  An analyst observes the frequency with which an agent takes actions, but not
the frequency with which she takes actions conditional on a payoff relevant
state. In this setting, we ask when the analyst can rationalize the agent's
choices as the outcome of the agent learning something about the state before
taking action. Our characterization marries the obedience approach in
information design (Bergemann and Morris, 2016) and the belief approach in
Bayesian persuasion (Kamenica and Gentzkow, 2011) relying on a theorem by
Strassen (1965) and Hall's marriage theorem. We apply our results to
ring-network games and to identify conditions under which a data set is
consistent with a public information structure in first-order Bayesian
persuasion games.
