The presented work considers the problem of evaluating upper and lower probabilities in systems characterized by imprecise random variables. According to the authors, “Methods of imprecise probability have increasingly attracted interest in the engineering community” in relation to the development of fuzzy sets and fuzzy logic concepts. At the same time, the examination of complex systems also involves the analysis of mathematical methods of fuzzy interval arithmetic to establish bounds on the evaluation of system parameters.

The paper sets up an explanation and gives sufficient conditions for making the formation of a random set possible. Two interpretations of lower/upper probabilities are investigated. Besides scrupulous mathematical analysis, the paper also presents a concrete example in engineering reliability, which is used for simulating random sets and the corresponding lower/upper probabilities. The authors “hope that the paper stimulates further research into computational [approaches for] imprecise probability.”

An important application of approximate reasoning is associated with the nowadays popular subject of big data. Actually, the implicit knowledge in big data often evaluates the imprecise understanding of the situations by subjective estimates of lower and upper probabilities as belief and plausibility, rather than using a more accurate frequentist definition approach. Thus, data projects should be treated as scientific experiments.