Fast estimation of expected information gains for Bayesian experimental designs based on Laplace approximations
KAUST DepartmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Applied Mathematics and Computational Science Program
Stochastic Numerics Research Group
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AbstractShannon-type expected information gain can be used to evaluate the relevance of a proposed experiment subjected to uncertainty. The estimation of such gain, however, relies on a double-loop integration. Moreover, its numerical integration in multi-dimensional cases, e.g., when using Monte Carlo sampling methods, is therefore computationally too expensive for realistic physical models, especially for those involving the solution of partial differential equations. In this work, we present a new methodology, based on the Laplace approximation for the integration of the posterior probability density function (pdf), to accelerate the estimation of the expected information gains in the model parameters and predictive quantities of interest. We obtain a closed-form approximation of the inner integral and the corresponding dominant error term in the cases where parameters are determined by the experiment, such that only a single-loop integration is needed to carry out the estimation of the expected information gain. To deal with the issue of dimensionality in a complex problem, we use a sparse quadrature for the integration over the prior pdf. We demonstrate the accuracy, efficiency and robustness of the proposed method via several nonlinear numerical examples, including the designs of the scalar parameter in a one-dimensional cubic polynomial function, the design of the same scalar in a modified function with two indistinguishable parameters, the resolution width and measurement time for a blurred single peak spectrum, and the boundary source locations for impedance tomography in a square domain. © 2013 Elsevier B.V.
SponsorsWe thank the referees for their helpful comments and suggestions that led to an improved version of this paper. We are also thankful for support from the Academic Excellency Alliance UT Austin-KAUST project "Predictability and uncertainty quantification for models of porous media" and the Institute of Applied Mathematics and Computational Sciences at TAMU. Part of this work was carried out while M. Scavino and S. Wang were Visiting Professors at KAUST. S. Wang's research was also partially supported by Award Number KUS-CI-016-04, made by King Abdullah University of Science and Technology (KAUST). M. Scavino and R. Tempone are members of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering.