Lattices of Graphical Gaussian Models with Symmetries

We study the structure of graphical Gaussian models which place symmetry restrictions on concentrations or partial correlations. The models can be represented by vertex and edge coloured graphs, in which parameters which are associated with equally coloured vertices or edges are restricted to being identical. We are particularly interested in two symmetry model classes within the model types with desirable model properties which express themselves in regularity of graph colouring. We demonstrate that each model class forms a complete lattice with respect to model inclusion and therefore qualifies for the Edwards-Havr\'anek model selection procedure, giving a first model selection algorithm for graphical Gaussian models with symmetries.