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

This paper studies posterior concentration behavior of the base probability measure of a Dirichlet process $\mathcal{D}$, given observations associated with $m$ random measures sampled from the Dirichlet process, as $m$ and the number of observations $m\times n$ tend to infinity. The base measure itself may be endowed with a prior, (another) Dirichlet process, a construction popularly known as the hierarchical Dirichlet process (Teh et al, 2006). The random measures sampled from Dirichlet process $\mathcal{D}$ serve as mixing measures for an exchangeable collection of $m$ mixture distributions, posterior concentration behavior of which is also investigated. Convergence rates are established in transportation distances (i.e. Wasserstein metrics), under the assumption that the true base measure has a finite but unknown number of support points in $\mathbb{R}^d$. This theory quantifies the benefit of "borrowing strength" in the inference of groups of data with small sample size --- a heuristic argument commonly used to motivate hierarchical modeling. In certain settings, the gain in efficiency can be dramatic, improving from a standard nonparametric rate to a parametric rate of convergence. Tools developed include transportation distances for nonparametric Bayesian hierarchies, the existence of tests for Dirichlet processes, and concentration properties of Dirichlet measures.