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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 | import Mathlib
import OAI.Probability.LogConcave.Analysis.Law
namespace OAI
section
noncomputable section
namespace LogConcaveSampling
open MeasureTheory ProbabilityTheory Function
open scoped Classical ENNReal
lemma Coupling.DistanceCoupled.integral_lipschitz {E : Type*}
[MetricSpace E] [CompleteSpace E] [SecondCountableTopology E]
[MeasurableSpace E] [BorelSpace E] {μ ν : Measure E} {R L C : ℝ}
(h : Coupling.DistanceCoupled μ ν R) (hL : 0≤L) {f : E → ℝ}
(hf : Measurable f) (hb : ∀x,|f x|≤C)
(hl : ∀x y,|f x-f y|≤L*dist x y) :
|(∫x,f x ∂μ)-(∫x,f x ∂ν)|≤L*R := by
obtain ⟨κ,hκ,hx,hy,hd⟩ := h
let := hκ
have hix := bounded_integrable (hf.comp measurable_fst) (fun p : E×E => hb p.1) (μ:=κ)
have hiy := bounded_integrable (hf.comp measurable_snd) (fun p : E×E => hb p.2) (μ:=κ)
simp only [Function.comp_def] at hix hiy
rw [←hx,←hy]
change |(∫x,f x ∂κ.map Prod.fst)-(∫x,f x ∂κ.map Prod.snd)|≤_
rw [integral_map measurable_fst.aemeasurable hf.aestronglyMeasurable,
integral_map measurable_snd.aemeasurable hf.aestronglyMeasurable,←integral_sub hix hiy,
←Real.norm_eq_abs]
exact (norm_integral_le_of_norm_le_const (hd.mono (fun p hp =>
(show ‖f p.1-f p.2‖≤L*dist p.1 p.2 by simpa only [Real.norm_eq_abs] using hl p.1 p.2).trans
(mul_le_mul_of_nonneg_left hp hL)))).trans (by simp)
lemma TVAtMost.integral_unit {E : Type*} [MeasurableSpace E]
{μ ν : Measure E} [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
{ε : ℝ} (h : TVAtMost μ ν ε) {f : E → ℝ} (hf : Measurable f)
(hb : ∀x,0≤f x ∧ f x≤1) : |(∫x,f x ∂μ)-(∫x,f x ∂ν)|≤ε := by
have hbf (x) : |f x|≤1 := by rw [abs_of_nonneg (hb x).1]; exact (hb x).2
have hiμ := bounded_integrable hf hbf (μ:=μ)
have hiν := bounded_integrable hf hbf (μ:=ν)
have hlayer (ρ : Measure E) [IsProbabilityMeasure ρ] :
Integrable (fun t : ℝ => ρ.real {x | t≤f x}) (volume.restrict (Set.Ioc 0 1)) := by
apply bounded_integrable (C:=1)
· exact (Antitone.measurable (show Antitone (fun t : ℝ => ρ {x | t≤f x}) from
fun a b hab => measure_mono (fun x hx => hab.trans hx))).ennreal_toReal
· intro t
rw [abs_of_nonneg measureReal_nonneg]
exact measureReal_le_one
rw [hiμ.integral_eq_integral_Ioc_meas_le (Filter.Eventually.of_forall (fun x => (hb x).1))
(Filter.Eventually.of_forall (fun x => (hb x).2)),
hiν.integral_eq_integral_Ioc_meas_le (Filter.Eventually.of_forall (fun x => (hb x).1))
(Filter.Eventually.of_forall (fun x => (hb x).2)),←integral_sub (hlayer μ) (hlayer ν),←Real.norm_eq_abs]
have hh := norm_integral_le_of_norm_le_const (μ:=volume.restrict (Set.Ioc (0:ℝ) 1))
(f:=fun t : ℝ => μ.real {x | t≤f x}-ν.real {x | t≤f x}) (C:=ε)
(Filter.Eventually.of_forall (fun t => by
simpa only [Real.norm_eq_abs] using h {x | t≤f x} (measurableSet_le measurable_const hf)))
simpa using hh
lemma kernel_integral_comp {E F : Type*} [MeasurableSpace E] [MeasurableSpace F]
(μ : Measure E) [IsProbabilityMeasure μ] (κ : Kernel E F) [IsMarkovKernel κ]
{f : F → ℝ} (hf : Measurable f) {C : ℝ} (hb : ∀x,|f x|≤C) :
(∫z,f z ∂κ ∘ₘ μ)=∫x,(∫z,f z ∂κ x) ∂μ := by
rw [Measure.comp_eq_comp_const_apply]
exact Kernel.integral_comp (bounded_integrable hf hb)
lemma kernel_event_real {E F : Type*} [MeasurableSpace E] [MeasurableSpace F]
(μ : Measure E) [IsProbabilityMeasure μ] (κ : Kernel E F) [IsMarkovKernel κ]
{s : Set F} (hs : MeasurableSet s) :
(κ ∘ₘ μ).real s=∫x,(κ x).real s ∂μ := by
rw [measureReal_def,Measure.bind_apply hs κ.aemeasurable]
exact (integral_toReal (κ.measurable_coe hs).aemeasurable
(Filter.Eventually.of_forall (fun x => measure_lt_top _ _))).symm
lemma kernel_event_integrable {E F : Type*} [MeasurableSpace E] [MeasurableSpace F]
(μ : Measure E) [IsProbabilityMeasure μ] (κ : Kernel E F) [IsMarkovKernel κ]
{s : Set F} (hs : MeasurableSet s) : Integrable (fun x => (κ x).real s) μ :=
bounded_integrable ((κ.measurable_coe hs).ennreal_toReal)
(fun x => by rw [abs_of_nonneg measureReal_nonneg]; exact measureReal_le_one)
lemma TVAtMost.kernel {E F : Type*} [MeasurableSpace E] [MeasurableSpace F]
{μ ν : Measure E} [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
{ε : ℝ} (h : TVAtMost μ ν ε) (κ : Kernel E F) [IsMarkovKernel κ] :
TVAtMost (κ ∘ₘ μ) (κ ∘ₘ ν) ε := by
intro s hs
rw [kernel_event_real μ κ hs,kernel_event_real ν κ hs]
exact h.integral_unit ((κ.measurable_coe hs).ennreal_toReal)
(fun x => ⟨measureReal_nonneg,measureReal_le_one⟩)
lemma TVAtMost.kernel_mixture {E F : Type*} [MeasurableSpace E] [MeasurableSpace F]
(μ : Measure E) [IsProbabilityMeasure μ] (κ η : Kernel E F)
[IsMarkovKernel κ] [IsMarkovKernel η] {e : E → ℝ}
(hi : Integrable e μ) (h : ∀x,TVAtMost (κ x) (η x) (e x)) :
TVAtMost (κ ∘ₘ μ) (η ∘ₘ μ) (∫x,e x ∂μ) := by
intro s hs
rw [kernel_event_real μ κ hs,kernel_event_real μ η hs,
←integral_sub (kernel_event_integrable μ κ hs) (kernel_event_integrable μ η hs),←Real.norm_eq_abs]
apply (norm_integral_le_integral_norm _).trans
apply integral_mono ((kernel_event_integrable μ κ hs).sub (kernel_event_integrable μ η hs)).norm hi
intro x
simpa only [Pi.sub_apply,Real.norm_eq_abs] using h x s hs
end LogConcaveSampling
end
end
end OAI
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