The question posed by Haipeng Guo and subsequent comments by Rina Dechter, Marek Druzdzel and Eugene Santos touch upon a basic problem. What is not widely recognized is that the problem in question is an instance of a class of problems which have no crisp solutions within the conceptual structure of classical logic and probability theory. The culprit is what may be called the dilemma of "it is possible but not probable." More specifically, to compute an upper bound on the error of approximation it is necessary to be able to assess the probability of the worst case scenario-- a scenario which, in general, is possible but not probable. The problem is that the probability of the worst case scenario does not lend itself to crisp assessment. The same problem arises in dealing with imprecise probabilities, especially in the context of Bayesian networks. When imprecisely known probabilities are treated as if they were precise, validity of analysis is open to question. What this points to is an imperative need to develop a better understanding of computing with imprecise probabilities-- as most real-world probabilities are. What is said above does not detact from the usefulness of results which are alluded to in the comments. What it means is that in the analysis of complex systems it is hard, and frequently impossible, to achieve precision, rigor and usefulness at the same time. -- Professor in the Graduate School, Computer Science Division Department of Electrical Engineering and Computer Sciences University of California Berkeley, CA 94720 -1776 Director, Berkeley Initiative in Soft Computing (BISC) Address: Computer Science Division University of California Berkeley, CA 94720-1776 Tel(office): (510) 642-4959 Fax(office): (510) 642-1712 Tel(home): (510) 526-2569 Fax(home): (510) 526-2433, (510) 526-5181 [EMAIL PROTECTED] http://www.cs.berkeley.edu/People/Faculty/Homepages/zadeh.html
