Relative entropy bounds for sampling with and without replacement

TL;DR

This paper derives sharp, nonasymptotic bounds on the relative entropy between sampling with and without replacement from an urn, with bounds depending on the urn's composition and connecting to finite de Finetti theorems.

Contribution

It introduces new bounds for relative entropy in sampling scenarios that depend on urn composition and links these bounds to finite de Finetti theorems, improving asymptotic convergence insights.

Findings

Bounds are asymptotically tight in certain regimes.

Bounds depend on the number of balls of each color.

A new finite de Finetti bound in relative entropy is derived.

Abstract

Sharp, nonasymptotic bounds are obtained for the relative entropy between the distributions of sampling with and without replacement from an urn with balls of $c\geq 2$ colors. Our bounds are asymptotically tight in certain regimes and, unlike previous results, they depend on the number of balls of each colour in the urn. The connection of these results with finite de Finetti-style theorems is explored, and it is observed that a sampling bound due to Stam (1978) combined with the convexity of relative entropy yield a new finite de Finetti bound in relative entropy, which achieves the optimal asymptotic convergence rate.