A Third Information-Theoretic Approach to Finite de Finetti Theorems
Mario Berta, Lampros Gavalakis, Ioannis Kontoyiannis · 2024
A new finite form of de Finetti's representation theorem is established using elementary information-theoretic tools. The distribution of the first$k$random variables in an exchangeable vector of$n\geq k$random variables is close to a mixture of product distributions. Closeness is measured in terms of the relative entropy and an explicit bound is provided. This bound is tighter than those obtained via earlier information-theoretic proofs, and its utility extends to random variables taking values in general spaces. The core argument employed has its origins in the quantum information-theoretic literature.