Empirical Distributions of Beliefs Under Imperfect Observation

Source abstract

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 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 y 1 ,…,y n holds a belief e n = P(p n ∣ x 1 ,…,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 .