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

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