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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 79 80 81 82 83 84 85 86 87 88 89 | import OAI.Probability.FreeIsing.Estimates.FilteredFutureIndependence
import OAI.Probability.FreeIsing.Posterior.ContinuousObservationHistory
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.ConstructedConditionalCopies
open WienerSampler ContinuousObservationHistory
theorem indepFun_map_of_comp {Ω Z A B : Type*}
[MeasurableSpace Ω] [MeasurableSpace Z] [MeasurableSpace A] [MeasurableSpace B]
{P : Measure Ω} [IsProbabilityMeasure P]
{z : Ω → Z} (hz : Measurable z) {f : Z → A} {g : Z → B}
(hf : Measurable f) (hg : Measurable g)
(hind : IndepFun (f ∘ z) (g ∘ z) P) : IndepFun f g (P.map z) := by
let : IsProbabilityMeasure (P.map z) := inferInstance
rw [indepFun_iff_map_prod_eq_prod_map_map hf.aemeasurable hg.aemeasurable]
rw [Measure.map_map (hf.prodMk hg) hz, Measure.map_map hf hz, Measure.map_map hg hz]
exact hind.map_prod_eq_prod_map_map (hf.comp hz).aemeasurable (hg.comp hz).aemeasurable
theorem exists_compatible_conditionalCopies
{ι Ω : Type*} [Countable ι] [mΩ : MeasurableSpace Ω]
(P : Measure Ω) [IsProbabilityMeasure P]
(F : Filtration ℝ≥0 mΩ) (X : ℝ≥0 → Ω → ι → ℝ)
(hX : ∀ t, Measurable[F t] (X t))
(hB : ∀ i, IsPreBrownianReal (fun t ω => X t ω i) P)
(hind : ∀ s t, s ≤ t → Indep
(MeasurableSpace.comap (fun ω => X t ω - X s ω) inferInstance) (F s) P)
(Y : Ω → ι → Path) (hY : Measurable Y)
(hYad : ∀ i t, Measurable[F t] (fun ω => Y ω i t)) :
∃ (rho : Measure (ℝ≥0 → ι → ℝ)) (eta : Kernel (ℝ≥0 → ι → ℝ) (ι → Path)),
IsProbabilityMeasure rho ∧ IsMarkovKernel eta ∧
rho = P.map (fun ω t => X t ω) ∧
rho ⊗ₘ eta = P.map (fun ω => ((fun t => X t ω), Y ω)) ∧
(∀ i, IsPreBrownianReal
(fun t (q : (ℝ≥0 → ι → ℝ) × ((ι → Path) × (ι → Path))) => q.1 t i)
(rho ⊗ₘ (eta ×ₖ eta))) ∧
(∀ i, Martingale
(fun t (q : (ℝ≥0 → ι → ℝ) × ((ι → Path) × (ι → Path))) => q.1 t i)
ContinuousObservationHistory.joinedFiltration (rho ⊗ₘ (eta ×ₖ eta))) := by
let W : Ω → ℝ≥0 → ι → ℝ := fun ω t => X t ω
have hW : Measurable W := Measurable.of_eval fun t =>
(hX t).mono (F.le t) le_rfl
let Z : Ω → (ℝ≥0 → ι → ℝ) × (ι → Path) := fun ω => (W ω, Y ω)
have hZ : Measurable Z := hW.prodMk hY
let R := P.map Z
have : IsProbabilityMeasure R := inferInstance
let rho := R.fst
let eta := R.condKernel
have hrho : rho = P.map W := Measure.fst_map_prodMk hW hY
have hdecomp : rho ⊗ₘ eta = R := Measure.disintegrate R eta
have hBrho (i : ι) : IsPreBrownianReal (fun t (w : ℝ≥0 → ι → ℝ) => w t i) rho := by
rw [hrho]
constructor
intro I
have hm : Measurable (fun w : ℝ≥0 → ι → ℝ => I.restrict (fun t => w t i)) :=
Measurable.of_eval fun t => (measurable_pi_apply i).comp (measurable_pi_apply t.1)
refine ⟨hm.aemeasurable, ?_⟩
rw [Measure.map_map hm hW]
exact (hB i).hasLaw I |>.map_eq
have hOriginal (s : ℝ≥0) : IndepFun
(fun q : (ℝ≥0 → ι → ℝ) × (ι → Path) =>
(PathSplit.pastPath s q.1, stoppedArray s q.2))
(fun q => PathSplit.futureIncrements s q.1) (rho ⊗ₘ eta) := by
rw [hdecomp]
have hPast : Measurable[F s] (fun ω => stoppedArray s (Y ω)) :=
measurable_stoppedArray_of_eval (mΩ := F s) s Y
(fun i t ht => (hYad i t).mono (F.mono ht) le_rfl)
apply indepFun_map_of_comp hZ
(((PathSplit.measurable_pastPath s).comp measurable_fst).prodMk
((measurable_stoppedArray s).comp measurable_snd))
((PathSplit.measurable_futureIncrements s).comp measurable_fst)
exact FilteredFutureIndependence.indepFun_past_and_observable_strict_future
hX hind s hPast
exact ⟨rho, eta, inferInstance, inferInstance, hrho, hdecomp,
(fun i => ConditionalCopyArrayBrownian.isPreBrownian_commonNoise i rho (hBrho i) eta),
(fun i => ContinuousObservationHistory.commonNoise_martingale i rho (hBrho i) eta hOriginal)⟩
end Problem367.ConstructedConditionalCopies
end
end OAI
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