Sparse Bayesian variable selection in high-dimensional logistic regression models with correlated priors

In this paper, we propose a sparse Bayesian procedure with global and local (GL) shrinkage priors for the problems of variable selection and classification in high-dimensional logistic regression models. In particular, we consider two types of GL shrinkage priors for the regression coefficients, the horseshoe (HS) prior and the normal-gamma (NG) prior, and then specify a correlated prior for the binary vector to distinguish models with the same size. The GL priors are then combined with mixture representations of logistic distribution to construct a hierarchical Bayes model that allows efficient implementation of a Markov chain Monte Carlo (MCMC) to generate samples from posterior distribution. We carry out simulations to compare the finite sample performances of the proposed Bayesian method with the existing Bayesian methods in terms of the accuracy of variable selection and prediction. Finally, two real-data applications are provided for illustrative purposes.