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f0f05de | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 | import OAI.Combinatorics.BalancedRyser.Probability.PermutationSubsetLaw
import OAI.Combinatorics.BalancedRyser.Probability.UncoveredSubsetVariance
namespace OAI
open scoped BigOperators
namespace Balanced.ConditionalOrderVariance
open WithoutReplacement PermutationSubsetLaw UncoveredSampling
theorem variance_two_images_conditioned_target
{α β : Type*} [Fintype α] [DecidableEq α] [Fintype β] [DecidableEq β]
(I : Finset α) (J : Finset β) (i x : α) (hi : i ∉ I)
(g : Finset α → Finset β → ℝ) :
finiteVariance
((Finset.univ.filter (fun σ : Equiv.Perm α => σ i = x)) ×ˢ
(Finset.univ : Finset (Equiv.Perm β)))
(fun p => g (I.map p.1.toEmbedding) (J.map p.2.toEmbedding)) =
finiteVariance ((Finset.univ.erase x).powersetCard I.card ×ˢ
(Finset.univ : Finset β).powersetCard J.card) (fun p => g p.1 p.2) := by
unfold finiteVariance
simp_rw [Finset.expect_product]
rw [expect_two_images_conditioned_target I J i x hi g]
exact expect_two_images_conditioned_target I J i x hi
(fun S T => (g S T - (𝔼 A ∈ (Finset.univ.erase x).powersetCard I.card,
𝔼 B ∈ (Finset.univ : Finset β).powersetCard J.card, g A B))^2)
theorem conditioned_order_variance_le
{α β : Type*} [Fintype α] [DecidableEq α] [Fintype β] [DecidableEq β]
(I : Finset α) (J : Finset β) (i x : α) (hi : i ∉ I)
(g : Finset α → Finset β → ℝ) {c d : ℝ} (hc : 0 ≤ c) (hd : 0 ≤ d)
(hfirst : ∀ S : Finset α, S ⊆ Finset.univ.erase x → S.card = I.card →
∀ T : Finset β, T ⊆ Finset.univ → T.card = J.card →
∀ a ∈ S, ∀ b ∈ Finset.univ.erase x, b ∉ S →
|g S T - g (insert b (S.erase a)) T| ≤ c)
(hsecond : ∀ S : Finset α, S ⊆ Finset.univ.erase x → S.card = I.card →
∀ T : Finset β, T ⊆ Finset.univ → T.card = J.card →
∀ a ∈ T, ∀ b ∈ Finset.univ, b ∉ T →
|g S T - g S (insert b (T.erase a))| ≤ d) :
finiteVariance
((Finset.univ.filter (fun σ : Equiv.Perm α => σ i = x)) ×ˢ
(Finset.univ : Finset (Equiv.Perm β)))
(fun p => g (I.map p.1.toEmbedding) (J.map p.2.toEmbedding)) ≤
(I.card : ℝ) * c^2 + (J.card : ℝ) * d^2 := by
rw [variance_two_images_conditioned_target I J i x hi]
have hI : I ⊆ Finset.univ.erase i := by
intro a ha
exact Finset.mem_erase.mpr ⟨ne_of_mem_of_not_mem ha hi, Finset.mem_univ _⟩
have hm : I.card ≤ (Finset.univ.erase x).card := by
simpa using Finset.card_le_card hI
exact finiteVariance_powersetCard_product_le _ _ _ _ g hc hd hm
(Finset.card_le_card (Finset.subset_univ J)) hfirst hsecond
section Affine
variable {F : Type*} [Field F] [Fintype F] [DecidableEq F]
def populationLines (U : Finset (Line F)) (S : Finset U) : Finset (Line F) :=
S.map ⟨Subtype.val, Subtype.val_injective⟩
omit [Field F] [Fintype F] in
@[simp] theorem populationLines_insert (U : Finset (Line F)) (S : Finset U) (a : U) :
populationLines U (insert a S) = insert a.val (populationLines U S) := by
simp [populationLines]
omit [Field F] [Fintype F] in
@[simp] theorem populationLines_erase (U : Finset (Line F)) (S : Finset U) (a : U) :
populationLines U (S.erase a) = (populationLines U S).erase a.val := by
simp [populationLines, Finset.map_erase]
omit [Field F] [Fintype F] [DecidableEq F] in
theorem mem_populationLines (U : Finset (Line F)) (S : Finset U) {a : U}
(ha : a ∈ S) : a.val ∈ populationLines U S :=
Finset.mem_map.mpr ⟨a, ha, rfl⟩
theorem plane_conditioned_order_variance_le
(U V : Finset (Line F)) (I : Finset U) (J : Finset V)
(i x : U) (hi : i ∉ I) :
finiteVariance
((Finset.univ.filter (fun σ : Equiv.Perm U => σ i = x)) ×ˢ
(Finset.univ : Finset (Equiv.Perm V)))
(fun p => planeFraction
(populationLines U (I.map p.1.toEmbedding))
(populationLines V (J.map p.2.toEmbedding))) ≤
4 * ((I.card : ℝ) + J.card) / (Fintype.card F : ℝ)^2 := by
have hc : 0 ≤ 2 / (Fintype.card F : ℝ) := by positivity
have h := conditioned_order_variance_le I J i x hi
(fun S T => planeFraction (populationLines U S) (populationLines V T)) hc hc
(by
intro S _ _ T _ _ a ha b _ _
simpa using planeFraction_exchange_left (populationLines U S)
(populationLines V T) (mem_populationLines U S ha) b.val)
(by
intro S _ _ T _ _ a ha b _ _
simpa using planeFraction_exchange_right (populationLines U S)
(populationLines V T) (mem_populationLines V T ha) b.val)
exact h.trans_eq (by ring)
theorem target_conditioned_order_variance_le
(U V : Finset (Line F)) (hUV : Disjoint U V)
(I : Finset U) (J : Finset V) (i x : U) (hi : i ∉ I) :
finiteVariance
((Finset.univ.filter (fun σ : Equiv.Perm U => σ i = x)) ×ˢ
(Finset.univ : Finset (Equiv.Perm V)))
(fun p => targetFraction x.val
(populationLines U (I.map p.1.toEmbedding))
(populationLines V (J.map p.2.toEmbedding))) ≤
4 * ((I.card : ℝ) + J.card) / (Fintype.card F : ℝ)^2 := by
have hc : 0 ≤ 2 / (Fintype.card F : ℝ) := by positivity
have hxV (a : V) : x.val ≠ a.val := by
intro h
exact Finset.disjoint_left.mp hUV x.property (h ▸ a.property)
have h := conditioned_order_variance_le I J i x hi
(fun S T => targetFraction x.val (populationLines U S) (populationLines V T)) hc hc
(by
intro S hS _ T _ _ a ha b hb _
have hxa : x.val ≠ a.val := by
intro h
exact (Finset.mem_erase.mp (hS ha)).1 (Subtype.ext h.symm)
have hxb : x.val ≠ b.val := by
intro h
exact (Finset.mem_erase.mp hb).1 (Subtype.ext h.symm)
simpa using targetFraction_exchange_left x.val (populationLines U S)
(populationLines V T) (mem_populationLines U S ha) b.val hxa hxb)
(by
intro S _ _ T _ _ a ha b _ _
simpa using targetFraction_exchange_right x.val (populationLines U S)
(populationLines V T) (mem_populationLines V T ha) b.val (hxV a) (hxV b))
exact h.trans_eq (by ring)
abbrev OrderPair (U V : Finset (Line F)) := Equiv.Perm U × Equiv.Perm V
noncomputable def conditionedOrderLaw (U V : Finset (Line F)) (i x : U) :
PMF (OrderPair U V) :=
PMF.uniformOfFinset
((Finset.univ.filter (fun σ : Equiv.Perm U => σ i = x)) ×ˢ
(Finset.univ : Finset (Equiv.Perm V)))
⟨(Equiv.swap i x, Equiv.refl V), by simp⟩
theorem plane_conditioned_order_measure_variance_le
(U V : Finset (Line F))
[MeasurableSpace (OrderPair U V)] [MeasurableSingletonClass (OrderPair U V)]
(I : Finset U) (J : Finset V) (i x : U) (hi : i ∉ I) :
ProbabilityTheory.variance
(fun p : OrderPair U V => planeFraction
(populationLines U (I.map p.1.toEmbedding))
(populationLines V (J.map p.2.toEmbedding)))
(conditionedOrderLaw U V i x).toMeasure ≤
4 * ((I.card : ℝ) + J.card) / (Fintype.card F : ℝ)^2 := by
erw [conditionedOrderLaw, variance_uniformOfFinset]
exact plane_conditioned_order_variance_le U V I J i x hi
theorem target_conditioned_order_measure_variance_le
(U V : Finset (Line F))
[MeasurableSpace (OrderPair U V)] [MeasurableSingletonClass (OrderPair U V)]
(hUV : Disjoint U V) (I : Finset U) (J : Finset V) (i x : U) (hi : i ∉ I) :
ProbabilityTheory.variance
(fun p : OrderPair U V => targetFraction x.val
(populationLines U (I.map p.1.toEmbedding))
(populationLines V (J.map p.2.toEmbedding)))
(conditionedOrderLaw U V i x).toMeasure ≤
4 * ((I.card : ℝ) + J.card) / (Fintype.card F : ℝ)^2 := by
erw [conditionedOrderLaw, variance_uniformOfFinset]
exact target_conditioned_order_variance_le U V hUV I J i x hi
end Affine
end Balanced.ConditionalOrderVariance
end OAI
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