Partial correlation hypersurfaces in Gaussian graphical models
Algebraic Combinatorics, Volume 2 (2019) no. 3, p. 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.

Received : 2018-06-13
Revised : 2018-09-22
Accepted : 2018-09-24
Published online : 2019-06-06
DOI : https://doi.org/10.5802/alco.44
Classification:  62H05,  62H20
Keywords: partial correlation, Gaussian graphical models, trek separation
@article{ALCO_2019__2_3_439_0,
     author = {Draisma, Jan},
     title = {Partial correlation hypersurfaces in Gaussian graphical models},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {2},
     number = {3},
     year = {2019},
     pages = {439-446},
     doi = {10.5802/alco.44},
     zbl = {07066883},
     language = {en},
     url = {https://alco.centre-mersenne.org/item/ALCO_2019__2_3_439_0}
}
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/item/ALCO_2019__2_3_439_0/

[1] Draisma, Jan; Sullivant, Seth; Talaska, Kelli Positivity for Gaussian graphical models, Adv. Appl. Math., Volume 50 (2013) no. 5, pp. 661-674 | Article | MR 3044565 | Zbl 1279.62135

[2] Drton, Mathias; Sturmfels, Bernd; Sullivant, Seth Lectures on algebraic statistics, Birkhäuser, Oberwolfach Seminars, Volume 39 (2009), viii+271 pages | MR 2723140 | Zbl 1166.13001

[3] Gessel, Ira; Viennot, Gérard Binomial determinants, paths, and hook length formulae, Adv. Math., Volume 58 (1985), pp. 300-321 | Article | MR 815360 | Zbl 0579.05004

[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 3260260 | Zbl 1308.62110

[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 0981.62001

[6] Sullivant, Seth; Talaska, Kelli; Draisma, Jan Trek separation for Gaussian graphical models, Ann. Stat., Volume 38 (2010) no. 3, pp. 1665-1685 | Article | MR 2662356 | Zbl 1189.62091

[7] Wright, Sewall The method of path coefficients, Ann. Math. Stat., Volume 5 (1934), pp. 161-215 | Article | Zbl 0010.31305