MODEL SELECTION IN THE SPACE OF GAUSSIAN MODELS INVARIANT BY SYMMETRY

• 成果类型: Article
• 署名作者: Graczyk, Piotr; Ishi, Hideyuki; Kolodziejek, Bartosz; Massam, Helene
• 署名单位: Universite d'Angers; Osaka Metropolitan University; Warsaw University of Technology; York University - Canada
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/22-AOS2174
• 发表日期: 2022
• 页码: 1747-1774
• 关键词: Graphical models; covariance; edge

摘要

We consider multivariate centered Gaussian models for the random variable Z = (Z(1),..., Z(p)), 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 Sigma and also the analytic expression of the normalizing constant of the Diaconis-Ylvisaker 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.