| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.TwoStopSplice |
|
|
| namespace OAI |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory InformationTheory |
| open scoped ENNReal |
|
|
| namespace DirectionalTransience.StoppedWindow |
|
|
| variable {A B Z : Type*} [MeasurableSpace A] [MeasurableSpace B] |
| [MeasurableSpace Z] [StandardBorelSpace B] [Nonempty B] |
|
|
| noncomputable def ownLowerReference (Q P : Measure (A × B)) [IsFiniteMeasure P] : |
| Measure (A × B) := Q.fst.compProd P.condKernel |
|
|
| theorem stopped_chain (Q P : Measure (A × B)) [IsFiniteMeasure Q] [IsFiniteMeasure P] : |
| klDiv Q P = klDiv Q.fst P.fst + klDiv Q (ownLowerReference Q P) := by |
| calc |
| klDiv Q P = klDiv (Q.fst.compProd Q.condKernel) (P.fst.compProd P.condKernel) := by |
| rw [Measure.disintegrate, Measure.disintegrate] |
| _ = _ := by |
| simpa only [Measure.disintegrate, ownLowerReference] using |
| klDiv_compProd_eq_add Q.fst P.fst Q.condKernel P.condKernel |
|
|
| theorem stopped_kernel_increment (Q P : Measure (A × B)) [IsFiniteMeasure Q] |
| [IsFiniteMeasure P] (K : Kernel (A × B) Z) [IsMarkovKernel K] : |
| klDiv Q.fst P.fst + |
| klDiv (K ∘ₘ Q) (K ∘ₘ ownLowerReference Q P) ≤ klDiv Q P := by |
| have : IsFiniteMeasure (ownLowerReference Q P) := by |
| unfold ownLowerReference |
| infer_instance |
| rw [stopped_chain Q P] |
| exact add_le_add le_rfl (klDiv_comp_right_le Q (ownLowerReference Q P) K) |
|
|
| theorem fresh_prod_chain (Q : Measure (A × B)) [IsFiniteMeasure Q] |
| (Pdata : Measure A) [IsFiniteMeasure Pdata] |
| (Pfield : Measure B) [IsProbabilityMeasure Pfield] : |
| klDiv Q (Pdata.prod Pfield) = |
| klDiv Q.fst Pdata + klDiv Q (Q.fst.prod Pfield) := by |
| have h := klDiv_compProd_eq_add Q.fst Pdata Q.condKernel (Kernel.const A Pfield) |
| simpa only [Measure.disintegrate, Measure.compProd_const] using h |
|
|
| omit [StandardBorelSpace B] [Nonempty B] in |
| theorem same_auxiliary_cost (Q P : Measure A) [IsFiniteMeasure Q] [IsFiniteMeasure P] |
| (R : Measure B) [IsProbabilityMeasure R] : |
| klDiv (Q.prod R) (P.prod R) = klDiv Q P := by |
| simpa only [Measure.compProd_const] using klDiv_compProd_left Q P (Kernel.const A R) |
|
|
| end DirectionalTransience.StoppedWindow |
|
|
| end |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory InformationTheory |
| open scoped ENNReal Classical |
| namespace DirectionalTransience |
|
|
| noncomputable def stoppedWindowOutput {d : ℕ} (e : Direction d) |
| (τ σ : Environment d → ℕ) (filler : Environment d) |
| (p : Environment d × Environment d) : |
| Environment d × ((ℕ × HorizontalSpace e) → Row d) := |
| (stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) τ filler p.1, |
| upperField e (τ p.1) |
| (stoppedRowGraft (fun n => BelowHeight (realPosition (step e)) n) σ p.1 p.2)) |
|
|
| lemma stoppedWindowOutput_measurable {d : ℕ} (e : Direction d) |
| (τ σ : Environment d → ℕ) |
| (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) : Measurable (stoppedWindowOutput e τ σ filler) := by |
| let S : ℕ → Set (Lattice d) := fun n => BelowHeight (realPosition (step e)) n |
| exact ((stoppedObservation_measurable S τ hτ filler).comp measurable_fst).prodMk |
| ((upperField_joint_measurable e).comp |
| ((stoppedRowGraft_measurable S σ (measurable_of_stop_events S σ hσ)).prodMk |
| ((measurable_of_stop_events S τ hτ).comp measurable_fst))) |
|
|
| lemma stoppedWindowOutput_factor {d : ℕ} (e : Direction d) |
| (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) : |
| stoppedWindowOutput e τ σ filler ∘ |
| Prod.map (stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) σ filler) id = |
| stoppedWindowOutput e τ σ filler := by |
| funext p |
| let S : ℕ → Set (Lattice d) := fun n => BelowHeight (realPosition (step e)) n |
| change (stoppedObservation S τ filler (stoppedRowGraft S σ p.1 filler), |
| upperField e (τ (stoppedObservation S σ filler p.1)) |
| (stoppedRowGraft S σ (stoppedObservation S σ filler p.1) p.2)) = _ |
| rw [stoppedObservation_nested S (belowHeight_mono e) τ σ hτσ hτ hσ, |
| stoppedObservation_nested_stop S (belowHeight_mono e) τ σ hτσ hτ hσ, |
| stoppedObservation_regraft S σ hσ] |
| rfl |
|
|
| noncomputable def stoppedWindowLaw {d : ℕ} (e : Direction d) (ν : Measure (Row d)) |
| (Q : Measure (Environment d)) (τ σ : Environment d → ℕ) (filler : Environment d) := |
| (Q.prod (environmentLaw ν)).map (stoppedWindowOutput e τ σ filler) |
|
|
| lemma stoppedWindowLaw_upper_observation {d : ℕ} (e : Direction d) |
| (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (Q : Measure (Environment d)) [IsFiniteMeasure Q] |
| (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) : |
| stoppedWindowLaw e ν Q τ σ filler = |
| ((Q.map (stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) σ filler)).prod |
| (environmentLaw ν)).map (stoppedWindowOutput e τ σ filler) := by |
| let S : ℕ → Set (Lattice d) := fun n => BelowHeight (realPosition (step e)) n |
| have hG := stoppedObservation_measurable S σ hσ filler |
| have hp := Measure.map_prod_map Q (environmentLaw ν) hG measurable_id |
| simp only [Measure.map_id] at hp |
| rw [hp, Measure.map_map (stoppedWindowOutput_measurable e τ σ hτ hσ filler) |
| (hG.prodMap measurable_id), stoppedWindowOutput_factor e τ σ hτσ hτ hσ filler] |
| rfl |
|
|
| lemma stoppedWindowLaw_fst {d : ℕ} (e : Direction d) |
| (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (Q : Measure (Environment d)) [IsFiniteMeasure Q] |
| (τ σ : Environment d → ℕ) |
| (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) : |
| (stoppedWindowLaw e ν Q τ σ filler).fst = |
| Q.map (stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) τ filler) := by |
| let G := stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) τ filler |
| have hG : Measurable G := stoppedObservation_measurable _ τ hτ filler |
| have hF : Measurable (fun p : Environment d × Environment d => |
| upperField e (τ p.1) (stoppedRowGraft |
| (fun n => BelowHeight (realPosition (step e)) n) σ p.1 p.2)) := |
| measurable_snd.comp (stoppedWindowOutput_measurable e τ σ hτ hσ filler) |
| change ((Q.prod (environmentLaw ν)).map (fun p => ((G ∘ Prod.fst) p, _))).fst = _ |
| rw [Measure.fst_map_prodMk (hG.comp measurable_fst) hF] |
| rw [← Measure.map_map hG measurable_fst, Measure.map_fst_prod, measure_univ, one_smul] |
|
|
| lemma stoppedWindowLaw_reference {d : ℕ} (e : Direction d) |
| (ν : Measure (Row d)) [IsProbabilityMeasure 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) : |
| stoppedWindowLaw e ν (environmentLaw ν) τ σ filler = |
| ((environmentLaw ν).map |
| (stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) τ filler)).prod |
| (Measure.infinitePi (fun _ : ℕ × HorizontalSpace e => ν)) := by |
| exact twoStopSplice_fresh e ν τ σ hτσ hτ hσ _ |
| (stoppedObservation_measurable _ τ hτ filler) |
| (stoppedObservation_observable _ τ hτ filler) |
|
|
| theorem actual_stopped_window_entropy_increment {d : ℕ} (e : Direction d) |
| (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (Q : Measure (Environment d)) [IsProbabilityMeasure Q] |
| (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) : |
| let Gτ := stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) τ filler |
| let Gσ := stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) σ filler |
| klDiv (Q.map Gτ) ((environmentLaw ν).map Gτ) + |
| klDiv (stoppedWindowLaw e ν Q τ σ filler) |
| ((Q.map Gτ).prod (Measure.infinitePi (fun _ : ℕ × HorizontalSpace e => ν))) ≤ |
| klDiv (Q.map Gσ) ((environmentLaw ν).map Gσ) := by |
| let : Nonempty (Row d) := nonempty_of_isProbabilityMeasure ν |
| let Gσ := stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) σ filler |
| have h := klDiv_map_le ((Q.map Gσ).prod (environmentLaw ν)) |
| (((environmentLaw ν).map Gσ).prod (environmentLaw ν)) |
| (stoppedWindowOutput_measurable e τ σ hτ hσ filler) |
| rw [← stoppedWindowLaw_upper_observation e ν Q τ σ hτσ hτ hσ filler, |
| ← stoppedWindowLaw_upper_observation e ν (environmentLaw ν) τ σ hτσ hτ hσ filler, |
| stoppedWindowLaw_reference e ν τ σ hτσ hτ hσ filler, |
| StoppedWindow.same_auxiliary_cost] at h |
| have : IsFiniteMeasure (stoppedWindowLaw e ν Q τ σ filler) := by |
| unfold stoppedWindowLaw |
| infer_instance |
| rw [StoppedWindow.fresh_prod_chain, |
| stoppedWindowLaw_fst e ν Q τ σ hτ hσ filler] at h |
| exact h |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| end OAI |
|
|
