Calculating the expected information gain in optimal Bayesian experimental design typically relies on nested Monte Carlo sampling. When the model also contains nuisance parameters, which are parameters that contribute to the overall uncertainty of the system but are of no interest in the Bayesian design framework, this introduces a second inner loop. We propose and derive a small-noise approximation for this additional inner loop. The computational cost of our method can be further reduced by applying a Laplace approximation to the remaining inner loop. Thus, we present two methods, the small-noise Double-loop Monte Carlo and small-noise Monte Carlo Laplace methods. Moreover, we demonstrate that the total complexity of these two approaches remains comparable to the case without nuisance uncertainty. To assess the efficiency of these methods, we present three examples, and the last example includes the partial differential equation for the electrical impedance tomography experiment for composite laminate materials.

翻译：在最优贝叶斯实验设计中计算期望信息增益通常依赖于嵌套蒙特卡洛采样。当模型还包含干扰参数（即对系统整体不确定性有贡献但在贝叶斯设计框架中不感兴趣的参数）时，这会引入第二个内层循环。我们提出并推导了针对这一额外内层循环的小噪声近似方法。通过将拉普拉斯近似应用于剩余的内层循环，可进一步降低我们方法的计算成本。因此，我们提出了两种方法：小噪声双环蒙特卡洛法和小噪声蒙特卡洛拉普拉斯法。此外，我们证明了这两种方法的总复杂度与不存在干扰不确定性时的情形相当。为评估这些方法的效率，我们给出了三个示例，其中最后一个示例包含了复合材料层合板电阻抗层析成像实验的偏微分方程。