Anytime-Valid Tests of Group Invariance through Conformal Prediction

The assumption that data are invariant under the action of a compact group is implicit in many statistical modeling assumptions such as normality, or the assumption of independence and identical distributions. Hence, testing for the presence of such invariances offers a principled way to falsify various statistical models. In this article, we develop sequential, anytime-valid tests of distributional symmetry under the action of general compact groups. The tests that are developed allow for the continuous monitoring of data as it is collected while keeping type-I error guarantees, and include tests for exchangeability and rotational symmetry as special examples. The main tool to this end is the machinery developed for conformal prediction. The resulting test statistic, called a conformal martingale, can be interpreted as a likelihood ratio. We use this interpretation to show that the test statistics are optimal -- in a specific log-optimality sense -- against certain alternatives. Furthermore, we draw a connection between conformal prediction, anytime-valid tests of distributional invariance, and current developments on anytime-valid testing. In particular, we extend existing anytime-valid tests of independence, which leverage exchangeability, to work under general group invariances. Additionally, we discuss testing for invariance under subgroups of the permutation group and orthogonal group, the latter of which corresponds to testing the assumptions behind linear regression models.