Post-hoc Valid Inference

A pervasive methodological error is the post-hoc interpretation of $p$-values. A $p$-value $p$ is not the level at which we reject the null, it is the level at which we would have rejected the null had we chosen level $p$. We introduce post-hoc $p$-values, that do admit this interpretation. We show that $p$ is a post-hoc $p$-value if and only if $1/p$ is an $e$-value. This implies that the product of independent post-hoc $p$-values is a post-hoc $p$-value, making them easy to combine. If we permit external randomization, we find any non-randomized post-hoc $p$-value can be trivially improved. However, we find only (essentially) non-randomized post-hoc $p$-values can be arbitrarily merged through multiplication. Our results extend to post-hoc anytime validity in a sequential setting. Moreover, we introduce two-way post-hoc $p$-values, whose reciprocal is also post-hoc under the alternative. Likelihood ratios are two-way post-hoc $p$-values, which supports their 'direct' interpretation often purported in the context of Bayes factors and links their interpretation to post-hoc $p$-values. Finally, we extend to geometric post-hoc validity and show that GRO $e$-values are the reciprocal of post-hoc $p$-values that minimize the geometric post-hoc error under the alternative.