A simple proof of Blackwell’s theorem on the comparison of experiments for a general state space

Abstract

This paper offers, for a general state space, a simple proof of the equivalence between Blackwell sufficiency and the Bohnenblust–Shapley–Sherman criterion of more-informativeness. The proof relies on nothing more than the finite intersection property of compact sets. While several proofs exist for finite state spaces, infinite spaces, as necessitated in applications with continuous distributions, is explored by Boll (1955), Amershi (1988) (but for a finite-dimensional action set), and reviewed in LeCam’s foundational rubric for the subject. We offer two examples to show the fragility of Boll’s definition of the second criterion, and the necessity of his assumption of absolute continuity.