A New Look at Bayesian Testing

2026-02-11

AI Summary

This study addresses the theoretical gap between classical hypothesis testing with fixed significance levels and Bayesian methods, particularly in light of the Lindley paradox. By leveraging moderate deviation theory, the authors develop a unified Bayesian framework for hypothesis testing. Through Bayesian risk analysis and asymptotic expansions, they show that the optimal test threshold operates on the scale of √(log n / n), naturally yielding Jeffreys’ threshold, the BIC penalty term, and the Chernoff–Stein error exponent. This framework not only resolves the Lindley paradox but also extends Rubin’s (1965) program to modern settings such as high-dimensional sparse inference, goodness-of-fit testing, and model selection. Moreover, it establishes the superiority of Bayesian procedures over classical Neyman–Pearson tests in terms of statistical risk.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Game Theory and Economic Paradigms: Behavioral Game Theory

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 develop a unified framework for Bayesian hypothesis testing through the theory of moderate deviations, providing explicit asymptotic expansions for Bayes risk and optimal test statistics. Our analysis reveals that Bayesian test cutoffs operate on the moderate deviation scale $\sqrt{\log n/n}$, in sharp contrast to the sample-size-invariant calibrations of classical testing. This fundamental difference explains the Lindley paradox and establishes the risk-theoretic superiority of Bayesian procedures over fixed-$\alpha$ Neyman-Pearson tests. We extend the seminal Rubin (1965) program to contemporary settings including high-dimensional sparse inference, goodness-of-fit testing, and model selection. The framework unifies several classical results: Jeffreys'$\sqrt{\log n}$ threshold, the BIC penalty $(d/2)\log n$, and the Chernoff-Stein error exponents all emerge naturally from moderate deviation analysis of Bayes risk. Our results provide theoretical foundations for adaptive significance levels and connect Bayesian testing to information theory through gambling-based interpretations.

Problem

• Bayesian hypothesis testing
• Lindley paradox
• moderate deviations
• Bayes risk
• Neyman-Pearson tests

Innovation

• Bayesian hypothesis testing
• moderate deviations
• Bayes risk
• adaptive significance levels
• information theory