Tree-Based Inference for Dirichlet Process Mixtures
; Proceedings of the Twelth International Conference on Artificial Intelligence and Statistics, PMLR 5:623-630, 2009.
The Dirichlet process mixture (DPM) is a widely used model for clustering and for general nonparametric Bayesian density estimation. Unfortunately, like in many statistical models, exact inference in a DPM is intractable, and approximate methods are needed to perform efficient inference. While most attention in the literature has been placed on Markov chain Monte Carlo (MCMC), variational Bayesian (VB) and collapsed variational methods, Heller and Ghahramani (2005) recently introduced a novel class of approximation for DPMs based on Bayesian hierarchical clustering (BHC). These tree-based combinatorial approximations efficiently sum over exponentially many ways of partitioning the data and offer a novel lower bound on the marginal likelihood of the DPM. In this paper we make the following contributions: (1) We show empirically that the BHC lower bounds are substantially tighter than the bounds given by VB and by collapsed variational methods on synthetic and real datasets. (2) We also show that BHC offers a more accurate predictive performance on these datasets. (3) We further improve the tree-based lower bounds with an algorithm that efficiently sums contributions from alternative trees. (4) We present a fast approximate method for BHC. Our results suggest that our combinatorial approximate inference methods and lower bounds may be useful not only in DPMs but in other models as well.