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 $p\llα$ vs $p\leq α$ has no (statistical) relevance for any downstream decision-making. Following recent work of Grünwald (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 $Γ$-admissibility, where $Γ$ is a set of adversaries which map the data to a significance level. A test is $Γ$-admissible if, roughly speaking, there is no other test which performs at least as well and sometimes better across all adversaries in $Γ$. For point nulls and alternatives, we prove general properties of any $Γ$-admissible test for any $Γ$ and show that they must be based on e-values. We also classify the set of admissible tests for various specific $Γ$.