File size: 3,509 Bytes
c4cdbbd | 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 | import Mathlib.InformationTheory.KullbackLeibler.ChainRule
import Mathlib.InformationTheory.KullbackLeibler.DataProcessing
import Mathlib.Probability.Kernel.Composition.AbsolutelyContinuous
namespace OAI
/-! The conditional entropy identity needed for the finite Gaussian
sampling path. Its nonnegative integral form also covers zero variances. -/
noncomputable section
open MeasureTheory ProbabilityTheory InformationTheory
open scoped ENNReal
namespace InvariantIsing
variable {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
[MeasurableSpace.CountableOrCountablyGenerated X Y]
/-- A measurable version of conditional relative entropy, for dominated kernels. -/
def kernelRelativeEntropy (κ η : Kernel X Y) [IsMarkovKernel κ] [IsMarkovKernel η]
(x : X) : ℝ≥0∞ :=
∫⁻ y, ENNReal.ofReal (klFun ((κ.rnDeriv η x y).toReal)) ∂η x
lemma measurable_kernelRelativeEntropy (κ η : Kernel X Y)
[IsMarkovKernel κ] [IsMarkovKernel η] : Measurable (kernelRelativeEntropy κ η) := by
unfold kernelRelativeEntropy
apply Measurable.lintegral_kernel_prod_right' (κ := η)
(f := fun p : X × Y => ENNReal.ofReal (klFun ((κ.rnDeriv η p.1 p.2).toReal)))
exact ENNReal.measurable_ofReal.comp
(continuous_klFun.measurable.comp (Kernel.measurable_rnDeriv κ η).ennreal_toReal)
lemma kernelRelativeEntropy_eq (κ η : Kernel X Y)
[IsMarkovKernel κ] [IsMarkovKernel η] (x : X) (hac : κ x ≪ η x) :
kernelRelativeEntropy κ η x = klDiv (κ x) (η x) := by
rw [klDiv_eq_lintegral_klFun_of_ac hac]
apply lintegral_congr_ae
filter_upwards [Kernel.rnDeriv_eq_rnDeriv_measure (κ := κ) (η := η) (a := x)] with y hy
rw [hy]
lemma klDiv_compProd_same_left (μ : Measure X) [IsProbabilityMeasure μ]
(κ η : Kernel X Y) [IsMarkovKernel κ] [IsMarkovKernel η]
(hac : ∀ x, κ x ≪ η x) :
klDiv (μ ⊗ₘ κ) (μ ⊗ₘ η) = ∫⁻ x, kernelRelativeEntropy κ η x ∂μ := by
have hac' : μ ⊗ₘ κ ≪ μ ⊗ₘ η :=
Measure.AbsolutelyContinuous.compProd_right (Filter.Eventually.of_forall hac)
rw [klDiv_eq_lintegral_klFun_of_ac hac']
calc
_ = ∫⁻ p, ENNReal.ofReal (klFun ((κ.rnDeriv η p.1 p.2).toReal)) ∂(μ ⊗ₘ η) := by
apply lintegral_congr_ae
filter_upwards [rnDeriv_measure_compProd_right μ κ η] with p hp
rw [hp]
_ = _ := Measure.lintegral_compProd (by
exact ENNReal.measurable_ofReal.comp
(continuous_klFun.measurable.comp (Kernel.measurable_rnDeriv κ η).ennreal_toReal))
lemma klDiv_compProd_integral (μ ν : Measure X)
[IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
(κ η : Kernel X Y) [IsMarkovKernel κ] [IsMarkovKernel η]
(hac : ∀ x, κ x ≪ η x) :
klDiv (μ ⊗ₘ κ) (ν ⊗ₘ η) = klDiv μ ν + ∫⁻ x, kernelRelativeEntropy κ η x ∂μ := by
rw [klDiv_compProd_eq_add, klDiv_compProd_same_left μ κ η hac]
lemma klDiv_comp_le_integral (μ ν : Measure X)
[IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
(κ η : Kernel X Y) [IsMarkovKernel κ] [IsMarkovKernel η]
(hac : ∀ x, κ x ≪ η x) :
klDiv (κ ∘ₘ μ) (η ∘ₘ ν) ≤ klDiv μ ν + ∫⁻ x, kernelRelativeEntropy κ η x ∂μ := by
have he := klDiv_map_le (μ ⊗ₘ κ) (ν ⊗ₘ η) measurable_snd
change klDiv (μ ⊗ₘ κ).snd (ν ⊗ₘ η).snd ≤ _ at he
rw [Measure.snd_compProd, Measure.snd_compProd,
klDiv_compProd_integral μ ν κ η hac] at he
exact he
end InvariantIsing
end
end OAI
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