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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 | import OAI.Combinatorics.BooleanFunctions.Blocks
namespace OAI
open scoped BigOperators
noncomputable section
namespace SquareRootDegree
attribute [local instance] Classical.propDecidable
namespace FiniteMoments
variable {ι α : Type*} [Fintype ι] [Fintype α]
theorem average_normalized_weight (w : α → ℝ) (hw : average w ≠ 0) :
average (fun x => w x / average w) = 1 := by
simp only [div_eq_mul_inv,average_mul_const,mul_inv_cancel₀ hw]
theorem normalized_product_eval (r : ι → α → ℝ) (hr : ∀ i, average (r i) = 1)
(j : ι) (g : α → ℝ) :
average (fun x : ι → α => (∏ i, r i (x i))*g (x j)) = average (fun a => r j a*g a) := by
have he : (fun x : ι → α => (∏ i, r i (x i))*g (x j)) =
fun x => ∏ i, (if i = j then r i (x i)*g (x i) else r i (x i)) := by
funext x
have hi (i : ι) : (if i = j then r i (x i)*g (x i) else r i (x i)) =
r i (x i)*(if i = j then g (x i) else 1) := by split_ifs <;> simp
simp_rw [hi]
rw [Finset.prod_mul_distrib]
simp
rw [he,average_pi_prod (fun i a => if i = j then r i a*g a else r i a)]
have hf (i : ι) : average (fun a => if i = j then r i a*g a else r i a) =
if i = j then average (fun a => r i a*g a) else 1 := by split_ifs <;> simp_all
simp_rw [hf]
simp
theorem normalized_product_two_eval (r : ι → α → ℝ) (hr : ∀ i, average (r i) = 1)
(j k : ι) (hjk : j ≠ k) (g h : α → ℝ) :
average (fun x : ι → α => (∏ i, r i (x i))*g (x j)*h (x k)) =
average (fun a => r j a*g a) * average (fun a => r k a*h a) := by
have he : (fun x : ι → α => (∏ i, r i (x i))*g (x j)*h (x k)) =
fun x => ∏ i, r i (x i)*(if i = j then g (x i) else 1)*(if i = k then h (x i) else 1) := by
funext x
rw [Finset.prod_mul_distrib,Finset.prod_mul_distrib]
simp
rw [he,average_pi_prod (fun i a => r i a*(if i = j then g a else 1)*(if i = k then h a else 1))]
have hf (i : ι) : average (fun a => r i a*(if i = j then g a else 1)*(if i = k then h a else 1)) =
(if i = j then average (fun a => r j a*g a) else 1)*
(if i = k then average (fun a => r k a*h a) else 1) := by
by_cases hj : i = j
· subst i; simp [hjk]
by_cases hk : i = k
· subst i; simp [hj]
simp [hj,hk,hr]
simp_rw [hf]
rw [Finset.prod_mul_distrib]
simp
theorem normalized_product_sum (r : ι → α → ℝ) (hr : ∀ i, average (r i) = 1)
(T : ι → α → ℝ) :
average (fun x : ι → α => (∏ i, r i (x i))*(∑ i, T i (x i))) =
∑ i, average (fun a => r i a*T i a) := by
simp_rw [Finset.mul_sum]
rw [average_sum]
exact Finset.sum_congr rfl (fun _ _ => normalized_product_eval r hr _ _)
theorem normalized_product_centered_sq (r : ι → α → ℝ) (hr : ∀ i, average (r i) = 1)
(T : ι → α → ℝ) (hT : ∀ i, average (fun a => r i a*T i a) = 0) :
average (fun x : ι → α => (∏ i, r i (x i))*(∑ i, T i (x i))^2) =
∑ i, average (fun a => r i a*(T i a)^2) := by
have he : (fun x : ι → α => (∏ i, r i (x i))*(∑ i, T i (x i))^2) =
fun x => ∑ i, ∑ j, (∏ k, r k (x k))*T i (x i)*T j (x j) := by
funext x
simp only [sq,Finset.sum_mul,Finset.mul_sum]
rw [Finset.sum_comm]
simp only [mul_assoc]
rw [he,average_sum]
simp_rw [average_sum]
apply Finset.sum_congr rfl
intro i _
rw [Finset.sum_eq_single i]
· have he' : (fun x : ι → α => (∏ k, r k (x k))*T i (x i)*T i (x i)) =
fun x => (∏ k, r k (x k))*(T i (x i))^2 := by funext x; ring
rw [he',normalized_product_eval r hr i (fun a => (T i a)^2)]
· intro j _ hij
rw [normalized_product_two_eval r hr i j (Ne.symm hij),hT,hT,mul_zero]
· simp
theorem normalized_product_shifted_sq (r : ι → α → ℝ) (hr : ∀ i, average (r i) = 1)
(T : ι → α → ℝ) (μ : ι → ℝ) (hμ : ∀ i, average (fun a => r i a*T i a) = μ i)
(c : ℝ) :
average (fun x : ι → α => (∏ i, r i (x i))*(c+∑ i, T i (x i))^2) =
(c+∑ i, μ i)^2 + ∑ i, (average (fun a => r i a*(T i a)^2)-(μ i)^2) := by
have hc (i : ι) : average (fun a => r i a*(T i a-μ i)) = 0 := by
simp only [mul_sub,average_sub,average_mul_const,hμ,hr,one_mul,sub_self]
have hfirst := normalized_product_sum r hr (fun i a => T i a-μ i)
simp only [hc,Finset.sum_const_zero] at hfirst
have hsecond := normalized_product_centered_sq r hr (fun i a => T i a-μ i) hc
have hp : average (fun x : ι → α => ∏ i, r i (x i)) = 1 := by rw [average_pi_prod]; simp [hr]
have he : (fun x : ι → α => (∏ i, r i (x i))*(c+∑ i, T i (x i))^2) =
fun x => (∏ i, r i (x i))*(∑ i, (T i (x i)-μ i))^2 +
2*(c+∑ i, μ i)*((∏ i, r i (x i))*(∑ i, (T i (x i)-μ i))) +
(c+∑ i, μ i)^2*(∏ i, r i (x i)) := by
funext x; rw [Finset.sum_sub_distrib]; ring
rw [he,average_add,average_add,average_const_mul,average_const_mul,hfirst,hsecond,hp]
have hd (i : ι) : average (fun a => r i a*(T i a-μ i)^2) =
average (fun a => r i a*(T i a)^2)-(μ i)^2 := by
have he' : (fun a => r i a*(T i a-μ i)^2) =
fun a => r i a*(T i a)^2-2*μ i*(r i a*T i a)+(μ i)^2*r i a := by funext a; ring
rw [he',average_add,average_sub,average_const_mul,average_const_mul,hμ,hr]
ring
simp_rw [hd]
ring
end FiniteMoments
end SquareRootDegree
end
end OAI
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