Response to Cho and Liu, “Sampling from complicated and unknown distributions: Monte Carlo and Markov chain Monte Carlo methods for redistricting”

A question of legal significance is whether an enacted map of political districts is “typical.” Recent work has used Markov chain Monte Carlo (MCMC) methods to produce null distributions of maps in order to answer this question. A recent article by Cho and Liu critiques one particular implementation of MCMC for redistricting, that of Fifield et al. The goal of the present commentary is to draw attention to two facts omitted by Cho and Liu that, if included, would have severely weakened their conclusions. In particular, Cho and Liu point out that Fifield et al.’s algorithm fails to approximate a known target distribution, but neglect Fifield et al.’s use of parallel and simulated tempering, which greatly improves the approximation. Secondly, Cho and Liu argue that it is overly difficult to detect when Markov chains have mixed; they neglect to mention diagnostics used for this exact purpose in Fifield et al.