File size: 3,030 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 79 | import OAI.Probability.FreeIsing.Posterior.ContinuousObservationHistory
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.ContinuousNoiseSplit
open WienerSampler ContinuousObservationHistory
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⟩
@[simp]
theorem futurePath_apply (s : ℝ≥0) (f : Path) (t : ℝ≥0) :
futurePath s f t = f (s + t) - f s := rfl
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
variable {ι : Type*}
def futureArray (s : ℝ≥0) (f : ι → Path) : ι → Path := fun i => futurePath s (f i)
theorem measurable_futureArray (s : ℝ≥0) : Measurable (futureArray (ι := ι) s) :=
Measurable.of_eval fun i => (measurable_futurePath s).comp (measurable_pi_apply i)
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
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)
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
end Problem367.ContinuousNoiseSplit
end
end OAI
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