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a2ee19a | 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 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
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