In Bayesian hypothesis testing, where one actually commits to estimates of the probabilities of each hypothesis, an analog of the Bonferroni correction is easy to justify. Namely, in the multiple comparison scenario, we might ask when Bayesian hypothesis testing: "what is the probability that at least one of these hypotheses is true, already knowing the probability of each individual hypothesis separately?". Then the union bound should give something analogous to Bonferroni.
Question: Is there an analog of Benjamini-Hochberg for Bayesian hypothesis testing?
More generally I am interested in the extent to which the theory for the multiple comparison problem is analogous between Bayesian and non-Bayesian hypothesis testing. But for the sake of specificity, and since that is the method I am most interested in right now, I refer to the Benjamini-Hochberg procedure. In particular, it is unclear to me whether the absence of p-values in the Bayesian setting makes procedures or considerations like Benjamini-Hochberg irrelevant.
Pointers to references will more than suffice for an answer. But of course if you want to share your unique insights into these issues, please do not hesitate.
