Measuring Evidence against Exchangeability and Group Invariance with E-values

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 exchangeabilit