Before revision:  We use our characterization to study joint belief distributions in which agents are informed regarding the state, and yet receive no information regarding the other's posterior. We study a related class of Bayesian persuasion problems with a single sender and multiple receivers, and explore the extreme points of the set of feasible distributions.
After revision:  We use these characterizations to study information structures with independent posteriors. We also study persuasion problems with a single sender and multiple receivers, and explore the extreme points of the set of feasible distributions.