AI 中文总结
该研究提出正性约束下贝叶斯图模型的可扩展广义似然方法，通过改进采样算法提升高维精度矩阵估计效率，在合成与金融数据上表现出计算优势。
AI 中文摘要
我们开发了一种计算上可扩展的贝叶斯框架，用于在完全正性约束下估计高斯图模型中的精度矩阵。为克服高斯似然的高计算成本，我们采用基于$D$-迹损失的广义贝叶斯方法，该方法消除了对数行列式项，实现高效优化，同时允许在采样过程中放松正定性。通过尖峰-板条先验诱导稀疏性，所得广义后验在温和条件下被证明是正常的。我们的主要贡献是一套针对高维场景定制的高效后验采样算法：从分量式吉布斯采样器出发，引入新颖的数据增广方案，诱导精度矩阵元素间的条件独立性，实现联合更新；利用样本协方差矩阵的Gram结构，进一步开发快速矩阵正态采样器，大幅降低高维场景下的每次迭代复杂度；交织策略结合增广与直接更新，在不损失可扩展性的前提下提升混合性。在合成数据和金融数据上的实验表明，与现有方法相比，我们的方法获得了显著的计算增益，同时保持了有竞争力的估计精度，并改进了结构化依赖关系的恢复。
英文摘要
We develop a computationally scalable Bayesian framework for precision matrix estimation in Gaussian graphical models under total positivity constraints. To overcome the high computational cost of the Gaussian likelihood, we adopt a generalized Bayesian approach based on the $D$-trace loss, which eliminates the log-determinant term and enables efficient optimization while allowing relaxation of positive definiteness during sampling. Sparsity is induced via spike-and-slab priors, and the resulting generalized posterior is shown to be proper under mild conditions. Our primary contribution is a suite of efficient posterior sampling algorithms tailored to high-dimensional settings. Starting from a component-wise Gibbs sampler, we introduce a novel data augmentation scheme that induces conditional independence among precision matrix entries, enabling joint updates. By exploiting the Gram structure of the sample covariance matrix, we further develop a fast matrix-normal sampler that significantly reduces per-iteration complexity in high-dimensional settings. An interweaving strategy combines augmented and direct updates to improve mixing without sacrificing scalability. Experiments on synthetic and financial data demonstrate substantial computational gains over existing methods, while maintaining competitive estimation accuracy and improved recovery of structured dependencies.
Comments27 pages, 3 figures, 5 tables
