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

4.2 Understanding the Impact on Margins of Deterministic vs. Probabilistic Formulations

Chapters 1 and 2 have shown the abundant use of mixed deterministic-probabilistic settings in natural or industrial risk, whereby some sources of uncertainty (e.g. epistemic, error,. . .) are ‘covered’ by the use of ‘penalized’ values, conditional on which probabilistic risk measures are built with the remaining uncertain inputs. This section will discuss in more depth the key properties that motivate such treatment implicitly, in order to clarify the associated assumptions and impacts on decision-making.

4.2.1 Understanding Probabilistic Averaging, Dependence Issues and Deterministic Maximisation and in the Linear Case

An essential feature of probabilistic uncertainty modelling is related to its fundamental averaging effect, that is error compensation, or risk mutualisation. To start with, a linear system model will clarify its meaning and help understand to what extent it is preferable (or not) to a deterministic alternative.

Error Compensation in the Linear Model

Consider the following linear system model predicting a scalar output of interest z from a given uncertain input vector x and set of actions d:

(4.1)

Although simple enough, note that such a model already covers a large domain: it does not need to be linear with respect to actions d, and may even provide a fair approximation of a ...