arXiv 2406.016642024-06hep-phhep-exhep-th

一种用于部分子分布拟合的广义统计模型

A generalized statistical model for fits to parton distributions

Mengshi Yan, Tie-Jiun Hou, Zhao Li, Kirtimaan Mohan, C. -P. Yuan

备注：: 52 pages, 14 figures. Updated to match journal version. Includes new appendices with a pedagogical examples and discussion
DOI：: 10.1103/zcsy-34cc

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AI总结：

针对PDF拟合中数据集张力导致结果依赖数据集、不确定性易被低估的问题，提出基于贝叶斯层次模型、通过GMM实现的广义统计模型，可准确估计含张力数据的PDF不确定性，且一致数据集下退化为卡方似然。

摘要：

部分子分布函数（PDFs）是粒子物理计算的核心组成部分。目前，针对这些非微扰函数的最精确预言是通过对全球数据的拟合生成的。多个PDF拟合团队遇到的一个问题是数据集中存在张力，这些张力似乎会将拟合拉向不同方向。换言之，最佳拟合结果取决于数据集的选择。目前已有多种用于在拟合结果看似不一致的情况下捕捉PDF不确定性的方法被提出并投入使用，这些方法对于确保PDF的不确定性不被低估至关重要。本文提出一种新的不确定性估计方法，该方法引入了一种基于贝叶斯层次模型的广义统计模型，并通过高斯混合模型（GMM）实现。该方法受无监督机器学习技术启发，与集成学习（Ensemble learning）和贝叶斯模型平均的统计方法密切相关。我们利用一个PDF玩具模型演示了如何使用GMM忠实地重构与PDF拟合相关的似然，该似然反过来可用于准确确定PDF的不确定性，尤其是在拟合数据集存在张力的情况下。我们进一步展示了对于一致的数据集，该统计模型如何退化为通常的卡方似然函数，并提供了优化GMM中高斯分量数量的度量方法。

英文摘要：

Parton distribution functions (PDFs) form an essential part of particle physics calculations. Currently, the most precise predictions for these non-perturbative functions are generated through fits to global data. A problem that several PDF fitting groups encounter is the presence of tension in data sets that appear to pull the fits in different directions. In other words, the best fit depends on the choice of data set. Several methods to capture the uncertainty in PDFs in presence of seemingly inconsistent fits have been proposed and are currently in use. These methods are important to ensure that uncertainty in PDFs are not underestimated. Here we propose a novel method for estimating the uncertainty by introducing a generalized statistical model based on Bayesian Hierarchical models which is implemented via the Gaussian Mixture Model (GMM). The methodology is inspired by unsupervised machine learning techniques and is closely related to the statistical methods of Ensemble learning and Bayesian model averaging. Using a toy model of PDFs, we demonstrate how the GMM can be used to faithfully reconstruct the likelihood associated with PDF fits, which can in turn be used to accurately determine the uncertainty on PDFs, especially in the presence of tension in the fitted data sets. We further show how this statistical model reduces to the usual chi-squared likelihood function for a consistent data set and provide measures to optimize the number of Gaussians in the GMM.

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