Bayesian Optimal Experiment Design for Artificial Intelligence Agriculture Fertilizer Trials Under Smallholder Plot Heterogeneity: A Computer Information-Gain Regret Analysis

Fertilizer trials in smallholder agriculture are expensive, and the plots available for experimentation are heterogeneous in ways that standard randomised designs do not accommodate. The framework treats the trial as a decision problem in which the designer chooses a treatment allocation to maximise expected information gain about a response surface that maps nitrogen application to yield, subject to a budget on the number of experimental plots. Because the response surface is unknown, the design is evaluated by its regret: the expected loss in information relative to an oracle design that knows the surface. The analysis is theoretical and computational. It derives the expected information gain under a Gaussian process prior with a separable kernel that decomposes plot heterogeneity into a spatially correlated component and a management-practice component, and it shows that the optimal allocation under this prior is a compromise between spreading treatments across heterogeneous plots and replicating treatments within homogeneous blocks. A regret bound is established that separates the contribution of prior misspecification from the contribution of finite-sample noise. The framework is then applied to a design problem with four candidate plot counts and three candidate effect sizes, and the information-gain regret is computed across the design grid. The results indicate that the regret of the recommended design is bounded and that the bound tightens as the number of plots increases, but the rate of tightening depends on the ratio of spatial to management variance. The contribution is a formal account of when heterogeneity-aware allocation outperforms blocked randomisation in smallholder fertilizer trials, and a regret-based criterion that a computer system can compute before a trial is funded.