Quantifying knowledge with a new calculus for belief functions - a generalization of probability theory
This work addresses uncertainty modeling for fields like forensics and gambling by proposing a more flexible alternative to probability theory, though it is incremental as it builds on existing belief function theory.
The paper tackles the limitations of classical probability theory in handling uncertainty in forensic and gambling contexts by introducing a new calculus for Shafer belief functions, which generalizes probability distributions and avoids Dempster's rule, and demonstrates its application with examples and proofs like a law of large numbers.
We first show that there are practical situations in for instance forensic and gambling settings, in which applying classical probability theory, that is, based on the axioms of Kolmogorov, is problematic. We then introduce and discuss Shafer belief functions. Technically, Shafer belief functions generalize probability distributions. Philosophically, they pertain to individual or shared knowledge of facts, rather than to facts themselves, and therefore can be interpreted as generalizing epistemic probability, that is, probability theory interpreted epistemologically. Belief functions are more flexible and better suited to deal with certain types of uncertainty than classical probability distributions. We develop a new calculus for belief functions which does not use the much criticized Dempster's rule of combination, by generalizing the classical notions of conditioning and independence in a natural and uncontroversial way. Using this calculus, we explain our rejection of Dempster's rule in detail. We apply the new theory to a number of examples, including a gambling example and an example in a forensic setting. We prove a law of large numbers for belief functions and offer a betting interpretation similar to the Dutch Book Theorem for probability distributions.