Bayesian precision matrix estimation for graphical Gaussian models with edge and vertex symmetries
Bayesian precision matrix estimation for graphical Gaussian models with edge and vertex symmetries
复制标题
具有边缘和顶点对称性的图形高斯模型的贝叶斯精度矩阵估计
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Xin Gao
中科院分区：
文献类型：
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作者：
H. Massam;Qiong Li;Xin Gao
Citet{HojLaur：2008}介绍了具有边和顶点对称性的图形高斯模型，他还给出了计算此类模型精度矩阵的最大似然估计的算法。在本文中，我们采取贝叶斯方法的精度矩阵的估计。我们只考虑那些模型的对称性约束施加在精度矩阵，从而形成一个自然的指数家庭的精度矩阵作为规范参数。
我们首先确定了Diaconis-Ylvisaker共轭先验这些模型，并制定了一个计划，从先验和后验分布的样本。因此，我们得到的精度矩阵的后验均值的估计。
其次，为了验证我们的估计的精度，我们推导出的精度矩阵的期望值的显式解析表达式时，我们的模型的图是一棵树，一个完全的图上的三个顶点和一个可分解的图上的四个顶点具有各种对称性。在这些情况下，我们将我们的估计值与先验分布均值的精确值进行比较。我们还验证了我们的后验均值的估计的准确性与多达30个顶点和各种对称性的图的模拟数据。
Graphical Gaussian models with edge and vertex symmetries were introduced by citet{HojLaur:2008} who also gave an algorithm to compute the maximum likelihood estimate of the precision matrix for such models. In this paper, we take a Bayesian approach to the estimation of the precision matrix. We consider only those models where the symmetry constraints are imposed on the precision matrix and which thus form a natural exponential family with the precision matrix as the canonical parameter.
We first identify the Diaconis-Ylvisaker conjugate prior for these models and develop a scheme to sample from the prior and posterior distributions. We thus obtain estimates of the posterior mean of the precision matrix.
Second, in order to verify the precision of our estimate, we derive the explicit analytic expression of the expected value of the precision matrix when the graph underlying our model is a tree, a complete graph on three vertices and a decomposable graph on four vertices with various symmetries. In those cases, we compare our estimates with the exact value of the mean of the prior distribution. We also verify the accuracy of our estimates of the posterior mean on simulated data for graphs with up to thirty vertices and various symmetries.
影响因子：
3.7
作者：
Dobra A;Lenkoski A;Rodriguez A
通讯作者：
Rodriguez A