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Gaussian graphical models with toric vanishing ideals

Author

Listed:
  • Pratik Misra

    (North Carolina State University)

  • Seth Sullivant

    (North Carolina State University)

Abstract

Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. They are widely used throughout natural sciences, computational biology and many other fields. Computing the vanishing ideal of the model gives us an implicit description of the model. In this paper, we resolve two conjectures given by Sturmfels and Uhler. In particular, we characterize those graphs for which the vanishing ideal of the Gaussian graphical model is generated in degree 1 and 2. These turn out to be the Gaussian graphical models whose ideals are toric ideals, and the resulting graphs are the 1-clique sums of complete graphs.

Suggested Citation

  • Pratik Misra & Seth Sullivant, 2021. "Gaussian graphical models with toric vanishing ideals," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(4), pages 757-785, August.
    • Handle: RePEc:spr:aistmt:v:73:y:2021:i:4:d:10.1007_s10463-020-00765-0
    DOI: 10.1007/s10463-020-00765-0
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    References listed on IDEAS

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    1. Bernd Sturmfels & Caroline Uhler, 2010. "Multivariate Gaussians, semidefinite matrix completion, and convex algebraic geometry," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(4), pages 603-638, August.
    2. Beatrix Jones & Mike West, 2005. "Covariance decomposition in undirected Gaussian graphical models," Biometrika, Biometrika Trust, vol. 92(4), pages 779-786, December.
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