Titel_SS06

and old information is performed through the sample likelihood Px ˆ:i and the prior

probability P
:i. The likelihood is the probability of obtaining the observation xˆ given the

true states of nature :i.

:

:

:

f:(:)

Prior

Posterior Likelihood

Likelihood

Prior

Posterior

Prior Posterior Likelihood

f;():

:

:

:

Figure 12.7: Illustration of updating of uncertainty models.

In Figure 12.7 an illustration is given of corresponding prior and posterior probability density
functions together with likelihood functions. In the first case the prior information is strong
and the likelihood is weak (small sample size). In the second case the prior information is
weak and the likelihood is strong. Finally in the last case the prior information and the
likelihood are of comparable strength.

It is seen from Figure 12.7 that the modelling of both the prior probabilistic models and the
likelihood is of utmost importance. The modelling of the likelihood and the evaluation of the
posterior probabilistic models will be discussed in the following.

As mentioned the likelihood is a measure for the probability of the observation given the true
state of nature.

Example 12.2 – Reassessment analysis based on new data - posterior analysis

In order to demonstrate what this actually means consider again the example with the steel
bar. The prior probabilistic model for the yield stress of the steel bar is assumed to be normal
distributed with known (deterministic) standard deviation fyequal to 17.5 MPa and uncertain

mean value. The mean value fyis assumed Normal distributed with known mean value

@equal to 350 MPa and standard deviation @ equal to 10 MPa. The loading is assumed

deterministic equal to 3041.5 MPa.

Assume that one test of the yield stress fy is performed on a specimen taken from the same