Default Bayesian Model Selection of Constrained Multivariate Normal Linear Models

A default Bayes factor is proposed for evaluating multivariate normal linear models with competing sets of equality and order constraints on the parameters of interest. The default Bayes factor is based on generalized fractional Bayes methodology where different fractions are used for different observations and where the default prior is centered on the boundary of the constrained space under investigation. First, the method is fully automatic and therefore can be applied when prior information is weak or completely unavailable. Second, using group specific fractions, the same amount of information is used from each group resulting in a minimally informative default prior having a matrix Cauchy distribution. This results in a consistent default Bayes factor. Third, numerical computation can be done using parallelization which makes it computationally cheap. Fourth, the evidence can be updated in a relatively simple manner when observing new data. Fifth, the selection criterion can be applied relatively straightforwardly in the presence of missing data that are missing at random. Finally the methodology can be used for default model selection and hypothesis testing of commonly used models such as (M)AN(C)OVA, (multivariate) multiple regression, or repeated measures.