Abstract
This paper is concerned with improving the performance of certain Markov chain algorithms for Monte Carlo simulation. We propose a new algorithm for simulating from multivariate Gaussian densities. This algorithm combines ideas from coupled Markov chain methods and from an existing algorithm based only on over-relaxation. The rate of convergence of the proposed and existing algorithms can be measured in terms of the square of the spectral radius of certain matrices. We present examples in which the proposed algorithm converges faster than the existing algorithm and the Gibbs sampler. We also derive an expression for the asymptotic variance of any linear combination of the variables simulated by the proposed algorithm. We outline how the proposed algorithm can be extended to non-Gaussian densities.
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References
Adler S.L. 1981. Over-relaxation method for the Monte Carlo evalua-tion of the partition function for multiquadratic actions. Physical Review D 23: 2901-2904.
Barone P. and Frigessi A. 1990. Improving stochastic relaxation for Gaussian random fields. Probability in the Engineering and Informational Sciences 4: 369-389.
Barone P., Sebastiani G., and Stander J. 1998. Metropolis coupled Markov chains and over-relaxation methods for Monte Carlo simulation.Istituto per le Applicazioni del Calcolo, CNR, Rome, Italy, Report 10/1998.
Barone P., Sebastiani G., and Stander J. 2001. General over-relaxation Markov chain Monte Carlo algorithms for Gaussian densities.Statistics and Probability Letters, 52: 115-124.
Brooks S.P. 1998. Markov chain Monte Carlo method and its applications.The Statistician 47: 69-100.
Brooks S.P. and Roberts G.O. 1998. Convergence assessment techniques for Markov chain Monte Carlo. Statistics and Computing 8: 319-335.
Cowles M.K. and Carlin B.P. 1996. Markov chain Monte Carlo convergence diagnostics: A comparative review. Journal of the American Statistical Association 91: 883-904.
Foster M.A. and Hutchison J.M.S. (Eds.). 1987. Practical NMR Imaging.IRL Press Limited, Oxford.
Geman S. and Geman D. 1984. Stochastic relaxation, Gibbs distribu-tions and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence PAMI-6: 721-741.
Geyer C.J. 1991. Markov chain Monte Carlo maximum likelihood. In: Keramidas E.M. (Ed.), Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface. Interface Foundation, Fairfax Station, pp. 156-163.
Gilks W.R., Richardson S., and Spiegelhalter, D.J. (Eds.). 1996. Markov Chain Monte Carlo in Practice. Chapman and Hall, London.
Gilks W.R. and Roberts G.O. 1996. Strategies for improving MCMC.In: Gilks W.R., Richardson S., and Spiegelhalter D.J. (Eds.), Markov Chain Monte Carlo in Practice. Chapman and Hall, London, ch. 6, pp. 89-114.
Golub G.H. and Van Loan C.F. 1989. Matrix Computations, 2nd edn. The Johns Hopkins University Press, Baltimore.
Green P.J. and Han X.-L. 1992. Metropolis methods, Gaussian proposals and antithetic variables. In: Barone P., Frigessi A., and Piccioni M.(Eds.), Stochastic Models, Statistical Methods, and Algorithms in Image Analysis. Springer-Verlag, Berlin, pp. 142-164.
Hastings W.K. 1970. Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57: 97-109.
Mengersen K.L., Robert C.P., and Guihenneuc-Jouyaux Ch. 1999.MCMC convergence diagnostics: A review ww. In: Bernardo J.M., Berger J.O., Dawid A.P., and Smith A.F.M. (Eds.), Bayesian Statistics 6. Oxford University Press, Oxford, pp. 415-440.
Metropolis N. Rosenbluth A.W., Rosenbluth M.N., Teller A.H., and Teller E. 1953. Equations of state calculations by fast computing machines. Journal of Chemical Physics 21: 1087-1091.
Neal R.M. 1995. Suppressing random walks in Markov chain Monte Carlo using ordered overrelaxation. Department of Statistics, University of Toronto, Technical Report 9508. Also available from http://www.statslab.cam.ac.uk/∼mcmc/
Robert C. 1996. Convergence assessments for Markov chain Monte Carlo methods. Statistical Science 10: 231-253.
Roberts G.O. and Sahu S.K. 1997. Updating schemes, correlation structure, blocking and parameterization for the Gibbs sampler. Journal of the Royal Statistical Society, Series B 59: 291-317.
Rue H. 2001. Fast sampling of Gaussian Markov random fields. Journal of the Royal Statistical Society Series B 63: 325-338. Also available from http://www.statslab.cam.ac.uk/∼mcmc/
Stewart G.W. 1973. Introduction to Matrix Computation. Academic Press, New York.
Tierney L. 1994. Markov chains for exploring posterior distributions (with discussion). Annals of Statistics 22: 1701-1762.
Tierney L. 1996. Introduction to general state-space Markov chain theory. In: Gilks W.R., Richardson S., and Spiegelhalter D.J.(Eds.), Markov Chain Monte Carlo in Practice. Chapman and Hall, London, ch. 4, pp. 59-88.
Varga R.S. 2000. Matrix Iterative Analysis, 2nd ed. Springer-Verlag, Berlin.
Wang Y. and Lee T. 1994. Statistical analysis of MR imaging and its applications in image modeling. In: Proc. IEEE Int. Conf. Image Processing Neural Networks, Vol. I, pp. 866-870.
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Barone, P., Sebastiani, G. & Stander, J. Over-relaxation methods and coupled Markov chains for Monte Carlo simulation. Statistics and Computing 12, 17–26 (2002). https://doi.org/10.1023/A:1013112103963
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DOI: https://doi.org/10.1023/A:1013112103963