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It is often claimed that Bayesian methods, in particular Bayes factor methodsfor hypothesis testing, can deal with optional stopping. We first give anoverview, using elementary probability theory, of three different mathematicalmeanings that various authors give to this claim: (1) stopping ruleindependence, (2) posterior calibration and (3) (semi-) frequentist robustnessto optional stopping. We then prove theorems to the effect that these claims doindeed hold in a general measure-theoretic setting. For claims of type (2) and(3), such results are new. By allowing for non-integrable measures based onimproper priors, we obtain particularly strong results for the practicallyimportant case of models with nuisance parameters satisfying a group invariance(such as location or scale). We also discuss the practical relevance of(1)--(3), and conclude that whether Bayes factor methods actually perform wellunder optional stopping crucially depends on details of models, priors and thegoal of the analysis.

Optional Stopping with Bayes Factors: a categorization and extension of folklore results, with an application to invariant situations