An Information-Theoretic Proof of a Finite de Finetti Theorem

TL;DR

This paper provides an information-theoretic proof of a finite de Finetti theorem, showing that the distribution of initial variables in an exchangeable binary sequence approximates a mixture of product distributions with explicit bounds.

Contribution

It introduces an elementary information-theoretic approach to establish a finite de Finetti theorem with explicit bounds on distribution closeness.

Findings

Distribution of initial variables is close to a mixture of product distributions

Closeness measured using relative entropy with explicit bounds

Applicable to exchangeable binary vectors of finite length

Abstract

A 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 binary vector of length $n\geq k$ is close to a mixture of product distributions. Closeness is measured in terms of the relative entropy and an explicit bound is provided.