The Annals of Statistics

Geometry of maximum likelihood estimation in Gaussian graphical models

Caroline Uhler

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Abstract

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.

Article information

Source
Ann. Statist., Volume 40, Number 1 (2012), 238-261.

Dates
First available in Project Euclid: 29 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.aos/1333029964

Digital Object Identifier
doi:10.1214/11-AOS957

Mathematical Reviews number (MathSciNet)
MR3014306

Zentralblatt MATH identifier
1246.62140

Subjects
Primary: 62H12: Estimation 14Q10: Surfaces, hypersurfaces 90C25: Convex programming

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

Citation

Uhler, Caroline. Geometry of maximum likelihood estimation in Gaussian graphical models. Ann. Statist. 40 (2012), no. 1, 238--261. doi:10.1214/11-AOS957. https://projecteuclid.org/euclid.aos/1333029964


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