Inclusion–exclusion principle for belief functions

F. Aguirre, S. Destercke, D. Dubois, M. Sallak, C. Jacob
2014 International Journal of Approximate Reasoning  
The inclusion-exclusion principle is a well-known property of set cardinality and probability measures, that is instrumental to solve some problems such as the evaluation of systems reliability or of uncertainty over Boolean formulas. However, when using sets and probabilities conjointly, this principle no longer holds in general. It is therefore useful to know in which cases it is still valid. This paper investigates this question when uncertainty is modelled by belief functions. After
more » ... ng necessary and sufficient conditions for the principle to hold, we illustrate its use on some applications, i.e. reliability analysis and uncertainty over Boolean formulas. 1 General Additivity Conditions for Belief Functions After introducing notations, Section 2.2 provides general conditions for families of subsets for which the inclusionexclusion principle holds for belief functions. We then interest ourselves to the specific case where focal sets of belief functions are Cartesian products of subsets. Setting A mass distribution [16] defined on a (finite) space X is a mapping m : 2 X → [0, 1] from the power set of X to the
doi:10.1016/j.ijar.2014.04.018 fatcat:ziwmfbyuv5dazenop4gpodzlzy