File size: 8,724 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 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 | import OAI.Probability.FreeIsing.Conditioning.Copies
import OAI.Probability.FreeIsing.Conditioning.KernelExtensionConditionalExpectation
import OAI.Probability.FreeIsing.Conditioning.Pullback
import OAI.Probability.FreeIsing.Estimates.KernelCompatibility
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
namespace Problem367.ConditionalCopyImmersion
variable {W O P X : Type*}
[mW : MeasurableSpace W] [mO : MeasurableSpace O]
[mP : MeasurableSpace P] [mX : MeasurableSpace X]
theorem map_onePast
(ρ : Measure W) [IsProbabilityMeasure ρ]
(η : Kernel W O) [IsMarkovKernel η]
(p : W → P) (hp : Measurable p)
(past : O → X) (hpast : Measurable past)
(κ : Kernel P X) [IsMarkovKernel κ]
(hκ : (fun w => (η.map past) w) =ᵐ[ρ] (fun w => κ (p w))) :
(ρ ⊗ₘ (η ×ₖ η)).map
(fun q : W × (O × O) => ((q.1, q.2.1), past q.2.2)) =
(ρ ⊗ₘ η) ⊗ₘ κ.comap (fun q : W × O => p q.1) (hp.comp measurable_fst) := by
apply Measure.ext_of_lintegral
intro f hf
rw [lintegral_map hf (by fun_prop), Measure.lintegral_compProd (by fun_prop),
Measure.lintegral_compProd hf, Measure.lintegral_compProd (by fun_prop)]
apply lintegral_congr_ae
filter_upwards [hκ] with w hw
rw [Kernel.prod_apply, lintegral_prod _ (by fun_prop)]
apply lintegral_congr
intro o
simp only [Kernel.comap_apply]
rw [← hw, Kernel.map_apply _ hpast, lintegral_map (by fun_prop) hpast]
theorem swap_measurePreserving
(ρ : Measure W) [IsProbabilityMeasure ρ]
(η : Kernel W O) [IsMarkovKernel η] :
MeasurePreserving (fun q : W × (O × O) => (q.1, (q.2.2, q.2.1)))
(ρ ⊗ₘ (η ×ₖ η)) (ρ ⊗ₘ (η ×ₖ η)) := by
refine ⟨by fun_prop, ?_⟩
change (ρ ⊗ₘ (η ×ₖ η)).map (Prod.map id Prod.swap) = _
rw [← Measure.compProd_map measurable_swap, Kernel.map_prod_swap]
theorem map_onePast_right
(ρ : Measure W) [IsProbabilityMeasure ρ]
(η : Kernel W O) [IsMarkovKernel η]
(p : W → P) (hp : Measurable p)
(past : O → X) (hpast : Measurable past)
(κ : Kernel P X) [IsMarkovKernel κ]
(hκ : (fun w => (η.map past) w) =ᵐ[ρ] (fun w => κ (p w))) :
(ρ ⊗ₘ (η ×ₖ η)).map
(fun q : W × (O × O) => ((q.1, q.2.2), past q.2.1)) =
(ρ ⊗ₘ η) ⊗ₘ κ.comap (fun q : W × O => p q.1) (hp.comp measurable_fst) := by
have hs := (swap_measurePreserving ρ η).map_eq
have he := congrArg (fun m : Measure (W × (O × O)) =>
m.map (fun q => ((q.1, q.2.1), past q.2.2))) hs
rw [Measure.map_map (by fun_prop) (by fun_prop)] at he
exact he.trans (map_onePast ρ η p hp past hpast κ hκ)
theorem exists_past_kernel_of_reconstructible_split
{Z : Type*} [MeasurableSpace Z] [StandardBorelSpace X] [Nonempty X]
(ρ : Measure W) [IsProbabilityMeasure ρ]
(p : W → P) (hp : Measurable p) (future : W → Z) (hf : Measurable future)
(hsplit : MeasurableEmbedding (fun w => (p w, future w)))
(η : Kernel W O) [IsMarkovKernel η]
(past : O → X) (hpast : Measurable past)
(hind : IndepFun (fun q : W × O => (p q.1, past q.2))
(fun q => future q.1) (ρ ⊗ₘ η)) :
∃ κ : Kernel P X, IsMarkovKernel κ ∧
(fun w => (η.map past) w) =ᵐ[ρ] (fun w => κ (p w)) := by
let : IsProbabilityMeasure (ρ.map p) := inferInstance
let : IsProbabilityMeasure (ρ.map future) := inferInstance
have hbase : IndepFun (fun q : W × O => p q.1)
(fun q => future q.1) (ρ ⊗ₘ η) := hind.comp measurable_fst measurable_id
have hlaw := hbase.map_prod_eq_prod_map_map (by fun_prop) (by fun_prop)
change (ρ ⊗ₘ η).map ((fun w => (p w, future w)) ∘ Prod.fst) =
((ρ ⊗ₘ η).map (p ∘ Prod.fst)).prod
((ρ ⊗ₘ η).map (future ∘ Prod.fst)) at hlaw
rw [← Measure.map_map (hp.prodMk hf) measurable_fst,
← Measure.map_map hp measurable_fst, ← Measure.map_map hf measurable_fst] at hlaw
change ((ρ ⊗ₘ η).fst.map (fun w => (p w, future w))) =
((ρ ⊗ₘ η).fst.map p).prod ((ρ ⊗ₘ η).fst.map future) at hlaw
rw [Measure.fst_compProd] at hlaw
exact KernelCompatibility.exists_past_kernel_of_indep_of_embedding ρ
(ρ.map p) (ρ.map future) p hp future hf hsplit hlaw η past hpast hind
theorem condExp_joinedPast_left_of_kernel_extension
(ρ : Measure W) [IsProbabilityMeasure ρ]
(η : Kernel W O) [IsMarkovKernel η]
(p : W → P) (hp : Measurable p)
(past : O → X) (hpast : Measurable past)
(κ : Kernel P X) [IsMarkovKernel κ]
(hκ : (fun w => (η.map past) w) =ᵐ[ρ] (fun w => κ (p w)))
{m : MeasurableSpace (W × O)} (hm : m ≤ mW.prod mO)
{f : W × O → ℝ} (hf : Integrable f (ρ ⊗ₘ η))
(hCE : let Q := (ρ ⊗ₘ η) ⊗ₘ
κ.comap (fun q : W × O => p q.1) (hp.comp measurable_fst)
Q[(fun z : (W × O) × X => f z.1) | m.prod inferInstance] =ᵐ[Q]
fun z => (ρ ⊗ₘ η)[f | m] z.1) :
(ρ ⊗ₘ (η ×ₖ η))[(fun q : W × (O × O) => f (q.1, q.2.1)) |
m.comap (fun q : W × (O × O) => (q.1, q.2.1)) ⊔
MeasurableSpace.comap (fun q : W × (O × O) => past q.2.2) inferInstance] =ᵐ[ρ ⊗ₘ (η ×ₖ η)]
fun q => (ρ ⊗ₘ η)[f | m] (q.1, q.2.1) := by
let : MeasurableSpace (W × O) := mW.prod mO
let Q := (ρ ⊗ₘ η) ⊗ₘ κ.comap (fun q : W × O => p q.1) (hp.comp measurable_fst)
let ψ : W × (O × O) → (W × O) × X := fun q => ((q.1, q.2.1), past q.2.2)
have hψ : MeasurePreserving ψ (ρ ⊗ₘ (η ×ₖ η)) Q :=
⟨by fun_prop, map_onePast ρ η p hp past hpast κ hκ⟩
have hfst : MeasurePreserving (Prod.fst : (W × O) × X → W × O) Q (ρ ⊗ₘ η) :=
⟨measurable_fst, Measure.fst_compProd _ _⟩
have hprod : m.prod (inferInstance : MeasurableSpace X) ≤
(inferInstance : MeasurableSpace ((W × O) × X)) := by
exact sup_le_sup (MeasurableSpace.comap_mono hm) le_rfl
have hj := condExp_comp_measurePreserving (ρ ⊗ₘ (η ×ₖ η)) Q ψ hψ
(m.prod inferInstance) hprod (hfst.integrable_comp_of_integrable hf)
have he := hψ.quasiMeasurePreserving.ae hCE
have htrans := hj.trans he
simpa only [MeasurableSpace.prod, MeasurableSpace.comap_sup,
MeasurableSpace.comap_comp, Function.comp_def, ψ] using htrans
theorem condExp_joinedPast_left
(ρ : Measure W) [IsProbabilityMeasure ρ]
(η : Kernel W O) [IsMarkovKernel η]
(p : W → P) (hp : Measurable p)
(past : O → X) (hpast : Measurable past)
(κ : Kernel P X) [IsMarkovKernel κ]
(hκ : (fun w => (η.map past) w) =ᵐ[ρ] (fun w => κ (p w)))
{m : MeasurableSpace (W × O)} (hm : m ≤ mW.prod mO)
(hpM : Measurable[m] (fun q : W × O => p q.1))
{f : W × O → ℝ} (hf : Integrable f (ρ ⊗ₘ η)) :
(ρ ⊗ₘ (η ×ₖ η))[(fun q : W × (O × O) => f (q.1, q.2.1)) |
m.comap (fun q : W × (O × O) => (q.1, q.2.1)) ⊔
MeasurableSpace.comap (fun q : W × (O × O) => past q.2.2) mX] =ᵐ[ρ ⊗ₘ (η ×ₖ η)]
fun q => (ρ ⊗ₘ η)[f | m] (q.1, q.2.1) := by
let : MeasurableSpace (W × O) := mW.prod mO
apply condExp_joinedPast_left_of_kernel_extension ρ η p hp past hpast κ hκ hm hf
exact KernelExtensionConditionalExpectation.condExp_fst_compProd hm hpM κ hf
theorem martingale_left
{I : Type*} [Preorder I]
(ρ : Measure W) [IsProbabilityMeasure ρ]
(η : Kernel W O) [IsMarkovKernel η]
(p : I → W → P) (hp : ∀ t, Measurable (p t))
(past : I → O → X) (hpast : ∀ t, Measurable (past t))
(F : Filtration I (mW.prod mO))
(G : Filtration I (mW.prod (mO.prod mO)))
(hG : ∀ t, G t =
(F t).comap (fun q : W × (O × O) => (q.1, q.2.1)) ⊔
MeasurableSpace.comap (fun q : W × (O × O) => past t q.2.2) mX)
(hpF : ∀ t, Measurable[F t] (fun q : W × O => p t q.1))
(hcompatible : ∀ t, ∃ κ : Kernel P X, IsMarkovKernel κ ∧
(fun w => (η.map (past t)) w) =ᵐ[ρ] fun w => κ (p t w))
{M : I → W × O → ℝ} (hM : Martingale M F (ρ ⊗ₘ η)) :
Martingale (fun t (q : W × (O × O)) => M t (q.1, q.2.1)) G (ρ ⊗ₘ (η ×ₖ η)) := by
have hpair := measurePreserving_conditionalCopy_fst ρ η
refine ⟨?_, ?_⟩
· intro t
apply (hM.stronglyAdapted t).comp_measurable
rw [hG t]
exact Measurable.of_comap_le le_sup_left
· intro s t hst
obtain ⟨κ, hκ, hcompat⟩ := hcompatible s
let := hκ
rw [hG s]
exact (condExp_joinedPast_left ρ η (p s) (hp s) (past s) (hpast s)
κ hcompat (F.le s) (hpF s) (hM.integrable t)).trans
(hpair.quasiMeasurePreserving.ae (hM.condExp_ae_eq hst))
end Problem367.ConditionalCopyImmersion
end
end OAI
|