Equality and inequality constrained multivariate linear models: objective model selection using constrained posterior priors

Abstract

In objective Bayesian model selection, a well-known problem is that standard non-informative prior distributions cannot be used to obtain a sensible outcome of the Bayes factor because these priors are improper. The use of a small part of the data, i.e., a training sample, to obtain a proper posterior prior distribution has become a popular method to resolve this issue and seems to result in reasonable outcomes of default Bayes factors, such as the intrinsic Bayes factor or a Bayes factor based on the empirical expected-posterior prior. In this paper, it will be illustrated that such default methods may not result in sensible outcomes when evaluating inequality constrained models that are supported by the data. To resolve this issue, a default method is proposed for constructing so-called constrained posterior priors, which are inspired by the symmetrical intrinsic priors discussed by Berger and Mortera (1999) for a simple inequality constrained model selection problem. The resulting Bayes factors can be called “balanced” because model complexity of inequality constrained models is incorporated according to a specific definition that is presented in this paper.