  In distributionally robust optimization the probability distribution of the
uncertain problem parameters is itself uncertain, and a fictitious adversary,
e.g., nature, chooses the worst distribution from within a known ambiguity set.
A common shortcoming of most existing distributionally robust optimization
models is that their ambiguity sets contain pathological discrete distribution
that give nature too much freedom to inflict damage. We thus introduce a new
class of ambiguity sets that contain only distributions with sum-of-squares
polynomial density functions of known degrees. We show that these ambiguity
sets are highly expressive as they conveniently accommodate distributional
information about higher-order moments, conditional probabilities, conditional
moments or marginal distributions. Exploiting the theoretical properties of a
measure-based hierarchy for polynomial optimization due to Lasserre [SIAM J.
Optim. 21(3) (2011), pp. 864--885], we prove that certain worst-case
expectation constraints are computationally tractable under these new ambiguity
sets. We showcase the practical applicability of the proposed approach in the
context of a stylized portfolio optimization problem and a risk aggregation
problem of an insurance company.
