Accelerating reversible Markov chains

Reversibility is usually applied in most popular Markov chain Monte Carlo algorithms, such as the Metropolis–Hastings algorithm and the Gibbs sampler. However, several researchers have shown that non-reversible Markov chains are better than reversible ones. In this paper, we present a method for accelerating a reversible Markov chain. For any reversible Markov chain with a cycle on the corresponding graph, we construct a non-reversible Markov chain by adding some antisymmetric perturbations to the original chain. We prove that this non-reversible Markov chain is uniformly better than the original one in the sense of having a smaller asymptotic variance. Furthermore, we propose a conjecture that no uniformly better chain exists for the acyclic case.