We consider problems involving groups of data where each observation within a group is a draw from a mixture model and where it is desirable to share mixture components between groups. We assume that the number of mixture components is unknown a priori and is to be inferred from the data. In this setting it is natural to consider sets of Dirichlet processes, one for each group, where the well-known clustering property of the Dirichlet process provides a nonparametric prior for the number of mixture components within each group. Given our desire to tie the mixture models in the various groups, we consider a hierarchical model, specifically one in which the base measure for the child Dirichlet processes is itself distributed according to a Dirichlet process. Such a base measure being discrete, the child Dirichlet processes necessarily share atoms. Thus, as desired, the mixture models in the different groups necessarily share mixture components. We discuss representations of hierarchical Dirichlet processes in terms of a stick-breaking process, and a generalization of the Chinese restaurant process that we refer to as the "Chinese restaurant franchise." We present Markov chain Monte Carlo algorithms for posterior inference in hierarchical Dirichlet process mixtures and describe applications to problems in information retrieval and text modeling.

One of the core problems of modern statistics is to approximate difficult-to-compute probability densities. This problem is especially important in Bayesian statistics, which frames all inference about unknown quantities as a calculation involving the posterior density. In this paper, we review variational inference (VI), a method from machine learning that approximates probability densities through optimization. VI has been used in many applications and tends to be faster than classical methods, such as Markov chain Monte Carlo sampling. The idea behind VI is to first posit a family of densities and then to find the member of that family which is close to the target. Closeness is measured by Kullback-Leibler divergence. We review the ideas behind mean-field variational inference, discuss the special case of VI applied to exponential family models, present a full example with a Bayesian mixture of Gaussians, and derive a variant that uses stochastic optimization to scale up to massive data. We discuss modern research in VI and highlight important open problems. VI is powerful, but it is not yet well understood. Our hope in writing this paper is to catalyze statistical research on this class of algorithms.

Surveying a suite of algorithms that offer a solution to managing large document archives.