Posterior uncertainty, asymptotic law and Cramér-Rao bound

In a globally identifiable Bayesian system identification problem, the uncertainty of model parameters can be quantified by their “posterior covariance matrix” calculated for a particular data set. When the data is modeled to be distributed as the likelihood function (i.e., no modeling error), a statistical law analogous to the law of large numbers results, where the posterior covariance matrix is asymptotic to a deterministic quantity that depends on the “information content” of data rather than its particular (stochastic) details. This was referred as the “uncertainty law” in a recent study of the achievable precision of modal parameters in operational modal analysis (OMA). Deriving the uncertainty law involves asymptotics techniques and leveraging on the mathematical structure of the likelihood function, which was found to be tedious. As a sequel to the development, this work shows that for long data and up to a Gaussian approximation of the posterior distribution, the uncertainty law is asymptotic to the inverse of the Fisher information matrix, which coincides with the tightest Cramér-Rao bound in classical statistics. A parametric study is presented to illustrate the theoretical results in the context of OMA. As a direct application with practical relevance, the relationship provides a systematic means for deriving the uncertainty laws in OMA. Applied and interpreted properly, the posterior covariance matrix (for given data), uncertainty law, and Cramér-Rao bound can provide a powerful means for quantifying and managing the uncertainties in structural health monitoring.