232 C. Argyris et al. 26.3 Global Sampling-Based Approach Since we do not know the measured data before the experiment takes place, we follow the strategy of averaging over the unknown, using the prior and likelihood in order to draw possible “data” samples. Following the theory of expected objective function for optimal experimental design [3] the expected objective is the average determinant over all possible data sets that can result from design m: Vm =E[Um] = p( dm)Umd dm (26.6) Next we demonstrate how the posterior covariance matrix of (26.5) can be evaluated. In the following we temporarily drop the dependence of the posterior covariance matrix of gon the specific set of data dmfor notational convenience. Similarly for the dependence of g onθ. Then the element (i,j) of g, representing the posterior covariance between the i andj elements of the predicted vector g is given by: i,j g =cov(gi,gj) =E[(gi −E[gi])(gj −E[gj])]=E[gigj]−E[gi]E[gj] (26.7) where the expectations are evaluated using the known theorem for the expectation of a function of a random variable as: E[gigj] = ! gi gj p(θ| d)dθ (26.8) E[gi] = ! gi p(θ| d)dθ (26.9) where the posterior is used as the density for θ in order to reflect our updated state of knowledge about the parameters after having observed the data. Substituting the posterior from Bayes’ rule (26.2) results in the following integrals over the prior: E[gigj]= 1 p( d) ! gi gj p( d|θ)p(θ)dθ (26.10) E[gi]= 1 p( d) ! gi p( d|θ)p(θ)dθ (26.11) where the evidence term is given by the total probability theorem as: p( d) = ! p( d|θ)p(θ)dθ (26.12) Analytic calculation of the above integrals is possible only for very simple models. Numerical quadrature is efficient only for a very small number of parameters and data points which result in low-dimensional integrals. For the general case of multiple parameters and data points, Monte Carlo integration is the only possible solution. Herein we introduce Monte Carlo importance sampling similar to [9] to approximate the objective function. 26.4 Local Sensitivity-Based Approach In the sensitivity-based method the posterior covariance matrix of θ is derived from the Gaussian approximation of the posterior: −1 θ| dm = [∇θ dm(θ)] T −1 em [∇θ dm(θ)] + −1 θ | θ=θ∗ (26.13) where ∇θ dm(θ) is the Nm×Nθ matrix of sensitivities of the measured quantity with respect to model parameters (Jacobian matrix). The second term, θ, is the covariance matrix of the Gaussian prior for θ.
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