Combination and Propagation of Uncertainty with Belief Functions - A Reexamination

The behavior of Dempster's rule of combination in typical situations is examined. Particularly, it is shown that assessing a zero value or a very small value may lead to very different results. Moreover a comparison with a possibility theory-based approach in case of conf l ict ing information is provided. The general problem of representing uncertainty with one or several numbers is addressed. Lastly, the propagation of uncertainty from a fact and "if . . . then..." rule is discussed in the framework of belief functions. I INTRODUCTION The treatment of uncertain information in knowledge engineering has encountered an increasing interest among AI researchers in the recent years. Roughly speaking, there are at least two basic problems when reasoning with uncertain facts or rules ; namely the combination problem and the propagation problem. The combination problem refers to the aggregation of uncertain pieces of information issued from different sources dealing with the same matter. The propagation problem deals with the aggregation of the uncertainty concerning the satisfaction of the condi t ion-part of a rule with the uncertainty of the rule itself in order to deduce the uncertainty pervading the conclusion of the rule. Two theoretical frameworks have recently emerged for discussing these problems : the Dempster-Shafer theory of evidence [ 4 ] and Zadeh's possibility theory ; see Prade [ 3 ] for a comparative overview. In the following we examine how Dempster's rule of combination behaves in typical situations and how the result given by this rule may depend on the assessment of the values of the basic assignment. Then the propagation problem is briefly discussed in Shafer's framework.Comparisons are made with the possibilistic approach as well as with the treatment of uncertainty in inference systems such as MYCIN. The key question of the representation of the uncertainty of a fact or of a rule by more than one number is discussed throughout the whole paper. II COMBINATION PROBLEM In Shafer's approach an uncertain body of evidence is represented by a so-called basic probability assignment m which is a set-function from the set of possible elementary events to the real interval [0,1] such that