Applications of belief functions in business decisions: A review <s> SUMMARY AND CONCLUSION <s> The auditing profession: the auditing profession audit report professional ethics legal liability. The auditing process: audit responsibilities and objectives audit evidence audit planning and documentation materiality and risk the study of internal control and assessment of control risk overall audit plan and audit program audit of the sales and collection cycle audit sampling for tests of transactions completing the tests in the sales and collection cycle - accounts receivable audit sampling for tests of details of balances the effect of complex EDP systems on auditing. Application of the auditing process to other cycles: audit of the payroll and personal cycle audit of the acquisition and payment cycle: tests of controls substantive tests of transactions and accounts payable completing the tests in the acquisitions and payments cycle: verification of selected accounts audit of the inventory and warehousing cycle audit of the capital acquisition and repayment cycle audit of cash balances. Completing the audit and offering other services: completing the audit other audit attestation services and compilation engagements internal and governmental financial auditing and operational auditing. <s> BIB001 </s> Applications of belief functions in business decisions: A review <s> SUMMARY AND CONCLUSION <s> The main goal of this paper is to demonstrate how the theory of belief functions [Dempster 1967, Shafer 1976, Kohlas and Monney 1995, Smets 1998] can be used to model financial portfolios. In particular, we are interested in modeling how a portfolio return distribution changes as we learn new information about the different factors that impact the portfolio. <s> BIB002
In this paper, we walked through the recent applications of Dempster-Shafer theory (DST) of belief functions in auditing and other business decision-making problems. We introduced how a recent addition to the family of DST-the theory of linear belief functions, could provide opportunities for a wide range of real applications. We showed how DST could better model real-world uncertainties than its competing approach-Bayesian theory of probabilities. From both classic and real-world examples, we learned that the advantage of DST in representing ignorance and partial ignorance over its competing theory is its ability to distinct lack of belief from disbelief and to allow people to withhold their belief from a statement without according that belief to the negation of the statement. From the review of existing empirical evidence, we learned that, descriptively, people tend to make judgments about uncertainties in terms of belief functions. Following the audit process defined by BIB001 , we described three types of audit evidence, namely hierarchical evidence, versatile evidence, and statistical evidence, that have to be collected and assembled to make a judgment whether a financial statement is fairly presented. Hierarchical evidence arises due to SAS 31 that stipulates auditing standards at three hierarchical levels: the financial statement level, the account level, and the objective level. Versatile evidence arises when a piece of information implies two or more belief mass allocations, usually, on several different variables. Statistical evidence is prevalent in auditing because, according to the audit risk model, a planned level of detection risk has to be attained through substantive audit procedures including statistical tests. We reviewed the applications of DST in auditing around how they handled the three types of evidence. We introduced with care the work of  on convert-ing audit risks into m-values through their re-interpretation as plausibilities and on attaching the audit risk model to all three levels of an audit problem. We showed how they represented hierarchical evidence using a tree-structured graphical model. We illustrated the Srivastava and Cogger approach to extending marginal belief functions into joint ones and how existing applications have applied it to coping with versatile evidence. We reviewed the applications that convert statistical evidence in variable and attribute sampling to evidence in belief functions. Finally, we looked at an extended framework of audit planning that considered explicitly the risk of fraud due to the inception of SAS 82. We presented an evidential network with hierarchical evidence, which propagates not only through "AND" gates but also "OR" gates. For the applications in other non-auditing areas, we grouped them into two loose categories: judgment and decision-making under ambiguity, and business model combination. We showed how the proponents of DST looked at the Ellsberg paradox and how empirical evidence was in support of their perspective. Then we briefly reviewed the remaining applications under the theme of combining business models. These applications are in miscellaneous areas including WebTrust, information quality, business mergers and acquisitions, mobile satellite service, and portfolio modeling, etc. Finally, we reviewed the applications of linear belief functions (LBF)-a new extension to DST, in statistical modeling and information systems management. Since the theory of LBFs is relatively new, we introduced its conception originally envisioned by  . We elaborated the idea of the matrix representation by  and presented an elementary approach to encoding the most common types of evidence using an audit example. We described the rationale of how the notion of LBFs was used in the existing applications. In this paper, although we reviewed the applications of DST in many business areas, we singled out those in audit decisions and thoroughly examined them at a level of technical details. Our emphasis on audit decisions is partially due to the abundance of such applications. It is also partially due to the fact that the technical challenge in mapping audit evidence also challenges other business applications. By synthesizing the applications in both audit and non-audit decisions, we found that the core of mapping real-world uncertainties lies at encoding hierarchical evidence, versatile evidence, and statistical evidence, into belief functions. The challenge in dealing with hierarchical evidence lies at how we combine items of evidence that are present at different levels of a hierarchy, especially when the items of evidence are connected through logi-cal 'AND' and 'OR' gates. The challenge of handling versatile evidence lies at how to convert marginal belief functions into ones in a joint space while preserving the interdependencies among the relevant variables. The challenge of handling statistical evidence is at how to convert sampling data into belief functions so that they can be combined with subjective opinions. In addition to the applications in auditing, we also reviewed the conception and applications of LBFs at a level of technical details. Our emphasis here is to demonstrate how this new addition to DST can give impetus to a wider range of applications of DST in business decisions. We showed that, like finite belief functions, the theory of LBFs possesses the usual advantages of the belief function modeling-the representation of ignorance by vacuous belief functions, the resolution of complex representations of uncertainty into components by graphical models, and the combination of independent models by Dempster's rule. As  argued, the notion of LBFs generalizes Bayesian inference of posterior distributions while abandoning its most controversial component: improper priors. It extends, unifies, and clarifies Fisher's fiducial method of posterior reasoning while filling the void of a prior distribution in the logical structure with a vacuous belief function. Its join-tree computational architecture (see Liu 1999) extends a parallel concept for probabilistic models  , but eliminates the requirement of conceiving and imposing directional structure on the graph whereby earlier components of uncertain knowledge are judged asymmetrically prior to later components that are described as conditional likelihood. The specification of graphical belief function models is based on symmetric evidential independence assumptions that are simpler and more directly susceptible to empirical checking than are typical Bayesian conditional probability assumptions that are assumed to hold given arbitrary choices of often long lists of unknown parameter values specifying families of both distributions and conditions. Also, the conception of an LBF treats all the components of a statistical model such as observations, model assumptions, and subjective beliefs, not as separate concepts, but as manifestations of a single concept. It allows people to concentrate modeling efforts on recognizing and incorporating independent components of real information that can be combined into a graphical model. Linear equations that link random variables are prevalent in many scientific and applied fields. The conception of an LBF allows one to capture them directly as an LBF. This property of an LBF alone can generate tremendous opportunities for its applications in the fields. For example, as BIB002 showed, the return on an individual security is linearly de-termined by factors such as stock market, gold price, firm-specific effects, and a residue. Furthermore, the portfolio return is the weighted average of the returns on individual securities. Both relationships are linear equations and therefore can be captured as LBFs. We can encode expert opinions and historical data on the factors that affect individual securities as additional LBFs. Then the whole evidential reasoning problem of portfolio prediction described in BIB002 becomes the one of combining those LBFs.