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This paper introduces a class of asymptotically most powerful knockoffstatistics based on a simple principle: that we should prioritize variables inorder of our ability to distinguish them from their knockoffs. Our contributionis threefold. First, we argue that feature statistics should estimate "oraclemasked likelihood ratios," which are Neyman-Pearson statistics fordiscriminating between features and knockoffs using partially observed (masked)data. Second, we introduce the masked likelihood ratio (MLR) statistic, aknockoff statistic that estimates the oracle MLR. We show that MLR statisticsare asymptotically average-case optimal, i.e., they maximize the expectednumber of discoveries made by knockoffs when averaging over a user-specifiedprior on unknown parameters. Our optimality result places no explicitrestrictions on the problem dimensions or the unknown relationship between theresponse and covariates; instead, we assume a "local dependence" conditionwhich depends only on simple quantities that can be calculated from the data.Third, in simulations and three real data applications, we show that MLRstatistics outperform state-of-the-art feature statistics, including insettings where the prior is highly misspecified. We implement MLR statistics inthe open-source python package knockpy; our implementation is often (althoughnot always) faster than computing a cross-validated lasso.

Asymptotically Optimal Knockoff Statistics via the Masked Likelihood Ratio