Environmental Engineering Reference
In-Depth Information
necessary to collect further information and thus to reduce the uncertainty before the final
design and construction can be decided. Exceptions from this low reliability can occur only
when the random variables with weakly informative priors have no or only limited influence
on the limit state.
The a priori reliability can also provide an indication on how much measurements are nec-
essary and optimal. This is formalized in the value-of-information concept of the Bayesian
decision analysis (Straub 2014).
5.3.3 Describing observations and data: the likelihood
Measurements (as well as other observations)
Z
are described by the likelihood. It is defined
as the conditional probability of making the measurement given a particular system state,
Pr(
Z
|system state). The system state can be, for example, the failure event
F
. In case the sys-
tem state is defined by the continuous random variables
X
, the likelihood function is defined
as being proportional to the probability of making the measurement when the uncertain
parameters
X
take a value
x
:
L
()
x
∝
Pr(
Z
|
X
=
x
).
(5.15)
Note that the likelihood is a function of
x
, even though it describes the probability of the
measurement outcome. Sometimes the likelihood is also denoted as
L
(
x
|
d
), where
d
is the
measured data (it is
Z
==
dd
).
The likelihood is not only used for Bayesian analysis, but is also a cornerstone of classical
statistics, where the maximum likelihood estimator (MLE) is the most common approach
to statistical inference (Fisher 1922); see also
Chapter 4
of this topic.
If multiple measurements or observations
Z
1
,
Z
2
, … ,
Z
m
are available, likelihood functions
L
i
,
i
= 1, … ,
m
, can be established for all of them individually. If these measurements are sta-
tistically independent for the given model parameters
X
, then it follows that the joint proba-
bility of all measurements is Pr(
Z
|
X
=
x
) = Pr(
{
}
m
ZZ
∩∩∩…= =∏
Z
|
Xx
)
Pr(
Z
|
Xx
=
)
1
2
m
i
=
1
i
and hence the joint likelihood is
m
∏
1
L
()
x
=
L
( .
x
(5.16)
i
i
=
The assumption of independence is commonly made in practice. Situations where this
assumption does not hold are discussed later.
In the following, the derivation of the likelihood function for single measurements
Z
i
is
described for different classes of measurements.
5.3.3.1 Measurement x
i
of a parameter X
In some applications, it is possible to directly measure a model parameter
X
. If the measure-
ment was perfect, then there would be no more uncertainty on
X
. In this case, the random vari-
able would become a deterministic parameter with value*
X
=
x
i
. However, in almost all real
applications, measurements are subject to measurement error or uncertainty, often because
the measurement is only indirect. The measurement error ε
is modeled probabilistically by
*
The corresponding likelihood function would be the Dirac delta function with argument
x
i
−
x
: δ(
x
i
−
x
).
However, there is no need to perform Bayesian updating, and it is sufficient and straightforward to replace
X
with
x
i
in this case.
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