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dd14c26 | 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 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import Mathlib
namespace OAI
open MeasureTheory ProbabilityTheory Set
open scoped BigOperators ENNReal
namespace RepeatedLocalization
namespace FiniteEntropy
lemma llr_restrict_left {α : Type*} [MeasurableSpace α] (P ν : Measure α)
[IsFiniteMeasure P] [IsFiniteMeasure ν] (hac : P ≪ ν) {E : Set α} (hE : MeasurableSet E) :
llr (P.restrict E) ν =ᵐ[P.restrict E] llr P ν := by
have ha : P.restrict E ≪ ν := Measure.absolutelyContinuous_restrict.trans hac
filter_upwards [ha.ae_le (Measure.rnDeriv_restrict P ν hE),ae_restrict_mem hE] with x hx hxE
simp only [llr,Set.indicator_of_mem hxE] at hx ⊢
rw [hx]
lemma llr_chain {α : Type*} [MeasurableSpace α] (P ν ρ : Measure α)
[IsFiniteMeasure P] [IsFiniteMeasure ν] [IsFiniteMeasure ρ]
(hPν : P ≪ ν) (hνρ : ν ≪ ρ) :
llr P ρ =ᵐ[P] fun x => llr P ν x + llr ν ρ x := by
filter_upwards [(hPν.trans hνρ).ae_le (Measure.rnDeriv_mul_rnDeriv hPν),
Measure.rnDeriv_pos hPν,hPν.ae_le (Measure.rnDeriv_lt_top P ν),
hPν.ae_le (Measure.rnDeriv_pos hνρ),(hPν.trans hνρ).ae_le (Measure.rnDeriv_lt_top ν ρ)]
with x hx hp hp' hν hν'
simp only [llr,Pi.mul_apply] at hx ⊢
rw [← hx,ENNReal.toReal_mul,Real.log_mul]
· exact ENNReal.toReal_ne_zero.mpr ⟨hp.ne',hp'.ne⟩
· exact ENNReal.toReal_ne_zero.mpr ⟨hν.ne',hν'.ne⟩
lemma llr_cond_left {α : Type*} [MeasurableSpace α] (P ν : Measure α)
[IsFiniteMeasure P] [IsFiniteMeasure ν] (hac : P ≪ ν)
{E : Set α} (hE : MeasurableSet E) (hp : P E ≠ 0) :
llr (ProbabilityTheory.cond P E) ν =ᵐ[ProbabilityTheory.cond P E]
fun x => llr P ν x - Real.log (P E).toReal := by
have ha : P.restrict E ≪ ν := Measure.absolutelyContinuous_restrict.trans hac
have hleft := llr_smul_left ha (P E)⁻¹ (ENNReal.inv_ne_zero.mpr (measure_ne_top P E))
(ENNReal.inv_ne_top.mpr hp)
have hrestrict := llr_restrict_left P ν hac hE
have hcond : ProbabilityTheory.cond P E ≪ P.restrict E := Measure.smul_absolutelyContinuous
filter_upwards [hcond.ae_le hleft,hcond.ae_le hrestrict] with x hx hy
change llr ((P E)⁻¹ • P.restrict E) ν x = _
rw [hx,hy,ENNReal.toReal_inv,Real.log_inv]
ring
lemma cond_ac_cond {α : Type*} [MeasurableSpace α] (P ν : Measure α)
[IsFiniteMeasure ν] (hac : P ≪ ν) {E R : Set α} (hER : E ⊆ R) (hνR : ν R ≠ 0) :
ProbabilityTheory.cond P E ≪ ProbabilityTheory.cond ν R := by
have normalize : ν R • ProbabilityTheory.cond ν R = ν.restrict R := by
rw [ProbabilityTheory.cond, smul_smul,
ENNReal.mul_inv_cancel hνR (measure_ne_top ν R), one_smul]
have restrict_ac : ν.restrict R ≪ ProbabilityTheory.cond ν R := by
rw [← normalize]
exact Measure.smul_absolutelyContinuous
exact (Measure.smul_absolutelyContinuous.trans
((Measure.absolutelyContinuous_of_le (Measure.restrict_mono hER le_rfl)).trans (hac.restrict R))).trans
restrict_ac
lemma llr_cond_cond {α : Type*} [MeasurableSpace α] (P ν : Measure α)
[IsFiniteMeasure P] [IsFiniteMeasure ν] (hac : P ≪ ν)
{E R : Set α} (hE : MeasurableSet E) (hR : MeasurableSet R) (hER : E ⊆ R)
(hp : P E ≠ 0) (hνR : ν R ≠ 0) :
llr (ProbabilityTheory.cond P E) (ProbabilityTheory.cond ν R) =ᵐ[ProbabilityTheory.cond P E]
fun x => llr P ν x - Real.log (P E).toReal + Real.log (ν R).toReal := by
let := ProbabilityTheory.cond_isProbabilityMeasure (μ := P) hp
let := ProbabilityTheory.cond_isProbabilityMeasure (μ := ν) hνR
have ha := cond_ac_cond P ν hac hER hνR
have he := llr_chain (ProbabilityTheory.cond P E) (ProbabilityTheory.cond ν R) ν ha
ProbabilityTheory.cond_absolutelyContinuous
have hpν := llr_cond_left P ν hac hE hp
have hνν := llr_cond_left ν ν Measure.AbsolutelyContinuous.rfl hR hνR
have hs := (ProbabilityTheory.cond_absolutelyContinuous (μ := ν) (s := R)).ae_le (llr_self ν)
filter_upwards [he,hpν,ha.ae_le hνν,ha.ae_le hs] with x he hpν hνν hs
simp only [Pi.zero_apply] at hs
rw [hs] at hνν
linarith
lemma integral_cond {α : Type*} [MeasurableSpace α] (P : Measure α) (E : Set α) (f : α → ℝ) :
(∫ x,f x ∂ProbabilityTheory.cond P E) = (P E).toReal⁻¹ * ∫ x in E,f x ∂P := by
rw [ProbabilityTheory.cond,integral_smul_measure,ENNReal.toReal_inv,smul_eq_mul]
lemma klDiv_cond_cond {α : Type*} [MeasurableSpace α] (P ν : Measure α)
[IsProbabilityMeasure P] [IsProbabilityMeasure ν] (hac : P ≪ ν)
(hK : InformationTheory.klDiv P ν ≠ ∞) {E R : Set α}
(hE : MeasurableSet E) (hR : MeasurableSet R) (hER : E ⊆ R)
(hp : P E ≠ 0) (hνR : ν R ≠ 0) :
InformationTheory.klDiv (ProbabilityTheory.cond P E) (ProbabilityTheory.cond ν R) ≠ ∞ ∧
(P E).toReal * (InformationTheory.klDiv (ProbabilityTheory.cond P E) (ProbabilityTheory.cond ν R)).toReal =
(∫ x in E,llr P ν x ∂P) - (P E).toReal * Real.log (P E).toReal +
(P E).toReal * Real.log (ν R).toReal := by
let := ProbabilityTheory.cond_isProbabilityMeasure (μ := P) hp
let := ProbabilityTheory.cond_isProbabilityMeasure (μ := ν) hνR
have ha := cond_ac_cond P ν hac hER hνR
have hint := (InformationTheory.klDiv_ne_top_iff.mp hK).2
have hint' : Integrable (llr P ν) (ProbabilityTheory.cond P E) :=
hint.integrableOn.smul_measure (ENNReal.inv_ne_top.mpr hp)
have he := llr_cond_cond P ν hac hE hR hER hp hνR
have hi : Integrable (llr (ProbabilityTheory.cond P E) (ProbabilityTheory.cond ν R))
(ProbabilityTheory.cond P E) :=
((hint'.sub (integrable_const _)).add (integrable_const _)).congr he.symm
refine ⟨InformationTheory.klDiv_ne_top ha hi,?_⟩
rw [InformationTheory.toReal_klDiv_of_measure_eq ha (by simp),integral_congr_ae he]
have hsub := integral_sub hint' (integrable_const (Real.log (P E).toReal))
have hadd := integral_add (hint'.sub (integrable_const (Real.log (P E).toReal)))
(integrable_const (Real.log (ν R).toReal))
simp only [Pi.sub_apply] at hsub hadd
rw [hadd,hsub]
simp only [integral_const,Measure.real,measure_univ,ENNReal.toReal_one,one_smul]
rw [integral_cond]
have hp' : (P E).toReal ≠ 0 := ENNReal.toReal_ne_zero.mpr ⟨hp,measure_ne_top _ _⟩
field_simp
end FiniteEntropy
end RepeatedLocalization
end OAI
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