On conditional and intrinsic autoregressions

Gaussian conditional autoregressions have been widely used in spatial statistics and Bayesian image analysis, where they are intended to describe interactions between random variables at fixed sites in Euclidean space. The main appeal of these distributions is in the Markovian interpretation of their full conditionals. Intrinsic autoregressions are limiting forms that retain the Markov property. Despite being improper, they can have advantages over the standard autoregressions, both conceptually and in practice. For example, they often avoid difficulties in parameter estimation, without apparent loss, or exhibit appealing invariances, as in texture analysis. However, on small arrays and in nonlattice applications, both forms of autoregression can lead to undesirable second-order characteristics, either in the variables themselves or in contrasts among them. This paper discusses standard and intrinsic autoregressions and describes how the problems that arise can be alleviated using Dempster's (1972) algorithm or an appropriate modification. The approach represents a partial synthesis of standard geostatistical and Gaussian Markov random field formulations. Some nonspatial applications are also mentioned.