Calculation of Hessian under constraints with applications to Bayesian system identification

In Bayesian system identification with globally identifiable models, the posterior (i.e., given data) probability density function (PDF) of model parameters can be approximated by a Gaussian PDF. The most probable value (MPV) of the parameters is equal to the mean of the Gaussian PDF. It maximises the posterior PDF, or equivalently, minimises the negative of logarithm (NL) of the posterior PDF. The covariance matrix of the Gaussian PDF is equal to the Hessian of the NL at the MPV. Model parameters can be subjected to constraints, which must be accounted for in the calculation of the posterior covariance matrix. In applications such as modal identification, existing strategies define a set of free parameters and map them to the model parameters so that the constraints are always satisfied. The Hessian of the NL with respect to the free parameters is obtained and then transformed to give the posterior covariance matrix of the model parameters where constraints are accounted for. Analytical expressions for this Hessian are complicated because of the composite actions of the NL and the mapping; and this creates significant burden in computer coding. In this work, a theoretical framework is developed for evaluating the Hessian of a function under constraints in a systematic manner. It is applied to obtain new analytical expressions for evaluating the posterior covariance matrix in Bayesian operational modal analysis. The resulting expressions are simpler than existing ones based on direct differentiation. They allow problems with similar mathematical structures to be computer-coded in a coherent manner. Numerical examples are presented to illustrate consistency and computational aspects.