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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
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. 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.
Keywords: hypothesis testing; weighted Chernoff information; weighted Bhattacharyya coefficient; exponential family; information geometry; context-sensitive loss
College: Faculty of Science and Engineering
Funders: 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).
Issue: 5

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