In this paper, we propose a generic Bayesian framework for inference in distributional 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 is then 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. We study properties of the resulting model class and provide detailed guidance on practical aspects of model choice including selecting an apropriate response distribution and predictor specification. The importance and flexibility of Bayesian structured additive distributional regression to estimate all parameters as functions of explanatory variables and therefore to obtain more realistic models, is exemplified in two applications with complex response distributions.
