Sampling hierarchies of discrete random structures

Hierarchical normalized discrete random measures identify a general class of priors that is suited to flexibly learn how the distribution of a response variable changes across groups of observations. A special case widely used in practice is the hierarchical Dirichlet process. Although current theory on hierarchies of nonparametric priors yields all relevant tools for drawing posterior inference, their implementation comes at a high computational cost. We fill this gap by proposing an approximation for a general class of hierarchical processes, which leads to an efficient conditional Gibbs sampling algorithm. The key idea consists of a deterministic truncation of the underlying random probability measures leading to a finite dimensional approximation of the original prior law. We provide both empirical and theoretical support for such a procedure.