A New Look at Bayesian Testing

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-$α$ 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.