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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 | import OAI.Probability.InvariantIsing.Arrays.CountableReplicaRestriction
namespace OAI
/-! Bounded two- and three-replica Ward terms pass through leaf exhaustion.
A finite Ward bound is preserved in the countable-leaf limit. -/
noncomputable section
open MeasureTheory IsingPerceptron Filter
open scoped Topology
namespace InvariantIsing
def expectedReplicaWard {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
(P : Measure Ω) (ν : Measure X) (H : Ω × X → ℝ)
(D₂ : Ω × (Fin 2 → X) → ℝ) (D₃ : Ω × (Fin 3 → X) → ℝ) : ℝ :=
(∫ ω, referenceReplicaMean ν (fun x => H (ω,x)) (fun σ => D₂ (ω,σ)) ∂P) -
2 * ∫ ω, referenceReplicaMean ν (fun x => H (ω,x)) (fun σ => D₃ (ω,σ)) ∂P
theorem expectedReplicaWard_restriction_tendsto {Ω X : Type*}
[MeasurableSpace Ω] [MeasurableSpace X] [Countable X] [MeasurableSingletonClass X]
(P : Measure Ω) [IsProbabilityMeasure P] (ν : Measure X) [IsProbabilityMeasure ν]
(H : Ω × X → ℝ) (hH : Measurable H)
(he : ∀ᵐ ω ∂P, Integrable (fun x => Real.exp (H (ω,x))) ν)
(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)
{S : ℕ → Set X} (hS : ∀ n, MeasurableSet (S n))
(hexh : ∀ x, ∀ᶠ n in atTop, x ∈ S n) (hpos : ∀ n, ν (S n) ≠ 0) :
Tendsto (fun n => expectedReplicaWard P (normalizedRestriction ν (S n)) H D₂ D₃) atTop
(𝓝 (expectedReplicaWard P ν H D₂ D₃)) := by
exact (expected_referenceReplicaMean_restriction_tendsto P ν H hH he D₂ hD₂ hB₂ hb₂
hS hexh hpos).sub
((expected_referenceReplicaMean_restriction_tendsto P ν H hH he D₃ hD₃ hB₃ hb₃
hS hexh hpos).const_mul 2)
/-- Uniform finite Ward residual bounds are preserved for the full countable
Gibbs prior. No convergence or integrability of a Gaussian correction is needed. -/
theorem expectedReplicaWard_abs_le_of_restriction {Ω X : Type*}
[MeasurableSpace Ω] [MeasurableSpace X] [Countable X] [MeasurableSingletonClass X]
(P : Measure Ω) [IsProbabilityMeasure P] (ν : Measure X) [IsProbabilityMeasure ν]
(H : Ω × X → ℝ) (hH : Measurable H)
(he : ∀ᵐ ω ∂P, Integrable (fun x => Real.exp (H (ω,x))) ν)
(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)
{S : ℕ → Set X} (hS : ∀ n, MeasurableSet (S n))
(hexh : ∀ x, ∀ᶠ n in atTop, x ∈ S n) (hpos : ∀ n, ν (S n) ≠ 0)
{ε : ℝ} (hfinite : ∀ n,
|expectedReplicaWard P (normalizedRestriction ν (S n)) H D₂ D₃| ≤ ε) :
|expectedReplicaWard P ν H D₂ D₃| ≤ ε := by
exact le_of_tendsto (expectedReplicaWard_restriction_tendsto P ν H hH he
D₂ hD₂ D₃ hD₃ hB₂ hB₃ hb₂ hb₃ hS hexh hpos).abs
(Filter.Eventually.of_forall hfinite)
end InvariantIsing
end
end OAI
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