  Recent deep learning approaches focus on improving quantitative scores of
dedicated benchmarks, and therefore only reduce the observation-related
(aleatoric) uncertainty. However, the model-immanent (epistemic) uncertainty is
less frequently systematically analyzed. In this work, we introduce a Bayesian
variational framework to quantify the epistemic uncertainty. To this end, we
solve the linear inverse problem of undersampled MRI reconstruction in a
variational setting. The associated energy functional is composed of a data
fidelity term and the total deep variation (TDV) as a learned parametric
regularizer. To estimate the epistemic uncertainty we draw the parameters of
the TDV regularizer from a multivariate Gaussian distribution, whose mean and
covariance matrix are learned in a stochastic optimal control problem. In
several numerical experiments, we demonstrate that our approach yields
competitive results for undersampled MRI reconstruction. Moreover, we can
accurately quantify the pixelwise epistemic uncertainty, which can serve
radiologists as an additional resource to visualize reconstruction reliability.
