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February 2012 Geometry of maximum likelihood estimation in Gaussian graphical models
Caroline Uhler
Ann. Statist. 40(1): 238-261 (February 2012). DOI: 10.1214/11-AOS957


We study maximum likelihood estimation in Gaussian graphical models from a geometric point of view. An algebraic elimination criterion allows us to find exact lower bounds on the number of observations needed to ensure that the maximum likelihood estimator (MLE) exists with probability one. This is applied to bipartite graphs, grids and colored graphs. We also study the ML degree, and we present the first instance of a graph for which the MLE exists with probability one, even when the number of observations equals the treewidth.


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Caroline Uhler. "Geometry of maximum likelihood estimation in Gaussian graphical models." Ann. Statist. 40 (1) 238 - 261, February 2012.


Published: February 2012
First available in Project Euclid: 29 March 2012

zbMATH: 1246.62140
MathSciNet: MR3014306
Digital Object Identifier: 10.1214/11-AOS957

Primary: 14Q10 , 62H12 , 90C25

Keywords: Algebraic statistics , algebraic variety , bipartite graphs , Duality , elimination ideal , Gaussian graphical model , matrix completion problems , maximum likelihood estimation , ML degree , number of observations , sufficient statistics , treewidth

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 1 • February 2012
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