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b317718 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 | import OAI.Probability.InvariantIsing.Arrays.CountableWardFunctional
namespace OAI
/-! Uniform fixed-prior Ward bounds survive a measurable mixture of
countable reference measures. -/
noncomputable section
open MeasureTheory IsingPerceptron
namespace InvariantIsing
lemma measurable_mixtureReplicaMean {T Ω X : Type*}
[MeasurableSpace T] [MeasurableSpace Ω] [MeasurableSpace X]
[Countable X] [MeasurableSingletonClass X]
(ν : T → Measure X) [∀ t, IsProbabilityMeasure (ν t)] (hν : Measurable ν)
(H : Ω × X → ℝ) (hH : Measurable H) {r : ℕ}
(D : Ω × (Fin r → X) → ℝ) (hD : Measurable D) :
Measurable (fun p : T × Ω =>
referenceReplicaMean (ν p.1) (fun x => H (p.2,x)) (fun σ => D (p.2,σ))) := by
have : ∀ p : T × Ω, IsProbabilityMeasure ((ν ∘ Prod.fst) p) := fun p => by
change IsProbabilityMeasure (ν p.1)
infer_instance
exact measurable_random_referenceReplicaMean (hν.comp measurable_fst)
(hH.comp (measurable_fst.snd.prodMk measurable_snd))
(hD.comp (measurable_fst.snd.prodMk measurable_snd))
lemma measurable_integral_mixtureReplicaMean {T Ω X : Type*}
[MeasurableSpace T] [MeasurableSpace Ω] [MeasurableSpace X]
[Countable X] [MeasurableSingletonClass X]
(P : Measure Ω) [SFinite P]
(ν : T → Measure X) [∀ t, IsProbabilityMeasure (ν t)] (hν : Measurable ν)
(H : Ω × X → ℝ) (hH : Measurable H) {r : ℕ}
(D : Ω × (Fin r → X) → ℝ) (hD : Measurable D) :
Measurable (fun t => ∫ ω,
referenceReplicaMean (ν t) (fun x => H (ω,x)) (fun σ => D (ω,σ)) ∂P) :=
(measurable_mixtureReplicaMean ν hν H hH D hD).stronglyMeasurable.integral_prod_right'.measurable
lemma integrable_mixtureReplicaMean {T Ω X : Type*}
[MeasurableSpace T] [MeasurableSpace Ω] [MeasurableSpace X]
[Countable X] [MeasurableSingletonClass X]
(ξ : Measure T) [IsProbabilityMeasure ξ] (P : Measure Ω) [IsProbabilityMeasure P]
(ν : T → Measure X) [∀ t, IsProbabilityMeasure (ν t)] (hν : Measurable ν)
(H : Ω × X → ℝ) (hH : Measurable H) {r : ℕ}
(D : Ω × (Fin r → X) → ℝ) (hD : Measurable D)
{B : ℝ} (hB : 0 ≤ B) (hb : ∀ z, |D z| ≤ B) :
Integrable (fun t => ∫ ω,
referenceReplicaMean (ν t) (fun x => H (ω,x)) (fun σ => D (ω,σ)) ∂P) ξ := by
have hm := measurable_integral_mixtureReplicaMean P ν hν H hH D hD
apply integrable_of_measurable_abs_le (c := B) hm
intro t
apply abs_integral_le_const_of_bound (c := B)
((measurable_mixtureReplicaMean ν hν H hH D hD).comp (measurable_const.prodMk measurable_id))
intro ω
change |referenceReplicaMean (ν t) (fun x => H (ω,x)) (fun σ => D (ω,σ))| ≤ B
exact referenceReplicaMean_abs_le (ν t) _ _ (measurable_of_countable _) hB (fun σ => hb (ω,σ))
theorem expectedReplicaWard_mixture_abs_le {T Ω X : Type*}
[MeasurableSpace T] [MeasurableSpace Ω] [MeasurableSpace X]
[Countable X] [MeasurableSingletonClass X]
(ξ : Measure T) [IsProbabilityMeasure ξ] (P : Measure Ω) [IsProbabilityMeasure P]
(ν : T → Measure X) [∀ t, IsProbabilityMeasure (ν t)] (hν : Measurable ν)
(H : Ω × X → ℝ) (hH : Measurable H)
(D₂ : Ω × (Fin 2 → X) → ℝ) (hD₂ : Measurable D₂)
(D₃ : Ω × (Fin 3 → X) → ℝ) (hD₃ : Measurable D₃)
{B₂ B₃ : ℝ} (hB₂ : 0 ≤ B₂) (hB₃ : 0 ≤ B₃)
(hb₂ : ∀ z, |D₂ z| ≤ B₂) (hb₃ : ∀ z, |D₃ z| ≤ B₃)
{ε : ℝ} (hward : ∀ t, |expectedReplicaWard P (ν t) H D₂ D₃| ≤ ε) :
|(∫ t, (∫ ω, referenceReplicaMean (ν t) (fun x => H (ω,x)) (fun σ => D₂ (ω,σ)) ∂P) ∂ ξ) -
2 * (∫ t, (∫ ω, referenceReplicaMean (ν t) (fun x => H (ω,x)) (fun σ => D₃ (ω,σ)) ∂P) ∂ ξ)| ≤ ε := by
have hi₂ := integrable_mixtureReplicaMean ξ P ν hν H hH D₂ hD₂ hB₂ hb₂
have hi₃ := integrable_mixtureReplicaMean ξ P ν hν H hH D₃ hD₃ hB₃ hb₃
rw [← integral_const_mul, ← integral_sub hi₂ (hi₃.const_mul 2)]
apply abs_integral_le_const_of_bound
((measurable_integral_mixtureReplicaMean P ν hν H hH D₂ hD₂).sub
((measurable_integral_mixtureReplicaMean P ν hν H hH D₃ hD₃).const_mul 2))
exact hward
end InvariantIsing
end
end OAI
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