Finite de Finetti bounds in relative entropy

TL;DR

This paper reviews finite de Finetti theorems in total variation and relative entropy, introduces new tight bounds for exchangeable vectors in arbitrary spaces, and explores their relation to sampling methods.

Contribution

It provides new finite de Finetti theorems with bounds independent of space size and dimension, extending previous results.

Findings

Bounds are tight and dimension-independent.

New theorems apply to exchangeable vectors in arbitrary spaces.

Connections between de Finetti theorems and sampling bounds are clarified.

Abstract

We review old and recent finite de Finetti theorems in total variation distance and in relative entropy, and we highlight their connections with bounds on the difference between sampling with and without replacement. We also establish two new finite de Finetti theorems for exchangeable random vectors taking values in arbitrary spaces. These bounds are tight, and they are independent of the size and the dimension of the underlying space.