Bayes Factors and Hierarchical Models

Consider m sets of data x(h), each with distribution depending on an unknown parameter theta(h), h = 1,2,..., m, and assume an exchangeable prior for the conditional distribution of theta(1), theta(2), ..., theta(m), given theta(h) not equal 0, h = 1,2, ..., m. In this paper we consider the problem of comparing the hypotheses {theta(i)=0} against {theta(i)not equal 0} with the Bayes factor (BF) and using all the data. This problem has practical interest in a variety of situations. Suppose, for instance, that the data are observed treatment effects in m groups of patients, all affected by the same disease, and theta(h) is the unknown treatment effect size for the hth group. If the ith group consists with patients still under treatment, then the decision of suspending it can be based on the above BF. We show that the BF is the product of two factors, both expressing the evidence of the data in favor of {theta(i) not equal 0} and against {theta(i) = 0}. The first of the two factors is the BF based on data xi only, the second factor gives the evidence coming from the joint knowledge of x(1),x(2), ...,x(m). Suitable alternative BFs are then defined to assign a non-informative prior to the hyperparameter at the third stage of the model. An application to hierarchical normal models is considered and the extension to the nuisance parameters case is outlined.