| import OAI.Probability.PosteriorReplicas.Information.ConditionalKL | |
| import OAI.Probability.PosteriorReplicas.Information.PiKL | |
| namespace OAI | |
| open MeasureTheory ProbabilityTheory InformationTheory Set Filter | |
| open scoped ENNReal NNReal | |
| noncomputable section | |
| namespace PosteriorReplicas | |
| section ConditionalPi | |
| variable {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] | |
| [StandardBorelSpace Y] [Nonempty Y] {n : ℕ} | |
| (μ : Measure X) [IsProbabilityMeasure μ] | |
| (κ : Kernel X (Fin n → Y)) [IsMarkovKernel κ] | |
| def coordKernel (i : Fin n) : Kernel X Y := κ.map (fun u => u i) | |
| instance coordKernel_markov (i : Fin n) : IsMarkovKernel (coordKernel κ i) := | |
| Kernel.IsMarkovKernel.map _ (measurable_pi_apply i) | |
| omit [StandardBorelSpace Y] [Nonempty Y] [IsMarkovKernel κ] in | |
| lemma coordKernel_apply (i : Fin n) (x : X) : coordKernel κ i x = coordinateMarginal (κ x) i := | |
| Kernel.map_apply _ (measurable_pi_apply i) _ | |
| def conditionalTotalCorrelation : ℝ≥0∞ := | |
| klDiv (μ ⊗ₘ κ) (μ ⊗ₘ piKernel (coordKernel κ)) | |
| omit [Nonempty Y] in | |
| lemma conditionalTotalCorrelation_integral : | |
| conditionalTotalCorrelation μ κ = ∫⁻ x, klDiv (κ x) (marginalProduct (κ x)) ∂μ := by | |
| rw [conditionalTotalCorrelation, klDiv_compProd_kernel] | |
| apply lintegral_congr | |
| intro x | |
| rw [piKernel_apply] | |
| simp_rw [coordKernel_apply, marginalProduct] | |
| lemma klDiv_conditional_pi_decomposition (ρ : Fin n → Measure Y) [∀ i, IsProbabilityMeasure (ρ i)] : | |
| klDiv (μ ⊗ₘ κ) (μ.prod (Measure.pi ρ)) = conditionalTotalCorrelation μ κ + | |
| ∑ i, klDiv (μ ⊗ₘ coordKernel κ i) (μ.prod (ρ i)) := by | |
| simp_rw [← Measure.compProd_const, klDiv_compProd_kernel] | |
| rw [conditionalTotalCorrelation_integral] | |
| calc | |
| _ = ∫⁻ x, klDiv (κ x) (marginalProduct (κ x)) + | |
| ∑ i, klDiv (coordinateMarginal (κ x) i) (ρ i) ∂μ := by | |
| exact lintegral_congr fun x => klDiv_pi_decomposition (κ x) ρ | |
| _ = _ := by | |
| have hc : Measurable (fun x => klDiv (κ x) (marginalProduct (κ x))) := by | |
| simpa only [piKernel_apply, coordKernel_apply, marginalProduct] using | |
| (measurable_kernel_klDiv κ (piKernel (coordKernel κ))) | |
| have hi : ∀ i, Measurable (fun x => klDiv (coordinateMarginal (κ x) i) (ρ i)) := by | |
| intro i | |
| simpa only [coordKernel_apply, Kernel.const_apply] using | |
| (measurable_kernel_klDiv (coordKernel κ i) (Kernel.const X (ρ i))) | |
| rw [lintegral_add_aux hc (Finset.measurable_sum _ (fun i _ => hi i)), | |
| lintegral_finsetSum _ (fun i _ => hi i)] | |
| simp_rw [coordKernel_apply, Kernel.const_apply] | |
| lemma conditionalTotalCorrelation_identity [StandardBorelSpace X] [Nonempty X] : | |
| conditionalTotalCorrelation μ κ + | |
| ∑ i, klDiv (μ ⊗ₘ coordKernel κ i) | |
| (μ.prod (coordinateMarginal (μ ⊗ₘ κ).snd i)) = | |
| klDiv (μ ⊗ₘ κ).snd (marginalProduct (μ ⊗ₘ κ).snd) + | |
| klDiv (μ ⊗ₘ κ) (μ.prod (μ ⊗ₘ κ).snd) := by | |
| rw [← klDiv_conditional_pi_decomposition] | |
| rw [klDiv_prod_decomposition, Measure.fst_compProd, klDiv_self, zero_add] | |
| rfl | |
| end ConditionalPi | |
| section Projection | |
| variable {X Y Z : Type*} [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace Z] | |
| [StandardBorelSpace Y] [Nonempty Y] [StandardBorelSpace Z] [Nonempty Z] | |
| {n : ℕ} (μ : Measure X) [IsProbabilityMeasure μ] | |
| (κ : Kernel X (Fin n → Y)) [IsMarkovKernel κ] | |
| {g : Y → Z} (hg : Measurable g) | |
| include hg | |
| omit [StandardBorelSpace Y] [Nonempty Y] [StandardBorelSpace Z] [Nonempty Z] in | |
| lemma marginalProduct_map (ν : Measure (Fin n → Y)) [IsProbabilityMeasure ν] : | |
| marginalProduct (ν.map (fun u i => g (u i))) = | |
| (marginalProduct ν).map (fun u i => g (u i)) := by | |
| unfold marginalProduct coordinateMarginal | |
| have he : (fun i : Fin n => (ν.map (fun u j => g (u j))).map (fun u => u i)) = | |
| fun i => (ν.map (fun u => u i)).map g := by | |
| funext i | |
| rw [Measure.map_map (measurable_pi_apply i) (by fun_prop), | |
| Measure.map_map hg (measurable_pi_apply i)] | |
| rfl | |
| rw [he] | |
| exact (Measure.pi_map_pi (fun _ : Fin n => hg.aemeasurable)).symm | |
| omit [Nonempty Y] [Nonempty Z] in | |
| lemma conditionalTotalCorrelation_map_le : | |
| conditionalTotalCorrelation μ (κ.map (fun u i => g (u i))) ≤ conditionalTotalCorrelation μ κ := by | |
| let : IsMarkovKernel (κ.map (fun u i => g (u i))) := Kernel.IsMarkovKernel.map _ (by fun_prop) | |
| rw [conditionalTotalCorrelation_integral, conditionalTotalCorrelation_integral] | |
| apply lintegral_mono | |
| intro x | |
| dsimp only | |
| rw [Kernel.map_apply _ (show Measurable (fun u : Fin n → Y => fun i => g (u i)) by fun_prop), marginalProduct_map hg] | |
| exact klDiv_map_le _ _ (by fun_prop) | |
| end Projection | |
| end PosteriorReplicas | |
| end | |
| end OAI | |