| 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 |
|
|
