Full Gibbs Sampling Procedure for Bayesian System Identification Incorporating Sparse Bayesian Learning with Automatic Relevance Determination

Bayesian system identification has attracted substantial interest in recent years for inferring structural models and quantifying their uncertainties based on measured dynamic response in a structure. The relative plausibility of each structural model in a specified model class is quantified by its posterior probability from Bayes' Theorem. The relative plausibility of each model class within a set of candidate model classes for the structure can also be assessed via Bayes' Theorem. Computation of this posterior probability over all candidate model classes automatically applies a quantitative Ockham's razor that trades off a data-fit measure with an information-theoretic measure of model complexity, which penalizes model classes that over-fit the data. In this article, we first present a general Bayesian system identification framework and point out that combining it with sparse Bayesian learning (SBL) is an effective strategy to implement the Bayesian Ockham razor. Then we review our recent progress in exploring SBL with the automatic relevance determination likelihood concept to detect and quantify spatially sparse substructure stiffness reductions. To characterize the full posterior uncertainty for this problem, an improved Gibbs sampling procedure for SBL is then developed. Finally, illustrative results are provided to compare the performance and validate the capability of the presented SBL algorithms for structural system identification.