Bayesian model selection methods for nonnested models

In the Bayesian approach to model selection and prediction, the posterior probability of each model under consideration must be computed. In the presence of weak prior information we need using default or automatic priors, that are typically improper, for the parameters of the models. However this leads to ill-defined posterior probabilities.Several methods have recently been proposed to overcome this difficulty. Of particular interest are the intrinsic and fractional methodologies introduced by Berger and Pericchi (1996) and O’Hagan (1995), respectively. In the specific case of comparing nested models, a significant feature of these methods is that they allow to derive proper priors to be used in the analysis.Unfortunately, the above methods do not apply to nonnested models: either the priors derived depend on the label we assign to models under comparison, and this may give two possible values to the posterior probability of each model, or they depends on the form of “encompassing”.We first consider a particular nonnested model selection problem: the one-sided testing problem. The encompassing approach for converting the non-nested problem into a nested one is discussed, and an alternative solution is proposed. Its behavior is illustrated on exponential distributions.For more general nonnested model selection problems we argue that an accessible piece of prior information on the observable random variable helps to find the posterior probability of the models. The way to deal with such a prior information is developed and illustrated on the comparison of separate normal and double exponential families of distributions.