Optimal experiment design (OED) aims to optimize the information content of experimental observations by designing the experimental conditions. In Bayesian OED for parameter estimation, the design selection is based on an expected utility metric that accounts for the joint probability distribution of the uncertain parameters and the observations. This work presents solution methods for two approximate formulations of the Bayesian OED problem based on Kullback–Leibler divergence for the particular case of Gaussian prior and observation noise distributions and the general case of arbitrary prior distributions and arbitrary observation noise distributions when the observation noise corresponds to arbitrary functions of the states and random variables with an arbitrary multivariate distribution. The proposed methods also allow satisfying path constraints with a specified probability. The solution approach relies on the reformulation of the approximate Bayesian OED problem as an optimal control problem (OCP), for which a parsimonious input parameterization is adopted to reduce the number of decision variables. An efficient global solution method for OCPs via sum-of-squares polynomials and parallel computing is then applied, which is based on approximating the cost of the OCP by a polynomial function of the decision variables and solving the resulting polynomial optimization problem to global optimality in a tractable way via semidefinite programming. It is established that the difference between the cost obtained by solving the polynomial optimization problem and the globally optimal cost of the OCP is bounded and depends on the polynomial approximation error.

优化实验设计（OED）旨在通过设计实验条件来优化实验观察的信息内容。在用于参数估计的贝叶斯 OED 中，设计选择基于预期效用度量，该度量考虑了不确定参数和观测值的联合概率分布。这项工作针对高斯先验和观测噪声分布的特定情况以及任意先验分布和任意观测噪声分布的一般情况（当观测噪声对应于具有任意多元分布的状态和随机变量的任意函数。所提出的方法还允许以指定的概率满足路径约束。求解方法依赖于将近似贝叶斯 OED 问题重新表述为最优控制问题 (OCP)，为此采用简约的输入参数化来减少决策变量的数量。然后应用通过平方和多项式和并行计算的 OCP 的有效全局求解方法，该方法基于通过决策变量的多项式函数逼近 OCP 的成本，并将生成的多项式优化问题求解为全局最优通过半定编程的一种易于处理的方式。