Spatial regression modeling via the R2D2 framework

Spatially dependent data arises in many applications, and Gaussian processes are a popular modeling 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 dataset. 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.