Titel_SS06

measurement errors should be modelled as random variables, having means (zero for unbiased
estimates), standard deviations and, if necessary some correlation pattern. The standard
deviation is a property of the measurement technique, but may also depend on the
circumstances. An important but difficult modelling part is the degree of correlation between
observations at different places and different points in time.

The information of the inequality type refers to observations where it is only known that the
observed variable is greater than or less than some limit: a crack may be less than the
observation threshold, a limit state of collapse may be reached (or not). Uncertainty in the
threshold value should be taken into account. The distribution function for the minimum
threshold level is often referred to as the Probability of Detection curve (POD curve). Also
here, correlations for the probability of detection in various observations should be known.

Mathematically the two types of information can be denoted as:

 equality type: h() 0 x 

 inequality type: h( ) x  0

where is a vector of the realizations of the basic random variables X. In this notation
measurement values and threshold values are considered as components of the vector x.

x

Updating of Random Variables

Inspection or test results relating directly to realisations of random variables may be used in
the updating. This is done by assuming the distribution parameters of the distributions used in
the probabilistic modelling to be uncertain themselves. New samples or observations of
realisations of the random variables are then used to update the probability distribution
functions of these distribution parameters.

The distribution parameters are initially (and prior to any update) modelled by prior
distribution functions. The prior distribution functions is best updated by Bayesian reasoning
which, however, requires that a weight is given to the information contained in the prior
distribution functions e.g. in terms of equivalent sample sizes if conjugate priors are used.
Unfortunately the latter are only available for some distribution functions which nevertheless
belong to the set of those models most commonly in use. By application of Bayes theorem,
see e.g. Madsen et al. (1986), the prior distribution functions, assessed by any mixture of
frequentistic and subjective information, are updated and transformed into posterior
distribution functions.

Assume that a random variable X has the probability distribution function and density

function

FxX()

fX()x

Q()

. Furthermore assume that one or more of the distribution parameters, e.g. the

mean value and standard deviation of X are uncertain themselves with probability density
function f q. Then the probability distribution function for Q may be updated on the basis

of observations of X, i.e. xˆ.

The general scheme for the updating is: