Marginalising posterior covariance matrix with application to Bayesian operational modal analysis

Consider making Bayesian inference of vector-valued model parameters x,y based on observed data D. When the ‘posterior’ (i.e., given data) probability density function (PDF) of x,y has a centralised shape, it can be approximated in the spirit of Laplace integral asymptotics by a Gaussian PDF centred at the ‘most probable value’ (MPV) that minimises the objective function Lx,y=-lnpD|x,yp(x,y), where pD|x,y is the likelihood function and px,y is the prior PDF. The ‘posterior covariance matrix’ of x,y that reflects the remaining uncertainty after using data is then equal to the inverse of the Hessian of L(x,y) at the MPV. Suppose the ‘partial MPV’ y^(x) is available, so that ∂L/∂y=0 for any x as long as y=y^(x). Correspondingly, the ‘partially minimised’ objective function that depends only on x is defined as L^x=Lx,y^(x). In the above context, this article shows that the posterior covariance matrix of x can be obtained as the inverse of the Hessian of L^(x) at the MPV. That is, the marginalisation of y in MPV can be carried over to the covariance matrix. The result can also be extended to the inverse of Fisher information matrix, which gives the large-sample asymptotic form of the posterior covariance matrix when there is no modelling error. The theory is applied to operational modal analysis with well-separated modes, providing an alternative means to conventional approach for evaluating the posterior covariance matrix of spectral parameters (e.g., frequency, damping) after marginalising out spatial parameters such as the mode shape. Issues of theoretical and computational nature are discussed and verified by synthetic, laboratory and field data.