| import OAI.NumberTheory.PrimePackets.ConditionalSigmaEntropy |
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| namespace OAI |
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| /-! # Conditional information in disintegration coordinates -/ |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped Topology |
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| namespace PrimeDegreePackets |
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| variable {S T α : Type*} [MeasurableSpace S] [MeasurableSpace T] |
| [StandardBorelSpace T] [Nonempty T] [Fintype α] |
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| omit [StandardBorelSpace T] [Nonempty T] in |
| lemma measurable_fiber_observationEntropy (κ : Kernel S T) [IsMarkovKernel κ] |
| (P : S × T → α) (hP : ∀ a, MeasurableSet {z | P z = a}) : |
| Measurable (fun s => observationEntropy (κ s) (fun t => P (s,t))) := by |
| apply continuous_finiteShannon.measurable.comp |
| apply measurable_pi_iff.mpr |
| intro a |
| exact (Kernel.measurable_kernel_prodMk_left (κ := κ) (hP a)).ennreal_toReal |
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| omit [Fintype α] in |
| lemma condExp_fst_observation_probability |
| (μ : Measure (S × T)) [IsFiniteMeasure μ] |
| (P : S × T → α) (hP : ∀ a, MeasurableSet {z | P z = a}) (a : α) : |
| μ[({z | P z = a}.indicator (fun _ => (1 : ℝ))) | |
| MeasurableSpace.comap Prod.fst (inferInstance : MeasurableSpace S)] =ᵐ[μ] |
| fun z => (μ.condKernel z.1).real {t | P (z.1,t) = a} := by |
| let f : S × T → ℝ := {z | P z = a}.indicator (fun _ => 1) |
| have hid : (fun z : S × T => (z.1,z.2)) = id := rfl |
| have hf : Integrable f (μ.map (fun z : S × T => (z.1,z.2))) := by |
| rw [hid, Measure.map_id] |
| exact (integrable_const (1 : ℝ)).indicator (hP a) |
| have H := condExp_prod_ae_eq_integral_condDistrib' |
| (μ := μ) (X := (Prod.fst : S × T → S)) (Y := (Prod.snd : S × T → T)) |
| measurable_fst measurable_snd.aemeasurable hf |
| have hκ : condDistrib (Prod.snd : S × T → T) Prod.fst μ = μ.condKernel := by |
| simp only [condDistrib, hid, Measure.map_id] |
| rw [hκ] at H |
| have hI (s : S) : (∫ t, f (s,t) ∂μ.condKernel s) = |
| (μ.condKernel s).real {t | P (s,t) = a} := by |
| change (∫ t, {t | P (s,t) = a}.indicator (fun _ => (1 : ℝ)) t ∂μ.condKernel s) = _ |
| have hs : MeasurableSet {t | P (s,t) = a} := (hP a).preimage measurable_prodMk_left |
| rw [integral_indicator_const _ hs] |
| simp only [smul_eq_mul, mul_one] |
| simpa only [Prod.eta, hI] using H |
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| theorem conditionalSigmaEntropy_fst |
| (μ : Measure (S × T)) [IsFiniteMeasure μ] |
| (P : S × T → α) (hP : ∀ a, MeasurableSet {z | P z = a}) : |
| conditionalSigmaEntropy μ P |
| (MeasurableSpace.comap Prod.fst (inferInstance : MeasurableSpace S)) = |
| ∫ s, observationEntropy (μ.condKernel s) (fun t => P (s,t)) ∂μ.fst := by |
| have H := ae_all_iff.mpr (fun a => condExp_fst_observation_probability μ P hP a) |
| unfold conditionalSigmaEntropy |
| calc |
| _ = ∫ z, observationEntropy (μ.condKernel z.1) (fun t => P (z.1,t)) ∂μ := by |
| apply integral_congr_ae |
| filter_upwards [H] with z hz |
| unfold observationEntropy |
| congr 1 |
| exact funext hz |
| _ = _ := by |
| exact (integral_map measurable_fst.aemeasurable |
| (measurable_fiber_observationEntropy μ.condKernel P hP).aestronglyMeasurable).symm |
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| end PrimeDegreePackets |
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| end |
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| end OAI |
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