This paper uses semidefinite programming (SDP) to construct Bayesian optimal design for nonlinear regression models. The setup here extends the formulation of the optimal designs problem as an SDP problem from linear to nonlinear models. Gaussian quadrature formulas (GQF) are used to compute the expectation in the Bayesian design criterion, such as D-, A- or E-optimality. As an illustrative example, we demonstrate the approach using the power-logistic model and compare results in the literature. Additionally, we investigate how the optimal design is impacted by different discretising schemes for the design space, different amounts of uncertainty in the parameter values, different choices of GQF and different prior distributions for the vector of model parameters, including normal priors with and without correlated components. Further applications to find Bayesian D-optimal designs with two regressors for a logistic model and a two-variable generalised linear model with a gamma distributed response are discussed, and some limitations of our approach are noted.

中文翻译：

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寻找非线性模型的贝叶斯最优设计：一种基于半定规划的方法

本文使用半定规划（SDP）来构建非线性回归模型的贝叶斯最优设计。此处的设置将优化设计问题的公式扩展为从线性模型到非线性模型的 SDP 问题。高斯求积公式 (GQF) 用于计算贝叶斯设计准则中的期望值，例如 D-、A- 或 E-最优性。作为说明性示例，我们使用幂逻辑模型演示了该方法并比较了文献中的结果。此外，我们研究了优化设计如何受到设计空间的不同离散方案、参数值中不同数量的不确定性、GQF 的不同选择和模型参数向量的不同先验分布的影响，包括具有和不相关的正态先验组件。