Stein’s method, Metropolis-Hastings reversiblizations and Markov chains mixing time

This talk consists of two parts that share a common theme: Markov chains mixing time.

In the first part of the talk, entitled “Hitting time and mixing time bounds of Stein’s factors”, we exploit the connection between Markov chains and Stein’s method via the generator approach and express the solution of Stein’s equation in terms of expected hitting time. This yields new upper bounds of Stein’s factors in terms of the parameters of the Markov chain, such as mixing time and the gradient of expected hitting time. We compare the performance of these bounds with those in the literature, and in particular we consider Stein’s method for discrete uniform, binomial, geometric and hypergeometric distribution. As another application, the same methodology applies to bound expected hitting time via Stein’s factors.

In the second part of the talk, entitled “Metropolis-Hastings reversiblizations of non-reversible Markov chains”, we study two types of Metropolis-Hastings (MH) reversiblizations for non-reversible Markov chains with transition 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. Compared with reversiblizations proposed in Fill '91 and Paulin '15, this approach has four attractive features: a version of Weyl's inequality for bounding the spectral gap of P, a comparison theorem between P and two MH kernels, a new pseudo-spectral gap based on the two MH kernels for bounding the total variation distance to stationarity, and finally a possibly tighter variance bound.

Bio:

Michael Choi joins the Institute of Data and Decision Analytics at the Chinese University of Hong Kong, Shenzhen as Assistant Professor in Fall 2018. He was a research fellow in the Department of Mathematics and Statistics at Hang Seng Management College in Hong Kong in 2017-2018. He graduated with a Ph.D. in Operations Research from Cornell University in the summer of 2017,

advised by Prof. Pierre Patie. He received his B.Sc. in Actuarial Science from The University of Hong Kong in 2013 and his M.Sc. in Operations Research from Cornell University in 2016. His research interest lies in the area of Markov chains theory and their applications.