Bayesian Integrals on Toric Varieties

This page contains auxiliary files to the paper:
Michael Borinsky, Anna-Laura Sattelberger, Bernd Sturmfels and Simon Telen: Bayesian integrals on toric varieties
In: SIAM journal on applied algebra and geometry, 7 (2023) 1, p. 77-103
DOI: 10.1137/22M1490569 ARXIV: https://arxiv.org/abs/2204.06414 CODE: https://mathrepo.mis.mpg.de/BayesianIntegrals

ABSTRACT: We explore the positive geometry of statistical models in the setting of toric varieties. Our focus lies on models for discrete data that are parameterized in terms of Cox coordinates. We develop a geometric theory for computations in Bayesian statistics, such as evaluating marginal likelihood integrals and sampling from posterior distributions. These are based on a tropical sampling method for evaluating Feynman integrals in physics. We here extend that method from projective spaces to arbitrary toric varieties.

As an illustration, the following figure shows the sector decomposition of a pentagon as seen in Example 4.5.

../_images/pentagon.png

The paper also features a polytope arising from a three-dimensional conditional independence model. Its Schlegel diagram is shown below.

../_images/schlegel.png

Our algorithms are implemented julia. The code can be downloaded here julia_code.zip. It is illustrated in the following notebook:

Project page created: 15/04/2022

Project contributors: Michael Borinsky, Anna-Laura Sattelberger, Bernd Sturmfels, Simon Telen

Corresponding author of this page: Simon Telen, Simon.Telen@mis.mpg.de

Software used: Julia (Version 1.7.1)