Some results on the Gaussian Markov Random Field construction problem based on the use of invariant subgraphs

Juan Baz, Irene Díaz, Susana Montes and Raúl Pérez-Fernández ()
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Juan Baz: University of Oviedo
Irene Díaz: University of Oviedo
Susana Montes: University of Oviedo
Raúl Pérez-Fernández: University of Oviedo

TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, 2022, vol. 31, issue 3, No 12, 856-874

Abstract: Abstract The study of Gaussian Markov Random Fields has attracted the attention of a large number of scientific areas due to its increasing usage in several fields of application. Here, we consider the construction of Gaussian Markov Random Fields from a graph and a positive-definite matrix, which is closely related to the problem of finding the Maximum Likelihood Estimator of the covariance matrix of the underlying distribution. In particular, it is simultaneously required that the variances and the covariances between variables associated with adjacent nodes in the graph are fixed by the positive-definite matrix and that pairs of variables associated with non-adjacent nodes in the graph are conditionally independent given all other variables. The solution to this construction problem exists and is unique up to the choice of a vector of means. In this paper, some results focusing on a certain type of subgraphs (invariant subgraphs) and a representation of the Gaussian Markov Random Field as a Multivariate Gaussian Markov Random Field are presented. These results ease the computation of the solution to the aforementioned construction problem.

Keywords: Gaussian Markov Random Field; Multivariate Gaussian Markov Random Field; Covariance matrix; Graph theory; 60G60 (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s11749-022-00804-3

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