| 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 |
|
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| end |
|
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| end OAI |
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