The use of weights provides an effective strategy to incorporate prior domain knowledge in large-scale inference. This article studies weighted multiple testing in a decision-theoretical framework. We develop oracle and data-driven procedures that aim to maximize the expected number of true positives subject to a constraint on the weighted false discovery rate. The asymptotic validity and optimality of the proposed methods are established. The results demonstrate that incorporating informative domain knowledge enhances the interpretability of results and precision of inference. Simulation studies show that the proposed method controls the error rate at the nominal level, and the gain in power over existing methods is substantial in many settings. An application to a genome-wide association study is discussed. Supplementary materials for this article are available online.
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