  Dirichlet process mixtures are flexible non-parametric models, particularly
suited to density estimation and probabilistic clustering. In this work we
study the posterior distribution induced by Dirichlet process mixtures as the
sample size increases, and more specifically focus on consistency for the
unknown number of clusters when the observed data are generated from a finite
mixture. Crucially, we consider the situation where a prior is placed on the
concentration parameter of the underlying Dirichlet process. Previous findings
in the literature suggest that Dirichlet process mixtures are typically not
consistent for the number of clusters if the concentration parameter is held
fixed and data come from a finite mixture. Here we show that consistency for
the number of clusters can be achieved if the concentration parameter is
adapted in a fully Bayesian way, as commonly done in practice. Our results are
derived for data coming from a class of finite mixtures, with mild assumptions
on the prior for the concentration parameter and for a variety of choices of
likelihood kernels for the mixture.
