Risk analysis involves modeling, explicitly or implicitly. As such, it is subject to error and uncertainty. Model error is the difference between reality and its representation in the form of a model. Symbolically, Δ = *r* − *m,* where Δ is the error, *r* is the reality (actual behavior), and *m* is its representation (or prediction) by the model. Models are usually created for a purpose, and the level of accuracy or the correspondence between the model and reality should be reviewed in the context of that purpose. A model could be accurate with respect to one aspect of reality and in relation to a certain purpose but inadequate in a different context. Any risk model has a “form” (or structure) and a set of “parameters.” Therefore, the assessment of the magnitude of Δ is subject to two sources of uncertainty—one stemming from the structure and form of the model, generally referred to as *model uncertainty,* and another due to uncertainty in the model parameters, generally known as *parameter uncertainty.*

Uncertainties exist for many reasons, including randomness, incomplete knowledge with regard to phenomena, inaccuracies in determination of the values of quantities and parameters (e.g., a probability value), high sensitivities of system performance to specific conditions, and omission of important factors (e.g., a basic event) from an analysis. Such sources of uncertainty should be

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APPENDIX E
Uncertainty, Sensitivity,
and Bayesian Methods
SOURCES OF UNCERTAINTY
Risk analysis involves modeling, explicitly or implicitly. As such, it
is subject to error and uncertainty. Model error is the difference
between reality and its representation in the form of a model. Sym-
bolically, Δ = r − m, where Δ is the error, r is the reality (actual
behavior), and m is its representation (or prediction) by the model.
Models are usually created for a purpose, and the level of accu-
racy or the correspondence between the model and reality should
be reviewed in the context of that purpose. A model could be
accurate with respect to one aspect of reality and in relation to a cer-
tain purpose but inadequate in a different context. Any risk model
has a “form” (or structure) and a set of “parameters.” Therefore, the
assessment of the magnitude of Δ is subject to two sources of
uncertainty—one stemming from the structure and form of the
model, generally referred to as model uncertainty, and another
due to uncertainty in the model parameters, generally known as
parameter uncertainty.
Uncertainties exist for many reasons, including randomness,
incomplete knowledge with regard to phenomena, inaccuracies in
determination of the values of quantities and parameters (e.g.,
a probability value), high sensitivities of system performance to
speciﬁc conditions, and omission of important factors (e.g., a basic
event) from an analysis. Such sources of uncertainty should be
194

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Uncertainty, Sensitivity, and Bayesian Methods • 195
documented in the risk analysis, whether qualitative or quanti-
tative. 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 normal-
ized 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 subjec-
tive 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

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196 • Risk of Vessel Accidents and Spills in the Aleutian Islands
simpliﬁed) elicitation-based quantiﬁcation process. This beneﬁt arises
from the subjective framework’s distinction between aleatory and
epistemic uncertainties, which requires a careful examination of the
factors contributing to uncertainty, resulting in a clearer deﬁnition
of the issues being addressed during the elicitation. It provides a
clear way to tell the truth about what is and is not known. For this
reason and with clear documentation, the analysis or elicitation can
be well defended.
The implications of the different values the uncertain variable
might take and their corresponding probabilities of being true can
be propagated to the ultimate answer obtained concerning the risk
level (see the discussion of uncertainty propagation below).
The main way in which incompleteness-related uncertainty can be
treated by using a pdf relates to the magnitude by which an analysis
could be in error because of omission of an important structural
element. The possibility of an omission is treated in terms of the
magnitudes of the consequences within an analysis to which such
an omission is considered likely to contribute.
SENSITIVITY ANALYSES
AND UNCERTAINTY PROPAGATION
Once the magnitudes of uncertainties of the factors of a risk analy-
sis have been determined, they can be used either to examine the
sensitivities of the results to variations in the factors within the
stated domain or to propagate the effects of the uncertainties to
ultimate risk results. Both types of analysis are important in per-
forming risk assessment, since uncertainties typically cannot be
eliminated (although their magnitudes can be reduced) and should
be addressed.
In sensitivity analyses, the inﬂuences of different values of a fac-
tor on ultimate answers are estimated. This is done to identify
the factors of greatest sensitivity and to obtain a rank ordering of
such sensitive factors. This knowledge can be valuable in iden-
tifying the factors for which improvement in performance would
be most valuable and those for which degradation of performance
is most important to avoid. This knowledge is also valuable to

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Uncertainty, Sensitivity, and Bayesian Methods • 197
the analyst in helping to resolve or understand the epistemic uncer-
tainty involved. In fact, there is a trade-off between the cost of
an analysis or experiment and the potential impacts of living with
resolvable epistemic uncertainty.
In propagation of uncertainties, the pdfs discussed above are
used in selecting the individual values of the uncertain factors
on the basis of the Monte Carlo technique in order to propagate
to the ultimate answers the inﬂuences of the important uncer-
tain factors, considered in combination. In one such sample, a
value of each factor is selected randomly, reﬂecting the inﬂu-
ences of its pdf. Then, the set of such values obtained from all
factors is used to obtain the values of the individual performance
measures of the system. When this procedure is repeated N times,
it will produce a set of N values of each performance measure.
Then, the relative abundance of values of a particular measure
found within a speciﬁed interval of measure values provides an
approximate estimate of the magnitude of the pdf of that mea-
sure corresponding to the interval speciﬁed. As the magnitude of
N increases, the error of this estimate will decrease asymptotically
toward zero.
Thus, one can obtain an approximate estimate of the range of the
measure consistent with a stated conﬁdence value. Since this estimate
reﬂects both subjective and objective factors, it should be viewed as a
statement of the estimator’s belief about the ﬁgure of merit (e.g., the
level of risk).
BAYESIAN STATISTICAL ANALYSIS
For risk assessment, the Bayesian approach offers important advan-
tages: all kinds of evidence are used, uncertainty bands are narrower,
and evaluating data with zero occurrences of events in N trials is
straightforward. The foundation of Bayesian statistical inference
is Bayes’ theorem. The basic idea is simple. From the calculus of
probability, we know that the probability of the joint occurrence
of two events A and E is the following:
P ( A ∩ E ) = P ( A) i P ( E A) = P ( E ) i P ( A E )

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198 • Risk of Vessel Accidents and Spills in the Aleutian Islands
which is a simple, uncontroversial statement. The Rev. Thomas Bayes
(1958) went a step further by rearranging the above as
P ( A) i P ( E A)
P ( A E) =
P (E)
and interpreting these terms as follows:
P(A⏐E) = “posterior” probability of A, after collecting evidence
E (say, the result of an experiment);
P(A) = “prior” probability of A; and
P(E⏐A) = “likelihood” of the evidence (if the evidence can
take on several values E1, E2, . . . , the likelihood is the
probability of getting evidence Ei, given the prior).
This Bayesian switch (called inverse probability) allows the use of
something that is known or can be calculated (the likelihood of the
evidence) to determine the value of P(A) (given all the evidence at
hand). It leads to a meaningful deﬁnition of subjective or state-of-
knowledge probability. Details of the approach can be found in many
standard sources (De Finetti 1975; Jefferies 1961).
E can be historical data, results of experiments, or expert opinion,
all of which may be subject to uncertainty and ambiguity. General-
ized forms of Bayesian inference developed in recent years provide
ways of using such information in developing epistemic uncertainty
distributions for risk model parameters.
REFERENCES
Apostolakis, G. 1990. The Concept of Probability in Safety Assessments
of Technological Systems. Science, Vol. 250, pp. 1359–1364.
Bayes, T. 1958. An Essay Towards Solving a Problem in the Doctrine of Chances.
Philosophical Transactions of the Royal Society, Vol. 53, pp. 370–418; Vol. 54,
pp. 296–325. Reprinted in Biometrika, 1958, Vol. 45, pp. 293–315.
De Finetti, B. 1975. Theory of Probability: A Critical Introductory Treatment,
Volumes 1 and 2. John Wiley and Sons, New York.
Jefferies, H. 1961. Theory of Probability, 3rd ed. Oxford University Press, New York.