标题： 图形证据 显示英文标题

标题： Graphical Evidence

摘要： 边缘似然，也称为模型证据，是贝叶斯统计中的一个基本量。它用于使用贝叶斯因子进行模型选择，或用于经验贝叶斯调整先验超参数。然而，在高斯图模型中，证据的计算一直是一个长期未解的问题。目前，存在的可行解决方案仅适用于某些特殊情况，如Wishart或G-Wishart，在中等维度下。我们提出了一种Chib技术的应用，该技术在温和要求下适用于非常广泛的先验类别。具体而言，要求是：(a) 精度矩阵对角线项上的先验可以写成伽马分布或伽马随机变量的尺度混合形式，(b) 非对角线项上的先验可以表示为正态分布或正态分布的尺度混合形式。这包括结构化先验，如Wishart或G-Wishart，以及最近引入的逐元素先验，如贝叶斯图Lasso和图马蹄先验。其中，对于Wishart，真实的边缘似然具有解析闭合形式，为我们的方法提供了一个有用的验证。对于其他三种情况以及满足上述条件(a)和(b)的几种先验，证据的计算仍然是一个开放问题，本文旨在解决这一问题。 显示英文摘要

摘要： 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 present an application of Chib's technique that is applicable to 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 seeks to resolve.