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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 theory was originally proposed by Arthur P. Dempster〔A. P. Dempster, "Normal belief functions and the Kalman filter," in ''Data Analysis from Statistical Foundations'', A. K. M. E. Saleh, Ed.: Nova Science Publishers, 2001, pp. 65-84.〕 in the context of Kalman Filters and later was reelaborated, refined, and applied to knowledge representation in artificial intelligence and decision making in finance and accounting by Liping Liu.〔Liu, Liping, Catherine Shenoy, and Prakash P. Shenoy, “Knowledge Representation and Integration for Portfolio Evaluation Using Linear Belief Functions,” IEEE Transactions on Systems, Man, and Cybernetics, Series A, vol. 36 (4), 2006, pp. 774-785.〕
== Concept ==
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. 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 function 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〔G. Shafer, "A note on Dempster's Gaussian belief functions," School of Business, University of Kansas, Lawrence, KS, Technical Report 1992.〕 and Liu〔L. Liu, "A theory of Gaussian belief functions," ''International Journal of Approximate Reasoning'', vol. 14, pp. 95-126, 1996〕 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 〔P. A. Monney, ''A Mathematical Theory of Arguments for Statistical Evidence''. New York, NY: Springer, 2003.〕 proposes still another structure called Gaussian hints. Although these representations are mathematically neat, they tend to be unsuitable for knowledge representation in expert systems.
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