Borrowing strength in hierarchical Bayes: convergence of the Dirichlet base measure

This paper studies posterior concentration behavior of the base probability measure $G$ of a Dirichlet measure $\mathcal{D}_{\alpha G}$, given observations associated with $m$ Dirichlet processes sampled from $\mathcal{D}_{\alpha G}$, as $m$ and the number of observations $m\times n$ tend to infinity. The base measure itself is endowed with another Dirichlet prior, a construction known as the hierarchical Dirichlet processes (Teh et al, 2006). Convergence rates are established in transportation distances (i.e. Wasserstein metrics) under various geometrically sparse conditions on the support of the true base measure. As a consequence of the theory we demonstrate the benefit of "borrowing strength" in the inference of multiple groups of data --- a heuristic argument commonly used to motivate hierarchical modeling. In certain settings, the gain in efficiency due to the latent hierarchy can be dramatic, improving from a standard nonparametric rate to a parametric rate of convergence. Tools developed include transportation distances for nonparametric Bayesian hierarchies of random measures, the existence of tests for Dirichlet measures, and geometric properties of the support of Dirichlet measures.