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180bc03 | 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 | import Mathlib
import OAI.Probability.Ballisticity.Estimates.ReferenceContinuity
import OAI.Probability.Ballisticity.Entropy.WindowProjectionEntropy
namespace OAI
section
open MeasureTheory ProbabilityTheory InformationTheory
open scoped ENNReal Classical
namespace DirectionalTransience.ReferenceClasses
variable {H : Type*} [AddCommGroup H]
noncomputable def anchorOffsets (U : ℕ → H) : Data H :=
⟨fun p => ((U p.2-U p.1 : H) : OnePoint H), by
constructor
· intro a; simp
· intro a b z h
simp only [OnePoint.coe_eq_coe] at h ⊢
rw [← h]; abel
· intro a b c z w h h'
simp only [OnePoint.coe_eq_coe] at h h' ⊢
rw [← h,← h']; abel⟩
lemma anchorOffsets_fibers (U : ℕ → H) (p q : Site H) :
inputSite (anchorOffsets U) p = inputSite (anchorOffsets U) q ↔
(p.2.1,U p.1+p.2.2) = (q.2.1,U q.1+q.2.2) := by
rcases p with ⟨a,l,z⟩
rcases q with ⟨b,l',z'⟩
rw [inputSite_eq_iff]
change (l=l' ∧ ((U b-U a:H):OnePoint H)=((z-z':H):OnePoint H)) ↔ _
simp only [OnePoint.coe_eq_coe,Prod.mk.injEq]
constructor
· rintro ⟨hl,h⟩
refine ⟨hl,?_⟩
calc U a+z = U a+(z-z')+z' := by abel
_ = U a+(U b-U a)+z' := by rw [h]
_ = U b+z' := by abel
· rintro ⟨hl,h⟩
refine ⟨hl,?_⟩
calc U b-U a = (U b+z')-(U a+z)+z-z' := by abel
_ = z-z' := by rw [h]; abel
noncomputable def anchorFields {R : Type*} (U : ℕ → H) (ξ : UpperField H R) : AllFields H R :=
fun p => ξ (p.2.1,U p.1+p.2.2)
section Measurable
variable [Countable H] [TopologicalSpace H] [DiscreteTopology H]
[MeasurableSpace H] [BorelSpace H]
[MeasurableSpace (OnePoint H)] [BorelSpace (OnePoint H)]
{R : Type*} [MeasurableSpace R]
lemma anchorOffsets_measurable : Measurable (anchorOffsets : (ℕ → H) → Data H) := by
apply Measurable.subtype_mk
apply Measurable.of_eval
intro p
exact (OnePoint.continuous_coe.measurable).comp
((measurable_pi_apply p.2).sub (measurable_pi_apply p.1))
omit [MeasurableSpace (OnePoint H)] [BorelSpace (OnePoint H)] in
lemma anchorFields_measurable :
Measurable (fun p : (ℕ → H) × UpperField H R => anchorFields p.1 p.2) := by
apply Measurable.of_eval
intro p
have hev : Measurable (fun q : UpperField H R × (ℕ × H) => q.1 q.2) :=
measurable_from_prod_countable_left fun q => measurable_pi_apply q
exact hev.comp (measurable_snd.prodMk (measurable_const.prodMk
(((measurable_pi_apply p.1).comp measurable_fst).add measurable_const)))
theorem anchorFields_reference (ν : Measure R) [IsProbabilityMeasure ν] (U : ℕ → H) :
(Measure.infinitePi (fun _ : ℕ × H => ν)).map (anchorFields U) = reference ν (anchorOffsets U) := by
rw [reference_apply]
exact map_iid_eq_of_same_fibers ν (fun p : Site H => (p.2.1,U p.1+p.2.2))
(inputSite (anchorOffsets U)) (fun p q => (anchorOffsets_fibers U p q).symm)
theorem anchorFrame_kl_le {Z P : Type*} [MeasurableSpace Z] [MeasurableSpace P]
(μ : Measure (Z × UpperField H R)) [IsFiniteMeasure μ]
(ν : Measure R) [IsProbabilityMeasure ν]
(U : Z → ℕ → H) (hU : Measurable U)
(profile : Z → P) (hp : Measurable profile) :
let out := μ.map (fun p => ((anchorOffsets (U p.1),profile p.1),anchorFields (U p.1) p.2))
klDiv out (out.fst.compProd ((reference (H:=H) ν).comap Prod.fst measurable_fst)) ≤
klDiv μ (μ.fst.prod (Measure.infinitePi (fun _ : ℕ × H => ν))) := by
apply StoppedWindow.own_marginal_projection μ _
(fun z => (anchorOffsets (U z),profile z)) ((anchorOffsets_measurable.comp hU).prodMk hp)
(fun z ξ => anchorFields (U z) ξ)
(anchorFields_measurable.comp ((hU.comp measurable_fst).prodMk measurable_snd))
((reference (H:=H) ν).comap Prod.fst measurable_fst)
intro z
exact anchorFields_reference ν (U z)
end Measurable
end DirectionalTransience.ReferenceClasses
end
end OAI
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