program. The authors wish to acknowledge helpful discussions with Lancelot James and Jim Pitman and the referees for useful comments. 1 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 necessar-ily 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

We define a probability distribution over equivalence classes of binary matrices with a finite number of rows and an unbounded number of columns. This distribution

Abstract. Dirichlet process (DP) mixture models are the cornerstone of nonparametric Bayesian statistics, and the development of Monte-Carlo Markov chain (MCMC) sampling methods for DP mixtures has enabled the application of nonparametric Bayesian methods to a variety of practical data analysis problems. However, MCMC sampling can be prohibitively slow, and it is important to explore alternatives. One class of alternatives is provided by variational methods, a class of deterministic algorithms that convert inference problems into optimization problems (Opper and Saad 2001; Wainwright and Jordan 2003). Thus far, variational methods have mainly been explored in the parametric setting, in particular within the formalism of the exponential family (Attias 2000; Ghahramani and Beal 2001; Blei et al. 2003). In this paper, we present a variational inference algorithm for DP mixtures. We present experiments that compare the algorithm to Gibbs sampling algorithms for DP mixtures of Gaussians and present an application to a large-scale image analysis problem.

Multi-task learning (MTL) is considered for logistic-regression classifiers, based on a Dirichlet process (DP) formulation. A symmetric MTL (SMTL) formulation is considered in which classifiers for multiple tasks are learned jointly, with a variational Bayesian (VB) solution. We also consider an asymmetric MTL (AMTL) formulation in which the posterior density function from the SMTL model parameters, from previous tasks, is used as a prior for a new task; this approach has the significant advantage of not requiring storage and use of all previous data from prior tasks. The AMTL formulation is solved with a simple Markov Chain Monte Carlo (MCMC) construction. Comparisons are also made to simpler approaches, such as single-task learning, pooling of data across tasks, and simplified approximations to DP. A comprehensive analysis of algorithm performance is addressed through consideration of two data sets that are matched to the MTL problem.

We propose a new hierarchical Bayesian n-gram model of natural languages. Our model makes use of a generalization of the commonly used Dirichlet distributions called Pitman-Yor processes which produce power-law distributions more closely resembling those in natural languages. We show that an approximation to the hierarchical Pitman-Yor language model recovers the exact formulation of interpolated Kneser-Ney, one of the best smoothing methods for n-gram language models. Experiments verify that our model gives cross entropy results superior to interpolated Kneser-Ney and comparable to modified Kneser-Ney. 1

We present a nonparametric Bayesian model of tree structures based on the hierarchical Dirichlet process (HDP). Our HDP-PCFG model allows the complexity of the grammar to grow as more training data is available. In addition to presenting a fully Bayesian model for the PCFG, we also develop an efficient variational inference procedure. On synthetic data, we recover the correct grammar without having to specify its complexity in advance. We also show that our techniques can be applied to full-scale parsing applications by demonstrating its effectiveness in learning state-split grammars. 1

Abstract: The class of species sampling mixture models is introduced as an extension of semiparametric models based on the Dirichlet process to models based on the general class of species sampling priors, or equivalently the class of all exchangeable urn distributions. Using Fubini calculus in conjunction with Pitman (1995, 1996), we derive characterizations of the posterior distribution in terms of a posterior partition distribution that extend the results of Lo (1984) for the Dirichlet process. These results provide a better understanding of models and have both theoretical and practical applications. To facilitate the use of our models we generalize the work in Brunner, Chan, James and Lo (2001) by extending their weighted Chinese restaurant (WCR) Monte Carlo procedure, an i.i.d. sequential importance sampling (SIS) procedure for approximating posterior mean functionals based on the Dirichlet process, to the case of approximation of mean functionals and additionally their posterior laws in species sampling mixture models. We also discuss collapsed Gibbs sampling, Pólya urn Gibbs sampling and a Pólya urn SIS scheme. Our framework allows for numerous applications, including multiplicative counting process models subject to weighted gamma processes, as well as nonparametric and semiparametric hierarchical models based on the Dirichlet process, its two-parameter extension, the Pitman-Yor process and finite dimensional Dirichlet priors. Key words and phrases: Dirichlet process, exchangeable partition, finite dimensional Dirichlet prior, two-parameter Poisson-Dirichlet process, prediction rule, random probability measure, species sampling sequence.

The Indian buffet process (IBP) is a Bayesian nonparametric distribution whereby objects are modelled using an unbounded number of latent features. In this paper we derive a stick-breaking representation for the IBP. Based on this new representation, we develop slice samplers for the IBP that are efficient, easy to implement and are more generally applicable than the currently available Gibbs sampler. This representation, along with the work of Thibaux and Jordan [17], also illuminates interesting theoretical connections between the IBP, Chinese restaurant processes, Beta processes and Dirichlet processes. 1

Variational inference methods, including mean field methods and loopy belief propagation, have been widely used for approximate probabilistic inference in graphical models. While often less accurate than MCMC, variational methods provide a fast deterministic approximation to marginal and conditional probabilities. Such approximations can be particularly useful in high dimensional problems where sampling methods are too slow to be effective. A limitation of current methods, however, is that they are restricted to parametric probabilistic models. MCMC does not have such a limitation; indeed, MCMC samplers have been developed for the Dirichlet process (DP), a nonparametric distribution on distributions (Ferguson, 1973) that is the cornerstone of Bayesian nonparametric statistics (Escobar & West, 1995; Neal, 2000). In this paper, we develop a meanfield variational approach to approximate inference for the Dirichlet process, where the approximate posterior is based on the truncated stick-breaking construction (Ishwaran & James, 2001). We compare our approach to DP samplers for Gaussian DP mixture models. 1.

Inference for Dirichlet process hierarchical models is typically performed using Markov chain Monte Carlo methods, which can be roughly categorised into marginal and conditional methods. The former integrate out analytically the infinite-dimensional component of the hierarchical model and sample from the marginal distribution of the remaining variables using the Gibbs sampler. Conditional methods impute the Dirichlet process and update it as a component of the Gibbs sampler. Since this requires imputation of an infinite-dimensional process, implementation of the conditional method has relied on finite approximations. In this paper we show how to avoid such approximations by designing two novel Markov chain Monte Carlo algorithms which sample from the exact posterior distribution of quantities of interest. The approximations are avoided by the new technique of retrospective sampling. We also show how the algorithms can obtain samples from functionals of the Dirichlet process. The marginal and the conditional methods are compared and a careful simulation study is included, which involves a non-conjugate model, different datasets and prior specifications.