openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Entropy /CountableConditionalInformation.lean
| import OAI.NumberTheory.DukePrimeDegree.Entropy.CountableFutureInformation | |
| import OAI.NumberTheory.DukePrimeDegree.Entropy.CountableObservationInformation | |
| namespace OAI | |
| /-! # Information of the countable present given its future | |
| A binary recoding gives each individual conditional probability. Since | |
| there are only countably many present labels, all of them hold together | |
| outside one null set. | |
| -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets | |
| variable {S V Y : Type*} [MeasurableSpace S] [MeasurableSpace V] | |
| [StandardBorelSpace V] [Nonempty V] [MeasurableSpace Y] | |
| theorem conditionalProbability_countablePresentFutureFiber | |
| (ρ : Measure (S × V)) [IsFiniteMeasure ρ] | |
| {Q : Y → ℕ} (hQ : Measurable Q) {T : Y → Y} (hT : Measurable T) | |
| {F : S × V → Y} (hF : Measurable F) | |
| (hpos : ∀ᵐ z ∂ρ, ρ.condKernel z.1 (countableFutureFiber T Q F z) ≠ 0) (a : ℕ) : | |
| ρ[({z | Q (F z)=a}.indicator (fun _ => (1:ℝ))) | | |
| ⨆ m, transverseLabelSigma (fun z => countableFutureCode T Q m (F z))] =ᵐ[ρ] | |
| fun z => (ρ.condKernel z.1).real ({t | Q (F (z.1,t))=a} ∩ countableFutureFiber T Q F z) / | |
| (ρ.condKernel z.1).real (countableFutureFiber T Q F z) := by | |
| let P : Y → Fin 2 := fun y => if Q y = a then 0 else 1 | |
| have hP : Measurable P := measurable_const.piecewise | |
| (hQ (measurableSet_singleton a)) measurable_const | |
| have H := conditionalProbability_countableFutureFiber ρ hP hQ hT hF hpos (0 : Fin 2) | |
| have he (y : Y) : P y = 0 ↔ Q y = a := by simp [P] | |
| simpa only [he] using H | |
| theorem conditionalInformation_countablePresentFutureFiber | |
| (ρ : Measure (S × V)) [IsFiniteMeasure ρ] | |
| {Q : Y → ℕ} (hQ : Measurable Q) {T : Y → Y} (hT : Measurable T) | |
| {F : S × V → Y} (hF : Measurable F) | |
| (hpos : ∀ᵐ z ∂ρ, ρ.condKernel z.1 (countableFutureFiber T Q F z) ≠ 0) : | |
| conditionalInformation ρ (Q ∘ F) | |
| (⨆ m, transverseLabelSigma (fun z => countableFutureCode T Q m (F z))) =ᵐ[ρ] | |
| fun z => -Real.log | |
| ((ρ.condKernel z.1).real ({t | Q (F (z.1,t))=Q (F z)} ∩ countableFutureFiber T Q F z) / | |
| (ρ.condKernel z.1).real (countableFutureFiber T Q F z)) := by | |
| filter_upwards [ae_all_iff.mpr (fun a => | |
| conditionalProbability_countablePresentFutureFiber ρ hQ hT hF hpos a)] with z hz | |
| exact congrArg (-Real.log ·) (hz (Q (F z))) | |
| end DukePrimeDegree | |
| end | |
| end OAI | |