使用 $\gamma$-散度的稳健贝叶斯图模型
Robust Bayesian graphical modeling using $γ$-divergence
AI总结：
本文提出一种基于 $\gamma$-散度的稳健贝叶斯高斯图模型推断方法，通过贝叶斯图形套索实现后验稳健性，并开发了加权贝叶斯自助法高效计算算法。
摘要：
高斯图模型是分析多元高斯分布观测中两个变量之间条件独立性的强大工具之一。当数据维度为中等或较高时，惩罚似然方法（如图形套索）可用于检测显著的条件独立性结构。然而，由于高斯假设，估计结果会受到异常值的影响。本文提出了一种新颖的稳健后验分布，用于使用 $\gamma$-散度（一种稳健散度）进行高斯图模型的推断。特别地，我们通过假设逆协方差矩阵元素服从拉普拉斯型先验，聚焦于贝叶斯图形套索。所提出的后验分布的最大后验估计与频率学派惩罚方法提供的最小 $\gamma$-散度估计相匹配。我们证明了所提出的方法满足后验稳健性，这是贝叶斯分析中一种稳健性度量。该性质意味着，只要异常值极大，后验分布就会自动忽略异常值的信息。我们还推导了所提出后验分布的后验适定性的充分条件。此外，提出了一种通过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出方法的性能。
英文摘要：
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 $γ$-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 $γ$-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 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. A sufficient condition for the posterior propriety of the proposed posterior distribution is also derived. 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.
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