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：在高维回归模型对相对稀疏数据进行训练时，这是一个重要且复杂的课题，尤其是在数据中模型参数远多于观测值的情况下。从贝叶斯视角看，借助收缩先验分布（至少对于广义线性模型而言）可以实现此类情况下的推断。然而，现实世界的数据通常具有多层结构，例如重复测量或个体的自然分组，而现有的收缩先验并不适用于处理此类结构。我们对Zhang等人（2020）提出的R2D2先验进行推广与扩展，将其应用于线性多层模型，从而提出所谓的R2D2M2先验。该先验能够同时实现模型参数的局部与全局收缩。它包含可解释的超参数，我们证明这些超参数与先验的关键性质（如原点附近的集中速率、尾部行为以及先验施加的收缩量）本质相关。通过推导收缩因子并测量非零模型系数的有效数量，我们提供了如何选择先验超参数的指导准则。因此，用户可以便捷地评估并解释特定超参数选择所隐含的收缩量。最后，我们在模拟数据和真实数据上进行了广泛实验，结果表明该先验的推断过程具有良好的校准性、理想的全局与局部正则化性质，并能够比以往更可靠且可解释地估计更复杂的贝叶斯多层模型。