  In the study of natural and artificial complex systems, responses that are
not completely determined by the considered decision variables are commonly
modelled probabilistically, resulting in response distributions varying across
decision space. We consider cases where the spatial variation of these response
distributions does not only concern their mean and/or variance but also other
features including for instance shape or uni-modality versus multi-modality.
Our contributions build upon a non-parametric Bayesian approach to modelling
the thereby induced fields of probability distributions, and in particular to a
spatial extension of the logistic Gaussian model.
  The considered models deliver probabilistic predictions of response
distributions at candidate points, allowing for instance to perform
(approximate) posterior simulations of probability density functions, to
jointly predict multiple moments and other functionals of target distributions,
as well as to quantify the impact of collecting new samples on the state of
knowledge of the distribution field of interest. In particular, we introduce
adaptive sampling strategies leveraging the potential of the considered random
distribution field models to guide system evaluations in a goal-oriented way,
with a view towards parsimoniously addressing calibration and related problems
from non-linear (stochastic) inversion and global optimisation.
