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 and Shafer's `e-values' or `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 supermartingales are known to be sufficient for valid sequential inference. Our central contribution is to show, in a very concrete sense, that martingales are also universal---all 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 nontrivial examples, with special focus on the nonparametric problem of testing if a distribution is symmetric, where our techniques render past methods inadmissible.