Expected Improvement (EI) is arguably the most widely used acquisition function in Bayesian optimization. However, it is often challenging to enhance the performance with EI due to its sensitivity to numerical precision. Previously, Hutter et al. (2009) tackled this problem by using Gaussian process trained on the log-transformed objective function and it was reported that this trick improves the predictive accuracy of GP, leading to substantially better performance. Although Hutter et al. (2009) offered the closed form of their EI, its intermediate derivation has not been provided so far. In this paper, we give a friendly derivation of their proposition.

翻译：期望改进（EI）可以说是贝叶斯优化中应用最广泛的采集函数。然而，由于其数值精度的敏感性，利用EI提升性能通常具有挑战性。先前，Hutter等人（2009）通过使用在对数变换目标函数上训练的高斯过程来解决此问题，据报告该技巧提升了高斯过程的预测精度，从而显著改善了性能。尽管Hutter等人（2009）给出了其EI的闭式表达式，但其推导过程迄今尚未公开。本文中，我们对该命题给出了清晰的推导过程。