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351
Hierarchical Dirichlet processes
 Journal of the American Statistical Association
, 2004
"... 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 comp ..."
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Cited by 543 (56 self)
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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 wellknown 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
On Bayesian analysis of mixtures with an unknown number of components
 INSTITUTE OF INTERNATIONAL ECONOMICS PROJECT ON INTERNATIONAL COMPETITION POLICY," COM/DAFFE/CLP/TD(94)42
, 1997
"... ..."
ModelBased Clustering, Discriminant Analysis, and Density Estimation
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 2000
"... Cluster analysis is the automated search for groups of related observations in a data set. Most clustering done in practice is based largely on heuristic but intuitively reasonable procedures and most clustering methods available in commercial software are also of this type. However, there is little ..."
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Cited by 270 (24 self)
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Cluster analysis is the automated search for groups of related observations in a data set. Most clustering done in practice is based largely on heuristic but intuitively reasonable procedures and most clustering methods available in commercial software are also of this type. However, there is little systematic guidance associated with these methods for solving important practical questions that arise in cluster analysis, such as \How many clusters are there?", "Which clustering method should be used?" and \How should outliers be handled?". We outline a general methodology for modelbased clustering that provides a principled statistical approach to these issues. We also show that this can be useful for other problems in multivariate analysis, such as discriminant analysis and multivariate density estimation. We give examples from medical diagnosis, mineeld detection, cluster recovery from noisy data, and spatial density estimation. Finally, we mention limitations of the methodology, a...
An Introduction to MCMC for Machine Learning
, 2003
"... This purpose of this introductory paper is threefold. First, it introduces the Monte Carlo method with emphasis on probabilistic machine learning. Second, it reviews the main building blocks of modern Markov chain Monte Carlo simulation, thereby providing and introduction to the remaining papers of ..."
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Cited by 226 (2 self)
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This purpose of this introductory paper is threefold. First, it introduces the Monte Carlo method with emphasis on probabilistic machine learning. Second, it reviews the main building blocks of modern Markov chain Monte Carlo simulation, thereby providing and introduction to the remaining papers of this special issue. Lastly, it discusses new interesting research horizons.
Gibbs Sampling Methods for StickBreaking Priors
"... ... In this paper we present two general types of Gibbs samplers that can be used to fit posteriors of Bayesian hierarchical models based on stickbreaking priors. The first type of Gibbs sampler, referred to as a Polya urn Gibbs sampler, is a generalized version of a widely used Gibbs sampling meth ..."
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Cited by 216 (17 self)
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... In this paper we present two general types of Gibbs samplers that can be used to fit posteriors of Bayesian hierarchical models based on stickbreaking priors. The first type of Gibbs sampler, referred to as a Polya urn Gibbs sampler, is a generalized version of a widely used Gibbs sampling method currently employed for Dirichlet process computing. This method applies to stickbreaking priors with a known P'olya urn characterization; that is priors with an explicit and simple prediction rule. Our second method, the blocked Gibbs sampler, is based on a entirely different approach that works by directly sampling values from the posterior of the random measure. The blocked Gibbs sampler can be viewed as a more general approach as it works without requiring an explicit prediction rule. We find that the blocked Gibbs avoids some of the limitations seen with the Polya urn approach and should be simpler for nonexperts to use.
Infinite Latent Feature Models and the Indian Buffet Process
, 2005
"... 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 ..."
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Cited by 185 (38 self)
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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
Mixed membership stochastic block models for relational data with application to proteinprotein interactions
 In Proceedings of the International Biometrics Society Annual Meeting
, 2006
"... We develop a model for examining data that consists of pairwise measurements, for example, presence or absence of links between pairs of objects. Examples include protein interactions and gene regulatory networks, collections of authorrecipient email, and social networks. Analyzing such data with p ..."
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Cited by 180 (31 self)
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We develop a model for examining data that consists of pairwise measurements, for example, presence or absence of links between pairs of objects. Examples include protein interactions and gene regulatory networks, collections of authorrecipient email, and social networks. Analyzing such data with probabilistic models requires special assumptions, since the usual independence or exchangeability assumptions no longer hold. We introduce a class of latent variable models for pairwise measurements: mixed membership stochastic blockmodels. Models in this class combine a global model of dense patches of connectivity (blockmodel) and a local model to instantiate nodespecific variability in the connections (mixed membership). We develop a general variational inference algorithm for fast approximate posterior inference. We demonstrate the advantages of mixed membership stochastic blockmodels with applications to social networks and protein interaction networks.
Variational inference for Dirichlet process mixtures
 Bayesian Analysis
, 2005
"... Abstract. Dirichlet process (DP) mixture models are the cornerstone of nonparametric Bayesian statistics, and the development of MonteCarlo Markov chain (MCMC) sampling methods for DP mixtures has enabled the application of nonparametric Bayesian methods to a variety of practical data analysis prob ..."
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Cited by 130 (16 self)
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Abstract. Dirichlet process (DP) mixture models are the cornerstone of nonparametric Bayesian statistics, and the development of MonteCarlo 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 largescale image analysis problem.
Practical Bayesian Density Estimation Using Mixtures Of Normals
 Journal of the American Statistical Association
, 1995
"... this paper, we propose some solutions to these problems. Our goal is to come up with a simple, practical method for estimating the density. This is an interesting problem in its own right, as well as a first step towards solving other inference problems, such as providing more flexible distributions ..."
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Cited by 115 (2 self)
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this paper, we propose some solutions to these problems. Our goal is to come up with a simple, practical method for estimating the density. This is an interesting problem in its own right, as well as a first step towards solving other inference problems, such as providing more flexible distributions in hierarchical models. To see why the posterior is improper under the usual reference prior, we write the model in the following way. Let Z = (Z 1 ; : : : ; Z n ) and X = (X 1 ; : : : ; X n ). The Z
Dealing with label switching in mixture models
 Journal of the Royal Statistical Society, Series B
, 2000
"... In a Bayesian analysis of finite mixture models, parameter estimation and clustering are sometimes less straightforward that might be expected. In particular, the common practice of estimating parameters by their posterior mean, and summarising joint posterior distributions by marginal distributions ..."
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Cited by 111 (0 self)
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In a Bayesian analysis of finite mixture models, parameter estimation and clustering are sometimes less straightforward that might be expected. In particular, the common practice of estimating parameters by their posterior mean, and summarising joint posterior distributions by marginal distributions, often leads to nonsensical answers. This is due to the socalled “labelswitching” problem, which is caused by symmetry in the likelihood of the model parameters. A frequent response to this problem is to remove the symmetry using artificial identifiability constraints. We demonstrate that this fails in general to solve the problem, and describe an alternative class of approaches, relabelling algorithms, which arise from attempting to minimise the posterior expected loss under a class of loss functions. We describe in detail one particularly simple and general relabelling algorithm, and illustrate its success in dealing with the labelswitching problem on two examples.