  In this paper, we address the problem of bounding conditional expectations
when moment information of the underlying distribution and the random event
conditioned upon are given. To this end, we propose an adapted version of the
generalized moment problem which deals with this conditional information
through a simple transformation. By exploiting conic duality, we obtain sharp
bounds that can be used for distribution-free decision-making under
uncertainty. Additionally, we derive computationally tractable mathematical
programs for distributionally robust optimization (DRO) with side information
by leveraging core ideas from ambiguity-averse uncertainty quantification and
robust optimization, establishing a moment-based DRO framework for prescriptive
stochastic programming.
