Admissible anytime-valid sequential inference must rely on nonnegative martingales

Wald's anytime-valid $p$-values and Robbins' confidence sequences enable sequential inference for composite and nonparametric classes of distributions at arbitrary stopping times, as do more recent proposals involving Vovk's `$e$-values' or Shafer's `betting scores'. Examining the literature, one finds that at the heart of all these (quite different) approaches has been the identification of composite nonnegative (super)martingales. Thus, informally, nonnegative (super)martingales are known to be sufficient for \emph{valid} sequential inference. Our central contribution is to show that martingales are also universal---all \emph{admissible} constructions of (composite) anytime $p$-values, confidence sequences, or $e$-values must necessarily utilize nonnegative martingales (or so-called max-martingales in the case of $p$-values). Sufficient conditions for composite admissibility are also provided. Our proofs utilize a plethora of modern mathematical tools for composite testing and estimation problems: max-martingales, Snell envelopes, and new Doob-L\'evy martingales make appearances in previously unencountered ways. Informally, if one wishes to perform anytime-valid sequential inference, then any existing approach can be recovered or dominated using martingales. We provide several sophisticated examples, with special focus on the nonparametric problem of testing if a distribution is symmetric, where our new constructions render past methods inadmissible.