  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}
