Julian Besag

BAYESIAN IMAGE RESTORATION, WITH TWO APPLICATIONS IN SPATIAL STATISTICS* **

There has been much recent interest in Bayesian image analysis, including such topics as removal of blur and noise, detection of object boundaries, classification of textures, and reconstruction of two- or three-dimensional scenes from noisy lower-dimensional views. Perhaps the most straightforward task is that of image restoration, though it is often suggested that this is an area of relatively minor practical importance. The present paper argues the contrary, since many problems in the analysis of spatial data can be interpreted as problems of image restoration. Furthermore, the amounts of data involved allow routine use of computer intensive methods, such as the Gibbs sampler, that are not yet practicable for conventional images. Two examples are given, one in archeology, the other in epidemiology. These are preceded by a partial review of pixel-based Bayesian image analysis.

Statistical Analysis of Non-Lattice Data

In rather formal terms, the situation with which this paper is concerned may be described as follows. We are given a fixed system of n sites, labelled by the first n positive integers, and an associated vector x of observations, Xi, . . ., Xn, which, in turn, is presumed to be a realization of a vector X of (dependent) random variables, Xi, . . ., X.. In practice, the sites may represent points or regions in space and the random variables may be either continuous or discrete. The main statistical objectives are the following: firstly, to provide a means of using the available concomitant information, particularly the configuration of the sites, to attach a plausible probability distribution to the random vector X; secondly, to estimate any unknown parameters in the distribution from the realization x; thirdly, where possible, to quantify the extent of disagreement between hypothesis and observation.

Modelling risk from a disease in time and space

This paper combines existing models for longitudinal and spatial data in a hierarchical Bayesian framework, with particular emphasis on the role of time- and space-varying covariate effects. Data analysis is implemented via Markov chain Monte Carlo methods. The methodology is illustrated by a tentative re-analysis of Ohio lung cancer data 1968-1988. Two approaches that adjust for unmeasured spatial covariates, particularly tobacco consumption, are described. The first includes random effects in the model to account for unobserved heterogeneity; the second adds a simple urbanization measure as a surrogate for smoking behaviour. The Ohio data set has been of particular interest because of the suggestion that a nuclear facility in the southwest of the state may have caused increased levels of lung cancer there. However, we contend here that the data are inadequate for a proper investigation of this issue.

Digital Image Processing: Towards Bayesian image analysis

Many of the tasks encountered in image processing can be considered as problems in statistical inference. In particular, they fit naturally into a subjectivist Bayesian framework. In this paper, we describe the Bayesian approach to image analysis. Numerical examples are not included but can be found among the references, in the previous Special Issue of this Journal and elsewhere. It is argued that the Bayesian approach, still in its infancy, has considerable potential for future development.

Towards Bayesian image analysis

Many of the tasks encountered in image processing can be considered as problems in statistical inference. In particular, they fit naturally into a subjectivist Bayesis framework. In this paper, we describe the Bayesian approach to image analysis. Numerical examples are not included but references are given. It is argued that the Bayesian approach, still in its infancy, has considerable potential for future development.

Markov Random Fields* with Higher-order Interactions

Discrete-state Markov random fields on regular arrays have played a signifi- cant role in spatial statistics and image analysis. For example, they are used to represent objects against background in computer vision and pixel-based classification of a region into different crop types in remote sensing. Convenience has generally favoured formulations that involve only pairwise interactions. Such models are in themselves unrealistic and, although they often perform surprisingly well in tasks such as the restoration of degraded images, they are unsatisfactory for many other purposes. In this paper, we consider particular forms of Markov random fields that involve higher-order interactions and therefore are better able to represent the large-scale properties of typical spatial scenes. Interpretations of the para- meters are given and realizations from a variety of models are produced via Markov chain Monte Carlo. Potential applications are illustrated in two examples. The first concerns Bayesian image analysis and confirms that pairwise-interaction priors may perform very poorly for image functionals such as number of objects, even when restoration apparently works well. The second example describes a model for a geological dataset and obtains maximum-likelihood parameter estimates using Markov chain Monte Carlo. Despite the complexity of the formulation, realizations of the estimated model suggest that the represen- tation is quite realistic.

Exact Goodness-of-Fit Tests for Markov Chains

Goodness-of-fit tests are useful in assessing whether a statistical model is consistent with available data. However, the usual χ² asymptotics often fail, either because of the paucity of the data or because a nonstandard test statistic is of interest. In this article, we describe exact goodness-of-fit tests for first- and higher order Markov chains, with particular attention given to time-reversible ones. The tests are obtained by conditioning on the sufficient statistics for the transition probabilities and are implemented by simple Monte Carlo sampling or by Markov chain Monte Carlo. They apply both to single and to multiple sequences and allow a free choice of test statistic. Three examples are given. The first concerns multiple sequences of dry and wet January days for the years 1948-1983 at Snoqualmie Falls, Washington State, and suggests that standard analysis may be misleading. The second one is for a four-state DNA sequence and lends support to the original conclusion that a second-order Markov chain provides an adequate fit to the data. The last one is six-state atomistic data arising in molecular conformational dynamics simulation of solvated alanine dipeptide and points to strong evidence against a first-order reversible Markov chain at 6 picosecond time steps.

Inference on a collapsed margin in disease mapping.

This paper describes a method for estimating the risk from a disease over a set of contiguous geographical regions, when data on a potentially important covariate, such as race, are not available. Conditions under which the extra margin can be recovered are suggested. An application to prostate cancer mortality among the non-white population in the counties of the U.S.A. is discussed.

On Spatial-Temporal Models and Markov Fields

A short account of the Hammersley-Clifford theorem in the theory of Markov random fields is given. A class of time-reversible spatial-temporal processes is defined and is shown to generate the class of positive discrete Markov fields as limiting spatial distribution. Some examples are given. Spatial-temporal autoregressions for Gaussian variâtes are discussed and it is shown that these are often consistent with the auto-normal spatial fields introduced previously by the author.

An Introduction to Markov Chain Monte Carlo Methods

This article provides an introduction to Markov chain Monte Carlo methods in statistical inference. Over the past twelve years or so, these have revolutionized what can be achieved computationally, especially in the Bayesian paradigm. Markov chain Monte Carlo has exactly the same goals as ordinary Monte Carlo and both are intended to exploit the fact that one can learn about a complex probability distribution if one can sample from it. Although the ordinary version can only rarely be implemented, it is convenient initially to presume otherwise and to focus on the rationale of the sampling approach, rather than computational details. The article then moves on to describe implementation via Markov chains, especially the Hastings algorithm, including the Metropolis method and the Gibbs sampler as special cases. Hidden Markov models and the autologistic distribution receive some emphasis, with the noisy binary channel used in some toy examples. A brief description of perfect simulation is also given. The account concludes with some discussion.