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。该泛函包含一个外层积分和一个内层积分，二者通过对数函数相连接。当实验的数学模型同时包含目标参数不确定性与干扰参数不确定性（即对模型有影响但实验者并不直接关注的参数所产生的不确定性）时，需要估计两个内层积分。这使得原本已相当庞大的、用于确定期望信息增益良好近似值的计算负担进一步加重。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量，可显著减轻两个内层积分的计算负担。第一种估计量先采用拉普拉斯方法，再实施拉普拉斯近似，会引入估计偏差。第二种估计量则使用双重拉普拉斯近似作为重要性采样测度，对内层积分进行蒙特卡洛近似。两种估计量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展示了所提估计量的适用性与有效性。