Post-hoc $\alpha$ Hypothesis Testing and the Post-hoc $p$-value

An unfortunate feature of traditional hypothesis testing is the necessity to pre-specify a significance level $\alpha$ to bound the size of the test: its probability to falsely reject the hypothesis. Indeed, a data-dependent selection of $\alpha$ would generally distort the size, possibly making it larger than the selected level $\alpha$. We develop post-hoc $\alpha$ hypothesis testing, which guarantees that there is no such size distortion in expectation, even if the level $\alpha$ is arbitrarily selected based on the data. Unlike regular $p$-values, resulting 'post-hoc $p$-values' allow us to 'reject at level $p$' and still provide this guarantee. Moreover, they can easily be combined since the product of independent post-hoc $p$-values is also a post-hoc $p$-value. Interestingly, we find that $p$ is a post-hoc $p$-value if and only if $1/p$ is an $e$-value, a recently introduced measure of evidence. This reveals what $e$-values truly are in the context of a hypothesis testing problem. Post-hoc $\alpha$ hypothesis testing eliminates the need for standardized levels such as $\alpha = 0.05$, which takes away incentives for $p$-hacking and contributes to solving the file-drawer problem.