The Dempster-Shafer Theory

The Dempster-Shafer Theory

Malcolm J. Beynon (Cardiff University, UK)
Copyright: © 2009 |Pages: 6
DOI: 10.4018/978-1-59904-849-9.ch068
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Abstract

The initial work introducing Dempster-Shafer (D-S) theory is found in Dempster (1967) and Shafer (1976). Since its introduction the very name causes confusion, a more general term often used is belief functions (both used intermittently here). Nguyen (1978) points out, soon after its introduction, that the rudiments of D-S theory can be considered through distributions of random sets. More furtive comparison has been with the traditional Bayesian theory, where D-S theory has been considered a generalisation of it (Schubert, 1994). Cobb and Shenoy (2003) direct its attention to the comparison of D-S theory and the Bayesian formulisation. Their conclusions are that they have the same expressive power, but that one technique cannot simply take the role of the other. The association with artificial intelligence (AI) is clearly outlined in Smets (1990), who at the time, acknowledged the AI community has started to show interest for what they call the Dempster-Shafer model. It is of interest that even then, they highlight that there is confusion on what type of version of D-S theory is considered. D-S theory was employed in an event driven integration reasoning scheme in Xia et al. (1997), associated with automated route planning, which they view as a very important branch in applications of AI. Liu (1999) investigated Gaussian belief functions and specifically considered their proposed computation scheme and its potential usage in AI and statistics. Huang and Lees (2005) apply a D-S theory model in natural-resource classification, comparing with it with two other AI models. Wadsworth and Hall (2007) considered D-S theory in a combination with other techniques to investigate site-specific critical loads for conservation agencies. Pertinently, they outline its positioning with respect to AI (p. 400); The approach was developed in the AI (artificial intelligence) community in an attempt to develop systems that could reason in a more human manner and particularly the ability of human experts to “diagnose” situations with limited information. This statement is pertinent here, since emphasis within the examples later given is more towards the general human decision making problem and the handling of ignorance in AI. Dempster and Kong (1988) investigated how D-S theory fits in with being an artificial analogy for human reasoning under uncertainty. An example problem is considered, the murder of Mr. White, where witness evidence is used to classify the belief in the identification of an assassin from considered suspects. The numerical analyses presented exposit a role played by D-S theory, including the different ways it can act on incomplete knowledge.
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Introduction

The initial work introducing Dempster-Shafer (D-S) theory is found in Dempster (1967) and Shafer (1976). Since its introduction the very name causes confusion, a more general term often used is belief functions (both used intermittently here). Nguyen (1978) points out, soon after its introduction, that the rudiments of D-S theory can be considered through distributions of random sets. More furtive comparison has been with the traditional Bayesian theory, where D-S theory has been considered a generalisation of it (Schubert, 1994). Cobb and Shenoy (2003) direct its attention to the comparison of D-S theory and the Bayesian formulisation. Their conclusions are that they have the same expressive power, but that one technique cannot simply take the role of the other.

The association with artificial intelligence (AI) is clearly outlined in Smets (1990), who at the time, acknowledged the AI community has started to show interest for what they call the Dempster-Shafer model. It is of interest that even then, they highlight that there is confusion on what type of version of D-S theory is considered. D-S theory was employed in an event driven integration reasoning scheme in Xia et al. (1997), associated with automated route planning, which they view as a very important branch in applications of AI. Liu (1999) investigated Gaussian belief functions and specifically considered their proposed computation scheme and its potential usage in AI and statistics. Huang and Lees (2005) apply a D-S theory model in natural-resource classification, comparing with it with two other AI models.

Wadsworth and Hall (2007) considered D-S theory in a combination with other techniques to investigate site-specific critical loads for conservation agencies. Pertinently, they outline its positioning with respect to AI (p. 400);

The approach was developed in the AI (artificial intelligence) community in an attempt to develop systems that could reason in a more human manner and particularly the ability of human experts to “diagnose” situations with limited information.

This statement is pertinent here, since emphasis within the examples later given is more towards the general human decision making problem and the handling of ignorance in AI. Dempster and Kong (1988) investigated how D-S theory fits in with being an artificial analogy for human reasoning under uncertainty.

An example problem is considered, the murder of Mr. White, where witness evidence is used to classify the belief in the identification of an assassin from considered suspects. The numerical analyses presented exposit a role played by D-S theory, including the different ways it can act on incomplete knowledge.

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Background

The background section to this article covers the basic formulisations of D-S theory, as well as certain developments. Formally, D-S theory is based on a finite set of p elements Θ = {s1, s2, ..., sp}, called a frame of discernment. A mass value is a function m: 2Θ → [0, 1] such that m(∅) = 0 (∅ - the empty set) and: = 1 (2Θ - the power set of Θ). Any proper subset s of the frame of discernment Θ, for which m(s) is non-zero, is called a focal element and represents the exact belief in the proposition depicted by s. The notion of a proposition here being the collection of the hypotheses represented by the elements in a focal element.

Key Terms in this Chapter

Dempster-Shafer Theory: General methodology, also known as the theory of belief functions, its rudiments are closely associated with uncertain reasoning.

Body of Evidence: In Dempster-Shafer theory, a series of focal elements and associated mass values.

Ignorance: In Dempster-Shafer theory, the level of mass value not discernible among the hypotheses.

Mass Value: In Dempster-Shafer theory, the level of exact belief in a focal element.

Focal Element: In Dempster-Shafer theory, a set of hypotheses with positive mass value in a body of evidence.

Frame of Discernment: In Dempster-Shafer theory, the set of all hypotheses considered.

Non-Specificity: In Dempster-Shafer theory, the weighted average of the focal elements’ mass values in a body of evidence, viewed as a species of a higher uncertainty type, encapsulated by the term ambiguity.

Plausibility: In Dempster-Shafer theory, the extent to which we fail to disbelieve a proposition lies in a focal element.

Belief: In Dempster-Shafer theory, the level of representing the confidence that a proposition lies in a focal element or any subset of it.

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