On admissibility in post-hoc hypothesis testing

Ben Chugg; Tyron Lardy; Aaditya Ramdas; Peter Grünwald

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arXiv · 2508.00770

On admissibility in post-hoc hypothesis testing

Ben Chugg·Tyron Lardy·Aaditya Ramdas·Peter Grünwald

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Abstract

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 $p= α- ε$ for tiny $ε> 0$ 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. We classify the set of $Γ$-admissible rules for various sets $Γ$, showing they must be based on e-values, and recover the Neyman-Pearson lemma when $Γ$ is the constant map.

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Ben Chugg, Tyron Lardy, Aaditya Ramdas, Peter Grünwald. 2026-01-19. On admissibility in post-hoc hypothesis testing. https://arxiv.org/abs/2508.00770

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