New Zealand Statistical Association 2024 Conference

Linus Fromm

University of Otago

Poster display: The algebra and geometry of Markov bases

This is joint work with Martin Hazelton

Linear statistical inverse problems can be found in many branches of science. This includes but is not limited to ecology, genetics and network tomography. The aim is to conduct inference on the distribution of some latent variable of interest conditional on corrupted or aggregated observations. This involves the evaluation of a relatively large sum which is infeasible in most circumstances. For that reason, we turn to Markov chain Monte-Carlo (MCMC) sampling. MCMC samplers over discrete data require the use of sets of moves called Markov bases. It is described how to find Markov bases and how the process of finding Markov bases is fundamentally connected to division of multivariate polynomials. We aim to give a geometric intuition for the fundamental theorem of Markov bases which was first proven by Diaconis and Sturmfels in 1998. We will then show that the division algorithm of polynomials is susceptible to changes in monomial orderings. These differences carry over to the bases found using the fundamental theorem of Markov bases. Finally, we will explore how different monomial orderings influence the convergence of samplers.