  This paper provides a non-robust interpretation of the distributionally
robust optimization (DRO) problem by relating the distributional uncertainties
to the chance probabilities. Our analysis allows a decision-maker to interpret
the size of the ambiguity set, which is often lack of business meaning, through
the chance parameters constraining the objective function. We first show that,
for general $\phi$-divergences, a DRO problem is asymptotically equivalent to a
class of mean-deviation problems. These mean-deviation problems are not subject
to uncertain distributions, and the ambiguity radius in the original DRO
problem now plays the role of controlling the risk preference of the
decision-maker. We then demonstrate that a DRO problem can be cast as a
chance-constrained optimization (CCO) problem when a boundedness constraint is
added to the decision variables. Without the boundedness constraint, the CCO
problem is shown to perform uniformly better than the DRO problem, irrespective
of the radius of the ambiguity set, the choice of the divergence measure, or
the tail heaviness of the center distribution. Thanks to our high-order
expansion result, a notable feature of our analysis is that it applies to
divergence measures that accommodate well heavy tail distributions such as the
student $t$-distribution and the lognormal distribution, besides the
widely-used Kullback-Leibler (KL) divergence, which requires the distribution
of the objective function to be exponentially bounded. Using the portfolio
selection problem as an example, our comprehensive testings on multivariate
heavy-tail datasets, both synthetic and real-world, shows that this
business-interpretation approach is indeed useful and insightful.
