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There is a fundamental limitation in the prediction performance that amachine learning model can achieve due to the inevitable uncertainty of theprediction target. In classification problems, this can be characterized by theBayes error, which is the best achievable error with any classifier. The Bayeserror can be used as a criterion to evaluate classifiers with state-of-the-artperformance and can be used to detect test set overfitting. We propose a simpleand direct Bayes error estimator, where we just take the mean of the labelsthat show \emph{uncertainty} of the class assignments. Our flexible approachenables us to perform Bayes error estimation even for weakly supervised data.In contrast to others, our method is model-free and even instance-free.Moreover, it has no hyperparameters and gives a more accurate estimate of theBayes error than several baselines empirically. Experiments using our methodsuggest that recently proposed deep networks such as the Vision Transformer mayhave reached, or is about to reach, the Bayes error for benchmark datasets.Finally, we discuss how we can study the inherent difficulty of theacceptance/rejection decision for scientific articles, by estimating the Bayeserror of the ICLR papers from 2017 to 2023.

Is the Performance of My Deep Network Too Good to Be True? A Direct Approach to Estimating the Bayes Error in Binary Classification