# Probabilistic Methods for Uncertainty Quantification

N. Chugunov (Institute for Systems Analysis – Russian Academy of Sciences, Russia), G. Shepelyov (Institute for Systems Analysis – Russian Academy of Sciences, Russia) and M. Sternin (Institute for Systems Analysis – Russian Academy of Sciences, Russia)
DOI: 10.4018/978-1-59904-843-7.ch082

## Abstract

The complexity and interdisciplinary nature of modern problems are often coupled with uncertainty inherent to real-life situations. There is a wide class of real-world problems described by well-formulated quantitative models for which a decision maker (DM) has to deal with uncertainty in values of initial parameters for these models. A good example of such a problem is hydrocarbon reservoir assessment in the exploration stage, which requires the involvement and joint consideration of geological, petroleum engineering, and financial models of reservoir exploration. The consequences of some unreasonable decisions can lead to millions of dollars in loss to the companies as it happens in the oil business, where industry sources on investment decision analysis continue to report surprise values (outside the [P10;P90] range) far more than the 20% indicated by this interval (Welsh, Begg, Bratvold, & Lee, 2004).

## Key Terms in this Chapter

Probability Bound: A probability bound is a pair of probability distribution functions Fmin(x) and Fmax(x) bounding the true or estimated distribution function F(x) such that Fmin(x)=F(x)=Fmax(x) for all x values.

Generalized Interval Estimation (GIE): GIE is a special case of poly-interval estimation, when each interval scenario from PIE is described by a probability distribution function and assigned a certain weight, with the sum of all weights being equal to 1.

Uncertainty: It is “the absence of perfectly detailed knowledge. Uncertainty includes incertitude (the exact value is not known) and variability (the value is changing). Uncertainty may also include other forms such as vagueness, ambiguity and fuzziness (in the sense of border-line cases)” (Ferson et al., 2004, p. 130).

Uncertainty Propagation: Uncertainty propagation is the calculation of the uncertainty in the model outputs, which is due to the uncertainties in the model’s inputs.

Generalized Probability Box (GPB): GPB is a decision-oriented interpretation of GIE that transforms GIE into a nested set of probability bounds, each of which corresponds to a certain confidence level of the expert.

Expert Knowledge: This refers to estimates, judgments, and patterns that are elicited from experts to describe values of quantitative parameters being analyzed and their interrelations.

Imprecise Probability: This is “the subject of any of several theories involving models of uncertainty that do not assume a unique underlying probability distribution, but instead correspond to a set of probability distributions….Theories of imprecise probabilities are often expressed in terms of a lower probability measure giving the lower probability for every possible event from some universal set, or in terms of closed convex sets of probability distributions. Interval probabilities, Dempster-Shafer structures and probability boxes can be regarded as special-cases of imprecise probabilities” (Ferson et al., 2004, pp. 124-125).

Dempster-Shafer Structure: This is a set of focal elements or bodies of evidence (in the context of this article, closed intervals of the real line), each of which is associated with a nonnegative value m (basic probability assignment), with the sum of all m values being equal to 1.

Poly-Interval Estimation (PIE): A PIE is a set of intervals that represents an expert’s judgments on possible ranges of values for estimated quantity.

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