Mixtures of g-priors in Generalized Linear Models

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.

References in corpus (4)

• Bayes and empirical-Bayes multiplicity adjustment in the variable-selection problem
• Criteria for Bayesian model choice with application to variable selection
• Bayesian variable selection using cost-adjusted BIC, with application to cost-effective measurement of quality of health care
• Methods and Tools for Bayesian Variable Selection and Model Averaging in Univariate Linear Regression

Cited by in corpus (11)

• A novel algorithmic approach to Bayesian Logic Regression
• Flexible Bayesian Nonlinear Model Configuration
• Combining Model and Parameter Uncertainty in Bayesian Neural Networks
• Predicting competitions by combining conditional logistic regression and subjective Bayes: An Academy Awards case study
• Bayesian graphical compositional regression for microbiome data
• Model Averaging and its Use in Economics
• Empirical Bayes Model Averaging with Influential Observations: Tuning Zellner's g Prior for Predictive Robustness
• Power-Expected-Posterior Priors as Mixtures of g-Priors
• Bayesian model selection in additive partial linear models via locally adaptive splines
• Revisiting High Dimensional Bayesian Model Selection for Gaussian Regression
• Rejoinder for the discussion of the paper "A novel algorithmic approach to Bayesian Logic Regression"