| import Mathlib |
| import OAI.Computability.VertexCover.Information.Finite |
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| namespace OAI |
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| namespace UniqueGames.Foundations.Information |
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|
| open scoped BigOperators |
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|
| variable {α β : Type*} [Fintype α] [Fintype β] |
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| theorem totalVariation_joint_common_weights (w : α → ℝ) (p q : α → β → ℝ) |
| (hw : ∀ a, 0 ≤ w a) : |
| totalVariation (fun ab : α × β => w ab.1 * p ab.1 ab.2) |
| (fun ab : α × β => w ab.1 * q ab.1 ab.2) = |
| ∑ a, w a * totalVariation (p a) (q a) := by |
| have hpoint : ∀ a b, |w a * p a b - w a * q a b| = w a * |p a b - q a b| := by |
| intro a b |
| rw [← mul_sub, abs_mul, abs_of_nonneg (hw a)] |
| simp only [totalVariation, Fintype.sum_prod_type, hpoint, ← Finset.mul_sum, |
| div_eq_mul_inv, Finset.sum_mul, mul_assoc] |
|
|
| theorem totalVariation_joint_common_kernel (p q : α → ℝ) (k : α → β → ℝ) |
| (hk : ∀ a, IsProbability (k a)) : |
| totalVariation (fun ab : α × β => p ab.1 * k ab.1 ab.2) |
| (fun ab : α × β => q ab.1 * k ab.1 ab.2) = totalVariation p q := by |
| have hpoint : ∀ a b, |p a * k a b - q a * k a b| = |p a - q a| * k a b := by |
| intro a b |
| rw [← sub_mul, abs_mul, abs_of_nonneg ((hk a).1 b)] |
| simp only [totalVariation, Fintype.sum_prod_type, hpoint, ← Finset.mul_sum, |
| (hk _).2, mul_one] |
|
|
| theorem totalVariation_condition_le (p q : α → ℝ) (event : α → Prop) |
| [DecidablePred event] {«c» : ℝ} (hc : 0 < «c») : |
| totalVariation (fun a => if event a then p a / «c» else 0) |
| (fun a => if event a then q a / «c» else 0) ≤ totalVariation p q / «c» := by |
| have hpoint : ∀ a, |(if event a then p a / «c» else 0) - |
| (if event a then q a / «c» else 0)| ≤ |p a - q a| / «c» := by |
| intro a |
| by_cases he : event a |
| · simp only [ite_eq_left he, ← sub_div, abs_div, abs_of_pos hc, le_refl] |
| · simp only [ite_eq_right he, sub_self, abs_zero] |
| exact div_nonneg (abs_nonneg _) hc.le |
| have hsum := Finset.sum_le_sum (fun a (_ : a ∈ (Finset.univ : Finset α)) => hpoint a) |
| simp only [div_eq_mul_inv, ← Finset.sum_mul] at hsum |
| have hd := div_le_div_of_nonneg_right hsum (show (0 : ℝ) ≤ 2 by norm_num) |
| have hreorder : ((∑ a, |p a - q a|) * «c»⁻¹) / 2 = |
| ((∑ a, |p a - q a|) / 2) / «c» := by ring |
| rw [hreorder] at hd |
| simpa only [totalVariation, div_eq_mul_inv] using hd |
|
|
| theorem weighted_coordinate_sum_sq_le |
| (w : α → ℝ) (d : α → β → ℝ) (C : α → ℝ) |
| (hw : IsProbability w) |
| (hbudget : ∀ a, w a * (∑ b, d a b ^ 2) ≤ w a * C a) : |
| (∑ b, ∑ a, w a * d a b)^2 ≤ |
| (Fintype.card β : ℝ) * (∑ a, w a * C a) := by |
| have hv := Finset.sum_le_sum |
| (fun b (_ : b ∈ (Finset.univ : Finset β)) => |
| weighted_sum_sq_le w (fun a => d a b) hw) |
| have he : |
| (∑ b, ∑ a, w a * d a b ^ 2) = |
| ∑ a, w a * (∑ b, d a b ^ 2) := by |
| rw [Finset.sum_comm] |
| simp only [Finset.mul_sum] |
| rw [he] at hv |
| have hb := Finset.sum_le_sum |
| (fun a (_ : a ∈ (Finset.univ : Finset α)) => hbudget a) |
| have hs := hv.trans hb |
| have hcs := Finset.sum_mul_sq_le_sq_mul_sq |
| (Finset.univ : Finset β) |
| (fun b => ∑ a, w a * d a b) (fun _ => (1 : ℝ)) |
| simp only [mul_one, one_pow, Finset.sum_const, |
| Finset.card_univ, nsmul_eq_mul, mul_one] at hcs |
| calc |
| (∑ b, ∑ a, w a * d a b)^2 |
| ≤ (∑ b, (∑ a, w a * d a b)^2) * |
| (Fintype.card β : ℝ) := hcs |
| _ = (Fintype.card β : ℝ) * |
| (∑ b, (∑ a, w a * d a b)^2) := mul_comm _ _ |
| _ ≤ (Fintype.card β : ℝ) * (∑ a, w a * C a) := |
| mul_le_mul_of_nonneg_left hs (Nat.cast_nonneg _) |
|
|
| end UniqueGames.Foundations.Information |
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| end |
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| end |
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| end |
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|
| end OAI |
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|
