Statistics-based Bayesian Modeling Framework for Uncertainty Quantification and Propagation

A new Bayesian modeling framework is proposed to account for the uncertainty in the model parameters arising from model and measurements errors, as well as experimental, operational, environmental and manufacturing variabilities. Uncertainty is embedded in the model parameters using a single level hierarchy where the uncertainties are quantified by Normal distributions with the mean and the covariance treated as hyperparameters. Unlike existing hierarchical Bayesian modelling frameworks, the likelihood function for each observed quantity is built based on the Kullback–Leibler divergence used to quantify the discrepancy between the probability density functions (PDFs) of the model predictions and measurements. The likelihood function is constructed assuming that this discrepancy for each measured quantity follows a truncated normal distribution. For Gaussian PDFs of measurements and response predictions, the posterior PDF of the model parameters depends on the lower two moments of the respective PDFs. This representation of the posterior is also used for non-Gaussian PDFs of measurements and model predictions to approximate the uncertainty in the model parameters. The proposed framework can tackle the situation where only PDFs or statistical characteristics are available for measurements. The propagation of uncertainties is accomplished through sampling. Two applications demonstrate the use and effectiveness of the proposed framework. In the first one, structural model parameter inference is considered using simulated statistics for the modal frequencies and mode shapes. In the second one, uncertainties in the parameters of the probabilistic S-N curves used in fatigue are quantified based on experimental data.