Bayesian analysis for learning and updating geotechnical parameters and models with measurements - Risk and Reliability in Geotechnical Engineering - page 220

illustration 1: Bayesian updating of a soil parameter

A geotechnical site consists of a silty soil, whose mean friction angle μ φ is to be estimated. It
has been found that the cohesion is close to zero and can be neglected. It is assumed that the
normal distribution describes the variability of the friction angle well (this choice is for illus-
tration purposes). On the basis of the previous measurements on similar soils in the vicinity,
it has been found that the mean value is commonly between 25° and 31°. We assume that
these values correspond to the 10 and 90% quantiles and we fit a normal distribution to
these values. It follows that the prior distribution of μ φ is the normal distribution with mean
′ =°

µ µ 28 and standard deviation σ µ

′ =

234 . ° :

2

µ 28

234

−°
°

1
2342

1
2

ϕ

f µϕ ′ () µ = exp − . (5.5)

.

ϕ

. ° π

Additionally, the engineer decides to take three samples from the site and carry out direct
shear tests. These result in the following observed values: φ 1 = 25.6°, φ 2 = 25.5°, φ 3 = 24°.
These values are taken from Oberguggenberger and Fellin (2002). The likelihood function
describing these three measurements is*

2

3

ϕµ −

°

1 1

2

∑

i ϕ

L () µ = exp − .

(5.6)

ϕ

( )

3

3

32 ° π

i = 1

The constant a can now be computed numerically following Equation 5.4 to a = 1313.4.
Following Equation 5.3 , the resulting posterior distribution is

f ′ () µ = aLf () () µ ′ µ

µϕ ϕµ ϕ

ϕ ϕ

2 2

3

µ 28

234

−°
°

i ϕµ

−
°

1 1

2342

1
2

1
2 ∑

ϕ ϕ

= a exp − −

( )

3

. 3

. ° π

32 ° π

i = 1

2

1
1392

1
2

µ − 26 . 08 °

ϕ

= exp − . (5.7)

139 . °

. ° π

As seen from the last line, the resulting posterior distribution of μ φ is again a normal dis-
tribution, with posterior mean σ µ 13. . Such spe-

cial cases, where the prior and the posterior distributions are of the same type, are known
as conjugate priors and are discussed in Section 5.3.4.

µ µ ′′ = 26 08 . ° ′′ = °

and standard deviation

The prior, the likelihood, and the posterior are shown in Figure 5.2 . It can be observed that
the posterior PDF is in-between the prior PDF and the likelihood. In this example, the prior
and the measurements have approximately equal influence on the posterior PDF. In many
instances, the posterior is mainly determined by the measurements, that is, the posterior
PDF is very similar to the likelihood, as discussed later.

* This likelihood is based on assuming that the variability of φ at the site is described by a normal distribution
with standard deviation σ φ = 3°. The derivation of this likelihood function is explained in Section 5.3.3.