E-Values, Bayes Risk, Dual Role of Markov's Inequality

TL;DR

This paper explores the duality between e-value testing and Bayes risk minimization, highlighting the role of Markov's inequality and the symmetry in their error control mechanisms.

Contribution

It clarifies the role-reversal symmetry in hypothesis testing approaches and connects them to information-theoretic and evidence calculus frameworks.

Findings

E-value testing and Bayes risk minimization both use Markov's inequality for error control.

Likelihood ratios are e-values only relative to specific experiments.

The duality is situated within a broader evidence calculus framework.

Abstract

Two approaches to hypothesis testing, e-value testing and Bayes risk minimisation, both invoke Markov's inequality to control error probabilities. They differ in which distribution certifies the unit-moment condition: the null for Type I error, the alternative for Type II error. The likelihood ratio is not intrinsically an e-value; it acquires that status only relative to the experiment under which its expectation is certified. This note makes the resulting role-reversal symmetry explicit, traces its asymptotic sharpening through the information-theoretic arguments of Barron and Clarke (1994), and situates the duality within the typed evidence calculus of Polson, Sokolov, and Zantedeschi (2026).