Consistency of hyper-$g$-prior-based Bayesian variable selection for generalized linear models

We study the consistency of a Bayesian variable selection procedure for generalized linear models. Specifically, we consider the consistency of a Bayes factor based on g-priors proposed by Sabanes Bove and Held (2011). The integrals necessary for the computation of this Bayes factor are performed with Laplace approximation and Gaussian quadrature. We show that, under certain regularity conditions, the resulting Bayes factor is consistent. Furthermore, a simulation study confirms our theoretical results. Finally, we illustrate this model selection procedure with an application to a real ecological dataset.