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A SplitMerge Markov Chain Monte Carlo Procedure for the Dirichlet Process Mixture Model
 Journal of Computational and Graphical Statistics
, 2000
"... . We propose a splitmerge Markov chain algorithm to address the problem of inefficient sampling for conjugate Dirichlet process mixture models. Traditional Markov chain Monte Carlo methods for Bayesian mixture models, such as Gibbs sampling, can become trapped in isolated modes corresponding to an ..."
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Cited by 91 (0 self)
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. We propose a splitmerge Markov chain algorithm to address the problem of inefficient sampling for conjugate Dirichlet process mixture models. Traditional Markov chain Monte Carlo methods for Bayesian mixture models, such as Gibbs sampling, can become trapped in isolated modes corresponding to an inappropriate clustering of data points. This article describes a MetropolisHastings procedure that can escape such local modes by splitting or merging mixture components. Our MetropolisHastings algorithm employs a new technique in which an appropriate proposal for splitting or merging components is obtained by using a restricted Gibbs sampling scan. We demonstrate empirically that our method outperforms the Gibbs sampler in situations where two or more components are similar in structure. Key words: Dirichlet process mixture model, Markov chain Monte Carlo, MetropolisHastings algorithm, Gibbs sampler, splitmerge updates 1 Introduction Mixture models are often applied to density estim...
Sequential Importance Sampling for Nonparametric Bayes Models: The Next Generation
 Journal of Statistics
, 1998
"... this paper, we exploit the similarities between the Gibbs sampler and the SIS, bringing over the improvements for Gibbs sampling algorithms to the SIS setting for nonparametric Bayes problems. These improvements result in an improved sampler and help satisfy questions of Diaconis (1995) pertaining t ..."
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Cited by 70 (6 self)
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this paper, we exploit the similarities between the Gibbs sampler and the SIS, bringing over the improvements for Gibbs sampling algorithms to the SIS setting for nonparametric Bayes problems. These improvements result in an improved sampler and help satisfy questions of Diaconis (1995) pertaining to convergence. Such an effort can see wide applications in many other problems related to dynamic systems where the SIS is useful (Berzuini et al. 1996; Liu and Chen 1996). Section 2 describes the specific model that we consider. For illustration we focus discussion on the betabinomial model, although the methods are applicable to other conjugate families. In Section 3, we describe the first generation of the SIS and Gibbs sampler in this context, and present the necessary conditional distributions upon which the techniques rely. Section 4 describes the alterations that create the second generation techniques, and provides specific algorithms for the model we consider. Section 5 presents a comparison of the techniques on a large set of data. Section 6 provides theory that ensures the proposed methods work and that is generally applicable to many other problems using importance sampling approaches. The final section presents discussion. 2 The Model
More Aspects of Polya Tree Distributions for Statistical Modelling
 Ann. Statist
, 1994
"... : The definition and elementary properties of Polya tree distributions are reviewed. Two theorems are presented showing that Polya trees can be constructed to concentrate arbitrarily closely about any desired pdf, and that Polya tree priors can put positive mass in every relative entropy neighborhoo ..."
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Cited by 57 (1 self)
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: The definition and elementary properties of Polya tree distributions are reviewed. Two theorems are presented showing that Polya trees can be constructed to concentrate arbitrarily closely about any desired pdf, and that Polya tree priors can put positive mass in every relative entropy neighborhood of every positive density with finite entropy, thereby satisfying a consistency condition. Such theorems are false for Dirichlet processes. Models are constructed combining partially specified Polya trees with other information like monotonicity or unimodality. It is shown how to compute bounds on posterior expectations over the class of all priors with the given specifications. A numerical example is given. A theorem of Diaconis and Freedman about Dirichlet processes is generalized to Polya trees, allowing Polya trees to be the models for errors in regression problems. Finally, empirical Bayes models using Dirichlet processes are generalized to Polya trees. An example from Berry and Chris...
Generalized weighted Chinese restaurant processes for species sampling mixture models
 Statistica Sinica
, 2003
"... 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 conj ..."
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Cited by 53 (8 self)
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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 twoparameter extension, the PitmanYor process and finite dimensional Dirichlet priors. Key words and phrases: Dirichlet process, exchangeable partition, finite dimensional Dirichlet prior, twoparameter PoissonDirichlet process, prediction rule, random probability measure, species sampling sequence.
Dirichlet Prior Sieves in Finite Normal Mixtures
 Statistica Sinica
, 2002
"... Abstract: The use of a finite dimensional Dirichlet prior in the finite normal mixture model has the effect of acting like a Bayesian method of sieves. Posterior consistency is directly related to the dimension of the sieve and the choice of the Dirichlet parameters in the prior. We find that naive ..."
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Cited by 40 (1 self)
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Abstract: The use of a finite dimensional Dirichlet prior in the finite normal mixture model has the effect of acting like a Bayesian method of sieves. Posterior consistency is directly related to the dimension of the sieve and the choice of the Dirichlet parameters in the prior. We find that naive use of the popular uniform Dirichlet prior leads to an inconsistent posterior. However, a simple adjustment to the parameters in the prior induces a random probability measure that approximates the Dirichlet process and yields a posterior that is strongly consistent for the density and weakly consistent for the unknown mixing distribution. The dimension of the resulting sieve can be selected easily in practice and a simple and efficient Gibbs sampler can be used to sample the posterior of the mixing distribution. Key words and phrases: BoseEinstein distribution, Dirichlet process, identification, method of sieves, random probability measure, relative entropy, weak convergence.
Bayesian density regression
 JOURNAL OF THE ROYAL STATISTICAL SOCIETY B
, 2007
"... This article considers Bayesian methods for density regression, allowing a random probability distribution to change flexibly with multiple predictors. The conditional response distribution is expressed as a nonparametric mixture of parametric densities, with the mixture distribution changing acc ..."
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Cited by 40 (23 self)
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This article considers Bayesian methods for density regression, allowing a random probability distribution to change flexibly with multiple predictors. The conditional response distribution is expressed as a nonparametric mixture of parametric densities, with the mixture distribution changing according to location in the predictor space. A new class of priors for dependent random measures is proposed for the collection of random mixing measures at each location. The conditional prior for the random measure at a given location is expressed as a mixture of a Dirichlet process (DP) distributed innovation measure and neighboring random measures. This specification results in a coherent prior for the joint measure, with the marginal random measure at each location being a finite mixture of DP basis measures. Integrating out the infinitedimensional collection of mixing measures, we obtain a simple expression for the conditional distribution of the subjectspecific random variables, which generalizes the Pólya urn scheme. Properties are considered and a simple Gibbs sampling algorithm is developed for posterior computation. The methods are illustrated using simulated data examples and epidemiologic studies.
Methods for Approximating Integrals in Statistics with Special Emphasis on Bayesian Integration Problems
 Statistical Science
"... This paper is a survey of the major techniques and approaches available for the numerical approximation of integrals in statistics. We classify these into five broad categories; namely, asymptotic methods, importance sampling, adaptive importance sampling, multiple quadrature and Markov chain method ..."
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Cited by 32 (4 self)
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This paper is a survey of the major techniques and approaches available for the numerical approximation of integrals in statistics. We classify these into five broad categories; namely, asymptotic methods, importance sampling, adaptive importance sampling, multiple quadrature and Markov chain methods. Each method is discussed giving an outline of the basic supporting theory and particular features of the technique. Conclusions are drawn concerning the relative merits of the methods based on the discussion and their application to three examples. The following broad recommendations are made. Asymptotic methods should only be considered in contexts where the integrand has a dominant peak with approximate ellipsoidal symmetry. Importance sampling, and preferably adaptive importance sampling, based on a multivariate Student should be used instead of asymptotics methods in such a context. Multiple quadrature, and in particular subregion adaptive integration, are the algorithms of choice for...
Poisson process partition calculus with an application to Bayesian . . .
, 2005
"... This article develops, and describes how to use, results concerning disintegrations of Poisson random measures. These results are fashioned as simple tools that can be tailormade to address inferential questions arising in a wide range of Bayesian nonparametric and spatial statistical models. The P ..."
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Cited by 32 (10 self)
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This article develops, and describes how to use, results concerning disintegrations of Poisson random measures. These results are fashioned as simple tools that can be tailormade to address inferential questions arising in a wide range of Bayesian nonparametric and spatial statistical models. The Poisson disintegration method is based on the formal statement of two results concerning a Laplace functional change of measure and a Poisson Palm/Fubini calculus in terms of random partitions of the integers {1,...,n}. The techniques are analogous to, but much more general than, techniques for the Dirichlet process and weighted gamma process developed in [Ann. Statist. 12
A computational approach for full nonparametric Bayesian inference under Dirichlet process mixture models
 Journal of Computational and Graphical Statistics
, 2002
"... Widely used parametric generalizedlinear models are, unfortunately,a somewhat limited class of speci � cations. Nonparametric aspects are often introduced to enrich this class, resultingin semiparametricmodels. Focusing on single or ksample problems,many classical nonparametricapproachesare limited ..."
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Cited by 27 (7 self)
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Widely used parametric generalizedlinear models are, unfortunately,a somewhat limited class of speci � cations. Nonparametric aspects are often introduced to enrich this class, resultingin semiparametricmodels. Focusing on single or ksample problems,many classical nonparametricapproachesare limited to hypothesistesting. Those that allow estimation are limited to certain functionals of the underlying distributions. Moreover, the associated inference often relies upon asymptotics when nonparametric speci � cations are often most appealing for smaller sample sizes. Bayesian nonparametricapproachesavoid asymptotics but have, to date, been limited in the range of inference. Working with Dirichlet process priors, we overcome the limitations of existing simulationbasedmodel � tting approaches which yield inference that is con � ned to posterior moments of linear functionals of the population distribution.This article provides a computationalapproach to obtain the entire posterior distribution for more general functionals. We illustrate with three applications: investigation of extreme value distributions associated with a single population, comparison of medians in a ksample problem, and comparison of survival times from different populations under fairly heavy censoring.