This paper develops a data-driven approach to Bayesian persuasion. The receiver is privately informed about the prior distribution of the state of the world, the sender knows the receiver's preferences but does not know the distribution of the state variable, and the sender's payoffs depend on the receiver's action but not on the state. Prior to interacting with the receiver, the sender observes the distribution of actions taken by a population of decision makers who share the receiver's preferences in best response to an unobserved distribution of messages generated by an unknown and potentially heterogeneous signal. The sender views any prior that rationalizes this data as plausible and seeks a signal that maximizes her worst-case payoff against the set of all such distributions. We show positively that the two-state many-action problem has a saddle point and negatively that the two-action many-state problem does not. In the former case, we identify adversarial priors and optimal signals. In the latter, we characterize the set of robustly optimal Blackwell experiments.