  Non-equilibrium systems have long-ranged spatial correlations even far away
from critical points. This implies that the likelihoods of spatial steady state
profiles of physical observables are nonlocal functionals. In this letter, it
is shown that these properties are essential to a successful analysis of a
functional level inverse problem, in which a macroscopic non-equilibrium
fluctuation field is estimated from limited but spatially scattered
information. To exemplify this, we dilute an out-of-equilibrium fluid flowing
through random media with a marker, which can be observed in an experiment. We
see that the hidden variables describing the random environment result in
spatial long-range correlations in the marker signal. Two types of statistical
estimators for the structure of the underlying media are then constructed: a
linear estimator provides unbiased and asymptotically precise information on
the particle density profiles, but yields negative estimates for the effective
resistances of the media in some cases. A nonlinear, maximum likelihood
estimator, on the other hand, results in a faithful media structure, but has a
small bias. These two approaches complement each other. Finally, estimation of
non-equilibrium fluctuation fields evolving in time is discussed.
