## ABSTRACT

This chapter discusses random field models with a discrete index set. Such models are useful when observations are collected over areal units such as pixels, census districts or tomographic bins. The log probability mass function of the Ising model can be interpreted as a sum of contributions from single sites and pairs of sites. Breaking up a high-dimensional joint probability mass function in lower dimensional components in this way is often useful, both from a computational point of view and conceptually. The Monte Carlo maximum likelihood estimation method requires samples from the model of interest. The chapter introduces the modern hierarchical modelling approach by means of two concrete examples: image segmentation and disease mapping. Inference is usually based on the posterior distribution of the process and/or the parameters conditional on the observations, which can be approximated by Monte Carlo methods. Sometimes a reconstruction of the process is required.