openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Entropy /CountableObservationInformation.lean
| import OAI.NumberTheory.DukePrimeDegree.Entropy.ObservationInformation | |
| import OAI.NumberTheory.DukePrimeDegree.Entropy.HeightObservationEntropy | |
| namespace OAI | |
| /-! # Finite clipping of a countable observation | |
| At a sampled point, clipping above its observed value leaves its atom | |
| unchanged. Positive atom mass also gives a common information bound. | |
| -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set Filter | |
| open scoped Topology | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets | |
| lemma heightClip_atom_eq_of_lt {Y : Type*} (Q : Y → ℕ) (y : Y) | |
| {N : ℕ} (hN : Q y < N) : | |
| {z | heightClip Q N z = heightClip Q N y} = {z | Q z = Q y} := by | |
| ext z | |
| simp only [mem_ofPred_eq, heightClip, Fin.mk.injEq] | |
| omega | |
| lemma observationInformation_heightClip_eventually {Y : Type*} [MeasurableSpace Y] | |
| (ν : Measure Y) (Q : Y → ℕ) (y : Y) : | |
| ∀ᶠ N : ℕ in atTop, | |
| observationInformation ν (heightClip Q N) y = observationInformation ν Q y := by | |
| filter_upwards [eventually_gt_atTop (Q y)] with N hN | |
| unfold observationInformation | |
| rw [heightClip_atom_eq_of_lt Q y hN] | |
| lemma observationInformation_heightClip_le {Y : Type*} [MeasurableSpace Y] | |
| (ν : Measure Y) [IsProbabilityMeasure ν] (Q : Y → ℕ) (y : Y) | |
| (hpos : 0 < ν {z | Q z = Q y}) (N : ℕ) : | |
| observationInformation ν (heightClip Q N) y ≤ observationInformation ν Q y := by | |
| have hsub : {z | Q z = Q y} ⊆ {z | heightClip Q N z = heightClip Q N y} := by | |
| intro z hz | |
| exact congrArg (fun a => (⟨min a N, Nat.lt_succ_of_le (min_le_right a N)⟩ : Fin (N+1))) hz | |
| have hreal : 0 < ν.real {z | Q z = Q y} := | |
| ENNReal.toReal_pos hpos.ne' (measure_ne_top ν _) | |
| exact neg_le_neg (Real.log_le_log hreal (measureReal_mono hsub)) | |
| /-- Integrable countable information is recovered by its finite clips, | |
| without requiring finite Shannon entropy of the countable observation. -/ | |
| theorem integral_kernel_heightClip_information_tendsto | |
| {S V : Type*} [MeasurableSpace S] [MeasurableSpace V] | |
| (ρ : Measure (S × V)) [IsFiniteMeasure ρ] | |
| (κ : Kernel S V) [IsMarkovKernel κ] | |
| (Q : S × V → ℕ) (hQ : Measurable Q) | |
| (hpos : ∀ᵐ z ∂ρ, 0 < κ z.1 {v | Q (z.1,v) = Q z}) | |
| (hint : Integrable | |
| (fun z => observationInformation (κ z.1) (fun v => Q (z.1,v)) z.2) ρ) : | |
| Tendsto (fun N => ∫ z, | |
| observationInformation (κ z.1) (fun v => heightClip Q N (z.1,v)) z.2 ∂ρ) | |
| atTop (𝓝 (∫ z, observationInformation (κ z.1) (fun v => Q (z.1,v)) z.2 ∂ρ)) := by | |
| let I := fun z : S × V => observationInformation (κ z.1) (fun v => Q (z.1,v)) z.2 | |
| apply tendsto_integral_of_dominated_convergence I | |
| · intro N | |
| exact (measurable_kernel_observationInformation κ (heightClip Q N) | |
| (fun a => (heightClip_measurable hQ N) (measurableSet_singleton a))).aestronglyMeasurable | |
| · exact hint | |
| · intro N | |
| filter_upwards [hpos] with z hz | |
| rw [Real.norm_eq_abs, abs_of_nonneg (observationInformation_nonneg _ _ _)] | |
| exact observationInformation_heightClip_le (κ z.1) (fun v => Q (z.1,v)) z.2 hz N | |
| · apply ae_of_all | |
| intro z | |
| apply tendsto_const_nhds.congr' | |
| filter_upwards [observationInformation_heightClip_eventually | |
| (κ z.1) (fun v => Q (z.1,v)) z.2] with N hN | |
| exact hN.symm | |
| end DukePrimeDegree | |
| end | |
| end OAI | |