We study e-values for quantifying evidence against exchangeability and general invariance of a random variable under a compact group. We start by characterizing such e-values, and explaining how they nest traditional group invariance tests as a special case. We show they can be easily designed for an arbitrary test statistic, and computed through Monte Carlo sampling. We prove a result that characterizes optimal e-values for group invariance against optimality targets that satisfy a mild orbit-wise decomposition property. We apply this to design expected-utility-optimal e-values for group invariance, which include both Neyman-Pearson-optimal tests and log-optimal e-values. Moreover, we generalize the notion of rank- and sign-based testing to compact groups, by using a representative inversion kernel. In addition, we characterize e-processes for group invariance for arbitrary filtrations, and provide tools to construct them. We also describe test martingales under a natural filtration, which are simpler to construct. Peeking beyond compact groups, we encounter e-values and e-processes based on ergodic theorems. These nest e-processes based on de Finetti's theorem for testing exchangeability.

翻译：本研究探讨了用于量化随机变量在紧致群下可交换性与一般不变性证据的E值。我们首先刻画此类E值的特征，并阐释其如何将传统群不变性检验作为特例纳入框架。研究表明，针对任意检验统计量均可便捷设计E值，并通过蒙特卡洛采样进行计算。我们证明了一个刻画群不变性最优E值的结果，该结果针对满足温和轨道分解性质的最优性目标成立。基于此，我们设计了群不变性的期望效用最优E值，其同时包含Neyman-Pearson最优检验与对数最优E值。此外，通过引入代表性逆核方法，我们将基于秩与符号的检验概念推广至紧致群情形。进一步地，我们刻画了任意滤子下群不变性的E过程，并提供了相应的构造工具。同时描述了自然滤子下的检验鞅，这类构造更为简洁。超越紧致群范畴的探索中，我们发现了基于遍历定理的E值与E过程，其中包含了基于de Finetti定理检验可交换性的E过程。