Exploration of the search space of Gaussian graphical models for paired data
Alberto Roverato, Dung Ngoc Nguyen
TL;DR
This work addresses structure learning for Gaussian graphical models when observations come from two dependent groups sharing the same variables. It introduces the twin lattice, a distributive partial order on paired-data coloured GGM models (pdRCON), enabling efficient, coherence-consistent exploration via a coherent stepwise backward elimination procedure. The approach refines the model-search space beyond the traditional model inclusion lattice, yielding substantial computational gains while preserving or enhancing recovery of both intra- and inter-group symmetries. Applications to simulated data and real brain fMRI data demonstrate improved efficiency and interpretable symmetric structures, with comparisons to penalized likelihood methods highlighting robustness to scaling and practical advantages in paired-data settings.
Abstract
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.