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Relative entropy bounds for sampling with and without replacement

Part of: Stochastic processes Distribution theory - Probability Communication, information

Published online by Cambridge University Press:  28 July 2025

Oliver Johnson Open the ORCID record for Oliver Johnson [Opens in a new window] ,
Lampros Gavalakis Open the ORCID record for Lampros Gavalakis [Opens in a new window]  and
Ioannis Kontoyiannis Open the ORCID record for Ioannis Kontoyiannis [Opens in a new window]
Oliver Johnson*
Affiliation:
University of Bristol
Lampros Gavalakis*
Affiliation:
Université Gustave Eiffel
Ioannis Kontoyiannis*
Affiliation:
University of Cambridge
*
*Postal address: School of Mathematics, Fry Building, Woodland Road, Bristol, BS8 1UG, UK. Email: O.Johnson@bristol.ac.uk
**Postal address: Université Gustave Eiffel, Université Paris Est Creteil, CNRS, LAMA UMR8050 F-77447 Marne-la-Vallée, France. Email: lg560@cam.ac.uk
***Postal address: Statistical Laboratory, DPMMS, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK. Email: yiannis@maths.cam.ac.uk
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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 color 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.

Keywords

Relative entropysamplingreplacementhypergeometricbinomialde Finetti

MSC classification

Primary: 60E05: Distributions
Secondary: 60G09: Exchangeability 94A17: Measures of information, entropy

Information

Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 16
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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