File size: 3,896 Bytes
a2ee19a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 | import Mathlib
namespace OAI
open MeasureTheory ProbabilityTheory InformationTheory Set Filter
open scoped ENNReal NNReal
noncomputable section
namespace PosteriorReplicas
variable {X : Type*} [MeasurableSpace X]
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
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
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
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
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
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)
end PosteriorReplicas
end
end OAI
|