An information-theoretic proof of a finite de Finetti theorem

Lampros Gavalakis,Ioannis Kontoyiannis

doi:10.1214/21-ecp428

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.

Highlights

A finite sequence of random variables (X1, X2, . . . , Xn) is exchangeable if it has the same distribution as (Xπ(1), Xπ(2), . . . , Xπ(n)) for every permutation π of {1, 2, . . . , n}
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 ≥ k is close to a mixture of product distributions
Closeness is measured in terms of the relative entropy and an explicit bound is provided

Summary

Introduction

A finite sequence of random variables (X1, X2, . . . , Xn) is exchangeable if it has the same distribution as (Xπ(1), Xπ(2), . . . , Xπ(n)) for every permutation π of {1, 2, . . . , n}. 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 ≥ k is close to a mixture of product distributions. Closeness is measured in terms of the relative entropy and an explicit bound is provided.

Results

Conclusion