Applications of belief functions in business decisions: A review <s> INTRODUCTION <s> In uncertainty situations where knowledge is described by a Dempster-Shafer belief function (which is more general than a probability measure), von Neumann-Morgenstern linear utility theory applies and leads to a generalized expected utility representation of preference which allows for risk-attitude and ambiguity-attitude (pessimism/optimism). <s> BIB001 </s> Applications of belief functions in business decisions: A review <s> INTRODUCTION <s> Abstract A primary motivation for reasoning under uncertainty is to derive decisions in the face of inconclusive evidence. Shafer's theory of belief functions, which explicity represents the underconstrained nature of many reasoning problems, lacks a formal procedure for making decisions. Clearly, when sufficient information is not available, no theory can prescribe actions without making additional assumptions. Faced with this situation, some assumption must be made if a clearly superior choice is to emerge. This paper offers a probabilistic interpretation of a simple assumption that disambiguates decision problems represented with belief functions. It is proved that it yields expected values identical to those obtained by a probabilistic analysis that makes the same assumption. A strict separation is maintained between evidence that carries information about a situation and assumptions that may be made for disambiguation of choices. In addition, it is shown how the decision analysis methodology frequently employed in probabilistic reasoning can be extended for use with belief functions. This generalization of decision analysis allows the use of belief functions within the familiar framework of decision trees. <s> BIB002 </s> Applications of belief functions in business decisions: A review <s> INTRODUCTION <s> Today, different formalisms exist to solve reasoning problems under uncertainty. For most of the known formalisms, corresponding computer implementations are available. The problem is that each of the existing systems has its own user interface and an individual language to model the knowledge and the queries. <s> BIB003 </s> Applications of belief functions in business decisions: A review <s> INTRODUCTION <s> Foundations.- to Belief Functions.- Decision Making in a Context where Uncertainty is Represented by Belief Functions.- Empirical Models for the Dempster-Shafer-Theory.- Systems and Auditing Applications.- The Descriptive Ability of Models of Audit Risk.- The Effectiveness and Efficiency of Belief Based Audit Procedures.- Auditors' Evaluations of Uncertain Audit Evidence: Belief Functions versus Probabilities.- Conflict, Consistency and Consonance in Belief Functions: Coherence and Integrity of Belief Systems.- Operations Management, Finance and Economics Applications.- Evaluating Mergers and Acquisitions: A Belief Function Approach.- Possibilistic Belief Network Constructed by Operators of Composition and its Application to Financial Analysis.- Using Belief Functions to Forecast Demand for Mobile Satellite Services.- Modeling Financial Portfolios Using Belief Functions.- Futures Hedging under Prospect Utility and Knightian Uncertainty. <s> BIB004 </s> Applications of belief functions in business decisions: A review <s> INTRODUCTION <s> This paper redefines the concept of sampling from a population with a given parametric form, and thus leads up to some proposed alternatives to the existing Bayesian and fiducial arguments for deriving posterior distributions. Section 2 spells out the basic assumptions of the suggested class of sampling models, and Sect. 3 suggests a mode of inference appropriate to the sampling models adopted. A novel property of these inferences is that they generally assign upper and lower probabilities to events concerning unknowns rather than precise probabilities as given by Bayesian or fiducial arguments. Sections 4 and 5 present details of the new arguments for binomial sampling with a continuous parameter p and for general multinomial sampling with a finite number of contemplated hypotheses. Among the concluding remarks, it is pointed out that the methods of Sect. 5 include as limiting cases situations with discrete or continuous observables and continuously ranging parameters. <s> BIB005 </s> Applications of belief functions in business decisions: A review <s> INTRODUCTION <s> A multivalued mapping from a space X to a space S carries a probability measure defined over subsets of X into a system of upper and lower probabilities over subsets of S. Some basic properties of such systems are explored in Sects. 1 and 2. Other approaches to upper and lower probabilities are possible and some of these are related to the present approach in Sect. 3. A distinctive feature of the present approach is a rule for conditioning, or more generally, a rule for combining sources of information, as discussed in Sects. 4 and 5. Finally, the context in statistical inference from which the present theory arose is sketched briefly in Sect. 6. <s> BIB006 </s> Applications of belief functions in business decisions: A review <s> INTRODUCTION <s> We present PULCinella and its use in comparing uncertainty theories. PULCinella is a general tool for Propagating Uncertainty based on the Local Computation technique of Shafer and Shenoy. It may be specialized to different uncertainty theories: at the moment, Pulcinella can propagate probabilities, belief functions, Boolean values, and possibilities. Moreover, Pulcinella allows the user to easily define his own specializations. To illustrate Pulcinella, we analyze two examples by using each of the four theories above. In the first one, we mainly focus on intrinsic differences between theories. In the second one, we take a knowledge engineer viewpoint, and check the adequacy of each theory to a given problem. <s> BIB007 </s> Applications of belief functions in business decisions: A review <s> INTRODUCTION <s> Valuation-based system (VBS) provides a general framework for representing knowledge and drawing inferences under uncertainty. Recent studies have shown that the semantics of VBS can represent and solve Bayesian decision problems (Shenoy, 1991a). The purpose of this paper is to propose a decision calculus for Dempster-Shafer (D-S) theory in the framework of VBS. The proposed calculus uses a weighting factor whose role is similar to the probabilistic interpretation of an assumption that disambiguates decision problems represented with belief functions (Start 1990). It will be shown that with the presented calculus, if the decision problems are represented in the valuation network properly, we can solve the problems by using fusion algorithm (Shenoy 1991a). It will also be shown the presented decision calculus can be reduced to the calculus for Bayesian probability theory when probabilities, instead of belief functions, are given. <s> BIB008
The notion of belief functions can be traced to its ancient root, the work of Jakob Bernoulli on pooling pure evidence. In modern language, an item of pure evidence proves a statement with a certain probability but has no opinions on its negation. Probabilities in accordance with pure evidence are non-additive. Bernoulli's idea of non-additive probabilities has now been well developed by BIB005 BIB006 ,  , and many others, under the name of DempsterShafer theory (DST) of belief functions. In recent years, there has been a significant surge in applications of belief functions to business decisions ranging from audit decisions and managing stock portfolios to forecasting demand for mobile satellite services (see, e.g., BIB004 . One can attribute the reasons for this surge to (1) the ease with which belief functions map uncertainties present in the real world, (2) the theoretical developments in local computations for propagating belief functions in a network (see. e.g., Shafer 1986, 1990) , and (3) availability of friendly software incorporating the local computations such as 'Auditor's Assistant' by  , 'Pulcinella' by BIB007 , and "ABEL' by BIB003 . In this paper, we focus on the review of recent applications in auditing and other business decision problems and emphasize their techniques of mapping real-world uncertainties using belief functions. Our choice of the review scope answers a renewed call for more applications of DST to real business and engineering problems  . It also reflects our basic position that, when putting belief functions into practice, the most challenging task is how to encode evidence using the belief function formalism, e.g., focal elements and belief mass assignments (see  . We realize that our choice will inevitably exclude many related contributions from the review, notably the ones on the methods of embedding belief functions into the normative decision model BIB001 BIB002  ) and on the implementation of belief functions in decision support systems BIB008  . We organize this review using two threads. First, we review the applications in two broad application domains, namely, audit decisions and other business decisions. Second, within each domain, we review the existing studies by focusing on how they encode evidence and which features of DST they take advantage of. An outline of this paper is as follows. Section 2 shows how belief function can better represent uncertainties than probabilities in auditing and business decisions. Section 3 presents the audit process and how belief functions are applied to audit decisions including the work done in audit decisions under the risk of fraud. Section 4 discusses the other business applications of belief functions. Section 5 shows how the notion of linear belief functions, a recent extension to DST, is applied in auditing and other business decisions. Finally, Section 6 presents a summary and conclusion.