Open Access
February 2018 The maximum likelihood threshold of a graph
Elizabeth Gross, Seth Sullivant
Bernoulli 24(1): 386-407 (February 2018). DOI: 10.3150/16-BEJ881


The maximum likelihood threshold of a graph is the smallest number of data points that guarantees that maximum likelihood estimates exist almost surely in the Gaussian graphical model associated to the graph. We show that this graph parameter is connected to the theory of combinatorial rigidity. In particular, if the edge set of a graph $G$ is an independent set in the $(n-1)$-dimensional generic rigidity matroid, then the maximum likelihood threshold of $G$ is less than or equal to $n$. This connection allows us to prove many results about the maximum likelihood threshold. We conclude by showing that these methods give exact bounds on the number of observations needed for the score matching estimator to exist with probability one.


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Elizabeth Gross. Seth Sullivant. "The maximum likelihood threshold of a graph." Bernoulli 24 (1) 386 - 407, February 2018.


Received: 1 September 2015; Revised: 1 June 2016; Published: February 2018
First available in Project Euclid: 27 July 2017

zbMATH: 06778333
MathSciNet: MR3706762
Digital Object Identifier: 10.3150/16-BEJ881

Keywords: algebraic matroids , Gaussian graphical models , Matrix completion , maximum likelihood estimation , maximum likelihood threshold , rigidity theory , score matching estimator , weak maximum likelihood threshold

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 1 • February 2018
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