openAI-math-dataset / lean /OAI /Probability /FreeIsing /Conditioning /ConstructedConditionalCopies.lean
| 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 | |