Spatial regression modeling via the R2D2 framework

TL;DR

This paper introduces a Bayesian spatial regression framework called R2D2 that employs a prior on model fit to improve prior specification and shrinkage, demonstrated through marine biodiversity data analysis.

Contribution

The paper develops a novel prior distribution for spatial regression models based on the distribution of the coefficient of determination, enhancing prior setting and model shrinkage capabilities.

Findings

No-take marine restrictions slightly increase biodiversity.

Most variance in the model is due to spatial effects.

The R2D2 framework effectively estimates spatial model parameters.

Abstract

Spatially dependent data arises in many applications, and Gaussian processes are a popular modelling choice for these scenarios. While Bayesian analyses of these problems have proven to be successful, selecting prior distributions for these complex models remains a difficult task. In this work, we propose a principled approach for setting prior distributions on model variance components by placing a prior distribution on a measure of model fit. In particular, we derive the distribution of the prior coefficient of determination. Placing a beta prior distribution on this measure induces a generalized beta prime prior distribution on the global variance of the linear predictor in the model. This method can also be thought of as shrinking the fit towards the intercept-only (null) model. We derive an efficient Gibbs sampler for the majority of the parameters and use Metropolis-Hasting updates for the others. Finally, the method is applied to a marine protection area data set. We estimate the effect of marine policies on biodiversity and conclude that no-take restrictions lead to a slight increase in biodiversity and that the majority of the variance in the linear predictor comes from the spatial effect.\vspace{12pt}