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.