Multivariate Dirichlet Moments and a Polychromatic Ewens Sampling Formula

We present an elementary non-recursive formula for the multivariate moments of the Dirichlet distribution on the standard simplex, in terms of the pattern inventory of the moments' exponents. We obtain analog formulas for the multivariate moments of the Dirichlet-Ferguson and Gamma measures. We further introduce a polychromatic analogue of Ewens sampling formula on colored integer partitions, discuss its relation with suitable extensions of Hoppe's urn model and of the Chinese restaurant process, and prove that it satisfies an adapted notion of consistency in the sense of Kingman.Comment: 22 page

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

In many applications, a finite mixture is a natural model, but it can be difficult to choose an appropriate number of components. To circumvent this choice, investigators are increasingly turning to Dirichlet process mixtures (DPMs), and Pitman-Yor process mixtures (PYMs), more generally. While these models may be well-suited for Bayesian density estimation, many investigators are using them for inferences about the number of components, by considering the posterior on the number of components represented in the observed data. We show that this posterior is not consistent --- that is, on data from a finite mixture, it does not concentrate at the true number of components. This result applies to a large class of nonparametric mixtures, including DPMs and PYMs, over a wide variety of families of component distributions, including essentially all discrete families, as well as continuous exponential families satisfying mild regularity conditions (such as multivariate Gaussians).Comment: This is a general treatment of the problem discussed in our related article, "A simple example of Dirichlet process mixture inconsistency for the number of components", Miller and Harrison (2013) arXiv:1301.270