Vol. 12, No. 1, 2021

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Gröbner bases for staged trees

Lamprini Ananiadi and Eliana Duarte

Vol. 12 (2021), No. 1, 1–20
Abstract

We consider the problem of finding generators of the toric ideal associated to a combinatorial object called a staged tree. Our motivation to consider this problem originates from the use of staged trees to represent discrete statistical models such as conditional independence models and discrete Bayesian networks. The main theorem in this article states that toric ideals of staged trees that are balanced and stratified are generated by a quadratic Gröbner basis whose initial ideal is square-free. We apply this theorem to construct Gröbner bases of a subclass of discrete statistical models represented by staged trees. The proof of the main result is based on Sullivant’s toric fiber product construction (J. Algebra 316:2 (2007), 560–577).

Keywords
graphical model, toric ideals, staged tree, Markov bases, toric fiber product
Mathematical Subject Classification 2010
Primary: 13P10, 13P25
Secondary: 14M25, 05E40, 68W30
Milestones
Received: 7 October 2019
Revised: 21 December 2020
Accepted: 5 January 2021
Published: 9 April 2021
Authors
Lamprini Ananiadi
Otto-Von-Guericke Universität
Magdeburg
Germany
Eliana Duarte
Otto-Von-Guericke Universität
Magdeburg
Germany
Max-Planck-Institute for Mathematics in the Sciences
Leipzig
Germany