In this work we consider a general notion of *distributional sensitivity*, which
measures the variation in solutions of a given physical/mathematical system with respect
to the variation of probability distribution of the inputs. This is distinctively
different from the classical sensitivity analysis, which studies the changes of solutions
with respect to the values of the inputs. The general idea is measurement of sensitivity
of outputs with respect to probability distributions, which is a well-studied concept
in related disciplines. We adapt these ideas to present a quantitative framework in
the context of uncertainty quantification for measuring such a kind of sensitivity and
a set of efficient algorithms to approximate the distributional sensitivity numerically.
A remarkable feature of the algorithms is that they do not incur additional computational
effort in addition to a one-time stochastic solver. Therefore, an accurate stochastic
computation with respect to a prior input distribution is needed only once, and
the ensuing distributional sensitivity computation for different input distributions is a
post-processing step. We prove that an accurate numerical model leads to accurate calculations
of this sensitivity, which applies not just to slowly-converging Monte-Carlo
estimates, but also to exponentially convergent spectral approximations. We provide
computational examples to demonstrate the ease of applicability and verify the convergence
claims.