| import OAI.NumberTheory.PrimePackets.GeneratingObservationEntropy |
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| namespace OAI |
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| /-! # Finite observation entropy conditioned on a sigma algebra -/ |
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| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped BigOperators Topology |
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| namespace PrimeDegreePackets |
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| variable {Y α : Type*} [mY : MeasurableSpace Y] [Fintype α] |
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| def conditionalSigmaEntropy (μ : Measure Y) (P : Y → α) (m : MeasurableSpace Y) : ℝ := |
| ∫ y, finiteShannon (fun a => |
| (μ[({z | P z = a}.indicator (fun _ => (1 : ℝ))) | m]) y) ∂μ |
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| theorem conditionalObservationEntropy_eq_sigma |
| {β : Type*} [Fintype β] (μ : Measure Y) [IsFiniteMeasure μ] |
| (P : Y → α) (Q : Y → β) |
| (hP : ∀ a, MeasurableSet {y | P y = a}) |
| (hQ : ∀ b, MeasurableSet {y | Q y = b}) : |
| conditionalObservationEntropy μ P Q = conditionalSigmaEntropy μ P (observationSigma Q) := by |
| rw [conditionalObservationEntropy_eq_integral μ P Q hP hQ] |
| apply integral_congr_ae |
| have H := ae_all_iff.mpr (fun a => observationConditionalProbability_ae_eq_condExp μ P Q hP hQ a) |
| filter_upwards [H] with y hy |
| congr 1 |
| exact funext hy |
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| theorem conditionalObservationEntropy_tendsto_sigma |
| {β : ℕ → Type*} [∀ n, Fintype (β n)] |
| (μ : Measure Y) [IsProbabilityMeasure μ] |
| (P : Y → α) (hP : ∀ a, MeasurableSet {y | P y = a}) |
| (Q : ∀ n, Y → β n) (hQ : ∀ n b, MeasurableSet {y | Q n y = b}) |
| (hmono : Monotone (fun n => observationSigma (Q n))) : |
| Tendsto (fun n => conditionalObservationEntropy μ P (Q n)) atTop |
| (𝓝 (conditionalSigmaEntropy μ P (⨆ n, observationSigma (Q n)))) := by |
| let ℱ : Filtration ℕ mY := |
| ⟨fun n => observationSigma (Q n), hmono, fun n => observationSigma_le (Q n) (hQ n)⟩ |
| let F : ℕ → Y → ℝ := fun n y => |
| finiteShannon (fun a => observationConditionalProbability μ P (Q n) a y) |
| let G : Y → ℝ := fun y => finiteShannon (fun a => |
| (μ[({z | P z = a}.indicator (fun _ => (1 : ℝ))) | ⨆ n, ℱ n]) y) |
| have hFint n : Integrable (F n) μ := |
| integrable_finite_values μ (Q n) (hQ n) (fun b => finiteShannon (fun a => |
| μ.real {y | (Q n y,P y) = (b,a)} / μ.real {y | Q n y = b})) |
| have hbound n : ∀ᵐ y ∂μ, ‖F n y‖ ≤ Real.log (Fintype.card α) := by |
| filter_upwards [observationConditionalProbability_ae_sum μ P (Q n) hP (hQ n)] with y hy |
| have hn := fun a => observationConditionalProbability_nonneg μ P (Q n) a y |
| change ‖finiteShannon (fun a => observationConditionalProbability μ P (Q n) a y)‖ ≤ _ |
| rw [Real.norm_eq_abs, abs_of_nonneg (finiteShannon_nonneg hn hy)] |
| exact finiteShannon_le_log_card hn hy |
| have hcoord (a : α) : ∀ᵐ y ∂μ, |
| Tendsto (fun n => observationConditionalProbability μ P (Q n) a y) atTop |
| (𝓝 ((μ[({z | P z = a}.indicator (fun _ => (1 : ℝ))) | ⨆ n, ℱ n]) y)) := by |
| have H := MeasureTheory.tendsto_ae_condExp (μ := μ) (ℱ := ℱ) |
| ({z | P z = a}.indicator (fun _ => (1 : ℝ))) |
| have heq := ae_all_iff.mpr (fun n => |
| observationConditionalProbability_ae_eq_condExp μ P (Q n) hP (hQ n) a) |
| filter_upwards [H,heq] with y hy he |
| simpa only [he] using hy |
| have hlim : ∀ᵐ y ∂μ, Tendsto (fun n => F n y) atTop (𝓝 (G y)) := by |
| filter_upwards [ae_all_iff.mpr hcoord] with y hy |
| have H := continuous_finiteShannon.continuousAt.tendsto.comp (tendsto_pi_nhds.mpr hy) |
| simpa only [F,G,Function.comp_def] using H |
| have H := tendsto_integral_of_dominated_convergence |
| (fun _ : Y => Real.log (Fintype.card α)) (fun n => (hFint n).aestronglyMeasurable) |
| (integrable_const _) hbound hlim |
| have heq n : conditionalObservationEntropy μ P (Q n) = ∫ y, F n y ∂μ := |
| conditionalObservationEntropy_eq_integral μ P (Q n) hP (hQ n) |
| simpa only [← heq, conditionalSigmaEntropy, G] using H |
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| end PrimeDegreePackets |
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| end |
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| end OAI |
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