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 four numerical examples demonstrating the applicability and effectiveness of our proposed estimators.

翻译：在贝叶斯框架下寻找实验的最优设计通常需要对期望信息增益泛函进行估计和优化。该泛函包含一个外层积分和一个内层积分，其间通过对数函数作用于内层积分。当实验的数学模型同时包含目标参数不确定性与混杂参数不确定性（即影响模型但并非实验者直接关注参数的不确定性）时，需要估计两个内层积分。这使得原本已相当庞大的期望信息增益近似计算量进一步增加。拉普拉斯近似已在实验设计领域以多种方式成功应用，本文提出两种融合拉普拉斯近似的新型估计器，可显著减轻两个内层积分的计算负担。第一种估计器先采用拉普拉斯方法再进行拉普拉斯近似，会引入估计偏差。第二种估计器将双重拉普拉斯近似作为重要性采样测度，用于内层积分的蒙特卡洛近似。两种估计器均采用蒙特卡洛方法处理剩余的外层积分估计。我们通过四个数值算例验证了所提估计器的适用性与有效性。