Precision matrices are crucial in many fields such as social networks, neuroscience, and economics, representing the edge structure of Gaussian graphical models (GGMs), where a zero in an off-diagonal position of the precision matrix indicates conditional independence between nodes. In high-dimensional settings where the dimension of the precision matrix \( p \) exceeds the sample size \( n \) and the matrix is sparse, methods like graphical Lasso, graphical SCAD, and CLIME are popular for estimating GGMs. While frequentist methods are well-studied, Bayesian approaches for (unstructured) sparse precision matrices are less explored. The graphical horseshoe estimate by \cite{li2019graphical}, applying the global-local horseshoe prior, shows superior empirical performance, but theoretical work for sparse precision matrix estimations using shrinkage priors is limited. This paper addresses these gaps by providing concentration results for the tempered posterior with the fully specified horseshoe prior in high-dimensional settings. Moreover, we also provide novel theoretical results for model misspecification, offering a general oracle inequality for the posterior. A concise set of simulations is performed to validate our theoretical findings.

翻译：精度矩阵在社交网络、神经科学和经济学等诸多领域中至关重要，它表征了高斯图模型（GGMs）的边结构，其中精度矩阵非对角线位置的零元素表示节点间的条件独立性。在精度矩阵维度 \( p \) 超过样本量 \( n \) 且矩阵稀疏的高维设定下，诸如图Lasso、图SCAD和CLIME等方法常用于估计GGMs。尽管频率学派方法已得到充分研究，针对（非结构化）稀疏精度矩阵的贝叶斯方法则探索较少。\cite{li2019graphical}提出的图马蹄估计通过应用全局-局部马蹄先验展现了优越的实证性能，但使用收缩先验进行稀疏精度矩阵估计的理论研究仍较为有限。本文通过在高维设定下为带完全指定马蹄先验的调和后验提供集中性结果，以填补这些空白。此外，我们还针对模型误设提供了新的理论结果，给出了后验的一般oracle不等式。我们进行了一组简洁的模拟实验以验证理论发现。