On the Value of derivative evaluations and random walk suppression in Markov Chain Monte Carlo algorithms

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.

[1]  A. Gelman,et al.  Weak convergence and optimal scaling of random walk Metropolis algorithms , 1997 .

[2]  P. Gustafson.  Flexible Bayesian Modelling for Survival Data , 1998, Lifetime data analysis.

[3]  J. Besag,et al.  Bayesian image restoration, with two applications in spatial statistics , 1991 .

[4]  A. Kennedy,et al.  Hybrid Monte Carlo , 1988 .

[5]  Geoffrey E. Hinton,et al.  Bayesian Learning for Neural Networks , 1995 .

[6]  J. Rosenthal,et al.  Optimal scaling for various Metropolis-Hastings algorithms , 2001 .

[7]  R. Tweedie,et al.  Exponential convergence of Langevin distributions and their discrete approximations , 1996 .

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

[9]  Ying C MacNab,et al.  Hierarchical Bayesian spatial modelling of small‐area rates of non‐rare disease , 2003, Statistics in medicine.

[10]  Norman E. Breslow,et al.  Estimation of Disease Rates in Small Areas: A new Mixed Model for Spatial Dependence , 2000 .

[11]  H. Rue.  Fast Sampling of Gaussian Markov Random Fields with Applications , 2000 .

[12]  J. Møller,et al.  Geometric ergodicity of Metropolis-Hastings algorithms for conditional simulation in generalised lin , 2001 .

[13]  B G Leroux.  Modelling spatial disease rates using maximum likelihood. , 2000, Statistics in medicine.

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

[15]  Radford M. Neal.  Bayesian Learning via Stochastic Dynamics , 1992, NIPS.

[16]  A. Horowitz.  A generalized guided Monte Carlo algorithm , 1991 .

[17]  P Gustafson,et al.  Large hierarchical Bayesian analysis of multivariate survival data. , 1997, Biometrics.

[18]  Y. MacNab.  Hierarchical Bayesian Modeling of Spatially Correlated Health Service Outcome and Utilization Rates , 2003, Biometrics.

[19]  Paul Gustafson,et al.  A guided walk Metropolis algorithm , 1998, Stat. Comput..

[20]  H. Ishwaran.  Applications of Hybrid Monte Carlo to Bayesian Generalized Linear Models: Quasicomplete Separation and Neural Networks , 1999 .

[21]  Dani Gamerman,et al.  Sampling from the posterior distribution in generalized linear mixed models , 1997, Stat. Comput..

[22]  J. Rosenthal,et al.  Optimal scaling of discrete approximations to Langevin diffusions , 1998 .

[23]  C B Dean,et al.  Parametric bootstrap and penalized quasi-likelihood inference in conditional autoregressive models. , 2000, Statistics in medicine.

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

[25]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .

[26]  Radford M. Neal,et al.  Suppressing Random Walks in Markov Chain Monte Carlo Using Ordered Overrelaxation , 1995, Learning in Graphical Models.