  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.
