Abstract
Two strategies that can potentially improve Markov Chain Monte Carlo algorithms are to use derivative evaluations of the target density, and to suppress random walk behaviour in the chain. The use of one or both of these strategies has been investigated in a few specific applications, but neither is used routinely. We undertake a broader evaluation of these techniques, with a view to assessing their utility for routine use. In addition to comparing different algorithms, we also compare two different ways in which the algorithms can be applied to a multivariate target distribution. Specifically, the univariate version of an algorithm can be applied repeatedly to one-dimensional conditional distributions, or the multivariate version can be applied directly to the target distribution.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Besag J., York J., and Mollié A. 1991. Bayesian image restoration, with two applications in spatial statistics (with discussion). Annals of the Institute of Statistical Mathematics 43: 1-20.
Christensen O.F., Moller J., and Waagepetersen R. 2001. Geometric ergodicity of Metropolis-Hastings algorithms for conditional simulation in generalised linear mixed models. Methodology and Computing in Applied Probability 3: 309-327.
Duane S., Kennedy A.D., Pendleton B.J., and Roweth D. 1987. Hybrid Monte Carlo. Physics Letters B 195: 216-222.
Gamerman D. 1997. Sampling from the posterior distribution in generalized linear mixed models. Statistics and Computing 7: 57-68.
Gelfand A.E. and Smith A.F.M. 1990. Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association 85: 398-409.
Gelman A., Roberts G.O., and Gilks W. 1996. Efficient Metropolis jumping rules. In: Berger J.O., Bernardo J.M., Dawid A.P., and Smith A.F.M.) Bayesian Statistics 5. Oxford University Press, pp. 599-607.
Gelman A. and Rubin D.B. 1992. Inference from iterative simulation using multiple sequences. Statistical Science 7: 457-472.
Gustafson P. 1997. Large hierarchical Bayesian analysis of multivariate survival data. Biometrics 53: 230-242.
Gustafson P. 1998a. A guided walk Metropolis algorithm. Statistics and Computing 8: 357-364.
Gustafson P. 1998b. Flexible Bayesian modelling for survival data. Lifetime Data Analysis 4: 281-299.
Hastings W.K. 1970. Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57: 97-109.
Horowitz A.M. 1991. A generalized guided Monte Carlo algorithm. Physics Letters B 268: 247-252.
Ishwaran H. 1999. Applications of hybrid Monte Carlo to Bayesian generalized linear models: Quasicomplete separation and neural networks. Journal of Computational and Graphical Statistics 8: 779-799.
Leroux B.G., Lei X., and Breslow N. 1999. Estimation of disease rates in small areas: A new mixed model for spatial dependence. In: Halloran M.E. and Berry D.A., (Eds.), Statistical Models in Epidemiology, the Environment and Clinical Trials. New York: Springer, pp. 179-192.
Leroux B.G. 2000. Modelling spatial disease rates using maximum likelihood. Statistics in Medicine 19: 2321-2332.
MacNab Y.C. 2003a. Hierarchical Bayesian spatial modelling of smallarea rates of non-rare disease. Statistics in Medicine 22: 1761-1773.
MacNab Y.C. 2003b. Hierarchical Bayesian modelling of spatially correlated health service outcome and utilization rates. Biometrics 59: 305-316.
MacNab Y.C. and Dean C.B. 2000. Parametric bootstrap and penalized quasi-likelihood inference in conditional autoregressive models. Statistics in Medicine 19: 2421-2435.
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: 1087-1092.
Neal R.M. 1993. Bayesian learning via stochastic dynamics. In: Giles C.L., Hanson S.J., and Cowan J.D. (Eds.), Advances in Neural Information Processing Systems 5. San Mateo, California, Morgan Kaufmann, pp. 475-482.
Neal R.M. 1996. Bayesian learning for neural networks. Lecture Notes in Statistics No. 118, New York, Springer-Verlag.
Neal R.M. 1998. Suppressing random walks in Markov chain Monte Carlo using ordered overrelaxation. In: Jordan M.I. (Ed.), Learning in Graphical Models. Dordrecht, Kluwer Academic Publishers, pp. 205-225.
Roberts G.O., Gelman A., and Gilks W.R. 1997.Weak convergence and optimal scaling of random walk Metropolis algorithms. Annals of Applied Probability 7: 110-120.
Roberts G.O. and Rosenthal J.S. 1998. Optimal scaling of discrete approximations to Langevin diffusions. Journal of the Royal Statistical Society B 60: 255-268.
Roberts G.O. and Rosenthal J.S. 2001. Optimal scaling for various Metropolis-Hastings algorithms. Statistical Science 16: 351-367.
Roberts G.O. and Tweedie R.L. 1996. Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli 2: 341-363.
Rue H. 2001. Fast sampling of Gaussian Markov random fields with applications. Journal of the Royal Statistical Society B 63: 325-338.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gustafson, P., MacNab, Y.C. & Wen, S. On the Value of derivative evaluations and random walk suppression in Markov Chain Monte Carlo algorithms. Statistics and Computing 14, 23–38 (2004). https://doi.org/10.1023/B:STCO.0000009413.87656.ef
Issue Date:
DOI: https://doi.org/10.1023/B:STCO.0000009413.87656.ef