Geometry of maximum likelihood estimation in Gaussian graphical models Journal Article


Author(s): Uhler, Caroline
Article Title: Geometry of maximum likelihood estimation in Gaussian graphical models
Affiliation IST Austria
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.
Keywords: maximum likelihood estimation; Duality; Gaussian graphical model; matrix completion problems; algebraic statistics; algebraic variety; number of observations; sufficient statistics; treewidth; elimination ideal; ML degree; bipartite graphs
Journal Title: Annals of Statistics
Volume: 40
Issue 1
ISSN: 0090-5364
Publisher: Institute of Mathematical Statistics  
Date Published: 2012-01-01
Start Page: 238
End Page: 261
URL:
DOI: 10.1214/11-AOS957
Notes: I wish to thank Bernd Sturmfels for many helpful discus- sions and Steffen Lauritzen for introducing me to the problem of the existence of the MLE in Gaussian graphical models. I would also like to thank two referees who provided helpful comments on the original version of this paper.
Open access: yes (repository)
IST Austria Authors
  1. Caroline Uhler
    26 Uhler