  We propose a novel a-posteriori error estimation technique where the target
quantities of interest are ratios of high-dimensional integrals, as occur e.g.
in PDE constrained Bayesian inversion and PDE constrained optimal control
subject to an entropic risk measure. We consider in particular parametric,
elliptic PDEs with affine-parametric diffusion coefficient, on high-dimensional
parameter spaces. We combine our recent a-posteriori Quasi-Monte Carlo (QMC)
error analysis, with Finite Element a-posteriori error estimation. The proposed
approach yields a computable a-posteriori estimator which is reliable, up to
higher order terms. The estimator's reliability is uniform with respect to the
PDE discretization, and robust with respect to the parametric dimension of the
uncertain PDE input.
