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Complexity bounds for Markov chain Monte Carlo algorithms via diffusion limits

Part of: Parametric inference Markov processes

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

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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

Information

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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