Prediction-error variance in Bayesian model updating: a comparative study

In Bayesian model updating, the likelihood function is commonly formulated by stochastic embedding in which the maximum information entropy probability model of prediction-error variances plays an important role and it is Gaussian distribution subject to the first two moments as constraints. The selection of prediction-error variances can be formulated as a model class selection problem, which automatically involves a trade-off between the average data-fit of the model class and the information it extracts from the data. Therefore, it is critical for the robustness in the updating of the structural model in the presence of modeling errors. To date, three models of prediction-error variances have been adopted in the literature, namely, the prediction-error variances can be 1) set as constant values empirically, 2) estimated from the measured data and output of the analytical model, and 3) updated as uncertain parameters by applying Bayes' Theorem at the model class level. In this paper, the three different strategies to deal with the prediction-error variances are investigated and compared. A six-story shear building model with six uncertain stiffness parameters is employed as an illustrative example. Transitional Markov Chain Monte Carlo method is used to draw samples of the posterior probability density function of the structural model parameters as well as the uncertain prediction-error variances. Different model classes for the structural model parameters are considered through three FE models, including a reference model, a model with too many unnecessary uncertain parameters, and a model with too few uncertain parameters. Bayesian updating is performed for the three FE model classes considering the three aforementioned treatments of the prediction-error variances. The effect of number of measurements on the model updating performance is also examined in the study. The results indicate that updating the prediction-error variances as uncertain parameters at the model class level produces more robust results especially when the number of measurements is small.