Arithmetic with uncertain numbers: rigorous and (often) best possible answers

A variety of complex arithmetic problems can be solved using a single—and fairly simple—approach based on probability bounds analysis. The inputs are first expressed as interval bounds on cumulative distribution functions. Each uncertain input variable is then decomposed into a list of pairs of the form (interval, probability). A Cartesian product of these lists, reflecting both the independence among inputs and the mathematical expression that binds them together, creates another list, which is recomposed to form the resulting uncertain number as upper and lower bounds on a cumulative distribution function. Ancillary techniques are also employed, such as condensation, which is necessary to keep the length of the list from growing inordinately in sequential operations, and subinterval reconstitution, which is needed to solve interval arithmetic problems involving repeated parameters. Moment propagation formulas are simultaneously used to bound mean and variance estimates accompanying the bounds on the cumulative distribution function. Generalizations of this approach are also described that allow for dependencies other than independence, completely unknown dependence, and model uncertainty more generally.