Fast and Scalable Likelihood-Free Bayesian Model Updating for Multiple-Measurement Data via Conditional Invertible Neural Network and Posterior Stacking

Bayesian model updating (BMU) is widely used in structural health monitoring (SHM) to calibrate model parameters and quantify uncertainties. However, a key challenge in BMU is the intractability of the likelihood function, which leads to high computational costs, particularly in long-term SHM involving multiple measurements. To address this, we propose a fast and scalable likelihood-free inference framework for BMU that effectively handles multiple measurements. In the proposed approach, the normalizing flow is employed to establish a bidirectional mapping between the posterior distributions of model parameters and multivariate Gaussian latent variables. A conditional invertible neural network (cINN) is utilized to construct the invertible transformation. Once trained on synthetic data generated from the structural model using prior knowledge, the cINN enables posterior approximation within seconds by sampling from a latent Gaussian distribution and applying the learned invertible transformation. To handle multiple measurements, we propose a posterior stacking strategy, where the posterior for each individual data set is first estimated using the trained cINN. Each individual posterior is then represented using a Gaussian mixture model fitted to its posterior samples. The final aggregated posterior is then obtained through a linear mixture of these posterior densities, where the optimal weights are determined by maximizing the log predictive density of the target posterior. The proposed approach circumvents the need to evaluate an intractable likelihood function while efficiently estimating posteriors from multiple measurements. We validate the proposed method using two application examples: a 4-story shear building and a reinforced concrete slab. Comparative analysis with traditional Markov chain Monte Carlo (MCMC)–based BMU demonstrates that our method achieves superior accuracy and efficiency in posterior estimation, particularly in the presence of noisy and multiple measurements. These results highlight the practical advantages of the proposed approach for long-term SHM.