Epistemic Uncertainty and Excess Risk in Variational Inference

Bayesian inference is widely used in practice due to its ability to assess epistemic uncertainty (EU) in predictions. However, its computational complexity necessitates the use of approximation methods, such as variational inference (VI). When estimating EU within the VI framework, metrics such as the variance of the posterior predictive distribution and conditional mutual information are commonly employed. Despite their practical importance, these metrics lack comprehensive theoretical analysis. In this paper, we investigate these EU metrics by providing their novel relationship to excess risk, which allows for a convergence analysis based on PAC-Bayesian theory. Based on these analyses, we then demonstrate that some existing objective functions of VI regularize EU metrics in different ways leading to different performance in EU evaluation. Finally, we propose a novel objective function for VI that directly optimizes both prediction and EU under the PAC-Bayesian framework. Experimental results indicate that our algorithm significantly improves EU estimation compared to existing VI methods.