基于精度矩阵伸缩块分解的高斯图模型证据估计
Evidence Estimation in Gaussian Graphical Models Using a Telescoping Block Decomposition of the Precision Matrix
AI总结：
本文提出基于精度矩阵伸缩块分解的新方法，结合Chib技术，在统一框架下解决了高斯图模型中Wishart、G-Wishart、贝叶斯图lasso和图horseshoe等广泛先验下的模型证据估计问题。
摘要：
边际似然，也称为模型证据，是贝叶斯统计中的一个基本量。它用于通过贝叶斯因子进行模型选择，或用于先验超参数的经验贝叶斯调优。然而，证据的计算在高斯图模型中一直是一个长期悬而未决的问题。目前，唯一可行的解决方案仅适用于中等维度下的特殊情况，如Wishart或G-Wishart先验。我们基于精度矩阵的一种新颖伸缩块分解开发了一种方法，该方法允许在非常广泛的先验类别和温和要求下，通过应用Chib技术来估计证据。具体而言，要求是：(a) 精度矩阵对角项上的先验可以写成gamma随机变量或gamma随机变量的尺度混合；(b) 非对角项上的先验可以表示为正态随机变量或正态随机变量的尺度混合。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horseshoe。在这些先验中，对于Wishart，真实边际具有解析闭式形式，为我们的方法提供了有用的验证。对于其他三种先验的一般设置，以及满足上述条件(a)和(b)的更多先验，证据的计算一直是一个悬而未决的问题，本文在一个统一框架下解决了这一问题。
英文摘要：
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.
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