Bayesian clustering and product partition models

We present a decision theoretic formulation of product partition models (PPMs) that allows a formal treatment of different decision problems such as estimation or hypothesis testing and clustering methods simultaneously. A key observation in our construction is the fact that PPMs can be formulated in the context of model selection. The underlying partition structure in these models is closely related to that arising in connection with Dirichlet processes. This allows a straightforward adaptation of some computational strategiesoriginally devised for nonparametric Bayesian problems-to our framework. The resulting algorithms are more flexible than other competing alternatives that are used for problems involving PPMs. We propose an algorithm that yields Bayes estimates of the quantities of interest and the groups of experimental units. We explore the application of our methods to the detection of outliers in normal and Student t regression models, with clustering structure equivalent to that induced by a Dirichlet process prior. We also discuss the sensitivity of the results considering different prior distributions for the partitions.