Maximum likelihood principle and its application in soil liquefaction assessment - Risk and Reliability in Geotechnical Engineering - page 189

in the calibration database should be as close to the true ratio in the region of interest as pos-
sible. The random sampling requirement can be compromised during the data collection
process in practice, however. For example, in that it is possible to place more emphasis on
the failed than stable slopes in a postevent investigation, the proportion of failed cases is
larger than actual real-world failures in the calibration database. In such a case, the effect
of the failed slopes on model calibration will be overrepresented in that database. Based
on Equation 4.14 , the average effect of a failed slope is ( 1 n F ) ∑

n

ln[ 1 − Φβ ( + ε β )], and

F

ci

i = 1

the average effect of a stable slope is (

= Φβ ε β If there was no sampling
bias in the calibration database, the number of failed and stable slopes should have been
( n F + n NF ) Q p and ( n F + n NF )(1 − Q p ), respectively, where Q p is the true proportion of failed
slopes. Thus, to remove this sampling bias, the contribution of likelihood regarding failed
slopes and stable slopes should be adjusted to [(

1 n NF ) ∑

n

ln ( + ).

NF

j 1 cj

n

nnQn
F

+ ) / ] ∑ ln[ 1 Φβ − ( + ε β )]

F

FpF ci

i = 1

and [( 1 / Φβ ε β respectively. Therefore, the likelihood

function of an unbiased calibration database with a sample size of n F + n NF is estimated as

nn + )( − Qn ) ] ∑

n NF

ln ( + ),

F F p NF j = cj

n

n

F NF

∑ ∑

(4.15)

L ( θ | d = ) w ln 1 − Φ ( β + ε ) + w ln Φ ( β + ε )

F ci β NF cj β

i = 1 j = 1

Q
Q

p
s

=

wn = ( + nQn ) (4.16)

F F NF pF

1
1

−
−

Q
Q

p
s

wn =+ − ( n )( 1 Qn ) = (4.17)

NF F F p NF

where Q s is the proportion of failed slopes in the database (i.e., Qnnn = ( /( + ))( = nn / )).

s F F F F

Equation 4.15 is often called the weighted likelihood function. The reasonableness of the
above equations can be checked with the special case of Q s = Q p , where the problem of
choice-based sampling bias vanishes, as w F = w NF = 1, and Equation 4.15 is reversed to
the likelihood function of Equation 4.14.
Originally proposed by Manski and Lerman (1977) for use in social science research,
this weighed likelihood method was later used in liquefaction probability analysis con-
ducted by Cetin et al. (2002). Zhang et al. (2007) also used such a weighted likelihood
method for calibrating several models for a reliability analysis of soil slopes in Hong Kong.

eXaMPLe 4.5
To illustrate this weighted likelihood method, assume that the problem described in
Example 4.4 occurs in a region where the proportion of failed slopes is 5% ( Q p = 0.05).
In the database shown in Figure 4.5 , Q s = 0.4, and based on Equations 4.16 and 4.17,
w F = 0.125, and w NF = 1.583. The spreadsheet template for implementing the weighted
likelihood method is shown in Figure 4.5 . Maximizing the weighted log-likelihood
function with respect to ε β , the best estimate of ε β in this example is 0.275. A positive
value of ε β indicates that the calculated reliability index underestimates the actual
reliability index. As the conventional un-weighted likelihood function is only a spe-
cial case of the weighted likelihood function with Q s = Q p , the spreadsheet template
in Figure 4.5 can also be used to maximize the conventional un-weighted likelihood
function.

4.2.4 ranking of competing models
In reality, it is possible that several competing probabilistic models can be developed for
the same problem. When it becomes difficult to justify which model is the best based on