openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /HierarchicalBackgroundVariation.lean
| import OAI.Computability.PerfectCompleteness.Construction.HiddenBucketBias | |
| import OAI.Computability.PerfectCompleteness.Foundations.HierarchicalAgreementMean | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.HierarchicalBackgroundVariation | |
| open TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype) | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| attribute [local instance] linearMapFintype | |
| private theorem common_right_variation {A B : Type*} [Fintype A] [Fintype B] | |
| (P Q : FiniteDistribution A) (ν : FiniteDistribution B) : | |
| (P.product ν).totalVariation (Q.product ν) = P.totalVariation Q := by | |
| change (∑ z : A × B, | |
| |P.weight z.1 * ν.weight z.2 - Q.weight z.1 * ν.weight z.2|) / 2 = | |
| (∑ a : A, |P.weight a - Q.weight a|) / 2 | |
| rw [Fintype.sum_prod_type] | |
| simp_rw [← sub_mul, abs_mul, abs_of_nonneg (ν.nonnegative _), ← Finset.mul_sum, | |
| ν.normalized, mul_one] | |
| private theorem variation_triangle {A : Type*} [Fintype A] | |
| (P U Q : FiniteDistribution A) : | |
| P.totalVariation Q ≤ P.totalVariation U + U.totalVariation Q := by | |
| unfold FiniteDistribution.totalVariation | |
| have h := Finset.sum_le_sum (s := Finset.univ) | |
| (fun x _ => abs_sub_le (P.weight x) (U.weight x) (Q.weight x)) | |
| rw [Finset.sum_add_distrib] at h | |
| linarith | |
| variable {branch rows : Nat → Nat} {n t : Nat} | |
| (slots : RecursiveSpaces.Slots branch n → Fin t → MixedSupport.Slot) | |
| (upper : Nodes branch n) | |
| local instance backgroundFintype : | |
| Fintype (HierarchicalAgreementMean.Background (rows := rows) slots upper) := Fintype.ofFinite _ | |
| local instance rowSpaceFintype : Fintype (NodeEmbedding.RowSpace slots upper) := Fintype.ofFinite _ | |
| theorem reference_background | |
| (backgroundLaw : FiniteDistribution (HierarchicalAgreementMean.Background (rows := rows) slots upper)) | |
| (hrows : 0 < rows (Nodes.height upper)) : | |
| (HierarchicalAgreementMean.referenceLaw slots upper backgroundLaw hrows).pushforward Prod.fst = | |
| backgroundLaw := by | |
| unfold HierarchicalAgreementMean.referenceLaw | |
| rw [FiniteDistribution.pushforward_comp] | |
| change (backgroundLaw.product (HierarchicalAgreementMean.bucketLaw slots upper hrows)).pushforward | |
| Prod.fst = backgroundLaw | |
| exact HiddenBucketBias.product_fst _ _ | |
| theorem reference_variation | |
| (P Q : FiniteDistribution (HierarchicalAgreementMean.Background (rows := rows) slots upper)) | |
| (hrows : 0 < rows (Nodes.height upper)) : | |
| (HierarchicalAgreementMean.referenceLaw slots upper P hrows).totalVariation | |
| (HierarchicalAgreementMean.referenceLaw slots upper Q hrows) ≤ P.totalVariation Q := | |
| (DensityVariation.variation_pushforward_le | |
| (P.product (HierarchicalAgreementMean.bucketLaw slots upper hrows)) | |
| (Q.product (HierarchicalAgreementMean.bucketLaw slots upper hrows)) | |
| (HierarchicalAgreementMean.pairRecord slots upper)).trans_eq | |
| (common_right_variation P Q (HierarchicalAgreementMean.bucketLaw slots upper hrows)) | |
| theorem original_background_variation | |
| (P : FiniteDistribution (HierarchicalAgreementMean.PairRecord (rows := rows) slots upper)) | |
| (referenceBackground : FiniteDistribution | |
| (HierarchicalAgreementMean.Background (rows := rows) slots upper)) | |
| (hrows : 0 < rows (Nodes.height upper)) : | |
| P.totalVariation | |
| (HierarchicalAgreementMean.referenceLaw slots upper (P.pushforward Prod.fst) hrows) ≤ | |
| 2 * P.totalVariation | |
| (HierarchicalAgreementMean.referenceLaw slots upper referenceBackground hrows) := by | |
| have hmarginal := DensityVariation.variation_pushforward_le P | |
| (HierarchicalAgreementMean.referenceLaw slots upper referenceBackground hrows) Prod.fst | |
| rw [reference_background] at hmarginal | |
| have href := reference_variation slots upper (P.pushforward Prod.fst) referenceBackground hrows | |
| rw [FiniteDistribution.totalVariation_comm] at href | |
| have htriangle := variation_triangle P | |
| (HierarchicalAgreementMean.referenceLaw slots upper referenceBackground hrows) | |
| (HierarchicalAgreementMean.referenceLaw slots upper (P.pushforward Prod.fst) hrows) | |
| linarith | |
| theorem original_background_le_of_bound | |
| (P : FiniteDistribution (HierarchicalAgreementMean.PairRecord (rows := rows) slots upper)) | |
| (referenceBackground : FiniteDistribution | |
| (HierarchicalAgreementMean.Background (rows := rows) slots upper)) | |
| (hrows : 0 < rows (Nodes.height upper)) {error : ℝ} | |
| (herror : P.totalVariation | |
| (HierarchicalAgreementMean.referenceLaw slots upper referenceBackground hrows) ≤ error) : | |
| P.totalVariation | |
| (HierarchicalAgreementMean.referenceLaw slots upper (P.pushforward Prod.fst) hrows) ≤ | |
| 2 * error := by | |
| have h := original_background_variation slots upper P referenceBackground hrows | |
| linarith | |
| end | |
| end PerfectCompleteness.HierarchicalBackgroundVariation | |
| end | |
| end OAI | |