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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 213 (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.
Retrospective Markov chain Monte Carlo methods for Dirichlet process hierarchical models
 PROC. IEEE
, 2008
"... 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 infinitedimensional component of the hierarchical model and sample fro ..."
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Cited by 41 (5 self)
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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 infinitedimensional 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 infinitedimensional 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 nonconjugate model, different datasets and prior specifications.
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.
Kernel stickbreaking processes
, 2007
"... Summary. This article proposes a class of kernel stickbreaking processes (KSBP) for uncountable collections of dependent random probability measures. The KSBP is constructed by first introducing an infinite sequence of random locations. Independent random probability measures and betadistributed ..."
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Cited by 39 (11 self)
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Summary. This article proposes a class of kernel stickbreaking processes (KSBP) for uncountable collections of dependent random probability measures. The KSBP is constructed by first introducing an infinite sequence of random locations. Independent random probability measures and betadistributed random weights are assigned to each location. Predictordependent random probability measures are then constructed by mixing over the locations, with stickbreaking probabilities expressed as a kernel multiplied by the beta weights. Some theoretical properties of the KSBP are described, including a covariatedependent prediction rule. A retrospective MCMC algorithm is developed for posterior computation, and the methods are illustrated using a simulated example and an epidemiologic application.
Approximate Dirichlet Process Computing in Finite Normal Mixtures: Smoothing and Prior Information
 JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS
, 2000
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Bayesian Model Selection in Finite Mixtures by Marginal Density Decompositions
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 2001
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Learning Multiscale Representations of Natural Scenes Using Dirichlet Processes
"... We develop nonparametric Bayesian models for multiscale representations of images depicting natural scene categories. Individual features or wavelet coefficients are marginally described by Dirichlet process (DP) mixtures, yielding the heavytailed marginal distributions characteristic of natural im ..."
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Cited by 21 (3 self)
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We develop nonparametric Bayesian models for multiscale representations of images depicting natural scene categories. Individual features or wavelet coefficients are marginally described by Dirichlet process (DP) mixtures, yielding the heavytailed marginal distributions characteristic of natural images. Dependencies between features are then captured with a hidden Markov tree, and Markov chain Monte Carlo methods used to learn models whose latent state space grows in complexity as more images are observed. By truncating the potentially infinite set of hidden states, we are able to exploit efficient belief propagation methods when learning these hierarchical Dirichlet process hidden Markov trees (HDPHMTs) from data. We show that our generative models capture interesting qualitative structure in natural scenes, and more accurately categorize novel images than models which ignore spatial relationships among features. 1.
Some Further Developments for StickBreaking Priors: Finite and Infinite Clustering and Classification
 Sankhya Series A
, 2003
"... this paper will be to develop new surrounding theory for the hierarchical model (7) and show how these may be used to develop computational algorithms for computing posterior quantities. Our theoretical contributions include developing key properties for the class of extended stickbreaking measures ..."
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Cited by 15 (0 self)
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this paper will be to develop new surrounding theory for the hierarchical model (7) and show how these may be used to develop computational algorithms for computing posterior quantities. Our theoretical contributions include developing key properties for the class of extended stickbreaking measures, which includes establishing a conjugacy property of their random weights to i.i.d sampling, and a characterization of the posterior for the extended stickbreaking prior under i.i.d sampling. See Section 3. These properties then lead us in Section 4 to a general characterization for the posterior of (7). In Section 5 we outline a collapsed Gibbs sampling algorithm and an i.i.d SIS (sequential importance sampling) algorithm that can be used for inference in (7). One important implication is our ability to t the posterior of (6) subject to in nite dimensional stickbreaking measures. The paper begins with a brief discussion of stickbreaking priors in Section 2
A General Framework for the Parametrization of Hierarchical Models
, 708
"... Abstract. In this paper, we describe centering and noncentering methodology as complementary techniques for use in parametrization of broad classes of hierarchical models, with a view to the construction of effective MCMC algorithms for exploring posterior distributions from these models. We give a ..."
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Cited by 7 (0 self)
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Abstract. In this paper, we describe centering and noncentering methodology as complementary techniques for use in parametrization of broad classes of hierarchical models, with a view to the construction of effective MCMC algorithms for exploring posterior distributions from these models. We give a clear qualitative understanding as to when centering and noncentering work well, and introduce theory concerning the convergence time complexity of Gibbs samplers using centered and noncentered parametrizations. We give general recipes for the construction of noncentered parametrizations, including an auxiliary variable technique called the statespace expansion technique. We also describe partially noncentered methods, and demonstrate their use in constructing robust Gibbs sampler algorithms whose convergence properties are not overly sensitive to the data. Key words and phrases: Parametrization, hierarchical models, latent stochastic processes, MCMC.
Hyperspectral image unmixing using a multiresolution sticky HDP
 IEEE Trans. Signal Processing
, 2012
"... Abstract—This paper is concerned with joint Bayesian endmember extraction and linear unmixing of hyperspectral images using a spatial prior on the abundance vectors. We propose a generative model for hyperspectral images in which the abundances are sampled from a Dirichlet distribution (DD) mixture ..."
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Cited by 7 (2 self)
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Abstract—This paper is concerned with joint Bayesian endmember extraction and linear unmixing of hyperspectral images using a spatial prior on the abundance vectors. We propose a generative model for hyperspectral images in which the abundances are sampled from a Dirichlet distribution (DD) mixture model, whose parameters depend on a latent label process. The label process is then used to enforces a spatial prior which encourages adjacent pixels to have the same label. A Gibbs sampling framework is used to generate samples from the posterior distributions of the abundances and the parameters of the DD mixture model. The spatial prior that is used is a treestructured sticky hierarchical Dirichlet process (SHDP) and, when used to determine the posterior endmember and abundance distributions, results in a new unmixing algorithm called spatially constrained unmixing (SCU). The directed Markov model facilitates the use of scalerecursive estimation algorithms, and is therefore more computationally efficient as compared to standard Markov random field (MRF) models. Furthermore, the proposed SCU algorithm estimates the number of regions in the image in an unsupervised fashion. The effectiveness of the proposed SCU algorithm is illustrated using synthetic and real data. Index Terms—Bayesian inference, hidden Markov trees, hyperspectral unmixing, image segmentation, spatially constrained unmixing, sticky hierarchical Dirichlet process. I.