Dempster–Shafer theory

A mathematical theory of evidence

Arthur P. Dempster, one of the developers of the Dempster-Shafer theory

The Dempster–Shafer theory is a mathematical theory of evidence that allows one to combine evidence from different sources and arrive at a degree of belief (represented by a mathematical object called a belief function) that takes into account all the available evidence. It is a generalization of the Bayesian probability theory and is used in various fields such as artificial intelligence, statistics, and decision theory.

Overview

The Dempster–Shafer theory, also known as the theory of belief functions, was developed by Arthur P. Dempster and Glenn Shafer. It provides a framework for modeling epistemic uncertainty, which is uncertainty about the state of the world due to incomplete or ambiguous information.

In this theory, evidence is represented by a set of propositions, and each proposition is assigned a belief mass. The belief mass is a number between 0 and 1, and the sum of the belief masses for all propositions is 1. The belief mass represents the degree of belief that a particular proposition is true, given the available evidence.

Belief Functions

A belief function is a function that assigns a belief mass to each subset of a given set of propositions, called the frame of discernment. The frame of discernment is a finite set of mutually exclusive and exhaustive propositions that represent all possible states of the world.

The belief function is defined as:

- Belief (Bel): The belief in a proposition is the sum of the belief masses of all subsets of the proposition. - Plausibility (Pl): The plausibility of a proposition is the sum of the belief masses of all subsets that intersect with the proposition.

The relationship between belief and plausibility is given by:

\[ \text{Belief}(A) \leq \text{Plausibility}(A) \]

Dempster's Rule of Combination

Dempster's rule of combination is a method for combining multiple belief functions into a single belief function. It is used to aggregate evidence from different sources.

The rule is defined as follows:

\[ m(A) = \frac{\sum_{B \cap C = A} m_1(B) \cdot m_2(C)}{1 - \sum_{B \cap C = \emptyset} m_1(B) \cdot m_2(C)} \]

where \( m_1 \) and \( m_2 \) are the belief functions to be combined, and \( A \) is a subset of the frame of discernment.

Applications

The Dempster–Shafer theory is used in various applications, including:

- Sensor fusion: Combining data from multiple sensors to improve the accuracy of information. - Expert systems: Aggregating expert opinions in decision-making processes. - Risk assessment: Evaluating the likelihood of different outcomes based on uncertain information.

Criticism and Limitations

While the Dempster–Shafer theory provides a flexible framework for dealing with uncertainty, it has been criticized for its computational complexity and the potential for counterintuitive results when combining conflicting evidence.

Related pages

• Bayesian probability
• Fuzzy logic
• Probability theory
• Artificial intelligence