How e-values generalize hypothesis testing: a Neyman-Pearson lemma for the e-value

While e-values are swiftly rising in prominence, their formal relationship to classical hypothesis testing remains unsettled. We develop a unified decision-theoretic framework by viewing the e-value as a multi-decision generalization of a hypothesis test. Replacing power by the expected utility of evidence, we derive a Neyman-Pearson lemma for e-values, with classical Neyman-Pearson tests and log-optimal e-values arising from particular utility functions. Our main technical contribution is to study such expected utility-optimal e-values without assumptions on the composite null hypothesis and simple alternative. As a corollary, we obtain an assumption-free form of the classical Neyman-Pearson lemma for composite null hypotheses, showing power-optimal tests exist and have a likelihood-ratio form.