Bayesian Controlled FDR Variable Selection via Knockoffs

In many research fields, researchers aim to identify significant associations between a set of explanatory variables and a response while controlling the false discovery rate (FDR). To this aim, we develop a fully Bayesian generalization of the classical model-X knockoff filter. Knockoff filter introduces controlled noise in the model in the form of cleverly constructed copies of the predictors as auxiliary variables. In our approach we consider the joint model of the covariates and the response and incorporate the conditional independence structure of the covariates into the prior distribution of the auxiliary knockoff variables. We further incorporate the estimation of a graphical model among the covariates, which in turn aids knockoffs generation and improves the estimation of the covariate effects on the response. We use a modified spike-and-slab prior on the regression coefficients, which avoids the increase of the model dimension as typical in the classical knockoff filter. Our model performs variable selection using an upper bound on the posterior probability of non-inclusion. We show how our model construction leads to valid model-X knockoffs and demonstrate that the proposed characterization is sufficient for controlling the BFDR at an arbitrary level, in finite samples. We also show that the model selection is robust to the estimation of the precision matrix. We use simulated data to demonstrate that our proposal increases the stability of the selection with respect to classical knockoff methods, as it relies on the entire posterior distribution of the knockoff variables instead of a single sample. With respect to Bayesian variable selection methods, we show that our selection procedure achieves comparable or better performances, while maintaining control over the FDR. Finally, we show the usefulness of the proposed model with an application to real data.