Empirical Distributions of Beliefs Under Imperfect Observation

Let ( x n ) n be a process with values in a finite set X and law P , and let y n = f ( x n ) be a function of the process. At stage n , the conditional distribution p n = P ( x n | x 1 ,..., x n -1 ), element of (Pi) = (Delta)( X ), is the belief that a perfect observer, who observes the process online, holds on its realization at stage n . A statistician observing the signals y 1 ,..., y n holds a belief e n = P ( p n | x 1 ,..., x n ) (in) (Delta)((Pi)) on the possible predictions of the perfect observer. Given X and f , we characterize the set of limits of expected empirical distributions of the process ( e n ) when P ranges over all possible laws of ( x n ) n .