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The Annals of Statistics
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.
graphical models, Markov random fields, model selection, inverse covariance estimation, high-dimensional statistics, exponential families, Legendre duality
Loh, P., & Wainwright, M. J. (2013). Structure Estimation for Discrete Graphical Models: Generalized Covariance Matrices and Their Inverses. The Annals of Statistics, 41 (6), 3022-3049. http://dx.doi.org/10.1214/13-AOS1162
Date Posted: 27 November 2017
This document has been peer reviewed.