| import Mathlib |
| import OAI.Probability.Ballisticity.Entropy.EntropyVariational |
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| namespace OAI |
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| section |
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| open MeasureTheory ProbabilityTheory InformationTheory TopologicalSpace |
| open scoped ENNReal Classical Topology BoundedContinuousFunction |
| namespace DirectionalTransience.Entropy |
|
|
| lemma exp_sub_le_exp_mul {x y : ℝ} (h : x≤y) : |
| Real.exp y-Real.exp x ≤ Real.exp y*(y-x) := by |
| rcases h.eq_or_lt with rfl | _ |
| · simp |
| · have hh := mul_le_mul_of_nonneg_left (Real.add_one_le_exp (x-y)) (Real.exp_pos y).le |
| rw [← Real.exp_add,add_sub_cancel] at hh |
| nlinarith |
|
|
| lemma abs_exp_sub_le {x y B : ℝ} (hx : x≤B) (hy : y≤B) : |
| |Real.exp x-Real.exp y| ≤ Real.exp B*|x-y| := by |
| rcases le_total x y with h|h |
| · rw [abs_of_nonpos (sub_nonpos.mpr (Real.exp_le_exp.mpr h)), |
| abs_of_nonpos (sub_nonpos.mpr h),neg_sub,neg_sub] |
| exact (exp_sub_le_exp_mul h).trans |
| (mul_le_mul_of_nonneg_right (Real.exp_le_exp.mpr hy) (sub_nonneg.mpr h)) |
| · rw [abs_of_nonneg (sub_nonneg.mpr (Real.exp_le_exp.mpr h)),abs_of_nonneg (sub_nonneg.mpr h)] |
| exact (exp_sub_le_exp_mul h).trans |
| (mul_le_mul_of_nonneg_right (Real.exp_le_exp.mpr hx) (sub_nonneg.mpr h)) |
|
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| lemma abs_sub_min_le {x y B : ℝ} (hx : x≤B) : |x-min y B| ≤ |x-y| := by |
| by_cases hy : y≤B |
| · simp [min_eq_left hy] |
| · have hy : B≤y := le_of_not_ge hy |
| rw [min_eq_right hy,abs_of_nonpos (sub_nonpos.mpr hx), |
| abs_of_nonpos (sub_nonpos.mpr (hx.trans hy))] |
| linarith |
|
|
| section Approximation |
| variable {X : Type*} [TopologicalSpace X] [MetrizableSpace X] [MeasurableSpace X] [BorelSpace X] |
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| lemma continuous_exp_test_approx (μ ν : Measure X) [IsProbabilityMeasure μ] |
| [IsProbabilityMeasure ν] (F : X → ℝ) (hF : Measurable F) |
| (hB : ∃ B : ℝ, ∀ x, |F x|≤B) {ε : ℝ} (hε : 0<ε) : |
| ∃ g : X →ᵇ ℝ, |
| |(∫ x, F x ∂μ)-(∫ x, g x ∂μ)| ≤ ε ∧ |
| |(∫ x, Real.exp (F x) ∂ν)-(∫ x, Real.exp (g x) ∂ν)| ≤ ε := by |
| obtain ⟨B,hB⟩ := hB |
| let δ := ε/(Real.exp B+1) |
| have hδ : 0<δ := div_pos hε (by positivity) |
| have hi : Integrable F (μ+ν) := integrable_of_bounded hF ⟨B,hB⟩ |
| obtain ⟨g,hg,hgi⟩ := hi.exists_boundedContinuous_integral_sub_le hδ |
| let k : X →ᵇ ℝ := min g (BoundedContinuousFunction.const X B) |
| have hk : ∀ x, k x≤B := fun x => min_le_right (g x) B |
| have hkB : ∃ b : ℝ, ∀ x, |k x|≤b := |
| ⟨‖k‖,fun x => by simpa only [Real.norm_eq_abs] using k.norm_coe_le_norm x⟩ |
| have hμF : Integrable F μ := integrable_of_bounded hF ⟨B,hB⟩ |
| have hνF : Integrable F ν := integrable_of_bounded hF ⟨B,hB⟩ |
| have herr (x : X) : |F x-k x|≤|F x-g x| := abs_sub_min_le ((le_abs_self _).trans (hB x)) |
| have hgμ : (∫ x, |F x-g x| ∂μ)≤δ := by |
| have h1 := integral_add_measure ((hμF.sub (g.integrable μ)).norm) |
| ((hνF.sub (g.integrable ν)).norm) |
| simp only [Real.norm_eq_abs, Pi.sub_apply] at h1 hg |
| rw [h1] at hg |
| have hn : 0 ≤ ∫ x, |F x-g x| ∂ν := integral_nonneg (fun x => abs_nonneg (F x-g x)) |
| linarith |
| have hgν : (∫ x, |F x-g x| ∂ν)≤δ := by |
| have h1 := integral_add_measure ((hμF.sub (g.integrable μ)).norm) |
| ((hνF.sub (g.integrable ν)).norm) |
| simp only [Real.norm_eq_abs, Pi.sub_apply] at h1 hg |
| rw [h1] at hg |
| have hn : 0 ≤ ∫ x, |F x-g x| ∂μ := integral_nonneg (fun x => abs_nonneg (F x-g x)) |
| linarith |
| have hkμ : (∫ x, |F x-k x| ∂μ)≤δ := |
| (integral_mono (hμF.sub (k.integrable μ)).abs (hμF.sub (g.integrable μ)).abs herr).trans hgμ |
| have hkν : (∫ x, |F x-k x| ∂ν)≤δ := |
| (integral_mono (hνF.sub (k.integrable ν)).abs (hνF.sub (g.integrable ν)).abs herr).trans hgν |
| have hδle : δ≤ε := by |
| apply (div_le_iff₀ (show 0<Real.exp B+1 by positivity)).mpr |
| nlinarith [Real.exp_pos B] |
| refine ⟨k,?_,?_⟩ |
| · rw [← integral_sub hμF (k.integrable μ)] |
| exact (abs_integral_le_integral_abs).trans (hkμ.trans hδle) |
| · have heF : Integrable (fun x => Real.exp (F x)) ν := exp_integrable_of_bounded hF ⟨B,hB⟩ |
| have hek : Integrable (fun x => Real.exp (k x)) ν := exp_integrable_of_bounded k.continuous.measurable hkB |
| rw [← integral_sub heF hek] |
| calc |
| _ ≤ ∫ x, |Real.exp (F x)-Real.exp (k x)| ∂ν := abs_integral_le_integral_abs |
| _ ≤ ∫ x, Real.exp B*|F x-k x| ∂ν := integral_mono (heF.sub hek).abs |
| (((hνF.sub (k.integrable ν)).abs).const_mul _) (fun x => |
| abs_exp_sub_le ((le_abs_self _).trans (hB x)) (hk x)) |
| _ = Real.exp B*(∫ x, |F x-k x| ∂ν) := integral_const_mul _ _ |
| _ ≤ Real.exp B*δ := mul_le_mul_of_nonneg_left hkν (Real.exp_pos B).le |
| _ ≤ ε := by |
| dsimp only [δ] |
| rw [← mul_div_assoc] |
| apply (div_le_iff₀ (show 0<Real.exp B+1 by positivity)).mpr |
| nlinarith |
|
|
| theorem expBudget_of_continuous (μ ν : Measure X) [IsProbabilityMeasure μ] |
| [IsProbabilityMeasure ν] {C : ℝ} |
| (h : ∀ g : X →ᵇ ℝ, (∫ x, g x ∂μ)-(∫ x, Real.exp (g x) ∂ν)+1≤C) : |
| ExpBudget μ ν C := by |
| intro F hF hB |
| apply le_of_forall_pos_le_add |
| intro ε hε |
| obtain ⟨g,hg,hge⟩ := continuous_exp_test_approx μ ν F hF hB (half_pos hε) |
| have h1 := (abs_le.mp hg).2 |
| have h2 := (abs_le.mp hge).1 |
| have h3 := h g |
| linarith |
|
|
| end Approximation |
| end DirectionalTransience.Entropy |
|
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| end |
|
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| section |
|
|
| open MeasureTheory ProbabilityTheory InformationTheory TopologicalSpace Filter |
| open scoped ENNReal Topology BoundedContinuousFunction |
| namespace DirectionalTransience.Entropy |
|
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| variable {X I : Type*} [TopologicalSpace X] [CompactSpace X] [MetrizableSpace X] |
| [MeasurableSpace X] [BorelSpace X] |
|
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| lemma kl_le_of_weak_limit {l : Filter I} [l.NeBot] |
| (μs νs : I → ProbabilityMeasure X) (μ ν : ProbabilityMeasure X) |
| (hμ : Tendsto μs l (𝓝 μ)) (hν : Tendsto νs l (𝓝 ν)) |
| {C : ℝ} (hC : 0≤C) |
| (hbudget : ∀ᶠ i in l, klDiv (μs i : Measure X) (νs i : Measure X) ≤ ENNReal.ofReal C) : |
| klDiv (μ : Measure X) (ν : Measure X) ≤ ENNReal.ofReal C := by |
| apply kl_le_of_expBudget |
| apply expBudget_of_continuous |
| intro f |
| let ef : X →ᵇ ℝ := BoundedContinuousFunction.mkOfCompact ⟨fun x => Real.exp (f x), Real.continuous_exp.comp f.continuous⟩ |
| have h1 := ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hμ f |
| have h2 := ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hν ef |
| apply le_of_tendsto ((h1.sub h2).add_const 1) |
| filter_upwards [hbudget] with i hi |
| exact expBudget_of_kl_le (μs i : Measure X) (νs i : Measure X) hC hi |
| f f.continuous.measurable ⟨‖f‖,fun x => by simpa only [Real.norm_eq_abs] using f.norm_coe_le_norm x⟩ |
|
|
| end DirectionalTransience.Entropy |
|
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| end |
|
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| end OAI |
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