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60
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.
On Bayesian analysis of mixtures with an unknown number of components
 INSTITUTE OF INTERNATIONAL ECONOMICS PROJECT ON INTERNATIONAL COMPETITION POLICY," COM/DAFFE/CLP/TD(94)42
, 1997
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Dealing with label switching in mixture models
 Journal of the Royal Statistical Society, Series B
, 2000
"... In a Bayesian analysis of finite mixture models, parameter estimation and clustering are sometimes less straightforward that might be expected. In particular, the common practice of estimating parameters by their posterior mean, and summarising joint posterior distributions by marginal distributions ..."
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Cited by 109 (0 self)
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In a Bayesian analysis of finite mixture models, parameter estimation and clustering are sometimes less straightforward that might be expected. In particular, the common practice of estimating parameters by their posterior mean, and summarising joint posterior distributions by marginal distributions, often leads to nonsensical answers. This is due to the socalled “labelswitching” problem, which is caused by symmetry in the likelihood of the model parameters. A frequent response to this problem is to remove the symmetry using artificial identifiability constraints. We demonstrate that this fails in general to solve the problem, and describe an alternative class of approaches, relabelling algorithms, which arise from attempting to minimise the posterior expected loss under a class of loss functions. We describe in detail one particularly simple and general relabelling algorithm, and illustrate its success in dealing with the labelswitching problem on two examples.
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...
Bayesian Methods for Hidden Markov Models  Recursive Computing in the 21st Century
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 2002
"... Markov chain Monte Carlo (MCMC) sampling strategies can be used to simulate hidden Markov model (HMM) parameters from their posterior distribution given observed data. Some MCMC methods (for computing likelihood, conditional probabilities of hidden states, and the most likely sequence of states) use ..."
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Cited by 86 (8 self)
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Markov chain Monte Carlo (MCMC) sampling strategies can be used to simulate hidden Markov model (HMM) parameters from their posterior distribution given observed data. Some MCMC methods (for computing likelihood, conditional probabilities of hidden states, and the most likely sequence of states) used in practice can be improved by incorporating established recursive algorithms. The most important is a set of forwardbackward recursions calculating conditional distributions of the hidden states given observed data and model parameters. We show how to use the recursive algorithms in an MCMC context and demonstrate mathematical and empirical results showing a Gibbs sampler using the forwardbackward recursions mixes more rapidly than another sampler often used for HMM's. We introduce an augmented variables technique for obtaining unique state labels in HMM's and finite mixture models. We show how recursive computing allows statistically efficient use of MCMC output when estimating the hidden states. We directly calculate the posterior distribution of the hidden chain's state space size by MCMC, circumventing asymptotic arguments underlying the Bayesian information criterion, which is shown to be inappropriate for a frequently analyzed data set in the HMM literature. The use of loglikelihood for assessing MCMC convergence is illustrated, and posterior predictive checks are used to investigate application specific questions of model adequacy.
The practical implementation of Bayesian model selection
 Institute of Mathematical Statistics
, 2001
"... In principle, the Bayesian approach to model selection is straightforward. Prior probability distributions are used to describe the uncertainty surrounding all unknowns. After observing the data, the posterior distribution provides a coherent post data summary of the remaining uncertainty which is r ..."
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Cited by 85 (3 self)
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In principle, the Bayesian approach to model selection is straightforward. Prior probability distributions are used to describe the uncertainty surrounding all unknowns. After observing the data, the posterior distribution provides a coherent post data summary of the remaining uncertainty which is relevant for model selection. However, the practical implementation of this approach often requires carefully tailored priors and novel posterior calculation methods. In this article, we illustrate some of the fundamental practical issues that arise for two different model selection problems: the variable selection problem for the linear model and the CART model selection problem.
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.
Optimal scaling of discrete approximations to Langevin diffusions
 J. R. Statist. Soc. B
, 1997
"... . We consider the optimal scaling problem for proposal distributions in HastingsMetropolis algorithms derived from Langevin diffusions. We prove an asymptotic diffusion limit theorem and show that the relative efficiency of the algorithm can be characterised by its overall acceptance rate, independ ..."
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Cited by 66 (22 self)
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. We consider the optimal scaling problem for proposal distributions in HastingsMetropolis algorithms derived from Langevin diffusions. We prove an asymptotic diffusion limit theorem and show that the relative efficiency of the algorithm can be characterised by its overall acceptance rate, independently of the target distribution. The asymptotically optimal acceptance rate is 0:574. We show that as a function of dimension n, the complexity of the algorithm is O(n 1=3 ), which compares favourably with the O(n) complexity of randomwalk Metropolis algorithms. We illustrate this comparison with a number of example simulations. Keywords. Langevin algorithm, HastingsMetropolis, Markov chain Monte Carlo, weak convergence. * Statistical Laboratory, University of Cambridge, Cambridge CB2 1SB, U.K. Internet: G.O.Roberts@statslab.cam.ac.uk. ** Department of Statistics, University of Toronto, Toronto, Ontario, Canada M5S 1A1. Internet: jeff@utstat.toronto.edu. Supported in part by NSERC o...
Bayesian Analysis of Mixture Models with an Unknown Number of Components  an alternative to reversible jump methods
, 1998
"... Richardson and Green (1997) present a method of performing a Bayesian analysis of data from a finite mixture distribution with an unknown number of components. Their method is a Markov Chain Monte Carlo (MCMC) approach, which makes use of the "reversible jump" methodology described by Green (1995). ..."
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Cited by 62 (0 self)
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Richardson and Green (1997) present a method of performing a Bayesian analysis of data from a finite mixture distribution with an unknown number of components. Their method is a Markov Chain Monte Carlo (MCMC) approach, which makes use of the "reversible jump" methodology described by Green (1995). We describe an alternative MCMC method which views the parameters of the model as a (marked) point process, extending methods suggested by Ripley (1977) to create a Markov birthdeath process with an appropriate stationary distribution. Our method is easy to implement, even in the case of data in more than one dimension, and we illustrate it on both univariate and bivariate data. Keywords: Bayesian analysis, Birthdeath process, Markov process, MCMC, Mixture model, Model Choice, Reversible Jump, Spatial point process 1 Introduction Finite mixture models are typically used to model data where each observation is assumed to have arisen from one of k groups, each group being suitably modelle...
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.