A general decision theoretic formulation of procedures controlling FDR and FNR from a Bayesian perspective

A general decision theoretic formulation is given to multiple testing, allowing descriptions of measures of false discoveries and false non-discoveries in terms of certain loss functions even when randomized decisions are made on the hypotheses. Randomized as well as non-randomized procedures controlling the Bayes false discovery rate (BFDR) and Bayes false non-discovery rate (BFNR) are developed. These are applicable in any situation, unlike the corresponding frequentist procedures that control the BFDR or BFNR but do so under certain dependence structures of the test statistics. Even in the presence of such dependence., as simulations show, the proposed procedures perform much better than the corresponding frequentist procedures. They provide better control of the BFDR or BFNR than those for which control is achieved through local FDR or local FNR.