High-dimensional linear models have been extensively studied in the recent
literature, but the developments in high-dimensional generalized linear models,
or GLMs, have been much slower. In this paper, we propose the use an empirical
or data-driven prior specification leading to an empirical Bayes posterior
distribution which can be used for estimation of and inference on the
coefficient vector in a high-dimensional GLM, as well as for variable
selection. For our proposed method, we prove that the posterior distribution
concentrates around the true/sparse coefficient vector at the optimal rate and,
furthermore, provide conditions under which the posterior can achieve variable
selection consistency. Computation of the proposed empirical Bayes posterior is
simple and efficient, and, in terms of variable selection in logistic and
Poisson regression, is shown to perform well in simulations compared to
existing Bayesian and non-Bayesian methods.Comment: 30 pages, 2 table