Model selection in the space of Gaussian models invariant by symmetry
Piotr Graczyk, Hideyuki Ishi, Bartosz Kołodziejek, Hélène Massam
The Annals of Statistics · 2022 · Volume 50 · Issue 3
Abstract
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=Σ−1. 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.
Keywords
20C35, 62F15, 62H99, Colored graph, conjugate prior, covariance selection, Invariance, permutation symmetry
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