On conditional and intrinsic autoregressions

Gaussian conditional autoregressions have been widely used inspatial statistics and Bayesian image analysis, where they areintended to describe interactions between random variables atfixed sites in Euclidean space. The main appeal of these distributionsis in the Markovian interpretation of their full conditionals.Intrinsic autoregressions are limiting forms that retain theMarkov property. Despite being improper, they can have advantagesover the standard autoregressions, both conceptually and inpractice. For example, they often avoid difficulties in parameterestimation, without apparent loss, or exhibit appealing invariances,as in texture analysis. However, on small arrays and in nonlatticeapplications, both forms of autoregression can lead to undesirablesecond-order characteristics, either in the variables themselvesor in contrasts among them. This paper discusses standard andintrinsic autoregressions and describes how the problems thatarise can be alleviated using Dempster's (1972) algorithm oran appropriate modification. The approach represents a partialsynthesis of standard geostatistical and Gaussian Markov randomfield formulations. Some nonspatial applications are also mentioned.