| import OAI.Probability.DirectionalWalk.SumLaws |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory Filter Preorder |
| open scoped ENNReal BigOperators Topology |
|
|
| namespace DirectionalZeroOne |
|
|
| open scoped Classical |
|
|
| section MarkedSums |
| variable {G : Type*} [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] |
| [AddCommGroup G] |
|
|
| noncomputable def sumKernel (ν : Measure G) : Kernel ℕ G where |
| toFun := sumLaw ν |
| measurable' := measurable_of_countable _ |
| |
| instance sumKernel_markov (ν : Measure G) [IsProbabilityMeasure ν] : IsMarkovKernel (sumKernel ν) := |
| ⟨fun n => sumLaw_probability ν n⟩ |
| |
| noncomputable def markedSumLaw (ν : Measure G) (θ : Measure ℕ) : Measure (ℕ × G) := |
| θ ⊗ₘ sumKernel ν |
| |
| instance markedSumLaw_probability (ν : Measure G) [IsProbabilityMeasure ν] |
| (θ : Measure ℕ) [IsProbabilityMeasure θ] : IsProbabilityMeasure (markedSumLaw ν θ) := |
| inferInstanceAs (IsProbabilityMeasure (θ ⊗ₘ sumKernel ν)) |
| |
| lemma markedSumLaw_atom (ν : Measure G) [IsProbabilityMeasure ν] |
| (θ : Measure ℕ) [IsProbabilityMeasure θ] (n : ℕ) (x : G) : |
| markedSumLaw ν θ {(n,x)} = θ {n} * sumLaw ν n {x} := compProd_atom θ (sumKernel ν) (n,x) |
| |
| lemma markedSumLaw_fst (ν : Measure G) [IsProbabilityMeasure ν] |
| (θ : Measure ℕ) [IsProbabilityMeasure θ] : (markedSumLaw ν θ).fst = θ := |
| Measure.fst_compProd θ (sumKernel ν) |
| |
| lemma markedSumLaw_conv (ν : Measure G) [IsProbabilityMeasure ν] |
| (θ η : Measure ℕ) [IsProbabilityMeasure θ] [IsProbabilityMeasure η] : |
| (markedSumLaw ν θ).conv (markedSumLaw ν η) = markedSumLaw ν (θ.conv η) := by |
| apply Measure.ext_of_lintegral |
| intro f hf |
| rw [Measure.lintegral_conv hf] |
| change (∫⁻ x, ∫⁻ y, f (x+y) ∂η ⊗ₘ sumKernel ν ∂θ ⊗ₘ sumKernel ν) = |
| ∫⁻ x, f x ∂(θ.conv η) ⊗ₘ sumKernel ν |
| rw [Measure.lintegral_compProd (measurable_of_countable _), |
| Measure.lintegral_compProd hf] |
| change (∫⁻ n, ∫⁻ x, ∫⁻ y, f ((n,x)+y) ∂η ⊗ₘ sumKernel ν ∂sumLaw ν n ∂θ) = |
| ∫⁻ n, ∫⁻ x, f (n,x) ∂sumLaw ν n ∂θ.conv η |
| rw [Measure.lintegral_conv (measurable_of_countable _)] |
| apply lintegral_congr |
| intro n |
| simp_rw [Measure.lintegral_compProd (measurable_of_countable _)] |
| rw [lintegral_lintegral_swap (measurable_of_countable _).aemeasurable] |
| apply lintegral_congr |
| intro m |
| change (∫⁻ x, ∫⁻ y, f (n+m,x+y) ∂sumLaw ν m ∂sumLaw ν n) = |
| ∫⁻ x, f (n+m,x) ∂sumLaw ν (n+m) |
| rw [sumLaw_add,Measure.lintegral_conv (measurable_of_countable _)] |
| |
| lemma conditionalInfo_markedSum (ν : Measure G) [IsProbabilityMeasure ν] |
| (θ : Measure ℕ) [IsProbabilityMeasure θ] : |
| conditionalInfo (markedSumLaw ν θ) =ᵐ[markedSumLaw ν θ] |
| fun p => atomInfo (sumLaw ν p.1) p.2 := by |
| filter_upwards [ae_atom_real_pos (markedSumLaw ν θ)] with p hp |
| have hf : 0 < θ.real {p.1} := by |
| have hh := atom_le_fst (markedSumLaw ν θ) p |
| rw [markedSumLaw_fst] at hh |
| exact hp.trans_le (ENNReal.toReal_mono (by finiteness) hh) |
| dsimp [conditionalInfo] |
| rw [markedSumLaw_fst] |
| have hx : (markedSumLaw ν θ).real {p} = θ.real {p.1} * (sumLaw ν p.1).real {p.2} := by |
| rw [measureReal_def,show p = (p.1,p.2) from rfl,markedSumLaw_atom,ENNReal.toReal_mul] |
| rfl |
| rw [hx,mul_div_cancel_left₀ _ hf.ne'] |
| rfl |
| |
| lemma markedSumLaw_conditional_entropy (ν : Measure G) [IsProbabilityMeasure ν] |
| (θ : Measure ℕ) [IsProbabilityMeasure θ] |
| (hi : ∀ n, Integrable (informationOf (sumLaw ν n) id) (sumLaw ν n)) |
| (hh : Integrable (fun n => entropyOf (sumLaw ν n) id) θ) : |
| Integrable (conditionalInfo (markedSumLaw ν θ)) (markedSumLaw ν θ) ∧ |
| conditionalEntropy (markedSumLaw ν θ) = ∫ n, entropyOf (sumLaw ν n) id ∂θ := by |
| have hin (n : ℕ) : Integrable (atomInfo (sumLaw ν n)) (sumLaw ν n) := by |
| simpa only [Measure.map_id] using (integrable_informationOf_iff _ id).mp (hi n) |
| have ha : Integrable (fun p : ℕ × G => atomInfo (sumLaw ν p.1) p.2) (markedSumLaw ν θ) := by |
| apply (Measure.integrable_compProd_iff (measurable_of_countable _).aestronglyMeasurable).mpr |
| constructor |
| · exact Eventually.of_forall hin |
| · change Integrable (fun n => ∫ y, ‖atomInfo (sumLaw ν n) y‖ ∂sumLaw ν n) θ |
| simpa only [Real.norm_eq_abs,abs_of_nonneg (atomInfo_nonneg _ _), |
| entropyOf_eq_discreteEntropy,Measure.map_id,discreteEntropy] using hh |
| refine ⟨ha.congr (conditionalInfo_markedSum ν θ).symm,?_⟩ |
| change (∫ p, conditionalInfo (markedSumLaw ν θ) p ∂markedSumLaw ν θ) = _ |
| rw [integral_congr_ae (conditionalInfo_markedSum ν θ)] |
| change (∫ p, atomInfo (sumLaw ν p.1) p.2 ∂θ ⊗ₘ sumKernel ν) = _ |
| rw [Measure.integral_compProd ha] |
| change (∫ n, discreteEntropy (sumLaw ν n) ∂θ) = _ |
| simp only [entropyOf_eq_discreteEntropy,Measure.map_id] |
| end MarkedSums |
| section CommonMoments |
| variable {α : Type*} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α] |
| |
| lemma commonBlock_integrable_total (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)) |
| (b : Bool) (D : α → ℝ) (hi : Integrable D (ν b)) : |
| Integrable (fun a => listTotal D (a b)) (commonBlockLaw ν L) := by |
| rw [commonBlockLaw,integrable_map_measure (measurable_of_countable _).aestronglyMeasurable |
| (measurable_firstCommonBlock L).aemeasurable] |
| refine ⟨((measurable_of_countable _).comp (measurable_firstCommonBlock L)).aestronglyMeasurable,?_⟩ |
| apply lt_of_le_of_lt _ ((commonBlock_cost_bound ν L hL he b (fun a => ‖D a‖ₑ)).trans_lt ?_) |
| · apply lintegral_mono |
| intro Z |
| exact enorm_sum_le Finset.univ (fun i => D ((firstCommonBlock L Z b).2 i)) |
| · apply ENNReal.mul_lt_top |
| · have hw := (hasFiniteIntegral_iff_ofReal (Eventually.of_forall |
| (fun Z => Nat.cast_nonneg (α := ℝ) (firstCommonWidth L Z)))).mp hW.hasFiniteIntegral |
| simpa only [ENNReal.ofReal_natCast] using hw |
| · exact hi.hasFiniteIntegral |
| |
| lemma commonGapLaw_lattice_integrable {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)) |
| (D : Bool → α → (Fin q → ℤ)) |
| (hi : ∀ b i, Integrable (fun a => (D b a i : ℝ)) (ν b)) (i : Fin q) : |
| Integrable (fun x => |(x i : ℝ)|) (commonGapLaw ν L D) := by |
| rw [commonGapLaw,integrable_map_measure (measurable_of_countable _).aestronglyMeasurable |
| (measurable_of_countable _).aemeasurable] |
| have h0 := commonBlock_integrable_total ν L hL he hW false (fun a => (D false a i : ℝ)) (hi false i) |
| have h1 := commonBlock_integrable_total ν L hL he hW true (fun a => (D true a i : ℝ)) (hi true i) |
| simpa only [Function.comp_def,pairTotal,listTotal,Pi.add_apply,Finset.sum_apply,Int.cast_add,Int.cast_sum] using (h0.add h1).abs |
| end CommonMoments |
| |
| section MarkedEntropy |
| variable {G : Type*} [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] |
| [AddCommGroup G] |
| |
| lemma markedSumLaw_entropy (ν : Measure G) [IsProbabilityMeasure ν] |
| (θ : Measure ℕ) [IsProbabilityMeasure θ] |
| (hi : ∀ n, Integrable (informationOf (sumLaw ν n) id) (sumLaw ν n)) |
| (hθ : Integrable (atomInfo θ) θ) |
| (hh : Integrable (fun n => entropyOf (sumLaw ν n) id) θ) : |
| Integrable (informationOf (markedSumLaw ν θ) id) (markedSumLaw ν θ) ∧ |
| entropyOf (markedSumLaw ν θ) id = discreteEntropy θ + ∫ n, entropyOf (sumLaw ν n) id ∂θ := by |
| have hc := markedSumLaw_conditional_entropy ν θ hi hh |
| have hfst : Integrable (informationOf (markedSumLaw ν θ) Prod.fst) (markedSumLaw ν θ) := by |
| rw [integrable_informationOf_iff] |
| change Integrable (atomInfo (markedSumLaw ν θ).fst) (markedSumLaw ν θ).fst |
| rwa [markedSumLaw_fst] |
| have he : conditionedInformation (markedSumLaw ν θ) Prod.snd Prod.fst =ᵐ[markedSumLaw ν θ] |
| conditionalInfo (markedSumLaw ν θ) := by |
| have h := conditionedInformation_eq (markedSumLaw ν θ) Prod.snd Prod.fst |
| simpa only [show (fun p : ℕ × G => (p.1,p.2)) = id from rfl,Measure.map_id] using h |
| have hcond : Integrable (conditionedInformation (markedSumLaw ν θ) Prod.snd Prod.fst) (markedSumLaw ν θ) := |
| hc.1.congr he.symm |
| constructor |
| · change Integrable (informationOf (markedSumLaw ν θ) (fun p => (p.1,p.2))) (markedSumLaw ν θ) |
| rw [informationOf_pair_chain] |
| exact hfst.add hcond |
| · change entropyOf (markedSumLaw ν θ) (fun p => (p.1,p.2)) = _ |
| rw [entropyOf_pair_chain _ _ _ hfst hcond,entropyOf_eq_discreteEntropy] |
| change discreteEntropy (markedSumLaw ν θ).fst + _ = _ |
| rw [markedSumLaw_fst] |
| congr 1 |
| change (∫ p, conditionedInformation (markedSumLaw ν θ) Prod.snd Prod.fst p ∂markedSumLaw ν θ) = _ |
| rw [integral_congr_ae he] |
| exact hc.2 |
| |
| lemma entropyOf_nonneg {Ω : Type*} [Countable Ω] [MeasurableSpace Ω] [MeasurableSingletonClass Ω] |
| {A : Type*} [Countable A] [MeasurableSpace A] [MeasurableSingletonClass A] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → A) : 0 ≤ entropyOf μ X := |
| integral_nonneg (informationOf_nonneg μ X) |
| |
| lemma sum_entropy_integrable_bounded (ν : Measure G) [IsProbabilityMeasure ν] |
| (θ : Measure ℕ) [IsProbabilityMeasure θ] |
| (hi : ∀ n, Integrable (informationOf (sumLaw ν n) id) (sumLaw ν n)) |
| (M : ℕ) (hM : ∀ᵐ n ∂θ, n ≤ M) : |
| Integrable (fun n => entropyOf (sumLaw ν n) id) θ := by |
| apply (integrable_const (entropyOf (sumLaw ν M) id)).mono' (measurable_of_countable _).aestronglyMeasurable |
| filter_upwards [hM] with n hn |
| rw [Real.norm_eq_abs,abs_of_nonneg (entropyOf_nonneg _ _)] |
| exact sumLaw_entropy_monotone ν hi hn |
| |
| lemma markedSumLaw_entropy_bounds (ν : Measure G) [IsProbabilityMeasure ν] |
| (θ : Measure ℕ) [IsProbabilityMeasure θ] |
| (hi : ∀ n, Integrable (informationOf (sumLaw ν n) id) (sumLaw ν n)) |
| (hθ : Integrable (atomInfo θ) θ) |
| (m M : ℕ) (hM : ∀ᵐ n ∂θ, m ≤ n ∧ n ≤ M) : |
| Integrable (informationOf (markedSumLaw ν θ) id) (markedSumLaw ν θ) ∧ |
| discreteEntropy θ + entropyOf (sumLaw ν m) id ≤ entropyOf (markedSumLaw ν θ) id ∧ |
| entropyOf (markedSumLaw ν θ) id ≤ discreteEntropy θ + entropyOf (sumLaw ν M) id := by |
| have hb := sum_entropy_integrable_bounded ν θ hi M (hM.mono (fun _ h => h.2)) |
| have hh := markedSumLaw_entropy ν θ hi hθ hb |
| refine ⟨hh.1,?_,?_⟩ |
| · rw [hh.2] |
| apply add_le_add le_rfl |
| calc |
| _ = ∫ _ : ℕ, entropyOf (sumLaw ν m) id ∂θ := by simp |
| _ ≤ _ := integral_mono_ae (integrable_const _) hb (hM.mono (fun _ h => sumLaw_entropy_monotone ν hi h.1)) |
| · rw [hh.2] |
| apply add_le_add le_rfl |
| calc |
| _ ≤ ∫ _ : ℕ, entropyOf (sumLaw ν M) id ∂θ := |
| integral_mono_ae hb (integrable_const _) (hM.mono (fun _ h => sumLaw_entropy_monotone ν hi h.2)) |
| _ = _ := by simp |
| end MarkedEntropy |
| |
| end DirectionalZeroOne |
| |
| end OAI |
| |
