Interval-type and affine arithmetic-type techniques for handling uncertainty in expert systems

Expert knowledge consists of statements SjSj (facts and rules). The facts and rules are often only true with some probability. For example, if we are interested in oil, we should look at seismic data. If in 90% of the cases, the seismic data were indeed helpful in locating oil, then we can say that if we are interested in oil, then with probability 90% it is helpful to look at the seismic data. In more formal terms, we can say that the implication “if oil then seismic” holds with probability 90%. Another example: a bank A trusts a client B, so if we trust the bank A, we should trust B too; if statistically this trust was justified in 99% of the cases, we can conclude that the corresponding implication holds with probability 99%.If a query Q   is deducible from facts and rules, what is the resulting probability p(Q)p(Q) in Q? We can describe the truth of Q as a propositional formula F   in terms of SjSj, i.e., as a combination of statements SjSj linked by operators like &, ∨∨, and ¬¬; computing p(Q)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 FjFj; hence intervals are too wide. Example: the estimate for P(A∨¬A)P(A∨¬A) is not 1. Solution: similar to affine arithmetic, besides P(Fj)P(Fj), we also compute P(Fj&Fi)P(Fj&Fi) (or P(Fj1&⋯&Fjd)P(Fj1&⋯&Fjd)), and on each step, use all combinations of l   such probabilities to get new estimates. Results: e.g., P(A∨¬A)P(A∨¬A) is estimated as 1.