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December 2013 Structure estimation for discrete graphical models: Generalized covariance matrices and their inverses
Po-Ling Loh, Martin J. Wainwright
Ann. Statist. 41(6): 3022-3049 (December 2013). DOI: 10.1214/13-AOS1162

Abstract

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.

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Po-Ling Loh. Martin J. Wainwright. "Structure estimation for discrete graphical models: Generalized covariance matrices and their inverses." Ann. Statist. 41 (6) 3022 - 3049, December 2013. https://doi.org/10.1214/13-AOS1162

Information

Published: December 2013
First available in Project Euclid: 1 January 2014

zbMATH: 1288.62081
MathSciNet: MR3161456
Digital Object Identifier: 10.1214/13-AOS1162

Subjects:
Primary: 62F12
Secondary: 68W25

Keywords: exponential families , graphical models , High-dimensional statistics , inverse covariance estimation , Legendre duality , Markov random fields , Model selection

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 6 • December 2013
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