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18
Reversible jump Markov chain Monte Carlo computation and Bayesian model determination
 Biometrika
, 1995
"... Markov chain Monte Carlo methods for Bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some xed standard underlying measure. They have therefore not been available for application to Bayesian model determi ..."
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Cited by 827 (19 self)
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Markov chain Monte Carlo methods for Bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some xed standard underlying measure. They have therefore not been available for application to Bayesian model determination, where the dimensionality of the parameter vector is typically not xed. This article proposes a new framework for the construction of reversible Markov chain samplers that jump between parameter subspaces of di ering dimensionality, which is exible and entirely constructive. It should therefore have wide applicability in model determination problems. The methodology is illustrated with applications to multiple changepoint analysis in one and two dimensions, and toaBayesian comparison of binomial experiments.
Modelling heterogeneity with and without the Dirichlet process
, 2001
"... We investigate the relationships between Dirichlet process (DP) based models and allocation models for a variable number of components, based on exchangeable distributions. It is shown that the DP partition distribution is a limiting case of a Dirichlet± multinomial allocation model. Comparisons of ..."
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Cited by 68 (3 self)
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We investigate the relationships between Dirichlet process (DP) based models and allocation models for a variable number of components, based on exchangeable distributions. It is shown that the DP partition distribution is a limiting case of a Dirichlet± multinomial allocation model. Comparisons of posterior performance of DP and allocation models are made in the Bayesian paradigm and illustrated in the context of univariate mixture models. It is shown in particular that the unbalancedness of the allocation distribution, present in the prior DP model, persists a posteriori. Exploiting the model connections, a new MCMC sampler for general DP based models is introduced, which uses split/merge moves in a reversible jump framework. Performance of this new sampler relative to that of some traditional samplers for DP processes is then explored.
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.
A Bayesian Model for Collaborative Filtering
, 2002
"... Consider the general setup where a set of items have been partially rated by a set of judges, in the sense that not every item has been rated by every judge. For this setup, we propose a Bayesian approach for the problem of predicting the missing ratings from the observed ratings. This approach inco ..."
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Cited by 21 (0 self)
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Consider the general setup where a set of items have been partially rated by a set of judges, in the sense that not every item has been rated by every judge. For this setup, we propose a Bayesian approach for the problem of predicting the missing ratings from the observed ratings. This approach incorporates similarity by assuming the set of judges can be partitioned into groups which share the same ratings probability distribution. This leads to a predictive distribution of missing ratings based on the posterior distribution of the groupings and associated ratings probabilities. Markov chain Monte Carlo methods and a hybrid search algorithm are then used to obtain predictions of the missing ratings. 1
Bayesian Tests And Model Diagnostics In Conditionally Independent Hierarchical Models
 Journal of the American Statistical Association
, 1994
"... Consider the conditionally independent hierarchical model (CIHM) where observations y i are independently distributed from f(y i j` i ), the parameters ` i are independently distributed from distributions g(`j), and the hyperparameters are distributed according to a distribution h(). The posterior ..."
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Cited by 16 (1 self)
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Consider the conditionally independent hierarchical model (CIHM) where observations y i are independently distributed from f(y i j` i ), the parameters ` i are independently distributed from distributions g(`j), and the hyperparameters are distributed according to a distribution h(). The posterior distribution of all parameters of the CIHM can be efficiently simulated by Monte Carlo Markov Chain (MCMC) algorithms. Although these simulation algorithms have facilitated the application of CIHM's, they generally have not addressed the problem of computing quantities useful in model selection. This paper explores how MCMC simulation algorithms and other related computational algorithms can be used to compute Bayes factors that are useful in criticizing a particular CIHM. In the case where the CIHM models a belief that the parameters are exchangeable or lie on a regression surface, the Bayes factor can measure the consistency of the data with the structural prior belief. Bayes factors can ...
Clustering Using Objective Functions and Stochastic Search
, 2007
"... Summary. A new approach to clustering multivariate data, based on a multilevel linear mixed model, is proposed. A key feature of the model is that observations from the same cluster are correlated, because they share clusterspecific random effects. The inclusion of clusterspecific random effects a ..."
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Cited by 16 (3 self)
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Summary. A new approach to clustering multivariate data, based on a multilevel linear mixed model, is proposed. A key feature of the model is that observations from the same cluster are correlated, because they share clusterspecific random effects. The inclusion of clusterspecific random effects allows parsimonious departure from an assumed base model for cluster mean profiles. This departure is captured statistically via the posterior expectation, or best linear unbiased predictor. One of the parameters in the model is the true underlying partition of the data, and the posterior distribution of this parameter, which is known up to a normalizing constant, is used to cluster the data. The problem of finding partitions with high posterior probability is not amenable to deterministic methods such as the EM algorithm. Thus, we propose a stochastic search algorithm that is driven by a Markov chain that is a mixture of two Metropolis–Hastings algorithms—one that makes small scale changes to individual objects and another that performs large scale moves involving entire clusters. The methodology proposed is fundamentally different from the wellknown finite mixture model approach to clustering, which does not explicitly include the partition as a parameter, and involves an independent and identically distributed structure.
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.
Bayesian analysis of extreme values by mixture modeling
 Extremes
, 2003
"... Modeling of extreme values in the presence of heterogeneity is still a relatively unexplored area. We consider losses pertaining to several related categories. For each category, we view exceedances over a given threshold as generated by a Poisson process whose intensity is regulated by a specific l ..."
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Cited by 8 (1 self)
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Modeling of extreme values in the presence of heterogeneity is still a relatively unexplored area. We consider losses pertaining to several related categories. For each category, we view exceedances over a given threshold as generated by a Poisson process whose intensity is regulated by a specific location, shape and scale parameter. Using a Bayesian approach, we develop a hierarchical mixture prior, with an unknown number of components, for each of the above parameters. Computations are performed using Reversible Jump MCMC. Our model accounts for possible grouping effects and takes advantage of the similarity across categories, both for estimation and prediction purposes. Some guidance on the specification of the prior distribution is provided, together with an assessment of inferential robustness. The method is illustrated throughout using a data set on large claims against a wellknown insurance company over a 15year period.
A note on the Dirichlet process prior in Bayesian nonparametric inference with partial exchangeability
 Statist. Prob. Letters
, 1997
"... We consider Bayesian nonparametric inference for continuousvalued partially exchangeable data, when the partition of the observations into groups is unknown. This includes changepoint problems and mixture models. As the prior, we consider a mixture of products of Dirichlet processes. We show that ..."
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Cited by 6 (1 self)
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We consider Bayesian nonparametric inference for continuousvalued partially exchangeable data, when the partition of the observations into groups is unknown. This includes changepoint problems and mixture models. As the prior, we consider a mixture of products of Dirichlet processes. We show that the discreteness of the Dirichlet process can have a large effect on inference (posterior distributions and Bayes factors), leading to conclusions that can be different from those that result from a reasonable parametric model. When the observed data are all distinct, the effect of the prior on the posterior is to favor more evenly balanced partitions, and its effect on Bayes factors is to favor more groups. In a hierarchical model with a Dirichlet process as the secondstage prior, the prior can also have a large effect on inference, but in the opposite direction, towards more unbalanced partitions. (~) 1997 Elsevier Science B.V.
Relaxing the Local Independence Assumption for Quantitative Learning in Acyclic Directed Graphical Models through Hierarchical Partition Models
 Proceedings of Artificial Intelligence and Statistics ’99
, 1999
"... The simplest method proposed by Spiegelhalter and Lauritzen (1990) to perform quantitative learning in ADG presents a potential weakness: the local independence assumption. We propose to alleviate this problem through the use of Hierarchical Partition Models. Our approach is compared with the previo ..."
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Cited by 6 (0 self)
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The simplest method proposed by Spiegelhalter and Lauritzen (1990) to perform quantitative learning in ADG presents a potential weakness: the local independence assumption. We propose to alleviate this problem through the use of Hierarchical Partition Models. Our approach is compared with the previous one from an interpretative and predictive point of view. 1 INTRODUCTION Spiegelhalter and Lauritzen (1990) (SL) proposed a Bayesian model for Acyclic Directed Graphical Models (ADG) (also known as Bayesian Networks) that has become somewhat standard in the burgeoning literature on learning discrete graphical models. The basic idea is to treat the conditional probabilities of the random variables at each vertex in the graph as unknowns and associate a prior distribution on each one (the conditioning in each case is on the random variables associated with the parent vertices in the graph). The simplest approach of SL introduces strong assumptions on the unknown conditional probabilities ...