Empirical Comparison of Two Methods for the Bayesian Update of the Parameters of Probability Distributions in a Two-Level Hybrid Probabilistic-Possibilistic Uncertainty Framework for Risk Assessment

Abstract

In this paper, the authors address the issue of updating in a Bayesian framework, the possibilistic representation of the epistemically uncertain parameters of (aleatory) probability distributions, as new information (e.g., data) becomes available. Two approaches are considered: the first is based on a purely possibilistic counterpart of the classical, well-grounded probabilistic Bayes’ theorem; the second relies on the hybrid combination of (1) fuzzy interval analysis (FIA) to process the uncertainty described by possibility distributions, and (2) repeated Bayesian updating of the uncertainty represented by probability distributions. The feasibility of the two methods is shown on a literature case study involving the risk-based design of a flood protection dike

Hybrid Probabilistic–Possibilistic Treatment of Uncertainty in Building Energy Models: A Case Study of Sizing Peak Cooling Loads

“…598 hospitals), where quantifying the exact number of occupants is extremely difficult to quantify. Therefore, opting 599 for a hybrid probabilistic-possibilistic framework for Bayesian calibration of building energy models is a potential 600 for future studies (Pedroni et al 2015). The hybrid treatment of uncertainty could also be evaluated when dealing 601 with control regimes of autonomous building components (shading and lighting systems), where incomplete 602 knowledge over occupant behavior and a system's state may have dependencies.…”

A Critical Discussion and Practical Recommendations on Some Issues Relevant to the Nonprobabilistic Treatment of Uncertainty in Engineering Risk Assessment

“…Among the alternative approaches mentioned above, that based on possibility theory is by many considered one of the most attractive for extending the risk assessment framework in practice. In this article, we focus on this approach for the following reasons: (i) the power it offers for the coherent representation of uncertainty under poor information (as testified to by the large amount of literature in the field, see above); (ii) its relative mathematical simplicity; (iii) its connection with fuzzy sets and fuzzy logic, as conceptualized and put forward by Zadeh: actually, in his original view possibility distributions were meant to provide a graded semantics to natural language statements, which makes them particularly suitable for quantitatively translating (possibly vague, qualitative, and imprecise) expert opinions; and, finally, (iv) the experience of the authors themselves in dealing and computing with possibility distributions . One the other hand, it is worth remembering that possibility theory is only one of the possible “alternatives” to the incorporation of uncertainty into an analysis (see the approaches mentioned above).…”

“…For completeness, we report (and extend) some of the results obtained in a previous work by the authors, in which the purely possibilistic Bayes theorem described above is applied for updating the possibilistic parameters of aleatory PDFs in a simple literature case study involving the risk‐based design of a flood protection dike . In the risk model considered, the maximal water level of a river (i.e., the output variable Z of the model) is given as a function of several (uncertain) parameters (the inputs Y to the model), i.e., , with Y 1 = Q ∼ , Y 2 = ∼ , Y 3 = ∼ and Y 4 = ∼ ; the “priors” of the parameters of the aleatory PDFs of the inputs are represented by triangular possibility distributions: see Ref.…”

“…By way of example, Fig. illustrates the possibility distributions of the epistemically uncertain parameters γ and σ Ks of the aleatory PDFs (left) and (right) of Y 1 = Q and Y 4 = K s , respectively: in particular, the prior possibility distributions are shown as solid lines, whereas the marginal posterior possibility distributions obtained by Approaches A (Section ) and B using D 1 = 149 and D 4 = 5 pieces of data are shown in dashed and dot‐dashed lines, respectively; the point estimates produced by the classical MLE method are also shown for comparison (dots) …”

“…In this context, the main objective of this article is to show in a systematic and comprehensive framework how some conceptual and technical issues on the treatment of uncertainty in risk assessment (items (1)‒(4) above) can be effectively tackled outside the probabilistic setting: practically speaking, different approaches and methods will be recommended for efficiently addressing each of issues (1)‒(4) listed above; classical probability theory tools as well as alternative, nonprobabilistic ones (in particular, possibility theory) are considered. The recommendations are “informed” by (i) a critical review of the literature approaches to solving the specific issues and (ii) the research work of the authors on addressing these issues: with respect to the latter item (ii), some of the considerations are based on results contained in articles previously published by the authors; other conclusions are instead drawn from analyses originally presented in this article (e.g., part of the work related to the issue of Bayesian updating).…”