Independence and Bayesian Updating Methods
  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.
