Exact Bayesian Formulation and its Computational Difficulties

Consider a single-degree-of-freedom (SDOF) system with the equation of motion  

It is well known that the stationary response x is a zero-mean Gaussian random process with the auto-correlation function  

Discrete data are sampled with a time step At and y are then used to denote the measured response at time t = nAt. Due to measurement noise and modeling error, there is a difference between the measured response y and the model response x nAt), referred to hereafter as prediction error. It is assumed that the prediction error can be adequately represented by a 

Furthermore, the stochastic response x and the prediction error c are assumed statistically independent. 

Therefore, the set of measurements V include the data points yi, y2, , TAf - Define a column vector Y  

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