  Efficiently quantifying predictive uncertainty in medical images remains a
challenge. While Bayesian neural networks (BNN) offer predictive uncertainty,
they require substantial computational resources to train. Although Bayesian
approximations such as ensembles have shown promise, they still suffer from
high training and inference costs. Existing approaches mainly address the costs
of BNN inference post-training, with little focus on improving training
efficiency and reducing parameter complexity. This study introduces a training
procedure for a sparse (partial) Bayesian network. Our method selectively
assigns a subset of parameters as Bayesian by assessing their deterministic
saliency through gradient sensitivity analysis. The resulting network combines
deterministic and Bayesian parameters, exploiting the advantages of both
representations to achieve high task-specific performance and minimize
predictive uncertainty. Demonstrated on multi-label ChestMNIST for
classification and ISIC, LIDC-IDRI for segmentation, our approach achieves
competitive performance and predictive uncertainty estimation by reducing
Bayesian parameters by over 95\%, significantly reducing computational expenses
compared to fully Bayesian and ensemble methods.
