Testing exchangeability: fork-convexity, supermartingales, and e-processes

Suppose we observe an infinite series of coin flips $X_1,X_2,\ldots$, and wish to sequentially test the null that these binary random variables are exchangeable. Nonnegative supermartingales (NSMs) are a workhorse of sequential inference, but we prove that they are powerless for this problem. First, utilizing a geometric concept called fork-convexity (a sequential analog of convexity), we show that any process that is an NSM under a set of distributions, is also necessarily an NSM under their "fork-convex hull". Second, we demonstrate that the fork-convex hull of the exchangeable null consists of all possible laws over binary sequences; this implies that any NSM under exchangeability is necessarily nonincreasing, hence always yields a powerless test for any alternative. Since testing arbitrary deviations from exchangeability is information theoretically impossible, we focus on Markovian alternatives. We combine ideas from universal inference and the method of mixtures to derive a "safe e-process", which is a nonnegative process with expectation at most one under the null at any stopping time, and is upper bounded by a martingale, but is not itself an NSM. This in turn yields a level $\alpha$ sequential test that is consistent; regret bounds from universal coding also demonstrate rate-optimal power. We present ways to extend these results to any finite alphabet and to Markovian alternatives of any order using a "double mixture" approach. We provide an array of simulations, and give general approaches based on betting for unstructured or ill-specified alternatives. Finally, inspired by Shafer, Vovk, and Ville, we provide game-theoretic interpretations of our e-processes and pathwise results.