Dear Friends,

This is just FYI, to let you know that there is now an informative and well maintained webpage on p-boxes.

WHAT ARE P-BOXES: BRIEF REMINDER. As many of you know, one of the most efficient ways to deal with combined interval and probabilistic uncertainty is to use probability boxes (p-boxes, for short).

In contrast to an exactly known probability distribution, which can be characterized by a cumulative distribution function (cdf) F(x)=Prob(X <= x), in the case of a partial knowledge about the probability distribution, we do not know the exact values of F(x) for all x. Instead, for each x, we often only know the bounds on the actual (unknown) value of F(x), i.e., we know the interval [F(x)] that contains the exact (unknown) value F(x). The mapping that assigns, to each real number x, such an interval [F(x)], is known as a p-box.

Of course, not all information is included in a p-box: for example, if we know that the distribution is normal, with bounds on parameters, then we can form the corresponding p-box, but that would mean considering all functions for which F(x) is in [F(x)] for all x, and most of these functions do not correspond to normal distributions.

However, many types of uncertainty are well-represented by p-boxes. For example, an interval [a,b] corresponds to a p-box in which [F(x)]=[0,0] for x<a, [F(x)]=[0,1] for x between a and b, and [F(x)] = [1,1] for x>b.

P-boxes were introduced by Scott Ferson, who also developed many algorithms for processing p-boxes, and who maintains Ramas software that incorporates these algorithms and algorithms invented by other folks.

GOOD NEWS. I have just learned from Scott that he has been actively maintaining a Wikipedia page on probability boxes

http://en.wikipedia.org/wiki/Probability_box