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b317718 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 | import Mathlib
import OAI.Probability.GaussianInformation.Kernel.Bayes
namespace OAI
open MeasureTheory ProbabilityTheory Metric Set
open scoped ENNReal
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal
open MeasureTheory ProbabilityTheory
namespace KernelPotential.DriftTools
namespace FiniteReference
variable {α β ι : Type*} [MeasurableSpace α] [MeasurableSpace β]
[Fintype ι] [MeasurableSpace ι] [MeasurableSingletonClass ι]
noncomputable def joint (ν : Measure α) (Λ : Measure β)
(K : β → α → ℝ≥0∞) (κ : β → ι → ℝ≥0∞) : Measure (α × β × ι) :=
(ν.prod (Λ.prod Measure.count)).withDensity (fun x => K x.2.1 x.1 * κ x.2.1 x.2.2)
lemma joint_integral_nonneg (ν : Measure α) (Λ : Measure β) [SigmaFinite ν] [SigmaFinite Λ]
(K : β → α → ℝ≥0∞) (κ : β → ι → ℝ≥0∞)
(hK : Measurable (Function.uncurry K))
(hκ : Measurable (Function.uncurry κ))
(f : α × β × ι → ℝ≥0∞) (hf : Measurable f) :
(∫⁻ x,f x ∂joint ν Λ K κ) = ∫⁻ s,∫⁻ z,∑ w,K z s*κ z w*f (s,z,w) ∂Λ ∂ν := by
rw [joint,lintegral_withDensity_eq_lintegral_mul]
· rw [lintegral_prod]
· apply lintegral_congr; intro s
rw [lintegral_prod]
· apply lintegral_congr; intro z
simp only [lintegral_count,tsum_fintype]
rfl
· fun_prop
· fun_prop
· fun_prop
· exact hf
lemma joint_probability (ν : Measure α) (Λ : Measure β) [IsProbabilityMeasure ν] [SigmaFinite Λ]
(K : β → α → ℝ≥0∞) (κ : β → ι → ℝ≥0∞)
(hK : Measurable (Function.uncurry K)) (hκ : Measurable (Function.uncurry κ))
(hK1 : ∀ s,∫⁻ z,K z s ∂Λ=1) (hκ1 : ∀ z,∑ w,κ z w=1) :
IsProbabilityMeasure (joint ν Λ K κ) := by
constructor
rw [← lintegral_one,(joint_integral_nonneg ν Λ K κ hK hκ _ measurable_const)]
simp only [mul_one,← Finset.mul_sum,hκ1,mul_one,hK1,lintegral_one,measure_univ]
noncomputable def C (ν : Measure α) (K : β → α → ℝ≥0∞) (x : ι → α → ℝ≥0∞)
(z : β) (w : ι) : ℝ≥0∞ := ∫⁻ s,K z s*x w s ∂ν
lemma measurable_C (ν : Measure α) [SFinite ν]
(K : β → α → ℝ≥0∞) (x : ι → α → ℝ≥0∞)
(hK : Measurable (Function.uncurry K)) (hx : Measurable (Function.uncurry x)) :
Measurable (Function.uncurry (C ν K x)) := by
apply measurable_from_prod_countable_left
intro index
unfold C
apply Measurable.lintegral_prod_right
fun_prop
lemma joint_compensation (ν : Measure α) (Λ : Measure β) [IsProbabilityMeasure ν] [SigmaFinite Λ]
(K : β → α → ℝ≥0∞) (κ : β → ι → ℝ≥0∞)
(hK : Measurable (Function.uncurry K)) (hκ : Measurable (Function.uncurry κ))
(hκ1 : ∀ z,∑ w,κ z w=1)
(x : ι → α → ℝ≥0∞) (hx : Measurable (Function.uncurry x))
(q : β → ℝ≥0∞) (hq : Measurable q) (hq1 : ∫⁻ z,q z ∂Λ=1) :
(∫⁻ t,q t.2.1*x t.2.2 t.1/(1+C ν K x t.2.1 t.2.2) ∂joint ν Λ K κ) ≤ 1 := by
rw [joint_integral_nonneg ν Λ K κ hK hκ]
· have hm : Measurable (fun t : α × β => ∑ w,K t.2 t.1*κ t.2 w*
(q t.2*x w t.1/(1+C ν K x t.2 w))) := by
have hC := measurable_C ν K x hK hx
fun_prop
rw [lintegral_lintegral_swap hm.aemeasurable]
have he : (∫⁻ z,∫⁻ s,∑ w,K z s*κ z w*(q z*x w s/(1+C ν K x z w)) ∂ν ∂Λ) =
∫⁻ z,∑ w,∫⁻ s,K z s*κ z w*q z*x w s/(1+C ν K x z w) ∂ν ∂Λ := by
apply lintegral_congr; intro z
rw [lintegral_finsetSum]
· apply Finset.sum_congr rfl; intro w _
apply lintegral_congr; intro s
dsimp only [C]; simp only [div_eq_mul_inv]; ring
· intro w _; fun_prop
rw [he]
exact compensation_lintegral ν Λ K (fun z => hK.comp (measurable_const.prodMk measurable_id))
κ hκ1 x (fun w => hx.comp (measurable_const.prodMk measurable_id)) q hq1
· have hC := measurable_C ν K x hK hx
fun_prop
lemma log_comparison (ν : Measure α) (Λ : Measure β) [IsProbabilityMeasure ν] [SigmaFinite Λ]
(K : β → α → ℝ≥0∞) (κ : β → ι → ℝ≥0∞)
(hK : Measurable (Function.uncurry K)) (hκ : Measurable (Function.uncurry κ))
(hK1 : ∀ s,∫⁻ z,K z s ∂Λ=1) (hκ1 : ∀ z,∑ w,κ z w=1)
(x : ι → α → ℝ≥0∞) (hx : Measurable (Function.uncurry x))
(hxpos : ∀ w s,0 < x w s) (hxfin : ∀ w s,x w s ≠ ∞)
(q : β → ℝ≥0∞) (hq : Measurable q) (hqpos : ∀ z,0 < q z)
(hqfin : ∀ z,q z ≠ ∞) (hq1 : ∫⁻ z,q z ∂Λ=1)
(hCfin : ∀ z w,C ν K x z w ≠ ∞)
(hlogx : Integrable (fun t => Real.log (x t.2.2 t.1).toReal) (joint ν Λ K κ))
(hlogq : Integrable (fun t => Real.log (q t.2.1).toReal) (joint ν Λ K κ))
(hlogC : Integrable (fun t => Real.log (1+(C ν K x t.2.1 t.2.2).toReal)) (joint ν Λ K κ)) :
(∫ t,Real.log (x t.2.2 t.1).toReal ∂joint ν Λ K κ) ≤
(∫ t,Real.log (1+(C ν K x t.2.1 t.2.2).toReal) ∂joint ν Λ K κ) -
∫ t,Real.log (q t.2.1).toReal ∂joint ν Λ K κ := by
have := joint_probability ν Λ K κ hK hκ hK1 hκ1
have hC := measurable_C ν K x hK hx
apply compensated_log_bound (joint ν Λ K κ)
(fun t => (x t.2.2 t.1).toReal) (fun t => (q t.2.1).toReal)
(fun t => (C ν K x t.2.1 t.2.2).toReal)
(Filter.Eventually.of_forall (fun t => ENNReal.toReal_pos (hxpos _ _).ne' (hxfin _ _)))
(Filter.Eventually.of_forall (fun t => ENNReal.toReal_pos (hqpos _).ne' (hqfin _)))
(Filter.Eventually.of_forall (fun _ => ENNReal.toReal_nonneg))
(by
have hm : Measurable (fun t : α × β × ι => (q t.2.1).toReal*(x t.2.2 t.1).toReal /
(1+(C ν K x t.2.1 t.2.2).toReal)) := by fun_prop
exact hm.aestronglyMeasurable) _ hlogx hlogq hlogC
have he (t : α × β × ι) : ENNReal.ofReal ((q t.2.1).toReal*(x t.2.2 t.1).toReal /
(1+(C ν K x t.2.1 t.2.2).toReal)) =
q t.2.1*x t.2.2 t.1/(1+C ν K x t.2.1 t.2.2) := by
rw [ENNReal.ofReal_div_of_pos (by positivity),ENNReal.ofReal_mul ENNReal.toReal_nonneg,
ENNReal.ofReal_toReal (hqfin _),ENNReal.ofReal_toReal (hxfin _ _),
ENNReal.ofReal_add (by norm_num) ENNReal.toReal_nonneg,
ENNReal.ofReal_one,ENNReal.ofReal_toReal (hCfin _ _)]
simp_rw [he]
exact joint_compensation ν Λ K κ hK hκ hκ1 x hx q hq hq1
end FiniteReference
end KernelPotential.DriftTools
end OAI
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