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507
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 817 (71 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
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 304 (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 242 (42 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
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 207 (21 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.
Hierarchical beta processes and the Indian buffet process. This volume
 In Practical Nonparametric and Semiparametric Bayesian Statistics
, 2007
"... We show that the beta process is the de Finetti mixing distribution underlying the Indian buffet process of [2]. This result shows that the beta process plays the role for the Indian buffet process that the Dirichlet process plays for Chinese restaurant process, a parallel that guides us in deriving ..."
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Cited by 123 (19 self)
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We show that the beta process is the de Finetti mixing distribution underlying the Indian buffet process of [2]. This result shows that the beta process plays the role for the Indian buffet process that the Dirichlet process plays for Chinese restaurant process, a parallel that guides us in deriving analogs for the beta process of the many known extensions of the Dirichlet process. In particular we define Bayesian hierarchies of beta processes and use the connection to the beta process to develop posterior inference algorithms for the Indian buffet process. We also present an application to document classification, exploring a relationship between the hierarchical beta process and smoothed naive Bayes models. 1 1
Multitask learning for classification with dirichlet process priors
 Journal of Machine Learning Research
, 2007
"... Multitask learning (MTL) is considered for logisticregression 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 asy ..."
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Cited by 118 (10 self)
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Multitask learning (MTL) is considered for logisticregression 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 singletask 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.
The nested chinese restaurant process and bayesian inference of topic hierarchies
, 2007
"... We present the nested Chinese restaurant process (nCRP), a stochastic process which assigns probability distributions to infinitelydeep, infinitelybranching trees. We show how this stochastic process can be used as a prior distribution in a Bayesian nonparametric model of document collections. Spe ..."
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Cited by 112 (15 self)
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We present the nested Chinese restaurant process (nCRP), a stochastic process which assigns probability distributions to infinitelydeep, infinitelybranching trees. We show how this stochastic process can be used as a prior distribution in a Bayesian nonparametric model of document collections. Specifically, we present an application to information retrieval in which documents are modeled as paths down a random tree, and the preferential attachment dynamics of the nCRP leads to clustering of documents according to sharing of topics at multiple levels of abstraction. Given a corpus of documents, a posterior inference algorithm finds an approximation to a posterior distribution over trees, topics and allocations of words to levels of the tree. We demonstrate this algorithm on collections of scientific abstracts from several journals. This model exemplifies a recent trend in statistical machine learning—the use of Bayesian nonparametric methods to infer distributions on flexible data structures.
The infinite PCFG using hierarchical Dirichlet processes
 In EMNLP ’07
, 2007
"... We present a nonparametric Bayesian model of tree structures based on the hierarchical Dirichlet process (HDP). Our HDPPCFG 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 effici ..."
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Cited by 106 (6 self)
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We present a nonparametric Bayesian model of tree structures based on the hierarchical Dirichlet process (HDP). Our HDPPCFG 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 fullscale parsing applications by demonstrating its effectiveness in learning statesplit grammars. 1
Orderbased dependent dirichlet processes
 Journal of the American Statistical Association
"... In this paper we propose a new framework for Bayesian nonparametric modelling with continuous covariates. In particular, we allow the nonparametric distribution to depend on covariates through ordering the random variables building the weights in the stickbreaking representation. We focus mostly o ..."
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Cited by 102 (5 self)
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In this paper we propose a new framework for Bayesian nonparametric modelling with continuous covariates. In particular, we allow the nonparametric distribution to depend on covariates through ordering the random variables building the weights in the stickbreaking representation. We focus mostly on the class of random distributions which induces a Dirichlet process at each covariate value. We derive the correlation between distributions at different covariate values, and use a point process to implement a practically useful type of ordering. Two main constructions with analytically known correlation structures are proposed. Practical and efficient computational methods are introduced. We apply our framework, though mixtures of these processes, to regression modelling, the modelling of stochastic volatility in time series data and spatial geostatistical modelling.
Stochastic Variational Inference
 JOURNAL OF MACHINE LEARNING RESEARCH (2013, IN PRESS)
, 2013
"... We develop stochastic variational inference, a scalable algorithm for approximating posterior distributions. We develop this technique for a large class of probabilistic models and we demonstrate it with two probabilistic topic models, latent Dirichlet allocation and the hierarchical Dirichlet proce ..."
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Cited by 88 (22 self)
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We develop stochastic variational inference, a scalable algorithm for approximating posterior distributions. We develop this technique for a large class of probabilistic models and we demonstrate it with two probabilistic topic models, latent Dirichlet allocation and the hierarchical Dirichlet process topic model. Using stochastic variational inference, we analyze several large collections of documents: 300K articles from Nature, 1.8M articles from The New York Times, and 3.8M articles from Wikipedia. Stochastic inference can easily handle data sets of this size and outperforms traditional variational inference, which can only handle a smaller subset. (We also show that the Bayesian nonparametric topic model outperforms its parametric counterpart.) Stochastic variational inference lets us apply complex Bayesian models to massive data sets.