Maximum likelihood principle and its application in soil liquefaction assessment - Risk and Reliability in Geotechnical Engineering - page 180

4.2.1 Independent observations
Let d 1 , d 2 , …, and d n denote n observed data. Assuming these observations are statistically
independent, the likelihood of θ, denoted as l (θ| d ), which is the chance to observe the data
d = { d 1 , d 2 , …, d n }, can be written as follows:

n

∏ 1

l ( θ | d ) = f ( d | θ )

(4.1)

i

i =

n

d |θ is the joint probability of observing d given θ, which is the product of n
probability terms [ f ( d i |θ), i = 1, n ] since d 1 , d 2 , …, and d n are statistically independent obser-
vations. Based on the maximum likelihood principle, the optimal values of θ can be obtained
by maximizing the above likelihood function, or equivalently, the logarithm of the likeli-
hood function is as follows:

where i f ( )

i

=1

n

∑ 1

L ( θ | d ) = ln ( f d | θ )

(4.2)

i

i =

The same optimal values of θ will be obtained regardless of whether the likelihood func-
tion ( Equation 4.1 ) o r the logarithm of the likelihood function ( Equation 4.2 ) is maximized.
However, maximizing the latter is more efficient computationally.

eXaMPLe 4.1
Table 4.1 lists the shear strength data of a clay core of a gravity dam. Suppose the friction
angle (φ) follows the normal distribution and we are interested in its mean and standard
deviation, which are denoted herein as μ and σ, respectively.
Let φ 1 , φ 2 , …, φ n denote n observed values of the friction angle φ, respectively. In this
example, d i = φ i , and d = { d 1 , d 2 , …, d n } = {φ 1 , φ 2 , …, φ n }. Based on the probability density
function of a normal distribution (e.g., Ang and Tang 2007), the chance to observe the
data d given θ, which is the likelihood of interest, can be written as follows:

Table 4.1 Test data from the Ankang Hydropower site
Test no. c (kPa) φ ( ° ) Test no. c (kPa) φ ( ° )

1 165 11.86 14 85 20.81

2 127 14.04 15 18 25.64

3 253 13.50 16 15 22.29

4 427 10.20 17 78 24.70

5 106 11.31 18 12 26.10

6 242 12.95 19 34 22.78

7 209 12.41 20 70 19.80

8 328 13.50 21 20 17.74

9 98 12.95 22 20 20.81

10 10 15.64 23 217 20.30

11 213 16.17 24 221 20.30

12 365 17.22 25 254 27.47

13 324 20.81
Source: Data from Tang, X.S. 2013. Computers and Geotechnics , 49, 264-278.