Bayesian analysis for learning and updating geotechnical parameters and models with measurements - Risk and Reliability in Geotechnical Engineering - page 243

illustration 11: Bayesian updating of friction angle with MCMC

We consider again the Bayesian updating problem of Illustration 1, where measurements of
the spatially variable friction angle of a silty soil are used to update the distribution of its
mean μ φ . We apply here the random-walk Metropolis-Hastings algorithm to obtain samples
of the posterior distribution of μ φ . As the proposal distribution, we choose the normal dis-
tribution centered at the current sample. Figure 5.9 shows the obtained samples from three
different choices of the standard deviation of the proposal PDF σ q , namely σ q = 0.5°, 2°,
and 8°. We choose the same starting point µ (0 =° for all three cases. It is shown that all
three cases include an initial burn-in period, whose length increases with decreasing σ q .
Moreover, if the variance of the proposal distribution is chosen too large (Case σ q = 8° in
Figure 5.9 ) , then the generated Markov chain is characterized by long flat periods that cor-
respond to low acceptance probabilities of the candidate states. Conversely, if the variance
is chosen too small (Case σ q = 0.5° in Figure 5.9 ) , then the acceptance probability increases

30

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μ φ (°)

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σ q =8

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0 50 100 150 200 250 300 350 400 450 500

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μ φ (°)

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σ q =2

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0 50 100 150 200 250 300 350 400 450 500

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μ φ (°)

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σ q = 0.5

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0 50 100 150 200 250 300 350 400 450 500

Figure 5.9 MCMC samples of the posterior distribution of the friction angle: Influence of the standard devia-
tion of the proposal PDF σ q .