Bayesian optimal experimental design provides a principled framework for selecting experimental settings that maximize obtained information. In this work, we focus on estimating the expected information gain in the setting where the differential entropy of the likelihood is either independent of the design or can be evaluated explicitly. This reduces the problem to maximum entropy estimation, alleviating several challenges inherent in expected information gain computation. Our study is motivated by large-scale inference problems, such as inverse problems, where the computational cost is dominated by expensive likelihood evaluations. We propose a computational approach in which the evidence density is approximated by a Monte Carlo or quasi-Monte Carlo surrogate, while the differential entropy is evaluated using standard methods without additional likelihood evaluations. We prove that this strategy achieves convergence rates that are comparable to, or better than, state-of-the-art methods for full expected information gain estimation, particularly when the cost of entropy evaluation is negligible. Moreover, our approach relies only on mild smoothness of the forward map and avoids stronger technical assumptions required in earlier work. We also present numerical experiments, which confirm our theoretical findings.

翻译：贝叶斯最优实验设计为选择能最大化获取信息的实验设置提供了原则性框架。本研究聚焦于在似然函数的微分熵与设计无关或可显式计算的情形下，估计期望信息增益。这将问题简化为最大熵估计，缓解了期望信息增益计算中固有的若干挑战。我们的研究受到大规模推断问题（如反问题）的推动，这类问题的计算成本主要来自昂贵的似然函数求值。我们提出一种计算方法：通过蒙特卡洛或拟蒙特卡洛代理近似证据密度，同时使用标准方法计算微分熵而无需额外的似然函数求值。我们证明该策略能达到与最先进的完整期望信息增益估计方法相当或更优的收敛速率，尤其在熵计算成本可忽略时。此外，我们的方法仅需前向映射满足温和光滑性条件，避免了早期工作中所需的更强技术假设。数值实验也验证了我们的理论结果。