Liang et al. (2008) Mixtures of g-Priors for Bayesian Variable Selection

Summary

Zellner's g-prior is the popular conventional prior for Bayesian variable selection because its marginal likelihoods are closed-form functions of the model $R^2$ — but a single fixed $g$ creates consistency pathologies. Liang, Paulo, Molina, Clyde and Berger study mixtures of g-priors (a prior on $g$, integrated out) as a default that resolves these pathologies while keeping tractability. They show the Zellner-Siow Cauchy prior is a special case, introduce the hyper-g and hyper-g/n prior families with marginal likelihoods in closed form via the Gaussian hypergeometric function ${}_2F_1$, and establish the theoretical properties (paradox resolution, model-selection consistency) with real and simulated comparisons against fixed-$g$, empirical-Bayes, and other default procedures.

Key Claims

• Bartlett's paradox. With a single $g$ taken very large to be "noninformative," the Bayes factor comparing a model $M_\gamma$ to the null goes to favour the null as $g\to\infty$ (with $n,p_\gamma$ fixed) — so a diffuse $g$-prior is a bad idea for model selection even though it is fine for estimation within a model.
• Information paradox. For fixed $g$, as the data's evidence for $M_\gamma$ grows without bound ($R^2\to1$), the Bayes factor stays bounded instead of diverging — an incoherent response to overwhelming evidence.
• Mixtures fix both. Placing a hyperprior on $g$ and integrating it out resolves both paradoxes and restores model-selection consistency, while retaining the computational convenience of the g-prior.
• Zellner-Siow and hyper-g. The Zellner-Siow (1980) Cauchy prior is a mixture of g-priors (an inverse-gamma mixing density) but lacks closed-form marginals; the paper's hyper-g / hyper-g/n family gives closed-form marginals via ${}_2F_1$, with the adaptivity and nonlinear shrinkage of empirical-Bayes procedures but a fully Bayesian justification and robustness to $g$ misspecification. (Independently proposed by Cui-George 2007.)

Concepts Introduced or Extended

• Zellner's g-Prior
• Variable Selection
• Bayes Factor

Entities Mentioned

• Feng Liang
• James O. Berger
• Merlise Clyde
• Rui Paulo (ISEG, Lisbon); German Molina (Vega Capital) — coauthors

Quotes

"Zellner's g-prior remains a popular conventional prior for use in Bayesian variable selection, despite several undesirable consistency issues … we study mixtures of g-priors as an alternative … that resolve many of the problems with the original formulation, while maintaining the computational tractability that has made the g-prior so popular."

My Take

This is the paper that made the g-prior safe to use as a default for model selection rather than just estimation. Its contribution is precise diagnosis plus a tractable cure: it names the two failure modes of fixed $g$ (Bartlett's and the information paradox), shows they are exactly what a mixture repairs, and — crucially for adoption — delivers the hyper-g family with closed-form marginals so practitioners lose nothing computationally. It underwrites the marginal-likelihood machinery in BAS (same authorship lineage — Clyde, Berger) and slots directly beside Maruyama-George (2011), which pushes the same fully-Bayes-$g$ idea into the $p>n$ regime. The honest caveat is that "default" still involves a choice among hyper-g, hyper-g/n and Zellner-Siow, whose differences matter most exactly when $n$ is small.