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a2ee19a | 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 | import Mathlib
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal
noncomputable section
namespace PosteriorReplicas
@[fun_prop] lemma measurable_fin_cons {Y : Type*} [MeasurableSpace Y] {n : ℕ} :
Measurable (fun z : Y × (Fin n → Y) => (Fin.cons z.1 z.2 : Fin (n+1) → Y)) := by
apply measurable_pi_iff.mpr
intro i
refine Fin.cases ?_ (fun j => ?_) i
· simpa only [Fin.cons_zero] using (measurable_fst : Measurable (Prod.fst : Y × (Fin n → Y) → Y))
· simpa only [Fin.cons_succ, Function.comp_def] using (measurable_pi_apply j).comp measurable_snd
def piKernel {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] :
{n : ℕ} → (Fin n → Kernel X Y) → Kernel X (Fin n → Y)
| 0, _ => Kernel.deterministic (fun _ i => Fin.elim0 i) (by fun_prop)
| n+1, κ => ((κ 0) ×ₖ piKernel (fun i : Fin n => κ i.succ)).map
(fun z => Fin.cons z.1 z.2)
instance piKernel_markov {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
{n : ℕ} (κ : Fin n → Kernel X Y) [∀ i, IsMarkovKernel (κ i)] :
IsMarkovKernel (piKernel κ) := by
induction n with
| zero => unfold piKernel; infer_instance
| succ n ih =>
have : IsMarkovKernel (piKernel (fun i : Fin n => κ i.succ)) := ih _
unfold piKernel
exact Kernel.IsMarkovKernel.map _ (by fun_prop)
lemma piKernel_apply {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
{n : ℕ} (κ : Fin n → Kernel X Y) [∀ i, IsMarkovKernel (κ i)] (x : X) :
piKernel κ x = Measure.pi (fun i => κ i x) := by
induction n with
| zero =>
rw [piKernel, Kernel.deterministic_apply, Measure.pi_of_empty _ (fun i => Fin.elim0 i)]
| succ n ih =>
have hκ : ∀ i : Fin n, IsMarkovKernel (κ i.succ) := inferInstance
rw [piKernel, Kernel.map_apply _ (by fun_prop), Kernel.prod_apply, ih]
let e := (MeasurableEquiv.piFinSuccAbove (fun _ : Fin (n+1) => Y) 0).symm
have he : (fun z : Y × (Fin n → Y) => Fin.cons z.1 z.2) = e := by
funext z
simp only [e, MeasurableEquiv.piFinSuccAbove_symm_apply, Fin.insertNthEquiv,
Fin.insertNth_zero, Fin.zero_succAbove, Equiv.coe_fn_mk, cast_eq]
rw [he]
exact (measurePreserving_piFinSuccAbove (fun i : Fin (n+1) => κ i x) 0).symm.map_eq
@[fun_prop] lemma measurable_piKernel {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
{n : ℕ} (κ : Fin n → Kernel X Y) [∀ i, IsMarkovKernel (κ i)] :
Measurable (fun x => Measure.pi (fun i => κ i x)) := by
simpa only [← piKernel_apply] using (piKernel κ).measurable
end PosteriorReplicas
end
end OAI
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