Historically time-reversibility of the transitions or processes underpinning Markov chain Monte Carlo methods (MCMC) has played a key role in their development, while the self-adjointness of associated operators together with the use of classical functional analysis techniques on Hilbert spaces have led to powerful and practically successful tools to characterise and compare their performance. Similar results for algorithms relying on nonreversible Markov processes are scarce. We show that for a type of nonreversible Monte Carlo Markov chains and processes, of current or renewed interest in the physics and statistical literatures, it is possible to develop comparison results which closely mirror those available in the reversible scenario. We show that these results shed light on earlier literature, proving some conjectures and strengthening some earlier results.
The authors acknowledge support from EPSRC “Intractable Likelihood: New Challenges from Modern Applications (ILike)”, (EP/K014463/1). CA acknowledges support from EPSRC “Computational Statistical Inference for Engineering and Security (CoSInES)”, (EP/R034710/1).
The authors would like to thank Florian Maire for pointing out an error in Remark 5 and suggesting the correct formulation of the result. The authors are grateful to Anthony Lee for sharing his LaTeX code to handle Supplementary Material efficiently.
"Peskun–Tierney ordering for Markovian Monte Carlo: Beyond the reversible scenario." Ann. Statist. 49 (4) 1958 - 1981, August 2021. https://doi.org/10.1214/20-AOS2008