Bayesian knockoff filter for controlling false discovery rate

In many scientific fields, researchers are interested in discovering features with substantial effects on the response from a large number of candidate features while controlling the proportion of false discoveries. By incorporating the knockoff procedure in a Bayesian framework, we develop the Bayesian knockoff filter (BKF) for selecting important features. The BKF is built upon a tight upper bound for the posterior probability of the null hypothesis, which is derived under the Bayesian "flip-sign" property implied by our procedure. In contrast to the standard knockoff methods in the frequentist paradigm, we allow the knockoff variables to be generated in every iteration of the Markov chain Monte Carlo procedure, which removes the unwanted variability of the single-knockoff generation (knockoff is typically generated only once before the model fitting). Compared with multi-knockoffs, the BKF is more principled and natural in incorporating side information through prior distributions under a fully Bayesian framework. We further develop an e-value BKF approach to theoretically guarantee the frequentist false discovery rate (FDR) control. Numerical experiments on both synthetic and real data demonstrate the advantages of the BKF over existing knockoff methods and Bayesian variable selection approaches, i.e., the BKF possesses comparable power, yields a lower FDR and more stable selection sets.