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

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. s 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.