From p-boxes to p-ellipsoids: Towards an optimal representation of imprecise probabilities

One of the most widely used ways to represent a probability distribution is by describing its cumulative distribution function (cdf) F(x). In practice, we rarely know the exact values of F(x): for each x, we only know F(x) with uncertainty. In such situations, it is reasonable to describe, for each x, the interval [F(x), F(x)] of possible values of x. This representation of imprecise probabilities is known as a p-box; it is effectively used in many applications. Similar interval bounds are possible for probability density function, for moments, etc. The problem is that when we transform from one of such representations to another one, we lose information. It is therefore desirable to come up with a universal representation of imprecise probabilities in which we do not lose information when we move from one representation to another. We show that under reasonable objective functions, the optimal representation is an ellipsoid. In particular, ellipsoids lead to faster computations, to narrower bounds, etc.