Page 222 - Numerical Analysis and Modelling in Geomechanics

PARAMETER ESTIMATION USING EXTENDED BAYESIAN METHOD 203
where x=arbitrary input vector, (x|θ )=probability distribution function of the
f
k
k
kth alternative model, and dim (θ )=number of model parameters for the kth model.
k
The best model among the various alternatives can be identified when the AIC(x)
value in Equation (7.3) is minimized.
The posterior distribution of model θ can be expressed as Equation (7.4) in the
Bayesian approach:

(7.4)

where θ is a model parameter vector, x is the input data vector, and Π (θ) is a
prior distribution of θ. The denominator is independent of θ; this is simply a
normalizing constant required to make g(θ|x) a proper density function.
Therefore, a Bayesian estimator of θ can be obtained by θ value that maximizes
Equation (7.5):
(7.5)

The major problem in employing the Bayesian approach is the selection of an
appropriate prior distribution Π(θ). Since this selection is subjective, Akaike
(1973) proposed a likelihood function, Equation (7.6):

(7.6)

where Π (θ) is an alternative prior distribution. Sometimes, a family of prior
k
distributions Π (θ|β) is employed instead of Π (θ) to indicate the possible prior
k
k
distribution, where β is called the ‘hyperparameter’, being generally less than the
model parameter θ. To select the best hyperparameter β, the AIC concept is again
used as follows:
(7.7a)

where

(7.7b)

The first term in Equation (7.7a) indicates the degree of model fitness to the
observed data, and dim β of the second term means the number of model
parameters. Equation (7.7a) is called the Bayesian version of the AIC, and it is
used to select the most appropriate prior information and model among the
various alternative models. Application of Equation (7.7a) to geotechnical
parameter estimation will be shown in the later section ‘Parameter estimation’.