| 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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|
