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 the notion of a post-hoc $p$-value, that does admit this interpretation. We show that $p$ is a post-hoc $p$-value if and only if $1/p$ is an $e$-value. Among other things, this implies that the product of independent post-hoc $p$-values is a post-hoc $p$-value. Moreover, we generalize post-hoc validity to a sequential setting and find that $(p_t)_{t \geq 1}$ is a post-hoc anytime valid $p$-process if and only if $(1/p_t)_{t \geq 1}$ is an $e$-process. In addition, we show that if we admit randomized procedures, any non-randomized post-hoc $p$-value can be trivially improved. In fact, we find that this in some sense characterizes non-randomized post-hoc $p$-values. Finally, we argue that we need to go beyond $e$-values if we want to consider randomized post-hoc inference in its full generality.

翻译：一种普遍存在的方法论错误是对$p$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显著性水平。我们引入了事后$p$值的概念，该概念确实允许这种解释。我们证明$p$是事后$p$值当且仅当$1/p$是$e$值。这尤其意味着独立事后$p$值的乘积仍然是事后$p$值。此外，我们将事后有效性推广到序列设定，发现$(p_t)_{t \geq 1}$是事后任意有效$p$过程当且仅当$(1/p_t)_{t \geq 1}$是$e$过程。进一步研究表明，若允许随机化程序，任何非随机化事后$p$值均可被平凡改进。事实上，我们发现这在某种意义上刻画了非随机化事后$p$值的特征。最后，我们论证若要在完全一般性下考虑随机化事后推断，则需超越$e$值的框架。