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Weighted Chernoff Information and Optimal Loss Exponent in Context-Sensitive Hypothesis Testing

Mark Kelbert, El’mira Yu. Kalimulina Orcid Logo

Entropy, Volume: 28, Issue: 5

Swansea University Author: Mark Kelbert

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DOI (Published version): 10.3390/e28050536

Abstract

We study binary hypothesis testing for i.i.d. observations under a multiplicative context weight. For the optimal weighted total loss, defined as the sum of weighted type-I and type-II losses, we prove the logarithmic asymptotic *=exp⁡{−⁢w⁡(ℙ,ℚ)+⁡()},→∞, where w is the weighted Chernoff information....

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Published in: Entropy
ISSN: 1099-4300
Published: MDPI AG 2026
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URI: https://cronfa.swan.ac.uk/Record/cronfa71943
first_indexed 2026-05-18T18:41:59Z
last_indexed 2026-05-19T11:19:17Z
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spelling 2026-05-18T19:44:09.7907580 v2 71943 2026-05-18 Weighted Chernoff Information and Optimal Loss Exponent in Context-Sensitive Hypothesis Testing 3a81821a33f0e13488bd6a81cdc30f9d Mark Kelbert Mark Kelbert true false 2026-05-18 We study binary hypothesis testing for i.i.d. observations under a multiplicative context weight. For the optimal weighted total loss, defined as the sum of weighted type-I and type-II losses, we prove the logarithmic asymptotic *=exp⁡{−⁢w⁡(ℙ,ℚ)+⁡()},→∞, where w is the weighted Chernoff information. The single-letter form of the exponent relies on a structural assumption that the weight factorises across observations, ⁡(1)=∏=1⁡(); this restriction is essential for the single-letter representation and should be distinguished from the weaker qualitative description “multiplicative context weight”. The proof embeds the weighted geometric mixtures ⁢⁢1− into a likelihood-ratio exponential family and identifies the rate through its log-normaliser. We also derive concentration bounds for the tilted weighted log-likelihood, obtain closed forms for Gaussian, Poisson, and exponential models, and extend the exponent characterisation to finitely many hypotheses. Journal Article Entropy 28 5 MDPI AG 1099-4300 hypothesis testing; weighted Chernoff information; weighted Bhattacharyya coefficient; exponential family; information geometry; context-sensitive loss 8 5 2026 2026-05-08 10.3390/e28050536 COLLEGE NANME COLLEGE CODE Swansea University Another institution paid the OA fee The work by MK was carried out in the framework of a research project HSE-BR-2025- 039 implemented as part of the Basic Research Program at HSE University. The second author (E.Yu. Kalimulina) was supported by the Ministry of Science and Higher Education of the Russian Federation under the state assignment (project FFNU-2025-0029). 2026-05-18T19:44:09.7907580 2026-05-18T19:36:43.5265264 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Mark Kelbert 1 El’mira Yu. Kalimulina 0000-0001-7158-040x 2 71943__36789__818a6a6216be4790bab35588533b2b5c.pdf 71943.VoR.pdf 2026-05-18T19:43:16.2292228 Output 485017 application/pdf Version of Record true © 2026 by the authors. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license. true eng https://creativecommons.org/licenses/by/4.0/
title Weighted Chernoff Information and Optimal Loss Exponent in Context-Sensitive Hypothesis Testing
spellingShingle Weighted Chernoff Information and Optimal Loss Exponent in Context-Sensitive Hypothesis Testing
Mark Kelbert
title_short Weighted Chernoff Information and Optimal Loss Exponent in Context-Sensitive Hypothesis Testing
title_full Weighted Chernoff Information and Optimal Loss Exponent in Context-Sensitive Hypothesis Testing
title_fullStr Weighted Chernoff Information and Optimal Loss Exponent in Context-Sensitive Hypothesis Testing
title_full_unstemmed Weighted Chernoff Information and Optimal Loss Exponent in Context-Sensitive Hypothesis Testing
title_sort Weighted Chernoff Information and Optimal Loss Exponent in Context-Sensitive Hypothesis Testing
author_id_str_mv 3a81821a33f0e13488bd6a81cdc30f9d
author_id_fullname_str_mv 3a81821a33f0e13488bd6a81cdc30f9d_***_Mark Kelbert
author Mark Kelbert
author2 Mark Kelbert
El’mira Yu. Kalimulina
format Journal article
container_title Entropy
container_volume 28
container_issue 5
publishDate 2026
institution Swansea University
issn 1099-4300
doi_str_mv 10.3390/e28050536
publisher MDPI AG
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description We study binary hypothesis testing for i.i.d. observations under a multiplicative context weight. For the optimal weighted total loss, defined as the sum of weighted type-I and type-II losses, we prove the logarithmic asymptotic *=exp⁡{−⁢w⁡(ℙ,ℚ)+⁡()},→∞, where w is the weighted Chernoff information. The single-letter form of the exponent relies on a structural assumption that the weight factorises across observations, ⁡(1)=∏=1⁡(); this restriction is essential for the single-letter representation and should be distinguished from the weaker qualitative description “multiplicative context weight”. The proof embeds the weighted geometric mixtures ⁢⁢1− into a likelihood-ratio exponential family and identifies the rate through its log-normaliser. We also derive concentration bounds for the tilted weighted log-likelihood, obtain closed forms for Gaussian, Poisson, and exponential models, and extend the exponent characterisation to finitely many hypotheses.
published_date 2026-05-08T17:20:50Z
_version_ 1866631008411975680
score 11.106612

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