Uncertainty Estimates of Predictions via a General Bias-Variance Decomposition

Reliably estimating the uncertainty of a prediction throughout the model lifecycle is crucial in many safety-critical applications. The most common way to measure this uncertainty is via the predicted confidence. While this tends to work well for in-domain samples, these estimates are unreliable under domain drift. Alternatively, a bias-variance decomposition allows to directly measure the predictive uncertainty across the entire input space. But, such a decomposition for proper scores does not exist in current literature, and for exponential families it is convoluted. In this work, we introduce a general bias-variance decomposition for proper scores and reformulate the exponential family case, giving rise to the Bregman Information as the variance term in both cases. This allows us to prove that the Bregman Information for classification measures the uncertainty in the logit space. We showcase the practical relevance of this decomposition on two downstream tasks. First, we show how to construct confidence intervals for predictions on the instance-level based on the Bregman Information. Second, we demonstrate how different approximations of the instance-level Bregman Information allow reliable out-of-distribution detection for all degrees of domain drift.