Independence and Bayesian Updating Methods

TL;DR

This paper examines the limitations of independence assumptions in Bayesian updating within rule-based inference systems, showing that multiple evidence items cannot simultaneously update a hypothesis's probability under certain conditions.

Contribution

It refutes previous claims that independence assumptions prevent any updating and clarifies that only one piece of evidence can influence a hypothesis at a time.

Findings

Multiple evidence items cannot simultaneously update a hypothesis under the specified conditions.

The result holds even with added assumptions to exclude special cases.

Updating remains possible but is limited to a single evidence item per hypothesis.

Abstract

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.