Oftentimes, no distinction is made between aleatory and epistemic uncertainty, and the inputs are simply defined using probability density functions that capture the overall state of knowledge about possible values for that variable. In other cases, the distinction may be important. For example, a reliability analysis aims to estimate the probability that a device, component, or system will meet some requirement by using probabilistic analysis to compute the probability of a model output exceeding a critical level. This type of analysis lends itself to a Frequentist interpretation of probability, in which case reliability describes the proportion of systems that will meet the requirement over the long run, as many such systems are put into service with random realizations of the loads, geometries, boundary conditions, material properties, etc. As such, it may be important to develop model input distributions that are intended to capture only aleatory uncertainty. In addition to substantiating a Frequentist interpretation of reliability, maintaining the distinction between the two types of uncertainty enables the analyst to explore the possibility of reducing epistemic uncertainty by obtaining additional information (Der Kiureghian and Ditlevsen, 2009). For example, the view may be taken that the model output statistics, such as the mean, standard deviation, and probability levels, have fixed values. These fixed values depend on the actual probability distributions associated with the model inputs. For model inputs that represent physical quantities, these probability distributions may be estimated by collecting data, but unless it is possible to observe the entire population, perfect information can never be obtained. As a result, there is lack-of-knowledge uncertainty associated with the description of the probabilistic inputs. This is one component of epistemic uncertainty that exists in a probabilistic analysis. Other components may include uncertainty associated with the deterministic model and uncertainty associated with the probabilistic analysis method itself (for example, the finite sample error introduced by Monte Carlo simulation). To allow some formality, we will introduce the following notation for describing a model output statistic: γ=H(θ), (1) where γ is a model output statistic that is functionally related through H(·) to a set of parameters θ, which may be subject to epistemic uncertainty. For example, γ could be the standard deviation of the deterministic model output, and θ could contain parameters describing the model input distributions, such as the means and standard deviations. The vector θmight also contain error terms associated with the deterministic model and/or probabilistic method. The purpose of H(·) is to emphasize that the model output statistic depends on the parameters θ. The idea is that there is a true value of γ, but that it is subject to epistemic uncertainty because the parameters θare not known exactly. A probabilistic analysis that addresses epistemic uncertainty and aleatory uncertainty separately allows the analyst to make statements about the confidence in the model output statistic. If the epistemic uncertainty is also modeled using probability Variance Decomposition in the Presence of Epistemic and Aleatory Uncertainty 419
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