Wasserstein distances for discrete measures and convergence in nonparametric mixture models

We consider Wasserstein distance functionals for comparing between and assessing the convergence of latent discrete measures, which serve as mixing distributions in hierarchical and nonparametric mixture models. We explore the space of discrete probability measures metrized by Wasserstein distances, clarify the relationships between Wasserstein distances of mixing distributions and $f$-divergence functionals such as Hellinger and Kullback-Leibler distances on the space of mixture distributions. The convergence in Wasserstein metrics has a useful interpretation of the convergence of individual atoms that provide support for the discrete measure. It can be shown to be stronger than the weak convergence induced by standard $f$-divergence metrics, while the conditions for establishing the convergence can be formulated in terms of the metric space of the supporting atoms. These results are applied to establish rates of convergence of posterior distributions for latent discrete measures in several mixture models, including finite mixtures of multivariate distributions, finite mixtures of Gaussian processes and infinite mixtures based on the Dirichlet process.