Bayesian updating may be used as an inverse method to identify stochastic models. It is applicable when experimental data is available, which consist of a set of realizations of the response of the studied system. Bayesian updating is applicable in case the uncertain parameters cannot be directly computed using the experimental data. A numerical model is used instead to propagate the uncertain parameters. The prior distribution accounts for the initial knowledge of these uncertain parameters; it is subsequently updated to define the posterior distribution, with an improved match between the output of the numerical model and the experimental data. This contribution focuses on problems with a numerical model involving multiple outputs, in the very particular case where the experimental data includes a small number of realizations.

The traditional approach consists of identifying the joint probability density function associated with the experimental data; which subsequently defines the likelihood term of Bayes' equation (with the definition of the likelihood function widely used in statistics). This joint probability density function is directly identified from the experimental data, for instance using kernel density estimation [1] or by assuming that these data follow a predefined distribution (normal, lognormal, uniform, etc.). However, such strategies require the covariance matrix associated with the experimental data, which cannot be identified if the number of realizations available is too reduced (recall that at least N+1 realizations are required to identify the covariance matrix of a set of N random variables).

It is here assumed that the experimental data follows a predefined distribution; some of its parameters are obtained directly from the experimental data (e.g. using the method of the maximum of likelihood). The remaining parameters are accounted for using the prior distribution involved in the definition of the Bayesian updating problem.