  A new empirical Bayes approach to variable selection in the context of
generalized linear models is developed. The proposed algorithm scales to
situations in which the number of putative explanatory variables is very large,
possibly much larger than the number of responses. The coefficients in the
linear predictor are modeled as a three-component mixture allowing the
explanatory variables to have a random positive effect on the response, a
random negative effect, or no effect. A key assumption is that only a small
(but unknown) fraction of the candidate variables have a non-zero effect. This
assumption, in addition to treating the coefficients as random effects
facilitates an approach that is computationally efficient. In particular, the
number of parameters that have to be estimated is small, and remains constant
regardless of the number of explanatory variables. The model parameters are
estimated using a Generalized Alternating Maximization algorithm which is
scalable, and leads to significantly faster convergence compared with
simulation-based fully Bayesian methods.
