Applications of belief functions in business decisions: A review <s> MAPPING UNCERTAINTIES <s> A description of the transferable belief model, which is used to quantify degrees of belief based on belief functions, is given. The impact of open- and closed-world assumption on conditioning is discussed. The nature of the frame of discernment on which a degree of belief will be established is discussed. A set of axioms justifying Dempster's rule for the combination of belief functions induced by two distinct evidences is presented. > <s> BIB001 </s> Applications of belief functions in business decisions: A review <s> MAPPING UNCERTAINTIES <s> Recently, Shafer and Srivastava [1], Srivastava and Shafer [2], Srivastava [3]–[4], and Van den Acker [5] have identified appealing features of belief function evidential networks. These networks can express the support that audit evidence provides for assertions, accounts and financial statements. These networks can also aggregate many pieces of evidence into an overall level of support for a particular assertion, account or an entire set of financial statements. <s> BIB002
The essence of the belief function formalism is to encode evidence in terms of its evidential support for propositions involving uncertainty and to represent the degree of the support through the belief mass assignment (BMA)  BIB001   . Let X be a set of finite discrete variables, Ω be its frame of discernment  , and A be a subset of Ω. We interpret A to be the statement that the true value of X is in A. We represent the degree of eviden-  Semantically, m(A) represents a portion of our total belief mass allocated to A in accordance with a piece of evidence. It expresses the belief that one commits exactly to A. To obtain the total belief committed to A including that to any subset of A, one may use a belief function Bel(A) and a plausibility function Pl(A) as follows: Here, Bel(A) expresses the total belief that one commits to A. On the other hand, Pl(A) expresses the total belief that one does not commit to the negation of A since A statement is called a focal element if there exists a piece of evidence in support of a positive BMA to it, i.e., m(A) > 0. Note that, if there is lack of evidence justifying a more specific allocation, a portion of our total belief allocated to a focal element A does not necessitate the allocation of any partial belief to its subsets. In other words, a positive m(A) does not automatically imply a finer re-allocation of it into any subsets of A. This property of the belief mass assignments makes the notion of belief function distinct from that of probabilities. As a matter of fact, if all the focal elements are singletons, the belief function is reduced to a Bayesian probability function. Consequently, one may think of Bayesian theory of probabilities as a special case of DST  . A most important advantage of the belief function modeling is its ability to represent ignorance and partial ignorance. In the extreme case, if one is completely ignorant about variable X, DST represents the ignorance by assigning the whole belief mass to Ω, i.e., using a vacuous belief function. In contrast, Bayesian theory would represent it as a uniform distribution, which Shafter (1976) considers to mix lack of belief with disbelief. In essence, Bayesian theory does not allow one to withhold belief from a statement without according that belief to the negation of the statement. To demonstrate the advantage of DST in representing partial ignorance, let us consider the classic example of non-additive probabilities  . Suppose I find a scrap of newspaper predicting a blizzard (b), which I regard as infallible. Also, suppose I am 75% certain that the newspaper is today's. Then, I am 75% sure of a upcoming blizzard, i.e., m(b) = 0.75. However, if the newspaper is not today's, anything can happen since the newspaper carries no information on future weather. Accordingly, we allocate the remaining belief mass 25% to both "blizzard" and "no blizzard" as a whole, i.e., m({b, ~b}) = 0.25. Note that, in this case, it is insensible to divide the belief mass 0.25 and assign partial beliefs to b and ~b because there exists no evidence in support of such a re-allocation. Therefore, different from Bayesian theory, DST assigns no positive belief to ~b. Now let us consider an auditing example to illustrate how well belief functions map uncertainties faced in the real world. Suppose that the auditor is conducting an audit of the financial statements of a company and has gathered evidence to support that the management has integrity and honesty, and also recalls that there has been no errors or irregularities in the financial statements in prior years. This piece of information, in the auditor's judgment, provides a positive but a low level of support that the financial statements are fairly presented. Let us say, on a scale of 0-1, this low level of support is 0.2, i.e., m(f) = 0.2, where f represents the state that the financial statements are fairly presented. However, based on the management integrity, honesty, and prior experience, the auditor thinks that there is no evidence in support that the financial statements are materially misstated. That is, the support that the financial statements are materially misstated is zero, i.e., m(~f) = 0, where ~f represents the negation of f. Therefore, the remaining belief mass 0.8 is best allocated to f and ~f as a whole, i.e., m({f, ~f}) = 0.8. It is impossible to represent such an item of evidence in the probability framework, because by doing so, one would have to assign a positive portion of the belief mass to ~f, i.e., m(~f) > 0, which contradicts the fact that there is no evidence in support of ~f.  and  argue that belief functions provide a more flexible and adaptable way to encode evidence from a variety of sources. One aspect of this flexibility is that the belief-function framework reduces to probability framework under a special condition (Shafer and Srivastava 1993) . Moreover, there is empirical evidence suggesting that the belief-function framework is a better representation for uncertainties present in the real world than probabilities. Recently, BIB002 , see also, Harrison 1999 ) conducted a study dealing with auditors' assessment of the strength of audit evidence and found that a statistically significant percentage of the auditors' responses could be modeled only using belief functions and not using probabilities (see also, Curley and Golden 1994).