A sensitivity-based one-at-a-time estimation of posterior marginal probability distributions in Bayesian model updating

Abstract The standard Bayesian updating method consists in calculating the joint posterior probability distribution of some parameters of a computational model using the Bayes rule and experimental data. When the model output has large dimension and its probability distribution is not assumed to belong to a prescribed family, the estimation of the likelihood function and the storage of the posterior joint probability density function for point estimation of the parameters or for subsequent stochastic analysis of the quantity of interests become very challenging, requiring the use of MCMC sampling methods with a large number of samples. In addition, when the number of parameters to be updated is large, the computational cost associated with the estimation of their posterior marginal distributions becomes significantly high. In this paper, a one-parameter-at-a-time Bayesian updating method is formulated. This novel approach consists in calculating the posterior marginal distribution of each parameter separately by constructing for each parameter a one-dimensional output that is sensitive to this parameter only, enabling the corresponding (1) likelihood function to be estimated easily and (2) the marginal posteriors to be build and stored directly and independently of the other parameters. This approach is illustrated through a numerical example consisting of the Bayesian updating of a three-storey structure with model-form uncertainties and for which the mass and stiffness parameters are calibrated.