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Gaussian Markov Random Fields and totally positive matrices

Published: 01 October 2023 Publication History

Abstract

The present paper focuses on the study of the conditions under which the covariance matrix of a multivariate Gaussian distribution is totally positive, paying particular attention to multivariate Gaussian distributions that are Gaussian Markov Random Fields. More specifically, it is proven that, if the graph over which the Gaussian Markov Random Field is defined consists of path graphs and the covariances between adjacent variables on the graph are non-negative, then there always exists a reordering of the variables that renders the resulting covariance matrix totally positive. Moreover, this reordering is identified and some cases for which the conditions for the covariance matrix of a multivariate Gaussian distribution to be totally positive are necessary and sufficient are provided.

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Published In

cover image Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics  Volume 430, Issue C
Oct 2023
182 pages

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Elsevier Science Publishers B. V.

Netherlands

Publication History

Published: 01 October 2023

Author Tags

  1. 15A09
  2. 65F05

Author Tags

  1. Gaussian Markov Random Field
  2. Graph
  3. Totally positive matrices

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