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Marginal transfer of common affine-slice tests

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

proven

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    Natural Language Statement

    Theorem

    A bounded marginal density transfers raw empirical variance to a quantitative comparison with the common reference mean. Near-certain empirical success therefore forces near-certain reference success, with the original density cost retained. This is the probability transfer used in Lemma 9.5.

    Concept map
    15 concepts; 1 descendant hidden
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 mean_comparison proven

    2 success_transfer proven

    Lean source view on GitHub

    1import Lax342547.EmpiricalVariance
    2
    3/-!
    4---
    5title: Marginal transfer of common affine-slice tests
    6type: theorem
    7---
    8A bounded marginal density transfers raw empirical variance to a quantitative
    9comparison with the common reference mean. Near-certain empirical success
    10therefore forces near-certain reference success, with the original density
    11cost retained. This is the probability transfer used in Lemma 9.5.
    12-/
    13
    14namespace Lax342547.CommonTestTransfer
    15open Lax342547.RelativeEntropy
    16open scoped BigOperators
    17
    18axiom mean_comparison {Ω : Type} [Fintype Ω] (ρ σ F : Ω → ℝ)
    19 (hσ : Probability σ) (L ε m : ℝ) (hL : 0 ≤ L)
    20 (hcap : ∀ o,σ o ≤ L*ρ o)
    21 (hvar : (∑ o,ρ o*(F o-m)^2) ≤ ε) :
    22 |(∑ o,σ o*F o)-m| ≤ Real.sqrt (L*ε)
    23
    24axiom success_transfer {Ω : Type} [Fintype Ω] (ρ σ F : Ω → ℝ)
    25 (hσ : Probability σ) (L ε m η : ℝ) (hL : 0 ≤ L)
    26 (hcap : ∀ o,σ o ≤ L*ρ o)
    27 (hvar : (∑ o,ρ o*(F o-m)^2) ≤ ε)
    28 (hsuccess : 1-η ≤ ∑ o,σ o*F o) :
    29 1-η-Real.sqrt (L*ε) ≤ m
    30
    31end Lax342547.CommonTestTransfer
    32
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