Admissible anytime-valid sequential inference must rely on nonnegative martingales

Abstract: Confidence sequences, anytime p-values (called p-processes in this paper), and e-processes all enable sequential inference for composite and nonparametric classes of distributions at arbitrary stopping times. Examining the literature, one finds that at the heart of all these (quite different) approaches has been the identification of nonnegative (super)martingales. Thus, informally, nonnegative (super)martingales are known to be sufficient for \emph{anytime-valid} sequential inference, even in composite and nonparametric settings. Our central contribution is to show that nonnegative martingales are also universal -- after appropriately defining \emph{admissibility}, we show that all admissible constructions of confidence sequences, p-processes, or e-processes must necessarily utilize nonnegative martingales. Our proofs utilize several modern mathematical tools for composite testing and estimation problems: max-martingales, Snell envelopes, transfinite induction, 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 nonnegative martingales. We provide several nontrivial examples, with special focus on testing symmetry, where our new constructions render past methods inadmissible. We also prove the subGaussian supermartingale to be admissible.