| import OAI.Combinatorics.ContingencyTables.Rooted.ForestPenaltyApplication |
| import OAI.Combinatorics.ContingencyTables.Rooted.LargePairEnumeration |
| import OAI.Combinatorics.ContingencyTables.Transport.EqualityReindex |
|
|
| namespace OAI |
|
|
| /-! |
| # The actual companion large-cell sum has its removal signature |
|
|
| Small coordinates are the actual two views of small cells, and every |
| large cell is summed once at its physical balance width. All line and |
| unary factors have the manuscript's row/column and forest parameters. |
| -/ |
|
|
| namespace ContingencyTables.ForestLargeMarginal |
|
|
| open scoped BigOperators Classical |
| open SimpleGraph |
| open ForestPenaltyApplication LargePairEnumeration RowColumnPenalties BalancedBox |
|
|
| variable {I J : Type*} [Fintype I] [Fintype J] |
|
|
| noncomputable def largeMarginal (small : I×J → Bool) |
| (T : SimpleGraph (I⊕J)) (b : I → J → ℕ) (ell : I → J → ℤ) |
| (M : I → J → ℕ) (r : I → ℤ) (c : J → ℤ) (alpha : ℝ) |
| {n : ℕ} (e : Fin n ≃ LargeCell small) (z : SmallView small → ℕ) : ℝ := |
| ∑ a : ((k : Fin n) → Fin (M (e k).val.1 (e k).val.2+1)), |
| physicalWeight T b ell M r c alpha |
| (joinedProfile (fun k => M (e k).val.1 (e k).val.2) z a ∘ |
| (physicalJoinEquiv small e).symm) |
|
|
| theorem largeMarginal_nonnegative (small : I×J → Bool) |
| (T : SimpleGraph (I⊕J)) (b : I → J → ℕ) (ell : I → J → ℤ) |
| (M : I → J → ℕ) (r : I → ℤ) (c : J → ℤ) (alpha : ℝ) |
| {n : ℕ} (e : Fin n ≃ LargeCell small) (z : SmallView small → ℕ) : |
| 0≤largeMarginal small T b ell M r c alpha e z := by |
| apply Finset.sum_nonneg |
| intro a _ |
| exact physicalWeight_nonnegative T b ell M r c alpha _ |
|
|
| private theorem largeMarginal_eq_eliminate (small : I×J → Bool) |
| (T : SimpleGraph (I⊕J)) (b : I → J → ℕ) (ell : I → J → ℤ) |
| (M : I → J → ℕ) (r : I → ℤ) (c : J → ℤ) (alpha : ℝ) |
| {n : ℕ} (e : Fin n ≃ LargeCell small) : |
| largeMarginal small T b ell M r c alpha e= |
| BalancedFamily.eliminate (widths (fun k => M (e k).val.1 (e k).val.2)) (fun q => |
| physicalWeight T b ell M r c alpha |
| (q ∘ (physicalCoordinateEquiv small e (fun k => M (e k).val.1 (e k).val.2)).symm)) := by |
| funext z |
| exact (eliminate_eq_actual_cells small e _ (physicalWeight T b ell M r c alpha) z).symm |
|
|
| / |
| with the physical widths, retained small views, and coordinate map made |
| explicit. -/ |
| theorem largeMarginal_removal (small : I×J → Bool) |
| (T : SimpleGraph (I⊕J)) (b : I → J → ℕ) (ell : I → J → ℤ) |
| (M : I → J → ℕ) (r : I → ℤ) (c : J → ℤ) (alpha : ℝ) |
| {n : ℕ} (e : Fin n ≃ LargeCell small) (m : SmallView small → ℕ) (p : ℕ) |
| (ha : 0≤alpha) (hwidth : ∀ k, 2≤M (e k).val.1 (e k).val.2) : |
| RemovalSlice (largeMarginal small T b ell M r c alpha e) m p := by |
| let N : Fin n → ℕ := fun k => M (e k).val.1 (e k).val.2 |
| have hw : ∀ w ∈ widths N, 2≤w := widths_lower N hwidth |
| have hu : ∀ a, 0≤unaryRate T b alpha a := by |
| intro a |
| unfold unaryRate |
| split_ifs |
| · exact div_nonneg ha (Nat.cast_nonneg _) |
| · exact le_refl 0 |
| have hs := EqualityReindex.physical_marginal_removal (widths N) m p |
| (physicalCoordinateEquiv small e N) cellLine (rootFlag T) (fun _ => alpha) |
| (lineOffset ell M r c) viewSlope (unaryRate T b alpha) (unaryEll ell) unarySlope (unaryCap b) |
| hw (fun _ => ha) hu |
| rw [largeMarginal_eq_eliminate small T b ell M r c alpha e] |
|
|
| exact hs |
|
|
| / |
| the complementary factorial lift's actual addition input. -/ |
| theorem largeMarginal_reflected (small : I×J → Bool) |
| (T : SimpleGraph (I⊕J)) (b : I → J → ℕ) (ell : I → J → ℤ) |
| (M : I → J → ℕ) (r : I → ℤ) (c : J → ℤ) (alpha : ℝ) |
| {n : ℕ} (e : Fin n ≃ LargeCell small) (m : SmallView small → ℕ) (p : ℕ) |
| (ha : 0≤alpha) (hwidth : ∀ k, 2≤M (e k).val.1 (e k).val.2) : |
| RemovalSlice (fun z => largeMarginal small T b ell M r c alpha e |
| (IntegerReflection.reflect m z)) m p := by |
| let N : Fin n → ℕ := fun k => M (e k).val.1 (e k).val.2 |
| have hu : ∀ a, 0≤unaryRate T b alpha a := by |
| intro a |
| unfold unaryRate |
| split_ifs |
| · exact div_nonneg ha (Nat.cast_nonneg _) |
| · exact le_refl 0 |
| rw [largeMarginal_eq_eliminate small T b ell M r c alpha e] |
| exact EqualityReindex.physical_marginal_reflected (widths N) m p |
| (physicalCoordinateEquiv small e N) cellLine (rootFlag T) (fun _ => alpha) |
| (lineOffset ell M r c) viewSlope (unaryRate T b alpha) (unaryEll ell) unarySlope (unaryCap b) |
| (widths_lower N hwidth) (fun _ => ha) hu |
|
|
| end ContingencyTables.ForestLargeMarginal |
|
|
| end OAI |
|
|
