  We present a unified duality approach to Bayesian persuasion. The optimal
dual variable, interpreted as a price function on the state space, is shown to
be a supergradient of the concave closure of the objective function at the
prior belief. Strong duality holds when the objective function is Lipschitz
continuous.
  When the objective depends on the posterior belief through a set of moments,
the price function induces prices for posterior moments that solve the
corresponding dual problem. Thus, our general approach unifies known results
for one-dimensional moment persuasion, while yielding new results for the
multi-dimensional case. In particular, we provide a necessary and sufficient
condition for the optimality of convex-partitional signals, derive structural
properties of solutions, and characterize the optimal persuasion scheme in the
case when the state is two-dimensional and the objective is quadratic.
