In (Q. Long, M. Scavino, R. Tempone, S. Wang. Fast estimation of expected information gains for Bayesian experimental designs based on Laplace approximation. Computer Methods in Applied Mechanics and Engineering 2013; 259:24-39.), a new method based on the Laplace approximation was developed to accelerate the estimation of the post–experimental expected information gains (Kullback–Leiblervergence) in model parameters and predictive quantities of interest in the Bayesian framework.closed–form asymptotic approximation of the inner integral and the order of the corresponding dominant error term were obtained in the cases where the parameters are determined by the experiment.
In this work, we extend that method to the general case where the model parameters can not be determined completely by the data from the proposed experiments. We carry out the Laplace approximations in the directions orthogonal to the null space of the Jacobian matrix of the model with respect to the parameters, so that the information gain can be reduced to an integration against the marginal density of the transformed parameters which are not determined by the experiments. Furthermore, the expected information gain can be approximated by an integration over the prior, where the integrand is a function of the posterior covariance matrix projected over the aforementioned orthogonal directions. To deal with the issue of dimensionality in a complex problem, weeither Monte Carlo sampling or sparse quadratures for the integration over the prior probability density function, depending on the regularity of the integrand function. We demonstrate the accuracy, efficiency and robustness of the proposed method via several nonlinear under determined test cases.
They include the designs of the scalar parameter in a one dimensional cubic polynomial function with two indistinguishable parameters forming a linear manifold, respectively, and the boundary source locations for impedance tomography in a square domain, where the unknown parameter, the conductivity, which is represented as a random field.