June 2022 Model selection in the space of Gaussian models invariant by symmetry
Piotr Graczyk, Hideyuki Ishi, Bartosz Kołodziejek, Hélène Massam
Author Affiliations +
Ann. Statist. 50(3): 1747-1774 (June 2022). DOI: 10.1214/22-AOS2174


We consider multivariate centered Gaussian models for the random variable Z=(Z1,,Zp), invariant under the action of a subgroup of the group of permutations on {1,,p}. Using the representation theory of the symmetric group on the field of reals, we derive the distribution of the maximum likelihood estimate of the covariance parameter Σ and also the analytic expression of the normalizing constant of the Diaconis–Ylvisaker conjugate prior for the precision parameter K=Σ1. We can thus perform Bayesian model selection in the class of complete Gaussian models invariant by the action of a subgroup of the symmetric group, which we could also call complete RCOP models. We illustrate our results with a toy example of dimension 4 and several examples for selection within cyclic groups, including a high- dimensional example with p=100.

Funding Statement

The second author was supported by JSPS KAKENHI Grant Number 16K05174, 20K03657 and JST PRESTO.
The third author was supported by Grant 2016/21/B/ST1/00005 of the National Science Center, Poland.
The fourth author was supported by an NSERC Discovery Grant.


The authors would like to thank Steffen Lauritzen for his interest and encouragements. We also thank M. Bogdan from Wrocław University, A. Descatha from INSERM and Centre Hospitalier Universitaire Angers and V. Seegers from Institut de Cancerologie de l’Ouest Nantes for explaining the specific nature of medical and genetic data. The paper benefited from the comments of an anonymous referee to whom the authors are grateful.


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Piotr Graczyk. Hideyuki Ishi. Bartosz Kołodziejek. Hélène Massam. "Model selection in the space of Gaussian models invariant by symmetry." Ann. Statist. 50 (3) 1747 - 1774, June 2022. https://doi.org/10.1214/22-AOS2174


Received: 1 April 2020; Revised: 1 January 2022; Published: June 2022
First available in Project Euclid: 16 June 2022

MathSciNet: MR4441139
zbMATH: 07547949
Digital Object Identifier: 10.1214/22-AOS2174

Primary: 62F15 , 62H99
Secondary: 20C35

Keywords: Colored graph , conjugate prior , covariance selection , Invariance , permutation symmetry

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 3 • June 2022
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