Journal
JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
Volume 113, Issue 524, Pages 1828-1845Publisher
TAYLOR & FRANCIS INC
DOI: 10.1080/01621459.2018.1469992
Keywords
Bayesian model averaging; Bayesian model selection; Hyper-g priors; Linear regression; Variable selection
Categories
Funding
- National Science Foundation [DMS-11060891]
- National Institute of Health [1-R21-ES020796-01]
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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 article, 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 etal., 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. Supplementary materials for this article are available online.
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