documented in the risk analysis, whether qualitative or quantitative. Given these multiple sources, discussions of uncertainty can easily become complex.

All of these sources of uncertainty can be portrayed in an analysis by means of a probability density function (pdf). This is a normalized function that portrays the relative likelihood that an uncertain variable will be observed within a particular interval.

It is helpful in a practical way to characterize several classes of uncertainty, because careful thinking about the character of sources of uncertainty leads to a better understanding of the problem and better representation of the associated pdfs. Three classes of uncertainty should be considered:

• Deterministic case—there is no variability or there is no imperfect state of knowledge that leads to variability in the results;

• Aleatory uncertainty—there is random variability in any of the factors that leads to variability in the results; and

• Epistemic uncertainty—the state of knowledge about the effects of specific factors is less than perfect.

To help understand these terms, a more operational point of view is that uncertainty is aleatory if

• It is (or is modeled as) irreducible;

• The uncertainty is observable (i.e., repeated trials yield different results); or

• Repeated trials of an idealized thought experiment will lead to a distribution of outcomes for the variable, and thus this distribution is a measure of the aleatory uncertainties in the variable.

The uncertainty is epistemic if

• One is dealing with uncertainties in a deterministic variable whose true value is unknown;

• Repeated trials of a thought experiment involving the variable will result in a single outcome, the true value of the variable; or

• It is reducible (at least in principle).

The approach for treating uncertainty implements the subjective framework for treating probabilities in analysis described by Apostolakis (1990) (see the discussion of Bayesian methods below). This approach provides the benefit of a clearer (and potentially