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Merging sequential e-values via martingales

Vladimir Vovk, Ruodu Wang

TL;DR

A class of e-value merging functions via martingales is described, and it is shown that all merging methods for sequential e-values are dominated by such a class.

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.

Merging sequential e-values via martingales

TL;DR

A class of e-value merging functions via martingales is described, and it is shown that all merging methods for sequential e-values are dominated by such a class.

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.

Paper Structure

This paper contains 7 sections, 10 theorems, 50 equations, 1 figure.

Table of Contents

  1. Introduction
  2. E-variables
  3. Sequential and independent e-values
  4. Merging sequential e-values
  5. E-processes and anytime validity
  6. Merging independent e-values
  7. Conclusion

Key Result

Proposition 1

A function $F:[0,\infty)^K \to [0,\infty)$ is an ie-merging function if and only if $F(E_1,\dots,E_K)\le 1$ for all independent e-variables $E_1,\dots,E_K$ each taking at most two values.

Figures (1)

  • Figure 1: A few ways of constructing supermartingales from likelihood ratio. Left: one run. Right: the average of 1000 runs (with average taken on the log values).

Theorems & Definitions (25)

  • Proposition 1
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • ...and 15 more