Duality and Policy Evaluation in Distributionally Robust Bayesian Diffusion Control

Data-driven control often represents unknown dynamics through a Bayesian prior, yet the resulting policy can be brittle when that prior is misspecified. We propose distributionally robust Bayesian control (DRBC), in which an adversary perturbs the latent-parameter prior once at time zero while preserving the conditional diffusion dynamics. For Kullback--Leibler ambiguity, a strong-duality representation reduces fixed-policy robust evaluation to a scalar optimization. We combine this representation with randomized multilevel Monte Carlo to obtain simulation-based policy evaluation with canonical root-$n$ accuracy, and with structure-aware policy learning for linear--quadratic control and Bayesian portfolio selection. Synthetic and real-data experiments show that DRBC improves robustness without the severe conservatism of classical dynamic robust control.