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 | x₁. . . . , x_n-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, . . ., y_n holds a belief e_n = P(P_n | x₁, . . . , x_n) ∈ 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 (e _n) when P ranges over all possible laws of (x_n) _n.