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