Lifting -- A Nonreversible Markov Chain Monte Carlo Algorithm

Marija Vucelja

Published 2014-12-30Version 1

Markov Chain Monte Carlo algorithms are invaluable numerical tools for exploring stationary properties of physical systems -- in particular when direct sampling is not feasible. They are widely used in many areas of physics and other sciences. Most common implementations are done with reversible Markov chains -- Markov chains that obey detailed balance. Reversible Markov chains are sufficient in order for the physical system to relax to equilibrium, but it is not necessary. Here we review several works that use "lifted" or nonreversible Markov chains, which violate detailed balance, yet still converge to the correct stationary distribution (they obey the global balance condition). In certain cases, the acceleration is a square root improvement at most, to the conventional reversible Markov chains. We introduce the problem in a way that makes it accessible to non-specialists. We illustrate the method on several representative examples (sampling on a ring, sampling on a torus, an Ising model on a complete graph and 1d Ising model). We also provide a pseudocode implementation, review related works and discuss the applicability of such methods.