FDR and Bayesian Multiple Comparisons Rules

We discuss Bayesian approaches to multiple comparison problems, using a decision theoretic perspective to critically compare competing approaches. We set up decision problems that lead to the use of FDR-based rules and generalizations. Alternative definitions of the probability model and the utility function lead to different rules and problem-specific adjustments. Using a loss function that controls realized FDR we derive an optimal Bayes rule that is a variation of the Benjamini and Hochberg (1995) procedure. The cutoff is based on increments in ordered posterior probabilities instead of ordered p-values. Throughout the discussion we take a Bayesian perspective. In particular, we focus on conditional expected FDR, conditional on the data. Variations of the probability model include explicit modeling for dependence. Variations of the utility function include weighting by the extent of a true negative and accounting for the impact in the final decision.