| import Mathlib |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory InformationTheory Set Filter |
| open scoped ENNReal NNReal |
| noncomputable section |
| namespace PosteriorReplicas |
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| variable {X : Type*} [MeasurableSpace X] |
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| lemma integrable_exp_neg_llr (μ ν : Measure X) [IsFiniteMeasure μ] [IsFiniteMeasure ν] |
| (hac : μ ≪ ν) : Integrable (fun x => Real.exp (-llr μ ν x)) μ := by |
| exact ν.integrable_toReal_rnDeriv.congr (exp_neg_llr hac).symm |
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| lemma entropy_test_positive_integrable (μ ν : Measure X) |
| [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (hkl : klDiv μ ν ≠ ∞) {g : X → ℝ} (hg : Measurable g) |
| (he : Integrable (fun x => Real.exp (g x)) ν) : |
| Integrable (fun x => max (g x) 0) μ := by |
| obtain ⟨hac,hllr⟩ := klDiv_ne_top_iff.mp hkl |
| have : IsProbabilityMeasure (ν.tilted g) := isProbabilityMeasure_tilted he |
| have hac' : μ ≪ ν.tilted g := hac.trans (absolutelyContinuous_tilted he) |
| have hi := (hllr.norm.add (integrable_exp_neg_llr μ (ν.tilted g) hac')).add |
| (integrable_const |Real.log (∫ x, Real.exp (g x) ∂ν)|) |
| apply hi.mono' (by fun_prop) |
| filter_upwards [llr_tilted_right hac he] with x hx |
| rw [Real.norm_eq_abs, abs_of_nonneg (le_max_right _ 0)] |
| simp only [Pi.add_apply, Real.norm_eq_abs] |
| apply max_le |
| · have h₁ := Real.add_one_le_exp (-llr μ (ν.tilted g) x) |
| have h₂ := le_abs_self (llr μ ν x) |
| have h₃ := le_abs_self (Real.log (∫ x, Real.exp (g x) ∂ν)) |
| linarith |
| · positivity |
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| lemma entropy_test_integral (μ ν : Measure X) |
| [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (hkl : klDiv μ ν ≠ ∞) {g : X → ℝ} (hg : Integrable g μ) |
| (he : Integrable (fun x => Real.exp (g x)) ν) : |
| (∫ x, g x ∂μ) ≤ (klDiv μ ν).toReal + Real.log (∫ x, Real.exp (g x) ∂ν) := by |
| obtain ⟨hac,hllr⟩ := klDiv_ne_top_iff.mp hkl |
| have : IsProbabilityMeasure (ν.tilted g) := isProbabilityMeasure_tilted he |
| have hnon := integral_llr_add_sub_measure_univ_nonneg |
| (hac.trans (absolutelyContinuous_tilted he)) (integrable_llr_tilted_right hac hg hllr he) |
| rw [integral_llr_tilted_right hac hg he hllr] at hnon |
| rw [toReal_klDiv_of_measure_eq hac (by simp)] |
| simp only [probReal_univ, add_sub_cancel_right] at hnon |
| linarith |
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| lemma entropy_test_normalized (μ ν : Measure X) |
| [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (hkl : klDiv μ ν ≠ ∞) {g : X → ℝ} (hg : Integrable g μ) |
| (he : Integrable (fun x => Real.exp (g x)) ν) |
| (hmean : (∫ x, Real.exp (g x) ∂ν) ≤ 1) : |
| (∫ x, g x ∂μ) ≤ (klDiv μ ν).toReal := by |
| have h := entropy_test_integral μ ν hkl hg he |
| have hlog : Real.log (∫ x, Real.exp (g x) ∂ν) ≤ 0 := |
| Real.log_nonpos (integral_nonneg (fun _ => (Real.exp_pos _).le)) hmean |
| linarith |
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| lemma llr_withDensity_ae (ν : Measure X) [SigmaFinite ν] {g : X → ℝ≥0∞} |
| (hg : Measurable g) [IsFiniteMeasure (ν.withDensity g)] : |
| llr (ν.withDensity g) ν =ᵐ[ν.withDensity g] (fun x => Real.log (g x).toReal) := by |
| filter_upwards [(withDensity_absolutelyContinuous ν g).ae_le (ν.rnDeriv_withDensity hg)] with x hx |
| exact congrArg (fun t : ℝ≥0∞ => Real.log t.toReal) hx |
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| lemma klDiv_withDensity_eq (ν : Measure X) [IsProbabilityMeasure ν] {g : X → ℝ≥0∞} |
| (hg : Measurable g) [IsProbabilityMeasure (ν.withDensity g)] |
| (hi : Integrable (fun x => Real.log (g x).toReal) (ν.withDensity g)) : |
| klDiv (ν.withDensity g) ν = ENNReal.ofReal (∫ x, Real.log (g x).toReal ∂ν.withDensity g) := by |
| have hi' : Integrable (llr (ν.withDensity g) ν) (ν.withDensity g) := |
| hi.congr (llr_withDensity_ae ν hg).symm |
| rw [klDiv_of_ac_of_integrable (withDensity_absolutelyContinuous ν g) hi'] |
| simp only [probReal_univ, add_sub_cancel_right] |
| congr 1 |
| exact integral_congr_ae (llr_withDensity_ae ν hg) |
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| end PosteriorReplicas |
| end |
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| end OAI |
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