We develop an extension of the classical Zellner's $g$-prior to generalized linear models. Any continuous proper hyperprior $f(g)$ can be used, giving rise to a large class of hyper-$g$ priors. Connections with the literature are described in detail. A fast and accurate integrated Laplace approximation of the marginal likelihood makes inference in large model spaces feasible. For posterior parameter estimation we propose an efficient and tuning-free Metropolis-Hastings sampler. The methodology is illustrated with variable selection and automatic covariate transformation in the Pima Indians diabetes data set.
"Hyper-$g$ priors for generalized linear models." Bayesian Anal. 6 (3) 387 - 410, September 2011. https://doi.org/10.1214/11-BA615