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53
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
Bayesian PSplines
 Journal of Computational and Graphical Statistics
, 2004
"... Psplines are an attractive approach for modelling nonlinear smooth effects of covariates within the generalized additive and varying coefficient models framework. In this paper we propose a Bayesian version for Psplines and generalize the approach for one dimensional curves to two dimensional surf ..."
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Cited by 68 (21 self)
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Psplines are an attractive approach for modelling nonlinear smooth effects of covariates within the generalized additive and varying coefficient models framework. In this paper we propose a Bayesian version for Psplines and generalize the approach for one dimensional curves to two dimensional surface fitting for modelling interactions between metrical covariates. A Bayesian approach to Psplines has the advantage of allowing for simultaneous estimation of smooth functions and smoothing parameters. Moreover, it can easily be extended to more complex formulations, for example to mixed models with random effects for serially or spatially correlated response. Additionally, the assumption of constant smoothing parameters can be replaced by allowing the smoothing parameters to be locally adaptive. This is particularly useful in situations with changing curvature of the underlying smooth function or where the function is highly oscillating. Inference is fully Bayesian and uses recent MCMC techniques for drawing random samples from the posterior. In a couple of simulation studies the performance of Bayesian Psplines is studied and compared to other approaches in the literature. We illustrate the approach by a complex application on rents for flats in Munich.
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 51 (7 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.
Bayesian Semiparametric Regression Analysis of Multicategorical TimeSpace Data
 Annals of the Institute of Statistical Mathematics
, 2000
"... this paper, we consider multicategorical timespace data (Y it ; x it ; s i ); i = 1; : : : ; n; t = 1; : : : ; T; where the spatial location or site s i of individual i is given as an additional information. A typical example are monthly register data from the German Employment Office 1 for the ..."
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Cited by 30 (18 self)
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this paper, we consider multicategorical timespace data (Y it ; x it ; s i ); i = 1; : : : ; n; t = 1; : : : ; T; where the spatial location or site s i of individual i is given as an additional information. A typical example are monthly register data from the German Employment Office 1 for the years 19801995, where Y it is the employment status (e.g. unemployed, part time job, full time job, others) of individual i during month t and s i is the district in Germany where i has its domicile. Data from surveys on forest damage are a further example: Damage state Y it of tree i in year t, indicated by the defoliation degree, is measured in ordered categories (none to severe) and s i is the site of the tree on a lattice map. In both examples, covariates can be categorical or continuous, and possibly timevarying
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 27 (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
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
Penalized structured additive regression for spacetime data: a Bayesian perspective
 STATISTICA SINICA
, 2004
"... We propose extensions of penalized spline generalized additive models for analyzing spacetime regression data and study them from a Bayesian perspective. Nonlinear effects of continuous covariates and time trends are modelled through Bayesian versions of penalized splines, while correlated spati ..."
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Cited by 16 (11 self)
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We propose extensions of penalized spline generalized additive models for analyzing spacetime regression data and study them from a Bayesian perspective. Nonlinear effects of continuous covariates and time trends are modelled through Bayesian versions of penalized splines, while correlated spatial effects follow a Markov random field prior. This allows to treat all functions and effects within a unified general framework by assigning appropriate priors with different forms and degrees of smoothness. Inference can be performed either with full (FB) or empirical Bayes (EB) posterior analysis. FB inference using MCMC techniques is a slight extension of previous work. For EB inference, a computationally efficient solution is developed on the basis of a generalized linear mixed model representation. The second approach can be viewed as posterior mode estimation and is closely related to penalized likelihood estimation in a frequentist setting. Variance components, corresponding to inverse smoothing parameters, are then estimated by marginal likelihood. We carefully compare both inferential procedures in simulation studies and illustrate them through data applications. The methodology is available in the open domain statistical package BayesX and as an Splus/R function.
Function Estimation With Locally Adaptive Dynamic Models
 Computational Statistics
, 1998
"... this paper, we present a Bayesian nonparametric approach, which is more closely related to spline fitting with locally adaptive penalties. Abramovich and Steinberg (1996) generalize the common penalized least squares criterion for smoothing splines with a global smoothing parameter by introducing a ..."
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Cited by 11 (8 self)
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this paper, we present a Bayesian nonparametric approach, which is more closely related to spline fitting with locally adaptive penalties. Abramovich and Steinberg (1996) generalize the common penalized least squares criterion for smoothing splines with a global smoothing parameter by introducing a variable smoothing parameter into the roughness penalty. For estimation, they propose a twostep procedure: First a smoothing spline is fitted with a constant smoothing parameter chosen by generalized crossvalidation. Then an estimate for the variable smoothing parameter is constructed, based on the derivatives of this pilot estimate, and is plugged into their locally adaptive penalty to fit the smoothing spline in a second step. Ruppert and Carroll (2000) propose Psplines based on a truncated power series basis and di#erence penalties on the regression coe#cients with locally adaptive smoothing parameters. The latter are obtained by linear interpolation from a smaller number of smoothing parameters, defined for a subset of knots and estimated by generalized crossvalidation
Bayesian Varyingcoefficient Models using Adaptive Regression Splines
, 2000
"... Varying{coecient models provide a exible framework for semi{ and nonparametric generalized regression analysis. We present a fully Bayesian B{spline basis function approach with adaptive knot selection. For each of the unknown regression functions or varying coecients, the number and location of ..."
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Cited by 6 (1 self)
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Varying{coecient models provide a exible framework for semi{ and nonparametric generalized regression analysis. We present a fully Bayesian B{spline basis function approach with adaptive knot selection. For each of the unknown regression functions or varying coecients, the number and location of knots and the B{spline coecients are estimated simultaneously using reversible jump Markov chain Monte Carlo sampling. The overall procedure can therefore be viewed as a kind of Bayesian model averaging. Although Gaussian responses are covered by the general framework, the method is particularly useful for fundamentally non{Gaussian responses, where less alternatives are available. We illustrate the approach with a thorough application to two data sets analyzed previously in the literature: the kyphosis data set with a binary response and survival data from the Veteran's Administration lung cancer trial. Keywords: B{spline basis; knot selection; non{Gaussian response; non{ and sem...
Bayesian Functional ANOVA Modeling Using Gaussian Process Prior Distributions
"... Functional analysis of variance (ANOVA) models partition a functional response according to the main effects and interactions of various factors. This article develops a general framework for functional ANOVA modeling from a Bayesian viewpoint, assigning Gaussian process prior distributions to each ..."
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Cited by 6 (1 self)
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Functional analysis of variance (ANOVA) models partition a functional response according to the main effects and interactions of various factors. This article develops a general framework for functional ANOVA modeling from a Bayesian viewpoint, assigning Gaussian process prior distributions to each batch of functional effects. We discuss the choices to be made in specifying such a model, advocating the treatment of levels within a given factor as dependent but exchangeable quantities, and we suggest weakly informative prior distributions for higher level parameters that may be appropriate in many situations. We discuss computationally efficient strategies for posterior sampling using Markov Chain Monte Carlo algorithms, and we emphasize useful graphical summaries based on the posterior distribution of modelbased analogues of traditional ANOVA decompositions of variance. We illustrate this process of model specification, posterior sampling, and graphical posterior summaries in two examples. The first considers the effect of geographic region on the temperature profiles at weather stations in Canada. The second example examines sources of variability in the output of regional climate models from a designed experiment.