The Algebra of Reversible Markov Chains

TL;DR

This paper explores the algebraic structure of reversible Markov chains using tools from Algebraic Statistics, providing an algebraic parameterization of their transitions and invariant distributions.

Contribution

It introduces an algebraic framework for reversible Markov chains, connecting detailed balance and cycle conditions with toric models and algebraic binomials.

Findings

Algebraic binomials characterize detailed balance and cycle conditions.

Provides an algebraic parameterization of reversible Markov transitions.

Links Markov chain properties to toric statistical models.

Abstract

For a Markov chain both the detailed balance condition and the cycle Kolmogorov condition are algebraic binomials. This remark suggests to study reversible Markov chains with the tool of Algebraic Statistics, such as toric statistical models. One of the results of this study in an algebraic parameterization of reversible Markov transitions and their invariant probability.