A primer on Bayesian distributional regression

Bayesian methods have become increasingly popular in the past two decades. With the constant rise of computational power even very complex models can be estimated on virtually any modern computer. Moreover, interest has shifted from conditional mean models to probabilistic distributional models capturing location, scale, shape and other aspects of a response distribution, where covariate effects can have flexible forms, e.g., linear, nonlinear, spatial or random effects. This tutorial paper discusses how to select models in the Bayesian distributional regression setting, how to monitor convergence of the Markov chains, evaluate relevance of effects using simultaneous credible intervals and how to use simulation-based inference also for quantities derived from the original model parameterisation. We exemplify the work flow using daily weather data on (i) temperatures on Germany's highest mountain and (ii) extreme values of precipitation all over Germany.