The only admissible way of merging arbitrary e-values

TL;DR

This paper establishes that the only admissible method for merging arbitrary e-values is a weighted arithmetic average, extending previous results and employing advanced mathematical tools like optimal transport duality.

Contribution

It proves that weighted arithmetic averaging is the unique admissible merging method for e-values, generalizing prior symmetric merging results.

Findings

Weighted arithmetic average is the only admissible merging method for e-values.

The proof employs optimal transport duality and minimax theorem techniques.

Completes the theoretical understanding of e-value merging 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.