  We consider the problem of providing optimal uncertainty quantification (UQ)
--- and hence rigorous certification --- for partially-observed functions. We
present a UQ framework within which the observations may be small or large in
number, and need not carry information about the probability distribution of
the system in operation. The UQ objectives are posed as optimization problems,
the solutions of which are optimal bounds on the quantities of interest; we
consider two typical settings, namely parameter sensitivities (McDiarmid
diameters) and output deviation (or failure) probabilities. The solutions of
these optimization problems depend non-trivially (even non-monotonically and
discontinuously) upon the specified legacy data. Furthermore, the extreme
values are often determined by only a few members of the data set; in our
principal physically-motivated example, the bounds are determined by just 2 out
of 32 data points, and the remainder carry no information and could be
neglected without changing the final answer. We propose an analogue of the
simplex algorithm from linear programming that uses these observations to offer
efficient and rigorous UQ for high-dimensional systems with high-cardinality
legacy data. These findings suggest natural methods for selecting optimal
(maximally informative) next experiments.
