Metropolis–Hastings reversiblizations of non-reversible Markov chains

Abstract

We study two types of Metropolis-Hastings (MH) reversiblizations for non-reversible Markov chains with Markov kernel P . While the first type is the classical Metropolised version of P , we introduce a new self-adjoint kernel which captures the opposite transition effect of the first type, that we call the second MH kernel. We investigate the spectral relationship between P and the two MH kernels. Along the way, we state a version of Weyl's inequality for the spectral gap of P (and hence its additive reversiblization), as well as an expansion of P . Both results are expressed in terms of the spectrum of the two MH kernels. In the spirit of Fill (1991) and Paulin (2015), we define a new pseudo-spectral gap based on the two MH kernels, and show that the total variation distance from stationarity can be bounded by this gap. We give variance bounds of the Markov chain in terms of the proposed gap, and offer spectral bounds in metastability and Cheeger's inequality in terms of the two MH kernels by comparison of Dirichlet form and Peskun ordering.

On Hitting Time, Mixing Time and Geometric Interpretations of Metropolis–Hastings Reversiblizations

“…It is what we meant by M 2 mirroring the transition effect of M 1 . As another remark, we note that in the discrete-time setting, M 2 as defined in Choi (2017) may not be a Markov kernel. In the continuous-time setting however, M 2 as defined in Definition 2.2 is a valid Markov generator.…”

“…In this paper, we study the so-called Metropolis-Hastings reversiblizations in a continuous-time and finite state space setting. This work is largely motivated by Choi (2017), in which the author introduced two Metropolis-Hastings (MH) kernels M 1 and M 2 to study non-reversible Markov chains in discretetime. While M 2 is a self-adjoint kernel, M 2 may not be Markovian, which makes further probabilistic analysis of M 2 to be difficult.…”

An improved variant of simulated annealing that converges under fast cooling

“…Recent advances in simulated annealing includes investigating piecewise deterministic Markov processes and their annealing variants, see for example Monmarché (2016). Inspired by the recent work by the author Choi (2019); Choi and Huang (2019), we would like to introduce a variant of X M 1 that we call X M 2 . It can be constructed by mirroring the transition effect of X M 1 to capture the opposite movement.…”

“…In this paper, inspired by the recent work of the author Choi (2019); Choi and Huang (2019) who studied a new variant of the Metropolis-Hastings algorithm, we propose a promising accelerated variant of simulated annealing X M 2 = (X M 2 t ) t 0 that enjoys superior mixing properties, provably converges under fast cooling and in some cases does not suffer from the drawbacks of X M 1 as mentioned above. Precisely, our contributions are the following:…”

Mixing Times and Hitting Times for General Markov Processes

“…) ) is itself π-reversible (see e.g. the introduction of [14] for a careful presentation of the additive reversibilization and a general argument as to why it is reversible). Thus, G 1 is π-reversible.…”