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

Abstract 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 m 1 , … , m n assigned to different sets S 1 , … , S n , but we do not know the distribution within each set S i . 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, Langewisch and 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 2  · 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 )).