| import OAI.Probability.DirectionalWalk.SignedInformation |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Filter Preorder |
| open scoped ENNReal BigOperators Topology |
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| namespace DirectionalZeroOne |
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| open scoped Classical |
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| section ActualComparison |
| variable {α G : Type*} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α] |
| [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] |
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| noncomputable def experimentLaw (ν : Bool → Measure α) [∀ b, IsProbabilityMeasure (ν b)] |
| (L : Bool → α → ℕ) (D : Bool → α → G) (n : ℕ) : |
| Measure (((ℕ × (ℕ × ℕ)) × G) × (ThreeLists α × ThreeLists α)) := |
| ((markedSumLaw (commonGapLaw ν L D) (countWindowLaw n n)).prod (signedChunkLaw ν L n)).map |
| (fun p => ((threeHeight L (joinThree p.2),p.1.2+signedTotal (D false) p.2.1),p.2)) |
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| noncomputable def experimentReference (ν : Bool → Measure α) [∀ b, IsProbabilityMeasure (ν b)] |
| (L : Bool → α → ℕ) (D : Bool → α → G) (n : ℕ) : |
| Measure (((ℕ × (ℕ × ℕ)) × G) × (ThreeLists α × ThreeLists α)) := |
| (experimentLaw ν L D n).fst ⊗ₘ |
| (signedThreeKernel (ν false) (L false) ×ₖ signedThreeKernel (ν true) (L true)).comap Prod.fst measurable_fst |
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| lemma experiment_kl_comparison (ν : Bool → Measure α) [∀ b, IsProbabilityMeasure (ν b)] |
| (L : Bool → α → ℕ) (hL : ∀ b, ∀ᵐ a ∂ν b, 0 < L b a) |
| (he : ∀ᵐ Z ∂twoTapeLaw ν, ∃ H, 0 < H ∧ Z ∈ commonCut L H) |
| (hW : Integrable (fun Z => (firstCommonWidth L Z : ℝ)) (twoTapeLaw ν)) |
| (hu : ∀ H b, Measure.infinitePi (fun _ : ℕ => ν b) (renewalCut (L b) H) ≠ 0) |
| (c : ℝ) (hc : 0 < c) (hlo : ∀ H, ENNReal.ofReal c ≤ renewalPairMass ν L H) |
| (D : Bool → α → G) |
| (hgap : ∀ r, Integrable (informationOf (sumLaw (commonGapLaw ν L D) r) id) (sumLaw (commonGapLaw ν L D) r)) |
| (n : ℕ) [NeZero n] |
| (hD : Integrable (informationOf (signedChunkLaw ν L n) (fun a => signedTotal (D false) a.1)) (signedChunkLaw ν L n)) : |
| InformationTheory.klDiv (experimentLaw ν L D n) (experimentReference ν L D n) ≠ ∞ ∧ |
| (InformationTheory.klDiv (experimentLaw ν L D n) (experimentReference ν L D n)).toReal ≤ |
| 6*(1+Real.log (8*((∫ Z, (firstCommonWidth L Z : ℝ) ∂twoTapeLaw ν)+1)+1)) + |
| 1+Real.log 16 + entropyOf (sumLaw (commonGapLaw ν L D) (14*n)) id - |
| entropyOf (sumLaw (commonGapLaw ν L D) n) id := by |
| let π := markedSumLaw (commonGapLaw ν L D) (countWindowLaw n n) |
| let ρ := signedChunkLaw ν L n |
| let T : (ℕ × G) × (ThreeLists α × ThreeLists α) → ℕ × G := |
| fun p => (p.1.1+threeCount L (joinThree p.2), |
| p.1.2+signedTotal (D false) p.2.1+signedTotal (D true) p.2.2) |
| let : IsMarkovKernel (signedThreeKernel (ν false) (L false)) := signedThreeKernel_markov _ _ (fun H => hu H false) |
| let : IsMarkovKernel (signedThreeKernel (ν true) (L true)) := signedThreeKernel_markov _ _ (fun H => hu H true) |
| have hmark : (π.prod ρ).map T = markedSumLaw (commonGapLaw ν L D) (totalCountLaw n) := |
| signed_buffer_marked ν L hL he D n |
| have hent := marked_count_entropy_difference (commonGapLaw ν L D) hgap n |
| have hpair : Integrable (informationOf (π.prod ρ) T) (π.prod ρ) := by |
| have ht := hent.2.1 |
| rw [← hmark,integrable_informationOf_map_iff] at ht |
| exact ht |
| have hT := integrable_informationOf_comp (π.prod ρ) T Prod.fst hpair |
| have hS := integrable_informationOf_comp (π.prod ρ) T Prod.snd hpair |
| have hU := integrable_informationOf_comp π id Prod.fst hent.1 |
| have hd := integrable_informationOf_comp π id Prod.snd hent.1 |
| have hzpair := integrable_informationOf_pair (π.prod ρ) (fun p => p.1.2) |
| (fun p => signedTotal (D false) p.2.1) |
| (integrable_informationOf_prod_fst π ρ Prod.snd hd) |
| (integrable_informationOf_prod_snd π ρ (fun a => signedTotal (D false) a.1) hD) |
| have hZ := integrable_informationOf_comp (π.prod ρ) _ (fun p : G × G => p.1+p.2) hzpair |
| have hkl := threeChunk_kl_uniform ν L hL he hW hu c hc hlo n |
| dsimp only at hkl |
| rw [← signedChunk_kl ν L hu n] at hkl |
| have hb := buffered_reference_comparison π ρ (threeHeight L ∘ joinThree) |
| (signedTotal (D false)) (signedTotal (D true)) (threeCount L ∘ joinThree) |
| (signedThreeKernel (ν false) (L false)) (signedThreeKernel (ν true) (L true)) |
| hkl.1 hU hd hZ hT hS |
| have ht : entropyOf (π.prod ρ) (fun p => |
| (p.1.2+signedTotal (D false) p.2.1+signedTotal (D true) p.2.2, |
| p.1.1+threeCount L (joinThree p.2))) = |
| entropyOf (markedSumLaw (commonGapLaw ν L D) (totalCountLaw n)) id := by |
| rw [entropyOf_pair_comm] |
| change entropyOf (π.prod ρ) T = _ |
| rw [← hmark,entropyOf_map] |
| rfl |
| have hπ : entropyOf π (fun p => (p.2,p.1)) = entropyOf π id := by |
| rw [entropyOf_pair_comm] |
| rfl |
| change InformationTheory.klDiv (experimentLaw ν L D n) (experimentReference ν L D n) ≠ ∞ ∧ _ |
| refine ⟨hb.1,?_⟩ |
| dsimp only [Function.comp_apply] at hb |
| rw [ht,hπ] at hb |
| have hh := hent.2.2 |
| dsimp only [π,ρ] at hb hh ⊢ |
| dsimp only [experimentLaw,experimentReference] |
| linarith only [hb.2,hkl.2,hh] |
| end ActualComparison |
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|
| lemma sumLaw_lattice_information {q : ℕ} (ν : Measure (Fin q → ℤ)) [IsProbabilityMeasure ν] |
| (hi : ∀ i, Integrable (fun x => |(x i : ℝ)|) ν) (n : ℕ) : |
| Integrable (informationOf (sumLaw ν n) id) (sumLaw ν n) := by |
| let a := 1+∑ i, ∫ x, |(x i : ℝ)| ∂ν |
| have hm (i : Fin q) : (∫ x, |(x i : ℝ)| ∂ν) ≤ a := by |
| have h : (∫ x, |(x i : ℝ)| ∂ν) ≤ ∑ j : Fin q, ∫ x, |(x j : ℝ)| ∂ν := |
| Finset.single_le_sum (f := fun j : Fin q => ∫ x, |(x j : ℝ)| ∂ν) (fun j _ => integral_nonneg (fun x : Fin q → ℤ => abs_nonneg (x j : ℝ))) (Finset.mem_univ i) |
| dsimp [a] |
| linarith |
| exact (sumLaw_lattice_entropy ν hi a (by |
| dsimp [a] |
| have h : 0 ≤ ∑ i : Fin q, ∫ x, |(x i : ℝ)| ∂ν := Finset.sum_nonneg (fun i _ => integral_nonneg (fun x => abs_nonneg _)) |
| linarith) hm n).1 |
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| lemma lattice_experiment_kl_comparison {α : Type*} [Countable α] [MeasurableSpace α] |
| [MeasurableSingletonClass α] {q : ℕ} |
| (ν : Bool → Measure α) [∀ b, IsProbabilityMeasure (ν b)] |
| (L : Bool → α → ℕ) (hL : ∀ b, ∀ᵐ a ∂ν b, 0 < L b a) |
| (he : ∀ᵐ Z ∂twoTapeLaw ν, ∃ H, 0 < H ∧ Z ∈ commonCut L H) |
| (hW : Integrable (fun Z => (firstCommonWidth L Z : ℝ)) (twoTapeLaw ν)) |
| (hu : ∀ H b, Measure.infinitePi (fun _ : ℕ => ν b) (renewalCut (L b) H) ≠ 0) |
| (c : ℝ) (hc : 0 < c) (hlo : ∀ H, ENNReal.ofReal c ≤ renewalPairMass ν L H) |
| (D : Bool → α → Fin q → ℤ) (hi : ∀ b i, Integrable (fun a => (D b a i : ℝ)) (ν b)) |
| (n : ℕ) [NeZero n] : |
| InformationTheory.klDiv (experimentLaw ν L D n) (experimentReference ν L D n) ≠ ∞ ∧ |
| (InformationTheory.klDiv (experimentLaw ν L D n) (experimentReference ν L D n)).toReal ≤ |
| 6*(1+Real.log (8*((∫ Z, (firstCommonWidth L Z : ℝ) ∂twoTapeLaw ν)+1)+1)) + |
| 1+Real.log 16 + entropyOf (sumLaw (commonGapLaw ν L D) (14*n)) id - |
| entropyOf (sumLaw (commonGapLaw ν L D) n) id := by |
| exact experiment_kl_comparison ν L hL he hW hu c hc hlo D |
| (sumLaw_lattice_information _ (commonGapLaw_lattice_integrable ν L hL he hW D hi)) n |
| (signed_chunk_positive_info ν L hL he hW (D false) (hi false) n) |
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| end DirectionalZeroOne |
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| end OAI |
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