File size: 4,504 Bytes
67aee1d | 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 | import Mathlib
import OAI.Analysis.HardSphere.Conditioning.SymmetricTag
namespace OAI
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal
namespace HardSphere.Comparison
variable {D X Y : Type*} [MeasurableSpace D] [MeasurableSpace X] [MeasurableSpace Y]
def retainedObserver (κ : Kernel (D×X) Y) : Kernel (D×X) (D×Y) :=
(Kernel.deterministic Prod.fst measurable_fst) ×ₖ κ
instance retainedObserver_markov (κ : Kernel (D×X) Y) [IsMarkovKernel κ] :
IsMarkovKernel (retainedObserver κ) := by unfold retainedObserver; infer_instance
lemma retainedObserver_apply (κ : Kernel (D×X) Y) [IsSFiniteKernel κ] (z : D×X) :
retainedObserver κ z=(κ z).map (Prod.mk z.1) := by
rw [retainedObserver,Kernel.prod_apply,Kernel.deterministic_apply,Measure.dirac_prod]
theorem retainedObserver_compProd (μ : Measure D) [SFinite μ]
(χ : Kernel D X) [IsSFiniteKernel χ] (κ : Kernel (D×X) Y) [IsSFiniteKernel κ] :
retainedObserver κ ∘ₘ (μ.compProd χ)=μ.compProd (χ⊗ₖκ).snd := by
ext A hA
rw [Measure.bind_apply hA (retainedObserver κ).aemeasurable,
Measure.lintegral_compProd ((retainedObserver κ).measurable_coe hA),Measure.compProd_apply hA]
apply lintegral_congr
intro d
rw [Kernel.snd_apply' _ _ (hA.preimage measurable_prodMk_left),Kernel.compProd_apply]
· apply lintegral_congr
intro w
rw [retainedObserver_apply,Measure.map_apply measurable_prodMk_left hA]
rfl
· exact (hA.preimage measurable_prodMk_left).preimage measurable_snd
variable {Q I W : Type*} [MeasurableSpace Q] [MeasurableSpace I] [MeasurableSpace W]
lemma conditional_tag_observation (μ : Measure D) [SFinite μ]
(σ : Kernel D Q) [IsSFiniteKernel σ] (χ : Kernel (D×Q) X) [IsSFiniteKernel χ]
(tag : Q→I) (ht : Measurable tag) (obs : Kernel (I×(D×X)) Y) [IsSFiniteKernel obs] :
((Kernel.deterministic (fun z : I×(D×X) ↦ z.2.1) measurable_snd.fst) ×ₖ obs) ∘ₘ
(((μ.compProd (σ⊗ₖχ)).map (fun z ↦ (tag z.2.1,(z.1,z.2.2))))) =
μ.compProd (((σ⊗ₖχ)⊗ₖ(obs.comap (fun z ↦ (tag z.2.1,(z.1,z.2.2)))
((ht.comp measurable_snd.fst).prodMk (measurable_fst.prodMk measurable_snd.snd)))).snd) := by
have hm : Measurable (fun z : D×(Q×X) ↦ (tag z.2.1,(z.1,z.2.2))) :=
(ht.comp measurable_snd.fst).prodMk (measurable_fst.prodMk measurable_snd.snd)
rw [←comp_comap_map _ _ hm]
have he : (((Kernel.deterministic (fun z : I×(D×X) ↦ z.2.1) measurable_snd.fst) ×ₖ obs).comap
(fun z : D×(Q×X) ↦ (tag z.2.1,(z.1,z.2.2))) hm)=
retainedObserver (obs.comap (fun z : D×(Q×X) ↦ (tag z.2.1,(z.1,z.2.2))) hm) := by
ext z A hA
simp only [Kernel.comap_apply, retainedObserver, Kernel.prod_apply, Kernel.deterministic_apply]
rw [he,retainedObserver_compProd]
lemma conditional_observation_map (σ : Kernel D Q) [IsSFiniteKernel σ]
(ψ : Kernel (D×Q) W) [IsSFiniteKernel ψ] (v : W→X) (hv : Measurable v)
(tag : Q→I) (ht : Measurable tag) (obs : Kernel (I×(D×X)) Y) [IsSFiniteKernel obs] :
(((σ⊗ₖψ.map v)⊗ₖobs.comap (fun z : D×(Q×X) ↦ (tag z.2.1,(z.1,z.2.2)))
((ht.comp measurable_snd.fst).prodMk (measurable_fst.prodMk measurable_snd.snd))).snd)=
(((σ⊗ₖψ)⊗ₖobs.comap (fun z : D×(Q×W) ↦ (tag z.2.1,(z.1,v z.2.2)))
((ht.comp measurable_snd.fst).prodMk (measurable_fst.prodMk (hv.comp measurable_snd.snd)))).snd) := by
ext d A hA
rw [Kernel.snd_apply' _ _ hA,Kernel.snd_apply' _ _ hA,
Kernel.compProd_apply (measurable_snd hA),Kernel.compProd_apply (measurable_snd hA)]
have hleft : Measurable (fun z : Q×X ↦ obs (tag z.1,(d,z.2)) A) :=
(obs.measurable_coe hA).comp ((ht.comp measurable_fst).prodMk (measurable_const.prodMk measurable_snd))
have hright : Measurable (fun z : Q×W ↦ obs (tag z.1,(d,v z.2)) A) :=
(obs.measurable_coe hA).comp ((ht.comp measurable_fst).prodMk (measurable_const.prodMk (hv.comp measurable_snd)))
change (∫⁻z,obs (tag z.1,(d,z.2)) A ∂(σ⊗ₖψ.map v) d)=
∫⁻z,obs (tag z.1,(d,v z.2)) A ∂(σ⊗ₖψ) d
rw [Kernel.lintegral_compProd σ (ψ.map v) d hleft,Kernel.lintegral_compProd σ ψ d hright]
apply lintegral_congr
intro q
rw [Kernel.map_apply _ hv]
change (∫⁻ c, obs (tag q,(d,c)) A ∂Measure.map v (ψ (d,q))) =
∫⁻ c, obs (tag q,(d,v c)) A ∂ψ (d,q)
exact lintegral_map (hleft.comp (measurable_prodMk_left (x:=q))) hv
end HardSphere.Comparison
end
end
end OAI
|