Civil Engineering Reference
In-Depth Information
•
Level 4
: develop methodological aspects of the theory, including proce-
dures for making various uncertainty principles operational within the
theory.
Rigorous and probabilistic applications of information theories are
reported in the literature (e.g. Kaplan 1981), and the need for different
uncertainty quantifi cation techniques is debatable (e.g. North 2010; Aven
2011). To illustrate different uncertainty quantifi cation techniques, the fol-
lowing theories, imprecise (interval) probability (Cui and Blockley 1990;
Walley 1991), possibility theory (Zadeh 1978; Dubois and Parade 1988), and
evidence theory (Dempster 1967; Shafer 1976), which may be useful for
dealing with uncertainties related to seismic risk assessment, are discussed
below. This is not a comprehensive coverage of the uncertainty theories;
interested readers should consult with Klir (2004) and Walley (1991) for
more detailed information.
6.2.1 Imprecise (interval) probability
The theory of imprecise (interval) probability was fi rst developed by Walley
(1996) as a generalization of probability. It uses an interval [
P
(
A
),
-
(
A
)] to
represent uncertainty about an event A, with lower probability
P
(
A
) and
upper probability
-
(
A
) where 0
-
(
A
)
≤
P
(
A
)
≤
≤
1. The imprecision in the
representation of the event
A
is defi ned by
()
=
()
−
()
Δ
PA
PA
PA
.
[6.1]
Evolution of the theory of imprecise probability is discussed by Aven
(2011). Walley (1991, 1996) and Klir (2004) have provided the following
justifi cation for the theory:
• Imprecision of probabilities is needed to refl ect the amount of informa-
tion on which they are based. The imprecision should decrease with the
amount of statistical information.
• Total ignorance can be properly modeled by vacuous probabilities,
which are maximally imprecise (i.e. each covers the whole range [0,1]),
but not by any precise probabilities.
• Imprecise probabilities are easier to assess and elicit than precise ones.
• We may be unable to assess probabilities precisely in practice, even if
that is possible in principle, because we lack the time or computational
ability.
• A precise probability model that is defi ned on some class of events
determines only imprecise probabilities for events outside the class.
•
When several sources of information are combined, the extent to which
they are inconsistent can be expressed by the imprecision of the com-
bined model.
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