| import OAI.Probability.FreeIsing.Estimates.ObservableDisintegration |
| import OAI.Probability.FreeIsing.Posterior.ProjectedObservationRecovery |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
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| namespace Problem367 |
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| private theorem measurable_time_state |
| {T W X Y : Type*} [MeasurableSpace T] [MeasurableSpace W] |
| [MeasurableSpace X] [MeasurableSpace Y] {F : T × X → Y} |
| (hF : Measurable F) : |
| Measurable (fun q : T × (W × X) => F (q.1, q.2.2)) := |
| hF.comp (by fun_prop) |
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| theorem isFactorOfIID_of_stationary_innovation_observable |
| {d : ℕ} {theta : ℝ} {A X : Type*} [MeasurableSpace A] |
| [StandardBorelSpace A] [Nonempty A] [MeasurableSpace X] |
| [StandardBorelSpace X] [Nonempty X] |
| (nu : Measure A) [IsProbabilityMeasure nu] |
| (P : Measure X) [IsProbabilityMeasure P] |
| (Z : X → TreeVertex d → A) (hZ : Measurable Z) |
| (hinput : P.map Z = Measure.infinitePi (fun _ : TreeVertex d => nu)) |
| (E : X → TreeVertex d → ℕ → ℝ) (hE : Measurable E) |
| (S : TreeAut d → X → X) (hS : ∀ g, MeasurePreserving (S g) P P) |
| (hEeq : ∀ g x, E (S g x) = relabel g (E x)) |
| (hZeq : ∀ g, ∀ᵐ x ∂P, Z (S g x) = relabel g (Z x)) |
| (o : TreeVertex d) |
| (hLaw : HasFreeIsingLaw theta (P.map (fun x => SpinRecovery.decodeArray (E x)))) |
| (H : ℝ≥0 → X → ℝ) (W : ℕ → (TreeVertex d → A) → ℝ) |
| (hm : Measurable (Function.uncurry (fun (t : ℝ) x => Real.tanh (H t.toNNReal x)))) |
| (hInt : ∀ t, Integrable (fun p => (H t p.2.1 - H t p.2.2) ^ 2) |
| ((P.map Z) ⊗ₘ ((ObservableDisintegration.kernel P Z) ×ₖ |
| (ObservableDisintegration.kernel P Z)))) |
| (hzero : ∀ t, (∫ p, (H t p.2.1 - H t p.2.2) ^ 2 |
| ∂(P.map Z) ⊗ₘ ((ObservableDisintegration.kernel P Z) ×ₖ |
| (ObservableDisintegration.kernel P Z))) = 0) |
| (hInnovation : ∀ n, ∀ᵐ x ∂P, |
| E x o n = W n (Z x) + nnrealTimePrimitive (fun t x => Real.tanh (H t x)) n x) : |
| IsFactorOfIID d theta := by |
| let kappa := ObservableDisintegration.kernel P Z |
| have hproj : MeasurePreserving Prod.snd |
| ((Measure.infinitePi (fun _ : TreeVertex d => nu)) ⊗ₘ kappa) P := by |
| simpa only [hinput] using ObservableDisintegration.measurePreserving_snd_kernel P hZ |
| refine isFactorOfIID_of_innovation_comparison nu kappa E hE S |
| (fun g => (hS g).measurable) hEeq o ?_ ?_ |
| (fun t p => H t p.2) W ?_ ?_ ?_ ?_ |
| · have hdecode : Measurable (fun x => SpinRecovery.decodeArray (E x)) := |
| SpinRecovery.measurable_decodeArray.comp hE |
| change HasFreeIsingLaw theta |
| (Measure.map ((fun x => SpinRecovery.decodeArray (E x)) ∘ Prod.snd) |
| ((Measure.infinitePi (fun _ : TreeVertex d => nu)) ⊗ₘ kappa)) |
| rw [← Measure.map_map hdecode measurable_snd, hproj.map_eq] |
| exact hLaw |
| · intro g |
| simpa only [hinput] using ObservableDisintegration.joint_kernel_invariant P hZ |
| (measurable_relabel g) (hS g) (hZeq g) |
| · exact measurable_time_state hm |
| · simpa only [hinput] using hInt |
| · simpa only [hinput] using hzero |
| · intro n |
| have hsupp : ∀ᵐ p ∂(Measure.infinitePi (fun _ : TreeVertex d => nu)) ⊗ₘ kappa, |
| Z p.2 = p.1 := by |
| simpa only [hinput] using ObservableDisintegration.ae_observable_eq_input P hZ |
| filter_upwards [hproj.quasiMeasurePreserving.ae (hInnovation n), hsupp] with p hp hs |
| rw [hs] at hp |
| exact hp |
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| end Problem367 |
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| end |
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| end OAI |
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