测试交换机：分叉凸性，上鞅和e-过程
Testing exchangeability: fork-convexity, supermartingales, and e-processes
摘要：
假设我们观察到一个无穷级数的硬币翻转$X_1,X_2,\ldots$，并希望顺序地测试这些二进制随机变量是可交换的。非负上鞅（NSM）是序列推理的主力，但我们证明了他们是无能为力的这个问题。首先，利用一个几何概念称为叉凸（凸的顺序模拟），我们表明，任何过程，是一个NSM下的一组分布，也必然是一个NSM下的“叉凸包“。第二，我们证明了叉凸包的可交换的空组成的所有可能的法律，在二进制序列，这意味着任何NSM下的可交换的必然是非增的，因此总是产生一个无能为力的测试任何替代品。由于测试任意偏离交换量的信息在理论上是不可能的，我们专注于马尔可夫替代方案。我们结合泛推理和混合方法的思想，得到了一个“安全e-过程”，它是一个非负过程，在任意停时，期望在零下至多为1，上界为鞅，但本身不是NSM.这反过来又产生了一个水平$α$序贯检验是一致的，从通用编码的遗憾界也证明了速率最优功率。我们提出的方法来扩展这些结果的任何有限的字母表和马尔可夫替代品的任何顺序使用“双混合”的方法。我们提供了一系列的模拟，并给出了一般的方法的基础上投注非结构化或指定不良的替代品。最后，受到Shafer，Vovk和Ville的启发，我们提供了我们的e-过程和路径结果的博弈论解释。
英文摘要：
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 $α$ 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 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.
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