J. ltal. Statist. Soc. (1993) 2, p. 213-232 EXPRESSING IMPRECISION IN PROBABILISTIC KNOWLEDGE* Gernot D. Kleiter 1 University of Salzburg, Department of Psychology, Austria Summary The representation of imprecise knowledge in probabilistic expert systems is investi- gated. It is argued that Bayesian statistics offers a multitude of models and methods that may be employed to express probabilistic knowledge by first and second order probability distributions. Distributions offer a much richer repertoire to express un- certainty that point probabilities. Special attention is payed to Dirichletian weights of evidence in discrete systems and to predictions and inverse predictions in linear re- gression. Approximate methods for non-conjugate distributions and for incomplete data are discussed. An important property of imprecision in complex systems is that it explodes rapidly when the conclusions are derived from long chains of premises. Thus, imprecision is a pruning criterion is a pruning criterion in complex knowledge systems. 1. Introduction The Bayesian theory was - and sometimes still is - misrepresented in the literature on expert and knowledge based systems. Here are a few typical misconceptions: Bayesian theory is equivalent to elementary probability theory. SHAFER [1.7], for example, identifies Bayesian theory with three rules: empy sets obtain zero probabilites, the sure event has probability one, and the probabi- lities of sets with empty intersections are additive. The rules are called 'Baye- sian rules' despite there is nothing Bayesian about these rules. Bayesian theory cannot express 'no evidence, 'conflicting evidence', and "uncommitted belief. It is argued that if Bayesians assign a degree of belief in 1. Address for correspondence: G. D. Kleiter, University of Salzburg, Department of Psychology, Herrbrunnerstr. 34, A-5020 Salzburg, Austria. This research was supported by the Fonds zur Frrderung der wissenschftlichen Forschung, Vienna. 213