Decision Making under the Exponential Family: Distributionally Robust Optimisation with Bayesian Ambiguity Sets

Decision making under uncertainty is challenging as the data-generating process (DGP) is often unknown. Bayesian inference proceeds by estimating the DGP through posterior beliefs on the model’s parameters. However, minimising the expected risk under these beliefs can lead to suboptimal decisions due to model uncertainty or limited, noisy observations. To address this, we introduce Distributionally Robust Optimisation with Bayesian Ambiguity Sets (DRO-BAS) which hedges against model uncertainty by optimising the worst-case risk over a posterior-informed ambiguity set. We provide two such sets, based on the posterior expectation (DRO-BAS(PE)) or the posterior predictive (DRO-BAS(PP)) and prove that both admit, under conditions, strong dual formulations leading to efficient single-stage stochastic programs which are solved with a sample average approximation. For DRO-BAS(PE), this covers all conjugate exponential family members while for DRO-BAS(PP) this is shown under conditions on the predictive's moment generating function. Our DRO-BAS formulations outperform existing Bayesian DRO on the Newsvendor problem and achieve faster solve times with comparable robustness on the Portfolio problem.

We consider risk-averse decision making under uncertainty about the data-generating process. This is common in settings like finance, healthcare, or inventory planning, where data may be limited or unreliable, and the goal is to be protected from worst-case scenarios. Traditional methods often assume the data perfectly describes reality, which can lead to sub-optimal decisions.In this paper, we introduce a new method, DRO-BAS, that uses Bayesian inference (used to update beliefs based on evidence) with a robust decision-making strategy that plans for a range of plausible outcomes. Instead of relying on a single estimator of reality, we consider a range of possible situations, informed by the Bayesian posterior, that are most consistent with the data. This helps ensure decisions are reliable even in worst-case scenarios.We achieve this through the formulation of two computationally efficient optimisation problems suitable for exponential family models, which are widely used in practical applications. We demonstrate the method’s improved robustness and efficiency compared to existing methods through decision-making examples such as the stock portfolio problem. Our method can be applied across a range of areas, such as resource management, where it’s important to avoid overly optimistic assumptions and safeguard against unexpected outcomes.