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Fast Sampling Of Gaussian Markov Random Fields With Applications
 Journal of the Royal Statistical Society, Series B
, 2000
"... This report has URL http://www.math.ntnu.no/preprint/statistics/2000/S12000.ps ..."
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Cited by 110 (6 self)
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This report has URL http://www.math.ntnu.no/preprint/statistics/2000/S12000.ps
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 89 (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.
On Block Updating in Markov Random Field Models For . . .
 SCANDINAVIAN JOURNAL OF STATISTICS
, 2002
"... Gaussian Markov random field (GMRF) models are commonlyufz to model spatial correlation in disease mapping applications. For Bayesian inference by MCMC, so far mainly singlesiteuinglealgorithms have been considered. However, convergence and mixing properties ofsuD algorithms can be extremely ..."
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Cited by 85 (8 self)
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Gaussian Markov random field (GMRF) models are commonlyufz to model spatial correlation in disease mapping applications. For Bayesian inference by MCMC, so far mainly singlesiteuinglealgorithms have been considered. However, convergence and mixing properties ofsuD algorithms can be extremely poordu to strong dependencies ofparameters in the posteriordistribuQ84K In this paper, we propose variou block sampling algorithms in order to improve the MCMC performance. The methodology is rather general, allows for nonstandardfu6 conditionals, and can be applied in amoduzK fashion in a large nugef of di#erent scenarios. For illu##Kzf0 n we consider three di#erent applications: twoformu8Df0z3 for spatial modelling of a single disease (with andwithou additionaluditionalfL parameters respectively), and one formu## ion for the joint analysis of two diseases. TheresuKK indicate that the largest benefits are obtained ifparameters and the corresponding hyperparameter areuefz#L jointly in one large block. Implementation ofsuQ block algorithms is relatively easy usyf methods for fast sampling ofGaungf3 Markov random fields (Rus 2001). By comparison, Monte Carlo estimates based on singlesiteungles can be rather misleading, even for very long rugfOu resuL6 may have wider relevance for efficient MCMCsimu6z8#f in hierarchical models with Markov random field components.
Modelling spatially correlated data via mixtures: a Bayesian approach
 Journal of the Royal Statistical Society, Series B
, 2002
"... This paper develops mixture models for spatially indexed data. We confine attention to the case of finite, typically irregular, patterns of points or regions with prescribed spatial relationships, and to problems where it is only the weights in the mixture that vary from one location to another. Our ..."
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Cited by 40 (2 self)
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This paper develops mixture models for spatially indexed data. We confine attention to the case of finite, typically irregular, patterns of points or regions with prescribed spatial relationships, and to problems where it is only the weights in the mixture that vary from one location to another. Our specific focus is on Poisson distributed data, and applications in disease mapping. We work in a Bayesian framework, with the Poisson parameters drawn from gamma priors, and an unknown number of components. We propose two alternative models for spatiallydependent weights, based on transformations of autoregressive gaussian processes: in one (the Logistic normal model), the mixture component labels are exchangeable, in the other (the Grouped continuous model), they are ordered. Reversible jump Markov chain Monte Carlo algorithms for posterior inference are developed. Finally, the performance of both of these formulations is examined on synthetic data and real data on mortality from rare disease.
Approximate Bayesian inference for hierarchical Gaussian Markov random fields models
 Journal of Statistical Planning and Inference
, 2007
"... Many commonly used models in statistics can be formulated as (Bayesian) hierarchical Gaussian Markov random field models. These are characterised by assuming a (often large) Gaussian Markov random field (GMRF) as the second stage in the hierarchical structure and a few hyperparameters at the third s ..."
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Cited by 40 (8 self)
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Many commonly used models in statistics can be formulated as (Bayesian) hierarchical Gaussian Markov random field models. These are characterised by assuming a (often large) Gaussian Markov random field (GMRF) as the second stage in the hierarchical structure and a few hyperparameters at the third stage. Markov chain Monte Carlo is the common approach for Bayesian inference in such models. The variance of the Monte Carlo estimates is Op(M −1/2) where M is the number of samples in the chain so, in order to obtain precise estimates of marginal densities, say, we need M to be very large. Inspired by the fact that often oneblock and independence samplers can be constructed for hierarchical GMRF models, we will in this work investigate whether MCMC is really needed to estimate marginal densities, which often is the goal of the analysis. By making use of GMRFapproximations, we show by typical examples that marginal densities can indeed be very precisely estimated by deterministic schemes. The methodological and practical consequence of these findings are indeed positive. We conjecture that for many hierarchical GMRFmodels there is really no need for MCMC based inference to estimate marginal densities. Further, by making use of numerical methods for sparse matrices the computational costs of these deterministic schemes are nearly instant compared to the MCMC alternative. In particular, we discuss in detail the issue of computing marginal variances for GMRFs.
MCMC Methods for Computing Bayes Factors: A Comparative Review
 Journal of the American Statistical Association
, 2000
"... this paper we review several of these methods, and subsequently compare them in the context of two examples, the first a simple regression example, and the second a much more challenging hierarchical longitudinal model of the kind often encountered in biostatistical practice. We find that the joint ..."
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Cited by 38 (1 self)
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this paper we review several of these methods, and subsequently compare them in the context of two examples, the first a simple regression example, and the second a much more challenging hierarchical longitudinal model of the kind often encountered in biostatistical practice. We find that the joint modelparameter space search methods perform adequately but can be difficult to program and tune, while the marginal likelihood methods are often less troublesome and require less in the way of additional coding. Our results suggest that the latter methods may be most appropriate for practitioners working in many standard model choice settings, while the former remain important for comparing large numbers of models, or models whose parameters cannot be easily updated in relatively few blocks. We caution however that all of the methods we compare require significant human and computer effort, suggesting that less formal Bayesian model choice methods may offer a more realistic alternative in many cases.
A Shared Component Model for Detecting Joint and Selective Clustering of Two Diseases
"... The study of spatial variations in disease rates is a common epidemiological approach used to describe geographical clustering of disease and to generate hypotheses about the possible `causes' which could explain apparent differences in risk. Recent statistical and computational developments ha ..."
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Cited by 36 (0 self)
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The study of spatial variations in disease rates is a common epidemiological approach used to describe geographical clustering of disease and to generate hypotheses about the possible `causes' which could explain apparent differences in risk. Recent statistical and computational developments have led to the use of realistically complex models to account for overdispersion and spatial correlation. However, these developments have focused almost exclusively on spatial modelling of a single disease. Many diseases share common risk factors (smoking being an obvious example) and if similar patterns of geographical variation of related diseases can be identified, this may provide more convincing evidence of real clustering in the underlying risk surface. In this paper, we propose a shared component model for the joint spatial analysis of two diseases. The key idea is to separate the underlying risk surface for each disease into a shared and a diseasespecific component. The various component...
Comparison of a spatial perspective with the multilevel analytical approach in neighborhood studies: The case of mental and behavioral disorders due to psychoactive substance use in
 American Journal of Epidemiology
, 2005
"... Most studies of neighborhood effects on health have used the multilevel approach. However, since this methodology does not incorporate any notion of space, it may not provide optimal epidemiologic information when modeling variations or when investigating associations between contextual factors and ..."
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Cited by 29 (7 self)
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Most studies of neighborhood effects on health have used the multilevel approach. However, since this methodology does not incorporate any notion of space, it may not provide optimal epidemiologic information when modeling variations or when investigating associations between contextual factors and health. Investigating mental disorders due to psychoactive substance use among all 65,830 individuals aged 40–59 years in 2001 in Malmö, Sweden, geolocated at their place of residence, the authors compared a spatial analytical perspective, which builds notions of space into hypotheses and methods, with the multilevel approach. Geoadditive models provided precise cartographic information on spatial variations in prevalence independent of administrative boundaries. The multilevel model showed significant neighborhood variations in the prevalence of substancerelated disorders. However, hierarchical geostatistical models provided information on not only the magnitude but also the scale of neighborhood variations, indicating a significant correlation between neighborhoods in close proximity to each other. The prevalence of disorders increased with neighborhood deprivation. Far stronger associations were observed when using indicators measured in spatially adaptive areas, centered on residences of individuals, smaller in size than administrative neighborhoods. In neighborhood studies, building notions of space into analytical procedures may yield more comprehensive information than heretofore has been gathered on the spatial distribution of outcomes.
Approximating Hidden Gaussian Markov Random Fields
 JOURNAL OF THE ROYAL STATISTICAL SOCIETY, SERIES B
, 2003
"... This paper discusses how to construct approximations to a unimodal hidden Gaussian Markov random field on a graph of dimension n when the likelihood consists of mutually independent data. We demonstrate that a class of nonGaussian approximations can be constructed for a wide range of likelihood ..."
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Cited by 24 (4 self)
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This paper discusses how to construct approximations to a unimodal hidden Gaussian Markov random field on a graph of dimension n when the likelihood consists of mutually independent data. We demonstrate that a class of nonGaussian approximations can be constructed for a wide range of likelihood models. They have the appealing properties that exact samples can be drawn from them, the normalisation constant is computable, and the computational complexity is only O(n 2 ) in the spatial case. The nonGaussian approximations are refined versions of a Gaussian approximation. The latter serves well if the likelihood is nearGaussian, but it is not sufficiently accurate when the likelihood is not nearGaussian or if n is large. The accuracy of our approximations can be tuned by intuitive parameters to near any precision. We apply
Partition Modelling
"... Introduction This chapter serves as an introduction to the use of partition models to estimate a spatial process z(x) over some pdimensional region of interest X . Partition models can be useful modelling tools as, unlike standard spatial models (e.g. kriging) they allow the correlation structure ..."
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Cited by 23 (5 self)
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Introduction This chapter serves as an introduction to the use of partition models to estimate a spatial process z(x) over some pdimensional region of interest X . Partition models can be useful modelling tools as, unlike standard spatial models (e.g. kriging) they allow the correlation structure between points to vary over the space of interest. Typically, the correlation between points is assumed to be a xed function which is most likely to be parameterised by a few variables that can be estimated from the data (see, for example, Diggle, Tawn and Moyeed (1998)). Partition models avoid the need for preexamination of the data to nd a suitable correlation function to use. This removes the bias necessarily introduced by picking the correlation function and estimating its parameters using the same set of data. Spatial clusters are, by their nature, regions which are not representative of the entire space of intere