| import OAI.Probability.PosteriorReplicas.Information.ConditionalKL |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory InformationTheory Set Filter |
| open scoped ENNReal NNReal |
| noncomputable section |
| namespace PosteriorReplicas |
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| lemma klDiv_map_leftInverse {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] |
| (μ ν : Measure X) [IsFiniteMeasure μ] [IsFiniteMeasure ν] |
| {f : X → Y} {g : Y → X} (hf : Measurable f) (hg : Measurable g) |
| (hgf : Function.LeftInverse g f) : klDiv (μ.map f) (ν.map f) = klDiv μ ν := by |
| apply le_antisymm (klDiv_map_le _ _ hf) |
| have h := klDiv_map_le (μ.map f) (ν.map f) hg |
| have he : g ∘ f = id := funext hgf |
| simpa only [Measure.map_map hg hf, he, Measure.map_id] using h |
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| lemma compProd_prod_right {Z X Y : Type*} [MeasurableSpace Z] [MeasurableSpace X] [MeasurableSpace Y] |
| (μ : Measure Z) [SFinite μ] (κ : Kernel Z X) [IsSFiniteKernel κ] |
| (ν : Measure Y) [SFinite ν] : |
| ((μ ⊗ₘ κ).prod ν).map MeasurableEquiv.prodAssoc = μ ⊗ₘ (κ ×ₖ Kernel.const Z ν) := by |
| have hk : κ ×ₖ Kernel.const Z ν = κ ⊗ₖ Kernel.const (Z × X) ν := by |
| ext z s hs |
| rw [Kernel.prod_apply' _ _ _ hs, Kernel.compProd_apply hs] |
| rfl |
| rw [hk, ← Measure.compProd_const, Measure.compProd_assoc'] |
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| def attachSide {X Y Z : Type*} [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace Z] |
| (g : X → Z) (q : Measure (X × Y)) : Measure (Z × (X × Y)) := |
| q.map (fun z => (g z.1, z)) |
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| instance attachSide_finite {X Y Z : Type*} [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace Z] |
| (g : X → Z) (q : Measure (X × Y)) [IsFiniteMeasure q] : |
| IsFiniteMeasure (attachSide g q) := by |
| unfold attachSide |
| infer_instance |
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| lemma attachSide_probability {X Y Z : Type*} [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace Z] |
| {g : X → Z} (hg : Measurable g) (q : Measure (X × Y)) [IsProbabilityMeasure q] : |
| IsProbabilityMeasure (attachSide g q) := |
| (Measure.isProbabilityMeasure_map_iff (by fun_prop)).mpr inferInstance |
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| lemma attachSide_first_pair {X Y Z : Type*} [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace Z] |
| {g : X → Z} (hg : Measurable g) (q : Measure (X × Y)) : |
| (attachSide g q).map (fun z => (z.1,z.2.1)) = q.fst.map (fun x => (g x,x)) := by |
| unfold attachSide Measure.fst |
| rw [Measure.map_map (by fun_prop) (by fun_prop), Measure.map_map (by fun_prop) measurable_fst] |
| rfl |
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| lemma compProd_condKernel_fst {Z X Y : Type*} |
| [MeasurableSpace Z] [MeasurableSpace X] [MeasurableSpace Y] |
| [StandardBorelSpace X] [StandardBorelSpace Y] [Nonempty X] [Nonempty Y] |
| (q : Measure (Z × (X × Y))) [IsFiniteMeasure q] : |
| q.fst ⊗ₘ q.condKernel.fst = q.map (fun z => (z.1,z.2.1)) := by |
| rw [Kernel.fst_eq, Measure.compProd_map measurable_fst, Measure.disintegrate q q.condKernel] |
| rfl |
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| lemma attachSide_conditional_reference {X Y Z : Type*} |
| [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace Z] |
| [StandardBorelSpace X] [StandardBorelSpace Y] [Nonempty X] [Nonempty Y] |
| {g : X → Z} (hg : Measurable g) (q : Measure (X × Y)) [IsProbabilityMeasure q] |
| (ν : Measure Y) [IsProbabilityMeasure ν] : |
| let r := attachSide g q |
| r.fst ⊗ₘ (r.condKernel.fst ×ₖ Kernel.const Z ν) = attachSide g (q.fst.prod ν) := by |
| let := attachSide_probability hg q |
| dsimp only |
| rw [← compProd_prod_right, compProd_condKernel_fst, attachSide_first_pair hg] |
| have hp : (q.fst.map (fun x => (g x,x))).prod ν = |
| (q.fst.prod ν).map (Prod.map (fun x => (g x,x)) id) := by |
| simpa only [Measure.map_id] using Measure.map_prod_map q.fst ν |
| (show Measurable (fun x => (g x,x)) by fun_prop) measurable_id |
| rw [hp] |
| rw [Measure.map_map (by fun_prop) (by fun_prop)] |
| rfl |
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|
| lemma mutualInformation_attachSide_chain {X Y Z : Type*} |
| [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace Z] |
| [StandardBorelSpace X] [StandardBorelSpace Y] [Nonempty X] [Nonempty Y] |
| {g : X → Z} (hg : Measurable g) (q : Measure (X × Y)) [IsProbabilityMeasure q] : |
| let r := attachSide g q |
| klDiv q (q.fst.prod q.snd) = cmi r + |
| klDiv (q.map (fun z => (g z.1,z.2))) ((q.fst.map g).prod q.snd) := by |
| let := attachSide_probability hg q |
| dsimp only |
| have h := klDiv_conditional_chain (attachSide g q) q.snd |
| rw [attachSide_conditional_reference hg] at h |
| have he : klDiv (attachSide g q) (attachSide g (q.fst.prod q.snd)) = klDiv q (q.fst.prod q.snd) := |
| klDiv_map_leftInverse _ _ (by fun_prop) measurable_snd (fun _ => rfl) |
| rw [he] at h |
| have hf : (attachSide g q).fst = q.fst.map g := by |
| unfold attachSide Measure.fst |
| rw [Measure.map_map measurable_fst (by fun_prop), Measure.map_map hg measurable_fst] |
| rfl |
| have hy : (attachSide g q).map (fun z => (z.1,z.2.2)) = |
| q.map (fun z => (g z.1,z.2)) := by |
| rw [attachSide, Measure.map_map (by fun_prop) (by fun_prop)] |
| rfl |
| rwa [hf,hy] at h |
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|
| end PosteriorReplicas |
| end |
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| end OAI |
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