P-splines 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 P-splines and generalize the approach for one dimensional curves to two dimensional surface fitting for modelling interactions between metrical covariates. A Bayesian approach to P-splines 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 P-splines is studied and compared to other approaches in the literature. We illustrate the approach by a complex application on rents for flats in Munich.

Generalized additive models (GAM) for modeling nonlinear effects of continuous covariates are now well established tools for the applied statistician. A Bayesian version of GAM’s and extensions to generalized structured additive regression (STAR) are developed. One or two dimensional P-splines are used as the main building block. 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. The approach covers the most common univariate response distributions, e.g. the binomial, Poisson or gamma distribution, as well as multicategorical responses. For the first time, Bayesian semiparametric inference for the widely used multinomial logit model is presented. Two applications on the forest health status of trees and a space-time analysis of health insurance data demonstrate the potential of the approach for realistic modeling of complex problems. Software for the methodology is provided within the public domain package BayesX. Key words: geoadditive models, IWLS proposals, multicategorical response, structured additive predictors, surface smoothing

In this paper we present a nonparametric Bayesian approach for £tting unsmooth or highly oscillating functions in regression models with binary responses. The approach extends previous work by Lang et al. (2002) for Gaussian responses. Nonlinear functions are modelled by £rst or second order random walk priors with locally varying variances or smoothing parameters. Estimation is fully Bayesian and uses latent utility representations of binary regression models for ef£cient block sampling from the full conditionals of nonlinear functions. Key words: adaptive smoothing, forest health data, highly oscillating functions, MCMC, random walk priors, unsmooth functions, variable smoothing parameter. 1

Common nonparametric curve tting methods such as spline smoothing, local polynomial regression and basis function approaches are now well developed and widely applied. More recently, Bayesian function estimation has become a useful supplementary or alternative tool for practical data analysis, mainly due to breakthroughs in computerintensive inference via Markov chain Monte Carlo simulation. This paper surveys recent developments in semiparametric Bayesian inference for generalized regression and outlines some directions in current research.

Magnetic resonance diffusion tensor imaging (DTI) allows to infere the ultrastructure of living tissue. In brain mapping, neural fiber trajectories can be identified by exploiting the anisotropy of diffusion processes. Manifold statistical methods may be linked into the comprehensive processing chain that is spanned between DTI raw images and the reliable visualization of fibers. In this work, a space varying coefficients model (SVCM) using penalized B-splines was developed to integrate diffusion tensor estimation, regularization and interpolation into a unified framework. The implementation challenges originating in multiple 3d space varying coefficient surfaces and the large dimensions of realistic datasets were met by incorporating matrix sparsity and efficient model approximation. Superiority of B-spline based SVCM to the standard approach was demonstrable from simulation studies in terms of the precision and accuracy of the individual tensor elements. The integration with a probabilistic fiber tractography algorithm and application on real brain data revealed that the unified approach is at least equivalent to the serial application of voxelwise estimation, smoothing and interpolation. From the error analysis using boxplots and visual inspection the conclusion was drawn that both the standard approach and the