Model selection for coloured graphical models for paired data

Abstract

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. Many model search algorithms require the exploration of the search space that is typically carried out by means of local moves between neighbouring models. It is therefore crucial to be able to rely on procedures that allow us to explore efficiently the space of models. However, the exploration of the space of coloured graphical models is much more challenging than for classical 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 both under the model inclusion order and under a novel order that we call the twin order. We show that our novel order allows a more efficient exploration of the search space. This is then used to implement a stepwise model search procedure and an application to the identification of a brain network from fMRI data is given.