Bayesian analysis for learning and updating geotechnical parameters and models with measurements - Risk and Reliability in Geotechnical Engineering - page 240

The key assumptions that are made in a Bayesian analysis and which should be
addressed are

- The selected deterministic geotechnical model used in the analysis . A Bayesian analy-
sis must always be seen in the context of the underlying deterministic geotechnical
model used for the prediction.

- The selected probabilistic model framework . This concerns mainly the representation
of the spatial distribution of parameters, through a random field model or a popula-
tion approach (see Section 5.3.1).

- The selected prior probability distributions of parameters . In the case of a probabilistic
reliability analysis, this also includes the choice of those parameters that are not updated.

- The representation of the model error . As discussed in Section 5.3.3, the represen-
tation of the model error can be crucial, in particular, when updating models with
observed performances, for example, measured deformations.

A common approach to investigating and documenting the effect of assumptions on the results
are sensitivity analyses or parameter studies. These are particularly helpful in communicating
and also facilitate checking the plausibility of the results. Some care is needed in selecting the
parameter cases to be analyzed and the way they are presented. When presented with parameter
studies, clients often tend to select the most conservative results. In this way, when performing
an extensive parameter study, one runs some risk of ending up with a conservative suboptimal
solution. In this way, advanced probabilistic approaches are punished compared to simplistic
code-based approaches, where even crude assumptions are often not investigated further.

illustration 10: effect of the probability distribution model on the reliability

We reconsider the reliability problem described in Illustrations 1 and 2. There it was origi-
nally assumed that the friction angle follows a normal distribution with standard deviation
σ φ = 3°. We now consider additionally the lognormal and the β-distribution, and we test
the effect of changing σ φ = 3° to σ φ = 2.5° and σ φ = 3.5°. We assume a weakly informative
prior in all cases, a normal distribution with ′ =°

σ µ 100 (see
Illustration 3). The resulting posterior reliability indexes β ″ are presented in Table 5.2; they
are computed following Illustration 2 as ′′ =−

′ =° µ µ 30 , and standard deviation

−
β Φ µ ϕ

1

F
Evidently, the choice of the distribution type for the friction angle φ has a minor effect on
the reliability. This is due to the fact that the capacity is (assumed to be) determined by the
mean friction angle. On the other hand, the standard deviation σ φ has a significant effect on
the posterior reliability, since it determines the sampling uncertainty (the width of the likeli-
hood function).

[(. ′′ 21 36 ° )].

It is possible to formalize the analysis of uncertainties in model choices by adding an
additional layer of random variables. These additional random variables represent the

Table 5.2 Resulting posterior reliability indexes for different model
assumptions on the distribution of φ
Normal Lognormal Beta

σ φ = 2.5° 2.54 2.56 2.53

2.12 2.22 2.15

σ φ = 3°

1.82 1.99 1.87 σ φ = 3.5°