| import OAI.Probability.PosteriorReplicas.Information.CmiKernel |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory InformationTheory Set Filter |
| open scoped ENNReal NNReal |
| noncomputable section |
| namespace PosteriorReplicas |
|
|
| section |
| variable {T U V : Type*} [MeasurableSpace T] [MeasurableSpace U] [MeasurableSpace V] |
|
|
| def parametricMapKernel (q : Measure U) (h : T × U → V) : Kernel T V := |
| (Kernel.id ×ₖ Kernel.const T q).map h |
| |
| instance parametricMapKernel_sfinite (q : Measure U) [SFinite q] |
| (h : T × U → V) : IsSFiniteKernel (parametricMapKernel q h) := by |
| unfold parametricMapKernel |
| infer_instance |
| |
| lemma parametricMapKernel_markov (q : Measure U) [IsProbabilityMeasure q] |
| (h : T × U → V) (hh : Measurable h) : IsMarkovKernel (parametricMapKernel q h) := |
| Kernel.IsMarkovKernel.map _ hh |
| |
| lemma parametricMapKernel_apply (q : Measure U) [SFinite q] |
| {h : T × U → V} (hh : Measurable h) (t : T) : |
| parametricMapKernel q h t = q.map (fun u => h (t,u)) := by |
| rw [parametricMapKernel, Kernel.map_apply _ hh, Kernel.prod_apply, |
| Kernel.id_apply, Kernel.const_apply, Measure.dirac_prod, |
| Measure.map_map hh (by fun_prop)] |
| rfl |
| |
| lemma compProd_parametricMapKernel (μ : Measure T) [SFinite μ] (q : Measure U) [SFinite q] |
| {h : T × U → V} (hh : Measurable h) : |
| μ ⊗ₘ parametricMapKernel q h = (μ.prod q).map (fun z => (z.1,h z)) := by |
| apply Measure.ext_of_lintegral |
| intro f hf |
| rw [Measure.lintegral_compProd hf, lintegral_map hf (by fun_prop), |
| lintegral_prod _ (by fun_prop)] |
| apply lintegral_congr |
| intro t |
| rw [parametricMapKernel_apply q hh, lintegral_map (by fun_prop) (by fun_prop)] |
| end |
| |
| section RandomSide |
| variable {T X Y Z : Type*} [MeasurableSpace T] [MeasurableSpace X] |
| [MeasurableSpace Y] [MeasurableSpace Z] |
| [StandardBorelSpace X] [StandardBorelSpace Y] [StandardBorelSpace Z] |
| [Nonempty X] [Nonempty Y] |
| |
| def randomSideLaw (μ : Measure T) (g : T × X → Z) (q : Measure (X × Y)) : |
| Measure ((T × Z) × (X × Y)) := |
| (μ.prod q).map (fun z => ((z.1,g (z.1,z.2.1)),z.2)) |
| |
| instance randomSideLaw_finite (μ : Measure T) [IsFiniteMeasure μ] |
| (g : T × X → Z) (q : Measure (X × Y)) [IsFiniteMeasure q] : |
| IsFiniteMeasure (randomSideLaw μ g q) := by unfold randomSideLaw; infer_instance |
| |
| lemma cmi_randomSideLaw (μ : Measure T) [IsFiniteMeasure μ] |
| {g : T × X → Z} (hg : Measurable g) (q : Measure (X × Y)) [IsProbabilityMeasure q] : |
| cmi (randomSideLaw μ g q) = ∫⁻ t, cmi (attachSide (fun x => g (t,x)) q) ∂μ := by |
| let h : T × (X × Y) → Z × (X × Y) := fun z => (g (z.1,z.2.1),z.2) |
| have hh : Measurable h := by fun_prop |
| let κ := parametricMapKernel q h |
| have : IsMarkovKernel κ := parametricMapKernel_markov q h hh |
| have he : (μ ⊗ₘ κ).map (MeasurableEquiv.prodAssoc.symm : |
| T × (Z × (X × Y)) ≃ᵐ (T × Z) × (X × Y)) = randomSideLaw μ g q := by |
| rw [compProd_parametricMapKernel μ q hh, Measure.map_map (by fun_prop) (by fun_prop)] |
| rfl |
| have ht (t : T) : κ t = attachSide (fun x => g (t,x)) q := by |
| rw [parametricMapKernel_apply q hh] |
| rfl |
| have hc := cmi_compProd_assoc μ κ |
| simpa only [he, ht] using hc |
| |
| lemma measurable_cmi_attachSide {g : T × X → Z} (hg : Measurable g) |
| (q : Measure (X × Y)) [IsProbabilityMeasure q] : |
| Measurable (fun t => cmi (attachSide (fun x => g (t,x)) q)) := by |
| let h : T × (X × Y) → Z × (X × Y) := fun z => (g (z.1,z.2.1),z.2) |
| have hh : Measurable h := by fun_prop |
| let κ := parametricMapKernel q h |
| have : IsMarkovKernel κ := parametricMapKernel_markov q h hh |
| have ht (t : T) : κ t = attachSide (fun x => g (t,x)) q := by |
| rw [parametricMapKernel_apply q hh] |
| rfl |
| simpa only [ht] using cmi_kernel_measurable κ |
| end RandomSide |
| |
| end PosteriorReplicas |
| end |
| |
| end OAI |
| |
