## Bayesian Inference for Generalized Additive Mixed Models Based on Markov Random Field Priors (2000)

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Venue: | C |

Citations: | 64 - 19 self |

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@ARTICLE{Fahrmeir00bayesianinference,

author = {Ludwig Fahrmeir and Stefan Lang},

title = {Bayesian Inference for Generalized Additive Mixed Models Based on Markov Random Field Priors},

journal = {C},

year = {2000},

volume = {50},

pages = {201--220}

}

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### Abstract

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 nonGaussian responses, where Gibbs sampling techniques developed for Gaussian m...

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Citation Context ...promising. Our approach is based on the close relationship between dynamic generalized linear models (see, e.g., Fahrmeir and Tutz, 1997, ch.8) and generalized additive or varying coefficient models (=-=Hastie and Tibshirani, 1990, 1993). T-=-his relationship is well known for the classical smoothing problem, where observations y = (y(1); : : : ; y(n)) are assumed to be the sum y(t) = f(t) + "(t); "(t)sN(0; oe 2 ) (2) of a smooth... |

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Citation Context ...y the Kalman filter and smoother. For a full Bayesian analysis with hyperpriors for the variances oe 2 and 2 , the Kalman filter and smoother can be exploited for efficient, blockwise Gibbs sampling (=-=Carter and Kohn, 1994-=-; Fruhwirth-Schnatter, 1994). Due to the Gaussian observation model (2), the posterior mean estimates f(t) and posterior mode estimates coincide, and are equivalent to the solution of a corresponding ... |

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Citation Context ...ms to have better convergence properties, and an extension to GAMMS and VCMMs might be promising. Our approach is based on the close relationship between dynamic generalized linear models (see, e.g., =-=Fahrmeir and Tutz, 1997-=-, ch.8) and generalized additive or varying coefficient models (Hastie and Tibshirani, 1990, 1993). This relationship is well known for the classical smoothing problem, where observations y = (y(1); :... |

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