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Emerging Concepts and Approaches for Efficient and Realistic Uncertainty Quantification
Published in Dan Frangopol, Yiannis Tsompanakis, Maintenance and Safety of Aging Infrastructure, 2014
Michael Beer,, Ioannis A. Kougioumtzoglou, Edoardo Patelli
can be constructed, and imprecise probabilities allow statistical estimations and tests with imprecise sample elements. Results from robust statistics in the form of solution domains of statistical estimators can also be considered directly to formulate imprecise probabilistic models. A key feature of imprecise probabilities is the identification of bounds on probabilities for events of interest; the uncertainty of an event is characterised with two measure values a lower probability and an upper probability. The distance between the probability bounds reflects the indeterminacy in model specifications expressed as imprecision of the models. This imprecision is the concession for not introducing artificial model assumptions. Such model assumptions based on expert knowledge are often too narrow, which is known as expert overconfidence. In imprecise probabilities, this problem is circumvented by implementing set-valued descriptors in the specification of a probabilistic model. The model description is thereby limited to some domain, and no further specific characteristics are ascribed. This introduces significantly less information in comparison with a specific subjective distribution function as used in the Bayesian approach. Imprecision in the model description expressed in a set-theoretical form does not migrate into probabilities, but it is reflected in the result as a set of probabilities which contains the true probability. This feature is particularly important when the calculated probabilities provide the basis for critical decisions. With imprecise probabilities the analysis may be performed with various relevant models to obtain a set of relevant results and associated decisions. This helps to avoid wrong decisions due to artificial restrictions in the modelling. Imprecise probabilities include a large variety of specific theories and mathematical models associated with an entire class of measures. This variety is discussed in (Klir 2006) in a unifying context; the diversity of model choices is highlighted, and arguments for imprecise probabilities are summarised. Imprecise probabilities have a close relationship to the theory of random sets and cover, for example, the concept of upper and lower probabilities, sets of probability measures, distribution envelopes, probability bounds analysis using p-boxes, interval probabilities, Choquet capacities of various orders, and evidence theory (or Dempster-Shafer Theory). Moreover, fuzzy probabilities, with their roots in the theory of fuzzy random variables, are also covered under the framework of imprecise probabilities and possess strong ties to several of the aforementioned concepts. 5.3.4 Engineering Applications of Imprecise Probability
A possibilistic finite element method for sparse data
Published in Safety and Reliability, 2018
A. Dridger, I. Caylak, R. Mahnken, E. Penner
Remarks 1The upper and lower bound characteristic as described in Equation (3) can be viewed as a particular case of nested random sets (De Cooman & Aeyels, 2000; Dubois et al., 2004; Dubois et al., 2000; Dubois & Prade, 1986; Grabisch, 2016).A possibility theory can be seen as a particular case of the Dempster-Shafer theory of evidence, which connects it to upper and lower probabilities. Detailed description can be found in Grabisch (2016).