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102
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 1331 (24 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.
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 Gree ..."
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Cited by 114 (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 91 (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.
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 63 (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.
The spectral gap for a Glaubertype dynamics in a continuous gas
, 2000
"... . We consider a continuous gas in a d dimensional rectangular box with a nite range, positive pair potential, and we construct a Markov process in which particles appear and disappear with appropriate rates so that the process is reversible w.r.t. the Gibbs measure. If the thermodynamical paramenter ..."
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Cited by 54 (8 self)
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. We consider a continuous gas in a d dimensional rectangular box with a nite range, positive pair potential, and we construct a Markov process in which particles appear and disappear with appropriate rates so that the process is reversible w.r.t. the Gibbs measure. If the thermodynamical paramenters are such that the Gibbs specication satises a certain mixing condition, then the spectral gap of the generator is strictly positive uniformly in the volume and boundary condition. The required mixing condition holds if, for instance, there is a convergent cluster expansion. Key Words: Spectral gap, Gibbs measures, continuous systems, birth and death processes Mathematics Subject Classication: 82C21, 60K35, 82C22, 60J75 This work was partially supported by GNAFA and by \Conanziamento Murst" v1.4 1. Introduction We consider a continuous gas in a bounded volume R d , distributed according the Gibbs probability measure associated to a nite range pair potential '. The Gibbs measu...
Glauber dynamics of continuous particle systems
"... This paper is devoted to the construction and study of an equilibrium Glaubertype dynamics of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process. On the space Γ of all locally finite subsets (configurations) in Rd, we fix a Gibbs measure µ cor ..."
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Cited by 51 (14 self)
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This paper is devoted to the construction and study of an equilibrium Glaubertype dynamics of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process. On the space Γ of all locally finite subsets (configurations) in Rd, we fix a Gibbs measure µ corresponding to a general pair potential φ and activity z> 0. We consider a Dirichlet form E on L2 (Γ,µ) which corresponds to the generator H of the Glauber dynamics. We prove the existence of a Markov process M on Γ that is properly associated with E. In the case of a positive potential φ which satisfies δ: = ∫ Rd(1 − e−φ(x))z dx < 1, we also prove that the generator H has a spectral gap ≥ 1−δ. Furthermore, for any pure Gibbs state µ, we derive a Poincaré inequality. The results about the spectral gap and the Poincaré inequality are a generalization and a refinement of a recent result from [6].
Spatial birth and death processes as solutions of stochastic equations
 ALEA Lat. Am. J. Probab. Math. Stat
, 2006
"... Abstract. Spatial birth and death processes are obtained as solutions of a system of stochastic equations. The processes are required to be locally finite, but may involve an infinite population over the full (noncompact) type space. Conditions are given for existence and uniqueness of such solution ..."
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Cited by 41 (3 self)
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Abstract. Spatial birth and death processes are obtained as solutions of a system of stochastic equations. The processes are required to be locally finite, but may involve an infinite population over the full (noncompact) type space. Conditions are given for existence and uniqueness of such solutions, and for temporal and spatial ergodicity. For birth and death processes with constant death rate, a subcriticality condition on the birth rate implies that the process is ergodic and converges exponentially fast to the stationary distribution. 1.
Markov evolutions and hierarchical equations in the continuum II. Multicomponent systems
 In preparation
, 2007
"... General birthanddeath as well as hopping stochastic dynamics of infinite particle systems in the continuum are considered. We derive corresponding evolution equations for correlation functions and generating functionals. General considerations are illustrated in a number of concrete examples of Ma ..."
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Cited by 38 (17 self)
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General birthanddeath as well as hopping stochastic dynamics of infinite particle systems in the continuum are considered. We derive corresponding evolution equations for correlation functions and generating functionals. General considerations are illustrated in a number of concrete examples of Markov evolutions appearing in applications. Keywords: Birthanddeath process; Hopping particles; Continuous system;
Phase Transition in Continuum Potts Models
 Commun. Math. Phys
, 1996
"... We establish phase transitions for a class of continuum multitype particle systems with finite range repulsive pair interaction between particles of different type. This proves an old conjecture of Lebowitz and Lieb. A phase transition still occurs when we allow a background pair interaction (betw ..."
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Cited by 35 (10 self)
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We establish phase transitions for a class of continuum multitype particle systems with finite range repulsive pair interaction between particles of different type. This proves an old conjecture of Lebowitz and Lieb. A phase transition still occurs when we allow a background pair interaction (between all particles) which is superstable and has sufficiently short range of repulsion. Our approach involves a randomcluster representation analogous to the FortuinKasteleyn representation of the Potts model. In the course of our argument, we establish the existence of a percolation transition for Gibbsian particle systems with random edges between the particles, and also give an alternative proof for the existence of Gibbs measures with superstable interaction. 1 Introduction Although the study of phase transitions for Gibbsian systems is one of the main subjects of statistical mechanics, examples of models exhibiting phase transition are mainly restricted to lattice systems. For syste...
Transdimensional Markov Chains: A Decade of Progress and Future Perspectives
, 2005
"... The last 10 years have witnessed the development of sampling frameworks that permit the construction of Markov chains that simultaneously traverse both parameter and model space. Substantial methodological progress has been made during this period. In this article we present a survey of the current ..."
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Cited by 31 (3 self)
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The last 10 years have witnessed the development of sampling frameworks that permit the construction of Markov chains that simultaneously traverse both parameter and model space. Substantial methodological progress has been made during this period. 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.