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Gaussian graphical models with toric vanishing ideals

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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.

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Acknowledgements

Pratik Misra and Seth Sullivant were partially supportedby the US National Science Foundation (DMS 1615660).

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  1. Department of Mathematics, North Carolina State University, 2108 SAS Hall Box 8205, Raleigh, NC, 27695, USA

    Pratik Misra & Seth Sullivant

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  1. Pratik Misra
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  2. Seth Sullivant
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Correspondence to Pratik Misra.

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Misra, P., Sullivant, S. Gaussian graphical models with toric vanishing ideals. Ann Inst Stat Math 73, 757–785 (2021). https://doi.org/10.1007/s10463-020-00765-0

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  • DOI: https://doi.org/10.1007/s10463-020-00765-0

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