Results 1  10
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16
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 67 (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.
Auxiliary Variable Methods for Markov Chain Monte Carlo with Applications
 Journal of the American Statistical Association
, 1997
"... Suppose one wishes to sample from the density ß(x) using Markov chain Monte Carlo (MCMC). An auxiliary variable u and its conditional distribution ß(ujx) can be defined, giving the joint distribution ß(x; u) = ß(x)ß(ujx). A MCMC scheme which samples over this joint distribution can lead to substanti ..."
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Cited by 63 (1 self)
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Suppose one wishes to sample from the density ß(x) using Markov chain Monte Carlo (MCMC). An auxiliary variable u and its conditional distribution ß(ujx) can be defined, giving the joint distribution ß(x; u) = ß(x)ß(ujx). A MCMC scheme which samples over this joint distribution can lead to substantial gains in efficiency compared to standard approaches. The revolutionary algorithm of Swendsen and Wang (1987) is one such example. In addition to reviewing the SwendsenWang algorithm and its generalizations, this paper introduces a new auxiliary variable method called partial decoupling. Two applications in Bayesian image analysis are considered. The first is a binary classification problem in which partial decoupling out performs SW and single site Metropolis. The second is a PET reconstruction which uses the gray level prior of Geman and McClure (1987). A generalized SwendsenWang algorithm is developed for this problem, which reduces the computing time to the point that MCMC is a viabl...
Prediction via Orthogonalized Model Mixing
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 1994
"... In this paper we introduce an approach and algorithms for model mixing in large prediction problems with correlated predictors. We focus on the choice of predictors in linear models, and mix over possible subsets of candidate predictors. Our approach is based on expressing the space of models in ter ..."
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Cited by 50 (9 self)
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In this paper we introduce an approach and algorithms for model mixing in large prediction problems with correlated predictors. We focus on the choice of predictors in linear models, and mix over possible subsets of candidate predictors. Our approach is based on expressing the space of models in terms of an orthogonalization of the design matrix. Advantages are both statistical and computational. Statistically, orthogonalization often leads to a reduction in the number of competing models by eliminating correlations. Computationally, large model spaces cannot be enumerated; recent approaches are based on sampling models with high posterior probability via Markov chains. Based on orthogonalization of the space of candidate predictors, we can approximate the posterior probabilities of models by products of predictorspecific terms. This leads to an importance sampling function for sampling directly from the joint distribution over the model space, without resorting to Markov chains. Comp...
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.
Bayesian SpatioTemporal Inference in Functional Magnetic Resonance Imaging
, 2001
"... this article is to present hierarchical Bayesian approaches that allow to simultaneously incorporate temporal and spatial dependencies between pixels directly in the model formulation. For reasons of computational feasibility, models have to be comparatively parsimonious, without oversimplifying. We ..."
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Cited by 20 (2 self)
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this article is to present hierarchical Bayesian approaches that allow to simultaneously incorporate temporal and spatial dependencies between pixels directly in the model formulation. For reasons of computational feasibility, models have to be comparatively parsimonious, without oversimplifying. We introduce parametric and semiparametric spatial and spatiotemporal models that proved appropriate and illustrate their performance by application to fMRI data from a visual stimulation experiment.
Bayesian smoothing in the estimation of the pair potential function of Gibbs point processes
, 1999
"... This paper introduces a method which can be viewed as the first step towards a truly nonparametric Bayesian estimation of Gibbs processes with pairwise interactions. The pair potential is approximated by a step function having a large number of fixed jump points. The induced high dimension of the pa ..."
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Cited by 14 (4 self)
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This paper introduces a method which can be viewed as the first step towards a truly nonparametric Bayesian estimation of Gibbs processes with pairwise interactions. The pair potential is approximated by a step function having a large number of fixed jump points. The induced high dimension of the parameter space causes two kinds of problems. First, each component of the sufficient statistic is typically a function of a small number of point locations, which causes instability in the estimation. Secondly, the computational complexity increases rapidly with the dimension. To combat the first problem we apply Bayesian smoothing by choosing a Markov chain prior which penalises large differences between nearby values of the pair potential function. This idea originates in Bayesian image analysis; see Besag (1986). As regards the computational complexity, we have found the full posterior analysis to be too demanding with the currently available machinery. Consequently, we have concentrated on the task of locating the posterior mode, which is computationally equivalent to that of ønding the maximum likelihood estimate (MLE). Starting from the Monte Carlo NewtonRaphson algorithm of Penttinen (1984) and the Monte Carlo likelihood approach of Geyer and Thompson (1992), we arrived at an efficient algorithm by modifying the former into an MCMC approximation of the Marquardt algorithm (Marquardt 1963) and then combining the two: The first approximation to the posterior mode is obtained using the Monte Carlo Marquardt algorithm, where the first two differentials of the logposterior are approximated by MCMC as in Penttinen (1984), and the final estimate is calculated using the Monte Carlo likelihood approximation. (The naming conventions applied here were introduced by Geyer 1998). Our appr...
Adaptive Langevin sampler for separation of tdistribution modelled astrophysical maps
 IEEE Trans. Image Process
, 2010
"... Abstract—We propose to model the image differentials of astrophysical source maps by Student’s tdistribution and to use them in the Bayesian source separation method as priors. We introduce an efficient Markov Chain Monte Carlo (MCMC) sampling scheme to unmix the astrophysical sources and describe ..."
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Cited by 6 (1 self)
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Abstract—We propose to model the image differentials of astrophysical source maps by Student’s tdistribution and to use them in the Bayesian source separation method as priors. We introduce an efficient Markov Chain Monte Carlo (MCMC) sampling scheme to unmix the astrophysical sources and describe the derivation details. In this scheme, we use the Langevin stochastic equation for transitions, which enables parallel drawing of random samples from the posterior, and reduces the computation time significantly (by two orders of magnitude). In addition, Student’s tdistribution parameters are updated throughout the iterations. The results on astrophysical source separation are assessed with two performance criteria defined in the pixel and the frequency domains. Index Terms—Astrophysical images, Bayesian source separation,
Fully Bayesian Image Segmentation  An Engineering Perspective
 INRIA Research Report No 3017
, 1996
"... Developments in Markov Chain Monte Carlo procedures have made it possible to perform fully Bayesian image segmentation. By this we mean that all the parameters are treated identically, be they the segmentation labels, the class parameters or the Markov Random Field prior parameters. We perform the a ..."
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Cited by 4 (1 self)
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Developments in Markov Chain Monte Carlo procedures have made it possible to perform fully Bayesian image segmentation. By this we mean that all the parameters are treated identically, be they the segmentation labels, the class parameters or the Markov Random Field prior parameters. We perform the analysis by sampling from the posterior distribution of all the parameters. Sampling from the MRF parameters has traditionally been considered if not intractable then at least computationally prohibitive. In the statistics literature there are descriptions of experiments showing that the MRF parameters may be sampled by approximating the partition function. These experiments are all, however, on `toy' problems for the typical size of image encountered in engineering applications phase transition behaviour of the models becomes a major limiting factor in the estimation of the partition function. Nevertheless, we show that, with some care, fully Bayesian segmentation can be performed on realist...
Cluster priors in the Bayesian modelling of fMRI data
, 2001
"... Functional magnetic resonance imaging (fMRI) is a scanning technique for revealing haemodynamic changes connected with brain processing on the neuronal level. In neuropsychology, fMRI has been used in designed experiments together with controlled stimulation. fMRI data are temporal series of digital ..."
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Cited by 4 (0 self)
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Functional magnetic resonance imaging (fMRI) is a scanning technique for revealing haemodynamic changes connected with brain processing on the neuronal level. In neuropsychology, fMRI has been used in designed experiments together with controlled stimulation. fMRI data are temporal series of digital images corrupted by spatiotemporally correlated physiological processes and scanner noise. The statistical challenge in analysing fMRI data is to localize stimulusrelated brain activation and estimate its characteristics. In this thesis, the focus is on spatial aspects of activations. A Bayesian approach is proposed and an a priori model which describes the clustering of activations is suggested. The prior is used to control the spatial extent, coherence and locations of clusters. Marked Gibbs point processes have been used to construct the prior. The prior is designed so that expert knowledge on the neuronal processing of interest can be incorporated into statistical analysis. To model the conditional distribution of observations, given the activations, Gaussian conditional autoregressive processes have been applied. Using these processes, heteroskedasticity and spatial autocorrelation in noise is accounted for. Inference is based on Markov chain Monte Carlo (MCMC) simulations of the posterior distribution. A modified version of an existing general simulation method for Gibbs point processes is devised to sample the posterior. Real fMRI data are analysed and the influence of different amounts of prior information on the uncertainty inactivations is illustrated. An example of analysing synthetic data is provided to compare the new method with conventional nonparametric techniques. The conclusion is that, by adopting a structural approach, relevant features of activations can be accounted for leading to a potentially more efficient inference.
D.: \Multiscale iterative techniques and adaptive mesh re for in porous media," Invited by Adv
 in Water Resources
, 2001
"... ..."