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arXiv · 2604.00337

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

Nicholas G. Polson·Daniel Zantedeschi
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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).

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Nicholas G. Polson, Daniel Zantedeschi. 2026-04-01. E-Values, Bayes Risk, Dual Role of Markov's Inequality. https://arxiv.org/abs/2604.00337

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