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.

翻译：我们识别了在正则参数检验中支配贝叶斯证据积累的关键偏离尺度。在零一损失的综合贝叶斯风险下，风险最优拒绝边界位于中等偏离区间，其平方根对数膨胀相对于通常的局部渐近正态尺度。在Cramer正则性、原假设处局部先验光滑性及对称损失条件下，我们推导了尖锐阈值，并表明其主导对数项在正则先验下具有普适性，而低阶常数则取决于局部先验密度、Fisher信息及先验模型比。该结果通过局部渐近正态性扩展至单参数指数族，并将杰弗里斯检验阈值、贝叶斯信息准则惩罚项及Chernoff-Stein型误差指数论证统一于一个共同的中等偏离渐近框架内。