Bayesian analysis for learning and updating geotechnical parameters and models with measurements - Risk and Reliability in Geotechnical Engineering - page 234

with the posterior distributions. That is, the posterior distribution is represented through
these samples, and a is not needed in these cases. This is described in the next section.

5.3.4.3 Advanced sampling methods
When Bayesian updating is to be performed for multiple random variables jointly, advanced
sampling methods are often the only computationally feasible approach. Arguably, the most
popular approach is the MCMC method, which allows to sample directly from the poste-
rior distribution without the need to compute the proportionally constant a , thus avoiding
the integration (Gilks et al. 1998; Gelman 2004). Many authors have applied and adopted
MCMC to Bayesian updating of mechanical models in general and geotechnical models in
particular, including Beck and Au (2002), Cheung and Beck (2009), Zhang et al. (2010),
and Sundar and Manohar (2013). The main problem of MCMC methods is that it can-
not generally be ensured that the samples have reached the stationary distribution of the
Markov chain, that is, the posterior distribution (Plummer et al. 2006). Various alternatives
to MCMC exist, which are mostly based on rejection sampling and importance sampling, for
example, the adaptive rejection sampling from a log-concave envelope distribution, which is
effective for updating single random variables (Gilks and Wild 1992), or generalized sequen-
tial particle filter methods for updating arbitrary static or dynamic systems (Chopin 2002;
Ching and Chen 2007). These methods are often combined with MCMC. The authors
have recently proposed a novel approach termed Bayesian updating with structural reli-
ability methods (BUS), which is based on principles from structural reliability (Straub and
Papaioannou 2014), but may also be considered as an extension of the rejection-sampling
approach, and which is particularly efficient for high-dimensional applications.
In Section 5.4, the three main strategies are presented in more detail: the MCMC approach,
the particle filter or sequential Monte Carlo approach, and the BUS approach.

5.3.4.4 Multinormal approximation of the posterior
For cases where large numbers of measurements are available, which contain strong infor-
mation relative to the prior distribution, the posterior can be approximated in terms of an
asymptotic expression (Beck and Katafygiotis 1998; Papadimitriou et al. 2001). Thereby,
the posterior PDF is approximated around its mode

x = argmax f ′′ (), x (5.32)

PM X

where the index PM stands for posterior maximum. Let a be the negative Hessian of the
logarithm of the posterior PDF evaluated at x PM , that is, a

=− ∇∇ln f

′′

(). x

Noting that the

X

f X () at x PM are zero, the second-order Taylor expansion of the log
posterior around x PM is obtained as

′′

first partial derivatives of

1
2

T

(5.33)

ln f ′′ () x ≈ ln f ′′ ( x ) − ( xx − ) ax ( − x ).

X X PM PM PM

Therefore,

1
2

T

f ′′ () x ≈ ′′ −− − f ( x )exp( xx )( ax x ) (5.34)

X X PM PM PM