Mixtures of g-priors in Generalized Linear Models

TL;DR

This paper unifies and extends mixtures of g-priors for Bayesian variable selection in Generalized Linear Models using a flexible distribution, providing analytical tools for model comparison and demonstrating their effectiveness through simulations and real data.

Contribution

It introduces a unified framework for mixtures of g-priors in GLMs using the tCCH distribution, enabling analytical marginal likelihoods and model selection criteria.

Findings

Analytical marginal likelihoods derived for the proposed priors.

Demonstrated model selection consistency and invariance properties.

Validated approach with simulations and real data examples.

Abstract

Mixtures of Zellner's g-priors have been studied extensively in linear models and have been shown to have numerous desirable properties for Bayesian variable selection and model averaging. Several extensions of g-priors to Generalized Linear Models (GLMs) have been proposed in the literature; however, the choice of prior distribution of g and resulting properties for inference have received considerably less attention. In this paper, we unify mixtures of g-priors in GLMs by assigning the truncated Compound Confluent Hypergeometric (tCCH) distribution to 1/(1 + g), which encompasses as special cases several mixtures of g-priors in the literature, such as the hyper-g, Beta-prime, truncated Gamma, incomplete inverse-Gamma, benchmark, robust, hyper-g/n, and intrinsic priors. Through an integrated Laplace approximation, the posterior distribution of 1/(1 + g) is in turn a tCCH distribution, and approximate marginal likelihoods are thus available analytically, leading to "Compound Hypergeometric Information Criteria" for model selection. We discuss the local geometric properties of the g-prior in GLMs and show how the desiderata for model selection proposed by Bayarri et al, such as asymptotic model selection consistency, intrinsic consistency, and measurement invariance may be used to justify the prior and specific choices of the hyper parameters. We illustrate inference using these priors and contrast them to other approaches via simulation and real data examples. The methodology is implemented in the R package BAS and freely available on CRAN.