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a2ee19a | 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 | 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
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