Computing Best-Possible Bounds for the Distribution of a Sum of Several Variables is NP-Hard

Authors

Vladik Kreinovich, The University of Texas at El PasoFollow
Scott Ferson

Publication Date

6-2005

Comments

UTEP-CS-04-25a.

Published in International Journal of Approximate Reasoning, 2006, Vol. 41, pp. 331-342.

Abstract

In many real-life situations, we know the probability distribution of two random variables x1 and x2, but we have no information about the correlation between x1 and x2; what are the possible probability distributions for the sum x1+x2? This question was originally raised by A. N. Kolmogorov. Algorithms exist that provide best-possible bounds for the distribution of x1+x2; these algorithms have been implemented as a part of the efficient software for handling probabilistic uncertainty. A natural question is: what if we have several (n>2) variables with known distribution, we have no information about their correlation, and we are interested in possible probability distribution for the sum y=x_1+...+xn? Known formulas for the case n=2 can be (and have been) extended to this case. However, as we prove in this paper, not only are these formulas not best-possible anymore, but in general, computing the best-possible bounds for arbitrary n is an NP-hard (computationally intractable) problem.