Environmental Engineering Reference
In-Depth Information
26.5
26
25.5
25
24.5
24
−10
−8
−6
−4
−2
0
2
4
6
8
10
4
3
2
1
0
−10
−8
−6
−4
−2
0
2
4
6
8
10
Location
z
(m)
Figure 5.7
Posterior mean and standard deviation of the friction angle {
φ
}.
a correlation length of zero instead of 2 m in
Equations 5.36
and
5.37
,
the results would be
identical to the random variable case.
As a last note, it is pointed out that Bayesian analysis can be carried out sequentially,
as long as measurements are independent for given
X
=
x
. The model can first be updated
with one measurement alone, the resulting posterior can then be updated with the sec-
ond measurement, and so on. This becomes obvious from noting that, for independent
measurements, the likelihood
L(
x
) is defined as the product of the likelihoods describing
the individual measurements,
Equation 5.16
,
L
= Π
1
Considering the case of two
measurements,
m
= 2, and inserting this expression in the general Bayesian updating formu-
lation,
Equation 5.3
, it follows
()
x
m
L
().
x
i
i
2
∏
f
()
x
=
aLf
() ()
x
x
=
aLf
() ()
x
x
=
a L
()
x
[
Lf
( )(
xx
)].
′′
′
′
′
X
X
i
X
22
11
X
(5.39)
i
=
1
First updating
Second updating
In other words, it is possible to initially compute a first posterior conditional on the
first measurement,
f
1
1
=
′
This posterior then becomes the new prior and
is again updated with the second measurement to
′′
()
x
a Lf
() ().
x
x
X
|
Z
X
22
1
This result can
be extended to
m
measurements. Therefore, it is always possible to consider measurements
sequentially, as long as they are independent for given
X
=
x
. The ordering of the measure-
ments is irrelevant. This possibility to do sequential updating is of particular use for moni-
toring geotechnical structures in-service. The posterior distribution at time
t
is a sufficient
description of all measurements made up to time
t
and it is not necessary to store all mea-
surement results explicitly.
f
()
x
=
a Lf
()
x
().
x
X
X
|
Z
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