Inclusion–exclusion principle for belief functions

• Published in: International journal of approximate reasoning Vol. 55; no. 8; pp. 1708 - 1727
• Main Authors: Aguirre, F., Destercke, S., Dubois, D., Sallak, M., Jacob, C.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01-11-2014; Elsevier
• Subjects: Applied sciences; Approximation; Artificial Intelligence; Belief function; Boolean algebra; Boolean formula; Civil Engineering; Combinatorial probability; Computation; Computer Science; Computer science; control theory; systems; Diagnosis; Engineering Sciences; Exact sciences and technology; Inclusion–exclusion principle; Independence; Information theory; Information, signal and communications theory; Learning and adaptive systems; Logic in Computer Science; Mathematical analysis; Mathematical models; Mathematics; Operational research and scientific management; Operational research. Management science; Probability and statistics; Probability theory; Probability theory and stochastic processes; Reliability analysis; Reliability theory. Replacement problems; Risques; Sciences and techniques of general use; Statistics; System reliability; Telecommunications and information theory; Uncertainty; Reliability analysis; Inclusion–exclusion principle; Boolean formula; Belief function; Independence; Belief; Probabilistic approach; Dependability; Necessary and sufficient condition; Inclusion-exclusion principle; Probability theory; System reliability; Inclusion exclusion method; Uncertain system; Boolean logic; Diagnosis; exclusion-inclusion principle; reliability analysis; belief functions; boolean formulas
• ISSN: 0888-613X; 1873-4731
• DOI: 10.1016/j.ijar.2014.04.018

The inclusion–exclusion principle is a well-known property in probability theory, and is instrumental in some computational problems such as the evaluation of system reliability or the calculation of the probability of a Boolean formula in diagnosis. However, in the setting of uncertainty theories more general than probability theory, this principle no longer holds in general. It is therefore useful to know for which families of events it continues to hold. This paper investigates this question in the setting of belief functions. After exhibiting original sufficient and necessary conditions for the principle to hold, we illustrate its use on the uncertainty analysis of Boolean and non-Boolean systems in reliability.
•We give conditions under which the principle of exclusion/inclusion applies to belief functions.•We discuss its interest for evaluating uncertainty on Boolean formulas and monotone functions.•We illustrate the applicability of our results on reliability analysis problems.•We discuss the connection with other types of independence (strong independence).