  We consider a population of Bayesian agents who share a common prior over
some finite state space and each agent is exposed to some information about the
state. We ask which distributions over empirical distributions of posteriors
beliefs in the population are feasible. We provide a necessary and sufficient
condition for feasibility. We apply this result in several domains. First, we
study the problem of maximizing the polarization of beliefs in a population.
Second, we provide a characterization of the feasible agent-symmetric product
distributions of posteriors. Finally, we study an instance of a private
Bayesian persuasion problem and provide a clean formula for the sender's
optimal value.
