this article we propose three criteria that can be used to address model selection. These emphasize observables rather than parameters and are based on a certain Bayesian predictive density. They have a unifying basis that is simple and interpretable,are free of asymptotic de#nitions,and allow the incorporation of prior information. Moreover,two of these criteria are readily calibrated.

Data heterogeneity appears when the sample comes from at least two different populations. We analyze three types of situations. In the first and simplest case the majority of the data come from a central model and a few isolated observations come from a contaminating distribution. The data from the contaminating distribution are called outliers and they have been studied in depth in the statistical literature. In the second case we still have a central model but the heterogeneous data may appear in clusters of outliers which mask each other. This is the multiple outlier problem which is much more difficult to handle and it has been analyzed and understood in the last few years. The few Bayesian contributions to this problem are presented. In the third case we do not have a central model but instead different groups of data have been generated by different models. For multivariate normal this problem has been analyzed by mixture models under the name of cluster analysis, but a challenging area of research is to develop a general methodology for applying this multiple model approach to other statistical problems. Heterogeneity implies in general an increase in the uncertainty of predictions, and we present in this paper a procedure to measure this effect.