Bounds on Belief and Plausibility of Functionally Propagated Random Sets

We are interested in improving risk and reliability analysis of complex systems where our knowledge of system performance is provided by large simulation codes, and where moreover input parameters are known only imprecisely. Such imprecision lends itself to interval representations of parameter values, and thence to quantifying our uncertinay through Dempster-Shafer or Probability Bounds representations on the input space. In this context, the simulation code acts as a large “black box” function f , transforming one input Dempster-Shafer structure on the line (also known as a random interval A) into an output random interval f(A). Our quantification of output uncertainty is then based on this output random interval. If some properties of f (perhaps monotonicity or other analytical properties) are known, then some information about f(A) can be determined. But when f is a pure black box, we must resort to sampling approaches. In this paper, we present the basic formalism of a Monte Carlo approach to sampling a functionally propagated general random set, as opposed to a random interval. We show that the results of straightforward formal definitions are mathematically coherent, in the sense that bounding and convergence properties are achieved. Keywords—Random sets, Dempster-Shafer theory, Monte Carlo sampling.