Bayesian analysis for learning and updating geotechnical parameters and models with measurements - Risk and Reliability in Geotechnical Engineering - page 241

uncertainty in model choices. These random variables are not included in the reliability
analysis; instead, the reliability is computed conditional on these variables. These random
variables then lead to the reliability index being a random variable itself (Kiureghian and
Ditlevsen 2009).
One example of this approach is found in Bayesian model class selection (Yuen 2010).
Thereby, different models are assigned prior probabilities, which can be updated if measure-
ments of system performances are available. The resulting prediction is made by combining
predictions from different models through the total probability theorem. However, it is also
possible to explicitly communicate the results obtained with the different models.
It has been proposed to model the uncertainty on model choices by nonprobabilistic
approaches, including, for example, interval theory or fuzzy set theory (Rubio et al. 2004).
This is not advocated here, since we believe that probabilistic methods are entirely capable
of dealing with these uncertainties without the need to resort to another technique. Also,
while the researchers developing such alternative methods are mostly aware of the underly-
ing limitations these model, other engineers are not. Hence, we believe that these methods
just add to the confusion. However, independent of which method one uses to represent
the uncertainty on model choices, the main difficulty in explicitly showing the uncertainty
associated with the model lies in selecting which parameters are model choices and which
are uncertainties inherent to the problem. As an example, reconsider the simple example of
Illustrations 1, 2, and 10. The parameter σ φ , the standard deviation of φ, can be selected by
the modeler, as we have done here; in this case, it is a model choice. Alternatively, the param-
eter σ φ can also be considered as a random variable, as we have done in Illustration 7; in this
case, it is no longer a model choice. Therefore, in the second case, the model uncertainty
would be reduced, without really changing the computations. For this reason, for practical
applications, it is advocated that the uncertainty on model choices is not further formalized.
Instead, the crucial model selections should be made explicit by parameter studies.
5.4 aDVanCeD algorIthMS For eFFICIent anD eFFeCtIVe
baYeSIan uPDatIng oF geoteChnICal MoDelS
In this section, three main strategies for the numerical solution of the Bayesian updating
problem are outlined. These are the MCMC approach (in Section 5.4.1), the particle filter
or sequential Monte Carlo approach (in Section 5.4.2), and the BUS approach, including
the basic rejection sampling (in Section 5.4.3). The principles of the methods are introduced
through basic algorithms, and more advanced and efficient algorithms are briefly described.
A detailed presentation of these methods is well beyond the scope of this chapter, but the
relevant references are given for readers who want to implement them.
5.4.1 Markov chain Monte Carlo
MCMC is a powerful approach for generating samples from distributions that are difficult
to sample from directly. The main advantage of MCMC methods is that they do not require
complete specification of the distribution from which one wants to sample. This is particu-
larly useful for Bayesian updating, whereby the posterior distribution is known only up to
a normalizing constant.
The basic idea of MCMC is to construct a stationary Markov chain with invariant dis-
tribution equal to the target distribution (Tierney 1994; Gelman 2004). MCMC methods
have their roots in the Metropolis algorithm (Metropolis et al. 1953) that was developed
for computing complex integrals with application to statistical physics. Hastings (1970)
presented a generalization of the original Metropolis algorithm that encompasses several