theory1, then there is a probability distribution associated with γ, which represents the state of knowledge. One might be interested in computing confidence intervals or conservative bounds on γ, for the purpose of conveying the state of knowledge to the decision maker. In practice, computing the uncertainty distribution on a model output statistic can be computationally very demanding. In fact, for applications in which the deterministic performance model is expensive to evaluate, computing a single value γ for a particular vector of parameters θ can be challenging (for example, performing a reliability analysis using a finite element model). The use of computationally efficient surrogate models can go a long way towards making such analyses feasible. However, the focus of this paper will not be practical approaches for computing uncertainty distributions on model output statistics due to epistemic uncertainty. This has been previously addressed by the authors (Bichon et al, 2008b; McFarland and Bichon, 2009). Alternatives to constructing the complete uncertainty distribution have also been proposed (for example, Hofer et al, 2002). Formulation of the epistemic uncertainty distributions is also of interest, and here three cases are considered: 1. Model input distribution parameter uncertainty 2. Deterministic performance model uncertainty 3. Probabilistic method uncertainty For the first case, distribution parameters defining model input random variables are subject to epistemic uncertainty. A typical example of this is when the probability density function of a model input is estimated based on a finite set of sample data (for example, a collection of material tests to estimate the distribution of yield strength). Since only a limited amount of data can be collected, the true distribution parameters are subject to uncertainty. A natural framework for quantifying distribution parameter uncertainty is Bayesian statistics, which is particularly powerful because the Bayesian approach models uncertainty using probability distributions. Thus, a Bayesian analysis allows one to compute a so-called posterior distribution of the uncertain distribution parameters, conditional on the observed sample data. Bayesian analysis also allows for the incorporation of prior knowledge, through the prior distribution, but in practice it is common to use non-informative prior distributions (also known as vague priors or reference priors) to capture the notion that no information is available before observing the data. Further information on Bayesian inference is available in Lee (2004), and example applications of distribution parameter estimation are given in Bichon et al (2008b); McFarland and Bichon (2009); Marhadi et al (2008). The second case of epistemic uncertainty listed above refers to uncertainty associated with the deterministic model itself. This may be in the form of model parameter uncertainty, model form uncertainty, convergence error (which may also 1 Alternatives to probability theory have been documented in the literature and include the Dempster-Shafer theory of evidence (Shafer, 1976), possibility theory (Dubois and Prade, 2001; Zadeh, 1978), and many others (Ferson et al, 2004) John McFarland and David Riha 420
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