  We present Posterior Temperature Optimized Bayesian Inverse Models (POTOBIM),
an unsupervised Bayesian approach to inverse problems in medical imaging using
mean-field variational inference with a fully tempered posterior. Bayesian
methods exhibit useful properties for approaching inverse tasks, such as
tomographic reconstruction or image denoising. A suitable prior distribution
introduces regularization, which is needed to solve the ill-posed problem and
reduces overfitting the data. In practice, however, this often results in a
suboptimal posterior temperature, and the full potential of the Bayesian
approach is not being exploited. In POTOBIM, we optimize both the parameters of
the prior distribution and the posterior temperature with respect to
reconstruction accuracy using Bayesian optimization with Gaussian process
regression. Our method is extensively evaluated on four different inverse tasks
on a variety of modalities with images from public data sets and we demonstrate
that an optimized posterior temperature outperforms both non-Bayesian and
Bayesian approaches without temperature optimization. The use of an optimized
prior distribution and posterior temperature leads to improved accuracy and
uncertainty estimation and we show that it is sufficient to find these
hyperparameters per task domain. Well-tempered posteriors yield calibrated
uncertainty, which increases the reliability in the predictions. Our source
code is publicly available at github.com/Cardio-AI/mfvi-dip-mia.
