A Unified Treatment of Multiple Testing with Prior Knowledge

A significant literature studies ways of employing prior knowledge to improve power and precision of multiple testing procedures. Some common forms of prior knowledge may include (a) \emph{a priori} beliefs about which hypotheses are null, modeled by non-uniform prior weights; (b) differing importances of hypotheses, modeled by differing penalties for false discoveries; (c) multiple arbitrary partitions of the hypotheses into known (possibly overlapping) groups, indicating (dis)similarity of hypotheses; and (d) knowledge of independence, positive dependence or arbitrary dependence between hypotheses or groups, allowing for more aggressive or conservative procedures. We unify a number of existing procedures, generalize the conditions under which they are known to work, and simplify their proofs of FDR control. Then, we present a unified algorithmic framework for global null testing and false discovery rate (FDR) control that allows the scientist to incorporate all four types of prior knowledge (a)--(d) simultaneously, recovering a wide variety of common algorithms as special cases.