Bayesian Integrals on Toric Varieties

TL;DR

This paper develops a geometric framework for Bayesian statistical computations on toric varieties, extending tropical sampling methods from physics to facilitate marginal likelihood evaluation and posterior sampling.

Contribution

It introduces a novel extension of tropical sampling techniques from projective spaces to general toric varieties for Bayesian analysis.

Findings

Extended tropical sampling method to arbitrary toric varieties

Enabled efficient evaluation of Bayesian marginal likelihoods

Facilitated sampling from posterior distributions in complex models

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.