  A paradigm put forth by E. Schr\"odinger in 1931/32, known as Schr\"odinger
bridges, represents a formalism to pose and solve control and estimation
problems seeking a perturbation from an initial control schedule (in the case
of control), or from a prior probability law (in the case of estimation),
sufficient to reconcile data in the form of marginal distributions and minimal
in the sense of relative entropy to the prior. In the same spirit, we consider
traffic-flow and apply a Schr\"odinger-type dictum, to perturb minimally with
respect to a suitable relative entropy functional a prior schedule/law so as to
reconcile the traffic flow with scarce aggregate distributions on families of
indistinguishable individuals. Specifically, we consider the problem to
regulate/estimate multi-commodity network flow rates based only on empirical
distributions of commodities being transported (e.g., types of vehicles through
a network, in motion) at two given times. Thus, building on Schr\"odinger's
large deviation rationale, we develop a method to identify {\em the most likely
flow rates (traffic flow)}, given prior information and aggregate observations.
Our method further extends the Schr\"odinger bridge formalism to the
multi-commodity setting, allowing commodities to exit or enter the flow field
as well (e.g., vehicles to enter and stop and park) at any time. The behavior
of entering or exiting the flow field, by commodities or vehicles, is modeled
by a Markov chains with killing and creation states. Our method is illustrated
with a numerical experiment.
