Applications of belief functions in business decisions: A review <s> Evidential Reasoning with Versatile Evidence <s> The framework of evidence theory is used to represent uncertainty pervading a set of statements which refer to subsets of a universe. Grades of credibility and plausibility attached to statements specify a class of bodies of evidence. Using newly appeared measures of specificity, a principle is stated in order to select, among these bodies of evidence, the one which suitably represents the available information in the least arbitrary way. It is shown that this principle, which is similar to the maximum entropy principle, leads to a deductive reasoning approach under uncertainty, and also provides a rule of combination which does not presuppose any independence assumption. Particularly, it is more general than Dempster's. <s> BIB001 </s> Applications of belief functions in business decisions: A review <s> Evidential Reasoning with Versatile Evidence <s> In this article, we present the belief-function approach to aggregating audit evidence. The approach uses an evidential network to represent the structure of audit evidence. In turn, it allows us to treat all types of dependencies and relationships among accounts and items of evidence, and thus the approach should help the auditor conduct an efficient and effective audit. Aggregation of evidence is equivalent to propagation of beliefs in an evidential network. The paper describes in detail the three major steps involved in the propagation process. The first step deals with drawing the evidential network representing the connections among variables and items of evidence, based on the experience and judgment of the auditor. We then use the evidential network to determine the clusters of variables over which we have belief functions. The second step deals with constructing a Markov tree from the clusters of variables determined in step one. The third step deals with the propagation of belief functions in the Markov tree. We use a moderately complex example to illustrate the details of the aggregation process. <s> BIB002 </s> Applications of belief functions in business decisions: A review <s> Evidential Reasoning with Versatile Evidence <s> A magnetic head flying controlling device in a hard disk drive including a swing arm which is pivoted by a voice coil motor, a suspension combined with the swing arm for supporting a magnetic head, a first sensor installed on the body of the suspension, for detecting a vertical displacement of the suspension as a signal, a second sensor installed on the body of the suspension, for detecting a horizontal displacement of the suspension as a signal, a control circuit portion for halting the recording or reproduction of the magnetic head based on signals output from the first and second sensors while at the same time storing the halted position of the magnetic head, and generating a control signal for controlling the operation of the magnetic head, and a magnetic head flying actuator for controlling the driving of the magnetic head based on the control signal. <s> BIB003
Some audit evidence pertains to only one variable but some pertains to more than one variable. For example, confirmations of accounts receivable from customers pertain to two audit objectives: 'existence' and 'valuation'. Consequently, the evidential diagram becomes a network BIB003  . In contrast,  assumed a tree type structure in order to develop the analytical audit risk model based on plausibility values at the three levels of the financial statement. If we can encode all the audit evidence in terms of belief functions, combining various items of evidence, whether in a tree or network structure, becomes essentially a problem of propagating beliefs in a network (see, e.g.,   BIB002 .  have developed propositions that make it easier to aggregate audit evidence in an "and" tree. However, a challenge with a network is how we can encode versatile evidence that pertains to more than one variable. As mentioned earlier, confirmations of accounts receivable may provide support to both 'Existence' and 'Valuation' objectives of accounts receivable but at different levels. The initial judgment can be expressed in terms of m-values on individual objectives. However, since the evidence is coming from a single source, these m-values need to be expressed in terms of the variables on the joint space in order to (1) combine all the evidence in the network, and (2) preserve the interdependencies among the evidence. In general, the conversion process can become quite complex depending on the number of variables involved and the nature of the individual support whether it is positive or negative (see, e.g., BIB001 BIB001 ). Srivastava and Cogger developed a heuristic approach to convert such individual beliefs into beliefs on the joint space (see, e.g., Srivastava 1997b). Srivastava and his co-researchers use this approach for evidential reasoning in auditing along with a computer program 'Auditor's Assistant' developed by  . Let us consider a simple illustration of the above issue where one item of evidence supports multiple objectives at different levels. Suppose the auditor has mailed 100 confirmations to the client's customers asking them to respond whether they owe the stated amount to the company. Consider that the auditor has received all the confirmations stating that they owed the company but with disputes in some cases regarding the amount owed. Based on this evidence, the auditor thinks he or she has, say, 0.8 level of support that 'Existence' objective is met, and a 