Bayesian Effect Selection in Structured Additive Distributional Regression Models

TL;DR

This paper introduces a new Bayesian effect selection method for structured additive distributional regression models, enabling flexible modeling of effects across all distribution parameters with improved computational efficiency.

Contribution

It proposes a novel spike and slab prior with scaled beta prime marginals, enhancing effect selection and shrinkage in complex distributional regression models.

Findings

Effective effect selection across various distribution parameters

Improved sampling performance over classical priors

Applicable to large-scale, hierarchical, and non-standard data distributions

Abstract

We propose a novel spike and slab prior specification with scaled beta prime marginals for the importance parameters of regression coefficients to allow for general effect selection within the class of structured additive distributional regression. This enables us to model effects on all distributional parameters for arbitrary parametric distributions, and to consider various effect types such as non-linear or spatial effects as well as hierarchical regression structures. 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. Hence, we can work with a scalar quantity for effect selection instead of a possibly high-dimensional effect vector, which yields improved shrinkage and sampling performance compared to the classical normal-inverse-gamma prior. We investigate the propriety of the posterior, show that the prior yields desirable shrinkage properties, propose a way of eliciting prior parameters and provide efficient Markov Chain Monte Carlo sampling. Using both simulated and three large-scale data sets, we show that our approach is applicable for data with a potentially large number of covariates, multilevel predictors accounting for hierarchically nested data and non-standard response distributions, such as bivariate normal or zero-inflated Poisson.