  We introduce a new framework, Bayesian Distributionally Robust Optimization
(Bayesian-DRO), for data-driven stochastic optimization where the underlying
distribution is unknown. Bayesian-DRO contrasts with most of the existing DRO
approaches in the use of Bayesian estimation of the unknown distribution. To
make computation of Bayesian updating tractable, Bayesian-DRO first assumes the
underlying distribution takes a parametric form with unknown parameter and then
computes the posterior distribution of the parameter. To address the model
uncertainty brought by the assumed parametric distribution, Bayesian-DRO
constructs an ambiguity set of distributions with the assumed parametric
distribution as the reference distribution and then optimizes with respect to
the worst case in the ambiguity set. We show the consistency of the Bayesian
posterior distribution and subsequently the convergence of objective functions
and optimal solutions of Bayesian-DRO. Our consistency result of the Bayesian
posterior requires simpler assumptions than the classical literature on
Bayesian consistency. We also consider several approaches for selecting the
ambiguity set size in Bayesian-DRO and compare them numerically. Our numerical
experiments demonstrate the out-of-sample performance of Bayesian-DRO in
comparison with Kullback-Leibler-based (KL-) and Wasserstein-based empirical
DRO as well as risk-neutral Bayesian Risk Optimization. Our numerical results
shed light on how to choose the modeling framework (Bayesian-DRO, KL-DRO,
Wasserstein-DRO) for specific problems, but the choice for general problems
still remains an important and open question.
