Bayesian Controlled FDR Variable Selection via Knockoffs (2411.03304v2)

Abstract: 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). The Knockoff filter has been recently proposed in the frequentist paradigm to introduce controlled noise in a model by cleverly constructing copies of the predictors as auxiliary variables. In this paper, we develop a fully Bayesian generalization of the classical model-X knockoff filter for normally distributed covariates. In our approach we consider a joint model of the covariates and the response variables, 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,leading to improved knockoffs generation and 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 construction leads to valid model-X knockoffs and demonstrate that the proposed variable selection procedure leads to controlling the Bayesian FDR at an arbitrary level, in finite samples, if the distribution of the covariates is fully known, and asymptotically if estimated as in the proposed model. We use simulated data to demonstrate that our proposal increases the stability of the selection with respect to classical knockoff methods. 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.