Open Access
2023 Symmetries in directed Gaussian graphical models
Visu Makam, Philipp Reichenbach, Anna Seigal
Author Affiliations +
Electron. J. Statist. 17(2): 3969-4010 (2023). DOI: 10.1214/23-EJS2192


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.

Funding Statement

VM was supported by the University of Melbourne and NSF Grant CCF 1900460. PR was funded by the European Research Council (ERC) under the European’s Horizon 2020 research and innovation programme (grant agreement no. 787840).


We are grateful to Dominic Bunnett, Mathias Drton, Robin Evans, Caroline Uhler, and Piotr Zwiernik for helpful discussions. We thank an anonymous referee for helpful comments that improved the paper.


Download Citation

Visu Makam. Philipp Reichenbach. Anna Seigal. "Symmetries in directed Gaussian graphical models." Electron. J. Statist. 17 (2) 3969 - 4010, 2023.


Received: 1 May 2022; Published: 2023
First available in Project Euclid: 12 December 2023

Digital Object Identifier: 10.1214/23-EJS2192

Primary: 62F10 , 62H22 , 62R01
Secondary: 05C15 , 14L24 , 15A23 , 20G45

Keywords: Gaussian models , graph colouring , graphical models , group actions , maximum likelihood estimation , maximum likelihood thresholds

Vol.17 • No. 2 • 2023
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