Combining Uncertain Estimates

In an expert system, it is necessary to supply the values of various parameters. Ideally, an absolutely reliable source is available to supply an exact value for any parameter. In reality, one may have unreliable, unconfident, conflicting estimates of the value for a particular parameter. This paper is a consideration of how to represent and combine imperfect estimates. It is assumed that the knowledge from each source takes the form of an estimate of the parameter value, paired with an associated measure of uncertainty. This uncertainty may arise from lack of confidence on the part of a human source, from some appraisal of the (un)reliability of a source, or both. Functions are proposed that combine several such value-and-uncertainty pairs to produce a resultant of the same form, that is, a resultant value and a resultant uncer­ tainty. The proposed combination functions are most comfortably understood in terms of a parameter with values on the reals, but there is some consideration of alternative scales. The functions are developed in terms of an analogy to statistical sampling. They conform with vari­ ous intuitions about the way estimates should combine. Three related matters are worth mentioning, though space constraints preclude prop. er dis­ cussion. First, in addition to representing uncertain values it is also necessary to be able to com­ pute with them, to determine both values and appropriate uncertainties for related, derived parameters in the system. (On this point, see [Garv81], and [Quin83]) . Next, if there is sharp disagreement among confident and ostensibly reliable sources, one may think of reevaluating their degree of reliability. Last, there is the possibility of using information about uncertainty for deciding when it is worthwhile to seek more information that might reduce uncertainty. (Ideas related to this point are present in [Slag84 ] and [Duda79]).