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Geometry of maximum likelihood estimation in Gaussian graphical models. (English) Zbl 1246.62140
Summary: 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.

MSC:
62H12 Estimation in multivariate analysis
05C90 Applications of graph theory
14Q99 Computational aspects in algebraic geometry
Software:
Macaulay2; CVX; gRc
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References:
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