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We develop a flexible online version of the permutation test. This allows usto test exchangeability as the data is arriving, where we can choose to stop orcontinue without invalidating the size of the test. Our methods generalizebeyond exchangeability to other forms of invariance under a compact group. Ourapproach relies on constructing an $e$-process that is the running product ofmultiple conditional $e$-values. To construct $e$-values, we first develop anessentially complete class of admissible $e$-values in which one can flexibly`plug in' almost any desired test statistic. To make the $e$-valuesconditional, we explore the intersection between the concepts of conditionalinvariance and sequential invariance, and find that the appropriate conditionaldistribution can be captured by a compact subgroup. To find powerful $e$-valuesfor given alternatives, we develop the theory of likelihood ratios for testinggroup invariance yielding new optimality results for group invariance tests.These statistics turn out to exist in three different flavors, depending on thespace on which we specify our alternative. We apply these statistics to testagainst a Gaussian location shift, which yields connections to the $t$-testwhen testing sphericity, connections to the softmax function and itstemperature when testing exchangeability, and yields an improved version of aknown $e$-value for testing sign-symmetry. Moreover, we introduce an impatienceparameter that allows users to obtain more power now in exchange for less powerin the long run.

Online Permutation Tests: $e$-values and Likelihood Ratios for Testing Group Invariance