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

10.2 Annex 2 – Comments About the Probabilistic Foundations of the Uncertainty Models

The probabilistic setting is a central feature of the book. In this section, the formal foundations of the underlying model are given more detail although it does not aim to provide a full discussion of the foundations of probability and statistics as well as their interpretation in the risk and uncertainty litterature. The resulting setting is comparable with the well-publicised views of Kaplan and Garrick (1981) or Helton (1993) albeit with different notation and a few additional features.

10.2.1 The Overall Space of System States and the Output Space

Suppose that the whole state of the system (states of natural or internal initiating events, true values of the mesurand in metrology, existing design characteristics or actions taken, as well as all consequences of interest such as fatalities, costs, etc.) can be described in a point ω of a space Ω which may be very complex: ω may contain an infinite (even non-countable) collection of real values characterising all components of the systems, all material properties and so on, or even of real functions characterising the trajectories in time of flows, loadings, and so on. Containing virtually any possible state of the system (including maybe any state in any choice of design variables), Ω may be referred to as the underlying overall sample space, space of system state, or ω -space.

In the decision-making process, we generally wish to control essentially ...