Marginal likelihood, also known as model evidence, is a fundamental quantity in Bayesian statistics. It is used for model selection using Bayes factors or for empirical Bayes tuning of prior hyper-parameters. Yet, the calculation of evidence has remained a longstanding open problem in Gaussian graphical models. Currently, the only feasible solutions that exist are for special cases such as the Wishart or G-Wishart, in moderate dimensions. We develop an approach based on a novel telescoping block decomposition of the precision matrix that allows the estimation of evidence by application of Chib's technique under a very broad class of priors under mild requirements. Specifically, the requirements are: (a) the priors on the diagonal terms on the precision matrix can be written as gamma or scale mixtures of gamma random variables and (b) those on the off-diagonal terms can be represented as normal or scale mixtures of normal. This includes structured priors such as the Wishart or G-Wishart, and more recently introduced element-wise priors, such as the Bayesian graphical lasso and the graphical horseshoe. Among these, the true marginal is known in an analytically closed form for Wishart, providing a useful validation of our approach. For the general setting of the other three, and several more priors satisfying conditions (a) and (b) above, the calculation of evidence has remained an open question that this article resolves under a unifying framework.

翻译：边际似然，又称模型证据，是贝叶斯统计学中的基本量。它用于通过贝叶斯因子进行模型选择，或通过经验贝叶斯方法调整先验超参数。然而，在高斯图模型中，证据的计算一直是一个长期未解决的开放问题。目前，唯一可行的解决方案仅适用于特殊情形，例如中等维度下的Wishart分布或G-Wishart分布。本文提出了一种基于精度矩阵的新型伸缩块分解方法，通过应用Chib技术，在非常宽泛的先验类别和温和条件下实现了证据的估计。具体而言，所需条件为：（a）精度矩阵对角元素的先验可表示为伽马分布或伽马随机变量的尺度混合；（b）非对角元素的先验可表示为正态分布或正态随机变量的尺度混合。这包括结构化先验（如Wishart分布或G-Wishart分布）以及近期引入的元素级先验（如贝叶斯图套索和图马铁索）。其中，Wishart分布的边际似然具有解析闭式解，可为我们的方法提供有效验证。对于其他三种通用设定以及更多满足上述条件（a）和（b）的先验，证据的计算一直是未解之谜，而本文在统一框架下解决了这一问题。