Model selection in the space of Gaussian models invariant by symmetry
Abstract
We consider multivariate centered Gaussian models for the random variable $Z=(Z_1,\ldots, Z_p)$, invariant under the action of a subgroup of the group of permutations on $\{1,\ldots, p\}$. 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 $\Sigma$ and also the analytic expression of the normalizing constant of the DiaconisYlvisaker conjugate prior for the precision parameter $K=\Sigma^{1}$. 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 $p=100$.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 DOI:
 10.48550/arXiv.2004.03503
 arXiv:
 arXiv:2004.03503
 Bibcode:
 2020arXiv200403503G
 Keywords:

 Mathematics  Statistics Theory;
 Mathematical Physics;
 Mathematics  Representation Theory;
 Primary: 62H99;
 62F15;
 Secondary: 20C35
 EPrint:
 28 pages of the main text, 15 pages of the Supplementary material, 6 figures, 5 tables. Accepted to Annals of Statistics