Hyper Nonlocal Priors for Variable Selection in Generalized Linear Models

We propose two novel hyper nonlocal priors for variable selection in generalized linear models. To obtain these priors, we first derive two new priors for generalized linear models that combine the Fisher information matrix with the Johnson-Rossell moment and inverse moment priors. We then obtain our hyper nonlocal priors from our nonlocal Fisher information priors by assigning hyperpriors to their scale parameters. As a consequence, the hyper nonlocal priors bring less information on the effect sizes than the Fisher information priors, and thus are very useful in practice whenever the prior knowledge of effect size is lacking. We develop a Laplace integration procedure to compute posterior model probabilities, and we show that under certain regularity conditions the proposed methods are variable selection consistent. We also show that, when compared to local priors, our hyper nonlocal priors lead to faster accumulation of evidence in favor of a true null hypothesis. Simulation studies that consider binomial, Poisson, and negative binomial regression models indicate that our methods select true models with higher success rates than other existing Bayesian methods. Furthermore, the simulation studies show that our methods lead to mean posterior probabilities for the true models that are closer to their empirical success rates. Finally, we illustrate the application of our methods with an analysis of the Pima Indians diabetes dataset.