In this paper, we consider high-dimensional Gaussian graphical models where the true underlying graph is decomposable. A hierarchical $G$-Wishart prior is proposed to conduct a Bayesian inference for the precision matrix and its graph structure. Although the posterior asymptotics using the $G$-Wishart prior has received increasing attention in recent years, most of results assume moderate high-dimensional settings, where the number of variables $p$ is smaller than the sample size $n$. However, this assumption might not hold in many real applications such as genomics, speech recognition and climatology. Motivated by this gap, we investigate asymptotic properties of posteriors under the high-dimensional setting where $p$ can be much larger than $n$. The pairwise Bayes factor consistency, posterior ratio consistency and graph selection consistency are obtained in this high-dimensional setting. Furthermore, the posterior convergence rate for precision matrices under the matrix $\ell_1$-norm is derived, which turns out to coincide with the minimax convergence rate for sparse precision matrices. A simulation study confirms that the proposed Bayesian procedure outperforms competitors.

翻译：在本文中,我们考虑的是真实基本图形可以解析的高维高斯图形模型。 之前的一个等级值$G$- Wishart建议对精确矩阵及其图形结构进行巴伊西亚推断。 虽然使用美元-Wishart之前的后等无症状学近年来受到越来越多的注意, 但大多数结果都假设中等高维环境, 变量的数量比样本规模小, 美元。 然而, 这一假设可能在许多实际应用中无法维持, 如基因组学、 语音识别和气候学。 受这一差距的驱使, 我们调查高方位设置下的后等离子体无症状特性, 在那里, 美元可能比美元大得多。 双向贝伊系数一致性、 后等比率一致性和图表选择一致性是在这个高维环境中取得的。 此外, 基质 $\ ell_ 1 美元- 诺姆 下的精度矩阵的后等相趋合率, 与低层竞争者平面矩阵的微分子趋同率一致。 模拟研究证实了拟议采用的方法。