How to Evaluate Uncertainty Estimates in Machine Learning for Regression?

As neural networks become more popular, the need for accompanying uncertainty estimates increases. The current testing methodology focusses on how good the predictive uncertainty estimates explain the differences between predictions and observations in a previously unseen test set. Intuitively this is a logical approach. The current setup of benchmark data sets also allows easy comparison between the different methods. We demonstrate, however, through both theoretical arguments and simulations that this way of evaluating the quality of uncertainty estimates has serious flaws. Firstly, it cannot disentangle the aleatoric from the epistemic uncertainty. Secondly, the current methodology considers the uncertainty averaged over all test samples, implicitly averaging out overconfident and underconfident predictions. When checking if the correct fraction of test points falls inside prediction intervals, a good score on average gives no guarantee that the intervals are sensible for individual points. We demonstrate through practical examples that these effects can result in favoring a method, based on the predictive uncertainty, that has undesirable behaviour of the confidence intervals. Finally, we propose a simulation-based testing approach that addresses these problems while still allowing easy comparison between different methods.