Gaussian graphical model is one of the powerful tools to analyze conditional independence between two variables for multivariate Gaussian-distributed observations. When the dimension of data is moderate or high, penalized likelihood methods such as the graphical lasso are useful to detect significant conditional independence structures. However, the estimates are affected by outliers due to the Gaussian assumption. This paper proposes a novel robust posterior distribution for inference of Gaussian graphical models using the $\gamma$-divergence which is one of the robust divergences. In particular, we focus on the Bayesian graphical lasso by assuming the Laplace-type prior for elements of the inverse covariance matrix. The proposed posterior distribution matches its maximum a posteriori estimate with the minimum $\gamma$-divergence estimate provided by the frequentist penalized method. We show that the proposed method satisfies the posterior robustness which is a kind of measure of robustness in the Bayesian analysis. The property means that the information of outliers is automatically ignored in the posterior distribution as long as the outliers are extremely large, which also provides theoretical robustness of point estimate for the existing frequentist method. A sufficient condition for the posterior propriety of the proposed posterior distribution is also shown. Furthermore, an efficient posterior computation algorithm via the weighted Bayesian bootstrap method is proposed. The performance of the proposed method is illustrated through simulation studies and real data analysis.

翻译：高斯图模型是分析多元高斯分布观测数据中两个变量间条件独立性的强大工具之一。当数据维度中等或较高时，图套索等惩罚似然方法有助于检测显著的条件独立结构。然而，由于高斯假设，这些估计易受异常值影响。本文提出一种利用鲁棒散度之一——$γ$-散度的新型鲁棒后验分布，用于高斯图模型的推断。具体而言，我们聚焦于贝叶斯图套索方法，对逆协方差矩阵元素采用拉普拉斯型先验。所提出的后验分布的最大后验估计与频率学派惩罚方法给出的最小$γ$-散度估计一致。我们证明该方法满足后验鲁棒性，这是贝叶斯分析中一种鲁棒性度量。该性质意味着，只要异常值极大，后验分布会自动忽略异常值信息，这也为现有频率学派方法的点估计提供了理论鲁棒性。此外，本文还给出了所提后验分布适后性的充分条件。进一步地，提出了一种基于加权贝叶斯自助法的高效后验计算算法。通过模拟研究和真实数据分析展示了所提方法的性能。