A Bayesian Discovery Procedure

We discuss a Bayesian discovery procedure for multiple comparison problems. We show that under a coherent decision theoretic framework, a loss function combining true positive and false positive counts leads to a decision rule based on a threshold of the posterior probability of the alternative. Under a semi-parametric model for the data, we show that the Bayes rule can be approximated by the optimal discovery procedure (ODP), recently introduced by Storey (2007a). Improving the approximation leads us to a Bayesian discovery procedure (BDP), which exploits the multiple shrinkage in clusters implied by the assumed nonparametric model. We compare the BDP and the ODP estimates in a simple simulation study and in an assessment of differential gene expression based on microarray data from tumor samples. We extend the setting of the ODP by discussing modifications of the loss function that lead to different single thresholding statistics. Finally, we provide an application of the previous arguments to dependent (spatial) data.