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c4cdbbd | 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 | import OAI.Probability.InvariantIsing.Arrays.TensorLabeledTerminal
namespace OAI
/-! The finite reference replica mean as the initial coordinates of iid Gibbs replicas. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
namespace InvariantIsing
lemma referenceReplicaMean_infinite {X : Type*} [MeasurableSpace X]
[Countable X] [MeasurableSingletonClass X] (ν : Measure X) [IsProbabilityMeasure ν]
(H : X → ℝ) (he : Integrable (fun x => Real.exp (H x)) ν)
{r : ℕ} (D : (Fin r → X) → ℝ) (hD : Measurable D) :
referenceReplicaMean ν H D =
∫ σ : ℕ → X, D (fun i => σ i) ∂Measure.infinitePi (fun _ : ℕ => gibbsProbability ν H) := by
let := isProbabilityMeasure_tilted he
rw [referenceReplicaMean_eq_tilted ν H he]
simp_rw [gibbsProbability_eq_tilted ν H he]
have hp : MeasurePreserving (fun σ : ℕ → X => fun i : Fin r => σ i)
(Measure.infinitePi (fun _ : ℕ => ν.tilted H)) (Measure.pi (fun _ : Fin r => ν.tilted H)) := by
refine ⟨by fun_prop,?_⟩
rw [Measure.map_infinitePi_infinitePi_of_inj Fin.val_injective,Measure.infinitePi_eq_pi]
exact (hp.hasLaw.integral_comp hD.aestronglyMeasurable).symm
end InvariantIsing
end
end OAI
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