A proof of the Shepp-Olkin entropy monotonicity conjecture

Erwan Hillion; Oliver T Johnson

doi:10.1214/19-EJP380

A proof of the Shepp-Olkin entropy monotonicity conjecture

Erwan Hillion, Oliver T Johnson

Research output: Contribution to journal › Article (Academic Journal) › peer-review

3 Citations (Scopus)

179 Downloads (Pure)

Abstract

Consider tossing a collection of coins, each fair or biased towards heads, and take the distribution of the total number of heads that result. It is natural to suppose that this distribution should be ‘more random’ when each coin is fairer. In this paper, we prove a 40 year old conjecture of Shepp and Olkin, by showing that the Shannon entropy is monotonically increasing in this case, using a construction inspired by optimal transport theory. We discuss whether this result can be generalized to q-R´enyi and q-Tsallis entropies, for a range of values of q. MSC 2010

Original language	English
Article number	126
Number of pages	14
Journal	Electronic Journal of Probability
Volume	24
DOIs	https://doi.org/10.1214/19-EJP380
Publication status	Published - 9 Nov 2019

Keywords

entropy
functional inequalities
mixing coefficients
Poisson–binomial distribution

Access to Document

10.1214/19-EJP380Licence: Unspecified

Full-text PDF (final published version)
This is the final published version of the article (version of record). It first appeared online via Institute of Mathematical Statistics at https://projecteuclid.org/euclid.ejp/1573268590. Please refer to any applicable terms of use of the publisher.
Final published version, 232 KBLicence: CC BY

https://arxiv.org/abs/1810.09791

Handle.net

Persistent link

Embargoed Document

Full-text PDF (author’s accepted manuscript)
Accepted author manuscript, 331 KB
Request copy

Cite this

@article{f0bc771cb63e42e4988d8f87914a0632,

title = "A proof of the Shepp-Olkin entropy monotonicity conjecture",

abstract = "Consider tossing a collection of coins, each fair or biased towards heads, and take the distribution of the total number of heads that result. It is natural to suppose that this distribution should be {\textquoteleft}more random{\textquoteright} when each coin is fairer. In this paper, we prove a 40 year old conjecture of Shepp and Olkin, by showing that the Shannon entropy is monotonically increasing in this case, using a construction inspired by optimal transport theory. We discuss whether this result can be generalized to q-R´enyi and q-Tsallis entropies, for a range of values of q. MSC 2010",

keywords = "entropy, functional inequalities, mixing coefficients, Poisson–binomial distribution",

author = "Erwan Hillion and Johnson, {Oliver T}",

year = "2019",

month = nov,

day = "9",

doi = "10.1214/19-EJP380",

language = "English",

volume = "24",

journal = "Electronic Journal of Probability",

issn = "1083-6489",

publisher = "Institute of Mathematical Statistics",

}

TY - JOUR

T1 - A proof of the Shepp-Olkin entropy monotonicity conjecture

AU - Hillion, Erwan

AU - Johnson, Oliver T

PY - 2019/11/9

Y1 - 2019/11/9

N2 - Consider tossing a collection of coins, each fair or biased towards heads, and take the distribution of the total number of heads that result. It is natural to suppose that this distribution should be ‘more random’ when each coin is fairer. In this paper, we prove a 40 year old conjecture of Shepp and Olkin, by showing that the Shannon entropy is monotonically increasing in this case, using a construction inspired by optimal transport theory. We discuss whether this result can be generalized to q-R´enyi and q-Tsallis entropies, for a range of values of q. MSC 2010

AB - Consider tossing a collection of coins, each fair or biased towards heads, and take the distribution of the total number of heads that result. It is natural to suppose that this distribution should be ‘more random’ when each coin is fairer. In this paper, we prove a 40 year old conjecture of Shepp and Olkin, by showing that the Shannon entropy is monotonically increasing in this case, using a construction inspired by optimal transport theory. We discuss whether this result can be generalized to q-R´enyi and q-Tsallis entropies, for a range of values of q. MSC 2010

KW - entropy

KW - functional inequalities

KW - mixing coefficients

KW - Poisson–binomial distribution

U2 - 10.1214/19-EJP380

DO - 10.1214/19-EJP380

M3 - Article (Academic Journal)

SN - 1083-6489

VL - 24

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

M1 - 126

ER -

A proof of the Shepp-Olkin entropy monotonicity conjecture

Abstract

Keywords

Access to Document

Handle.net

Embargoed Document

Fingerprint

Cite this