| import OAI.Analysis.FiniteFactor.Coupling.FiniteDisintegration |
| import OAI.Analysis.FiniteFactor.BiasedPairs.PrefixLimit |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory ProbabilityTheory CouplingParameters |
| open scoped ENNReal |
| noncomputable section |
|
|
| def sourceDiagonalHaarKernel : Kernel ℝ ℝ where |
| toFun := diagonalHaarReference |
| measurable' := by |
| apply Measure.measurable_of_measurable_coe |
| intro S hS |
| simp only [diagonalHaarReference,Measure.smul_apply,Measure.add_apply,smul_eq_mul] |
| exact measurable_const.mul (((Measure.measurable_coe hS).comp Measure.measurable_dirac).add measurable_const) |
| |
| instance sourceDiagonalHaarKernel_markov : IsMarkovKernel sourceDiagonalHaarKernel := |
| ⟨fun t => diagonalHaarReference_probability t⟩ |
| |
| / |
| def sourceFiniteReferenceKernel (F : Finset SourceCoordinate) : |
| Kernel (SourceCoordinate → ℝ) (SourceCoordinate → ℝ) := by |
| letI : ∀ i, IsMarkovKernel (if i ∈ F then sourceDiagonalHaarKernel else Kernel.id) := |
| fun i => by split_ifs <;> infer_instance |
| exact independentKernel (fun i => if i ∈ F then sourceDiagonalHaarKernel else Kernel.id) |
| |
| instance sourceFiniteReferenceKernel_markov (F : Finset SourceCoordinate) : |
| IsMarkovKernel (sourceFiniteReferenceKernel F) := by |
| unfold sourceFiniteReferenceKernel |
| infer_instance |
| |
| / |
| theorem sourceMixedPosterior_ac (j : ℕ) (F : Finset SourceCoordinate) (x : SourceCoordinate → ℝ) : |
| mixedPosteriorKernel (fun i : SourceCoordinate => stagedCouplingParameters j i.1) |
| (fun i => decide (i ∈ F)) x ≪ sourceFiniteReferenceKernel F x := by |
| have hh := vectorPosterior_finite_ac |
| (fun i : SourceCoordinate => stagedCouplingParameters j i.1) F x |
| rw [vectorPosterior_finite_map] at hh |
| convert hh using 1 <;> |
| dsimp [mixedPosteriorKernel,sourceFiniteReferenceKernel,independentKernel] <;> |
| congr 1 <;> funext i <;> by_cases hi : i ∈ F <;> |
| simp [hi,sourceDiagonalHaarKernel] <;> rfl |
| |
| / |
| The x marginal is replaced by Haar only through its proved density, so the |
| infinite diagonal tail is not assumed Haar-distributed. -/ |
| theorem sourceBiasedPrefix_ac_reference (j n : ℕ) : |
| sourceBiasedRealMeasure j (sourceBiasedPrefixMask n) ≪ |
| sourceSamplingHaar ⊗ₘ sourceFiniteReferenceKernel (blockPrefix sourceBlockSize n) := by |
| let p := fun i : SourceCoordinate => stagedCouplingParameters j i.1 |
| have he : (fun i : SourceCoordinate => decide (i ∈ blockPrefix sourceBlockSize n)) = |
| sourceBiasedPrefixMask n := by |
| funext i |
| simp [sourceBiasedPrefixMask,mem_blockPrefix] |
| have hdis := mixedPosterior_disintegration p (sourceBiasedPrefixMask n) |
| have hbase : (vectorCoupling p).map (vectorEndpoint (fun _ => false)) ≪ sourceSamplingHaar := by |
| obtain ⟨g,_,_,_,hg,_⟩ := sourceBiasedEndpoint_density j (fun _ => false) |
| change (sourceBiasedLaw j).map (sourceBiasedEndpoint (fun _ => false)) ≪ sourceSamplingHaar |
| rw [hg] |
| exact withDensity_absolutelyContinuous _ _ |
| have hcond : ∀ᵐ x ∂((vectorCoupling p).map (vectorEndpoint (fun _ => false))), |
| mixedPosteriorKernel p (sourceBiasedPrefixMask n) x ≪ |
| sourceFiniteReferenceKernel (blockPrefix sourceBlockSize n) x := by |
| apply Filter.Eventually.of_forall |
| intro x |
| rw [← he] |
| exact sourceMixedPosterior_ac j _ x |
| have hh := hbase.compProd hcond |
| rw [hdis] at hh |
| exact hh |
| |
| end |
| end FiniteFactor |
| |
| end OAI |
| |
