Belief functions

These two independent sources of information can be described by the following belief functions ... 

In such situations where empty conjunctions of focal elements are present e.g., L3 n (Li U L2) = 0), a renormalization is carried over the complete total mass not assigned to the empty set. The normalized belief function of previous... 

Finally, the last step is the decision making process which is supported by the results provided by the combination rules. Indeed, as previously highlighted, the combination of the available sources of information provides us with a new belief function which represents the most reliable and complete information. However, if the choice of the most likely h3rpothesis is straightforward in the probabilistic framework, it can become quite complex in the Evidence theory. 

As it has been explained in section 2.2, only belief functions defined on a same frame of discernment can be combined. This implies that all the possible states of the system should be listed in f2. This can be long to write for... 

Applying this relation to the Cartesian product of all frames of discernment of each input i.e., Q = i i x 172 x. .. x if there are n sensors), it is possible to refine all available belief functions into a common reference set and then to combine them to produce a unique belief structure that, in addition, is able to indicate the conflict between information sources. 

The hrst choice does not change the structure of the belief assignment but it affects the value of the term d. On the contrary, the second one leads to the following belief function ... 

D. Dubois and H. Prade. Representation and combination of uncertainty with belief functions and possibility measures. Computational Intelligence, 4 244-264, 1998. 

E. Lefevre, O. Colot, and P. Vaimoorenberghe. Belief function combination and conflict management. Information Fusion, 3(2) 149-162, 2002. 

As the analysis progresses, evidence is accumulated supporting the presence or absence of defined substructures. The evidence is combined by the Reasoner module to form a belief function, which describes the degree to which each substructure is currently believed. This information is stored in the chemical database, where it is available to the Expert modules and to the Controller as it decides the course of the analysis. As the belief function evolves, the current state is displayed graphically to the user, who may halt the analysis, query the current state, and redirect the course of the analysis by supplying evidence for or against a substructure. 

The only satisfactory description of uncertainty is probability. By this I mean that every uncertainty statement must be in the form of a probability that several uncertainties must be combined using the rules of probability and that the calculus of probabilities is adequate to handle all situations involving uncertainty. Probability is the only sensible description of uncertainty and is adequate for all problems involving uncertainty. All other methods are inadequate... that can be done with fuzzy logic, belief functions, upper and lower probabilities, or any other alternative to probability, can better be done with probability. 

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. 

Shafer, G., Perspectives on the Theory and Practice of Belief Functions, International Journal of Approximate Reasoning, 3, 1, 1990. 

A model was proposed in Magnani et al. [2005] for combining semantic and probabilistic attribute correspondences using constructs of uncertain semantic relationships in an ER model. An uncertain semantic relationship is a distribution of beliefs over the set of all possible semantic relationships, using belief functions [Shafer 1976], The set of possible semantic relations serve as the frame of discernment (marked ), based on which two functions are defined, namely belief and plausability. Both functions assign a value to a subset of the frame of discernment, starting from the basic probability mass that is assigned with each element in the frame of discernment. Belief of a set A e sums the probability mass of all subsets B c A. Plausability of a set A is the sum of all subsets that intersect with A, i.e., all B such that A n B 0. 

Shafer, G. Perspectives on the theory and practice of belief functions. Int. J. Approx. Reasoning... 

Strut, T. M. (1994), Decision Analysis Using Belief Functions, in Decision Making under Dempster-Shafer Uncertainties, M. Fedrizzi, J. Kacprzyk, R. R. Yager, Eds., John Wiley Sons, New York. 

In reality (e.g., the assessment of safety cases), decision problems tend be structured certain factors may feed-in to each other, and can form more complex, hierarchical belief structures. ER [2] is an extension of DS-theory that enables the aggregation of belief functions, where the factors are arranged in a hierarchical structure. The root-node represents the final decision one wishes to make. Branch nodes represent contributory factors. Branches can be given different weights, indicating the extent to which they contribute to the overall decision. Leaf-nodes represent points at which one can present ones own belief functions. ER then provides the mathematically sound basis by which to combine the belief-functions provided in the leaf-nodes, and to propagate them up to the root. 

This RQ aims to understand the expert s decision-making process. It attempts to identify the various factors and criteria for individual evidence types that influence the confidence of the expert. The key challenge here would be to identify through systematic examination the specific questions to establish the underlying belief functions in ER. Different evidence types are likely to have specific factors that influence the expert s confidence and these needs to be identified. An initial attempt to answer this RQ was through interviews with experts [10]. 

In general, the considerations reported for the statistical approach in sub-Section 3.3 hold also in this case. However, attention must be paid to the fact that the statistical approach considers probability distributions, whereas the DSTE approach refers to Plausibility and Belief functions, and Pignistic probability, as discussed in the following Sections. 

Therefore, for every degradation state the output distributions provided by MUSTADEPT as specific solutions at time t, bracket the unknown probability of occupying at time / corresponding to the true value of 6. Generally speaking, the closer the Plausibility and Belief functions, the smaller the uncertainty in the solution, which however, heavily depends on the width of the intervals provided by the experts. 

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. 

Belieffunction and plausibility function The analysis results are demonstrated by an interval which describes the degree of support for a hypothesis. The lower bound is called belief function, denoted by... 

The value of the function m (X) refers only to the set X, and does not give any information about the subsets of X, for which individual belief functions are defined ... 

In case of information conflict, i.e. the product of belief functions is an empty set, the conflict indicator Kf, is defined ... 

The highest values of the resulting belief function mE(l-2)(rn), the combination of two experts hypotheses indicates the level of failure risk analysis of water supply network. Measures of beliefs belE(l-2)(rn) and the presumption plE(l-2)(rn) characterized by the so-called interval of uncertainty as to the hypotheses interpretation. The uncertainty factor AF is the difference between belief and presumption in the case of hypothesis. 

DSTE with different descriptions of epistemic variables 95% quantile of Pp are evaluated on the pignistic function and on the belief function (pessimistic assumption, between brackets in Table 2). 

DSTE approach gives results less penalizing even if the 95% quantile is taken on the belief function and not on the pignistic function. 

Epistemic variables are modeled with 95% intervals and medians known. The results obtained on the epistemic uncertainty of the quantile MF g are shown in Eigure 7. The risk indicator proposed MF g g) is equal to 1.19 on the plausibility function, 1.22 on the pignistic function and 1.26 on the belief function. 

The combination rule affects the belief function and it can presented in such a way ... 

Belief measures (also conmiOTly referred to as belief functions) aim to generalize the well-known interpretation of subjective probability theory - i.e., the Bayesian probability for a subjective measure of uncertainty - to a broader concept of evidence. Like possibility and necessity measures, evidence theory is developed from dual measures of belief and plausibility (KJir and Yuan 1995). These measures express beliefs or judgments formulated by available evidence (Yager and Liu 2008). Although Dempster s original woiks were closely linked to classical probability theory, there are some significant distinctions between classical probability theory and evidence theory. 

Belief functions are based on the following axioms (Klir 2006) ... 

If there is limited evidence available for A and no evidence available for A, then the amount of ignorance is large (Ross 2010). Total ignorance, the special case where Bel(A) = 0 and P1(A) = 1, is characterized by evidence that neither supports nor refutes A (Shafer 1987). Vacuous belief functions are those belief functions defined by complete ignorance (Shafer 1976). Conversely, in the special case of complete evidence (zero ignorance) such that the sum of the belief and the complement of the belief equal unity, the belief measure is a probability measure. 

Special Cases of Evidence Theory At a fundamental level, belief functions are expressions of subjective degrees of belief (Shafer 1976). In this regard, subjective probabilities can be viewed as a special case of belief measures defined on singletons. When aU available evidence is defined only on individual elements, belief and plausibility measures become probability measures (proof in Klir 2006). The evidence may therefore be described probabilistically (Ross 2010). Shafer (1976) refers to this special class of belief functions as Bayesian Belief Functions. 

Princeton University Press, Princeton Shafer G (1987) Belief functions and possibility measures. In Bezdek JC (ed) Analysis of fuzzy information, vol 1, Mathematics and logic. CRC Press, Boca Raton Shafer G (2012) A mathematical theory of evidence. Glenn Shafer, http //www.glennshafer.com/books/ amte.html. 7 Nov 2012... 

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... 

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