×

zbMATH — the first resource for mathematics

Optimal scaling of the random walk Metropolis on elliptically symmetric unimodal targets. (English) Zbl 1215.60047
Summary: Scaling of proposals for Metropolis algorithms is an important practical problem in MCMC implementation. Criteria for scaling based on empirical acceptance rates of algorithms have been found to work consistently well across a broad range of problems. Essentially, proposal jump sizes are increased when acceptance rates are high and decreased when rates are low. In recent years, considerable theoretical support has been given for rules of this type which work on the basis that acceptance rates around 0.234 should be preferred. This has been based on asymptotic results that approximate high dimensional algorithm trajectories by diffusions. In this paper, we develop a novel approach to understanding 0.234 which avoids the need for diffusion limits. We derive explicit formulae for algorithm efficiency and acceptance rates as functions of the scaling parameter. We apply these to the family of elliptically symmetric target densities, where further illuminating explicit results are possible. Under suitable conditions, we verify the 0.234 rule for a new class of target densities. Moreover, we can characterise cases where 0.234 fails to hold, either because the target density is too diffuse in a sense we make precise, or because the eccentricity of the target density is too severe, again in a sense we make precise. We provide numerical verifications of our results.

MSC:
60J22 Computational methods in Markov chains
65C40 Numerical analysis or methods applied to Markov chains
65C05 Monte Carlo methods
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Apostol, T.M. (1974). Mathematical Analysis . Reading, MA: Addison-Wesley. · Zbl 0309.26002
[2] Bedard, M. (2007). Weak convergence of Metropolis algorithms for non-iid target distributions. Ann. Appl. Probab. 17 1222-1244. · Zbl 1144.60016 · doi:10.1214/105051607000000096
[3] Breyer, L.A. and Roberts, G.O. (2000). From Metropolis to diffusions: Gibbs states and optimal scaling. Stochastic Process. Appl. 90 181-206. · Zbl 1047.60065 · doi:10.1016/S0304-4149(00)00041-7
[4] Fang, K.T., Kotz, S. and Ng, K.W. (1990). Symmetric Multivariate and Related Distributions. Monographs on Statistics and Applied Probability 36 . London: Chapman and Hall. · Zbl 0699.62048
[5] Gelman, A., Roberts, G.O. and Gilks, W.R. (1996). Efficient Metropolis jumping rules. In Bayesian Statistics, 5 (Alicante, 1994) 599-607. New York: Oxford Univ. Press.
[6] Krzanowski, W.J. (2000). Principles of Multivariate Analysis: A User’s Perspective , 2nd ed. Oxford Statistical Science Series 22 . New York: The Clarendon Press Oxford Univ. Press. · Zbl 0678.62001
[7] Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. and Teller, E. (1953). Equations of state calculations by fast computing machine. J. Chem. Phys. 21 1087-1091.
[8] Roberts, G.O. (1998). Optimal metropolis algorithms for product measures on the vertices of a hypercube. Stochastics Stochastic Rep. 62 275-283. · Zbl 0904.60021
[9] Roberts, G.O., Gelman, A. and Gilks, W.R. (1997). Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Probab. 7 110-120. · Zbl 0876.60015 · doi:10.1214/aoap/1034625254
[10] Roberts, G.O. and Rosenthal, J.S. (2001). Optimal scaling for various Metropolis-Hastings algorithms. Statist. Sci. 16 351-367. · Zbl 1127.65305 · doi:10.1214/ss/1015346320
[11] Sherlock, C. (2006). Methodology for inference on the Markov modulated Poisson process and theory for optimal scaling of the random walk Metropolis. Ph.D. thesis, Lancaster University. Available at http://eprints.lancs.ac.uk/850/.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.