Bayesian analysis for learning and updating geotechnical parameters and models with measurements - Risk and Reliability in Geotechnical Engineering - page 221

0.35

0.3

Posterior f μ ′′ (μ φ )

0.25

Likelihood L (μ φ )

0.2

Prior f μ ′ (μ φ )

0.15

0.1

0.05

0

10 15 20 25 30 35 40 45 50

μ φ (°)

Figure 5.2 Prior PDF of the mean friction angle, likelihood function describing the joint samples, and pos-
terior PDF of the mean friction angle. The likelihood function is scaled by [

∞

µ ϕϕ 1 ∫ −

L () d ]

−

.

∞

The difficulty in Bayesian updating stems from the need to compute the integral

∫ ∞

X d Analytical solutions are available only for special cases, namely
for the so-called conjugate priors as discussed in Section 5.3.4. Efficient numerical integra-
tion using quadrature rules, as implemented in standard mathematical software, is appli-
cable only when the number of random variables in X is small, in the order of 3. For most
realistic problems, tailor-made numerical methods are required to solve the Bayesian updat-
ing problem. The most popular among these is the family of Markov chain Monte Carlo
(MCMC) methods, which allow sampling directly from ′′

Pr() Z = L () () . xxx f ′

−∞

f X without determining the con-
stant a, but many alternative methods have been proposed. Methods for the numerical solu-
tion are presented in Section 5.4.
Ultimately, the goal of the analysis is to make predictions on the performance of the geo-
technical system and, sometimes, on individual parameters of the model. In case the interest
is in the performance expressed through one or more failure events F , the prediction is in
terms of the updated Pr( F|Z ). This can be obtained through direct application of Equations
5.1 or 5.2 , which is the preferred approach for many problems and which is presented later
in Section 5.3.5. Alternatively, it is possible to first update the probability distribution of
the model parameters X to ′′

f X following Equation 5.3 and then perform a probabilistic reli-

ability analysis with ′′

f X . This approach is illustrated in the following.

illustration 2: Bayesian updating of the geotechnical reliability

We compute the reliability of a centrically loaded square footing with dimensions l = 3 m
and t = 1 m, illustrated in Figure 5.3. This example is modified from Oberguggenberger and
Fellin (2002). The footing will be in the silty soil described in Illustration 1, with a mean
friction angle μ φ and cohesion c = 0.