Dempster–Shafer theory: Difference between revisions

Latest revision as of 05:15, 16 February 2025

A mathematical theory of evidence

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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-'''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.
+{{short description|A mathematical theory of evidence}}
+[[File:Dempster in Brest.JPG|thumb|right|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==
-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.
+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.
-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.
+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.
-==Mathematical Formulation==
+==Belief Functions==
-The mathematical framework of DST is built around three main concepts: the frame of discernment, belief functions, and Dempster's Rule of Combination.
+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.
-===Frame of Discernment===
+The belief function is defined as:
-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===
+- '''Belief (Bel)''': The belief in a proposition is the sum of the belief masses of all subsets of the proposition.
-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.
+- '''Plausibility (Pl)''': The plausibility of a proposition is the sum of the belief masses of all subsets that intersect with the proposition.
-===Dempster's Rule of Combination===
+The relationship between belief and plausibility is given by:
-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.
+\[ \text{Belief}(A) \leq \text{Plausibility}(A) \]
+==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==
-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.
+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 Comparison to Bayesian Theory==
+==Criticism and Limitations==
-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.
+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.
-==See Also==
+==Related pages==
 * [[Bayesian probability]]
-* [[Evidence theory]]
+* [[Fuzzy logic]]
-* [[Information fusion]]
+* [[Probability theory]]
-* [[Decision making]]
+* [[Artificial intelligence]]
-[[Category:Mathematical and Quantitative Methods in Economics]]
-[[Category:Artificial Intelligence]]
-[[Category:Information Theory]]
-{{Mathematics-stub}}
+[[Category:Mathematical theories]]
-{{AI-stub}}
+[[Category:Uncertainty]]
+[[Category:Artificial intelligence]]