## Electronic Communications in Probability

### Fast mixing of metropolis-hastings with unimodal targets

#### Abstract

A well-known folklore result in the MCMC community is that the Metropolis-Hastings algorithm mixes quickly for any unimodal target, as long as the tails are not too heavy. Although we’ve heard this fact stated many times in conversation, we are not aware of any quantitative statement of this result in the literature, and we are not aware of any quick derivation from well-known results. The present paper patches this small gap in the literature, providing a generic bound based on the popular “drift-and-minorization” framework of [19]. Our main contribution is to study two sublevel sets of the Lyapunov function and use path arguments in order to obtain a sharper bound than what can typically be obtained from multistep minorization arguments.

#### Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 71, 9 pp.

Dates
Accepted: 23 September 2018
First available in Project Euclid: 16 October 2018

https://projecteuclid.org/euclid.ecp/1539655259

Digital Object Identifier
doi:10.1214/18-ECP170

Mathematical Reviews number (MathSciNet)
MR3866044

Zentralblatt MATH identifier
06964414

Subjects
Primary: 60J05: Discrete-time Markov processes on general state spaces

#### Citation

Johndrow, James; Smith, Aaron. Fast mixing of metropolis-hastings with unimodal targets. Electron. Commun. Probab. 23 (2018), paper no. 71, 9 pp. doi:10.1214/18-ECP170. https://projecteuclid.org/euclid.ecp/1539655259

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