  This paper considers Bayesian persuasion for routing games where information
about the uncertain state of the network is provided by a traffic information
system (TIS) using public signals. In this setup, the TIS commits to a
signalling scheme and participants form a posterior belief about the state of
the network based on prior beliefs and the received signal. They subsequently
select routes minimizing their individual expected travel time under their
posterior beliefs, giving rise to a Wardrop equilibrium. We investigate how the
TIS can infer the prior beliefs held by the participants by designing suitable
signalling schemes, and observing the equilibrium flows under different
signals. We show that under mild conditions a signalling scheme that allows for
exact inference of the prior exists. We then provide an iterative algorithm
that finds such a scheme in a finite number of steps. We show that schemes
designed by our algorithm are robust, in the sense that they can still identify
the prior after a small enough perturbation. We also investigate the case where
the population is divided among multiple priors, and give conditions under
which the fraction associated to each prior can be identified. Several examples
illustrate our results.
