Bayesian Effect Selection in Structured Additive Distributional Regression Models

Abstract

We establish Bayesian effect selection for the broad class of structured additive distributional regression models using a spike and slab prior specification with scaled beta prime marginals for the importance parameters of blocks of regression coefficients. This enables us to model and select effects in all distributional parameters, such as location, scale, skewness or correlation parameters, for arbitrary distributions. The regression specifications encompass various effect types such as non-linear or spatial effects. Our spike and slab prior relies on a parameter expansion that separates blocks of regression coefficients into overall scalar importance parameters and vectors of standardised coefficients, and yields effective shrinkage and good sampling performance. Using constrained priors, it is possible to implement effect decompositions, where, for example, a non-linear effect can be decomposed into a linear component and the non-linear deviation from this linear effect; and to select both separately. We investigate some shrinkage properties, propose a way of eliciting prior hyperparameters and provide full posterior inference through Markov Chain Monte Carlo simulations. Using both simulated and real data sets, we show that our approach is applicable for data with various functional covariate effects, multilevel predictors and non-standard response distributions, such as bivariate Gaussian or zero-inflated Poisson.