#### Publication Date

4-2006

#### Abstract

In most real-life situations, we have uncertainty: we do not know the exact state of the world, there are several (n) different states which are consistent with our knowledge. In such situations, it is desirable to gauge how much information we need to gain to determine the actual state of the world. A natural measure of this amount of information is the average number of "yes"-"no" questions that we need to ask to find the exact state. When we know the probabilities p1,...,pn of different states, then, as Shannon has shown, this number of questions can be determined as S=-p1 log(p1)-...-pn log(pn).

In many real-life situations, we only have partial information about the probabilities; for example, we may only know intervals [pi]=[p-i,p+i] of possible values of pi. For different values pi from Pi, we get different values S. So, to gauge the corresponding uncertainty, we must find the range [S]=[S-,S+] of possible values of S. In this paper, we show that the problem of computing [S] is, in general, NP-hard, and we provide algorithms that efficiently compute [S] in many practically important situations.

## Comments

Updated version UTEP-CS-06-10a.

Short version published in the

Proceedings of the International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems IPMU'06, Paris, France, July 2-7, 2006, pp. 840-847.