Expert knowledge consists of statements Sj (facts and rules). The expert's degree of confidence in each statement Sj can be described as a (subjective) probability (some probabilities are known to be independent). Examples: if we are interested in oil, we should look at seismic data (confidence 90%); a bank A trusts a client B, so if we trust A, we should trust B too (confidence 99%). If a query Q is deducible from facts and rules, what is our confidence p(Q) in Q? We can describe Q as a propositional formula F in terms of Sj; computing p(Q) exactly is NP-hard, so heuristics are needed.
Traditionally, expert systems use technique similar to straightforward interval computations: we parse F and replace each computation step with corresponding probability operation. Problem: at each step, we ignore the dependence between the intermediate results Fj; hence intervals are too wide. Example: the estimate for P(A\/~A) is not 1. Solution: similarly to affine arithmetic, besides P(Fj), we also compute P(Fj & Fi) (or P(Fj & ... & Fk)), and on each step, use all combinations of l such probabilities to get new estimates. Results: e.g., P(A\/~A) is estimated as 1.