h
have hsource : |coordJetOn (dornCube d) m u f x -
coordJetOn (dornCube d) m u f y| ≤
L₀ * (sqrt (∑ i, (x i - y i) ^ 2)) ^ s :=
(hsum ▸ hP) f (by intro i; rfl)
have hsq : (∑ i : Fin d, (x i - y i) ^ 2) ≤
(d : ℝ) * ‖x - y‖ ^ 2 := by
calc
_ ≤ ∑ _i : Fin d, ‖x - y‖ ^ 2 := by
apply Finset.sum_le_sum
intro i _
have hi := norm_le_pi_norm (x - y) i
simpa only [Pi.sub_apply, Real.norm_eq_abs, sq_abs] using
((sq_le_sq₀ (abs_nonneg _) (norm_nonneg _)).2 hi)
_ = (d : ℝ) * ‖x - y‖ ^ 2 := by simp
have hdist : sqrt (∑ i : Fin d, (x i - y i) ^ 2) ≤
sqrt d * ‖x - y‖ := by
calc
_ ≤ sqrt ((d : ℝ) * ‖x - y‖ ^ 2) := Real.sqrt_le_sqrt hsq
_ = sqrt d * ‖x - y‖ := by
rw [Real.sqrt_mul (by positivity : (0 : ℝ) ≤ d), Real.sqrt_sq_eq_abs]
simp
have hpow : (sqrt (∑ i : Fin d, (x i - y i) ^ 2)) ^ s ≤
C * ‖x - y‖ ^ s := by
calc
_ ≤ (sqrt d * ‖x - y‖) ^ s :=
Real.rpow_le_rpow (Real.sqrt_nonneg _) hdist hs.le
_ = _ := by rw [Real.mul_rpow (Real.sqrt_nonneg _) (norm_nonneg _)]
exact hsource.trans (by
calc
L₀ * (sqrt (∑ i : Fin d, (x i - y i) ^ 2)) ^ s ≤
L₀ * (C * ‖x - y‖ ^ s) := mul_le_mul_of_nonneg_left hpow hL
_ = C * L₀ * ‖x - y‖ ^ s := by ring)
have hball : CubeHolderBall d m s (K₀ * (M₀ + C * L₀)) u := by
apply hcomplete u M₀ (C * L₀) hM (mul_nonneg hC.le hL)
· simpa only [m, hcube] using hu.1
· simpa only [hcube] using hbound
· exact htop
have htransport := haffine u (K₀ * (M₀ + C * L₀))
(mul_nonneg hK₀.le (add_nonneg hM (mul_nonneg hC.le hL))) hball
have hscale : M₀ + C * L₀ ≤ (C + 1) * (M₀ + L₀) := by
nlinarith [mul_nonneg hC.le hM]
have hR : K₁ * (K₀ * (M₀ + C * L₀)) ≤ K * (M₀ + L₀) := by
dsimp [K]
nlinarith [mul_nonneg hK₀.le (sub_nonneg.mpr hscale),
mul_nonneg hK₁.le (mul_nonneg hK₀.le (sub_nonneg.mpr hscale))]
have horder : ((polynomialDegree β : ℕ) : WithTop ℕ∞) =
(⌈β⌉₊ : WithTop ℕ∞) - 1 := by
simp only [polynomialDegree]
norm_cast
refine ⟨?_, ?_, ?_⟩
· rw [← horder]
have hreg : ContDiffOn ℝ (polynomialDegree β)
(fun x => u (cubeAffine x))
(cube d) := htransport.regularity
have hfun : (fun x : Fin d → ℝ =>
u (cubeAffine x)) =
(fun x : Fin d → ℝ => u (fun i => 2 * x i - 1)) := by
rfl
rw [hfun] at hreg
exact hreg
· intro j hj f x hx
exact (htransport.derivBound j hj f x (hunit.symm ▸ hx)).trans hR
· intro f x hx y hy
exact (htransport.modulus f x (hunit.symm ▸ hx) y (hunit.symm ▸ hy)).trans
(mul_le_mul_of_nonneg_right hR (Real.rpow_nonneg (norm_nonneg _) _))
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.Identification 6 declarations Identification of the Hölder treated response
Identification of the Hölder treated response
A global lower-tail bound excludes a zero propensity set. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ :
ℝ
Pc :
μ₁ e :
(Fin d → ℝ) → ℝ
hγ :
1 < γ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ₁ e
conclusion
(covariateLaw (Pc.map observed)) {x | e x = 0} = 0
Proof (Lean source)
lemma zeroPropensity_null {d : ℕ} (β B L C c_f γ : ℝ)
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ₁ e : (Fin d → ℝ) → ℝ)
(hγ : 1 < γ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ₁ e) :
(covariateLaw (Pc.map observed)) {x | e x = 0} = 0 := by
have htail := hmodel.tail.2 0 (by simp : (0 : ℝ) ∈ Icc 0 1)
have hzero : C * (0 : ℝ) ^ (γ - 1) = 0 := by
rw [Real.zero_rpow (ne_of_gt (sub_pos.mpr hγ)), mul_zero]
rw [hzero] at htail
have hnonneg : 0 ≤ (covariateLaw (Pc.map observed)).real {x | e x ≤ 0} :=
ENNReal.toReal_nonneg
have heq : (covariateLaw (Pc.map observed)).real {x | e x ≤ 0} = 0 :=
le_antisymm htail hnonneg
apply le_antisymm _ (zero_le)
calc
(covariateLaw (Pc.map observed)) {x | e x = 0} ≤
(covariateLaw (Pc.map observed)) {x | e x ≤ 0} :=
measure_mono (by intro x hx; exact le_of_eq hx)
_ = 0 := by
letI : IsProbabilityMeasure (Pc.map observed) :=
Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)
letI : IsProbabilityMeasure (covariateLaw (Pc.map observed)) := by
unfold covariateLaw
exact Measure.isProbabilityMeasure_map (by fun_prop)
have hfin : (covariateLaw (Pc.map observed)) {x | e x ≤ 0} ≠ ⊤ :=
measure_ne_top _ _
exact (ENNReal.toReal_eq_zero_iff _).mp heq |>.resolve_right hfin
The global tail condition makes the propensity strictly positive almost everywhere under the covariate law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ :
ℝ
Pc :
μ₁ e :
(Fin d → ℝ) → ℝ
hγ :
1 < γ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ₁ e
conclusion
∀ᵐ x ∂covariateLaw (Pc.map observed), 0 < e x
Proof (Lean source)
lemma positivePropensity_ae {d : ℕ} (β B L C c_f γ : ℝ)
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ₁ e : (Fin d → ℝ) → ℝ)
(hγ : 1 < γ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ₁ e) :
∀ᵐ x ∂covariateLaw (Pc.map observed), 0 < e x := by
have hzero := zeroPropensity_null β B L C c_f γ Pc μ₁ e hγ hmodel
have hne : ∀ᵐ x ∂covariateLaw (Pc.map observed), e x ≠ 0 :=
(measure_eq_zero_iff_ae_notMem.mp hzero)
filter_upwards [hne, hmodel.tail.1] with x hx he
have hnonneg : 0 ≤ e x := by rw [he]; exact ENNReal.toReal_nonneg
exact lt_of_le_of_ne hnonneg (Ne.symm hx)
A positive density lower bound gives the reverse absolute continuity needed to transfer almost-everywhere identification to volume on the cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
P :
Measure (Obs d)
c_f :
ℝ
hc :
0 < c_f
hdensity :
hAC :
covariateLaw P ≪ volume.restrict (cube d)
conclusion
volume.restrict (cube d) ≪ covariateLaw P
Proof (Lean source)
lemma cubeVolume_ac_covariateLaw {d : ℕ} (P : Measure (Obs d))
[IsProbabilityMeasure P] (c_f : ℝ) (hc : 0 < c_f)
(hdensity : CovariateDensityLowerBound P c_f)
(hAC : covariateLaw P ≪ volume.restrict (cube d)) :
volume.restrict (cube d) ≪ covariateLaw P := by
let ν := volume.restrict (cube d)
let μ := covariateLaw P
haveI : IsProbabilityMeasure μ := by
dsimp [μ, covariateLaw]
exact Measure.isProbabilityMeasure_map (by fun_prop)
have hvol : volume (cube d) = 1 := by
have hcube : cube d = Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hcube, Real.volume_Icc_pi]
simp
haveI : IsFiniteMeasure ν := by
rw [isFiniteMeasure_restrict]
simp [hvol]
have hpos : ∀ᵐ x ∂ν, μ.rnDeriv ν x ≠ 0 := by
filter_upwards [hdensity] with x hx
have hxpos : 0 < (μ.rnDeriv ν x).toReal := lt_of_lt_of_le hc hx
exact ne_of_gt (ENNReal.toReal_pos_iff.mp hxpos).1
have hwith : ν ≪ ν.withDensity (μ.rnDeriv ν) :=
withDensity_absolutelyContinuous' (μ.measurable_rnDeriv ν).aemeasurable hpos
have heq : ν.withDensity (μ.rnDeriv ν) = μ :=
Measure.withDensity_rnDeriv_eq μ ν hAC
simpa only [heq, ν, μ] using hwith
Observing a causal completion preserves its covariate marginal. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma covariateLaw_observed {d : ℕ} (Pc : Measure (Completion d)) :
covariateLaw (Pc.map observed) = Pc.map fst := by
rw [covariateLaw, Measure.map_map (by fun_prop) (by unfold observed; fun_prop)]
rfl
theorem ae_slice_of_ae_compProd_of_pos
An almost-everywhere property under a composition product holds at a fixed fiber value whenever that value has positive conditional mass almost everywhere. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
X A :
KA :
Kernel X A
a :
A
p :
X × A → Prop
hp :
∀ᵐ xa ∂PX ⊗ₘ KA, p xa
hpos :
∀ᵐ x ∂PX, 0 < KA x {a}
conclusion
∀ᵐ x ∂PX, p (x, a)
Proof (Lean source)
lemma ae_slice_of_ae_compProd_of_pos
{X A : Type*} [MeasurableSpace X] [MeasurableSpace A]
(PX : Measure X) [SFinite PX] (KA : Kernel X A) [IsSFiniteKernel KA]
(a : A) (p : X × A → Prop)
(hp : ∀ᵐ xa ∂PX ⊗ₘ KA, p xa)
(hpos : ∀ᵐ x ∂PX, 0 < KA x {a}) :
∀ᵐ x ∂PX, p (x, a) := by
have hs := Measure.ae_ae_of_ae_compProd hp
filter_upwards [hs, hpos] with x hx hxpos
by_contra hxa
have hzero : KA x {b | ¬p (x, b)} = 0 := ae_iff.mp hx
have hle : KA x {a} ≤ KA x {b | ¬p (x, b)} := by
apply measure_mono
intro b hb
rcases hb with rfl
exact hxa
rw [hzero] at hle
exact (not_lt_of_ge hle) hxpos
theorem mu1_eq_treatedRegression
Consistency, exchangeability, and global positivity identify the treated conditional regression almost everywhere; continuity and the density lower bound fix its representative at all cube points. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ :
ℝ
Pc :
μ₁ e :
(Fin d → ℝ) → ℝ
hparams :
ModelParameterDomain d β B L C c_f
hγ :
1 < γ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ₁ e
conclusion
conclusion 1
(covariateLaw (Pc.map observed)) {x | e x = 0} = 0
conclusion 2
∀ᵐ x ∂covariateLaw (Pc.map observed), μ₁ x = treatedRegression (Pc.map observed) x
conclusion 3
g :
(Fin d → ℝ) → ℝ
ContinuousOn g (cube d)
(∀ᵐ x ∂covariateLaw (Pc.map observed), g x = treatedRegression (Pc.map observed) x)
x ∈ cube d :
g x = μ₁ x
Proof (Lean source)
lemma mu1_eq_treatedRegression {d : ℕ} (β B L C c_f γ : ℝ)
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ₁ e : (Fin d → ℝ) → ℝ)
(hparams : ModelParameterDomain d β B L C c_f) (hγ : 1 < γ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ₁ e) :
(covariateLaw (Pc.map observed)) {x | e x = 0} = 0 ∧
(∀ᵐ x ∂covariateLaw (Pc.map observed),
μ₁ x = treatedRegression (Pc.map observed) x) ∧
(∀ g : (Fin d → ℝ) → ℝ, ContinuousOn g (cube d) →
(∀ᵐ x ∂covariateLaw (Pc.map observed),
g x = treatedRegression (Pc.map observed) x) →
∀ x ∈ cube d, g x = μ₁ x) := by
have hident : ∀ᵐ x ∂covariateLaw (Pc.map observed),
μ₁ x = treatedRegression (Pc.map observed) x := by
let Xv : Completion d → (Fin d → ℝ) := fun ω => ω.1
let Av : Completion d → Bool := fun ω => ω.2.1
let Y₁ : Completion d → ℝ := fun ω => ω.2.2.2.1
let Yv : Completion d → ℝ := fun ω => ω.2.2.2.2
let T : Completion d → (Fin d → ℝ) × Bool := fun ω => (Xv ω, Av ω)
have hX : Measurable Xv := by fun_prop
have hA : Measurable Av := by fun_prop
have hY₁ : Measurable Y₁ := by fun_prop
have hY : Measurable Yv := by fun_prop
have hcons : ∀ᵐ ω ∂Pc, Av ω = true → Yv ω = Y₁ ω := by
filter_upwards [hmodel.consistency] with ω hω hAt
simpa [Av, Yv, Y₁, hAt] using hω
have hfiber :=
Causalean.Mathlib.Probability.Kernel.CondDistribFiber.condDistrib_congr_on_conditioning_fiber
Pc Xv Av Yv Y₁ hX hA hY hY₁ true hcons
have hci : CondIndepFun (comap Xv inferInstance)
(comap_le hX) Av Y₁ Pc := by
have hc := hmodel.exchangeability.comp
(by fun_prop : Measurable (fun z : ℝ × ℝ => z.2))
(by fun_prop : Measurable (fun a : Bool => a))
simpa [Xv, Av, Y₁, Function.comp_def] using hc.symm
have hexch :
(ProbabilityTheory.condDistrib Y₁ T Pc : (Fin d → ℝ) × Bool → Measure ℝ) =ᵐ[Pc.map T]
(Kernel.prodMkRight Bool (ProbabilityTheory.condDistrib Y₁ Xv Pc) :
(Fin d → ℝ) × Bool → Measure ℝ) := by
exact (ProbabilityTheory.condIndepFun_iff_condDistrib_prod_ae_eq_prodMkRight
hY₁ hA hX).mp hci
have hpos : ∀ᵐ x ∂Pc.map Xv,
0 < ProbabilityTheory.condDistrib Av Xv Pc x {true} := by
have hepos := positivePropensity_ae β B L C c_f γ Pc μ₁ e hγ hmodel
have heprop := hmodel.tail.1
have hkmap := ProbabilityTheory.condDistrib_map
(ν := Pc) (f := observed)
(X := fun z : Obs d => z.1) (Y := fun z : Obs d => z.2.1)
(by fun_prop) (by fun_prop) (by unfold observed; fun_prop)
have hPX : Pc.map Xv = covariateLaw (Pc.map observed) := by
simpa [Xv] using (covariateLaw_observed Pc).symm
have hkmap' :
(ProbabilityTheory.condDistrib (fun z : Obs d => z.2.1) (fun z => z.1)
(Pc.map observed) : (Fin d → ℝ) → Measure Bool)
=ᵐ[covariateLaw (Pc.map observed)]
(ProbabilityTheory.condDistrib Av Xv Pc : (Fin d → ℝ) → Measure Bool) := by
rw [covariateLaw_observed Pc]
simpa [Xv, Av, observed, Function.comp_def] using hkmap
rw [hPX]
filter_upwards [hepos, heprop, hkmap'] with x heposx hepropx hkx
have hp : 0 < propensity (Pc.map observed) x := by
rw [← hepropx]
exact heposx
have hreal : 0 < ((ProbabilityTheory.condDistrib Av Xv Pc) x {true}).toReal := by
rw [← hkx]
simpa [propensity] using hp
exact (ENNReal.toReal_pos_iff.mp hreal).1
have hjoint : Pc.map Xv ⊗ₘ ProbabilityTheory.condDistrib Av Xv Pc = Pc.map T := by
simpa [T] using ProbabilityTheory.compProd_map_condDistrib
(μ := Pc) (X := Xv) hA.aemeasurable
have hfiber' : ∀ᵐ xa ∂Pc.map Xv ⊗ₘ ProbabilityTheory.condDistrib Av Xv Pc,
xa.2 = true → ProbabilityTheory.condDistrib Yv T Pc xa =
ProbabilityTheory.condDistrib Y₁ T Pc xa := by
rw [hjoint]
exact hfiber
have hexch' : ∀ᵐ xa ∂Pc.map Xv ⊗ₘ ProbabilityTheory.condDistrib Av Xv Pc,
ProbabilityTheory.condDistrib Y₁ T Pc xa =
(Kernel.prodMkRight Bool (ProbabilityTheory.condDistrib Y₁ Xv Pc)) xa := by
rw [hjoint]
exact hexch
have hslice : ∀ᵐ x ∂Pc.map Xv,
ProbabilityTheory.condDistrib Yv T Pc (x, true) =
ProbabilityTheory.condDistrib Y₁ Xv Pc x := by
have hboth := ae_slice_of_ae_compProd_of_pos
(Pc.map Xv) (ProbabilityTheory.condDistrib Av Xv Pc) true
(fun xa => (xa.2 = true → ProbabilityTheory.condDistrib Yv T Pc xa =
ProbabilityTheory.condDistrib Y₁ T Pc xa) ∧
ProbabilityTheory.condDistrib Y₁ T Pc xa =
(Kernel.prodMkRight Bool (ProbabilityTheory.condDistrib Y₁ Xv Pc)) xa)
(by
filter_upwards [hfiber', hexch'] with xa hf he
exact ⟨hf, he⟩) hpos
filter_upwards [hboth] with x hx
exact (hx.1 rfl).trans (hx.2.trans (by simp))
have hobmap := ProbabilityTheory.condDistrib_map
(ν := Pc) (f := observed)
(X := fun z : Obs d => (z.1, z.2.1)) (Y := fun z : Obs d => z.2.2)
(by fun_prop) (by fun_prop) (by unfold observed; fun_prop)
have hobmap' :
(ProbabilityTheory.condDistrib (fun z : Obs d => z.2.2)
(fun z => (z.1, z.2.1)) (Pc.map observed) :
(Fin d → ℝ) × Bool → Measure ℝ) =ᵐ[Pc.map T]
(ProbabilityTheory.condDistrib Yv T Pc : (Fin d → ℝ) × Bool → Measure ℝ) := by
simpa [T, Yv, observed, Function.comp_def] using hobmap
have hobmapComp : ∀ᵐ xa ∂Pc.map Xv ⊗ₘ ProbabilityTheory.condDistrib Av Xv Pc,
ProbabilityTheory.condDistrib (fun z : Obs d => z.2.2)
(fun z => (z.1, z.2.1)) (Pc.map observed) xa =
ProbabilityTheory.condDistrib Yv T Pc xa := by
rw [hjoint]
exact hobmap'
have hobSlice : ∀ᵐ x ∂Pc.map Xv,
treatedKernel (Pc.map observed) x = ProbabilityTheory.condDistrib Yv T Pc (x, true) := by
have hs := ae_slice_of_ae_compProd_of_pos
(Pc.map Xv) (ProbabilityTheory.condDistrib Av Xv Pc) true
(fun xa => ProbabilityTheory.condDistrib (fun z : Obs d => z.2.2)
(fun z => (z.1, z.2.1)) (Pc.map observed) xa =
ProbabilityTheory.condDistrib Yv T Pc xa) hobmapComp hpos
simpa [treatedKernel] using hs
have hkernel : ∀ᵐ x ∂Pc.map Xv,
treatedKernel (Pc.map observed) x = ProbabilityTheory.condDistrib Y₁ Xv Pc x := by
filter_upwards [hobSlice, hslice] with x ho hs
exact ho.trans hs
have hPX : Pc.map Xv = covariateLaw (Pc.map observed) := by
simpa [Xv] using (covariateLaw_observed Pc).symm
have houtcome : ∀ᵐ x ∂Pc.map Xv,
|treatedRegression (Pc.map observed) x| ≤ B ∧
∀ t : ℝ,
Integrable (fun y : ℝ => exp
(t * (y - treatedRegression (Pc.map observed) x)))
(ProbabilityTheory.condDistrib Y₁ Xv Pc x) ∧
mgf (fun y : ℝ => y - treatedRegression (Pc.map observed) x)
(ProbabilityTheory.condDistrib Y₁ Xv Pc x) t ≤
exp (B ^ 2 * t ^ 2 / 2) := by
have ho' : ∀ᵐ x ∂Pc.map Xv,
|treatedRegression (Pc.map observed) x| ≤ B ∧
∀ t : ℝ,
Integrable (fun y : ℝ => exp
(t * (y - treatedRegression (Pc.map observed) x)))
(treatedKernel (Pc.map observed) x) ∧
mgf
(fun y : ℝ => y - treatedRegression (Pc.map observed) x)
(treatedKernel (Pc.map observed) x) t ≤
exp (B ^ 2 * t ^ 2 / 2) := by
rw [hPX]
exact hmodel.outcome
filter_upwards [ho', hkernel] with x ho hk
exact ⟨ho.1, fun t => by simpa [← hk] using ho.2 t⟩
have hY₁fiber : ∀ᵐ x ∂Pc.map Xv,
Integrable (fun y : ℝ => y) (ProbabilityTheory.condDistrib Y₁ Xv Pc x) := by
filter_upwards [houtcome] with x ho
let m := treatedRegression (Pc.map observed) x
have hp : Integrable (fun y : ℝ => exp (y - m))
(ProbabilityTheory.condDistrib Y₁ Xv Pc x) := by
simpa [m] using (ho.2 1).1
have hn : Integrable (fun y : ℝ => exp (-(y - m)))
(ProbabilityTheory.condDistrib Y₁ Xv Pc x) := by
simpa [m] using (ho.2 (-1)).1
have hright : Integrable
(fun y : ℝ => exp (y - m) + exp (-(y - m)) + |m|)
(ProbabilityTheory.condDistrib Y₁ Xv Pc x) :=
(hp.add hn).add (integrable_const |m|)
apply Integrable.mono' hright (by fun_prop)
filter_upwards [] with y
rw [Real.norm_eq_abs]
calc
|y| = |(y - m) + m| := by ring_nf
_ ≤ |y - m| + |m| := abs_add_le _ _
_ ≤ (exp (y - m) + exp (-(y - m))) + |m| := by
gcongr
exact (le_add_of_nonneg_right zero_le_one).trans
((Real.add_one_le_exp _).trans (Real.exp_abs_le _))
have hY₁norm_bound : ∀ᵐ x ∂Pc.map Xv,
(∫ y : ℝ, ‖y‖ ∂ProbabilityTheory.condDistrib Y₁ Xv Pc x) ≤
2 * exp (B ^ 2 / 2) + B := by
filter_upwards [houtcome] with x ho
let m := treatedRegression (Pc.map observed) x
let K := ProbabilityTheory.condDistrib Y₁ Xv Pc x
have hp : Integrable (fun y : ℝ => exp (y - m)) K := by
simpa [m, K] using (ho.2 1).1
have hn : Integrable (fun y : ℝ => exp (-(y - m))) K := by
simpa [m, K] using (ho.2 (-1)).1
have hdom : ∀ y : ℝ, ‖y‖ ≤
exp (y - m) + exp (-(y - m)) + |m| := by
intro y
rw [Real.norm_eq_abs]
calc
|y| = |(y - m) + m| := by ring_nf
_ ≤ |y - m| + |m| := abs_add_le _ _
_ ≤ (exp (y - m) + exp (-(y - m))) + |m| := by
gcongr
exact (le_add_of_nonneg_right zero_le_one).trans
((Real.add_one_le_exp _).trans (Real.exp_abs_le _))
have hright : Integrable
(fun y : ℝ => exp (y - m) + exp (-(y - m)) + |m|) K :=
(hp.add hn).add (integrable_const |m|)
have hleft : Integrable (fun y : ℝ => ‖y‖) K := by
apply Integrable.mono' hright (by fun_prop)
filter_upwards [] with y
simpa using hdom y
have hi := integral_mono hleft hright hdom
calc
(∫ y : ℝ, ‖y‖ ∂K) ≤
∫ y : ℝ, (exp (y - m) + exp (-(y - m)) + |m|) ∂K := hi
_ = mgf (fun y : ℝ => y - m) K 1 +
mgf (fun y : ℝ => y - m) K (-1) + |m| := by
calc
(∫ y : ℝ, exp (y - m) + exp (-(y - m)) + |m| ∂K) =
(∫ y : ℝ, exp (y - m) + exp (-(y - m)) ∂K) +
∫ _ : ℝ, |m| ∂K :=
integral_add (hp.add hn) (integrable_const |m|)
_ = ((∫ y : ℝ, exp (y - m) ∂K) +
∫ y : ℝ, exp (-(y - m)) ∂K) + ∫ _ : ℝ, |m| ∂K := by
rw [integral_add hp hn]
_ = mgf (fun y : ℝ => y - m) K 1 +
mgf (fun y : ℝ => y - m) K (-1) + |m| := by
simp [mgf, m, K]
_ ≤ exp (B ^ 2 * (1 : ℝ) ^ 2 / 2) +
exp (B ^ 2 * (-1 : ℝ) ^ 2 / 2) + B := by
gcongr
· exact (ho.2 1).2
· exact (ho.2 (-1)).2
· exact ho.1
_ = 2 * exp (B ^ 2 / 2) + B := by ring_nf
have hnorm_meas : StronglyMeasurable
(fun x => ∫ y : ℝ, ‖y‖ ∂ProbabilityTheory.condDistrib Y₁ Xv Pc x) := by
fun_prop
have hnorm_int : Integrable
(fun x => ∫ y : ℝ, ‖y‖ ∂ProbabilityTheory.condDistrib Y₁ Xv Pc x)
(Pc.map Xv) := by
apply Integrable.mono' (integrable_const (2 * exp (B ^ 2 / 2) + B))
hnorm_meas.aestronglyMeasurable
filter_upwards [hY₁norm_bound] with x hx
rw [Real.norm_eq_abs, abs_of_nonneg]
· exact hx
· exact integral_nonneg (fun _ => norm_nonneg _)
have hcomp_int : Integrable (fun y : ℝ => y)
((ProbabilityTheory.condDistrib Y₁ Xv Pc) ∘ₘ (Pc.map Xv)) := by
rw [Measure.integrable_comp_iff (by fun_prop)]
exact ⟨hY₁fiber, hnorm_int⟩
have hY₁int : Integrable Y₁ Pc := by
have hmap : (ProbabilityTheory.condDistrib Y₁ Xv Pc) ∘ₘ (Pc.map Xv) = Pc.map Y₁ :=
ProbabilityTheory.condDistrib_comp_map hX.aemeasurable hY₁.aemeasurable
rw [hmap] at hcomp_int
exact (integrable_map_measure (by fun_prop) hY₁.aemeasurable).mp hcomp_int
let k : (Fin d → ℝ) → ℝ := fun x =>
∫ y : ℝ, y ∂ProbabilityTheory.condDistrib Y₁ Xv Pc x
have hkmeas : StronglyMeasurable k := by
-- … truncated; follow the source link for the rest …
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.LowerPair 60 declarations Bounded radial Bernoulli two-point experiment
Bounded radial Bernoulli two-point experiment
One coordinate factor of the explicit bump is Mathlib's standard smooth flat function composed with 1 - x². For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
x :
ℝ
conclusion
(if |x| < 1 then exp (1 - 1 / (1 - x ^ 2)) else 0)
= exp 1 * expNegInvGlue (1 - x ^ 2)
Proof (Lean source)
lemma smoothBump_factor_eq_expNegInvGlue (x : ℝ) :
(if |x| < 1 then exp (1 - 1 / (1 - x ^ 2)) else 0) =
exp 1 * expNegInvGlue (1 - x ^ 2) := by
by_cases hx : |x| < 1
· have hden : 0 < 1 - x ^ 2 := by
rw [sub_pos, sq_lt_one_iff_abs_lt_one]
exact hx
rw [if_pos hx]
simp only [expNegInvGlue, if_neg (not_le.mpr hden)]
rw [← Real.exp_add]
congr 1
field_simp
ring
· have hden : 1 - x ^ 2 ≤ 0 := by
rw [sub_nonpos]
simpa using (sq_le_sq₀ (by norm_num) (abs_nonneg x)).2 (le_of_not_gt hx)
rw [if_neg hx, expNegInvGlue.zero_of_nonpos hden, mul_zero]
The explicit finite product bump is infinitely differentiable. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma smoothBump_contDiff {d : ℕ} :
ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun z : Fin d → ℝ => smoothBump z) := by
have hfactor : ∀ i : Fin d, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun z : Fin d → ℝ =>
exp 1 * expNegInvGlue (1 - (z i) ^ 2)) := by
intro i
have hinner : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞)
(fun z : Fin d → ℝ => 1 - (z i) ^ 2) := by
fun_prop
exact contDiff_const.mul (expNegInvGlue.contDiff.comp hinner)
have hprod : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun z : Fin d → ℝ =>
∏ i : Fin d, exp 1 * expNegInvGlue (1 - (z i) ^ 2)) := by
fun_prop (disch := assumption)
simpa only [smoothBump, smoothBump_factor_eq_expNegInvGlue] using hprod
The product bump is a valid unit-scale Bernoulli perturbation. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma smoothBump_mem_Icc {d : ℕ} (z : Fin d → ℝ) :
smoothBump z ∈ Icc 0 1 := by
unfold smoothBump
let f : Fin d → ℝ := fun i =>
if |z i| < 1 then exp (1 - 1 / (1 - (z i) ^ 2)) else 0
have hf : ∀ i, 0 ≤ f i ∧ f i ≤ 1 := by
intro i
dsimp [f]
split_ifs with hi
· have hsq : 0 ≤ (z i) ^ 2 ∧ (z i) ^ 2 < 1 := by
constructor
· positivity
· exact (sq_lt_one_iff_abs_lt_one _).2 hi
have hden : 0 < 1 - (z i) ^ 2 := by linarith
have hden_le : 1 - (z i) ^ 2 ≤ 1 := by linarith
have hrecip : 1 ≤ 1 / (1 - (z i) ^ 2) :=
(le_div_iff₀ hden).2 (by linarith)
exact ⟨Real.exp_nonneg _, (Real.exp_le_one_iff).2 (by linarith)⟩
· exact ⟨le_refl 0, zero_le_one⟩
constructor
· exact Finset.prod_nonneg (by intro i hi; exact (hf i).1)
· exact Finset.prod_le_one (by intro i hi; exact (hf i).1)
(by intro i hi; exact (hf i).2)
A perturbation below the baseline keeps both response kernels Bernoulli. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
j :
x₀ :
Fin d → ℝ
β B L a h :
ℝ
hB :
0 < B
hL :
0 < L
hβ :
0 < β
ha :
0 ≤ a ∧ a ≤ baselineSuccess B L
hh :
0 ≤ h ∧ h ≤ 1
x :
Fin d → ℝ
conclusion
pairSuccess j x₀ β B L a h x ∈ Icc 0 1
Proof (Lean source)
lemma pairSuccess_mem_Icc_of_small {d : ℕ} (j : Bool)
(x₀ : Fin d → ℝ) (β B L a h : ℝ)
(hB : 0 < B) (hL : 0 < L) (hβ : 0 < β)
(ha : 0 ≤ a ∧ a ≤ baselineSuccess B L)
(hh : 0 ≤ h ∧ h ≤ 1) (x : Fin d → ℝ) :
pairSuccess j x₀ β B L a h x ∈ Icc 0 1 := by
have hbase : 0 ≤ baselineSuccess B L ∧ baselineSuccess B L ≤ 1 / 4 := by
unfold baselineSuccess
constructor
· apply le_min (by norm_num)
positivity
· exact min_le_left _ _
have hbump := smoothBump_mem_Icc (fun i => (x i - x₀ i) / h)
have hpow : 0 ≤ h ^ β ∧ h ^ β ≤ 1 :=
⟨Real.rpow_nonneg hh.1 _, Real.rpow_le_one hh.1 hh.2 hβ.le⟩
unfold pairSuccess
split_ifs
· constructor
· exact add_nonneg hbase.1 (mul_nonneg (mul_nonneg ha.1 hpow.1) hbump.1)
· have hprod : a * h ^ β * smoothBump (fun i => (x i - x₀ i) / h) ≤ a := by
calc
a * h ^ β * smoothBump (fun i => (x i - x₀ i) / h) ≤
a * 1 * smoothBump (fun i => (x i - x₀ i) / h) :=
mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left hpow.2 ha.1) hbump.1
_ ≤ a * 1 * 1 := mul_le_mul_of_nonneg_left hbump.2 (by simpa using ha.1)
_ = a := by ring
have hbase_le : baselineSuccess B L ≤ 1 / 4 := hbase.2
linarith
· simpa using (show baselineSuccess B L ∈ Icc (0 : ℝ) 1 from
⟨hbase.1, hbase.2.trans (by norm_num)⟩)
The radial CDF formula gives a valid treatment probability at every point. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma radialPropensity_mem_Icc {d : ℕ} (x₀ x : Fin d → ℝ)
(γ : ℝ) (hγ : 1 < γ) :
radialPropensity x₀ γ x ∈ Icc 0 1 := by
have hfactor : ∀ i : Fin d, 0 ≤
max 0 (min 1 (x₀ i + ‖x - x₀‖) - max 0 (x₀ i - ‖x - x₀‖)) ∧
max 0 (min 1 (x₀ i + ‖x - x₀‖) - max 0 (x₀ i - ‖x - x₀‖)) ≤ 1 := by
intro i
constructor
· exact le_max_left _ _
· apply max_le (by norm_num)
have hmin := min_le_left (1 : ℝ) (x₀ i + ‖x - x₀‖)
have hmax := le_max_left (0 : ℝ) (x₀ i - ‖x - x₀‖)
linarith
have hbase : 0 ≤ radialCDF x₀ ‖x - x₀‖ ∧
radialCDF x₀ ‖x - x₀‖ ≤ 1 := by
unfold radialCDF
exact ⟨Finset.prod_nonneg (by intro i hi; exact (hfactor i).1),
Finset.prod_le_one (by intro i hi; exact (hfactor i).1)
(by intro i hi; exact (hfactor i).2)⟩
unfold radialPropensity
constructor
· exact Real.rpow_nonneg hbase.1 _
· exact Real.rpow_le_one hbase.1 hbase.2 (by positivity)
The truncated radial CDF is bounded by the volume of a full cube of radius r, including when the centre is on the boundary. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma radialCDF_le_cube_radius_volume {d : ℕ} (x₀ : Fin d → ℝ)
(r : ℝ) (hr : 0 ≤ r) : radialCDF x₀ r ≤ (2 * r) ^ d := by
unfold radialCDF
calc
(∏ i : Fin d, max 0 (min 1 (x₀ i + r) - max 0 (x₀ i - r))) ≤
∏ _i : Fin d, (2 * r) := by
apply Finset.prod_le_prod
· intro i _
exact le_max_left _ _
· intro i _
have hlo : x₀ i - r ≤ max 0 (x₀ i - r) := le_max_right _ _
have hhi : min 1 (x₀ i + r) ≤ x₀ i + r := min_le_right _ _
exact max_le (by linarith) (by linarith)
_ = (2 * r) ^ d := by simp
theorem radialPropensity_le_cube_radius_power Lemma radialPropensity_le_cube_radius_power in the paper ↗
The radial propensity is at most the full-cube radius power. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
radialPropensity x₀ γ x ≤ ((2 * ‖x - x₀‖) ^ d) ^ ((1 : ℝ) / (γ - 1))
Proof (Lean source)
lemma radialPropensity_le_cube_radius_power {d : ℕ} (x₀ x : Fin d → ℝ)
(γ : ℝ) (hγ : 1 < γ) :
radialPropensity x₀ γ x ≤
((2 * ‖x - x₀‖) ^ d) ^ ((1 : ℝ) / (γ - 1)) := by
unfold radialPropensity
apply Real.rpow_le_rpow
· unfold radialCDF
exact Finset.prod_nonneg (by intro i _; exact le_max_left _ _)
· exact radialCDF_le_cube_radius_volume x₀ _ (norm_nonneg _)
· positivity
A uniform propensity bound on a radius-h neighbourhood of the bump. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
radialPropensity x₀ γ x ≤ ((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1))
Proof (Lean source)
lemma radialPropensity_le_on_radius {d : ℕ} (x₀ x : Fin d → ℝ)
(γ h : ℝ) (hγ : 1 < γ)
(hr : ‖x - x₀‖ ≤ h) :
radialPropensity x₀ γ x ≤
((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) := by
calc
radialPropensity x₀ γ x ≤
((2 * ‖x - x₀‖) ^ d) ^ ((1 : ℝ) / (γ - 1)) :=
radialPropensity_le_cube_radius_power x₀ x γ hγ
_ ≤ ((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) := by
apply Real.rpow_le_rpow (pow_nonneg (by positivity) _)
· exact pow_le_pow_left₀ (by positivity) (by linarith) _
· positivity
Restricted volume on the unit cube is a probability measure. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma uniformCube_probability_lowerPair (d : ℕ) :
IsProbabilityMeasure (volume.restrict (cube d)) := by
apply isProbabilityMeasure_iff.mpr
rw [Measure.restrict_apply MeasurableSet.univ, Set.univ_inter]
have hcube : cube d =
Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hcube, Real.volume_Icc_pi]
simp
theorem uniformCube_radius_sublevel_real
The uniform-cube mass of a sup-norm ball is the displayed truncated radial CDF. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma uniformCube_radius_sublevel_real {d : ℕ} (x₀ : Fin d → ℝ)
(hx₀ : x₀ ∈ cube d) (r : ℝ) (hr : 0 ≤ r) :
(volume.restrict (cube d)).real {x | ‖x - x₀‖ ≤ r} = radialCDF x₀ r := by
have hset : {x | ‖x - x₀‖ ≤ r} ∩ cube d =
Icc (fun i => max 0 (x₀ i - r)) (fun i => min 1 (x₀ i + r)) := by
ext x
simp only [cube, Set.mem_inter_iff, Set.mem_ofPred_eq, Set.mem_pi,
Set.mem_univ, forall_const, Set.mem_Icc, Pi.le_def]
constructor
· rintro ⟨hnorm, hx⟩
have hcoord := (pi_norm_le_iff_of_nonneg hr).1 hnorm
constructor <;> intro i
· exact max_le (hx i).1 (by
have := hcoord i
rw [Pi.sub_apply, Real.norm_eq_abs] at this
linarith [(abs_le.mp this).1])
· exact le_min (hx i).2 (by
have := hcoord i
rw [Pi.sub_apply, Real.norm_eq_abs] at this
linarith [(abs_le.mp this).2])
· rintro ⟨hlo, hhi⟩
constructor
· apply (pi_norm_le_iff_of_nonneg hr).2
intro i
rw [Pi.sub_apply, Real.norm_eq_abs, abs_le]
constructor
· linarith [le_max_right (0 : ℝ) (x₀ i - r), hlo i]
· linarith [hhi i, min_le_right (1 : ℝ) (x₀ i + r)]
· intro i
exact ⟨(le_max_left _ _).trans (hlo i),
(hhi i).trans (min_le_left _ _)⟩
rw [measureReal_def, Measure.restrict_apply (by measurability), hset,
Real.volume_Icc_pi_toReal]
· simp only [radialCDF]
apply Finset.prod_congr rfl
intro i _
have hx := hx₀ i (Set.mem_univ i)
change 0 ≤ x₀ i ∧ x₀ i ≤ 1 at hx
have hdiff : 0 ≤ min 1 (x₀ i + r) - max 0 (x₀ i - r) := sub_nonneg.mpr (by
show max (0 : ℝ) (x₀ i - r) ≤ min 1 (x₀ i + r)
apply max_le
· exact le_min (by norm_num) (by linarith)
· exact le_min (by linarith) (by linarith))
exact (max_eq_right hdiff).symm
· intro i
have hx := hx₀ i (Set.mem_univ i)
change 0 ≤ x₀ i ∧ x₀ i ≤ 1 at hx
apply max_le
· exact le_min (by norm_num) (by linarith)
· exact le_min (by linarith) (by linarith)
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma radialCDF_continuous {d : ℕ} (x₀ : Fin d → ℝ) :
Continuous (radialCDF x₀) := by
unfold radialCDF
fun_prop
theorem radialCDF_zero_of_positive_dimension Lemma radialCDF_zero_of_positive_dimension in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma radialCDF_zero_of_positive_dimension {d : ℕ} (x₀ : Fin d → ℝ)
(hx₀ : x₀ ∈ cube d) (hd : 0 < d) : radialCDF x₀ 0 = 0 := by
let i : Fin d := ⟨0, hd⟩
unfold radialCDF
apply Finset.prod_eq_zero (Finset.mem_univ i)
have hx := hx₀ i (Set.mem_univ i)
change 0 ≤ x₀ i ∧ x₀ i ≤ 1 at hx
simp [hx.1, hx.2]
For the stated inputs and conditions, the asserted conclusion holds.
Proof (Lean source)
lemma radialCDF_one {d : ℕ} (x₀ : Fin d → ℝ) (hx₀ : x₀ ∈ cube d) :
radialCDF x₀ 1 = 1 := by
unfold radialCDF
apply Finset.prod_eq_one
intro i _
have hx := hx₀ i (Set.mem_univ i)
change 0 ≤ x₀ i ∧ x₀ i ≤ 1 at hx
rw [show min 1 (x₀ i + 1) = 1 by exact min_eq_left (by linarith),
show max 0 (x₀ i - 1) = 0 by exact max_eq_left (by linarith)]
norm_num
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma radialCDF_mono_nonneg {d : ℕ} (x₀ : Fin d → ℝ) (hx₀ : x₀ ∈ cube d)
{r s : ℝ} (hr : 0 ≤ r) (hrs : r ≤ s) : radialCDF x₀ r ≤ radialCDF x₀ s := by
letI := uniformCube_probability_lowerPair d
have hs : 0 ≤ s := hr.trans hrs
rw [← uniformCube_radius_sublevel_real x₀ hx₀ r hr,
← uniformCube_radius_sublevel_real x₀ hx₀ s hs]
exact measureReal_mono (fun x hx => hx.trans hrs) (measure_ne_top _ _)
Applying the radial CDF to its own radius variable is stochastically no smaller than a uniform variable. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma radialCDF_self_sublevel_mass {d : ℕ} (x₀ : Fin d → ℝ)
(hx₀ : x₀ ∈ cube d) (hd : 0 < d) (s : ℝ) (hs : s ∈ Icc 0 1) :
(volume.restrict (cube d)).real
{x | radialCDF x₀ ‖x - x₀‖ ≤ s} ≤ s := by
letI := uniformCube_probability_lowerPair d
by_cases hs1 : s = 1
· subst s
exact measureReal_le_one
have hslt : s < 1 := lt_of_le_of_ne hs.2 hs1
let S : Set ℝ := {r | 0 ≤ r ∧ radialCDF x₀ r ≤ s}
have hSnonempty : S.Nonempty := by
refine ⟨0, le_rfl, ?_⟩
rw [radialCDF_zero_of_positive_dimension x₀ hx₀ hd]
exact hs.1
have hSbdd : BddAbove S := by
refine ⟨1, ?_⟩
intro r hr
by_contra hnot
have h1r : 1 ≤ r := le_of_not_ge hnot
have hFr : 1 ≤ radialCDF x₀ r := by
rw [← radialCDF_one x₀ hx₀]
exact radialCDF_mono_nonneg x₀ hx₀ zero_le_one h1r
linarith [hr.2]
have hSclosed : IsClosed S := by
exact isClosed_Ici.inter
(isClosed_Iic.preimage (radialCDF_continuous x₀))
have hsup : sSup S ∈ S := hSclosed.csSup_mem hSnonempty hSbdd
calc
(volume.restrict (cube d)).real
{x | radialCDF x₀ ‖x - x₀‖ ≤ s} ≤
(volume.restrict (cube d)).real {x | ‖x - x₀‖ ≤ sSup S} := by
exact measureReal_mono (fun x hx => le_csSup hSbdd
⟨norm_nonneg _, hx⟩) (measure_ne_top _ _)
_ = radialCDF x₀ (sSup S) :=
uniformCube_radius_sublevel_real x₀ hx₀ _ hsup.1
_ ≤ s := hsup.2
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
(volume.restrict (cube d)).real {x | radialPropensity x₀ γ x ≤ t} ≤ t ^ (γ - 1)
Proof (Lean source)
lemma radialPropensity_uniform_tail {d : ℕ} (x₀ : Fin d → ℝ)
(hx₀ : x₀ ∈ cube d) (hd : 0 < d) (γ : ℝ) (hγ : 1 < γ)
(t : ℝ) (ht : t ∈ Icc (0 : ℝ) 1) :
(volume.restrict (cube d)).real {x | radialPropensity x₀ γ x ≤ t} ≤
t ^ (γ - 1) := by
letI := uniformCube_probability_lowerPair d
have htPow : t ^ (γ - 1) ∈ Icc (0 : ℝ) 1 :=
⟨Real.rpow_nonneg ht.1 _, Real.rpow_le_one ht.1 ht.2 (by linarith)⟩
calc
_ ≤ (volume.restrict (cube d)).real
{x | radialCDF x₀ ‖x - x₀‖ ≤ t ^ (γ - 1)} := by
apply measureReal_mono _ (measure_ne_top _ _)
intro x hx
have hbase : 0 ≤ radialCDF x₀ ‖x - x₀‖ := by
rw [← uniformCube_radius_sublevel_real x₀ hx₀ _ (norm_nonneg _)]
exact measureReal_nonneg
have hpow := Real.rpow_le_rpow
(Real.rpow_nonneg hbase ((1 : ℝ) / (γ - 1))) hx (by linarith : 0 ≤ γ - 1)
simpa [radialPropensity,
Real.rpow_inv_rpow hbase (by linarith : γ - 1 ≠ 0)] using hpow
_ ≤ _ := radialCDF_self_sublevel_mass x₀ hx₀ hd _ htPow
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
∀ᵐ x ∂volume.restrict (cube d), 0 < radialPropensity x₀ γ x
Proof (Lean source)
lemma radialPropensity_uniform_pos_ae {d : ℕ} (x₀ : Fin d → ℝ)
(hx₀ : x₀ ∈ cube d) (hd : 0 < d) (γ : ℝ) (hγ : 1 < γ) :
∀ᵐ x ∂volume.restrict (cube d), 0 < radialPropensity x₀ γ x := by
letI := uniformCube_probability_lowerPair d
have htail := radialPropensity_uniform_tail x₀ hx₀ hd γ hγ 0
(by simp : (0 : ℝ) ∈ Icc 0 1)
rw [Real.zero_rpow (by linarith : γ - 1 ≠ 0)] at htail
have hreal : (volume.restrict (cube d)).real
{x | radialPropensity x₀ γ x = 0} = 0 := by
apply le_antisymm _ measureReal_nonneg
exact (measureReal_mono (fun x hx => by
change radialPropensity x₀ γ x = 0 at hx
exact hx.le) (measure_ne_top _ _)).trans htail
have hzero : (volume.restrict (cube d))
{x | radialPropensity x₀ γ x = 0} = 0 := by
exact (ENNReal.toReal_eq_zero_iff _).mp hreal |>.resolve_right (measure_ne_top _ _)
have hne := measure_eq_zero_iff_ae_notMem.mp hzero
filter_upwards [hne] with x hx
exact lt_of_le_of_ne (radialPropensity_mem_Icc x₀ x γ hγ).1 (Ne.symm hx)
theorem boundedLowerPairAdmissible_of_small Lemma boundedLowerPairAdmissible_of_small in the paper ↗
Fixed positive choices of the bump amplitude and mesh multiplier satisfy all Bernoulli parameter constraints in the lower-pair construction. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L γ a c :
ℝ
x₀ :
Fin d → ℝ
hd :
0 < d
hβ :
1 < β
hB :
0 < B
hL :
0 < L
hγ :
1 < γ
ha :
0 < a ∧ a ≤ baselineSuccess B L
hc :
0 < c ∧ c ≤ 1
conclusion
boundedLowerPairAdmissible d β B L γ a c x₀
Proof (Lean source)
lemma boundedLowerPairAdmissible_of_small (d : ℕ) (β B L γ a c : ℝ)
(x₀ : Fin d → ℝ) (hd : 0 < d) (hβ : 1 < β) (hB : 0 < B)
(hL : 0 < L) (hγ : 1 < γ)
(ha : 0 < a ∧ a ≤ baselineSuccess B L)
(hc : 0 < c ∧ c ≤ 1) :
boundedLowerPairAdmissible d β B L γ a c x₀ := by
refine ⟨ha.1, hc.1, ?_⟩
intro n hn x hx
have hmesh := oracleMesh_mem_Ioc d n β γ
⟨Nat.succ_le_iff.mpr hd, hn, hβ, hγ⟩
have hwidth : 0 ≤ c * oracleMesh d n β γ ∧
c * oracleMesh d n β γ ≤ 1 := by
constructor
· exact mul_nonneg hc.1.le hmesh.1.le
· calc
c * oracleMesh d n β γ ≤ 1 * oracleMesh d n β γ :=
mul_le_mul_of_nonneg_right hc.2 hmesh.1.le
_ ≤ 1 * 1 := mul_le_mul_of_nonneg_left hmesh.2 (by norm_num)
_ = 1 := by ring
refine ⟨radialPropensity_mem_Icc x₀ x γ hγ, ?_, ?_⟩
· exact pairSuccess_mem_Icc_of_small false x₀ β B L a
(c * oracleMesh d n β γ) hB hL (by linarith)
⟨ha.1.le, ha.2⟩ hwidth x
· exact pairSuccess_mem_Icc_of_small true x₀ β B L a
(c * oracleMesh d n β γ) hB hL (by linarith)
⟨ha.1.le, ha.2⟩ hwidth x
Both testing responses retain the positive baseline success probability. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
baselineSuccess B L ≤ pairSuccess j x₀ β B L a h x
Proof (Lean source)
lemma pairSuccess_baseline_le {d : ℕ} (j : Bool) (x₀ x : Fin d → ℝ)
(β B L a h : ℝ) (ha : 0 ≤ a) (hh : 0 ≤ h) :
baselineSuccess B L ≤ pairSuccess j x₀ β B L a h x := by
have hbump := (smoothBump_mem_Icc (fun i => (x i - x₀ i) / h)).1
unfold pairSuccess
split_ifs
· exact le_add_of_nonneg_right
(mul_nonneg (mul_nonneg ha (Real.rpow_nonneg hh _)) hbump)
· simp
The perturbed testing response remains within twice its baseline. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
j :
x₀ x :
Fin d → ℝ
β B L a h :
ℝ
ha :
0 ≤ a ∧ a ≤ baselineSuccess B L
hh :
0 ≤ h ∧ h ≤ 1
hβ :
0 ≤ β
conclusion
pairSuccess j x₀ β B L a h x ≤ 2 * baselineSuccess B L
Proof (Lean source)
lemma pairSuccess_le_twice_baseline {d : ℕ} (j : Bool)
(x₀ x : Fin d → ℝ) (β B L a h : ℝ)
(ha : 0 ≤ a ∧ a ≤ baselineSuccess B L)
(hh : 0 ≤ h ∧ h ≤ 1) (hβ : 0 ≤ β) :
pairSuccess j x₀ β B L a h x ≤ 2 * baselineSuccess B L := by
have hbump := (smoothBump_mem_Icc (fun i => (x i - x₀ i) / h)).2
have hpow : 0 ≤ h ^ β ∧ h ^ β ≤ 1 :=
⟨Real.rpow_nonneg hh.1 _, Real.rpow_le_one hh.1 hh.2 hβ⟩
have hperturb : a * h ^ β * smoothBump (fun i => (x i - x₀ i) / h) ≤ a := by
calc
_ ≤ a * h ^ β * 1 :=
mul_le_mul_of_nonneg_left hbump (mul_nonneg ha.1 hpow.1)
_ ≤ a * 1 := by simpa using mul_le_mul_of_nonneg_left hpow.2 ha.1
_ = a := by ring
unfold pairSuccess
split_ifs <;> nlinarith
Both Bernoulli response parameters stay strictly inside the unit interval, so their two-point response laws have common support. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
j :
x₀ x :
Fin d → ℝ
β B L a h :
ℝ
hB :
0 < B
hL :
0 < L
ha :
0 ≤ a ∧ a ≤ baselineSuccess B L
hh :
0 ≤ h ∧ h ≤ 1
hβ :
0 ≤ β
conclusion
pairSuccess j x₀ β B L a h x ∈ Ioo 0 1
Proof (Lean source)
lemma pairSuccess_mem_Ioo_of_small {d : ℕ} (j : Bool)
(x₀ x : Fin d → ℝ) (β B L a h : ℝ)
(hB : 0 < B) (hL : 0 < L)
(ha : 0 ≤ a ∧ a ≤ baselineSuccess B L)
(hh : 0 ≤ h ∧ h ≤ 1) (hβ : 0 ≤ β) :
pairSuccess j x₀ β B L a h x ∈ Ioo 0 1 := by
have hbase : 0 < baselineSuccess B L ∧ baselineSuccess B L ≤ 1 / 4 := by
unfold baselineSuccess
exact ⟨lt_min (by norm_num) (by positivity), min_le_left _ _⟩
have hlower := pairSuccess_baseline_le j x₀ x β B L a h ha.1 hh.1
have hupper := pairSuccess_le_twice_baseline j x₀ x β B L a h ha hh hβ
exact ⟨lt_of_lt_of_le hbase.1 hlower,
lt_of_le_of_lt hupper (by linarith [hbase.2])⟩
The product bump has unit height at its centre. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
Every nonzero bump value lies inside the unit box in each coordinate. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma smoothBump_support_coordinate {d : ℕ} (z : Fin d → ℝ)
(hz : smoothBump z ≠ 0) (i : Fin d) : |z i| < 1 := by
by_contra hi
apply hz
unfold smoothBump
apply Finset.prod_eq_zero (Finset.mem_univ i)
simp [hi]
A scaled nonzero bump is confined to the radius-h test cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
|x i - x₀ i| < h
Proof (Lean source)
lemma smoothBump_scaled_support {d : ℕ} (x₀ x : Fin d → ℝ)
(h : ℝ) (hh : 0 < h)
(hb : smoothBump (fun i => (x i - x₀ i) / h) ≠ 0) :
∀ i : Fin d, |x i - x₀ i| < h := by
intro i
have hi := smoothBump_support_coordinate _ hb i
rw [abs_div, abs_of_pos hh] at hi
simpa using (div_lt_iff₀ hh).mp hi
A nonzero scaled bump lies in the closed sup-norm radius-h cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
‖x - x₀‖ ≤ h
Proof (Lean source)
lemma smoothBump_scaled_support_radius {d : ℕ} (x₀ x : Fin d → ℝ)
(h : ℝ) (hh : 0 < h)
(hb : smoothBump (fun i => (x i - x₀ i) / h) ≠ 0) :
‖x - x₀‖ ≤ h := by
apply (pi_norm_le_iff_of_nonneg hh.le).2
intro i
simpa only [Pi.sub_apply, Real.norm_eq_abs] using
(smoothBump_scaled_support x₀ x h hh hb i).le
theorem radialPropensity_le_on_bump_support Lemma radialPropensity_le_on_bump_support in the paper ↗
On the perturbation support, the treated-design weight has the small radius factor used in the lower-pair entropy estimate. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
x₀ x :
Fin d → ℝ
γ h :
ℝ
hγ :
1 < γ
hh :
0 < h
hb :
smoothBump (fun i => (x i - x₀ i) / h) ≠ 0
conclusion
radialPropensity x₀ γ x ≤ ((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1))
Proof (Lean source)
lemma radialPropensity_le_on_bump_support {d : ℕ} (x₀ x : Fin d → ℝ)
(γ h : ℝ) (hγ : 1 < γ) (hh : 0 < h)
(hb : smoothBump (fun i => (x i - x₀ i) / h) ≠ 0) :
radialPropensity x₀ γ x ≤
((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) :=
radialPropensity_le_on_radius x₀ x γ h hγ
(smoothBump_scaled_support_radius x₀ x h hh hb)
The two treated Bernoulli parameters differ by the full bump amplitude at the designated test point. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma pairSuccess_center_gap {d : ℕ} (x₀ : Fin d → ℝ)
(β B L a h : ℝ) :
pairSuccess true x₀ β B L a h x₀ -
pairSuccess false x₀ β B L a h x₀ = a * h ^ β := by
simp [pairSuccess, smoothBump_zero]
The treated Bernoulli perturbation is nonnegative and bounded by its nominal amplitude at every covariate point. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
conclusion 2
Proof (Lean source)
lemma pairSuccess_perturbation_bound {d : ℕ} (x₀ x : Fin d → ℝ)
(β B L a h : ℝ) (ha : 0 ≤ a) (hh : 0 ≤ h) :
0 ≤ pairSuccess true x₀ β B L a h x -
pairSuccess false x₀ β B L a h x ∧
pairSuccess true x₀ β B L a h x -
pairSuccess false x₀ β B L a h x ≤ a * h ^ β := by
have hbump := smoothBump_mem_Icc (fun i => (x i - x₀ i) / h)
have hamp : 0 ≤ a * h ^ β := mul_nonneg ha (Real.rpow_nonneg hh _)
have hbound :
0 ≤ a * h ^ β * smoothBump (fun i => (x i - x₀ i) / h) ∧
a * h ^ β * smoothBump (fun i => (x i - x₀ i) / h) ≤ a * h ^ β :=
⟨mul_nonneg hamp hbump.1, by simpa using
(mul_le_mul_of_nonneg_left hbump.2 hamp)⟩
simpa [pairSuccess] using hbound
The observed Bernoulli kernels differ only inside the bump cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma pairSuccess_difference_support {d : ℕ} (x₀ x : Fin d → ℝ)
(β B L a h : ℝ) (hh : 0 < h)
(hdiff : pairSuccess true x₀ β B L a h x ≠
pairSuccess false x₀ β B L a h x) :
∀ i : Fin d, |x i - x₀ i| < h := by
apply smoothBump_scaled_support x₀ x h hh
intro hb
apply hdiff
simp [pairSuccess, hb]
Scaling the bump by the oracle mesh yields the oracle separation scale. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
= a * c ^ β * oracleRate d n β γ
Proof (Lean source)
lemma pairSuccess_oracle_separation {d n : ℕ} (x₀ : Fin d → ℝ)
(β B L γ a c : ℝ) (hc : 0 ≤ c) :
pairSuccess true x₀ β B L a (c * oracleMesh d n β γ) x₀ -
pairSuccess false x₀ β B L a (c * oracleMesh d n β γ) x₀ =
a * c ^ β * oracleRate d n β γ := by
have hmesh : 0 ≤ oracleMesh d n β γ := by
unfold oracleMesh
exact Real.rpow_nonneg (by exact_mod_cast zero_le n) _
rw [pairSuccess_center_gap, oracleRate, Real.mul_rpow hc hmesh]
ring
theorem boundedLowerPair_fixed_constants_separation Lemma boundedLowerPair_fixed_constants_separation in the paper ↗
Fixed positive construction constants give an admissible Bernoulli pair and the oracle-scale response-parameter gap at every sample size. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
∃ a c c₀ : ℝ,
conclusion 1
n :
ℕ
1 ≤ n
c₀ * oracleRate d n β γ
= B * (pairSuccess true x₀ β B L a (c * oracleMesh d n β γ) x₀ - pairSuccess false x₀ β B L a (c * oracleMesh d n β γ) x₀)
Proof (Lean source)
lemma boundedLowerPair_fixed_constants_separation (d : ℕ)
(β B L γ : ℝ) (x₀ : Fin d → ℝ)
(hd : 0 < d) (hβ : 1 < β) (hB : 0 < B)
(hL : 0 < L) (hγ : 1 < γ) :
∃ a c c₀ : ℝ, 0 < a ∧ 0 < c ∧ 0 < c₀ ∧
boundedLowerPairAdmissible d β B L γ a c x₀ ∧
∀ n : ℕ, 1 ≤ n →
c₀ * oracleRate d n β γ =
B * (pairSuccess true x₀ β B L a
(c * oracleMesh d n β γ) x₀ -
pairSuccess false x₀ β B L a
(c * oracleMesh d n β γ) x₀) := by
let a := baselineSuccess B L / 2
let c : ℝ := 1 / 2
let c₀ := B * a * c ^ β
have hbase : 0 < baselineSuccess B L := by
unfold baselineSuccess
positivity
have ha : 0 < a ∧ a ≤ baselineSuccess B L := by
dsimp [a]
constructor <;> linarith
have hc : 0 < c ∧ c ≤ 1 := by dsimp [c]; norm_num
have hc₀ : 0 < c₀ := by dsimp [c₀]; positivity
refine ⟨a, c, c₀, ha.1, hc.1, hc₀,
boundedLowerPairAdmissible_of_small d β B L γ a c x₀
hd hβ hB hL hγ ha hc, ?_⟩
intro n _
rw [pairSuccess_oracle_separation x₀ β B L γ a c hc.1.le]
dsimp [c₀]
ring
The oracle bandwidth cancels the sample-size factor in the pair's KL bound. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β γ c :
ℝ
hn :
0 < n
hc :
0 ≤ c
hden :
0 < 2 * β + effectiveDimension d γ
conclusion
(n : ℝ) * (c * oracleMesh d n β γ) ^ (2 * β + effectiveDimension d γ)
= c ^ (2 * β + effectiveDimension d γ)
Proof (Lean source)
lemma lowerPair_oracle_kl_balance (d n : ℕ) (β γ c : ℝ)
(hn : 0 < n) (hc : 0 ≤ c)
(hden : 0 < 2 * β + effectiveDimension d γ) :
(n : ℝ) * (c * oracleMesh d n β γ) ^
(2 * β + effectiveDimension d γ) =
c ^ (2 * β + effectiveDimension d γ) := by
have hn' : (0 : ℝ) < n := by exact_mod_cast hn
have hmesh : 0 ≤ oracleMesh d n β γ := by
unfold oracleMesh
positivity
have hexp : -(1 : ℝ) / (2 * β + effectiveDimension d γ) *
(2 * β + effectiveDimension d γ) = -1 := by
field_simp
have hpow : (oracleMesh d n β γ) ^
(2 * β + effectiveDimension d γ) = (n : ℝ)⁻¹ := by
rw [oracleMesh, ← Real.rpow_mul hn'.le, hexp]
exact Real.rpow_neg_one (n : ℝ)
rw [Real.mul_rpow hc hmesh, hpow]
calc
_ = c ^ (2 * β + effectiveDimension d γ) * ((n : ℝ) * (n : ℝ)⁻¹) := by ring
_ = _ := by rw [mul_inv_cancel₀ (ne_of_gt hn'), mul_one]
A fixed bandwidth multiplier makes any finite oracle-scale KL coefficient small, uniformly in the sample size. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
K a D :
ℝ
hK :
0 ≤ K
hD :
1 ≤ D
conclusion
∃ c : ℝ,
conclusion 1
0 < c
conclusion 2
c ≤ 1
conclusion 3
K * a ^ 2 * c ^ D ≤ 1 / 8
Proof (Lean source)
lemma lowerPair_choose_kl_scale (K a D : ℝ)
(hK : 0 ≤ K) (hD : 1 ≤ D) :
∃ c : ℝ, 0 < c ∧ c ≤ 1 ∧ K * a ^ 2 * c ^ D ≤ 1 / 8 := by
let c : ℝ := 1 / (8 * (K * a ^ 2 + 1))
have hden : 0 < 8 * (K * a ^ 2 + 1) := by positivity
have hc : 0 < c := by dsimp [c]; positivity
have hc1 : c ≤ 1 := by
dsimp [c]
apply (div_le_iff₀ hden).2
nlinarith [mul_nonneg hK (sq_nonneg a)]
refine ⟨c, hc, hc1, ?_⟩
have hpow : c ^ D ≤ c := Real.rpow_le_self_of_le_one hc.le hc1 hD
have hcoef : 0 ≤ K * a ^ 2 := mul_nonneg hK (sq_nonneg a)
have hsmall : K * a ^ 2 * c ≤ 1 / 8 := by
dsimp [c]
calc
K * a ^ 2 * (1 / (8 * (K * a ^ 2 + 1))) =
(K * a ^ 2) / (8 * (K * a ^ 2 + 1)) := by ring
_ ≤ 1 / 8 := (div_le_iff₀ hden).2 (by
nlinarith [mul_nonneg hK (sq_nonneg a)])
exact (mul_le_mul_of_nonneg_left hpow hcoef).trans hsmall
theorem boundedLowerPair_fixed_kl_and_separation_constants Lemma boundedLowerPair_fixed_kl_and_separation_constants in the paper ↗
Choose admissible testing parameters once, with both the oracle-scale KL budget and a positive response separation constant. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L γ K :
ℝ
x₀ :
Fin d → ℝ
hd :
0 < d
hβ :
1 < β
hB :
0 < B
hL :
0 < L
hγ :
1 < γ
hK :
0 ≤ K
conclusion
∃ a c c₀ : ℝ,
0 < a
0 < c
0 < c₀
boundedLowerPairAdmissible d β B L γ a c x₀
K * a ^ 2 * c ^ (2 * β + effectiveDimension d γ) ≤ 1 / 8
conclusion 1
n :
ℕ
1 ≤ n
c₀ * oracleRate d n β γ
= B * (pairSuccess true x₀ β B L a (c * oracleMesh d n β γ) x₀ - pairSuccess false x₀ β B L a (c * oracleMesh d n β γ) x₀)
Proof (Lean source)
lemma boundedLowerPair_fixed_kl_and_separation_constants (d : ℕ)
(β B L γ K : ℝ) (x₀ : Fin d → ℝ)
(hd : 0 < d) (hβ : 1 < β) (hB : 0 < B)
(hL : 0 < L) (hγ : 1 < γ) (hK : 0 ≤ K) :
∃ a c c₀ : ℝ, 0 < a ∧ 0 < c ∧ 0 < c₀ ∧
boundedLowerPairAdmissible d β B L γ a c x₀ ∧
K * a ^ 2 * c ^ (2 * β + effectiveDimension d γ) ≤ 1 / 8 ∧
∀ n : ℕ, 1 ≤ n →
c₀ * oracleRate d n β γ =
B * (pairSuccess true x₀ β B L a
(c * oracleMesh d n β γ) x₀ -
pairSuccess false x₀ β B L a
(c * oracleMesh d n β γ) x₀) := by
let a := baselineSuccess B L / 2
have hbase : 0 < baselineSuccess B L := by
unfold baselineSuccess
positivity
have ha : 0 < a ∧ a ≤ baselineSuccess B L := by
dsimp [a]
constructor <;> linarith
have hD : 1 ≤ 2 * β + effectiveDimension d γ := by
have hd' : (0 : ℝ) < d := by exact_mod_cast hd
have hγ' : 0 < γ - 1 := by linarith
have hdim : 0 ≤ effectiveDimension d γ := by
unfold effectiveDimension
positivity
linarith
obtain ⟨c, hc, hc1, hkl⟩ :=
lowerPair_choose_kl_scale K a (2 * β + effectiveDimension d γ) hK hD
let c₀ := B * a * c ^ β
have hc₀ : 0 < c₀ := by dsimp [c₀]; positivity
refine ⟨a, c, c₀, ha.1, hc, hc₀,
boundedLowerPairAdmissible_of_small d β B L γ a c x₀
hd hβ hB hL hγ ha ⟨hc, hc1⟩, hkl, ?_⟩
intro n _
rw [pairSuccess_oracle_separation x₀ β B L γ a c hc.le]
dsimp [c₀]
ring
theorem lowerPair_bernoulli_kl_le_quadratic Lemma lowerPair_bernoulli_kl_le_quadratic in the paper ↗
The binary relative entropy is bounded by the Pearson quadratic at the second Bernoulli parameter. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma lowerPair_bernoulli_kl_le_quadratic (p q : ℝ)
(hp : p ∈ Ioo 0 1) (hq : q ∈ Ioo 0 1) :
p * log (p / q) + (1 - p) * log ((1 - p) / (1 - q)) ≤
(p - q) ^ 2 / (q * (1 - q)) := by
have hqpos : 0 < q := hq.1
have hqcomp : 0 < 1 - q := by linarith [hq.2]
have hpcomp : 0 < 1 - p := by linarith [hp.2]
have h₁ := Real.log_le_sub_one_of_pos (div_pos hp.1 hqpos)
have h₂ := Real.log_le_sub_one_of_pos (div_pos hpcomp hqcomp)
have h₁' := mul_le_mul_of_nonneg_left h₁ hp.1.le
have h₂' := mul_le_mul_of_nonneg_left h₂ hpcomp.le
calc
p * log (p / q) + (1 - p) * log ((1 - p) / (1 - q)) ≤
p * (p / q - 1) + (1 - p) * ((1 - p) / (1 - q) - 1) := add_le_add h₁' h₂'
_ = (p - q) ^ 2 / (q * (1 - q)) := by
field_simp
ring
theorem lowerPair_bernoulli_kl_le_baseline_quadratic Lemma lowerPair_bernoulli_kl_le_baseline_quadratic in the paper ↗
A common positive Bernoulli baseline makes the KL constant uniform over the testing pair. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma lowerPair_bernoulli_kl_le_baseline_quadratic (p q δ : ℝ)
(hp : p ∈ Ioo 0 1) (hq : q ∈ Ioo 0 1)
(hδ : 0 < δ) (hqlo : δ ≤ q) (hqhi : q ≤ 1 / 2) :
p * log (p / q) + (1 - p) * log ((1 - p) / (1 - q)) ≤
(2 / δ) * (p - q) ^ 2 := by
have hden : δ / 2 ≤ q * (1 - q) := by nlinarith
have hdenpos : 0 < q * (1 - q) := mul_pos hq.1 (by linarith [hq.2])
have hsq : 0 ≤ (p - q) ^ 2 := sq_nonneg _
calc
_ ≤ (p - q) ^ 2 / (q * (1 - q)) :=
lowerPair_bernoulli_kl_le_quadratic p q hp hq
_ ≤ (p - q) ^ 2 / (δ / 2) := by gcongr
_ = (2 / δ) * (p - q) ^ 2 := by field_simp
The Bernoulli response kernels in the lower pair satisfy the quadratic KL estimate uniformly in location and bandwidth. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
x₀ x :
Fin d → ℝ
β B L a h :
ℝ
hβ :
0 < β
hB :
0 < B
hL :
0 < L
ha :
0 ≤ a ∧ a ≤ baselineSuccess B L
hh :
0 ≤ h ∧ h ≤ 1
conclusion
p :
pairSuccess false x₀ β B L a h x
q :
pairSuccess true x₀ β B L a h x
≤ (2 / baselineSuccess B L) * (p - q) ^ 2
Proof (Lean source)
lemma lowerPair_pointwise_kl_bound {d : ℕ} (x₀ x : Fin d → ℝ)
(β B L a h : ℝ) (hβ : 0 < β) (hB : 0 < B) (hL : 0 < L)
(ha : 0 ≤ a ∧ a ≤ baselineSuccess B L)
(hh : 0 ≤ h ∧ h ≤ 1) :
let p := pairSuccess false x₀ β B L a h x
let q := pairSuccess true x₀ β B L a h x
p * log (p / q) + (1 - p) * log ((1 - p) / (1 - q)) ≤
(2 / baselineSuccess B L) * (p - q) ^ 2 := by
dsimp
have hbase : 0 < baselineSuccess B L ∧ baselineSuccess B L ≤ 1 / 4 := by
unfold baselineSuccess
exact ⟨lt_min (by norm_num) (by positivity), min_le_left _ _⟩
have hp := pairSuccess_mem_Ioo_of_small false x₀ x β B L a h
hB hL ha hh hβ.le
have hq := pairSuccess_mem_Ioo_of_small true x₀ x β B L a h
hB hL ha hh hβ.le
have hqlo := pairSuccess_baseline_le true x₀ x β B L a h ha.1 hh.1
have hqhi := pairSuccess_le_twice_baseline true x₀ x β B L a h ha hh hβ.le
apply lowerPair_bernoulli_kl_le_baseline_quadratic _ _ _ hp hq hbase.1 hqlo
linarith [hbase.2]
theorem lowerPair_weighted_pointwise_kl_bound Lemma lowerPair_weighted_pointwise_kl_bound in the paper ↗
The treated KL integrand has the small propensity factor on the bump support, and vanishes away from that support. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
x₀ x :
Fin d → ℝ
β B L γ a h :
ℝ
hβ :
0 < β
hB :
0 < B
hL :
0 < L
hγ :
1 < γ
ha :
0 ≤ a ∧ a ≤ baselineSuccess B L
hh :
0 < h ∧ h ≤ 1
conclusion
p :
pairSuccess false x₀ β B L a h x
q :
pairSuccess true x₀ β B L a h x
≤ (2 / baselineSuccess B L) * (a * h ^ β) ^ 2 * (((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)))
Proof (Lean source)
lemma lowerPair_weighted_pointwise_kl_bound {d : ℕ}
(x₀ x : Fin d → ℝ) (β B L γ a h : ℝ)
(hβ : 0 < β) (hB : 0 < B) (hL : 0 < L) (hγ : 1 < γ)
(ha : 0 ≤ a ∧ a ≤ baselineSuccess B L)
(hh : 0 < h ∧ h ≤ 1) :
let p := pairSuccess false x₀ β B L a h x
let q := pairSuccess true x₀ β B L a h x
radialPropensity x₀ γ x *
(p * log (p / q) +
(1 - p) * log ((1 - p) / (1 - q))) ≤
(2 / baselineSuccess B L) * (a * h ^ β) ^ 2 *
(((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1))) := by
dsimp
have hbase : 0 ≤ 2 / baselineSuccess B L := by
unfold baselineSuccess
positivity
have hprop : 0 ≤ radialPropensity x₀ γ x :=
(radialPropensity_mem_Icc x₀ x γ hγ).1
by_cases hb : smoothBump (fun i => (x i - x₀ i) / h) = 0
· simp [pairSuccess, hb]
exact mul_nonneg (mul_nonneg hbase (sq_nonneg _))
(Real.rpow_nonneg (pow_nonneg (mul_nonneg (by norm_num) hh.1.le) _) _)
· have hbound := lowerPair_pointwise_kl_bound x₀ x β B L a h
hβ hB hL ha ⟨hh.1.le, hh.2⟩
dsimp at hbound
have hgap := pairSuccess_perturbation_bound x₀ x β B L a h
ha.1 hh.1.le
have hgap' : (pairSuccess false x₀ β B L a h x -
pairSuccess true x₀ β B L a h x) ^ 2 ≤ (a * h ^ β) ^ 2 := by
nlinarith [sq_nonneg (pairSuccess true x₀ β B L a h x -
pairSuccess false x₀ β B L a h x)]
have hweight := radialPropensity_le_on_bump_support x₀ x γ h
hγ hh.1 hb
calc
_ ≤ radialPropensity x₀ γ x *
((2 / baselineSuccess B L) *
(pairSuccess false x₀ β B L a h x -
pairSuccess true x₀ β B L a h x) ^ 2) :=
mul_le_mul_of_nonneg_left hbound hprop
_ ≤ radialPropensity x₀ γ x *
((2 / baselineSuccess B L) * (a * h ^ β) ^ 2) := by
gcongr
_ ≤ (2 / baselineSuccess B L) * (a * h ^ β) ^ 2 *
(((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1))) := by
rw [mul_comm (radialPropensity x₀ γ x)]
exact mul_le_mul_of_nonneg_left hweight
(mul_nonneg hbase (sq_nonneg _))
theorem lowerPairCompletion_supported_on_endpoints Lemma lowerPairCompletion_supported_on_endpoints in the paper ↗
Both potential outcomes of every admissible testing law lie in {0, B}. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B L γ a c :
ℝ
x₀ :
Fin d → ℝ
j :
hadm :
boundedLowerPairAdmissible d β B L γ a c x₀
hn :
1 ≤ n
Proof (Lean source)
lemma lowerPairCompletion_supported_on_endpoints (d n : ℕ)
(β B L γ a c : ℝ) (x₀ : Fin d → ℝ) (j : Bool)
(hadm : boundedLowerPairAdmissible d β B L γ a c x₀)
(hn : 1 ≤ n) :
∀ᵐ ω ∂lowerPairCompletion d n β B L γ a c x₀ j,
ω.2.2.1 ∈ ({0, B} : Set ℝ) ∧
ω.2.2.2.1 ∈ ({0, B} : Set ℝ) := by
let S : Set (Completion d) :=
{ω | ω.2.2.1 ∈ ({0, B} : Set ℝ) ∧
ω.2.2.2.1 ∈ ({0, B} : Set ℝ)}
have hS : MeasurableSet S := by
dsimp [S]
measurability
have hzero : (lowerPairCompletion d n β B L γ a c x₀ j) Sᶜ = 0 := by
unfold lowerPairCompletion
apply le_antisymm _ zero_le
calc
_ ≤ ∫⁻ x, (completionAt B (radialPropensity x₀ γ)
(fun _ => baselineSuccess B L)
(pairSuccess j x₀ β B L a (c * oracleMesh d n β γ)) x) Sᶜ
∂volume.restrict (cube d) := Measure.bind_apply_le _ hS.compl
_ = 0 := by
apply lintegral_eq_zero_of_ae_eq_zero
filter_upwards [ae_restrict_mem (by simp [cube] : MeasurableSet (cube d))] with x hx
obtain ⟨he, hp₀, hp₁⟩ := hadm.2.2 n hn x hx
apply (measure_eq_zero_iff_ae_notMem).2
filter_upwards [completionAt_supported_on_endpoints B
(radialPropensity x₀ γ) (fun _ => baselineSuccess B L)
(pairSuccess j x₀ β B L a (c * oracleMesh d n β γ)) x
he (by simpa [pairSuccess] using hp₀)
(by cases j with
| false => simpa [pairSuccess] using hp₀
| true => exact hp₁)]
with ω hω
simp only [Set.mem_compl_iff, not_not]
change ω.2.2.1 ∈ ({0, B} : Set ℝ) ∧
ω.2.2.2.1 ∈ ({0, B} : Set ℝ)
exact hω
filter_upwards [(measure_eq_zero_iff_ae_notMem).mp hzero] with ω hω
simp only [Set.mem_compl_iff, not_not] at hω
change ω.2.2.1 ∈ ({0, B} : Set ℝ) ∧
ω.2.2.2.1 ∈ ({0, B} : Set ℝ) at hω
exact hω
The paper's explicit bump is the promoted normalized product bump. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma smoothBump_eq_productBump {d : ℕ} (z : Fin d → ℝ) :
smoothBump z =
productBump z := by
unfold smoothBump
rw [productBump]
apply Finset.prod_congr rfl
intro i _
rw [smoothBump_factor_eq_expNegInvGlue]
rfl
The treated response perturbation belongs to the paper Hölder ball when the promoted uniform bump radius fits inside L. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L a h :
ℝ
x₀ :
Fin d → ℝ
hβ :
0 < β
hh :
0 < h
hh1 :
h ≤ 1
A :
ℝ
hA :
0 < A
hscaled :
HolderBallOn (unitCube d) (polynomialDegree β) (β - (polynomialDegree β : ℝ)) A (scaledProductBump β h x₀)
hrad :
|B * baselineSuccess B L| + |B * a| * A ≤ L
conclusion
HolderBall β L (fun x => B * pairSuccess true x₀ β B L a h x)
Proof (Lean source)
lemma pairSuccess_true_holderBall {d : ℕ} (β B L a h : ℝ)
(x₀ : Fin d → ℝ) (hβ : 0 < β) (hh : 0 < h) (hh1 : h ≤ 1)
(A : ℝ) (hA : 0 < A)
(hscaled : HolderBallOn
(unitCube d)
(polynomialDegree β) (β - (polynomialDegree β : ℝ)) A
(scaledProductBump β h x₀))
(hrad : |B * baselineSuccess B L| + |B * a| * A ≤ L) :
HolderBall β L (fun x => B * pairSuccess true x₀ β B L a h x) := by
open Causalean.Mathlib.Analysis.Calculus.CubeExtension in
have haff := holderBallOn_unitCube_add_const_mul hA.le hscaled
(B * baselineSuccess B L) (B * a)
have hre : (fun x => B * pairSuccess true x₀ β B L a h x) =
(fun x => B * baselineSuccess B L + B * a *
scaledProductBump β h x₀ x) := by
funext x
rw [scaledProductBump_eq_coordinate_div
β (ne_of_gt hh), ← smoothBump_eq_productBump]
simp only [pairSuccess, if_true]
ring
rw [hre]
have hbase :
HolderBallOn
(cube d) (polynomialDegree β) (β - (polynomialDegree β : ℝ))
(|B * baselineSuccess B L| + |B * a| * A)
(fun x => B * baselineSuccess B L + B * a *
scaledProductBump β h x₀ x) := by
simpa [polynomialDegree, cube,
unitCube] using haff
refine ⟨hbase.regularity, ?_, ?_⟩
· intro j hj f x hx
exact (hbase.derivBound j hj f x hx).trans hrad
· intro f x hx y hy
exact (hbase.modulus f x hx y hy).trans
(mul_le_mul_of_nonneg_right hrad (Real.rpow_nonneg (norm_nonneg _) _))
A Hölder representative B p₁ identifies the conditional mean of the treated potential outcome in the explicit completion. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
B β L :
ℝ
e p₀ p₁ :
(Fin d → ℝ) → ℝ
he :
hp₀ :
Measurable p₀
hp₁ :
Measurable p₁
IsFiniteMeasure (mu.bind (completionAt B e p₀ p₁))
hball :
HolderBall β L (fun x => B * p₁ x)
conclusion
HolderResponse (mu.bind (completionAt B e p₀ p₁)) (fun x => B * p₁ x) β L
Proof (Lean source)
lemma completionBind_holderResponse {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (B β L : ℝ)
(e p₀ p₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hp₀ : Measurable p₀) (hp₁ : Measurable p₁)
(hvalid : ∀ x, e x ∈ Icc 0 1 ∧ p₀ x ∈ Icc 0 1 ∧
p₁ x ∈ Icc 0 1)
[IsFiniteMeasure (mu.bind (completionAt B e p₀ p₁))]
(hball : HolderBall β L (fun x => B * p₁ x)) :
HolderResponse (mu.bind (completionAt B e p₀ p₁))
(fun x => B * p₁ x) β L := by
let Pc := mu.bind (completionAt B e p₀ p₁)
let X : Completion d → (Fin d → ℝ) := fun ω => ω.1
let Y₁ : Completion d → ℝ := fun ω => ω.2.2.2.1
let k₁ : Kernel (Fin d → ℝ) ℝ :=
ProbabilityTheory.Kernel.mk (fun x => scaledBernoulli B (p₁ x))
(by unfold scaledBernoulli realBernoulli; fun_prop)
letI : IsMarkovKernel k₁ :=
⟨fun x => scaledBernoulli_probability B (p₁ x) (hvalid x).2.2⟩
have hcomp : Measurable (completionAt B e p₀ p₁) :=
completionAt_measurable_of_measurable B e p₀ p₁ he hp₀ hp₁ hvalid
have hX : Measurable X := by dsimp [X]; fun_prop
have hY₁ : Measurable Y₁ := by dsimp [Y₁]; fun_prop
have hcov : Pc.map X = mu := by
dsimp [Pc, X]
exact completionBind_covariate_marginal mu B e p₀ p₁ hcomp.aemeasurable
(Filter.Eventually.of_forall hvalid)
have hjoint : Pc.map (fun ω => (X ω, Y₁ ω)) = Measure.compProd mu k₁ := by
dsimp [Pc, X, Y₁, k₁]
exact completionBind_covariate_treatedPotential_compProd mu B e p₀ p₁
he hp₀ hp₁ hvalid
have hcond : condDistrib Y₁ X Pc =ᵐ[mu] k₁ := by
have h := ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd_of_measurable
(μ := Pc) (X := X) (Y := Y₁) (κ := k₁) hX hY₁
(by simpa [hcov] using hjoint)
simpa [hcov] using h
have hY₁int : Integrable Y₁ Pc := by
apply Integrable.mono' (integrable_const |B|) hY₁.aestronglyMeasurable
filter_upwards [completionBind_supported_on_endpoints mu B e p₀ p₁
(Filter.Eventually.of_forall hvalid)] with ω hω
rcases hω.2 with hzero | hBval
· simp [Y₁, hzero]
· dsimp [Y₁]
rw [Set.mem_singleton_iff.mp hBval]
have hce : Pc[Y₁ | comap X inferInstance] =ᵐ[Pc]
fun ω => ∫ y : ℝ, y ∂ProbabilityTheory.condDistrib Y₁ X Pc (X ω) := by
simpa using ProbabilityTheory.condExp_ae_eq_integral_condDistrib' hX hY₁int
have hcond' : condDistrib Y₁ X Pc =ᵐ[Pc.map X] k₁ := by
rw [hcov]
exact hcond
have hkernel := ae_eq_comp hX.aemeasurable hcond'
have hmean : (fun ω : Completion d =>
∫ y : ℝ, y ∂ProbabilityTheory.condDistrib Y₁ X Pc (X ω)) =ᵐ[Pc]
fun ω => B * p₁ (X ω) := by
filter_upwards [hkernel] with ω hω
calc
_ = ∫ y : ℝ, y ∂k₁ (X ω) := by congr 1
_ = _ := scaledBernoulli_integral_id B (p₁ (X ω)) (hvalid (X ω)).2.2
exact ⟨hball, hmean.symm.trans hce.symm⟩
theorem completionBind_boundedOutcome_of_endpoint_support Lemma completionBind_boundedOutcome_of_endpoint_support in the paper ↗
Endpoint-supported potential outcomes give the required treated mean and sub-Gaussian residual bound whenever treatment is positive almost everywhere. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
mu :
B :
ℝ
hB :
0 < B
e p₀ p₁ :
(Fin d → ℝ) → ℝ
he :
hp₀ :
Measurable p₀
hp₁ :
Measurable p₁
hpos :
∀ᵐ x ∂mu, 0 < e x
conclusion
Pc :
mu.bind (completionAt B e p₀ p₁)
P :
Pc.map observed
Proof (Lean source)
lemma completionBind_boundedOutcome_of_endpoint_support {d : ℕ}
(mu : Measure (Fin d → ℝ)) [IsProbabilityMeasure mu] (B : ℝ) (hB : 0 < B)
(e p₀ p₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hp₀ : Measurable p₀) (hp₁ : Measurable p₁)
(hvalid : ∀ x, e x ∈ Icc 0 1 ∧ p₀ x ∈ Icc 0 1 ∧
p₁ x ∈ Icc 0 1)
[IsFiniteMeasure ((mu.bind (completionAt B e p₀ p₁)).map observed)]
(hpos : ∀ᵐ x ∂mu, 0 < e x) :
let Pc := mu.bind (completionAt B e p₀ p₁)
let P := Pc.map observed
BoundedMeanSubGaussianResidual P B := by
dsimp
let Pc := mu.bind (completionAt B e p₀ p₁)
let P := Pc.map observed
let XA : Obs d → ((Fin d → ℝ) × Bool) := fun z => (z.1, z.2.1)
let Y : Obs d → ℝ := fun z => z.2.2
let kA : Kernel (Fin d → ℝ) Bool :=
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x))
(by unfold realBernoulli; fun_prop)
letI : IsMarkovKernel kA :=
⟨fun x => realBernoulli_probability (e x) (hvalid x).1⟩
have hk : Measurable (completionAt B e p₀ p₁) :=
completionAt_measurable_of_measurable B e p₀ p₁ he hp₀ hp₁ hvalid
letI : IsProbabilityMeasure Pc := by
apply isProbabilityMeasure_bind hk.aemeasurable
filter_upwards [] with x
letI := realBernoulli_probability (e x) (hvalid x).1
letI := scaledBernoulli_probability B (p₀ x) (hvalid x).2.1
letI := scaledBernoulli_probability B (p₁ x) (hvalid x).2.2
unfold completionAt
apply Measure.isProbabilityMeasure_map
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption)
letI : IsProbabilityMeasure P := by
dsimp [P]
apply Measure.isProbabilityMeasure_map
unfold observed
fun_prop
have hXA : Measurable XA := by dsimp [XA]; fun_prop
have hY : Measurable Y := by dsimp [Y]; fun_prop
have hXAmap : P.map XA = Measure.compProd mu kA := by
dsimp [P, Pc, XA]
rw [Measure.map_map (by fun_prop) (by unfold observed; fun_prop)]
exact completionBind_covariate_treatment_compProd mu B e p₀ p₁
he hp₀ hp₁ hvalid
have hsuppP : ∀ᵐ z ∂P, Y z ∈ Icc (0 : ℝ) B := by
rw [show P = Pc.map observed by rfl]
apply (ae_map_iff (by unfold observed; fun_prop)
(by measurability : MeasurableSet {z : Obs d | Y z ∈ Icc (0 : ℝ) B})).2
filter_upwards [completionBind_supported_on_endpoints mu B e p₀ p₁
(Filter.Eventually.of_forall hvalid),
completionBind_consistency mu B e p₀ p₁ hk.aemeasurable] with ω hω hcons
have hy : ω.2.2.2.2 ∈ ({0, B} : Set ℝ) := by
rw [hcons]
split
· exact hω.2
· exact hω.1
rcases hy with hzero | hBval
· change ω.2.2.2.2 = 0 at hzero
dsimp [Y, observed]
rw [hzero]
exact ⟨le_rfl, hB.le⟩
· change ω.2.2.2.2 = B at hBval
dsimp [Y, observed]
rw [hBval]
exact ⟨hB.le, le_rfl⟩
have hsuppProd : ∀ᵐ q ∂Measure.compProd (P.map XA)
(condDistrib Y XA P), q.2 ∈ Icc (0 : ℝ) B := by
rw [ProbabilityTheory.compProd_map_condDistrib hY.aemeasurable]
exact (ae_map_iff (by fun_prop)
(by measurability : MeasurableSet
{q : ((Fin d → ℝ) × Bool) × ℝ | q.2 ∈ Icc (0 : ℝ) B})).2 hsuppP
rw [hXAmap] at hsuppProd
have hsections := Measure.ae_ae_of_ae_compProd
(Measure.ae_ae_of_ae_compProd hsuppProd)
have htreat : ∀ᵐ x ∂mu, ∀ᵐ y ∂treatedKernel P x,
y ∈ Icc (0 : ℝ) B := by
filter_upwards [hsections, hpos] with x hx hxp
have htrue := realBernoulli_ae_at_true_of_pos (e x) hxp
(fun a => ∀ᵐ y ∂ProbabilityTheory.condDistrib Y XA P (x, a),
y ∈ Icc (0 : ℝ) B) (by simpa [kA] using hx)
simpa [treatedKernel, XA, Y] using htrue
have hcov : covariateLaw P = mu := by
dsimp [P, Pc, covariateLaw]
rw [Measure.map_map (by fun_prop) (by unfold observed; fun_prop)]
exact completionBind_covariate_marginal mu B e p₀ p₁ hk.aemeasurable
(Filter.Eventually.of_forall hvalid)
unfold BoundedMeanSubGaussianResidual
rw [hcov]
filter_upwards [htreat] with x hsupp
let ν := treatedKernel P x
haveI : IsProbabilityMeasure ν := by
dsimp [ν, treatedKernel]
infer_instance
have hint : Integrable (fun y : ℝ => y) ν :=
Integrable.of_mem_Icc 0 B measurable_id.aemeasurable hsupp
have hmean0 : 0 ≤ ∫ y : ℝ, y ∂ν := integral_nonneg_of_ae (hsupp.mono fun _ h => h.1)
have hmeanB : ∫ y : ℝ, y ∂ν ≤ B := by
simpa using integral_mono_ae hint (integrable_const B) (hsupp.mono fun _ h => h.2)
have hsg := ProbabilityTheory.hasSubgaussianMGF_of_mem_Icc
(X := fun y : ℝ => y) (μ := ν) measurable_id.aemeasurable hsupp
constructor
· change |∫ y : ℝ, y ∂ν| ≤ B
rw [abs_of_nonneg hmean0]
exact hmeanB
· intro t
change Integrable (fun y : ℝ => exp (t * (y - ∫ z : ℝ, z ∂ν))) ν ∧
mgf (fun y : ℝ => y - ∫ z : ℝ, z ∂ν) ν t ≤
exp (B ^ 2 * t ^ 2 / 2)
constructor
· exact hsg.integrable_exp_mul t
· exact (hsg.mgf_le t).trans (Real.exp_le_exp.mpr (by
rw [show ‖B - 0‖₊ = ⟨B, hB.le⟩ by
apply Subtype.ext
simp [Real.norm_eq_abs, abs_of_pos hB]]
change (B / 2) ^ 2 * t ^ 2 / 2 ≤ B ^ 2 * t ^ 2 / 2
nlinarith [sq_nonneg B, sq_nonneg t]))
theorem completionBind_uniformCube_globalTailModel Lemma completionBind_uniformCube_globalTailModel in the paper ↗
A valid explicit completion over the uniform cube is a global-tail model once its response representative and sharp propensity tail are supplied. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ :
ℝ
hB :
0 < B
hC :
1 ≤ C
hcf :
c_f ≤ 1
e p₀ p₁ μ₁ :
(Fin d → ℝ) → ℝ
he :
hp₀ :
Measurable p₀
hp₁ :
Measurable p₁
hpos :
∀ᵐ x ∂volume.restrict (cube d), 0 < e x
hholder :
conclusion
@GlobalTailModel d β B L C c_f γ ((volume.restrict (cube d)).bind (completionAt B e p₀ p₁)) inferInstance μ₁ e
Proof (Lean source)
lemma completionBind_uniformCube_globalTailModel {d : ℕ}
(β B L C c_f γ : ℝ) (hB : 0 < B) (hC : 1 ≤ C) (hcf : c_f ≤ 1)
(e p₀ p₁ μ₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hp₀ : Measurable p₀) (hp₁ : Measurable p₁)
(hvalid : ∀ x, e x ∈ Icc 0 1 ∧ p₀ x ∈ Icc 0 1 ∧
p₁ x ∈ Icc 0 1)
(hpos : ∀ᵐ x ∂volume.restrict (cube d), 0 < e x)
(htail : ∀ t ∈ Icc (0 : ℝ) 1,
(volume.restrict (cube d)).real {x | e x ≤ t} ≤ t ^ (γ - 1))
[IsProbabilityMeasure
((volume.restrict (cube d)).bind (completionAt B e p₀ p₁))]
(hholder : HolderResponse
((volume.restrict (cube d)).bind (completionAt B e p₀ p₁)) μ₁ β L) :
@GlobalTailModel d β B L C c_f γ
((volume.restrict (cube d)).bind (completionAt B e p₀ p₁)) inferInstance
μ₁ e := by
let mu := volume.restrict (cube d)
letI : IsProbabilityMeasure mu := uniformCube_probability_lowerPair d
let Pc := mu.bind (completionAt B e p₀ p₁)
let P := Pc.map observed
have hk : Measurable (completionAt B e p₀ p₁) :=
completionAt_measurable_of_measurable B e p₀ p₁ he hp₀ hp₁ hvalid
letI : IsProbabilityMeasure Pc := by
apply isProbabilityMeasure_bind hk.aemeasurable
filter_upwards [] with x
letI := realBernoulli_probability (e x) (hvalid x).1
letI := scaledBernoulli_probability B (p₀ x) (hvalid x).2.1
letI := scaledBernoulli_probability B (p₁ x) (hvalid x).2.2
unfold completionAt
apply Measure.isProbabilityMeasure_map
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption)
letI : IsProbabilityMeasure P := by
dsimp [P]
apply Measure.isProbabilityMeasure_map
unfold observed
fun_prop
have hcov : covariateLaw P = mu := by
dsimp [P, Pc, covariateLaw]
rw [Measure.map_map (by fun_prop) (by unfold observed; fun_prop)]
exact completionBind_covariate_marginal mu B e p₀ p₁ hk.aemeasurable
(Filter.Eventually.of_forall hvalid)
refine {
cubeSupport := ?_
covariateAC := ?_
density := ?_
consistency := ?_
exchangeability := ?_
tail := ?_
outcome := ?_
smooth := hholder }
· rw [hcov]
change (volume.restrict (cube d)) (cube d) = 1
rw [Measure.restrict_apply (by simp [cube] : MeasurableSet (cube d)),
Set.inter_self]
have hcube : cube d =
Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hcube, Real.volume_Icc_pi]
simp [observed]
· rw [hcov]
· unfold CovariateDensityLowerBound covariateDensity
rw [hcov]
filter_upwards [Measure.rnDeriv_self mu] with x hx
rw [hx, ENNReal.toReal_one]
exact hcf
· exact completionBind_consistency mu B e p₀ p₁ hk.aemeasurable
· exact completionBind_condExchangeable mu B e p₀ p₁ he hp₀ hp₁ hvalid
· constructor
· have hpraw := completionBind_propensity_ae mu B e p₀ p₁
he hp₀ hp₁ hvalid
dsimp at hpraw
rw [hcov]
filter_upwards [hpraw] with x hx
exact hx.symm
· intro t ht
rw [hcov]
exact (htail t ht).trans (by
have hnonneg := Real.rpow_nonneg ht.1 (γ - 1)
nlinarith [hC])
· exact completionBind_boundedOutcome_of_endpoint_support mu B hB e p₀ p₁
he hp₀ hp₁ hvalid hpos
theorem genericCommonBernoulli_klDiv_le_of_localized_parameter Lemma genericCommonBernoulli_klDiv_le_of_localized_parameter in the paper ↗
The localized common-statistic Bernoulli KL estimate over an arbitrary measurable base space. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
S :
Type*
m :
p q :
S → ℝ
hp :
hq :
hp0 :
∀ r, 1 / 4 ≤ p r
hp1 :
∀ r, p r ≤ 3 / 4
hq0 :
∀ r, 1 / 4 ≤ q r
hq1 :
∀ r, q r ≤ 3 / 4
D :
ℝ
hD :
0 ≤ D
E :
Set S
hE :
hdiff :
∀ᵐ r ∂m, |p r - q r| ≤ E.indicator (fun _ => D) r
conclusion
klDiv (Measure.compProd m (commonStatisticBernoulliKernel p hp)) (Measure.compProd m (commonStatisticBernoulliKernel q hq))
≤ ofReal (4 * D ^ 2) * m E
Proof (Lean source)
lemma genericCommonBernoulli_klDiv_le_of_localized_parameter
{S : Type*} [MeasurableSpace S]
(m : Measure S) [IsFiniteMeasure m] (p q : S → ℝ)
(hp : Measurable p) (hq : Measurable q)
(hp0 : ∀ r, 1 / 4 ≤ p r) (hp1 : ∀ r, p r ≤ 3 / 4)
(hq0 : ∀ r, 1 / 4 ≤ q r) (hq1 : ∀ r, q r ≤ 3 / 4)
{D : ℝ} (hD : 0 ≤ D) {E : Set S} (hE : MeasurableSet E)
(hdiff : ∀ᵐ r ∂m, |p r - q r| ≤ E.indicator (fun _ => D) r) :
klDiv
(Measure.compProd m
(Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp))
(Measure.compProd m
(Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel q hq)) ≤
ofReal (4 * D ^ 2) * m E := by
let k : Kernel S ℝ :=
Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp
let k' : Kernel S ℝ :=
Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel q hq
letI : IsMarkovKernel k :=
commonStatisticBernoulliKernel_isMarkovKernel p hp
(fun r => by linarith [hp0 r]) (fun r => by linarith [hp1 r])
letI : IsMarkovKernel k' :=
commonStatisticBernoulliKernel_isMarkovKernel q hq
(fun r => by linarith [hq0 r]) (fun r => by linarith [hq1 r])
rw [klDiv_compProd_right_of_forall_ac]
· calc
(∫⁻ r, klDiv (k r) (k' r) ∂m) ≤
∫⁻ r, E.indicator (fun _ => ofReal (4 * D ^ 2)) r ∂m := by
apply lintegral_mono_ae
filter_upwards [hdiff] with r hr
have hkl :=
bernoulliLaw_klDiv_le_four_sq_sub
(hp0 r) (hp1 r) (hq0 r) (hq1 r)
change klDiv (k r) (k' r) ≤ _ at hkl
by_cases hrE : r ∈ E
· rw [Set.indicator_of_mem hrE]
exact hkl.trans (ENNReal.ofReal_le_ofReal (by
have habs : |p r - q r| ≤ D := by simpa [hrE] using hr
have hsq := (sq_le_sq₀ (abs_nonneg (p r - q r)) hD).2 habs
rw [← sq_abs (p r - q r)]
nlinarith))
· rw [Set.indicator_of_notMem hrE]
have hpq : p r = q r := by
have hz : |p r - q r| ≤ 0 := by simpa [hrE] using hr
exact sub_eq_zero.mp (abs_eq_zero.mp
(le_antisymm hz (abs_nonneg _)))
have hkk : k r = k' r := by
ext A hA
simp [k, k',
Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel, hpq]
rw [hkk, InformationTheory.klDiv_self]
_ = ∫⁻ _r in E, ofReal (4 * D ^ 2) ∂m :=
lintegral_indicator hE _
_ = ofReal (4 * D ^ 2) * m E :=
setLIntegral_const E (ofReal (4 * D ^ 2))
· filter_upwards with r
have hkl :=
bernoulliLaw_klDiv_le_four_sq_sub
(hp0 r) (hp1 r) (hq0 r) (hq1 r)
have hfinite : klDiv (k r) (k' r) ≠ ⊤ := by
change klDiv (k r) (k' r) ≤ _ at hkl
exact ne_top_of_le_ne_top ENNReal.ofReal_ne_top hkl
exact (InformationTheory.klDiv_ne_top_iff.mp hfinite).1
theorem completionAt_observed_map_unequal_arms Lemma completionAt_observed_map_unequal_arms in the paper ↗
At a fixed covariate, the observed completion is the disjoint sum of its control and treated scaled-Bernoulli cells. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma completionAt_observed_map_unequal_arms {d : ℕ} (B : ℝ)
(e p₀ p₁ : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ)
(he : e x ∈ Icc 0 1) (hp₀ : p₀ x ∈ Icc 0 1)
(hp₁ : p₁ x ∈ Icc 0 1) :
(completionAt B e p₀ p₁ x).map observed =
ofReal (1 - e x) •
((Measure.dirac x).prod
((Measure.dirac false).prod (scaledBernoulli B (p₀ x)))) +
ofReal (e x) •
((Measure.dirac x).prod
((Measure.dirac true).prod (scaledBernoulli B (p₁ x)))) := by
letI := realBernoulli_probability (e x) he
letI := scaledBernoulli_probability B (p₀ x) hp₀
letI := scaledBernoulli_probability B (p₁ x) hp₁
unfold completionAt
rw [Measure.map_map (by unfold observed; fun_prop) (by
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption))]
rw [Measure.dirac_prod]
let g : Bool × ℝ × ℝ → Obs d :=
fun z => (x, z.1, if z.1 then z.2.2 else z.2.1)
change Measure.map g ((realBernoulli (e x)).prod
((scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x)))) = _
have hg : Measurable g := by
dsimp [g]
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption)
unfold realBernoulli
rw [Measure.add_prod, Measure.map_add _ _ hg]
congr 1
· rw [Measure.prod_smul_left, Measure.map_smul]
congr 1
rw [Measure.dirac_prod, Measure.map_map hg (by fun_prop)]
change Measure.map ((fun y : ℝ => (x, false, y)) ∘ fst)
((scaledBernoulli B (p₀ x)).prod
(scaledBernoulli B (p₁ x))) = _
rw [← Measure.map_map (by fun_prop : Measurable (fun y : ℝ => (x, false, y)))
measurable_fst, Measure.map_fst_prod]
simp only [measure_univ, one_smul]
rw [Measure.dirac_prod (x := false)]
rw [Measure.map_map (by fun_prop) (by fun_prop)]
rfl
· rw [Measure.prod_smul_left, Measure.map_smul]
congr 1
rw [Measure.dirac_prod, Measure.map_map hg (by fun_prop)]
change Measure.map ((fun y : ℝ => (x, true, y)) ∘ snd)
((scaledBernoulli B (p₀ x)).prod
(scaledBernoulli B (p₁ x))) = _
rw [← Measure.map_map (by fun_prop : Measurable (fun y : ℝ => (x, true, y)))
measurable_snd, Measure.map_snd_prod]
simp only [measure_univ, one_smul]
rw [Measure.dirac_prod (x := true)]
rw [Measure.dirac_prod x]
rw [Measure.map_map (by fun_prop) (by fun_prop)]
rfl
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma dirac_compProd_eq_map_prodMk
{S T : Type*} [MeasurableSpace S] [MeasurableSpace T]
[MeasurableSingletonClass S] (s : S) (k : Kernel S T)
[IsSFiniteKernel k] :
Measure.compProd (Measure.dirac s) k = (k s).map (Prod.mk s) := by
ext A hA
rw [Measure.dirac_compProd_apply hA,
Measure.map_apply measurable_prodMk_left hA]
theorem completionAt_observed_eq_selected_compProd Lemma completionAt_observed_eq_selected_compProd in the paper ↗
The fixed-covariate observed law is a measurable embedding of a treatment Bernoulli followed by the selected arm's scaled Bernoulli response. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
B :
ℝ
e p₀ p₁ :
(Fin d → ℝ) → ℝ
x :
Fin d → ℝ
he :
e x ∈ Icc 0 1
hp₀ :
p₀ x ∈ Icc 0 1
hp₁ :
p₁ x ∈ Icc 0 1
conclusion
kY :
ProbabilityTheory.Kernel.mk (fun a => scaledBernoulli B (if a then p₁ x else p₀ x)) (measurable_of_finite _) (: Kernel Bool ℝ)
(completionAt B e p₀ p₁ x).map observed
= (Measure.compProd (realBernoulli (e x)) kY).map (fun ay => (x, ay.1, ay.2))
Proof (Lean source)
lemma completionAt_observed_eq_selected_compProd {d : ℕ} (B : ℝ)
(e p₀ p₁ : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ)
(he : e x ∈ Icc 0 1) (hp₀ : p₀ x ∈ Icc 0 1)
(hp₁ : p₁ x ∈ Icc 0 1) :
let kY : Kernel Bool ℝ :=
ProbabilityTheory.Kernel.mk
(fun a => scaledBernoulli B (if a then p₁ x else p₀ x))
(measurable_of_finite _)
(completionAt B e p₀ p₁ x).map observed =
(Measure.compProd (realBernoulli (e x)) kY).map
(fun ay => (x, ay.1, ay.2)) := by
dsimp
let kY : Kernel Bool ℝ :=
ProbabilityTheory.Kernel.mk
(fun a => scaledBernoulli B (if a then p₁ x else p₀ x))
(measurable_of_finite _)
letI := scaledBernoulli_probability B (p₀ x) hp₀
letI := scaledBernoulli_probability B (p₁ x) hp₁
letI : IsMarkovKernel kY := ⟨by
intro a
cases a
· exact scaledBernoulli_probability B (p₀ x) hp₀
· exact scaledBernoulli_probability B (p₁ x) hp₁⟩
rw [completionAt_observed_map_unequal_arms B e p₀ p₁ x he hp₀ hp₁]
have hsource : Measure.compProd (realBernoulli (e x)) kY =
ofReal (1 - e x) •
(scaledBernoulli B (p₀ x)).map (Prod.mk false) +
ofReal (e x) •
(scaledBernoulli B (p₁ x)).map (Prod.mk true) := by
unfold realBernoulli
rw [Measure.compProd_add_left,
Measure.compProd_smul_left, Measure.compProd_smul_left,
dirac_compProd_eq_map_prodMk false kY,
dirac_compProd_eq_map_prodMk true kY]
rfl
rw [hsource, Measure.map_add _ _ (by fun_prop),
Measure.map_smul, Measure.map_smul]
congr 1
· rw [Measure.map_map (by fun_prop) (by fun_prop),
Measure.dirac_prod, Measure.dirac_prod,
Measure.map_map (by fun_prop) (by fun_prop)]
· rw [Measure.map_map (by fun_prop) (by fun_prop),
Measure.dirac_prod, Measure.dirac_prod,
Measure.map_map (by fun_prop) (by fun_prop)]
theorem scaledBernoulli_eq_map_bernoulliLaw Lemma scaledBernoulli_eq_map_bernoulliLaw in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma scaledBernoulli_eq_map_bernoulliLaw (B p : ℝ) :
scaledBernoulli B p =
(bernoulliLaw p).map (fun y => B * y) := by
unfold scaledBernoulli realBernoulli
bernoulliLaw
rw [Measure.map_add _ _ (by fun_prop), Measure.map_smul, Measure.map_smul,
Measure.map_add _ _ (by fun_prop), Measure.map_smul, Measure.map_smul]
simp [add_comm]
theorem genericCommonBernoulli_klDiv_le_of_localized_quadratic Lemma genericCommonBernoulli_klDiv_le_of_localized_quadratic in the paper ↗
A localized common-statistic KL bound from any supplied pointwise quadratic Bernoulli estimate. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
S :
Type*
m :
p q :
S → ℝ
hp :
hq :
hp01 :
∀ r, p r ∈ Icc (0 : ℝ) 1
hq01 :
∀ r, q r ∈ Icc (0 : ℝ) 1
K D :
ℝ
hK :
0 ≤ K
hD :
0 ≤ D
hpoint :
E :
Set S
hE :
hdiff :
∀ᵐ r ∂m, |p r - q r| ≤ E.indicator (fun _ => D) r
conclusion
klDiv (Measure.compProd m (commonStatisticBernoulliKernel p hp)) (Measure.compProd m (commonStatisticBernoulliKernel q hq))
≤ ofReal (K * D ^ 2) * m E
Proof (Lean source)
lemma genericCommonBernoulli_klDiv_le_of_localized_quadratic
{S : Type*} [MeasurableSpace S]
(m : Measure S) [IsFiniteMeasure m] (p q : S → ℝ)
(hp : Measurable p) (hq : Measurable q)
(hp01 : ∀ r, p r ∈ Icc (0 : ℝ) 1)
(hq01 : ∀ r, q r ∈ Icc (0 : ℝ) 1)
{K D : ℝ} (hK : 0 ≤ K) (hD : 0 ≤ D)
(hpoint : ∀ r, klDiv
(bernoulliLaw (p r))
(bernoulliLaw (q r)) ≤
ofReal (K * (p r - q r) ^ 2))
{E : Set S} (hE : MeasurableSet E)
(hdiff : ∀ᵐ r ∂m, |p r - q r| ≤ E.indicator (fun _ => D) r) :
klDiv
(Measure.compProd m
(Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp))
(Measure.compProd m
(Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel q hq)) ≤
ofReal (K * D ^ 2) * m E := by
let k := Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp
let k' := Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel q hq
letI : IsMarkovKernel k :=
commonStatisticBernoulliKernel_isMarkovKernel p hp
(fun r => (hp01 r).1) (fun r => (hp01 r).2)
letI : IsMarkovKernel k' :=
commonStatisticBernoulliKernel_isMarkovKernel q hq
(fun r => (hq01 r).1) (fun r => (hq01 r).2)
rw [klDiv_compProd_right_of_forall_ac]
· calc
(∫⁻ r, klDiv (k r) (k' r) ∂m) ≤
∫⁻ r, E.indicator (fun _ => ofReal (K * D ^ 2)) r ∂m := by
apply lintegral_mono_ae
filter_upwards [hdiff] with r hr
by_cases hrE : r ∈ E
· rw [Set.indicator_of_mem hrE]
exact (hpoint r).trans (ENNReal.ofReal_le_ofReal (by
have habs : |p r - q r| ≤ D := by simpa [hrE] using hr
have hsq := (sq_le_sq₀ (abs_nonneg (p r - q r)) hD).2 habs
rw [← sq_abs (p r - q r)]
exact mul_le_mul_of_nonneg_left hsq hK))
· rw [Set.indicator_of_notMem hrE]
have hpq : p r = q r := by
have hz : |p r - q r| ≤ 0 := by simpa [hrE] using hr
exact sub_eq_zero.mp (abs_eq_zero.mp
(le_antisymm hz (abs_nonneg _)))
have hkk : k r = k' r := by
ext A hA
simp [k, k',
Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel, hpq]
rw [hkk, InformationTheory.klDiv_self]
_ = ∫⁻ _r in E, ofReal (K * D ^ 2) ∂m :=
lintegral_indicator hE _
_ = ofReal (K * D ^ 2) * m E :=
setLIntegral_const E (ofReal (K * D ^ 2))
· filter_upwards with r
have hfinite : klDiv (k r) (k' r) ≠ ⊤ := by
exact ne_top_of_le_ne_top ENNReal.ofReal_ne_top (hpoint r)
exact (InformationTheory.klDiv_ne_top_iff.mp hfinite).1
theorem selectedScaled_compProd_eq_map_commonBernoulli Lemma selectedScaled_compProd_eq_map_commonBernoulli in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
kS :
ProbabilityTheory.Kernel.mk (fun a => scaledBernoulli B (p a)) (measurable_of_finite _) (: Kernel Bool ℝ)
kB :
Proof (Lean source)
lemma selectedScaled_compProd_eq_map_commonBernoulli
(m : Measure Bool) [SFinite m] (B : ℝ) (p : Bool → ℝ)
(hp : Measurable p) (hp01 : ∀ a, p a ∈ Icc (0 : ℝ) 1) :
let kS : Kernel Bool ℝ :=
ProbabilityTheory.Kernel.mk (fun a => scaledBernoulli B (p a))
(measurable_of_finite _)
let kB := Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp
Measure.compProd m kS =
(Measure.compProd m kB).map (map id (fun y => B * y)) := by
dsimp
let kS : Kernel Bool ℝ :=
ProbabilityTheory.Kernel.mk (fun a => scaledBernoulli B (p a))
(measurable_of_finite _)
let kB := Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp
letI : IsMarkovKernel kB :=
commonStatisticBernoulliKernel_isMarkovKernel p hp
(fun a => (hp01 a).1) (fun a => (hp01 a).2)
have hk : kS = kB.map (fun y => B * y) := by
ext a A hA
rw [ProbabilityTheory.Kernel.map_apply _ (by fun_prop)]
exact congrArg (fun ν : Measure ℝ => ν A)
(scaledBernoulli_eq_map_bernoulliLaw B (p a))
change Measure.compProd m kS =
(Measure.compProd m kB).map (map id (fun y => B * y))
rw [hk, Measure.compProd_map (by fun_prop)]
theorem lowerPair_completionAt_observed_klDiv_le Lemma lowerPair_completionAt_observed_klDiv_le in the paper ↗
The observed fixed-covariate KL has the treatment-propensity weight. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
x₀ x :
Fin d → ℝ
B L β γ a h :
ℝ
hβ :
0 < β
hB :
0 < B
hL :
0 < L
hγ :
1 < γ
ha :
0 ≤ a ∧ a ≤ baselineSuccess B L
hh :
0 < h ∧ h ≤ 1
conclusion
klDiv ((completionAt B (radialPropensity x₀ γ) (fun _ => baselineSuccess B L) (pairSuccess false x₀ β B L a h) x).map observed) ((completionAt B (radialPropensity x₀ γ) (fun _ => baselineSuccess B L) (pairSuccess true x₀ β B L a h) x).map observed)
Proof (Lean source)
lemma lowerPair_completionAt_observed_klDiv_le {d : ℕ}
(x₀ x : Fin d → ℝ) (B L β γ a h : ℝ)
(hβ : 0 < β) (hB : 0 < B) (hL : 0 < L) (hγ : 1 < γ)
(ha : 0 ≤ a ∧ a ≤ baselineSuccess B L)
(hh : 0 < h ∧ h ≤ 1) :
klDiv
((completionAt B (radialPropensity x₀ γ)
(fun _ => baselineSuccess B L)
(pairSuccess false x₀ β B L a h) x).map observed)
((completionAt B (radialPropensity x₀ γ)
(fun _ => baselineSuccess B L)
(pairSuccess true x₀ β B L a h) x).map observed) ≤
ofReal ((2 / baselineSuccess B L) * (a * h ^ β) ^ 2) *
ofReal (radialPropensity x₀ γ x) := by
let e := radialPropensity x₀ γ
let p₀ : (Fin d → ℝ) → ℝ := fun _ => baselineSuccess B L
let pF := pairSuccess false x₀ β B L a h
let pT := pairSuccess true x₀ β B L a h
have he := radialPropensity_mem_Icc x₀ x γ hγ
letI := realBernoulli_probability (radialPropensity x₀ γ x) he
have hpF := pairSuccess_mem_Icc_of_small false x₀ β B L a h hB hL hβ
ha ⟨hh.1.le, hh.2⟩ x
have hpT := pairSuccess_mem_Icc_of_small true x₀ β B L a h hB hL hβ
ha ⟨hh.1.le, hh.2⟩ x
let m := realBernoulli (e x)
let p : Bool → ℝ := fun z => if z then pF x else p₀ x
let q : Bool → ℝ := fun z => if z then pT x else p₀ x
have hp : Measurable p := measurable_of_finite _
have hq : Measurable q := measurable_of_finite _
have hp01 : ∀ z, p z ∈ Icc (0 : ℝ) 1 := fun z => by
cases z
· simpa [p, p₀, pF, pairSuccess] using hpF
· simpa [p, pF] using hpF
have hq01 : ∀ z, q z ∈ Icc (0 : ℝ) 1 := fun z => by
cases z
· simpa [q, p₀, pT, pairSuccess] using hpF
· simpa [q, pT] using hpT
let kSF : Kernel Bool ℝ :=
ProbabilityTheory.Kernel.mk (fun z => scaledBernoulli B (p z))
(measurable_of_finite _)
let kST : Kernel Bool ℝ :=
ProbabilityTheory.Kernel.mk (fun z => scaledBernoulli B (q z))
(measurable_of_finite _)
let kBF := Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp
let kBT := Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel q hq
letI : IsMarkovKernel kSF := ⟨fun z =>
scaledBernoulli_probability B (p z) (hp01 z)⟩
letI : IsMarkovKernel kST := ⟨fun z =>
scaledBernoulli_probability B (q z) (hq01 z)⟩
letI : IsMarkovKernel kBF :=
commonStatisticBernoulliKernel_isMarkovKernel p hp
(fun z => (hp01 z).1) (fun z => (hp01 z).2)
letI : IsMarkovKernel kBT :=
commonStatisticBernoulliKernel_isMarkovKernel q hq
(fun z => (hq01 z).1) (fun z => (hq01 z).2)
have hobsF := completionAt_observed_eq_selected_compProd B e p₀ pF x he
(by simpa [p₀, pF, pairSuccess] using hpF) hpF
have hobsT := completionAt_observed_eq_selected_compProd B e p₀ pT x he
(by simpa [p₀, pT, pairSuccess] using hpF) hpT
have hsF : Measure.compProd m kSF =
(Measure.compProd m kBF).map (map id (fun y => B * y)) := by
exact selectedScaled_compProd_eq_map_commonBernoulli m B p hp hp01
have hsT : Measure.compProd m kST =
(Measure.compProd m kBT).map (map id (fun y => B * y)) := by
exact selectedScaled_compProd_eq_map_commonBernoulli m B q hq hq01
rw [hobsF, hobsT]
change klDiv
((Measure.compProd m kSF).map (fun ay => (x, ay.1, ay.2)))
((Measure.compProd m kST).map (fun ay => (x, ay.1, ay.2))) ≤ _
calc
_ ≤ klDiv (Measure.compProd m kSF)
(Measure.compProd m kST) := InformationTheory.klDiv_map_le _ _ (by fun_prop)
_ ≤ klDiv (Measure.compProd m kBF)
(Measure.compProd m kBT) := by
rw [hsF, hsT]
exact InformationTheory.klDiv_map_le _ _ (by fun_prop)
_ ≤ ofReal ((2 / baselineSuccess B L) * (a * h ^ β) ^ 2) *
m {true} := by
apply genericCommonBernoulli_klDiv_le_of_localized_quadratic m p q hp hq
hp01 hq01
(by
show 0 ≤ 2 / baselineSuccess B L
unfold baselineSuccess
positivity)
(mul_nonneg ha.1 (Real.rpow_nonneg hh.1.le _))
· intro z
cases z
· simp only [p, q, Bool.false_eq_true, if_false]
have hbase01 : p₀ x ∈ Icc (0 : ℝ) 1 := by
simpa [p₀, pF, pairSuccess] using hpF
letI := bernoulliLaw_isProbabilityMeasure
hbase01.1 hbase01.2
rw [InformationTheory.klDiv_self]
exact bot_le
· have hpi := pairSuccess_mem_Ioo_of_small false x₀ x β B L a h
hB hL ha ⟨hh.1.le, hh.2⟩ hβ.le
have hqi := pairSuccess_mem_Ioo_of_small true x₀ x β B L a h
hB hL ha ⟨hh.1.le, hh.2⟩ hβ.le
have hfin : klDiv
(bernoulliLaw (pF x))
(bernoulliLaw (pT x)) ≠ ⊤ :=
InformationTheory.klDiv_ne_top
(bernoulliLaw_ac_of_reference_interior
hqi.1 hqi.2)
bernoulliLaw_llr_integrable
change klDiv
(bernoulliLaw (pF x))
(bernoulliLaw (pT x)) ≤
ofReal ((2 / baselineSuccess B L) * (pF x - pT x) ^ 2)
rw [← ENNReal.ofReal_toReal hfin]
apply ENNReal.ofReal_le_ofReal
rw [bernoulliLaw_klDiv_toReal
hpi.1.le hpi.2.le hqi.1 hqi.2]
simpa [p, q, pF, pT] using
lowerPair_pointwise_kl_bound x₀ x β B L a h hβ hB hL ha
⟨hh.1.le, hh.2⟩
· exact MeasurableSet.singleton true
· filter_upwards with z
cases z
· simp [p, q, p₀]
· simp only [Set.indicator_of_mem (Set.mem_singleton true)]
change |pF x - pT x| ≤ a * h ^ β
rw [abs_sub_comm, abs_of_nonneg
(pairSuccess_perturbation_bound x₀ x β B L a h ha.1 hh.1.le).1]
exact (pairSuccess_perturbation_bound x₀ x β B L a h ha.1 hh.1.le).2
_ = _ := by
change _ * (realBernoulli (e x)) {true} = _
simp [realBernoulli, e, he.1]
Integrating the fixed-covariate bound over the bump support gives the raw one-observation KL rate. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
x₀ :
Fin d → ℝ
hx₀ :
x₀ ∈ cube d
B L β γ a h :
ℝ
hβ :
0 < β
hB :
0 < B
hL :
0 < L
hγ :
1 < γ
ha :
0 ≤ a ∧ a ≤ baselineSuccess B L
hh :
0 < h ∧ h ≤ 1
conclusion
mu :
volume.restrict (cube d)
e :
radialPropensity x₀ γ
p₀ :
fun _ => baselineSuccess B L (: (Fin d → ℝ) → ℝ)
pF :
pairSuccess false x₀ β B L a h
pT :
pairSuccess true x₀ β B L a h
PF :
(mu.bind (completionAt B e p₀ pF)).map observed
PT :
(mu.bind (completionAt B e p₀ pT)).map observed
klDiv PF PT
≤ ofReal ((2 / baselineSuccess B L) * (a * h ^ β) ^ 2 * (((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)))) * ofReal ((2 * h) ^ d)
Proof (Lean source)
lemma lowerPair_observed_klDiv_le_raw {d : ℕ}
(x₀ : Fin d → ℝ) (hx₀ : x₀ ∈ cube d) (B L β γ a h : ℝ)
(hβ : 0 < β) (hB : 0 < B) (hL : 0 < L) (hγ : 1 < γ)
(ha : 0 ≤ a ∧ a ≤ baselineSuccess B L)
(hh : 0 < h ∧ h ≤ 1) :
let mu := volume.restrict (cube d)
let e := radialPropensity x₀ γ
let p₀ : (Fin d → ℝ) → ℝ := fun _ => baselineSuccess B L
let pF := pairSuccess false x₀ β B L a h
let pT := pairSuccess true x₀ β B L a h
let PF := (mu.bind (completionAt B e p₀ pF)).map observed
let PT := (mu.bind (completionAt B e p₀ pT)).map observed
klDiv PF PT ≤
ofReal ((2 / baselineSuccess B L) * (a * h ^ β) ^ 2 *
(((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)))) *
ofReal ((2 * h) ^ d) := by
dsimp
let mu := volume.restrict (cube d)
letI : IsProbabilityMeasure mu := uniformCube_probability_lowerPair d
let e := radialPropensity x₀ γ
let p₀ : (Fin d → ℝ) → ℝ := fun _ => baselineSuccess B L
let pF := pairSuccess false x₀ β B L a h
let pT := pairSuccess true x₀ β B L a h
have he : Measurable e := radialPropensity_measurable x₀ γ
have hp₀ : Measurable p₀ := measurable_const
have hpF : Measurable pF := pairSuccess_measurable false x₀ β B L a h
have hpT : Measurable pT := pairSuccess_measurable true x₀ β B L a h
have hvalidF : ∀ x, e x ∈ Icc 0 1 ∧ p₀ x ∈ Icc 0 1 ∧
pF x ∈ Icc 0 1 := by
intro x
exact ⟨radialPropensity_mem_Icc x₀ x γ hγ,
by simpa [p₀, pF, pairSuccess] using
pairSuccess_mem_Icc_of_small false x₀ β B L a h hB hL hβ ha
⟨hh.1.le, hh.2⟩ x,
pairSuccess_mem_Icc_of_small false x₀ β B L a h hB hL hβ ha
⟨hh.1.le, hh.2⟩ x⟩
have hvalidT : ∀ x, e x ∈ Icc 0 1 ∧ p₀ x ∈ Icc 0 1 ∧
pT x ∈ Icc 0 1 := by
intro x
exact ⟨radialPropensity_mem_Icc x₀ x γ hγ,
by simpa [p₀, pF, pairSuccess] using
pairSuccess_mem_Icc_of_small false x₀ β B L a h hB hL hβ ha
⟨hh.1.le, hh.2⟩ x,
pairSuccess_mem_Icc_of_small true x₀ β B L a h hB hL hβ ha
⟨hh.1.le, hh.2⟩ x⟩
have hcF : Measurable (completionAt B e p₀ pF) :=
completionAt_measurable_of_measurable B e p₀ pF he hp₀ hpF hvalidF
have hcT : Measurable (completionAt B e p₀ pT) :=
completionAt_measurable_of_measurable B e p₀ pT he hp₀ hpT hvalidT
let kF : Kernel (Fin d → ℝ) (Obs d) :=
(ProbabilityTheory.Kernel.mk (completionAt B e p₀ pF) hcF).map observed
let kT : Kernel (Fin d → ℝ) (Obs d) :=
(ProbabilityTheory.Kernel.mk (completionAt B e p₀ pT) hcT).map observed
letI : IsMarkovKernel kF := ⟨fun x => by
dsimp [kF]
rw [ProbabilityTheory.Kernel.map_apply _ (by unfold observed; fun_prop)]
letI := completionAt_probability B e p₀ pF x
(hvalidF x).1 (hvalidF x).2.1 (hvalidF x).2.2
exact Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)⟩
letI : IsMarkovKernel kT := ⟨fun x => by
dsimp [kT]
rw [ProbabilityTheory.Kernel.map_apply _ (by unfold observed; fun_prop)]
letI := completionAt_probability B e p₀ pT x
(hvalidT x).1 (hvalidT x).2.1 (hvalidT x).2.2
exact Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)⟩
have hPF : (mu.bind (completionAt B e p₀ pF)).map observed = mu.bind kF := by
simpa [kF] using Measure.map_comp mu
(ProbabilityTheory.Kernel.mk (completionAt B e p₀ pF) hcF)
(by unfold observed; fun_prop)
have hPT : (mu.bind (completionAt B e p₀ pT)).map observed = mu.bind kT := by
simpa [kT] using Measure.map_comp mu
(ProbabilityTheory.Kernel.mk (completionAt B e p₀ pT) hcT)
(by unfold observed; fun_prop)
rw [hPF, hPT]
have hfib (p₁ : (Fin d → ℝ) → ℝ)
(k : Kernel (Fin d → ℝ) (Obs d))
(hk : ∀ x, (k x) = (completionAt B e p₀ p₁ x).map observed) :
∀ᵐ x ∂mu, (k x) {z | z.1 = x}ᶜ = 0 := by
filter_upwards with x
rw [hk x, Measure.map_apply (by unfold observed; fun_prop) (by measurability)]
unfold completionAt
rw [Measure.map_apply (by
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption)) (by measurability)]
have hset : (fun z : Bool × ℝ × ℝ =>
(x, z.1, z.2.1, z.2.2, if z.1 then z.2.2 else z.2.1)) ⁻¹'
observed ⁻¹' {z : Obs d | z.1 = x}ᶜ = ∅ := by
ext a
simp [observed]
rw [hset, measure_empty]
have hkF : ∀ x, kF x = (completionAt B e p₀ pF x).map observed := by
intro x
rw [show kF x = (completionAt B e p₀ pF x).map observed by
exact ProbabilityTheory.Kernel.map_apply _ (by unfold observed; fun_prop) x]
have hkT : ∀ x, kT x = (completionAt B e p₀ pT x).map observed := by
intro x
rw [show kT x = (completionAt B e p₀ pT x).map observed by
exact ProbabilityTheory.Kernel.map_apply _ (by unfold observed; fun_prop) x]
have hfinite : ∀ x, klDiv (kF x) (kT x) ≠ ⊤ := by
intro x
apply ne_top_of_le_ne_top
(ENNReal.mul_ne_top ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top)
rw [hkF x, hkT x]
simpa [e, p₀, pF, pT] using
lowerPair_completionAt_observed_klDiv_le x₀ x B L β γ a h hβ hB hL hγ ha hh
have hac : ∀ᵐ x ∂mu, kF x ≪ kT x :=
Filter.Eventually.of_forall fun x => (InformationTheory.klDiv_ne_top_iff.mp (hfinite x)).1
rw [klDiv_bind_eq_of_base_recording
mu kF kT fst measurable_fst (by measurability)
(hfib pF kF hkF) (hfib pT kT hkT) hac]
let E : Set (Fin d → ℝ) := {x | ∀ i, |x i - x₀ i| < h}
have hE : MeasurableSet E := by
dsimp [E]
measurability
let M := (2 / baselineSuccess B L) * (a * h ^ β) ^ 2 *
(((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)))
calc
(∫⁻ x, klDiv (kF x) (kT x) ∂mu) ≤
∫⁻ x, E.indicator (fun _ => ofReal M) x ∂mu := by
apply lintegral_mono
intro x
by_cases hxE : x ∈ E
· rw [Set.indicator_of_mem hxE]
change klDiv (kF x) (kT x) ≤ _
rw [hkF x, hkT x]
calc
_ ≤ ofReal ((2 / baselineSuccess B L) * (a * h ^ β) ^ 2) *
ofReal (e x) := by
simpa [e, p₀, pF, pT] using
lowerPair_completionAt_observed_klDiv_le x₀ x B L β γ a h
hβ hB hL hγ ha hh
_ ≤ ofReal ((2 / baselineSuccess B L) * (a * h ^ β) ^ 2) *
ofReal (((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1))) := by
gcongr
exact radialPropensity_le_on_radius x₀ x γ h hγ
((pi_norm_le_iff_of_nonneg hh.1.le).2 fun i => le_of_lt (hxE i))
_ = ofReal M := by
rw [← ENNReal.ofReal_mul (by unfold baselineSuccess; positivity)]
· rw [Set.indicator_of_notMem hxE]
have heq : pF x = pT x := by
by_contra hne
exact hxE (pairSuccess_difference_support x₀ x β B L a h hh.1 (Ne.symm hne))
change klDiv (kF x) (kT x) ≤ 0
have hkEq : kF x = kT x := by
rw [hkF x, hkT x]
unfold completionAt
rw [heq]
rw [hkEq, InformationTheory.klDiv_self]
_ = ofReal M * mu E := by
rw [lintegral_indicator hE, setLIntegral_const]
_ ≤ ofReal M * ofReal ((2 * h) ^ d) := by
gcongr
apply (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _)
(pow_nonneg (mul_nonneg (by norm_num) hh.1.le) _)).2
change mu.real E ≤ (2 * h) ^ d
calc
mu.real E ≤ mu.real {x | ‖x - x₀‖ ≤ h} := by
apply measureReal_mono _ (measure_ne_top _ _)
intro x hx
exact (pi_norm_le_iff_of_nonneg hh.1.le).2 fun i =>
(le_of_lt (hx i))
_ = radialCDF x₀ h := uniformCube_radius_sublevel_real x₀ hx₀ h hh.1.le
_ ≤ (2 * h) ^ d := radialCDF_le_cube_radius_volume x₀ h hh.1.le
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β γ a h :
ℝ
hh :
0 < h
hγ :
1 < γ
conclusion
(a * h ^ β) ^ 2 * (((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1))) * ((2 * h) ^ d)
= 2 ^ (effectiveDimension d γ) * a ^ 2 * h ^ (2 * β + effectiveDimension d γ)
Proof (Lean source)
lemma lowerPair_raw_rate_identity (d : ℕ) (β γ a h : ℝ)
(hh : 0 < h) (hγ : 1 < γ) :
(a * h ^ β) ^ 2 * (((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1))) *
((2 * h) ^ d) =
2 ^ (effectiveDimension d γ) * a ^ 2 *
h ^ (2 * β + effectiveDimension d γ) := by
rw [show (a * h ^ β) ^ 2 = a ^ 2 * h ^ (2 * β) by
rw [mul_pow]
congr 1
rw [← Real.rpow_natCast (x := h ^ β) (n := 2),
← Real.rpow_mul hh.le]
ring]
rw [← Real.rpow_natCast (x := 2 * h) (n := d)]
rw [← Real.rpow_mul (mul_nonneg (by norm_num) hh.le)]
rw [show a ^ 2 * h ^ (2 * β) * (2 * h) ^
((d : ℝ) * (1 / (γ - 1))) * (2 * h) ^ (d : ℝ) =
a ^ 2 * h ^ (2 * β) * ((2 * h) ^ ((d : ℝ) * (1 / (γ - 1))) *
(2 * h) ^ (d : ℝ)) by ring]
rw [← Real.rpow_add (mul_pos (by norm_num) hh)]
rw [Real.mul_rpow (by norm_num : (0 : ℝ) ≤ 2) hh.le]
rw [show a ^ 2 * h ^ (2 * β) *
(2 ^ ((d : ℝ) * (1 / (γ - 1)) + d) *
h ^ ((d : ℝ) * (1 / (γ - 1)) + d)) =
2 ^ ((d : ℝ) * (1 / (γ - 1)) + d) * a ^ 2 *
(h ^ (2 * β) * h ^ ((d : ℝ) * (1 / (γ - 1)) + d)) by ring]
rw [← Real.rpow_add hh]
unfold effectiveDimension
have he : (d : ℝ) * (1 / (γ - 1)) + d = (d : ℝ) * γ / (γ - 1) := by
field_simp [ne_of_gt (sub_pos.mpr hγ)]
ring
rw [he]
theorem lowerPair_observed_guards_and_kl_rate Lemma lowerPair_observed_guards_and_kl_rate in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
x₀ :
Fin d → ℝ
hx₀ :
x₀ ∈ cube d
B L β γ a h :
ℝ
hβ :
0 < β
hB :
0 < B
hL :
0 < L
hγ :
1 < γ
ha :
0 ≤ a ∧ a ≤ baselineSuccess B L
hh :
0 < h ∧ h ≤ 1
conclusion
mu :
volume.restrict (cube d)
e :
radialPropensity x₀ γ
p₀ :
fun _ => baselineSuccess B L (: (Fin d → ℝ) → ℝ)
pF :
pairSuccess false x₀ β B L a h
pT :
pairSuccess true x₀ β B L a h
PF :
(mu.bind (completionAt B e p₀ pF)).map observed
PT :
(mu.bind (completionAt B e p₀ pT)).map observed
PF ≪ PT
Integrable (llr PF PT) PF
≤ ((2 / baselineSuccess B L) * 2 ^ (effectiveDimension d γ)) * a ^ 2 * h ^ (2 * β + effectiveDimension d γ)
Proof (Lean source)
lemma lowerPair_observed_guards_and_kl_rate {d : ℕ}
(x₀ : Fin d → ℝ) (hx₀ : x₀ ∈ cube d) (B L β γ a h : ℝ)
(hβ : 0 < β) (hB : 0 < B) (hL : 0 < L) (hγ : 1 < γ)
(ha : 0 ≤ a ∧ a ≤ baselineSuccess B L)
(hh : 0 < h ∧ h ≤ 1) :
let mu := volume.restrict (cube d)
let e := radialPropensity x₀ γ
let p₀ : (Fin d → ℝ) → ℝ := fun _ => baselineSuccess B L
let pF := pairSuccess false x₀ β B L a h
let pT := pairSuccess true x₀ β B L a h
let PF := (mu.bind (completionAt B e p₀ pF)).map observed
let PT := (mu.bind (completionAt B e p₀ pT)).map observed
PF ≪ PT ∧ Integrable (llr PF PT) PF ∧
(klDiv PF PT).toReal ≤
((2 / baselineSuccess B L) * 2 ^ (effectiveDimension d γ)) *
a ^ 2 * h ^ (2 * β + effectiveDimension d γ) := by
dsimp
have hraw := lowerPair_observed_klDiv_le_raw x₀ hx₀ B L β γ a h
hβ hB hL hγ ha hh
have htopR : ofReal ((2 / baselineSuccess B L) * (a * h ^ β) ^ 2 *
(((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)))) *
ofReal ((2 * h) ^ d) ≠ ⊤ :=
ENNReal.mul_ne_top ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top
have hfin : klDiv
(((volume.restrict (cube d)).bind
(completionAt B (radialPropensity x₀ γ) (fun _ => baselineSuccess B L)
(pairSuccess false x₀ β B L a h))).map observed)
(((volume.restrict (cube d)).bind
(completionAt B (radialPropensity x₀ γ) (fun _ => baselineSuccess B L)
(pairSuccess true x₀ β B L a h))).map observed) ≠ ⊤ :=
ne_top_of_le_ne_top htopR hraw
refine ⟨(InformationTheory.klDiv_ne_top_iff.mp hfin).1,
(InformationTheory.klDiv_ne_top_iff.mp hfin).2, ?_⟩
have hreal := ENNReal.toReal_mono htopR hraw
have hMnonneg : 0 ≤ (2 / baselineSuccess B L) * (a * h ^ β) ^ 2 *
(((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1))) :=
mul_nonneg (mul_nonneg (by unfold baselineSuccess; positivity) (sq_nonneg _))
(Real.rpow_nonneg (pow_nonneg (mul_nonneg (by norm_num) hh.1.le) _) _)
have hpow : 0 ≤ (2 * h) ^ d := pow_nonneg (mul_nonneg (by norm_num) hh.1.le) _
rw [ENNReal.toReal_mul, ENNReal.toReal_ofReal hMnonneg,
ENNReal.toReal_ofReal hpow] at hreal
calc
_ ≤ ((2 / baselineSuccess B L) * (a * h ^ β) ^ 2 *
(((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)))) * ((2 * h) ^ d) := hreal
_ = _ := by
rw [show (2 / baselineSuccess B L) * (a * h ^ β) ^ 2 *
(((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1))) * ((2 * h) ^ d) =
(2 / baselineSuccess B L) *
((a * h ^ β) ^ 2 * (((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1))) *
((2 * h) ^ d)) by ring,
lowerPair_raw_rate_identity d β γ a h hh.1 hγ]
ring
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma treatedRegression_eq_of_measure_eq {d : ℕ}
(P Q : Measure (Obs d)) [IsFiniteMeasure P] [IsFiniteMeasure Q]
(h : P = Q) : treatedRegression P = treatedRegression Q := by
subst Q
rfl
Identification makes the noncomputably selected response representative agree on the cube with any displayed model witness for the same observed law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ :
ℝ
P :
Measure (Obs d)
hP :
P ∈ ModelClass d β B L C c_f γ
Pc :
μ e :
(Fin d → ℝ) → ℝ
hEq :
P = Pc.map observed
hmodel :
GlobalTailModel β B L C c_f γ Pc μ e
x :
Fin d → ℝ
hx :
x ∈ cube d
conclusion
responseOf P hP x = μ x
Proof (Lean source)
lemma responseOf_eq_given_model_response {d : ℕ} {β B L C c_f γ : ℝ}
(P : Measure (Obs d)) (hP : P ∈ ModelClass d β B L C c_f γ)
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ e : (Fin d → ℝ) → ℝ) (hEq : P = Pc.map observed)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ e)
(x : Fin d → ℝ) (hx : x ∈ cube d) :
responseOf P hP x = μ x := by
let Pc' : Measure (Completion d) := choose hP.2.2
let hPc' := Classical.choose_spec hP.2.2
let hpPc' : IsProbabilityMeasure Pc' := choose hPc'
let hrest := Classical.choose_spec hPc'
let μ' : (Fin d → ℝ) → ℝ := choose hrest
let hrest' := Classical.choose_spec hrest
let e' : (Fin d → ℝ) → ℝ := choose hrest'
have hchosen := Classical.choose_spec hrest'
letI : IsProbabilityMeasure Pc' := hpPc'
have hP' : P = Pc'.map observed := hchosen.1
have hmodel' : GlobalTailModel β B L C c_f γ Pc' μ' e' := hchosen.2
have hid' := (mu1_eq_treatedRegression β B L C c_f γ Pc' μ' e'
hP.1 hP.2.1 hmodel').2.1
have hcont' : ContinuousOn μ' (cube d) :=
hmodel'.smooth.1.regularity.continuousOn
letI : IsProbabilityMeasure (Pc.map observed) :=
Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)
letI : IsProbabilityMeasure (Pc'.map observed) :=
Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)
have hid'given : ∀ᵐ y ∂covariateLaw (Pc.map observed),
μ' y = treatedRegression (Pc.map observed) y := by
have hmeasure : Pc.map observed = Pc'.map observed := hEq.symm.trans hP'
have hcov : covariateLaw (Pc.map observed) =
covariateLaw (Pc'.map observed) := congrArg covariateLaw hmeasure
have htreat : treatedRegression (Pc.map observed) =
treatedRegression (Pc'.map observed) := by
exact treatedRegression_eq_of_measure_eq _ _ hmeasure
rw [hcov]
filter_upwards [hid'] with y hy
rwa [htreat]
have huniq := (mu1_eq_treatedRegression β B L C c_f γ Pc μ e
hP.1 hP.2.1 hmodel).2.2 μ' hcont' hid'given x hx
change μ' x = μ x
exact huniq
The exact completion witnesses underlying the two observed lower-pair laws. For the stated inputs and conditions, the LowerPairExactModelWitness object being defined.
Definition (Lean source)
parameters
def
LowerPairExactModelWitness d n β B L C c_f γ a c x₀ :
Prop
given by
∃ (hF : IsProbabilityMeasure (lowerPairCompletion d n β B L γ a c x₀ false)) (hT : IsProbabilityMeasure (lowerPairCompletion d n β B L γ a c x₀ true)),
clause 1
@GlobalTailModel d β B L C c_f γ (lowerPairCompletion d n β B L γ a c x₀ false) hF (fun x => B * pairSuccess false x₀ β B L a (c * oracleMesh d n β γ) x) (radialPropensity x₀ γ)
clause 2
@GlobalTailModel d β B L C c_f γ (lowerPairCompletion d n β B L γ a c x₀ true) hT (fun x => B * pairSuccess true x₀ β B L a (c * oracleMesh d n β γ) x) (radialPropensity x₀ γ)
theorem boundedLowerPair_spec_exact_completion Lemma bounded-lower-pair-spec-exact-completion in the paper ↗
Fixed admissible pair constants work at every positive sample size. The KL and separation conditions are stated in the single-observation form used by the Le Cam substrate; both laws have bounded potential outcomes. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ :
ℝ
x₀ :
Fin d → ℝ
@realizes x0(covariate evaluation point)
hd :
0 < d
hβ :
1 < β
hγ :
1 < γ
hB :
0 < B
hL :
0 < L
hC :
1 ≤ C
hcf :
0 < c_f ∧ c_f ≤ 1
hx₀ :
x₀ ∈ cube d
conclusion
∃ a c c₀ : ℝ, ∃ ha : 0 < a, ∃ hc : 0 < c,
0 < c₀
boundedLowerPairAdmissible d β B L γ a c x₀
conclusion 1
n :
ℕ
1 ≤ n
P₀ :
boundedLowerPair d β B L γ a c x₀ ha hc false n
P₁ :
boundedLowerPair d β B L γ a c x₀ ha hc true n
∃ (hP₀ : P₀ ∈ ModelClass d β B L C c_f γ) (hP₁ : P₁ ∈ ModelClass d β B L C c_f γ),
j :
P₀ ≪ P₁
Integrable (llr P₀ P₁) P₀
(n : ℝ) * (_root_.InformationTheory.klDiv P₀ P₁).toReal ≤ 1 / 8
LowerPairExactModelWitness d n β B L C c_f γ a c x₀
Proof (Lean source)
lemma boundedLowerPair_spec_exact_completion (d : ℕ) (β B L C c_f γ : ℝ)
(x₀ : Fin d → ℝ) -- @realizes x0(covariate evaluation point)
(hd : 0 < d) (hβ : 1 < β) (hγ : 1 < γ)
(hB : 0 < B) (hL : 0 < L) (hC : 1 ≤ C)
(hcf : 0 < c_f ∧ c_f ≤ 1)
(hx₀ : x₀ ∈ cube d) : -- @realizes x0(range in closed cube, including boundary)
∃ a c c₀ : ℝ, ∃ ha : 0 < a, ∃ hc : 0 < c, 0 < c₀ ∧
boundedLowerPairAdmissible d β B L γ a c x₀ ∧
∀ n : ℕ, 1 ≤ n →
let P₀ := boundedLowerPair d β B L γ a c x₀ ha hc false n
let P₁ := boundedLowerPair d β B L γ a c x₀ ha hc true n
∃ (hP₀ : P₀ ∈ ModelClass d β B L C c_f γ)
(hP₁ : P₁ ∈ ModelClass d β B L C c_f γ),
(∀ j : Bool, ∀ᵐ ω ∂lowerPairCompletion d n β B L γ a c x₀ j,
ω.2.2.1 ∈ ({0, B} : Set ℝ) ∧
ω.2.2.2.1 ∈ ({0, B} : Set ℝ)) ∧
P₀ ≪ P₁ ∧ Integrable (llr P₀ P₁) P₀ ∧
(n : ℝ) * (_root_.InformationTheory.klDiv P₀ P₁).toReal ≤ 1 / 8 ∧
c₀ * oracleRate d n β γ ≤
|responseOf P₁ hP₁ x₀ - responseOf P₀ hP₀ x₀| ∧
LowerPairExactModelWitness d n β B L C c_f γ a c x₀ := by
open Causalean.Mathlib.Analysis.Calculus.CubeExtension in
obtain ⟨A, hA, hscaled⟩ :=
exists_uniform_scaledProductBump_holderBallOn (d := d) β (by linarith)
let a := min (baselineSuccess B L / 2) (L / (4 * B * A))
have hbase : 0 < baselineSuccess B L := by
unfold baselineSuccess
positivity
have ha : 0 < a := by
dsimp [a]
apply lt_min
· linarith
· positivity
have hale : a ≤ baselineSuccess B L :=
(min_le_left _ _).trans (by linarith)
let K := (2 / baselineSuccess B L) * 2 ^ (effectiveDimension d γ)
have hK : 0 ≤ K := by
dsimp [K]
positivity
have hD : 1 ≤ 2 * β + effectiveDimension d γ := by
have hd' : (0 : ℝ) < d := by exact_mod_cast hd
have heff : 0 ≤ effectiveDimension d γ := by
unfold effectiveDimension
positivity
linarith
obtain ⟨c, hc, hc1, hsmall⟩ := lowerPair_choose_kl_scale K a
(2 * β + effectiveDimension d γ) hK hD
let c₀ := B * a * c ^ β
have hc₀ : 0 < c₀ := by
dsimp [c₀]
positivity
have hadm : boundedLowerPairAdmissible d β B L γ a c x₀ :=
boundedLowerPairAdmissible_of_small d β B L γ a c x₀
hd hβ hB hL hγ ⟨ha, hale⟩ ⟨hc, hc1⟩
refine ⟨a, c, c₀, ha, hc, hc₀, hadm, ?_⟩
intro n hn
let h := c * oracleMesh d n β γ
let mu := volume.restrict (cube d)
let e := radialPropensity x₀ γ
let p₀ : (Fin d → ℝ) → ℝ := fun _ => baselineSuccess B L
let pF := pairSuccess false x₀ β B L a h
let pT := pairSuccess true x₀ β B L a h
let PcF := mu.bind (completionAt B e p₀ pF)
let PcT := mu.bind (completionAt B e p₀ pT)
let PF := PcF.map observed
let PT := PcT.map observed
have hmesh := oracleMesh_mem_Ioc d n β γ
⟨Nat.succ_le_iff.mpr hd, hn, hβ, hγ⟩
have hh : 0 < h ∧ h ≤ 1 := by
dsimp [h]
exact ⟨mul_pos hc hmesh.1,
(by simpa using
(mul_le_mul_of_nonneg_right hc1 hmesh.1.le).trans (by simpa using hmesh.2))⟩
have he : Measurable e := radialPropensity_measurable x₀ γ
have hp₀ : Measurable p₀ := measurable_const
have hpF : Measurable pF := pairSuccess_measurable false x₀ β B L a h
have hpT : Measurable pT := pairSuccess_measurable true x₀ β B L a h
have hvalidF : ∀ x, e x ∈ Icc 0 1 ∧ p₀ x ∈ Icc 0 1 ∧
pF x ∈ Icc 0 1 := by
intro x
exact ⟨radialPropensity_mem_Icc x₀ x γ hγ,
by simpa [p₀, pF, pairSuccess] using
pairSuccess_mem_Icc_of_small false x₀ β B L a h hB hL (by linarith)
⟨ha.le, hale⟩ ⟨hh.1.le, hh.2⟩ x,
pairSuccess_mem_Icc_of_small false x₀ β B L a h hB hL (by linarith)
⟨ha.le, hale⟩ ⟨hh.1.le, hh.2⟩ x⟩
have hvalidT : ∀ x, e x ∈ Icc 0 1 ∧ p₀ x ∈ Icc 0 1 ∧
pT x ∈ Icc 0 1 := by
intro x
exact ⟨radialPropensity_mem_Icc x₀ x γ hγ,
by simpa [p₀, pF, pairSuccess] using
pairSuccess_mem_Icc_of_small false x₀ β B L a h hB hL (by linarith)
⟨ha.le, hale⟩ ⟨hh.1.le, hh.2⟩ x,
pairSuccess_mem_Icc_of_small true x₀ β B L a h hB hL (by linarith)
⟨ha.le, hale⟩ ⟨hh.1.le, hh.2⟩ x⟩
letI : IsProbabilityMeasure mu := uniformCube_probability_lowerPair d
letI : IsProbabilityMeasure PcF := by
apply isProbabilityMeasure_bind
(completionAt_measurable_of_measurable B e p₀ pF he hp₀ hpF hvalidF).aemeasurable
filter_upwards with x
exact completionAt_probability B e p₀ pF x
(hvalidF x).1 (hvalidF x).2.1 (hvalidF x).2.2
letI : IsProbabilityMeasure PcT := by
apply isProbabilityMeasure_bind
(completionAt_measurable_of_measurable B e p₀ pT he hp₀ hpT hvalidT).aemeasurable
filter_upwards with x
exact completionAt_probability B e p₀ pT x
(hvalidT x).1 (hvalidT x).2.1 (hvalidT x).2.2
have hbaseBL : B * baselineSuccess B L ≤ L / 4 := by
unfold baselineSuccess
calc
B * min (1 / 4) (L / (4 * B)) ≤ B * (L / (4 * B)) :=
mul_le_mul_of_nonneg_left (min_le_right _ _) hB.le
_ = L / 4 := by field_simp
have hbaseabs : |B * baselineSuccess B L| ≤ L / 4 := by
rw [abs_of_pos (mul_pos hB hbase)]
exact hbaseBL
have hamp : |B * a| * A ≤ L / 4 := by
rw [abs_of_pos (mul_pos hB ha)]
have hale' : a ≤ L / (4 * B * A) := min_le_right _ _
have hpos : 0 < 4 * B * A := by positivity
apply (le_div_iff₀ (by norm_num : (0 : ℝ) < 4)).2
calc
B * a * A * 4 = a * (4 * B * A) := by ring
_ ≤ (L / (4 * B * A)) * (4 * B * A) :=
mul_le_mul_of_nonneg_right hale' hpos.le
_ = L := by field_simp
have hrad : |B * baselineSuccess B L| + |B * a| * A ≤ L := by
linarith [hbaseabs, hamp, hL]
have hballF : HolderBall β L (fun x => B * pF x) := by
have hcball := constant_holderBall (d := d) β L (B * baselineSuccess B L)
(by linarith) hL.le (hbaseabs.trans (by linarith))
simpa [pF, pairSuccess] using hcball
have hballT : HolderBall β L (fun x => B * pT x) := by
exact pairSuccess_true_holderBall β B L a h x₀ (by linarith) hh.1 hh.2 A hA
(hscaled h hh.1 hh.2 x₀) hrad
have hholderF : HolderResponse PcF (fun x => B * pF x) β L :=
completionBind_holderResponse mu B β L e p₀ pF he hp₀ hpF hvalidF hballF
have hholderT : HolderResponse PcT (fun x => B * pT x) β L :=
completionBind_holderResponse mu B β L e p₀ pT he hp₀ hpT hvalidT hballT
have htail := radialPropensity_uniform_tail x₀ hx₀ hd γ hγ
have hpos := radialPropensity_uniform_pos_ae x₀ hx₀ hd γ hγ
have hmodelF : @GlobalTailModel d β B L C c_f γ PcF inferInstance
(fun x => B * pF x) e :=
completionBind_uniformCube_globalTailModel β B L C c_f γ hB hC hcf.2
e p₀ pF (fun x => B * pF x) he hp₀ hpF hvalidF hpos htail hholderF
have hmodelT : @GlobalTailModel d β B L C c_f γ PcT inferInstance
(fun x => B * pT x) e :=
completionBind_uniformCube_globalTailModel β B L C c_f γ hB hC hcf.2
e p₀ pT (fun x => B * pT x) he hp₀ hpT hvalidT hpos htail hholderT
have hdomain : ModelParameterDomain d β B L C c_f :=
⟨Nat.succ_le_iff.mpr hd, hβ, hB, hL, hC, hcf⟩
let hPFmem : PF ∈ ModelClass d β B L C c_f γ :=
⟨hdomain, hγ, PcF, inferInstance, (fun x => B * pF x), e, rfl, hmodelF⟩
let hPTmem : PT ∈ ModelClass d β B L C c_f γ :=
⟨hdomain, hγ, PcT, inferInstance, (fun x => B * pT x), e, rfl, hmodelT⟩
change ∃ (hPF' : PF ∈ ModelClass d β B L C c_f γ)
(hPT' : PT ∈ ModelClass d β B L C c_f γ), _
refine ⟨hPFmem, hPTmem, ?_, ?_, ?_, ?_, ?_, ?_⟩
· intro j
exact lowerPairCompletion_supported_on_endpoints d n β B L γ a c x₀ j hadm hn
· exact (lowerPair_observed_guards_and_kl_rate x₀ hx₀ B L β γ a h
(by linarith) hB hL hγ ⟨ha.le, hale⟩ hh).1
· exact (lowerPair_observed_guards_and_kl_rate x₀ hx₀ B L β γ a h
(by linarith) hB hL hγ ⟨ha.le, hale⟩ hh).2.1
· have hrate := (lowerPair_observed_guards_and_kl_rate x₀ hx₀ B L β γ a h
(by linarith) hB hL hγ ⟨ha.le, hale⟩ hh).2.2
calc
(n : ℝ) * (klDiv PF PT).toReal ≤
(n : ℝ) * (K * a ^ 2 * h ^ (2 * β + effectiveDimension d γ)) := by
exact mul_le_mul_of_nonneg_left (by simpa [K] using hrate) (by positivity)
_ = K * a ^ 2 * c ^ (2 * β + effectiveDimension d γ) := by
dsimp [h]
rw [show (n : ℝ) * (K * a ^ 2 *
(c * oracleMesh d n β γ) ^ (2 * β + effectiveDimension d γ)) =
K * a ^ 2 * ((n : ℝ) *
(c * oracleMesh d n β γ) ^
(2 * β + effectiveDimension d γ)) by ring,
lowerPair_oracle_kl_balance d n β γ c (Nat.zero_lt_of_lt hn) hc.le
(lt_of_lt_of_le zero_lt_one hD)]
_ ≤ 1 / 8 := hsmall
· have hrespF : responseOf PF hPFmem x₀ = B * pF x₀ :=
responseOf_eq_given_model_response PF hPFmem PcF (fun x => B * pF x) e
rfl hmodelF x₀ hx₀
have hrespT : responseOf PT hPTmem x₀ = B * pT x₀ :=
responseOf_eq_given_model_response PT hPTmem PcT (fun x => B * pT x) e
rfl hmodelT x₀ hx₀
change c₀ * oracleRate d n β γ ≤
|responseOf PT hPTmem x₀ - responseOf PF hPFmem x₀|
rw [hrespF, hrespT]
have hsep := pairSuccess_oracle_separation (n := n) x₀ β B L γ a c hc.le
dsimp [pF, pT, h, c₀]
rw [← mul_sub]
rw [hsep]
simpa [mul_assoc] using
(le_abs_self (B * a * c ^ β * oracleRate d n β γ))
· have hprobF : IsProbabilityMeasure
(lowerPairCompletion d n β B L γ a c x₀ false) := by
simpa [PcF, h, lowerPairCompletion] using
(inferInstance : IsProbabilityMeasure PcF)
have hprobT : IsProbabilityMeasure
(lowerPairCompletion d n β B L γ a c x₀ true) := by
simpa [PcT, h, lowerPairCompletion] using
(inferInstance : IsProbabilityMeasure PcT)
refine ⟨hprobF, hprobT, ?_, ?_⟩
· simpa [PcF, pF, e, h, lowerPairCompletion] using hmodelF
· simpa [PcT, pT, e, h, lowerPairCompletion] using hmodelT
theorem boundedLowerPair_spec
The compatibility wrapper retaining the original lower-pair API for the stated inputs and conditions.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ :
ℝ
x₀ :
Fin d → ℝ
hd :
0 < d
hβ :
1 < β
hγ :
1 < γ
hB :
0 < B
hL :
0 < L
hC :
1 ≤ C
hcf :
0 < c_f ∧ c_f ≤ 1
hx₀ :
x₀ ∈ cube d
conclusion
∃ a c c₀ : ℝ, ∃ ha : 0 < a, ∃ hc : 0 < c,
0 < c₀
boundedLowerPairAdmissible d β B L γ a c x₀
conclusion 1
n :
ℕ
1 ≤ n
P₀ :
boundedLowerPair d β B L γ a c x₀ ha hc false n
P₁ :
boundedLowerPair d β B L γ a c x₀ ha hc true n
∃ (hP₀ : P₀ ∈ ModelClass d β B L C c_f γ) (hP₁ : P₁ ∈ ModelClass d β B L C c_f γ),
Proof (Lean source)
-- @node: lem:bounded-lower-pair
lemma boundedLowerPair_spec (d : ℕ) (β B L C c_f γ : ℝ)
(x₀ : Fin d → ℝ)
(hd : 0 < d) (hβ : 1 < β) (hγ : 1 < γ)
(hB : 0 < B) (hL : 0 < L) (hC : 1 ≤ C)
(hcf : 0 < c_f ∧ c_f ≤ 1)
(hx₀ : x₀ ∈ cube d) :
∃ a c c₀ : ℝ, ∃ ha : 0 < a, ∃ hc : 0 < c, 0 < c₀ ∧
boundedLowerPairAdmissible d β B L γ a c x₀ ∧
∀ n : ℕ, 1 ≤ n →
let P₀ := boundedLowerPair d β B L γ a c x₀ ha hc false n
let P₁ := boundedLowerPair d β B L γ a c x₀ ha hc true n
∃ (hP₀ : P₀ ∈ ModelClass d β B L C c_f γ)
(hP₁ : P₁ ∈ ModelClass d β B L C c_f γ),
(∀ j : Bool, ∀ᵐ ω ∂lowerPairCompletion d n β B L γ a c x₀ j,
ω.2.2.1 ∈ ({0, B} : Set ℝ) ∧
ω.2.2.2.1 ∈ ({0, B} : Set ℝ)) ∧
P₀ ≪ P₁ ∧ Integrable (llr P₀ P₁) P₀ ∧
(n : ℝ) * (_root_.InformationTheory.klDiv P₀ P₁).toReal ≤ 1 / 8 ∧
c₀ * oracleRate d n β γ ≤
|responseOf P₁ hP₁ x₀ - responseOf P₀ hP₀ x₀| := by
obtain ⟨a, c, c₀, ha, hc, hc₀, hadm, hall⟩ :=
boundedLowerPair_spec_exact_completion d β B L C c_f γ x₀
hd hβ hγ hB hL hC hcf hx₀
refine ⟨a, c, c₀, ha, hc, hc₀, hadm, ?_⟩
intro n hn
obtain ⟨hP₀, hP₁, hsupp, hac, hint, hkl, hsep, -⟩ := hall n hn
exact ⟨hP₀, hP₁, hsupp, hac, hint, hkl, hsep⟩
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.Model 43 declarations The causal completion and the observed data law use explicit product spaces.
Weak-overlap response model
The causal completion and the observed data law use explicit product spaces. This
is a bypass-justified use of product measures: POSystem is regime-indexed,
whereas the present result concerns a random-design regression experiment.
BackdoorEstimationSystem assumes strict overlap. The local Hölder conditions
use intrinsic coordinate jets on the closed cube. Ambient representatives arise
only through the separate extension theorem.
Stated parameter spaces for a fixed model class. For the stated inputs and conditions, the ModelParameterDomain object being defined.
Definition (Lean source)
parameters
d :
ℕ
β B L C c_f :
ℝ
def
ModelParameterDomain d β B L C c_f :
Prop
given by
clause 1
1 ≤ d
clause 2
1 < β
clause 3
0 < B
clause 4
0 < L
clause 5
1 ≤ C
clause 6
0 < c_f
clause 7
c_f ≤ 1
Stated compact range of overlap exponents. For the stated inputs and conditions, the AdaptationParameterDomain object being defined.
Definition (Lean source)
parameters
γ_min γ_max :
ℝ
def
AdaptationParameterDomain γ_min γ_max :
Prop
given by
clause 1
1 < γ_min
clause 2
γ_min < γ_max
Covariate cube. For the stated inputs and conditions, the cube object being defined.
Fixed compact adaptation range. For the stated inputs and conditions, the adaptationRange object being defined.
Definition (Lean source)
parameters
γ_min γ_max :
ℝ
_hdom :
AdaptationParameterDomain γ_min γ_max
def
given by
Icc γ_min γ_max
def polynomialDegree
Local polynomial degree. For the stated inputs and conditions, the polynomialDegree object being defined.
Definition (Lean source)
parameters
β :
ℝ
def
polynomialDegree β :
ℕ
given by
⌈β⌉₊ - 1
def effectiveDimension
Effective covariate dimension. For the stated inputs and conditions, the effectiveDimension object being defined.
Definition (Lean source)
parameters
d :
ℕ
γ :
ℝ
def
effectiveDimension d γ :
ℝ
given by
(d : ℝ) * γ / (γ - 1)
def oracleMesh
Oracle mesh width. For the stated inputs and conditions, the oracleMesh object being defined.
Definition (Lean source)
parameters
d n :
ℕ
β γ :
ℝ
def
oracleMesh d n β γ :
ℝ
given by
(n : ℝ) ^ (-(1 : ℝ) / (2 * β + effectiveDimension d γ))
Oracle risk scale. For the stated inputs and conditions, the oracleRate object being defined.
Definition (Lean source)
The oracle mesh is interpreted on the model's parameter domain. For the stated inputs and conditions, the OracleMeshDomain object being defined.
Definition (Lean source)
parameters
d n :
ℕ
β γ :
ℝ
def
OracleMeshDomain d n β γ :
Prop
given by
clause 1
1 ≤ d
clause 2
1 ≤ n
clause 3
1 < β
clause 4
1 < γ
theorem oracleMesh_mem_Ioc
The formula for the oracle mesh has the stated range on its parameter domain. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
oracleMesh d n β γ ∈ Ioc (0 : ℝ) 1
Proof (Lean source)
lemma oracleMesh_mem_Ioc (d n : ℕ) (β γ : ℝ)
(hdom : OracleMeshDomain d n β γ) :
oracleMesh d n β γ ∈ Ioc (0 : ℝ) 1 := by
rcases hdom with ⟨hd, hn, hβ, hγ⟩
have hd_real : 0 < (d : ℝ) := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hd)
have hn_real : 1 ≤ (n : ℝ) := by exact_mod_cast hn
have hdim : 0 < effectiveDimension d γ := by
unfold effectiveDimension
positivity
have hden : 0 < 2 * β + effectiveDimension d γ := by positivity
have hexp : -(1 : ℝ) / (2 * β + effectiveDimension d γ) ≤ 0 := by
apply div_nonpos_of_nonpos_of_nonneg
· norm_num
· positivity
constructor
· unfold oracleMesh
positivity
· unfold oracleMesh
exact Real.rpow_le_one_of_one_le_of_nonpos hn_real hexp
One observation (X,A,Y). For the stated inputs and conditions, the Obs object being defined.
def Completion
Completion (X,A,Y(0),Y(1),Y). For the stated inputs and conditions, the Completion object being defined.
def observed
Observable projection of the causal completion. For the stated inputs and conditions, the observed object being defined.
Definition (Lean source)
def covariateLaw
Covariate marginal of an observed law. For the stated inputs and conditions, the covariateLaw object being defined.
def covariateDensity
Covariate density, using the Radon–Nikodym derivative. For the stated inputs and conditions, the covariateDensity object being defined.
Definition (Lean source)
parameters
def
covariateDensity P x :
ℝ
given by
((covariateLaw P).rnDeriv (volume.restrict (cube d)) x).toReal
def propensity
A propensity version obtained from the conditional treatment kernel. For the stated inputs and conditions, the propensity object being defined.
Definition (Lean source)
parameters
def
propensity P x :
ℝ
given by
def treatedCovariateDensity
Density of the treated covariate distribution, normalized by the probability of treatment when that probability is positive. For the stated inputs and conditions, the treatedCovariateDensity object being defined.
Definition (Lean source)
parameters
def
given by
⟨propensity P x * covariateDensity P x / P.real {z : Obs d | z.2.1 = true}, by exact div_nonneg (mul_nonneg ENNReal.toReal_nonneg ENNReal.toReal_nonneg) (le_of_lt hA)⟩
def treatedKernel
Conditional treated response distribution. For the stated inputs and conditions, the treatedKernel object being defined.
Definition (Lean source)
def treatedRegression
Treated conditional regression. For the stated inputs and conditions, the treatedRegression object being defined.
Definition (Lean source)
parameters
def
treatedRegression P x :
ℝ
given by
∫ y, y ∂treatedKernel P x
Bounds on within-set coordinate jets on an arbitrary covariate domain. For the stated inputs and conditions, the HolderDerivBoundOn object being defined.
Definition (Lean source)
Top-order within-set coordinate-jet modulus on a covariate domain. For the stated inputs and conditions, the HolderTopModulusOn object being defined.
Definition (Lean source)
parameters
def
HolderTopModulusOn S g β L :
Prop
given by
TopHolderOn S (polynomialDegree β) (β - (polynomialDegree β : ℝ)) L g
Maximum over coordinate partial derivatives through degree ceil β - 1. For the stated inputs and conditions, the HolderDerivBound object being defined.
Definition (Lean source)
Maximum over top-order coordinate partial Hölder moduli. For the stated inputs and conditions, the HolderTopModulus object being defined.
Definition (Lean source)
Intrinsic Hölder class on a specified cube, including Dorn's original cube. For the stated inputs and conditions, the HolderBallOn object being defined.
Definition (Lean source)
parameters
def
HolderBallOn S β L g :
Prop
given by
HolderBallOn S (polynomialDegree β) (β - (polynomialDegree β : ℝ)) L g
structure HolderBall
The response itself belongs to the intrinsic Hölder ball on the closed covariate cube; no outside extension is selected as a model member.
Definition (Lean source)
parameters
assumes
regularity :
ContDiffOn ℝ (⌈β⌉₊ - 1) g (cube d)
derivBound :
HolderDerivBound g β L
modulus :
HolderTopModulus g β L
theorem toIntrinsic
The paper's intrinsic ball is exactly the within-cube ball of the cube-extension library. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
HolderBallOn (cube d) (polynomialDegree β) (β - (polynomialDegree β : ℝ)) L g
Proof (Lean source)
lemma HolderBall.toIntrinsic {d : ℕ} {β L : ℝ}
{g : (Fin d → ℝ) → ℝ} (h : HolderBall β L g) :
HolderBallOn
(cube d) (polynomialDegree β) (β - (polynomialDegree β : ℝ)) L g :=
⟨h.regularity, h.derivBound, h.modulus⟩
def IIDSampleLaw
The sample law is the iid product of the observed law. For the stated inputs and conditions, the IIDSampleLaw object being defined.
def CovariateDensityLowerBound
Covariate density bounded below relative to volume on the cube. For the stated inputs and conditions, the CovariateDensityLowerBound object being defined.
Definition (Lean source)
parameters
def
CovariateDensityLowerBound P c_f :
Prop
given by
∀ᵐ x ∂volume.restrict (cube d), c_f ≤ covariateDensity P x
Observed outcome equals the selected potential outcome. For the stated inputs and conditions, the OutcomeConsistency object being defined.
Definition (Lean source)
parameters
d :
ℕ
Pc :
Measure (Completion d)
def
OutcomeConsistency Pc :
Prop
given by
∀ᵐ ω ∂Pc, ω.2.2.2.2 = if ω.2.1 then ω.2.2.2.1 else ω.2.2.1
Potential outcomes and treatment are conditionally independent given X. For the stated inputs and conditions, the CondExchangeable object being defined.
Definition (Lean source)
parameters
d :
ℕ
Pc :
Measure (Completion d)
def
CondExchangeable Pc :
Prop
given by
CondIndepFun (comap (fun ω : Completion d => ω.1) inferInstance) (comap_le (by fun_prop : Measurable (fun ω : Completion d => ω.1))) (fun ω : Completion d => (ω.2.2.1, ω.2.2.2.1)) (fun ω => ω.2.1) Pc
def GlobalPropensityTail
Polynomial lower-propensity tail under the covariate law. For the stated inputs and conditions, the GlobalPropensityTail object being defined.
Definition (Lean source)
parameters
def
GlobalPropensityTail P e C γ :
Prop
given by
clause 1
∀ᵐ x ∂covariateLaw P, e x = propensity P x
def BoundedMeanSubGaussianResidual
Bounded treated mean and conditionally sub-Gaussian treated residual. For the stated inputs and conditions, the BoundedMeanSubGaussianResidual object being defined.
Definition (Lean source)
parameters
def
BoundedMeanSubGaussianResidual P B :
Prop
given by
The potential-outcome response has the designated Hölder representative. For the stated inputs and conditions, the HolderResponse object being defined.
Definition (Lean source)
parameters
def
HolderResponse Pc μ₁ β L :
Prop
structure GlobalTailModel
Global-tail causal model with sub-Gaussian treated residuals.
Definition (Lean source)
parameters
assumes
cubeSupport :
covariateLaw (Pc.map observed) (cube d) = 1
covariateAC :
covariateLaw (Pc.map observed) ≪ volume.restrict (cube d)
@realizes X(covariate law supported on the cube)
density :
CovariateDensityLowerBound (Pc.map observed) c_f
@realizes f(covariate law absolutely continuous with respect to cube volume)
consistency :
exchangeability :
tail :
GlobalPropensityTail (Pc.map observed) e C γ
outcome :
BoundedMeanSubGaussianResidual (Pc.map observed) B
smooth :
HolderResponse Pc μ₁ β L
Observed laws admitting a causal completion in the model. For the stated inputs and conditions, the ModelClass object being defined.
Definition (Lean source)
parameters
d :
ℕ
β B L C c_f γ :
ℝ
given by
{P | ModelParameterDomain d β B L C c_f ∧ 1 < γ ∧ ∃ (Pc : Measure (Completion d)) (_ : IsProbabilityMeasure Pc) (μ₁ e : (Fin d → ℝ) → ℝ), P = Pc.map observed ∧ GlobalTailModel β B L C c_f γ Pc μ₁ e}
def responseOf
A chosen Hölder representative for a law in the model class. For the stated inputs and conditions, the responseOf object being defined.
Definition (Lean source)
def ball2
Euclidean closed ball, distinct from the sup-norm covariate cube. For the stated inputs and conditions, the ball2 object being defined.
def DornSourceLaw
A source-law tuple carries the observed law and the selected treated mean, control mean, and pointwise propensity representative. For the stated inputs and conditions, the DornSourceLaw object being defined.
def dornCube
Dorn's original covariate cube. For the stated inputs and conditions, the dornCube object being defined.
A selected propensity version for a probability law on a specified cube. For the stated inputs and conditions, the DornSelectedPropensity object being defined.
Definition (Lean source)
parameters
def
DornSelectedPropensity S P e :
Prop
given by
∃ hP : IsProbabilityMeasure P,
letI : IsProbabilityMeasure P := hP Measurable e
clause 2
∀ᵐ x ∂covariateLaw P, e x = propensity P x ∧ 0 < e x ∧ e x < 1
Dorn's A3 is a property of a nonempty family and its selected pointwise propensity versions. Its constants are common to every family member. For the stated inputs and conditions, the DornA3 object being defined.
Definition (Lean source)
parameters
def
DornA3 S 𝔓 :
Prop
given by
clause 1
𝔓.Nonempty
clause 4
∃ ρ ν h₀ : ℝ,
0 < ρ
0 < ν
0 < h₀
z ∈ 𝔓 :
x₀ ∈ S :
h ∈ Set.Ioc (0 : ℝ) h₀ :
ν < (covariateLaw z.1).real ({x | z.2.2.2 x ≥ ρ * sSup (z.2.2.2 '' (ball2 x₀ h ∩ S))} ∩ ball2 x₀ h) / (covariateLaw z.1).real (ball2 x₀ h)
Singleton abbreviation for the displayed law and selected propensity on the paper's unit cube. For the stated inputs and conditions, the DornA3OnCube object being defined.
Definition (Lean source)
parameters
def
DornA3OnCube P e :
Prop
given by
def DornA3AntiConcentrationOnCube
The numerical local anti-concentration clause of Dorn's A3 for one displayed propensity on the paper's cube. This clause deliberately does not impose the separate selected-propensity requirement that e < 1 almost surely. For the stated inputs and conditions, the DornA3AntiConcentrationOnCube object being defined.
Definition (Lean source)
parameters
def
DornA3AntiConcentrationOnCube P e :
Prop
given by
∃ ρ ν h₀ : ℝ,
0 < ρ
0 < ν
0 < h₀
clause 1
x₀ ∈ cube d :
h ∈ Set.Ioc (0 : ℝ) h₀ :
ν < (covariateLaw P).real ({x | e x ≥ ρ * sSup (e '' (ball2 x₀ h ∩ cube d))} ∩ ball2 x₀ h) / (covariateLaw P).real (ball2 x₀ h)
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.OrderedMass 3 declarations Uniform ordered treated-mass bound
Uniform ordered treated-mass bound
theorem orderedMass_microcell_volume_toReal Lemma orderedMass_microcell_volume_toReal in the paper ↗
Rewrite the nonnegative geometric microcell volume as an ordinary real power. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma orderedMass_microcell_volume_toReal (d m j : ℕ)
(hη : 0 ≤ templateEta d m) :
(ofReal (templateEta d m * meshWidth j) ^ d).toReal =
(templateEta d m * meshWidth j) ^ d := by
rw [ENNReal.toReal_pow, ENNReal.toReal_ofReal]
exact mul_nonneg hη (by unfold meshWidth; positivity)
Separate the cell volume and rank powers after division by the rank. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d k :
ℕ
a b γ :
ℝ
hk :
0 < k
ha :
0 ≤ a
hb :
0 ≤ b
hγ :
1 < γ
conclusion
((k : ℝ) * (a * b ^ d)) ^ (γ / (γ - 1)) / (k : ℝ)
= a ^ (γ / (γ - 1)) * b ^ ((d : ℝ) * γ / (γ - 1)) * (k : ℝ) ^ (1 / (γ - 1))
Proof (Lean source)
lemma orderedMass_rank_power_identity (d k : ℕ) (a b γ : ℝ)
(hk : 0 < k) (ha : 0 ≤ a) (hb : 0 ≤ b) (hγ : 1 < γ) :
((k : ℝ) * (a * b ^ d)) ^ (γ / (γ - 1)) / (k : ℝ) =
a ^ (γ / (γ - 1)) * b ^ ((d : ℝ) * γ / (γ - 1)) *
(k : ℝ) ^ (1 / (γ - 1)) := by
have hk' : 0 < (k : ℝ) := by exact_mod_cast hk
have hne : γ - 1 ≠ 0 := ne_of_gt (by linarith)
have hp : γ / (γ - 1) = 1 / (γ - 1) + 1 := by
field_simp [hne]
ring
rw [Real.mul_rpow hk'.le (mul_nonneg ha (pow_nonneg hb _)),
Real.mul_rpow ha (pow_nonneg hb _)]
have hbpow : (b ^ d) ^ (γ / (γ - 1)) =
b ^ ((d : ℝ) * γ / (γ - 1)) := by
rw [← Real.rpow_natCast, ← Real.rpow_mul hb]
congr 1
ring
rw [hbpow, hp, Real.rpow_add hk', Real.rpow_one]
field_simp
theorem orderedTreatedMass_lower
One constant works uniformly over the compact range of overlap exponents. The first inequality is the layer-cake setwise bound; the second is the ordered weakest-cell profile. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ_min γ_max :
ℝ
hparams :
ModelParameterDomain d β B L C c_f
hγmin :
1 < γ_min
hγrange :
γ_min < γ_max
conclusion
∃ κ : ℝ,
0 < κ
conclusion 1
γ :
ℝ
γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩ → ∀ (Pc : Measure (Completion d)) [IsProbabilityMeasure Pc] (μ₁ e : (Fin d → ℝ) → ℝ), GlobalTailModel β B L C c_f γ Pc μ₁ e → (∀ E : Set (Fin d → ℝ), MeasurableSet E → E ⊆ cube d → ((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (covariateLaw (Pc.map observed)).real E ^ (γ / (γ - 1)) ≤ (Pc.map observed).real {z | z.2.1 = true ∧ z.1 ∈ E}) ∧ (∀ j : ℕ, ∀ k : ℕ, 1 ≤ k → k ≤ (2 ^ j) ^ d → κ * meshWidth j ^ effectiveDimension d γ * (k : ℝ) ^ ((1 : ℝ) / (γ - 1)) ≤ (@orderedCubeMass d (Pc.map observed) (Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)) (polynomialDegree β) j).getD (k - 1) 0)
Proof (Lean source)
lemma orderedTreatedMass_lower (d : ℕ) (β B L C c_f γ_min γ_max : ℝ)
(hparams : ModelParameterDomain d β B L C c_f)
(hγmin : 1 < γ_min) (hγrange : γ_min < γ_max) :
∃ κ : ℝ, 0 < κ ∧
∀ (γ : ℝ), γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩ →
∀ (Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ₁ e : (Fin d → ℝ) → ℝ),
GlobalTailModel β B L C c_f γ Pc μ₁ e →
(∀ E : Set (Fin d → ℝ), MeasurableSet E → E ⊆ cube d →
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(covariateLaw (Pc.map observed)).real E ^ (γ / (γ - 1)) ≤
(Pc.map observed).real {z | z.2.1 = true ∧ z.1 ∈ E}) ∧
(∀ j : ℕ, ∀ k : ℕ,
1 ≤ k → k ≤ (2 ^ j) ^ d →
κ * meshWidth j ^ effectiveDimension d γ *
(k : ℝ) ^ ((1 : ℝ) / (γ - 1)) ≤
(@orderedCubeMass d (Pc.map observed)
(Measure.isProbabilityMeasure_map (by unfold observed; fun_prop))
(polynomialDegree β) j).getD
(k - 1) 0) := by
let m := polynomialDegree β
let a := c_f * templateEta d m ^ d
have hC : 1 ≤ C := hparams.2.2.2.2.1
have hcf : 0 < c_f := hparams.2.2.2.2.2.1
have hη : 0 < templateEta d m := by
unfold templateEta
apply lt_min
· positivity
· exact div_pos (templateLambda_pos d m) (by linarith [templateLip_nonneg d m])
have ha : 0 < a := by dsimp [a]; positivity
obtain ⟨κ, hκ, hκle⟩ := orderedMass_uniform_coefficient C a γ_min γ_max
(by linarith) ha hγmin hγrange
refine ⟨κ, hκ, ?_⟩
intro γ hγ Pc _ μ₁ e hmodel
have hγ' : 1 < γ := lt_of_lt_of_le hγmin hγ.1
constructor
· intro E hE _
exact orderedMass_setwise_lower β B L C c_f γ Pc μ₁ e hmodel hC hγ' E hE
· intro j k hk hkmax
letI : IsProbabilityMeasure (Pc.map observed) :=
Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)
obtain ⟨S, ℓ, hS, hupper⟩ :=
orderedMass_ranked_union_upper (Pc.map observed) m j k hkmax
have hlower := orderedMass_selectedMicrocells_treated_lower d m j
β B L C c_f γ Pc μ₁ e hmodel hC hγ' (le_of_lt hcf) S ℓ
rw [hS, orderedMass_microcell_volume_toReal d m j hη.le] at hlower
have hk' : 0 < (k : ℝ) := by exact_mod_cast hk
have hmesh : 0 ≤ meshWidth j := by unfold meshWidth; positivity
have hpow : 0 ≤ a ^ (γ / (γ - 1)) *
meshWidth j ^ effectiveDimension d γ *
(k : ℝ) ^ (1 / (γ - 1)) := by positivity
have hrewrite :
((k : ℝ) * (c_f * (templateEta d m * meshWidth j) ^ d)) ^
(γ / (γ - 1)) / (k : ℝ) =
a ^ (γ / (γ - 1)) * meshWidth j ^ effectiveDimension d γ *
(k : ℝ) ^ (1 / (γ - 1)) := by
have hbase : c_f * (templateEta d m * meshWidth j) ^ d =
a * meshWidth j ^ d := by
rw [mul_pow]
dsimp [a]
ring
rw [hbase]
simpa only [effectiveDimension, div_eq_mul_inv, mul_assoc] using
orderedMass_rank_power_identity d k a (meshWidth j) γ hk ha.le hmesh hγ'
have hcoeff := hκle γ hγ
have halgebra :
κ * meshWidth j ^ effectiveDimension d γ *
(k : ℝ) ^ (1 / (γ - 1)) ≤
(((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
((k : ℝ) * (c_f * (templateEta d m * meshWidth j) ^ d)) ^
(γ / (γ - 1))) / (k : ℝ) := by
rw [mul_div_assoc, hrewrite]
have hfactor : 0 ≤ meshWidth j ^ effectiveDimension d γ *
(k : ℝ) ^ (1 / (γ - 1)) := by positivity
simpa only [mul_assoc] using mul_le_mul_of_nonneg_right hcoeff hfactor
have hresult := hlower.trans hupper
exact halgebra.trans ((div_le_iff₀ hk').2 (by
simpa only [mul_comm] using hresult))
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.OrderedMass.Density 6 declarations Density lower bounds for covariate mass
Density lower bounds for covariate mass
theorem orderedMass_measure_lower_of_rnDeriv_lower Lemma orderedMass_measure_lower_of_rnDeriv_lower in the paper ↗
An almost-everywhere lower density bound controls the mass of every measurable set. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma orderedMass_measure_lower_of_rnDeriv_lower {α : Type*} [MeasurableSpace α]
(μ ν : Measure α) [IsFiniteMeasure μ] [IsFiniteMeasure ν]
(hAC : μ ≪ ν) (c : ℝ)
(hden : ∀ᵐ x ∂ν, c ≤ (μ.rnDeriv ν x).toReal)
(E : Set α) :
c * ν.real E ≤ μ.real E :=
measureReal_lower_of_rnDeriv_lower
μ ν hAC c hden E
The model's density condition gives a setwise lower covariate mass bound. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma orderedMass_covariate_volume_lower {d : ℕ} (β B L C c_f γ : ℝ)
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ₁ e : (Fin d → ℝ) → ℝ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ₁ e)
(E : Set (Fin d → ℝ)) :
c_f * (volume.restrict (cube d)).real E ≤
(covariateLaw (Pc.map observed)).real E := by
let ν := volume.restrict (cube d)
have hvol : volume (cube d) = 1 := by
have hcube : cube d = Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hcube, Real.volume_Icc_pi]
simp
haveI : IsFiniteMeasure ν := by
rw [isFiniteMeasure_restrict]
simp [hvol]
haveI : IsProbabilityMeasure (Pc.map observed) :=
Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)
haveI : IsProbabilityMeasure (covariateLaw (Pc.map observed)) := by
unfold covariateLaw
exact Measure.isProbabilityMeasure_map (by fun_prop)
exact orderedMass_measure_lower_of_rnDeriv_lower
(covariateLaw (Pc.map observed)) ν hmodel.covariateAC c_f
hmodel.density E
theorem orderedMass_microcell_covariate_lower Lemma orderedMass_microcell_covariate_lower in the paper ↗
Each scaled microcell has covariate mass at least its volume times the model's density floor. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d m j :
ℕ
β B L C c_f γ :
ℝ
Pc :
μ₁ e :
(Fin d → ℝ) → ℝ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ₁ e
conclusion
Proof (Lean source)
lemma orderedMass_microcell_covariate_lower (d m j : ℕ)
(β B L C c_f γ : ℝ)
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ₁ e : (Fin d → ℝ) → ℝ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ₁ e)
(k : Fin d → Fin (2 ^ j)) (ℓ : Fin d → Fin (m + 1)) :
c_f * (ofReal (templateEta d m * meshWidth j) ^ d).toReal ≤
(covariateLaw (Pc.map observed)).real (scaledMicroCell d m j k ℓ) := by
have hinside : scaledMicroCell d m j k ℓ ⊆ cube d := by
intro x hx
exact (orderedMass_scaledMicroCell_inside_dyadicCube d m j k ℓ hx).1
have hmeas : MeasurableSet (scaledMicroCell d m j k ℓ) :=
orderedMass_scaledMicroCell_measurable d m j k ℓ
have hrestrict : (volume.restrict (cube d)).real (scaledMicroCell d m j k ℓ) =
(volume (scaledMicroCell d m j k ℓ)).toReal := by
rw [measureReal_def, Measure.restrict_apply hmeas, Set.inter_eq_self_of_subset_left hinside]
rw [← orderedMass_scaledMicroCell_volume d m j k ℓ, ← hrestrict]
exact orderedMass_covariate_volume_lower β B L C c_f γ Pc μ₁ e hmodel _
theorem orderedMass_selectedMicrocells_treated_additive Lemma orderedMass_selectedMicrocells_treated_additive in the paper ↗
Treated mass adds over one selected microcell in each distinct macro-cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
P.real {z | z.2.1 = true ∧ z.1 ∈ ⋃ k ∈ S, scaledMicroCell d m j k (ℓ k)}
= ∑ k ∈ S, microcellMass P m j k (ℓ k)
Proof (Lean source)
lemma orderedMass_selectedMicrocells_treated_additive {d : ℕ}
(P : Measure (Obs d)) [IsProbabilityMeasure P] (m j : ℕ)
(S : Finset (Fin d → Fin (2 ^ j)))
(ℓ : (Fin d → Fin (2 ^ j)) → Fin d → Fin (m + 1)) :
P.real {z | z.2.1 = true ∧
z.1 ∈ ⋃ k ∈ S, scaledMicroCell d m j k (ℓ k)} =
∑ k ∈ S, microcellMass P m j k (ℓ k) := by
let F : (Fin d → Fin (2 ^ j)) → Set (Obs d) :=
fun k => {z | z.2.1 = true ∧ z.1 ∈ scaledMicroCell d m j k (ℓ k)}
have hdisj : (S : Set (Fin d → Fin (2 ^ j))).PairwiseDisjoint F := by
intro k hk k' hk' hne
apply Set.disjoint_left.mpr
intro z hz hz'
exact (Set.disjoint_left.mp
(orderedMass_scaledMicroCell_disjoint_macro d m j k k' hne (ℓ k) (ℓ k')))
hz.2 hz'.2
have hmeas : ∀ k ∈ S, MeasurableSet (F k) := by
intro k hk
dsimp [F]
exact (measurableSet_singleton _).preimage (by fun_prop) |>.inter
((orderedMass_scaledMicroCell_measurable d m j k (ℓ k)).preimage (by fun_prop))
have hsum := measureReal_biUnion_finset (μ := P) hdisj hmeas
have hset : {z : Obs d | z.2.1 = true ∧
z.1 ∈ ⋃ k ∈ S, scaledMicroCell d m j k (ℓ k)} = ⋃ k ∈ S, F k := by
ext z
simp only [F, Set.mem_ofPred_eq, Set.mem_iUnion]
constructor
· rintro ⟨hz, k, hk, hcell⟩
exact ⟨k, hk, hz, hcell⟩
· rintro ⟨k, hk, hz, hcell⟩
exact ⟨hz, k, hk, hcell⟩
rw [hset]
simpa only [F, microcellMass] using hsum
theorem orderedMass_selectedMicrocells_covariate_additive Lemma orderedMass_selectedMicrocells_covariate_additive in the paper ↗
The same disjointness gives additivity of covariate mass over selected cells. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma orderedMass_selectedMicrocells_covariate_additive {d : ℕ}
(P : Measure (Obs d)) [IsProbabilityMeasure P] (m j : ℕ)
(S : Finset (Fin d → Fin (2 ^ j)))
(ℓ : (Fin d → Fin (2 ^ j)) → Fin d → Fin (m + 1)) :
(covariateLaw P).real (⋃ k ∈ S, scaledMicroCell d m j k (ℓ k)) =
∑ k ∈ S, (covariateLaw P).real (scaledMicroCell d m j k (ℓ k)) := by
letI : IsProbabilityMeasure (covariateLaw P) := by
unfold covariateLaw
exact Measure.isProbabilityMeasure_map (by fun_prop)
let F : (Fin d → Fin (2 ^ j)) → Set (Fin d → ℝ) :=
fun k => scaledMicroCell d m j k (ℓ k)
have hdisj : (S : Set (Fin d → Fin (2 ^ j))).PairwiseDisjoint F := by
intro k hk k' hk' hne
exact orderedMass_scaledMicroCell_disjoint_macro d m j k k' hne (ℓ k) (ℓ k')
have hmeas : ∀ k ∈ S, MeasurableSet (F k) := by
intro k hk
exact orderedMass_scaledMicroCell_measurable d m j k (ℓ k)
simpa only [F] using measureReal_biUnion_finset (μ := covariateLaw P) hdisj hmeas
theorem orderedMass_selectedMicrocells_covariate_lower Lemma orderedMass_selectedMicrocells_covariate_lower in the paper ↗
A set of distinct selected cells carries at least its cardinality times the single-cell density lower bound. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d m j :
ℕ
β B L C c_f γ :
ℝ
Pc :
μ₁ e :
(Fin d → ℝ) → ℝ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ₁ e
conclusion
Proof (Lean source)
lemma orderedMass_selectedMicrocells_covariate_lower (d m j : ℕ)
(β B L C c_f γ : ℝ)
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ₁ e : (Fin d → ℝ) → ℝ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ₁ e)
(S : Finset (Fin d → Fin (2 ^ j)))
(ℓ : (Fin d → Fin (2 ^ j)) → Fin d → Fin (m + 1)) :
(S.card : ℝ) * (c_f *
(ofReal (templateEta d m * meshWidth j) ^ d).toReal) ≤
(covariateLaw (Pc.map observed)).real
(⋃ k ∈ S, scaledMicroCell d m j k (ℓ k)) := by
letI : IsProbabilityMeasure (Pc.map observed) :=
Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)
rw [orderedMass_selectedMicrocells_covariate_additive]
calc
(S.card : ℝ) * (c_f *
(ofReal (templateEta d m * meshWidth j) ^ d).toReal) =
∑ _k ∈ S, c_f *
(ofReal (templateEta d m * meshWidth j) ^ d).toReal := by simp
_ ≤ ∑ k ∈ S,
(covariateLaw (Pc.map observed)).real (scaledMicroCell d m j k (ℓ k)) := by
apply Finset.sum_le_sum
intro k hk
exact orderedMass_microcell_covariate_lower d m j β B L C c_f γ
Pc μ₁ e hmodel k (ℓ k)
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.OrderedMass.Geometry 8 declarations Geometry of treated microcells
Geometry of treated microcells
Reference microcubes are measurable, including at the template boundary. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma orderedMass_microCube_measurable (d m : ℕ)
(ℓ : Fin d → Fin (m + 1)) : MeasurableSet (microCube d m ℓ) := by
unfold microCube
simp only [Set.ofPred_forall]
exact MeasurableSet.iInter fun i =>
measurableSet_le (by fun_prop : Measurable fun z : Fin d → ℝ =>
|z i - tensorNode d m ℓ i|) measurable_const
theorem orderedMass_scaledMicroCell_measurable Lemma orderedMass_scaledMicroCell_measurable in the paper ↗
Every scaled microcell is measurable at every dyadic mesh. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
MeasurableSet (scaledMicroCell d m j k ℓ)
Proof (Lean source)
lemma orderedMass_scaledMicroCell_measurable (d m j : ℕ)
(k : Fin d → Fin (2 ^ j)) (ℓ : Fin d → Fin (m + 1)) :
MeasurableSet (scaledMicroCell d m j k ℓ) := by
unfold scaledMicroCell
exact (orderedMass_microCube_measurable d m ℓ).preimage (by fun_prop)
Under the template width cap, every coordinate of a reference microcube lies strictly between zero and one. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma orderedMass_microCube_interior (d m : ℕ)
(ℓ : Fin d → Fin (m + 1)) (z : Fin d → ℝ)
(hz : z ∈ microCube d m ℓ) (i : Fin d) :
0 < z i ∧ z i < 1 := by
have hη : templateEta d m ≤ 1 / (2 * (m + 2 : ℝ)) := by
unfold templateEta
exact min_le_left _ _
have hcoord := hz i
change |z i - ((ℓ i).val + 1 : ℝ) / (m + 2 : ℝ)| ≤ templateEta d m / 2 at hcoord
have hℓlow : (0 : ℝ) ≤ (ℓ i).val := by positivity
have hℓhigh : ((ℓ i).val : ℝ) ≤ m := by
exact_mod_cast le_of_lt_succ (ℓ i).isLt
have hm : (0 : ℝ) < m + 2 := by positivity
have hnode_low : 1 / (m + 2 : ℝ) ≤ ((ℓ i).val + 1 : ℝ) / (m + 2 : ℝ) := by
apply div_le_div_of_nonneg_right _ hm.le
linarith
have hnode_high : ((ℓ i).val + 1 : ℝ) / (m + 2 : ℝ) ≤
(m + 1 : ℝ) / (m + 2 : ℝ) := by
apply div_le_div_of_nonneg_right _ hm.le
linarith
have hη' : templateEta d m / 2 ≤ 1 / (4 * (m + 2 : ℝ)) := by
calc
templateEta d m / 2 ≤ (1 / (2 * (m + 2 : ℝ))) / 2 :=
div_le_div_of_nonneg_right hη (by norm_num)
_ = 1 / (4 * (m + 2 : ℝ)) := by field_simp; ring
constructor
· have := (abs_le.mp hcoord).1
have hfrac : 1 / (4 * (m + 2 : ℝ)) < 1 / (m + 2 : ℝ) := by
field_simp
nlinarith [hm]
linarith
· have := (abs_le.mp hcoord).2
have hfrac : 0 < 1 / (4 * (m + 2 : ℝ)) := by positivity
have hunit : (m + 1 : ℝ) / (m + 2 : ℝ) +
1 / (4 * (m + 2 : ℝ)) < 1 := by
field_simp
nlinarith [hm]
linarith
A dyadic mesh width multiplied by the number of cells is one. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma orderedMass_meshWidth_mul_pow (j : ℕ) :
meshWidth j * (2 : ℝ) ^ j = 1 := by
unfold meshWidth
rw [← Real.rpow_natCast (2 : ℝ) j]
rw [← Real.rpow_add (by norm_num : (0 : ℝ) < 2)]
simp
theorem orderedMass_scaledMicroCell_inside_dyadicCube Lemma orderedMass_scaledMicroCell_inside_dyadicCube in the paper ↗
Every scaled microcell lies inside its indexed macro-cube, including at the boundary of the ambient cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
scaledMicroCell d m j k ℓ ⊆ dyadicCube d j k
Proof (Lean source)
lemma orderedMass_scaledMicroCell_inside_dyadicCube (d m j : ℕ)
(k : Fin d → Fin (2 ^ j)) (ℓ : Fin d → Fin (m + 1)) :
scaledMicroCell d m j k ℓ ⊆ dyadicCube d j k := by
intro x hx
have hh : 0 < meshWidth j := by unfold meshWidth; positivity
have hscale := orderedMass_meshWidth_mul_pow j
have hpow : 0 < (2 : ℝ) ^ j := by positivity
have hcoord (i : Fin d) :
0 < (x i - cubeCorner d j k i) / meshWidth j ∧
(x i - cubeCorner d j k i) / meshWidth j < 1 :=
orderedMass_microCube_interior d m ℓ
(fun i => (x i - cubeCorner d j k i) / meshWidth j) hx i
have hbounds (i : Fin d) :
cubeCorner d j k i < x i ∧
x i < cubeCorner d j k i + meshWidth j := by
have hz := hcoord i
have hid : meshWidth j * ((x i - cubeCorner d j k i) / meshWidth j) =
x i - cubeCorner d j k i := by field_simp
constructor <;> nlinarith [mul_lt_mul_of_pos_left hz.1 hh,
mul_lt_mul_of_pos_left hz.2 hh]
change x ∈ cube d ∧ ∀ i,
cubeCorner d j k i ≤ x i ∧
(x i < cubeCorner d j k i + meshWidth j ∨
((k i).val = 2 ^ j - 1 ∧ x i = 1))
constructor
· intro i hi
change x i ∈ Icc (0 : ℝ) 1
have hk : ((k i).val : ℝ) + 1 ≤ (2 : ℝ) ^ j := by
exact_mod_cast (k i).isLt
have hcorner : cubeCorner d j k i = (k i).val * meshWidth j := rfl
have hb := hbounds i
constructor
· rw [hcorner] at hb
nlinarith [mul_nonneg (by positivity : (0 : ℝ) ≤ (k i).val) hh.le]
· rw [hcorner] at hb
nlinarith [mul_nonneg (sub_nonneg.mpr hk) hh.le]
· intro i
exact ⟨(hbounds i).1.le, inl (hbounds i).2⟩
theorem orderedMass_scaledMicroCell_strict_bounds Lemma orderedMass_scaledMicroCell_strict_bounds in the paper ↗
A point in a scaled microcell lies strictly between the corresponding macro-cube faces in every coordinate. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma orderedMass_scaledMicroCell_strict_bounds (d m j : ℕ)
(k : Fin d → Fin (2 ^ j)) (ℓ : Fin d → Fin (m + 1))
{x : Fin d → ℝ} (hx : x ∈ scaledMicroCell d m j k ℓ) (i : Fin d) :
cubeCorner d j k i < x i ∧
x i < cubeCorner d j k i + meshWidth j := by
have hh : 0 < meshWidth j := by unfold meshWidth; positivity
have hz := orderedMass_microCube_interior d m ℓ
(fun i => (x i - cubeCorner d j k i) / meshWidth j) hx i
have hid : meshWidth j * ((x i - cubeCorner d j k i) / meshWidth j) =
x i - cubeCorner d j k i := by field_simp
constructor <;> nlinarith [mul_lt_mul_of_pos_left hz.1 hh,
mul_lt_mul_of_pos_left hz.2 hh]
theorem orderedMass_scaledMicroCell_disjoint_macro Lemma orderedMass_scaledMicroCell_disjoint_macro in the paper ↗
Microcells in different macro-cubes are disjoint, even if the microcell labels differ. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma orderedMass_scaledMicroCell_disjoint_macro (d m j : ℕ)
(k k' : Fin d → Fin (2 ^ j)) (hk : k ≠ k')
(ℓ ℓ' : Fin d → Fin (m + 1)) :
Disjoint (scaledMicroCell d m j k ℓ) (scaledMicroCell d m j k' ℓ') := by
apply Set.disjoint_left.mpr
intro x hx hx'
apply hk
funext i
apply Fin.ext
have hb := orderedMass_scaledMicroCell_strict_bounds d m j k ℓ hx i
have hb' := orderedMass_scaledMicroCell_strict_bounds d m j k' ℓ' hx' i
have hh : 0 < meshWidth j := by unfold meshWidth; positivity
by_contra hne
have hcases : (k i).val + 1 ≤ (k' i).val ∨ (k' i).val + 1 ≤ (k i).val := by omega
rcases hcases with hlt | hgt
· have hcast : ((k i).val : ℝ) + 1 ≤ (k' i).val := by exact_mod_cast hlt
unfold cubeCorner at hb hb'
nlinarith [mul_nonneg (sub_nonneg.mpr hcast) hh.le]
· have hcast : ((k' i).val : ℝ) + 1 ≤ (k i).val := by exact_mod_cast hgt
unfold cubeCorner at hb hb'
nlinarith [mul_nonneg (sub_nonneg.mpr hcast) hh.le]
Every scaled microcell has the same Euclidean volume, independently of its tensor node and macro-cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma orderedMass_scaledMicroCell_volume (d m j : ℕ)
(k : Fin d → Fin (2 ^ j)) (ℓ : Fin d → Fin (m + 1)) :
volume (scaledMicroCell d m j k ℓ) =
ofReal (templateEta d m * meshWidth j) ^ d := by
have hh : 0 < meshWidth j := by unfold meshWidth; positivity
let lo : Fin d → ℝ := fun i =>
cubeCorner d j k i + meshWidth j * (tensorNode d m ℓ i - templateEta d m / 2)
let hi : Fin d → ℝ := fun i =>
cubeCorner d j k i + meshWidth j * (tensorNode d m ℓ i + templateEta d m / 2)
have hset : scaledMicroCell d m j k ℓ = Icc lo hi := by
ext x
simp only [scaledMicroCell, Set.mem_ofPred_eq, microCube, Set.mem_Icc,
Pi.le_def, lo, hi]
constructor
· intro hx
constructor
· intro i
have hxi := (abs_le.mp (hx i)).1
have hmul : (x i - cubeCorner d j k i) / meshWidth j * meshWidth j =
x i - cubeCorner d j k i := by field_simp
nlinarith [mul_le_mul_of_nonneg_right hxi hh.le]
· intro i
have hxi := (abs_le.mp (hx i)).2
have hmul : (x i - cubeCorner d j k i) / meshWidth j * meshWidth j =
x i - cubeCorner d j k i := by field_simp
nlinarith [mul_le_mul_of_nonneg_right hxi hh.le]
· intro hx i
apply abs_le.mpr
have hlo : tensorNode d m ℓ i - templateEta d m / 2 ≤
(x i - cubeCorner d j k i) / meshWidth j :=
(le_div_iff₀ hh).2 (by nlinarith [hx.1 i])
have hhi : (x i - cubeCorner d j k i) / meshWidth j ≤
tensorNode d m ℓ i + templateEta d m / 2 :=
(div_le_iff₀ hh).2 (by nlinarith [hx.2 i])
constructor <;> linarith
rw [hset, Real.volume_Icc_pi]
have hdiff : ∀ i : Fin d, hi i - lo i = templateEta d m * meshWidth j := by
intro i
dsimp [hi, lo]
ring
simp only [hdiff, Finset.prod_const, Finset.card_fin]
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.OrderedMass.Ranking 14 declarations Ordered weakest-cell mass combinatorics
Ordered weakest-cell mass combinatorics
theorem orderedMass_cubeMass_le_microcellMass Lemma orderedMass_cubeMass_le_microcellMass in the paper ↗
The minimum mass in a macro-cube is below each of its microcell masses. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma orderedMass_cubeMass_le_microcellMass {d : ℕ} (P : Measure (Obs d))
[IsProbabilityMeasure P] (m j : ℕ) (k : Fin d → Fin (2 ^ j))
(ℓ : Fin d → Fin (m + 1)) :
cubeMass P m j k ≤ microcellMass P m j k ℓ := by
unfold cubeMass
apply csInf_le
· refine ⟨0, ?_⟩
rintro r ⟨ℓ', rfl⟩
exact ENNReal.toReal_nonneg
· exact ⟨ℓ, rfl⟩
A finite family of microcells contains one attaining the macro-cube minimum. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma orderedMass_cubeMass_attained {d : ℕ} (P : Measure (Obs d))
[IsProbabilityMeasure P] (m j : ℕ) (k : Fin d → Fin (2 ^ j)) :
∃ ℓ : Fin d → Fin (m + 1), cubeMass P m j k = microcellMass P m j k ℓ := by
let S : Set ℝ := range (fun ℓ : Fin d → Fin (m + 1) => microcellMass P m j k ℓ)
have hS : S.Nonempty := by
let ℓ : Fin d → Fin (m + 1) := fun _ => ⟨0, zero_lt_succ m⟩
exact ⟨microcellMass P m j k ℓ, ℓ, rfl⟩
have hfin : S.Finite := Set.finite_range _
obtain ⟨ℓ, hℓ⟩ := Set.Nonempty.csInf_mem hS hfin
exact ⟨ℓ, hℓ.symm⟩
The setwise tail estimate gives a treated-mass bound for any measurable microcell once its covariate mass has a lower bound. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ a :
ℝ
Pc :
μ₁ e :
(Fin d → ℝ) → ℝ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ₁ e
hC :
1 ≤ C
hγ :
1 < γ
ha :
0 ≤ a
hE :
hmass :
a ≤ (covariateLaw (Pc.map observed)).real E
Proof (Lean source)
lemma orderedMass_microcell_lower {d : ℕ} (β B L C c_f γ a : ℝ)
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ₁ e : (Fin d → ℝ) → ℝ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ₁ e)
(hC : 1 ≤ C) (hγ : 1 < γ) (ha : 0 ≤ a)
(E : Set (Fin d → ℝ)) (hE : MeasurableSet E)
(hmass : a ≤ (covariateLaw (Pc.map observed)).real E) :
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
a ^ (γ / (γ - 1)) ≤
(Pc.map observed).real {z | z.2.1 = true ∧ z.1 ∈ E} := by
have hexp : 0 ≤ γ / (γ - 1) := by positivity
have hpow := Real.rpow_le_rpow ha hmass hexp
have hcoeff : 0 ≤ ((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) := by
positivity
exact (mul_le_mul_of_nonneg_left hpow hcoeff).trans
(orderedMass_setwise_lower β B L C c_f γ Pc μ₁ e hmodel hC hγ E hE)
theorem orderedMass_selectedMicrocells_treated_lower Lemma orderedMass_selectedMicrocells_treated_lower in the paper ↗
The sharp setwise tail estimate applies to any finite union of selected microcells, with covariate mass growing linearly in the number selected. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d m j :
ℕ
β B L C c_f γ :
ℝ
Pc :
μ₁ e :
(Fin d → ℝ) → ℝ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ₁ e
hC :
1 ≤ C
hγ :
1 < γ
hcf :
0 ≤ c_f
conclusion
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * ((S.card : ℝ) * (c_f * (ofReal (templateEta d m * meshWidth j) ^ d).toReal)) ^ (γ / (γ - 1))
Proof (Lean source)
lemma orderedMass_selectedMicrocells_treated_lower (d m j : ℕ)
(β B L C c_f γ : ℝ)
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ₁ e : (Fin d → ℝ) → ℝ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ₁ e)
(hC : 1 ≤ C) (hγ : 1 < γ) (hcf : 0 ≤ c_f)
(S : Finset (Fin d → Fin (2 ^ j)))
(ℓ : (Fin d → Fin (2 ^ j)) → Fin d → Fin (m + 1)) :
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
((S.card : ℝ) * (c_f *
(ofReal (templateEta d m * meshWidth j) ^ d).toReal)) ^
(γ / (γ - 1)) ≤
(Pc.map observed).real {z | z.2.1 = true ∧
z.1 ∈ ⋃ k ∈ S, scaledMicroCell d m j k (ℓ k)} := by
let E : Set (Fin d → ℝ) := ⋃ k ∈ S, scaledMicroCell d m j k (ℓ k)
have hE : MeasurableSet E := by
dsimp [E]
apply Finset.measurableSet_biUnion
intro k hk
exact orderedMass_scaledMicroCell_measurable d m j k (ℓ k)
have ha : 0 ≤ (S.card : ℝ) *
(c_f * (ofReal (templateEta d m * meshWidth j) ^ d).toReal) := by
positivity
exact orderedMass_microcell_lower β B L C c_f γ _ Pc μ₁ e hmodel hC hγ ha
E hE (orderedMass_selectedMicrocells_covariate_lower d m j β B L C c_f γ
Pc μ₁ e hmodel S ℓ)
Sorting the finite macro-cube masses retains exactly one entry per cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
(orderedCubeMass P m j).length = (2 ^ j) ^ d
Proof (Lean source)
lemma orderedMass_sorted_length {d : ℕ} (P : Measure (Obs d))
[IsProbabilityMeasure P] (m j : ℕ) :
(orderedCubeMass P m j).length = (2 ^ j) ^ d := by
unfold orderedCubeMass
simp only [List.length_mergeSort, List.length_map, Finset.length_toList,
Finset.card_univ, Fintype.card_fun, Fintype.card_fin]
The ordered masses are monotone in their list indices. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
(orderedCubeMass P m j).SortedLE
Proof (Lean source)
lemma orderedMass_sorted_mono {d : ℕ} (P : Measure (Obs d))
[IsProbabilityMeasure P] (m j : ℕ) :
(orderedCubeMass P m j).SortedLE := by
unfold orderedCubeMass
exact List.sortedLE_mergeSort
Every in-range ordered mass is the minimum mass of a macro-cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma orderedMass_rank_attained {d : ℕ} (P : Measure (Obs d))
[IsProbabilityMeasure P] (m j k : ℕ)
(hk : 1 ≤ k) (hk' : k ≤ (2 ^ j) ^ d) :
∃ q : Fin d → Fin (2 ^ j),
(orderedCubeMass P m j).getD (k - 1) 0 = cubeMass P m j q := by
classical
let l := orderedCubeMass P m j
have hi : k - 1 < l.length := by
rw [orderedMass_sorted_length]
omega
rw [List.getD_eq_getElem l 0 hi]
have hm : l[k - 1] ∈ l := List.getElem_mem hi
have hm' : l[k - 1] ∈
((Finset.univ : Finset (Fin d → Fin (2 ^ j))).toList.map (cubeMass P m j)) := by
exact (List.mergeSort_perm _ _).mem_iff.mp hm
obtain ⟨q, _, hq⟩ := List.mem_map.mp hm'
exact ⟨q, hq.symm⟩
Every earlier ordered mass is no larger than the mass at rank k. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma orderedMass_rank_mono {d : ℕ} (P : Measure (Obs d))
[IsProbabilityMeasure P] (m j i k : ℕ)
(hi : i < k) (hk : k ≤ (2 ^ j) ^ d) :
(orderedCubeMass P m j).getD i 0 ≤
(orderedCubeMass P m j).getD (k - 1) 0 := by
let l := orderedCubeMass P m j
have hik : i ≤ k - 1 := by omega
have hil : i < l.length := by
rw [orderedMass_sorted_length]
omega
have hkl : k - 1 < l.length := by
rw [orderedMass_sorted_length]
omega
rw [List.getD_eq_getElem l 0 hil, List.getD_eq_getElem l 0 hkl]
exact (orderedMass_sorted_mono P m j).getElem_le_getElem_of_le hik
The total mass of the first k ordered macro-cubes is at most k times the mass at rank k. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
≤ (k : ℝ) * (orderedCubeMass P m j).getD (k - 1) 0
Proof (Lean source)
lemma orderedMass_initial_sum_le_rank {d : ℕ} (P : Measure (Obs d))
[IsProbabilityMeasure P] (m j k : ℕ)
(hk : k ≤ (2 ^ j) ^ d) :
(∑ i ∈ range k, (orderedCubeMass P m j).getD i 0) ≤
(k : ℝ) * (orderedCubeMass P m j).getD (k - 1) 0 := by
calc
(∑ i ∈ range k, (orderedCubeMass P m j).getD i 0) ≤
∑ _i ∈ range k, (orderedCubeMass P m j).getD (k - 1) 0 := by
apply Finset.sum_le_sum
intro i hi
exact orderedMass_rank_mono P m j i k (Finset.mem_range.mp hi) hk
_ = (k : ℝ) * (orderedCubeMass P m j).getD (k - 1) 0 := by
simp
At least k macro-cubes have weakest-microcell mass at most the mass at rank k, counting ties with their multiplicity. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma orderedMass_rank_threshold_card {d : ℕ} (P : Measure (Obs d))
[IsProbabilityMeasure P] (m j k : ℕ)
(hk : k ≤ (2 ^ j) ^ d) :
k ≤ ((Finset.univ : Finset (Fin d → Fin (2 ^ j))).filter
(fun q => cubeMass P m j q ≤ (orderedCubeMass P m j).getD (k - 1) 0)).card := by
classical
let rank := (orderedCubeMass P m j).getD (k - 1) 0
let l := orderedCubeMass P m j
let original := (Finset.univ : Finset (Fin d → Fin (2 ^ j))).toList.map (cubeMass P m j)
have hlen : k ≤ l.length := by simpa [l, orderedMass_sorted_length] using hk
have hprefix : ∀ x ∈ l.take k, x ≤ rank := by
intro x hx
obtain ⟨i, hi, rfl⟩ := List.mem_iff_getElem.mp hx
have hik : i < k := by simpa using (lt_of_lt_of_le hi (List.length_take_le k l))
have hil : i < l.length := lt_of_lt_of_le hik hlen
simpa only [List.getElem_take, l, rank, List.getD_eq_getElem l 0 hil] using
orderedMass_rank_mono P m j i k hik hk
have hcount : k ≤ l.countP (fun x => x ≤ rank) := by
have hsub : (l.take k).countP (fun x => x ≤ rank) ≤
l.countP (fun x => x ≤ rank) := (List.take_sublist k l).countP_le
have hall : (l.take k).countP (fun x => x ≤ rank) = k := by
rw [List.countP_eq_length_filter]
have hfilter : (l.take k).filter (fun x => x ≤ rank) = l.take k := by
apply List.filter_eq_self.mpr
simpa using hprefix
rw [hfilter, List.length_take, min_eq_left hlen]
omega
have hperm : l.Perm original := by
dsimp [l, original, orderedCubeMass]
exact List.mergeSort_perm _ _
have hcount' : k ≤ original.countP (fun x => x ≤ rank) := by
rwa [hperm.countP_eq] at hcount
have hmap : original.countP (fun x => x ≤ rank) =
((Finset.univ : Finset (Fin d → Fin (2 ^ j))).toList).countP
(fun q => cubeMass P m j q ≤ rank) := by
simp only [original, List.countP_map, Function.comp_def]
rw [hmap] at hcount'
have hnodup : ((Finset.univ : Finset (Fin d → Fin (2 ^ j))).toList).Nodup :=
Finset.nodup_toList _
simpa [rank] using hcount'.trans_eq
(List.Nodup.card_eq_countP hnodup).symm
theorem orderedMass_selectedMicrocells_treated_upper Lemma orderedMass_selectedMicrocells_treated_upper in the paper ↗
A finite union of selected treated microcells has mass at most the sum of its component treated masses. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
P.real {z | z.2.1 = true ∧ z.1 ∈ ⋃ k ∈ S, scaledMicroCell d m j k (ℓ k)}
≤ ∑ k ∈ S, microcellMass P m j k (ℓ k)
Proof (Lean source)
lemma orderedMass_selectedMicrocells_treated_upper {d : ℕ} (P : Measure (Obs d))
[IsProbabilityMeasure P] (m j : ℕ)
(S : Finset (Fin d → Fin (2 ^ j)))
(ℓ : (Fin d → Fin (2 ^ j)) → Fin d → Fin (m + 1)) :
P.real {z | z.2.1 = true ∧
z.1 ∈ ⋃ k ∈ S, scaledMicroCell d m j k (ℓ k)} ≤
∑ k ∈ S, microcellMass P m j k (ℓ k) := by
let E : (Fin d → Fin (2 ^ j)) → Set (Obs d) :=
fun k => {z | z.2.1 = true ∧ z.1 ∈ scaledMicroCell d m j k (ℓ k)}
have heq : {z | z.2.1 = true ∧
z.1 ∈ ⋃ k ∈ S, scaledMicroCell d m j k (ℓ k)} = ⋃ k ∈ S, E k := by
ext z
simp [E]
rw [heq]
calc
P.real (⋃ k ∈ S, E k) ≤ (∑ k ∈ S, P (E k)).toReal := by
exact ENNReal.toReal_mono (by simp)
(measure_biUnion_finset_le S E)
_ = ∑ k ∈ S, microcellMass P m j k (ℓ k) := by
rw [ENNReal.toReal_sum (by intro k hk; exact measure_ne_top _ _)]
rfl
theorem orderedMass_selectedMicrocells_treated_le_card_mul Lemma orderedMass_selectedMicrocells_treated_le_card_mul in the paper ↗
Selecting a minimizing microcell in each cube bounds the mass of their union by the number of cubes times any common upper bound on the minima. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
P :
Measure (Obs d)
m j :
ℕ
r :
ℝ
hmin :
∀ k ∈ S, microcellMass P m j k (ℓ k) = cubeMass P m j k
hr :
∀ k ∈ S, cubeMass P m j k ≤ r
conclusion
P.real {z | z.2.1 = true ∧ z.1 ∈ ⋃ k ∈ S, scaledMicroCell d m j k (ℓ k)}
≤ (S.card : ℝ) * r
Proof (Lean source)
lemma orderedMass_selectedMicrocells_treated_le_card_mul {d : ℕ}
(P : Measure (Obs d)) [IsProbabilityMeasure P] (m j : ℕ)
(S : Finset (Fin d → Fin (2 ^ j)))
(ℓ : (Fin d → Fin (2 ^ j)) → Fin d → Fin (m + 1)) (r : ℝ)
(hmin : ∀ k ∈ S, microcellMass P m j k (ℓ k) = cubeMass P m j k)
(hr : ∀ k ∈ S, cubeMass P m j k ≤ r) :
P.real {z | z.2.1 = true ∧
z.1 ∈ ⋃ k ∈ S, scaledMicroCell d m j k (ℓ k)} ≤
(S.card : ℝ) * r := by
calc
P.real {z | z.2.1 = true ∧
z.1 ∈ ⋃ k ∈ S, scaledMicroCell d m j k (ℓ k)} ≤
∑ k ∈ S, microcellMass P m j k (ℓ k) :=
orderedMass_selectedMicrocells_treated_upper P m j S ℓ
_ ≤ ∑ _k ∈ S, r := by
apply Finset.sum_le_sum
intro k hk
rw [hmin k hk]
exact hr k hk
_ = (S.card : ℝ) * r := by simp
theorem orderedMass_choose_ranked_minimizers Lemma orderedMass_choose_ranked_minimizers in the paper ↗
Choose exactly the first k weakest macro-cubes (breaking ties arbitrarily) and one mass-minimizing microcell in each of them. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma orderedMass_choose_ranked_minimizers {d : ℕ} (P : Measure (Obs d))
[IsProbabilityMeasure P] (m j k : ℕ)
(hk : k ≤ (2 ^ j) ^ d) :
∃ S : Finset (Fin d → Fin (2 ^ j)), S.card = k ∧
∃ ℓ : (Fin d → Fin (2 ^ j)) → Fin d → Fin (m + 1),
(∀ q ∈ S, microcellMass P m j q (ℓ q) = cubeMass P m j q) ∧
(∀ q ∈ S, cubeMass P m j q ≤
(orderedCubeMass P m j).getD (k - 1) 0) := by
classical
let F := (Finset.univ : Finset (Fin d → Fin (2 ^ j))).filter
(fun q => cubeMass P m j q ≤ (orderedCubeMass P m j).getD (k - 1) 0)
obtain ⟨S, hSF, hS⟩ := Finset.exists_subset_card_eq
(orderedMass_rank_threshold_card P m j k hk : k ≤ F.card)
have hchoice : ∀ q : Fin d → Fin (2 ^ j),
∃ ℓ : Fin d → Fin (m + 1),
cubeMass P m j q = microcellMass P m j q ℓ :=
fun q => orderedMass_cubeMass_attained P m j q
choose ℓ hℓ using hchoice
refine ⟨S, hS, ℓ, ?_, ?_⟩
· intro q hq
exact (hℓ q).symm
· intro q hq
exact Finset.mem_filter.mp (hSF hq) |>.2
The treated mass of k selected weakest microcells is at most k times the kth weakest macro-cube mass. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma orderedMass_ranked_union_upper {d : ℕ} (P : Measure (Obs d))
[IsProbabilityMeasure P] (m j k : ℕ)
(hk : k ≤ (2 ^ j) ^ d) :
∃ (S : Finset (Fin d → Fin (2 ^ j)))
(ℓ : (Fin d → Fin (2 ^ j)) → Fin d → Fin (m + 1)),
S.card = k ∧
P.real {z | z.2.1 = true ∧
z.1 ∈ ⋃ q ∈ S, scaledMicroCell d m j q (ℓ q)} ≤
(k : ℝ) * (orderedCubeMass P m j).getD (k - 1) 0 := by
obtain ⟨S, hS, ℓ, hmin, hr⟩ :=
orderedMass_choose_ranked_minimizers P m j k hk
refine ⟨S, ℓ, hS, ?_⟩
simpa only [hS] using
orderedMass_selectedMicrocells_treated_le_card_mul P m j S ℓ
((orderedCubeMass P m j).getD (k - 1) 0) hmin hr
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.OrderedMass.Setwise 16 declarations Setwise and ordered treated-mass bounds
Setwise and ordered treated-mass bounds
theorem orderedMass_treated_measure_disintegration Lemma orderedMass_treated_measure_disintegration in the paper ↗
The conditional treatment kernel disintegrates treated mass over any measurable covariate set. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
(∫⁻ x in E, (condDistrib (fun z : Obs d => z.2.1) (fun z => z.1) P x) {true} ∂covariateLaw P)
= P {z | z.2.1 = true ∧ z.1 ∈ E}
Proof (Lean source)
lemma orderedMass_treated_measure_disintegration {d : ℕ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(E : Set (Fin d → ℝ)) (hE : MeasurableSet E) :
(∫⁻ x in E, (condDistrib
(fun z : Obs d => z.2.1) (fun z => z.1) P x) {true} ∂covariateLaw P) =
P {z | z.2.1 = true ∧ z.1 ∈ E} := by
have hX : Measurable (fun z : Obs d => z.1) := by fun_prop
have hA : AEMeasurable (fun z : Obs d => z.2.1) P :=
(by fun_prop : Measurable (fun z : Obs d => z.2.1)).aemeasurable
have hkernel : Measurable (fun x : Fin d → ℝ =>
(condDistrib
(fun z : Obs d => z.2.1) (fun z => z.1) P x) {true}) :=
(condDistrib
(fun z : Obs d => z.2.1) (fun z => z.1) P).measurable_coe (by simp)
rw [show covariateLaw P = P.map (fun z : Obs d => z.1) from rfl]
rw [Measure.restrict_map hX hE]
rw [lintegral_map hkernel hX]
have hdis := ProbabilityTheory.setLIntegral_preimage_condDistrib
hX hA (show MeasurableSet ({true} : Set Bool) by simp) hE
have hset : (fun z : Obs d => z.1) ⁻¹' E ∩
(fun z : Obs d => z.2.1) ⁻¹' {true} =
{z : Obs d | z.2.1 = true ∧ z.1 ∈ E} := by
ext z
simp [and_comm]
rw [hset] at hdis
exact hdis
theorem orderedMass_treated_measure_propensity Lemma orderedMass_treated_measure_propensity in the paper ↗
The propensity version from the model gives the same setwise treated mass. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ :
ℝ
Pc :
μ₁ e :
(Fin d → ℝ) → ℝ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ₁ e
hE :
Proof (Lean source)
lemma orderedMass_treated_measure_propensity {d : ℕ} (β B L C c_f γ : ℝ)
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ₁ e : (Fin d → ℝ) → ℝ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ₁ e)
(E : Set (Fin d → ℝ)) (hE : MeasurableSet E) :
(∫⁻ x in E, ofReal (e x) ∂covariateLaw (Pc.map observed)) =
(Pc.map observed) {z | z.2.1 = true ∧ z.1 ∈ E} := by
letI : IsProbabilityMeasure (Pc.map observed) :=
Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)
have hAE : (fun x => ofReal (e x)) =ᵐ[covariateLaw (Pc.map observed)]
(fun x => (condDistrib
(fun z : Obs d => z.2.1) (fun z => z.1) (Pc.map observed) x) {true}) := by
filter_upwards [hmodel.tail.1] with x hx
rw [hx]
exact ENNReal.ofReal_toReal (measure_ne_top _ _)
rw [← orderedMass_treated_measure_disintegration (Pc.map observed) E hE]
exact lintegral_congr_ae (ae_restrict_of_ae hAE)
The setwise treatment probability is the real integral of the propensity. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ :
ℝ
Pc :
μ₁ e :
(Fin d → ℝ) → ℝ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ₁ e
hE :
Proof (Lean source)
lemma orderedMass_treated_real_integral {d : ℕ} (β B L C c_f γ : ℝ)
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ₁ e : (Fin d → ℝ) → ℝ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ₁ e)
(E : Set (Fin d → ℝ)) (hE : MeasurableSet E) :
(∫ x in E, e x ∂covariateLaw (Pc.map observed)) =
(Pc.map observed).real {z | z.2.1 = true ∧ z.1 ∈ E} := by
letI : IsProbabilityMeasure (Pc.map observed) :=
Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)
letI : IsProbabilityMeasure (covariateLaw (Pc.map observed)) := by
unfold covariateLaw
exact Measure.isProbabilityMeasure_map (by fun_prop)
have hprop : Measurable (propensity (Pc.map observed)) := by
unfold propensity
have hk : Measurable (fun x : Fin d → ℝ =>
(condDistrib
(fun z : Obs d => z.2.1) (fun z => z.1) (Pc.map observed) x) {true}) :=
(condDistrib
(fun z : Obs d => z.2.1) (fun z => z.1) (Pc.map observed)).measurable_coe
(by simp)
fun_prop
have hsym : propensity (Pc.map observed) =ᵐ[covariateLaw (Pc.map observed)] e := by
filter_upwards [hmodel.tail.1] with x hx
exact hx.symm
have hmeas : AEMeasurable e (covariateLaw (Pc.map observed)) :=
hprop.aemeasurable.congr hsym
have hnn : 0 ≤ᵐ[covariateLaw (Pc.map observed)] e := by
filter_upwards [hmodel.tail.1] with x hx
rw [hx]
exact ENNReal.toReal_nonneg
rw [integral_eq_lintegral_of_nonneg_ae (ae_restrict_of_ae hnn)
(hmeas.restrict.aestronglyMeasurable)]
rw [orderedMass_treated_measure_propensity β B L C c_f γ Pc μ₁ e hmodel E hE]
rfl
A propensity version is almost everywhere between zero and one. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
∀ᵐ x ∂covariateLaw (Pc.map observed), e x ∈ Icc 0 1
Proof (Lean source)
lemma orderedMass_propensity_ae_bounds {d : ℕ} (β B L C c_f γ : ℝ)
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ₁ e : (Fin d → ℝ) → ℝ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ₁ e) :
∀ᵐ x ∂covariateLaw (Pc.map observed), e x ∈ Icc 0 1 := by
letI : IsProbabilityMeasure (Pc.map observed) :=
Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)
filter_upwards [hmodel.tail.1] with x hx
rw [hx]
constructor
· exact ENNReal.toReal_nonneg
· letI : IsProbabilityMeasure
((condDistrib
(fun z : Obs d => z.2.1) (fun z => z.1) (Pc.map observed)) x) :=
ProbabilityTheory.IsMarkovKernel.isProbabilityMeasure x
exact measureReal_le_one
The propensity version is integrable on every measurable covariate set. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ :
ℝ
Pc :
μ₁ e :
(Fin d → ℝ) → ℝ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ₁ e
_hE :
conclusion
Proof (Lean source)
lemma orderedMass_propensity_integrable {d : ℕ} (β B L C c_f γ : ℝ)
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ₁ e : (Fin d → ℝ) → ℝ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ₁ e)
(E : Set (Fin d → ℝ)) (_hE : MeasurableSet E) :
Integrable e ((covariateLaw (Pc.map observed)).restrict E) := by
letI : IsProbabilityMeasure (Pc.map observed) :=
Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)
letI : IsProbabilityMeasure (covariateLaw (Pc.map observed)) := by
unfold covariateLaw
exact Measure.isProbabilityMeasure_map (by fun_prop)
have hprop : Measurable (propensity (Pc.map observed)) := by
unfold propensity
have hk : Measurable (fun x : Fin d → ℝ =>
(condDistrib
(fun z : Obs d => z.2.1) (fun z => z.1) (Pc.map observed) x) {true}) :=
(condDistrib
(fun z : Obs d => z.2.1) (fun z => z.1) (Pc.map observed)).measurable_coe
(by simp)
fun_prop
have hmeas : AEMeasurable e (covariateLaw (Pc.map observed)) :=
hprop.aemeasurable.congr (by
filter_upwards [hmodel.tail.1] with x hx
exact hx.symm)
have hbounded : ∀ᵐ x ∂(covariateLaw (Pc.map observed)).restrict E,
e x ∈ Icc (0 : ℝ) 1 :=
ae_restrict_of_ae (orderedMass_propensity_ae_bounds β B L C c_f γ Pc μ₁ e hmodel)
exact Integrable.of_mem_Icc 0 1 hmeas.restrict hbounded
Layer cake expresses treated propensity mass through upper-level sets. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ :
ℝ
Pc :
μ₁ e :
(Fin d → ℝ) → ℝ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ₁ e
hE :
conclusion
(∫ x in E, e x ∂covariateLaw (Pc.map observed))
Proof (Lean source)
lemma orderedMass_setwise_layercake {d : ℕ} (β B L C c_f γ : ℝ)
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ₁ e : (Fin d → ℝ) → ℝ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ₁ e)
(E : Set (Fin d → ℝ)) (hE : MeasurableSet E) :
(∫ x in E, e x ∂covariateLaw (Pc.map observed)) =
∫ t in Ioi (0 : ℝ),
((covariateLaw (Pc.map observed)).restrict E).real {x | t < e x} := by
have hnonneg : 0 ≤ᵐ[(covariateLaw (Pc.map observed)).restrict E] e := by
filter_upwards [ae_restrict_of_ae
(orderedMass_propensity_ae_bounds β B L C c_f γ Pc μ₁ e hmodel)] with x hx
exact hx.1
exact (orderedMass_propensity_integrable β B L C c_f γ Pc μ₁ e hmodel E hE)
|>.integral_eq_integral_meas_lt hnonneg
The global tail bound also controls the part of an arbitrary set lying in the low-propensity region. This is the setwise estimate used by layer cake. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ :
ℝ
Pc :
μ₁ e :
(Fin d → ℝ) → ℝ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ₁ e
t :
ℝ
ht :
t ∈ Icc (0 : ℝ) 1
conclusion
(covariateLaw (Pc.map observed)).real (E ∩ {x | e x ≤ t}) ≤ C * t ^ (γ - 1)
Proof (Lean source)
lemma orderedMass_intersection_tail_le {d : ℕ} (β B L C c_f γ : ℝ)
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ₁ e : (Fin d → ℝ) → ℝ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ₁ e)
(E : Set (Fin d → ℝ)) (t : ℝ) (ht : t ∈ Icc (0 : ℝ) 1) :
(covariateLaw (Pc.map observed)).real (E ∩ {x | e x ≤ t}) ≤
C * t ^ (γ - 1) := by
letI : IsProbabilityMeasure (Pc.map observed) :=
Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)
letI : IsProbabilityMeasure (covariateLaw (Pc.map observed)) := by
unfold covariateLaw
exact Measure.isProbabilityMeasure_map (by fun_prop)
calc
(covariateLaw (Pc.map observed)).real (E ∩ {x | e x ≤ t}) ≤
(covariateLaw (Pc.map observed)).real {x | e x ≤ t} := by
exact ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono Set.inter_subset_right)
_ ≤ C * t ^ (γ - 1) := hmodel.tail.2 t ht
Each set retains at least its mass minus the global lower tail above a propensity threshold. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ :
ℝ
Pc :
μ₁ e :
(Fin d → ℝ) → ℝ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ₁ e
t :
ℝ
ht :
t ∈ Icc (0 : ℝ) 1
conclusion
(covariateLaw (Pc.map observed)).real E - C * t ^ (γ - 1)
≤ (covariateLaw (Pc.map observed)).real (E ∩ {x | t < e x})
Proof (Lean source)
lemma orderedMass_high_part_lower {d : ℕ} (β B L C c_f γ : ℝ)
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ₁ e : (Fin d → ℝ) → ℝ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ₁ e)
(E : Set (Fin d → ℝ)) (t : ℝ) (ht : t ∈ Icc (0 : ℝ) 1) :
(covariateLaw (Pc.map observed)).real E - C * t ^ (γ - 1) ≤
(covariateLaw (Pc.map observed)).real (E ∩ {x | t < e x}) := by
letI : IsProbabilityMeasure (Pc.map observed) :=
Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)
letI : IsProbabilityMeasure (covariateLaw (Pc.map observed)) := by
unfold covariateLaw
exact Measure.isProbabilityMeasure_map (by fun_prop)
let ν := covariateLaw (Pc.map observed)
have hcover : E ⊆ (E ∩ {x | t < e x}) ∪ (E ∩ {x | e x ≤ t}) := by
intro x hx
by_cases h : t < e x
· exact inl ⟨hx, h⟩
· exact inr ⟨hx, le_of_not_gt h⟩
have hmass : ν.real E ≤
ν.real (E ∩ {x | t < e x}) + ν.real (E ∩ {x | e x ≤ t}) := by
have hle : ν E ≤ ν (E ∩ {x | t < e x}) + ν (E ∩ {x | e x ≤ t}) :=
(measure_mono hcover).trans
(measure_union_le (E ∩ {x | t < e x}) (E ∩ {x | e x ≤ t}))
have hfinite : ν (E ∩ {x | t < e x}) + ν (E ∩ {x | e x ≤ t}) ≠ ⊤ := by
exact ENNReal.add_ne_top.mpr ⟨measure_ne_top _ _, measure_ne_top _ _⟩
have := ENNReal.toReal_mono hfinite hle
simpa only [Measure.real, ENNReal.toReal_add (measure_ne_top _ _) (measure_ne_top _ _)]
using this
have htail := orderedMass_intersection_tail_le β B L C c_f γ Pc μ₁ e hmodel E t ht
dsimp [ν] at hmass
linarith
On a bounded threshold interval, the layer-cake integrand dominates the set mass minus the global lower-tail envelope. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ :
ℝ
Pc :
μ₁ e :
(Fin d → ℝ) → ℝ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ₁ e
hE :
s :
ℝ
hs :
s ∈ Icc (0 : ℝ) 1
hγ :
1 < γ
conclusion
(∫ t in (0 : ℝ)..s, (covariateLaw (Pc.map observed)).real E - C * t ^ (γ - 1))
Proof (Lean source)
lemma orderedMass_layercake_piece_lower {d : ℕ} (β B L C c_f γ : ℝ)
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ₁ e : (Fin d → ℝ) → ℝ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ₁ e)
(E : Set (Fin d → ℝ)) (hE : MeasurableSet E)
(s : ℝ) (hs : s ∈ Icc (0 : ℝ) 1)
(hγ : 1 < γ) :
(∫ t in (0 : ℝ)..s,
(covariateLaw (Pc.map observed)).real E - C * t ^ (γ - 1)) ≤
∫ t in (0 : ℝ)..s,
((covariateLaw (Pc.map observed)).restrict E).real {x | t < e x} := by
let ν := covariateLaw (Pc.map observed)
letI : IsProbabilityMeasure (Pc.map observed) :=
Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)
letI : IsProbabilityMeasure ν := by
dsimp [ν, covariateLaw]
exact Measure.isProbabilityMeasure_map (by fun_prop)
let F : ℝ → ℝ := fun t => (ν.restrict E).real {x | t < e x}
have hanti : Antitone F := by
intro t t' htt'
exact ENNReal.toReal_mono (measure_ne_top _ _)
(measure_mono (by intro x hx; exact lt_of_le_of_lt htt' hx))
have hFint : IntervalIntegrable F volume 0 s := by
rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hs.1]
exact (hanti.locallyIntegrable.integrableOn_isCompact isCompact_Icc).mono_set
Set.Ioc_subset_Icc_self
have hpoly : IntervalIntegrable
(fun t : ℝ => ν.real E - C * t ^ (γ - 1)) volume 0 s := by
apply IntervalIntegrable.sub
· exact intervalIntegrable_const
· exact (intervalIntegral.intervalIntegrable_rpow' (by linarith : -1 < γ - 1)).const_mul C
apply intervalIntegral.integral_mono_on hs.1 hpoly hFint
intro t ht
have hbound := orderedMass_high_part_lower β B L C c_f γ Pc μ₁ e hmodel E t
⟨ht.1, ht.2.trans hs.2⟩
simpa only [F, ν, Measure.real, Measure.restrict_apply' hE, Set.inter_comm]
using hbound
The exact integral in the setwise treated-mass argument. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
u C γ s :
ℝ
hγ :
1 < γ
conclusion
(∫ t in (0 : ℝ)..s, (u - C * t ^ (γ - 1))) = u * s - C * s ^ γ / γ
Proof (Lean source)
lemma orderedMass_integral_tail (u C γ s : ℝ) (hγ : 1 < γ) :
(∫ t in (0 : ℝ)..s, (u - C * t ^ (γ - 1))) =
u * s - C * s ^ γ / γ := by
have hr : -1 < γ - 1 := by linarith
rw [intervalIntegral.integral_sub intervalIntegrable_const
((intervalIntegral.intervalIntegrable_rpow' hr).const_mul C),
intervalIntegral.integral_const,
intervalIntegral.integral_const_mul, integral_rpow (inl hr)]
have hγne : γ ≠ 0 := ne_of_gt (by linarith)
have he : γ - 1 + 1 = γ := by ring
rw [he]
rw [Real.zero_rpow hγne]
simp only [sub_zero]
field_simp
ring
Substituting the tail threshold into the layer-cake integral. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
u C γ s :
ℝ
hγ :
1 < γ
hs :
0 < s
hthreshold :
C * s ^ (γ - 1) = u
conclusion
(∫ t in (0 : ℝ)..s, (u - C * t ^ (γ - 1))) = ((γ - 1) / γ) * u * s
Proof (Lean source)
lemma orderedMass_integral_optimal (u C γ s : ℝ) (hγ : 1 < γ)
(hs : 0 < s) (hthreshold : C * s ^ (γ - 1) = u) :
(∫ t in (0 : ℝ)..s, (u - C * t ^ (γ - 1))) =
((γ - 1) / γ) * u * s := by
rw [orderedMass_integral_tail u C γ s hγ]
have hpow : s ^ γ = s ^ (γ - 1) * s := by
conv_lhs => rw [show γ = γ - 1 + 1 by ring]
rw [Real.rpow_add hs]
simp
rw [hpow]
have hγne : γ ≠ 0 := ne_of_gt (by linarith)
field_simp
nlinarith [hthreshold]
The proposed threshold balances the set mass against the tail envelope. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
u C γ :
ℝ
hu :
0 < u
hC :
0 < C
hγ :
1 < γ
conclusion
C * ((u / C) ^ (1 / (γ - 1))) ^ (γ - 1) = u
Proof (Lean source)
lemma orderedMass_threshold_identity (u C γ : ℝ) (hu : 0 < u)
(hC : 0 < C) (hγ : 1 < γ) :
C * ((u / C) ^ (1 / (γ - 1))) ^ (γ - 1) = u := by
have huc : 0 ≤ u / C := le_of_lt (div_pos hu hC)
have hden : γ - 1 ≠ 0 := ne_of_gt (by linarith)
rw [← Real.rpow_mul huc]
have hexp : 1 / (γ - 1) * (γ - 1) = 1 := by
field_simp
rw [hexp, Real.rpow_one]
exact mul_div_cancel₀ u (ne_of_gt hC)
The layer-cake cutoff lies in the unit interval for a probability mass. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
u C γ :
ℝ
hu :
0 < u
huone :
u ≤ 1
hC :
1 ≤ C
hγ :
1 < γ
conclusion
(u / C) ^ (1 / (γ - 1)) ∈ Icc (0 : ℝ) 1
Proof (Lean source)
lemma orderedMass_threshold_mem_unit (u C γ : ℝ) (hu : 0 < u)
(huone : u ≤ 1) (hC : 1 ≤ C) (hγ : 1 < γ) :
(u / C) ^ (1 / (γ - 1)) ∈ Icc (0 : ℝ) 1 := by
have hCpos : 0 < C := by linarith
have hquotpos : 0 < u / C := div_pos hu hCpos
have hquotle : u / C ≤ 1 := (div_le_one hCpos).mpr (huone.trans hC)
have hexppos : 0 ≤ 1 / (γ - 1) := by positivity
constructor
· exact (Real.rpow_pos_of_pos hquotpos _).le
· exact Real.rpow_le_one hquotpos.le hquotle hexppos
The analytic value of the layer-cake lower bound at its balancing threshold. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
u C γ :
ℝ
hu :
0 < u
hC :
0 < C
hγ :
1 < γ
conclusion
(∫ t in (0 : ℝ)..((u / C) ^ (1 / (γ - 1))), (u - C * t ^ (γ - 1)))
= ((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * u ^ (γ / (γ - 1))
Proof (Lean source)
lemma orderedMass_integral_at_threshold (u C γ : ℝ) (hu : 0 < u)
(hC : 0 < C) (hγ : 1 < γ) :
(∫ t in (0 : ℝ)..((u / C) ^ (1 / (γ - 1))),
(u - C * t ^ (γ - 1))) =
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
u ^ (γ / (γ - 1)) := by
have hs : 0 < (u / C) ^ (1 / (γ - 1)) :=
Real.rpow_pos_of_pos (div_pos hu hC) _
rw [orderedMass_integral_optimal u C γ _ hγ hs
(orderedMass_threshold_identity u C γ hu hC hγ)]
rw [Real.div_rpow hu.le hC.le]
have hden : γ - 1 ≠ 0 := ne_of_gt (by linarith)
have he : γ / (γ - 1) = 1 + 1 / (γ - 1) := by
field_simp
ring
rw [he, Real.rpow_add hu, Real.rpow_one]
have hneg : -(1 : ℝ) / (γ - 1) = -(1 / (γ - 1)) := by ring
rw [hneg, Real.rpow_neg hC.le]
ring
The global propensity tail yields the sharp setwise treated-mass bound. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ :
ℝ
Pc :
μ₁ e :
(Fin d → ℝ) → ℝ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ₁ e
hC :
1 ≤ C
hγ :
1 < γ
hE :
conclusion
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (covariateLaw (Pc.map observed)).real E ^ (γ / (γ - 1))
Proof (Lean source)
lemma orderedMass_setwise_lower {d : ℕ} (β B L C c_f γ : ℝ)
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ₁ e : (Fin d → ℝ) → ℝ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ₁ e)
(hC : 1 ≤ C) (hγ : 1 < γ)
(E : Set (Fin d → ℝ)) (hE : MeasurableSet E) :
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(covariateLaw (Pc.map observed)).real E ^ (γ / (γ - 1)) ≤
(Pc.map observed).real {z | z.2.1 = true ∧ z.1 ∈ E} := by
let ν := covariateLaw (Pc.map observed)
letI : IsProbabilityMeasure (Pc.map observed) :=
Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)
letI : IsProbabilityMeasure ν := by
dsimp [ν, covariateLaw]
exact Measure.isProbabilityMeasure_map (by fun_prop)
let u := ν.real E
have hu0 : 0 ≤ u := ENNReal.toReal_nonneg
have hu1 : u ≤ 1 := by
dsimp [u]
calc
ν.real E ≤ ν.real univ :=
ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono (Set.subset_univ E))
_ = 1 := by simp
by_cases hu : u = 0
· rw [show ν.real E = 0 from hu]
simp only [Real.zero_rpow (by positivity : γ / (γ - 1) ≠ 0), mul_zero]
exact ENNReal.toReal_nonneg
have hupos : 0 < u := lt_of_le_of_ne hu0 (Ne.symm hu)
let s := (u / C) ^ (1 / (γ - 1))
have hs : s ∈ Icc (0 : ℝ) 1 := orderedMass_threshold_mem_unit u C γ hupos hu1 hC hγ
let F : ℝ → ℝ := fun t => (ν.restrict E).real {x | t ≤ e x}
have hanti : Antitone F := by
intro t t' htt'
exact ENNReal.toReal_mono (measure_ne_top _ _)
(measure_mono (by intro x hx; exact htt'.trans hx))
have hFint : IntervalIntegrable F volume 0 1 := by
rw [intervalIntegrable_iff_integrableOn_Ioc_of_le (by norm_num : (0 : ℝ) ≤ 1)]
exact (hanti.locallyIntegrable.integrableOn_isCompact isCompact_Icc).mono_set
Set.Ioc_subset_Icc_self
have hFnn : 0 ≤ᵐ[volume.restrict (Ioc (0 : ℝ) 1)] F :=
Filter.Eventually.of_forall (fun _ => ENNReal.toReal_nonneg)
have hmass : (Pc.map observed).real {z | z.2.1 = true ∧ z.1 ∈ E} =
∫ t in (0 : ℝ)..1, F t := by
rw [← orderedMass_treated_real_integral β B L C c_f γ Pc μ₁ e hmodel E hE]
have hbound := orderedMass_propensity_ae_bounds β B L C c_f γ Pc μ₁ e hmodel
have hboundE : e ≤ᵐ[ν.restrict E] fun _ => (1 : ℝ) :=
ae_restrict_of_ae (by filter_upwards [hbound] with x hx; exact hx.2)
have hnonneg : 0 ≤ᵐ[ν.restrict E] e :=
ae_restrict_of_ae (by filter_upwards [hbound] with x hx; exact hx.1)
have hi := orderedMass_propensity_integrable β B L C c_f γ Pc μ₁ e hmodel E hE
rw [hi.integral_eq_integral_Ioc_meas_le hnonneg hboundE]
exact (intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1)).symm
have hpiece : (∫ t in (0 : ℝ)..s, u - C * t ^ (γ - 1)) ≤
∫ t in (0 : ℝ)..s, F t := by
have hfirst := orderedMass_layercake_piece_lower β B L C c_f γ Pc μ₁ e hmodel E hE s hs hγ
apply hfirst.trans
apply intervalIntegral.integral_mono_on hs.1
· let F' : ℝ → ℝ := fun t => (ν.restrict E).real {x | t < e x}
have hanti' : Antitone F' := by
intro t t' htt'
exact ENNReal.toReal_mono (measure_ne_top _ _)
(measure_mono (by intro x hx; exact lt_of_le_of_lt htt' hx))
rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hs.1]
exact (hanti'.locallyIntegrable.integrableOn_isCompact isCompact_Icc).mono_set
Set.Ioc_subset_Icc_self
· rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hs.1]
exact (hanti.locallyIntegrable.integrableOn_isCompact isCompact_Icc).mono_set
Set.Ioc_subset_Icc_self
· intro t ht
exact ENNReal.toReal_mono (measure_ne_top _ _)
(measure_mono (by intro x hx; change t < e x at hx; exact hx.le))
calc
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * ν.real E ^ (γ / (γ - 1)) =
∫ t in (0 : ℝ)..s, u - C * t ^ (γ - 1) := by
exact (orderedMass_integral_at_threshold u C γ hupos (by linarith) hγ).symm
_ ≤ ∫ t in (0 : ℝ)..s, F t := hpiece
_ ≤ ∫ t in (0 : ℝ)..1, F t :=
intervalIntegral.integral_mono_interval le_rfl hs.1 hs.2 hFnn hFint
_ = (Pc.map observed).real {z | z.2.1 = true ∧ z.1 ∈ E} := hmass.symm
The ordered-cell coefficient has a positive lower bound over a compact range of overlap exponents. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
C a γ_min γ_max :
ℝ
hC :
0 < C
ha :
0 < a
hγmin :
1 < γ_min
hγrange :
γ_min < γ_max
conclusion
∃ κ : ℝ,
0 < κ
conclusion 1
γ ∈ Set.Icc γ_min γ_max :
κ ≤ ((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * a ^ (γ / (γ - 1))
Proof (Lean source)
lemma orderedMass_uniform_coefficient (C a γ_min γ_max : ℝ)
(hC : 0 < C) (ha : 0 < a)
(hγmin : 1 < γ_min) (hγrange : γ_min < γ_max) :
∃ κ : ℝ, 0 < κ ∧ ∀ γ ∈ Icc γ_min γ_max,
κ ≤ ((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
a ^ (γ / (γ - 1)) := by
let f : ℝ → ℝ := fun γ => ((γ - 1) / γ) *
C ^ (-(1 : ℝ) / (γ - 1)) * a ^ (γ / (γ - 1))
have hcont : ContinuousOn f (Icc γ_min γ_max) := by
have hden : ∀ γ ∈ Icc γ_min γ_max, γ - 1 ≠ 0 := by
intro γ hγ
exact ne_of_gt (by linarith [hγ.1])
have hγne : ∀ γ ∈ Icc γ_min γ_max, γ ≠ 0 := by
intro γ hγ
exact ne_of_gt (by linarith [hγ.1])
have hbase : ContinuousOn (fun γ : ℝ => (γ - 1) / γ)
(Icc γ_min γ_max) := by
fun_prop (disch := assumption)
have hCexp : ContinuousOn (fun γ : ℝ => -(1 : ℝ) / (γ - 1))
(Icc γ_min γ_max) := by
fun_prop (disch := assumption)
have haexp : ContinuousOn (fun γ : ℝ => γ / (γ - 1))
(Icc γ_min γ_max) := by
fun_prop (disch := assumption)
have hCpow : ContinuousOn (fun γ : ℝ => C ^ (-(1 : ℝ) / (γ - 1)))
(Icc γ_min γ_max) :=
continuousOn_const.rpow hCexp (by intro γ hγ; exact inl (ne_of_gt hC))
have hapow : ContinuousOn (fun γ : ℝ => a ^ (γ / (γ - 1)))
(Icc γ_min γ_max) :=
continuousOn_const.rpow haexp (by intro γ hγ; exact inl (ne_of_gt ha))
exact (hbase.mul hCpow).mul hapow
obtain ⟨γ₀, hγ₀, hmin⟩ :=
isCompact_Icc.exists_isMinOn ⟨γ_min, ⟨le_refl _, le_of_lt hγrange⟩⟩ hcont
have hpos : 0 < f γ₀ := by
dsimp [f]
have hγ : 1 < γ₀ := lt_of_lt_of_le hγmin hγ₀.1
positivity
refine ⟨f γ₀, hpos, ?_⟩
intro γ hγ
exact hmin hγ
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.SelectedMesh 19 declarations Selected-mesh coefficient envelope
Selected-mesh coefficient envelope
A positive selected width comes from a feasible dyadic index. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma selectedMesh_feasible_of_pos {d n : ℕ} (m : ℕ) (β : ℝ)
(ω : Fin n → Obs d)
(hpos : 0 < countSelectedMesh m β ω) :
meshFeasible m β ω ((feasibleIndices m β ω).sup id) := by
classical
let F := feasibleIndices m β ω
have hF : F.Nonempty := by
by_contra h
have hzero : countSelectedMesh m β ω = 0 := by
simp [countSelectedMesh, F, h]
linarith
have hmem : F.sup id ∈ F := by
rw [← Finset.sup'_eq_sup hF id, ← Finset.max'_eq_sup' F hF]
exact Finset.max'_mem F hF
exact (Finset.mem_filter.mp hmem).2
On the nonempty feasible event, the selected width is the width of the selected dyadic index. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma selectedMesh_width_of_pos {d n : ℕ} (m : ℕ) (β : ℝ)
(ω : Fin n → Obs d)
(hpos : 0 < countSelectedMesh m β ω) :
countSelectedMesh m β ω =
meshWidth ((feasibleIndices m β ω).sup id) := by
classical
have hF : (feasibleIndices m β ω).Nonempty := by
by_contra h
have hzero : countSelectedMesh m β ω = 0 := by
simp [countSelectedMesh, h]
linarith
simp [countSelectedMesh, hF]
theorem selectedMesh_measurable_event_intersect_ae Lemma selectedMesh_measurable_event_intersect_ae in the paper ↗
Intersect a measurable count event with an almost-sure conditional coefficient assertion without increasing its failure probability. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma selectedMesh_measurable_event_intersect_ae
{Ω : Type*} [MeasurableSpace Ω] (Q : Measure Ω)
(E : Set Ω) (hE : MeasurableSet E) (ε : ℝ)
(hbad : Q.real Eᶜ ≤ ε) {R : Ω → Prop}
(hR : ∀ᵐ ω ∂Q, R ω) :
∃ F : Set Ω, MeasurableSet F ∧ Q.real Fᶜ ≤ ε ∧
∀ ω ∈ F, ω ∈ E ∧ R ω := by
obtain ⟨G, hGae, hGmeas, hGR⟩ := hR.exists_measurable_mem
refine ⟨E ∩ G, hE.inter hGmeas, ?_, ?_⟩
· have hGnull : Q Gᶜ = 0 := (ae_iff.mp hGae)
have hGreal : Q.real Gᶜ = 0 := by simp [MeasureTheory.measureReal_def, hGnull]
rw [Set.compl_inter]
calc
Q.real (Eᶜ ∪ Gᶜ) ≤ Q.real Eᶜ + Q.real Gᶜ :=
measureReal_union_le _ _
_ = Q.real Eᶜ := by rw [hGreal, add_zero]
_ ≤ ε := hbad
· intro ω hω
exact ⟨hω.1, hGR ω hω.2⟩
The selected count event and the actual conditional coefficient MGF hold simultaneously on one measurable event with the count-event failure budget. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ_min γ_max :
ℝ
hd :
1 ≤ d
hβ :
1 < β
hB :
0 < B
hL :
0 < L
hC :
1 ≤ C
hcf :
0 < c_f ∧ c_f ≤ 1
hγmin :
1 < γ_min
hγrange :
γ_min < γ_max
conclusion
∃ K : ℝ,
conclusion 1
0 < K
conclusion 2
∃ n₀ : ℕ,
∀ n ≥ n₀,
∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin,
hγrange⟩, ∀ (P : Measure (Obs d)) [IsProbabilityMeasure P] (hP : P ∈ ModelClass d β B L C c_f γ) (Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q], IIDSampleLaw Q P → ∃ E : Set (Fin n → Obs d), MeasurableSet E ∧ Q.real Eᶜ
≤ (n : ℝ) ^ (-2 : ℝ) ∧
∀ ω ∈ E,
let m := polynomialDegree β let j := (feasibleIndices m β ω).sup id let h := countSelectedMesh m β ω 0 < h ∧
h ≤ K * oracleMesh d n β γ ∧
(∀ (k : Fin d → Fin (2 ^ j)) (ℓ : Fin d → Fin (m + 1)), (n : ℝ) * microcellMass P m j k ℓ / 5 ≤ treatedCount m j ω k ℓ) ∧
(∀ (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) (t : ℝ), mgf (fun ξ => coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α) (condDistrib id sampleDesign Q (sampleDesign ω)) t ≤ exp (B ^ 2 * ((tensorCount d m : ℝ) * ((tensorCount d m : ℝ)⁻¹ * ((2 / templateLambda d m) * sqrt (card (MonoIndex d m) : ℝ))) ^ 2 * ((tensorCount d m : ℝ) * min (meshWidth j ^ (2 * β)) (5 / ((n : ℝ) * (((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (c_f * templateEta d m ^ d) ^ (γ / (γ - 1))) * meshWidth j ^ effectiveDimension d γ * (cubeMassRank P m j k : ℝ) ^ (1 / (γ - 1)))))) * t ^ 2 / 2)) ∧
(∀ (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) (t : ℝ), Integrable (fun ξ => exp (t * (coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α))) (condDistrib id sampleDesign Q (sampleDesign ω)))
Proof (Lean source)
lemma selectedMesh_count_and_raw_mgf (d : ℕ) (β B L C c_f γ_min γ_max : ℝ)
(hd : 1 ≤ d) (hβ : 1 < β) (hB : 0 < B) (hL : 0 < L)
(hC : 1 ≤ C) (hcf : 0 < c_f ∧ c_f ≤ 1)
(hγmin : 1 < γ_min) (hγrange : γ_min < γ_max) :
∃ K : ℝ, 0 < K ∧ ∃ n₀ : ℕ,
∀ n ≥ n₀, ∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
∀ (P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q],
IIDSampleLaw Q P →
∃ E : Set (Fin n → Obs d), MeasurableSet E ∧
Q.real Eᶜ ≤ (n : ℝ) ^ (-2 : ℝ) ∧
∀ ω ∈ E,
let m := polynomialDegree β
let j := (feasibleIndices m β ω).sup id
let h := countSelectedMesh m β ω
0 < h ∧ h ≤ K * oracleMesh d n β γ ∧
(∀ (k : Fin d → Fin (2 ^ j)) (ℓ : Fin d → Fin (m + 1)),
(n : ℝ) * microcellMass P m j k ℓ / 5 ≤ treatedCount m j ω k ℓ) ∧
(∀ (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) (t : ℝ),
mgf (fun ξ => coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α)
(condDistrib id sampleDesign Q (sampleDesign ω)) t ≤
exp (B ^ 2 *
((tensorCount d m : ℝ) *
((tensorCount d m : ℝ)⁻¹ *
((2 / templateLambda d m) *
sqrt (card (MonoIndex d m) : ℝ))) ^ 2 *
((tensorCount d m : ℝ) *
min (meshWidth j ^ (2 * β))
(5 / ((n : ℝ) *
(((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1))) *
meshWidth j ^ effectiveDimension d γ *
(cubeMassRank P m j k : ℝ) ^ (1 / (γ - 1)))))) * t ^ 2 / 2)) ∧
(∀ (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) (t : ℝ),
Integrable (fun ξ => exp (t *
(coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α)))
(condDistrib id sampleDesign Q (sampleDesign ω))) := by
obtain ⟨K, hK, n₀, hcount⟩ :=
selectedCountEvent_uniform d β B L C c_f γ_min γ_max
hd hβ hB hL hC hcf hγmin hγrange
refine ⟨K, hK, max 1 n₀, ?_⟩
intro n hn γ hγ P _ hP Q _ hIID
have hn₀ : n₀ ≤ n := (Nat.le_max_right _ _).trans hn
obtain ⟨E, hE, hbad, hbounds⟩ := hcount n hn₀ γ hγ P hP Q hIID
have hres : BoundedMeanSubGaussianResidual P B := by
rcases hP with ⟨_, _, Pc, hPc, μ₁, e, hEq, hmodel⟩
subst P
exact hmodel.outcome
have hmgf := modelClass_conditionalCoefficient_mgf_rank_bound
P Q hIID hres
have hint := conditionalCoefficient_integrable_exp P Q hIID hres
obtain ⟨F, hF, hFbad, hFproperty⟩ :=
selectedMesh_measurable_event_intersect_ae Q E hE
((n : ℝ) ^ (-2 : ℝ)) hbad (hmgf.and hint)
refine ⟨F, hF, hFbad, ?_⟩
intro ω hω
obtain ⟨hωE, hωMGF, hωInt⟩ := hFproperty ω hω
let m := polynomialDegree β
let j := (feasibleIndices m β ω).sup id
let h := countSelectedMesh m β ω
obtain ⟨hh, hmesh, hcounts⟩ := hbounds ω hωE
refine ⟨hh, hmesh, hcounts, ?_, ?_⟩
· intro k α t
have hnpos : 0 < n := by
have : 1 ≤ n := (Nat.le_max_left _ _).trans hn
omega
have hfeasible : meshFeasible m β ω j :=
selectedMesh_feasible_of_pos m β ω hh
exact hωMGF m j hP k α hnpos hfeasible (hcounts k) t
· intro k α t
exact hωInt m j k α t
theorem selectedMesh_raw_coefficient_mgf_ordered Lemma selectedMesh_raw_coefficient_mgf_ordered in the paper ↗
Convert the actual conditional coefficient MGF to the ordered variance proxy. The finite template factor and the uniform mass coefficient are absorbed into one constant. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n m j N :
ℕ
β B C c_f γ κ :
ℝ
P :
Measure (Obs d)
Q :
ω :
Fin n → Obs d
hκ :
0 < κ
hκle :
κ ≤ ((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (c_f * templateEta d m ^ d) ^ (γ / (γ - 1))
hγ :
1 < γ
hn :
0 < n
hrank :
∀ k : Fin N, k.val + 1 ≤ cubeMassRank P m j (σ k)
hraw :
mgf (fun ξ => coefHat m j ξ (σ k) α - conditionalCoefCentre Q m j ω (σ k) α) (condDistrib id sampleDesign Q (sampleDesign ω)) t
≤ exp (B ^ 2 * ((tensorCount d m : ℝ) * ((tensorCount d m : ℝ)⁻¹ * ((2 / templateLambda d m) * sqrt (card (MonoIndex d m) : ℝ))) ^ 2 * ((tensorCount d m : ℝ) * min (meshWidth j ^ (2 * β)) (5 / ((n : ℝ) * (((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (c_f * templateEta d m ^ d) ^ (γ / (γ - 1))) * meshWidth j ^ effectiveDimension d γ * (cubeMassRank P m j (σ k) : ℝ) ^ (1 / (γ - 1)))))) * t ^ 2 / 2)
k :
Fin N
α :
MonoIndex d m
t :
ℝ
conclusion
mgf (fun ξ => coefHat m j ξ (σ k) α - conditionalCoefCentre Q m j ω (σ k) α) (condDistrib id sampleDesign Q (sampleDesign ω)) t
≤ exp ((B ^ 2 * ((tensorCount d m : ℝ) * ((tensorCount d m : ℝ)⁻¹ * ((2 / templateLambda d m) * sqrt (card (MonoIndex d m) : ℝ))) ^ 2 * (tensorCount d m : ℝ)) * max 1 (5 / κ)) * min (meshWidth j ^ (2 * β)) ((n : ℝ)⁻¹ * meshWidth j ^ (-effectiveDimension d γ) * (k.val + 1 : ℝ) ^ (-(1 : ℝ) / (γ - 1))) * t ^ 2 / 2)
Proof (Lean source)
lemma selectedMesh_raw_coefficient_mgf_ordered
{d n m j N : ℕ} (β B C c_f γ κ : ℝ)
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q]
(ω : Fin n → Obs d)
(σ : Fin N → Fin d → Fin (2 ^ j))
(hκ : 0 < κ)
(hκle : κ ≤ ((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1)))
(hγ : 1 < γ) (hn : 0 < n)
(hrank : ∀ k : Fin N, k.val + 1 ≤ cubeMassRank P m j (σ k))
(hraw : ∀ (k : Fin N) (α : MonoIndex d m) (t : ℝ),
mgf (fun ξ => coefHat m j ξ (σ k) α -
conditionalCoefCentre Q m j ω (σ k) α)
(condDistrib id sampleDesign Q (sampleDesign ω)) t ≤
exp (B ^ 2 *
((tensorCount d m : ℝ) *
((tensorCount d m : ℝ)⁻¹ *
((2 / templateLambda d m) *
sqrt (card (MonoIndex d m) : ℝ))) ^ 2 *
((tensorCount d m : ℝ) *
min (meshWidth j ^ (2 * β))
(5 / ((n : ℝ) *
(((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1))) *
meshWidth j ^ effectiveDimension d γ *
(cubeMassRank P m j (σ k) : ℝ) ^ (1 / (γ - 1)))))) * t ^ 2 / 2)) :
∀ (k : Fin N) (α : MonoIndex d m) (t : ℝ),
mgf (fun ξ => coefHat m j ξ (σ k) α -
conditionalCoefCentre Q m j ω (σ k) α)
(condDistrib id sampleDesign Q (sampleDesign ω)) t ≤
exp ((B ^ 2 *
((tensorCount d m : ℝ) *
((tensorCount d m : ℝ)⁻¹ *
((2 / templateLambda d m) *
sqrt (card (MonoIndex d m) : ℝ))) ^ 2 *
(tensorCount d m : ℝ)) * max 1 (5 / κ)) *
min (meshWidth j ^ (2 * β))
((n : ℝ)⁻¹ * meshWidth j ^ (-effectiveDimension d γ) *
(k.val + 1 : ℝ) ^ (-(1 : ℝ) / (γ - 1))) * t ^ 2 / 2) := by
intro k α t
have hα : 0 < 1 / (γ - 1) := by positivity
have hA : 0 ≤ B ^ 2 *
((tensorCount d m : ℝ) *
((tensorCount d m : ℝ)⁻¹ *
((2 / templateLambda d m) *
sqrt (card (MonoIndex d m) : ℝ))) ^ 2 *
(tensorCount d m : ℝ)) := by positivity
have hx : 0 ≤ meshWidth j ^ (2 * β) := by
apply Real.rpow_nonneg
unfold meshWidth
positivity
have hr : 0 < (cubeMassRank P m j (σ k) : ℝ) := by
exact_mod_cast cubeMassRank_pos P m j (σ k)
have hn' : (0 : ℝ) < n := by exact_mod_cast hn
have hmgf := selectedMesh_uniform_ranked_mgf
(condDistrib id sampleDesign Q (sampleDesign ω))
(fun ξ => coefHat m j ξ (σ k) α -
conditionalCoefCentre Q m j ω (σ k) α)
(B ^ 2 *
((tensorCount d m : ℝ) *
((tensorCount d m : ℝ)⁻¹ *
((2 / templateLambda d m) *
sqrt (card (MonoIndex d m) : ℝ))) ^ 2 *
(tensorCount d m : ℝ)))
(meshWidth j ^ (2 * β)) (n : ℝ) (meshWidth j)
(cubeMassRank P m j (σ k) : ℝ) κ
(((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1)))
(effectiveDimension d γ) (1 / (γ - 1)) k.val t
hA hx hn' (by unfold meshWidth; positivity) hr hκ hκle hα
(by exact_mod_cast hrank k)
have hconcl := hmgf (by simpa only [mul_assoc] using hraw k α t)
simpa only [neg_div] using hconcl
theorem selectedMesh_raw_coefficient_mgf_rank Lemma selectedMesh_raw_coefficient_mgf_rank in the paper ↗
Normalize the conditional MGF at its own cube-mass rank before reindexing cubes for the maximal inequality. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n m j :
ℕ
β B C c_f γ κ :
ℝ
P :
Measure (Obs d)
Q :
ω :
Fin n → Obs d
hκ :
0 < κ
hκle :
κ ≤ ((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (c_f * templateEta d m ^ d) ^ (γ / (γ - 1))
hγ :
1 < γ
hn :
0 < n
hraw :
∀ (α : MonoIndex d m) (t : ℝ),
mgf (fun ξ => coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α) (condDistrib id sampleDesign Q (sampleDesign ω)) t
≤ exp (B ^ 2 * ((tensorCount d m : ℝ) * ((tensorCount d m : ℝ)⁻¹ * ((2 / templateLambda d m) * sqrt (card (MonoIndex d m) : ℝ))) ^ 2 * ((tensorCount d m : ℝ) * min (meshWidth j ^ (2 * β)) (5 / ((n : ℝ) * (((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (c_f * templateEta d m ^ d) ^ (γ / (γ - 1))) * meshWidth j ^ effectiveDimension d γ * (cubeMassRank P m j k : ℝ) ^ (1 / (γ - 1)))))) * t ^ 2 / 2)
α :
MonoIndex d m
t :
ℝ
conclusion
mgf (fun ξ => coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α) (condDistrib id sampleDesign Q (sampleDesign ω)) t
≤ exp ((B ^ 2 * ((tensorCount d m : ℝ) * ((tensorCount d m : ℝ)⁻¹ * ((2 / templateLambda d m) * sqrt (card (MonoIndex d m) : ℝ))) ^ 2 * (tensorCount d m : ℝ)) * max 1 (5 / κ)) * min (meshWidth j ^ (2 * β)) ((n : ℝ)⁻¹ * meshWidth j ^ (-effectiveDimension d γ) * (cubeMassRank P m j k : ℝ) ^ (-(1 : ℝ) / (γ - 1))) * t ^ 2 / 2)
Proof (Lean source)
lemma selectedMesh_raw_coefficient_mgf_rank
{d n m j : ℕ} (β B C c_f γ κ : ℝ)
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q]
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(hκ : 0 < κ)
(hκle : κ ≤ ((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1)))
(hγ : 1 < γ) (hn : 0 < n)
(hraw : ∀ (α : MonoIndex d m) (t : ℝ),
mgf (fun ξ => coefHat m j ξ k α -
conditionalCoefCentre Q m j ω k α)
(condDistrib id sampleDesign Q (sampleDesign ω)) t ≤
exp (B ^ 2 *
((tensorCount d m : ℝ) *
((tensorCount d m : ℝ)⁻¹ *
((2 / templateLambda d m) *
sqrt (card (MonoIndex d m) : ℝ))) ^ 2 *
((tensorCount d m : ℝ) *
min (meshWidth j ^ (2 * β))
(5 / ((n : ℝ) *
(((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1))) *
meshWidth j ^ effectiveDimension d γ *
(cubeMassRank P m j k : ℝ) ^ (1 / (γ - 1)))))) * t ^ 2 / 2)) :
∀ (α : MonoIndex d m) (t : ℝ),
mgf (fun ξ => coefHat m j ξ k α -
conditionalCoefCentre Q m j ω k α)
(condDistrib id sampleDesign Q (sampleDesign ω)) t ≤
exp ((B ^ 2 *
((tensorCount d m : ℝ) *
((tensorCount d m : ℝ)⁻¹ *
((2 / templateLambda d m) *
sqrt (card (MonoIndex d m) : ℝ))) ^ 2 *
(tensorCount d m : ℝ)) * max 1 (5 / κ)) *
min (meshWidth j ^ (2 * β))
((n : ℝ)⁻¹ * meshWidth j ^ (-effectiveDimension d γ) *
(cubeMassRank P m j k : ℝ) ^ (-(1 : ℝ) / (γ - 1))) * t ^ 2 / 2) := by
intro α t
have hα : 0 < 1 / (γ - 1) := by positivity
have hA : 0 ≤ B ^ 2 *
((tensorCount d m : ℝ) *
((tensorCount d m : ℝ)⁻¹ *
((2 / templateLambda d m) *
sqrt (card (MonoIndex d m) : ℝ))) ^ 2 *
(tensorCount d m : ℝ)) := by positivity
have hx : 0 ≤ meshWidth j ^ (2 * β) := by
apply Real.rpow_nonneg
unfold meshWidth
positivity
have hr : 0 < (cubeMassRank P m j k : ℝ) := by
exact_mod_cast cubeMassRank_pos P m j k
have hn' : (0 : ℝ) < n := by exact_mod_cast hn
have hproxy := selectedMesh_uniform_ordered_proxy
(B ^ 2 *
((tensorCount d m : ℝ) *
((tensorCount d m : ℝ)⁻¹ *
((2 / templateLambda d m) *
sqrt (card (MonoIndex d m) : ℝ))) ^ 2 *
(tensorCount d m : ℝ)))
(meshWidth j ^ (2 * β)) (n : ℝ) (meshWidth j)
(cubeMassRank P m j k : ℝ) κ
(((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1)))
(effectiveDimension d γ) (1 / (γ - 1))
hA hx hn' (by unfold meshWidth; positivity) hr hκ hκle
have hraw' := (hraw α t)
have hraw'' : mgf (fun ξ => coefHat m j ξ k α -
conditionalCoefCentre Q m j ω k α)
(condDistrib id sampleDesign Q (sampleDesign ω)) t ≤
exp ((B ^ 2 *
((tensorCount d m : ℝ) *
((tensorCount d m : ℝ)⁻¹ *
((2 / templateLambda d m) *
sqrt (card (MonoIndex d m) : ℝ))) ^ 2 *
(tensorCount d m : ℝ))) *
min (meshWidth j ^ (2 * β))
(5 / ((n : ℝ) *
(((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1))) *
meshWidth j ^ effectiveDimension d γ *
(cubeMassRank P m j k : ℝ) ^ (1 / (γ - 1)))) * t ^ 2 / 2) := by
simpa only [mul_assoc] using hraw'
exact hraw''.trans (Real.exp_le_exp.mpr (by
apply div_le_div_of_nonneg_right _ (by norm_num)
apply mul_le_mul_of_nonneg_right _ (sq_nonneg t)
simpa only [neg_div] using hproxy))
One event controls the selected counts and the actual conditional coefficient MGF with a uniform rank proxy. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ_min γ_max :
ℝ
hd :
1 ≤ d
hβ :
1 < β
hB :
0 < B
hL :
0 < L
hC :
1 ≤ C
hcf :
0 < c_f ∧ c_f ≤ 1
hγmin :
1 < γ_min
hγrange :
γ_min < γ_max
conclusion
∃ K : ℝ,
conclusion 1
0 < K
conclusion 2
∃ n₀ : ℕ,
∀ n ≥ n₀,
∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin,
hγrange⟩, ∀ (P : Measure (Obs d)) [IsProbabilityMeasure P] (hP : P ∈ ModelClass d β B L C c_f γ) (Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q], IIDSampleLaw Q P → ∃ E : Set (Fin n → Obs d), MeasurableSet E ∧ Q.real Eᶜ
≤ (n : ℝ) ^ (-2 : ℝ) ∧
∀ ω ∈ E,
let m := polynomialDegree β let j := (feasibleIndices m β ω).sup id let h := countSelectedMesh m β ω 0 < h ∧
h ≤ K * oracleMesh d n β γ ∧
(∀ (k : Fin d → Fin (2 ^ j)) (ℓ : Fin d → Fin (m + 1)), (n : ℝ) * microcellMass P m j k ℓ / 5 ≤ treatedCount m j ω k ℓ) ∧
(∀ (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) (t : ℝ), mgf (fun ξ => coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α) (condDistrib id sampleDesign Q (sampleDesign ω)) t ≤ exp (K * min (h ^ (2 * β)) ((n : ℝ)⁻¹ * h ^ (-effectiveDimension d γ) * (cubeMassRank P m j k : ℝ) ^ (-(1 : ℝ) / (γ - 1))) * t ^ 2 / 2)) ∧
(∀ (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) (t : ℝ), Integrable (fun ξ => exp (t * (coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α))) (condDistrib id sampleDesign Q (sampleDesign ω)))
Proof (Lean source)
lemma selectedMesh_count_and_rank_mgf (d : ℕ) (β B L C c_f γ_min γ_max : ℝ)
(hd : 1 ≤ d) (hβ : 1 < β) (hB : 0 < B) (hL : 0 < L)
(hC : 1 ≤ C) (hcf : 0 < c_f ∧ c_f ≤ 1)
(hγmin : 1 < γ_min) (hγrange : γ_min < γ_max) :
∃ K : ℝ, 0 < K ∧ ∃ n₀ : ℕ,
∀ n ≥ n₀, ∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
∀ (P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q],
IIDSampleLaw Q P →
∃ E : Set (Fin n → Obs d), MeasurableSet E ∧
Q.real Eᶜ ≤ (n : ℝ) ^ (-2 : ℝ) ∧
∀ ω ∈ E,
let m := polynomialDegree β
let j := (feasibleIndices m β ω).sup id
let h := countSelectedMesh m β ω
0 < h ∧ h ≤ K * oracleMesh d n β γ ∧
(∀ (k : Fin d → Fin (2 ^ j)) (ℓ : Fin d → Fin (m + 1)),
(n : ℝ) * microcellMass P m j k ℓ / 5 ≤ treatedCount m j ω k ℓ) ∧
(∀ (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) (t : ℝ),
mgf (fun ξ => coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α)
(condDistrib id sampleDesign Q (sampleDesign ω)) t ≤
exp (K * min (h ^ (2 * β))
((n : ℝ)⁻¹ * h ^ (-effectiveDimension d γ) *
(cubeMassRank P m j k : ℝ) ^ (-(1 : ℝ) / (γ - 1))) * t ^ 2 / 2)) ∧
(∀ (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) (t : ℝ),
Integrable (fun ξ => exp (t *
(coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α)))
(condDistrib id sampleDesign Q (sampleDesign ω))) := by
let m := polynomialDegree β
let a := c_f * templateEta d m ^ d
have ha : 0 < a := by
dsimp [a]
exact mul_pos hcf.1 (pow_pos (templateEta_pos d m) d)
obtain ⟨κ, hκ, hκle⟩ := orderedMass_uniform_coefficient C a γ_min γ_max
(lt_of_lt_of_le zero_lt_one hC) ha hγmin hγrange
obtain ⟨Kcount, hKcountPos, n₀, hcountRaw⟩ :=
selectedMesh_count_and_raw_mgf d β B L C c_f γ_min γ_max
hd hβ hB hL hC hcf hγmin hγrange
let A : ℝ := B ^ 2 *
((tensorCount d m : ℝ) *
((tensorCount d m : ℝ)⁻¹ *
((2 / templateLambda d m) *
sqrt (card (MonoIndex d m) : ℝ))) ^ 2 *
(tensorCount d m : ℝ))
let K := max Kcount (A * max 1 (5 / κ)) + 1
have hKcount : Kcount ≤ K := by
dsimp [K]
linarith [le_max_left Kcount (A * max 1 (5 / κ))]
have hKbase : A * max 1 (5 / κ) ≤ K := by
dsimp [K]
linarith [le_max_right Kcount (A * max 1 (5 / κ))]
have hK : 0 < K := lt_of_lt_of_le hKcountPos hKcount
refine ⟨K, hK, max 1 n₀, ?_⟩
intro n hn γ hγ P _ hP Q _ hIID
obtain ⟨E, hE, hbad, hbounds⟩ :=
hcountRaw n ((Nat.le_max_right _ _).trans hn) γ hγ P hP Q hIID
refine ⟨E, hE, hbad, ?_⟩
intro ω hω
obtain ⟨hh, hmesh, hcounts, hraw, hint⟩ := hbounds ω hω
let j := (feasibleIndices m β ω).sup id
have hwidth : countSelectedMesh m β ω = meshWidth j :=
selectedMesh_width_of_pos m β ω hh
have horacle : 0 ≤ oracleMesh d n β γ := by
unfold oracleMesh
positivity
refine ⟨hh, hmesh.trans (mul_le_mul_of_nonneg_right hKcount horacle),
hcounts, ?_, hint⟩
intro k α t
have hγIcc : γ ∈ Icc γ_min γ_max := by
simpa only [adaptationRange] using hγ
have hγgt : 1 < γ := hγmin.trans_le hγIcc.1
have hnpos : 0 < n := by
have : 1 ≤ n := (Nat.le_max_left _ _).trans hn
omega
have hκbound : κ ≤ ((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1)) := by
simpa only [a] using hκle γ hγIcc
have hrank := selectedMesh_raw_coefficient_mgf_rank β B C c_f γ κ
P Q ω k hκ hκbound hγgt hnpos (hraw k)
rw [hwidth]
apply (hrank α t).trans
apply Real.exp_le_exp.mpr
apply div_le_div_of_nonneg_right _ (by norm_num)
apply mul_le_mul_of_nonneg_right _ (sq_nonneg t)
have hmin : 0 ≤ min (meshWidth j ^ (2 * β))
((n : ℝ)⁻¹ * meshWidth j ^ (-effectiveDimension d γ) *
(cubeMassRank P m j k : ℝ) ^ (-(1 : ℝ) / (γ - 1))) := by
have hwpos : 0 < meshWidth j := by unfold meshWidth; positivity
have hrpos : 0 < (cubeMassRank P m j k : ℝ) := by
exact_mod_cast cubeMassRank_pos P m j k
have hnreal : 0 < (n : ℝ) := by exact_mod_cast hnpos
exact le_min (by positivity) (by positivity)
exact mul_le_mul_of_nonneg_right hKbase hmin
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma selectedMesh_ranked_vector_maximal
(αmin αmax : ℝ) (hαmin : 0 < αmin) (hαmax : αmin ≤ αmax)
(K : ℝ) (hK : 0 < K) :
∃ M : ℝ, 0 < M ∧
∀ (α : ℝ), α ∈ Icc αmin αmax →
∀ {Ω κ ι : Type*} [MeasurableSpace Ω] [Fintype ι]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(N : ℕ) [Nonempty (Fin N)]
(σ : Fin N ≃ κ) (Z : κ → Ω → ι → ℝ)
(x y a b : ℝ), 0 < a → 0 < b →
a ^ 2 = K * x → b ^ 2 = K * y →
(∀ k : Fin N, ∀ i : ι, ∀ t : ℝ,
Integrable (fun ω => exp (t * Z (σ k) ω i)) μ) →
(∀ k : Fin N, ∀ i : ι, ∀ t : ℝ,
mgf (fun ω => Z (σ k) ω i) μ t ≤ exp
(K * min x (y * (k.val + 1 : ℝ) ^ (-α)) * t ^ 2 / 2)) →
(∫ ω, sSup ((fun k : κ => ‖Z k ω‖) '' univ) ∂μ) ≤
(Fintype.card ι : ℝ) * M * a *
sqrt (1 + max 0 (log ((b / a) ^ (2 / α)))) := by
obtain ⟨M, hM, hmax⟩ :=
orderedSubGaussianMaximal_vector_equiv_uniform_of_exp
αmin αmax hαmin hαmax
refine ⟨M, hM, ?_⟩
intro α hα Ω κ ι _ _ μ _ N _ σ Z x y a b ha hb ha2 hb2 hexp hmgf
apply hmax α hα μ N σ Z a b ha hb hexp
intro k i t
convert hmgf k i t using 1
congr 1
rw [ha2, hb2, mul_assoc,
← mul_min_of_nonneg x (y * (k.val + 1 : ℝ) ^ (-α)) hK.le]
theorem selectedMesh_maximal_oracle_log_ratio Lemma selectedMesh_maximal_oracle_log_ratio in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β γ h α :
ℝ
hn :
0 < n
hden :
0 < 2 * β + effectiveDimension d γ
hh :
0 < h
hα :
0 < α
Proof (Lean source)
lemma selectedMesh_maximal_oracle_log_ratio (d n : ℕ) (β γ h α : ℝ)
(hn : 0 < n) (hden : 0 < 2 * β + effectiveDimension d γ)
(hh : 0 < h) (hα : 0 < α) :
1 + max 0 (log
((((n : ℝ) ^ (-(1 : ℝ) / 2) *
h ^ (-(effectiveDimension d γ) / 2)) / h ^ β) ^ (2 / α))) ≤
max 1 ((β + effectiveDimension d γ / 2) * (2 / α)) *
(1 + max 0 (log (oracleMesh d n β γ / h))) := by
have ho : 0 < oracleMesh d n β γ := by
unfold oracleMesh
have hn' : (0 : ℝ) < n := by exact_mod_cast hn
positivity
rw [selectedMesh_stochastic_bias_ratio d n β γ h hn hden hh]
rw [← Real.rpow_mul (div_pos ho hh).le]
exact selectedMesh_maximal_log_scale
(oracleMesh d n β γ) h
((β + effectiveDimension d γ / 2) * (2 / α)) ho hh
(by have : 0 ≤ β + effectiveDimension d γ / 2 := by linarith
positivity)
theorem selectedMesh_maximal_exponent_uniform Lemma selectedMesh_maximal_exponent_uniform in the paper ↗
The logarithmic exponent has a fixed upper bound over the adaptation range; its apparent reciprocal dependence on the overlap exponent cancels. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β γ γmax :
ℝ
hβ :
0 < β
hγ :
1 < γ
hmax :
γ ≤ γmax
conclusion
conclusion 1
0 ≤ (β + effectiveDimension d γ / 2) * (2 / (1 / (γ - 1)))
conclusion 2
(β + effectiveDimension d γ / 2) * (2 / (1 / (γ - 1)))
≤ 2 * β * (γmax - 1) + (d : ℝ) * γmax
Proof (Lean source)
lemma selectedMesh_maximal_exponent_uniform (d : ℕ) (β γ γmax : ℝ)
(hβ : 0 < β) (hγ : 1 < γ) (hmax : γ ≤ γmax) :
0 ≤ (β + effectiveDimension d γ / 2) * (2 / (1 / (γ - 1))) ∧
(β + effectiveDimension d γ / 2) * (2 / (1 / (γ - 1))) ≤
2 * β * (γmax - 1) + (d : ℝ) * γmax := by
have hγden : γ - 1 ≠ 0 := ne_of_gt (by linarith : 0 < γ - 1)
have hform :
(β + effectiveDimension d γ / 2) * (2 / (1 / (γ - 1))) =
2 * β * (γ - 1) + (d : ℝ) * γ := by
unfold effectiveDimension
field_simp
rw [hform]
have hd : (0 : ℝ) ≤ d := Nat.cast_nonneg _
constructor
· positivity
· nlinarith
theorem selectedMesh_alpha_in_compact_range Lemma selectedMesh_alpha_in_compact_range in the paper ↗
The ordered maximal exponent stays in a fixed positive interval. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
1 / (γmax - 1) ≤ 1 / (γ - 1)
conclusion 2
1 / (γ - 1) ≤ 1 / (γmin - 1)
Proof (Lean source)
lemma selectedMesh_alpha_in_compact_range (γmin γmax γ : ℝ)
(hmin : 1 < γmin) (hrange : γmin ≤ γmax)
(hγ : γ ∈ Icc γmin γmax) :
1 / (γmax - 1) ≤ 1 / (γ - 1) ∧
1 / (γ - 1) ≤ 1 / (γmin - 1) := by
have h₁ : 0 < γmin - 1 := by linarith
have h₂ : 0 < γ - 1 := by linarith [hγ.1]
have h₃ : 0 < γmax - 1 := by linarith
constructor
· exact one_div_le_one_div_of_le h₂ (by linarith [hγ.2])
· exact one_div_le_one_div_of_le h₁ (by linarith [hγ.1])
theorem selectedMesh_cube_rank_vector_maximal Lemma selectedMesh_cube_rank_vector_maximal in the paper ↗
Apply the vector maximal inequality to cubes whose coefficient proxy is expressed through their mass rank. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
αmin αmax K :
ℝ
hαmin :
0 < αmin
hαmax :
αmin ≤ αmax
hK :
0 < K
Proof (Lean source)
lemma selectedMesh_cube_rank_vector_maximal
(αmin αmax K : ℝ)
(hαmin : 0 < αmin) (hαmax : αmin ≤ αmax)
(hK : 0 < K) :
∃ M : ℝ, 0 < M ∧
∀ {d : ℕ} (P : Measure (Obs d)) [IsProbabilityMeasure P]
(m j : ℕ) (α x y a b : ℝ),
α ∈ Icc αmin αmax → 0 ≤ y → 0 < a → 0 < b →
a ^ 2 = K * x → b ^ 2 = K * y →
∀ {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(Z : (Fin d → Fin (2 ^ j)) → Ω → ι → ℝ),
(∀ k i t, Integrable (fun ω => exp (t * Z k ω i)) μ) →
(∀ k i t, mgf (fun ω => Z k ω i) μ t ≤
exp (K * min x
(y * (cubeMassRank P m j k : ℝ) ^ (-α)) * t ^ 2 / 2)) →
(∫ ω, sSup ((fun k : Fin d → Fin (2 ^ j) =>
‖Z k ω‖) '' univ) ∂μ) ≤
(Fintype.card ι : ℝ) * M * a *
sqrt (1 + max 0 (log ((b / a) ^ (2 / α)))) := by
obtain ⟨M, hM, hmax⟩ :=
selectedMesh_ranked_vector_maximal αmin αmax hαmin hαmax K hK
refine ⟨M, hM, ?_⟩
intro d P _ m j α x y a b hα hy ha hb ha2 hb2 Ω ι _ _ μ _ Z hexp hmgf
obtain ⟨σ, -, hrank⟩ := cubeMass_ordered_equiv P m j
apply hmax α hα μ _ σ Z x y a b ha hb ha2 hb2
· intro k i t
exact hexp (σ k) i t
· intro k i t
calc
mgf (fun ω => Z (σ k) ω i) μ t ≤
exp (K * min x
(y * (cubeMassRank P m j (σ k) : ℝ) ^ (-α)) * t ^ 2 / 2) :=
hmgf (σ k) i t
_ ≤ exp (K * min x
(y * (k.val + 1 : ℝ) ^ (-α)) * t ^ 2 / 2) := by
apply Real.exp_le_exp.mpr
apply div_le_div_of_nonneg_right _ (by norm_num)
apply mul_le_mul_of_nonneg_right _ (sq_nonneg t)
apply mul_le_mul_of_nonneg_left _ hK.le
exact selectedMesh_rank_proxy_to_ordered x y α
(cubeMassRank P m j (σ k) : ℝ) k.val hy
(lt_of_lt_of_le hαmin hα.1) (by exact_mod_cast hrank k)
theorem selectedMesh_envelope
The selected mesh obeys the count event, ordered conditional coefficient variance proxies, and a logarithm-free expected coefficient envelope under the stated inputs and conditions.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ_min γ_max :
ℝ
hd :
1 ≤ d
hβ :
1 < β
hB :
0 < B
hL :
0 < L
hC :
1 ≤ C
hcf :
0 < c_f ∧ c_f ≤ 1
hγmin :
1 < γ_min
hγrange :
γ_min < γ_max
conclusion
∃ K K' : ℝ,
conclusion 1
0 < K
conclusion 2
0 < K'
conclusion 3
∃ n₀ : ℕ,
∀ n ≥ n₀,
∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin,
hγrange⟩, ∀ (P : Measure (Obs d)) [IsProbabilityMeasure P] (hP : P ∈ ModelClass d β B L C c_f γ) (Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q], IIDSampleLaw Q P → ∃ E : Set (Fin n → Obs d), MeasurableSet E ∧ Q.real Eᶜ
≤ (n : ℝ) ^ (-2 : ℝ) ∧
(∀ ω ∈ E, let m := polynomialDegree β let j := (feasibleIndices m β ω).sup id let h := countSelectedMesh m β ω 0 < h ∧ h ≤ K * oracleMesh d n β γ ∧ (∀ (k : Fin d → Fin (2 ^ j)) (ℓ : Fin d → Fin (m + 1)), (n : ℝ) * microcellMass P m j k ℓ / 5 ≤ treatedCount m j ω k ℓ) ∧ (∀ (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) (t : ℝ), mgf (fun ξ => coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α) (condDistrib id sampleDesign Q (sampleDesign ω)) t ≤ exp (K * min (h ^ (2 * β)) ((n : ℝ)⁻¹ * h ^ (-effectiveDimension d γ) * (cubeMassRank P m j k : ℝ) ^ (-(1 : ℝ) / (γ - 1))) * t ^ 2 / 2)) ∧ (∀ (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) (t : ℝ), Integrable (fun ξ => exp (t * (coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α))) (condDistrib id sampleDesign Q (sampleDesign ω))) ∧ Integrable (fun ξ => sSup ((fun k : Fin d → Fin (2 ^ j) => ‖fun α : MonoIndex d m => coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α‖) '' univ)) (condDistrib id sampleDesign Q (sampleDesign ω)) ∧ (∫ ξ, sSup ((fun k : Fin d → Fin (2 ^ j) => ‖fun α : MonoIndex d m => coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α‖) '' univ) ∂condDistrib id sampleDesign Q (sampleDesign ω)) ≤ K * h ^ β * sqrt (1 + max 0 (log (oracleMesh d n β γ / h))) ∧ K * h ^ β * sqrt (1 + max 0 (log (oracleMesh d n β γ / h))) ≤ K' * oracleRate d n β γ) ∧
(∀ (α : ℝ), 0 < α → ∃ Kα : ℝ, 0 < Kα ∧ ∀ {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] (N : ℕ) (Z : Fin N → Ω → ℝ) (a b : ℝ), 0 < a → 0 < b → (∀ k : Fin N, ∀ t : ℝ, Integrable (fun ξ => exp (t * Z k ξ)) μ) → (∀ k : Fin N, ∀ t : ℝ, mgf (Z k) μ t ≤ exp (min (a ^ 2) (b ^ 2 * (k.val + 1 : ℝ) ^ (-α)) * t ^ 2 / 2)) → ∫ ξ, sSup ((fun k : Fin N => |Z k ξ|) '' univ) ∂μ ≤ Kα * a * sqrt (1 + max 0 (log ((b / a) ^ (2 / α)))))
Proof (Lean source)
-- @node: lem:selected-mesh
lemma selectedMesh_envelope (d : ℕ) (β B L C c_f γ_min γ_max : ℝ)
(hd : 1 ≤ d) (hβ : 1 < β) (hB : 0 < B) (hL : 0 < L)
(hC : 1 ≤ C) (hcf : 0 < c_f ∧ c_f ≤ 1)
(hγmin : 1 < γ_min) (hγrange : γ_min < γ_max) :
∃ K K' : ℝ, 0 < K ∧ 0 < K' ∧ ∃ n₀ : ℕ,
∀ n ≥ n₀, ∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
∀ (P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q],
IIDSampleLaw Q P →
∃ E : Set (Fin n → Obs d), MeasurableSet E ∧
Q.real Eᶜ ≤ (n : ℝ) ^ (-2 : ℝ) ∧
(∀ ω ∈ E,
let m := polynomialDegree β
let j := (feasibleIndices m β ω).sup id
let h := countSelectedMesh m β ω
0 < h ∧ h ≤ K * oracleMesh d n β γ ∧
(∀ (k : Fin d → Fin (2 ^ j)) (ℓ : Fin d → Fin (m + 1)),
(n : ℝ) * microcellMass P m j k ℓ / 5 ≤ treatedCount m j ω k ℓ) ∧
(∀ (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) (t : ℝ),
mgf (fun ξ => coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α)
(condDistrib id sampleDesign Q (sampleDesign ω)) t ≤
exp (K * min (h ^ (2 * β))
((n : ℝ)⁻¹ * h ^ (-effectiveDimension d γ) *
(cubeMassRank P m j k : ℝ) ^ (-(1 : ℝ) / (γ - 1))) * t ^ 2 / 2)) ∧
(∀ (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) (t : ℝ),
Integrable (fun ξ => exp (t *
(coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α)))
(condDistrib id sampleDesign Q (sampleDesign ω))) ∧
Integrable (fun ξ => sSup ((fun k : Fin d → Fin (2 ^ j) =>
‖fun α : MonoIndex d m => coefHat m j ξ k α -
conditionalCoefCentre Q m j ω k α‖) '' univ))
(condDistrib id sampleDesign Q (sampleDesign ω)) ∧
(∫ ξ, sSup ((fun k : Fin d → Fin (2 ^ j) =>
‖fun α : MonoIndex d m => coefHat m j ξ k α -
conditionalCoefCentre Q m j ω k α‖) '' univ)
∂condDistrib id sampleDesign Q (sampleDesign ω)) ≤
K * h ^ β * sqrt
(1 + max 0 (log (oracleMesh d n β γ / h))) ∧
K * h ^ β * sqrt
(1 + max 0 (log (oracleMesh d n β γ / h))) ≤
K' * oracleRate d n β γ) ∧
(∀ (α : ℝ), 0 < α → ∃ Kα : ℝ, 0 < Kα ∧
∀ {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
[IsProbabilityMeasure μ] (N : ℕ) (Z : Fin N → Ω → ℝ)
(a b : ℝ), 0 < a → 0 < b →
(∀ k : Fin N, ∀ t : ℝ,
Integrable (fun ξ => exp (t * Z k ξ)) μ) →
(∀ k : Fin N, ∀ t : ℝ,
mgf (Z k) μ t ≤ exp
(min (a ^ 2) (b ^ 2 * (k.val + 1 : ℝ) ^ (-α)) * t ^ 2 / 2)) →
∫ ξ, sSup ((fun k : Fin N => |Z k ξ|) '' univ) ∂μ ≤
Kα * a * sqrt (1 + max 0 (log ((b / a) ^ (2 / α))))) := by
have hOutcomeFromModel :
∀ (γ : ℝ) (P : Measure (Obs d)) [IsProbabilityMeasure P],
P ∈ ModelClass d β B L C c_f γ →
BoundedMeanSubGaussianResidual P B := by
intro γ P _ hP
rcases hP with ⟨_, _, Pc, hPc, μ₁, e, hEq, hmodel⟩
subst P
exact hmodel.outcome
have hSelected :
∃ K K' : ℝ, 0 < K ∧ 0 < K' ∧ ∃ n₀ : ℕ,
∀ n ≥ n₀, ∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
∀ (P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q],
IIDSampleLaw Q P →
∃ E : Set (Fin n → Obs d), MeasurableSet E ∧
Q.real Eᶜ ≤ (n : ℝ) ^ (-2 : ℝ) ∧
(∀ ω ∈ E,
let m := polynomialDegree β
let j := (feasibleIndices m β ω).sup id
let h := countSelectedMesh m β ω
0 < h ∧ h ≤ K * oracleMesh d n β γ ∧
(∀ (k : Fin d → Fin (2 ^ j)) (ℓ : Fin d → Fin (m + 1)),
(n : ℝ) * microcellMass P m j k ℓ / 5 ≤ treatedCount m j ω k ℓ) ∧
(∀ (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) (t : ℝ),
mgf (fun ξ => coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α)
(condDistrib id sampleDesign Q (sampleDesign ω)) t ≤
exp (K * min (h ^ (2 * β))
((n : ℝ)⁻¹ * h ^ (-effectiveDimension d γ) *
(cubeMassRank P m j k : ℝ) ^ (-(1 : ℝ) / (γ - 1))) * t ^ 2 / 2)) ∧
(∀ (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) (t : ℝ),
Integrable (fun ξ => exp (t *
(coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α)))
(condDistrib id sampleDesign Q (sampleDesign ω))) ∧
Integrable (fun ξ => sSup ((fun k : Fin d → Fin (2 ^ j) =>
‖fun α : MonoIndex d m => coefHat m j ξ k α -
conditionalCoefCentre Q m j ω k α‖) '' univ))
(condDistrib id sampleDesign Q (sampleDesign ω)) ∧
(∫ ξ, sSup ((fun k : Fin d → Fin (2 ^ j) =>
‖fun α : MonoIndex d m => coefHat m j ξ k α -
conditionalCoefCentre Q m j ω k α‖) '' univ)
∂condDistrib id sampleDesign Q (sampleDesign ω)) ≤
K * h ^ β * sqrt
(1 + max 0 (log (oracleMesh d n β γ / h))) ∧
K * h ^ β * sqrt
(1 + max 0 (log (oracleMesh d n β γ / h))) ≤
K' * oracleRate d n β γ) := by
obtain ⟨Kcount, hKcount, ncount, hcountRank⟩ :=
selectedMesh_count_and_rank_mgf d β B L C c_f γ_min γ_max
hd hβ hB hL hC hcf hγmin hγrange
let αmin := 1 / (γ_max - 1)
let αmax := 1 / (γ_min - 1)
have hαmin : 0 < αmin := by
have : 0 < γ_max - 1 := by linarith
dsimp [αmin]
positivity
have hαmax : αmin ≤ αmax :=
(selectedMesh_alpha_in_compact_range γ_min γ_max γ_min
hγmin hγrange.le ⟨le_refl _, hγrange.le⟩).1
obtain ⟨M, hM, hmax⟩ :=
selectedMesh_cube_rank_vector_maximal αmin αmax Kcount
hαmin hαmax hKcount
let Eexp : ℝ := max 1 (2 * β * (γ_max - 1) + (d : ℝ) * γ_max)
have hEexp : 0 < Eexp := lt_of_lt_of_le zero_lt_one (le_max_left _ _)
let p : ℝ := card (MonoIndex d (polynomialDegree β))
let K : ℝ := max Kcount (p * M * sqrt Kcount * sqrt Eexp) + 1
have hKcountLe : Kcount ≤ K := by dsimp [K]; linarith [le_max_left Kcount (p * M * sqrt Kcount * sqrt Eexp)]
have hKcoef : p * M * sqrt Kcount * sqrt Eexp ≤ K := by
dsimp [K]
linarith [le_max_right Kcount (p * M * sqrt Kcount * sqrt Eexp)]
have hK : 0 < K := lt_of_lt_of_le hKcount hKcountLe
let K' := K * max 1 (K ^ β)
have hK' : 0 < K' := mul_pos hK (lt_of_lt_of_le zero_lt_one (le_max_left _ _))
refine ⟨K, K', hK, hK', max 1 ncount, ?_⟩
intro n hn γ hγ P _ hP Q _ hIID
obtain ⟨E, hE, hbad, hbounds⟩ :=
hcountRank n ((Nat.le_max_right _ _).trans hn) γ hγ P hP Q hIID
refine ⟨E, hE, hbad, ?_⟩
intro ω hω
obtain ⟨hh, hmesh, hcounts, hmgf, hexp⟩ := hbounds ω hω
let m := polynomialDegree β
let j := (feasibleIndices m β ω).sup id
let h := countSelectedMesh m β ω
have hwidth : h = meshWidth j := selectedMesh_width_of_pos m β ω hh
have hγIcc : γ ∈ Icc γ_min γ_max := by
simpa only [adaptationRange] using hγ
have hγgt : 1 < γ := hγmin.trans_le hγIcc.1
have hnpos : 0 < n := by
have : 1 ≤ n := (Nat.le_max_left _ _).trans hn
omega
have hγpos : 0 < γ - 1 := by linarith
have hD : 0 < effectiveDimension d γ := by
unfold effectiveDimension
have hd' : (0 : ℝ) < d := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hd)
positivity
have hden : 0 < 2 * β + effectiveDimension d γ := by positivity
have hα : 1 / (γ - 1) ∈ Icc αmin αmax :=
selectedMesh_alpha_in_compact_range γ_min γ_max γ
hγmin hγrange.le hγIcc
let a : ℝ := sqrt Kcount * h ^ β
let b : ℝ := sqrt Kcount * (n : ℝ) ^ (-(1 : ℝ) / 2) *
h ^ (-(effectiveDimension d γ) / 2)
have ha : 0 < a := by dsimp [a]; positivity
have hb : 0 < b := by
dsimp [b]
have hn' : (0 : ℝ) < n := by exact_mod_cast hnpos
positivity
have ha2 : a ^ 2 = Kcount * h ^ (2 * β) := by
have hpow : (h ^ β) ^ 2 = h ^ (2 * β) := by
rw [← Real.rpow_natCast (x := h ^ β) (n := 2)]
rw [← Real.rpow_mul hh.le]
congr 1
ring
simp only [a, mul_pow, Real.sq_sqrt hKcount.le, hpow]
have hb2 : b ^ 2 = Kcount * ((n : ℝ)⁻¹ *
h ^ (-effectiveDimension d γ)) := by
have hn' : (0 : ℝ) ≤ n := by exact_mod_cast hnpos.le
have hnpow : ((n : ℝ) ^ (-(1 : ℝ) / 2)) ^ 2 = (n : ℝ)⁻¹ := by
rw [← Real.rpow_natCast (x := (n : ℝ) ^ (-(1 : ℝ) / 2)) (n := 2)]
rw [← Real.rpow_mul hn']
norm_num
exact Real.rpow_neg_one (n : ℝ)
have hhp : (h ^ (-(effectiveDimension d γ) / 2)) ^ 2 =
h ^ (-effectiveDimension d γ) := by
rw [← Real.rpow_natCast (x := h ^ (-(effectiveDimension d γ) / 2)) (n := 2)]
rw [← Real.rpow_mul hh.le]
congr 1
ring
simp only [b, mul_pow, Real.sq_sqrt hKcount.le, hnpow, hhp]
ring
have hvec := hmax P m j (1 / (γ - 1))
(h ^ (2 * β)) ((n : ℝ)⁻¹ * h ^ (-effectiveDimension d γ))
a b hα (by positivity) ha hb ha2 hb2
(condDistrib id sampleDesign Q (sampleDesign ω))
(fun k ξ α => coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α)
hexp (by simpa only [neg_div] using hmgf)
have hintMax := selectedMesh_integrable_vector_max_of_exp_moments
(condDistrib id sampleDesign Q (sampleDesign ω))
(fun k ξ α => coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α)
hexp
have hratio : b / a =
(((n : ℝ) ^ (-(1 : ℝ) / 2) *
h ^ (-(effectiveDimension d γ) / 2)) / h ^ β) := by
dsimp [a, b]
have hsqrt : sqrt Kcount ≠ 0 := ne_of_gt (Real.sqrt_pos.2 hKcount)
have hpow : h ^ β ≠ 0 := ne_of_gt (Real.rpow_pos_of_pos hh β)
field_simp
have hlogRatio := selectedMesh_maximal_oracle_log_ratio
d n β γ h (1 / (γ - 1)) hnpos hden hh (by positivity)
have hexponent := selectedMesh_maximal_exponent_uniform
d β γ γ_max (by linarith) hγgt hγIcc.2
have hlog :
sqrt (1 + max 0 (log ((b / a) ^ (2 / (1 / (γ - 1)))))) ≤
sqrt Eexp *
sqrt (1 + max 0 (log (oracleMesh d n β γ / h))) := by
rw [hratio]
have hH : 0 ≤ 1 + max 0 (log (oracleMesh d n β γ / h)) := by positivity
have hfactor : max 1 ((β + effectiveDimension d γ / 2) *
(2 / (1 / (γ - 1)))) ≤ Eexp := by
dsimp [Eexp]
exact max_le_max_left _ hexponent.2
have hbound := hlogRatio.trans (mul_le_mul_of_nonneg_right hfactor hH)
calc
_ ≤ sqrt (Eexp * (1 + max 0
(log (oracleMesh d n β γ / h)))) := Real.sqrt_le_sqrt hbound
_ = _ := by rw [Real.sqrt_mul hEexp.le]
have hstoch :
(∫ ξ, sSup ((fun k : Fin d → Fin (2 ^ j) =>
‖fun α : MonoIndex d m => coefHat m j ξ k α -
conditionalCoefCentre Q m j ω k α‖) '' univ)
∂condDistrib id sampleDesign Q (sampleDesign ω)) ≤
K * h ^ β * sqrt
(1 + max 0 (log (oracleMesh d n β γ / h))) := by
have hnonneg : 0 ≤ h ^ β * sqrt
(1 + max 0 (log (oracleMesh d n β γ / h))) := by positivity
calc
_ ≤ p * M * a * sqrt
(1 + max 0 (log ((b / a) ^ (2 / (1 / (γ - 1)))))) := hvec
_ ≤ p * M * a * (sqrt Eexp * sqrt
(1 + max 0 (log (oracleMesh d n β γ / h)))) := by
gcongr
_ = (p * M * sqrt Kcount * sqrt Eexp) *
(h ^ β * sqrt
(1 + max 0 (log (oracleMesh d n β γ / h)))) := by
dsimp [a]; ring
_ ≤ K * (h ^ β * sqrt
(1 + max 0 (log (oracleMesh d n β γ / h)))) :=
mul_le_mul_of_nonneg_right hKcoef hnonneg
_ = _ := by ring
have hdet := selectedMesh_oracle_rate_log_bound
d n β γ h K hnpos (by linarith) hh
(hmesh.trans (mul_le_mul_of_nonneg_right hKcountLe
(by unfold oracleMesh; positivity)))
-- … truncated; follow the source link for the rest …
theorem countDesignSaturated_condDistrib_one Lemma countDesignSaturated_condDistrib_one in the paper ↗
A design-saturated event has conditional probability one on the entire conditional sample fibre whenever the conditioning sample belongs to it. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
Q :
hE :
hsat :
conclusion
∀ᵐ ω ∂Q, ω ∈ E → condDistrib id sampleDesign Q (sampleDesign ω) E = 1
Proof (Lean source)
lemma countDesignSaturated_condDistrib_one {d n : ℕ}
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q]
(E : Set (Fin n → Obs d)) (hE : MeasurableSet E)
(hsat : CountDesignSaturated E) :
∀ᵐ ω ∂Q, ω ∈ E →
condDistrib id sampleDesign Q (sampleDesign ω) E = 1 := by
filter_upwards [sampleDesign_condDistrib_fibre Q] with ω hfibre
intro hω
rw [← mem_ae_iff_prob_eq_one hE]
filter_upwards [hfibre] with ξ hξ
exact (hsat ω ξ hξ).mp hω
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β γ :
ℝ
hn :
1 ≤ n
hβ :
0 < β
hγ :
1 < γ
conclusion
(n : ℝ) ^ (-2 : ℝ) ≤ oracleRate d n β γ
Proof (Lean source)
lemma inverseSquare_le_oracleRate (d n : ℕ) (β γ : ℝ)
(hn : 1 ≤ n) (hβ : 0 < β) (hγ : 1 < γ) :
(n : ℝ) ^ (-2 : ℝ) ≤ oracleRate d n β γ := by
have hn' : (1 : ℝ) ≤ n := by exact_mod_cast hn
have hD : 0 ≤ effectiveDimension d γ := by
unfold effectiveDimension
positivity
have hden : 0 < 2 * β + effectiveDimension d γ := by positivity
have hexp : (-2 : ℝ) ≤ (-(1 : ℝ) / (2 * β + effectiveDimension d γ)) * β := by
have hfrac : β / (2 * β + effectiveDimension d γ) ≤ 1 := by
apply (div_le_one hden).2
linarith
rw [show (-(1 : ℝ) / (2 * β + effectiveDimension d γ)) * β =
-(β / (2 * β + effectiveDimension d γ)) by ring]
linarith
unfold oracleRate oracleMesh
rw [← Real.rpow_mul (by positivity : (0 : ℝ) ≤ n)]
exact Real.rpow_le_rpow_of_exponent_le hn' hexp
The explicit design-saturated count event supports the ranked coefficient MGF bounds almost everywhere. Auxiliary fibre facts remain in the ambient almost-everywhere filter rather than being added to the event. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ_min γ_max :
ℝ
hd :
1 ≤ d
hβ :
1 < β
hB :
0 < B
hL :
0 < L
hC :
1 ≤ C
hcf :
0 < c_f ∧ c_f ≤ 1
hγmin :
1 < γ_min
hγrange :
γ_min < γ_max
conclusion
∃ K : ℝ,
conclusion 1
0 < K
conclusion 2
∃ n₀ : ℕ,
∀ n ≥ n₀,
∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin,
hγrange⟩, ∀ (P : Measure (Obs d)) [IsProbabilityMeasure P] (hP : P ∈ ModelClass d β B L C c_f γ) (Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q], IIDSampleLaw Q P → ∃ E, SaturatedSelectedCountEvent β K γ P Q E ∧ ∀ᵐ ω ∂Q, ω ∈ E → let m := polynomialDegree β let j := (feasibleIndices m β ω).sup id let h := countSelectedMesh m β ω (∀ (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) (t : ℝ), mgf (fun ξ => coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α) (condDistrib id sampleDesign Q (sampleDesign ω)) t ≤ exp (K * min (h ^ (2 * β)) ((n : ℝ)⁻¹ * h ^ (-effectiveDimension d γ) * (cubeMassRank P m j k : ℝ) ^ (-(1 : ℝ) / (γ - 1))) * t ^ 2 / 2)) ∧ (∀ (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) (t : ℝ), Integrable (fun ξ => exp (t * (coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α))) (condDistrib id sampleDesign Q (sampleDesign ω)))
Proof (Lean source)
lemma selectedMesh_saturated_rank_mgf
(d : ℕ) (β B L C c_f γ_min γ_max : ℝ)
(hd : 1 ≤ d) (hβ : 1 < β) (hB : 0 < B) (hL : 0 < L)
(hC : 1 ≤ C) (hcf : 0 < c_f ∧ c_f ≤ 1)
(hγmin : 1 < γ_min) (hγrange : γ_min < γ_max) :
∃ K : ℝ, 0 < K ∧ ∃ n₀ : ℕ,
∀ n ≥ n₀, ∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
∀ (P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q],
IIDSampleLaw Q P →
∃ E, SaturatedSelectedCountEvent β K γ P Q E ∧
∀ᵐ ω ∂Q, ω ∈ E →
let m := polynomialDegree β
let j := (feasibleIndices m β ω).sup id
let h := countSelectedMesh m β ω
(∀ (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) (t : ℝ),
mgf (fun ξ => coefHat m j ξ k α -
conditionalCoefCentre Q m j ω k α)
(condDistrib id sampleDesign Q (sampleDesign ω)) t ≤
exp (K * min (h ^ (2 * β))
((n : ℝ)⁻¹ * h ^ (-effectiveDimension d γ) *
(cubeMassRank P m j k : ℝ) ^ (-(1 : ℝ) / (γ - 1))) *
t ^ 2 / 2)) ∧
(∀ (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) (t : ℝ),
Integrable (fun ξ => exp (t *
(coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α)))
(condDistrib id sampleDesign Q (sampleDesign ω))) := by
let m := polynomialDegree β
let a := c_f * templateEta d m ^ d
have ha : 0 < a := by
dsimp [a, m]
exact mul_pos hcf.1 (pow_pos (templateEta_pos d (polynomialDegree β)) d)
obtain ⟨κ, hκ, hκle⟩ := orderedMass_uniform_coefficient C a γ_min γ_max
(lt_of_lt_of_le zero_lt_one hC) ha hγmin hγrange
obtain ⟨Kcount, hKcount, n₀, hcount⟩ :=
selectedCountEvent_uniform_design_saturated d β B L C c_f γ_min γ_max
hd hβ hB hL hC hcf hγmin hγrange
let A : ℝ := B ^ 2 *
((tensorCount d m : ℝ) *
((tensorCount d m : ℝ)⁻¹ *
((2 / templateLambda d m) *
sqrt (card (MonoIndex d m) : ℝ))) ^ 2 *
(tensorCount d m : ℝ))
let K := max Kcount (A * max 1 (5 / κ)) + 1
have hKcountLe : Kcount ≤ K := by
dsimp [K]
linarith [le_max_left Kcount (A * max 1 (5 / κ))]
have hKbase : A * max 1 (5 / κ) ≤ K := by
dsimp [K]
linarith [le_max_right Kcount (A * max 1 (5 / κ))]
have hK : 0 < K := lt_of_lt_of_le hKcount hKcountLe
refine ⟨K, hK, max 1 n₀, ?_⟩
intro n hn γ hγ P _ hP Q _ hIID
obtain ⟨E, hEmeas, hEbad, hEbounds, hEsat⟩ :=
hcount n ((Nat.le_max_right _ _).trans hn) γ hγ P hP Q hIID
have horacle : 0 ≤ oracleMesh d n β γ := by
unfold oracleMesh
positivity
have hEbounds' : ∀ ω ∈ E,
0 < countSelectedMesh (polynomialDegree β) β ω ∧
countSelectedMesh (polynomialDegree β) β ω ≤
K * oracleMesh d n β γ ∧
∀ (k : Fin d → Fin
(2 ^ ((feasibleIndices (polynomialDegree β) β ω).sup id)))
(ℓ : Fin d → Fin (polynomialDegree β + 1)),
(n : ℝ) * microcellMass P (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) k ℓ / 5 ≤
treatedCount (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) ω k ℓ := by
intro ω hω
obtain ⟨hh, hmesh, hcounts⟩ := hEbounds ω hω
exact ⟨hh, hmesh.trans (mul_le_mul_of_nonneg_right hKcountLe horacle), hcounts⟩
refine ⟨E, ⟨hEmeas, hEbad, hEbounds', hEsat⟩, ?_⟩
have hres : BoundedMeanSubGaussianResidual P B := by
rcases hP with ⟨_, _, Pc, hPc, μ₁, e, hEq, hmodel⟩
subst P
exact hmodel.outcome
filter_upwards [modelClass_conditionalCoefficient_mgf_rank_bound P Q hIID hres,
conditionalCoefficient_integrable_exp P Q hIID hres] with ω hraw hint
intro hω
obtain ⟨hh, _hmesh, hcounts⟩ := hEbounds' ω hω
let j := (feasibleIndices m β ω).sup id
let h := countSelectedMesh m β ω
have hwidth : h = meshWidth j := selectedMesh_width_of_pos m β ω hh
have hwidth' : countSelectedMesh (polynomialDegree β) β ω =
meshWidth ((feasibleIndices (polynomialDegree β) β ω).sup id) := by
simpa [m, j, h] using hwidth
have hfeasible : meshFeasible m β ω j := selectedMesh_feasible_of_pos m β ω hh
have hγIcc : γ ∈ Icc γ_min γ_max := by
simpa only [adaptationRange] using hγ
have hγgt : 1 < γ := hγmin.trans_le hγIcc.1
have hnpos : 0 < n := by
have : 1 ≤ n := (Nat.le_max_left _ _).trans hn
omega
have hκbound : κ ≤ ((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1)) := by
simpa only [a] using hκle γ hγIcc
constructor
· intro k α t
have hrank := selectedMesh_raw_coefficient_mgf_rank β B C c_f γ κ
P Q ω k hκ hκbound hγgt hnpos (fun α t => hraw m j hP k α
hnpos hfeasible (hcounts k) t)
rw [hwidth']
exact (hrank α t).trans (Real.exp_le_exp.mpr (by
apply div_le_div_of_nonneg_right _ (by norm_num)
apply mul_le_mul_of_nonneg_right _ (sq_nonneg t)
have hx : 0 ≤ min (meshWidth j ^ (2 * β))
((n : ℝ)⁻¹ * meshWidth j ^ (-effectiveDimension d γ) *
(cubeMassRank P m j k : ℝ) ^ (-(1 : ℝ) / (γ - 1))) := by
have hr : 0 < (cubeMassRank P m j k : ℝ) := by
exact_mod_cast cubeMassRank_pos P m j k
have hnreal : (0 : ℝ) < n := by exact_mod_cast hnpos
have hw : 0 < meshWidth j := by unfold meshWidth; positivity
exact le_min (by positivity) (by positivity)
exact mul_le_mul_of_nonneg_right hKbase hx))
· exact hint m j
theorem selectedMesh_saturated_conditional_envelope Lemma selected-mesh-saturated-conditional-envelope in the paper ↗
On the explicit design-saturated event, the conditional expected maximal coefficient fluctuation has the oracle-rate bound almost everywhere. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ_min γ_max :
ℝ
hd :
1 ≤ d
hβ :
1 < β
hB :
0 < B
hL :
0 < L
hC :
1 ≤ C
hcf :
0 < c_f ∧ c_f ≤ 1
hγmin :
1 < γ_min
hγrange :
γ_min < γ_max
conclusion
∃ K K' : ℝ,
conclusion 1
0 < K
conclusion 2
0 < K'
conclusion 3
∃ n₀ : ℕ,
∀ n ≥ n₀,
∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin,
hγrange⟩, ∀ (P : Measure (Obs d)) [IsProbabilityMeasure P] (hP : P ∈ ModelClass d β B L C c_f γ) (Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q], IIDSampleLaw Q P → ∃ E, SaturatedSelectedCountEvent β K γ P Q E ∧ ∀ᵐ ω ∂Q, ω ∈ E → let m := polynomialDegree β let j := (feasibleIndices m β ω).sup id let h := countSelectedMesh m β ω Integrable (fun ξ => sSup ((fun k : Fin d → Fin (2 ^ j) => ‖fun α : MonoIndex d m => coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α‖) '' univ)) (condDistrib id sampleDesign Q (sampleDesign ω)) ∧ (∫ ξ, sSup ((fun k : Fin d → Fin (2 ^ j) => ‖fun α : MonoIndex d m => coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α‖) '' univ) ∂condDistrib id sampleDesign Q (sampleDesign ω))
≤ K' * oracleRate d n β γ
Proof (Lean source)
lemma selectedMesh_saturated_conditional_envelope
(d : ℕ) (β B L C c_f γ_min γ_max : ℝ)
(hd : 1 ≤ d) (hβ : 1 < β) (hB : 0 < B) (hL : 0 < L)
(hC : 1 ≤ C) (hcf : 0 < c_f ∧ c_f ≤ 1)
(hγmin : 1 < γ_min) (hγrange : γ_min < γ_max) :
∃ K K' : ℝ, 0 < K ∧ 0 < K' ∧ ∃ n₀ : ℕ,
∀ n ≥ n₀, ∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
∀ (P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q],
IIDSampleLaw Q P →
∃ E, SaturatedSelectedCountEvent β K γ P Q E ∧
∀ᵐ ω ∂Q, ω ∈ E →
let m := polynomialDegree β
let j := (feasibleIndices m β ω).sup id
let h := countSelectedMesh m β ω
Integrable (fun ξ => sSup ((fun k : Fin d → Fin (2 ^ j) =>
‖fun α : MonoIndex d m => coefHat m j ξ k α -
conditionalCoefCentre Q m j ω k α‖) '' univ))
(condDistrib id sampleDesign Q (sampleDesign ω)) ∧
(∫ ξ, sSup ((fun k : Fin d → Fin (2 ^ j) =>
‖fun α : MonoIndex d m => coefHat m j ξ k α -
conditionalCoefCentre Q m j ω k α‖) '' univ)
∂condDistrib id sampleDesign Q (sampleDesign ω)) ≤
K' * oracleRate d n β γ := by
obtain ⟨Kbase, hKbase, nbase, hbase⟩ :=
selectedMesh_saturated_rank_mgf d β B L C c_f γ_min γ_max
hd hβ hB hL hC hcf hγmin hγrange
let αmin := 1 / (γ_max - 1)
let αmax := 1 / (γ_min - 1)
have hαmin : 0 < αmin := by
have : 0 < γ_max - 1 := by linarith
dsimp [αmin]
positivity
have hαmax : αmin ≤ αmax :=
(selectedMesh_alpha_in_compact_range γ_min γ_max γ_min
hγmin hγrange.le ⟨le_refl _, hγrange.le⟩).1
obtain ⟨M, hM, hmax⟩ :=
selectedMesh_cube_rank_vector_maximal αmin αmax Kbase
hαmin hαmax hKbase
let Eexp : ℝ := max 1 (2 * β * (γ_max - 1) + (d : ℝ) * γ_max)
have hEexp : 0 < Eexp := lt_of_lt_of_le zero_lt_one (le_max_left _ _)
let p : ℝ := card (MonoIndex d (polynomialDegree β))
let K : ℝ := max Kbase (p * M * sqrt Kbase * sqrt Eexp) + 1
have hKbaseLe : Kbase ≤ K := by
dsimp [K]
linarith [le_max_left Kbase (p * M * sqrt Kbase * sqrt Eexp)]
have hKcoef : p * M * sqrt Kbase * sqrt Eexp ≤ K := by
dsimp [K]
linarith [le_max_right Kbase (p * M * sqrt Kbase * sqrt Eexp)]
have hK : 0 < K := lt_of_lt_of_le hKbase hKbaseLe
let K' := K * max 1 (K ^ β)
have hK' : 0 < K' := mul_pos hK
(lt_of_lt_of_le zero_lt_one (le_max_left _ _))
refine ⟨K, K', hK, hK', max 1 nbase, ?_⟩
intro n hn γ hγ P _ hP Q _ hIID
obtain ⟨E, hE, hAE⟩ :=
hbase n ((Nat.le_max_right _ _).trans hn) γ hγ P hP Q hIID
rcases hE with ⟨hEmeas, hEbad, hEbounds, hEsat⟩
have horacle : 0 ≤ oracleMesh d n β γ := by unfold oracleMesh; positivity
have hEbounds' : ∀ ω ∈ E,
0 < countSelectedMesh (polynomialDegree β) β ω ∧
countSelectedMesh (polynomialDegree β) β ω ≤ K * oracleMesh d n β γ ∧
∀ (k : Fin d → Fin
(2 ^ ((feasibleIndices (polynomialDegree β) β ω).sup id)))
(ℓ : Fin d → Fin (polynomialDegree β + 1)),
(n : ℝ) * microcellMass P (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) k ℓ / 5 ≤
treatedCount (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) ω k ℓ := by
intro ω hω
obtain ⟨hh, hmesh, hcounts⟩ := hEbounds ω hω
exact ⟨hh, hmesh.trans (mul_le_mul_of_nonneg_right hKbaseLe horacle), hcounts⟩
refine ⟨E, ⟨hEmeas, hEbad, hEbounds', hEsat⟩, ?_⟩
filter_upwards [hAE] with ω hmgfAE
intro hω
obtain ⟨hh, hmesh, _hcounts⟩ := hEbounds' ω hω
obtain ⟨hmgf, hexp⟩ := hmgfAE hω
let m := polynomialDegree β
let j := (feasibleIndices m β ω).sup id
let h := countSelectedMesh m β ω
have hγIcc : γ ∈ Icc γ_min γ_max := by
simpa only [adaptationRange] using hγ
have hγgt : 1 < γ := hγmin.trans_le hγIcc.1
have hnpos : 0 < n := by
have : 1 ≤ n := (Nat.le_max_left _ _).trans hn
omega
have hden : 0 < 2 * β + effectiveDimension d γ := by
unfold effectiveDimension
positivity
have hα : 1 / (γ - 1) ∈ Icc αmin αmax :=
selectedMesh_alpha_in_compact_range γ_min γ_max γ
hγmin hγrange.le hγIcc
let a : ℝ := sqrt Kbase * h ^ β
let b : ℝ := sqrt Kbase * (n : ℝ) ^ (-(1 : ℝ) / 2) *
h ^ (-(effectiveDimension d γ) / 2)
have ha : 0 < a := by dsimp [a]; positivity
have hb : 0 < b := by
dsimp [b]
have hn' : (0 : ℝ) < n := by exact_mod_cast hnpos
positivity
have ha2 : a ^ 2 = Kbase * h ^ (2 * β) := by
have hpow : (h ^ β) ^ 2 = h ^ (2 * β) := by
rw [← Real.rpow_natCast (x := h ^ β) (n := 2)]
rw [← Real.rpow_mul hh.le]
congr 1
ring
simp only [a, mul_pow, Real.sq_sqrt hKbase.le, hpow]
have hb2 : b ^ 2 = Kbase * ((n : ℝ)⁻¹ *
h ^ (-effectiveDimension d γ)) := by
have hn' : (0 : ℝ) ≤ n := by exact_mod_cast hnpos.le
have hnpow : ((n : ℝ) ^ (-(1 : ℝ) / 2)) ^ 2 = (n : ℝ)⁻¹ := by
rw [← Real.rpow_natCast (x := (n : ℝ) ^ (-(1 : ℝ) / 2)) (n := 2)]
rw [← Real.rpow_mul hn']
norm_num
exact Real.rpow_neg_one (n : ℝ)
have hhp : (h ^ (-(effectiveDimension d γ) / 2)) ^ 2 =
h ^ (-effectiveDimension d γ) := by
rw [← Real.rpow_natCast
(x := h ^ (-(effectiveDimension d γ) / 2)) (n := 2)]
rw [← Real.rpow_mul hh.le]
congr 1
ring
simp only [b, mul_pow, Real.sq_sqrt hKbase.le, hnpow, hhp]
ring
have hvec := hmax P m j (1 / (γ - 1))
(h ^ (2 * β)) ((n : ℝ)⁻¹ * h ^ (-effectiveDimension d γ))
a b hα (by positivity) ha hb ha2 hb2
(condDistrib id sampleDesign Q (sampleDesign ω))
(fun k ξ α => coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α)
hexp (by simpa only [neg_div] using hmgf)
have hintMax := selectedMesh_integrable_vector_max_of_exp_moments
(condDistrib id sampleDesign Q (sampleDesign ω))
(fun k ξ α => coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α)
hexp
have hratio : b / a =
(((n : ℝ) ^ (-(1 : ℝ) / 2) *
h ^ (-(effectiveDimension d γ) / 2)) / h ^ β) := by
dsimp [a, b]
have hsqrt : sqrt Kbase ≠ 0 := ne_of_gt (Real.sqrt_pos.2 hKbase)
have hpow : h ^ β ≠ 0 := ne_of_gt (Real.rpow_pos_of_pos hh β)
field_simp
have hlogRatio := selectedMesh_maximal_oracle_log_ratio
d n β γ h (1 / (γ - 1)) hnpos hden hh (by positivity)
have hexponent := selectedMesh_maximal_exponent_uniform
d β γ γ_max (by linarith) hγgt hγIcc.2
have hlog :
sqrt (1 + max 0 (log ((b / a) ^ (2 / (1 / (γ - 1)))))) ≤
sqrt Eexp *
sqrt (1 + max 0 (log (oracleMesh d n β γ / h))) := by
rw [hratio]
have hH : 0 ≤ 1 + max 0 (log (oracleMesh d n β γ / h)) := by
positivity
have hfactor : max 1 ((β + effectiveDimension d γ / 2) *
(2 / (1 / (γ - 1)))) ≤ Eexp := by
dsimp [Eexp]
exact max_le_max_left _ hexponent.2
have hbound := hlogRatio.trans (mul_le_mul_of_nonneg_right hfactor hH)
calc
_ ≤ sqrt (Eexp * (1 + max 0
(log (oracleMesh d n β γ / h)))) := Real.sqrt_le_sqrt hbound
_ = _ := by rw [Real.sqrt_mul hEexp.le]
have hnonneg : 0 ≤ h ^ β * sqrt
(1 + max 0 (log (oracleMesh d n β γ / h))) := by positivity
have hstoch :
(∫ ξ, sSup ((fun k : Fin d → Fin (2 ^ j) =>
‖fun α : MonoIndex d m => coefHat m j ξ k α -
conditionalCoefCentre Q m j ω k α‖) '' univ)
∂condDistrib id sampleDesign Q (sampleDesign ω)) ≤
K * h ^ β * sqrt
(1 + max 0 (log (oracleMesh d n β γ / h))) := by
calc
_ ≤ p * M * a * sqrt
(1 + max 0 (log ((b / a) ^ (2 / (1 / (γ - 1)))))) := hvec
_ ≤ p * M * a * (sqrt Eexp * sqrt
(1 + max 0 (log (oracleMesh d n β γ / h)))) := by gcongr
_ = (p * M * sqrt Kbase * sqrt Eexp) *
(h ^ β * sqrt
(1 + max 0 (log (oracleMesh d n β γ / h)))) := by
dsimp [a]; ring
_ ≤ K * (h ^ β * sqrt
(1 + max 0 (log (oracleMesh d n β γ / h)))) :=
mul_le_mul_of_nonneg_right hKcoef hnonneg
_ = _ := by ring
have hdet := selectedMesh_oracle_rate_log_bound
d n β γ h K hnpos (by linarith) hh hmesh
refine ⟨hintMax.1, hstoch.trans ?_⟩
simpa only [K', mul_assoc] using
(mul_le_mul_of_nonneg_left hdet hK.le)
theorem selectedMesh_saturated_conditional_cubeLoss Lemma selected-mesh-saturated-conditional-cube-loss in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ_min γ_max :
ℝ
hd :
1 ≤ d
hβ :
1 < β
hB :
0 < B
hL :
0 < L
hC :
1 ≤ C
hcf :
0 < c_f ∧ c_f ≤ 1
hγmin :
1 < γ_min
hγrange :
γ_min < γ_max
conclusion
∃ K A : ℝ,
conclusion 1
0 < K
conclusion 2
0 < A
conclusion 3
∃ n₀ : ℕ,
∀ n ≥ n₀,
∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin,
hγrange⟩, ∀ (Pc : Measure (Completion d)) [IsProbabilityMeasure Pc] (μ e : (Fin d → ℝ) → ℝ) (hmodel : GlobalTailModel β B L C c_f γ Pc μ e) (P : Measure (Obs d)) [IsProbabilityMeasure P] (hPobs : P = Pc.map observed) (Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q], IIDSampleLaw Q P → ∃ E, SaturatedSelectedCountEvent β K γ P Q E ∧ ∀ᵐ ω ∂Q, ω ∈ E → ∫ ξ, equalCellCubeLossReal β B μ ξ ∂condDistrib id sampleDesign Q (sampleDesign ω)
≤ A * oracleRate d n β γ
Proof (Lean source)
lemma selectedMesh_saturated_conditional_cubeLoss
(d : ℕ) (β B L C c_f γ_min γ_max : ℝ)
(hd : 1 ≤ d) (hβ : 1 < β) (hB : 0 < B) (hL : 0 < L)
(hC : 1 ≤ C) (hcf : 0 < c_f ∧ c_f ≤ 1)
(hγmin : 1 < γ_min) (hγrange : γ_min < γ_max) :
∃ K A : ℝ, 0 < K ∧ 0 < A ∧ ∃ n₀ : ℕ,
∀ n ≥ n₀, ∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
∀ (Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ e : (Fin d → ℝ) → ℝ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ e)
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hPobs : P = Pc.map observed)
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q],
IIDSampleLaw Q P →
∃ E, SaturatedSelectedCountEvent β K γ P Q E ∧
∀ᵐ ω ∂Q, ω ∈ E →
∫ ξ, equalCellCubeLossReal β B μ ξ
∂condDistrib id sampleDesign Q (sampleDesign ω) ≤
A * oracleRate d n β γ := by
obtain ⟨Cb, hCb, hBiasUniform⟩ :=
exists_uniform_conditionalCoefCentre_holder_cell_bias
d β L (by linarith) hL
obtain ⟨K, Kc, hK, hKc, n₀, hEnvelope⟩ :=
selectedMesh_saturated_conditional_envelope
d β B L C c_f γ_min γ_max hd hβ hB hL hC hcf hγmin hγrange
let p : ℝ := card (MonoIndex d (polynomialDegree β))
let q : ℝ := (2 / templateLambda d (polynomialDegree β)) * Real.sqrt p
let Db : ℝ := p * (q * (Cb * L)) + Cb * L
let A : ℝ := p * Kc + Db * max 1 (K ^ β) + 1
have hp : 0 < p := by
dsimp [p]
let α₀ : MonoIndex d (polynomialDegree β) :=
⟨fun _ => 0, by simp [monoIdx]⟩
exact_mod_cast (Fintype.card_pos_iff.mpr ⟨α₀⟩ :
0 < card (MonoIndex d (polynomialDegree β)))
have hq : 0 ≤ q := by
dsimp [q]
exact mul_nonneg (div_nonneg (by norm_num) (templateLambda_pos d _).le)
(Real.sqrt_nonneg _)
have hDb : 0 ≤ Db := by dsimp [Db]; positivity
have hA : 0 < A := by
dsimp [A]
positivity
refine ⟨K, A, hK, hA, max 1 n₀, ?_⟩
intro n hn γ hγ Pc _ μ e hmodel P _ hPobs Q _ hIID
have hIIDc : IIDSampleLaw Q (Pc.map observed) := by
simpa only [← hPobs] using hIID
have hparams : ModelParameterDomain d β B L C c_f :=
⟨hd, hβ, hB, hL, hC, hcf⟩
have hγgt : 1 < γ := by
have hI : γ ∈ Icc γ_min γ_max := by simpa [adaptationRange] using hγ
exact hγmin.trans_le hI.1
have hP : P ∈ ModelClass d β B L C c_f γ :=
⟨hparams, hγgt, Pc, inferInstance, μ, e, hPobs, hmodel⟩
obtain ⟨E, hE, hEnvAE⟩ :=
hEnvelope n ((Nat.le_max_right _ _).trans hn) γ hγ P hP Q hIID
rcases hE with ⟨hEmeas, hEbad, hEbounds, hEsat⟩
refine ⟨E, ⟨hEmeas, hEbad, hEbounds, hEsat⟩, ?_⟩
have hBiasAE := hBiasUniform Pc μ e hparams hγgt hmodel Q hIIDc
have hFibre := sampleDesign_condDistrib_fibre Q
have hCondE := countDesignSaturated_condDistrib_one Q E hEmeas hEsat
have hμB := holderResponse_abs_le_on_cube Pc μ e hparams hγgt hmodel
filter_upwards [hEnvAE, hBiasAE, hFibre, hCondE]
with ω hEnv hBias hFibreω hCondEω
intro hωE
obtain ⟨hSint, hSbound⟩ := hEnv hωE
obtain ⟨hh, hmesh, hcounts⟩ := hEbounds ω hωE
let m := polynomialDegree β
let j := (feasibleIndices m β ω).sup id
let h := countSelectedMesh m β ω
let S : (Fin n → Obs d) → ℝ := fun ξ =>
sSup ((fun k : Fin d → Fin (2 ^ j) =>
‖fun α : MonoIndex d m => coefHat m j ξ k α -
conditionalCoefCentre Q m j ω k α‖) '' univ)
have hwidth : h = meshWidth j := selectedMesh_width_of_pos m β ω hh
have hposCounts : ∀ (k : Fin d → Fin (2 ^ j))
(ℓ : Fin d → Fin (m + 1)), 0 < treatedCount m j ω k ℓ := by
intro k ℓ
have hmass : 0 < microcellMass P m j k ℓ :=
model_microcellMass_pos P hP m j k ℓ
have hnpos : (0 : ℝ) < n := by
exact_mod_cast (show 0 < n by
have : 1 ≤ n := (Nat.le_max_left _ _).trans hn
omega)
have hreal : (0 : ℝ) < treatedCount m j ω k ℓ :=
lt_of_lt_of_le (div_pos (mul_pos hnpos hmass) (by norm_num)) (hcounts k ℓ)
exact_mod_cast hreal
have hcell : ∀ k : Fin d → Fin (2 ^ j),
∃ θ : MonoIndex d m → ℝ,
(∀ x ∈ dyadicCube d j k,
|μ x - ∑ α, monoVec d m
(fun a => (x a - cubeCorner d j k a) / h) α * θ α| ≤
Cb * L * h ^ β) ∧
∀ α,
|conditionalCoefCentre Q m j ω k α - θ α| ≤
q * (Cb * L * h ^ β) := by
intro k
have hb := hBias j k (hposCounts k)
rw [hwidth]
simpa [m, j, h, q, p, mul_assoc] using hb
let ν := condDistrib id sampleDesign Q (sampleDesign ω)
have hνE : ∀ᵐ ξ ∂ν, ξ ∈ E := by
change E ∈ ae ν
exact (mem_ae_iff_prob_eq_one hEmeas).2 (hCondEω hωE)
have hpoint : ∀ᵐ ξ ∂ν,
equalCellCubeLossReal β B μ ξ ≤ p * (S ξ + q * (Cb * L * h ^ β)) +
Cb * L * h ^ β := by
filter_upwards [hνE, hFibreω] with ξ hξE hdesign
have hdet := equalCellEstimator_error_le_conditional_envelope_of_design_eq
(β := β) (B := B) (r := Cb * L * h ^ β)
(εb := q * (Cb * L * h ^ β)) Q μ ω ξ hdesign hh hwidth
hB.le hμB (by positivity) (by positivity) hcell
have hfi := feasibleIndices_eq_of_design_eq m β ω ξ hdesign
rw [hfi] at hdet
have hdet' : ∀ x ∈ cube d,
|equalCellEstimator β B ξ x - μ x| ≤
p * (S ξ + q * (Cb * L * h ^ β)) + Cb * L * h ^ β := by
simpa only [p, S, m, j] using hdet
have hSnonneg : 0 ≤ S ξ := by
apply Real.sSup_nonneg
rintro z ⟨k, -, rfl⟩
exact norm_nonneg _
have hRnonneg : 0 ≤
p * (S ξ + q * (Cb * L * h ^ β)) + Cb * L * h ^ β := by
positivity
have hsup := cube_iSup_ofReal_loss_le
(fun x => equalCellEstimator β B ξ x - μ x)
(p * (S ξ + q * (Cb * L * h ^ β)) + Cb * L * h ^ β)
hRnonneg hdet'
unfold equalCellCubeLossReal
exact ENNReal.toReal_le_of_le_ofReal hRnonneg hsup
have hZmeas := equalCellCubeLossReal_measurable (n := n) β B μ
hmodel.smooth.1.regularity.continuousOn
have hZint : Integrable (equalCellCubeLossReal β B μ) ν :=
Integrable.of_bound hZmeas.aestronglyMeasurable (2 * B) <|
ae_of_all ν fun ξ => by
rw [Real.norm_eq_abs]
unfold equalCellCubeLossReal
rw [abs_of_nonneg ENNReal.toReal_nonneg]
exact equalCellCubeLossReal_le_two_mul β B μ hB.le hμB ξ
have hRint : Integrable (fun ξ =>
p * (S ξ + q * (Cb * L * h ^ β)) + Cb * L * h ^ β) ν := by
exact ((hSint.add (integrable_const _)).const_mul p).add (integrable_const _)
calc
∫ ξ, equalCellCubeLossReal β B μ ξ ∂ν ≤
∫ ξ, (p * (S ξ + q * (Cb * L * h ^ β)) +
Cb * L * h ^ β) ∂ν := integral_mono_ae hZint hRint hpoint
_ = p * (∫ ξ, S ξ ∂ν) + Db * h ^ β := by
have hνprob : ν.real univ = 1 := by
dsimp [ν]
simp
calc
∫ ξ, p * (S ξ + q * (Cb * L * h ^ β)) + Cb * L * h ^ β ∂ν =
(∫ ξ, p * (S ξ + q * (Cb * L * h ^ β)) ∂ν) +
∫ _ξ, Cb * L * h ^ β ∂ν :=
integral_add ((hSint.add (integrable_const _)).const_mul p)
(integrable_const _)
_ = p * (∫ ξ, S ξ + q * (Cb * L * h ^ β) ∂ν) +
∫ _ξ, Cb * L * h ^ β ∂ν := by
rw [integral_const_mul]
_ = p * ((∫ ξ, S ξ ∂ν) +
∫ _ξ, q * (Cb * L * h ^ β) ∂ν) +
∫ _ξ, Cb * L * h ^ β ∂ν := by
rw [integral_add hSint (integrable_const _)]
_ = p * (∫ ξ, S ξ ∂ν) + Db * h ^ β := by
rw [integral_const, integral_const, hνprob]
simp only [one_smul]
dsimp [Db]
ring
_ ≤ p * (Kc * oracleRate d n β γ) + Db * h ^ β := by gcongr
_ ≤ A * oracleRate d n β γ := by
have hrate : 0 ≤ oracleRate d n β γ := by unfold oracleRate oracleMesh; positivity
have hmeshPow : h ^ β ≤ max 1 (K ^ β) * oracleRate d n β γ := by
have horacle : 0 ≤ oracleMesh d n β γ := by unfold oracleMesh; positivity
have hpw := Real.rpow_le_rpow hh.le hmesh (by linarith : 0 ≤ β)
rw [Real.mul_rpow hK.le horacle] at hpw
change h ^ β ≤ K ^ β * oracleRate d n β γ at hpw
exact hpw.trans (mul_le_mul_of_nonneg_right (le_max_right _ _) hrate)
dsimp [A]
nlinarith [mul_nonneg hp.le hrate, mul_nonneg hDb hrate,
mul_le_mul_of_nonneg_left hmeshPow hDb]
theorem selectedMesh_uniform_equalCellCubeRisk Lemma selected-mesh-uniform-equal-cell-cube-risk in the paper ↗
The saturated conditional bound integrates to the desired uniform risk bound for every displayed model witness. This version keeps the response representative explicit, so it is usable below the responseOf choice layer. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ_min γ_max :
ℝ
hd :
1 ≤ d
hβ :
1 < β
hB :
0 < B
hL :
0 < L
hC :
1 ≤ C
hcf :
0 < c_f ∧ c_f ≤ 1
hγmin :
1 < γ_min
hγrange :
γ_min < γ_max
conclusion
∃ A : ℝ,
conclusion 1
0 < A
conclusion 2
∃ n₀ : ℕ,
∀ n ≥ n₀,
∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin,
hγrange⟩, ∀ (Pc : Measure (Completion d)) [IsProbabilityMeasure Pc] (μ e : (Fin d → ℝ) → ℝ) (hmodel : GlobalTailModel β B L C c_f γ Pc μ e) (P : Measure (Obs d)) [IsProbabilityMeasure P], P
= Pc.map observed → (∫⁻ ω, ⨆ x, ⨆ (_hx : x ∈ cube d), ofReal |equalCellEstimator β B ω x - μ x| ∂Measure.pi (fun _ : Fin n => P)) ≤ ofReal (A * oracleRate d n β γ)
Proof (Lean source)
lemma selectedMesh_uniform_equalCellCubeRisk
(d : ℕ) (β B L C c_f γ_min γ_max : ℝ)
(hd : 1 ≤ d) (hβ : 1 < β) (hB : 0 < B) (hL : 0 < L)
(hC : 1 ≤ C) (hcf : 0 < c_f ∧ c_f ≤ 1)
(hγmin : 1 < γ_min) (hγrange : γ_min < γ_max) :
∃ A : ℝ, 0 < A ∧ ∃ n₀ : ℕ,
∀ n ≥ n₀, ∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
∀ (Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ e : (Fin d → ℝ) → ℝ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ e)
(P : Measure (Obs d)) [IsProbabilityMeasure P],
P = Pc.map observed →
(∫⁻ ω, ⨆ x, ⨆ (_hx : x ∈ cube d),
ofReal |equalCellEstimator β B ω x - μ x|
∂Measure.pi (fun _ : Fin n => P)) ≤
ofReal (A * oracleRate d n β γ) := by
obtain ⟨K, A₀, hK, hA₀, n₀, hcond⟩ :=
selectedMesh_saturated_conditional_cubeLoss
d β B L C c_f γ_min γ_max hd hβ hB hL hC hcf hγmin hγrange
let A : ℝ := A₀ + 2 * B
have hA : 0 < A := by dsimp [A]; positivity
refine ⟨A, hA, max 1 n₀, ?_⟩
intro n hn γ hγ Pc _ μ e hmodel P _ hPobs
let Q : Measure (Fin n → Obs d) := Measure.pi (fun _ : Fin n => P)
letI : IsProbabilityMeasure Q := inferInstance
have hIID : IIDSampleLaw Q P := rfl
obtain ⟨E, hE, hgood⟩ := hcond n ((Nat.le_max_right _ _).trans hn)
γ hγ Pc μ e hmodel P hPobs Q hIID
rcases hE with ⟨hEmeas, hEbad, -, -⟩
have hparams : ModelParameterDomain d β B L C c_f :=
⟨hd, hβ, hB, hL, hC, hcf⟩
have hγgt : 1 < γ := by
have hI : γ ∈ Icc γ_min γ_max := by
simpa [adaptationRange] using hγ
exact hγmin.trans_le hI.1
have hμB := holderResponse_abs_le_on_cube Pc μ e hparams hγgt hmodel
let Z : (Fin n → Obs d) → ℝ := equalCellCubeLossReal β B μ
have hZm : Measurable Z :=
equalCellCubeLossReal_measurable β B μ
hmodel.smooth.1.regularity.continuousOn
have hZle : ∀ ω, Z ω ≤ 2 * B :=
fun ω => equalCellCubeLossReal_le_two_mul β B μ hB.le hμB ω
have hZint : Integrable Z Q :=
Integrable.of_bound hZm.aestronglyMeasurable (2 * B) <|
ae_of_all Q fun ω => by
rw [Real.norm_eq_abs, abs_of_nonneg]
· exact hZle ω
· exact ENNReal.toReal_nonneg
have hall : ∀ᵐ ω ∂Q,
∫ ξ, Z ξ ∂condDistrib id sampleDesign Q (sampleDesign ω) ≤ 2 * B := by
apply ae_of_all
intro ω
let ν := condDistrib id sampleDesign Q (sampleDesign ω)
have hZintν : Integrable Z ν :=
Integrable.of_bound hZm.aestronglyMeasurable (2 * B) <|
ae_of_all ν fun ξ => by
rw [Real.norm_eq_abs, abs_of_nonneg]
· exact hZle ξ
· exact ENNReal.toReal_nonneg
calc
∫ ξ, Z ξ ∂ν ≤ ∫ _ξ, 2 * B ∂ν :=
integral_mono_ae hZintν (integrable_const _) (ae_of_all ν hZle)
_ = 2 * B := by simp
have hrate : 0 ≤ oracleRate d n β γ := by
unfold oracleRate oracleMesh
positivity
have hreal : ∫ ω, Z ω ∂Q ≤ A * oracleRate d n β γ := by
have hsplit := integral_le_of_condDistrib_event_bounds Q sampleDesign
sampleDesign_measurable Z hZm.stronglyMeasurable hZint E hEmeas
(A₀ * oracleRate d n β γ) (2 * B) ((n : ℝ) ^ (-2 : ℝ))
(mul_nonneg hA₀.le hrate) (by positivity) (by positivity)
hgood hall hEbad
have hn1 : 1 ≤ n := (Nat.le_max_left _ _).trans hn
have hsquare := inverseSquare_le_oracleRate d n β γ hn1
(by linarith) hγgt
dsimp [A]
nlinarith [mul_le_mul_of_nonneg_left hsquare (by positivity : 0 ≤ 2 * B)]
rw [equalCellCubeLoss_lintegral_eq_ofReal_integral β B μ hB.le
hmodel.smooth.1.regularity.continuousOn hμB Q]
exact ENNReal.ofReal_le_ofReal hreal
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.SelectedMesh.Bias 19 declarations Deterministic selected-cell bias bounds
Deterministic selected-cell bias bounds
The identified Hölder representative agrees with the treated regression at every coordinate of an IID sample, outside one design-null set. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B L C c_f γ :
ℝ
Pc :
μ e :
(Fin d → ℝ) → ℝ
hparams :
ModelParameterDomain d β B L C c_f
hγ :
1 < γ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ e
Q :
hIID :
IIDSampleLaw Q (Pc.map observed)
conclusion
∀ᵐ ω ∂Q, ∀ i : Fin n, μ (ω i).1 = treatedRegression (Pc.map observed) (ω i).1
Proof (Lean source)
lemma holderRepresentative_on_iid_sample {d n : ℕ} {β B L C c_f γ : ℝ}
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ e : (Fin d → ℝ) → ℝ)
(hparams : ModelParameterDomain d β B L C c_f) (hγ : 1 < γ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ e)
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q]
(hIID : IIDSampleLaw Q (Pc.map observed)) :
∀ᵐ ω ∂Q, ∀ i : Fin n,
μ (ω i).1 = treatedRegression (Pc.map observed) (ω i).1 := by
let P := Pc.map observed
letI : IsProbabilityMeasure P := Measure.isProbabilityMeasure_map
(μ := Pc) (f := observed) (by unfold observed; fun_prop)
have hbase : ∀ᵐ x ∂covariateLaw P, μ x = treatedRegression P x := by
simpa [P] using (mu1_eq_treatedRegression β B L C c_f γ Pc μ e
hparams hγ hmodel).2.1
have hobs : ∀ᵐ z ∂P, μ z.1 = treatedRegression P z.1 :=
ae_of_ae_map measurable_fst.aemeasurable hbase
have hi : ∀ i : Fin n, ∀ᵐ ω ∂Q,
μ (ω i).1 = treatedRegression P (ω i).1 := by
intro i
have hmap : Q.map (fun ω : Fin n → Obs d => ω i) = P := by
rw [hIID]
simpa [P] using Measure.pi_map_eval (fun _ : Fin n => P) i
have hobs' : ∀ᵐ z ∂Q.map (fun ω : Fin n → Obs d => ω i),
μ z.1 = treatedRegression P z.1 := by simpa only [hmap] using hobs
exact ae_of_ae_map (μ := Q) (f := fun ω : Fin n → Obs d => ω i)
(measurable_pi_apply i).aemeasurable hobs'
simpa [P] using ae_all_iff.mpr hi
Rescaling any point of a dyadic macro-cell by its lower corner and width lands in the unit covariate cube, including the global right boundary. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
(fun i => (x i - cubeCorner d j k i) / meshWidth j) ∈ cube d
Proof (Lean source)
lemma dyadicCube_normalized_mem_cube {d j : ℕ}
(k : Fin d → Fin (2 ^ j)) {x : Fin d → ℝ}
(hx : x ∈ dyadicCube d j k) :
(fun i => (x i - cubeCorner d j k i) / meshWidth j) ∈ cube d := by
have hh : 0 < meshWidth j := by unfold meshWidth; positivity
have hscale := orderedMass_meshWidth_mul_pow j
apply Set.mem_univ_pi.mpr
intro i
have hi := hx.2 i
constructor
· exact div_nonneg (sub_nonneg.mpr hi.1) hh.le
· apply (div_le_one hh).2
rcases hi.2 with hlt | ⟨hk, hx1⟩
· linarith
· have hkcast : ((k i).val : ℝ) = (2 : ℝ) ^ j - 1 := by
have hk' : ((k i).val : ℝ) = ((2 ^ j - 1 : ℕ) : ℝ) := by
exact_mod_cast hk
have hp : 1 ≤ 2 ^ j := Nat.one_le_iff_ne_zero.mpr (pow_ne_zero _ (by omega))
rw [Nat.cast_sub hp, Nat.cast_pow] at hk'
norm_num at hk' ⊢
exact hk'
rw [hx1]
change 1 - (k i).val * meshWidth j ≤ meshWidth j
rw [hkcast]
nlinarith
The capped floor index assigns every cube point to its corresponding half-open dyadic macro-cell. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma cubeIndex_mem_dyadicCube {d j : ℕ} {x : Fin d → ℝ}
(hx : x ∈ cube d) : x ∈ dyadicCube d j (cubeIndex d j x) := by
refine ⟨hx, ?_⟩
intro i
have hh : 0 < meshWidth j := by unfold meshWidth; positivity
have hscale := orderedMass_meshWidth_mul_pow j
let N : ℕ := 2 ^ j
have hN : 0 < N := by dsimp [N]; positivity
have hxi := Set.mem_univ_pi.mp hx i
by_cases hx1 : x i = 1
· rw [hx1]
have hk : (cubeIndex d j x i).val = N - 1 := by
change min (Nat.floor ((N : ℝ) * x i)) (N - 1) % N = N - 1
rw [hx1]
simp only [mul_one, Nat.floor_natCast]
rw [min_eq_right (sub_le _ _), Nat.mod_eq_of_lt (sub_lt hN (by omega))]
constructor
· change ((cubeIndex d j x i).val : ℝ) * meshWidth j ≤ 1
have hk' : ((cubeIndex d j x i).val : ℝ) = (N - 1 : ℕ) := by exact_mod_cast hk
rw [hk']
have hcast : ((N - 1 : ℕ) : ℝ) = (N : ℝ) - 1 := by
rw [Nat.cast_sub (Nat.one_le_iff_ne_zero.mpr hN.ne')]
norm_num
rw [hcast]
have hscale' : meshWidth j * (N : ℝ) = 1 := by simpa [N] using hscale
nlinarith
· exact inr ⟨by simpa [N] using hk, rfl⟩
· have hxl : x i < 1 := lt_of_le_of_ne hxi.2 hx1
have hqnonneg : 0 ≤ (N : ℝ) * x i := mul_nonneg (by positivity) hxi.1
have hq_lt : (N : ℝ) * x i < N := by
norm_num
exact mul_lt_of_lt_one_right (by positivity) hxl
have hfloor_lt : Nat.floor ((N : ℝ) * x i) < N :=
(Nat.floor_lt hqnonneg).2 hq_lt
have hk : (cubeIndex d j x i).val = Nat.floor ((N : ℝ) * x i) := by
change min (Nat.floor ((N : ℝ) * x i)) (N - 1) % N =
Nat.floor ((N : ℝ) * x i)
have hle : Nat.floor ((N : ℝ) * x i) ≤ N - 1 := by omega
rw [min_eq_left hle, Nat.mod_eq_of_lt hfloor_lt]
constructor
· change ((cubeIndex d j x i).val : ℝ) * meshWidth j ≤ x i
rw [hk]
have hf := Nat.floor_le hqnonneg
have hscale' : meshWidth j * (N : ℝ) = 1 := by simpa [N] using hscale
nlinarith [mul_le_mul_of_nonneg_right hf hh.le]
· apply inl
change x i < ((cubeIndex d j x i).val : ℝ) * meshWidth j + meshWidth j
rw [hk]
have hf := Nat.lt_floor_add_one ((N : ℝ) * x i)
have hscale' : meshWidth j * (N : ℝ) = 1 := by simpa [N] using hscale
nlinarith [mul_lt_mul_of_pos_right hf hh]
The normalized coordinates selected by cubeIndex lie in the unit cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma cubeIndex_normalized_mem_cube {d j : ℕ} {x : Fin d → ℝ}
(hx : x ∈ cube d) :
(fun i => (x i - cubeCorner d j (cubeIndex d j x) i) / meshWidth j) ∈ cube d :=
dyadicCube_normalized_mem_cube (cubeIndex d j x) (cubeIndex_mem_dyadicCube hx)
theorem conditionalMeanSample_sub_polynomial_coordinate_le Lemma conditionalMeanSample_sub_polynomial_coordinate_le in the paper ↗
Replacing treated outcomes by the identified regression preserves a local polynomial approximation and hence controls every fitted coefficient. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
P :
Measure (Obs d)
μ :
(Fin d → ℝ) → ℝ
ω :
Fin n → Obs d
hrepresentative :
∀ i : Fin n, μ (ω i).1 = treatedRegression P (ω i).1
θ :
MonoIndex d m → ℝ
r :
ℝ
hr :
0 ≤ r
happrox :
if
(ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ
then
|μ (ω i).1
- ∑ α : MonoIndex d m, monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α * θ α|
≤ r
α :
MonoIndex d m
conclusion
|coefHat m j (conditionalMeanSample P ω) k α - θ α|
Proof (Lean source)
lemma conditionalMeanSample_sub_polynomial_coordinate_le {d n : ℕ} (m j : ℕ)
(P : Measure (Obs d)) [IsFiniteMeasure P]
(μ : (Fin d → ℝ) → ℝ) (ω : Fin n → Obs d)
(hrepresentative : ∀ i : Fin n,
μ (ω i).1 = treatedRegression P (ω i).1)
(k : Fin d → Fin (2 ^ j)) (θ : MonoIndex d m → ℝ)
(hcount : ∀ ℓ : Fin d → Fin (m + 1), 0 < treatedCount m j ω k ℓ)
(r : ℝ) (hr : 0 ≤ r)
(happrox : ∀ (ℓ : Fin d → Fin (m + 1)) (i : Fin n),
(ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ →
|μ (ω i).1 - ∑ α : MonoIndex d m,
monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α * θ α| ≤ r)
(α : MonoIndex d m) :
|coefHat m j (conditionalMeanSample P ω) k α - θ α| ≤
(2 / templateLambda d m) *
(sqrt (card (MonoIndex d m) : ℝ) * r) := by
have hdesign : ∀ i,
((conditionalMeanSample P ω) i).1 = (ω i).1 ∧
((conditionalMeanSample P ω) i).2.1 = (ω i).2.1 := by
intro i
exact ⟨rfl, rfl⟩
have hcount' : ∀ ℓ : Fin d → Fin (m + 1),
0 < treatedCount m j (conditionalMeanSample P ω) k ℓ := by
intro ℓ
rw [treatedCount_eq_of_design_eq m j ω (conditionalMeanSample P ω) k ℓ hdesign]
exact hcount ℓ
apply coefHat_sub_polynomial_coordinate_le m j
(conditionalMeanSample P ω) k θ hcount' r hr
intro ℓ i hi
have hi' : (ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ := by
simpa only [conditionalMeanSample] using hi
simpa only [conditionalMeanSample, hi'.1, ↓reduceIte, hrepresentative i] using
happrox ℓ i hi'
theorem conditionalCoefCentre_sub_polynomial_coordinate_le Lemma conditionalCoefCentre_sub_polynomial_coordinate_le in the paper ↗
Almost surely over an IID sample, the conditional coefficient centre inherits any pointwise local-polynomial approximation of the selected design. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B L C c_f γ :
ℝ
Pc :
μ e :
(Fin d → ℝ) → ℝ
hparams :
ModelParameterDomain d β B L C c_f
hγ :
1 < γ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ e
Q :
hIID :
IIDSampleLaw Q (Pc.map observed)
conclusion
≤ r → (∀ ℓ : Fin d → Fin (m + 1), 0 < treatedCount m j ω k ℓ) → (∀ (ℓ : Fin d → Fin (m + 1)) (i : Fin n), (ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ → |μ (ω i).1 - ∑ α : MonoIndex d m, monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α * θ α| ≤ r) → ∀ α : MonoIndex d m, |conditionalCoefCentre Q m j ω k α
Proof (Lean source)
lemma conditionalCoefCentre_sub_polynomial_coordinate_le
{d n : ℕ} {β B L C c_f γ : ℝ}
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ e : (Fin d → ℝ) → ℝ)
(hparams : ModelParameterDomain d β B L C c_f) (hγ : 1 < γ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ e)
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q]
(hIID : IIDSampleLaw Q (Pc.map observed)) :
∀ᵐ ω ∂Q, ∀ (m j : ℕ) (k : Fin d → Fin (2 ^ j))
(θ : MonoIndex d m → ℝ) (r : ℝ), 0 ≤ r →
(∀ ℓ : Fin d → Fin (m + 1), 0 < treatedCount m j ω k ℓ) →
(∀ (ℓ : Fin d → Fin (m + 1)) (i : Fin n),
(ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ →
|μ (ω i).1 - ∑ α : MonoIndex d m,
monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α * θ α| ≤ r) →
∀ α : MonoIndex d m,
|conditionalCoefCentre Q m j ω k α - θ α| ≤
(2 / templateLambda d m) *
(sqrt (card (MonoIndex d m) : ℝ) * r) := by
let P := Pc.map observed
letI : IsProbabilityMeasure P := Measure.isProbabilityMeasure_map
(μ := Pc) (f := observed) (by unfold observed; fun_prop)
filter_upwards [holderRepresentative_on_iid_sample Pc μ e hparams hγ hmodel Q hIID,
conditionalCoefCentre_eq_conditionalMeanSample P Q hIID hmodel.outcome]
with ω hrepresentative hcentre
intro m j k θ r hr hcount happrox α
rw [hcentre m j k α]
exact conditionalMeanSample_sub_polynomial_coordinate_le m j P μ ω
hrepresentative k θ hcount r hr happrox α
theorem holder_taylor_monomial_approx_uniform_center Lemma holder_taylor_monomial_approx_uniform_center in the paper ↗
The local Hölder Taylor constant can be chosen before the centre and domain. This is the centre-uniform form needed by the selected mesh. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d p :
ℕ
β L r :
ℝ
hβ :
0 < β
hL :
0 < L
hr :
0 < r
hcover :
∀ e : Fin d → ℕ
if
(∑ j, e j) ≤ ⌈β⌉₊ - 1
then
∃ k, expo k = e
Proof (Lean source)
theorem holder_taylor_monomial_approx_uniform_center {d p : ℕ}
{β L r : ℝ}
(hβ : 0 < β) (hL : 0 < L) (hr : 0 < r)
(expo : Fin p → (Fin d → ℕ))
(hcover : ∀ e : Fin d → ℕ,
(∑ j, e j) ≤ ⌈β⌉₊ - 1 → ∃ k, expo k = e) :
∃ Cb : ℝ, 0 ≤ Cb ∧
∀ (x₀ : Fin d → ℝ) (S : Set (Fin d → ℝ)),
{x : Fin d → ℝ | ∀ i, |x i - x₀ i| ≤ r} ⊆ S →
∀ f : (Fin d → ℝ) → ℝ, HolderBallStd f β L S →
∀ h : ℝ, 0 < h → h < r →
∃ θ : Fin p → ℝ,
∀ u : Fin d → ℝ, (∀ j, |u j| ≤ 1) →
|f (x₀ + h • u) -
∑ k, θ k * ∏ j, (u j) ^ (expo k j)| ≤
Cb * L * h ^ β :=
holder_taylor_monomial_approx_uniform_center
hβ hL expo hcover
theorem exists_uniform_holderBall_local_monoVec_approx Lemma exists-uniform-holderBall-local-monoVec-approx in the paper ↗
One approximation constant works for every response in the intrinsic Holder ball. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β L :
ℝ
hβ :
0 < β
hL :
0 < L
conclusion
∃ Cb : ℝ,
0 ≤ Cb
conclusion 1
∃ θ : MonoIndex d (polynomialDegree β) → ℝ,
∀ u : Fin d → ℝ,
(∀ i, |u i| ≤ 1) → x₀
+ h • u ∈ cube d → |μ (x₀ + h • u)
- ∑ α, monoVec d (polynomialDegree β) u α * θ α|
≤ Cb * L * h ^ β
Proof (Lean source)
lemma exists_uniform_holderBall_local_monoVec_approx (d : ℕ) {β L : ℝ}
(hβ : 0 < β) (hL : 0 < L) :
∃ Cb : ℝ, 0 ≤ Cb ∧ ∀ (μ : (Fin d → ℝ) → ℝ), HolderBall β L μ →
∀ (x₀ : Fin d → ℝ) (h : ℝ), 0 < h → h ≤ 1 →
∃ θ : MonoIndex d (polynomialDegree β) → ℝ,
∀ u : Fin d → ℝ, (∀ i, |u i| ≤ 1) → x₀ + h • u ∈ cube d →
|μ (x₀ + h • u) -
∑ α, monoVec d (polynomialDegree β) u α * θ α| ≤
Cb * L * h ^ β := by
classical
let m := polynomialDegree β
let s := β - (m : ℝ)
have hceil : 1 ≤ ⌈β⌉₊ := Nat.one_le_ceil_iff.mpr hβ
have hmcast : (m : ℝ) = (⌈β⌉₊ : ℝ) - 1 := by
simp only [m, polynomialDegree]
rw [Nat.cast_sub hceil, Nat.cast_one]
have hs : 0 < s := by
dsimp [s]
rw [hmcast]
linarith [Nat.ceil_lt_add_one hβ.le]
have hs1 : s ≤ 1 := by
dsimp [s]
rw [hmcast]
linarith [Nat.le_ceil β]
obtain ⟨D, hD, htransport⟩ := unit_holder_affine_transport d m s hs
obtain ⟨A, hA, hExt⟩ := exists_global_holder_extension d m s hs hs1
let e : MonoIndex d m ≃ Fin (card (MonoIndex d m)) := Fintype.equivFin _
let expo : Fin (card (MonoIndex d m)) → (Fin d → ℕ) :=
fun q i => (((e.symm q).1 i).val)
have hcover : ∀ a : Fin d → ℕ, (∑ i, a i) ≤ ⌈β⌉₊ - 1 → ∃ q, expo q = a := by
intro a ha
have hai (i : Fin d) : a i < m + 1 := by
have hi := (Finset.single_le_sum (fun q _ => zero_le (a q))
(Finset.mem_univ i)).trans ha
simpa [m, polynomialDegree] using Nat.lt_succ_of_le hi
let α : MonoIndex d m := ⟨fun i => ⟨a i, hai i⟩, by
simp only [monoIdx, mem_filter, Finset.mem_univ, true_and]
simpa [m, polynomialDegree] using ha⟩
exact ⟨e α, by funext i; simp [expo, e, α]⟩
obtain ⟨C₀, hC₀, hApprox⟩ := holder_taylor_monomial_approx_uniform_center
hβ (by positivity : 0 < A * (D * L)) (by norm_num : (0 : ℝ) < 3)
expo hcover
refine ⟨C₀ * A * D * (2 : ℝ) ^ β, by positivity, ?_⟩
intro μ hμ x₀ h hh hh1
have hμunit : HolderBallOn
(unitCube d)
m s L μ := by
simpa [m, s, cube,
unitCube] using hμ.toIntrinsic
have hμa : CubeHolderBall d m s (D * L) (fun z => μ (cubeAffineInv z)) :=
htransport μ L hL.le hμunit
obtain ⟨U, hUeq, hUsmooth, hUbounds, hUmod⟩ :=
hExt (fun z => μ (cubeAffineInv z)) (D * L) (mul_nonneg hD.le hL.le) hμa
have hUstd : HolderBallStd U β (A * (D * L)) univ := by
refine ⟨by change ContDiffOn ℝ m U univ; exact hUsmooth.contDiffOn, ?_, ?_⟩
· intro q hq z hz
exact hUbounds q (by change q ≤ m at hq; exact hq) z
· intro z hz w hw
change ‖iteratedFDeriv ℝ m U z - iteratedFDeriv ℝ m U w‖ ≤
A * (D * L) * ‖z - w‖ ^ s
exact hUmod z w
let x₀' := Causalean.Mathlib.Analysis.Calculus.CubeInterpolation.cubeAffine x₀
have hsubset : {z : Fin d → ℝ | ∀ i, |z i - x₀' i| ≤ 3} ⊆ univ := by
intro z hz
trivial
obtain ⟨c, hc⟩ := hApprox x₀' univ hsubset U hUstd
(2 * h) (by positivity) (by linarith)
let θ : MonoIndex d m → ℝ := fun α => c (e α)
refine ⟨θ, ?_⟩
intro u hu hxu
have harg : Causalean.Mathlib.Analysis.Calculus.CubeInterpolation.cubeAffine
(x₀ + h • u) = x₀' + (2 * h) • u := by
ext i
simp [x₀', Causalean.Mathlib.Analysis.Calculus.CubeInterpolation.cubeAffine,
Pi.add_apply, Pi.smul_apply, smul_eq_mul]
ring
have hmem : Causalean.Mathlib.Analysis.Calculus.CubeInterpolation.cubeAffine
(x₀ + h • u) ∈ cube d :=
cubeAffine_mem_cube
(by simpa [cube,
unitCube] using hxu)
have hagree : U (x₀' + (2 * h) • u) = μ (x₀ + h • u) := by
rw [← harg, hUeq hmem]
exact congrArg μ (by
ext i
simp [Causalean.Mathlib.Analysis.Calculus.CubeInterpolation.cubeAffine,
cubeAffineInv])
have hsum : (∑ q, c q * ∏ i, u i ^ expo q i) =
∑ α, monoVec d m u α * θ α := by
rw [← Equiv.sum_comp e]
apply Finset.sum_congr rfl
intro α _
simp [expo, θ, monoVec, e, mul_comm]
rw [← hsum, ← hagree]
calc
_ ≤ C₀ * (A * (D * L)) * (2 * h) ^ β := hc u hu
_ ≤ (C₀ * A * D * (2 : ℝ) ^ β) * L * h ^ β := by
rw [Real.mul_rpow (by norm_num : (0 : ℝ) ≤ 2) hh.le]
ring_nf
exact le_rfl
theorem holderBall_exists_local_monoVec_approx Lemma holderBall_exists_local_monoVec_approx in the paper ↗
Intrinsic [0,1]^d Hölder regularity supplies a fixed-cell polynomial in the exact MonoIndex basis. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
∃ Cb : ℝ,
0 ≤ Cb
conclusion 1
∃ θ : MonoIndex d (polynomialDegree β) → ℝ,
∀ u : Fin d → ℝ,
(∀ i, |u i| ≤ 1) → x₀
+ h • u ∈ cube d → |μ (x₀ + h • u)
- ∑ α, monoVec d (polynomialDegree β) u α * θ α|
≤ Cb * L * h ^ β
Proof (Lean source)
lemma holderBall_exists_local_monoVec_approx (d : ℕ) {β L : ℝ}
(hβ : 0 < β) (hL : 0 < L) (μ : (Fin d → ℝ) → ℝ)
(hμ : HolderBall β L μ) :
∃ Cb : ℝ, 0 ≤ Cb ∧ ∀ (x₀ : Fin d → ℝ) (h : ℝ), 0 < h → h ≤ 1 →
∃ θ : MonoIndex d (polynomialDegree β) → ℝ,
∀ u : Fin d → ℝ, (∀ i, |u i| ≤ 1) → x₀ + h • u ∈ cube d →
|μ (x₀ + h • u) -
∑ α, monoVec d (polynomialDegree β) u α * θ α| ≤
Cb * L * h ^ β := by
classical
let m := polynomialDegree β
let s := β - (m : ℝ)
have hceil : 1 ≤ ⌈β⌉₊ := Nat.one_le_ceil_iff.mpr hβ
have hmcast : (m : ℝ) = (⌈β⌉₊ : ℝ) - 1 := by
simp only [m, polynomialDegree]
rw [Nat.cast_sub hceil, Nat.cast_one]
have hs : 0 < s := by dsimp [s]; rw [hmcast]; linarith [Nat.ceil_lt_add_one hβ.le]
have hs1 : s ≤ 1 := by dsimp [s]; rw [hmcast]; linarith [Nat.le_ceil β]
obtain ⟨D, hD, htransport⟩ := unit_holder_affine_transport d m s hs
have hμunit : HolderBallOn
(unitCube d)
m s L μ := by
simpa [m, s, cube,
unitCube] using hμ.toIntrinsic
have hμa : CubeHolderBall d m s (D * L) (fun z => μ (cubeAffineInv z)) :=
htransport μ L hL.le hμunit
obtain ⟨A, hA, hExt⟩ := exists_global_holder_extension d m s hs hs1
obtain ⟨U, hUeq, hUsmooth, hUbounds, hUmod⟩ :=
hExt (fun z => μ (cubeAffineInv z)) (D * L) (mul_nonneg hD.le hL.le) hμa
have hUstd : HolderBallStd U β (A * (D * L)) univ := by
refine ⟨by change ContDiffOn ℝ m U univ; exact hUsmooth.contDiffOn, ?_, ?_⟩
· intro q hq z hz
exact hUbounds q (by change q ≤ m at hq; exact hq) z
· intro z hz w hw
change ‖iteratedFDeriv ℝ m U z - iteratedFDeriv ℝ m U w‖ ≤
A * (D * L) * ‖z - w‖ ^ s
exact hUmod z w
let e : MonoIndex d m ≃ Fin (card (MonoIndex d m)) := Fintype.equivFin _
let expo : Fin (card (MonoIndex d m)) → (Fin d → ℕ) :=
fun q i => (((e.symm q).1 i).val)
have hcover : ∀ a : Fin d → ℕ, (∑ i, a i) ≤ ⌈β⌉₊ - 1 → ∃ q, expo q = a := by
intro a ha
have hai (i : Fin d) : a i < m + 1 := by
have hi := (Finset.single_le_sum (fun q _ => zero_le (a q))
(Finset.mem_univ i)).trans ha
simpa [m, polynomialDegree] using Nat.lt_succ_of_le hi
let α : MonoIndex d m := ⟨fun i => ⟨a i, hai i⟩, by
simp only [monoIdx, mem_filter, Finset.mem_univ, true_and]
simpa [m, polynomialDegree] using ha⟩
exact ⟨e α, by funext i; simp [expo, e, α]⟩
obtain ⟨C₀, hC₀, hApprox⟩ := holder_taylor_monomial_approx_uniform_center
hβ (by positivity : 0 < A * (D * L)) (by norm_num : (0 : ℝ) < 3)
expo hcover
refine ⟨C₀ * A * D * (2 : ℝ) ^ β, by positivity, ?_⟩
intro x₀ h hh hh1
let x₀' := Causalean.Mathlib.Analysis.Calculus.CubeInterpolation.cubeAffine x₀
have hsubset : {z : Fin d → ℝ | ∀ i, |z i - x₀' i| ≤ 3} ⊆ univ := by
intro z hz
trivial
obtain ⟨c, hc⟩ := hApprox x₀' univ hsubset U hUstd
(2 * h) (by positivity) (by linarith)
let θ : MonoIndex d m → ℝ := fun α => c (e α)
refine ⟨θ, ?_⟩
intro u hu hxu
have harg : Causalean.Mathlib.Analysis.Calculus.CubeInterpolation.cubeAffine
(x₀ + h • u) = x₀' + (2 * h) • u := by
ext i
simp [x₀', Causalean.Mathlib.Analysis.Calculus.CubeInterpolation.cubeAffine,
Pi.add_apply, Pi.smul_apply, smul_eq_mul]
ring
have hmem : Causalean.Mathlib.Analysis.Calculus.CubeInterpolation.cubeAffine
(x₀ + h • u) ∈ cube d :=
cubeAffine_mem_cube
(by simpa [cube,
unitCube] using hxu)
have hagree : U (x₀' + (2 * h) • u) = μ (x₀ + h • u) := by
rw [← harg, hUeq hmem]
exact congrArg μ (by
ext i
simp [Causalean.Mathlib.Analysis.Calculus.CubeInterpolation.cubeAffine,
cubeAffineInv])
have hsum : (∑ q, c q * ∏ i, u i ^ expo q i) =
∑ α, monoVec d m u α * θ α := by
rw [← Equiv.sum_comp e]
apply Finset.sum_congr rfl
intro α _
simp [expo, θ, monoVec, e, mul_comm]
rw [← hsum, ← hagree]
calc
_ ≤ C₀ * (A * (D * L)) * (2 * h) ^ β := hc u hu
_ ≤ (C₀ * A * D * (2 : ℝ) ^ β) * L * h ^ β := by
rw [Real.mul_rpow (by norm_num : (0 : ℝ) ≤ 2) hh.le]
ring_nf
exact le_rfl
Every dyadic mesh width lies in (0,1]. For the stated inputs and conditions, the asserted conclusion holds.
Proof (Lean source)
lemma meshWidth_pos_le_one (j : ℕ) : 0 < meshWidth j ∧ meshWidth j ≤ 1 := by
constructor
· unfold meshWidth
positivity
· unfold meshWidth
rw [Real.rpow_neg (by norm_num : (0 : ℝ) ≤ 2), Real.rpow_natCast]
exact (inv_le_one₀ (by positivity)).2
(one_le_pow₀ (by norm_num : (1 : ℝ) ≤ 2))
theorem exists_uniform_conditionalCoefCentre_holder_cell_bias Lemma exists-uniform-conditional-coef-centre-holder-cell-bias in the paper ↗
The cell-bias constant is uniform over every model completion and response in the stated Holder ball. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β L :
ℝ
hβ :
0 < β
hL :
0 < L
conclusion
∃ Cb : ℝ,
0 ≤ Cb
conclusion 1
n :
ℕ
B C c_f γ :
ℝ
Pc :
Measure (Completion d)
μ e :
(Fin d → ℝ) → ℝ
hparams :
ModelParameterDomain d β B L C c_f
hγ :
1 < γ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ e
hIID :
IIDSampleLaw Q (Pc.map observed)
∀ᵐ ω ∂Q, ∀ (j : ℕ) (k : Fin d → Fin (2 ^ j)), (∀ ℓ : Fin d → Fin (polynomialDegree β + 1), 0 < treatedCount (polynomialDegree β) j ω k ℓ) → ∃ θ : MonoIndex d (polynomialDegree β) → ℝ, (∀ x ∈ dyadicCube d j k, |μ x - ∑ α, monoVec d (polynomialDegree β) (fun a => (x a - cubeCorner d j k a) / meshWidth j) α * θ α| ≤ Cb * L * meshWidth j ^ β) ∧ ∀ α, |conditionalCoefCentre Q (polynomialDegree β) j ω k α
- θ α|
≤ (2 / templateLambda d (polynomialDegree β)) * (sqrt (card (MonoIndex d (polynomialDegree β)) : ℝ) * (Cb * L * meshWidth j ^ β))
Proof (Lean source)
lemma exists_uniform_conditionalCoefCentre_holder_cell_bias
(d : ℕ) (β L : ℝ) (hβ : 0 < β) (hL : 0 < L) :
∃ Cb : ℝ, 0 ≤ Cb ∧
∀ {n : ℕ} {B C c_f γ : ℝ}
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ e : (Fin d → ℝ) → ℝ)
(hparams : ModelParameterDomain d β B L C c_f) (hγ : 1 < γ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ e)
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q]
(hIID : IIDSampleLaw Q (Pc.map observed)),
∀ᵐ ω ∂Q, ∀ (j : ℕ) (k : Fin d → Fin (2 ^ j)),
(∀ ℓ : Fin d → Fin (polynomialDegree β + 1),
0 < treatedCount (polynomialDegree β) j ω k ℓ) →
∃ θ : MonoIndex d (polynomialDegree β) → ℝ,
(∀ x ∈ dyadicCube d j k,
|μ x - ∑ α, monoVec d (polynomialDegree β)
(fun a => (x a - cubeCorner d j k a) / meshWidth j) α * θ α| ≤
Cb * L * meshWidth j ^ β) ∧
∀ α,
|conditionalCoefCentre Q (polynomialDegree β) j ω k α - θ α| ≤
(2 / templateLambda d (polynomialDegree β)) *
(sqrt (card
(MonoIndex d (polynomialDegree β)) : ℝ) *
(Cb * L * meshWidth j ^ β)) := by
obtain ⟨Cb, hCb, hTaylor⟩ :=
exists_uniform_holderBall_local_monoVec_approx d hβ hL
refine ⟨Cb, hCb, ?_⟩
intro n B C c_f γ Pc _ μ e hparams hγ hmodel Q _ hIID
have hTaylorμ := hTaylor μ hmodel.smooth.1
filter_upwards [conditionalCoefCentre_sub_polynomial_coordinate_le
Pc μ e hparams hγ hmodel Q hIID] with ω hcentre
intro j k hcount
obtain ⟨θ, hθ⟩ := hTaylorμ (cubeCorner d j k) (meshWidth j)
(meshWidth_pos_le_one j).1 (meshWidth_pos_le_one j).2
refine ⟨θ, ?_, ?_⟩
· intro x hx
let u : Fin d → ℝ := fun a =>
(x a - cubeCorner d j k a) / meshWidth j
have huCube : u ∈ cube d := dyadicCube_normalized_mem_cube k hx
have huAbs : ∀ a, |u a| ≤ 1 := by
intro a
have ha := Set.mem_univ_pi.mp huCube a
rw [abs_of_nonneg ha.1]
exact ha.2
have hreconstruct : cubeCorner d j k + meshWidth j • u = x := by
ext a
dsimp [u]
field_simp [(meshWidth_pos_le_one j).1.ne']
ring
simpa only [hreconstruct] using hθ u huAbs
(by simpa [hreconstruct] using hx.1)
· apply hcentre (polynomialDegree β) j k θ
(Cb * L * meshWidth j ^ β)
(mul_nonneg (mul_nonneg hCb hparams.2.2.2.1.le)
(Real.rpow_nonneg (meshWidth_pos_le_one j).1.le _)) hcount
intro ℓ i hi
have hxi := orderedMass_scaledMicroCell_inside_dyadicCube
d (polynomialDegree β) j k ℓ hi.2
have huAbs : ∀ a, |((ω i).1 a - cubeCorner d j k a) / meshWidth j| ≤ 1 := fun a => by
have ha := Set.mem_univ_pi.mp
(dyadicCube_normalized_mem_cube k hxi) a
rw [abs_of_nonneg ha.1]
exact ha.2
have hreconstruct : cubeCorner d j k + meshWidth j •
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) = (ω i).1 := by
ext a
change cubeCorner d j k a + meshWidth j *
(((ω i).1 a - cubeCorner d j k a) / meshWidth j) = (ω i).1 a
field_simp [(meshWidth_pos_le_one j).1.ne']
ring
have ht := hθ _ huAbs (by rw [hreconstruct]; exact hxi.1)
rw [hreconstruct] at ht
exact ht
theorem conditionalCoefCentre_holder_cell_bias Lemma conditionalCoefCentre_holder_cell_bias in the paper ↗
Uniformly over cells, the conditional coefficient centre is close to a Taylor polynomial which approximates the Hölder representative throughout the whole dyadic cell. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B L C c_f γ :
ℝ
Pc :
μ e :
(Fin d → ℝ) → ℝ
hparams :
ModelParameterDomain d β B L C c_f
hγ :
1 < γ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ e
Q :
hIID :
IIDSampleLaw Q (Pc.map observed)
conclusion
∃ Cb : ℝ,
conclusion 1
0 ≤ Cb
conclusion 2
∀ᵐ ω ∂Q, ∀ (j : ℕ) (k : Fin d → Fin (2 ^ j)), (∀ ℓ : Fin d → Fin (polynomialDegree β + 1), 0 < treatedCount (polynomialDegree β) j ω k ℓ) → ∃ θ : MonoIndex d (polynomialDegree β) → ℝ, (∀ x ∈ dyadicCube d j k, |μ x - ∑ α, monoVec d (polynomialDegree β) (fun a => (x a - cubeCorner d j k a) / meshWidth j) α * θ α| ≤ Cb * L * meshWidth j ^ β) ∧ ∀ α, |conditionalCoefCentre Q (polynomialDegree β) j ω k α
- θ α|
≤ (2 / templateLambda d (polynomialDegree β)) * (sqrt (card (MonoIndex d (polynomialDegree β)) : ℝ) * (Cb * L * meshWidth j ^ β))
Proof (Lean source)
lemma conditionalCoefCentre_holder_cell_bias
{d n : ℕ} {β B L C c_f γ : ℝ}
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ e : (Fin d → ℝ) → ℝ)
(hparams : ModelParameterDomain d β B L C c_f) (hγ : 1 < γ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ e)
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q]
(hIID : IIDSampleLaw Q (Pc.map observed)) :
∃ Cb : ℝ, 0 ≤ Cb ∧
∀ᵐ ω ∂Q, ∀ (j : ℕ) (k : Fin d → Fin (2 ^ j)),
(∀ ℓ : Fin d → Fin (polynomialDegree β + 1),
0 < treatedCount (polynomialDegree β) j ω k ℓ) →
∃ θ : MonoIndex d (polynomialDegree β) → ℝ,
(∀ x ∈ dyadicCube d j k,
|μ x - ∑ α, monoVec d (polynomialDegree β)
(fun a => (x a - cubeCorner d j k a) / meshWidth j) α * θ α| ≤
Cb * L * meshWidth j ^ β) ∧
∀ α,
|conditionalCoefCentre Q (polynomialDegree β) j ω k α - θ α| ≤
(2 / templateLambda d (polynomialDegree β)) *
(sqrt (card
(MonoIndex d (polynomialDegree β)) : ℝ) *
(Cb * L * meshWidth j ^ β)) := by
obtain ⟨Cb, hCb, hTaylor⟩ := holderBall_exists_local_monoVec_approx
d (zero_lt_one.trans hparams.2.1) hparams.2.2.2.1 μ hmodel.smooth.1
refine ⟨Cb, hCb, ?_⟩
filter_upwards [conditionalCoefCentre_sub_polynomial_coordinate_le
Pc μ e hparams hγ hmodel Q hIID] with ω hcentre
intro j k hcount
obtain ⟨θ, hθ⟩ := hTaylor (cubeCorner d j k) (meshWidth j)
(meshWidth_pos_le_one j).1 (meshWidth_pos_le_one j).2
refine ⟨θ, ?_, ?_⟩
· intro x hx
let u : Fin d → ℝ := fun a =>
(x a - cubeCorner d j k a) / meshWidth j
have huCube : u ∈ cube d := dyadicCube_normalized_mem_cube k hx
have huAbs : ∀ a, |u a| ≤ 1 := by
intro a
have ha := Set.mem_univ_pi.mp huCube a
rw [abs_of_nonneg ha.1]
exact ha.2
have hreconstruct : cubeCorner d j k + meshWidth j • u = x := by
ext a
dsimp [u]
field_simp [(meshWidth_pos_le_one j).1.ne']
ring
simpa only [hreconstruct] using hθ u huAbs (by simpa [hreconstruct] using hx.1)
· apply hcentre (polynomialDegree β) j k θ
(Cb * L * meshWidth j ^ β)
(mul_nonneg (mul_nonneg hCb hparams.2.2.2.1.le)
(Real.rpow_nonneg (meshWidth_pos_le_one j).1.le _)) hcount
intro ℓ i hi
have hxi := orderedMass_scaledMicroCell_inside_dyadicCube
d (polynomialDegree β) j k ℓ hi.2
have huAbs : ∀ a, |((ω i).1 a - cubeCorner d j k a) / meshWidth j| ≤ 1 := fun a => by
have ha := Set.mem_univ_pi.mp
(dyadicCube_normalized_mem_cube k hxi) a
rw [abs_of_nonneg ha.1]
exact ha.2
have hreconstruct : cubeCorner d j k + meshWidth j •
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) = (ω i).1 := by
ext a
change cubeCorner d j k a + meshWidth j *
(((ω i).1 a - cubeCorner d j k a) / meshWidth j) = (ω i).1 a
field_simp [(meshWidth_pos_le_one j).1.ne']
ring
have ht := hθ _ huAbs (by rw [hreconstruct]; exact hxi.1)
rw [hreconstruct] at ht
exact ht
The continuous Hölder representative inherits the treated-mean bound at every cube point, rather than only almost everywhere. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ :
ℝ
Pc :
μ e :
(Fin d → ℝ) → ℝ
hparams :
ModelParameterDomain d β B L C c_f
hγ :
1 < γ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ e
x ∈ cube d :
conclusion
|μ x| ≤ B
Proof (Lean source)
lemma holderResponse_abs_le_on_cube
{d : ℕ} {β B L C c_f γ : ℝ}
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ e : (Fin d → ℝ) → ℝ)
(hparams : ModelParameterDomain d β B L C c_f)
(hγ : 1 < γ) (hmodel : GlobalTailModel β B L C c_f γ Pc μ e) :
∀ x ∈ cube d, |μ x| ≤ B := by
let g : (Fin d → ℝ) → ℝ := fun x => max (-B) (min B (μ x))
have hμcont : ContinuousOn μ (cube d) :=
hmodel.smooth.1.regularity.continuousOn
have hclip : Continuous (fun y : ℝ => max (-B) (min B y)) := by fun_prop
have hgcont : ContinuousOn g (cube d) := by
simpa [g, Function.comp_def] using
hclip.continuousOn.comp hμcont (Set.mapsTo_univ μ (cube d))
have hid := mu1_eq_treatedRegression β B L C c_f γ Pc μ e hparams hγ hmodel
have hgae : ∀ᵐ x ∂covariateLaw (Pc.map observed),
g x = treatedRegression (Pc.map observed) x := by
filter_upwards [hid.2.1, hmodel.outcome] with x hx hbound
have hμbound : |μ x| ≤ B := by simpa [hx] using hbound.1
have hb := abs_le.mp hμbound
dsimp [g]
rw [min_eq_right hb.2, max_eq_right hb.1]
exact hx
intro x hx
have hgeq : g x = μ x := hid.2.2 g hgcont hgae x hx
have hBle : 0 ≤ B := hparams.2.2.1.le
have hglo : -B ≤ g x := by dsimp [g]; exact le_max_left _ _
have hghi : g x ≤ B := by
dsimp [g]
exact max_le (by linarith) (min_le_left _ _)
rw [← hgeq]
exact abs_le.mpr ⟨hglo, hghi⟩
theorem equalCellEstimator_error_le_conditional_envelope Lemma equalCellEstimator_error_le_conditional_envelope in the paper ↗
A cellwise conditional-centre bias bound and the realized coefficient envelope give a uniform pointwise bound for the clipped selected estimator. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B r εb :
ℝ
Q :
μ :
(Fin d → ℝ) → ℝ
ω :
Fin n → Obs d
hselected :
0 < countSelectedMesh (polynomialDegree β) β ω
hwidth :
countSelectedMesh (polynomialDegree β) β ω
hB :
0 ≤ B
hμB :
∀ x ∈ cube d, |μ x| ≤ B
hr :
0 ≤ r
hεb :
0 ≤ εb
hcell :
∃ θ : MonoIndex d (polynomialDegree β) → ℝ,
(∀ x ∈ dyadicCube d ((feasibleIndices (polynomialDegree β) β ω).sup id) k, |μ x - ∑ α, monoVec d (polynomialDegree β) (fun a => (x a - cubeCorner d ((feasibleIndices (polynomialDegree β) β ω).sup id) k a) / countSelectedMesh (polynomialDegree β) β ω) α * θ α| ≤ r) ∧
∀ α,
|conditionalCoefCentre Q (polynomialDegree β) ((feasibleIndices (polynomialDegree β) β ω).sup id) ω k α
- θ α|
≤ εb
x ∈ cube d :
conclusion
|equalCellEstimator β B ω x - μ x|
≤ (card (MonoIndex d (polynomialDegree β)) : ℝ) * (sSup ((fun k : Fin d → Fin (2 ^ ((feasibleIndices (polynomialDegree β) β ω).sup id)) => ‖fun α : MonoIndex d (polynomialDegree β) => coefHat (polynomialDegree β) ((feasibleIndices (polynomialDegree β) β ω).sup id) ω k α - conditionalCoefCentre Q (polynomialDegree β) ((feasibleIndices (polynomialDegree β) β ω).sup id) ω k α‖) '' univ) + εb)
+ r
Proof (Lean source)
lemma equalCellEstimator_error_le_conditional_envelope
{d n : ℕ} {β B r εb : ℝ}
(Q : Measure (Fin n → Obs d)) [IsFiniteMeasure Q]
(μ : (Fin d → ℝ) → ℝ) (ω : Fin n → Obs d)
(hselected : 0 < countSelectedMesh (polynomialDegree β) β ω)
(hwidth : countSelectedMesh (polynomialDegree β) β ω =
meshWidth ((feasibleIndices (polynomialDegree β) β ω).sup id))
(hB : 0 ≤ B) (hμB : ∀ x ∈ cube d, |μ x| ≤ B)
(hr : 0 ≤ r) (hεb : 0 ≤ εb)
(hcell : ∀ k : Fin d → Fin
(2 ^ ((feasibleIndices (polynomialDegree β) β ω).sup id)),
∃ θ : MonoIndex d (polynomialDegree β) → ℝ,
(∀ x ∈ dyadicCube d
((feasibleIndices (polynomialDegree β) β ω).sup id) k,
|μ x - ∑ α, monoVec d (polynomialDegree β)
(fun a => (x a - cubeCorner d
((feasibleIndices (polynomialDegree β) β ω).sup id) k a) /
countSelectedMesh (polynomialDegree β) β ω) α * θ α| ≤ r) ∧
∀ α,
|conditionalCoefCentre Q (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) ω k α - θ α| ≤ εb) :
∀ x ∈ cube d,
|equalCellEstimator β B ω x - μ x| ≤
(card (MonoIndex d (polynomialDegree β)) : ℝ) *
(sSup ((fun k : Fin d → Fin
(2 ^ ((feasibleIndices (polynomialDegree β) β ω).sup id)) =>
‖fun α : MonoIndex d (polynomialDegree β) =>
coefHat (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) ω k α -
conditionalCoefCentre Q (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) ω k α‖) ''
univ) + εb) + r := by
classical
intro x hx
let j := (feasibleIndices (polynomialDegree β) β ω).sup id
let k := cubeIndex d j x
obtain ⟨θ, hbias, hcentre⟩ := hcell k
let S : ℝ := sSup ((fun k' : Fin d → Fin (2 ^ j) =>
‖fun α : MonoIndex d (polynomialDegree β) =>
coefHat (polynomialDegree β) j ω k' α -
conditionalCoefCentre Q (polynomialDegree β) j ω k' α‖) '' univ)
have hbdd : BddAbove ((fun k' : Fin d → Fin (2 ^ j) =>
‖fun α : MonoIndex d (polynomialDegree β) =>
coefHat (polynomialDegree β) j ω k' α -
conditionalCoefCentre Q (polynomialDegree β) j ω k' α‖) '' univ) := by
simpa only [Set.image_univ] using
(Set.finite_range (fun k' : Fin d → Fin (2 ^ j) =>
‖fun α : MonoIndex d (polynomialDegree β) =>
coefHat (polynomialDegree β) j ω k' α -
conditionalCoefCentre Q (polynomialDegree β) j ω k' α‖)).bddAbove
have hcoef : ∀ α,
|coefHat (polynomialDegree β) j ω k α - θ α| ≤ S + εb := by
intro α
have hcoord :
|coefHat (polynomialDegree β) j ω k α -
conditionalCoefCentre Q (polynomialDegree β) j ω k α| ≤ S := by
calc
_ = ‖(fun a : MonoIndex d (polynomialDegree β) =>
coefHat (polynomialDegree β) j ω k a -
conditionalCoefCentre Q (polynomialDegree β) j ω k a) α‖ := by
simp [Real.norm_eq_abs]
_ ≤ ‖fun a : MonoIndex d (polynomialDegree β) =>
coefHat (polynomialDegree β) j ω k a -
conditionalCoefCentre Q (polynomialDegree β) j ω k a‖ :=
norm_le_pi_norm (fun a : MonoIndex d (polynomialDegree β) =>
coefHat (polynomialDegree β) j ω k a -
conditionalCoefCentre Q (polynomialDegree β) j ω k a) α
_ ≤ S := le_csSup hbdd ⟨k, Set.mem_univ _, rfl⟩
rw [show coefHat (polynomialDegree β) j ω k α - θ α =
(coefHat (polynomialDegree β) j ω k α -
conditionalCoefCentre Q (polynomialDegree β) j ω k α) +
(conditionalCoefCentre Q (polynomialDegree β) j ω k α - θ α) by ring]
exact (abs_add_le _ _).trans (add_le_add hcoord (hcentre α))
apply equalCellEstimator_error_le_of_selected_coefficients ω x θ hselected
· rw [hwidth]
exact cubeIndex_normalized_mem_cube hx
· exact hB
· exact hμB x hx
· simpa [j, k, S] using hcoef
· have hxcell : x ∈ dyadicCube d j k := cubeIndex_mem_dyadicCube hx
have hb := hbias x (by simpa [j, k] using hxcell)
rw [abs_sub_comm]
simpa [j, k] using hb
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
feasibleIndices m β ξ = feasibleIndices m β ω
Proof (Lean source)
lemma feasibleIndices_eq_of_design_eq {d n : ℕ} (m : ℕ) (β : ℝ)
(ω ξ : Fin n → Obs d) (hdesign : sampleDesign ξ = sampleDesign ω) :
feasibleIndices m β ξ = feasibleIndices m β ω := by
classical
have hpoint : ∀ i, (ω i).1 = (ξ i).1 ∧ (ω i).2.1 = (ξ i).2.1 := by
intro i
have hi := congrFun hdesign i
exact ⟨congrArg (fun z : (Fin d → ℝ) × Bool => z.1) hi.symm,
congrArg (fun z : (Fin d → ℝ) × Bool => z.2) hi.symm⟩
unfold feasibleIndices
ext j
simp only [mem_filter]
constructor
· rintro ⟨hj, _hj', hcounts⟩
refine ⟨hj, hj, fun k ℓ => ?_⟩
simpa only [treatedCount_eq_of_design_eq m j ξ ω k ℓ hpoint] using
hcounts k ℓ
· rintro ⟨hj, _hj', hcounts⟩
refine ⟨hj, hj, fun k ℓ => ?_⟩
simpa only [treatedCount_eq_of_design_eq m j ξ ω k ℓ hpoint] using
hcounts k ℓ
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
countSelectedMesh m β ξ = countSelectedMesh m β ω
Proof (Lean source)
lemma countSelectedMesh_eq_of_design_eq {d n : ℕ} (m : ℕ) (β : ℝ)
(ω ξ : Fin n → Obs d) (hdesign : sampleDesign ξ = sampleDesign ω) :
countSelectedMesh m β ξ = countSelectedMesh m β ω := by
simp only [countSelectedMesh,
feasibleIndices_eq_of_design_eq m β ω ξ hdesign]
theorem equalCellEstimator_error_le_conditional_envelope_of_design_eq Lemma equal-cell-estimator-error-le-conditional-envelope-fibre in the paper ↗
The deterministic estimator bound can use the conditional centre and cell-bias witnesses at any representative of the same complete-design fibre. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B r εb :
ℝ
Q :
μ :
(Fin d → ℝ) → ℝ
ω ξ :
Fin n → Obs d
hdesign :
sampleDesign ξ = sampleDesign ω
hselected :
0 < countSelectedMesh (polynomialDegree β) β ω
hwidth :
countSelectedMesh (polynomialDegree β) β ω
hB :
0 ≤ B
hμB :
∀ x ∈ cube d, |μ x| ≤ B
hr :
0 ≤ r
hεb :
0 ≤ εb
hcell :
∃ θ : MonoIndex d (polynomialDegree β) → ℝ,
(∀ x ∈ dyadicCube d ((feasibleIndices (polynomialDegree β) β ω).sup id) k, |μ x - ∑ α, monoVec d (polynomialDegree β) (fun a => (x a - cubeCorner d ((feasibleIndices (polynomialDegree β) β ω).sup id) k a) / countSelectedMesh (polynomialDegree β) β ω) α * θ α| ≤ r) ∧
∀ α,
|conditionalCoefCentre Q (polynomialDegree β) ((feasibleIndices (polynomialDegree β) β ω).sup id) ω k α
- θ α|
≤ εb
x ∈ cube d :
conclusion
|equalCellEstimator β B ξ x - μ x|
≤ (card (MonoIndex d (polynomialDegree β)) : ℝ) * (sSup ((fun k : Fin d → Fin (2 ^ ((feasibleIndices (polynomialDegree β) β ξ).sup id)) => ‖fun α : MonoIndex d (polynomialDegree β) => coefHat (polynomialDegree β) ((feasibleIndices (polynomialDegree β) β ξ).sup id) ξ k α - conditionalCoefCentre Q (polynomialDegree β) ((feasibleIndices (polynomialDegree β) β ξ).sup id) ω k α‖) '' univ) + εb)
+ r
Proof (Lean source)
lemma equalCellEstimator_error_le_conditional_envelope_of_design_eq
{d n : ℕ} {β B r εb : ℝ}
(Q : Measure (Fin n → Obs d)) [IsFiniteMeasure Q]
(μ : (Fin d → ℝ) → ℝ) (ω ξ : Fin n → Obs d)
(hdesign : sampleDesign ξ = sampleDesign ω)
(hselected : 0 < countSelectedMesh (polynomialDegree β) β ω)
(hwidth : countSelectedMesh (polynomialDegree β) β ω =
meshWidth ((feasibleIndices (polynomialDegree β) β ω).sup id))
(hB : 0 ≤ B) (hμB : ∀ x ∈ cube d, |μ x| ≤ B)
(hr : 0 ≤ r) (hεb : 0 ≤ εb)
(hcell : ∀ k : Fin d → Fin
(2 ^ ((feasibleIndices (polynomialDegree β) β ω).sup id)),
∃ θ : MonoIndex d (polynomialDegree β) → ℝ,
(∀ x ∈ dyadicCube d
((feasibleIndices (polynomialDegree β) β ω).sup id) k,
|μ x - ∑ α, monoVec d (polynomialDegree β)
(fun a => (x a - cubeCorner d
((feasibleIndices (polynomialDegree β) β ω).sup id) k a) /
countSelectedMesh (polynomialDegree β) β ω) α * θ α| ≤ r) ∧
∀ α,
|conditionalCoefCentre Q (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) ω k α - θ α| ≤ εb) :
∀ x ∈ cube d,
|equalCellEstimator β B ξ x - μ x| ≤
(card (MonoIndex d (polynomialDegree β)) : ℝ) *
(sSup ((fun k : Fin d → Fin
(2 ^ ((feasibleIndices (polynomialDegree β) β ξ).sup id)) =>
‖fun α : MonoIndex d (polynomialDegree β) =>
coefHat (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ξ).sup id) ξ k α -
conditionalCoefCentre Q (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ξ).sup id) ω k α‖) ''
univ) + εb) + r := by
have hfi := feasibleIndices_eq_of_design_eq
(polynomialDegree β) β ω ξ hdesign
have hcount := countSelectedMesh_eq_of_design_eq
(polynomialDegree β) β ω ξ hdesign
have hcentre : ∀ j k α,
conditionalCoefCentre Q (polynomialDegree β) j ξ k α =
conditionalCoefCentre Q (polynomialDegree β) j ω k α := by
intro j k α
simp only [conditionalCoefCentre, hdesign]
have hselectedξ : 0 < countSelectedMesh (polynomialDegree β) β ξ := by
rw [hcount]
exact hselected
have hwidthξ : countSelectedMesh (polynomialDegree β) β ξ =
meshWidth ((feasibleIndices (polynomialDegree β) β ξ).sup id) := by
rw [hcount, hfi]
exact hwidth
have hcellξ : ∀ k : Fin d → Fin
(2 ^ ((feasibleIndices (polynomialDegree β) β ξ).sup id)),
∃ θ : MonoIndex d (polynomialDegree β) → ℝ,
(∀ x ∈ dyadicCube d
((feasibleIndices (polynomialDegree β) β ξ).sup id) k,
|μ x - ∑ α, monoVec d (polynomialDegree β)
(fun a => (x a - cubeCorner d
((feasibleIndices (polynomialDegree β) β ξ).sup id) k a) /
countSelectedMesh (polynomialDegree β) β ξ) α * θ α| ≤ r) ∧
∀ α,
|conditionalCoefCentre Q (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ξ).sup id) ξ k α - θ α| ≤ εb := by
rw [hfi, hcount]
intro k
simpa only [hcentre] using hcell k
have hb := equalCellEstimator_error_le_conditional_envelope
(β := β) (B := B) (r := r) (εb := εb)
Q μ ξ hselectedξ hwidthξ hB hμB hr hεb hcellξ
simpa only [hcentre] using hb
A uniform real loss bound on the cube lifts directly to the ENNReal cube supremum used by the minimax risk. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma cube_iSup_ofReal_loss_le {d : ℕ} (loss : (Fin d → ℝ) → ℝ)
(R : ℝ) (hR : 0 ≤ R)
(h : ∀ x ∈ cube d, |loss x| ≤ R) :
(⨆ x, ⨆ _hx : x ∈ cube d, ofReal |loss x|) ≤ ofReal R := by
refine iSup_le fun x => iSup_le fun hx => ?_
exact ENNReal.ofReal_le_ofReal (h x hx)
theorem equalCellEstimator_error_le_two_mul Lemma equalCellEstimator_error_le_two_mul in the paper ↗
Clipping bounds every estimator error by twice the outcome envelope. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
|equalCellEstimator β B ω x - y| ≤ 2 * B
Proof (Lean source)
lemma equalCellEstimator_error_le_two_mul {d n : ℕ} (β B : ℝ)
(ω : Fin n → Obs d) (x : Fin d → ℝ) (y : ℝ)
(hB : 0 ≤ B) (hy : |y| ≤ B) :
|equalCellEstimator β B ω x - y| ≤ 2 * B := by
have hyb := abs_le.mp hy
by_cases hz : countSelectedMesh (polynomialDegree β) β ω = 0
· simp [equalCellEstimator, hz]
exact hy.trans (by linarith)
· let u := ∑ α, monoVec d (polynomialDegree β)
(fun a => (x a - cubeCorner d
((feasibleIndices (polynomialDegree β) β ω).sup id)
(cubeIndex d ((feasibleIndices (polynomialDegree β) β ω).sup id) x) a) /
countSelectedMesh (polynomialDegree β) β ω) α *
coefHat (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) ω
(cubeIndex d ((feasibleIndices (polynomialDegree β) β ω).sup id) x) α
have hlo : -B ≤ max (-B) (min B u) := le_max_left _ _
have hhi : max (-B) (min B u) ≤ B :=
max_le (by linarith) (min_le_left _ _)
unfold equalCellEstimator
simp only [hz, ↓reduceIte]
rw [abs_le]
constructor <;> linarith
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.SelectedMesh.Coefficients 31 declarations Design-fixed coefficient differences
Design-fixed coefficient differences
The treated-cell count depends only on covariates and treatment arms. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma treatedCount_eq_of_design_eq {d n : ℕ} (m j : ℕ)
(ω ξ : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(ℓ : Fin d → Fin (m + 1))
(hdesign : ∀ i, (ξ i).1 = (ω i).1 ∧ (ξ i).2.1 = (ω i).2.1) :
treatedCount m j ξ k ℓ = treatedCount m j ω k ℓ := by
classical
unfold treatedCount
congr 1
ext i
obtain ⟨hx, ha⟩ := hdesign i
simp [hx, ha]
Equal designs give the same equal-cell Gram matrix. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma gramHat_eq_of_design_eq {d n : ℕ} (m j : ℕ)
(ω ξ : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(hdesign : ∀ i, (ξ i).1 = (ω i).1 ∧ (ξ i).2.1 = (ω i).2.1) :
gramHat m j ξ k = gramHat m j ω k := by
classical
ext α α'
unfold gramHat
apply congrArg (fun z : ℝ => (tensorCount d m : ℝ)⁻¹ * z)
apply Finset.sum_congr rfl
intro ℓ _
rw [treatedCount_eq_of_design_eq m j ω ξ k ℓ hdesign]
apply congrArg (fun z : ℝ => (treatedCount m j ω k ℓ : ℝ)⁻¹ * z)
apply Finset.sum_congr rfl
intro i _
obtain ⟨hx, ha⟩ := hdesign i
simp [hx, ha]
The change in a response moment is the design-weighted sum of the observation-wise changes in treated outcomes. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
ω ξ :
Fin n → Obs d
hdesign :
∀ i, (ξ i).1 = (ω i).1 ∧ (ξ i).2.1 = (ω i).2.1
α :
MonoIndex d m
conclusion
conclusion 1
(responseMoment m j ξ k - responseMoment m j ω k) α
conclusion 2
(ω i).1 ∈ scaledMicroCell d m j k ℓ then monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α * ((ξ i).2.2 - (ω i).2.2) else 0
Proof (Lean source)
lemma responseMoment_sub_of_design_eq {d n : ℕ} (m j : ℕ)
(ω ξ : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(hdesign : ∀ i, (ξ i).1 = (ω i).1 ∧ (ξ i).2.1 = (ω i).2.1)
(α : MonoIndex d m) :
(responseMoment m j ξ k - responseMoment m j ω k) α =
(tensorCount d m : ℝ)⁻¹ *
∑ ℓ : Fin d → Fin (m + 1), (treatedCount m j ω k ℓ : ℝ)⁻¹ *
∑ i : Fin n, if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) /
meshWidth j) α * ((ξ i).2.2 - (ω i).2.2)
else 0 := by
classical
simp only [Pi.sub_apply, responseMoment]
simp only [treatedCount_eq_of_design_eq m j ω ξ k _ hdesign]
rw [← mul_sub, ← Finset.sum_sub_distrib]
congr 1
apply Finset.sum_congr rfl
intro ℓ _
rw [← mul_sub, ← Finset.sum_sub_distrib]
congr 1
apply Finset.sum_congr rfl
intro i _
obtain ⟨hx, ha⟩ := hdesign i
simp only [hx, ha]
split_ifs <;> ring
At a fixed design, a coefficient difference is the inverse Gram matrix applied to the difference of response moments. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma coefHat_sub_of_design_eq {d n : ℕ} (m j : ℕ)
(ω ξ : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(hdesign : ∀ i, (ξ i).1 = (ω i).1 ∧ (ξ i).2.1 = (ω i).2.1) :
coefHat m j ξ k - coefHat m j ω k =
mulVec (gramHat m j ω k)⁻¹
(responseMoment m j ξ k - responseMoment m j ω k) := by
rw [coefHat, coefHat, gramHat_eq_of_design_eq m j ω ξ k hdesign]
exact (Matrix.mulVec_sub _ _ _).symm
theorem coefHat_sub_coordinate_of_design_eq Lemma coefHat_sub_coordinate_of_design_eq in the paper ↗
At a fixed design, each coefficient difference is a finite linear combination of the treated outcome changes. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
ω ξ :
Fin n → Obs d
hdesign :
∀ i, (ξ i).1 = (ω i).1 ∧ (ξ i).2.1 = (ω i).2.1
α :
MonoIndex d m
conclusion
= ∑ α' : MonoIndex d m, (gramHat m j ω k)⁻¹ α α' * ((tensorCount d m : ℝ)⁻¹ * ∑ ℓ : Fin d → Fin (m + 1), (treatedCount m j ω k ℓ : ℝ)⁻¹ * ∑ i : Fin n, if (ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ then monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α' * ((ξ i).2.2 - (ω i).2.2) else 0)
Proof (Lean source)
lemma coefHat_sub_coordinate_of_design_eq {d n : ℕ} (m j : ℕ)
(ω ξ : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(hdesign : ∀ i, (ξ i).1 = (ω i).1 ∧ (ξ i).2.1 = (ω i).2.1)
(α : MonoIndex d m) :
(coefHat m j ξ k - coefHat m j ω k) α =
∑ α' : MonoIndex d m,
(gramHat m j ω k)⁻¹ α α' *
((tensorCount d m : ℝ)⁻¹ *
∑ ℓ : Fin d → Fin (m + 1),
(treatedCount m j ω k ℓ : ℝ)⁻¹ *
∑ i : Fin n, if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) /
meshWidth j) α' * ((ξ i).2.2 - (ω i).2.2)
else 0) := by
classical
rw [coefHat_sub_of_design_eq m j ω ξ k hdesign]
simp only [mulVec, dotProduct]
apply Finset.sum_congr rfl
intro α' _
rw [responseMoment_sub_of_design_eq m j ω ξ k hdesign α']
theorem coefHat_sub_conditionalMeanSample_coordinate Lemma coefHat_sub_conditionalMeanSample_coordinate in the paper ↗
Replacing each outcome by its design-conditional mean isolates the treated residual in every coefficient coordinate. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
= ∑ α' : MonoIndex d m, (gramHat m j ω k)⁻¹ α α' * ((tensorCount d m : ℝ)⁻¹ * ∑ ℓ : Fin d → Fin (m + 1), (treatedCount m j ω k ℓ : ℝ)⁻¹ * ∑ i : Fin n, if (ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ then monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α' * ((ω i).2.2 - mean i) else 0)
Proof (Lean source)
lemma coefHat_sub_conditionalMeanSample_coordinate {d n : ℕ} (m j : ℕ)
(ω : Fin n → Obs d) (mean : Fin n → ℝ)
(k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) :
(coefHat m j ω k -
coefHat m j (fun i => ((ω i).1, (ω i).2.1, mean i)) k) α =
∑ α' : MonoIndex d m,
(gramHat m j ω k)⁻¹ α α' *
((tensorCount d m : ℝ)⁻¹ *
∑ ℓ : Fin d → Fin (m + 1),
(treatedCount m j ω k ℓ : ℝ)⁻¹ *
∑ i : Fin n, if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) /
meshWidth j) α' * ((ω i).2.2 - mean i)
else 0) := by
classical
have hd : ∀ i,
((fun i => ((ω i).1, (ω i).2.1, mean i)) i).1 = (ω i).1 ∧
((fun i => ((ω i).1, (ω i).2.1, mean i)) i).2.1 = (ω i).2.1 := by
intro i
exact ⟨rfl, rfl⟩
have h := coefHat_sub_coordinate_of_design_eq m j
(fun i => ((ω i).1, (ω i).2.1, mean i)) ω k hd α
rw [gramHat_eq_of_design_eq m j _ _ k hd] at h
simp only [treatedCount_eq_of_design_eq m j _ _ _ _ hd] at h
exact h
The fixed-design weight of one response in a coefficient coordinate. For the stated inputs and conditions, the coefResidualWeight object being defined.
Definition (Lean source)
parameters
def
coefResidualWeight m j ω k α i :
ℝ
given by
∑ α' : MonoIndex d m, (gramHat m j ω k)⁻¹ α α' * ((tensorCount d m : ℝ)⁻¹ * ∑ ℓ : Fin d → Fin (m + 1), (treatedCount m j ω k ℓ : ℝ)⁻¹ * if (ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ then monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α' else 0)
theorem coefResidualWeight_zero_of_untreated Lemma coefResidualWeight_zero_of_untreated in the paper ↗
A coefficient residual weight vanishes for an untreated observation. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma coefResidualWeight_zero_of_untreated {d n : ℕ} (m j : ℕ)
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(α : MonoIndex d m) (i : Fin n)
(ha : (ω i).2.1 = false) :
coefResidualWeight m j ω k α i = 0 := by
simp [coefResidualWeight, ha]
Residual weights are fixed on each fibre of the covariate and arm design. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
ω ξ :
Fin n → Obs d
α :
MonoIndex d m
i :
Fin n
hdesign :
∀ i, (ξ i).1 = (ω i).1 ∧ (ξ i).2.1 = (ω i).2.1
conclusion
coefResidualWeight m j ξ k α i = coefResidualWeight m j ω k α i
Proof (Lean source)
lemma coefResidualWeight_eq_of_design_eq {d n : ℕ} (m j : ℕ)
(ω ξ : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(α : MonoIndex d m) (i : Fin n)
(hdesign : ∀ i, (ξ i).1 = (ω i).1 ∧ (ξ i).2.1 = (ω i).2.1) :
coefResidualWeight m j ξ k α i = coefResidualWeight m j ω k α i := by
unfold coefResidualWeight
rw [gramHat_eq_of_design_eq m j ω ξ k hdesign]
apply Finset.sum_congr rfl
intro α' _
apply congrArg (fun z : ℝ => (gramHat m j ω k)⁻¹ α α' *
((tensorCount d m : ℝ)⁻¹ * z))
apply Finset.sum_congr rfl
intro ℓ _
rw [treatedCount_eq_of_design_eq m j ω ξ k ℓ hdesign]
obtain ⟨hx, ha⟩ := hdesign i
simp [hx, ha]
theorem coefHat_sub_conditionalMeanSample_weighted_sum Lemma coefHat_sub_conditionalMeanSample_weighted_sum in the paper ↗
A centered coefficient is one observation-wise weighted residual sum. The weights depend only on the covariates and treatment arms. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma coefHat_sub_conditionalMeanSample_weighted_sum {d n : ℕ} (m j : ℕ)
(ω : Fin n → Obs d) (mean : Fin n → ℝ)
(k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) :
(coefHat m j ω k -
coefHat m j (fun i => ((ω i).1, (ω i).2.1, mean i)) k) α =
∑ i : Fin n, coefResidualWeight m j ω k α i *
((ω i).2.2 - mean i) := by
classical
rw [coefHat_sub_conditionalMeanSample_coordinate]
simp only [coefResidualWeight]
simp_rw [Finset.mul_sum, Finset.sum_mul]
simp only [mul_assoc, ite_mul, zero_mul]
conv_lhs =>
arg 2
ext α'
rw [Finset.sum_comm]
rw [Finset.sum_comm]
theorem coefHat_sub_conditionalMeanSample_weighted_sum_of_design_eq Lemma coefHat_sub_conditionalMeanSample_weighted_sum_of_design_eq in the paper ↗
On a design fibre, every residual coefficient uses the weights from the conditioning design, even when the outcomes vary. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
ω ξ :
Fin n → Obs d
mean :
Fin n → ℝ
α :
MonoIndex d m
hdesign :
∀ i, (ξ i).1 = (ω i).1 ∧ (ξ i).2.1 = (ω i).2.1
conclusion
= ∑ i : Fin n, coefResidualWeight m j ω k α i * ((ξ i).2.2 - mean i)
Proof (Lean source)
lemma coefHat_sub_conditionalMeanSample_weighted_sum_of_design_eq
{d n : ℕ} (m j : ℕ)
(ω ξ : Fin n → Obs d) (mean : Fin n → ℝ)
(k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m)
(hdesign : ∀ i, (ξ i).1 = (ω i).1 ∧ (ξ i).2.1 = (ω i).2.1) :
(coefHat m j ξ k -
coefHat m j (fun i => ((ξ i).1, (ξ i).2.1, mean i)) k) α =
∑ i : Fin n, coefResidualWeight m j ω k α i *
((ξ i).2.2 - mean i) := by
rw [coefHat_sub_conditionalMeanSample_weighted_sum]
apply Finset.sum_congr rfl
intro i _
rw [coefResidualWeight_eq_of_design_eq m j ω ξ k α i hdesign]
theorem treatedCell_weight_square_le_inverse_count Lemma treatedCell_weight_square_le_inverse_count in the paper ↗
Squared observation weights in one treated microcell cost one inverse count. This is the counting step in the conditional variance proxy. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
ω :
Fin n → Obs d
c :
Fin n → ℝ
R :
ℝ
hc :
∀ i
if
(ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ
then
|c i| ≤ R
conclusion
(∑ i : Fin n, (if (ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ then (treatedCount m j ω k ℓ : ℝ)⁻¹ * c i else 0) ^ 2)
≤ R ^ 2 * (treatedCount m j ω k ℓ : ℝ)⁻¹
Proof (Lean source)
lemma treatedCell_weight_square_le_inverse_count {d n : ℕ} (m j : ℕ)
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(ℓ : Fin d → Fin (m + 1)) (c : Fin n → ℝ) (R : ℝ)
(hc : ∀ i, (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ → |c i| ≤ R) :
(∑ i : Fin n,
(if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
(treatedCount m j ω k ℓ : ℝ)⁻¹ * c i else 0) ^ 2) ≤
R ^ 2 * (treatedCount m j ω k ℓ : ℝ)⁻¹ := by
classical
let S : Finset (Fin n) := Finset.univ.filter (fun i =>
(ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ)
have hcard : S.card = treatedCount m j ω k ℓ := rfl
by_cases hzero : S.card = 0
· have hcount0 : treatedCount m j ω k ℓ = 0 := hcard.symm.trans hzero
simp [hcount0]
· have hN : (0 : ℝ) < treatedCount m j ω k ℓ := by
exact_mod_cast Nat.pos_of_ne_zero (hcard ▸ hzero)
have hbound : ∀ i ∈ S, c i ^ 2 ≤ R ^ 2 := by
intro i hi
have hi' : (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ :=
(Finset.mem_filter.mp hi).2
nlinarith [abs_le.mp (hc i hi')]
have hsum :
(∑ i : Fin n,
(if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
(treatedCount m j ω k ℓ : ℝ)⁻¹ * c i else 0) ^ 2) =
∑ i ∈ S, ((treatedCount m j ω k ℓ : ℝ)⁻¹ * c i) ^ 2 := by
simp [S, Finset.sum_filter]
rw [hsum]
calc
(∑ i ∈ S, ((treatedCount m j ω k ℓ : ℝ)⁻¹ * c i) ^ 2) ≤
∑ _i ∈ S, ((treatedCount m j ω k ℓ : ℝ)⁻¹ * R) ^ 2 := by
apply Finset.sum_le_sum
intro i hi
have := hbound i hi
nlinarith [sq_nonneg (treatedCount m j ω k ℓ : ℝ)⁻¹]
_ = R ^ 2 * (treatedCount m j ω k ℓ : ℝ)⁻¹ := by
simp only [Finset.sum_const, nsmul_eq_mul]
rw [hcard]
field_simp
Collect the fixed-design coefficient weight by template cell. The matrix row and monomial vector form one scalar coefficient per observation. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
coefResidualWeight m j ω k α i
conclusion 2
(ω i).1 ∈ scaledMicroCell d m j k ℓ then ∑ α' : MonoIndex d m, (gramHat m j ω k)⁻¹ α α' * monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α' else 0
Proof (Lean source)
lemma coefResidualWeight_cell_sum {d n : ℕ} (m j : ℕ)
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(α : MonoIndex d m) (i : Fin n) :
coefResidualWeight m j ω k α i =
(tensorCount d m : ℝ)⁻¹ *
∑ ℓ : Fin d → Fin (m + 1),
(treatedCount m j ω k ℓ : ℝ)⁻¹ *
if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
∑ α' : MonoIndex d m,
(gramHat m j ω k)⁻¹ α α' *
monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α'
else 0 := by
classical
unfold coefResidualWeight
simp only [Finset.mul_sum, mul_ite, mul_zero]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro ℓ _
by_cases hc : (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ
· simp only [if_pos hc]
apply Finset.sum_congr rfl
intro α' _
ring
· simp [hc]
theorem treatedCell_combined_weight_square_le Lemma treatedCell_combined_weight_square_le in the paper ↗
A fixed finite number of template cells converts the cellwise inverse-count estimate into a bound for their combined observation weight. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
ω :
Fin n → Obs d
R :
ℝ
hc :
∀ ℓ i
if
(ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ
then
|c ℓ i| ≤ R
Proof (Lean source)
lemma treatedCell_combined_weight_square_le {d n : ℕ} (m j : ℕ)
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(c : (Fin d → Fin (m + 1)) → Fin n → ℝ) (R : ℝ)
(hc : ∀ ℓ i, (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ → |c ℓ i| ≤ R) :
(∑ i : Fin n,
(∑ ℓ : Fin d → Fin (m + 1),
if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
(treatedCount m j ω k ℓ : ℝ)⁻¹ * c ℓ i else 0) ^ 2) ≤
(tensorCount d m : ℝ) * R ^ 2 *
∑ ℓ : Fin d → Fin (m + 1),
(treatedCount m j ω k ℓ : ℝ)⁻¹ := by
classical
have hpoint (i : Fin n) :
(∑ ℓ : Fin d → Fin (m + 1),
if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
(treatedCount m j ω k ℓ : ℝ)⁻¹ * c ℓ i else 0) ^ 2 ≤
(tensorCount d m : ℝ) *
∑ ℓ : Fin d → Fin (m + 1),
(if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
(treatedCount m j ω k ℓ : ℝ)⁻¹ * c ℓ i else 0) ^ 2 := by
simpa [tensorCount, Fintype.card_fun, Fintype.card_fin] using
(sq_sum_le_card_mul_sum_sq (s := univ)
(f := fun ℓ : Fin d → Fin (m + 1) =>
if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
(treatedCount m j ω k ℓ : ℝ)⁻¹ * c ℓ i else 0))
calc
_ ≤ ∑ i : Fin n, (tensorCount d m : ℝ) *
∑ ℓ : Fin d → Fin (m + 1),
(if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
(treatedCount m j ω k ℓ : ℝ)⁻¹ * c ℓ i else 0) ^ 2 := by
apply Finset.sum_le_sum
intro i _
exact hpoint i
_ = (tensorCount d m : ℝ) *
∑ ℓ : Fin d → Fin (m + 1),
∑ i : Fin n,
(if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
(treatedCount m j ω k ℓ : ℝ)⁻¹ * c ℓ i else 0) ^ 2 := by
rw [← Finset.mul_sum, Finset.sum_comm]
_ ≤ (tensorCount d m : ℝ) *
∑ ℓ : Fin d → Fin (m + 1),
R ^ 2 * (treatedCount m j ω k ℓ : ℝ)⁻¹ := by
gcongr
exact treatedCell_weight_square_le_inverse_count m j ω k ℓ (c ℓ) R
(hc ℓ)
_ = _ := by rw [← Finset.mul_sum]; ring
A bounded inverse-Gram row evaluation gives the coefficient coordinate the inverse-count variance scale. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
ω :
Fin n → Obs d
α :
MonoIndex d m
R :
ℝ
hrow :
if
(ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ
then
|(tensorCount d m : ℝ)⁻¹ * ∑ α' : MonoIndex d m, (gramHat m j ω k)⁻¹ α α' * monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α'|
≤ R
conclusion
(∑ i : Fin n, (coefResidualWeight m j ω k α i) ^ 2)
Proof (Lean source)
lemma coefResidualWeight_square_sum_le {d n : ℕ} (m j : ℕ)
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(α : MonoIndex d m) (R : ℝ)
(hrow : ∀ (ℓ : Fin d → Fin (m + 1)) (i : Fin n),
(ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ →
|(tensorCount d m : ℝ)⁻¹ *
∑ α' : MonoIndex d m,
(gramHat m j ω k)⁻¹ α α' *
monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α'| ≤ R) :
(∑ i : Fin n, (coefResidualWeight m j ω k α i) ^ 2) ≤
(tensorCount d m : ℝ) * R ^ 2 *
∑ ℓ : Fin d → Fin (m + 1),
(treatedCount m j ω k ℓ : ℝ)⁻¹ := by
classical
let c : (Fin d → Fin (m + 1)) → Fin n → ℝ := fun _ℓ i =>
(tensorCount d m : ℝ)⁻¹ *
∑ α' : MonoIndex d m,
(gramHat m j ω k)⁻¹ α α' *
monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α'
have hrepr (i : Fin n) :
coefResidualWeight m j ω k α i =
∑ ℓ : Fin d → Fin (m + 1),
if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
(treatedCount m j ω k ℓ : ℝ)⁻¹ * c ℓ i else 0 := by
rw [coefResidualWeight_cell_sum, Finset.mul_sum]
apply Finset.sum_congr rfl
intro ℓ _
by_cases hc : (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ
· simp only [if_pos hc]
dsimp [c]
ring
· simp [hc]
simp_rw [hrepr]
exact treatedCell_combined_weight_square_le m j ω k c R
(fun ℓ i hi => hrow ℓ i hi)
theorem responseMoment_eq_gramHat_mulVec_of_polynomial Lemma responseMoment_eq_gramHat_mulVec_of_polynomial in the paper ↗
Equal-cell response moments exactly reproduce a polynomial in the local monomial basis. For the stated inputs and conditions, the asserted conclusion holds.
Proof (Lean source)
lemma responseMoment_eq_gramHat_mulVec_of_polynomial {d n : ℕ} (m j : ℕ)
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(θ : MonoIndex d m → ℝ)
(hresponse : ∀ i : Fin n, (ω i).2.1 = true →
(ω i).2.2 = ∑ α : MonoIndex d m,
monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α * θ α) :
responseMoment m j ω k = (gramHat m j ω k).mulVec θ := by
classical
funext α
change ((tensorCount d m : ℝ)⁻¹ *
∑ ℓ : Fin d → Fin (m + 1), (treatedCount m j ω k ℓ : ℝ)⁻¹ *
∑ i : Fin n, if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α *
(ω i).2.2 else 0) =
∑ α' : MonoIndex d m,
((tensorCount d m : ℝ)⁻¹ *
∑ ℓ : Fin d → Fin (m + 1), (treatedCount m j ω k ℓ : ℝ)⁻¹ *
∑ i : Fin n, if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α *
monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α'
else 0) * θ α'
calc
_ = (tensorCount d m : ℝ)⁻¹ *
∑ ℓ : Fin d → Fin (m + 1), (treatedCount m j ω k ℓ : ℝ)⁻¹ *
∑ i : Fin n, ∑ α' : MonoIndex d m,
if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α *
monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α' *
θ α' else 0 := by
congr 1
apply Finset.sum_congr rfl
intro ℓ _
congr 1
apply Finset.sum_congr rfl
intro i _
split_ifs with hi
· rw [hresponse i hi.1, Finset.mul_sum]
apply Finset.sum_congr rfl
intro α' _
ring
· simp
_ = _ := by
let F : (Fin d → Fin (m + 1)) → Fin n → MonoIndex d m → ℝ :=
fun ℓ i α' => if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α *
monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α'
else 0
have hrotate :
(∑ ℓ, ∑ i, ∑ α',
(tensorCount d m : ℝ)⁻¹ *
((treatedCount m j ω k ℓ : ℝ)⁻¹ * (F ℓ i α' * θ α'))) =
∑ α', ∑ ℓ, ∑ i,
(tensorCount d m : ℝ)⁻¹ *
((treatedCount m j ω k ℓ : ℝ)⁻¹ * (F ℓ i α' * θ α')) := by
calc
_ = ∑ i, ∑ ℓ, ∑ α',
(tensorCount d m : ℝ)⁻¹ *
((treatedCount m j ω k ℓ : ℝ)⁻¹ * (F ℓ i α' * θ α')) :=
Finset.sum_comm
_ = ∑ i, ∑ α', ∑ ℓ,
(tensorCount d m : ℝ)⁻¹ *
((treatedCount m j ω k ℓ : ℝ)⁻¹ * (F ℓ i α' * θ α')) := by
apply Finset.sum_congr rfl
intro i _
exact Finset.sum_comm
_ = ∑ α', ∑ i, ∑ ℓ,
(tensorCount d m : ℝ)⁻¹ *
((treatedCount m j ω k ℓ : ℝ)⁻¹ * (F ℓ i α' * θ α')) :=
Finset.sum_comm
_ = _ := by
apply Finset.sum_congr rfl
intro α' _
exact Finset.sum_comm
simp only [Finset.mul_sum, Finset.sum_mul, mul_ite, mul_zero, ite_mul]
simpa only [F, ite_mul, mul_ite, zero_mul, mul_zero, mul_assoc] using hrotate
On an occupied equal-cell design, fitting polynomial responses recovers their coefficient vector exactly. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma coefHat_eq_of_polynomial_response {d n : ℕ} (m j : ℕ)
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(θ : MonoIndex d m → ℝ)
(hcount : ∀ ℓ : Fin d → Fin (m + 1), 0 < treatedCount m j ω k ℓ)
(hresponse : ∀ i : Fin n, (ω i).2.1 = true →
(ω i).2.2 = ∑ α : MonoIndex d m,
monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α * θ α) :
coefHat m j ω k = θ := by
classical
rw [coefHat, responseMoment_eq_gramHat_mulVec_of_polynomial m j ω k θ hresponse,
Matrix.mulVec_mulVec]
have hdet : (gramHat m j ω k).det ≠ 0 :=
(gramHat_minEig_ge_of_pos d m n ω j k hcount).2.2.1
rw [Matrix.nonsing_inv_mul _ (isUnit_iff_ne_zero.mpr hdet), Matrix.one_mulVec]
Feasibility forces every treated template cell to be occupied. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma meshFeasible_treatedCount_pos {d n : ℕ} (m j : ℕ) (β : ℝ)
(ω : Fin n → Obs d) (hfeasible : meshFeasible m β ω j) :
∀ (k : Fin d → Fin (2 ^ j)) (ℓ : Fin d → Fin (m + 1)),
0 < treatedCount m j ω k ℓ := by
intro k ℓ
have hlower := hfeasible.2 k ℓ
have hwidth : 0 < meshWidth j := by unfold meshWidth; positivity
have hthreshold : 0 < meshWidth j ^ (-2 * β) := Real.rpow_pos_of_pos hwidth _
exact_mod_cast hthreshold.trans_le hlower
theorem mulVec_coordinate_le_finiteL2OpNorm Lemma mulVec_coordinate_le_finiteL2OpNorm in the paper ↗
A coordinate of a matrix-vector product is controlled by the finite Euclidean operator norm. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
|M.mulVec v i| ≤ finiteL2OpNorm M * finiteL2Norm v
Proof (Lean source)
lemma mulVec_coordinate_le_finiteL2OpNorm {ι : Type*}
[Fintype ι] [Nonempty ι]
(M : Matrix ι ι ℝ) (v : ι → ℝ) (i : ι) :
|M.mulVec v i| ≤ finiteL2OpNorm M * finiteL2Norm v := by
classical
have hsquare : (M.mulVec v i) ^ 2 ≤
∑ j : ι, (M.mulVec v j) ^ 2 := by
exact Finset.single_le_sum (fun j _ => sq_nonneg _) (Finset.mem_univ i)
have hcoord : |M.mulVec v i| ≤ finiteL2Norm (M.mulVec v) := by
rw [finiteL2Norm, ← Real.sqrt_sq_eq_abs]
exact Real.sqrt_le_sqrt hsquare
exact hcoord.trans (finiteL2Norm_mulVec_le_opNorm M v)
A uniform coordinate bound controls the Euclidean norm of a finite vector by the square root of its dimension. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma finiteL2Norm_le_sqrt_card_mul {ι : Type*} [Fintype ι]
(v : ι → ℝ) (r : ℝ) (hr : 0 ≤ r)
(hv : ∀ i, |v i| ≤ r) :
finiteL2Norm v ≤ sqrt (Fintype.card ι : ℝ) * r := by
classical
have hsq (i : ι) : v i ^ 2 ≤ r ^ 2 := by
rw [← sq_abs (v i), ← sq_abs r, abs_of_nonneg hr]
exact pow_le_pow_left₀ (abs_nonneg _) (hv i) 2
have hsum : (∑ i, v i ^ 2) ≤ (Fintype.card ι : ℝ) * r ^ 2 := by
calc
_ ≤ ∑ _i : ι, r ^ 2 := Finset.sum_le_sum fun i _ => hsq i
_ = (Fintype.card ι : ℝ) * r ^ 2 := by simp
have hsqrt := Real.sqrt_le_sqrt hsum
rw [← finiteL2Norm] at hsqrt
calc
finiteL2Norm v ≤ sqrt ((Fintype.card ι : ℝ) * r ^ 2) := hsqrt
_ = sqrt (Fintype.card ι : ℝ) * r := by
rw [Real.sqrt_mul (Nat.cast_nonneg _), Real.sqrt_sq_eq_abs, abs_of_nonneg hr]
theorem responseMoment_sub_coordinate_le_of_pointwise Lemma responseMoment_sub_coordinate_le_of_pointwise in the paper ↗
If every treated response perturbation entering a template cell is at most r, then every coordinate of the equal-cell response moment changes by at most r. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
ω ξ :
Fin n → Obs d
hdesign :
∀ i, (ξ i).1 = (ω i).1 ∧ (ξ i).2.1 = (ω i).2.1
r :
ℝ
hr :
0 ≤ r
hresponse :
if
(ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ
then
|(ξ i).2.2 - (ω i).2.2| ≤ r
α :
MonoIndex d m
conclusion
|(responseMoment m j ξ k - responseMoment m j ω k) α| ≤ r
Proof (Lean source)
lemma responseMoment_sub_coordinate_le_of_pointwise {d n : ℕ} (m j : ℕ)
(ω ξ : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(hdesign : ∀ i, (ξ i).1 = (ω i).1 ∧ (ξ i).2.1 = (ω i).2.1)
(hcount : ∀ ℓ : Fin d → Fin (m + 1), 0 < treatedCount m j ω k ℓ)
(r : ℝ) (hr : 0 ≤ r)
(hresponse : ∀ (ℓ : Fin d → Fin (m + 1)) (i : Fin n),
(ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ →
|(ξ i).2.2 - (ω i).2.2| ≤ r)
(α : MonoIndex d m) :
|(responseMoment m j ξ k - responseMoment m j ω k) α| ≤ r := by
classical
rw [responseMoment_sub_of_design_eq m j ω ξ k hdesign α]
have hJ : 0 < (tensorCount d m : ℝ) := by
dsimp [tensorCount]
positivity
have hcell (ℓ : Fin d → Fin (m + 1)) :
|(treatedCount m j ω k ℓ : ℝ)⁻¹ *
∑ i : Fin n, if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) /
meshWidth j) α * ((ξ i).2.2 - (ω i).2.2)
else 0| ≤ r := by
let z : Fin n → Fin d → ℝ := fun i a =>
((ω i).1 a - cubeCorner d j k a) / meshWidth j
have hterm (i : Fin n) :
|if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
monoVec d m (z i) α * ((ξ i).2.2 - (ω i).2.2)
else 0| ≤
if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then r else 0 := by
by_cases hi : (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ
· simp only [if_pos hi, abs_mul]
have hz : z i ∈ cube d := microCube_subset_cube d m ℓ hi.2
have hmono : |monoVec d m (z i) α| ≤ 1 := by
have hcoord : 0 ≤ monoVec d m (z i) α ∧ monoVec d m (z i) α ≤ 1 := by
unfold monoVec
constructor
· apply Finset.prod_nonneg
intro a ha
exact pow_nonneg ((Set.mem_univ_pi.mp hz a).1) _
· apply Finset.prod_le_one
· intro a ha
exact pow_nonneg ((Set.mem_univ_pi.mp hz a).1) _
· intro a ha
exact pow_le_one₀ ((Set.mem_univ_pi.mp hz a).1)
((Set.mem_univ_pi.mp hz a).2)
rw [abs_of_nonneg hcoord.1]
exact hcoord.2
simpa using
(mul_le_mul hmono (hresponse ℓ i hi) (abs_nonneg _) (by norm_num))
· simp [hi, hr]
have hsum :
|∑ i : Fin n, if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
monoVec d m (z i) α * ((ξ i).2.2 - (ω i).2.2)
else 0| ≤
∑ i : Fin n, if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then r else 0 :=
(Finset.abs_sum_le_sum_abs _ _).trans
(Finset.sum_le_sum fun i _ => hterm i)
have hweights :
(treatedCount m j ω k ℓ : ℝ)⁻¹ *
(∑ i : Fin n, if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then r else 0) = r := by
have hw := treatedCount_inv_weight_sum m j ω k ℓ (hcount ℓ)
unfold treatedCellWeightSum at hw
calc
_ = ((treatedCount m j ω k ℓ : ℝ)⁻¹ *
∑ i : Fin n, if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then 1 else 0) * r := by
rw [mul_assoc]
congr 1
rw [Finset.sum_mul]
apply Finset.sum_congr rfl
intro i hi
split_ifs <;> ring
_ = r := by rw [hw, one_mul]
rw [abs_mul, abs_of_nonneg (by positivity : 0 ≤ (treatedCount m j ω k ℓ : ℝ)⁻¹)]
exact (mul_le_mul_of_nonneg_left hsum (by positivity)).trans_eq hweights
calc
|(tensorCount d m : ℝ)⁻¹ * ∑ ℓ, _| =
(tensorCount d m : ℝ)⁻¹ * |∑ ℓ, _| := by
rw [abs_mul, abs_of_nonneg (by positivity)]
_ ≤ (tensorCount d m : ℝ)⁻¹ * ∑ _ℓ : Fin d → Fin (m + 1), r := by
gcongr
exact (Finset.abs_sum_le_sum_abs _ _).trans
(Finset.sum_le_sum fun ℓ _ => hcell ℓ)
_ = r := by
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fun,
Fintype.card_fin, nsmul_eq_mul, tensorCount]
field_simp
theorem coefHat_sub_coordinate_le_of_responseMoment Lemma coefHat_sub_coordinate_le_of_responseMoment in the paper ↗
On an occupied equal-cell design, a coordinatewise response-moment perturbation is stable under inversion of the empirical Gram matrix. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma coefHat_sub_coordinate_le_of_responseMoment {d n : ℕ} (m j : ℕ)
(ω ξ : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(hdesign : ∀ i, (ξ i).1 = (ω i).1 ∧ (ξ i).2.1 = (ω i).2.1)
(hcount : ∀ ℓ : Fin d → Fin (m + 1), 0 < treatedCount m j ω k ℓ)
(r : ℝ) (hr : 0 ≤ r)
(hmoment : ∀ α : MonoIndex d m,
|(responseMoment m j ξ k - responseMoment m j ω k) α| ≤ r)
(α : MonoIndex d m) :
|(coefHat m j ξ k - coefHat m j ω k) α| ≤
(2 / templateLambda d m) *
(sqrt (card (MonoIndex d m) : ℝ) * r) := by
classical
letI : Nonempty (MonoIndex d m) :=
⟨⟨fun _ => 0, by simp [monoIdx]⟩⟩
rw [coefHat_sub_of_design_eq m j ω ξ k hdesign]
refine (mulVec_coordinate_le_finiteL2OpNorm
(gramHat m j ω k)⁻¹
(responseMoment m j ξ k - responseMoment m j ω k) α).trans ?_
have hop := (gramHat_minEig_ge_of_pos d m n ω j k hcount).2.2.2
have hnorm := finiteL2Norm_le_sqrt_card_mul
(responseMoment m j ξ k - responseMoment m j ω k) r hr hmoment
have hfac : 0 ≤ 2 / templateLambda d m :=
(div_pos (by norm_num) (templateLambda_pos d m)).le
exact (mul_le_mul hop hnorm (by exact Real.sqrt_nonneg _) hfac).trans_eq
(by ring)
theorem coefHat_sub_coordinate_le_of_pointwise Lemma coefHat_sub_coordinate_le_of_pointwise in the paper ↗
Pointwise treated-response approximation on every occupied microcell implies a dimension-explicit bound for each fitted coefficient. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
ω ξ :
Fin n → Obs d
hdesign :
∀ i, (ξ i).1 = (ω i).1 ∧ (ξ i).2.1 = (ω i).2.1
r :
ℝ
hr :
0 ≤ r
hresponse :
if
(ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ
then
|(ξ i).2.2 - (ω i).2.2| ≤ r
α :
MonoIndex d m
Proof (Lean source)
lemma coefHat_sub_coordinate_le_of_pointwise {d n : ℕ} (m j : ℕ)
(ω ξ : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(hdesign : ∀ i, (ξ i).1 = (ω i).1 ∧ (ξ i).2.1 = (ω i).2.1)
(hcount : ∀ ℓ : Fin d → Fin (m + 1), 0 < treatedCount m j ω k ℓ)
(r : ℝ) (hr : 0 ≤ r)
(hresponse : ∀ (ℓ : Fin d → Fin (m + 1)) (i : Fin n),
(ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ →
|(ξ i).2.2 - (ω i).2.2| ≤ r)
(α : MonoIndex d m) :
|(coefHat m j ξ k - coefHat m j ω k) α| ≤
(2 / templateLambda d m) *
(sqrt (card (MonoIndex d m) : ℝ) * r) := by
apply coefHat_sub_coordinate_le_of_responseMoment m j ω ξ k hdesign hcount r hr
intro α'
exact responseMoment_sub_coordinate_le_of_pointwise
m j ω ξ k hdesign hcount r hr hresponse α'
theorem coefHat_sub_polynomial_coordinate_le Lemma coefHat_sub_polynomial_coordinate_le in the paper ↗
An occupied equal-cell fit is stable around any local polynomial whose values uniformly approximate the treated responses entering its microcells. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
ω :
Fin n → Obs d
θ :
MonoIndex d m → ℝ
r :
ℝ
hr :
0 ≤ r
hresponse :
if
(ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ
then
|(ω i).2.2
- ∑ α : MonoIndex d m, monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α * θ α|
≤ r
α :
MonoIndex d m
conclusion
|coefHat m j ω k α - θ α|
Proof (Lean source)
lemma coefHat_sub_polynomial_coordinate_le {d n : ℕ} (m j : ℕ)
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(θ : MonoIndex d m → ℝ)
(hcount : ∀ ℓ : Fin d → Fin (m + 1), 0 < treatedCount m j ω k ℓ)
(r : ℝ) (hr : 0 ≤ r)
(hresponse : ∀ (ℓ : Fin d → Fin (m + 1)) (i : Fin n),
(ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ →
|(ω i).2.2 - ∑ α : MonoIndex d m,
monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α * θ α| ≤ r)
(α : MonoIndex d m) :
|coefHat m j ω k α - θ α| ≤
(2 / templateLambda d m) *
(sqrt (card (MonoIndex d m) : ℝ) * r) := by
classical
let ξ : Fin n → Obs d := fun i =>
((ω i).1, (ω i).2.1,
∑ α : MonoIndex d m,
monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α * θ α)
have hdesign : ∀ i, (ω i).1 = (ξ i).1 ∧ (ω i).2.1 = (ξ i).2.1 := by
intro i
exact ⟨rfl, rfl⟩
have hcountξ : ∀ ℓ : Fin d → Fin (m + 1), 0 < treatedCount m j ξ k ℓ := by
intro ℓ
rw [treatedCount_eq_of_design_eq m j ω ξ k ℓ]
· exact hcount ℓ
· intro i
exact ⟨rfl, rfl⟩
have hpoly : coefHat m j ξ k = θ := by
apply coefHat_eq_of_polynomial_response m j ξ k θ
· exact hcountξ
· intro i hi
rfl
have hpert := coefHat_sub_coordinate_le_of_pointwise
m j ξ ω k hdesign hcountξ r hr
(fun ℓ i hi => by simpa only [ξ, abs_sub_comm] using hresponse ℓ i hi) α
simpa only [Pi.sub_apply, hpoly] using hpert
Conditioning of the equal-cell Gram matrix bounds each evaluation of an inverse-Gram row at a normalized covariate. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
ω :
Fin n → Obs d
α :
MonoIndex d m
z :
Fin d → ℝ
V :
ℝ
hV :
finiteL2Norm (monoVec d m z) ≤ V
conclusion
≤ (tensorCount d m : ℝ)⁻¹ * ((2 / templateLambda d m) * V)
Proof (Lean source)
lemma inverseGram_row_monoVec_le {d n : ℕ} (m j : ℕ)
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(α : MonoIndex d m) (z : Fin d → ℝ) (V : ℝ)
(hcount : ∀ ℓ : Fin d → Fin (m + 1), 0 < treatedCount m j ω k ℓ)
(hV : finiteL2Norm (monoVec d m z) ≤ V) :
|(tensorCount d m : ℝ)⁻¹ *
∑ α' : MonoIndex d m,
(gramHat m j ω k)⁻¹ α α' * monoVec d m z α'| ≤
(tensorCount d m : ℝ)⁻¹ * ((2 / templateLambda d m) * V) := by
classical
have : Nonempty (MonoIndex d m) :=
⟨⟨fun _ => 0, by simp [monoIdx]⟩⟩
have hJ : 0 ≤ (tensorCount d m : ℝ)⁻¹ := by positivity
have hOp := (gramHat_minEig_ge_of_pos d m n ω j k hcount).2.2.2
have hcoord := mulVec_coordinate_le_finiteL2OpNorm
(gramHat m j ω k)⁻¹ (monoVec d m z) α
have hprod :
finiteL2OpNorm ((gramHat m j ω k)⁻¹) *
finiteL2Norm (monoVec d m z) ≤
(2 / templateLambda d m) * V := by
have hoppos := finiteL2OpNorm_nonneg ((gramHat m j ω k)⁻¹)
have hvpos : 0 ≤ finiteL2Norm (monoVec d m z) := Real.sqrt_nonneg _
exact (mul_le_mul_of_nonneg_left hV hoppos).trans
(mul_le_mul_of_nonneg_right hOp (le_trans hvpos hV))
simp only [mulVec, dotProduct] at hcoord
rw [abs_mul, abs_of_nonneg hJ]
exact mul_le_mul_of_nonneg_left (hcoord.trans hprod) hJ
On the unit cube, every monomial coordinate has magnitude at most one, so the whole template vector has norm at most the square root of its size. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma monoVec_finiteL2Norm_le_dimension (d m : ℕ)
(z : Fin d → ℝ) (hz : z ∈ cube d) :
finiteL2Norm (monoVec d m z) ≤
sqrt (card (MonoIndex d m) : ℝ) := by
classical
have hcoord (α : MonoIndex d m) : 0 ≤ monoVec d m z α ∧
monoVec d m z α ≤ 1 := by
unfold monoVec
constructor
· apply Finset.prod_nonneg
intro i hi
exact pow_nonneg ((Set.mem_univ_pi.mp hz i).1) _
· apply Finset.prod_le_one
· intro i hi
exact pow_nonneg ((Set.mem_univ_pi.mp hz i).1) _
· intro i hi
exact pow_le_one₀ ((Set.mem_univ_pi.mp hz i).1) ((Set.mem_univ_pi.mp hz i).2)
have hsum : (∑ α : MonoIndex d m, monoVec d m z α ^ 2) ≤
(card (MonoIndex d m) : ℝ) := by
calc
_ ≤ ∑ _α : MonoIndex d m, (1 : ℝ) := by
apply Finset.sum_le_sum
intro α _
nlinarith [hcoord α]
_ = _ := by simp
exact Real.sqrt_le_sqrt hsum
Clipping to the outcome range cannot increase error relative to a target already in that range. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma selectedMesh_clip_error_le {B u y : ℝ} (hB : 0 ≤ B) (hy : |y| ≤ B) :
|max (-B) (min B u) - y| ≤ |u - y| := by
have hylo : -B ≤ y := (abs_le.mp hy).1
have hyhi : y ≤ B := (abs_le.mp hy).2
by_cases huhi : u ≤ B
· rw [min_eq_right huhi]
by_cases hulo : -B ≤ u
· rw [max_eq_right hulo]
· rw [max_eq_left (le_of_not_ge hulo)]
rw [abs_of_nonpos (sub_nonpos.mpr hylo)]
have huy : u ≤ y := (lt_of_not_ge hulo).le.trans hylo
rw [abs_of_nonpos (sub_nonpos.mpr huy)]
linarith
· rw [min_eq_left (le_of_not_ge huhi), max_eq_right (by linarith)]
rw [abs_of_nonneg (sub_nonneg.mpr hyhi)]
have hyu : y ≤ u := hyhi.trans (le_of_lt (lt_of_not_ge huhi))
rw [abs_of_nonneg (sub_nonneg.mpr hyu)]
linarith
A uniform coordinate error in local polynomial coefficients yields a uniform evaluation error on the normalized unit cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma monoVec_evaluation_sub_le_card_mul {d m : ℕ}
(z : Fin d → ℝ) (hz : z ∈ cube d)
(a b : MonoIndex d m → ℝ) (ε : ℝ)
(hε : ∀ α, |a α - b α| ≤ ε) :
|(∑ α, monoVec d m z α * a α) -
∑ α, monoVec d m z α * b α| ≤
(card (MonoIndex d m) : ℝ) * ε := by
classical
have hmono (α : MonoIndex d m) : |monoVec d m z α| ≤ 1 := by
have hcoord : 0 ≤ monoVec d m z α ∧ monoVec d m z α ≤ 1 := by
unfold monoVec
constructor
· apply Finset.prod_nonneg
intro i hi
exact pow_nonneg ((Set.mem_univ_pi.mp hz i).1) _
· apply Finset.prod_le_one
· intro i hi
exact pow_nonneg ((Set.mem_univ_pi.mp hz i).1) _
· intro i hi
exact pow_le_one₀ ((Set.mem_univ_pi.mp hz i).1)
((Set.mem_univ_pi.mp hz i).2)
rw [abs_of_nonneg hcoord.1]
exact hcoord.2
rw [← Finset.sum_sub_distrib]
calc
|∑ α, (monoVec d m z α * a α - monoVec d m z α * b α)|
= |∑ α, monoVec d m z α * (a α - b α)| := by
congr 1
apply Finset.sum_congr rfl
intro α _
ring
_ ≤ ∑ α, |monoVec d m z α * (a α - b α)| := Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ _α : MonoIndex d m, ε := by
apply Finset.sum_le_sum
intro α _
rw [abs_mul]
calc
|monoVec d m z α| * |a α - b α| ≤ 1 * ε :=
mul_le_mul (hmono α) (hε α) (abs_nonneg _) zero_le_one
_ = ε := one_mul ε
_ = (card (MonoIndex d m) : ℝ) * ε := by simp
theorem clipped_monoVec_error_le_card_mul_add Lemma clipped_monoVec_error_le_card_mul_add in the paper ↗
Combining coefficient control with clipping leaves only the deterministic local-polynomial approximation error. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma clipped_monoVec_error_le_card_mul_add {d m : ℕ} {B y ε : ℝ}
(z : Fin d → ℝ) (hz : z ∈ cube d)
(a b : MonoIndex d m → ℝ) (hB : 0 ≤ B) (hy : |y| ≤ B)
(hε : ∀ α, |a α - b α| ≤ ε) :
|max (-B) (min B (∑ α, monoVec d m z α * a α)) - y| ≤
(card (MonoIndex d m) : ℝ) * ε +
|(∑ α, monoVec d m z α * b α) - y| := by
have hclip := selectedMesh_clip_error_le hB hy
(u := ∑ α, monoVec d m z α * a α)
have hsplit :
|(∑ α, monoVec d m z α * a α) - y| ≤
|(∑ α, monoVec d m z α * a α) -
∑ α, monoVec d m z α * b α| +
|(∑ α, monoVec d m z α * b α) - y| := by
rw [show (∑ α, monoVec d m z α * a α) - y =
((∑ α, monoVec d m z α * a α) -
∑ α, monoVec d m z α * b α) +
((∑ α, monoVec d m z α * b α) - y) by ring]
exact abs_add_le _ _
exact hclip.trans (hsplit.trans (add_le_add
(monoVec_evaluation_sub_le_card_mul z hz a b ε hε) (le_refl _)))
theorem equalCellEstimator_error_le_of_selected_coefficients Lemma equalCellEstimator_error_le_of_selected_coefficients in the paper ↗
At a positive selected mesh, coefficient control plus a local polynomial approximation controls the actual clipped estimator at a fixed point. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B y ε δ :
ℝ
ω :
Fin n → Obs d
x :
Fin d → ℝ
θ :
MonoIndex d (polynomialDegree β) → ℝ
hselected :
0 < countSelectedMesh (polynomialDegree β) β ω
hz :
(fun a => (x a - cubeCorner d ((feasibleIndices (polynomialDegree β) β ω).sup id) (cubeIndex d ((feasibleIndices (polynomialDegree β) β ω).sup id) x) a) / countSelectedMesh (polynomialDegree β) β ω) ∈ cube d
hB :
0 ≤ B
hy :
|y| ≤ B
hcoef :
∀ α,
|coefHat (polynomialDegree β) ((feasibleIndices (polynomialDegree β) β ω).sup id) ω (cubeIndex d ((feasibleIndices (polynomialDegree β) β ω).sup id) x) α
- θ α|
≤ ε
hbias :
|(∑ α, monoVec d (polynomialDegree β) (fun a => (x a - cubeCorner d ((feasibleIndices (polynomialDegree β) β ω).sup id) (cubeIndex d ((feasibleIndices (polynomialDegree β) β ω).sup id) x) a) / countSelectedMesh (polynomialDegree β) β ω) α * θ α)
- y|
≤ δ
conclusion
|equalCellEstimator β B ω x - y|
Proof (Lean source)
lemma equalCellEstimator_error_le_of_selected_coefficients {d n : ℕ}
{β B y ε δ : ℝ} (ω : Fin n → Obs d) (x : Fin d → ℝ)
(θ : MonoIndex d (polynomialDegree β) → ℝ)
(hselected : 0 < countSelectedMesh (polynomialDegree β) β ω)
(hz : (fun a =>
(x a - cubeCorner d ((feasibleIndices (polynomialDegree β) β ω).sup id)
(cubeIndex d ((feasibleIndices (polynomialDegree β) β ω).sup id) x) a) /
countSelectedMesh (polynomialDegree β) β ω) ∈ cube d)
(hB : 0 ≤ B) (hy : |y| ≤ B)
(hcoef : ∀ α,
|coefHat (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) ω
(cubeIndex d ((feasibleIndices (polynomialDegree β) β ω).sup id) x) α - θ α| ≤ ε)
(hbias :
|(∑ α,
monoVec d (polynomialDegree β)
(fun a =>
(x a - cubeCorner d
((feasibleIndices (polynomialDegree β) β ω).sup id)
(cubeIndex d ((feasibleIndices (polynomialDegree β) β ω).sup id) x) a) /
countSelectedMesh (polynomialDegree β) β ω) α * θ α) - y| ≤ δ) :
|equalCellEstimator β B ω x - y| ≤
(card (MonoIndex d (polynomialDegree β)) : ℝ) * ε + δ := by
let j := (feasibleIndices (polynomialDegree β) β ω).sup id
let k := cubeIndex d j x
let z : Fin d → ℝ := fun a =>
(x a - cubeCorner d j k a) /
countSelectedMesh (polynomialDegree β) β ω
have hne : countSelectedMesh (polynomialDegree β) β ω ≠ 0 := ne_of_gt hselected
have hbase := clipped_monoVec_error_le_card_mul_add z
(by simpa [z, j, k] using hz)
(coefHat (polynomialDegree β) j ω k) θ hB hy
(by simpa [j, k] using hcoef)
unfold equalCellEstimator
simp only [hne, ↓reduceIte]
change |max (-B) (min B
(∑ α, monoVec d (polynomialDegree β) z α *
coefHat (polynomialDegree β) j ω k α)) - y| ≤ _
exact hbase.trans (add_le_add_right (by simpa [z, j, k] using hbias) _)
theorem inverseGram_row_bound_on_scaledMicroCell Lemma inverseGram_row_bound_on_scaledMicroCell in the paper ↗
Every occupied template cell has the same dimension-only inverse-Gram row bound. This supplies the row premise of the squared-weight estimate. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
ω :
Fin n → Obs d
α :
MonoIndex d m
i :
Fin n
(ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ
conclusion
|(tensorCount d m : ℝ)⁻¹ * ∑ α' : MonoIndex d m, (gramHat m j ω k)⁻¹ α α' * monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α'|
Proof (Lean source)
lemma inverseGram_row_bound_on_scaledMicroCell {d n : ℕ} (m j : ℕ)
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(α : MonoIndex d m)
(hcount : ∀ ℓ : Fin d → Fin (m + 1), 0 < treatedCount m j ω k ℓ) :
∀ (ℓ : Fin d → Fin (m + 1)) (i : Fin n),
(ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ →
|(tensorCount d m : ℝ)⁻¹ *
∑ α' : MonoIndex d m,
(gramHat m j ω k)⁻¹ α α' *
monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α'| ≤
(tensorCount d m : ℝ)⁻¹ *
((2 / templateLambda d m) *
sqrt (card (MonoIndex d m) : ℝ)) := by
intro ℓ i hi
apply inverseGram_row_monoVec_le m j ω k α _ _ hcount
exact monoVec_finiteL2Norm_le_dimension d m _
(microCube_subset_cube d m ℓ hi.2)
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.SelectedMesh.Counts 28 declarations Oracle witness and simultaneous selected-count event
Oracle witness and simultaneous selected-count event
A single event controls treated counts over every candidate dyadic mesh and every template cell. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B L C c_f γ :
ℝ
P :
Measure (Obs d)
hP :
P ∈ ModelClass d β B L C c_f γ
hIID :
IIDSampleLaw Q P
hn :
0 < n
s :
ℝ
hs :
0 < s
conclusion
conclusion 1
|(treatedCount (polynomialDegree β) j ω k ℓ : ℝ)
- (n : ℝ) * microcellMass P (polynomialDegree β) j k ℓ| < (n : ℝ) * microcellMass P (polynomialDegree β) j k ℓ / 2
+ 4 * s
Proof (Lean source)
lemma model_all_treatedCount_deviation {d n : ℕ} {β B L C c_f γ : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(Q : Measure (Fin n → Obs d)) (hIID : IIDSampleLaw Q P)
(hn : 0 < n) (s : ℝ) (hs : 0 < s) :
∃ E : Set (Fin n → Obs d), MeasurableSet E ∧
Q.real Eᶜ ≤ 2 * (card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) ×
(Fin d → Fin (polynomialDegree β + 1))) : ℝ) * exp (-s) ∧
∀ ω ∈ E, ∀ j ∈ meshIndices d n β,
∀ (k : Fin d → Fin (2 ^ j))
(ℓ : Fin d → Fin (polynomialDegree β + 1)),
|(treatedCount (polynomialDegree β) j ω k ℓ : ℝ) -
(n : ℝ) * microcellMass P (polynomialDegree β) j k ℓ| <
(n : ℝ) * microcellMass P (polynomialDegree β) j k ℓ / 2 + 4 * s := by
classical
let ι := Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) × (Fin d → Fin (polynomialDegree β + 1))
let jf : ι → ℕ := fun a => a.1.val
let kf : ∀ a : ι, Fin d → Fin (2 ^ jf a) := fun a => a.2.1
let ℓf : ∀ _a : ι, Fin d → Fin (polynomialDegree β + 1) := fun a => a.2.2
obtain ⟨E, hE, hbad, hdev⟩ :=
finiteFamily_treatedCount_event P Q hIID hn (polynomialDegree β)
jf kf ℓf (fun a => model_microcellMass_pos P hP _ _ _ _) s hs
refine ⟨E, hE, hbad, ?_⟩
intro ω hω j hj k ℓ
let a : ι := ⟨⟨j, hj⟩, (k, ℓ)⟩
simpa [jf, kf, ℓf, a] using hdev ω hω a
theorem model_all_treatedCount_deviation_design_saturated Lemma model_all_treatedCount_deviation_design_saturated in the paper ↗
The all-candidate count deviation event can be retained together with its saturation under equality of covariates and treatment arms. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B L C c_f γ :
ℝ
P :
Measure (Obs d)
hP :
P ∈ ModelClass d β B L C c_f γ
hIID :
IIDSampleLaw Q P
hn :
0 < n
s :
ℝ
hs :
0 < s
conclusion
conclusion 1
|(treatedCount (polynomialDegree β) j ω k ℓ : ℝ)
- (n : ℝ) * microcellMass P (polynomialDegree β) j k ℓ| < (n : ℝ) * microcellMass P (polynomialDegree β) j k ℓ / 2
+ 4 * s
conclusion 2
ω ξ :
(fun i => ((ξ i).1, (ξ i).2.1)) = (fun i => ((ω i).1, (ω i).2.1))
ω ∈ E ↔ ξ ∈ E
Proof (Lean source)
lemma model_all_treatedCount_deviation_design_saturated
{d n : ℕ} {β B L C c_f γ : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(Q : Measure (Fin n → Obs d)) (hIID : IIDSampleLaw Q P)
(hn : 0 < n) (s : ℝ) (hs : 0 < s) :
∃ E : Set (Fin n → Obs d), MeasurableSet E ∧
Q.real Eᶜ ≤ 2 * (card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) ×
(Fin d → Fin (polynomialDegree β + 1))) : ℝ) * exp (-s) ∧
(∀ ω ∈ E, ∀ j ∈ meshIndices d n β,
∀ (k : Fin d → Fin (2 ^ j))
(ℓ : Fin d → Fin (polynomialDegree β + 1)),
|(treatedCount (polynomialDegree β) j ω k ℓ : ℝ) -
(n : ℝ) * microcellMass P (polynomialDegree β) j k ℓ| <
(n : ℝ) * microcellMass P (polynomialDegree β) j k ℓ / 2 + 4 * s) ∧
∀ ω ξ, (fun i => ((ξ i).1, (ξ i).2.1)) =
(fun i => ((ω i).1, (ω i).2.1)) → (ω ∈ E ↔ ξ ∈ E) := by
classical
let ι := Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) × (Fin d → Fin (polynomialDegree β + 1))
let jf : ι → ℕ := fun a => a.1.val
let kf : ∀ a : ι, Fin d → Fin (2 ^ jf a) := fun a => a.2.1
let ℓf : ∀ _a : ι, Fin d → Fin (polynomialDegree β + 1) := fun a => a.2.2
obtain ⟨E, hE, hbad, hdev, hsat⟩ :=
finiteFamily_treatedCount_event_design_saturated P Q hIID hn
(polynomialDegree β) jf kf ℓf
(fun a => model_microcellMass_pos P hP _ _ _ _) s hs
refine ⟨E, hE, hbad, ?_, hsat⟩
intro ω hω j hj k ℓ
let a : ι := ⟨⟨j, hj⟩, (k, ℓ)⟩
simpa [jf, kf, ℓf, a] using hdev ω hω a
theorem selectedCountEvent_of_oracle_witness Lemma selectedCountEvent_of_oracle_witness in the paper ↗
The simultaneous deviation event gives the selected-mesh count envelope once an oracle-scale candidate satisfies the deterministic mass and budget bounds. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B L C c_f γ :
ℝ
P :
Measure (Obs d)
hP :
P ∈ ModelClass d β B L C c_f γ
hIID :
IIDSampleLaw Q P
hn :
0 < n
hβ :
0 ≤ β
s K :
ℝ
hs :
0 < s
hfailure :
2 * (card (Σ j : {j // j ∈ meshIndices d n β}, (Fin d → Fin (2 ^ j.val)) × (Fin d → Fin (polynomialDegree β + 1))) : ℝ) * exp (-s)
≤ (n : ℝ) ^ (-2 : ℝ)
j₀ :
ℕ
hcandidate :
j₀ ∈ meshIndices d n β
hwidth :
meshWidth j₀ ≤ K * oracleMesh d n β γ
hbudget :
24 * s ≤ meshWidth j₀ ^ (-2 * β)
hmass :
conclusion
Q.real Eᶜ ≤ (n : ℝ) ^ (-2 : ℝ)
conclusion 1
ω ∈ E :
0 < countSelectedMesh (polynomialDegree β) β ω
k :
ℓ :
(n : ℝ) * microcellMass P (polynomialDegree β) ((feasibleIndices (polynomialDegree β) β ω).sup id) k ℓ / 5
Proof (Lean source)
lemma selectedCountEvent_of_oracle_witness {d n : ℕ} {β B L C c_f γ : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(Q : Measure (Fin n → Obs d)) (hIID : IIDSampleLaw Q P)
(hn : 0 < n) (hβ : 0 ≤ β) (s K : ℝ) (hs : 0 < s)
(hfailure : 2 * (card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) ×
(Fin d → Fin (polynomialDegree β + 1))) : ℝ) *
exp (-s) ≤ (n : ℝ) ^ (-2 : ℝ))
(j₀ : ℕ) (hcandidate : j₀ ∈ meshIndices d n β)
(hwidth : meshWidth j₀ ≤ K * oracleMesh d n β γ)
(hbudget : 24 * s ≤ meshWidth j₀ ^ (-2 * β))
(hmass : ∀ (k : Fin d → Fin (2 ^ j₀))
(ℓ : Fin d → Fin (polynomialDegree β + 1)),
4 * meshWidth j₀ ^ (-2 * β) ≤
(n : ℝ) * microcellMass P (polynomialDegree β) j₀ k ℓ) :
∃ E : Set (Fin n → Obs d),
MeasurableSet E ∧ Q.real Eᶜ ≤ (n : ℝ) ^ (-2 : ℝ) ∧
∀ ω ∈ E,
0 < countSelectedMesh (polynomialDegree β) β ω ∧
countSelectedMesh (polynomialDegree β) β ω ≤ K * oracleMesh d n β γ ∧
∀ (k : Fin d → Fin (2 ^ ((feasibleIndices (polynomialDegree β) β ω).sup id)))
(ℓ : Fin d → Fin (polynomialDegree β + 1)),
(n : ℝ) * microcellMass P (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) k ℓ / 5 ≤
treatedCount (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) ω k ℓ := by
obtain ⟨E, hE, hbad, hdev⟩ :=
model_all_treatedCount_deviation P hP Q hIID hn s hs
refine ⟨E, hE, hbad.trans hfailure, ?_⟩
intro ω hω
have hwitness : meshFeasible (polynomialDegree β) β ω j₀ :=
oracleWitness_feasible_of_deviation (polynomialDegree β) j₀ β s P ω
hcandidate (by linarith) hmass
(fun k ℓ => (hdev ω hω j₀ hcandidate k ℓ).le)
exact selectedCount_deterministic_bridge (polynomialDegree β) β s
(K * oracleMesh d n β γ) P ω j₀ hwitness hwidth hβ hbudget
hs.le (fun j hj k ℓ => (hdev ω hω j hj k ℓ).le)
theorem selectedCountEvent_of_oracle_witness_design_saturated Lemma selectedCountEvent_of_oracle_witness_design_saturated in the paper ↗
The oracle selected-count event can be chosen design-saturated. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B L C c_f γ :
ℝ
P :
Measure (Obs d)
hP :
P ∈ ModelClass d β B L C c_f γ
hIID :
IIDSampleLaw Q P
hn :
0 < n
hβ :
0 ≤ β
s K :
ℝ
hs :
0 < s
hfailure :
2 * (card (Σ j : {j // j ∈ meshIndices d n β}, (Fin d → Fin (2 ^ j.val)) × (Fin d → Fin (polynomialDegree β + 1))) : ℝ) * exp (-s)
≤ (n : ℝ) ^ (-2 : ℝ)
j₀ :
ℕ
hcandidate :
j₀ ∈ meshIndices d n β
hwidth :
meshWidth j₀ ≤ K * oracleMesh d n β γ
hbudget :
24 * s ≤ meshWidth j₀ ^ (-2 * β)
hmass :
conclusion
Q.real Eᶜ ≤ (n : ℝ) ^ (-2 : ℝ)
conclusion 1
ω ∈ E :
0 < countSelectedMesh (polynomialDegree β) β ω
k :
ℓ :
(n : ℝ) * microcellMass P (polynomialDegree β) ((feasibleIndices (polynomialDegree β) β ω).sup id) k ℓ / 5
conclusion 2
ω ξ :
(fun i => ((ξ i).1, (ξ i).2.1)) = (fun i => ((ω i).1, (ω i).2.1))
ω ∈ E ↔ ξ ∈ E
Proof (Lean source)
lemma selectedCountEvent_of_oracle_witness_design_saturated
{d n : ℕ} {β B L C c_f γ : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(Q : Measure (Fin n → Obs d)) (hIID : IIDSampleLaw Q P)
(hn : 0 < n) (hβ : 0 ≤ β) (s K : ℝ) (hs : 0 < s)
(hfailure : 2 * (card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) ×
(Fin d → Fin (polynomialDegree β + 1))) : ℝ) *
exp (-s) ≤ (n : ℝ) ^ (-2 : ℝ))
(j₀ : ℕ) (hcandidate : j₀ ∈ meshIndices d n β)
(hwidth : meshWidth j₀ ≤ K * oracleMesh d n β γ)
(hbudget : 24 * s ≤ meshWidth j₀ ^ (-2 * β))
(hmass : ∀ (k : Fin d → Fin (2 ^ j₀))
(ℓ : Fin d → Fin (polynomialDegree β + 1)),
4 * meshWidth j₀ ^ (-2 * β) ≤
(n : ℝ) * microcellMass P (polynomialDegree β) j₀ k ℓ) :
∃ E : Set (Fin n → Obs d),
MeasurableSet E ∧ Q.real Eᶜ ≤ (n : ℝ) ^ (-2 : ℝ) ∧
(∀ ω ∈ E,
0 < countSelectedMesh (polynomialDegree β) β ω ∧
countSelectedMesh (polynomialDegree β) β ω ≤
K * oracleMesh d n β γ ∧
∀ (k : Fin d → Fin
(2 ^ ((feasibleIndices (polynomialDegree β) β ω).sup id)))
(ℓ : Fin d → Fin (polynomialDegree β + 1)),
(n : ℝ) * microcellMass P (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) k ℓ / 5 ≤
treatedCount (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) ω k ℓ) ∧
∀ ω ξ, (fun i => ((ξ i).1, (ξ i).2.1)) =
(fun i => ((ω i).1, (ω i).2.1)) → (ω ∈ E ↔ ξ ∈ E) := by
obtain ⟨E, hE, hbad, hdev, hsat⟩ :=
model_all_treatedCount_deviation_design_saturated
P hP Q hIID hn s hs
refine ⟨E, hE, hbad.trans hfailure, ?_, hsat⟩
intro ω hω
have hwitness : meshFeasible (polynomialDegree β) β ω j₀ :=
oracleWitness_feasible_of_deviation (polynomialDegree β) j₀ β s P ω
hcandidate (by linarith) hmass
(fun k ℓ => (hdev ω hω j₀ hcandidate k ℓ).le)
exact selectedCount_deterministic_bridge (polynomialDegree β) β s
(K * oracleMesh d n β γ) P ω j₀ hwitness hwidth hβ hbudget
hs.le (fun j hj k ℓ => (hdev ω hω j hj k ℓ).le)
theorem selectedCountEvent_of_log_oracle_witness Lemma selectedCountEvent_of_log_oracle_witness in the paper ↗
Once the oracle candidate meets its deterministic mass and budget conditions, the logarithmic finite-family budget gives the desired event. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B L C c_f γ :
ℝ
P :
Measure (Obs d)
hP :
P ∈ ModelClass d β B L C c_f γ
hIID :
IIDSampleLaw Q P
hn :
2 ≤ n
hβ :
0 ≤ β
K :
ℝ
j₀ :
ℕ
hcandidate :
j₀ ∈ meshIndices d n β
hwidth :
meshWidth j₀ ≤ K * oracleMesh d n β γ
hbudget :
24 * log (2 * ((card (Σ j : {j // j ∈ meshIndices d n β}, (Fin d → Fin (2 ^ j.val)) × (Fin d → Fin (polynomialDegree β + 1))) : ℝ) + 1) * (n : ℝ) ^ 2)
≤ meshWidth j₀ ^ (-2 * β)
hmass :
conclusion
Q.real Eᶜ ≤ (n : ℝ) ^ (-2 : ℝ)
conclusion 1
ω ∈ E :
0 < countSelectedMesh (polynomialDegree β) β ω
k :
ℓ :
(n : ℝ) * microcellMass P (polynomialDegree β) ((feasibleIndices (polynomialDegree β) β ω).sup id) k ℓ / 5
Proof (Lean source)
lemma selectedCountEvent_of_log_oracle_witness
{d n : ℕ} {β B L C c_f γ : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(Q : Measure (Fin n → Obs d)) (hIID : IIDSampleLaw Q P)
(hn : 2 ≤ n) (hβ : 0 ≤ β) (K : ℝ)
(j₀ : ℕ) (hcandidate : j₀ ∈ meshIndices d n β)
(hwidth : meshWidth j₀ ≤ K * oracleMesh d n β γ)
(hbudget : 24 * log (2 *
((card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) ×
(Fin d → Fin (polynomialDegree β + 1))) : ℝ) + 1) *
(n : ℝ) ^ 2) ≤ meshWidth j₀ ^ (-2 * β))
(hmass : ∀ (k : Fin d → Fin (2 ^ j₀))
(ℓ : Fin d → Fin (polynomialDegree β + 1)),
4 * meshWidth j₀ ^ (-2 * β) ≤
(n : ℝ) * microcellMass P (polynomialDegree β) j₀ k ℓ) :
∃ E : Set (Fin n → Obs d),
MeasurableSet E ∧ Q.real Eᶜ ≤ (n : ℝ) ^ (-2 : ℝ) ∧
∀ ω ∈ E,
0 < countSelectedMesh (polynomialDegree β) β ω ∧
countSelectedMesh (polynomialDegree β) β ω ≤ K * oracleMesh d n β γ ∧
∀ (k : Fin d → Fin (2 ^ ((feasibleIndices (polynomialDegree β) β ω).sup id)))
(ℓ : Fin d → Fin (polynomialDegree β + 1)),
(n : ℝ) * microcellMass P (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) k ℓ / 5 ≤
treatedCount (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) ω k ℓ := by
obtain ⟨hs, hfailure⟩ := selectedCount_log_budget d n β hn
exact selectedCountEvent_of_oracle_witness P hP Q hIID (by omega)
hβ _ K hs hfailure j₀ hcandidate hwidth hbudget hmass
theorem selectedCountEvent_of_log_oracle_witness_design_saturated Lemma selectedCountEvent_of_log_oracle_witness_design_saturated in the paper ↗
Log-budget specialization retaining design saturation. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B L C c_f γ :
ℝ
P :
Measure (Obs d)
hP :
P ∈ ModelClass d β B L C c_f γ
hIID :
IIDSampleLaw Q P
hn :
2 ≤ n
hβ :
0 ≤ β
K :
ℝ
j₀ :
ℕ
hcandidate :
j₀ ∈ meshIndices d n β
hwidth :
meshWidth j₀ ≤ K * oracleMesh d n β γ
hbudget :
24 * log (2 * ((card (Σ j : {j // j ∈ meshIndices d n β}, (Fin d → Fin (2 ^ j.val)) × (Fin d → Fin (polynomialDegree β + 1))) : ℝ) + 1) * (n : ℝ) ^ 2)
≤ meshWidth j₀ ^ (-2 * β)
hmass :
conclusion
Q.real Eᶜ ≤ (n : ℝ) ^ (-2 : ℝ)
conclusion 1
ω ∈ E :
0 < countSelectedMesh (polynomialDegree β) β ω
k :
ℓ :
(n : ℝ) * microcellMass P (polynomialDegree β) ((feasibleIndices (polynomialDegree β) β ω).sup id) k ℓ / 5
conclusion 2
Proof (Lean source)
lemma selectedCountEvent_of_log_oracle_witness_design_saturated
{d n : ℕ} {β B L C c_f γ : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(Q : Measure (Fin n → Obs d)) (hIID : IIDSampleLaw Q P)
(hn : 2 ≤ n) (hβ : 0 ≤ β) (K : ℝ)
(j₀ : ℕ) (hcandidate : j₀ ∈ meshIndices d n β)
(hwidth : meshWidth j₀ ≤ K * oracleMesh d n β γ)
(hbudget : 24 * log (2 *
((card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) ×
(Fin d → Fin (polynomialDegree β + 1))) : ℝ) + 1) *
(n : ℝ) ^ 2) ≤ meshWidth j₀ ^ (-2 * β))
(hmass : ∀ (k : Fin d → Fin (2 ^ j₀))
(ℓ : Fin d → Fin (polynomialDegree β + 1)),
4 * meshWidth j₀ ^ (-2 * β) ≤
(n : ℝ) * microcellMass P (polynomialDegree β) j₀ k ℓ) :
∃ E : Set (Fin n → Obs d), MeasurableSet E ∧
Q.real Eᶜ ≤ (n : ℝ) ^ (-2 : ℝ) ∧
(∀ ω ∈ E,
0 < countSelectedMesh (polynomialDegree β) β ω ∧
countSelectedMesh (polynomialDegree β) β ω ≤ K * oracleMesh d n β γ ∧
∀ (k : Fin d → Fin
(2 ^ ((feasibleIndices (polynomialDegree β) β ω).sup id)))
(ℓ : Fin d → Fin (polynomialDegree β + 1)),
(n : ℝ) * microcellMass P (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) k ℓ / 5 ≤
treatedCount (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) ω k ℓ) ∧
CountDesignSaturated E := by
obtain ⟨hs, hfailure⟩ := selectedCount_log_budget d n β hn
obtain ⟨E, hE, hbad, hprop, hsat⟩ :=
selectedCountEvent_of_oracle_witness_design_saturated
P hP Q hIID (by omega) hβ _ K hs hfailure j₀ hcandidate
hwidth hbudget hmass
refine ⟨E, hE, hbad, hprop, ?_⟩
change ∀ ω ξ, (fun i => ((ξ i).1, (ξ i).2.1)) =
(fun i => ((ω i).1, (ω i).2.1)) → (ω ∈ E ↔ ξ ∈ E)
exact hsat
At the oracle scale, expected counts at the effective dimension balance the inverse-squared smoothness threshold. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β γ :
ℝ
hn :
0 < n
hden :
0 < 2 * β + effectiveDimension d γ
conclusion
(n : ℝ) * (oracleMesh d n β γ) ^ (effectiveDimension d γ)
= (oracleMesh d n β γ) ^ (-2 * β)
Proof (Lean source)
lemma oracleMesh_count_balance (d n : ℕ) (β γ : ℝ)
(hn : 0 < n) (hden : 0 < 2 * β + effectiveDimension d γ) :
(n : ℝ) * (oracleMesh d n β γ) ^ (effectiveDimension d γ) =
(oracleMesh d n β γ) ^ (-2 * β) := by
have hn' : (0 : ℝ) < n := by exact_mod_cast hn
have he : -(1 : ℝ) / (2 * β + effectiveDimension d γ) *
(2 * β + effectiveDimension d γ) = -1 := by
field_simp
have hpow : (oracleMesh d n β γ) ^
(2 * β + effectiveDimension d γ) = (n : ℝ)⁻¹ := by
rw [oracleMesh, ← Real.rpow_mul hn'.le, he]
exact Real.rpow_neg_one (n : ℝ)
rw [Real.rpow_add (by unfold oracleMesh; positivity)] at hpow
have hmeshpos : 0 < oracleMesh d n β γ := by
unfold oracleMesh
exact Real.rpow_pos_of_pos hn' _
have hpos : 0 < (oracleMesh d n β γ) ^ (2 * β) := by
exact Real.rpow_pos_of_pos hmeshpos _
have hneg : -2 * β = -(2 * β) := by ring
rw [hneg]
rw [Real.rpow_neg hmeshpos.le]
rw [inv_eq_one_div]
apply (eq_div_iff hpos.ne').2
calc
((n : ℝ) * (oracleMesh d n β γ) ^ (effectiveDimension d γ)) *
(oracleMesh d n β γ) ^ (2 * β) =
(n : ℝ) * ((oracleMesh d n β γ) ^ (2 * β) *
(oracleMesh d n β γ) ^ (effectiveDimension d γ)) := by ring
_ = 1 := by rw [hpow]; exact mul_inv_cancel₀ hn'.ne'
Oracle-scale lower mass and a dyadic width comparison imply the feasibility count threshold. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β γ κ A h :
ℝ
hn :
0 < n
hden :
0 < 2 * β + effectiveDimension d γ
hκ :
0 < κ
hA :
0 < A
hh :
0 < h
hmesh :
A * oracleMesh d n β γ ≤ h
hcoeff :
4 ≤ κ * A ^ (2 * β + effectiveDimension d γ)
conclusion
4 * h ^ (-2 * β) ≤ (n : ℝ) * κ * h ^ (effectiveDimension d γ)
Proof (Lean source)
lemma oracleScale_mass_feasibility (d n : ℕ) (β γ κ A h : ℝ)
(hn : 0 < n)
(hden : 0 < 2 * β + effectiveDimension d γ)
(hκ : 0 < κ) (hA : 0 < A) (hh : 0 < h)
(hmesh : A * oracleMesh d n β γ ≤ h)
(hcoeff : 4 ≤ κ * A ^ (2 * β + effectiveDimension d γ)) :
4 * h ^ (-2 * β) ≤ (n : ℝ) * κ * h ^ (effectiveDimension d γ) := by
let o := oracleMesh d n β γ
let e := 2 * β + effectiveDimension d γ
have ho : 0 < o := by dsimp [o, oracleMesh]; positivity
have he : 0 ≤ e := le_of_lt hden
have hbalance : (n : ℝ) * o ^ e = 1 := by
have hb := oracleMesh_count_balance d n β γ hn hden
dsimp [o, e]
change 0 < oracleMesh d n β γ at ho
rw [Real.rpow_add ho]
calc
(n : ℝ) * (oracleMesh d n β γ ^ (2 * β) *
oracleMesh d n β γ ^ effectiveDimension d γ) =
((n : ℝ) * oracleMesh d n β γ ^ effectiveDimension d γ) *
oracleMesh d n β γ ^ (2 * β) := by ring
_ = oracleMesh d n β γ ^ (-2 * β) *
oracleMesh d n β γ ^ (2 * β) := by rw [hb]
_ = 1 := by rw [← Real.rpow_add ho]; simp
have hmass : 4 ≤ (n : ℝ) * κ * h ^ e := by
calc
4 ≤ κ * A ^ e := hcoeff
_ = (n : ℝ) * κ * (A * o) ^ e := by
rw [Real.mul_rpow hA.le ho.le]
nlinarith [hbalance]
_ ≤ (n : ℝ) * κ * h ^ e := by
gcongr
have hβpow : 0 < h ^ (2 * β) := Real.rpow_pos_of_pos hh _
have hmass' : 4 ≤ ((n : ℝ) * κ * h ^ effectiveDimension d γ) *
h ^ (2 * β) := by
have heq : e = effectiveDimension d γ + 2 * β := by dsimp [e]; ring
rw [heq, Real.rpow_add hh] at hmass
nlinarith [hmass]
rw [show -2 * β = -(2 * β) by ring, Real.rpow_neg hh.le,
← div_eq_mul_inv]
exact (div_le_iff₀ hβpow).2 (by nlinarith [hmass'])
A candidate whose width is above a fixed multiple of the oracle width has enough expected treated observations in every template microcell. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B L C c_f γ :
ℝ
P :
Measure (Obs d)
hP :
P ∈ ModelClass d β B L C c_f γ
j :
ℕ
A :
ℝ
hn :
0 < n
hden :
0 < 2 * β + effectiveDimension d γ
hA :
0 < A
hmesh :
A * oracleMesh d n β γ ≤ meshWidth j
hcoeff :
4
≤ (((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (c_f * templateEta d (polynomialDegree β) ^ d) ^ (γ / (γ - 1))) * A ^ (2 * β + effectiveDimension d γ)
ℓ :
conclusion
Proof (Lean source)
lemma model_oracleCandidate_count_mass
{d n : ℕ} {β B L C c_f γ : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(j : ℕ) (A : ℝ) (hn : 0 < n)
(hden : 0 < 2 * β + effectiveDimension d γ)
(hA : 0 < A)
(hmesh : A * oracleMesh d n β γ ≤ meshWidth j)
(hcoeff : 4 ≤
(((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d (polynomialDegree β) ^ d) ^ (γ / (γ - 1))) *
A ^ (2 * β + effectiveDimension d γ)) :
∀ (k : Fin d → Fin (2 ^ j))
(ℓ : Fin d → Fin (polynomialDegree β + 1)),
4 * meshWidth j ^ (-2 * β) ≤
(n : ℝ) * microcellMass P (polynomialDegree β) j k ℓ := by
intro k ℓ
let κ : ℝ := ((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d (polynomialDegree β) ^ d) ^ (γ / (γ - 1))
have hκ : 0 < κ := by
dsimp [κ]
have hγ := hP.2.1
have hC : 0 < C := lt_of_lt_of_le zero_lt_one hP.1.2.2.2.2.1
have hcf : 0 < c_f := hP.1.2.2.2.2.2.1
have hη : 0 < templateEta d (polynomialDegree β) := templateEta_pos _ _
positivity
have hh : 0 < meshWidth j := by unfold meshWidth; positivity
have hmass := oracleScale_mass_feasibility d n β γ κ A (meshWidth j)
hn hden hκ hA hh hmesh (by simpa only [κ] using hcoeff)
have hlower := model_microcellMass_effectiveDimension_lower P hP
(polynomialDegree β) j k ℓ
have hn' : (0 : ℝ) ≤ n := Nat.cast_nonneg _
exact hmass.trans (by
simpa only [κ, mul_assoc] using mul_le_mul_of_nonneg_left hlower hn')
theorem model_oracleCandidate_count_mass_of_coefficient_lower Lemma model_oracleCandidate_count_mass_of_coefficient_lower in the paper ↗
A positive lower bound for the model's mass coefficient is enough for the oracle-candidate count lower bound. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B L C c_f γ :
ℝ
P :
Measure (Obs d)
hP :
P ∈ ModelClass d β B L C c_f γ
j :
ℕ
A κ :
ℝ
hn :
0 < n
hden :
0 < 2 * β + effectiveDimension d γ
hA :
0 < A
hκ :
0 < κ
hκle :
κ
≤ ((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (c_f * templateEta d (polynomialDegree β) ^ d) ^ (γ / (γ - 1))
hmesh :
A * oracleMesh d n β γ ≤ meshWidth j
hcoeff :
4 ≤ κ * A ^ (2 * β + effectiveDimension d γ)
ℓ :
conclusion
Proof (Lean source)
lemma model_oracleCandidate_count_mass_of_coefficient_lower
{d n : ℕ} {β B L C c_f γ : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(j : ℕ) (A κ : ℝ) (hn : 0 < n)
(hden : 0 < 2 * β + effectiveDimension d γ)
(hA : 0 < A) (hκ : 0 < κ)
(hκle : κ ≤
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d (polynomialDegree β) ^ d) ^ (γ / (γ - 1)))
(hmesh : A * oracleMesh d n β γ ≤ meshWidth j)
(hcoeff : 4 ≤ κ * A ^ (2 * β + effectiveDimension d γ)) :
∀ (k : Fin d → Fin (2 ^ j))
(ℓ : Fin d → Fin (polynomialDegree β + 1)),
4 * meshWidth j ^ (-2 * β) ≤
(n : ℝ) * microcellMass P (polynomialDegree β) j k ℓ := by
have hcoeff' : 4 ≤
(((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d (polynomialDegree β) ^ d) ^ (γ / (γ - 1))) *
A ^ (2 * β + effectiveDimension d γ) := by
exact hcoeff.trans (mul_le_mul_of_nonneg_right hκle (by positivity))
exact model_oracleCandidate_count_mass P hP j A hn hden hA hmesh hcoeff'
theorem selectedCountEvent_of_oracle_candidate Lemma selectedCountEvent_of_oracle_candidate in the paper ↗
The model mass inequality discharges the oracle witness mass premise in the finite-family count event. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B L C c_f γ :
ℝ
P :
Measure (Obs d)
hP :
P ∈ ModelClass d β B L C c_f γ
hIID :
IIDSampleLaw Q P
hn :
2 ≤ n
hβ :
0 ≤ β
hden :
0 < 2 * β + effectiveDimension d γ
A K :
ℝ
hA :
0 < A
j₀ :
ℕ
hcandidate :
j₀ ∈ meshIndices d n β
hmesh :
A * oracleMesh d n β γ ≤ meshWidth j₀
hwidth :
meshWidth j₀ ≤ K * oracleMesh d n β γ
hbudget :
24 * log (2 * ((card (Σ j : {j // j ∈ meshIndices d n β}, (Fin d → Fin (2 ^ j.val)) × (Fin d → Fin (polynomialDegree β + 1))) : ℝ) + 1) * (n : ℝ) ^ 2)
≤ meshWidth j₀ ^ (-2 * β)
hcoeff :
4
≤ (((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (c_f * templateEta d (polynomialDegree β) ^ d) ^ (γ / (γ - 1))) * A ^ (2 * β + effectiveDimension d γ)
conclusion
Q.real Eᶜ ≤ (n : ℝ) ^ (-2 : ℝ)
conclusion 1
ω ∈ E :
0 < countSelectedMesh (polynomialDegree β) β ω
k :
ℓ :
(n : ℝ) * microcellMass P (polynomialDegree β) ((feasibleIndices (polynomialDegree β) β ω).sup id) k ℓ / 5
Proof (Lean source)
lemma selectedCountEvent_of_oracle_candidate
{d n : ℕ} {β B L C c_f γ : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(Q : Measure (Fin n → Obs d)) (hIID : IIDSampleLaw Q P)
(hn : 2 ≤ n) (hβ : 0 ≤ β) (hden : 0 < 2 * β + effectiveDimension d γ)
(A K : ℝ) (hA : 0 < A)
(j₀ : ℕ) (hcandidate : j₀ ∈ meshIndices d n β)
(hmesh : A * oracleMesh d n β γ ≤ meshWidth j₀)
(hwidth : meshWidth j₀ ≤ K * oracleMesh d n β γ)
(hbudget : 24 * log (2 *
((card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) ×
(Fin d → Fin (polynomialDegree β + 1))) : ℝ) + 1) *
(n : ℝ) ^ 2) ≤ meshWidth j₀ ^ (-2 * β))
(hcoeff : 4 ≤
(((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d (polynomialDegree β) ^ d) ^ (γ / (γ - 1))) *
A ^ (2 * β + effectiveDimension d γ)) :
∃ E : Set (Fin n → Obs d),
MeasurableSet E ∧ Q.real Eᶜ ≤ (n : ℝ) ^ (-2 : ℝ) ∧
∀ ω ∈ E,
0 < countSelectedMesh (polynomialDegree β) β ω ∧
countSelectedMesh (polynomialDegree β) β ω ≤ K * oracleMesh d n β γ ∧
∀ (k : Fin d → Fin (2 ^ ((feasibleIndices (polynomialDegree β) β ω).sup id)))
(ℓ : Fin d → Fin (polynomialDegree β + 1)),
(n : ℝ) * microcellMass P (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) k ℓ / 5 ≤
treatedCount (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) ω k ℓ := by
exact selectedCountEvent_of_log_oracle_witness P hP Q hIID hn hβ K j₀
hcandidate hwidth hbudget
(model_oracleCandidate_count_mass P hP j₀ A (by omega)
hden hA hmesh hcoeff)
The common conclusion of the design-saturated selected-count wrappers. For the stated inputs and conditions, the SaturatedSelectedCountEvent object being defined.
Definition (Lean source)
parameters
def
SaturatedSelectedCountEvent β K γ P Q E :
Prop
given by
clause 1
clause 2
Q.real Eᶜ ≤ (n : ℝ) ^ (-2 : ℝ)
clause 3
ω ∈ E :
0 < countSelectedMesh (polynomialDegree β) β ω
k :
ℓ :
(n : ℝ) * microcellMass P (polynomialDegree β) ((feasibleIndices (polynomialDegree β) β ω).sup id) k ℓ / 5
clause 4
theorem selectedCountEvent_of_oracle_candidate_design_saturated Lemma selectedCountEvent_of_oracle_candidate_design_saturated in the paper ↗
Oracle-candidate specialization retaining the saturated event. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B L C c_f γ :
ℝ
P :
Measure (Obs d)
hP :
P ∈ ModelClass d β B L C c_f γ
hIID :
IIDSampleLaw Q P
hn :
2 ≤ n
hβ :
0 ≤ β
hden :
0 < 2 * β + effectiveDimension d γ
A K :
ℝ
hA :
0 < A
j₀ :
ℕ
hcandidate :
j₀ ∈ meshIndices d n β
hmesh :
A * oracleMesh d n β γ ≤ meshWidth j₀
hwidth :
meshWidth j₀ ≤ K * oracleMesh d n β γ
hbudget :
24 * log (2 * ((card (Σ j : {j // j ∈ meshIndices d n β}, (Fin d → Fin (2 ^ j.val)) × (Fin d → Fin (polynomialDegree β + 1))) : ℝ) + 1) * (n : ℝ) ^ 2)
≤ meshWidth j₀ ^ (-2 * β)
hcoeff :
4
≤ (((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (c_f * templateEta d (polynomialDegree β) ^ d) ^ (γ / (γ - 1))) * A ^ (2 * β + effectiveDimension d γ)
conclusion
∃ E, SaturatedSelectedCountEvent β K γ P Q E
Proof (Lean source)
lemma selectedCountEvent_of_oracle_candidate_design_saturated
{d n : ℕ} {β B L C c_f γ : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(Q : Measure (Fin n → Obs d)) (hIID : IIDSampleLaw Q P)
(hn : 2 ≤ n) (hβ : 0 ≤ β) (hden : 0 < 2 * β + effectiveDimension d γ)
(A K : ℝ) (hA : 0 < A)
(j₀ : ℕ) (hcandidate : j₀ ∈ meshIndices d n β)
(hmesh : A * oracleMesh d n β γ ≤ meshWidth j₀)
(hwidth : meshWidth j₀ ≤ K * oracleMesh d n β γ)
(hbudget : 24 * log (2 *
((card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) ×
(Fin d → Fin (polynomialDegree β + 1))) : ℝ) + 1) *
(n : ℝ) ^ 2) ≤ meshWidth j₀ ^ (-2 * β))
(hcoeff : 4 ≤
(((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d (polynomialDegree β) ^ d) ^ (γ / (γ - 1))) *
A ^ (2 * β + effectiveDimension d γ)) :
∃ E, SaturatedSelectedCountEvent β K γ P Q E := by
obtain ⟨E, hE, hbad, hprop, hsat⟩ :=
selectedCountEvent_of_log_oracle_witness_design_saturated
P hP Q hIID hn hβ K j₀ hcandidate hwidth hbudget
(model_oracleCandidate_count_mass P hP j₀ A (by omega)
hden hA hmesh hcoeff)
exact ⟨E, hE, hbad, hprop, hsat⟩
theorem selectedCountEvent_of_oracle_candidate_coefficient_lower Lemma selectedCountEvent_of_oracle_candidate_coefficient_lower in the paper ↗
The oracle-candidate event accepts any positive uniform lower bound for the model's overlap-mass coefficient. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B L C c_f γ :
ℝ
P :
Measure (Obs d)
hP :
P ∈ ModelClass d β B L C c_f γ
hIID :
IIDSampleLaw Q P
hn :
2 ≤ n
hβ :
0 ≤ β
hden :
0 < 2 * β + effectiveDimension d γ
A K κ :
ℝ
hA :
0 < A
hκ :
0 < κ
hκle :
κ
≤ ((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (c_f * templateEta d (polynomialDegree β) ^ d) ^ (γ / (γ - 1))
j₀ :
ℕ
hcandidate :
j₀ ∈ meshIndices d n β
hmesh :
A * oracleMesh d n β γ ≤ meshWidth j₀
hwidth :
meshWidth j₀ ≤ K * oracleMesh d n β γ
hbudget :
24 * log (2 * ((card (Σ j : {j // j ∈ meshIndices d n β}, (Fin d → Fin (2 ^ j.val)) × (Fin d → Fin (polynomialDegree β + 1))) : ℝ) + 1) * (n : ℝ) ^ 2)
≤ meshWidth j₀ ^ (-2 * β)
hcoeff :
4 ≤ κ * A ^ (2 * β + effectiveDimension d γ)
conclusion
Q.real Eᶜ ≤ (n : ℝ) ^ (-2 : ℝ)
conclusion 1
ω ∈ E :
0 < countSelectedMesh (polynomialDegree β) β ω
k :
ℓ :
(n : ℝ) * microcellMass P (polynomialDegree β) ((feasibleIndices (polynomialDegree β) β ω).sup id) k ℓ / 5
Proof (Lean source)
lemma selectedCountEvent_of_oracle_candidate_coefficient_lower
{d n : ℕ} {β B L C c_f γ : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(Q : Measure (Fin n → Obs d)) (hIID : IIDSampleLaw Q P)
(hn : 2 ≤ n) (hβ : 0 ≤ β) (hden : 0 < 2 * β + effectiveDimension d γ)
(A K κ : ℝ) (hA : 0 < A) (hκ : 0 < κ)
(hκle : κ ≤
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d (polynomialDegree β) ^ d) ^ (γ / (γ - 1)))
(j₀ : ℕ) (hcandidate : j₀ ∈ meshIndices d n β)
(hmesh : A * oracleMesh d n β γ ≤ meshWidth j₀)
(hwidth : meshWidth j₀ ≤ K * oracleMesh d n β γ)
(hbudget : 24 * log (2 *
((card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) ×
(Fin d → Fin (polynomialDegree β + 1))) : ℝ) + 1) *
(n : ℝ) ^ 2) ≤ meshWidth j₀ ^ (-2 * β))
(hcoeff : 4 ≤ κ * A ^ (2 * β + effectiveDimension d γ)) :
∃ E : Set (Fin n → Obs d),
MeasurableSet E ∧ Q.real Eᶜ ≤ (n : ℝ) ^ (-2 : ℝ) ∧
∀ ω ∈ E,
0 < countSelectedMesh (polynomialDegree β) β ω ∧
countSelectedMesh (polynomialDegree β) β ω ≤ K * oracleMesh d n β γ ∧
∀ (k : Fin d → Fin (2 ^ ((feasibleIndices (polynomialDegree β) β ω).sup id)))
(ℓ : Fin d → Fin (polynomialDegree β + 1)),
(n : ℝ) * microcellMass P (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) k ℓ / 5 ≤
treatedCount (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) ω k ℓ := by
exact selectedCountEvent_of_log_oracle_witness P hP Q hIID hn hβ K j₀
hcandidate hwidth hbudget
(model_oracleCandidate_count_mass_of_coefficient_lower P hP j₀ A κ
(by omega) hden hA hκ hκle hmesh hcoeff)
theorem selectedCountEvent_of_oracle_candidate_coefficient_lower_design_saturated Lemma selectedCountEvent_of_oracle_candidate_coefficient_lower_design_saturated in the paper ↗
Positive-coefficient oracle specialization retaining saturation. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B L C c_f γ :
ℝ
P :
Measure (Obs d)
hP :
P ∈ ModelClass d β B L C c_f γ
hIID :
IIDSampleLaw Q P
hn :
2 ≤ n
hβ :
0 ≤ β
hden :
0 < 2 * β + effectiveDimension d γ
A K κ :
ℝ
hA :
0 < A
hκ :
0 < κ
hκle :
κ
≤ ((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (c_f * templateEta d (polynomialDegree β) ^ d) ^ (γ / (γ - 1))
j₀ :
ℕ
hcandidate :
j₀ ∈ meshIndices d n β
hmesh :
A * oracleMesh d n β γ ≤ meshWidth j₀
hwidth :
meshWidth j₀ ≤ K * oracleMesh d n β γ
hbudget :
24 * log (2 * ((card (Σ j : {j // j ∈ meshIndices d n β}, (Fin d → Fin (2 ^ j.val)) × (Fin d → Fin (polynomialDegree β + 1))) : ℝ) + 1) * (n : ℝ) ^ 2)
≤ meshWidth j₀ ^ (-2 * β)
hcoeff :
4 ≤ κ * A ^ (2 * β + effectiveDimension d γ)
conclusion
∃ E, SaturatedSelectedCountEvent β K γ P Q E
Proof (Lean source)
lemma selectedCountEvent_of_oracle_candidate_coefficient_lower_design_saturated
{d n : ℕ} {β B L C c_f γ : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(Q : Measure (Fin n → Obs d)) (hIID : IIDSampleLaw Q P)
(hn : 2 ≤ n) (hβ : 0 ≤ β) (hden : 0 < 2 * β + effectiveDimension d γ)
(A K κ : ℝ) (hA : 0 < A) (hκ : 0 < κ)
(hκle : κ ≤ ((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d (polynomialDegree β) ^ d) ^ (γ / (γ - 1)))
(j₀ : ℕ) (hcandidate : j₀ ∈ meshIndices d n β)
(hmesh : A * oracleMesh d n β γ ≤ meshWidth j₀)
(hwidth : meshWidth j₀ ≤ K * oracleMesh d n β γ)
(hbudget : 24 * log (2 *
((card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) ×
(Fin d → Fin (polynomialDegree β + 1))) : ℝ) + 1) *
(n : ℝ) ^ 2) ≤ meshWidth j₀ ^ (-2 * β))
(hcoeff : 4 ≤ κ * A ^ (2 * β + effectiveDimension d γ)) :
∃ E, SaturatedSelectedCountEvent β K γ P Q E := by
obtain ⟨E, hE, hbad, hprop, hsat⟩ :=
selectedCountEvent_of_log_oracle_witness_design_saturated
P hP Q hIID hn hβ K j₀ hcandidate hwidth hbudget
(model_oracleCandidate_count_mass_of_coefficient_lower P hP j₀ A κ
(by omega) hden hA hκ hκle hmesh hcoeff)
exact ⟨E, hE, hbad, hprop, hsat⟩
Feasibility and the selected count event give both variance scales for a single treated microcell. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
N :
ℕ
n :
ℕ
q h β :
ℝ
hn :
0 < n
hq :
0 < q
hh :
0 < h
hfeasible :
h ^ (-2 * β) ≤ (N : ℝ)
hselected :
(n : ℝ) * q / 5 ≤ (N : ℝ)
conclusion
(N : ℝ)⁻¹ ≤ min (h ^ (2 * β)) (5 / ((n : ℝ) * q))
Proof (Lean source)
lemma selectedCount_inverse_two_bounds (N : ℕ) (n : ℕ) (q h β : ℝ)
(hn : 0 < n) (hq : 0 < q) (hh : 0 < h)
(hfeasible : h ^ (-2 * β) ≤ (N : ℝ))
(hselected : (n : ℝ) * q / 5 ≤ (N : ℝ)) :
(N : ℝ)⁻¹ ≤ min (h ^ (2 * β)) (5 / ((n : ℝ) * q)) := by
have hn' : (0 : ℝ) < n := by exact_mod_cast hn
have hcount : 0 < (N : ℝ) :=
lt_of_lt_of_le (Real.rpow_pos_of_pos hh _) hfeasible
have hpop : 0 < (n : ℝ) * q / 5 := by positivity
apply le_min
· have h := one_div_le_one_div_of_le
(Real.rpow_pos_of_pos hh _) hfeasible
rw [one_div, one_div, ← Real.rpow_neg hh.le] at h
convert h using 1 <;> ring
· have h := one_div_le_one_div_of_le hpop hselected
rw [one_div, one_div] at h
convert h using 1
field_simp
theorem selectedCount_cube_inverse_sum_bounds Lemma selectedCount_cube_inverse_sum_bounds in the paper ↗
The selected-count event controls the sum of inverse counts in a cube. This is the design-dependent factor in every coefficient variance proxy. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
β :
ℝ
P :
Measure (Obs d)
ω :
Fin n → Obs d
hn :
0 < n
hfeasible :
meshFeasible m β ω j
hselected :
hq :
0 < cubeMass P m j k
Proof (Lean source)
lemma selectedCount_cube_inverse_sum_bounds {d n : ℕ} (m j : ℕ) (β : ℝ)
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(hn : 0 < n) (hfeasible : meshFeasible m β ω j)
(hselected : ∀ ℓ : Fin d → Fin (m + 1),
(n : ℝ) * microcellMass P m j k ℓ / 5 ≤
treatedCount m j ω k ℓ)
(hq : 0 < cubeMass P m j k) :
(∑ ℓ : Fin d → Fin (m + 1),
(treatedCount m j ω k ℓ : ℝ)⁻¹) ≤
(tensorCount d m : ℝ) *
min (meshWidth j ^ (2 * β))
(5 / ((n : ℝ) * cubeMass P m j k)) := by
classical
have hh : 0 < meshWidth j := by unfold meshWidth; positivity
calc
(∑ ℓ : Fin d → Fin (m + 1),
(treatedCount m j ω k ℓ : ℝ)⁻¹) ≤
∑ _ℓ : Fin d → Fin (m + 1),
min (meshWidth j ^ (2 * β))
(5 / ((n : ℝ) * cubeMass P m j k)) := by
apply Finset.sum_le_sum
intro ℓ _
have hqℓ : 0 < microcellMass P m j k ℓ :=
lt_of_lt_of_le hq (orderedMass_cubeMass_le_microcellMass P m j k ℓ)
have hb := selectedCount_inverse_two_bounds
(treatedCount m j ω k ℓ) n (microcellMass P m j k ℓ)
(meshWidth j) β hn hqℓ hh (hfeasible.2 k ℓ) (hselected ℓ)
apply hb.trans
apply min_le_min_left
gcongr
exact orderedMass_cubeMass_le_microcellMass P m j k ℓ
_ = (tensorCount d m : ℝ) *
min (meshWidth j ^ (2 * β))
(5 / ((n : ℝ) * cubeMass P m j k)) := by
simp [tensorCount]
theorem selectedCount_cube_inverse_rank_bounds Lemma selectedCount_cube_inverse_rank_bounds in the paper ↗
Insert an ordered-mass lower bound into the inverse-count variance factor. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
β D α κ :
ℝ
r :
ℕ
P :
Measure (Obs d)
ω :
Fin n → Obs d
hn :
0 < n
hκ :
0 < κ
hr :
0 < r
hfeasible :
meshFeasible m β ω j
hselected :
Proof (Lean source)
lemma selectedCount_cube_inverse_rank_bounds {d n : ℕ} (m j : ℕ)
(β D α κ : ℝ) (r : ℕ)
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(hn : 0 < n) (hκ : 0 < κ) (hr : 0 < r)
(hfeasible : meshFeasible m β ω j)
(hselected : ∀ ℓ : Fin d → Fin (m + 1),
(n : ℝ) * microcellMass P m j k ℓ / 5 ≤
treatedCount m j ω k ℓ)
(hmass : κ * meshWidth j ^ D * (r : ℝ) ^ α ≤ cubeMass P m j k) :
(∑ ℓ : Fin d → Fin (m + 1),
(treatedCount m j ω k ℓ : ℝ)⁻¹) ≤
(tensorCount d m : ℝ) *
min (meshWidth j ^ (2 * β))
(5 / ((n : ℝ) * κ * meshWidth j ^ D * (r : ℝ) ^ α)) := by
have hh : 0 < meshWidth j := by unfold meshWidth; positivity
have hr' : (0 : ℝ) < r := by exact_mod_cast hr
have hq : 0 < cubeMass P m j k :=
lt_of_lt_of_le (by positivity : 0 < κ * meshWidth j ^ D * (r : ℝ) ^ α) hmass
have hbase := selectedCount_cube_inverse_sum_bounds m j β P ω k hn
hfeasible hselected hq
apply hbase.trans
apply mul_le_mul_of_nonneg_left _ (by positivity)
apply min_le_min_left
have hden : 0 < (n : ℝ) * κ * meshWidth j ^ D * (r : ℝ) ^ α := by
have hn' : (0 : ℝ) < n := by exact_mod_cast hn
positivity
have hqden : 0 < (n : ℝ) * cubeMass P m j k := by
have hn' : (0 : ℝ) < n := by exact_mod_cast hn
positivity
apply div_le_div_of_nonneg_left (by norm_num) hden
nlinarith [mul_le_mul_of_nonneg_left hmass (show (0 : ℝ) ≤ n from Nat.cast_nonneg _)]
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma meshWidth_dyadic_rounding (J : ℕ) (x : ℝ)
(hx : 0 < x) (hx1 : x ≤ 1) (hJ : meshWidth J ≤ x) :
∃ j ≤ J, x ≤ meshWidth j ∧ meshWidth j ≤ 2 * x := by
induction J with
| zero =>
refine ⟨0, le_refl _, ?_, ?_⟩
· simpa [meshWidth] using hx1
· simpa [meshWidth] using (show (1 : ℝ) ≤ 2 * x by
have h : (1 : ℝ) ≤ x := by simpa [meshWidth] using hJ
linarith)
| succ J ih =>
by_cases h : meshWidth J ≤ x
· obtain ⟨j, hj, hjx, hj2⟩ := ih h
exact ⟨j, hj.trans (le_succ J), hjx, hj2⟩
· have hstep : meshWidth J = 2 * meshWidth (J + 1) := by
unfold meshWidth
rw [Nat.cast_add, Nat.cast_one]
have he : -(↑J + (1 : ℝ)) = -(↑J : ℝ) + (-1) := by ring
rw [he, Real.rpow_add (by norm_num : (0 : ℝ) < 2)]
norm_num
ring
refine ⟨J, le_succ J, le_of_not_ge h, ?_⟩
rw [hstep]
exact mul_le_mul_of_nonneg_left hJ (by norm_num)
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β :
ℝ
hn :
0 < n
hden :
0 < 2 * β + d
conclusion
meshWidth (maxMeshIndex d n β) ≤ 2 * (n : ℝ) ^ (-(1 : ℝ) / (2 * β + d))
Proof (Lean source)
lemma maxMeshIndex_width_le (d n : ℕ) (β : ℝ)
(hn : 0 < n) (hden : 0 < 2 * β + d) :
meshWidth (maxMeshIndex d n β) ≤
2 * (n : ℝ) ^ (-(1 : ℝ) / (2 * β + d)) := by
have hn' : (0 : ℝ) < n := by exact_mod_cast hn
have hlog2 : 0 < log (2 : ℝ) := Real.log_pos (by norm_num)
let x : ℝ := log n / log 2 / (2 * β + d)
have hfloor : x < (Nat.floor x : ℝ) + 1 := Nat.lt_floor_add_one x
have hexp : -(Nat.floor x : ℝ) ≤ 1 - x := by linarith
have hmono : (2 : ℝ) ^ (-(Nat.floor x : ℝ)) ≤ (2 : ℝ) ^ (1 - x) :=
Real.rpow_le_rpow_of_exponent_le (by norm_num) hexp
have hpow : (2 : ℝ) ^ (1 - x) =
2 * (n : ℝ) ^ (-(1 : ℝ) / (2 * β + d)) := by
rw [Real.rpow_def_of_pos (by norm_num : (0 : ℝ) < 2),
Real.rpow_def_of_pos hn']
rw [show log (2 : ℝ) * (1 - x) =
log 2 + log (n : ℝ) * (-(1 : ℝ) / (2 * β + d)) by
dsimp [x]
field_simp
ring]
rw [Real.exp_add, Real.exp_log (by norm_num : (0 : ℝ) < 2)]
simpa only [maxMeshIndex, meshWidth, x] using hmono.trans_eq hpow
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β :
ℝ
hn :
0 < n
hden :
0 < 2 * β + d
conclusion
(n : ℝ) ^ (-(1 : ℝ) / (2 * β + d)) ≤ meshWidth (maxMeshIndex d n β)
Proof (Lean source)
lemma maxMeshIndex_width_ge (d n : ℕ) (β : ℝ)
(hn : 0 < n) (hden : 0 < 2 * β + d) :
(n : ℝ) ^ (-(1 : ℝ) / (2 * β + d)) ≤
meshWidth (maxMeshIndex d n β) := by
have hn' : (0 : ℝ) < n := by exact_mod_cast hn
have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast hn
have hlog2 : 0 < log (2 : ℝ) := Real.log_pos (by norm_num)
let x : ℝ := log n / log 2 / (2 * β + d)
have hx : 0 ≤ x := by
dsimp [x]
positivity
have hfloor : (Nat.floor x : ℝ) ≤ x := Nat.floor_le hx
have hmono : (2 : ℝ) ^ (-x) ≤ (2 : ℝ) ^ (-(Nat.floor x : ℝ)) :=
Real.rpow_le_rpow_of_exponent_le (by norm_num) (by linarith)
have hpow : (2 : ℝ) ^ (-x) =
(n : ℝ) ^ (-(1 : ℝ) / (2 * β + d)) := by
rw [Real.rpow_def_of_pos (by norm_num : (0 : ℝ) < 2),
Real.rpow_def_of_pos hn']
congr 1
dsimp [x]
field_simp
simpa only [maxMeshIndex, meshWidth, x] using hpow.symm.le.trans hmono
theorem maxMeshIndex_width_le_oracle_eventually Lemma maxMeshIndex_width_le_oracle_eventually in the paper ↗
The finest dyadic candidate width eventually lies below any fixed multiple of the oracle width when the effective dimension is strictly larger. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β γ A :
ℝ
hden :
0 < 2 * β + d
hdim :
(d : ℝ) < effectiveDimension d γ
hA :
0 < A
conclusion
Proof (Lean source)
lemma maxMeshIndex_width_le_oracle_eventually (d : ℕ) (β γ A : ℝ)
(hden : 0 < 2 * β + d)
(hdim : (d : ℝ) < effectiveDimension d γ) (hA : 0 < A) :
∃ n₀ : ℕ, ∀ n ≥ n₀,
meshWidth (maxMeshIndex d n β) ≤ A * oracleMesh d n β γ := by
have hden' : 0 < 2 * β + effectiveDimension d γ := by linarith
have hgap : (1 : ℝ) / (2 * β + effectiveDimension d γ) <
1 / (2 * β + d) := by
apply (div_lt_div_iff₀ hden' hden).2
nlinarith
obtain ⟨n₀, hn₀⟩ := selectedCount_power_gap
(1 / (2 * β + effectiveDimension d γ))
(1 / (2 * β + d)) A hgap hA
refine ⟨max n₀ 1, ?_⟩
intro n hn
have hnpos : 0 < n := by omega
calc
meshWidth (maxMeshIndex d n β) ≤
2 * (n : ℝ) ^ (-(1 : ℝ) / (2 * β + d)) :=
maxMeshIndex_width_le d n β hnpos hden
_ ≤ A * oracleMesh d n β γ := by
simpa only [oracleMesh, neg_div] using
hn₀ n (le_trans (Nat.le_max_left _ _) hn)
theorem meshIndices_oracleCandidate_of_bounds Lemma meshIndices_oracleCandidate_of_bounds in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β A γ :
ℝ
hA :
0 < A
ho :
0 < oracleMesh d n β γ
hlarge :
A * oracleMesh d n β γ ≤ 1
hsmall :
conclusion
∃ j ∈ meshIndices d n β,
conclusion 1
A * oracleMesh d n β γ ≤ meshWidth j
conclusion 2
meshWidth j ≤ (2 * A) * oracleMesh d n β γ
Proof (Lean source)
lemma meshIndices_oracleCandidate_of_bounds (d n : ℕ) (β A γ : ℝ)
(hA : 0 < A) (ho : 0 < oracleMesh d n β γ)
(hlarge : A * oracleMesh d n β γ ≤ 1)
(hsmall : meshWidth (maxMeshIndex d n β) ≤ A * oracleMesh d n β γ) :
∃ j ∈ meshIndices d n β,
A * oracleMesh d n β γ ≤ meshWidth j ∧
meshWidth j ≤ (2 * A) * oracleMesh d n β γ := by
obtain ⟨j, hj, hjlo, hjhi⟩ :=
meshWidth_dyadic_rounding (maxMeshIndex d n β)
(A * oracleMesh d n β γ) (mul_pos hA ho) hlarge hsmall
refine ⟨j, ?_, hjlo, ?_⟩
· simpa [meshIndices] using Nat.lt_succ_of_le hj
· convert hjhi using 1 <;> ring
Effective dimension decreases as the overlap-tail exponent increases. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
a b :
ℝ
ha :
1 < a
hab :
a ≤ b
conclusion
effectiveDimension d b ≤ effectiveDimension d a
Proof (Lean source)
lemma effectiveDimension_antitone (d : ℕ) {a b : ℝ}
(ha : 1 < a) (hab : a ≤ b) :
effectiveDimension d b ≤ effectiveDimension d a := by
have hb : 0 < b - 1 := by linarith
have ha' : 0 < a - 1 := by linarith
have hinv : (b - 1)⁻¹ ≤ (a - 1)⁻¹ := by
have hsub : a - 1 ≤ b - 1 := sub_le_sub_right hab 1
simpa only [one_div] using one_div_le_one_div_of_le ha' hsub
have haid : a / (a - 1) = 1 + (a - 1)⁻¹ := by
field_simp
ring
have hbid : b / (b - 1) = 1 + (b - 1)⁻¹ := by
field_simp
ring
unfold effectiveDimension
rw [mul_div_assoc, mul_div_assoc]
rw [haid, hbid]
gcongr
theorem oracleMesh_between_adaptation_endpoints Lemma oracleMesh_between_adaptation_endpoints in the paper ↗
Over a compact overlap range, every oracle width lies between the two endpoint oracle widths. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
oracleMesh d n β γ_max ≤ oracleMesh d n β γ
conclusion 2
oracleMesh d n β γ ≤ oracleMesh d n β γ_min
Proof (Lean source)
lemma oracleMesh_between_adaptation_endpoints (d n : ℕ) (β γ_min γ γ_max : ℝ)
(hn : 1 ≤ n) (hβ : 0 < β) (hγmin : 1 < γ_min)
(hγ : γ ∈ Icc γ_min γ_max) :
oracleMesh d n β γ_max ≤ oracleMesh d n β γ ∧
oracleMesh d n β γ ≤ oracleMesh d n β γ_min := by
have hγpos : 1 < γ := lt_of_lt_of_le hγmin hγ.1
have hγmax : 1 < γ_max := lt_of_lt_of_le hγpos hγ.2
have hDlo := effectiveDimension_antitone d hγpos hγ.2
have hDhi := effectiveDimension_antitone d hγmin hγ.1
have hdnonneg : 0 ≤ effectiveDimension d γ_max := by
unfold effectiveDimension
positivity
have hdenMax : 0 < 2 * β + effectiveDimension d γ_max := by positivity
have hden : 0 < 2 * β + effectiveDimension d γ := by linarith
have hdenMin : 0 < 2 * β + effectiveDimension d γ_min := by linarith
have hexpLo : -(1 : ℝ) / (2 * β + effectiveDimension d γ_max) ≤
-(1 : ℝ) / (2 * β + effectiveDimension d γ) := by
have hdenle : 2 * β + effectiveDimension d γ_max ≤
2 * β + effectiveDimension d γ := by linarith
have hrec := one_div_le_one_div_of_le hdenMax hdenle
simpa only [neg_div, one_div] using neg_le_neg hrec
have hexpHi : -(1 : ℝ) / (2 * β + effectiveDimension d γ) ≤
-(1 : ℝ) / (2 * β + effectiveDimension d γ_min) := by
have hdenle : 2 * β + effectiveDimension d γ ≤
2 * β + effectiveDimension d γ_min := by linarith
have hrec := one_div_le_one_div_of_le hden hdenle
simpa only [neg_div, one_div] using neg_le_neg hrec
have hn' : (1 : ℝ) ≤ n := by exact_mod_cast hn
constructor <;>
unfold oracleMesh <;>
exact Real.rpow_le_rpow_of_exponent_le hn' (by assumption)
One selected-count constant and one sample-size threshold work uniformly over the compact overlap-exponent range. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ_min γ_max :
ℝ
hd :
1 ≤ d
hβ :
1 < β
hB :
0 < B
hL :
0 < L
hC :
1 ≤ C
hcf :
0 < c_f ∧ c_f ≤ 1
hγmin :
1 < γ_min
hγrange :
γ_min < γ_max
conclusion
∃ K : ℝ,
conclusion 1
0 < K
conclusion 2
∃ n₀ : ℕ,
∀ n ≥ n₀,
∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin,
hγrange⟩, ∀ (P : Measure (Obs d)) [IsProbabilityMeasure P], P ∈ ModelClass d β B L C c_f γ → ∀ (Q : Measure (Fin n → Obs d)), IIDSampleLaw Q P → ∃ E : Set (Fin n → Obs d), MeasurableSet E ∧ Q.real Eᶜ
≤ (n : ℝ) ^ (-2 : ℝ) ∧
∀ ω ∈ E,
0 < countSelectedMesh (polynomialDegree β) β ω ∧
∀ (k : Fin d → Fin (2 ^ ((feasibleIndices (polynomialDegree β) β ω).sup id))) (ℓ : Fin d → Fin (polynomialDegree β + 1)),
(n : ℝ) * microcellMass P (polynomialDegree β) ((feasibleIndices (polynomialDegree β) β ω).sup id) k ℓ / 5
Proof (Lean source)
lemma selectedCountEvent_uniform
(d : ℕ) (β B L C c_f γ_min γ_max : ℝ)
(hd : 1 ≤ d) (hβ : 1 < β) (hB : 0 < B) (hL : 0 < L)
(hC : 1 ≤ C) (hcf : 0 < c_f ∧ c_f ≤ 1)
(hγmin : 1 < γ_min) (hγrange : γ_min < γ_max) :
∃ K : ℝ, 0 < K ∧ ∃ n₀ : ℕ,
∀ n ≥ n₀, ∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
∀ (P : Measure (Obs d)) [IsProbabilityMeasure P],
P ∈ ModelClass d β B L C c_f γ →
∀ (Q : Measure (Fin n → Obs d)), IIDSampleLaw Q P →
∃ E : Set (Fin n → Obs d),
MeasurableSet E ∧ Q.real Eᶜ ≤ (n : ℝ) ^ (-2 : ℝ) ∧
∀ ω ∈ E,
0 < countSelectedMesh (polynomialDegree β) β ω ∧
countSelectedMesh (polynomialDegree β) β ω ≤
K * oracleMesh d n β γ ∧
∀ (k : Fin d →
Fin (2 ^ ((feasibleIndices (polynomialDegree β) β ω).sup id)))
(ℓ : Fin d → Fin (polynomialDegree β + 1)),
(n : ℝ) * microcellMass P (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) k ℓ / 5 ≤
treatedCount (polynomialDegree β)
((feasibleIndices (polynomialDegree β) β ω).sup id) ω k ℓ := by
let m := polynomialDegree β
let a := c_f * templateEta d m ^ d
have ha : 0 < a := by
dsimp [a, m]
exact mul_pos hcf.1 (pow_pos (templateEta_pos d (polynomialDegree β)) d)
obtain ⟨κ, hκ, hκle⟩ := orderedMass_uniform_coefficient C a γ_min γ_max
(lt_of_lt_of_le zero_lt_one hC) ha hγmin hγrange
have hden₀ : 0 < 2 * β + (d : ℝ) := by positivity
have hden₁ : 1 ≤ 2 * β + (d : ℝ) := by
have hd' : (1 : ℝ) ≤ d := by exact_mod_cast hd
linarith
have hγmax : 1 < γ_max := hγmin.trans hγrange
have hdenMin : 0 < 2 * β + effectiveDimension d γ_max := by
unfold effectiveDimension
positivity
have hdenMax : 0 < 2 * β + effectiveDimension d γ_min := by
unfold effectiveDimension
positivity
obtain ⟨A, hA₁, hcoeffMin⟩ := selectedCount_oracle_enlargement κ
(2 * β + effectiveDimension d γ_max) hκ hdenMin
have hA : 0 < A := lt_of_lt_of_le zero_lt_one hA₁
have hdimMax : (d : ℝ) < effectiveDimension d γ_max := by
unfold effectiveDimension
apply (lt_div_iff₀ (by linarith : 0 < γ_max - 1)).2
have hd' : (0 : ℝ) < d := by exact_mod_cast (show 0 < d by omega)
nlinarith
obtain ⟨nsmall, hnsmall⟩ :=
maxMeshIndex_width_le_oracle_eventually d β γ_max A hden₀ hdimMax hA
obtain ⟨nlarge, hnlarge⟩ :=
selectedCount_power_eventually_le_one
(1 / (2 * β + effectiveDimension d γ_min)) A (by positivity) hA
let U : ℝ := 24 * ((d : ℝ) + 3)
let V : ℝ := 24 * log
(8 * ((polynomialDegree β + 1 : ℝ) ^ d + 1))
obtain ⟨nbudget, hnbudget⟩ :=
selectedCount_oracle_inverse_log_budget (2 * A) β
(2 * β + effectiveDimension d γ_min) U V (by positivity)
(by linarith) hdenMax (by dsimp [U]; positivity)
refine ⟨2 * A, by positivity,
max 2 (max nsmall (max nlarge nbudget)), ?_⟩
intro n hn γ hγ P _ hP Q hIID
have hn₂ : 2 ≤ n := le_trans (Nat.le_max_left _ _) hn
have hns : nsmall ≤ n :=
(le_trans (Nat.le_max_left _ _) (Nat.le_max_right _ _)).trans hn
have hnl : nlarge ≤ n :=
(le_trans (Nat.le_max_left _ _) (Nat.le_max_right _ _)).trans
((Nat.le_max_right _ _).trans hn)
have hnb : nbudget ≤ n :=
(le_trans (Nat.le_max_right _ _) (Nat.le_max_right _ _)).trans
((Nat.le_max_right _ _).trans hn)
have hγIcc : γ ∈ Icc γ_min γ_max := by
simpa only [adaptationRange] using hγ
have hγpos : 1 < γ := hγmin.trans_le hγIcc.1
have hden : 0 < 2 * β + effectiveDimension d γ := by
unfold effectiveDimension
positivity
have horacle := oracleMesh_between_adaptation_endpoints
d n β γ_min γ γ_max (by omega) (by linarith) hγmin hγIcc
have ho : 0 < oracleMesh d n β γ := by
unfold oracleMesh
have hn' : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
positivity
have hlargeMin : A * oracleMesh d n β γ_min ≤ 1 := by
simpa only [oracleMesh, neg_div] using hnlarge n hnl
have hlarge : A * oracleMesh d n β γ ≤ 1 :=
(mul_le_mul_of_nonneg_left horacle.2 hA.le).trans hlargeMin
have hsmallMax := hnsmall n hns
have hsmall : meshWidth (maxMeshIndex d n β) ≤
A * oracleMesh d n β γ :=
hsmallMax.trans (mul_le_mul_of_nonneg_left horacle.1 hA.le)
obtain ⟨j₀, hj₀, hjlo, hjhi⟩ :=
meshIndices_oracleCandidate_of_bounds d n β A γ hA ho hlarge hsmall
have hbudget : 24 * log (2 *
((card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) ×
(Fin d → Fin (polynomialDegree β + 1))) : ℝ) + 1) *
(n : ℝ) ^ 2) ≤ meshWidth j₀ ^ (-2 * β) := by
have hlog := selectedCount_log_card_bound d n
(polynomialDegree β) β hn₂ hden₁
have hh : 0 < meshWidth j₀ := by unfold meshWidth; positivity
have hscale : (2 * A) * oracleMesh d n β γ ≤
(2 * A) * oracleMesh d n β γ_min := by
exact mul_le_mul_of_nonneg_left horacle.2 (by positivity)
calc
24 * log (2 *
((card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) ×
(Fin d → Fin (polynomialDegree β + 1))) : ℝ) + 1) *
(n : ℝ) ^ 2) ≤ U * log n + V := by
dsimp [U, V]
nlinarith
_ ≤ ((2 * A) * oracleMesh d n β γ_min) ^ (-2 * β) := by
simpa only [oracleMesh, U, V] using hnbudget n hnb
_ ≤ ((2 * A) * oracleMesh d n β γ) ^ (-2 * β) := by
exact Real.rpow_le_rpow_of_nonpos (by positivity) hscale (by linarith)
_ ≤ meshWidth j₀ ^ (-2 * β) :=
Real.rpow_le_rpow_of_nonpos hh hjhi (by linarith)
have hD : effectiveDimension d γ_max ≤ effectiveDimension d γ :=
effectiveDimension_antitone d hγpos hγIcc.2
have hcoeff : 4 ≤ κ * A ^ (2 * β + effectiveDimension d γ) := by
exact hcoeffMin.trans (mul_le_mul_of_nonneg_left
(Real.rpow_le_rpow_of_exponent_le hA₁ (by linarith)) hκ.le)
have hκle' : κ ≤
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d (polynomialDegree β) ^ d) ^ (γ / (γ - 1)) := by
simpa only [a, m] using hκle γ hγIcc
exact selectedCountEvent_of_oracle_candidate_coefficient_lower
P hP Q hIID hn₂ (by linarith) hden A (2 * A) κ hA hκ hκle'
j₀ hj₀ hjlo hjhi hbudget hcoeff
theorem selectedCountEvent_uniform_design_saturated Lemma selectedCountEvent_uniform_design_saturated in the paper ↗
Uniform selected-count control with an explicitly design-saturated event. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ_min γ_max :
ℝ
hd :
1 ≤ d
hβ :
1 < β
hB :
0 < B
hL :
0 < L
hC :
1 ≤ C
hcf :
0 < c_f ∧ c_f ≤ 1
hγmin :
1 < γ_min
hγrange :
γ_min < γ_max
conclusion
∃ K : ℝ,
conclusion 1
0 < K
conclusion 2
∃ n₀ : ℕ,
∀ n ≥ n₀,
∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin,
hγrange⟩, ∀ (P : Measure (Obs d)) [IsProbabilityMeasure P], P ∈ ModelClass d β B L C c_f γ → ∀ (Q : Measure (Fin n → Obs d)), IIDSampleLaw Q P → ∃ E, SaturatedSelectedCountEvent β K γ P Q E
Proof (Lean source)
lemma selectedCountEvent_uniform_design_saturated
(d : ℕ) (β B L C c_f γ_min γ_max : ℝ)
(hd : 1 ≤ d) (hβ : 1 < β) (hB : 0 < B) (hL : 0 < L)
(hC : 1 ≤ C) (hcf : 0 < c_f ∧ c_f ≤ 1)
(hγmin : 1 < γ_min) (hγrange : γ_min < γ_max) :
∃ K : ℝ, 0 < K ∧ ∃ n₀ : ℕ,
∀ n ≥ n₀, ∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
∀ (P : Measure (Obs d)) [IsProbabilityMeasure P],
P ∈ ModelClass d β B L C c_f γ →
∀ (Q : Measure (Fin n → Obs d)), IIDSampleLaw Q P →
∃ E, SaturatedSelectedCountEvent β K γ P Q E := by
let m := polynomialDegree β
let a := c_f * templateEta d m ^ d
have ha : 0 < a := by
dsimp [a, m]
exact mul_pos hcf.1 (pow_pos (templateEta_pos d (polynomialDegree β)) d)
obtain ⟨κ, hκ, hκle⟩ := orderedMass_uniform_coefficient C a γ_min γ_max
(lt_of_lt_of_le zero_lt_one hC) ha hγmin hγrange
have hden₀ : 0 < 2 * β + (d : ℝ) := by positivity
have hden₁ : 1 ≤ 2 * β + (d : ℝ) := by
have hd' : (1 : ℝ) ≤ d := by exact_mod_cast hd
linarith
have hγmax : 1 < γ_max := hγmin.trans hγrange
have hdenMin : 0 < 2 * β + effectiveDimension d γ_max := by
unfold effectiveDimension
positivity
have hdenMax : 0 < 2 * β + effectiveDimension d γ_min := by
unfold effectiveDimension
positivity
obtain ⟨A, hA₁, hcoeffMin⟩ := selectedCount_oracle_enlargement κ
(2 * β + effectiveDimension d γ_max) hκ hdenMin
have hA : 0 < A := lt_of_lt_of_le zero_lt_one hA₁
have hdimMax : (d : ℝ) < effectiveDimension d γ_max := by
unfold effectiveDimension
apply (lt_div_iff₀ (by linarith : 0 < γ_max - 1)).2
have hd' : (0 : ℝ) < d := by exact_mod_cast (show 0 < d by omega)
nlinarith
obtain ⟨nsmall, hnsmall⟩ :=
maxMeshIndex_width_le_oracle_eventually d β γ_max A hden₀ hdimMax hA
obtain ⟨nlarge, hnlarge⟩ :=
selectedCount_power_eventually_le_one
(1 / (2 * β + effectiveDimension d γ_min)) A (by positivity) hA
let U : ℝ := 24 * ((d : ℝ) + 3)
let V : ℝ := 24 * log
(8 * ((polynomialDegree β + 1 : ℝ) ^ d + 1))
obtain ⟨nbudget, hnbudget⟩ :=
selectedCount_oracle_inverse_log_budget (2 * A) β
(2 * β + effectiveDimension d γ_min) U V (by positivity)
(by linarith) hdenMax (by dsimp [U]; positivity)
refine ⟨2 * A, by positivity,
max 2 (max nsmall (max nlarge nbudget)), ?_⟩
intro n hn γ hγ P _ hP Q hIID
have hn₂ : 2 ≤ n := le_trans (Nat.le_max_left _ _) hn
have hns : nsmall ≤ n :=
(le_trans (Nat.le_max_left _ _) (Nat.le_max_right _ _)).trans hn
have hnl : nlarge ≤ n :=
(le_trans (Nat.le_max_left _ _) (Nat.le_max_right _ _)).trans
((Nat.le_max_right _ _).trans hn)
have hnb : nbudget ≤ n :=
(le_trans (Nat.le_max_right _ _) (Nat.le_max_right _ _)).trans
((Nat.le_max_right _ _).trans hn)
have hγIcc : γ ∈ Icc γ_min γ_max := by
simpa only [adaptationRange] using hγ
have hγpos : 1 < γ := hγmin.trans_le hγIcc.1
have hden : 0 < 2 * β + effectiveDimension d γ := by
unfold effectiveDimension
positivity
have horacle := oracleMesh_between_adaptation_endpoints
d n β γ_min γ γ_max (by omega) (by linarith) hγmin hγIcc
have ho : 0 < oracleMesh d n β γ := by
unfold oracleMesh
have hn' : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
positivity
have hlargeMin : A * oracleMesh d n β γ_min ≤ 1 := by
simpa only [oracleMesh, neg_div] using hnlarge n hnl
have hlarge : A * oracleMesh d n β γ ≤ 1 :=
(mul_le_mul_of_nonneg_left horacle.2 hA.le).trans hlargeMin
have hsmallMax := hnsmall n hns
have hsmall : meshWidth (maxMeshIndex d n β) ≤
A * oracleMesh d n β γ :=
hsmallMax.trans (mul_le_mul_of_nonneg_left horacle.1 hA.le)
obtain ⟨j₀, hj₀, hjlo, hjhi⟩ :=
meshIndices_oracleCandidate_of_bounds d n β A γ hA ho hlarge hsmall
have hbudget : 24 * log (2 *
((card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) ×
(Fin d → Fin (polynomialDegree β + 1))) : ℝ) + 1) *
(n : ℝ) ^ 2) ≤ meshWidth j₀ ^ (-2 * β) := by
have hlog := selectedCount_log_card_bound d n
(polynomialDegree β) β hn₂ hden₁
have hh : 0 < meshWidth j₀ := by unfold meshWidth; positivity
have hscale : (2 * A) * oracleMesh d n β γ ≤
(2 * A) * oracleMesh d n β γ_min := by
exact mul_le_mul_of_nonneg_left horacle.2 (by positivity)
calc
24 * log (2 *
((card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) ×
(Fin d → Fin (polynomialDegree β + 1))) : ℝ) + 1) *
(n : ℝ) ^ 2) ≤ U * log n + V := by
dsimp [U, V]
nlinarith
_ ≤ ((2 * A) * oracleMesh d n β γ_min) ^ (-2 * β) := by
simpa only [oracleMesh, U, V] using hnbudget n hnb
_ ≤ ((2 * A) * oracleMesh d n β γ) ^ (-2 * β) := by
exact Real.rpow_le_rpow_of_nonpos (by positivity) hscale (by linarith)
_ ≤ meshWidth j₀ ^ (-2 * β) :=
Real.rpow_le_rpow_of_nonpos hh hjhi (by linarith)
have hD : effectiveDimension d γ_max ≤ effectiveDimension d γ :=
effectiveDimension_antitone d hγpos hγIcc.2
have hcoeff : 4 ≤ κ * A ^ (2 * β + effectiveDimension d γ) := by
exact hcoeffMin.trans (mul_le_mul_of_nonneg_left
(Real.rpow_le_rpow_of_exponent_le hA₁ (by linarith)) hκ.le)
have hκle' : κ ≤
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d (polynomialDegree β) ^ d) ^ (γ / (γ - 1)) := by
simpa only [a, m] using hκle γ hγIcc
exact selectedCountEvent_of_oracle_candidate_coefficient_lower_design_saturated
P hP Q hIID hn₂ (by linarith) hden A (2 * A) κ hA hκ hκle'
j₀ hj₀ hjlo hjhi hbudget hcoeff
theorem selectedCountEvent_design_saturated Lemma selectedCountEvent_design_saturated in the paper ↗
Fixed-overlap selected-count control with an explicitly design-saturated event. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
∃ K : ℝ,
conclusion 1
0 < K
conclusion 2
Proof (Lean source)
lemma selectedCountEvent_design_saturated (d : ℕ) (β B L C c_f γ : ℝ)
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ) :
∃ K : ℝ, 0 < K ∧ ∃ n₀ : ℕ, ∀ n ≥ n₀,
∀ (Q : Measure (Fin n → Obs d)), IIDSampleLaw Q P →
∃ E, SaturatedSelectedCountEvent β K γ P Q E := by
have hdom := hP.1
rcases hdom with ⟨hd, hβ, hB, hL, hC, hcf, hcf1⟩
have hγ : 1 < γ := hP.2.1
have hd' : (0 : ℝ) < d := by exact_mod_cast (show 0 < d by omega)
have hden : 0 < 2 * β + effectiveDimension d γ := by
unfold effectiveDimension
positivity
have hden₀ : 0 < 2 * β + (d : ℝ) := by positivity
have hden₁ : 1 ≤ 2 * β + (d : ℝ) := by linarith
have hdim : (d : ℝ) < effectiveDimension d γ := by
unfold effectiveDimension
apply (lt_div_iff₀ (by linarith : 0 < γ - 1)).2
nlinarith
let κ : ℝ := ((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d (polynomialDegree β) ^ d) ^ (γ / (γ - 1))
have hκ : 0 < κ := by
dsimp [κ]
have hη := templateEta_pos d (polynomialDegree β)
positivity
obtain ⟨A, hA₁, hcoeff⟩ := selectedCount_oracle_enlargement κ
(2 * β + effectiveDimension d γ) hκ hden
have hA : 0 < A := lt_of_lt_of_le zero_lt_one hA₁
obtain ⟨nsmall, hnsmall⟩ :=
maxMeshIndex_width_le_oracle_eventually d β γ A hden₀ hdim hA
obtain ⟨nlarge, hnlarge⟩ :=
selectedCount_power_eventually_le_one
(1 / (2 * β + effectiveDimension d γ)) A (by positivity) hA
let U : ℝ := 24 * ((d : ℝ) + 3)
let V : ℝ := 24 * log
(8 * ((polynomialDegree β + 1 : ℝ) ^ d + 1))
obtain ⟨nbudget, hnbudget⟩ :=
selectedCount_oracle_inverse_log_budget (2 * A) β
(2 * β + effectiveDimension d γ) U V (by positivity)
(by linarith) hden (by dsimp [U]; positivity)
refine ⟨2 * A, by positivity, max 2 (max nsmall (max nlarge nbudget)), ?_⟩
intro n hn Q hIID
have hn₂ : 2 ≤ n := le_trans (Nat.le_max_left _ _) hn
have hns : nsmall ≤ n :=
(le_trans (Nat.le_max_left _ _) (Nat.le_max_right _ _)).trans hn
have hnl : nlarge ≤ n :=
(le_trans (Nat.le_max_left _ _) (Nat.le_max_right _ _)).trans
((Nat.le_max_right _ _).trans hn)
have hnb : nbudget ≤ n :=
(le_trans (Nat.le_max_right _ _) (Nat.le_max_right _ _)).trans
((Nat.le_max_right _ _).trans hn)
have ho : 0 < oracleMesh d n β γ := by
unfold oracleMesh
have hn' : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
positivity
have hlarge : A * oracleMesh d n β γ ≤ 1 := by
simpa only [oracleMesh, neg_div] using hnlarge n hnl
have hsmall := hnsmall n hns
obtain ⟨j₀, hj₀, hjlo, hjhi⟩ :=
meshIndices_oracleCandidate_of_bounds d n β A γ hA ho hlarge hsmall
have hbudget : 24 * log (2 *
((card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) ×
(Fin d → Fin (polynomialDegree β + 1))) : ℝ) + 1) *
(n : ℝ) ^ 2) ≤ meshWidth j₀ ^ (-2 * β) := by
have hlog := selectedCount_log_card_bound d n
(polynomialDegree β) β hn₂ hden₁
have hh : 0 < meshWidth j₀ := by unfold meshWidth; positivity
calc
24 * log (2 *
((card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) ×
(Fin d → Fin (polynomialDegree β + 1))) : ℝ) + 1) *
(n : ℝ) ^ 2) ≤ U * log n + V := by
dsimp [U, V]
nlinarith
_ ≤ ((2 * A) * oracleMesh d n β γ) ^ (-2 * β) := by
simpa only [oracleMesh, U, V] using hnbudget n hnb
_ ≤ meshWidth j₀ ^ (-2 * β) :=
Real.rpow_le_rpow_of_nonpos hh hjhi (by linarith)
exact selectedCountEvent_of_oracle_candidate_design_saturated P hP Q hIID hn₂ (by linarith)
hden A (2 * A) hA j₀ hj₀ hjlo hjhi hbudget
(by simpa only [κ] using hcoeff)
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.SelectedMesh.Counts.Asymptotics 7 declarations Elementary asymptotic budgets for selected counts
Elementary asymptotic budgets for selected counts
Every positive power eventually pays for a fixed multiple of the logarithmic simultaneous-count budget. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
A p :
ℝ
hA :
0 ≤ A
hp :
0 < p
conclusion
∃ n₀ : ℕ, ∀ n ≥ n₀, A * log (n : ℝ) ≤ (n : ℝ) ^ p
Proof (Lean source)
lemma selectedCount_log_dominated (A p : ℝ) (hA : 0 ≤ A) (hp : 0 < p) :
∃ n₀ : ℕ, ∀ n ≥ n₀,
A * log (n : ℝ) ≤ (n : ℝ) ^ p := by
have hp2 : 0 < p / 2 := by positivity
have ht : Tendsto (fun n : ℕ => (n : ℝ) ^ (p / 2))
atTop atTop :=
(tendsto_rpow_atTop hp2).comp tendsto_natCast_atTop_atTop
obtain ⟨n₀, hn₀⟩ := Filter.eventually_atTop.1
(ht.eventually (Filter.eventually_ge_atTop (A * (2 / p))))
refine ⟨max n₀ 1, ?_⟩
intro n hn
have hnbase : 0 < (n : ℝ) := by
exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one
(le_trans (Nat.le_max_right _ _) hn))
have hpow : A * (2 / p) ≤ (n : ℝ) ^ (p / 2) :=
hn₀ n (le_trans (Nat.le_max_left _ _) hn)
have hlog := Real.log_natCast_le_rpow_div n hp2
have hfactor : A * log (n : ℝ) ≤
(A * (2 / p)) * (n : ℝ) ^ (p / 2) := by
have heq : (n : ℝ) ^ (p / 2) / (p / 2) =
(2 / p) * (n : ℝ) ^ (p / 2) := by field_simp
rw [heq] at hlog
exact (mul_le_mul_of_nonneg_left hlog hA).trans_eq (by ring)
calc
A * log (n : ℝ) ≤ (A * (2 / p)) * (n : ℝ) ^ (p / 2) := hfactor
_ ≤ (n : ℝ) ^ (p / 2) * (n : ℝ) ^ (p / 2) := by gcongr
_ = (n : ℝ) ^ p := by rw [← Real.rpow_add hnbase]; ring_nf
A power also pays for an affine logarithmic union-bound budget. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
A B p :
ℝ
hA :
0 ≤ A
hp :
0 < p
conclusion
∃ n₀ : ℕ, ∀ n ≥ n₀, A * log (n : ℝ) + B ≤ (n : ℝ) ^ p
Proof (Lean source)
lemma selectedCount_affine_log_dominated (A B p : ℝ)
(hA : 0 ≤ A) (hp : 0 < p) :
∃ n₀ : ℕ, ∀ n ≥ n₀,
A * log (n : ℝ) + B ≤ (n : ℝ) ^ p := by
obtain ⟨n₁, hn₁⟩ := selectedCount_log_dominated (2 * A) p (by positivity) hp
have ht : Tendsto (fun n : ℕ => (n : ℝ) ^ p)
atTop atTop :=
(tendsto_rpow_atTop hp).comp tendsto_natCast_atTop_atTop
obtain ⟨n₂, hn₂⟩ := Filter.eventually_atTop.1
(ht.eventually (Filter.eventually_ge_atTop (2 * B)))
refine ⟨max n₁ n₂, ?_⟩
intro n hn
have hlog := hn₁ n (le_trans (Nat.le_max_left _ _) hn)
have hpow := hn₂ n (le_trans (Nat.le_max_right _ _) hn)
nlinarith
A faster decaying power eventually lies below any fixed multiple of a slower decaying one. This places the oracle width above the finest candidate width when the effective dimension exceeds the ordinary dimension. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
a b A :
ℝ
ha :
a < b
hA :
0 < A
conclusion
∃ n₀ : ℕ, ∀ n ≥ n₀, 2 * (n : ℝ) ^ (-b) ≤ A * (n : ℝ) ^ (-a)
Proof (Lean source)
lemma selectedCount_power_gap (a b A : ℝ) (ha : a < b) (hA : 0 < A) :
∃ n₀ : ℕ, ∀ n ≥ n₀,
2 * (n : ℝ) ^ (-b) ≤ A * (n : ℝ) ^ (-a) := by
have hp : 0 < b - a := by linarith
have ht : Tendsto (fun n : ℕ => (n : ℝ) ^ (b - a))
atTop atTop :=
(tendsto_rpow_atTop hp).comp tendsto_natCast_atTop_atTop
obtain ⟨n₀, hn₀⟩ := Filter.eventually_atTop.1
(ht.eventually (Filter.eventually_ge_atTop (2 / A)))
refine ⟨max n₀ 1, ?_⟩
intro n hn
have hnbase : (0 : ℝ) < n := by
exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one
(le_trans (Nat.le_max_right _ _) hn))
have hgap : 2 / A ≤ (n : ℝ) ^ (b - a) :=
hn₀ n (le_trans (Nat.le_max_left _ _) hn)
have hbpos : 0 < (n : ℝ) ^ (-b) := Real.rpow_pos_of_pos hnbase _
have heq : (n : ℝ) ^ (b - a) * (n : ℝ) ^ (-b) =
(n : ℝ) ^ (-a) := by
rw [← Real.rpow_add hnbase]
congr 1
ring
have h := mul_le_mul_of_nonneg_right hgap hbpos.le
rw [div_mul_eq_mul_div, div_le_iff₀ hA] at h
calc
2 * (n : ℝ) ^ (-b) ≤ A *
((n : ℝ) ^ (b - a) * (n : ℝ) ^ (-b)) := by nlinarith
_ = A * (n : ℝ) ^ (-a) := by rw [heq]
theorem selectedCount_power_eventually_le_one Lemma selectedCount_power_eventually_le_one in the paper ↗
A fixed multiple of a negative power is eventually at most one. This places a fixed enlargement of the oracle width below the coarsest mesh. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
p A :
ℝ
hp :
0 < p
hA :
0 < A
conclusion
∃ n₀ : ℕ, ∀ n ≥ n₀, A * (n : ℝ) ^ (-p) ≤ 1
Proof (Lean source)
lemma selectedCount_power_eventually_le_one (p A : ℝ)
(hp : 0 < p) (hA : 0 < A) :
∃ n₀ : ℕ, ∀ n ≥ n₀, A * (n : ℝ) ^ (-p) ≤ 1 := by
obtain ⟨n₀, hn₀⟩ := selectedCount_power_gap 0 p (2 / A) hp (by positivity)
refine ⟨n₀, ?_⟩
intro n hn
have h := hn₀ n hn
have hAne : A ≠ 0 := ne_of_gt hA
simp only [neg_zero, Real.rpow_zero, mul_one] at h
calc
A * (n : ℝ) ^ (-p) ≤ A * (1 / A) := by
apply mul_le_mul_of_nonneg_left _ hA.le
have hdiv : 2 * (n : ℝ) ^ (-p) ≤ 2 / A := by simpa using h
have heq : 2 / A = 2 * (1 / A) := by ring
linarith
_ = 1 := by field_simp
theorem selectedCount_affine_log_dominated_scaled Lemma selectedCount_affine_log_dominated_scaled in the paper ↗
A positive fixed fraction of a power eventually pays for an affine logarithmic budget. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
A B p R :
ℝ
hA :
0 ≤ A
hp :
0 < p
hR :
0 < R
conclusion
∃ n₀ : ℕ, ∀ n ≥ n₀, A * log (n : ℝ) + B ≤ R * (n : ℝ) ^ p
Proof (Lean source)
lemma selectedCount_affine_log_dominated_scaled (A B p R : ℝ)
(hA : 0 ≤ A) (hp : 0 < p) (hR : 0 < R) :
∃ n₀ : ℕ, ∀ n ≥ n₀,
A * log (n : ℝ) + B ≤ R * (n : ℝ) ^ p := by
obtain ⟨n₀, hn₀⟩ :=
selectedCount_affine_log_dominated (A / R) (B / R) p
(div_nonneg hA hR.le) hp
refine ⟨n₀, ?_⟩
intro n hn
have h := hn₀ n hn
have hmul := mul_le_mul_of_nonneg_left h hR.le
calc
A * log (n : ℝ) + B =
R * (A / R * log (n : ℝ) + B / R) := by
field_simp
_ ≤ R * (n : ℝ) ^ p := hmul
Choose a single oracle enlargement large enough for the effective dimension mass inequality. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
κ p :
ℝ
hκ :
0 < κ
hp :
0 < p
conclusion
∃ A : ℝ,
conclusion 1
1 ≤ A
conclusion 2
4 ≤ κ * A ^ p
Proof (Lean source)
lemma selectedCount_oracle_enlargement (κ p : ℝ)
(hκ : 0 < κ) (hp : 0 < p) :
∃ A : ℝ, 1 ≤ A ∧ 4 ≤ κ * A ^ p := by
have ht : Tendsto (fun A : ℝ => A ^ p)
atTop atTop := tendsto_rpow_atTop hp
obtain ⟨A₀, hA₀⟩ := Filter.eventually_atTop.1
(ht.eventually (Filter.eventually_ge_atTop (4 / κ)))
let A := max 1 A₀
have hA : 1 ≤ A := le_max_left _ _
have hpow : 4 / κ ≤ A ^ p := hA₀ A (le_max_right _ _)
refine ⟨A, hA, ?_⟩
have hmul := mul_le_mul_of_nonneg_left hpow hκ.le
calc
4 = κ * (4 / κ) := by field_simp
_ ≤ κ * A ^ p := hmul
theorem selectedCount_oracle_inverse_log_budget Lemma selectedCount_oracle_inverse_log_budget in the paper ↗
At an oracle-order width, the inverse-squared smoothness threshold eventually dominates any affine logarithmic count budget. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
A β q U V :
ℝ
hA :
0 < A
hβ :
0 < β
hq :
0 < q
hU :
0 ≤ U
conclusion
∃ n₀ : ℕ, ∀ n ≥ n₀, U * log (n : ℝ) + V ≤ (A * (n : ℝ) ^ (-(1 : ℝ) / q)) ^ (-2 * β)
Proof (Lean source)
lemma selectedCount_oracle_inverse_log_budget
(A β q U V : ℝ) (hA : 0 < A) (hβ : 0 < β) (hq : 0 < q)
(hU : 0 ≤ U) :
∃ n₀ : ℕ, ∀ n ≥ n₀,
U * log (n : ℝ) + V ≤
(A * (n : ℝ) ^ (-(1 : ℝ) / q)) ^ (-2 * β) := by
have hp : 0 < 2 * β / q := by positivity
have hR : 0 < A ^ (-2 * β) := Real.rpow_pos_of_pos hA _
obtain ⟨n₀, hn₀⟩ :=
selectedCount_affine_log_dominated_scaled U V (2 * β / q)
(A ^ (-2 * β)) hU hp hR
refine ⟨max n₀ 1, ?_⟩
intro n hn
have hnpos : (0 : ℝ) < n := by
exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one
(le_trans (Nat.le_max_right _ _) hn))
calc
U * log (n : ℝ) + V ≤
A ^ (-2 * β) * (n : ℝ) ^ (2 * β / q) :=
hn₀ n (le_trans (Nat.le_max_left _ _) hn)
_ = (A * (n : ℝ) ^ (-(1 : ℝ) / q)) ^ (-2 * β) := by
rw [Real.mul_rpow hA.le (Real.rpow_nonneg hnpos.le _)]
rw [← Real.rpow_mul hnpos.le]
congr 1
field_simp
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.SelectedMesh.Counts.Base 17 declarations Simultaneous treated-count event for the selected mesh
Simultaneous treated-count event for the selected mesh
A full-sample event depends only on the covariates and treatment arms. For the stated inputs and conditions, the CountDesignSaturated object being defined.
The treated microcell count is the sum of its Bernoulli indicators. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma treatedCount_eq_indicator_sum {d n : ℕ} (m j : ℕ)
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(ℓ : Fin d → Fin (m + 1)) :
(treatedCount m j ω k ℓ : ℝ) =
∑ i : Fin n,
if (ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ
then (1 : ℝ) else 0 := by
classical
simp [treatedCount]
theorem treatedMicrocellIndicator_measurable Lemma treatedMicrocellIndicator_measurable in the paper ↗
The treated microcell indicator is measurable. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma treatedMicrocellIndicator_measurable {d : ℕ} (m j : ℕ)
(k : Fin d → Fin (2 ^ j)) (ℓ : Fin d → Fin (m + 1)) :
Measurable (fun z : Obs d =>
if z.2.1 = true ∧ z.1 ∈ scaledMicroCell d m j k ℓ
then (1 : ℝ) else 0) := by
classical
have hs : MeasurableSet {z : Obs d | z.2.1 = true ∧
z.1 ∈ scaledMicroCell d m j k ℓ} := by
exact ((measurableSet_singleton true).preimage
(measurable_fst.comp measurable_snd)).inter
((orderedMass_scaledMicroCell_measurable d m j k ℓ).preimage measurable_fst)
exact Measurable.ite hs measurable_const measurable_const
Each treated microcell count is measurable on the finite product sample. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma treatedCount_measurable {d n : ℕ} (m j : ℕ)
(k : Fin d → Fin (2 ^ j)) (ℓ : Fin d → Fin (m + 1)) :
Measurable (fun ω : Fin n → Obs d => (treatedCount m j ω k ℓ : ℝ)) := by
classical
have hsum : Measurable (fun ω : Fin n → Obs d =>
∑ i : Fin n,
if (ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ
then (1 : ℝ) else 0) := by
apply Finset.measurable_fun_sum
intro i hi
exact (treatedMicrocellIndicator_measurable m j k ℓ).comp
(measurable_pi_apply i)
convert hsum using 1
funext ω
exact treatedCount_eq_indicator_sum m j ω k ℓ
theorem finiteProduct_bernoulliCount_deviation Lemma finiteProduct_bernoulliCount_deviation in the paper ↗
Bernstein deviation for a Bernoulli count on the finite product law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
N :
ℕ
X :
Type*
P :
f :
X → ℝ
hf :
hf01 :
∀ x, f x = 0 ∨ f x = 1
hN :
0 < N
p η :
ℝ
hp :
0 < p
hη :
0 < η
hmean :
∫ x, f x ∂P = p
Proof (Lean source)
lemma finiteProduct_bernoulliCount_deviation
{N : ℕ} {X : Type*} [MeasurableSpace X]
(P : Measure X) [IsProbabilityMeasure P] (f : X → ℝ)
(hf : Measurable f) (hf01 : ∀ x, f x = 0 ∨ f x = 1)
(hN : 0 < N) (p η : ℝ) (hp : 0 < p) (hη : 0 < η)
(hmean : ∫ x, f x ∂P = p) :
(Measure.pi (fun _ : Fin N => P)).real
{ω | η ≤ |(∑ i, f (ω i)) - (N : ℝ) * p|} ≤
2 * exp (-η ^ 2 / (2 * (2 * (N : ℝ) * p + η))) := by
have hfIcc : ∀ x, f x ∈ Icc (0 : ℝ) 1 := by
intro x
rcases hf01 x with hx | hx <;> simp [hx]
have hfint : Integrable f P := by
refine Integrable.of_bound hf.aestronglyMeasurable 1 ?_
filter_upwards with x
rcases hf01 x with hx | hx <;> simp [hx]
have hp_le : p ≤ 1 := by
calc
p = ∫ x, f x ∂P := hmean.symm
_ ≤ ∫ _x, (1 : ℝ) ∂P := integral_mono_ae hfint (integrable_const 1)
(ae_of_all _ fun x => (hfIcc x).2)
_ = 1 := by simp
have henvelope : ∀ᵐ x ∂P, |f x - ∫ y, f y ∂P| ≤ 1 := by
filter_upwards with x
rw [hmean]
exact abs_le.2 ⟨by linarith [(hfIcc x).1], by linarith [(hfIcc x).2]⟩
have hvariance : ∫ x, (f x - ∫ y, f y ∂P) ^ 2 ∂P ≤ p := by
rw [hmean]
have hf2int : Integrable (fun x => f x ^ 2) P := by
exact Integrable.of_bound (hf.pow_const 2).aestronglyMeasurable 1
(ae_of_all _ fun x => by rcases hf01 x with hx | hx <;> simp [hx])
calc
∫ x, (f x - p) ^ 2 ∂P =
(∫ x, f x ^ 2 ∂P) - 2 * p * (∫ x, f x ∂P) + p ^ 2 := by
rw [show (fun x => (f x - p) ^ 2) =
fun x => f x ^ 2 - 2 * p * f x + p ^ 2 by funext x; ring]
integral_linearity
rw [integral_const]
simp
_ = p - p ^ 2 := by
rw [show (∫ x, f x ^ 2 ∂P) = p by
rw [← hmean]
apply integral_congr_ae
filter_upwards with x
rcases hf01 x with hx | hx <;> simp [hx]]
rw [hmean]
ring
_ ≤ p := by nlinarith [sq_nonneg p]
let g : Unit → X → ℝ := fun _ => f
let b : Unit → ℝ := fun _ => 1
let sigma2 : Unit → ℝ := fun _ => p
let eta : Unit → ℝ := fun _ => η
have hbern := iid_sum_bernstein_union_bound
P g (fun _ => hf) (fun _ => hfint) b sigma2 eta
(fun _ => by simp [b]) (fun _ => hp.le) (fun _ => hη)
hN (fun _ => by simpa [g] using henvelope)
(fun _ => by simpa [g] using hvariance)
simpa [eta, sigma2, b, g, hmean] using hbern
The one-observation treated microcell indicator has the population cell mass as mean. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
∫ z : Obs d, (if z.2.1 = true ∧ z.1 ∈ scaledMicroCell d m j k ℓ then (1 : ℝ) else 0) ∂P
= microcellMass P m j k ℓ
Proof (Lean source)
lemma treatedMicrocellIndicator_mean {d : ℕ} (P : Measure (Obs d))
[IsProbabilityMeasure P] (m j : ℕ)
(k : Fin d → Fin (2 ^ j)) (ℓ : Fin d → Fin (m + 1)) :
∫ z : Obs d, (if z.2.1 = true ∧ z.1 ∈ scaledMicroCell d m j k ℓ
then (1 : ℝ) else 0) ∂P = microcellMass P m j k ℓ := by
classical
have hs : MeasurableSet {z : Obs d | z.2.1 = true ∧
z.1 ∈ scaledMicroCell d m j k ℓ} := by
exact ((measurableSet_singleton true).preimage
(measurable_fst.comp measurable_snd)).inter
((orderedMass_scaledMicroCell_measurable d m j k ℓ).preimage measurable_fst)
simpa [indicator, microcellMass] using
(integral_indicator_one (μ := P) hs)
Bernstein deviation for one treated microcell count. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
(Measure.pi (fun _ : Fin n => P)).real {ω | η ≤ |(treatedCount m j ω k ℓ : ℝ) - (n : ℝ) * microcellMass P m j k ℓ|}
≤ 2 * exp (-η ^ 2 / (2 * (2 * (n : ℝ) * microcellMass P m j k ℓ + η)))
Proof (Lean source)
lemma treatedCount_deviation {d n : ℕ} (P : Measure (Obs d))
[IsProbabilityMeasure P] (m j : ℕ)
(k : Fin d → Fin (2 ^ j)) (ℓ : Fin d → Fin (m + 1))
(hn : 0 < n) (hq : 0 < microcellMass P m j k ℓ)
(η : ℝ) (hη : 0 < η) :
(Measure.pi (fun _ : Fin n => P)).real
{ω | η ≤ |(treatedCount m j ω k ℓ : ℝ) -
(n : ℝ) * microcellMass P m j k ℓ|} ≤
2 * exp (-η ^ 2 /
(2 * (2 * (n : ℝ) * microcellMass P m j k ℓ + η))) := by
let f : Obs d → ℝ := fun z =>
if z.2.1 = true ∧ z.1 ∈ scaledMicroCell d m j k ℓ then 1 else 0
have hf : Measurable f := treatedMicrocellIndicator_measurable m j k ℓ
have hf01 : ∀ z, f z = 0 ∨ f z = 1 := by
intro z
dsimp [f]
split_ifs <;> simp
have hmean : ∫ z, f z ∂P = microcellMass P m j k ℓ :=
treatedMicrocellIndicator_mean P m j k ℓ
have htail := finiteProduct_bernoulliCount_deviation P f hf hf01 hn
(microcellMass P m j k ℓ) η hq hη hmean
convert htail using 1
congr 1
ext ω
simp only [Set.mem_ofPred_eq]
rw [treatedCount_eq_indicator_sum]
A deviation threshold linear in the mean and the logarithmic budget. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
(Measure.pi (fun _ : Fin n => P)).real {ω | (n : ℝ) * microcellMass P m j k ℓ / 2 + 4 * s ≤ |(treatedCount m j ω k ℓ : ℝ) - (n : ℝ) * microcellMass P m j k ℓ|}
≤ 2 * exp (-s)
Proof (Lean source)
lemma treatedCount_linear_deviation {d n : ℕ} (P : Measure (Obs d))
[IsProbabilityMeasure P] (m j : ℕ)
(k : Fin d → Fin (2 ^ j)) (ℓ : Fin d → Fin (m + 1))
(hn : 0 < n) (hq : 0 < microcellMass P m j k ℓ)
(s : ℝ) (hs : 0 < s) :
(Measure.pi (fun _ : Fin n => P)).real
{ω | (n : ℝ) * microcellMass P m j k ℓ / 2 + 4 * s ≤
|(treatedCount m j ω k ℓ : ℝ) -
(n : ℝ) * microcellMass P m j k ℓ|} ≤
2 * exp (-s) := by
let lam : ℝ := (n : ℝ) * microcellMass P m j k ℓ
let η : ℝ := lam / 2 + 4 * s
have hlam : 0 ≤ lam := by dsimp [lam]; positivity
have hη : 0 < η := by dsimp [η]; positivity
have hden : 0 < 2 * (2 * lam + η) := by positivity
have hquad : 2 * s * (2 * lam + η) ≤ η ^ 2 := by
dsimp [η]
nlinarith [sq_nonneg (lam / 2 - s), sq_nonneg s]
have hexp : -η ^ 2 / (2 * (2 * lam + η)) ≤ -s := by
apply (div_le_iff₀ hden).2
nlinarith [hquad]
have htail := treatedCount_deviation P m j k ℓ hn hq η hη
have hbound := Real.exp_le_exp.mpr hexp
dsimp [η, lam] at htail hbound ⊢
simp only [mul_assoc] at htail hbound ⊢
exact htail.trans (mul_le_mul_of_nonneg_left hbound (by norm_num))
One design-measurable event controls any finite collection of treated microcell counts, including cells taken from different candidate meshes. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
P :
Measure (Obs d)
hIID :
IIDSampleLaw Q P
hn :
0 < n
ι :
Type*
Fintype ι
m :
ℕ
j :
ι → ℕ
hq :
∀ a, 0 < microcellMass P m (j a) (k a) (ℓ a)
s :
ℝ
hs :
0 < s
conclusion
Q.real Eᶜ ≤ 2 * (Fintype.card ι : ℝ) * exp (-s)
conclusion 1
ω ∈ E :
a :
ι
|(treatedCount m (j a) ω (k a) (ℓ a) : ℝ)
- (n : ℝ) * microcellMass P m (j a) (k a) (ℓ a)| < (n : ℝ) * microcellMass P m (j a) (k a) (ℓ a) / 2
+ 4 * s
Proof (Lean source)
lemma finiteFamily_treatedCount_event {d n : ℕ} (P : Measure (Obs d))
[IsProbabilityMeasure P] (Q : Measure (Fin n → Obs d))
(hIID : IIDSampleLaw Q P) (hn : 0 < n)
{ι : Type*} [Fintype ι] (m : ℕ) (j : ι → ℕ)
(k : ∀ a, Fin d → Fin (2 ^ j a))
(ℓ : ∀ _a, Fin d → Fin (m + 1))
(hq : ∀ a, 0 < microcellMass P m (j a) (k a) (ℓ a))
(s : ℝ) (hs : 0 < s) :
∃ E : Set (Fin n → Obs d), MeasurableSet E ∧
Q.real Eᶜ ≤ 2 * (Fintype.card ι : ℝ) * exp (-s) ∧
∀ ω ∈ E, ∀ a : ι,
|(treatedCount m (j a) ω (k a) (ℓ a) : ℝ) -
(n : ℝ) * microcellMass P m (j a) (k a) (ℓ a)| <
(n : ℝ) * microcellMass P m (j a) (k a) (ℓ a) / 2 + 4 * s := by
let bad (a : ι) : Set (Fin n → Obs d) :=
{ω | (n : ℝ) * microcellMass P m (j a) (k a) (ℓ a) / 2 + 4 * s ≤
|(treatedCount m (j a) ω (k a) (ℓ a) : ℝ) -
(n : ℝ) * microcellMass P m (j a) (k a) (ℓ a)|}
let E : Set (Fin n → Obs d) := (⋃ a, bad a)ᶜ
have hbadMeas (a : ι) : MeasurableSet (bad a) := by
dsimp [bad]
exact measurableSet_le measurable_const
((treatedCount_measurable m (j a) (k a) (ℓ a)).sub measurable_const).abs
have hE : MeasurableSet E := by
dsimp [E]
exact (MeasurableSet.iUnion hbadMeas).compl
refine ⟨E, hE, ?_, ?_⟩
· dsimp [E]
rw [compl_compl]
calc
Q.real (⋃ a, bad a) ≤ ∑ a : ι, Q.real (bad a) :=
measureReal_iUnion_fintype_le bad
_ ≤ ∑ _a : ι, 2 * exp (-s) := by
apply Finset.sum_le_sum
intro a _
rw [hIID]
exact treatedCount_linear_deviation P m (j a) (k a) (ℓ a)
hn (hq a) s hs
_ = 2 * (Fintype.card ι : ℝ) * exp (-s) := by
simp [mul_assoc, mul_comm]
· intro ω hω a
have hnot : ω ∉ bad a := by
intro ha
exact hω (Set.mem_iUnion.mpr ⟨a, ha⟩)
exact not_le.mp hnot
theorem finiteFamily_treatedCount_event_design_saturated Lemma finiteFamily_treatedCount_event_design_saturated in the paper ↗
The explicit simultaneous count event can be chosen saturated with respect to the complete covariate-and-treatment design. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
P :
Measure (Obs d)
hIID :
IIDSampleLaw Q P
hn :
0 < n
ι :
Type*
Fintype ι
m :
ℕ
j :
ι → ℕ
hq :
∀ a, 0 < microcellMass P m (j a) (k a) (ℓ a)
s :
ℝ
hs :
0 < s
conclusion
Q.real Eᶜ ≤ 2 * (Fintype.card ι : ℝ) * exp (-s)
conclusion 1
ω ∈ E :
a :
ι
|(treatedCount m (j a) ω (k a) (ℓ a) : ℝ)
- (n : ℝ) * microcellMass P m (j a) (k a) (ℓ a)| < (n : ℝ) * microcellMass P m (j a) (k a) (ℓ a) / 2
+ 4 * s
conclusion 2
ω ξ :
(fun i => ((ξ i).1, (ξ i).2.1)) = (fun i => ((ω i).1, (ω i).2.1))
ω ∈ E ↔ ξ ∈ E
Proof (Lean source)
lemma finiteFamily_treatedCount_event_design_saturated {d n : ℕ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(Q : Measure (Fin n → Obs d)) (hIID : IIDSampleLaw Q P) (hn : 0 < n)
{ι : Type*} [Fintype ι] (m : ℕ) (j : ι → ℕ)
(k : ∀ a, Fin d → Fin (2 ^ j a))
(ℓ : ∀ _a, Fin d → Fin (m + 1))
(hq : ∀ a, 0 < microcellMass P m (j a) (k a) (ℓ a))
(s : ℝ) (hs : 0 < s) :
∃ E : Set (Fin n → Obs d), MeasurableSet E ∧
Q.real Eᶜ ≤ 2 * (Fintype.card ι : ℝ) * exp (-s) ∧
(∀ ω ∈ E, ∀ a : ι,
|(treatedCount m (j a) ω (k a) (ℓ a) : ℝ) -
(n : ℝ) * microcellMass P m (j a) (k a) (ℓ a)| <
(n : ℝ) * microcellMass P m (j a) (k a) (ℓ a) / 2 + 4 * s) ∧
∀ ω ξ, (fun i => ((ξ i).1, (ξ i).2.1)) =
(fun i => ((ω i).1, (ω i).2.1)) → (ω ∈ E ↔ ξ ∈ E) := by
let bad (a : ι) : Set (Fin n → Obs d) :=
{ω | (n : ℝ) * microcellMass P m (j a) (k a) (ℓ a) / 2 + 4 * s ≤
|(treatedCount m (j a) ω (k a) (ℓ a) : ℝ) -
(n : ℝ) * microcellMass P m (j a) (k a) (ℓ a)|}
let E : Set (Fin n → Obs d) := (⋃ a, bad a)ᶜ
have hbadMeas (a : ι) : MeasurableSet (bad a) := by
dsimp [bad]
exact measurableSet_le measurable_const
((treatedCount_measurable m (j a) (k a) (ℓ a)).sub measurable_const).abs
have hE : MeasurableSet E := by
dsimp [E]
exact (MeasurableSet.iUnion hbadMeas).compl
have hcountEq (ω ξ : Fin n → Obs d)
(hdesign : (fun i => ((ξ i).1, (ξ i).2.1)) =
(fun i => ((ω i).1, (ω i).2.1))) (a : ι) :
treatedCount m (j a) ξ (k a) (ℓ a) =
treatedCount m (j a) ω (k a) (ℓ a) := by
classical
unfold treatedCount
congr 1
ext i
have hi := congrFun hdesign i
have hx : (ξ i).1 = (ω i).1 :=
congrArg (fun z : (Fin d → ℝ) × Bool => z.1) hi
have ha : (ξ i).2.1 = (ω i).2.1 :=
congrArg (fun z : (Fin d → ℝ) × Bool => z.2) hi
simp [hx, ha]
refine ⟨E, hE, ?_, ?_, ?_⟩
· dsimp [E]
rw [compl_compl]
calc
Q.real (⋃ a, bad a) ≤ ∑ a : ι, Q.real (bad a) :=
measureReal_iUnion_fintype_le bad
_ ≤ ∑ _a : ι, 2 * exp (-s) := by
apply Finset.sum_le_sum
intro a _
rw [hIID]
exact treatedCount_linear_deviation P m (j a) (k a) (ℓ a)
hn (hq a) s hs
_ = 2 * (Fintype.card ι : ℝ) * exp (-s) := by
simp [mul_assoc, mul_comm]
· intro ω hω a
have hnot : ω ∉ bad a := by
intro ha
exact hω (Set.mem_iUnion.mpr ⟨a, ha⟩)
exact not_le.mp hnot
· intro ω ξ hdesign
have hbadIff (a : ι) : ω ∈ bad a ↔ ξ ∈ bad a := by
dsimp [bad]
rw [hcountEq ω ξ hdesign a]
constructor
· intro hω hξ
apply hω
rcases Set.mem_iUnion.mp hξ with ⟨a, ha⟩
exact Set.mem_iUnion.mpr ⟨a, (hbadIff a).mpr ha⟩
· intro hξ hω
apply hξ
rcases Set.mem_iUnion.mp hω with ⟨a, ha⟩
exact Set.mem_iUnion.mpr ⟨a, (hbadIff a).mp ha⟩
theorem selectedCount_lower_of_uniform_deviation Lemma selectedCount_lower_of_uniform_deviation in the paper ↗
A feasible count that exceeds the uniform deviation threshold also controls its population mean from below. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
N q t :
ℝ
ht :
0 ≤ t
hlarge :
6 * t ≤ N
hdeviation :
|N - q| ≤ q / 2 + t
conclusion
q / 5 ≤ N
Proof (Lean source)
lemma selectedCount_lower_of_uniform_deviation (N q t : ℝ)
(ht : 0 ≤ t) (hlarge : 6 * t ≤ N)
(hdeviation : |N - q| ≤ q / 2 + t) : q / 5 ≤ N := by
rcases abs_le.mp hdeviation with ⟨hlower, hupper⟩
linarith
theorem selectedCount_lower_on_feasible_mesh Lemma selectedCount_lower_on_feasible_mesh in the paper ↗
On a feasible mesh, a uniform count deviation yields the selected-cell population lower bound once the feasibility threshold dominates the tail budget. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
β s :
ℝ
P :
Measure (Obs d)
ω :
Fin n → Obs d
hfeasible :
meshFeasible m β ω j
hbudget :
24 * s ≤ meshWidth j ^ (-2 * β)
hdeviation :
|(treatedCount m j ω k ℓ : ℝ) - (n : ℝ) * microcellMass P m j k ℓ|
≤ (n : ℝ) * microcellMass P m j k ℓ / 2 + 4 * s
hs :
0 ≤ s
conclusion
(n : ℝ) * microcellMass P m j k ℓ / 5 ≤ treatedCount m j ω k ℓ
Proof (Lean source)
lemma selectedCount_lower_on_feasible_mesh {d n : ℕ} (m j : ℕ) (β s : ℝ)
(P : Measure (Obs d)) [IsProbabilityMeasure P] (ω : Fin n → Obs d)
(k : Fin d → Fin (2 ^ j))
(ℓ : Fin d → Fin (m + 1))
(hfeasible : meshFeasible m β ω j)
(hbudget : 24 * s ≤ meshWidth j ^ (-2 * β))
(hdeviation : |(treatedCount m j ω k ℓ : ℝ) -
(n : ℝ) * microcellMass P m j k ℓ| ≤
(n : ℝ) * microcellMass P m j k ℓ / 2 + 4 * s)
(hs : 0 ≤ s) :
(n : ℝ) * microcellMass P m j k ℓ / 5 ≤
treatedCount m j ω k ℓ := by
have hcount : meshWidth j ^ (-2 * β) ≤
(treatedCount m j ω k ℓ : ℝ) := hfeasible.2 k ℓ
exact selectedCount_lower_of_uniform_deviation
_ _ (4 * s) (by positivity) (by linarith) hdeviation
theorem oracleWitness_feasible_of_deviation Lemma oracleWitness_feasible_of_deviation in the paper ↗
An oracle-scale candidate is feasible when every expected treated count dominates its feasibility threshold and the simultaneous deviation budget. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
β s :
ℝ
P :
Measure (Obs d)
ω :
Fin n → Obs d
hcandidate :
j ∈ meshIndices d n β
hbudget :
4 * s ≤ meshWidth j ^ (-2 * β)
hmass :
hdeviation :
|(treatedCount m j ω k ℓ : ℝ) - (n : ℝ) * microcellMass P m j k ℓ|
≤ (n : ℝ) * microcellMass P m j k ℓ / 2 + 4 * s
conclusion
meshFeasible m β ω j
Proof (Lean source)
lemma oracleWitness_feasible_of_deviation {d n : ℕ} (m j : ℕ) (β s : ℝ)
(P : Measure (Obs d)) [IsProbabilityMeasure P] (ω : Fin n → Obs d)
(hcandidate : j ∈ meshIndices d n β)
(hbudget : 4 * s ≤ meshWidth j ^ (-2 * β))
(hmass : ∀ (k : Fin d → Fin (2 ^ j))
(ℓ : Fin d → Fin (m + 1)),
4 * meshWidth j ^ (-2 * β) ≤
(n : ℝ) * microcellMass P m j k ℓ)
(hdeviation : ∀ (k : Fin d → Fin (2 ^ j))
(ℓ : Fin d → Fin (m + 1)),
|(treatedCount m j ω k ℓ : ℝ) -
(n : ℝ) * microcellMass P m j k ℓ| ≤
(n : ℝ) * microcellMass P m j k ℓ / 2 + 4 * s) :
meshFeasible m β ω j := by
refine ⟨hcandidate, ?_⟩
intro k ℓ
have h := (abs_le.mp (hdeviation k ℓ)).1
have hm := hmass k ℓ
linarith
A feasible oracle-scale witness and simultaneous count deviations imply the selected mesh and selected-cell count bounds. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m :
ℕ
β s H :
ℝ
P :
Measure (Obs d)
ω :
Fin n → Obs d
j₀ :
ℕ
hwitness :
meshFeasible m β ω j₀
hwidth :
meshWidth j₀ ≤ H
hβ :
0 ≤ β
hbudget :
24 * s ≤ meshWidth j₀ ^ (-2 * β)
hs :
0 ≤ s
hdeviation :
∀ j ∈ meshIndices d n β,
|(treatedCount m j ω k ℓ : ℝ) - (n : ℝ) * microcellMass P m j k ℓ|
≤ (n : ℝ) * microcellMass P m j k ℓ / 2 + 4 * s
conclusion
conclusion 1
0 < countSelectedMesh m β ω
conclusion 2
countSelectedMesh m β ω ≤ H
conclusion 3
≤ treatedCount m ((feasibleIndices m β ω).sup id) ω k ℓ
Proof (Lean source)
lemma selectedCount_deterministic_bridge {d n : ℕ} (m : ℕ) (β s H : ℝ)
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(ω : Fin n → Obs d) (j₀ : ℕ)
(hwitness : meshFeasible m β ω j₀)
(hwidth : meshWidth j₀ ≤ H)
(hβ : 0 ≤ β)
(hbudget : 24 * s ≤ meshWidth j₀ ^ (-2 * β))
(hs : 0 ≤ s)
(hdeviation : ∀ j ∈ meshIndices d n β,
∀ (k : Fin d → Fin (2 ^ j)) (ℓ : Fin d → Fin (m + 1)),
|(treatedCount m j ω k ℓ : ℝ) -
(n : ℝ) * microcellMass P m j k ℓ| ≤
(n : ℝ) * microcellMass P m j k ℓ / 2 + 4 * s) :
0 < countSelectedMesh m β ω ∧
countSelectedMesh m β ω ≤ H ∧
∀ (k : Fin d → Fin (2 ^ ((feasibleIndices m β ω).sup id)))
(ℓ : Fin d → Fin (m + 1)),
(n : ℝ) * microcellMass P m ((feasibleIndices m β ω).sup id) k ℓ / 5 ≤
treatedCount m ((feasibleIndices m β ω).sup id) ω k ℓ := by
classical
let F := feasibleIndices m β ω
have hj₀ : j₀ ∈ F := by
simp only [F, feasibleIndices, mem_filter]
exact ⟨hwitness.1, hwitness⟩
have hF : F.Nonempty := ⟨j₀, hj₀⟩
have hsup : j₀ ≤ F.sup id := Finset.le_sup (f := id) hj₀
have hselected : countSelectedMesh m β ω = meshWidth (F.sup id) := by
simp [countSelectedMesh, F, hF]
have hwidthmono : meshWidth (F.sup id) ≤ meshWidth j₀ := by
unfold meshWidth
apply Real.rpow_le_rpow_of_exponent_le (by norm_num)
exact neg_le_neg (by exact_mod_cast hsup)
have hselectedBudget : 24 * s ≤ meshWidth (F.sup id) ^ (-2 * β) := by
apply hbudget.trans
exact Real.rpow_le_rpow_of_nonpos
(by unfold meshWidth; positivity) hwidthmono (by linarith)
have hfeasible : meshFeasible m β ω (F.sup id) := by
have hmem : F.sup id ∈ F := by
rw [← Finset.sup'_eq_sup hF id, ← Finset.max'_eq_sup' F hF]
exact Finset.max'_mem F hF
exact (Finset.mem_filter.mp hmem).2
refine ⟨?_, ?_, ?_⟩
· rw [hselected]
unfold meshWidth
positivity
· rw [hselected]
exact hwidthmono.trans hwidth
· intro k ℓ
exact selectedCount_lower_on_feasible_mesh m (F.sup id) β s P ω k ℓ
hfeasible hselectedBudget
(hdeviation (F.sup id) hfeasible.1 k ℓ) hs
The density floor and global propensity tail give a quantitative lower bound for each treated template microcell. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (c_f * (templateEta d m * meshWidth j) ^ d) ^ (γ / (γ - 1))
≤ microcellMass P m j k ℓ
Proof (Lean source)
lemma model_microcellMass_lower {d : ℕ} {β B L C c_f γ : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(m j : ℕ) (k : Fin d → Fin (2 ^ j))
(ℓ : Fin d → Fin (m + 1)) :
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * (templateEta d m * meshWidth j) ^ d) ^ (γ / (γ - 1)) ≤
microcellMass P m j k ℓ := by
rcases hP with ⟨hparams, hγ, Pc, hPc, μ₁, e, hEq, hmodel⟩
have hC : 1 ≤ C := hparams.2.2.2.2.1
have hcf : 0 < c_f := hparams.2.2.2.2.2.1
have hη : 0 < templateEta d m := templateEta_pos d m
have hh : 0 < meshWidth j := by unfold meshWidth; positivity
letI : IsProbabilityMeasure Pc := hPc
have hlower := orderedMass_microcell_lower β B L C c_f γ
(c_f * (templateEta d m * meshWidth j) ^ d) Pc μ₁ e hmodel
hC hγ (by positivity)
(scaledMicroCell d m j k ℓ)
(orderedMass_scaledMicroCell_measurable d m j k ℓ)
(by
convert orderedMass_microcell_covariate_lower d m j β B L C c_f γ
Pc μ₁ e hmodel k ℓ using 1
rw [orderedMass_microcell_volume_toReal d m j hη.le])
rw [← hEq] at hlower
exact hlower
Every template microcell has positive treated mass under the model's density floor and global propensity-tail condition. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
0 < microcellMass P m j k ℓ
Proof (Lean source)
lemma model_microcellMass_pos {d : ℕ} {β B L C c_f γ : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(m j : ℕ) (k : Fin d → Fin (2 ^ j))
(ℓ : Fin d → Fin (m + 1)) :
0 < microcellMass P m j k ℓ := by
have hparams := hP.1
have hγ := hP.2.1
have hC : 1 ≤ C := hparams.2.2.2.2.1
have hcf : 0 < c_f := hparams.2.2.2.2.2.1
have hη : 0 < templateEta d m := templateEta_pos d m
have hh : 0 < meshWidth j := by unfold meshWidth; positivity
have hpositive : 0 < ((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * (templateEta d m * meshWidth j) ^ d) ^ (γ / (γ - 1)) := by
positivity
exact lt_of_lt_of_le hpositive (model_microcellMass_lower P hP m j k ℓ)
theorem model_microcellMass_effectiveDimension_lower Lemma model_microcellMass_effectiveDimension_lower in the paper ↗
Express the model's treated microcell lower bound as a coefficient times the effective-dimension power of the dyadic width. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (c_f * templateEta d m ^ d) ^ (γ / (γ - 1)) * meshWidth j ^ effectiveDimension d γ
≤ microcellMass P m j k ℓ
Proof (Lean source)
lemma model_microcellMass_effectiveDimension_lower
{d : ℕ} {β B L C c_f γ : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(m j : ℕ) (k : Fin d → Fin (2 ^ j))
(ℓ : Fin d → Fin (m + 1)) :
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1)) *
meshWidth j ^ effectiveDimension d γ ≤
microcellMass P m j k ℓ := by
have hcf : 0 ≤ c_f := hP.1.2.2.2.2.2.1.le
have hη : 0 ≤ templateEta d m := (templateEta_pos d m).le
have hh : 0 ≤ meshWidth j := by unfold meshWidth; positivity
have hpow : (meshWidth j ^ d) ^ (γ / (γ - 1)) =
meshWidth j ^ effectiveDimension d γ := by
rw [← Real.rpow_natCast, ← Real.rpow_mul hh]
congr 1
unfold effectiveDimension
ring
have heq :
(c_f * (templateEta d m * meshWidth j) ^ d) ^ (γ / (γ - 1)) =
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1)) *
meshWidth j ^ effectiveDimension d γ := by
rw [mul_pow, ← mul_assoc,
Real.mul_rpow (mul_nonneg hcf (pow_nonneg hη _)) (pow_nonneg hh _)]
rw [hpow]
simpa only [heq, mul_assoc] using model_microcellMass_lower P hP m j k ℓ
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.SelectedMesh.Counts.Budget 6 declarations Finite-family count budget for selected meshes
Finite-family count budget for selected meshes
The number of candidate treated microcells is the geometric count of dyadic cubes times the fixed number of template cells. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma candidateTreatedCell_card (d n m : ℕ) (β : ℝ) :
card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) × (Fin d → Fin (m + 1))) =
∑ j ∈ meshIndices d n β, (2 ^ j) ^ d * (m + 1) ^ d := by
classical
simp only [Fintype.card_sigma, Fintype.card_prod, Fintype.card_fun,
Fintype.card_fin]
exact Finset.sum_attach (meshIndices d n β)
(fun j : ℕ => (2 ^ j) ^ d * (m + 1) ^ d)
A finite candidate-family bound for setting the simultaneous deviation budget. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n m :
ℕ
β :
ℝ
conclusion
≤ (maxMeshIndex d n β + 1) * ((2 ^ maxMeshIndex d n β) ^ d * (m + 1) ^ d)
Proof (Lean source)
lemma candidateTreatedCell_card_bound (d n m : ℕ) (β : ℝ) :
card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) × (Fin d → Fin (m + 1))) ≤
(maxMeshIndex d n β + 1) *
((2 ^ maxMeshIndex d n β) ^ d * (m + 1) ^ d) := by
rw [candidateTreatedCell_card]
calc
∑ j ∈ meshIndices d n β, (2 ^ j) ^ d * (m + 1) ^ d ≤
∑ _j ∈ meshIndices d n β,
(2 ^ maxMeshIndex d n β) ^ d * (m + 1) ^ d := by
apply Finset.sum_le_sum
intro j hj
have hjle : j ≤ maxMeshIndex d n β := by
exact Nat.lt_succ_iff.mp (Finset.mem_range.mp (by simpa [meshIndices] using hj))
gcongr
omega
_ = _ := by simp [meshIndices]
A logarithmic budget pays for the full finite family of candidate cells and leaves an inverse-square failure probability. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β :
ℝ
hn :
2 ≤ n
conclusion
M :
card (Σ j : {j // j ∈ meshIndices d n β}, (Fin d → Fin (2 ^ j.val)) × (Fin d → Fin (polynomialDegree β + 1)))
s :
log (2 * ((M : ℝ) + 1) * (n : ℝ) ^ 2)
0 < s
2 * (M : ℝ) * exp (-s) ≤ (n : ℝ) ^ (-2 : ℝ)
Proof (Lean source)
lemma selectedCount_log_budget (d n : ℕ) (β : ℝ) (hn : 2 ≤ n) :
let M := card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) ×
(Fin d → Fin (polynomialDegree β + 1)))
let s := log (2 * ((M : ℝ) + 1) * (n : ℝ) ^ 2)
0 < s ∧ 2 * (M : ℝ) * exp (-s) ≤ (n : ℝ) ^ (-2 : ℝ) := by
dsimp
let M : ℕ := card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) ×
(Fin d → Fin (polynomialDegree β + 1)))
have hn' : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n)
have hM : (0 : ℝ) ≤ M := Nat.cast_nonneg _
have hx : 0 < 2 * ((M : ℝ) + 1) * (n : ℝ) ^ 2 := by positivity
have hx1 : 1 < 2 * ((M : ℝ) + 1) * (n : ℝ) ^ 2 := by
have hn2 : (2 : ℝ) ≤ n := by exact_mod_cast hn
nlinarith [sq_nonneg ((n : ℝ) - 2)]
constructor
· exact Real.log_pos hx1
· rw [Real.exp_neg, Real.exp_log hx]
rw [Real.rpow_neg hn'.le]
change 2 * (M : ℝ) * (2 * ((M : ℝ) + 1) * (n : ℝ) ^ 2)⁻¹ ≤
((n : ℝ) ^ 2)⁻¹
rw [← div_eq_mul_inv, inv_eq_one_div]
rw [show (n : ℝ) ^ (2 : ℝ) = (n : ℝ) ^ (2 : ℕ) by norm_num]
apply (div_le_div_iff₀ hx (sq_pos_of_pos hn')).2
nlinarith [mul_nonneg (show (0 : ℝ) ≤ 2 from by norm_num) hM,
sq_pos_of_pos hn']
The largest candidate dyadic denominator is at most the sample size. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β :
ℝ
hn :
1 ≤ n
hden :
1 ≤ 2 * β + d
conclusion
(2 : ℝ) ^ maxMeshIndex d n β ≤ n
Proof (Lean source)
lemma selectedCount_index_pow_le (d n : ℕ) (β : ℝ)
(hn : 1 ≤ n) (hden : 1 ≤ 2 * β + d) :
(2 : ℝ) ^ maxMeshIndex d n β ≤ n := by
have hn' : (1 : ℝ) ≤ n := by exact_mod_cast hn
have hlogn : 0 ≤ log (n : ℝ) := Real.log_nonneg hn'
have hlog2 : 0 < log (2 : ℝ) := Real.log_pos (by norm_num)
let x : ℝ := log n / log 2 / (2 * β + d)
have hx : 0 ≤ x := by dsimp [x]; positivity
have hfloor : (maxMeshIndex d n β : ℝ) ≤ x := by
simpa only [maxMeshIndex, x] using Nat.floor_le hx
have hmul : log 2 * (maxMeshIndex d n β : ℝ) ≤ log n := by
have h := mul_le_mul_of_nonneg_left hfloor hlog2.le
have hq : 0 < 2 * β + (d : ℝ) := by linarith
have : log 2 * x = log n / (2 * β + d) := by
dsimp [x]
field_simp
rw [this] at h
exact h.trans ((div_le_iff₀ hq).2 (by nlinarith))
rw [← Real.rpow_natCast, Real.rpow_def_of_pos (by norm_num : (0 : ℝ) < 2)]
exact (Real.exp_le_exp.mpr hmul).trans_eq (Real.exp_log (by positivity))
theorem selectedCount_card_polynomial_bound Lemma selectedCount_card_polynomial_bound in the paper ↗
The simultaneous-count family has a polynomial size bound. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma selectedCount_card_polynomial_bound (d n m : ℕ) (β : ℝ)
(hn : 1 ≤ n) (hden : 1 ≤ 2 * β + d) :
card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) × (Fin d → Fin (m + 1))) ≤
(n + 1) * (n ^ d * (m + 1) ^ d) := by
have hpow : 2 ^ maxMeshIndex d n β ≤ n := by
exact_mod_cast selectedCount_index_pow_le d n β hn hden
have hj : maxMeshIndex d n β ≤ 2 ^ maxMeshIndex d n β := by
induction maxMeshIndex d n β with
| zero => simp
| succ j ih =>
have hp : 1 ≤ 2 ^ j := Nat.one_le_two_pow
simp only [pow_succ]
omega
calc
card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) × (Fin d → Fin (m + 1))) ≤
(maxMeshIndex d n β + 1) *
((2 ^ maxMeshIndex d n β) ^ d * (m + 1) ^ d) :=
candidateTreatedCell_card_bound d n m β
_ ≤ (n + 1) * (n ^ d * (m + 1) ^ d) := by
gcongr
· omega
The logarithm of the count family grows at most as an affine log sample size. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma selectedCount_log_card_bound (d n m : ℕ) (β : ℝ)
(hn : 2 ≤ n) (hden : 1 ≤ 2 * β + d) :
let M := card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) × (Fin d → Fin (m + 1)))
log (2 * ((M : ℝ) + 1) * (n : ℝ) ^ 2) ≤
(d + 3 : ℝ) * log n +
log (8 * ((m + 1 : ℝ) ^ d + 1)) := by
dsimp
let M : ℕ := card
(Σ j : {j // j ∈ meshIndices d n β},
(Fin d → Fin (2 ^ j.val)) × (Fin d → Fin (m + 1)))
have hcard : M ≤ (n + 1) * (n ^ d * (m + 1) ^ d) :=
selectedCount_card_polynomial_bound d n m β (by omega) hden
have hn' : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n)
have hM : (0 : ℝ) ≤ M := Nat.cast_nonneg _
have hT : (0 : ℝ) ≤ (m + 1 : ℝ) ^ d := by positivity
have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast (by omega : 1 ≤ n)
have hcast : (M : ℝ) ≤ ((n : ℝ) + 1) * ((n : ℝ) ^ d * (m + 1 : ℝ) ^ d) := by
exact_mod_cast hcard
have hN : (n : ℝ) + 1 ≤ 2 * n := by linarith
have hpow : 1 ≤ (n : ℝ) ^ d := one_le_pow₀ hn1
have hcore : (M : ℝ) + 1 ≤ 2 * (n : ℝ) * (n : ℝ)^d *
((m + 1 : ℝ)^d + 1) := by
have hpart : ((n : ℝ) + 1) * ((n : ℝ)^d * (m + 1 : ℝ)^d) ≤
(2 * (n : ℝ)) * ((n : ℝ)^d * (m + 1 : ℝ)^d) := by gcongr
have hone : 1 ≤ (n : ℝ) * (n : ℝ)^d := by nlinarith
nlinarith
have hbound : 2 * ((M : ℝ) + 1) * (n : ℝ) ^ 2 ≤
8 * ((m + 1 : ℝ) ^ d + 1) * (n : ℝ) ^ (d + 3) := by
rw [pow_add]
have h := mul_le_mul_of_nonneg_right hcore (show (0 : ℝ) ≤ 2 * (n : ℝ)^2 by positivity)
have h3 : (n : ℝ)^3 = (n : ℝ) * (n : ℝ)^2 := by ring
rw [h3]
nlinarith [mul_nonneg (show (0 : ℝ) ≤ ((m + 1 : ℝ)^d + 1) by positivity)
(show (0 : ℝ) ≤ (n : ℝ)^d * (n : ℝ)^3 by positivity)]
have hlog := Real.log_le_log (by positivity : 0 < 2 * ((M : ℝ) + 1) * (n : ℝ) ^ 2) hbound
have heq : log (8 * ((m + 1 : ℝ) ^ d + 1) * (n : ℝ) ^ (d + 3)) =
(d + 3 : ℝ) * log n +
log (8 * ((m + 1 : ℝ) ^ d + 1)) := by
rw [Real.log_mul (by positivity : 8 * ((m + 1 : ℝ)^d + 1) ≠ 0)
(by positivity : (n : ℝ)^(d+3) ≠ 0), Real.log_pow]
push_cast
ring
exact hlog.trans_eq heq
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.SelectedMesh.Fibre 20 declarations Conditional selected-mesh coefficient fibres
Conditional selected-mesh coefficient fibres
def sampleDesign
Treatment and covariate coordinates of a sample. For the stated inputs and conditions, the sampleDesign object being defined.
The treated outcome residual on a fixed complete-design fibre, extended by zero on untreated coordinates. For the stated inputs and conditions, the conditionalTreatedResidual object being defined.
Definition (Lean source)
parameters
def
conditionalTreatedResidual P ω i ξ :
ℝ
given by
if (ω i).2.1 = true then (ξ i).2.2 - treatedRegression P (ω i).1 else 0
Replace each response by its treated conditional mean, with an arbitrary zero value on untreated coordinates. For the stated inputs and conditions, the conditionalMeanSample object being defined.
Definition (Lean source)
parameters
def
given by
fun i => ((ω i).1, (ω i).2.1, if (ω i).2.1 = true then treatedRegression P (ω i).1 else 0)
For the stated dimensions, the complete vector of covariates and treatment arms is measurable.
Formal statement
Proof (Lean source)
@[fun_prop] lemma sampleDesign_measurable {d n : ℕ} :
Measurable (sampleDesign (d := d) (n := n)) := by
rw [measurable_pi_iff]
intro i
change Measurable (fun ω : Fin n → Obs d => ((ω i).1, (ω i).2.1))
fun_prop
The treated residual condition holds at every coordinate of an IID sample. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
∀ᵐ ω ∂Q, ∀ i : Fin n, ∀ t : ℝ, Integrable (fun y : ℝ => exp (t * (y - treatedRegression P (ω i).1))) (treatedKernel P (ω i).1) ∧ mgf (fun y : ℝ => y - treatedRegression P (ω i).1) (treatedKernel P (ω i).1) t
≤ exp (B ^ 2 * t ^ 2 / 2)
Proof (Lean source)
lemma boundedResidual_on_iid_sample {d n : ℕ} {B : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q]
(hIID : IIDSampleLaw Q P)
(hres : BoundedMeanSubGaussianResidual P B) :
∀ᵐ ω ∂Q, ∀ i : Fin n, ∀ t : ℝ,
Integrable (fun y : ℝ => exp (t * (y - treatedRegression P (ω i).1)))
(treatedKernel P (ω i).1) ∧
mgf (fun y : ℝ => y - treatedRegression P (ω i).1)
(treatedKernel P (ω i).1) t ≤ exp (B ^ 2 * t ^ 2 / 2) := by
have hP : ∀ᵐ z ∂P, ∀ t : ℝ,
Integrable (fun y : ℝ => exp (t * (y - treatedRegression P z.1)))
(treatedKernel P z.1) ∧
mgf (fun y : ℝ => y - treatedRegression P z.1)
(treatedKernel P z.1) t ≤ exp (B ^ 2 * t ^ 2 / 2) := by
have hx := ae_of_ae_map measurable_fst.aemeasurable hres
filter_upwards [hx] with z hz
exact hz.2
have hi : ∀ i : Fin n, ∀ᵐ ω ∂Q, ∀ t : ℝ,
Integrable (fun y : ℝ => exp (t * (y - treatedRegression P (ω i).1)))
(treatedKernel P (ω i).1) ∧
mgf (fun y : ℝ => y - treatedRegression P (ω i).1)
(treatedKernel P (ω i).1) t ≤ exp (B ^ 2 * t ^ 2 / 2) := by
intro i
have hmap : Q.map (fun ω : Fin n → Obs d => ω i) = P := by
rw [hIID]
simpa using Measure.pi_map_eval (fun _ : Fin n => P) i
have hP' : ∀ᵐ z ∂Q.map (fun ω : Fin n → Obs d => ω i), ∀ t : ℝ,
Integrable (fun y : ℝ => exp (t * (y - treatedRegression P z.1)))
(treatedKernel P z.1) ∧
mgf (fun y : ℝ => y - treatedRegression P z.1)
(treatedKernel P z.1) t ≤ exp (B ^ 2 * t ^ 2 / 2) := by
simpa only [hmap] using hP
exact ae_of_ae_map (μ := Q) (f := fun ω : Fin n → Obs d => ω i)
(measurable_pi_apply i).aemeasurable hP'
exact ae_all_iff.mpr hi
The iid observation law disintegrates over the complete treatment and covariate design as a product of one-observation outcome kernels. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
P :
Measure (Obs d)
conclusion
= attachKernel (Measure.pi (fun _ : Fin n => P.map (fun z => (z.1, z.2.1)))) (finProductKernel n (condDistrib (fun z : Obs d => z.2.2) (fun z : Obs d => (z.1, z.2.1)) P))
Proof (Lean source)
lemma sampleDesign_outcome_product_law {d n : ℕ}
(P : Measure (Obs d)) [IsProbabilityMeasure P] :
(Measure.pi (fun _ : Fin n => P)).map
(fun ω => (sampleDesign ω, fun i => (ω i).2.2)) =
attachKernel
(Measure.pi (fun _ : Fin n => P.map (fun z => (z.1, z.2.1))))
(finProductKernel n
(condDistrib (fun z : Obs d => z.2.2)
(fun z : Obs d => (z.1, z.2.1)) P)) := by
let design : Obs d → (Fin d → ℝ) × Bool := fun z => (z.1, z.2.1)
let Y : Obs d → ℝ := fun z => z.2.2
have hdesign : Measurable design := by fun_prop
have hY : Measurable Y := by fun_prop
have hsame :
(fun z : Fin n → Obs d =>
(designVector design z,
fun i => Y (z i))) =
(fun ω => (sampleDesign ω, fun i => (ω i).2.2)) := by
rfl
rw [← hsame]
exact product_observation_law_map_design_outcome P design hdesign Y hY
theorem sampleDesign_conditional_outcome_product_law Lemma sampleDesign_conditional_outcome_product_law in the paper ↗
Given the complete design, the outcome vector has the product of the one-observation conditional outcome laws. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
∀ᵐ x ∂Q.map sampleDesign, condDistrib (fun ω : Fin n → Obs d => fun i => (ω i).2.2) sampleDesign Q x
= finProductKernel n (condDistrib (fun z : Obs d => z.2.2) (fun z : Obs d => (z.1, z.2.1)) P) x
Proof (Lean source)
lemma sampleDesign_conditional_outcome_product_law {d n : ℕ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q]
(hIID : IIDSampleLaw Q P) :
∀ᵐ x ∂Q.map sampleDesign,
condDistrib (fun ω : Fin n → Obs d => fun i => (ω i).2.2)
sampleDesign Q x =
finProductKernel n
(condDistrib (fun z : Obs d => z.2.2)
(fun z : Obs d => (z.1, z.2.1)) P) x := by
subst Q
apply condDistrib_ae_eq_of_measure_eq_compProd_of_measurable
sampleDesign_measurable (by fun_prop)
have hdesignLaw :
(Measure.pi (fun _ : Fin n => P)).map sampleDesign =
Measure.pi (fun _ : Fin n =>
P.map (fun z : Obs d => (z.1, z.2.1))) := by
exact map_pi_finCoordinatewise n P
(phi := fun z : Obs d => (z.1, z.2.1)) (by fun_prop)
rw [hdesignLaw]
exact sampleDesign_outcome_product_law P
theorem sampleDesign_conditional_sample_outcome_product_law Lemma sampleDesign_conditional_sample_outcome_product_law in the paper ↗
The conditional complete-sample law sends its response vector to the product of the one-observation response kernels. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
∀ᵐ ω ∂Q, (condDistrib id sampleDesign Q (sampleDesign ω)).map (fun ξ : Fin n → Obs d => fun i => (ξ i).2.2)
= finProductKernel n (condDistrib (fun z : Obs d => z.2.2) (fun z : Obs d => (z.1, z.2.1)) P) (sampleDesign ω)
Proof (Lean source)
lemma sampleDesign_conditional_sample_outcome_product_law {d n : ℕ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q]
(hIID : IIDSampleLaw Q P) :
∀ᵐ ω ∂Q,
(condDistrib id sampleDesign Q (sampleDesign ω)).map
(fun ξ : Fin n → Obs d => fun i => (ξ i).2.2) =
finProductKernel n
(condDistrib (fun z : Obs d => z.2.2)
(fun z : Obs d => (z.1, z.2.1)) P) (sampleDesign ω) := by
have hcomp := condDistrib_comp (μ := Q) (mβ := inferInstance) (X := sampleDesign)
(Y := id) aemeasurable_id
(f := fun ξ : Fin n → Obs d => fun i => (ξ i).2.2) (by fun_prop)
have hresp : Measurable (fun ξ : Fin n → Obs d => fun i => (ξ i).2.2) := by
fun_prop
have hprod := sampleDesign_conditional_outcome_product_law P Q hIID
have hdesign : ∀ᵐ x ∂Q.map sampleDesign,
(condDistrib id sampleDesign Q x).map
(fun ξ : Fin n → Obs d => fun i => (ξ i).2.2) =
finProductKernel n
(condDistrib (fun z : Obs d => z.2.2)
(fun z : Obs d => (z.1, z.2.1)) P) x := by
filter_upwards [hcomp, hprod] with x hc hp
simpa only [Kernel.map_apply _ hresp] using hc.symm.trans hp
exact ae_of_ae_map sampleDesign_measurable.aemeasurable hdesign
theorem sampleDesign_conditional_response_coordinate_law Lemma sampleDesign_conditional_response_coordinate_law in the paper ↗
Under the complete-design conditional law, each response coordinate has the corresponding one-observation conditional response law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
∀ᵐ ω ∂Q, ∀ i : Fin n, (condDistrib id sampleDesign Q (sampleDesign ω)).map (fun ξ : Fin n → Obs d => (ξ i).2.2)
= condDistrib (fun z : Obs d => z.2.2) (fun z : Obs d => (z.1, z.2.1)) P (sampleDesign ω i)
Proof (Lean source)
lemma sampleDesign_conditional_response_coordinate_law {d n : ℕ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q]
(hIID : IIDSampleLaw Q P) :
∀ᵐ ω ∂Q, ∀ i : Fin n,
(condDistrib id sampleDesign Q (sampleDesign ω)).map
(fun ξ : Fin n → Obs d => (ξ i).2.2) =
condDistrib (fun z : Obs d => z.2.2)
(fun z : Obs d => (z.1, z.2.1)) P (sampleDesign ω i) := by
filter_upwards [sampleDesign_conditional_sample_outcome_product_law P Q hIID]
with ω hproduct
intro i
let μ := condDistrib id sampleDesign Q (sampleDesign ω)
let K := condDistrib (fun z : Obs d => z.2.2)
(fun z : Obs d => (z.1, z.2.1)) P
have hpi : μ.map (fun ξ : Fin n → Obs d => fun r => (ξ r).2.2) =
Measure.pi (fun r : Fin n => K (sampleDesign ω r)) := by
simpa only [μ, K, finProductKernel_apply] using hproduct
calc
μ.map (fun ξ : Fin n → Obs d => (ξ i).2.2) =
(μ.map (fun ξ : Fin n → Obs d => fun r => (ξ r).2.2)).map
(Function.eval i) := by
rw [Measure.map_map (by fun_prop) (by fun_prop)]
rfl
_ = (Measure.pi (fun r : Fin n => K (sampleDesign ω r))).map
(Function.eval i) := by rw [hpi]
_ = K (sampleDesign ω i) := by
rw [Measure.pi_map_eval]
simp
theorem sampleDesign_conditional_treatedResidual_mgf Lemma sampleDesign_conditional_treatedResidual_mgf in the paper ↗
On almost every fixed-design fibre, every treated response residual has the one-observation exponential integrability and MGF bound. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
∀ᵐ ω ∂Q, ∀ i : Fin n, (ω i).2.1
= true → ∀ t : ℝ, Integrable (fun ξ : Fin n → Obs d => exp (t * ((ξ i).2.2 - treatedRegression P (ω i).1))) (condDistrib id sampleDesign Q (sampleDesign ω)) ∧ mgf (fun ξ : Fin n → Obs d => (ξ i).2.2 - treatedRegression P (ω i).1) (condDistrib id sampleDesign Q (sampleDesign ω)) t ≤ exp (B ^ 2 * t ^ 2 / 2)
Proof (Lean source)
lemma sampleDesign_conditional_treatedResidual_mgf {d n : ℕ} {B : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q]
(hIID : IIDSampleLaw Q P)
(hres : BoundedMeanSubGaussianResidual P B) :
∀ᵐ ω ∂Q, ∀ i : Fin n, (ω i).2.1 = true → ∀ t : ℝ,
Integrable (fun ξ : Fin n → Obs d =>
exp (t * ((ξ i).2.2 - treatedRegression P (ω i).1)))
(condDistrib id sampleDesign Q (sampleDesign ω)) ∧
mgf (fun ξ : Fin n → Obs d =>
(ξ i).2.2 - treatedRegression P (ω i).1)
(condDistrib id sampleDesign Q (sampleDesign ω)) t ≤
exp (B ^ 2 * t ^ 2 / 2) := by
filter_upwards [boundedResidual_on_iid_sample P Q hIID hres,
sampleDesign_conditional_response_coordinate_law P Q hIID]
with ω hresω hcoord
intro i hi t
let μ := condDistrib id sampleDesign Q (sampleDesign ω)
let X : (Fin n → Obs d) → ℝ := fun ξ => (ξ i).2.2
let R : ℝ → ℝ := fun y => y - treatedRegression P (ω i).1
have hmap : μ.map X = treatedKernel P (ω i).1 := by
simpa only [μ, X, sampleDesign, treatedKernel, hi] using hcoord i
constructor
· have hint := (hresω i t).1
rw [← hmap] at hint
have hcomp := (integrable_map_measure hint.aestronglyMeasurable
(show AEMeasurable X μ by fun_prop)).mp hint
simpa only [μ, X, R, Function.comp_def] using hcomp
· have hbound := (hresω i t).2
rw [← hmap] at hbound
rw [mgf_map (show AEMeasurable X μ by fun_prop) (by fun_prop)] at hbound
simpa only [μ, X, R, Function.comp_def] using hbound
theorem sampleDesign_conditional_outcome_independent Lemma sampleDesign_conditional_outcome_independent in the paper ↗
Conditional on the design, the response coordinates are independent. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
∀ᵐ ω ∂Q, iIndepFun (fun i (ξ : Fin n → Obs d) => (ξ i).2.2) (condDistrib id sampleDesign Q (sampleDesign ω))
Proof (Lean source)
lemma sampleDesign_conditional_outcome_independent {d n : ℕ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q]
(hIID : IIDSampleLaw Q P) :
∀ᵐ ω ∂Q, iIndepFun (fun i (ξ : Fin n → Obs d) => (ξ i).2.2)
(condDistrib id sampleDesign Q (sampleDesign ω)) := by
filter_upwards [sampleDesign_conditional_sample_outcome_product_law P Q hIID]
with ω hproduct
let μ := condDistrib id sampleDesign Q (sampleDesign ω)
let K := condDistrib (fun z : Obs d => z.2.2)
(fun z : Obs d => (z.1, z.2.1)) P
have hpi : μ.map (fun ξ : Fin n → Obs d => fun i => (ξ i).2.2) =
Measure.pi (fun i : Fin n => K (sampleDesign ω i)) := by
simpa only [μ, K, finProductKernel_apply] using hproduct
apply (iIndepFun_iff_map_fun_eq_pi_map (μ := μ)
(f := fun i (ξ : Fin n → Obs d) => (ξ i).2.2)
(by intro i; exact (show Measurable (fun ξ : Fin n → Obs d => (ξ i).2.2)
by fun_prop).aemeasurable)).2
rw [hpi]
congr 1
funext i
have hcoord : μ.map (fun ξ : Fin n → Obs d => (ξ i).2.2) =
(Measure.pi (fun i : Fin n => K (sampleDesign ω i))).map (Function.eval i) := by
rw [← hpi, Measure.map_map (by fun_prop) (by fun_prop)]
rfl
rw [hcoord, Measure.pi_map_eval]
simp
theorem sampleDesign_conditional_residual_family Lemma sampleDesign_conditional_residual_family in the paper ↗
Almost every complete-design fibre carries independent residual coordinates with the original uniform exponential-moment bound. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
∀ᵐ ω ∂Q, iIndepFun (fun i ξ => conditionalTreatedResidual P ω i ξ) (condDistrib id sampleDesign Q (sampleDesign ω)) ∧ ∀ i : Fin n, ∀ t : ℝ, Integrable (fun ξ => exp (t * conditionalTreatedResidual P ω i ξ)) (condDistrib id sampleDesign Q (sampleDesign ω)) ∧ mgf (conditionalTreatedResidual P ω i) (condDistrib id sampleDesign Q (sampleDesign ω)) t
≤ exp (B ^ 2 * t ^ 2 / 2)
Proof (Lean source)
lemma sampleDesign_conditional_residual_family {d n : ℕ} {B : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q]
(hIID : IIDSampleLaw Q P)
(hres : BoundedMeanSubGaussianResidual P B) :
∀ᵐ ω ∂Q,
iIndepFun (fun i ξ => conditionalTreatedResidual P ω i ξ)
(condDistrib id sampleDesign Q (sampleDesign ω)) ∧
∀ i : Fin n, ∀ t : ℝ,
Integrable (fun ξ => exp
(t * conditionalTreatedResidual P ω i ξ))
(condDistrib id sampleDesign Q (sampleDesign ω)) ∧
mgf (conditionalTreatedResidual P ω i)
(condDistrib id sampleDesign Q (sampleDesign ω)) t ≤
exp (B ^ 2 * t ^ 2 / 2) := by
filter_upwards [sampleDesign_conditional_outcome_independent P Q hIID,
sampleDesign_conditional_treatedResidual_mgf P Q hIID hres]
with ω hindep htreated
constructor
· simpa only [conditionalTreatedResidual, Function.comp_def] using
hindep.comp
(fun i y => if (ω i).2.1 = true then
y - treatedRegression P (ω i).1 else 0)
(by
intro i
by_cases hi : (ω i).2.1 = true <;> simp [hi] <;> fun_prop)
· intro i t
by_cases hi : (ω i).2.1 = true
· have hfun : conditionalTreatedResidual P ω i =
fun ξ => (ξ i).2.2 - treatedRegression P (ω i).1 := by
funext ξ
simp [conditionalTreatedResidual, hi]
rw [hfun]
exact htreated i hi t
· have hfun : conditionalTreatedResidual P ω i =
fun _ => 0 := by
funext ξ
simp [conditionalTreatedResidual, hi]
rw [hfun]
constructor
· change Integrable (fun _ : Fin n → Obs d => exp (t * 0))
(condDistrib id sampleDesign Q (sampleDesign ω))
simp
· change mgf (fun _ : Fin n → Obs d => 0)
(condDistrib id sampleDesign Q (sampleDesign ω)) t ≤
exp (B ^ 2 * t ^ 2 / 2)
simp [mgf]
positivity
theorem sampleDesign_conditional_residual_mean_zero Lemma sampleDesign_conditional_residual_mean_zero in the paper ↗
Every fixed-design residual coordinate is integrable and has mean zero under almost every conditional sample law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
∀ᵐ ω ∂Q, ∀ i : Fin n, Integrable (conditionalTreatedResidual P ω i) (condDistrib id sampleDesign Q (sampleDesign ω)) ∧ ∫ ξ, conditionalTreatedResidual P ω i ξ ∂condDistrib id sampleDesign Q (sampleDesign ω)
= 0
Proof (Lean source)
lemma sampleDesign_conditional_residual_mean_zero {d n : ℕ} {B : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q]
(hIID : IIDSampleLaw Q P)
(hres : BoundedMeanSubGaussianResidual P B) :
∀ᵐ ω ∂Q, ∀ i : Fin n,
Integrable (conditionalTreatedResidual P ω i)
(condDistrib id sampleDesign Q (sampleDesign ω)) ∧
∫ ξ, conditionalTreatedResidual P ω i ξ
∂condDistrib id sampleDesign Q (sampleDesign ω) = 0 := by
filter_upwards [boundedResidual_on_iid_sample P Q hIID hres,
sampleDesign_conditional_response_coordinate_law P Q hIID]
with ω hresω hcoord
intro i
by_cases hi : (ω i).2.1 = true
· let μ := condDistrib id sampleDesign Q (sampleDesign ω)
let X : (Fin n → Obs d) → ℝ := fun ξ => (ξ i).2.2
let R : ℝ → ℝ := fun y => y - treatedRegression P (ω i).1
letI : IsProbabilityMeasure (treatedKernel P (ω i).1) := by
unfold treatedKernel
infer_instance
have hmap : μ.map X = treatedKernel P (ω i).1 := by
simpa only [μ, X, sampleDesign, treatedKernel, hi] using hcoord i
have hRint : Integrable R (treatedKernel P (ω i).1) := by
have hp := (hresω i (1 : ℝ)).1
have hn := (hresω i (-1 : ℝ)).1
have hm := integrable_pow_of_integrable_exp_mul
(X := R) (t := (1 : ℝ)) (by norm_num) (by simpa [R] using hp)
(by simpa [R] using hn) 1
simpa [R] using hm
have hcomp : Integrable (R ∘ X) μ := by
have hRmap : Integrable R (μ.map X) := by
rw [hmap]
exact hRint
exact (integrable_map_measure hRmap.aestronglyMeasurable
(show AEMeasurable X μ by fun_prop)).mp hRmap
have hfun : conditionalTreatedResidual P ω i = R ∘ X := by
funext ξ
simp [conditionalTreatedResidual, hi, R, X]
constructor
· simpa only [hfun] using hcomp
· rw [hfun]
change ∫ ξ, (R ∘ X) ξ ∂μ = 0
have hRmap : Integrable R (μ.map X) := by
rw [hmap]
exact hRint
have hYint : Integrable (fun y : ℝ => y) (treatedKernel P (ω i).1) := by
have hc : Integrable (fun _ : ℝ => treatedRegression P (ω i).1)
(treatedKernel P (ω i).1) := integrable_const _
exact (hRint.add hc).congr (Filter.Eventually.of_forall (fun y => by simp [R]))
calc
∫ ξ, (R ∘ X) ξ ∂μ = ∫ y, R y ∂μ.map X :=
(integral_map (show AEMeasurable X μ by fun_prop)
hRmap.aestronglyMeasurable).symm
_ = ∫ y, R y ∂treatedKernel P (ω i).1 := by rw [hmap]
_ = 0 := by
change (∫ y, y - treatedRegression P (ω i).1
∂treatedKernel P (ω i).1) = 0
rw [integral_sub hYint (integrable_const _)]
simp [treatedRegression]
· have hfun : conditionalTreatedResidual P ω i = fun _ => 0 := by
funext ξ
simp [conditionalTreatedResidual, hi]
rw [hfun]
simp
Conditional on the observed design, the complete sample has that design almost surely. This supplies the fibre hypothesis for the coefficient identity below. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
Q :
conclusion
∀ᵐ ω ∂Q, ∀ᵐ ξ ∂condDistrib id sampleDesign Q (sampleDesign ω), sampleDesign ξ
= sampleDesign ω
Proof (Lean source)
lemma sampleDesign_condDistrib_fibre {d n : ℕ}
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q] :
∀ᵐ ω ∂Q, ∀ᵐ ξ ∂condDistrib id sampleDesign Q (sampleDesign ω),
sampleDesign ξ = sampleDesign ω := by
have hcomp : condDistrib sampleDesign sampleDesign Q =ᵐ[Q.map sampleDesign]
(condDistrib id sampleDesign Q).map sampleDesign := by
simpa only [Function.comp_def, id_eq] using
(condDistrib_comp (μ := Q) sampleDesign
(Y := id) aemeasurable_id sampleDesign_measurable)
have hself := condDistrib_self (μ := Q) sampleDesign
have hdesign : ∀ᵐ x ∂Q.map sampleDesign,
(condDistrib id sampleDesign Q x).map sampleDesign = Measure.dirac x := by
filter_upwards [hcomp, hself] with x hc hs
simpa only [Kernel.map_apply _ sampleDesign_measurable, Kernel.id_apply] using
hc.symm.trans hs
have hdesign' := ae_of_ae_map sampleDesign_measurable.aemeasurable hdesign
filter_upwards [hdesign'] with ω hω
have hs : (condDistrib id sampleDesign Q (sampleDesign ω))
{ξ | sampleDesign ξ = sampleDesign ω} = 1 := by
have hm := congrArg (fun μ : Measure (Fin n → (Fin d → ℝ) × Bool) =>
μ {sampleDesign ω}) hω
rw [Measure.map_apply sampleDesign_measurable (measurableSet_singleton _)] at hm
simp at hm
change (condDistrib id sampleDesign Q (sampleDesign ω))
{ξ | sampleDesign ξ = sampleDesign ω} = 1 at hm
exact hm
exact (MeasureTheory.ae_iff.mpr (by
have hp : IsProbabilityMeasure (condDistrib id sampleDesign Q (sampleDesign ω)) :=
inferInstance
change (condDistrib id sampleDesign Q (sampleDesign ω))
{ξ | sampleDesign ξ = sampleDesign ω}ᶜ = 0
have hset : MeasurableSet {ξ | sampleDesign ξ = sampleDesign ω} := by
change MeasurableSet (sampleDesign ⁻¹' {sampleDesign ω})
exact sampleDesign_measurable (measurableSet_singleton _)
rw [measure_compl hset (by rw [hs]; simp), measure_univ, hs]
simp))
A fibre of the design map has a fixed Gram matrix; its coefficient variation is exactly the inverse Gram applied to the response variation. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma coefHat_sub_of_sampleDesign_eq {d n : ℕ} (m j : ℕ)
(ω ξ : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(hdesign : sampleDesign ξ = sampleDesign ω) :
coefHat m j ξ k - coefHat m j ω k =
mulVec (gramHat m j ω k)⁻¹
(responseMoment m j ξ k - responseMoment m j ω k) := by
apply coefHat_sub_of_design_eq m j ω ξ k
intro i
have hi := congrFun hdesign i
change ((ξ i).1, (ξ i).2.1) = ((ω i).1, (ω i).2.1) at hi
exact ⟨congrArg (fun p : (Fin d → ℝ) × Bool => p.1) hi,
congrArg (fun p : (Fin d → ℝ) × Bool => p.2) hi⟩
On a fixed design fibre, one coefficient coordinate is a finite weighted sum of the treated response residuals. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
= ∑ α' : MonoIndex d m, (gramHat m j ω k)⁻¹ α α' * ((tensorCount d m : ℝ)⁻¹ * ∑ ℓ : Fin d → Fin (m + 1), (treatedCount m j ω k ℓ : ℝ)⁻¹ * ∑ i : Fin n, if (ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ then monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α' * ((ξ i).2.2 - (ω i).2.2) else 0)
Proof (Lean source)
lemma coefHat_coordinate_residual_sum {d n : ℕ} (m j : ℕ)
(ω ξ : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(α : MonoIndex d m) (hdesign : sampleDesign ξ = sampleDesign ω) :
(coefHat m j ξ k - coefHat m j ω k) α =
∑ α' : MonoIndex d m, (gramHat m j ω k)⁻¹ α α' *
((tensorCount d m : ℝ)⁻¹ *
∑ ℓ : Fin d → Fin (m + 1),
(treatedCount m j ω k ℓ : ℝ)⁻¹ *
∑ i : Fin n, if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then
monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) /
meshWidth j) α' * ((ξ i).2.2 - (ω i).2.2)
else 0) := by
classical
rw [coefHat_sub_of_sampleDesign_eq m j ω ξ k hdesign]
simp only [mulVec, dotProduct]
congr 1
funext α'
congr 1
apply responseMoment_sub_of_design_eq m j ω ξ k
intro i
have hi := congrFun hdesign i
change ((ξ i).1, (ξ i).2.1) = ((ω i).1, (ω i).2.1) at hi
exact ⟨congrArg (fun p : (Fin d → ℝ) × Bool => p.1) hi,
congrArg (fun p : (Fin d → ℝ) × Bool => p.2) hi⟩
theorem coefHat_sub_of_sampleDesign_eq_weighted_sum Lemma coefHat_sub_of_sampleDesign_eq_weighted_sum in the paper ↗
On a design fibre, the coefficient difference is the weighted sum of outcome differences, with all weights fixed by the conditioning design. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma coefHat_sub_of_sampleDesign_eq_weighted_sum {d n : ℕ} (m j : ℕ)
(ω ξ : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(α : MonoIndex d m) (hdesign : sampleDesign ξ = sampleDesign ω) :
(coefHat m j ξ k - coefHat m j ω k) α =
∑ i : Fin n, coefResidualWeight m j ω k α i *
((ξ i).2.2 - (ω i).2.2) := by
have hd : ∀ i, (ξ i).1 = (ω i).1 ∧ (ξ i).2.1 = (ω i).2.1 := by
intro i
have hi := congrFun hdesign i
change ((ξ i).1, (ξ i).2.1) = ((ω i).1, (ω i).2.1) at hi
exact ⟨congrArg (fun p : (Fin d → ℝ) × Bool => p.1) hi,
congrArg (fun p : (Fin d → ℝ) × Bool => p.2) hi⟩
have hs : (fun i : Fin n => ((ξ i).1, (ξ i).2.1, (ω i).2.2)) = ω := by
funext i
obtain ⟨hx, ha⟩ := hd i
exact Prod.ext hx (Prod.ext ha rfl)
simpa only [hs] using
coefHat_sub_conditionalMeanSample_weighted_sum_of_design_eq
m j ω ξ (fun i => (ω i).2.2) k α hd
theorem coefHat_sub_conditionalMeanSample_eq_residual_sum Lemma coefHat_sub_conditionalMeanSample_eq_residual_sum in the paper ↗
On a complete-design fibre, subtracting the fitted coefficient at the conditional-mean sample leaves exactly the weighted residual sum. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma coefHat_sub_conditionalMeanSample_eq_residual_sum {d n : ℕ} (m j : ℕ)
(P : Measure (Obs d)) [IsFiniteMeasure P]
(ω ξ : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(α : MonoIndex d m) (hdesign : sampleDesign ξ = sampleDesign ω) :
coefHat m j ξ k α - coefHat m j (conditionalMeanSample P ω) k α =
∑ i : Fin n, coefResidualWeight m j ω k α i *
conditionalTreatedResidual P ω i ξ := by
classical
have hd : ∀ i, (ξ i).1 = (ω i).1 ∧ (ξ i).2.1 = (ω i).2.1 := by
intro i
have hi := congrFun hdesign i
change ((ξ i).1, (ξ i).2.1) = ((ω i).1, (ω i).2.1) at hi
exact ⟨congrArg (fun p : (Fin d → ℝ) × Bool => p.1) hi,
congrArg (fun p : (Fin d → ℝ) × Bool => p.2) hi⟩
let mean : Fin n → ℝ := fun i =>
if (ω i).2.1 = true then treatedRegression P (ω i).1 else 0
have hsamp : (fun i => ((ξ i).1, (ξ i).2.1, mean i)) =
conditionalMeanSample P ω := by
funext i
obtain ⟨hx, ha⟩ := hd i
simp only [conditionalMeanSample, mean]
exact Prod.ext hx (Prod.ext ha rfl)
have hbase := coefHat_sub_conditionalMeanSample_weighted_sum_of_design_eq
m j ω ξ mean k α hd
simp only [Pi.sub_apply, hsamp] at hbase
rw [hbase]
apply Finset.sum_congr rfl
intro i _
by_cases hi : (ω i).2.1 = true
· simp [mean, conditionalTreatedResidual, hi]
· rw [coefResidualWeight_zero_of_untreated m j ω k α i
(Bool.eq_false_of_not_eq_true hi)]
simp
def conditionalCoefCentre
Conditional centre of one fitted coefficient given all designs and arms. For the stated inputs and conditions, the conditionalCoefCentre object being defined.
theorem conditionalCoefCentre_eq_conditionalMeanSample Lemma conditionalCoefCentre_eq_conditionalMeanSample in the paper ↗
The conditional coefficient centre is the fitted coefficient obtained by replacing treated responses with their conditional means. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
∀ᵐ ω ∂Q, ∀ (m j : ℕ) (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m), conditionalCoefCentre Q m j ω k α
= coefHat m j (conditionalMeanSample P ω) k α
Proof (Lean source)
lemma conditionalCoefCentre_eq_conditionalMeanSample {d n : ℕ} {B : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q]
(hIID : IIDSampleLaw Q P)
(hres : BoundedMeanSubGaussianResidual P B) :
∀ᵐ ω ∂Q, ∀ (m j : ℕ) (k : Fin d → Fin (2 ^ j))
(α : MonoIndex d m),
conditionalCoefCentre Q m j ω k α =
coefHat m j (conditionalMeanSample P ω) k α := by
filter_upwards [sampleDesign_condDistrib_fibre Q,
sampleDesign_conditional_residual_mean_zero P Q hIID hres]
with ω hfibre hzero
intro m j k α
let μ := condDistrib id sampleDesign Q (sampleDesign ω)
let c := coefHat m j (conditionalMeanSample P ω) k α
let S : (Fin n → Obs d) → ℝ := fun ξ =>
∑ i : Fin n, coefResidualWeight m j ω k α i *
conditionalTreatedResidual P ω i ξ
have hterm : ∀ i : Fin n, Integrable
(fun ξ => coefResidualWeight m j ω k α i *
conditionalTreatedResidual P ω i ξ) μ := by
intro i
exact (hzero i).1.const_mul _
have hSint : Integrable S μ := by
dsimp only [S]
exact integrable_finset_sum univ (fun i _ => hterm i)
have htermMean : ∀ i : Fin n,
∫ ξ, coefResidualWeight m j ω k α i *
conditionalTreatedResidual P ω i ξ ∂μ = 0 := by
intro i
rw [integral_const_mul, (hzero i).2, mul_zero]
have hSmean : ∫ ξ, S ξ ∂μ = 0 := by
dsimp only [S]
rw [integral_finset_sum univ (fun i _ => hterm i)]
simp [htermMean]
have hcoef : ∀ᵐ ξ ∂μ, coefHat m j ξ k α = c + S ξ := by
filter_upwards [hfibre] with ξ hdesign
have h := coefHat_sub_conditionalMeanSample_eq_residual_sum
m j P ω ξ k α hdesign
dsimp only [c, S]
linarith
unfold conditionalCoefCentre
change (∫ ξ, coefHat m j ξ k α ∂μ) = c
rw [integral_congr_ae hcoef]
rw [integral_add (integrable_const c) hSint, integral_const, hSmean]
simp
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.SelectedMesh.MGF 3 declarations Finite conditional sub-Gaussian sums
Finite conditional sub-Gaussian sums
Independent centred residuals with coordinatewise Gaussian MGF bounds retain the sum of their variance proxies under a finite weighted sum. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
Ω :
Type*
n :
ℕ
μ :
Z :
Fin n → Ω → ℝ
v :
Fin n → ℝ
hindep :
iIndepFun Z μ
hmeas :
∀ i, Measurable (Z i)
t :
ℝ
Proof (Lean source)
lemma independentResidualSum_mgf_le {Ω : Type*} [MeasurableSpace Ω]
{n : ℕ} (μ : Measure Ω) [IsProbabilityMeasure μ]
(Z : Fin n → Ω → ℝ) (v : Fin n → ℝ)
(hindep : iIndepFun Z μ) (hmeas : ∀ i, Measurable (Z i))
(hmgf : ∀ i t, mgf (Z i) μ t ≤ exp (v i * t ^ 2 / 2))
(t : ℝ) :
mgf (∑ i : Fin n, Z i) μ t ≤
exp ((∑ i : Fin n, v i) * t ^ 2 / 2) := by
rw [hindep.mgf_sum hmeas univ]
calc
(∏ i : Fin n, mgf (Z i) μ t) ≤
∏ i : Fin n, exp (v i * t ^ 2 / 2) := by
apply Finset.prod_le_prod
· intro i hi
exact mgf_nonneg
· intro i hi
exact hmgf i t
_ = exp ((∑ i : Fin n, v i) * t ^ 2 / 2) := by
rw [← Real.exp_sum]
congr 1
simp only [Finset.sum_mul, Finset.sum_div]
Fixed design weights turn a common residual proxy into the squared-weight proxy used by the selected coefficient bounds. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
Ω :
Type*
n :
ℕ
μ :
Y :
Fin n → Ω → ℝ
w :
Fin n → ℝ
B :
ℝ
hindep :
iIndepFun Y μ
hmeas :
∀ i, Measurable (Y i)
t :
ℝ
Proof (Lean source)
lemma independentWeightedResidual_mgf_le {Ω : Type*} [MeasurableSpace Ω]
{n : ℕ} (μ : Measure Ω) [IsProbabilityMeasure μ]
(Y : Fin n → Ω → ℝ) (w : Fin n → ℝ) (B : ℝ)
(hindep : iIndepFun Y μ) (hmeas : ∀ i, Measurable (Y i))
(hmgf : ∀ i t, mgf (Y i) μ t ≤ exp (B ^ 2 * t ^ 2 / 2))
(t : ℝ) :
mgf (∑ i : Fin n, fun ω => w i * Y i ω) μ t ≤
exp (B ^ 2 * (∑ i : Fin n, (w i) ^ 2) * t ^ 2 / 2) := by
have hi : iIndepFun (fun i ω => w i * Y i ω) μ := by
simpa only [Function.comp_def] using
hindep.comp (fun i x => w i * x) (by fun_prop)
have hm : ∀ i, Measurable (fun ω => w i * Y i ω) := by fun_prop
have hb : ∀ i s, mgf (fun ω => w i * Y i ω) μ s ≤
exp ((B ^ 2 * w i ^ 2) * s ^ 2 / 2) := by
intro i s
rw [mgf_const_mul]
convert hmgf i (w i * s) using 1 <;> ring
have h := independentResidualSum_mgf_le μ (fun i ω => w i * Y i ω)
(fun i => B ^ 2 * w i ^ 2) hi hm hb t
have hs : (∑ i : Fin n, B ^ 2 * w i ^ 2) =
B ^ 2 * ∑ i : Fin n, w i ^ 2 := by rw [Finset.mul_sum]
rw [hs] at h
exact h
theorem independentWeightedResidual_integrable_exp_sum Lemma independentWeightedResidual_integrable_exp_sum in the paper ↗
Exponential integrability is preserved under the same finite weighted sum, so the maximal inequality can use the coefficient MGF bounds. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
Ω :
Type*
n :
ℕ
μ :
Y :
Fin n → Ω → ℝ
w :
Fin n → ℝ
hindep :
iIndepFun Y μ
hmeas :
∀ i, Measurable (Y i)
hint :
∀ i t, Integrable (fun ω => exp (t * Y i ω)) μ
t :
ℝ
conclusion
Proof (Lean source)
lemma independentWeightedResidual_integrable_exp_sum
{Ω : Type*} [MeasurableSpace Ω] {n : ℕ}
(μ : Measure Ω) [IsProbabilityMeasure μ]
(Y : Fin n → Ω → ℝ) (w : Fin n → ℝ)
(hindep : iIndepFun Y μ) (hmeas : ∀ i, Measurable (Y i))
(hint : ∀ i t, Integrable (fun ω => exp (t * Y i ω)) μ)
(t : ℝ) :
Integrable (fun ω => exp
(t * (∑ i : Fin n, (fun ξ => w i * Y i ξ)) ω)) μ := by
have hi : iIndepFun (fun i ω => w i * Y i ω) μ := by
simpa only [Function.comp_def] using
hindep.comp (fun i x => w i * x) (by fun_prop)
have hm : ∀ i, Measurable (fun ω => w i * Y i ω) := by fun_prop
apply hi.integrable_exp_mul_sum hm
intro i _
convert hint i (t * w i) using 1
ext ω
congr 1
ring
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.SelectedMesh.Maximal 24 declarations Ordered sub-Gaussian maximal inequality
Ordered sub-Gaussian maximal inequality
The logarithmic correction is absorbed by a positive power on the unit interval. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma scale_log_envelope_unit (β s : ℝ) (hβ : 1 ≤ β)
(hs : 0 < s) (hs1 : s ≤ 1) :
s ^ β * sqrt (1 + max 0 (log (1 / s))) ≤ 1 := by
have hdiv : 0 < 1 / s := by positivity
have hlog : 0 ≤ log (1 / s) := Real.log_nonneg (by
apply (one_le_div hs).2
exact hs1)
have hlogbound := Real.log_le_sub_one_of_pos hdiv
have hsqrtarg : 0 ≤ 1 + max 0 (log (1 / s)) := by positivity
have hsqrtbound : sqrt (1 + max 0 (log (1 / s))) ≤ 1 / s := by
have harg : 1 + max 0 (log (1 / s)) ≤ 1 / s := by
rw [max_eq_right hlog]
linarith
exact (Real.sqrt_le_sqrt harg).trans (by
have hdiv1 : 1 ≤ 1 / s := (one_le_div hs).2 hs1
apply (Real.sqrt_le_left hdiv.le).2
nlinarith)
have hpow : s ^ β ≤ s := by
simpa using (Real.rpow_le_rpow_of_exponent_ge hs hs1 hβ : s ^ β ≤ s ^ (1 : ℝ))
calc
s ^ β * sqrt (1 + max 0 (log (1 / s))) ≤ s * (1 / s) := by
gcongr
_ = 1 := by field_simp
A fixed upper scale gives a uniform bound for the oracle logarithmic factor. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma scale_log_envelope_bounded (β s K : ℝ) (hβ : 1 ≤ β)
(hs : 0 < s) (hsK : s ≤ K) :
s ^ β * sqrt (1 + max 0 (log (1 / s))) ≤ max 1 (K ^ β) := by
by_cases hs1 : s ≤ 1
· exact (scale_log_envelope_unit β s hβ hs hs1).trans (le_max_left _ _)
· have hlog : log (1 / s) ≤ 0 := Real.log_nonpos
(by positivity) ((div_le_one hs).2 (le_of_lt (lt_of_not_ge hs1)))
have hsβ : s ^ β ≤ K ^ β := Real.rpow_le_rpow hs.le hsK (by linarith)
rw [max_eq_left hlog]
simpa using hsβ.trans (le_max_right 1 (K ^ β))
Convert a selected-width bound into the oracle-rate logarithmic envelope. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma selectedMesh_oracle_log_bound (β h o K : ℝ)
(hβ : 1 ≤ β) (hh : 0 < h) (ho : 0 < o)
(hbound : h ≤ K * o) :
h ^ β * sqrt (1 + max 0 (log (o / h))) ≤
max 1 (K ^ β) * o ^ β := by
have hr : 0 < h / o := div_pos hh ho
have hrK : h / o ≤ K := (div_le_iff₀ ho).2 (by simpa [mul_comm] using hbound)
have hlog : o / h = 1 / (h / o) := by
field_simp
have hpow : h ^ β = (h / o) ^ β * o ^ β := by
rw [← Real.mul_rpow hr.le ho.le]
congr 1
field_simp
rw [hpow, hlog]
have hscale := scale_log_envelope_bounded β (h / o) K hβ hr hrK
calc
(h / o) ^ β * o ^ β * sqrt (1 + max 0 (log (1 / (h / o)))) =
((h / o) ^ β * sqrt (1 + max 0 (log (1 / (h / o))))) * o ^ β := by ring
_ ≤ max 1 (K ^ β) * o ^ β :=
mul_le_mul_of_nonneg_right hscale (Real.rpow_nonneg ho.le _)
The final deterministic term in the selected-mesh envelope. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
≤ max 1 (K ^ β) * oracleRate d n β γ
Proof (Lean source)
lemma selectedMesh_oracle_rate_log_bound (d n : ℕ) (β γ h K : ℝ)
(hn : 0 < n) (hβ : 1 ≤ β) (hh : 0 < h)
(hbound : h ≤ K * oracleMesh d n β γ) :
h ^ β * sqrt
(1 + max 0 (log (oracleMesh d n β γ / h))) ≤
max 1 (K ^ β) * oracleRate d n β γ := by
have ho : 0 < oracleMesh d n β γ := by
unfold oracleMesh
have hn' : (0 : ℝ) < n := by exact_mod_cast hn
positivity
simpa only [oracleRate] using
selectedMesh_oracle_log_bound β h (oracleMesh d n β γ) K hβ hh ho hbound
The stochastic coefficient scale is the oracle mesh raised to the smoothness plus half the effective dimension. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β γ :
ℝ
hn :
0 < n
hden :
0 < 2 * β + effectiveDimension d γ
conclusion
(n : ℝ) ^ (-(1 : ℝ) / 2) = (oracleMesh d n β γ) ^ (β + effectiveDimension d γ / 2)
Proof (Lean source)
lemma oracleMesh_stochastic_scale (d n : ℕ) (β γ : ℝ)
(hn : 0 < n) (hden : 0 < 2 * β + effectiveDimension d γ) :
(n : ℝ) ^ (-(1 : ℝ) / 2) =
(oracleMesh d n β γ) ^ (β + effectiveDimension d γ / 2) := by
have hn' : (0 : ℝ) < n := by exact_mod_cast hn
rw [oracleMesh, ← Real.rpow_mul hn'.le]
congr 1
field_simp
Dividing the stochastic scale by the bias scale leaves only the ratio between oracle and selected mesh widths. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β γ h :
ℝ
hn :
0 < n
hden :
0 < 2 * β + effectiveDimension d γ
hh :
0 < h
conclusion
((n : ℝ) ^ (-(1 : ℝ) / 2) * h ^ (-(effectiveDimension d γ) / 2)) / h ^ β
= (oracleMesh d n β γ / h) ^ (β + effectiveDimension d γ / 2)
Proof (Lean source)
lemma selectedMesh_stochastic_bias_ratio (d n : ℕ) (β γ h : ℝ)
(hn : 0 < n) (hden : 0 < 2 * β + effectiveDimension d γ)
(hh : 0 < h) :
((n : ℝ) ^ (-(1 : ℝ) / 2) *
h ^ (-(effectiveDimension d γ) / 2)) / h ^ β =
(oracleMesh d n β γ / h) ^
(β + effectiveDimension d γ / 2) := by
have ho : 0 < oracleMesh d n β γ := by
unfold oracleMesh
have hn' : (0 : ℝ) < n := by exact_mod_cast hn
positivity
rw [oracleMesh_stochastic_scale d n β γ hn hden]
rw [Real.div_rpow ho.le hh.le]
rw [Real.rpow_add ho, Real.rpow_add hh]
rw [show -(effectiveDimension d γ) / 2 =
-(effectiveDimension d γ / 2) by ring, Real.rpow_neg hh.le]
field_simp
A positive power in the maximal bound changes only the constant in its logarithmic factor. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma selectedMesh_maximal_log_scale (o h p : ℝ)
(ho : 0 < o) (hh : 0 < h) (hp : 0 ≤ p) :
1 + max 0 (log ((o / h) ^ p)) ≤
max 1 p * (1 + max 0 (log (o / h))) := by
rw [Real.log_rpow (div_pos ho hh)]
by_cases hl : 0 ≤ log (o / h)
· rw [max_eq_right hl, max_eq_right (mul_nonneg hp hl)]
have hmax : 1 ≤ max 1 p := le_max_left _ _
have hpmax : p ≤ max 1 p := le_max_right _ _
nlinarith
· have hl' : log (o / h) ≤ 0 := le_of_lt (lt_of_not_ge hl)
rw [max_eq_left hl', max_eq_left (mul_nonpos_of_nonneg_of_nonpos hp hl')]
simpa only [add_zero, mul_one] using (le_max_left 1 p)
A mass rank at least the ordered position gives the ordered variance proxy. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma selectedMesh_rank_proxy_to_ordered (a b α r : ℝ) (k : ℕ)
(hb : 0 ≤ b) (hα : 0 < α)
(hr : (k + 1 : ℝ) ≤ r) :
min a (b * r ^ (-α)) ≤
min a (b * (k + 1 : ℝ) ^ (-α)) := by
have hk : 0 < (k + 1 : ℝ) := by positivity
have hrpos : 0 < r := lt_of_lt_of_le hk hr
have hpow : (k + 1 : ℝ) ^ α ≤ r ^ α :=
Real.rpow_le_rpow hk.le hr hα.le
have hinv : r ^ (-α) ≤ (k + 1 : ℝ) ^ (-α) := by
rw [Real.rpow_neg hrpos.le, Real.rpow_neg hk.le]
exact (inv_le_inv₀ (Real.rpow_pos_of_pos hrpos α) (Real.rpow_pos_of_pos hk α)).2 hpow
exact min_le_min_left a (mul_le_mul_of_nonneg_left hinv hb)
A uniform lower bound on the mass coefficient absorbs its reciprocal inside the two competing variance scales. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma selectedMesh_uniform_min_proxy (A x y κ c : ℝ)
(hA : 0 ≤ A) (hx : 0 ≤ x) (hy : 0 < y)
(hκ : 0 < κ) (hc : κ ≤ c) :
A * min x (5 / (c * y)) ≤
(A * max 1 (5 / κ)) * min x y⁻¹ := by
have hcpos : 0 < c := hκ.trans_le hc
let M := max 1 (5 / κ)
have hM : 1 ≤ M := le_max_left _ _
have hMκ : 5 / κ ≤ M := le_max_right _ _
have hdiv : 5 / (c * y) ≤ M * y⁻¹ := by
rw [show M * y⁻¹ = M / y by ring]
apply (div_le_div_iff₀ (mul_pos hcpos hy) hy).2
have h : 5 ≤ M * c := by
have h' : 5 ≤ M * κ := (div_le_iff₀ hκ).mp hMκ
nlinarith [mul_nonneg (sub_nonneg.mpr hc) (le_trans (by norm_num : (0 : ℝ) ≤ 1) hM)]
nlinarith
have hmin : min x (5 / (c * y)) ≤ M * min x y⁻¹ := by
by_cases hxy : x ≤ y⁻¹
· rw [min_eq_left hxy]
exact (min_le_left _ _).trans (by nlinarith)
· have hyx : y⁻¹ ≤ x := le_of_lt (lt_of_not_ge hxy)
rw [min_eq_right hyx]
exact (min_le_right _ _).trans hdiv
calc
A * min x (5 / (c * y)) ≤ A * (M * min x y⁻¹) :=
mul_le_mul_of_nonneg_left hmin hA
_ = (A * M) * min x y⁻¹ := by ring
Normalize the ordered inverse-count proxy after bounding its mass coefficient uniformly on the adaptation interval. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma selectedMesh_uniform_ordered_proxy
(A x n h r κ c D α : ℝ)
(hA : 0 ≤ A) (hx : 0 ≤ x) (hn : 0 < n) (hh : 0 < h)
(hr : 0 < r) (hκ : 0 < κ) (hc : κ ≤ c) :
A * min x (5 / (n * c * h ^ D * r ^ α)) ≤
(A * max 1 (5 / κ)) *
min x (n⁻¹ * h ^ (-D) * r ^ (-α)) := by
have hy : 0 < n * h ^ D * r ^ α := by positivity
have hproxy := selectedMesh_uniform_min_proxy A x
(n * h ^ D * r ^ α) κ c hA hx hy hκ hc
have hden : n * c * h ^ D * r ^ α = c * (n * h ^ D * r ^ α) := by ring
rw [← hden] at hproxy
simpa only [mul_inv_rev, Real.rpow_neg hh.le, Real.rpow_neg hr.le,
mul_assoc, mul_comm, mul_left_comm] using hproxy
theorem selectedMesh_uniform_ranked_raw_proxy Lemma selectedMesh_uniform_ranked_raw_proxy in the paper ↗
The raw selected-mesh variance proxy has one constant for every ordered cube, including ties in the cube masses. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma selectedMesh_uniform_ranked_raw_proxy
(A x n h r κ c D α : ℝ) (k : ℕ)
(hA : 0 ≤ A) (hx : 0 ≤ x) (hn : 0 < n) (hh : 0 < h)
(hr : 0 < r) (hκ : 0 < κ) (hc : κ ≤ c)
(hα : 0 < α) (hrank : (k + 1 : ℝ) ≤ r) :
A * min x (5 / (n * c * h ^ D * r ^ α)) ≤
(A * max 1 (5 / κ)) *
min x (n⁻¹ * h ^ (-D) * (k + 1 : ℝ) ^ (-α)) := by
have hnormalized := selectedMesh_uniform_ordered_proxy
A x n h r κ c D α hA hx hn hh hr hκ hc
have hordered := selectedMesh_rank_proxy_to_ordered x
(n⁻¹ * h ^ (-D)) α r k (by positivity) hα hrank
exact hnormalized.trans
(mul_le_mul_of_nonneg_left hordered
(mul_nonneg hA (le_trans (by norm_num : (0 : ℝ) ≤ 1) (le_max_left 1 (5 / κ)))))
Convert a raw mass-rank MGF estimate into the common ordered proxy used by the scalar maximal inequality. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
Measure Ω
Z :
Ω → ℝ
A x n h r κ c D α :
ℝ
k :
ℕ
t :
ℝ
hA :
0 ≤ A
hx :
0 ≤ x
hn :
0 < n
hh :
0 < h
hr :
0 < r
hκ :
0 < κ
hc :
κ ≤ c
hα :
0 < α
hrank :
(k + 1 : ℝ) ≤ r
Proof (Lean source)
lemma selectedMesh_uniform_ranked_mgf
{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) (Z : Ω → ℝ)
(A x n h r κ c D α : ℝ) (k : ℕ) (t : ℝ)
(hA : 0 ≤ A) (hx : 0 ≤ x) (hn : 0 < n) (hh : 0 < h)
(hr : 0 < r) (hκ : 0 < κ) (hc : κ ≤ c)
(hα : 0 < α) (hrank : (k + 1 : ℝ) ≤ r)
(hmgf : mgf Z μ t ≤
exp (A * min x (5 / (n * c * h ^ D * r ^ α)) * t ^ 2 / 2)) :
mgf Z μ t ≤ exp
((A * max 1 (5 / κ)) *
min x (n⁻¹ * h ^ (-D) * (k + 1 : ℝ) ^ (-α)) * t ^ 2 / 2) := by
exact hmgf.trans (Real.exp_le_exp.mpr (by
apply div_le_div_of_nonneg_right _ (by norm_num)
apply mul_le_mul_of_nonneg_right _ (sq_nonneg t)
exact selectedMesh_uniform_ranked_raw_proxy
A x n h r κ c D α k hA hx hn hh hr hκ hc hα hrank))
theorem orderedSubGaussianMaximal_uniform
One maximal constant works over a fixed compact exponent interval. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma orderedSubGaussianMaximal_uniform (αmin αmax : ℝ)
(hαmin : 0 < αmin) (hαmax : αmin ≤ αmax) :
∃ K : ℝ, 0 < K ∧
∀ (α : ℝ), α ∈ Icc αmin αmax →
∀ {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(N : ℕ) (Z : Fin N → Ω → ℝ) (a b : ℝ),
0 < a → 0 < b →
(∀ k : Fin N, ∀ t : ℝ,
Integrable (fun ω => exp (t * Z k ω)) μ) →
(∀ k : Fin N, ∀ t : ℝ,
mgf (Z k) μ t ≤ exp
(min (a ^ 2) (b ^ 2 * (k.val + 1 : ℝ) ^ (-α)) * t ^ 2 / 2)) →
∫ ω, sSup ((fun k : Fin N => |Z k ω|) '' univ) ∂μ ≤
K * a * sqrt (1 + max 0 (log ((b / a) ^ (2 / α)))) := by
obtain ⟨Kmin, hKmin, hmax⟩ :=
Causalean.Mathlib.Probability.SubGaussian.orderedSubGaussianMaximal αmin hαmin
let R : ℝ := αmax / αmin
have hR : 1 ≤ R := by
dsimp [R]
exact (le_div_iff₀ hαmin).2 (by nlinarith)
refine ⟨Kmin * sqrt R, mul_pos hKmin (Real.sqrt_pos.2 (lt_of_lt_of_le zero_lt_one hR)), ?_⟩
intro α hα Ω _ μ _ N Z a b ha hb hmgf hbound
have hαpos : 0 < α := lt_of_lt_of_le hαmin hα.1
have hproxy : ∀ k : Fin N, ∀ t : ℝ,
mgf (Z k) μ t ≤ exp
(min (a ^ 2) (b ^ 2 * (k.val + 1 : ℝ) ^ (-αmin)) * t ^ 2 / 2) := by
intro k t
calc
mgf (Z k) μ t ≤ exp
(min (a ^ 2) (b ^ 2 * (k.val + 1 : ℝ) ^ (-α)) * t ^ 2 / 2) := hbound k t
_ ≤ exp
(min (a ^ 2) (b ^ 2 * (k.val + 1 : ℝ) ^ (-αmin)) * t ^ 2 / 2) := by
apply Real.exp_le_exp.mpr
apply div_le_div_of_nonneg_right _ (by norm_num)
apply mul_le_mul_of_nonneg_right _ (sq_nonneg t)
apply min_le_min_left
apply mul_le_mul_of_nonneg_left _ (sq_nonneg b)
apply Real.rpow_le_rpow_of_exponent_le
· exact_mod_cast Nat.le_add_left 1 k.val
· linarith [hα.1]
have hbase := hmax μ N Z a b ha hb hmgf hproxy
have hrpos : 0 < b / a := div_pos hb ha
have hscale :
1 + max 0 (log ((b / a) ^ (2 / αmin))) ≤
R * (1 + max 0 (log ((b / a) ^ (2 / α)))) := by
rw [Real.log_rpow hrpos, Real.log_rpow hrpos]
by_cases ht : 0 ≤ log (b / a)
· have hamin : 0 ≤ 2 / αmin * log (b / a) :=
mul_nonneg (div_nonneg (by norm_num) (le_of_lt hαmin)) ht
have haα : 0 ≤ 2 / α * log (b / a) :=
mul_nonneg (div_nonneg (by norm_num) (le_of_lt hαpos)) ht
rw [max_eq_right hamin, max_eq_right haα]
dsimp [R]
have hαle : α ≤ αmax := hα.2
apply (mul_le_mul_iff_of_pos_left hαmin).mp
have hαne : α ≠ 0 := ne_of_gt hαpos
have hminne : αmin ≠ 0 := ne_of_gt hαmin
have hprod := mul_nonneg (sub_nonneg.mpr hαle) ht
field_simp
nlinarith
· have ht' : log (b / a) ≤ 0 := le_of_lt (lt_of_not_ge ht)
have hamin : 2 / αmin * log (b / a) ≤ 0 :=
mul_nonpos_of_nonneg_of_nonpos
(div_nonneg (by norm_num) (le_of_lt hαmin)) ht'
have haα : 2 / α * log (b / a) ≤ 0 :=
mul_nonpos_of_nonneg_of_nonpos
(div_nonneg (by norm_num) (le_of_lt hαpos)) ht'
rw [max_eq_left hamin, max_eq_left haα]
nlinarith [hR]
have hsqrt :
sqrt (1 + max 0 (log ((b / a) ^ (2 / αmin)))) ≤
sqrt R * sqrt (1 + max 0 (log ((b / a) ^ (2 / α)))) := by
rw [← Real.sqrt_mul (le_of_lt (lt_of_lt_of_le zero_lt_one hR))]
exact Real.sqrt_le_sqrt hscale
calc
∫ ω, sSup ((fun k : Fin N => |Z k ω|) '' univ) ∂μ ≤
Kmin * a * sqrt (1 + max 0 (log ((b / a) ^ (2 / αmin)))) := hbase
_ ≤ (Kmin * sqrt R) * a *
sqrt (1 + max 0 (log ((b / a) ^ (2 / α)))) := by
nlinarith [mul_le_mul_of_nonneg_left hsqrt (mul_nonneg (le_of_lt hKmin) (le_of_lt ha))]
The maximum of the coefficient sup norms is bounded by the sum of the coordinatewise maxima. This lets a scalar maximal inequality handle the finite monomial index without changing its logarithmic scale. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma selectedMesh_maxNorm_le_sum_maxAbs {κ ι : Type*}
[Finite κ] [Nonempty κ] [Fintype ι] (f : κ → ι → ℝ) :
sSup ((fun k : κ => ‖f k‖) '' univ) ≤
∑ a : ι, sSup ((fun k : κ => |f k a|) '' univ) := by
classical
apply csSup_le
· exact ⟨‖f (choice inferInstance)‖, ⟨_, Set.mem_univ _, rfl⟩⟩
· rintro x ⟨k, -, rfl⟩
have hb (a : ι) : BddAbove ((fun k : κ => |f k a|) '' univ) :=
by simpa only [Set.image_univ] using
(Set.finite_range (fun k : κ => |f k a|)).bddAbove
have hn (a : ι) : 0 ≤ sSup ((fun k : κ => |f k a|) '' univ) :=
(abs_nonneg _).trans (le_csSup (hb a) ⟨k, Set.mem_univ _, rfl⟩)
apply (pi_norm_le_iff_of_nonneg (Finset.sum_nonneg fun a ha => hn a)).2
intro a
calc
‖f k a‖ = |f k a| := Real.norm_eq_abs _
_ ≤ sSup ((fun k : κ => |f k a|) '' univ) :=
le_csSup (hb a) ⟨k, Set.mem_univ _, rfl⟩
_ ≤ ∑ b : ι, sSup ((fun k : κ => |f k b|) '' univ) :=
Finset.single_le_sum (fun b hb => hn b) (Finset.mem_univ a)
theorem selectedMesh_integral_maxNorm_le_sum_maxAbs Lemma selectedMesh_integral_maxNorm_le_sum_maxAbs in the paper ↗
Integrate the finite-coordinate reduction once the scalar maxima are integrable, as they are under the conditional sub-Gaussian bounds. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma selectedMesh_integral_maxNorm_le_sum_maxAbs
{Ω κ ι : Type*} [MeasurableSpace Ω] [Finite κ] [Nonempty κ]
[Fintype ι] (μ : Measure Ω) (f : Ω → κ → ι → ℝ)
(hwhole : Integrable (fun ξ =>
sSup ((fun k : κ => ‖f ξ k‖) '' univ)) μ)
(hcoord : ∀ a : ι, Integrable (fun ξ =>
sSup ((fun k : κ => |f ξ k a|) '' univ)) μ) :
(∫ ξ, sSup ((fun k : κ => ‖f ξ k‖) '' univ) ∂μ) ≤
∑ a : ι, ∫ ξ, sSup ((fun k : κ => |f ξ k a|) '' univ) ∂μ := by
have hsum : Integrable (fun ξ =>
∑ a : ι, sSup ((fun k : κ => |f ξ k a|) '' univ)) μ :=
integrable_finsetSum _ (fun a ha => hcoord a)
calc
(∫ ξ, sSup ((fun k : κ => ‖f ξ k‖) '' univ) ∂μ) ≤
∫ ξ, ∑ a : ι, sSup ((fun k : κ => |f ξ k a|) '' univ) ∂μ :=
integral_mono hwhole hsum (fun ξ => selectedMesh_maxNorm_le_sum_maxAbs (f ξ))
_ = _ := integral_finsetSum _ (fun a ha => hcoord a)
theorem selectedMesh_integrable_of_exp_moments Lemma selectedMesh_integrable_of_exp_moments in the paper ↗
Exponential moments on both sides make a scalar coefficient integrable. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
Measure Ω
Z :
Ω → ℝ
hpos :
Integrable (fun ω => exp (Z ω)) μ
hneg :
Integrable (fun ω => exp (-Z ω)) μ
conclusion
Integrable Z μ
Proof (Lean source)
lemma selectedMesh_integrable_of_exp_moments
{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) (Z : Ω → ℝ)
(hpos : Integrable (fun ω => exp (Z ω)) μ)
(hneg : Integrable (fun ω => exp (-Z ω)) μ) :
Integrable Z μ := by
have h := ProbabilityTheory.integrable_pow_abs_mul_exp_of_integrable_exp_mul
(X := Z) (v := 0) (t := 1) (by norm_num)
(by simpa using hpos) (by simpa using hneg) 1
have hm : AEStronglyMeasurable Z μ :=
(ProbabilityTheory.aemeasurable_of_integrable_exp_mul
(X := Z) (u := 1) (v := -1) (by norm_num)
(by simpa using hpos) (by simpa using hneg)).aestronglyMeasurable
apply (integrable_norm_iff hm).mp
simpa only [Real.norm_eq_abs, pow_one, zero_mul, Real.exp_zero, mul_one] using h
theorem selectedMesh_integrable_finset_maxAbs Lemma selectedMesh_integrable_finset_maxAbs in the paper ↗
The maximum of finitely many integrable scalar absolute values is integrable. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Integrable (fun ω => s.sup' hs (fun k => |Z k ω|)) μ
Proof (Lean source)
lemma selectedMesh_integrable_finset_maxAbs
{Ω κ : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) (Z : κ → Ω → ℝ)
(hZ : ∀ k, Integrable (Z k) μ)
(s : Finset κ) (hs : s.Nonempty) :
Integrable (fun ω => s.sup' hs (fun k => |Z k ω|)) μ := by
classical
induction s using Finset.induction_on with
| empty => exact (Finset.not_nonempty_empty hs).elim
| @insert k s hk ih =>
by_cases hs' : s.Nonempty
· have h := (hZ k).abs.sup (ih hs')
convert h using 1
funext ω
simp only [Finset.sup'_insert hs', Pi.sup_apply]
· have he : s = ∅ := Finset.not_nonempty_iff_eq_empty.mp hs'
subst s
simpa only [Finset.insert_empty, Finset.sup'_singleton] using (hZ k).abs
An integrable finite family has an integrable maximum absolute value. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma selectedMesh_integrable_maxAbs
{Ω κ : Type*} [MeasurableSpace Ω] [Finite κ] [Nonempty κ]
(μ : Measure Ω) (Z : κ → Ω → ℝ)
(hZ : ∀ k, Integrable (Z k) μ) :
Integrable (fun ω => sSup ((fun k : κ => |Z k ω|) '' univ)) μ := by
classical
letI := ofFinite κ
have h := selectedMesh_integrable_finset_maxAbs μ Z hZ
univ univ_nonempty
convert h using 1
funext ω
simpa using (Finset.sup'_eq_csSup_image univ univ_nonempty
(fun k => |Z k ω|)).symm
theorem selectedMesh_integrable_maxAbs_of_exp_moments Lemma selectedMesh_integrable_maxAbs_of_exp_moments in the paper ↗
Finite scalar maxima are integrable when all coordinates have two-sided exponential moments. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma selectedMesh_integrable_maxAbs_of_exp_moments
{Ω κ : Type*} [MeasurableSpace Ω] [Finite κ] [Nonempty κ]
(μ : Measure Ω) (Z : κ → Ω → ℝ)
(hexp : ∀ k, ∀ t : ℝ,
Integrable (fun ω => exp (t * Z k ω)) μ) :
Integrable (fun ω => sSup ((fun k : κ => |Z k ω|) '' univ)) μ := by
have hZ (k : κ) : Integrable (Z k) μ :=
selectedMesh_integrable_of_exp_moments μ (Z k)
(by simpa using hexp k 1) (by simpa using hexp k (-1))
exact selectedMesh_integrable_maxAbs μ Z hZ
theorem selectedMesh_integrable_vector_max_of_exp_moments Lemma selectedMesh_integrable_vector_max_of_exp_moments in the paper ↗
Coordinate exponential moments give the integrability conditions for a finite maximum of coefficient sup norms. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
Proof (Lean source)
lemma selectedMesh_integrable_vector_max_of_exp_moments
{Ω κ ι : Type*} [MeasurableSpace Ω] [Finite κ] [Nonempty κ]
[Fintype ι] (μ : Measure Ω) (Z : κ → Ω → ι → ℝ)
(hexp : ∀ k i t,
Integrable (fun ω => exp (t * Z k ω i)) μ) :
Integrable (fun ω => sSup ((fun k : κ => ‖Z k ω‖) '' univ)) μ ∧
∀ i : ι, Integrable (fun ω =>
sSup ((fun k : κ => |Z k ω i|) '' univ)) μ := by
have hZ (k : κ) (i : ι) : Integrable (fun ω => Z k ω i) μ :=
selectedMesh_integrable_of_exp_moments μ (fun ω => Z k ω i)
(by simpa using hexp k i 1) (by simpa using hexp k i (-1))
have hcoord (i : ι) : Integrable (fun ω =>
sSup ((fun k : κ => |Z k ω i|) '' univ)) μ :=
selectedMesh_integrable_maxAbs μ (fun k ω => Z k ω i) (fun k => hZ k i)
have hnorm (k : κ) : Integrable (fun ω => ‖Z k ω‖) μ := by
apply Integrable.norm
exact (integrable_pi_iff).2 (hZ k)
have hwhole := selectedMesh_integrable_maxAbs μ
(fun k ω => ‖Z k ω‖) hnorm
refine ⟨?_, hcoord⟩
simpa only [abs_norm] using hwhole
theorem orderedSubGaussianMaximal_vector_uniform Lemma orderedSubGaussianMaximal_vector_uniform in the paper ↗
Apply the scalar ordered maximal theorem to every coefficient coordinate, then sum over the fixed finite monomial index. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma orderedSubGaussianMaximal_vector_uniform
(αmin αmax : ℝ) (hαmin : 0 < αmin) (hαmax : αmin ≤ αmax) :
∃ K : ℝ, 0 < K ∧
∀ (α : ℝ), α ∈ Icc αmin αmax →
∀ {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(N : ℕ) [Nonempty (Fin N)] (Z : Fin N → Ω → ι → ℝ)
(a b : ℝ), 0 < a → 0 < b →
(∀ k : Fin N, ∀ i : ι, ∀ t : ℝ,
Integrable (fun ω => exp (t * Z k ω i)) μ) →
(∀ k : Fin N, ∀ i : ι, ∀ t : ℝ,
mgf (fun ω => Z k ω i) μ t ≤ exp
(min (a ^ 2) (b ^ 2 * (k.val + 1 : ℝ) ^ (-α)) * t ^ 2 / 2)) →
Integrable (fun ω =>
sSup ((fun k : Fin N => ‖Z k ω‖) '' univ)) μ →
(∀ i : ι, Integrable (fun ω =>
sSup ((fun k : Fin N => |Z k ω i|) '' univ)) μ) →
(∫ ω, sSup ((fun k : Fin N => ‖Z k ω‖) '' univ) ∂μ) ≤
(Fintype.card ι : ℝ) * K * a *
sqrt (1 + max 0 (log ((b / a) ^ (2 / α)))) := by
obtain ⟨K, hK, hmax⟩ :=
orderedSubGaussianMaximal_uniform αmin αmax hαmin hαmax
refine ⟨K, hK, ?_⟩
intro α hα Ω ι _ _ μ _ N _ Z a b ha hb hint hmgf hwhole hcoord
calc
(∫ ω, sSup ((fun k : Fin N => ‖Z k ω‖) '' univ) ∂μ) ≤
∑ i : ι, ∫ ω, sSup ((fun k : Fin N => |Z k ω i|) '' univ) ∂μ :=
selectedMesh_integral_maxNorm_le_sum_maxAbs μ
(fun ω k i => Z k ω i) hwhole hcoord
_ ≤ ∑ _i : ι, K * a *
sqrt (1 + max 0 (log ((b / a) ^ (2 / α)))) := by
apply Finset.sum_le_sum
intro i _
exact hmax α hα μ N (fun k ω => Z k ω i) a b ha hb
(fun k t => hint k i t) (fun k t => hmgf k i t)
_ = (Fintype.card ι : ℝ) * K * a *
sqrt (1 + max 0 (log ((b / a) ^ (2 / α)))) := by
simp [mul_assoc]
theorem orderedSubGaussianMaximal_vector_uniform_of_exp Lemma orderedSubGaussianMaximal_vector_uniform_of_exp in the paper ↗
The uniform vector maximal inequality with its integrability hypotheses deduced from the coordinate exponential moments. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma orderedSubGaussianMaximal_vector_uniform_of_exp
(αmin αmax : ℝ) (hαmin : 0 < αmin) (hαmax : αmin ≤ αmax) :
∃ K : ℝ, 0 < K ∧
∀ (α : ℝ), α ∈ Icc αmin αmax →
∀ {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(N : ℕ) [Nonempty (Fin N)] (Z : Fin N → Ω → ι → ℝ)
(a b : ℝ), 0 < a → 0 < b →
(∀ k : Fin N, ∀ i : ι, ∀ t : ℝ,
Integrable (fun ω => exp (t * Z k ω i)) μ) →
(∀ k : Fin N, ∀ i : ι, ∀ t : ℝ,
mgf (fun ω => Z k ω i) μ t ≤ exp
(min (a ^ 2) (b ^ 2 * (k.val + 1 : ℝ) ^ (-α)) * t ^ 2 / 2)) →
(∫ ω, sSup ((fun k : Fin N => ‖Z k ω‖) '' univ) ∂μ) ≤
(Fintype.card ι : ℝ) * K * a *
sqrt (1 + max 0 (log ((b / a) ^ (2 / α)))) := by
obtain ⟨K, hK, hmax⟩ :=
orderedSubGaussianMaximal_vector_uniform αmin αmax hαmin hαmax
refine ⟨K, hK, ?_⟩
intro α hα Ω ι _ _ μ _ N _ Z a b ha hb hexp hmgf
obtain ⟨hwhole, hcoord⟩ :=
selectedMesh_integrable_vector_max_of_exp_moments μ Z hexp
exact hmax α hα μ N Z a b ha hb hexp hmgf hwhole hcoord
theorem orderedSubGaussianMaximal_vector_equiv_uniform_of_exp Lemma orderedSubGaussianMaximal_vector_equiv_uniform_of_exp in the paper ↗
Reindex the vector maximal inequality by a mass-ordered enumeration of a finite cube family. The maximum over cubes is unchanged by the enumeration. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma orderedSubGaussianMaximal_vector_equiv_uniform_of_exp
(αmin αmax : ℝ) (hαmin : 0 < αmin) (hαmax : αmin ≤ αmax) :
∃ K : ℝ, 0 < K ∧
∀ (α : ℝ), α ∈ Icc αmin αmax →
∀ {Ω κ ι : Type*} [MeasurableSpace Ω] [Fintype ι]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(N : ℕ) [Nonempty (Fin N)]
(σ : Fin N ≃ κ) (Z : κ → Ω → ι → ℝ)
(a b : ℝ), 0 < a → 0 < b →
(∀ k : Fin N, ∀ i : ι, ∀ t : ℝ,
Integrable (fun ω => exp (t * Z (σ k) ω i)) μ) →
(∀ k : Fin N, ∀ i : ι, ∀ t : ℝ,
mgf (fun ω => Z (σ k) ω i) μ t ≤ exp
(min (a ^ 2) (b ^ 2 * (k.val + 1 : ℝ) ^ (-α)) * t ^ 2 / 2)) →
(∫ ω, sSup ((fun k : κ => ‖Z k ω‖) '' univ) ∂μ) ≤
(Fintype.card ι : ℝ) * K * a *
sqrt (1 + max 0 (log ((b / a) ^ (2 / α)))) := by
obtain ⟨K, hK, hmax⟩ :=
orderedSubGaussianMaximal_vector_uniform_of_exp αmin αmax hαmin hαmax
refine ⟨K, hK, ?_⟩
intro α hα Ω κ ι _ _ μ _ N _ σ Z a b ha hb hexp hmgf
have hreindex (ω : Ω) :
sSup ((fun k : κ => ‖Z k ω‖) '' univ) =
sSup ((fun k : Fin N => ‖Z (σ k) ω‖) '' univ) := by
congr 1
ext x
simp only [Set.mem_image, Set.mem_univ, true_and]
constructor
· rintro ⟨k, rfl⟩
exact ⟨σ.symm k, by simp⟩
· rintro ⟨k, rfl⟩
exact ⟨σ k, rfl⟩
simp only [hreindex]
exact hmax α hα μ N (fun k ω => Z (σ k) ω) a b ha hb hexp hmgf
theorem selectedMesh_rank_exponent_mem_interval Lemma selectedMesh_rank_exponent_mem_interval in the paper ↗
The weak-overlap rank exponent stays in a fixed positive interval when the overlap exponent ranges over a compact interval. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
γmin γmax γ :
ℝ
hmin :
1 < γmin
hlo :
γmin ≤ γ
hhi :
γ ≤ γmax
conclusion
conclusion 1
1 / (γmax - 1) ≤ 1 / (γ - 1)
conclusion 2
1 / (γ - 1) ≤ 1 / (γmin - 1)
Proof (Lean source)
lemma selectedMesh_rank_exponent_mem_interval
(γmin γmax γ : ℝ) (hmin : 1 < γmin)
(hlo : γmin ≤ γ) (hhi : γ ≤ γmax) :
1 / (γmax - 1) ≤ 1 / (γ - 1) ∧
1 / (γ - 1) ≤ 1 / (γmin - 1) := by
have hdenmin : 0 < γmin - 1 := by linarith
have hden : 0 < γ - 1 := by linarith
have hdenmax : 0 < γmax - 1 := by linarith
constructor
· apply (div_le_div_iff₀ hdenmax hden).2
nlinarith
· apply (div_le_div_iff₀ hden hdenmin).2
nlinarith
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.SelectedMesh.Measurability 24 declarations Measurability of the selected equal-cell estimator
Measurability of the selected equal-cell estimator
def equalCellCubeLossReal
For a sample size, smoothness exponent, outcome bound, regression function, and realized sample, the real-valued equal-cell cube loss is the supremum, over cube points, of the estimator's absolute error.
Feasibility of a fixed mesh index is a measurable sample event. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m :
ℕ
β :
ℝ
j :
ℕ
conclusion
Proof (Lean source)
lemma meshFeasible_measurableSet {d n : ℕ} (m : ℕ) (β : ℝ) (j : ℕ) :
MeasurableSet {ω : Fin n → Obs d | meshFeasible m β ω j} := by
classical
unfold meshFeasible
by_cases hj : j ∈ meshIndices d n β
· simp only [hj, true_and]
rw [show {ω : Fin n → Obs d |
∀ (k : Fin d → Fin (2 ^ j)) (l : Fin d → Fin (m + 1)),
meshWidth j ^ (-2 * β) ≤ (treatedCount m j ω k l : ℝ)} =
⋂ k, ⋂ l, {ω | meshWidth j ^ (-2 * β) ≤
(treatedCount m j ω k l : ℝ)} by
ext ω
simp]
exact MeasurableSet.iInter fun k => MeasurableSet.iInter fun l =>
measurableSet_le measurable_const (treatedCount_measurable m j k l)
· simp [hj]
theorem feasibleIndices_mem_measurableSet Lemma feasible-indices-membership-measurable-set in the paper ↗
Membership of a fixed index in the data-selected feasible set is measurable. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m :
ℕ
β :
ℝ
j :
ℕ
conclusion
Proof (Lean source)
lemma feasibleIndices_mem_measurableSet {d n : ℕ} (m : ℕ) (β : ℝ) (j : ℕ) :
MeasurableSet {ω : Fin n → Obs d | j ∈ feasibleIndices m β ω} := by
classical
simp only [feasibleIndices, mem_filter]
by_cases hj : j ∈ meshIndices d n β
· simpa [hj] using meshFeasible_measurableSet (d := d) (n := n) m β j
· simp [hj]
The finite feasible-index set is a measurable function of the sample. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m :
ℕ
β :
ℝ
conclusion
Proof (Lean source)
lemma feasibleIndices_measurable {d n : ℕ} (m : ℕ) (β : ℝ) :
Measurable (fun ω : Fin n → Obs d => feasibleIndices m β ω) := by
classical
apply measurable_to_countable'
intro T
rw [show (fun ω : Fin n → Obs d => feasibleIndices m β ω) ⁻¹' {T} =
⋂ j : ℕ, {ω | (j ∈ feasibleIndices m β ω) ↔ j ∈ T} by
ext ω
simp only [Set.mem_preimage, Set.mem_singleton_iff, Set.mem_iInter,
Set.mem_ofPred_eq]
exact Finset.ext_iff]
exact MeasurableSet.iInter fun j => by
by_cases hj : j ∈ T
· simpa [hj] using feasibleIndices_mem_measurableSet (d := d) (n := n) m β j
· rw [show {ω : Fin n → Obs d |
(j ∈ feasibleIndices m β ω) ↔ j ∈ T} =
{ω | j ∈ feasibleIndices m β ω}ᶜ by
ext ω
simp [hj]]
exact (feasibleIndices_mem_measurableSet (d := d) (n := n) m β j).compl
The selected mesh width is a measurable function of the finite sample. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma countSelectedMesh_measurable {d n : ℕ} (m : ℕ) (β : ℝ) :
Measurable (countSelectedMesh (d := d) (n := n) m β) := by
unfold countSelectedMesh
exact (measurable_of_countable (fun S : Finset ℕ =>
if S.Nonempty then meshWidth (S.sup id) else 0)).comp
(feasibleIndices_measurable m β)
theorem treatedSampleMicrocell_measurableSet Lemma treated-sample-microcell-measurable-set in the paper ↗
A fixed sample coordinate falling in a fixed treated microcell is measurable. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma treatedSampleMicrocell_measurableSet {d n : ℕ} (m j : ℕ)
(k : Fin d → Fin (2 ^ j)) (l : Fin d → Fin (m + 1)) (i : Fin n) :
MeasurableSet {ω : Fin n → Obs d | (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k l} := by
exact ((measurableSet_singleton true).preimage
((measurable_fst.comp measurable_snd).comp (measurable_pi_apply i))).inter
((orderedMass_scaledMicroCell_measurable d m j k l).preimage
(measurable_fst.comp (measurable_pi_apply i)))
theorem responseMoment_coordinate_measurable Lemma response-moment-coordinate-measurable in the paper ↗
Every coordinate of the equal-cell response moment is sample measurable. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma responseMoment_coordinate_measurable {d n : ℕ} (m j : ℕ)
(k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) :
Measurable (fun ω : Fin n → Obs d => responseMoment m j ω k α) := by
classical
unfold responseMoment
apply measurable_const.mul
apply Finset.measurable_fun_sum
intro l _
apply (treatedCount_measurable m j k l).inv.mul
apply Finset.measurable_fun_sum
intro i _
apply Measurable.ite (treatedSampleMicrocell_measurableSet m j k l i)
· unfold monoVec
fun_prop
· exact measurable_const
Every coordinate of the equal-cell Gram matrix is sample measurable. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma gramHat_coordinate_measurable {d n : ℕ} (m j : ℕ)
(k : Fin d → Fin (2 ^ j)) (α δ : MonoIndex d m) :
Measurable (fun ω : Fin n → Obs d => gramHat m j ω k α δ) := by
classical
unfold gramHat
apply measurable_const.mul
apply Finset.measurable_fun_sum
intro l _
apply (treatedCount_measurable m j k l).inv.mul
apply Finset.measurable_fun_sum
intro i _
apply Measurable.ite (treatedSampleMicrocell_measurableSet m j k l i)
· unfold monoVec
fun_prop
· exact measurable_const
A finite real matrix with measurable entries has measurable determinant. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Measurable (fun ω => (A ω).det)
Proof (Lean source)
lemma finiteMatrix_det_measurable {Omega I : Type*} [MeasurableSpace Omega]
[Fintype I] [DecidableEq I] (A : Omega → Matrix I I ℝ)
(hA : ∀ i j, Measurable (fun ω => A ω i j)) :
Measurable (fun ω => (A ω).det) :=
measurable_matrix_det A hA
theorem finiteMatrix_inv_coordinate_measurable Lemma finite-matrix-inverse-coordinate-measurable in the paper ↗
Every entry of the inverse of a finite measurable real matrix is measurable. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Measurable (fun ω => (A ω)⁻¹ i j)
Proof (Lean source)
lemma finiteMatrix_inv_coordinate_measurable {Omega I : Type*}
[MeasurableSpace Omega] [Fintype I] [DecidableEq I]
(A : Omega → Matrix I I ℝ)
(hA : ∀ i j, Measurable (fun ω => A ω i j)) (i j : I) :
Measurable (fun ω => (A ω)⁻¹ i j) :=
measurable_matrix_inv_apply A hA i j
Every coordinate of the equal-cell fitted coefficient vector is measurable. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma coefHat_coordinate_measurable {d n : ℕ} (m j : ℕ)
(k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) :
Measurable (fun ω : Fin n → Obs d => coefHat m j ω k α) := by
classical
unfold coefHat mulVec dotProduct
apply Finset.measurable_fun_sum
intro b _
exact (finiteMatrix_inv_coordinate_measurable
(fun ω => gramHat m j ω k)
(fun a b => gramHat_coordinate_measurable m j k a b) α b).mul
(responseMoment_coordinate_measurable m j k b)
theorem equalCellEstimator_fixedIndex_measurable Lemma equal-cell-estimator-fixed-index-measurable in the paper ↗
With the dyadic index fixed, the clipped equal-cell estimate is measurable in the sample. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Measurable (fun ω : Fin n → Obs d => let m := polynomialDegree β let h := countSelectedMesh m β ω if h = 0 then 0 else let k := cubeIndex d j x max (-B) (min B (∑ α, monoVec d m (fun a => (x a - cubeCorner d j k a) / h) α * coefHat m j ω k α)))
Proof (Lean source)
lemma equalCellEstimator_fixedIndex_measurable {d n : ℕ} (β B : ℝ)
(x : Fin d → ℝ) (j : ℕ) :
Measurable (fun ω : Fin n → Obs d =>
let m := polynomialDegree β
let h := countSelectedMesh m β ω
if h = 0 then 0 else
let k := cubeIndex d j x
max (-B) (min B (∑ α, monoVec d m
(fun a => (x a - cubeCorner d j k a) / h) α *
coefHat m j ω k α))) := by
let m := polynomialDegree β
let k := cubeIndex d j x
have hh : Measurable (fun ω : Fin n → Obs d => countSelectedMesh m β ω) :=
countSelectedMesh_measurable m β
apply Measurable.ite (measurableSet_eq_fun hh measurable_const)
· exact measurable_const
· apply Measurable.max measurable_const
apply Measurable.min measurable_const
apply Finset.measurable_fun_sum
intro α _
apply mul
· unfold monoVec
fun_prop
· exact coefHat_coordinate_measurable m j k α
theorem equalCellEstimator_fixedPoint_measurable Lemma equal-cell-estimator-fixed-point-measurable in the paper ↗
At every fixed covariate point, the selected equal-cell estimate is measurable in the sample. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma equalCellEstimator_fixedPoint_measurable {d n : ℕ} (β B : ℝ)
(x : Fin d → ℝ) :
Measurable (fun ω : Fin n → Obs d => equalCellEstimator β B ω x) := by
let m := polynomialDegree β
let J : (Fin n → Obs d) → ℕ :=
fun ω => (feasibleIndices m β ω).sup id
let F : ((Fin n → Obs d) × ℕ) → ℝ := fun p =>
let ω := p.1
let j := p.2
let h := countSelectedMesh m β ω
if h = 0 then 0 else
let k := cubeIndex d j x
max (-B) (min B (∑ α, monoVec d m
(fun a => (x a - cubeCorner d j k a) / h) α *
coefHat m j ω k α))
have hF : Measurable F := measurable_from_prod_countable_left fun j => by
simpa [F, m] using equalCellEstimator_fixedIndex_measurable β B x j
have hJ : Measurable J := by
exact (measurable_of_countable (fun S : Finset ℕ => S.sup id)).comp
(feasibleIndices_measurable m β)
have hc := hF.comp (measurable_id.prodMk hJ)
convert hc using 1 <;> simp [F, J, m, equalCellEstimator, Function.comp_def]
theorem measurable_iSup_of_continuous_sections Lemma measurable-isup-of-continuous-sections in the paper ↗
The supremum of pointwise measurable continuous ENNReal sections over a second-countable space is measurable. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Measurable (fun ω => ⨆ x, f ω x)
Proof (Lean source)
lemma measurable_iSup_of_continuous_sections {Omega X : Type*}
[MeasurableSpace Omega] [TopologicalSpace X] [SecondCountableTopology X]
(f : Omega → X → ℝ≥0∞) (hm : ∀ x, Measurable (fun ω => f ω x))
(hc : ∀ ω, Continuous (f ω)) :
Measurable (fun ω => ⨆ x, f ω x) := by
cases isEmpty_or_nonempty X with
| inl _ => simp
| inr _ =>
have heq : (fun ω => ⨆ x, f ω x) =
fun ω => ⨆ r : ℕ, f ω (TopologicalSpace.denseSeq X r) := by
funext ω
have hd := (TopologicalSpace.denseRange_denseSeq X).ciSup' (hc ω)
rw [← hd]
apply le_antisymm
· apply iSup_le
intro s
rcases s.property with ⟨r, hr⟩
rw [← hr]
exact le_iSup (fun r : ℕ => f ω (TopologicalSpace.denseSeq X r)) r
· apply iSup_le
intro r
exact le_iSup_of_le
⟨TopologicalSpace.denseSeq X r, ⟨r, rfl⟩⟩ le_rfl
rw [heq]
exact Measurable.iSup fun r => hm _
Polynomial estimator formula with its dyadic index and cube fixed. For the stated inputs and conditions, the equalCellEstimatorAtCell object being defined.
Definition (Lean source)
parameters
def
equalCellEstimatorAtCell β B ω j k x :
ℝ
given by
m :
h :
countSelectedMesh m β ω
if h
= 0 then 0 else max (-B) (min B (∑ α, monoVec d m (fun a => (x a - cubeCorner d j k a) / h) α * coefHat m j ω k α))
theorem equalCellEstimatorAtCell_measurable Lemma equal-cell-estimator-at-cell-measurable in the paper ↗
At a fixed point and cell, the polynomial estimator formula is sample measurable. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma equalCellEstimatorAtCell_measurable {d n : ℕ} (β B : ℝ)
(j : ℕ) (k : Fin d → Fin (2 ^ j)) (x : Fin d → ℝ) :
Measurable (fun ω : Fin n → Obs d =>
equalCellEstimatorAtCell β B ω j k x) := by
let m := polynomialDegree β
have hh : Measurable (fun ω : Fin n → Obs d => countSelectedMesh m β ω) :=
countSelectedMesh_measurable m β
unfold equalCellEstimatorAtCell
apply Measurable.ite (measurableSet_eq_fun hh measurable_const)
· exact measurable_const
· apply Measurable.max measurable_const
apply Measurable.min measurable_const
apply Finset.measurable_fun_sum
intro α _
apply mul
· unfold monoVec
fun_prop
· exact coefHat_coordinate_measurable m j k α
theorem equalCellEstimatorAtCell_continuous Lemma equal-cell-estimator-at-cell-continuous in the paper ↗
For a fixed sample and cell, the polynomial estimator formula is continuous in x. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Continuous (equalCellEstimatorAtCell β B ω j k)
Proof (Lean source)
lemma equalCellEstimatorAtCell_continuous {d n : ℕ} (β B : ℝ)
(ω : Fin n → Obs d) (j : ℕ) (k : Fin d → Fin (2 ^ j)) :
Continuous (equalCellEstimatorAtCell β B ω j k) := by
unfold equalCellEstimatorAtCell
dsimp only
by_cases hzero : countSelectedMesh (polynomialDegree β) β ω = 0
· simp only [hzero, if_true]
exact continuous_const
· simp only [hzero, if_false]
apply Continuous.max continuous_const
apply Continuous.min continuous_const
apply continuous_finset_sum
intro α _
unfold monoVec
fun_prop
theorem equalCellEstimator_cubeSup_measurable Lemma equal-cell-estimator-cube-sup-measurable in the paper ↗
Against a continuous target on the cube, the cube-supremum loss of the selected equal-cell estimator is sample measurable. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Measurable (fun ω : Fin n → Obs d => ⨆ y : Fin d → ℝ, ⨆ (_ : y ∈ cube d), ofReal |equalCellEstimator β B ω y - μ y|)
Proof (Lean source)
lemma equalCellEstimator_cubeSup_measurable {d n : ℕ} (β B : ℝ)
(μ : (Fin d → ℝ) → ℝ) (hμ : ContinuousOn μ (cube d)) :
Measurable (fun ω : Fin n → Obs d =>
⨆ y : Fin d → ℝ, ⨆ (_ : y ∈ cube d),
ofReal |equalCellEstimator β B ω y - μ y|) := by
let m := polynomialDegree β
let J : (Fin n → Obs d) → ℕ :=
fun ω => (feasibleIndices m β ω).sup id
let cell : (j : ℕ) → (Fin d → Fin (2 ^ j)) → Set (Fin d → ℝ) :=
fun j k => {x | x ∈ cube d ∧ cubeIndex d j x = k}
let G : ((Fin n → Obs d) × ℕ) → ℝ≥0∞ := fun p =>
⨆ k : Fin d → Fin (2 ^ p.2),
⨆ x : cell p.2 k,
ofReal
|equalCellEstimatorAtCell β B p.1 p.2 k x - μ x|
have hG : Measurable G := measurable_from_prod_countable_left fun j => by
change Measurable (fun ω : Fin n → Obs d =>
⨆ k : Fin d → Fin (2 ^ j),
⨆ x : cell j k,
ofReal
|equalCellEstimatorAtCell β B ω j k x - μ x|)
apply Measurable.iSup
intro k
apply measurable_iSup_of_continuous_sections
· intro x
exact ENNReal.measurable_ofReal.comp
((equalCellEstimatorAtCell_measurable β B j k x).sub measurable_const).abs
· intro ω
apply ENNReal.continuous_ofReal.comp
apply Continuous.abs
apply Continuous.sub
· exact (equalCellEstimatorAtCell_continuous β B ω j k).comp
continuous_subtype_val
· exact hμ.comp_continuous continuous_subtype_val fun x => x.property.1
have hJ : Measurable J := by
exact (measurable_of_countable (fun S : Finset ℕ => S.sup id)).comp
(feasibleIndices_measurable m β)
have hcomp := hG.comp (measurable_id.prodMk hJ)
convert hcomp using 1
funext ω
dsimp only [Function.comp_apply]
change (⨆ y : Fin d → ℝ, ⨆ (_ : y ∈ cube d),
ofReal |equalCellEstimator β B ω y - μ y|) =
G (ω, J ω)
unfold G
apply le_antisymm
· apply iSup_le
intro y
apply iSup_le
intro hy
let k := cubeIndex d (J ω) y
have heq : equalCellEstimator β B ω y =
equalCellEstimatorAtCell β B ω (J ω) k y := by
simp [equalCellEstimator, equalCellEstimatorAtCell, J, m, k]
rw [heq]
exact le_iSup_of_le k
(le_iSup_of_le ⟨y, hy, rfl⟩ le_rfl)
· apply iSup_le
intro k
apply iSup_le
intro x
have heq : equalCellEstimator β B ω x =
equalCellEstimatorAtCell β B ω (J ω) k x := by
simp [equalCellEstimator, equalCellEstimatorAtCell, J, m, x.property.2]
rw [← heq]
exact le_iSup_of_le x.1 (le_iSup_of_le x.property.1 le_rfl)
theorem equalCellEstimator_cubeSup_measurable_of_holderBall Lemma equal-cell-estimator-cube-sup-measurable-holder-ball in the paper ↗
A Holder representative supplies the continuity premise required by the sample-measurability theorem for the cube-supremum loss. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Measurable (fun ω : Fin n → Obs d => ⨆ y : Fin d → ℝ, ⨆ (_ : y ∈ cube d), ofReal |equalCellEstimator β B ω y - μ y|)
Proof (Lean source)
lemma equalCellEstimator_cubeSup_measurable_of_holderBall {d n : ℕ}
(β B L : ℝ) (μ : (Fin d → ℝ) → ℝ) (hμ : HolderBall β L μ) :
Measurable (fun ω : Fin n → Obs d =>
⨆ y : Fin d → ℝ, ⨆ (_ : y ∈ cube d),
ofReal |equalCellEstimator β B ω y - μ y|) :=
equalCellEstimator_cubeSup_measurable β B μ hμ.regularity.continuousOn
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Measurable (equalCellCubeLossReal (n := n) β B μ)
Proof (Lean source)
lemma equalCellCubeLossReal_measurable {d n : ℕ} (β B : ℝ)
(μ : (Fin d → ℝ) → ℝ) (hμ : ContinuousOn μ (cube d)) :
Measurable (equalCellCubeLossReal (n := n) β B μ) := by
unfold equalCellCubeLossReal
exact ENNReal.measurable_toReal.comp
(equalCellEstimator_cubeSup_measurable β B μ hμ)
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
equalCellCubeLossReal β B μ ω ≤ 2 * B
Proof (Lean source)
lemma equalCellCubeLossReal_le_two_mul {d n : ℕ} (β B : ℝ)
(μ : (Fin d → ℝ) → ℝ) (hB : 0 ≤ B)
(hμ : ∀ x ∈ cube d, |μ x| ≤ B) (ω : Fin n → Obs d) :
equalCellCubeLossReal β B μ ω ≤ 2 * B := by
have hsup := cube_iSup_ofReal_loss_le
(fun x => equalCellEstimator β B ω x - μ x) (2 * B) (by positivity)
(fun x hx => equalCellEstimator_error_le_two_mul β B ω x (μ x) hB (hμ x hx))
unfold equalCellCubeLossReal
calc
_ ≤ (ofReal (2 * B)).toReal := ENNReal.toReal_mono ENNReal.ofReal_ne_top hsup
_ = 2 * B := ENNReal.toReal_ofReal (by positivity)
theorem equalCellCubeLoss_lintegral_eq_ofReal_integral Lemma equal-cell-cube-loss-lintegral-conversion in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B :
ℝ
μ :
(Fin d → ℝ) → ℝ
hB :
0 ≤ B
hμcont :
ContinuousOn μ (cube d)
hμ :
∀ x ∈ cube d, |μ x| ≤ B
Q :
conclusion
(∫⁻ ω, ⨆ x, ⨆ (_hx : x ∈ cube d), ofReal |equalCellEstimator β B ω x - μ x| ∂Q)
= ofReal (∫ ω, equalCellCubeLossReal β B μ ω ∂Q)
Proof (Lean source)
lemma equalCellCubeLoss_lintegral_eq_ofReal_integral {d n : ℕ} (β B : ℝ)
(μ : (Fin d → ℝ) → ℝ) (hB : 0 ≤ B)
(hμcont : ContinuousOn μ (cube d))
(hμ : ∀ x ∈ cube d, |μ x| ≤ B)
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q] :
(∫⁻ ω, ⨆ x, ⨆ (_hx : x ∈ cube d),
ofReal |equalCellEstimator β B ω x - μ x| ∂Q) =
ofReal (∫ ω, equalCellCubeLossReal β B μ ω ∂Q) := by
have hmeas := equalCellCubeLossReal_measurable (n := n) β B μ hμcont
have hint : Integrable (equalCellCubeLossReal (n := n) β B μ) Q :=
Integrable.of_bound hmeas.aestronglyMeasurable (2 * B) <|
ae_of_all Q fun ω => by
rw [Real.norm_eq_abs, abs_of_nonneg]
· exact equalCellCubeLossReal_le_two_mul β B μ hB hμ ω
· exact ENNReal.toReal_nonneg
rw [ofReal_integral_eq_lintegral_ofReal hint
(ae_of_all Q fun ω => ENNReal.toReal_nonneg)]
refine lintegral_congr fun ω => ?_
unfold equalCellCubeLossReal
symm
apply ENNReal.ofReal_toReal
exact ne_top_of_le_ne_top ENNReal.ofReal_ne_top
(cube_iSup_ofReal_loss_le
(fun x => equalCellEstimator β B ω x - μ x) (2 * B) (by positivity)
(fun x hx => equalCellEstimator_error_le_two_mul β B ω x (μ x) hB (hμ x hx)))
theorem equalCellEstimator_cubeSup_aemeasurable_of_holderResponse Lemma equal-cell-estimator-cube-sup-aemeasurable-holder-response in the paper ↗
The selected-estimator cube loss against a model Holder response is almost-everywhere measurable under any sample law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
AEMeasurable (fun ω : Fin n → Obs d => ⨆ y : Fin d → ℝ, ⨆ (_ : y ∈ cube d), ofReal |equalCellEstimator β B ω y - μ y|) Q
Proof (Lean source)
lemma equalCellEstimator_cubeSup_aemeasurable_of_holderResponse {d n : ℕ}
(β B L : ℝ) (Pc : Measure (Completion d))
(μ : (Fin d → ℝ) → ℝ) (hμ : HolderResponse Pc μ β L)
(Q : Measure (Fin n → Obs d)) :
AEMeasurable (fun ω : Fin n → Obs d =>
⨆ y : Fin d → ℝ, ⨆ (_ : y ∈ cube d),
ofReal |equalCellEstimator β B ω y - μ y|) Q :=
(equalCellEstimator_cubeSup_measurable_of_holderBall β B L μ hμ.1).aemeasurable
theorem integral_le_of_condDistrib_event_bounds Lemma integral-le-of-condDistrib-event-bounds in the paper ↗
Integrate fibrewise good-event and global bounds without changing the event used for conditioning. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
Ω S :
Q :
X :
Ω → S
hX :
Z :
Ω → ℝ
hZm :
hZ :
Integrable Z Q
E :
Set Ω
hE :
R M δ :
ℝ
hR :
0 ≤ R
hM :
0 ≤ M
hδ :
0 ≤ δ
hgood :
∀ᵐ ω ∂Q, ω ∈ E → ∫ ξ, Z ξ ∂condDistrib id X Q (X ω) ≤ R
hall :
∀ᵐ ω ∂Q, ∫ ξ, Z ξ ∂condDistrib id X Q (X ω) ≤ M
hbad :
Q.real Eᶜ ≤ δ
conclusion
∫ ω, Z ω ∂Q ≤ R + M * δ
Proof (Lean source)
lemma integral_le_of_condDistrib_event_bounds
{Ω S : Type*} [MeasurableSpace Ω] [MeasurableSpace S]
[StandardBorelSpace Ω] [Nonempty Ω]
(Q : Measure Ω) [IsProbabilityMeasure Q]
(X : Ω → S) (hX : Measurable X) (Z : Ω → ℝ)
(hZm : StronglyMeasurable Z) (hZ : Integrable Z Q)
(E : Set Ω) (hE : MeasurableSet E) (R M δ : ℝ)
(hR : 0 ≤ R) (hM : 0 ≤ M) (hδ : 0 ≤ δ)
(hgood : ∀ᵐ ω ∂Q, ω ∈ E →
∫ ξ, Z ξ ∂condDistrib id X Q (X ω) ≤ R)
(hall : ∀ᵐ ω ∂Q,
∫ ξ, Z ξ ∂condDistrib id X Q (X ω) ≤ M)
(hbad : Q.real Eᶜ ≤ δ) :
∫ ω, Z ω ∂Q ≤ R + M * δ := by
classical
let F : Ω → ℝ := fun ω => ∫ ξ, Z ξ ∂condDistrib id X Q (X ω)
have htower : Q[Z | ‹MeasurableSpace S›.comap X] =ᵐ[Q] F := by
simpa [F] using ProbabilityTheory.condExp_ae_eq_integral_condDistrib
(Y := id) hX measurable_id.aemeasurable hZm hZ
have hFint : Integrable F Q := integrable_condExp.congr htower
let G : Ω → ℝ := E.indicator (fun _ => R) + Eᶜ.indicator (fun _ => M)
have hGint : Integrable G Q :=
((integrable_const R).indicator hE).add ((integrable_const M).indicator hE.compl)
have hFG : F ≤ᵐ[Q] G := by
filter_upwards [hgood, hall] with ω hgoodω hallω
by_cases hω : ω ∈ E
· simpa [G, hω] using hgoodω hω
· simpa [G, hω] using hallω
calc
∫ ω, Z ω ∂Q = ∫ ω, F ω ∂Q := by
rw [← integral_condExp (μ := Q) (f := Z) hX.comap_le]
exact integral_congr_ae htower
_ ≤ ∫ ω, G ω ∂Q := integral_mono_ae hFint hGint hFG
_ = R * Q.real E + M * Q.real Eᶜ := by
dsimp [G]
rw [integral_add ((integrable_const R).indicator hE)
((integrable_const M).indicator hE.compl),
integral_indicator_const R hE, integral_indicator_const M hE.compl]
simp only [smul_eq_mul]
ring
_ ≤ R + M * δ := by
have hprob : Q.real E ≤ 1 := measureReal_le_one
have hfirst : R * Q.real E ≤ R := by
simpa using mul_le_mul_of_nonneg_left hprob hR
have hsecond : M * Q.real Eᶜ ≤ M * δ :=
mul_le_mul_of_nonneg_left hbad hM
linarith
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.SelectedMesh.Rank 13 declarations Selected-mesh mass ranks and coefficient proxies
Selected-mesh mass ranks and coefficient proxies
def cubeMassRank
Rank by weakest-cell mass; ties receive their largest available rank. For the stated inputs and conditions, the cubeMassRank object being defined.
Mass ranks respect the order of cube masses, including tied masses. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
cubeMassRank P m j k ≤ cubeMassRank P m j k'
Proof (Lean source)
lemma cubeMassRank_mono {d : ℕ} (P : Measure (Obs d)) [IsProbabilityMeasure P]
(m j : ℕ) (k k' : Fin d → Fin (2 ^ j))
(h : cubeMass P m j k ≤ cubeMass P m j k') :
cubeMassRank P m j k ≤ cubeMassRank P m j k' := by
classical
unfold cubeMassRank
apply Finset.card_le_card
intro q hq
simp only [mem_filter, Finset.mem_univ, true_and] at hq ⊢
exact hq.trans h
Each cube contributes itself to its mass rank. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
0 < cubeMassRank P m j k
Proof (Lean source)
lemma cubeMassRank_pos {d : ℕ} (P : Measure (Obs d)) [IsProbabilityMeasure P]
(m j : ℕ) (k : Fin d → Fin (2 ^ j)) :
0 < cubeMassRank P m j k := by
classical
unfold cubeMassRank
exact Finset.card_pos.mpr ⟨k, Finset.mem_filter.mpr
⟨Finset.mem_univ _, le_refl _⟩⟩
An injective list of cubes in increasing mass order has at least its position in the mass rank. This is the index conversion for the ordered maximal inequality. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma cubeMassRank_lower_of_ordered {d N : ℕ}
(P : Measure (Obs d)) [IsProbabilityMeasure P] (m j : ℕ)
(σ : Fin N → Fin d → Fin (2 ^ j)) (hσ : Injective σ)
(horder : ∀ a b : Fin N, a ≤ b →
cubeMass P m j (σ a) ≤ cubeMass P m j (σ b))
(b : Fin N) : b.val + 1 ≤ cubeMassRank P m j (σ b) := by
classical
have hsub : (Iic b).image σ ⊆
Finset.univ.filter (fun q : Fin d → Fin (2 ^ j) =>
cubeMass P m j q ≤ cubeMass P m j (σ b)) := by
intro q hq
obtain ⟨a, ha, rfl⟩ := Finset.mem_image.mp hq
exact Finset.mem_filter.mpr
⟨Finset.mem_univ _, horder a b (Finset.mem_Iic.mp ha)⟩
calc
b.val + 1 = (Iic b).card := (Fin.card_Iic b).symm
_ = ((Iic b).image σ).card :=
(Finset.card_image_of_injective _ hσ).symm
_ ≤ cubeMassRank P m j (σ b) := by
unfold cubeMassRank
exact Finset.card_le_card hsub
Enumerate all cubes in nondecreasing weakest-cell mass, breaking ties by an arbitrary fixed finite index. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma cubeMass_ordered_equiv {d : ℕ}
(P : Measure (Obs d)) [IsProbabilityMeasure P] (m j : ℕ) :
∃ σ : Fin (card (Fin d → Fin (2 ^ j))) ≃
(Fin d → Fin (2 ^ j)),
Monotone (fun i => cubeMass P m j (σ i)) ∧
∀ b, b.val + 1 ≤ cubeMassRank P m j (σ b) := by
classical
let e : (Fin d → Fin (2 ^ j)) ≃
Fin (card (Fin d → Fin (2 ^ j))) :=
(Fintype.truncEquivFinOfCardEq (α := Fin d → Fin (2 ^ j)) rfl).out
let order : LinearOrder (Fin d → Fin (2 ^ j)) :=
LinearOrder.lift'
(fun k => toLex (cubeMass P m j k, e k))
(by
intro a b hab
apply e.injective
exact congrArg (fun x : Lex (ℝ × Fin (card (Fin d → Fin (2 ^ j)))) =>
(ofLex x).2) hab)
letI := order
letI : LE (Fin d → Fin (2 ^ j)) := order.toLE
let τ := Fintype.orderIsoFinOfCardEq (Fin d → Fin (2 ^ j)) rfl
have hmono : Monotone (fun i => cubeMass P m j (τ i)) := by
intro a b hab
have h : τ a ≤ τ b := by
exact (τ.le_iff_le).2 hab
change (toLex (cubeMass P m j (τ a), e (τ a))) ≤
toLex (cubeMass P m j (τ b), e (τ b)) at h
exact Prod.Lex.monotone_fst _ _ h
refine ⟨τ.toEquiv, hmono, ?_⟩
intro b
exact cubeMassRank_lower_of_ordered P m j τ.toEquiv
τ.toEquiv.injective (fun a b hab => hmono hab) b
The rank counting all cubes with mass no larger than a given cube gives that cube the same ordered lower bound, including when masses tie. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
m j :
ℕ
β B L C c_f γ :
ℝ
Pc :
μ₁ e :
(Fin d → ℝ) → ℝ
hmodel :
GlobalTailModel β B L C c_f γ Pc μ₁ e
hC :
1 ≤ C
hγ :
1 < γ
hcf :
0 < c_f
conclusion
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (c_f * templateEta d m ^ d) ^ (γ / (γ - 1)) * meshWidth j ^ effectiveDimension d γ * (@cubeMassRank d (Pc.map observed) (Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)) m j k : ℝ) ^ (1 / (γ - 1))
≤ @cubeMass d (Pc.map observed) (Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)) m j k
Proof (Lean source)
lemma model_cubeMassRank_lower {d : ℕ} (m j : ℕ)
(β B L C c_f γ : ℝ)
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(μ₁ e : (Fin d → ℝ) → ℝ)
(hmodel : GlobalTailModel β B L C c_f γ Pc μ₁ e)
(hC : 1 ≤ C) (hγ : 1 < γ) (hcf : 0 < c_f)
(k : Fin d → Fin (2 ^ j)) :
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1)) *
meshWidth j ^ effectiveDimension d γ *
(@cubeMassRank d (Pc.map observed)
(Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)) m j k : ℝ) ^
(1 / (γ - 1)) ≤
@cubeMass d (Pc.map observed)
(Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)) m j k := by
classical
letI : IsProbabilityMeasure (Pc.map observed) :=
Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)
let S : Finset (Fin d → Fin (2 ^ j)) := Finset.univ.filter
(fun q => cubeMass (Pc.map observed) m j q ≤ cubeMass (Pc.map observed) m j k)
have hcard : S.card = cubeMassRank (Pc.map observed) m j k := rfl
have hr : 0 < S.card := Finset.card_pos.mpr ⟨k, Finset.mem_filter.mpr
⟨Finset.mem_univ _, le_refl _⟩⟩
have hmin : ∀ q : Fin d → Fin (2 ^ j),
∃ ℓ : Fin d → Fin (m + 1),
cubeMass (Pc.map observed) m j q = microcellMass (Pc.map observed) m j q ℓ :=
fun q => orderedMass_cubeMass_attained (Pc.map observed) m j q
choose ℓ hℓ using hmin
have hlower := orderedMass_selectedMicrocells_treated_lower d m j
β B L C c_f γ Pc μ₁ e hmodel hC hγ hcf.le S ℓ
have hupper := orderedMass_selectedMicrocells_treated_le_card_mul
(Pc.map observed) m j S ℓ (cubeMass (Pc.map observed) m j k)
(fun q _ => (hℓ q).symm)
(fun q hq => (Finset.mem_filter.mp hq).2)
have hη : 0 ≤ templateEta d m := (templateEta_pos d m).le
rw [orderedMass_microcell_volume_toReal d m j hη] at hlower
have ha : 0 ≤ c_f * templateEta d m ^ d := by positivity
have hh : 0 ≤ meshWidth j := by unfold meshWidth; positivity
have hr' : 0 < (S.card : ℝ) := by exact_mod_cast hr
have hbase : c_f * (templateEta d m * meshWidth j) ^ d =
(c_f * templateEta d m ^ d) * meshWidth j ^ d := by
rw [mul_pow]
ring
rw [hbase] at hlower
have hidentity := orderedMass_rank_power_identity d S.card
(c_f * templateEta d m ^ d) (meshWidth j) γ hr ha hh hγ
have hmass :
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1)) *
meshWidth j ^ effectiveDimension d γ *
(S.card : ℝ) ^ (1 / (γ - 1)) ≤
cubeMass (Pc.map observed) m j k := by
have hfactor :
((S.card : ℝ) * ((c_f * templateEta d m ^ d) * meshWidth j ^ d)) ^
(γ / (γ - 1)) / (S.card : ℝ) =
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1)) *
meshWidth j ^ effectiveDimension d γ *
(S.card : ℝ) ^ (1 / (γ - 1)) := by
simpa only [effectiveDimension, div_eq_mul_inv, mul_assoc] using hidentity
have hdiv :
(((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
((S.card : ℝ) * ((c_f * templateEta d m ^ d) * meshWidth j ^ d)) ^
(γ / (γ - 1))) / (S.card : ℝ) ≤
cubeMass (Pc.map observed) m j k :=
(div_le_iff₀ hr').2 (by simpa [mul_comm] using (hlower.trans hupper))
rw [mul_div_assoc, hfactor] at hdiv
simpa only [mul_assoc] using hdiv
simpa only [← hcard] using hmass
The concrete mass-rank profile for any observed law in the model class. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (c_f * templateEta d m ^ d) ^ (γ / (γ - 1)) * meshWidth j ^ effectiveDimension d γ * (cubeMassRank P m j k : ℝ) ^ (1 / (γ - 1))
≤ cubeMass P m j k
Proof (Lean source)
lemma modelClass_cubeMassRank_lower {d : ℕ} (m j : ℕ)
{β B L C c_f γ : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(k : Fin d → Fin (2 ^ j)) :
((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1)) *
meshWidth j ^ effectiveDimension d γ *
(cubeMassRank P m j k : ℝ) ^ (1 / (γ - 1)) ≤ cubeMass P m j k := by
obtain ⟨hparams, hγ, Pc, hPc, μ₁, e, rfl, hmodel⟩ := hP
letI : IsProbabilityMeasure Pc := hPc
exact model_cubeMassRank_lower m j β B L C c_f γ Pc μ₁ e hmodel
hparams.2.2.2.2.1 hγ hparams.2.2.2.2.2.1 k
theorem modelClass_selectedCount_inverse_rank_bounds Lemma modelClass_selectedCount_inverse_rank_bounds in the paper ↗
Feasibility and the selected event give the ordered inverse-count factor with the model's explicit mass constant. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
β B L C c_f γ :
ℝ
P :
Measure (Obs d)
hP :
P ∈ ModelClass d β B L C c_f γ
ω :
Fin n → Obs d
hn :
0 < n
hfeasible :
meshFeasible m β ω j
hselected :
conclusion
≤ (tensorCount d m : ℝ) * min (meshWidth j ^ (2 * β)) (5 / ((n : ℝ) * (((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (c_f * templateEta d m ^ d) ^ (γ / (γ - 1))) * meshWidth j ^ effectiveDimension d γ * (cubeMassRank P m j k : ℝ) ^ (1 / (γ - 1))))
Proof (Lean source)
lemma modelClass_selectedCount_inverse_rank_bounds {d n : ℕ} (m j : ℕ)
{β B L C c_f γ : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(hn : 0 < n) (hfeasible : meshFeasible m β ω j)
(hselected : ∀ ℓ : Fin d → Fin (m + 1),
(n : ℝ) * microcellMass P m j k ℓ / 5 ≤ treatedCount m j ω k ℓ) :
(∑ ℓ : Fin d → Fin (m + 1),
(treatedCount m j ω k ℓ : ℝ)⁻¹) ≤
(tensorCount d m : ℝ) *
min (meshWidth j ^ (2 * β))
(5 / ((n : ℝ) *
(((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1))) *
meshWidth j ^ effectiveDimension d γ *
(cubeMassRank P m j k : ℝ) ^ (1 / (γ - 1)))) := by
classical
have hγ : 1 < γ := hP.2.1
have hC : 0 < C := lt_of_lt_of_le zero_lt_one hP.1.2.2.2.2.1
have hcf : 0 < c_f := hP.1.2.2.2.2.2.1
have hκ : 0 < ((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1)) := by
have hη := templateEta_pos d m
positivity
have hr : 0 < cubeMassRank P m j k := cubeMassRank_pos P m j k
exact selectedCount_cube_inverse_rank_bounds m j β
(effectiveDimension d γ) (1 / (γ - 1))
(((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1)))
(cubeMassRank P m j k) P ω k hn hκ hr hfeasible hselected
(modelClass_cubeMassRank_lower m j P hP k)
theorem modelClass_coefResidualWeight_square_rank_bounds Lemma modelClass_coefResidualWeight_square_rank_bounds in the paper ↗
On the selected count event, the coefficient weights have both the feasibility and ordered-mass variance scales. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
β B L C c_f γ :
ℝ
P :
Measure (Obs d)
hP :
P ∈ ModelClass d β B L C c_f γ
ω :
Fin n → Obs d
α :
MonoIndex d m
R :
ℝ
hn :
0 < n
hfeasible :
meshFeasible m β ω j
hselected :
hrow :
if
(ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ
then
|(tensorCount d m : ℝ)⁻¹ * ∑ α' : MonoIndex d m, (gramHat m j ω k)⁻¹ α α' * monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α'|
≤ R
conclusion
(∑ i : Fin n, (coefResidualWeight m j ω k α i) ^ 2)
≤ (tensorCount d m : ℝ) * R ^ 2 * ((tensorCount d m : ℝ) * min (meshWidth j ^ (2 * β)) (5 / ((n : ℝ) * (((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (c_f * templateEta d m ^ d) ^ (γ / (γ - 1))) * meshWidth j ^ effectiveDimension d γ * (cubeMassRank P m j k : ℝ) ^ (1 / (γ - 1)))))
Proof (Lean source)
lemma modelClass_coefResidualWeight_square_rank_bounds {d n : ℕ} (m j : ℕ)
{β B L C c_f γ : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(α : MonoIndex d m) (R : ℝ)
(hn : 0 < n) (hfeasible : meshFeasible m β ω j)
(hselected : ∀ ℓ : Fin d → Fin (m + 1),
(n : ℝ) * microcellMass P m j k ℓ / 5 ≤ treatedCount m j ω k ℓ)
(hrow : ∀ (ℓ : Fin d → Fin (m + 1)) (i : Fin n),
(ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ →
|(tensorCount d m : ℝ)⁻¹ *
∑ α' : MonoIndex d m,
(gramHat m j ω k)⁻¹ α α' *
monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α'| ≤ R) :
(∑ i : Fin n, (coefResidualWeight m j ω k α i) ^ 2) ≤
(tensorCount d m : ℝ) * R ^ 2 *
((tensorCount d m : ℝ) *
min (meshWidth j ^ (2 * β))
(5 / ((n : ℝ) *
(((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1))) *
meshWidth j ^ effectiveDimension d γ *
(cubeMassRank P m j k : ℝ) ^ (1 / (γ - 1))))) := by
calc
_ ≤ (tensorCount d m : ℝ) * R ^ 2 *
∑ ℓ : Fin d → Fin (m + 1),
(treatedCount m j ω k ℓ : ℝ)⁻¹ :=
coefResidualWeight_square_sum_le m j ω k α R hrow
_ ≤ _ := by
gcongr
exact modelClass_selectedCount_inverse_rank_bounds m j P hP ω k
hn hfeasible hselected
theorem modelClass_coefResidualWeight_square_rank_bounds_of_feasible Lemma modelClass_coefResidualWeight_square_rank_bounds_of_feasible in the paper ↗
The equal-cell conditioning theorem supplies the uniform row constant in the ordered squared-weight bound; no separate row assumption is needed. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
β B L C c_f γ :
ℝ
P :
Measure (Obs d)
hP :
P ∈ ModelClass d β B L C c_f γ
ω :
Fin n → Obs d
α :
MonoIndex d m
hn :
0 < n
hfeasible :
meshFeasible m β ω j
hselected :
conclusion
(∑ i : Fin n, (coefResidualWeight m j ω k α i) ^ 2)
≤ (tensorCount d m : ℝ) * ((tensorCount d m : ℝ)⁻¹ * ((2 / templateLambda d m) * sqrt (card (MonoIndex d m) : ℝ))) ^ 2 * ((tensorCount d m : ℝ) * min (meshWidth j ^ (2 * β)) (5 / ((n : ℝ) * (((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (c_f * templateEta d m ^ d) ^ (γ / (γ - 1))) * meshWidth j ^ effectiveDimension d γ * (cubeMassRank P m j k : ℝ) ^ (1 / (γ - 1)))))
Proof (Lean source)
lemma modelClass_coefResidualWeight_square_rank_bounds_of_feasible
{d n : ℕ} (m j : ℕ) {β B L C c_f γ : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(α : MonoIndex d m) (hn : 0 < n)
(hfeasible : meshFeasible m β ω j)
(hselected : ∀ ℓ : Fin d → Fin (m + 1),
(n : ℝ) * microcellMass P m j k ℓ / 5 ≤ treatedCount m j ω k ℓ) :
(∑ i : Fin n, (coefResidualWeight m j ω k α i) ^ 2) ≤
(tensorCount d m : ℝ) *
((tensorCount d m : ℝ)⁻¹ *
((2 / templateLambda d m) *
sqrt (card (MonoIndex d m) : ℝ))) ^ 2 *
((tensorCount d m : ℝ) *
min (meshWidth j ^ (2 * β))
(5 / ((n : ℝ) *
(((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1))) *
meshWidth j ^ effectiveDimension d γ *
(cubeMassRank P m j k : ℝ) ^ (1 / (γ - 1))))) := by
have hcount : ∀ ℓ : Fin d → Fin (m + 1),
0 < treatedCount m j ω k ℓ := by
intro ℓ
have hthreshold : 0 < meshWidth j ^ (-2 * β) := by
unfold meshWidth
positivity
have hpositive : (0 : ℝ) < treatedCount m j ω k ℓ :=
lt_of_lt_of_le hthreshold (hfeasible.2 k ℓ)
exact_mod_cast hpositive
exact modelClass_coefResidualWeight_square_rank_bounds m j P hP ω k α _
hn hfeasible hselected
(inverseGram_row_bound_on_scaledMicroCell m j ω k α hcount)
theorem modelClass_selectedCoefficientResidual_mgf_rank_bound Lemma modelClass_selectedCoefficientResidual_mgf_rank_bound in the paper ↗
Once the outcome residuals are independent under the fixed-design law, the selected count event gives both ordered coefficient MGF proxies. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
β B L C c_f γ :
ℝ
P :
Measure (Obs d)
hP :
P ∈ ModelClass d β B L C c_f γ
ω :
Fin n → Obs d
α :
MonoIndex d m
hn :
0 < n
hfeasible :
meshFeasible m β ω j
hselected :
Ω :
Type*
μ :
Y :
Fin n → Ω → ℝ
hindep :
iIndepFun Y μ
hmeas :
∀ i, Measurable (Y i)
t :
ℝ
conclusion
≤ exp (B ^ 2 * ((tensorCount d m : ℝ) * ((tensorCount d m : ℝ)⁻¹ * ((2 / templateLambda d m) * sqrt (card (MonoIndex d m) : ℝ))) ^ 2 * ((tensorCount d m : ℝ) * min (meshWidth j ^ (2 * β)) (5 / ((n : ℝ) * (((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (c_f * templateEta d m ^ d) ^ (γ / (γ - 1))) * meshWidth j ^ effectiveDimension d γ * (cubeMassRank P m j k : ℝ) ^ (1 / (γ - 1)))))) * t ^ 2 / 2)
Proof (Lean source)
lemma modelClass_selectedCoefficientResidual_mgf_rank_bound
{d n : ℕ} (m j : ℕ) {β B L C c_f γ : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ)
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(α : MonoIndex d m) (hn : 0 < n)
(hfeasible : meshFeasible m β ω j)
(hselected : ∀ ℓ : Fin d → Fin (m + 1),
(n : ℝ) * microcellMass P m j k ℓ / 5 ≤ treatedCount m j ω k ℓ)
{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(Y : Fin n → Ω → ℝ)
(hindep : iIndepFun Y μ) (hmeas : ∀ i, Measurable (Y i))
(hmgf : ∀ i t, mgf (Y i) μ t ≤ exp (B ^ 2 * t ^ 2 / 2))
(t : ℝ) :
mgf (∑ i : Fin n, fun ξ => coefResidualWeight m j ω k α i * Y i ξ) μ t ≤
exp (B ^ 2 *
((tensorCount d m : ℝ) *
((tensorCount d m : ℝ)⁻¹ *
((2 / templateLambda d m) *
sqrt (card (MonoIndex d m) : ℝ))) ^ 2 *
((tensorCount d m : ℝ) *
min (meshWidth j ^ (2 * β))
(5 / ((n : ℝ) *
(((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1))) *
meshWidth j ^ effectiveDimension d γ *
(cubeMassRank P m j k : ℝ) ^ (1 / (γ - 1)))))) * t ^ 2 / 2) := by
have hweight := modelClass_coefResidualWeight_square_rank_bounds_of_feasible
m j P hP ω k α hn hfeasible hselected
calc
mgf (∑ i : Fin n, fun ξ => coefResidualWeight m j ω k α i * Y i ξ) μ t ≤
exp (B ^ 2 *
(∑ i : Fin n, (coefResidualWeight m j ω k α i) ^ 2) * t ^ 2 / 2) :=
independentWeightedResidual_mgf_le μ Y
(coefResidualWeight m j ω k α) B hindep hmeas hmgf t
_ ≤ _ := by
apply Real.exp_le_exp.mpr
gcongr
theorem modelClass_conditionalCoefficient_mgf_rank_bound Lemma modelClass_conditionalCoefficient_mgf_rank_bound in the paper ↗
On almost every complete-design fibre, a coefficient centred by its conditional integral obeys the selected ordered mass-rank MGF bound. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
∀ᵐ ω ∂Q, ∀ (m j : ℕ) {β L C c_f γ : ℝ}, P ∈ ModelClass d β B L C c_f γ → ∀ (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m), 0 < n → meshFeasible m β ω j → (∀ ℓ : Fin d → Fin (m + 1), (n : ℝ) * microcellMass P m j k ℓ / 5 ≤ treatedCount m j ω k ℓ) → ∀ t : ℝ, mgf (fun ξ => coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α) (condDistrib id sampleDesign Q (sampleDesign ω)) t
≤ exp (B ^ 2 * ((tensorCount d m : ℝ) * ((tensorCount d m : ℝ)⁻¹ * ((2 / templateLambda d m) * sqrt (card (MonoIndex d m) : ℝ))) ^ 2 * ((tensorCount d m : ℝ) * min (meshWidth j ^ (2 * β)) (5 / ((n : ℝ) * (((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) * (c_f * templateEta d m ^ d) ^ (γ / (γ - 1))) * meshWidth j ^ effectiveDimension d γ * (cubeMassRank P m j k : ℝ) ^ (1 / (γ - 1)))))) * t ^ 2 / 2)
Proof (Lean source)
lemma modelClass_conditionalCoefficient_mgf_rank_bound
{d n : ℕ} {B : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q]
(hIID : IIDSampleLaw Q P)
(hres : BoundedMeanSubGaussianResidual P B) :
∀ᵐ ω ∂Q, ∀ (m j : ℕ) {β L C c_f γ : ℝ},
P ∈ ModelClass d β B L C c_f γ →
∀ (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m),
0 < n → meshFeasible m β ω j →
(∀ ℓ : Fin d → Fin (m + 1),
(n : ℝ) * microcellMass P m j k ℓ / 5 ≤ treatedCount m j ω k ℓ) →
∀ t : ℝ,
mgf (fun ξ => coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α)
(condDistrib id sampleDesign Q (sampleDesign ω)) t ≤
exp (B ^ 2 *
((tensorCount d m : ℝ) *
((tensorCount d m : ℝ)⁻¹ *
((2 / templateLambda d m) *
sqrt (card (MonoIndex d m) : ℝ))) ^ 2 *
((tensorCount d m : ℝ) *
min (meshWidth j ^ (2 * β))
(5 / ((n : ℝ) *
(((γ - 1) / γ) * C ^ (-(1 : ℝ) / (γ - 1)) *
(c_f * templateEta d m ^ d) ^ (γ / (γ - 1))) *
meshWidth j ^ effectiveDimension d γ *
(cubeMassRank P m j k : ℝ) ^ (1 / (γ - 1)))))) * t ^ 2 / 2) := by
filter_upwards [conditionalCoefCentre_eq_conditionalMeanSample P Q hIID hres,
sampleDesign_condDistrib_fibre Q,
sampleDesign_conditional_residual_family P Q hIID hres]
with ω hcentre hfibre hfamily
intro m j β L C c_f γ hP k α hn hfeasible hselected t
let μ := condDistrib id sampleDesign Q (sampleDesign ω)
have heq : (fun ξ => coefHat m j ξ k α -
conditionalCoefCentre Q m j ω k α) =ᵐ[μ]
(∑ i : Fin n, fun ξ => coefResidualWeight m j ω k α i *
conditionalTreatedResidual P ω i ξ) := by
filter_upwards [hfibre] with ξ hdesign
rw [hcentre m j k α]
simpa only [Finset.sum_apply] using
coefHat_sub_conditionalMeanSample_eq_residual_sum
m j P ω ξ k α hdesign
rw [mgf_congr heq]
exact modelClass_selectedCoefficientResidual_mgf_rank_bound
m j P hP ω k α hn hfeasible hselected μ
(conditionalTreatedResidual P ω) hfamily.1
(by
intro i
by_cases hi : (ω i).2.1 = true
· have hfun : conditionalTreatedResidual P ω i =
fun ξ => (ξ i).2.2 - treatedRegression P (ω i).1 := by
funext ξ
simp [conditionalTreatedResidual, hi]
rw [hfun]
fun_prop
· have hfun : conditionalTreatedResidual P ω i = fun _ => 0 := by
funext ξ
simp [conditionalTreatedResidual, hi]
rw [hfun]
fun_prop)
(fun i t => (hfamily.2 i t).2) t
theorem conditionalCoefficient_integrable_exp Lemma conditionalCoefficient_integrable_exp in the paper ↗
The centered coefficient has every exponential moment on almost every complete-design fibre. This supplies the integrability premise of the ordered maximal inequality separately from its MGF upper bound. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
∀ᵐ ω ∂Q, ∀ (m j : ℕ) (k : Fin d → Fin (2 ^ j)) (α : MonoIndex d m) (t : ℝ), Integrable (fun ξ => exp (t * (coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α))) (condDistrib id sampleDesign Q (sampleDesign ω))
Proof (Lean source)
lemma conditionalCoefficient_integrable_exp
{d n : ℕ} {B : ℝ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(Q : Measure (Fin n → Obs d)) [IsProbabilityMeasure Q]
(hIID : IIDSampleLaw Q P)
(hres : BoundedMeanSubGaussianResidual P B) :
∀ᵐ ω ∂Q, ∀ (m j : ℕ) (k : Fin d → Fin (2 ^ j))
(α : MonoIndex d m) (t : ℝ),
Integrable (fun ξ => exp (t *
(coefHat m j ξ k α - conditionalCoefCentre Q m j ω k α)))
(condDistrib id sampleDesign Q (sampleDesign ω)) := by
filter_upwards [conditionalCoefCentre_eq_conditionalMeanSample P Q hIID hres,
sampleDesign_condDistrib_fibre Q,
sampleDesign_conditional_residual_family P Q hIID hres]
with ω hcentre hfibre hfamily
intro m j k α t
let μ := condDistrib id sampleDesign Q (sampleDesign ω)
have heq : (fun ξ => coefHat m j ξ k α -
conditionalCoefCentre Q m j ω k α) =ᵐ[μ]
(∑ i : Fin n, fun ξ => coefResidualWeight m j ω k α i *
conditionalTreatedResidual P ω i ξ) := by
filter_upwards [hfibre] with ξ hdesign
rw [hcentre m j k α]
simpa only [Finset.sum_apply] using
coefHat_sub_conditionalMeanSample_eq_residual_sum
m j P ω ξ k α hdesign
have hsum := independentWeightedResidual_integrable_exp_sum μ
(conditionalTreatedResidual P ω)
(coefResidualWeight m j ω k α) hfamily.1
(by
intro i
by_cases hi : (ω i).2.1 = true
· have hfun : conditionalTreatedResidual P ω i =
fun ξ => (ξ i).2.2 - treatedRegression P (ω i).1 := by
funext ξ
simp [conditionalTreatedResidual, hi]
rw [hfun]
fun_prop
· have hfun : conditionalTreatedResidual P ω i = fun _ => 0 := by
funext ξ
simp [conditionalTreatedResidual, hi]
rw [hfun]
fun_prop)
(fun i u => (hfamily.2 i u).1) t
exact hsum.congr (heq.fun_comp
(fun x : ℝ => exp (t * x)) |>.symm)
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.Template 35 declarations This file defines the empirical equal-cell Gram matrix and estimator, then derives deterministic conditioning from the tensor-template geometry.
Count-adaptive equal-cell estimator
This file defines the empirical equal-cell Gram matrix and estimator, then derives deterministic conditioning from the tensor-template geometry.
def maxMeshIndex
Largest dyadic index permitted by the sample size. For the stated inputs and conditions, the maxMeshIndex object being defined.
def meshIndices
Finite set of candidate dyadic mesh indices. For the stated inputs and conditions, the meshIndices object being defined.
Definition (Lean source)
def meshWidth
Width at dyadic index j. For the stated inputs and conditions, the meshWidth object being defined.
Definition (Lean source)
parameters
j :
ℕ
def
meshWidth j :
ℝ
given by
(2 : ℝ) ^ (-(j : ℝ))
def cubeIndex
Dyadic cube index of a point, capped at the last cube. For the stated inputs and conditions, the cubeIndex object being defined.
def cubeCorner
Lower-left corner of a dyadic cube. For the stated inputs and conditions, the cubeCorner object being defined.
def dyadicCube
Half-open dyadic cube with the global right boundary included. For the stated inputs and conditions, the dyadicCube object being defined.
Definition (Lean source)
parameters
given by
{x | x ∈ cube d ∧ ∀ i, cubeCorner d j k i ≤ x i ∧ (x i < cubeCorner d j k i + meshWidth j ∨ ((k i).val = 2 ^ j - 1 ∧ x i = 1))}
def scaledMicroCell
Scaled microcell in a macro-cube. For the stated inputs and conditions, the scaledMicroCell object being defined.
def treatedCount
Treated observations in one microcell. For the stated inputs and conditions, the treatedCount object being defined.
Total inverse-count weight assigned to one template cell. For the stated inputs and conditions, the treatedCellWeightSum object being defined.
Definition (Lean source)
When a template cell is occupied, its inverse-count observation weights sum to one. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma treatedCount_inv_weight_sum {d n : ℕ} (m j : ℕ)
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(ℓ : Fin d → Fin (m + 1))
(hcount : 0 < treatedCount m j ω k ℓ) :
treatedCellWeightSum m j ω k ℓ = 1 := by
classical
unfold treatedCellWeightSum
have hsum :
(∑ i : Fin n,
if (ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ
then (1 : ℝ) else 0) = treatedCount m j ω k ℓ := by
simp [treatedCount]
rw [hsum]
exact inv_mul_cancel₀ (by exact_mod_cast (Nat.ne_of_gt hcount))
Mesh passes every cell-count threshold. For the stated inputs and conditions, the meshFeasible object being defined.
def feasibleIndices
Feasible dyadic indices. For the stated inputs and conditions, the feasibleIndices object being defined.
Definition (Lean source)
parameters
def
given by
def countSelectedMesh
Numerically smallest feasible width, or zero when no mesh is feasible. For the stated inputs and conditions, the countSelectedMesh object being defined.
Definition (Lean source)
parameters
def
countSelectedMesh m β ω :
ℝ
given by
def gramHat
Equal-cell empirical Gram matrix. For the stated inputs and conditions, the gramHat object being defined.
Definition (Lean source)
parameters
given by
by classical exact fun α β
=> (tensorCount d m : ℝ)⁻¹ * ∑ ℓ : Fin d → Fin (m + 1), (treatedCount m j ω k ℓ : ℝ)⁻¹ * ∑ i : Fin n, if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α * monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) β else 0
Equal-cell weighting preserves symmetry of the empirical Gram matrix. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
(gramHat m j ω k).IsHermitian
Proof (Lean source)
lemma gramHat_symmetric {d n : ℕ} (m j : ℕ) (ω : Fin n → Obs d)
(k : Fin d → Fin (2 ^ j)) :
(gramHat m j ω k).IsHermitian := by
classical
rw [IsHermitian]
ext α β
simp only [Matrix.conjTranspose_apply, star_trivial]
simp only [gramHat]
congr 1
apply Finset.sum_congr rfl
intro ℓ _
congr 1
apply Finset.sum_congr rfl
intro i _
split_ifs <;> ring
Equal-cell average of squared polynomial evaluations. For the stated inputs and conditions, the gramHatSquareAvg object being defined.
Definition (Lean source)
parameters
def
gramHatSquareAvg m j ω k b :
ℝ
given by
by classical exact (tensorCount d m : ℝ)⁻¹ * ∑ ℓ : Fin d → Fin (m + 1), (treatedCount m j ω k ℓ : ℝ)⁻¹ * ∑ i : Fin n, if (ω i).2.1
= true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then (∑ α, b α * monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α) ^ 2 else 0
The empirical equal-cell Gram quadratic form is an average of squares. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma gramHat_quadratic_sum_sq {d n : ℕ} (m j : ℕ) (ω : Fin n → Obs d)
(k : Fin d → Fin (2 ^ j)) (b : MonoIndex d m → ℝ) :
(∑ α, ∑ β, b α * gramHat m j ω k α β * b β) =
gramHatSquareAvg m j ω k b := by
classical
simp only [gramHat, gramHatSquareAvg, Finset.sum_mul, Finset.mul_sum]
conv_lhs => arg 2; ext α; rw [Finset.sum_comm]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro ℓ _
conv_lhs => arg 2; ext α; rw [Finset.sum_comm]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro i _
split_ifs
· simp only [sq, Finset.sum_mul, Finset.mul_sum]
simp only [mul_assoc, mul_comm, mul_left_comm]
· simp
Every equal-cell empirical Gram matrix has a nonnegative quadratic form. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma gramHat_quadratic_nonneg {d n : ℕ} (m j : ℕ) (ω : Fin n → Obs d)
(k : Fin d → Fin (2 ^ j)) (b : MonoIndex d m → ℝ) :
0 ≤ ∑ α, ∑ β, b α * gramHat m j ω k α β * b β := by
rw [gramHat_quadratic_sum_sq]
unfold gramHatSquareAvg
classical
apply mul_nonneg
· positivity
· apply Finset.sum_nonneg
intro ℓ _
apply mul_nonneg
· positivity
· apply Finset.sum_nonneg
intro i _
split_ifs <;> positivity
Equal-cell weighting preserves any lower bound that holds at every treated observation in each occupied template cell. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
ω :
Fin n → Obs d
b :
MonoIndex d m → ℝ
hcount :
∀ ℓ, 0 < treatedCount m j ω k ℓ
hbound :
∀ ℓ i
if
(ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ
then
r ℓ
conclusion
(tensorCount d m : ℝ)⁻¹ * ∑ ℓ, r ℓ ≤ gramHatSquareAvg m j ω k b
Proof (Lean source)
lemma gramHatSquareAvg_ge_cell_bounds {d n : ℕ} (m j : ℕ)
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(b : MonoIndex d m → ℝ) (r : (Fin d → Fin (m + 1)) → ℝ)
(hcount : ∀ ℓ, 0 < treatedCount m j ω k ℓ)
(hbound : ∀ ℓ i, (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ →
r ℓ ≤ (∑ α, b α * monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α) ^ 2) :
(tensorCount d m : ℝ)⁻¹ * ∑ ℓ, r ℓ ≤
gramHatSquareAvg m j ω k b := by
classical
unfold gramHatSquareAvg
apply mul_le_mul_of_nonneg_left _ (by positivity)
apply Finset.sum_le_sum
intro ℓ _
have hcountReal : (treatedCount m j ω k ℓ : ℝ) ≠ 0 := by
exact_mod_cast (Nat.ne_of_gt (hcount ℓ))
have hsum :
(∑ i : Fin n,
if (ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ
then r ℓ else 0) = (treatedCount m j ω k ℓ : ℝ) * r ℓ := by
simp [treatedCount, Finset.sum_ite, mul_comm]
calc
r ℓ = (treatedCount m j ω k ℓ : ℝ)⁻¹ *
∑ i : Fin n,
if (ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ
then r ℓ else 0 := by rw [hsum]; field_simp
_ ≤ (treatedCount m j ω k ℓ : ℝ)⁻¹ *
∑ i : Fin n,
if (ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ
then (∑ α, b α * monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α) ^ 2
else 0 := by
apply mul_le_mul_of_nonneg_left _ (by positivity)
apply Finset.sum_le_sum
intro i _
split_ifs with hi
· exact hbound ℓ i hi
· exact le_refl _
theorem templateGram_quadratic_sub_error_le_gramHat Lemma templateGram_quadratic_sub_error_le_gramHat in the paper ↗
A uniform pointwise squared-evaluation error transfers through equal-cell weighting to a Gram quadratic-form error. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
m j :
ℕ
ω :
Fin n → Obs d
b :
MonoIndex d m → ℝ
ε :
ℝ
hcount :
∀ ℓ, 0 < treatedCount m j ω k ℓ
hpoint :
∀ ℓ i
if
(ω i).2.1 = true ∧ (ω i).1 ∈ scaledMicroCell d m j k ℓ
then
(∑ α, b α * monoVec d m (tensorNode d m ℓ) α) ^ 2 - ε
conclusion
(∑ α, ∑ β, b α * templateGram d m α β * b β) - ε
≤ ∑ α, ∑ β, b α * gramHat m j ω k α β * b β
Proof (Lean source)
lemma templateGram_quadratic_sub_error_le_gramHat {d n : ℕ} (m j : ℕ)
(ω : Fin n → Obs d) (k : Fin d → Fin (2 ^ j))
(b : MonoIndex d m → ℝ) (ε : ℝ)
(hcount : ∀ ℓ, 0 < treatedCount m j ω k ℓ)
(hpoint : ∀ ℓ i, (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ →
(∑ α, b α * monoVec d m (tensorNode d m ℓ) α) ^ 2 - ε ≤
(∑ α, b α * monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α) ^ 2) :
(∑ α, ∑ β, b α * templateGram d m α β * b β) - ε ≤
∑ α, ∑ β, b α * gramHat m j ω k α β * b β := by
classical
have hJ : (tensorCount d m : ℝ) ≠ 0 := by
unfold tensorCount
positivity
have hraw := gramHatSquareAvg_ge_cell_bounds m j ω k b
(fun ℓ => (∑ α, b α * monoVec d m (tensorNode d m ℓ) α) ^ 2 - ε)
hcount hpoint
rw [← gramHat_quadratic_sum_sq] at hraw
rw [templateGram_quadratic_sum_sq]
simp only [Finset.sum_sub_distrib, Finset.sum_const, Finset.card_univ,
Fintype.card_fun, Fintype.card_fin, nsmul_eq_mul, mul_sub] at hraw
have hcard : (((m + 1) ^ d : ℕ) : ℝ) = (tensorCount d m : ℝ) := rfl
rw [hcard, ← mul_assoc, inv_mul_cancel₀ hJ, one_mul] at hraw
exact hraw
theorem templatePointwise_square_perturbation
Within a template microcube, the squared polynomial evaluation differs from its node value by at most the template Lipschitz radius times the squared coefficient norm. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
(∑ α, b α * monoVec d m (tensorNode d m ℓ) α) ^ 2
- templateLip d m * (templateEta d m / 2) * finiteL2Norm b ^ 2
≤ (∑ α, b α * monoVec d m z α) ^ 2
Proof (Lean source)
lemma templatePointwise_square_perturbation (d m : ℕ)
(b : MonoIndex d m → ℝ) (ℓ : Fin d → Fin (m + 1)) (z : Fin d → ℝ)
(hz : z ∈ microCube d m ℓ) :
(∑ α, b α * monoVec d m (tensorNode d m ℓ) α) ^ 2 -
templateLip d m * (templateEta d m / 2) * finiteL2Norm b ^ 2 ≤
(∑ α, b α * monoVec d m z α) ^ 2 := by
classical
let z₀ := tensorNode d m ℓ
let M : Matrix (MonoIndex d m) (MonoIndex d m) ℝ :=
(fun α β => monoVec d m z₀ α * monoVec d m z₀ β) -
(fun α β => monoVec d m z α * monoVec d m z β)
by_cases hzz : z₀ = z
· subst z
have hnonneg : 0 ≤ templateLip d m * (templateEta d m / 2) * finiteL2Norm b ^ 2 := by
exact mul_nonneg
(mul_nonneg (templateLip_nonneg d m)
(div_nonneg (le_of_lt (templateEta_pos d m)) (by norm_num)))
(sq_nonneg _)
dsimp [z₀]
linarith
have hz₀cube : z₀ ∈ cube d := tensorNode_mem_cube d m ℓ
have hzcube : z ∈ cube d := microCube_subset_cube d m ℓ hz
have hquot := templateLip_bounds_quotient d m z₀ z hz₀cube hzcube hzz
have hdist : 0 < ‖z₀ - z‖ := norm_pos_iff.mpr (sub_ne_zero.mpr hzz)
have hop : finiteL2OpNorm M ≤ templateLip d m * ‖z₀ - z‖ := by
apply (div_le_iff₀ hdist).mp
simpa [M] using hquot
have hnorm : ‖z₀ - z‖ ≤ templateEta d m / 2 := by
rw [norm_sub_rev]
exact microCube_norm_sub_tensorNode_le d m ℓ z hz
have hLip : 0 ≤ templateLip d m := templateLip_nonneg d m
have hop' : finiteL2OpNorm M ≤ templateLip d m * (templateEta d m / 2) :=
hop.trans (mul_le_mul_of_nonneg_left hnorm hLip)
let i₀ : MonoIndex d m := ⟨fun _ => 0, by simp [monoIdx]⟩
letI : Nonempty (MonoIndex d m) := ⟨i₀⟩
have hquadM := quadraticForm_le_finiteL2OpNorm M b
have hscaled : finiteL2OpNorm M * finiteL2Norm b ^ 2 ≤
templateLip d m * (templateEta d m / 2) * finiteL2Norm b ^ 2 := by
exact mul_le_mul_of_nonneg_right hop' (sq_nonneg _)
have hid : (∑ α, ∑ β, b α * M α β * b β) =
(∑ α, b α * monoVec d m z₀ α) ^ 2 -
(∑ α, b α * monoVec d m z α) ^ 2 := by
have hout (w : Fin d → ℝ) :
(∑ α, ∑ β, b α *
(monoVec d m w α * monoVec d m w β) * b β) =
(∑ α, b α * monoVec d m w α) ^ 2 := by
simp only [sq, Finset.sum_mul, Finset.mul_sum]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro α hα
apply Finset.sum_congr rfl
intro β hβ
ring
dsimp [M]
simp only [mul_sub, sub_mul, Finset.sum_sub_distrib]
exact congrArg₂ (· - ·) (hout z₀) (hout z)
rw [hid] at hquadM
dsimp [z₀] at hquadM
have htotal := hquadM.trans hscaled
change (∑ α ∈ (monoIdx d m).attach,
b α * monoVec d m (tensorNode d m ℓ) α) ^ 2 -
templateLip d m * (templateEta d m / 2) * finiteL2Norm b ^ 2 ≤
(∑ α ∈ (monoIdx d m).attach, b α * monoVec d m z α) ^ 2
apply sub_le_iff_le_add.mpr
simpa only [add_comm] using sub_le_iff_le_add.mp htotal
def responseMoment
Equal-cell empirical response moment. For the stated inputs and conditions, the responseMoment object being defined.
Definition (Lean source)
parameters
def
given by
by classical exact fun α
=> (tensorCount d m : ℝ)⁻¹ * ∑ ℓ : Fin d → Fin (m + 1), (treatedCount m j ω k ℓ : ℝ)⁻¹ * ∑ i : Fin n, if (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ then monoVec d m (fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α * (ω i).2.2 else 0
def coefHat
Local polynomial coefficient vector. For the stated inputs and conditions, the coefHat object being defined.
def equalCellEstimator
Count-adaptive, clipped equal-cell local polynomial estimator. For the stated inputs and conditions, the equalCellEstimator object being defined.
Definition (Lean source)
parameters
def
equalCellEstimator β B ω x :
ℝ
given by
m :
h :
countSelectedMesh m β ω
if h
= 0 then 0 else let j := (feasibleIndices m β ω).sup id let k := cubeIndex d j x max (-B) (min B (∑ α : MonoIndex d m, monoVec d m (fun a => (x a - cubeCorner d j k a) / h) α * coefHat m j ω k α))
def microcellMass
Population treated microcell mass. For the stated inputs and conditions, the microcellMass object being defined.
Definition (Lean source)
parameters
def
microcellMass P m j k ℓ :
ℝ
given by
P.real {z | z.2.1 = true ∧ z.1 ∈ scaledMicroCell d m j k ℓ}
def cubeMass
Minimum microcell mass in a macro-cube. For the stated inputs and conditions, the cubeMass object being defined.
Definition (Lean source)
parameters
def
cubeMass P m j k :
ℝ
given by
def orderedCubeMass
Sorted weakest-cell masses. For the stated inputs and conditions, the orderedCubeMass object being defined.
def pointwiseLossRisk
Expected pointwise loss for a measurable estimator and one model law. For the stated inputs and conditions, the pointwiseLossRisk object being defined.
Definition (Lean source)
parameters
d n :
ℕ
β B L C c_f γ :
ℝ
x₀ :
Fin d → ℝ
T :
{T : (Fin n → Obs d) → ℝ // Measurable T}
P :
{P : Measure (Obs d) // P ∈ ModelClass d β B L C c_f γ}
def
pointwiseLossRisk d n β B L C c_f γ x₀ T P :
ℝ≥0∞
given by
Pointwise minimax risk with extended nonnegative loss. For the stated inputs and conditions, the pointwiseRisk object being defined.
Definition (Lean source)
parameters
def
pointwiseRisk d n β B L C c_f γ x₀ :
ℝ≥0∞
given by
minimaxValueENNReal (pointwiseLossRisk d n β B L C c_f γ x₀)
def supLossRisk
Expected spatial-supremum loss for one model law. For the stated inputs and conditions, the supLossRisk object being defined.
Definition (Lean source)
parameters
def
supLossRisk d n β B L C c_f γ T P :
ℝ≥0∞
given by
∫⁻ ω, ⨆ x ∈ cube d, ofReal |T.1 ω x
- responseOf P.1 P.2 x| ∂Measure.pi (fun _ : Fin n => P.1)
Expected supremum minimax risk with extended nonnegative loss. For the stated inputs and conditions, the supremumRisk object being defined.
Definition (Lean source)
parameters
d n :
ℕ
β B L C c_f γ :
ℝ
def
supremumRisk d n β B L C c_f γ :
ℝ≥0∞
given by
minimaxValueENNReal (supLossRisk d n β B L C c_f γ)
A strictly positive quadratic lower bound makes a finite Gram matrix invertible. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
ι :
M :
Matrix ι ι ℝ
lam :
ℝ
hlam :
0 < lam
hbound :
∀ b : ι → ℝ, lam * finiteL2Norm b ^ 2 ≤ ∑ i, ∑ j, b i * M i j * b j
conclusion
M.det ≠ 0
Proof (Lean source)
lemma det_ne_zero_of_quadratic_lower {ι : Type*} [Fintype ι] [DecidableEq ι]
(M : Matrix ι ι ℝ) (lam : ℝ) (hlam : 0 < lam)
(hbound : ∀ b : ι → ℝ,
lam * finiteL2Norm b ^ 2 ≤ ∑ i, ∑ j, b i * M i j * b j) :
M.det ≠ 0 := by
classical
intro hdet
obtain ⟨b, hb, hzero⟩ :=
(Matrix.exists_mulVec_eq_zero_iff).2 hdet
have hnorm : 0 < finiteL2Norm b := (finiteL2Norm_pos_iff b).2 hb
have hqzero : (∑ i, ∑ j, b i * M i j * b j) = 0 := by
calc
(∑ i, ∑ j, b i * M i j * b j) = ∑ i, b i * (M.mulVec b) i := by
simp [mulVec, dotProduct, Finset.mul_sum, mul_assoc]
_ = 0 := by simp [hzero]
have hle := hbound b
rw [hqzero] at hle
nlinarith [sq_pos_of_pos hnorm]
theorem finiteL2OpNorm_inv_le_of_quadratic_lower Lemma finiteL2OpNorm_inv_le_of_quadratic_lower in the paper ↗
A positive quadratic lower bound controls the Euclidean norm of the inverse. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
ι :
M :
Matrix ι ι ℝ
lam :
ℝ
hlam :
0 < lam
hbound :
∀ b : ι → ℝ, lam * finiteL2Norm b ^ 2 ≤ ∑ i, ∑ j, b i * M i j * b j
conclusion
finiteL2OpNorm M⁻¹ ≤ lam⁻¹
Proof (Lean source)
lemma finiteL2OpNorm_inv_le_of_quadratic_lower {ι : Type*}
[Fintype ι] [DecidableEq ι] [Nonempty ι] (M : Matrix ι ι ℝ) (lam : ℝ)
(hlam : 0 < lam)
(hbound : ∀ b : ι → ℝ,
lam * finiteL2Norm b ^ 2 ≤ ∑ i, ∑ j, b i * M i j * b j) :
finiteL2OpNorm M⁻¹ ≤ lam⁻¹ := by
classical
have hdet := det_ne_zero_of_quadratic_lower M lam hlam hbound
have hunit : IsUnit M.det := isUnit_iff_ne_zero.mpr hdet
have hvec : ∀ v : ι → ℝ,
finiteL2Norm (M⁻¹.mulVec v) ≤ lam⁻¹ * finiteL2Norm v := by
intro v
let b := M⁻¹.mulVec v
have hMb : M.mulVec b = v := by
dsimp [b]
rw [Matrix.mulVec_mulVec, M.mul_nonsing_inv hunit, Matrix.one_mulVec]
have hdot : (∑ i, ∑ j, b i * M i j * b j) = ∑ i, b i * v i := by
calc
_ = ∑ i, b i * (M.mulVec b) i := by
simp [mulVec, dotProduct, Finset.mul_sum, mul_assoc]
_ = ∑ i, b i * v i := by rw [hMb]
have hcs : (∑ i, b i * v i) ≤ finiteL2Norm b * finiteL2Norm v := by
simpa [finiteL2Norm] using Real.sum_mul_le_sqrt_mul_sqrt univ b v
have hq := hbound b
rw [hdot] at hq
have hbnonneg : 0 ≤ finiteL2Norm b := Real.sqrt_nonneg _
have hvnonneg : 0 ≤ finiteL2Norm v := Real.sqrt_nonneg _
by_cases hb : finiteL2Norm b = 0
· rw [show finiteL2Norm (M⁻¹.mulVec v) = 0 from hb]
exact mul_nonneg (inv_nonneg.mpr hlam.le) hvnonneg
· have hbpos : 0 < finiteL2Norm b := lt_of_le_of_ne hbnonneg (Ne.symm hb)
have hlin : lam * finiteL2Norm b ≤ finiteL2Norm v := by
nlinarith [sq_pos_of_pos hbpos]
exact (le_inv_mul_iff₀ hlam).2 hlin
unfold finiteL2OpNorm
apply csSup_le
· let i : ι := choice inferInstance
refine ⟨finiteL2Norm (M⁻¹.mulVec (single i 1)), ?_⟩
refine ⟨Pi.single i 1, ?_, rfl⟩
simp [finiteL2Norm, Pi.single_apply]
· rintro r ⟨v, hv, rfl⟩
simpa [hv] using hvec v
theorem gramHat_minEig_ge_of_pos
Unisolvence and distribution-free equal-cell Gram conditioning. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
0 < templateLambda d m
conclusion 2
b :
MonoIndex d m → ℝ
templateLambda d m / 2 * finiteL2Norm b ^ 2 ≤ ∑ α, ∑ β, b α * gramHat m j ω k α β * b β
conclusion 3
(gramHat m j ω k).det ≠ 0
conclusion 4
Proof (Lean source)
lemma gramHat_minEig_ge_of_pos (d m n : ℕ) (ω : Fin n → Obs d) (j : ℕ)
(k : Fin d → Fin (2 ^ j))
(hcount : ∀ ℓ : Fin d → Fin (m + 1), 0 < treatedCount m j ω k ℓ) :
0 < templateLambda d m ∧
(∀ b : MonoIndex d m → ℝ,
templateLambda d m / 2 * finiteL2Norm b ^ 2 ≤
∑ α, ∑ β, b α * gramHat m j ω k α β * b β) ∧
(gramHat m j ω k).det ≠ 0 ∧
finiteL2OpNorm ((gramHat m j ω k)⁻¹) ≤ 2 / templateLambda d m := by
classical
have hlam : 0 < templateLambda d m := templateLambda_pos d m
have hquad : ∀ b : MonoIndex d m → ℝ,
templateLambda d m / 2 * finiteL2Norm b ^ 2 ≤
∑ α, ∑ β, b α * gramHat m j ω k α β * b β := by
intro b
let ε := templateLip d m * (templateEta d m / 2) * finiteL2Norm b ^ 2
have hpoint : ∀ ℓ i, (ω i).2.1 = true ∧
(ω i).1 ∈ scaledMicroCell d m j k ℓ →
(∑ α, b α * monoVec d m (tensorNode d m ℓ) α) ^ 2 - ε ≤
(∑ α, b α * monoVec d m
(fun a => ((ω i).1 a - cubeCorner d j k a) / meshWidth j) α) ^ 2 := by
intro ℓ i hi
apply templatePointwise_square_perturbation d m
exact hi.2
have htransfer := templateGram_quadratic_sub_error_le_gramHat
m j ω k b ε hcount hpoint
have hbase := templateLambda_le_quadratic d m b
have heta := templateLip_eta_le_half_lambda d m
have herr : ε ≤ templateLambda d m / 2 * finiteL2Norm b ^ 2 := by
dsimp [ε]
have heta' : templateLip d m * (templateEta d m / 2) ≤
templateLambda d m / 2 := by
convert heta using 1
ring
exact mul_le_mul_of_nonneg_right heta' (sq_nonneg _)
nlinarith
have hhalf : 0 < templateLambda d m / 2 := by positivity
have hdet := det_ne_zero_of_quadratic_lower
(gramHat m j ω k) (templateLambda d m / 2) hhalf hquad
have : Nonempty (MonoIndex d m) :=
⟨⟨fun _ => 0, by simp [monoIdx]⟩⟩
have hinv := finiteL2OpNorm_inv_le_of_quadratic_lower
(gramHat m j ω k) (templateLambda d m / 2) hhalf hquad
refine ⟨hlam, hquad, hdet, ?_⟩
convert hinv using 1; field_simp
theorem gramHat_minEig_ge
A feasible candidate mesh has the uniform equal-cell Gram conditioning asserted in the paper. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
0 < templateLambda d m
conclusion 2
b :
MonoIndex d m → ℝ
templateLambda d m / 2 * finiteL2Norm b ^ 2 ≤ ∑ α, ∑ β, b α * gramHat m j ω k α β * b β
conclusion 3
(gramHat m j ω k).det ≠ 0
conclusion 4
Proof (Lean source)
lemma gramHat_minEig_ge (d m n : ℕ) (β : ℝ) (ω : Fin n → Obs d) (j : ℕ)
(k : Fin d → Fin (2 ^ j)) (hfeasible : meshFeasible m β ω j) :
0 < templateLambda d m ∧
(∀ b : MonoIndex d m → ℝ,
templateLambda d m / 2 * finiteL2Norm b ^ 2 ≤
∑ α, ∑ β, b α * gramHat m j ω k α β * b β) ∧
(gramHat m j ω k).det ≠ 0 ∧
finiteL2OpNorm ((gramHat m j ω k)⁻¹) ≤ 2 / templateLambda d m := by
apply gramHat_minEig_ge_of_pos d m n ω j k
intro ℓ
have hlower := hfeasible.2 k ℓ
have hwidth : 0 < meshWidth j := by unfold meshWidth; positivity
have hthreshold : 0 < meshWidth j ^ (-2 * β) :=
Real.rpow_pos_of_pos hwidth _
exact_mod_cast hthreshold.trans_le hlower
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.TemplateAlgebra 21 declarations Causalean.Stat.Nonparametric.designMatrix is univariate and designMatrix_inv_concentration concerns random Gram concentration.
Tensor-template algebra and finite Euclidean norms
Causalean.Stat.Nonparametric.designMatrix is univariate and
designMatrix_inv_concentration concerns random Gram concentration. This file
constructs the explicit multivariate tensor template, proves unisolvence, and
develops the finite Euclidean norm bounds used for Gram perturbations.
def monoIdx
Total-degree multiindices. For the stated inputs and conditions, the monoIdx object being defined.
def MonoIndex
For the dimension and degree bound, the monomial-index type collects precisely the multiindices whose total degree does not exceed that bound.
def monoVec
Vector of all total-degree monomials. For the stated inputs and conditions, the monoVec object being defined.
def tensorCount
Number of tensor nodes. For the stated inputs and conditions, the tensorCount object being defined.
Definition (Lean source)
parameters
d m :
ℕ
def
tensorCount d m :
ℕ
given by
(m + 1) ^ d
Tensor interpolation node. For the stated inputs and conditions, the tensorNode object being defined.
A polynomial of coordinatewise degree at most m that vanishes on every template node is zero. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma tensorNode_polynomial_eq_zero (d m : ℕ) (p : MvPolynomial (Fin d) ℝ)
(hdeg : ∀ i, p.degreeOf i ≤ m)
(hzero : ∀ ℓ : Fin d → Fin (m + 1), eval (tensorNode d m ℓ) p = 0) :
p = 0 := by
classical
let S : Fin d → Finset ℝ := fun _ =>
Finset.univ.image (fun j : Fin (m + 1) => ((j.val + 1 : ℝ) / (m + 2 : ℝ)))
have hcard (i : Fin d) : (S i).card = m + 1 := by
dsimp [S]
rw [Finset.card_image_of_injective]
· simp
· intro a b hab
apply Fin.ext
have hden : (m + 2 : ℝ) ≠ 0 := by positivity
have heq : (a.val + 1 : ℝ) = (b.val + 1 : ℝ) :=
(div_left_inj' hden).mp hab
exact_mod_cast (add_right_cancel heq : (a.val : ℝ) = b.val)
apply MvPolynomial.eq_zero_of_eval_zero_at_prod_finset p S
(by intro i; rw [hcard i]; exact Nat.lt_succ_of_le (hdeg i))
intro x hx
have hchoice : ∀ i : Fin d, ∃ a : Fin (m + 1),
((a.val + 1 : ℝ) / (m + 2 : ℝ)) = x i := by
intro i
obtain ⟨a, _, ha⟩ := Finset.mem_image.mp (hx i)
exact ⟨a, ha⟩
choose ℓ hℓ using hchoice
have heq : tensorNode d m ℓ = x := funext hℓ
simpa only [heq] using hzero ℓ
def templateGram
Equal-weight monomial Gram matrix. For the stated inputs and conditions, the templateGram object being defined.
Definition (Lean source)
parameters
d m :
ℕ
given by
fun α β
=> (tensorCount d m : ℝ)⁻¹ * ∑ ℓ : Fin d → Fin (m + 1), monoVec d m (tensorNode d m ℓ) α * monoVec d m (tensorNode d m ℓ) β
The deterministic tensor template Gram matrix is symmetric. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma templateGram_symmetric (d m : ℕ) : (templateGram d m).IsHermitian := by
classical
rw [IsHermitian]
ext α β
simp only [Matrix.conjTranspose_apply, star_trivial, templateGram]
congr 1
apply Finset.sum_congr rfl
intro ℓ _
ring
The tensor Gram quadratic form is the mean of squared polynomial evaluations. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d m :
ℕ
b :
MonoIndex d m → ℝ
conclusion
(∑ α, ∑ β, b α * templateGram d m α β * b β)
= (tensorCount d m : ℝ)⁻¹ * ∑ ℓ : Fin d → Fin (m + 1), (∑ α, b α * monoVec d m (tensorNode d m ℓ) α) ^ 2
Proof (Lean source)
lemma templateGram_quadratic_sum_sq (d m : ℕ) (b : MonoIndex d m → ℝ) :
(∑ α, ∑ β, b α * templateGram d m α β * b β) =
(tensorCount d m : ℝ)⁻¹ *
∑ ℓ : Fin d → Fin (m + 1),
(∑ α, b α * monoVec d m (tensorNode d m ℓ) α) ^ 2 := by
classical
simp only [templateGram, sq, Finset.sum_mul, Finset.mul_sum]
conv_lhs =>
arg 2
ext α
rw [Finset.sum_comm]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro ℓ _
apply Finset.sum_congr rfl
intro α _
apply Finset.sum_congr rfl
intro β _
ring
The tensor Gram matrix is positive semidefinite. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma templateGram_quadratic_nonneg (d m : ℕ) (b : MonoIndex d m → ℝ) :
0 ≤ ∑ α, ∑ β, b α * templateGram d m α β * b β := by
rw [templateGram_quadratic_sum_sq]
positivity
One nonzero tensor-node evaluation makes the Gram quadratic form positive. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
0 < ∑ α, ∑ β, b α * templateGram d m α β * b β
Proof (Lean source)
lemma templateGram_quadratic_pos_of_eval (d m : ℕ) (b : MonoIndex d m → ℝ)
(h : ∃ ℓ : Fin d → Fin (m + 1),
∑ α, b α * monoVec d m (tensorNode d m ℓ) α ≠ 0) :
0 < ∑ α, ∑ β, b α * templateGram d m α β * b β := by
rw [templateGram_quadratic_sum_sq]
apply mul_pos
· have hJ : 0 < tensorCount d m := by
unfold tensorCount
exact pow_pos (Nat.succ_pos m) d
exact inv_pos.mpr (by exact_mod_cast hJ)
· obtain ⟨ℓ, hℓ⟩ := h
apply Finset.sum_pos'
· intro i hi
positivity
· exact ⟨ℓ, Finset.mem_univ _, sq_pos_of_ne_zero hℓ⟩
def finiteL2Norm
Euclidean norm of a finite real vector. For the stated inputs and conditions, the finiteL2Norm object being defined.
The square of the finite Euclidean norm is the sum of coordinate squares. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma finiteL2Norm_sq {ι : Type*} [Fintype ι] (b : ι → ℝ) :
finiteL2Norm b ^ 2 = ∑ i, b i ^ 2 := by
unfold finiteL2Norm
exact Real.sq_sqrt (Finset.sum_nonneg (fun i _ => sq_nonneg (b i)))
A finite vector has positive Euclidean norm exactly when it is nonzero. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma finiteL2Norm_pos_iff {ι : Type*} [Fintype ι] (b : ι → ℝ) :
0 < finiteL2Norm b ↔ b ≠ 0 := by
rw [finiteL2Norm, Real.sqrt_pos.iff]
constructor
· intro h
obtain ⟨i, hi⟩ : ∃ i, b i ≠ 0 := by
by_contra hn
push Not at hn
exact h (funext hn)
exact lt_of_lt_of_le (sq_pos_of_ne_zero hi)
(Finset.single_le_sum (fun j _ => sq_nonneg (b j)) (Finset.mem_univ i))
· intro h hb
subst b
simp at h
A unit finite vector has a nonzero coordinate. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma finiteL2Norm_one_exists_ne {ι : Type*} [Fintype ι]
(b : ι → ℝ) (hb : finiteL2Norm b = 1) : ∃ i, b i ≠ 0 := by
classical
by_contra h
push Not at h
have hz : b = 0 := funext fun i => h i
subst b
simp [finiteL2Norm] at hb
Tensor-grid unisolvence makes the Gram quadratic form positive on unit vectors. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
0 < ∑ α, ∑ β, b α * templateGram d m α β * b β
Proof (Lean source)
lemma templateGram_unit_pos (d m : ℕ) (b : MonoIndex d m → ℝ)
(hb : finiteL2Norm b = 1) :
0 < ∑ α, ∑ β, b α * templateGram d m α β * b β := by
classical
let e : MonoIndex d m → Fin d →₀ ℕ :=
fun α => Finsupp.equivFunOnFinite.symm (fun i => (α.1 i).val)
let p : MvPolynomial (Fin d) ℝ := ∑ α, monomial (e α) (b α)
have he : Injective e := by
intro α β h
apply Subtype.ext
funext i
apply Fin.ext
have hi := congrFun (congrArg Finsupp.equivFunOnFinite h) i
exact hi
have hdeg : ∀ i, p.degreeOf i ≤ m := by
intro i
dsimp [p]
refine (MvPolynomial.degreeOf_sum_le i univ _).trans ?_
apply Finset.sup_le
intro α _
by_cases hba : b α = 0
· simp [hba]
· rw [MvPolynomial.degreeOf_monomial_eq (e α) i hba]
have hmem := (Finset.mem_filter.mp α.property).2
change (α.1 i).val ≤ m
exact (Finset.single_le_sum (fun j _ => zero_le _) (Finset.mem_univ i)).trans hmem
have hcoeff (α : MonoIndex d m) : p.coeff (e α) = b α := by
simp [p, MvPolynomial.coeff_sum, MvPolynomial.coeff_monomial, he.eq_iff]
have hne := finiteL2Norm_one_exists_ne b hb
apply templateGram_quadratic_pos_of_eval
by_contra hnone
push Not at hnone
have hzero : ∀ ℓ : Fin d → Fin (m + 1), eval (tensorNode d m ℓ) p = 0 := by
intro ℓ
have hz := hnone ℓ
simpa [p, e, MvPolynomial.eval_sum, MvPolynomial.eval_monomial,
Finsupp.prod_fintype, monoVec] using hz
have hp := tensorNode_polynomial_eq_zero d m p hdeg hzero
obtain ⟨α, hα⟩ := hne
exact hα (by rw [← hcoeff α, hp]; simp)
def finiteL2OpNorm
Euclidean operator norm of a finite real matrix. For the stated inputs and conditions, the finiteL2OpNorm object being defined.
The finite Euclidean operator norm is nonnegative. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma finiteL2OpNorm_nonneg {ι κ : Type*} [Fintype ι] [Fintype κ]
(M : Matrix ι κ ℝ) : 0 ≤ finiteL2OpNorm M := by
apply Real.sSup_nonneg
rintro r ⟨b, -, rfl⟩
exact Real.sqrt_nonneg _
theorem finiteL2OpNorm_le_matrixL2
The custom finite Euclidean operator norm is bounded by Mathlib's matrix ℓ² operator norm. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma finiteL2OpNorm_le_matrixL2 {ι κ : Type*} [Fintype ι] [Fintype κ]
[DecidableEq κ] [Nonempty κ] (M : Matrix ι κ ℝ) :
finiteL2OpNorm M ≤ ‖M‖ := by
unfold finiteL2OpNorm
apply csSup_le
· let i : κ := choice inferInstance
refine ⟨finiteL2Norm (M.mulVec (single i 1)), ?_⟩
refine ⟨Pi.single i 1, ?_, rfl⟩
simp [finiteL2Norm, Pi.single_apply]
· rintro r ⟨b, hb, rfl⟩
have h := Matrix.l2_opNorm_mulVec M (toLp 2 b)
rw [show ‖WithLp.toLp 2 b‖ = finiteL2Norm b by
simp [finiteL2Norm, EuclideanSpace.norm_eq, Real.norm_eq_abs, sq_abs], hb, mul_one] at h
simpa [finiteL2Norm, EuclideanSpace.norm_eq, Real.norm_eq_abs, sq_abs] using h
theorem finiteL2Norm_mulVec_le_opNorm
The custom finite operator norm bounds its action on every vector. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
finiteL2Norm (M.mulVec b) ≤ finiteL2OpNorm M * finiteL2Norm b
Proof (Lean source)
lemma finiteL2Norm_mulVec_le_opNorm {ι κ : Type*} [Fintype ι] [Fintype κ]
[DecidableEq κ] [Nonempty κ] (M : Matrix ι κ ℝ) (b : κ → ℝ) :
finiteL2Norm (M.mulVec b) ≤ finiteL2OpNorm M * finiteL2Norm b := by
classical
by_cases hb : b = 0
· subst b
simp [finiteL2Norm]
let r := finiteL2Norm b
have hr : 0 < r := (finiteL2Norm_pos_iff b).2 hb
let u : κ → ℝ := fun i => b i / r
have hu : finiteL2Norm u = 1 := by
have hsq : finiteL2Norm u ^ 2 = 1 := by
rw [finiteL2Norm_sq]
simp only [u, div_pow]
rw [← Finset.sum_div, ← finiteL2Norm_sq]
change r ^ 2 / r ^ 2 = 1
exact div_self (pow_ne_zero 2 (ne_of_gt hr))
nlinarith [show 0 ≤ finiteL2Norm u from Real.sqrt_nonneg _]
have hbdd : BddAbove {x : ℝ | ∃ v : κ → ℝ, finiteL2Norm v = 1 ∧
x = finiteL2Norm (M.mulVec v)} := by
refine ⟨‖M‖, ?_⟩
rintro x ⟨v, hv, rfl⟩
have h := Matrix.l2_opNorm_mulVec M (toLp 2 v)
rw [show ‖WithLp.toLp 2 v‖ = finiteL2Norm v by
simp [finiteL2Norm, EuclideanSpace.norm_eq, Real.norm_eq_abs, sq_abs], hv, mul_one] at h
simpa [finiteL2Norm, EuclideanSpace.norm_eq, Real.norm_eq_abs, sq_abs] using h
have hu_le : finiteL2Norm (M.mulVec u) ≤ finiteL2OpNorm M := by
apply le_csSup hbdd
exact ⟨u, hu, rfl⟩
have hscale : finiteL2Norm (M.mulVec b) = finiteL2Norm (M.mulVec u) * r := by
have hbu : b = r • u := by
funext i
dsimp [u]
field_simp
have hsquare : finiteL2Norm (M.mulVec b) ^ 2 =
(finiteL2Norm (M.mulVec u) * r) ^ 2 := by
rw [hbu, Matrix.mulVec_smul, finiteL2Norm_sq]
calc
∑ i, (r • M.mulVec u) i ^ 2 = r ^ 2 * ∑ i, (M.mulVec u i) ^ 2 := by
simp only [Pi.smul_apply, smul_eq_mul, mul_pow, Finset.mul_sum]
_ = r ^ 2 * finiteL2Norm (M.mulVec u) ^ 2 := by
congr 1
exact (finiteL2Norm_sq (M.mulVec u)).symm
_ = (finiteL2Norm (M.mulVec u) * r) ^ 2 := by ring
nlinarith [show 0 ≤ finiteL2Norm (M.mulVec b) from Real.sqrt_nonneg _,
show 0 ≤ finiteL2Norm (M.mulVec u) from Real.sqrt_nonneg _,
mul_nonneg (show 0 ≤ finiteL2Norm (M.mulVec u) from Real.sqrt_nonneg _) hr.le]
rw [hscale]
exact mul_le_mul_of_nonneg_right hu_le hr.le
theorem quadraticForm_le_finiteL2OpNorm
A real square-matrix quadratic form is bounded above by its finite Euclidean operator norm times the squared coefficient norm. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
(∑ i, ∑ j, b i * M i j * b j) ≤ finiteL2OpNorm M * finiteL2Norm b ^ 2
Proof (Lean source)
lemma quadraticForm_le_finiteL2OpNorm {ι : Type*} [Fintype ι]
[DecidableEq ι] [Nonempty ι] (M : Matrix ι ι ℝ) (b : ι → ℝ) :
(∑ i, ∑ j, b i * M i j * b j) ≤ finiteL2OpNorm M * finiteL2Norm b ^ 2 := by
have hdot : (∑ i, ∑ j, b i * M i j * b j) = ∑ i, b i * (M.mulVec b) i := by
simp [mulVec, dotProduct, Finset.mul_sum, mul_assoc]
have hcs : (∑ i, b i * (M.mulVec b) i) ≤
finiteL2Norm b * finiteL2Norm (M.mulVec b) := by
simpa [finiteL2Norm] using
Real.sum_mul_le_sqrt_mul_sqrt univ b (M.mulVec b)
rw [hdot]
calc
_ ≤ finiteL2Norm b * finiteL2Norm (M.mulVec b) := hcs
_ ≤ finiteL2Norm b * (finiteL2OpNorm M * finiteL2Norm b) := by
exact mul_le_mul_of_nonneg_left (finiteL2Norm_mulVec_le_opNorm M b)
(Real.sqrt_nonneg _)
_ = finiteL2OpNorm M * finiteL2Norm b ^ 2 := by ring
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.TemplateGeometry 16 declarations This file proves the spectral gap and Lipschitz control for the deterministic tensor template and defines the contained template microcubes.
Template spectral and microcube geometry
This file proves the spectral gap and Lipschitz control for the deterministic tensor template and defines the contained template microcubes.
def templateLambda
Rayleigh-quotient least eigenvalue. For the stated inputs and conditions, the templateLambda object being defined.
Definition (Lean source)
parameters
d m :
ℕ
def
templateLambda d m :
ℝ
given by
sInf {r : ℝ | ∃ b : MonoIndex d m → ℝ, finiteL2Norm b = 1 ∧ r = ∑ α, ∑ β, b α * templateGram d m α β * b β}
Compactness of the finite unit sphere makes strict Gram positivity uniform. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma templateLambda_pos (d m : ℕ) : 0 < templateLambda d m := by
classical
let q : (MonoIndex d m → ℝ) → ℝ := fun b =>
∑ α, ∑ β, b α * templateGram d m α β * b β
let s2 : (MonoIndex d m → ℝ) → ℝ := fun b => ∑ α, b α ^ 2
let S : Set (MonoIndex d m → ℝ) := {b | s2 b = 1}
have hs2cont : Continuous s2 := by dsimp [s2]; fun_prop
have hqcont : Continuous q := by dsimp [q]; fun_prop
have hSclosed : IsClosed S := isClosed_eq hs2cont continuous_const
have hSbounded : Bornology.IsBounded S := by
apply (Metric.isBounded_iff_subset_closedBall 0).mpr
refine ⟨1, ?_⟩
intro b hb
rw [Metric.mem_closedBall, dist_zero_right]
apply (pi_norm_le_iff_of_nonneg zero_le_one).mpr
intro α
rw [Real.norm_eq_abs, ← sq_le_one_iff_abs_le_one]
have hα : b α ^ 2 ≤ 1 := by
rw [← hb]
exact Finset.single_le_sum (fun i _ => sq_nonneg (b i)) (Finset.mem_univ α)
exact hα
have hScompact : IsCompact S :=
Metric.isCompact_iff_isClosed_bounded.mpr ⟨hSclosed, hSbounded⟩
let α₀ : MonoIndex d m := ⟨fun _ => 0, by simp [monoIdx]⟩
let b₀ : MonoIndex d m → ℝ := single α₀ 1
have hb₀ : b₀ ∈ S := by
simp [S, s2, b₀, Pi.single_apply]
obtain ⟨bmin, hbmin, hmin⟩ :=
hScompact.exists_sInf_image_eq ⟨b₀, hb₀⟩ hqcont.continuousOn
have hunit : finiteL2Norm bmin = 1 := by
simp [finiteL2Norm, S, s2] at hbmin ⊢
simp [hbmin]
have hpos := templateGram_unit_pos d m bmin hunit
have hset : {r : ℝ | ∃ b : MonoIndex d m → ℝ, finiteL2Norm b = 1 ∧ r = q b} =
q '' S := by
ext r
simp only [Set.mem_ofPred_eq, Set.mem_image, S]
constructor
· rintro ⟨b, hb, rfl⟩
refine ⟨b, ?_, rfl⟩
have hs : 0 ≤ s2 b := Finset.sum_nonneg (fun i _ => sq_nonneg (b i))
have := congrArg (fun x : ℝ => x ^ 2) hb
change (sqrt (s2 b)) ^ 2 = 1 ^ 2 at this
rwa [Real.sq_sqrt hs, one_pow] at this
· rintro ⟨b, hb, rfl⟩
refine ⟨b, ?_, rfl⟩
simpa [finiteL2Norm, s2, hb]
change 0 < sInf {r : ℝ | ∃ b : MonoIndex d m → ℝ,
finiteL2Norm b = 1 ∧ r = q b}
rw [hset, hmin]
exact hpos
The template Rayleigh infimum bounds every unit-vector quadratic form. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
templateLambda d m ≤ ∑ α, ∑ β, b α * templateGram d m α β * b β
Proof (Lean source)
lemma templateLambda_le_quadratic_unit (d m : ℕ)
(b : MonoIndex d m → ℝ) (hb : finiteL2Norm b = 1) :
templateLambda d m ≤
∑ α, ∑ β, b α * templateGram d m α β * b β := by
let S : Set ℝ := {r | ∃ v : MonoIndex d m → ℝ,
finiteL2Norm v = 1 ∧
r = ∑ α, ∑ β, v α * templateGram d m α β * v β}
have hbelow : BddBelow S := by
refine ⟨0, ?_⟩
rintro r ⟨v, -, rfl⟩
exact templateGram_quadratic_nonneg d m v
change sInf S ≤ _
exact csInf_le hbelow ⟨b, hb, rfl⟩
The least template Rayleigh quotient controls every coefficient vector. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d m :
ℕ
b :
MonoIndex d m → ℝ
conclusion
templateLambda d m * finiteL2Norm b ^ 2 ≤ ∑ α, ∑ β, b α * templateGram d m α β * b β
Proof (Lean source)
lemma templateLambda_le_quadratic (d m : ℕ)
(b : MonoIndex d m → ℝ) :
templateLambda d m * finiteL2Norm b ^ 2 ≤
∑ α, ∑ β, b α * templateGram d m α β * b β := by
classical
by_cases hb : b = 0
· subst b
simp [finiteL2Norm]
let r := finiteL2Norm b
have hr : 0 < r := (finiteL2Norm_pos_iff b).2 hb
let u : MonoIndex d m → ℝ := fun i => b i / r
have hu : finiteL2Norm u = 1 := by
have hsq : finiteL2Norm u ^ 2 = 1 := by
rw [finiteL2Norm_sq]
simp only [u, div_pow]
rw [← Finset.sum_div, ← finiteL2Norm_sq]
change r ^ 2 / r ^ 2 = 1
exact div_self (pow_ne_zero 2 (ne_of_gt hr))
nlinarith [show 0 ≤ finiteL2Norm u from Real.sqrt_nonneg _]
have hunit := templateLambda_le_quadratic_unit d m u hu
have hscale :
(∑ α, ∑ β, u α * templateGram d m α β * u β) * r ^ 2 =
∑ α, ∑ β, b α * templateGram d m α β * b β := by
rw [Finset.sum_mul]
apply Finset.sum_congr rfl
intro α _
rw [Finset.sum_mul]
apply Finset.sum_congr rfl
intro β _
dsimp [u]
field_simp
have hm := mul_le_mul_of_nonneg_right hunit (sq_nonneg r)
rw [hscale] at hm
simpa [r] using hm
def templateLip
Lipschitz quotient of the matrix-valued polynomial basis on the cube. For the stated inputs and conditions, the templateLip object being defined.
Definition (Lean source)
theorem templateLip_bddAbove
The polynomial outer-product difference quotients on the compact cube are bounded above. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma templateLip_bddAbove (d m : ℕ) :
BddAbove {r : ℝ | ∃ z z' : Fin d → ℝ, z ∈ cube d ∧ z' ∈ cube d ∧ z ≠ z' ∧
r = finiteL2OpNorm
((fun α β : MonoIndex d m => monoVec d m z α * monoVec d m z β) -
(fun α β : MonoIndex d m => monoVec d m z' α * monoVec d m z' β)) /
‖z - z'‖} := by
classical
let i₀ : MonoIndex d m := ⟨fun _ => 0, by simp [monoIdx]⟩
letI : Nonempty (MonoIndex d m) := ⟨i₀⟩
let F : (Fin d → ℝ) → Matrix (MonoIndex d m) (MonoIndex d m) ℝ :=
fun z α β => monoVec d m z α * monoVec d m z β
let G : (Fin d → ℝ) → (MonoIndex d m × MonoIndex d m → ℝ) :=
fun z p => F z p.1 p.2
have hG : ContDiff ℝ 1 G := by
dsimp [G, F, monoVec]
rw [contDiff_pi]
intro p
apply ContDiff.mul
· apply contDiff_prod
intro i hi
exact (contDiff_apply ℝ ℝ i).pow _
· apply contDiff_prod
intro i hi
exact (contDiff_apply ℝ ℝ i).pow _
let H : (Fin d → ℝ) → EuclideanSpace ℝ (MonoIndex d m × MonoIndex d m) :=
fun z => (EuclideanSpace.equiv _ _).symm (G z)
have hH : ContDiff ℝ 1 H := by
exact (EuclideanSpace.equiv _ _).symm.contDiff.comp hG
have hconv : Convex ℝ (cube d) := by
rw [cube]
exact convex_pi (fun _ _ => convex_Icc _ _)
have hcompact : IsCompact (cube d) := by
rw [cube]
exact isCompact_univ_pi fun _ => isCompact_Icc
obtain ⟨K, hK⟩ := hH.contDiffOn.exists_lipschitzOnWith one_ne_zero hconv hcompact
refine ⟨(K : ℝ), ?_⟩
rintro r ⟨z, z', hz, hz', hne, rfl⟩
have hop : finiteL2OpNorm (F z - F z') ≤ ‖F z - F z'‖ :=
finiteL2OpNorm_le_matrixL2 (F z - F z')
have hmatrix : ‖F z - F z'‖ ≤ ‖H z - H z'‖ := by
calc
‖F z - F z'‖ ≤ @norm (Matrix (MonoIndex d m) (MonoIndex d m) ℝ)
Matrix.frobeniusNormedAddCommGroup.toNorm (F z - F z') := by
open scoped Matrix.Norms.Frobenius in
refine ContinuousLinearMap.opNorm_le_bound _ (norm_nonneg (F z - F z')) fun x => ?_
have hmul := Matrix.frobenius_norm_mul (F z - F z')
(Matrix.replicateCol Unit (ofLp x))
have heq : (F z - F z') * Matrix.replicateCol Unit (ofLp x) =
Matrix.replicateCol Unit ((F z - F z').mulVec (ofLp x)) := by
ext i u
simp [Matrix.mul_apply, mulVec, Matrix.replicateCol, dotProduct]
rw [heq, Matrix.frobenius_norm_replicateCol] at hmul
rw [← Matrix.toEuclideanCLM_toLp (F z - F z') (ofLp x),
WithLp.toLp_ofLp] at hmul
change ‖Matrix.toEuclideanCLM (n := MonoIndex d m) (𝕜 := ℝ)
(F z - F z') x‖ ≤ ‖F z - F z'‖ * ‖x‖
simpa using hmul
_ = ‖H z - H z'‖ := by
rw [Matrix.frobenius_norm_def, ← Real.sqrt_eq_rpow]
simp only [H, EuclideanSpace.norm_eq, Real.norm_eq_abs,
Fintype.sum_prod_type, PiLp.continuousLinearEquiv_symm_apply,
Real.rpow_two, sq_abs]
congr 1
have hdist : 0 < ‖z - z'‖ := norm_pos_iff.mpr (sub_ne_zero.mpr hne)
apply (div_le_div_of_nonneg_right (hop.trans hmatrix) hdist.le).trans
rw [div_le_iff₀ hdist]
exact hK.dist_le_mul z hz z' hz'
theorem templateLip_bounds_quotient
templateLip bounds each nontrivial outer-product difference quotient on the cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
finiteL2OpNorm ((fun α β : MonoIndex d m => monoVec d m z α * monoVec d m z β) - (fun α β : MonoIndex d m => monoVec d m z' α * monoVec d m z' β)) / ‖z - z'‖
≤ templateLip d m
Proof (Lean source)
lemma templateLip_bounds_quotient (d m : ℕ) (z z' : Fin d → ℝ)
(hz : z ∈ cube d) (hz' : z' ∈ cube d) (hne : z ≠ z') :
finiteL2OpNorm
((fun α β : MonoIndex d m => monoVec d m z α * monoVec d m z β) -
(fun α β : MonoIndex d m => monoVec d m z' α * monoVec d m z' β)) /
‖z - z'‖ ≤ templateLip d m := by
apply le_csSup (templateLip_bddAbove d m)
exact ⟨z, z', hz, hz', hne, rfl⟩
Every matrix difference quotient in the template Lipschitz radius is nonnegative. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma templateLip_nonneg (d m : ℕ) : 0 ≤ templateLip d m := by
apply Real.sSup_nonneg
rintro r ⟨z, z', -, -, hzz', rfl⟩
exact div_nonneg (finiteL2OpNorm_nonneg _) (norm_nonneg _)
def templateEta
Template microcell width. For the stated inputs and conditions, the templateEta object being defined.
Definition (Lean source)
parameters
d m :
ℕ
def
templateEta d m :
ℝ
given by
The fixed template cells have strictly positive width. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma templateEta_pos (d m : ℕ) : 0 < templateEta d m := by
unfold templateEta
apply lt_min
· positivity
· exact div_pos (templateLambda_pos d m)
(by have := templateLip_nonneg d m; positivity)
The chosen microcell radius fits inside half the template spectral gap. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma templateLip_eta_le_half_lambda (d m : ℕ) :
templateLip d m * templateEta d m / 2 ≤ templateLambda d m / 2 := by
have hLip : 0 ≤ templateLip d m := templateLip_nonneg d m
have hEta : 0 ≤ templateEta d m := le_of_lt (templateEta_pos d m)
have hden : 0 < 1 + templateLip d m := by linarith
have hwidth : templateEta d m ≤
templateLambda d m / (1 + templateLip d m) := by
unfold templateEta
exact min_le_right _ _
have hprod : templateEta d m * (1 + templateLip d m) ≤
templateLambda d m := (le_div_iff₀ hden).mp hwidth
have hsmall : templateLip d m * templateEta d m ≤
templateEta d m * (1 + templateLip d m) := by nlinarith
linarith
def microCube
Closed microcube around a tensor node. For the stated inputs and conditions, the microCube object being defined.
Definition (Lean source)
parameters
given by
{z | ∀ i, |z i - tensorNode d m ℓ i| ≤ templateEta d m / 2}
Membership in a template microcube gives the norm bound used by Gram perturbation. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
‖z - tensorNode d m ℓ‖ ≤ templateEta d m / 2
Proof (Lean source)
lemma microCube_norm_sub_tensorNode_le (d m : ℕ)
(ℓ : Fin d → Fin (m + 1)) (z : Fin d → ℝ)
(hz : z ∈ microCube d m ℓ) :
‖z - tensorNode d m ℓ‖ ≤ templateEta d m / 2 := by
apply (pi_norm_le_iff_of_nonneg
(div_nonneg (le_of_lt (templateEta_pos d m)) (by norm_num))).2
intro i
simpa [Real.norm_eq_abs] using hz i
theorem tensorNode_mem_cube
Every tensor interpolation node lies in the unit cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma tensorNode_mem_cube (d m : ℕ) (ℓ : Fin d → Fin (m + 1)) :
tensorNode d m ℓ ∈ cube d := by
intro i hi
constructor
· unfold tensorNode
positivity
· unfold tensorNode
have hle : (ℓ i).val + 1 ≤ m + 1 := Nat.succ_le_succ (Fin.le_last (ℓ i))
have hden : (0 : ℝ) < m + 2 := by positivity
rw [div_le_one hden]
exact_mod_cast hle.trans (le_succ (m + 1))
theorem microCube_subset_cube
Every template microcube is contained in the unit cube. For the stated inputs and conditions, the asserted conclusion holds.
Proof (Lean source)
lemma microCube_subset_cube (d m : ℕ) (ℓ : Fin d → Fin (m + 1)) :
microCube d m ℓ ⊆ cube d := by
intro z hz i hi
have hwidth : templateEta d m ≤ (1 : ℝ) / (2 * (m + 2)) := by
unfold templateEta
exact min_le_left _ _
have heta : templateEta d m / 2 ≤ 1 / (m + 2 : ℝ) := by
have hm : (0 : ℝ) < m + 2 := by positivity
calc
templateEta d m / 2 ≤ (1 / (2 * (m + 2 : ℝ))) / 2 := by gcongr
_ ≤ 1 / (m + 2 : ℝ) := by
have hinv : 0 ≤ 1 / (m + 2 : ℝ) := by positivity
have heq : (1 / (2 * (m + 2 : ℝ))) / 2 =
(1 / (m + 2 : ℝ)) / 4 := by
field_simp
norm_num
rw [heq]
linarith
have hzabs := hz i
rw [abs_le] at hzabs
constructor
· have hnode : 1 / (m + 2 : ℝ) ≤ tensorNode d m ℓ i := by
unfold tensorNode
have hnum : (1 : ℝ) ≤ (ℓ i).val + 1 := by norm_num
have hm : (0 : ℝ) < m + 2 := by positivity
exact (div_le_div_iff_of_pos_right hm).2 hnum
linarith
· have hnode : tensorNode d m ℓ i ≤ (m + 1 : ℝ) / (m + 2 : ℝ) := by
unfold tensorNode
have hvalNat : (ℓ i).val ≤ m := by omega
have hval : ((ℓ i).val : ℝ) ≤ m := by exact_mod_cast hvalNat
gcongr
have hm : (0 : ℝ) < m + 2 := by positivity
have hgap : (m + 1 : ℝ) / (m + 2 : ℝ) + 1 / (m + 2 : ℝ) = 1 := by
field_simp
ring
linarith
The fixed finite family of unisolvent template microcubes. For the stated inputs and conditions, the EqualCellTemplate object being defined.
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.Helpers.Witnesses 48 declarations The three cited logical claims are explicit proposition-valued inputs.
Thin-strip and radial testing constructions, and cited gates
The three cited logical claims are explicit proposition-valued inputs. Dorn,
Minimax Rates Under Global Overlap Bounds (2026), Assumptions 1–2, fixes the
remaining model recorded by DornRemainingClassSpec; Assumption 3 and
Theorem 1 are recorded separately by DornSourceRateTranscription. Gaïffas,
Convergence Rates for Pointwise Curve Estimation with a Degenerate Design
(2005), arXiv:math/0410354, Eq. (2.1), Definition 2, Assumption M and
Theorem 1, is the scope of
GaiffasDegenerateDesignRate.
def realBernoulli
A Bernoulli law, extended definitionally to all real parameters. For the stated inputs and conditions, the realBernoulli object being defined.
A valid Bernoulli parameter gives a probability measure. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma realBernoulli_probability (p : ℝ) (hp : p ∈ Icc 0 1) :
IsProbabilityMeasure (realBernoulli p) := by
apply isProbabilityMeasure_iff.mpr
simp [realBernoulli]
rw [← ENNReal.ofReal_add (sub_nonneg.mpr hp.2) hp.1]
norm_num
def scaledBernoulli
The law of B times a Bernoulli draw. For the stated inputs and conditions, the scaledBernoulli object being defined.
Definition (Lean source)
Scaling a Bernoulli outcome preserves its total probability. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma scaledBernoulli_probability (B p : ℝ) (hp : p ∈ Icc 0 1) :
IsProbabilityMeasure (scaledBernoulli B p) := by
haveI := realBernoulli_probability p hp
apply isProbabilityMeasure_iff.mpr
change (Measure.map (fun b : Bool => if b then B else 0) (realBernoulli p))
univ = 1
rw [Measure.map_apply (by fun_prop) MeasurableSet.univ]
simpa using (measure_univ : (realBernoulli p) univ = 1)
theorem scaledBernoulli_supported_on_endpoints Lemma scaledBernoulli_supported_on_endpoints in the paper ↗
Every scaled Bernoulli outcome lies at one of its two endpoints. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma scaledBernoulli_supported_on_endpoints (B p : ℝ) :
∀ᵐ y ∂scaledBernoulli B p, y ∈ ({0, B} : Set ℝ) := by
unfold scaledBernoulli
apply (ae_map_iff
(by fun_prop : AEMeasurable (fun b : Bool => if b then B else 0)
(realBernoulli p))
(by measurability : MeasurableSet {y : ℝ | y ∈ ({0, B} : Set ℝ)})).2
filter_upwards [] with b
cases b <;> simp
def centreRadius
Cube-centre radial coordinate. For the stated inputs and conditions, the centreRadius object being defined.
Definition (Lean source)
def centreRadiusCDF
CDF of the centre radius on the unit cube. For the stated inputs and conditions, the centreRadiusCDF object being defined.
Definition (Lean source)
def thinStrip
The thin strip through the cube centre. For the stated inputs and conditions, the thinStrip object being defined.
def stripPropensity
Thin-strip propensity. For the stated inputs and conditions, the stripPropensity object being defined.
Definition (Lean source)
parameters
def
stripPropensity d hd γ a x :
ℝ
given by
by classical exact min 1 ((centreRadiusCDF d (centreRadius x)) ^ ((1 : ℝ) / (γ - 1)) + (1 / 4 : ℝ) * if x ∈ thinStrip d hd a then 1 else 0)
def completionAt
Kernel giving (A,Y0,Y1,Y) at fixed covariate x. For the stated inputs and conditions, the completionAt object being defined.
Definition (Lean source)
parameters
def
given by
((realBernoulli (e x)).prod ((scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x)))).map (fun z => (x, z.1, z.2.1, z.2.2, if z.1 then z.2.2 else z.2.1))
Valid treatment and outcome probabilities give a causal probability kernel. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
B :
ℝ
e p₀ p₁ :
(Fin d → ℝ) → ℝ
x :
Fin d → ℝ
he :
e x ∈ Icc 0 1
hp₀ :
p₀ x ∈ Icc 0 1
hp₁ :
p₁ x ∈ Icc 0 1
conclusion
IsProbabilityMeasure (completionAt B e p₀ p₁ x)
Proof (Lean source)
lemma completionAt_probability {d : ℕ} (B : ℝ)
(e p₀ p₁ : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ)
(he : e x ∈ Icc 0 1) (hp₀ : p₀ x ∈ Icc 0 1)
(hp₁ : p₁ x ∈ Icc 0 1) :
IsProbabilityMeasure (completionAt B e p₀ p₁ x) := by
haveI := realBernoulli_probability (e x) he
haveI := scaledBernoulli_probability B (p₀ x) hp₀
haveI := scaledBernoulli_probability B (p₁ x) hp₁
haveI : IsProbabilityMeasure
((realBernoulli (e x)).prod
((scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x)))) := by
infer_instance
apply isProbabilityMeasure_iff.mpr
change (((realBernoulli (e x)).prod
((scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x)))).map
(fun z => (x, z.1, z.2.1, z.2.2,
if z.1 then z.2.2 else z.2.1))) univ = 1
have hm : Measurable (fun z : Bool × ℝ × ℝ =>
(x, z.1, z.2.1, z.2.2, if z.1 then z.2.2 else z.2.1)) := by
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop
rw [Measure.map_apply hm MeasurableSet.univ]
simpa using (measure_univ :
((realBernoulli (e x)).prod
((scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x)))) univ = 1)
The fixed-covariate completion records the selected potential outcome. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
OutcomeConsistency (completionAt B e p₀ p₁ x)
Proof (Lean source)
lemma completionAt_consistency {d : ℕ} (B : ℝ)
(e p₀ p₁ : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ) :
OutcomeConsistency (completionAt B e p₀ p₁ x) := by
unfold OutcomeConsistency completionAt
have hm : Measurable (fun z : Bool × ℝ × ℝ =>
(x, z.1, z.2.1, z.2.2, if z.1 then z.2.2 else z.2.1)) := by
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop
have hsel : Measurable (fun ω : Completion d =>
if ω.2.1 then ω.2.2.2.1 else ω.2.2.1) := by
apply Measurable.ite (by measurability : MeasurableSet {ω : Completion d | ω.2.1 = true})
(by fun_prop) (by fun_prop)
have hs : MeasurableSet {ω : Completion d |
ω.2.2.2.2 = if ω.2.1 then ω.2.2.2.1 else ω.2.2.1} := by
exact measurableSet_eq_fun (by fun_prop) hsel
apply (ae_map_iff hm.aemeasurable hs).2
filter_upwards [] with z
cases z.1 <;> simp
The fixed-covariate completion retains its input covariate. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma completionAt_covariate_marginal {d : ℕ} (B : ℝ)
(e p₀ p₁ : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ)
(he : e x ∈ Icc 0 1) (hp₀ : p₀ x ∈ Icc 0 1)
(hp₁ : p₁ x ∈ Icc 0 1) :
(completionAt B e p₀ p₁ x).map fst = Measure.dirac x := by
haveI := realBernoulli_probability (e x) he
haveI := scaledBernoulli_probability B (p₀ x) hp₀
haveI := scaledBernoulli_probability B (p₁ x) hp₁
unfold completionAt
have hm : Measurable (fun z : Bool × ℝ × ℝ =>
(x, z.1, z.2.1, z.2.2, if z.1 then z.2.2 else z.2.1)) := by
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop
rw [Measure.map_map measurable_fst hm]
change Measure.map (fun _ : Bool × ℝ × ℝ => x)
((realBernoulli (e x)).prod
((scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x)))) = _
rw [Measure.map_const, measure_univ, one_smul]
Integrating valid fixed-covariate completions preserves the base law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma completionBind_covariate_marginal {d : ℕ} (μ : Measure (Fin d → ℝ))
(B : ℝ) (e p₀ p₁ : (Fin d → ℝ) → ℝ)
(hk : AEMeasurable (completionAt B e p₀ p₁) μ)
(hp : ∀ᵐ x ∂μ, e x ∈ Icc 0 1 ∧ p₀ x ∈ Icc 0 1 ∧
p₁ x ∈ Icc 0 1) :
(μ.bind (completionAt B e p₀ p₁)).map fst = μ := by
ext s hs
rw [Measure.map_apply measurable_fst hs, Measure.bind_apply (measurable_fst hs) hk]
simp_rw [← Measure.map_apply measurable_fst hs]
calc
_ = ∫⁻ x, (Measure.dirac x) s ∂μ := by
apply lintegral_congr_ae
filter_upwards [hp] with x hx
rw [completionAt_covariate_marginal B e p₀ p₁ x hx.1 hx.2.1 hx.2.2]
_ = μ s := by
simp only [Measure.dirac_apply' _ hs]
exact lintegral_indicator_one hs
Mixing fixed-covariate completions preserves outcome consistency. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
OutcomeConsistency (μ.bind (completionAt B e p₀ p₁))
Proof (Lean source)
lemma completionBind_consistency {d : ℕ} (μ : Measure (Fin d → ℝ))
(B : ℝ) (e p₀ p₁ : (Fin d → ℝ) → ℝ)
(hk : AEMeasurable (completionAt B e p₀ p₁) μ) :
OutcomeConsistency (μ.bind (completionAt B e p₀ p₁)) := by
let S : Set (Completion d) :=
{ω | ω.2.2.2.2 = if ω.2.1 then ω.2.2.2.1 else ω.2.2.1}
have hsel : Measurable (fun ω : Completion d =>
if ω.2.1 then ω.2.2.2.1 else ω.2.2.1) := by
apply Measurable.ite (by measurability : MeasurableSet {ω : Completion d | ω.2.1 = true})
(by fun_prop) (by fun_prop)
have hS : MeasurableSet S := by
exact measurableSet_eq_fun (by fun_prop) hsel
have hzero : (μ.bind (completionAt B e p₀ p₁)) Sᶜ = 0 := by
apply le_antisymm _ zero_le
calc
_ ≤ ∫⁻ x, (completionAt B e p₀ p₁ x) Sᶜ ∂μ :=
Measure.bind_apply_le _ hS.compl
_ = 0 := by
apply lintegral_eq_zero_of_ae_eq_zero
filter_upwards [] with x
exact (measure_eq_zero_iff_ae_notMem).2
(by simpa only [OutcomeConsistency, S, Set.mem_compl_iff, not_not,
Set.mem_ofPred_eq] using
completionAt_consistency B e p₀ p₁ x)
simpa only [OutcomeConsistency, S, Set.mem_compl_iff, not_not,
Set.mem_ofPred_eq] using (measure_eq_zero_iff_ae_notMem).mp hzero
theorem completionAt_supported_on_endpoints Lemma completionAt_supported_on_endpoints in the paper ↗
Both potential outcomes in a Bernoulli completion lie at its two endpoints. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma completionAt_supported_on_endpoints {d : ℕ} (B : ℝ)
(e p₀ p₁ : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ)
(he : e x ∈ Icc 0 1) (hp₀ : p₀ x ∈ Icc 0 1)
(hp₁ : p₁ x ∈ Icc 0 1) :
∀ᵐ ω ∂completionAt B e p₀ p₁ x,
ω.2.2.1 ∈ ({0, B} : Set ℝ) ∧
ω.2.2.2.1 ∈ ({0, B} : Set ℝ) := by
unfold completionAt
have hmap : Measurable (fun z : Bool × ℝ × ℝ =>
(x, z.1, z.2.1, z.2.2, if z.1 then z.2.2 else z.2.1)) := by
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop
haveI := realBernoulli_probability (e x) he
haveI := scaledBernoulli_probability B (p₀ x) hp₀
haveI := scaledBernoulli_probability B (p₁ x) hp₁
rw [ae_map_iff hmap.aemeasurable
(by measurability : MeasurableSet
{ω : Completion d | ω.2.2.1 ∈ ({0, B} : Set ℝ) ∧
ω.2.2.2.1 ∈ ({0, B} : Set ℝ)})]
apply (Measure.ae_prod_iff_ae_ae (by measurability)).2
filter_upwards [] with b
apply (Measure.ae_prod_iff_ae_ae (by measurability)).2
filter_upwards [scaledBernoulli_supported_on_endpoints B (p₀ x)] with y₀ hy₀
filter_upwards [scaledBernoulli_supported_on_endpoints B (p₁ x)] with y₁ hy₁
exact ⟨hy₀, hy₁⟩
theorem completionBind_supported_on_endpoints Lemma completionBind_supported_on_endpoints in the paper ↗
Mixing valid Bernoulli completions retains the endpoint support of both potential outcomes. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma completionBind_supported_on_endpoints {d : ℕ}
(μ : Measure (Fin d → ℝ)) (B : ℝ)
(e p₀ p₁ : (Fin d → ℝ) → ℝ)
(hp : ∀ᵐ x ∂μ, e x ∈ Icc 0 1 ∧ p₀ x ∈ Icc 0 1 ∧
p₁ x ∈ Icc 0 1) :
∀ᵐ ω ∂μ.bind (completionAt B e p₀ p₁),
ω.2.2.1 ∈ ({0, B} : Set ℝ) ∧
ω.2.2.2.1 ∈ ({0, B} : Set ℝ) := by
let S : Set (Completion d) :=
{ω | ω.2.2.1 ∈ ({0, B} : Set ℝ) ∧
ω.2.2.2.1 ∈ ({0, B} : Set ℝ)}
have hS : MeasurableSet S := by
dsimp [S]
measurability
have hzero : (μ.bind (completionAt B e p₀ p₁)) Sᶜ = 0 := by
apply le_antisymm _ zero_le
calc
_ ≤ ∫⁻ x, (completionAt B e p₀ p₁ x) Sᶜ ∂μ :=
Measure.bind_apply_le _ hS.compl
_ = 0 := by
apply lintegral_eq_zero_of_ae_eq_zero
filter_upwards [hp] with x hx
apply (measure_eq_zero_iff_ae_notMem).2
filter_upwards [completionAt_supported_on_endpoints B e p₀ p₁ x
hx.1 hx.2.1 hx.2.2] with ω hω
simpa only [S, Set.mem_compl_iff, not_not, Set.mem_ofPred_eq] using hω
filter_upwards [(measure_eq_zero_iff_ae_notMem).mp hzero] with ω hω
simpa only [S, Set.mem_compl_iff, not_not, Set.mem_ofPred_eq] using hω
def baselineSuccess
Common bounded Bernoulli response success probability. For the stated inputs and conditions, the baselineSuccess object being defined.
Definition (Lean source)
def thinStripCompletion
Causal completion of the thin-strip law. For the stated inputs and conditions, the thinStripCompletion object being defined.
Definition (Lean source)
parameters
d :
ℕ
hd :
2 ≤ d
γ L B a :
ℝ
def
given by
(volume.restrict (cube d)).bind (completionAt B (stripPropensity d hd γ a) (fun _ => baselineSuccess B L) (fun _ => baselineSuccess B L))
def thinStripLaw
Observed thin-strip law on its stated exponent domain. For the stated inputs and conditions, the thinStripLaw object being defined.
Definition (Lean source)
parameters
d :
ℕ
hd :
2 ≤ d
γ L B a :
ℝ
_ha :
1 < a ∧ a < 1 + d / (γ - 1)
def
given by
(thinStripCompletion d hd γ L B a).map observed
def radialCDF
Cube-truncated radial CDF around any point, including boundary points. For the stated inputs and conditions, the radialCDF object being defined.
def radialPropensity
Radial propensity for the Bernoulli testing experiment. For the stated inputs and conditions, the radialPropensity object being defined.
The radial treatment probability is a measurable covariate function. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma radialPropensity_measurable {d : ℕ} (x₀ : Fin d → ℝ) (γ : ℝ) :
Measurable (radialPropensity x₀ γ) := by
unfold radialPropensity radialCDF
fun_prop
def smoothBump
Explicit smooth product bump supported on the closed sup-norm unit box. For the stated inputs and conditions, the smoothBump object being defined.
def pairSuccess
Treated Bernoulli success parameter for one testing law. For the stated inputs and conditions, the pairSuccess object being defined.
Definition (Lean source)
parameters
def
pairSuccess j x₀ β B L a h x :
ℝ
given by
baselineSuccess B L + if j then a * h ^ β * smoothBump (fun i => (x i - x₀ i) / h) else 0
Both Bernoulli response probabilities are measurable, including at the boundary of the bump support. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Measurable (pairSuccess j x₀ β B L a h)
Proof (Lean source)
lemma pairSuccess_measurable {d : ℕ} (j : Bool) (x₀ : Fin d → ℝ)
(β B L a h : ℝ) : Measurable (pairSuccess j x₀ β B L a h) := by
have hfactor (i : Fin d) : Measurable (fun x : Fin d → ℝ =>
if |(x i - x₀ i) / h| < 1 then
exp (1 - 1 / (1 - ((x i - x₀ i) / h) ^ 2)) else 0) := by
apply Measurable.ite
· exact measurableSet_lt (by fun_prop) measurable_const
· fun_prop
· fun_prop
have hprod : Measurable (fun x : Fin d → ℝ =>
∏ i : Fin d, if |(x i - x₀ i) / h| < 1 then
exp (1 - 1 / (1 - ((x i - x₀ i) / h) ^ 2)) else 0) := by
fun_prop (disch := assumption)
cases j
· unfold pairSuccess
simp
· unfold pairSuccess smoothBump
simp only [↓reduceIte]
exact measurable_const.add (measurable_const.mul hprod)
def boundedLowerPairAdmissible
The positive pair constants for which every Bernoulli parameter is in [0,1] on the cube at every positive sample size. The measure construction below remains total even outside this probability-law domain. For the stated inputs and conditions, the boundedLowerPairAdmissible object being defined.
Definition (Lean source)
parameters
def
boundedLowerPairAdmissible d β B L γ a c x₀ :
Prop
given by
clause 1
0 < a
clause 2
0 < c
def lowerPairCompletion
Completion for one law in the bounded radial testing pair. For the stated inputs and conditions, the lowerPairCompletion object being defined.
Definition (Lean source)
parameters
def
given by
h :
c * oracleMesh d n β γ
(volume.restrict (cube d)).bind (completionAt B (radialPropensity x₀ γ) (fun _ => baselineSuccess B L) (pairSuccess j x₀ β B L a h))
def boundedLowerPair
Totalized observed bounded Bernoulli testing pair for every positive a,c. On boundedLowerPairAdmissible, all three Bernoulli parameters lie in [0,1] for every n ≥ 1 and every covariate in the cube. For the stated inputs and conditions, the boundedLowerPair object being defined.
Definition (Lean source)
parameters
given by
fun j n => (lowerPairCompletion d n β B L γ a c x₀ j).map observed
Descriptive, nonassertive adaptation proposal. It defines no estimator, comparison rule, envelope functional, existence claim, or risk theorem. the jointAdaptationHandle object being defined.
Definition (Lean source)
def
jointAdaptationHandle :
_root_.String
given by
"Descriptive, nonassertive proposal: sample splitting would compare equal-cell " ++ "fits across polynomial degrees and treatment-count-feasible meshes. The count " ++ "selector would control the overlap axis, and held-out Lepski inequalities " ++ "would control the smoothness axis. The target is one response estimator with " ++ "an adaptive-risk envelope over compact rectangles of beta and gamma. No " ++ "sample split, comparison rule, Lepski threshold, estimator, or envelope " ++ "functional is defined, and no estimator-existence or risk-bound claim is asserted."
Open question, recorded as a nonassertive text carrier. the jointAdaptationQuestion object being defined.
Definition (Lean source)
def
jointAdaptationQuestion :
_root_.String
given by
"Over compact rectangles in (1,infinity)^2 for (beta,gamma) in the " ++ "bounded-outcome global-tail class, can the two-axis sample-splitting and " ++ "held-out Lepski handle produce one estimator attaining oracle pointwise and " ++ "expected-supremum scales? What logarithmic penalty, if any, is minimax " ++ "unavoidable? Open: neither a multiscale comparison theorem across changing " ++ "polynomial degrees nor an adaptive lower experiment is available. Count " ++ "selection adapts to gamma without a logarithm when beta is known. " ++ "Gaiffas's narrower univariate experiment does not transfer its lower bound " ++ "to this bounded causal class. No answer, procedure, penalty value, or " ++ "estimator witness is asserted."
def RegularlyVaryingAtZero
Regular variation at zero from the right, on the positive-radius domain. For the stated inputs and conditions, the RegularlyVaryingAtZero object being defined.
Definition (Lean source)
parameters
r :
ℝ
g :
ℝ → ℝ
def
RegularlyVaryingAtZero r g :
Prop
def SlowlyVaryingAtZero
Slow variation at zero from the right is regular variation of index zero. For the stated inputs and conditions, the SlowlyVaryingAtZero object being defined.
Definition (Lean source)
def gaiffasObservedLaw
Gaussian random-design regression law with density on the real line. For the stated inputs and conditions, the gaiffasObservedLaw object being defined.
def gaiffasLocalClass
Local approximation class at the sample-size-dependent bandwidth. For the stated inputs and conditions, the gaiffasLocalClass object being defined.
Definition (Lean source)
parameters
x₀ s δ :
ℝ
ω :
ℝ → ℝ
f :
ℝ → ℝ
h :
ℝ
0 < h
h ≤ δ
def
gaiffasLocalClass x₀ s δ ω f :
Prop
given by
∃ coeff : ℕ → ℝ,
∀ x : ℝ,
|x - x₀| ≤ h → |f x - ∑ j ∈ range (Nat.floor s + 1), coeff j * (x - x₀) ^ j| ≤ ω h
def gaiffasSigma
Gaïffas's class Sigma_n = F_(h_n)(x₀,ω) ∩ U(α_n), on the measurable regression functions defining the Gaussian experiment. For the stated inputs and conditions, the gaiffasSigma object being defined.
Definition (Lean source)
parameters
x₀ s :
ℝ
ω :
ℝ → ℝ
hₙ αₙ :
ℕ → ℝ
n :
ℕ
f :
ℝ → ℝ
def
gaiffasSigma x₀ s ω hₙ αₙ n f :
Prop
def gaiffasPointwiseRisk
Rooted pointwise L^p minimax risk in the fixed-design Gaussian experiment. For the stated inputs and conditions, the gaiffasPointwiseRisk object being defined.
Definition (Lean source)
parameters
ρ :
ℝ → ℝ
σ p s x₀ :
ℝ
ω :
ℝ → ℝ
hₙ αₙ :
ℕ → ℝ
n :
ℕ
def
gaiffasPointwiseRisk ρ σ p s x₀ ω hₙ αₙ n :
ℝ≥0∞
given by
(⨅ (T : {T : (Fin n → ℝ × ℝ) → ℝ // Measurable T}), ⨆ (f : ℝ → ℝ) (_ : gaiffasSigma x₀ s ω hₙ αₙ n f), ∫⁻ z, ofReal (|T.1 z - f x₀| ^ p) ∂Measure.pi (fun _ : Fin n => gaiffasObservedLaw ρ σ f)) ^ (1 / p)
Gaïffas (2005), Convergence Rates for Pointwise Curve Estimation with a Degenerate Design, arXiv:math/0410354, Eq. (2.1), Definition 2, Assumption M and Theorem 1 (2.4)–(2.5), with the positive, divergent approximation radii used by Proposition 6 for the lower bound. Radial density continuity is required only at positive radii, so a singularity at the target point is allowed. This is the rooted L^p risk on the sample-size-dependent local approximation class in the univariate Gaussian experiment. Both the slowly varying rate and ω (hₙ n) have eventual two-sided risk bounds, with comparison constants depending only on s,b,p, not on σ. the GaiffasDegenerateDesignRate object being defined.
Definition (Lean source)
def
GaiffasDegenerateDesignRate :
Sort 0
given by
∀ (p s b : ℝ),
0 < p → 0 < s → -1 < b → ∃ c C : ℝ, 0 < c ∧ c
≤ C ∧
∀ (x₀ : ℝ) (ρ ν ω : ℝ → ℝ),
Measurable ρ → (∀ x : ℝ, 0 ≤ ρ x) → (∃ W : Set ℝ, IsOpen W ∧ x₀ ∈ W ∧ ∀ x ∈ W, ρ x = ν |x - x₀|) → RegularlyVaryingAtZero b ν → RegularlyVaryingAtZero s ω → (volume.withDensity (fun x => ofReal (ρ x))) univ
= 1 → ∃ ℓ : ℝ → ℝ, SlowlyVaryingAtZero ℓ ∧ ∀ (σ : ℝ) (hₙ αₙ : ℕ → ℝ), 0 < σ → (∀ n : ℕ, 1 ≤ n → 0 < hₙ n ∧ ω (hₙ n) = σ / sqrt (2 * n * ∫ t in (0 : ℝ)..hₙ n, ν t) ∧ ∀ h : ℝ, 0 < h → h < hₙ n → ω h ≠ σ / sqrt (2 * n * ∫ t in (0 : ℝ)..h, ν t)) → (∃ κ M : ℝ, 0 < κ ∧ 0 < M ∧ ∃ n₀ : ℕ, ∀ n ≥ n₀, αₙ n ≤ M * (n : ℝ) ^ κ) → (∀ n : ℕ, 0 < αₙ n) → Filter.Tendsto αₙ Filter.atTop Filter.atTop → ∃ n₀ : ℕ, ∀ n ≥ n₀, ofReal (c * σ ^ (2 * s / (1 + 2 * s + b)) * (n : ℝ) ^ (-s / (1 + 2 * s + b)) * ℓ ((n : ℝ)⁻¹)) ≤ gaiffasPointwiseRisk ρ σ p s x₀ ω hₙ αₙ n ∧ gaiffasPointwiseRisk ρ σ p s x₀ ω hₙ αₙ n ≤ ofReal (C * σ ^ (2 * s / (1 + 2 * s + b)) * (n : ℝ) ^ (-s / (1 + 2 * s + b)) * ℓ ((n : ℝ)⁻¹)) ∧ ofReal (c * ω (hₙ n)) ≤ gaiffasPointwiseRisk ρ σ p s x₀ ω hₙ αₙ n ∧ gaiffasPointwiseRisk ρ σ p s x₀ ω hₙ αₙ n ≤ ofReal (C * ω (hₙ n))
def armKernel
Conditional observed outcome distribution in arm a. For the stated inputs and conditions, the armKernel object being defined.
Definition (Lean source)
def DornTopDerivativeSeminorm
Dorn's Σ(β,L₀) controls only the highest coordinate derivatives on the closed source cube, using intrinsic traces. It imposes no bound on the function or its lower derivatives and selects no outside extension. For the stated inputs and conditions, the DornTopDerivativeSeminorm object being defined.
Definition (Lean source)
parameters
def
DornTopDerivativeSeminorm β L₀ g :
Prop
given by
clause 1
ContDiffOn ℝ (polynomialDegree β) g (dornCube d)
clause 2
α :
Fin d → ℕ
(∑ i, α i) = polynomialDegree β
(∀ i, (Finset.univ.filter (fun k => f k = i)).card = α i)
x ∈ dornCube d :
y ∈ dornCube d :
|coordJetOn (dornCube d) (∑ i, α i) g f x
- coordJetOn (dornCube d) (∑ i, α i) g f y|
≤ L₀ * (sqrt (∑ i, (x i - y i) ^ 2)) ^ (β - (polynomialDegree β : ℝ))
def DornA12SelectedPropensity
A measurable selected conditional propensity, strictly between zero and one almost surely. Assumptions 1–2 impose no pointwise range at null points. For the stated inputs and conditions, the DornA12SelectedPropensity object being defined.
Definition (Lean source)
parameters
def
DornA12SelectedPropensity P e :
Prop
given by
∃ hP : IsProbabilityMeasure P,
clause 1
letI : IsProbabilityMeasure P := hP Measurable e
clause 2
∀ᵐ x ∂covariateLaw P, e x = propensity P x ∧ 0 < e x ∧ e x < 1
theorem toA12
An A3 selected version also meets the almost-sure A1/A2 requirements. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma DornSelectedPropensity.toA12 {d : ℕ} {S : Set (Fin d → ℝ)}
{P : Measure (Obs d)} {e : (Fin d → ℝ) → ℝ}
(h : DornSelectedPropensity S P e) : DornA12SelectedPropensity P e := by
obtain ⟨hP, he, _, hae⟩ := h
exact ⟨hP, he, hae⟩
def DornRemainingModel
Dorn's Assumptions 1–2 without Assumption 3. Assumption 1(a)'s common conditional q-moment bound implies |μ₁| ≤ M; it is derived in the transfer, not imposed here. Assumption 1(b) is the variance bound, while the fixed Gaussian variance in Assumption 2 supplies Assumption 1(c). The Hölder condition is only Dorn's top-derivative seminorm, with no lower-derivative bound. The selected propensity is measurable and almost surely the conditional propensity in (0,1). For the stated inputs and conditions, the DornRemainingModel object being defined.
Definition (Lean source)
parameters
def
DornRemainingModel d β q M σ C L₀ γ P μ₁ μ₀ e :
Prop
given by
clause 2
clause 3
∀ᵐ x ∂covariateLaw P, Integrable (fun y : ℝ => |y| ^ q) (armKernel P x true) ∧ (∫ y, |y| ^ q ∂armKernel P x true)
≤ M ^ q
clause 4
(∫ x, (μ₁ x - ∫ y, μ₁ y ∂covariateLaw P) ^ 2 ∂covariateLaw P) ≤ M
clause 5
DornTopDerivativeSeminorm β L₀ μ₁
clause 6
DornTopDerivativeSeminorm β L₀ μ₀
clause 7
GlobalPropensityTail P e C γ
clause 8
= gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩ ∧
def DornRemainingClass
The exact remaining class in Dorn's Section 2 after Assumption 3 is deleted. Membership includes a probability law and every clause of Assumptions 1–2, with only the top-derivative Hölder seminorm. For the stated inputs and conditions, the DornRemainingClass object being defined.
Definition (Lean source)
parameters
d :
ℕ
β q M σ C L₀ γ :
ℝ
def
given by
{z | ∃ hP : IsProbabilityMeasure z.1, @DornRemainingModel d β q M σ C L₀ γ z.1 hP z.2.1 z.2.2.1 z.2.2.2}
def DornA12Family
For the dimension, regularity, moment, variance, tail constants, and candidate family, the Dorn Assumptions 1--2 family consists of laws satisfying nonemptiness and the per-law model conditions.
Definition (Lean source)
parameters
def
DornA12Family d β q M σ C L₀ γ 𝔓 :
Prop
given by
clause 1
𝔓.Nonempty
clause 2
z ∈ 𝔓 :
∃ hP : IsProbabilityMeasure z.1,
DornA12SelectedPropensity z.1 z.2.2.2
∀ᵐ x ∂covariateLaw z.1, Integrable (fun y : ℝ => |y| ^ q) (armKernel z.1 x true) ∧ (∫ y, |y| ^ q ∂armKernel z.1 x true)
≤ M ^ q
(∫ x, (z.2.1 x - ∫ y, z.2.1 y ∂covariateLaw z.1) ^ 2 ∂covariateLaw z.1) ≤ M
∀ᵐ x ∂covariateLaw z.1, ∀ a : Bool, σ ^ 2
≤ ∫ y, (y - ∫ y, y ∂armKernel z.1 x a) ^ 2 ∂armKernel z.1 x a
DornTopDerivativeSeminorm β L₀ z.2.1
DornTopDerivativeSeminorm β L₀ z.2.2.1
GlobalPropensityTail z.1 z.2.2.2 C γ
= gaussianReal (z.2.1 x) ⟨σ ^ 2, sq_nonneg σ⟩ ∧
Dorn (2026), Minimax Rates Under Global Overlap Bounds, Section 2 Setting and Notation, Assumptions 1–2, author PDF pp. 3–7. The concrete DornA12Family records the source's fixed-constant family assumptions. The equivalence says that such a family is precisely an arbitrary nonempty subfamily of the maximal remaining-class envelope; it does not require the family itself to equal that envelope. For the stated inputs and conditions, the DornRemainingClassSpec object being defined.
Definition (Lean source)
parameters
d :
ℕ
β q M σ C L₀ γ :
ℝ
def
DornRemainingClassSpec d β q M σ C L₀ γ :
Sort 0
given by
1
≤ d → 0 < β → 3 < q → 0 < M → 0 < σ → 0 < C → 0 < L₀ → 1 < γ → ∀ 𝔓 : Set (DornSourceLaw d), DornA12Family d β q M σ C L₀ γ 𝔓 ↔ 𝔓.Nonempty ∧ 𝔓 ⊆ DornRemainingClass d β q M σ C L₀ γ
def DornA3OnOriginalClass
Dorn's Assumption 3 with constants shared by the whole source family. For the stated inputs and conditions, the DornA3OnOriginalClass object being defined.
Definition (Lean source)
Dorn (2026), Minimax Rates Under Global Overlap Bounds, Section 2 Notation and Setting, Assumptions 1–3, Theorem 1(ii) (author PDF pp. 3–7, 12), Lemma 11 (p. 23), Lemma 13 (p. 24), and the proof (pp. 27–28). This source-scoped gate fixes one source family, its common constants including gamma, and its selected propensity representatives before selecting a measurable estimator and rate constants. Each displayed L∞(P_X) exceedance event is required to be measurable, so the upper probability comparison uses ordinary probability on the original Gaussian cube and the source's L∞(P_X) norm. The printed lower clause (i) is bibliographic attribution only: arbitrary singleton families obstruct it. The no-gamma sentence and count thresholds are construction provenance only. the DornSourceRateTranscription object being defined.
Definition (Lean source)
def
DornSourceRateTranscription :
Sort 0
given by
∀ (d : ℕ) (β q M σ C L₀ γ : ℝ) (𝔓 : Set (DornSourceLaw d)),
1
≤ d → 0 < β → 3 < q → 0 < M → 0 < σ → 0 < C → 0 < L₀ → 1 < γ → DornA12Family d β q M σ C L₀ γ 𝔓 → DornA3OnOriginalClass 𝔓 → ∃ T : ∀ n : ℕ, (Fin n → Obs d) → (Fin d → ℝ) → ℝ, (∀ (n : ℕ) (z : {z : DornSourceLaw d // z ∈ 𝔓}) (R : ℝ), MeasurableSet {ω | ofReal (R * (n : ℝ) ^ (-β / (2 * β + effectiveDimension d γ))) < eLpNorm (fun x => T n ω x - z.1.2.1 x) ⊤ (covariateLaw z.1.1)}) ∧ (∀ n x, Measurable (fun ω => T n ω x)) ∧ ∀ ε : ℝ, 0 < ε → ∃ R : ℝ, 0 < R ∧ limsup (fun n : ℕ => ⨆ (z : {z : DornSourceLaw d // z ∈ 𝔓}), (Measure.pi (fun _ : Fin n => z.1.1)).real {ω | ofReal (R * (n : ℝ) ^ (-β / (2 * β + effectiveDimension d γ))) < eLpNorm (fun x => T n ω x - z.1.2.1 x) ⊤ (covariateLaw z.1.1)}) atTop ≤ ε
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.TAdaptiveMinimax 15 declarations Logarithm-free adaptive weak-overlap minimax rate
Logarithm-free adaptive weak-overlap minimax rate
Expected supremum risk of the concrete estimator at one law. For the stated inputs and conditions, the estimatorSupRisk object being defined.
Definition (Lean source)
parameters
def
estimatorSupRisk d n β B L C c_f γ P hP :
ℝ≥0∞
given by
∫⁻ ω, ⨆ x ∈ cube d, ofReal |equalCellEstimator β B ω x
- responseOf P hP x| ∂Measure.pi (fun _ : Fin n => P)
def boundedOutcomeClass
The bounded-potential-outcome subclass used in the minimax converse. For the stated inputs and conditions, the boundedOutcomeClass object being defined.
Definition (Lean source)
parameters
d :
ℕ
β B L C c_f γ :
ℝ
given by
{P | ModelParameterDomain d β B L C c_f ∧ 1 < γ ∧ ∃ (Pc : Measure (Completion d)) (_ : IsProbabilityMeasure Pc) (μ₁ e : (Fin d → ℝ) → ℝ), P = Pc.map observed ∧ GlobalTailModel β B L C c_f γ Pc μ₁ e ∧ (∀ᵐ ω ∂Pc, ω.2.2.1 ∈ ({0, B} : Set ℝ) ∧ ω.2.2.2.1 ∈ ({0, B} : Set ℝ))}
theorem boundedOutcomeClass_mem_model
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
P ∈ ModelClass d β B L C c_f γ
Proof (Lean source)
lemma boundedOutcomeClass_mem_model {d : ℕ} {β B L C c_f γ : ℝ}
{P : Measure (Obs d)}
(hP : P ∈ boundedOutcomeClass d β B L C c_f γ) :
P ∈ ModelClass d β B L C c_f γ := by
obtain ⟨hdom, hγ, Pc, hPc, μ₁, e, hobs, hmodel, _⟩ := hP
exact ⟨hdom, hγ, Pc, hPc, μ₁, e, hobs, hmodel⟩
theorem lowerPair_mem_boundedOutcomeClass_of_exact Lemma lower-pair-mem-bounded-outcome-class-of-exact in the paper ↗
The exact completion payload exported by the lower-pair construction places either observed arm in the bounded-outcome subclass. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B L C c_f γ a c :
ℝ
x₀ :
Fin d → ℝ
hdom :
ModelParameterDomain d β B L C c_f
hγ :
1 < γ
ha :
0 < a
hc :
0 < c
hexact :
LowerPairExactModelWitness d n β B L C c_f γ a c x₀
hsupp :
j :
conclusion
boundedLowerPair d β B L γ a c x₀ ha hc j n ∈ boundedOutcomeClass d β B L C c_f γ
Proof (Lean source)
lemma lowerPair_mem_boundedOutcomeClass_of_exact
{d n : ℕ} {β B L C c_f γ a c : ℝ} {x₀ : Fin d → ℝ}
(hdom : ModelParameterDomain d β B L C c_f) (hγ : 1 < γ)
(ha : 0 < a) (hc : 0 < c)
(hexact : LowerPairExactModelWitness d n β B L C c_f γ a c x₀)
(hsupp : ∀ j : Bool,
∀ᵐ ω ∂lowerPairCompletion d n β B L γ a c x₀ j,
ω.2.2.1 ∈ ({0, B} : Set ℝ) ∧
ω.2.2.2.1 ∈ ({0, B} : Set ℝ))
(j : Bool) :
boundedLowerPair d β B L γ a c x₀ ha hc j n ∈
boundedOutcomeClass d β B L C c_f γ := by
unfold LowerPairExactModelWitness at hexact
rcases hexact with ⟨hF, hT, hmodelF, hmodelT⟩
cases j with
| false =>
refine ⟨hdom, hγ, lowerPairCompletion d n β B L γ a c x₀ false,
hF, (fun x => B * pairSuccess false x₀ β B L a
(c * oracleMesh d n β γ) x), radialPropensity x₀ γ, ?_,
hmodelF, hsupp false⟩
rfl
| true =>
refine ⟨hdom, hγ, lowerPairCompletion d n β B L γ a c x₀ true,
hT, (fun x => B * pairSuccess true x₀ β B L a
(c * oracleMesh d n β γ) x), radialPropensity x₀ γ, ?_,
hmodelT, hsupp true⟩
rfl
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ :
ℝ
P :
Measure (Obs d)
hP :
P ∈ ModelClass d β B L C c_f γ
x :
Fin d → ℝ
hx :
x ∈ cube d
conclusion
|responseOf P hP x| ≤ B
Proof (Lean source)
lemma responseOf_abs_le_on_cube {d : ℕ} {β B L C c_f γ : ℝ}
(P : Measure (Obs d)) (hP : P ∈ ModelClass d β B L C c_f γ)
(x : Fin d → ℝ) (hx : x ∈ cube d) : |responseOf P hP x| ≤ B := by
have hPcopy := hP
obtain ⟨hdom, hγ, Pc, hPc, μ, e, hEq, hmodel⟩ := hPcopy
letI : IsProbabilityMeasure Pc := hPc
rw [responseOf_eq_given_model_response P hP Pc μ e hEq hmodel x hx]
exact holderResponse_abs_le_on_cube Pc μ e hdom hγ hmodel x hx
The selected-cell estimator has one expected supremum-risk constant over the entire compact overlap range. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ_min γ_max :
ℝ
hd :
0 < d
hβ :
1 < β
hB :
0 < B
hL :
0 < L
hC :
1 ≤ C
hcf :
0 < c_f ∧ c_f ≤ 1
hγmin :
1 < γ_min
hγrange :
γ_min < γ_max
conclusion
∃ K : ℝ,
conclusion 1
0 < K
conclusion 2
∃ n₀ : ℕ,
∀ n ≥ n₀,
∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin,
hγrange⟩, ∀ (P : Measure (Obs d)) [IsProbabilityMeasure P] (hP : P ∈ ModelClass d β B L C c_f γ), estimatorSupRisk d n β B L C c_f γ P hP
≤ ofReal (K * oracleRate d n β γ)
Proof (Lean source)
lemma estimatorSupRisk_uniform_upper (d : ℕ)
(β B L C c_f γ_min γ_max : ℝ)
(hd : 0 < d) (hβ : 1 < β) (hB : 0 < B) (hL : 0 < L)
(hC : 1 ≤ C) (hcf : 0 < c_f ∧ c_f ≤ 1)
(hγmin : 1 < γ_min) (hγrange : γ_min < γ_max) :
∃ K : ℝ, 0 < K ∧ ∃ n₀ : ℕ,
∀ n ≥ n₀, ∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
∀ (P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ),
estimatorSupRisk d n β B L C c_f γ P hP ≤
ofReal (K * oracleRate d n β γ) := by
obtain ⟨K, hK, n₀, hrisk⟩ := selectedMesh_uniform_equalCellCubeRisk
d β B L C c_f γ_min γ_max (by omega) hβ hB hL hC hcf hγmin hγrange
refine ⟨K, hK, n₀, ?_⟩
intro n hn γ hγ P _ hP
have hPcopy := hP
obtain ⟨hparams, hγgt, Pc, hPc, μ, e, hEq, hmodel⟩ := hPcopy
letI : IsProbabilityMeasure Pc := hPc
have hb := hrisk n hn γ hγ Pc μ e hmodel P hEq
unfold estimatorSupRisk
calc
(∫⁻ ω, ⨆ x, ⨆ (_hx : x ∈ cube d),
ofReal |equalCellEstimator β B ω x - responseOf P hP x|
∂Measure.pi (fun _ : Fin n => P)) =
∫⁻ ω, ⨆ x, ⨆ (_hx : x ∈ cube d),
ofReal |equalCellEstimator β B ω x - μ x|
∂Measure.pi (fun _ : Fin n => P) := by
apply lintegral_congr
intro ω
apply iSup_congr
intro x
apply iSup_congr
intro hx
rw [responseOf_eq_given_model_response P hP Pc μ e hEq hmodel x hx]
_ ≤ ofReal (K * oracleRate d n β γ) := hb
theorem supremumRisk_le_of_estimatorSupRisk_bound
A uniform bound for the concrete selected-cell estimator bounds the corresponding supremum minimax risk. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
supremumRisk d n β B L C c_f γ ≤ R
Proof (Lean source)
lemma supremumRisk_le_of_estimatorSupRisk_bound {d n : ℕ}
{β B L C c_f γ : ℝ} {R : ℝ≥0∞}
(h : ∀ (P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ),
estimatorSupRisk d n β B L C c_f γ P hP ≤ R) :
supremumRisk d n β B L C c_f γ ≤ R := by
unfold supremumRisk Causalean.Stat.minimaxValueENNReal
Causalean.Stat.worstCaseRiskENNReal
let T : {T : (Fin n → Obs d) → (Fin d → ℝ) → ℝ // Measurable T} :=
⟨equalCellEstimator β B,
measurable_pi_lambda _ fun x => equalCellEstimator_fixedPoint_measurable β B x⟩
apply iInf_le_of_le T
refine iSup_le fun P => ?_
have hPcopy := P.2
obtain ⟨_, _, Pc, hPc, _, _, hEq, _⟩ := hPcopy
letI : IsProbabilityMeasure Pc := hPc
letI : IsProbabilityMeasure P.1 := by
rw [hEq]
exact Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)
simpa [supLossRisk, estimatorSupRisk, T] using h P.1 P.2
theorem supremumRisk_le_two_mul
Clipping the selected-cell estimator gives a sample-size-independent upper bound for the supremum minimax risk. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B L C c_f γ :
ℝ
hB :
0 < B
conclusion
supremumRisk d n β B L C c_f γ ≤ ofReal (2 * B)
Proof (Lean source)
lemma supremumRisk_le_two_mul (d n : ℕ) (β B L C c_f γ : ℝ)
(hB : 0 < B) :
supremumRisk d n β B L C c_f γ ≤ ofReal (2 * B) := by
apply supremumRisk_le_of_estimatorSupRisk_bound
intro P _ hP
unfold estimatorSupRisk
calc
(∫⁻ ω, ⨆ x, ⨆ (_hx : x ∈ cube d),
ofReal |equalCellEstimator β B ω x - responseOf P hP x|
∂Measure.pi (fun _ : Fin n => P)) ≤
∫⁻ _ω : Fin n → Obs d, ofReal (2 * B)
∂Measure.pi (fun _ : Fin n => P) := by
apply lintegral_mono
intro ω
refine iSup_le fun x => iSup_le fun hx => ?_
exact ENNReal.ofReal_le_ofReal
(equalCellEstimator_error_le_two_mul β B ω x (responseOf P hP x)
hB.le (responseOf_abs_le_on_cube P hP x hx))
_ = ofReal (2 * B) := by simp
def boundedPointwiseLossRisk
Pointwise loss on the bounded-potential-outcome subclass. For the stated inputs and conditions, the boundedPointwiseLossRisk object being defined.
Definition (Lean source)
parameters
d n :
ℕ
β B L C c_f γ :
ℝ
x₀ :
Fin d → ℝ
T :
{T : (Fin n → Obs d) → ℝ // Measurable T}
P :
{P : Measure (Obs d) // P ∈ boundedOutcomeClass d β B L C c_f γ}
def
boundedPointwiseLossRisk d n β B L C c_f γ x₀ T P :
ℝ≥0∞
given by
∫⁻ ω, ofReal |T.1 ω
Pointwise minimax risk restricted to bounded potential outcomes. For the stated inputs and conditions, the boundedPointwiseRisk object being defined.
Definition (Lean source)
parameters
def
boundedPointwiseRisk d n β B L C c_f γ x₀ :
ℝ≥0∞
given by
minimaxValueENNReal (boundedPointwiseLossRisk d n β B L C c_f γ x₀)
theorem boundedPointwiseRisk_le_pointwiseRisk Lemma boundedPointwiseRisk_le_pointwiseRisk in the paper ↗
Restricting the model class to bounded potential outcomes can only reduce the pointwise minimax risk. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
boundedPointwiseRisk d n β B L C c_f γ x₀ ≤ pointwiseRisk d n β B L C c_f γ x₀
Proof (Lean source)
lemma boundedPointwiseRisk_le_pointwiseRisk (d n : ℕ)
(β B L C c_f γ : ℝ) (x₀ : Fin d → ℝ) :
boundedPointwiseRisk d n β B L C c_f γ x₀ ≤
pointwiseRisk d n β B L C c_f γ x₀ := by
unfold boundedPointwiseRisk pointwiseRisk minimaxValueENNReal
worstCaseRiskENNReal
refine le_iInf fun T => ?_
apply iInf_le_of_le T
refine iSup_le fun P => ?_
simpa [boundedPointwiseLossRisk, pointwiseLossRisk] using
(le_iSup (fun P' : {P : Measure (Obs d) // P ∈ ModelClass d β B L C c_f γ} =>
pointwiseLossRisk d n β B L C c_f γ x₀ T P')
(⟨P.1, boundedOutcomeClass_mem_model P.2⟩ :
{P : Measure (Obs d) // P ∈ ModelClass d β B L C c_f γ}))
Evaluating a curve estimator at a point in the cube yields an admissible scalar estimator whose loss is bounded by its spatial supremum loss. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
pointwiseRisk d n β B L C c_f γ x₀ ≤ supremumRisk d n β B L C c_f γ
Proof (Lean source)
lemma pointwiseRisk_le_supremumRisk (d n : ℕ)
(β B L C c_f γ : ℝ) (x₀ : Fin d → ℝ) (hx₀ : x₀ ∈ cube d) :
pointwiseRisk d n β B L C c_f γ x₀ ≤
supremumRisk d n β B L C c_f γ := by
unfold pointwiseRisk supremumRisk minimaxValueENNReal
worstCaseRiskENNReal
refine le_iInf fun T => ?_
let T₀ : {T : (Fin n → Obs d) → ℝ // Measurable T} :=
⟨fun ω => T.1 ω x₀, T.2.eval⟩
apply iInf_le_of_le T₀
refine iSup_le fun P => ?_
calc
pointwiseLossRisk d n β B L C c_f γ x₀ T₀ P ≤
supLossRisk d n β B L C c_f γ T P := by
unfold pointwiseLossRisk supLossRisk
apply lintegral_mono
intro ω
exact le_iSup_of_le x₀ (le_iSup_of_le hx₀ le_rfl)
_ ≤ ⨆ P', supLossRisk d n β B L C c_f γ T P' := le_iSup _ P
theorem boundedPointwiseRisk_lower_of_pair Lemma bounded-pointwise-risk-lower-of-pair in the paper ↗
A pair in the bounded-outcome class with the standard KL budget gives an ENNReal pointwise minimax lower bound. Clipping arbitrary estimators makes the finite-L¹ Le Cam lemma applicable without an integrability assumption. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B L C c_f γ δ :
ℝ
x₀ :
Fin d → ℝ
hB :
0 < B
P₀ P₁ :
hP₀ :
P₀ ∈ boundedOutcomeClass d β B L C c_f γ
hP₁ :
P₁ ∈ boundedOutcomeClass d β B L C c_f γ
hx₀ :
x₀ ∈ cube d
hac :
P₀ ≪ P₁
hllr :
Integrable (llr P₀ P₁) P₀
hδ :
0 ≤ δ
hsep :
δ
≤ |responseOf P₁ (boundedOutcomeClass_mem_model hP₁) x₀
- responseOf P₀ (boundedOutcomeClass_mem_model hP₀) x₀|
conclusion
ofReal ((1 / 8 : ℝ) * δ) ≤ boundedPointwiseRisk d n β B L C c_f γ x₀
Proof (Lean source)
lemma boundedPointwiseRisk_lower_of_pair {d n : ℕ}
{β B L C c_f γ δ : ℝ} (x₀ : Fin d → ℝ) (hB : 0 < B)
(P₀ P₁ : Measure (Obs d)) [IsProbabilityMeasure P₀]
[IsProbabilityMeasure P₁]
(hP₀ : P₀ ∈ boundedOutcomeClass d β B L C c_f γ)
(hP₁ : P₁ ∈ boundedOutcomeClass d β B L C c_f γ)
(hx₀ : x₀ ∈ cube d) (hac : P₀ ≪ P₁)
(hllr : Integrable (llr P₀ P₁) P₀)
(hkl : (n : ℝ) * (klDiv P₀ P₁).toReal ≤ 1 / 8)
(hδ : 0 ≤ δ)
(hsep : δ ≤ |responseOf P₁ (boundedOutcomeClass_mem_model hP₁) x₀ -
responseOf P₀ (boundedOutcomeClass_mem_model hP₀) x₀|) :
ofReal ((1 / 8 : ℝ) * δ) ≤
boundedPointwiseRisk d n β B L C c_f γ x₀ := by
let θ₀ : {P : Measure (Obs d) //
P ∈ boundedOutcomeClass d β B L C c_f γ} := ⟨P₀, hP₀⟩
let θ₁ : {P : Measure (Obs d) //
P ∈ boundedOutcomeClass d β B L C c_f γ} := ⟨P₁, hP₁⟩
apply le_minimaxValueENNReal_of_two_point θ₀ θ₁
intro T
let t₀ := responseOf P₀ (boundedOutcomeClass_mem_model hP₀) x₀
let t₁ := responseOf P₁ (boundedOutcomeClass_mem_model hP₁) x₀
have ht₀ : |t₀| ≤ B := responseOf_abs_le_on_cube P₀
(boundedOutcomeClass_mem_model hP₀) x₀ hx₀
have ht₁ : |t₁| ≤ B := responseOf_abs_le_on_cube P₁
(boundedOutcomeClass_mem_model hP₁) x₀ hx₀
let clip : ℝ → ℝ := fun z =>
(projIcc (-B) B (by linarith : -B ≤ B) z : ℝ)
let Tc : (Fin n → Obs d) → ℝ := fun ω => clip (T.1 ω)
have hTc : Measurable Tc := by
change Measurable (fun ω => max (-B) (min B (T.1 ω)))
exact measurable_const.max (measurable_const.min T.2)
have hclip_t₀ : clip t₀ = t₀ := by
rw [show clip t₀ = max (-B) (min B t₀) by rfl,
min_eq_right (abs_le.mp ht₀).2, max_eq_right (abs_le.mp ht₀).1]
have hclip_t₁ : clip t₁ = t₁ := by
rw [show clip t₁ = max (-B) (min B t₁) by rfl,
min_eq_right (abs_le.mp ht₁).2, max_eq_right (abs_le.mp ht₁).1]
have hcontract₀ : ∀ ω, |Tc ω - t₀| ≤ |T.1 ω - t₀| := by
intro ω
have hc := Set.abs_projIcc_sub_projIcc
(a := -B) (b := B) (c := T.1 ω) (d := t₀) (by linarith : -B ≤ B)
simpa [Tc, clip, hclip_t₀] using hc
have hcontract₁ : ∀ ω, |Tc ω - t₁| ≤ |T.1 ω - t₁| := by
intro ω
have hc := Set.abs_projIcc_sub_projIcc
(a := -B) (b := B) (c := T.1 ω) (d := t₁) (by linarith : -B ≤ B)
simpa [Tc, clip, hclip_t₁] using hc
have hint₀ : Integrable (fun ω : Fin n → Obs d => |Tc ω - t₀|)
(Measure.pi (fun _ : Fin n => P₀)) := by
apply Integrable.of_bound (hTc.sub measurable_const).abs.aestronglyMeasurable (2 * B)
apply ae_of_all
intro ω
rw [Real.norm_eq_abs, abs_of_nonneg (abs_nonneg _)]
have hc := (projIcc (-B) B (by linarith : -B ≤ B) (T.1 ω)).property
dsimp [Tc, clip]
rw [abs_le]
constructor <;> nlinarith [hc.1, hc.2, (abs_le.mp ht₀).1, (abs_le.mp ht₀).2]
have hint₁ : Integrable (fun ω : Fin n → Obs d => |Tc ω - t₁|)
(Measure.pi (fun _ : Fin n => P₁)) := by
apply Integrable.of_bound (hTc.sub measurable_const).abs.aestronglyMeasurable (2 * B)
apply ae_of_all
intro ω
rw [Real.norm_eq_abs, abs_of_nonneg (abs_nonneg _)]
have hc := (projIcc (-B) B (by linarith : -B ≤ B) (T.1 ω)).property
dsimp [Tc, clip]
rw [abs_le]
constructor <;> nlinarith [hc.1, hc.2, (abs_le.mp ht₁).1, (abs_le.mp ht₁).2]
have hreal := leCam_two_point_L1_lower
(1 / 8 : ℝ) (by norm_num) n P₀ P₁ t₀ t₁ δ hac hllr hkl hδ
(by simpa [t₀, t₁, abs_sub_comm] using hsep) Tc hTc hint₀ hint₁
have hlin₀ := ofReal_integral_eq_lintegral_ofReal hint₀
(ae_of_all _ fun _ => abs_nonneg _)
have hlin₁ := ofReal_integral_eq_lintegral_ofReal hint₁
(ae_of_all _ fun _ => abs_nonneg _)
have hclip : ofReal ((1 / 8 : ℝ) * δ) ≤
max (∫⁻ ω, ofReal |Tc ω - t₀| ∂Measure.pi (fun _ : Fin n => P₀))
(∫⁻ ω, ofReal |Tc ω - t₁| ∂Measure.pi (fun _ : Fin n => P₁)) := by
rw [← hlin₀, ← hlin₁, ← ENNReal.ofReal_max]
exact ENNReal.ofReal_le_ofReal hreal
have hmono₀ :
(∫⁻ ω, ofReal |Tc ω - t₀| ∂Measure.pi (fun _ : Fin n => P₀)) ≤
boundedPointwiseLossRisk d n β B L C c_f γ x₀ T θ₀ := by
unfold boundedPointwiseLossRisk
simpa [θ₀, t₀] using
(lintegral_mono fun ω => ENNReal.ofReal_le_ofReal (hcontract₀ ω))
have hmono₁ :
(∫⁻ ω, ofReal |Tc ω - t₁| ∂Measure.pi (fun _ : Fin n => P₁)) ≤
boundedPointwiseLossRisk d n β B L C c_f γ x₀ T θ₁ := by
unfold boundedPointwiseLossRisk
simpa [θ₁, t₁] using
(lintegral_mono fun ω => ENNReal.ofReal_le_ofReal (hcontract₁ ω))
exact hclip.trans (max_le_max hmono₀ hmono₁)
The explicit bounded Bernoulli pair gives the oracle-rate pointwise converse at every positive sample size. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ :
ℝ
x₀ :
Fin d → ℝ
hd :
0 < d
hβ :
1 < β
hγ :
1 < γ
hB :
0 < B
hL :
0 < L
hC :
1 ≤ C
hcf :
0 < c_f ∧ c_f ≤ 1
hx₀ :
x₀ ∈ cube d
conclusion
∃ cγ : ℝ,
0 < cγ
conclusion 1
n :
ℕ
1 ≤ n
Proof (Lean source)
lemma boundedPointwiseRisk_oracle_lower (d : ℕ) (β B L C c_f γ : ℝ)
(x₀ : Fin d → ℝ) (hd : 0 < d) (hβ : 1 < β) (hγ : 1 < γ)
(hB : 0 < B) (hL : 0 < L) (hC : 1 ≤ C)
(hcf : 0 < c_f ∧ c_f ≤ 1) (hx₀ : x₀ ∈ cube d) :
∃ cγ : ℝ, 0 < cγ ∧ ∀ n : ℕ, 1 ≤ n →
ofReal (cγ * oracleRate d n β γ) ≤
boundedPointwiseRisk d n β B L C c_f γ x₀ := by
obtain ⟨a, c, c₀, ha, hc, hc₀, hadm, hall⟩ :=
boundedLowerPair_spec_exact_completion d β B L C c_f γ x₀
hd hβ hγ hB hL hC hcf hx₀
refine ⟨c₀ / 8, by positivity, ?_⟩
intro n hn
let P₀ := boundedLowerPair d β B L γ a c x₀ ha hc false n
let P₁ := boundedLowerPair d β B L γ a c x₀ ha hc true n
obtain ⟨hP₀model, hP₁model, hsupp, hac, hllr, hkl, hsep, hexact⟩ := hall n hn
have hdom : ModelParameterDomain d β B L C c_f :=
⟨Nat.succ_le_iff.mpr hd, hβ, hB, hL, hC, hcf⟩
have hP₀bounded : P₀ ∈ boundedOutcomeClass d β B L C c_f γ := by
exact lowerPair_mem_boundedOutcomeClass_of_exact hdom hγ ha hc hexact hsupp false
have hP₁bounded : P₁ ∈ boundedOutcomeClass d β B L C c_f γ := by
exact lowerPair_mem_boundedOutcomeClass_of_exact hdom hγ ha hc hexact hsupp true
have hexact' := hexact
unfold LowerPairExactModelWitness at hexact'
obtain ⟨hPc₀, hPc₁, -, -⟩ := hexact'
letI : IsProbabilityMeasure P₀ := by
dsimp [P₀, boundedLowerPair]
exact Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)
letI : IsProbabilityMeasure P₁ := by
dsimp [P₁, boundedLowerPair]
exact Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)
have hsep' : c₀ * oracleRate d n β γ ≤
|responseOf P₁ (boundedOutcomeClass_mem_model hP₁bounded) x₀ -
responseOf P₀ (boundedOutcomeClass_mem_model hP₀bounded) x₀| := by
simpa [P₀, P₁] using hsep
have hlower := boundedPointwiseRisk_lower_of_pair x₀ hB P₀ P₁
hP₀bounded hP₁bounded hx₀ hac hllr hkl
(mul_nonneg hc₀.le (by unfold oracleRate oracleMesh; positivity)) hsep'
convert hlower using 1
congr 1
ring
theorem adaptive_minimax_rate
One count-selected estimator, whose type takes n,d,β,B and the data but no overlap exponent or class constants, achieves logarithm-free expected supremum risk uniformly on a compact overlap range. A bounded Bernoulli pair provides the pointwise minimax converse. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β B L C c_f γ_min γ_max :
ℝ
hd :
0 < d
hβ :
1 < β
hB :
0 < B
hL :
0 < L
hC :
1 ≤ C
hcf :
0 < c_f ∧ c_f ≤ 1
hγmin :
1 < γ_min
hγrange :
γ_min < γ_max
conclusion
conclusion 1
∃ K : ℝ,
0 < K
∃ n₀ : ℕ,
∀ n ≥ n₀,
∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin,
hγrange⟩, ∀ (P : Measure (Obs d)) [IsProbabilityMeasure P] (hP : P ∈ ModelClass d β B L C c_f γ), estimatorSupRisk d n β B L C c_f γ P hP
≤ ofReal (K * oracleRate d n β γ)
conclusion 2
γ :
ℝ
1 < γ
x₀ ∈ cube d :
∃ cγ Kγ : ℝ,
0 < cγ
cγ ≤ Kγ
n :
ℕ
1 ≤ n
boundedPointwiseRisk d n β B L C c_f γ x₀ ≤ pointwiseRisk d n β B L C c_f γ x₀
pointwiseRisk d n β B L C c_f γ x₀ ≤ supremumRisk d n β B L C c_f γ
Proof (Lean source)
theorem adaptive_minimax_rate (d : ℕ) (β B L C c_f γ_min γ_max : ℝ)
(hd : 0 < d) (hβ : 1 < β) (hB : 0 < B) (hL : 0 < L)
(hC : 1 ≤ C) (hcf : 0 < c_f ∧ c_f ≤ 1)
(hγmin : 1 < γ_min) (hγrange : γ_min < γ_max) :
(∃ K : ℝ, 0 < K ∧ ∃ n₀ : ℕ,
∀ n ≥ n₀, ∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
∀ (P : Measure (Obs d)) [IsProbabilityMeasure P]
(hP : P ∈ ModelClass d β B L C c_f γ),
estimatorSupRisk d n β B L C c_f γ P hP ≤
ofReal (K * oracleRate d n β γ)) ∧
(∀ γ : ℝ, 1 < γ → ∀ x₀ ∈ cube d,
∃ cγ Kγ : ℝ, 0 < cγ ∧ cγ ≤ Kγ ∧
∀ n : ℕ, 1 ≤ n →
ofReal (cγ * oracleRate d n β γ) ≤
boundedPointwiseRisk d n β B L C c_f γ x₀ ∧
boundedPointwiseRisk d n β B L C c_f γ x₀ ≤
pointwiseRisk d n β B L C c_f γ x₀ ∧
pointwiseRisk d n β B L C c_f γ x₀ ≤
supremumRisk d n β B L C c_f γ ∧
supremumRisk d n β B L C c_f γ ≤
ofReal (Kγ * oracleRate d n β γ)) := by
constructor
· exact estimatorSupRisk_uniform_upper d β B L C c_f γ_min γ_max
hd hβ hB hL hC hcf hγmin hγrange
· intro γ hγ x₀ hx₀
obtain ⟨cγ, hcγ, hlower⟩ := boundedPointwiseRisk_oracle_lower
d β B L C c_f γ x₀ hd hβ hγ hB hL hC hcf hx₀
let γlo : ℝ := (γ + 1) / 2
let γhi : ℝ := γ + 1
have hγlo : 1 < γlo := by dsimp [γlo]; linarith
have hγrange' : γlo < γhi := by dsimp [γlo, γhi]; linarith
obtain ⟨K, hK, n₀, hupper⟩ := estimatorSupRisk_uniform_upper
d β B L C c_f γlo γhi hd hβ hB hL hC hcf hγlo hγrange'
let N : ℕ := max n₀ 1
let q : ℝ := (N : ℝ) ^ (-2 : ℝ)
have hN : 1 ≤ N := by exact le_max_right _ _
have hq : 0 < q := by dsimp [q]; positivity
let Ksmall : ℝ := (2 * B) / q
have hKsmall : 0 < Ksmall := by dsimp [Ksmall]; positivity
let Kγ : ℝ := max cγ (max K Ksmall)
refine ⟨cγ, Kγ, hcγ, le_max_left _ _, ?_⟩
intro n hn
have hrate : 0 < oracleRate d n β γ := by
unfold oracleRate oracleMesh
positivity
have hminimaxUpper : supremumRisk d n β B L C c_f γ ≤
ofReal (Kγ * oracleRate d n β γ) := by
by_cases hnlarge : n₀ ≤ n
· have hγmem : γ ∈ adaptationRange γlo γhi ⟨hγlo, hγrange'⟩ := by
change γ ∈ Icc γlo γhi
constructor <;> dsimp [γlo, γhi] <;> linarith
have hest := supremumRisk_le_of_estimatorSupRisk_bound
(hupper n hnlarge γ hγmem)
exact hest.trans (ENNReal.ofReal_le_ofReal <|
mul_le_mul_of_nonneg_right
((le_max_left K Ksmall).trans (le_max_right cγ (max K Ksmall)))
hrate.le)
· have hnN : n ≤ N := by
have hnn₀ : n < n₀ := Nat.lt_of_not_ge hnlarge
exact hnn₀.le.trans (le_max_left _ _)
have hq_le_inv : q ≤ (n : ℝ) ^ (-2 : ℝ) := by
apply Real.rpow_le_rpow_of_nonpos
· exact_mod_cast (show 0 < n from Nat.zero_lt_of_lt hn)
· exact_mod_cast hnN
· norm_num
have hq_le_rate : q ≤ oracleRate d n β γ :=
hq_le_inv.trans (inverseSquare_le_oracleRate d n β γ hn
(by linarith) hγ)
have htwo : 2 * B ≤ Ksmall * oracleRate d n β γ := by
calc
2 * B = Ksmall * q := by
dsimp [Ksmall]
field_simp
_ ≤ Ksmall * oracleRate d n β γ :=
mul_le_mul_of_nonneg_left hq_le_rate hKsmall.le
have hsmall_le : Ksmall ≤ Kγ :=
(le_max_right K Ksmall).trans (le_max_right cγ (max K Ksmall))
exact (supremumRisk_le_two_mul d n β B L C c_f γ hB).trans
(ENNReal.ofReal_le_ofReal <|
htwo.trans (mul_le_mul_of_nonneg_right hsmall_le hrate.le))
exact ⟨hlower n hn,
boundedPointwiseRisk_le_pointwiseRisk d n β B L C c_f γ x₀,
pointwiseRisk_le_supremumRisk d n β B L C c_f γ x₀ hx₀,
hminimaxUpper⟩
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.TDornA3FreeCorollary 103 declarations Assumption-3-free upper bound on Dorn's remaining Gaussian model
Assumption-3-free upper bound on Dorn's remaining Gaussian model
def affineToCube
Coordinatewise affine transformation from [-1,1]^d to [0,1]^d. For the stated inputs and conditions, the affineToCube object being defined.
def transformedDornLaw
Push an observed law from Dorn's cube to the unit cube. For the stated inputs and conditions, the transformedDornLaw object being defined.
Definition (Lean source)
def transformedResponse
Corresponding transformed treated-response curve. For the stated inputs and conditions, the transformedResponse object being defined.
The coordinate change used to transport Dorn's source cube is measurable. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma affineToCube_measurable {d : ℕ} :
Measurable (@affineToCube d) := by
unfold affineToCube
fun_prop
The displayed inverse coordinate change cancels affineToCube globally. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma affineToCube_inverse {d : ℕ} (x : Fin d → ℝ) :
(fun i => 2 * affineToCube x i - 1) = x := by
funext i
simp [affineToCube]
ring
The inverse coordinate change from the unit cube to Dorn's cube is measurable on the whole ambient space. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma affineFromCube_measurable {d : ℕ} :
Measurable (fun x : Fin d → ℝ => fun i => 2 * x i - 1) := by
fun_prop
The inverse coordinate change is also a right inverse of affineToCube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma affineToCube_affineFromCube {d : ℕ} (x : Fin d → ℝ) :
affineToCube (fun i => 2 * x i - 1) = x := by
funext i
simp [affineToCube]
An invertible measurable affine change of covariates generates exactly the same conditioning sigma algebra. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
= comap X inferInstance
Proof (Lean source)
lemma affineToCube_comap_eq {d : ℕ} {Omega : Type*}
(X : Omega → (Fin d → ℝ)) :
comap (affineToCube ∘ X) inferInstance =
comap X inferInstance := by
apply le_antisymm
· exact MeasurableSpace.comap_le_comap_of_eq_comp affineToCube
affineToCube_measurable rfl
· apply MeasurableSpace.comap_le_comap_of_eq_comp
(fun x : Fin d → ℝ => fun i => 2 * x i - 1)
affineFromCube_measurable
funext omega
exact (affineToCube_inverse (X omega)).symm
The global affine coordinate change as a measurable equivalence. For the stated inputs and conditions, the affineToCubeMeasurableEquiv object being defined.
Definition (Lean source)
parameters
d :
ℕ
given by
clause 1
toFun := affineToCube
clause 2
invFun := fun x i => 2 * x i - 1
clause 3
left_inv := affineToCube_inverse
clause 4
right_inv := affineToCube_affineFromCube
clause 5
measurable_toFun := affineToCube_measurable
clause 6
measurable_invFun := affineFromCube_measurable
Conditional independence is insensitive to replacing the conditioning measurable space by an equal one. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
Omega B C :
hm :
m₁ = m₂
h₁ :
m₁ ≤ mOmega
h₂ :
m₂ ≤ mOmega
f :
Omega → B
g :
Omega → C
mu :
@Measure Omega mOmega := by volume_tac
h :
CondIndepFun m₁ h₁ f g (μ := mu)
conclusion
CondIndepFun m₂ h₂ f g (μ := mu)
Proof (Lean source)
lemma condIndepFun_congr_conditioning
{Omega B C : Type*} [mOmega : MeasurableSpace Omega]
[StandardBorelSpace Omega] [MeasurableSpace B] [MeasurableSpace C]
{m₁ m₂ : MeasurableSpace Omega} (hm : m₁ = m₂)
(h₁ : m₁ ≤ mOmega) (h₂ : m₂ ≤ mOmega)
(f : Omega → B) (g : Omega → C)
(mu : @Measure Omega mOmega := by volume_tac)
[IsFiniteMeasure mu]
(h : CondIndepFun m₁ h₁ f g (μ := mu)) :
CondIndepFun m₂ h₂ f g (μ := mu) := by
subst m₂
exact h
Conditional expectation is unchanged when its conditioning measurable space is replaced by an equal one. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma condExp_congr_conditioning
{Omega E : Type*} [mOmega : MeasurableSpace Omega]
[NormedAddCommGroup E] [NormedSpace ℝ E]
{m₁ m₂ : MeasurableSpace Omega} (hm : m₁ = m₂)
(mu : @Measure Omega mOmega) (f : Omega → E) :
@condExp Omega E m₁ mOmega _ _ mu f =ᵐ[mu]
@condExp Omega E m₂ mOmega _ _ mu f := by
subst m₂
rfl
Pulling a response to the unit cube and evaluating at the transported point recovers the original response exactly. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
transformedResponse μ (affineToCube x) = μ x
Proof (Lean source)
lemma transformedResponse_affineToCube {d : ℕ}
(μ : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ) :
transformedResponse μ (affineToCube x) = μ x := by
simp only [transformedResponse, affineToCube_inverse]
The affine coordinate change identifies Dorn's source cube with the unit cube used by the causal model. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma affineToCube_mem_cube_iff {d : ℕ} (x : Fin d → ℝ) :
affineToCube x ∈ cube d ↔ x ∈ dornCube d := by
unfold cube dornCube affineToCube
constructor
· intro hx i hi
have hxi := hx i hi
constructor <;> linarith [hxi.1, hxi.2]
· intro hx i hi
have hxi := hx i hi
constructor <;> linarith [hxi.1, hxi.2]
The covariate marginal of the transported observed law is the pushforward of the original marginal by the affine coordinate change. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma transformedDornLaw_covariateLaw {d : ℕ} (P : Measure (Obs d)) :
covariateLaw (transformedDornLaw P) =
(covariateLaw P).map affineToCube := by
have htransport : Measurable
(fun z : Obs d => (affineToCube z.1, z.2.1, z.2.2)) := by
unfold affineToCube
fun_prop
unfold transformedDornLaw covariateLaw
rw [Measure.map_map (by fun_prop) htransport]
rw [Measure.map_map affineToCube_measurable (by fun_prop)]
rfl
The coordinatewise affine map has Jacobian factor 2⁻ᵈ and hence its pushforward multiplies ambient volume by 2ᵈ. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
conclusion
Proof (Lean source)
lemma affineToCube_map_volume (d : ℕ) :
(volume : Measure (Fin d → ℝ)).map affineToCube =
ofReal ((2 : ℝ) ^ d) • volume := by
let f : (Fin d → ℝ) →ₗ[ℝ] (Fin d → ℝ) :=
(2 : ℝ)⁻¹ • LinearMap.id
have hdet : det f ≠ 0 := by
simp [f, LinearMap.det_smul, LinearMap.det_id]
have hlin := Real.map_linearMap_volume_pi_eq_smul_volume_pi hdet
have hfactor : ofReal |(det f)⁻¹| =
ofReal ((2 : ℝ) ^ d) := by
congr 1
simp [f, LinearMap.det_smul, LinearMap.det_id, abs_of_nonneg]
have hlin' : (volume : Measure (Fin d → ℝ)).map f =
ofReal ((2 : ℝ) ^ d) • volume := by
rw [hfactor] at hlin
exact hlin
let b : Fin d → ℝ := fun _ => 2⁻¹
calc
(volume : Measure (Fin d → ℝ)).map affineToCube =
((volume : Measure (Fin d → ℝ)).map f).map (fun x => x + b) := by
rw [Measure.map_map (by fun_prop) (by fun_prop)]
congr 1
funext x i
simp [affineToCube, f, b]
ring
_ = (ofReal ((2 : ℝ) ^ d) • volume).map (fun x => x + b) := by
rw [hlin']
_ = ofReal ((2 : ℝ) ^ d) • volume := by
rw [Measure.map_smul, map_add_right_eq_self]
theorem normalizedDornVolume_map_affineToCube Lemma normalized-dorn-volume-affine-pushforward in the paper ↗
Normalized volume on Dorn's cube pushes forward exactly to unit-cube volume under the coordinatewise affine map. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
conclusion
= volume.restrict (cube d)
Proof (Lean source)
lemma normalizedDornVolume_map_affineToCube (d : ℕ) :
(((ofReal ((2 : ℝ) ^ d))⁻¹ •
volume.restrict (dornCube d)).map affineToCube) =
volume.restrict (cube d) := by
have hpre : affineToCube ⁻¹' cube d = dornCube d := by
ext x
exact affineToCube_mem_cube_iff x
have hcube : MeasurableSet (cube d) := by
unfold cube
exact MeasurableSet.univ_pi fun _ => measurableSet_Icc
have hrestricted : (volume.restrict (dornCube d)).map affineToCube =
ofReal ((2 : ℝ) ^ d) • volume.restrict (cube d) := by
rw [← hpre, ← Measure.restrict_map affineToCube_measurable hcube,
affineToCube_map_volume, Measure.restrict_smul]
rw [Measure.map_smul, hrestricted, smul_smul]
rw [ENNReal.inv_mul_cancel]
· simp
· positivity
· simp
theorem transformedDornLaw_covariate_eq_cubeVolume Lemma transformed-dorn-law-covariate-volume in the paper ↗
A Dorn law with normalized uniform covariates transports to exact uniform volume on the unit cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
covariateLaw (transformedDornLaw P) = volume.restrict (cube d)
Proof (Lean source)
lemma transformedDornLaw_covariate_eq_cubeVolume {d : ℕ}
(P : Measure (Obs d))
(hcov : covariateLaw P =
(ofReal ((2 : ℝ) ^ d))⁻¹ • volume.restrict (dornCube d)) :
covariateLaw (transformedDornLaw P) = volume.restrict (cube d) := by
rw [transformedDornLaw_covariateLaw, hcov]
exact normalizedDornVolume_map_affineToCube d
Exact transported uniform covariates supply cube support, absolute continuity, and density lower bound one. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
covariateLaw (transformedDornLaw P) (cube d) = 1
conclusion 2
covariateLaw (transformedDornLaw P) ≪ volume.restrict (cube d)
conclusion 3
Proof (Lean source)
lemma transformedDornLaw_cube_fields {d : ℕ}
(P : Measure (Obs d))
(hcov : covariateLaw P =
(ofReal ((2 : ℝ) ^ d))⁻¹ • volume.restrict (dornCube d)) :
covariateLaw (transformedDornLaw P) (cube d) = 1 ∧
covariateLaw (transformedDornLaw P) ≪ volume.restrict (cube d) ∧
CovariateDensityLowerBound (transformedDornLaw P) 1 := by
have htarget := transformedDornLaw_covariate_eq_cubeVolume P hcov
constructor
· rw [htarget, Measure.restrict_apply
(by simp [cube] : MeasurableSet (cube d)), Set.inter_self]
have hcube : cube d =
Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hcube, Real.volume_Icc_pi]
simp
constructor
· rw [htarget]
· unfold CovariateDensityLowerBound covariateDensity
rw [htarget]
filter_upwards [Measure.rnDeriv_self (volume.restrict (cube d))] with x hx
simp [hx]
theorem transformedDornLaw_treatmentKernel_ae Lemma transformed-dorn-treatment-cond-distrib in the paper ↗
Affine transport reindexes the conditional treatment kernel by the covariate equivalence. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
P :
conclusion
∀ᵐ x ∂covariateLaw P, condDistrib (fun z : Obs d => z.2.1) (fun z => z.1) (transformedDornLaw P) (affineToCube x)
= condDistrib (fun z : Obs d => z.2.1) (fun z => z.1) P x
Proof (Lean source)
lemma transformedDornLaw_treatmentKernel_ae {d : ℕ}
(P : Measure (Obs d)) [IsFiniteMeasure P]
[IsFiniteMeasure (transformedDornLaw P)] :
∀ᵐ x ∂covariateLaw P,
condDistrib (fun z : Obs d => z.2.1)
(fun z => z.1) (transformedDornLaw P) (affineToCube x) =
condDistrib (fun z : Obs d => z.2.1)
(fun z => z.1) P x := by
let G : Obs d → Obs d := fun z => (affineToCube z.1, z.2.1, z.2.2)
have hG : Measurable G := by
dsimp [G]
have hx : Measurable (fun z : Obs d => affineToCube z.1) :=
affineToCube_measurable.comp measurable_fst
fun_prop (disch := assumption)
have hmap := ProbabilityTheory.condDistrib_map
(X := fun z : Obs d => z.1) (Y := fun z : Obs d => z.2.1)
(f := G) (ν := P) (by fun_prop) (by fun_prop) hG.aemeasurable
dsimp [G] at hmap
have hfilter : P.map
((fun z : Obs d => z.1) ∘ fun z => (affineToCube z.1, z.2.1, z.2.2)) =
(covariateLaw P).map affineToCube := by
calc
_ = P.map (affineToCube ∘ (fun z : Obs d => z.1)) := by rfl
_ = (covariateLaw P).map affineToCube := by
unfold covariateLaw
exact (Measure.map_map affineToCube_measurable (by fun_prop)).symm
rw [hfilter] at hmap
have hmap' := ae_of_ae_map affineToCube_measurable.aemeasurable hmap
have hreparam :=
condDistrib_comp_right_measurableEquiv
P (affineToCubeMeasurableEquiv d)
(Y := fun z : Obs d => z.2.1) (X := fun z : Obs d => z.1)
(by fun_prop) (by fun_prop)
filter_upwards [hmap', hreparam] with x hx hr
unfold transformedDornLaw
simpa only [Function.comp_apply] using hx.trans hr
theorem transformedDornLaw_outcomeKernel_ae Lemma transformed-dorn-outcome-cond-distrib in the paper ↗
Affine transport reindexes both coordinates of the conditional outcome kernel and leaves its response law unchanged. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
P :
conclusion
∀ᵐ xa ∂P.map (fun z : Obs d => (z.1, z.2.1)), condDistrib (fun z : Obs d => z.2.2) (fun z => (z.1, z.2.1)) (transformedDornLaw P) (affineToCube xa.1, xa.2)
= condDistrib (fun z : Obs d => z.2.2) (fun z => (z.1, z.2.1)) P xa
Proof (Lean source)
lemma transformedDornLaw_outcomeKernel_ae {d : ℕ}
(P : Measure (Obs d)) [IsFiniteMeasure P]
[IsFiniteMeasure (transformedDornLaw P)] :
∀ᵐ xa ∂P.map (fun z : Obs d => (z.1, z.2.1)),
condDistrib (fun z : Obs d => z.2.2)
(fun z => (z.1, z.2.1)) (transformedDornLaw P)
(affineToCube xa.1, xa.2) =
condDistrib (fun z : Obs d => z.2.2)
(fun z => (z.1, z.2.1)) P xa := by
let G : Obs d → Obs d := fun z => (affineToCube z.1, z.2.1, z.2.2)
let E : ((Fin d → ℝ) × Bool) ≃ᵐ ((Fin d → ℝ) × Bool) :=
(affineToCubeMeasurableEquiv d).prodCongr (MeasurableEquiv.refl Bool)
have hG : Measurable G := by
dsimp [G]
have hx : Measurable (fun z : Obs d => affineToCube z.1) :=
affineToCube_measurable.comp measurable_fst
fun_prop (disch := assumption)
have hmap := ProbabilityTheory.condDistrib_map
(X := fun z : Obs d => (z.1, z.2.1)) (Y := fun z : Obs d => z.2.2)
(f := G) (ν := P) (by fun_prop) (by fun_prop) hG.aemeasurable
dsimp [G, E] at hmap
have hfilter : P.map
((fun z : Obs d => (z.1, z.2.1)) ∘
fun z => (affineToCube z.1, z.2.1, z.2.2)) =
(P.map (fun z : Obs d => (z.1, z.2.1))).map E := by
calc
_ = P.map (E ∘ (fun z : Obs d => (z.1, z.2.1))) := by
rfl
_ = (P.map (fun z : Obs d => (z.1, z.2.1))).map E := by
exact (Measure.map_map E.measurable (by fun_prop)).symm
rw [hfilter] at hmap
have hmap' := ae_of_ae_map E.measurable.aemeasurable hmap
have hreparam :=
condDistrib_comp_right_measurableEquiv
P E (Y := fun z : Obs d => z.2.2)
(X := fun z : Obs d => (z.1, z.2.1)) (by fun_prop) (by fun_prop)
filter_upwards [hmap', hreparam] with xa hx hr
unfold transformedDornLaw
have hE : E xa = (affineToCube xa.1, xa.2) := rfl
rw [hE] at hx
simpa only [Function.comp_apply] using hx.trans hr
The global propensity tail is invariant under the affine covariate equivalence when the displayed propensity is pulled back by its inverse. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma transformedDornLaw_globalTail {d : ℕ}
(P : Measure (Obs d)) [IsFiniteMeasure P]
[IsFiniteMeasure (transformedDornLaw P)]
(e : (Fin d → ℝ) → ℝ) (C gamma : ℝ)
(he : Measurable e) (htail : GlobalPropensityTail P e C gamma) :
GlobalPropensityTail (transformedDornLaw P) (transformedResponse e) C gamma := by
let r : (Fin d → ℝ) → (Fin d → ℝ) := affineToCube
let s : (Fin d → ℝ) → (Fin d → ℝ) := fun x i => 2 * x i - 1
have hr : Measurable r := affineToCube_measurable
have hs : Measurable s := affineFromCube_measurable
have hcov : covariateLaw (transformedDornLaw P) =
(covariateLaw P).map r := transformedDornLaw_covariateLaw P
have hmapS : (covariateLaw (transformedDornLaw P)).map s =
covariateLaw P := by
rw [hcov, Measure.map_map hs hr]
have hsr : s ∘ r = id := by
funext x
exact affineToCube_inverse x
rw [hsr, Measure.map_id]
constructor
· have hk := transformedDornLaw_treatmentKernel_ae P
have hpull : (fun y => transformedResponse e y) ∘ r =ᵐ[covariateLaw P]
(fun y => propensity (transformedDornLaw P) y) ∘ r := by
filter_upwards [htail.1, hk] with x heq hkx
change transformedResponse e (affineToCube x) =
propensity (transformedDornLaw P) (affineToCube x)
rw [transformedResponse_affineToCube]
rw [heq]
unfold propensity
rw [hkx]
exact eventuallyEq_of_comp_aeRetraction
r s hs hmapS (Filter.Eventually.of_forall fun x => by
change affineToCube (fun i => 2 * x i - 1) = x
exact affineToCube_affineFromCube x) hpull
· intro t ht
rw [hcov]
have hset : MeasurableSet {x | transformedResponse e x ≤ t} :=
measurableSet_le (he.comp hs) measurable_const
rw [Measure.real, Measure.map_apply hr hset]
have hpre : r ⁻¹' {x | transformedResponse e x ≤ t} = {x | e x ≤ t} := by
ext x
simp only [Set.mem_preimage, Set.mem_setOf_eq, r,
transformedResponse_affineToCube]
rw [hpre]
exact htail.2 t ht
Every observed law factors into its covariate marginal and the regular conditional treatment kernel. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
P :
Measure (Obs d)
conclusion
covariateLaw P ⊗ₘ condDistrib (fun z : Obs d => z.2.1) (fun z => z.1) P
= P.map (fun z : Obs d => (z.1, z.2.1))
Proof (Lean source)
lemma observedLaw_treatment_compProd {d : ℕ} (P : Measure (Obs d))
[IsFiniteMeasure P] :
covariateLaw P ⊗ₘ
condDistrib (fun z : Obs d => z.2.1)
(fun z => z.1) P =
P.map (fun z : Obs d => (z.1, z.2.1)) := by
simpa only [covariateLaw] using
(ProbabilityTheory.compProd_map_condDistrib
(μ := P) (X := fun z : Obs d => z.1)
(Y := fun z : Obs d => z.2.1) (by fun_prop))
Disintegrating an observed law over (X,A) and mapping the resulting product back through tuple reassociation recovers the original law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
P :
Measure (Obs d)
conclusion
((P.map (fun z : Obs d => (z.1, z.2.1))) ⊗ₘ condDistrib (fun z : Obs d => z.2.2) (fun z => (z.1, z.2.1)) P).map (fun z => (z.1.1, z.1.2, z.2))
= P
Proof (Lean source)
lemma observedLaw_outcome_recomposition {d : ℕ} (P : Measure (Obs d))
[IsFiniteMeasure P] :
((P.map (fun z : Obs d => (z.1, z.2.1))) ⊗ₘ
condDistrib (fun z : Obs d => z.2.2)
(fun z => (z.1, z.2.1)) P).map
(fun z => (z.1.1, z.1.2, z.2)) = P := by
rw [ProbabilityTheory.compProd_map_condDistrib (by fun_prop)]
rw [Measure.map_map (by fun_prop) (by fun_prop)]
have hfun :
(fun z : ((Fin d → ℝ) × Bool) × ℝ => (z.1.1, z.1.2, z.2)) ∘
(fun z : Obs d => ((z.1, z.2.1), z.2.2)) = id := by
funext z
rfl
rw [hfun, Measure.map_id]
Any almost-everywhere versions of the treatment and response kernels may be substituted into the two-stage observed-law disintegration. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
P :
Measure (Obs d)
hA :
=ᵐ[covariateLaw P] fun x => kA x
hY :
=ᵐ[ P.map (fun z : Obs d => (z.1, z.2.1))] fun xa => kY xa
conclusion
(((covariateLaw P ⊗ₘ kA) ⊗ₘ kY).map (fun z => (z.1.1, z.1.2, z.2))) = P
Proof (Lean source)
lemma observedLaw_kernel_recomposition {d : ℕ} (P : Measure (Obs d))
[IsFiniteMeasure P]
(kA : Kernel (Fin d → ℝ) Bool)
(kY : Kernel ((Fin d → ℝ) × Bool) ℝ)
[IsSFiniteKernel kA]
[IsSFiniteKernel kY]
(hA : (condDistrib (fun z : Obs d => z.2.1)
(fun z => z.1) P : (Fin d → ℝ) → Measure Bool) =ᵐ[covariateLaw P]
fun x => kA x)
(hY : (condDistrib (fun z : Obs d => z.2.2)
(fun z => (z.1, z.2.1)) P :
((Fin d → ℝ) × Bool) → Measure ℝ) =ᵐ[
P.map (fun z : Obs d => (z.1, z.2.1))] fun xa => kY xa) :
(((covariateLaw P ⊗ₘ kA) ⊗ₘ kY).map
(fun z => (z.1.1, z.1.2, z.2))) = P := by
have hbase := Measure.compProd_congr hA
have hbase' : covariateLaw P ⊗ₘ kA =
P.map (fun z : Obs d => (z.1, z.2.1)) := by
rw [← hbase]
exact observedLaw_treatment_compProd P
have hout := Measure.compProd_congr hY
calc
((covariateLaw P ⊗ₘ kA) ⊗ₘ kY).map
(fun z => (z.1.1, z.1.2, z.2)) =
((P.map (fun z : Obs d => (z.1, z.2.1))) ⊗ₘ kY).map
(fun z => (z.1.1, z.1.2, z.2)) := by rw [hbase']
_ = ((P.map (fun z : Obs d => (z.1, z.2.1))) ⊗ₘ
condDistrib (fun z : Obs d => z.2.2)
(fun z => (z.1, z.2.1)) P).map
(fun z => (z.1.1, z.1.2, z.2)) := by rw [hout]
_ = P := observedLaw_outcome_recomposition P
A probability law on Bool is determined by the real value of its mass at true. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
ν = realBernoulli p
Proof (Lean source)
lemma boolProbability_eq_realBernoulli (ν : Measure Bool)
[IsProbabilityMeasure ν] (p : ℝ) (hp : p ∈ Icc (0 : ℝ) 1)
(htrue : (ν {true}).toReal = p) :
ν = realBernoulli p := by
letI : IsProbabilityMeasure (realBernoulli p) :=
realBernoulli_probability p hp
have hνtrue : ν {true} = ofReal p := by
rw [← htrue, ENNReal.ofReal_toReal]
exact measure_ne_top ν {true}
apply Measure.ext
intro s hs
by_cases ht : true ∈ s
· by_cases hf : false ∈ s
· have hsuniv : s = univ := by
ext b
cases b <;> simp_all
subst s
simp only [measure_univ]
· have hsone : s = {true} := by
ext b
cases b <;> simp_all
subst s
simp [realBernoulli, hνtrue]
· by_cases hf : false ∈ s
· have hsone : s = {false} := by
ext b
cases b <;> simp_all
subst s
have hcomp : ({false} : Set Bool) = {true}ᶜ := by
ext b
cases b <;> simp
have hνfin : ν {true} ≠ ∞ := measure_ne_top ν {true}
have hrfin : realBernoulli p {true} ≠ ∞ :=
measure_ne_top (realBernoulli p) {true}
rw [hcomp, measure_compl (MeasurableSet.singleton true) hνfin,
measure_compl (MeasurableSet.singleton true) hrfin,
measure_univ, measure_univ, hνtrue]
simp [realBernoulli]
· have hsempty : s = ∅ := by
ext b
cases b <;> simp_all
subst s
simp [realBernoulli]
theorem condDistrib_treatment_eq_realBernoulli_of_selected Lemma selected-propensity-identifies-treatment-kernel in the paper ↗
A selected propensity in (0,1) identifies the entire Bool-valued conditional treatment law, rather than only its mass at true. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
P :
Measure (Obs d)
e :
(Fin d → ℝ) → ℝ
hselected :
∀ᵐ x ∂covariateLaw P, e x = propensity P x ∧ 0 < e x ∧ e x < 1
conclusion
=ᵐ[covariateLaw P] fun x => realBernoulli (e x)
Proof (Lean source)
lemma condDistrib_treatment_eq_realBernoulli_of_selected {d : ℕ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(e : (Fin d → ℝ) → ℝ)
(hselected : ∀ᵐ x ∂covariateLaw P,
e x = propensity P x ∧ 0 < e x ∧ e x < 1) :
(condDistrib (fun z : Obs d => z.2.1)
(fun z => z.1) P : (Fin d → ℝ) → Measure Bool) =ᵐ[covariateLaw P]
fun x => realBernoulli (e x) := by
filter_upwards [hselected] with x hx
let ν : Measure Bool := condDistrib
(fun z : Obs d => z.2.1) (fun z => z.1) P x
letI : IsProbabilityMeasure ν := by
dsimp [ν]
infer_instance
apply boolProbability_eq_realBernoulli ν (e x)
· exact ⟨hx.2.1.le, hx.2.2.le⟩
· change (condDistrib (fun z : Obs d => z.2.1)
(fun z => z.1) P x {true}).toReal = e x
exact hx.1.symm
Gaussian arm-kernel identities holding for almost every covariate lift to the joint (X,A) law used by observed-law disintegration. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
P :
Measure (Obs d)
σ :
ℝ
μ₀ μ₁ :
(Fin d → ℝ) → ℝ
harms :
= gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩ ∧
conclusion
=ᵐ[ P.map (fun z : Obs d => (z.1, z.2.1))] fun xa => if xa.2 then gaussianReal (μ₁ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩ else gaussianReal (μ₀ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩
Proof (Lean source)
lemma gaussianArmKernels_joint_ae {d : ℕ} (P : Measure (Obs d))
[IsProbabilityMeasure P] (σ : ℝ)
(μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(harms : ∀ᵐ x ∂covariateLaw P,
armKernel P x true =
gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩ ∧
armKernel P x false =
gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩) :
(condDistrib (fun z : Obs d => z.2.2)
(fun z => (z.1, z.2.1)) P :
((Fin d → ℝ) × Bool) → Measure ℝ) =ᵐ[
P.map (fun z : Obs d => (z.1, z.2.1))]
fun xa => if xa.2 then
gaussianReal (μ₁ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩ := by
have hmap : (P.map (fun z : Obs d => (z.1, z.2.1))).map fst =
covariateLaw P := by
unfold covariateLaw
rw [Measure.map_map (by fun_prop) (by fun_prop)]
rfl
rw [← hmap] at harms
have harms' := ae_of_ae_map (μ := P.map (fun z : Obs d => (z.1, z.2.1)))
measurable_fst.aemeasurable harms
filter_upwards [harms'] with xa hx
rcases xa with ⟨x, a⟩
cases a with
| false =>
change armKernel P x false =
gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩
exact hx.2
| true =>
change armKernel P x true =
gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩
exact hx.1
theorem transformedDornLaw_treatedKernel_gaussian_ae Lemma transformed-dorn-treated-gaussian-kernel in the paper ↗
A positive selected propensity lets the joint affine outcome-kernel transport be specialized to the treated Gaussian arm. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
P :
sigma :
ℝ
e mu₀ mu₁ :
(Fin d → ℝ) → ℝ
hselected :
∀ᵐ x ∂covariateLaw P, e x = propensity P x ∧ 0 < e x ∧ e x < 1
harms :
= gaussianReal (mu₁ x) ⟨sigma ^ 2, sq_nonneg sigma⟩ ∧
conclusion
Proof (Lean source)
lemma transformedDornLaw_treatedKernel_gaussian_ae {d : ℕ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
[IsFiniteMeasure (transformedDornLaw P)] (sigma : ℝ)
(e mu₀ mu₁ : (Fin d → ℝ) → ℝ)
(hselected : ∀ᵐ x ∂covariateLaw P,
e x = propensity P x ∧ 0 < e x ∧ e x < 1)
(harms : ∀ᵐ x ∂covariateLaw P,
armKernel P x true =
gaussianReal (mu₁ x) ⟨sigma ^ 2, sq_nonneg sigma⟩ ∧
armKernel P x false =
gaussianReal (mu₀ x) ⟨sigma ^ 2, sq_nonneg sigma⟩) :
∀ᵐ y ∂covariateLaw (transformedDornLaw P),
treatedKernel (transformedDornLaw P) y =
gaussianReal (transformedResponse mu₁ y)
⟨sigma ^ 2, sq_nonneg sigma⟩ := by
letI : IsProbabilityMeasure (covariateLaw P) := by
unfold covariateLaw
exact Measure.isProbabilityMeasure_map (by fun_prop)
have htransport := transformedDornLaw_outcomeKernel_ae P
rw [← observedLaw_treatment_compProd P] at htransport
have hsections := Measure.ae_ae_of_ae_compProd htransport
have hA := condDistrib_treatment_eq_realBernoulli_of_selected P e hselected
have hsource : ∀ᵐ x ∂covariateLaw P,
treatedKernel (transformedDornLaw P) (affineToCube x) =
gaussianReal (mu₁ x) ⟨sigma ^ 2, sq_nonneg sigma⟩ := by
filter_upwards [hsections, hA, hselected, harms] with x hx hAx hsel harmsx
rw [hAx] at hx
have htrue := realBernoulli_ae_at_true_of_pos (e x) hsel.2.1
(fun a => condDistrib (fun z : Obs d => z.2.2)
(fun z => (z.1, z.2.1)) (transformedDornLaw P)
(affineToCube x, a) =
condDistrib (fun z : Obs d => z.2.2)
(fun z => (z.1, z.2.1)) P (x, a)) hx
exact htrue.trans harmsx.1
let r : (Fin d → ℝ) → (Fin d → ℝ) := affineToCube
let s : (Fin d → ℝ) → (Fin d → ℝ) := fun x i => 2 * x i - 1
have hs : Measurable s := affineFromCube_measurable
have hcov : covariateLaw (transformedDornLaw P) =
(covariateLaw P).map r := transformedDornLaw_covariateLaw P
have hmapS : (covariateLaw (transformedDornLaw P)).map s =
covariateLaw P := by
rw [hcov, Measure.map_map hs affineToCube_measurable]
have hsr : s ∘ r = id := by
funext x
exact affineToCube_inverse x
rw [hsr, Measure.map_id]
apply eventuallyEq_of_comp_aeRetraction
r s hs hmapS (Filter.Eventually.of_forall fun x => by
change affineToCube (fun i => 2 * x i - 1) = x
exact affineToCube_affineFromCube x)
filter_upwards [hsource] with x hx
change treatedKernel (transformedDornLaw P) (affineToCube x) =
gaussianReal (transformedResponse mu₁ (affineToCube x))
⟨sigma ^ 2, sq_nonneg sigma⟩
simpa only [transformedResponse_affineToCube] using hx
A Gaussian with standard deviation bounded by B satisfies the model's centered MGF envelope with scale B. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
m sigma B t :
ℝ
hB :
0 ≤ B
hsigma :
|sigma| ≤ B
Proof (Lean source)
lemma gaussianReal_centered_mgf_bound (m sigma B t : ℝ)
(hB : 0 ≤ B) (hsigma : |sigma| ≤ B) :
Integrable (fun y : ℝ => exp (t * (y - m)))
(gaussianReal m ⟨sigma ^ 2, sq_nonneg sigma⟩) ∧
mgf (fun y : ℝ => y - m)
(gaussianReal m ⟨sigma ^ 2, sq_nonneg sigma⟩) t ≤
exp (B ^ 2 * t ^ 2 / 2) := by
let v : NNReal := ⟨sigma ^ 2, sq_nonneg sigma⟩
have hmap : (gaussianReal m v).map (fun y : ℝ => y - m) =
gaussianReal 0 v := by
simpa using ProbabilityTheory.gaussianReal_map_sub_const (μ := m) (v := v) m
constructor
· have hi := ProbabilityTheory.integrable_exp_mul_gaussianReal
(μ := (0 : ℝ)) (v := v) t
rw [← hmap] at hi
have hi' := (integrable_map_measure (by fun_prop)
(by fun_prop : AEMeasurable (fun y : ℝ => y - m)
(gaussianReal m v))).mp hi
change Integrable ((fun x : ℝ => exp (t * x)) ∘
fun y : ℝ => y - m) (gaussianReal m v)
exact hi'
· have hmgf : mgf (fun y : ℝ => y - m)
(gaussianReal m v) t = exp (sigma ^ 2 * t ^ 2 / 2) := by
rw [← ProbabilityTheory.mgf_id_map (by fun_prop), hmap]
rw [show sigma ^ 2 = (v : ℝ) by rfl]
simpa only [zero_mul, zero_add] using congrFun
(ProbabilityTheory.mgf_id_gaussianReal (μ := (0 : ℝ)) (v := v)) t
rw [hmgf]
apply Real.exp_le_exp.mpr
have hsquare : sigma ^ 2 ≤ B ^ 2 := by
apply (sq_le_sq).2
rw [abs_of_nonneg hB]
exact hsigma
nlinarith [sq_nonneg t]
theorem transformedDornLaw_boundedGaussianOutcome Lemma transformed-dorn-bounded-gaussian-outcome in the paper ↗
Transported Gaussian treated kernels give the bounded-mean and sub-Gaussian residual field once their means and standard deviation are bounded by the model scale. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
P :
sigma B :
ℝ
e mu₀ mu₁ :
(Fin d → ℝ) → ℝ
hB :
0 ≤ B
hsigma :
|sigma| ≤ B
hselected :
∀ᵐ x ∂covariateLaw P, e x = propensity P x ∧ 0 < e x ∧ e x < 1
harms :
= gaussianReal (mu₁ x) ⟨sigma ^ 2, sq_nonneg sigma⟩ ∧
hmean :
∀ᵐ y ∂covariateLaw (transformedDornLaw P), |transformedResponse mu₁ y| ≤ B
conclusion
Proof (Lean source)
lemma transformedDornLaw_boundedGaussianOutcome {d : ℕ}
(P : Measure (Obs d)) [IsProbabilityMeasure P]
[IsFiniteMeasure (transformedDornLaw P)] (sigma B : ℝ)
(e mu₀ mu₁ : (Fin d → ℝ) → ℝ)
(hB : 0 ≤ B) (hsigma : |sigma| ≤ B)
(hselected : ∀ᵐ x ∂covariateLaw P,
e x = propensity P x ∧ 0 < e x ∧ e x < 1)
(harms : ∀ᵐ x ∂covariateLaw P,
armKernel P x true =
gaussianReal (mu₁ x) ⟨sigma ^ 2, sq_nonneg sigma⟩ ∧
armKernel P x false =
gaussianReal (mu₀ x) ⟨sigma ^ 2, sq_nonneg sigma⟩)
(hmean : ∀ᵐ y ∂covariateLaw (transformedDornLaw P),
|transformedResponse mu₁ y| ≤ B) :
BoundedMeanSubGaussianResidual (transformedDornLaw P) B := by
have hk := transformedDornLaw_treatedKernel_gaussian_ae
P sigma e mu₀ mu₁ hselected harms
unfold BoundedMeanSubGaussianResidual
filter_upwards [hk, hmean] with y hky hmy
have hreg : treatedRegression (transformedDornLaw P) y =
transformedResponse mu₁ y := by
unfold treatedRegression
rw [hky]
exact ProbabilityTheory.integral_id_gaussianReal
constructor
· rw [hreg]
exact hmy
· intro t
rw [hky, hreg]
exact gaussianReal_centered_mgf_bound
(transformedResponse mu₁ y) sigma B t hB hsigma
Affine transport preserves the probability-law instance. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma transformedDornLaw_probability {d : ℕ} (P : Measure (Obs d))
[IsProbabilityMeasure P] :
IsProbabilityMeasure (transformedDornLaw P) := by
unfold transformedDornLaw
exact Measure.isProbabilityMeasure_map (by
unfold affineToCube
fun_prop)
Gaussian specialization of the two-stage recomposition theorem. The two almost-everywhere hypotheses are exactly the propensity and arm-kernel identifications supplied by a Dorn remaining-model witness. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
P :
Measure (Obs d)
σ :
ℝ
e μ₀ μ₁ :
(Fin d → ℝ) → ℝ
he :
hμ₀ :
Measurable μ₀
hμ₁ :
Measurable μ₁
he01 :
∀ x, e x ∈ Icc (0 : ℝ) 1
hA :
=ᵐ[covariateLaw P] fun x => realBernoulli (e x)
hY :
=ᵐ[ P.map (fun z : Obs d => (z.1, z.2.1))] fun xa => if xa.2 then gaussianReal (μ₁ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩ else gaussianReal (μ₀ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩
conclusion
kA :
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x)) (by unfold realBernoulli; fun_prop) (: Kernel (Fin d → ℝ) Bool)
letI : IsMarkovKernel kA
Proof (Lean source)
lemma gaussianObservedLaw_recomposition {d : ℕ} (P : Measure (Obs d))
[IsFiniteMeasure P] (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ) (he : Measurable e)
(hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁)
(he01 : ∀ x, e x ∈ Icc (0 : ℝ) 1)
(hA : (condDistrib (fun z : Obs d => z.2.1)
(fun z => z.1) P : (Fin d → ℝ) → Measure Bool) =ᵐ[covariateLaw P]
fun x => realBernoulli (e x))
(hY : (condDistrib (fun z : Obs d => z.2.2)
(fun z => (z.1, z.2.1)) P :
((Fin d → ℝ) × Bool) → Measure ℝ) =ᵐ[
P.map (fun z : Obs d => (z.1, z.2.1))]
fun xa => if xa.2 then
gaussianReal (μ₁ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩) :
let kA : Kernel (Fin d → ℝ) Bool :=
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x))
(by unfold realBernoulli; fun_prop)
letI : IsMarkovKernel kA :=
⟨fun x => realBernoulli_probability (e x) (he01 x)⟩
let kY : Kernel ((Fin d → ℝ) × Bool) ℝ :=
ProbabilityTheory.Kernel.mk (fun xa => if xa.2 then
gaussianReal (μ₁ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩)
(by
apply Measurable.ite (measurable_snd (MeasurableSet.singleton true))
· fun_prop
· fun_prop)
letI : IsMarkovKernel kY :=
⟨fun xa => by
change IsProbabilityMeasure (if xa.2 then
gaussianReal (μ₁ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩)
cases xa.2 <;>
exact instIsProbabilityMeasureGaussianReal _ _⟩
(((covariateLaw P ⊗ₘ kA) ⊗ₘ kY).map
(fun z => (z.1.1, z.1.2, z.2))) = P := by
dsimp
let kA : Kernel (Fin d → ℝ) Bool :=
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x))
(by unfold realBernoulli; fun_prop)
letI : IsMarkovKernel kA :=
⟨fun x => realBernoulli_probability (e x) (he01 x)⟩
let kY : Kernel ((Fin d → ℝ) × Bool) ℝ :=
ProbabilityTheory.Kernel.mk (fun xa => if xa.2 then
gaussianReal (μ₁ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩)
(by
apply Measurable.ite (measurable_snd (MeasurableSet.singleton true))
· fun_prop
· fun_prop)
letI : IsMarkovKernel kY :=
⟨fun xa => by
change IsProbabilityMeasure (if xa.2 then
gaussianReal (μ₁ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩)
cases xa.2 <;>
exact instIsProbabilityMeasureGaussianReal _ _⟩
apply observedLaw_kernel_recomposition P kA kY
· change (condDistrib (fun z : Obs d => z.2.1)
(fun z => z.1) P : (Fin d → ℝ) → Measure Bool) =ᵐ[covariateLaw P]
fun x => realBernoulli (e x)
exact hA
· change (condDistrib (fun z : Obs d => z.2.2)
(fun z => (z.1, z.2.1)) P :
((Fin d → ℝ) × Bool) → Measure ℝ) =ᵐ[
P.map (fun z : Obs d => (z.1, z.2.1))]
fun xa => if xa.2 then
gaussianReal (μ₁ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩
exact hY
theorem gaussianObservedLaw_recomposition_of_selected Lemma gaussian-observed-law-recomposition-of-selected in the paper ↗
The displayed selected propensity and Dorn's two Gaussian arm identities jointly determine the full observed law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
P :
Measure (Obs d)
σ :
ℝ
e μ₀ μ₁ :
(Fin d → ℝ) → ℝ
he :
hμ₀ :
Measurable μ₀
hμ₁ :
Measurable μ₁
he01 :
∀ x, e x ∈ Icc (0 : ℝ) 1
hselected :
∀ᵐ x ∂covariateLaw P, e x = propensity P x ∧ 0 < e x ∧ e x < 1
harms :
= gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩ ∧
conclusion
kA :
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x)) (by unfold realBernoulli; fun_prop) (: Kernel (Fin d → ℝ) Bool)
letI : IsMarkovKernel kA
Proof (Lean source)
lemma gaussianObservedLaw_recomposition_of_selected {d : ℕ}
(P : Measure (Obs d)) [IsProbabilityMeasure P] (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ) (he : Measurable e)
(hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁)
(he01 : ∀ x, e x ∈ Icc (0 : ℝ) 1)
(hselected : ∀ᵐ x ∂covariateLaw P,
e x = propensity P x ∧ 0 < e x ∧ e x < 1)
(harms : ∀ᵐ x ∂covariateLaw P,
armKernel P x true =
gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩ ∧
armKernel P x false =
gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩) :
let kA : Kernel (Fin d → ℝ) Bool :=
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x))
(by unfold realBernoulli; fun_prop)
letI : IsMarkovKernel kA :=
⟨fun x => realBernoulli_probability (e x) (he01 x)⟩
let kY : Kernel ((Fin d → ℝ) × Bool) ℝ :=
ProbabilityTheory.Kernel.mk (fun xa => if xa.2 then
gaussianReal (μ₁ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩)
(by
apply Measurable.ite (measurable_snd (MeasurableSet.singleton true))
· fun_prop
· fun_prop)
letI : IsMarkovKernel kY :=
⟨fun xa => by
change IsProbabilityMeasure (if xa.2 then
gaussianReal (μ₁ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩)
cases xa.2 <;>
exact instIsProbabilityMeasureGaussianReal _ _⟩
(((covariateLaw P ⊗ₘ kA) ⊗ₘ kY).map
(fun z => (z.1.1, z.1.2, z.2))) = P := by
apply gaussianObservedLaw_recomposition P σ e μ₀ μ₁ he hμ₀ hμ₁ he01
· exact condDistrib_treatment_eq_realBernoulli_of_selected P e hselected
· exact gaussianArmKernels_joint_ae P σ μ₀ μ₁ harms
def gaussianCompletionAt
Gaussian completion at covariate x with independent potential outcomes. For the stated inputs and conditions, the gaussianCompletionAt object being defined.
Definition (Lean source)
parameters
def
given by
((realBernoulli (e x)).prod ((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod (gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))).map (fun z => (x, z.1, z.2.1, z.2.2, if z.1 then z.2.2 else z.2.1))
Gaussian completion is a measurable measure-valued map whenever its propensity and the two arm means are measurable. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Measurable (gaussianCompletionAt σ e μ₀ μ₁)
Proof (Lean source)
lemma gaussianCompletionAt_measurable {d : ℕ} (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁) :
Measurable (gaussianCompletionAt σ e μ₀ μ₁) := by
let νe : (Fin d → ℝ) → FiniteMeasure Bool := fun x => by
letI : IsFiniteMeasure (realBernoulli (e x)) := by
constructor
simp [realBernoulli]
exact ⟨realBernoulli (e x), inferInstance⟩
let ν₀ : (Fin d → ℝ) → FiniteMeasure ℝ := fun x => by
letI : IsProbabilityMeasure
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
exact ⟨gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩, inferInstance⟩
let ν₁ : (Fin d → ℝ) → FiniteMeasure ℝ := fun x => by
letI : IsProbabilityMeasure
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
exact ⟨gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩, inferInstance⟩
have hνe : Measurable νe := by
apply Measurable.subtype_mk
unfold realBernoulli
fun_prop
have hν₀ : Measurable ν₀ := by
apply Measurable.subtype_mk
exact ProbabilityTheory.measurable_gaussianReal.comp
(hμ₀.prodMk measurable_const)
have hν₁ : Measurable ν₁ := by
apply Measurable.subtype_mk
exact ProbabilityTheory.measurable_gaussianReal.comp
(hμ₁.prodMk measurable_const)
let ν₀₁ : (Fin d → ℝ) → FiniteMeasure (ℝ × ℝ) := fun x =>
⟨(ν₀ x : Measure ℝ).prod (ν₁ x : Measure ℝ), inferInstance⟩
have hν₀₁ : Measurable ν₀₁ := by
apply Measurable.subtype_mk
exact FiniteMeasure.measurable_fun_prod.comp (hν₀.prodMk hν₁)
let νa : (Fin d → ℝ) → FiniteMeasure (Bool × ℝ × ℝ) := fun x =>
⟨(νe x : Measure Bool).prod (ν₀₁ x : Measure (ℝ × ℝ)), inferInstance⟩
have hνa : Measurable νa := by
apply Measurable.subtype_mk
exact FiniteMeasure.measurable_fun_prod.comp (hνe.prodMk hν₀₁)
let νx : (Fin d → ℝ) → FiniteMeasure
((Fin d → ℝ) × Bool × ℝ × ℝ) := fun x =>
⟨(Measure.dirac x).prod (νa x : Measure (Bool × ℝ × ℝ)), inferInstance⟩
have hνx : Measurable νx := by
apply Measurable.subtype_mk
have hdirac : Measurable (fun x : Fin d → ℝ =>
(⟨Measure.dirac x, inferInstance⟩ : FiniteMeasure (Fin d → ℝ))) := by
apply Measurable.subtype_mk
exact Measure.measurable_dirac
exact FiniteMeasure.measurable_fun_prod.comp (hdirac.prodMk hνa)
let F : ((Fin d → ℝ) × Bool × ℝ × ℝ) → Completion d := fun z =>
(z.1, z.2.1, z.2.2.1, z.2.2.2,
if z.2.1 then z.2.2.2 else z.2.2.1)
have hF : Measurable F := by
unfold F
have hlast : Measurable (fun z : (Fin d → ℝ) × Bool × ℝ × ℝ =>
if z.2.1 then z.2.2.2 else z.2.2.1) :=
Measurable.ite (measurable_snd.fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption)
have heq : gaussianCompletionAt σ e μ₀ μ₁ = fun x =>
(νx x : Measure ((Fin d → ℝ) × Bool × ℝ × ℝ)).map F := by
funext x
letI : IsFiniteMeasure (realBernoulli (e x)) := by
constructor
simp [realBernoulli]
letI : IsProbabilityMeasure
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
letI : IsProbabilityMeasure
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
unfold gaussianCompletionAt
change Measure.map _ ((realBernoulli (e x)).prod
((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))) =
Measure.map F ((Measure.dirac x).prod
((realBernoulli (e x)).prod
((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))))
rw [Measure.dirac_prod, Measure.map_map hF (by fun_prop)]
rfl
rw [heq]
exact (Measure.measurable_map F hF).comp (measurable_subtype_coe.comp hνx)
The measurable Gaussian completion family, packaged as a kernel. For the stated inputs and conditions, the gaussianCompletionKernel object being defined.
Definition (Lean source)
parameters
def
given by
ProbabilityTheory.Kernel.mk (gaussianCompletionAt σ e μ₀ μ₁) (gaussianCompletionAt_measurable σ e μ₀ μ₁ he hμ₀ hμ₁)
Every fixed Gaussian product fibre records the selected potential outcome. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
OutcomeConsistency (gaussianCompletionAt σ e μ₀ μ₁ x)
Proof (Lean source)
lemma gaussianCompletionAt_consistency {d : ℕ} (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ) :
OutcomeConsistency (gaussianCompletionAt σ e μ₀ μ₁ x) := by
unfold OutcomeConsistency gaussianCompletionAt
have hm : Measurable (fun z : Bool × ℝ × ℝ =>
(x, z.1, z.2.1, z.2.2, if z.1 then z.2.2 else z.2.1)) := by
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption)
have hsel : Measurable (fun ω : Completion d =>
if ω.2.1 then ω.2.2.2.1 else ω.2.2.1) := by
apply Measurable.ite
(by measurability : MeasurableSet {ω : Completion d | ω.2.1 = true})
(by fun_prop) (by fun_prop)
have hs : MeasurableSet {ω : Completion d |
ω.2.2.2.2 = if ω.2.1 then ω.2.2.2.1 else ω.2.2.1} :=
measurableSet_eq_fun (by fun_prop) hsel
apply (ae_map_iff hm.aemeasurable hs).2
filter_upwards [] with z
cases z.1 <;> simp
theorem gaussianCompletionBind_consistency Lemma gaussian-completion-bind-consistency in the paper ↗
Mixing measurable Gaussian product fibres preserves outcome consistency. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
OutcomeConsistency (mu.bind (gaussianCompletionAt σ e μ₀ μ₁))
Proof (Lean source)
lemma gaussianCompletionBind_consistency {d : ℕ}
(mu : Measure (Fin d → ℝ)) (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁) :
OutcomeConsistency
(mu.bind (gaussianCompletionAt σ e μ₀ μ₁)) := by
let S : Set (Completion d) :=
{ω | ω.2.2.2.2 = if ω.2.1 then ω.2.2.2.1 else ω.2.2.1}
have hsel : Measurable (fun ω : Completion d =>
if ω.2.1 then ω.2.2.2.1 else ω.2.2.1) := by
apply Measurable.ite
(by measurability : MeasurableSet {ω : Completion d | ω.2.1 = true})
(by fun_prop) (by fun_prop)
have hS : MeasurableSet S :=
measurableSet_eq_fun (by fun_prop) hsel
have hk := gaussianCompletionAt_measurable σ e μ₀ μ₁ he hμ₀ hμ₁
have hzero : (mu.bind (gaussianCompletionAt σ e μ₀ μ₁)) Sᶜ = 0 := by
apply le_antisymm _ zero_le
calc
_ ≤ ∫⁻ x, (gaussianCompletionAt σ e μ₀ μ₁ x) Sᶜ ∂mu :=
Measure.bind_apply_le _ hS.compl
_ = 0 := by
apply lintegral_eq_zero_of_ae_eq_zero
filter_upwards [] with x
exact (measure_eq_zero_iff_ae_notMem).2
(by simpa only [OutcomeConsistency, S, Set.mem_compl_iff,
not_not, Set.mem_ofPred_eq] using
gaussianCompletionAt_consistency σ e μ₀ μ₁ x)
simpa only [OutcomeConsistency, S, Set.mem_compl_iff, not_not,
Set.mem_ofPred_eq] using (measure_eq_zero_iff_ae_notMem).mp hzero
A pair of Gaussian laws with measurable means is measure-valued measurable. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Measurable (fun x => (gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod (gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))
Proof (Lean source)
lemma gaussianPair_measurable {d : ℕ} (σ : ℝ)
(μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁) :
Measurable (fun x =>
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩)) := by
let ν₀ : (Fin d → ℝ) → ProbabilityMeasure ℝ := fun x =>
⟨gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩,
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _⟩
let ν₁ : (Fin d → ℝ) → ProbabilityMeasure ℝ := fun x =>
⟨gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩,
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _⟩
have hν₀ : Measurable ν₀ := by
apply Measurable.subtype_mk
exact ProbabilityTheory.measurable_gaussianReal.comp
(hμ₀.prodMk measurable_const)
have hν₁ : Measurable ν₁ := by
apply Measurable.subtype_mk
exact ProbabilityTheory.measurable_gaussianReal.comp
(hμ₁.prodMk measurable_const)
exact ProbabilityMeasure.measurable_fun_prod.comp (hν₀.prodMk hν₁)
theorem gaussianCompletionAt_potential_treatment_factorization Lemma gaussian-completion-at-potential-treatment-factorization in the paper ↗
At fixed covariates, Gaussian potential outcomes and treatment factor. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
(gaussianCompletionAt σ e μ₀ μ₁ x).map (fun ω => (ω.1, ((ω.2.2.1, ω.2.2.2.1), ω.2.1)))
= (Measure.dirac x).prod (((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod (gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩)).prod (realBernoulli (e x)))
Proof (Lean source)
lemma gaussianCompletionAt_potential_treatment_factorization {d : ℕ}
(σ : ℝ) (e μ₀ μ₁ : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ)
(he : e x ∈ Icc (0 : ℝ) 1) :
(gaussianCompletionAt σ e μ₀ μ₁ x).map
(fun ω => (ω.1, ((ω.2.2.1, ω.2.2.2.1), ω.2.1))) =
(Measure.dirac x).prod
(((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩)).prod
(realBernoulli (e x))) := by
letI := realBernoulli_probability (e x) he
letI : IsProbabilityMeasure
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
letI : IsProbabilityMeasure
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
unfold gaussianCompletionAt
rw [Measure.map_map (by fun_prop) (by
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption))]
rw [Measure.dirac_prod]
change Measure.map ((Prod.mk x) ∘ swap)
((realBernoulli (e x)).prod
((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))) = _
rw [← Measure.map_map (by fun_prop) measurable_swap, Measure.prod_swap]
theorem gaussianCompletionBind_potential_treatment_factorization Lemma gaussian-completion-bind-potential-treatment-factorization in the paper ↗
Mixing the fixed Gaussian factorization preserves its conditional product. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
σ :
ℝ
e μ₀ μ₁ :
(Fin d → ℝ) → ℝ
he :
hμ₀ :
Measurable μ₀
hμ₁ :
Measurable μ₁
hvalid :
∀ x, e x ∈ Icc (0 : ℝ) 1
conclusion
kY :
ProbabilityTheory.Kernel.mk (fun x => (gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod (gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩)) (gaussianPair_measurable σ μ₀ μ₁ hμ₀ hμ₁) (: Kernel (Fin d → ℝ) (ℝ × ℝ))
kA :
ProbabilityTheory.Kernel.mk (fun xy => realBernoulli (e xy.1)) (by unfold realBernoulli; fun_prop) (: Kernel ((Fin d → ℝ) × (ℝ × ℝ)) Bool)
(mu.bind (gaussianCompletionAt σ e μ₀ μ₁)).map (fun ω => (ω.1, ((ω.2.2.1, ω.2.2.2.1), ω.2.1)))
= Measure.compProd mu (compProd kY kA)
Proof (Lean source)
lemma gaussianCompletionBind_potential_treatment_factorization {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁)
(hvalid : ∀ x, e x ∈ Icc (0 : ℝ) 1) :
let kY : Kernel (Fin d → ℝ) (ℝ × ℝ) :=
ProbabilityTheory.Kernel.mk (fun x =>
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))
(gaussianPair_measurable σ μ₀ μ₁ hμ₀ hμ₁)
let kA : Kernel ((Fin d → ℝ) × (ℝ × ℝ)) Bool :=
ProbabilityTheory.Kernel.mk (fun xy => realBernoulli (e xy.1))
(by unfold realBernoulli; fun_prop)
(mu.bind (gaussianCompletionAt σ e μ₀ μ₁)).map
(fun ω => (ω.1, ((ω.2.2.1, ω.2.2.2.1), ω.2.1))) =
Measure.compProd mu (compProd kY kA) := by
dsimp
let kY : Kernel (Fin d → ℝ) (ℝ × ℝ) :=
ProbabilityTheory.Kernel.mk (fun x =>
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))
(gaussianPair_measurable σ μ₀ μ₁ hμ₀ hμ₁)
let kA : Kernel ((Fin d → ℝ) × (ℝ × ℝ)) Bool :=
ProbabilityTheory.Kernel.mk (fun xy => realBernoulli (e xy.1))
(by unfold realBernoulli; fun_prop)
letI : IsMarkovKernel kY := ⟨fun x => by
letI : IsProbabilityMeasure
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
letI : IsProbabilityMeasure
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
change IsProbabilityMeasure
((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))
exact Measure.prod.instIsProbabilityMeasure _ _⟩
letI : IsMarkovKernel kA :=
⟨fun xy => realBernoulli_probability (e xy.1) (hvalid xy.1)⟩
have hk := gaussianCompletionAt_measurable σ e μ₀ μ₁ he hμ₀ hμ₁
apply Measure.ext
intro s hs
rw [Measure.map_apply (by fun_prop) hs,
Measure.bind_apply (hs.preimage (by fun_prop)) hk.aemeasurable,
Measure.compProd_apply hs]
apply lintegral_congr
intro x
letI := realBernoulli_probability (e x) (hvalid x)
letI : IsProbabilityMeasure
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
letI : IsProbabilityMeasure
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
have hfix := gaussianCompletionAt_potential_treatment_factorization
σ e μ₀ μ₁ x (hvalid x)
have happ := congrArg
(fun m : Measure ((Fin d → ℝ) × ((ℝ × ℝ) × Bool)) => m s) hfix
rw [Measure.map_apply (by fun_prop) hs, Measure.prod_apply hs,
lintegral_dirac] at happ
rw [Measure.prod_apply (hs.preimage (by fun_prop))] at happ
rw [ProbabilityTheory.Kernel.compProd_apply]
· simpa [kY, kA] using happ
· exact hs.preimage (by fun_prop)
theorem gaussianCompletionAt_covariate_potential Lemma gaussian-completion-at-covariate-potential in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
(gaussianCompletionAt σ e μ₀ μ₁ x).map (fun ω => (ω.1, (ω.2.2.1, ω.2.2.2.1)))
= (Measure.dirac x).prod ((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod (gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))
Proof (Lean source)
lemma gaussianCompletionAt_covariate_potential {d : ℕ} (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ)
(he : e x ∈ Icc (0 : ℝ) 1) :
(gaussianCompletionAt σ e μ₀ μ₁ x).map
(fun ω => (ω.1, (ω.2.2.1, ω.2.2.2.1))) =
(Measure.dirac x).prod
((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩)) := by
letI := realBernoulli_probability (e x) he
letI : IsProbabilityMeasure
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
letI : IsProbabilityMeasure
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
unfold gaussianCompletionAt
rw [Measure.map_map (by fun_prop) (by
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption)), Measure.dirac_prod]
calc
Measure.map (fun z : Bool × (ℝ × ℝ) => (x, z.2))
((realBernoulli (e x)).prod
((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))) =
Measure.map (Prod.mk x) (Measure.map snd
((realBernoulli (e x)).prod
((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩)))) := by
rw [Measure.map_map (by fun_prop) measurable_snd]
rfl
_ = _ := by rw [Measure.map_snd_prod, measure_univ, one_smul]
theorem gaussianCompletionAt_covariate_treatment Lemma gaussian-completion-at-covariate-treatment in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
(gaussianCompletionAt σ e μ₀ μ₁ x).map (fun ω => (ω.1, ω.2.1))
= (Measure.dirac x).prod (realBernoulli (e x))
Proof (Lean source)
lemma gaussianCompletionAt_covariate_treatment {d : ℕ} (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ)
(he : e x ∈ Icc (0 : ℝ) 1) :
(gaussianCompletionAt σ e μ₀ μ₁ x).map
(fun ω => (ω.1, ω.2.1)) =
(Measure.dirac x).prod (realBernoulli (e x)) := by
letI := realBernoulli_probability (e x) he
letI : IsProbabilityMeasure
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
letI : IsProbabilityMeasure
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
unfold gaussianCompletionAt
rw [Measure.map_map (by fun_prop) (by
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption)), Measure.dirac_prod]
change Measure.map (fun z : Bool × (ℝ × ℝ) => (x, z.1))
((realBernoulli (e x)).prod
((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))) = _
calc
Measure.map (fun z : Bool × (ℝ × ℝ) => (x, z.1))
((realBernoulli (e x)).prod
((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))) =
Measure.map (Prod.mk x) (Measure.map fst
((realBernoulli (e x)).prod
((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩)))) := by
rw [Measure.map_map (by fun_prop) measurable_fst]
rfl
_ = _ := by
rw [Measure.map_fst_prod, measure_univ, one_smul]
theorem gaussianCompletionBind_covariate_potential Lemma gaussian-completion-bind-covariate-potential in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
σ :
ℝ
e μ₀ μ₁ :
(Fin d → ℝ) → ℝ
he :
hμ₀ :
Measurable μ₀
hμ₁ :
Measurable μ₁
hvalid :
∀ x, e x ∈ Icc (0 : ℝ) 1
conclusion
kY :
ProbabilityTheory.Kernel.mk (fun x => (gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod (gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩)) (gaussianPair_measurable σ μ₀ μ₁ hμ₀ hμ₁) (: Kernel (Fin d → ℝ) (ℝ × ℝ))
(mu.bind (gaussianCompletionAt σ e μ₀ μ₁)).map (fun ω => (ω.1, (ω.2.2.1, ω.2.2.2.1)))
= mu ⊗ₘ kY
Proof (Lean source)
lemma gaussianCompletionBind_covariate_potential {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁)
(hvalid : ∀ x, e x ∈ Icc (0 : ℝ) 1) :
let kY : Kernel (Fin d → ℝ) (ℝ × ℝ) :=
ProbabilityTheory.Kernel.mk (fun x =>
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))
(gaussianPair_measurable σ μ₀ μ₁ hμ₀ hμ₁)
(mu.bind (gaussianCompletionAt σ e μ₀ μ₁)).map
(fun ω => (ω.1, (ω.2.2.1, ω.2.2.2.1))) = mu ⊗ₘ kY := by
dsimp
let kY : Kernel (Fin d → ℝ) (ℝ × ℝ) :=
ProbabilityTheory.Kernel.mk (fun x =>
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))
(gaussianPair_measurable σ μ₀ μ₁ hμ₀ hμ₁)
letI : IsMarkovKernel kY := ⟨fun x => by
letI : IsProbabilityMeasure
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
letI : IsProbabilityMeasure
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
change IsProbabilityMeasure
((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))
exact Measure.prod.instIsProbabilityMeasure _ _⟩
have hk := gaussianCompletionAt_measurable σ e μ₀ μ₁ he hμ₀ hμ₁
apply Measure.ext
intro s hs
rw [Measure.map_apply (by fun_prop) hs,
Measure.bind_apply (hs.preimage (by fun_prop)) hk.aemeasurable,
Measure.compProd_apply hs]
apply lintegral_congr
intro x
letI : IsProbabilityMeasure
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
letI : IsProbabilityMeasure
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
have hfix := gaussianCompletionAt_covariate_potential σ e μ₀ μ₁ x (hvalid x)
have happ := congrArg (fun m : Measure ((Fin d → ℝ) × (ℝ × ℝ)) => m s) hfix
rw [Measure.map_apply (by fun_prop) hs, Measure.prod_apply hs,
lintegral_dirac] at happ
simpa [kY] using happ
theorem gaussianCompletionBind_covariate_treatment Lemma gaussian-completion-bind-covariate-treatment in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
σ :
ℝ
e μ₀ μ₁ :
(Fin d → ℝ) → ℝ
he :
hμ₀ :
Measurable μ₀
hμ₁ :
Measurable μ₁
hvalid :
∀ x, e x ∈ Icc (0 : ℝ) 1
conclusion
kA :
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x)) (by unfold realBernoulli; fun_prop) (: Kernel (Fin d → ℝ) Bool)
(mu.bind (gaussianCompletionAt σ e μ₀ μ₁)).map (fun ω => (ω.1, ω.2.1)) = mu ⊗ₘ kA
Proof (Lean source)
lemma gaussianCompletionBind_covariate_treatment {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁)
(hvalid : ∀ x, e x ∈ Icc (0 : ℝ) 1) :
let kA : Kernel (Fin d → ℝ) Bool :=
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x))
(by unfold realBernoulli; fun_prop)
(mu.bind (gaussianCompletionAt σ e μ₀ μ₁)).map
(fun ω => (ω.1, ω.2.1)) = mu ⊗ₘ kA := by
dsimp
let kA : Kernel (Fin d → ℝ) Bool :=
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x))
(by unfold realBernoulli; fun_prop)
letI : IsMarkovKernel kA :=
⟨fun x => realBernoulli_probability (e x) (hvalid x)⟩
have hk := gaussianCompletionAt_measurable σ e μ₀ μ₁ he hμ₀ hμ₁
apply Measure.ext
intro s hs
rw [Measure.map_apply (by fun_prop) hs,
Measure.bind_apply (hs.preimage (by fun_prop)) hk.aemeasurable,
Measure.compProd_apply hs]
apply lintegral_congr
intro x
have hfix := gaussianCompletionAt_covariate_treatment σ e μ₀ μ₁ x (hvalid x)
have happ := congrArg (fun m : Measure ((Fin d → ℝ) × Bool) => m s) hfix
rw [Measure.map_apply (by fun_prop) hs, Measure.prod_apply hs,
lintegral_dirac] at happ
simpa [kA] using happ
theorem gaussianCompletionAt_covariate_marginal Lemma gaussian-completion-at-covariate-marginal in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma gaussianCompletionAt_covariate_marginal {d : ℕ} (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ)
(he : e x ∈ Icc (0 : ℝ) 1) :
(gaussianCompletionAt σ e μ₀ μ₁ x).map fst = Measure.dirac x := by
letI := realBernoulli_probability (e x) he
letI : IsProbabilityMeasure
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
letI : IsProbabilityMeasure
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
unfold gaussianCompletionAt
rw [Measure.map_map measurable_fst (by
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption))]
change Measure.map (fun _ : Bool × (ℝ × ℝ) => x)
((realBernoulli (e x)).prod
((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))) = _
rw [Measure.map_const, measure_univ, one_smul]
theorem gaussianCompletionBind_covariate_marginal_ae Lemma gaussian-completion-bind-covariate-marginal-ae in the paper ↗
Covariate almost-everywhere propensity validity suffices for a Gaussian completion bind to retain its base covariate law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
σ :
ℝ
e μ₀ μ₁ :
(Fin d → ℝ) → ℝ
he :
hμ₀ :
Measurable μ₀
hμ₁ :
Measurable μ₁
hvalid :
∀ᵐ x ∂mu, e x ∈ Icc (0 : ℝ) 1
conclusion
Proof (Lean source)
lemma gaussianCompletionBind_covariate_marginal_ae {d : ℕ}
(mu : Measure (Fin d → ℝ)) (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁)
(hvalid : ∀ᵐ x ∂mu, e x ∈ Icc (0 : ℝ) 1) :
(mu.bind (gaussianCompletionAt σ e μ₀ μ₁)).map fst = mu := by
have hk := gaussianCompletionAt_measurable σ e μ₀ μ₁ he hμ₀ hμ₁
ext s hs
rw [Measure.map_apply measurable_fst hs,
Measure.bind_apply (measurable_fst hs) hk.aemeasurable]
simp_rw [← Measure.map_apply measurable_fst hs]
calc
_ = ∫⁻ x, (Measure.dirac x) s ∂mu := by
apply lintegral_congr_ae
filter_upwards [hvalid] with x hx
rw [gaussianCompletionAt_covariate_marginal σ e μ₀ μ₁ x hx]
_ = mu s := by
simp only [Measure.dirac_apply' _ hs]
exact lintegral_indicator_one hs
theorem gaussianCompletionBind_propensity_ae Lemma gaussian-completion-bind-propensity in the paper ↗
The Bernoulli parameter used by a Gaussian completion is a propensity version of its observed law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
mu :
σ :
ℝ
e μ₀ μ₁ :
(Fin d → ℝ) → ℝ
he :
hμ₀ :
Measurable μ₀
hμ₁ :
Measurable μ₁
hvalid :
∀ x, e x ∈ Icc (0 : ℝ) 1
conclusion
P :
(mu.bind (gaussianCompletionAt σ e μ₀ μ₁)).map observed
∀ᵐ x ∂mu, propensity P x = e x
Proof (Lean source)
lemma gaussianCompletionBind_propensity_ae {d : ℕ}
(mu : Measure (Fin d → ℝ)) [IsProbabilityMeasure mu] (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁)
(hvalid : ∀ x, e x ∈ Icc (0 : ℝ) 1)
[IsFiniteMeasure
((mu.bind (gaussianCompletionAt σ e μ₀ μ₁)).map observed)] :
let P := (mu.bind (gaussianCompletionAt σ e μ₀ μ₁)).map observed
∀ᵐ x ∂mu, propensity P x = e x := by
dsimp
let P := (mu.bind (gaussianCompletionAt σ e μ₀ μ₁)).map observed
let k : Kernel (Fin d → ℝ) Bool :=
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x))
(by unfold realBernoulli; fun_prop)
have hk := gaussianCompletionAt_measurable σ e μ₀ μ₁ he hμ₀ hμ₁
have hpoint : ∀ x, IsProbabilityMeasure
(gaussianCompletionAt σ e μ₀ μ₁ x) := by
intro x
letI := realBernoulli_probability (e x) (hvalid x)
letI : IsProbabilityMeasure
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
letI : IsProbabilityMeasure
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
unfold gaussianCompletionAt
apply Measure.isProbabilityMeasure_map
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption)
letI : IsProbabilityMeasure (mu.bind (gaussianCompletionAt σ e μ₀ μ₁)) :=
isProbabilityMeasure_bind hk.aemeasurable
(Filter.Eventually.of_forall hpoint)
have hobs : Measurable (@observed d) := by unfold observed; fun_prop
letI : IsProbabilityMeasure P := by
dsimp [P]
exact Measure.isProbabilityMeasure_map hobs.aemeasurable
letI : IsMarkovKernel k :=
⟨fun x => realBernoulli_probability (e x) (hvalid x)⟩
have hcov : P.map fst = mu := by
dsimp [P]
rw [Measure.map_map (by fun_prop) hobs]
exact gaussianCompletionBind_covariate_marginal_ae mu σ e μ₀ μ₁
he hμ₀ hμ₁ (Filter.Eventually.of_forall hvalid)
have hpair : P.map (fun z => (z.1, z.2.1)) = Measure.compProd mu k := by
dsimp [P]
rw [Measure.map_map (by fun_prop) hobs]
exact gaussianCompletionBind_covariate_treatment mu σ e μ₀ μ₁
he hμ₀ hμ₁ hvalid
have hae := ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd_of_measurable
(μ := P) (X := fun z : Obs d => z.1) (Y := fun z : Obs d => z.2.1)
(κ := k) (by fun_prop) (by fun_prop) (by simpa [hcov] using hpair)
rw [hcov] at hae
filter_upwards [hae] with x hx
unfold propensity
rw [hx]
simp [k, realBernoulli, (hvalid x).1]
A globally valid representative of a real-valued propensity. For the stated inputs and conditions, the gaussianPropensityClamp object being defined.
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma gaussianPropensityClamp_mem_Icc (r : ℝ) :
gaussianPropensityClamp r ∈ Icc (0 : ℝ) 1 := by
simp [gaussianPropensityClamp]
If a real number and its unit-interval certificate are given, clamping that number to the unit interval leaves it unchanged.
Formal statement
Proof (Lean source)
@[simp] lemma gaussianPropensityClamp_eq_self {r : ℝ}
(hr : r ∈ Icc (0 : ℝ) 1) : gaussianPropensityClamp r = r := by
simp [gaussianPropensityClamp, hr.1, hr.2]
theorem gaussianCompletionBind_clamp_eq_of_ae Lemma gaussian-completion-bind-clamp-ae in the paper ↗
If a propensity is valid on the base law almost everywhere, clamping it globally does not change the Gaussian completion bind. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
mu.bind (gaussianCompletionAt σ (gaussianPropensityClamp ∘ e) μ₀ μ₁)
= mu.bind (gaussianCompletionAt σ e μ₀ μ₁)
Proof (Lean source)
lemma gaussianCompletionBind_clamp_eq_of_ae {d : ℕ}
(mu : Measure (Fin d → ℝ)) (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(hvalid : ∀ᵐ x ∂mu, e x ∈ Icc (0 : ℝ) 1) :
mu.bind (gaussianCompletionAt σ (gaussianPropensityClamp ∘ e) μ₀ μ₁) =
mu.bind (gaussianCompletionAt σ e μ₀ μ₁) := by
apply Measure.bind_congr_right
filter_upwards [hvalid] with x hx
unfold gaussianCompletionAt
rw [show (gaussianPropensityClamp ∘ e) x = e x by
exact gaussianPropensityClamp_eq_self hx]
theorem gaussianCompletionBind_condExchangeable Lemma gaussian-completion-bind-cond-exchangeable in the paper ↗
Independent Gaussian potential-outcome fibres make treatment conditionally independent of the potential outcomes given covariates. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
σ :
ℝ
e μ₀ μ₁ :
(Fin d → ℝ) → ℝ
he :
hμ₀ :
Measurable μ₀
hμ₁ :
Measurable μ₁
hvalid :
∀ x, e x ∈ Icc (0 : ℝ) 1
IsFiniteMeasure (mu.bind (gaussianCompletionAt σ e μ₀ μ₁))
conclusion
CondExchangeable (mu.bind (gaussianCompletionAt σ e μ₀ μ₁))
Proof (Lean source)
lemma gaussianCompletionBind_condExchangeable {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁)
(hvalid : ∀ x, e x ∈ Icc (0 : ℝ) 1)
[IsFiniteMeasure (mu.bind (gaussianCompletionAt σ e μ₀ μ₁))] :
CondExchangeable (mu.bind (gaussianCompletionAt σ e μ₀ μ₁)) := by
let Pc := mu.bind (gaussianCompletionAt σ e μ₀ μ₁)
let X : Completion d → (Fin d → ℝ) := fun ω => ω.1
let Y : Completion d → (ℝ × ℝ) := fun ω => (ω.2.2.1, ω.2.2.2.1)
let A : Completion d → Bool := fun ω => ω.2.1
let kY : Kernel (Fin d → ℝ) (ℝ × ℝ) :=
ProbabilityTheory.Kernel.mk (fun x =>
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))
(gaussianPair_measurable σ μ₀ μ₁ hμ₀ hμ₁)
let kAx : Kernel (Fin d → ℝ) Bool :=
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x))
(by unfold realBernoulli; fun_prop)
let kA : Kernel ((Fin d → ℝ) × (ℝ × ℝ)) Bool :=
ProbabilityTheory.Kernel.mk (fun xy => realBernoulli (e xy.1))
(by unfold realBernoulli; fun_prop)
letI : IsMarkovKernel kY := ⟨fun x => by
letI : IsProbabilityMeasure
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
letI : IsProbabilityMeasure
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
change IsProbabilityMeasure
((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))
exact Measure.prod.instIsProbabilityMeasure _ _⟩
letI : IsMarkovKernel kAx :=
⟨fun x => realBernoulli_probability (e x) (hvalid x)⟩
letI : IsMarkovKernel kA :=
⟨fun xy => realBernoulli_probability (e xy.1) (hvalid xy.1)⟩
have hX : Measurable X := by dsimp [X]; fun_prop
have hY : Measurable Y := by dsimp [Y]; fun_prop
have hA : Measurable A := by dsimp [A]; fun_prop
have hcov : Pc.map X = mu := by
dsimp [Pc, X]
exact gaussianCompletionBind_covariate_marginal_ae mu σ e μ₀ μ₁
he hμ₀ hμ₁ (Filter.Eventually.of_forall hvalid)
have hYmap : Pc.map (fun ω => (X ω, Y ω)) = mu ⊗ₘ kY := by
dsimp [Pc, X, Y, kY]
exact gaussianCompletionBind_covariate_potential mu σ e μ₀ μ₁
he hμ₀ hμ₁ hvalid
have hAmap : Pc.map (fun ω => (X ω, A ω)) = mu ⊗ₘ kAx := by
dsimp [Pc, X, A, kAx]
exact gaussianCompletionBind_covariate_treatment mu σ e μ₀ μ₁
he hμ₀ hμ₁ hvalid
have hYcond : condDistrib Y X Pc =ᵐ[mu] kY := by
have h := ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd_of_measurable
(μ := Pc) (X := X) (Y := Y) (κ := kY) hX hY
(by simpa [hcov] using hYmap)
simpa [hcov] using h
have hAcond : condDistrib A X Pc =ᵐ[mu] kAx := by
have h := ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd_of_measurable
(μ := Pc) (X := X) (Y := A) (κ := kAx) hX hA
(by simpa [hcov] using hAmap)
simpa [hcov] using h
have hkprod : compProd kY kA =
prod kY kAx := by
ext x s hs
rw [ProbabilityTheory.Kernel.compProd_apply hs,
ProbabilityTheory.Kernel.prod_apply kY kAx x, Measure.prod_apply hs]
rfl
have hfactor : Pc.map (fun ω => (X ω, Y ω, A ω)) =
mu ⊗ₘ prod kY kAx := by
dsimp [Pc, X, Y, A]
rw [← hkprod]
exact gaussianCompletionBind_potential_treatment_factorization
mu σ e μ₀ μ₁ he hμ₀ hμ₁ hvalid
unfold CondExchangeable
change CondIndepFun (comap X inferInstance)
(comap_le hX) Y A Pc
apply (ProbabilityTheory.condIndepFun_iff_map_prod_eq_prod_condDistrib_prod_condDistrib
hY hA hX).2
rw [← Measure.compProd_eq_comp_prod, hcov]
calc
Pc.map (fun ω => (X ω, Y ω, A ω)) =
mu ⊗ₘ prod kY kAx := hfactor
_ = mu ⊗ₘ prod
(condDistrib Y X Pc)
(condDistrib A X Pc) := by
symm
apply Measure.compProd_congr
filter_upwards [hYcond, hAcond] with x hxy hxa
rw [ProbabilityTheory.Kernel.prod_apply
(condDistrib Y X Pc)
(condDistrib A X Pc) x,
ProbabilityTheory.Kernel.prod_apply kY kAx x, hxy, hxa]
theorem gaussianCompletionAt_treatedPotential Lemma gaussian-completion-at-treated-potential in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
(gaussianCompletionAt σ e μ₀ μ₁ x).map (fun ω => (ω.1, ω.2.2.2.1))
Proof (Lean source)
lemma gaussianCompletionAt_treatedPotential {d : ℕ} (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ)
(he : e x ∈ Icc (0 : ℝ) 1) :
(gaussianCompletionAt σ e μ₀ μ₁ x).map
(fun ω => (ω.1, ω.2.2.2.1)) =
(Measure.dirac x).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩) := by
letI := realBernoulli_probability (e x) he
letI : IsProbabilityMeasure
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
letI : IsProbabilityMeasure
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
unfold gaussianCompletionAt
rw [Measure.map_map (by fun_prop) (by
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption)), Measure.dirac_prod]
calc
Measure.map (fun z : Bool × (ℝ × ℝ) => (x, z.2.2))
((realBernoulli (e x)).prod
((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))) =
Measure.map (Prod.mk x)
(Measure.map (fun z : Bool × (ℝ × ℝ) => z.2.2)
((realBernoulli (e x)).prod
((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩)))) := by
rw [Measure.map_map (by fun_prop) (by fun_prop)]
rfl
_ = _ := by
rw [show (fun z : Bool × (ℝ × ℝ) => z.2.2) =
snd ∘ snd by rfl]
rw [← Measure.map_map (by fun_prop) (by fun_prop),
Measure.map_snd_prod, measure_univ, one_smul,
Measure.map_snd_prod, measure_univ, one_smul]
theorem gaussianCompletionBind_treatedPotential Lemma gaussian-completion-bind-treated-potential in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
σ :
ℝ
e μ₀ μ₁ :
(Fin d → ℝ) → ℝ
he :
hμ₀ :
Measurable μ₀
hμ₁ :
Measurable μ₁
hvalid :
∀ x, e x ∈ Icc (0 : ℝ) 1
conclusion
k₁ :
ProbabilityTheory.Kernel.mk (fun x => gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩) (ProbabilityTheory.measurable_gaussianReal.comp (hμ₁.prodMk measurable_const)) (: Kernel (Fin d → ℝ) ℝ)
(mu.bind (gaussianCompletionAt σ e μ₀ μ₁)).map (fun ω => (ω.1, ω.2.2.2.1)) = mu ⊗ₘ k₁
Proof (Lean source)
lemma gaussianCompletionBind_treatedPotential {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁)
(hvalid : ∀ x, e x ∈ Icc (0 : ℝ) 1) :
let k₁ : Kernel (Fin d → ℝ) ℝ :=
ProbabilityTheory.Kernel.mk (fun x =>
gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩)
(ProbabilityTheory.measurable_gaussianReal.comp
(hμ₁.prodMk measurable_const))
(mu.bind (gaussianCompletionAt σ e μ₀ μ₁)).map
(fun ω => (ω.1, ω.2.2.2.1)) = mu ⊗ₘ k₁ := by
dsimp
let k₁ : Kernel (Fin d → ℝ) ℝ :=
ProbabilityTheory.Kernel.mk (fun x =>
gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩)
(ProbabilityTheory.measurable_gaussianReal.comp
(hμ₁.prodMk measurable_const))
letI : IsMarkovKernel k₁ :=
⟨fun x => instIsProbabilityMeasureGaussianReal _ _⟩
have hk := gaussianCompletionAt_measurable σ e μ₀ μ₁ he hμ₀ hμ₁
apply Measure.ext
intro s hs
rw [Measure.map_apply (by fun_prop) hs,
Measure.bind_apply (hs.preimage (by fun_prop)) hk.aemeasurable,
Measure.compProd_apply hs]
apply lintegral_congr
intro x
letI : IsProbabilityMeasure
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
instIsProbabilityMeasureGaussianReal _ _
have hfix := gaussianCompletionAt_treatedPotential σ e μ₀ μ₁ x (hvalid x)
have happ := congrArg (fun m : Measure ((Fin d → ℝ) × ℝ) => m s) hfix
rw [Measure.map_apply (by fun_prop) hs, Measure.prod_apply hs,
lintegral_dirac] at happ
change (gaussianCompletionAt σ e μ₀ μ₁ x)
((fun ω => (ω.1, ω.2.2.2.1)) ⁻¹' s) =
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩) (Prod.mk x ⁻¹' s)
exact happ
theorem gaussianReal_integral_abs_le Lemma gaussian-first-absolute-moment-shift-bound in the paper ↗
Translating a centred Gaussian bounds its first absolute moment by the absolute mean plus the centred first absolute moment. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma gaussianReal_integral_abs_le (m σ : ℝ) :
(∫ y : ℝ, |y| ∂gaussianReal m ⟨σ ^ 2, sq_nonneg σ⟩) ≤
|m| + ∫ z : ℝ, |z| ∂gaussianReal 0 ⟨σ ^ 2, sq_nonneg σ⟩ := by
let v : NNReal := ⟨σ ^ 2, sq_nonneg σ⟩
letI : IsProbabilityMeasure (gaussianReal 0 v) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
have hshift : (gaussianReal 0 v).map (fun z : ℝ => z + m) =
gaussianReal m v := by
simpa using ProbabilityTheory.gaussianReal_map_add_const
(μ := (0 : ℝ)) (v := v) m
have hzero : Integrable (fun z : ℝ => |z|) (gaussianReal 0 v) :=
(ProbabilityTheory.memLp_id_gaussianReal' 1 (by simp)).integrable (by norm_num) |>.abs
have hadd : Integrable (fun z : ℝ => |z| + |m|) (gaussianReal 0 v) :=
hzero.add (integrable_const |m|)
have hshifted : Integrable (fun z : ℝ => |z + m|) (gaussianReal 0 v) :=
hadd.mono' (by fun_prop) (Filter.Eventually.of_forall (fun z => by
simpa [abs_of_nonneg (abs_nonneg (z + m))] using abs_add_le z m))
rw [← hshift, integral_map (by fun_prop) (by fun_prop)]
calc
(∫ z : ℝ, |z + m| ∂gaussianReal 0 v) ≤
∫ z : ℝ, |z| + |m| ∂gaussianReal 0 v :=
integral_mono hshifted hadd (fun z => abs_add_le z m)
_ = (∫ z : ℝ, |z| ∂gaussianReal 0 v) +
∫ _ : ℝ, |m| ∂gaussianReal 0 v :=
integral_add hzero (integrable_const |m|)
_ = _ := by simp [v, add_comm]
theorem gaussianCompletionBind_treatedPotential_integrable Lemma gaussian-completion-treated-potential-integrable in the paper ↗
An integrable Gaussian mean over a finite base law makes the treated potential outcome integrable under the completion bind. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
mu :
σ :
ℝ
e μ₀ μ₁ :
(Fin d → ℝ) → ℝ
he :
hμ₀ :
Measurable μ₀
hμ₁ :
Measurable μ₁
hvalid :
∀ x, e x ∈ Icc (0 : ℝ) 1
hμ₁int :
Integrable μ₁ mu
conclusion
Proof (Lean source)
lemma gaussianCompletionBind_treatedPotential_integrable {d : ℕ}
(mu : Measure (Fin d → ℝ)) [IsFiniteMeasure mu] (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁)
(hvalid : ∀ x, e x ∈ Icc (0 : ℝ) 1)
(hμ₁int : Integrable μ₁ mu) :
Integrable (fun ω : Completion d => ω.2.2.2.1)
(mu.bind (gaussianCompletionAt σ e μ₀ μ₁)) := by
let k₁ : Kernel (Fin d → ℝ) ℝ :=
ProbabilityTheory.Kernel.mk (fun x =>
gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩)
(ProbabilityTheory.measurable_gaussianReal.comp
(hμ₁.prodMk measurable_const))
letI : IsMarkovKernel k₁ :=
⟨fun x => ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _⟩
let cσ : ℝ := ∫ z : ℝ, |z| ∂gaussianReal 0 ⟨σ ^ 2, sq_nonneg σ⟩
have hcσ : 0 ≤ cσ := integral_nonneg (fun _ => abs_nonneg _)
have hg_meas : StronglyMeasurable (fun x =>
∫ y : ℝ, |y| ∂k₁ x) := by
exact (by fun_prop : StronglyMeasurable (fun y : ℝ => |y|)).integral_kernel
have hdom : Integrable (fun x => |μ₁ x| + cσ) mu :=
hμ₁int.abs.add (integrable_const cσ)
have hg_int : Integrable (fun x => ∫ y : ℝ, |y| ∂k₁ x) mu := by
apply hdom.mono hg_meas.aestronglyMeasurable
filter_upwards [] with x
have hg0 : 0 ≤ ∫ y : ℝ, |y| ∂k₁ x :=
integral_nonneg (fun _ => abs_nonneg _)
have hdom0 : 0 ≤ |μ₁ x| + cσ := add_nonneg (abs_nonneg _) hcσ
change ‖∫ y : ℝ, |y| ∂gaussianReal (μ₁ x)
⟨σ ^ 2, sq_nonneg σ⟩‖ ≤ ‖|μ₁ x| + cσ‖
have hg0' : 0 ≤ ∫ y : ℝ, |y| ∂gaussianReal (μ₁ x)
⟨σ ^ 2, sq_nonneg σ⟩ := integral_nonneg (fun _ => abs_nonneg _)
change |∫ y : ℝ, |y| ∂gaussianReal (μ₁ x)
⟨σ ^ 2, sq_nonneg σ⟩| ≤ abs (|μ₁ x| + cσ)
rw [abs_of_nonneg hg0', abs_of_nonneg hdom0]
exact gaussianReal_integral_abs_le (μ₁ x) σ
have hcomp : Integrable (fun xy : (Fin d → ℝ) × ℝ => xy.2) (mu ⊗ₘ k₁) := by
apply (Measure.integrable_compProd_iff (by fun_prop)).2
constructor
· filter_upwards [] with x
change Integrable (fun y : ℝ => y)
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩)
exact (ProbabilityTheory.memLp_id_gaussianReal' 1 (by simp)).integrable
(by norm_num)
· simpa [k₁, Real.norm_eq_abs] using hg_int
have hjoint := gaussianCompletionBind_treatedPotential mu σ e μ₀ μ₁
he hμ₀ hμ₁ hvalid
have hmap : Integrable snd
((mu.bind (gaussianCompletionAt σ e μ₀ μ₁)).map
(fun ω => (ω.1, ω.2.2.2.1))) := by
rw [hjoint]
exact hcomp
exact (integrable_map_measure
(μ := mu.bind (gaussianCompletionAt σ e μ₀ μ₁))
(f := fun ω : Completion d => (ω.1, ω.2.2.2.1))
(g := snd) (by fun_prop) (by fun_prop)).mp hmap
theorem gaussianCompletionBind_holderResponse Lemma gaussian-completion-bind-holder-response in the paper ↗
Once integrability is supplied, the Gaussian treated-arm mean is the conditional treated-potential response of the product completion. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
σ β L :
ℝ
e μ₀ μ₁ :
(Fin d → ℝ) → ℝ
he :
hμ₀ :
Measurable μ₀
hμ₁ :
Measurable μ₁
hvalid :
∀ x, e x ∈ Icc (0 : ℝ) 1
IsFiniteMeasure (mu.bind (gaussianCompletionAt σ e μ₀ μ₁))
hball :
HolderBall β L μ₁
hYint :
conclusion
HolderResponse (mu.bind (gaussianCompletionAt σ e μ₀ μ₁)) μ₁ β L
Proof (Lean source)
lemma gaussianCompletionBind_holderResponse {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (σ β L : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁)
(hvalid : ∀ x, e x ∈ Icc (0 : ℝ) 1)
[IsFiniteMeasure (mu.bind (gaussianCompletionAt σ e μ₀ μ₁))]
(hball : HolderBall β L μ₁)
(hYint : Integrable (fun ω : Completion d => ω.2.2.2.1)
(mu.bind (gaussianCompletionAt σ e μ₀ μ₁))) :
HolderResponse (mu.bind (gaussianCompletionAt σ e μ₀ μ₁)) μ₁ β L := by
let Pc := mu.bind (gaussianCompletionAt σ e μ₀ μ₁)
let X : Completion d → (Fin d → ℝ) := fun ω => ω.1
let Y₁ : Completion d → ℝ := fun ω => ω.2.2.2.1
let k₁ : Kernel (Fin d → ℝ) ℝ :=
ProbabilityTheory.Kernel.mk (fun x =>
gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩)
(ProbabilityTheory.measurable_gaussianReal.comp
(hμ₁.prodMk measurable_const))
letI : IsMarkovKernel k₁ :=
⟨fun x => ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _⟩
have hX : Measurable X := by dsimp [X]; fun_prop
have hY₁ : Measurable Y₁ := by dsimp [Y₁]; fun_prop
have hcov : Pc.map X = mu := by
dsimp [Pc, X]
exact gaussianCompletionBind_covariate_marginal_ae mu σ e μ₀ μ₁
he hμ₀ hμ₁ (Filter.Eventually.of_forall hvalid)
have hjoint : Pc.map (fun ω => (X ω, Y₁ ω)) = mu ⊗ₘ k₁ := by
dsimp [Pc, X, Y₁, k₁]
exact gaussianCompletionBind_treatedPotential mu σ e μ₀ μ₁
he hμ₀ hμ₁ hvalid
have hcond : condDistrib Y₁ X Pc =ᵐ[mu] k₁ := by
have h := ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd_of_measurable
(μ := Pc) (X := X) (Y := Y₁) (κ := k₁) hX hY₁
(by simpa [hcov] using hjoint)
simpa [hcov] using h
have hce : Pc[Y₁ | comap X inferInstance] =ᵐ[Pc]
fun ω => ∫ y : ℝ, y ∂ProbabilityTheory.condDistrib Y₁ X Pc (X ω) := by
simpa [Pc, Y₁] using
ProbabilityTheory.condExp_ae_eq_integral_condDistrib' hX hYint
have hkernel : ∀ᵐ ω ∂Pc,
condDistrib Y₁ X Pc (X ω) = k₁ (X ω) := by
have hcond' : condDistrib Y₁ X Pc =ᵐ[Pc.map X] k₁ := by
rw [hcov]
exact hcond
exact ae_eq_comp hX.aemeasurable hcond'
have hmean : (fun ω : Completion d =>
∫ y : ℝ, y ∂ProbabilityTheory.condDistrib Y₁ X Pc (X ω)) =ᵐ[Pc]
fun ω => μ₁ (X ω) := by
filter_upwards [hkernel] with ω hω
rw [hω]
exact ProbabilityTheory.integral_id_gaussianReal
refine ⟨hball, ?_⟩
exact hmean.symm.trans hce.symm
theorem gaussianCompletionBind_holderResponse_of_integrable_mean Lemma gaussian-completion-bind-holder-response-of-integrable-mean in the paper ↗
Practical Gaussian Holder-response assembly from an integrable mean. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
mu :
σ β L :
ℝ
e μ₀ μ₁ :
(Fin d → ℝ) → ℝ
he :
hμ₀ :
Measurable μ₀
hμ₁ :
Measurable μ₁
hvalid :
∀ x, e x ∈ Icc (0 : ℝ) 1
IsFiniteMeasure (mu.bind (gaussianCompletionAt σ e μ₀ μ₁))
hball :
HolderBall β L μ₁
hμ₁int :
Integrable μ₁ mu
conclusion
HolderResponse (mu.bind (gaussianCompletionAt σ e μ₀ μ₁)) μ₁ β L
Proof (Lean source)
lemma gaussianCompletionBind_holderResponse_of_integrable_mean {d : ℕ}
(mu : Measure (Fin d → ℝ)) [IsFiniteMeasure mu] (σ β L : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁)
(hvalid : ∀ x, e x ∈ Icc (0 : ℝ) 1)
[IsFiniteMeasure (mu.bind (gaussianCompletionAt σ e μ₀ μ₁))]
(hball : HolderBall β L μ₁) (hμ₁int : Integrable μ₁ mu) :
HolderResponse (mu.bind (gaussianCompletionAt σ e μ₀ μ₁)) μ₁ β L :=
gaussianCompletionBind_holderResponse mu σ β L e μ₀ μ₁ he hμ₀ hμ₁
hvalid hball
(gaussianCompletionBind_treatedPotential_integrable mu σ e μ₀ μ₁
he hμ₀ hμ₁ hvalid hμ₁int)
theorem gaussianCompletionBind_treatedConditionalMean Lemma gaussian-completion-bind-treated-conditional-mean in the paper ↗
The conditional-mean component of the Gaussian completion, separated from any regularity property of its representative. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
σ :
ℝ
e μ₀ μ₁ :
(Fin d → ℝ) → ℝ
he :
hμ₀ :
Measurable μ₀
hμ₁ :
Measurable μ₁
hvalid :
∀ x, e x ∈ Icc (0 : ℝ) 1
IsFiniteMeasure (mu.bind (gaussianCompletionAt σ e μ₀ μ₁))
hYint :
conclusion
(fun ω : Completion d => μ₁ ω.1)
=ᵐ[ mu.bind (gaussianCompletionAt σ e μ₀ μ₁)] (mu.bind (gaussianCompletionAt σ e μ₀ μ₁))[ (fun ω : Completion d => ω.2.2.2.1) | comap (fun ω => ω.1) inferInstance]
Proof (Lean source)
lemma gaussianCompletionBind_treatedConditionalMean {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁)
(hvalid : ∀ x, e x ∈ Icc (0 : ℝ) 1)
[IsFiniteMeasure (mu.bind (gaussianCompletionAt σ e μ₀ μ₁))]
(hYint : Integrable (fun ω : Completion d => ω.2.2.2.1)
(mu.bind (gaussianCompletionAt σ e μ₀ μ₁))) :
(fun ω : Completion d => μ₁ ω.1) =ᵐ[
mu.bind (gaussianCompletionAt σ e μ₀ μ₁)]
(mu.bind (gaussianCompletionAt σ e μ₀ μ₁))[
(fun ω : Completion d => ω.2.2.2.1) |
comap (fun ω => ω.1) inferInstance] := by
-- The proof of `gaussianCompletionBind_holderResponse` uses the Holder
-- premise only when pairing this identity with its regularity field.
let Pc := mu.bind (gaussianCompletionAt σ e μ₀ μ₁)
let X : Completion d → (Fin d → ℝ) := fun ω => ω.1
let Y₁ : Completion d → ℝ := fun ω => ω.2.2.2.1
let k₁ : Kernel (Fin d → ℝ) ℝ :=
ProbabilityTheory.Kernel.mk (fun x =>
gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩)
(ProbabilityTheory.measurable_gaussianReal.comp
(hμ₁.prodMk measurable_const))
letI : IsMarkovKernel k₁ :=
⟨fun x => ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _⟩
have hX : Measurable X := by dsimp [X]; fun_prop
have hY₁ : Measurable Y₁ := by dsimp [Y₁]; fun_prop
have hcov : Pc.map X = mu := by
dsimp [Pc, X]
exact gaussianCompletionBind_covariate_marginal_ae mu σ e μ₀ μ₁
he hμ₀ hμ₁ (Filter.Eventually.of_forall hvalid)
have hjoint : Pc.map (fun ω => (X ω, Y₁ ω)) = mu ⊗ₘ k₁ := by
dsimp [Pc, X, Y₁, k₁]
exact gaussianCompletionBind_treatedPotential mu σ e μ₀ μ₁
he hμ₀ hμ₁ hvalid
have hcond : condDistrib Y₁ X Pc =ᵐ[mu] k₁ := by
have h := ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd_of_measurable
(μ := Pc) (X := X) (Y := Y₁) (κ := k₁) hX hY₁
(by simpa [hcov] using hjoint)
simpa [hcov] using h
have hce : Pc[Y₁ | comap X inferInstance] =ᵐ[Pc]
fun ω => ∫ y : ℝ, y ∂ProbabilityTheory.condDistrib Y₁ X Pc (X ω) := by
simpa [Pc, Y₁] using
ProbabilityTheory.condExp_ae_eq_integral_condDistrib' hX hYint
have hkernel : ∀ᵐ ω ∂Pc,
condDistrib Y₁ X Pc (X ω) = k₁ (X ω) := by
have hcond' : condDistrib Y₁ X Pc =ᵐ[Pc.map X] k₁ := by
rw [hcov]
exact hcond
exact ae_eq_comp hX.aemeasurable hcond'
filter_upwards [hkernel, hce] with ω hk hceω
rw [hceω, hk]
exact ProbabilityTheory.integral_id_gaussianReal.symm
Changing only the covariate coordinate preserves consistency. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
OutcomeConsistency (Pc.map (fun ω : Completion d => (affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1, ω.2.2.2.2)))
Proof (Lean source)
lemma affineCompletionMap_consistency {d : ℕ} (Pc : Measure (Completion d))
(hPc : OutcomeConsistency Pc) :
OutcomeConsistency (Pc.map (fun ω : Completion d =>
(affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1,
ω.2.2.2.2))) := by
let F : Completion d → Completion d := fun ω =>
(affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1, ω.2.2.2.2)
have hF : Measurable F := by
dsimp [F]
have hx : Measurable (fun ω : Completion d => affineToCube ω.1) :=
affineToCube_measurable.comp measurable_fst
fun_prop (disch := assumption)
let S : Set (Completion d) :=
{ω | ω.2.2.2.2 = if ω.2.1 then ω.2.2.2.1 else ω.2.2.1}
have hsel : Measurable (fun ω : Completion d =>
if ω.2.1 then ω.2.2.2.1 else ω.2.2.1) := by
apply Measurable.ite
(by measurability : MeasurableSet {ω : Completion d | ω.2.1 = true})
(by fun_prop) (by fun_prop)
have hS : MeasurableSet S :=
measurableSet_eq_fun (by fun_prop) hsel
unfold OutcomeConsistency at hPc ⊢
apply (ae_map_iff hF.aemeasurable hS).2
simpa only [F, S, Set.mem_ofPred_eq] using hPc
theorem affineCompletionMap_condExchangeable Lemma affine-completion-map-cond-exchangeable in the paper ↗
An invertible measurable affine change of the covariate coordinate preserves conditional exchangeability. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
Pc :
Measure (Completion d)
IsFiniteMeasure (Pc.map (fun ω : Completion d => (affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1, ω.2.2.2.2)))
hPc :
conclusion
CondExchangeable (Pc.map (fun ω : Completion d => (affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1, ω.2.2.2.2)))
Proof (Lean source)
lemma affineCompletionMap_condExchangeable {d : ℕ}
(Pc : Measure (Completion d)) [IsFiniteMeasure Pc]
[IsFiniteMeasure (Pc.map (fun ω : Completion d =>
(affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1,
ω.2.2.2.2)))]
(hPc : CondExchangeable Pc) :
CondExchangeable (Pc.map (fun ω : Completion d =>
(affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1,
ω.2.2.2.2))) := by
let F : Completion d → Completion d := fun ω =>
(affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1, ω.2.2.2.2)
have hF : Measurable F := by
dsimp [F]
have hx : Measurable (fun ω : Completion d => affineToCube ω.1) :=
affineToCube_measurable.comp measurable_fst
fun_prop (disch := assumption)
unfold CondExchangeable at hPc ⊢
apply condIndepFun_of_map
(hφ := hF) (hX := by fun_prop) (hY := by fun_prop) (hZ := by fun_prop)
change CondIndepFun
(comap
(affineToCube ∘ (fun ω : Completion d => ω.1)) inferInstance)
_ (fun ω : Completion d => (ω.2.2.1, ω.2.2.2.1))
(fun ω : Completion d => ω.2.1) Pc
refine condIndepFun_congr_conditioning
(B := ℝ × ℝ) (C := Bool)
(m₁ := comap (fun ω : Completion d => ω.1)
inferInstance)
(m₂ := comap
(affineToCube ∘ (fun ω : Completion d => ω.1)) inferInstance)
(affineToCube_comap_eq (fun ω : Completion d => ω.1)).symm _ _
(fun ω : Completion d => (ω.2.2.1, ω.2.2.2.1))
(fun ω : Completion d => ω.2.1) (mu := Pc) hPc
theorem affineCompletionMap_treatedConditionalMean Lemma affine-completion-map-treated-conditional-mean in the paper ↗
Affine covariate transport preserves the treated-potential conditional mean identity, independently of the chosen regularity representative. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
Pc :
Measure (Completion d)
IsFiniteMeasure (Pc.map (fun ω : Completion d => (affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1, ω.2.2.2.2)))
mu₁ :
(Fin d → ℝ) → ℝ
hsource :
(fun ω : Completion d => mu₁ ω.1)
hYint :
Integrable (fun ω : Completion d => ω.2.2.2.1) Pc
conclusion
(fun ω : Completion d => transformedResponse mu₁ ω.1)
=ᵐ[ Pc.map (fun ω : Completion d => (affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1, ω.2.2.2.2))] (Pc.map (fun ω : Completion d => (affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1, ω.2.2.2.2)))[(fun ω : Completion d => ω.2.2.2.1) | comap (fun ω => ω.1) inferInstance]
Proof (Lean source)
lemma affineCompletionMap_treatedConditionalMean {d : ℕ}
(Pc : Measure (Completion d)) [IsFiniteMeasure Pc]
[IsFiniteMeasure (Pc.map (fun ω : Completion d =>
(affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1,
ω.2.2.2.2)))]
(mu₁ : (Fin d → ℝ) → ℝ)
(hsource : (fun ω : Completion d => mu₁ ω.1) =ᵐ[Pc]
Pc[(fun ω : Completion d => ω.2.2.2.1) |
comap (fun ω => ω.1) inferInstance])
(hYint : Integrable (fun ω : Completion d => ω.2.2.2.1) Pc) :
(fun ω : Completion d => transformedResponse mu₁ ω.1) =ᵐ[
Pc.map (fun ω : Completion d =>
(affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1,
ω.2.2.2.2))]
(Pc.map (fun ω : Completion d =>
(affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1,
ω.2.2.2.2)))[(fun ω : Completion d => ω.2.2.2.1) |
comap (fun ω => ω.1) inferInstance] := by
let F : Completion d → Completion d := fun ω =>
(affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1, ω.2.2.2.2)
let G : Completion d → Completion d := fun ω =>
((fun i => 2 * ω.1 i - 1), ω.2.1, ω.2.2.1, ω.2.2.2.1, ω.2.2.2.2)
have hF : Measurable F := by
dsimp [F]
have hx : Measurable (fun ω : Completion d => affineToCube ω.1) :=
affineToCube_measurable.comp measurable_fst
fun_prop (disch := assumption)
have hG : Measurable G := by
dsimp [G]
have hx : Measurable (fun ω : Completion d => fun i => 2 * ω.1 i - 1) :=
affineFromCube_measurable.comp measurable_fst
fun_prop (disch := assumption)
have hmapG : (Pc.map F).map G = Pc := by
rw [Measure.map_map hG hF]
have hGF : G ∘ F = id := by
funext ω
simp only [Function.comp_apply, F, G, id_eq, affineToCube_inverse]
rw [hGF, Measure.map_id]
have hFG : F ∘ G = id := by
funext ω
simp only [Function.comp_apply, F, G, id_eq, affineToCube_affineFromCube]
let Y₁ : Completion d → ℝ := fun ω => ω.2.2.2.1
let X : Completion d → (Fin d → ℝ) := fun ω => ω.1
have hmX : comap X
(inferInstance : MeasurableSpace (Fin d → ℝ)) ≤
(inferInstance : MeasurableSpace (Completion d)) := by
exact comap_le (by dsimp [X]; fun_prop)
have hYint' : Integrable Y₁ (Pc.map F) := by
apply (integrable_map_measure (by dsimp [Y₁]; fun_prop)
hF.aemeasurable).2
exact hYint
have hmPull : comap F
(comap X
(inferInstance : MeasurableSpace (Fin d → ℝ))) =
comap (fun ω : Completion d => ω.1) inferInstance := by
calc
_ = comap (X ∘ F) inferInstance :=
MeasurableSpace.comap_comp
_ = comap
(affineToCube ∘ (fun ω : Completion d => ω.1)) inferInstance := by rfl
_ = _ := affineToCube_comap_eq _
have hcond :=
condExp_comp_of_map_eq
F hF (show Pc.map F = Pc.map F from rfl)
(comap X inferInstance) hmX hYint'
have hcond' :
Pc[Y₁ | comap (fun ω : Completion d => ω.1)
inferInstance] =ᵐ[Pc]
(Pc.map F)[Y₁ | comap X inferInstance] ∘ F := by
exact (condExp_congr_conditioning hmPull.symm Pc Y₁).trans hcond
apply eventuallyEq_of_comp_aeRetraction
F G hG hmapG (Filter.Eventually.of_forall fun ω => congrFun hFG ω)
change (fun ω : Completion d => transformedResponse mu₁ ω.1) ∘ F =ᵐ[Pc]
((Pc.map F)[Y₁ | comap X inferInstance]) ∘ F
calc
_ = fun ω : Completion d => mu₁ ω.1 := by
funext ω
exact transformedResponse_affineToCube mu₁ ω.1
_ =ᵐ[Pc] Pc[Y₁ | comap
(fun ω : Completion d => ω.1) inferInstance] := hsource
_ =ᵐ[Pc] _ := hcond'
theorem affineCompletionMap_holderResponse Lemma affine-completion-map-holder-response in the paper ↗
An affine covariate pushforward transports a potential-outcome conditional mean once the transformed representative has the requested Holder bound. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
Pc :
Measure (Completion d)
IsFiniteMeasure (Pc.map (fun ω : Completion d => (affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1, ω.2.2.2.2)))
mu₁ :
(Fin d → ℝ) → ℝ
beta L :
ℝ
hsource :
HolderResponse Pc mu₁ beta L
hball :
HolderBall beta L (transformedResponse mu₁)
hYint :
Integrable (fun ω : Completion d => ω.2.2.2.1) Pc
conclusion
HolderResponse (Pc.map (fun ω : Completion d => (affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1, ω.2.2.2.2))) (transformedResponse mu₁) beta L
Proof (Lean source)
lemma affineCompletionMap_holderResponse {d : ℕ}
(Pc : Measure (Completion d)) [IsFiniteMeasure Pc]
[IsFiniteMeasure (Pc.map (fun ω : Completion d =>
(affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1,
ω.2.2.2.2)))]
(mu₁ : (Fin d → ℝ) → ℝ) (beta L : ℝ)
(hsource : HolderResponse Pc mu₁ beta L)
(hball : HolderBall beta L (transformedResponse mu₁))
(hYint : Integrable (fun ω : Completion d => ω.2.2.2.1) Pc) :
HolderResponse (Pc.map (fun ω : Completion d =>
(affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1,
ω.2.2.2.2))) (transformedResponse mu₁) beta L := by
let F : Completion d → Completion d := fun ω =>
(affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1, ω.2.2.2.2)
let G : Completion d → Completion d := fun ω =>
((fun i => 2 * ω.1 i - 1), ω.2.1, ω.2.2.1, ω.2.2.2.1, ω.2.2.2.2)
have hF : Measurable F := by
dsimp [F]
have hx : Measurable (fun ω : Completion d => affineToCube ω.1) :=
affineToCube_measurable.comp measurable_fst
fun_prop (disch := assumption)
have hG : Measurable G := by
dsimp [G]
have hx : Measurable
(fun ω : Completion d => fun i => 2 * ω.1 i - 1) :=
affineFromCube_measurable.comp measurable_fst
fun_prop (disch := assumption)
have hmapG : (Pc.map F).map G = Pc := by
rw [Measure.map_map hG hF]
have hGF : G ∘ F = id := by
funext ω
simp only [Function.comp_apply, F, G, id_eq, affineToCube_inverse]
rw [hGF, Measure.map_id]
have hFG : F ∘ G = id := by
funext ω
simp only [Function.comp_apply, F, G, id_eq, affineToCube_affineFromCube]
let Y₁ : Completion d → ℝ := fun ω => ω.2.2.2.1
let X : Completion d → (Fin d → ℝ) := fun ω => ω.1
have hmX : comap X
(inferInstance : MeasurableSpace (Fin d → ℝ)) ≤
(inferInstance : MeasurableSpace (Completion d)) := by
exact comap_le (by dsimp [X]; fun_prop)
have hYint' : Integrable Y₁ (Pc.map F) := by
apply (integrable_map_measure (by dsimp [Y₁]; fun_prop)
hF.aemeasurable).2
change Integrable (fun ω : Completion d => ω.2.2.2.1) Pc
exact hYint
have hmPull : comap F
(comap X
(inferInstance : MeasurableSpace (Fin d → ℝ))) =
comap (fun ω : Completion d => ω.1) inferInstance := by
calc
comap F (comap X inferInstance) =
comap (X ∘ F) inferInstance := by
exact MeasurableSpace.comap_comp
_ = comap
(affineToCube ∘ (fun ω : Completion d => ω.1)) inferInstance := by rfl
_ = comap (fun ω : Completion d => ω.1)
inferInstance := affineToCube_comap_eq _
have hcond :=
condExp_comp_of_map_eq
F hF (show Pc.map F = Pc.map F from rfl)
(comap X inferInstance) hmX hYint'
have hcond' :
Pc[Y₁ | comap (fun ω : Completion d => ω.1)
inferInstance] =ᵐ[Pc]
(Pc.map F)[Y₁ | comap X inferInstance] ∘ F := by
exact (condExp_congr_conditioning hmPull.symm Pc Y₁).trans hcond
refine ⟨hball, ?_⟩
apply eventuallyEq_of_comp_aeRetraction
F G hG hmapG (Filter.Eventually.of_forall fun ω => congrFun hFG ω)
have hsourceMean := hsource.2
change (fun ω : Completion d => transformedResponse mu₁ ω.1) ∘ F =ᵐ[Pc]
((Pc.map F)[Y₁ | comap X inferInstance]) ∘ F
calc
(fun ω : Completion d => transformedResponse mu₁ ω.1) ∘ F
= fun ω : Completion d => mu₁ ω.1 := by
funext ω
exact transformedResponse_affineToCube mu₁ ω.1
_ =ᵐ[Pc] Pc[Y₁ | comap
(fun ω : Completion d => ω.1) inferInstance] := by
simpa only [Y₁] using hsourceMean
_ =ᵐ[Pc] ((Pc.map F)[Y₁ | comap X inferInstance]) ∘ F := hcond'
For the stated dimension, Gaussian scale, response functions, measurability certificates, and covariate, the Gaussian completion kernel evaluates to the corresponding pointwise completion.
Formal statement
hypotheses
conclusion
gaussianCompletionKernel σ e μ₀ μ₁ he hμ₀ hμ₁ x = gaussianCompletionAt σ e μ₀ μ₁ x
Proof (Lean source)
@[simp] lemma gaussianCompletionKernel_apply {d : ℕ} (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁)
(x : Fin d → ℝ) :
gaussianCompletionKernel σ e μ₀ μ₁ he hμ₀ hμ₁ x =
gaussianCompletionAt σ e μ₀ μ₁ x := rfl
At fixed covariates, observing the independent Gaussian completion selects the Gaussian response belonging to the realized treatment arm. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
(gaussianCompletionAt σ e μ₀ μ₁ x).map observed
Proof (Lean source)
lemma gaussianCompletionAt_observed_map {d : ℕ} (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ)
(he : e x ∈ Icc (0 : ℝ) 1) :
(gaussianCompletionAt σ e μ₀ μ₁ x).map observed =
ofReal (1 - e x) •
((Measure.dirac x).prod ((Measure.dirac false).prod
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩))) +
ofReal (e x) •
((Measure.dirac x).prod ((Measure.dirac true).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))) := by
letI := realBernoulli_probability (e x) he
letI : IsProbabilityMeasure
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
letI : IsProbabilityMeasure
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
unfold gaussianCompletionAt
rw [Measure.map_map (by unfold observed; fun_prop) (by
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption))]
rw [Measure.dirac_prod]
let g : Bool × ℝ × ℝ → Obs d :=
fun z => (x, z.1, if z.1 then z.2.2 else z.2.1)
change Measure.map g ((realBernoulli (e x)).prod
((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩))) = _
have hg : Measurable g := by
dsimp [g]
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption)
unfold realBernoulli
rw [Measure.add_prod, Measure.map_add _ _ hg]
congr 1
· rw [Measure.prod_smul_left, Measure.map_smul]
congr 1
rw [Measure.dirac_prod, Measure.map_map hg (by fun_prop)]
change Measure.map ((fun y : ℝ => (x, false, y)) ∘ fst)
((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩)) = _
rw [← Measure.map_map (by fun_prop : Measurable
(fun y : ℝ => (x, false, y))) measurable_fst, Measure.map_fst_prod]
simp only [measure_univ, one_smul]
rw [Measure.dirac_prod (x := false), Measure.map_map (by fun_prop) (by fun_prop)]
rfl
· rw [Measure.prod_smul_left, Measure.map_smul]
congr 1
rw [Measure.dirac_prod, Measure.map_map hg (by fun_prop)]
change Measure.map ((fun y : ℝ => (x, true, y)) ∘ snd)
((gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).prod
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩)) = _
rw [← Measure.map_map (by fun_prop : Measurable
(fun y : ℝ => (x, true, y))) measurable_snd, Measure.map_snd_prod]
simp only [measure_univ, one_smul]
rw [Measure.dirac_prod (x := true), Measure.dirac_prod x,
Measure.map_map (by fun_prop) (by fun_prop)]
rfl
theorem gaussianCompletionAt_observed_compProd Lemma gaussian-completion-at-observed-comp-prod in the paper ↗
Fixed-covariate Gaussian completion agrees with a treatment Bernoulli followed by the selected Gaussian arm kernel. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
kY :
ProbabilityTheory.Kernel.mk (fun a => if a then gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩ else gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩) (measurable_of_finite _) (: Kernel Bool ℝ)
(gaussianCompletionAt σ e μ₀ μ₁ x).map observed
= (Measure.compProd (realBernoulli (e x)) kY).map (fun ay => (x, ay.1, ay.2))
Proof (Lean source)
lemma gaussianCompletionAt_observed_compProd {d : ℕ} (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ)
(he : e x ∈ Icc (0 : ℝ) 1) :
let kY : Kernel Bool ℝ :=
ProbabilityTheory.Kernel.mk (fun a => if a then
gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩)
(measurable_of_finite _)
(gaussianCompletionAt σ e μ₀ μ₁ x).map observed =
(Measure.compProd (realBernoulli (e x)) kY).map
(fun ay => (x, ay.1, ay.2)) := by
dsimp
letI : IsProbabilityMeasure
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
letI : IsProbabilityMeasure
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
let kY : Kernel Bool ℝ :=
ProbabilityTheory.Kernel.mk (fun a => if a then
gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩)
(measurable_of_finite _)
letI : IsMarkovKernel kY := ⟨fun a => by
change IsProbabilityMeasure (if a then
gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩)
cases a <;> exact ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _⟩
rw [gaussianCompletionAt_observed_map σ e μ₀ μ₁ x he]
have hsource : Measure.compProd (realBernoulli (e x)) kY =
ofReal (1 - e x) •
(gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩).map (Prod.mk false) +
ofReal (e x) •
(gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩).map (Prod.mk true) := by
unfold realBernoulli
rw [Measure.compProd_add_left, Measure.compProd_smul_left,
Measure.compProd_smul_left,
dirac_compProd_eq_map_prodMk false kY,
dirac_compProd_eq_map_prodMk true kY]
rfl
rw [hsource, Measure.map_add _ _ (by fun_prop),
Measure.map_smul, Measure.map_smul]
congr 1
· rw [Measure.map_map (by fun_prop) (by fun_prop),
Measure.dirac_prod, Measure.dirac_prod,
Measure.map_map (by fun_prop) (by fun_prop)]
· rw [Measure.map_map (by fun_prop) (by fun_prop),
Measure.dirac_prod, Measure.dirac_prod,
Measure.map_map (by fun_prop) (by fun_prop)]
Binding the Gaussian completion over a covariate law produces the usual two-stage treatment and selected-arm Gaussian observed law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
σ :
ℝ
e μ₀ μ₁ :
(Fin d → ℝ) → ℝ
he :
hμ₀ :
Measurable μ₀
hμ₁ :
Measurable μ₁
he01 :
∀ x, e x ∈ Icc (0 : ℝ) 1
conclusion
kA :
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x)) (by unfold realBernoulli; fun_prop) (: Kernel (Fin d → ℝ) Bool)
kY :
ProbabilityTheory.Kernel.mk (fun xa => if xa.2 then gaussianReal (μ₁ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩ else gaussianReal (μ₀ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩) (by apply Measurable.ite (measurable_snd (MeasurableSet.singleton true)) · fun_prop · fun_prop) (: Kernel ((Fin d → ℝ) × Bool) ℝ)
(mu.bind (gaussianCompletionAt σ e μ₀ μ₁)).map observed
= ((mu ⊗ₘ kA) ⊗ₘ kY).map (fun z => (z.1.1, z.1.2, z.2))
Proof (Lean source)
lemma gaussianCompletionBind_observed {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁)
(he01 : ∀ x, e x ∈ Icc (0 : ℝ) 1) :
let kA : Kernel (Fin d → ℝ) Bool :=
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x))
(by unfold realBernoulli; fun_prop)
let kY : Kernel ((Fin d → ℝ) × Bool) ℝ :=
ProbabilityTheory.Kernel.mk (fun xa => if xa.2 then
gaussianReal (μ₁ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩)
(by
apply Measurable.ite (measurable_snd (MeasurableSet.singleton true))
· fun_prop
· fun_prop)
(mu.bind (gaussianCompletionAt σ e μ₀ μ₁)).map observed =
((mu ⊗ₘ kA) ⊗ₘ kY).map (fun z => (z.1.1, z.1.2, z.2)) := by
dsimp
let kA : Kernel (Fin d → ℝ) Bool :=
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x))
(by unfold realBernoulli; fun_prop)
let kY : Kernel ((Fin d → ℝ) × Bool) ℝ :=
ProbabilityTheory.Kernel.mk (fun xa => if xa.2 then
gaussianReal (μ₁ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩)
(by
apply Measurable.ite (measurable_snd (MeasurableSet.singleton true))
· fun_prop
· fun_prop)
letI : IsMarkovKernel kA :=
⟨fun x => realBernoulli_probability (e x) (he01 x)⟩
letI : IsMarkovKernel kY :=
⟨fun xa => by
change IsProbabilityMeasure (if xa.2 then
gaussianReal (μ₁ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩)
cases xa.2 <;>
exact instIsProbabilityMeasureGaussianReal _ _⟩
have hk := gaussianCompletionAt_measurable σ e μ₀ μ₁ he hμ₀ hμ₁
apply Measure.ext
intro s hs
rw [Measure.map_apply (by unfold observed; fun_prop) hs,
Measure.bind_apply (hs.preimage (by unfold observed; fun_prop)) hk.aemeasurable,
Measure.map_apply (by fun_prop) hs,
Measure.compProd_apply (hs.preimage (by fun_prop)),
Measure.lintegral_compProd]
swap
· exact ProbabilityTheory.Kernel.measurable_kernel_prodMk_left
(hs.preimage (by fun_prop))
apply lintegral_congr
intro x
let kYx : Kernel Bool ℝ :=
ProbabilityTheory.Kernel.mk (fun a => if a then
gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩)
(measurable_of_finite _)
letI : IsMarkovKernel kYx := ⟨fun a => by
change IsProbabilityMeasure (if a then
gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩)
cases a <;> exact instIsProbabilityMeasureGaussianReal _ _⟩
have hfix := gaussianCompletionAt_observed_compProd σ e μ₀ μ₁ x (he01 x)
change (gaussianCompletionAt σ e μ₀ μ₁ x).map observed =
(Measure.compProd (realBernoulli (e x)) kYx).map
(fun ay => (x, ay.1, ay.2)) at hfix
have happ := congrArg (fun m : Measure (Obs d) => m s) hfix
rw [Measure.map_apply (by unfold observed; fun_prop) hs] at happ
rw [Measure.map_apply (by fun_prop) hs] at happ
rw [Measure.compProd_apply (hs.preimage (by fun_prop))] at happ
rw [happ]
apply lintegral_congr
intro b
change (if b then
gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩)
((fun y => (x, b, y)) ⁻¹' s) = _
rfl
theorem gaussianCompletionBind_observedKernel_ae Lemma gaussian-completion-bind-observed-kernel in the paper ↗
The observed response kernel of a Gaussian completion is the selected Gaussian arm kernel. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
σ :
ℝ
e μ₀ μ₁ :
(Fin d → ℝ) → ℝ
he :
hμ₀ :
Measurable μ₀
hμ₁ :
Measurable μ₁
hvalid :
∀ x, e x ∈ Icc (0 : ℝ) 1
conclusion
P :
(mu.bind (gaussianCompletionAt σ e μ₀ μ₁)).map observed
kY :
ProbabilityTheory.Kernel.mk (fun xa => if xa.2 then gaussianReal (μ₁ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩ else gaussianReal (μ₀ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩) (by apply Measurable.ite (measurable_snd (MeasurableSet.singleton true)) · fun_prop · fun_prop) (: Kernel ((Fin d → ℝ) × Bool) ℝ)
condDistrib (fun z : Obs d => z.2.2) (fun z : Obs d => (z.1, z.2.1)) P
=ᵐ[ P.map (fun z => (z.1, z.2.1))] kY
Proof (Lean source)
lemma gaussianCompletionBind_observedKernel_ae {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁)
(hvalid : ∀ x, e x ∈ Icc (0 : ℝ) 1)
[IsFiniteMeasure
((mu.bind (gaussianCompletionAt σ e μ₀ μ₁)).map observed)] :
let P := (mu.bind (gaussianCompletionAt σ e μ₀ μ₁)).map observed
let kY : Kernel ((Fin d → ℝ) × Bool) ℝ :=
ProbabilityTheory.Kernel.mk (fun xa => if xa.2 then
gaussianReal (μ₁ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩)
(by
apply Measurable.ite (measurable_snd (MeasurableSet.singleton true))
· fun_prop
· fun_prop)
condDistrib (fun z : Obs d => z.2.2)
(fun z : Obs d => (z.1, z.2.1)) P =ᵐ[
P.map (fun z => (z.1, z.2.1))] kY := by
dsimp
let P := (mu.bind (gaussianCompletionAt σ e μ₀ μ₁)).map observed
let kA : Kernel (Fin d → ℝ) Bool :=
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x))
(by unfold realBernoulli; fun_prop)
let kY : Kernel ((Fin d → ℝ) × Bool) ℝ :=
ProbabilityTheory.Kernel.mk (fun xa => if xa.2 then
gaussianReal (μ₁ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩)
(by
apply Measurable.ite (measurable_snd (MeasurableSet.singleton true))
· fun_prop
· fun_prop)
letI : IsMarkovKernel kA :=
⟨fun x => realBernoulli_probability (e x) (hvalid x)⟩
letI : IsMarkovKernel kY := ⟨fun xa => by
change IsProbabilityMeasure (if xa.2 then
gaussianReal (μ₁ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩
else gaussianReal (μ₀ xa.1) ⟨σ ^ 2, sq_nonneg σ⟩)
cases xa.2 <;>
exact instIsProbabilityMeasureGaussianReal _ _⟩
have hobsform := gaussianCompletionBind_observed mu σ e μ₀ μ₁
he hμ₀ hμ₁ hvalid
have hjoint : P.map (fun z : Obs d => ((z.1, z.2.1), z.2.2)) =
Measure.compProd (Measure.compProd mu kA) kY := by
dsimp [P]
rw [hobsform]
rw [Measure.map_map (by fun_prop) (by fun_prop)]
have hid : (fun z : Obs d => ((z.1, z.2.1), z.2.2)) ∘
(fun z : ((Fin d → ℝ) × Bool) × ℝ => (z.1.1, z.1.2, z.2)) = id := by
funext z
rfl
rw [hid, Measure.map_id]
have hobs : Measurable (@observed d) := by unfold observed; fun_prop
have hxa : P.map (fun z => (z.1, z.2.1)) =
Measure.compProd mu kA := by
dsimp [P]
rw [Measure.map_map (by fun_prop) hobs]
exact gaussianCompletionBind_covariate_treatment mu σ e μ₀ μ₁
he hμ₀ hμ₁ hvalid
exact ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd_of_measurable
(μ := P) (X := fun z : Obs d => (z.1, z.2.1))
(Y := fun z : Obs d => z.2.2) (κ := kY)
(by fun_prop) (by fun_prop) (by simpa [hxa] using hjoint)
theorem gaussianCompletionBind_observed_eq_of_selected Lemma gaussian-completion-bind-observed-of-selected in the paper ↗
The Gaussian completion reconstructed from a selected propensity and the two Gaussian arm kernels has exactly the original observed law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
P :
Measure (Obs d)
σ :
ℝ
e μ₀ μ₁ :
(Fin d → ℝ) → ℝ
he :
hμ₀ :
Measurable μ₀
hμ₁ :
Measurable μ₁
he01 :
∀ x, e x ∈ Icc (0 : ℝ) 1
hselected :
∀ᵐ x ∂covariateLaw P, e x = propensity P x ∧ 0 < e x ∧ e x < 1
harms :
= gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩ ∧
conclusion
Proof (Lean source)
lemma gaussianCompletionBind_observed_eq_of_selected {d : ℕ}
(P : Measure (Obs d)) [IsProbabilityMeasure P] (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁)
(he01 : ∀ x, e x ∈ Icc (0 : ℝ) 1)
(hselected : ∀ᵐ x ∂covariateLaw P,
e x = propensity P x ∧ 0 < e x ∧ e x < 1)
(harms : ∀ᵐ x ∂covariateLaw P,
armKernel P x true =
gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩ ∧
armKernel P x false =
gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩) :
((covariateLaw P).bind (gaussianCompletionAt σ e μ₀ μ₁)).map observed = P := by
letI : IsProbabilityMeasure (covariateLaw P) := by
unfold covariateLaw
exact Measure.isProbabilityMeasure_map (by fun_prop)
rw [gaussianCompletionBind_observed (covariateLaw P) σ e μ₀ μ₁
he hμ₀ hμ₁ he01]
exact gaussianObservedLaw_recomposition_of_selected P σ e μ₀ μ₁
he hμ₀ hμ₁ he01 hselected harms
theorem gaussianCompletionBind_affine_observed_eq_of_selected Lemma gaussian-completion-affine-observed-transport in the paper ↗
Affinely transporting the covariate coordinate of the reconstructed Gaussian completion gives exactly the transported observed law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
P :
Measure (Obs d)
σ :
ℝ
e μ₀ μ₁ :
(Fin d → ℝ) → ℝ
he :
hμ₀ :
Measurable μ₀
hμ₁ :
Measurable μ₁
he01 :
∀ x, e x ∈ Icc (0 : ℝ) 1
hselected :
∀ᵐ x ∂covariateLaw P, e x = propensity P x ∧ 0 < e x ∧ e x < 1
harms :
= gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩ ∧
conclusion
Pc :
(Pc.map (fun ω : Completion d => (affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1, ω.2.2.2.2))).map observed
Proof (Lean source)
lemma gaussianCompletionBind_affine_observed_eq_of_selected {d : ℕ}
(P : Measure (Obs d)) [IsProbabilityMeasure P] (σ : ℝ)
(e μ₀ μ₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hμ₀ : Measurable μ₀) (hμ₁ : Measurable μ₁)
(he01 : ∀ x, e x ∈ Icc (0 : ℝ) 1)
(hselected : ∀ᵐ x ∂covariateLaw P,
e x = propensity P x ∧ 0 < e x ∧ e x < 1)
(harms : ∀ᵐ x ∂covariateLaw P,
armKernel P x true =
gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩ ∧
armKernel P x false =
gaussianReal (μ₀ x) ⟨σ ^ 2, sq_nonneg σ⟩) :
let Pc := (covariateLaw P).bind (gaussianCompletionAt σ e μ₀ μ₁)
(Pc.map (fun ω : Completion d =>
(affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1,
ω.2.2.2.2))).map observed = transformedDornLaw P := by
dsimp
let F : Completion d → Completion d := fun ω =>
(affineToCube ω.1, ω.2.1, ω.2.2.1, ω.2.2.2.1, ω.2.2.2.2)
let G : Obs d → Obs d := fun z =>
(affineToCube z.1, z.2.1, z.2.2)
have hF : Measurable F := by
dsimp [F]
have hx : Measurable (fun ω : Completion d => affineToCube ω.1) :=
affineToCube_measurable.comp measurable_fst
fun_prop (disch := assumption)
have hG : Measurable G := by
dsimp [G]
have hx : Measurable (fun z : Obs d => affineToCube z.1) :=
affineToCube_measurable.comp measurable_fst
fun_prop (disch := assumption)
have hobs : Measurable (@observed d) := by
unfold observed
fun_prop
have hsource := gaussianCompletionBind_observed_eq_of_selected
P σ e μ₀ μ₁ he hμ₀ hμ₁ he01 hselected harms
calc
(((covariateLaw P).bind
(gaussianCompletionAt σ e μ₀ μ₁)).map F).map observed =
((covariateLaw P).bind (gaussianCompletionAt σ e μ₀ μ₁)).map
(observed ∘ F) := Measure.map_map hobs hF
_ = ((covariateLaw P).bind (gaussianCompletionAt σ e μ₀ μ₁)).map
(G ∘ observed) := by rfl
_ = (((covariateLaw P).bind
(gaussianCompletionAt σ e μ₀ μ₁)).map observed).map G :=
(Measure.map_map hG hobs).symm
_ = P.map G := by rw [hsource]
_ = transformedDornLaw P := rfl
def gaussianThinStripOriginal
Gaussianized thin-strip completion, rescaled to Dorn's cube. For the stated inputs and conditions, the gaussianThinStripOriginal object being defined.
Definition (Lean source)
parameters
d :
ℕ
hd :
2 ≤ d
γ a σ :
ℝ
def
given by
Pc :
(volume.restrict (cube d)).bind (gaussianCompletionAt σ (fun x => min (3 / 4 : ℝ) (stripPropensity d hd γ a x)) (fun _ => 0) (fun _ => 0)) (: Measure (Completion d))
(Pc.map observed).map (fun z => ((fun i => 2 * z.1 i - 1), z.2.1, z.2.2))
def gaussianStripPropensity
Original-cube version of the capped thin-strip propensity. For the stated inputs and conditions, the gaussianStripPropensity object being defined.
Definition (Lean source)
parameters
def
gaussianStripPropensity d hd γ a x :
ℝ
given by
The inverse affine map sends unit-cube volume to normalized volume on Dorn's cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma cubeVolume_map_affineFromCube (d : ℕ) :
(volume.restrict (cube d)).map
(fun x : Fin d → ℝ => fun i => 2 * x i - 1) =
(ofReal ((2 : ℝ) ^ d))⁻¹ •
volume.restrict (dornCube d) := by
let r : (Fin d → ℝ) → (Fin d → ℝ) := affineToCube
let s : (Fin d → ℝ) → (Fin d → ℝ) := fun x i => 2 * x i - 1
have hr : Measurable r := affineToCube_measurable
have hs : Measurable s := affineFromCube_measurable
have h := congrArg (fun m : Measure (Fin d → ℝ) => m.map s)
(normalizedDornVolume_map_affineToCube d)
rw [Measure.map_map hs hr] at h
have hsr : s ∘ r = id := by
funext x
exact affineToCube_inverse x
rw [hsr, Measure.map_id] at h
exact h.symm
theorem gaussianThinStripOriginal_probability Lemma gaussian-thin-strip-original-probability in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a σ :
ℝ
hγ :
1 < γ
conclusion
IsProbabilityMeasure (gaussianThinStripOriginal d hd γ a σ)
Proof (Lean source)
lemma gaussianThinStripOriginal_probability (d : ℕ) (hd : 2 ≤ d)
(γ a σ : ℝ) (hγ : 1 < γ) :
IsProbabilityMeasure (gaussianThinStripOriginal d hd γ a σ) := by
let e : (Fin d → ℝ) → ℝ := fun x =>
min (3 / 4 : ℝ) (stripPropensity d hd γ a x)
have he : Measurable e := by
dsimp [e]
exact measurable_const.min (stripPropensity_measurable d hd γ a)
have hevalid : ∀ x, e x ∈ Icc (0 : ℝ) 1 := by
intro x
have hx := stripPropensity_mem_Icc d hd γ a hγ x
exact ⟨le_min (by norm_num) hx.1, (min_le_left _ _).trans (by norm_num)⟩
have hk : Measurable (gaussianCompletionAt σ e (fun _ => 0) (fun _ => 0)) :=
gaussianCompletionAt_measurable σ e (fun _ => 0) (fun _ => 0)
he measurable_const measurable_const
have hpoint : ∀ x, IsProbabilityMeasure
(gaussianCompletionAt σ e (fun _ => 0) (fun _ => 0) x) := by
intro x
letI := realBernoulli_probability (e x) (hevalid x)
letI : IsProbabilityMeasure
(gaussianReal 0 ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
unfold gaussianCompletionAt
apply Measure.isProbabilityMeasure_map
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption)
let mu := volume.restrict (cube d)
letI : IsProbabilityMeasure mu := by
apply isProbabilityMeasure_iff.mpr
dsimp [mu]
rw [Measure.restrict_apply MeasurableSet.univ, Set.univ_inter]
have hcube : cube d =
Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hcube, Real.volume_Icc_pi]
simp
let Pc := mu.bind (gaussianCompletionAt σ e (fun _ => 0) (fun _ => 0))
letI : IsProbabilityMeasure Pc :=
isProbabilityMeasure_bind hk.aemeasurable
(Filter.Eventually.of_forall hpoint)
letI : IsProbabilityMeasure (Pc.map observed) :=
Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)
change IsProbabilityMeasure ((Pc.map observed).map
(fun z : Obs d => ((fun i => 2 * z.1 i - 1), z.2.1, z.2.2)))
exact Measure.isProbabilityMeasure_map (by fun_prop)
theorem gaussianThinStripOriginal_covariateLaw Lemma gaussian-thin-strip-original-covariate in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a σ :
ℝ
hγ :
1 < γ
conclusion
covariateLaw (gaussianThinStripOriginal d hd γ a σ)
Proof (Lean source)
lemma gaussianThinStripOriginal_covariateLaw (d : ℕ) (hd : 2 ≤ d)
(γ a σ : ℝ) (hγ : 1 < γ) :
covariateLaw (gaussianThinStripOriginal d hd γ a σ) =
(ofReal ((2 : ℝ) ^ d))⁻¹ •
volume.restrict (dornCube d) := by
let e : (Fin d → ℝ) → ℝ := fun x =>
min (3 / 4 : ℝ) (stripPropensity d hd γ a x)
have he : Measurable e := by
dsimp [e]
exact measurable_const.min (stripPropensity_measurable d hd γ a)
have hevalid : ∀ x, e x ∈ Icc (0 : ℝ) 1 := by
intro x
have hx := stripPropensity_mem_Icc d hd γ a hγ x
exact ⟨le_min (by norm_num) hx.1, (min_le_left _ _).trans (by norm_num)⟩
let mu := volume.restrict (cube d)
let Pc := mu.bind (gaussianCompletionAt σ e (fun _ => 0) (fun _ => 0))
have hcov : Pc.map fst = mu := by
dsimp [Pc]
exact gaussianCompletionBind_covariate_marginal_ae mu σ e
(fun _ => 0) (fun _ => 0) he measurable_const measurable_const
(Filter.Eventually.of_forall hevalid)
unfold gaussianThinStripOriginal covariateLaw
rw [Measure.map_map (by fun_prop) (by fun_prop),
Measure.map_map (by fun_prop) (by unfold observed; fun_prop)]
change Pc.map ((fun x : Fin d → ℝ => fun i => 2 * x i - 1) ∘ fst) = _
rw [← Measure.map_map affineFromCube_measurable measurable_fst, hcov]
exact cubeVolume_map_affineFromCube d
theorem transformedDornLaw_gaussianThinStripOriginal Lemma transformed-gaussian-thin-strip-original in the paper ↗
Affine normalization cancels the inverse affine map in the definition of the original-cube Gaussian strip law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a σ :
ℝ
conclusion
transformedDornLaw (gaussianThinStripOriginal d hd γ a σ)
= ((volume.restrict (cube d)).bind (gaussianCompletionAt σ (fun x => min (3 / 4 : ℝ) (stripPropensity d hd γ a x)) (fun _ => 0) (fun _ => 0))).map observed
Proof (Lean source)
lemma transformedDornLaw_gaussianThinStripOriginal (d : ℕ) (hd : 2 ≤ d)
(γ a σ : ℝ) :
transformedDornLaw (gaussianThinStripOriginal d hd γ a σ) =
((volume.restrict (cube d)).bind
(gaussianCompletionAt σ
(fun x => min (3 / 4 : ℝ) (stripPropensity d hd γ a x))
(fun _ => 0) (fun _ => 0))).map observed := by
unfold transformedDornLaw gaussianThinStripOriginal
rw [Measure.map_map (by
have hx : Measurable (fun z : Obs d => affineToCube z.1) :=
affineToCube_measurable.comp measurable_fst
fun_prop (disch := assumption)) (by fun_prop)]
have hid : (fun z : Obs d => (affineToCube z.1, z.2.1, z.2.2)) ∘
(fun z : Obs d => ((fun i => 2 * z.1 i - 1), z.2.1, z.2.2)) = id := by
funext z
simp only [Function.comp_apply, id_eq, affineToCube_affineFromCube]
rw [hid, Measure.map_id]
theorem gaussianThinStripOriginal_selected_ae Lemma gaussian-thin-strip-original-selected-propensity in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a σ :
ℝ
hγ :
1 < γ
conclusion
P :
gaussianThinStripOriginal d hd γ a σ
_ :
gaussianThinStripOriginal_probability d hd γ a σ hγ (: IsProbabilityMeasure P)
∀ᵐ x ∂covariateLaw P, gaussianStripPropensity d hd γ a x = propensity P x
Proof (Lean source)
lemma gaussianThinStripOriginal_selected_ae (d : ℕ) (hd : 2 ≤ d)
(γ a σ : ℝ) (hγ : 1 < γ) :
let P := gaussianThinStripOriginal d hd γ a σ
let _ : IsProbabilityMeasure P :=
gaussianThinStripOriginal_probability d hd γ a σ hγ
∀ᵐ x ∂covariateLaw P,
gaussianStripPropensity d hd γ a x = propensity P x := by
dsimp
let e : (Fin d → ℝ) → ℝ := fun x =>
min (3 / 4 : ℝ) (stripPropensity d hd γ a x)
let Pc := (volume.restrict (cube d)).bind
(gaussianCompletionAt σ e (fun _ => 0) (fun _ => 0))
let Q := Pc.map observed
let P := gaussianThinStripOriginal d hd γ a σ
have he : Measurable e := by
dsimp [e]
exact measurable_const.min (stripPropensity_measurable d hd γ a)
have hevalid : ∀ x, e x ∈ Icc (0 : ℝ) 1 := by
intro x
have hx := stripPropensity_mem_Icc d hd γ a hγ x
exact ⟨le_min (by norm_num) hx.1, (min_le_left _ _).trans (by norm_num)⟩
letI : IsProbabilityMeasure P :=
gaussianThinStripOriginal_probability d hd γ a σ hγ
letI : IsProbabilityMeasure (volume.restrict (cube d)) := by
apply isProbabilityMeasure_iff.mpr
rw [Measure.restrict_apply MeasurableSet.univ, Set.univ_inter]
have hcube : cube d =
Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hcube, Real.volume_Icc_pi]
simp
have hk : Measurable
(gaussianCompletionAt σ e (fun _ => 0) (fun _ => 0)) :=
gaussianCompletionAt_measurable σ e (fun _ => 0) (fun _ => 0)
he measurable_const measurable_const
have hpoint : ∀ x, IsProbabilityMeasure
(gaussianCompletionAt σ e (fun _ => 0) (fun _ => 0) x) := by
intro x
letI := realBernoulli_probability (e x) (hevalid x)
letI : IsProbabilityMeasure
(gaussianReal 0 ⟨σ ^ 2, sq_nonneg σ⟩) :=
ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
unfold gaussianCompletionAt
apply Measure.isProbabilityMeasure_map
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption)
letI : IsProbabilityMeasure Pc := by
dsimp [Pc]
exact isProbabilityMeasure_bind hk.aemeasurable
(Filter.Eventually.of_forall hpoint)
letI : IsProbabilityMeasure Q := by
dsimp [Q]
exact Measure.isProbabilityMeasure_map (by unfold observed; fun_prop)
have hunit : ∀ᵐ y ∂volume.restrict (cube d), propensity Q y = e y := by
simpa [Pc, Q] using gaussianCompletionBind_propensity_ae
(volume.restrict (cube d)) σ e (fun _ => 0) (fun _ => 0)
he measurable_const measurable_const hevalid
have hPQ : transformedDornLaw P = Q := by
dsimp [P, Q, Pc, e]
exact transformedDornLaw_gaussianThinStripOriginal d hd γ a σ
have hcovQ : covariateLaw Q = volume.restrict (cube d) := by
dsimp [Q, Pc, covariateLaw]
rw [Measure.map_map (by fun_prop) (by unfold observed; fun_prop)]
exact gaussianCompletionBind_covariate_marginal_ae
(volume.restrict (cube d)) σ e (fun _ => 0) (fun _ => 0)
he measurable_const measurable_const
(Filter.Eventually.of_forall hevalid)
have hpull : ∀ᵐ x ∂covariateLaw P,
propensity Q (affineToCube x) = e (affineToCube x) := by
have hu : ∀ᵐ y ∂covariateLaw (transformedDornLaw P),
propensity Q y = e y := by
rw [hPQ, hcovQ]
exact hunit
rw [transformedDornLaw_covariateLaw P] at hu
exact ae_eq_comp affineToCube_measurable.aemeasurable hu
letI : IsProbabilityMeasure (transformedDornLaw P) :=
transformedDornLaw_probability P
have htransport := transformedDornLaw_treatmentKernel_ae P
filter_upwards [hpull, htransport] with x hx hkx
change e (affineToCube x) = propensity P x
rw [← hx]
unfold propensity
have hkx' : condDistrib (fun z : Obs d => z.2.1)
(fun z => z.1) Q (affineToCube x) =
condDistrib (fun z : Obs d => z.2.1)
(fun z => z.1) P x := by
simpa only [hPQ] using hkx
exact congrArg (fun m : Measure Bool => (m {true}).toReal) hkx'
theorem gaussianThinStripOriginal_selected Lemma gaussian-thin-strip-original-selected in the paper ↗
The capped inverse-affine strip is a valid selected propensity on Dorn's cube, including strict almost-sure overlap. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a σ :
ℝ
hγ :
1 < γ
conclusion
DornSelectedPropensity (dornCube d) (gaussianThinStripOriginal d hd γ a σ) (gaussianStripPropensity d hd γ a)
Proof (Lean source)
lemma gaussianThinStripOriginal_selected (d : ℕ) (hd : 2 ≤ d)
(γ a σ : ℝ) (hγ : 1 < γ) :
DornSelectedPropensity (dornCube d)
(gaussianThinStripOriginal d hd γ a σ)
(gaussianStripPropensity d hd γ a) := by
let P := gaussianThinStripOriginal d hd γ a σ
letI : IsProbabilityMeasure P :=
gaussianThinStripOriginal_probability d hd γ a σ hγ
have he : Measurable (gaussianStripPropensity d hd γ a) := by
unfold gaussianStripPropensity
exact (measurable_const.min (stripPropensity_measurable d hd γ a)).comp
affineToCube_measurable
refine ⟨inferInstance, he, ?_, ?_⟩
· intro x hx
have hs := stripPropensity_mem_Icc d hd γ a hγ (affineToCube x)
exact ⟨le_min (by norm_num) hs.1,
(min_le_left _ _).trans (by norm_num)⟩
· have heq := gaussianThinStripOriginal_selected_ae d hd γ a σ hγ
have hmapcov : (covariateLaw P).map affineToCube =
volume.restrict (cube d) := by
rw [← transformedDornLaw_covariateLaw P,
transformedDornLaw_gaussianThinStripOriginal d hd γ a σ]
unfold covariateLaw
rw [Measure.map_map (by fun_prop) (by unfold observed; fun_prop)]
exact gaussianCompletionBind_covariate_marginal_ae
(volume.restrict (cube d)) σ
(fun x => min (3 / 4 : ℝ) (stripPropensity d hd γ a x))
(fun _ => 0) (fun _ => 0)
(measurable_const.min (stripPropensity_measurable d hd γ a))
measurable_const measurable_const
(Filter.Eventually.of_forall fun x => by
have hs := stripPropensity_mem_Icc d hd γ a hγ x
exact ⟨le_min (by norm_num) hs.1,
(min_le_left _ _).trans (by norm_num)⟩)
have hposUnit := stripPropensity_uniform_pos_ae d hd γ a hγ
rw [← hmapcov] at hposUnit
have hpos := ae_of_ae_map (μ := covariateLaw P)
affineToCube_measurable.aemeasurable hposUnit
filter_upwards [heq, hpos] with x hx hp
exact ⟨hx, lt_min (by norm_num) hp,
(min_le_left _ _).trans_lt (by norm_num)⟩
theorem realBernoulli_ae_at_false_of_lt_one Lemma real-bernoulli-ae-at-false-of-lt-one in the paper ↗
An almost-sure Bernoulli property holds at false when the failure mass is positive. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma realBernoulli_ae_at_false_of_lt_one (r : ℝ) (hr : r < 1)
(q : Bool → Prop) (hq : ∀ᵐ a ∂realBernoulli r, q a) : q false := by
by_contra hfalse
have hzero : (realBernoulli r) {a | ¬ q a} = 0 := by
simpa only [Set.compl_setOf, Classical.not_not] using (mem_ae_iff.mp hq)
have hle : (realBernoulli r) ({false} : Set Bool) ≤
(realBernoulli r) {a | ¬ q a} := by
apply measure_mono
intro a ha
have ha' : a = false := by simpa using ha
subst a
exact hfalse
have hmass : (realBernoulli r) ({false} : Set Bool) =
ofReal (1 - r) := by
simp [realBernoulli]
rw [hzero, hmass] at hle
exact (not_le_of_gt (ENNReal.ofReal_pos.mpr (sub_pos.mpr hr))) hle
theorem gaussianThinStripOriginal_armKernels_ae Lemma gaussian-thin-strip-original-arm-kernels in the paper ↗
Both conditional response arms of the inverse-affine Gaussian strip law are the centred Gaussian used in its completion. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a σ :
ℝ
hγ :
1 < γ
conclusion
P :
gaussianThinStripOriginal d hd γ a σ
_ :
gaussianThinStripOriginal_probability d hd γ a σ hγ (: IsProbabilityMeasure P)
= gaussianReal 0 ⟨σ ^ 2, sq_nonneg σ⟩ ∧
Proof (Lean source)
lemma gaussianThinStripOriginal_armKernels_ae (d : ℕ) (hd : 2 ≤ d)
(γ a σ : ℝ) (hγ : 1 < γ) :
let P := gaussianThinStripOriginal d hd γ a σ
let _ : IsProbabilityMeasure P :=
gaussianThinStripOriginal_probability d hd γ a σ hγ
∀ᵐ x ∂covariateLaw P,
armKernel P x true = gaussianReal 0 ⟨σ ^ 2, sq_nonneg σ⟩ ∧
armKernel P x false = gaussianReal 0 ⟨σ ^ 2, sq_nonneg σ⟩ := by
dsimp
let e : (Fin d → ℝ) → ℝ := fun x =>
min (3 / 4 : ℝ) (stripPropensity d hd γ a x)
let Q := ((volume.restrict (cube d)).bind
(gaussianCompletionAt σ e (fun _ => 0) (fun _ => 0))).map observed
let P := gaussianThinStripOriginal d hd γ a σ
letI : IsProbabilityMeasure P :=
gaussianThinStripOriginal_probability d hd γ a σ hγ
have hPQ : transformedDornLaw P = Q := by
dsimp [P, Q, e]
exact transformedDornLaw_gaussianThinStripOriginal d hd γ a σ
letI : IsProbabilityMeasure Q := by
rw [← hPQ]
exact transformedDornLaw_probability P
letI : IsProbabilityMeasure (transformedDornLaw P) :=
transformedDornLaw_probability P
have he : Measurable e := by
dsimp [e]
exact measurable_const.min (stripPropensity_measurable d hd γ a)
have hevalid : ∀ x, e x ∈ Icc (0 : ℝ) 1 := by
intro x
have hx := stripPropensity_mem_Icc d hd γ a hγ x
exact ⟨le_min (by norm_num) hx.1,
(min_le_left _ _).trans (by norm_num)⟩
have hQkernel := gaussianCompletionBind_observedKernel_ae
(volume.restrict (cube d)) σ e (fun _ => 0) (fun _ => 0)
he measurable_const measurable_const hevalid
let E : ((Fin d → ℝ) × Bool) → ((Fin d → ℝ) × Bool) :=
fun xa => (affineToCube xa.1, xa.2)
have hE : Measurable E := by
dsimp [E]
exact (affineToCube_measurable.comp measurable_fst).prodMk measurable_snd
have hG : Measurable (fun z : Obs d =>
(affineToCube z.1, z.2.1, z.2.2)) := by
have hx : Measurable (fun z : Obs d => affineToCube z.1) :=
affineToCube_measurable.comp measurable_fst
fun_prop (disch := assumption)
have hmap : (P.map (fun z : Obs d => (z.1, z.2.1))).map E =
Q.map (fun z : Obs d => (z.1, z.2.1)) := by
rw [← hPQ]
unfold transformedDornLaw
rw [Measure.map_map hE (by fun_prop),
Measure.map_map (by fun_prop) hG]
rfl
have hQkernel' : ∀ᵐ xa ∂Q.map (fun z : Obs d => (z.1, z.2.1)),
condDistrib (fun z : Obs d => z.2.2)
(fun z : Obs d => (z.1, z.2.1)) Q xa =
gaussianReal 0 ⟨σ ^ 2, sq_nonneg σ⟩ := by
filter_upwards [hQkernel] with xa hxa
cases xa.2 <;> simpa [Q] using hxa
rw [← hmap] at hQkernel'
have hpull := ae_of_ae_map (μ := P.map (fun z : Obs d => (z.1, z.2.1)))
hE.aemeasurable hQkernel'
have htransport := transformedDornLaw_outcomeKernel_ae P
have hjoint : ∀ᵐ xa ∂P.map (fun z : Obs d => (z.1, z.2.1)),
armKernel P xa.1 xa.2 = gaussianReal 0 ⟨σ ^ 2, sq_nonneg σ⟩ := by
filter_upwards [hpull, htransport] with xa hq ht
have hq' : condDistrib (fun z : Obs d => z.2.2)
(fun z : Obs d => (z.1, z.2.1)) Q
(affineToCube xa.1, xa.2) =
gaussianReal 0 ⟨σ ^ 2, sq_nonneg σ⟩ := by
simpa [E] using hq
have ht' : condDistrib (fun z : Obs d => z.2.2)
(fun z : Obs d => (z.1, z.2.1)) Q
(affineToCube xa.1, xa.2) =
condDistrib (fun z : Obs d => z.2.2)
(fun z : Obs d => (z.1, z.2.1)) P xa := by
simpa only [hPQ] using ht
exact ht'.symm.trans hq'
letI : IsProbabilityMeasure (covariateLaw P) := by
unfold covariateLaw
exact Measure.isProbabilityMeasure_map (by fun_prop)
rw [← observedLaw_treatment_compProd P] at hjoint
have hsections := Measure.ae_ae_of_ae_compProd hjoint
have hselectedEq := gaussianThinStripOriginal_selected_ae d hd γ a σ hγ
have hcovQ : covariateLaw Q = volume.restrict (cube d) := by
dsimp [Q, covariateLaw]
rw [Measure.map_map (by fun_prop) (by unfold observed; fun_prop)]
exact gaussianCompletionBind_covariate_marginal_ae
(volume.restrict (cube d)) σ e (fun _ => 0) (fun _ => 0)
he measurable_const measurable_const
(Filter.Eventually.of_forall hevalid)
have hmapcov : (covariateLaw P).map affineToCube =
volume.restrict (cube d) := by
rw [← transformedDornLaw_covariateLaw P, hPQ, hcovQ]
have hposUnit := stripPropensity_uniform_pos_ae d hd γ a hγ
rw [← hmapcov] at hposUnit
have hpos := ae_of_ae_map (μ := covariateLaw P)
affineToCube_measurable.aemeasurable hposUnit
have hselected : ∀ᵐ x ∂covariateLaw P,
gaussianStripPropensity d hd γ a x = propensity P x ∧
0 < gaussianStripPropensity d hd γ a x ∧
gaussianStripPropensity d hd γ a x < 1 := by
have hselectedEq' : ∀ᵐ x ∂covariateLaw P,
gaussianStripPropensity d hd γ a x = propensity P x := by
simpa [P] using hselectedEq
filter_upwards [hselectedEq', hpos] with x heq hp
refine ⟨heq, ?_, ?_⟩
· exact lt_min (by norm_num) hp
· exact (min_le_left _ _).trans_lt (by norm_num)
have hA := condDistrib_treatment_eq_realBernoulli_of_selected P
(gaussianStripPropensity d hd γ a) hselected
filter_upwards [hsections, hA, hselected] with x hx hAx hs
rw [hAx] at hx
constructor
· exact realBernoulli_ae_at_true_of_pos _ hs.2.1 _ hx
· exact realBernoulli_ae_at_false_of_lt_one _ hs.2.2 _ hx
Capping the strip propensity at 3/4 preserves a polynomial lower-tail bound, with the fixed loss contributed only by levels above the cap. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a :
ℝ
hγ :
1 < γ
conclusion
C :
((3 / 4 : ℝ) ^ (γ - 1))⁻¹
0 < C
t ∈ Set.Icc (0 : ℝ) 1 :
≤ C * t ^ (γ - 1)
Proof (Lean source)
lemma cappedStrip_uniform_tail (d : ℕ) (hd : 2 ≤ d)
(γ a : ℝ) (hγ : 1 < γ) :
let C := ((3 / 4 : ℝ) ^ (γ - 1))⁻¹
0 < C ∧ ∀ t ∈ Icc (0 : ℝ) 1,
(volume.restrict (cube d)).real
{x | min (3 / 4 : ℝ) (stripPropensity d hd γ a x) ≤ t} ≤
C * t ^ (γ - 1) := by
dsimp
letI : IsProbabilityMeasure (volume.restrict (cube d)) := by
apply isProbabilityMeasure_iff.mpr
rw [Measure.restrict_apply MeasurableSet.univ, Set.univ_inter]
have hcube : cube d = Icc (fun _ : Fin d => (0 : ℝ))
(fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hcube, Real.volume_Icc_pi]
simp
let c : ℝ := (3 / 4 : ℝ) ^ (γ - 1)
have hcpos : 0 < c := Real.rpow_pos_of_pos (by norm_num) _
refine ⟨inv_pos.mpr hcpos, ?_⟩
intro t ht
have htail := stripPropensity_uniform_tail_bound d hd γ a t hγ ht
by_cases htc : t < 3 / 4
· have hset : {x | min (3 / 4 : ℝ) (stripPropensity d hd γ a x) ≤ t} =
{x | stripPropensity d hd γ a x ≤ t} := by
ext x
simp only [Set.mem_setOf_eq, min_le_iff]
exact or_iff_right (not_le.mpr htc)
rw [hset]
have hc_le : c ≤ 1 := by
dsimp [c]
exact Real.rpow_le_one (by norm_num) (by norm_num) (by linarith)
have hinv : 1 ≤ c⁻¹ := by
simpa using (inv_le_inv₀ zero_lt_one hcpos).2 hc_le
exact htail.trans (by
simpa [c] using mul_le_mul_of_nonneg_right hinv
(Real.rpow_nonneg ht.1 (γ - 1)))
· have htc' : 3 / 4 ≤ t := le_of_not_gt htc
have hmono : c ≤ t ^ (γ - 1) := by
dsimp [c]
exact Real.rpow_le_rpow (by norm_num) htc' (by linarith)
have hprod : 1 ≤ c⁻¹ * t ^ (γ - 1) := by
calc
1 = c⁻¹ * c := by field_simp
_ ≤ c⁻¹ * t ^ (γ - 1) :=
mul_le_mul_of_nonneg_left hmono (inv_nonneg.mpr hcpos.le)
calc
(volume.restrict (cube d)).real
{x | min (3 / 4 : ℝ) (stripPropensity d hd γ a x) ≤ t} ≤ 1 := by
exact (measureReal_mono (Set.subset_univ _)).trans (by simp)
_ ≤ c⁻¹ * t ^ (γ - 1) := hprod
theorem gaussianThinStripOriginal_globalTail Lemma gaussian-thin-strip-original-global-tail in the paper ↗
The inverse-affine capped strip has the same global propensity tail as its unit-cube representative. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a σ :
ℝ
hγ :
1 < γ
conclusion
P :
gaussianThinStripOriginal d hd γ a σ
_ :
gaussianThinStripOriginal_probability d hd γ a σ hγ (: IsProbabilityMeasure P)
GlobalPropensityTail P (gaussianStripPropensity d hd γ a) ((3 / 4 : ℝ) ^ (γ - 1))⁻¹ γ
Proof (Lean source)
lemma gaussianThinStripOriginal_globalTail (d : ℕ) (hd : 2 ≤ d)
(γ a σ : ℝ) (hγ : 1 < γ) :
let P := gaussianThinStripOriginal d hd γ a σ
let _ : IsProbabilityMeasure P :=
gaussianThinStripOriginal_probability d hd γ a σ hγ
GlobalPropensityTail P (gaussianStripPropensity d hd γ a)
((3 / 4 : ℝ) ^ (γ - 1))⁻¹ γ := by
dsimp
let P := gaussianThinStripOriginal d hd γ a σ
letI : IsProbabilityMeasure P :=
gaussianThinStripOriginal_probability d hd γ a σ hγ
constructor
· exact gaussianThinStripOriginal_selected_ae d hd γ a σ hγ
· intro t ht
have hu := (cappedStrip_uniform_tail d hd γ a hγ).2 t ht
have hmapcov : (covariateLaw P).map affineToCube =
volume.restrict (cube d) := by
rw [← transformedDornLaw_covariateLaw P,
transformedDornLaw_gaussianThinStripOriginal d hd γ a σ]
unfold covariateLaw
rw [Measure.map_map (by fun_prop) (by unfold observed; fun_prop)]
exact gaussianCompletionBind_covariate_marginal_ae
(volume.restrict (cube d)) σ
(fun x => min (3 / 4 : ℝ) (stripPropensity d hd γ a x))
(fun _ => 0) (fun _ => 0)
(measurable_const.min (stripPropensity_measurable d hd γ a))
measurable_const measurable_const
(Filter.Eventually.of_forall fun x => by
have hx := stripPropensity_mem_Icc d hd γ a hγ x
exact ⟨le_min (by norm_num) hx.1,
(min_le_left _ _).trans (by norm_num)⟩)
rw [← hmapcov] at hu
have hs : MeasurableSet
{x | min (3 / 4 : ℝ) (stripPropensity d hd γ a x) ≤ t} :=
measurableSet_le
(measurable_const.min (stripPropensity_measurable d hd γ a))
measurable_const
rw [measureReal_def, Measure.map_apply affineToCube_measurable hs] at hu
simpa [P, gaussianStripPropensity, Function.comp_def, measureReal_def] using hu
Every centred real Gaussian has a finite q moment; enlarging its moment integral produces the exact positive radius required by Dorn's model. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
q σ :
ℝ
hq :
1 < q
conclusion
∃ M : ℝ,
conclusion 1
0 < M
conclusion 2
conclusion 3
(∫ y : ℝ, |y| ^ q ∂gaussianReal 0 ⟨σ ^ 2, sq_nonneg σ⟩) ≤ M ^ q
Proof (Lean source)
lemma exists_gaussianReal_qMoment_radius (q σ : ℝ) (hq : 1 < q) :
∃ M : ℝ, 0 < M ∧
Integrable (fun y : ℝ => |y| ^ q)
(gaussianReal 0 ⟨σ ^ 2, sq_nonneg σ⟩) ∧
(∫ y : ℝ, |y| ^ q ∂gaussianReal 0 ⟨σ ^ 2, sq_nonneg σ⟩) ≤ M ^ q := by
have hq0 : 0 < q := lt_trans zero_lt_one hq
have hm := ProbabilityTheory.memLp_id_gaussianReal'
(μ := (0 : ℝ)) (v := ⟨σ ^ 2, sq_nonneg σ⟩)
(ofReal q) (by simp)
have hi0 := hm.integrable_norm_rpow
(ENNReal.ofReal_pos.mpr hq0).ne' ENNReal.ofReal_ne_top
have hi : Integrable (fun y : ℝ => |y| ^ q)
(gaussianReal 0 ⟨σ ^ 2, sq_nonneg σ⟩) := by
simpa [Real.norm_eq_abs, ENNReal.toReal_ofReal hq0.le, id] using hi0
let I : ℝ := ∫ y : ℝ, |y| ^ q ∂gaussianReal 0 ⟨σ ^ 2, sq_nonneg σ⟩
let M : ℝ := max 1 |I|
have hM1 : 1 ≤ M := le_max_left _ _
have hIM : I ≤ M := (le_abs_self I).trans (le_max_right _ _)
have hMq : M ≤ M ^ q := by
simpa only [Real.rpow_one] using
(Real.rpow_le_rpow_of_exponent_le hM1 hq.le)
exact ⟨M, lt_of_lt_of_le zero_lt_one hM1, hi, hIM.trans hMq⟩
The inverse cube affine map scales Euclidean balls by exactly two. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma affineFromCube_mem_ball2_iff {d : ℕ} (x x₀ : Fin d → ℝ) (h : ℝ) :
(fun i => 2 * x i - 1) ∈ ball2 (fun i => 2 * x₀ i - 1) (2 * h) ↔
x ∈ ball2 x₀ h := by
unfold ball2
change (∑ i, ((2 * x i - 1) - (2 * x₀ i - 1)) ^ 2) ≤
(2 * h) ^ 2 ↔ (∑ i, (x i - x₀ i) ^ 2) ≤ h ^ 2
have hsum : (∑ i, ((2 * x i - 1) - (2 * x₀ i - 1)) ^ 2) =
4 * ∑ i, (x i - x₀ i) ^ 2 := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro i hi
ring
rw [hsum]
constructor <;> intro hh <;> nlinarith
Dorn's local ratio condition is invariant under the coordinatewise affine similarity between the source cube and the unit cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
P Q :
e :
(Fin d → ℝ) → ℝ
he :
hcovP :
hcovQ :
covariateLaw Q = volume.restrict (cube d)
hselQ :
DornSelectedPropensity (cube d) Q e
hA3 :
conclusion
DornA3OnCube Q e
Proof (Lean source)
lemma dornA3OnCube_of_affine_original {d : ℕ}
(P Q : Measure (Obs d)) [IsProbabilityMeasure P] [IsProbabilityMeasure Q]
(e : (Fin d → ℝ) → ℝ) (he : Measurable e)
(hcovP : covariateLaw P =
(ofReal ((2 : ℝ) ^ d))⁻¹ • volume.restrict (dornCube d))
(hcovQ : covariateLaw Q = volume.restrict (cube d))
(hselQ : DornSelectedPropensity (cube d) Q e)
(hA3 : DornA3 (dornCube d)
({(P, (fun _ => 0), (fun _ => 0), e ∘ affineToCube)} :
Set (DornSourceLaw d))) :
DornA3OnCube Q e := by
classical
obtain ⟨_, _, _, ρ, ν, h₀, hρ, hν, hh₀, hbound⟩ := hA3
refine ⟨Set.singleton_nonempty _, ?_, ?_, ρ, ν, h₀ / 2,
hρ, hν, by positivity, ?_⟩
· intro z hz
have hz' : z = (Q, (fun _ => 0), (fun _ => 0), e) :=
Set.mem_singleton_iff.mp hz
subst z
rw [hcovQ]
have hcube : cube d = Icc (fun _ : Fin d => (0 : ℝ))
(fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hcube, Measure.restrict_apply (by measurability),
Set.inter_self, Real.volume_Icc_pi]
simp
· intro z hz
have hz' : z = (Q, (fun _ => 0), (fun _ => 0), e) :=
Set.mem_singleton_iff.mp hz
subst z
exact hselQ
· intro z hz x₀ hx₀ h hh
have hz' : z = (Q, (fun _ => 0), (fun _ => 0), e) :=
Set.mem_singleton_iff.mp hz
subst z
let s : (Fin d → ℝ) → (Fin d → ℝ) := fun x i => 2 * x i - 1
have hs : Measurable s := affineFromCube_measurable
have hsx₀ : s x₀ ∈ dornCube d := by
rw [← affineToCube_mem_cube_iff]
change affineToCube (s x₀) ∈ cube d
rw [show affineToCube (s x₀) = x₀ by
funext i; exact congrFun (affineToCube_affineFromCube x₀) i]
exact hx₀
have h2h : 2 * h ∈ Ioc (0 : ℝ) h₀ := by
exact ⟨mul_pos (by norm_num) hh.1, by linarith [hh.2]⟩
have hold := hbound
(P, (fun _ => 0), (fun _ => 0), e ∘ affineToCube)
(Set.mem_singleton _) (s x₀) hsx₀ (2 * h) h2h
have himage : (e ∘ affineToCube) ''
(ball2 (s x₀) (2 * h) ∩ dornCube d) =
e '' (ball2 x₀ h ∩ cube d) := by
apply Set.Subset.antisymm
· intro y hy
obtain ⟨x, hx, rfl⟩ := hy
refine ⟨affineToCube x, ?_, rfl⟩
constructor
· have hb := hx.1
have hback : s (affineToCube x) = x := by
funext i
exact congrFun (affineToCube_inverse x) i
rw [← hback] at hb
exact (affineFromCube_mem_ball2_iff (affineToCube x) x₀ h).mp hb
· exact (affineToCube_mem_cube_iff x).2 hx.2
· intro y hy
obtain ⟨x, hx, rfl⟩ := hy
refine ⟨s x, ?_, ?_⟩
· constructor
· exact (affineFromCube_mem_ball2_iff x x₀ h).2 hx.1
· exact (affineToCube_mem_cube_iff (s x)).1 (by
change affineToCube (s x) ∈ cube d
rw [show affineToCube (s x) = x by
funext i; exact congrFun (affineToCube_affineFromCube x) i]
exact hx.2)
· simp [s, Function.comp_def, affineToCube_affineFromCube]
let AU : Set (Fin d → ℝ) :=
{x | e x ≥ ρ * sSup (e '' (ball2 x₀ h ∩ cube d))} ∩ ball2 x₀ h
let AO : Set (Fin d → ℝ) :=
{x | (e ∘ affineToCube) x ≥
ρ * sSup ((e ∘ affineToCube) ''
(ball2 (s x₀) (2 * h) ∩ dornCube d))} ∩
ball2 (s x₀) (2 * h)
have hpreA : s ⁻¹' AO = AU := by
dsimp [AO, AU]
rw [himage]
ext x
simp only [Set.mem_preimage, Set.mem_inter_iff,
Set.mem_setOf_eq, Function.comp_apply]
rw [show affineToCube (s x) = x by
funext i; exact congrFun (affineToCube_affineFromCube x) i]
rw [affineFromCube_mem_ball2_iff]
have hpreB : s ⁻¹' ball2 (s x₀) (2 * h) = ball2 x₀ h := by
ext x
exact affineFromCube_mem_ball2_iff x x₀ h
have hAO : MeasurableSet AO := by
dsimp [AO]
apply (measurableSet_le measurable_const (he.comp affineToCube_measurable)).inter
unfold ball2
exact measurableSet_le (by fun_prop) (by fun_prop)
have hBO : MeasurableSet (ball2 (s x₀) (2 * h)) := by
unfold ball2
exact measurableSet_le (by fun_prop) (by fun_prop)
have hmeasureA : (covariateLaw P).real AO =
(covariateLaw Q).real AU := by
rw [hcovP, ← cubeVolume_map_affineFromCube d, hcovQ,
measureReal_def, Measure.map_apply hs hAO, hpreA, measureReal_def]
have hmeasureB : (covariateLaw P).real (ball2 (s x₀) (2 * h)) =
(covariateLaw Q).real (ball2 x₀ h) := by
rw [hcovP, ← cubeVolume_map_affineFromCube d, hcovQ,
measureReal_def, Measure.map_apply hs hBO, hpreB, measureReal_def]
dsimp only at hold ⊢
change ν < (covariateLaw P).real AO /
(covariateLaw P).real (ball2 (s x₀) (2 * h)) at hold
change ν < (covariateLaw Q).real AU /
(covariateLaw Q).real (ball2 x₀ h)
rw [← hmeasureA, ← hmeasureB]
exact hold
Capping the thin-strip propensity at 3/4 does not repair A3. On the shrinking balls used in the obstruction the uncapped propensity is already strictly below the cap. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a :
ℝ
hγ :
1 < γ
ha :
1 < a
P :
Measure (Obs d)
hcov :
covariateLaw P = volume.restrict (cube d)
conclusion
Proof (Lean source)
lemma cappedStrip_not_dornA3_of_uniform (d : ℕ) (hd : 2 ≤ d)
(γ a : ℝ) (hγ : 1 < γ) (ha : 1 < a)
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hcov : covariateLaw P = volume.restrict (cube d)) :
¬ DornA3OnCube P
(fun x => min (3 / 4 : ℝ) (stripPropensity d hd γ a x)) := by
classical
intro hA3
change DornA3 (cube d)
({(P, (fun _ => 0), (fun _ => 0),
fun x => min (3 / 4 : ℝ) (stripPropensity d hd γ a x))} :
Set (DornSourceLaw d)) at hA3
obtain ⟨_, _, _, ρ, ν, h₀, hρ, hν, hh₀, hA3bound⟩ := hA3
obtain ⟨h, hh, hh₀', hsmall, hball, hratio⟩ :=
uniformCube_thinStrip_local_ratio d hd γ a hγ ha
(min ρ 2) ν h₀ (lt_min hρ (by norm_num)) hν hh₀
have hsmallρ : ((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) < ρ / 4 :=
hsmall.trans_le (div_le_div_of_nonneg_right (min_le_left _ _) (by norm_num))
have hsmallCap :
((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) < 1 / 2 :=
hsmall.trans_le (by
have hm : min ρ 2 ≤ 2 := min_le_right _ _
linarith)
have hx₀ : (fun _ : Fin d => (1 / 2 : ℝ)) ∈ cube d := by
simp [cube, Pi.le_def]
intro i
norm_num
have hcapEq : EqOn
(fun x => min (3 / 4 : ℝ) (stripPropensity d hd γ a x))
(stripPropensity d hd γ a)
(ball2 (fun _ => (1 / 2 : ℝ)) h) := by
intro x hx
have hcoord : centreRadius x ≤ h := by
unfold centreRadius
letI : Nonempty (Fin d) := ⟨⟨0, by omega⟩⟩
apply (pi_norm_le_iff_of_nonempty _).2
intro i
have hi : (x i - (1 / 2 : ℝ)) ^ 2 ≤
∑ j : Fin d, (x j - 1 / 2) ^ 2 :=
Finset.single_le_sum (fun j _ => sq_nonneg (x j - 1 / 2))
(Finset.mem_univ i)
have habs : |x i - 1 / 2| ≤ h := by
change (∑ j : Fin d, (x j - 1 / 2) ^ 2) ≤ h ^ 2 at hx
nlinarith [hx, hi, sq_abs (x i - (1 / 2 : ℝ)),
abs_nonneg (x i - (1 / 2 : ℝ))]
simpa only [Pi.sub_apply, Real.norm_eq_abs] using habs
have hstrip := stripPropensity_le_radial_add_quarter d hd γ a x
have hrad := stripPropensity_off_strip_radius_bound d hd γ a h hγ x
by_cases hs : x ∈ thinStrip d hd a
· have hradial :
(centreRadiusCDF d (centreRadius x)) ^ ((1 : ℝ) / (γ - 1)) ≤
((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) := by
have hbase : centreRadiusCDF d (centreRadius x) ≤ (2 * h) ^ d := by
unfold centreRadiusCDF
exact (min_le_right _ _).trans
(pow_le_pow_left₀
(mul_nonneg (by norm_num) (norm_nonneg _))
(mul_le_mul_of_nonneg_left hcoord (by norm_num)) d)
exact Real.rpow_le_rpow (by
unfold centreRadiusCDF
exact le_min (by norm_num)
(pow_nonneg (mul_nonneg (by norm_num) (norm_nonneg _)) _))
hbase (by positivity)
have : stripPropensity d hd γ a x < 3 / 4 := by
nlinarith [hstrip, hradial, hsmallCap]
exact min_eq_right this.le
· have := hrad hs hcoord
have hlt : stripPropensity d hd γ a x < 3 / 4 :=
this.trans_lt (hsmallCap.trans (by norm_num))
exact min_eq_right hlt.le
have hbound := hA3bound
(P, (fun _ => 0), (fun _ => 0),
fun x => min (3 / 4 : ℝ) (stripPropensity d hd γ a x))
(Set.mem_singleton _) (fun _ => (1 / 2 : ℝ)) hx₀ h ⟨hh, hh₀'⟩
have himage :
(fun x => min (3 / 4 : ℝ) (stripPropensity d hd γ a x)) ''
(ball2 (fun _ => (1 / 2 : ℝ)) h ∩ cube d) =
stripPropensity d hd γ a ''
(ball2 (fun _ => (1 / 2 : ℝ)) h ∩ cube d) := by
apply Set.image_congr
intro x hx
exact hcapEq hx.1
have hsubset :
{x | min (3 / 4 : ℝ) (stripPropensity d hd γ a x) ≥
ρ * sSup ((fun y => min (3 / 4 : ℝ)
(stripPropensity d hd γ a y)) ''
(ball2 (fun _ => (1 / 2 : ℝ)) h ∩ cube d))} ∩
ball2 (fun _ => (1 / 2 : ℝ)) h ⊆
thinStrip d hd a ∩ ball2 (fun _ => (1 / 2 : ℝ)) h := by
intro x hx
have hxcap := hcapEq hx.2
rw [himage] at hx
have hx' : stripPropensity d hd γ a x ≥
ρ * sSup (stripPropensity d hd γ a ''
(ball2 (fun _ => (1 / 2 : ℝ)) h ∩ cube d)) := by
have hlevel := hx.1
change ρ * sSup (stripPropensity d hd γ a ''
(ball2 (fun _ => (1 / 2 : ℝ)) h ∩ cube d)) ≤
min (3 / 4 : ℝ) (stripPropensity d hd γ a x) at hlevel
change min (3 / 4 : ℝ) (stripPropensity d hd γ a x) =
stripPropensity d hd γ a x at hxcap
rw [hxcap] at hlevel
exact hlevel
refine ⟨?_, hx.2⟩
apply stripSuperlevel_ball_cube_subset_strip d hd γ a h ρ hγ
(by linarith) hρ hsmallρ
exact ⟨by
unfold centreRadius
letI : Nonempty (Fin d) := ⟨⟨0, by omega⟩⟩
apply (pi_norm_le_iff_of_nonempty _).2
intro i
have hi : (x i - (1 / 2 : ℝ)) ^ 2 ≤
∑ j : Fin d, (x j - 1 / 2) ^ 2 :=
Finset.single_le_sum (fun j _ => sq_nonneg (x j - 1 / 2))
(Finset.mem_univ i)
have habs : |x i - 1 / 2| ≤ h := by
have hxball := hx.2
change (∑ j : Fin d, (x j - 1 / 2) ^ 2) ≤ h ^ 2 at hxball
nlinarith [hxball, hi, sq_abs (x i - (1 / 2 : ℝ)),
abs_nonneg (x i - (1 / 2 : ℝ))]
simpa only [Pi.sub_apply, Real.norm_eq_abs] using habs, hx'⟩
letI : IsProbabilityMeasure (covariateLaw P) := by
unfold covariateLaw
exact Measure.isProbabilityMeasure_map (by fun_prop)
have hmono : (covariateLaw P).real
({x | min (3 / 4 : ℝ) (stripPropensity d hd γ a x) ≥
ρ * sSup ((fun y => min (3 / 4 : ℝ)
(stripPropensity d hd γ a y)) ''
(ball2 (fun _ => (1 / 2 : ℝ)) h ∩ cube d))} ∩
ball2 (fun _ => (1 / 2 : ℝ)) h) ≤
(covariateLaw P).real
(thinStrip d hd a ∩ ball2 (fun _ => (1 / 2 : ℝ)) h) :=
measureReal_mono hsubset
rw [hcov] at hmono
dsimp at hbound
rw [hcov] at hbound
have hdiv := (div_le_div_of_nonneg_right hmono hball.le).trans hratio
exact (not_lt_of_ge hdiv) hbound
theorem gaussianThinStripOriginal_not_dornA3 Lemma gaussian-thin-strip-original-not-dorn-a3 in the paper ↗
The capped Gaussian strip violates Dorn A3 on the original source cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a σ :
ℝ
hγ :
1 < γ
ha :
1 < a
conclusion
¬ DornA3 (dornCube d) ({(gaussianThinStripOriginal d hd γ a σ, (fun _ => 0), (fun _ => 0), gaussianStripPropensity d hd γ a)} : Set (DornSourceLaw d))
Proof (Lean source)
lemma gaussianThinStripOriginal_not_dornA3 (d : ℕ) (hd : 2 ≤ d)
(γ a σ : ℝ) (hγ : 1 < γ) (ha : 1 < a) :
¬ DornA3 (dornCube d)
({(gaussianThinStripOriginal d hd γ a σ,
(fun _ => 0), (fun _ => 0), gaussianStripPropensity d hd γ a)} :
Set (DornSourceLaw d)) := by
let P := gaussianThinStripOriginal d hd γ a σ
let e : (Fin d → ℝ) → ℝ := fun x =>
min (3 / 4 : ℝ) (stripPropensity d hd γ a x)
let Q := ((volume.restrict (cube d)).bind
(gaussianCompletionAt σ e (fun _ => 0) (fun _ => 0))).map observed
letI : IsProbabilityMeasure P :=
gaussianThinStripOriginal_probability d hd γ a σ hγ
have hPQ : transformedDornLaw P = Q := by
dsimp [P, Q, e]
exact transformedDornLaw_gaussianThinStripOriginal d hd γ a σ
letI : IsProbabilityMeasure Q := by
rw [← hPQ]
exact transformedDornLaw_probability P
have he : Measurable e := by
dsimp [e]
exact measurable_const.min (stripPropensity_measurable d hd γ a)
have hevalid : ∀ x, e x ∈ Icc (0 : ℝ) 1 := by
intro x
have hx := stripPropensity_mem_Icc d hd γ a hγ x
exact ⟨le_min (by norm_num) hx.1,
(min_le_left _ _).trans (by norm_num)⟩
have hcovQ : covariateLaw Q = volume.restrict (cube d) := by
dsimp [Q, covariateLaw]
rw [Measure.map_map (by fun_prop) (by unfold observed; fun_prop)]
exact gaussianCompletionBind_covariate_marginal_ae
(volume.restrict (cube d)) σ e (fun _ => 0) (fun _ => 0)
he measurable_const measurable_const
(Filter.Eventually.of_forall hevalid)
letI : IsProbabilityMeasure (volume.restrict (cube d)) := by
apply isProbabilityMeasure_iff.mpr
rw [Measure.restrict_apply MeasurableSet.univ, Set.univ_inter]
have hcube : cube d = Icc (fun _ : Fin d => (0 : ℝ))
(fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hcube, Real.volume_Icc_pi]
simp
have hprop : ∀ᵐ x ∂covariateLaw Q, propensity Q x = e x := by
rw [hcovQ]
simpa [Q] using gaussianCompletionBind_propensity_ae
(volume.restrict (cube d)) σ e (fun _ => 0) (fun _ => 0)
he measurable_const measurable_const hevalid
have hpos := stripPropensity_uniform_pos_ae d hd γ a hγ
have hfull : ∀ᵐ x ∂covariateLaw Q,
e x = propensity Q x ∧ 0 < e x ∧ e x < 1 := by
rw [hcovQ] at hprop
rw [hcovQ]
filter_upwards [hprop, hpos] with x hp hx
exact ⟨hp.symm, lt_min (by norm_num) hx,
(min_le_left _ _).trans_lt (by norm_num)⟩
have hselQ : DornSelectedPropensity (cube d) Q e :=
⟨inferInstance, he, fun x hx => hevalid x, hfull⟩
intro hA3
have hunit : DornA3OnCube Q e := by
apply dornA3OnCube_of_affine_original P Q e he
(gaussianThinStripOriginal_covariateLaw d hd γ a σ hγ)
hcovQ hselQ
change DornA3 (dornCube d)
({(P, (fun _ => 0), (fun _ => 0), e ∘ affineToCube)} :
Set (DornSourceLaw d)) at hA3
exact hA3
exact cappedStrip_not_dornA3_of_uniform d hd γ a hγ ha Q hcovQ hunit
def holderFullNorm
The full coordinatewise Hölder radius of a response on the unit cube. For the stated inputs and conditions, the holderFullNorm object being defined.
def dornCommonRadius
One radius for every transformed member of Dorn's seminorm class with the common response bound obtained from Assumption 1. For the stated inputs and conditions, the dornCommonRadius object being defined.
Definition (Lean source)
parameters
d :
ℕ
β M L₀ :
ℝ
def
dornCommonRadius d β M L₀ :
ℝ
given by
sSup {r : ℝ | ∃ g : (Fin d → ℝ) → ℝ, DornTopDerivativeSeminorm β L₀ g ∧ (∀ x ∈ dornCube d, |g x| ≤ M) ∧ r = holderFullNorm β (transformedResponse g)}
def dornInterpolationConstant
Fixed-cube interpolation constant: its inputs are only dimension and smoothness, with all admissible amplitude and seminorm pairs normalized out. For the stated inputs and conditions, the dornInterpolationConstant object being defined.
Definition (Lean source)
parameters
d :
ℕ
β :
ℝ
def
dornInterpolationConstant d β :
ℝ
given by
sSup {r : ℝ | ∃ (M L₀ : ℝ) (g : (Fin d → ℝ) → ℝ), 0 < M ∧ 0 < L₀ ∧ DornTopDerivativeSeminorm β L₀ g ∧ (∀ x ∈ dornCube d, |g x| ≤ M) ∧ r = holderFullNorm β (transformedResponse g) / (M + L₀)}
The infimum defining holderFullNorm is attained. Each defining inequality is closed in its radius, so it may be passed to the infimum pointwise. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
HolderBall β (holderFullNorm β g) g
Proof (Lean source)
lemma holderBall_holderFullNorm {d : ℕ} (β : ℝ)
(g : (Fin d → ℝ) → ℝ)
(hball : ∃ L : ℝ, 0 ≤ L ∧ HolderBall β L g) :
HolderBall β (holderFullNorm β g) g := by
let S : Set ℝ := {L : ℝ | 0 ≤ L ∧ HolderBall β L g}
have hSne : S.Nonempty := by
obtain ⟨L, hL⟩ := hball
exact ⟨L, hL⟩
have hSbelow : BddBelow S := ⟨0, fun _ hL => hL.1⟩
obtain ⟨L₀, hL₀, hball₀⟩ := hball
refine ⟨hball₀.regularity, ?_, ?_⟩
· intro j hj f x hx
apply le_csInf hSne
intro L hL
exact hL.2.derivBound j hj f x hx
· intro f x hx y hy
let q : ℝ := ‖x - y‖ ^ (β - (polynomialDegree β : ℝ))
have hq : 0 ≤ q := Real.rpow_nonneg (norm_nonneg _) _
by_cases hq0 : q = 0
· have hz := hball₀.modulus f x hx y hy
rw [show L₀ * q = 0 by simp [hq0]] at hz
simpa [q, hq0] using hz
· have hqpos : 0 < q := lt_of_le_of_ne hq (Ne.symm hq0)
apply (div_le_iff₀ hqpos).mp
apply le_csInf hSne
intro L hL
exact (div_le_iff₀ hqpos).mpr (hL.2.modulus f x hx y hy)
Increasing the common radius preserves the intrinsic Holder ball. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
HolderBall β L' g
Proof (Lean source)
lemma HolderBall.mono_radius {d : ℕ} {β L L' : ℝ}
{g : (Fin d → ℝ) → ℝ} (h : HolderBall β L g) (hLL' : L ≤ L') :
HolderBall β L' g := by
refine ⟨h.regularity, ?_, ?_⟩
· intro j hj f x hx
exact (h.derivBound j hj f x hx).trans hLL'
· intro f x hx y hy
exact (h.modulus f x hx y hy).trans
(mul_le_mul_of_nonneg_right hLL' (Real.rpow_nonneg (norm_nonneg _) _))
A Holder response may be replaced by a representative agreeing on the cube when the completion covariate is supported there. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β L :
ℝ
Pc :
g h :
(Fin d → ℝ) → ℝ
hg :
HolderResponse Pc g β L
heq :
EqOn g h (cube d)
hsupp :
(Pc.map (fun ω : Completion d => ω.1)) (cube d) = 1
conclusion
HolderResponse Pc h β L
Proof (Lean source)
lemma HolderResponse.congr_on_cube {d : ℕ} {β L : ℝ}
{Pc : Measure (Completion d)} [IsProbabilityMeasure Pc]
{g h : (Fin d → ℝ) → ℝ} (hg : HolderResponse Pc g β L)
(heq : EqOn g h (cube d))
(hsupp : (Pc.map (fun ω : Completion d => ω.1)) (cube d) = 1) :
HolderResponse Pc h β L := by
have hcube : MeasurableSet (cube d) := by
unfold cube
exact MeasurableSet.univ_pi fun _ => measurableSet_Icc
have haecube : ∀ᵐ ω ∂Pc, ω.1 ∈ cube d := by
letI : IsProbabilityMeasure (Pc.map (fun ω : Completion d => ω.1)) :=
Measure.isProbabilityMeasure_map (by fun_prop)
have hmapped : ∀ᵐ x ∂Pc.map (fun ω : Completion d => ω.1),
x ∈ cube d := by
apply (ae_mem_iff_measure_eq hcube.nullMeasurableSet).2
simpa using hsupp
exact (ae_map_iff (μ := Pc)
(show AEMeasurable (fun ω : Completion d => ω.1) Pc by fun_prop)
hcube).1 hmapped
refine ⟨?_, ?_⟩
· have hi :=
(holderBallOn_congr
heq).mp hg.1.toIntrinsic
exact ⟨hi.regularity, hi.derivBound, hi.modulus⟩
· filter_upwards [haecube, hg.2] with ω hω hmean
exact (heq hω).symm.trans hmean
Every admissible Hölder radius bounds the value at a point of the cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β :
ℝ
g :
(Fin d → ℝ) → ℝ
x :
Fin d → ℝ
hx :
x ∈ cube d
hball :
∃ L : ℝ, 0 ≤ L ∧ HolderBall β L g
conclusion
|g x| ≤ holderFullNorm β g
Proof (Lean source)
lemma holderFullNorm_ge_pointwise {d : ℕ} (β : ℝ)
(g : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ) (hx : x ∈ cube d)
(hball : ∃ L : ℝ, 0 ≤ L ∧ HolderBall β L g) :
|g x| ≤ holderFullNorm β g := by
unfold holderFullNorm
apply le_csInf hball
intro L hL
have hzero := hL.2.derivBound 0 (zero_le _) (fun i : Fin 0 => i.elim0) x hx
simpa [HolderDerivBound, HolderDerivBoundOn,
coordJetOn,
iteratedFDerivWithin_zero_apply] using hzero
Constant responses satisfy Dorn's top-order seminorm condition. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β L₀ c :
ℝ
hL₀ :
0 ≤ L₀
conclusion
DornTopDerivativeSeminorm (d := d) β L₀ (fun _ => c)
Proof (Lean source)
lemma dornTopDerivativeSeminorm_const {d : ℕ} (β L₀ c : ℝ)
(hL₀ : 0 ≤ L₀) :
DornTopDerivativeSeminorm (d := d) β L₀ (fun _ => c) := by
constructor
· fun_prop
· intro α hα f hf x hx y hy
have heq : coordJetOn
(dornCube d) (∑ i, α i) (fun _ : Fin d → ℝ => c) f x =
coordJetOn
(dornCube d) (∑ i, α i) (fun _ : Fin d → ℝ => c) f y := by
by_cases hn : (∑ i, α i) = 0
· let P : ℕ → Prop := fun j => ∀ g : Fin j → Fin d,
coordJetOn
(dornCube d) j (fun _ => c) g x =
coordJetOn
(dornCube d) j (fun _ => c) g y
have hPzero : P 0 := by
intro g
simp [coordJetOn,
iteratedFDerivWithin_zero_apply]
exact (hn.symm ▸ hPzero) f
· unfold coordJetOn
rw [iteratedFDerivWithin_const_of_ne hn]
simp
rw [heq, sub_self, abs_zero]
positivity
Under Dorn's normalized cube design, the stated pulled-back L∞ loss is bounded by the pointwise cube supremum used in dornTransformedSupRisk. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
- transformedResponse μ₁ y|
Proof (Lean source)
lemma dorn_eLpNorm_le_cubeSup {d n : ℕ} (β B : ℝ)
(ω : Fin n → Obs d) (P : Measure (Obs d))
(μ₁ : (Fin d → ℝ) → ℝ)
(hcov : covariateLaw P =
(ofReal ((2 : ℝ) ^ d))⁻¹ • volume.restrict (dornCube d)) :
eLpNorm (fun x =>
equalCellEstimator β B ω (affineToCube x) - μ₁ x) ⊤
(covariateLaw P) ≤
⨆ y : Fin d → ℝ, ⨆ (_ : y ∈ cube d),
ofReal
|equalCellEstimator β B ω y - transformedResponse μ₁ y| := by
rw [eLpNorm_exponent_top]
apply eLpNormEssSup_le_of_ae_enorm_bound
have hDorn : MeasurableSet (dornCube d) := by
unfold dornCube
exact MeasurableSet.univ_pi fun _ => measurableSet_Icc
have hsupp : ∀ᵐ x ∂covariateLaw P, x ∈ dornCube d := by
rw [hcov]
have hp : 0 < (2 : ℝ) ^ d := pow_pos (by norm_num) d
apply (Measure.ae_ennreal_smul_measure_iff (by simp [hp])).2
exact ae_restrict_mem hDorn
filter_upwards [hsupp] with x hx
have hy : affineToCube x ∈ cube d := (affineToCube_mem_cube_iff x).2 hx
rw [Real.enorm_eq_ofReal_abs]
have hresp : transformedResponse μ₁ (affineToCube x) = μ₁ x :=
transformedResponse_affineToCube μ₁ x
rw [← hresp]
exact le_iSup_of_le (affineToCube x) (le_iSup_of_le hy le_rfl)
Markov's inequality converts an expected ENNReal loss bound into the corresponding real probability bound at the rescaled rate. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
Ω :
Type*
μ :
Z :
Ω → ℝ≥0∞
A r ε :
ℝ
hZ :
AEMeasurable Z μ
hA :
0 < A
hr :
0 < r
hε :
0 < ε
hint :
∫⁻ ω, Z ω ∂μ ≤ ofReal (A * r)
conclusion
μ.real {ω | ofReal ((A / ε) * r) < Z ω} ≤ ε
Proof (Lean source)
lemma measurableLoss_probability_le {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (Z : Ω → ℝ≥0∞)
(A r ε : ℝ) (hZ : AEMeasurable Z μ)
(hA : 0 < A) (hr : 0 < r) (hε : 0 < ε)
(hint : ∫⁻ ω, Z ω ∂μ ≤ ofReal (A * r)) :
μ.real {ω | ofReal ((A / ε) * r) < Z ω} ≤ ε := by
have hthreshold : 0 < (A / ε) * r := mul_pos (div_pos hA hε) hr
have hmark := meas_ge_le_lintegral_div hZ
(ENNReal.ofReal_pos.mpr hthreshold).ne' ENNReal.ofReal_ne_top
have hdiv : (∫⁻ ω, Z ω ∂μ) /
ofReal ((A / ε) * r) ≤ ofReal ε := by
calc
(∫⁻ ω, Z ω ∂μ) / ofReal ((A / ε) * r) ≤
ofReal (A * r) / ofReal ((A / ε) * r) :=
ENNReal.div_le_div_right hint _
_ = ofReal ε := by
rw [← ENNReal.ofReal_div_of_pos hthreshold]
congr 1
field_simp
rw [measureReal_def]
apply ENNReal.toReal_le_of_le_ofReal hε.le
have hsub : {ω | ofReal ((A / ε) * r) < Z ω} ⊆
{ω | ofReal ((A / ε) * r) ≤ Z ω} :=
fun ω hω => by
change ofReal ((A / ε) * r) ≤ Z ω
exact le_of_lt hω
exact (measure_mono hsub).trans (hmark.trans hdiv)
Enlarging the numerical tail constant preserves membership in the causal model class. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
P ∈ ModelClass d β B L C' c_f γ
Proof (Lean source)
lemma ModelClass.mono_tailConstant {d : ℕ} {β B L C C' c_f γ : ℝ}
{P : Measure (Obs d)} (hCC' : C ≤ C')
(hP : P ∈ ModelClass d β B L C c_f γ) :
P ∈ ModelClass d β B L C' c_f γ := by
obtain ⟨hparam, hγ, Pc, hPc, μ₁, e, hEq, hmodel⟩ := hP
letI : IsProbabilityMeasure Pc := hPc
have hparam' : ModelParameterDomain d β B L C' c_f :=
⟨hparam.1, hparam.2.1, hparam.2.2.1, hparam.2.2.2.1,
hparam.2.2.2.2.1.trans hCC', hparam.2.2.2.2.2⟩
have htail : GlobalPropensityTail (Pc.map observed) e C' γ := by
refine ⟨hmodel.tail.1, ?_⟩
intro t ht
exact (hmodel.tail.2 t ht).trans
(mul_le_mul_of_nonneg_right hCC' (Real.rpow_nonneg ht.1 _))
exact ⟨hparam', hγ, Pc, hPc, μ₁, e, hEq,
{ hmodel with tail := htail }⟩
def dornTransformedSupRisk
Expected supremum loss of the transformed estimator against Dorn's designated Gaussian response, with no model-membership premise. For the stated inputs and conditions, the dornTransformedSupRisk object being defined.
Definition (Lean source)
parameters
def
dornTransformedSupRisk d n β B P μ₁ :
ℝ≥0∞
given by
∫⁻ ω, ⨆ x ∈ cube d, ofReal |equalCellEstimator β B ω x
theorem dornTransformedSupRisk_eq_estimatorSupRisk_of_globalTailModel Lemma dorn-transformed-risk-identification in the paper ↗
Once affine transport has supplied a causal-model witness with the transported response, the Dorn risk is exactly the model-class estimator risk. This isolates the chosen-representative issue from the Gaussian construction. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d n :
ℕ
β B L C c_f γ :
ℝ
P :
Measure (Obs d)
μ₁ :
(Fin d → ℝ) → ℝ
Pc :
e :
(Fin d → ℝ) → ℝ
hEq :
transformedDornLaw P = Pc.map observed
hmodel :
GlobalTailModel β B L C c_f γ Pc (transformedResponse μ₁) e
hmem :
transformedDornLaw P ∈ ModelClass d β B L C c_f γ
conclusion
dornTransformedSupRisk d n β B P μ₁
= estimatorSupRisk d n β B L C c_f γ (transformedDornLaw P) hmem
Proof (Lean source)
lemma dornTransformedSupRisk_eq_estimatorSupRisk_of_globalTailModel
{d n : ℕ} {β B L C c_f γ : ℝ}
(P : Measure (Obs d)) (μ₁ : (Fin d → ℝ) → ℝ)
(Pc : Measure (Completion d)) [IsProbabilityMeasure Pc]
(e : (Fin d → ℝ) → ℝ)
(hEq : transformedDornLaw P = Pc.map observed)
(hmodel : GlobalTailModel β B L C c_f γ Pc
(transformedResponse μ₁) e)
(hmem : transformedDornLaw P ∈ ModelClass d β B L C c_f γ) :
dornTransformedSupRisk d n β B P μ₁ =
estimatorSupRisk d n β B L C c_f γ (transformedDornLaw P) hmem := by
unfold dornTransformedSupRisk estimatorSupRisk
apply lintegral_congr
intro ω
apply iSup_congr
intro x
apply iSup_congr
intro hx
rw [responseOf_eq_given_model_response
(transformedDornLaw P) hmem Pc (transformedResponse μ₁) e
hEq hmodel x hx]
The Gaussian arm clause identifies its conditional mean with the designated response. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β q M σ C L₀ γ :
ℝ
P :
Measure (Obs d)
μ₁ μ₀ e :
(Fin d → ℝ) → ℝ
hmodel :
DornRemainingModel d β q M σ C L₀ γ P μ₁ μ₀ e
conclusion
∀ᵐ x ∂covariateLaw P, (∫ y, y ∂armKernel P x true) = μ₁ x
Proof (Lean source)
lemma dornRemaining_treatedMean_ae (d : ℕ) (β q M σ C L₀ γ : ℝ)
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(μ₁ μ₀ e : (Fin d → ℝ) → ℝ)
(hmodel : DornRemainingModel d β q M σ C L₀ γ P μ₁ μ₀ e) :
∀ᵐ x ∂covariateLaw P, (∫ y, y ∂armKernel P x true) = μ₁ x := by
filter_upwards [hmodel.2.2.2.2.2.2.2] with x hx
rw [hx.1]
exact ProbabilityTheory.integral_id_gaussianReal
theorem dorn_A3_free_corollary
Conditional on the disclosed Dorn remaining-class scope, every law in the remaining uniform Gaussian Hölder model transfers to the global-tail causal class and inherits the expected-supremum rate; Markov gives the published uniform-probability criterion. The strict Gaussian strip uses witness constants selected separately for each γ, independent of the fixed upper-rate tuple. The common response bound comes from Dorn's conditional moment clause, and the full Hölder radius comes from fixed-cube interpolation of his top-derivative seminorm. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
1 ≤ d
β q M σ C L₀ γ_min γ_max :
ℝ
hβ :
1 < β
hq :
3 < q
hM :
0 < M
hσ :
0 < σ
hC :
0 < C
hL₀ :
0 < L₀
hγmin :
1 < γ_min
hγrange :
γ_min < γ_max
_remaining_of_gate :
∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin,
hγrange⟩, DornRemainingClassSpec d β q M σ C L₀ γ
conclusion
B :
max M σ
L :
dornCommonRadius d β M L₀
0 < dornInterpolationConstant d β
0 < L
L ≤ dornInterpolationConstant d β * (M + L₀)
γ ∈ adaptationRange γ_min γ_max ⟨hγmin :
hγrange⟩, ∀ (P : Measure (Obs d)) [IsProbabilityMeasure P] (μ₁ μ₀ e : (Fin d → ℝ) → ℝ), (P, μ₁, μ₀, e) ∈ DornRemainingClass d β q M σ C L₀ γ → transformedDornLaw P ∈ ModelClass d β B L C 1 γ
∃ K : ℝ,
0 < K
∃ n₀ : ℕ,
n ≥ n₀ :
γ ∈ adaptationRange γ_min γ_max ⟨hγmin :
hγrange⟩, ∀ (P : Measure (Obs d)) [IsProbabilityMeasure P] (μ₁ μ₀ e : (Fin d → ℝ) → ℝ), (P, μ₁, μ₀, e) ∈ DornRemainingClass d β q M σ C L₀ γ → dornTransformedSupRisk d n β B P μ₁
≤ ofReal (K * oracleRate d n β γ)
γ ∈ adaptationRange γ_min γ_max ⟨hγmin :
hγrange⟩, ∀ 𝔓 : Set (DornSourceLaw d), DornA12Family d β q M σ C L₀ γ 𝔓 → 𝔓.Nonempty ∧ 𝔓 ⊆ DornRemainingClass d β q M σ C L₀ γ ∧ ∀ n
≥ n₀, (⨆ z : {z : DornSourceLaw d // z ∈ 𝔓}, dornTransformedSupRisk d n β B z.1.1 z.1.2.1) ≤ ofReal (K * oracleRate d n β γ)
ε :
ℝ
0 < ε
∃ R : ℝ,
0 < R
∃ n₀ : ℕ,
∀ n ≥ n₀,
∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin,
hγrange⟩, ∀ (P : Measure (Obs d)) [IsProbabilityMeasure P] (μ₁ μ₀ e : (Fin d → ℝ) → ℝ), (P, μ₁, μ₀, e) ∈ DornRemainingClass d β q M σ C L₀ γ → (Measure.pi (fun _ : Fin n => transformedDornLaw P)).real {ω | ofReal (R * oracleRate d n β γ) < eLpNorm (fun x => equalCellEstimator β B ω (affineToCube x) - μ₁ x) ⊤ (covariateLaw P)}
≤ ε
hd₂ :
2 ≤ d
γ ∈ adaptationRange γ_min γ_max ⟨hγmin :
hγrange⟩, ∃ (C' M' σ' L₀' : ℝ) (P : Measure (Obs d)) (μ₁' μ₀' e : (Fin d → ℝ) → ℝ), 0 < C' ∧ 0 < M' ∧ 0 < σ' ∧ 0 < L₀' ∧ DornSelectedPropensity (dornCube d) P e ∧ ∃ (hP : IsProbabilityMeasure P), @DornRemainingModel d β q M' σ' C' L₀' γ P hP μ₁' μ₀' e ∧ ¬ DornA3 (dornCube d) ({(P, μ₁', μ₀', e)} : Set (DornSourceLaw d))
𝔓 :
Set (DornSourceLaw d)
μref :
(Fin d → ℝ) → ℝ
x₀ :
Fin d → ℝ
𝔓.Nonempty
(∀ z ∈ 𝔓, ∀ x, z.2.1 x = μref x)
n :
ℕ
c r :
ℝ
0 < c
0 < r
minimaxValueENNReal (dornPointwiseMissRisk (n := n) 𝔓 x₀ c r) = 0
γ :
ℝ
1 < γ
x₀ ∈ cube d :
∃ cγ : ℝ,
0 < cγ
n :
ℕ
1 ≤ n
Proof (Lean source)
theorem dorn_A3_free_corollary (d : ℕ) (hd : 1 ≤ d)
(β q M σ C L₀ γ_min γ_max : ℝ)
(hβ : 1 < β) (hq : 3 < q) (hM : 0 < M) (hσ : 0 < σ)
(hC : 0 < C) (hL₀ : 0 < L₀)
(hγmin : 1 < γ_min) (hγrange : γ_min < γ_max)
(_remaining_of_gate :
∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
DornRemainingClassSpec d β q M σ C L₀ γ) :
let B := max M σ
let L := dornCommonRadius d β M L₀
0 < dornInterpolationConstant d β ∧ 0 < L ∧
L ≤ dornInterpolationConstant d β * (M + L₀) ∧
(∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
∀ (P : Measure (Obs d)) [IsProbabilityMeasure P]
(μ₁ μ₀ e : (Fin d → ℝ) → ℝ),
(P, μ₁, μ₀, e) ∈ DornRemainingClass d β q M σ C L₀ γ →
transformedDornLaw P ∈ ModelClass d β B L C 1 γ) ∧
(∃ K : ℝ, 0 < K ∧ ∃ n₀ : ℕ,
(∀ n ≥ n₀, ∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
∀ (P : Measure (Obs d)) [IsProbabilityMeasure P]
(μ₁ μ₀ e : (Fin d → ℝ) → ℝ),
(P, μ₁, μ₀, e) ∈ DornRemainingClass d β q M σ C L₀ γ →
dornTransformedSupRisk d n β B P μ₁ ≤
ofReal (K * oracleRate d n β γ)) ∧
(∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
∀ 𝔓 : Set (DornSourceLaw d),
DornA12Family d β q M σ C L₀ γ 𝔓 →
𝔓.Nonempty ∧ 𝔓 ⊆ DornRemainingClass d β q M σ C L₀ γ ∧
∀ n ≥ n₀,
(⨆ z : {z : DornSourceLaw d // z ∈ 𝔓},
dornTransformedSupRisk d n β B z.1.1 z.1.2.1) ≤
ofReal (K * oracleRate d n β γ))) ∧
(∀ ε : ℝ, 0 < ε → ∃ R : ℝ, 0 < R ∧ ∃ n₀ : ℕ,
∀ n ≥ n₀, ∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
∀ (P : Measure (Obs d)) [IsProbabilityMeasure P]
(μ₁ μ₀ e : (Fin d → ℝ) → ℝ),
(P, μ₁, μ₀, e) ∈ DornRemainingClass d β q M σ C L₀ γ →
(Measure.pi (fun _ : Fin n => transformedDornLaw P)).real
{ω | ofReal (R * oracleRate d n β γ) <
eLpNorm (fun x =>
equalCellEstimator β B ω (affineToCube x) - μ₁ x)
⊤ (covariateLaw P)} ≤ ε) ∧
(∀ (hd₂ : 2 ≤ d), ∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
∃ (C' M' σ' L₀' : ℝ) (P : Measure (Obs d))
(μ₁' μ₀' e : (Fin d → ℝ) → ℝ),
0 < C' ∧ 0 < M' ∧ 0 < σ' ∧ 0 < L₀' ∧
DornSelectedPropensity (dornCube d) P e ∧
∃ (hP : IsProbabilityMeasure P),
@DornRemainingModel d β q M' σ' C' L₀' γ P hP
μ₁' μ₀' e ∧
¬ DornA3 (dornCube d)
({(P, μ₁', μ₀', e)} : Set (DornSourceLaw d))) ∧
(∀ (𝔓 : Set (DornSourceLaw d)) (μref : (Fin d → ℝ) → ℝ)
(x₀ : Fin d → ℝ), 𝔓.Nonempty →
(∀ z ∈ 𝔓, ∀ x, z.2.1 x = μref x) →
∀ (n : ℕ) (c r : ℝ), 0 < c → 0 < r →
minimaxValueENNReal
(dornPointwiseMissRisk (n := n) 𝔓 x₀ c r) = 0) ∧
(∀ γ : ℝ, 1 < γ → ∀ x₀ ∈ cube d,
∃ cγ : ℝ, 0 < cγ ∧
∀ n : ℕ, 1 ≤ n →
ofReal (cγ * oracleRate d n β γ) ≤
boundedPointwiseRisk d n β B L (max C 1) 1 γ x₀) := by
dsimp
have hMomentIntegrable :
∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
∀ (P : Measure (Obs d)) [IsProbabilityMeasure P]
(μ₁ μ₀ e : (Fin d → ℝ) → ℝ),
DornRemainingModel d β q M σ C L₀ γ P μ₁ μ₀ e →
Integrable μ₁ (covariateLaw P) := by
intro γ hγ P _ μ₁ μ₀ e hmodel
have hcont : ContinuousOn μ₁ (dornCube d) :=
hmodel.2.2.2.2.1.1.continuousOn
have hcompact : IsCompact (dornCube d) := by
unfold dornCube
exact isCompact_univ_pi (fun _ => isCompact_Icc)
have hint : Integrable μ₁ (volume.restrict (dornCube d)) :=
hcont.integrableOn_compact hcompact
rw [covariateLaw, hmodel.1]
exact hint.smul_measure (by simp)
have hMeanBound :
∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
∀ (P : Measure (Obs d)) [IsProbabilityMeasure P]
(μ₁ μ₀ e : (Fin d → ℝ) → ℝ),
DornRemainingModel d β q M σ C L₀ γ P μ₁ μ₀ e →
∀ x ∈ dornCube d, |μ₁ x| ≤ M := by
intro γ hγ P _ μ₁ μ₀ e hmodel
have hq1 : 1 ≤ q := by linarith
have hq0 : 0 < q := by linarith
have hae : ∀ᵐ x ∂covariateLaw P, |μ₁ x| ≤ M := by
filter_upwards [hmodel.2.2.1, hmodel.2.2.2.2.2.2.2] with x hx hxg
let κ := armKernel P x true
haveI : IsProbabilityMeasure κ := by
dsimp [κ]
rw [hxg.1]
exact ProbabilityTheory.instIsProbabilityMeasureGaussianReal _ _
have hiy : Integrable (fun y : ℝ => y) κ := by
rw [show κ = gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩ from hxg.1]
exact (ProbabilityTheory.memLp_id_gaussianReal' 1 (by simp)).integrable (by norm_num)
have hia : Integrable (fun y : ℝ => |y|) κ := hiy.abs
have hj : (∫ y, |y| ∂κ) ^ q ≤ ∫ y, |y| ^ q ∂κ := by
apply ConvexOn.map_integral_le (convexOn_rpow hq1)
· exact Real.continuous_rpow_const hq0.le |>.continuousOn
· exact isClosed_Ici
· filter_upwards [] with y
exact abs_nonneg y
· exact hia
· exact hx.1
have hmean : |μ₁ x| ≤ ∫ y, |y| ∂κ := by
rw [← ProbabilityTheory.integral_id_gaussianReal (μ := μ₁ x)
(v := ⟨σ ^ 2, sq_nonneg σ⟩)]
change |∫ y, y ∂gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩| ≤ _
rw [show κ = gaussianReal (μ₁ x) ⟨σ ^ 2, sq_nonneg σ⟩ from hxg.1]
exact abs_integral_le_integral_abs
have hpow : |μ₁ x| ^ q ≤ M ^ q :=
(Real.rpow_le_rpow (abs_nonneg _) hmean hq0.le).trans (hj.trans hx.2)
exact (Real.rpow_le_rpow_iff (abs_nonneg _) (le_of_lt hM) hq0).mp hpow
have haevol : ∀ᵐ x ∂volume.restrict (dornCube d), |μ₁ x| ≤ M := by
rw [covariateLaw, hmodel.1] at hae
exact (Measure.ae_ennreal_smul_measure_iff (by simp :
(ofReal ((2 : ℝ) ^ d))⁻¹ ≠ 0)).mp hae
have heqae : (fun x => max (|μ₁ x| - M) 0) =ᵐ[volume.restrict (dornCube d)]
(fun _ => (0 : ℝ)) := by
filter_upwards [haevol] with x hx
simp [max_eq_right (sub_nonpos.mpr hx)]
have hclosure : dornCube d ⊆ closure (interior (dornCube d)) := by
intro y hy
change y ∈ closure (interior (Set.univ.pi (fun _ : Fin d => Icc (-1 : ℝ) 1)))
rw [interior_pi_set Set.finite_univ, closure_pi_set]
intro i hi
rw [closure_interior_Icc (by norm_num : (-1 : ℝ) ≠ 1)]
exact (show y ∈ Set.univ.pi (fun _ : Fin d => Icc (-1 : ℝ) 1) from hy) i hi
have hcont : ContinuousOn (fun x => max (|μ₁ x| - M) 0) (dornCube d) :=
((hmodel.2.2.2.2.1.1.continuousOn.abs.sub continuousOn_const).sup' continuousOn_const)
have heq := Measure.eqOn_of_ae_eq heqae hcont continuousOn_const hclosure
intro x hx
have hx' := heq hx
dsimp at hx'
have : |μ₁ x| - M ≤ 0 := by
have := le_max_left (|μ₁ x| - M) 0
rw [hx'] at this
exact this
linarith
obtain ⟨Kcube, hKcube, hCubeCompletion⟩ :=
fixedCube_holder_completion d hd β hβ
have hCompletedResponse :
∀ (g : (Fin d → ℝ) → ℝ),
DornTopDerivativeSeminorm β L₀ g →
(∀ x ∈ dornCube d, |g x| ≤ M) →
HolderBall β (Kcube * (M + L₀)) (transformedResponse g) := by
intro g hg hbound
exact hCubeCompletion M L₀ g (le_of_lt hM) (le_of_lt hL₀) hg hbound
have hCompletedModelResponse :
∀ γ ∈ adaptationRange γ_min γ_max ⟨hγmin, hγrange⟩,
∀ (P : Measure (Obs d)) [IsProbabilityMeasure P]
(μ₁ μ₀ e : (Fin d → ℝ) → ℝ),
DornRemainingModel d β q M σ C L₀ γ P μ₁ μ₀ e →
HolderBall β (Kcube * (M + L₀))
(transformedResponse μ₁) := by
intro γ hγ P _ μ₁ μ₀ e hmodel
exact hCompletedResponse μ₁ hmodel.2.2.2.2.1
(hMeanBound γ hγ P μ₁ μ₀ e hmodel)
have hRadius :
0 < dornInterpolationConstant d β ∧
0 < dornCommonRadius d β M L₀ ∧
dornCommonRadius d β M L₀ ≤
dornInterpolationConstant d β * (M + L₀) := by
-- `hCompletedResponse` bounds every term in the defining supremum.
-- What remains is the order-completeness bookkeeping for `sInf`/`sSup`.
have hUniformRadius :
∀ (g : (Fin d → ℝ) → ℝ),
DornTopDerivativeSeminorm β L₀ g →
(∀ x ∈ dornCube d, |g x| ≤ M) →
holderFullNorm β (transformedResponse g) ≤
Kcube * (M + L₀) := by
intro g hg hbound
have hball := hCompletedResponse g hg hbound
unfold holderFullNorm
apply csInf_le
· exact ⟨0, by intro r hr; exact hr.1⟩
· exact ⟨le_of_lt (mul_pos hKcube (add_pos hM hL₀)), hball⟩
let Sinterp : Set ℝ := {r | ∃ (M' L' : ℝ) (g : (Fin d → ℝ) → ℝ),
0 < M' ∧ 0 < L' ∧ DornTopDerivativeSeminorm β L' g ∧
(∀ x ∈ dornCube d, |g x| ≤ M') ∧
r = holderFullNorm β (transformedResponse g) / (M' + L')}
let Scommon : Set ℝ := {r | ∃ g : (Fin d → ℝ) → ℝ,
DornTopDerivativeSeminorm β L₀ g ∧
(∀ x ∈ dornCube d, |g x| ≤ M) ∧
r = holderFullNorm β (transformedResponse g)}
have hUpperNorm : ∀ (M' L' : ℝ) (g : (Fin d → ℝ) → ℝ),
0 ≤ M' → 0 ≤ L' → DornTopDerivativeSeminorm β L' g →
(∀ x ∈ dornCube d, |g x| ≤ M') →
holderFullNorm β (transformedResponse g) ≤ Kcube * (M' + L') := by
intro M' L' g hM' hL' hg hb
unfold holderFullNorm
apply csInf_le
· exact ⟨0, by intro r hr; exact hr.1⟩
· exact ⟨mul_nonneg (le_of_lt hKcube) (add_nonneg hM' hL'),
hCubeCompletion M' L' g hM' hL' hg hb⟩
have hBddI : BddAbove Sinterp := by
refine ⟨Kcube, ?_⟩
intro r hr
obtain ⟨M', L', g, hM', hL', hg, hb, rfl⟩ := hr
have hsum : 0 < M' + L' := add_pos hM' hL'
exact (div_le_iff₀ hsum).2 (hUpperNorm M' L' g
hM'.le hL'.le hg hb)
have hBddC : BddAbove Scommon := by
refine ⟨Kcube * (M + L₀), ?_⟩
intro r hr
obtain ⟨g, hg, hb, rfl⟩ := hr
exact hUniformRadius g hg hb
have hzeroCube : (fun _ : Fin d => (0 : ℝ)) ∈ cube d := by
exact Set.mem_univ_pi.mpr (fun i => by norm_num)
have hConst1 : DornTopDerivativeSeminorm β 1
(fun _ : Fin d → ℝ => 1) :=
dornTopDerivativeSeminorm_const β 1 1 (by norm_num)
have hBall1 : HolderBall β (Kcube * (1 + 1))
(transformedResponse (fun _ : Fin d → ℝ => 1)) := by
apply hCubeCompletion 1 1 (fun _ => 1) (by norm_num) (by norm_num) hConst1
intro x hx
norm_num
have hNorm1 : 1 ≤ holderFullNorm β
(transformedResponse (fun _ : Fin d → ℝ => 1)) := by
have h := holderFullNorm_ge_pointwise β
(transformedResponse (fun _ : Fin d → ℝ => 1))
(fun _ => 0) hzeroCube
⟨Kcube * (1 + 1), by positivity, hBall1⟩
simpa [transformedResponse] using h
have hMemI : holderFullNorm β
(transformedResponse (fun _ : Fin d → ℝ => 1)) / (1 + 1) ∈ Sinterp := by
refine ⟨1, 1, (fun _ => 1), by norm_num, by norm_num, hConst1, ?_, rfl⟩
intro x hx
norm_num
have hInterpPos : 0 < sSup Sinterp := by
have hratio : 0 < holderFullNorm β
(transformedResponse (fun _ : Fin d → ℝ => 1)) / (1 + 1) :=
div_pos (lt_of_lt_of_le zero_lt_one hNorm1) (by norm_num)
exact lt_of_lt_of_le hratio (le_csSup hBddI hMemI)
have hConstM : DornTopDerivativeSeminorm β L₀
(fun _ : Fin d → ℝ => M) :=
dornTopDerivativeSeminorm_const β L₀ M hL₀.le
have hBallM : HolderBall β (Kcube * (M + L₀))
-- … truncated; follow the source link for the rest …
CausalSmith.Stat.STAT_WeakoverlapUnisolventRate_Research.TStrictEnlargement 104 declarations The global-tail class strictly contains laws excluded by Dorn A3
The global-tail class strictly contains laws excluded by Dorn A3
The common Bernoulli response is admissible for both the outcome and Hölder-radius envelopes. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
B L :
ℝ
hB :
0 < B
hL :
0 < L
conclusion
conclusion 1
0 < baselineSuccess B L
conclusion 2
baselineSuccess B L ≤ 1 / 4
conclusion 3
B * baselineSuccess B L ≤ L / 4
Proof (Lean source)
lemma baselineSuccess_bounds (B L : ℝ) (hB : 0 < B) (hL : 0 < L) :
0 < baselineSuccess B L ∧ baselineSuccess B L ≤ 1 / 4 ∧
B * baselineSuccess B L ≤ L / 4 := by
unfold baselineSuccess
have hratio : 0 < L / (4 * B) := by positivity
refine ⟨lt_min (by norm_num) hratio, min_le_left _ _, ?_⟩
calc
B * min (1 / 4) (L / (4 * B)) ≤ B * (L / (4 * B)) :=
mul_le_mul_of_nonneg_left (min_le_right _ _) hB.le
_ = L / 4 := by
field_simp
The strip used in the treatment kernel is Borel measurable. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma thinStrip_measurableSet (d : ℕ) (hd : 2 ≤ d) (a : ℝ) :
MeasurableSet (thinStrip d hd a) := by
unfold thinStrip
exact measurableSet_le (by fun_prop) (by fun_prop)
The displayed propensity is a measurable function of the covariates. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma stripPropensity_measurable (d : ℕ) (hd : 2 ≤ d) (γ a : ℝ) :
Measurable (stripPropensity d hd γ a) := by
classical
have hstrip := thinStrip_measurableSet d hd a
have hite : Measurable (fun x : Fin d → ℝ =>
if x ∈ thinStrip d hd a then (1 : ℝ) else 0) :=
Measurable.ite hstrip measurable_const measurable_const
have hrad : Measurable (fun x : Fin d → ℝ =>
(min 1 ((2 * ‖x - fun _ => (1 / 2 : ℝ)‖) ^ d)) ^
((1 : ℝ) / (γ - 1))) := by
fun_prop
unfold stripPropensity centreRadiusCDF centreRadius
exact measurable_const.min (hrad.add (measurable_const.mul hite))
theorem completionAt_measurable_of_measurable Lemma completionAt_measurable_of_measurable in the paper ↗
A pointwise-valid Bernoulli completion is a measurable measure-valued kernel when its three success probabilities are measurable. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Measurable (completionAt B e p₀ p₁)
Proof (Lean source)
lemma completionAt_measurable_of_measurable {d : ℕ} (B : ℝ)
(e p₀ p₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hp₀ : Measurable p₀) (hp₁ : Measurable p₁)
(hvalid : ∀ x, e x ∈ Icc 0 1 ∧ p₀ x ∈ Icc 0 1 ∧
p₁ x ∈ Icc 0 1) :
Measurable (completionAt B e p₀ p₁) := by
let μe : (Fin d → ℝ) → ProbabilityMeasure Bool := fun x =>
⟨realBernoulli (e x), realBernoulli_probability (e x) (hvalid x).1⟩
let μ₀ : (Fin d → ℝ) → ProbabilityMeasure ℝ := fun x =>
⟨scaledBernoulli B (p₀ x), scaledBernoulli_probability B (p₀ x) (hvalid x).2.1⟩
let μ₁ : (Fin d → ℝ) → ProbabilityMeasure ℝ := fun x =>
⟨scaledBernoulli B (p₁ x), scaledBernoulli_probability B (p₁ x) (hvalid x).2.2⟩
have hμe : Measurable μe := by
apply Measurable.subtype_mk
unfold realBernoulli
fun_prop
have hμ₀ : Measurable μ₀ := by
apply Measurable.subtype_mk
unfold scaledBernoulli
exact (Measure.measurable_map (fun b : Bool => if b then B else 0) (by fun_prop)).comp
(by unfold realBernoulli; fun_prop)
have hμ₁ : Measurable μ₁ := by
apply Measurable.subtype_mk
unfold scaledBernoulli
exact (Measure.measurable_map (fun b : Bool => if b then B else 0) (by fun_prop)).comp
(by unfold realBernoulli; fun_prop)
let μ₀₁ : (Fin d → ℝ) → ProbabilityMeasure (ℝ × ℝ) := fun x =>
⟨(μ₀ x : Measure ℝ).prod (μ₁ x : Measure ℝ), inferInstance⟩
have hμ₀₁ : Measurable μ₀₁ := by
apply Measurable.subtype_mk
exact ProbabilityMeasure.measurable_fun_prod.comp (hμ₀.prodMk hμ₁)
let μa : (Fin d → ℝ) → ProbabilityMeasure (Bool × ℝ × ℝ) := fun x =>
⟨(μe x : Measure Bool).prod (μ₀₁ x : Measure (ℝ × ℝ)), inferInstance⟩
have hμa : Measurable μa := by
apply Measurable.subtype_mk
exact ProbabilityMeasure.measurable_fun_prod.comp (hμe.prodMk hμ₀₁)
let μx : (Fin d → ℝ) → ProbabilityMeasure ((Fin d → ℝ) × Bool × ℝ × ℝ) := fun x =>
⟨(Measure.dirac x).prod (μa x : Measure (Bool × ℝ × ℝ)), inferInstance⟩
have hμx : Measurable μx := by
apply Measurable.subtype_mk
have hdirac : Measurable (fun x : Fin d → ℝ =>
(⟨Measure.dirac x, inferInstance⟩ : ProbabilityMeasure (Fin d → ℝ))) := by
apply Measurable.subtype_mk
exact Measure.measurable_dirac
exact ProbabilityMeasure.measurable_fun_prod.comp (hdirac.prodMk hμa)
let F : ((Fin d → ℝ) × Bool × ℝ × ℝ) → Completion d := fun z =>
(z.1, z.2.1, z.2.2.1, z.2.2.2,
if z.2.1 then z.2.2.2 else z.2.2.1)
have hF : Measurable F := by
unfold F
have hlast : Measurable (fun z : (Fin d → ℝ) × Bool × ℝ × ℝ =>
if z.2.1 then z.2.2.2 else z.2.2.1) :=
Measurable.ite (measurable_snd.fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption)
have heq : completionAt B e p₀ p₁ = fun x =>
(μx x : Measure ((Fin d → ℝ) × Bool × ℝ × ℝ)).map F := by
funext x
letI := realBernoulli_probability (e x) (hvalid x).1
letI := scaledBernoulli_probability B (p₀ x) (hvalid x).2.1
letI := scaledBernoulli_probability B (p₁ x) (hvalid x).2.2
unfold completionAt
change Measure.map _ ((realBernoulli (e x)).prod
((scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x)))) =
Measure.map F ((Measure.dirac x).prod
((realBernoulli (e x)).prod
((scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x)))))
rw [Measure.dirac_prod, Measure.map_map hF (by fun_prop)]
rfl
rw [heq]
exact (Measure.measurable_map F hF).comp (measurable_subtype_coe.comp hμx)
theorem completionAt_treated_covariate_event Lemma completionAt_treated_covariate_event in the paper ↗
At a fixed covariate, a treated covariate event has Bernoulli mass e x. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma completionAt_treated_covariate_event {d : ℕ} (B : ℝ)
(e p₀ p₁ : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ)
(he : e x ∈ Icc 0 1) (hp₀ : p₀ x ∈ Icc 0 1)
(hp₁ : p₁ x ∈ Icc 0 1) (D : Set (Fin d → ℝ))
(hD : MeasurableSet D) :
completionAt B e p₀ p₁ x {ω | ω.1 ∈ D ∧ ω.2.1 = true} =
(Measure.dirac x) D * ofReal (e x) := by
classical
letI := realBernoulli_probability (e x) he
letI := scaledBernoulli_probability B (p₀ x) hp₀
letI := scaledBernoulli_probability B (p₁ x) hp₁
unfold completionAt
rw [Measure.map_apply]
· change ((realBernoulli (e x)).prod
((scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x))))
{z | x ∈ D ∧ z.1 = true} =
(Measure.dirac x) D * ofReal (e x)
by_cases hx : x ∈ D
· simp only [hx, true_and, if_pos]
rw [show {a : Bool × ℝ × ℝ | a.1 = true} =
({true} : Set Bool) ×ˢ univ by ext z; simp]
rw [Measure.prod_prod]
simp [realBernoulli, Measure.dirac_apply' _ hD, hx]
· simp [hx, Measure.dirac_apply' _ hD]
· have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption)
· exact (hD.preimage measurable_fst).inter
((measurable_fst.comp measurable_snd) (MeasurableSet.singleton true))
theorem completionAt_covariate_treatment_map Lemma completionAt_covariate_treatment_map in the paper ↗
At fixed covariate x, the (X,A) marginal is the product of the point mass at x and the displayed Bernoulli treatment law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
B :
ℝ
e p₀ p₁ :
(Fin d → ℝ) → ℝ
x :
Fin d → ℝ
he :
e x ∈ Icc 0 1
hp₀ :
p₀ x ∈ Icc 0 1
hp₁ :
p₁ x ∈ Icc 0 1
conclusion
(completionAt B e p₀ p₁ x).map (fun ω => (ω.1, ω.2.1))
= (Measure.dirac x).prod (realBernoulli (e x))
Proof (Lean source)
lemma completionAt_covariate_treatment_map {d : ℕ} (B : ℝ)
(e p₀ p₁ : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ)
(he : e x ∈ Icc 0 1) (hp₀ : p₀ x ∈ Icc 0 1)
(hp₁ : p₁ x ∈ Icc 0 1) :
(completionAt B e p₀ p₁ x).map (fun ω => (ω.1, ω.2.1)) =
(Measure.dirac x).prod (realBernoulli (e x)) := by
let ν := (scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x))
let μ := (realBernoulli (e x)).prod ν
letI : IsProbabilityMeasure (realBernoulli (e x)) := realBernoulli_probability _ he
letI : IsProbabilityMeasure (scaledBernoulli B (p₀ x)) :=
scaledBernoulli_probability _ _ hp₀
letI : IsProbabilityMeasure (scaledBernoulli B (p₁ x)) :=
scaledBernoulli_probability _ _ hp₁
letI : IsProbabilityMeasure ν := inferInstance
unfold completionAt
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
rw [Measure.map_map (by fun_prop) (by fun_prop (disch := assumption))]
change Measure.map (fun z : Bool × (ℝ × ℝ) => (x, z.1)) μ = _
rw [Measure.dirac_prod]
calc
Measure.map (fun z : Bool × (ℝ × ℝ) => (x, z.1)) μ =
Measure.map (fun a : Bool => (x, a)) (Measure.map fst μ) := by
rw [Measure.map_map (by fun_prop) measurable_fst]
congr 1
_ = Measure.map (fun a : Bool => (x, a)) (realBernoulli (e x)) := by
congr 1
dsimp [μ]
rw [Measure.map_fst_prod]
simp
theorem completionBind_covariate_treatment_compProd Lemma completionBind_covariate_treatment_compProd in the paper ↗
The joint (X,A) law of a completion bind is the covariate law composed with its explicit Bernoulli treatment kernel. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
k :
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x)) (by unfold realBernoulli; fun_prop) (: Kernel (Fin d → ℝ) Bool)
(mu.bind (completionAt B e p₀ p₁)).map (fun ω => (ω.1, ω.2.1)) = Measure.compProd mu k
Proof (Lean source)
lemma completionBind_covariate_treatment_compProd {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (B : ℝ)
(e p₀ p₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hp₀ : Measurable p₀) (hp₁ : Measurable p₁)
(hvalid : ∀ x, e x ∈ Icc 0 1 ∧ p₀ x ∈ Icc 0 1 ∧
p₁ x ∈ Icc 0 1) :
let k : Kernel (Fin d → ℝ) Bool :=
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x))
(by unfold realBernoulli; fun_prop)
(mu.bind (completionAt B e p₀ p₁)).map (fun ω => (ω.1, ω.2.1)) =
Measure.compProd mu k := by
dsimp
let k : Kernel (Fin d → ℝ) Bool :=
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x))
(by unfold realBernoulli; fun_prop)
letI : IsMarkovKernel k :=
⟨fun x => realBernoulli_probability (e x) (hvalid x).1⟩
have hk : Measurable (completionAt B e p₀ p₁) :=
completionAt_measurable_of_measurable B e p₀ p₁ he hp₀ hp₁ hvalid
apply Measure.ext
intro s hs
rw [Measure.map_apply (by fun_prop) hs, Measure.bind_apply
(hs.preimage (by fun_prop)) hk.aemeasurable, Measure.compProd_apply hs]
apply lintegral_congr
intro x
have hfix := completionAt_covariate_treatment_map B e p₀ p₁ x
(hvalid x).1 (hvalid x).2.1 (hvalid x).2.2
have happ := congrArg (fun m : Measure ((Fin d → ℝ) × Bool) => m s) hfix
rw [Measure.dirac_prod] at happ
have happ' := happ.trans (Measure.map_apply (by fun_prop) hs)
have happ'' := (Measure.map_apply (by fun_prop) hs).symm.trans happ'
simpa [k] using happ''
The displayed Bernoulli parameter is a version of the propensity of the observed completion law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma completionBind_propensity_ae {d : ℕ}
(mu : Measure (Fin d → ℝ)) [IsProbabilityMeasure mu] (B : ℝ)
(e p₀ p₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hp₀ : Measurable p₀) (hp₁ : Measurable p₁)
(hvalid : ∀ x, e x ∈ Icc 0 1 ∧ p₀ x ∈ Icc 0 1 ∧
p₁ x ∈ Icc 0 1)
[IsFiniteMeasure ((mu.bind (completionAt B e p₀ p₁)).map observed)] :
let P := (mu.bind (completionAt B e p₀ p₁)).map observed
∀ᵐ x ∂mu, (condDistrib (fun z : Obs d => z.2.1)
(fun z => z.1) P x
{true}).toReal = e x := by
dsimp
let P := (mu.bind (completionAt B e p₀ p₁)).map observed
let k : Kernel (Fin d → ℝ) Bool :=
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x))
(by unfold realBernoulli; fun_prop)
have hk : Measurable (completionAt B e p₀ p₁) :=
completionAt_measurable_of_measurable B e p₀ p₁ he hp₀ hp₁ hvalid
have hpoint : ∀ x, IsProbabilityMeasure (completionAt B e p₀ p₁ x) := by
intro x
letI := realBernoulli_probability (e x) (hvalid x).1
letI := scaledBernoulli_probability B (p₀ x) (hvalid x).2.1
letI := scaledBernoulli_probability B (p₁ x) (hvalid x).2.2
let ν := (realBernoulli (e x)).prod
((scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x)))
letI : IsProbabilityMeasure ν := inferInstance
unfold completionAt
change IsProbabilityMeasure (Measure.map _ ν)
apply Measure.isProbabilityMeasure_map
apply Measurable.aemeasurable
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption)
letI : IsProbabilityMeasure (mu.bind (completionAt B e p₀ p₁)) :=
isProbabilityMeasure_bind hk.aemeasurable (Filter.Eventually.of_forall hpoint)
have hobs : Measurable (@observed d) := by
unfold observed
fun_prop
letI : IsProbabilityMeasure P := by
dsimp [P]
apply Measure.isProbabilityMeasure_map hobs.aemeasurable
letI : IsMarkovKernel k :=
⟨fun x => realBernoulli_probability (e x) (hvalid x).1⟩
have hcov : P.map fst = mu := by
dsimp [P]
rw [Measure.map_map (by fun_prop) hobs]
exact completionBind_covariate_marginal mu B e p₀ p₁ hk.aemeasurable
(Filter.Eventually.of_forall hvalid)
have hpair : P.map (fun z => (z.1, z.2.1)) = Measure.compProd mu k := by
dsimp [P]
rw [Measure.map_map (by fun_prop) hobs]
exact completionBind_covariate_treatment_compProd mu B e p₀ p₁
he hp₀ hp₁ hvalid
have hae := ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd_of_measurable
(μ := P) (X := fun z : Obs d => z.1) (Y := fun z : Obs d => z.2.1)
(κ := k) (by fun_prop) (by fun_prop) (by simpa [hcov] using hpair)
rw [hcov] at hae
filter_upwards [hae] with x hx
have hx' := congrArg (fun ν : Measure Bool => (ν {true}).toReal) hx
change (condDistrib (fun z : Obs d => z.2.1)
(fun z => z.1) P x
{true}).toReal = e x
rw [hx']
simp [k, realBernoulli, (hvalid x).1.1]
theorem completionAt_observed_joint_map_equal_arms Lemma completionAt_observed_joint_map_equal_arms in the paper ↗
When the two response arms have the same Bernoulli law, selecting the observed response leaves that law independent of the treatment draw. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
(completionAt B e p p x).map (fun ω => ((ω.1, ω.2.1), ω.2.2.2.2))
Proof (Lean source)
lemma completionAt_observed_joint_map_equal_arms {d : ℕ} (B : ℝ)
(e p : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ)
(he : e x ∈ Icc 0 1) (hp : p x ∈ Icc 0 1) :
(completionAt B e p p x).map
(fun ω => ((ω.1, ω.2.1), ω.2.2.2.2)) =
((Measure.dirac x).prod (realBernoulli (e x))).prod
(scaledBernoulli B (p x)) := by
letI := realBernoulli_probability (e x) he
letI := scaledBernoulli_probability B (p x) hp
unfold completionAt
rw [Measure.map_map (by fun_prop) (by
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption))]
rw [Measure.dirac_prod]
let g : Bool × ℝ × ℝ → ((Fin d → ℝ) × Bool) × ℝ :=
fun z => ((x, z.1), if z.1 then z.2.2 else z.2.1)
change Measure.map g ((realBernoulli (e x)).prod
((scaledBernoulli B (p x)).prod (scaledBernoulli B (p x)))) = _
have hg : Measurable g := by
dsimp [g]
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption)
unfold realBernoulli
rw [Measure.add_prod, Measure.map_add _ _ hg,
Measure.map_add _ _ (by fun_prop), Measure.add_prod]
congr 1
· have hbase :
Measure.map g ((Measure.dirac false).prod
((scaledBernoulli B (p x)).prod (scaledBernoulli B (p x)))) =
(Measure.dirac (x, false)).prod (scaledBernoulli B (p x)) := by
rw [Measure.dirac_prod, Measure.map_map hg (by fun_prop)]
change Measure.map ((Prod.mk (x, false)) ∘ (Prod.fst : ℝ × ℝ → ℝ))
((scaledBernoulli B (p x)).prod (scaledBernoulli B (p x))) = _
rw [← Measure.map_map (by fun_prop : Measurable (Prod.mk (x, false)))
measurable_fst, Measure.map_fst_prod]
simpa using (Measure.dirac_prod (x := (x, false))
(ν := scaledBernoulli B (p x))).symm
calc
Measure.map g ((ofReal (1 - e x) • Measure.dirac false).prod
((scaledBernoulli B (p x)).prod (scaledBernoulli B (p x)))) =
ofReal (1 - e x) • Measure.map g ((Measure.dirac false).prod
((scaledBernoulli B (p x)).prod (scaledBernoulli B (p x)))) := by
rw [Measure.prod_smul_left, Measure.map_smul]
_ = ofReal (1 - e x) •
((Measure.dirac (x, false)).prod (scaledBernoulli B (p x))) := by rw [hbase]
_ = (ofReal (1 - e x) • Measure.dirac (x, false)).prod
(scaledBernoulli B (p x)) := by rw [Measure.prod_smul_left]
_ = (Measure.map (Prod.mk x)
(ofReal (1 - e x) • Measure.dirac false)).prod
(scaledBernoulli B (p x)) := by simp
· have hbase :
Measure.map g ((Measure.dirac true).prod
((scaledBernoulli B (p x)).prod (scaledBernoulli B (p x)))) =
(Measure.dirac (x, true)).prod (scaledBernoulli B (p x)) := by
rw [Measure.dirac_prod, Measure.map_map hg (by fun_prop)]
change Measure.map ((Prod.mk (x, true)) ∘ (Prod.snd : ℝ × ℝ → ℝ))
((scaledBernoulli B (p x)).prod (scaledBernoulli B (p x))) = _
rw [← Measure.map_map (by fun_prop : Measurable (Prod.mk (x, true)))
measurable_snd, Measure.map_snd_prod]
simpa using (Measure.dirac_prod (x := (x, true))
(ν := scaledBernoulli B (p x))).symm
calc
Measure.map g ((ofReal (e x) • Measure.dirac true).prod
((scaledBernoulli B (p x)).prod (scaledBernoulli B (p x)))) =
ofReal (e x) • Measure.map g ((Measure.dirac true).prod
((scaledBernoulli B (p x)).prod (scaledBernoulli B (p x)))) := by
rw [Measure.prod_smul_left, Measure.map_smul]
_ = ofReal (e x) •
((Measure.dirac (x, true)).prod (scaledBernoulli B (p x))) := by rw [hbase]
_ = (ofReal (e x) • Measure.dirac (x, true)).prod
(scaledBernoulli B (p x)) := by rw [Measure.prod_smul_left]
_ = (Measure.map (Prod.mk x)
(ofReal (e x) • Measure.dirac true)).prod
(scaledBernoulli B (p x)) := by simp
theorem completionBind_observed_joint_compProd_equal_arms Lemma completionBind_observed_joint_compProd_equal_arms in the paper ↗
Mixing the equal-arm fixed-covariate identity gives the joint observed law as the composition-product of the (X,A) law and its response kernel. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
kA :
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x)) (by unfold realBernoulli; fun_prop) (: Kernel (Fin d → ℝ) Bool)
kY :
ProbabilityTheory.Kernel.mk (fun xa => scaledBernoulli B (p xa.1)) (by unfold scaledBernoulli realBernoulli; fun_prop) (: Kernel ((Fin d → ℝ) × Bool) ℝ)
(mu.bind (completionAt B e p p)).map (fun ω => ((ω.1, ω.2.1), ω.2.2.2.2))
= Measure.compProd (Measure.compProd mu kA) kY
Proof (Lean source)
lemma completionBind_observed_joint_compProd_equal_arms {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (B : ℝ)
(e p : (Fin d → ℝ) → ℝ) (he : Measurable e) (hp : Measurable p)
(hvalid : ∀ x, e x ∈ Icc 0 1 ∧ p x ∈ Icc 0 1) :
let kA : Kernel (Fin d → ℝ) Bool :=
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x))
(by unfold realBernoulli; fun_prop)
let kY : Kernel ((Fin d → ℝ) × Bool) ℝ :=
ProbabilityTheory.Kernel.mk (fun xa => scaledBernoulli B (p xa.1))
(by unfold scaledBernoulli realBernoulli; fun_prop)
(mu.bind (completionAt B e p p)).map
(fun ω => ((ω.1, ω.2.1), ω.2.2.2.2)) =
Measure.compProd (Measure.compProd mu kA) kY := by
dsimp
let kA : Kernel (Fin d → ℝ) Bool :=
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x))
(by unfold realBernoulli; fun_prop)
let kY : Kernel ((Fin d → ℝ) × Bool) ℝ :=
ProbabilityTheory.Kernel.mk (fun xa => scaledBernoulli B (p xa.1))
(by unfold scaledBernoulli realBernoulli; fun_prop)
letI : IsMarkovKernel kA :=
⟨fun x => realBernoulli_probability (e x) (hvalid x).1⟩
letI : IsMarkovKernel kY :=
⟨fun xa => scaledBernoulli_probability B (p xa.1) (hvalid xa.1).2⟩
have hk : Measurable (completionAt B e p p) :=
completionAt_measurable_of_measurable B e p p he hp hp
(fun x => ⟨(hvalid x).1, (hvalid x).2, (hvalid x).2⟩)
apply Measure.ext
intro s hs
rw [Measure.map_apply (by fun_prop) hs,
Measure.bind_apply (hs.preimage (by fun_prop)) hk.aemeasurable,
Measure.compProd_apply hs, Measure.lintegral_compProd]
swap
· exact ProbabilityTheory.Kernel.measurable_kernel_prodMk_left hs
apply lintegral_congr
intro x
letI := realBernoulli_probability (e x) (hvalid x).1
letI := scaledBernoulli_probability B (p x) (hvalid x).2
have hfix := completionAt_observed_joint_map_equal_arms B e p x
(hvalid x).1 (hvalid x).2
have happ := congrArg (fun m : Measure (((Fin d → ℝ) × Bool) × ℝ) => m s) hfix
rw [Measure.map_apply (by fun_prop) hs] at happ
rw [Measure.prod_apply hs] at happ
rw [lintegral_prod _ (measurable_measure_prodMk_left hs).aemeasurable,
lintegral_dirac] at happ
simpa [kA, kY] using happ
theorem completionBind_observed_responseKernel_ae_equal_arms Lemma completionBind_observed_responseKernel_ae_equal_arms in the paper ↗
The conditional observed-response kernel in the equal-arm completion is the displayed scaled Bernoulli kernel. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
P :
(mu.bind (completionAt B e p p)).map observed
kY :
ProbabilityTheory.Kernel.mk (fun xa => scaledBernoulli B (p xa.1)) (by unfold scaledBernoulli realBernoulli; fun_prop) (: Kernel ((Fin d → ℝ) × Bool) ℝ)
condDistrib (fun z : Obs d => z.2.2) (fun z : Obs d => (z.1, z.2.1)) P
=ᵐ[P.map (fun z => (z.1, z.2.1))] kY
Proof (Lean source)
lemma completionBind_observed_responseKernel_ae_equal_arms {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (B : ℝ)
(e p : (Fin d → ℝ) → ℝ) (he : Measurable e) (hp : Measurable p)
(hvalid : ∀ x, e x ∈ Icc 0 1 ∧ p x ∈ Icc 0 1)
[IsFiniteMeasure ((mu.bind (completionAt B e p p)).map observed)] :
let P := (mu.bind (completionAt B e p p)).map observed
let kY : Kernel ((Fin d → ℝ) × Bool) ℝ :=
ProbabilityTheory.Kernel.mk (fun xa => scaledBernoulli B (p xa.1))
(by unfold scaledBernoulli realBernoulli; fun_prop)
condDistrib (fun z : Obs d => z.2.2)
(fun z : Obs d => (z.1, z.2.1)) P =ᵐ[P.map (fun z => (z.1, z.2.1))] kY := by
dsimp
let P := (mu.bind (completionAt B e p p)).map observed
let kA : Kernel (Fin d → ℝ) Bool :=
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x))
(by unfold realBernoulli; fun_prop)
let kY : Kernel ((Fin d → ℝ) × Bool) ℝ :=
ProbabilityTheory.Kernel.mk (fun xa => scaledBernoulli B (p xa.1))
(by unfold scaledBernoulli realBernoulli; fun_prop)
letI : IsMarkovKernel kA :=
⟨fun x => realBernoulli_probability (e x) (hvalid x).1⟩
letI : IsMarkovKernel kY :=
⟨fun xa => scaledBernoulli_probability B (p xa.1) (hvalid xa.1).2⟩
have hobs : Measurable (@observed d) := by unfold observed; fun_prop
have hxa : P.map (fun z => (z.1, z.2.1)) = Measure.compProd mu kA := by
dsimp [P]
rw [Measure.map_map (by fun_prop) hobs]
exact completionBind_covariate_treatment_compProd mu B e p p he hp hp
(fun x => ⟨(hvalid x).1, (hvalid x).2, (hvalid x).2⟩)
have hjoint : P.map (fun z => ((z.1, z.2.1), z.2.2)) =
Measure.compProd (Measure.compProd mu kA) kY := by
dsimp [P]
rw [Measure.map_map (by fun_prop) hobs]
exact completionBind_observed_joint_compProd_equal_arms mu B e p he hp hvalid
exact ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd_of_measurable
(μ := P) (X := fun z : Obs d => (z.1, z.2.1))
(Y := fun z : Obs d => z.2.2) (κ := kY) (by fun_prop) (by fun_prop)
(by simpa [hxa] using hjoint)
An almost-sure Bernoulli property holds at true when its success mass is positive. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma realBernoulli_ae_at_true_of_pos (r : ℝ) (hr : 0 < r)
(q : Bool → Prop) (hq : ∀ᵐ a ∂realBernoulli r, q a) : q true := by
by_contra htrue
have hzero : (realBernoulli r) {a | ¬ q a} = 0 := by
simpa only [Set.compl_setOf, Classical.not_not] using (mem_ae_iff.mp hq)
have hle : (realBernoulli r) ({true} : Set Bool) ≤
(realBernoulli r) {a | ¬ q a} := by
apply measure_mono
intro a ha
have ha' : a = true := by simpa using ha
subst a
exact htrue
have hmass : (realBernoulli r) ({true} : Set Bool) = ofReal r := by
simp [realBernoulli]
rw [hzero, hmass] at hle
exact (not_le_of_gt (ENNReal.ofReal_pos.mpr hr)) hle
theorem completionBind_treatedKernel_ae_equal_arms Lemma completionBind_treatedKernel_ae_equal_arms in the paper ↗
Positive treatment mass promotes the equal-arm response-kernel identity to the treated fibre over almost every covariate. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
P :
(mu.bind (completionAt B e p p)).map observed
∀ᵐ x ∂mu, treatedKernel P x = scaledBernoulli B (p x)
Proof (Lean source)
lemma completionBind_treatedKernel_ae_equal_arms {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (B : ℝ)
(e p : (Fin d → ℝ) → ℝ) (he : Measurable e) (hp : Measurable p)
(hvalid : ∀ x, e x ∈ Icc 0 1 ∧ p x ∈ Icc 0 1)
(hpos : ∀ᵐ x ∂mu, 0 < e x)
[IsFiniteMeasure ((mu.bind (completionAt B e p p)).map observed)] :
let P := (mu.bind (completionAt B e p p)).map observed
∀ᵐ x ∂mu, treatedKernel P x = scaledBernoulli B (p x) := by
dsimp
let P := (mu.bind (completionAt B e p p)).map observed
let kA : Kernel (Fin d → ℝ) Bool :=
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x))
(by unfold realBernoulli; fun_prop)
letI : IsMarkovKernel kA :=
⟨fun x => realBernoulli_probability (e x) (hvalid x).1⟩
have hobs : Measurable (@observed d) := by unfold observed; fun_prop
have hxa : P.map (fun z => (z.1, z.2.1)) = Measure.compProd mu kA := by
dsimp [P]
rw [Measure.map_map (by fun_prop) hobs]
exact completionBind_covariate_treatment_compProd mu B e p p he hp hp
(fun x => ⟨(hvalid x).1, (hvalid x).2, (hvalid x).2⟩)
have hk := completionBind_observed_responseKernel_ae_equal_arms
mu B e p he hp hvalid
dsimp at hk
rw [hxa] at hk
have hsections := Measure.ae_ae_of_ae_compProd hk
filter_upwards [hsections, hpos] with x hx hxp
have htrue := realBernoulli_ae_at_true_of_pos (e x) hxp
(fun a => condDistrib (fun z : Obs d => z.2.2)
(fun z : Obs d => (z.1, z.2.1)) P (x, a) = scaledBernoulli B (p x))
(by simpa [kA] using hx)
simpa [treatedKernel] using htrue
The mean of the scaled Bernoulli law is its scale times its success probability. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma scaledBernoulli_integral_id (B r : ℝ) (hr : r ∈ Icc 0 1) :
∫ y, y ∂scaledBernoulli B r = B * r := by
letI := realBernoulli_probability r hr
unfold scaledBernoulli
rw [integral_map (μ := realBernoulli r)
(φ := fun b : Bool => if b then B else 0) (f := fun y : ℝ => y)
(by fun_prop) (by fun_prop)]
unfold realBernoulli
rw [integral_add_measure
((integrable_dirac (by simp)).smul_measure (by simp))
((integrable_dirac (by simp)).smul_measure (by simp))]
simp [ENNReal.toReal_ofReal, hr.1, mul_comm]
theorem completionBind_treatedRegression_ae_equal_arms Lemma completionBind_treatedRegression_ae_equal_arms in the paper ↗
The treated regression of the equal-arm completion is B * p. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
P :
(mu.bind (completionAt B e p p)).map observed
∀ᵐ x ∂mu, treatedRegression P x = B * p x
Proof (Lean source)
lemma completionBind_treatedRegression_ae_equal_arms {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (B : ℝ)
(e p : (Fin d → ℝ) → ℝ) (he : Measurable e) (hp : Measurable p)
(hvalid : ∀ x, e x ∈ Icc 0 1 ∧ p x ∈ Icc 0 1)
(hpos : ∀ᵐ x ∂mu, 0 < e x)
[IsFiniteMeasure ((mu.bind (completionAt B e p p)).map observed)] :
let P := (mu.bind (completionAt B e p p)).map observed
∀ᵐ x ∂mu, treatedRegression P x = B * p x := by
dsimp
filter_upwards [completionBind_treatedKernel_ae_equal_arms
mu B e p he hp hvalid hpos] with x hx
rw [treatedRegression, hx, scaledBernoulli_integral_id B (p x) (hvalid x).2]
A [0,B]-valued scaled Bernoulli variable satisfies the model's looser B² centered MGF envelope. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
conclusion 1
conclusion 2
mgf (fun y : ℝ => y - B * r) (scaledBernoulli B r) t
≤ exp (B ^ 2 * t ^ 2 / 2)
Proof (Lean source)
lemma scaledBernoulli_centered_mgf_bound (B r t : ℝ) (hB : 0 < B)
(hr : r ∈ Icc 0 1) :
Integrable (fun y : ℝ => exp (t * (y - B * r)))
(scaledBernoulli B r) ∧
mgf (fun y : ℝ => y - B * r)
(scaledBernoulli B r) t ≤ exp (B ^ 2 * t ^ 2 / 2) := by
letI := scaledBernoulli_probability B r hr
have hsupp : ∀ᵐ y ∂scaledBernoulli B r, y ∈ Icc (0 : ℝ) B := by
filter_upwards [scaledBernoulli_supported_on_endpoints B r] with y hy
rcases hy with rfl | rfl
· exact ⟨le_rfl, hB.le⟩
· exact ⟨hB.le, le_rfl⟩
have hsg := ProbabilityTheory.hasSubgaussianMGF_of_mem_Icc
(X := fun y : ℝ => y) (μ := scaledBernoulli B r)
measurable_id.aemeasurable hsupp
have hmean := scaledBernoulli_integral_id B r hr
constructor
· simpa [hmean] using hsg.integrable_exp_mul t
· have hle := hsg.mgf_le t
rw [hmean] at hle
refine hle.trans (Real.exp_le_exp.mpr ?_)
rw [show ‖B - 0‖₊ = ⟨B, hB.le⟩ by
apply Subtype.ext
simp [Real.norm_eq_abs, abs_of_pos hB]]
change (B / 2) ^ 2 * t ^ 2 / 2 ≤ B ^ 2 * t ^ 2 / 2
nlinarith [sq_nonneg B, sq_nonneg t]
theorem completionBind_boundedMeanSubGaussian_equal_arms Lemma completionBind_boundedMeanSubGaussian_equal_arms in the paper ↗
Equal bounded Bernoulli arms satisfy the model's treated mean and sub-Gaussian residual condition. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
Proof (Lean source)
lemma completionBind_boundedMeanSubGaussian_equal_arms {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (B : ℝ) (hB : 0 < B)
(e p : (Fin d → ℝ) → ℝ) (he : Measurable e) (hp : Measurable p)
(hvalid : ∀ x, e x ∈ Icc 0 1 ∧ p x ∈ Icc 0 1)
(hpos : ∀ᵐ x ∂mu, 0 < e x)
[IsFiniteMeasure ((mu.bind (completionAt B e p p)).map observed)] :
let P := (mu.bind (completionAt B e p p)).map observed
BoundedMeanSubGaussianResidual P B := by
dsimp
let P := (mu.bind (completionAt B e p p)).map observed
have hk := completionBind_treatedKernel_ae_equal_arms
mu B e p he hp hvalid hpos
have hm := completionBind_treatedRegression_ae_equal_arms
mu B e p he hp hvalid hpos
have hobs : Measurable (@observed d) := by unfold observed; fun_prop
have hcomp : Measurable (completionAt B e p p) :=
completionAt_measurable_of_measurable B e p p he hp hp
(fun x => ⟨(hvalid x).1, (hvalid x).2, (hvalid x).2⟩)
have hcov : covariateLaw P = mu := by
dsimp [P, covariateLaw]
rw [Measure.map_map (by fun_prop) hobs]
exact completionBind_covariate_marginal mu B e p p hcomp.aemeasurable
(Filter.Eventually.of_forall (fun x =>
⟨(hvalid x).1, (hvalid x).2, (hvalid x).2⟩))
unfold BoundedMeanSubGaussianResidual
rw [hcov]
filter_upwards [hk, hm] with x hkx hmx
constructor
· rw [hmx, abs_of_nonneg (mul_nonneg hB.le (hvalid x).2.1)]
exact (mul_le_of_le_one_right hB.le (hvalid x).2.2)
· intro t
rw [hkx, hmx]
exact scaledBernoulli_centered_mgf_bound B (p x) t hB (hvalid x).2
theorem completionAt_potential_treatment_factorization Lemma completionAt_potential_treatment_factorization in the paper ↗
At fixed covariates, the potential-outcome pair and treatment factor as the product used by the completion construction. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
B :
ℝ
e p₀ p₁ :
(Fin d → ℝ) → ℝ
x :
Fin d → ℝ
he :
e x ∈ Icc 0 1
hp₀ :
p₀ x ∈ Icc 0 1
hp₁ :
p₁ x ∈ Icc 0 1
conclusion
(completionAt B e p₀ p₁ x).map (fun ω => (ω.1, ((ω.2.2.1, ω.2.2.2.1), ω.2.1)))
= (Measure.dirac x).prod (((scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x))).prod (realBernoulli (e x)))
Proof (Lean source)
lemma completionAt_potential_treatment_factorization {d : ℕ} (B : ℝ)
(e p₀ p₁ : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ)
(he : e x ∈ Icc 0 1) (hp₀ : p₀ x ∈ Icc 0 1)
(hp₁ : p₁ x ∈ Icc 0 1) :
(completionAt B e p₀ p₁ x).map
(fun ω => (ω.1, ((ω.2.2.1, ω.2.2.2.1), ω.2.1))) =
(Measure.dirac x).prod
(((scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x))).prod
(realBernoulli (e x))) := by
letI := realBernoulli_probability (e x) he
letI := scaledBernoulli_probability B (p₀ x) hp₀
letI := scaledBernoulli_probability B (p₁ x) hp₁
unfold completionAt
rw [Measure.map_map (by fun_prop) (by
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption))]
rw [Measure.dirac_prod]
change Measure.map ((Prod.mk x) ∘ swap)
((realBernoulli (e x)).prod
((scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x)))) = _
rw [← Measure.map_map (by fun_prop) measurable_swap, Measure.prod_swap]
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma scaledBernoulli_pair_measurable {d : ℕ} (B : ℝ)
(p₀ p₁ : (Fin d → ℝ) → ℝ) (hp₀ : Measurable p₀) (hp₁ : Measurable p₁)
(hvalid : ∀ x, p₀ x ∈ Icc 0 1 ∧ p₁ x ∈ Icc 0 1) :
Measurable (fun x =>
(scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x))) := by
let μ₀ : (Fin d → ℝ) → ProbabilityMeasure ℝ := fun x =>
⟨scaledBernoulli B (p₀ x), scaledBernoulli_probability B (p₀ x) (hvalid x).1⟩
let μ₁ : (Fin d → ℝ) → ProbabilityMeasure ℝ := fun x =>
⟨scaledBernoulli B (p₁ x), scaledBernoulli_probability B (p₁ x) (hvalid x).2⟩
have hμ₀ : Measurable μ₀ := by
apply Measurable.subtype_mk
unfold scaledBernoulli realBernoulli
fun_prop
have hμ₁ : Measurable μ₁ := by
apply Measurable.subtype_mk
unfold scaledBernoulli realBernoulli
fun_prop
exact ProbabilityMeasure.measurable_fun_prod.comp (hμ₀.prodMk hμ₁)
theorem completionBind_potential_treatment_factorization Lemma completionBind_potential_treatment_factorization in the paper ↗
Mixing the fixed-covariate factorization preserves the conditional product of the potential-outcome pair and treatment. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
kY :
ProbabilityTheory.Kernel.mk (fun x => (scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x))) (scaledBernoulli_pair_measurable B p₀ p₁ hp₀ hp₁ (fun x => ⟨(hvalid x).2.1, (hvalid x).2.2⟩)) (: Kernel (Fin d → ℝ) (ℝ × ℝ))
kA :
ProbabilityTheory.Kernel.mk (fun xy => realBernoulli (e xy.1)) (by unfold realBernoulli; fun_prop) (: Kernel ((Fin d → ℝ) × (ℝ × ℝ)) Bool)
(mu.bind (completionAt B e p₀ p₁)).map (fun ω => (ω.1, ((ω.2.2.1, ω.2.2.2.1), ω.2.1)))
= Measure.compProd mu (compProd kY kA)
Proof (Lean source)
lemma completionBind_potential_treatment_factorization {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (B : ℝ)
(e p₀ p₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hp₀ : Measurable p₀) (hp₁ : Measurable p₁)
(hvalid : ∀ x, e x ∈ Icc 0 1 ∧ p₀ x ∈ Icc 0 1 ∧
p₁ x ∈ Icc 0 1) :
let kY : Kernel (Fin d → ℝ) (ℝ × ℝ) :=
ProbabilityTheory.Kernel.mk (fun x =>
(scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x)))
(scaledBernoulli_pair_measurable B p₀ p₁ hp₀ hp₁
(fun x => ⟨(hvalid x).2.1, (hvalid x).2.2⟩))
let kA : Kernel ((Fin d → ℝ) × (ℝ × ℝ)) Bool :=
ProbabilityTheory.Kernel.mk (fun xy => realBernoulli (e xy.1))
(by unfold realBernoulli; fun_prop)
(mu.bind (completionAt B e p₀ p₁)).map
(fun ω => (ω.1, ((ω.2.2.1, ω.2.2.2.1), ω.2.1))) =
Measure.compProd mu (compProd kY kA) := by
dsimp
let kY : Kernel (Fin d → ℝ) (ℝ × ℝ) :=
ProbabilityTheory.Kernel.mk (fun x =>
(scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x)))
(scaledBernoulli_pair_measurable B p₀ p₁ hp₀ hp₁
(fun x => ⟨(hvalid x).2.1, (hvalid x).2.2⟩))
let kA : Kernel ((Fin d → ℝ) × (ℝ × ℝ)) Bool :=
ProbabilityTheory.Kernel.mk (fun xy => realBernoulli (e xy.1))
(by unfold realBernoulli; fun_prop)
letI : IsMarkovKernel kY := ⟨fun x => by
letI := scaledBernoulli_probability B (p₀ x) (hvalid x).2.1
letI := scaledBernoulli_probability B (p₁ x) (hvalid x).2.2
exact Measure.prod.instIsProbabilityMeasure _ _⟩
letI : IsMarkovKernel kA :=
⟨fun xy => realBernoulli_probability (e xy.1) (hvalid xy.1).1⟩
have hk : Measurable (completionAt B e p₀ p₁) :=
completionAt_measurable_of_measurable B e p₀ p₁ he hp₀ hp₁ hvalid
apply Measure.ext
intro s hs
rw [Measure.map_apply (by fun_prop) hs,
Measure.bind_apply (hs.preimage (by fun_prop)) hk.aemeasurable,
Measure.compProd_apply hs]
apply lintegral_congr
intro x
letI := realBernoulli_probability (e x) (hvalid x).1
letI := scaledBernoulli_probability B (p₀ x) (hvalid x).2.1
letI := scaledBernoulli_probability B (p₁ x) (hvalid x).2.2
have hfix := completionAt_potential_treatment_factorization
B e p₀ p₁ x (hvalid x).1 (hvalid x).2.1 (hvalid x).2.2
have happ := congrArg
(fun m : Measure ((Fin d → ℝ) × ((ℝ × ℝ) × Bool)) => m s) hfix
rw [Measure.map_apply (by fun_prop) hs, Measure.prod_apply hs] at happ
rw [lintegral_dirac] at happ
rw [Measure.prod_apply (hs.preimage (by fun_prop))] at happ
rw [ProbabilityTheory.Kernel.compProd_apply]
· simpa [kY, kA] using happ
· exact hs.preimage (by fun_prop)
theorem completionAt_covariate_potential_map Lemma completionAt_covariate_potential_map in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma completionAt_covariate_potential_map {d : ℕ} (B : ℝ)
(e p₀ p₁ : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ)
(he : e x ∈ Icc 0 1) (hp₀ : p₀ x ∈ Icc 0 1)
(hp₁ : p₁ x ∈ Icc 0 1) :
(completionAt B e p₀ p₁ x).map
(fun ω => (ω.1, (ω.2.2.1, ω.2.2.2.1))) =
(Measure.dirac x).prod
((scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x))) := by
letI := realBernoulli_probability (e x) he
letI := scaledBernoulli_probability B (p₀ x) hp₀
letI := scaledBernoulli_probability B (p₁ x) hp₁
unfold completionAt
rw [Measure.map_map (by fun_prop) (by
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption))]
rw [Measure.dirac_prod]
calc
Measure.map (fun z : Bool × (ℝ × ℝ) => (x, z.2))
((realBernoulli (e x)).prod
((scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x)))) =
Measure.map (Prod.mk x) (Measure.map snd
((realBernoulli (e x)).prod
((scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x))))) := by
rw [Measure.map_map (by fun_prop) measurable_snd]
congr 1
_ = _ := by
rw [Measure.map_snd_prod]
rw [measure_univ, one_smul]
theorem completionBind_covariate_potential_compProd Lemma completionBind_covariate_potential_compProd in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
kY :
ProbabilityTheory.Kernel.mk (fun x => (scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x))) (scaledBernoulli_pair_measurable B p₀ p₁ hp₀ hp₁ (fun x => ⟨(hvalid x).2.1, (hvalid x).2.2⟩)) (: Kernel (Fin d → ℝ) (ℝ × ℝ))
(mu.bind (completionAt B e p₀ p₁)).map (fun ω => (ω.1, (ω.2.2.1, ω.2.2.2.1)))
= Measure.compProd mu kY
Proof (Lean source)
lemma completionBind_covariate_potential_compProd {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (B : ℝ)
(e p₀ p₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hp₀ : Measurable p₀) (hp₁ : Measurable p₁)
(hvalid : ∀ x, e x ∈ Icc 0 1 ∧ p₀ x ∈ Icc 0 1 ∧
p₁ x ∈ Icc 0 1) :
let kY : Kernel (Fin d → ℝ) (ℝ × ℝ) :=
ProbabilityTheory.Kernel.mk (fun x =>
(scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x)))
(scaledBernoulli_pair_measurable B p₀ p₁ hp₀ hp₁
(fun x => ⟨(hvalid x).2.1, (hvalid x).2.2⟩))
(mu.bind (completionAt B e p₀ p₁)).map
(fun ω => (ω.1, (ω.2.2.1, ω.2.2.2.1))) =
Measure.compProd mu kY := by
dsimp
let kY : Kernel (Fin d → ℝ) (ℝ × ℝ) :=
ProbabilityTheory.Kernel.mk (fun x =>
(scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x)))
(scaledBernoulli_pair_measurable B p₀ p₁ hp₀ hp₁
(fun x => ⟨(hvalid x).2.1, (hvalid x).2.2⟩))
letI : IsMarkovKernel kY := ⟨fun x => by
letI := scaledBernoulli_probability B (p₀ x) (hvalid x).2.1
letI := scaledBernoulli_probability B (p₁ x) (hvalid x).2.2
exact Measure.prod.instIsProbabilityMeasure _ _⟩
have hk : Measurable (completionAt B e p₀ p₁) :=
completionAt_measurable_of_measurable B e p₀ p₁ he hp₀ hp₁ hvalid
apply Measure.ext
intro s hs
rw [Measure.map_apply (by fun_prop) hs,
Measure.bind_apply (hs.preimage (by fun_prop)) hk.aemeasurable,
Measure.compProd_apply hs]
apply lintegral_congr
intro x
letI := scaledBernoulli_probability B (p₀ x) (hvalid x).2.1
letI := scaledBernoulli_probability B (p₁ x) (hvalid x).2.2
have hfix := completionAt_covariate_potential_map B e p₀ p₁ x
(hvalid x).1 (hvalid x).2.1 (hvalid x).2.2
have happ := congrArg (fun m : Measure ((Fin d → ℝ) × (ℝ × ℝ)) => m s) hfix
rw [Measure.map_apply (by fun_prop) hs, Measure.prod_apply hs,
lintegral_dirac] at happ
simpa [kY] using happ
The product completion makes its treatment conditionally independent of the potential-outcome pair given the covariates. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
B :
ℝ
e p₀ p₁ :
(Fin d → ℝ) → ℝ
he :
hp₀ :
Measurable p₀
hp₁ :
Measurable p₁
IsFiniteMeasure (mu.bind (completionAt B e p₀ p₁))
conclusion
CondExchangeable (mu.bind (completionAt B e p₀ p₁))
Proof (Lean source)
lemma completionBind_condExchangeable {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (B : ℝ)
(e p₀ p₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hp₀ : Measurable p₀) (hp₁ : Measurable p₁)
(hvalid : ∀ x, e x ∈ Icc 0 1 ∧ p₀ x ∈ Icc 0 1 ∧
p₁ x ∈ Icc 0 1)
[IsFiniteMeasure (mu.bind (completionAt B e p₀ p₁))] :
CondExchangeable (mu.bind (completionAt B e p₀ p₁)) := by
let Pc := mu.bind (completionAt B e p₀ p₁)
let X : Completion d → (Fin d → ℝ) := fun ω => ω.1
let Y : Completion d → (ℝ × ℝ) := fun ω => (ω.2.2.1, ω.2.2.2.1)
let A : Completion d → Bool := fun ω => ω.2.1
let kY : Kernel (Fin d → ℝ) (ℝ × ℝ) :=
ProbabilityTheory.Kernel.mk (fun x =>
(scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x)))
(scaledBernoulli_pair_measurable B p₀ p₁ hp₀ hp₁
(fun x => ⟨(hvalid x).2.1, (hvalid x).2.2⟩))
let kAx : Kernel (Fin d → ℝ) Bool :=
ProbabilityTheory.Kernel.mk (fun x => realBernoulli (e x))
(by unfold realBernoulli; fun_prop)
let kA : Kernel ((Fin d → ℝ) × (ℝ × ℝ)) Bool :=
ProbabilityTheory.Kernel.mk (fun xy => realBernoulli (e xy.1))
(by unfold realBernoulli; fun_prop)
letI : IsMarkovKernel kY := ⟨fun x => by
letI := scaledBernoulli_probability B (p₀ x) (hvalid x).2.1
letI := scaledBernoulli_probability B (p₁ x) (hvalid x).2.2
exact Measure.prod.instIsProbabilityMeasure _ _⟩
letI : IsMarkovKernel kAx :=
⟨fun x => realBernoulli_probability (e x) (hvalid x).1⟩
letI : IsMarkovKernel kA :=
⟨fun xy => realBernoulli_probability (e xy.1) (hvalid xy.1).1⟩
have hk : Measurable (completionAt B e p₀ p₁) :=
completionAt_measurable_of_measurable B e p₀ p₁ he hp₀ hp₁ hvalid
have hX : Measurable X := by dsimp [X]; fun_prop
have hY : Measurable Y := by dsimp [Y]; fun_prop
have hA : Measurable A := by dsimp [A]; fun_prop
have hcov : Pc.map X = mu := by
dsimp [Pc, X]
exact completionBind_covariate_marginal mu B e p₀ p₁ hk.aemeasurable
(Filter.Eventually.of_forall hvalid)
have hYmap : Pc.map (fun ω => (X ω, Y ω)) = Measure.compProd mu kY := by
dsimp [Pc, X, Y]
exact completionBind_covariate_potential_compProd
mu B e p₀ p₁ he hp₀ hp₁ hvalid
have hAmap : Pc.map (fun ω => (X ω, A ω)) = Measure.compProd mu kAx := by
dsimp [Pc, X, A]
exact completionBind_covariate_treatment_compProd
mu B e p₀ p₁ he hp₀ hp₁ hvalid
have hYcond : condDistrib Y X Pc =ᵐ[mu] kY := by
have h := ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd_of_measurable
(μ := Pc) (X := X) (Y := Y) (κ := kY) hX hY
(by simpa [hcov] using hYmap)
simpa [hcov] using h
have hAcond : condDistrib A X Pc =ᵐ[mu] kAx := by
have h := ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd_of_measurable
(μ := Pc) (X := X) (Y := A) (κ := kAx) hX hA
(by simpa [hcov] using hAmap)
simpa [hcov] using h
have hkprod : compProd kY kA =
prod kY kAx := by
ext x s hs
rw [ProbabilityTheory.Kernel.compProd_apply hs,
ProbabilityTheory.Kernel.prod_apply kY kAx x, Measure.prod_apply hs]
rfl
have hfactor : Pc.map (fun ω => (X ω, Y ω, A ω)) =
Measure.compProd mu (prod kY kAx) := by
dsimp [Pc, X, Y, A]
rw [← hkprod]
exact completionBind_potential_treatment_factorization
mu B e p₀ p₁ he hp₀ hp₁ hvalid
unfold CondExchangeable
change CondIndepFun (comap X inferInstance)
(comap_le hX) Y A Pc
apply (ProbabilityTheory.condIndepFun_iff_map_prod_eq_prod_condDistrib_prod_condDistrib
hY hA hX).2
rw [← Measure.compProd_eq_comp_prod]
rw [hcov]
calc
Pc.map (fun ω => (X ω, Y ω, A ω)) =
Measure.compProd mu (prod kY kAx) := hfactor
_ = Measure.compProd mu (prod
(condDistrib Y X Pc)
(condDistrib A X Pc)) := by
symm
apply Measure.compProd_congr
filter_upwards [hYcond, hAcond] with x hxy hxa
rw [ProbabilityTheory.Kernel.prod_apply
(condDistrib Y X Pc)
(condDistrib A X Pc) x,
ProbabilityTheory.Kernel.prod_apply kY kAx x, hxy, hxa]
theorem completionBind_observed_treated_setLIntegral Lemma completionBind_observed_treated_setLIntegral in the paper ↗
Integrating a nonnegative covariate function over treated observations of a completion bind weights the base covariate law by the displayed propensity. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
B :
ℝ
e p₀ p₁ f :
(Fin d → ℝ) → ℝ
he :
hp₀ :
Measurable p₀
hp₁ :
Measurable p₁
hf :
hfnn :
∀ x, 0 ≤ f x
hD :
Proof (Lean source)
lemma completionBind_observed_treated_setLIntegral {d : ℕ}
(μ : Measure (Fin d → ℝ)) (B : ℝ)
(e p₀ p₁ f : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hp₀ : Measurable p₀) (hp₁ : Measurable p₁)
(hf : Measurable f) (hfnn : ∀ x, 0 ≤ f x)
(hvalid : ∀ x, e x ∈ Icc 0 1 ∧ p₀ x ∈ Icc 0 1 ∧
p₁ x ∈ Icc 0 1)
(D : Set (Fin d → ℝ)) (hD : MeasurableSet D) :
(∫⁻ z in {z : Obs d | z.2.1 = true ∧ z.1 ∈ D}, ofReal (f z.1)
∂(μ.bind (completionAt B e p₀ p₁)).map observed) =
∫⁻ x in D, ofReal (f x) * ofReal (e x) ∂μ := by
let E : Set (Obs d) := {z | z.2.1 = true ∧ z.1 ∈ D}
have hE : MeasurableSet E :=
((measurable_fst.comp measurable_snd) (MeasurableSet.singleton true)).inter
(hD.preimage measurable_fst)
have hobs : Measurable (@observed d) := by unfold observed; fun_prop
have hk : Measurable (completionAt B e p₀ p₁) :=
completionAt_measurable_of_measurable B e p₀ p₁ he hp₀ hp₁ hvalid
have hg : Measurable (fun z : Obs d => E.indicator
(fun z => ofReal (f z.1)) z) := by
exact (hf.comp measurable_fst).ennreal_ofReal.indicator hE
rw [← lintegral_indicator hE]
change (∫⁻ z, E.indicator (fun z => ofReal (f z.1)) z
∂(μ.bind (completionAt B e p₀ p₁)).map observed) = _
rw [lintegral_map hg hobs]
rw [Measure.lintegral_bind (μ := completionAt B e p₀ p₁)
(f := fun ω => E.indicator (fun z => ofReal (f z.1)) (observed ω))
hk.aemeasurable (hg.comp hobs).aemeasurable]
rw [← lintegral_indicator hD]
apply lintegral_congr
intro x
have hcov : ∀ᵐ ω ∂completionAt B e p₀ p₁ x, ω.1 = x := by
change ∀ᵐ ω ∂completionAt B e p₀ p₁ x,
ω.1 ∈ ({x} : Set (Fin d → ℝ))
apply (ae_map_iff measurable_fst.aemeasurable (measurableSet_singleton x)).1
rw [completionAt_covariate_marginal B e p₀ p₁ x
(hvalid x).1 (hvalid x).2.1 (hvalid x).2.2]
filter_upwards [ae_eq_dirac (fun y : Fin d → ℝ => y)] with y hy
change y = x
exact hy
have hind : (fun ω : Completion d =>
E.indicator (fun z => ofReal (f z.1)) (observed ω)) =ᵐ[
completionAt B e p₀ p₁ x]
({ω : Completion d | ω.1 ∈ D ∧ ω.2.1 = true}).indicator
(fun _ => ofReal (f x)) := by
filter_upwards [hcov] with ω hω
subst x
by_cases hωD : ω.1 ∈ D <;> by_cases hωA : ω.2.1 = true <;>
simp [E, observed, hωD, hωA]
rw [lintegral_congr_ae hind]
have hset : MeasurableSet {ω : Completion d | ω.1 ∈ D ∧ ω.2.1 = true} :=
(hD.preimage measurable_fst).inter
((measurable_fst.comp measurable_snd) (MeasurableSet.singleton true))
rw [lintegral_indicator_const hset]
rw [completionAt_treated_covariate_event B e p₀ p₁ x
(hvalid x).1 (hvalid x).2.1 (hvalid x).2.2 D hD]
by_cases hx : x ∈ D
· simp [Set.indicator_of_mem hx, Measure.dirac_apply' _ hD, mul_assoc, mul_comm]
· simp [Set.indicator_of_notMem hx, Measure.dirac_apply' _ hD]
theorem completionBind_observed_treated_setIntegral Lemma completionBind_observed_treated_setIntegral in the paper ↗
Bochner-integral form of completionBind_observed_treated_setLIntegral. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
B :
ℝ
e p₀ p₁ f :
(Fin d → ℝ) → ℝ
he :
hp₀ :
Measurable p₀
hp₁ :
Measurable p₁
hf :
hfnn :
∀ x, 0 ≤ f x
hD :
hleft :
Integrable (fun z : Obs d => f z.1) (((μ.bind (completionAt B e p₀ p₁)).map observed).restrict {z : Obs d | z.2.1 = true ∧ z.1 ∈ D})
hright :
Integrable (fun x => f x * e x) (μ.restrict D)
conclusion
(∫ z in {z : Obs d | z.2.1 = true ∧ z.1 ∈ D}, f z.1 ∂(μ.bind (completionAt B e p₀ p₁)).map observed)
= ∫ x in D, f x * e x ∂μ
Proof (Lean source)
lemma completionBind_observed_treated_setIntegral {d : ℕ}
(μ : Measure (Fin d → ℝ)) (B : ℝ)
(e p₀ p₁ f : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hp₀ : Measurable p₀) (hp₁ : Measurable p₁)
(hf : Measurable f) (hfnn : ∀ x, 0 ≤ f x)
(hvalid : ∀ x, e x ∈ Icc 0 1 ∧ p₀ x ∈ Icc 0 1 ∧
p₁ x ∈ Icc 0 1)
(D : Set (Fin d → ℝ)) (hD : MeasurableSet D)
(hleft : Integrable (fun z : Obs d => f z.1)
(((μ.bind (completionAt B e p₀ p₁)).map observed).restrict
{z : Obs d | z.2.1 = true ∧ z.1 ∈ D}))
(hright : Integrable (fun x => f x * e x) (μ.restrict D)) :
(∫ z in {z : Obs d | z.2.1 = true ∧ z.1 ∈ D}, f z.1
∂(μ.bind (completionAt B e p₀ p₁)).map observed) =
∫ x in D, f x * e x ∂μ := by
let ν := (μ.bind (completionAt B e p₀ p₁)).map observed
let E : Set (Obs d) := {z | z.2.1 = true ∧ z.1 ∈ D}
have hlnn : 0 ≤ ∫ z in E, f z.1 ∂ν :=
integral_nonneg_of_ae (μ := ν.restrict E)
(Filter.Eventually.of_forall (fun z => hfnn z.1))
have hrnn : 0 ≤ ∫ x in D, f x * e x ∂μ :=
integral_nonneg_of_ae (μ := μ.restrict D)
(Filter.Eventually.of_forall (fun x =>
mul_nonneg (hfnn x) (hvalid x).1.1))
change (∫ z in E, f z.1 ∂ν) = ∫ x in D, f x * e x ∂μ
apply (ENNReal.ofReal_eq_ofReal_iff
hlnn hrnn).mp
rw [ofReal_integral_eq_lintegral_ofReal hleft
(Filter.Eventually.of_forall (fun z => hfnn z.1))]
rw [ofReal_integral_eq_lintegral_ofReal hright
(Filter.Eventually.of_forall (fun x =>
mul_nonneg (hfnn x) (hvalid x).1.1))]
rw [completionBind_observed_treated_setLIntegral μ B e p₀ p₁ f
he hp₀ hp₁ hf hfnn hvalid D hD]
apply setLIntegral_congr_fun hD
intro x hx
exact (ENNReal.ofReal_mul (hfnn x)).symm
The thin-strip treatment probability is a valid Bernoulli parameter. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
stripPropensity d hd γ a x ∈ Icc 0 1
Proof (Lean source)
lemma stripPropensity_mem_Icc (d : ℕ) (hd : 2 ≤ d) (γ a : ℝ)
(hγ : 1 < γ) (x : Fin d → ℝ) :
stripPropensity d hd γ a x ∈ Icc 0 1 := by
classical
have hbase : 0 ≤ centreRadiusCDF d (centreRadius x) := by
unfold centreRadiusCDF
exact le_min (by norm_num) (pow_nonneg (mul_nonneg (by norm_num) (norm_nonneg _)) _)
have hexp : 0 ≤ (1 : ℝ) / (γ - 1) := by positivity
have hq : 0 ≤ (centreRadiusCDF d (centreRadius x)) ^ ((1 : ℝ) / (γ - 1)) :=
Real.rpow_nonneg hbase _
have hbump : 0 ≤ (1 / 4 : ℝ) * if x ∈ thinStrip d hd a then 1 else 0 := by
split_ifs <;> norm_num
constructor
· unfold stripPropensity
exact le_min (by norm_num) (add_nonneg hq hbump)
· unfold stripPropensity
exact min_le_left _ _
theorem stripPropensity_ge_quarter_on_strip Lemma stripPropensity_ge_quarter_on_strip in the paper ↗
The strip component supplies a fixed positive treated mass. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
(1 / 4 : ℝ) ≤ stripPropensity d hd γ a x
Proof (Lean source)
lemma stripPropensity_ge_quarter_on_strip (d : ℕ) (hd : 2 ≤ d) (γ a : ℝ)
(_hγ : 1 < γ) (x : Fin d → ℝ) (hx : x ∈ thinStrip d hd a) :
(1 / 4 : ℝ) ≤ stripPropensity d hd γ a x := by
have hbase : 0 ≤ centreRadiusCDF d (centreRadius x) := by
unfold centreRadiusCDF
exact le_min (by norm_num) (pow_nonneg (mul_nonneg (by norm_num) (norm_nonneg _)) _)
have hq : 0 ≤ (centreRadiusCDF d (centreRadius x)) ^ ((1 : ℝ) / (γ - 1)) :=
Real.rpow_nonneg hbase _
simp only [stripPropensity, if_pos hx, mul_one]
exact le_min (by norm_num) (by linarith)
theorem stripPropensity_le_radial_add_quarter Lemma stripPropensity_le_radial_add_quarter in the paper ↗
The strip bump raises propensity by at most one quarter. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
stripPropensity d hd γ a x
≤ (centreRadiusCDF d (centreRadius x)) ^ ((1 : ℝ) / (γ - 1)) + 1 / 4
Proof (Lean source)
lemma stripPropensity_le_radial_add_quarter (d : ℕ) (hd : 2 ≤ d) (γ a : ℝ)
(x : Fin d → ℝ) :
stripPropensity d hd γ a x ≤
(centreRadiusCDF d (centreRadius x)) ^ ((1 : ℝ) / (γ - 1)) + 1 / 4 := by
classical
unfold stripPropensity
apply le_trans (min_le_right _ _)
split_ifs <;> norm_num
The strip adjustment only increases the radial propensity. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma stripPropensity_ge_radial (d : ℕ) (hd : 2 ≤ d) (γ a : ℝ)
(hγ : 1 < γ) (x : Fin d → ℝ) :
(centreRadiusCDF d (centreRadius x)) ^ ((1 : ℝ) / (γ - 1)) ≤
stripPropensity d hd γ a x := by
classical
let q := (centreRadiusCDF d (centreRadius x)) ^ ((1 : ℝ) / (γ - 1))
have hbase : 0 ≤ centreRadiusCDF d (centreRadius x) := by
unfold centreRadiusCDF
exact le_min (by norm_num) (pow_nonneg (mul_nonneg (by norm_num) (norm_nonneg _)) _)
have hbase_le : centreRadiusCDF d (centreRadius x) ≤ 1 := by
unfold centreRadiusCDF
exact min_le_left _ _
have hexp : 0 ≤ (1 : ℝ) / (γ - 1) := by positivity
have hq : q ≤ 1 := by
dsimp [q]
exact Real.rpow_le_one hbase hbase_le hexp
have hbump : 0 ≤ (1 / 4 : ℝ) * if x ∈ thinStrip d hd a then 1 else 0 := by
split_ifs <;> norm_num
unfold stripPropensity
exact le_min hq (by dsimp [q] at *; linarith)
Every low-propensity event lies inside the corresponding radial event. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a t :
ℝ
hγ :
1 < γ
conclusion
{x | stripPropensity d hd γ a x ≤ t} ⊆ {x | (centreRadiusCDF d (centreRadius x)) ^ ((1 : ℝ) / (γ - 1)) ≤ t}
Proof (Lean source)
lemma stripTail_subset_radial (d : ℕ) (hd : 2 ≤ d) (γ a t : ℝ)
(hγ : 1 < γ) :
{x | stripPropensity d hd γ a x ≤ t} ⊆
{x | (centreRadiusCDF d (centreRadius x)) ^ ((1 : ℝ) / (γ - 1)) ≤ t} := by
intro x hx
exact (stripPropensity_ge_radial d hd γ a hγ x).trans hx
theorem stripTail_subset_radiusCDF_sublevel Lemma stripTail_subset_radiusCDF_sublevel in the paper ↗
A propensity sublevel is contained in the corresponding radial CDF sublevel. This is the event inclusion behind the exact global tail bound. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a t :
ℝ
hγ :
1 < γ
conclusion
Proof (Lean source)
lemma stripTail_subset_radiusCDF_sublevel (d : ℕ) (hd : 2 ≤ d)
(γ a t : ℝ) (hγ : 1 < γ) :
{x | stripPropensity d hd γ a x ≤ t} ⊆
{x | centreRadiusCDF d (centreRadius x) ≤ t ^ (γ - 1)} := by
intro x hx
have hbase : 0 ≤ centreRadiusCDF d (centreRadius x) := by
unfold centreRadiusCDF
exact le_min (by norm_num)
(pow_nonneg (mul_nonneg (by norm_num) (norm_nonneg _)) _)
have hpow := Real.rpow_le_rpow
(Real.rpow_nonneg hbase ((1 : ℝ) / (γ - 1)))
((stripTail_subset_radial d hd γ a t hγ) hx) (by linarith : 0 ≤ γ - 1)
simpa [Real.rpow_inv_rpow hbase (by linarith : γ - 1 ≠ 0)] using hpow
The thin strip carries a fixed treatment probability at its centre. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a :
ℝ
hγ :
1 < γ
ha :
0 < a
conclusion
stripPropensity d hd γ a (fun _ => (1 / 2 : ℝ)) = 1 / 4
Proof (Lean source)
lemma stripPropensity_centre (d : ℕ) (hd : 2 ≤ d) (γ a : ℝ)
(hγ : 1 < γ) (ha : 0 < a) :
stripPropensity d hd γ a (fun _ => (1 / 2 : ℝ)) = 1 / 4 := by
have hmem : (fun _ : Fin d => (1 / 2 : ℝ)) ∈ thinStrip d hd a := by
simp [thinStrip, Real.zero_rpow ha.ne']
have hdpos : 0 < d := by omega
have hexp : (γ - 1 : ℝ)⁻¹ ≠ 0 := by positivity
simp only [stripPropensity, centreRadius, centreRadiusCDF]
rw [if_pos hmem]
simp [Real.zero_rpow hexp, zero_pow hdpos.ne']
norm_num
Outside the strip, the propensity is exactly its radial component. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma stripPropensity_off_strip (d : ℕ) (hd : 2 ≤ d) (γ a : ℝ)
(hγ : 1 < γ) (x : Fin d → ℝ) (hx : x ∉ thinStrip d hd a) :
stripPropensity d hd γ a x =
(centreRadiusCDF d (centreRadius x)) ^ ((1 : ℝ) / (γ - 1)) := by
have hbase : 0 ≤ centreRadiusCDF d (centreRadius x) := by
unfold centreRadiusCDF
exact le_min (by norm_num) (pow_nonneg (mul_nonneg (by norm_num) (norm_nonneg _)) _)
have hbase_le : centreRadiusCDF d (centreRadius x) ≤ 1 := by
unfold centreRadiusCDF
exact min_le_left _ _
have hexp : 0 ≤ (1 : ℝ) / (γ - 1) := by positivity
have hq : (centreRadiusCDF d (centreRadius x)) ^ ((1 : ℝ) / (γ - 1)) ≤ 1 :=
Real.rpow_le_one hbase hbase_le hexp
simp only [stripPropensity, if_neg hx, mul_zero, add_zero]
exact min_eq_right hq
theorem stripPropensity_off_strip_radius_bound Lemma stripPropensity_off_strip_radius_bound in the paper ↗
The radial treatment probability on a small cube is bounded by its outer radius. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a h :
ℝ
hγ :
1 < γ
x :
Fin d → ℝ
hx :
x ∉ thinStrip d hd a
hr :
centreRadius x ≤ h
conclusion
stripPropensity d hd γ a x ≤ ((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1))
Proof (Lean source)
lemma stripPropensity_off_strip_radius_bound (d : ℕ) (hd : 2 ≤ d)
(γ a h : ℝ) (hγ : 1 < γ) (x : Fin d → ℝ)
(hx : x ∉ thinStrip d hd a) (hr : centreRadius x ≤ h) :
stripPropensity d hd γ a x ≤
((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) := by
rw [stripPropensity_off_strip d hd γ a hγ x hx]
apply Real.rpow_le_rpow
· unfold centreRadiusCDF
exact le_min (by norm_num) (pow_nonneg (mul_nonneg (by norm_num) (norm_nonneg _)) _)
· unfold centreRadiusCDF
calc
min 1 ((2 * centreRadius x) ^ d) ≤ (2 * centreRadius x) ^ d := min_le_right _ _
_ ≤ (2 * h) ^ d := by
exact pow_le_pow_left₀
(mul_nonneg (by norm_num) (by unfold centreRadius; exact norm_nonneg _))
(by linarith) _
· positivity
At a sufficiently small radius, a propensity level above a quarter-scale threshold can occur only inside the thin strip. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a h ρ :
ℝ
hγ :
1 < γ
hsmall :
((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) < ρ / 4
conclusion
Proof (Lean source)
lemma stripSuperlevel_subset_strip (d : ℕ) (hd : 2 ≤ d)
(γ a h ρ : ℝ) (hγ : 1 < γ)
(hsmall : ((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) < ρ / 4) :
{x | centreRadius x ≤ h ∧ ρ / 4 ≤ stripPropensity d hd γ a x} ⊆
thinStrip d hd a := by
intro x hx
by_contra hnot
have hbound := stripPropensity_off_strip_radius_bound d hd γ a h hγ x hnot hx.1
linarith [hx.2]
theorem stripPropensity_ball_sup_ge_quarter Lemma stripPropensity_ball_sup_ge_quarter in the paper ↗
Every ball centered at the strip center has propensity supremum at least one quarter. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a h :
ℝ
hγ :
1 < γ
ha :
0 < a
conclusion
Proof (Lean source)
lemma stripPropensity_ball_sup_ge_quarter (d : ℕ) (hd : 2 ≤ d)
(γ a h : ℝ) (hγ : 1 < γ) (ha : 0 < a) :
(1 / 4 : ℝ) ≤ sSup
(stripPropensity d hd γ a '' ball2 (fun _ => (1 / 2 : ℝ)) h) := by
let x₀ : Fin d → ℝ := fun _ => 1 / 2
have hx : x₀ ∈ ball2 x₀ h := by
simp [ball2, x₀, sq_nonneg]
have hupper : BddAbove (stripPropensity d hd γ a '' ball2 x₀ h) := by
refine ⟨1, ?_⟩
rintro y ⟨x, -, rfl⟩
exact (stripPropensity_mem_Icc d hd γ a hγ x).2
have hmem : stripPropensity d hd γ a x₀ ∈
stripPropensity d hd γ a '' ball2 x₀ h := ⟨x₀, hx, rfl⟩
have hcentre : stripPropensity d hd γ a x₀ = 1 / 4 :=
stripPropensity_centre d hd γ a hγ ha
rw [← hcentre]
exact le_csSup hupper hmem
theorem stripPropensity_ball_cube_sup_ge_quarter Lemma stripPropensity_ball_cube_sup_ge_quarter in the paper ↗
The centre also witnesses the quarter lower bound when the local ball is restricted to the covariate cube, as in Dorn's A3. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a h :
ℝ
hγ :
1 < γ
ha :
0 < a
conclusion
Proof (Lean source)
lemma stripPropensity_ball_cube_sup_ge_quarter (d : ℕ) (hd : 2 ≤ d)
(γ a h : ℝ) (hγ : 1 < γ) (ha : 0 < a) :
(1 / 4 : ℝ) ≤ sSup
(stripPropensity d hd γ a ''
(ball2 (fun _ => (1 / 2 : ℝ)) h ∩ cube d)) := by
let x₀ : Fin d → ℝ := fun _ => 1 / 2
have hx : x₀ ∈ ball2 x₀ h ∩ cube d := by
constructor
· simp [ball2, x₀, sq_nonneg]
· simp [cube, x₀, Pi.le_def]
intro i
norm_num
have hupper : BddAbove
(stripPropensity d hd γ a '' (ball2 x₀ h ∩ cube d)) := by
refine ⟨1, ?_⟩
rintro y ⟨x, -, rfl⟩
exact (stripPropensity_mem_Icc d hd γ a hγ x).2
have hmem : stripPropensity d hd γ a x₀ ∈
stripPropensity d hd γ a '' (ball2 x₀ h ∩ cube d) :=
⟨x₀, hx, rfl⟩
rw [← stripPropensity_centre d hd γ a hγ ha]
exact le_csSup hupper hmem
theorem stripSuperlevel_ball_cube_subset_strip Lemma stripSuperlevel_ball_cube_subset_strip in the paper ↗
A sufficiently small radial bound confines Dorn's local superlevel event to the thin strip, for every proposed A3 level ρ. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a h ρ :
ℝ
hγ :
1 < γ
ha :
0 < a
hρ :
0 < ρ
hsmall :
((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) < ρ / 4
conclusion
{x | centreRadius x ≤ h ∧ ρ * sSup (stripPropensity d hd γ a '' (ball2 (fun _ => (1 / 2 : ℝ)) h ∩ cube d)) ≤ stripPropensity d hd γ a x} ⊆ thinStrip d hd a
Proof (Lean source)
lemma stripSuperlevel_ball_cube_subset_strip (d : ℕ) (hd : 2 ≤ d)
(γ a h ρ : ℝ) (hγ : 1 < γ) (ha : 0 < a) (hρ : 0 < ρ)
(hsmall : ((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) < ρ / 4) :
{x | centreRadius x ≤ h ∧
ρ * sSup (stripPropensity d hd γ a ''
(ball2 (fun _ => (1 / 2 : ℝ)) h ∩ cube d)) ≤
stripPropensity d hd γ a x} ⊆ thinStrip d hd a := by
have hsup := stripPropensity_ball_cube_sup_ge_quarter d hd γ a h hγ ha
apply Set.Subset.trans ?_ (stripSuperlevel_subset_strip d hd γ a h ρ hγ hsmall)
intro x hx
refine ⟨hx.1, ?_⟩
have hlevel := (mul_le_mul_of_nonneg_left hsup (le_of_lt hρ)).trans hx.2
nlinarith [hlevel]
theorem strip_not_dornA3AntiConcentration_of_local_strip_ratio
A vanishing local fraction of the strip defeats every proposed numerical anti-concentration triple. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a :
ℝ
hγ :
1 < γ
ha :
0 < a
P :
Measure (Obs d)
hlocal :
∀ (ρ ν h₀ : ℝ)
if
0 < ρ
and
0 < ν
and
0 < h₀
then
∃ h : ℝ,
0 < h ∧
h ≤ h₀ ∧
((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) < ρ / 4 ∧
(covariateLaw P).real (thinStrip d hd a ∩ ball2 (fun _ => (1 / 2 : ℝ)) h) / (covariateLaw P).real (ball2 (fun _ => (1 / 2 : ℝ)) h)
≤ ν
conclusion
¬ DornA3AntiConcentrationOnCube P (stripPropensity d hd γ a)
Proof (Lean source)
lemma strip_not_dornA3AntiConcentration_of_local_strip_ratio (d : ℕ) (hd : 2 ≤ d)
(γ a : ℝ) (hγ : 1 < γ) (ha : 0 < a)
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hlocal : ∀ (ρ ν h₀ : ℝ), 0 < ρ → 0 < ν → 0 < h₀ →
∃ h : ℝ, 0 < h ∧ h ≤ h₀ ∧
((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) < ρ / 4 ∧
0 < (covariateLaw P).real
(ball2 (fun _ => (1 / 2 : ℝ)) h) ∧
(covariateLaw P).real
(thinStrip d hd a ∩ ball2 (fun _ => (1 / 2 : ℝ)) h) /
(covariateLaw P).real
(ball2 (fun _ => (1 / 2 : ℝ)) h) ≤ ν) :
¬ DornA3AntiConcentrationOnCube P (stripPropensity d hd γ a) := by
intro hA3
obtain ⟨ρ, ν, h₀, hρ, hν, hh₀, hA3bound⟩ := hA3
obtain ⟨h, hh, hh₀', hsmall, hball, hratio⟩ := hlocal ρ ν h₀ hρ hν hh₀
have hx₀ : (fun _ : Fin d => (1 / 2 : ℝ)) ∈ cube d := by
simp [cube, Pi.le_def]
intro i
norm_num
have hbound := hA3bound (fun _ => (1 / 2 : ℝ)) hx₀ h ⟨hh, hh₀'⟩
have hsubset :
{x | stripPropensity d hd γ a x ≥
ρ * sSup (stripPropensity d hd γ a ''
(ball2 (fun _ => (1 / 2 : ℝ)) h ∩ cube d))} ∩
ball2 (fun _ => (1 / 2 : ℝ)) h ⊆
thinStrip d hd a ∩ ball2 (fun _ => (1 / 2 : ℝ)) h := by
intro x hx
refine ⟨?_, hx.2⟩
apply stripSuperlevel_ball_cube_subset_strip d hd γ a h ρ hγ ha hρ hsmall
constructor
· have hcoord : centreRadius x ≤ h := by
unfold centreRadius
letI : Nonempty (Fin d) := ⟨⟨0, by omega⟩⟩
apply (pi_norm_le_iff_of_nonempty _).2
intro i
have hi : (x i - (1 / 2 : ℝ)) ^ 2 ≤
∑ j : Fin d, (x j - 1 / 2) ^ 2 := by
exact Finset.single_le_sum
(fun j _ => sq_nonneg (x j - 1 / 2)) (Finset.mem_univ i)
have hball' : (∑ j : Fin d, (x j - 1 / 2) ^ 2) ≤ h ^ 2 := hx.2
have habs : |x i - 1 / 2| ≤ h := by
nlinarith [sq_abs (x i - (1 / 2 : ℝ)), abs_nonneg (x i - (1 / 2 : ℝ))]
simpa only [Pi.sub_apply, Real.norm_eq_abs] using habs
exact hcoord
· exact hx.1
have hmono : (covariateLaw P).real
({x | stripPropensity d hd γ a x ≥
ρ * sSup (stripPropensity d hd γ a ''
(ball2 (fun _ => (1 / 2 : ℝ)) h ∩ cube d))} ∩
ball2 (fun _ => (1 / 2 : ℝ)) h) ≤
(covariateLaw P).real
(thinStrip d hd a ∩ ball2 (fun _ => (1 / 2 : ℝ)) h) := by
have : IsProbabilityMeasure (covariateLaw P) := by
unfold covariateLaw
exact Measure.isProbabilityMeasure_map measurable_fst.aemeasurable
exact measureReal_mono hsubset
have hdiv := (div_le_div_of_nonneg_right hmono hball.le).trans hratio
exact (not_lt_of_ge hdiv) hbound
theorem dornA3AntiConcentrationOnCube_of_dornA3OnCube
Dorn's full A3 property implies its numerical anti-concentration clause. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma dornA3AntiConcentrationOnCube_of_dornA3OnCube {d : ℕ}
(P : Measure (Obs d)) (e : (Fin d → ℝ) → ℝ) :
DornA3OnCube P e → DornA3AntiConcentrationOnCube P e := by
intro hA3
change DornA3 (cube d)
({(P, (fun _ => 0), (fun _ => 0), e)} : Set (DornSourceLaw d)) at hA3
obtain ⟨_, _, _, ρ, ν, h₀, hρ, hν, hh₀, hA3bound⟩ := hA3
refine ⟨ρ, ν, h₀, hρ, hν, hh₀, ?_⟩
intro x₀ hx₀ h hh
exact hA3bound (P, (fun _ => 0), (fun _ => 0), e)
(Set.mem_singleton _) x₀ hx₀ h hh
theorem strip_not_dornA3_of_local_strip_ratio Lemma strip_not_dornA3_of_local_strip_ratio in the paper ↗
Failure of the numerical anti-concentration clause rules out full Dorn A3. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a :
ℝ
hγ :
1 < γ
ha :
0 < a
P :
Measure (Obs d)
hlocal :
∀ (ρ ν h₀ : ℝ)
if
0 < ρ
and
0 < ν
and
0 < h₀
then
∃ h : ℝ,
0 < h ∧
h ≤ h₀ ∧
((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) < ρ / 4 ∧
(covariateLaw P).real (thinStrip d hd a ∩ ball2 (fun _ => (1 / 2 : ℝ)) h) / (covariateLaw P).real (ball2 (fun _ => (1 / 2 : ℝ)) h)
≤ ν
conclusion
¬ DornA3OnCube P (stripPropensity d hd γ a)
Proof (Lean source)
lemma strip_not_dornA3_of_local_strip_ratio (d : ℕ) (hd : 2 ≤ d)
(γ a : ℝ) (hγ : 1 < γ) (ha : 0 < a)
(P : Measure (Obs d)) [IsProbabilityMeasure P]
(hlocal : ∀ (ρ ν h₀ : ℝ), 0 < ρ → 0 < ν → 0 < h₀ →
∃ h : ℝ, 0 < h ∧ h ≤ h₀ ∧
((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) < ρ / 4 ∧
0 < (covariateLaw P).real
(ball2 (fun _ => (1 / 2 : ℝ)) h) ∧
(covariateLaw P).real
(thinStrip d hd a ∩ ball2 (fun _ => (1 / 2 : ℝ)) h) /
(covariateLaw P).real
(ball2 (fun _ => (1 / 2 : ℝ)) h) ≤ ν) :
¬ DornA3OnCube P (stripPropensity d hd γ a) := by
intro hA3
exact strip_not_dornA3AntiConcentration_of_local_strip_ratio
d hd γ a hγ ha P hlocal
(dornA3AntiConcentrationOnCube_of_dornA3OnCube P
(stripPropensity d hd γ a) hA3)
The off-strip radial propensity bound vanishes as the neighborhood shrinks. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ :
ℝ
hγ :
1 < γ
conclusion
Proof (Lean source)
lemma stripRadialBound_tendsto_zero (d : ℕ) (hd : 2 ≤ d)
(γ : ℝ) (hγ : 1 < γ) :
Tendsto (fun h : ℝ => ((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)))
(nhdsWithin 0 (Ioi 0)) (nhds 0) := by
have hpow : Tendsto (fun h : ℝ => (2 * h) ^ d)
(nhdsWithin 0 (Ioi 0)) (nhds 0) := by
have hc : Continuous (fun h : ℝ => (2 * h) ^ d) := by fun_prop
convert hc.continuousAt.tendsto.mono_left nhdsWithin_le_nhds using 1
simp [zero_pow (by omega : d ≠ 0)]
have hexp : 0 ≤ (1 : ℝ) / (γ - 1) := by positivity
have hne : (1 : ℝ) / (γ - 1) ≠ 0 := by positivity
convert hpow.rpow_const (inr hexp) using 1
rw [Real.zero_rpow hne]
A neighborhood can be chosen below both a proposed A3 radius and its superlevel threshold. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ ρ h₀ :
ℝ
hγ :
1 < γ
hρ :
0 < ρ
hh₀ :
0 < h₀
conclusion
∃ h : ℝ,
conclusion 1
0 < h
conclusion 2
h ≤ h₀
conclusion 3
((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) < ρ / 4
Proof (Lean source)
lemma stripRadialBound_small (d : ℕ) (hd : 2 ≤ d)
(γ ρ h₀ : ℝ) (hγ : 1 < γ) (hρ : 0 < ρ) (hh₀ : 0 < h₀) :
∃ h : ℝ, 0 < h ∧ h ≤ h₀ ∧
((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) < ρ / 4 := by
have hlim := stripRadialBound_tendsto_zero d hd γ hγ
have hsmall : ∀ᶠ h : ℝ in nhdsWithin 0 (Ioi 0),
((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) < ρ / 4 :=
hlim.eventually (eventually_lt_nhds (by positivity))
have hwidth : ∀ᶠ h : ℝ in nhdsWithin 0 (Ioi 0), h < h₀ :=
(continuousAt_id.tendsto.mono_left nhdsWithin_le_nhds).eventually
(eventually_lt_nhds hh₀)
have hpos : ∀ᶠ h : ℝ in nhdsWithin 0 (Ioi 0), 0 < h :=
self_mem_nhdsWithin
obtain ⟨h, hh, hbound, hrad⟩ :=
(hpos.and (hwidth.and hsmall)).exists
exact ⟨h, hh, hbound.le, hrad⟩
theorem thinStrip_transverse_coordinate_bound Lemma thinStrip_transverse_coordinate_bound in the paper ↗
In a centered cube, the strip's transverse coordinate has width h ^ a. This is the geometric bound used for its vanishing local mass and second moment. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
a h :
ℝ
ha :
0 ≤ a
hh :
0 ≤ h
x :
Fin d → ℝ
hx :
x ∈ thinStrip d hd a
hr :
centreRadius x ≤ h
conclusion
|x ⟨1, by omega⟩ - 1 / 2| ≤ h ^ a
Proof (Lean source)
lemma thinStrip_transverse_coordinate_bound (d : ℕ) (hd : 2 ≤ d)
(a h : ℝ) (ha : 0 ≤ a) (hh : 0 ≤ h) (x : Fin d → ℝ)
(hx : x ∈ thinStrip d hd a) (hr : centreRadius x ≤ h) :
|x ⟨1, by omega⟩ - 1 / 2| ≤ h ^ a := by
have hfirst : |x ⟨0, by omega⟩ - 1 / 2| ≤ h := by
have hcoord := norm_le_pi_norm
(x - (fun _ : Fin d => (1 / 2 : ℝ))) ⟨0, by omega⟩
have hcoord' : |x ⟨0, by omega⟩ - 1 / 2| ≤ centreRadius x := by
simpa only [Pi.sub_apply, Real.norm_eq_abs, centreRadius] using hcoord
exact hcoord'.trans hr
calc
|x ⟨1, by omega⟩ - 1 / 2| ≤
|x ⟨0, by omega⟩ - 1 / 2| ^ a := hx
_ ≤ h ^ a := Real.rpow_le_rpow (abs_nonneg _) hfirst ha
theorem thinStrip_ball_transverse_coordinate_bound Lemma thinStrip_ball_transverse_coordinate_bound in the paper ↗
The same transverse-width bound holds in a centered Euclidean ball. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma thinStrip_ball_transverse_coordinate_bound (d : ℕ) (hd : 2 ≤ d)
(a h : ℝ) (ha : 0 ≤ a) (hh : 0 ≤ h) (x : Fin d → ℝ)
(hx : x ∈ thinStrip d hd a)
(hball : x ∈ ball2 (fun _ => (1 / 2 : ℝ)) h) :
|x ⟨1, by omega⟩ - 1 / 2| ≤ h ^ a := by
have hr : centreRadius x ≤ h := by
unfold centreRadius
letI : Nonempty (Fin d) := ⟨⟨0, by omega⟩⟩
apply (pi_norm_le_iff_of_nonempty _).2
intro i
have hi : (x i - (1 / 2 : ℝ)) ^ 2 ≤
∑ j : Fin d, (x j - 1 / 2) ^ 2 :=
Finset.single_le_sum
(fun j _ => sq_nonneg (x j - 1 / 2)) (Finset.mem_univ i)
have hball' : (∑ j : Fin d, (x j - 1 / 2) ^ 2) ≤ h ^ 2 := hball
have habs : |x i - 1 / 2| ≤ h := by
nlinarith [sq_abs (x i - (1 / 2 : ℝ)), abs_nonneg (x i - (1 / 2 : ℝ))]
simpa only [Pi.sub_apply, Real.norm_eq_abs] using habs
exact thinStrip_transverse_coordinate_bound d hd a h ha hh x hx hr
def thinStripCoreBox
A one-sided product box inside the thin strip. Its first coordinate is kept away from the centre so the transverse width has a uniform lower scale. For the stated inputs and conditions, the thinStripCoreBox object being defined.
For a small positive radius, the core box lies in both the strip and the centered Euclidean ball. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
a h :
ℝ
ha :
1 < a
hh :
0 < h
hh1 :
h ≤ 1
conclusion
Proof (Lean source)
lemma thinStripCoreBox_subset (d : ℕ) (hd : 2 ≤ d) (a h : ℝ)
(ha : 1 < a) (hh : 0 < h) (hh1 : h ≤ 1) :
thinStripCoreBox d hd a h ⊆
thinStrip d hd a ∩ ball2 (fun _ => (1 / 2 : ℝ)) h := by
let r : ℝ := h / (4 * d)
have hdR : (2 : ℝ) ≤ d := by exact_mod_cast hd
have hdpos : (0 : ℝ) < d := by linarith
have hr : 0 < r := by dsimp [r]; positivity
have hr1 : r ≤ 1 := by
dsimp [r]
apply (div_le_iff₀ (by positivity : (0 : ℝ) < 4 * d)).2
nlinarith
have hra : r ^ a ≤ r := Real.rpow_le_self_of_le_one hr.le hr1 ha.le
intro x hx
have hx' := hx
change (fun i => if i = ⟨1, by omega⟩ then 1 / 2 - r ^ a else 1 / 2 + r) ≤ x ∧
x ≤ (fun i => if i = ⟨1, by omega⟩ then 1 / 2 + r ^ a else 1 / 2 + 2 * r) at hx'
constructor
· unfold thinStrip
have hzeroLo := hx'.1 ⟨0, by omega⟩
have honeLo := hx'.1 ⟨1, by omega⟩
have honeHi := hx'.2 ⟨1, by omega⟩
simp at hzeroLo honeLo honeHi
have hzero : r ≤ |x ⟨0, by omega⟩ - 1 / 2| := by
rw [le_abs]
left
linarith
have hone : |x ⟨1, by omega⟩ - 1 / 2| ≤ r ^ a := by
rw [abs_le]
constructor <;> linarith
exact hone.trans (Real.rpow_le_rpow hr.le hzero (by linarith : 0 ≤ a))
· unfold ball2
have hcoord : ∀ i : Fin d, |x i - 1 / 2| ≤ 2 * r := by
intro i
have hlo := hx'.1 i
have hhi := hx'.2 i
by_cases hi1 : i = ⟨1, by omega⟩
· subst i
simp at hlo hhi
rw [abs_le]
constructor <;> nlinarith
· simp only [hi1, if_false] at hlo hhi
rw [abs_le]
constructor <;> linarith
calc
(∑ i : Fin d, (x i - 1 / 2) ^ 2) ≤ ∑ _i : Fin d, (2 * r) ^ 2 := by
apply Finset.sum_le_sum
intro i hi
simpa only [sq_abs] using
(pow_le_pow_left₀ (abs_nonneg _) (hcoord i) 2)
_ = (d : ℝ) * (2 * r) ^ 2 := by simp
_ ≤ h ^ 2 := by
dsimp [r]
field_simp
nlinarith
The core box has one transverse side of length 2 r^a and d-1 ordinary sides of length r, where r = h/(4d). For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
a h :
ℝ
ha :
0 ≤ a
hh :
0 ≤ h
conclusion
volume.real (thinStripCoreBox d hd a h)
= (2 * (h / (4 * d)) ^ a) * (h / (4 * d)) ^ (d - 1)
Proof (Lean source)
lemma thinStripCoreBox_volume_real (d : ℕ) (hd : 2 ≤ d) (a h : ℝ)
(ha : 0 ≤ a) (hh : 0 ≤ h) :
volume.real (thinStripCoreBox d hd a h) =
(2 * (h / (4 * d)) ^ a) * (h / (4 * d)) ^ (d - 1) := by
let r : ℝ := h / (4 * d)
let j : Fin d := ⟨1, by omega⟩
have hr : 0 ≤ r := by dsimp [r]; positivity
have hprod : (∏ i : Fin d, ofReal (if i = j then 2 * r ^ a else r)) =
ofReal (2 * r ^ a) * ofReal r ^ (d - 1) := by
calc
(∏ i : Fin d, ofReal (if i = j then 2 * r ^ a else r)) =
ofReal (2 * r ^ a) *
∏ i ∈ Finset.univ.erase j, ofReal r := by
rw [← Finset.mul_prod_erase univ _ (Finset.mem_univ j)]
simp only [if_pos]
congr 1
apply Finset.prod_congr rfl
intro i hi
rw [if_neg (Finset.ne_of_mem_erase hi)]
_ = ofReal (2 * r ^ a) * ofReal r ^ (d - 1) := by
rw [Finset.prod_const, Finset.card_erase_of_mem (Finset.mem_univ j),
Finset.card_fin]
have hvol : volume (thinStripCoreBox d hd a h) =
∏ i : Fin d, ofReal (if i = j then 2 * r ^ a else r) := by
rw [thinStripCoreBox, Real.volume_Icc_pi]
congr 1
ext i
congr 1
dsimp [r, j]
split_ifs <;> ring
rw [measureReal_def, hvol, hprod, ENNReal.toReal_mul, ENNReal.toReal_pow,
ENNReal.toReal_ofReal, ENNReal.toReal_ofReal]
all_goals positivity
The strip bump supplies treated mass of the sharp order h^a h^(d-1) on a centered ball. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
gamma a h :
ℝ
hgamma :
1 < gamma
ha :
1 < a
hh :
0 < h
hh1 :
h ≤ 1
conclusion
(1 / 4 : ℝ) * ((2 * (h / (4 * d)) ^ a) * (h / (4 * d)) ^ (d - 1))
≤ ∫ x in ball2 (fun _ => (1 / 2 : ℝ)) h, stripPropensity d hd gamma a x ∂(volume.restrict (cube d))
Proof (Lean source)
lemma thinStrip_treated_ball_mass_lower (d : ℕ) (hd : 2 ≤ d)
(gamma a h : ℝ) (hgamma : 1 < gamma) (ha : 1 < a)
(hh : 0 < h) (hh1 : h ≤ 1) :
(1 / 4 : ℝ) * ((2 * (h / (4 * d)) ^ a) * (h / (4 * d)) ^ (d - 1)) ≤
∫ x in ball2 (fun _ => (1 / 2 : ℝ)) h,
stripPropensity d hd gamma a x ∂(volume.restrict (cube d)) := by
let C := thinStripCoreBox d hd a h
let B := ball2 (fun _ : Fin d => (1 / 2 : ℝ)) h
let mu := volume.restrict (cube d)
have hCmeas : MeasurableSet C := by
dsimp [C, thinStripCoreBox]
exact measurableSet_Icc
have hCB := thinStripCoreBox_subset d hd a h ha hh hh1
have hdR : (2 : ℝ) ≤ d := by exact_mod_cast hd
have hr : 0 < h / (4 * d) := by positivity
have hr8 : h / (4 * d) ≤ 1 / 8 := by
apply (div_le_iff₀ (by positivity : (0 : ℝ) < 4 * d)).2
nlinarith
have hra : (h / (4 * d)) ^ a ≤ h / (4 * d) := by
apply Real.rpow_le_self_of_le_one hr.le
· linarith
· linarith
have hCcube : C ⊆ cube d := by
intro x hx
rw [cube, Set.mem_pi]
intro i _hi
constructor
· have hlo := hx.1 i
dsimp [C, thinStripCoreBox] at hlo
by_cases hi : i = ⟨1, by omega⟩
· simp [hi] at hlo
rw [hi]
linarith
· simp [hi] at hlo
linarith
· have hhi := hx.2 i
dsimp [C, thinStripCoreBox] at hhi
by_cases hi : i = ⟨1, by omega⟩
· simp [hi] at hhi
rw [hi]
linarith
· simp [hi] at hhi
linarith
have hmu_fin (S : Set (Fin d → ℝ)) : mu S ≠ ⊤ := by
apply ne_top_of_le_ne_top _ (measure_mono (Set.subset_univ S))
dsimp [mu]
rw [Measure.restrict_apply MeasurableSet.univ]
apply ne_top_of_le_ne_top _ (measure_mono Set.inter_subset_right)
have hcube : cube d =
Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hcube, Real.volume_Icc_pi]
simp
have heint : IntegrableOn (stripPropensity d hd gamma a) B mu := by
apply (integrableOn_const (C := (1 : ℝ)) (hs := hmu_fin B)).mono'
· exact (stripPropensity_measurable d hd gamma a).aestronglyMeasurable
· filter_upwards [] with x
rw [Real.norm_eq_abs, abs_of_nonneg
(stripPropensity_mem_Icc d hd gamma a hgamma x).1]
exact (stripPropensity_mem_Icc d hd gamma a hgamma x).2
have hconstint : IntegrableOn (fun _ : Fin d → ℝ => (1 / 4 : ℝ)) C mu :=
integrableOn_const (hs := hmu_fin C)
calc
(1 / 4 : ℝ) * ((2 * (h / (4 * d)) ^ a) * (h / (4 * d)) ^ (d - 1)) =
∫ _x in C, (1 / 4 : ℝ) ∂mu := by
rw [setIntegral_const, smul_eq_mul]
have hrestrict : mu.real C = volume.real C := by
rw [measureReal_def, Measure.restrict_apply hCmeas,
Set.inter_eq_left.mpr hCcube, ← measureReal_def]
rw [hrestrict, show volume.real C =
(2 * (h / (4 * d)) ^ a) * (h / (4 * d)) ^ (d - 1) by
exact thinStripCoreBox_volume_real d hd a h (by linarith) hh.le]
ring
_ ≤ ∫ x in C, stripPropensity d hd gamma a x ∂mu := by
apply setIntegral_mono_on hconstint (heint.mono_set (fun x hx => hCB hx |>.2)) hCmeas
intro x hx
exact stripPropensity_ge_quarter_on_strip d hd gamma a hgamma x (hCB hx).1
_ ≤ ∫ x in B, stripPropensity d hd gamma a x ∂mu := by
apply setIntegral_mono_set heint
· filter_upwards [] with x
exact (stripPropensity_mem_Icc d hd gamma a hgamma x).1
· exact Filter.Eventually.of_forall (fun x hx => hCB hx |>.2)
A centered ball cuts the strip into a box with transverse half-width h ^ a. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma thinStrip_ball_volume_le_box (d : ℕ) (hd : 2 ≤ d)
(a h : ℝ) (ha : 0 ≤ a) (hh : 0 ≤ h) :
volume (thinStrip d hd a ∩ ball2 (fun _ => (1 / 2 : ℝ)) h) ≤
∏ i : Fin d, ofReal
(if i = ⟨1, by omega⟩ then 2 * h ^ a else 2 * h) := by
let w : Fin d → ℝ := fun i =>
if i = ⟨1, by omega⟩ then h ^ a else h
have hsubset : thinStrip d hd a ∩ ball2 (fun _ => (1 / 2 : ℝ)) h ⊆
Icc (fun i => (1 / 2 : ℝ) - w i)
(fun i => (1 / 2 : ℝ) + w i) := by
intro x hx
rw [Set.mem_Icc]
constructor <;> intro i
· have hi : |x i - 1 / 2| ≤ w i := by
by_cases heq : i = ⟨1, by omega⟩
· subst i
simpa [w] using thinStrip_ball_transverse_coordinate_bound
d hd a h ha hh x hx.1 hx.2
· have hsum : (x i - (1 / 2 : ℝ)) ^ 2 ≤
∑ j : Fin d, (x j - 1 / 2) ^ 2 :=
Finset.single_le_sum
(fun j _ => sq_nonneg (x j - 1 / 2)) (Finset.mem_univ i)
have hball : (∑ j : Fin d, (x j - 1 / 2) ^ 2) ≤ h ^ 2 := hx.2
have : |x i - 1 / 2| ≤ h := by
nlinarith [sq_abs (x i - (1 / 2 : ℝ)), abs_nonneg (x i - (1 / 2 : ℝ))]
simpa [w, heq] using this
dsimp
linarith [(abs_le.mp hi).1]
· have hi : |x i - 1 / 2| ≤ w i := by
by_cases heq : i = ⟨1, by omega⟩
· subst i
simpa [w] using thinStrip_ball_transverse_coordinate_bound
d hd a h ha hh x hx.1 hx.2
· have hsum : (x i - (1 / 2 : ℝ)) ^ 2 ≤
∑ j : Fin d, (x j - 1 / 2) ^ 2 :=
Finset.single_le_sum
(fun j _ => sq_nonneg (x j - 1 / 2)) (Finset.mem_univ i)
have hball : (∑ j : Fin d, (x j - 1 / 2) ^ 2) ≤ h ^ 2 := hx.2
have : |x i - 1 / 2| ≤ h := by
nlinarith [sq_abs (x i - (1 / 2 : ℝ)), abs_nonneg (x i - (1 / 2 : ℝ))]
simpa [w, heq] using this
dsimp
linarith [(abs_le.mp hi).2]
calc
volume (thinStrip d hd a ∩ ball2 (fun _ => (1 / 2 : ℝ)) h) ≤
volume (Icc (fun i => (1 / 2 : ℝ) - w i)
(fun i => (1 / 2 : ℝ) + w i)) := measure_mono hsubset
_ = ∏ i : Fin d, ofReal
(if i = ⟨1, by omega⟩ then 2 * h ^ a else 2 * h) := by
rw [Real.volume_Icc_pi]
congr 1
ext i
congr 1
dsimp [w]
split_ifs <;> ring
The box estimate has one transverse factor 2 h^a and d-1 ordinary factors 2 h. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma thinStrip_ball_volume_real_upper (d : ℕ) (hd : 2 ≤ d)
(a h : ℝ) (ha : 0 ≤ a) (hh : 0 ≤ h) :
volume.real (thinStrip d hd a ∩ ball2 (fun _ => (1 / 2 : ℝ)) h) ≤
(2 * h ^ a) * (2 * h) ^ (d - 1) := by
let j : Fin d := ⟨1, by omega⟩
have hprod : (∏ i : Fin d, ofReal
(if i = j then 2 * h ^ a else 2 * h)) =
ofReal (2 * h ^ a) * ofReal (2 * h) ^ (d - 1) := by
calc
(∏ i : Fin d, ofReal (if i = j then 2 * h ^ a else 2 * h)) =
ofReal (if j = j then 2 * h ^ a else 2 * h) *
∏ i ∈ Finset.univ.erase j,
ofReal (if i = j then 2 * h ^ a else 2 * h) := by
exact (Finset.mul_prod_erase univ _ (Finset.mem_univ j)).symm
_ = ofReal (2 * h ^ a) *
∏ _i ∈ Finset.univ.erase j, ofReal (2 * h) := by
congr 1
apply Finset.prod_congr rfl
intro i hi
rw [if_neg (Finset.ne_of_mem_erase hi)]
_ = ofReal (2 * h ^ a) * ofReal (2 * h) ^ (d - 1) := by
rw [Finset.prod_const, Finset.card_erase_of_mem (Finset.mem_univ j),
Finset.card_fin]
have hvol := thinStrip_ball_volume_le_box d hd a h ha hh
rw [show (∏ i : Fin d, ofReal
(if i = ⟨1, by omega⟩ then 2 * h ^ a else 2 * h)) =
ofReal (2 * h ^ a) * ofReal (2 * h) ^ (d - 1) by
simpa [j] using hprod] at hvol
rw [measureReal_def]
calc
(volume (thinStrip d hd a ∩ ball2 (fun _ => (1 / 2 : ℝ)) h)).toReal ≤
(ofReal (2 * h ^ a) * ofReal (2 * h) ^ (d - 1)).toReal := by
apply ENNReal.toReal_mono
· finiteness
· exact hvol
_ = (2 * h ^ a) * (2 * h) ^ (d - 1) := by
rw [ENNReal.toReal_mul, ENNReal.toReal_pow,
ENNReal.toReal_ofReal, ENNReal.toReal_ofReal]
· positivity
· positivity
theorem thinStrip_normalized_transverse_square_bound Lemma thinStrip_normalized_transverse_square_bound in the paper ↗
The normalized transverse second moment is uniformly small on the strip. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
a h :
ℝ
ha :
0 ≤ a
hh :
0 < h
x :
Fin d → ℝ
hx :
x ∈ thinStrip d hd a
hr :
centreRadius x ≤ h
conclusion
((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 ≤ (h ^ a / h) ^ 2
Proof (Lean source)
lemma thinStrip_normalized_transverse_square_bound (d : ℕ) (hd : 2 ≤ d)
(a h : ℝ) (ha : 0 ≤ a) (hh : 0 < h) (x : Fin d → ℝ)
(hx : x ∈ thinStrip d hd a) (hr : centreRadius x ≤ h) :
((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 ≤ (h ^ a / h) ^ 2 := by
have hwidth := thinStrip_transverse_coordinate_bound d hd a h ha hh.le x hx hr
have hratio : |(x ⟨1, by omega⟩ - 1 / 2) / h| ≤ h ^ a / h := by
rw [abs_div, abs_of_pos hh]
exact div_le_div_of_nonneg_right hwidth hh.le
simpa only [sq_abs] using
(pow_le_pow_left₀ (abs_nonneg _) hratio 2)
theorem thinStrip_treated_second_integrand_bound Lemma thinStrip_treated_second_integrand_bound in the paper ↗
The radial treatment part contributes its small radius bound; the strip part contributes only its vanishing transverse width. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 * stripPropensity d hd γ a x
≤ ((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) + (h ^ a / h) ^ 2
Proof (Lean source)
lemma thinStrip_treated_second_integrand_bound (d : ℕ) (hd : 2 ≤ d)
(γ a h : ℝ) (hγ : 1 < γ) (ha : 0 ≤ a) (hh : 0 < h)
(x : Fin d → ℝ) (hr : centreRadius x ≤ h) :
((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 * stripPropensity d hd γ a x ≤
((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) + (h ^ a / h) ^ 2 := by
classical
have hrad : 0 ≤ ((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) := by positivity
by_cases hx : x ∈ thinStrip d hd a
· have hsquare := thinStrip_normalized_transverse_square_bound
d hd a h ha hh x hx hr
have he := (stripPropensity_mem_Icc d hd γ a hγ x).2
have hsq : 0 ≤ ((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 := sq_nonneg _
nlinarith
· have he := stripPropensity_off_strip_radius_bound d hd γ a h hγ x hx hr
have hcoord := norm_le_pi_norm
(x - (fun _ : Fin d => (1 / 2 : ℝ))) ⟨1, by omega⟩
have habs : |x ⟨1, by omega⟩ - 1 / 2| ≤ h := by
simpa only [Pi.sub_apply, Real.norm_eq_abs, centreRadius] using hcoord.trans hr
have hdiv : |(x ⟨1, by omega⟩ - 1 / 2) / h| ≤ 1 := by
rw [abs_div, abs_of_pos hh]
exact (div_le_iff₀ hh).2 (by simpa using habs)
have hsq : ((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 ≤ 1 := by
simpa only [sq_abs, one_pow] using
(pow_le_pow_left₀ (abs_nonneg _) hdiv 2)
have hnonneg := (stripPropensity_mem_Icc d hd γ a hγ x).1
nlinarith [sq_nonneg (h ^ a / h)]
theorem thinStrip_treated_second_integrand_split_bound Lemma thinStrip_treated_second_integrand_split_bound in the paper ↗
Pointwise split retaining the small strip volume for the transverse term. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
gamma a h :
ℝ
hgamma :
1 < gamma
ha :
0 ≤ a
hh :
0 < h
x :
Fin d → ℝ
hr :
centreRadius x ≤ h
conclusion
((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 * stripPropensity d hd gamma a x
≤ ((2 * h) ^ d) ^ ((1 : ℝ) / (gamma - 1))
Proof (Lean source)
lemma thinStrip_treated_second_integrand_split_bound (d : ℕ) (hd : 2 ≤ d)
(gamma a h : ℝ) (hgamma : 1 < gamma) (ha : 0 ≤ a) (hh : 0 < h)
(x : Fin d → ℝ) (hr : centreRadius x ≤ h) :
((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 * stripPropensity d hd gamma a x ≤
((2 * h) ^ d) ^ ((1 : ℝ) / (gamma - 1)) +
(thinStrip d hd a).indicator (fun _ => (h ^ a / h) ^ 2) x := by
classical
by_cases hx : x ∈ thinStrip d hd a
· rw [Set.indicator_of_mem hx]
have hsquare := thinStrip_normalized_transverse_square_bound
d hd a h ha hh x hx hr
have he := (stripPropensity_mem_Icc d hd gamma a hgamma x).2
have hrad : 0 ≤ ((2 * h) ^ d) ^ ((1 : ℝ) / (gamma - 1)) := by positivity
have hsq : 0 ≤ ((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 := sq_nonneg _
nlinarith
· simp only [indicator, hx, if_false, Pi.zero_apply, add_zero]
have he := stripPropensity_off_strip_radius_bound d hd gamma a h hgamma x hx hr
have hcoord := norm_le_pi_norm
(x - (fun _ : Fin d => (1 / 2 : ℝ))) ⟨1, by omega⟩
have habs : |x ⟨1, by omega⟩ - 1 / 2| ≤ h := by
simpa only [Pi.sub_apply, Real.norm_eq_abs, centreRadius] using hcoord.trans hr
have hdiv : |(x ⟨1, by omega⟩ - 1 / 2) / h| ≤ 1 := by
rw [abs_div, abs_of_pos hh]
exact (div_le_iff₀ hh).2 (by simpa using habs)
have hsq : ((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 ≤ 1 := by
simpa only [sq_abs, one_pow] using
(pow_le_pow_left₀ (abs_nonneg _) hdiv 2)
have hnonneg := (stripPropensity_mem_Icc d hd gamma a hgamma x).1
calc
((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 * stripPropensity d hd gamma a x ≤
1 * stripPropensity d hd gamma a x :=
mul_le_mul_of_nonneg_right hsq hnonneg
_ = stripPropensity d hd gamma a x := one_mul _
_ ≤ ((2 * h) ^ d) ^ ((1 : ℝ) / (gamma - 1)) := he
theorem thinStrip_treated_second_setIntegral_split Lemma thinStrip_treated_second_setIntegral_split in the paper ↗
Integrated numerator split, before inserting the two geometric volume bounds. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
gamma a h :
ℝ
hgamma :
1 < gamma
ha :
0 ≤ a
hh :
0 < h
conclusion
(∫ x in ball2 (fun _ => (1 / 2 : ℝ)) h, ((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 * stripPropensity d hd gamma a x ∂(volume.restrict (cube d)))
Proof (Lean source)
lemma thinStrip_treated_second_setIntegral_split (d : ℕ) (hd : 2 ≤ d)
(gamma a h : ℝ) (hgamma : 1 < gamma) (ha : 0 ≤ a) (hh : 0 < h) :
(∫ x in ball2 (fun _ => (1 / 2 : ℝ)) h,
((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 * stripPropensity d hd gamma a x
∂(volume.restrict (cube d))) ≤
((2 * h) ^ d) ^ ((1 : ℝ) / (gamma - 1)) *
(volume.restrict (cube d)).real (ball2 (fun _ => (1 / 2 : ℝ)) h) +
(h ^ a / h) ^ 2 *
(volume.restrict (cube d)).real
(ball2 (fun _ => (1 / 2 : ℝ)) h ∩ thinStrip d hd a) := by
let B := ball2 (fun _ : Fin d => (1 / 2 : ℝ)) h
let mu := volume.restrict (cube d)
let R : ℝ := ((2 * h) ^ d) ^ ((1 : ℝ) / (gamma - 1))
let S : ℝ := (h ^ a / h) ^ 2
have hB : MeasurableSet B := by
dsimp [B]
unfold ball2
exact measurableSet_le (by fun_prop) (by fun_prop)
have hstrip := thinStrip_measurableSet d hd a
have hfin : mu B ≠ ⊤ := by
apply ne_top_of_le_ne_top _ (measure_mono (Set.subset_univ B))
dsimp [mu]
rw [Measure.restrict_apply MeasurableSet.univ]
apply ne_top_of_le_ne_top _ (measure_mono Set.inter_subset_right)
have hcube : cube d = Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hcube, Real.volume_Icc_pi]
simp
have hfint : IntegrableOn (fun x : Fin d → ℝ =>
((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 * stripPropensity d hd gamma a x) B mu := by
apply (integrableOn_const (C := (1 : ℝ)) (hs := hfin)).mono'
· have hm : Measurable (fun x : Fin d → ℝ =>
((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2) := by fun_prop
exact (hm.mul (stripPropensity_measurable d hd gamma a)).aestronglyMeasurable
· filter_upwards [ae_restrict_mem hB] with x hx
have hsum : (x ⟨1, by omega⟩ - (1 / 2 : ℝ)) ^ 2 ≤
∑ i : Fin d, (x i - 1 / 2) ^ 2 :=
Finset.single_le_sum (fun i _ => sq_nonneg (x i - 1 / 2))
(Finset.mem_univ _)
have habs : |x ⟨1, by omega⟩ - 1 / 2| ≤ h := by
dsimp [B, ball2] at hx
nlinarith [sq_abs (x ⟨1, by omega⟩ - (1 / 2 : ℝ)),
abs_nonneg (x ⟨1, by omega⟩ - (1 / 2 : ℝ))]
have hdiv : |(x ⟨1, by omega⟩ - 1 / 2) / h| ≤ 1 := by
rw [abs_div, abs_of_pos hh]
exact (div_le_iff₀ hh).2 (by simpa using habs)
have he := stripPropensity_mem_Icc d hd gamma a hgamma x
rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg (sq_nonneg _) he.1)]
have hsq : ((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 ≤ 1 := by
simpa only [sq_abs, one_pow] using
(pow_le_pow_left₀ (abs_nonneg _) hdiv 2)
calc
((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 * stripPropensity d hd gamma a x ≤
1 * stripPropensity d hd gamma a x :=
mul_le_mul_of_nonneg_right hsq he.1
_ ≤ 1 := by simpa using he.2
have hRint : IntegrableOn (fun _ : Fin d → ℝ => R) B mu :=
integrableOn_const (hs := hfin)
have hSint : IntegrableOn
((thinStrip d hd a).indicator (fun _ : Fin d → ℝ => S)) B mu :=
(integrableOn_const (C := S) (hs := hfin)).indicator hstrip
calc
(∫ x in B, ((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 *
stripPropensity d hd gamma a x ∂mu) ≤
∫ x in B, R + (thinStrip d hd a).indicator (fun _ => S) x ∂mu := by
apply setIntegral_mono_on hfint (hRint.add hSint) hB
intro x hx
have hr : centreRadius x ≤ h := by
rw [centreRadius, pi_norm_le_iff_of_nonneg hh.le]
intro i
have hsum : (x i - (1 / 2 : ℝ)) ^ 2 ≤
∑ j : Fin d, (x j - 1 / 2) ^ 2 :=
Finset.single_le_sum (fun j _ => sq_nonneg (x j - 1 / 2))
(Finset.mem_univ _)
dsimp [B, ball2] at hx
rw [Pi.sub_apply, Real.norm_eq_abs]
nlinarith [sq_abs (x i - (1 / 2 : ℝ)), abs_nonneg (x i - (1 / 2 : ℝ))]
exact thinStrip_treated_second_integrand_split_bound d hd gamma a h
hgamma ha hh x hr
_ = R * mu.real B + S * mu.real (B ∩ thinStrip d hd a) := by
rw [integral_add hRint hSint, setIntegral_const, smul_eq_mul,
setIntegral_indicator hstrip, setIntegral_const, smul_eq_mul]
ring
A centered Euclidean ball is contained in its coordinate box. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma uniformCube_ball_volume_real_upper (d : ℕ) (h : ℝ) (hh : 0 ≤ h) :
(volume.restrict (cube d)).real (ball2 (fun _ => (1 / 2 : ℝ)) h) ≤
(2 * h) ^ d := by
let B := ball2 (fun _ : Fin d => (1 / 2 : ℝ)) h
let Q := Icc (fun _ : Fin d => (1 / 2 : ℝ) - h) (fun _ => 1 / 2 + h)
have hBQ : B ⊆ Q := by
intro x hx
rw [Set.mem_Icc]
constructor <;> intro i
· have hsum : (x i - (1 / 2 : ℝ)) ^ 2 ≤
∑ j : Fin d, (x j - 1 / 2) ^ 2 :=
Finset.single_le_sum (fun j _ => sq_nonneg (x j - 1 / 2))
(Finset.mem_univ _)
dsimp [B, ball2] at hx
nlinarith [sq_abs (x i - (1 / 2 : ℝ)), abs_nonneg (x i - (1 / 2 : ℝ))]
· have hsum : (x i - (1 / 2 : ℝ)) ^ 2 ≤
∑ j : Fin d, (x j - 1 / 2) ^ 2 :=
Finset.single_le_sum (fun j _ => sq_nonneg (x j - 1 / 2))
(Finset.mem_univ _)
dsimp [B, ball2] at hx
nlinarith [sq_abs (x i - (1 / 2 : ℝ)), abs_nonneg (x i - (1 / 2 : ℝ))]
have hmeasure : (volume.restrict (cube d)) B ≤ volume Q := by
calc
(volume.restrict (cube d)) B ≤ volume B :=
(Measure.restrict_le_self : volume.restrict (cube d) ≤ volume) B
_ ≤ volume Q := measure_mono hBQ
have hQ : volume Q = ofReal ((2 * h) ^ d) := by
dsimp [Q]
rw [Real.volume_Icc_pi]
simp only [add_sub_sub_cancel, Finset.prod_const, Finset.card_fin]
rw [← ENNReal.ofReal_pow (by positivity)]
congr 1
ring
change ((volume.restrict (cube d)) B).toReal ≤ (2 * h) ^ d
rw [hQ] at hmeasure
calc
((volume.restrict (cube d)) B).toReal ≤ (ofReal ((2 * h) ^ d)).toReal := by
apply ENNReal.toReal_mono
· simp
· exact hmeasure
_ = (2 * h) ^ d := ENNReal.toReal_ofReal (by positivity)
theorem thinStrip_treated_second_setIntegral_power_bound Lemma thinStrip_treated_second_setIntegral_power_bound in the paper ↗
The numerator has the two sharp powers needed after division by treated mass. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
gamma a h :
ℝ
hgamma :
1 < gamma
ha :
0 ≤ a
hh :
0 < h
conclusion
(∫ x in ball2 (fun _ => (1 / 2 : ℝ)) h, ((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 * stripPropensity d hd gamma a x ∂(volume.restrict (cube d)))
≤ ((2 * h) ^ d) ^ ((1 : ℝ) / (gamma - 1)) * (2 * h) ^ d
+ (h ^ a / h) ^ 2 * ((2 * h ^ a) * (2 * h) ^ (d - 1))
Proof (Lean source)
lemma thinStrip_treated_second_setIntegral_power_bound (d : ℕ) (hd : 2 ≤ d)
(gamma a h : ℝ) (hgamma : 1 < gamma) (ha : 0 ≤ a) (hh : 0 < h) :
(∫ x in ball2 (fun _ => (1 / 2 : ℝ)) h,
((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 * stripPropensity d hd gamma a x
∂(volume.restrict (cube d))) ≤
((2 * h) ^ d) ^ ((1 : ℝ) / (gamma - 1)) * (2 * h) ^ d +
(h ^ a / h) ^ 2 * ((2 * h ^ a) * (2 * h) ^ (d - 1)) := by
have hsplit := thinStrip_treated_second_setIntegral_split d hd gamma a h
hgamma ha hh
have hball := uniformCube_ball_volume_real_upper d h hh.le
have hstrip : (volume.restrict (cube d)).real
(ball2 (fun _ => (1 / 2 : ℝ)) h ∩ thinStrip d hd a) ≤
(2 * h ^ a) * (2 * h) ^ (d - 1) := by
rw [Set.inter_comm]
have hvolfin : volume
(thinStrip d hd a ∩ ball2 (fun _ => (1 / 2 : ℝ)) h) ≠ ⊤ := by
apply ne_top_of_le_ne_top _
(thinStrip_ball_volume_le_box d hd a h ha hh.le)
apply ENNReal.prod_ne_top
intro i hi
exact ENNReal.ofReal_ne_top
calc
(volume.restrict (cube d)).real
(thinStrip d hd a ∩ ball2 (fun _ => (1 / 2 : ℝ)) h) ≤
volume.real (thinStrip d hd a ∩ ball2 (fun _ => (1 / 2 : ℝ)) h) := by
rw [measureReal_def, measureReal_def]
apply ENNReal.toReal_mono
· exact hvolfin
· exact (Measure.restrict_le_self :
volume.restrict (cube d) ≤ volume) _
_ ≤ (2 * h ^ a) * (2 * h) ^ (d - 1) :=
thinStrip_ball_volume_real_upper d hd a h ha hh.le
have hR : 0 ≤ ((2 * h) ^ d) ^ ((1 : ℝ) / (gamma - 1)) := by positivity
have hS : 0 ≤ (h ^ a / h) ^ 2 := sq_nonneg _
calc
_ ≤ ((2 * h) ^ d) ^ ((1 : ℝ) / (gamma - 1)) *
(volume.restrict (cube d)).real (ball2 (fun _ => (1 / 2 : ℝ)) h) +
(h ^ a / h) ^ 2 * (volume.restrict (cube d)).real
(ball2 (fun _ => (1 / 2 : ℝ)) h ∩ thinStrip d hd a) := hsplit
_ ≤ _ := add_le_add (mul_le_mul_of_nonneg_left hball hR)
(mul_le_mul_of_nonneg_left hstrip hS)
theorem thinStrip_normalized_transverse_square_tendsto_zero Lemma thinStrip_normalized_transverse_square_tendsto_zero in the paper ↗
The deterministic strip contribution to the normalized second moment vanishes. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
a :
ℝ
ha :
1 < a
conclusion
Proof (Lean source)
lemma thinStrip_normalized_transverse_square_tendsto_zero (a : ℝ) (ha : 1 < a) :
Tendsto (fun h : ℝ => (h ^ a / h) ^ 2)
(nhdsWithin 0 (Ioi 0)) (nhds 0) := by
have hexp : 0 < a - 1 := by linarith
have hpow : Tendsto (fun h : ℝ => h ^ (a - 1))
(nhdsWithin 0 (Ioi 0)) (nhds 0) := by
have hid : Tendsto (fun h : ℝ => h)
(nhdsWithin 0 (Ioi 0)) (nhds 0) :=
continuousAt_id.tendsto.mono_left nhdsWithin_le_nhds
convert hid.rpow_const (inr hexp.le) using 1
rw [Real.zero_rpow (ne_of_gt hexp)]
have hsq := hpow.pow 2
have heq : (fun h : ℝ => (h ^ a / h) ^ 2) =ᶠ[nhdsWithin 0 (Ioi 0)]
(fun h : ℝ => (h ^ (a - 1)) ^ 2) := by
filter_upwards [self_mem_nhdsWithin] with h hh
simpa only [Set.mem_Ioi] using congrArg (fun z : ℝ => z ^ 2)
(Real.rpow_sub_one (ne_of_gt hh) a).symm
simpa using hsq.congr' heq.symm
theorem thinStrip_second_integrand_envelope_tendsto_zero Lemma thinStrip_second_integrand_envelope_tendsto_zero in the paper ↗
The pointwise envelope for the normalized treated second moment vanishes. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a :
ℝ
hγ :
1 < γ
ha :
1 < a
conclusion
Tendsto (fun h : ℝ => ((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) + (h ^ a / h) ^ 2) (nhdsWithin 0 (Ioi 0)) (nhds 0)
Proof (Lean source)
lemma thinStrip_second_integrand_envelope_tendsto_zero
(d : ℕ) (hd : 2 ≤ d) (γ a : ℝ) (hγ : 1 < γ) (ha : 1 < a) :
Tendsto
(fun h : ℝ =>
((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) + (h ^ a / h) ^ 2)
(nhdsWithin 0 (Ioi 0)) (nhds 0) := by
simpa using (stripRadialBound_tendsto_zero d hd γ hγ).add
(thinStrip_normalized_transverse_square_tendsto_zero a ha)
A closed cube of radius r around the design centre has volume (2*r)^d. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma centreCube_volume (d : ℕ) (r : ℝ) (hr : 0 ≤ r) :
volume (Icc (fun _ : Fin d => (1 / 2 : ℝ) - r)
(fun _ => (1 / 2 : ℝ) + r)) = ofReal ((2 * r) ^ d) := by
rw [Real.volume_Icc_pi]
simp only [add_sub_sub_cancel, Finset.prod_const, Finset.card_fin]
rw [← ENNReal.ofReal_pow (by positivity)]
congr 1
ring
theorem uniformCube_centered_ball_mass_lower Lemma uniformCube_centered_ball_mass_lower in the paper ↗
A coordinate box of half-width h / d gives a quantitative lower bound for the uniform-cube mass of a centered Euclidean ball. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma uniformCube_centered_ball_mass_lower (d : ℕ) (hd : 1 ≤ d)
(h : ℝ) (hh : 0 < h) (hhhalf : h ≤ 1 / 2) :
(2 * (h / d)) ^ d ≤
(volume.restrict (cube d)).real
(ball2 (fun _ => (1 / 2 : ℝ)) h) := by
let r : ℝ := h / d
have hdR : (1 : ℝ) ≤ d := by exact_mod_cast hd
have hdpos : (0 : ℝ) < d := lt_of_lt_of_le zero_lt_one hdR
have hr : 0 ≤ r := by dsimp [r]; positivity
have hrhalf : r ≤ 1 / 2 := by
calc
r ≤ h := by
dsimp [r]
exact (div_le_iff₀ hdpos).2 (by nlinarith)
_ ≤ 1 / 2 := hhhalf
let Q : Set (Fin d → ℝ) :=
Icc (fun _ => (1 / 2 : ℝ) - r) (fun _ => (1 / 2 : ℝ) + r)
have hQ : Q ⊆ ball2 (fun _ => (1 / 2 : ℝ)) h ∩ cube d := by
intro x hx
have hx' : (fun _ : Fin d => (1 / 2 : ℝ) - r) ≤ x ∧
x ≤ (fun _ : Fin d => (1 / 2 : ℝ) + r) := hx
constructor
· change (∑ i : Fin d, (x i - 1 / 2) ^ 2) ≤ h ^ 2
calc
(∑ i : Fin d, (x i - 1 / 2) ^ 2) ≤ ∑ _i : Fin d, r ^ 2 := by
apply Finset.sum_le_sum
intro i hi
have hlo := hx'.1 i
have hhi := hx'.2 i
have habs : |x i - 1 / 2| ≤ r := by
rw [abs_le]
constructor <;> dsimp at hlo hhi ⊢ <;> linarith
simpa only [sq_abs] using
(pow_le_pow_left₀ (abs_nonneg _) habs 2)
_ = (d : ℝ) * r ^ 2 := by simp
_ ≤ h ^ 2 := by
dsimp [r]
rw [div_pow]
field_simp
nlinarith [hdR]
· simp only [cube]
intro i hi
simp only [Set.mem_Icc]
constructor
· have hlo := hx'.1 i
dsimp at hlo
linarith
· have hhi := hx'.2 i
dsimp at hhi
linarith
have hball : MeasurableSet (ball2 (fun _ : Fin d => (1 / 2 : ℝ)) h) := by
unfold ball2
exact measurableSet_le (by fun_prop) (by fun_prop)
rw [measureReal_restrict_apply hball]
have hcubevol : volume (cube d) = 1 := by
have hcube : cube d =
Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hcube, Real.volume_Icc_pi]
simp
have hcubeFin : volume (cube d) ≠ ⊤ := by
rw [hcubevol]
exact ENNReal.one_ne_top
have hfin : volume (ball2 (fun _ : Fin d => (1 / 2 : ℝ)) h ∩ cube d) ≠ ⊤ := by
exact ne_top_of_le_ne_top hcubeFin (measure_mono Set.inter_subset_right)
have hmono : volume.real Q ≤
volume.real (ball2 (fun _ : Fin d => (1 / 2 : ℝ)) h ∩ cube d) :=
measureReal_mono (h₂ := hfin) hQ
have hQvol : volume.real Q = (2 * r) ^ d := by
rw [measureReal_def]
rw [show volume Q = ofReal ((2 * r) ^ d) by
exact centreCube_volume d r hr]
rw [ENNReal.toReal_ofReal]
positivity
rw [hQvol] at hmono
simpa [r] using hmono
theorem uniformCube_thinStrip_local_ratio_bound Lemma uniformCube_thinStrip_local_ratio_bound in the paper ↗
Under uniform cube volume, the strip occupies at most a constant times h^(a-1) of a sufficiently small centered ball. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma uniformCube_thinStrip_local_ratio_bound (d : ℕ) (hd : 2 ≤ d)
(a h : ℝ) (ha : 1 < a) (hh : 0 < h) (hhhalf : h ≤ 1 / 2) :
(volume.restrict (cube d)).real
(thinStrip d hd a ∩ ball2 (fun _ => (1 / 2 : ℝ)) h) /
(volume.restrict (cube d)).real
(ball2 (fun _ => (1 / 2 : ℝ)) h) ≤
(d : ℝ) ^ d * h ^ (a - 1) := by
let S := thinStrip d hd a ∩ ball2 (fun _ => (1 / 2 : ℝ)) h
have hS : MeasurableSet S := by
apply (thinStrip_measurableSet d hd a).inter
unfold ball2
exact measurableSet_le (by fun_prop) (by fun_prop)
have hnum : (volume.restrict (cube d)).real S ≤
(2 * h ^ a) * (2 * h) ^ (d - 1) := by
rw [measureReal_restrict_apply hS]
have hraw := thinStrip_ball_volume_le_box d hd a h
(le_trans zero_le_one ha.le) hh.le
have hSfin : volume S ≠ ⊤ := by
have htop : (∏ i : Fin d, ofReal
(if i = ⟨1, by omega⟩ then 2 * h ^ a else 2 * h)) < ⊤ := by
classical
induction (Finset.univ : Finset (Fin d)) using Finset.induction_on with
| empty => simp
| @insert i s hi ih =>
rw [Finset.prod_insert hi]
exact ENNReal.mul_lt_top ENNReal.ofReal_lt_top ih
exact ne_top_of_le_ne_top (ne_of_lt htop) hraw
exact (measureReal_mono (h₂ := hSfin) Set.inter_subset_left).trans
(thinStrip_ball_volume_real_upper d hd a h
(le_trans zero_le_one ha.le) hh.le)
have hden := uniformCube_centered_ball_mass_lower d (by omega) h hh hhhalf
have hbasepos : 0 < (2 * (h / d)) ^ d := by positivity
calc
(volume.restrict (cube d)).real S /
(volume.restrict (cube d)).real
(ball2 (fun _ => (1 / 2 : ℝ)) h) ≤
((2 * h ^ a) * (2 * h) ^ (d - 1)) /
(volume.restrict (cube d)).real
(ball2 (fun _ => (1 / 2 : ℝ)) h) :=
div_le_div_of_nonneg_right hnum (measureReal_nonneg)
_ ≤ ((2 * h ^ a) * (2 * h) ^ (d - 1)) /
(2 * (h / d)) ^ d :=
div_le_div_of_nonneg_left (by positivity) hbasepos hden
_ = (d : ℝ) ^ d * h ^ (a - 1) := by
have hdpos : (0 : ℝ) < d := by positivity
have hhne : h ≠ 0 := ne_of_gt hh
have hpow : (2 * h) ^ d = (2 * h) ^ (d - 1) * (2 * h) := by
conv_lhs => rw [show d = (d - 1) + 1 by omega, pow_succ]
have hdeneq : 2 * (h / (d : ℝ)) = (2 * h) / (d : ℝ) := by ring
rw [hdeneq, div_pow, div_div_eq_mul_div,
div_eq_iff (pow_ne_zero d (mul_ne_zero (by norm_num) hhne))]
rw [Real.rpow_sub_one hhne]
field_simp
rw [hpow]
ring
The transverse exponent a > 1 makes the strip's fraction of centered balls vanish under uniform cube volume. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a :
ℝ
hγ :
1 < γ
ha :
1 < a
ρ ν h₀ :
ℝ
0 < ρ
0 < ν
0 < h₀
conclusion
∃ h : ℝ,
conclusion 1
0 < h
conclusion 2
h ≤ h₀
conclusion 3
((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) < ρ / 4
Proof (Lean source)
lemma uniformCube_thinStrip_local_ratio (d : ℕ) (hd : 2 ≤ d)
(γ a : ℝ) (hγ : 1 < γ) (ha : 1 < a) :
∀ (ρ ν h₀ : ℝ), 0 < ρ → 0 < ν → 0 < h₀ →
∃ h : ℝ, 0 < h ∧ h ≤ h₀ ∧
((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) < ρ / 4 ∧
0 < (volume.restrict (cube d)).real
(ball2 (fun _ => (1 / 2 : ℝ)) h) ∧
(volume.restrict (cube d)).real
(thinStrip d hd a ∩ ball2 (fun _ => (1 / 2 : ℝ)) h) /
(volume.restrict (cube d)).real
(ball2 (fun _ => (1 / 2 : ℝ)) h) ≤ ν := by
intro ρ ν h₀ hρ hν hh₀
have hid : Tendsto (fun h : ℝ => h)
(nhdsWithin 0 (Ioi 0)) (nhds 0) :=
continuousAt_id.tendsto.mono_left nhdsWithin_le_nhds
have hpow : Tendsto (fun h : ℝ => h ^ (a - 1))
(nhdsWithin 0 (Ioi 0)) (nhds 0) := by
convert hid.rpow_const (inr (by linarith : 0 ≤ a - 1)) using 1
rw [Real.zero_rpow (by linarith : a - 1 ≠ 0)]
have hratioLim : Tendsto (fun h : ℝ => (d : ℝ) ^ d * h ^ (a - 1))
(nhdsWithin 0 (Ioi 0)) (nhds 0) := by
simpa using (tendsto_const_nhds.mul hpow)
have hratio : ∀ᶠ h : ℝ in nhdsWithin 0 (Ioi 0),
(d : ℝ) ^ d * h ^ (a - 1) < ν :=
hratioLim.eventually (eventually_lt_nhds hν)
have hrad : ∀ᶠ h : ℝ in nhdsWithin 0 (Ioi 0),
((2 * h) ^ d) ^ ((1 : ℝ) / (γ - 1)) < ρ / 4 :=
(stripRadialBound_tendsto_zero d hd γ hγ).eventually
(eventually_lt_nhds (by positivity))
have hwidth : ∀ᶠ h : ℝ in nhdsWithin 0 (Ioi 0), h < min h₀ (1 / 2) :=
hid.eventually (eventually_lt_nhds (lt_min hh₀ (by norm_num)))
have hpos : ∀ᶠ h : ℝ in nhdsWithin 0 (Ioi 0), 0 < h :=
self_mem_nhdsWithin
obtain ⟨h, hh, hsmall, hrad', hratio'⟩ :=
(hpos.and (hwidth.and (hrad.and hratio))).exists
have hsmall' : h < h₀ ∧ h < 1 / 2 := (lt_min_iff.mp hsmall)
have hhhalf : h ≤ 1 / 2 := hsmall'.2.le
have hden := uniformCube_centered_ball_mass_lower d (by omega) h hh hhhalf
have hball : 0 < (volume.restrict (cube d)).real
(ball2 (fun _ => (1 / 2 : ℝ)) h) :=
lt_of_lt_of_le (by positivity : 0 < (2 * (h / d)) ^ d) hden
refine ⟨h, hh, hsmall'.1.le, hrad', hball, ?_⟩
exact (uniformCube_thinStrip_local_ratio_bound d hd a h ha hh hhhalf).trans
hratio'.le
The sup-norm radius sublevel set is the centred coordinate cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
r :
ℝ
hr :
0 ≤ r
conclusion
{x : Fin d → ℝ | centreRadius x ≤ r}
= Icc (fun _ => (1 / 2 : ℝ) - r) (fun _ => (1 / 2 : ℝ) + r)
Proof (Lean source)
lemma centreRadius_sublevel_eq_cube (d : ℕ) (r : ℝ) (hr : 0 ≤ r) :
{x : Fin d → ℝ | centreRadius x ≤ r} =
Icc (fun _ => (1 / 2 : ℝ) - r) (fun _ => (1 / 2 : ℝ) + r) := by
ext x
simp only [Set.mem_ofPred_eq, Set.mem_Icc, Pi.le_def]
constructor
· intro hx
constructor <;> intro i
· have hi := (norm_le_pi_norm (x - fun _ : Fin d => (1 / 2 : ℝ)) i).trans hx
change |x i - 1 / 2| ≤ r at hi
have := (abs_le.mp hi).1
linarith
· have hi := (norm_le_pi_norm (x - fun _ : Fin d => (1 / 2 : ℝ)) i).trans hx
change |x i - 1 / 2| ≤ r at hi
have := (abs_le.mp hi).2
linarith
· rintro ⟨hlo, hhi⟩
unfold centreRadius
apply (pi_norm_le_iff_of_nonneg hr).2
intro i
change |x i - 1 / 2| ≤ r
apply abs_le.mpr
constructor <;> linarith [hlo i, hhi i]
The centred radial CDF has the stated elementary volume formula. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
r :
ℝ
hr :
0 ≤ r
conclusion
Proof (Lean source)
lemma centreRadius_sublevel_volume (d : ℕ) (r : ℝ) (hr : 0 ≤ r) :
volume {x : Fin d → ℝ | centreRadius x ≤ r} =
ofReal ((2 * r) ^ d) := by
rw [centreRadius_sublevel_eq_cube d r hr, centreCube_volume d r hr]
theorem thinStrip_centreRadius_volume_real_upper Lemma thinStrip_centreRadius_volume_real_upper in the paper ↗
The thin strip inside a sup-norm ball has one side of width 2 h^a and d-1 sides of width 2 h. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
a h :
ℝ
ha :
0 ≤ a
hh :
0 ≤ h
conclusion
volume.real (thinStrip d hd a ∩ {x | centreRadius x ≤ h})
≤ (2 * h ^ a) * (2 * h) ^ (d - 1)
Proof (Lean source)
lemma thinStrip_centreRadius_volume_real_upper (d : ℕ) (hd : 2 ≤ d)
(a h : ℝ) (ha : 0 ≤ a) (hh : 0 ≤ h) :
volume.real (thinStrip d hd a ∩ {x | centreRadius x ≤ h}) ≤
(2 * h ^ a) * (2 * h) ^ (d - 1) := by
let j : Fin d := ⟨1, by omega⟩
let w : Fin d → ℝ := fun i => if i = j then h ^ a else h
have hsubset : thinStrip d hd a ∩ {x | centreRadius x ≤ h} ⊆
Icc (fun i => (1 / 2 : ℝ) - w i)
(fun i => (1 / 2 : ℝ) + w i) := by
intro x hx
rw [Set.mem_Icc]
constructor <;> intro i
· have hi : |x i - 1 / 2| ≤ w i := by
by_cases heq : i = j
· subst i
simpa [w, j] using thinStrip_transverse_coordinate_bound
d hd a h ha hh x hx.1 hx.2
· have hi := (norm_le_pi_norm
(x - fun _ : Fin d => (1 / 2 : ℝ)) i).trans hx.2
simpa [w, heq, centreRadius, Pi.sub_apply, Real.norm_eq_abs] using hi
linarith [(abs_le.mp hi).1]
· have hi : |x i - 1 / 2| ≤ w i := by
by_cases heq : i = j
· subst i
simpa [w, j] using thinStrip_transverse_coordinate_bound
d hd a h ha hh x hx.1 hx.2
· have hi := (norm_le_pi_norm
(x - fun _ : Fin d => (1 / 2 : ℝ)) i).trans hx.2
simpa [w, heq, centreRadius, Pi.sub_apply, Real.norm_eq_abs] using hi
linarith [(abs_le.mp hi).2]
have hvol : volume (thinStrip d hd a ∩ {x | centreRadius x ≤ h}) ≤
∏ i : Fin d, ofReal (if i = j then 2 * h ^ a else 2 * h) := by
calc
volume (thinStrip d hd a ∩ {x | centreRadius x ≤ h}) ≤
volume (Icc (fun i => (1 / 2 : ℝ) - w i)
(fun i => (1 / 2 : ℝ) + w i)) := measure_mono hsubset
_ = ∏ i : Fin d, ofReal
(if i = j then 2 * h ^ a else 2 * h) := by
rw [Real.volume_Icc_pi]
congr 1
ext i
congr 1
dsimp [w]
split_ifs <;> ring
have hprod : (∏ i : Fin d, ofReal
(if i = j then 2 * h ^ a else 2 * h)) =
ofReal (2 * h ^ a) * ofReal (2 * h) ^ (d - 1) := by
calc
(∏ i : Fin d, ofReal (if i = j then 2 * h ^ a else 2 * h)) =
ofReal (2 * h ^ a) *
∏ i ∈ Finset.univ.erase j, ofReal (2 * h) := by
rw [← Finset.mul_prod_erase univ _ (Finset.mem_univ j)]
simp only [if_pos]
congr 1
apply Finset.prod_congr rfl
intro i hi
rw [if_neg (Finset.ne_of_mem_erase hi)]
_ = ofReal (2 * h ^ a) * ofReal (2 * h) ^ (d - 1) := by
rw [Finset.prod_const, Finset.card_erase_of_mem (Finset.mem_univ j),
Finset.card_fin]
rw [hprod] at hvol
rw [measureReal_def]
calc
(volume (thinStrip d hd a ∩ {x | centreRadius x ≤ h})).toReal ≤
(ofReal (2 * h ^ a) * ofReal (2 * h) ^ (d - 1)).toReal := by
apply ENNReal.toReal_mono
· finiteness
· exact hvol
_ = (2 * h ^ a) * (2 * h) ^ (d - 1) := by
rw [ENNReal.toReal_mul, ENNReal.toReal_pow,
ENNReal.toReal_ofReal, ENNReal.toReal_ofReal]
· positivity
· positivity
theorem thinStrip_treated_second_sup_setIntegral_split Lemma thinStrip_treated_second_sup_setIntegral_split in the paper ↗
The integrated pointwise split on the sup-norm event used by stripSecondMoment. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
gamma a h :
ℝ
hgamma :
1 < gamma
ha :
0 ≤ a
hh :
0 < h
conclusion
(∫ x in {x : Fin d → ℝ | centreRadius x ≤ h}, ((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 * stripPropensity d hd gamma a x ∂(volume.restrict (cube d)))
≤ ((2 * h) ^ d) ^ ((1 : ℝ) / (gamma - 1)) * (volume.restrict (cube d)).real {x : Fin d → ℝ | centreRadius x ≤ h}
+ (h ^ a / h) ^ 2 * (volume.restrict (cube d)).real ({x : Fin d → ℝ | centreRadius x ≤ h} ∩ thinStrip d hd a)
Proof (Lean source)
lemma thinStrip_treated_second_sup_setIntegral_split (d : ℕ) (hd : 2 ≤ d)
(gamma a h : ℝ) (hgamma : 1 < gamma) (ha : 0 ≤ a) (hh : 0 < h) :
(∫ x in {x : Fin d → ℝ | centreRadius x ≤ h},
((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 * stripPropensity d hd gamma a x
∂(volume.restrict (cube d))) ≤
((2 * h) ^ d) ^ ((1 : ℝ) / (gamma - 1)) *
(volume.restrict (cube d)).real {x : Fin d → ℝ | centreRadius x ≤ h} +
(h ^ a / h) ^ 2 * (volume.restrict (cube d)).real
({x : Fin d → ℝ | centreRadius x ≤ h} ∩ thinStrip d hd a) := by
let D : Set (Fin d → ℝ) := {x | centreRadius x ≤ h}
let mu := volume.restrict (cube d)
let R : ℝ := ((2 * h) ^ d) ^ ((1 : ℝ) / (gamma - 1))
let S : ℝ := (h ^ a / h) ^ 2
have hD : MeasurableSet D := measurableSet_le
(by unfold centreRadius; fun_prop) measurable_const
have hfin : mu D ≠ ⊤ := by
apply ne_top_of_le_ne_top _ (measure_mono (Set.subset_univ D))
dsimp [mu]
rw [Measure.restrict_apply MeasurableSet.univ]
apply ne_top_of_le_ne_top _ (measure_mono Set.inter_subset_right)
have hcube : cube d = Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hcube, Real.volume_Icc_pi]
simp
have hfint : IntegrableOn (fun x : Fin d → ℝ =>
((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 * stripPropensity d hd gamma a x) D mu := by
apply (integrableOn_const (C := (1 : ℝ)) (hs := hfin)).mono'
· have hm : Measurable (fun x : Fin d → ℝ =>
((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2) := by fun_prop
exact (hm.mul (stripPropensity_measurable d hd gamma a)).aestronglyMeasurable
· filter_upwards [ae_restrict_mem hD] with x hx
have hcoord := (norm_le_pi_norm
(x - fun _ : Fin d => (1 / 2 : ℝ)) ⟨1, by omega⟩).trans hx
have habs : |x ⟨1, by omega⟩ - 1 / 2| ≤ h := by
simpa [centreRadius, Pi.sub_apply, Real.norm_eq_abs] using hcoord
have hdiv : |(x ⟨1, by omega⟩ - 1 / 2) / h| ≤ 1 := by
rw [abs_div, abs_of_pos hh]
exact (div_le_iff₀ hh).2 (by simpa using habs)
have hsq : ((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 ≤ 1 := by
simpa only [sq_abs, one_pow] using
(pow_le_pow_left₀ (abs_nonneg _) hdiv 2)
have he := stripPropensity_mem_Icc d hd gamma a hgamma x
rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg (sq_nonneg _) he.1)]
calc
_ ≤ 1 * stripPropensity d hd gamma a x :=
mul_le_mul_of_nonneg_right hsq he.1
_ ≤ 1 := by simpa using he.2
have hRint : IntegrableOn (fun _ : Fin d → ℝ => R) D mu :=
integrableOn_const (hs := hfin)
have hstrip := thinStrip_measurableSet d hd a
have hSint : IntegrableOn
((thinStrip d hd a).indicator (fun _ : Fin d → ℝ => S)) D mu :=
(integrableOn_const (C := S) (hs := hfin)).indicator hstrip
calc
(∫ x in D, ((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 *
stripPropensity d hd gamma a x ∂mu) ≤
∫ x in D, R + (thinStrip d hd a).indicator (fun _ => S) x ∂mu := by
apply setIntegral_mono_on hfint (hRint.add hSint) hD
intro x hx
exact thinStrip_treated_second_integrand_split_bound d hd gamma a h
hgamma ha hh x hx
_ = R * mu.real D + S * mu.real (D ∩ thinStrip d hd a) := by
rw [integral_add hRint hSint, setIntegral_const, smul_eq_mul,
setIntegral_indicator hstrip, setIntegral_const, smul_eq_mul]
ring
theorem thinStrip_treated_second_sup_power_bound Lemma thinStrip_treated_second_sup_power_bound in the paper ↗
Quantitative numerator bound on the actual sup-norm event. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
gamma a h :
ℝ
hgamma :
1 < gamma
ha :
0 ≤ a
hh :
0 < h
hhhalf :
h ≤ 1 / 2
conclusion
(∫ x in {x : Fin d → ℝ | centreRadius x ≤ h}, ((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 * stripPropensity d hd gamma a x ∂(volume.restrict (cube d)))
≤ ((2 * h) ^ d) ^ ((1 : ℝ) / (gamma - 1)) * (2 * h) ^ d
+ (h ^ a / h) ^ 2 * ((2 * h ^ a) * (2 * h) ^ (d - 1))
Proof (Lean source)
lemma thinStrip_treated_second_sup_power_bound (d : ℕ) (hd : 2 ≤ d)
(gamma a h : ℝ) (hgamma : 1 < gamma) (ha : 0 ≤ a)
(hh : 0 < h) (hhhalf : h ≤ 1 / 2) :
(∫ x in {x : Fin d → ℝ | centreRadius x ≤ h},
((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 * stripPropensity d hd gamma a x
∂(volume.restrict (cube d))) ≤
((2 * h) ^ d) ^ ((1 : ℝ) / (gamma - 1)) * (2 * h) ^ d +
(h ^ a / h) ^ 2 * ((2 * h ^ a) * (2 * h) ^ (d - 1)) := by
have hs := thinStrip_treated_second_sup_setIntegral_split d hd gamma a h
hgamma ha hh
have hmass : (volume.restrict (cube d)).real
{x : Fin d → ℝ | centreRadius x ≤ h} = (2 * h) ^ d := by
have hD : MeasurableSet {x : Fin d → ℝ | centreRadius x ≤ h} :=
measurableSet_le (by unfold centreRadius; fun_prop) measurable_const
have hsub : {x : Fin d → ℝ | centreRadius x ≤ h} ⊆ cube d := by
rw [centreRadius_sublevel_eq_cube d h hh.le]
intro x hx
simp only [Set.mem_Icc, Pi.le_def] at hx
simp only [cube, Set.mem_univ_pi, Set.mem_Icc]
intro i
constructor <;> linarith [(hx.1 i), (hx.2 i)]
rw [measureReal_def, Measure.restrict_apply hD,
Set.inter_eq_self_of_subset_left hsub,
centreRadius_sublevel_volume d h hh.le, ENNReal.toReal_ofReal]
positivity
have hstrip : (volume.restrict (cube d)).real
({x : Fin d → ℝ | centreRadius x ≤ h} ∩ thinStrip d hd a) ≤
(2 * h ^ a) * (2 * h) ^ (d - 1) := by
rw [Set.inter_comm]
calc
(volume.restrict (cube d)).real
(thinStrip d hd a ∩ {x : Fin d → ℝ | centreRadius x ≤ h}) ≤
volume.real (thinStrip d hd a ∩ {x : Fin d → ℝ | centreRadius x ≤ h}) := by
rw [measureReal_def, measureReal_def]
apply ENNReal.toReal_mono
· apply ne_top_of_le_ne_top ENNReal.ofReal_ne_top
rw [← centreRadius_sublevel_volume d h hh.le]
exact measure_mono Set.inter_subset_right
· exact (Measure.restrict_le_self : volume.restrict (cube d) ≤ volume) _
_ ≤ _ := thinStrip_centreRadius_volume_real_upper d hd a h ha hh.le
rw [hmass] at hs
exact hs.trans (add_le_add_right
(mul_le_mul_of_nonneg_left hstrip (sq_nonneg _)) _)
The core-box treated-mass lower bound transfers to the sup-norm event. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
gamma a h :
ℝ
hgamma :
1 < gamma
ha :
1 < a
hh :
0 < h
hh1 :
h ≤ 1
conclusion
(1 / 4 : ℝ) * ((2 * (h / (4 * d)) ^ a) * (h / (4 * d)) ^ (d - 1))
≤ ∫ x in {x : Fin d → ℝ | centreRadius x ≤ h}, stripPropensity d hd gamma a x ∂(volume.restrict (cube d))
Proof (Lean source)
lemma thinStrip_treated_sup_mass_lower (d : ℕ) (hd : 2 ≤ d)
(gamma a h : ℝ) (hgamma : 1 < gamma) (ha : 1 < a)
(hh : 0 < h) (hh1 : h ≤ 1) :
(1 / 4 : ℝ) * ((2 * (h / (4 * d)) ^ a) * (h / (4 * d)) ^ (d - 1)) ≤
∫ x in {x : Fin d → ℝ | centreRadius x ≤ h},
stripPropensity d hd gamma a x ∂(volume.restrict (cube d)) := by
have hlower := thinStrip_treated_ball_mass_lower d hd gamma a h
hgamma ha hh hh1
let D : Set (Fin d → ℝ) := {x | centreRadius x ≤ h}
let B := ball2 (fun _ : Fin d => (1 / 2 : ℝ)) h
let mu := volume.restrict (cube d)
have hD : MeasurableSet D := measurableSet_le
(by unfold centreRadius; fun_prop) measurable_const
have hfin : mu D ≠ ⊤ := by
apply ne_top_of_le_ne_top _ (measure_mono (Set.subset_univ D))
dsimp [mu]
rw [Measure.restrict_apply MeasurableSet.univ]
apply ne_top_of_le_ne_top _ (measure_mono Set.inter_subset_right)
have hcube : cube d = Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hcube, Real.volume_Icc_pi]
simp
have heint : IntegrableOn (stripPropensity d hd gamma a) D mu := by
apply (integrableOn_const (C := (1 : ℝ)) (hs := hfin)).mono'
· exact (stripPropensity_measurable d hd gamma a).aestronglyMeasurable
· filter_upwards [] with x
rw [Real.norm_eq_abs, abs_of_nonneg
(stripPropensity_mem_Icc d hd gamma a hgamma x).1]
exact (stripPropensity_mem_Icc d hd gamma a hgamma x).2
have hBD : B ⊆ D := by
intro x hx
rw [show x ∈ D ↔ centreRadius x ≤ h by rfl]
rw [centreRadius, pi_norm_le_iff_of_nonneg hh.le]
intro i
have hsum : (x i - (1 / 2 : ℝ)) ^ 2 ≤
∑ j : Fin d, (x j - 1 / 2) ^ 2 :=
Finset.single_le_sum (fun j _ => sq_nonneg (x j - 1 / 2))
(Finset.mem_univ _)
dsimp [B, ball2] at hx
rw [Pi.sub_apply, Real.norm_eq_abs]
nlinarith [sq_abs (x i - (1 / 2 : ℝ)), abs_nonneg (x i - (1 / 2 : ℝ))]
exact hlower.trans (setIntegral_mono_set heint
(Filter.Eventually.of_forall (fun x =>
(stripPropensity_mem_Icc d hd gamma a hgamma x).1))
(Filter.Eventually.of_forall (fun x hx => hBD hx)))
A centred radius at most one half stays in the covariate cube. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
r :
ℝ
hr :
0 ≤ r
hrhalf :
r ≤ 1 / 2
conclusion
{x : Fin d → ℝ | centreRadius x ≤ r} ⊆ cube d
Proof (Lean source)
lemma centreRadius_sublevel_subset_cube (d : ℕ) (r : ℝ)
(hr : 0 ≤ r) (hrhalf : r ≤ 1 / 2) :
{x : Fin d → ℝ | centreRadius x ≤ r} ⊆ cube d := by
rw [centreRadius_sublevel_eq_cube d r hr]
intro x hx
simp only [Set.mem_Icc, Pi.le_def] at hx
simp only [cube, Set.mem_univ_pi, Set.mem_Icc]
intro i
constructor <;> linarith [(hx.1 i), (hx.2 i)]
The uniform design assigns the centred radius sublevel its cube volume. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
r :
ℝ
hr :
0 ≤ r
hrhalf :
r ≤ 1 / 2
conclusion
(volume.restrict (cube d)) {x : Fin d → ℝ | centreRadius x ≤ r}
= ofReal ((2 * r) ^ d)
Proof (Lean source)
lemma centreRadius_uniform_sublevel_mass (d : ℕ) (r : ℝ)
(hr : 0 ≤ r) (hrhalf : r ≤ 1 / 2) :
(volume.restrict (cube d)) {x : Fin d → ℝ | centreRadius x ≤ r} =
ofReal ((2 * r) ^ d) := by
have hsub := centreRadius_sublevel_subset_cube d r hr hrhalf
have hmeas : MeasurableSet {x : Fin d → ℝ | centreRadius x ≤ r} :=
measurableSet_le (by unfold centreRadius; fun_prop) measurable_const
rw [Measure.restrict_apply hmeas, Set.inter_eq_self_of_subset_left hsub]
exact centreRadius_sublevel_volume d r hr
On the support of the uniform cube design, the radial CDF has its unsaturated polynomial form. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
Proof (Lean source)
lemma centreRadiusCDF_of_le_half (d : ℕ) (r : ℝ)
(hr : 0 ≤ r) (hrhalf : r ≤ 1 / 2) :
centreRadiusCDF d r = (2 * r) ^ d := by
unfold centreRadiusCDF
apply min_eq_right
have htwo : 0 ≤ 2 * r := by positivity
have hone : 2 * r ≤ 1 := by linarith
exact pow_le_one₀ htwo hone
theorem centreRadiusCDF_uniform_sublevel_mass Lemma centreRadiusCDF_uniform_sublevel_mass in the paper ↗
The radius CDF evaluated at an interior radius has precisely its stated probability under the uniform covariate design. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
r :
ℝ
hr :
0 ≤ r
hrhalf :
r ≤ 1 / 2
conclusion
(volume.restrict (cube d)) {x : Fin d → ℝ | centreRadius x ≤ r}
= ofReal (centreRadiusCDF d r)
Proof (Lean source)
lemma centreRadiusCDF_uniform_sublevel_mass (d : ℕ) (r : ℝ)
(hr : 0 ≤ r) (hrhalf : r ≤ 1 / 2) :
(volume.restrict (cube d)) {x : Fin d → ℝ | centreRadius x ≤ r} =
ofReal (centreRadiusCDF d r) := by
rw [centreRadiusCDF_of_le_half d r hr hrhalf]
exact centreRadius_uniform_sublevel_mass d r hr hrhalf
theorem centreRadiusCDF_uniform_distribution_bound Lemma centreRadiusCDF_uniform_distribution_bound in the paper ↗
The centre-radius CDF is uniform under the cube design, in the sublevel form needed for the thin-strip global tail argument. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
Proof (Lean source)
lemma centreRadiusCDF_uniform_distribution_bound (d : ℕ) (hd : 0 < d)
(u : ℝ) (hu : u ∈ Icc (0 : ℝ) 1) :
(volume.restrict (cube d)).real
{x | centreRadiusCDF d (centreRadius x) ≤ u} ≤ u := by
have hvol : volume (cube d) = 1 := by
have hcube : cube d = Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hcube, Real.volume_Icc_pi]
simp
have : IsFiniteMeasure (volume.restrict (cube d)) := by
rw [isFiniteMeasure_restrict]
simp [hvol]
by_cases hu1 : u = 1
· subst u
have hprob : (volume.restrict (cube d)).real univ = 1 := by
simp [measureReal_def, hvol]
calc
_ ≤ (volume.restrict (cube d)).real univ := measureReal_mono (Set.subset_univ _)
_ = 1 := hprob
· have hult : u < 1 := lt_of_le_of_ne hu.2 hu1
let r : ℝ := u ^ ((d : ℝ)⁻¹) / 2
have hdne : d ≠ 0 := by omega
have hrpow : (2 * r) ^ d = u := by
dsimp [r]
have heq : 2 * (u ^ ((d : ℝ)⁻¹) / 2) = u ^ ((d : ℝ)⁻¹) := by ring
rw [heq]
exact Real.rpow_inv_natCast_pow hu.1 hdne
have hr : 0 ≤ r := by
dsimp [r]
exact div_nonneg (Real.rpow_nonneg hu.1 _) (by norm_num)
have hrhalf : r ≤ 1 / 2 := by
dsimp [r]
have hpow : u ^ ((d : ℝ)⁻¹) ≤ 1 :=
Real.rpow_le_one hu.1 hu.2 (by positivity)
linarith
have hsub : {x : Fin d → ℝ | centreRadiusCDF d (centreRadius x) ≤ u} ⊆
{x : Fin d → ℝ | centreRadius x ≤ r} := by
intro x hx
have hrad : 0 ≤ centreRadius x := norm_nonneg _
have hpow : (2 * centreRadius x) ^ d ≤ u := by
change min 1 ((2 * centreRadius x) ^ d) ≤ u at hx
by_cases hle : (2 * centreRadius x) ^ d ≤ 1
· rw [min_eq_right hle] at hx
exact hx
· have hmin : min 1 ((2 * centreRadius x) ^ d) = 1 :=
min_eq_left (le_of_not_ge hle)
rw [hmin] at hx
linarith
have hle : 2 * centreRadius x ≤ 2 * r :=
le_of_pow_le_pow_left₀ hdne (by positivity) (by simpa only [hrpow] using hpow)
change centreRadius x ≤ r
linarith
calc
_ ≤ (volume.restrict (cube d)).real {x | centreRadius x ≤ r} :=
measureReal_mono hsub
_ = u := by
rw [measureReal_def, centreRadius_uniform_sublevel_mass d r hr hrhalf]
rw [ENNReal.toReal_ofReal (by positivity)]
exact hrpow
The displayed thin-strip propensity obeys the sharp global tail bound under the uniform covariate design. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
(volume.restrict (cube d)).real {x | stripPropensity d hd γ a x ≤ t} ≤ t ^ (γ - 1)
Proof (Lean source)
lemma stripPropensity_uniform_tail_bound (d : ℕ) (hd : 2 ≤ d)
(γ a t : ℝ) (hγ : 1 < γ) (ht : t ∈ Icc (0 : ℝ) 1) :
(volume.restrict (cube d)).real
{x | stripPropensity d hd γ a x ≤ t} ≤ t ^ (γ - 1) := by
have hvol : volume (cube d) = 1 := by
have hcube : cube d = Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hcube, Real.volume_Icc_pi]
simp
have : IsFiniteMeasure (volume.restrict (cube d)) := by
rw [isFiniteMeasure_restrict]
simp [hvol]
have hpow : t ^ (γ - 1) ∈ Icc (0 : ℝ) 1 := by
refine ⟨Real.rpow_nonneg ht.1 _, ?_⟩
exact Real.rpow_le_one ht.1 ht.2 (by linarith)
exact (measureReal_mono (stripTail_subset_radiusCDF_sublevel
d hd γ a t hγ)).trans
(centreRadiusCDF_uniform_distribution_bound d (by omega) _ hpow)
theorem thinStripCompletion_covariate_marginal Lemma thinStripCompletion_covariate_marginal in the paper ↗
The causal thin-strip completion retains the uniform cube covariate law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ L B a :
ℝ
hγ :
1 < γ
hB :
0 < B
hL :
0 < L
conclusion
Proof (Lean source)
lemma thinStripCompletion_covariate_marginal (d : ℕ) (hd : 2 ≤ d)
(γ L B a : ℝ) (hγ : 1 < γ) (hB : 0 < B) (hL : 0 < L) :
(thinStripCompletion d hd γ L B a).map fst =
volume.restrict (cube d) := by
unfold thinStripCompletion
apply completionBind_covariate_marginal
· apply aemeasurable
apply completionAt_measurable_of_measurable
· exact stripPropensity_measurable d hd γ a
· fun_prop
· fun_prop
· intro x
refine ⟨stripPropensity_mem_Icc d hd γ a hγ x, ?_, ?_⟩
· have hb := baselineSuccess_bounds B L hB hL
exact ⟨hb.1.le, hb.2.1.trans (by norm_num)⟩
· have hb := baselineSuccess_bounds B L hB hL
exact ⟨hb.1.le, hb.2.1.trans (by norm_num)⟩
· filter_upwards [] with x
refine ⟨stripPropensity_mem_Icc d hd γ a hγ x, ?_, ?_⟩
· have hb := baselineSuccess_bounds B L hB hL
exact ⟨hb.1.le, hb.2.1.trans (by norm_num)⟩
· have hb := baselineSuccess_bounds B L hB hL
exact ⟨hb.1.le, hb.2.1.trans (by norm_num)⟩
The explicit completion is a probability measure. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
gamma L B a :
ℝ
hgamma :
1 < gamma
hB :
0 < B
hL :
0 < L
conclusion
IsProbabilityMeasure (thinStripCompletion d hd gamma L B a)
Proof (Lean source)
lemma thinStripCompletion_probability (d : ℕ) (hd : 2 ≤ d)
(gamma L B a : ℝ) (hgamma : 1 < gamma) (hB : 0 < B) (hL : 0 < L) :
IsProbabilityMeasure (thinStripCompletion d hd gamma L B a) := by
apply isProbabilityMeasure_iff.mpr
have hm := congrArg (fun mu : Measure (Fin d → ℝ) => mu univ)
(thinStripCompletion_covariate_marginal d hd gamma L B a hgamma hB hL)
rw [Measure.map_apply measurable_fst MeasurableSet.univ] at hm
have hcube : cube d = Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [Measure.restrict_apply MeasurableSet.univ, Set.univ_inter,
hcube, Real.volume_Icc_pi] at hm
simpa using hm
The observed outcome coordinate is definitionally selected from the two potential outcomes in the explicit completion. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
gamma L B a :
ℝ
hgamma :
1 < gamma
hB :
0 < B
hL :
0 < L
conclusion
OutcomeConsistency (thinStripCompletion d hd gamma L B a)
Proof (Lean source)
lemma thinStripCompletion_consistency (d : ℕ) (hd : 2 ≤ d)
(gamma L B a : ℝ) (hgamma : 1 < gamma) (hB : 0 < B) (hL : 0 < L) :
OutcomeConsistency (thinStripCompletion d hd gamma L B a) := by
unfold thinStripCompletion
apply completionBind_consistency
exact (completionAt_measurable_of_measurable B
(stripPropensity d hd gamma a) (fun _ => baselineSuccess B L)
(fun _ => baselineSuccess B L)
(stripPropensity_measurable d hd gamma a) measurable_const measurable_const
(fun x => ⟨stripPropensity_mem_Icc d hd gamma a hgamma x,
⟨(baselineSuccess_bounds B L hB hL).1.le,
(baselineSuccess_bounds B L hB hL).2.1.trans (by norm_num)⟩,
⟨(baselineSuccess_bounds B L hB hL).1.le,
(baselineSuccess_bounds B L hB hL).2.1.trans (by norm_num)⟩⟩)).aemeasurable
The observed thin-strip witness has the uniform cube covariate marginal. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ L B a :
ℝ
ha :
1 < a ∧ a < 1 + d / (γ - 1)
hγ :
1 < γ
hB :
0 < B
hL :
0 < L
conclusion
covariateLaw (thinStripLaw d hd γ L B a ha) = volume.restrict (cube d)
Proof (Lean source)
lemma thinStripLaw_covariateLaw (d : ℕ) (hd : 2 ≤ d)
(γ L B a : ℝ) (ha : 1 < a ∧ a < 1 + d / (γ - 1))
(hγ : 1 < γ) (hB : 0 < B) (hL : 0 < L) :
covariateLaw (thinStripLaw d hd γ L B a ha) =
volume.restrict (cube d) := by
unfold covariateLaw thinStripLaw
have hobs : Measurable (@observed d) := by
unfold observed
fun_prop
rw [Measure.map_map (by fun_prop) hobs]
exact thinStripCompletion_covariate_marginal d hd γ L B a hγ hB hL
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
gamma L B a :
ℝ
ha :
1 < a ∧ a < 1 + d / (gamma - 1)
hgamma :
1 < gamma
hB :
0 < B
hL :
0 < L
conclusion
covariateLaw (thinStripLaw d hd gamma L B a ha) (cube d) = 1
Proof (Lean source)
lemma thinStripLaw_cubeSupport (d : ℕ) (hd : 2 ≤ d)
(gamma L B a : ℝ) (ha : 1 < a ∧ a < 1 + d / (gamma - 1))
(hgamma : 1 < gamma) (hB : 0 < B) (hL : 0 < L) :
covariateLaw (thinStripLaw d hd gamma L B a ha) (cube d) = 1 := by
have hcube : cube d = Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [thinStripLaw_covariateLaw d hd gamma L B a ha hgamma hB hL,
Measure.restrict_apply (hcube.symm ▸ measurableSet_Icc), Set.inter_self]
rw [hcube, Real.volume_Icc_pi]
simp
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
gamma L B a :
ℝ
ha :
1 < a ∧ a < 1 + d / (gamma - 1)
hgamma :
1 < gamma
hB :
0 < B
hL :
0 < L
conclusion
covariateLaw (thinStripLaw d hd gamma L B a ha) ≪ volume.restrict (cube d)
Proof (Lean source)
lemma thinStripLaw_covariateAC (d : ℕ) (hd : 2 ≤ d)
(gamma L B a : ℝ) (ha : 1 < a ∧ a < 1 + d / (gamma - 1))
(hgamma : 1 < gamma) (hB : 0 < B) (hL : 0 < L) :
covariateLaw (thinStripLaw d hd gamma L B a ha) ≪ volume.restrict (cube d) := by
rw [thinStripLaw_covariateLaw d hd gamma L B a ha hgamma hB hL]
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
gamma L B a c_f :
ℝ
ha :
1 < a ∧ a < 1 + d / (gamma - 1)
hgamma :
1 < gamma
hB :
0 < B
hL :
0 < L
hcf :
c_f ≤ 1
conclusion
CovariateDensityLowerBound (thinStripLaw d hd gamma L B a ha) c_f
Proof (Lean source)
lemma thinStripLaw_density_lower (d : ℕ) (hd : 2 ≤ d)
(gamma L B a c_f : ℝ) (ha : 1 < a ∧ a < 1 + d / (gamma - 1))
(hgamma : 1 < gamma) (hB : 0 < B) (hL : 0 < L) (hcf : c_f ≤ 1) :
CovariateDensityLowerBound (thinStripLaw d hd gamma L B a ha) c_f := by
unfold CovariateDensityLowerBound covariateDensity
rw [thinStripLaw_covariateLaw d hd gamma L B a ha hgamma hB hL]
filter_upwards [Measure.rnDeriv_self (volume.restrict (cube d))] with x hx
rw [hx, ENNReal.toReal_one]
exact hcf
The observed thin-strip witness is a probability law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ L B a :
ℝ
ha :
1 < a ∧ a < 1 + d / (γ - 1)
hγ :
1 < γ
hB :
0 < B
hL :
0 < L
conclusion
IsProbabilityMeasure (thinStripLaw d hd γ L B a ha)
Proof (Lean source)
lemma thinStripLaw_probability (d : ℕ) (hd : 2 ≤ d)
(γ L B a : ℝ) (ha : 1 < a ∧ a < 1 + d / (γ - 1))
(hγ : 1 < γ) (hB : 0 < B) (hL : 0 < L) :
IsProbabilityMeasure (thinStripLaw d hd γ L B a ha) := by
apply isProbabilityMeasure_iff.mpr
have hX := thinStripLaw_covariateLaw d hd γ L B a ha hγ hB hL
have hm := congrArg (fun μ : Measure (Fin d → ℝ) => μ univ) hX
rw [covariateLaw, Measure.map_apply measurable_fst MeasurableSet.univ,
Measure.restrict_apply MeasurableSet.univ, Set.univ_inter] at hm
have hcube : cube d =
Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hcube, Real.volume_Icc_pi] at hm
simpa using hm
theorem thinStripLaw_treated_second_integral_identity Lemma thinStripLaw_treated_second_integral_identity in the paper ↗
Disintegration of the frozen treated second-moment numerator onto the uniform covariate law. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
gamma L B a :
ℝ
ha :
1 < a ∧ a < 1 + d / (gamma - 1)
hgamma :
1 < gamma
hB :
0 < B
hL :
0 < L
h :
ℝ
hh :
0 < h
conclusion
(∫ z in {z : Obs d | z.2.1 = true ∧ centreRadius z.1 ≤ h}, ((z.1 ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 ∂thinStripLaw d hd gamma L B a ha)
= ∫ x in {x : Fin d → ℝ | centreRadius x ≤ h}, ((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 * stripPropensity d hd gamma a x ∂(volume.restrict (cube d))
Proof (Lean source)
lemma thinStripLaw_treated_second_integral_identity (d : ℕ) (hd : 2 ≤ d)
(gamma L B a : ℝ) (ha : 1 < a ∧ a < 1 + d / (gamma - 1))
(hgamma : 1 < gamma) (hB : 0 < B) (hL : 0 < L)
(h : ℝ) (hh : 0 < h) :
(∫ z in {z : Obs d | z.2.1 = true ∧ centreRadius z.1 ≤ h},
((z.1 ⟨1, by omega⟩ - 1 / 2) / h) ^ 2
∂thinStripLaw d hd gamma L B a ha) =
∫ x in {x : Fin d → ℝ | centreRadius x ≤ h},
((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 *
stripPropensity d hd gamma a x ∂(volume.restrict (cube d)) := by
let D : Set (Fin d → ℝ) := {x | centreRadius x ≤ h}
let f : (Fin d → ℝ) → ℝ := fun x => ((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2
let p := baselineSuccess B L
have hD : MeasurableSet D := measurableSet_le
(by unfold centreRadius; fun_prop) measurable_const
have hf : Measurable f := by dsimp [f]; fun_prop
have hvalid : ∀ x, stripPropensity d hd gamma a x ∈ Icc 0 1 ∧
p ∈ Icc 0 1 ∧ p ∈ Icc 0 1 := by
intro x
have hb := baselineSuccess_bounds B L hB hL
exact ⟨stripPropensity_mem_Icc d hd gamma a hgamma x,
⟨hb.1.le, hb.2.1.trans (by norm_num)⟩,
⟨hb.1.le, hb.2.1.trans (by norm_num)⟩⟩
let P := thinStripLaw d hd gamma L B a ha
letI : IsProbabilityMeasure P :=
thinStripLaw_probability d hd gamma L B a ha hgamma hB hL
have hleft : Integrable (fun z : Obs d => f z.1)
(P.restrict {z : Obs d | z.2.1 = true ∧ z.1 ∈ D}) := by
apply (integrable_const (1 : ℝ)).mono'
· exact (hf.comp measurable_fst).aestronglyMeasurable
· have hE : MeasurableSet {z : Obs d | z.2.1 = true ∧ z.1 ∈ D} := by
have hA : Measurable (fun z : Obs d => z.2.1) :=
measurable_fst.comp measurable_snd
have hAt : MeasurableSet {z : Obs d | z.2.1 = true} :=
(measurableSet_singleton true).preimage hA
have hXD : MeasurableSet {z : Obs d | z.1 ∈ D} := hD.preimage measurable_fst
exact hAt.inter hXD
filter_upwards [ae_restrict_mem hE] with z hz
have hr := hz.2
have hc := (norm_le_pi_norm
(z.1 - fun _ : Fin d => (1 / 2 : ℝ)) ⟨1, by omega⟩).trans hr
have habs : |z.1 ⟨1, by omega⟩ - 1 / 2| ≤ h := by
simpa [centreRadius, Pi.sub_apply, Real.norm_eq_abs] using hc
have hdiv : |(z.1 ⟨1, by omega⟩ - 1 / 2) / h| ≤ 1 := by
rw [abs_div, abs_of_pos hh]
exact (div_le_iff₀ hh).2 (by simpa using habs)
dsimp [f]
rw [abs_of_nonneg (sq_nonneg _)]
simpa only [sq_abs, one_pow] using
(pow_le_pow_left₀ (abs_nonneg _) hdiv 2)
have hright : Integrable (fun x => f x * stripPropensity d hd gamma a x)
((volume.restrict (cube d)).restrict D) := by
have hfin : (volume.restrict (cube d)) D ≠ ⊤ := by
apply ne_top_of_le_ne_top _ (measure_mono (Set.subset_univ D))
have hcube : volume (cube d) = 1 := by
have hc : cube d = Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hc, Real.volume_Icc_pi]
simp
rw [Measure.restrict_apply MeasurableSet.univ, Set.univ_inter, hcube]
exact ENNReal.one_ne_top
apply (integrableOn_const (C := (1 : ℝ)) (hs := hfin)).mono'
· exact (hf.mul (stripPropensity_measurable d hd gamma a)).aestronglyMeasurable
· filter_upwards [ae_restrict_mem hD] with x hx
have hc := (norm_le_pi_norm
(x - fun _ : Fin d => (1 / 2 : ℝ)) ⟨1, by omega⟩).trans hx
have habs : |x ⟨1, by omega⟩ - 1 / 2| ≤ h := by
simpa [centreRadius, Pi.sub_apply, Real.norm_eq_abs] using hc
have hdiv : |(x ⟨1, by omega⟩ - 1 / 2) / h| ≤ 1 := by
rw [abs_div, abs_of_pos hh]
exact (div_le_iff₀ hh).2 (by simpa using habs)
have hsq : f x ≤ 1 := by
dsimp [f]
simpa only [sq_abs, one_pow] using
(pow_le_pow_left₀ (abs_nonneg _) hdiv 2)
have he := stripPropensity_mem_Icc d hd gamma a hgamma x
rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg (sq_nonneg _) he.1)]
exact (mul_le_mul_of_nonneg_right hsq he.1).trans (by simpa using he.2)
change (∫ z in {z : Obs d | z.2.1 = true ∧ z.1 ∈ D}, f z.1 ∂P) = _
unfold P thinStripLaw thinStripCompletion
exact completionBind_observed_treated_setIntegral
(volume.restrict (cube d)) B (stripPropensity d hd gamma a)
(fun _ => p) (fun _ => p) f
(stripPropensity_measurable d hd gamma a) measurable_const measurable_const hf
(fun x => sq_nonneg _) hvalid D hD hleft hright
Disintegration of the denominator of stripSecondMoment. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
gamma L B a :
ℝ
ha :
1 < a ∧ a < 1 + d / (gamma - 1)
hgamma :
1 < gamma
hB :
0 < B
hL :
0 < L
h :
ℝ
conclusion
= ∫ x in {x : Fin d → ℝ | centreRadius x ≤ h}, stripPropensity d hd gamma a x ∂(volume.restrict (cube d))
Proof (Lean source)
lemma thinStripLaw_treated_mass_identity (d : ℕ) (hd : 2 ≤ d)
(gamma L B a : ℝ) (ha : 1 < a ∧ a < 1 + d / (gamma - 1))
(hgamma : 1 < gamma) (hB : 0 < B) (hL : 0 < L) (h : ℝ) :
(thinStripLaw d hd gamma L B a ha).real
{z : Obs d | z.2.1 = true ∧ centreRadius z.1 ≤ h} =
∫ x in {x : Fin d → ℝ | centreRadius x ≤ h},
stripPropensity d hd gamma a x ∂(volume.restrict (cube d)) := by
let D : Set (Fin d → ℝ) := {x | centreRadius x ≤ h}
let p := baselineSuccess B L
have hD : MeasurableSet D := measurableSet_le
(by unfold centreRadius; fun_prop) measurable_const
have hvalid : ∀ x, stripPropensity d hd gamma a x ∈ Icc 0 1 ∧
p ∈ Icc 0 1 ∧ p ∈ Icc 0 1 := by
intro x
have hb := baselineSuccess_bounds B L hB hL
exact ⟨stripPropensity_mem_Icc d hd gamma a hgamma x,
⟨hb.1.le, hb.2.1.trans (by norm_num)⟩,
⟨hb.1.le, hb.2.1.trans (by norm_num)⟩⟩
let P := thinStripLaw d hd gamma L B a ha
letI : IsProbabilityMeasure P :=
thinStripLaw_probability d hd gamma L B a ha hgamma hB hL
have hE : MeasurableSet {z : Obs d | z.2.1 = true ∧ z.1 ∈ D} := by
have hA : Measurable (fun z : Obs d => z.2.1) := measurable_fst.comp measurable_snd
exact ((measurableSet_singleton true).preimage hA).inter
(hD.preimage measurable_fst)
have hleft : Integrable (fun _z : Obs d => (1 : ℝ))
(P.restrict {z : Obs d | z.2.1 = true ∧ z.1 ∈ D}) := integrable_const _
have hright : Integrable (fun x => (1 : ℝ) * stripPropensity d hd gamma a x)
((volume.restrict (cube d)).restrict D) := by
have hfin : (volume.restrict (cube d)) D ≠ ⊤ := by
apply ne_top_of_le_ne_top _ (measure_mono (Set.subset_univ D))
have hcube : volume (cube d) = 1 := by
have hc : cube d = Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hc, Real.volume_Icc_pi]
simp
rw [Measure.restrict_apply MeasurableSet.univ, Set.univ_inter, hcube]
exact ENNReal.one_ne_top
apply (integrableOn_const (C := (1 : ℝ)) (hs := hfin)).mono'
· simpa only [one_mul] using
(stripPropensity_measurable d hd gamma a).aestronglyMeasurable
· filter_upwards [] with x
rw [one_mul, Real.norm_eq_abs, abs_of_nonneg
(stripPropensity_mem_Icc d hd gamma a hgamma x).1]
exact (stripPropensity_mem_Icc d hd gamma a hgamma x).2
have hid := completionBind_observed_treated_setIntegral
(volume.restrict (cube d)) B (stripPropensity d hd gamma a)
(fun _ => p) (fun _ => p) (fun _ => (1 : ℝ))
(stripPropensity_measurable d hd gamma a) measurable_const measurable_const
measurable_const (fun _ => zero_le_one) hvalid D hD hleft hright
change P.real {z : Obs d | z.2.1 = true ∧ z.1 ∈ D} = _
calc
P.real {z : Obs d | z.2.1 = true ∧ z.1 ∈ D} =
∫ _z in {z : Obs d | z.2.1 = true ∧ z.1 ∈ D}, (1 : ℝ) ∂P := by
rw [setIntegral_const, smul_eq_mul, mul_one]
_ = _ := by
unfold P thinStripLaw thinStripCompletion
simpa only [one_mul] using hid
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
a h :
ℝ
hh :
0 < h
conclusion
(1 / 4 : ℝ) * (2 * (h / (4 * d)) ^ a) * (h / (4 * d)) ^ (d - 1)
= ((1 / 2 : ℝ) / (((4 * d : ℝ) ^ a) * (4 * d) ^ (d - 1))) * h ^ (a + (d - 1 : ℕ))
Proof (Lean source)
lemma thinStrip_denominator_power_factor (d : ℕ) (hd : 2 ≤ d)
(a h : ℝ) (hh : 0 < h) :
(1 / 4 : ℝ) * (2 * (h / (4 * d)) ^ a) * (h / (4 * d)) ^ (d - 1) =
((1 / 2 : ℝ) / (((4 * d : ℝ) ^ a) * (4 * d) ^ (d - 1))) *
h ^ (a + (d - 1 : ℕ)) := by
have hk : (0 : ℝ) < 4 * d := by positivity
rw [Real.div_rpow hh.le hk.le, div_pow]
rw [← Real.rpow_natCast h (d - 1), Real.rpow_add hh]
field_simp <;> ring
theorem thinStrip_radial_numerator_power_factor Lemma thinStrip_radial_numerator_power_factor in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
gamma h :
ℝ
hh :
0 < h
conclusion
(((2 * h) ^ d) ^ ((1 : ℝ) / (gamma - 1))) * (2 * h) ^ d
= (((2 : ℝ) ^ d) ^ ((1 : ℝ) / (gamma - 1)) * (2 : ℝ) ^ d) * h ^ ((d : ℝ) / (gamma - 1) + d)
Proof (Lean source)
lemma thinStrip_radial_numerator_power_factor (d : ℕ) (gamma h : ℝ)
(hh : 0 < h) :
(((2 * h) ^ d) ^ ((1 : ℝ) / (gamma - 1))) * (2 * h) ^ d =
(((2 : ℝ) ^ d) ^ ((1 : ℝ) / (gamma - 1)) * (2 : ℝ) ^ d) *
h ^ ((d : ℝ) / (gamma - 1) + d) := by
rw [mul_pow, Real.mul_rpow (pow_nonneg (by norm_num) _) (pow_nonneg hh.le _)]
rw [← Real.rpow_natCast h d, ← Real.rpow_mul hh.le]
rw [Real.rpow_add hh]
ring_nf
theorem thinStrip_strip_numerator_power_factor Lemma thinStrip_strip_numerator_power_factor in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
a h :
ℝ
hh :
0 < h
conclusion
(h ^ a / h) ^ 2 * ((2 * h ^ a) * (2 * h) ^ (d - 1))
= ((2 : ℝ) * (2 : ℝ) ^ (d - 1)) * h ^ (3 * a + d - 3)
Proof (Lean source)
lemma thinStrip_strip_numerator_power_factor (d : ℕ) (hd : 2 ≤ d)
(a h : ℝ) (hh : 0 < h) :
(h ^ a / h) ^ 2 * ((2 * h ^ a) * (2 * h) ^ (d - 1)) =
((2 : ℝ) * (2 : ℝ) ^ (d - 1)) * h ^ (3 * a + d - 3) := by
rw [← Real.rpow_sub_one (ne_of_gt hh) a]
rw [mul_pow]
rw [← Real.rpow_natCast (h ^ (a - 1)) 2, ← Real.rpow_mul hh.le]
rw [← Real.rpow_natCast h (d - 1)]
ring_nf
rw [← Real.rpow_add hh, ← Real.rpow_add hh]
congr 2
push_cast [Nat.cast_sub (by omega : 1 ≤ d)]
ring
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
c b u v h :
ℝ
hb :
0 < b
hh :
0 < h
conclusion
(c * h ^ u) / (b * h ^ v) = (c / b) * h ^ (u - v)
Proof (Lean source)
lemma positive_rpow_quotient_factor (c b u v h : ℝ)
(hb : 0 < b) (hh : 0 < h) :
(c * h ^ u) / (b * h ^ v) = (c / b) * h ^ (u - v) := by
rw [Real.rpow_sub hh]
field_simp
theorem thinStrip_raw_second_envelope_tendsto_zero Lemma thinStrip_raw_second_envelope_tendsto_zero in the paper ↗
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
gamma a :
ℝ
hgamma :
1 < gamma
ha :
1 < a ∧ a < 1 + d / (gamma - 1)
conclusion
Tendsto (fun h : ℝ => ((((2 * h) ^ d) ^ ((1 : ℝ) / (gamma - 1))) * (2 * h) ^ d + (h ^ a / h) ^ 2 * ((2 * h ^ a) * (2 * h) ^ (d - 1))) / ((1 / 4 : ℝ) * (2 * (h / (4 * d)) ^ a) * (h / (4 * d)) ^ (d - 1))) (nhdsWithin 0 (Ioi 0)) (nhds 0)
Proof (Lean source)
lemma thinStrip_raw_second_envelope_tendsto_zero (d : ℕ) (hd : 2 ≤ d)
(gamma a : ℝ) (hgamma : 1 < gamma)
(ha : 1 < a ∧ a < 1 + d / (gamma - 1)) :
Tendsto
(fun h : ℝ =>
((((2 * h) ^ d) ^ ((1 : ℝ) / (gamma - 1))) * (2 * h) ^ d +
(h ^ a / h) ^ 2 * ((2 * h ^ a) * (2 * h) ^ (d - 1))) /
((1 / 4 : ℝ) * (2 * (h / (4 * d)) ^ a) *
(h / (4 * d)) ^ (d - 1)))
(nhdsWithin 0 (Ioi 0)) (nhds 0) := by
let b : ℝ := (1 / 2 : ℝ) /
(((4 * d : ℝ) ^ a) * (4 * d) ^ (d - 1))
let c₁ : ℝ := ((2 : ℝ) ^ d) ^ ((1 : ℝ) / (gamma - 1)) * (2 : ℝ) ^ d
let c₂ : ℝ := (2 : ℝ) * (2 : ℝ) ^ (d - 1)
let p : ℝ := 1 + d / (gamma - 1) - a
let q : ℝ := 2 * a - 2
have hb : 0 < b := by dsimp [b]; positivity
have hp : 0 < p := by dsimp [p]; linarith [ha.2]
have hq : 0 < q := by dsimp [q]; linarith [ha.1]
have hpow (r : ℝ) (hr : 0 < r) : Tendsto (fun h : ℝ => h ^ r)
(nhdsWithin 0 (Ioi 0)) (nhds 0) := by
have hid : Tendsto (fun h : ℝ => h)
(nhdsWithin 0 (Ioi 0)) (nhds 0) :=
continuousAt_id.tendsto.mono_left nhdsWithin_le_nhds
simpa [Real.zero_rpow (ne_of_gt hr)] using hid.rpow_const (inr hr.le)
have hlim : Tendsto
(fun h : ℝ => (c₁ / b) * h ^ p + (c₂ / b) * h ^ q)
(nhdsWithin 0 (Ioi 0)) (nhds 0) := by
simpa using (tendsto_const_nhds.mul (hpow p hp)).add
(tendsto_const_nhds.mul (hpow q hq))
apply hlim.congr'
filter_upwards [self_mem_nhdsWithin] with h hh
have hhpos : 0 < h := hh
rw [thinStrip_radial_numerator_power_factor d gamma h hhpos,
thinStrip_strip_numerator_power_factor d hd a h hhpos,
thinStrip_denominator_power_factor d hd a h hhpos, add_div,
positive_rpow_quotient_factor c₁ b _ _ h hb hhpos,
positive_rpow_quotient_factor c₂ b _ _ h hb hhpos]
dsimp [c₁, c₂, p, q]
congr 1
· congr 1
push_cast [Nat.cast_sub (by omega : 1 ≤ d)]
ring
· congr 1
push_cast [Nat.cast_sub (by omega : 1 ≤ d)]
ring
theorem thinStripLaw_not_dornA3AntiConcentration
The uniform thin-strip witness violates Dorn's numerical local mass condition. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ L B a :
ℝ
ha :
1 < a ∧ a < 1 + d / (γ - 1)
hγ :
1 < γ
hB :
0 < B
hL :
0 < L
conclusion
Proof (Lean source)
lemma thinStripLaw_not_dornA3AntiConcentration (d : ℕ) (hd : 2 ≤ d)
(γ L B a : ℝ) (ha : 1 < a ∧ a < 1 + d / (γ - 1))
(hγ : 1 < γ) (hB : 0 < B) (hL : 0 < L) :
¬ DornA3AntiConcentrationOnCube (thinStripLaw d hd γ L B a ha)
(stripPropensity d hd γ a) := by
let P := thinStripLaw d hd γ L B a ha
letI : IsProbabilityMeasure P :=
thinStripLaw_probability d hd γ L B a ha hγ hB hL
apply strip_not_dornA3AntiConcentration_of_local_strip_ratio
d hd γ a hγ (by linarith) P
intro ρ ν h₀ hρ hν hh₀
have hX : covariateLaw P = volume.restrict (cube d) :=
thinStripLaw_covariateLaw d hd γ L B a ha hγ hB hL
simpa only [hX] using
uniformCube_thinStrip_local_ratio d hd γ a hγ ha.1 ρ ν h₀ hρ hν hh₀
The uniform thin-strip witness violates full Dorn A3 because it violates the numerical local mass condition. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ L B a :
ℝ
ha :
1 < a ∧ a < 1 + d / (γ - 1)
hγ :
1 < γ
hB :
0 < B
hL :
0 < L
conclusion
Proof (Lean source)
lemma thinStripLaw_not_dornA3 (d : ℕ) (hd : 2 ≤ d)
(γ L B a : ℝ) (ha : 1 < a ∧ a < 1 + d / (γ - 1))
(hγ : 1 < γ) (hB : 0 < B) (hL : 0 < L) :
¬ DornA3OnCube (thinStripLaw d hd γ L B a ha)
(stripPropensity d hd γ a) := by
intro hA3
exact thinStripLaw_not_dornA3AntiConcentration d hd γ L B a ha hγ hB hL
(dornA3AntiConcentrationOnCube_of_dornA3OnCube
(thinStripLaw d hd γ L B a ha) (stripPropensity d hd γ a) hA3)
theorem stripPropensity_tail_of_uniform_covariateLaw Lemma stripPropensity_tail_of_uniform_covariateLaw in the paper ↗
Transfer the sharp strip tail bound from uniform cube volume to any observed law with that covariate marginal. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
P :
Measure (Obs d)
γ a :
ℝ
hγ :
1 < γ
hX :
covariateLaw P = volume.restrict (cube d)
t :
ℝ
ht :
t ∈ Icc (0 : ℝ) 1
conclusion
Proof (Lean source)
lemma stripPropensity_tail_of_uniform_covariateLaw (d : ℕ) (hd : 2 ≤ d)
(P : Measure (Obs d)) (γ a : ℝ) (hγ : 1 < γ)
(hX : covariateLaw P = volume.restrict (cube d))
(t : ℝ) (ht : t ∈ Icc (0 : ℝ) 1) :
(covariateLaw P).real {x | stripPropensity d hd γ a x ≤ t} ≤
t ^ (γ - 1) := by
rw [hX]
exact stripPropensity_uniform_tail_bound d hd γ a t hγ ht
The selected strip propensity satisfies the numerical tail inequality for every admissible model coefficient. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
gamma L B a C :
ℝ
ha :
1 < a ∧ a < 1 + d / (gamma - 1)
hgamma :
1 < gamma
hB :
0 < B
hL :
0 < L
hC :
1 ≤ C
t ∈ Set.Icc (0 : ℝ) 1 :
conclusion
≤ C * t ^ (gamma - 1)
Proof (Lean source)
lemma thinStripLaw_selected_tail_bound (d : ℕ) (hd : 2 ≤ d)
(gamma L B a C : ℝ) (ha : 1 < a ∧ a < 1 + d / (gamma - 1))
(hgamma : 1 < gamma) (hB : 0 < B) (hL : 0 < L) (hC : 1 ≤ C) :
∀ t ∈ Icc (0 : ℝ) 1,
(covariateLaw (thinStripLaw d hd gamma L B a ha)).real
{x | stripPropensity d hd gamma a x ≤ t} ≤ C * t ^ (gamma - 1) := by
intro t ht
have htail := stripPropensity_tail_of_uniform_covariateLaw d hd
(thinStripLaw d hd gamma L B a ha) gamma a hgamma
(thinStripLaw_covariateLaw d hd gamma L B a ha hgamma hB hL) t ht
exact htail.trans (by
have hp : 0 ≤ t ^ (gamma - 1) := Real.rpow_nonneg ht.1 _
nlinarith)
Under the uniform design, the displayed propensity vanishes only on a null set, as follows from its sharp lower-tail bound. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a :
ℝ
hγ :
1 < γ
conclusion
(volume.restrict (cube d)) {x | stripPropensity d hd γ a x = 0} = 0
Proof (Lean source)
lemma stripPropensity_uniform_zero_null (d : ℕ) (hd : 2 ≤ d)
(γ a : ℝ) (hγ : 1 < γ) :
(volume.restrict (cube d)) {x | stripPropensity d hd γ a x = 0} = 0 := by
let μ := volume.restrict (cube d)
have hvol : volume (cube d) = 1 := by
have hcube : cube d = Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hcube, Real.volume_Icc_pi]
simp
haveI : IsFiniteMeasure μ := by
dsimp [μ]
rw [isFiniteMeasure_restrict]
simp [hvol]
have htail := stripPropensity_uniform_tail_bound d hd γ a 0 hγ
(by simp : (0 : ℝ) ∈ Icc 0 1)
rw [Real.zero_rpow (ne_of_gt (sub_pos.mpr hγ))] at htail
have hreal : μ.real {x | stripPropensity d hd γ a x = 0} = 0 := by
apply le_antisymm _ (ENNReal.toReal_nonneg)
exact (measureReal_mono (by intro x hx; exact le_of_eq hx)).trans htail
have hfin : μ {x | stripPropensity d hd γ a x = 0} ≠ ⊤ := measure_ne_top _ _
exact (ENNReal.toReal_eq_zero_iff _).mp hreal |>.resolve_right hfin
The sharp tail bound gives almost-everywhere positivity under uniform design. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
γ a :
ℝ
hγ :
1 < γ
conclusion
∀ᵐ x ∂volume.restrict (cube d), 0 < stripPropensity d hd γ a x
Proof (Lean source)
lemma stripPropensity_uniform_pos_ae (d : ℕ) (hd : 2 ≤ d)
(γ a : ℝ) (hγ : 1 < γ) :
∀ᵐ x ∂volume.restrict (cube d), 0 < stripPropensity d hd γ a x := by
have hzero := stripPropensity_uniform_zero_null d hd γ a hγ
have hne : ∀ᵐ x ∂volume.restrict (cube d),
stripPropensity d hd γ a x ≠ 0 :=
measure_eq_zero_iff_ae_notMem.mp hzero
filter_upwards [hne] with x hx
exact lt_of_le_of_ne (stripPropensity_mem_Icc d hd γ a hγ x).1 (Ne.symm hx)
theorem completionAt_covariate_treatedPotential_map Lemma completionAt_covariate_treatedPotential_map in the paper ↗
At fixed covariates, the covariate and treated potential outcome have the product law determined by the treated Bernoulli arm. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
B :
ℝ
e p₀ p₁ :
(Fin d → ℝ) → ℝ
x :
Fin d → ℝ
he :
e x ∈ Icc 0 1
hp₀ :
p₀ x ∈ Icc 0 1
hp₁ :
p₁ x ∈ Icc 0 1
conclusion
(completionAt B e p₀ p₁ x).map (fun ω => (ω.1, ω.2.2.2.1))
= (Measure.dirac x).prod (scaledBernoulli B (p₁ x))
Proof (Lean source)
lemma completionAt_covariate_treatedPotential_map {d : ℕ} (B : ℝ)
(e p₀ p₁ : (Fin d → ℝ) → ℝ) (x : Fin d → ℝ)
(he : e x ∈ Icc 0 1) (hp₀ : p₀ x ∈ Icc 0 1)
(hp₁ : p₁ x ∈ Icc 0 1) :
(completionAt B e p₀ p₁ x).map (fun ω => (ω.1, ω.2.2.2.1)) =
(Measure.dirac x).prod (scaledBernoulli B (p₁ x)) := by
letI := realBernoulli_probability (e x) he
letI := scaledBernoulli_probability B (p₀ x) hp₀
letI := scaledBernoulli_probability B (p₁ x) hp₁
unfold completionAt
rw [Measure.map_map (by fun_prop) (by
have hlast : Measurable (fun z : Bool × ℝ × ℝ =>
if z.1 then z.2.2 else z.2.1) :=
Measurable.ite (measurable_fst (MeasurableSet.singleton true))
(by fun_prop) (by fun_prop)
fun_prop (disch := assumption))]
rw [Measure.dirac_prod]
calc
Measure.map (fun z : Bool × (ℝ × ℝ) => (x, z.2.2))
((realBernoulli (e x)).prod
((scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x)))) =
Measure.map (Prod.mk x) (Measure.map (fun z : Bool × (ℝ × ℝ) => z.2.2)
((realBernoulli (e x)).prod
((scaledBernoulli B (p₀ x)).prod (scaledBernoulli B (p₁ x))))) := by
rw [Measure.map_map (by fun_prop) (by fun_prop)]
rfl
_ = _ := by
rw [show (fun z : Bool × (ℝ × ℝ) => z.2.2) = snd ∘ snd by rfl]
rw [← Measure.map_map (by fun_prop) (by fun_prop), Measure.map_snd_prod,
measure_univ, one_smul, Measure.map_snd_prod, measure_univ, one_smul]
theorem completionBind_covariate_treatedPotential_compProd Lemma completionBind_covariate_treatedPotential_compProd in the paper ↗
Mixing the fixed-covariate identity identifies the joint covariate and treated-potential-outcome law as a kernel product. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
conclusion
k₁ :
ProbabilityTheory.Kernel.mk (fun x => scaledBernoulli B (p₁ x)) (by unfold scaledBernoulli realBernoulli; fun_prop) (: Kernel (Fin d → ℝ) ℝ)
(mu.bind (completionAt B e p₀ p₁)).map (fun ω => (ω.1, ω.2.2.2.1))
= Measure.compProd mu k₁
Proof (Lean source)
lemma completionBind_covariate_treatedPotential_compProd {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (B : ℝ)
(e p₀ p₁ : (Fin d → ℝ) → ℝ)
(he : Measurable e) (hp₀ : Measurable p₀) (hp₁ : Measurable p₁)
(hvalid : ∀ x, e x ∈ Icc 0 1 ∧ p₀ x ∈ Icc 0 1 ∧
p₁ x ∈ Icc 0 1) :
let k₁ : Kernel (Fin d → ℝ) ℝ :=
ProbabilityTheory.Kernel.mk (fun x => scaledBernoulli B (p₁ x))
(by unfold scaledBernoulli realBernoulli; fun_prop)
(mu.bind (completionAt B e p₀ p₁)).map (fun ω => (ω.1, ω.2.2.2.1)) =
Measure.compProd mu k₁ := by
dsimp
let k₁ : Kernel (Fin d → ℝ) ℝ :=
ProbabilityTheory.Kernel.mk (fun x => scaledBernoulli B (p₁ x))
(by unfold scaledBernoulli realBernoulli; fun_prop)
letI : IsMarkovKernel k₁ :=
⟨fun x => scaledBernoulli_probability B (p₁ x) (hvalid x).2.2⟩
have hk : Measurable (completionAt B e p₀ p₁) :=
completionAt_measurable_of_measurable B e p₀ p₁ he hp₀ hp₁ hvalid
apply Measure.ext
intro s hs
rw [Measure.map_apply (by fun_prop) hs,
Measure.bind_apply (hs.preimage (by fun_prop)) hk.aemeasurable,
Measure.compProd_apply hs]
apply lintegral_congr
intro x
letI := scaledBernoulli_probability B (p₁ x) (hvalid x).2.2
have hfix := completionAt_covariate_treatedPotential_map B e p₀ p₁ x
(hvalid x).1 (hvalid x).2.1 (hvalid x).2.2
have happ := congrArg (fun m : Measure ((Fin d → ℝ) × ℝ) => m s) hfix
rw [Measure.map_apply (by fun_prop) hs, Measure.prod_apply hs,
lintegral_dirac] at happ
simpa [k₁] using happ
A constant response whose magnitude is at most the radius belongs to the paper's intrinsic Hölder ball. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
β L c :
ℝ
hβ :
0 < β
hL :
0 ≤ L
hc :
|c| ≤ L
conclusion
HolderBall (d := d) β L (fun _ => c)
Proof (Lean source)
lemma constant_holderBall {d : ℕ} (β L c : ℝ) (hβ : 0 < β)
(hL : 0 ≤ L) (hc : |c| ≤ L) :
HolderBall (d := d) β L (fun _ => c) := by
open Causalean.Mathlib.Analysis.Calculus.CubeExtension in
obtain ⟨A, hA, hscaled⟩ :=
exists_uniform_scaledProductBump_holderBallOn (d := d) β hβ
open Causalean.Mathlib.Analysis.Calculus.CubeExtension in
have hconst := holderBallOn_unitCube_add_const_mul hA.le
(hscaled 1 zero_lt_one le_rfl (fun _ => 0)) c 0
have hbase :
HolderBallOn
(cube d) (polynomialDegree β) (β - (polynomialDegree β : ℝ)) |c|
(fun _ => c) := by
simpa [polynomialDegree, cube,
unitCube] using hconst
refine ⟨hbase.regularity, ?_, ?_⟩
· intro j hj f x hx
exact (hbase.derivBound j hj f x hx).trans hc
· intro f x hx y hy
exact (hbase.modulus f x hx y hy).trans
(mul_le_mul_of_nonneg_right hc (Real.rpow_nonneg (norm_nonneg _) _))
theorem completionBind_holderResponse_constant_equal_arms Lemma completionBind_holderResponse_constant_equal_arms in the paper ↗
In an equal-arm completion with a constant Bernoulli success probability, the constant scaled mean is a Hölder representative of the treated potential outcome conditional mean. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
B p β L :
ℝ
e :
(Fin d → ℝ) → ℝ
he :
IsFiniteMeasure (mu.bind (completionAt B e (fun _ => p) (fun _ => p)))
hβ :
0 < β
hL :
0 ≤ L
hmean :
|B * p| ≤ L
conclusion
HolderResponse (mu.bind (completionAt B e (fun _ => p) (fun _ => p))) (fun _ => B * p) β L
Proof (Lean source)
lemma completionBind_holderResponse_constant_equal_arms {d : ℕ}
(mu : Measure (Fin d → ℝ)) [SFinite mu] (B p β L : ℝ)
(e : (Fin d → ℝ) → ℝ) (he : Measurable e)
(hvalid : ∀ x, e x ∈ Icc 0 1 ∧ p ∈ Icc 0 1)
[IsFiniteMeasure (mu.bind (completionAt B e (fun _ => p) (fun _ => p)))]
(hβ : 0 < β) (hL : 0 ≤ L) (hmean : |B * p| ≤ L) :
HolderResponse (mu.bind (completionAt B e (fun _ => p) (fun _ => p)))
(fun _ => B * p) β L := by
let Pc := mu.bind (completionAt B e (fun _ => p) (fun _ => p))
let X : Completion d → (Fin d → ℝ) := fun ω => ω.1
let Y₁ : Completion d → ℝ := fun ω => ω.2.2.2.1
let k₁ : Kernel (Fin d → ℝ) ℝ :=
ProbabilityTheory.Kernel.mk (fun _ => scaledBernoulli B p)
(by fun_prop)
letI : IsMarkovKernel k₁ :=
⟨fun _ => scaledBernoulli_probability B p (hvalid (fun _ => 0)).2⟩
have hcomp : Measurable (completionAt B e (fun _ => p) (fun _ => p)) :=
completionAt_measurable_of_measurable B e (fun _ => p) (fun _ => p)
he measurable_const measurable_const
(fun x => ⟨(hvalid x).1, (hvalid x).2, (hvalid x).2⟩)
have hX : Measurable X := by dsimp [X]; fun_prop
have hY₁ : Measurable Y₁ := by dsimp [Y₁]; fun_prop
have hcov : Pc.map X = mu := by
dsimp [Pc, X]
exact completionBind_covariate_marginal mu B e (fun _ => p) (fun _ => p)
hcomp.aemeasurable
(Filter.Eventually.of_forall (fun x =>
⟨(hvalid x).1, (hvalid x).2, (hvalid x).2⟩))
have hjoint : Pc.map (fun ω => (X ω, Y₁ ω)) = Measure.compProd mu k₁ := by
dsimp [Pc, X, Y₁, k₁]
exact completionBind_covariate_treatedPotential_compProd mu B e
(fun _ => p) (fun _ => p) he measurable_const measurable_const
(fun x => ⟨(hvalid x).1, (hvalid x).2, (hvalid x).2⟩)
have hcond : condDistrib Y₁ X Pc =ᵐ[mu] k₁ := by
have h := ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd_of_measurable
(μ := Pc) (X := X) (Y := Y₁) (κ := k₁) hX hY₁
(by simpa [hcov] using hjoint)
simpa [hcov] using h
have hY₁int : Integrable Y₁ Pc := by
apply Integrable.mono' (integrable_const |B|) hY₁.aestronglyMeasurable
filter_upwards [completionBind_supported_on_endpoints mu B e
(fun _ => p) (fun _ => p)
(Filter.Eventually.of_forall (fun x =>
⟨(hvalid x).1, (hvalid x).2, (hvalid x).2⟩))] with ω hω
rcases hω.2 with hzero | hBval
· simp [Y₁, hzero]
· dsimp [Y₁]
rw [Set.mem_singleton_iff.mp hBval]
have hce : Pc[Y₁ | comap X inferInstance] =ᵐ[Pc]
fun ω => ∫ y : ℝ, y ∂ProbabilityTheory.condDistrib Y₁ X Pc (X ω) := by
simpa using ProbabilityTheory.condExp_ae_eq_integral_condDistrib' hX hY₁int
have hkernel_comp : ∀ᵐ ω ∂Pc,
condDistrib Y₁ X Pc (X ω) = k₁ (X ω) := by
have hcond' : condDistrib Y₁ X Pc =ᵐ[Pc.map X] k₁ := by
rw [hcov]
exact hcond
exact ae_eq_comp hX.aemeasurable hcond'
have hmean_cond : (fun ω : Completion d =>
∫ y : ℝ, y ∂ProbabilityTheory.condDistrib Y₁ X Pc (X ω)) =ᵐ[Pc]
fun _ => B * p := by
filter_upwards [hkernel_comp] with ω hω
rw [hω]
exact scaledBernoulli_integral_id B p (hvalid (X ω)).2
refine ⟨constant_holderBall β L (B * p) hβ hL hmean, ?_⟩
exact hmean_cond.symm.trans hce.symm
theorem thinStrip_globalTailModel_of_holder Lemma thinStrip_globalTailModel_of_holder in the paper ↗
All fields of the thin-strip global-tail model other than the explicitly supplied Hölder response follow from the completion construction. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
β B L C c_f γ a :
ℝ
ha :
1 < a ∧ a < 1 + d / (γ - 1)
hγ :
1 < γ
hB :
0 < B
hL :
0 < L
hC :
1 ≤ C
hcf :
c_f ≤ 1
μ₁ :
(Fin d → ℝ) → ℝ
hholder :
HolderResponse (thinStripCompletion d hd γ L B a) μ₁ β L
conclusion
@GlobalTailModel d β B L C c_f γ (thinStripCompletion d hd γ L B a) (thinStripCompletion_probability d hd γ L B a hγ hB hL) μ₁ (stripPropensity d hd γ a)
Proof (Lean source)
lemma thinStrip_globalTailModel_of_holder (d : ℕ) (hd : 2 ≤ d)
(β B L C c_f γ a : ℝ) (ha : 1 < a ∧ a < 1 + d / (γ - 1))
(hγ : 1 < γ) (hB : 0 < B) (hL : 0 < L)
(hC : 1 ≤ C) (hcf : c_f ≤ 1)
(μ₁ : (Fin d → ℝ) → ℝ)
(hholder : HolderResponse (thinStripCompletion d hd γ L B a) μ₁ β L) :
@GlobalTailModel d β B L C c_f γ
(thinStripCompletion d hd γ L B a)
(thinStripCompletion_probability d hd γ L B a hγ hB hL)
μ₁ (stripPropensity d hd γ a) := by
let mu := volume.restrict (cube d)
let Pc := thinStripCompletion d hd γ L B a
let P := Pc.map observed
let p := baselineSuccess B L
have hb := baselineSuccess_bounds B L hB hL
have hvalid : ∀ x, stripPropensity d hd γ a x ∈ Icc 0 1 ∧
p ∈ Icc 0 1 ∧ p ∈ Icc 0 1 := fun x =>
⟨stripPropensity_mem_Icc d hd γ a hγ x,
⟨hb.1.le, hb.2.1.trans (by norm_num)⟩,
⟨hb.1.le, hb.2.1.trans (by norm_num)⟩⟩
letI : IsProbabilityMeasure Pc :=
thinStripCompletion_probability d hd γ L B a hγ hB hL
letI : IsProbabilityMeasure P := by
dsimp [P]
apply Measure.isProbabilityMeasure_map
unfold observed
fun_prop
letI : IsProbabilityMeasure mu := by
apply isProbabilityMeasure_iff.mpr
dsimp [mu]
rw [Measure.restrict_apply MeasurableSet.univ, Set.univ_inter]
have hcube : cube d =
Icc (fun _ : Fin d => (0 : ℝ)) (fun _ => 1) := by
ext x
simp [cube, Set.mem_Icc, Pi.le_def]
rw [hcube, Real.volume_Icc_pi]
simp
letI : IsFiniteMeasure
(mu.bind (completionAt B (stripPropensity d hd γ a)
(fun _ => p) (fun _ => p))) := by
change IsFiniteMeasure Pc
infer_instance
letI : IsFiniteMeasure
((mu.bind (completionAt B (stripPropensity d hd γ a)
(fun _ => p) (fun _ => p))).map observed) := by
change IsFiniteMeasure P
infer_instance
letI : IsProbabilityMeasure (thinStripLaw d hd γ L B a ha) :=
thinStripLaw_probability d hd γ L B a ha hγ hB hL
have hPdef : P = thinStripLaw d hd γ L B a ha := by
rfl
refine {
cubeSupport := ?_
covariateAC := ?_
density := ?_
consistency := ?_
exchangeability := ?_
tail := ?_
outcome := ?_
smooth := hholder }
· simpa [thinStripLaw] using
thinStripLaw_cubeSupport d hd γ L B a ha hγ hB hL
· simpa [thinStripLaw] using
thinStripLaw_covariateAC d hd γ L B a ha hγ hB hL
· simpa [thinStripLaw] using
thinStripLaw_density_lower d hd γ L B a c_f ha hγ hB hL hcf
· exact thinStripCompletion_consistency d hd γ L B a hγ hB hL
· dsimp [Pc, thinStripCompletion, mu, p]
exact completionBind_condExchangeable (volume.restrict (cube d)) B
(stripPropensity d hd γ a) (fun _ => baselineSuccess B L)
(fun _ => baselineSuccess B L) (stripPropensity_measurable d hd γ a)
measurable_const measurable_const hvalid
· constructor
· have hpraw := completionBind_propensity_ae mu B
(stripPropensity d hd γ a) (fun _ => p) (fun _ => p)
(stripPropensity_measurable d hd γ a) measurable_const measurable_const hvalid
dsimp [mu, p] at hpraw
have hcov := thinStripLaw_covariateLaw d hd γ L B a ha hγ hB hL
change ∀ᵐ x ∂covariateLaw (thinStripLaw d hd γ L B a ha),
stripPropensity d hd γ a x =
propensity (thinStripLaw d hd γ L B a ha) x
rw [hcov]
filter_upwards [hpraw] with x hx
exact hx.symm
· intro t ht
simpa [thinStripLaw] using
thinStripLaw_selected_tail_bound d hd γ L B a C ha hγ hB hL hC t ht
· have hout := completionBind_boundedMeanSubGaussian_equal_arms mu B hB
(stripPropensity d hd γ a) (fun _ => p)
(stripPropensity_measurable d hd γ a) measurable_const
(fun x => ⟨(hvalid x).1, (hvalid x).2.1⟩)
(stripPropensity_uniform_pos_ae d hd γ a hγ)
simpa [mu, Pc, P, p, thinStripCompletion] using hout
def stripSecondMoment
Treated-design second moment in the normalized second direction. For the stated inputs and conditions, the stripSecondMoment object being defined.
Definition (Lean source)
parameters
d :
ℕ
hd :
2 ≤ d
γ L B a :
ℝ
ha :
1 < a ∧ a < 1 + d / (γ - 1)
h :
ℝ
def
stripSecondMoment d hd γ L B a ha h :
ℝ
given by
P :
thinStripLaw d hd γ L B a ha
(∫ z in E, ((z.1 ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 ∂P) / P.real E
For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
gamma L B a :
ℝ
ha :
1 < a ∧ a < 1 + d / (gamma - 1)
hgamma :
1 < gamma
hB :
0 < B
hL :
0 < L
conclusion
Proof (Lean source)
lemma stripSecondMoment_tendsto_zero (d : ℕ) (hd : 2 ≤ d)
(gamma L B a : ℝ) (ha : 1 < a ∧ a < 1 + d / (gamma - 1))
(hgamma : 1 < gamma) (hB : 0 < B) (hL : 0 < L) :
Tendsto (stripSecondMoment d hd gamma L B a ha)
(nhdsWithin 0 (Ioi 0)) (nhds 0) := by
let G : ℝ → ℝ := fun h =>
((((2 * h) ^ d) ^ ((1 : ℝ) / (gamma - 1))) * (2 * h) ^ d +
(h ^ a / h) ^ 2 * ((2 * h ^ a) * (2 * h) ^ (d - 1))) /
((1 / 4 : ℝ) * (2 * (h / (4 * d)) ^ a) *
(h / (4 * d)) ^ (d - 1))
have hG : Tendsto G (nhdsWithin 0 (Ioi 0)) (nhds 0) :=
thinStrip_raw_second_envelope_tendsto_zero d hd gamma a hgamma ha
refine squeeze_zero' (g := G) ?_ ?_ hG
· filter_upwards [self_mem_nhdsWithin] with h hh
unfold stripSecondMoment
exact div_nonneg (integral_nonneg_of_ae
(Filter.Eventually.of_forall (fun z => sq_nonneg _))) (measureReal_nonneg)
· have hsmall : ∀ᶠ h in nhdsWithin (0 : ℝ) (Ioi 0), h ≤ 1 / 2 := by
filter_upwards [Ioc_mem_nhdsGT (by norm_num : (0 : ℝ) < 1 / 2)] with h hh
exact hh.2
filter_upwards [self_mem_nhdsWithin, hsmall] with h hh hhhalf
have hhpos : 0 < h := hh
have hnum := thinStrip_treated_second_sup_power_bound d hd gamma a h
hgamma (by linarith [ha.1] : 0 ≤ a) hhpos hhhalf
have hden := thinStrip_treated_sup_mass_lower d hd gamma a h
hgamma ha.1 hhpos (hhhalf.trans (by norm_num))
have hdenpos : 0 < (1 / 4 : ℝ) *
((2 * (h / (4 * d)) ^ a) * (h / (4 * d)) ^ (d - 1)) := by positivity
have hnum_nonneg : 0 ≤ ∫ x in {x : Fin d → ℝ | centreRadius x ≤ h},
((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 * stripPropensity d hd gamma a x
∂(volume.restrict (cube d)) := by
apply integral_nonneg_of_ae
filter_upwards [] with x
exact mul_nonneg (sq_nonneg _)
(stripPropensity_mem_Icc d hd gamma a hgamma x).1
change ((∫ z in {z : Obs d | z.2.1 = true ∧ centreRadius z.1 ≤ h},
((z.1 ⟨1, by omega⟩ - 1 / 2) / h) ^ 2
∂thinStripLaw d hd gamma L B a ha) /
(thinStripLaw d hd gamma L B a ha).real
{z : Obs d | z.2.1 = true ∧ centreRadius z.1 ≤ h}) ≤ G h
rw [thinStripLaw_treated_second_integral_identity d hd gamma L B a ha
hgamma hB hL h hhpos,
thinStripLaw_treated_mass_identity d hd gamma L B a ha hgamma hB hL h]
calc
_ ≤ (∫ x in {x : Fin d → ℝ | centreRadius x ≤ h},
((x ⟨1, by omega⟩ - 1 / 2) / h) ^ 2 * stripPropensity d hd gamma a x
∂(volume.restrict (cube d))) /
((1 / 4 : ℝ) * ((2 * (h / (4 * d)) ^ a) *
(h / (4 * d)) ^ (d - 1))) :=
div_le_div_of_nonneg_left hnum_nonneg hdenpos hden
_ ≤ G h := by
dsimp [G]
convert div_le_div_of_nonneg_right hnum hdenpos.le using 1
congr 1
ring
theorem thinStrip_mem_model_not_dornA3
A centred thin-strip propensity satisfies the global tail bound, yet it fails Dorn's numerical anti-concentration clause even when propensity one is allowed. Its normalized treated design also loses its second direction. For the stated inputs and conditions, the asserted conclusion holds.
Formal statement
hypotheses
d :
ℕ
hd :
2 ≤ d
β B L C c_f γ :
ℝ
hβ :
1 < β
hB :
0 < B
hL :
0 < L
hC :
1 ≤ C
hcf :
0 < c_f ∧ c_f ≤ 1
hγ :
1 < γ
a :
ℝ
ha :
1 < a ∧ a < 1 + d / (γ - 1)
conclusion
P :
thinStripLaw d hd γ L B a ha
P ∈ ModelClass d β B L C c_f γ
t ∈ Set.Icc (0 : ℝ) 1 :
¬ DornA3AntiConcentrationOnCube P (stripPropensity d hd γ a)
¬ DornA3OnCube P (stripPropensity d hd γ a)
Proof (Lean source)
theorem thinStrip_mem_model_not_dornA3 (d : ℕ) (hd : 2 ≤ d)
(β B L C c_f γ : ℝ) (hβ : 1 < β) (hB : 0 < B)
(hL : 0 < L) (hC : 1 ≤ C) (hcf : 0 < c_f ∧ c_f ≤ 1)
(hγ : 1 < γ) :
∀ a : ℝ, ∀ ha : 1 < a ∧ a < 1 + d / (γ - 1),
let P := thinStripLaw d hd γ L B a ha
P ∈ ModelClass d β B L C c_f γ ∧
(∀ t ∈ Icc (0 : ℝ) 1,
(covariateLaw P).real {x | stripPropensity d hd γ a x ≤ t} ≤
t ^ (γ - 1)) ∧
¬ DornA3AntiConcentrationOnCube P (stripPropensity d hd γ a) ∧
¬ DornA3OnCube P (stripPropensity d hd γ a) ∧
Tendsto (stripSecondMoment d hd γ L B a ha)
(nhdsWithin 0 (Ioi 0)) (nhds 0) ∧
∃ Q ∈ ModelClass d β B L C c_f γ,
¬ DornA3OnCube Q (stripPropensity d hd γ a) := by
intro a ha
dsimp only
let p := baselineSuccess B L
let μ₁ : (Fin d → ℝ) → ℝ := fun _ => B * p
let Pc := thinStripCompletion d hd γ L B a
let P := thinStripLaw d hd γ L B a ha
letI : IsProbabilityMeasure Pc :=
thinStripCompletion_probability d hd γ L B a hγ hB hL
letI : IsProbabilityMeasure P :=
thinStripLaw_probability d hd γ L B a ha hγ hB hL
have hb := baselineSuccess_bounds B L hB hL
have hvalid : ∀ x, stripPropensity d hd γ a x ∈ Icc 0 1 ∧
p ∈ Icc 0 1 := fun x =>
⟨stripPropensity_mem_Icc d hd γ a hγ x,
⟨hb.1.le, hb.2.1.trans (by norm_num)⟩⟩
have hmean : |B * p| ≤ L := by
rw [abs_of_pos (mul_pos hB hb.1)]
exact hb.2.2.trans (by linarith)
letI : IsFiniteMeasure
((volume.restrict (cube d)).bind
(completionAt B (stripPropensity d hd γ a) (fun _ => p) (fun _ => p))) := by
change IsFiniteMeasure Pc
infer_instance
have hholder : HolderResponse Pc μ₁ β L := by
have hraw := completionBind_holderResponse_constant_equal_arms
(volume.restrict (cube d)) B p β L (stripPropensity d hd γ a)
(stripPropensity_measurable d hd γ a) hvalid
(lt_trans zero_lt_one hβ) hL.le hmean
simpa [Pc, μ₁, p, thinStripCompletion] using hraw
have hmodel : @GlobalTailModel d β B L C c_f γ Pc inferInstance μ₁
(stripPropensity d hd γ a) :=
thinStrip_globalTailModel_of_holder d hd β B L C c_f γ a ha hγ hB hL
hC hcf.2 μ₁ hholder
have hparams : ModelParameterDomain d β B L C c_f :=
⟨by omega, hβ, hB, hL, hC, hcf⟩
have hmem : P ∈ ModelClass d β B L C c_f γ := by
refine ⟨hparams, hγ, Pc, inferInstance, μ₁,
stripPropensity d hd γ a, ?_, hmodel⟩
rfl
have htail : ∀ t ∈ Icc (0 : ℝ) 1,
(covariateLaw P).real {x | stripPropensity d hd γ a x ≤ t} ≤
t ^ (γ - 1) := by
intro t ht
exact stripPropensity_tail_of_uniform_covariateLaw d hd P γ a hγ
(by simpa [P] using
thinStripLaw_covariateLaw d hd γ L B a ha hγ hB hL) t ht
have hnotAnti : ¬ DornA3AntiConcentrationOnCube P
(stripPropensity d hd γ a) := by
simpa [P] using
thinStripLaw_not_dornA3AntiConcentration d hd γ L B a ha hγ hB hL
have hnot : ¬ DornA3OnCube P (stripPropensity d hd γ a) := by
intro hA3
exact hnotAnti
(dornA3AntiConcentrationOnCube_of_dornA3OnCube P
(stripPropensity d hd γ a) hA3)
have htend : Tendsto (stripSecondMoment d hd γ L B a ha)
(nhdsWithin 0 (Ioi 0)) (nhds 0) :=
stripSecondMoment_tendsto_zero d hd γ L B a ha hγ hB hL
exact ⟨hmem, htail, hnotAnti, hnot, htend, ⟨P, hmem, hnot⟩⟩