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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.