  A large class of stochastic programs involve optimizing an expectation taken
with respect to an underlying distribution that is unknown in practice. One
popular approach to addressing the distributional uncertainty, known as the
distributionally robust optimization (DRO), is to hedge against the worst case
over an uncertainty set of candidate distributions. However, it has been
observed that inappropriate construction of the uncertainty set can sometimes
result in over-conservative solutions. To explore the middle ground between
optimistically ignoring the distributional uncertainty and pessimistically
fixating on the worst-case scenario, we propose a Bayesian risk optimization
(BRO) framework for parametric underlying distributions, which is to optimize a
risk functional applied to the posterior distribution of an unknown
distribution parameter. Of our particular interest are four risk functionals:
mean, mean-variance, value-at-risk, and conditional value-at-risk. To unravel
the implication of BRO, we establish the consistency of objective functions and
optimal solutions, as well as the asymptotic normality of objective functions
and optimal values. More importantly, our analysis reveals a hidden
interpretation: the objectives of BRO can be approximately viewed as a weighted
sum of posterior mean objective and the (squared) half-width of the true
objective's confidence interval.
