A Monte Carlo-based method for the estimation of lower and upper probabilities of events using infinite random sets of indexable type

Random set theory is a useful tool to quantify lower and upper bounds on the probability of the occurrence of events given uncertain information represented for example by possibility distributions, probability boxes, or Dempster–Shafer structures, among others. In this paper it is shown that the belief and plausibility estimated by Dempster–Shafer evidence theory are basically approximations by Riemann–Stieltjes sums of the integrals of the lower and upper probability employed when using infinite random sets of indexable type. In addition, it is shown that the evaluation of the lower and upper probability is more efficient if it is done by pseudo-Monte Carlo strategies. This discourages the use of Dempster–Shafer evidence theory and suggests the use of infinite random sets of indexable type specially in high dimensions, not only because the initial discretization step of the basic variables is not required anymore, but also because the evaluation of the lower and upper probability of events is much more efficient using the different techniques for multidimensional integration like Monte Carlo simulation.