The training of high-dimensional regression models on comparably sparse data is an important yet complicated topic, especially when there are many more model parameters than observations in the data. From a Bayesian perspective, inference in such cases can be achieved with the help of shrinkage prior distributions, at least for generalized linear models. However, real-world data usually possess multilevel structures, such as repeated measurements or natural groupings of individuals, which existing shrinkage priors are not built to deal with. We generalize and extend one of these priors, the R2D2 prior by Zhang et al. (2020), to linear multilevel models leading to what we call the R2D2M2 prior. The proposed prior enables both local and global shrinkage of the model parameters. It comes with interpretable hyperparameters, which we show to be intrinsically related to vital properties of the prior, such as rates of concentration around the origin, tail behavior, and amount of shrinkage the prior exerts. We offer guidelines on how to select the prior's hyperparameters by deriving shrinkage factors and measuring the effective number of non-zero model coefficients. Hence, the user can readily evaluate and interpret the amount of shrinkage implied by a specific choice of hyperparameters. Finally, we perform extensive experiments on simulated and real data, showing that our inference procedure for the prior is well calibrated, has desirable global and local regularization properties and enables the reliable and interpretable estimation of much more complex Bayesian multilevel models than was previously possible.

翻译：在相对稀疏数据上训练高维回归模型是一个重要但复杂的课题，尤其当模型参数远超数据观测数时。从贝叶斯视角，此类推断可通过收缩先验分布实现（至少对广义线性模型而言）。然而，现实数据通常具有多层结构（如重复测量或个体自然分组），现有收缩先验并未针对此类场景设计。我们将Zhang等人(2020)提出的R2D2先验进行泛化与扩展，将其推广至线性多层模型，从而形成R2D2M2先验。该先验能同时实现模型参数的局部与全局收缩，其超参数具有可解释性，且与先验的关键性质（如原点处集中速率、尾部行为及收缩强度）存在本质关联。通过推导收缩因子并测量有效非零模型系数数量，我们提供了超参数选择指南，使用户能直接评估特定超参数选择所隐含的收缩程度。最后，我们在模拟与真实数据上开展广泛实验，结果表明：该先验的推断过程校准良好，兼具理想的全局与局部正则化特性，能够比以往更可靠、可解释地估计复杂得多的贝叶斯多层模型。