  Inverse problems constrained by partial differential equations (PDEs) play a
critical role in model development and calibration. In many applications, there
are multiple uncertain parameters in a model which must be estimated. Although
the Bayesian formulation is attractive for such problems, computational cost
and high dimensionality frequently prohibit a thorough exploration of the
parametric uncertainty. A common approach is to reduce the dimension by fixing
some parameters (which we will call auxiliary parameters) to a best estimate
and use techniques from PDE-constrained optimization to approximate properties
of the Bayesian posterior distribution. For instance, the maximum a posteriori
probability (MAP) and the Laplace approximation of the posterior covariance can
be computed. In this article, we propose using hyper-differential sensitivity
analysis (HDSA) to assess the sensitivity of the MAP point to changes in the
auxiliary parameters. We establish an interpretation of HDSA as correlations in
the posterior distribution. Our proposed framework is demonstrated on the
inversion of bedrock topography for the Greenland ice sheet with uncertainties
arising from the basal friction coefficient and climate forcing (ice
accumulation rate)
