  For many important problems the quantity of interest is an unknown function
of the parameters, which is a random vector with known statistics. Since the
dependence of the output on this random vector is unknown, the challenge is to
identify its statistics, using the minimum number of function evaluations. This
problem can been seen in the context of active learning or optimal experimental
design. We employ Bayesian regression to represent the derived model
uncertainty due to finite and small number of input-output pairs. In this
context we evaluate existing methods for optimal sample selection, such as
model error minimization and mutual information maximization. We show that for
the case of known output variance, the commonly employed criteria in the
literature do not take into account the output values of the existing
input-output pairs, while for the case of unknown output variance this
dependence can be very weak. We introduce a criterion that takes into account
the values of the output for the existing samples and adaptively selects inputs
from regions of the parameter space which have important contribution to the
output. The new method allows for application to high-dimensional inputs,
paving the way for optimal experimental design in high-dimensions.
