# Staged tree models with toric structure

The theoretical results of this work explore the algebraic geometry of discrete statistical models for asymmetric conditional independence relationships that can be represented by staged trees. Background literature and information on the applications of these models is cited in the article. Sometimes, as explored in previous work and enhanced in various new technical results, these statistical models are toric varieties intersected with the probability simplex. Whilst in special cases the toric structure is apparent from the tree graph representation itself, oftentimes it is not. We here present two computational approaches to complement our theoretical work.

The first approach is based on the software \(\verb|Macaulay2|\). It provides a simple tool which allows the user to input a staged-tree parametrisation and output its ideal of model invariants. The given staged tree has toric structure whenever this ideal is generated by binomials.

The second approach uses the software \(\verb|Mathematica|\). It provides functions which randomly generate transformations of a given model specification and try out whether these give rise to binomial generators as above. The code is presented in the first subpage below and discussed and applied in examples listed in the second.

Project page created: 09/07/2021

Project contributors: Christiane Görgen, Aida Maraj, and Lisa Nicklasson

Code written by: Christiane Görgen, Aida Maraj, and Lisa Nicklasson 09/07/2021

Software used: Macaulay2 (Version 1.15), Mathematica (Version 12.1.1.0)

System setup used: MacBook Pro with macOS Catalina Version 10.15.7, Processor 3,1 GHz Intel Core i5, Memory 16 GB 2133 MHz LPDDR3, Graphics Intel Iris Graphics 550 1536 MB.

Responsible person for project page on https://mathrepo.mis.mpg.de: Christiane Görgen

Corresponding author of this page: Christiane Görgen, goergen@mis.mpg.de