Measuring Evidence against Exchangeability and Group Invariance with E-values

TL;DR

This paper develops a framework for designing and analyzing e-values and e-processes to test exchangeability and invariance under compact groups, unifying and extending traditional invariance tests with optimality properties.

Contribution

It introduces a general method for constructing e-values for group invariance, characterizes their optimality, and extends the concept to e-processes and ergodic theorems for broader applicability.

Findings

Characterization of e-values for group invariance.

Design of optimal e-values including Neyman-Pearson and log-optimal.

Development of e-processes for arbitrary filtrations and ergodic settings.

Abstract

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.