Is Shafer General Bayes?

TL;DR

This paper explores the relationship between Shafer's belief functions and convex sets of probability distributions, highlighting their equivalence in certain cases and discussing implications for decision making and rule comparisons.

Contribution

It demonstrates that many convex sets of probabilities produce identical belief functions and compares Dempster's rule with Bayes' rule in this context.

Findings

Belief functions can be generated by multiple convex probability sets

Many convex sets support the same belief function, affecting decision theory

Dempster's rule differs from Bayes' rule in combining evidence

Abstract

This paper examines the relationship between Shafer's belief functions and convex sets of probability distributions. Kyburg's (1986) result showed that belief function models form a subset of the class of closed convex probability distributions. This paper emphasizes the importance of Kyburg's result by looking at simple examples involving Bernoulli trials. Furthermore, it is shown that many convex sets of probability distributions generate the same belief function in the sense that they support the same lower and upper values. This has implications for a decision theoretic extension. Dempster's rule of combination is also compared with Bayes' rule of conditioning.