Spatial Statistics and Bayesian Computation

on Wednesday, May 6th, 1992, Professor B. W. Silverman in the Chair] SUMMARY Markov chain Monte Carlo (MCMC) algorithms, such as the Gibbs sampler, have provided a Bayesian inference machine in image analysis and in other areas of spatial statistics for several years, founded on the pioneering ideas of Ulf Grenander. More recently, the observation that hyperparameters can be included as part of the updating schedule and the fact that almost any multivariate distribution is equivalently a Markov random field has opened the way to the use of MCMC in general Bayesian computation. In this paper, we trace the early development of MCMC in Bayesian inference, review some recent computational progress in statistical physics, based on the introduction of auxiliary variables, and discuss its current and future relevance in Bayesian applications. We briefly describe a simple MCMC implementation for the Bayesian analysis of agricultural field experiments, with which we have some practical experience.

[1]  R. B. Potts.  Some generalized order-disorder transformations , 1952, Mathematical Proceedings of the Cambridge Philosophical Society.

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

[3]  P. Peskun,et al.  Optimum Monte-Carlo sampling using Markov chains , 1973 .

[4]  J. Besag.  Spatial Interaction and the Statistical Analysis of Lattice Systems , 1974 .

[5]  H. D. Patterson,et al.  A new class of resolvable incomplete block designs , 1976 .

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

[7]  A. Seheult,et al.  Analysis of Field Experiments by Least Squares Smoothing , 1985 .

[8]  Julian Besag,et al.  Statistical Analysis of Field Experiments Using Neighbouring Plots , 1986 .

[9]  Wang,et al.  Nonuniversal critical dynamics in Monte Carlo simulations. , 1987, Physical review letters.

[10]  Stuart Geman,et al.  Statistical methods for tomographic image reconstruction , 1987 .

[11]  K. Fredenhagen,et al.  A modified heat bath method suitable for Monte Carlo simulations on vector and parallel processors , 1987 .

[12]  Adrian F. M. Smith,et al.  Bayesian computation via the gibbs sampler and related markov chain monte carlo methods (with discus , 1993 .

[13]  Ron,et al.  Simulations without critical slowing down. , 1988, Physical review letters.

[14]  A. Sokal,et al.  Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. , 1988, Physical review. D, Particles and fields.

[15]  A. Sokal,et al.  Absence of mass gap for a class of stochastic contour models , 1988 .

[16]  Julian Besag,et al.  Digital Image Processing: Towards Bayesian image analysis , 1989 .

[17]  Peter Clifford,et al.  Reconstruction of polygonal images , 1989 .

[18]  D. M. Keenan,et al.  Towards automated image understanding , 1989 .

[19]  Basilis Gidas,et al.  A Renormalization Group Approach to Image Processing Problems , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[20]  W. A. Wright.  A Markov random field approach to data fusion and colour segmentation , 1989, Image Vis. Comput..

[21]  J. Besag,et al.  Generalized Monte Carlo significance tests , 1989 .

[22]  Brandt,et al.  Simulations without critical slowing down: Ising and three-state Potts models. , 1989, Physical review. B, Condensed matter.

[23]  A. Sokal,et al.  Exponential convergence to equilibrium for a class of random-walk models , 1989 .

[24]  Wolff,et al.  Collective Monte Carlo updating for spin systems. , 1989, Physical review letters.

[25]  D. Geman.  Random fields and inverse problems in imaging , 1990 .

[26]  B. D. Riley,et al.  Iterative simulation methods , 1990 .

[27]  P. Green.  Bayesian reconstructions from emission tomography data using a modified EM algorithm. , 1990, IEEE transactions on medical imaging.

[28]  Donald Geman,et al.  Boundary Detection by Constrained Optimization , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[29]  Ulf Grenander,et al.  Hands: A Pattern Theoretic Study of Biological Shapes , 1990 .

[30]  A. L. Sutherland,et al.  Finding spiral structures in images of galaxies , 1990, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[31]  P. Barone,et al.  Improving Stochastic Relaxation for Gussian Random Fields , 1990, Probability in the Engineering and Informational Sciences.

[32]  C. Geyer.  Markov Chain Monte Carlo Maximum Likelihood , 1991 .

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

[34]  J. Besag,et al.  Sequential Monte Carlo p-values , 1991 .

[35]  U. Grenander,et al.  Structural Image Restoration through Deformable Templates , 1991 .

[36]  D G Clayton,et al.  A Monte Carlo method for Bayesian inference in frailty models. , 1991, Biometrics.

[37]  The empirical efficiency and validity of two neighbour models , 1991 .

[38]  P. Diaconis,et al.  Geometric Bounds for Eigenvalues of Markov Chains , 1991 .

[39]  P. Green,et al.  Global and local priors, and the location of lesions using gamma-camera imagery , 1991, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[40]  Y. Amit.  On rates of convergence of stochastic relaxation for Gaussian and non-Gaussian distributions , 1991 .

[41]  Kanti V. Mardia,et al.  Statistical Shape Models in Image Analysis , 1992 .

[42]  P. Green,et al.  Metropolis Methods, Gaussian Proposals and Antithetic Variables , 1992 .

[43]  C. Geyer,et al.  Constrained Monte Carlo Maximum Likelihood for Dependent Data , 1992 .

[44]  C. Hwang,et al.  Optimal Spectral Structure of Reversible Stochastic Matrices, Monte Carlo Methods and the Simulation of Markov Random Fields , 1992 .

[45]  L. Tierney.  Exploring Posterior Distributions Using Markov Chains , 1992 .

[46]  N. Sheehan,et al.  On the irreducibility of a Markov chain defined on a space of genotype configurations by a sampling scheme. , 1993, Biometrics.

[47]  Michael I. Miller,et al.  REPRESENTATIONS OF KNOWLEDGE IN COMPLEX SYSTEMS , 1994 .