Model inclusion lattice of coloured Gaussian graphical models for paired data

We consider the problem of learning a graphical model when the observations come from two groups sharing the same variables but, unlike the usual approach to the joint learning of graphical models, the two groups do not correspond to different populations and therefore produce dependent samples. A Gaussian graphical model for paired data may be implemented by applying the methodology developed for the family of graphical models with edge and vertex symmetries, also known as coloured graphical models. We identify a family of coloured graphical models suited for the paired data problem and investigate the structure of the corresponding model space. More specifically, we provide a comprehensive description of the lattice structure formed by this family of models under the model inclusion order. Furthermore, we give rules for the computation of the join and meet operations between models, which are useful in the exploration of the model space. These are then applied to implement a stepwise model search procedure and an application to the identification of a brain network from fMRI data is given.