Bayesian design can be used for efficient data collection over time when the process can be described by the solution to an ordinary differential equation (ODE). Typically, Bayesian designs in such settings are obtained by maximising the expected value of a utility function that is derived from the joint probability distribution of the parameters and the response, given prior information about an appropriate ODE. However, in practice, appropriately defining such information \textit{a priori} can be difficult due to incomplete knowledge about the mechanisms that govern how the process evolves over time. In this paper, we propose a method for finding Bayesian designs based on a flexible class of ODEs. Specifically, we consider the inclusion of spline terms into ODEs to provide flexibility in modelling how the process changes over time. We then propose to leverage this flexibility to form designs that are efficient even when the prior information is misspecified. Our approach is motivated by a sampling problem in agriculture where the goal is to provide a better understanding of fruit growth where prior information is based on studies conducted overseas, and therefore is potentially misspecified.

翻译：当过程可由常微分方程（ODE）的解描述时，贝叶斯设计可用于随时间推移的高效数据收集。通常，此类背景下的贝叶斯设计是通过最大化效用函数的期望值而获得的，该效用函数源自给定适当ODE先验信息时参数与响应的联合概率分布。然而在实践中，由于对过程随时间演化机制的认识不完整，先验恰当地定义此类信息可能较为困难。本文提出了一种基于灵活ODE类寻找贝叶斯设计的方法。具体而言，我们考虑在ODE中引入样条项，从而为过程随时间变化的建模提供灵活性。随后我们建议利用这种灵活性来构建即使在先验信息误设时仍能保持高效的设计方案。本方法的提出源于农业领域的采样问题，其目标是在先验信息基于海外研究（因而可能存在误设）的情况下，更好地理解果实生长规律。