| import OAI.NumberTheory.DukePrimeDegree.Entropy.ConditionalNameCovers | |
| namespace OAI | |
| /-! # Finite entropy conditioned on a measurable observation -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped BigOperators | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets | |
| theorem conditionalSigmaEntropy_comap_eq_integral | |
| {S Y α : Type*} [MeasurableSpace S] [MeasurableSpace Y] | |
| [StandardBorelSpace Y] [Nonempty Y] [Fintype α] | |
| (μ : Measure Y) [IsProbabilityMeasure μ] | |
| (F : Y → S) (hF : Measurable F) | |
| (P : Y → α) (hP : ∀ a, MeasurableSet {y | P y=a}) : | |
| conditionalSigmaEntropy μ P (MeasurableSpace.comap F inferInstance) = | |
| ∫ s, observationEntropy (condDistrib id F μ s) P ∂μ.map F := by | |
| have hc (a : α) : | |
| μ[({y | P y=a}.indicator (fun _ => (1 : ℝ))) | | |
| MeasurableSpace.comap F inferInstance] =ᵐ[μ] | |
| fun y => (condDistrib id F μ (F y)).real {z | P z=a} := by | |
| have H := condExp_ae_eq_integral_condDistrib_id (μ := μ) hF | |
| ((integrable_const (μ := μ) (1 : ℝ)).indicator (hP a)) | |
| have hi (s : S) : (∫ z, {y | P y=a}.indicator (fun _ => (1 : ℝ)) z | |
| ∂condDistrib id F μ s) = (condDistrib id F μ s).real {z | P z=a} := by | |
| rw [integral_indicator_const _ (hP a)] | |
| simp only [smul_eq_mul,mul_one] | |
| simpa only [hi] using H | |
| have hall := ae_all_iff.mpr hc | |
| unfold conditionalSigmaEntropy | |
| calc | |
| _ = ∫ y, observationEntropy (condDistrib id F μ (F y)) P ∂μ := by | |
| apply integral_congr_ae | |
| filter_upwards [hall] with y hy | |
| unfold observationEntropy | |
| congr 1 | |
| exact funext hy | |
| _ = _ := (integral_map hF.aemeasurable | |
| (integrable_kernel_entropy (μ.map F) (condDistrib id F μ) P hP).aestronglyMeasurable).symm | |
| theorem conditionalSigmaEntropy_comap_nonneg | |
| {S Y α : Type*} [MeasurableSpace S] [MeasurableSpace Y] | |
| [StandardBorelSpace Y] [Nonempty Y] [Fintype α] | |
| (μ : Measure Y) [IsProbabilityMeasure μ] | |
| (F : Y → S) (hF : Measurable F) | |
| (P : Y → α) (hP : ∀ a, MeasurableSet {y | P y=a}) : | |
| 0 ≤ conditionalSigmaEntropy μ P (MeasurableSpace.comap F inferInstance) := by | |
| rw [conditionalSigmaEntropy_comap_eq_integral μ F hF P hP] | |
| exact integral_nonneg fun s => observationEntropy_nonneg (condDistrib id F μ s) P hP | |
| theorem conditionalSigmaEntropy_comap_le_reference_bad_sets | |
| {S Y α : Type*} [MeasurableSpace S] [MeasurableSpace Y] | |
| [StandardBorelSpace Y] [Nonempty Y] | |
| [MeasurableSpace α] [MeasurableSingletonClass α] | |
| [Fintype α] [Nonempty α] [DecidableEq α] | |
| (μ : Measure Y) [IsProbabilityMeasure μ] | |
| (F : Y → S) (hF : Measurable F) (N M : ℕ) | |
| (P : Y → Fin N → α) (hP : Measurable P) | |
| (r : S → Y → Fin M) (hr : ∀ s, Measurable (r s)) | |
| (v : S → Fin M → Fin N → α) (B : Fin N → Set Y) | |
| (hB : ∀ i, MeasurableSet (B i)) | |
| (hbad : ∀ᵐ s ∂μ.map F, ∀ i, ∀ᵐ y ∂condDistrib id F μ s, | |
| P y i ≠ v s (r s y) i → y ∈ B i) | |
| {δ : ℝ} (hδ : 0 < δ) (hδ1 : δ < 1) : | |
| conditionalSigmaEntropy μ P (MeasurableSpace.comap F inferInstance) ≤ | |
| Real.log M+(N : ℝ)*(-Real.log (1-δ))+ | |
| (∑ i : Fin N, μ.real (B i)) * | |
| (Real.log (Fintype.card α)-Real.log δ) := by | |
| rw [conditionalSigmaEntropy_comap_eq_integral μ F hF P | |
| (fun a => hP (measurableSet_singleton a))] | |
| have H := integral_nameEntropy_le_reference_bad_sets (μ.map F) (condDistrib id F μ) | |
| N M P hP r hr v B hB hbad hδ hδ1 | |
| have hm : condDistrib id F μ ∘ₘ (μ.map F)=μ := by | |
| simpa only [Measure.map_id] using condDistrib_comp_map hF.aemeasurable aemeasurable_id | |
| simpa only [hm] using H | |
| end DukePrimeDegree | |
| end | |
| end OAI | |