Maximum likelihood principle and its application in soil liquefaction assessment - Risk and Reliability in Geotechnical Engineering - page 190

geotechnical knowledge, the Bayesian Information Criterion (BIC) (Schwarz 1978) is often
used for comparing and ranking these models. Suppose there are r models under investiga-
tion, which are denoted as M 1 , M 2 , …, and M r , respectively. The statistical BIC of the i th
model M i can be calculated as follows (Schwarz 1978):

θ

BIC =− 2( L |M ,) d + k ln n (4.18)

i i

where θ* is the point where the likelihood function is maximized, L (θ ∗ | M i , d ) is the value of
log-likelihood function of model M i at point θ ∗ , k is the number of parameters of the model,
and n is the number of measurements. In Equation 4.18, the first term accounts for the
effect of fitting the model to the data and the second term for the complexity of the model.
If the model fits the data better, the likelihood function will increase, and the first term on
the right side of Equation 4.18 will decrease. Conversely, it is possible to raise the model it
by adding more parameters to the model, which will cause an increase in the second term
of the BIC, however. The BIC value will decrease if the reduction of the value of the first
term in Equation 4.18 overcomes the increase of the value of the second term i n Equation
4.18, and vice versa. The smaller the BIC is, the stronger a model is supported by the data.
The BIC can also be used to define a more intuitive quantity for model comparison, that is,
the probability that a model is true if the true model is among the candidate models under
consideration. The model probability of model M i given data d , P ( M i | d ), can be computed
as follows (e.g., Burnham and Anderson 2004):

[ ]

exp( − ∆ BIC/ ) 2

i

P ( M i | d = )

r

(4.19)

∑

exp( − ∆ BIC/ ) 2

j

j = 1

where

∆ i ( BIC IC ) = − min{ BIC } (4.20)

i j

j =12 , ,..., r

If the true model is not among the candidate models, then P ( M i | d ) may be interpreted as
the chance that model M i is the best among the candidate models considered.

eXaMPLe 4.6
In Example 4.1, the friction angle is assumed to follow the normal distribution.
Alternatively, the friction angle may be assumed to follow the lognormal distribution. As
shown in Figure 4.1, the maximum value of the log-likelihood function for the normal
distribution model is −75.781. In the normal distribution model, there are two uncertain
parameters, that is, k = 2. The number of observations is n = 25. Hence, its BIC value
is 158.000. Similarly, it can be shown that for the lognormal distribution model, the
maximum value of the log-likelihood function is −75.474, the number of parameter is
k = 2, and the number of observations is n = 25. Hence, its BIC value is 157.384. Based
on Equations 4.19 and 4.20, the probability of the normal and lognormal distribution
models as the best fit to the given data d are 0.424 and 0.576, respectively. In this case,
the lognormal assumption is more strongly supported by the data.