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38
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
Hidden Markov models and disease mapping
 Journal of the American Statistical Association
, 2001
"... We present new methodology to extend Hidden Markov models to the spatial domain, and use this class of models to analyse spatial heterogeneity of count data on a rare phenomenon. This situation occurs commonly in many domains of application, particularly in disease mapping. We assume that the counts ..."
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Cited by 56 (4 self)
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We present new methodology to extend Hidden Markov models to the spatial domain, and use this class of models to analyse spatial heterogeneity of count data on a rare phenomenon. This situation occurs commonly in many domains of application, particularly in disease mapping. We assume that the counts follow a Poisson model at the lowest level of the hierarchy, and introduce a finite mixture model for the Poisson rates at the next level. The novelty lies in the model for allocation to the mixture components, which follows a spatially correlated process, the Potts model, and in treating the number of components of the spatial mixture as unknown. Inference is performed in a Bayesian framework using reversible jump MCMC. The model introduced can be viewed as a Bayesian semiparametric approach to specifying exible spatial distribution in hierarchical models. Performance of the model and comparison with an alternative wellknown Markov random field specification for the Poisson rates are demonstrated on synthetic data sets. We show that our allocation model avoids the problem of oversmoothing in cases where the underlying rates exhibit discontinuities, while giving equally good results in cases of smooth gradientlike or highly autocorrelated rates. The methodology is illustrated on an epidemiological application to data on a rare cancer in France.
Lang S: Generalized structured additive regression based on Bayesian P splines
 Computational Statistics & Data Analysis
"... Generalized additive models (GAM) for modelling nonlinear effects of continuous covariates are now well established tools for the applied statistician. In this paper we develop Bayesian GAM’s and extensions to generalized structured additive regression based on one or two dimensional Psplines as th ..."
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Cited by 26 (7 self)
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Generalized additive models (GAM) for modelling nonlinear effects of continuous covariates are now well established tools for the applied statistician. In this paper we develop Bayesian GAM’s and extensions to generalized structured additive regression based on one or two dimensional Psplines as the main building block. The approach extends previous work by Lang and Brezger (2003) for Gaussian responses. Inference relies on Markov chain Monte Carlo (MCMC) simulation techniques, and is either based on iteratively weighted least squares (IWLS) proposals or on latent utility representations of (multi)categorical regression models. Our approach covers the most common univariate response distributions, e.g. the Binomial, Poisson or Gamma distribution, as well as multicategorical responses. As we will demonstrate through two applications on the forest health status of trees and a spacetime analysis of health insurance data, the approach allows realistic modelling of complex problems. We consider the enormous flexibility and extendability of our approach as a main advantage of Bayesian inference based on MCMC techniques compared to more traditional approaches. Software for the methodology presented in the paper is provided within the public domain package BayesX. Key words: geoadditive models, IWLS proposals, multicategorical response, structured additive
Double Markov Random Fields and Bayesian Image Segmentation
, 2002
"... Markov random fields are used extensively in modelbased approaches to image segmentation and, under the Bayesian paradigm, are implemented through Markov chain Monte Carlo (MCMC) methods. In this paper, we describe a class of such models (the double Markov random field) for images composed of severa ..."
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Cited by 23 (0 self)
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Markov random fields are used extensively in modelbased approaches to image segmentation and, under the Bayesian paradigm, are implemented through Markov chain Monte Carlo (MCMC) methods. In this paper, we describe a class of such models (the double Markov random field) for images composed of several textures, which we consider to be the natural hierarchical model for such a task. We show how several of the Bayesian approaches in the literature can be viewed as modifications of this model, made in order to make MCMC implementation possible. From a simulation study, conclusions are made concerning the performance of these modified models.
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 19 (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
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 17 (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.
GMRFLib: a Clibrary for fast and exact simulation of Gaussian Markov random fields
"... This manual describes the library GMRFLib of Croutines for fast and exact simulation of Gaussian Markov Random Fields (GMRF) on graphs. The library performs . ..."
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Cited by 12 (5 self)
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This manual describes the library GMRFLib of Croutines for fast and exact simulation of Gaussian Markov Random Fields (GMRF) on graphs. The library performs .
A General Framework for the Parametrization of Hierarchical Models
, 708
"... Abstract. In this paper, we describe centering and noncentering methodology as complementary techniques for use in parametrization of broad classes of hierarchical models, with a view to the construction of effective MCMC algorithms for exploring posterior distributions from these models. We give a ..."
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Cited by 7 (0 self)
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Abstract. In this paper, we describe centering and noncentering methodology as complementary techniques for use in parametrization of broad classes of hierarchical models, with a view to the construction of effective MCMC algorithms for exploring posterior distributions from these models. We give a clear qualitative understanding as to when centering and noncentering work well, and introduce theory concerning the convergence time complexity of Gibbs samplers using centered and noncentered parametrizations. We give general recipes for the construction of noncentered parametrizations, including an auxiliary variable technique called the statespace expansion technique. We also describe partially noncentered methods, and demonstrate their use in constructing robust Gibbs sampler algorithms whose convergence properties are not overly sensitive to the data. Key words and phrases: Parametrization, hierarchical models, latent stochastic processes, MCMC.
Spatially Correlated Allocation Models for Count Data
, 2000
"... Spatial heterogeneity of count data on a rare phenomenon occurs commonly in many domains of application, in particularly in disease mapping. We present new methodology to analyse such data, based on a hierarchical allocation model. We assume that the counts follow a Poisson model at the lowest le ..."
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Cited by 5 (0 self)
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Spatial heterogeneity of count data on a rare phenomenon occurs commonly in many domains of application, in particularly in disease mapping. We present new methodology to analyse such data, based on a hierarchical allocation model. We assume that the counts follow a Poisson model at the lowest level of the hierarchy, and introduce a finite mixture model for the Poisson rates at the next level. The novelty lies in the allocation model to the mixture components, which follows a spatially correlated process, the Potts model, and in treating the number of components of the spatial mixture as unknown. Inference is performed in a Bayesian framework using reversible jump MCMC. The model introduced can be viewed as a Bayesian semiparametric approach to specifying flexible spatial distribution in hierarchical models. It could also be used in contexts where the spatial mixture subgroups are themselves of interest, as in health care monitoring. Performance of the model and comparison wi...
Accelerating computation in Markov random field models for spatial data via structured MCMC
 Journal of Computational and Graphical Statistics
, 2003
"... ..."