Bayesian analysis for learning and updating geotechnical parameters and models with measurements - Risk and Reliability in Geotechnical Engineering - page 229

typically due to measurement error, but sometimes also includes model errors;* it is mod-
eled through the PDF f ∈ (). Assuming an additive error, the following relationship holds:
y i − h i ( X ) = ε i . The likelihood function L i ( x ) describing this observation is therefore

L () x = f ( yh − ( )). x

(5.21) i ε i i

i

More generally, the likelihood function for measurements y i of continuous quantities is

L () x = f ( y |, x )

(5.22)

i Y | X i

i

where f Y i | X is the conditional PDF of the measured quantity given X = x , and, as in Equation
5.21 , involves the outcome h i ( x ) of the geotechnical model.
In case the measurements or observations Z are on discrete quantities or events, they are
characterized by a finite probability of occurrence, for example, observations of categorical
values or observations of system performances such as failure/survival, and also censored
data. One then refers to inequality information (Madsen et al. 1985; Straub 2011). Such
information can be described through

n

Z =∈ ≤ { x R : h () x 0 }, (5.23)

i i

where h i is a function that describes the relation between the observed event and the model
parameters X . In structural reliability, h i is known as an LSF. Inequality information can be
interpreted as an observation that X must be in the domain {()

h i x ≤ 0 }. The corresponding

likelihood function is then

( ) =

L () Pr x = Z | X = x Ih (() x ≤ 0 ). (5.24)

i i i

where I is the indicator function. This likelihood thus takes on values 0 or 1, because for
given X = x , the event Z i either occurs, L i ( x ) = 1, or does not occur, L i ( x ) = 0.

illustration 4: Likelihood of a spatially distributed soil parameter

Reconsider the example given in Illustration 1, where the likelihood function was introduced
without a detailed explanation. In Illustration 1, the measurements are samples of the spatially
variable friction angle φ. In constructing the likelihood function, it was assumed that the vari-
ability of φ within a site could be described by a normal distribution with fixed standard devi-
ation σ φ = 3° and mean value μ φ . Therefore, following Equation 5.19 , the likelihood function
for learning μ φ is the conditional PDF of φ given μ φ , that is, L i (μ φ ) = f φ (φ|μ φ ), which is the nor-
mal distribution with parameters μ φ and σ φ = 3°: L i () (

µ =° 13 / 2 π )exp[ ( ( − 1 2 / ϕµ − / 3 ° ) ].

2

ϕ i ϕ

The combined likelihood of m samples is given by Equation 5.20 :

2

m m

ϕµ −

1 1

2 ∑

∏

i ϕ

L () µ = L () µ = exp −

.

(5.25)

ϕ i ϕ

( )

m

3 °

32 ° π

i = 1 i = 1

The full measurements reported in Oberguggenberger and Fellin (2002) consist of 20 sam-
ples, of which only three were considered in Illustration 1. The full samples are reported in
Table 5.1 . To illustrate the effect of the numbers of measurements on the likelihood function,

* The representation of model errors is further discussed at the end of this section.