  This paper studies posterior concentration behavior of the base probability
measure of a Dirichlet measure, given observations associated with the sampled
Dirichlet processes, as the number of observations tends to infinity. The base
measure itself is endowed with another Dirichlet prior, a construction known as
the hierarchical Dirichlet processes (Teh et al. [J. Amer. Statist. Assoc. 101
(2006) 1566-1581]). Convergence rates are established in transportation
distances (i.e., Wasserstein metrics) under various conditions on the geometry
of 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 powerful insight often invoked 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.
