Metropolis–Hastings reversiblizations of non-reversible Markov chains

摘要

We study two types of Metropolis-Hastings (MH) reversiblizations fornon-reversible Markov chains with transition kernel $P$. While the first typeis the classical Metropolised version of $P$, we introduce a new self-adjointkernel which captures the opposite transition effect of the first type, that wecall 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 inequalityfor the spectral gap of $P$ (and hence its additive reversiblization), as wellas an expansion of $P$. Both results are expressed in terms of the spectrum ofthe two MH kernels. In the spirit of cite{Fill91} and cite{Paulin15}, wedefine a new pseudo-spectral gap based on the two MH kernels, and show that thetotal variation distance from stationarity can be bounded by this gap. We givevariance bounds of the Markov chain in terms of the proposed gap, and offerspectral bounds in metastability and Cheeger's inequality in terms of the twoMH kernels by comparison of Dirichlet form and Peskun ordering.