  Structured additive distributional regression models offer a versatile
framework for estimating complete conditional distributions by relating all
parameters of a parametric distribution to covariates. Although these models
efficiently leverage information in vast and intricate data sets, they often
result in highly-parameterized models with many unknowns. Standard estimation
methods, like Bayesian approaches based on Markov chain Monte Carlo methods,
face challenges in estimating these models due to their complexity and
costliness. To overcome these issues, we suggest a fast and scalable
alternative based on variational inference. Our approach combines a
parsimonious parametric approximation for the posteriors of regression
coefficients, with the exact conditional posterior for hyperparameters. For
optimization, we use a stochastic gradient ascent method combined with an
efficient strategy to reduce the variance of estimators. We provide theoretical
properties and investigate global and local annealing to enhance robustness,
particularly against data outliers. Our implementation is very general,
allowing us to include various functional effects like penalized splines or
complex tensor product interactions. In a simulation study, we demonstrate the
efficacy of our approach in terms of accuracy and computation time. Lastly, we
present two real examples illustrating the modeling of infectious COVID-19
outbreaks and outlier detection in brain activity.
