  Variable selection methods with nonlocal priors have been widely studied in
linear regression models, and their theoretical and empirical performances have
been reported. However, the crucial model selection properties for hierarchical
nonlocal priors in high-dimensional generalized linear regression have rarely
been investigated. In this paper, we consider a hierarchical nonlocal prior for
high-dimensional logistic regression models and investigate theoretical
properties of the posterior distribution. Specifically, a product moment (pMOM)
nonlocal prior is imposed over the regression coefficients with an
Inverse-Gamma prior on the tuning parameter. Under standard regularity
assumptions, we establish strong model selection consistency in a
high-dimensional setting, where the number of covariates is allowed to increase
at a sub-exponential rate with the sample size. We implement the Laplace
approximation for computing the posterior probabilities, and a modified shotgun
stochastic search procedure is suggested for efficiently exploring the model
space. We demonstrate the validity of the proposed method through simulation
studies and an RNA-sequencing dataset for stratifying disease risk.
