# Metropolis-Hastings reversiblizations of non-reversible Markov chains

## TL;DR

This paper introduces two Metropolis-Hastings reversiblizations for non-reversible Markov chains, analyzing their spectral properties and establishing bounds on convergence and variance using these new kernels.

## Contribution

It proposes a novel second MH kernel capturing opposite transitions, and develops spectral and convergence bounds based on both MH kernels for non-reversible Markov chains.

## Key findings

- Spectral relationships between original and MH kernels are established.
- A new pseudo-spectral gap bounds total variation distance to stationarity.
- Variance and spectral bounds are derived using the two MH kernels.

## 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 \cite{Fill91} and \cite{Paulin15}, 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.