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.

翻译：一种普遍的方法论错误是对$p$值的事后解释。$p$值$p$并非我们拒绝原假设的显著性水平，而是若我们选择水平$p$时本应拒绝原假设的水平。我们引入了事后$p$值，它确实允许这种解释。我们证明$p$是事后$p$值当且仅当$1/p$是一个$e$值。这意味着独立事后$p$值的乘积仍为事后$p$值，从而便于合并。如果允许外部随机化，我们发现任何非随机化的事后$p$值都可以被平凡地改进。然而，我们发现只有（本质上）非随机化的事后$p$值才能通过乘法任意合并。我们的结果扩展到了序列设定中的事后任意有效性。此外，我们引入了双向事后$p$值，其倒数在备择假设下也是事后有效的。似然比是双向事后$p$值，这支持了其在贝叶斯因子背景下常被宣称的“直接”解释，并将其解释与事后$p$值联系起来。最后，我们将之扩展到几何事后有效性，并证明GRO $e$值是在备择假设下最小化几何事后误差的事后$p$值的倒数。