A New Look at Bayesian Testing

TL;DR

This paper investigates the critical deviation scale in Bayesian hypothesis testing, revealing a universal moderate deviation threshold that extends classical asymptotic results and unifies various testing criteria.

Contribution

It derives the sharp Bayesian testing threshold under regularity conditions, showing its universality and connection to classical error exponents within a moderate deviation framework.

Findings

The Bayesian rejection boundary lies in a moderate deviation regime.

The threshold's leading logarithmic term is universal across regular priors.

Connections are established between Bayesian testing, information criteria, and error exponents.

Abstract

We identify the critical deviation scale governing Bayesian evidence accumulation in regular parametric testing. Under integrated Bayes risk with zero-one loss, the risk-optimal rejection boundary lies in a moderate deviation regime, with a square-root logarithmic inflation relative to the usual local asymptotic normal scale. Under Cramer regularity, local prior smoothness at the null, and symmetric loss, we derive the sharp threshold and show that its leading logarithmic term is universal across regular priors, while lower-order constants depend on the local prior density, Fisher information, and prior model odds. The result extends to one-parameter exponential families through local asymptotic normality and places Jeffreys' testing threshold, the Bayesian information criterion penalty, and Chernoff-Stein type error-exponent arguments within a common asymptotic moderate deviation framework.