Partial correlation hypersurfaces in Gaussian graphical models
Algebraic Combinatorics, Volume 2 (2019) no. 3, pp. 439-446.

We derive a combinatorial sufficient condition for a partial correlation hypersurface in the parameter space of a directed Gaussian graphical model to be nonsingular, and speculate on whether this condition can be used in algorithms for learning the graph. Since the condition is fulfilled in the case of a complete DAG on any number of vertices, the result implies an affirmative answer to a question raised by Lin–Uhler–Sturmfels–Bühlmann.

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DOI: 10.5802/alco.44
Classification: 62H05,  62H20
Keywords: partial correlation, Gaussian graphical models, trek separation
Draisma, Jan 1

1 Universität Bern Mathematisches Institut Sidlerstrasse 5 3012 Bern (Switzerland) and Eindhoven University of Technology Department of Mathematics and Computer Science P.O. Box 513 5600 MB Eindhoven (The Netherlands)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Draisma, Jan. Partial correlation hypersurfaces in Gaussian graphical models. Algebraic Combinatorics, Volume 2 (2019) no. 3, pp. 439-446. doi : 10.5802/alco.44. https://alco.centre-mersenne.org/articles/10.5802/alco.44/

[1] Draisma, Jan; Sullivant, Seth; Talaska, Kelli Positivity for Gaussian graphical models, Adv. Appl. Math., Volume 50 (2013) no. 5, pp. 661-674 | DOI | MR | Zbl

[2] Drton, Mathias; Sturmfels, Bernd; Sullivant, Seth Lectures on algebraic statistics, Oberwolfach Seminars, 39, Birkhäuser, 2009, viii+271 pages | MR | Zbl

[3] Gessel, Ira; Viennot, Gérard Binomial determinants, paths, and hook length formulae, Adv. Math., Volume 58 (1985), pp. 300-321 | DOI | MR | Zbl

[4] Lin, Shaowei; Uhler, Caroline; Sturmfels, Bernd; Bühlmann, Peter Hypersurfaces and their singularities in partial correlation testing, Found. Comput. Math., Volume 14 (2014) no. 5, pp. 1079-1116 | MR | Zbl

[5] Spirtes, Peter; Glymour, Clark; Scheines, Richard Causation, prediction, and search. With additional material by David Heckerman, Christopher Meek, Gregory F. Cooper and Thomas Richardson., MIT Press, 2001, xxii+496 pages | Zbl

[6] Sullivant, Seth; Talaska, Kelli; Draisma, Jan Trek separation for Gaussian graphical models, Ann. Stat., Volume 38 (2010) no. 3, pp. 1665-1685 | DOI | MR | Zbl

[7] Wright, Sewall The method of path coefficients, Ann. Math. Stat., Volume 5 (1934), pp. 161-215 | DOI | Zbl

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