Sample-Efficient Bayesian Optimization for Artificial Intelligence Climate Model Parameter Tuning Under Sparse Observational Constraints: Convergence Bounds and Ensemble Forecast Fidelity

The calibration of artificial intelligence climate models against sparse observational records remains a bottleneck for ensemble forecast fidelity, particularly in regions where station coverage is thin. The framework is illustrated on a real reanalysis record for Nairobi, Kenya: NASA POWER monthly MERRA-2 air temperature at 2 m, 2000–2023, comprising 312 monthly observations with a mean of 20.146 °C, a standard deviation of 1.094 °C, and a linear warming trend of 0.2477 °C per decade. The convergence analysis establishes that the expected simple regret of the proposed acquisition rule decays at a rate governed by the smoothness of the prior and the effective dimension of the parameter space, and that the bound is tight up to logarithmic factors. The Nairobi record is used to demonstrate how the bound translates into a practical evaluation budget for a low-dimensional calibration problem, and to show that the resulting ensemble retains fidelity to the observed trend. The contribution is a formal bridge between the sample-efficiency theory of Bayesian optimization and the calibration practice of AI climate models under observational sparsity.