有限规模加权蒙特卡罗数据不确定性的概率处理
Probabilistic treatment of the uncertainty from the finite size of weighted Monte Carlo data
摘要：
HEP实验中的参数估计通常涉及蒙特卡罗模拟来模拟实验响应函数。一个典型的应用是前向折叠似然分析与重新加权，或耗时的最小化计划与一个新的模拟设置为每个参数值。问题是，这种Monte Carlo样本的有限大小具有内在的不确定性，如果忽略该不确定性并且样本量较小，则可能导致参数估计的实质性偏倚。我们引入了一个概率处理这个问题，通过取代通常的似然函数与新的广义概率分布，通过适当的边缘化，将有限的统计。这些新的PDF是分析性的，可用于取代Poisson、多项式和基于样本的未分组可能性，其中涵盖了高能物理中的许多用例。在无限统计的极限下，它们还原为各自的标准概率分布。在一般情况下的任意蒙特卡罗权重，表达式涉及第四劳瑞塞拉函数$F_D$，我们找到了一个新的有限和表示在一定的参数设置。该结果还表示了具有$n>0$的Carlson Dirichlet平均$R_n$的精确形式，从而为计算Dirichlet-多项式分布的概率母函数、单项函数的扩展除差或一元B样条的任意矩提供了一种有效的方法.我们证明了我们的方法与一个典型的玩具蒙特卡罗问题的偏差减少，估计下降的能量谱中的峰的归一化，并比较结果与以前发表的方法从文献中。
英文摘要：
Parameter estimation in HEP experiments often involves Monte-Carlo simulation to model the experimental response function. A typical application are forward-folding likelihood analyses with re-weighting, or time-consuming minimization schemes with a new simulation set for each parameter value. Problematically, the finite size of such Monte Carlo samples carries intrinsic uncertainty that can lead to a substantial bias in parameter estimation if it is neglected and the sample size is small. We introduce a probabilistic treatment of this problem by replacing the usual likelihood functions with novel generalized probability distributions that incorporate the finite statistics via suitable marginalization. These new PDFs are analytic, and can be used to replace the Poisson, multinomial, and sample-based unbinned likelihoods, which covers many use cases in high-energy physics. In the limit of infinite statistics, they reduce to the respective standard probability distributions. In the general case of arbitrary Monte Carlo weights, the expressions involve the fourth Lauricella function $F_D$, for which we find a new finite-sum representation in a certain parameter setting. The result also represents an exact form for Carlson's Dirichlet average $R_n$ with $n>0$, and thereby an efficient way to calculate the probability generating function of the Dirichlet-multinomial distribution, the extended divided difference of a monomial, or arbitrary moments of univariate B-splines. We demonstrate the bias reduction of our approach with a typical toy Monte Carlo problem, estimating the normalization of a peak in a falling energy spectrum, and compare the results with previously published methods from the literature.
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