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Fahrmeir, Ludwig and Lang, S. (1999): Bayesian Inference for Generalized Additive Mixed Models Based on Markov Random Field Priors. Collaborative Research Center 386, Discussion Paper 134

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Most regression problems in practice require flexible semiparametric forms of the predictor for modelling the dependence of responses on covariates. Moreover, it is often necessary to add random effects accounting for overdispersion caused by unobserved heterogeneity or for correlation in longitudinal or spatial data. We present a unified approach for Bayesian inference via Markov chain Monte Carlo (MCMC) simulation in generalized additive and semiparametric mixed models. Different types of covariates, such as usual covariates with fixed effects, metrical covariates with nonlinear effects, unstructured random effects, trend and seasonal components in longitudinal data and spatial covariates are all treated within the same general framework by assigning appropriate priors with different forms and degrees of smoothness. The approach is particularly appropriate for discrete and other fundamentally non-Gaussian responses, where Gibbs sampling techniques developed for Gaussian models cannot be applied, but it also works well for Gaussian responses. We use the close relation between nonparametric regression and dynamic or state space models to develop posterior sampling procedures, based on Markov random field priors. They include recent Metropolis-Hastings block move algorithms for dynamic generalized linear models and extensions for spatial covariates as building blocks. We illustrate the approach with a number of applications that arose out of consulting cases, showing that the methods are computionally feasible also in problems with many covariates and large data sets.

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