Environmental Engineering Reference
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x
2
f
X
´
(
x
)
F
:
g
(
x
) < 0
g
(
x
) = 0
Z
e
:
h
(
x
,
p
) < 0
h
(
x
,
p
) = 0
x
1
Figure 5.8
Illustration of the Bayesian updating of the reliability. The observation
Z
e
reduces the sample
space to the domain {( ,)
hxp
≤
0 The posterior probability of failure Pr(
e
is com-
}.
FZ
|
)
=
Pr(
FZ
|
)
puted as Pr(
=∩
by solving the integrals in
Equation 5.41
.
(After Straub,
D. 2011.
Probabilistic Engineering Mechanics
26
(2): 254-258.)
FZ
|
e
)
Pr(
FZ
) /Pr(
Z
)
e
e
this approach does not necessitate to first compute the posterior PDF of
X
. Therefore, the
complete Bayesian updating and reliability computation is performed by solving two stan-
dard structural reliability problems.
An intuitive interpretation of the solution provided in
Equation 5.41
is as follows. The LSF
sense that Pr(
F
|
Z
) = Pr(
F
|
Z
e
). According to the definition of the conditional probability, it
is Pr(
FZ
|
)
= ∩ The numerator in
Equation 5.41
is Pr(
Pr(
FZ
) Pr( .
/
Z
F
∩
)
and the
e
e
e
e
5.3.6 Communicating the results
One of the difficulties in applying Bayesian analysis—as with any probabilistic analysis—is
the communication of the results and their interpretation. The client or other stakeholders
are not generally adept at probabilistic analysis. It is therefore important that significant
efforts are made to communicate the results and the assumptions underlying these results in
a clear manner. Graphical representations of inputs and results can help in this task, as can
sensitivity studies.
A problem that is specific to Bayesian analysis is that the theory is often criticized for
being “subjective,” due to the need for selecting a prior distribution.* This criticism is
an artifact from the time when the frequentist interpretation of probability was the lead-
ing school of thought. In the mathematical community, Bayesian analysis has long since
become an accepted method for probabilistic inference. It has been recognized that, when
solving practical problems, subjective model choices cannot be avoided. The advantage of
Bayesian analysis is that these choices are made more explicit. It should also be noted that
the Bayesian interpretation of probability is much closer to the intuitive understanding of
engineers than the frequentist interpretation. Nevertheless, as with any method, it is impor-
tant to not only communicate the results, but also the limitations of the modeling and the
assumptions underlying the obtained results.
*
Interestingly, those who make this criticism often have no difficulty in making deterministic model choices
based on their subjective experience.
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