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 exhibiting 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
In fault-tree analysis, probabilities of failure of components are often assumed to be precise and the events are assumed to be independent, but this is not always verified in practice. By giving up some of these assumptions, results can still be computed, even though it may require more expensive algorithms, or provide more imprecise results. Once compared to those obtained with the simplified model, the impact of these assumptions can be evaluated. This paper investigates the case when probability intervals of atomic propositions come from independent sources of information. In this case, the problem is solved by means of belief functions. We provide the general framework, discuss computation methods, and compare this setting with other approaches to evaluating the uncertainty of formulas.
Abstract. In many real-life applications (e.g., in aircraft maintenance), we need to estimate the probability of failure of a complex system (such as an aircraft as a whole or one of its subsystems). Complex systems are usually built with redundancy allowing them to withstand the failure of a small number of components. In this paper, we assume that we know the structure of the system, and, as a result, for each possible set of failed components, we can tell whether this set will lead to a system failure. For each component A, we know the probability P (A) of its failure with some uncertainty: e.g., we know the lower and upper bounds P (A) and P (A) for this probability. Usually, it is assumed that failures of different components are independent events. Our objective is to use all this information to estimate the probability of failure of the entire the complex system. In this paper, we describe a new efficient method for such estimation based on Cauchy deviates.
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