Belief Calculus

TL;DR

This paper introduces belief calculus, a mathematical framework for representing and manipulating beliefs using opinions, which extend classical probability calculus to subjective and Dempster-Shafer theories, facilitating practical belief reasoning.

Contribution

It formalizes belief calculus based on opinions, linking subjective logic with Beta distributions, and demonstrates how classical operators apply to belief representations.

Findings

Belief calculus enables algebraic operations on beliefs.

Opinions are mathematically equivalent to Beta distributions.

The framework bridges subjective logic and probability theory.

Abstract

In Dempster-Shafer belief theory, general beliefs are expressed as belief mass distribution functions over frames of discernment. In Subjective Logic beliefs are expressed as belief mass distribution functions over binary frames of discernment. Belief representations in Subjective Logic, which are called opinions, also contain a base rate parameter which express the a priori belief in the absence of evidence. Philosophically, beliefs are quantitative representations of evidence as perceived by humans or by other intelligent agents. The basic operators of classical probability calculus, such as addition and multiplication, can be applied to opinions, thereby making belief calculus practical. Through the equivalence between opinions and Beta probability density functions, this also provides a calculus for Beta probability density functions. This article explains the basic elements of belief calculus.