| import Mathlib |
| import OAI.Probability.LogConcave.Dynamics.Squared |
| import OAI.Probability.LogConcave.Dynamics.NormSubSqThree |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace LogConcaveSampling.Coupling |
| open MeasureTheory ProbabilityTheory Function |
| open scoped Classical |
|
|
| variable {Ω Γ : Type*} [MeasurableSpace Ω] [MeasurableSpace Γ] |
| [StandardBorelSpace Γ] [Nonempty Γ] {d : ℕ} |
|
|
| theorem SquaredAt.of_same_law (μ : Measure Ω) [IsProbabilityMeasure μ] |
| (ν : Measure Γ) [IsProbabilityMeasure ν] (f : Ω → Point d) (g : Γ → Point d) |
| (hf : Measurable f) (hg : Measurable g) (hlaw : μ.map f=ν.map g) : |
| SquaredAt μ ν f g 0 := by |
| let J : Γ → Point d × Γ := fun z => (g z,z) |
| have hJ : Measurable J := hg.prodMk measurable_id |
| let ι := ν.map J |
| have : IsProbabilityMeasure ι := inferInstance |
| have hl : ι.fst=ν.map g := by |
| change (ν.map J).map Prod.fst=_ |
| rw [Measure.map_map measurable_fst hJ] |
| rfl |
| have hr : ι.snd=ν := by |
| change (ν.map J).map Prod.snd=_ |
| rw [Measure.map_map measurable_snd hJ] |
| exact Measure.map_id |
| let Λ := lift μ ι f hf |
| have hlink : μ.map f=ι.fst := hlaw.trans hl.symm |
| have hm : Measurable (fun z : Point d × Γ => ‖z.1-g z.2‖^2) := by fun_prop |
| have hi : Integrable (fun z : Point d × Γ => ‖z.1-g z.2‖^2) ι := by |
| apply (integrable_map_measure hm.aestronglyMeasurable hJ.aemeasurable).mpr |
| change Integrable (fun z => ‖g z-g z‖^2) ν |
| simpa only [sub_self,norm_zero,zero_pow (by decide : 2≠0)] using |
| (integrable_const (0 : ℝ) : Integrable (fun _ : Γ => (0 : ℝ)) ν) |
| have he : (∫z : Point d × Γ,‖z.1-g z.2‖^2 ∂ι)=0 := by |
| rw [integral_map hJ.aemeasurable hm.aestronglyMeasurable] |
| simp [J] |
| obtain ⟨hI,hE⟩ := lift_integrable μ ι f hf hlink _ hi |
| refine ⟨Λ,inferInstance,lift_fst μ ι f hf,?_,hI,hE.trans_le he.le⟩ |
| have hh := congrArg (Measure.map Prod.snd) (lift_map μ ι f hf hlink) |
| rw [Measure.map_map measurable_snd (by fun_prop)] at hh |
| exact hh.trans hr |
|
|
| lemma SquaredAt.change_ideal {W : Type*} [MeasurableSpace W] |
| {μ : Measure Ω} {ν : Measure W} [IsProbabilityMeasure ν] |
| (θ : Measure Γ) [IsProbabilityMeasure θ] |
| {f : Ω → Point d} {g : W → Point d} (h : Γ → Point d) |
| (hf : Measurable f) (hg : Measurable g) (hh : Measurable h) |
| {B : ℝ} (hc : SquaredAt μ ν f g B) (hlaw : ν.map g=θ.map h) : |
| SquaredAt μ θ f h (2*B) := by |
| have hzero := SquaredAt.of_same_law ν θ g h hg hh hlaw |
| simpa only [mul_zero,add_zero] using hc.trans hf hg hh hzero |
|
|
| end LogConcaveSampling.Coupling |
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| end |
|
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| end |
|
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| end OAI |
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|
