An analytically tractable solution for hierarchical Bayesian model updating with variational inference scheme. (15th April 2023) - Jia, Xinyu; Yan, Wang-Ji; Papadimitriou, Costas; Yuen, Ka-Veng
Highlights: Variational inference scheme is proposed for hierarchical Bayesian model updating framework. Closed-from formula of the hyper parameters and model parameters are derived. Analytical solutions shed new insights into the MPVs and covariance of the hyper and model parameters. Analytical solutions provide constructive physical interpretations for uncertainties involved in HBM updating. Examples demonstrate the computationally efficient performance and accuracy. Abstract: The hierarchical Bayesian modelling (HBM) framework has recently been proposed to properly account for the model parameter uncertainty in structural dynamics. This framework postulates a hierarchical prior for the model parameters that depend on the hyper parameters to be identified using the multiple datasets. However, the number of hyper parameters increases linearly with the number of model parameters as well as the number of datasets, making the framework computationally challenging and analytically intractable. To this end, this study employs a variational inference scheme within the framework, deriving explicit expressions for the posterior distributions of the hyper parameters and further the predictive distribution of the model parameters so as to avoid resorting to sampling-based strategy, thus efficiently enhancing the computational efficiency of the HBM framework. In particular, the posterior distribution of the hyper parameters is derived as a Normal-Inverse-Wishart (NIW) distribution, and the predictive distribution of the model parameters are expressed as a normal distribution conditional on the samples of its hyper parameters. Such derivations offer valuable insights into the interpretations of different sources of uncertainties existing in the procedure of model updating. Specifically, it reveals that the ensemble uncertainty of the model parameters consists of both the identification uncertainty obtained from each dataset as well as the test-to-test variability across the overall datasets. Additionally, it suggests that, when the test-to-test variability dominates the uncertainty of the model parameters, the HBM framework with negligible identification uncertainty can reduce to the same expression with a frequentist perspective. Linear models with modal properties data and nonlinear models with time histories data of building systems are utilized to verify and demonstrate the effectiveness of the proposed framework. Results indicate that the proposed formulations provide sufficiently accurate solutions with efficient computational performance, compared to the full sampling approach regarded as the reference of the HBM framework.