In this paper, we consider that observations Y come from a general normal linear model and that it is desired to test a simplifying (null) hypothesis about the parameters. We approach this problem from an objective Bayesian, model selection perspective. Crucial ingredients for this approach are ‘proper objective priors’ to be used for deriving the Bayes factors. Jeffreys-Zellner-Siow priors have shown to have good properties for testing null hypotheses defined by specific values of the parameters in full rank linear models. We extend these priors to deal with general hypotheses in general linear models, not necessarily full rank. The resulting priors, which we call ‘conventional priors’, are expressed as a generalization of recently introduced ‘partially informative distributions’. The corresponding Bayes factors are fully automatic, easy to compute and very reasonable. The methodology is illustrated for two popular problems: the change point problem and the equality of treatments effects problem. We compare the conventional priors derived for these problems with other objective Bayesian proposals like the intrinsic priors. It is concluded that both priors behave similarly although interesting subtle differences arise. Finally, we accommodate the conventional priors to deal with non nested model selection as well as multiple model comparison.

Keywords: ANOVA models; Change Point problem; Model Selection; Objective Bayesian methods; Partially Informative Distributions; Regression models.

The manuscript is available in PDF format.