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

Nicholas G. Polson, Daniel Zantedeschi

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).

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

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).

Paper Structure

This paper contains 7 sections, 6 equations, 2 tables.

Table of Contents

  1. The Duality at a Glance
  2. Composite Alternatives and Mixtures
  3. The marginal likelihood as an e-value
  4. Barron--Clarke asymptotics: pointwise sharpening
  5. Three levels of the same skeleton
  6. Information-theoretic interpretation
  7. Closing Remarks

Theorems & Definitions (3)

  • Remark 1: Bayes risk threshold and the Fubini decomposition
  • Remark 2: Asymmetry of the certification burden
  • Remark 3: Dawid's supermartingale and pathwise optimality