Dense orbits of the Bayesian updating group action

We study dynamic properties of the group action on the simplex that is induced by Bayesian updating. We show that generically the orbits are dense in the simplex, although one must make use of the entire group, hence departing from straightforward Bayesian updating. We demonstrate also the necessity of the genericity of the signalling structure, a relationship to descriptive set theoretical concepts and applications thereof to repeated games of incomplete information, as well a strengthening concerning the group action on itself.