Scale-Dependent Priors for Variance Parameters in Structured Additive Distributional Regression

The selection of appropriate hyperpriors for variance parameters is an important and sensible topic in all kinds of Bayesian regression models involving the specification of (conditionally) Gaussian prior structures where the variance parameters determine a data-driven, adaptive amount of prior variability or precision. We consider the special case of structured additive distributional regression where Gaussian priors are used to enforce specific properties such as smoothness or shrinkage on various effect types combined in predictors for multiple parameters related to the distribution of the response. Relying on a recently proposed class of penalised complexity priors motivated from a general set of construction principles, we derive a hyperprior structure where prior elicitation is facilitated by assumptions on the scaling of the different effect types. The posterior distribution is assessed with an adaptive Markov chain Monte Carlo scheme and conditions for its propriety are studied theoretically. We investigate the new type of scale-dependent priors in simulations and two challenging applications, in particular in comparison to the standard inverse gamma priors but also alternatives such as half-normal, half-Cauchy and proper uniform priors for standard deviations.