We consider the problem of learning a Gaussian graphical model in the case where the observations come from two dependent groups sharing the same variables. We focus on a family of coloured Gaussian graphical models specifically suited for the paired data problem. Commonly, graphical models are ordered by the submodel relationship so that the search space is a lattice, called the model inclusion lattice. We introduce a novel order between models, named the twin order. We show that, embedded with this order, the model space is a lattice that, unlike the model inclusion lattice, is distributive. Furthermore, we provide the relevant rules for the computation of the neighbours of a model. The latter are more efficient than the same operations in the model inclusion lattice, and are then exploited to achieve a more efficient exploration of the search space. These results can be applied to improve the efficiency of both greedy and Bayesian model search procedures. Here we implement a stepwise backward elimination procedure and evaluate its performance by means of simulations. Finally, the procedure is applied to learn a brain network from fMRI data where the two groups correspond to the left and right hemispheres, respectively.

翻译：我们考虑在观测数据来自共享相同变量的两个依赖组的情况下学习高斯图模型的问题。我们聚焦于一类特别适用于配对数据问题的彩色高斯图模型家族。通常，图模型通过子模型关系进行排序，使得搜索空间成为一个格，称为模型包含格。我们引入了一种新的模型顺序，称为孪生顺序。我们证明，在该顺序下，模型空间是一个格，与模型包含格不同，该格是分配格。此外，我们提供了计算模型邻居的相关规则。这些规则比模型包含格中的相同操作更高效，并因此被用于实现更有效的搜索空间探索。这些结果可应用于提高贪婪搜索和贝叶斯模型搜索过程的效率。本文实现了一种逐步向后消除过程，并通过模拟评估其性能。最后，将该过程应用于从fMRI数据中学习脑网络，其中两个组分别对应左半球和右半球。