2006 HierarchicalDirichletProcesses

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Subject Headings: Dirichlet Process, Clustering, Mixture Model, Nonparametric Bayesian Analysis, Hierarchical Bayesian Model, Markov-Chain Monte Carlo Algorithm.

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Author Keywords

clustering, mixture models, nonparametric Bayesian statistics, hierarchical models, Markov chain Monte Carlo

Abstract

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 modelling.

1. Introduction

A recurring theme in statistics is the need to separate observations into groups, and yet allow the groups to remain linked — to "share statistical strength." In the Bayesian formalism such sharing is achieved naturally via hierarchical modeling; parameters are shared among groups, and the randomness of the parameters induces dependencies among the groups. Estimates based on the posterior distribution exhibit “shrinkage.”

In the current paper we explore a hierarchical approach to the problem of model-based clustering of grouped data. We assume that the data are subdivided into a set of groups, and that within each group we wish to find clusters that capture latent structure in the data assigned to that group. The number of clusters within each group is unknown and is to be inferred. Moreover, in a sense that we make precise, we wish to allow clusters to be shared among the groups.

...

Moreover, we may want to extend the model to allow for multiple corpora. For example, documents in scientific journals are often grouped into themes (e.g., “empirical process theory,” “multivariate statistics,” “survival analysis”), and it would be of interest to discover to what extent the latent topics that are shared among documents are also shared across these groupings. Thus in general we wish to consider the sharing of clusters across multiple, nested groupings of data

Our approach to the problem of sharing clusters among multiple, related groups is a nonparametric Bayesian approach, reposing on the Dirichlet process (Ferguson 1973). The Dirichlet process [math]\operatorname{DP}(\alpha_0,G_0)[/math] is a measure on measures. It has two parameters, a scaling parameter [math]\alpha_0 \gt 0[/math] and a base probability measure [math]G_0[/math]. An explicit representation of a draw from a Dirichlet process (DP) was given by Sethuraman (1994), who showed that if [math]G ~ \operatorname{DP}(\alpha_0,G_0)[/math], then with probability one:

[math]G = \Sigma_{k=1}^{\infty} \beta_k \delta_{\phi_k} , \text{(1)}[/math]

where the [math]\phi_k[/math] are independent random variables distributed according to [math]G0[/math], where [math]\delta_{\phi_k}[/math] is an atom at [math]\phi_k[/math], and where the “stick-breaking weights” [math]\beta_k[/math] are also random and depend on the parameter [math]\alpha_0[/math] (the definition of the [math]\beta_k[/math] is provided in Section 3.1).

The representation in (1) shows that draws from a DP are discrete (with probability one). The discrete nature of the DP makes it unsuitable for general applications in Bayesian nonparametrics, but it is well suited for the problem of placing priors on mixture components in mixture modeling. The idea is basically to associate a mixture component with each atom in [math]G[/math]. Introducing indicator variables to associate data points with mixture components, the posterior distribution yields a probability distribution on partitions of the data. A number of authors have studied such Dirichlet process mixture models (Antoniak 1974; Escobar and West 1995; MacEachern and M ̈uller 1998). These models provide an alternative to methods that attempt to select a particular number of mixture components, or methods that place an explicit parametric prior on the number of components.

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 AuthorvolumeDate ValuetitletypejournaltitleUrldoinoteyear
2006 HierarchicalDirichletProcessesYee Whye Teh
Michael I. Jordan
Matthew J. Beal
David M. Blei
Hierarchical Dirichlet ProcessesJournal of the American Statistical Associationhttp://www.gatsby.ucl.ac.uk/~ywteh/research/npbayes/jasa2006.pdf2006