| import OAI.Probability.PosteriorReplicas.Information.EntropyTest |
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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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| lemma entropy_loss_test {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] |
| (μ ν : Measure X) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (hkl : klDiv μ ν ≠ ∞) {ψ : X → Y} (hψ : Measurable ψ) |
| {g : Y → ℝ} (hg : Measurable g) |
| (he : Integrable (fun y => Real.exp (g y)) (ν.map ψ)) |
| (hmean : (∫ y, Real.exp (g y) ∂ν.map ψ) ≤ 1) |
| (herr : Integrable (fun x => max (llr μ ν x - g (ψ x)) 0) μ) : |
| Integrable (fun x => g (ψ x)) μ ∧ |
| Integrable (fun x => llr μ ν x - g (ψ x)) μ ∧ |
| (klDiv μ ν).toReal ≤ (klDiv (μ.map ψ) (ν.map ψ)).toReal + |
| ∫ x, llr μ ν x - g (ψ x) ∂μ := by |
| have hkl' : klDiv (μ.map ψ) (ν.map ψ) ≠ ∞ := ne_top_of_le_ne_top hkl (klDiv_map_le μ ν hψ) |
| obtain ⟨hac,hllr⟩ := klDiv_ne_top_iff.mp hkl |
| have hp := (entropy_test_positive_integrable (μ.map ψ) (ν.map ψ) hkl' hg he).comp_measurable hψ |
| have hb := (hllr.norm.add hp).add herr |
| have hgi : Integrable (fun x => g (ψ x)) μ := by |
| apply hb.mono' (hg.comp hψ).aestronglyMeasurable |
| apply ae_of_all |
| intro x |
| simp only [Pi.add_apply, Function.comp_def, Real.norm_eq_abs] |
| apply abs_le.mpr |
| constructor |
| · have h₁ := neg_abs_le (llr μ ν x) |
| have h₂ := le_max_left (llr μ ν x-g (ψ x)) 0 |
| have h₃ := le_max_right (g (ψ x)) 0 |
| linarith |
| · have h₁ := le_max_left (g (ψ x)) 0 |
| have h₂ := abs_nonneg (llr μ ν x) |
| have h₃ := le_max_right (llr μ ν x-g (ψ x)) 0 |
| linarith |
| refine ⟨hgi,hllr.sub hgi,?_⟩ |
| have hgm : Integrable g (μ.map ψ) := (integrable_map_measure hg.aestronglyMeasurable hψ.aemeasurable).mpr hgi |
| have ht := entropy_test_normalized (μ.map ψ) (ν.map ψ) hkl' hgm he hmean |
| rw [integral_map hψ.aemeasurable hg.aestronglyMeasurable] at ht |
| rw [toReal_klDiv_of_measure_eq hac (by simp), integral_sub hllr hgi] |
| linarith |
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| end PosteriorReplicas |
| end |
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| end OAI |
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