Adding Unimodality or Independence Makes Interval Probability Problems NP-Hard

In many real-life situations, we only have partial information about probabilities. This information is usually described by bounds on moments, on probabilities of certain events, etc. – i.e., by characteristics c(p) which are linear in terms of the unknown probabilities pj . If we know interval bounds on some such characteristics ai ≤ ci(p) ≤ ai, and we are interested in a characteristic c(p), then we can find the bounds on c(p) by solving a linear programming problem. In some situations, we also have additional conditions on the probability distribution – e.g., we may know that the two variables x1 and x2 are independent, or that for each value of x2, the corresponding conditional distribution for x1 is unimodal. We show that adding each of these conditions makes the corresponding interval probability problem NP-hard.