On admissibility in post-hoc hypothesis testing

The validity of classical hypothesis testing requires the significance level α be fixed before any statistical analysis takes place. This is a stringent requirement. For instance, it prohibits updating α during (or after) an experiment due to changing concern about the cost of false positives, or to reflect unexpectedly strong evidence against the null. Perhaps most disturbingly, witnessing a p-value pllα vs p= α- ε for tiny ε > 0 has no (statistical) relevance for any downstream decision-making. Following recent work of Gr"unwald (2024), we develop a theory of post-hoc hypothesis testing, enabling α to be chosen after seeing and analyzing the data. To study "good" post-hoc tests we introduce Gamma-admissibility, where Gamma is a set of adversaries which map the data to a significance level. We classify the set of Gamma-admissible rules for various sets Gamma, showing they must be based on e-values, and recover the Neyman-Pearson lemma when Gamma is the constant map.