File size: 6,263 Bytes
e320e82 | 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 | import OAI.Combinatorics.GraphThreshold.BernoulliCoupling
import OAI.Combinatorics.GraphThreshold.AdaptiveTrials
import OAI.Combinatorics.GraphThreshold.GraphProbability
namespace OAI
open scoped BigOperators
noncomputable section
attribute [local instance] Classical.propDecidable
namespace LeanBlast.SecondKahnKalai
variable {X : Type*} [Fintype X] [DecidableEq X]
theorem bernoulli_union_product (p q : ℝ) (f : X → Bool → ℝ) :
bernoulliExpectation p (fun U => bernoulliExpectation q
(fun V => ∏ x, f x (decide (x ∈ U ∪ V)))) =
∏ x, ((p + q - p * q) * f x true + (1 - (p + q - p * q)) * f x false) := by
have hi (U : Finset X) :
bernoulliExpectation q (fun V => ∏ x, f x (decide (x ∈ U ∪ V))) =
∏ x, (q * f x true + (1 - q) * f x (decide (x ∈ U))) := by
have hfun : (fun V : Finset X => ∏ x, f x (decide (x ∈ U ∪ V))) =
(fun V => ∏ x, f x (decide (x ∈ U) || decide (x ∈ V))) := by
funext V
apply Finset.prod_congr rfl
intro x _
by_cases hu : x ∈ U <;> by_cases hv : x ∈ V <;> simp [Finset.mem_union, hu, hv]
rw [hfun]
simpa using
bernoulliExpectation_product q (fun x b => f x (decide (x ∈ U) || b))
calc
_ = bernoulliExpectation p (fun U =>
∏ x, (q * f x true + (1 - q) * f x (decide (x ∈ U)))) :=
bernoulliExpectation_congr p hi
_ = _ := by
rw [bernoulliExpectation_product p (fun x b => q * f x true + (1 - q) * f x b)]
apply Finset.prod_congr rfl
intro x _
ring
theorem bernoulli_union_mass (p q : ℝ) (Y : Finset X) :
bernoulliExpectation p (fun U => bernoulliExpectation q
(fun V => if U ∪ V = Y then 1 else 0)) = bernoulliWeight (p + q - p * q) Y := by
let f : X → Bool → ℝ := fun x b => if (b = true ↔ x ∈ Y) then 1 else 0
have hi (U V : Finset X) :
(if U ∪ V = Y then (1 : ℝ) else 0) = ∏ x, f x (decide (x ∈ U ∪ V)) := by
simpa [f] using finset_eq_indicator_product (U ∪ V) Y
calc
_ = bernoulliExpectation p (fun U => bernoulliExpectation q
(fun V => ∏ x, f x (decide (x ∈ U ∪ V)))) :=
bernoulliExpectation_congr p fun U => bernoulliExpectation_congr q (hi U)
_ = _ := by
rw [bernoulli_union_product]
apply Finset.prod_congr rfl
intro x _
by_cases hy : x ∈ Y <;> simp [f, hy]
/-- The union of two independent Bernoulli subsets has the stated Bernoulli law. -/
theorem bernoulliExpectation_union (p q : ℝ) (F : Finset X → ℝ) :
bernoulliExpectation p (fun U => bernoulliExpectation q (fun V => F (U ∪ V))) =
bernoulliExpectation (p + q - p * q) F := by
calc
_ = bernoulliExpectation p (fun U => ∑ Y : Finset X,
bernoulliExpectation q (fun V => if U ∪ V = Y then 1 else 0) * F Y) :=
bernoulliExpectation_congr p fun U =>
bernoulliExpectation_pushforward q (fun V => U ∪ V) F
_ = ∑ Y : Finset X, bernoulliExpectation p (fun U =>
bernoulliExpectation q (fun V => if U ∪ V = Y then 1 else 0) * F Y) :=
bernoulliExpectation_sum p _
_ = ∑ Y : Finset X, bernoulliWeight (p + q - p * q) Y * F Y := by
apply Finset.sum_congr rfl
intro Y _
have hfactor : bernoulliExpectation p (fun U =>
bernoulliExpectation q (fun V => if U ∪ V = Y then 1 else 0) * F Y) =
bernoulliExpectation p (fun U =>
bernoulliExpectation q (fun V => if U ∪ V = Y then 1 else 0)) * F Y := by
simp only [bernoulliExpectation, Finset.sum_mul, mul_assoc]
rw [hfactor, bernoulli_union_mass]
_ = _ := rfl
omit [Fintype X] in
theorem biUnion_fin_cons {n : ℕ} (A : Finset X) (draws : Fin n → Finset X) :
Finset.univ.biUnion (Fin.cases A draws) = A ∪ Finset.univ.biUnion draws := by
ext x
simp only [Finset.mem_biUnion, Finset.mem_univ, true_and, Finset.mem_union]
constructor
· rintro ⟨i, hi⟩
revert hi
refine Fin.cases ?_ (fun j => ?_) i
· intro hi
exact Or.inl hi
· intro hi
exact Or.inr ⟨j, hi⟩
· rintro (hx | ⟨j, hj⟩)
· exact ⟨0, hx⟩
· exact ⟨j.succ, hj⟩
/-- Exact law of the union of a finite history of independent draws. -/
theorem iid_bernoulli_union_expectation (p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1)
(n : ℕ) (F : Finset X → ℝ) :
((bernoulliLaw p hp0 hp1).iid n).expectation (fun draws => F (Finset.univ.biUnion draws)) =
bernoulliExpectation (1 - (1 - p) ^ n) F := by
induction n generalizing F with
| zero => simp
| succ n ih =>
rw [FiniteLaw.iid_expectation_succ]
simp_rw [biUnion_fin_cons]
have hi (A : Finset X) := ih (fun W => F (A ∪ W))
have hrewrite : (fun A => ((bernoulliLaw p hp0 hp1).iid n).expectation
(fun draws => F (A ∪ Finset.univ.biUnion draws))) =
(fun A => bernoulliExpectation (1 - (1 - p) ^ n) (fun W => F (A ∪ W))) := by
funext A
exact hi A
rw [hrewrite, bernoulliLaw_expectation, bernoulliExpectation_union]
congr 1
rw [pow_succ]
ring
theorem iid_bernoulli_union_probability (p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1)
(n : ℕ) (P : Finset X → Prop) :
((bernoulliLaw p hp0 hp1).iid n).prob (fun draws => P (Finset.univ.biUnion draws)) =
bernoulliProbability (1 - (1 - p) ^ n) P := by
rw [FiniteLaw.prob_eq_expectation, bernoulliProbability_eq_expectation]
exact iid_bernoulli_union_expectation p hp0 hp1 n (fun W => if P W then 1 else 0)
theorem bernoulli_union_parameter_mem {p : ℝ} (hp0 : 0 ≤ p) (hp1 : p ≤ 1) (n : ℕ) :
0 ≤ 1 - (1 - p) ^ n ∧ 1 - (1 - p) ^ n ≤ 1 := by
have hpow0 : 0 ≤ (1 - p) ^ n := pow_nonneg (sub_nonneg.mpr hp1) _
have hpow1 : (1 - p) ^ n ≤ 1 := pow_le_one₀ (sub_nonneg.mpr hp1) (by linarith)
constructor <;> linarith
theorem bernoulli_union_parameter_le {p : ℝ} (hp0 : 0 ≤ p) (hp1 : p ≤ 1) (n : ℕ) :
1 - (1 - p) ^ n ≤ (n : ℝ) * p := by
induction n with
| zero => simp
| succ n ih =>
have hpow1 : (1 - p) ^ n ≤ 1 := pow_le_one₀ (sub_nonneg.mpr hp1) (by linarith)
have hmul := mul_le_mul_of_nonneg_left hpow1 hp0
rw [pow_succ, Nat.cast_succ]
nlinarith
end LeanBlast.SecondKahnKalai
end
end OAI
|