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 \citet{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.

翻译：精度矩阵在社会网络、神经科学和经济学等诸多领域中至关重要，它代表了高斯图模型（GGMs）的边结构，其中精度矩阵非对角线位置上的零值表示节点间的条件独立性。在高维设定下，即精度矩阵的维度 $p$ 超过样本量 $n$ 且矩阵是稀疏的，诸如 graphical Lasso、graphical SCAD 和 CLIME 等方法被广泛用于估计 GGMs。虽然频率主义方法已得到深入研究，但针对（非结构化）稀疏精度矩阵的贝叶斯方法探索较少。\citet{li2019graphical} 提出的 graphical horseshoe 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然有限。本文通过在高维设定下为具有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，我们还为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。