Bayesian maximum entropy method for stochastic model updating using measurement data and statistical information

The presence of summarized statistical information, such as some statistics of the system response, is not rare in practical engineering as the acquisition of precisely measured point data is expensive and may not be always accessible. In this paper, we integrate the Bayesian framework with the maximum entropy theory and develop a Bayesian Maximum Entropy (BME) approach for model updating in a scenario where measurement data and statistical information are simultaneously available. Within the scope of this contribution, it is assumed that measurement data denote direct observations, e.g. point data, representing system response measurements while statistical information involves summarized information, e.g. moment and/or reliability information, of the system response. The basic principle of our approach is to convert point data and various statistical information into constraints under the BME framework and use the method of Lagrange multipliers to find the optimal posterior distributions. We then extend this approach to imprecise probabilistic models which have not been addressed so far. The approximate Bayesian computation is employed to facilitate the estimation of cumbersome likelihood functions which results from the involvement of entropy terms accounting for statistical information. Furthermore, a Wasserstein distance-based metric is proposed and embedded into the framework to capture the divergence information in an effective and efficient way. The effectiveness of the proposed approach is verified by a numerical case of simply supported beam and an engineering problem of fatigue crack growth. It shows some promising aspects of this research as better calibration results are produced with less uncertainty, and hence potential of our approach for engineering applications.