| 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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|
