Numerical Methods for Propagating Imprecise Uncertainty

Since engineering design requires decision making under uncertainty, the degree to which good decisions can be made depends upon the degree to which the decision maker has expressive and accurate representations of his or her uncertain beliefs. Whereas traditional decision analysis uses precise probability distributions to represent uncertain beliefs, recent research has examined the effects of relaxing this assumption of precision. A specific example of this is the theory of imprecise probability. Imprecise probabilities are more expressive than precise probabilities, but they are also more computationally expensive to propagate through mathematical models. The probability box (p-box) is an alternative representation that is both more expressive than precise probabilities, and less computationally expensive than general imprecise probabilities. In this paper, we introduce a method for propagating p-boxes through black box models. Based on two example models, a new method, called p-box convolution sampling (PCS), is compared with three other p-box propagation methods. It is found that, although PCS is less expensive than the alternatives, it is still relatively expensive and therefore only justifiable when the expected benefits are large. Several directions for further improving the efficiency of PCS are discussed.