| import Mathlib |
| import OAI.Probability.Ballisticity.Model |
|
|
| namespace OAI |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory InformationTheory Filter |
| open scoped ENNReal NNReal Topology |
| namespace DirectionalTransience.Entropy |
|
|
| variable {X : Type*} [MeasurableSpace X] |
|
|
| theorem entropy_test_le (μ ν : Measure X) [IsProbabilityMeasure μ] |
| [IsProbabilityMeasure ν] (hkl : klDiv μ ν ≠ ∞) |
| (F : X → ℝ) (hFμ : Integrable F μ) |
| (he : Integrable (fun x => Real.exp (F x)) ν) : |
| (∫ x, F x ∂μ) - Real.log (∫ x, Real.exp (F x) ∂ν) ≤ |
| (klDiv μ ν).toReal := by |
| obtain ⟨hac,hllr⟩ := klDiv_ne_top_iff.mp hkl |
| have hac' : μ ≪ ν.tilted F := hac.trans (absolutelyContinuous_tilted he) |
| have hh := integrable_llr_tilted_right hac hFμ hllr he |
| have hnon := integral_llr_add_sub_measure_univ_nonneg hac' hh |
| have : IsProbabilityMeasure (ν.tilted F) := isProbabilityMeasure_tilted he |
| rw [probReal_univ,probReal_univ,add_sub_cancel_right] at hnon |
| rw [integral_llr_tilted_right hac hFμ he hllr] at hnon |
| rw [toReal_klDiv_of_measure_eq hac (by simp)] |
| linarith |
|
|
| theorem entropy_exp_test_le (μ ν : Measure X) [IsProbabilityMeasure μ] |
| [IsProbabilityMeasure ν] (hkl : klDiv μ ν ≠ ∞) |
| (F : X → ℝ) (hFμ : Integrable F μ) |
| (he : Integrable (fun x => Real.exp (F x)) ν) : |
| (∫ x, F x ∂μ) - (∫ x, Real.exp (F x) ∂ν) + 1 ≤ |
| (klDiv μ ν).toReal := by |
| have h := entropy_test_le μ ν hkl F hFμ he |
| have hp : 0 < ∫ x, Real.exp (F x) ∂ν := integral_exp_pos he |
| have hl := Real.log_le_sub_one_of_pos hp |
| linarith |
|
|
| lemma entropy_young_exact {u : ℝ} (hu : 0 ≤ u) (v : ℝ) : |
| u*v - Real.exp v + 1 ≤ klFun u := by |
| by_cases h0 : u = 0 |
| · simp [h0,klFun,Real.exp_nonneg] |
| have hp : 0 < u := lt_of_le_of_ne hu (Ne.symm h0) |
| have he := mul_le_mul_of_nonneg_left (Real.add_one_le_exp (v-Real.log u)) hu |
| have heq : u*Real.exp (v-Real.log u) = Real.exp v := by |
| rw [Real.exp_sub,Real.exp_log hp] |
| field_simp |
| rw [heq] at he |
| unfold klFun |
| nlinarith |
|
|
| def ExpBudget (μ ν : Measure X) (C : ℝ) : Prop := |
| ∀ F : X → ℝ, Measurable F → (∃ B : ℝ, ∀ x, |F x| ≤ B) → |
| (∫ x, F x ∂μ) - (∫ x, Real.exp (F x) ∂ν) + 1 ≤ C |
|
|
| lemma integrable_of_bounded {μ : Measure X} [IsFiniteMeasure μ] {F : X → ℝ} |
| (hF : Measurable F) (hB : ∃ B : ℝ, ∀ x, |F x| ≤ B) : Integrable F μ := by |
| obtain ⟨B,hB⟩ := hB |
| exact (integrable_const B).mono' hF.aestronglyMeasurable |
| (ae_of_all _ fun x => by simpa only [Real.norm_eq_abs] using hB x) |
| |
| lemma exp_integrable_of_bounded {μ : Measure X} [IsFiniteMeasure μ] {F : X → ℝ} |
| (hF : Measurable F) (hB : ∃ B : ℝ, ∀ x, |F x| ≤ B) : |
| Integrable (fun x => Real.exp (F x)) μ := by |
| obtain ⟨B,hB⟩ := hB |
| exact (integrable_const (Real.exp B)).mono' hF.exp.aestronglyMeasurable |
| (ae_of_all _ fun x => by |
| rw [Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)] |
| exact Real.exp_le_exp.mpr ((le_abs_self _).trans (hB x))) |
|
|
| lemma expBudget_of_kl_le (μ ν : Measure X) [IsProbabilityMeasure μ] |
| [IsProbabilityMeasure ν] {C : ℝ} (hC : 0 ≤ C) (hkl : klDiv μ ν ≤ ENNReal.ofReal C) : |
| ExpBudget μ ν C := by |
| intro F hF hB |
| have hf : klDiv μ ν ≠ ∞ := ne_top_of_le_ne_top ENNReal.ofReal_ne_top hkl |
| exact (entropy_exp_test_le μ ν hf F (integrable_of_bounded hF hB) |
| (exp_integrable_of_bounded hF hB)).trans (ENNReal.toReal_le_of_le_ofReal hC hkl) |
|
|
| lemma ExpBudget.nonneg (μ ν : Measure X) [IsProbabilityMeasure μ] |
| [IsProbabilityMeasure ν] {C : ℝ} (h : ExpBudget μ ν C) : 0 ≤ C := by |
| have hh := h (fun _ => 0) measurable_const ⟨0,fun _ => by simp⟩ |
| simpa using hh |
|
|
| lemma ExpBudget.absolutelyContinuous (μ ν : Measure X) [IsProbabilityMeasure μ] |
| [IsProbabilityMeasure ν] {C : ℝ} (h : ExpBudget μ ν C) : μ ≪ ν := by |
| apply Measure.AbsolutelyContinuous.mk |
| intro s hs hνs |
| by_contra hμs |
| have hp : 0 < μ.real s := ENNReal.toReal_pos hμs (measure_ne_top _ _) |
| let t : ℝ := (C+1) / μ.real s |
| have ht : 0 < t := div_pos (by linarith [h.nonneg μ ν]) hp |
| let F : X → ℝ := s.indicator (fun _ => t) |
| have hF : Measurable F := measurable_const.indicator hs |
| have hh := h F hF ⟨t,fun x => by |
| by_cases hx : x ∈ s |
| · simp [F,hx,abs_of_pos ht] |
| · simp [F,hx,ht.le]⟩ |
| have he : (∫ x, Real.exp (F x) ∂ν) = 1 := by |
| have heq : (fun x => Real.exp (F x)) =ᵐ[ν] fun _ => (1:ℝ) := by |
| filter_upwards [show ∀ᵐ x ∂ν, x ∉ s from by simpa only [ae_iff,not_not,Set.ofPred_mem_eq] using hνs] with x hx |
| simp [F,hx] |
| simpa using integral_congr_ae heq |
| rw [he,show (∫ x, F x ∂μ) = μ.real s * t by |
| exact integral_indicator_const t hs] at hh |
| dsimp [t] at hh |
| rw [mul_div_cancel₀ _ hp.ne'] at hh |
| linarith |
| |
| lemma integrable_of_bounded_truncated_integrals (μ : Measure X) [IsFiniteMeasure μ] |
| (F : X → ℝ) (hF : Measurable F) (hF0 : ∀ x, 0 ≤ F x) |
| (C : ℝ) (hbound : ∀ n : ℕ, (∫ x, min (n:ℝ) (F x) ∂μ) ≤ C) : |
| Integrable F μ := by |
| have hi (n : ℕ) : Integrable (fun x => min (n:ℝ) (F x)) μ := by |
| apply integrable_of_bounded (measurable_const.min hF) |
| exact ⟨n,fun x => by |
| rw [abs_of_nonneg (le_min (Nat.cast_nonneg _) (hF0 x))] |
| exact min_le_left _ _⟩ |
| have heq (x : X) : ENNReal.ofReal (F x) = |
| ⨆ n : ℕ, ENNReal.ofReal (min (n:ℝ) (F x)) := by |
| apply le_antisymm |
| · obtain ⟨n,hn⟩ := exists_nat_ge (F x) |
| exact (by rw [min_eq_right hn] : ENNReal.ofReal (F x) ≤ |
| ENNReal.ofReal (min (n:ℝ) (F x))).trans (le_iSup (fun n : ℕ => ENNReal.ofReal (min (n:ℝ) (F x))) n) |
| · exact iSup_le fun n => ENNReal.ofReal_le_ofReal (min_le_right _ _) |
| have hl : (∫⁻ x, ENNReal.ofReal (F x) ∂μ) ≤ ENNReal.ofReal C := by |
| simp_rw [heq] |
| rw [lintegral_iSup (fun n => (measurable_const.min hF).ennreal_ofReal) |
| (fun n m hnm x => ENNReal.ofReal_le_ofReal |
| (min_le_min_right _ (Nat.cast_le.mpr hnm)))] |
| apply iSup_le |
| intro n |
| rw [←ofReal_integral_eq_lintegral_ofReal (hi n) |
| (ae_of_all _ fun x => le_min (Nat.cast_nonneg _) (hF0 x))] |
| exact ENNReal.ofReal_le_ofReal (hbound n) |
| exact (lintegral_ofReal_ne_top_iff_integrable hF.aestronglyMeasurable |
| (ae_of_all _ hF0)).mp (ne_top_of_le_ne_top ENNReal.ofReal_ne_top hl) |
| |
| lemma ExpBudget.finite_kl (μ ν : Measure X) [IsProbabilityMeasure μ] |
| [IsProbabilityMeasure ν] {C : ℝ} (h : ExpBudget μ ν C) : klDiv μ ν ≠ ∞ := by |
| have hac := h.absolutelyContinuous μ ν |
| 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 |
| 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))) |
| let F : X → ℝ := 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 _ _) |
| have hFi : Integrable F μ := by |
| apply integrable_of_bounded_truncated_integrals μ F hF hF0 (C+1) |
| intro n |
| have hFn : Measurable (fun x => min (n:ℝ) (F x)) := measurable_const.min hF |
| have hBn : ∃ B : ℝ, ∀ x, |min (n:ℝ) (F x)| ≤ B := |
| ⟨n,fun x => by |
| rw [abs_of_nonneg (le_min (Nat.cast_nonneg _) (hF0 x))] |
| exact min_le_left _ _⟩ |
| have he := h _ hFn hBn |
| have he₂ : (∫ x, Real.exp (min (n:ℝ) (F x)) ∂ν) ≤ 2 := by |
| have hp := integral_mono (exp_integrable_of_bounded hFn hBn) |
| ((integrable_const 1).add hgi) (fun x => by |
| calc Real.exp (min (n:ℝ) (F 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 hp |
| linarith |
| have hpos : Integrable (fun x => g x * F x) ν := |
| (integrable_toReal_rnDeriv_mul_iff hac).mpr hFi |
| have hlog : Integrable (fun x => g x * Real.log (g x)) ν := by |
| apply (hpos.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 hl := mul_le_mul_of_nonneg_left (Real.neg_inv_le_log (hg0 x)) (hg0 x) |
| rw [mul_neg,mul_inv_cancel₀ hx] at hl |
| exact hl |
| have hn : 0 ≤ g x * F x := mul_nonneg (hg0 x) (hF0 x) |
| linarith |
| · by_cases hx : g x = 0 |
| · simp [hx] |
| have hl := 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 hl |
| linarith |
| exact klDiv_ne_top hac ((integrable_rnDeriv_mul_log_iff hac).mp hlog) |
|
|
| noncomputable def logClip (n : ℕ) (x : ℝ) : ℝ := max (-(n:ℝ)) (min (n:ℝ) x) |
|
|
| lemma abs_logClip_le (n : ℕ) (x : ℝ) : |logClip n x| ≤ (n:ℝ) := by |
| apply abs_le.mpr |
| exact ⟨le_max_left _ _,max_le (by linarith [Nat.cast_nonneg (α := ℝ) n]) (min_le_left _ _)⟩ |
|
|
| lemma abs_logClip_le_abs (n : ℕ) (x : ℝ) : |logClip n x| ≤ |x| := by |
| unfold logClip |
| rcases le_total x (n:ℝ) with hx|hx |
| · rw [min_eq_right hx] |
| rcases le_total (-(n:ℝ)) x with hn|hn |
| · rw [max_eq_right hn] |
| · rw [max_eq_left hn,abs_neg,abs_of_nonneg (Nat.cast_nonneg n)] |
| exact (by linarith : (n:ℝ) ≤ -x).trans (neg_le_abs _) |
| · rw [min_eq_left hx,max_eq_right (by linarith [Nat.cast_nonneg (α := ℝ) n]),abs_of_nonneg (Nat.cast_nonneg n)] |
| exact hx.trans (le_abs_self _) |
|
|
| lemma logClip_eventually (x : ℝ) : ∀ᶠ n : ℕ in atTop, logClip n x = x := by |
| obtain ⟨N,hN⟩ := exists_nat_ge |x| |
| filter_upwards [eventually_ge_atTop N] with n hn |
| have h : |x| ≤ (n:ℝ) := hN.trans (Nat.cast_le.mpr hn) |
| rw [logClip,min_eq_right ((le_abs_self _).trans h),max_eq_right] |
| linarith [neg_le_abs x] |
|
|
| lemma exp_logClip_log_le {u : ℝ} (hu : 0 < u) (n : ℕ) : |
| Real.exp (logClip n (Real.log u)) ≤ 1+u := by |
| by_cases h : 0 ≤ Real.log u |
| · calc Real.exp (logClip n (Real.log u)) ≤ Real.exp (Real.log u) := by |
| apply Real.exp_le_exp.mpr |
| exact max_le (by linarith [Nat.cast_nonneg (α := ℝ) n]) (min_le_right _ _) |
| _ = u := Real.exp_log hu |
| _ ≤ _ := by linarith |
| · calc Real.exp (logClip n (Real.log u)) ≤ Real.exp 0 := by |
| apply Real.exp_le_exp.mpr |
| exact max_le (neg_nonpos.mpr (Nat.cast_nonneg n)) |
| ((min_le_right _ _).trans (le_of_not_ge h)) |
| _ = 1 := Real.exp_zero |
| _ ≤ _ := by linarith |
|
|
| theorem kl_le_of_expBudget (μ ν : Measure X) [IsProbabilityMeasure μ] |
| [IsProbabilityMeasure ν] {C : ℝ} (h : ExpBudget μ ν C) : |
| klDiv μ ν ≤ ENNReal.ofReal C := by |
| have hfinite := h.finite_kl μ ν |
| obtain ⟨hac,hllr⟩ := klDiv_ne_top_iff.mp hfinite |
| 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 |
| have hgp : ∀ᵐ x ∂μ, 0 < g x := by |
| filter_upwards [Measure.rnDeriv_pos hac,hac.ae_le (Measure.rnDeriv_lt_top μ ν)] with x hx ht |
| exact ENNReal.toReal_pos hx.ne' ht.ne |
| 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))) |
| let F : ℕ → X → ℝ := fun n x => |
| if g x = 0 then -(n:ℝ) else logClip n (Real.log (g x)) |
| have hFm (n : ℕ) : Measurable (F n) := by |
| exact measurable_const.ite (measurableSet_eq_fun hg measurable_const) |
| (measurable_const.max (measurable_const.min hg.log)) |
| have hFb (n : ℕ) (x : X) : |F n x| ≤ (n:ℝ) := by |
| dsimp only [F] |
| split_ifs |
| · rw [abs_neg,abs_of_nonneg (Nat.cast_nonneg n)] |
| · exact abs_logClip_le n _ |
| have hFdom (n : ℕ) : ∀ᵐ x ∂μ, ‖F n x‖ ≤ ‖llr μ ν x‖ := by |
| filter_upwards [hgp] with x hx |
| simp only [F,ite_eq_right hx.ne',llr,Real.norm_eq_abs] |
| exact abs_logClip_le_abs n _ |
| have hFpt : ∀ᵐ x ∂μ, Tendsto (fun n => F n x) atTop (𝓝 (llr μ ν x)) := by |
| filter_upwards [hgp] with x hx |
| apply tendsto_const_nhds.congr' |
| filter_upwards [logClip_eventually (Real.log (g x))] with n hn |
| simpa only [F,ite_eq_right hx.ne',llr] using hn.symm |
| have hFlim : Tendsto (fun n => ∫ x, F n x ∂μ) atTop (𝓝 (∫ x, llr μ ν x ∂μ)) := |
| tendsto_integral_of_dominated_convergence (fun x => ‖llr μ ν x‖) |
| (fun n => (hFm n).aestronglyMeasurable) hllr.norm hFdom hFpt |
| have heDom (n : ℕ) (x : X) : ‖Real.exp (F n x)‖ ≤ 1+g x := by |
| rw [Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)] |
| dsimp only [F] |
| split_ifs with hx |
| · have he : Real.exp (-(n:ℝ)) ≤ 1 := by |
| rw [←Real.exp_zero] |
| exact Real.exp_le_exp.mpr (neg_nonpos.mpr (Nat.cast_nonneg n)) |
| linarith [hg0 x] |
| · exact exp_logClip_log_le (lt_of_le_of_ne (hg0 x) (Ne.symm hx)) n |
| have hePt (x : X) : Tendsto (fun n => Real.exp (F n x)) atTop (𝓝 (g x)) := by |
| by_cases hx : g x = 0 |
| · simp only [F,hx,ite_eq_left] |
| exact Real.tendsto_exp_atBot.comp |
| (tendsto_neg_atTop_atBot.comp tendsto_natCast_atTop_atTop) |
| · apply tendsto_const_nhds.congr' |
| filter_upwards [logClip_eventually (Real.log (g x))] with n hn |
| simp only [F,ite_eq_right hx,hn,Real.exp_log (lt_of_le_of_ne (hg0 x) (Ne.symm hx))] |
| have heLim : Tendsto (fun n => ∫ x, Real.exp (F n x) ∂ν) atTop (𝓝 1) := by |
| rw [←hg1] |
| exact tendsto_integral_of_dominated_convergence (fun x => 1+g x) |
| (fun n => (hFm n).exp.aestronglyMeasurable) ((integrable_const 1).add hgi) |
| (fun n => ae_of_all _ (heDom n)) (ae_of_all _ hePt) |
| have hbound (n : ℕ) := h (F n) (hFm n) ⟨n,hFb n⟩ |
| have hlimit := (hFlim.sub heLim).add (tendsto_const_nhds (x := (1:ℝ))) |
| have hh := le_of_tendsto hlimit (Eventually.of_forall hbound) |
| have hh' : (klDiv μ ν).toReal ≤ C := by |
| rw [toReal_klDiv_of_measure_eq hac (by simp)] |
| linarith |
| rw [←ENNReal.ofReal_toReal hfinite] |
| exact ENNReal.ofReal_le_ofReal hh' |
| |
| end DirectionalTransience.Entropy |
| |
| end |
| |
| end OAI |
| |
