标题： 高斯图模型中精度矩阵分块分解的证据估计 显示英文标题

标题： Evidence Estimation in Gaussian Graphical Models Using a Telescoping Block Decomposition of the Precision Matrix

摘要： 边际似然，也称为模型证据，在贝叶斯统计中是一个基本量。它用于通过贝叶斯因子进行模型选择，或者用于经验贝叶斯的先验超参数调整。然而，在高斯图模型中计算证据一直是一个长期存在的开放性问题。目前，唯一可行的解决方案仅适用于一些特殊情况，例如Wishart分布或G-Wishart分布，并且这些情况是在中等维度下的。 我们提出了一种基于精确矩阵的新颖望远镜块分解的方法，这种方法允许在非常广泛的先验条件下使用Chib的技术来估计证据。具体来说，这些条件包括：(a) 精确矩阵对角线元素上的先验可以表示为伽马随机变量或伽马随机变量的比例混合；(b) 非对角线元素上的先验可以表示为正态分布或正态分布的比例混合。这包括结构化先验，如Wishart分布或G-Wishart分布，以及最近引入的元素级先验，如贝叶斯图形lasso和图形马蹄铁先验。在这之中，对于Wishart分布，真正的边缘似然以解析闭合形式已知，为我们提供了方法的有效验证。对于其他三种先验的通用设置，以及其他满足上述条件(a)和(b)的若干先验，计算证据一直是一个未解的问题，本文在统一框架下解决了这一问题。 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 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.