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The only admissible way of merging arbitrary e-values

Ruodu Wang

TL;DR

This paper addresses the problem of characterizing admissible ways to merge multiple e-values under arbitrary dependence. It develops a maximin / optimal transport duality framework to show that any admissible e-merging function F must be bounded above by a weighted average M_lambda for some lambda in $Delta_{K+1}$, and F is admissible if and only if F equals M_lambda. Consequently, the weighted arithmetic averages (with appropriate weights) are the unique admissible e-merging rules for arbitrary dependence, extending prior symmetric results to the general setting. The findings provide a complete, dependence-robust justification for using weighted averages in e-value merging and have implications for constructing e-processes and related p-value methods.

Abstract

We prove that the only admissible way of merging arbitrary e-values is to use a weighted arithmetic average. This result completes the picture of merging methods for e-values, and generalizes the result of Vovk and Wang (2021, Annals of Statistics, 49(3), 1736--1754) that the only admissible way of symmetrically merging e-values is to use the arithmetic average combined with a constant. Although the proved statement is naturally anticipated, its proof relies on a sophisticated application of optimal transport duality and a minimax theorem.

The only admissible way of merging arbitrary e-values

TL;DR

This paper addresses the problem of characterizing admissible ways to merge multiple e-values under arbitrary dependence. It develops a maximin / optimal transport duality framework to show that any admissible e-merging function F must be bounded above by a weighted average M_lambda for some lambda in , and F is admissible if and only if F equals M_lambda. Consequently, the weighted arithmetic averages (with appropriate weights) are the unique admissible e-merging rules for arbitrary dependence, extending prior symmetric results to the general setting. The findings provide a complete, dependence-robust justification for using weighted averages in e-value merging and have implications for constructing e-processes and related p-value methods.

Abstract

We prove that the only admissible way of merging arbitrary e-values is to use a weighted arithmetic average. This result completes the picture of merging methods for e-values, and generalizes the result of Vovk and Wang (2021, Annals of Statistics, 49(3), 1736--1754) that the only admissible way of symmetrically merging e-values is to use the arithmetic average combined with a constant. Although the proved statement is naturally anticipated, its proof relies on a sophisticated application of optimal transport duality and a minimax theorem.
Paper Structure (5 sections, 5 theorems, 19 equations)

This paper contains 5 sections, 5 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. The main result
  3. Proof of Theorem \ref{['th:admissible']}
  4. Proof of Lemma \ref{['lem:mean2']}
  5. Merging functions of subclasses of e-values

Key Result

Theorem 1

For a function $F:\mathbb{R}_+^K\to\mathbb{R}_+$,

Theorems & Definitions (9)

  • Theorem 1
  • Lemma 2
  • Remark 3
  • Lemma 4
  • proof
  • Lemma 5: Optimal transport duality
  • Lemma 6
  • proof : of Lemma \ref{['lem:sion2']}
  • proof