Bayesian subset selection and model averaging using a centred and dispersed prior for the error variance

This article proposes a new data-based prior distribution for the error vari-ance in a Gaussian linear regression model, when the model is used for Bayesian variable selection and model averaging. For a given subset of vari-ables in the model, this prior has a mode that is an unbiased estimator of the error variance but is suitably dispersed to make it uninformative rela-tive to the marginal likelihood. The advantage of this empirical Bayes prior for the error variance is that it is centred and dispersed sensibly and avoids the arbitrary specification of hyperparameters. The performance of the new prior is compared to that of a prior proposed previously in the literature using several simulated examples and two loss functions. For each example our paper also reports results for the model that orthogonalizes the pre-dictor variables before performing subset selection. A real example is also investigated. The empirical results suggest that for both the simulated and real data, the performance of the estimators based on the prior proposed in our article compares favourably with that of a prior used previously in the literature. Key words: empirical Bayes prior; Markov chain Monte Carlo; nonparametric regression; orthogonal predictors. ∗Author to whom correspondence should be addressed