Merging sequential e-values via martingales

TL;DR

This paper introduces a martingale-based framework for merging sequential and independent e-values into a single e-value or e-process, providing a unified approach that dominates existing methods and encompasses all admissible e-process constructions.

Contribution

It develops a new class of e-value merging functions using martingales, advancing the theoretical understanding of e-value combination methods for sequential and independent data.

Findings

Martingale-based merging functions dominate existing methods for sequential e-values.

All admissible e-process construction methods can be derived from this martingale framework.

The approach extends to merging independent e-values with data reordering.

Abstract

We study the problem of merging sequential or independent e-values into one e-value or e-process. We describe a class of e-value merging functions via martingales and show that it dominates all merging methods for sequential e-values. All admissible methods for constructing e-processes can also be obtained in this way. In the case of merging independent e-values, the situation becomes much more complicated, and we provide a general class of such merging functions based on martingales applied to reordered data.