As machine learning models are increasingly deployed in safety-critical and industrial applications, the need for reliable uncertainty estimation alongside predictions becomes essential. Uncertainty estimates not only foster trust in model outputs but also support downstream tasks such as active learning and out-of-distribution detection.

Two types of uncertainty are typically distinguished: aleatoric uncertainty, which captures irreducible noise inherent in the data, and epistemic uncertainty, which arises from limited knowledge and can be reduced with more data. Accurately estimating and disentangling these uncertainty types is critical. For instance, in active learning, one should prioritize samples with high epistemic but low aleatoric uncertainty.

Bayesian methods provide a principled framework for modeling epistemic uncertainty by learning distributions over model parameters. In particular, Bayesian inference in function space has gained attention due to its robustness against overparameterization. However, approximating distributions in infinite-dimensional function spaces introduces additional challenges.

This thesis aims to address two key goals: (1) developing efficient and scalable methods to improve epistemic uncertainty estimation in deep learning, and (2) applying these methods to real-world industrial problems, such as the optimization of carbide-free bainitic steel.