Dempster-Shafer Theory of Evidence

The Dempster-Shafer theory, also known as the theory of belief functions, is a generalization of the Bayesian theory of subjective probability [30,42]. Whereas the Bayesian theory requires probabilities for each question of interest, belief functions allow us to base degrees of belief for one question on probabilities for a related question. These degrees of belief may or may not have the mathematical properties of probabilities how much they differ from probabilities will depend on how closely the two questions are related. 

In summary, we obtain degrees of belief for one question (i.e.. Will John reach the bus in time ) from probabilities for another question (i.e.. Is the alarm clock reliable ). Dempster s rule begins with the assumption that the questions for which we have probabilities are independent with respect to our subjective probability judgments, but this independence is only a priori it disappears when conflict is discerned between the different items of evidence. 

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Dempster-Shafer Theory of Evidence is a generalization of the Bayesian theory based on degrees of belief rather than probabilities. 

A Dempster-Shafer theory of evidence approach to model uncertainty analysis... 

Irrespective of the taxonomy used, epistemic and aleatory uncertainty, or uncertainty and variation, alternatives to probability have been suggested for the representation of the epistemic concept. These include interval or imprecise probabihty (Coolen 2004, Coolen Utkin 2007, Utkin Coolen 2007, Weichselberger 2000), fuzzy set theory and the associated theory of possibility (Zadeh 1965, Zadeh 1978, Unwin 1986), and the theory of behef functions (Shafer 1976), also known as evidence theory or the Dempster-Shafer theory of evidence. 

Park, N.W. 2010. Application of Dempster-Shafer theory of evidence to GIS-based landsUde susceptibility analysis. Environmental Earth Sciences 62(2) 367-376. 

The theory of evidence was first generated by Dempster (1968) and further developed by Shafer (1976). It is often referred to as the Dempster-Shafer theory of evidence or D-S theory. The D-S theory was originally used for information aggregation in expert systems as an approximate reasoning tool (Mantaras, 1990). Subsequently, it has been used in decision making under uncertainty (Yager, 2004). 

Baraldi, P, Compare, M., Zio, E. 2013a. Maintenance policy performance assessment in presence of imprecision based on Dempster-Shafer Theory of evidence. Information Sciences 245 112-113. 

Smets P (1994) What is Dempster-Shafer s model Advances in the Dempster-Shafer theory of evidence. Wiley, New York, pp 5-34... 

Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Set Syst 1 3-28 Zadeh LA (1986) A simple view of the Dempster-Shafer theory of evidence and its implication for the rule of combination. AI Mag 7(2) 85-90... 

Jones, L, Beynon, MJ, Hoh, CA, Roy, S (2006) An application of the Dempster Shafer theory of evidence to the classification of knee function and detection of improvement due to total knee replacement surgery. Journal of Biomechanics, 39, pp. 2512-2520... 

Evidence theory also known as the Dempster-Shafer theory, has been first introduced by Dempster [5], then formalized by [25] and finally axiomized later into the framework of the Transferable Belief Model (Evidence theory) by Smets [28]. The Evidence theory can be understood as an alternative to probability theory for the representation of uncertainty It allows one to manipulate non-necessarily exclusive events and thus to represent explicitly information uncertainty... 

Another important advantage of Evidence Theory is that the Dempster-Shafer law of combination (the orthogonal sum) allows us to combine data from different independent sources. Thus, if we have the same frame of discernment for two mass functions which have been derived independently from different data, we may obtain a unified mass assignment. 

In this Section, the Dempster-Shafer Theory (DST) of Evidence (Shafer, 1976) is considered for the representation of the epistemic uncertainty affecting the expert knowledge of the probability P Mi) that the alternative model Mi, I = 1,..., be correct. In the DS framework, a lower and an upper bound are introduced for representing the uncertainty associated to P (Ml). The lower bound, called behef, Bel (Mi), represents the amount of belief that directly supports M at least in part, whereas the upper bound, called plausibility, Pl Mi), measures the fact that M could be the correct model up to that value because there is only so much evidence that contradicts it. 

Sentz, K. Ferson, S. 2002. Combination of Evidence in Dempster-Shafer Theory. Report of Sandia National Laboratories Technical. Report no. SAND2002-0835. Albuquerque, New Mexico. 

The reliability of data communication elements is an important constituent part on the quality of transferred information (IQ). It may be stated that IQ consists of accessibility, actual value, completeness, credibility, flexibility, form, meaning over time, relevance, reliability, selectivity, validity (Olaisen 1990, MITIQ). All the above components of IQ tend to be encumbered by some uncertainty. Whether associated with measurement or caused by the unreliability of equipment or software. Hence, the proposal to determine IQ using uncertainty modelling. There are several major methods of uncertainty modelling. The best known are Bayes networks based on conditional probability, the theory mathematical evidence based on the Dempster-Shafer theory, the method making use of the Certainty Factor (Clf) created by Buchanan, Shortliffe for the MYCIN expert system, the theory of fuzzy sets or the theory of rough sets. 

The theory of evidence assumes that it is possible to conduct a synthesis of information for particular elementary measures of probability. Information can be synthesised even if it is contradictory or comes from various sources (Dempster 1967, Shafer 1976). Such synthesis can be described by the following formula ... 

The theory of belief functions or the theory of evidence has been known as Dempster-Shafer theory [2], [3]. The interpretation of belief theory as generalization of probability has been published in the 1990s [4], [5]. Many real-world problems were solved using this universal formalism such as valuation-based systems for the oil... 

Bayesian approach and one applying Dempster-Shafer evidence theory to quantify uncertainty about model parameters. The precautionary view is only found once, in the early 2000s, and hence represents somewhat of an outlier view in the application area. There is relatively more work in line with the scientific proceduralism view on risk assessment, from the late 1990s onwards. The argument view, where expert judgment is taken as a core element of the risk description, represents a minority of the perspectives on risk assessment. 

In order to estimate the information quality of a selected system you should model the uncertainty the reverse of which could indicate IQ. Uncertainty can be modelled by using, among others, Bayesian networks. However, in this case, the data come from various sources and to simplify the modelling, and thereby calculation, could be used the theory mathematical evidence created by Dempster and Shafer. However, since we assess information from computer systems (modern detectors and... 

Zhang, R, Zhou, Y., Liao, T., Research on quantitative risk assessment model and failure probability of oil/gas pipeline based on dempster-shafer evidence and intuitionist fuzzy theory. Proceedings of the International Conference on Quality, Reliability, Risk, Maintenance, and Safety Engineering, 2013, pp. 145-150. 

The axioms in Eq. 23 are motivated by the rule for combining evidence in evidence theory, referred to as Dempster s rule of combination (Shafer 1976). This rule is formally introduced later in section Dempster s Rule of Combination. ... 

Within evidence theory, Dempster s rule of combination presents a tmique mathematical framework for combining evidence from distinct sources (Smets 1994). Dempster s rule of combination intuitively combines evidence so that the combination of two beliefs - both supporting A -yields more support than either piece of evidence alone (Shafer 1987). 

The D-S evidence theory, is developed by Dempster (Dempster 1967) in the I960 . Shafer developed the original theory by introducing the concept of belief function. Some basics of D-S evidence theory are given as follows. 

Formal quantitative evaluation of judgement requires means of elicitation that will provide numerical results in forms that will facilitate the composition of this evidence with that from other sources. We believe that Bayesian probability might be an appropriate mechanism here, but will also investigate other avenues such as possibility theory, Shafer-Dempster theory. 

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