Distributionally Robust Bayesian Experimental Design via Confidence-Band Ambiguity Sets

Model misspecification in Bayesian Experimental Design (BED) can lead to overconfident posterior inference and unreliable design selection. This limitation hinders the use of the approach in safety-critical applications, such as healthcare, where experimental outcomes must be accompanied by guarantees at a specified confidence level. To address this challenge, we propose a Distributionally Robust BED (DR-BED) framework that introduces a robust optimisation problem with a novel ambiguity set, constructed from a pair of lower-bound and upper-bound functions as confidence bands for the true Data Generating Process (DGP). We establish theoretical guarantees showing that, under standard regularity conditions on the experimental setting, namely Hölder continuity and compactness of the design space and bounded support of the experimental outcome, the constructed ambiguity set contains the true DGP with the experimenter-specified confidence level $1-\alpha$. Notably, the confidence level can be updated dynamically throughout the sequential experimentation process, allowing experts to explore different robustness-efficiency trade-offs without repeating the experimental workflow. Building on the theoretical guarantees, we reformulate the DR-BED optimisation problem in its dual form and develop a stochastic subgradient algorithm for its practical implementation. Experiments on controlled and benchmark problems demonstrate that, under model misspecification, DR-BED consistently produces improved posterior beliefs and achieves lower distances to the true DGP compared to the baseline across multiple distributional metrics.