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 ($R^2)$, which then induces a prior on the individual parameters. We achieve this by placing a beta prior on $R^2$ 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 and real-world data, where the method particularly shines in high-dimensional settings, as well as modeling 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。虽然这种方法较为便捷，但在许多场景下，研究者可能更希望将先验置于模型的汇总度量上。本文提出了一种基于模型拟合优度（以贝叶斯决定系数$R^2$衡量）的先验方案，该方案进而诱导出各个参数的先验分布。我们通过为$R^2$设置贝塔先验，推导出广义线性混合模型中全局方差参数的诱导先验。针对多种情形推导了闭式表达式，并在无法获得解析形式时（或为简化计算）提出了若干近似策略。在这些情况下，我们建议采用广义贝塔主分布近似先验，并提供简洁的默认先验构建方案。该方法具有高度灵活性，便于在标准贝叶斯软件中实现。最后，我们在模拟数据和真实数据上验证了该方法的表现——该方法尤其在处理高维数据与随机效应建模时展现出显著优势。