Some properties of a random set approximation to upper and lower distribution functions
Information on an uncertain real variable is oftentimes conveyed using upper and lower distribution functions, which define a credal set, X. The paper explores the properties of a random set (random interval) approximation, R, to the upper and lower distribution functions carried out using the outer discretization method (ODM) introduced by the author as a generalization to an algorithm proposed in Williamson and Downs [R.C. Williamson, T. Downs, Probabilistic arithmetic. i. Numerical methods for calculating convolutions and dependency bounds, Int. J. Approx. Reason. 4 (1990) 89-158]. It is shown that probability bounds calculated using the ODM random sets, R, always contain the probability bounds calculated using M. This result holds even in the multivariate case (when each marginal is ODM discretized into a random set, R) regardless of the concept of dependence or independence adopted. The bound inclusion is also true for the image of a function defined on those variables. Finer discretizations of the original credal sets yield tighter or equal probability bounds. Since the ODM yields a random set, R, the information can be modeled either using probability measures of the measurable selections or the credal set of the belief and plausibility of R. It is proven that both models yield the same probability bounds and that the Choquet integrals of the belief and plausibility of R are the inferior and superior, respectively, of the expectations calculated using the measurable selections. However, the probabilistic information conveyed by the measurable selections may be more restrictive than the information contained in the credal set of the belief and plausibility of R. (C) 2007 Elsevier Inc. All rights reserved.
random sets credal sets random intervals outer discretization method independence measurable selections dempster-shafer upper and lower probabilities