Geometry of maximum likelihood estimation in Gaussian graphical models
The Annals of Statistics · 2012 · Volume 40 · Issue 1
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
14Q10, 62H12, 90C25, 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
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