  In order to anticipate rare and impactful events, we propose to quantify the
worst-case risk under distributional ambiguity using a recent development in
kernel methods -- the kernel mean embedding. Specifically, we formulate the
generalized moment problem whose ambiguity set (i.e., the moment constraint) is
described by constraints in the associated reproducing kernel Hilbert space in
a nonparametric manner. We then present the tractable approximation and its
theoretical justification. As a concrete application, we numerically test the
proposed method in characterizing the worst-case constraint violation
probability in the context of a constrained stochastic control system.
