Evidential Reasoning in a Categorial Perspective: Conjunction and Disjunction of Belief Functions

TL;DR

This paper explores the categorical foundations of evidential reasoning, specifically within Dempster-Shafer theory, introducing new logical connectives for belief functions that are cautious and do not require source distinctness.

Contribution

It introduces Dempster's category and explicit formulas for conjunction and disjunction of belief functions, advancing the theoretical understanding of evidential reasoning.

Findings

Defined conjunction as the most cautious belief combination

Proved the existence of conjunction and disjunction in Dempster's category

Provided explicit formulas for these logical connectives

Abstract

The categorial approach to evidential reasoning can be seen as a combination of the probability kinematics approach of Richard Jeffrey (1965) and the maximum (cross-) entropy inference approach of E. T. Jaynes (1957). As a consequence of that viewpoint, it is well known that category theory provides natural definitions for logical connectives. In particular, disjunction and conjunction are modelled by general categorial constructions known as products and coproducts. In this paper, I focus mainly on Dempster-Shafer theory of belief functions for which I introduce a category I call Dempster?s category. I prove the existence of and give explicit formulas for conjunction and disjunction in the subcategory of separable belief functions. In Dempster?s category, the new defined conjunction can be seen as the most cautious conjunction of beliefs, and thus no assumption about distinctness (of the sources) of beliefs is needed as opposed to Dempster?s rule of combination, which calls for distinctness (of the sources) of beliefs.