openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Entropy /ConditioningFiniteLabels.lean
| import OAI.NumberTheory.PrimePackets.ConditionalFibers | |
| namespace OAI | |
| /-! # Conditioning a leaf kernel on a finite observed label | |
| Adding a finite label to the transverse coordinate restricts the actual | |
| conditional leaf to the corresponding label fiber and normalizes it. | |
| -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ENNReal | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets | |
| variable {S T : Type*} [MeasurableSpace S] [MeasurableSpace T] | |
| [StandardBorelSpace T] [Nonempty T] | |
| def finiteLabelLift {N : ℕ} (Q : S × T → Fin N) (z : S × T) : (S × Fin N) × T := | |
| ((z.1,Q z),z.2) | |
| omit [StandardBorelSpace T] [Nonempty T] in | |
| lemma finiteLabelLift_measurable {N : ℕ} {Q : S × T → Fin N} (hQ : Measurable Q) : | |
| Measurable (finiteLabelLift Q) := | |
| (measurable_fst.prodMk hQ).prodMk measurable_snd | |
| omit [StandardBorelSpace T] [Nonempty T] in | |
| lemma finiteLabelLift_restrict {N : ℕ} (ρ : Measure (S × T)) | |
| {Q : S × T → Fin N} (hQ : Measurable Q) (b : Fin N) : | |
| (ρ.map (finiteLabelLift Q)).restrict ({s : S × Fin N | s.2=b} ×ˢ univ) = | |
| (ρ.restrict {z | Q z=b}).map (fun z : S × T => ((z.1,b),z.2)) := by | |
| rw [Measure.restrict_map (finiteLabelLift_measurable hQ) | |
| ((measurableSet_eq_fun measurable_snd measurable_const).prod MeasurableSet.univ)] | |
| have he : finiteLabelLift Q ⁻¹' ({s : S × Fin N | s.2=b} ×ˢ univ) = {z | Q z=b} := by | |
| ext z | |
| simp [finiteLabelLift] | |
| rw [he] | |
| apply Measure.map_congr | |
| filter_upwards [ae_restrict_mem (measurableSet_eq_fun hQ measurable_const)] with z hz | |
| simp only [finiteLabelLift,hz] | |
| theorem finiteLabelLift_condKernel_on_label {N : ℕ} | |
| (ρ : Measure (S × T)) [IsFiniteMeasure ρ] | |
| {Q : S × T → Fin N} (hQ : Measurable Q) (b : Fin N) : | |
| ∀ᵐ s ∂(ρ.restrict {z | Q z=b}).fst, | |
| (ρ.map (finiteLabelLift Q)).condKernel (s,b) = | |
| ((ρ.restrict {z | Q z=b}).condKernel s) := by | |
| let σ := ρ.map (finiteLabelLift Q) | |
| let U : Set (S × Fin N) := {s | s.2=b} | |
| let ρb := ρ.restrict {z | Q z=b} | |
| have hU : MeasurableSet U := measurableSet_eq_fun measurable_snd measurable_const | |
| have hs : σ.restrict (U ×ˢ univ) = | |
| ρb.map (fun z : S × T => ((z.1,b),z.2)) := finiteLabelLift_restrict ρ hQ b | |
| have hm : (σ.restrict (U ×ˢ univ)).fst = σ.fst.restrict U := by | |
| rw [restricted_conditional_marginal σ hU MeasurableSet.univ] | |
| simp only [measure_univ,withDensity_const,one_smul] | |
| have hmap : (σ.restrict (U ×ˢ univ)).fst = ρb.fst.map (fun s : S => (s,b)) := by | |
| rw [hs] | |
| exact fst_map_fibers_of_measurable ρb | |
| (fun s : S => (s,b)) (measurable_id.prodMk measurable_const) Prod.snd measurable_snd | |
| have hrestr := restricted_conditional_kernel σ hU MeasurableSet.univ | |
| simp only [measure_univ,one_smul,Measure.restrict_univ] at hrestr | |
| rw [←hm,hmap] at hrestr | |
| have hrestr' := ae_of_ae_map (measurable_id.prodMk measurable_const).aemeasurable hrestr | |
| have hfixed := conditional_map_fibers_leftInverse ρb | |
| (fun s : S => (s,b)) (measurable_id.prodMk measurable_const) | |
| Prod.fst measurable_fst (fun _ => rfl) Prod.snd measurable_snd | |
| filter_upwards [hrestr',hfixed] with s hs' hf | |
| simp only [hs,id_eq] at hs' | |
| change _ = (ρb.condKernel s).map id at hf | |
| rw [Measure.map_id] at hf | |
| exact hs'.symm.trans hf | |
| theorem finiteLabelLift_condKernel {N : ℕ} | |
| (ρ : Measure (S × T)) [IsFiniteMeasure ρ] | |
| {Q : S × T → Fin N} (hQ : Measurable Q) : | |
| ∀ᵐ z ∂ρ, | |
| ρ.condKernel z.1 {t | Q (z.1,t)=Q z} ≠ 0 ∧ | |
| (ρ.map (finiteLabelLift Q)).condKernel (z.1,Q z) = | |
| (anchorNormalize (ρ.condKernel z.1) {t | Q (z.1,t)=Q z}).restrict | |
| {t | Q (z.1,t)=Q z} := by | |
| have H (b : Fin N) : ∀ᵐ z ∂ρ.restrict {z | Q z=b}, | |
| ρ.condKernel z.1 {t | Q (z.1,t)=b} ≠ 0 ∧ | |
| (ρ.map (finiteLabelLift Q)).condKernel (z.1,b) = | |
| (anchorNormalize (ρ.condKernel z.1) {t | Q (z.1,t)=b}).restrict | |
| {t | Q (z.1,t)=b} := by | |
| have hb := (finiteLabelLift_condKernel_on_label ρ hQ b).and | |
| (conditional_restrict_normalized ρ {z | Q z=b} (measurableSet_eq_fun hQ measurable_const)) | |
| have hb' := ae_of_ae_map measurable_fst.aemeasurable hb | |
| filter_upwards [hb'] with z hz | |
| exact ⟨hz.2.1,hz.1.trans hz.2.2⟩ | |
| have H' (b : Fin N) : ∀ᵐ z ∂ρ, Q z=b → | |
| ρ.condKernel z.1 {t | Q (z.1,t)=b} ≠ 0 ∧ | |
| (ρ.map (finiteLabelLift Q)).condKernel (z.1,b) = | |
| (anchorNormalize (ρ.condKernel z.1) {t | Q (z.1,t)=b}).restrict | |
| {t | Q (z.1,t)=b} := | |
| (ae_restrict_iff' (measurableSet_eq_fun hQ measurable_const)).mp (H b) | |
| filter_upwards [ae_all_iff.mpr H'] with z hz | |
| exact hz (Q z) rfl | |
| end DukePrimeDegree | |
| end | |
| end OAI | |