GROUPS ACTING ON GAUSSIAN GRAPHICAL MODELS

• 成果类型: Article
• 署名作者: Draisma, Jan; Kuhnt, Sonja; Zwiernik, Piotr
• 署名单位: Eindhoven University of Technology; Dortmund University of Technology; University of California System; University of California Berkeley
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/13-AOS1130
• 发表日期: 2013
• 页码: 1944-1969
• 关键词: maximum-likelihood estimator; conditional-independence; Covariance matrices; breakdown; distributions; inference

摘要

Gaussian graphical models have become a well-recognized tool for the analysis of conditional independencies within a set of continuous random variables. From an inferential point of view, it is important to realize that they are composite exponential transformation families. We reveal this structure by explicitly describing, for any undirected graph, the (maximal) matrix group acting on the space of concentration matrices in the model. The continuous part of this group is captured by a poset naturally associated to the graph, while automorphisms of the graph account for the discrete part of the group. We compute the dimension of the space of orbits of this group on concentration matrices, in terms of the combinatorics of the graph; and for dimension zero we recover the characterization by Letac and Massam of models that are transformation families. Furthermore, we describe the maximal invariant of this group on the sample space, and we give a sharp lower bound on the sample size needed for the existence of equivariant estimators of the concentration matrix. Finally, we address the issue of robustness of these estimators by computing upper bounds on finite sample breakdown points.