Dempster–Shafer theory

Dempster–Shafer Theory (DST), also known as the theory of belief functions, is a mathematical theory of evidence that allows one to combine evidence from different sources and arrive at a degree of belief (represented as a belief function) that takes into account all the available evidence. Unlike Bayesian probability theory, which requires probabilities for each event, DST works with degrees of belief for events, which can be more general.

Overview

Dempster–Shafer Theory is based on two ideas: the concept of a belief function and the Dempster's Rule of Combination. A belief function is a function that assigns to each subset of a frame of discernment (a set of all possible hypotheses) a number between 0 and 1, representing how much a given piece of evidence supports that subset. The belief assigned to a set represents the total belief that the truth is in that set, without specifying exactly where within the set the truth lies.

Dempster's Rule of Combination is a rule for combining such belief functions when they are based on independent pieces of evidence. It provides a way to update beliefs in the light of new evidence.

Mathematical Formulation

The mathematical framework of DST is built around three main concepts: the frame of discernment, belief functions, and Dempster's Rule of Combination.

Frame of Discernment

The frame of discernment, denoted by Θ, is the set of all possible outcomes or hypotheses. For example, in a medical diagnosis problem, Θ could be the set of all possible diseases.

Belief Functions

A belief function, Bel, on a frame of discernment Θ is a function that maps subsets of Θ to the interval [0,1]. For any subset A of Θ, Bel(A) represents the total belief that the true state of the world is in A.

Dempster's Rule of Combination

Dempster's Rule of Combination provides a method for combining two belief functions, Bel1 and Bel2, into a new belief function, Bel. This rule is applied when the belief functions are based on independent pieces of evidence.

Applications

Dempster–Shafer Theory has been applied in various fields, including artificial intelligence, computer science, and engineering, for tasks such as decision making, information fusion, and risk assessment. Its ability to handle uncertain and incomplete information makes it a valuable tool in these areas.

Criticism and Comparison to Bayesian Theory

DST has been both praised for its flexibility in handling uncertainty and criticized for its computational complexity and for certain counterintuitive results that can arise. Compared to Bayesian probability theory, DST does not require precise probabilities for each hypothesis, which can be an advantage in situations where such probabilities are hard to estimate. However, the interpretation of belief functions and the results of Dempster's Rule of Combination can sometimes be less intuitive than probabilities.

See Also

• Bayesian probability
• Evidence theory
• Information fusion
• Decision making