Page 224 - Numerical Analysis and Modelling in Geomechanics

PARAMETER ESTIMATION USING EXTENDED BAYESIAN METHOD 205
where β=a positive scalar adjustment of the relative importance of the observed
data to the prior information, p=prior mean of the model parameter θ, and V =
p
prior covariance matrix of θ.
The main difference between the EBM and conventional Bayesian analysis is
the introduction of the scalar, β. The β parameter can be estimated again by the
Bayesian theorem maximizing the following function:

(7.14)

u* can be linearized as

(7.15a)

where
(7.15b)

(7.15c)

By employing Equation (7.14), the log-likelihood function can be obtained as

(7.16)

β should be chosen to maximize Equation (7.16). The AIC value can be
expressed, for the present study, as

(7.17)

where dim β is one for the proposed model.
Once we obtain β from Equation (7.16), parameter θ is to be estimated. Either
the Gauss-Newton method or a modified Box-Kanemasu iteration method can be
used to estimate θ by minimizing Equation (7.13) (Beck and Arnold 1977).

Uncertainty evaluation of model parameters
It is possible to reduce uncertainties by comparing the uncertainty of initially
estimated parameters (the prior estimation) with the uncertainty of the estimated