anytime-valid

Anytime validity is a property of inferential tasks. In particular, we say that something is anytime-valid if its properties hold at all stopping times, not just at some fixed-time determined a priori. (Note that by “time” we usually mean the number of observations).

Anytime-validity has a close relationship to time uniformity. For probability statements, the two concepts are identical. Formally, for a set of events $\{A_t\}$ ,

$$\mathbb{P}(\exists t:A_t)\le \alpha \iff \mathbb{P}(A_{\tau })\le \alpha ,\forall \tau ,$$

where $\tau$ is a stopping time. But they are not equivalent for statements concerning expected values. Suppose for instance that $\{E_t\}$ is a collection of random variables. Then the statement $\mathbb{E}[{\sup }_t{E}_t]\le c$ implies that for all $\tau$, $\mathbb{E}[E_{\tau }]\le c$ but not vice versa. The first statement is actually equivalent to the statement $\mathbb{E}[E_T]\le c$ for all random times $c$ (not only for stopping times).

The equivalence when discussing probability statements means that confidence sequences are objects that are both time-uniform and anytime-valid. But e-processes, which are defined in terms of stopping times, are not necessarily time-uniform.

Refs

• Admissible anytime-valid sequential inference must rely on nonnegative martingales by Ramdas et al.