Modelling Under Risk and Uncertainty: An Introduction to Statistical, Phenomenological and Computational Methods

2.5 Combining Probabilistic and Deterministic Settings

Probability is central to the approach taken in this book, although it proves necessary to mix it with deterministic settings. This section will first comment on the meaning of probabilistic uncertainty models and introduce the mixed probabilistic-deterministic settings.

2.5.1 Preliminary Comments About the Interpretations of Probabilistic Uncertainty Models

Different epistemological interpretations correspond to the various probabilistic structures available for uncertainty modelling. This choice is closely linked to that of the various natures of uncertainty about the state of the system which may be represented, such as natural time or space variability, lack of knowledge and so on: Chapter 4 will return to that discussion. At this stage, only preliminary observations will be made.

The first interpretation, frequentist or classical, would consider x, e and z as observable realisations of uncertain (or variable) events (or properties of the system) occurring several times independently so that, at least in theory, the frequency records of both variables would allow for the inference and validation of the probability distribution functions (for both inputs and output). In that context, modelling probabilistic distribution functions in the inputs may be seen as a basis for the inference of some output quantities of interest, such as a probability to exceed a regulatory threshold or an expected cost. Taking such a frequentist ...