How can one test if a binary sequence is exchangeable? Fork-convex hulls, supermartingales, and Snell envelopes

Suppose we observe an infinite series of coin flips X1, X2, . . . , and wish to sequentially test the null that these binary random variables are exchangeable against Markovian alternatives. We utilize a geometric concept called “fork-convexity” (an adapted analog of convexity) that lies at the heart of this problem, and relate it to other concepts like Snell envelopes that are absent in the sequential testing literature. By demonstrating that the alternative lies within the fork-convex hull of the null, we prove that any nonnegative supermartingale under the exchangeable null is necessarily also a supermartingale under the alternative, and thus yields a powerless test. We then combine ideas from universal inference (maximum likelihood under the null) and the method of mixtures (Jeffreys’ prior over the alternative) to derive a nonnegative process that is upper bounded by a martingale, but is not itself a supermartingale. We show that this process yields safe e-values, which in turn yield sequential level-α tests that are consistent (power one), using regret bounds from universal coding to demonstrate their 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 also discuss their power against change point alternatives, and give general approaches based on betting for unstructured or ill-specified alternatives.