The Multiple-Try Metropolis is a recent extension of the Metropolis algorithm in which the next state of the chain is selected among a set of proposals. We propose a modification of the Multiple-Try Metropolis algorithm which allows for the use of correlated proposals, particularly antithetic and stratified proposals. The method is particularly useful for random walk Metropolis in high dimensional spaces and can be used easily when the proposal distribution is Gaussian. We explore the use of quasi Monte Carlo (QMC) methods to generate highly stratified samples. A series of examples is presented to evaluate the potential of the method.
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Atchade Y.F. and Perron F. 2005. Improving on the independent Metropolis-Hastings algorithm. Statistica Sinica 5: 3–18.
Atchade Y.F. and Rosenthal J. 2005. On adaptive Markov Chain Monte Carlo algorithms. Bernoulli 11: 815–828.
Bliss C.I. 1935. The calculation of the dosage-mortality curve. Ann. Appl. Biology 22: 134–167.
Caflisch R.E., Morokoff W., and Owen A.B. 1997. Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension. The Journal of Computational Finance 1: 27–46.
Carlin B.P. and Louis T.A. 2000. Bayes and Emprirical Bayes Methods for Data Analysis, Chapman and Hall, London.
Chen M.-H. and Schmeiser B.W. 1993. Performances of the Gibbs, hit-and-run, and Metropolis samplers. J. Computat. Graph. Statist. 2: 251–272.
Craiu R.V. and Meng X.L. 2005. Multi-process parallel antithetic coupling for backward and forward MCMC. Ann. Statist. 33: 661–697.
Cranley R. and Patterson T. 1976. Randomization of number theoretic methods for multiple integration. SIAM J. Numer. Anal. 13: 904–914.
Draper N.R. and Smith H. 1981. Applied Regression Analysis and Its Applications, Wiley, New York.
Frigessi A., Gåsemyr J., and Rue H. 2000. Antithetic coupling of two Gibbs sampler chains. Ann. Statist. 28: 1128–1149.
Gelman A. and Meng X.L. 1991. A note on bivariate distributions that are conditionally normal. Am. Statistician 125–126.
Gilks W.R., Roberts R.O., and George E.I. 1994. Adaptive direction sampling. Statistician 43: 179–189.
L'Ecuyer P. and Lemieux C. 2002. Recent advances in randomized quasi-Monte Carlo methods. In: Dror M., L'Ecuyer P., and F. Szidarovszki (Eds.), Modelling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications, Kluwer Academic Publishers, Boston, pp. 419–474.
Lemieux C. and Sidorsky P. 2006. Exact sampling with highly-uniform point sets. Mathematical and Computer Modelling 43: 339–349.
Lindstrom M.J. and Bates D.M. 1990. Nonlinear mixed effects models for repeated measures data. Biometrics 46: 673–687.
Liu J.S., Liang F., and Wong W.H. 2000. The use of Multiple-Try method and local optimization in Metropolis sampling. J. Amer. Statist. Assoc. 95: 121–134.
Loh W.L. 1996. On Latin hypercube sampling. Ann. Statist. 24: 2058–2080.
Marsaglia G. 1962. Random variables and computers. In: Koseznik J. (Ed.), Information Theory and Statistical Decision Functions Random Processes: Transactions of the Third Prague Conference, Czechoslovak Academy of Sciences, Prague, pp. 499–510.
McKay M.D., Beckman R.J., and Conover W.J. 1979. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21: 239–245.
Niederreiter H. 1992. Random Number Generation and Quasi-Monte Carlo Methods. SIAM CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63, SIAM, Philadelphia.
Owen A.B. 1992. A central limit theorem for Latin hypercube sampling. J. Roy. Statist. Soc. Ser. B 54: 541–551.
Owen A.B. 2003. Quasi Monte Carlo. Presented at the First Cape Cod Workshop on Monte Carlo Methods.
Owen A.B. and Tribble S.D. 2004. A quasi-Monte Carlo Metropolis algorithm. Technical Report 2004-35, Department of Statistics, Stanford University.
Paskov S. and Traub J. 1995. Faster valuation of financial derivatives. Journal of Portfolio Management 22: 113–120.
Ritter C. and Tanner M.A. 1992. Facilitating the Gibbs sampler: the Gibbs stopper and the griddy-Gibbs sampler. J. Amer. Statist. Assoc. 87: 861–868.
Stein M. 1987. Large sample properties of simulations using Latin hypercube sampling. Technometrics 29: 143–151, (Correction: 32, 367).
Sobol' I.M. 1967. The distribution of points in a cube and the approximate evaluation of integrals. U.S.S.R. Comput. Math. and Math. Phys. 7: 86–112.
Tan Z. 2006. Monte Carlo integration with acceptance-rejection. J. Comput. Graphical Statist., to appear.
van Dyk D. and Meng X.L. 2001. The art of data augmentation (with discussion). J. Comput. Graphical Statist. 10: 1–111.
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Craiu, R.V., Lemieux, C. Acceleration of the Multiple-Try Metropolis algorithm using antithetic and stratified sampling. Stat Comput 17, 109 (2007). https://doi.org/10.1007/s11222-006-9009-4
- Antithetic variates
- Markov Chain Monte Carlo
- Extreme antithesis
- Korobov rule
- Latin Hypercube sampling
- Quasi Monte Carlo
- Sobol’ sequence
- Multiple-Try Metropolis
- Random-Ray Monte Carlo