| import Mathlib |
| import OAI.Probability.Perceptron.Cascade.DecoratedWeightedTotal |
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| namespace OAI |
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| noncomputable section |
| namespace SphericalPerceptronFreeEnergy |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal ENNReal BigOperators |
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| variable {X S : Type} [MeasurableSpace X] [MeasurableSpace S] [Nonempty S] |
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| omit [Nonempty S] in |
| lemma decoratedWeightedTotal_measurable (step : X×S → X) (hs : Measurable step) |
| (n : ℕ) (F : Fin n → X×S → ℝ) (hF : ∀ i, Measurable (F i)) : |
| Measurable (decoratedWeightedTotal step n F) := by |
| have he : decoratedWeightedTotal step n F = |
| fun p => cascadeTotal n (decoratedCascadeTransform step n F p) := by |
| funext p; exact decoratedWeightedTotal_transform step hs n F hF p |
| rw [he] |
| exact (cascadeTotal_measurable n).comp (decoratedCascadeTransform_measurable step hs n F hF) |
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| def decoratedBiasedLaw (ν : ProbabilityMeasure S) (step : X×S → X) |
| (n : ℕ) (z : Fin n → ℝ) (F : Fin n → X×S → ℝ) (x : X) (a : ℝ) : |
| Measure (DecoratedCascade S n) := |
| normalizedMeasure ((decoratedCascadeLaw ν n z : Measure (DecoratedCascade S n)).withDensity |
| (fun η => ENNReal.ofReal ((decoratedWeightedTotal step n F (x,η))^a))) |
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| lemma decoratedWeightedTotal_pos (ν : ProbabilityMeasure S) (step : X×S → X) |
| (hs : Measurable step) (n : ℕ) (z : Fin n → ℝ) (hz : StrictMono z) |
| (hz0 : ∀ i, 0 < z i) (hz1 : ∀ i, z i < 1) |
| (F : Fin n → X×S → ℝ) (hF : ∀ i, Measurable (F i)) |
| (hI : ∀ i x, Integrable (fun s => Real.exp (z i*F i (x,s))) ν) |
| (hM : ∀ i x, (∫ s, Real.exp (z i*F i (x,s)) ∂ν) = 1) (x : X) : |
| ∀ᵐ η ∂(decoratedCascadeLaw ν n z : Measure (DecoratedCascade S n)), |
| 0 < decoratedWeightedTotal step n F (x,η) := |
| (decoratedWeightedTotal_identDistrib ν step hs n z hz0 hz1 F hF hI hM x).symm.ae_snd |
| measurableSet_Ioi (cascadeTotal_regular n z hz hz0 hz1).1 |
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| lemma decoratedLogTotalMoment_integrable (ν : ProbabilityMeasure S) (step : X×S → X) |
| (hs : Measurable step) (n : ℕ) (z : Fin n → ℝ) (hz : StrictMono z) |
| (hz0 : ∀ i, 0 < z i) (hz1 : ∀ i, z i < 1) |
| (F : Fin n → X×S → ℝ) (hF : ∀ i, Measurable (F i)) |
| (hI : ∀ i x, Integrable (fun s => Real.exp (z i*F i (x,s))) ν) |
| (hM : ∀ i x, (∫ s, Real.exp (z i*F i (x,s)) ∂ν) = 1) |
| (x : X) (a : ℝ) (ha : ∀ i, a < z i) : |
| Integrable (fun η => Real.exp (a*Real.log (decoratedWeightedTotal step n F (x,η)))) |
| (decoratedCascadeLaw ν n z : Measure (DecoratedCascade S n)) := by |
| have hid := (decoratedWeightedTotal_identDistrib ν step hs n z hz0 hz1 F hF hI hM x).comp |
| (Real.measurable_exp.comp (Real.measurable_log.const_mul a)) |
| exact hid.integrable_iff.mpr (cascadeLogTotalMoment_integrable n z hz hz0 hz1 a ha) |
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| lemma decoratedLogTotalMoment_eq (ν : ProbabilityMeasure S) (step : X×S → X) |
| (hs : Measurable step) (n : ℕ) (z : Fin n → ℝ) |
| (hz0 : ∀ i, 0 < z i) (hz1 : ∀ i, z i < 1) |
| (F : Fin n → X×S → ℝ) (hF : ∀ i, Measurable (F i)) |
| (hI : ∀ i x, Integrable (fun s => Real.exp (z i*F i (x,s))) ν) |
| (hM : ∀ i x, (∫ s, Real.exp (z i*F i (x,s)) ∂ν) = 1) (x : X) (a : ℝ) : |
| (∫ η, Real.exp (a*Real.log (decoratedWeightedTotal step n F (x,η))) |
| ∂decoratedCascadeLaw ν n z) = |
| ∫ η, Real.exp (a*Real.log (cascadeTotal n η)) ∂cascadeLaw n z := by |
| exact ((decoratedWeightedTotal_identDistrib ν step hs n z hz0 hz1 F hF hI hM x).comp |
| (Real.measurable_exp.comp (Real.measurable_log.const_mul a))).integral_eq |
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| lemma decoratedBiasedLaw_eq_exponentialTilt (ν : ProbabilityMeasure S) (step : X×S → X) |
| (hs : Measurable step) (n : ℕ) (z : Fin n → ℝ) (hz : StrictMono z) |
| (hz0 : ∀ i, 0 < z i) (hz1 : ∀ i, z i < 1) |
| (F : Fin n → X×S → ℝ) (hF : ∀ i, Measurable (F i)) |
| (hI : ∀ i x, Integrable (fun s => Real.exp (z i*F i (x,s))) ν) |
| (hM : ∀ i x, (∫ s, Real.exp (z i*F i (x,s)) ∂ν) = 1) (x : X) (a : ℝ) : |
| decoratedBiasedLaw ν step n z F x a = |
| exponentialMarkTilt (decoratedCascadeLaw ν n z) a |
| (fun η => Real.log (decoratedWeightedTotal step n F (x,η))) := by |
| unfold decoratedBiasedLaw exponentialMarkTilt |
| congr 1 |
| apply withDensity_congr_ae |
| filter_upwards [decoratedWeightedTotal_pos ν step hs n z hz hz0 hz1 F hF hI hM x] with η hη |
| rw [Real.rpow_def_of_pos hη,mul_comm] |
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| lemma decoratedBiasedLaw_probability (ν : ProbabilityMeasure S) (step : X×S → X) |
| (hs : Measurable step) (n : ℕ) (z : Fin n → ℝ) (hz : StrictMono z) |
| (hz0 : ∀ i, 0 < z i) (hz1 : ∀ i, z i < 1) |
| (F : Fin n → X×S → ℝ) (hF : ∀ i, Measurable (F i)) |
| (hI : ∀ i x, Integrable (fun s => Real.exp (z i*F i (x,s))) ν) |
| (hM : ∀ i x, (∫ s, Real.exp (z i*F i (x,s)) ∂ν) = 1) |
| (x : X) (a : ℝ) (ha : ∀ i, a < z i) : |
| IsProbabilityMeasure (decoratedBiasedLaw ν step n z F x a) := by |
| rw [decoratedBiasedLaw_eq_exponentialTilt ν step hs n z hz hz0 hz1 F hF hI hM x a] |
| exact exponentialMarkTilt_probability _ |
| (decoratedLogTotalMoment_integrable ν step hs n z hz hz0 hz1 F hF hI hM x a ha) |
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| def decoratedRootPotential (step : X×S → X) (n : ℕ) (F : Fin (n+1) → X×S → ℝ) |
| (p : X×(S×DecoratedCascade S n)) : ℝ := |
| F 0 (p.1,p.2.1)+Real.log |
| (decoratedWeightedTotal step n (fun i => F i.succ) (step (p.1,p.2.1),p.2.2)) |
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| omit [Nonempty S] in |
| lemma decoratedRootPotential_measurable (step : X×S → X) (hs : Measurable step) |
| (n : ℕ) (F : Fin (n+1) → X×S → ℝ) (hF : ∀ i, Measurable (F i)) : |
| Measurable (decoratedRootPotential step n F) := |
| ((hF 0).comp (measurable_fst.prodMk measurable_snd.fst)).add |
| (((decoratedWeightedTotal_measurable step hs n _ (fun i => hF i.succ)).comp |
| ((hs.comp (measurable_fst.prodMk measurable_snd.fst)).prodMk measurable_snd.snd)).log) |
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| lemma decoratedRootPotential_integrable (ν : ProbabilityMeasure S) (step : X×S → X) |
| (hs : Measurable step) (n : ℕ) (z : Fin (n+1) → ℝ) (hz : StrictMono z) |
| (hz0 : ∀ i, 0 < z i) (hz1 : ∀ i, z i < 1) |
| (F : Fin (n+1) → X×S → ℝ) (hF : ∀ i, Measurable (F i)) |
| (hI : ∀ i x, Integrable (fun s => Real.exp (z i*F i (x,s))) ν) |
| (hM : ∀ i x, (∫ s, Real.exp (z i*F i (x,s)) ∂ν) = 1) (x : X) : |
| Integrable (fun p => Real.exp (z 0*decoratedRootPotential step n F (x,p))) |
| ((ν : Measure S).prod (decoratedCascadeLaw ν n (fun i => z i.succ))) := by |
| let ρ : Measure (DecoratedCascade S n) := decoratedCascadeLaw ν n (fun i => z i.succ) |
| let θ : Measure (StableCascade n) := cascadeLaw n (fun i => z i.succ) |
| let T : S×DecoratedCascade S n → StableCascade n := fun p => |
| decoratedCascadeTransform step n (fun i => F i.succ) (step (x,p.1),p.2) |
| have hT : Measurable T := (decoratedCascadeTransform_measurable step hs n _ |
| (fun i => hF i.succ)).comp |
| ((hs.comp (measurable_const.prodMk measurable_fst)).prodMk measurable_snd) |
| have hLaw (s) : ρ.map (fun c => T (s,c)) = θ := by |
| exact decoratedCascadeTransform_law ν step hs n (fun i => z i.succ) |
| (fun i => hz0 i.succ) (fun i => hz1 i.succ) (fun i => F i.succ) |
| (fun i => hF i.succ) (fun i => hI i.succ) (fun i => hM i.succ) (step (x,s)) |
| have hp : MeasurePreserving (fun p : S×DecoratedCascade S n => (p.1,T p)) |
| ((ν : Measure S).prod ρ) ((ν : Measure S).prod θ) := |
| ⟨measurable_fst.prodMk hT,descendant_transform_independent _ _ _ T hT hLaw⟩ |
| have htail := cascadeLogTotalMoment_integrable n (fun i => z i.succ) |
| (fun i j h => hz (Fin.succ_lt_succ_iff.mpr h)) (fun i => hz0 i.succ) |
| (fun i => hz1 i.succ) (z 0) (fun i => hz (by simp)) |
| have hprod := (hI 0 x).mul_prod htail |
| have he := (hp.integrable_comp hprod.aestronglyMeasurable).mpr hprod |
| apply he.congr |
| filter_upwards [] with p |
| dsimp [Function.comp,T,decoratedRootPotential] |
| rw [decoratedWeightedTotal_transform step hs n _ (fun i => hF i.succ),mul_add,Real.exp_add] |
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| lemma decoratedWeightedTotal_succ_eq (ν : ProbabilityMeasure S) (step : X×S → X) |
| (hs : Measurable step) (n : ℕ) (z : Fin (n+1) → ℝ) (hz : StrictMono z) |
| (hz0 : ∀ i, 0 < z i) (hz1 : ∀ i, z i < 1) |
| (F : Fin (n+1) → X×S → ℝ) (hF : ∀ i, Measurable (F i)) |
| (hI : ∀ i x, Integrable (fun s => Real.exp (z i*F i (x,s))) ν) |
| (hM : ∀ i x, (∫ s, Real.exp (z i*F i (x,s)) ∂ν) = 1) (x : X) : |
| ∀ᵐ η ∂(decoratedCascadeLaw ν (n+1) z : Measure (DecoratedCascade S (n+1))), |
| decoratedWeightedTotal step (n+1) F (x,η) = |
| poissonWeightedTotal (fun p => decoratedRootPotential step n F (x,p)) η := by |
| have hT : Measurable (fun η : DecoratedCascade S (n+1) => |
| decoratedCascadeTransform step (n+1) F (x,η)) := |
| (decoratedCascadeTransform_measurable step hs (n+1) F hF).comp |
| (measurable_const.prodMk measurable_id) |
| have hp : MeasurePreserving (fun η => decoratedCascadeTransform step (n+1) F (x,η)) |
| (decoratedCascadeLaw ν (n+1) z : Measure (DecoratedCascade S (n+1))) (cascadeLaw (n+1) z) := |
| ⟨hT,decoratedCascadeTransform_law ν step hs (n+1) z hz0 hz1 F hF hI hM x⟩ |
| have htail := (cascadeTotal_regular n (fun i => z i.succ) |
| (fun i j h => hz (Fin.succ_lt_succ_iff.mpr h)) |
| (fun i => hz0 i.succ) (fun i => hz1 i.succ)).1 |
| have hae := hp.quasiMeasurePreserving.ae (cascadeTotal_succ_eq z htail) |
| have hreg := markedStable_regular |
| ((ν : Measure S).prod (decoratedCascadeLaw ν n (fun i => z i.succ) : Measure (DecoratedCascade S n))) (hz0 0) (hz1 0) |
| filter_upwards [hae,hreg] with η hη hr |
| rw [decoratedWeightedTotal_transform step hs (n+1) F hF,hη] |
| change poissonWeightedTotal (fun C => Real.log (cascadeTotal n C)) |
| ((markedStableCountKernel η).map (fun q : ℝ×(S×DecoratedCascade S n) => |
| (q.1+F 0 (x,q.2.1),decoratedCascadeTransform step n (fun i => F i.succ) |
| (step (x,q.2.1),q.2.2)))) = _ |
| rw [markedStableCountKernel_eq η hr.1 hr.2] |
| have hmap : Measurable (fun q : ℝ×(S×DecoratedCascade S n) => |
| (q.1+F 0 (x,q.2.1),decoratedCascadeTransform step n (fun i => F i.succ) |
| (step (x,q.2.1),q.2.2))) := |
| (measurable_fst.add ((hF 0).comp (measurable_const.prodMk measurable_snd.fst))).prodMk |
| ((decoratedCascadeTransform_measurable step hs n _ (fun i => hF i.succ)).comp |
| ((hs.comp (measurable_const.prodMk measurable_snd.fst)).prodMk measurable_snd.snd)) |
| unfold poissonWeightedTotal |
| have hobs : Measurable (fun p : ℝ×StableCascade n => |
| ENNReal.ofReal (Real.exp (p.1+Real.log (cascadeTotal n p.2)))) := |
| (measurable_fst.add ((cascadeTotal_measurable n).comp measurable_snd).log).exp.ennreal_ofReal |
| rw [lintegral_map hobs hmap] |
| congr 1 |
| apply lintegral_congr |
| intro q |
| dsimp [decoratedRootPotential] |
| rw [decoratedWeightedTotal_transform step hs n _ (fun i => hF i.succ),add_assoc] |
|
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| lemma decoratedBiasedLaw_succ_eq_weighted (ν : ProbabilityMeasure S) (step : X×S → X) |
| (hs : Measurable step) (n : ℕ) (z : Fin (n+1) → ℝ) (hz : StrictMono z) |
| (hz0 : ∀ i, 0 < z i) (hz1 : ∀ i, z i < 1) |
| (F : Fin (n+1) → X×S → ℝ) (hF : ∀ i, Measurable (F i)) |
| (hI : ∀ i x, Integrable (fun s => Real.exp (z i*F i (x,s))) ν) |
| (hM : ∀ i x, (∫ s, Real.exp (z i*F i (x,s)) ∂ν) = 1) (x : X) (a : ℝ) : |
| decoratedBiasedLaw ν step (n+1) z F x a = weightedTotalBiasedLaw |
| ((ν : Measure S).prod (decoratedCascadeLaw ν n (fun i => z i.succ))) a (z 0) |
| (fun p => decoratedRootPotential step n F (x,p)) := by |
| unfold decoratedBiasedLaw weightedTotalBiasedLaw |
| congr 1 |
| apply withDensity_congr_ae |
| filter_upwards [decoratedWeightedTotal_succ_eq ν step hs n z hz hz0 hz1 F hF hI hM x] with η hη |
| rw [hη] |
|
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| end SphericalPerceptronFreeEnergy |
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| end |
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| end OAI |
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