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2f887db | 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 | import Mathlib.Analysis.Real.Sqrt
import OAI.Computability.PerfectCompleteness.Foundations.RecursiveSpaceEquivLemmas
import OAI.Computability.UniqueGames.Foundations.MixtureLemmas
import OAI.Computability.UniqueGames.Foundations.ValueLemmas
namespace OAI
section
namespace PerfectCompleteness.DensityVariation
open scoped BigOperators
open UniqueGamesTheorem.Foundations.Games
open PerfectCompleteness.SmallBias
noncomputable section
variable {Ω Γ : Type*} [Fintype Ω] [Fintype Γ]
theorem expectation_sub (μ : FiniteDistribution Ω) (f g : Ω → ℝ) :
μ.expectation (fun x => f x - g x) = μ.expectation f - μ.expectation g := by
simp only [FiniteDistribution.expectation, mul_sub, Finset.sum_sub_distrib]
theorem expectation_mul_const (μ : FiniteDistribution Ω) (f : Ω → ℝ) (c : ℝ) :
μ.expectation (fun x => f x * c) = μ.expectation f * c := by
simp only [FiniteDistribution.expectation, ← mul_assoc, Finset.sum_mul]
theorem expectation_div (μ : FiniteDistribution Ω) (f : Ω → ℝ) (c : ℝ) :
μ.expectation (fun x => f x / c) = μ.expectation f / c := by
simpa only [div_eq_mul_inv] using expectation_mul_const μ f c⁻¹
def reweight (μ : FiniteDistribution Ω) (density : Ω → ℝ)
(nonnegative : ∀ x, 0 ≤ density x) (mean_one : μ.expectation density = 1) :
FiniteDistribution Ω where
weight x := μ.weight x * density x
nonnegative x := mul_nonneg (μ.nonnegative x) (nonnegative x)
normalized := mean_one
theorem variation_eq_density (μ ν : FiniteDistribution Ω) (density : Ω → ℝ)
(h : ∀ x, ν.weight x = μ.weight x * density x) :
ν.totalVariation μ = μ.expectation (fun x => |density x - 1|) / 2 := by
unfold FiniteDistribution.totalVariation FiniteDistribution.expectation
congr 1
apply Finset.sum_congr rfl
intro x _
rw [h x, show μ.weight x * density x - μ.weight x =
μ.weight x * (density x - 1) by ring, abs_mul,
abs_of_nonneg (μ.nonnegative x)]
theorem variation_le_sqrt (μ ν : FiniteDistribution Ω) (density : Ω → ℝ)
(h : ∀ x, ν.weight x = μ.weight x * density x) (v : ℝ)
(hv : μ.expectation (fun x => (density x - 1) ^ 2) ≤ v) :
ν.totalVariation μ ≤ Real.sqrt v / 2 := by
rw [variation_eq_density μ ν density h]
apply div_le_div_of_nonneg_right _ (by norm_num : (0 : ℝ) ≤ 2)
apply Real.le_sqrt_of_sq_le
have hcs := UniqueGamesTheorem.Foundations.Information.weighted_sum_sq_le μ.weight
(fun x => |density x - 1|) ⟨μ.nonnegative, μ.normalized⟩
change μ.expectation (fun x => |density x - 1|) ^ 2 ≤
μ.expectation (fun x => |density x - 1| ^ 2) at hcs
simpa only [sq_abs] using hcs.trans (by simpa only [sq_abs] using hv)
theorem variation_pushforward_le (μ ν : FiniteDistribution Ω) (observe : Ω → Γ) :
(μ.pushforward observe).totalVariation (ν.pushforward observe) ≤
μ.totalVariation ν := by
classical
unfold FiniteDistribution.totalVariation
apply div_le_div_of_nonneg_right _ (by norm_num : (0 : ℝ) ≤ 2)
have hdiff (y : Γ) :
(μ.pushforward observe).weight y - (ν.pushforward observe).weight y =
∑ x, if observe x = y then μ.weight x - ν.weight x else 0 := by
simp only [FiniteDistribution.pushforward, ← Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro x _
by_cases hx : observe x = y <;> simp [hx]
calc
_ ≤ ∑ y, ∑ x, if observe x = y then |μ.weight x - ν.weight x| else 0 := by
apply Finset.sum_le_sum
intro y _
rw [hdiff]
calc
_ ≤ ∑ x, |if observe x = y then μ.weight x - ν.weight x else 0| :=
Finset.abs_sum_le_sum_abs _ _
_ = _ := by
apply Finset.sum_congr rfl
intro x _
split <;> simp
_ = _ := by rw [Finset.sum_comm]; simp
theorem observed_variation_le_sqrt (μ ν : FiniteDistribution Ω) (density : Ω → ℝ)
(h : ∀ x, ν.weight x = μ.weight x * density x) (v : ℝ)
(hv : μ.expectation (fun x => (density x - 1) ^ 2) ≤ v)
(observe : Ω → Γ) :
(ν.pushforward observe).totalVariation (μ.pushforward observe) ≤ Real.sqrt v / 2 :=
(variation_pushforward_le ν μ observe).trans (variation_le_sqrt μ ν density h v hv)
end
end PerfectCompleteness.DensityVariation
end
end OAI
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