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Optimized combination of independent or simultaneous e-values

Jiahao Ming, Yi Shen, Ruodu Wang

Abstract

We show that a class of optimized e-value combinations, arising from a standard construction of e-processes, remains valid even when the tuning parameter is optimized based on the data. This result holds for independent e-values, and, more generally, for a new class called simultaneous e-variables, whose dependence structure lies between independence and sequential validity. We further propose an improved combination test for such e-values based on elementary symmetric polynomials.

Optimized combination of independent or simultaneous e-values

Abstract

We show that a class of optimized e-value combinations, arising from a standard construction of e-processes, remains valid even when the tuning parameter is optimized based on the data. This result holds for independent e-values, and, more generally, for a new class called simultaneous e-variables, whose dependence structure lies between independence and sequential validity. We further propose an improved combination test for such e-values based on elementary symmetric polynomials.
Paper Structure (4 sections, 3 theorems, 34 equations)

This paper contains 4 sections, 3 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Simultaneous e-variables
  3. Probability bounds for simultaneous e-variables
  4. Tests with simultaneous e-values

Key Result

Proposition 1

Suppose that $E_1,\dots,E_n$ are conditionally independent and conditionally valid e-variables on a common factor $Z$. Then $E_1,\dots,E_n$ are simultaneous e-variables.

Theorems & Definitions (8)

  • Definition 1
  • Proposition 1
  • proof
  • Theorem 1: Optimized betting inequality
  • proof
  • Corollary 1
  • Remark 1
  • Example 1