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International Journal for Uncertainty Quantification

Impact factor: 1.000

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2012003641
pages 271-288


James L. Beck
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125, USA
Alexandros Taflanidis
University of Notre Dame


An overview is given of a powerful unifying probabilistic framework for treating modeling uncertainty, along with input uncertainty, when using dynamic models to predict the response of a system during its design or operation. This framework uses probability as a multivalued conditional logic for quantitative plausible reasoning in the presence of uncertainty due to incomplete information. The fundamental probability models that represent the system's uncertain behavior are specified by the choice of a stochastic system model class: a set of input–output probability models for the system and a prior probability distribution over this set that quantifies the relative plausibility of each model. A model class can be constructed from a parametrized deterministic system model by stochastic embedding which utilizes Jaynes' principle of maximum information entropy. Robust predictive analyses use the entire model class with the probabilistic predictions of each model being weighted by its prior probability, or if response data are available, by its posterior probability from Bayes' theorem for the model class. Additional robustness to modeling uncertainty comes from combining the robust predictions of each model class in a set of competing candidates weighted by the prior or posterior probability of the model class, the latter being computed from Bayes' theorem. This higher-level application of Bayes' theorem automatically applies a quantitative Ockham razor that penalizes the data-fit of more complex model classes that extract more information from the data. Robust predictive analyses involve integrals over high-dimensional spaces that usually must be evaluated numerically by Laplace's method of asymptotic approximation or by Markov chain Monte Carlo methods. These computational tools are demonstrated in an illustrative example involving the vertical dynamic response of a car being driven along a rough road.