Bayesian precision and covariance matrix estimation for graphical Gaussian models with edge and vertex symmetries

• 成果类型: Article
• 署名作者: Massam, H.; Li, Q.; Gao, X.
• 署名单位: York University - Canada; Sun Yat Sen University
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/asx084
• 发表日期: 2018
• 页码: 371388
• 关键词: exponential-families; wishart distribution; GIBBS SAMPLER; distributions; inference

摘要

Graphical Gaussian models with edge and vertex symmetries were introduced by Hojsgaard & Lauritzen (2008), who gave an algorithm for computing the maximum likelihood estimate of the precision matrix for such models. In this paper, we take a Bayesian approach to its estimation. We consider only models with symmetry constraints and which thus form a natural exponential family with the precision matrix as the canonical parameter. We identify the Diaconis-Ylvisaker conjugate prior for these models, develop a scheme to sample from the prior and posterior distributions, and thus obtain estimates of the posterior mean of the precision and covariance matrices. Such a sampling scheme is essential for model selection in coloured graphical Gaussian models. In order to verify the precision of our estimates, we derive an analytic expression for the expected value of the precision matrix when the graph underlying our model is a tree, a complete graph on three vertices, or a decomposable graph on four vertices with various symmetries, and we compare our estimates with the posterior mean of the precision matrix and the expected mean of the coloured graphical Gaussian model, that is, of the covariance matrix. We also verify the accuracy of our estimates on simulated data.