Non-reversible Markov chain Monte Carlo for sampling of districting maps

TL;DR

This paper introduces non-reversible MCMC algorithms for sampling districting maps, improving mixing efficiency over traditional reversible methods, and extends the theoretical framework for non-reversible Markov chains on discrete spaces.

Contribution

The paper develops novel non-reversible MCMC algorithms for districting plan sampling, enhancing mixing properties and extending the theoretical framework with a generalized skew detailed balance.

Findings

Improved mixing performance in numerical experiments

Extension of non-reversible Markov chain theory to discrete spaces

Demonstrated efficiency gains over reversible MCMC methods

Abstract

Evaluating the degree of partisan districting (Gerrymandering) in a statistical framework typically requires an ensemble of districting plans which are drawn from a prescribed probability distribution that adheres to a realistic and non-partisan criteria. In this article we introduce novel non-reversible Markov chain Monte-Carlo (MCMC) methods for the sampling of such districting plans which have improved mixing properties in comparison to previously used (reversible) MCMC algorithms. In doing so we extend the current framework for construction of non-reversible Markov chains on discrete sampling spaces by considering a generalization of skew detailed balance. We provide a detailed description of the proposed algorithms and evaluate their performance in numerical experiments.