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}

翻译：空间相关数据在许多应用中出现，高斯过程是这些场景下流行的建模选择。虽然贝叶斯分析在解决这些问题上已被证明是成功的，但为这些复杂模型选择先验分布仍然是一项困难的任务。在这项工作中，我们提出了一种基于模型拟合度度量来设置模型方差分量先验分布的原则性方法。具体地，我们推导了先验决定系数的分布。在此度量上放置一个贝塔先验分布，会在模型线性预测器的全局方差上诱导出一个广义贝塔主先验分布。该方法也可以被视为将拟合度向仅含截距（零）模型收缩。我们推导了针对大多数参数的高效吉布斯采样器，并使用Metropolis-Hasting更新处理其余参数。最后，该方法应用于一个海洋保护区数据集。我们估计了海洋政策对生物多样性的影响，并得出结论：禁捕限制导致生物多样性略有增加，且线性预测器中的大部分方差来自空间效应。