import OAI.Probability.GaussianReplacement.ActualRow namespace OAI open MeasureTheory ProbabilityTheory Set Filter open scoped ENNReal NNReal RealInnerProductSpace Topology noncomputable section namespace CurrentProjection open InnerProductSpace open InformationTheory def conditionalMutualInformation {Z A B : Type*} [MeasurableSpace Z] [MeasurableSpace A] [MeasurableSpace B] [StandardBorelSpace A] [StandardBorelSpace B] (P : Measure (Z×(A×B))) [IsFiniteMeasure P] : ℝ≥0∞ := by classical exact if h : Nonempty (A×B) then letI := h InformationTheory.klDiv P (P.fst ⊗ₘ (P.condKernel.fst ×ₖ P.condKernel.snd)) else 0 lemma kernel_prod_eq_compProd {Z A B : Type*} [MeasurableSpace Z] [MeasurableSpace A] [MeasurableSpace B] (κ : Kernel Z A) (η : Kernel Z B) [IsSFiniteKernel κ] [IsSFiniteKernel η] : κ ×ₖ η = κ ⊗ₖ η.comap Prod.fst measurable_fst := by ext z s hs rw [Kernel.prod_apply' _ _ _ hs, Kernel.compProd_apply hs] rfl lemma compProd_prod_assoc {Z A B : Type*} [MeasurableSpace Z] [MeasurableSpace A] [MeasurableSpace B] (μ : Measure Z) [SFinite μ] (κ : Kernel Z A) (η : Kernel Z B) [IsSFiniteKernel κ] [IsSFiniteKernel η] : (μ ⊗ₘ (κ ×ₖ η)).map MeasurableEquiv.prodAssoc.symm = (μ ⊗ₘ κ) ⊗ₘ η.comap Prod.fst measurable_fst := by rw [kernel_prod_eq_compProd, Measure.compProd_assoc] def swapAssoc {Z A B : Type*} [MeasurableSpace Z] [MeasurableSpace A] [MeasurableSpace B] : Z×(A×B) ≃ᵐ (Z×B)×A := ((MeasurableEquiv.refl Z).prodCongr MeasurableEquiv.prodComm).trans MeasurableEquiv.prodAssoc.symm lemma compProd_prod_swapAssoc {Z A B : Type*} [MeasurableSpace Z] [MeasurableSpace A] [MeasurableSpace B] (μ : Measure Z) [SFinite μ] (κ : Kernel Z A) (η : Kernel Z B) [IsSFiniteKernel κ] [IsSFiniteKernel η] : (μ ⊗ₘ (κ ×ₖ η)).map swapAssoc = (μ ⊗ₘ η) ⊗ₘ κ.comap Prod.fst measurable_fst := by change (μ ⊗ₘ (κ ×ₖ η)).map (MeasurableEquiv.prodAssoc.symm ∘ Prod.map id Prod.swap) = _ rw [← Measure.map_map MeasurableEquiv.prodAssoc.symm.measurable (measurable_id.prodMap measurable_swap), ← Measure.compProd_map measurable_swap, Kernel.map_prod_swap, compProd_prod_assoc] lemma fst_map_swapAssoc_compProd {Z A B : Type*} [MeasurableSpace Z] [MeasurableSpace A] [MeasurableSpace B] (μ : Measure Z) [SFinite μ] (κ : Kernel Z (A×B)) [IsSFiniteKernel κ] : ((μ ⊗ₘ κ).map swapAssoc).fst = μ ⊗ₘ κ.snd := by rw [Kernel.snd_eq] rw [Measure.compProd_map measurable_snd] rw [Measure.fst, Measure.map_map measurable_fst swapAssoc.measurable] rfl lemma conditionalKL_chain {Z A B : Type*} [MeasurableSpace Z] [MeasurableSpace A] [MeasurableSpace B] [StandardBorelSpace A] [Nonempty A] (μ : Measure Z) [IsProbabilityMeasure μ] (κ : Kernel Z (A×B)) [IsMarkovKernel κ] (π : Measure B) [IsProbabilityMeasure π] : klDiv (μ ⊗ₘ κ) (μ ⊗ₘ (κ.fst ×ₖ Kernel.const Z π)) = klDiv (μ ⊗ₘ κ.snd) (μ.prod π) + klDiv (μ ⊗ₘ κ) (μ ⊗ₘ (κ.fst ×ₖ κ.snd)) := by let R := (μ ⊗ₘ κ).map swapAssoc have hR : R.fst = μ ⊗ₘ κ.snd := fst_map_swapAssoc_compProd μ κ have hchain := klDiv_compProd_eq_add R.fst (μ.prod π) R.condKernel (κ.fst.comap Prod.fst measurable_fst) rw [R.disintegrate R.condKernel, hR] at hchain have hleft := klDiv_map_equiv (μ ⊗ₘ κ) (μ ⊗ₘ (κ.fst ×ₖ Kernel.const Z π)) swapAssoc rw [compProd_prod_swapAssoc, Measure.compProd_const] at hleft have hright := klDiv_map_equiv (μ ⊗ₘ κ) (μ ⊗ₘ (κ.fst ×ₖ κ.snd)) swapAssoc rw [compProd_prod_swapAssoc] at hright exact hleft.symm.trans (hchain.trans (congrArg (klDiv (μ ⊗ₘ κ.snd) (μ.prod π) + ·) hright)) def mutualInformation {A B : Type*} [MeasurableSpace A] [MeasurableSpace B] (P : Measure (A×B)) : ℝ≥0∞ := klDiv P (P.fst.prod P.snd) lemma fst_map_assoc_compProd {Z A B : Type*} [MeasurableSpace Z] [MeasurableSpace A] [MeasurableSpace B] (μ : Measure Z) [SFinite μ] (κ : Kernel Z (A×B)) [IsSFiniteKernel κ] : ((μ ⊗ₘ κ).map MeasurableEquiv.prodAssoc.symm).fst = μ ⊗ₘ κ.fst := by rw [Kernel.fst_eq, Measure.compProd_map measurable_fst, Measure.fst, Measure.map_map measurable_fst MeasurableEquiv.prodAssoc.symm.measurable] rfl lemma mutualInformation_chain {Z A B : Type*} [MeasurableSpace Z] [MeasurableSpace A] [MeasurableSpace B] [StandardBorelSpace A] [StandardBorelSpace B] [Nonempty A] [Nonempty B] (P : Measure (Z×(A×B))) [IsProbabilityMeasure P] : mutualInformation (P.map MeasurableEquiv.prodAssoc.symm) = mutualInformation (P.map (Prod.map id Prod.snd)) + conditionalMutualInformation P := by let μ := P.fst let κ := P.condKernel let Q := P.map MeasurableEquiv.prodAssoc.symm let R := P.map (Prod.map id Prod.snd) let : IsProbabilityMeasure Q := inferInstance let : IsProbabilityMeasure R := inferInstance have hP : μ ⊗ₘ κ = P := P.disintegrate P.condKernel have hQ : Q.fst = μ ⊗ₘ κ.fst := by change (P.map MeasurableEquiv.prodAssoc.symm).fst = _ rw [← hP] exact fst_map_assoc_compProd μ κ have hR : R = μ ⊗ₘ κ.snd := by rw [Kernel.snd_eq,Measure.compProd_map measurable_snd,hP] have hRf : R.fst = μ := by rw [hR,Measure.fst_compProd] have hRs : R.snd = Q.snd := by dsimp only [R,Q,Measure.snd] rw [Measure.map_map measurable_snd (measurable_id.prodMap measurable_snd), Measure.map_map measurable_snd MeasurableEquiv.prodAssoc.symm.measurable] rfl have hleft := klDiv_map_equiv P (μ ⊗ₘ (κ.fst ×ₖ Kernel.const Z Q.snd)) MeasurableEquiv.prodAssoc.symm rw [compProd_prod_assoc] at hleft have hconst : (Kernel.const Z Q.snd).comap (Prod.fst : Z×A → Z) measurable_fst = Kernel.const (Z×A) Q.snd := by ext x t ht; rfl rw [hconst,Measure.compProd_const,← hQ] at hleft have hc := conditionalKL_chain μ κ Q.snd have hSmall : klDiv (μ ⊗ₘ κ.snd) (μ.prod Q.snd) = mutualInformation R := by rw [← hR,← hRf,← hRs] rfl rw [hP,hSmall] at hc have hCM : conditionalMutualInformation P = klDiv P (μ ⊗ₘ (κ.fst ×ₖ κ.snd)) := by simp only [conditionalMutualInformation,dite_eq_left (inferInstance : Nonempty (A×B))] rfl rw [hCM] exact hleft.trans hc lemma conditionalMutualInformation_le_entropy {Z A B : Type*} [MeasurableSpace Z] [MeasurableSpace A] [MeasurableSpace B] [StandardBorelSpace A] [Nonempty A] [Nonempty B] [Countable B] [MeasurableSingletonClass B] (P : Measure (Z×(A×B))) [IsProbabilityMeasure P] (hH : shannonEntropy (P.map MeasurableEquiv.prodAssoc.symm).snd ≠ ∞) : conditionalMutualInformation P ≠ ∞ ∧ (conditionalMutualInformation P).toReal ≤ (shannonEntropy (P.map MeasurableEquiv.prodAssoc.symm).snd).toReal := by let : IsProbabilityMeasure (P.map MeasurableEquiv.prodAssoc.symm) := inferInstance have h := countable_information_le_entropy (P.map MeasurableEquiv.prodAssoc.symm) hH have hle : conditionalMutualInformation P ≤ mutualInformation (P.map MeasurableEquiv.prodAssoc.symm) := by rw [mutualInformation_chain P] exact le_add_left le_rfl exact ⟨ne_top_of_le_ne_top h.1 hle,(ENNReal.toReal_mono h.1 hle).trans h.2⟩ lemma conditionalMutualInformation_compProd {Z A B : Type*} [MeasurableSpace Z] [MeasurableSpace A] [MeasurableSpace B] [StandardBorelSpace A] [StandardBorelSpace B] [Nonempty A] [Nonempty B] (μ : Measure Z) [IsFiniteMeasure μ] (κ : Kernel Z (A×B)) [IsMarkovKernel κ] : conditionalMutualInformation (μ ⊗ₘ κ) = klDiv (μ ⊗ₘ κ) (μ ⊗ₘ (κ.fst ×ₖ κ.snd)) := by simp only [conditionalMutualInformation,dite_eq_left (inferInstance : Nonempty (A×B)),Measure.fst_compProd] congr 1 apply Measure.compProd_congr filter_upwards [condKernel_compProd μ κ] with z hz simp only [Kernel.prod_apply,Kernel.fst_apply,Kernel.snd_apply,hz] lemma conditionalMutualInformation_map_right_le {Z A B C : Type*} [MeasurableSpace Z] [MeasurableSpace A] [MeasurableSpace B] [MeasurableSpace C] [StandardBorelSpace A] [StandardBorelSpace B] [StandardBorelSpace C] [Nonempty A] [Nonempty B] [Nonempty C] (P : Measure (Z×(A×B))) [IsProbabilityMeasure P] (f : B → C) (hf : Measurable f) : conditionalMutualInformation (P.map (Prod.map id (Prod.map id f))) ≤ conditionalMutualInformation P := by let μ := P.fst let κ := P.condKernel let g : A×B → A×C := Prod.map id f have hg : Measurable g := measurable_id.prodMap hf let : IsMarkovKernel (κ.map g) := Kernel.IsMarkovKernel.map κ hg have hT : Measurable (Prod.map (id : Z → Z) g) := measurable_id.prodMap hg have hP : μ ⊗ₘ κ = P := P.disintegrate P.condKernel have hmap : P.map (Prod.map id g) = μ ⊗ₘ κ.map g := by rw [Measure.compProd_map hg,hP] have hfst : (κ.map g).fst = κ.fst := by rw [Kernel.fst_eq,←Kernel.map_comp_right _ hg measurable_fst,Kernel.fst_eq] rfl have hsnd : (κ.map g).snd = κ.snd.map f := by rw [Kernel.snd_eq,Kernel.snd_eq,←Kernel.map_comp_right _ hg measurable_snd, ←Kernel.map_comp_right _ measurable_snd hf] rfl have hRef : μ ⊗ₘ ((κ.map g).fst ×ₖ (κ.map g).snd) = (μ ⊗ₘ (κ.fst ×ₖ κ.snd)).map (Prod.map id g) := by have hk : κ.fst ×ₖ κ.snd.map f = (κ.fst ×ₖ κ.snd).map g := by simpa only [Kernel.map_id] using Kernel.map_prod_map κ.fst κ.snd measurable_id hf rw [hfst,hsnd,hk,Measure.compProd_map hg] change conditionalMutualInformation (P.map (Prod.map id g)) ≤ _ have hCM := conditionalMutualInformation_compProd μ (κ.map g) have hCM' : conditionalMutualInformation (P.map (Prod.map id g)) = klDiv (P.map (Prod.map id g)) ((μ ⊗ₘ (κ.fst ×ₖ κ.snd)).map (Prod.map id g)) := by simpa only [←hmap,hRef] using hCM rw [hCM'] have h := klDiv_map_le P (μ ⊗ₘ (κ.fst ×ₖ κ.snd)) hT have hCM0 : conditionalMutualInformation P = klDiv P (μ ⊗ₘ (κ.fst ×ₖ κ.snd)) := by simpa only [hP] using conditionalMutualInformation_compProd μ κ rwa [hCM0] lemma rowInformation_eq_conditionalMutualInformation {S W X : Type*} [MeasurableSpace S] [MeasurableSpace W] [MeasurableSpace X] [StandardBorelSpace W] [StandardBorelSpace X] [Nonempty W] [Nonempty X] (P : Measure ((S×W)×X)) [IsProbabilityMeasure P] (ν : Measure S) (γ : Measure X) [IsProbabilityMeasure ν] [IsProbabilityMeasure γ] (hInd : (P.map swapLast).fst = ν.prod γ) : rowInformation P γ = conditionalMutualInformation (P.map MeasurableEquiv.prodAssoc) := by let Q := P.map (MeasurableEquiv.prodAssoc : (S×W)×X ≃ᵐ S×(W×X)) let : IsProbabilityMeasure Q := inferInstance have hX : P.snd = γ := by have h := congrArg Measure.snd hInd rw [Measure.snd_prod] at h convert h using 1 simp only [Measure.snd, Measure.fst, Measure.map_map measurable_fst swapLast.measurable, Measure.map_map measurable_snd (measurable_fst.comp swapLast.measurable)] rfl have hQ : Q.map MeasurableEquiv.prodAssoc.symm = P := MeasurableEquiv.prodAssoc.map_symm_map have hR : Q.map (Prod.map id Prod.snd) = ν.prod γ := by trans (P.map swapLast).fst · dsimp only [Q,Measure.fst] rw [Measure.map_map (measurable_id.prodMap measurable_snd) MeasurableEquiv.prodAssoc.measurable, Measure.map_map measurable_fst swapLast.measurable] rfl · exact hInd have hc := mutualInformation_chain Q rw [hQ,hR] at hc have hzero : mutualInformation (ν.prod γ) = 0 := by simp [mutualInformation, klDiv_self] rw [hzero,zero_add] at hc simpa only [rowInformation,mutualInformation,hX] using hc lemma conditionalMutualInformation_swap {S A B : Type*} [MeasurableSpace S] [MeasurableSpace A] [MeasurableSpace B] [StandardBorelSpace A] [StandardBorelSpace B] [Nonempty A] [Nonempty B] (P : Measure (S×(A×B))) [IsProbabilityMeasure P] : conditionalMutualInformation (P.map (Prod.map id Prod.swap)) = conditionalMutualInformation P := by let μ := P.fst let κ := P.condKernel let e : S×(A×B) ≃ᵐ S×(B×A) := (MeasurableEquiv.refl S).prodCongr MeasurableEquiv.prodComm have hP : μ ⊗ₘ κ = P := P.disintegrate P.condKernel let : IsMarkovKernel (κ.map Prod.swap) := Kernel.IsMarkovKernel.map κ measurable_swap have hf : (κ.map Prod.swap).fst = κ.snd := by rw [Kernel.fst_eq, ← Kernel.map_comp_right _ measurable_swap measurable_fst, Kernel.snd_eq] rfl have hs : (κ.map Prod.swap).snd = κ.fst := by rw [Kernel.snd_eq, ← Kernel.map_comp_right _ measurable_swap measurable_snd, Kernel.fst_eq] rfl have href : μ ⊗ₘ ((κ.map Prod.swap).fst ×ₖ (κ.map Prod.swap).snd) = (μ ⊗ₘ (κ.fst ×ₖ κ.snd)).map e := by rw [hf,hs, ← Kernel.map_prod_swap, Measure.compProd_map measurable_swap] rfl have hQ : μ ⊗ₘ κ.map Prod.swap = P.map (Prod.map id Prod.swap) := by rw [Measure.compProd_map measurable_swap,hP] have hcm : conditionalMutualInformation (P.map (Prod.map id Prod.swap)) = klDiv (P.map (Prod.map id Prod.swap)) ((μ ⊗ₘ (κ.fst ×ₖ κ.snd)).map e) := by simpa only [hQ,href] using conditionalMutualInformation_compProd μ (κ.map Prod.swap) have hcm0 : conditionalMutualInformation P = klDiv P (μ ⊗ₘ (κ.fst ×ₖ κ.snd)) := by simpa only [hP] using conditionalMutualInformation_compProd μ κ rw [hcm,hcm0] exact klDiv_map_equiv P (μ ⊗ₘ (κ.fst ×ₖ κ.snd)) e lemma rowInformation_eq_sourceConditionalInformation {S W X : Type*} [MeasurableSpace S] [MeasurableSpace W] [MeasurableSpace X] [StandardBorelSpace W] [StandardBorelSpace X] (P : Measure ((S×W)×X)) [IsProbabilityMeasure P] (ν : Measure S) (γ : Measure X) [IsProbabilityMeasure ν] [IsProbabilityMeasure γ] (hInd : (P.map swapLast).fst = ν.prod γ) : rowInformation P γ = conditionalMutualInformation (P.map (fun p => (p.1.1,(p.2,p.1.2)))) := by let := nonempty_of_isProbabilityMeasure P let : Nonempty W := ⟨(Classical.choice (inferInstance : Nonempty ((S×W)×X))).1.2⟩ let : Nonempty X := ⟨(Classical.choice (inferInstance : Nonempty ((S×W)×X))).2⟩ let : IsProbabilityMeasure (P.map (MeasurableEquiv.prodAssoc : (S×W)×X ≃ᵐ S×(W×X))) := inferInstance rw [rowInformation_eq_conditionalMutualInformation P ν γ hInd] have he := conditionalMutualInformation_swap (P.map (MeasurableEquiv.prodAssoc : (S×W)×X ≃ᵐ S×(W×X))) have hfun : (Prod.map id Prod.swap ∘ (MeasurableEquiv.prodAssoc : (S×W)×X ≃ᵐ S×(W×X))) = (fun p : (S×W)×X => (p.1.1,(p.2,p.1.2))) := rfl simpa only [Measure.map_map (measurable_id.prodMap measurable_swap) MeasurableEquiv.prodAssoc.measurable, hfun] using he.symm def actualConditionalRowInformation {d m : ℕ} {W : Type*} [MeasurableSpace W] [StandardBorelSpace W] (P : Measure ((Sphere d×W)×Rows m d)) [IsFiniteMeasure P] : ℝ≥0∞ := conditionalMutualInformation (P.map (fun p => (p.1.1,(p.2,p.1.2)))) structure ActualRowSourceConclusion {d m r q : ℕ} {W : Type*} [MeasurableSpace W] [StandardBorelSpace W] (P : Measure ((Sphere d×W)×Rows m d)) [IsProbabilityMeasure P] (ρ : W → Measure (Sphere d)) (B D : W → ℝ) : Prop where row_finite : actualConditionalRowInformation P ≠ ∞ row_le_entropy : (actualConditionalRowInformation P).toReal ≤ (shannonEntropy (P.map swapLast).snd).toReal conditional_density : ∀ᵐ gw ∂((actualExperiment (r:=r) P).map actualObservation).fst, ((actualExperiment (r:=r) P).map actualObservation).condKernel gw ≪ volume actual_limit : ∀ᵐ p ∂actualExperiment (r:=r) P, Tendsto (fun j : ℕ => selectedCube ρ (1/(j:ℝ)) (actualObservation p)) atTop (𝓝 (selectedExact ρ (actualObservation p))) smoothed_bound : ∀ δ : ℝ, 0 < δ → Integrable (fun p => Real.log (1+(selectedCube ρ δ (actualObservation p)).toReal)) (actualExperiment (r:=r) P) ∧ (∫ p, Real.log (1+(selectedCube ρ δ (actualObservation p)).toReal) ∂actualExperiment (r:=r) P) ≤ (actualConditionalRowInformation P).toReal/q + ∫ sw : Sphere d×W, Real.log (1+momentBound 9 d (B sw.2) (D sw.2)) ∂P.fst exact_integrable : Integrable (fun p => Real.log (1+(selectedExact ρ (actualObservation p)).toReal)) (actualExperiment (r:=r) P) exact_bound : (∫ p, Real.log (1+(selectedExact ρ (actualObservation p)).toReal) ∂actualExperiment (r:=r) P) ≤ (actualConditionalRowInformation P).toReal/q + ∫ sw : Sphere d×W, Real.log (1+momentBound 9 d (B sw.2) (D sw.2)) ∂P.fst entropy_bound : (actualConditionalRowInformation P).toReal/q + ∫ sw : Sphere d×W, Real.log (1+momentBound 9 d (B sw.2) (D sw.2)) ∂P.fst ≤ (shannonEntropy (P.map swapLast).snd).toReal/q + ∫ sw : Sphere d×W, Real.log (1+momentBound 9 d (B sw.2) (D sw.2)) ∂P.fst positive_log_integrable : Integrable (fun p => logPositive (selectedExact ρ (actualObservation p))) (actualExperiment (r:=r) P) extended_log_bound : extendedLogExpectation (actualExperiment (r:=r) P) (fun p => selectedExact ρ (actualObservation p)) ≤ (((actualConditionalRowInformation P).toReal/q + ∫ sw : Sphere d×W, Real.log (1+momentBound 9 d (B sw.2) (D sw.2)) ∂P.fst) : EReal) theorem actualRowSourceMain {d m r q : ℕ} {W : Type*} [MeasurableSpace W] [Countable W] [MeasurableSingletonClass W] (hdim : AdmissibleDimensions d m r q) (P : Measure ((Sphere d×W)×Rows m d)) [IsProbabilityMeasure P] (ν : Measure (Sphere d)) [IsProbabilityMeasure ν] (hInd : (P.map swapLast).fst = ν.prod (gaussianRows m d)) (H : ℝ) (hν : ν ≤ ENNReal.ofReal H • uniformSphere d) (hEntropy : shannonEntropy (P.map swapLast).snd ≠ ∞) (ρ : PositiveMessage (P.map swapLast).snd → Measure (Sphere d)) [∀ w, IsFiniteMeasure (ρ w)] (_hρac : ∀ w, ρ w ≪ uniformSphere d) (B D : PositiveMessage (P.map swapLast).snd → ℝ) (hB : ∀ w, 0 < B w) (hD : ∀ w, 0 < D w) (hgB : ∀ w, LocalGrowth (ρ w) (m+r) (B w)) (hgD : ∀ w, LocalGrowth (ρ w) (d-1) (D w)) (hCI : Integrable (fun sw : Sphere d×W => Real.log (1+momentBound 9 d (completeFamily (P.map swapLast).snd B 1 sw.2) (completeFamily (P.map swapLast).snd D 1 sw.2))) P.fst) : ActualRowSourceConclusion (r:=r) (q:=q) P (completeFamily (P.map swapLast).snd ρ 0) (completeFamily (P.map swapLast).snd B 1) (completeFamily (P.map swapLast).snd D 1) := by have h := actualRowMain hdim P ν hInd H hν hEntropy ρ _hρac B D hB hD hgB hgD hCI have hi := rowInformation_eq_sourceConditionalInformation P ν (gaussianRows m d) hInd refine { row_finite := by simpa only [actualConditionalRowInformation, ← hi] using h.row_finite row_le_entropy := by simpa only [actualConditionalRowInformation, ← hi] using h.row_le_entropy conditional_density := by simpa only [actualConditionalRowInformation, ← hi] using h.conditional_density actual_limit := by simpa only [actualConditionalRowInformation, ← hi] using h.actual_limit smoothed_bound := by simpa only [actualConditionalRowInformation, ← hi] using h.smoothed_bound exact_integrable := by simpa only [actualConditionalRowInformation, ← hi] using h.exact_integrable exact_bound := by simpa only [actualConditionalRowInformation, ← hi] using h.exact_bound entropy_bound := by simpa only [actualConditionalRowInformation, ← hi] using h.entropy_bound positive_log_integrable := by simpa only [actualConditionalRowInformation, ← hi] using h.positive_log_integrable extended_log_bound := by simpa only [actualConditionalRowInformation, ← hi] using h.extended_log_bound } theorem actualRowSourceRefinement {d m : ℕ} {W E : Type*} [MeasurableSpace W] [MeasurableSpace E] [Countable W] [MeasurableSingletonClass W] [Countable E] [MeasurableSingletonClass E] (P : Measure ((Sphere d×W)×Rows m d)) [IsProbabilityMeasure P] (ν : Measure (Sphere d)) [IsProbabilityMeasure ν] (hInd : (P.map swapLast).fst = ν.prod (gaussianRows m d)) (e : Sphere d×W → E) (he : Measurable e) (hEntropy : shannonEntropy ((P.map (fun p => ((p.1.1,(p.1.2,e p.1)),p.2))).map swapLast).snd ≠ ∞) : actualConditionalRowInformation (P.map (fun p => ((p.1.1,(p.1.2,e p.1)),p.2))) = actualConditionalRowInformation P := by let F : (Sphere d×W)×Rows m d → (Sphere d×(W×E))×Rows m d := fun p => ((p.1.1,(p.1.2,e p.1)),p.2) have hF : Measurable F := (measurable_fst.fst.prodMk (measurable_fst.snd.prodMk (he.comp measurable_fst))).prodMk measurable_snd let Q := P.map F let : IsProbabilityMeasure Q := inferInstance have hIndQ : (Q.map swapLast).fst = ν.prod (gaussianRows m d) := by trans (P.map swapLast).fst · dsimp only [Q,Measure.fst] rw [Measure.map_map swapLast.measurable hF, Measure.map_map measurable_fst (swapLast.measurable.comp hF), Measure.map_map measurable_fst swapLast.measurable] rfl · exact hInd have hiQ := rowInformation_eq_sourceConditionalInformation Q ν (gaussianRows m d) hIndQ have hi := rowInformation_eq_sourceConditionalInformation P ν (gaussianRows m d) hInd change actualConditionalRowInformation Q = actualConditionalRowInformation P calc _ = rowInformation Q (gaussianRows m d) := hiQ.symm _ = rowInformation P (gaussianRows m d) := actualRowRefinement P e he hEntropy _ = _ := hi def alignedExperiment {d m r : ℕ} {W : Type*} [MeasurableSpace W] (P : Measure ((Sphere d×W)×Rows m d)) : Measure ((Sphere d×W)×Rows (m+r) d) := (actualExperiment (r:=r) P).map (fun p => (p.1.1,Fin.append p.1.2 p.2)) lemma measurable_alignedMap {d m r : ℕ} {W : Type*} [MeasurableSpace W] : Measurable (fun p : ((Sphere d×W)×Rows m d)×Rows r d => (p.1.1,Fin.append p.1.2 p.2)) := measurable_fst.fst.prodMk ((measurable_append (E:=Vector d) m r).comp (measurable_fst.snd.prodMk measurable_snd)) instance alignedExperiment_probability {d m r : ℕ} {W : Type*} [MeasurableSpace W] (P : Measure ((Sphere d×W)×Rows m d)) [IsProbabilityMeasure P] : IsProbabilityMeasure (alignedExperiment (r:=r) P) := by unfold alignedExperiment infer_instance def exposedJointMap {d k : ℕ} {W : Type*} (p : (Sphere d×W)×Rows k d) : (Rows k d×(Fin k → ℝ))×(Sphere d×W) := ((p.2,label p.2 (p.1.1:Vector d)),p.1) lemma measurable_exposedJointMap {d k : ℕ} {W : Type*} [MeasurableSpace W] : Measurable (exposedJointMap (d:=d) (k:=k) (W:=W)) := by have hY : Measurable (fun p : (Sphere d×W)×Rows k d => label p.2 (p.1.1:Vector d)) := (measurable_own_label (d:=d) (k:=k)).comp (measurable_snd.prodMk measurable_fst.fst) exact (measurable_snd.prodMk hY).prodMk measurable_fst def exposedInformation {d k : ℕ} {W : Type*} [MeasurableSpace W] [StandardBorelSpace W] (P : Measure ((Sphere d×W)×Rows k d)) [IsFiniteMeasure P] : ℝ≥0∞ := conditionalMutualInformation (P.map exposedJointMap) end CurrentProjection end end OAI