A problem in sequential hypothesis testing where we want to test whether a sequence of observations is exchangeable. That is, we observe and test
In some sense, this the same probability as testing the iid assumption. Ramdas et al. (Corollary 16) show that a sequence is an e-process under exchangeability iff it is an e-process under the iid assumption.
Vovk tests exchangeability with conformal prediction: Testing randomness online This works in a coarsened filtration, by examining only the conformal p-values. This is an instance when coarsened filtrations can increase power.
Ramdas, Ruf, Larsson, Koolen study testing exchangeability against Markovian alternatives. Interestingly, they also show that every test-martingale under exchangeability is non-increasing, therefore powerless. To employ the framework of game-theoretic hypothesis testing, one must therefore study e-processes instead of supermartingales.
Saha and Ramdas also test exchangeability under Markovian alternatives. They also work in a coarsened filtration, examining the data in pairs instead of one at a time.