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50
Transdimensional Markov chain Monte Carlo
 in Highly Structured Stochastic Systems
, 2003
"... In the context of samplebased computation of Bayesian posterior distributions in complex stochastic systems, this chapter discusses some of the uses for a Markov chain with a prescribed invariant distribution whose support is a union of euclidean spaces of differing dimensions. This leads into a re ..."
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Cited by 56 (0 self)
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In the context of samplebased computation of Bayesian posterior distributions in complex stochastic systems, this chapter discusses some of the uses for a Markov chain with a prescribed invariant distribution whose support is a union of euclidean spaces of differing dimensions. This leads into a reformulation of the reversible jump MCMC framework for constructing such ‘transdimensional ’ Markov chains. This framework is compared to alternative approaches for the same task, including methods that involve separate sampling within different fixeddimension models. We consider some of the difficulties researchers have encountered with obtaining adequate performance with some of these methods, attributing some of these to misunderstandings, and offer tentative recommendations about algorithm choice for various classes of problem. The chapter concludes with a look towards desirable future developments.
Hidden Markov models and disease mapping
 Journal of the American Statistical Association
, 2001
"... We present new methodology to extend Hidden Markov models to the spatial domain, and use this class of models to analyse spatial heterogeneity of count data on a rare phenomenon. This situation occurs commonly in many domains of application, particularly in disease mapping. We assume that the counts ..."
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Cited by 56 (4 self)
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We present new methodology to extend Hidden Markov models to the spatial domain, and use this class of models to analyse spatial heterogeneity of count data on a rare phenomenon. This situation occurs commonly in many domains of application, particularly in disease mapping. We assume that the counts follow a Poisson model at the lowest level of the hierarchy, and introduce a finite mixture model for the Poisson rates at the next level. The novelty lies in the model for allocation to the mixture components, which follows a spatially correlated process, the Potts model, and in treating the number of components of the spatial mixture as unknown. Inference is performed in a Bayesian framework using reversible jump MCMC. The model introduced can be viewed as a Bayesian semiparametric approach to specifying exible spatial distribution in hierarchical models. Performance of the model and comparison with an alternative wellknown Markov random field specification for the Poisson rates are demonstrated on synthetic data sets. We show that our allocation model avoids the problem of oversmoothing in cases where the underlying rates exhibit discontinuities, while giving equally good results in cases of smooth gradientlike or highly autocorrelated rates. The methodology is illustrated on an epidemiological application to data on a rare cancer in France.
Markov Chain Monte Carlo methods and the label switching problem in Bayesian mixture modelling
 Statistical Science
"... Abstract. In the past ten years there has been a dramatic increase of interest in the Bayesian analysis of finite mixture models. This is primarily because of the emergence of Markov chain Monte Carlo (MCMC) methods. While MCMC provides a convenient way to draw inference from complicated statistical ..."
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Cited by 51 (4 self)
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Abstract. In the past ten years there has been a dramatic increase of interest in the Bayesian analysis of finite mixture models. This is primarily because of the emergence of Markov chain Monte Carlo (MCMC) methods. While MCMC provides a convenient way to draw inference from complicated statistical models, there are many, perhaps underappreciated, problems associated with the MCMC analysis of mixtures. The problems are mainly caused by the nonidentifiability of the components under symmetric priors, which leads to socalled label switching in the MCMC output. This means that ergodic averages of component specific quantities will be identical and thus useless for inference. We review the solutions to the label switching problem, such as artificial identifiability constraints, relabelling algorithms and label invariant loss functions. We also review various MCMC sampling schemes that have been suggested for mixture models and discuss posterior sensitivity to prior specification.
Efficient construction of reversible jump markov chain monte carlo proposal distributions
 Journal of the Royal Statistical Society: Series B (Statistical Methodology
"... Summary. The major implementational problem for reversible jump Markov chain Monte Carlo methods is that there is commonly no natural way to choose jump proposals since there is no Euclidean structure in the parameter space to guide our choice. We consider mechanisms for guiding the choice of propos ..."
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Cited by 38 (2 self)
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Summary. The major implementational problem for reversible jump Markov chain Monte Carlo methods is that there is commonly no natural way to choose jump proposals since there is no Euclidean structure in the parameter space to guide our choice. We consider mechanisms for guiding the choice of proposal. The first group of methods is based on an analysis of acceptance probabilities for jumps. Essentially, these methods involve a Taylor series expansion of the acceptance probability around certain canonical jumps and turn out to have close connections to Langevin algorithms.The second group of methods generalizes the reversible jump algorithm by using the socalled saturated space approach. These allow the chain to retain some degree of memory so that, when proposing to move from a smaller to a larger model, information is borrowed from the last time that the reverse move was performed. The main motivation for this paper is that, in complex problems, the probability that the Markov chain moves between such spaces may be prohibitively small, as the probability mass can be very thinly spread across the space. Therefore, finding reasonable jump proposals becomes extremely important. We illustrate the procedure by using several examples of reversible jump Markov chain Monte Carlo applications including the analysis of autoregressive time series, graphical Gaussian modelling and mixture modelling.
Bayesian variable selection in clustering highdimensional data
 Journal of the American Statistical Association
, 2005
"... Over the last decade, technological advances have generated an explosion of data with substantially smaller sample size relative to the number of covariates ( p ≫ n). A common goal in the analysis of such data involves uncovering the group structure of the observations and identifying the discrimina ..."
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Cited by 34 (4 self)
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Over the last decade, technological advances have generated an explosion of data with substantially smaller sample size relative to the number of covariates ( p ≫ n). A common goal in the analysis of such data involves uncovering the group structure of the observations and identifying the discriminating variables. In this article we propose a methodology for addressing these problems simultaneously. Given a set of variables, we formulate the clustering problem in terms of a multivariate normal mixture model with an unknown number of components and use the reversiblejump Markov chain Monte Carlo technique to define a sampler that moves between different dimensional spaces. We handle the problem of selecting a few predictors among the prohibitively vast number of variable subsets by introducing a binary exclusion/inclusion latent vector, which gets updated via stochastic search techniques. We specify conjugate priors and exploit the conjugacy by integrating out some of the parameters. We describe strategies for posterior inference and explore the performance of the methodology with simulated and real datasets.
Variable selection in clustering via Dirichlet process mixture models
, 2006
"... The increased collection of highdimensional data in various fields has raised a strong interest in clustering algorithms and variable selection procedures. In this paper, we propose a modelbased method that addresses the two problems simultaneously. We introduce a latent binary vector to identify ..."
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Cited by 25 (3 self)
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The increased collection of highdimensional data in various fields has raised a strong interest in clustering algorithms and variable selection procedures. In this paper, we propose a modelbased method that addresses the two problems simultaneously. We introduce a latent binary vector to identify discriminating variables and use Dirichlet process mixture models to define the cluster structure. We update the variable selection index using a Metropolis algorithm and obtain inference on the cluster structure via a splitmerge Markov chain Monte Carlo technique. We explore the performance of the methodology on simulated data and illustrate an application with a dna microarray study.
Transdimensional Markov Chains: A Decade of Progress and Future Perspectives
 Journal of the American Statistical Association
, 2005
"... The last ten years have witnessed the development of sampling frameworks that permit the construction of Markov chains which simultaneously traverse both parameter and model space. In this time substantial methodological progress has been made. In this article we present a survey of the current stat ..."
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Cited by 18 (2 self)
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The last ten years have witnessed the development of sampling frameworks that permit the construction of Markov chains which simultaneously traverse both parameter and model space. In this time substantial methodological progress has been made. In this article we present a survey of the current state of the art and evaluate some of the most recent advances in this field. We also discuss future research perspectives in the context of the drive to develop sampling mechanisms with high degrees of both efficiency and automation. 1
Hierarchical Gaussian Process Mixtures for Regression
, 2002
"... this paper, a mixture regression model of Gaussian processes is proposed, and a hybrid Markov chain Monte Carlo (MCMC) algorithm is used for the implementation. If we use this model and algorithm, the computational burden decreases dramatically. A real application is used to illustrate the mixture m ..."
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Cited by 16 (7 self)
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this paper, a mixture regression model of Gaussian processes is proposed, and a hybrid Markov chain Monte Carlo (MCMC) algorithm is used for the implementation. If we use this model and algorithm, the computational burden decreases dramatically. A real application is used to illustrate the mixture model and its implementation
Computing Normalizing Constants for Finite Mixture Models via Incremental Mixture Importance Sampling (IMIS)
, 2003
"... We propose a method for approximating integrated likelihoods in finite mixture models. We formulate the model in terms of the unobserved group memberships, z, and make them the variables of integration. The integral is then evaluated using importance sampling over the z. We propose an adaptive imp ..."
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Cited by 14 (5 self)
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We propose a method for approximating integrated likelihoods in finite mixture models. We formulate the model in terms of the unobserved group memberships, z, and make them the variables of integration. The integral is then evaluated using importance sampling over the z. We propose an adaptive importance sampling function which is itself a mixture, with two types of component distributions, one concentrated and one diffuse. The more concentrated type of component serves the usual purpose of an importance sampling function, sampling mostly group assignments of high posterior probability. The less concentrated type of component allows for the importance sampling function to explore the space in a controlled way to find other, unvisited assignments with high posterior probability. Components are added adaptively, one at a time, to cover areas of high posterior probability not well covered by the current important sampling function. The method is called Incremental Mixture Importance Sampling (IMIS). IMIS is easy to implement and to monitor for convergence. It scales easily for higher dimensional
Multivariate mixtures of normals with an unknown number of components
 Statist. Comp
, 2006
"... Multivariate techniques and especially cluster analysis have been commonly used in Archaeometry. Exploratory and modelbased techniques of clustering have been applied in geochemical (continuous) data of archaeological artifacts for provenance studies. Modelbased clustering techniques like classifi ..."
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Cited by 13 (0 self)
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Multivariate techniques and especially cluster analysis have been commonly used in Archaeometry. Exploratory and modelbased techniques of clustering have been applied in geochemical (continuous) data of archaeological artifacts for provenance studies. Modelbased clustering techniques like classification maximumlikelihood and mixture maximum likelihood had been used in a lesser extent in this context and although they seem to be suitable for such data, they either present practical difficultieslike high dimensionality of the data or their performance give no evidence to support that they prevail on the standard methods (Papageorgiou et al., 2001). In this paper standard statistical methods (hierarchical clustering, principal components analysis) and the recently developed one of the multivariate mixture of normals with unknown number of components (see Dellaportas and Papageorgiou, 2005) in the category of the model– based ones, are applied and compared. The data set comprises of chemical compositions in 188 ceramic samples derived from the Aegean islands and surrounding areas.