| import OAI.Probability.FreeIsing.Posterior.ContinuousObservationHistory |
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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.ContinuousNoiseSplit |
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| open WienerSampler ContinuousObservationHistory |
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| def futurePath (s : ℝ≥0) (f : Path) : Path := |
| ⟨fun t => f (s + t) - f s, |
| (f.continuous.comp (continuous_const.add continuous_id)).sub continuous_const⟩ |
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| @[simp] |
| theorem futurePath_apply (s : ℝ≥0) (f : Path) (t : ℝ≥0) : |
| futurePath s f t = f (s + t) - f s := rfl |
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| theorem measurable_futurePath (s : ℝ≥0) : Measurable (futurePath s) := by |
| apply measurable_path_of_eval |
| intro t |
| exact (continuous_eval_const (s + t)).measurable.sub (continuous_eval_const s).measurable |
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| variable {ι : Type*} |
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| def futureArray (s : ℝ≥0) (f : ι → Path) : ι → Path := fun i => futurePath s (f i) |
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| theorem measurable_futureArray (s : ℝ≥0) : Measurable (futureArray (ι := ι) s) := |
| Measurable.of_eval fun i => (measurable_futurePath s).comp (measurable_pi_apply i) |
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| theorem injective_split (s : ℝ≥0) : Function.Injective |
| (fun f : ι → Path => (stoppedArray s f, futureArray s f)) := by |
| intro f g h |
| have hp := congrArg Prod.fst h |
| have hf := congrArg Prod.snd h |
| funext i |
| apply ContinuousMap.ext |
| intro t |
| have hs : f i s = g i s := by |
| have hh := congrArg (fun a : ι → Path => a i s) hp |
| simpa only [stoppedArray, stoppedPath_apply, min_self] using hh |
| by_cases hts : t ≤ s |
| · have hh := congrArg (fun a : ι → Path => a i t) hp |
| simpa only [stoppedArray, stoppedPath_apply, min_eq_left hts] using hh |
| · have hst : s ≤ t := (lt_of_not_ge hts).le |
| have hh := congrArg (fun a : ι → Path => a i (t - s)) hf |
| simp only [futureArray, futurePath_apply, add_tsub_cancel_of_le hst, hs] at hh |
| linarith |
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| theorem measurableEmbedding_split [Countable ι] (s : ℝ≥0) : MeasurableEmbedding |
| (fun f : ι → Path => (stoppedArray s f, futureArray s f)) := |
| ((measurable_stoppedArray s).prodMk (measurable_futureArray s)).measurableEmbedding |
| (injective_split s) |
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| theorem indepFun_future_conditionalCopies [Countable ι] |
| {O X : Type*} [MeasurableSpace O] [MeasurableSpace X] |
| [StandardBorelSpace X] [Nonempty X] |
| (s : ℝ≥0) (rho : Measure (ι → Path)) [IsProbabilityMeasure rho] |
| (eta : Kernel (ι → Path) O) [IsMarkovKernel eta] |
| (past : O → X) (hpast : Measurable past) |
| (hind : IndepFun (fun q : (ι → Path) × O => (stoppedArray s q.1, past q.2)) |
| (fun q => futureArray s q.1) (rho ⊗ₘ eta)) : |
| IndepFun (fun q : (ι → Path) × (O × O) => |
| (stoppedArray s q.1, (past q.2.1, past q.2.2))) |
| (fun q => futureArray s q.1) (rho ⊗ₘ (eta ×ₖ eta)) := |
| KernelCompatibility.indepFun_future_conditionalCopies_of_reconstructible_split |
| rho (stoppedArray s) (measurable_stoppedArray s) |
| (futureArray s) (measurable_futureArray s) (measurableEmbedding_split s) |
| eta past hpast hind |
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| end Problem367.ContinuousNoiseSplit |
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| end |
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| end OAI |
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