Bayesian structured additive distributional regression provides a generic framework for inference in regression models in which each parameter of a potentially complex response distribution and not only the mean is related to a structured additive predictor. The latter is composed additively of a variety of different functional effect types such as nonlinear effects, spatial effects, random coefficients, interaction surfaces or other (possibly non-standard) basis function representations. To enforce specific properties of the functional effects such as smoothness, informative multivariate Gaussian priors are assigned to the basis function coefficients. Inference can then be based on efficient Markov chain Monte Carlo simulation techniques where a generic procedure makes use of distribution-specific iteratively weighted least squares approximations to the full conditionals. The framework of distributional regression encompasses many special cases relevant for treating non-standard response structures such as highly skewed nonnegative responses, overdispersed and zero-inflated counts, shares including the possibility for zero- and one-inflation, or multivariate responses. We discuss structured additive distributional regression along applications illustrating such special cases and provide detailed guidance on practical aspects of model choice including selecting an appropriate response distribution and predictor specification.

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