高维精度矩阵估计中具有Horseshoe先验的稀疏贝叶斯模型的集中性
Concentration of a sparse Bayesian model with Horseshoe prior in estimating high-dimensional precision matrix
AI总结：
本文针对高维稀疏精度矩阵估计，提出基于Horseshoe先验的回火后验集中性理论，并给出模型误设下的oracle不等式，通过模拟验证了方法的有效性。
摘要：
精度矩阵在社交网络、神经科学和经济学等许多领域中都至关重要，它代表了高斯图模型（GGMs）的边结构，其中精度矩阵的非对角位置为零表示节点之间的条件独立性。在高维设置中，当精度矩阵的维度 \\( p \\) 超过样本量 \\( n \\) 且矩阵是稀疏的时，诸如图形Lasso、图形SCAD和CLIME等方法常用于估计GGMs。虽然频率派方法已被充分研究，但针对（非结构化）稀疏精度矩阵的贝叶斯方法探索较少。\n\n\cite{li2019graphical}提出的图形Horseshoe估计，应用全局-局部Horseshoe先验，显示出优越的经验性能，但使用收缩先验进行稀疏精度矩阵估计的理论工作有限。本文通过在高维设置下提供具有完全指定Horseshoe先验的回火后验的集中性结果来解决这些空白。此外，我们还为模型误设提供了新颖的理论结果，为后验提供了一个通用的oracle不等式。我们进行了一组简明的模拟实验来验证我们的理论发现。
英文摘要：
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.
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