On the calculation of the bounds of probability of events using infinite random sets

This paper presents an extension of the theory of finite random sets to infinite random sets, that is useful for estimating the bounds of probability of events, when there is both aleatory and epistemic uncertainty in the representation of the basic variables. In particular, the basic variables can be modelled as CDFs, probability boxes, possibility distributions or as families of intervals provided by experts. These four representations are special cases of an infinite random set. The method introduces a new geometrical representation of the space of basic variables, where many of the methods for the estimation of probabilities using Monte Carlo simulation can be employed. This method is an appropriate technique to model the bounds of the probability of failure of structural systems when there is parameter uncertainty in the representation of the basic variables. A benchmark example is used to demonstrate the advantages and differences of the proposed method compared with the finite approach.