Algorithms for Dempster-Shafer Theory

The method of reasoning with uncertain information known as Dempster-Shafer theory arose from the reinterpretation and development of work of Arthur Dempster [Dempster, 67; 68] by Glenn Shafer in his book a mathematical theory of evidence [Shafer, 76], and further publications e.g., [Shafer, 81; 90]. More recent variants of Dempster-Shafer theory include the Transferable Belief Model see e.g., [Smets, 88; Smets and Kennes, 94] and the Theory of Hints e.g., [Kohlas and Monney, 95]. Use of the method involves collecting a number of pieces of uncertain information, which are judged to be ‘independent’. Each individual piece of information is represented within the formalism as what is known as a mass function, these are combined using Dempster’s rule, and the degrees of belief for various propositions of interest are then calculated. Propositions are expressed as subsets of a set of possibilities, known as the frame of discernment. Two major problems with the theory are (i) understanding what the calculated values of belief mean; this issue is briefly discussed in section 2.5; and (ii) the computational problems of Dempster’s rule, to which most of this chapter is addressed. The obvious algorithm for calculating the effect of Dempster’s rule, as sketched in [Shafer, 76], is (at worst) exponential and [Orponen, 90] has shown that the problem is #P-complete. However Monte-Carlo methods can be used to approximate very closely a value of combined belief, and these have much better complexity: some methods have complexity almost linearly related to the size of the frame, but with a high constant factor because of having to use a large number of trials in order to ensure that the approximation is a good one. In many applications the frame of discernment is expressed in terms of a product set generated by the values of a number of variables (see section 9), so the size of the frame is itself exponential in the number of variables. Glenn Shafer and Prakash Shenoy have devised techniques for this situation. The exact