The Annals of Statistics
- Ann. Statist.
- Volume 41, Number 6 (2013), 3022-3049.
Structure estimation for discrete graphical models: Generalized covariance matrices and their inverses
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.
Ann. Statist., Volume 41, Number 6 (2013), 3022-3049.
First available in Project Euclid: 1 January 2014
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Loh, Po-Ling; Wainwright, Martin J. Structure estimation for discrete graphical models: Generalized covariance matrices and their inverses. Ann. Statist. 41 (2013), no. 6, 3022--3049. doi:10.1214/13-AOS1162. https://projecteuclid.org/euclid.aos/1388545677
- Supplementary material: Supplementary material for “Structure estimation for discrete graphical models: Generalized covariance matrices and their inverses”. Due to space constraints, we have relegated technical details of the remaining proofs to the supplement .