  Data from experiments and theoretical arguments are the two pillars
sustaining the job of modelling physical systems through inference. In order to
solve the inference problem, the data should satisfy certain conditions that
depend also upon the particular questions addressed in a research. Here we
focus on the characterization of systems in terms of a distance from
equilibrium, typically the entropy production (time-reversal asymmetry) or the
violation of the Kubo fluctuation-dissipation relation. We show how general,
counter-intuitive and negative for inference, is the problem of the
impossibility to estimate the distance from equilibrium using a series of
scalar data which have a Gaussian statistics. This impossibility occurs also
when the data are correlated in time, and that is the most interesting case
because it usually stems from a multi-dimensional linear Markovian system where
there are many time-scales associated to different variables and, possibly,
thermal baths. Observing a single variable (or a linear combination of
variables) results in a one-dimensional process which is always
indistinguishable from an equilibrium one (unless a perturbation-response
experiment is available). In a setting where only data analysis (and not new
experiments) is allowed, we propose - as a way out - the combined use of
different series of data acquired with different parameters. This strategy
works when there is a sufficient knowledge of the connection between
experimental parameters and model parameters. We also briefly discuss how such
results emerge, similarly, in the context of Markov chains within certain
coarse-graining schemes. Our conclusion is that the distance from equilibrium
is related to quite a fine knowledge of the full phase space, and therefore
typically hard to approximate in real experiments.
