ON CONDITIONAL AND INTRINSIC AUTOREGRESSIONS

Gaussian conditional autoregressions have been widely used in spatial statistics and Bayesian image analysis, where they are intended to des cribe interactions between random variables at fixed sites in Euclidea n space. The main appeal of these distributions is in the Markovian in terpretation of their full conditionals. Intrinsic autoregressions are limiting forms that retain the Markov property. Despite being imprope r, they can have advantages over the standard autoregressions, both co nceptually 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 n onlattice applications, both forms of autoregression can lead to undes irable second-order characteristics, either in the variables themselve s or in contrasts among them. This paper discusses standard and intrin sic autoregressions and describes how the problems that arise can be a lleviated using Dempster's (1972) algorithm or an appropriate modifica tion. The approach represents a partial synthesis of standard geostati stical and Gaussian Markov random field formulations. Some nonspatial applications are also mentioned.