Inference From Intrinsic Bayes' Procedures Under Model Selection and Uncertainty

In this article, we present a fully coherent and consistent objective Bayesian analysis of the linear regression model using intrinsic priors. The intrinsic prior is a scaled mixture of g -priors and promotes shrinkage toward the subspace defined by a base (or null) model. While it has been established that the intrinsic prior provides consistent model selectors across a range of models, the posterior distribution of the model parameters has not previously been investigated. We prove that the posterior distribution of the model parameters is consistent under both model selection and model averaging when the number of regressors is fixed. Further, we derive tractable expressions for the intrinsic posterior distribution as well as sampling algorithms for both a selected model and model averaging. We compare the intrinsic prior to other mixtures of g -priors and provide details on the consistency properties of modified versions of the Zellner-Siow prior and hyper g -priors. Supplementary materials for this article are available online.