anytime-valid.md

created	2024-08-29
lastmod	2025-03-13

Anytime validity is a property of inferential tasks. In particular, we say that something is anytime-valid if its properties hold at all [[stopping-time|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-uniform|time uniformity]]. For probability statements, the two concepts are identical. Formally, for a set of events ${A_t}$ , $$ \Pr(\exists t: A_t) \leq \alpha \Leftrightarrow \Pr(A_\tau)\leq \alpha, \forall \tau, $$ 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 $\E[\sup_t E_t]\leq c$ implies that for all $\tau$, $\E[E_\tau]\leq c$ but not vice versa. The first statement is actually equivalent to the statement $\E[E_T]\leq c$ 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-process|e-processes]], which are defined in terms of stopping times, are not necessarily time-uniform.

Reading

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