Interval Infuence Dia g rams K. W. Fer a J. S. Breee Rockwell Interational Science Center Palo Alto Laboraor 444 High Street Palo Alto, CA 94301 Abtact We describe a mechanism for performing probabilistic reaoning in infuence diagram us ing interval rather than point valued probabilities. We derive the procedures for node removal ( corresponding to conditional expectation) and arc reversal ( corresponding to Bayesian condi tioning) in infuence diagrams where lower bounds on probabilities are stored at each node. The resulting bounds for the transformed diagram are shown to be optimal within the clas of constraints on probability distributions which can be expressed exclusively a lower bounds on the component probabilities of the diagram. Sequences of these operations can be performed to ase probabilistic queries idaiesi the input ad  performing sensitivity analysis on an infuence diagram. The storage requirements and computational complexity of this approach are comparable to those for point-valued probabilistic inference mechanisms, mak ing the approach attactive for performing sensitivity analysis and where probability information is not available. Limited empirical data on an implementation of the methodology is provided. 1 Introduction One of the most difcult tasks in constructing an infuence diagram is development of conditional and magina probabilities for each node in the network. In some instances probability information may not be readily avalable, and a reaoner wishes to determine what conclusions can be drawn with partial information on probabilities. In others cases, one may wish to assess the robustness  various conclusions to imprecision in the input. The subject of probabilty bounds has been a topic of interest for a number of years in artificial intelligence. Early users of Dempster-Shaer formalisms were orignally motivated by the ability to specify bounds on probabilties [4,6]. Inequaity bounds have aso been examined by those attempting to bridge between Dempster-Shafer theory and traditional probability theory [5,8,9]. A number of other reseachers have attempted to deal with bounds on probabilities within a traditiona Bayesian framework [2,18,12,14]. In this paper we develop and demonstrate a means of incorporating imprecision in probability vaues by specifying lower bounds on input probabilities and using infuence diagrams as a means of expressing conditiona independence. A number of authors have developed systems which derive probabilistic conclusions, given genera פa constaints on the inputs (18,20]. These systems typicaly use lnear programming methods repeatedly to propagate constraints through a set of probabilistic calculations. The characterization of constraints as lower bounds alows us to derve a relatively efcient procedure for probabilistic inference, based on successive transformations to the diagram, at the cost of some expressiveness. The implications of these transformations in terms of the sets of probability distributions admitted by the bounds are anayzed in detail. 102