Bayesian‐optimized experimental designs for estimating the economic optimum nitrogen rate: a model‐averaging approach

Abstract Field experiments play a crucial role in optimizing nutrient application strategies and determining the economic optimum nitrogen rate (EONR), aiding stakeholders in agricultural decision‐making. These experiments tailor agricultural input management to maximize efficiency and sustainability, ultimately improving farm economics. However, the optimal setup of field experiments remains an ongoing debate, particularly regarding economic considerations such as the selection of treatment levels (design points), their spatial arrangement, and the number of replications required for statistical validity and cost‐effectiveness. This study optimizes field experiments for estimating the EONR using a model‐averaging approach within a Bayesian framework. We employed Bayesian inference and the No‐U‐turn sampler to integrate model averaging across multiple yield response models, improving robustness in EONR estimation. Stochastic optimization, specifically simultaneous perturbation stochastic approximation, was used to optimize experimental designs, and their performance was evaluated through Monte Carlo simulations. Our results show that optimized experimental designs significantly improve the precision of EONR estimates. Designs incorporating higher number of nitrogen levels provided the best trade‐off between accuracy and efficiency, minimizing bias and mean squared error. Even with a fixed total number of plots (120), increasing the number of design points resulted in lower variance, demonstrating the efficiency of well‐structured experimental designs. This research lays the groundwork for future developments in experimental methodologies with wide‐ranging implications for agricultural economics and policymaking, ultimately supporting better‐informed decision‐making. Future work should integrate environmental constraints and account for real‐world variability in treatment replication to further refine experimental optimization strategies.