12.4.2 Bayesian methods
Juang et al.（2013）proposed a Bayesian framework using field observations for back⁃analysis and updating of soil parameters in a multistage braced excavation.The framework was adapted for the KJHH model（Kuang et al.，2007a），a semi⁃empirical equation for predicting the maximum wall deflection or maximum settlement.Both the updating methodology using one and two type response（wall deflection or maximum settlement）were given.
1）Updating soil parameters using one type of response observation
As for updating use of only one type of response，the implementation starts with expressing the model as follows：
where y＝predicted response；δ（θ）＝prediction model；θ＝vector of the soil parameters；and c＝model bias factor，which represents the model uncertainty.
Based on Eq.12.43，the likelihood that the prediction（y）is equal to the observation（Yobs）can be expressed as a conditional probability density function（PDF）ofθ：
where L
＝likelihood；and the notation N＝normal PDF that is function of[ Yobs／δ（θ）].It is noted that at a given Yobs，the term N [ Yobs／δ（θ）]is a function ofθonly.Recalling that c＝y／δ（θ），this normal PDF can be characterized with a mean ofμc and a standard deviationσc.In a Bayesian framework，given a prior PDF，f（θ），the posterior PDF of can be obtained as follows：
where m1 is a normalization factor that guarantees a unity for the cumulative probability over the entire range ofθ.
2）Updating soil parameters using two types of response observation
To update the soil parameters with the observation of both wall deflection and surface settlement，the likelihood equal to the corresponding observations is a conditional PDF ofθ：
N2＝PDF of a bivariate normal distribution with a mean vector［μ］＝[ μ1，μ2]and a covariance matrix of
where
，andρ＝correlation coefficient between the two model bias factor c1 and c2.The preceding formulation is simply an extension of the formulation presented in Eq.12.44 from using one type of observation to using two types of observations.Similarly，the posterior PDF ofμ updated with two types of observations can be obtained as follows：
where m1 is a normalization factor that guarantees a unity for the cumulative probability over the entire range ofθ.
The posterior distribution may be obtained through optimization or sampling techniques.Markov Chain Monte Carlo（MCMC）simulation method is an efficient sampling technique that can yield samples of a posterior distribution.MCMC performs a random walk within the domain defined by the uncertain soil parameters according to their prior distributions.At each random walk，if the likelihood of model predictions matching the observations is increased，then the candidate point is accepted.Otherwise，the candidate is rejected.One advantage of MCMC is that the computation of the normalization factor may be avoided，which is generally difficult for multiple dimensional problems.
3）Procedure of Markov chain Monte⁃Carlo simulation using the Metropolis⁃Hastings algorithm
Juang et al.（2013）adopted the Metropolis⁃Hastings algorithm（Metropolis et al.，1953；Hastings，1970）for its efficiency to implement MCMC sampling of the key parameterθfor its posterior PDF.The procedure can be summarized as follows：
①At Stage k＝1，determine the first pointθ1 in the Markov chain.This first instanceθ1 may be obtained by random selection from the prior distribution or may simply be assigned the mean value.
②At next Stage k（k starts from 2），randomly generate a newθp from a proposal distribution
，which is assumed to be a multivariate normal distribution where the mean is set to be the current pointθk－1 in the Markov chain and the covariance matrix is equal to sCθ，where s is a scaling factor and Cθis the covariance matrix of the prior distribution ofθ.The multivariate normal distribution is chosen for its good convergence properties in the Bayesian inference.
③Generate a random numberθfrom a uniform distribution U（0，1）
④Compute the ratio of densities r：
where 
is the unnormalized posterior PDF，which is essentially Eq.12.45 or 12.49 without the normalization factor.Note that
is q（θ）evaluated at y＝yobs.
⑤Determine whetherθp is acceptable（and thus yields a new point in the Markov chain）with the following acceptance rule：if u≤r，then up is acceptable and setθk＝θp；otherwise，setθk＝θk－1.Then go back to Step 2.
⑥Repeat Steps 2⁃5 until the target number of samples（i.e.，Markov chain length）is reached.
The Metropolis⁃Hastings algorithm randomly samples from the posterior distribution.Typically，initial samples are not completely valid because the Markov chain has not stabilized.These initial samples may be discarded as burn⁃in samples.Several factors influence the efficiency of sampling posterior distribution with the MCMC approach，such as the proposal distribution，Markov chain length，and number of burn⁃in samples.Therefore，the construction of a Markov chain is problem specific and needs to be examined case by case.It should be noted that other MCMC algorithms may be used in this procedure as long as efficient Markov chains can be achieved.
Results from Juang et al.（2013）suggested that the Bayesian updating is not much affected by the assumed prior distributions and the levels of the coefficient of variation of the soil parameters.Thus，while prior knowledge is important，the Bayesian updating with observations through stages of excavation can reduce the influence of this prior knowledge，and converged results can be obtained even if the prior knowledge is imperfect.