Network models are useful tools for modelling complex associations. If a Gaussian graphical model is assumed, conditional independence is determined by the non-zero entries of the inverse covariance (precision) matrix of the data. The Bayesian graphical horseshoe estimator provides a robust and flexible framework for precision matrix inference, as it introduces local, edge-specific parameters which prevent over-shrinkage of non-zero off-diagonal elements. However, for many applications such as statistical omics, the current implementation based on Gibbs sampling becomes computationally inefficient or even unfeasible in high dimensions. Moreover, the graphical horseshoe has only been formulated for a single network, whereas interest has grown in the network analysis of multiple data sets that might share common structures. We propose (i) a scalable expectation conditional maximisation (ECM) algorithm for obtaining the posterior mode of the precision matrix in the graphical horseshoe, and (ii) a novel joint graphical horseshoe estimator, which borrows information across multiple related networks to improve estimation. We show, on both simulated and real omics data, that our single-network ECM approach is more scalable than the existing graphical horseshoe Gibbs implementation, while achieving the same level of accuracy. We also show that our joint-network proposal successfully leverages shared edge-specific information between networks while still retaining differences, outperforming state-of-the-art methods at any level of network similarity.

翻译：网络模型是建模复杂关联的有效工具。若假定高斯图模型，条件独立性由数据协方差逆矩阵（精度矩阵）的非零元素决定。贝叶斯图形马靴估计器通过引入局部边特定参数，可防止非对角线非零元素的过度收缩，为精度矩阵推断提供了稳健且灵活的框架。然而，在统计组学等应用中，基于吉布斯采样的现有实现在高维场景下计算效率低下甚至不可行。此外，图形马靴方法仅针对单一网络设计，而当前对可能共享共同结构的多个数据集的网络分析需求日益增长。我们提出：（i）一种可扩展的期望条件最大化（ECM）算法，用于获取图形马靴模型中精度矩阵的后验众数；（ii）一种新型联合图形马靴估计器，通过跨多个相关网络借力信息以改进估计。基于模拟数据和真实组学数据的实验表明，我们的单网络ECM方法在达到相同精度水平的同时，比现有图形马靴吉布斯实现具有更高可扩展性。我们还证明，联合网络方案成功利用了网络间共享的边特定信息，同时保留差异，在所有网络相似度水平上均优于现有最优方法。