Bayes factors are widely computed by Monte Carlo, yet heavy-tailed sampling distributions can make numerical validation unreliable. The Turing--Good identities provide exact moment equalities for powers of a Bayes factor (a density ratio). When these identities are used as Good-check diagnostics, the power choice becomes a statistical design parameter. We develop a nonasymptotic variance theory for Monte Carlo evaluation of the identities and show that the half-order (square-root) power is uniquely minimax-stable: it equalizes variability across the two model orientations and is the only choice that guarantees finite second moments in a distribution-free worst-case sense over all mutually absolutely continuous model pairs. This yields a balanced two-sample half-order diagnostic that is symmetric in model labeling and has a uniform variance bound at fixed computational budget; in small-overlap regimes it is guaranteed to be no less efficient than the standard one-sided Turing check. Simulations for binomial Bayes factor workflows illustrate stable finite-sample behavior and sensitivity to simulator--evaluator mismatches. We further connect the half-order overlap viewpoint to stable primitives for normalizing-constant ratios and importance-sampling degeneracy summaries.

翻译：贝叶斯因子通常通过蒙特卡洛方法计算，但重尾采样分布可能导致数值验证不可靠。图灵-古德恒等式为贝叶斯因子（密度比）的幂次提供了精确的矩等式。当这些恒等式被用作古德检验诊断工具时，幂次选择便成为统计设计参数。我们建立了用于恒等式蒙特卡洛评估的非渐近方差理论，证明半阶（平方根）幂具有唯一的极小极大稳定性：它在两种模型方向上均衡变异性，并且是唯一能在所有互绝对连续模型对的分布自由最坏情况下保证二阶矩有限的选择。由此产生了一种平衡的双样本半阶诊断方法，该方法在模型标记上对称，且在固定计算预算下具有一致的方差上界；在低重叠区域，该方法保证不低于标准单侧图灵检验的效率。针对二项贝叶斯因子工作流程的模拟实验展示了稳定的有限样本特性及对模拟器-评估器失配的敏感性。我们进一步将半阶重叠视角与归一化常数比及重要性采样退化度度量的稳定原语相联系。