Quantification of margins and uncertainties: Alternative representations of epistemic uncertainty

Abstract

In 2001, the National Nuclear Security Administration of the U.S. Department of Energy in conjunction with the national security laboratories (i.e., Los Alamos National Laboratory, Lawrence Livermore National Laboratory and Sandia National Laboratories) initiated development of a process designated Quantification of Margins and Uncertainties (QMU) for the use of risk assessment methodologies in the certification of the reliability and safety of the nation's nuclear weapons stockpile. A previous presentation, “Quantification of Margins and Uncertainties: Conceptual and Computational Basis,” describes the basic ideas that underlie QMU and illustrates these ideas with two notional examples that employ probability for the representation of aleatory and epistemic uncertainty. The current presentation introduces and illustrates the use of interval analysis, possibility theory and evidence theory as alternatives to the use of probability theory for the representation of epistemic uncertainty in QMU-type analyses. The following topics are considered: the mathematical structure of alternative representations of uncertainty, alternative representations of epistemic uncertainty in QMU analyses involving only epistemic uncertainty, and alternative representations of epistemic uncertainty in QMU analyses involving a separation of aleatory and epistemic uncertainty. Analyses involving interval analysis, possibility theory and evidence theory are illustrated with the same two notional examples used in the presentation indicated above to illustrate the use of probability to represent aleatory and epistemic uncertainty in QMU analyses.

Introduction

In 2001, the National Nuclear Security Administration (NNSA) of the U.S. Department of Energy (DOE) in conjunction with the national security laboratories (i.e., Los Alamos National Laboratory, Lawrence Livermore National Laboratory, and Sandia National Laboratories) initiated development of a process designated Quantification of Margins and Uncertainties (QMU) for the use of risk assessment methodologies in the certification of the reliability and safety of the nation's nuclear weapons stockpile. A previous presentation describes the basic ideas that underlie QMU and illustrates these ideas with two notional examples [1]. An additional presentation illustrates the ideas underlying QMU with examples from three real analyses involving reactor safety and radioactive waste disposal that incorporate a separation of aleatory and epistemic uncertainty [2].

The analyses in Refs. [1], [2] employ probability as the mathematical structure used to represent both aleatory uncertainty and epistemic uncertainty. As a reminder, aleatory uncertainty arises from an inherent randomness in the properties or behavior of the system under study, and epistemic uncertainty derives from a lack of knowledge about the appropriate value to use for a quantity that is assumed to have a fixed value in the context of a particular analysis (Section 2 in Ref. [1]; also, Refs. [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]). Recently, several alternatives to probability for the representation of epistemic uncertainty have been introduced, including interval analysis [16–21], possibility theory [22], [23], [24], [25], [26] and evidence theory [27], [28], [29], [30], [31], [32], [33], [34].

The purpose of this presentation is to illustrate alternatives to probability for the representation of epistemic uncertainty in QMU-type analyses. The following topics are considered: the mathematical structure of alternative representations of uncertainty (Section 2), alternative representations of epistemic uncertainty in QMU analyses involving only epistemic uncertainty (Section 3), and alternative representations of epistemic uncertainty in QMU analyses involving a separation of aleatory and epistemic uncertainty (Section 4). The examples in 3 QMU with epistemic uncertainty: characterization with alternative uncertainty representations, 4 QMU with aleatory and epistemic uncertainty: characterization with alternative uncertainty representations use the same two notional examples previously used in Ref. [1]. The presentation then ends with a summary discussion (Section 5). The content of this presentation is adapted from Sections 8–10 of Ref. [35].

Of indicated representations, interval analysis and probability theory are the most widely used. The introduction of other uncertainty representations such as possibility theory and evidence theory has been accompanied by an extensive discussion of their potential use as alternatives to probability theory for the characterization of uncertainty, with some individuals maintaining that the use of these alternative representations is essential to an adequate representation of uncertainty in situations involving limited information and other individuals maintaining that probability provides the only appropriate structure for the representation of uncertainty [36], [37], [38], [39], [40], [41], [42], [43], [44]. For perspective, several comparative discussions of these different approaches to the representation of uncertainty are available [45], [46], [47], [48], [49], [50], [51], [52], [53]. The view of the authors is that interval analysis and probability theory will remain the dominant structures used in the representation of uncertainty but there will be analysis situations involving limited information where the use of other representations for uncertainty, especially evidence theory, could be appropriate and beneficial.

It is important to recognize that, at least at an intuitive level, the need for alternatives to probability theory for the representation of epistemic uncertainty has already been introduced into the discussion of QMU. In particular, the following statement appears on p. 47 of Ref. [54]: “we require positive evidence that a nuclear weapon will work; absence of evidence that it will not work is not sufficient.”. The preceding quote is an informal expression of the uncertainty information that evidence theory is intended to formally capture and communicate. As discussed in Section 2, evidence theory provides two measures of uncertainty: belief and plausibility. In the context of the preceding quote, belief provides a measure of the amount of “positive evidence” that supports the truth of a proposition, and plausibility provides a measure of the “absence of evidence” that refutes the truth of a proposition. Related concepts of necessity and possibility are present in possibility theory as discussed in Section 2. Thus, it should not be assumed that probability provides the only possible mathematical structure for the representation of uncertainty in QMU analyses. The possible use of alternatives to probability for the characterization of epistemic uncertainty is recognized in the National Academy of Science/National Research Council (NAS/NRC) report on QMU (p. 29, Ref. [55]).