Symmetries in Directed Gaussian Graphical Models

We define Gaussian graphical models on directed acyclic graphs with coloured vertices and edges, calling them RDAG (or restricted directed acyclic graph) models. If two vertices or edges have the same colour, their parameters in the model must be the same. We present an algorithm to find the maximum likelihood estimate (MLE) in an RDAG model, and characterise when the MLE exists, via linear independence conditions. We relate properties of a graph, and its colouring, to the number of samples needed for the MLE to exist and to be unique. We also characterise when an RDAG model is equal to an associated undirected graphical model. Finally, we consider connections between RDAG models and Gaussian group models.