We propose Bayesian methods for Gaussian graphical models that lead to sparse and adaptively shrunk estimators of the precision (inverse covariance) matrix. Our methods are based on lasso‐type regularization priors leading to parsimonious parameterization of the precision matrix, which is essential in several applications involving learning relationships among the variables. In this context, we introduce a novel type of selection prior that develops a sparse structure on the precision matrix by making most of the elements exactly zero, in addition to ensuring positive definiteness—thus conducting model selection and estimation simultaneously. More importantly, we extend these methods to analyse clustered data using finite mixtures of Gaussian graphical model and infinite mixtures of Gaussian graphical models. We discuss appropriate posterior simulation schemes to implement posterior inference in the proposed models, including the evaluation of normalizing constants that are functions of parameters of interest, which result from the restriction of positive definiteness on the correlation matrix. We evaluate the operating characteristics of our method via several simulations and demonstrate the application to real‐data examples in genomics. Copyright © 2014 John Wiley & Sons, Ltd

中文翻译：

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贝叶斯稀疏图形模型及其混合。

我们为高斯图形模型提出了贝叶斯方法，该方法导致精度（逆协方差）矩阵的稀疏和自适应收缩估计量。我们的方法基于套索型正则化先验，导致精度矩阵的简约参数化，这在涉及变量之间学习关系的几个应用中是必不可少的。在这种情况下，我们引入了一种新型的先验选择，除了确保正定性之外，还通过使大多数元素精确为零来在精度矩阵上开发稀疏结构，从而同时进行模型选择和估计。更重要的是，我们将这些方法扩展到使用高斯图形模型的有限混合和高斯图形模型的无限混合来分析聚类数据。我们讨论了在所提出的模型中实现后验推理的适当后验模拟方案，包括对作为感兴趣参数的函数的归一化常数的评估，这是由于相关矩阵的正定性的限制而产生的。我们通过几次模拟评估了我们方法的操作特性，并演示了在基因组学中的真实数据示例中的应用。版权所有 © 2014 John Wiley & Sons, Ltd 我们通过几次模拟评估了我们方法的操作特性，并演示了在基因组学中的真实数据示例中的应用。版权所有 © 2014 John Wiley & Sons, Ltd 我们通过几次模拟评估了我们方法的操作特性，并演示了在基因组学中的真实数据示例中的应用。版权所有 © 2014 John Wiley & Sons, Ltd