| import Mathlib |
| import OAI.Probability.Perceptron.Interpolation.CompactArrayMap |
| import OAI.Probability.Perceptron.Interpolation.GibbsOverlapArray |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology BigOperators BoundedContinuousFunction |
| namespace SphericalPerceptronFreeEnergy |
| section |
| variable {A S K : Type*} [MeasurableSpace A] [MeasurableSpace S] |
| variable [TopologicalSpace K] [MeasurableSpace K] [BorelSpace K] |
| [SecondCountableTopology K] [CompactSpace K] |
| variable (κ : Kernel A S) [IsMarkovKernel κ] (P : Measure A) [IsProbabilityMeasure P] |
| variable (H : A → S → ℝ) (hH : Measurable (Function.uncurry H)) |
| variable (Q : S → S → K) (hQ : Measurable (Function.uncurry Q)) |
|
|
| omit [CompactSpace K] in |
| lemma gibbsOverlapArray_exchangeable (e : Equiv.Perm ℕ) : |
| MeasurePreserving (compactRelabel (K:=K) e) |
| (gibbsOverlapArrayLaw κ P H hH Q hQ : Measure _) |
| (gibbsOverlapArrayLaw κ P H hH Q hQ : Measure _) := by |
| have hr := annealedInfiniteReplica_reindex κ P H hH e |
| have hm : Measurable (fun a : A×(ℕ→S) => overlapArray Q a.2) := |
| (overlapArray_measurable Q hQ).comp measurable_snd |
| refine ⟨(compactRelabel_continuous e).measurable,?_⟩ |
| change Measure.map _ (Measure.map (fun a : A×(ℕ→S) => overlapArray Q a.2) _) = _ |
| rw [Measure.map_map (compactRelabel_continuous e).measurable hm] |
| change Measure.map ((fun a : A×(ℕ→S) => overlapArray Q a.2) ∘ |
| (fun a : A×(ℕ→S) => (a.1,fun i => a.2 (e i)))) _ = _ |
| rw [← Measure.map_map hm hr.measurable,hr.map_eq] |
| rfl |
|
|
| omit [CompactSpace K] in |
| lemma gibbsOverlapArray_closed_full {F : Set (CompactArray K)} (hF : IsClosed F) |
| (hfull : ∀ x : ℕ→S, overlapArray Q x ∈ F) : |
| gibbsOverlapArrayLaw κ P H hH Q hQ F=1 := by |
| have hp : (fun a : A×(ℕ→S) => overlapArray Q a.2) ⁻¹' F=univ := by |
| ext a |
| simp only [mem_preimage,mem_univ,iff_true] |
| exact hfull a.2 |
| apply ENNReal.coe_injective |
| rw [ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure] |
| change (Measure.map (fun a : A×(ℕ→S) => overlapArray Q a.2) _) F=1 |
| have hm : Measurable (fun a : A×(ℕ→S) => overlapArray Q a.2) := |
| (overlapArray_measurable Q hQ).comp measurable_snd |
| rw [Measure.map_apply hm hF.measurableSet,hp] |
| exact measure_univ |
| end |
|
|
| lemma compactMapLaw_exchangeable {K L : Type*} [TopologicalSpace K] [TopologicalSpace L] |
| [MeasurableSpace K] [MeasurableSpace L] [BorelSpace K] [BorelSpace L] |
| [SecondCountableTopology K] [SecondCountableTopology L] |
| (f : K→L) (hf : Continuous f) (μ : ProbabilityMeasure (CompactArray K)) |
| (e : Equiv.Perm ℕ) |
| (he : MeasurePreserving (compactRelabel (K:=K) e) (μ : Measure _) (μ : Measure _)) : |
| MeasurePreserving (compactRelabel (K:=L) e) (compactMapLaw f hf μ : Measure _) |
| (compactMapLaw f hf μ : Measure _) := by |
| refine ⟨(compactRelabel_continuous e).measurable,?_⟩ |
| change Measure.map _ (Measure.map (compactMapArray f) _) = _ |
| rw [Measure.map_map (compactRelabel_continuous e).measurable (compactMapArray_continuous hf).measurable] |
| change Measure.map (compactMapArray f ∘ compactRelabel e) _ = _ |
| rw [← Measure.map_map (compactMapArray_continuous hf).measurable he.measurable,he.map_eq] |
| rfl |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
|
|
| end OAI |
|
|
