A note on the Dirichlet process prior in Bayesian nonparametric inference with partial exchangeability

We consider Bayesian nonparametric inference for continuous-valued partially exchangeable data, when the partition of the observations into groups is unknown. This includes change-point problems and mixture models. As the prior, we consider a mixture of products of Dirichlet processes. We show that the discreteness of the Dirichlet process can have a substantial effect on inference (posterior distributions and Bayes factors), leading to conclusions that can be different from those that result from a reasonable parametric model. When the observed data are all distinct, the effect of the prior on the posterior is to favor more evenly balanced partitions, and its effect on Bayes factors is to favor more groups. In a hierarchical model with a Dirichlet process as the second-stage prior, the prior can also have a large effect on inference, but in the opposite direction, towards more unbalanced partitions.