Symmetries in Directed Gaussian Graphical Models

TL;DR

This paper introduces RDAG models, a class of Gaussian graphical models on coloured directed acyclic graphs, providing algorithms for MLE, conditions for existence and uniqueness, and exploring their relation to undirected models and invariance properties.

Contribution

It defines RDAG models with colour-based parameter constraints, develops an MLE algorithm, and characterizes conditions for MLE existence, uniqueness, and model equivalence to undirected graphs.

Findings

MLE algorithm for RDAG models

Conditions for MLE existence and uniqueness

RDAGs can outperform uncoloured DAGs in certain scenarios

Abstract

We define Gaussian graphical models on directed acyclic graphs with coloured vertices and edges, calling them RDAG (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 and study connections to groups and invariant theory. We provide examples and simulations to study the benefits of RDAGs over uncoloured DAGs.