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d5b7741 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | import OAI.Combinatorics.ContingencyTables.Transport.ContextAdjacentTransport
import OAI.Combinatorics.ContingencyTables.Transport.FiniteExposureVariance
namespace OAI
/-! # Between-child variance for actual physical exposure contexts -/
namespace ContingencyTables.ExcessContextProfiles
open scoped BigOperators Classical
open SmallContextCoordinates SmallContextEnumeration SmallContextFibres
open PhysicalCompletionFibres LargePairEnumeration BalancedFamily BalancedBox
open IntegerWeightedTransport FirstPaperProfiles FirstPaperContextTransport
open FirstPaperPhysicalMarginal PaddedCompletions CompletionCounts
universe u
lemma graphEnergy_mono {C : Type*} [DecidableEq C]
(V W : Finset (C → ℕ)) (f H : (C → ℕ) → ℝ)
(hVW : V ⊆ W) (hf : ∀ x, 0 ≤ f x) :
integerGraphEnergy V f H ≤ integerGraphEnergy W f H := by
unfold integerGraphEnergy
apply div_le_div_of_nonneg_right _ (by norm_num)
apply Finset.sum_le_sum_of_subset_of_nonneg
· intro e he
obtain ⟨he,hadj⟩ := Finset.mem_filter.mp he
obtain ⟨hx,hy⟩ := Finset.mem_product.mp he
exact Finset.mem_filter.mpr ⟨Finset.mem_product.mpr ⟨hVW hx,hVW hy⟩,hadj⟩
· intro e _ _
exact mul_nonneg (le_min (hf _) (hf _)) (sq_nonneg _)
noncomputable def contextVertices (U n : ℕ) :
Finset (Coordinates Bool (widths (fun _ : Fin n => U)) → ℕ) :=
Finset.univ.biUnion (fun l : Fin (U+1) => leafVertices _ (rootDisplay U l.val))
lemma leafVertices_subset_context (U n l : ℕ) (hl : l ≤ U) :
leafVertices (widths (fun _ : Fin n => U)) (rootDisplay U l) ⊆ contextVertices U n := by
intro x hx
exact Finset.mem_biUnion.mpr ⟨⟨l,Nat.lt_succ_of_le hl⟩,Finset.mem_univ _,hx⟩
variable {I J : Type u} [Fintype I] [Fintype J] [LinearOrder I] [LinearOrder J]
/-- The physical conditional masses and means satisfy the paper's
between-child estimate, allowing arbitrary zero-mass children. -/
theorem physical_child_variance (small : I → J → Bool) (B : I → J → ℕ)
(r : I → ℕ) (c : J → ℕ) (L : ℕ) (s : I×J) (hs : small s.1 s.2=true)
(σ : I×J → ℕ) (U : ℕ) (hU : 2 ≤ U)
(hB : ∀ i j, small i j=true → B i j=U)
(hlarge : ∀ i j, small i j=false → 2 ≤ B i j)
(hσ : ∀ a : Earlier small s, σ a.val.val ≤ U) {n : ℕ} (e : Fin n ≃ Later small s)
(hcap : ∀ T : ResidualTables small r c
(fun i => r i+rowLargeCount small i*L) (fun j => c j+columnLargeCount small j*L),
∀ i j, T.val.val i j ≤ B i j)
(H : (Coordinates Bool (widths (fun _ : Fin n => U)) → ℕ) → ℝ)
(hZ : 0 < ∑ j : Fin (U+1), eliminate (widths (fun _ : Fin n => U))
(contextHard small B r c L _ (contextEquiv small s hs e U) (prefixRaw small s σ U))
(childProfile U j.val)) :
let Ms := widths (fun _ : Fin n => U)
let f := contextHard small B r c L Ms (contextEquiv small s hs e U) (prefixRaw small s σ U)
let z := fun j => eliminate Ms f (childProfile U j)
let μ := fun j => eliminate Ms (fun p => f p*H p) (childProfile U j)/z j
let E := integerGraphEnergy ((contextVertices U n).filter
(fun x => InBox (bounds (fun _ : Bool => U) Ms) x ∧ (∑ i, x i)=U+Ms.sum)) f H
(∑ j : Fin (U+1), z j.val*(μ j.val-
(∑ k : Fin (U+1),z k.val*μ k.val)/(∑ k : Fin (U+1),z k.val))^2) ≤
((U:ℝ)+1)^4*(2+2*n*((U:ℝ)+1)^2)*E := by
dsimp only
let Ms := widths (fun _ : Fin n => U)
let f := contextHard small B r c L Ms (contextEquiv small s hs e U) (prefixRaw small s σ U)
let z := fun j => eliminate Ms f (childProfile U j)
let μ := fun j => eliminate Ms (fun p => f p*H p) (childProfile U j)/z j
let V := (contextVertices U n).filter
(fun x => InBox (bounds (fun _ : Bool => U) Ms) x ∧ (∑ i, x i)=U+Ms.sum)
have hf : ∀ x, 0 ≤ f x := contextHard_nonnegative small B r c L Ms _ _
have hz (j : ℕ) : 0 ≤ z j := eliminate_nonnegative Ms f hf _
have hE : 0 ≤ integerGraphEnergy V f H := graphEnergy_nonnegative V f H hf
apply child_variance_bound z μ U (2+2*n*((U:ℝ)+1)^2) (integerGraphEnergy V f H)
(fun j _ => hz j) hZ (by positivity) hE
· intro a j b haj hjb hbU
exact context_child_minimum small B r c L Ms (contextEquiv small s hs e U)
(prefixRaw small s σ U) (fun _ => U) U (prefixRaw_inBox small s σ U hσ) hlarge
(fun w hw => by rw [widths_constant_mem U w hw]; exact hU) a j b haj hjb hbU
· intro j hj
by_cases hl : 0 < z j
· by_cases hr : 0 < z (j+1)
· have htransport := adjacent_context_transport small B r c L s hs σ U (j+1) hU
(by omega) (by omega) hB hlarge hσ e hcap H (by simpa only [Nat.add_sub_cancel] using hl) hr
dsimp only at htransport
simp only [Nat.add_sub_cancel] at htransport
have hsub : (leafVertices Ms (rootDisplay U (j+1))).filter
(fun x => InBox (bounds (fun _ : Bool => U) Ms) x ∧ (∑ i, x i)=U+Ms.sum) ⊆ V := by
intro x hx
obtain ⟨hx,hbox⟩ := Finset.mem_filter.mp hx
exact Finset.mem_filter.mpr ⟨leafVertices_subset_context U n (j+1) (by omega) hx,hbox⟩
exact htransport.trans (mul_le_mul_of_nonneg_left (graphEnergy_mono _ V f H hsub hf) (by positivity))
· have hr0 : z (j+1)=0 := le_antisymm (le_of_not_gt hr) (hz (j+1))
rw [hr0,min_eq_right (hz j),zero_mul]
positivity
· have hl0 : z j=0 := le_antisymm (le_of_not_gt hl) (hz j)
rw [hl0,min_eq_left (hz (j+1)),zero_mul]
positivity
end ContingencyTables.ExcessContextProfiles
end OAI
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