**Mathematics of Operations Research**, 31: 13-30, 2006

**Abstract**: Let (x_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] ∈ 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).