File size: 3,439 Bytes
f61e547 | 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 72 73 74 75 76 77 78 | import OAI.Probability.FreeIsing.Estimates.ObservableDisintegration
import OAI.Probability.FreeIsing.Posterior.ProjectedObservationRecovery
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367
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)
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
end Problem367
end
end OAI
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