Finding the optimal design of experiments in the Bayesian setting typically requires estimation and optimization of the expected information gain functional. This functional consists of one outer and one inner integral, separated by the logarithm function applied to the inner integral. When the mathematical model of the experiment contains uncertainty about the parameters of interest and nuisance uncertainty, (i.e., uncertainty about parameters that affect the model but are not themselves of interest to the experimenter), two inner integrals must be estimated. Thus, the already considerable computational effort required to determine good approximations of the expected information gain is increased further. The Laplace approximation has been applied successfully in the context of experimental design in various ways, and we propose two novel estimators featuring the Laplace approximation to alleviate the computational burden of both inner integrals considerably. The first estimator applies Laplace's method followed by a Laplace approximation, introducing a bias. The second estimator uses two Laplace approximations as importance sampling measures for Monte Carlo approximations of the inner integrals. Both estimators use Monte Carlo approximation for the remaining outer integral estimation. We provide three numerical examples demonstrating the applicability and effectiveness of our proposed estimators.

翻译：在贝叶斯框架下寻找最优实验设计通常需要估计并优化期望信息增益泛函。该泛函由一个外积分和一个内积分构成，两者通过对内积分应用对数函数相分离。当实验的数学模型同时包含感兴趣参数的不确定性和干扰不确定性（即影响模型但并非实验者所关注参数的不确定性）时，必须估计两个内积分。因此，确定期望信息增益良好近似所需的计算量本就相当可观，而这一情况进一步加剧了计算负担。拉普拉斯近似已在实验设计领域以多种方式成功应用，我们提出两种基于拉普拉斯近似的新型估计器，可显著减轻两个内积分的计算负担。第一种估计器先应用拉普拉斯方法，再实施拉普拉斯近似，会引入一定偏差。第二种估计器使用两次拉普拉斯近似作为蒙特卡罗近似内积分的重要性采样测度。两种估计器均采用蒙特卡罗近似进行剩余外积分的估计。我们通过三个数值算例展示了所提出估计器的适用性和有效性。