Suppose we observe an infinite series of coin flips $X_1,X_2,\dots$, 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 two 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 α 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 a wide 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.

假设我们观察到无限系列的硬币翻转 $X_1,X_2,\dots$，并希望依次测试这些二进制随机变量是否可交换的空值。非负超鞅 (NSM) 是顺序推理的主力，但我们证明它们对这个问题无能为力。首先，利用称为分叉凸性（凸性的顺序类比）的几何概念，我们表明任何在两个分布下是 NSM 的过程，也必然是在其“分叉-凸包”下的 NSM。其次，我们证明了可交换空值的叉凸包由二进制序列上的所有可能定律组成；这意味着在可交换性下的任何 NSM 必然是非递增的，因此对于任何替代方案总是产生无能为力的测试。由于测试与可交换性的任意偏差在理论上是不可能的，因此我们专注于马尔可夫替代方案。我们结合通用推理和混合方法的思想推导出“安全电子过程”，这是一个非负过程，在任何停止时间，期望最多为零，并且上限为鞅，但不是本身就是一个 NSM。这反过来又产生了一个水平α序贯检验是一致的；通用编码的遗憾边界也证明了速率最优能力。我们提出了使用“双重混合”方法将这些结果扩展到任何有限字母表和任何顺序的马尔可夫替代方案的方法。我们提供了广泛的模拟，并给出了基于对非结构化或不明确的替代品进行投注的一般方法。最后，受 Shafer、Vovk 和 Ville 的启发，我们提供了对电子流程和路径结果的博弈论解释。