Markov's Equality and Post-hoc (Anytime) Valid Inference

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.