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113
Monte Carlo Statistical Methods
, 1998
"... This paper is also the originator of the Markov Chain Monte Carlo methods developed in the following chapters. The potential of these two simultaneous innovations has been discovered much latter by statisticians (Hastings 1970; Geman and Geman 1984) than by of physicists (see also Kirkpatrick et al. ..."
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Cited by 900 (23 self)
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This paper is also the originator of the Markov Chain Monte Carlo methods developed in the following chapters. The potential of these two simultaneous innovations has been discovered much latter by statisticians (Hastings 1970; Geman and Geman 1984) than by of physicists (see also Kirkpatrick et al. 1983). 5.5.5 ] PROBLEMS 211
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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Implementing approximate Bayesian inference for latent Gaussian models using integrated nested Laplace approximations: A manual for the inlaprogram
, 2008
"... Structured additive regression models are perhaps the most commonly used class of models in statistical applications. It includes, among others, (generalised) linear models, (generalised) additive models, smoothingspline models, statespace models, semiparametric regression, spatial and spatiotemp ..."
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Cited by 79 (16 self)
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Structured additive regression models are perhaps the most commonly used class of models in statistical applications. It includes, among others, (generalised) linear models, (generalised) additive models, smoothingspline models, statespace models, semiparametric regression, spatial and spatiotemporal models, logGaussian Coxprocesses, geostatistical and geoadditive models. In this paper we consider approximate Bayesian inference in a popular subset of structured additive regression models, latent Gaussian models, where the latent field is Gaussian, controlled by a few hyperparameters and with nonGaussian response variables. The posterior marginals are not available in closed form due to the nonGaussian response variables. For such models, Markov chain Monte Carlo methods can be implemented, but they are not without problems, both in terms of convergence and computational time. In some practical applications, the extent of these problems is such that Markov chain Monte Carlo is simply not an appropriate tool for routine analysis. We show that, by using an integrated nested Laplace approximation and its simplified version, we can directly compute very accurate approximations to the posterior marginals. The main benefit of these approximations
Honest Exploration of Intractable Probability Distributions Via Markov Chain Monte Carlo
 STATISTICAL SCIENCE
, 2001
"... Two important questions that must be answered whenever a Markov chain Monte Carlo (MCMC) algorithm is used are (Q1) What is an appropriate burnin? and (Q2) How long should the sampling continue after burnin? Developing rigorous answers to these questions presently requires a detailed study of the ..."
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Cited by 74 (19 self)
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Two important questions that must be answered whenever a Markov chain Monte Carlo (MCMC) algorithm is used are (Q1) What is an appropriate burnin? and (Q2) How long should the sampling continue after burnin? Developing rigorous answers to these questions presently requires a detailed study of the convergence properties of the underlying Markov chain. Consequently, in most practical applications of MCMC, exact answers to (Q1) and (Q2) are not sought. The goal of this paper is to demystify the analysis that leads to honest answers to (Q1) and (Q2). The authors hope that this article will serve as a bridge between those developing Markov chain theory and practitioners using MCMC to solve practical problems. The ability to formally address (Q1) and (Q2) comes from establishing a drift condition and an associated minorization condition, which together imply that the underlying Markov chain is geometrically ergodic. In this paper, we explain exactly what drift and minorization are as well as how and why these conditions can be used to form rigorous answers to (Q1) and (Q2). The basic ideas are as follows. The results of Rosenthal (1995) and Roberts and Tweedie (1999) allow one to use drift and minorization conditions to construct a formula giving an analytic upper bound on the distance to stationarity. A rigorous answer to (Q1) can be calculated using this formula. The desired characteristics of the target distribution are typically estimated using ergodic averages. Geometric ergodicity of the underlying Markov chain implies that there are central limit theorems available for ergodic averages (Chan and Geyer 1994). The regenerative simulation technique (Mykland, Tierney and Yu 1995, Robert 1995) can be used to get a consistent estimate of the variance of the asymptotic nor...
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.
BUGS  Bayesian inference Using Gibbs Sampling Version 0.50
, 1995
"... e wrong, which is even worse. Please let us know of any successes or failures. Beware  Gibbs sampling can be dangerous!. BUGS c flcopyright MRC Biostatistics Unit 1995. ALL RIGHTS RESERVED. The support of the Economic and Social Research Council (UK) is gratefully acknowledged. The work was funde ..."
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Cited by 64 (0 self)
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e wrong, which is even worse. Please let us know of any successes or failures. Beware  Gibbs sampling can be dangerous!. BUGS c flcopyright MRC Biostatistics Unit 1995. ALL RIGHTS RESERVED. The support of the Economic and Social Research Council (UK) is gratefully acknowledged. The work was funded in part by ESRC (UK) Award Number H519 25 5023. 1 2 Contents 1 Introduction 5 1.1 What is BUGS? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 For what kind of problems is BUGS best suited? . . . . . . . . . . . . . . . . . . . . . 5 1.3 Markov Chain Monte Carlo (MCMC) techniques . . . . . . . . . . . . . . . . . . . . 5 1.4 A simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Hardware platforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 Software . . .
Bayesian inference for generalized additive mixed models based on markov random field priors
 C
, 2001
"... Summary. Most regression problems in practice require ¯exible semiparametric forms of the predictor for modelling the dependence of responses on covariates. Moreover, it is often necessary to add random effects accounting for overdispersion caused by unobserved heterogeneity or for correlation in lo ..."
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Cited by 63 (19 self)
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Summary. Most regression problems in practice require ¯exible semiparametric forms of the predictor for modelling the dependence of responses on covariates. Moreover, it is often necessary to add random effects accounting for overdispersion caused by unobserved heterogeneity or for correlation in longitudinal or spatial data. We present a uni®ed approach for Bayesian inference via Markov chain Monte Carlo simulation in generalized additive and semiparametric mixed models. Different types of covariates, such as the usual covariates with ®xed effects, metrical covariates with nonlinear effects, unstructured random effects, trend and seasonal components in longitudinal data and spatial covariates, are all treated within the same general framework by assigning appropriate Markov random ®eld priors with different forms and degrees of smoothness. We applied the approach in several casestudies and consulting cases, showing that the methods are also computationally feasible in problems with many covariates and large data sets. In this paper, we choose two typical applications.
Geometric ergodicity of Metropolis algorithms
 STOCHASTIC PROCESSES AND THEIR APPLICATIONS
, 1998
"... In this paper we derive conditions for geometric ergodicity of the random walkbased Metropolis algorithm on R k . We show that at least exponentially light tails of the target density is a necessity. This extends the onedimensional result of (Mengersen and Tweedie, 1996). For subexponential targe ..."
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Cited by 57 (2 self)
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In this paper we derive conditions for geometric ergodicity of the random walkbased Metropolis algorithm on R k . We show that at least exponentially light tails of the target density is a necessity. This extends the onedimensional result of (Mengersen and Tweedie, 1996). For subexponential target densities we characterize the geometrically ergodic algorithms and we derive a practical sufficient condition which is stable under addition and multiplication. This condition is especially satisfied for the class of densities considered in (Roberts and Tweedie, 1996).
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.