| 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 |
| |
