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On New Cautious Structural Reliability Models in the Framework of imprecise Probabilities

Authors
Publisher
Taylor & Francis
Publication Date
Keywords
  • Bayesian Inference
  • Imprecise Probabilities
  • Structural Reliability
Disciplines
  • Computer Science
  • Design
  • Engineering
  • Logic
  • Mathematics

Abstract

Uncertainty of parameters in engineering design has been modeled in different frameworks such as inter-val analysis, fuzzy set and possibility theories, ran-dom set theory and imprecise probability theory. The authors of this paper for many years have been de-veloping new imprecise reliability models and gen-eralizing conventional ones to imprecise probabili-ties. The theoretical setup employed for this purpose is imprecise statistical reasoning (Walley 1991), whose general framework is provided by upper and lower previsions (expectations). The appeal of this theory is its ability to capture both aleatory (stochas-tic) and epistemic uncertainty and the flexibility with which information can be represented. The previous research of the authors related to generalizing structural reliability models to impre-cise statistical measures is summarized in Utkin & Kozine (2002) and Utkin (2004). The presupposed input for the imprecise structural reliability models was some probabilistic measures (precise or impre-cise) of strength and stress. While the accepted pre-mises are meaningful and practical in some applica-tions, they do not cover many other cases the reliability analyst faces in practice. Often the above mentioned inputs do not exist and the analyst has on-ly some judgments or measurements (observations) of values of stress and strength. How to utilize this available information for computing the structural reliability and what to do if the number of judgments or measurements is very small? Developing models enabling to answer these two questions has been in the focus of the new research the results of which are described in the paper. In this paper we describe new models for com-puting structural reliability based on measurements of values of stress and strength and taking account of the fact that the number of observations may be ra-ther small. The approach to developing the models is based on using the imprecise Bayesian inference models (Walley 1996). These models provide a rich supply of coherent imprecise inferences that are ex-pressed in terms of posterior upper and lower prob-abilities. The probabilities are initially vacuous, re-flecting prior ignorance, become more precise as the number of observations increase. The new imprecise structural reliability models are based on imprecise Bayesian inference and are imprecise Dirichlet, imprecise negative binomial, gamma-exponential and normal models. The models are applied to computing cautious structural reliabil-ity measures when the number of events of interest or observations is very small. The main feature of the models is that prior ignorance is not modeled by a fixed single prior distribution, but by a class of pri-ors which is defined by upper and lower probabili-ties that can converge as statistical data accumulate. Numerical examples illustrate some features of the proposed approach.

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