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

5.4 Controlling Epistemic Uncertainty Through Classical or Bayesian Estimators

So far the estimation techniques have been devoted to the estimation of the conditional (or single-probabilistic) distribution f_X (·|θ_X) of uncertain inputs X. As indicated in Chapter 2, a careful description of the extent of uncertainty should also account for sample limitations. The full aleatory and epistemic uncertainty model involves an additional distribution designed to represent (to some extent) those additional estimation uncertainty sources:

(5.65)

Such an estimation will be discussed first with a classical approach and then a Bayesian one. The Bayesian approach (see Section 5.4.3) offers a more natural statistical interpretation of the double-level probabilistic structure and of its updating through the introduction of a data sample. However, classical statistics are easier to manipulate in a number of cases, particularly in the Gaussian case where closed-form expressions are available even for very small samples (see Section 5.4.2), or when the sample is large enough for the asymptotic theory to apply (see Section 5.4.3).

5.4.1 Epistemic Uncertainty in the Classical Approach

It is necessary to be more specific with the definition of the associated distributions that will be manipulated. The conditional distribution f_X (·|θ_X) should first be understood as a model for each observation X_j of ...