Applications of belief functions in business decisions: A review <s> APPLICATIONS OF LINEAR BELIEF FUNCTIONS <s> Nature of Bayesian Inference Standard Normal Theory Inference Problems Bayesian Assessment of Assumptions: Effect of Non-Normality on Inferences About a Population Mean with Generalizations Bayesian Assessment of Assumptions: Comparison of Variances Random Effect Models Analysis of Cross Classification Designs Inference About Means with Information from More than One Source: One-Way Classification and Block Designs Some Aspects of Multivariate Analysis Estimation of Common Regression Coefficients Transformation of Data Tables References Indexes. <s> BIB001 </s> Applications of belief functions in business decisions: A review <s> APPLICATIONS OF LINEAR BELIEF FUNCTIONS <s> PART I: INTRODUCTION TO META-ANALYSIS 1. Integrating Research Finding Across Studies 2. Study Artifacts and Their Impact on Study Outcomes PART II: META-ANALYSIS OF CORRELATIONS 3. Meta-Analysis of Correlations Corrected Individually for Artifacts 4. Meta-Analysis of Correlations Using Artifact Distributions 5. Technical Questions In Meta-Analysis of Correlations PART III: META-ANALYSIS OF EXPERIMENTAL EFFECTS AND OTHER DICHOTOMOUS COMPARISONS 6. Treatment Effects: Experimental Artifacts and Their Impact 7. Meta-Analysis Methods for d Values 8. Technical Questions in Meta-Analysis of d Values PART IV: GENERAL ISSUES IN META-ANALYSIS 9. General Technical Issues in Meta-Analysis 10. Cumulation of Findings Within Studies 11. Different Meta-Analysis Methods and Related Software 12. Locating, Evaluating, Selecting, and Coding Studies and Presentation of Meta-Analysis Results 13. Availability Bias, Source Bias, and Publication Bias in Meta-Analysis 14. Summary of Psychometric Meta-Analysis <s> BIB002
The notion of linear (or normal) belief functions (LBF) extends DST to the case when variables of interest are continuous. As we introduced before, a belief function in general is defined by a BMA over a class of focal elements, which may have nonempty intersections. An LBF is a special type of belief functions in the sense that its focal elements are exclusive, parallel subhyperplanes over a hyperplane and its BMA is a normal distribution across the sub-hyperplanes. An LBF intends to represent our belief regarding the location of the true value as follows: We are certain that the truth is on a so-called certainty hyperplane but we do not know its exact location; along some dimensions of the certainty hyperplane, we believe the true value could be anywhere from -∞ to +∞ and the probability of being at a particular location is described by a normal distribution; along other dimensions, our knowledge is vacuous, i.e., we believe the location of the true value is somewhere from -∞ to +∞ but the associated probability is unknown. An LBF can represent both logical and probabilistic knowledge for three types of variables: deterministic such as an observable or controllable, random whose distribution is normal, and vacuous on which no knowledge bears. Logical knowledge is represented by linear equations, or geometrically, a certainty hyperplane. Probabilistic knowledge is represented by a normal distribution across all parallel focal elements. We may use an audit problem to illustrate the three types of variables as follows. Suppose we want to audit the ending balance of accounts receivable (E). As we saw earlier, E is equal to the beginning balance (B) plus the sales (S) for the period minus the cash receipts (C) on the sales plus a residual (R) that represents insignificant sales returns and cash discounts. Thus, we can represent the logical relation as a linear equation: Furthermore, if the auditor believes E and B are 100 on the average with a standard deviation 5 thousand dollars and the covariance 15, we can represent the belief as a multivariate normal distribution. If historical data indicate that the residual R is zero on the average with a standard deviation of 0.5 thousand dollars, we can summarize the historical data by normal distribution R ~ N(0, 0.5 2 ). If there is a direct observation on cash receipts, we can represent the evidence as an equation say, C = 50 (thousand dollars). If the auditor knows nothing about the beginning balance of accounts receivable, we can represent his or her ignorance by a vacuous LBF. Finally, if historical data suggests that, given cash receipts C, the sales S is on the average 8C + 4 and has a standard deviation 4 thousand dollars, we can represent the knowledge as a linear regression model S ~ N(4 + 8C, 16).  claims that an LBF in its general form is the combination of its degenerate components such as a linear equation, a vacuous belief function, etc. Therefore, he pro-poses representing an LBF indirectly by representing each of its degenerate components as a moment matrix. However, due to the many unproved claims,  was never fully understood (see  . Based on the geometrical description by  ,  and  propose two mathematical representations of an LBF: a widesense inner product and a linear functional in the variable space, and as their duals over a hyperplane in the frame of discernment. Monney (2002a) proposes a still another structure called Gaussian hints. By relating sweeping operations to multivariate statistics,  further elaborates on  and clarifies the moment matrix representation. In this approach, to encode a piece of evidence using an LBF, the key is to represent the mean and covariance, if exist, of variables into a moment matrix. For example, we can use the matrices M(R) and M(C) to represent distribution R ~ N(0, 0.5) and direct observation C = 50 respectively as follows by noting that the variance of C is zero: On the other hand, if one is ignorant about a variable, its mean and variance do not exist. Then we need to represent the evidence by a swept matrix. For a one-variable matrix like M(R), a sweeping involves a negative inverse on the variance element and a multiplication of the inverse with the mean element. For example, the sweeping of M(R) is performed as follows:  Of course, M(C) cannot be swept because the variance is zero. However, we can temporarily imagine the variance as a small number ε in order to do the sweeping. Eventually, we can let ε → 0 after it is combined with other LBFs. Therefore, M(C) can be swept as follows: With the notion of sweeping, a vacuous belief function is represented as a swept matrix with all elements being set zero. For example, if the auditor is completely ignorant about B, we can represent the evidence as follows: Note that variables B and C are semantically at two extremes; C means complete certainty whereas B means complete ignorance. Mathematically, we can see that Equations (8) and (9) are also opposite in the sense that (9) is the result of (8) by letting ε → ∞. In other words, we can imagine complete ignorance as the limiting case when the variance increases to ∞. When an LBF involves multiple variables, we will use their means and covariance matrix, if they exist, to form a moment matrix. For example, we can represent the auditor's belief on E and B as the following multivariate moment matrix: If both means and covariance matrix do not exit as in the case of complete ignorance, we can represent it by a fully swept moment matrix with all elements being zero like Equation (9). However, in the case we have partial ignorance, i.e., some but not all variables have means, variances, and covariance, we need to represent it as a partially swept matrix 3 . Note that, for a multivariate moment matrix, a full sweeping is performed through a se-  In the special case when S = 8C + 4, i.e., S is completely determined given C through a linear equation, the conditional variance of S will be zero in its partial swept matrix: Thus, a linear equation is considered as a special case of a linear regression model. Therefore, to represent Equation (5), we use the following partially swept matrix: By adapting Dempster's rule to the continuous case,  derives a rule of combination in a continuous frame of discernment and proves its equivalence to its geometric description in  .  further showed that both combination and marginalization of LBFs could be done using the swept matrix representations; the combination corresponds to the sum of fully swept matrices whereas marginalization corresponds to restricting a non-swept matrix into a subset of variables. For example, to combine M(E, B) in Equation (10) with the vacuous LBF on B, we can simply add the corresponding elements in ) , ( B E M of Equation (11) and ) (B M of Equation (9). Of course, the resulting matrix is the same as ) , ( B E M since ) (B M = 0. This result makes sense because the complete ignorance does not contribute to our knowledge about the variables of interest. For another example, to combine C = 50 with S ~ N(4 + 8C, 16), we first transform and then sum it with M( C ) of Equation (8)  Note that, during combinations, we can leave the intermediate matrices like Equation (15) swept. However, when we need to interpret a combination or make an inference using marginalization, we have to do a full reverse sweeping through a series of partial reverse sweepings 4 . For example, by a full reverse sweeping and then letting ε → 0, we can transform Equation (15) into: Then by marginalizing the combined LBF to variables S and C respectively, we found that C is still 50 with certainty but S is now distributed with mean 404 and variance 16. Furthermore, suppose through testing a sample of sales transactions occurred in the period, we obtain an independent piece of evidence that is encoded as a normal distribution of S say, S ~ N(500, 25). Then we can integrate the new evidence into the above combined belief and obtain the newly updated belief on S as S ~ N(441.46, 9.76). The notion of LBFs turns out to have a wide range of applications in statistical analysis and business decision-making. In Bayesian statistics, it has been a classic problem that how we 4 To do a partial reverse sweeping on an element say, m ij , in a multivariate moment matrix, we first replace the element m ij by its negative inverse ( can infer a belief distribution of the parameter θ for a parametric distribution f(x, θ) given a set of observations. In the framework of DST, Shafer (1982) made an attempt in the case of simple Binomial parametric distributions and demonstrated a tremendous challenge in applying finite belief functions. In contrast, using the theory of LBFs,  showed that the problem of statistical inference on linear models is reduced to the one of combining LBFs. His result is identical to Bayesian posterior distribution of the same parameters (see BIB001 . Recently, Monney (2002b) derived the same distribution using Fisher's fiducial arguments.   ) also showed how the Kalman filter could be understood in terms of LBFs. As he showed, the recursion involved in the filter can be regarded as a special case of the recursion involved in the computation of LBF marginals in a join tree. The full Kalman filter model results from judging all these component belief functions to be independent, and combining them into a single belief function according to Dempster's rule. Monney (2002a) did similar work in this regard. Based on the work of  in statistical inferences,  attempted to apply LBFs to the combination of linear statistical models. His idea is as follows. The data sets used for inferring individual statistical models can be considered as independent items of evidence. The statistical model inferred from each data set is then represented as an LBF. Then the models from different data sets are combined in a fashion we combine LBFs. Combined predictions or inferences are then made based on the combined model. Obviously, this approach is consistent with the spirit of Dempster's rule of combination. The proposed method generalizes the meta-analysis for integrating independent statistical findings (see BIB002 ) and the Bayesian method of estimating common regression coefficients (see BIB001 . As we see easily, the LBF-based method can combine models of different kinds that may involve different variables. In contrast, the models to be combined in the meta-analysis and the Bayesian method must be the same and the parameters to be estimated must be common. The notion of LBFs is relatively new. However, there have been some attempts to apply it to real business decisions. In an attempt to shorten the amount of time it takes to perform the modeling, measurement, and analysis of the business processes, which usually must be accomplished before the systems development life cycle takes place,  describes a method for rapidly producing a performance model of business processes through the use of LBFs. According to him, the performance model can be used to determine the optimal placement of in-formation systems to achieve the maximum benefit for the organization. In another application to information systems management,  analyzed existing empirical studies by  and others on the two important constructs, perceived ease of use and perceived usefulness, and their relationships to user acceptance and actual usage behavior of emerging information technologies. They synthesized the mixed empirical findings using both the traditional meta-analysis method BIB002 ) and the LBF-based approach as described by  .