Weighted Chernoff information and optimal loss exponent in context-sensitive hypothesis testing

2026-03-09

AI Summary

This study addresses the context-sensitive binary hypothesis testing problem under i.i.d. observations with a multiplicative weight function, aiming to characterize the asymptotic optimality of the total risk. By embedding the weighted geometric mixture into an exponential family framework, the work proposes a weighted Chernoff information as the optimal error exponent and expresses it as the maximum of the log-normalizer of the associated exponential family. Leveraging exponential family modeling, weighted likelihood analysis, and concentration inequalities, the authors establish a logarithmic asymptotic expression for the total risk, derive explicit solutions for canonical models such as Gaussian and Poisson, and provide concentration bounds for the tilted weighted log-likelihood. These results offer both theoretical foundations and computational tools for weighted hypothesis testing.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider context-sensitive (binary) hypothesis testing for i.i.d. observations under a multiplicative weight function. We establish the logarithmic asymptotic, as the sample size grows, of the optimal total loss (sum of type-I and type-II losses) and express the corresponding error exponent through a weighted Chernoff information between the competing distributions. Our approach embeds weighted geometric mixtures into an exponential family and identifies the exponent as the maximizer of its log-normaliser. We also provide concentration bounds for a tilted weighted log-likelihood and derive explicit expressions for Gaussian and Poisson models, as well as further parametric examples.

Problem

• context-sensitive hypothesis testing
• weighted Chernoff information
• optimal loss exponent
• error exponent
• i.i.d. observations

Innovation

• weighted Chernoff information
• context-sensitive hypothesis testing
• exponential family
• error exponent
• log-normaliser