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Complexity bounds for Markov chain Monte Carlo algorithms via diffusion limits

Part of: Markov processes Parametric inference

Published online by Cambridge University Press:  21 June 2016

Gareth O. Roberts  and
Jeffrey S. Rosenthal
Gareth O. Roberts*
Affiliation:
University of Warwick
Jeffrey S. Rosenthal*
Affiliation:
University of Toronto
*
* Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email address: gareth.o.roberts@warwick.ac.uk
** Postal address: Department of Statistics, University of Toronto, Toronto, Ontario, M5S 3G3, Canada. Email address: jeff@math.toronto.edu
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Abstract

We connect known results about diffusion limits of Markov chain Monte Carlo (MCMC) algorithms to the computer science notion of algorithm complexity. Our main result states that any weak limit of a Markov process implies a corresponding complexity bound (in an appropriate metric). We then combine this result with previously-known MCMC diffusion limit results to prove that under appropriate assumptions, the random-walk Metropolis algorithm in d dimensions takes O(d) iterations to converge to stationarity, while the Metropolis-adjusted Langevin algorithm takes O(d1/3) iterations to converge to stationarity.

Keywords

MCMCconvergencecomplexitydiffusion limitrandom-walk Metropolis algorithmMetropolis-adjusted Langevin algorithm

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces 60J25: Continuous-time Markov processes on general state spaces
Secondary: 62F10: Point estimation 62F15: Bayesian inference
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 2 , June 2016 , pp. 410 - 420
Copyright
Copyright © Applied Probability Trust 2016 

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