We present Markov's equality: a tight version of Markov's inequality, that does not impose further assumptions on the on the random variable. We show that this equality, as well as Markov's inequality and its randomized improvement, are directly implied by a set of deterministic inequalities. We apply Markov's equality to show that standard tests based on $e$-values and $e$-processes are post-hoc (anytime) valid: the tests remain valid, even if the level $\alpha$ is selected after observing the data. In fact, we show that this property characterizes $e$-values and $e$-processes.

翻译：我们提出马尔可夫等式：马尔可夫不等式的一个紧凑版本，且不对随机变量施加额外假设。我们证明该等式、马尔可夫不等式及其随机化改进，均直接由一组确定性不等式推导得出。应用马尔可夫等式，我们表明基于$e$-值和$e$-过程的标准检验具有事后（任意时间）有效性：即使观测数据后选择显著性水平$\alpha$，这些检验仍然有效。事实上，我们证明该性质刻画了$e$-值和$e$-过程。