Giovanni Punzi
Unbinned likelihood fits are frequent in Physics, and often involve complex functions with several components. We discuss the potential pitfalls of situations where the templates used in the fit are not fixed but depend on the event observables, as it happens when the resolution of the measurement is event--dependent, and the procedure to avoid them.
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Parameter estimation via unbinned maximum likelihood fits is central for many analyses performed in high energy physics. Unbinned maximum likelihood fits using event weights, for example to statistically subtract background contributions via the sPlot formalism, or to correct for acceptance effects, have recently seen increasing use in the community. However, it is well known that the naive approach to the estimation of parameter uncertainties via the second derivative of the logarithmic likelihood does not yield confidence intervals with the correct coverage in the presence of event weights. …
Uncertainty components in profile likelihood fits
本文针对轮廓似然拟合中系统不确定性量化不准确的问题，区分了“影响”与实际不确定性分量，并提出了在多种统计模型中确定实际分量的方法。
A common goal in an experimental physics analysis is to extract information from a reaction with multi-dimensional kinematics. The preferred method for such a task is typically the unbinned maximum likelihood method. In fits using this method, the likelihood is a goodness-of-fit quantity in that it effectively discriminates between available hypotheses; however, it does not provide any information as to how well the best hypothesis describes the data. In this paper, we present an {\em ad-hoc} procedure for obtaining chi**2/n.d.f. values from unbinned maximum likelihood fits. This method does n…
The method of extended maximum likelihood is a well known concept of parameter estimation. One can implement external knowledge on the unknown parameters by multiplying the likelihood by constraint terms. In this note, we emphasize that this is also true for yield parameters in an extended maximum likelihood fit, which is widely used in the particle physics community. We recommend a way to generate pseudo-experiments in presence of constraint terms on yield parameters, and point to pitfalls inside the RooFit framework.
Maximum likelihood fits to data can be performed using binned data and unbinned data. The likelihood fits in either case produce only the fitted quantities but not the goodness of fit. With binned data, one can obtain a measure of the goodness of fit by using the $χ^2$ method, after the maximum likelihood fitting is performed. With unbinned data, currently, the fitted parameters are obtained but no measure of goodness of fit is available. This remains, to date, an unsolved problem in statistics. By considering the transformation properties of likelihood functions with respect to change of vari…
Maximum likelihood fits to data can be done using binned data (histograms) and unbinned data. With binned data, one gets not only the fitted parameters but also a measure of the goodness of fit. With unbinned data, currently, the fitted parameters are obtained but no measure of goodness of fit is available. This remains, to date, an unsolved problem in statistics. Using Bayes' theorem and likelihood ratios, we provide a method by which both the fitted quantities and a measure of the goodness of fit are obtained for unbinned likelihood fits, as well as errors in the fitted quantities. The quant…
Maximum likelihood fits to data can be done using binned data (histograms) and unbinned data. With binned data, one gets not only the fitted parameters but also a measure of the goodness of fit. With unbinned data, currently, the fitted parameters are obtained but no measure of goodness of fit is available. This remains, to date, an unsolved problem in statistics. Using Bayes theorem and likelihood ratios, we provide a method by which both the fitted quantities and a measure of the goodness of fit are obtained for unbinned likelihood fits, as well as errors in the fitted quantities. We provide…
A fast and stable approximate maximum-likelihood method for template fits
本文从精确的Barlow-Beeston似然推导出一种新的快速稳定近似似然，并将其与Conway似然推广到加权模板拟合场景，实验表明其性能与精确似然相当且计算更快。
The possibility of strong biases in a multicomponent Maximum Likelihood fits with component-dependent templates has been demonstrated in some toy problems. We discuss here in detail a problem of practical interest, particle identification based on time-of-flight or dE/dx information. We show that large biases can occur in estimating particle fractions in a sample if differences between the momentum spectra of particles are ignored, and we present a more robust fit technique, allowing bias-free estimation even when the particle spectra in the sample are unknown.
Straightforward methods for adapting the familiar chi^2 statistic to histograms of discrete events and other Poisson distributed data generally yield biased estimates of the parameters of a model. The bias can be important even when the total number of events is large. For the case of estimating a microcalorimeter's energy resolution at 6 keV from the observed shape of the Mn K-alpha fluorescence spectrum, a poor choice of chi^2 can lead to biases of at least 10% in the estimated resolution when up to thousands of photons are observed. The best remedy is a Poisson maximum-likelihood fit, throu…
Combinations of measurements that are simultaneous fits of parameters of interest and systematic uncertainties using the BLUE method
本文提出用BLUE方法同时组合感兴趣参数与系统不确定性参数，通过示例和伪实验证明包含干扰参数可提高精度，并指出Convino软件近似似然组合低估不确定性，方法已实现于开源工具Combiner。
The geometry of uncertainty decomposition in profile-likelihood fits
该研究通过耦合纤维丛的几何方法，阐明轮廓似然拟合中两种不确定性分解方法的区别，解释了高能物理中系统不确定性估计方法间的关系，明确了冗余参数影响与重复实验源方差不重合的原因。
Binned maximum likelihood fits are an attractive option when analysing large datasets, but require care when computing likelihoods of continuous PDFs in bins. For many years the widely used statistical modelling package RooFit evaluated probabilities at the bin centre, leading to significant biases for strongly curved probability density functions. We demonstrate the biases with real-world examples, and introduce a PDF class to RooFit that removes these biases. The physics and computation performance of this new class are discussed.
The incorporation of systematic uncertainties into confidence interval calculations has been addressed recently in a paper by Conrad et al. (Physical Review D 67 (2003) 012002). In their work, systematic uncertainities in detector efficiencies and background flux predictions were incorporated following the hybrid frequentist-Bayesian prescription of Cousins and Highland, but using the likelihood ratio ordering of Feldman and Cousins in order to produce "unified" confidence intervals. In general, the resulting intervals behaved as one would intuitively expect, i.e. increased with increasing unc…
Maximum-likelihood fits of piece-wise Pareto distributions with finite and non-zero core
本文针对具有有限非零核心和幂律尾部的分段帕累托分布，提出三类核心形式并推导最大似然估计及数值方法，提供Python包实现。
Likelihood ratio tests are performed for the hypothesis that charged-particle multiplicities measured in proton-(anti)proton collisions at $\sqrt{s}$ = 0.9 and 2.36 TeV are distributed according to the negative binomial form. Results indicate that the hypothesis should be rejected in the all cases of ALICE-LHC measurements in the limited pseudo-rapidity windows, whereas should be accepted in the corresponding cases of UA5 data. Possible explanations of that and of the disagreement with the least-squares fitting method are given.
Comment on Frank Porter, "Confidence intervals for the Poisson distribution"
本文评论 Porter 对泊松分布置信区间的综述，指出其关于 FC 似然比检验区间的讨论存在缺陷，核心分歧在于模型不存在的参数空间部分似然函数是否成立。
MoreFit: A More Optimised, Rapid and Efficient Fit
MoreFit通过优化计算图和并行计算，提供高效的未分桶最大似然拟合解决方案，适用于多种异构平台。
Multivariate analyses play an important role in high energy physics. Such analyses often involve performing an unbinned maximum likelihood fit of a probability density function (p.d.f.) to the data. This paper explores a variety of unbinned methods for determining the goodness of fit of the p.d.f. to the data. The application and performance of each method is discussed in the context of a real-life high energy physics analysis (a Dalitz-plot analysis). Several of the methods presented in this paper can also be used for the non-parametric determination of whether two samples originate from the …
A pressing problem for supersymmetry (SUSY) phenomenologists is how to incorporate Large Hadron Collider search results into parameter fits designed to measure or constrain the SUSY parameters. Owing to the computational expense of fully simulating lots of points in a generic SUSY space to aid the calculation of the likelihoods, the limits published by experimental collaborations are frequently interpreted in slices of reduced parameter spaces. For example, both ATLAS and CMS have presented results in the Constrained Minimal Supersymmetric Model (CMSSM) by fixing two of four parameters, and ge…
A new fitting method is explored for momentum reconstruction of tracks in a constant magnetic field for a silicon-strip tracker. Substantial increases of momentum resolution respect to standard fit is obtained. The key point is the use of a realistic probability distribution for each hit (heteroscedasticity). Two different methods are used for the fits, the first method introduces an effective variance for each hit, the second method implements the maximum likelihood search. The tracker model is similar to the PAMELA tracker. Each side, of the two sided of the PAMELA detectors, is simulated as…
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