Relating Imprecise Representations of imprecise Probabilities

International audienceThere exist many practical representations of probability families that make them easier to handle. Among them are random sets, possibility distributions, probability intervals, Ferson's p-boxes and Neumaier's clouds. Both for theoretical and practical considerations, it is important to know whether one representation has the same expressive power than other ones, or can be approximated by other ones. In this paper, we mainly study the relationships between the two latter representations and the three other ones

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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UNIFYING PRACTICAL UNCERTAINTY REPRESENTATIONS: I. GENERALIZED P-BOXES

Pre-print of final version.International audienceThere exist several simple representations of uncertainty that are easier to handle than more general ones. Among them are random sets, possibility distributions, probability intervals, and more recently Ferson's p-boxes and Neumaier's clouds. Both for theoretical and practical considerations, it is very useful to know whether one representation is equivalent to or can be approximated by other ones. In this paper, we define a generalized form of usual p-boxes. These generalized p-boxes have interesting connections with other previously known representations. In particular, we show that they are equivalent to pairs of possibility distributions, and that they are special kinds of random sets. They are also the missing link between p-boxes and clouds, which are the topic of the second part of this study

Interval linear systems as a necessary step in fuzzy linear systems

International audienceThis article clarifies what it means to solve a system of fuzzy linear equations, relying on the fact that they are a direct extension of interval linear systems of equations, already studied in a specific interval mathematics literature. We highlight four distinct definitions of a systems of linear equations where coefficients are replaced by intervals, each of which based on a generalization of scalar equality to intervals. Each of the four extensions of interval linear systems has a corresponding solution set whose calculation can be carried out by a general unified method based on a relatively new concept of constraint intervals. We also consider the smallest multidimensional intervals containing the solution sets. We propose several extensions of the interval setting to systems of linear equations where coefficients are fuzzy intervals. This unified setting clarifies many of the anomalous or inconsistent published results in various fuzzy interval linear systems studies

Special Cases

International audienceThis chapter reviews special cases of lower previsions, that are instrumental in practical applications. We emphasize their various advantages and drawbacks, as well as the kind of problems in which they can be the most useful

Stochastic simulation methods for structural reliability under mixed uncertainties

Uncertainty quantification (UQ) has been widely recognized as one of the most important, yet challenging task in both structural engineering and system engineering, and the current researches are mainly on the proper treatment of different types of uncertainties, resulting from either natural randomness or lack of information, in all related sub-problems of UQ such as uncertainty characterization, uncertainty propagation, sensitivity analysis, model updating, model validation, risk and reliability analysis, etc. It has been widely accepted that those uncertainties can be grouped as either aleatory uncertainty or epistemic uncertainty, depending on whether they are reducible or not. For dealing with the above challenge, many non-traditional uncertainty characterization models have been developed, and those models can be grouped as either imprecise probability models (e.g., probability-box model, evidence theory, second-order probability model and fuzzy probability model) or non-probabilistic models (e.g., interval/convex model and fuzzy set theory). This thesis concerns the efficient numerical propagation of the three kinds of uncertainty characterization models, and for simplicity, the precise probability model, the distribution probability-box model, and the interval model are taken as examples. The target is to develop efficient numerical algorithms for learning the functional behavior of the probabilistic responses (e.g., response moments and failure probability) with respect to the epistemic parameters of model inputs, which is especially useful for making reliable decisions even when the available information on model inputs is imperfect. To achieve the above target, my thesis presents three main developments for improving the Non-intrusive Imprecise Stochastic Simulation (NISS), which is a general methodology framework for propagating the imprecise probability models with only one stochastic simulation. The first development is on generalizing the NISS methods to the problems with inputs including both imprecise probability models and non-probability models. The algorithm is established by combining Bayes rule and kernel density estimation. The sensitivity indices of the epistemic parameters are produced as by-products. The NASA Langley UQ challenge is then successfully solved by using the generalized NISS method. The second development is to inject the classical line sampling to the NISS framework so as to substantially improve the efficiency of the algorithm for rare failure event analysis, and two strategies, based on different interpretations of line sampling, are developed. The first strategy is based on the hyperplane approximations, while the second-strategy is derived based on the one-dimensional integrals. Both strategies can be regarded as post-processing of the classical line sampling, while the results show that their resultant NISS estimators have different performance. The third development aims at further substantially improving the efficiency and suitability to highly nonlinear problems of line sampling, for complex structures and systems where one deterministic simulation may take hours. For doing this, the active learning strategy based on Gaussian process regression is embedded into the line sampling procedure for accurately estimating the interaction point for each sample line, with only a small number of deterministic simulations. The above three developments have largely improved the suitability and efficiency of the NISS methods, especially for real-world engineering applications. The efficiency and effectiveness of those developments are clearly interpreted with toy examples and sufficiently demonstrated by real-world test examples in system engineering, civil engineering, and mechanical engineering

Probabilistic constraint reasoning

Dissertação apresentada para obtenção do Grau de Doutor em Engenharia Informática, pela Universidade Nova de Lisboa, Faculdade de Ciências e TecnologiaThe continuous constraint paradigm has been often used to model safe reasoning in applications where uncertainty arises. Constraint propagation propagates intervals of uncertainty among the variables of the problem, eliminating values that do not belong to any solution. However, constraint programming is very conservative: if initial intervals are wide (reflecting large uncertainty), the obtained safe enclosure of all consistent scenarios may be inadequately wide for decision support. Since all scenarios are considered equally likely, insufficient pruning leads to great inefficiency if some costly decisions may be justified by very unlikely scenarios. Even when probabilistic information is available for the variables of the problem, the continuous constraint paradigm is unable to incorporate and reason with such information. Therefore, it is incapable of distinguishing between different scenarios, based on their likelihoods. This thesis presents a probabilistic continuous constraint paradigm that associates a probabilistic space to the variables of the problem, enabling probabilistic reasoning to complement the underlying constraint reasoning. Such reasoning is used to address probabilistic queries and requires the computation of multi-dimensional integrals on possibly non linear integration regions. Suitable algorithms for such queries are developed, using safe or approximate integration techniques and relying on methods from continuous constraint programming in order to compute safe covers of the integration region. The thesis illustrates the adequacy of the probabilistic continuous constraint framework for decision support in nonlinear continuous problems with uncertain information, namely on inverse and reliability problems, two different types of engineering problems where the developed framework is particularly adequate to support decision makers