Independence and Bayesian Updating Methods
This clarifies theoretical limitations in Bayesian updating methods for AI and probabilistic reasoning, though it is incremental as it refines prior claims.
The paper addresses the claim by Pednault et al. that independence assumptions in rule-based inference systems prevent updating of hypothesis probabilities based on multiple evidence items, showing that updating is possible but limited to at most one item of evidence per hypothesis.
Duda, Hart, and Nilsson have set forth a method for rule-based inference systems to use in updating the probabilities of hypotheses on the basis of multiple items of new evidence. Pednault, Zucker, and Muresan claimed to give conditions under which independence assumptions made by Duda et al. preclude updating-that is, prevent the evidence from altering the probabilities of the hypotheses. Glymour refutes Pednault et al.'s claim with a counterexample of a rather special form (one item of evidence is incompatible with all but one of the hypotheses); he raises, but leaves open, the question whether their result would be true with an added assumption to rule out such special cases. We show that their result does not hold even with the added assumption, but that it can nevertheless be largely salvaged. Namely, under the conditions assumed by Pednault et al., at most one of the items of evidence can alter the probability of any given hypothesis; thus, although updating is possible, multiple updating for any of the hypotheses is precluded.