In the Bayesian variable selection framework, a common prior distribution for
the regression coefficients is the g-prior of Zellner (1986). However, there
are two standard cases in which the associated covariance matrix does not
exist, and the conventional prior of Zellner can not be used: if the number of
observations is lower than the number of variables (large p and small n
paradigm), or if some variables are linear combinations of others. In such
situations we propose a prior distribution derived from the prior of Zellner,
by introducing a ridge parameter. The prior obtained is a flexible and simple
adaptation of the g-prior. This adaptation is a compromise between the
conditional independent case of the coefficient regressors and the automatic
scaling advantage offered by the g-prior. A full variable selection method
using this prior is developed for probit mixed models, with a
Metropolis-within-Gibbs algorithm using the grouping technique of Liu (1994).
The method is then applied to both simulated and real datasets.