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import OAI.Probability.Perceptron.Interpolation.TripleGaussianReplica
import OAI.Probability.Perceptron.Variational.SingleRestorationLaw
namespace OAI
noncomputable section
namespace SphericalPerceptronFreeEnergy
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal NNReal BigOperators
variable {I J K : Type} [Fintype I] [Fintype J] [Fintype K]
def tripleGaussianMean (k : ℕ)
(A : ℕ→EuclideanSpace ℝ I →L[ℝ] EuclideanSpace ℝ I)
(B : ℕ→EuclideanSpace ℝ J →L[ℝ] EuclideanSpace ℝ J)
(C : ℕ→EuclideanSpace ℝ K →L[ℝ] EuclideanSpace ℝ K)
(R : EuclideanSpace ℝ I →L[ℝ] EuclideanSpace ℝ I)
(Q : EuclideanSpace ℝ J →L[ℝ] EuclideanSpace ℝ J)
(T : EuclideanSpace ℝ K →L[ℝ] EuclideanSpace ℝ K)
(H : EuclideanSpace ℝ I→ℝ) (G : EuclideanSpace ℝ J→ℝ) (F : EuclideanSpace ℝ K→ℝ)
(f : EuclideanSpace ℝ I→ℝ) (g : EuclideanSpace ℝ J→ℝ)
(p : TripleGaussianData I J K k) : ℝ :=
indexedTerminalMean (tripleGaussianStep A B C) k (tripleGaussianTerminal H G F)
(tripleGaussianRoot R Q T p.1) (tripleGaussianObservable f g)
(p.2.1,indexedMarksZip k p.2.2)
lemma tripleGaussianMean_measurable (k : ℕ)
(A : ℕ→EuclideanSpace ℝ I →L[ℝ] EuclideanSpace ℝ I)
(B : ℕ→EuclideanSpace ℝ J →L[ℝ] EuclideanSpace ℝ J)
(C : ℕ→EuclideanSpace ℝ K →L[ℝ] EuclideanSpace ℝ K)
(R : EuclideanSpace ℝ I →L[ℝ] EuclideanSpace ℝ I)
(Q : EuclideanSpace ℝ J →L[ℝ] EuclideanSpace ℝ J)
(T : EuclideanSpace ℝ K →L[ℝ] EuclideanSpace ℝ K)
(H : EuclideanSpace ℝ I→ℝ) (G : EuclideanSpace ℝ J→ℝ) (F : EuclideanSpace ℝ K→ℝ)
(hH : Measurable H) (hG : Measurable G) (hF : Measurable F)
(f : EuclideanSpace ℝ I→ℝ) (g : EuclideanSpace ℝ J→ℝ)
(hf : Measurable f) (hg : Measurable g) :
Measurable (tripleGaussianMean k A B C R Q T H G F f g) := by
have ht : Measurable (tripleGaussianTerminal H G F) := by unfold tripleGaussianTerminal; fun_prop
have ho : Measurable (tripleGaussianObservable (K:=K) f g) := by unfold tripleGaussianObservable; fun_prop
have hr : Measurable (tripleGaussianRoot R Q T) := by unfold tripleGaussianRoot; fun_prop
have hm := indexedTerminalMean_joint_measurable (tripleGaussianStep A B C)
(tripleGaussianStep_measurable A B C) k (tripleGaussianTerminal H G F) ht
(tripleGaussianObservable f g) ho
have hp : Measurable (fun p : TripleGaussianData I J K k =>
(tripleGaussianRoot R Q T p.1,(p.2.1,indexedMarksZip k p.2.2))) :=
(hr.comp measurable_fst).prodMk (measurable_snd.fst.prodMk
((indexedMarksZip_measurable k).comp measurable_snd.snd))
have hc := hm.comp hp
exact hc
theorem countable_three_terminal_single_value (k : ℕ) (z : Fin k→ℝ)
(hz : StrictMono z) (hz0 : ∀ i, 0<z i) (hz1 : ∀ i, z i<1)
(A : ℕ→EuclideanSpace ℝ I →L[ℝ] EuclideanSpace ℝ I)
(B : ℕ→EuclideanSpace ℝ J →L[ℝ] EuclideanSpace ℝ J)
(C : ℕ→EuclideanSpace ℝ K →L[ℝ] EuclideanSpace ℝ K)
(R : EuclideanSpace ℝ I →L[ℝ] EuclideanSpace ℝ I)
(Q : EuclideanSpace ℝ J →L[ℝ] EuclideanSpace ℝ J)
(T : EuclideanSpace ℝ K →L[ℝ] EuclideanSpace ℝ K)
(H : EuclideanSpace ℝ I→ℝ) (G : EuclideanSpace ℝ J→ℝ) (F : EuclideanSpace ℝ K→ℝ)
{L₁ L₂ L₀ : ℝ≥0} (hH : LipschitzWith L₁ H) (hG : LipschitzWith L₂ G) (hF : LipschitzWith L₀ F)
(f : EuclideanSpace ℝ I→ℝ) (g : EuclideanSpace ℝ J→ℝ)
(hf : Measurable f) (hg : Measurable g) (D E : ℝ) (hD : 0≤D) (hE : 0≤E)
(hfB : ∀ x, |f x|≤D) (hgB : ∀ y, |g y|≤E) :
(∫ t : IndexedCascadeBase k×(((ℕ→ℝ)×(ℕ→ℝ))×(ℕ→ℝ)),
tripleGaussianMean k A B C R Q T H G F f g
((((indexedGaussianDisorder k I t.2.1.1).1,(indexedGaussianDisorder k J t.2.1.2).1),
(indexedGaussianDisorder k K t.2.2).1),
(t.1,(indexedMarksZip k ((indexedGaussianDisorder k I t.2.1.1).2,
(indexedGaussianDisorder k J t.2.1.2).2),(indexedGaussianDisorder k K t.2.2).2)))
∂(indexedCascadeBaseLaw k z : Measure (IndexedCascadeBase k)).prod
((countableGaussianLaw.prod countableGaussianLaw).prod countableGaussianLaw)) =
(rootPathMean (stdGaussian (EuclideanSpace ℝ I)) (fun r => fun _ => R r) k
(fun i => tiltedStateStep (gaussianMarkLaw (E:=EuclideanSpace ℝ I)) (gaussianLinearMarkStep A) (z i)
(finiteCascadeShifts (gaussianMarkLaw (E:=EuclideanSpace ℝ I)) (gaussianLinearMarkStep A) k z (fun x => H (x 0)) i))
(fun x => f (x 0)) *
rootPathMean (stdGaussian (EuclideanSpace ℝ J)) (fun r => fun _ => Q r) k
(fun i => tiltedStateStep (gaussianMarkLaw (E:=EuclideanSpace ℝ J)) (gaussianLinearMarkStep B) (z i)
(finiteCascadeShifts (gaussianMarkLaw (E:=EuclideanSpace ℝ J)) (gaussianLinearMarkStep B) k z (fun x => G (x 0)) i))
(fun x => g (x 0))) := by
let : IsGaussian (gaussianMarkLaw (E:=EuclideanSpace ℝ I) : Measure (EuclideanSpace ℝ I)) := (inferInstance : IsGaussian (stdGaussian (EuclideanSpace ℝ I)))
let : IsGaussian (gaussianMarkLaw (E:=EuclideanSpace ℝ J) : Measure (EuclideanSpace ℝ J)) := (inferInstance : IsGaussian (stdGaussian (EuclideanSpace ℝ J)))
let : IsGaussian (gaussianMarkLaw (E:=EuclideanSpace ℝ K) : Measure (EuclideanSpace ℝ K)) := (inferInstance : IsGaussian (stdGaussian (EuclideanSpace ℝ K)))
have hI := gaussianLinearRecursion_realization (gaussianMarkLaw (E:=EuclideanSpace ℝ I)) A hH k z hz0
have hJ := gaussianLinearRecursion_realization (gaussianMarkLaw (E:=EuclideanSpace ℝ J)) B hG k z hz0
have hK := gaussianLinearRecursion_realization (gaussianMarkLaw (E:=EuclideanSpace ℝ K)) C hF k z hz0
have hv := indexedTerminalMean_two_fresh_old
(gaussianMarkLaw (E:=EuclideanSpace ℝ I)) (gaussianMarkLaw (E:=EuclideanSpace ℝ J))
(gaussianMarkLaw (E:=EuclideanSpace ℝ K))
(gaussianMarkLaw (E:=EuclideanSpace ℝ I)) (gaussianMarkLaw (E:=EuclideanSpace ℝ J))
(gaussianMarkLaw (E:=EuclideanSpace ℝ K))
(gaussianLinearMarkStep A) (gaussianLinearMarkStep B) (gaussianLinearMarkStep C)
(gaussianLinearMarkStep_measurable A) (gaussianLinearMarkStep_measurable B) (gaussianLinearMarkStep_measurable C)
k z hz hz0 hz1 (fun x => H (x 0)) (fun x => G (x 0)) (fun x => F (x 0))
(hH.continuous.measurable.comp (measurable_pi_apply 0))
(hG.continuous.measurable.comp (measurable_pi_apply 0))
(hF.continuous.measurable.comp (measurable_pi_apply 0)) hI.2 hJ.2 hK.2
(fun r => fun _ => R r) (fun r => fun _ => Q r) (fun r => fun _ => T r)
(by fun_prop) (by fun_prop) (by fun_prop) (fun x => f (x 0)) (fun x => g (x 0))
(hf.comp (measurable_pi_apply 0)) (hg.comp (measurable_pi_apply 0)) D E hD hE (fun x => hfB _) (fun x => hgB _)
have hp := indexedGaussian_three_regroup_law k z I J K
have hm := tripleGaussianMean_measurable k A B C R Q T H G F hH.continuous.measurable
hG.continuous.measurable hF.continuous.measurable f g hf hg
have he := integral_map (μ:=(indexedCascadeBaseLaw k z : Measure (IndexedCascadeBase k)).prod
((countableGaussianLaw.prod countableGaussianLaw).prod countableGaussianLaw))
hp.measurable.aemeasurable hm.aestronglyMeasurable
rw [hp.map_eq] at he
exact he.symm.trans hv
lemma tripleGaussianMean_replica (k : ℕ)
(A : ℕ→EuclideanSpace ℝ I →L[ℝ] EuclideanSpace ℝ I)
(B : ℕ→EuclideanSpace ℝ J →L[ℝ] EuclideanSpace ℝ J)
(C : ℕ→EuclideanSpace ℝ K →L[ℝ] EuclideanSpace ℝ K)
(R : EuclideanSpace ℝ I →L[ℝ] EuclideanSpace ℝ I)
(Q : EuclideanSpace ℝ J →L[ℝ] EuclideanSpace ℝ J)
(T : EuclideanSpace ℝ K →L[ℝ] EuclideanSpace ℝ K)
(H : EuclideanSpace ℝ I→ℝ) (G : EuclideanSpace ℝ J→ℝ) (F : EuclideanSpace ℝ K→ℝ)
(hH : Measurable H) (hG : Measurable G) (hF : Measurable F)
(f : EuclideanSpace ℝ I→ℝ) (g : EuclideanSpace ℝ J→ℝ)
(b : IndexedCascadeBase k) (t : ((ℕ→ℝ)×(ℕ→ℝ))×(ℕ→ℝ))
(hi : Integrable (fun l => Real.exp ((H (gaussianLeafValue k A R t.1.1 l)+
G (gaussianLeafValue k B Q t.1.2 l))+F (gaussianLeafValue k C T t.2 l))) (indexedLeafProbability k b)) :
tripleGaussianMean k A B C R Q T H G F f g (tripleGaussianCountableData (I:=I) (J:=J) (K:=K) k b t) =
gibbsReplicaMean (indexedLeafProbability k b)
(fun l => (H (gaussianLeafValue k A R t.1.1 l)+G (gaussianLeafValue k B Q t.1.2 l))+
F (gaussianLeafValue k C T t.2 l)) 1
(fun l => f (gaussianLeafValue k A R t.1.1 (l 0))*g (gaussianLeafValue k B Q t.1.2 (l 0))) := by
have ht : Measurable (tripleGaussianTerminal H G F) := by unfold tripleGaussianTerminal; fun_prop
unfold tripleGaussianMean
have he := indexedTerminalMean_tiltMean (tripleGaussianStep A B C) (tripleGaussianStep_measurable A B C)
k (tripleGaussianTerminal H G F) ht
(tripleGaussianRoot R Q T (tripleGaussianCountableData (I:=I) (J:=J) (K:=K) k b t).1) (tripleGaussianObservable f g)
((tripleGaussianCountableData (I:=I) (J:=J) (K:=K) k b t).2.1,indexedMarksZip k (tripleGaussianCountableData (I:=I) (J:=J) (K:=K) k b t).2.2)
have hstate (l : IndexedLeaf k) := tripleGaussianLeafState k A B C
(fun _ => R (indexedGaussianDisorder k I t.1.1).1)
(fun _ => Q (indexedGaussianDisorder k J t.1.2).1)
(fun _ => T (indexedGaussianDisorder k K t.2).1)
(indexedGaussianDisorder k I t.1.1).2 (indexedGaussianDisorder k J t.1.2).2
(indexedGaussianDisorder k K t.2).2 l
simp only [tripleGaussianCountableData,tripleGaussianRoot] at he ⊢
simp only [tripleGaussianTerminal,hstate] at he
have hv := he hi
simp only [tripleGaussianObservable] at hv
rw [hv]
symm
refine (gibbsReplicaMean_coordinate_of_integrable (indexedLeafProbability k b)
(F:=fun l => f (gaussianLeafValue k A R t.1.1 l)*g (gaussianLeafValue k B Q t.1.2 l))
(measurable_of_countable _) (measurable_of_countable _) hi 1 0).trans ?_
congr 1
funext l
simp only [indexedTerminalEnergy,tripleGaussianTerminal,hstate,gaussianLeafValue]
end SphericalPerceptronFreeEnergy
end
end OAI
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