| import Mathlib |
| import OAI.Probability.Ballisticity.Entropy.EntropyVariational |
|
|
| namespace OAI |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory InformationTheory |
| open scoped ENNReal Classical BigOperators |
| namespace DirectionalTransience.Entropy |
| variable {X I : Type*} [MeasurableSpace X] |
|
|
| lemma restricted_exp_test (μ ν : Measure X) [IsProbabilityMeasure μ] |
| [IsProbabilityMeasure ν] (hkl : klDiv μ ν ≠ ∞) |
| (E : Set X) (hE : MeasurableSet E) |
| (F : X → ℝ) (hF : Measurable F) (hB : ∃ B : ℝ, ∀ x, |F x| ≤ B) : |
| (∫ x, F x ∂μ.restrict E) - (∫ x, Real.exp (F x) ∂ν.restrict E) + ν.real E ≤ |
| (klDiv μ ν).toReal := by |
| let G : X → ℝ := E.indicator F |
| have hG : Measurable G := hF.indicator hE |
| have hGB : ∃ B : ℝ, ∀ x, |G x| ≤ B := by |
| obtain ⟨B,hB⟩ := hB |
| refine ⟨max B 0,fun x => ?_⟩ |
| by_cases hx : x ∈ E |
| · simpa [G,hx] using (hB x).trans (le_max_left B 0) |
| · simp [G,hx] |
| have h := entropy_exp_test_le μ ν hkl G (integrable_of_bounded hG hGB) |
| (exp_integrable_of_bounded hG hGB) |
| have heq : (fun x => Real.exp (G x)) = |
| fun x => E.indicator (fun x => Real.exp (F x)) x + Eᶜ.indicator (fun _ => (1:ℝ)) x := by |
| funext x |
| by_cases hx : x ∈ E <;> simp [G,hx] |
| have hei := (exp_integrable_of_bounded (μ:=ν) hF hB).indicator hE |
| have hci := (integrable_const (μ:=ν) (1:ℝ)).indicator hE.compl |
| rw [heq, integral_add hei hci, integral_indicator hE, |
| integral_indicator_const 1 hE.compl, smul_eq_mul, mul_one, |
| probReal_compl_eq_one_sub hE] at h |
| rw [integral_indicator hE] at h |
| linarith |
|
|
| noncomputable def activeSum (s : Finset I) (μ : I → Measure X) (E : I → Set X) : Measure X := |
| ∑ i ∈ s, (μ i).restrict (E i) |
| |
| instance activeSum_finite (s : Finset I) (μ : I → Measure X) |
| [∀ i, IsFiniteMeasure (μ i)] (E : I → Set X) : IsFiniteMeasure (activeSum s μ E) := by |
| unfold activeSum |
| infer_instance |
| |
| lemma activeSum_integral (s : Finset I) (μ : I → Measure X) |
| [∀ i, IsFiniteMeasure (μ i)] (E : I → Set X) |
| (F : X → ℝ) (hF : Measurable F) (hB : ∃ B : ℝ, ∀ x, |F x| ≤ B) : |
| (∫ x, F x ∂activeSum s μ E) = ∑ i ∈ s, ∫ x, F x ∂(μ i).restrict (E i) := by |
| unfold activeSum |
| exact integral_finsetSum_measure (fun i _ => integrable_of_bounded hF hB) |
| |
| lemma activeSum_mass (s : Finset I) (μ : I → Measure X) |
| [∀ i, IsFiniteMeasure (μ i)] (E : I → Set X) : |
| (activeSum s μ E).real Set.univ = ∑ i ∈ s, (μ i).real (E i) := by |
| have h := activeSum_integral s μ E (fun _ => (1:ℝ)) measurable_const ⟨1,fun _ => by norm_num⟩ |
| simpa only [integral_const,smul_eq_mul,mul_one,measureReal_restrict_apply_univ] using h |
| |
| theorem activeSum_exp_test (s : Finset I) (μ ν : I → Measure X) |
| [∀ i, IsProbabilityMeasure (μ i)] [∀ i, IsProbabilityMeasure (ν i)] |
| (hkl : ∀ i ∈ s, klDiv (μ i) (ν i) ≠ ∞) |
| (E : I → Set X) (hE : ∀ i ∈ s, MeasurableSet (E i)) |
| (F : X → ℝ) (hF : Measurable F) (hB : ∃ B : ℝ, ∀ x, |F x| ≤ B) : |
| (∫ x, F x ∂activeSum s μ E) - (∫ x, Real.exp (F x) ∂activeSum s ν E) + |
| (activeSum s ν E).real Set.univ ≤ ∑ i ∈ s, (klDiv (μ i) (ν i)).toReal := by |
| have heB : ∃ B : ℝ, ∀ x, |Real.exp (F x)| ≤ B := by |
| obtain ⟨B,hB⟩ := hB |
| exact ⟨Real.exp B,fun x => by |
| rw [abs_of_pos (Real.exp_pos _)] |
| exact Real.exp_le_exp.mpr ((le_abs_self _).trans (hB x))⟩ |
| rw [activeSum_integral s μ E F hF hB,activeSum_integral s ν E _ hF.exp heB, |
| activeSum_mass,← Finset.sum_sub_distrib,← Finset.sum_add_distrib] |
| exact Finset.sum_le_sum fun i hi => restricted_exp_test (μ i) (ν i) (hkl i hi) |
| (E i) (hE i hi) F hF hB |
| |
| theorem normalized_activeSum_kl_le [Nonempty X] |
| (s : Finset I) (μ ν : I → Measure X) |
| [∀ i, IsProbabilityMeasure (μ i)] [∀ i, IsProbabilityMeasure (ν i)] |
| (hkl : ∀ i ∈ s, klDiv (μ i) (ν i) ≠ ∞) |
| (E : I → Set X) (hE : ∀ i ∈ s, MeasurableSet (E i)) |
| (hequal : ∀ i ∈ s, μ i (E i) = ν i (E i)) |
| (hpos : 0 < (activeSum s μ E).real Set.univ) : |
| let μs : FiniteMeasure X := ⟨activeSum s μ E,inferInstance⟩ |
| let νs : FiniteMeasure X := ⟨activeSum s ν E,inferInstance⟩ |
| klDiv (μs.normalize : Measure X) (νs.normalize : Measure X) ≤ |
| ENNReal.ofReal ((∑ i ∈ s, (klDiv (μ i) (ν i)).toReal) / |
| (activeSum s μ E).real Set.univ) := by |
| let μs : FiniteMeasure X := ⟨activeSum s μ E,inferInstance⟩ |
| let νs : FiniteMeasure X := ⟨activeSum s ν E,inferInstance⟩ |
| have hm : (activeSum s ν E).real Set.univ = (activeSum s μ E).real Set.univ := by |
| rw [activeSum_mass,activeSum_mass] |
| apply Finset.sum_congr rfl |
| intro i hi |
| exact congrArg ENNReal.toReal (hequal i hi).symm |
| have hμne : μs ≠ 0 := by |
| intro h |
| have hz : (activeSum s μ E).real Set.univ = 0 := by |
| change (μs : Measure X).real Set.univ = 0 |
| rw [h]; simp |
| linarith |
| have hνne : νs ≠ 0 := by |
| intro h |
| have hz : (activeSum s ν E).real Set.univ = 0 := by |
| change (νs : Measure X).real Set.univ = 0 |
| rw [h]; simp |
| linarith |
| apply kl_le_of_expBudget |
| intro F hF hB |
| have h := activeSum_exp_test s μ ν hkl E hE F hF hB |
| rw [hm] at h |
| rw [← μs.average_eq_integral_normalize hμne,← νs.average_eq_integral_normalize hνne, |
| average,average,integral_smul_measure,integral_smul_measure] |
| change ((activeSum s μ E) Set.univ)⁻¹.toReal * _ - |
| ((activeSum s ν E) Set.univ)⁻¹.toReal * _ + 1 ≤ _ |
| rw [ENNReal.toReal_inv,ENNReal.toReal_inv] |
| change ((activeSum s μ E).real Set.univ)⁻¹ * _ - |
| ((activeSum s ν E).real Set.univ)⁻¹ * _ + 1 ≤ _ |
| rw [hm] |
| apply (le_div_iff₀ hpos).mpr |
| field_simp |
| exact h |
| |
| end DirectionalTransience.Entropy |
| |
| end |
| |
| end OAI |
| |
