Efficient Algorithms for Computing Mean and Variance Under Dempster-Shafer Uncertainty

In many real-life situations, we only have partial information about the actual probability distribution. For example, under Dempster-Shafer uncertainty, we only know the masses m1, . . . , mn assigned to different sets S1, . . . , Sn, but we do not know the distribution within each set Si. Because of this uncertainty, there are many possible probability distributions consistent with our knowledge; different distributions have, in general, different values of standard statistical characteristics such as mean and variance. It is therefore desirable, given a Dempster-Shafer knowledge base, to compute the ranges of possible values of mean and of variance. The existing algorithms for computing the range for the variance require ≈ 2 computational steps, and therefore, cannot be used for large n. In this paper, we propose new efficient algorithms that work for large n as well. 1 Formulation of the Problem In many real-life situations, we only have partial information about the actual probability distribution. In many practical situations, this uncertainty is naturally described by a Dempster-Shafer (DS) approach (see, e.g., [14]), in which the knowledge consists of a finite collection of sets S1, . . . , Sn and non-negative “masses” (probabilities) m1, . . . , mn assigned to these sets in such a way that m1 + . . . + mn = 1, In particular, in the 1-D case, instead of the exact probability distribution, we have a finite collection of intervals x1 = [x1, x1], . . . , xn = [xn, xn], and we have non-negative “masses” (probabilities) m1, . . . , mn assigned to these intervals in such a way that m1 + . . . + mn = 1.