Bayesian precision and covariance matrix estimation for graphical Gaussian models with edge and vertex symmetries
针对具有对称性约束的图高斯模型，提出贝叶斯方法估计精度矩阵和协方差矩阵，开发了从共轭先验和后验分布采样的方案，并在树图、完全图等结构上验证了估计准确性。
Graphical Gaussian models with edge and vertex symmetries were introduced by Højsgaard & 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.