Much of uncertainty quantification to date has focused on determining the
effect of variables modeled probabilistically, and with a known distribution,
on some physical or engineering system. We develop methods to obtain
information on the system when the distributions of some variables are known
exactly, others are known only approximately, and perhaps others are not
modeled as random variables at all. The main tool used is the duality between
risk-sensitive integrals and relative entropy, and we obtain explicit bounds on
standard performance measures (variances, exceedance probabilities) over
families of distributions whose distance from a nominal distribution is
measured by relative entropy. The evaluation of the risk-sensitive expectations
is based on polynomial chaos expansions, which help keep the computational
aspects tractable.