## Statistical Science

- Statist. Sci.
- Volume 10, Number 1 (1995), 3-41.

### Bayesian Computation and Stochastic Systems

Julian Besag, Peter Green, David Higdon, and Kerrie Mengersen

#### Abstract

Markov chain Monte Carlo (MCMC) methods have been used extensively in statistical physics over the last 40 years, in spatial statistics for the past 20 and in Bayesian image analysis over the last decade. In the last five years, MCMC has been introduced into significance testing, general Bayesian inference and maximum likelihood estimation. This paper presents basic methodology of MCMC, emphasizing the Bayesian paradigm, conditional probability and the intimate relationship with Markov random fields in spatial statistics. Hastings algorithms are discussed, including Gibbs, Metropolis and some other variations. Pairwise difference priors are described and are used subsequently in three Bayesian applications, in each of which there is a pronounced spatial or temporal aspect to the modeling. The examples involve logistic regression in the presence of unobserved covariates and ordinal factors; the analysis of agricultural field experiments, with adjustment for fertility gradients; and processing of low-resolution medical images obtained by a gamma camera. Additional methodological issues arise in each of these applications and in the Appendices. The paper lays particular emphasis on the calculation of posterior probabilities and concurs with others in its view that MCMC facilitates a fundamental breakthrough in applied Bayesian modeling.

#### Article information

**Source**

Statist. Sci., Volume 10, Number 1 (1995), 3-41.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.ss/1177010123

**Digital Object Identifier**

doi:10.1214/ss/1177010123

**Mathematical Reviews number (MathSciNet)**

MR1349818

**Zentralblatt MATH identifier**

0955.62552

**JSTOR**

links.jstor.org

**Keywords**

Agricultural field experiments Bayesian inference conditional distributions deconvolution gamma-camera imaging Gibbs sampler Hastings algorithms image analysis logistic regression Markov chain Monte Carlo Markov random fields Metropolis method prostate cancer simultaneous credible regions spatial statistics time reversibility unobserved covariates variety trials

#### Citation

Besag, Julian; Green, Peter; Higdon, David; Mengersen, Kerrie. Bayesian Computation and Stochastic Systems. Statist. Sci. 10 (1995), no. 1, 3--41. doi:10.1214/ss/1177010123. https://projecteuclid.org/euclid.ss/1177010123

#### See also

- See Comment: Arnoldo Frigessi. [Bayesian Computation and Stochastic Systems]: Comment. Statist. Sci., Volume 10, Number 1 (1995), 41--43.Project Euclid: euclid.ss/1177010124
- See Comment: Alan E. Gelfand, Bradley P. Carlin. [Bayesian Computation and Stochastic Systems]: Comment. Statist. Sci., Volume 10, Number 1 (1995), 43--46.Project Euclid: euclid.ss/1177010125
- See Comment: Charles J. Geyer. [Bayesian Computation and Stochastic Systems]: Comment. Statist. Sci., Volume 10, Number 1 (1995), 46--48.Project Euclid: euclid.ss/1177010126
- See Comment: G. O. Roberts, S. K. Sahu, W. R. Gilks. [Bayesian Computation and Stochastic Systems]: Comment. Statist. Sci., Volume 10, Number 1 (1995), 49--51.Project Euclid: euclid.ss/1177010127
- See Comment: Wing Hung Wong. [Bayesian Computation and Stochastic Systems]: Comment. Statist. Sci., Volume 10, Number 1 (1995), 52--53.Project Euclid: euclid.ss/1177010128
- See Comment: Bin Yu. [Bayesian Computation and Stochastic Systems]: Comment: Extracting More Diagnostic Information from a Single Run Using Cusum Path Plot. Statist. Sci., Volume 10, Number 1 (1995), 54--58.Project Euclid: euclid.ss/1177010129
- See Comment: Julian Besag, Peter Green, David Higdon, Kerrie Mengersen. [Bayesian Computation and Stochastic Systems]: Rejoinder. Statist. Sci., Volume 10, Number 1 (1995), 58--66.Project Euclid: euclid.ss/1177010130