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In many applications, a finite mixture is a natural model, but it can bedifficult to choose an appropriate number of components. To circumvent thischoice, investigators are increasingly turning to Dirichlet process mixtures(DPMs), and Pitman-Yor process mixtures (PYMs), more generally. While thesemodels may be well-suited for Bayesian density estimation, many investigatorsare using them for inferences about the number of components, by consideringthe posterior on the number of components represented in the observed data. Weshow that this posterior is not consistent --- that is, on data from a finitemixture, it does not concentrate at the true number of components. This resultapplies to a large class of nonparametric mixtures, including DPMs and PYMs,over a wide variety of families of component distributions, includingessentially all discrete families, as well as continuous exponential familiessatisfying mild regularity conditions (such as multivariate Gaussians).

Inconsistency of Pitman-Yor process mixtures for the number of components