Maximum likelihood principle and its application in soil liquefaction assessment - Risk and Reliability in Geotechnical Engineering - page 179

in the geotechnical profession. Hence, the objectives of this chapter are to introduce the
maximum likelihood principle with an emphasis on its application in geotechnical engineer-
ing, and to exemplify its use in the development of various probabilistic models for liquefac-
tion probability prediction. This chapter thus consists of two parts. First, the principle of
maximum likelihood is introduced with a focus on its applications in geotechnical engineer-
ing. Several examples are then presented to illustrate the development and application of
the maximum likelihood-based models for the evaluation of the liquefaction potential and
liquefaction-induced settlement.
4.2 PrInCIPle oF MaXIMuM lIkelIhooD
Let θ denote the parameters of an intended model to be estimated or calibrated, which is a
vector, and let d denote the observed data, which is a vector or matrix. Let f ( d |θ) denote
the joint probability density function of d given θ, or equivalently, the chance to observe d
given θ. When viewed as a function of θ, f ( d |θ) can be denoted as l (θ| d ), which is known
as the likelihood function. The maximum likelihood principle (e.g., Givens and Hoeting
2005) states that the optimal value of θ can be estimated by maximizing the likelihood
function. In other words, a series of θ values are assumed and the corresponding values
of the likelihood function l (θ| d ) are computed. The θ value yielding the highest likelihood
value has the greatest chance for observing the given (known) d, and thus is the optimal
value of θ.
A maximization of the likelihood function l (θ| d ) is equivalent to the maximization of
the log-likelihood function L (θ |d ), which is defined as L (θ |d ) = ln l (θ| d ). In that the maxi-
mum of a log-likelihood function is more easily evaluated, the log-likelihood function is
often maximized for estimating the parameters (e.g., Givens and Hoeting 2005). Under
some general regular conditions and for most distributions used as models in practical
applications, the maximum likelihood estimate has the following properties (e.g., Barnett
1999; Gentle 2002):
1. Consistency—As the number of observations increases, the maximum likelihood esti-
mator will approach the true value.
2. Normality—As the number of observations increases, the estimate of θ , denoted as
θ*, tends toward a normal distribution with a mean of θ and a covariance matrix of
h −1 with h ij = E(−∂ 2 L (θ| d )/∂θ i ∂θ j ). A consistent estimator of h is the negative Hessian
matrix of the log-likelihood function evaluated at θ* (Givens and Hoeting 2005).
3. Invariance—If θ* is the maximum likelihood estimate of θ, then τ* = g(θ*) is also the
maximum likelihood estimate of τ = g(θ).
Interested readers are referred to Lehmann and Casella (1998) for further discussions
on the regular conditions where the maximum likelihood estimates exhibit good statisti-
cal properties, and to Cam (1990) and Cheng and Traylor (1995) for counterexamples
in nonregular conditions. Edwards (1974), Aldrich (1997), and Stigler (2007) provide a
review of the historical development of this maximum likelihood method as applicable
to statistics.
Two major challenges are often inherent in the application of the maximum likelihood
principle: (1) in constructing the likelihood function l (θ| d ) and (2) in solving the maximiza-
tion problem. In the following sections, we will illustrate how the likelihood function can
be constructed for different types of data, and how the maximum likelihood point can be
found for several typical geotechnical problems.