STRUCTURE ESTIMATION FOR DISCRETE GRAPHICAL MODELS: GENERALIZED COVARIANCE MATRICES AND THEIR INVERSES

• 成果类型: Article
• 署名作者: Loh, Po-Ling; Wainwright, Martin J.
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/13-AOS1162
• 发表日期: 2013
• 页码: 3022-3049
• 关键词: maximum-likelihood; selection; regression; Lasso; RECOVERY; systems

摘要

We investigate the relationship between the structure of a discrete graphical model and the support of the inverse of a generalized covariance matrix. We show that for certain graph structures, the support of the inverse covariance matrix of indicator variables on the vertices of a graph reflects the conditional independence structure of the graph. Our work extends results that have previously been established only in the context of multivariate Gaussian graphical models, thereby addressing an open question about the significance of the inverse covariance matrix of a non-Gaussian distribution. The proof exploits a combination of ideas from the geometry of exponential families, junction tree theory and convex analysis. These population-level results have various consequences for graph selection methods, both known and novel, including a novel method for structure estimation for missing or corrupted observations. We provide nonasymptotic guarantees for such methods and illustrate the sharpness of these predictions via simulations.