Consistent skinny Gibbs in probit regression

Spike and slab priors have emerged as effective and computationally scalable tools for Bayesian variable selection in high-dimensional linear regression. However, the crucial model selection consistency and efficient computational strategies using spike and slab priors in probit regression have rarely been investigated. A hierarchical probit model with continuous spike and slab priors over regression coefficients is considered, and a highly scalable Gibbs sampler with a computational complexity that grows only linearly in the dimension of predictors is proposed. Specifically, the “Skinny Gibbs” algorithm is adapted to the setting of probit and negative binomial regression and model selection consistency for the proposed method under probit model is established, when the number of covariates is allowed to grow much larger than the sample size. Through simulation studies, the method is shown to achieve superior empirical performance compared with other state-of-the art methods. Gene expression data from 51 asthmatic and 44 non-asthmatic samples are analyzed and the performance for predicting asthma using the proposed approach is compared with existing approaches.