AI 中文总结
研究针对粒子物理等领域模型错误设定时标准置信区间失效的问题，提出适度似然性方法，通过除以拟合优度统计量，使置信区间自我限制，给出分箱和不分箱公式并经实例验证。
AI 中文摘要
在粒子物理学及更广泛的物理科学中，基于似然性的推断依赖于模型准确描述数据这一假设。然而，当模型错误设定时，随着数据增加，标准置信区间宽度缩至零，产生过度自信且可能误导性的约束。我们提出适度似然性，即将似然比检验统计量除以在最佳拟合点评估的拟合优度（GOF）统计量。在正确设定下，GOF为\(\mathcal{O}(1)\)并恢复标准推断。在错误设定下，对于\(N\)个数据点，似然性和GOF都按\(\mathcal{O}(N)\)缩放，所以它们的比值仍为\(\mathcal{O}(1)\)，所得适度似然性置信区间在由模型不充分性决定的下限处自我限制，本质上减小了有效样本量。我们给出了适度似然性的分箱和不分箱公式，并在高斯示例以及使用合成电子 - 正电子碰撞对强耦合常数的测量中展示了我们的方法。
英文摘要
Likelihood-based inference in particle physics, and in the physical sciences more broadly, relies on the assumption that the model accurately describes the data. When the model is misspecified, though, standard confidence intervals shrink to zero width with increasing data, producing overconfident and potentially misleading constraints. We propose the well-tempered likelihood, which divides the likelihood-ratio test statistic by a goodness-of-fit (GOF) statistic evaluated at the best-fit point. Under correct specification, the GOF is $\mathcal{O}(1)$ and standard inference is recovered. Under misspecification, both the likelihood and the GOF scale as $\mathcal{O}(N)$ for $N$ data points, so their ratio remains $\mathcal{O}(1)$ and the resulting well-tempered confidence interval self-limits at a floor determined by the model's inadequacy, essentially reducing the effective sample size. In other words, \textit{all models are correct, as long as your dataset is small enough}. We present binned and unbinned formulations of the well-tempered likelihood---the latter based on a classifier two-sample test---and demonstrate our method on a Gaussian example and on a measurement of the strong coupling constant using synthetic electron-positron collisions.
Comments11 pages, 2 figures
