In Bayesian analysis, the selection of a prior distribution is typically done by considering each parameter in the model. While this can be convenient, in many scenarios it may be desirable to place a prior on a summary measure of the model instead. In this work, we propose a prior on the model fit, as measured by a Bayesian coefficient of determination (R2), which then induces a prior on the individual parameters. We achieve this by placing a beta prior on R2 and then deriving the induced prior on the global variance parameter for generalized linear mixed models. We derive closed-form expressions in many scenarios and present several approximation strategies when an analytic form is not possible and/or to allow for easier computation. In these situations, we suggest approximating the prior by using a generalized beta prime distribution and provide a simple default prior construction scheme. This approach is quite flexible and can be easily implemented in standard Bayesian software. Lastly, we demonstrate the performance of the method on simulated data, where it particularly shines in high-dimensional examples, as well as real-world data, which shows its ability to model spatial correlation in the random effects.

翻译：在贝叶斯分析中，先验分布的选择通常通过考虑模型中的每个参数来完成。虽然这种方法很方便，但在许多场景中，可能更希望将先验置于模型的汇总度量上。在本工作中，我们提出了一种基于模型拟合度的先验，该拟合度由贝叶斯决定系数（R2）衡量，进而推导出个体参数上的先验。我们通过将贝塔先验置于R2上，然后推导出广义线性混合模型中全局方差参数的诱导先验来实现这一点。我们在多种场景下推导了闭式表达式，并在无法获得解析形式或为简化计算时，提出了几种近似策略。在这些情况下，我们建议使用广义贝塔素数分布来近似先验，并提供了一种简单的默认先验构建方案。该方法非常灵活，可轻松在标准贝叶斯软件中实现。最后，我们在模拟数据上展示了该方法的性能，尤其是在高维示例中表现突出，同时也在真实数据上验证了其对随机效应中空间相关性建模的能力。