We develop a novel full-Bayesian approach for multiple correlated precision matrices, called multiple Graphical Horseshoe (mGHS). The proposed approach relies on a novel multivariate shrinkage prior based on the Horseshoe prior that borrows strength and shares sparsity patterns across groups, improving posterior edge selection when the precision matrices are similar. On the other hand, there is no loss of performance when the groups are independent. Moreover, mGHS provides a similarity matrix estimate, useful for understanding network similarities across groups. We implement an efficient Metropolis-within-Gibbs for posterior inference; specifically, local variance parameters are updated via a novel and efficient modified rejection sampling algorithm that samples from a three-parameter Gamma distribution. The method scales well with respect to the number of variables and provides one of the fastest full-Bayesian approaches for the estimation of multiple precision matrices. Finally, edge selection is performed with a novel approach based on model cuts. We empirically demonstrate that mGHS outperforms competing approaches through both simulation studies and an application to a bike-sharing dataset.

翻译：我们开发了一种新颖的全贝叶斯方法，用于处理多个相关精度矩阵，称为多重图形马靴（mGHS）。该方法基于马靴先验提出了一种新颖的多变量收缩先验，能够跨组借用信息并共享稀疏模式，从而在精度矩阵相似时改善后验边选择。另一方面，当组间独立时，性能不会损失。此外，mGHS提供了相似性矩阵估计，有助于理解组间的网络相似性。我们实现了一种高效的Metropolis-within-Gibbs算法进行后验推断；具体而言，局部方差参数通过一种新颖且高效的修正拒绝采样算法进行更新，该算法从三参数伽马分布中采样。该方法在变量数量方面具有良好的可扩展性，并提供了估计多个精度矩阵的最快全贝叶斯方法之一。最后，基于模型切割的新颖方法进行边选择。通过模拟研究和对共享单车数据集的应用，我们实证证明了mGHS优于现有竞争方法。