总体归一化为拟合参数时的卡方拟合
Chi-square Fitting When Overall Normalization is a Fit Parameter
摘要：
在实验研究中，不断出现当期望事件总数不固定时拟合事件分布的问题。在卡方拟合中，如果整体归一化是要拟合的参数之一，则拟合曲线相对于数据点可能非常低，有时低于所有数据点。这个问题和它的解决方案在统计界是众所周知的，但显然，在一些物理界并不为人所知。这篇笔记的目的是说教，解释问题的原因和简单而优雅的解决方案。解决方法是使用极大似然而不是卡方检验。这两种方法之间的本质区别是，极大似然使用卡方中每个项的归一化，假设它是正态分布，1/sqrt（2 pi sigma-square）。此外，归一化应用于理论期望而不是数据。在本说明中，我们说明了什么是错的，以及极大似然如何解决这个问题，在一个非常简单的玩具的例子，清楚地说明了这个问题，是适当的物理模型的事件直方图。然后，我们注意到如何一个简单的修改卡方方法给出了相同的结果的极大似然方法。
英文摘要：
The problem of fitting an event distribution when the total expected number of events is not fixed, keeps appearing in experimental studies. In a chi-square fit, if overall normalization is one of the parameters parameters to be fit, the fitted curve may be seriously low with respect to the data points, sometimes below all of them. This problem and the solution for it are well known within the statistics community, but, apparently, not well known among some of the physics community. The purpose of this note is didactic, to explain the cause of the problem and the easy and elegant solution. The solution is to use maximum likelihood instead of chi-square. The essential difference between the two approaches is that maximum likelihood uses the normalization of each term in the chi-square assuming it is a normal distribution, 1/sqrt(2 pi sigma-square). In addition, the normalization is applied to the theoretical expectation not to the data. In the present note we illustrate what goes wrong and how maximum likelihood fixes the problem in a very simple toy example which illustrates the problem clearly and is the appropriate physics model for event histograms. We then note how a simple modification to the chi-square method gives a result identical to the maximum likelihood method.
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