| import Mathlib | |
| import OAI.Probability.Ballisticity.Stationary.ArrayMap | |
| import OAI.Probability.Ballisticity.Estimates.TwoStopSplice | |
| namespace OAI | |
| section | |
| open MeasureTheory ProbabilityTheory InformationTheory | |
| open scoped ENNReal Classical | |
| namespace DirectionalTransience | |
| namespace StoppedWindow | |
| variable {A B F : Type*} [MeasurableSpace A] [MeasurableSpace B] [MeasurableSpace F] | |
| lemma parallel_lower_fst (μ : Measure (A × F)) [IsFiniteMeasure μ] | |
| (K : Kernel A B) [IsMarkovKernel K] : | |
| ((K ∥ₖ Kernel.id) ∘ₘ μ).fst = K ∘ₘ μ.fst := by | |
| ext s hs | |
| rw [Measure.fst_apply hs, Measure.bind_apply (measurable_fst hs) (Kernel.aemeasurable _), | |
| Measure.bind_apply hs K.aemeasurable, Measure.fst, lintegral_map (K.measurable_coe hs) measurable_fst] | |
| apply lintegral_congr | |
| intro p | |
| rw [Kernel.parallelComp_apply, Kernel.id_apply] | |
| rw [← Measure.map_apply measurable_fst hs, Measure.map_fst_prod, measure_univ, one_smul] | |
| theorem parallel_lower_kl_le (μ : Measure (A × F)) [IsFiniteMeasure μ] | |
| (π : Measure F) [IsProbabilityMeasure π] | |
| (K : Kernel A B) [IsMarkovKernel K] : | |
| let out := (K ∥ₖ Kernel.id) ∘ₘ μ | |
| klDiv out (out.fst.prod π) ≤ klDiv μ (μ.fst.prod π) := by | |
| have h := klDiv_comp_right_le μ (μ.fst.prod π) (K ∥ₖ Kernel.id) | |
| rw [← Measure.prod_comp_left, ← parallel_lower_fst μ K] at h | |
| exact h | |
| end StoppedWindow | |
| noncomputable def currentStoppedProfile {d : ℕ} (e : Direction d) | |
| (τ : Environment d → ℕ) (ω : Environment d) : Measure (HorizontalSpace e) := | |
| globalHorizontalProfile e (τ ω) ω | |
| instance currentStoppedProfile_probability {d : ℕ} (e : Direction d) | |
| (τ : Environment d → ℕ) (ω : Environment d) : | |
| IsProbabilityMeasure (currentStoppedProfile e τ ω) := by | |
| unfold currentStoppedProfile | |
| infer_instance | |
| lemma currentStoppedProfile_measurable {d : ℕ} (e : Direction d) | |
| (τ : Environment d → ℕ) (hτ : Measurable τ) : | |
| Measurable (currentStoppedProfile e τ) := by | |
| change Measurable (fun ω : Environment d => globalHorizontalProfile e (τ ω) ω) | |
| have h := (globalHorizontalProfile_joint_measurable e).comp (measurable_id.prodMk hτ) | |
| simpa only [Function.comp_def,id_eq] using h | |
| lemma currentStoppedProfile_observable {d : ℕ} (e : Direction d) | |
| (τ : Environment d → ℕ) | |
| (hτ : ∀ n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | τ ω=n}) | |
| (B : Set (Measure (HorizontalSpace e))) (hB : MeasurableSet B) : | |
| StoppedRowsEvent (fun n => BelowHeight (realPosition (step e)) n) τ | |
| (currentStoppedProfile e τ ⁻¹' B) := by | |
| intro n | |
| have he : (currentStoppedProfile e τ ⁻¹' B) ∩ {ω | τ ω=n} = | |
| (globalHorizontalProfile e n ⁻¹' B) ∩ {ω | τ ω=n} := by | |
| ext ω | |
| simp only [Set.mem_inter_iff,Set.mem_preimage,Set.mem_ofPred_eq] | |
| constructor <;> rintro ⟨hb,hn⟩ <;> exact ⟨by simpa only [currentStoppedProfile,hn] using hb,hn⟩ | |
| rw [he] | |
| exact (hB.preimage (globalHorizontalProfile_measurable_below e n)).inter (hτ n) | |
| lemma currentStoppedProfile_observation {d : ℕ} (e : Direction d) | |
| (τ : Environment d → ℕ) | |
| (hτ : ∀ n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | τ ω=n}) | |
| (filler ω : Environment d) : | |
| currentStoppedProfile e τ | |
| (stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) τ filler ω) = | |
| currentStoppedProfile e τ ω := by | |
| ext s hs | |
| apply stoppedRowGraft_observable _ τ hτ | |
| (fun ω => currentStoppedProfile e τ ω s) | |
| intro B hB | |
| exact currentStoppedProfile_observable e τ hτ _ | |
| (hB.preimage (Measure.measurable_coe hs)) | |
| noncomputable def currentAnchorKernel {d : ℕ} (e : Direction d) | |
| (τ : Environment d → ℕ) (hτ : Measurable τ) : | |
| Kernel (Environment d) (ℕ → HorizontalSpace e) where | |
| toFun ω := Measure.infinitePi (fun _ => currentStoppedProfile e τ ω) | |
| measurable' := AnchorSampling.measurable_infinitePi _ | |
| (fun _ => currentStoppedProfile_measurable e τ hτ) | |
| instance currentAnchorKernel_markov {d : ℕ} (e : Direction d) | |
| (τ : Environment d → ℕ) (hτ : Measurable τ) : | |
| IsMarkovKernel (currentAnchorKernel e τ hτ) where | |
| isProbabilityMeasure ω := by | |
| change IsProbabilityMeasure (Measure.infinitePi (fun _ : ℕ => currentStoppedProfile e τ ω)) | |
| infer_instance | |
| lemma currentAnchorKernel_observation {d : ℕ} (e : Direction d) | |
| (τ : Environment d → ℕ) | |
| (hτ : ∀ n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | τ ω=n}) | |
| (filler ω : Environment d) : | |
| currentAnchorKernel e τ (measurable_of_stop_events _ τ hτ) | |
| (stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) τ filler ω) = | |
| currentAnchorKernel e τ (measurable_of_stop_events _ τ hτ) ω := by | |
| change Measure.infinitePi _ = Measure.infinitePi _ | |
| congr 1 | |
| funext a | |
| exact currentStoppedProfile_observation e τ hτ filler ω | |
| lemma globalEpisodeAnchors_current {d : ℕ} (e : Direction d) | |
| (t : Environment d → ℤ → ℕ) (ht : ∀ i, Measurable fun ω => t ω i) | |
| (ω : Environment d) (i : ℤ) : | |
| ((globalEpisodeAnchors e t ht) ω).map (fun U a => U (i,a)) = | |
| currentAnchorKernel e (fun ω => t ω i) (ht i) ω := by | |
| exact Measure.map_infinitePi_infinitePi_of_inj (fun a b h => by exact (Prod.mk.inj h).2) | |
| end DirectionalTransience | |
| end | |
| end OAI | |