Environmental Engineering Reference
In-Depth Information
of the testing equipment. This is inequality information, which can be written according to
Lx
()
=
Ix
(
−
x
≤ 0
)
(5.27)
i
m
with
I
being the indicator function.
It is commonly assumed, and often reasonably so, that the measurement outcomes are inde-
pendent for given model parameters
X
=
x
or Θ = θ. However, situations may arise where this
does not hold. The accuracy of measurements is frequently influenced by factors that are com-
mon to multiple measurements, for example, temperature. In such cases, the measurement
errors will be correlated, and
Equation 5.16
is no longer valid. If this correlation is known, it is
possible to include it in the formulation of the joint likelihood function. For example, if the indi-
vidual measurement errors are represented by a normal distribution, the joint probability distri-
bution of the errors, and therefore the likelihood function, can be expressed by the multinormal
PDF. An example of such a correlated error model is given in Straub and Papaioannou (2014).
In many instances, the likelihood includes model errors, that is, it reflects deviations of the
model predictions from the true value. As an example, reconsider Illustration 5. There, it was
implicitly assumed that the model prediction of the deformations
h
i
(
x
) is exact for correct
values of
x
. As every engineer knows, no deterministic model is 100% correct; neverthe-
less, the assumption of a correct model can be reasonable in a probabilistic context if one
or more random variables are included in
X
to represent the model errors (Ditlevsen 1982;
Zhang et al. 2009). If the model does not include error terms, then this model error must
be included in the likelihood function. The same holds if the model does contain terms that
describe the errors in the prediction of the quantity of interest (e.g., of the ultimate limit state
describing failure), but which do not adequately describe the error in the prediction of the
measured quantity, such as a deformation. If model errors must be included in the likelihood,
the deviation ε
i
of the measurement
y
i
from the model prediction
h
i
(
x
) is due to a combination
of the measurement error and the model error. The problem with this representation is that
the model errors related to different predictions
h
i
(
x
),
i
= 1, … ,
m
, will generally be depen-
dent. Unfortunately, this dependence is difficult or even impossible to estimate. By neglecting
it, however, one generally overestimates the effect of the measurements, which can lead to
strong overconfidence in the model and its parameters. The appropriate representation of the
model error is probably the most critical challenge for Bayesian updating in practice and is
the subject of ongoing research (Beven 2010; Goulet and Smith 2013; Simoen et al. 2013).
5.3.4 updating the model
Once the prior distribution and the likelihood are established, Bayesian updating of the
model parameters
X
or their distribution parameters Θ is performed following the basic
Equations 5.2
o
r
5.3
.
However, in practice, the application of these equations is not generally
straightforward and is often computationally demanding. For this reason, it is necessary to
choose an effective computational strategy. This choice will be a function of the probabilistic
model, in particular the number of random variables and the type of their distribution, and
the computational cost of the geotechnical model. In the following section, different compu-
tational strategies for implementing Bayesian updating are presented.
5.3.4.1 Conjugate priors
When performing Bayesian analysis of parameters Θ based on soil samples, it is often pos-
sible to select the so-called conjugate priors for the parameters, which lead to analytical
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