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Optimal Scaling of the Random Walk Metropolis: General Criteria for the 0.234 Acceptance Rule

Part of: Limit theorems

Published online by Cambridge University Press:  30 January 2018

Chris Sherlock
Chris Sherlock*
Affiliation:
Lancaster University
*
∗ Postal address: Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK. Email address: [email protected]
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Abstract

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Scaling of proposals for Metropolis algorithms is an important practical problem in Markov chain Monte Carlo implementation. Analyses of the random walk Metropolis for high-dimensional targets with specific functional forms have shown that in many cases the optimal scaling is achieved when the acceptance rate is approximately 0.234, but that there are exceptions. We present a general set of sufficient conditions which are invariant to orthonormal transformation of the coordinate axes and which ensure that the limiting optimal acceptance rate is 0.234. The criteria are shown to hold for the joint distribution of successive elements of a stationary pth-order multivariate Markov process.

Keywords

random walk Metropolisoptimal scalingoptimal acceptance rate

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 1 - 15
Copyright
Copyright © Applied Probability Trust 2013 

References

Atchadé, Y. F., Roberts, G. O. and Rosenthal, J. S. (2011). Towards optimal scaling of Metropolis-coupled Markov chain Monte Carlo. Statist. Comput. 21, 555–568.CrossRefGoogle Scholar
Bédard, M. (2007). Weak convergence of Metropolis algorithms for non-i.i.d. target distributions. Ann. Appl. Prob. 17, 1222–1244.CrossRefGoogle Scholar
Bédard, M. (2008). Optimal acceptance rates for Metropolis algorithms: moving beyond 0.234. Stoch. Process. Appl. 118, 2198–2222.CrossRefGoogle Scholar
Beskos, A., Roberts, G. and Stuart, A. (2009). Optimal scalings for local Metropolis–Hastings chains on nonproduct targets in high dimensions. Ann. Appl. Prob. 19, 863–898.CrossRefGoogle Scholar
Breyer, L. A. and Roberts, G. O. (2000). From Metropolis to diffusions: Gibbs states and optimal scaling. Stoch. Process. Appl. 90, 181–206.CrossRefGoogle Scholar
Folland, G. B. (2002). Advanced Calculus. Prentice-Hall, Upper Saddle River, NJ.Google Scholar
Grimmett, G. R. and Stirzaker, D. R. (2001). Probability and Random Processes, 3rd edn. Oxford University Press, New York.CrossRefGoogle Scholar
Metropolis, N. et al. (1953). Equations of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1091.CrossRefGoogle Scholar
Neal, P. and Roberts, G. (2008). Optimal scaling for random walk Metropolis on spherically constrained target densities. Methodology Comput. Appl. Prob. 10, 277–297.CrossRefGoogle Scholar
Roberts, G. O. (1998). Optimal Metropolis algorithms for product measures on the vertices of a hypercube. Stoch. Stoch. 62, 275–283.CrossRefGoogle Scholar
Roberts, G. O. and Rosenthal, J. S. (2001). Optimal scaling for various Metropolis–Hastings algorithms. Statist. Sci. 16, 351–367.CrossRefGoogle Scholar
Roberts, G. O., Gelman, A. and Gilks, W. R. (1997). Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Prob. 7, 110–120.Google Scholar
Sherlock, C. (2006). Methodology for inference on the Markov modulated Poisson process and theory for optimal scaling of the random walk Metropolis. Doctoral Thesis, Lancaster University. Available at http://eprints.lancs.ac.uk/850/.Google Scholar
Sherlock, C. and Elton, D. (2012). A subclass of spherical and elliptical distributions with Gaussian-like limit properties. J. Prob. Statist. 2012, 17pp.CrossRefGoogle Scholar
Sherlock, C. and Roberts, G. (2009). Optimal scaling of the random walk Metropolis on elliptically symmetric unimodal targets. Bernoulli 15, 774–798.CrossRefGoogle Scholar