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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 | import Mathlib
import OAI.Probability.Ballisticity.Entropy.StoppedWindowEntropy
import OAI.Probability.Ballisticity.Stationary.EpisodeEntropyTelescoping
namespace OAI
section
open MeasureTheory ProbabilityTheory InformationTheory
open scoped ENNReal Classical BigOperators
namespace DirectionalTransience
noncomputable def layerInformation {d : ℕ} (e : Direction d)
(ν : Measure (Row d)) (Q : Measure (Environment d))
(τ : Environment d → ℕ) (filler : Environment d) : ℝ≥0∞ :=
klDiv (Q.map (stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) τ filler))
((environmentLaw ν).map
(stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) τ filler))
noncomputable def ownedWindowEntropy {d : ℕ} (e : Direction d)
(ν : Measure (Row d)) (Q : Measure (Environment d))
(τ σ : Environment d → ℕ) (filler : Environment d) : ℝ≥0∞ :=
klDiv (stoppedWindowLaw e ν Q τ σ filler)
((Q.map (stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) τ filler)).prod
(Measure.infinitePi (fun _ : ℕ × HorizontalSpace e => ν)))
lemma layerInformation_le {d : ℕ} (e : Direction d)
(ν : Measure (Row d)) [IsProbabilityMeasure ν]
(Q : Measure (Environment d)) [IsFiniteMeasure Q]
(τ : Environment d → ℕ)
(hτ : ∀ n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | τ ω=n})
(filler : Environment d) : layerInformation e ν Q τ filler ≤ klDiv Q (environmentLaw ν) :=
klDiv_map_le Q (environmentLaw ν) (stoppedObservation_measurable _ τ hτ filler)
lemma ownedWindowEntropy_finite_budget {d : ℕ} (e : Direction d)
(ν : Measure (Row d)) [IsProbabilityMeasure ν]
(Q : Measure (Environment d)) [IsProbabilityMeasure Q]
(hfin : klDiv Q (environmentLaw ν) ≠ ∞)
(τ σ : Environment d → ℕ) (hτσ : ∀ ω, τ ω≤σ ω)
(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) :
ownedWindowEntropy e ν Q τ σ filler ≠ ∞ ∧
(ownedWindowEntropy e ν Q τ σ filler).toReal ≤
(layerInformation e ν Q σ filler).toReal - (layerInformation e ν Q τ filler).toReal := by
have h : layerInformation e ν Q τ filler + ownedWindowEntropy e ν Q τ σ filler ≤
layerInformation e ν Q σ filler :=
actual_stopped_window_entropy_increment e ν Q τ σ hτσ hτ hσ filler
have hfσ : layerInformation e ν Q σ filler ≠ ∞ :=
ne_top_of_le_ne_top hfin (layerInformation_le e ν Q σ hσ filler)
have hf := ENNReal.add_ne_top.mp (ne_top_of_le_ne_top hfσ h)
refine ⟨hf.2, ?_⟩
have hr := ENNReal.toReal_mono hfσ h
rw [ENNReal.toReal_add hf.1 hf.2] at hr
linarith
theorem sum_actual_stopped_window_entropy {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)
(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) :
(∑ i ∈ Finset.range N, (ownedWindowEntropy e ν Q (t i) (t (i+m)) filler).toReal) ≤
(m:ℝ)*(klDiv Q (environmentLaw ν)).toReal := by
let s : ℕ → ℝ := fun i => (layerInformation e ν Q (t i) filler).toReal
calc
_ ≤ ∑ i ∈ Finset.range N, (s (i+m)-s i) := by
apply Finset.sum_le_sum
intro i _
exact (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).2
_ ≤ _ := StoppedWindow.sum_window_increments_le s N m _
(fun _ => ENNReal.toReal_nonneg)
(fun i => ENNReal.toReal_mono hfin (layerInformation_le e ν Q (t i) (ht i) filler))
end DirectionalTransience
end
end OAI
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