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Finite empirical variance from a two-coefficient comparison

Lax342547.EmpiricalVariance · concepts/Lax342547/EmpiricalVariance.lean · lax-342547

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    Natural Language Statement

    Theorem

    The affine-slice second-moment step keeps the independent coefficient draws and the actual bad-pair probability. This finite estimate is also valid for arbitrary bounded common tests, without polynomial assumptions on them.

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    1import Lax342547.FiniteSampling
    2
    3/-!
    4---
    5title: Finite empirical variance from a two-coefficient comparison
    6type: theorem
    7---
    8The affine-slice second-moment step keeps the independent coefficient draws
    9and the actual bad-pair probability. This finite estimate is also valid for
    10arbitrary bounded common tests, without polynomial assumptions on them.
    11-/
    12
    13namespace Lax342547.EmpiricalVariance
    14open Lax342547.RelativeEntropy Lax342547.RetainedImages
    15open scoped BigOperators
    16
    17axiom empirical_variance {Ω A : Type} [Fintype Ω] [Fintype A]
    18 (ρ : Ω → ℝ) (κ : A → ℝ) (F : Ω → A → ℝ) (good : A → A → Prop)
    19 (m ε δ : ℝ) (hρ : Probability ρ) (hκ : Probability κ)
    20 (hf : ∀ ω a,|F ω a| ≤ 1) (hm : |m| ≤ 1) (hε : 0 ≤ ε)
    21 (hmean : ∀ a b,good a b → (∑ ω,ρ ω*F ω a) = m ∧ (∑ ω,ρ ω*F ω b) = m)
    22 (hpair : ∀ a b,good a b → |(∑ ω,ρ ω*F ω a*F ω b)-m^2| ≤ ε)
    23 (hbad : cellMass (fun ab : A × A => κ ab.1*κ ab.2) (fun ab => ¬ good ab.1 ab.2) ≤ δ) :
    24 (∑ ω,ρ ω*((∑ a,κ a*F ω a)-m)^2) ≤ ε+4*δ
    25
    26end Lax342547.EmpiricalVariance
    27
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