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9b6e014 | 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 | import Mathlib
import OAI.Probability.Ballisticity.Estimates.AnchorFrameReference
import OAI.Probability.Ballisticity.Geometry.RawWindow
import OAI.Probability.Ballisticity.Estimates.InputMarginal
namespace OAI
section
open MeasureTheory ProbabilityTheory InformationTheory
open scoped ENNReal Classical
namespace DirectionalTransience
abbrev CurrentProfiles {d : ℕ} (e : Direction d) :=
(ℕ × HorizontalSpace e) → Set.Icc (0 : ℝ) 1
abbrev CurrentWindow {d : ℕ} (e : Direction d) :=
(ReferenceClasses.Data (HorizontalSpace e) × CurrentProfiles e) ×
ReferenceClasses.AllFields (HorizontalSpace e) (Row d)
noncomputable def rawCurrentProfiles {d : ℕ} (e : Direction d)
(z : RawCurrentData e) : CurrentProfiles e :=
fun p => globalHorizontalCoordinate e z.1 z.2.1 (z.2.2 p.1 + p.2)
lemma rawCurrentProfileCoordinate_measurable {d : ℕ} (e : Direction d)
(a : ℕ) (z : HorizontalSpace e) :
Measurable (fun p : RawCurrentData e => globalHorizontalCoordinate e p.1 p.2.1 (p.2.2 a+z)) := by
have hw : Measurable (fun p : RawCurrentData e => p.2.1) := measurable_fst.comp measurable_snd
have hn : Measurable (fun p : RawCurrentData e => p.1) := measurable_fst
have hu : Measurable (fun p : RawCurrentData e => p.2.2 a) :=
(measurable_pi_apply a).comp (measurable_snd.comp measurable_snd)
have ha : Measurable (fun p : RawCurrentData e => ((p.2.1,p.1),p.2.2 a+z)) :=
(hw.prodMk hn).prodMk (hu.add_const z)
have hm := (globalHorizontalCoordinate_joint_measurable e).comp ha
simpa only [Function.comp_def] using hm
lemma rawCurrentProfiles_measurable {d : ℕ} (e : Direction d) :
Measurable (rawCurrentProfiles e) :=
Measurable.of_eval fun p => rawCurrentProfileCoordinate_measurable e p.1 p.2
noncomputable def rawCurrentProjection {d : ℕ} (e : Direction d)
(p : RawCurrentData e × LocalUpperField e) : CurrentWindow e :=
((ReferenceClasses.anchorOffsets p.1.2.2,rawCurrentProfiles e p.1),
ReferenceClasses.anchorFields p.1.2.2 p.2)
lemma rawCurrentProjection_measurable {d : ℕ} (e : Direction d) :
Measurable (rawCurrentProjection e) := by
have hU : Measurable (fun p : RawCurrentData e × LocalUpperField e => p.1.2.2) :=
measurable_snd.comp (measurable_snd.comp measurable_fst)
exact ((ReferenceClasses.anchorOffsets_measurable.comp hU).prodMk
((rawCurrentProfiles_measurable e).comp measurable_fst)).prodMk
(ReferenceClasses.anchorFields_measurable.comp (hU.prodMk measurable_snd))
noncomputable def currentWindowReference {d : ℕ} (e : Direction d) (ν : Measure (Row d))
[IsProbabilityMeasure ν] :
Kernel (ReferenceClasses.Data (HorizontalSpace e) × CurrentProfiles e)
(ReferenceClasses.AllFields (HorizontalSpace e) (Row d)) :=
(ReferenceClasses.reference ν).comap Prod.fst measurable_fst
instance currentWindowReference_markov {d : ℕ} (e : Direction d) (ν : Measure (Row d))
[IsProbabilityMeasure ν] : IsMarkovKernel (currentWindowReference e ν) := by
unfold currentWindowReference
infer_instance
lemma rawCurrentProjection_kl_le {d : ℕ} (e : Direction d)
(ν : Measure (Row d)) [IsProbabilityMeasure ν]
(μ : Measure (RawCurrentData e × LocalUpperField e)) [IsFiniteMeasure μ] :
let out := μ.map (rawCurrentProjection e)
klDiv out (out.fst.compProd (currentWindowReference e ν)) ≤
klDiv μ (μ.fst.prod (Measure.infinitePi (fun _ : ℕ × HorizontalSpace e => ν))) := by
exact ReferenceClasses.anchorFrame_kl_le μ ν (fun z => z.2.2)
(measurable_snd.comp measurable_snd) (rawCurrentProfiles e) (rawCurrentProfiles_measurable e)
noncomputable def actualCurrentWindow {d : ℕ} (e : Direction d)
(t : Environment d → ℤ → ℕ) (i : ℤ) (m : ℕ) (X : EpisodeInput e) : CurrentWindow e :=
((ReferenceClasses.anchorOffsets (fun a => X.1.2 (i,a)),
fun p => globalHorizontalCoordinate e (t X.1.1 i) X.1.1 (X.1.2 (i,p.1)+p.2)),
fun p => episodeSplicedRow e t X i m p.1 p.2.1 p.2.2)
lemma actualCurrentWindow_measurable {d : ℕ} (e : Direction d)
(t : Environment d → ℤ → ℕ) (ht : ∀ i, Measurable fun ω => t ω i)
(i : ℤ) (m : ℕ) : Measurable (actualCurrentWindow e t i m) := by
have hu : Measurable (fun X : EpisodeInput e => fun a => X.1.2 (i,a)) :=
Measurable.of_eval fun a => (measurable_pi_apply (i,a)).comp
(measurable_snd.comp measurable_fst)
have ho := ReferenceClasses.anchorOffsets_measurable.comp hu
have hp : Measurable (fun X : EpisodeInput e => fun p : ℕ × HorizontalSpace e =>
globalHorizontalCoordinate e (t X.1.1 i) X.1.1 (X.1.2 (i,p.1)+p.2)) :=
Measurable.of_eval fun p => actualArrayCoordinate_measurable e t ht i (i,p.1) p.2
have hf : Measurable (fun X : EpisodeInput e => fun p : ReferenceClasses.Site (HorizontalSpace e) =>
episodeSplicedRow e t X i m p.1 p.2.1 p.2.2) :=
Measurable.of_eval fun p => episodeSplicedRow_measurable e t ht i m p.1 p.2.1 p.2.2
exact (ho.prodMk hp).prodMk hf
lemma globalHorizontalCoordinate_stoppedObservation {d : ℕ} (e : Direction d)
(τ : Environment d → ℕ)
(hτ : ∀ n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | τ ω=n})
(filler ω : Environment d) (z : HorizontalSpace e) :
globalHorizontalCoordinate e (τ ω)
(stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) τ filler ω) z =
globalHorizontalCoordinate e (τ ω) ω z := by
have h := currentStoppedProfile_observation e τ hτ filler ω
have hn : τ (stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) τ filler ω) = τ ω :=
stoppedRowGraft_stop (fun n => BelowHeight (realPosition (step e)) n) τ hτ ω filler
dsimp only [currentStoppedProfile] at h
rw [hn] at h
apply Subtype.ext
exact congrArg (fun μ : Measure (HorizontalSpace e) => (μ {z}).toReal) h
lemma actualCurrentWindow_raw {d : ℕ} (e : Direction d)
(t : Environment d → ℤ → ℕ) (i : ℤ) (m : ℕ)
(ht : ∀ n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)]
{ω | t ω i=n}) (filler : Environment d) (X : EpisodeInput e) :
rawCurrentProjection e ((t X.1.1 i,
(stoppedObservation (fun n => BelowHeight (realPosition (step e)) n)
(fun ω => t ω i) filler X.1.1,fun a => X.1.2 (i,a))),
localStoppedField e (fun ω => t ω i) (fun ω => t ω (i+m)) (X.1.1,X.2 (i,m))) =
actualCurrentWindow e t i m X := by
dsimp only [rawCurrentProjection,actualCurrentWindow,rawCurrentProfiles]
refine Prod.ext (Prod.ext rfl ?_) ?_
· funext p
exact globalHorizontalCoordinate_stoppedObservation e (fun ω => t ω i) ht filler X.1.1 _
· funext p
dsimp only [ReferenceClasses.anchorFields,localStoppedField,episodeSplicedRow]
theorem actualCurrentWindow_law {d : ℕ} (e : Direction d)
(ν : Measure (Row d)) [IsProbabilityMeasure ν]
(Q : Measure (Environment d)) [IsFiniteMeasure Q]
(t : Environment d → ℤ → ℕ) (ht : ∀ i, Measurable fun ω => t ω i)
(i : ℤ) (m : ℕ)
(hi : ∀ n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | t ω i=n})
(him : ∀ n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | t ω (i+m)=n})
(filler : Environment d) :
(rawCurrentWindow e ν Q (fun ω => t ω i) (fun ω => t ω (i+m)) (ht i) filler).map
(rawCurrentProjection e) =
(episodeInputLaw e ν Q t ht).map (actualCurrentWindow e t i m) := by
unfold rawCurrentWindow
rw [← episodeInputLaw_current e ν Q t ht i m]
rw [Measure.map_map (rawCurrentWindow_measurable e _ _ hi him filler)
(currentInput_measurable e i m)]
rw [Measure.map_map (rawCurrentProjection_measurable e)]
· congr 1
funext X
exact actualCurrentWindow_raw e t i m hi filler X
· exact (rawCurrentWindow_measurable e _ _ hi him filler).comp (currentInput_measurable e i m)
end DirectionalTransience
end
end OAI
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