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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 99 100 101 102 103 104 105 106 107 108 109 | import Mathlib
import OAI.Probability.Ballisticity.Geometry.WindowProjection
import OAI.Probability.Ballisticity.Estimates.OwnMarginalOccupation
namespace OAI
section
open MeasureTheory ProbabilityTheory InformationTheory
open scoped ENNReal Classical BigOperators
namespace DirectionalTransience
lemma rawCurrentWindow_probability {d : ℕ} (e : Direction d)
(ν : Measure (Row d)) [IsProbabilityMeasure ν]
(Q : Measure (Environment d)) [IsProbabilityMeasure Q]
(τ σ : Environment d → ℕ) (hτm : Measurable τ)
(hτ : ∀ n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | τ ω=n})
(hσ : ∀ n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | σ ω=n})
(filler : Environment d) :
IsProbabilityMeasure (rawCurrentWindow e ν Q τ σ hτm filler) := by
unfold rawCurrentWindow
exact (Measure.isProbabilityMeasure_map_iff
(rawCurrentWindow_measurable e τ σ hτ hσ filler).aemeasurable).2 inferInstance
noncomputable def rawCurrentOccupation {d : ℕ} (e : Direction d)
(ν : Measure (Row d)) (Q : Measure (Environment d))
(t : ℕ → Environment d → ℕ) (ht : ∀ i, Measurable (t i))
(filler : Environment d) (N m : ℕ) : Measure (RawCurrentData e × LocalUpperField e) :=
Entropy.activeSum (Finset.range N)
(fun i => rawCurrentWindow e ν Q (t i) (t (i+m)) (ht i) filler)
(fun _ => {p | p.1.1 < N})
instance rawCurrentOccupation_finite {d : ℕ} (e : Direction d)
(ν : Measure (Row d)) [IsProbabilityMeasure ν]
(Q : Measure (Environment d)) [IsFiniteMeasure Q]
(t : ℕ → Environment d → ℕ) (ht : ∀ i, Measurable (t i))
(filler : Environment d) (N m : ℕ) :
IsFiniteMeasure (rawCurrentOccupation e ν Q t ht filler N m) := by
unfold rawCurrentOccupation rawCurrentWindow
infer_instance
theorem rawCurrentOccupation_kl_le {d : ℕ} (e : Direction d)
(ν : Measure (Row d)) [IsProbabilityMeasure ν]
(Q : Measure (Environment d)) [IsProbabilityMeasure Q]
(hfin : klDiv Q (environmentLaw ν) ≠ ∞)
(t : ℕ → Environment d → ℕ) (htm : ∀ i, Measurable (t i))
(ht : ∀ i n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | t i ω=n})
(hmono : ∀ ω, Monotone (fun i => t i ω))
(filler : Environment d) (N m : ℕ)
(hpos : 0 < (rawCurrentOccupation e ν Q t htm filler N m).real Set.univ) :
haveI : Nonempty (Row d) := nonempty_of_isProbabilityMeasure ν
let μs : FiniteMeasure (RawCurrentData e × LocalUpperField e) :=
⟨rawCurrentOccupation e ν Q t htm filler N m,inferInstance⟩
klDiv (μs.normalize : Measure (RawCurrentData e × LocalUpperField e))
((μs.normalize : Measure (RawCurrentData e × LocalUpperField e)).fst.prod
(Measure.infinitePi (fun _ : ℕ × HorizontalSpace e => ν))) ≤
ENNReal.ofReal ((m:ℝ)*(klDiv Q (environmentLaw ν)).toReal /
(rawCurrentOccupation e ν Q t htm filler N m).real Set.univ) := by
have : Nonempty (Row d) := nonempty_of_isProbabilityMeasure ν
let μ : ℕ → Measure (RawCurrentData e × LocalUpperField e) :=
fun i => rawCurrentWindow e ν Q (t i) (t (i+m)) (htm i) filler
let π := Measure.infinitePi (fun _ : ℕ × HorizontalSpace e => ν)
let A : ℕ → Set (RawCurrentData e) := fun _ => {z | z.1 < N}
have (i : ℕ) : IsProbabilityMeasure (μ i) :=
rawCurrentWindow_probability e ν Q (t i) (t (i+m)) (htm i) (ht i) (ht (i+m)) filler
have hle (i : ℕ) : klDiv (μ i) ((μ i).fst.prod π) ≤
ownedWindowEntropy e ν Q (t i) (t (i+m)) filler :=
rawCurrentWindow_kl_le e ν Q (t i) (t (i+m))
(fun ω => hmono ω (Nat.le_add_right i m)) (ht i) (ht (i+m)) filler
have hf (i : ℕ) := (ownedWindowEntropy_finite_budget e ν Q hfin (t i) (t (i+m))
(fun ω => hmono ω (Nat.le_add_right i m)) (ht i) (ht (i+m)) filler).1
have hkl (i : ℕ) : klDiv (μ i) ((μ i).fst.prod π) ≠ ∞ := ne_top_of_le_ne_top (hf i) (hle i)
have h := Entropy.own_normalized_activeSum_kl_le (Finset.range N) μ π A
(fun _ _ => measurableSet_lt measurable_fst measurable_const) (fun i _ => hkl i) hpos
refine h.trans (ENNReal.ofReal_le_ofReal ?_)
apply div_le_div_of_nonneg_right _ hpos.le
calc
_ ≤ ∑ i ∈ Finset.range N, (ownedWindowEntropy e ν Q (t i) (t (i+m)) filler).toReal :=
Finset.sum_le_sum fun i _ => ENNReal.toReal_mono (hf i) (hle i)
_ ≤ _ := sum_actual_stopped_window_entropy e ν Q hfin t ht hmono filler N m
theorem currentOccupation_kl_le {d : ℕ} (e : Direction d)
(ν : Measure (Row d)) [IsProbabilityMeasure ν]
(Q : Measure (Environment d)) [IsProbabilityMeasure Q]
(hfin : klDiv Q (environmentLaw ν) ≠ ∞)
(t : ℕ → Environment d → ℕ) (htm : ∀ i, Measurable (t i))
(ht : ∀ i n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | t i ω=n})
(hmono : ∀ ω, Monotone (fun i => t i ω))
(filler : Environment d) (N m : ℕ)
(hpos : 0 < (rawCurrentOccupation e ν Q t htm filler N m).real Set.univ) :
let μs : FiniteMeasure (RawCurrentData e × LocalUpperField e) :=
⟨rawCurrentOccupation e ν Q t htm filler N m,inferInstance⟩
haveI : Nonempty (Row d) := nonempty_of_isProbabilityMeasure ν
let out := (μs.normalize : Measure (RawCurrentData e × LocalUpperField e)).map (rawCurrentProjection e)
klDiv out (out.fst.compProd (currentWindowReference e ν)) ≤
ENNReal.ofReal ((m:ℝ)*(klDiv Q (environmentLaw ν)).toReal /
(rawCurrentOccupation e ν Q t htm filler N m).real Set.univ) := by
have : Nonempty (Row d) := nonempty_of_isProbabilityMeasure ν
let μs : FiniteMeasure (RawCurrentData e × LocalUpperField e) :=
⟨rawCurrentOccupation e ν Q t htm filler N m,inferInstance⟩
exact (rawCurrentProjection_kl_le e ν (μs.normalize : Measure _)).trans
(rawCurrentOccupation_kl_le e ν Q hfin t htm ht hmono filler N m hpos)
end DirectionalTransience
end
end OAI
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