The Multiple-Try Method and Local Optimization in Metropolis Sampling

Abstract This article describes a new Metropolis-like transition rule, the multiple-try Metropolis, for Markov chain Monte Carlo (MCMC) simulations. By using this transition rule together with adaptive direction sampling, we propose a novel method for incorporating local optimization steps into a MCMC sampler in continuous state-space. Numerical studies show that the new method performs significantly better than the traditional Metropolis-Hastings (M-H) sampler. With minor tailoring in using the rule, the multiple-try method can also be exploited to achieve the effect of a griddy Gibbs sampler without having to bear with griddy approximations, and the effect of a hit-and-run algorithm without having to figure out the required conditional distribution in a random direction.

[1]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[2]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[3]  Mihaly Mezei,et al.  A cavity-biased (T, V, μ) Monte Carlo method for the computer simulation of fluids , 1980 .

[4]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[5]  William H. Press,et al.  Numerical recipes in C. The art of scientific computing , 1987 .

[6]  E. Loh.  Multigrid Monte Carlo methods , 1988 .

[7]  Goodman,et al.  Multigrid Monte Carlo method. Conceptual foundations. , 1989, Physical review. D, Particles and fields.

[8]  R. Cracknell,et al.  Rotational insertion bias: a novel method for simulating dense phases of structured particles, with particular application to water , 1990 .

[9]  D. Rubin,et al.  Inference from Iterative Simulation Using Multiple Sequences , 1992 .

[10]  M. Tanner,et al.  Facilitating the Gibbs Sampler: The Gibbs Stopper and the Griddy-Gibbs Sampler , 1992 .

[11]  B. Schmeiser,et al.  Performance of the Gibbs, Hit-and-Run, and Metropolis Samplers , 1993 .

[12]  Walter R. Gilks,et al.  Adaptive Direction Sampling , 1994 .

[13]  C. Robert,et al.  Estimation of Finite Mixture Distributions Through Bayesian Sampling , 1994 .

[14]  G. Roberts,et al.  Convergence of adaptive direction sampling , 1994 .

[15]  R. Tweedie,et al.  Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms , 1996 .

[16]  Jun S. Liu,et al.  Predictive updating methods with application to Bayesian classification , 1996 .

[17]  L. Wasserman,et al.  Practical Bayesian Density Estimation Using Mixtures of Normals , 1997 .

[18]  G. Roberts,et al.  Adaptive Markov Chain Monte Carlo through Regeneration , 1998 .

[19]  Xiao-Li Meng,et al.  Modeling covariance matrices in terms of standard deviations and correlations, with application to shrinkage , 2000 .

[20]  Berend Smit,et al.  Understanding Molecular Simulation , 2001 .