| 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 | |