We analyze the best achievable performance of Bayesian learning under generative models by defining and upper-bounding the minimum excess risk (MER): the gap between the minimum expected loss attainable by learning from data and the minimum expected loss that could be achieved if the model realization were known. The definition of MER provides a principled way to define different notions of uncertainties in Bayesian learning, including the aleatoric uncertainty and the minimum epistemic uncertainty. Two methods for deriving upper bounds for the MER are presented. The first method, generally suitable for Bayesian learning with a parametric generative model, upper-bounds the MER by the conditional mutual information between the model parameters and the quantity being predicted given the observed data. It allows us to quantify the rate at which the MER decays to zero as more data becomes available. The second method, particularly suitable for Bayesian learning with a parametric predictive model, relates the MER to the deviation of the posterior predictive distribution from the true predictive model, and further to the minimum estimation error of the model parameters from data. It explicitly shows how the uncertainty in model parameter estimation translates to the MER and to the final prediction uncertainty. We also extend the definition and analysis of MER to the setting with multiple parametric model families and the setting with nonparametric models. Along the discussions we draw some comparisons between the MER in Bayesian learning and the excess risk in frequentist learning.

本文探讨了基于生成模型下贝叶斯学习的最佳性能，并通过定义和上界最小超额风险（MER）来说明不同不确定性的概念，包括 aleatoric 不确定性和最小认知不确定性。

贝叶斯学习中的最小超额风险