Bayesian Updating of Structural Models and Reliability using Markov Chain Monte Carlo Simulation

In a full Bayesian probabilistic framework for “robust” system identification, structural response predictions and performance reliability are updated using structural test data 𝒟 by considering the predictions of a whole set of possible structural models that are weighted by their updated probability. This involves integrating $h\left(\mathbf{\theta }\right)p\left(\mathbf{\theta }\right|𝒟)$ over the whole parameter space, where θ is a parameter vector defining each model within the set of possible models of the structure, $h\left(\mathbf{\theta }\right)$ is a model prediction of a response quantity of interest, and $p\left(\mathbf{\theta }\right|𝒟)$ is the updated probability density for θ, which provides a measure of how plausible each model is given the data 𝒟. The evaluation of this integral is difficult because the dimension of the parameter space is usually too large for direct numerical integration and $p\left(\mathbf{\theta }\right|𝒟)$ is concentrated in a small region in the parameter space and only known up to a scaling constant. An adaptive Markov chain Monte Carlo simulation approach is proposed to evaluate the desired integral that is based on the Metropolis-Hastings algorithm and a concept similar to simulated annealing. By carrying out a series of Markov chain simulations with limiting stationary distributions equal to a sequence of intermediate probability densities that converge on $p\left(\mathbf{\theta }\right|𝒟)\text{,}$ the region of concentration of $p\left(\mathbf{\theta }\right|𝒟)$ is gradually portrayed. The Markov chain samples are used to estimate the desired integral by statistical averaging. The method is illustrated using simulated dynamic test data to update the robust response variance and reliability of a moment-resisting frame for two cases: one where the model is only locally identifiable based on the data and the other where it is unidentifiable.