Linear belief function
Linear Belief Function is an extension of the Dempster-Shafer theory of belief functions to the case when variables of interest are continuous. Examples of such variables include financial asset prices, portfolio performance, and other antecedent and consequent variables.
THE CONCEPT OF LINEAR BELIEF FUNCTIONS
A linear belief function 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., the true value is somewhere from –∞ to +∞ but the associated probability is unknown. As we know, a belief function in general is defined by a mass function over a class of focal elements, which may have nonempty intersections. A linear belief function is a special type of belief functions in the sense that its focal elements are exclusive, parallel sub-hyperplanes over the certainty hyperplane and its mass function is a normal distribution across the sub-hyperplanes.
Based on the above geometrical description, Shafer [23] and Liu [15] propose two mathematical representations of a LBF: a wide-sense inner product and a linear functional in the variable space, and as their duals over a hyperplane in the sample space. Monney [24] proposes a still another structure called Gaussian hints. Although these representations are mathemati-cally neat, they tend to be unsuitable for knowledge represen-tation in expert systems.
Knowledge Representation
A linear belief function 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 thousand dollars on the average with a standard deviation 5 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, 0.52). 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 bal-ance 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).