Computing mean and variance under Dempster-Shafer uncertainty: Towards faster algorithms

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 [E, E] and [V , V ] of possible values of mean E and of variance V . In their recent paper, A. T. Langewisch and F. F. Choobineh show how to compute these ranges in polynomial time. In particular, they reduce the problem of computing V to the problem of minimizing a convex quadratic function, a problem which can be solved in time O(n · log(n)). We show that the corresponding quadratic optimization problem can be actually solved faster, in time O(n · log(n)); thus, we can compute the bounds V and V in time O(n · log(n)). 1 Formulation of the Problem Computing mean and variance under Dempster-Shafer uncertainty: an important practical 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., [9]). In the 1-D case, instead of the exact probability distribution, we have a finite collection of intervals x1 = [x1, x1], . . . , xn = [xn, xn],