Let (xn)n be a process with values in a finite set X and law P, and let yn = f(xn) be a function of the process. At stage n, the conditional distribution pn = P(xn ∣ x1,…,xn-1), element of Π = Δ(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 y1,…,yn holds a belief en = P(pn ∣ x1,…,xn) ∈ Δ(Π) 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 (en) when P ranges over all possible laws of (xn)n.
