| import Mathlib |
| import OAI.Probability.Ballisticity.Entropy.StoppedWindowEntropy |
| import OAI.Probability.Ballisticity.Entropy.EntropyVariational |
|
|
| namespace OAI |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory InformationTheory Filter |
| open scoped ENNReal |
| namespace DirectionalTransience.Entropy |
| variable {X : Type*} [MeasurableSpace X] |
|
|
| lemma finite_kl_of_integrable_poslog (μ ν : Measure X) [IsProbabilityMeasure μ] |
| [IsProbabilityMeasure ν] (hac : μ≪ν) |
| (hF : Integrable (fun x => Real.log (max 1 ((μ.rnDeriv ν x).toReal))) μ) : |
| klDiv μ ν≠∞ := by |
| let g : X → ℝ := fun x => (μ.rnDeriv ν x).toReal |
| have hg : Measurable g := (Measure.measurable_rnDeriv μ ν).ennreal_toReal |
| have hg0 (x : X) : 0≤g x := ENNReal.toReal_nonneg |
| let F : X → ℝ := fun x => Real.log (max 1 (g x)) |
| have hF0 (x : X) : 0≤F x := Real.log_nonneg (le_max_left _ _) |
| have hp : Integrable (fun x => g x*F x) ν := (integrable_toReal_rnDeriv_mul_iff hac).mpr hF |
| have hl : Integrable (fun x => g x*Real.log (g x)) ν := by |
| apply (hp.add (integrable_const 1)).mono' (hg.mul hg.log).aestronglyMeasurable |
| exact ae_of_all _ fun x => by |
| change |g x*Real.log (g x)|≤g x*F x+1 |
| apply abs_le.mpr |
| constructor |
| · have hlow : -1≤g x*Real.log (g x) := by |
| by_cases hx : g x=0 |
| · simp [hx] |
| have h := mul_le_mul_of_nonneg_left (Real.neg_inv_le_log (hg0 x)) (hg0 x) |
| rw [mul_neg,mul_inv_cancel₀ hx] at h |
| exact h |
| have hn := mul_nonneg (hg0 x) (hF0 x) |
| linarith |
| · by_cases hx : g x=0 |
| · simp [hx] |
| have h := mul_le_mul_of_nonneg_left |
| (Real.log_le_log (lt_of_le_of_ne (hg0 x) (Ne.symm hx)) (le_max_right 1 (g x))) (hg0 x) |
| change g x*Real.log (g x)≤g x*F x at h |
| linarith |
| exact klDiv_ne_top hac ((integrable_rnDeriv_mul_log_iff hac).mp hl) |
|
|
| lemma finite_kl_of_domination (Q μ ν : Measure X) [IsProbabilityMeasure Q] |
| [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] (hkl : klDiv μ ν≠∞) |
| (K : ℝ) (hK : 0≤K) (hdom : Q≤ENNReal.ofReal K • μ) : klDiv Q ν≠∞ := by |
| have hac : Q≪ν := (Measure.absolutelyContinuous_of_le hdom).trans |
| (Measure.smul_absolutelyContinuous.trans (klDiv_ne_top_iff.mp hkl).1) |
| let g : X → ℝ := fun x => (Q.rnDeriv ν x).toReal |
| have hg : Measurable g := (Measure.measurable_rnDeriv Q ν).ennreal_toReal |
| have hg0 (x : X) : 0≤g x := ENNReal.toReal_nonneg |
| have hgi : Integrable g ν := by |
| simpa only [mul_one] using |
| (integrable_toReal_rnDeriv_mul_iff hac (f := fun _ => (1:ℝ))).mpr (integrable_const 1) |
| have hg1 : (∫ x, g x ∂ν)=1 := by |
| simpa only [mul_one,integral_const,probReal_univ,one_smul] using |
| (integral_toReal_rnDeriv_mul hac (f := fun _ => (1:ℝ))) |
| apply finite_kl_of_integrable_poslog Q ν hac |
| let F := fun x => Real.log (max 1 (g x)) |
| have hF : Measurable F := (measurable_const.max hg).log |
| have hF0 (x : X) : 0≤F x := Real.log_nonneg (le_max_left _ _) |
| apply integrable_of_bounded_truncated_integrals Q F hF hF0 (K*((klDiv μ ν).toReal+1)) |
| intro n |
| let Fn : X → ℝ := fun x => min (n:ℝ) (F x) |
| have hFn : Measurable Fn := measurable_const.min hF |
| have hBn : ∃ B : ℝ, ∀ x, |Fn x|≤B := ⟨n,fun x => by |
| rw [abs_of_nonneg (le_min (Nat.cast_nonneg _) (hF0 x))] |
| exact min_le_left _ _⟩ |
| have hi : Integrable Fn μ := integrable_of_bounded hFn hBn |
| have he : Integrable (fun x => Real.exp (Fn x)) ν := exp_integrable_of_bounded hFn hBn |
| have htest := entropy_exp_test_le μ ν hkl Fn hi he |
| have he₂ : (∫ x, Real.exp (Fn x) ∂ν)≤2 := by |
| have hb := integral_mono he ((integrable_const 1).add hgi) (fun x => by |
| calc Real.exp (Fn x)≤Real.exp (F x) := Real.exp_le_exp.mpr (min_le_right _ _) |
| _ = max 1 (g x) := Real.exp_log (lt_of_lt_of_le zero_lt_one (le_max_left _ _)) |
| _ ≤ 1+g x := max_le (by linarith [hg0 x]) (by linarith)) |
| simpa only [Pi.add_apply,integral_add (integrable_const 1) hgi,integral_const, |
| probReal_univ,one_smul,hg1,one_add_one_eq_two] using hb |
| have hd := integral_mono_measure hdom |
| (ae_of_all _ fun x => le_min (Nat.cast_nonneg n) (hF0 x)) |
| (hi.smul_measure ENNReal.ofReal_ne_top) |
| rw [integral_smul_measure,ENNReal.toReal_ofReal hK,smul_eq_mul] at hd |
| exact hd.trans (mul_le_mul_of_nonneg_left (by linarith) hK) |
|
|
| end DirectionalTransience.Entropy |
|
|
| end |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory InformationTheory |
| open scoped ENNReal |
| namespace DirectionalTransience.StoppedWindow |
| variable {X A B F G : Type*} [MeasurableSpace X] [MeasurableSpace A] |
| [MeasurableSpace B] [MeasurableSpace F] [MeasurableSpace G] |
|
|
| lemma sample_data_map (μ : Measure X) [IsFiniteMeasure μ] |
| (g : X → A) (hg : Measurable g) (f : X → F) (hf : Measurable f) |
| (K : Kernel A B) [IsMarkovKernel K] : |
| (μ.compProd (K.comap g hg)).map (fun p => ((g p.1,p.2),f p.1)) = |
| (((Kernel.id.prod K) ∥ₖ Kernel.id) ∘ₘ (μ.map (fun x => (g x,f x)))) := by |
| apply Measure.ext_of_lintegral |
| intro h hh |
| have hw : Measurable (fun p : X × B => ((g p.1,p.2),f p.1)) := |
| ((hg.comp measurable_fst).prodMk measurable_snd).prodMk (hf.comp measurable_fst) |
| have hk : Measurable (fun z : A × F => ∫⁻ y, h y ∂((Kernel.id.prod K) ∥ₖ Kernel.id) z) := |
| hh.lintegral_kernel |
| rw [lintegral_map hh hw,Measure.lintegral_compProd (show Measurable (fun p : X × B => h ((g p.1,p.2),f p.1)) from hh.comp hw), |
| Measure.lintegral_bind (Kernel.aemeasurable _) hh.aemeasurable, |
| lintegral_map hk (hg.prodMk hf)] |
| apply lintegral_congr |
| intro x |
| rw [Kernel.lintegral_parallelComp _ hh] |
| simp_rw [Kernel.id_apply] |
| simp only [Kernel.comap_apply] |
| have he (p : A × B) : (∫⁻ z : F, h (p,z) ∂Measure.dirac (f x))=h (p,f x) := |
| lintegral_dirac' _ (hh.comp measurable_prodMk_left) |
| simp_rw [he] |
| simpa only [Kernel.id,id_eq,Function.comp_def] using (Kernel.lintegral_deterministic_prod |
| (f:=id) measurable_id K (g x) (hh.comp measurable_prodMk_right)).symm |
|
|
| noncomputable def selectedKernel (K : Kernel A B) (u : B × F → G) |
| (_ : Measurable u) : Kernel (A × F) ((A × B) × G) := |
| (((Kernel.id.prod K) ∥ₖ Kernel.id)).map (fun p => (p.1,u (p.1.2,p.2))) |
| instance selectedKernel_markov (K : Kernel A B) [IsMarkovKernel K] |
| (u : B × F → G) (hu : Measurable u) : IsMarkovKernel (selectedKernel K u hu) := by |
| unfold selectedKernel |
| apply Kernel.IsMarkovKernel.map |
| exact measurable_fst.prodMk (hu.comp ((measurable_snd.comp measurable_fst).prodMk measurable_snd)) |
|
|
| lemma selectedKernel_lintegral (K : Kernel A B) [IsMarkovKernel K] |
| (u : B × F → G) (hu : Measurable u) (h : (A × B) × G → ℝ≥0∞) |
| (hh : Measurable h) (a : A) (f : F) : |
| (∫⁻ q, h q ∂selectedKernel K u hu (a,f)) = ∫⁻ b, h ((a,b),u (b,f)) ∂K a := by |
| have hp : Measurable (fun p : (A × B) × F => (p.1,u (p.1.2,p.2))) := |
| measurable_fst.prodMk (hu.comp ((measurable_snd.comp measurable_fst).prodMk measurable_snd)) |
| rw [selectedKernel,Kernel.map_apply _ hp,lintegral_map hh hp, |
| Kernel.lintegral_parallelComp _ (show Measurable (fun p : (A × B) × F => h (p.1,u (p.1.2,p.2))) from hh.comp hp)] |
| simp_rw [Kernel.id_apply] |
| have he (p : A × B) : (∫⁻ z : F, h (p,u (p.2,z)) ∂Measure.dirac f)=h (p,u (p.2,f)) := |
| lintegral_dirac' _ (hh.comp (measurable_const.prodMk |
| (hu.comp (measurable_const.prodMk measurable_id)))) |
| simp_rw [he] |
| exact Kernel.lintegral_deterministic_prod measurable_id K a |
| (hh.comp (measurable_id.prodMk (hu.comp (measurable_snd.prodMk measurable_const)))) |
|
|
| lemma selected_reference (δ : Measure A) [IsFiniteMeasure δ] |
| (R : Kernel A F) [IsMarkovKernel R] (K : Kernel A B) [IsMarkovKernel K] |
| (P : Measure G) [IsProbabilityMeasure P] (u : B × F → G) (hu : Measurable u) |
| (hP : ∀ a b, (R a).map (fun f => u (b,f))=P) : |
| selectedKernel K u hu ∘ₘ δ.compProd R=(δ.compProd K).prod P := by |
| apply Measure.ext_of_lintegral |
| intro h hh |
| rw [Measure.lintegral_bind (Kernel.aemeasurable _) hh.aemeasurable, |
| Measure.lintegral_compProd hh.lintegral_kernel, |
| lintegral_prod _ hh.aemeasurable, |
| Measure.lintegral_compProd hh.lintegral_prod_right'] |
| apply lintegral_congr |
| intro a |
| simp_rw [selectedKernel_lintegral K u hu h hh] |
| have hm : Measurable (fun p : F × B => h ((a,p.2),u (p.2,p.1))) := |
| hh.comp ((measurable_const.prodMk measurable_snd).prodMk (hu.comp measurable_swap)) |
| rw [←lintegral_prod _ hm.aemeasurable,lintegral_prod_symm _ hm.aemeasurable] |
| apply lintegral_congr |
| intro b |
| rw [←hP a b] |
| exact (lintegral_map (show Measurable (fun y : G => h ((a,b),y)) from hh.comp measurable_prodMk_left) |
| (show Measurable (fun f : F => u (b,f)) from hu.comp measurable_prodMk_left)).symm |
|
|
| lemma selected_kl_finite [StandardBorelSpace G] [Nonempty G] |
| (Q : Measure (A × F)) [IsProbabilityMeasure Q] |
| (δ : Measure A) [IsProbabilityMeasure δ] |
| (R : Kernel A F) [IsMarkovKernel R] (K : Kernel A B) [IsMarkovKernel K] |
| (P : Measure G) [IsProbabilityMeasure P] (u : B × F → G) (hu : Measurable u) |
| (hP : ∀ a b, (R a).map (fun f => u (b,f))=P) |
| (hkl : klDiv Q (δ.compProd R)≠∞) : |
| let W := selectedKernel K u hu ∘ₘ Q |
| klDiv W (W.fst.prod P)≠∞ := by |
| have h := klDiv_comp_right_le Q (δ.compProd R) (selectedKernel K u hu) |
| rw [selected_reference δ R K P u hu hP] at h |
| have hf := ne_top_of_le_ne_top hkl h |
| rw [fresh_prod_chain] at hf |
| exact (ENNReal.add_ne_top.mp hf).2 |
|
|
| end DirectionalTransience.StoppedWindow |
|
|
| end |
|
|
| end OAI |
|
|
