This paper presents an efficient Bayesian framework for solving nonlinear,high-dimensional model calibration problems. It is based on a VariationalBayesian formulation that aims at approximating the exact posterior by means ofsolving an optimization problem over an appropriately selected family ofdistributions. The goal is two-fold. Firstly, to find lower-dimensionalrepresentations of the unknown parameter vector that capture as much aspossible of the associated posterior density, and secondly to enable thecomputation of the approximate posterior density with as few forward calls aspossible. We discuss how these objectives can be achieved by using a fullyBayesian argumentation and employing the marginal likelihood or evidence as theultimate model validation metric for any proposed dimensionality reduction. Wedemonstrate the performance of the proposed methodology for problems innonlinear elastography where the identification of the mechanical properties ofbiological materials can inform non-invasive, medical diagnosis. An ImportanceSampling scheme is finally employed in order to validate the results and assessthe efficacy of the approximations provided.
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