Abstract
We propose to combine two quite powerful ideas that have recently appeared in the Markov chain Monte Carlo literature: adaptive Metropolis samplers and delayed rejection. The ergodicity of the resulting non-Markovian sampler is proved, and the efficiency of the combination is demonstrated with various examples. We present situations where the combination outperforms the original methods: adaptation clearly enhances efficiency of the delayed rejection algorithm in cases where good proposal distributions are not available. Similarly, delayed rejection provides a systematic remedy when the adaptation process has a slow start.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrieu C. and Robert C.P. 2001. Controlled MCMC. Preprint.
Andrieu C. and Moulines E. 2002. On the ergodicity properties of some adaptive MCMC algorithms. To appear in Annals of Applied Probability.
Atchade Y.F. and Rosenthal J.S. 2005. On adaptive Markov chain Monte carlo algorithms. Bernoulli 11(5): 815–282.
Bowie G.L., Mills W.B., et al. 1985. Rates, constants, and kinetic formulations in surface water modeling. Technical Report EPA/600/3-85/040, U.S. Environmental Agency, ORD, Athens, GA, ERL.
Gelman A.G., Roberts G.O., and Gilks W.R. 1996. Efficient Metropolis jumping rules. In: Bernardo J.M., Berger J.O., David A.F., and Smith A.F.M. (Eds.), Bayesian Statistics V. Oxford University Press, pp. 599–608.
Green, P.J. and Mira, A. 2001 Delayed rejection in reversible jump Metropolis-Hastings. Biometrika 88: 1035–1053.
Haario H., Kalachev L., Lehtonen J., and Salmi T. 1999. Asymptotic analysis of chemical reactions. Chem. Eng. Sci. 54: 1131–1143.
Haario H., Saksman E., and Tamminen J. 1999. Adaptive proposal distribution for random walk Metropolis algorithm. Comp. Stat. 14: 375–395.
Haario H., Saksman E., and Tamminen J. 2001. An adaptive Metropolis algorithm. Bernoulli 7: 223–242.
Haario H., Saksman E., and Tamminen J. 2005. Componentwise adaptation for high dimensional MCMC. Computational Statistics 20(2): 265–274.
Malve O., Laine M., Haario H., Kirkkala T., and Sarvala J. Bayesian modeling of algae mass occurrences—using adaptive MCMC methods with a lake water quality model. To appear in Environmental Modelling and Software, 2006.
Mira A. 2001. On Metropolis-Hastings algorithms with delayed rejection. Metron, Vol. LIX, (3–4): 231–241.
Mira A. 2002. Ordering and improving the performance of Monte Carlo Markov Chains. Statistical Science 16: 340–350.
Peskun P.H. 1973. Optimum Monte Carlo sampling using markov chains. Biometrika 60: 607–612.
Sokal A.D. 1998. Monte carlo methods in statistical mechanics: Foundations and new algorithms. Cours de Troisième Cycle de la Physique en Suisse Romande. Lausanne.
Tierney L. 1994. Markov chains for exploring posterior distributions. Annals of Statistics 22: 1701–1762.
Tierney L. 1998. A note on Metropolis-Hastings kernels for general state spaces. Annals of Applied Probability 8: 1–9.
Tierney L. and Mira A. 1999. Some adaptive Monte Carlo methods for bayesian inference. Statistics in Medicine 18:2507–2515.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Haario, H., Laine, M., Mira, A. et al. DRAM: Efficient adaptive MCMC. Stat Comput 16, 339–354 (2006). https://doi.org/10.1007/s11222-006-9438-0
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11222-006-9438-0