openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Entropy /ConditionalInformation.lean
| import OAI.NumberTheory.DukePrimeDegree.Entropy.ObservationInformation | |
| namespace OAI | |
| /-! # Conditional entropy as the mean pointwise information | |
| The finite observed variable has a regular conditional distribution even | |
| when the conditioning sigma algebra is arbitrary. Lifting its underlying | |
| space keeps the ambient and conditioning measurable structures distinct. | |
| -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets | |
| def conditionalInformation {Y α : Type*} [MeasurableSpace Y] | |
| (μ : Measure Y) (P : Y → α) (m : MeasurableSpace Y) (y : Y) : ℝ := | |
| -Real.log ((μ[({z | P z=P y}.indicator (fun _ => (1:ℝ))) | m]) y) | |
| theorem conditionalInformation_integrable_integral {Y : Type*} (m : MeasurableSpace Y) | |
| [mY : MeasurableSpace Y] | |
| (μ : Measure Y) [IsFiniteMeasure μ] {N : ℕ} [NeZero N] | |
| (P : Y → Fin N) (hP : Measurable P) (hm : m ≤ mY) : | |
| Integrable (conditionalInformation μ P m) μ ∧ | |
| (∫ y, conditionalInformation μ P m y ∂μ) = conditionalSigmaEntropy μ P m := by | |
| let : MeasurableSpace (ULift.{0} Y) := m.comap ULift.down | |
| let X : Y → ULift.{0} Y := ULift.up | |
| have hmX : MeasurableSpace.comap X (inferInstance : MeasurableSpace (ULift.{0} Y)) = m := by | |
| change (m.comap ULift.down).comap ULift.up = m | |
| rw [MeasurableSpace.comap_comp] | |
| exact MeasurableSpace.comap_id | |
| have hX : Measurable X := by | |
| apply Measurable.of_comap_le | |
| simpa only [hmX] using hm | |
| let σ := μ.map (fun y => (X y,P y)) | |
| let κ := σ.condKernel | |
| have hκ : condDistrib P X μ = κ := by rw [condDistrib] | |
| have heq (a : Fin N) : | |
| μ[({z | P z=a}.indicator (fun _ => (1:ℝ))) | m] =ᵐ[μ] | |
| fun y => (κ (X y)).real {a} := by | |
| have H := condExp_ae_eq_integral_condDistrib (μ := μ) (X := X) (Y := P) | |
| hX hP.aemeasurable | |
| ((stronglyMeasurable_const.indicator (measurableSet_singleton a)) : | |
| StronglyMeasurable (({a} : Set (Fin N)).indicator (fun _ : Fin N => (1:ℝ)))) | |
| ((integrable_const (1:ℝ)).indicator (hP (measurableSet_singleton a))) | |
| rw [hκ,hmX] at H | |
| have hleft : (fun y => ({a} : Set (Fin N)).indicator (fun _ => (1:ℝ)) (P y)) = | |
| {z | P z=a}.indicator (fun _ => (1:ℝ)) := by | |
| funext y | |
| simp only [Set.indicator_apply,Set.mem_singleton_iff,Set.mem_ofPred_eq] | |
| simpa only [hleft,integral_indicator_const (1:ℝ) (measurableSet_singleton a), | |
| smul_eq_mul,mul_one] using H | |
| let I : ULift.{0} Y × Fin N → ℝ := | |
| fun z => observationInformation (κ z.1) (id : Fin N → Fin N) z.2 | |
| have hI : Measurable I := measurable_kernel_observationInformation κ Prod.snd | |
| (fun a => measurable_snd (measurableSet_singleton a)) | |
| have hIp : conditionalInformation μ P m =ᵐ[μ] fun y => I (X y,P y) := by | |
| filter_upwards [ae_all_iff.mpr heq] with y hy | |
| unfold conditionalInformation I observationInformation | |
| rw [hy (P y)] | |
| rfl | |
| have hIint : Integrable I σ := by | |
| have H := integrable_kernel_observationInformation σ.fst κ Prod.snd | |
| (fun a => measurable_snd (measurableSet_singleton a)) | |
| change Integrable I (σ.fst ⊗ₘ σ.condKernel) at H | |
| rwa [Measure.disintegrate] at H | |
| have hIf : Integrable (fun y => I (X y,P y)) μ := | |
| hIint.comp_measurable (hX.prodMk hP) | |
| refine ⟨hIf.congr hIp.symm,?_⟩ | |
| rw [integral_congr_ae hIp] | |
| have HI := conditionalEntropy_fst_information σ Prod.snd | |
| (fun a => measurable_snd (measurableSet_singleton a)) | |
| rw [conditionalSigmaEntropy_fst σ Prod.snd | |
| (fun a => measurable_snd (measurableSet_singleton a))] at HI | |
| have HE : conditionalSigmaEntropy μ P m = | |
| ∫ y, observationEntropy (κ (X y)) (id : Fin N → Fin N) ∂μ := by | |
| unfold conditionalSigmaEntropy observationEntropy | |
| apply integral_congr_ae | |
| filter_upwards [ae_all_iff.mpr heq] with y hy | |
| congr 1 | |
| funext a | |
| simpa only [id_eq,Set.ofPred_eq_eq_singleton] using hy a | |
| rw [HE] | |
| calc | |
| _ = ∫ z, I z ∂σ := | |
| (integral_map (hX.prodMk hP).aemeasurable hI.aestronglyMeasurable).symm | |
| _ = ∫ s, observationEntropy (κ s) (id : Fin N → Fin N) ∂σ.fst := HI.symm | |
| _ = ∫ z, observationEntropy (κ z.1) (id : Fin N → Fin N) ∂σ := | |
| integral_map measurable_fst.aemeasurable | |
| (measurable_fiber_observationEntropy κ Prod.snd | |
| (fun a => measurable_snd (measurableSet_singleton a))).aestronglyMeasurable | |
| _ = _ := integral_map (hX.prodMk hP).aemeasurable | |
| ((measurable_fiber_observationEntropy κ Prod.snd | |
| (fun a => measurable_snd (measurableSet_singleton a))).comp measurable_fst).aestronglyMeasurable | |
| end DukePrimeDegree | |
| end | |
| end OAI | |