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.