  Although instantaneous interactions are unphysical, a large variety of
maximum entropy statistical inference methods match the model-inferred and the
empirically-measured equal-time correlation functions. Focusing on collective
motion of active units, this constraint is reasonable when the interaction
timescale is much faster than that of the interacting units, as in starling
flocks, yet it fails in a number of counter examples, as in leukocyte
coordination (where signalling proteins diffuse among two cells). Here, we
relax this assumption and develop a path integral approach to maximum-entropy
framework, which includes delay in signalling. Our method is able to infer the
strength of couplings and fields, but also the time required by the couplings
to completely transfer information among the units. We demonstrate the validity
of our approach providing excellent results on synthetic datasets of
non-Markovian trajectories generated by the Heisenberg-Kuramoto and Vicsek
models equipped with delayed interactions. As a proof of concept, we also apply
the method to experiments on dendritic migration, where matching equal-time
correlations results in a significant information loss.
